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Index: trunk/src/models/parameters.SM_top_anom.f90
===================================================================
--- trunk/src/models/parameters.SM_top_anom.f90 (revision 4047)
+++ trunk/src/models/parameters.SM_top_anom.f90 (revision 4048)
@@ -1,555 +1,576 @@
! $Id: parameters.SM_top_anom.f90,v 1.4 2006/06/16 13:31:48 kilian Exp $
!
! Copyright (C) 1999-2013 by
! Wolfgang Kilian <kilian@physik.uni-siegen.de>
! Thorsten Ohl <ohl@physik.uni-wuerzburg.de>
! Juergen Reuter <juergen.reuter@desy.de>
! Christian Speckner <cnspeckn@googlemail.com>
! Fabian Bach <fabian.bach@cern.ch> (only this file)
!
! WHIZARD is free software; you can redistribute it and/or modify it
! under the terms of the GNU General Public License as published by
! the Free Software Foundation; either version 2, or (at your option)
! any later version.
!
! WHIZARD is distributed in the hope that it will be useful, but
! WITHOUT ANY WARRANTY; without even the implied warranty of
! MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
! GNU General Public License for more details.
!
! You should have received a copy of the GNU General Public License
! along with this program; if not, write to the Free Software
! Foundation, Inc., 675 Mass Ave, Cambridge, MA 02139, USA.
!
!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
module parameters_sm_top_anom
use kinds
use constants
use sm_physics !NODEP!
implicit none
private
real(default), dimension(27), public :: mass, width
real(default), public :: as
complex(default), public :: gs, igs, ig, unit, half
real(default), public :: e, g, e_em
real(default), public :: sinthw, costhw, sin2thw, tanthw
real(default), public :: qelep, qeup, qedwn
real(default), public :: ttop, tbot, tch, ttau, tw
real(default), public :: ltop, lbot, lc, ltau, lw
complex(default), public :: qlep, qup, qdwn, gcc, qw, &
gzww, gwww, ghww, ghhww, ghzz, ghhzz, &
ghbb, ghtt, ghcc, ghtautau, gh3, gh4, ghmm, &
iqw, igzww, igwww, gw4, gzzww, gazww, gaaww, &
gvl_qbub_n, gvl_qw, gvl_qw_u, gvl_qw_d, &
- gsl_dttr, gsr_dttr, gsl_dttl, gsl_dbtl
+ gsl_dttr, gsr_dttr, gsl_dttl, gsl_dbtl, &
+ c_quqd1_1, c_quqd1_2, c_quqd8_1, c_quqd8_2
real(default), public :: vev, lambda, gi_flag, norm_flag, norm, &
n_tvaa, n_vlrz, n_tvaz, n_vlrw, n_tlrw, n_tvag, n_sph
complex(default), dimension(2), public :: &
gncneu, gnclep, gncup, gncdwn, &
tvaa, tvaabb, vlrz, tvaz, tvazbb, vlrw, tlrw, tvag, sph, &
gvlr_qbub, gvlr_qbub_u, gvlr_qbub_d, gvlr_qbub_e, &
gvlr_qgug, gslr_dbtr
integer, public :: fun_flag
logical, public :: bz=.false., bw=.false., ba=.false.
public :: import_from_whizard, model_update_alpha_s, &
gmom, gtva_tta, gvlr_ttz, gtva_ttz, gvlr_btw, gvlr_tbw, &
gtlr_btw, gtrl_tbw, gtlr_btwz, gtrl_tbwz, gtlr_btwa, gtrl_tbwa, &
gtva_ttww, gtva_bba, gtva_bbz, gtva_bbww, &
gtva_ttg, gtva_ttgg, gsp_tth
contains
subroutine import_from_whizard (par_array)
- real(default), dimension(55), intent(in) :: par_array
+ real(default), dimension(63), intent(in) :: par_array
type :: parameter_set
real(default) :: gf
real(default) :: mZ
real(default) :: mW
real(default) :: mH
real(default) :: alphas
real(default) :: me
real(default) :: mmu
real(default) :: mtau
real(default) :: ms
real(default) :: mc
real(default) :: mb
real(default) :: mtop
real(default) :: wtop
real(default) :: wZ
real(default) :: wW
real(default) :: wH
real(default) :: khgaz
real(default) :: khgaga
real(default) :: khgg
real(default) :: xi0
real(default) :: xipm
real(default) :: tvA
real(default) :: taA
real(default) :: vlZ
real(default) :: vrZ
real(default) :: tvZ
real(default) :: taZ
real(default) :: vlWRe
real(default) :: vlWIm
real(default) :: vrWRe
real(default) :: vrWIm
real(default) :: tlWRe
real(default) :: tlWIm
real(default) :: trWRe
real(default) :: trWIm
real(default) :: tvG
real(default) :: taG
real(default) :: sH
real(default) :: pH
real(default) :: lam
real(default) :: fun
real(default) :: nrm
real(default) :: gi
real(default) :: re_CqG
real(default) :: re_CuG
real(default) :: re_CqB
real(default) :: re_CuB
real(default) :: re_CqW
real(default) :: re_CDu
real(default) :: re_CDd
real(default) :: im_CDd
+ real(default) :: re_Cquqd1_1
+ real(default) :: im_Cquqd1_1
+ real(default) :: re_Cquqd1_2
+ real(default) :: im_Cquqd1_2
+ real(default) :: re_Cquqd8_1
+ real(default) :: im_Cquqd8_1
+ real(default) :: re_Cquqd8_2
+ real(default) :: im_Cquqd8_2
real(default) :: v
real(default) :: cw
real(default) :: sw
real(default) :: ee
end type parameter_set
type(parameter_set) :: par
!!! This corresponds to 1/alpha = 137.03598949333
real(default), parameter :: &
alpha = 1.0_default/137.03598949333_default
e_em = sqrt(4.0_default * PI * alpha)
par%gf = par_array(1)
par%mZ = par_array(2)
par%mW = par_array(3)
par%mH = par_array(4)
par%alphas = par_array(5)
par%me = par_array(6)
par%mmu = par_array(7)
par%mtau = par_array(8)
par%ms = par_array(9)
par%mc = par_array(10)
par%mb = par_array(11)
par%mtop = par_array(12)
par%wtop = par_array(13)
par%wZ = par_array(14)
par%wW = par_array(15)
par%wH = par_array(16)
par%khgaz = par_array(17)
par%khgaga = par_array(18)
par%khgg = par_array(19)
par%xi0 = par_array(20)
par%xipm = par_array(21)
par%tvA = par_array(22)
par%taA = par_array(23)
par%vlZ = par_array(24)
par%vrZ = par_array(25)
par%tvZ = par_array(26)
par%taZ = par_array(27)
par%vlWRe = par_array(28)
par%vlWIm = par_array(29)
par%vrWRe = par_array(30)
par%vrWIm = par_array(31)
par%tlWRe = par_array(32)
par%tlWIm = par_array(33)
par%trWRe = par_array(34)
par%trWIm = par_array(35)
par%tvG = par_array(36)
par%taG = par_array(37)
par%sH = par_array(38)
par%pH = par_array(39)
par%lam = par_array(40)
par%fun = par_array(41)
par%nrm = par_array(42)
par%gi = par_array(43)
par%re_CqG = par_array(44)
par%re_CuG = par_array(45)
par%re_CqB = par_array(46)
par%re_CuB = par_array(47)
par%re_CqW = par_array(48)
par%re_CDu = par_array(49)
par%re_CDd = par_array(50)
par%im_CDd = par_array(51)
- par%v = par_array(52)
- par%cw = par_array(53)
- par%sw = par_array(54)
- par%ee = par_array(55)
+ par%re_Cquqd1_1 = par_array(52)
+ par%im_Cquqd1_1 = par_array(53)
+ par%re_Cquqd1_2 = par_array(54)
+ par%im_Cquqd1_2 = par_array(55)
+ par%re_Cquqd8_1 = par_array(56)
+ par%im_Cquqd8_1 = par_array(57)
+ par%re_Cquqd8_2 = par_array(58)
+ par%im_Cquqd8_2 = par_array(59)
+ par%v = par_array(60)
+ par%cw = par_array(61)
+ par%sw = par_array(62)
+ par%ee = par_array(63)
mass(1:27) = 0
width(1:27) = 0
mass(3) = par%ms
mass(4) = par%mc
mass(5) = par%mb
mass(6) = par%mtop
width(6) = par%wtop
mass(11) = par%me
mass(13) = par%mmu
mass(15) = par%mtau
mass(23) = par%mZ
width(23) = par%wZ
mass(24) = par%mW
width(24) = par%wW
mass(25) = par%mH
width(25) = par%wH
mass(26) = par%xi0 * mass(23)
width(26) = 0
mass(27) = par%xipm * mass(24)
width(27) = 0
ttop = 4.0_default * mass(6)**2 / mass(25)**2
tbot = 4.0_default * mass(5)**2 / mass(25)**2
tch = 4.0_default * mass(4)**2 / mass(25)**2
ttau = 4.0_default * mass(15)**2 / mass(25)**2
tw = 4.0_default * mass(24)**2 / mass(25)**2
ltop = 4.0_default * mass(6)**2 / mass(23)**2
lbot = 4.0_default * mass(5)**2 / mass(23)**2
lc = 4.0_default * mass(4)**2 / mass(23)**2
ltau = 4.0_default * mass(15)**2 / mass(23)**2
lw = 4.0_default * mass(24)**2 / mass(23)**2
vev = par%v
e = par%ee
sinthw = par%sw
sin2thw = par%sw**2
costhw = par%cw
tanthw = sinthw/costhw
qelep = - 1
qeup = 2.0_default / 3.0_default
qedwn = - 1.0_default / 3.0_default
g = e / sinthw
ig = cmplx (0.0_default, 1.0_default, kind=default) * g
unit = 1.0_default
half = 0.5_default
gcc = - g / 2 / sqrt (2.0_default)
gncneu(1) = - g / 2 / costhw * ( + 0.5_default)
gnclep(1) = - g / 2 / costhw * ( - 0.5_default - 2 * qelep * sin2thw)
gncup(1) = - g / 2 / costhw * ( + 0.5_default - 2 * qeup * sin2thw)
gncdwn(1) = - g / 2 / costhw * ( - 0.5_default - 2 * qedwn * sin2thw)
gncneu(2) = - g / 2 / costhw * ( + 0.5_default)
gnclep(2) = - g / 2 / costhw * ( - 0.5_default)
gncup(2) = - g / 2 / costhw * ( + 0.5_default)
gncdwn(2) = - g / 2 / costhw * ( - 0.5_default)
qlep = - e * qelep
qup = - e * qeup
qdwn = - e * qedwn
qw = e
iqw = (0,1)*qw
gzww = g * costhw
igzww = (0,1)*gzww
gwww = g
igwww = (0,1)*gwww
gw4 = gwww**2
gzzww = gzww**2
gazww = gzww * qw
gaaww = qw**2
ghww = mass(24) * g
ghhww = g**2 / 2.0_default
ghzz = mass(23) * g / costhw
ghhzz = g**2 / 2.0_default / costhw**2
ghtt = - mass(6) / vev
ghbb = - mass(5) / vev
ghcc = - mass(4) / vev
ghtautau = - mass(15) / vev
ghmm = - mass(13) / vev
gh3 = - 3 * mass(25)**2 / vev
gh4 = - 3 * mass(25)**2 / vev**2
!!! Color flow basis, divide by sqrt(2)
gs = sqrt(2.0_default*PI*par%alphas)
igs = cmplx (0.0_default, 1.0_default, kind=default) * gs
fun_flag = nint(par%fun)
lambda = par%lam
norm_flag = par%nrm
norm = vev / lambda**2
if ( norm_flag > 0. ) then
n_tvaa = e / mass(6)
n_vlrz = g / 2 / costhw / 2
n_tvaz = 1.0_default / vev
n_vlrw = g / sqrt(2.0_default) / 2
n_tlrw = g / sqrt(2.0_default) / mass(24) / 2
n_tvag = gs / mass(6)
n_sph = sqrt(0.5_default)
else
n_tvaa = e * norm
n_vlrz = mass(23) * norm / 2
n_tvaz = norm
n_vlrw = sqrt(2.0_default) * mass(24) * norm / 2
n_tlrw = sqrt(2.0_default) * norm / 2
n_tvag = gs * norm
n_sph = sqrt(0.5_default) * vev * norm
end if
tvaa(1) = n_tvaa * par%tvA
tvaa(2) = n_tvaa * par%taA * (0,1)
vlrz(1) = n_vlrz * par%vlZ
vlrz(2) = n_vlrz * par%vrZ
tvaz(1) = n_tvaz * par%tvZ
tvaz(2) = n_tvaz * par%taZ * (0,1)
vlrw(1) = n_vlrw * ( par%vlWRe + par%vlWIm * (0,1) )
vlrw(2) = n_vlrw * ( par%vrWRe + par%vrWIm * (0,1) )
tlrw(1) = n_tlrw * ( par%tlWRe + par%tlWIm * (0,1) )
tlrw(2) = n_tlrw * ( par%trWRe + par%trWIm * (0,1) )
tvag(1) = n_tvag * par%tvG
tvag(2) = n_tvag * par%taG * (0,1)
sph(1) = n_sph * par%sH
sph(2) = n_sph * par%pH * (0,1)
tvaabb(1) = 0.0_default
tvaabb(2) = 0.0_default
tvazbb(1) = 0.0_default
tvazbb(2) = 0.0_default
gi_flag = par%gi
if ( gi_flag > 0. ) then
if ( abs(par%vlWRe) > 0. ) then
print *, "WARNING: gauge invariance and vanishing anomalous"
print *, " bbZ vector coupling implies a relation"
print *, " vl_ttZ ~ vl_tbW_Re ."
print *, " Inferring vl_ttZ from vl_tbW_Re and IGNORING any"
print *, " inconsistent values set!"
vlrz(1) = real(vlrw(1)) * sqrt(2.0_default) / costhw
print *, " vl_ttZ = ", real(vlrz(1))/n_vlrz
end if
if ( ( abs(par%tlWRe) > 0. ).or.( abs(par%tlWIm) > 0. ) ) then
print *, "WARNING: anomalous tbW tensor couplings are related to anomalous"
print *, " bbZ and bbA tensor couplings by gauge invariance:"
print *, " Inferring bottom couplings from tl_tbW"
tvaabb(1) = real(tlrw(1)) * sinthw / sqrt(2.0_default)
tvaabb(2) = aimag(tlrw(1))*(0,1) * sinthw / sqrt(2.0_default)
tvazbb(1) = real(tlrw(1)) * costhw / sqrt(2.0_default)
tvazbb(2) = aimag(tlrw(1))*(0,1) * costhw / sqrt(2.0_default)
print *, " tv_bbA = ", real(tvaabb(1))/n_tvaa
print *, " ta_bbA = ", aimag(tvaabb(2))/n_tvaa
print *, " tv_bbZ = ", real(tvazbb(1))/n_tvaz
print *, " ta_bbZ = ", aimag(tvazbb(2))/n_tvaz
end if
if ( ( abs(par%tvZ) > 0. ).or.( abs(par%taZ) > 0. ) ) then
bz = .true.
end if
if ( ( abs(par%trWRe) > 0. ).or.( abs(par%trWIm) > 0. ) ) then
bw = .true.
end if
if ( ( abs(par%tvA) > 0. ).or.( abs(par%taA)> 0. ) ) then
ba = .true.
end if
if ( bz.or.bw.or.ba ) then
print *, "WARNING: top anomalous tensor couplings to W, A and Z"
print *, " are interrelated by gauge invariance:"
print *, " Inferring Z couplings from W/A couplings according to"
print *, " the relation in the model file and IGNORING any inconsistent"
print *, " values set manually! (Exception: only tX_ttZ != 0:"
print *, " tr_tbW ~ tv_ttZ + i*ta_ttZ and tX_ttA = 0)"
if ( ( bw.and.bz ).and..not.ba ) then
tvaa(1) = ( real(tlrw(2)) - sqrt(2.0_default)*costhw*tvaz(1) ) / ( 2.0_default*sinthw )
tvaa(2) = ( aimag(tlrw(2))*(0,1) - sqrt(2.0_default)*costhw*tvaz(2) ) / ( 2.0_default*sinthw )
else if ( bz.and..not.bw ) then
tlrw(2) = sqrt(2.0_default)*costhw*( tvaz(1) + tvaz(2) ) + 2.0_default*sinthw*( tvaa(1) + tvaa(2) )
else
tvaz(1) = ( real(tlrw(2)) - 2.0_default*sinthw*tvaa(1) ) / ( sqrt(2.0_default)*costhw )
tvaz(2) = ( aimag(tlrw(2))*(0,1) - 2.0_default*sinthw*tvaa(2) ) / ( sqrt(2.0_default)*costhw )
end if
print *, " tv_ttA = ", real(tvaa(1))/n_tvaa
print *, " ta_ttA = ", aimag(tvaa(2))/n_tvaa
print *, " tv_ttZ = ", real(tvaz(1))/n_tvaz
print *, " ta_ttZ = ", aimag(tvaz(2))/n_tvaz
print *, " tr_tbW = ", real(tlrw(2))/n_tlrw, " + ", aimag(tlrw(2))/n_tlrw, "I"
end if
end if
gvlr_qgug(1) = gs / 2 / (2*lambda**2) * par%re_CqG
gvlr_qgug(2) = gs / 2 / (2*lambda**2) * par%re_CuG
gvlr_qbub(1) = g / 2 * tanthw / (2*lambda**2) * par%re_CqB
gvlr_qbub(2) = g / 2 *tanthw / (2*lambda**2) * par%re_CuB
gvlr_qbub_u(1) = 1.0_default / 3.0_default / 2
gvlr_qbub_u(2) = 4.0_default / 3.0_default / 2
gvlr_qbub_d(1) = 1.0_default / 3.0_default / 2
gvlr_qbub_d(2) = - 2.0_default / 3.0_default / 2
gvlr_qbub_e(1) = - 1.0_default / 2
gvlr_qbub_e(2) = - 2.0_default / 2
gvl_qbub_n = - 1.0_default / 2
gvl_qw = g / 2 / (lambda**2) * par%re_CqW
gvl_qw_u = 0.5_default / 2
gvl_qw_d = - 0.5_default / 2
gsl_dttr = mass(6) / 2 / (sqrt(2.0_default)*vev*lambda**2) * 2.0_default * par%re_CDu
gsr_dttr = mass(6) / 2 / (sqrt(2.0_default)*vev*lambda**2) * (par%re_CDd + par%im_CDd*(0,1))
gsl_dttl = mass(6) / 2 / (sqrt(2.0_default)*vev*lambda**2) * (par%re_CDd - par%im_CDd*(0,1))
gslr_dbtr(1) = mass(6) / 2 / (sqrt(2.0_default)*vev*lambda**2) * 2.0_default * par%re_CDu
gslr_dbtr(2) = - mass(6) / 2 / (sqrt(2.0_default)*vev*lambda**2) * (par%re_CDd + par%im_CDd*(0,1))
gsl_dbtl = - mass(6) / 2 / (sqrt(2.0_default)*vev*lambda**2) * (par%re_CDd - par%im_CDd*(0,1))
+ c_quqd1_1 = - 1.0_default / 2 / (lambda**2) * (par%re_Cquqd1_1 + par%im_Cquqd1_1*(0,1))
+ c_quqd1_2 = - 1.0_default / 2 / (lambda**2) * (par%re_Cquqd1_2 + par%im_Cquqd1_2*(0,1))
+ c_quqd8_1 = - 1.0_default / 2 / (lambda**2) * (par%re_Cquqd8_1 + par%im_Cquqd8_1*(0,1))
+ c_quqd8_2 = - 1.0_default / 2 / (lambda**2) * (par%re_Cquqd8_2 + par%im_Cquqd8_2*(0,1))
end subroutine import_from_whizard
subroutine model_update_alpha_s (alpha_s)
real(default), intent(in) :: alpha_s
gs = sqrt(2.0_default*PI*alpha_s)
igs = cmplx (0.0_default, 1.0_default, kind=default) * gs
end subroutine model_update_alpha_s
pure function gmom(k2, i, coeff, lam) result (c)
complex(default) :: c
real(default), intent(in) :: k2
integer, intent(in) :: i
complex(default), dimension(2), intent(in) :: coeff
real(default), intent(in) :: lam
select case (fun_flag)
case (0)
c = coeff(i)
case (1)
c = coeff(i) / (1.0_default + k2/lam**2)
case (2)
c = coeff(i) / (1.0_default + k2/lam**2)**2
case (3)
c = coeff(i) * exp(-k2/lam**2)
end select
end function gmom
pure function gtva_tta(k2, i) result (c)
complex(default) :: c
real(default), intent(in) :: k2
integer, intent(in) :: i
c = - gmom(k2, i, tvaa, lambda)
end function gtva_tta
pure function gtva_bba(k2, i) result (c)
complex(default) :: c
real(default), intent(in) :: k2
integer, intent(in) :: i
c = - gmom(k2, i, tvaabb, lambda)
end function gtva_bba
pure function gvlr_ttz(k2, i) result (c)
complex(default) :: c
real(default), intent(in) :: k2
integer, intent(in) :: i
c = - gmom(k2, i, vlrz, lambda)
end function gvlr_ttz
pure function gtva_ttz(k2, i) result (c)
complex(default) :: c
real(default), intent(in) :: k2
integer, intent(in) :: i
c = - gmom(k2, i, tvaz, lambda)
end function gtva_ttz
pure function gtva_bbz(k2, i) result (c)
complex(default) :: c
real(default), intent(in) :: k2
integer, intent(in) :: i
c = - gmom(k2, i, tvazbb, lambda)
end function gtva_bbz
pure function gvlr_btw(k2, i) result (c)
complex(default) :: c
real(default), intent(in) :: k2
integer, intent(in) :: i
c = - gmom(k2, i, vlrw, lambda)
end function gvlr_btw
pure function gvlr_tbw(k2, i) result (c)
complex(default) :: c
real(default), intent(in) :: k2
integer, intent(in) :: i
c = - gmom(k2, i, conjg(vlrw), lambda)
end function gvlr_tbw
pure function gtlr_btw(k2, i) result (c)
complex(default) :: c
real(default), intent(in) :: k2
integer, intent(in) :: i
c = - gmom(k2, i, tlrw, lambda)
end function gtlr_btw
pure function gtrl_tbw(k2, i) result (c)
complex(default) :: c
real(default), intent(in) :: k2
integer, intent(in) :: i
c = - gmom(k2, i, conjg(tlrw), lambda)
end function gtrl_tbw
pure function gtlr_btwa(k2, i) result (c)
complex(default) :: c
real(default), intent(in) :: k2
integer, intent(in) :: i
c = - sinthw / 2 * gtlr_btw(k2, i)
end function gtlr_btwa
pure function gtrl_tbwa(k2, i) result (c)
complex(default) :: c
real(default), intent(in) :: k2
integer, intent(in) :: i
c = - sinthw / 2 * gtrl_tbw(k2, i)
end function gtrl_tbwa
pure function gtlr_btwz(k2, i) result (c)
complex(default) :: c
real(default), intent(in) :: k2
integer, intent(in) :: i
c = - costhw / 2 * gtlr_btw(k2, i)
end function gtlr_btwz
pure function gtrl_tbwz(k2, i) result (c)
complex(default) :: c
real(default), intent(in) :: k2
integer, intent(in) :: i
c = - costhw / 2 * gtrl_tbw(k2, i)
end function gtrl_tbwz
pure function gtva_ttww(k2, i) result (c)
complex(default) :: c
real(default), intent(in) :: k2
integer, intent(in) :: i
select case (i)
case (1)
c = - sqrt(0.5_default) * real(gtlr_btw(k2, 2))
case (2)
c = - sqrt(0.5_default) * aimag(gtlr_btw(k2, 2)) * (0, 1)
end select
end function gtva_ttww
pure function gtva_bbww(k2, i) result (c)
complex(default) :: c
real(default), intent(in) :: k2
integer, intent(in) :: i
select case (i)
case (1)
c = - sqrt(0.5_default) * real(gtlr_btw(k2, 1))
case (2)
c = sqrt(0.5_default) * aimag(gtlr_btw(k2, 1)) * (0, 1)
end select
end function gtva_bbww
pure function gtva_ttg(k2, i) result (c)
complex(default) :: c
real(default), intent(in) :: k2
integer, intent(in) :: i
c = - gmom(k2, i, tvag, lambda)
end function gtva_ttg
pure function gtva_ttgg(k2, i) result (c)
complex(default) :: c
real(default), intent(in) :: k2
integer, intent(in) :: i
c = - gtva_ttg(k2, i)
end function gtva_ttgg
pure function gsp_tth(k2, i) result (c)
complex(default) :: c
real(default), intent(in) :: k2
integer, intent(in) :: i
c = - gmom(k2, i, sph, lambda)
end function gsp_tth
end module parameters_sm_top_anom
Index: trunk/src/omega/src/colorize.ml
===================================================================
--- trunk/src/omega/src/colorize.ml (revision 4047)
+++ trunk/src/omega/src/colorize.ml (revision 4048)
@@ -1,1471 +1,1472 @@
(* $Id$
Copyright (C) 1999-2013 by
Wolfgang Kilian <kilian@physik.uni-siegen.de>
Thorsten Ohl <ohl@physik.uni-wuerzburg.de>
Juergen Reuter <juergen.reuter@desy.de>
Christian Speckner <cnspeckn@googlemail.com>
WHIZARD is free software; you can redistribute it and/or modify it
under the terms of the GNU General Public License as published by
the Free Software Foundation; either version 2, or (at your option)
any later version.
WHIZARD is distributed in the hope that it will be useful, but
WITHOUT ANY WARRANTY; without even the implied warranty of
MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
GNU General Public License for more details.
You should have received a copy of the GNU General Public License
along with this program; if not, write to the Free Software
Foundation, Inc., 675 Mass Ave, Cambridge, MA 02139, USA. *)
let rcs_file = RCS.parse "Colorize" ["Colorizing Monochrome Models"]
{ RCS.revision = "$Revision$";
RCS.date = "$Date$";
RCS.author = "$Author$";
RCS.source
= "$URL$" }
(* \thocwmodulesection{Colorizing a Monochrome Model} *)
module It (M : Model.T) =
struct
let rcs = RCS.rename rcs_file "Colorize.It()"
[ "Colorizing Generic Monochrome Models"]
open Coupling
module C = Color
let incomplete s =
failwith ("Colorize.It()." ^ s ^ " not done yet!")
let invalid s =
invalid_arg ("Colorize.It()." ^ s ^ " must not be evaluated!")
let impossible s =
invalid_arg ("Colorize.It()." ^ s ^ " can't happen! (but just did ...)")
let su0 s =
invalid_arg ("Colorize.It()." ^ s ^ ": found SU(0)!")
let colored_vertex s =
invalid_arg ("Colorize.It()." ^ s ^ ": colored vertex!")
let baryonic_vertex s =
invalid_arg ("Colorize.It()." ^ s ^
": baryonic (i.e. eps_ijk) vertices not supported yet!")
let color_flow_ambiguous s =
invalid_arg ("Colorize.It()." ^ s ^ ": ambiguous color flow!")
let color_flow_of_string s =
let c = int_of_string s in
if c < 1 then
invalid_arg ("Colorize.It()." ^ s ^ ": color flow # < 1!")
else
c
type cf_in = int
type cf_out = int
type flavor =
| White of M.flavor
| CF_in of M.flavor * cf_in
| CF_out of M.flavor * cf_out
| CF_io of M.flavor * cf_in * cf_out
| CF_aux of M.flavor
type flavor_sans_color = M.flavor
let flavor_sans_color = function
| White f -> f
| CF_in (f, _) -> f
| CF_out (f, _) -> f
| CF_io (f, _, _) -> f
| CF_aux f -> f
let pullback f arg1 =
f (flavor_sans_color arg1)
type gauge = M.gauge
type constant = M.constant
let options = M.options
let color = pullback M.color
let pdg = pullback M.pdg
let lorentz = pullback M.lorentz
module Ch = M.Ch
let charges = pullback M.charges
(* For the propagator we cannot use pullback because we have to add the case
of the color singlet propagator by hand. *)
let cf_aux_propagator = function
| Prop_Scalar -> Prop_Col_Scalar (* Spin 0 octets. *)
| Prop_Majorana -> Prop_Col_Majorana (* Spin 1/2 octets. *)
| Prop_Feynman -> Prop_Col_Feynman (* Spin 1 states, massless. *)
| Prop_Unitarity -> Prop_Col_Unitarity (* Spin 1 states, massive. *)
+ | Aux_Scalar -> Aux_Col_Scalar (* constant colored scalar propagator *)
| Aux_Vector -> Aux_Col_Vector (* constant colored vector propagator *)
| Aux_Tensor_1 -> Aux_Col_Tensor_1 (* constant colored tensor propagator *)
| Prop_Col_Scalar | Prop_Col_Feynman
| Prop_Col_Majorana | Prop_Col_Unitarity
- | Aux_Col_Vector | Aux_Col_Tensor_1
+ | Aux_Col_Scalar | Aux_Col_Vector | Aux_Col_Tensor_1
-> failwith ("Colorize.It().colorize_propagator: already colored particle!")
| _ -> failwith ("Colorize.It().colorize_propagator: impossible!")
let propagator = function
| CF_aux f -> cf_aux_propagator (M.propagator f)
| White f -> M.propagator f
| CF_in (f, _) -> M.propagator f
| CF_out (f, _) -> M.propagator f
| CF_io (f, _, _) -> M.propagator f
let width = pullback M.width
let goldstone = function
| White f ->
begin match M.goldstone f with
| None -> None
| Some (f', g) -> Some (White f', g)
end
| CF_in (f, c) ->
begin match M.goldstone f with
| None -> None
| Some (f', g) -> Some (CF_in (f', c), g)
end
| CF_out (f, c) ->
begin match M.goldstone f with
| None -> None
| Some (f', g) -> Some (CF_out (f', c), g)
end
| CF_io (f, c1, c2) ->
begin match M.goldstone f with
| None -> None
| Some (f', g) -> Some (CF_io (f', c1, c2), g)
end
| CF_aux f ->
begin match M.goldstone f with
| None -> None
| Some (f', g) -> Some (CF_aux f', g)
end
let conjugate = function
| White f -> White (M.conjugate f)
| CF_in (f, c) -> CF_out (M.conjugate f, c)
| CF_out (f, c) -> CF_in (M.conjugate f, c)
| CF_io (f, c1, c2) -> CF_io (M.conjugate f, c2, c1)
| CF_aux f -> CF_aux (M.conjugate f)
let conjugate_sans_color = M.conjugate
let fermion = pullback M.fermion
let max_degree = M.max_degree
let flavors () =
invalid "flavors"
let external_flavors () =
invalid "external_flavors"
let parameters = M.parameters
module ISet = Set.Make (struct type t = int let compare = compare end)
let nc_value =
let nc_set =
List.fold_left
(fun nc_set f ->
match M.color f with
| C.Singlet -> nc_set
| C.SUN nc -> ISet.add (abs nc) nc_set
| C.AdjSUN nc -> ISet.add (abs nc) nc_set)
ISet.empty (M.flavors ()) in
match ISet.elements nc_set with
| [] -> 0
| [n] -> n
| nc_list ->
invalid_arg
("Colorize.It(): more than one value of N_C: " ^
String.concat ", " (List.map string_of_int nc_list))
let nc () =
nc_value
let split_color_string s =
try
let i1 = String.index s '/' in
let i2 = String.index_from s (succ i1) '/' in
let sf = String.sub s 0 i1
and sc1 = String.sub s (succ i1) (i2 - i1 - 1)
and sc2 = String.sub s (succ i2) (String.length s - i2 - 1) in
(sf, sc1, sc2)
with
| Not_found -> (s, "", "")
let flavor_of_string s =
try
let sf, sc1, sc2 = split_color_string s in
let f = M.flavor_of_string sf in
match M.color f with
| C.Singlet -> White f
| C.SUN nc ->
if nc > 0 then
CF_in (f, color_flow_of_string sc1)
else
CF_out (f, color_flow_of_string sc2)
| C.AdjSUN _ ->
begin match sc1, sc2 with
| "", "" -> CF_aux f
| _, _ -> CF_io (f, color_flow_of_string sc1, color_flow_of_string sc2)
end
with
| Failure "int_of_string" ->
invalid_arg "Colorize().flavor_of_string: expecting integer"
let flavor_to_string = function
| White f ->
M.flavor_to_string f
| CF_in (f, c) ->
M.flavor_to_string f ^ "/" ^ string_of_int c ^ "/"
| CF_out (f, c) ->
M.flavor_to_string f ^ "//" ^ string_of_int c
| CF_io (f, c1, c2) ->
M.flavor_to_string f ^ "/" ^ string_of_int c1 ^ "/" ^ string_of_int c2
| CF_aux f ->
M.flavor_to_string f ^ "//"
let flavor_to_TeX = function
| White f ->
M.flavor_to_TeX f
| CF_in (f, c) ->
"{" ^ M.flavor_to_TeX f ^ "}_c" ^ string_of_int c
| CF_out (f, c) ->
"{" ^ M.flavor_to_TeX f ^ "}_a" ^ string_of_int c
| CF_io (f, c1, c2) ->
"{" ^ M.flavor_to_TeX f ^ "}_c" ^ string_of_int c1 ^ string_of_int c2
| CF_aux f ->
"{" ^ M.flavor_to_TeX f ^ "}_0"
let flavor_symbol = function
| White f ->
M.flavor_symbol f
| CF_in (f, c) ->
M.flavor_symbol f ^ "_" ^ string_of_int c ^ "_"
| CF_out (f, c) ->
M.flavor_symbol f ^ "__" ^ string_of_int c
| CF_io (f, c1, c2) ->
M.flavor_symbol f ^ "_" ^ string_of_int c1 ^ "_" ^ string_of_int c2
| CF_aux f ->
M.flavor_symbol f ^ "__"
let gauge_symbol = M.gauge_symbol
(* Masses and widths must not depend on the colors anyway! *)
let mass_symbol = pullback M.mass_symbol
let width_symbol = pullback M.width_symbol
let constant_symbol = M.constant_symbol
(* \thocwmodulesubsection{Vertices} *)
(* \thocwmodulesubsection{Auxiliary functions} *)
let mult_vertex3 x = function
| FBF (c, fb, coup, f) ->
FBF ((x * c), fb, coup, f)
| PBP (c, fb, coup, f) ->
PBP ((x * c), fb, coup, f)
| BBB (c, fb, coup, f) ->
BBB ((x * c), fb, coup, f)
| GBG (c, fb, coup, f) ->
GBG ((x * c), fb, coup, f)
| Gauge_Gauge_Gauge c ->
Gauge_Gauge_Gauge (x * c)
| Aux_Gauge_Gauge c ->
Aux_Gauge_Gauge (x * c)
| Scalar_Vector_Vector c ->
Scalar_Vector_Vector (x * c)
| Aux_Vector_Vector c ->
Aux_Vector_Vector (x * c)
| Aux_Scalar_Vector c ->
Aux_Scalar_Vector (x * c)
| Scalar_Scalar_Scalar c ->
Scalar_Scalar_Scalar (x * c)
| Aux_Scalar_Scalar c ->
Aux_Scalar_Scalar (x * c)
| Vector_Scalar_Scalar c ->
Vector_Scalar_Scalar (x * c)
| Graviton_Scalar_Scalar c ->
Graviton_Scalar_Scalar (x * c)
| Graviton_Vector_Vector c ->
Graviton_Vector_Vector (x * c)
| Graviton_Spinor_Spinor c ->
Graviton_Spinor_Spinor (x * c)
| Dim4_Vector_Vector_Vector_T c ->
Dim4_Vector_Vector_Vector_T (x * c)
| Dim4_Vector_Vector_Vector_L c ->
Dim4_Vector_Vector_Vector_L (x * c)
| Dim4_Vector_Vector_Vector_T5 c ->
Dim4_Vector_Vector_Vector_T5 (x * c)
| Dim4_Vector_Vector_Vector_L5 c ->
Dim4_Vector_Vector_Vector_L5 (x * c)
| Dim6_Gauge_Gauge_Gauge c ->
Dim6_Gauge_Gauge_Gauge (x * c)
| Dim6_Gauge_Gauge_Gauge_5 c ->
Dim6_Gauge_Gauge_Gauge_5 (x * c)
| Aux_DScalar_DScalar c ->
Aux_DScalar_DScalar (x * c)
| Aux_Vector_DScalar c ->
Aux_Vector_DScalar (x * c)
| Dim5_Scalar_Gauge2 c ->
Dim5_Scalar_Gauge2 (x * c)
| Dim5_Scalar_Gauge2_Skew c ->
Dim5_Scalar_Gauge2_Skew (x * c)
| Dim5_Scalar_Vector_Vector_T c ->
Dim5_Scalar_Vector_Vector_T (x * c)
| Dim5_Scalar_Vector_Vector_U c ->
Dim5_Scalar_Vector_Vector_U (x * c)
| Dim5_Scalar_Vector_Vector_TU c ->
Dim5_Scalar_Vector_Vector_TU (x * c)
| Dim6_Vector_Vector_Vector_T c ->
Dim6_Vector_Vector_Vector_T (x * c)
| Tensor_2_Vector_Vector c ->
Tensor_2_Vector_Vector (x * c)
| Dim5_Tensor_2_Vector_Vector_1 c ->
Dim5_Tensor_2_Vector_Vector_1 (x * c)
| Dim5_Tensor_2_Vector_Vector_2 c ->
Dim5_Tensor_2_Vector_Vector_2 (x * c)
| Dim7_Tensor_2_Vector_Vector_T c ->
Dim7_Tensor_2_Vector_Vector_T (x * c)
let mult_vertex4 x = function
| Scalar4 c ->
Scalar4 (x * c)
| Scalar2_Vector2 c ->
Scalar2_Vector2 (x * c)
| Vector4 ic4_list ->
Vector4 (List.map (fun (c, icl) -> (x * c, icl)) ic4_list)
| DScalar4 ic4_list ->
DScalar4 (List.map (fun (c, icl) -> (x * c, icl)) ic4_list)
| DScalar2_Vector2 ic4_list ->
DScalar2_Vector2 (List.map (fun (c, icl) -> (x * c, icl)) ic4_list)
| GBBG (c, fb, b2, f) ->
GBBG ((x * c), fb, b2, f)
| Vector4_K_Matrix_tho (c, ic4_list) ->
Vector4_K_Matrix_tho ((x * c), ic4_list)
| Vector4_K_Matrix_jr (c, ch2_list) ->
Vector4_K_Matrix_jr ((x * c), ch2_list)
let mult_vertexn x = function
| foo -> ignore (incomplete "mult_vertexn"); foo
let mult_vertex x = function
| V3 (v, fuse, c) -> V3 (mult_vertex3 x v, fuse, c)
| V4 (v, fuse, c) -> V4 (mult_vertex4 x v, fuse, c)
| Vn (v, fuse, c) -> Vn (mult_vertexn x v, fuse, c)
(* Below, we will need to permute Lorentz structures. The following
permutes the three possible contractions of four vectors. We permute
the first three indices, as they correspond to the particles entering
the fusion. *)
type permutation4 =
| P123 | P231 | P312
| P213 | P321 | P132
let permute_contract4 = function
| P123 ->
begin function
| C_12_34 -> C_12_34
| C_13_42 -> C_13_42
| C_14_23 -> C_14_23
end
| P231 ->
begin function
| C_12_34 -> C_14_23
| C_13_42 -> C_12_34
| C_14_23 -> C_13_42
end
| P312 ->
begin function
| C_12_34 -> C_13_42
| C_13_42 -> C_14_23
| C_14_23 -> C_12_34
end
| P213 ->
begin function
| C_12_34 -> C_12_34
| C_13_42 -> C_14_23
| C_14_23 -> C_13_42
end
| P321 ->
begin function
| C_12_34 -> C_14_23
| C_13_42 -> C_13_42
| C_14_23 -> C_12_34
end
| P132 ->
begin function
| C_12_34 -> C_13_42
| C_13_42 -> C_12_34
| C_14_23 -> C_14_23
end
let permute_contract4_list perm ic4_list =
List.map (fun (i, c4) -> (i, permute_contract4 perm c4)) ic4_list
let permute_vertex4' perm = function
| Scalar4 c ->
Scalar4 c
| Vector4 ic4_list ->
Vector4 (permute_contract4_list perm ic4_list)
| Vector4_K_Matrix_jr (c, ic4_list) ->
Vector4_K_Matrix_jr (c, permute_contract4_list perm ic4_list)
| Scalar2_Vector2 c ->
incomplete "permute_vertex4' Scalar2_Vector2"
| DScalar4 ic4_list ->
incomplete "permute_vertex4' DScalar4"
| DScalar2_Vector2 ic4_list ->
incomplete "permute_vertex4' DScalar2_Vector2"
| GBBG (c, fb, b2, f) ->
incomplete "permute_vertex4' GBBG"
| Vector4_K_Matrix_tho (c, ch2_list) ->
incomplete "permute_vertex4' Vector4_K_Matrix_tho"
let permute_vertex4 perm = function
| V3 (v, fuse, c) -> V3 (v, fuse, c)
| V4 (v, fuse, c) -> V4 (permute_vertex4' perm v, fuse, c)
| Vn (v, fuse, c) -> Vn (v, fuse, c)
(* [vertices] are \emph{only} used by functor applications and
for indexing a cache of precomputed fusion rules, which is not
used for colorized models. *)
let vertices () =
invalid "vertices"
(* \thocwmodulesubsection{Cubic Vertices} *)
(* \begin{dubious}
The following pattern matches could eventually become quite long.
The O'Caml compiler will (hopefully) optimize them aggressively
(\url{http://pauillac.inria.fr/~maranget/papers/opat/}).
\end{dubious} *)
let colorize_fusion2 f1 f2 (f, v) =
match M.color f with
| C.Singlet ->
begin match f1, f2 with
| White _, White _ ->
[White f, v]
| CF_in (_, c1), CF_out (_, c2')
| CF_out (_, c1), CF_in (_, c2') ->
if c1 = c2' then
[White f, v]
else
[]
| CF_io (f1, c1, c1'), CF_io (f2, c2, c2') ->
if c1 = c2' && c2 = c1' then
[White f, v]
else
[]
| CF_aux f1, CF_aux f2 ->
[White f, mult_vertex (- (nc ())) v]
| CF_aux _, CF_io _ | CF_io _, CF_aux _ ->
[]
| (CF_in _ | CF_out _ | CF_io _ | CF_aux _), White _
| White _, (CF_in _ | CF_out _ | CF_io _ | CF_aux _)
| (CF_io _ | CF_aux _), (CF_in _ | CF_out _)
| (CF_in _ | CF_out _), (CF_io _ | CF_aux _)
| CF_in _, CF_in _ | CF_out _, CF_out _ ->
colored_vertex "colorize_fusion2"
end
| C.SUN nc1 ->
begin match f1, f2 with
| CF_in (_, c1), (White _ | CF_aux _)
| (White _ | CF_aux _), CF_in (_, c1) ->
if nc1 > 0 then
[CF_in (f, c1), v]
else
colored_vertex "colorize_fusion2"
| CF_out (_, c1'), (White _ | CF_aux _)
| (White _ | CF_aux _), CF_out (_, c1') ->
if nc1 < 0 then
[CF_out (f, c1'), v]
else
colored_vertex "colorize_fusion2"
| CF_in (_, c1), CF_io (_, c2, c2')
| CF_io (_, c2, c2'), CF_in (_, c1) ->
if nc1 > 0 then begin
if c1 = c2' then
[CF_in (f, c2), v]
else
[]
end else
colored_vertex "colorize_fusion2"
| CF_out (_, c1'), CF_io (_, c2, c2')
| CF_io (_, c2, c2'), CF_out (_, c1') ->
if nc1 < 0 then begin
if c1' = c2 then
[CF_out (f, c2'), v]
else
[]
end else
colored_vertex "colorize_fusion2"
| CF_in _, CF_in _ ->
if nc1 > 0 then
baryonic_vertex "colorize_fusion2"
else
colored_vertex "colorize_fusion2"
| CF_out _, CF_out _ ->
if nc1 < 0 then
baryonic_vertex "colorize_fusion2"
else
colored_vertex "colorize_fusion2"
| CF_in _, CF_out _ | CF_out _, CF_in _
| (White _ | CF_io _ | CF_aux _),
(White _ | CF_io _ | CF_aux _) ->
colored_vertex "colorize_fusion2"
end
| C.AdjSUN _ ->
begin match f1, f2 with
| White _, CF_io (_, c1, c2') | CF_io (_, c1, c2'), White _ ->
[CF_io (f, c1, c2'), v]
| White _, CF_aux _ | CF_aux _, White _ ->
[CF_aux f, mult_vertex (- (nc ())) v]
| CF_in (_, c1), CF_out (_, c2')
| CF_out (_, c2'), CF_in (_, c1) ->
if c1 <> c2' then
[CF_io (f, c1, c2'), v]
else
[CF_aux f, v]
(* In the adjoint representation
\begin{subequations}
\begin{equation}
\parbox{28mm}{\fmfframe(2,2)(2,1){\begin{fmfgraph*}(24,24)
\fmfsurround{d1,e1,d2,e2,d3,e3}
\fmf{gluon}{v,e1}
\fmf{gluon}{v,e2}
\fmf{gluon}{v,e3}
\fmflabel{1}{e1}
\fmflabel{2}{e2}
\fmflabel{3}{e3}
\fmfdot{v}
\fmffreeze
\fmf{warrow_right}{v,e1}
\fmf{warrow_right}{v,e2}
\fmf{warrow_right}{v,e3}
\end{fmfgraph*}}} \,=
%begin{split}
g f_{a_1a_2a_3} C^{\mu_1\mu_2\mu_3} (k_1,k_2,k_3)
%end{split}
\end{equation}
with
\begin{multline}
\label{eq:C123}
C^{\mu_1\mu_2\mu_3}(k_1,k_2,k_3) = \\
( g^{\mu_1\mu_2} (k_1^{\mu_3}-k_2^{\mu_3})
+ g^{\mu_2\mu_3} (k_2^{\mu_1}-k_3^{\mu_1})
+ g^{\mu_3\mu_1} (k_3^{\mu_2}-k_1^{\mu_2}) )
\end{multline}
\end{subequations}
while in the color flow basis find from
\begin{equation}
\ii f_{a_1a_2a_3}
= \tr\left(T_{a_1}\left\lbrack T_{a_2},T_{a_3}\right\rbrack\right)
= \tr\left(T_{a_1}T_{a_2}T_{a_3}\right)
- \tr\left(T_{a_1}T_{a_3}T_{a_2}\right)
\end{equation}
the decomposition
\begin{equation}
\ii f_{a_1a_2a_3} T_{a_1}^{i_1j_1}T_{a_2}^{i_2j_2}T_{a_3}^{i_3j_3}
= \delta^{i_1j_2}\delta^{i_2j_3}\delta^{i_3j_1}
- \delta^{i_1j_3}\delta^{i_3j_2}\delta^{i_2j_1}\,.
\end{equation}
The resulting Feynman rule is
\begin{equation}
\parbox{28mm}{\fmfframe(2,2)(2,1){\begin{fmfgraph*}(24,24)
\fmfsurround{d1,e1,d2,e2,d3,e3}
\fmf{phantom}{v,e1}
\fmf{phantom}{v,e2}
\fmf{phantom}{v,e3}
\fmflabel{1}{e1}
\fmflabel{2}{e2}
\fmflabel{3}{e3}
\fmffreeze
\fmfi{phantom_arrow}{(reverse vpath (__e1, __v) sideways -thick)}
\fmfi{phantom_arrow}{( vpath (__e2, __v) sideways -thick)}
\fmfi{phantom_arrow}{(reverse vpath (__e2, __v) sideways -thick)}
\fmfi{phantom_arrow}{( vpath (__e3, __v) sideways -thick)}
\fmfi{phantom_arrow}{(reverse vpath (__e3, __v) sideways -thick)}
\fmfi{phantom_arrow}{( vpath (__e1, __v) sideways -thick)}
\fmfi{plain}{%
(reverse vpath (__e1, __v) sideways -thick)
join ( vpath (__e2, __v) sideways -thick)}
\fmfi{plain}{%
(reverse vpath (__e2, __v) sideways -thick)
join ( vpath (__e3, __v) sideways -thick)}
\fmfi{plain}{%
(reverse vpath (__e3, __v) sideways -thick)
join ( vpath (__e1, __v) sideways -thick)}
\end{fmfgraph*}}} \,=
\ii g
\left( \delta^{i_1j_3}\delta^{i_2j_1}\delta^{i_3j_2}
- \delta^{i_1j_2}\delta^{i_2j_3}\delta^{i_3j_1} \right)
C^{\mu_1\mu_2\mu_3} (k_1,k_2,k_3)
\end{equation} *)
(* \begin{dubious}
We have to generalize this for cases of three particles
in the adjoint that are not all gluons (gluinos, scalar octets):
\begin{itemize}
\item scalar-scalar-scalar
\item scalar-scalar-vector
\item scalar-vector-vector
\item scalar-fermion-fermion
\item vector-fermion-fermion
\end{itemize}
\end{dubious} *)
(* \begin{dubious}
We could use a better understanding of the signs for the
gaugino-gaugino-gaugeboson couplings!!!
\end{dubious} *)
| CF_io (f1, c1, c1'), CF_io (f2, c2, c2') ->
let sign =
begin match v with
| V3 (Gauge_Gauge_Gauge _, _, _)
| V3 (Aux_Gauge_Gauge _, _, _) -> 1
| V3 (FBF (_, _, _, _), fuse2, _) ->
begin match fuse2 with
| F12 -> 1 (* works, but needs theoretical underpinning *)
| F21 -> -1 (* dto. *)
| F31 -> 1 (* dto. *)
| F32 -> -1 (* transposition of [F12] (no testcase) *)
| F23 -> 1 (* transposition of [F21] (no testcase) *)
| F13 -> -1 (* transposition of [F12] (no testcase) *)
end
| V3 _ -> incomplete "colorize_fusion2 (V3 _)"
| V4 _ -> impossible "colorize_fusion2 (V4 _)"
| Vn _ -> impossible "colorize_fusion2 (Vn _)"
end in
if c1' = c2 then
[CF_io (f, c1, c2'), mult_vertex (-sign) v]
else if c2' = c1 then
[CF_io (f, c2, c1'), mult_vertex ( sign) v]
else
[]
| CF_aux _ , CF_io _
| CF_io _ , CF_aux _
| CF_aux _ , CF_aux _ ->
[]
| White _, White _
| (White _ | CF_io _ | CF_aux _), (CF_in _ | CF_out _)
| (CF_in _ | CF_out _), (White _ | CF_io _ | CF_aux _)
| CF_in _, CF_in _ | CF_out _, CF_out _ ->
colored_vertex "colorize_fusion2"
end
(* \thocwmodulesubsection{Quartic Vertices} *)
let colorize_fusion3 f1 f2 f3 (f, v) =
match M.color f with
| C.Singlet ->
begin match f1, f2, f3 with
| White _, White _, White _ ->
[White f, v]
| (White _ | CF_aux _), CF_in (_, c1), CF_out (_, c2')
| (White _ | CF_aux _), CF_out (_, c1), CF_in (_, c2')
| CF_in (_, c1), (White _ | CF_aux _), CF_out (_, c2')
| CF_out (_, c1), (White _ | CF_aux _), CF_in (_, c2')
| CF_in (_, c1), CF_out (_, c2'), (White _ | CF_aux _)
| CF_out (_, c1), CF_in (_, c2'), (White _ | CF_aux _) ->
if c1 = c2' then
[White f, v]
else
[]
| White _, CF_io (_, c1, c1'), CF_io (_, c2, c2')
| CF_io (_, c1, c1'), White _, CF_io (_, c2, c2')
| CF_io (_, c1, c1'), CF_io (_, c2, c2'), White _ ->
if c1 = c2' && c2 = c1' then
[White f, v]
else
[]
| White _, CF_aux _, CF_aux _
| CF_aux _, White _, CF_aux _
| CF_aux _, CF_aux _, White _ ->
[White f, mult_vertex (- (nc ())) v]
| White _, CF_io _, CF_aux _
| White _, CF_aux _, CF_io _
| CF_io _, White _, CF_aux _
| CF_aux _, White _, CF_io _
| CF_io _, CF_aux _, White _
| CF_aux _, CF_io _, White _ ->
[]
| CF_io (_, c1, c1'), CF_in (_, c2), CF_out (_, c3')
| CF_io (_, c1, c1'), CF_out (_, c3'), CF_in (_, c2)
| CF_in (_, c2), CF_io (_, c1, c1'), CF_out (_, c3')
| CF_out (_, c3'), CF_io (_, c1, c1'), CF_in (_, c2)
| CF_in (_, c2), CF_out (_, c3'), CF_io (_, c1, c1')
| CF_out (_, c3'), CF_in (_, c2), CF_io (_, c1, c1') ->
if c1 = c3' && c1' = c2 then
[White f, v]
else
[]
| CF_io (_, c1, c1'), CF_io (_, c2, c2'), CF_io (_, c3, c3') ->
if c1' = c2 && c2' = c3 && c3' = c1 then
[White f, mult_vertex (-1) v]
else if c1' = c3 && c2' = c1 && c3' = c2 then
[White f, mult_vertex ( 1) v]
else
[]
| CF_io _, CF_io _, CF_aux _
| CF_io _, CF_aux _, CF_io _
| CF_aux _, CF_io _, CF_io _
| CF_io _, CF_aux _, CF_aux _
| CF_aux _, CF_io _, CF_aux _
| CF_aux _, CF_aux _, CF_io _
| CF_aux _, CF_aux _, CF_aux _ ->
[]
| CF_in _, CF_in _, CF_in _
| CF_out _, CF_out _, CF_out _ ->
baryonic_vertex "colorize_fusion3"
| CF_in _, CF_in _, CF_out _
| CF_in _, CF_out _, CF_in _
| CF_out _, CF_in _, CF_in _
| CF_in _, CF_out _, CF_out _
| CF_out _, CF_in _, CF_out _
| CF_out _, CF_out _, CF_in _
| White _, White _, (CF_io _ | CF_aux _)
| White _, (CF_io _ | CF_aux _), White _
| (CF_io _ | CF_aux _), White _, White _
| (White _ | CF_io _ | CF_aux _), CF_in _, CF_in _
| CF_in _, (White _ | CF_io _ | CF_aux _), CF_in _
| CF_in _, CF_in _, (White _ | CF_io _ | CF_aux _)
| (White _ | CF_io _ | CF_aux _), CF_out _, CF_out _
| CF_out _, (White _ | CF_io _ | CF_aux _), CF_out _
| CF_out _, CF_out _, (White _ | CF_io _ | CF_aux _)
| (CF_in _ | CF_out _),
(White _ | CF_io _ | CF_aux _),
(White _ | CF_io _ | CF_aux _)
| (White _ | CF_io _ | CF_aux _),
(CF_in _ | CF_out _),
(White _ | CF_io _ | CF_aux _)
| (White _ | CF_io _ | CF_aux _),
(White _ | CF_io _ | CF_aux _),
(CF_in _ | CF_out _) ->
colored_vertex "colorize_fusion3"
end
| C.SUN nc1 ->
begin match f1, f2, f3 with
| CF_in (_, c1), CF_io (_, c2, c2'), CF_io (_, c3, c3')
| CF_io (_, c2, c2'), CF_in (_, c1), CF_io (_, c3, c3')
| CF_io (_, c2, c2'), CF_io (_, c3, c3'), CF_in (_, c1) ->
if nc1 > 0 then
if c1 = c2' && c2 = c3' then
[CF_in (f, c3), v]
else if c1 = c3' && c3 = c2' then
[CF_in (f, c2), v]
else
[]
else
colored_vertex "colorize_fusion3"
| CF_out (_, c1'), CF_io (_, c2, c2'), CF_io (_, c3, c3')
| CF_io (_, c2, c2'), CF_out (_, c1'), CF_io (_, c3, c3')
| CF_io (_, c2, c2'), CF_io (_, c3, c3'), CF_out (_, c1') ->
if nc1 < 0 then
if c1' = c2 && c2' = c3 then
[CF_out (f, c3'), v]
else if c1' = c3 && c3' = c2 then
[CF_out (f, c2'), v]
else
[]
else
colored_vertex "colorize_fusion3"
| CF_aux _, CF_in (_, c1), CF_io (_, c2, c2')
| CF_aux _, CF_io (_, c2, c2'), CF_in (_, c1)
| CF_in (_, c1), CF_aux _, CF_io (_, c2, c2')
| CF_io (_, c2, c2'), CF_aux _, CF_in (_, c1)
| CF_in (_, c1), CF_io (_, c2, c2'), CF_aux _
| CF_io (_, c2, c2'), CF_in (_, c1), CF_aux _ ->
if nc1 > 0 then
if c1 = c2' then
[CF_in (f, c2), mult_vertex ( 2) v]
else
[]
else
colored_vertex "colorize_fusion3"
| CF_aux _, CF_out (_, c1'), CF_io (_, c2, c2')
| CF_aux _, CF_io (_, c2, c2'), CF_out (_, c1')
| CF_out (_, c1'), CF_aux _, CF_io (_, c2, c2')
| CF_io (_, c2, c2'), CF_aux _, CF_out (_, c1')
| CF_out (_, c1'), CF_io (_, c2, c2'), CF_aux _
| CF_io (_, c2, c2'), CF_out (_, c1'), CF_aux _ ->
if nc1 < 0 then
if c1' = c2 then
[CF_out (f, c2'), mult_vertex ( 2) v]
else
[]
else
colored_vertex "colorize_fusion3"
| White _, CF_in (_, c1), CF_io (_, c2, c2')
| White _, CF_io (_, c2, c2'), CF_in (_, c1)
| CF_in (_, c1), White _, CF_io (_, c2, c2')
| CF_io (_, c2, c2'), White _, CF_in (_, c1)
| CF_in (_, c1), CF_io (_, c2, c2'), White _
| CF_io (_, c2, c2'), CF_in (_, c1), White _ ->
if nc1 > 0 then
if c1 = c2' then
[CF_in (f, c2), v]
else
[]
else
colored_vertex "colorize_fusion3"
| White _, CF_out (_, c1'), CF_io (_, c2, c2')
| White _, CF_io (_, c2, c2'), CF_out (_, c1')
| CF_out (_, c1'), White _, CF_io (_, c2, c2')
| CF_io (_, c2, c2'), White _, CF_out (_, c1')
| CF_out (_, c1'), CF_io (_, c2, c2'), White _
| CF_io (_, c2, c2'), CF_out (_, c1'), White _ ->
if nc1 < 0 then
if c2 = c1' then
[CF_out (f, c2'), v]
else
[]
else
colored_vertex "colorize_fusion3"
| CF_in (_, c1), CF_aux _, CF_aux _
| CF_aux _, CF_in (_, c1), CF_aux _
| CF_aux _, CF_aux _, CF_in (_, c1) ->
if nc1 > 0 then
[CF_in (f, c1), mult_vertex ( 2) v]
else
colored_vertex "colorize_fusion3"
| CF_in (_, c1), CF_aux _, White _
| CF_in (_, c1), White _, CF_aux _
| CF_in (_, c1), White _, White _
| CF_aux _, CF_in (_, c1), White _
| White _, CF_in (_, c1), CF_aux _
| White _, CF_in (_, c1), White _
| CF_aux _, White _, CF_in (_, c1)
| White _, CF_aux _, CF_in (_, c1)
| White _, White _, CF_in (_, c1) ->
if nc1 > 0 then
[CF_in (f, c1), v]
else
colored_vertex "colorize_fusion3"
| CF_out (_, c1'), CF_aux _, CF_aux _
| CF_aux _, CF_out (_, c1'), CF_aux _
| CF_aux _, CF_aux _, CF_out (_, c1') ->
if nc1 < 0 then
[CF_out (f, c1'), mult_vertex ( 2) v]
else
colored_vertex "colorize_fusion3"
| CF_out (_, c1'), CF_aux _, White _
| CF_out (_, c1'), White _, CF_aux _
| CF_out (_, c1'), White _, White _
| CF_aux _, CF_out (_, c1'), White _
| White _, CF_out (_, c1'), CF_aux _
| White _, CF_out (_, c1'), White _
| CF_aux _, White _, CF_out (_, c1')
| White _, CF_aux _, CF_out (_, c1')
| White _, White _, CF_out (_, c1') ->
if nc1 < 0 then
[CF_out (f, c1'), v]
else
colored_vertex "colorize_fusion3"
| CF_in _, CF_in _, CF_out _
| CF_in _, CF_out _, CF_in _
| CF_out _, CF_in _, CF_in _ ->
if nc1 > 0 then
color_flow_ambiguous "colorize_fusion3"
else
colored_vertex "colorize_fusion3"
| CF_in _, CF_out _, CF_out _
| CF_out _, CF_in _, CF_out _
| CF_out _, CF_out _, CF_in _ ->
if nc1 < 0 then
color_flow_ambiguous "colorize_fusion3"
else
colored_vertex "colorize_fusion3"
| CF_in _, CF_in _, CF_in _
| CF_out _, CF_out _, CF_out _
| (White _ | CF_io _ | CF_aux _),
(White _ | CF_io _ | CF_aux _),
(White _ | CF_io _ | CF_aux _)
| (CF_in _ | CF_out _),
(CF_in _ | CF_out _),
(White _ | CF_io _ | CF_aux _)
| (CF_in _ | CF_out _),
(White _ | CF_io _ | CF_aux _),
(CF_in _ | CF_out _)
| (White _ | CF_io _ | CF_aux _),
(CF_in _ | CF_out _),
(CF_in _ | CF_out _) ->
colored_vertex "colorize_fusion3"
end
| C.AdjSUN nc ->
begin match f1, f2, f3 with
| CF_in (_, c1), CF_out (_, c1'), White _
| CF_out (_, c1'), CF_in (_, c1), White _
| CF_in (_, c1), White _, CF_out (_, c1')
| CF_out (_, c1'), White _, CF_in (_, c1)
| White _, CF_in (_, c1), CF_out (_, c1')
| White _, CF_out (_, c1'), CF_in (_, c1) ->
if c1 <> c1' then
[CF_io (f, c1, c1'), v]
else
[CF_aux f, v]
| CF_in (_, c1), CF_out (_, c1'), CF_aux _
| CF_out (_, c1'), CF_in (_, c1), CF_aux _
| CF_in (_, c1), CF_aux _, CF_out (_, c1')
| CF_out (_, c1'), CF_aux _, CF_in (_, c1)
| CF_aux _, CF_in (_, c1), CF_out (_, c1')
| CF_aux _, CF_out (_, c1'), CF_in (_, c1) ->
if c1 <> c1' then
[CF_io (f, c1, c1'), mult_vertex ( 2) v]
else
[CF_aux f, mult_vertex ( 2) v]
| CF_in (_, c1), CF_out (_, c1'), CF_io (_, c2, c2')
| CF_out (_, c1'), CF_in (_, c1), CF_io (_, c2, c2')
| CF_in (_, c1), CF_io (_, c2, c2'), CF_out (_, c1')
| CF_out (_, c1'), CF_io (_, c2, c2'), CF_in (_, c1)
| CF_io (_, c2, c2'), CF_in (_, c1), CF_out (_, c1')
| CF_io (_, c2, c2'), CF_out (_, c1'), CF_in (_, c1) ->
if c1 = c2' && c2 = c1' then
[CF_aux f, mult_vertex ( 2) v]
else if c1 = c2' then
[CF_io (f, c2, c1'), v]
else if c2 = c1' then
[CF_io (f, c1, c2'), v]
else
[]
(* \begin{equation}
\parbox{28mm}{\fmfframe(2,2)(2,1){\begin{fmfgraph*}(24,24)
\fmfsurround{d1,e1,d2,e2,d3,e3,d4,e4}
\fmf{gluon}{v,e1}
\fmf{gluon}{v,e2}
\fmf{gluon}{v,e3}
\fmf{gluon}{v,e4}
\fmflabel{1}{e1}
\fmflabel{2}{e2}
\fmflabel{3}{e3}
\fmflabel{4}{e4}
\fmfdot{v}
\fmffreeze
\fmf{warrow_right}{v,e1}
\fmf{warrow_right}{v,e2}
\fmf{warrow_right}{v,e3}
\fmf{warrow_right}{v,e4}
\end{fmfgraph*}}} \,=
\begin{split}
\mbox{} - & \ii g^2 f_{a_1a_2b}f_{a_3a_4b}
(g_{\mu_1\mu_3} g_{\mu_4\mu_2} - g_{\mu_1\mu_4} g_{\mu_2\mu_3}) \\
\mbox{} - & \ii g^2 f_{a_1a_3b}f_{a_4a_2b}
(g_{\mu_1\mu_4} g_{\mu_2\mu_3} - g_{\mu_1\mu_2} g_{\mu_3\mu_4}) \\
\mbox{} - & \ii g^2 f_{a_1a_4b}f_{a_2a_3b}
(g_{\mu_1\mu_2} g_{\mu_3\mu_4} - g_{\mu_1\mu_3} g_{\mu_4\mu_2})
\end{split}
\end{equation} *)
(* Using
\begin{equation}
\mathcal{P}_4 = \left\{\{1,2,3,4\},\{1,3,4,2\},\{1,4,2,3\},
\{1,2,4,3\},\{1,4,3,2\},\{1,3,2,4\}\right\}
\end{equation}
as the set of permutations of~$\{1,2,3,4\}$ with the cyclic permutations
factored out, we have:
\begin{equation}
\parbox{28mm}{\fmfframe(2,2)(2,1){\begin{fmfgraph*}(24,24)
\fmfsurround{d1,e1,d2,e2,d3,e3,d4,e4}
\fmf{phantom}{v,e1}
\fmf{phantom}{v,e2}
\fmf{phantom}{v,e3}
\fmf{phantom}{v,e4}
\fmflabel{1}{e1}
\fmflabel{2}{e2}
\fmflabel{3}{e3}
\fmflabel{4}{e4}
\fmffreeze
\fmfi{phantom_arrow}{(reverse vpath (__e1, __v) sideways -thick)}
\fmfi{phantom_arrow}{( vpath (__e2, __v) sideways -thick)}
\fmfi{phantom_arrow}{(reverse vpath (__e2, __v) sideways -thick)}
\fmfi{phantom_arrow}{( vpath (__e3, __v) sideways -thick)}
\fmfi{phantom_arrow}{(reverse vpath (__e3, __v) sideways -thick)}
\fmfi{phantom_arrow}{( vpath (__e4, __v) sideways -thick)}
\fmfi{phantom_arrow}{(reverse vpath (__e4, __v) sideways -thick)}
\fmfi{phantom_arrow}{( vpath (__e1, __v) sideways -thick)}
\fmfi{plain}{%
(reverse vpath (__e1, __v) sideways -thick)
join ( vpath (__e2, __v) sideways -thick)}
\fmfi{plain}{%
(reverse vpath (__e2, __v) sideways -thick)
join ( vpath (__e3, __v) sideways -thick)}
\fmfi{plain}{%
(reverse vpath (__e3, __v) sideways -thick)
join ( vpath (__e4, __v) sideways -thick)}
\fmfi{plain}{%
(reverse vpath (__e4, __v) sideways -thick)
join ( vpath (__e1, __v) sideways -thick)}
\end{fmfgraph*}}} \,=
\begin{aligned}
\ii g^2 \sum_{\{\alpha_k\}_{k=1,2,3,4}\in\mathcal{P}_4}
\delta^{i_{\alpha_1}j_{\alpha_2}}\delta^{i_{\alpha_2}j_{\alpha_3}}
\delta^{i_{\alpha_3}j_{\alpha_4}}\delta^{i_{\alpha_4}j_{\alpha_1}}\qquad\qquad\\
\left( 2g_{\mu_{\alpha_1}\mu_{\alpha_3}} g_{\mu_{\alpha_4}\mu_{\alpha_2}}
- g_{\mu_{\alpha_1}\mu_{\alpha_4}} g_{\mu_{\alpha_2}\mu_{\alpha_3}}
- g_{\mu_{\alpha_1}\mu_{\alpha_2}} g_{\mu_{\alpha_3}\mu_{\alpha_4}}\right)
\end{aligned}
\end{equation} *)
(* The different color connections correspond to permutations of the
particles entering the fusion and have to be matched by a corresponding
permutation of the Lorentz structure: *)
(* \begin{dubious}
We have to generalize this for cases of four particles
in the adjoint that are not all gluons:
\begin{itemize}
\item scalar-scalar-scalar-scalar
\item scalar-scalar-vector-vector
\end{itemize}
and even ones including fermions (gluinos) if higher dimensional
operators are involved.
\end{dubious} *)
| CF_io (_, c1, c1'), CF_io (_, c2, c2'), CF_io (_, c3, c3') ->
if c1' = c2 && c2' = c3 then
[CF_io (f, c1, c3'), permute_vertex4 P123 v]
else if c1' = c3 && c3' = c2 then
[CF_io (f, c1, c2'), permute_vertex4 P132 v]
else if c2' = c3 && c3' = c1 then
[CF_io (f, c2, c1'), permute_vertex4 P231 v]
else if c2' = c1 && c1' = c3 then
[CF_io (f, c2, c3'), permute_vertex4 P213 v]
else if c3' = c1 && c1' = c2 then
[CF_io (f, c3, c2'), permute_vertex4 P312 v]
else if c3' = c2 && c2' = c1 then
[CF_io (f, c3, c1'), permute_vertex4 P321 v]
else
[]
| CF_io _, CF_io _, CF_aux _
| CF_io _, CF_aux _, CF_io _
| CF_aux _, CF_io _, CF_io _
| CF_io _, CF_aux _, CF_aux _
| CF_aux _, CF_aux _, CF_io _
| CF_aux _, CF_io _, CF_aux _
| CF_aux _, CF_aux _, CF_aux _ ->
[]
| CF_io (_, c1, c1'), CF_io (_, c2, c2'), White _
| CF_io (_, c1, c1'), White _, CF_io (_, c2, c2')
| White _, CF_io (_, c1, c1'), CF_io (_, c2, c2') ->
if c1' = c2 then
[CF_io (f, c1, c2'), mult_vertex (-1) v]
else if c2' = c1 then
[CF_io (f, c2, c1'), mult_vertex ( 1) v]
else
[]
| CF_io (_, c1, c1'), CF_aux _, White _
| CF_aux _, CF_io (_, c1, c1'), White _
| CF_io (_, c1, c1'), White _, CF_aux _
| CF_aux _, White _, CF_io (_, c1, c1')
| White _, CF_io (_, c1, c1'), CF_aux _
| White _, CF_aux _, CF_io (_, c1, c1') ->
[]
| CF_aux _, CF_aux _, White _
| CF_aux _, White _, CF_aux _
| White _, CF_aux _, CF_aux _ ->
[]
| White _, White _, CF_io (_, c1, c1')
| White _, CF_io (_, c1, c1'), White _
| CF_io (_, c1, c1'), White _, White _ ->
[CF_io (f, c1, c1'), v]
| White _, White _, CF_aux _
| White _, CF_aux _, White _
| CF_aux _, White _, White _ ->
[]
| White _, White _, White _
| (White _ | CF_io _ | CF_aux _),
(White _ | CF_io _ | CF_aux _),
(CF_in _ | CF_out _)
| (White _ | CF_io _ | CF_aux _),
(CF_in _ | CF_out _),
(White _ | CF_io _ | CF_aux _)
| (CF_in _ | CF_out _),
(White _ | CF_io _ | CF_aux _),
(White _ | CF_io _ | CF_aux _)
| CF_in _, CF_in _, (White _ | CF_io _ | CF_aux _)
| CF_in _, (White _ | CF_io _ | CF_aux _), CF_in _
| (White _ | CF_io _ | CF_aux _), CF_in _, CF_in _
| CF_out _, CF_out _, (White _ | CF_io _ | CF_aux _)
| CF_out _, (White _ | CF_io _ | CF_aux _), CF_out _
| (White _ | CF_io _ | CF_aux _), CF_out _, CF_out _
| (CF_in _ | CF_out _),
(CF_in _ | CF_out _),
(CF_in _ | CF_out _) ->
colored_vertex "colorize_fusion3"
end
(* \thocwmodulesubsection{Quintic and Higher Vertices} *)
let is_white = function
| White _ -> true
| _ -> false
let colorize_fusionn flist (f, v) =
let incomplete_match () =
incomplete
("colorize_fusionn { " ^
String.concat ", " (List.map flavor_to_string flist) ^
" } -> " ^ M.flavor_to_string f) in
match M.color f with
| C.Singlet ->
if List.for_all is_white flist then
[White f, v]
else
incomplete_match ()
| C.SUN _ ->
if List.for_all is_white flist then
colored_vertex "colorize_fusionn"
else
incomplete_match ()
| C.AdjSUN _ ->
if List.for_all is_white flist then
colored_vertex "colorize_fusionn"
else
incomplete_match ()
let fuse2 f1 f2 =
ThoList.flatmap
(colorize_fusion2 f1 f2)
(M.fuse2 (flavor_sans_color f1) (flavor_sans_color f2))
let fuse3 f1 f2 f3 =
ThoList.flatmap
(colorize_fusion3 f1 f2 f3)
(M.fuse3 (flavor_sans_color f1) (flavor_sans_color f2) (flavor_sans_color f3))
let fuse_list flist =
ThoList.flatmap
(colorize_fusionn flist)
(M.fuse (List.map flavor_sans_color flist))
let fuse = function
| [] | [_] -> invalid_arg "Colorize.It().fuse"
| [f1; f2] -> fuse2 f1 f2
| [f1; f2; f3] -> fuse3 f1 f2 f3
| flist -> fuse_list flist
let max_degree = M.max_degree
(* \thocwmodulesubsection{Adding Color to External Particles} *)
let count_color_strings f_list =
let rec count_color_strings' n_in n_out n_glue = function
| f :: rest ->
begin match M.color f with
| C.Singlet -> count_color_strings' n_in n_out n_glue rest
| C.SUN nc ->
if nc > 0 then
count_color_strings' (succ n_in) n_out n_glue rest
else if nc < 0 then
count_color_strings' n_in (succ n_out) n_glue rest
else
su0 "count_color_strings"
| C.AdjSUN _ ->
count_color_strings' (succ n_in) (succ n_out) (succ n_glue) rest
end
| [] -> (n_in, n_out, n_glue)
in
count_color_strings' 0 0 0 f_list
let external_color_flows f_list =
let n_in, n_out, n_glue = count_color_strings f_list in
if n_in <> n_out then
[]
else
let color_strings = ThoList.range 1 n_in in
List.rev_map
(fun permutation -> (color_strings, permutation))
(Combinatorics.permute color_strings)
(* If there are only adjoints \emph{and} there are no couplings of
adjoints to singlets, we can ignore the $\mathrm{U}(1)$-ghosts. *)
let pure_adjoints f_list =
List.for_all (fun f -> match M.color f with C.AdjSUN _ -> true | _ -> false) f_list
let two_adjoints_couple_to_singlets () =
let vertices3, vertices4, verticesn = M.vertices () in
List.exists (fun ((f1, f2, f3), _, _) ->
match M.color f1, M.color f2, M.color f3 with
| C.AdjSUN _, C.AdjSUN _, C.Singlet
| C.AdjSUN _, C.Singlet, C.AdjSUN _
| C.Singlet, C.AdjSUN _, C.AdjSUN _ -> true
| _ -> false) vertices3 ||
List.exists (fun ((f1, f2, f3, f4), _, _) ->
match M.color f1, M.color f2, M.color f3, M.color f4 with
| C.AdjSUN _, C.AdjSUN _, C.Singlet, C.Singlet
| C.AdjSUN _, C.Singlet, C.AdjSUN _, C.Singlet
| C.Singlet, C.AdjSUN _, C.AdjSUN _, C.Singlet
| C.AdjSUN _, C.Singlet, C.Singlet, C.AdjSUN _
| C.Singlet, C.AdjSUN _, C.Singlet, C.AdjSUN _
| C.Singlet, C.Singlet, C.AdjSUN _, C.AdjSUN _ -> true
| _ -> false) vertices4 ||
List.exists (fun (flist, _, g) -> true) verticesn
let external_ghosts f_list =
if pure_adjoints f_list then
two_adjoints_couple_to_singlets ()
else
true
(* We use [List.hd] and [List.tl] instead of pattern matching, because we
consume [ecf_in] and [ecf_out] at a different pace. *)
let tail_opt = function
| [] -> []
| _ :: tail -> tail
let head_req = function
| [] ->
invalid_arg "Colorize.It().colorize_crossed_amplitude1: insufficient flows"
| x :: _ -> x
let rec colorize_crossed_amplitude1 ghosts acc f_list (ecf_in, ecf_out) =
match f_list, ecf_in, ecf_out with
| [], [], [] -> [List.rev acc]
| [], _, _ ->
invalid_arg "Colorize.It().colorize_crossed_amplitude1: leftover flows"
| f :: rest, _, _ ->
begin match M.color f with
| C.Singlet ->
colorize_crossed_amplitude1 ghosts
(White f :: acc)
rest (ecf_in, ecf_out)
| C.SUN nc ->
if nc > 0 then
colorize_crossed_amplitude1 ghosts
(CF_in (f, head_req ecf_in) :: acc)
rest (tail_opt ecf_in, ecf_out)
else if nc < 0 then
colorize_crossed_amplitude1 ghosts
(CF_out (f, head_req ecf_out) :: acc)
rest (ecf_in, tail_opt ecf_out)
else
su0 "colorize_flavor"
| C.AdjSUN _ ->
let ecf_in' = head_req ecf_in
and ecf_out' = head_req ecf_out in
if ecf_in' = ecf_out' then begin
if ghosts then
colorize_crossed_amplitude1 ghosts
(CF_aux f :: acc)
rest (tail_opt ecf_in, tail_opt ecf_out)
else
[]
end else
colorize_crossed_amplitude1 ghosts
(CF_io (f, ecf_in', ecf_out') :: acc)
rest (tail_opt ecf_in, tail_opt ecf_out)
end
let colorize_crossed_amplitude1 ghosts f_list (ecf_in, ecf_out) =
colorize_crossed_amplitude1 ghosts [] f_list (ecf_in, ecf_out)
let colorize_crossed_amplitude f_list =
ThoList.rev_flatmap
(colorize_crossed_amplitude1 (external_ghosts f_list) f_list)
(external_color_flows f_list)
let cross_uncolored p_in p_out =
(List.map M.conjugate p_in) @ p_out
let uncross_colored n_in p_lists_colorized =
let p_in_out_colorized = List.map (ThoList.splitn n_in) p_lists_colorized in
List.map
(fun (p_in_colored, p_out_colored) ->
(List.map conjugate p_in_colored, p_out_colored))
p_in_out_colorized
let amplitude p_in p_out =
uncross_colored
(List.length p_in)
(colorize_crossed_amplitude (cross_uncolored p_in p_out))
(* The $-$-sign in the second component is redundant, but a Whizard convention. *)
let indices = function
| White _ -> Color.Flow.of_list [0; 0]
| CF_in (_, c) -> Color.Flow.of_list [c; 0]
| CF_out (_, c) -> Color.Flow.of_list [0; -c]
| CF_io (_, c1, c2) -> Color.Flow.of_list [c1; -c2]
| CF_aux f -> Color.Flow.ghost ()
let flow p_in p_out =
(List.map indices p_in, List.map indices p_out)
end
(* \thocwmodulesection{Colorizing a Monochrome Gauge Model} *)
module Gauge (M : Model.Gauge) =
struct
let rcs = RCS.rename rcs_file "Colorize.Gauge()"
[ "Colorizing Monochrome Gauge Models"]
module CM = It(M)
type flavor = CM.flavor
type flavor_sans_color = CM.flavor_sans_color
type gauge = CM.gauge
type constant = CM.constant
module Ch = CM.Ch
let charges = CM.charges
let flavor_sans_color = CM.flavor_sans_color
let color = CM.color
let pdg = CM.pdg
let lorentz = CM.lorentz
let propagator = CM.propagator
let width = CM.width
let conjugate = CM.conjugate
let conjugate_sans_color = CM.conjugate_sans_color
let fermion = CM.fermion
let max_degree = CM.max_degree
let vertices = CM.vertices
let fuse2 = CM.fuse2
let fuse3 = CM.fuse3
let fuse = CM.fuse
let flavors = CM.flavors
let nc = CM.nc
let external_flavors = CM.external_flavors
let goldstone = CM.goldstone
let parameters = CM.parameters
let flavor_of_string = CM.flavor_of_string
let flavor_to_string = CM.flavor_to_string
let flavor_to_TeX = CM.flavor_to_TeX
let flavor_symbol = CM.flavor_symbol
let gauge_symbol = CM.gauge_symbol
let mass_symbol = CM.mass_symbol
let width_symbol = CM.width_symbol
let constant_symbol = CM.constant_symbol
let options = CM.options
let incomplete s =
failwith ("Colorize.Gauge()." ^ s ^ " not done yet!")
type matter_field = M.matter_field
type gauge_boson = M.gauge_boson
type other = M.other
type field =
| Matter of matter_field
| Gauge of gauge_boson
| Other of other
let field f =
incomplete "field"
let matter_field f =
incomplete "matter_field"
let gauge_boson f =
incomplete "gauge_boson"
let other f =
incomplete "other"
let amplitude = CM.amplitude
let flow = CM.flow
end
(*i
* Local Variables:
* mode:caml
* indent-tabs-mode:nil
* page-delimiter:"^(\\* .*\n"
* End:
i*)
Index: trunk/src/omega/src/modellib_SM.ml
===================================================================
--- trunk/src/omega/src/modellib_SM.ml (revision 4047)
+++ trunk/src/omega/src/modellib_SM.ml (revision 4048)
@@ -1,3000 +1,3061 @@
(* $Id$
Copyright (C) 1999-2013 by
Wolfgang Kilian <kilian@physik.uni-siegen.de>
Thorsten Ohl <ohl@physik.uni-wuerzburg.de>
Juergen Reuter <juergen.reuter@desy.de>
Christian Speckner <cnspeckn@googlemail.com>
Fabian Bach <fabian.bach@cern.ch> (only parts of this file)
WHIZARD is free software; you can redistribute it and/or modify it
under the terms of the GNU General Public License as published by
the Free Software Foundation; either version 2, or (at your option)
any later version.
WHIZARD is distributed in the hope that it will be useful, but
WITHOUT ANY WARRANTY; without even the implied warranty of
MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
GNU General Public License for more details.
You should have received a copy of the GNU General Public License
along with this program; if not, write to the Free Software
Foundation, Inc., 675 Mass Ave, Cambridge, MA 02139, USA. *)
let rcs_file = RCS.parse "Modellib_SM" ["Lagragians"]
{ RCS.revision = "$Revision$";
RCS.date = "$Date$";
RCS.author = "$Author$";
RCS.source
= "$URL$" }
(* \thocwmodulesection{$\phi^3$} *)
module Phi3 =
struct
let rcs = RCS.rename rcs_file "Modellib.Phi3"
["phi**3 with a single flavor"]
open Coupling
let options = Options.empty
type flavor = Phi
let external_flavors () = [ "", [Phi]]
let flavors () = ThoList.flatmap snd (external_flavors ())
type gauge = unit
type constant = G
let lorentz _ = Scalar
let color _ = Color.Singlet
let propagator _ = Prop_Scalar
let width _ = Timelike
let goldstone _ = None
let conjugate f = f
let fermion _ = 0
module Ch = Charges.Null
let charges _ = ()
module F = Modeltools.Fusions (struct
type f = flavor
type c = constant
let compare = compare
let conjugate = conjugate
end)
let vertices () =
([(Phi, Phi, Phi), Scalar_Scalar_Scalar 1, G], [], [])
let table = F.of_vertices (vertices ())
let fuse2 = F.fuse2 table
let fuse3 = F.fuse3 table
let fuse = F.fuse table
let max_degree () = 3
let parameters () = { input = [G, 1.0]; derived = []; derived_arrays = [] }
let flavor_of_string = function
| "p" -> Phi
| _ -> invalid_arg "Modellib.Phi3.flavor_of_string"
let flavor_to_string Phi = "phi"
let flavor_to_TeX Phi = "\\phi"
let flavor_symbol Phi = "phi"
let gauge_symbol () =
failwith "Modellib.Phi3.gauge_symbol: internal error"
let pdg _ = 1
let mass_symbol _ = "m"
let width_symbol _ = "w"
let constant_symbol G = "g"
end
(* \thocwmodulesection{$\lambda_3\phi^3+\lambda_4\phi^4$} *)
module Phi4 =
struct
let rcs = RCS.rename rcs_file "Modellib.Phi4"
["phi**4 with a single flavor"]
open Coupling
let options = Options.empty
type flavor = Phi
let external_flavors () = [ "", [Phi]]
let flavors () = ThoList.flatmap snd (external_flavors ())
type gauge = unit
type constant = G3 | G4
let lorentz _ = Scalar
let color _ = Color.Singlet
let propagator _ = Prop_Scalar
let width _ = Timelike
let goldstone _ = None
let conjugate f = f
let fermion _ = 0
module Ch = Charges.Null
let charges _ = ()
module F = Modeltools.Fusions (struct
type f = flavor
type c = constant
let compare = compare
let conjugate = conjugate
end)
let vertices () =
([(Phi, Phi, Phi), Scalar_Scalar_Scalar 1, G3],
[(Phi, Phi, Phi, Phi), Scalar4 1, G4], [])
let fuse2 _ = failwith "Modellib.Phi4.fuse2"
let fuse3 _ = failwith "Modellib.Phi4.fuse3"
let fuse = function
| [] | [_] -> invalid_arg "Modellib.Phi4.fuse"
| [_; _] -> [Phi, V3 (Scalar_Scalar_Scalar 1, F23, G3)]
| [_; _; _] -> [Phi, V4 (Scalar4 1, F234, G4)]
| _ -> []
let max_degree () = 4
let parameters () =
{ input = [G3, 1.0; G4, 1.0]; derived = []; derived_arrays = [] }
let flavor_of_string = function
| "p" -> Phi
| _ -> invalid_arg "Modellib.Phi4.flavor_of_string"
let flavor_to_string Phi = "phi"
let flavor_to_TeX Phi = "\\phi"
let flavor_symbol Phi = "phi"
let gauge_symbol () =
failwith "Modellib.Phi4.gauge_symbol: internal error"
let pdg _ = 1
let mass_symbol _ = "m"
let width_symbol _ = "w"
let constant_symbol = function
| G3 -> "g3"
| G4 -> "g4"
end
(* \thocwmodulesection{Quantum Electro Dynamics} *)
module QED =
struct
let rcs = RCS.rename rcs_file "Modellib.QED"
["QED with two leptonic flavors"]
open Coupling
let options = Options.empty
type flavor =
| Electron | Positron
| Muon | AntiMuon
| Tau | AntiTau
| Photon
let external_flavors () =
[ "Leptons", [Electron; Positron; Muon; AntiMuon; Tau; AntiTau];
"Gauge Bosons", [Photon] ]
let flavors () = ThoList.flatmap snd (external_flavors ())
type gauge = unit
type constant = Q
let lorentz = function
| Electron | Muon | Tau -> Spinor
| Positron | AntiMuon | AntiTau -> ConjSpinor
| Photon -> Vector
let color _ = Color.Singlet
let propagator = function
| Electron | Muon | Tau -> Prop_Spinor
| Positron | AntiMuon | AntiTau -> Prop_ConjSpinor
| Photon -> Prop_Feynman
let width _ = Timelike
let goldstone _ =
None
let conjugate = function
| Electron -> Positron | Positron -> Electron
| Muon -> AntiMuon | AntiMuon -> Muon
| Tau -> AntiTau | AntiTau -> Tau
| Photon -> Photon
let fermion = function
| Electron | Muon | Tau -> 1
| Positron | AntiMuon | AntiTau -> -1
| Photon -> 0
(* Taking generation numbers makes electric charge redundant. *)
module Ch = Charges.ZZ
let charges = function
| Electron -> [1; 0; 0]
| Muon -> [0; 1; 0]
| Tau -> [0; 0; 1]
| Positron -> [-1;0; 0]
| AntiMuon -> [0;-1; 0]
| AntiTau -> [0; 0;-1]
| Photon -> [0; 0; 0]
module F = Modeltools.Fusions (struct
type f = flavor
type c = constant
let compare = compare
let conjugate = conjugate
end)
let vertices () =
([(Positron, Photon, Electron), FBF (1, Psibar, V, Psi), Q;
(AntiMuon, Photon, Muon), FBF (1, Psibar, V, Psi), Q;
(AntiTau, Photon, Tau), FBF (1, Psibar, V, Psi), Q], [], [])
let table = F.of_vertices (vertices ())
let fuse2 = F.fuse2 table
let fuse3 = F.fuse3 table
let fuse = F.fuse table
let max_degree () = 3
let parameters () = { input = [Q, 1.0]; derived = []; derived_arrays = [] }
let flavor_of_string = function
| "e-" -> Electron | "e+" -> Positron
| "m-" -> Muon | "m+" -> AntiMuon
| "t-" -> Tau | "t+" -> AntiTau
| "A" -> Photon
| _ -> invalid_arg "Modellib.QED.flavor_of_string"
let flavor_to_string = function
| Electron -> "e-" | Positron -> "e+"
| Muon -> "m-" | AntiMuon -> "m+"
| Tau -> "t-" | AntiTau -> "t+"
| Photon -> "A"
let flavor_to_TeX = function
| Electron -> "e^-" | Positron -> "e^+"
| Muon -> "\\mu^-" | AntiMuon -> "\\mu^+"
| Tau -> "^\\tau^-" | AntiTau -> "\\tau+^"
| Photon -> "\\gamma"
let flavor_symbol = function
| Electron -> "ele" | Positron -> "pos"
| Muon -> "muo" | AntiMuon -> "amu"
| Tau -> "tau" | AntiTau -> "ata"
| Photon -> "gam"
let gauge_symbol () =
failwith "Modellib.QED.gauge_symbol: internal error"
let pdg = function
| Electron -> 11 | Positron -> -11
| Muon -> 13 | AntiMuon -> -13
| Tau -> 15 | AntiTau -> -15
| Photon -> 22
let mass_symbol f =
"mass(" ^ string_of_int (abs (pdg f)) ^ ")"
let width_symbol f =
"width(" ^ string_of_int (abs (pdg f)) ^ ")"
let constant_symbol = function
| Q -> "qlep"
end
(* \thocwmodulesection{Quantum Chromo Dynamics} *)
module QCD =
struct
let rcs = RCS.rename rcs_file "Modellib.QCD"
["QCD"]
open Coupling
let options = Options.empty
type flavor =
| U | Ubar | D | Dbar
| C | Cbar | S | Sbar
| T | Tbar | B | Bbar
| Gl
let external_flavors () =
[ "Quarks", [U; D; C; S; T; B; Ubar; Dbar; Cbar; Sbar; Tbar; Bbar];
"Gauge Bosons", [Gl]]
let flavors () = ThoList.flatmap snd (external_flavors ())
type gauge = unit
type constant = Gs | G2 | I_Gs
let lorentz = function
| U | D | C | S | T | B -> Spinor
| Ubar | Dbar | Cbar | Sbar | Tbar | Bbar -> ConjSpinor
| Gl -> Vector
let color = function
| U | D | C | S | T | B -> Color.SUN 3
| Ubar | Dbar | Cbar | Sbar | Tbar | Bbar -> Color.SUN (-3)
| Gl -> Color.AdjSUN 3
let propagator = function
| U | D | C | S | T | B -> Prop_Spinor
| Ubar | Dbar | Cbar | Sbar | Tbar | Bbar -> Prop_ConjSpinor
| Gl -> Prop_Feynman
let width _ = Timelike
let goldstone _ =
None
let conjugate = function
| U -> Ubar
| D -> Dbar
| C -> Cbar
| S -> Sbar
| T -> Tbar
| B -> Bbar
| Ubar -> U
| Dbar -> D
| Cbar -> C
| Sbar -> S
| Tbar -> T
| Bbar -> B
| Gl -> Gl
let fermion = function
| U | D | C | S | T | B -> 1
| Ubar | Dbar | Cbar | Sbar | Tbar | Bbar -> -1
| Gl -> 0
module Ch = Charges.ZZ
let charges = function
| D -> [1; 0; 0; 0; 0; 0]
| U -> [0; 1; 0; 0; 0; 0]
| S -> [0; 0; 1; 0; 0; 0]
| C -> [0; 0; 0; 1; 0; 0]
| B -> [0; 0; 0; 0; 1; 0]
| T -> [0; 0; 0; 0; 0; 1]
| Dbar -> [-1; 0; 0; 0; 0; 0]
| Ubar -> [0; -1; 0; 0; 0; 0]
| Sbar -> [0; 0; -1; 0; 0; 0]
| Cbar -> [0; 0; 0; -1; 0; 0]
| Bbar -> [0; 0; 0; 0; -1; 0]
| Tbar -> [0; 0; 0; 0; 0; -1]
| Gl -> [0; 0; 0; 0; 0; 0]
module F = Modeltools.Fusions (struct
type f = flavor
type c = constant
let compare = compare
let conjugate = conjugate
end)
(* This is compatible with CD+. *)
let color_current =
[ ((Dbar, Gl, D), FBF ((-1), Psibar, V, Psi), Gs);
((Ubar, Gl, U), FBF ((-1), Psibar, V, Psi), Gs);
((Cbar, Gl, C), FBF ((-1), Psibar, V, Psi), Gs);
((Sbar, Gl, S), FBF ((-1), Psibar, V, Psi), Gs);
((Tbar, Gl, T), FBF ((-1), Psibar, V, Psi), Gs);
((Bbar, Gl, B), FBF ((-1), Psibar, V, Psi), Gs)]
let three_gluon =
[ ((Gl, Gl, Gl), Gauge_Gauge_Gauge 1, I_Gs)]
let gauge4 = Vector4 [(2, C_13_42); (-1, C_12_34); (-1, C_14_23)]
let four_gluon =
[ ((Gl, Gl, Gl, Gl), gauge4, G2)]
let vertices3 =
(color_current @ three_gluon)
let vertices4 = four_gluon
let vertices () =
(vertices3, vertices4, [])
let table = F.of_vertices (vertices ())
let fuse2 = F.fuse2 table
let fuse3 = F.fuse3 table
let fuse = F.fuse table
let max_degree () = 4
let parameters () = { input = [Gs, 1.0]; derived = []; derived_arrays = [] }
let flavor_of_string = function
| "u" -> U
| "d" -> D
| "c" -> C
| "s" -> S
| "t" -> T
| "b" -> B
| "ubar" -> Ubar
| "dbar" -> Dbar
| "cbar" -> Cbar
| "sbar" -> Sbar
| "tbar" -> Tbar
| "bbar" -> Bbar
| "gl" -> Gl
| _ -> invalid_arg "Modellib.QCD.flavor_of_string"
let flavor_to_string = function
| U -> "u"
| Ubar -> "ubar"
| D -> "d"
| Dbar -> "dbar"
| C -> "c"
| Cbar -> "cbar"
| S -> "s"
| Sbar -> "sbar"
| T -> "t"
| Tbar -> "tbar"
| B -> "b"
| Bbar -> "bbar"
| Gl -> "gl"
let flavor_to_TeX = function
| U -> "u"
| Ubar -> "\bar{u}"
| D -> "d"
| Dbar -> "\bar{d}"
| C -> "c"
| Cbar -> "bar{c}"
| S -> "s"
| Sbar -> "\bar{s}"
| T -> "t"
| Tbar -> "\bar{t}"
| B -> "b"
| Bbar -> "\bar{b}"
| Gl -> "g"
let flavor_symbol = function
| U -> "u"
| Ubar -> "ubar"
| D -> "d"
| Dbar -> "dbar"
| C -> "c"
| Cbar -> "cbar"
| S -> "s"
| Sbar -> "sbar"
| T -> "t"
| Tbar -> "tbar"
| B -> "b"
| Bbar -> "bbar"
| Gl -> "gl"
let gauge_symbol () =
failwith "Modellib.QCD.gauge_symbol: internal error"
let pdg = function
| D -> 1 | Dbar -> -1
| U -> 2 | Ubar -> -2
| S -> 3 | Sbar -> -3
| C -> 4 | Cbar -> -4
| B -> 5 | Bbar -> -5
| T -> 6 | Tbar -> -6
| Gl -> 21
let mass_symbol f =
"mass(" ^ string_of_int (abs (pdg f)) ^ ")"
let width_symbol f =
"width(" ^ string_of_int (abs (pdg f)) ^ ")"
let constant_symbol = function
| I_Gs -> "(0,1)*gs"
| Gs -> "gs"
| G2 -> "gs**2"
end
(* \thocwmodulesection{Complete Minimal Standard Model (Unitarity Gauge)} *)
module type SM_flags =
sig
val higgs_triangle : bool (* $H\gamma\gamma$, $Hg\gamma$ and $Hgg$ couplings *)
val higgs_hmm : bool
val triple_anom : bool
val quartic_anom : bool
val higgs_anom : bool
val k_matrix : bool
val ckm_present : bool
val top_anom : bool
val top_anom_4f : bool
end
module SM_no_anomalous : SM_flags =
struct
let higgs_triangle = false
let higgs_hmm = false
let triple_anom = false
let quartic_anom = false
let higgs_anom = false
let k_matrix = false
let ckm_present = false
let top_anom = false
let top_anom_4f = false
end
module SM_no_anomalous_ckm : SM_flags =
struct
let higgs_triangle = false
let higgs_hmm = false
let triple_anom = false
let quartic_anom = false
let higgs_anom = false
let k_matrix = false
let ckm_present = true
let top_anom = false
let top_anom_4f = false
end
module SM_anomalous : SM_flags =
struct
let higgs_triangle = false
let higgs_hmm = false
let triple_anom = true
let quartic_anom = true
let higgs_anom = true
let k_matrix = false
let ckm_present = false
let top_anom = false
let top_anom_4f = false
end
module SM_anomalous_ckm : SM_flags =
struct
let higgs_triangle = false
let higgs_hmm = false
let triple_anom = true
let quartic_anom = true
let higgs_anom = true
let k_matrix = false
let ckm_present = true
let top_anom = false
let top_anom_4f = false
end
module SM_k_matrix : SM_flags =
struct
let higgs_triangle = false
let higgs_hmm = false
let triple_anom = false
let quartic_anom = true
let higgs_anom = false
let k_matrix = true
let ckm_present = false
let top_anom = false
let top_anom_4f = false
end
module SM_Higgs : SM_flags =
struct
let higgs_triangle = true
let higgs_hmm = true
let triple_anom = false
let quartic_anom = false
let higgs_anom = false
let k_matrix = false
let ckm_present = false
let top_anom = false
let top_anom_4f = false
end
module SM_anomalous_top : SM_flags =
struct
let higgs_triangle = false
let higgs_hmm = false
let triple_anom = false
let quartic_anom = false
let higgs_anom = false
let k_matrix = false
let ckm_present = false
let top_anom = true
let top_anom_4f = true
end
(* \thocwmodulesection{Complete Minimal Standard Model (including some extensions)} *)
module SM (Flags : SM_flags) =
struct
let rcs = RCS.rename rcs_file "Modellib.SM"
[ "minimal electroweak standard model in unitarity gauge"]
open Coupling
let default_width = ref Timelike
let use_fudged_width = ref false
let options = Options.create
[ "constant_width", Arg.Unit (fun () -> default_width := Constant),
"use constant width (also in t-channel)";
"fudged_width", Arg.Set use_fudged_width,
"use fudge factor for charge particle width";
"custom_width", Arg.String (fun f -> default_width := Custom f),
"use custom width";
"cancel_widths", Arg.Unit (fun () -> default_width := Vanishing),
"use vanishing width"]
type f_aux_top = TTGG | TBWA | TBWZ | TTWW | BBWW | (*i top auxiliary field "flavors" *)
- QGUG | QBUB | QW | DL | DR
+ QGUG | QBUB | QW | DL | DR |
+ QUQD1L | QUQD1R | QUQD8L | QUQD8R
type matter_field = L of int | N of int | U of int | D of int
type gauge_boson = Ga | Wp | Wm | Z | Gl
type other = Phip | Phim | Phi0 | H
| Aux_top of int*int*int*bool*f_aux_top (*i lorentz*color*charge*top-side*flavor *)
type flavor = M of matter_field | G of gauge_boson | O of other
let matter_field f = M f
let gauge_boson f = G f
let other f = O f
type field =
| Matter of matter_field
| Gauge of gauge_boson
| Other of other
let field = function
| M f -> Matter f
| G f -> Gauge f
| O f -> Other f
type gauge = unit
let gauge_symbol () =
failwith "Modellib.SM.gauge_symbol: internal error"
let family n = List.map matter_field [ L n; N n; U n; D n ]
let rec aux_top_flavors (f,l,co,ch) = List.append
( List.map other [ Aux_top(l,co,ch/2,true,f); Aux_top(l,co,ch/2,false,f) ] )
( if ch > 1 then List.append
( List.map other [ Aux_top(l,co,-ch/2,true,f); Aux_top(l,co,-ch/2,false,f) ] )
( aux_top_flavors (f,l,co,(ch-2)) )
else [] )
let external_flavors () =
[ "1st Generation", ThoList.flatmap family [1; -1];
"2nd Generation", ThoList.flatmap family [2; -2];
"3rd Generation", ThoList.flatmap family [3; -3];
"Gauge Bosons", List.map gauge_boson [Ga; Z; Wp; Wm; Gl];
"Higgs", List.map other [H];
"Goldstone Bosons", List.map other [Phip; Phim; Phi0] ]
let flavors () = List.append
( ThoList.flatmap snd (external_flavors ()) )
( ThoList.flatmap aux_top_flavors
[ (TTGG,2,1,1); (TBWA,2,0,2); (TBWZ,2,0,2); (TTWW,2,0,1); (BBWW,2,0,1);
- (QGUG,1,1,1); (QBUB,1,0,1); (QW,1,0,3); (DL,0,0,3); (DR,0,0,3) ] )
+ (QGUG,1,1,1); (QBUB,1,0,1); (QW,1,0,3); (DL,0,0,3); (DR,0,0,3);
+ (QUQD1L,0,0,3); (QUQD1R,0,0,3); (QUQD8L,0,1,3); (QUQD8R,0,1,3) ] )
let spinor n =
if n >= 0 then
Spinor
else
ConjSpinor
let lorentz_aux = function
| 2 -> Tensor_1
| 1 -> Vector
| 0 -> Scalar
| _ -> invalid_arg ("SM.lorentz_aux: wrong value")
let lorentz = function
| M f ->
begin match f with
| L n -> spinor n | N n -> spinor n
| U n -> spinor n | D n -> spinor n
end
| G f ->
begin match f with
| Ga | Gl -> Vector
| Wp | Wm | Z -> Massive_Vector
end
| O f ->
begin match f with
| Aux_top (l,_,_,_,_) -> lorentz_aux l
| _ -> Scalar
end
let color = function
| M (U n) -> Color.SUN (if n > 0 then 3 else -3)
| M (D n) -> Color.SUN (if n > 0 then 3 else -3)
| G Gl -> Color.AdjSUN 3
| O (Aux_top (_,co,_,_,_)) -> if co == 0 then Color.Singlet else Color.AdjSUN 3
| _ -> Color.Singlet
let prop_spinor n =
if n >= 0 then
Prop_Spinor
else
Prop_ConjSpinor
let prop_aux = function
| 2 -> Aux_Tensor_1
| 1 -> Aux_Vector
| 0 -> Aux_Scalar
| _ -> invalid_arg ("SM.prop_aux: wrong value")
let propagator = function
| M f ->
begin match f with
| L n -> prop_spinor n | N n -> prop_spinor n
| U n -> prop_spinor n | D n -> prop_spinor n
end
| G f ->
begin match f with
| Ga | Gl -> Prop_Feynman
| Wp | Wm | Z -> Prop_Unitarity
end
| O f ->
begin match f with
| Phip | Phim | Phi0 -> Only_Insertion
| H -> Prop_Scalar
| Aux_top (l,_,_,_,_) -> prop_aux l
end
(* Optionally, ask for the fudge factor treatment for the widths of
charged particles. Currently, this only applies to $W^\pm$ and top. *)
let width f =
if !use_fudged_width then
match f with
| G Wp | G Wm | M (U 3) | M (U (-3)) -> Fudged
| _ -> !default_width
else
!default_width
let goldstone = function
| G f ->
begin match f with
| Wp -> Some (O Phip, Coupling.Const 1)
| Wm -> Some (O Phim, Coupling.Const 1)
| Z -> Some (O Phi0, Coupling.Const 1)
| _ -> None
end
| _ -> None
let conjugate = function
| M f ->
M (begin match f with
| L n -> L (-n) | N n -> N (-n)
| U n -> U (-n) | D n -> D (-n)
end)
| G f ->
G (begin match f with
| Gl -> Gl | Ga -> Ga | Z -> Z
| Wp -> Wm | Wm -> Wp
end)
| O f ->
O (begin match f with
| Phip -> Phim | Phim -> Phip | Phi0 -> Phi0
| H -> H
| Aux_top (l,co,ch,n,f) -> Aux_top (l,co,(-ch),(not n),f)
end)
let fermion = function
| M f ->
begin match f with
| L n -> if n > 0 then 1 else -1
| N n -> if n > 0 then 1 else -1
| U n -> if n > 0 then 1 else -1
| D n -> if n > 0 then 1 else -1
end
| G f ->
begin match f with
| Gl | Ga | Z | Wp | Wm -> 0
end
| O _ -> 0
(* Electrical charge, lepton number, baryon number. We could avoid the
rationals altogether by multiplying the first and last by 3 \ldots *)
module Ch = Charges.QQ
let ( // ) = Algebra.Small_Rational.make
let generation' = function
| 1 -> [ 1//1; 0//1; 0//1]
| 2 -> [ 0//1; 1//1; 0//1]
| 3 -> [ 0//1; 0//1; 1//1]
| -1 -> [-1//1; 0//1; 0//1]
| -2 -> [ 0//1; -1//1; 0//1]
| -3 -> [ 0//1; 0//1; -1//1]
| n -> invalid_arg ("SM.generation': " ^ string_of_int n)
let generation f =
if Flags.ckm_present then
[]
else
match f with
| M (L n | N n | U n | D n) -> generation' n
| G _ | O _ -> [0//1; 0//1; 0//1]
let charge = function
| M f ->
begin match f with
| L n -> if n > 0 then -1//1 else 1//1
| N n -> 0//1
| U n -> if n > 0 then 2//3 else -2//3
| D n -> if n > 0 then -1//3 else 1//3
end
| G f ->
begin match f with
| Gl | Ga | Z -> 0//1
| Wp -> 1//1
| Wm -> -1//1
end
| O f ->
begin match f with
| H | Phi0 -> 0//1
| Phip -> 1//1
| Phim -> -1//1
| Aux_top (_,_,ch,_,_) -> ch//1
end
let lepton = function
| M f ->
begin match f with
| L n | N n -> if n > 0 then 1//1 else -1//1
| U _ | D _ -> 0//1
end
| G _ | O _ -> 0//1
let baryon = function
| M f ->
begin match f with
| L _ | N _ -> 0//1
| U n | D n -> if n > 0 then 1//1 else -1//1
end
| G _ | O _ -> 0//1
let charges f =
[ charge f; lepton f; baryon f] @ generation f
type constant =
| Unit | Half | Pi | Alpha_QED | Sin2thw
| Sinthw | Costhw | E | G_weak | I_G_weak | Vev
| Q_lepton | Q_up | Q_down | G_CC | G_CCQ of int*int
| G_NC_neutrino | G_NC_lepton | G_NC_up | G_NC_down
| G_TVA_ttA | G_TVA_bbA
| G_VLR_ttZ | G_TVA_ttZ | G_TVA_bbZ
| G_VLR_btW | G_VLR_tbW
| G_TLR_btW | G_TRL_tbW
| G_TLR_btWZ | G_TRL_tbWZ
| G_TLR_btWA | G_TRL_tbWA
| G_TVA_ttWW | G_TVA_bbWW
| G_TVA_ttG | G_TVA_ttGG
| G_SP_ttH
| G_VLR_qGuG | G_VLR_qBuB
| G_VLR_qBuB_u | G_VLR_qBuB_d | G_VLR_qBuB_e | G_VL_qBuB_n
| G_VL_qW | G_VL_qW_u | G_VL_qW_d
| G_SL_DttR | G_SR_DttR | G_SL_DttL | G_SLR_DbtR | G_SL_DbtL
+ | C_quqd1R_bt | C_quqd1R_tb | C_quqd1L_bt | C_quqd1L_tb
+ | C_quqd8R_bt | C_quqd8R_tb | C_quqd8L_bt | C_quqd8L_tb
| I_Q_W | I_G_ZWW
| G_WWWW | G_ZZWW | G_AZWW | G_AAWW
| I_G1_AWW | I_G1_ZWW
| I_G1_plus_kappa_plus_G4_AWW
| I_G1_plus_kappa_plus_G4_ZWW
| I_G1_plus_kappa_minus_G4_AWW
| I_G1_plus_kappa_minus_G4_ZWW
| I_G1_minus_kappa_plus_G4_AWW
| I_G1_minus_kappa_plus_G4_ZWW
| I_G1_minus_kappa_minus_G4_AWW
| I_G1_minus_kappa_minus_G4_ZWW
| I_lambda_AWW | I_lambda_ZWW
| G5_AWW | G5_ZWW
| I_kappa5_AWW | I_kappa5_ZWW
| I_lambda5_AWW | I_lambda5_ZWW
| Alpha_WWWW0 | Alpha_ZZWW1 | Alpha_WWWW2
| Alpha_ZZWW0 | Alpha_ZZZZ
| D_Alpha_ZZWW0_S | D_Alpha_ZZWW0_T | D_Alpha_ZZWW1_S
| D_Alpha_ZZWW1_T | D_Alpha_ZZWW1_U | D_Alpha_WWWW0_S
| D_Alpha_WWWW0_T | D_Alpha_WWWW0_U | D_Alpha_WWWW2_S
| D_Alpha_WWWW2_T | D_Alpha_ZZZZ_S | D_Alpha_ZZZZ_T
| G_HWW | G_HHWW | G_HZZ | G_HHZZ
| G_Htt | G_Hbb | G_Hcc | G_Hmm | G_Htautau | G_H3 | G_H4
| G_HGaZ | G_HGaGa | G_Hgg
| G_HGaZ_anom | G_HGaGa_anom | G_HZZ_anom | G_HWW_anom
| G_HGaZ_u | G_HZZ_u | G_HWW_u
| Gs | I_Gs | G2
| Mass of flavor | Width of flavor
| K_Matrix_Coeff of int | K_Matrix_Pole of int
(* \begin{dubious}
The current abstract syntax for parameter dependencies is admittedly
tedious. Later, there will be a parser for a convenient concrete syntax
as a part of a concrete syntax for models. But as these examples show,
it should include simple functions.
\end{dubious} *)
(* \begin{subequations}
\begin{align}
\alpha_{\text{QED}} &= \frac{1}{137.0359895} \\
\sin^2\theta_w &= 0.23124
\end{align}
\end{subequations} *)
let input_parameters =
[ Alpha_QED, 1. /. 137.0359895;
Sin2thw, 0.23124;
Mass (G Z), 91.187;
Mass (M (N 1)), 0.0; Mass (M (L 1)), 0.51099907e-3;
Mass (M (N 2)), 0.0; Mass (M (L 2)), 0.105658389;
Mass (M (N 3)), 0.0; Mass (M (L 3)), 1.77705;
Mass (M (U 1)), 5.0e-3; Mass (M (D 1)), 3.0e-3;
Mass (M (U 2)), 1.2; Mass (M (D 2)), 0.1;
Mass (M (U 3)), 174.0; Mass (M (D 3)), 4.2 ]
(* \begin{subequations}
\begin{align}
e &= \sqrt{4\pi\alpha} \\
\sin\theta_w &= \sqrt{\sin^2\theta_w} \\
\cos\theta_w &= \sqrt{1-\sin^2\theta_w} \\
g &= \frac{e}{\sin\theta_w} \\
m_W &= \cos\theta_w m_Z \\
v &= \frac{2m_W}{g} \\
g_{CC} =
-\frac{g}{2\sqrt2} &= -\frac{e}{2\sqrt2\sin\theta_w} \\
Q_{\text{lepton}} =
-q_{\text{lepton}}e &= e \\
Q_{\text{up}} =
-q_{\text{up}}e &= -\frac{2}{3}e \\
Q_{\text{down}} =
-q_{\text{down}}e &= \frac{1}{3}e \\
\ii q_We =
\ii g_{\gamma WW} &= \ii e \\
\ii g_{ZWW} &= \ii g \cos\theta_w \\
\ii g_{WWW} &= \ii g
\end{align}
\end{subequations} *)
(* \begin{dubious}
\ldots{} to be continued \ldots{}
The quartic couplings can't be correct, because the dimensions are wrong!
\begin{subequations}
\begin{align}
g_{HWW} &= g m_W = 2 \frac{m_W^2}{v}\\
g_{HHWW} &= 2 \frac{m_W^2}{v^2} = \frac{g^2}{2} \\
g_{HZZ} &= \frac{g}{\cos\theta_w}m_Z \\
g_{HHZZ} &= 2 \frac{m_Z^2}{v^2} = \frac{g^2}{2\cos\theta_w} \\
g_{Htt} &= \lambda_t \\
g_{Hbb} &= \lambda_b=\frac{m_b}{m_t}\lambda_t \\
g_{H^3} &= - \frac{3g}{2}\frac{m_H^2}{m_W} = - 3 \frac{m_H^2}{v}
g_{H^4} &= - \frac{3g^2}{4} \frac{m_W^2}{v^2} = -3 \frac{m_H^2}{v^2}
\end{align}
\end{subequations}
\end{dubious} *)
let derived_parameters =
[ Real E, Sqrt (Prod [Const 4; Atom Pi; Atom Alpha_QED]);
Real Sinthw, Sqrt (Atom Sin2thw);
Real Costhw, Sqrt (Diff (Const 1, Atom Sin2thw));
Real G_weak, Quot (Atom E, Atom Sinthw);
Real (Mass (G Wp)), Prod [Atom Costhw; Atom (Mass (G Z))];
Real Vev, Quot (Prod [Const 2; Atom (Mass (G Wp))], Atom G_weak);
Real Q_lepton, Atom E;
Real Q_up, Prod [Quot (Const (-2), Const 3); Atom E];
Real Q_down, Prod [Quot (Const 1, Const 3); Atom E];
Real G_CC, Neg (Quot (Atom G_weak, Prod [Const 2; Sqrt (Const 2)]));
Complex I_Q_W, Prod [I; Atom E];
Complex I_G_weak, Prod [I; Atom G_weak];
Complex I_G_ZWW, Prod [I; Atom G_weak; Atom Costhw] ]
(* \begin{equation}
- \frac{g}{2\cos\theta_w}
\end{equation} *)
let g_over_2_costh =
Quot (Neg (Atom G_weak), Prod [Const 2; Atom Costhw])
(* \begin{subequations}
\begin{align}
- \frac{g}{2\cos\theta_w} g_V
&= - \frac{g}{2\cos\theta_w} (T_3 - 2 q \sin^2\theta_w) \\
- \frac{g}{2\cos\theta_w} g_A
&= - \frac{g}{2\cos\theta_w} T_3
\end{align}
\end{subequations} *)
let nc_coupling c t3 q =
(Real_Array c,
[Prod [g_over_2_costh; Diff (t3, Prod [Const 2; q; Atom Sin2thw])];
Prod [g_over_2_costh; t3]])
let half = Quot (Const 1, Const 2)
let derived_parameter_arrays =
[ nc_coupling G_NC_neutrino half (Const 0);
nc_coupling G_NC_lepton (Neg half) (Const (-1));
nc_coupling G_NC_up half (Quot (Const 2, Const 3));
nc_coupling G_NC_down (Neg half) (Quot (Const (-1), Const 3)) ]
let parameters () =
{ input = input_parameters;
derived = derived_parameters;
derived_arrays = derived_parameter_arrays }
module F = Modeltools.Fusions (struct
type f = flavor
type c = constant
let compare = compare
let conjugate = conjugate
end)
(* \begin{equation}
\mathcal{L}_{\textrm{EM}} =
- e \sum_i q_i \bar\psi_i\fmslash{A}\psi_i
\end{equation} *)
let mgm ((m1, g, m2), fbf, c) = ((M m1, G g, M m2), fbf, c)
let mom ((m1, o, m2), fbf, c) = ((M m1, O o, M m2), fbf, c)
let electromagnetic_currents n =
List.map mgm
[ ((L (-n), Ga, L n), FBF (1, Psibar, V, Psi), Q_lepton);
((U (-n), Ga, U n), FBF (1, Psibar, V, Psi), Q_up);
((D (-n), Ga, D n), FBF (1, Psibar, V, Psi), Q_down) ]
let color_currents n =
List.map mgm
[ ((U (-n), Gl, U n), FBF ((-1), Psibar, V, Psi), Gs);
((D (-n), Gl, D n), FBF ((-1), Psibar, V, Psi), Gs) ]
(* \begin{equation}
\mathcal{L}_{\textrm{NC}} =
- \frac{g}{2\cos\theta_W}
\sum_i \bar\psi_i\fmslash{Z}(g_V^i-g_A^i\gamma_5)\psi_i
\end{equation} *)
let neutral_currents n =
List.map mgm
[ ((L (-n), Z, L n), FBF (1, Psibar, VA, Psi), G_NC_lepton);
((N (-n), Z, N n), FBF (1, Psibar, VA, Psi), G_NC_neutrino);
((U (-n), Z, U n), FBF (1, Psibar, VA, Psi), G_NC_up);
((D (-n), Z, D n), FBF (1, Psibar, VA, Psi), G_NC_down) ]
(* \begin{equation}
\mathcal{L}_{\textrm{CC}} =
- \frac{g}{2\sqrt2} \sum_i \bar\psi_i
(T^+\fmslash{W}^+ + T^-\fmslash{W}^-)(1-\gamma_5)\psi_i
\end{equation} *)
let charged_currents' n =
List.map mgm
[ ((L (-n), Wm, N n), FBF (1, Psibar, VL, Psi), G_CC);
((N (-n), Wp, L n), FBF (1, Psibar, VL, Psi), G_CC) ]
let charged_currents'' n =
List.map mgm
[ ((D (-n), Wm, U n), FBF (1, Psibar, VL, Psi), G_CC);
((U (-n), Wp, D n), FBF (1, Psibar, VL, Psi), G_CC) ]
let charged_currents_triv =
ThoList.flatmap charged_currents' [1;2;3] @
ThoList.flatmap charged_currents'' [1;2;3]
let charged_currents_ckm =
let charged_currents_2 n1 n2 =
List.map mgm
[ ((D (-n1), Wm, U n2), FBF (1, Psibar, VL, Psi), G_CCQ (n2,n1));
((U (-n1), Wp, D n2), FBF (1, Psibar, VL, Psi), G_CCQ (n1,n2)) ] in
ThoList.flatmap charged_currents' [1;2;3] @
List.flatten (Product.list2 charged_currents_2 [1;2;3] [1;2;3])
let yukawa =
[ ((M (U (-3)), O H, M (U 3)), FBF (1, Psibar, S, Psi), G_Htt);
((M (D (-3)), O H, M (D 3)), FBF (1, Psibar, S, Psi), G_Hbb);
((M (U (-2)), O H, M (U 2)), FBF (1, Psibar, S, Psi), G_Hcc);
((M (L (-3)), O H, M (L 3)), FBF (1, Psibar, S, Psi), G_Htautau) ] @
if Flags.higgs_hmm then
[ ((M (L (-2)), O H, M (L 2)), FBF (1, Psibar, S, Psi), G_Hmm)]
else
[]
(* \begin{equation}
\mathcal{L}_{\textrm{TGC}} =
- e \partial_\mu A_\nu W_+^\mu W_-^\nu + \ldots
- e \cot\theta_w \partial_\mu Z_\nu W_+^\mu W_-^\nu + \ldots
\end{equation} *)
let tgc ((g1, g2, g3), t, c) = ((G g1, G g2, G g3), t, c)
let standard_triple_gauge =
List.map tgc
[ ((Ga, Wm, Wp), Gauge_Gauge_Gauge 1, I_Q_W);
((Z, Wm, Wp), Gauge_Gauge_Gauge 1, I_G_ZWW);
((Gl, Gl, Gl), Gauge_Gauge_Gauge 1, I_Gs)]
(* \begin{multline}
\mathcal{L}_{\textrm{TGC}}(g_1,\kappa)
= g_1 \mathcal{L}_T(V,W^+,W^-) \\
+ \frac{\kappa+g_1}{2} \Bigl(\mathcal{L}_T(W^-,V,W^+)
- \mathcal{L}_T(W^+,V,W^-)\Bigr)\\
+ \frac{\kappa-g_1}{2} \Bigl(\mathcal{L}_L(W^-,V,W^+)
- \mathcal{L}_T(W^+,V,W^-)\Bigr)
\end{multline} *)
(* \begin{dubious}
The whole thing in the LEP2 workshop notation:
\begin{multline}
\ii\mathcal{L}_{\textrm{TGC},V} / g_{WWV} = \\
g_1^V V^\mu (W^-_{\mu\nu}W^{+,\nu}-W^+_{\mu\nu}W^{-,\nu})
+ \kappa_V W^+_\mu W^-_\nu V^{\mu\nu}
+ \frac{\lambda_V}{m_W^2} V_{\mu\nu}
W^-_{\rho\mu} W^{+,\hphantom{\nu}\rho}_{\hphantom{+,}\nu} \\
+ \ii g_5^V \epsilon_{\mu\nu\rho\sigma}
\left( (\partial^\rho W^{-,\mu}) W^{+,\nu}
- W^{-,\mu}(\partial^\rho W^{+,\nu}) \right) V^\sigma \\
+ \ii g_4^V W^-_\mu W^+_\nu (\partial^\mu V^\nu + \partial^\nu V^\mu)
- \frac{\tilde\kappa_V}{2} W^-_\mu W^+_\nu \epsilon^{\mu\nu\rho\sigma}
V_{\rho\sigma}
- \frac{\tilde\lambda_V}{2m_W^2}
W^-_{\rho\mu} W^{+,\mu}_{\hphantom{+,\mu}\nu} \epsilon^{\nu\rho\alpha\beta}
V_{\alpha\beta}
\end{multline}
using the conventions of Itzykson and Zuber with $\epsilon^{0123} = +1$.
\end{dubious} *)
(* \begin{dubious}
This is equivalent to the notation of Hagiwara et al.~\cite{HPZH87}, if we
remember that they have opposite signs for~$g_{WWV}$:
\begin{multline}
\mathcal{L}_{WWV} / (-g_{WWV}) = \\
\ii g_1^V \left( W^\dagger_{\mu\nu} W^\mu
- W^\dagger_\mu W^\mu_{\hphantom{\mu}\nu} \right) V^\nu
+ \ii \kappa_V W^\dagger_\mu W_\nu V^{\mu\nu}
+ \ii \frac{\lambda_V}{m_W^2}
W^\dagger_{\lambda\mu} W^\mu_{\hphantom{\mu}\nu} V^{\nu\lambda} \\
- g_4^V W^\dagger_\mu W_\nu
\left(\partial^\mu V^\nu + \partial^\nu V^\mu \right)
+ g_5^V \epsilon^{\mu\nu\lambda\sigma}
\left( W^\dagger_\mu \stackrel{\leftrightarrow}{\partial_\lambda}
W_\nu \right) V_\sigma\\
+ \ii \tilde\kappa_V W^\dagger_\mu W_\nu \tilde{V}^{\mu\nu}
+ \ii\frac{\tilde\lambda_V}{m_W^2}
W^\dagger_{\lambda\mu} W^\mu_{\hphantom{\mu}\nu} \tilde{V}^{\nu\lambda}
\end{multline}
Here $V^\mu$ stands for either the photon or the~$Z$ field, $W^\mu$ is the
$W^-$ field, $W_{\mu\nu} = \partial_\mu W_\nu - \partial_\nu W_\mu$,
$V_{\mu\nu} = \partial_\mu V_\nu - \partial_\nu V_\mu$, and
$\tilde{V}_{\mu\nu} = \frac{1}{2} \epsilon_{\mu\nu\lambda\sigma}
V^{\lambda\sigma}$.
\end{dubious} *)
let anomalous_triple_gauge =
List.map tgc
[ ((Ga, Wm, Wp), Dim4_Vector_Vector_Vector_T (-1),
I_G1_AWW);
((Z, Wm, Wp), Dim4_Vector_Vector_Vector_T (-1),
I_G1_ZWW);
((Wm, Ga, Wp), Dim4_Vector_Vector_Vector_T 1,
I_G1_plus_kappa_minus_G4_AWW);
((Wm, Z, Wp), Dim4_Vector_Vector_Vector_T 1,
I_G1_plus_kappa_minus_G4_ZWW);
((Wp, Ga, Wm), Dim4_Vector_Vector_Vector_T (-1),
I_G1_plus_kappa_plus_G4_AWW);
((Wp, Z, Wm), Dim4_Vector_Vector_Vector_T (-1),
I_G1_plus_kappa_plus_G4_ZWW);
((Wm, Ga, Wp), Dim4_Vector_Vector_Vector_L (-1),
I_G1_minus_kappa_plus_G4_AWW);
((Wm, Z, Wp), Dim4_Vector_Vector_Vector_L (-1),
I_G1_minus_kappa_plus_G4_ZWW);
((Wp, Ga, Wm), Dim4_Vector_Vector_Vector_L 1,
I_G1_minus_kappa_minus_G4_AWW);
((Wp, Z, Wm), Dim4_Vector_Vector_Vector_L 1,
I_G1_minus_kappa_minus_G4_ZWW);
((Ga, Wm, Wp), Dim4_Vector_Vector_Vector_L5 (-1),
I_kappa5_AWW);
((Z, Wm, Wp), Dim4_Vector_Vector_Vector_L5 (-1),
I_kappa5_ZWW);
((Ga, Wm, Wp), Dim4_Vector_Vector_Vector_T5 (-1),
G5_AWW);
((Z, Wm, Wp), Dim4_Vector_Vector_Vector_T5 (-1),
G5_ZWW);
((Ga, Wp, Wm), Dim6_Gauge_Gauge_Gauge (-1),
I_lambda_AWW);
((Z, Wp, Wm), Dim6_Gauge_Gauge_Gauge (-1),
I_lambda_ZWW);
((Ga, Wp, Wm), Dim6_Gauge_Gauge_Gauge_5 (-1),
I_lambda5_AWW);
((Z, Wp, Wm), Dim6_Gauge_Gauge_Gauge_5 (-1),
I_lambda5_ZWW) ]
let triple_gauge =
if Flags.triple_anom then
anomalous_triple_gauge
else
standard_triple_gauge
(* \begin{equation}
\mathcal{L}_{\textrm{QGC}} =
- g^2 W_{+,\mu} W_{-,\nu} W_+^\mu W_-^\nu + \ldots
\end{equation} *)
(* Actually, quartic gauge couplings are a little bit more straightforward
using auxiliary fields. Here we have to impose the antisymmetry manually:
\begin{subequations}
\begin{multline}
(W^{+,\mu}_1 W^{-,\nu}_2 - W^{+,\nu}_1 W^{-,\mu}_2)
(W^+_{3,\mu} W^-_{4,\nu} - W^+_{3,\nu} W^-_{4,\mu}) \\
= 2(W^+_1W^+_3)(W^-_2W^-_4) - 2(W^+_1W^-_4)(W^-_2W^+_3)
\end{multline}
also ($V$ can be $A$ or $Z$)
\begin{multline}
(W^{+,\mu}_1 V^\nu_2 - W^{+,\nu}_1 V^\mu_2)
(W^-_{3,\mu} V_{4,\nu} - W^-_{3,\nu} V_{4,\mu}) \\
= 2(W^+_1W^-_3)(V_2V_4) - 2(W^+_1V_4)(V_2W^-_3)
\end{multline}
\end{subequations} *)
(* \begin{subequations}
\begin{multline}
W^{+,\mu} W^{-,\nu} W^+_\mu W^-_\nu
\end{multline}
\end{subequations} *)
let qgc ((g1, g2, g3, g4), t, c) = ((G g1, G g2, G g3, G g4), t, c)
let gauge4 = Vector4 [(2, C_13_42); (-1, C_12_34); (-1, C_14_23)]
let minus_gauge4 = Vector4 [(-2, C_13_42); (1, C_12_34); (1, C_14_23)]
let standard_quartic_gauge =
List.map qgc
[ (Wm, Wp, Wm, Wp), gauge4, G_WWWW;
(Wm, Z, Wp, Z), minus_gauge4, G_ZZWW;
(Wm, Z, Wp, Ga), minus_gauge4, G_AZWW;
(Wm, Ga, Wp, Ga), minus_gauge4, G_AAWW;
(Gl, Gl, Gl, Gl), gauge4, G2 ]
(* \begin{subequations}
\begin{align}
\mathcal{L}_4
&= \alpha_4 \left( \frac{g^4}{2}\left( (W^+_\mu W^{-,\mu})^2
+ W^+_\mu W^{+,\mu} W^-_\mu W^{-,\mu}
\right)\right.\notag \\
&\qquad\qquad\qquad \left.
+ \frac{g^4}{\cos^2\theta_w} W^+_\mu Z^\mu W^-_\nu Z^\nu
+ \frac{g^4}{4\cos^4\theta_w} (Z_\mu Z^\mu)^2 \right) \\
\mathcal{L}_5
&= \alpha_5 \left( g^4 (W^+_\mu W^{-,\mu})^2
+ \frac{g^4}{\cos^2\theta_w} W^+_\mu W^{-,\mu} Z_\nu Z^\nu
+ \frac{g^4}{4\cos^4\theta_w} (Z_\mu Z^\mu)^2 \right)
\end{align}
\end{subequations}
or
\begin{multline}
\mathcal{L}_4 + \mathcal{L}_5
= (\alpha_4+2\alpha_5) g^4 \frac{1}{2} (W^+_\mu W^{-,\mu})^2 \\
+ 2\alpha_4 g^4 \frac{1}{4} W^+_\mu W^{+,\mu} W^-_\mu W^{-,\mu}
+ \alpha_4 \frac{g^4}{\cos^2\theta_w} W^+_\mu Z^\mu W^-_\nu Z^\nu \\
+ 2\alpha_5 \frac{g^4}{\cos^2\theta_w} \frac{1}{2} W^+_\mu W^{-,\mu} Z_\nu Z^\nu
+ (2\alpha_4 + 2\alpha_5) \frac{g^4}{\cos^4\theta_w} \frac{1}{8} (Z_\mu Z^\mu)^2
\end{multline}
and therefore
\begin{subequations}
\begin{align}
\alpha_{(WW)_0} &= (\alpha_4+2\alpha_5) g^4 \\
\alpha_{(WW)_2} &= 2\alpha_4 g^4 \\
\alpha_{(WZ)_0} &= 2\alpha_5 \frac{g^4}{\cos^2\theta_w} \\
\alpha_{(WZ)_1} &= \alpha_4 \frac{g^4}{\cos^2\theta_w} \\
\alpha_{ZZ} &= (2\alpha_4 + 2\alpha_5) \frac{g^4}{\cos^4\theta_w}
\end{align}
\end{subequations} *)
let anomalous_quartic_gauge =
if Flags.quartic_anom then
List.map qgc
[ ((Wm, Wm, Wp, Wp),
Vector4 [(1, C_13_42); (1, C_14_23)], Alpha_WWWW0);
((Wm, Wm, Wp, Wp),
Vector4 [1, C_12_34], Alpha_WWWW2);
((Wm, Wp, Z, Z),
Vector4 [1, C_12_34], Alpha_ZZWW0);
((Wm, Wp, Z, Z),
Vector4 [(1, C_13_42); (1, C_14_23)], Alpha_ZZWW1);
((Z, Z, Z, Z),
Vector4 [(1, C_12_34); (1, C_13_42); (1, C_14_23)], Alpha_ZZZZ) ]
else
[]
(* In any diagonal channel~$\chi$, the scattering amplitude~$a_\chi(s)$ is
unitary iff\footnote{%
Trivial proof:
\begin{equation}
-1 = \textrm{Im}\left(\frac{1}{a_\chi(s)}\right)
= \frac{\textrm{Im}(a_\chi^*(s))}{|a_\chi(s)|^2}
= - \frac{\textrm{Im}(a_\chi(s))}{|a_\chi(s)|^2}
\end{equation}
i.\,e.~$\textrm{Im}(a_\chi(s)) = |a_\chi(s)|^2$.}
\begin{equation}
\textrm{Im}\left(\frac{1}{a_\chi(s)}\right) = -1
\end{equation}
For a real perturbative scattering amplitude~$r_\chi(s)$ this can be
enforced easily--and arbitrarily--by
\begin{equation}
\frac{1}{a_\chi(s)} = \frac{1}{r_\chi(s)} - \mathrm{i}
\end{equation}
*)
let k_matrix_quartic_gauge =
if Flags.k_matrix then
List.map qgc
[ ((Wm, Wp, Wm, Wp), Vector4_K_Matrix_jr (0,
[(1, C_12_34)]), D_Alpha_WWWW0_S);
((Wm, Wp, Wm, Wp), Vector4_K_Matrix_jr (0,
[(1, C_14_23)]), D_Alpha_WWWW0_T);
((Wm, Wp, Wm, Wp), Vector4_K_Matrix_jr (0,
[(1, C_13_42)]), D_Alpha_WWWW0_U);
((Wp, Wm, Wp, Wm), Vector4_K_Matrix_jr (0,
[(1, C_12_34)]), D_Alpha_WWWW0_S);
((Wp, Wm, Wp, Wm), Vector4_K_Matrix_jr (0,
[(1, C_14_23)]), D_Alpha_WWWW0_T);
((Wp, Wm, Wp, Wm), Vector4_K_Matrix_jr (0,
[(1, C_13_42)]), D_Alpha_WWWW0_U);
((Wm, Wm, Wp, Wp), Vector4_K_Matrix_jr (0,
[(1, C_12_34)]), D_Alpha_WWWW2_S);
((Wm, Wm, Wp, Wp), Vector4_K_Matrix_jr (0,
[(1, C_13_42); (1, C_14_23)]), D_Alpha_WWWW2_T);
((Wm, Wp, Z, Z), Vector4_K_Matrix_jr (0,
[(1, C_12_34)]), D_Alpha_ZZWW0_S);
((Wm, Wp, Z, Z), Vector4_K_Matrix_jr (0,
[(1, C_13_42); (1, C_14_23)]), D_Alpha_ZZWW0_T);
((Wm, Z, Wp, Z), Vector4_K_Matrix_jr (0,
[(1, C_12_34)]), D_Alpha_ZZWW1_S);
((Wm, Z, Wp, Z), Vector4_K_Matrix_jr (0,
[(1, C_13_42)]), D_Alpha_ZZWW1_T);
((Wm, Z, Wp, Z), Vector4_K_Matrix_jr (0,
[(1, C_14_23)]), D_Alpha_ZZWW1_U);
((Wp, Z, Z, Wm), Vector4_K_Matrix_jr (1,
[(1, C_12_34)]), D_Alpha_ZZWW1_S);
((Wp, Z, Z, Wm), Vector4_K_Matrix_jr (1,
[(1, C_13_42)]), D_Alpha_ZZWW1_U);
((Wp, Z, Z, Wm), Vector4_K_Matrix_jr (1,
[(1, C_14_23)]), D_Alpha_ZZWW1_T);
((Z, Wp, Wm, Z), Vector4_K_Matrix_jr (2,
[(1, C_12_34)]), D_Alpha_ZZWW1_S);
((Z, Wp, Wm, Z), Vector4_K_Matrix_jr (2,
[(1, C_13_42)]), D_Alpha_ZZWW1_U);
((Z, Wp, Wm, Z), Vector4_K_Matrix_jr (2,
[(1, C_14_23)]), D_Alpha_ZZWW1_T);
((Z, Z, Z, Z), Vector4_K_Matrix_jr (0,
[(1, C_12_34)]), D_Alpha_ZZZZ_S);
((Z, Z, Z, Z), Vector4_K_Matrix_jr (0,
[(1, C_13_42); (1, C_14_23)]), D_Alpha_ZZZZ_T);
((Z, Z, Z, Z), Vector4_K_Matrix_jr (3,
[(1, C_14_23)]), D_Alpha_ZZZZ_S);
((Z, Z, Z, Z), Vector4_K_Matrix_jr (3,
[(1, C_13_42); (1, C_12_34)]), D_Alpha_ZZZZ_T)]
else
[]
(*i Thorsten's original implementation of the K matrix, which we keep since
it still might be usefull for the future.
let k_matrix_quartic_gauge =
if Flags.k_matrix then
List.map qgc
[ ((Wm, Wp, Wm, Wp), Vector4_K_Matrix_tho (0, [K_Matrix_Coeff 0,
K_Matrix_Pole 0]), Alpha_WWWW0);
((Wm, Wm, Wp, Wp), Vector4_K_Matrix_tho (0, [K_Matrix_Coeff 2,
K_Matrix_Pole 2]), Alpha_WWWW2);
((Wm, Wp, Z, Z), Vector4_K_Matrix_tho (0, [(K_Matrix_Coeff 0,
K_Matrix_Pole 0); (K_Matrix_Coeff 2,
K_Matrix_Pole 2)]), Alpha_ZZWW0);
((Wm, Z, Wp, Z), Vector4_K_Matrix_tho (0, [K_Matrix_Coeff 1,
K_Matrix_Pole 1]), Alpha_ZZWW1);
((Z, Z, Z, Z), Vector4_K_Matrix_tho (0, [K_Matrix_Coeff 0,
K_Matrix_Pole 0]), Alpha_ZZZZ) ]
else
[]
i*)
let quartic_gauge =
standard_quartic_gauge @ anomalous_quartic_gauge @ k_matrix_quartic_gauge
let standard_gauge_higgs =
[ ((O H, G Wp, G Wm), Scalar_Vector_Vector 1, G_HWW);
((O H, G Z, G Z), Scalar_Vector_Vector 1, G_HZZ) ]
let standard_gauge_higgs4 =
[ (O H, O H, G Wp, G Wm), Scalar2_Vector2 1, G_HHWW;
(O H, O H, G Z, G Z), Scalar2_Vector2 1, G_HHZZ ]
let standard_higgs =
[ (O H, O H, O H), Scalar_Scalar_Scalar 1, G_H3 ]
let standard_higgs4 =
[ (O H, O H, O H, O H), Scalar4 1, G_H4 ]
(* WK's couplings (apparently, he still intends to divide by
$\Lambda^2_{\text{EWSB}}=16\pi^2v_{\mathrm{F}}^2$):
\begin{subequations}
\begin{align}
\mathcal{L}^{\tau}_4 &=
\left\lbrack (\partial_{\mu}H)(\partial^{\mu}H)
+ \frac{g^2v_{\mathrm{F}}^2}{4} V_{\mu} V^{\mu} \right\rbrack^2 \\
\mathcal{L}^{\tau}_5 &=
\left\lbrack (\partial_{\mu}H)(\partial_{\nu}H)
+ \frac{g^2v_{\mathrm{F}}^2}{4} V_{\mu} V_{\nu} \right\rbrack^2
\end{align}
\end{subequations}
with
\begin{equation}
V_{\mu} V_{\nu} =
\frac{1}{2} \left( W^+_{\mu} W^-_{\nu} + W^+_{\nu} W^-_{\mu} \right)
+ \frac{1}{2\cos^2\theta_{w}} Z_{\mu} Z_{\nu}
\end{equation}
(note the symmetrization!), i.\,e.
\begin{subequations}
\begin{align}
\mathcal{L}_4 &= \alpha_4 \frac{g^4v_{\mathrm{F}}^4}{16} (V_{\mu} V_{\nu})^2 \\
\mathcal{L}_5 &= \alpha_5 \frac{g^4v_{\mathrm{F}}^4}{16} (V_{\mu} V^{\mu})^2
\end{align}
\end{subequations} *)
(* Breaking thinks up
\begin{subequations}
\begin{align}
\mathcal{L}^{\tau,H^4}_4 &=
\left\lbrack (\partial_{\mu}H)(\partial^{\mu}H) \right\rbrack^2 \\
\mathcal{L}^{\tau,H^4}_5 &=
\left\lbrack (\partial_{\mu}H)(\partial^{\mu}H) \right\rbrack^2
\end{align}
\end{subequations}
and
\begin{subequations}
\begin{align}
\mathcal{L}^{\tau,H^2V^2}_4 &= \frac{g^2v_{\mathrm{F}}^2}{2}
(\partial_{\mu}H)(\partial^{\mu}H) V_{\mu}V^{\mu} \\
\mathcal{L}^{\tau,H^2V^2}_5 &= \frac{g^2v_{\mathrm{F}}^2}{2}
(\partial_{\mu}H)(\partial_{\nu}H) V_{\mu}V_{\nu}
\end{align}
\end{subequations}
i.\,e.
\begin{subequations}
\begin{align}
\mathcal{L}^{\tau,H^2V^2}_4 &=
\frac{g^2v_{\mathrm{F}}^2}{2}
\left\lbrack
(\partial_{\mu}H)(\partial^{\mu}H) W^+_{\nu}W^{-,\nu}
+ \frac{1}{2\cos^2\theta_{w}} (\partial_{\mu}H)(\partial^{\mu}H) Z_{\nu} Z^{\nu}
\right\rbrack \\
\mathcal{L}^{\tau,H^2V^2}_5 &=
\frac{g^2v_{\mathrm{F}}^2}{2}
\left\lbrack
(W^{+,\mu}\partial_{\mu}H) (W^{-,\nu}\partial_{\nu}H)
+ \frac{1}{2\cos^2\theta_{w}} (Z^{\mu}\partial_{\mu}H)(Z^{\nu}\partial_{\nu}H)
\right\rbrack
\end{align}
\end{subequations} *)
(* \begin{multline}
\tau^4_8 \mathcal{L}^{\tau,H^2V^2}_4 + \tau^5_8 \mathcal{L}^{\tau,H^2V^2}_5 = \\
- \frac{g^2v_{\mathrm{F}}^2}{2} \Biggl\lbrack
2\tau^4_8
\frac{1}{2}(\ii\partial_{\mu}H)(\ii\partial^{\mu}H) W^+_{\nu}W^{-,\nu}
+ \tau^5_8
(W^{+,\mu}\ii\partial_{\mu}H) (W^{-,\nu}\ii\partial_{\nu}H) \\
+ \frac{2\tau^4_8}{\cos^2\theta_{w}}
\frac{1}{4} (\ii\partial_{\mu}H)(\ii\partial^{\mu}H) Z_{\nu} Z^{\nu}
+ \frac{\tau^5_8}{\cos^2\theta_{w}}
\frac{1}{2} (Z^{\mu}\ii\partial_{\mu}H)(Z^{\nu}\ii\partial_{\nu}H)
\Biggr\rbrack
\end{multline}
where the two powers of $\ii$ make the sign conveniently negative,
i.\,e.
\begin{subequations}
\begin{align}
\alpha_{(\partial H)^2W^2}^2 &= \tau^4_8 g^2v_{\mathrm{F}}^2\\
\alpha_{(\partial HW)^2}^2 &= \frac{\tau^5_8 g^2v_{\mathrm{F}}^2}{2} \\
\alpha_{(\partial H)^2Z^2}^2 &= \frac{\tau^4_8 g^2v_{\mathrm{F}}^2}{\cos^2\theta_{w}} \\
\alpha_{(\partial HZ)^2}^2 &=\frac{\tau^5_8 g^2v_{\mathrm{F}}^2}{2\cos^2\theta_{w}}
\end{align}
\end{subequations} *)
let anomalous_gauge_higgs =
[ (O H, G Ga, G Ga), Dim5_Scalar_Gauge2 1, G_HGaGa_anom;
(O H, G Ga, G Z), Dim5_Scalar_Gauge2 1, G_HGaZ_anom;
(O H, G Z, G Z), Dim5_Scalar_Gauge2 1, G_HZZ_anom;
(O H, G Wp, G Wm), Dim5_Scalar_Gauge2 1, G_HWW_anom;
(O H, G Ga, G Z), Dim5_Scalar_Vector_Vector_TU 1, G_HGaZ_u;
(O H, G Z, G Z), Dim5_Scalar_Vector_Vector_U 1, G_HZZ_u;
(O H, G Wp, G Wm), Dim5_Scalar_Vector_Vector_U 1, G_HWW_u
]
let anomalous_gauge_higgs4 =
[]
let anomalous_higgs =
[]
let higgs_triangle_vertices =
if Flags.higgs_triangle then
[ (O H, G Ga, G Ga), Dim5_Scalar_Gauge2 1, G_HGaGa;
(O H, G Ga, G Z), Dim5_Scalar_Gauge2 1, G_HGaZ;
(O H, G Gl, G Gl), Dim5_Scalar_Gauge2 1, G_Hgg ]
else
[]
let anomalous_higgs4 =
[]
let gauge_higgs =
if Flags.higgs_anom then
standard_gauge_higgs @ anomalous_gauge_higgs
else
standard_gauge_higgs
let gauge_higgs4 =
if Flags.higgs_anom then
standard_gauge_higgs4 @ anomalous_gauge_higgs4
else
standard_gauge_higgs4
let higgs =
if Flags.higgs_anom then
standard_higgs @ anomalous_higgs
else
standard_higgs
let higgs4 =
if Flags.higgs_anom then
standard_higgs4 @ anomalous_higgs4
else
standard_higgs4
let goldstone_vertices =
[ ((O Phi0, G Wm, G Wp), Scalar_Vector_Vector 1, I_G_ZWW);
((O Phip, G Ga, G Wm), Scalar_Vector_Vector 1, I_Q_W);
((O Phip, G Z, G Wm), Scalar_Vector_Vector 1, I_G_ZWW);
((O Phim, G Wp, G Ga), Scalar_Vector_Vector 1, I_Q_W);
((O Phim, G Wp, G Z), Scalar_Vector_Vector 1, I_G_ZWW) ]
(* Anomalous trilinear interactions $f_i f_j V$ and $ttH$:
\begin{equation}
\Delta\mathcal{L}_{tt\gamma} =
- e \frac{\upsilon}{\Lambda^2}
\bar{t} i\sigma^{\mu\nu} k_\nu (d_V(k^2) + i d_A(k^2) \gamma_5) t A_\mu
\end{equation} *)
let anomalous_ttA =
if Flags.top_anom then
[ ((M (U (-3)), G Ga, M (U 3)), FBF (1, Psibar, TVAM, Psi), G_TVA_ttA) ]
else
[]
(* \begin{equation}
\Delta\mathcal{L}_{bb\gamma} =
- e \frac{\upsilon}{\Lambda^2}
\bar{b} i\sigma^{\mu\nu} k_\nu (d_V(k^2) + i d_A(k^2) \gamma_5) b A_\mu
\end{equation} *)
let anomalous_bbA =
if Flags.top_anom then
[ ((M (D (-3)), G Ga, M (D 3)), FBF (1, Psibar, TVAM, Psi), G_TVA_bbA) ]
else
[]
(* \begin{equation}
\Delta\mathcal{L}_{ttg} =
- g_s \frac{\upsilon}{\Lambda^2}
\bar{t}\lambda^a i\sigma^{\mu\nu}k_\nu
(d_V(k^2)+id_A(k^2)\gamma_5)tG^a_\mu
\end{equation} *)
let anomalous_ttG =
if Flags.top_anom then
[ ((M (U (-3)), G Gl, M (U 3)), FBF (1, Psibar, TVAM, Psi), G_TVA_ttG) ]
else
[]
(* \begin{equation}
\Delta\mathcal{L}_{ttZ} =
- \frac{g}{2 c_W} \frac{\upsilon^2}{\Lambda^2}\left\lbrack
\bar{t} \fmslash{Z} (X_L(k^2) P_L + X_R(k^2) P_R) t
+ \bar{t}\frac{i\sigma^{\mu\nu}k_\nu}{m_Z}
(d_V(k^2)+id_A(k^2)\gamma_5)tZ_\mu\right\rbrack
\end{equation} *)
let anomalous_ttZ =
if Flags.top_anom then
[ ((M (U (-3)), G Z, M (U 3)), FBF (1, Psibar, VLRM, Psi), G_VLR_ttZ);
((M (U (-3)), G Z, M (U 3)), FBF (1, Psibar, TVAM, Psi), G_TVA_ttZ) ]
else
[]
(* \begin{equation}
\Delta\mathcal{L}_{bbZ} =
- \frac{g}{2 c_W} \frac{\upsilon^2}{\Lambda^2}
\bar{b}\frac{i\sigma^{\mu\nu}k_\nu}{m_Z}
(d_V(k^2)+id_A(k^2)\gamma_5)bZ_\mu
\end{equation} *)
let anomalous_bbZ =
if Flags.top_anom then
[ ((M (D (-3)), G Z, M (D 3)), FBF (1, Psibar, TVAM, Psi), G_TVA_bbZ) ]
else
[]
(* \begin{equation}
\Delta\mathcal{L}_{tbW} =
- \frac{g}{\sqrt{2}} \frac{\upsilon^2}{\Lambda^2}\left\lbrack
\bar{b}\fmslash{W}^-(V_L(k^2) P_L+V_R(k^2) P_R) t
+ \bar{b}\frac{i\sigma^{\mu\nu}k_\nu}{m_W}
(g_L(k^2)P_L+g_R(k^2)P_R)tW^-_\mu\right\rbrack
+ \textnormal{H.c.}
\end{equation} *)
let anomalous_tbW =
if Flags.top_anom then
[ ((M (D (-3)), G Wm, M (U 3)), FBF (1, Psibar, VLRM, Psi), G_VLR_btW);
((M (U (-3)), G Wp, M (D 3)), FBF (1, Psibar, VLRM, Psi), G_VLR_tbW);
((M (D (-3)), G Wm, M (U 3)), FBF (1, Psibar, TLRM, Psi), G_TLR_btW);
((M (U (-3)), G Wp, M (D 3)), FBF (1, Psibar, TRLM, Psi), G_TRL_tbW) ]
else
[]
(* \begin{equation}
\Delta\mathcal{L}_{ttH} =
- \frac{1}{\sqrt{2}} \bar{t} (Y_V(k^2)+iY_A(k^2)\gamma_5)t H
\end{equation} *)
let anomalous_ttH =
if Flags.top_anom then
[ ((M (U (-3)), O H, M (U 3)), FBF (1, Psibar, SPM, Psi), G_SP_ttH) ]
else
[]
(* quartic fermion-gauge interactions $f_i f_j V_1 V_2$ emerging from gauge-invariant
effective operators:
\begin{equation}
\Delta\mathcal{L}_{ttgg} =
- \frac{g_s^2}{2} f_{abc} \frac{\upsilon}{\Lambda^2}
\bar{t} \lambda^a \sigma^{\mu\nu}
(d_V(k^2)+id_A(k^2)\gamma_5)t G^b_\mu G^c_\nu
\end{equation} *)
let anomalous_ttGG =
if Flags.top_anom then
[ ((M (U (-3)), O (Aux_top (2,1,0,true,TTGG)), M (U 3)), FBF (1, Psibar, TVA, Psi), G_TVA_ttGG);
((O (Aux_top (2,1,0,false,TTGG)), G Gl, G Gl), Aux_Gauge_Gauge 1, I_Gs) ]
else
[]
(* \begin{equation}
\Delta\mathcal{L}_{tbWA} =
- i\sin\theta_w \frac{g^2}{2\sqrt{2}} \frac{\upsilon^2}{\Lambda^2}\left\lbrack
\bar{b}\frac{\sigma^{\mu\nu}}{m_W}
(g_L(k^2)P_L+g_R(k^2)P_R)t A_\mu W^-_\nu \right\rbrack
+ \textnormal{H.c.}
\end{equation} *)
let anomalous_tbWA =
if Flags.top_anom then
[ ((M (D (-3)), O (Aux_top (2,0,-1,true,TBWA)), M (U 3)), FBF (1, Psibar, TLR, Psi), G_TLR_btWA);
((O (Aux_top (2,0,1,false,TBWA)), G Ga, G Wm), Aux_Gauge_Gauge 1, I_G_weak);
((M (U (-3)), O (Aux_top (2,0,1,true,TBWA)), M (D 3)), FBF (1, Psibar, TRL, Psi), G_TRL_tbWA);
((O (Aux_top (2,0,-1,false,TBWA)), G Wp, G Ga), Aux_Gauge_Gauge 1, I_G_weak) ]
else
[]
(* \begin{equation}
\Delta\mathcal{L}_{tbWZ} =
- i\cos\theta_w \frac{g^2}{2\sqrt{2}} \frac{\upsilon^2}{\Lambda^2}\left\lbrack
\bar{b}\frac{\sigma^{\mu\nu}}{m_W}
(g_L(k^2)P_L+g_R(k^2)P_R)t Z_\mu W^-_\nu \right\rbrack
+ \textnormal{H.c.}
\end{equation} *)
let anomalous_tbWZ =
if Flags.top_anom then
[ ((M (D (-3)), O (Aux_top (2,0,-1,true,TBWZ)), M (U 3)), FBF (1, Psibar, TLR, Psi), G_TLR_btWZ);
((O (Aux_top (2,0,1,false,TBWZ)), G Z, G Wm), Aux_Gauge_Gauge 1, I_G_weak);
((M (U (-3)), O (Aux_top (2,0,1,true,TBWZ)), M (D 3)), FBF (1, Psibar, TRL, Psi), G_TRL_tbWZ);
((O (Aux_top (2,0,-1,false,TBWZ)), G Wp, G Z), Aux_Gauge_Gauge 1, I_G_weak) ]
else
[]
(* \begin{equation}
\Delta\mathcal{L}_{ttWW} =
- i \frac{g^2}{2} \frac{\upsilon^2}{\Lambda^2}
\bar{t} \frac{\sigma^{\mu\nu}}{m_W}
(d_V(k^2)+id_A(k^2)\gamma_5)t W^-_\mu W^+_\nu
\end{equation} *)
let anomalous_ttWW =
if Flags.top_anom then
[ ((M (U (-3)), O (Aux_top (2,0,0,true,TTWW)), M (U 3)), FBF (1, Psibar, TVA, Psi), G_TVA_ttWW);
((O (Aux_top (2,0,0,false,TTWW)), G Wm, G Wp), Aux_Gauge_Gauge 1, I_G_weak) ]
else
[]
(* \begin{equation}
\Delta\mathcal{L}_{bbWW} =
- i \frac{g^2}{2} \frac{\upsilon^2}{\Lambda^2}
\bar{b} \frac{\sigma^{\mu\nu}}{m_W}
(d_V(k^2)+id_A(k^2)\gamma_5)b W^-_\mu W^+_\nu
\end{equation} *)
let anomalous_bbWW =
if Flags.top_anom then
[ ((M (D (-3)), O (Aux_top (2,0,0,true,BBWW)), M (D 3)), FBF (1, Psibar, TVA, Psi), G_TVA_bbWW);
((O (Aux_top (2,0,0,false,BBWW)), G Wm, G Wp), Aux_Gauge_Gauge 1, I_G_weak) ]
else
[]
(* 4-fermion contact terms emerging from operator rewriting: *)
let anomalous_top_qGuG_tt =
[ ((M (U (-3)), O (Aux_top (1,1,0,true,QGUG)), M (U 3)), FBF (1, Psibar, VLR, Psi), G_VLR_qGuG) ]
let anomalous_top_qGuG_ff n =
List.map mom
[ ((U (-n), Aux_top (1,1,0,false,QGUG), U n), FBF (1, Psibar, V, Psi), Unit);
((D (-n), Aux_top (1,1,0,false,QGUG), D n), FBF (1, Psibar, V, Psi), Unit) ]
let anomalous_top_qGuG =
if Flags.top_anom_4f then
anomalous_top_qGuG_tt @ ThoList.flatmap anomalous_top_qGuG_ff [1;2;3]
else
[]
let anomalous_top_qBuB_tt =
[ ((M (U (-3)), O (Aux_top (1,0,0,true,QBUB)), M (U 3)), FBF (1, Psibar, VLR, Psi), G_VLR_qBuB) ]
let anomalous_top_qBuB_ff n =
List.map mom
[ ((U (-n), Aux_top (1,0,0,false,QBUB), U n), FBF (1, Psibar, VLR, Psi), G_VLR_qBuB_u);
((D (-n), Aux_top (1,0,0,false,QBUB), D n), FBF (1, Psibar, VLR, Psi), G_VLR_qBuB_d);
((L (-n), Aux_top (1,0,0,false,QBUB), L n), FBF (1, Psibar, VLR, Psi), G_VLR_qBuB_e);
((N (-n), Aux_top (1,0,0,false,QBUB), N n), FBF (1, Psibar, VL, Psi), G_VL_qBuB_n) ]
let anomalous_top_qBuB =
if Flags.top_anom_4f then
anomalous_top_qBuB_tt @ ThoList.flatmap anomalous_top_qBuB_ff [1;2;3]
else
[]
let anomalous_top_qW_tq =
[ ((M (U (-3)), O (Aux_top (1,0,0,true,QW)), M (U 3)), FBF (1, Psibar, VL, Psi), G_VL_qW);
((M (D (-3)), O (Aux_top (1,0,-1,true,QW)), M (U 3)), FBF (1, Psibar, VL, Psi), G_VL_qW);
((M (U (-3)), O (Aux_top (1,0,1,true,QW)), M (D 3)), FBF (1, Psibar, VL, Psi), G_VL_qW) ]
let anomalous_top_qW_ff n =
List.map mom
[ ((U (-n), Aux_top (1,0,0,false,QW), U n), FBF (1, Psibar, VL, Psi), G_VL_qW_u);
((D (-n), Aux_top (1,0,0,false,QW), D n), FBF (1, Psibar, VL, Psi), G_VL_qW_d);
((N (-n), Aux_top (1,0,0,false,QW), N n), FBF (1, Psibar, VL, Psi), G_VL_qW_u);
((L (-n), Aux_top (1,0,0,false,QW), L n), FBF (1, Psibar, VL, Psi), G_VL_qW_d);
((D (-n), Aux_top (1,0,-1,false,QW), U n), FBF (1, Psibar, VL, Psi), Half);
((U (-n), Aux_top (1,0,1,false,QW), D n), FBF (1, Psibar, VL, Psi), Half);
((L (-n), Aux_top (1,0,-1,false,QW), N n), FBF (1, Psibar, VL, Psi), Half);
((N (-n), Aux_top (1,0,1,false,QW), L n), FBF (1, Psibar, VL, Psi), Half) ]
let anomalous_top_qW =
if Flags.top_anom_4f then
anomalous_top_qW_tq @ ThoList.flatmap anomalous_top_qW_ff [1;2;3]
else
[]
let anomalous_top_DuDd =
if Flags.top_anom_4f then
[ ((M (U (-3)), O (Aux_top (0,0,0,true,DR)), M (U 3)), FBF (1, Psibar, SR, Psi), Half);
((M (U (-3)), O (Aux_top (0,0,0,false,DR)), M (U 3)), FBF (1, Psibar, SL, Psi), G_SL_DttR);
((M (D (-3)), O (Aux_top (0,0,0,false,DR)), M (D 3)), FBF (1, Psibar, SR, Psi), G_SR_DttR);
((M (U (-3)), O (Aux_top (0,0,0,true,DL)), M (U 3)), FBF (1, Psibar, SL, Psi), Half);
((M (D (-3)), O (Aux_top (0,0,0,false,DL)), M (D 3)), FBF (1, Psibar, SL, Psi), G_SL_DttL);
((M (D (-3)), O (Aux_top (0,0,-1,true,DR)), M (U 3)), FBF (1, Psibar, SR, Psi), Half);
((M (U (-3)), O (Aux_top (0,0,1,false,DR)), M (D 3)), FBF (1, Psibar, SLR, Psi), G_SLR_DbtR);
((M (D (-3)), O (Aux_top (0,0,-1,true,DL)), M (U 3)), FBF (1, Psibar, SL, Psi), Half);
((M (U (-3)), O (Aux_top (0,0,1,false,DL)), M (D 3)), FBF (1, Psibar, SL, Psi), G_SL_DbtL) ]
else
[]
+ let anomalous_top_quqd1_tq =
+ [ ((M (D (-3)), O (Aux_top (0,0,-1,true,QUQD1R)), M (U 3)), FBF (1, Psibar, SR, Psi), C_quqd1R_bt);
+ ((M (U (-3)), O (Aux_top (0,0, 1,true,QUQD1R)), M (D 3)), FBF (1, Psibar, SL, Psi), C_quqd1R_tb);
+ ((M (D (-3)), O (Aux_top (0,0,-1,true,QUQD1L)), M (U 3)), FBF (1, Psibar, SL, Psi), C_quqd1L_bt);
+ ((M (U (-3)), O (Aux_top (0,0, 1,true,QUQD1L)), M (D 3)), FBF (1, Psibar, SR, Psi), C_quqd1L_tb) ]
+
+ let anomalous_top_quqd1_ff n =
+ List.map mom
+ [ ((U (-n), Aux_top (0,0, 1,false,QUQD1R), D n), FBF (1, Psibar, SR, Psi), Half);
+ ((D (-n), Aux_top (0,0,-1,false,QUQD1R), U n), FBF (1, Psibar, SL, Psi), Half);
+ ((U (-n), Aux_top (0,0, 1,false,QUQD1L), D n), FBF (1, Psibar, SL, Psi), Half);
+ ((D (-n), Aux_top (0,0,-1,false,QUQD1L), U n), FBF (1, Psibar, SR, Psi), Half) ]
+
+ let anomalous_top_quqd1 =
+ if Flags.top_anom_4f then
+ anomalous_top_quqd1_tq @ ThoList.flatmap anomalous_top_quqd1_ff [1;2;3]
+ else
+ []
+
+ let anomalous_top_quqd8_tq =
+ [ ((M (D (-3)), O (Aux_top (0,1,-1,true,QUQD8R)), M (U 3)), FBF (1, Psibar, SR, Psi), C_quqd8R_bt);
+ ((M (U (-3)), O (Aux_top (0,1, 1,true,QUQD8R)), M (D 3)), FBF (1, Psibar, SL, Psi), C_quqd8R_tb);
+ ((M (D (-3)), O (Aux_top (0,1,-1,true,QUQD8L)), M (U 3)), FBF (1, Psibar, SL, Psi), C_quqd8L_bt);
+ ((M (U (-3)), O (Aux_top (0,1, 1,true,QUQD8L)), M (D 3)), FBF (1, Psibar, SR, Psi), C_quqd8L_tb) ]
+
+ let anomalous_top_quqd8_ff n =
+ List.map mom
+ [ ((U (-n), Aux_top (0,1, 1,false,QUQD8R), D n), FBF (1, Psibar, SR, Psi), Half);
+ ((D (-n), Aux_top (0,1,-1,false,QUQD8R), U n), FBF (1, Psibar, SL, Psi), Half);
+ ((U (-n), Aux_top (0,1, 1,false,QUQD8L), D n), FBF (1, Psibar, SL, Psi), Half);
+ ((D (-n), Aux_top (0,1,-1,false,QUQD8L), U n), FBF (1, Psibar, SR, Psi), Half) ]
+
+ let anomalous_top_quqd8 =
+ if Flags.top_anom_4f then
+ anomalous_top_quqd8_tq @ ThoList.flatmap anomalous_top_quqd8_ff [1;2;3]
+ else
+ []
+
let vertices3 =
(ThoList.flatmap electromagnetic_currents [1;2;3] @
ThoList.flatmap color_currents [1;2;3] @
ThoList.flatmap neutral_currents [1;2;3] @
(if Flags.ckm_present then
charged_currents_ckm
else
charged_currents_triv) @
yukawa @ triple_gauge @
gauge_higgs @ higgs @ higgs_triangle_vertices
@ goldstone_vertices @
anomalous_ttA @ anomalous_bbA @
anomalous_ttZ @ anomalous_bbZ @
anomalous_tbW @ anomalous_tbWA @ anomalous_tbWZ @
anomalous_ttWW @ anomalous_bbWW @
anomalous_ttG @ anomalous_ttGG @
anomalous_ttH @
anomalous_top_qGuG @ anomalous_top_qBuB @
- anomalous_top_qW @ anomalous_top_DuDd)
+ anomalous_top_qW @ anomalous_top_DuDd @
+ anomalous_top_quqd1 @ anomalous_top_quqd8)
let vertices4 =
quartic_gauge @ gauge_higgs4 @ higgs4
let vertices () = (vertices3, vertices4, [])
(* For efficiency, make sure that [F.of_vertices vertices] is
evaluated only once. *)
let table = F.of_vertices (vertices ())
let fuse2 = F.fuse2 table
let fuse3 = F.fuse3 table
let fuse = F.fuse table
let max_degree () = 4
let flavor_of_string = function
| "e-" -> M (L 1) | "e+" -> M (L (-1))
| "mu-" -> M (L 2) | "mu+" -> M (L (-2))
| "tau-" -> M (L 3) | "tau+" -> M (L (-3))
| "nue" -> M (N 1) | "nuebar" -> M (N (-1))
| "numu" -> M (N 2) | "numubar" -> M (N (-2))
| "nutau" -> M (N 3) | "nutaubar" -> M (N (-3))
| "u" -> M (U 1) | "ubar" -> M (U (-1))
| "c" -> M (U 2) | "cbar" -> M (U (-2))
| "t" -> M (U 3) | "tbar" -> M (U (-3))
| "d" -> M (D 1) | "dbar" -> M (D (-1))
| "s" -> M (D 2) | "sbar" -> M (D (-2))
| "b" -> M (D 3) | "bbar" -> M (D (-3))
| "g" | "gl" -> G Gl
| "A" -> G Ga | "Z" | "Z0" -> G Z
| "W+" -> G Wp | "W-" -> G Wm
| "H" -> O H
| "Aux_t_ttGG0" -> O (Aux_top (2,1, 0,true,TTGG)) | "Aux_ttGG0" -> O (Aux_top (2,1, 0,false,TTGG))
| "Aux_t_tbWA+" -> O (Aux_top (2,0, 1,true,TBWA)) | "Aux_tbWA+" -> O (Aux_top (2,0, 1,false,TBWA))
| "Aux_t_tbWA-" -> O (Aux_top (2,0,-1,true,TBWA)) | "Aux_tbWA-" -> O (Aux_top (2,0,-1,false,TBWA))
| "Aux_t_tbWZ+" -> O (Aux_top (2,0, 1,true,TBWZ)) | "Aux_tbWZ+" -> O (Aux_top (2,0, 1,false,TBWZ))
| "Aux_t_tbWZ-" -> O (Aux_top (2,0,-1,true,TBWZ)) | "Aux_tbWZ-" -> O (Aux_top (2,0,-1,false,TBWZ))
| "Aux_t_ttWW0" -> O (Aux_top (2,0, 0,true,TTWW)) | "Aux_ttWW0" -> O (Aux_top (2,0, 0,false,TTWW))
| "Aux_t_bbWW0" -> O (Aux_top (2,0, 0,true,BBWW)) | "Aux_bbWW0" -> O (Aux_top (2,0, 0,false,BBWW))
| "Aux_t_qGuG0" -> O (Aux_top (1,1, 0,true,QGUG)) | "Aux_qGuG0" -> O (Aux_top (1,1, 0,false,QGUG))
| "Aux_t_qBuB0" -> O (Aux_top (1,0, 0,true,QBUB)) | "Aux_qBuB0" -> O (Aux_top (1,0, 0,false,QBUB))
| "Aux_t_qW0" -> O (Aux_top (1,0, 0,true,QW)) | "Aux_qW0" -> O (Aux_top (1,0, 0,false,QW))
| "Aux_t_qW+" -> O (Aux_top (1,0, 1,true,QW)) | "Aux_qW+" -> O (Aux_top (1,0, 1,false,QW))
| "Aux_t_qW-" -> O (Aux_top (1,0,-1,true,QW)) | "Aux_qW-" -> O (Aux_top (1,0,-1,false,QW))
| "Aux_t_dL0" -> O (Aux_top (0,0, 0,true,DL)) | "Aux_dL0" -> O (Aux_top (0,0, 0,false,DL))
| "Aux_t_dL+" -> O (Aux_top (0,0, 1,true,DL)) | "Aux_dL+" -> O (Aux_top (0,0, 1,false,DL))
| "Aux_t_dL-" -> O (Aux_top (0,0,-1,true,DL)) | "Aux_dL-" -> O (Aux_top (0,0,-1,false,DL))
| "Aux_t_dR0" -> O (Aux_top (0,0, 0,true,DR)) | "Aux_dR0" -> O (Aux_top (0,0, 0,false,DR))
| "Aux_t_dR+" -> O (Aux_top (0,0, 1,true,DR)) | "Aux_dR+" -> O (Aux_top (0,0, 1,false,DR))
| "Aux_t_dR-" -> O (Aux_top (0,0,-1,true,DR)) | "Aux_dR-" -> O (Aux_top (0,0,-1,false,DR))
+ | "Aux_t_quqd1L+" -> O (Aux_top (0,0, 1,true,QUQD1L)) | "Aux_quqd1L+" -> O (Aux_top (0,0, 1,false,QUQD1L))
+ | "Aux_t_quqd1L-" -> O (Aux_top (0,0,-1,true,QUQD1L)) | "Aux_quqd1L-" -> O (Aux_top (0,0,-1,false,QUQD1L))
+ | "Aux_t_quqd1R+" -> O (Aux_top (0,0, 1,true,QUQD1R)) | "Aux_quqd1R+" -> O (Aux_top (0,0, 1,false,QUQD1R))
+ | "Aux_t_quqd1R-" -> O (Aux_top (0,0,-1,true,QUQD1R)) | "Aux_quqd1R-" -> O (Aux_top (0,0,-1,false,QUQD1R))
+ | "Aux_t_quqd8L+" -> O (Aux_top (0,0, 1,true,QUQD8L)) | "Aux_quqd8L+" -> O (Aux_top (0,0, 1,false,QUQD8L))
+ | "Aux_t_quqd8L-" -> O (Aux_top (0,0,-1,true,QUQD8L)) | "Aux_quqd8L-" -> O (Aux_top (0,0,-1,false,QUQD8L))
+ | "Aux_t_quqd8R+" -> O (Aux_top (0,0, 1,true,QUQD8R)) | "Aux_quqd8R+" -> O (Aux_top (0,0, 1,false,QUQD8R))
+ | "Aux_t_quqd8R-" -> O (Aux_top (0,0,-1,true,QUQD8R)) | "Aux_quqd8R-" -> O (Aux_top (0,0,-1,false,QUQD8R))
| _ -> invalid_arg "Modellib.SM.flavor_of_string"
let flavor_to_string = function
| M f ->
begin match f with
| L 1 -> "e-" | L (-1) -> "e+"
| L 2 -> "mu-" | L (-2) -> "mu+"
| L 3 -> "tau-" | L (-3) -> "tau+"
| L _ -> invalid_arg
"Modellib.SM.flavor_to_string: invalid lepton"
| N 1 -> "nue" | N (-1) -> "nuebar"
| N 2 -> "numu" | N (-2) -> "numubar"
| N 3 -> "nutau" | N (-3) -> "nutaubar"
| N _ -> invalid_arg
"Modellib.SM.flavor_to_string: invalid neutrino"
| U 1 -> "u" | U (-1) -> "ubar"
| U 2 -> "c" | U (-2) -> "cbar"
| U 3 -> "t" | U (-3) -> "tbar"
| U _ -> invalid_arg
"Modellib.SM.flavor_to_string: invalid up type quark"
| D 1 -> "d" | D (-1) -> "dbar"
| D 2 -> "s" | D (-2) -> "sbar"
| D 3 -> "b" | D (-3) -> "bbar"
| D _ -> invalid_arg
"Modellib.SM.flavor_to_string: invalid down type quark"
end
| G f ->
begin match f with
| Gl -> "gl"
| Ga -> "A" | Z -> "Z"
| Wp -> "W+" | Wm -> "W-"
end
| O f ->
begin match f with
| Phip -> "phi+" | Phim -> "phi-" | Phi0 -> "phi0"
| H -> "H"
| Aux_top (_,_,ch,n,v) -> "Aux_" ^ (if n then "t_" else "") ^ (
begin match v with
| TTGG -> "ttGG" | TBWA -> "tbWA" | TBWZ -> "tbWZ"
| TTWW -> "ttWW" | BBWW -> "bbWW"
| QGUG -> "qGuG" | QBUB -> "qBuB"
| QW -> "qW" | DL -> "dL" | DR -> "dR"
+ | QUQD1L -> "quqd1L" | QUQD1R -> "quqd1R"
+ | QUQD8L -> "quqd8L" | QUQD8R -> "quqd8R"
end ) ^ ( if ch > 0 then "+" else if ch < 0 then "-" else "0" )
end
let flavor_to_TeX = function
| M f ->
begin match f with
| L 1 -> "e^-" | L (-1) -> "e^+"
| L 2 -> "\\mu^-" | L (-2) -> "\\mu^+"
| L 3 -> "\\tau^-" | L (-3) -> "\\tau^+"
| L _ -> invalid_arg
"Modellib.SM.flavor_to_TeX: invalid lepton"
| N 1 -> "\\nu_e" | N (-1) -> "\\bar{\\nu}_e"
| N 2 -> "\\nu_\\mu" | N (-2) -> "\\bar{\\nu}_\\mu"
| N 3 -> "\\nu_\\tau" | N (-3) -> "\\bar{\\nu}_\\tau"
| N _ -> invalid_arg
"Modellib.SM.flavor_to_TeX: invalid neutrino"
| U 1 -> "u" | U (-1) -> "\\bar{u}"
| U 2 -> "c" | U (-2) -> "\\bar{c}"
| U 3 -> "t" | U (-3) -> "\\bar{t}"
| U _ -> invalid_arg
"Modellib.SM.flavor_to_TeX: invalid up type quark"
| D 1 -> "d" | D (-1) -> "\\bar{d}"
| D 2 -> "s" | D (-2) -> "\\bar{s}"
| D 3 -> "b" | D (-3) -> "\\bar{b}"
| D _ -> invalid_arg
"Modellib.SM.flavor_to_TeX: invalid down type quark"
end
| G f ->
begin match f with
| Gl -> "g"
| Ga -> "\\gamma" | Z -> "Z"
| Wp -> "W^+" | Wm -> "W^-"
end
| O f ->
begin match f with
| Phip -> "\\phi^+" | Phim -> "\\phi^-" | Phi0 -> "\\phi^0"
| H -> "H"
| Aux_top (_,_,ch,n,v) -> "\\textnormal{Aux_" ^ (if n then "t_" else "") ^ (
begin match v with
| TTGG -> "ttGG" | TBWA -> "tbWA" | TBWZ -> "tbWZ"
| TTWW -> "ttWW" | BBWW -> "bbWW"
| QGUG -> "qGuG" | QBUB -> "qBuB"
| QW -> "qW" | DL -> "dL" | DR -> "dR"
+ | QUQD1L -> "quqd1L" | QUQD1R -> "quqd1R"
+ | QUQD8L -> "quqd8L" | QUQD8R -> "quqd8R"
end ) ^ ( if ch > 0 then "^+" else if ch < 0 then "^-" else "^0" ) ^ "}"
end
let flavor_symbol = function
| M f ->
begin match f with
| L n when n > 0 -> "l" ^ string_of_int n
| L n -> "l" ^ string_of_int (abs n) ^ "b"
| N n when n > 0 -> "n" ^ string_of_int n
| N n -> "n" ^ string_of_int (abs n) ^ "b"
| U n when n > 0 -> "u" ^ string_of_int n
| U n -> "u" ^ string_of_int (abs n) ^ "b"
| D n when n > 0 -> "d" ^ string_of_int n
| D n -> "d" ^ string_of_int (abs n) ^ "b"
end
| G f ->
begin match f with
| Gl -> "gl"
| Ga -> "a" | Z -> "z"
| Wp -> "wp" | Wm -> "wm"
end
| O f ->
begin match f with
| Phip -> "pp" | Phim -> "pm" | Phi0 -> "p0"
| H -> "h"
| Aux_top (_,_,ch,n,v) -> "aux_" ^ (if n then "t_" else "") ^ (
begin match v with
| TTGG -> "ttgg" | TBWA -> "tbwa" | TBWZ -> "tbwz"
| TTWW -> "ttww" | BBWW -> "bbww"
| QGUG -> "qgug" | QBUB -> "qbub"
| QW -> "qw" | DL -> "dl" | DR -> "dr"
+ | QUQD1L -> "quqd1l" | QUQD1R -> "quqd1r"
+ | QUQD8L -> "quqd8l" | QUQD8R -> "quqd8r"
end ) ^ "_" ^ ( if ch > 0 then "p" else if ch < 0 then "m" else "0" )
end
let pdg = function
| M f ->
begin match f with
| L n when n > 0 -> 9 + 2*n
| L n -> - 9 + 2*n
| N n when n > 0 -> 10 + 2*n
| N n -> - 10 + 2*n
| U n when n > 0 -> 2*n
| U n -> 2*n
| D n when n > 0 -> - 1 + 2*n
| D n -> 1 + 2*n
end
| G f ->
begin match f with
| Gl -> 21
| Ga -> 22 | Z -> 23
| Wp -> 24 | Wm -> (-24)
end
| O f ->
begin match f with
| Phip | Phim -> 27 | Phi0 -> 26
| H -> 25
| Aux_top (_,_,_,_,_) -> 81
end
let mass_symbol f =
"mass(" ^ string_of_int (abs (pdg f)) ^ ")"
let width_symbol f =
"width(" ^ string_of_int (abs (pdg f)) ^ ")"
let constant_symbol = function
| Unit -> "unit" | Half -> "half" | Pi -> "PI"
| Alpha_QED -> "alpha" | E -> "e" | G_weak -> "g" | Vev -> "vev"
| I_G_weak -> "ig"
| Sin2thw -> "sin2thw" | Sinthw -> "sinthw" | Costhw -> "costhw"
| Q_lepton -> "qlep" | Q_up -> "qup" | Q_down -> "qdwn"
| G_NC_lepton -> "gnclep" | G_NC_neutrino -> "gncneu"
| G_NC_up -> "gncup" | G_NC_down -> "gncdwn"
| G_TVA_ttA -> "gtva_tta" | G_TVA_bbA -> "gtva_bba"
| G_VLR_ttZ -> "gvlr_ttz" | G_TVA_ttZ -> "gtva_ttz" | G_TVA_bbZ -> "gtva_bbz"
| G_VLR_btW -> "gvlr_btw" | G_VLR_tbW -> "gvlr_tbw"
| G_TLR_btW -> "gtlr_btw" | G_TRL_tbW -> "gtrl_tbw"
| G_TLR_btWA -> "gtlr_btwa" | G_TRL_tbWA -> "gtrl_tbwa"
| G_TLR_btWZ -> "gtlr_btwz" | G_TRL_tbWZ -> "gtrl_tbwz"
| G_TVA_ttWW -> "gtva_ttww" | G_TVA_bbWW -> "gtva_bbww"
| G_TVA_ttG -> "gtva_ttg" | G_TVA_ttGG -> "gtva_ttgg"
| G_SP_ttH -> "gsp_tth"
| G_VLR_qGuG -> "gvlr_qgug"
| G_VLR_qBuB -> "gvlr_qbub"
| G_VLR_qBuB_u -> "gvlr_qbub_u" | G_VLR_qBuB_d -> "gvlr_qbub_d"
| G_VLR_qBuB_e -> "gvlr_qbub_e" | G_VL_qBuB_n -> "gvl_qbub_n"
| G_VL_qW -> "gvl_qw"
| G_VL_qW_u -> "gvl_qw_u" | G_VL_qW_d -> "gvl_qw_d"
| G_SL_DttR -> "gsl_dttr" | G_SR_DttR -> "gsr_dttr" | G_SL_DttL -> "gsl_dttl"
| G_SLR_DbtR -> "gslr_dbtr" | G_SL_DbtL -> "gsl_dbtl"
+ | C_quqd1R_bt -> "c_quqd1_1" | C_quqd1R_tb -> "conjg(c_quqd1_1)"
+ | C_quqd1L_bt -> "conjg(c_quqd1_2)" | C_quqd1L_tb -> "c_quqd1_2"
+ | C_quqd8R_bt -> "c_quqd8_1" | C_quqd8R_tb -> "conjg(c_quqd8_1)"
+ | C_quqd8L_bt -> "conjg(c_quqd8_2)" | C_quqd8L_tb -> "c_quqd8_2"
| G_CC -> "gcc"
| G_CCQ (n1,n2) -> "gccq" ^ string_of_int n1 ^ string_of_int n2
| I_Q_W -> "iqw" | I_G_ZWW -> "igzww"
| G_WWWW -> "gw4" | G_ZZWW -> "gzzww"
| G_AZWW -> "gazww" | G_AAWW -> "gaaww"
| I_G1_AWW -> "ig1a" | I_G1_ZWW -> "ig1z"
| I_G1_plus_kappa_plus_G4_AWW -> "ig1pkpg4a"
| I_G1_plus_kappa_plus_G4_ZWW -> "ig1pkpg4z"
| I_G1_plus_kappa_minus_G4_AWW -> "ig1pkmg4a"
| I_G1_plus_kappa_minus_G4_ZWW -> "ig1pkmg4z"
| I_G1_minus_kappa_plus_G4_AWW -> "ig1mkpg4a"
| I_G1_minus_kappa_plus_G4_ZWW -> "ig1mkpg4z"
| I_G1_minus_kappa_minus_G4_AWW -> "ig1mkmg4a"
| I_G1_minus_kappa_minus_G4_ZWW -> "ig1mkmg4z"
| I_lambda_AWW -> "ila"
| I_lambda_ZWW -> "ilz"
| G5_AWW -> "rg5a"
| G5_ZWW -> "rg5z"
| I_kappa5_AWW -> "ik5a"
| I_kappa5_ZWW -> "ik5z"
| I_lambda5_AWW -> "il5a" | I_lambda5_ZWW -> "il5z"
| Alpha_WWWW0 -> "alww0" | Alpha_WWWW2 -> "alww2"
| Alpha_ZZWW0 -> "alzw0" | Alpha_ZZWW1 -> "alzw1"
| Alpha_ZZZZ -> "alzz"
| D_Alpha_ZZWW0_S -> "dalzz0_s(gkm,mkm,"
| D_Alpha_ZZWW0_T -> "dalzz0_t(gkm,mkm,"
| D_Alpha_ZZWW1_S -> "dalzz1_s(gkm,mkm,"
| D_Alpha_ZZWW1_T -> "dalzz1_t(gkm,mkm,"
| D_Alpha_ZZWW1_U -> "dalzz1_u(gkm,mkm,"
| D_Alpha_WWWW0_S -> "dalww0_s(gkm,mkm,"
| D_Alpha_WWWW0_T -> "dalww0_t(gkm,mkm,"
| D_Alpha_WWWW0_U -> "dalww0_u(gkm,mkm,"
| D_Alpha_WWWW2_S -> "dalww2_s(gkm,mkm,"
| D_Alpha_WWWW2_T -> "dalww2_t(gkm,mkm,"
| D_Alpha_ZZZZ_S -> "dalz4_s(gkm,mkm,"
| D_Alpha_ZZZZ_T -> "dalz4_t(gkm,mkm,"
| G_HWW -> "ghww" | G_HZZ -> "ghzz"
| G_HHWW -> "ghhww" | G_HHZZ -> "ghhzz"
| G_Htt -> "ghtt" | G_Hbb -> "ghbb"
| G_Htautau -> "ghtautau" | G_Hcc -> "ghcc" | G_Hmm -> "ghmm"
| G_HGaZ -> "ghgaz" | G_HGaGa -> "ghgaga" | G_Hgg -> "ghgg"
| G_HGaGa_anom -> "ghgaga_ac" | G_HGaZ_anom -> "ghgaz_ac"
| G_HZZ_anom -> "ghzz_ac" | G_HWW_anom -> "ghww_ac"
| G_HGaZ_u -> "ghgaz_u" | G_HZZ_u -> "ghzz_u"
| G_HWW_u -> "ghww_u"
| G_H3 -> "gh3" | G_H4 -> "gh4"
| Gs -> "gs" | I_Gs -> "igs" | G2 -> "gs**2"
| Mass f -> "mass" ^ flavor_symbol f
| Width f -> "width" ^ flavor_symbol f
| K_Matrix_Coeff i -> "kc" ^ string_of_int i
| K_Matrix_Pole i -> "kp" ^ string_of_int i
end
(* \thocwmodulesection{Incomplete Standard Model in $R_\xi$ Gauge} *)
(* \begin{dubious}
At the end of the day, we want a functor mapping from gauge models
in unitarity gauge to $R_\xi$ gauge and vice versa. For this, we
will need a more abstract implementation of (spontaneously broken)
gauge theories.
\end{dubious} *)
module SM_Rxi =
struct
let rcs = RCS.rename rcs_file "Modellib.SM_Rxi"
[ "minimal electroweak standard model in R-xi gauge";
"NB: very incomplete still!, no CKM matrix" ]
open Coupling
module SM = SM(SM_no_anomalous)
let options = SM.options
type flavor = SM.flavor
let flavors = SM.flavors
let external_flavors = SM.external_flavors
type constant = SM.constant
let lorentz = SM.lorentz
let color = SM.color
let goldstone = SM.goldstone
let conjugate = SM.conjugate
let fermion = SM.fermion
(* \begin{dubious}
Check if it makes sense to have separate gauge fixing parameters
for each vector boson. There's probably only one independent
parameter for each group factor.
\end{dubious} *)
type gauge =
| XiA | XiZ | XiW
let gauge_symbol = function
| XiA -> "xia" | XiZ -> "xi0" | XiW -> "xipm"
(* Change the gauge boson propagators and make the Goldstone bosons
propagating. *)
let propagator = function
| SM.G SM.Ga -> Prop_Gauge XiA
| SM.G SM.Z -> Prop_Rxi XiZ
| SM.G SM.Wp | SM.G SM.Wm -> Prop_Rxi XiW
| SM.O SM.Phip | SM.O SM.Phim | SM.O SM.Phi0 -> Prop_Scalar
| f -> SM.propagator f
let width = SM.width
module Ch = Charges.QQ
let charges = SM.charges
module F = Modeltools.Fusions (struct
type f = flavor
type c = constant
let compare = compare
let conjugate = conjugate
end)
let vertices = SM.vertices
let table = F.of_vertices (vertices ())
let fuse2 = F.fuse2 table
let fuse3 = F.fuse3 table
let fuse = F.fuse table
let max_degree () = 3
let parameters = SM.parameters
let flavor_of_string = SM.flavor_of_string
let flavor_to_string = SM.flavor_to_string
let flavor_to_TeX = SM.flavor_to_TeX
let flavor_symbol = SM.flavor_symbol
let pdg = SM.pdg
let mass_symbol = SM.mass_symbol
let width_symbol = SM.width_symbol
let constant_symbol = SM.constant_symbol
end
(* \thocwmodulesection{QCD with electroweak insertions.} *)
module SM_QCD =
struct
let rcs = RCS.rename rcs_file "Modellib.SM_QCD"
[ "QCD with electroweak insertions"]
open Coupling
let default_width = ref Timelike
let use_fudged_width = ref false
let options = Options.create
[ "constant_width", Arg.Unit (fun () -> default_width := Constant),
"use constant width (also in t-channel)";
"fudged_width", Arg.Set use_fudged_width,
"use fudge factor for charge particle width";
"custom_width", Arg.String (fun f -> default_width := Custom f),
"use custom width";
"cancel_widths", Arg.Unit (fun () -> default_width := Vanishing),
"use vanishing width"]
type matter_field = L of int | N of int | U of int | D of int
type gauge_boson = Ga | GaX | Wp | Wm | Z | Gl
type other = H
type flavor = M of matter_field | G of gauge_boson | O of other
let matter_field f = M f
let gauge_boson f = G f
let other f = O f
type field =
| Matter of matter_field
| Gauge of gauge_boson
| Other of other
let field = function
| M f -> Matter f
| G f -> Gauge f
| O f -> Other f
type gauge = unit
let gauge_symbol () =
failwith "Modellib.SM_QCD.gauge_symbol: internal error"
let family n = List.map matter_field [ L n; N n; U n; D n ]
let external_flavors () =
[ "1st Generation", ThoList.flatmap family [1; -1];
"2nd Generation", ThoList.flatmap family [2; -2];
"3rd Generation", ThoList.flatmap family [3; -3];
"Gauge Bosons", List.map gauge_boson [Ga; GaX; Z; Wp; Wm; Gl];
"Higgs", List.map other [H] ]
let flavors () = ThoList.flatmap snd (external_flavors ())
let spinor n =
if n >= 0 then
Spinor
else
ConjSpinor
let lorentz_aux = function
| 2 -> Tensor_1
| 1 -> Vector
| 0 -> Scalar
| _ -> invalid_arg ("SM_QCD.lorentz_aux: wrong value")
let lorentz = function
| M f ->
begin match f with
| L n -> spinor n | N n -> spinor n
| U n -> spinor n | D n -> spinor n
end
| G f ->
begin match f with
| Ga | GaX | Gl -> Vector
| Wp | Wm | Z -> Massive_Vector
end
| O f ->
begin match f with
| _ -> Scalar
end
let color = function
| M (U n) -> Color.SUN (if n > 0 then 3 else -3)
| M (D n) -> Color.SUN (if n > 0 then 3 else -3)
| G Gl -> Color.AdjSUN 3
| _ -> Color.Singlet
let prop_spinor n =
if n >= 0 then
Prop_Spinor
else
Prop_ConjSpinor
let prop_aux = function
| 2 -> Aux_Tensor_1
| 1 -> Aux_Vector
| 0 -> Aux_Scalar
| _ -> invalid_arg ("SM_QCD.prop_aux: wrong value")
let propagator = function
| M f ->
begin match f with
| L n -> prop_spinor n | N n -> prop_spinor n
| U n -> prop_spinor n | D n -> prop_spinor n
end
| G f ->
begin match f with
| Ga | GaX | Gl -> Prop_Feynman
| Wp | Wm | Z -> Prop_Unitarity
end
| O f ->
begin match f with
| H -> Prop_Scalar
end
(* Optionally, ask for the fudge factor treatment for the widths of
charged particles. Currently, this only applies to $W^\pm$ and top. *)
let width f =
if !use_fudged_width then
match f with
| G Wp | G Wm | M (U 3) | M (U (-3)) -> Fudged
| _ -> !default_width
else
!default_width
let goldstone _ = None
let conjugate = function
| M f ->
M (begin match f with
| L n -> L (-n) | N n -> N (-n)
| U n -> U (-n) | D n -> D (-n)
end)
| G f ->
G (begin match f with
| Gl -> Gl | Ga -> GaX | Z -> Z
| GaX -> Ga | Wp -> Wm | Wm -> Wp
end)
| O f ->
O (begin match f with
| H -> H
end)
let fermion = function
| M f ->
begin match f with
| L n -> if n > 0 then 1 else -1
| N n -> if n > 0 then 1 else -1
| U n -> if n > 0 then 1 else -1
| D n -> if n > 0 then 1 else -1
end
| G f ->
begin match f with
| Gl | Ga | GaX | Z | Wp | Wm -> 0
end
| O _ -> 0
(* Electrical charge, lepton number, baryon number. We could avoid the
rationals altogether by multiplying the first and last by 3 \ldots *)
module Ch = Charges.QQ
let ( // ) = Algebra.Small_Rational.make
let generation' = function
| 1 -> [ 1//1; 0//1; 0//1]
| 2 -> [ 0//1; 1//1; 0//1]
| 3 -> [ 0//1; 0//1; 1//1]
| -1 -> [-1//1; 0//1; 0//1]
| -2 -> [ 0//1; -1//1; 0//1]
| -3 -> [ 0//1; 0//1; -1//1]
| n -> invalid_arg ("SM_QCD.generation': " ^ string_of_int n)
let generation f =
match f with
| M (L n | N n | U n | D n) -> generation' n
| G _ | O _ -> [0//1; 0//1; 0//1]
let charge = function
| M f ->
begin match f with
| L n -> if n > 0 then -1//1 else 1//1
| N n -> 0//1
| U n -> if n > 0 then 2//3 else -2//3
| D n -> if n > 0 then -1//3 else 1//3
end
| G f ->
begin match f with
| Gl | Ga | GaX | Z -> 0//1
| Wp -> 1//1
| Wm -> -1//1
end
| O f ->
begin match f with
| H -> 0//1
end
let lepton = function
| M f ->
begin match f with
| L n | N n -> if n > 0 then 1//1 else -1//1
| U _ | D _ -> 0//1
end
| G _ | O _ -> 0//1
let baryon = function
| M f ->
begin match f with
| L _ | N _ -> 0//1
| U n | D n -> if n > 0 then 1//1 else -1//1
end
| G _ | O _ -> 0//1
let charges f =
[ charge f; lepton f; baryon f] @ generation f
type constant =
| Unit | Half | Pi | Alpha_QED | Sin2thw
| Sinthw | Costhw | E | G_weak | I_G_weak | Vev
| Q_lepton | Q_up | Q_down | G_CC | G_CCQ of int*int
| G_NC_neutrino | G_NC_lepton | G_NC_up | G_NC_down
| I_Q_W | G_Htt | G_Hbb | G_Hcc | G_Hmm | G_Htautau
| Gs | I_Gs | G2
| Mass of flavor | Width of flavor
(* \begin{dubious}
The current abstract syntax for parameter dependencies is admittedly
tedious. Later, there will be a parser for a convenient concrete syntax
as a part of a concrete syntax for models. But as these examples show,
it should include simple functions.
\end{dubious} *)
(* \begin{subequations}
\begin{align}
\alpha_{\text{QED}} &= \frac{1}{137.0359895} \\
\sin^2\theta_w &= 0.23124
\end{align}
\end{subequations} *)
let input_parameters =
[ Alpha_QED, 1. /. 137.0359895;
Sin2thw, 0.23124;
Mass (G Z), 91.187;
Mass (M (N 1)), 0.0; Mass (M (L 1)), 0.51099907e-3;
Mass (M (N 2)), 0.0; Mass (M (L 2)), 0.105658389;
Mass (M (N 3)), 0.0; Mass (M (L 3)), 1.77705;
Mass (M (U 1)), 5.0e-3; Mass (M (D 1)), 3.0e-3;
Mass (M (U 2)), 1.2; Mass (M (D 2)), 0.1;
Mass (M (U 3)), 174.0; Mass (M (D 3)), 4.2 ]
let derived_parameters =
[ Real E, Sqrt (Prod [Const 4; Atom Pi; Atom Alpha_QED]);
Real Sinthw, Sqrt (Atom Sin2thw);
Real Costhw, Sqrt (Diff (Const 1, Atom Sin2thw));
Real G_weak, Quot (Atom E, Atom Sinthw);
Real (Mass (G Wp)), Prod [Atom Costhw; Atom (Mass (G Z))];
Real Vev, Quot (Prod [Const 2; Atom (Mass (G Wp))], Atom G_weak);
Real Q_lepton, Atom E;
Real Q_up, Prod [Quot (Const (-2), Const 3); Atom E];
Real Q_down, Prod [Quot (Const 1, Const 3); Atom E];
Real G_CC, Neg (Quot (Atom G_weak, Prod [Const 2; Sqrt (Const 2)]));
Complex I_Q_W, Prod [I; Atom E]]
let g_over_2_costh =
Quot (Neg (Atom G_weak), Prod [Const 2; Atom Costhw])
let nc_coupling c t3 q =
(Real_Array c,
[Prod [g_over_2_costh; Diff (t3, Prod [Const 2; q; Atom Sin2thw])];
Prod [g_over_2_costh; t3]])
let half = Quot (Const 1, Const 2)
let derived_parameter_arrays =
[ nc_coupling G_NC_neutrino half (Const 0);
nc_coupling G_NC_lepton (Neg half) (Const (-1));
nc_coupling G_NC_up half (Quot (Const 2, Const 3));
nc_coupling G_NC_down (Neg half) (Quot (Const (-1), Const 3)) ]
let parameters () =
{ input = input_parameters;
derived = derived_parameters;
derived_arrays = derived_parameter_arrays }
module F = Modeltools.Fusions (struct
type f = flavor
type c = constant
let compare = compare
let conjugate = conjugate
end)
let mgm ((m1, g, m2), fbf, c) = ((M m1, G g, M m2), fbf, c)
let mom ((m1, o, m2), fbf, c) = ((M m1, O o, M m2), fbf, c)
let electromagnetic_currents n =
List.map mgm
[ ((L (-n), GaX, L n), FBF (1, Psibar, V, Psi), Q_lepton);
((U (-n), Ga, U n), FBF (1, Psibar, V, Psi), Q_up);
((D (-n), Ga, D n), FBF (1, Psibar, V, Psi), Q_down) ]
let color_currents n =
List.map mgm
[ ((U (-n), Gl, U n), FBF ((-1), Psibar, V, Psi), Gs);
((D (-n), Gl, D n), FBF ((-1), Psibar, V, Psi), Gs) ]
(* \begin{equation}
\mathcal{L}_{\textrm{NC}} =
- \frac{g}{2\cos\theta_W}
\sum_i \bar\psi_i\fmslash{Z}(g_V^i-g_A^i\gamma_5)\psi_i
\end{equation} *)
let neutral_currents n =
List.map mgm
[ ((L (-n), Z, L n), FBF (1, Psibar, VA, Psi), G_NC_lepton);
((N (-n), Z, N n), FBF (1, Psibar, VA, Psi), G_NC_neutrino);
((U (-n), Z, U n), FBF (1, Psibar, VA, Psi), G_NC_up);
((D (-n), Z, D n), FBF (1, Psibar, VA, Psi), G_NC_down) ]
(* \begin{equation}
\mathcal{L}_{\textrm{CC}} =
- \frac{g}{2\sqrt2} \sum_i \bar\psi_i
(T^+\fmslash{W}^+ + T^-\fmslash{W}^-)(1-\gamma_5)\psi_i
\end{equation} *)
let charged_currents' n =
List.map mgm
[ ((L (-n), Wm, N n), FBF (1, Psibar, VL, Psi), G_CC);
((N (-n), Wp, L n), FBF (1, Psibar, VL, Psi), G_CC) ]
let charged_currents'' n =
List.map mgm
[ ((D (-n), Wm, U n), FBF (1, Psibar, VL, Psi), G_CC);
((U (-n), Wp, D n), FBF (1, Psibar, VL, Psi), G_CC) ]
let charged_currents_triv =
ThoList.flatmap charged_currents' [1;2;3] @
ThoList.flatmap charged_currents'' [1;2;3]
let charged_currents_ckm =
let charged_currents_2 n1 n2 =
List.map mgm
[ ((D (-n1), Wm, U n2), FBF (1, Psibar, VL, Psi), G_CCQ (n2,n1));
((U (-n1), Wp, D n2), FBF (1, Psibar, VL, Psi), G_CCQ (n1,n2)) ] in
ThoList.flatmap charged_currents' [1;2;3] @
List.flatten (Product.list2 charged_currents_2 [1;2;3] [1;2;3])
let yukawa =
[ ((M (U (-3)), O H, M (U 3)), FBF (1, Psibar, S, Psi), G_Htt);
((M (D (-3)), O H, M (D 3)), FBF (1, Psibar, S, Psi), G_Hbb);
((M (U (-2)), O H, M (U 2)), FBF (1, Psibar, S, Psi), G_Hcc);
((M (L (-2)), O H, M (L 2)), FBF (1, Psibar, S, Psi), G_Hmm);
((M (L (-3)), O H, M (L 3)), FBF (1, Psibar, S, Psi), G_Htautau) ]
(* \begin{equation}
\mathcal{L}_{\textrm{TGC}} =
- e \partial_\mu A_\nu W_+^\mu W_-^\nu + \ldots
- e \cot\theta_w \partial_\mu Z_\nu W_+^\mu W_-^\nu + \ldots
\end{equation} *)
let tgc ((g1, g2, g3), t, c) = ((G g1, G g2, G g3), t, c)
let triple_gauge =
List.map tgc
[ ((Gl, Gl, Gl), Gauge_Gauge_Gauge 1, I_Gs)]
let qgc ((g1, g2, g3, g4), t, c) = ((G g1, G g2, G g3, G g4), t, c)
let gauge4 = Vector4 [(2, C_13_42); (-1, C_12_34); (-1, C_14_23)]
let minus_gauge4 = Vector4 [(-2, C_13_42); (1, C_12_34); (1, C_14_23)]
let quartic_gauge =
List.map qgc
[ (Gl, Gl, Gl, Gl), gauge4, G2 ]
let vertices3 =
(ThoList.flatmap electromagnetic_currents [1;2;3] @
ThoList.flatmap color_currents [1;2;3] @
ThoList.flatmap neutral_currents [1;2;3] @
charged_currents_triv @
yukawa @ triple_gauge)
let vertices4 =
quartic_gauge
let vertices () = (vertices3, vertices4, [])
(* For efficiency, make sure that [F.of_vertices vertices] is
evaluated only once. *)
let table = F.of_vertices (vertices ())
let fuse2 = F.fuse2 table
let fuse3 = F.fuse3 table
let fuse = F.fuse table
let max_degree () = 4
let flavor_of_string = function
| "e-" -> M (L 1) | "e+" -> M (L (-1))
| "mu-" -> M (L 2) | "mu+" -> M (L (-2))
| "tau-" -> M (L 3) | "tau+" -> M (L (-3))
| "nue" -> M (N 1) | "nuebar" -> M (N (-1))
| "numu" -> M (N 2) | "numubar" -> M (N (-2))
| "nutau" -> M (N 3) | "nutaubar" -> M (N (-3))
| "u" -> M (U 1) | "ubar" -> M (U (-1))
| "c" -> M (U 2) | "cbar" -> M (U (-2))
| "t" -> M (U 3) | "tbar" -> M (U (-3))
| "d" -> M (D 1) | "dbar" -> M (D (-1))
| "s" -> M (D 2) | "sbar" -> M (D (-2))
| "b" -> M (D 3) | "bbar" -> M (D (-3))
| "g" | "gl" -> G Gl
| "A" -> G Ga | "Z" | "Z0" -> G Z
| "W+" -> G Wp | "W-" -> G Wm
| "H" -> O H
| _ -> invalid_arg "Modellib.SM_QCD.flavor_of_string"
let flavor_to_string = function
| M f ->
begin match f with
| L 1 -> "e-" | L (-1) -> "e+"
| L 2 -> "mu-" | L (-2) -> "mu+"
| L 3 -> "tau-" | L (-3) -> "tau+"
| L _ -> invalid_arg
"Modellib.SM_QCD.flavor_to_string: invalid lepton"
| N 1 -> "nue" | N (-1) -> "nuebar"
| N 2 -> "numu" | N (-2) -> "numubar"
| N 3 -> "nutau" | N (-3) -> "nutaubar"
| N _ -> invalid_arg
"Modellib.SM_QCD.flavor_to_string: invalid neutrino"
| U 1 -> "u" | U (-1) -> "ubar"
| U 2 -> "c" | U (-2) -> "cbar"
| U 3 -> "t" | U (-3) -> "tbar"
| U _ -> invalid_arg
"Modellib.SM_QCD.flavor_to_string: invalid up type quark"
| D 1 -> "d" | D (-1) -> "dbar"
| D 2 -> "s" | D (-2) -> "sbar"
| D 3 -> "b" | D (-3) -> "bbar"
| D _ -> invalid_arg
"Modellib.SM_QCD.flavor_to_string: invalid down type quark"
end
| G f ->
begin match f with
| Gl -> "gl"
| Ga | GaX -> "A" | Z -> "Z"
| Wp -> "W+" | Wm -> "W-"
end
| O f ->
begin match f with
| H -> "H"
end
let flavor_to_TeX = function
| M f ->
begin match f with
| L 1 -> "e^-" | L (-1) -> "e^+"
| L 2 -> "\\mu^-" | L (-2) -> "\\mu^+"
| L 3 -> "\\tau^-" | L (-3) -> "\\tau^+"
| L _ -> invalid_arg
"Modellib.SM_QCD.flavor_to_TeX: invalid lepton"
| N 1 -> "\\nu_e" | N (-1) -> "\\bar{\\nu}_e"
| N 2 -> "\\nu_\\mu" | N (-2) -> "\\bar{\\nu}_\\mu"
| N 3 -> "\\nu_\\tau" | N (-3) -> "\\bar{\\nu}_\\tau"
| N _ -> invalid_arg
"Modellib.SM_QCD.flavor_to_TeX: invalid neutrino"
| U 1 -> "u" | U (-1) -> "\\bar{u}"
| U 2 -> "c" | U (-2) -> "\\bar{c}"
| U 3 -> "t" | U (-3) -> "\\bar{t}"
| U _ -> invalid_arg
"Modellib.SM_QCD.flavor_to_TeX: invalid up type quark"
| D 1 -> "d" | D (-1) -> "\\bar{d}"
| D 2 -> "s" | D (-2) -> "\\bar{s}"
| D 3 -> "b" | D (-3) -> "\\bar{b}"
| D _ -> invalid_arg
"Modellib.SM_QCD.flavor_to_TeX: invalid down type quark"
end
| G f ->
begin match f with
| Gl -> "g"
| Ga | GaX -> "\\gamma" | Z -> "Z"
| Wp -> "W^+" | Wm -> "W^-"
end
| O f ->
begin match f with
| H -> "H"
end
let flavor_symbol = function
| M f ->
begin match f with
| L n when n > 0 -> "l" ^ string_of_int n
| L n -> "l" ^ string_of_int (abs n) ^ "b"
| N n when n > 0 -> "n" ^ string_of_int n
| N n -> "n" ^ string_of_int (abs n) ^ "b"
| U n when n > 0 -> "u" ^ string_of_int n
| U n -> "u" ^ string_of_int (abs n) ^ "b"
| D n when n > 0 -> "d" ^ string_of_int n
| D n -> "d" ^ string_of_int (abs n) ^ "b"
end
| G f ->
begin match f with
| Gl -> "gl"
| Ga -> "a" | Z -> "z"
| GaX -> "gax"
| Wp -> "wp" | Wm -> "wm"
end
| O f ->
begin match f with
| H -> "h"
end
let pdg = function
| M f ->
begin match f with
| L n when n > 0 -> 9 + 2*n
| L n -> - 9 + 2*n
| N n when n > 0 -> 10 + 2*n
| N n -> - 10 + 2*n
| U n when n > 0 -> 2*n
| U n -> 2*n
| D n when n > 0 -> - 1 + 2*n
| D n -> 1 + 2*n
end
| G f ->
begin match f with
| Gl -> 21
| Ga | GaX -> 22 | Z -> 23
| Wp -> 24 | Wm -> (-24)
end
| O f ->
begin match f with
| H -> 25
end
let mass_symbol f =
"mass(" ^ string_of_int (abs (pdg f)) ^ ")"
let width_symbol f =
"width(" ^ string_of_int (abs (pdg f)) ^ ")"
let constant_symbol = function
| Unit -> "unit" | Half -> "half" | Pi -> "PI"
| Alpha_QED -> "alpha" | E -> "e" | G_weak -> "g" | Vev -> "vev"
| I_G_weak -> "ig"
| Sin2thw -> "sin2thw" | Sinthw -> "sinthw" | Costhw -> "costhw"
| Q_lepton -> "qlep" | Q_up -> "qup" | Q_down -> "qdwn"
| G_NC_lepton -> "gnclep" | G_NC_neutrino -> "gncneu"
| G_NC_up -> "gncup" | G_NC_down -> "gncdwn"
| G_CC -> "gcc"
| G_CCQ (n1,n2) -> "gccq" ^ string_of_int n1 ^ string_of_int n2
| I_Q_W -> "iqw"
| G_Htt -> "ghtt" | G_Hbb -> "ghbb"
| G_Htautau -> "ghtautau" | G_Hcc -> "ghcc" | G_Hmm -> "ghmm"
| Gs -> "gs" | I_Gs -> "igs" | G2 -> "gs**2"
| Mass f -> "mass" ^ flavor_symbol f
| Width f -> "width" ^ flavor_symbol f
end
(* \thocwmodulesection{Groves} *)
module Groves (M : Model.Gauge) : Model.Gauge with module Ch = M.Ch =
struct
let max_generations = 5
let rcs = RCS.rename M.rcs
("Modellib.Groves(" ^ (RCS.name M.rcs) ^ ")")
([ "experimental Groves functor";
Printf.sprintf "for maximally %d flavored legs"
(2 * max_generations) ] @
RCS.description M.rcs)
let options = M.options
type matter_field = M.matter_field * int
type gauge_boson = M.gauge_boson
type other = M.other
type field =
| Matter of matter_field
| Gauge of gauge_boson
| Other of other
type flavor = M of matter_field | G of gauge_boson | O of other
let matter_field (f, g) = M (f, g)
let gauge_boson f = G f
let other f = O f
let field = function
| M f -> Matter f
| G f -> Gauge f
| O f -> Other f
let project = function
| M (f, _) -> M.matter_field f
| G f -> M.gauge_boson f
| O f -> M.other f
let inject g f =
match M.field f with
| M.Matter f -> M (f, g)
| M.Gauge f -> G f
| M.Other f -> O f
type gauge = M.gauge
let gauge_symbol = M.gauge_symbol
let color f = M.color (project f)
let pdg f = M.pdg (project f)
let lorentz f = M.lorentz (project f)
let propagator f = M.propagator (project f)
let fermion f = M.fermion (project f)
let width f = M.width (project f)
let mass_symbol f = M.mass_symbol (project f)
let width_symbol f = M.width_symbol (project f)
let flavor_symbol f = M.flavor_symbol (project f)
type constant = M.constant
let constant_symbol = M.constant_symbol
let max_degree = M.max_degree
let parameters = M.parameters
let conjugate = function
| M (_, g) as f -> inject g (M.conjugate (project f))
| f -> inject 0 (M.conjugate (project f))
let read_generation s =
try
let offset = String.index s '/' in
(int_of_string
(String.sub s (succ offset) (String.length s - offset - 1)),
String.sub s 0 offset)
with
| Not_found -> (1, s)
let format_generation c s =
s ^ "/" ^ string_of_int c
let flavor_of_string s =
let g, s = read_generation s in
inject g (M.flavor_of_string s)
let flavor_to_string = function
| M (_, g) as f -> format_generation g (M.flavor_to_string (project f))
| f -> M.flavor_to_string (project f)
let flavor_to_TeX = function
| M (_, g) as f -> format_generation g (M.flavor_to_TeX (project f))
| f -> M.flavor_to_TeX (project f)
let goldstone = function
| G _ as f ->
begin match M.goldstone (project f) with
| None -> None
| Some (f, c) -> Some (inject 0 f, c)
end
| M _ | O _ -> None
let clone generations flavor =
match M.field flavor with
| M.Matter f -> List.map (fun g -> M (f, g)) generations
| M.Gauge f -> [G f]
| M.Other f -> [O f]
let generations = ThoList.range 1 max_generations
let flavors () =
ThoList.flatmap (clone generations) (M.flavors ())
let external_flavors () =
List.map (fun (s, fl) -> (s, ThoList.flatmap (clone generations) fl))
(M.external_flavors ())
module Ch = M.Ch
let charges f = M.charges (project f)
module F = Modeltools.Fusions (struct
type f = flavor
type c = constant
let compare = compare
let conjugate = conjugate
end)
(* In the following functions, we might replace [_] by [(M.Gauge _ | M.Other _)],
in order to allow the compiler to check completeness. However, this
makes the code much less readable. *)
let clone3 ((f1, f2, f3), v, c) =
match M.field f1, M.field f2, M.field f3 with
| M.Matter _, M.Matter _, M.Matter _ ->
invalid_arg "Modellib.Groves().vertices: three matter fields!"
| M.Matter f1', M.Matter f2', _ ->
List.map (fun g -> ((M (f1', g), M (f2', g), inject 0 f3), v, c))
generations
| M.Matter f1', _, M.Matter f3' ->
List.map (fun g -> ((M (f1', g), inject 0 f2, M (f3', g)), v, c))
generations
| _, M.Matter f2', M.Matter f3' ->
List.map (fun g -> ((inject 0 f1, M (f2', g), M (f3', g)), v, c))
generations
| M.Matter _, _, _ | _, M.Matter _, _ | _, _, M.Matter _ ->
invalid_arg "Modellib.Groves().vertices: lone matter field!"
| _, _, _ ->
[(inject 0 f1, inject 0 f2, inject 0 f3), v, c]
let clone4 ((f1, f2, f3, f4), v, c) =
match M.field f1, M.field f2, M.field f3, M.field f4 with
| M.Matter _, M.Matter _, M.Matter _, M.Matter _ ->
invalid_arg "Modellib.Groves().vertices: four matter fields!"
| M.Matter _, M.Matter _, M.Matter _, _
| M.Matter _, M.Matter _, _, M.Matter _
| M.Matter _, _, M.Matter _, M.Matter _
| _, M.Matter _, M.Matter _, M.Matter _ ->
invalid_arg "Modellib.Groves().vertices: three matter fields!"
| M.Matter f1', M.Matter f2', _, _ ->
List.map (fun g ->
((M (f1', g), M (f2', g), inject 0 f3, inject 0 f4), v, c))
generations
| M.Matter f1', _, M.Matter f3', _ ->
List.map (fun g ->
((M (f1', g), inject 0 f2, M (f3', g), inject 0 f4), v, c))
generations
| M.Matter f1', _, _, M.Matter f4' ->
List.map (fun g ->
((M (f1', g), inject 0 f2, inject 0 f3, M (f4', g)), v, c))
generations
| _, M.Matter f2', M.Matter f3', _ ->
List.map (fun g ->
((inject 0 f1, M (f2', g), M (f3', g), inject 0 f4), v, c))
generations
| _, M.Matter f2', _, M.Matter f4' ->
List.map (fun g ->
((inject 0 f1, M (f2', g), inject 0 f3, M (f4', g)), v, c))
generations
| _, _, M.Matter f3', M.Matter f4' ->
List.map (fun g ->
((inject 0 f1, inject 0 f2, M (f3', g), M (f4', g)), v, c))
generations
| M.Matter _, _, _, _ | _, M.Matter _, _, _
| _, _, M.Matter _, _ | _, _, _, M.Matter _ ->
invalid_arg "Modellib.Groves().vertices: lone matter field!"
| _, _, _, _ ->
[(inject 0 f1, inject 0 f2, inject 0 f3, inject 0 f4), v, c]
let clonen (fl, v, c) =
match List.map M.field fl with
| _ -> failwith "Modellib.Groves().vertices: incomplete"
let vertices () =
let vertices3, vertices4, verticesn = M.vertices () in
(ThoList.flatmap clone3 vertices3,
ThoList.flatmap clone4 vertices4,
ThoList.flatmap clonen verticesn)
let table = F.of_vertices (vertices ())
let fuse2 = F.fuse2 table
let fuse3 = F.fuse3 table
let fuse = F.fuse table
(* \begin{dubious}
The following (incomplete) alternative implementations are
included for illustrative purposes only:
\end{dubious} *)
let injectl g fcl =
List.map (fun (f, c) -> (inject g f, c)) fcl
let alt_fuse2 f1 f2 =
match f1, f2 with
| M (f1', g1'), M (f2', g2') ->
if g1' = g2' then
injectl 0 (M.fuse2 (M.matter_field f1') (M.matter_field f2'))
else
[]
| M (f1', g'), _ -> injectl g' (M.fuse2 (M.matter_field f1') (project f2))
| _, M (f2', g') -> injectl g' (M.fuse2 (project f1) (M.matter_field f2'))
| _, _ -> injectl 0 (M.fuse2 (project f1) (project f2))
let alt_fuse3 f1 f2 f3 =
match f1, f2, f3 with
| M (f1', g1'), M (f2', g2'), M (f3', g3') ->
invalid_arg "Modellib.Groves().fuse3: three matter fields!"
| M (f1', g1'), M (f2', g2'), _ ->
if g1' = g2' then
injectl 0
(M.fuse3 (M.matter_field f1') (M.matter_field f2') (project f3))
else
[]
| M (f1', g1'), _, M (f3', g3') ->
if g1' = g3' then
injectl 0
(M.fuse3 (M.matter_field f1') (project f2) (M.matter_field f3'))
else
[]
| _, M (f2', g2'), M (f3', g3') ->
if g2' = g3' then
injectl 0
(M.fuse3 (project f1) (M.matter_field f2') (M.matter_field f3'))
else
[]
| M (f1', g'), _, _ ->
injectl g' (M.fuse3 (M.matter_field f1') (project f2) (project f3))
| _, M (f2', g'), _ ->
injectl g' (M.fuse3 (project f1) (M.matter_field f2') (project f3))
| _, _, M (f3', g') ->
injectl g' (M.fuse3 (project f1) (project f2) (M.matter_field f3'))
| _, _, _ -> injectl 0 (M.fuse3 (project f1) (project f2) (project f3))
end
(* \thocwmodulesection{MSM With Cloned Families} *)
module SM_clones = Groves(SM(SM_no_anomalous))
(*i
* Local Variables:
* mode:caml
* indent-tabs-mode:nil
* page-delimiter:"^(\\* .*\n"
* End:
i*)
Index: trunk/src/omega/src/coupling.mli
===================================================================
--- trunk/src/omega/src/coupling.mli (revision 4047)
+++ trunk/src/omega/src/coupling.mli (revision 4048)
@@ -1,2559 +1,2559 @@
(* $Id$
Copyright (C) 1999-2013 by
Wolfgang Kilian <kilian@physik.uni-siegen.de>
Thorsten Ohl <ohl@physik.uni-wuerzburg.de>
Juergen Reuter <juergen.reuter@desy.de>
Christian Speckner <cnspeckn@googlemail.com>
WHIZARD is free software; you can redistribute it and/or modify it
under the terms of the GNU General Public License as published by
the Free Software Foundation; either version 2, or (at your option)
any later version.
WHIZARD is distributed in the hope that it will be useful, but
WITHOUT ANY WARRANTY; without even the implied warranty of
MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
GNU General Public License for more details.
You should have received a copy of the GNU General Public License
along with this program; if not, write to the Free Software
Foundation, Inc., 675 Mass Ave, Cambridge, MA 02139, USA. *)
(* The enumeration types used for communication from [Models]
to [Targets]. On the physics side, the modules in [Models]
must implement the Feynman rules according to the conventions
set up here. On the numerics side, the modules in [Targets]
must handle all cases according to the same conventions. *)
(* \thocwmodulesection{Propagators}
The Lorentz representation of the particle. NB: O'Mega
treats all lines as \emph{outgoing} and particles are therefore
transforming as [ConjSpinor] and antiparticles as [Spinor]. *)
type lorentz =
| Scalar
| Spinor (* $\psi$ *)
| ConjSpinor (* $\bar\psi$ *)
| Majorana (* $\chi$ *)
| Maj_Ghost (* SUSY ghosts *)
| Vector
(*i | Ward_Vector i*)
| Massive_Vector
| Vectorspinor (* supersymmetric currents and gravitinos *)
| Tensor_1
| Tensor_2 (* massive gravitons (large extra dimensions) *)
| BRS of lorentz
(* \begin{table}
\begin{center}
\renewcommand{\arraystretch}{2.2}
\begin{tabular}{|r|l|l|}\hline
& only Dirac fermions & incl.~Majorana fermions \\\hline
[Prop_Scalar]
& \multicolumn{2}{l|}{%
$\displaystyle\phi(p)\leftarrow
\frac{\ii}{p^2-m^2+\ii m\Gamma}\phi(p)$} \\\hline
[Prop_Spinor]
& $\displaystyle\psi(p)\leftarrow
\frac{\ii(-\fmslash{p}+m)}{p^2-m^2+\ii m\Gamma}\psi(p)$
& $\displaystyle\psi(p)\leftarrow
\frac{\ii(-\fmslash{p}+m)}{p^2-m^2+\ii m\Gamma}\psi(p)$ \\\hline
[Prop_ConjSpinor]
& $\displaystyle\bar\psi(p)\leftarrow
\bar\psi(p)\frac{\ii(\fmslash{p}+m)}{p^2-m^2+\ii m\Gamma}$
& $\displaystyle\psi(p)\leftarrow
\frac{\ii(-\fmslash{p}+m)}{p^2-m^2+\ii m\Gamma}\psi(p)$ \\\hline
[Prop_Majorana]
& \multicolumn{1}{c|}{N/A}
& $\displaystyle\chi(p)\leftarrow
\frac{\ii(-\fmslash{p}+m)}{p^2-m^2+\ii m\Gamma}\chi(p)$ \\\hline
[Prop_Unitarity]
& \multicolumn{2}{l|}{%
$\displaystyle\epsilon_\mu(p)\leftarrow
\frac{\ii}{p^2-m^2+\ii m\Gamma}
\left(-g_{\mu\nu}+\frac{p_\mu p_\nu}{m^2}\right)\epsilon^\nu(p)$} \\\hline
[Prop_Feynman]
& \multicolumn{2}{l|}{%
$\displaystyle\epsilon^\nu(p)\leftarrow
\frac{-\ii}{p^2-m^2+\ii m\Gamma}\epsilon^\nu(p)$} \\\hline
[Prop_Gauge]
& \multicolumn{2}{l|}{%
$\displaystyle\epsilon_\mu(p)\leftarrow
\frac{\ii}{p^2}
\left(-g_{\mu\nu}+(1-\xi)\frac{p_\mu p_\nu}{p^2}\right)\epsilon^\nu(p)$} \\\hline
[Prop_Rxi]
& \multicolumn{2}{l|}{%
$\displaystyle\epsilon_\mu(p)\leftarrow
\frac{\ii}{p^2-m^2+\ii m\Gamma}
\left(-g_{\mu\nu}+(1-\xi)\frac{p_\mu p_\nu}{p^2-\xi m^2}\right)
\epsilon^\nu(p)$} \\\hline
\end{tabular}
\end{center}
\caption{\label{tab:propagators} Propagators. NB: The sign of the
momenta in the spinor propagators comes about because O'Mega
treats all momenta as \emph{outgoing} and the charge flow for
[Spinor] is therefore opposite to the momentum, while the charge
flow for [ConjSpinor] is parallel to the momentum.}
\end{table}
\begin{table}
\begin{center}
\renewcommand{\arraystretch}{1.5}
\begin{tabular}{|r|l|}\hline
[Aux_Scalar]
& $\displaystyle\phi(p)\leftarrow\ii\phi(p)$ \\\hline
[Aux_Spinor]
& $\displaystyle\psi(p)\leftarrow\ii\psi(p)$ \\\hline
[Aux_ConjSpinor]
& $\displaystyle\bar\psi(p)\leftarrow\ii\bar\psi(p)$ \\\hline
[Aux_Vector]
& $\displaystyle\epsilon^\mu(p)\leftarrow\ii\epsilon^\mu(p)$ \\\hline
[Aux_Tensor_1]
& $\displaystyle T^{\mu\nu}(p)\leftarrow\ii T^{\mu\nu}(p)$ \\\hline
[Only_Insertion]
& \multicolumn{1}{c|}{N/A} \\\hline
\end{tabular}
\end{center}
\caption{\label{tab:aux-propagators} Auxiliary and non propagating fields}
\end{table}
If there were no vectors or auxiliary fields, we could deduce the propagator from
the Lorentz representation. While we're at it, we can introduce
``propagators'' for the contact interactions of auxiliary fields
as well. [Prop_Gauge] and [Prop_Feynman] are redundant as special
cases of [Prop_Rxi].
The special case [Only_Insertion] corresponds to operator insertions
that do not correspond to a propagating field all. These are used
for checking Slavnov-Taylor identities
\begin{equation}
\partial_\mu\Braket{\text{out}|W^\mu(x)|\text{in}}
= m_W\Braket{\text{out}|\phi(x)|\text{in}}
\end{equation}
of gauge theories in unitarity gauge where the Goldstone bosons are
not propagating. Numerically, it would suffice to use a vanishing
propagator, but then superflous fusions would be calculated in
production code in which the Slavnov-Taylor identities are not tested. *)
type 'a propagator =
| Prop_Scalar | Prop_Ghost
| Prop_Spinor | Prop_ConjSpinor | Prop_Majorana
| Prop_Unitarity | Prop_Feynman | Prop_Gauge of 'a | Prop_Rxi of 'a
| Prop_Tensor_2 | Prop_Vectorspinor
| Prop_Col_Scalar | Prop_Col_Feynman | Prop_Col_Majorana
| Prop_Col_Unitarity
| Aux_Scalar | Aux_Vector | Aux_Tensor_1
- | Aux_Col_Vector | Aux_Col_Tensor_1
+ | Aux_Col_Scalar | Aux_Col_Vector | Aux_Col_Tensor_1
| Aux_Spinor | Aux_ConjSpinor | Aux_Majorana
| Only_Insertion
(* \begin{JR}
We don't need different fermionic propagators as supposed by the variable
names [Prop_Spinor], [Prop_ConjSpinor] or [Prop_Majorana]. The
propagator in all cases has to be multiplied on the left hand side of the
spinor out of which a new one should be built. All momenta are treated as
\emph{outgoing}, so for the propagation of the different fermions the
following table arises, in which the momentum direction is always downwards
and the arrows show whether the momentum and the fermion line,
respectively are parallel or antiparallel to the direction of calculation:
\begin{center}
\begin{tabular}{|l|c|c|c|c|}\hline
Fermion type & fermion arrow & mom. & calc. & sign \\\hline\hline
Dirac fermion & $\uparrow$ & $\uparrow~\downarrow$ &
$\uparrow~\uparrow$ & negative \\\hline
Dirac antifermion & $\downarrow$ & $\downarrow~\downarrow$ &
$\uparrow~\downarrow$ & negative \\\hline
Majorana fermion & - & $\uparrow~\downarrow$ & - & negative \\\hline
\end{tabular}
\end{center}
So the sign of the momentum is always negative and no further distinction
is needed.
\end{JR} *)
type width =
| Vanishing
| Constant
| Timelike
| Running
| Fudged
| Custom of string
(* \thocwmodulesection{Vertices}
The combined $S-P$ and $V-A$ couplings (see
tables~\ref{tab:dim4-fermions-SP}, \ref{tab:dim4-fermions-VA},
\ref{tab:dim4-fermions-SPVA-maj} and~\ref{tab:dim4-fermions-SPVA-maj2})
are redundant, of course, but they allow some targets to create
more efficient numerical code.\footnote{An additional benefit
is that the counting of Feynman diagrams is not upset by a splitting
of the vectorial and axial pieces of gauge bosons.} Choosing VA2 over
VA will cause the FORTRAN backend to pass the coupling as a whole array *)
type fermion = Psi | Chi | Grav
type fermionbar = Psibar | Chibar | Gravbar
type boson =
| SP | SPM | S | P | SL | SR | SLR | VA | V | A | VL | VR | VLR | VLRM
| TVA | TLR | TRL | TVAM | TLRM | TRLM
| POT | MOM | MOM5 | MOML | MOMR | LMOM | RMOM | VMOM | VA2
type boson2 = S2 | P2 | S2P | S2L | S2R | S2LR
| SV | PV | SLV | SRV | SLRV | V2 | V2LR
(* The integer is an additional coefficient that multiplies the respective
coupling constant. This allows to reduce the number of required coupling
constants in manifestly symmetrc cases. Most of times it will be equal
unity, though. *)
(* The two vertex types [PBP] and [BBB] for the couplings of two fermions or
two antifermions ("clashing arrows") is unavoidable in supersymmetric
theories.
\begin{dubious}
\ldots{} tho doesn't like the names and has promised to find a better
mnemonics!
\end{dubious} *)
type 'a vertex3 =
| FBF of int * fermionbar * boson * fermion
| PBP of int * fermion * boson * fermion
| BBB of int * fermionbar * boson * fermionbar
| GBG of int * fermionbar * boson * fermion (* gravitino-boson-fermion *)
| Gauge_Gauge_Gauge of int | Aux_Gauge_Gauge of int
| Scalar_Vector_Vector of int
| Aux_Vector_Vector of int | Aux_Scalar_Vector of int
| Scalar_Scalar_Scalar of int | Aux_Scalar_Scalar of int
| Vector_Scalar_Scalar of int
| Graviton_Scalar_Scalar of int
| Graviton_Vector_Vector of int
| Graviton_Spinor_Spinor of int
| Dim4_Vector_Vector_Vector_T of int
| Dim4_Vector_Vector_Vector_L of int
| Dim4_Vector_Vector_Vector_T5 of int
| Dim4_Vector_Vector_Vector_L5 of int
| Dim6_Gauge_Gauge_Gauge of int
| Dim6_Gauge_Gauge_Gauge_5 of int
| Aux_DScalar_DScalar of int | Aux_Vector_DScalar of int
| Dim5_Scalar_Gauge2 of int (* %
$\frac12 \phi F_{1,\mu\nu} F_2^{\mu\nu} = - \frac12
\phi (\ii \partial_{[\mu,} V_{1,\nu]})(\ii \partial^{[\mu,} V_2^{\nu]})$ *)
| Dim5_Scalar_Gauge2_Skew of int
(* %
$\frac12 \phi F_{1,\mu\nu} \tilde{F}_2^{\mu\nu} = -
\phi (\ii \partial_\mu V_{1,\nu})(\ii \partial_\rho V_{2,\sigma})\epsilon^{\mu\nu\rho\sigma}$ *)
| Dim5_Scalar_Vector_Vector_T of int (* %
$\phi(\ii\partial_\mu V_1^\nu)(\ii\partial_\nu V_2^\mu)$ *)
| Dim5_Scalar_Vector_Vector_TU of int (* %
$(\ii\partial_\nu\phi) (\ii\partial_\mu V_1^\nu) V_2^\mu$ *)
| Dim5_Scalar_Vector_Vector_U of int (* %
$(\ii\partial_\nu\phi) (\ii\partial_\mu V^\nu) V^\mu$ *)
| Dim6_Vector_Vector_Vector_T of int (* %
$V_1^\mu ((\ii\partial_\nu V_2^\rho)%
\ii\overleftrightarrow{\partial_\mu}(\ii\partial_\rho V_3^\nu))$ *)
| Tensor_2_Vector_Vector of int (* %
$T^{\mu\nu} (V_{1,\mu}V_{2,\nu} + V_{1,\nu}V_{2,\mu})$ *)
| Dim5_Tensor_2_Vector_Vector_1 of int (* %
$T^{\alpha\beta} (V_1^\mu
\ii\overleftrightarrow\partial_\alpha
\ii\overleftrightarrow\partial_\beta V_{2,\mu}$ *)
| Dim5_Tensor_2_Vector_Vector_2 of int
(* %
$T^{\alpha\beta}
( V_1^\mu \ii\overleftrightarrow\partial_\beta (\ii\partial_\mu V_{2,\alpha})
+ V_1^\mu \ii\overleftrightarrow\partial_\alpha (\ii\partial_\mu V_{2,\beta}))$ *)
| Dim7_Tensor_2_Vector_Vector_T of int (* %
$T^{\alpha\beta} ((\ii\partial^\mu V_1^\nu)
\ii\overleftrightarrow\partial_\alpha
\ii\overleftrightarrow\partial_\beta
(\ii\partial_\nu V_{2,\mu})) $ *)
(* As long as we stick to renormalizable couplings, there are only
three types of quartic couplings: [Scalar4], [Scalar2_Vector2]
and [Vector4]. However, there are three inequivalent contractions
for the latter and the general vertex will be a linear combination
with integer coefficients:
\begin{subequations}
\begin{align}
\ocwupperid{Scalar4}\,1 :&\;\;\;\;\;
\phi_1 \phi_2 \phi_3 \phi_4 \\
\ocwupperid{Scalar2\_Vector2}\,1 :&\;\;\;\;\;
\phi_1^{\vphantom{\mu}} \phi_2^{\vphantom{\mu}}
V_3^\mu V_{4,\mu}^{\vphantom{\mu}} \\
\ocwupperid{Vector4}\,\lbrack 1, \ocwupperid{C\_12\_34} \rbrack :&\;\;\;\;\;
V_1^\mu V_{2,\mu}^{\vphantom{\mu}}
V_3^\nu V_{4,\nu}^{\vphantom{\mu}} \\
\ocwupperid{Vector4}\,\lbrack 1, \ocwupperid{C\_13\_42} \rbrack :&\;\;\;\;\;
V_1^\mu V_2^\nu
V_{3,\mu}^{\vphantom{\mu}} V_{4,\nu}^{\vphantom{\mu}} \\
\ocwupperid{Vector4}\,\lbrack 1, \ocwupperid{C\_14\_23} \rbrack :&\;\;\;\;\;
V_1^\mu V_2^\nu
V_{3,\nu}^{\vphantom{\mu}} V_{4,\mu}^{\vphantom{\mu}}
\end{align}
\end{subequations} *)
type contract4 = C_12_34 | C_13_42 | C_14_23
(*i\begin{dubious}
CS objected to the polymorphic [type 'a vertex4], since it broke the
implementation of some of his extensions. Is there another way of
getting coupling constants into [Vector4_K_Matrix], besides the brute
force solution of declaring the possible coupling constants here?
\textit{I'd like to put the blame on CS for two reasons: it's not clear
that the brute force solution will actually work and everytime a new
vertex that depends non-linearly on coupling contanst pops up, the
problem will make another appearance.}
\end{dubious}i*)
type 'a vertex4 =
| Scalar4 of int
| Scalar2_Vector2 of int
| Vector4 of (int * contract4) list
| DScalar4 of (int * contract4) list
| DScalar2_Vector2 of (int * contract4) list
| GBBG of int * fermionbar * boson2 * fermion
(* In some applications, we have to allow for contributions outside of
perturbation theory. The most prominent example is heavy gauge boson
scattering at very high energies, where the perturbative expression
violates unitarity. *)
(* One solution is the `$K$-matrix' ansatz. Such unitarizations typically
introduce effective propagators and/or vertices that violate crossing
symmetry and vanish in the $t$-channel. This can be taken care of in
[Fusion] by filtering out vertices that have the wrong momenta. *)
(* In this case the ordering of the fields in a vertex of the Feynman
rules becomes significant. In particular, we assume that $(V_1,V_2,V_3,V_4)$
implies
\begin{equation}
\parbox{25mm}{\fmfframe(2,3)(2,3){\begin{fmfgraph*}(20,20)
\fmfleft{v1,v2}
\fmfright{v4,v3}
\fmflabel{$V_1$}{v1}
\fmflabel{$V_2$}{v2}
\fmflabel{$V_3$}{v3}
\fmflabel{$V_4$}{v4}
\fmf{plain}{v,v1}
\fmf{plain}{v,v2}
\fmf{plain}{v,v3}
\fmf{plain}{v,v4}
\fmfblob{.2w}{v}
\end{fmfgraph*}}}
\qquad\Longrightarrow\qquad
\parbox{45mm}{\fmfframe(2,3)(2,3){\begin{fmfgraph*}(40,20)
\fmfleft{v1,v2}
\fmfright{v4,v3}
\fmflabel{$V_1$}{v1}
\fmflabel{$V_2$}{v2}
\fmflabel{$V_3$}{v3}
\fmflabel{$V_4$}{v4}
\fmf{plain}{v1,v12,v2}
\fmf{plain}{v3,v34,v4}
\fmf{dots,label=$\Theta((p_1+p_2)^2)$,tension=0.7}{v12,v34}
\fmfdot{v12,v34}
\end{fmfgraph*}}}
\end{equation}
The list of pairs of parameters denotes the location and strengths
of the poles in the $K$-matrix ansatz:
\begin{equation}
(c_1,a_1,c_2,a_2,\ldots,c_n,a_n) \Longrightarrow
f(s) = \sum_{i=1}^{n} \frac{c_i}{s-a_i}
\end{equation} *)
| Vector4_K_Matrix_tho of int * ('a * 'a) list
| Vector4_K_Matrix_jr of int * (int * contract4) list
type 'a vertexn = unit
(* An obvious candidate for addition to [boson] is [T], of course. *)
(* \begin{dubious}
This list is sufficient for the minimal standard model, but not comprehensive
enough for most of its extensions, supersymmetric or otherwise.
In particular, we need a \emph{general} parameterization for all trilinear
vertices. One straightforward possibility are polynomials in the momenta for
each combination of fields.
\end{dubious}
\begin{JR}
Here we use the rules which can be found in~\cite{Denner:Majorana}
and are more properly described in [Targets] where the performing of the fusion
rules in analytical expressions is encoded.
\end{JR}
\begin{table}
\begin{center}
\renewcommand{\arraystretch}{1.2}
\begin{tabular}{|r|l|l|}\hline
& only Dirac fermions & incl.~Majorana fermions \\\hline
\multicolumn{3}{|l|}{[FBF (Psibar, S, Psi)]:
$\mathcal{L}_I=g_S\bar\psi_1 S\psi_2$}\\\hline
[F12] & $\bar\psi_2\leftarrow\ii\cdot g_S\bar\psi_1 S$
& $\psi_2\leftarrow\ii\cdot g_S\psi_1 S$ \\\hline
[F21] & $\bar\psi_2\leftarrow\ii\cdot g_S S \bar\psi_1$
& $\psi_2\leftarrow\ii\cdot g_SS\psi_1$ \\\hline
[F13] & $S\leftarrow\ii\cdot g_S\bar\psi_1\psi_2$
& $S\leftarrow\ii\cdot g_S\psi_1^T{\mathrm{C}}\psi_2$ \\\hline
[F31] & $S\leftarrow\ii\cdot g_S\psi_{2,\alpha}\bar\psi_{1,\alpha}$
& $S\leftarrow\ii\cdot g_S\psi_2^T{\mathrm{C}} \psi_1$\\\hline
[F23] & $\psi_1\leftarrow\ii\cdot g_SS\psi_2$
& $\psi_1\leftarrow\ii\cdot g_SS\psi_2$ \\\hline
[F32] & $\psi_1\leftarrow\ii\cdot g_S\psi_2 S$
& $\psi_1\leftarrow\ii\cdot g_S\psi_2 S$ \\\hline
\multicolumn{3}{|l|}{[FBF (Psibar, P, Psi)]:
$\mathcal{L}_I=g_P\bar\psi_1 P\gamma_5\psi_2$} \\\hline
[F12] & $\bar\psi_2\leftarrow\ii\cdot g_P\bar\psi_1\gamma_5 P$
& $\psi_2\leftarrow\ii\cdot g_P \gamma_5\psi_1 P$ \\\hline
[F21] & $\bar\psi_2\leftarrow\ii\cdot g_P P\bar\psi_1\gamma_5$
& $\psi_2\leftarrow\ii\cdot g_P P\gamma_5\psi_1$ \\\hline
[F13] & $P\leftarrow\ii\cdot g_P\bar\psi_1\gamma_5\psi_2$
& $P\leftarrow\ii\cdot g_P\psi_1^T {\mathrm{C}}\gamma_5\psi_2$ \\\hline
[F31] & $P\leftarrow\ii\cdot g_P[\gamma_5\psi_2]_\alpha\bar\psi_{1,\alpha}$
& $P\leftarrow\ii\cdot g_P\psi_2^T {\mathrm{C}}\gamma_5\psi_1$ \\\hline
[F23] & $\psi_1\leftarrow\ii\cdot g_P P\gamma_5\psi_2$
& $\psi_1\leftarrow\ii\cdot g_P P\gamma_5\psi_2$ \\\hline
[F32] & $\psi_1\leftarrow\ii\cdot g_P \gamma_5\psi_2 P$
& $\psi_1\leftarrow\ii\cdot g_P \gamma_5\psi_2 P$ \\\hline
\multicolumn{3}{|l|}{[FBF (Psibar, V, Psi)]:
$\mathcal{L}_I=g_V\bar\psi_1\fmslash{V}\psi_2$} \\\hline
[F12] & $\bar\psi_2\leftarrow\ii\cdot g_V\bar\psi_1\fmslash{V}$
& $\psi_{2,\alpha}\leftarrow\ii\cdot
(-g_V)\psi_{1,\beta}\fmslash{V}_{\alpha\beta}$ \\\hline
[F21] & $\bar\psi_{2,\beta}\leftarrow\ii\cdot
g_V\fmslash{V}_{\alpha\beta} \bar\psi_{1,\alpha}$
& $\psi_2\leftarrow\ii\cdot (-g_V)\fmslash{V}\psi_1$ \\\hline
[F13] & $V_\mu\leftarrow\ii\cdot g_V\bar\psi_1\gamma_\mu\psi_2$
& $V_\mu\leftarrow\ii\cdot
g_V (\psi_1)^T {\mathrm{C}}\gamma_{\mu}\psi_2$ \\\hline
[F31] & $V_\mu\leftarrow\ii\cdot g_V[\gamma_\mu\psi_2]_\alpha\bar\psi_{1,\alpha}$
& $V_\mu\leftarrow\ii\cdot
(-g_V)(\psi_2)^T {\mathrm{C}}\gamma_{\mu}\psi_1$ \\\hline
[F23] & $\psi_1\leftarrow\ii\cdot g_V\fmslash{V}\psi_2$
& $\psi_1\leftarrow\ii\cdot g_V\fmslash{V}\psi_2$ \\\hline
[F32] & $\psi_{1,\alpha}\leftarrow\ii\cdot
g_V\psi_{2,\beta}\fmslash{V}_{\alpha\beta}$
& $\psi_{1,\alpha}\leftarrow\ii\cdot
g_V\psi_{2,\beta}\fmslash{V}_{\alpha\beta}$ \\\hline
\multicolumn{3}{|l|}{[FBF (Psibar, A, Psi)]:
$\mathcal{L}_I=g_A\bar\psi_1\gamma_5\fmslash{A}\psi_2$} \\\hline
[F12] & $\bar\psi_2\leftarrow\ii\cdot g_A\bar\psi_1\gamma_5\fmslash{A}$
& $\psi_{2,\alpha}\leftarrow\ii\cdot
g_A\psi_{\beta}[\gamma_5\fmslash{A}]_{\alpha\beta}$ \\\hline
[F21] & $\bar\psi_{2,\beta}\leftarrow\ii\cdot g_A
[\gamma_5\fmslash{A}]_{\alpha\beta} \bar\psi_{1,\alpha}$
& $\psi_2\leftarrow\ii\cdot g_A \gamma_5\fmslash{A}\psi$ \\\hline
[F13] & $A_\mu\leftarrow\ii\cdot g_A\bar\psi_1\gamma_5\gamma_\mu\psi_2$
& $A_\mu\leftarrow\ii\cdot
g_A \psi_1^T {\textrm{C}}\gamma_5\gamma_{\mu}\psi_2$ \\\hline
[F31] & $A_\mu\leftarrow\ii\cdot
g_A[\gamma_5\gamma_\mu\psi_2]_\alpha\bar\psi_{1,\alpha}$
& $A_\mu\leftarrow\ii\cdot
g_A \psi_2^T {\textrm{C}}\gamma_5\gamma_{\mu}\psi_1$ \\\hline
[F23] & $\psi_1\leftarrow\ii\cdot g_A\gamma_5\fmslash{A}\psi_2$
& $\psi_1\leftarrow\ii\cdot g_A\gamma_5\fmslash{A}\psi_2$ \\\hline
[F32] & $\psi_{1,\alpha}\leftarrow\ii\cdot g_A
\psi_{2,\beta}[\gamma_5\fmslash{A}]_{\alpha\beta}$
& $\psi_{1,\alpha}\leftarrow\ii\cdot
g_A\psi_{2,\beta}[\gamma_5\fmslash{A}]_{\alpha\beta}$ \\\hline
\end{tabular}
\end{center}
\caption{\label{tab:dim4-fermions} Dimension-4 trilinear fermionic couplings.
The momenta are unambiguous, because there are no derivative couplings
and all participating fields are different.}
\end{table}
\begin{table}
\begin{center}
\renewcommand{\arraystretch}{1.3}
\begin{tabular}{|r|l|l|}\hline
& only Dirac fermions & incl.~Majorana fermions \\\hline
\multicolumn{3}{|l|}{[FBF (Psibar, T, Psi)]:
$\mathcal{L}_I=g_TT_{\mu\nu}\bar\psi_1
[\gamma^\mu,\gamma^\nu]_-\psi_2$}\\\hline
[F12] & $\bar\psi_2\leftarrow\ii\cdot g_T
\bar\psi_1[\gamma^\mu,\gamma^\nu]_-T_{\mu\nu}$
& $\bar\psi_2\leftarrow\ii\cdot g_T \cdots$ \\\hline
[F21] & $\bar\psi_2\leftarrow\ii\cdot g_T T_{\mu\nu}
\bar\psi_1[\gamma^\mu,\gamma^\nu]_-$
& $\bar\psi_2\leftarrow\ii\cdot g_T \cdots$ \\\hline
[F13] & $T_{\mu\nu}\leftarrow\ii\cdot g_T\bar\psi_1[\gamma_\mu,\gamma_\nu]_-\psi_2$
& $T_{\mu\nu}\leftarrow\ii\cdot g_T \cdots $ \\\hline
[F31] & $T_{\mu\nu}\leftarrow\ii\cdot g_T
[[\gamma_\mu,\gamma_\nu]_-\psi_2]_\alpha\bar\psi_{1,\alpha}$
& $T_{\mu\nu}\leftarrow\ii\cdot g_T \cdots $ \\\hline
[F23] & $\psi_1\leftarrow\ii\cdot g_T T_{\mu\nu}[\gamma^\mu,\gamma^\nu]_-\psi_2$
& $\psi_1\leftarrow\ii\cdot g_T \cdots$ \\\hline
[F32] & $\psi_1\leftarrow\ii\cdot g_T [\gamma^\mu,\gamma^\nu]_-\psi_2 T_{\mu\nu}$
& $\psi_1\leftarrow\ii\cdot g_T \cdots$ \\\hline
\end{tabular}
\end{center}
\caption{\label{tab:dim5-fermions} Dimension-5 trilinear fermionic couplings
(NB: the coefficients and signs are not fixed yet).
The momenta are unambiguous, because there are no derivative couplings
and all participating fields are different.}
\end{table}
\begin{table}
\begin{center}
\renewcommand{\arraystretch}{1.3}
\begin{tabular}{|r|l|l|}\hline
& only Dirac fermions & incl.~Majorana fermions \\\hline
\multicolumn{3}{|l|}{[FBF (Psibar, SP, Psi)]:
$\mathcal{L}_I=\bar\psi_1\phi(g_S+g_P\gamma_5)\psi_2$}\\\hline
[F12] & $\bar\psi_2\leftarrow\ii\cdot\bar\psi_1(g_S+g_P\gamma_5)\phi$
& $\psi_2\leftarrow\ii\cdot \cdots$ \\\hline
[F21] & $\bar\psi_2\leftarrow\ii\cdot\phi\bar\psi_1(g_S+g_P\gamma_5)$
& $\psi_2\leftarrow\ii\cdot \cdots$ \\\hline
[F13] & $\phi\leftarrow\ii\cdot\bar\psi_1(g_S+g_P\gamma_5)\psi_2$
& $\phi\leftarrow\ii\cdot\cdots$ \\\hline
[F31] & $\phi\leftarrow\ii\cdot[(g_S+g_P\gamma_5)\psi_2]_\alpha\bar\psi_{1,\alpha}$
& $\phi\leftarrow\ii\cdot\cdots$ \\\hline
[F23] & $\psi_1\leftarrow\ii\cdot \phi(g_S+g_P\gamma_5)\psi_2$
& $\psi_1\leftarrow\ii\cdot\cdots$ \\\hline
[F32] & $\psi_1\leftarrow\ii\cdot(g_S+g_P\gamma_5)\psi_2\phi$
& $\psi_1\leftarrow\ii\cdot\cdots$ \\\hline
\multicolumn{3}{|l|}{[FBF (Psibar, SL, Psi)]:
$\mathcal{L}_I=g_L\bar\psi_1\phi(1-\gamma_5)\psi_2$}\\\hline
[F12] & $\bar\psi_2\leftarrow\ii\cdot g_L\bar\psi_1(1-\gamma_5)\phi$
& $\psi_2\leftarrow\ii\cdot \cdots$ \\\hline
[F21] & $\bar\psi_2\leftarrow\ii\cdot g_L\phi\bar\psi_1(1-\gamma_5)$
& $\psi_2\leftarrow\ii\cdot \cdots$ \\\hline
[F13] & $\phi\leftarrow\ii\cdot g_L\bar\psi_1(1-\gamma_5)\psi_2$
& $\phi\leftarrow\ii\cdot\cdots$ \\\hline
[F31] & $\phi\leftarrow\ii\cdot g_L[(1-\gamma_5)\psi_2]_\alpha\bar\psi_{1,\alpha}$
& $\phi\leftarrow\ii\cdot\cdots$ \\\hline
[F23] & $\psi_1\leftarrow\ii\cdot g_L\phi(1-\gamma_5)\psi_2$
& $\psi_1\leftarrow\ii\cdot\cdots$ \\\hline
[F32] & $\psi_1\leftarrow\ii\cdot g_L(1-\gamma_5)\psi_2\phi$
& $\psi_1\leftarrow\ii\cdot\cdots$ \\\hline
\multicolumn{3}{|l|}{[FBF (Psibar, SR, Psi)]:
$\mathcal{L}_I=g_R\bar\psi_1\phi(1+\gamma_5)\psi_2$}\\\hline
[F12] & $\bar\psi_2\leftarrow\ii\cdot g_R\bar\psi_1(1+\gamma_5)\phi$
& $\psi_2\leftarrow\ii\cdot \cdots$ \\\hline
[F21] & $\bar\psi_2\leftarrow\ii\cdot g_R\phi\bar\psi_1(1+\gamma_5)$
& $\psi_2\leftarrow\ii\cdot \cdots$ \\\hline
[F13] & $\phi\leftarrow\ii\cdot g_R\bar\psi_1(1+\gamma_5)\psi_2$
& $\phi\leftarrow\ii\cdot\cdots$ \\\hline
[F31] & $\phi\leftarrow\ii\cdot g_R[(1+\gamma_5)\psi_2]_\alpha\bar\psi_{1,\alpha}$
& $\phi\leftarrow\ii\cdot\cdots$ \\\hline
[F23] & $\psi_1\leftarrow\ii\cdot g_R\phi(1+\gamma_5)\psi_2$
& $\psi_1\leftarrow\ii\cdot\cdots$ \\\hline
[F32] & $\psi_1\leftarrow\ii\cdot g_R(1+\gamma_5)\psi_2\phi$
& $\psi_1\leftarrow\ii\cdot\cdots$ \\\hline
\multicolumn{3}{|l|}{[FBF (Psibar, SLR, Psi)]:
$\mathcal{L}_I=g_L\bar\psi_1\phi(1-\gamma_5)\psi_2
+g_R\bar\psi_1\phi(1+\gamma_5)\psi_2$}\\\hline
\end{tabular}
\end{center}
\caption{\label{tab:dim4-fermions-SP} Combined dimension-4 trilinear fermionic couplings.}
\end{table}
\begin{table}
\begin{center}
\renewcommand{\arraystretch}{1.3}
\begin{tabular}{|r|l|l|}\hline
& only Dirac fermions & incl.~Majorana fermions \\\hline
\multicolumn{3}{|l|}{[FBF (Psibar, VA, Psi)]:
$\mathcal{L}_I=\bar\psi_1\fmslash{Z}(g_V-g_A\gamma_5)\psi_2$}\\\hline
[F12] & $\bar\psi_2\leftarrow\ii\cdot\bar\psi_1\fmslash{Z}(g_V-g_A\gamma_5)$
& $\psi_2\leftarrow\ii\cdot \cdots$ \\\hline
[F21] & $\bar\psi_{2,\beta}\leftarrow\ii\cdot
[\fmslash{Z}(g_V-g_A\gamma_5)]_{\alpha\beta}\bar\psi_{1,\alpha}$
& $\psi_2\leftarrow\ii\cdot \cdots$ \\\hline
[F13] & $Z_\mu\leftarrow\ii\cdot\bar\psi_1\gamma_\mu(g_V-g_A\gamma_5)\psi_2$
& $Z_\mu\leftarrow\ii\cdot \cdots$ \\\hline
[F31] & $Z_\mu\leftarrow\ii\cdot
[\gamma_\mu(g_V-g_A\gamma_5)\psi_2]_\alpha\bar\psi_{1,\alpha}$
& $Z_\mu\leftarrow\ii\cdot \cdots$ \\\hline
[F23] & $\psi_1\leftarrow\ii\cdot\fmslash{Z}(g_V-g_A\gamma_5)\psi_2$
& $\psi_1\leftarrow\ii\cdot\cdots$ \\\hline
[F32] & $\psi_{1,\alpha}\leftarrow\ii\cdot
\psi_{2,\beta}[\fmslash{Z}(g_V-g_A\gamma_5)]_{\alpha\beta}$
& $\psi_1\leftarrow\ii\cdot\cdots$ \\\hline
\multicolumn{3}{|l|}{[FBF (Psibar, VL, Psi)]:
$\mathcal{L}_I=g_L\bar\psi_1\fmslash{Z}(1-\gamma_5)\psi_2$}\\\hline
[F12] & $\bar\psi_2\leftarrow\ii\cdot g_L\bar\psi_1\fmslash{Z}(1-\gamma_5)$
& $\psi_2\leftarrow\ii\cdot \cdots$ \\\hline
[F21] & $\bar\psi_{2,\beta}\leftarrow\ii\cdot
g_L[\fmslash{Z}(1-\gamma_5)]_{\alpha\beta}\bar\psi_{1,\alpha}$
& $\psi_2\leftarrow\ii\cdot \cdots$ \\\hline
[F13] & $Z_\mu\leftarrow\ii\cdot g_L\bar\psi_1\gamma_\mu(1-\gamma_5)\psi_2$
& $Z_\mu\leftarrow\ii\cdot \cdots$ \\\hline
[F31] & $Z_\mu\leftarrow\ii\cdot
g_L[\gamma_\mu(1-\gamma_5)\psi_2]_\alpha\bar\psi_{1,\alpha}$
& $Z_\mu\leftarrow\ii\cdot \cdots$ \\\hline
[F23] & $\psi_1\leftarrow\ii\cdot g_L\fmslash{Z}(1-\gamma_5)\psi_2$
& $\psi_1\leftarrow\ii\cdot\cdots$ \\\hline
[F32] & $\psi_{1,\alpha}\leftarrow\ii\cdot
g_L\psi_{2,\beta}[\fmslash{Z}(1-\gamma_5)]_{\alpha\beta}$
& $\psi_1\leftarrow\ii\cdot\cdots$ \\\hline
\multicolumn{3}{|l|}{[FBF (Psibar, VR, Psi)]:
$\mathcal{L}_I=g_R\bar\psi_1\fmslash{Z}(1+\gamma_5)\psi_2$}\\\hline
[F12] & $\bar\psi_2\leftarrow\ii\cdot g_R\bar\psi_1\fmslash{Z}(1+\gamma_5)$
& $\psi_2\leftarrow\ii\cdot \cdots$ \\\hline
[F21] & $\bar\psi_{2,\beta}\leftarrow\ii\cdot
g_R[\fmslash{Z}(1+\gamma_5)]_{\alpha\beta}\bar\psi_{1,\alpha}$
& $\psi_2\leftarrow\ii\cdot \cdots$ \\\hline
[F13] & $Z_\mu\leftarrow\ii\cdot g_R\bar\psi_1\gamma_\mu(1+\gamma_5)\psi_2$
& $Z_\mu\leftarrow\ii\cdot \cdots$ \\\hline
[F31] & $Z_\mu\leftarrow\ii\cdot
g_R[\gamma_\mu(1+\gamma_5)\psi_2]_\alpha\bar\psi_{1,\alpha}$
& $Z_\mu\leftarrow\ii\cdot \cdots$ \\\hline
[F23] & $\psi_1\leftarrow\ii\cdot g_R\fmslash{Z}(1+\gamma_5)\psi_2$
& $\psi_1\leftarrow\ii\cdot\cdots$ \\\hline
[F32] & $\psi_{1,\alpha}\leftarrow\ii\cdot
g_R\psi_{2,\beta}[\fmslash{Z}(1+\gamma_5)]_{\alpha\beta}$
& $\psi_1\leftarrow\ii\cdot\cdots$ \\\hline
\multicolumn{3}{|l|}{[FBF (Psibar, VLR, Psi)]:
$\mathcal{L}_I=g_L\bar\psi_1\fmslash{Z}(1-\gamma_5)\psi_2
+g_R\bar\psi_1\fmslash{Z}(1+\gamma_5)\psi_2$}\\\hline
\end{tabular}
\end{center}
\caption{\label{tab:dim4-fermions-VA} Combined dimension-4 trilinear
fermionic couplings continued.}
\end{table}
\begin{table}
\begin{center}
\renewcommand{\arraystretch}{1.4}
\begin{tabular}{|>{\qquad}r<{:}l|r<{:}l|}\hline
\multicolumn{4}{|l|}{[FBF (Psibar, S, Chi)]: $\bar\psi S\chi$}\\\hline
[F12] & $\chi\leftarrow\psi S$
& [F21] & $\chi\leftarrow S \psi$ \\\hline
[F13] & $S\leftarrow \psi^T{\rm C}\chi$
& [F31] & $S\leftarrow \chi^T {\rm C}\psi$ \\\hline
[F23] & $\psi\leftarrow S\chi$
& [F32] & $\psi\leftarrow\chi S$ \\\hline
\multicolumn{4}{|l|}{[FBF (Psibar, P, Chi)]: $\bar\psi P\gamma_5\chi$}\\\hline
[F12] & $\chi\leftarrow \gamma_5 \psi P$
& [F21] & $\chi\leftarrow P \gamma_5 \psi$ \\\hline
[F13] & $P\leftarrow \psi^T {\rm C}\gamma_5\chi$
& [F31] & $P\leftarrow \chi^T {\rm C}\gamma_5\psi$ \\\hline
[F23] & $\psi\leftarrow P\gamma_5\chi$
& [F32] & $\psi\leftarrow\gamma_5\chi P$ \\\hline
\multicolumn{4}{|l|}{[FBF (Psibar, V, Chi)]: $\bar\psi\fmslash{V}\chi$}\\\hline
[F12] & $\chi_{\alpha}\leftarrow-\psi_{\beta}\fmslash{V}_{\alpha\beta}$
& [F21] & $\chi\leftarrow-\fmslash{V}\psi$ \\\hline
[F13] & $V_{\mu}\leftarrow \psi^T {\rm C}\gamma_{\mu}\chi$
& [F31] & $V_{\mu}\leftarrow \chi^T {\rm C}(-\gamma_{\mu}\psi)$ \\\hline
[F23] & $\psi\leftarrow\fmslash{V}\chi$
& [F32] & $\psi_\alpha\leftarrow\chi_\beta\fmslash{V}_{\alpha\beta}$ \\\hline
\multicolumn{4}{|l|}{[FBF (Psibar, A, Chi)]: $\bar\psi\gamma^5\fmslash{A}\chi$}\\\hline
[F12] & $\chi_{\alpha}\leftarrow\psi_{\beta}\lbrack \gamma^5 \fmslash{A} \rbrack_{\alpha\beta}$
& [F21] & $\chi\leftarrow\gamma^5\fmslash{A}\psi$ \\\hline
[F13] & $A_{\mu}\leftarrow \psi^T {\rm C}\gamma^5\gamma_{\mu}\chi$
& [F31] & $A_{\mu}\leftarrow \chi^T {\rm C}(\gamma^5 \gamma_{\mu}\psi)$ \\\hline
[F23] & $\psi\leftarrow\gamma^5\fmslash{A}\chi$
& [F32] & $\psi_\alpha\leftarrow\chi_\beta\lbrack \gamma^5 \fmslash{A} \rbrack_{\alpha\beta}$ \\\hline
\end{tabular}
\end{center}
\caption{\label{tab:dim4-fermions-maj} Dimension-4 trilinear couplings
including one Dirac and one Majorana fermion}
\end{table}
\begin{table}
\begin{center}
\renewcommand{\arraystretch}{1.4}
\begin{tabular}{|>{\qquad}r<{:}l|r<{:}l|}\hline
\multicolumn{4}{|l|}{[FBF (Psibar, SP, Chi)]:
$\bar\psi\phi(g_S+g_P\gamma_5)\chi$}\\\hline
[F12] & $\chi \leftarrow (g_S+g_P\gamma_5)\psi \phi$
& [F21] & $\chi\leftarrow\phi(g_S+g_P\gamma_5)\psi$ \\\hline
[F13] & $\phi\leftarrow \psi^T {\rm C}(g_S+g_P\gamma_5)\chi$
& [F31] & $\phi\leftarrow \chi^T {\rm C}(g_S+g_P\gamma_5) \chi$ \\\hline
[F23] & $\psi\leftarrow \phi(g_S+g_P\gamma_5)\chi$
& [F32] & $\psi\leftarrow(g_S+g_P\gamma_5)\chi\phi$ \\\hline
\multicolumn{4}{|l|}{[FBF (Psibar, VA, Chi)]:
$\bar\psi\fmslash{Z}(g_V - g_A\gamma_5)\chi$}\\\hline
[F12] & $\chi_\alpha\leftarrow
\psi_\beta[\fmslash{Z}(-g_V-g_A\gamma_5)]_{\alpha\beta}$
& [F21] & $\chi\leftarrow\fmslash{Z}(-g_V-g_A\gamma_5)]
\psi$ \\\hline
[F13] & $Z_\mu\leftarrow \psi^T {\rm C}\gamma_\mu(g_V-g_A\gamma_5)\chi$
& [F31] & $Z_\mu\leftarrow \chi^T {\rm C}\gamma_\mu(-g_V-g_A\gamma_5)\psi$ \\\hline
[F23] & $\psi\leftarrow\fmslash{Z}(g_V-g_A\gamma_5)\chi$
& [F32] & $\psi_\alpha\leftarrow
\chi_\beta[\fmslash{Z}(g_V-g_A\gamma_5)]_{\alpha\beta}$ \\\hline
\end{tabular}
\end{center}
\caption{\label{tab:dim4-fermions-SPVA-maj} Combined dimension-4 trilinear
fermionic couplings including one Dirac and one Majorana fermion.}
\end{table}
\begin{table}
\begin{center}
\renewcommand{\arraystretch}{1.4}
\begin{tabular}{|>{\qquad}r<{:}l|r<{:}l|}\hline
\multicolumn{4}{|l|}{[FBF (Chibar, S, Psi)]: $\bar\chi S\psi$}\\\hline
[F12] & $\psi\leftarrow\chi S$
& [F21] & $\psi\leftarrow S\chi$ \\\hline
[F13] & $S\leftarrow \chi^T {\rm C}\psi$
& [F31] & $S\leftarrow \psi^T {\rm C}\chi$ \\\hline
[F23] & $\chi\leftarrow S \psi$
& [F32] & $\chi\leftarrow\psi S$ \\\hline
\multicolumn{4}{|l|}{[FBF (Chibar, P, Psi)]: $\bar\chi P\gamma_5\psi$}\\\hline
[F12] & $\psi\leftarrow\gamma_5\chi P$
& [F21] & $\psi\leftarrow P\gamma_5\chi$ \\\hline
[F13] & $P\leftarrow \chi^T {\rm C}\gamma_5\psi$
& [F31] & $P\leftarrow \psi^T {\rm C}\gamma_5\chi$ \\\hline
[F23] & $\chi\leftarrow P \gamma_5 \psi$
& [F32] & $\chi\leftarrow \gamma_5 \psi P$ \\\hline
\multicolumn{4}{|l|}{[FBF (Chibar, V, Psi)]: $\bar\chi\fmslash{V}\psi$}\\\hline
[F12] & $\psi_\alpha\leftarrow-\chi_\beta\fmslash{V}_{\alpha\beta}$
& [F21] & $\psi\leftarrow-\fmslash{V}\chi$ \\\hline
[F13] & $V_{\mu}\leftarrow \chi^T {\rm C}\gamma_{\mu}\psi$
& [F31] & $V_{\mu}\leftarrow \psi^T {\rm C}(-\gamma_{\mu}\chi)$ \\\hline
[F23] & $\chi\leftarrow\fmslash{V}\psi$
& [F32] & $\chi_{\alpha}\leftarrow\psi_{\beta}\fmslash{V}_{\alpha\beta}$ \\\hline
\multicolumn{4}{|l|}{[FBF (Chibar, A, Psi)]: $\bar\chi\gamma^5\fmslash{A}\psi$}\\\hline
[F12] & $\psi_\alpha\leftarrow\chi_\beta\lbrack\gamma^5\fmslash{A} \rbrack_{\alpha\beta}$
& [F21] & $\psi\leftarrow\gamma^5\fmslash{A}\chi$ \\\hline
[F13] & $A_{\mu}\leftarrow \chi^T {\rm C}(\gamma^5\gamma_{\mu}\psi)$
& [F31] & $A_{\mu}\leftarrow \psi^T {\rm C}\gamma^5\gamma_{\mu}\chi$ \\\hline
[F23] & $\chi\leftarrow\gamma^5\fmslash{A}\psi$
& [F32] & $\chi_{\alpha}\leftarrow\psi_{\beta}\lbrack\gamma^5\fmslash{A} \rbrack_{\alpha\beta}$ \\\hline
\end{tabular}
\end{center}
\caption{\label{tab:dim4-fermions-maj'} Dimension-4 trilinear couplings
including one Dirac and one Majorana fermion}
\end{table}
\begin{table}
\begin{center}
\renewcommand{\arraystretch}{1.4}
\begin{tabular}{|>{\qquad}r<{:}l|r<{:}l|}\hline
\multicolumn{4}{|l|}{[FBF (Chibar, SP, Psi)]: $\bar\chi\phi(g_S+g_P\gamma_5)\psi$}\\\hline
[F12] & $\psi\leftarrow(g_S+g_P\gamma_5)\chi\phi$
& [F21] & $\psi\leftarrow \phi(g_S+g_P\gamma_5)\chi$ \\\hline
[F13] & $\phi\leftarrow \chi^T {\rm C}(g_S+g_P\gamma_5) \psi$
& [F31] & $\phi\leftarrow \psi^T {\rm C}(g_S+g_P\gamma_5)\chi$ \\\hline
[F23] & $\chi\leftarrow\phi(g_S+g_P\gamma_5)\psi$
& [F32] & $\chi \leftarrow (g_S+g_P\gamma_5)\psi \phi$ \\\hline
\multicolumn{4}{|l|}{[FBF (Chibar, VA, Psi)]:
$\bar\chi\fmslash{Z}(g_V - g_A\gamma_5)\psi$}\\\hline
[F12] & $\psi_\alpha\leftarrow
\chi_\beta[\fmslash{Z}(-g_V-g_A\gamma_5)]_{\alpha\beta}$
& [F21] & $\psi\leftarrow\fmslash{Z}(-g_V-g_A\gamma_5)\chi$ \\\hline
[F13] & $Z_\mu\leftarrow \chi^T {\rm C}\gamma_\mu(g_V-g_A\gamma_5)\psi$
& [F31] & $Z_\mu\leftarrow \psi^T {\rm C}\gamma_\mu(-g_V-g_A\gamma_5)\chi$ \\\hline
[F23] & $\chi\leftarrow\fmslash{Z}(g_V-g_A\gamma_5)]
\psi$
& [F32] & $\chi_\alpha\leftarrow\psi_\beta[\fmslash{Z}(g_V-g_A\gamma_5)]_{\alpha\beta}$ \\\hline
\end{tabular}
\end{center}
\caption{\label{tab:dim4-fermions-SPVA-maj'} Combined dimension-4 trilinear
fermionic couplings including one Dirac and one Majorana fermion.}
\end{table}
\begin{table}
\begin{center}
\renewcommand{\arraystretch}{1.4}
\begin{tabular}{|>{\qquad}r<{:}l|r<{:}l|}\hline
\multicolumn{4}{|l|}{[FBF (Chibar, S, Chi)]: $\bar\chi_a S\chi_b$}\\\hline
[F12] & $\chi_b\leftarrow\chi_a S$
& [F21] & $\chi_b\leftarrow S \chi_a$ \\\hline
[F13] & $S\leftarrow \chi^T_a {\rm C}\chi_b$
& [F31] & $S\leftarrow \chi^T_b {\rm C}\chi_a$ \\\hline
[F23] & $\chi_a\leftarrow S\chi_b$
& [F32] & $\chi_a\leftarrow\chi S_b$ \\\hline
\multicolumn{4}{|l|}{[FBF (Chibar, P, Chi)]: $\bar\chi_a P\gamma_5\psi_b$}\\\hline
[F12] & $\chi_b\leftarrow \gamma_5 \chi_a P$
& [F21] & $\chi_b\leftarrow P \gamma_5 \chi_a$ \\\hline
[F13] & $P\leftarrow \chi^T_a {\rm C}\gamma_5\chi_b$
& [F31] & $P\leftarrow \chi^T_b {\rm C}\gamma_5\chi_a$ \\\hline
[F23] & $\chi_a\leftarrow P\gamma_5\chi_b$
& [F32] & $\chi_a\leftarrow\gamma_5\chi_b P$ \\\hline
\multicolumn{4}{|l|}{[FBF (Chibar, V, Chi)]: $\bar\chi_a\fmslash{V}\chi_b$}\\\hline
[F12] & $\chi_{b,\alpha}\leftarrow-\chi_{a,\beta}\fmslash{V}_{\alpha\beta}$
& [F21] & $\chi_b\leftarrow-\fmslash{V}\chi_a$ \\\hline
[F13] & $V_{\mu}\leftarrow \chi^T_a {\rm C}\gamma_{\mu}\chi_b$
& [F31] & $V_{\mu}\leftarrow - \chi^T_b {\rm C}\gamma_{\mu}\chi_a$ \\\hline
[F23] & $\chi_a\leftarrow\fmslash{V}\chi_b$
& [F32] & $\chi_{a,\alpha}\leftarrow\chi_{b,\beta}\fmslash{V}_{\alpha\beta}$ \\\hline
\multicolumn{4}{|l|}{[FBF (Chibar, A, Chi)]: $\bar\chi_a\gamma^5\fmslash{A}\chi_b$}\\\hline
[F12] & $\chi_{b,\alpha}\leftarrow\chi_{a,\beta}\lbrack\gamma^5\fmslash{A} \rbrack_{\alpha\beta}$
& [F21] & $\chi_b\leftarrow\gamma^5\fmslash{A}\chi_a$ \\\hline
[F13] & $A_{\mu}\leftarrow \chi^T_a {\rm C}\gamma^5\gamma_{\mu}\chi_b$
& [F31] & $A_{\mu}\leftarrow \chi^T_b {\rm C}(\gamma^5\gamma_{\mu}\chi_a)$ \\\hline
[F23] & $\chi_a\leftarrow\gamma^5\fmslash{A}\chi_b$
& [F32] & $\chi_{a,\alpha}\leftarrow\chi_{b,\beta}\lbrack\gamma^5\fmslash{A} \rbrack_{\alpha\beta}$ \\\hline
\end{tabular}
\end{center}
\caption{\label{tab:dim4-fermions-maj2} Dimension-4 trilinear couplings
of two Majorana fermions}
\end{table}
\begin{table}
\begin{center}
\renewcommand{\arraystretch}{1.4}
\begin{tabular}{|>{\qquad}r<{:}l|r<{:}l|}\hline
\multicolumn{4}{|l|}{[FBF (Chibar, SP, Chi)]:
$\bar\chi\phi_a(g_S+g_P\gamma_5)\chi_b$}\\\hline
[F12] & $\chi_b \leftarrow (g_S+g_P\gamma_5)\chi_a \phi$
& [F21] & $\chi_b\leftarrow\phi(g_S+g_P\gamma_5)\chi_a$ \\\hline
[F13] & $\phi\leftarrow \chi^T_a {\rm C}(g_S+g_P\gamma_5)\chi_b$
& [F31] & $\phi\leftarrow \chi^T_b {\rm C}(g_S+g_P\gamma_5) \chi_a$ \\\hline
[F23] & $\chi_a\leftarrow \phi(g_S+g_P\gamma_5)\chi_b$
& [F32] & $\chi_a\leftarrow(g_S+g_P\gamma_5)\chi_b\phi$ \\\hline
\multicolumn{4}{|l|}{[FBF (Chibar, VA, Chi)]:
$\bar\chi_a\fmslash{Z}(g_V-g_A\gamma_5)\chi_b$}\\\hline
[F12] & $\chi_{b,\alpha}\leftarrow\chi_{a,\beta}[\fmslash{Z}(-g_V-g_A\gamma_5)]_{\alpha\beta}$
& [F21] & $\chi_b\leftarrow\fmslash{Z}(-g_V-g_A\gamma_5)]\chi_a$ \\\hline
[F13] & $Z_\mu\leftarrow \chi^T_a {\rm C}\gamma_\mu(g_V-g_A\gamma_5)\chi_b$
& [F31] & $Z_\mu\leftarrow \chi^T_b {\rm C}\gamma_\mu(-g_V-g_A\gamma_5)\chi_a$ \\\hline
[F23] & $\chi_a\leftarrow\fmslash{Z}(g_V-g_A\gamma_5)\chi_b$
& [F32] & $\chi_{a,\alpha}\leftarrow
\chi_{b,\beta}[\fmslash{Z}(g_V-g_A\gamma_5)]_{\alpha\beta}$ \\\hline
\end{tabular}
\end{center}
\caption{\label{tab:dim4-fermions-SPVA-maj2} Combined dimension-4 trilinear
fermionic couplings of two Majorana fermions.}
\end{table}
\begin{table}
\begin{center}
\renewcommand{\arraystretch}{1.3}
\begin{tabular}{|>{\qquad}r<{:}l|}\hline
\multicolumn{2}{|l|}{[Gauge_Gauge_Gauge]:
$\mathcal{L}_I=gf_{abc}
A_a^\mu A_b^\nu\partial_\mu A_{c,\nu}$}\\\hline
[_] & $A_a^\mu\leftarrow\ii\cdot
(-\ii g/2)\cdot C_{abc}^{\mu\rho\sigma}(-k_2-k_3,k_2,k_3)
A^b_\rho A^c_\sigma$\\\hline
\multicolumn{2}{|l|}{[Aux_Gauge_Gauge]:
$\mathcal{L}_I=gf_{abc}X_{a,\mu\nu}(k_1)
( A_b^{\mu}(k_2)A_c^{\nu}(k_3)
-A_b^{\nu}(k_2)A_c^{\mu}(k_3))$}\\\hline
[F23]$\lor$[F32] & $X_a^{\mu\nu}(k_2+k_3)\leftarrow\ii\cdot
gf_{abc}( A_b^\mu(k_2)A_c^\nu(k_3)
-A_b^\nu(k_2)A_c^\mu(k_3))$ \\\hline
[F12]$\lor$[F13] & $A_{a,\mu}(k_1+k_{2/3})\leftarrow\ii\cdot
gf_{abc}X_{b,\nu\mu}(k_1)A_c^\nu(k_{2/3})$ \\\hline
[F21]$\lor$[F31] & $A_{a,\mu}(k_{2/3}+k_1)\leftarrow\ii\cdot
gf_{abc}A_b^\nu(k_{2/3}) X_{c,\mu\nu}(k_1)$ \\\hline
\end{tabular}
\end{center}
\caption{\label{tab:dim4-bosons} Dimension-4 Vector Boson couplings with
\emph{outgoing} momenta.
See~(\ref{eq:C123}) and~(\ref{eq:C123'}) for the definition of the
antisymmetric tensor $C^{\mu_1\mu_2\mu_3}(k_1,k_2,k_3)$.}
\end{table}
\begin{table}
\begin{center}
\renewcommand{\arraystretch}{1.3}
\begin{tabular}{|>{\qquad}r<{:}l|r<{:}l|}\hline
\multicolumn{4}{|l|}{[Scalar_Vector_Vector]:
$\mathcal{L}_I=g\phi V_1^\mu V_{2,\mu}$}\\\hline
[F13] & $\leftarrow\ii\cdot g\cdots$
& [F31] & $\leftarrow\ii\cdot g\cdots$ \\\hline
[F12] & $\leftarrow\ii\cdot g\cdots$
& [F21] & $\leftarrow\ii\cdot g\cdots$ \\\hline
[F23] & $\phi\leftarrow\ii\cdot g V_1^\mu V_{2,\mu}$
& [F32] & $\phi\leftarrow\ii\cdot g V_{2,\mu} V_1^\mu$ \\\hline
\multicolumn{4}{|l|}{[Aux_Vector_Vector]:
$\mathcal{L}_I=gX V_1^\mu V_{2,\mu}$}\\\hline
[F13] & $\leftarrow\ii\cdot g\cdots$
& [F31] & $\leftarrow\ii\cdot g\cdots$ \\\hline
[F12] & $\leftarrow\ii\cdot g\cdots$
& [F21] & $\leftarrow\ii\cdot g\cdots$ \\\hline
[F23] & $X\leftarrow\ii\cdot g V_1^\mu V_{2,\mu}$
& [F32] & $X\leftarrow\ii\cdot g V_{2,\mu} V_1^\mu$ \\\hline
\multicolumn{4}{|l|}{[Aux_Scalar_Vector]:
$\mathcal{L}_I=gX^\mu \phi V_\mu$}\\\hline
[F13] & $\leftarrow\ii\cdot g\cdots$
& [F31] & $\leftarrow\ii\cdot g\cdots$ \\\hline
[F12] & $\leftarrow\ii\cdot g\cdots$
& [F21] & $\leftarrow\ii\cdot g\cdots$ \\\hline
[F23] & $\leftarrow\ii\cdot g\cdots$
& [F32] & $\leftarrow\ii\cdot g\cdots$ \\\hline
\end{tabular}
\end{center}
\caption{\label{tab:scalar-vector}
\ldots}
\end{table}
\begin{table}
\begin{center}
\renewcommand{\arraystretch}{1.3}
\begin{tabular}{|>{\qquad}r<{:}l|r<{:}l|}\hline
\multicolumn{4}{|l|}{[Scalar_Scalar_Scalar]:
$\mathcal{L}_I=g\phi_1\phi_2\phi_3$}\\\hline
[F13] & $\phi_2\leftarrow\ii\cdot g \phi_1\phi_3$
& [F31] & $\phi_2\leftarrow\ii\cdot g \phi_3\phi_1$ \\\hline
[F12] & $\phi_3\leftarrow\ii\cdot g \phi_1\phi_2$
& [F21] & $\phi_3\leftarrow\ii\cdot g \phi_2\phi_1$ \\\hline
[F23] & $\phi_1\leftarrow\ii\cdot g \phi_2\phi_3$
& [F32] & $\phi_1\leftarrow\ii\cdot g \phi_3\phi_2$ \\\hline
\multicolumn{4}{|l|}{[Aux_Scalar_Scalar]:
$\mathcal{L}_I=gX\phi_1\phi_2$}\\\hline
[F13] & $\leftarrow\ii\cdot g\cdots$
& [F31] & $\leftarrow\ii\cdot g\cdots$ \\\hline
[F12] & $\leftarrow\ii\cdot g\cdots$
& [F21] & $\leftarrow\ii\cdot g\cdots$ \\\hline
[F23] & $X\leftarrow\ii\cdot g \phi_1\phi_2$
& [F32] & $X\leftarrow\ii\cdot g \phi_2\phi_1$ \\\hline
\end{tabular}
\end{center}
\caption{\label{tab:scalars}
\ldots}
\end{table}
\begin{table}
\begin{center}
\renewcommand{\arraystretch}{1.3}
\begin{tabular}{|>{\qquad}r<{:}l|}\hline
\multicolumn{2}{|l|}{[Vector_Scalar_Scalar]:
$\mathcal{L}_I=gV^\mu\phi_1
\ii\overleftrightarrow{\partial_\mu}\phi_2$}\\\hline
[F23] & $V^\mu(k_2+k_3)\leftarrow\ii\cdot
g(k_2^\mu-k_3^\mu)\phi_1(k_2)\phi_2(k_3)$ \\\hline
[F32] & $V^\mu(k_2+k_3)\leftarrow\ii\cdot
g(k_2^\mu-k_3^\mu)\phi_2(k_3)\phi_1(k_2)$ \\\hline
[F12] & $\phi_2(k_1+k_2)\leftarrow\ii\cdot
g(k_1^\mu+2k_2^\mu)V_\mu(k_1)\phi_1(k_2)$ \\\hline
[F21] & $\phi_2(k_1+k_2)\leftarrow\ii\cdot
g(k_1^\mu+2k_2^\mu)\phi_1(k_2)V_\mu(k_1)$ \\\hline
[F13] & $\phi_1(k_1+k_3)\leftarrow\ii\cdot
g(-k_1^\mu-2k_3^\mu)V_\mu(k_1)\phi_2(k_3)$ \\\hline
[F31] & $\phi_1(k_1+k_3)\leftarrow\ii\cdot
g(-k_1^\mu-2k_3^\mu)\phi_2(k_3)V_\mu(k_1)$ \\\hline
\end{tabular}
\end{center}
\caption{\label{tab:scalar-current}
\ldots}
\end{table} *)
(* \begin{table}
\begin{center}
\renewcommand{\arraystretch}{1.3}
\begin{tabular}{|>{\qquad}r<{:}l|}\hline
\multicolumn{2}{|l|}{[Aux_DScalar_DScalar]:
$\mathcal{L}_I=g\chi
(\ii\partial_\mu\phi_1)(\ii\partial^\mu\phi_2)$}\\\hline
[F23] & $\chi(k_2+k_3)\leftarrow\ii\cdot
g (k_2\cdot k_3) \phi_1(k_2) \phi_2(k_3) $ \\\hline
[F32] & $\chi(k_2+k_3)\leftarrow\ii\cdot
g (k_3\cdot k_2) \phi_2(k_3) \phi_1(k_2) $ \\\hline
[F12] & $\phi_2(k_1+k_2)\leftarrow\ii\cdot
g ((-k_1-k_2) \cdot k_2) \chi(k_1) \phi_1(k_2) $ \\\hline
[F21] & $\phi_2(k_1+k_2)\leftarrow\ii\cdot
g (k_2 \cdot (-k_1-k_2)) \phi_1(k_2) \chi(k_1) $ \\\hline
[F13] & $\phi_1(k_1+k_3)\leftarrow\ii\cdot
g ((-k_1-k_3) \cdot k_3) \chi(k_1) \phi_2(k_3) $ \\\hline
[F31] & $\phi_1(k_1+k_3)\leftarrow\ii\cdot
g (k_3 \cdot (-k_1-k_3)) \phi_2(k_3) \chi(k_1) $ \\\hline
\end{tabular}
\end{center}
\caption{\label{tab:dscalar-dscalar}
\ldots}
\end{table}
\begin{table}
\begin{center}
\renewcommand{\arraystretch}{1.3}
\begin{tabular}{|>{\qquad}r<{:}l|}\hline
\multicolumn{2}{|l|}{[Aux_Vector_DScalar]:
$\mathcal{L}_I=g\chi V_\mu (\ii\partial^\mu\phi)$}\\\hline
[F23] & $\chi(k_2+k_3)\leftarrow\ii\cdot
g k_3^\mu V_\mu(k_2) \phi(k_3) $ \\\hline
[F32] & $\chi(k_2+k_3)\leftarrow\ii\cdot
g \phi(k_3) k_3^\mu V_\mu(k_2) $ \\\hline
[F12] & $\phi(k_1+k_2)\leftarrow\ii\cdot
g \chi(k_1) (-k_1-k_2)^\mu V_\mu(k_2) $ \\\hline
[F21] & $\phi(k_1+k_2)\leftarrow\ii\cdot
g (-k_1-k_2)^\mu V_\mu(k_2) \chi(k_1) $ \\\hline
[F13] & $V_\mu(k_1+k_3)\leftarrow\ii\cdot
g (-k_1-k_3)_\mu \chi(k_1) \phi(k_3) $ \\\hline
[F31] & $V_\mu(k_1+k_3)\leftarrow\ii\cdot
g (-k_1-k_3)_\mu \phi(k_3) \chi(k_1) $ \\\hline
\end{tabular}
\end{center}
\caption{\label{tab:vector-dscalar}
\ldots}
\end{table}
*)
(* Signify which two of three fields are fused: *)
type fuse2 = F23 | F32 | F31 | F13 | F12 | F21
(* Signify which three of four fields are fused: *)
type fuse3 =
| F123 | F231 | F312 | F132 | F321 | F213
| F124 | F241 | F412 | F142 | F421 | F214
| F134 | F341 | F413 | F143 | F431 | F314
| F234 | F342 | F423 | F243 | F432 | F324
(* Explicit enumeration types make no sense for higher degrees. *)
type fusen = int list
(* The third member of the triplet will contain the coupling constant: *)
type 'a t =
| V3 of 'a vertex3 * fuse2 * 'a
| V4 of 'a vertex4 * fuse3 * 'a
| Vn of 'a vertexn * fusen * 'a
(* \thocwmodulesection{Gauge Couplings}
Dimension-4 trilinear vector boson couplings
\begin{subequations}
\begin{multline}
f_{abc}\partial^{\mu}A^{a,\nu}A^b_{\mu}A^c_{\nu} \rightarrow
\ii f_{abc}k_1^\mu A^{a,\nu}(k_1)A^b_{\mu}(k_2)A^c_{\nu}(k_3) \\
= -\frac{\ii}{3!} f_{a_1a_2a_3} C^{\mu_1\mu_2\mu_3}(k_1,k_2,k_3)
A^{a_1}_{\mu_1}(k_1)A^{a_2}_{\mu_2}(k_2)A^{a_3}_{\mu_3}(k_3)
\end{multline}
with the totally antisymmetric tensor (under simultaneous permutations
of all quantum numbers $\mu_i$ and $k_i$) and all momenta \emph{outgoing}
\begin{equation}
\label{eq:C123}
C^{\mu_1\mu_2\mu_3}(k_1,k_2,k_3) =
( g^{\mu_1\mu_2} (k_1^{\mu_3}-k_2^{\mu_3})
+ g^{\mu_2\mu_3} (k_2^{\mu_1}-k_3^{\mu_1})
+ g^{\mu_3\mu_1} (k_3^{\mu_2}-k_1^{\mu_2}) )
\end{equation}
\end{subequations}
Since~$f_{a_1a_2a_3}C^{\mu_1\mu_2\mu_3}(k_1,k_2,k_3)$ is totally symmetric
(under simultaneous permutations of all quantum numbers $a_i$, $\mu_i$ and $k_i$),
it is easy to take the partial derivative
\begin{subequations}
\label{eq:AofAA}
\begin{equation}
A^{a,\mu}(k_2+k_3) =
- \frac{\ii}{2!} f_{abc}C^{\mu\rho\sigma}(-k_2-k_3,k_2,k_3) A^b_\rho(k_2)A^c_\sigma(k_3)
\end{equation}
with
\begin{equation}
\label{eq:C123'}
C^{\mu\rho\sigma}(-k_2-k_3,k_2,k_3) =
( g^{\rho\sigma} ( k_2^{\mu} -k_3^{\mu} )
+ g^{\mu\sigma} (2k_3^{\rho} +k_2^{\rho} )
- g^{\mu\rho} (2k_2^{\sigma}+k_3^{\sigma}) )
\end{equation}
i.\,e.
\begin{multline}
\label{eq:fuse-gauge}
A^{a,\mu}(k_2+k_3) = - \frac{\ii}{2!} f_{abc}
\bigl( (k_2^{\mu}-k_3^{\mu})A^b(k_2) \cdot A^c(k_3) \\
+ (2k_3+k_2)\cdot A^b(k_2)A^{c,\mu}(k_3)
- A^{b,\mu}(k_2)A^c(k_3)\cdot(2k_2+k_3) \bigr)
\end{multline}
\end{subequations}
\begin{dubious}
Investigate the rearrangements proposed in~\cite{HELAS} for improved
numerical stability.
\end{dubious} *)
(* \thocwmodulesubsection{Non-Gauge Vector Couplings}
As a basis for the dimension-4 couplings of three vector bosons, we
choose ``transversal'' and ``longitudinal'' (with respect to the first
vector field) tensors that are odd and even under permutation of the
second and third argument
\begin{subequations}
\begin{align}
\mathcal{L}_T(V_1,V_2,V_3)
&= V_1^\mu (V_{2,\nu}\ii\overleftrightarrow{\partial_\mu}V_3^\nu)
= - \mathcal{L}_T(V_1,V_3,V_2) \\
\mathcal{L}_L(V_1,V_2,V_3)
&= (\ii\partial_\mu V_1^\mu) V_{2,\nu}V_3^\nu
= \mathcal{L}_L(V_1,V_3,V_2)
\end{align}
\end{subequations}
Using partial integration in~$\mathcal{L}_L$, we find the
convenient combinations
\begin{subequations}
\begin{align}
\mathcal{L}_T(V_1,V_2,V_3) + \mathcal{L}_L(V_1,V_2,V_3)
&= - 2 V_1^\mu \ii\partial_\mu V_{2,\nu} V_3^\nu \\
\mathcal{L}_T(V_1,V_2,V_3) - \mathcal{L}_L(V_1,V_2,V_3)
&= 2 V_1^\mu V_{2,\nu} \ii\partial_\mu V_3^\nu
\end{align}
\end{subequations}
As an important example, we can rewrite the dimension-4 ``anomalous'' triple
gauge couplings
\begin{multline}
\ii\mathcal{L}_{\textrm{TGC}}(g_1,\kappa,g_4)/g_{VWW}
= g_1 V^\mu (W^-_{\mu\nu} W^{+,\nu} - W^+_{\mu\nu} W^{-,\nu}) \\
+ \kappa W^+_\mu W^-_\nu V^{\mu\nu}
+ g_4 W^+_\mu W^-_\nu (\partial^\mu V^\nu + \partial^\nu V^\mu)
\end{multline}
as
\begin{multline}
\mathcal{L}_{\textrm{TGC}}(g_1,\kappa,g_4)
= g_1 \mathcal{L}_T(V,W^-,W^+) \\
- \frac{\kappa+g_1-g_4}{2} \mathcal{L}_T(W^-,V,W^+)
+ \frac{\kappa+g_1+g_4}{2} \mathcal{L}_T(W^+,V,W^-) \\
- \frac{\kappa-g_1-g_4}{2} \mathcal{L}_L(W^-,V,W^+)
+ \frac{\kappa-g_1+g_4}{2} \mathcal{L}_L(W^+,V,W^-)
\end{multline}
\thocwmodulesubsection{$CP$ Violation}
\begin{subequations}
\begin{align}
\mathcal{L}_{\tilde T}(V_1,V_2,V_3)
&= V_{1,\mu}(V_{2,\rho}\ii\overleftrightarrow{\partial_\nu}
V_{3,\sigma})\epsilon^{\mu\nu\rho\sigma}
= + \mathcal{L}_T(V_1,V_3,V_2) \\
\mathcal{L}_{\tilde L}(V_1,V_2,V_3)
&= (\ii\partial_\mu V_{1,\nu})
V_{2,\rho}V_{3,\sigma}\epsilon^{\mu\nu\rho\sigma}
= - \mathcal{L}_L(V_1,V_3,V_2)
\end{align}
\end{subequations}
Here the notations~$\tilde T$ and~$\tilde L$ are clearly
\textit{abuse de langage}, because
$\mathcal{L}_{\tilde L}(V_1,V_2,V_3)$ is actually the
transversal combination, due to the antisymmetry of~$\epsilon$.
Using partial integration in~$\mathcal{L}_{\tilde L}$, we could again find
combinations
\begin{subequations}
\begin{align}
\mathcal{L}_{\tilde T}(V_1,V_2,V_3) + \mathcal{L}_{\tilde L}(V_1,V_2,V_3)
&= - 2 V_{1,\mu} V_{2,\nu} \ii\partial_\rho V_{3,\sigma}
\epsilon^{\mu\nu\rho\sigma} \\
\mathcal{L}_{\tilde T}(V_1,V_2,V_3) - \mathcal{L}_{\tilde L}(V_1,V_2,V_3)
&= - 2 V_{1,\mu} \ii\partial_\nu V_{2,\rho} V_{3,\sigma}
\epsilon^{\mu\nu\rho\sigma}
\end{align}
\end{subequations}
but we don't need them, since
\begin{multline}
\ii\mathcal{L}_{\textrm{TGC}}(g_5,\tilde\kappa)/g_{VWW}
= g_5 \epsilon_{\mu\nu\rho\sigma}
(W^{+,\mu} \ii\overleftrightarrow{\partial^\rho} W^{-,\nu}) V^\sigma \\
- \frac{\tilde\kappa_V}{2} W^-_\mu W^+_\nu \epsilon^{\mu\nu\rho\sigma}
V_{\rho\sigma}
\end{multline}
is immediately recognizable as
\begin{equation}
\mathcal{L}_{\textrm{TGC}}(g_5,\tilde\kappa) / g_{VWW}
= - \ii g_5 \mathcal{L}_{\tilde L}(V,W^-,W^+)
+ \tilde\kappa \mathcal{L}_{\tilde T}(V,W^-,W^+)
\end{equation}
%%% #procedure decl
%%% symbol g1, kappa;
%%% vector V, Wp, Wm, k0, kp, km;
%%% vector v, V1, V2, V3, k1, k2, k3;
%%% index mu, nu;
%%% #endprocedure
%%%
%%% #call decl
%%%
%%% global L_T(k1,V1,k2,V2,k3,V3)
%%% = (V1.k2 - V1.k3) * V2.V3;
%%%
%%% global L_L(k1,V1,k2,V2,k3,V3)
%%% = - V1.k1 * V2.V3;
%%%
%%% global L_g1(k1,V1,k2,V2,k3,V3)
%%% = - V1(mu) * ( (k2(mu)*V2(nu) - k2(nu)*V2(mu)) * V3(nu)
%%% - (k3(mu)*V3(nu) - k3(nu)*V3(mu)) * V2(nu) );
%%%
%%% global L_kappa(k1,V1,k2,V2,k3,V3)
%%% = (k1(mu)*V1(nu) - k1(nu)*V1(mu)) * V2(mu) * V3(nu);
%%%
%%% print;
%%% .sort
%%% .store
%%%
%%% #call decl
%%%
%%% local lp = L_T(k1,V1,k2,V2,k3,V3) + L_L(k1,V1,k2,V2,k3,V3);
%%% local lm = L_T(k1,V1,k2,V2,k3,V3) - L_L(k1,V1,k2,V2,k3,V3);
%%% print;
%%% .sort
%%% id k1.v? = - k2.v - k3.v;
%%% print;
%%% .sort
%%% .store
%%%
%%% #call decl
%%%
%%% local [sum(TL)-g1] = - L_g1(k0,V,km,Wm,kp,Wp)
%%% + L_T(k0,V,kp,Wp,km,Wm)
%%% + (L_T(km,Wm,k0,V,kp,Wp) - L_T(kp,Wp,k0,V,km,Wm)) / 2
%%% - (L_L(km,Wm,k0,V,kp,Wp) - L_L(kp,Wp,k0,V,km,Wm)) / 2;
%%%
%%% local [sum(TL)-kappa] = - L_kappa(k0,V,km,Wm,kp,Wp)
%%% + (L_T(km,Wm,k0,V,kp,Wp) - L_T(kp,Wp,k0,V,km,Wm)) / 2
%%% + (L_L(km,Wm,k0,V,kp,Wp) - L_L(kp,Wp,k0,V,km,Wm)) / 2;
%%%
%%% local delta =
%%% - (g1 * L_g1(k0,V,km,Wm,kp,Wp) + kappa * L_kappa(k0,V,km,Wm,kp,Wp))
%%% + g1 * L_T(k0,V,kp,Wp,km,Wm)
%%% + ( g1 + kappa) / 2 * (L_T(km,Wm,k0,V,kp,Wp) - L_T(kp,Wp,k0,V,km,Wm))
%%% + (- g1 + kappa) / 2 * (L_L(km,Wm,k0,V,kp,Wp) - L_L(kp,Wp,k0,V,km,Wm));
%%%
%%% print;
%%% .sort
%%%
%%% id k0.v? = - kp.v - km.v;
%%% print;
%%% .sort
%%% .store
%%%
%%% .end *)
(* \begin{table}
\begin{center}
\renewcommand{\arraystretch}{1.3}
\begin{tabular}{|>{\qquad}r<{:}l|}\hline
\multicolumn{2}{|l|}{[Dim4_Vector_Vector_Vector_T]:
$\mathcal{L}_I=gV_1^\mu
V_{2,\nu}\ii\overleftrightarrow{\partial_\mu}V_3^\nu$}\\\hline
[F23] & $V_1^\mu(k_2+k_3)\leftarrow\ii\cdot
g(k_2^\mu-k_3^\mu)V_{2,\nu}(k_2)V_3^\nu(k_3)$ \\\hline
[F32] & $V_1^\mu(k_2+k_3)\leftarrow\ii\cdot
g(k_2^\mu-k_3^\mu)V_3^\nu(k_3)V_{2,\nu}(k_2)$ \\\hline
[F12] & $V_3^\mu(k_1+k_2)\leftarrow\ii\cdot
g(2k_2^\nu+k_1^\nu)V_{1,\nu}(k_1)V_2^\mu(k_2)$ \\\hline
[F21] & $V_3^\mu(k_1+k_2)\leftarrow\ii\cdot
g(2k_2^\nu+k_1^\nu)V_2^\mu(k_2)V_{1,\nu}(k_1)$ \\\hline
[F13] & $V_2^\mu(k_1+k_3)\leftarrow\ii\cdot
g(-k_1^\nu-2k_3^\nu)V_1^\nu(k_1)V_3^\mu(k_3)$ \\\hline
[F31] & $V_2^\mu(k_1+k_3)\leftarrow\ii\cdot
g(-k_1^\nu-2k_3^\nu)V_3^\mu(k_3)V_1^\nu(k_1)$ \\\hline
\multicolumn{2}{|l|}{[Dim4_Vector_Vector_Vector_L]:
$\mathcal{L}_I=g\ii\partial_\mu V_1^\mu
V_{2,\nu}V_3^\nu$}\\\hline
[F23] & $V_1^\mu(k_2+k_3)\leftarrow\ii\cdot
g(k_2^\mu+k_3^\mu)V_{2,\nu}(k_2)V_3^\nu(k_3)$ \\\hline
[F32] & $V_1^\mu(k_2+k_3)\leftarrow\ii\cdot
g(k_2^\mu+k_3^\mu)V_3^\nu(k_3)V_{2,\nu}(k_2)$ \\\hline
[F12] & $V_3^\mu(k_1+k_2)\leftarrow\ii\cdot
g(-k_1^\nu)V_{1,\nu}(k_1)V_2^\mu(k_2)$ \\\hline
[F21] & $V_3^\mu(k_1+k_2)\leftarrow\ii\cdot
g(-k_1^\nu)V_2^\mu(k_2)V_{1,\nu}(k_1)$ \\\hline
[F13] & $V_2^\mu(k_1+k_3)\leftarrow\ii\cdot
g(-k_1^\nu)V_1^\nu(k_1)V_3^\mu(k_3)$ \\\hline
[F31] & $V_2^\mu(k_1+k_3)\leftarrow\ii\cdot
g(-k_1^\nu)V_3^\mu(k_3)V_1^\nu(k_1)$ \\\hline
\end{tabular}
\end{center}
\caption{\label{tab:dim4-TGC}
\ldots}
\end{table}
\begin{table}
\begin{center}
\renewcommand{\arraystretch}{1.3}
\begin{tabular}{|>{\qquad}r<{:}l|}\hline
\multicolumn{2}{|l|}{[Dim4_Vector_Vector_Vector_T5]:
$\mathcal{L}_I=gV_{1,\mu}
V_{2,\rho}\ii\overleftrightarrow{\partial_\nu}
V_{3,\sigma}\epsilon^{\mu\nu\rho\sigma}$}\\\hline
[F23] & $V_1^\mu(k_2+k_3)\leftarrow\ii\cdot
g\epsilon^{\mu\nu\rho\sigma}(k_{2,\nu}-k_{3,\nu})
V_{2,\rho}(k_2)V_{3,\sigma}(k_3)$ \\\hline
[F32] & $V_1^\mu(k_2+k_3)\leftarrow\ii\cdot
g\epsilon^{\mu\nu\rho\sigma}(k_{2,\nu}-k_{3,\nu})
V_{3,\sigma}(k_3)V_{2,\rho}(k_2)$ \\\hline
[F12] & $V_3^\mu(k_1+k_2)\leftarrow\ii\cdot
g\epsilon^{\mu\nu\rho\sigma}(2k_{2,\nu}+k_{1,\nu})
V_{1,\rho}(k_1)V_{2,\sigma}(k_2)$ \\\hline
[F21] & $V_3^\mu(k_1+k_2)\leftarrow\ii\cdot
g\epsilon^{\mu\nu\rho\sigma}(2k_{2,\nu}+k_{1,\nu})
V_{2,\sigma}(k_2)V_{1,\rho}(k_1)$ \\\hline
[F13] & $V_2^\mu(k_1+k_3)\leftarrow\ii\cdot
g\epsilon^{\mu\nu\rho\sigma}(-k_{1,\nu}-2k_{3,\nu})
V_{1,\rho}(k_1)V_{3,\sigma}(k_3)$ \\\hline
[F31] & $V_2^\mu(k_1+k_3)\leftarrow\ii\cdot
g\epsilon^{\mu\nu\rho\sigma}(-k_{1,\nu}-2k_{3,\nu})
V_{3,\sigma}(k_3)V_{1,\rho}(k_1)$ \\\hline
\multicolumn{2}{|l|}{[Dim4_Vector_Vector_Vector_L5]:
$\mathcal{L}_I=g\ii\partial_\mu V_{1,\nu}
V_{2,\nu}V_{3,\sigma}\epsilon^{\mu\nu\rho\sigma}$}\\\hline
[F23] & $V_1^\mu(k_2+k_3)\leftarrow\ii\cdot
g\epsilon^{\mu\nu\rho\sigma}(k_{2,\nu}+k_{3,\nu})
V_{2,\rho}(k_2)V_{3,\sigma}(k_3)$ \\\hline
[F32] & $V_1^\mu(k_2+k_3)\leftarrow\ii\cdot
g\epsilon^{\mu\nu\rho\sigma}(k_{2,\nu}+k_{3,\nu})
V_{2,\rho}(k_2)V_{3,\sigma}(k_3)$ \\\hline
[F12] & $V_3^\mu(k_1+k_2)\leftarrow\ii\cdot
g\epsilon^{\mu\nu\rho\sigma}(-k_{1,\nu})
V_{1,\rho}(k_1)V_{2,\sigma}(k_2)$ \\\hline
[F21] & $V_3^\mu(k_1+k_2)\leftarrow\ii\cdot
g\epsilon^{\mu\nu\rho\sigma}(-k_{1,\nu})
V_{2,\sigma}(k_2)V_{1,\rho}(k_1)$ \\\hline
[F13] & $V_2^\mu(k_1+k_3)\leftarrow\ii\cdot
g\epsilon^{\mu\nu\rho\sigma}(-k_{1,\nu})
V_{1,\rho}(k_1)V_{3,\sigma}(k_3)$ \\\hline
[F31] & $V_2^\mu(k_1+k_3)\leftarrow\ii\cdot
g\epsilon^{\mu\nu\rho\sigma}(-k_{1,\nu})
V_{3,\sigma}(k_3)V_{1,\rho}(k_1)$ \\\hline
\end{tabular}
\end{center}
\caption{\label{tab:dim4-TGC5}
\ldots}
\end{table}
\begin{table}
\begin{center}
\renewcommand{\arraystretch}{1.3}
\begin{tabular}{|>{\qquad}r<{:}l|}\hline
\multicolumn{2}{|l|}{[Dim6_Gauge_Gauge_Gauge]:
$\mathcal{L}_I=gF_1^{\mu\nu}F_{2,\nu\rho}
F_{3,\hphantom{\rho}\mu}^{\hphantom{3,}\rho}$}\\\hline
[_] & $A_1^\mu(k_2+k_3)\leftarrow-\ii\cdot
\Lambda^{\mu\rho\sigma}(-k_2-k_3,k_2,k_3)
A_{2,\rho} A_{c,\sigma}$\\\hline
\end{tabular}
\end{center}
\caption{\label{tab:dim6-TGC}
\ldots}
\end{table}
\begin{table}
\begin{center}
\renewcommand{\arraystretch}{1.3}
\begin{tabular}{|>{\qquad}r<{:}l|}\hline
\multicolumn{2}{|l|}{[Dim6_Gauge_Gauge_Gauge_5]:
$\mathcal{L}_I=g/2\cdot\epsilon^{\mu\nu\lambda\tau}
F_{1,\mu\nu}F_{2,\tau\rho}
F_{3,\hphantom{\rho}\lambda}^{\hphantom{3,}\rho}$}\\\hline
[F23] & $A_1^\mu(k_2+k_3)\leftarrow-\ii\cdot
\Lambda_5^{\mu\rho\sigma}(-k_2-k_3,k_2,k_3)
A_{2,\rho} A_{3,\sigma}$\\\hline
[F32] & $A_1^\mu(k_2+k_3)\leftarrow-\ii\cdot
\Lambda_5^{\mu\rho\sigma}(-k_2-k_3,k_2,k_3)
A_{3,\sigma} A_{2,\rho}$\\\hline
[F12] & $A_3^\mu(k_1+k_2)\leftarrow-\ii\cdot$\\\hline
[F21] & $A_3^\mu(k_1+k_2)\leftarrow-\ii\cdot$\\\hline
[F13] & $A_2^\mu(k_1+k_3)\leftarrow-\ii\cdot$\\\hline
[F31] & $A_2^\mu(k_1+k_3)\leftarrow-\ii\cdot$\\\hline
\end{tabular}
\end{center}
\caption{\label{tab:dim6-TGC5}
\ldots}
\end{table} *)
(* \thocwmodulesection{$\textrm{SU}(2)$ Gauge Bosons}
An important special case for table~\ref{tab:dim4-bosons} are the two
usual coordinates of~$\textrm{SU}(2)$
\begin{equation}
W_\pm = \frac{1}{\sqrt2} \left(W_1 \mp \ii W_2\right)
\end{equation}
i.\,e.
\begin{subequations}
\begin{align}
W_1 &= \frac{1}{\sqrt2} \left(W_+ + W_-\right) \\
W_2 &= \frac{\ii}{\sqrt2} \left(W_+ - W_-\right)
\end{align}
\end{subequations}
and
\begin{equation}
W_1^\mu W_2^\nu - W_2^\mu W_1^\nu
= \ii\left(W_-^\mu W_+^\nu - W_+^\mu W_-^\nu\right)
\end{equation}
Thus the symmtry remains after the change of basis:
\begin{multline}
\epsilon^{abc} W_a^{\mu_1}W_b^{\mu_2}W_c^{\mu_3}
= \ii W_-^{\mu_1} (W_+^{\mu_2}W_3^{\mu_3} - W_3^{\mu_2}W_+^{\mu_3}) \\
+ \ii W_+^{\mu_1} (W_3^{\mu_2}W_-^{\mu_3} - W_-^{\mu_2}W_3^{\mu_3})
+ \ii W_3^{\mu_1} (W_-^{\mu_2}W_+^{\mu_3} - W_+^{\mu_2}W_-^{\mu_3})
\end{multline} *)
(* \thocwmodulesection{Quartic Couplings and Auxiliary Fields}
Quartic couplings can be replaced by cubic couplings to a non-propagating
auxiliary field. The quartic term should get a negative sign so that it the
energy is bounded from below for identical fields. In the language of
functional integrals
\begin{subequations}
\label{eq:quartic-aux}
\begin{multline}
\mathcal{L}_{\phi^4} = - g^2\phi_1\phi_2\phi_3\phi_4
\Longrightarrow \\
\mathcal{L}_{X\phi^2}
= X^*X \pm gX\phi_1\phi_2 \pm gX^*\phi_3\phi_4
= (X^* \pm g\phi_1\phi_2)(X \pm g\phi_3\phi_4)
- g^2\phi_1\phi_2\phi_3\phi_4
\end{multline}
and in the language of Feynman diagrams
\begin{equation}
\parbox{21mm}{\begin{fmfgraph*}(20,20)
\fmfleft{e1,e2}
\fmfright{e3,e4}
\fmf{plain}{v,e1}
\fmf{plain}{v,e2}
\fmf{plain}{v,e3}
\fmf{plain}{v,e4}
\fmfv{d.sh=circle,d.si=dot_size,label=$-\ii g^2$}{v}
\end{fmfgraph*}}
\qquad\Longrightarrow\qquad
\parbox{21mm}{\begin{fmfgraph*}(20,20)
\fmfleft{e1,e2}
\fmfright{e3,e4}
\fmf{plain}{v12,e1}
\fmf{plain}{v12,e2}
\fmf{plain}{v34,e3}
\fmf{plain}{v34,e4}
\fmf{dashes,label=$+\ii$}{v12,v34}
\fmfv{d.sh=circle,d.si=dot_size,label=$\pm\ii g$}{v12}
\fmfv{d.sh=circle,d.si=dot_size,label=$\pm\ii g$}{v34}
\end{fmfgraph*}}
\end{equation}
\end{subequations}
The other choice of signs
\begin{equation}
\mathcal{L}_{X\phi^2}'
= - X^*X \pm gX\phi_1\phi_2 \mp gX^*\phi_3\phi_4
= - (X^* \pm g\phi_1\phi_2)(X \mp g\phi_3\phi_4)
- g^2\phi_1\phi_2\phi_3\phi_4
\end{equation}
can not be extended easily to identical particles and is therefore
not used. For identical particles we have
\begin{multline}
\mathcal{L}_{\phi^4} = - \frac{g^2}{4!}\phi^4
\Longrightarrow \\
\mathcal{L}_{X\phi^2}
= \frac{1}{2}X^2 \pm \frac{g}{2}X\phi^2 \pm \frac{g}{2}X\phi^2
= \frac{1}{2}\left(X \pm \frac{g}{2}\phi^2\right)
\left(X \pm \frac{g}{2}\phi^2\right)
- \frac{g^2}{4!}\phi^4
\end{multline}
\begin{dubious}
Explain the factor~$1/3$ in the functional setting and its
relation to the three diagrams in the graphical setting?
\end{dubious}
\thocwmodulesubsection{Quartic Gauge Couplings}
\begin{figure}
\begin{subequations}
\label{eq:Feynman-QCD}
\begin{align}
\parbox{28mm}{\fmfframe(2,2)(2,1){\begin{fmfgraph*}(24,24)
\threeexternal{k,,\mu,,a}{p}{p'}
\fmf{gluon}{v,e1}
\fmf{fermion}{e2,v,e3}
\fmfdot{v} \end{fmfgraph*}}} \,&=
\begin{split}
\mbox{} + & \ii g\gamma_\mu T_a
\end{split} \\
\label{eq:TGV}
\parbox{28mm}{\fmfframe(2,2)(2,1){\begin{fmfgraph*}(24,24)
\threeexternal{1}{2}{3}
\fmf{gluon}{v,e1}
\fmf{gluon}{v,e2}
\fmf{gluon}{v,e3}
\threeoutgoing
\end{fmfgraph*}}} \,&=
\begin{split}
& g f_{a_1a_2a_3} C^{\mu_1\mu_2\mu_3} (k_1,k_2,k_3)
\end{split} \\
\parbox{28mm}{\fmfframe(2,2)(2,1){\begin{fmfgraph*}(24,24)
\fmfsurround{d1,e1,d2,e2,d3,e3,d4,e4}
\fmf{gluon}{v,e1}
\fmf{gluon}{v,e2}
\fmf{gluon}{v,e3}
\fmf{gluon}{v,e4}
\fmflabel{1}{e1}
\fmflabel{2}{e2}
\fmflabel{3}{e3}
\fmflabel{4}{e4}
\fmfdot{v}
\fmffreeze
\fmf{warrow_right}{v,e1}
\fmf{warrow_right}{v,e2}
\fmf{warrow_right}{v,e3}
\fmf{warrow_right}{v,e4}
\end{fmfgraph*}}} \,&=
\begin{split}
\mbox{} - & \ii g^2 f_{a_1a_2b}f_{a_3a_4b}
(g_{\mu_1\mu_3} g_{\mu_4\mu_2} - g_{\mu_1\mu_4} g_{\mu_2\mu_3}) \\
\mbox{} - & \ii g^2 f_{a_1a_3b}f_{a_4a_2b}
(g_{\mu_1\mu_4} g_{\mu_2\mu_3} - g_{\mu_1\mu_2} g_{\mu_3\mu_4}) \\
\mbox{} - & \ii g^2 f_{a_1a_4b}f_{a_2a_3b}
(g_{\mu_1\mu_2} g_{\mu_3\mu_4} - g_{\mu_1\mu_3} g_{\mu_4\mu_2})
\end{split}
\end{align}
\end{subequations}
\caption{\label{fig:gauge-feynman-rules} Gauge couplings.
See~(\ref{eq:C123}) for the definition of the antisymmetric
tensor $C^{\mu_1\mu_2\mu_3}(k_1,k_2,k_3)$.}
\end{figure}
\begin{figure}
\begin{equation}
\label{eq:Feynman-QCD'}
\parbox{28mm}{\fmfframe(2,2)(2,1){\begin{fmfgraph*}(24,24)
\fmfsurround{d1,e1,d2,e2,d3,e3,d4,e4}
\fmf{gluon}{v12,e1}
\fmf{gluon}{v12,e2}
\fmf{gluon}{v34,e3}
\fmf{gluon}{v34,e4}
\fmf{dashes}{v12,v34}
\fmflabel{1}{e1}
\fmflabel{2}{e2}
\fmflabel{3}{e3}
\fmflabel{4}{e4}
\fmfdot{v12,v34}
\fmffreeze
\fmf{warrow_right}{v12,e1}
\fmf{warrow_right}{v12,e2}
\fmf{warrow_right}{v34,e3}
\fmf{warrow_right}{v34,e4}
\end{fmfgraph*}}} \,=
\mbox{} - \ii g^2 f_{a_1a_2b}f_{a_3a_4b}
(g_{\mu_1\mu_3} g_{\mu_4\mu_2} - g_{\mu_1\mu_4} g_{\mu_2\mu_3})
\end{equation}
\caption{\label{fig:gauge-feynman-rules'} Gauge couplings.}
\end{figure}
The three crossed versions of
figure~\ref{fig:gauge-feynman-rules'} reproduces the quartic coupling in
figure~\ref{fig:gauge-feynman-rules}, because
\begin{multline}
- \ii g^2 f_{a_1a_2b}f_{a_3a_4b}
(g_{\mu_1\mu_3} g_{\mu_4\mu_2} - g_{\mu_1\mu_4} g_{\mu_2\mu_3}) \\
= (\ii g f_{a_1a_2b} T_{\mu_1\mu_2,\nu_1\nu_2})
\left(\frac{\ii g^{\nu_1\nu_3} g^{\nu_2\nu_4}}{2}\right)
(\ii g f_{a_3a_4b} T_{\mu_3\mu_4,\nu_3\nu_4})
\end{multline}
with $T_{\mu_1\mu_2,\mu_3\mu_4} =
g_{\mu_1\mu_3}g_{\mu_4\mu_2}-g_{\mu_1\mu_4}g_{\mu_2\mu_3}$. *)
(* \thocwmodulesection{Gravitinos and supersymmetric currents}
In supergravity theories there is a fermionic partner of the graviton, the
gravitino. Therefore we have introduced the Lorentz type [Vectorspinor].
*)
(* \begin{table}
\begin{center}
\renewcommand{\arraystretch}{1.4}
\begin{tabular}{|>{\qquad}r<{:}l|r<{:}l|}\hline
\multicolumn{4}{|l|}{[GBG (Fermbar, MOM, Ferm)]:
$\bar\psi_1(\ii\fmslash{\partial}\pm m)\phi\psi_2$}\\\hline
[F12] & $\psi_2\leftarrow-(\fmslash{k}\mp m)\psi_1S$
& [F21] & $\psi_2\leftarrow-S(\fmslash{k}\mp m)\psi_1$ \\\hline
[F13] & $S\leftarrow \psi^T_1 {\rm C}(\fmslash{k}\pm m)\psi_2$
& [F31] & $S\leftarrow \psi^T_2 {\rm C}(-(\fmslash{k}\mp m)\psi_1)$ \\\hline
[F23] & $\psi_1\leftarrow S(\fmslash{k}\pm m)\psi_2$
& [F32] & $\psi_1\leftarrow(\fmslash{k}\pm m)\psi_2 S$ \\\hline
\multicolumn{4}{|l|}{[GBG (Fermbar, MOM5, Ferm)]:
$\bar\psi_1(\ii\fmslash{\partial}\pm m)\phi\gamma^5\psi_2$}\\\hline
[F12] & $\psi_2\leftarrow(\fmslash{k}\pm m)\gamma^5\psi_1P$
& [F21] & $\psi_2\leftarrow P(\fmslash{k}\pm m)\gamma^5\psi_1$ \\\hline
[F13] & $P\leftarrow \psi^T_1 {\rm C}(\fmslash{k}\pm m)\gamma^5\psi_2$
& [F31] & $P\leftarrow \psi^T_2 {\rm C}(\fmslash{k}\pm m)\gamma^5\psi_1$ \\\hline
[F23] & $\psi_1\leftarrow P(\fmslash{k}\pm m)\gamma^5\psi_2$
& [F32] & $\psi_1\leftarrow(\fmslash{k}\pm m)\gamma^5\psi_2 P$ \\\hline
\multicolumn{4}{|l|}{[GBG (Fermbar, MOML, Ferm)]:
$\bar\psi_1 (\ii\fmslash{\partial}\pm m)\phi(1-\gamma^5)\psi_2$}\\\hline
[F12] & $\psi_2\leftarrow-(1-\gamma^5)(\fmslash{k}\mp m)\psi_1\phi$
& [F21] & $\psi_2\leftarrow-\phi(1-\gamma^5)(\fmslash{k}\mp m)\psi_1$ \\\hline
[F13] & $\phi\leftarrow \psi^T_1 {\rm C}(\fmslash{k}\pm m)(1-\gamma^5)\psi_2$
& [F31] & $\phi\leftarrow \psi^T_2 {\rm C}(1-\gamma^5)(-(\fmslash{k}\mp m)\psi_1)$ \\\hline
[F23] & $\psi_1\leftarrow\phi(\fmslash{k}\pm m)(1-\gamma^5)\psi_2$
& [F32] & $\psi_1\leftarrow(\fmslash{k}\pm m)(1-\gamma^5)\psi_2 \phi$ \\\hline
\multicolumn{4}{|l|}{[GBG (Fermbar, LMOM, Ferm)]:
$\bar\psi_1 \phi(1-\gamma^5)(\ii\fmslash{\partial}\pm m)\psi_2$}\\\hline
[F12] & $\psi_2\leftarrow-(\fmslash{k}\mp m)\psi_1(1-\gamma^5)\phi$
& [F21] & $\psi_2\leftarrow-\phi(\fmslash{k}\mp m)(1-\gamma^5)\psi_1$ \\\hline
[F13] & $\phi\leftarrow \psi^T_1 {\rm C}(1-\gamma^5)(\fmslash{k}\pm m)\psi_2$
& [F31] & $\phi\leftarrow \psi^T_2 {\rm C}(-(\fmslash{k}\mp m)(1-\gamma^5)\psi_1)$ \\\hline
[F23] & $\psi_1\leftarrow\phi(1-\gamma^5)(\fmslash{k}\pm m)\psi_2$
& [F32] & $\psi_1\leftarrow(1-\gamma^5)(\fmslash{k}\pm m)\psi_2 \phi$ \\\hline
\multicolumn{4}{|l|}{[GBG (Fermbar, VMOM, Ferm)]:
$\bar\psi_1 \ii\fmslash{\partial}_\alpha V_\beta \lbrack \gamma^\alpha, \gamma^\beta \rbrack \psi_2$}\\\hline
[F12] & $\psi_2\leftarrow-\lbrack\fmslash{k},\gamma^\alpha\rbrack\psi_1 V_\alpha$
& [F21] & $\psi_2\leftarrow-\lbrack\fmslash{k},\fmslash{V}\rbrack\psi_1$ \\\hline
[F13] & $V_\alpha\leftarrow \psi^T_1 {\rm C}\lbrack\fmslash{k},\gamma_\alpha\rbrack\psi_2$
& [F31] & $V_\alpha\leftarrow \psi^T_2 {\rm C}(-\lbrack\fmslash{k}, \gamma_\alpha\rbrack\psi_1)$ \\\hline
[F23] & $\psi_1\leftarrow\rbrack\fmslash{k},\fmslash{V}\rbrack\psi_2$
& [F32] & $\psi_1\leftarrow\lbrack\fmslash{k},\gamma^\alpha\rbrack\psi_2 V_\alpha$ \\\hline
\end{tabular}
\end{center}
\caption{\label{tab:dim4-fermions-MOM} Combined dimension-4 trilinear
fermionic couplings including a momentum. $Ferm$ stands for $Psi$ and
$Chi$. The case of $MOMR$ is identical to $MOML$ if one substitutes
$1+\gamma^5$ for $1-\gamma^5$, as well as for $LMOM$ and $RMOM$. The
mass term forces us to keep the chiral projector always on the left
after "inverting the line" for $MOML$ while on the right for $LMOM$.}
\end{table}
\begin{table}
\begin{center}
\renewcommand{\arraystretch}{1.4}
\begin{tabular}{|>{\qquad}r<{:}l|r<{:}l|}\hline
\multicolumn{2}{|l|}{[GBBG (Fermbar, S2LR, Ferm)]: $\bar\psi_1 S_1 S_2
(g_L P_L + g_R P_R) \psi_2$}\\\hline
[F123] [F213] [F132] [F231] [F312] [F321] & $\psi_2\leftarrow S_1 S_2 (g_R P_L + g_L P_R) \psi_1$ \\ \hline
[F423] [F243] [F432] [F234] [F342] [F324] & $\psi_1 \leftarrow S_1 S_2 (g_L P_L + g_R P_R) \psi_2$ \\ \hline
[F134] [F143] [F314] & $S_1 \leftarrow \psi^T_1 C S_2 (g_L P_L + g_R P_R) \psi_2$ \\ \hline
[F124] [F142] [F214] & $S_2 \leftarrow \psi^T_1 C S_1 (g_L P_L + g_R P_R) \psi_2$ \\ \hline
[F413] [F431] [F341] & $S_1 \leftarrow \psi^T_2 C S_2 (g_R P_L + g_L P_R) \psi_1$ \\ \hline
[F412] [F421] [F241] & $S_2 \leftarrow \psi^T_2 C S_1 (g_R P_L + g_L P_R) \psi_1$ \\ \hline
\multicolumn{2}{|l|}{[GBBG (Fermbar, S2, Ferm)]: $\bar\psi_1 S_1 S_2
\gamma^5 \psi_2$}\\\hline
[F123] [F213] [F132] [F231] [F312] [F321] & $\psi_2\leftarrow S_1 S_2 \gamma^5 \psi_1$ \\ \hline
[F423] [F243] [F432] [F234] [F342] [F324] & $\psi_1 \leftarrow S_1 S_2 \gamma^5 \psi_2$ \\ \hline
[F134] [F143] [F314] & $S_1 \leftarrow \psi^T_1 C S_2 \gamma^5 \psi_2$ \\ \hline
[F124] [F142] [F214] & $S_2 \leftarrow \psi^T_1 C S_1 \gamma^5 \psi_2$ \\ \hline
[F413] [F431] [F341] & $S_1 \leftarrow \psi^T_2 C S_2 \gamma^5 \psi_1$ \\ \hline
[F412] [F421] [F241] & $S_2 \leftarrow \psi^T_2 C S_1 \gamma^5 \psi_1$ \\ \hline
\multicolumn{2}{|l|}{[GBBG (Fermbar, V2, Ferm)]: $\bar\psi_1 \lbrack \fmslash{V}_1 , \fmslash{V}_2 \rbrack \psi_2$}\\\hline
[F123] [F213] [F132] [F231] [F312] [F321] & $\psi_2\leftarrow - \lbrack \fmslash{V}_1 , \fmslash{V}_2 \rbrack \psi_1$ \\ \hline
[F423] [F243] [F432] [F234] [F342] [F324] & $\psi_1 \leftarrow \lbrack \fmslash{V}_1 , \fmslash{V}_2 \rbrack \psi_2$ \\ \hline
[F134] [F143] [F314] & $V_{1\:\alpha} \leftarrow \psi^T_1 C \lbrack \gamma_\alpha , \fmslash{V}_2 \rbrack \psi_2$ \\ \hline
[F124] [F142] [F214] & $V_{2\:\alpha} \leftarrow \psi^T_1 C (-\lbrack \gamma_\alpha , \fmslash{V}_1 \rbrack) \psi_2$ \\ \hline
[F413] [F431] [F341] & $V_{1\:\alpha} \leftarrow \psi^T_2 C (-\lbrack \gamma_\alpha , \fmslash{V}_2 \rbrack) \psi_1$ \\ \hline
[F412] [F421] [F241] & $V_{2\:\alpha} \leftarrow \psi^T_2 C \lbrack \gamma_\alpha , \fmslash{V}_1 \rbrack \psi_1$ \\ \hline
\end{tabular}
\end{center}
\caption{\label{tab:dim5-mom2} Vertices with two fermions ($Ferm$ stands
for $Psi$ and $Chi$, but not for $Grav$) and two bosons (two scalars,
scalar/vector, two vectors) for the BRST transformations. Part I}
\end{table}
\begin{table}
\begin{center}
\renewcommand{\arraystretch}{1.4}
\begin{tabular}{|>{\qquad}r<{:}l|r<{:}l|}\hline
\multicolumn{2}{|l|}{[GBBG (Fermbar, SV, Ferm)]: $\bar\psi_1 \fmslash{V} S \psi_2$}\\\hline
[F123] [F213] [F132] [F231] [F312] [F321] & $\psi_2\leftarrow - \fmslash{V} S \psi_1$ \\ \hline
[F423] [F243] [F432] [F234] [F342] [F324] & $\psi_1 \leftarrow \fmslash{V} S \psi_2$ \\ \hline
[F134] [F143] [F314] & $V_\alpha \leftarrow \psi^T_1 C \gamma_\alpha S \psi_2$ \\ \hline
[F124] [F142] [F214] & $S \leftarrow \psi^T_1 C \fmslash{V} \psi_2$ \\ \hline
[F413] [F431] [F341] & $V_\alpha \leftarrow \psi^T_2 C (- \gamma_\alpha S \psi_1)$ \\ \hline
[F412] [F421] [F241] & $S \leftarrow \psi^T_2 C (- \fmslash{V} \psi_1)$ \\ \hline
\multicolumn{2}{|l|}{[GBBG (Fermbar, PV, Ferm)]: $\bar\psi_1 \fmslash{V} \gamma^5 P \psi_2$}\\\hline
[F123] [F213] [F132] [F231] [F312] [F321] & $\psi_2\leftarrow \fmslash{V} \gamma^5 P \psi_1$ \\ \hline
[F423] [F243] [F432] [F234] [F342] [F324] & $\psi_1 \leftarrow \fmslash{V} \gamma^5 P \psi_2$ \\ \hline
[F134] [F143] [F314] & $V_\alpha \leftarrow \psi^T_1 C \gamma_\alpha \gamma^5 P \psi_2$ \\ \hline
[F124] [F142] [F214] & $P \leftarrow \psi^T_1 C \fmslash{V} \gamma^5 \psi_2$ \\ \hline
[F413] [F431] [F341] & $V_\alpha \leftarrow \psi^T_2 C \gamma_\alpha \gamma^5 P \psi_1$ \\ \hline
[F412] [F421] [F241] & $P \leftarrow \psi^T_2 C \fmslash{V} \gamma^5 \psi_1$ \\ \hline
\multicolumn{2}{|l|}{[GBBG (Fermbar, S(L/R)V, Ferm)]: $\bar\psi_1 \fmslash{V} (1 \mp\gamma^5) \phi \psi_2$}\\\hline
[F123] [F213] [F132] [F231] [F312] [F321] & $\psi_2\leftarrow - \fmslash{V} (1\pm\gamma^5) \phi \psi_1$ \\ \hline
[F423] [F243] [F432] [F234] [F342] [F324] & $\psi_1 \leftarrow \fmslash{V} (1\mp\gamma^5) \phi \psi_2$ \\ \hline
[F134] [F143] [F314] & $V_\alpha \leftarrow \psi^T_1 C \gamma_\alpha (1\mp\gamma^5) \phi \psi_2$ \\ \hline
[F124] [F142] [F214] & $\phi \leftarrow \psi^T_1 C \fmslash{V} (1\mp\gamma^5) \psi_2$ \\ \hline
[F413] [F431] [F341] & $V_\alpha \leftarrow \psi^T_2 C \gamma_\alpha (-(1\pm\gamma^5) \phi \psi_1)$ \\ \hline
[F412] [F421] [F241] & $\phi \leftarrow \psi^T_2 C \fmslash{V} (-(1\pm\gamma^5) \psi_1)$ \\ \hline
\end{tabular}
\end{center}
\caption{\label{tab:dim5-mom2} Vertices with two fermions ($Ferm$ stands
for $Psi$ and $Chi$, but not for $Grav$) and two bosons (two scalars,
scalar/vector, two vectors) for the BRST transformations. Part II}
\end{table}
\begin{table}
\begin{center}
\renewcommand{\arraystretch}{1.4}
\begin{tabular}{|>{\qquad}r<{:}l|r<{:}l|}\hline
\multicolumn{4}{|l|}{[GBG (Gravbar, POT, Psi)]: $\bar\psi_\mu S \gamma^\mu \psi$}\\\hline
[F12] & $\psi\leftarrow - \gamma^\mu \psi_\mu S$
& [F21] & $\psi\leftarrow - S\gamma^\mu \psi_\mu$ \\\hline
[F13] & $S\leftarrow \psi^T_\mu {\rm C} \gamma^\mu \psi$
& [F31] & $S\leftarrow \psi^T{\rm C} (-\gamma^\mu)\psi_\mu$ \\\hline
[F23] & $\psi_\mu\leftarrow S\gamma_\mu\psi$
& [F32] & $\psi_\mu\leftarrow \gamma_\mu \psi S$ \\\hline
\multicolumn{4}{|l|}{[GBG (Gravbar, S, Psi)]: $\bar\psi_\mu \fmslash{k}_S S \gamma^\mu \psi$}\\\hline
[F12] & $\psi\leftarrow \gamma^\mu \fmslash{k}_S \psi_\mu S$
& [F21] & $\psi\leftarrow S\gamma^\mu \fmslash{k}_S \psi_\mu$ \\\hline
[F13] & $S\leftarrow \psi^T_\mu {\rm C} \fmslash{k}_S \gamma^\mu \psi$
& [F31] & $S\leftarrow \psi^T{\rm C}\gamma^\mu\fmslash{k}_S \psi_\mu$ \\\hline
[F23] & $\psi_\mu\leftarrow S\fmslash{k}_S\gamma_\mu\psi$
& [F32] & $\psi_\mu\leftarrow \fmslash{k}_S \gamma_\mu \psi S$ \\\hline
\multicolumn{4}{|l|}{[GBG (Gravbar, P, Psi)]: $\bar\psi_\mu \fmslash{k}_P P \gamma^\mu \gamma_5 \psi$}\\\hline
[F12] & $\psi\leftarrow \gamma^\mu\fmslash{k}_P\gamma_5\psi_\mu P$
& [F21] & $\psi\leftarrow P\gamma^\mu\fmslash{k}_P\gamma_5\psi_\mu$ \\\hline
[F13] & $P\leftarrow \psi^T_\mu {\rm C}\fmslash{k}_P\gamma^\mu\gamma_5\psi$
& [F31] & $P\leftarrow \psi^T {\rm C}\gamma^\mu\fmslash{k}_P\gamma_5\psi_\mu$ \\\hline
[F23] & $\psi_\mu\leftarrow P\fmslash{k}_P \gamma_\mu \gamma_5 \psi$
& [F32] & $\psi_\mu\leftarrow \fmslash{k}_P \gamma_\mu \gamma_5 \psi P$ \\\hline
\multicolumn{4}{|l|}{[GBG (Gravbar, V, Psi)]: $\bar\psi_\mu\lbrack\fmslash{k}_V,\fmslash{V}\rbrack\gamma^\mu\gamma^5\psi$}\\\hline
[F12] & $\psi\leftarrow \gamma^5\gamma^\mu \lbrack \fmslash{k}_V , \gamma^\alpha \rbrack \psi_\mu V_\alpha$
& [F21] & $\psi\leftarrow \gamma^5\gamma^\mu \lbrack \fmslash{k}_V , \fmslash{V} \rbrack\psi_\mu$ \\\hline
[F13] & $V_{\mu}\leftarrow \psi^T_\rho {\rm C} \lbrack \fmslash{k}_V , \gamma_\mu \rbrack \gamma^\rho \gamma^5 \psi$
& [F31] & $V_{\mu}\leftarrow \psi^T {\rm C} \gamma^5 \gamma^{\rho} \lbrack \fmslash{k}_V , \gamma_\mu \rbrack \psi_\rho$ \\\hline
[F23] & $\psi_\mu\leftarrow\lbrack \fmslash{k}_V , \fmslash{V} \rbrack \gamma_\mu \gamma^5 \psi $
& [F32] & $\psi_\mu\leftarrow\lbrack \fmslash{k}_V , \gamma^\alpha \rbrack \gamma_\mu \gamma^5 \psi V_\alpha$ \\\hline
\end{tabular}
\end{center}
\caption{\label{tab:dim5-fermions-gravdirac} Dimension-5 trilinear
couplings including one Dirac, one Gravitino fermion and one additional particle.The option [POT] is for the coupling of the supersymmetric current to the derivative of the quadratic terms in the superpotential.}
\end{table}
\begin{table}
\begin{center}
\renewcommand{\arraystretch}{1.4}
\begin{tabular}{|>{\qquad}r<{:}l|r<{:}l|}\hline
\multicolumn{4}{|l|}{[GBG (Psibar, POT, Grav)]: $\bar\psi \gamma^\mu S \psi_\mu$}\\\hline
[F12] & $\psi_\mu\leftarrow - \gamma_\mu \psi S$
& [F21] & $\psi_\mu\leftarrow - S \gamma_\mu\psi$ \\\hline
[F13] & $S\leftarrow \psi^T{\rm C}\gamma^\mu\psi_\mu$
& [F31] & $S\leftarrow \psi^T_\mu {\rm C} (-\gamma^\mu) \psi$ \\\hline
[F23] & $\psi\leftarrow S\gamma^\mu\psi_\mu$
& [F32] & $\psi\leftarrow \gamma^\mu\psi_\mu S$ \\\hline
\multicolumn{4}{|l|}{[GBG (Psibar, S, Grav)]: $\bar\psi \gamma^\mu \fmslash{k}_S S \psi_\mu$}\\\hline
[F12] & $\psi_\mu\leftarrow \fmslash{k}_S \gamma_\mu \psi S$
& [F21] & $\psi_\mu\leftarrow S \fmslash{k}_S \gamma_\mu\psi$ \\\hline
[F13] & $S\leftarrow \psi^T{\rm C}\gamma^\mu\fmslash{k}_S \psi_\mu$
& [F31] & $S\leftarrow \psi^T_\mu {\rm C} \fmslash{k}_S \gamma^\mu \psi$ \\\hline
[F23] & $\psi\leftarrow S\gamma^\mu\fmslash{k}_S\psi_\mu$
& [F32] & $\psi\leftarrow \gamma^\mu\fmslash{k}_S\psi_\mu S$ \\\hline
\multicolumn{4}{|l|}{[GBG (Psibar, P, Grav)]: $\bar\psi \gamma^\mu\gamma^5 P\fmslash{k}_P \psi_\mu$}\\\hline
[F12] & $\psi_\mu\leftarrow -\fmslash{k}_P \gamma_\mu \gamma^5 \psi P$
& [F21] & $\psi_\mu\leftarrow -P\fmslash{k}_P \gamma_\mu \gamma^5 \psi$ \\\hline
[F13] & $P\leftarrow \psi^T {\rm C}\gamma^\mu\gamma^5\fmslash{k}_P\psi_\mu$
& [F31] & $P\leftarrow -\psi^T_\mu {\rm C}\fmslash{k}_P\gamma^\mu\gamma_5\psi$ \\\hline
[F23] & $\psi\leftarrow P \gamma^\mu\gamma^5\fmslash{k}_P\psi_\mu$
& [F32] & $\psi\leftarrow \gamma^\mu\gamma^5\fmslash{k}_P\psi_\mu P$ \\\hline
\multicolumn{4}{|l|}{[GBG (Psibar, V, Grav)]: $\bar\psi\gamma^5\gamma^\mu\lbrack\fmslash{k}_V,\fmslash{V}\rbrack\psi_\mu$}\\\hline
[F12] & $\psi_\mu\leftarrow \lbrack \fmslash{k}_V , \gamma^\alpha \rbrack \gamma_\mu \gamma^5 \psi V_\alpha$
& [F21] & $\psi_\mu\leftarrow \lbrack \fmslash{k}_V , \fmslash{V} \rbrack \gamma_\mu \gamma^5 \psi$ \\\hline
[F13] & $V_{\mu}\leftarrow \psi^T {\rm C} \gamma^5 \gamma^\rho \lbrack \fmslash{k}_V , \gamma_\mu \rbrack \psi_\rho$
& [F31] & $V_{\mu}\leftarrow \psi^T_\rho {\rm C} \lbrack \fmslash{k}_V , \gamma_\mu \rbrack \gamma^\rho \gamma^5 \psi$ \\\hline
[F23] & $\psi\leftarrow\gamma^5\gamma^\mu\lbrack \fmslash{k}_V , \fmslash{V} \rbrack\psi_\mu$
& [F32] & $\psi\leftarrow\gamma^5\gamma^\mu\lbrack \fmslash{k}_V , \gamma^\alpha \rbrack\psi_\mu V_\alpha$ \\\hline
\end{tabular}
\end{center}
\caption{\label{tab:dim5-fermions-diracgrav} Dimension-5 trilinear
couplings including one conjugated Dirac, one Gravitino fermion and one additional particle.}
\end{table}
\begin{table}
\begin{center}
\renewcommand{\arraystretch}{1.4}
\begin{tabular}{|>{\qquad}r<{:}l|r<{:}l|}\hline
\multicolumn{4}{|l|}{[GBG (Gravbar, POT, Chi)]: $\bar\psi_\mu S \gamma^\mu \chi$}\\\hline
[F12] & $\chi\leftarrow - \gamma^\mu \psi_\mu S$
& [F21] & $\chi\leftarrow - S\gamma^\mu \psi_\mu$ \\\hline
[F13] & $S\leftarrow \psi^T_\mu {\rm C} \gamma^\mu \chi$
& [F31] & $S\leftarrow \chi^T{\rm C} (-\gamma^\mu)\psi_\mu$ \\\hline
[F23] & $\psi_\mu\leftarrow S\gamma_\mu\chi$
& [F32] & $\psi_\mu\leftarrow \gamma_\mu \chi S$ \\\hline
\multicolumn{4}{|l|}{[GBG (Gravbar, S, Chi)]: $\bar\psi_\mu \fmslash{k}_S S \gamma^\mu \chi$}\\\hline
[F12] & $\chi\leftarrow \gamma^\mu \fmslash{k}_S \psi_\mu S$
& [F21] & $\chi\leftarrow S\gamma^\mu \fmslash{k}_S \psi_\mu$ \\\hline
[F13] & $S\leftarrow \psi^T_\mu {\rm C} \fmslash{k}_S \gamma^\mu \chi$
& [F31] & $S\leftarrow \chi^T{\rm C}\gamma^\mu\fmslash{k}_S \psi_\mu$ \\\hline
[F23] & $\psi_\mu\leftarrow S\fmslash{k}_S\gamma_\mu\chi$
& [F32] & $\psi_\mu\leftarrow \fmslash{k}_S \gamma_\mu \chi S$ \\\hline
\multicolumn{4}{|l|}{[GBG (Gravbar, P, Chi)]: $\bar\psi_\mu \fmslash{k}_P P \gamma^\mu \gamma_5 \chi$}\\\hline
[F12] & $\chi\leftarrow \gamma^\mu\fmslash{k}_P\gamma_5\psi_\mu P$
& [F21] & $\chi\leftarrow P\gamma^\mu\fmslash{k}_P\gamma_5\psi_\mu$ \\\hline
[F13] & $P\leftarrow \psi^T_\mu {\rm C}\fmslash{k}_P\gamma^\mu\gamma_5\chi$
& [F31] & $P\leftarrow \chi^T {\rm C}\gamma^\mu\fmslash{k}_P\gamma_5\psi_\mu$ \\\hline
[F23] & $\psi_\mu\leftarrow P\fmslash{k}_P \gamma_\mu \gamma_5 \chi$
& [F32] & $\psi_\mu\leftarrow \fmslash{k}_P \gamma_\mu \gamma_5 \chi P$ \\\hline
\multicolumn{4}{|l|}{[GBG (Gravbar, V, Chi)]: $\bar\psi_\mu\lbrack\fmslash{k}_V,\fmslash{V}\rbrack\gamma^\mu\gamma^5\chi$}\\\hline
[F12] & $\chi\leftarrow \gamma^5\gamma^\mu \lbrack \fmslash{k}_V , \gamma^\alpha \rbrack \psi_\mu V_\alpha$
& [F21] & $\chi\leftarrow \gamma^5\gamma^\mu \lbrack \fmslash{k}_V , \fmslash{V} \rbrack\psi_\mu$ \\\hline
[F13] & $V_{\mu}\leftarrow \psi^T_\rho {\rm C} \lbrack \fmslash{k}_V , \gamma_\mu \rbrack \gamma^\rho \gamma^5 \chi$
& [F31] & $V_{\mu}\leftarrow \chi^T {\rm C} \gamma^5 \gamma^{\rho} \lbrack \fmslash{k}_V , \gamma_\mu \rbrack \psi_\rho$ \\\hline
[F23] & $\psi_\mu\leftarrow\lbrack \fmslash{k}_V , \fmslash{V} \rbrack \gamma_\mu \gamma^5 \chi $
& [F32] & $\psi_\mu\leftarrow\lbrack \fmslash{k}_V , \gamma^\alpha \rbrack \gamma_\mu \gamma^5 \chi V_\alpha$ \\\hline
\end{tabular}
\end{center}
\caption{\label{tab:dim5-fermions-gravmajo} Dimension-5 trilinear
couplings including one Majorana, one Gravitino fermion and one
additional particle. The table is essentially the same as the one
with the Dirac fermion and only written for the sake of completeness.}
\end{table}
\begin{table}
\begin{center}
\renewcommand{\arraystretch}{1.4}
\begin{tabular}{|>{\qquad}r<{:}l|r<{:}l|}\hline
\multicolumn{4}{|l|}{[GBG (Chibar, POT, Grav)]: $\bar\chi \gamma^\mu S \psi_\mu$}\\\hline
[F12] & $\psi_\mu\leftarrow - \gamma_\mu \chi S$
& [F21] & $\psi_\mu\leftarrow - S \gamma_\mu\chi$ \\\hline
[F13] & $S\leftarrow \chi^T{\rm C}\gamma^\mu\psi_\mu$
& [F31] & $S\leftarrow \psi^T_\mu {\rm C} (-\gamma^\mu) \chi$ \\\hline
[F23] & $\chi\leftarrow S\gamma^\mu\psi_\mu$
& [F32] & $\chi\leftarrow \gamma^\mu\psi_\mu S$ \\\hline
\multicolumn{4}{|l|}{[GBG (Chibar, S, Grav)]: $\bar\chi \gamma^\mu \fmslash{k}_S S \psi_\mu$}\\\hline
[F12] & $\psi_\mu\leftarrow \fmslash{k}_S \gamma_\mu \chi S$
& [F21] & $\psi_\mu\leftarrow S \fmslash{k}_S \gamma_\mu\chi$ \\\hline
[F13] & $S\leftarrow \chi^T{\rm C}\gamma^\mu\fmslash{k}_S \psi_\mu$
& [F31] & $S\leftarrow \psi^T_\mu {\rm C} \fmslash{k}_S \gamma^\mu \chi$ \\\hline
[F23] & $\chi\leftarrow S\gamma^\mu\fmslash{k}_S\psi_\mu$
& [F32] & $\chi\leftarrow \gamma^\mu\fmslash{k}_S\psi_\mu S$ \\\hline
\multicolumn{4}{|l|}{[GBG (Chibar, P, Grav)]: $\bar\chi \gamma^\mu\gamma^5 P\fmslash{k}_P \psi_\mu$}\\\hline
[F12] & $\psi_\mu\leftarrow -\fmslash{k}_P \gamma_\mu \gamma^5 \chi P$
& [F21] & $\psi_\mu\leftarrow -P\fmslash{k}_P \gamma_\mu \gamma^5 \chi$ \\\hline
[F13] & $P\leftarrow \chi^T {\rm C}\gamma^\mu\gamma^5\fmslash{k}_P\psi_\mu$
& [F31] & $P\leftarrow -\psi^T_\mu {\rm C}\fmslash{k}_P\gamma^\mu\gamma_5\chi$ \\\hline
[F23] & $\chi\leftarrow P \gamma^\mu\gamma^5\fmslash{k}_P\psi_\mu$
& [F32] & $\chi\leftarrow \gamma^\mu\gamma^5\fmslash{k}_P\psi_\mu P$ \\\hline
\multicolumn{4}{|l|}{[GBG (Chibar, V, Grav)]: $\bar\chi\gamma^5\gamma^\mu\lbrack\fmslash{k}_V,\fmslash{V}\rbrack\psi_\mu$}\\\hline
[F12] & $\psi_\mu\leftarrow \lbrack \fmslash{k}_V , \gamma^\alpha \rbrack \gamma_\mu \gamma^5 \chi V_\alpha$
& [F21] & $\psi_\mu\leftarrow \lbrack \fmslash{k}_V , \fmslash{V} \rbrack \gamma_\mu \gamma^5 \chi$ \\\hline
[F13] & $V_{\mu}\leftarrow \chi^T {\rm C} \gamma^5 \gamma^\rho \lbrack \fmslash{k}_V , \gamma_\mu \rbrack \psi_\rho$
& [F31] & $V_{\mu}\leftarrow \psi^T_\rho {\rm C} \lbrack \fmslash{k}_V , \gamma_\mu \rbrack \gamma^\rho \gamma^5 \chi$ \\\hline
[F23] & $\chi\leftarrow\gamma^5\gamma^\mu\lbrack \fmslash{k}_V , \fmslash{V} \rbrack\psi_\mu$
& [F32] & $\chi\leftarrow\gamma^5\gamma^\mu\lbrack \fmslash{k}_V , \gamma^\alpha \rbrack\psi_\mu V_\alpha$ \\\hline
\end{tabular}
\end{center}
\caption{\label{tab:dim5-fermions-majograv} Dimension-5 trilinear
couplings including one conjugated Majorana, one Gravitino fermion and
one additional particle. This table is not only the same as the one
with the conjugated Dirac fermion but also the same part of the
Lagrangian density as the one with the Majorana particle on the right
of the gravitino.}
\end{table}
\begin{table}
\begin{center}
\renewcommand{\arraystretch}{1.4}
\begin{tabular}{|>{\qquad}r<{:}l|r<{:}l|}\hline
\multicolumn{2}{|l|}{[GBBG (Gravbar, S2, Psi)]: $\bar\psi_\mu S_1 S_2
\gamma^\mu \psi$}\\\hline
[F123] [F213] [F132] [F231] [F312] [F321] & $\psi\leftarrow - \gamma^\mu S_1 S_2 \psi_\mu$ \\ \hline
[F423] [F243] [F432] [F234] [F342] [F324] & $\psi_\mu \leftarrow \gamma_\mu S_1 S_2 \psi$ \\ \hline
[F134] [F143] [F314] & $S_1 \leftarrow \psi^T_\mu C S_2 \gamma^\mu \psi$ \\ \hline
[F124] [F142] [F214] & $S_2 \leftarrow \psi^T_\mu C S_1 \gamma^\mu \psi$ \\ \hline
[F413] [F431] [F341] & $S_1 \leftarrow - \psi^T C S_2 \gamma^\mu \psi_\mu$ \\ \hline
[F412] [F421] [F241] & $S_2 \leftarrow - \psi^T C S_1 \gamma^\mu \psi_\mu$ \\ \hline
\multicolumn{2}{|l|}{[GBBG (Gravbar, SV, Psi)]: $\bar\psi_\mu S \fmslash{V} \gamma^\mu \gamma^5 \psi$}\\\hline
[F123] [F213] [F132] [F231] [F312] [F321] & $\psi\leftarrow \gamma^5 \gamma^\mu S \fmslash{V} \psi_\mu$ \\ \hline
[F423] [F243] [F432] [F234] [F342] [F324] & $\psi_\mu \leftarrow \fmslash{V} S \gamma_\mu \gamma^5 \psi$ \\ \hline
[F134] [F143] [F314] & $S \leftarrow \psi^T_\mu C \fmslash{V} \gamma^\mu \gamma^5 \psi$ \\ \hline
[F124] [F142] [F214] & $V_\mu \leftarrow \psi^T_\rho C S \gamma_\mu \gamma^\rho \gamma^5 \psi$ \\ \hline
[F413] [F431] [F341] & $S \leftarrow \psi^T C \gamma^5 \gamma^\mu \fmslash{V} \psi_\mu$ \\ \hline
[F412] [F421] [F241] & $V_\mu \leftarrow \psi^T C S \gamma^5 \gamma^\rho \gamma_\mu \psi_\rho$ \\ \hline
\multicolumn{2}{|l|}{[GBBG (Gravbar, PV, Psi)]: $\bar\psi_\mu P \fmslash{V} \gamma^\mu \psi$}\\\hline
[F123] [F213] [F132] [F231] [F312] [F321] & $\psi\leftarrow \gamma^\mu P \fmslash{V} \psi_\mu$ \\ \hline
[F423] [F243] [F432] [F234] [F342] [F324] & $\psi_\mu \leftarrow \fmslash{V} P \gamma_\mu \psi$ \\ \hline
[F134] [F143] [F314] & $P \leftarrow \psi^T_\mu C \fmslash{V} \gamma^\mu \psi$ \\ \hline
[F124] [F142] [F214] & $V_\mu \leftarrow \psi^T_\rho C P \gamma_\mu \gamma^\rho \psi$ \\ \hline
[F413] [F431] [F341] & $P \leftarrow \psi^T C \gamma^\mu \fmslash{V} \psi_\mu$ \\ \hline
[F412] [F421] [F241] & $V_\mu \leftarrow \psi^T C P \gamma^\rho \gamma_\mu \psi_\rho$ \\ \hline
\multicolumn{2}{|l|}{[GBBG (Gravbar, V2, Psi)]: $\bar\psi_\mu f_{abc} \lbrack \fmslash{V}^a , \fmslash{V}^b \rbrack\gamma^\mu \gamma^5 \psi$}\\\hline
[F123] [F213] [F132] [F231] [F312] [F321] & $\psi\leftarrow f_{abc} \gamma^5 \gamma^\mu \lbrack \fmslash{V}^a , \fmslash{V}^b \rbrack \psi_\mu$ \\ \hline
[F423] [F243] [F432] [F234] [F342] [F324] & $\psi_\mu \leftarrow f_{abc} \lbrack \fmslash{V}^a , \fmslash{V}^b \rbrack \gamma_\mu \gamma^5 \psi$ \\ \hline
[F134] [F143] [F314] [F124] [F142] [F214] & $V_\mu^a \leftarrow\psi^T_\rho C f_{abc} \lbrack \gamma_\mu , \fmslash{V}^b \rbrack \gamma^\rho \gamma^5 \psi$ \\ \hline
[F413] [F431] [F341] [F412] [F421] [F241] & $V_\mu^a \leftarrow\psi^T C f_{abc} \gamma^5 \gamma^\rho\lbrack \gamma_\mu , \fmslash{V}^b \rbrack \psi_\rho$ \\ \hline
\end{tabular}
\end{center}
\caption{\label{tab:dim5-gravferm2boson} Dimension-5 trilinear
couplings including one Dirac, one Gravitino fermion and two additional bosons. In each lines we list the fusion possibilities with the same order of the fermions, but the order of the bosons is arbitrary (of course, one has to take care of this order in the mapping of the wave functions in [fusion]).}
\end{table}
\begin{table}
\begin{center}
\renewcommand{\arraystretch}{1.4}
\begin{tabular}{|>{\qquad}r<{:}l|r<{:}l|}\hline
\multicolumn{2}{|l|}{[GBBG (Psibar, S2, Grav)]: $\bar\psi S_1 S_2
\gamma^\mu \psi_\mu$}\\\hline
[F123] [F213] [F132] [F231] [F312] [F321] & $\psi_\mu\leftarrow - \gamma_\mu S_1 S_2 \psi$ \\ \hline
[F423] [F243] [F432] [F234] [F342] [F324] & $\psi \leftarrow \gamma^\mu S_1 S_2 \psi_\mu$ \\ \hline
[F134] [F143] [F314] & $S_1 \leftarrow \psi^T C S_2 \gamma^\mu \psi_\mu$ \\ \hline
[F124] [F142] [F214] & $S_2 \leftarrow \psi^T C S_1 \gamma^\mu \psi_\mu$ \\ \hline
[F413] [F431] [F341] & $S_1 \leftarrow - \psi^T_\mu C S_2 \gamma^\mu \psi$ \\ \hline
[F412] [F421] [F241] & $S_2 \leftarrow - \psi^T_\mu C S_1 \gamma^\mu \psi$ \\ \hline
\multicolumn{2}{|l|}{[GBBG (Psibar, SV, Grav)]: $\bar\psi S \gamma^\mu \gamma^5 \fmslash{V} \psi_\mu$}\\\hline
[F123] [F213] [F132] [F231] [F312] [F321] & $\psi_\mu\leftarrow \fmslash{V} S \gamma^5 \gamma^\mu \psi$ \\ \hline
[F423] [F243] [F432] [F234] [F342] [F324] & $\psi\leftarrow \gamma^\mu\gamma^5 S\fmslash{V}\psi_\mu$ \\ \hline
[F134] [F143] [F314] & $S \leftarrow \psi^T C \gamma^\mu \gamma^5 \fmslash{V}\psi$ \\ \hline
[F124] [F142] [F214] & $V_\mu \leftarrow \psi^T C \gamma^\rho \gamma^5 S \gamma_\mu \psi_\rho$ \\ \hline
[F413] [F431] [F341] & $S \leftarrow \psi^T_\mu C \fmslash{V} \gamma^5 \gamma^\mu \psi$ \\ \hline
[F412] [F421] [F241] & $V_\mu \leftarrow \psi^T_\rho C S \gamma_\mu \gamma^5 \gamma^\rho \psi$ \\ \hline
\multicolumn{2}{|l|}{[GBBG (Psibar, PV, Grav)]: $\bar\psi P \gamma^\mu \fmslash{V} \psi_\mu$}\\\hline
[F123] [F213] [F132] [F231] [F312] [F321] & $\psi_\mu\leftarrow \fmslash{V}\gamma_\mu P \psi$ \\ \hline
[F423] [F243] [F432] [F234] [F342] [F324] & $\psi\leftarrow \gamma^\mu\fmslash{V} P\psi_\mu$ \\ \hline
[F134] [F143] [F314] & $P \leftarrow \psi^T C \gamma^\mu\fmslash{V}\psi_\mu$ \\ \hline
[F124] [F142] [F214] & $V_\mu \leftarrow \psi^T C P \gamma^\rho \gamma_\mu \psi_\rho$ \\ \hline
[F413] [F431] [F341] & $P \leftarrow \psi^T_\mu C \fmslash{V}\gamma^\mu \psi$ \\ \hline
[F412] [F421] [F241] & $V_\mu \leftarrow \psi^T_\rho C P \gamma_\mu \gamma^\rho \psi$ \\ \hline
\multicolumn{2}{|l|}{[GBBG (Psibar, V2, Grav)]: $\bar\psi f_{abc} \gamma^5 \gamma^\mu \lbrack \fmslash{V}^a , \fmslash{V}^b \rbrack\psi_\mu$}\\\hline
[F123] [F213] [F132] [F231] [F312] [F321] & $\psi_\mu\leftarrow f_{abc} \lbrack \fmslash{V}^a , \fmslash{V}^b \rbrack \gamma_\mu \gamma^5 \psi$ \\ \hline
[F423] [F243] [F432] [F234] [F342] [F324] & $\psi\leftarrow f_{abc} \gamma^5\gamma^\mu\lbrack \fmslash{V}^a , \fmslash{V}^b \rbrack\psi_\mu$ \\ \hline
[F134] [F143] [F314] [F124] [F142] [F214] & $V_\mu^a \leftarrow\psi^T C f_{abc} \gamma^5\gamma^\rho\lbrack \gamma_\mu , \fmslash{V}^b \rbrack\psi_\rho$ \\ \hline
[F413] [F431] [F341] [F412] [F421] [F241] & $V_\mu^a \leftarrow\psi^T_\rho C f_{abc}\lbrack \gamma_\mu , \fmslash{V}^b \rbrack\gamma^\rho\gamma^5 \psi$ \\ \hline
\end{tabular}
\end{center}
\caption{\label{tab:dim5-gravferm2boson2} Dimension-5 trilinear
couplings including one conjugated Dirac, one Gravitino fermion and two additional bosons. The couplings of Majorana fermions to the gravitino and two bosons are essentially the same as for Dirac fermions and they are omitted here.}
\end{table}
*)
(* \thocwmodulesection{Perturbative Quantum Gravity and Kaluza-Klein Interactions}
The gravitational coupling constant and the relative strength of
the dilaton coupling are abbreviated as
\begin{subequations}
\begin{align}
\kappa &= \sqrt{16\pi G_N} \\
\omega &= \sqrt{\frac{2}{3(n+2)}} = \sqrt{\frac{2}{3(d-2)}}\,,
\end{align}
\end{subequations}
where~$n=d-4$ is the number of extra space dimensions. *)
(* In~(\ref{eq:graviton-feynman-rules3}-\ref{eq:dilaton-feynman-rules5}),
we use the notation of~\cite{Han/Lykken/Zhang:1999:Kaluza-Klein}:
\begin{subequations}
\begin{equation}
C_{\mu\nu,\rho\sigma} =
g_{\mu\rho} g_{\nu\sigma} + g_{\mu\sigma} g_{\nu\rho}
- g_{\mu\nu} g_{\rho\sigma}
\end{equation}
\begin{multline}
D_{\mu\nu,\rho\sigma}(k_1,k_2) =
g_{\mu\nu} k_{1,\sigma} k_{2,\rho} \\
\mbox{}
- ( g_{\mu\sigma} k_{1,\nu} k_{2,\rho}
+ g_{\mu\rho} k_{1,\sigma} k_{2,\nu}
- g_{\rho\sigma} k_{1,\mu} k_{2,\nu}
+ (\mu\leftrightarrow\nu))
\end{multline}
\begin{multline}
E_{\mu\nu,\rho\sigma}(k_1,k_2) =
g_{\mu\nu} (k_{1,\rho} k_{1,\sigma}
+ k_{2,\rho} k_{2,\sigma} + k_{1,\rho} k_{2,\sigma}) \\
\mbox{}
- ( g_{\nu\sigma} k_{1,\mu} k_{1,\rho}
+ g_{\nu\rho} k_{2,\mu} k_{2,\sigma}
+ (\mu\leftrightarrow\nu))
\end{multline}
\begin{multline}
F_{\mu\nu,\rho\sigma\lambda}(k_1,k_2,k_3) = \\
g_{\mu\rho} g_{\sigma\lambda} (k_2 - k_3)_{\nu}
+ g_{\mu\sigma} g_{\lambda\rho} (k_3 - k_1)_{\nu}
+ g_{\mu\lambda} g_{\rho\sigma} (k_1 - k_2)_{\nu}
+ (\mu\leftrightarrow\nu)
\end{multline}
\begin{multline}
G_{\mu\nu,\rho\sigma\lambda\delta} =
g_{\mu\nu} (g_{\rho\sigma}g_{\lambda\delta} - g_{\rho\delta}g_{\lambda\sigma})
\\ \mbox{}
+ ( g_{\mu\rho}g_{\nu\delta}g_{\lambda\sigma}
+ g_{\mu\lambda}g_{\nu\sigma}g_{\rho\delta}
- g_{\mu\rho}g_{\nu\sigma}g_{\lambda\delta}
- g_{\mu\lambda}g_{\nu\delta}g_{\rho\sigma}
+ (\mu\leftrightarrow\nu) )
\end{multline}
\end{subequations} *)
(* \begin{figure}
\begin{subequations}
\label{eq:graviton-feynman-rules3}
\begin{align}
\label{eq:graviton-scalar-scalar}
\parbox{28mm}{\fmfframe(2,2)(2,1){\begin{fmfgraph*}(24,22)
\Threeexternal{1}{2}{h_{\mu\nu}}
\fmf{plain}{v,e1}
\fmf{plain}{v,e2}
\fmf{dbl_dots}{v,e3}
\threeoutgoing
\end{fmfgraph*}}} \,&=
\begin{split}
\mbox{} & - \ii \frac{\kappa}{2} g_{\mu\nu} m^2
+ \ii \frac{\kappa}{2} C_{\mu\nu,\mu_1\mu_2}k^{\mu_1}_1k^{\mu_2}_2
\end{split} \\
\parbox{28mm}{\fmfframe(2,2)(2,1){\begin{fmfgraph*}(24,22)
\Threeexternal{1}{2}{h_{\mu\nu}}
\fmf{photon}{v,e1}
\fmf{photon}{v,e2}
\fmf{dbl_dots}{v,e3}
\threeoutgoing
\end{fmfgraph*}}} \,&=
\begin{split}
\mbox{} - \ii \frac{\kappa}{2} m^2 C_{\mu\nu,\mu_1\mu_2}
- \ii \frac{\kappa}{2}
(& k_1k_2 C_{\mu\nu,\mu_1\mu_2} \\
&\mbox{} + D_{\mu\nu,\mu_1\mu_2}(k_1,k_2) \\
&\mbox{} + \xi^{-1} E_{\mu\nu,\mu_1\mu_2}(k_1,k_2))
\end{split} \\
\parbox{28mm}{\fmfframe(2,2)(2,1){\begin{fmfgraph*}(24,22)
\Threeexternal{p}{p'}{h_{\mu\nu}}
\fmf{fermion}{e1,v,e2}
\fmf{dbl_dots}{v,e3}
\fmfdot{v}
\end{fmfgraph*}}} \,&=
\begin{split}
\mbox{} - \ii \frac{\kappa}{2} m g_{\mu\nu}
- \ii \frac{\kappa}{8}
(& \gamma_{\mu}(p+p')_{\nu} + \gamma_{\nu}(p+p')_{\mu} \\
& \mbox{} - 2 g_{\mu\nu} (\fmslash{p}+\fmslash{p}') )
\end{split}
\end{align}
\end{subequations}
\caption{\label{fig:graviton-feynman-rules3} Three-point graviton couplings.}
\end{figure}
\begin{table}
\begin{center}
\renewcommand{\arraystretch}{1.4}
\begin{tabular}{|>{\qquad}r<{:}l|}\hline
\multicolumn{2}{|l|}{[Graviton_Scalar_Scalar]:
$h_{\mu\nu} C^{\mu\nu}_{0}(k_1,k_2)\phi_1\phi_2$}\\\hline
[F12|F21]
& $\phi_2 \leftarrow \ii\cdot
h_{\mu\nu} C^{\mu\nu}_{0} (k_1, -k-k_1)\phi_1 $ \\\hline
[F13|F31]
& $\phi_1 \leftarrow \ii\cdot
h_{\mu\nu} C^{\mu\nu}_{0} (-k-k_2, k_2)\phi_2 $ \\\hline
[F23|F32]
& $h^{\mu\nu} \leftarrow \ii\cdot
C^{\mu\nu}_0 (k_1,k_2)\phi_1\phi_2 $ \\\hline
\multicolumn{2}{|l|}{[Graviton_Vector_Vector]:
$h_{\mu\nu} C^{\mu\nu,\mu_1\mu_2}_1(k_1,k_2,\xi)
V_{\mu_1}V_{\mu_2} $}\\\hline
[F12|F21] & $ V^\mu_2 \leftarrow \ii\cdot h_{\kappa\lambda}
C^{\kappa\lambda,\mu\nu}_1(-k-k_1,k_1\xi) V_{1,\nu}$ \\\hline
[F13|F31] & $ V^\mu_1 \leftarrow \ii\cdot h_{\kappa\lambda}
C^{\kappa\lambda,\mu\nu}_1(-k-k_2,k_2,\xi) V_{2,\nu}$ \\\hline
[F23|F32]
& $h^{\mu\nu} \leftarrow \ii\cdot
C^{\mu\nu,\mu_1\mu_2}_1(k_1,k_2,\xi)
V_{1,\mu_1}V_{2,\mu_2} $ \\\hline
\multicolumn{2}{|l|}{[Graviton_Spinor_Spinor]:
$h_{\mu\nu} \bar\psi_1
C^{\mu\nu}_{\frac{1}{2}}(k_1,k_2)\psi_2 $}\\\hline
[F12] & $ \bar\psi_2 \leftarrow \ii\cdot
h_{\mu\nu} \bar\psi_1 C^{\mu\nu}_{\frac{1}{2}}(k_1,-k-k_1) $ \\\hline
[F21] & $ \bar\psi_2 \leftarrow \ii\cdot\ldots $ \\\hline
[F13] & $ \psi_1 \leftarrow \ii\cdot
h_{\mu\nu}C^{\mu\nu}_{\frac{1}{2}}(-k-k_2,k_2)\psi_2$ \\\hline
[F31] & $ \psi_1 \leftarrow \ii\cdot\ldots $ \\\hline
[F23] & $ h^{\mu\nu} \leftarrow \ii\cdot
\bar\psi_1 C^{\mu\nu}_{\frac{1}{2}}(k_1,k_2)\psi_2 $ \\\hline
[F32] & $ h^{\mu\nu} \leftarrow \ii\cdot\ldots $ \\\hline
\end{tabular}
\end{center}
\caption{\label{tab:graviton-three-point} \ldots}
\end{table}
Derivation of~(\ref{eq:graviton-scalar-scalar})
\begin{subequations}
\begin{align}
L &= \frac{1}{2} (\partial_\mu \phi) (\partial^\mu \phi) - \frac{m^2}{2} \phi^2 \\
(\partial_\mu\phi) \frac{\partial L}{\partial(\partial^\nu\phi)}
&= (\partial_\mu\phi)(\partial_\nu\phi) \\
T_{\mu\nu} &= -g_{\mu\nu} L +
(\partial_\mu\phi) \frac{\partial L}{\partial(\partial^\nu\phi)}
+
\end{align}
\end{subequations}
\begin{subequations}
\begin{align}
C^{\mu\nu}_{0}(k_1,k_2)
&= C^{\mu\nu,\mu_1\mu_2} k_{1,\mu_1} k_{2,\mu_2} \\
C^{\mu\nu,\mu_1\mu_2}_1(k_1,k_2,\xi)
&= k_1k_2 C^{\mu\nu,\mu_1\mu_2}
+ D^{\mu\nu,\mu_1\mu_2}(k_1,k_2)
+ \xi^{-1} E^{\mu\nu,\mu_1\mu_2}(k_1,k_2) \\
C^{\mu\nu}_{\frac{1}{2},\alpha\beta}(p,p')
&= \gamma^{\mu}_{\alpha\beta}(p+p')^{\nu}
+ \gamma^{\nu}_{\alpha\beta}(p+p')^{\mu}
- 2 g^{\mu\nu} (\fmslash{p}+\fmslash{p}')_{\alpha\beta}
\end{align}
\end{subequations} *)
(* \begin{figure}
\begin{subequations}
\label{eq:dilaton-feynman-rules3}
\begin{align}
\parbox{28mm}{\fmfframe(2,2)(2,1){\begin{fmfgraph*}(24,22)
\Threeexternal{1}{2}{\phi(k)}
\fmf{plain}{v,e1}
\fmf{plain}{v,e2}
\fmf{dots}{v,e3}
\threeoutgoing
\end{fmfgraph*}}} \,&=
- \ii \omega \kappa 2m^2 - \ii \omega \kappa k_1k_2 \\
\parbox{28mm}{\fmfframe(2,2)(2,1){\begin{fmfgraph*}(24,22)
\Threeexternal{1}{2}{\phi(k)}
\fmf{photon}{v,e1}
\fmf{photon}{v,e2}
\fmf{dots}{v,e3}
\threeoutgoing
\end{fmfgraph*}}} \,&=
- \ii \omega \kappa g_{\mu_1\mu_2}m^2
- \ii \omega \kappa
\xi^{-1} (k_{1,\mu_1}k_{\mu_2} + k_{2,\mu_2}k_{\mu_1}) \\
\parbox{28mm}{\fmfframe(2,2)(2,1){\begin{fmfgraph*}(24,22)
\Threeexternal{p}{p'}{\phi(k)}
\fmf{fermion}{e1,v,e2}
\fmf{dots}{v,e3}
\fmfdot{v}
\end{fmfgraph*}}} \,&=
- \ii \omega \kappa 2m
+ \ii \omega \kappa \frac{3}{4}(\fmslash{p}+\fmslash{p}')
\end{align}
\end{subequations}
\caption{\label{fig:dilaton-feynman-rules3} Three-point dilaton couplings.}
\end{figure}
\begin{table}
\begin{center}
\renewcommand{\arraystretch}{1.4}
\begin{tabular}{|>{\qquad}r<{:}l|}\hline
\multicolumn{2}{|l|}{[Dilaton_Scalar_Scalar]:
$\phi \ldots k_1k_2\phi_1\phi_2 $}\\\hline
[F12|F21] & $ \phi_2 \leftarrow \ii\cdot k_1(-k-k_1)\phi\phi_1 $ \\\hline
[F13|F31] & $ \phi_1 \leftarrow \ii\cdot (-k-k_2)k_2\phi\phi_2 $ \\\hline
[F23|F32] & $ \phi \leftarrow \ii\cdot k_1k_2\phi_1\phi_2 $ \\\hline
\multicolumn{2}{|l|}{[Dilaton_Vector_Vector]:
$\phi \ldots $}\\\hline
[F12] & $ V_{2,\mu} \leftarrow \ii\cdot\ldots $ \\\hline
[F21] & $ V_{2,\mu} \leftarrow \ii\cdot\ldots $ \\\hline
[F13] & $ V_{1,\mu} \leftarrow \ii\cdot\ldots $ \\\hline
[F31] & $ V_{1,\mu} \leftarrow \ii\cdot\ldots $ \\\hline
[F23] & $ \phi \leftarrow \ii\cdot\ldots $ \\\hline
[F32] & $ \phi \leftarrow \ii\cdot\ldots $ \\\hline
\multicolumn{2}{|l|}{[Dilaton_Spinor_Spinor]:
$\phi \ldots $}\\\hline
[F12] & $ \bar\psi_2 \leftarrow \ii\cdot\ldots $ \\\hline
[F21] & $ \bar\psi_2 \leftarrow \ii\cdot\ldots $ \\\hline
[F13] & $ \psi_1 \leftarrow \ii\cdot\ldots $ \\\hline
[F31] & $ \psi_1 \leftarrow \ii\cdot\ldots $ \\\hline
[F23] & $ \phi \leftarrow \ii\cdot\ldots $ \\\hline
[F32] & $ \phi \leftarrow \ii\cdot\ldots $ \\\hline
\end{tabular}
\end{center}
\caption{\label{tab:dilaton-three-point} \ldots}
\end{table} *)
(* \begin{figure}
\begin{subequations}
\label{eq:graviton-feynman-rules4}
\begin{align}
\label{eq:graviton-scalar-scalar-scalar}
\parbox{28mm}{\fmfframe(2,2)(2,1){\begin{fmfgraph*}(24,22)
\Fourexternal{1}{2}{3}{h_{\mu\nu}}
\fmf{plain}{v,e1}
\fmf{plain}{v,e2}
\fmf{plain}{v,e3}
\fmf{dbl_dots}{v,e4}
\fouroutgoing
\end{fmfgraph*}}} \,&=
\begin{split}
\mbox{} & ???
\end{split} \\
\parbox{28mm}{\fmfframe(2,2)(2,1){\begin{fmfgraph*}(24,22)
\Fourexternal{1}{2}{3}{h_{\mu\nu}}
\fmf{plain}{v,e1}
\fmf{plain}{v,e2}
\fmf{photon}{v,e3}
\fmf{dbl_dots}{v,e4}
\fouroutgoing
\end{fmfgraph*}}} \,&=
\begin{split}
\mbox{} &
- \ii g\frac{\kappa}{2} C_{\mu\nu,\mu_3\rho}(k_1-k_2)^{\rho} T^{a_3}_{n_2n_1}
\end{split} \\
\label{eq:graviton-scalar-vector-vector}
\parbox{28mm}{\fmfframe(2,2)(2,1){\begin{fmfgraph*}(24,22)
\Fourexternal{1}{2}{3}{h_{\mu\nu}}
\fmf{plain}{v,e1}
\fmf{photon}{v,e2}
\fmf{photon}{v,e3}
\fmf{dbl_dots}{v,e4}
\fouroutgoing
\end{fmfgraph*}}} \,&=
\begin{split}
\mbox{} & ???
\end{split} \\
\parbox{28mm}{\fmfframe(2,2)(2,1){\begin{fmfgraph*}(24,22)
\Fourexternal{1}{2}{3}{h_{\mu\nu}}
\fmf{photon}{v,e1}
\fmf{photon}{v,e2}
\fmf{photon}{v,e3}
\fmf{dbl_dots}{v,e4}
\fouroutgoing
\end{fmfgraph*}}} \,&=
\begin{split}
\mbox{} - g \frac{\kappa}{2} f^{a_1a_2a_3}
(& C_{\mu\nu,\mu_1\mu_2} (k_1-k_2)_{\mu_3} \\
& \mbox{} + C_{\mu\nu,\mu_2\mu_3} (k_2-k_3)_{\mu_1} \\
& \mbox{} + C_{\mu\nu,\mu_3\mu_1} (k_3-k_1)_{\mu_2} \\
& \mbox{} + F_{\mu\nu,\mu_1\mu_2\mu_3}(k_1,k_2,k_3) )
\end{split} \\
\label{eq:graviton-yukawa}
\parbox{28mm}{\fmfframe(2,2)(2,1){\begin{fmfgraph*}(24,22)
\Fourexternal{1}{2}{3}{h_{\mu\nu}}
\fmf{fermion}{e1,v,e2}
\fmf{plain}{v,e3}
\fmf{dbl_dots}{v,e4}
\fmfdot{v}
\fmffreeze
\fmf{warrow_right}{v,e3}
\fmf{warrow_right}{v,e4}
\end{fmfgraph*}}} \,&=
\begin{split}
\mbox{} & ???
\end{split} \\
\parbox{28mm}{\fmfframe(2,2)(2,1){\begin{fmfgraph*}(24,22)
\Fourexternal{1}{2}{3}{h_{\mu\nu}}
\fmf{fermion}{e1,v,e2}
\fmf{photon}{v,e3}
\fmf{dbl_dots}{v,e4}
\fmfdot{v}
\fmffreeze
\fmf{warrow_right}{v,e3}
\fmf{warrow_right}{v,e4}
\end{fmfgraph*}}} \,&=
\begin{split}
\mbox{} & \ii g\frac{\kappa}{4}
(C_{\mu\nu,\mu_3\rho} - g_{\mu\nu}g_{\mu_3\rho})
\gamma^{\rho} T^{a_3}_{n_2n_1}
\end{split}
\end{align}
\end{subequations}
\caption{\label{fig:graviton-feynman-rules4} Four-point graviton couplings.
(\ref{eq:graviton-scalar-scalar-scalar}),
(\ref{eq:graviton-scalar-vector-vector}),
and~(\ref{eq:graviton-yukawa)} are missing
in~\cite{Han/Lykken/Zhang:1999:Kaluza-Klein}, but should be generated
by standard model Higgs selfcouplings, Higgs-gaugeboson couplings, and
Yukawa couplings.}
\end{figure} *)
(* \begin{figure}
\begin{subequations}
\label{eq:dilaton-feynman-rules4}
\begin{align}
\label{eq:dilaton-scalar-scalar-scalar}
\parbox{28mm}{\fmfframe(2,2)(2,1){\begin{fmfgraph*}(24,22)
\Fourexternal{1}{2}{3}{\phi(k)}
\fmf{plain}{v,e1}
\fmf{plain}{v,e2}
\fmf{plain}{v,e3}
\fmf{dots}{v,e4}
\fouroutgoing
\end{fmfgraph*}}} \,&= ??? \\
\parbox{28mm}{\fmfframe(2,2)(2,1){\begin{fmfgraph*}(24,22)
\Fourexternal{1}{2}{3}{\phi(k)}
\fmf{plain}{v,e1}
\fmf{plain}{v,e2}
\fmf{photon}{v,e3}
\fmf{dots}{v,e4}
\fouroutgoing
\end{fmfgraph*}}} \,&=
- \ii \omega \kappa (k_1 + k_2)_{\mu_3} T^{a_3}_{n_1,n_2} \\
\label{eq:dilaton-scalar-vector-vector}
\parbox{28mm}{\fmfframe(2,2)(2,1){\begin{fmfgraph*}(24,22)
\Fourexternal{1}{2}{3}{\phi(k)}
\fmf{plain}{v,e1}
\fmf{photon}{v,e2}
\fmf{photon}{v,e3}
\fmf{dots}{v,e4}
\fouroutgoing
\end{fmfgraph*}}} \,&= ??? \\
\parbox{28mm}{\fmfframe(2,2)(2,1){\begin{fmfgraph*}(24,22)
\Fourexternal{1}{2}{3}{\phi(k)}
\fmf{photon}{v,e1}
\fmf{photon}{v,e2}
\fmf{photon}{v,e3}
\fmf{dots}{v,e4}
\fouroutgoing
\end{fmfgraph*}}} \,&= 0 \\
\label{eq:dilaton-yukawa}
\parbox{28mm}{\fmfframe(2,2)(2,1){\begin{fmfgraph*}(24,22)
\Fourexternal{1}{2}{3}{h_{\mu\nu}}
\fmf{fermion}{e1,v,e2}
\fmf{plain}{v,e3}
\fmf{dots}{v,e4}
\fmfdot{v}
\fmffreeze
\fmf{warrow_right}{v,e3}
\fmf{warrow_right}{v,e4}
\end{fmfgraph*}}} \,&= ??? \\
\parbox{28mm}{\fmfframe(2,2)(2,1){\begin{fmfgraph*}(24,22)
\Fourexternal{1}{2}{3}{\phi(k)}
\fmf{fermion}{e1,v,e2}
\fmf{photon}{v,e3}
\fmf{dots}{v,e4}
\fmfdot{v}
\fmffreeze
\fmf{warrow_right}{v,e3}
\fmf{warrow_right}{v,e4}
\end{fmfgraph*}}} \,&=
- \ii \frac{3}{2} \omega g \kappa \gamma_{\mu_3} T^{a_3}_{n_1n_2}
\end{align}
\end{subequations}
\caption{\label{fig:dilaton-feynman-rules4} Four-point dilaton couplings.
(\ref{eq:dilaton-scalar-scalar-scalar}),
(\ref{eq:dilaton-scalar-vector-vector})
and~(\ref{eq:dilaton-yukawa}) are missing
in~\cite{Han/Lykken/Zhang:1999:Kaluza-Klein}, but could be generated
by standard model Higgs selfcouplings, Higgs-gaugeboson couplings,
and Yukawa couplings.}
\end{figure} *)
(* \begin{figure}
\begin{subequations}
\label{eq:graviton-feynman-rules5}
\begin{align}
\label{eq:graviton-scalar-scalar-scalar-scalar}
\parbox{28mm}{\fmfframe(2,2)(2,1){\begin{fmfgraph*}(24,22)
\Fiveexternal{1}{2}{3}{4}{h_{\mu\nu}}
\fmf{plain}{v,e1}
\fmf{plain}{v,e2}
\fmf{plain}{v,e3}
\fmf{plain}{v,e4}
\fmf{dots}{v,e5}
\fiveoutgoing
\end{fmfgraph*}}} \,&=
\begin{split}
\mbox{} & ???
\end{split} \\
\parbox{28mm}{\fmfframe(2,2)(2,1){\begin{fmfgraph*}(24,22)
\Fiveexternal{1}{2}{3}{4}{h_{\mu\nu}}
\fmf{plain}{v,e1}
\fmf{plain}{v,e2}
\fmf{photon}{v,e3}
\fmf{photon}{v,e4}
\fmf{dots}{v,e5}
\fiveoutgoing
\end{fmfgraph*}}} \,&=
\begin{split}
\mbox{} & - \ii g^2 \frac{\kappa}{2} C_{\mu\nu,\mu_3\mu_4}
(T^{a_3}T^{a_4} + T^{a_4}T^{a_3})_{n_2n_1}
\end{split} \\
\parbox{28mm}{\fmfframe(2,2)(2,1){\begin{fmfgraph*}(24,22)
\Fiveexternal{1}{2}{3}{4}{h_{\mu\nu}}
\fmf{photon}{v,e1}
\fmf{photon}{v,e2}
\fmf{photon}{v,e3}
\fmf{photon}{v,e4}
\fmf{dots}{v,e5}
\fiveoutgoing
\end{fmfgraph*}}} \,&=
\begin{split}
\mbox{} - \ii g^2 \frac{\kappa}{2}
(& f^{ba_1a_3} f^{ba_2a_4} G_{\mu\nu,\mu_1\mu_2\mu_3\mu_4} \\
& \mbox + f^{ba_1a_2} f^{ba_3a_4} G_{\mu\nu,\mu_1\mu_3\mu_2\mu_4} \\
& \mbox + f^{ba_1a_4} f^{ba_2a_3} G_{\mu\nu,\mu_1\mu_2\mu_4\mu_3} )
\end{split}
\end{align}
\end{subequations}
\caption{\label{fig:graviton-feynman-rules5} Five-point graviton couplings.
(\ref{eq:graviton-scalar-scalar-scalar-scalar}) is missing
in~\cite{Han/Lykken/Zhang:1999:Kaluza-Klein}, but should be generated
by standard model Higgs selfcouplings.}
\end{figure} *)
(* \begin{figure}
\begin{subequations}
\label{eq:dilaton-feynman-rules5}
\begin{align}
\label{eq:dilaton-scalar-scalar-scalar-scalar}
\parbox{28mm}{\fmfframe(2,2)(2,1){\begin{fmfgraph*}(24,22)
\Fiveexternal{1}{2}{3}{4}{\phi(k)}
\fmf{plain}{v,e1}
\fmf{plain}{v,e2}
\fmf{plain}{v,e3}
\fmf{plain}{v,e4}
\fmf{dots}{v,e5}
\fiveoutgoing
\end{fmfgraph*}}} \,&= ??? \\
\parbox{28mm}{\fmfframe(2,2)(2,1){\begin{fmfgraph*}(24,22)
\Fiveexternal{1}{2}{3}{4}{\phi(k)}
\fmf{plain}{v,e1}
\fmf{plain}{v,e2}
\fmf{photon}{v,e3}
\fmf{photon}{v,e4}
\fmf{dots}{v,e5}
\fiveoutgoing
\end{fmfgraph*}}} \,&=
\ii \omega g^2 \kappa g_{\mu_3\mu_4}
(T^{a_3}T^{a_4} + T^{a_4}T^{a_3})_{n_2n_1} \\
\parbox{28mm}{\fmfframe(2,2)(2,1){\begin{fmfgraph*}(24,22)
\Fiveexternal{1}{2}{3}{4}{\phi(k)}
\fmf{photon}{v,e1}
\fmf{photon}{v,e2}
\fmf{photon}{v,e3}
\fmf{photon}{v,e4}
\fmf{dots}{v,e5}
\fiveoutgoing
\end{fmfgraph*}}} \,&= 0
\end{align}
\end{subequations}
\caption{\label{fig:dilaton-feynman-rules5} Five-point dilaton couplings.
(\ref{eq:dilaton-scalar-scalar-scalar-scalar}) is missing
in~\cite{Han/Lykken/Zhang:1999:Kaluza-Klein}, but could be generated
by standard model Higgs selfcouplings.}
\end{figure} *)
(* \thocwmodulesection{Dependent Parameters}
This is a simple abstract syntax for parameter dependencies.
Later, there will be a parser for a convenient concrete syntax
as a part of a concrete syntax for models. There is no intention
to do \emph{any} symbolic manipulation with this. The expressions
will be translated directly by [Targets] to the target language. *)
type 'a expr =
| I | Const of int
| Atom of 'a
| Sum of 'a expr list
| Diff of 'a expr * 'a expr
| Neg of 'a expr
| Prod of 'a expr list
| Quot of 'a expr * 'a expr
| Rec of 'a expr
| Pow of 'a expr * int
| Sqrt of 'a expr
| Sin of 'a expr
| Cos of 'a expr
| Tan of 'a expr
| Cot of 'a expr
| Atan2 of 'a expr * 'a expr
| Conj of 'a expr
type 'a variable = Real of 'a | Complex of 'a
type 'a variable_array = Real_Array of 'a | Complex_Array of 'a
type 'a parameters =
{ input : ('a * float) list;
derived : ('a variable * 'a expr) list;
derived_arrays : ('a variable_array * 'a expr list) list }
(* \thocwmodulesection{More Exotic Couplings}
\begin{table}
\begin{center}
\renewcommand{\arraystretch}{1.3}
\begin{tabular}{|>{\qquad}r<{:}l|}\hline
\multicolumn{2}{|l|}{[Dim5_Scalar_Vector_Vector_T]:
$\mathcal{L}_I=g\phi
(\ii\partial_\mu V_1^\nu)(\ii\partial_\nu V_2^\mu)$}\\\hline
[F23] & $\phi(k_2+k_3)\leftarrow\ii\cdot g
k_3^\mu V_{1,\mu}(k_2) k_2^\nu V_{2,\nu}(k_3)$ \\\hline
[F32] & $\phi(k_2+k_3)\leftarrow\ii\cdot g
k_2^\mu V_{2,\mu}(k_3) k_3^\nu V_{1,\nu}(k_2)$ \\\hline
[F12] & $V_2^\mu(k_1+k_2)\leftarrow\ii\cdot g
k_2^\mu \phi(k_1) (-k_1^\nu-k_2^\nu) V_{1,\nu}(k_2)$ \\\hline
[F21] & $V_2^\mu(k_1+k_2)\leftarrow\ii\cdot g
k_2^\mu (-k_1^\nu-k_2^\nu)V_{1,\nu}(k_2) \phi(k_1)$ \\\hline
[F13] & $V_1^\mu(k_1+k_3)\leftarrow\ii\cdot g
k_3^\mu \phi(k_1) (-k_1^\nu-k_3^\nu)V_{2,\nu}(k_3)$ \\\hline
[F31] & $V_1^\mu(k_1+k_3)\leftarrow\ii\cdot g
k_3^\mu (-k_1^\nu-k_3^\nu)V_{2,\nu}(k_3) \phi(k_1)$ \\\hline
\end{tabular}
\end{center}
\caption{\label{tab:dim5-scalar-vector-vector}
\ldots}
\end{table}
\begin{table}
\begin{center}
\renewcommand{\arraystretch}{1.3}
\begin{tabular}{|>{\qquad}r<{:}l|}\hline
\multicolumn{2}{|l|}{[Dim6_Vector_Vector_Vector_T]:
$\mathcal{L}_I=gV_1^\mu
((\ii\partial_\nu V_2^\rho)%
\ii\overleftrightarrow{\partial_\mu}
(\ii\partial_\rho V_3^\nu))$}\\\hline
[F23] & $V_1^\mu(k_2+k_3)\leftarrow\ii\cdot g
(k_2^\mu - k_3^\mu) k_3^\nu V_{2,\nu} (k_2)
k_2^\rho V_{3,\rho}(k_3)$ \\\hline
[F32] & $V_1^\mu(k_2+k_3)\leftarrow\ii\cdot g
(k_2^\mu - k_3^\mu) k_2^\nu V_{3,\nu} (k_3)
k_3^\rho V_{2,\rho}(k_2)$ \\\hline
[F12] & $V_3^\mu(k_1+k_2)\leftarrow\ii\cdot g
k_2^\mu (k_1^\nu+2k_2^\nu) V_{1,\nu} (k_1)
(-k_1^\rho-k_2^\rho) V_{2,\rho}(k_2)$ \\\hline
[F21] & $V_3^\mu(k_1+k_2)\leftarrow\ii\cdot g
k_2^\mu (-k_1^\rho-k_2^\rho) V_{2,\rho}(k_2)
(k_1^\nu+2k_2^\nu) V_{1,\nu} (k_1)$ \\\hline
[F13] & $V_2^\mu(k_1+k_3)\leftarrow\ii\cdot g
k_3^\mu (k_1^\nu+2k_3^\nu) V_{1,\nu} (k_1)
(-k_1^\rho-k_3^\rho) V_{3,\rho}(k_3)$ \\\hline
[F31] & $V_2^\mu(k_1+k_3)\leftarrow\ii\cdot g
k_3^\mu (-k_1^\rho-k_3^\rho) V_{3,\rho}(k_3)
(k_1^\nu+2k_3^\nu) V_{1,\nu} (k_1)$ \\\hline
\end{tabular}
\end{center}
\caption{\label{tab:dim6-vector-vector-vector}
\ldots}
\end{table}
\begin{table}
\begin{center}
\renewcommand{\arraystretch}{1.3}
\begin{tabular}{|>{\qquad}r<{:}l|}\hline
\multicolumn{2}{|l|}{[Tensor_2_Vector_Vector]:
$\mathcal{L}_I=gT^{\mu\nu}
(V_{1,\mu}V_{2,\nu} + V_{1,\nu}V_{2,\mu})$}\\\hline
[F23] & $T^{\mu\nu}(k_2+k_3)\leftarrow\ii\cdot g
(V_{1,\mu}(k_2) V_{2,\nu}(k_3) + V_{1,\nu}(k_2) V_{2,\mu}(k_3))$ \\\hline
[F32] & $T^{\mu\nu}(k_2+k_3)\leftarrow\ii\cdot g
(V_{2,\nu}(k_3) V_{1,\mu}(k_2) + V_{2,\mu}(k_3) V_{1,\nu}(k_2))$ \\\hline
[F12] & $V_2^\mu(k_1+k_2)\leftarrow\ii\cdot g
(T^{\mu\nu}(k_1) + T^{\nu\mu}(k_1)) V_{1,\nu}(k_2)$ \\\hline
[F21] & $V_2^\mu(k_1+k_2)\leftarrow\ii\cdot g
V_{1,\nu}(k_2)(T^{\mu\nu}(k_1) + T^{\nu\mu}(k_1))$ \\\hline
[F13] & $V_1^\mu(k_1+k_3)\leftarrow\ii\cdot g
(T^{\mu\nu}(k_1) + T^{\nu\mu}(k_1)) V_{2,\nu}(k_3)$ \\\hline
[F31] & $V_1^\mu(k_1+k_3)\leftarrow\ii\cdot g
V_{2,\nu}(k_3) (T^{\mu\nu}(k_1) + T^{\nu\mu}(k_1))$ \\\hline
\end{tabular}
\end{center}
\caption{\label{tab:tensor2-vector-vector}
\ldots}
\end{table}
\begin{table}
\begin{center}
\renewcommand{\arraystretch}{1.3}
\begin{tabular}{|>{\qquad}r<{:}l|}\hline
\multicolumn{2}{|l|}{[Dim5_Tensor_2_Vector_Vector_1]:
$\mathcal{L}_I=gT^{\alpha\beta}
(V_1^\mu
\ii\overleftrightarrow\partial_\alpha
\ii\overleftrightarrow\partial_\beta V_{2,\mu})$}\\\hline
[F23] & $T^{\alpha\beta}(k_2+k_3)\leftarrow\ii\cdot g
(k_2^\alpha-k_3^\alpha)(k_2^\beta-k_3^\beta)
V_1^\mu(k_2)V_{2,\mu}(k_3)$ \\\hline
[F32] & $T^{\alpha\beta}(k_2+k_3)\leftarrow\ii\cdot g
(k_2^\alpha-k_3^\alpha)(k_2^\beta-k_3^\beta)
V_{2,\mu}(k_3)V_1^\mu(k_2)$ \\\hline
[F12] & $V_2^\mu(k_1+k_2)\leftarrow\ii\cdot g
(k_1^\alpha+2k_2^\alpha) (k_1^\beta+2k_2^\beta)
T_{\alpha\beta}(k_1) V_1^\mu(k_2)$ \\\hline
[F21] & $V_2^\mu(k_1+k_2)\leftarrow\ii\cdot g
(k_1^\alpha+2k_2^\alpha) (k_1^\beta+2k_2^\beta)
V_1^\mu(k_2) T_{\alpha\beta}(k_1)$ \\\hline
[F13] & $V_1^\mu(k_1+k_3)\leftarrow\ii\cdot g
(k_1^\alpha+2k_3^\alpha) (k_1^\beta+2k_3^\beta)
T_{\alpha\beta}(k_1) V_2^\mu(k_3)$ \\\hline
[F31] & $V_1^\mu(k_1+k_3)\leftarrow\ii\cdot g
(k_1^\alpha+2k_3^\alpha) (k_1^\beta+2k_3^\beta)
V_2^\mu(k_3) T_{\alpha\beta}(k_1)$ \\\hline
\end{tabular}
\end{center}
\caption{\label{tab:dim5-tensor2-vector-vector-1}
\ldots}
\end{table}
\begin{table}
\begin{center}
\renewcommand{\arraystretch}{1.3}
\begin{tabular}{|>{\qquad}r<{:}l|}\hline
\multicolumn{2}{|l|}{[Dim5_Tensor_2_Vector_Vector_2]:
$\mathcal{L}_I=gT^{\alpha\beta}
( V_1^\mu \ii\overleftrightarrow\partial_\beta (\ii\partial_\mu V_{2,\alpha})
+ V_1^\mu \ii\overleftrightarrow\partial_\alpha (\ii\partial_\mu V_{2,\beta}))
$}\\\hline
[F23] & $T^{\alpha\beta}(k_2+k_3)\leftarrow\ii\cdot g
(k_3^\beta-k_2^\beta) k_3^\mu V_{1,\mu}(k_2)V_2^\alpha(k_3)
+ (\alpha\leftrightarrow\beta)$ \\\hline
[F32] & $T^{\alpha\beta}(k_2+k_3)\leftarrow\ii\cdot g
(k_3^\beta-k_2^\beta) V_2^\alpha(k_3) k_3^\mu V_{1,\mu}(k_2)
+ (\alpha\leftrightarrow\beta)$ \\\hline
[F12] & $V_2^\alpha(k_1+k_2)\leftarrow\ii\cdot g
(k_1^\beta+2k_2^\beta)
(T^{\alpha\beta}(k_1)+T^{\beta\alpha}(k_1))
(k_1^\mu+k_2^\mu) V_{1,\mu}(k_2)$ \\\hline
[F21] & $V_2^\alpha(k_1+k_2)\leftarrow\ii\cdot g
(k_1^\mu+k_2^\mu) V_{1,\mu}(k_2)
(k_1^\beta+2k_2^\beta)
(T^{\alpha\beta}(k_1)+T^{\beta\alpha}(k_1))$ \\\hline
[F13] & $V_1^\alpha(k_1+k_3)\leftarrow\ii\cdot g
(k_1^\beta+2k_3^\beta)
(T^{\alpha\beta}(k_1)+T^{\beta\alpha}(k_1))
(k_1^\mu+k_3^\mu) V_{2,\mu}(k_3)$ \\\hline
[F31] & $V_1^\alpha(k_1+k_3)\leftarrow\ii\cdot g
(k_1^\mu+k_3^\mu) V_{2,\mu}(k_3)
(k_1^\beta+2k_3^\beta)
(T^{\alpha\beta}(k_1)+T^{\beta\alpha}(k_1))$ \\\hline
\end{tabular}
\end{center}
\caption{\label{tab:dim5-tensor2-vector-vector-1'}
\ldots}
\end{table}
\begin{table}
\begin{center}
\renewcommand{\arraystretch}{1.3}
\begin{tabular}{|>{\qquad}r<{:}l|}\hline
\multicolumn{2}{|l|}{[Dim7_Tensor_2_Vector_Vector_T]:
$\mathcal{L}_I=gT^{\alpha\beta}
((\ii\partial^\mu V_1^\nu)
\ii\overleftrightarrow\partial_\alpha
\ii\overleftrightarrow\partial_\beta
(\ii\partial_\nu V_{2,\mu}))$}\\\hline
[F23] & $T^{\alpha\beta}(k_2+k_3)\leftarrow\ii\cdot g
(k_2^\alpha-k_3^\alpha)(k_2^\beta-k_3^\beta)
k_3^\mu V_{1,\mu}(k_2) k_2^\nu V_{2,\nu}(k_3)$ \\\hline
[F32] & $T^{\alpha\beta}(k_2+k_3)\leftarrow\ii\cdot g
(k_2^\alpha-k_3^\alpha)(k_2^\beta-k_3^\beta)
k_2^\nu V_{2,\nu}(k_3) k_3^\mu V_{1,\mu}(k_2)$ \\\hline
[F12] & $V_2^\mu(k_1+k_2)\leftarrow\ii\cdot g
k_2^\mu
(k_1^\alpha+2k_2^\alpha) (k_1^\beta+2k_2^\beta)
T_{\alpha\beta}(k_1) (-k_1^\nu-k_2^\nu)V_{1,\nu}(k_2)$ \\\hline
[F21] & $V_2^\mu(k_1+k_2)\leftarrow\ii\cdot g
k_2^\mu (-k_1^\nu-k_2^\nu)V_{1,\nu}(k_2)
(k_1^\alpha+2k_2^\alpha) (k_1^\beta+2k_2^\beta)
T_{\alpha\beta}(k_1)$ \\\hline
[F13] & $V_1^\mu(k_1+k_3)\leftarrow\ii\cdot g
k_3^\mu
(k_1^\alpha+2k_3^\alpha) (k_1^\beta+2k_3^\beta)
T_{\alpha\beta}(k_1) (-k_1^\nu-k_3^\nu) V_{2,\nu}(k_3)$ \\\hline
[F31] & $V_1^\mu(k_1+k_3)\leftarrow\ii\cdot g
k_3^\mu (-k_1^\nu-k_3^\nu) V_{2,\nu}(k_3)
(k_1^\alpha+2k_3^\alpha) (k_1^\beta+2k_3^\beta)
T_{\alpha\beta}(k_1)$ \\\hline
\end{tabular}
\end{center}
\caption{\label{tab:dim7-tensor2-vector-vector-T}
\ldots}
\end{table} *)
(*i
* Local Variables:
* mode:caml
* indent-tabs-mode:nil
* page-delimiter:"^(\\* .*\n"
* End:
i*)
Index: trunk/src/omega/src/targets.ml
===================================================================
--- trunk/src/omega/src/targets.ml (revision 4047)
+++ trunk/src/omega/src/targets.ml (revision 4048)
@@ -1,4095 +1,4095 @@
(* $Id$
Copyright (C) 1999-2013 by
Wolfgang Kilian <kilian@physik.uni-siegen.de>
Thorsten Ohl <ohl@physik.uni-wuerzburg.de>
Juergen Reuter <juergen.reuter@desy.de>
Christian Speckner <cnspeckn@googlemail.com>
Fabian Bach <fabian.bach@cern.ch> (only parts of this file)
WHIZARD is free software; you can redistribute it and/or modify it
under the terms of the GNU General Public License as published by
the Free Software Foundation; either version 2, or (at your option)
any later version.
WHIZARD is distributed in the hope that it will be useful, but
WITHOUT ANY WARRANTY; without even the implied warranty of
MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
GNU General Public License for more details.
You should have received a copy of the GNU General Public License
along with this program; if not, write to the Free Software
Foundation, Inc., 675 Mass Ave, Cambridge, MA 02139, USA. *)
let rcs_file = RCS.parse "Targets" ["Code Generation"]
{ RCS.revision = "$Revision$";
RCS.date = "$Date$";
RCS.author = "$Author$";
RCS.source
= "$URL$" }
module Dummy (F : Fusion.Maker) (P : Momentum.T) (M : Model.T) =
struct
let rcs_list = []
type amplitudes = Fusion.Multi(F)(P)(M).amplitudes
type diagnostic = All | Arguments | Momenta | Gauge
let options = Options.empty
let amplitudes_to_channel cmdline oc amplitudes = failwith "Targets.Dummy"
let parameters_to_channel oc = failwith "Targets.Dummy"
end
(* \thocwmodulesection{\texttt{Fortran\,90/95}} *)
(* \thocwmodulesubsection{Dirac Fermions}
We factor out the code for fermions so that we can use the simpler
implementation for Dirac fermions if the model contains no Majorana
fermions. *)
module type Fermions =
sig
open Coupling
val psi_type : string
val psibar_type : string
val chi_type : string
val grav_type : string
val psi_incoming : string
val brs_psi_incoming : string
val psibar_incoming : string
val brs_psibar_incoming : string
val chi_incoming : string
val brs_chi_incoming : string
val grav_incoming : string
val psi_outgoing : string
val brs_psi_outgoing : string
val psibar_outgoing : string
val brs_psibar_outgoing : string
val chi_outgoing : string
val brs_chi_outgoing : string
val grav_outgoing : string
val psi_propagator : string
val psibar_propagator : string
val chi_propagator : string
val grav_propagator : string
val psi_projector : string
val psibar_projector : string
val chi_projector : string
val grav_projector : string
val psi_gauss : string
val psibar_gauss : string
val chi_gauss : string
val grav_gauss : string
val print_current : int * fermionbar * boson * fermion ->
string -> string -> string -> fuse2 -> unit
val print_current_mom : int * fermionbar * boson * fermion ->
string -> string -> string -> string -> string -> string
-> fuse2 -> unit
val print_current_p : int * fermion * boson * fermion ->
string -> string -> string -> fuse2 -> unit
val print_current_b : int * fermionbar * boson * fermionbar ->
string -> string -> string -> fuse2 -> unit
val print_current_g : int * fermionbar * boson * fermion ->
string -> string -> string -> string -> string -> string
-> fuse2 -> unit
val print_current_g4 : int * fermionbar * boson2 * fermion ->
string -> string -> string -> string -> fuse3 -> unit
val reverse_braket : lorentz -> bool
val use_module : string
val require_library : string list
val rcs : RCS.t
end
module Fortran_Fermions : Fermions =
struct
let rcs = RCS.rename rcs_file "Targets.Fortran_Fermions()"
[ "generates Fortran95 code for Dirac fermions";
"using revision 2000_10_A of module omega95" ]
open Coupling
open Format
let psi_type = "spinor"
let psibar_type = "conjspinor"
let chi_type = "???"
let grav_type = "???"
let psi_incoming = "u"
let brs_psi_incoming = "brs_u"
let psibar_incoming = "vbar"
let brs_psibar_incoming = "brs_vbar"
let chi_incoming = "???"
let brs_chi_incoming = "???"
let grav_incoming = "???"
let psi_outgoing = "v"
let brs_psi_outgoing = "brs_v"
let psibar_outgoing = "ubar"
let brs_psibar_outgoing = "brs_ubar"
let chi_outgoing = "???"
let brs_chi_outgoing = "???"
let grav_outgoing = "???"
let psi_propagator = "pr_psi"
let psibar_propagator = "pr_psibar"
let chi_propagator = "???"
let grav_propagator = "???"
let psi_projector = "pj_psi"
let psibar_projector = "pj_psibar"
let chi_projector = "???"
let grav_projector = "???"
let psi_gauss = "pg_psi"
let psibar_gauss = "pg_psibar"
let chi_gauss = "???"
let grav_gauss = "???"
let format_coupling coeff c =
match coeff with
| 1 -> c
| -1 -> "(-" ^ c ^")"
| coeff -> string_of_int coeff ^ "*" ^ c
let format_coupling_2 coeff c =
match coeff with
| 1 -> c
| -1 -> "-" ^ c
| coeff -> string_of_int coeff ^ "*" ^ c
(* \begin{dubious}
JR's coupling constant HACK, necessitated by tho's bad design descition.
\end{dubious} *)
let fastener s i ?p () =
try
let offset = (String.index s '(') in
if ((String.get s (String.length s - 1)) != ')') then
failwith "fastener: wrong usage of parentheses"
else
let func_name = (String.sub s 0 offset) and
tail =
(String.sub s (succ offset) (String.length s - offset - 2)) in
if (String.contains func_name ')') or
(String.contains tail '(') or
(String.contains tail ')') then
failwith "fastener: wrong usage of parentheses"
else
func_name ^ "(" ^ string_of_int i ^ "," ^ tail ^ ")"
with
| Not_found ->
if (String.contains s ')') then
failwith "fastener: wrong usage of parentheses"
else
match p with
| None -> s ^ "(" ^ string_of_int i ^ ")"
| Some p -> s ^ "(" ^ p ^ "*" ^ p ^ "," ^ string_of_int i ^ ")"
let print_fermion_current coeff f c wf1 wf2 fusion =
let c = format_coupling coeff c in
match fusion with
| F13 -> printf "%s_ff(%s,%s,%s)" f c wf1 wf2
| F31 -> printf "%s_ff(%s,%s,%s)" f c wf2 wf1
| F23 -> printf "f_%sf(%s,%s,%s)" f c wf1 wf2
| F32 -> printf "f_%sf(%s,%s,%s)" f c wf2 wf1
| F12 -> printf "f_f%s(%s,%s,%s)" f c wf1 wf2
| F21 -> printf "f_f%s(%s,%s,%s)" f c wf2 wf1
(* \begin{dubious}
Using a two element array for the combined vector-axial and scalar-pseudo
couplings helps to support HELAS as well. Since we will probably never
support general boson couplings with HELAS, it might be retired in favor
of two separate variables. For this [Model.constant_symbol] has to be
generalized.
\end{dubious} *)
(* \begin{dubious}
NB: passing the array instead of two separate constants would be a
\emph{bad} idea, because the support for Majorana spinors below will
have to flip signs!
\end{dubious} *)
let print_fermion_current2 coeff f c wf1 wf2 fusion =
let c = format_coupling_2 coeff c in
let c1 = fastener c 1 ()
and c2 = fastener c 2 () in
match fusion with
| F13 -> printf "%s_ff(%s,%s,%s,%s)" f c1 c2 wf1 wf2
| F31 -> printf "%s_ff(%s,%s,%s,%s)" f c1 c2 wf2 wf1
| F23 -> printf "f_%sf(%s,%s,%s,%s)" f c1 c2 wf1 wf2
| F32 -> printf "f_%sf(%s,%s,%s,%s)" f c1 c2 wf2 wf1
| F12 -> printf "f_f%s(%s,%s,%s,%s)" f c1 c2 wf1 wf2
| F21 -> printf "f_f%s(%s,%s,%s,%s)" f c1 c2 wf2 wf1
let print_fermion_current_mom1 coeff f c wf1 wf2 p1 p2 p12 fusion =
let c = format_coupling coeff c in
let c1 = fastener c 1 and
c2 = fastener c 2 in
match fusion with
| F13 -> printf "%s_ff(%s,%s,%s,%s)" f (c1 ~p:p12 ()) (c2 ~p:p12 ()) wf1 wf2
| F31 -> printf "%s_ff(%s,%s,%s,%s)" f (c1 ~p:p12 ()) (c2 ~p:p12 ()) wf2 wf1
| F23 -> printf "f_%sf(%s,%s,%s,%s)" f (c1 ~p:p1 ()) (c2 ~p:p1 ()) wf1 wf2
| F32 -> printf "f_%sf(%s,%s,%s,%s)" f (c1 ~p:p2 ()) (c2 ~p:p2 ()) wf2 wf1
| F12 -> printf "f_f%s(%s,%s,%s,%s)" f (c1 ~p:p2 ()) (c2 ~p:p2 ()) wf1 wf2
| F21 -> printf "f_f%s(%s,%s,%s,%s)" f (c1 ~p:p1 ()) (c2 ~p:p1 ()) wf2 wf1
let print_fermion_current_mom2 coeff f c wf1 wf2 p1 p2 p12 fusion =
let c = format_coupling coeff c in
let c1 = fastener c 1 and
c2 = fastener c 2 in
match fusion with
| F13 -> printf "%s_ff(%s,%s,@,%s,%s,%s)" f (c1 ~p:p12 ()) (c2 ~p:p12 ()) wf1 wf2 p12
| F31 -> printf "%s_ff(%s,%s,@,%s,%s,%s)" f (c1 ~p:p12 ()) (c2 ~p:p12 ()) wf2 wf1 p12
| F23 -> printf "f_%sf(%s,%s,@,%s,%s,%s)" f (c1 ~p:p1 ()) (c2 ~p:p1 ()) wf1 wf2 p1
| F32 -> printf "f_%sf(%s,%s,@,%s,%s,%s)" f (c1 ~p:p2 ()) (c2 ~p:p2 ()) wf2 wf1 p2
| F12 -> printf "f_f%s(%s,%s,@,%s,%s,%s)" f (c1 ~p:p2 ()) (c2 ~p:p2 ()) wf1 wf2 p2
| F21 -> printf "f_f%s(%s,%s,@,%s,%s,%s)" f (c1 ~p:p1 ()) (c2 ~p:p1 ()) wf2 wf1 p1
let print_current = function
| coeff, Psibar, VA, Psi -> print_fermion_current2 coeff "va"
| coeff, Psibar, VA2, Psi -> print_fermion_current coeff "va2"
| coeff, Psibar, V, Psi -> print_fermion_current coeff "v"
| coeff, Psibar, A, Psi -> print_fermion_current coeff "a"
| coeff, Psibar, VL, Psi -> print_fermion_current coeff "vl"
| coeff, Psibar, VR, Psi -> print_fermion_current coeff "vr"
| coeff, Psibar, VLR, Psi -> print_fermion_current2 coeff "vlr"
| coeff, Psibar, SP, Psi -> print_fermion_current2 coeff "sp"
| coeff, Psibar, S, Psi -> print_fermion_current coeff "s"
| coeff, Psibar, P, Psi -> print_fermion_current coeff "p"
| coeff, Psibar, SL, Psi -> print_fermion_current coeff "sl"
| coeff, Psibar, SR, Psi -> print_fermion_current coeff "sr"
| coeff, Psibar, SLR, Psi -> print_fermion_current2 coeff "slr"
| coeff, Psibar, _, Psi -> invalid_arg
"Targets.Fortran_Fermions: no superpotential here"
| _, Chibar, _, _ | _, _, _, Chi -> invalid_arg
"Targets.Fortran_Fermions: Majorana spinors not handled"
| _, Gravbar, _, _ | _, _, _, Grav -> invalid_arg
"Targets.Fortran_Fermions: Gravitinos not handled"
let print_current_mom = function
| coeff, Psibar, VLRM, Psi -> print_fermion_current_mom1 coeff "vlr"
| coeff, Psibar, SPM, Psi -> print_fermion_current_mom1 coeff "sp"
| coeff, Psibar, TVA, Psi -> print_fermion_current_mom1 coeff "tva"
| coeff, Psibar, TVAM, Psi -> print_fermion_current_mom2 coeff "tvam"
| coeff, Psibar, TLR, Psi -> print_fermion_current_mom1 coeff "tlr"
| coeff, Psibar, TLRM, Psi -> print_fermion_current_mom2 coeff "tlrm"
| coeff, Psibar, TRL, Psi -> print_fermion_current_mom1 coeff "trl"
| coeff, Psibar, TRLM, Psi -> print_fermion_current_mom2 coeff "trlm"
| coeff, Psibar, _, Psi -> invalid_arg
"Targets.Fortran_Fermions: only sigma tensor coupling here"
| _, Chibar, _, _ | _, _, _, Chi -> invalid_arg
"Targets.Fortran_Fermions: Majorana spinors not handled"
| _, Gravbar, _, _ | _, _, _, Grav -> invalid_arg
"Targets.Fortran_Fermions: Gravitinos not handled"
let print_current_p = function
| _, _, _, _ -> invalid_arg
"Targets.Fortran_Fermions: No clashing arrows here"
let print_current_b = function
| _, _, _, _ -> invalid_arg
"Targets.Fortran_Fermions: No clashing arrows here"
let print_current_g = function
| _, _, _, _ -> invalid_arg
"Targets.Fortran_Fermions: No gravitinos here"
let print_current_g4 = function
| _, _, _, _ -> invalid_arg
"Targets.Fortran_Fermions: No gravitinos here"
let reverse_braket= function
| Spinor -> true
| _ -> false
let use_module = "omega95"
let require_library =
["omega_spinors_2010_01_A"; "omega_spinor_cpls_2010_01_A"]
end
(* \thocwmodulesubsection{Main Functor} *)
module Make_Fortran (Fermions : Fermions)
(Fusion_Maker : Fusion.Maker) (P : Momentum.T) (M : Model.T) =
struct
let rcs_list =
[ RCS.rename rcs_file "Targets.Make_Fortran()"
[ "Interface for Whizard 2.X";
"NB: non-gauge vector couplings are not available yet" ];
Fermions.rcs ]
let require_library =
Fermions.require_library @
[ "omega_vectors_2010_01_A"; "omega_polarizations_2010_01_A";
"omega_couplings_2010_01_A"; "omega_color_2010_01_A";
"omega_utils_2010_01_A" ]
module CM = Colorize.It(M)
module F = Fusion_Maker(P)(M)
type amplitude = F.amplitude
module CF = Fusion.Multi(Fusion_Maker)(P)(M)
type amplitudes = CF.amplitudes
open Coupling
open Format
type output_mode =
| Single_Function
| Single_Module of int
| Single_File of int
| Multi_File of int
let line_length = ref 80
let continuation_lines = ref (-1) (* 255 *)
let kind = ref "default"
let fortran95 = ref true
let module_name = ref "omega_amplitude"
let output_mode = ref (Single_Module 10)
let use_modules = ref []
let whizard = ref false
let parameter_module = ref ""
let md5sum = ref None
let no_write = ref false
let km_write = ref false
let km_pure = ref false
let openmp = ref false
let options = Options.create
[ "90", Arg.Clear fortran95,
"don't use Fortran95 features that are not in Fortran90";
"kind", Arg.String (fun s -> kind := s),
"real and complex kind (default: " ^ !kind ^ ")";
"width", Arg.Int (fun w -> line_length := w), "maximum line length";
"continuation", Arg.Int (fun l -> continuation_lines := l),
"maximum # of continuation lines";
"module", Arg.String (fun s -> module_name := s), "module name";
"single_function", Arg.Unit (fun () -> output_mode := Single_Function),
"compute the matrix element(s) in a monolithis function";
"split_function", Arg.Int (fun n -> output_mode := Single_Module n),
"split the matrix element(s) into small functions [default, size = 10]";
"split_module", Arg.Int (fun n -> output_mode := Single_File n),
"split the matrix element(s) into small modules";
"split_file", Arg.Int (fun n -> output_mode := Multi_File n),
"split the matrix element(s) into small files";
"use", Arg.String (fun s -> use_modules := s :: !use_modules),
"use module";
"parameter_module", Arg.String (fun s -> parameter_module := s),
"parameter_module";
"md5sum", Arg.String (fun s -> md5sum := Some s),
"transfer MD5 checksum";
"whizard", Arg.Set whizard, "include WHIZARD interface";
"no_write", Arg.Set no_write, "no 'write' statements";
"kmatrix_write", Arg.Set km_write, "write K matrix functions";
"kmatrix_write_pure", Arg.Set km_pure, "write K matrix pure functions";
"openmp", Arg.Set openmp, "activate OpenMP support in generated code"]
(* Fortran style line continuation: *)
(* Default function to output spaces (copied from \texttt{format.ml}). *)
let blank_line = String.make 80 ' '
let rec display_blanks oc n =
if n > 0 then
if n <= 80 then
output oc blank_line 0 n
else begin
output oc blank_line 0 80;
display_blanks oc (n - 80)
end
(* Default function to output new lines (copied from \texttt{format.ml}). *)
let display_newline oc () =
output oc "\n" 0 1
(* [current_continuation_line]
\begin{itemize}
\item $\le0$: not continuing: print a straight newline,
\item $>0$: continuing: append [" &"] until we run up to [!continuation_lines].
NB: [!continuation_lines < 0] means \emph{unlimited} continuation lines.
\end{itemize} *)
let current_continuation_line = ref 1
exception Continuation_Lines of int
let fortran_newline oc () =
if !current_continuation_line > 0 then begin
if !continuation_lines >= 0 && !current_continuation_line > !continuation_lines then
raise (Continuation_Lines !current_continuation_line)
else begin
output oc " &" 0 2;
incr current_continuation_line
end
end;
display_newline oc ()
let nl () =
current_continuation_line := 0;
print_newline ();
current_continuation_line := 1
(* Make a formatter with default functions to output spaces and new lines. *)
let setup_fortran_formatter width oc =
set_all_formatter_output_functions
~out:(output oc)
~flush:(fun () -> flush oc)
~newline:(fortran_newline oc)
~spaces:(display_blanks oc);
set_margin (width - 2)
let print_list = function
| [] -> ()
| a :: rest ->
print_string a;
List.iter (fun s -> printf ",@ %s" s) rest
(* \thocwmodulesubsection{Variables and Declarations} *)
(* ["NC"] is already used up in the module ["constants"]: *)
let nc_parameter = "N_"
let omega_color_factor_abbrev = "OCF"
let openmp_tld_type = "thread_local_data"
let openmp_tld = "tld"
let flavors_symbol ?(decl = false) flavors =
(if !openmp & not decl then openmp_tld ^ "%" else "" ) ^
"oks_" ^ String.concat "" (List.map CM.flavor_symbol flavors)
let p2s p =
if p >= 0 && p <= 9 then
string_of_int p
else if p <= 36 then
String.make 1 (Char.chr (Char.code 'A' + p - 10))
else
"_"
let format_momentum p =
"p" ^ String.concat "" (List.map p2s p)
let format_p wf =
String.concat "" (List.map p2s (F.momentum_list wf))
let ext_momentum wf =
match F.momentum_list wf with
| [n] -> n
| _ -> invalid_arg "Targets.Fortran.ext_momentum"
module PSet = Set.Make (struct type t = int list let compare = compare end)
module WFSet = Set.Make (struct type t = F.wf let compare = compare end)
let add_tag wf name =
match F.wf_tag wf with
| None -> name
| Some tag -> name ^ "_" ^ tag
let variable ?(decl = false) wf =
(if !openmp & not decl then openmp_tld ^ "%" else "")
^ add_tag wf ("owf_" ^ CM.flavor_symbol (F.flavor wf) ^ "_" ^ format_p wf)
let momentum wf = "p" ^ format_p wf
let spin wf = "s(" ^ string_of_int (ext_momentum wf) ^ ")"
let format_multiple_variable ?(decl = false) wf i =
variable ~decl:decl wf ^ "_X" ^ string_of_int i
let multiple_variable ?(decl = false) amplitude dictionary wf =
try
format_multiple_variable ~decl:decl wf (dictionary amplitude wf)
with
| Not_found -> variable wf
let multiple_variables ?(decl = false) multiplicity wf =
try
List.map
(format_multiple_variable ~decl:decl wf)
(ThoList.range 1 (multiplicity wf))
with
| Not_found -> [variable ~decl:decl wf]
let declaration_chunk_size = 64
let declare_list_chunk multiplicity t = function
| [] -> ()
| wfs ->
printf " @[<2>%s :: " t;
print_list (ThoList.flatmap (multiple_variables ~decl:true multiplicity) wfs); nl ()
let declare_list multiplicity t = function
| [] -> ()
| wfs ->
List.iter
(declare_list_chunk multiplicity t)
(ThoList.chopn declaration_chunk_size wfs)
type declarations =
{ scalars : F.wf list;
spinors : F.wf list;
conjspinors : F.wf list;
realspinors : F.wf list;
ghostspinors : F.wf list;
vectorspinors : F.wf list;
vectors : F.wf list;
ward_vectors : F.wf list;
massive_vectors : F.wf list;
tensors_1 : F.wf list;
tensors_2 : F.wf list;
brs_scalars : F.wf list;
brs_spinors : F.wf list;
brs_conjspinors : F.wf list;
brs_realspinors : F.wf list;
brs_vectorspinors : F.wf list;
brs_vectors : F.wf list;
brs_massive_vectors : F.wf list }
let rec classify_wfs' acc = function
| [] -> acc
| wf :: rest ->
classify_wfs'
(match CM.lorentz (F.flavor wf) with
| Scalar -> {acc with scalars = wf :: acc.scalars}
| Spinor -> {acc with spinors = wf :: acc.spinors}
| ConjSpinor -> {acc with conjspinors = wf :: acc.conjspinors}
| Majorana -> {acc with realspinors = wf :: acc.realspinors}
| Maj_Ghost -> {acc with ghostspinors = wf :: acc.ghostspinors}
| Vectorspinor ->
{acc with vectorspinors = wf :: acc.vectorspinors}
| Vector -> {acc with vectors = wf :: acc.vectors}
(*i | Ward_Vector -> {acc with ward_vectors = wf :: acc.ward_vectors}
i*)
| Massive_Vector ->
{acc with massive_vectors = wf :: acc.massive_vectors}
| Tensor_1 -> {acc with tensors_1 = wf :: acc.tensors_1}
| Tensor_2 -> {acc with tensors_2 = wf :: acc.tensors_2}
| BRS Scalar -> {acc with brs_scalars = wf :: acc.brs_scalars}
| BRS Spinor -> {acc with brs_spinors = wf :: acc.brs_spinors}
| BRS ConjSpinor -> {acc with brs_conjspinors =
wf :: acc.brs_conjspinors}
| BRS Majorana -> {acc with brs_realspinors =
wf :: acc.brs_realspinors}
| BRS Vectorspinor -> {acc with brs_vectorspinors =
wf :: acc.brs_vectorspinors}
| BRS Vector -> {acc with brs_vectors = wf :: acc.brs_vectors}
| BRS Massive_Vector -> {acc with brs_massive_vectors =
wf :: acc.brs_massive_vectors}
| BRS _ -> invalid_arg "Targets.wfs_classify': not needed here")
rest
let classify_wfs wfs = classify_wfs'
{ scalars = []; spinors = []; conjspinors = []; realspinors = [];
ghostspinors = []; vectorspinors = []; vectors = [];
ward_vectors = [];
massive_vectors = []; tensors_1 = []; tensors_2 = [];
brs_scalars = [] ; brs_spinors = []; brs_conjspinors = [];
brs_realspinors = []; brs_vectorspinors = [];
brs_vectors = []; brs_massive_vectors = []}
wfs
(* \thocwmodulesubsection{Parameters} *)
type 'a parameters =
{ real_singles : 'a list;
real_arrays : ('a * int) list;
complex_singles : 'a list;
complex_arrays : ('a * int) list }
let rec classify_singles acc = function
| [] -> acc
| Real p :: rest -> classify_singles
{ acc with real_singles = p :: acc.real_singles } rest
| Complex p :: rest -> classify_singles
{ acc with complex_singles = p :: acc.complex_singles } rest
let rec classify_arrays acc = function
| [] -> acc
| (Real_Array p, rhs) :: rest -> classify_arrays
{ acc with real_arrays =
(p, List.length rhs) :: acc.real_arrays } rest
| (Complex_Array p, rhs) :: rest -> classify_arrays
{ acc with complex_arrays =
(p, List.length rhs) :: acc.complex_arrays } rest
let classify_parameters params =
classify_arrays
(classify_singles
{ real_singles = [];
real_arrays = [];
complex_singles = [];
complex_arrays = [] }
(List.map fst params.derived)) params.derived_arrays
(* \begin{dubious}
Unify this with the other code using [ThoList.chopn].
\end{dubious} *)
let rec schisma n l =
if List.length l <= n then
[l]
else
let a, b = ThoList.splitn n l in
[a] @ (schisma n b)
let rec schisma_num i n l =
if List.length l <= n then
[(i,l)]
else
let a, b = ThoList.splitn n l in
[(i,a)] @ (schisma_num (i+1) n b)
let declare_parameters' t = function
| [] -> ()
| plist ->
printf " @[<2>%s(kind=%s), public, save :: " t !kind;
print_list (List.map CM.constant_symbol plist); nl ()
let declare_parameters t plist =
List.iter (declare_parameters' t) plist
let declare_parameter_array t (p, n) =
printf " @[<2>%s(kind=%s), dimension(%d), public, save :: %s"
t !kind n (CM.constant_symbol p); nl ()
let default_parameter (x, v) =
printf "@ %s = %g_%s" (CM.constant_symbol x) v !kind
let declare_default_parameters t = function
| [] -> ()
| p :: plist ->
printf " @[<2>%s(kind=%s), public, save ::" t !kind;
default_parameter p;
List.iter (fun p' -> printf ","; default_parameter p') plist;
nl ()
let rec format_constant = function
| I -> sprintf "cmplx (0.0_%s, 1.0_%s)" !kind !kind
| Const c when c < 0 -> sprintf "(%d.0_%s)" c !kind
| Const c -> sprintf "%d.0_%s" c !kind
| _ -> invalid_arg "format_constant"
let rec eval_parameter' = function
| I -> printf "cmplx (0.0_%s, 1.0_%s)" !kind !kind
| Const c when c < 0 -> printf "(%d.0_%s)" c !kind
| Const c -> printf "%d.0_%s" c !kind
| Atom x -> printf "%s" (CM.constant_symbol x)
| Sum [] -> printf "0.0_%s" !kind
| Sum [x] -> eval_parameter' x
| Sum (x :: xs) ->
printf "@,("; eval_parameter' x;
List.iter (fun x -> printf "@, + "; eval_parameter' x) xs;
printf ")"
| Diff (x, y) ->
printf "@,("; eval_parameter' x;
printf " - "; eval_parameter' y; printf ")"
| Neg x -> printf "@,( - "; eval_parameter' x; printf ")"
| Prod [] -> printf "1.0_%s" !kind
| Prod [x] -> eval_parameter' x
| Prod (x :: xs) ->
printf "@,("; eval_parameter' x;
List.iter (fun x -> printf " * "; eval_parameter' x) xs;
printf ")"
| Quot (x, y) ->
printf "@,("; eval_parameter' x;
printf " / "; eval_parameter' y; printf ")"
| Rec x ->
printf "@, (1.0_%s / " !kind; eval_parameter' x; printf ")"
| Pow (x, n) ->
printf "@,("; eval_parameter' x; printf "**%d" n; printf ")"
| Sqrt x -> printf "@,sqrt ("; eval_parameter' x; printf ")"
| Sin x -> printf "@,sin ("; eval_parameter' x; printf ")"
| Cos x -> printf "@,cos ("; eval_parameter' x; printf ")"
| Tan x -> printf "@,tan ("; eval_parameter' x; printf ")"
| Cot x -> printf "@,cot ("; eval_parameter' x; printf ")"
| Atan2 (y, x) -> printf "@,atan2 ("; eval_parameter' y;
printf ",@ "; eval_parameter' x; printf ")"
| Conj x -> printf "@,conjg ("; eval_parameter' x; printf ")"
let strip_single_tag = function
| Real x -> x
| Complex x -> x
let strip_array_tag = function
| Real_Array x -> x
| Complex_Array x -> x
let eval_parameter (lhs, rhs) =
let x = CM.constant_symbol (strip_single_tag lhs) in
printf " @[<2>%s = " x; eval_parameter' rhs; nl ()
let eval_para_list n l =
printf " subroutine setup_parameters%s ()" (string_of_int n); nl();
List.iter eval_parameter l;
printf " end subroutine setup_parameters%s" (string_of_int n); nl()
let eval_parameter_pair (lhs, rhs) =
let x = CM.constant_symbol (strip_array_tag lhs) in
let _ = List.fold_left (fun i rhs' ->
printf " @[<2>%s(%d) = " x i; eval_parameter' rhs'; nl ();
succ i) 1 rhs in
()
let eval_para_pair_list n l =
printf " subroutine setup_parameters%s ()" (string_of_int n); nl();
List.iter eval_parameter_pair l;
printf " end subroutine setup_parameters%s" (string_of_int n); nl()
let print_echo fmt p =
let s = CM.constant_symbol p in
printf " write (unit = *, fmt = fmt_%s) \"%s\", %s"
fmt s s; nl ()
let print_echo_array fmt (p, n) =
let s = CM.constant_symbol p in
for i = 1 to n do
printf " write (unit = *, fmt = fmt_%s_array) " fmt ;
printf "\"%s\", %d, %s(%d)" s i s i; nl ()
done
let parameters_to_fortran oc params =
setup_fortran_formatter !line_length oc;
let declarations = classify_parameters params in
printf "module %s" !parameter_module; nl ();
printf " use kinds"; nl ();
printf " use constants"; nl ();
printf " implicit none"; nl ();
printf " private"; nl ();
printf " @[<2>public :: setup_parameters";
if !no_write then begin
printf "! No print_parameters"; nl();
end else begin
printf "@,, print_parameters"; nl ();
end;
declare_default_parameters "real" params.input;
declare_parameters "real" (schisma 69 declarations.real_singles);
List.iter (declare_parameter_array "real") declarations.real_arrays;
declare_parameters "complex" (schisma 69 declarations.complex_singles);
List.iter (declare_parameter_array "complex") declarations.complex_arrays;
printf "contains"; nl ();
printf " ! derived parameters:"; nl ();
let shredded = schisma_num 1 120 params.derived in
let shredded_arrays = schisma_num 1 120 params.derived_arrays in
let num_sub = List.length shredded in
let num_sub_arrays = List.length shredded_arrays in
printf " !length: %s" (string_of_int (List.length params.derived));
nl();
printf " !Num_Sub: %s" (string_of_int num_sub); nl();
List.iter (fun (i,l) -> eval_para_list i l) shredded;
List.iter (fun (i,l) -> eval_para_pair_list (num_sub + i) l)
shredded_arrays;
printf " subroutine setup_parameters ()"; nl();
let sum_sub = num_sub + num_sub_arrays in
for i = 1 to sum_sub do
printf " call setup_parameters%s" (string_of_int i); nl();
done;
printf " end subroutine setup_parameters"; nl();
if !no_write then begin
printf "! No print_parameters"; nl();
end else begin
printf " subroutine print_parameters ()"; nl();
printf " @[<2>character(len=*), parameter ::";
printf "@ fmt_real = \"(A12,4X,' = ',E25.18)\",";
printf "@ fmt_complex = \"(A12,4X,' = ',E25.18,' + i*',E25.18)\",";
printf "@ fmt_real_array = \"(A12,'(',I2.2,')',' = ',E25.18)\",";
printf "@ fmt_complex_array = ";
printf "\"(A12,'(',I2.2,')',' = ',E25.18,' + i*',E25.18)\""; nl ();
printf " @[<2>write (unit = *, fmt = \"(A)\") @,";
printf "\"default values for the input parameters:\""; nl ();
List.iter (fun (p, _) -> print_echo "real" p) params.input;
printf " @[<2>write (unit = *, fmt = \"(A)\") @,";
printf "\"derived parameters:\""; nl ();
List.iter (print_echo "real") declarations.real_singles;
List.iter (print_echo "complex") declarations.complex_singles;
List.iter (print_echo_array "real") declarations.real_arrays;
List.iter (print_echo_array "complex") declarations.complex_arrays;
printf " end subroutine print_parameters"; nl();
end;
printf "end module %s" !parameter_module; nl ();
printf "! O'Mega revision control information:"; nl ();
List.iter (fun s -> printf "! %s" s; nl ())
(ThoList.flatmap RCS.summary (M.rcs :: rcs_list));
printf "!!! program test_parameters"; nl();
printf "!!! use %s" !parameter_module; nl();
printf "!!! call setup_parameters ()"; nl();
printf "!!! call print_parameters ()"; nl();
printf "!!! end program test_parameters"; nl()
(* \thocwmodulesubsection{Run-Time Diagnostics} *)
type diagnostic = All | Arguments | Momenta | Gauge
type diagnostic_mode = Off | Warn | Panic
let warn mode =
match !mode with
| Off -> false
| Warn -> true
| Panic -> true
let panic mode =
match !mode with
| Off -> false
| Warn -> false
| Panic -> true
let suffix mode =
if panic mode then
"panic"
else
"warn"
let diagnose_arguments = ref Off
let diagnose_momenta = ref Off
let diagnose_gauge = ref Off
let rec parse_diagnostic = function
| All, panic ->
parse_diagnostic (Arguments, panic);
parse_diagnostic (Momenta, panic);
parse_diagnostic (Gauge, panic)
| Arguments, panic ->
diagnose_arguments := if panic then Panic else Warn
| Momenta, panic ->
diagnose_momenta := if panic then Panic else Warn
| Gauge, panic ->
diagnose_gauge := if panic then Panic else Warn
(* If diagnostics are required, we have to switch off
Fortran95 features like pure functions. *)
let parse_diagnostics = function
| [] -> ()
| diagnostics ->
fortran95 := false;
List.iter parse_diagnostic diagnostics
(* \thocwmodulesubsection{Amplitude} *)
let declare_momenta_chunk = function
| [] -> ()
| momenta ->
printf " @[<2>type(momentum) :: ";
print_list (List.map format_momentum momenta); nl ()
let declare_momenta = function
| [] -> ()
| momenta ->
List.iter
declare_momenta_chunk
(ThoList.chopn declaration_chunk_size momenta)
let declare_wavefunctions multiplicity wfs =
let wfs' = classify_wfs wfs in
declare_list multiplicity ("complex(kind=" ^ !kind ^ ")")
(wfs'.scalars @ wfs'.brs_scalars);
declare_list multiplicity ("type(" ^ Fermions.psi_type ^ ")")
(wfs'.spinors @ wfs'.brs_spinors);
declare_list multiplicity ("type(" ^ Fermions.psibar_type ^ ")")
(wfs'.conjspinors @ wfs'.brs_conjspinors);
declare_list multiplicity ("type(" ^ Fermions.chi_type ^ ")")
(wfs'.realspinors @ wfs'.brs_realspinors @ wfs'.ghostspinors);
declare_list multiplicity ("type(" ^ Fermions.grav_type ^ ")") wfs'.vectorspinors;
declare_list multiplicity "type(vector)" (wfs'.vectors @ wfs'.massive_vectors @
wfs'.brs_vectors @ wfs'.brs_massive_vectors @ wfs'.ward_vectors);
declare_list multiplicity "type(tensor2odd)" wfs'.tensors_1;
declare_list multiplicity "type(tensor)" wfs'.tensors_2
let flavors a = F.incoming a @ F.outgoing a
let declare_brakets_chunk = function
| [] -> ()
| amplitudes ->
printf " @[<2>complex(kind=%s) :: " !kind;
print_list (List.map (fun a -> flavors_symbol ~decl:true (flavors a)) amplitudes); nl ()
let declare_brakets = function
| [] -> ()
| amplitudes ->
List.iter
declare_brakets_chunk
(ThoList.chopn declaration_chunk_size amplitudes)
let print_variable_declarations amplitudes =
let multiplicity = CF.multiplicity amplitudes
and processes = CF.processes amplitudes in
declare_momenta
(PSet.elements
(List.fold_left
(fun set a ->
PSet.union set (List.fold_right
(fun wf -> PSet.add (F.momentum_list wf))
(F.externals a) PSet.empty))
PSet.empty processes));
declare_momenta
(PSet.elements
(List.fold_left
(fun set a ->
PSet.union set (List.fold_right
(fun wf -> PSet.add (F.momentum_list wf))
(F.variables a) PSet.empty))
PSet.empty processes));
if !openmp then begin
printf " type %s@[<2>" openmp_tld_type;
nl ();
end ;
declare_wavefunctions multiplicity
(WFSet.elements
(List.fold_left
(fun set a ->
WFSet.union set (List.fold_right WFSet.add (F.externals a) WFSet.empty))
WFSet.empty processes));
declare_wavefunctions multiplicity
(WFSet.elements
(List.fold_left
(fun set a ->
WFSet.union set (List.fold_right WFSet.add (F.variables a) WFSet.empty))
WFSet.empty processes));
declare_brakets processes;
if !openmp then begin
printf "@] end type %s\n" openmp_tld_type;
printf " type(%s) :: %s" openmp_tld_type openmp_tld;
nl ();
end
(* [print_current] is the most important function that has to match the functions
in \verb+omega95+ (see appendix~\ref{sec:fortran}). It offers plentiful
opportunities for making mistakes, in particular those related to signs.
We start with a few auxiliary functions: *)
let children2 rhs =
match F.children rhs with
| [wf1; wf2] -> (wf1, wf2)
| _ -> failwith "Targets.children2: can't happen"
let children3 rhs =
match F.children rhs with
| [wf1; wf2; wf3] -> (wf1, wf2, wf3)
| _ -> invalid_arg "Targets.children3: can't happen"
(* Note that it is (marginally) faster to multiply the two scalar products
with the coupling constant than the four vector components.
\begin{dubious}
This could be part of \verb+omegalib+ as well \ldots
\end{dubious} *)
let format_coeff = function
| 1 -> ""
| -1 -> "-"
| coeff -> "(" ^ string_of_int coeff ^ ")*"
let format_coupling coeff c =
match coeff with
| 1 -> c
| -1 -> "(-" ^ c ^")"
| coeff -> string_of_int coeff ^ "*" ^ c
(* \begin{dubious}
The following is error prone and should be generated automagically.
\end{dubious} *)
let print_vector4 c wf1 wf2 wf3 fusion (coeff, contraction) =
match contraction, fusion with
| C_12_34, (F341|F431|F342|F432|F123|F213|F124|F214)
| C_13_42, (F241|F421|F243|F423|F132|F312|F134|F314)
| C_14_23, (F231|F321|F234|F324|F142|F412|F143|F413) ->
printf "((%s%s)*(%s*%s))*%s" (format_coeff coeff) c wf1 wf2 wf3
| C_12_34, (F134|F143|F234|F243|F312|F321|F412|F421)
| C_13_42, (F124|F142|F324|F342|F213|F231|F413|F431)
| C_14_23, (F123|F132|F423|F432|F214|F241|F314|F341) ->
printf "((%s%s)*(%s*%s))*%s" (format_coeff coeff) c wf2 wf3 wf1
| C_12_34, (F314|F413|F324|F423|F132|F231|F142|F241)
| C_13_42, (F214|F412|F234|F432|F123|F321|F143|F341)
| C_14_23, (F213|F312|F243|F342|F124|F421|F134|F431) ->
printf "((%s%s)*(%s*%s))*%s" (format_coeff coeff) c wf1 wf3 wf2
let print_add_vector4 c wf1 wf2 wf3 fusion (coeff, contraction) =
printf "@ + ";
print_vector4 c wf1 wf2 wf3 fusion (coeff, contraction)
let print_vector4_km c pa pb wf1 wf2 wf3 fusion (coeff, contraction) =
match contraction, fusion with
| C_12_34, (F341|F431|F342|F432|F123|F213|F124|F214)
| C_13_42, (F241|F421|F243|F423|F132|F312|F134|F314)
| C_14_23, (F231|F321|F234|F324|F142|F412|F143|F413) ->
printf "((%s%s%s+%s))*(%s*%s))*%s"
(format_coeff coeff) c pa pb wf1 wf2 wf3
| C_12_34, (F134|F143|F234|F243|F312|F321|F412|F421)
| C_13_42, (F124|F142|F324|F342|F213|F231|F413|F431)
| C_14_23, (F123|F132|F423|F432|F214|F241|F314|F341) ->
printf "((%s%s%s+%s))*(%s*%s))*%s"
(format_coeff coeff) c pa pb wf2 wf3 wf1
| C_12_34, (F314|F413|F324|F423|F132|F231|F142|F241)
| C_13_42, (F214|F412|F234|F432|F123|F321|F143|F341)
| C_14_23, (F213|F312|F243|F342|F124|F421|F134|F431) ->
printf "((%s%s%s+%s))*(%s*%s))*%s"
(format_coeff coeff) c pa pb wf1 wf3 wf2
let print_add_vector4_km c pa pb wf1 wf2 wf3 fusion (coeff, contraction) =
printf "@ + ";
print_vector4_km c pa pb wf1 wf2 wf3 fusion (coeff, contraction)
let print_dscalar4 c wf1 wf2 wf3 p1 p2 p3 p123
fusion (coeff, contraction) =
match contraction, fusion with
| C_12_34, (F341|F431|F342|F432|F123|F213|F124|F214)
| C_13_42, (F241|F421|F243|F423|F132|F312|F134|F314)
| C_14_23, (F231|F321|F234|F324|F142|F412|F143|F413) ->
printf "((%s%s)*(%s*%s)*(%s*%s)*%s*%s*%s)"
(format_coeff coeff) c p1 p2 p3 p123 wf1 wf2 wf3
| C_12_34, (F134|F143|F234|F243|F312|F321|F412|F421)
| C_13_42, (F124|F142|F324|F342|F213|F231|F413|F431)
| C_14_23, (F123|F132|F423|F432|F214|F241|F314|F341) ->
printf "((%s%s)*(%s*%s)*(%s*%s)*%s*%s*%s)"
(format_coeff coeff) c p2 p3 p1 p123 wf1 wf2 wf3
| C_12_34, (F314|F413|F324|F423|F132|F231|F142|F241)
| C_13_42, (F214|F412|F234|F432|F123|F321|F143|F341)
| C_14_23, (F213|F312|F243|F342|F124|F421|F134|F431) ->
printf "((%s%s)*(%s*%s)*(%s*%s)*%s*%s*%s)"
(format_coeff coeff) c p1 p3 p2 p123 wf1 wf2 wf3
let print_add_dscalar4 c wf1 wf2 wf3 p1 p2 p3 p123
fusion (coeff, contraction) =
printf "@ + ";
print_dscalar4 c wf1 wf2 wf3 p1 p2 p3 p123 fusion (coeff, contraction)
let print_dscalar2_vector2 c wf1 wf2 wf3 p1 p2 p3 p123
fusion (coeff, contraction) =
failwith "Targets.Fortran.print_dscalar2_vector2: incomplete!";
match contraction, fusion with
| C_12_34, (F134|F143|F234|F243) ->
printf "((%s%s)*(%s*%s)*(%s*%s)*%s)"
(format_coeff coeff) c p123 p1 wf2 wf3 wf1
| C_12_34, (F312|F321|F412|F421) ->
printf "((%s%s)*((%s*%s)*%s*%s)*%s)"
(format_coeff coeff) c p2 p3 wf2 wf3 wf1
| C_12_34, (F341|F431|F342|F432|F123|F213|F124|F214)
| C_13_42, (F241|F421|F243|F423|F132|F312|F134|F314)
| C_14_23, (F231|F321|F234|F324|F142|F412|F143|F413) ->
printf "((%s%s)*(%s*%s)*(%s*%s)*%s*%s*%s)"
(format_coeff coeff) c p1 p2 p3 p123 wf1 wf2 wf3
| C_13_42, (F124|F142|F324|F342|F213|F231|F413|F431)
| C_14_23, (F123|F132|F423|F432|F214|F241|F314|F341) ->
printf "((%s%s)*(%s*%s)*(%s*%s)*%s*%s*%s)"
(format_coeff coeff) c p2 p3 p1 p123 wf1 wf2 wf3
| C_12_34, (F314|F413|F324|F423|F132|F231|F142|F241)
| C_13_42, (F214|F412|F234|F432|F123|F321|F143|F341)
| C_14_23, (F213|F312|F243|F342|F124|F421|F134|F431) ->
printf "((%s%s)*(%s*%s)*(%s*%s)*%s*%s*%s)"
(format_coeff coeff) c p1 p3 p2 p123 wf1 wf2 wf3
let print_add_dscalar2_vector2 c wf1 wf2 wf3 p1 p2 p3 p123
fusion (coeff, contraction) =
printf "@ + ";
print_dscalar2_vector2 c wf1 wf2 wf3 p1 p2 p3 p123
fusion (coeff, contraction)
let print_current amplitude dictionary rhs =
match F.coupling rhs with
| V3 (vertex, fusion, constant) ->
let ch1, ch2 = children2 rhs in
let wf1 = multiple_variable amplitude dictionary ch1
and wf2 = multiple_variable amplitude dictionary ch2
and p1 = momentum ch1
and p2 = momentum ch2
and m1 = CM.mass_symbol (F.flavor ch1)
and m2 = CM.mass_symbol (F.flavor ch2) in
let c = CM.constant_symbol constant in
printf "@, %s " (if (F.sign rhs) < 0 then "-" else "+");
begin match vertex with
(* Fermionic currents $\bar\psi\fmslash{A}\psi$ and $\bar\psi\phi\psi$
are handled by the [Fermions] module, since they depend on the
choice of Feynman rules: Dirac or Majorana. *)
| FBF (coeff, fb, b, f) ->
begin match coeff, fb, b, f with
| _, Psibar, VLRM, Psi | _, Psibar, SPM, Psi
| _, Psibar, TVA, Psi | _, Psibar, TVAM, Psi
| _, Psibar, TLR, Psi | _, Psibar, TLRM, Psi
| _, Psibar, TRL, Psi | _, Psibar, TRLM, Psi ->
let p12 = Printf.sprintf "(-%s-%s)" p1 p2 in
Fermions.print_current_mom (coeff, fb, b, f) c wf1 wf2 p1 p2
p12 fusion
| _, _, _, _ ->
Fermions.print_current (coeff, fb, b, f) c wf1 wf2 fusion
end
| PBP (coeff, f1, b, f2) ->
Fermions.print_current_p (coeff, f1, b, f2) c wf1 wf2 fusion
| BBB (coeff, fb1, b, fb2) ->
Fermions.print_current_b (coeff, fb1, b, fb2) c wf1 wf2 fusion
| GBG (coeff, fb, b, f) -> let p12 =
Printf.sprintf "(-%s-%s)" p1 p2 in
Fermions.print_current_g (coeff, fb, b, f) c wf1 wf2 p1 p2
p12 fusion
(* Table~\ref{tab:dim4-bosons} is a bit misleading, since if includes
totally antisymmetric structure constants. The space-time part alone
is also totally antisymmetric: *)
| Gauge_Gauge_Gauge coeff ->
let c = format_coupling coeff c in
begin match fusion with
| (F23|F31|F12) ->
printf "g_gg(%s,%s,%s,%s,%s)" c wf1 p1 wf2 p2
| (F32|F13|F21) ->
printf "g_gg(%s,%s,%s,%s,%s)" c wf2 p2 wf1 p1
end
(* In [Aux_Gauge_Gauge], we can not rely on antisymmetry alone, because of the
different Lorentz representations of the auxialiary and the gauge field.
Instead we have to provide the sign in
\begin{equation}
(V_2 \wedge V_3) \cdot T_1 =
\begin{cases}
V_2 \cdot (T_1 \cdot V_3) = - V_2 \cdot (V_3 \cdot T_1) & \\
V_3 \cdot (V_2 \cdot T_1) = - V_3 \cdot (T_1 \cdot V_2) &
\end{cases}
\end{equation}
ourselves. Alternatively, one could provide \verb+g_xg+ mirroring
\verb+g_gx+. *)
| Aux_Gauge_Gauge coeff ->
let c = format_coupling coeff c in
begin match fusion with
| F23 -> printf "x_gg(%s,%s,%s)" c wf1 wf2
| F32 -> printf "x_gg(%s,%s,%s)" c wf2 wf1
| F12 -> printf "g_gx(%s,%s,%s)" c wf2 wf1
| F21 -> printf "g_gx(%s,%s,%s)" c wf1 wf2
| F13 -> printf "(-1)*g_gx(%s,%s,%s)" c wf2 wf1
| F31 -> printf "(-1)*g_gx(%s,%s,%s)" c wf1 wf2
end
(* These cases are symmetric and we just have to juxtapose the correct fields
and provide parentheses to minimize the number of multiplications. *)
| Scalar_Vector_Vector coeff ->
let c = format_coupling coeff c in
begin match fusion with
| (F23|F32) -> printf "%s*(%s*%s)" c wf1 wf2
| (F12|F13) -> printf "(%s*%s)*%s" c wf1 wf2
| (F21|F31) -> printf "(%s*%s)*%s" c wf2 wf1
end
| Aux_Vector_Vector coeff ->
let c = format_coupling coeff c in
begin match fusion with
| (F23|F32) -> printf "%s*(%s*%s)" c wf1 wf2
| (F12|F13) -> printf "(%s*%s)*%s" c wf1 wf2
| (F21|F31) -> printf "(%s*%s)*%s" c wf2 wf1
end
(* Even simpler: *)
| Scalar_Scalar_Scalar coeff ->
printf "(%s*%s*%s)" (format_coupling coeff c) wf1 wf2
| Aux_Scalar_Scalar coeff ->
printf "(%s*%s*%s)" (format_coupling coeff c) wf1 wf2
| Aux_Scalar_Vector coeff ->
let c = format_coupling coeff c in
begin match fusion with
| (F13|F31) -> printf "%s*(%s*%s)" c wf1 wf2
| (F23|F21) -> printf "(%s*%s)*%s" c wf1 wf2
| (F32|F12) -> printf "(%s*%s)*%s" c wf2 wf1
end
| Vector_Scalar_Scalar coeff ->
let c = format_coupling coeff c in
begin match fusion with
| F23 -> printf "v_ss(%s,%s,%s,%s,%s)" c wf1 p1 wf2 p2
| F32 -> printf "v_ss(%s,%s,%s,%s,%s)" c wf2 p2 wf1 p1
| F12 -> printf "s_vs(%s,%s,%s,%s,%s)" c wf1 p1 wf2 p2
| F21 -> printf "s_vs(%s,%s,%s,%s,%s)" c wf2 p2 wf1 p1
| F13 -> printf "(-1)*s_vs(%s,%s,%s,%s,%s)" c wf1 p1 wf2 p2
| F31 -> printf "(-1)*s_vs(%s,%s,%s,%s,%s)" c wf2 p2 wf1 p1
end
| Graviton_Scalar_Scalar coeff ->
let c = format_coupling coeff c in
begin match fusion with
| F12 -> printf "s_gravs(%s,%s,-(%s+%s),%s,%s,%s)" c m2 p1 p2 p2 wf1 wf2
| F21 -> printf "s_gravs(%s,%s,-(%s+%s),%s,%s,%s)" c m1 p1 p2 p1 wf2 wf1
| F13 -> printf "s_gravs(%s,%s,%s,-(%s+%s),%s,%s)" c m2 p2 p1 p2 wf1 wf2
| F31 -> printf "s_gravs(%s,%s,%s,-(%s+%s),%s,%s)" c m1 p1 p1 p2 wf2 wf1
| F23 -> printf "grav_ss(%s,%s,%s,%s,%s,%s)" c m1 p1 p2 wf1 wf2
| F32 -> printf "grav_ss(%s,%s,%s,%s,%s,%s)" c m1 p2 p1 wf2 wf1
end
(* In producing a vector in the fusion we always contract the rightmost index with the
vector wavefunction from [rhs]. So the first momentum is always the one of the
vector boson produced in the fusion, while the second one is that from the [rhs].
This makes the cases [F12] and [F13] as well as [F21] and [F31] equal. In principle,
we could have already done this for the [Graviton_Scalar_Scalar] case. *)
| Graviton_Vector_Vector coeff ->
let c = format_coupling coeff c in
begin match fusion with
| (F12|F13) -> printf "v_gravv(%s,%s,-(%s+%s),%s,%s,%s)" c m2 p1 p2 p2 wf1 wf2
| (F21|F31) -> printf "v_gravv(%s,%s,-(%s+%s),%s,%s,%s)" c m1 p1 p2 p1 wf2 wf1
| F23 -> printf "grav_vv(%s,%s,%s,%s,%s,%s)" c m1 p1 p2 wf1 wf2
| F32 -> printf "grav_vv(%s,%s,%s,%s,%s,%s)" c m1 p2 p1 wf2 wf1
end
| Graviton_Spinor_Spinor coeff ->
let c = format_coupling coeff c in
begin match fusion with
| F23 -> printf "f_gravf(%s,%s,-(%s+%s),(-%s),%s,%s)" c m2 p1 p2 p2 wf1 wf2
| F32 -> printf "f_gravf(%s,%s,-(%s+%s),(-%s),%s,%s)" c m1 p1 p2 p1 wf2 wf1
| F12 -> printf "f_fgrav(%s,%s,%s,%s+%s,%s,%s)" c m1 p1 p1 p2 wf1 wf2
| F21 -> printf "f_fgrav(%s,%s,%s,%s+%s,%s,%s)" c m2 p2 p1 p2 wf2 wf1
| F13 -> printf "grav_ff(%s,%s,%s,(-%s),%s,%s)" c m1 p1 p2 wf1 wf2
| F31 -> printf "grav_ff(%s,%s,%s,(-%s),%s,%s)" c m1 p2 p1 wf2 wf1
end
| Dim4_Vector_Vector_Vector_T coeff ->
let c = format_coupling coeff c in
begin match fusion with
| F23 -> printf "tkv_vv(%s,%s,%s,%s,%s)" c wf1 p1 wf2 p2
| F32 -> printf "tkv_vv(%s,%s,%s,%s,%s)" c wf2 p2 wf1 p1
| F12 -> printf "tv_kvv(%s,%s,%s,%s,%s)" c wf1 p1 wf2 p2
| F21 -> printf "tv_kvv(%s,%s,%s,%s,%s)" c wf2 p2 wf1 p1
| F13 -> printf "(-1)*tv_kvv(%s,%s,%s,%s,%s)" c wf1 p1 wf2 p2
| F31 -> printf "(-1)*tv_kvv(%s,%s,%s,%s,%s)" c wf2 p2 wf1 p1
end
| Dim4_Vector_Vector_Vector_L coeff ->
let c = format_coupling coeff c in
begin match fusion with
| F23 -> printf "lkv_vv(%s,%s,%s,%s,%s)" c wf1 p1 wf2 p2
| F32 -> printf "lkv_vv(%s,%s,%s,%s,%s)" c wf2 p2 wf1 p1
| F12 | F13 -> printf "lv_kvv(%s,%s,%s,%s)" c wf1 p1 wf2
| F21 | F31 -> printf "lv_kvv(%s,%s,%s,%s)" c wf2 p2 wf1
end
| Dim6_Gauge_Gauge_Gauge coeff ->
let c = format_coupling coeff c in
begin match fusion with
| F23 | F31 | F12 ->
printf "kg_kgkg(%s,%s,%s,%s,%s)" c wf1 p1 wf2 p2
| F32 | F13 | F21 ->
printf "kg_kgkg(%s,%s,%s,%s,%s)" c wf2 p2 wf1 p1
end
| Dim4_Vector_Vector_Vector_T5 coeff ->
let c = format_coupling coeff c in
begin match fusion with
| F23 -> printf "t5kv_vv(%s,%s,%s,%s,%s)" c wf1 p1 wf2 p2
| F32 -> printf "t5kv_vv(%s,%s,%s,%s,%s)" c wf2 p2 wf1 p1
| F12 | F13 -> printf "t5v_kvv(%s,%s,%s,%s,%s)" c wf1 p1 wf2 p2
| F21 | F31 -> printf "t5v_kvv(%s,%s,%s,%s,%s)" c wf2 p2 wf1 p1
end
| Dim4_Vector_Vector_Vector_L5 coeff ->
let c = format_coupling coeff c in
begin match fusion with
| F23 -> printf "l5kv_vv(%s,%s,%s,%s,%s)" c wf1 p1 wf2 p2
| F32 -> printf "l5kv_vv(%s,%s,%s,%s,%s)" c wf2 p2 wf1 p1
| F12 -> printf "l5v_kvv(%s,%s,%s,%s)" c wf1 p1 wf2
| F21 -> printf "l5v_kvv(%s,%s,%s,%s)" c wf2 p2 wf1
| F13 -> printf "(-1)*l5v_kvv(%s,%s,%s,%s)" c wf1 p1 wf2
| F31 -> printf "(-1)*l5v_kvv(%s,%s,%s,%s)" c wf2 p2 wf1
end
| Dim6_Gauge_Gauge_Gauge_5 coeff ->
let c = format_coupling coeff c in
begin match fusion with
| F23 -> printf "kg5_kgkg(%s,%s,%s,%s,%s)" c wf1 p1 wf2 p2
| F32 -> printf "kg5_kgkg(%s,%s,%s,%s,%s)" c wf2 p2 wf1 p1
| F12 -> printf "kg_kg5kg(%s,%s,%s,%s,%s)" c wf1 p1 wf2 p2
| F21 -> printf "kg_kg5kg(%s,%s,%s,%s,%s)" c wf2 p2 wf1 p1
| F13 -> printf "(-1)*kg_kg5kg(%s,%s,%s,%s,%s)" c wf1 p1 wf2 p2
| F31 -> printf "(-1)*kg_kg5kg(%s,%s,%s,%s,%s)" c wf2 p2 wf1 p1
end
| Aux_DScalar_DScalar coeff ->
let c = format_coupling coeff c in
begin match fusion with
| (F23|F32) ->
printf "%s*(%s*%s)*(%s*%s)" c p1 p2 wf1 wf2
| (F12|F13) ->
printf "%s*(-((%s+%s)*%s))*(%s*%s)" c p1 p2 p2 wf1 wf2
| (F21|F31) ->
printf "%s*(-((%s+%s)*%s))*(%s*%s)" c p1 p2 p1 wf1 wf2
end
| Aux_Vector_DScalar coeff ->
let c = format_coupling coeff c in
begin match fusion with
| F23 -> printf "%s*(%s*%s)*%s" c wf1 p2 wf2
| F32 -> printf "%s*(%s*%s)*%s" c wf2 p1 wf1
| F12 -> printf "%s*(-((%s+%s)*%s))*%s" c p1 p2 wf2 wf1
| F21 -> printf "%s*(-((%s+%s)*%s))*%s" c p1 p2 wf1 wf2
| (F13|F31) -> printf "(-(%s+%s))*(%s*%s*%s)" p1 p2 c wf1 wf2
end
| Dim5_Scalar_Gauge2 coeff ->
let c = format_coupling coeff c in
begin match fusion with
| (F23|F32) -> printf "(%s)*((%s*%s)*(%s*%s) - (%s*%s)*(%s*%s))"
c p1 wf2 p2 wf1 p1 p2 wf2 wf1
| (F12|F13) -> printf "(%s)*%s*((-((%s+%s)*%s))*%s - ((-(%s+%s)*%s))*%s)"
c wf1 p1 p2 wf2 p2 p1 p2 p2 wf2
| (F21|F31) -> printf "(%s)*%s*((-((%s+%s)*%s))*%s - ((-(%s+%s)*%s))*%s)"
c wf2 p2 p1 wf1 p1 p1 p2 p1 wf1
end
| Dim5_Scalar_Gauge2_Skew coeff ->
let c = format_coupling coeff c in
begin match fusion with
| (F23|F32) -> printf "(- phi_vv (%s, %s, %s, %s, %s))" c p1 p2 wf1 wf2
| (F12|F13) -> printf "(- v_phiv (%s, %s, %s, %s, %s))" c wf1 p1 p2 wf2
| (F21|F31) -> printf "v_phiv (%s, %s, %s, %s, %s)" c wf2 p1 p2 wf1
end
| Dim5_Scalar_Vector_Vector_T coeff ->
let c = format_coupling coeff c in
begin match fusion with
| (F23|F32) -> printf "(%s)*(%s*%s)*(%s*%s)" c p1 wf2 p2 wf1
| (F12|F13) -> printf "(%s)*%s*(-((%s+%s)*%s))*%s" c wf1 p1 p2 wf2 p2
| (F21|F31) -> printf "(%s)*%s*(-((%s+%s)*%s))*%s" c wf2 p2 p1 wf1 p1
end
| Dim5_Scalar_Vector_Vector_U coeff ->
let c = format_coupling coeff c in
begin match fusion with
| (F23|F32) -> printf "phi_u_vv (%s, %s, %s, %s, %s)" c p1 p2 wf1 wf2
| (F12|F13) -> printf "v_u_phiv (%s, %s, %s, %s, %s)" c wf1 p1 p2 wf2
| (F21|F31) -> printf "v_u_phiv (%s, %s, %s, %s, %s)" c wf2 p2 p1 wf1
end
| Dim5_Scalar_Vector_Vector_TU coeff ->
let c = format_coupling coeff c in
begin match fusion with
| F23 -> printf "(%s)*((%s*%s)*(-(%s+%s)*%s) - (-(%s+%s)*%s)*(%s*%s))"
c p1 wf2 p1 p2 wf1 p1 p2 p1 wf1 wf2
| F32 -> printf "(%s)*((%s*%s)*(-(%s+%s)*%s) - (-(%s+%s)*%s)*(%s*%s))"
c p2 wf1 p1 p2 wf2 p1 p2 p2 wf1 wf2
| F12 -> printf "(%s)*%s*((%s*%s)*%s - (%s*%s)*%s)"
c wf1 p1 wf2 p2 p1 p2 wf2
| F21 -> printf "(%s)*%s*((%s*%s)*%s - (%s*%s)*%s)"
c wf2 p2 wf1 p1 p1 p2 wf1
| F13 -> printf "(%s)*%s*((-(%s+%s)*%s)*%s - (-(%s+%s)*%s)*%s)"
c wf1 p1 p2 wf2 p1 p1 p2 p1 wf2
| F31 -> printf "(%s)*%s*((-(%s+%s)*%s)*%s - (-(%s+%s)*%s)*%s)"
c wf2 p1 p2 wf1 p2 p1 p2 p2 wf1
end
| Dim6_Vector_Vector_Vector_T coeff ->
let c = format_coupling coeff c in
begin match fusion with
| F23 -> printf "(%s)*(%s*%s)*(%s*%s)*(%s-%s)" c p2 wf1 p1 wf2 p1 p2
| F32 -> printf "(%s)*(%s*%s)*(%s*%s)*(%s-%s)" c p1 wf2 p2 wf1 p2 p1
| (F12|F13) -> printf "(%s)*((%s+2*%s)*%s)*(-((%s+%s)*%s))*%s"
c p1 p2 wf1 p1 p2 wf2 p2
| (F21|F31) -> printf "(%s)*((-((%s+%s)*%s))*(%s+2*%s)*%s)*%s"
c p2 p1 wf1 p2 p1 wf2 p1
end
| Tensor_2_Vector_Vector coeff ->
let c = format_coupling coeff c in
begin match fusion with
| (F23|F32) -> printf "t2_vv(%s,%s,%s)" c wf1 wf2
| (F12|F13) -> printf "v_t2v(%s,%s,%s)" c wf1 wf2
| (F21|F31) -> printf "v_t2v(%s,%s,%s)" c wf2 wf1
end
| Dim5_Tensor_2_Vector_Vector_1 coeff ->
let c = format_coupling coeff c in
begin match fusion with
| (F23|F32) -> printf "t2_vv_d5_1(%s,%s,%s,%s,%s)" c wf1 p1 wf2 p2
| (F12|F13) -> printf "v_t2v_d5_1(%s,%s,%s,%s,%s)" c wf1 p1 wf2 p2
| (F21|F31) -> printf "v_t2v_d5_1(%s,%s,%s,%s,%s)" c wf2 p2 wf1 p1
end
| Dim5_Tensor_2_Vector_Vector_2 coeff ->
let c = format_coupling coeff c in
begin match fusion with
| F23 -> printf "t2_vv_d5_2(%s,%s,%s,%s,%s)" c wf1 p1 wf2 p2
| F32 -> printf "t2_vv_d5_2(%s,%s,%s,%s,%s)" c wf2 p2 wf1 p1
| (F12|F13) -> printf "v_t2v_d5_2(%s,%s,%s,%s,%s)" c wf1 p1 wf2 p2
| (F21|F31) -> printf "v_t2v_d5_2(%s,%s,%s,%s,%s)" c wf2 p2 wf1 p1
end
| Dim7_Tensor_2_Vector_Vector_T coeff ->
let c = format_coupling coeff c in
begin match fusion with
| F23 -> printf "t2_vv_d7(%s,%s,%s,%s,%s)" c wf1 p1 wf2 p2
| F32 -> printf "t2_vv_d7(%s,%s,%s,%s,%s)" c wf2 p2 wf1 p1
| (F12|F13) -> printf "v_t2v_d7(%s,%s,%s,%s,%s)" c wf1 p1 wf2 p2
| (F21|F31) -> printf "v_t2v_d7(%s,%s,%s,%s,%s)" c wf2 p2 wf1 p1
end
end
(* Flip the sign to account for the~$\mathrm{i}^2$ relative to diagrams
with only cubic couplings. *)
| V4 (vertex, fusion, constant) ->
let c = CM.constant_symbol constant
and ch1, ch2, ch3 = children3 rhs in
let wf1 = multiple_variable amplitude dictionary ch1
and wf2 = multiple_variable amplitude dictionary ch2
and wf3 = multiple_variable amplitude dictionary ch3
and p1 = momentum ch1
and p2 = momentum ch2
and p3 = momentum ch3 in
printf "@, %s " (if (F.sign rhs) < 0 then "+" else "-");
begin match vertex with
| Scalar4 coeff ->
printf "(%s*%s*%s*%s)" (format_coupling coeff c) wf1 wf2 wf3
| Scalar2_Vector2 coeff ->
let c = format_coupling coeff c in
begin match fusion with
| F134 | F143 | F234 | F243 ->
printf "%s*%s*(%s*%s)" c wf1 wf2 wf3
| F314 | F413 | F324 | F423 ->
printf "%s*%s*(%s*%s)" c wf2 wf1 wf3
| F341 | F431 | F342 | F432 ->
printf "%s*%s*(%s*%s)" c wf3 wf1 wf2
| F312 | F321 | F412 | F421 ->
printf "(%s*%s*%s)*%s" c wf2 wf3 wf1
| F231 | F132 | F241 | F142 ->
printf "(%s*%s*%s)*%s" c wf1 wf3 wf2
| F123 | F213 | F124 | F214 ->
printf "(%s*%s*%s)*%s" c wf1 wf2 wf3
end
| Vector4 contractions ->
begin match contractions with
| [] -> invalid_arg "Targets.print_current: Vector4 []"
| head :: tail ->
printf "(";
print_vector4 c wf1 wf2 wf3 fusion head;
List.iter (print_add_vector4 c wf1 wf2 wf3 fusion) tail;
printf ")"
end
| Vector4_K_Matrix_tho (disc, poles) ->
let pa, pb =
begin match fusion with
| (F341|F431|F342|F432|F123|F213|F124|F214) -> (p1, p2)
| (F134|F143|F234|F243|F312|F321|F412|F421) -> (p2, p3)
| (F314|F413|F324|F423|F132|F231|F142|F241) -> (p1, p3)
end in
printf "(%s*(%s*%s)*(%s*%s)*(%s*%s)@,*("
c p1 wf1 p2 wf2 p3 wf3;
List.iter (fun (coeff, pole) ->
printf "+%s/((%s+%s)*(%s+%s)-%s)"
(CM.constant_symbol coeff) pa pb pa pb
(CM.constant_symbol pole))
poles;
printf ")*(-%s-%s-%s))" p1 p2 p3
| Vector4_K_Matrix_jr (disc, contractions) ->
let pa, pb =
begin match disc, fusion with
| 3, (F143|F413|F142|F412|F321|F231|F324|F234) -> (p1, p2)
| 3, (F314|F341|F214|F241|F132|F123|F432|F423) -> (p2, p3)
| 3, (F134|F431|F124|F421|F312|F213|F342|F243) -> (p1, p3)
| _, (F341|F431|F342|F432|F123|F213|F124|F214) -> (p1, p2)
| _, (F134|F143|F234|F243|F312|F321|F412|F421) -> (p2, p3)
| _, (F314|F413|F324|F423|F132|F231|F142|F241) -> (p1, p3)
end in
begin match contractions with
| [] -> invalid_arg "Targets.print_current: Vector4_K_Matrix_jr []"
| head :: tail ->
printf "(";
print_vector4_km c pa pb wf1 wf2 wf3 fusion head;
List.iter (print_add_vector4_km c pa pb wf1 wf2 wf3 fusion)
tail;
printf ")"
end
| GBBG (coeff, fb, b, f) ->
Fermions.print_current_g4 (coeff, fb, b, f) c wf1 wf2 wf3
fusion
(* \begin{dubious}
In principle, [p4] could be obtained from the left hand side \ldots
\end{dubious} *)
| DScalar4 contractions ->
let p123 = Printf.sprintf "(-%s-%s-%s)" p1 p2 p3 in
begin match contractions with
| [] -> invalid_arg "Targets.print_current: DScalar4 []"
| head :: tail ->
printf "(";
print_dscalar4 c wf1 wf2 wf3 p1 p2 p3 p123 fusion head;
List.iter (print_add_dscalar4
c wf1 wf2 wf3 p1 p2 p3 p123 fusion) tail;
printf ")"
end
| DScalar2_Vector2 contractions ->
let p123 = Printf.sprintf "(-%s-%s-%s)" p1 p2 p3 in
begin match contractions with
| [] -> invalid_arg "Targets.print_current: DScalar4 []"
| head :: tail ->
printf "(";
print_dscalar2_vector2
c wf1 wf2 wf3 p1 p2 p3 p123 fusion head;
List.iter (print_add_dscalar2_vector2
c wf1 wf2 wf3 p1 p2 p3 p123 fusion) tail;
printf ")"
end
end
| Vn (_, _, _) ->
invalid_arg "Targets.print_current: n-ary fusion"
let print_propagator f p m gamma =
let minus_third = "(-1.0_" ^ !kind ^ "/3.0_" ^ !kind ^ ")" in
let w =
begin match CM.width f with
| Vanishing | Fudged -> "0.0_" ^ !kind
| Constant -> gamma
| Timelike -> "wd_tl(" ^ p ^ "," ^ gamma ^ ")"
| Running ->
failwith "Targets.Fortran: running width not yet available"
| Custom f -> f ^ "(" ^ p ^ "," ^ gamma ^ ")"
end in
match CM.propagator f with
| Prop_Scalar ->
printf "pr_phi(%s,%s,%s," p m w
| Prop_Col_Scalar ->
printf "%s * pr_phi(%s,%s,%s," minus_third p m w
| Prop_Ghost -> printf "(0,1) * pr_phi(%s, %s, %s," p m w
| Prop_Spinor ->
printf "%s(%s,%s,%s," Fermions.psi_propagator p m w
| Prop_ConjSpinor ->
printf "%s(%s,%s,%s," Fermions.psibar_propagator p m w
| Prop_Majorana ->
printf "%s(%s,%s,%s," Fermions.chi_propagator p m w
| Prop_Col_Majorana ->
printf "%s * %s(%s,%s,%s," minus_third Fermions.chi_propagator p m w
| Prop_Unitarity ->
printf "pr_unitarity(%s,%s,%s," p m w
| Prop_Col_Unitarity ->
printf "%s * pr_unitarity(%s,%s,%s," minus_third p m w
| Prop_Feynman ->
printf "pr_feynman(%s," p
| Prop_Col_Feynman ->
printf "%s * pr_feynman(%s," minus_third p
| Prop_Gauge xi ->
printf "pr_gauge(%s,%s," p (CM.gauge_symbol xi)
| Prop_Rxi xi ->
printf "pr_rxi(%s,%s,%s,%s," p m w (CM.gauge_symbol xi)
| Prop_Tensor_2 ->
printf "pr_tensor(%s,%s,%s," p m w
| Prop_Vectorspinor ->
printf "pr_grav(%s,%s,%s," p m w
| Aux_Scalar | Aux_Spinor | Aux_ConjSpinor | Aux_Majorana
| Aux_Vector | Aux_Tensor_1 -> printf "("
- | Aux_Col_Vector | Aux_Col_Tensor_1 -> printf "%s * (" minus_third
+ | Aux_Col_Scalar | Aux_Col_Vector | Aux_Col_Tensor_1 -> printf "%s * (" minus_third
| Only_Insertion -> printf "("
let print_projector f p m gamma =
let minus_third = "(-1.0_" ^ !kind ^ "/3.0_" ^ !kind ^ ")" in
match CM.propagator f with
| Prop_Scalar ->
printf "pj_phi(%s,%s," m gamma
| Prop_Col_Scalar ->
printf "%s * pj_phi(%s,%s," minus_third m gamma
| Prop_Ghost ->
printf "(0,1) * pj_phi(%s,%s," m gamma
| Prop_Spinor ->
printf "%s(%s,%s,%s," Fermions.psi_projector p m gamma
| Prop_ConjSpinor ->
printf "%s(%s,%s,%s," Fermions.psibar_projector p m gamma
| Prop_Majorana ->
printf "%s(%s,%s,%s," Fermions.chi_projector p m gamma
| Prop_Col_Majorana ->
printf "%s * %s(%s,%s,%s," minus_third Fermions.chi_projector p m gamma
| Prop_Unitarity ->
printf "pj_unitarity(%s,%s,%s," p m gamma
| Prop_Col_Unitarity ->
printf "%s * pj_unitarity(%s,%s,%s," minus_third p m gamma
| Prop_Feynman | Prop_Col_Feynman ->
invalid_arg "no on-shell Feynman propagator!"
| Prop_Gauge xi ->
invalid_arg "no on-shell massless gauge propagator!"
| Prop_Rxi xi ->
invalid_arg "no on-shell Rxi propagator!"
| Prop_Vectorspinor ->
printf "pj_grav(%s,%s,%s," p m gamma
| Prop_Tensor_2 ->
printf "pj_tensor(%s,%s,%s," p m gamma
| Aux_Scalar | Aux_Spinor | Aux_ConjSpinor | Aux_Majorana
| Aux_Vector | Aux_Tensor_1 -> printf "("
- | Aux_Col_Vector | Aux_Col_Tensor_1 -> printf "%s * (" minus_third
+ | Aux_Col_Scalar | Aux_Col_Vector | Aux_Col_Tensor_1 -> printf "%s * (" minus_third
| Only_Insertion -> printf "("
let print_gauss f p m gamma =
let minus_third = "(-1.0_" ^ !kind ^ "/3.0_" ^ !kind ^ ")" in
match CM.propagator f with
| Prop_Scalar ->
printf "pg_phi(%s,%s,%s," p m gamma
| Prop_Ghost ->
printf "(0,1) * pg_phi(%s,%s,%s," p m gamma
| Prop_Spinor ->
printf "%s(%s,%s,%s," Fermions.psi_projector p m gamma
| Prop_ConjSpinor ->
printf "%s(%s,%s,%s," Fermions.psibar_projector p m gamma
| Prop_Majorana ->
printf "%s(%s,%s,%s," Fermions.chi_projector p m gamma
| Prop_Col_Majorana ->
printf "%s * %s(%s,%s,%s," minus_third Fermions.chi_projector p m gamma
| Prop_Unitarity ->
printf "pg_unitarity(%s,%s,%s," p m gamma
| Prop_Feynman | Prop_Col_Feynman ->
invalid_arg "no on-shell Feynman propagator!"
| Prop_Gauge xi ->
invalid_arg "no on-shell massless gauge propagator!"
| Prop_Rxi xi ->
invalid_arg "no on-shell Rxi propagator!"
| Prop_Tensor_2 ->
printf "pg_tensor(%s,%s,%s," p m gamma
| Aux_Scalar | Aux_Spinor | Aux_ConjSpinor | Aux_Majorana
| Aux_Vector | Aux_Tensor_1 -> printf "("
| Only_Insertion -> printf "("
| _ -> invalid_arg "targets:print_gauss: not available"
let print_fusion_diagnostics amplitude dictionary fusion =
if warn diagnose_gauge then begin
let lhs = F.lhs fusion in
let f = F.flavor lhs
and v = variable lhs
and p = momentum lhs in
let mass = CM.mass_symbol f in
match CM.propagator f with
| Prop_Gauge _ | Prop_Feynman
| Prop_Rxi _ | Prop_Unitarity ->
printf " @[<2>%s =" v;
List.iter (print_current amplitude dictionary) (F.rhs fusion); nl();
begin match CM.goldstone f with
| None ->
printf " call omega_ward_%s(\"%s\",%s,%s,%s)"
(suffix diagnose_gauge) v mass p v; nl ()
| Some (g, phase) ->
let gv = add_tag lhs (CM.flavor_symbol g ^ "_" ^ format_p lhs) in
printf " call omega_slavnov_%s"
(suffix diagnose_gauge);
printf "(@[\"%s\",%s,%s,%s,@,%s*%s)"
v mass p v (format_constant phase) gv; nl ()
end
| _ -> ()
end
let print_fusion amplitude dictionary fusion =
let lhs = F.lhs fusion in
let f = F.flavor lhs in
printf " @[<2>%s =@, " (multiple_variable amplitude dictionary lhs);
if F.on_shell amplitude lhs then
print_projector f (momentum lhs)
(CM.mass_symbol f) (CM.width_symbol f)
else
if F.is_gauss amplitude lhs then
print_gauss f (momentum lhs)
(CM.mass_symbol f) (CM.width_symbol f)
else
print_propagator f (momentum lhs)
(CM.mass_symbol f) (CM.width_symbol f);
List.iter (print_current amplitude dictionary) (F.rhs fusion);
printf ")"; nl ()
let print_momenta seen_momenta amplitude =
List.fold_left (fun seen f ->
let wf = F.lhs f in
let p = F.momentum_list wf in
if not (PSet.mem p seen) then begin
let rhs1 = List.hd (F.rhs f) in
printf " %s = %s" (momentum wf)
(String.concat " + "
(List.map momentum (F.children rhs1))); nl ()
end;
PSet.add p seen)
seen_momenta (F.fusions amplitude)
let print_fusions dictionary fusions =
List.iter
(fun (f, amplitude) ->
print_fusion_diagnostics amplitude dictionary f;
print_fusion amplitude dictionary f)
fusions
let print_braket amplitude dictionary name braket =
let bra = F.bra braket
and ket = F.ket braket in
printf " @[<2>%s = %s@, + " name name;
begin match Fermions.reverse_braket (CM.lorentz (F.flavor bra)) with
| false ->
printf "%s*(@," (multiple_variable amplitude dictionary bra);
List.iter (print_current amplitude dictionary) ket;
printf ")"
| true ->
printf "(@,";
List.iter (print_current amplitude dictionary) ket;
printf ")*%s" (multiple_variable amplitude dictionary bra)
end; nl ()
(* \begin{equation}
\ii T = \ii^{\#\text{vertices}}\ii^{\#\text{propagators}} \cdots
= \ii^{n-2}\ii^{n-3} \cdots
= -\ii(-1)^n \cdots
\end{equation} *)
(* \begin{dubious}
[tho:] we write some brakets twice using different names. Is it useful
to cache them?
\end{dubious} *)
let print_brakets dictionary amplitude =
let name = flavors_symbol (flavors amplitude) in
printf " %s = 0" name; nl ();
List.iter (print_braket amplitude dictionary name) (F.brakets amplitude);
let n = List.length (F.externals amplitude) in
if n mod 2 = 0 then begin
printf " @[<2>%s =@, - %s ! %d vertices, %d propagators"
name name (n - 2) (n - 3); nl ()
end else begin
printf " ! %s = %s ! %d vertices, %d propagators"
name name (n - 2) (n - 3); nl ()
end;
let s = F.symmetry amplitude in
if s > 1 then
printf " @[<2>%s =@, %s@, / sqrt(%d.0_%s) ! symmetry factor" name name s !kind
else
printf " ! unit symmetry factor";
nl ()
let print_incoming wf =
let p = momentum wf
and s = spin wf
and f = F.flavor wf in
let m = CM.mass_symbol f in
match CM.lorentz f with
| Scalar -> printf "1"
| BRS Scalar -> printf "(0,-1) * (%s * %s - %s**2)" p p m
| Spinor ->
printf "%s (%s, - %s, %s)" Fermions.psi_incoming m p s
| BRS Spinor ->
printf "%s (%s, - %s, %s)" Fermions.brs_psi_incoming m p s
| ConjSpinor ->
printf "%s (%s, - %s, %s)" Fermions.psibar_incoming m p s
| BRS ConjSpinor ->
printf "%s (%s, - %s, %s)" Fermions.brs_psibar_incoming m p s
| Majorana ->
printf "%s (%s, - %s, %s)" Fermions.chi_incoming m p s
| Maj_Ghost -> printf "ghost (%s, - %s, %s)" m p s
| BRS Majorana ->
printf "%s (%s, - %s, %s)" Fermions.brs_chi_incoming m p s
| Vector | Massive_Vector ->
printf "eps (%s, - %s, %s)" m p s
(*i | Ward_Vector -> printf "%s" p i*)
| BRS Vector | BRS Massive_Vector -> printf
"(0,1) * (%s * %s - %s**2) * eps (%s, -%s, %s)" p p m m p s
| Vectorspinor | BRS Vectorspinor ->
printf "%s (%s, - %s, %s)" Fermions.grav_incoming m p s
| Tensor_1 -> invalid_arg "Tensor_1 only internal"
| Tensor_2 -> printf "eps2 (%s, - %s, %s)" m p s
| _ -> invalid_arg "no such BRST transformations"
let print_outgoing wf =
let p = momentum wf
and s = spin wf
and f = F.flavor wf in
let m = CM.mass_symbol f in
match CM.lorentz f with
| Scalar -> printf "1"
| BRS Scalar -> printf "(0,-1) * (%s * %s - %s**2)" p p m
| Spinor ->
printf "%s (%s, %s, %s)" Fermions.psi_outgoing m p s
| BRS Spinor ->
printf "%s (%s, %s, %s)" Fermions.brs_psi_outgoing m p s
| ConjSpinor ->
printf "%s (%s, %s, %s)" Fermions.psibar_outgoing m p s
| BRS ConjSpinor ->
printf "%s (%s, %s, %s)" Fermions.brs_psibar_outgoing m p s
| Majorana ->
printf "%s (%s, %s, %s)" Fermions.chi_outgoing m p s
| BRS Majorana ->
printf "%s (%s, %s, %s)" Fermions.brs_chi_outgoing m p s
| Maj_Ghost -> printf "ghost (%s, %s, %s)" m p s
| Vector | Massive_Vector ->
printf "conjg (eps (%s, %s, %s))" m p s
(*i | Ward_Vector -> printf "%s" p i*)
| BRS Vector | BRS Massive_Vector -> printf
"(0,1) * (%s*%s-%s**2) * (conjg (eps (%s, %s, %s)))" p p m m p s
| Vectorspinor | BRS Vectorspinor ->
printf "%s (%s, %s, %s)" Fermions.grav_incoming m p s
| Tensor_1 -> invalid_arg "Tensor_1 only internal"
| Tensor_2 -> printf "conjg (eps2 (%s, %s, %s))" m p s
| BRS _ -> invalid_arg "no such BRST transformations"
let twice_spin wf =
match CM.lorentz (F.flavor wf) with
| Scalar | BRS Scalar -> "0"
| Spinor | ConjSpinor | Majorana | Maj_Ghost | Vectorspinor
| BRS Spinor | BRS ConjSpinor | BRS Majorana | BRS Vectorspinor -> "1"
| Vector | BRS Vector | Massive_Vector | BRS Massive_Vector -> "2"
| Tensor_1 -> "2"
| Tensor_2 -> "4"
| BRS _ -> invalid_arg "Targets.twice_spin: no such BRST transformation"
let print_argument_diagnostics amplitude =
let externals = (F.externals amplitude) in
let n = List.length externals
and masses = List.map (fun wf -> CM.mass_symbol (F.flavor wf)) externals in
if warn diagnose_arguments then begin
printf " call omega_check_arguments_%s (%d, k)"
(suffix diagnose_arguments) n; nl ()
end;
if warn diagnose_momenta then begin
printf " @[<2>call omega_check_momenta_%s ((/ "
(suffix diagnose_momenta);
print_list masses;
printf " /), k)"; nl ()
end
let print_external_momenta amplitude =
let externals =
List.combine
(F.externals amplitude)
(List.map (fun _ -> true) (F.incoming amplitude) @
List.map (fun _ -> false) (F.outgoing amplitude)) in
List.iter (fun (wf, incoming) ->
if incoming then
printf " %s = - k(:,%d) ! incoming"
(momentum wf) (ext_momentum wf)
else
printf " %s = k(:,%d) ! outgoing"
(momentum wf) (ext_momentum wf); nl ()) externals
let print_externals seen_wfs amplitude =
let externals =
List.combine
(F.externals amplitude)
(List.map (fun _ -> true) (F.incoming amplitude) @
List.map (fun _ -> false) (F.outgoing amplitude)) in
List.fold_left (fun seen (wf, incoming) ->
if not (WFSet.mem wf seen) then begin
printf " @[<2>%s =@, " (variable wf);
(if incoming then print_incoming else print_outgoing) wf; nl ()
end;
WFSet.add wf seen) seen_wfs externals
let flavors_to_string flavors =
String.concat " " (List.map CM.flavor_to_string flavors)
let process_to_string amplitude =
flavors_to_string (F.incoming amplitude) ^ " -> " ^
flavors_to_string (F.outgoing amplitude)
let flavors_sans_color_to_string flavors =
String.concat " " (List.map M.flavor_to_string flavors)
let process_sans_color_to_string (fin, fout) =
flavors_sans_color_to_string fin ^ " -> " ^
flavors_sans_color_to_string fout
let print_fudge_factor amplitude =
let name = flavors_symbol (flavors amplitude) in
List.iter (fun wf ->
let p = momentum wf
and f = F.flavor wf in
match CM.width f with
| Fudged ->
let m = CM.mass_symbol f
and w = CM.width_symbol f in
printf " if (%s > 0.0_%s) then" w !kind; nl ();
printf " @[<2>%s = %s@ * (%s*%s - %s**2)"
name name p p m;
printf "@ / cmplx (%s*%s - %s**2, %s*%s, kind=%s)"
p p m m w !kind; nl();
printf " end if"; nl ()
| _ -> ()) (F.s_channel amplitude)
let num_helicities amplitudes =
List.length (CF.helicities amplitudes)
(* \thocwmodulesubsection{Spin, Flavor \&\ Color Tables} *)
(* The following abomination is required to keep the number of continuation
lines as low as possible. FORTRAN77-style \texttt{DATA} statements
are actually a bit nicer here, but they are nor available for
\emph{constant} arrays. *)
(* \begin{dubious}
We used to have a more elegent design with a sentinel~0 added to each
initializer, but some revisions of the Compaq/Digital Compiler have a
bug that causes it to reject this variant.
\end{dubious} *)
(* \begin{dubious}
The actual table writing code using \texttt{reshape} should be factored,
since it's the same algorithm every time.
\end{dubious} *)
let print_integer_parameter name value =
printf " @[<2>integer, parameter :: %s = %d" name value; nl ()
let print_real_parameter name value =
printf " @[<2>real(kind=%s), parameter :: %s = %d"
!kind name value; nl ()
let print_logical_parameter name value =
printf " @[<2>logical, parameter :: %s = .%s."
name (if value then "true" else "false"); nl ()
let num_particles_in amplitudes =
match CF.flavors amplitudes with
| [] -> 0
| (fin, _) :: _ -> List.length fin
let num_particles_out amplitudes =
match CF.flavors amplitudes with
| [] -> 0
| (_, fout) :: _ -> List.length fout
let num_particles amplitudes =
match CF.flavors amplitudes with
| [] -> 0
| (fin, fout) :: _ -> List.length fin + List.length fout
module CFlow = Color.Flow
let num_color_flows amplitudes =
List.length (CF.color_flows amplitudes)
let num_color_indices_default = 2 (* Standard model *)
let num_color_indices amplitudes =
try CFlow.rank (List.hd (CF.color_flows amplitudes)) with _ -> num_color_indices_default
let color_to_string c =
"(" ^ (String.concat "," (List.map (Printf.sprintf "%3d") c)) ^ ")"
let cflow_to_string cflow =
String.concat " " (List.map color_to_string (CFlow.in_to_lists cflow)) ^ " -> " ^
String.concat " " (List.map color_to_string (CFlow.out_to_lists cflow))
let protected = ""
let protected = ", protected" (* Fortran 2003! *)
let print_spin_table_old abbrev name = function
| [] ->
printf " @[<2>integer, dimension(n_prt,0) ::";
printf "@ table_spin_%s" name; nl ()
| _ :: tuples' as tuples ->
ignore (List.fold_left (fun i (tuple1, tuple2) ->
printf " @[<2>integer, dimension(n_prt), parameter, private ::";
printf "@ %s%04d = (/ %s /)" abbrev i
(String.concat ", " (List.map (Printf.sprintf "%2d") (tuple1 @ tuple2)));
nl (); succ i) 1 tuples);
printf
" @[<2>integer, dimension(n_prt,n_hel), parameter ::";
printf "@ table_spin_%s =@ reshape ( (/" name;
printf "@ %s%04d" abbrev 1;
ignore (List.fold_left (fun i tuple ->
printf ",@ %s%04d" abbrev i; succ i) 2 tuples');
printf "@ /), (/ n_prt, n_hel /) )"; nl ()
let print_spin_table name tuples =
printf " @[<2>integer, dimension(n_prt,n_hel), save%s :: table_spin_%s"
protected name; nl();
match tuples with
| [] -> ()
| _ ->
ignore (List.fold_left (fun i (tuple1, tuple2) ->
printf " @[<2>data table_spin_%s(:,%4d) / %s /" name i
(String.concat ", " (List.map (Printf.sprintf "%2d") (tuple1 @ tuple2)));
nl (); succ i) 1 tuples)
let print_spin_tables amplitudes =
(* [print_spin_table_old "s" "states_old" (CF.helicities amplitudes);] *)
print_spin_table "states" (CF.helicities amplitudes);
nl ()
let print_flavor_table_old n abbrev name = function
| [] ->
printf " @[<2>integer, dimension(n_prt,0) ::";
printf "@ table_flavor_%s" name; nl ()
| _ :: tuples' as tuples ->
ignore (List.fold_left (fun i tuple ->
printf
" @[<2>integer, dimension(n_prt), parameter, private ::";
printf "@ %s%04d = (/ %s /) ! %s" abbrev i
(String.concat ", "
(List.map (fun f -> Printf.sprintf "%3d" (M.pdg f)) tuple))
(String.concat " " (List.map M.flavor_to_string tuple));
nl (); succ i) 1 tuples);
printf
" @[<2>integer, dimension(n_prt,n_flv), parameter ::";
printf "@ table_flavor_%s =@ reshape ( (/" name;
printf "@ %s%04d" abbrev 1;
ignore (List.fold_left (fun i tuple ->
printf ",@ %s%04d" abbrev i; succ i) 2 tuples');
printf "@ /), (/ n_prt, n_flv /) )"; nl ()
let print_flavor_table n name tuples =
printf " @[<2>integer, dimension(n_prt,n_flv), save%s :: table_flavor_%s"
protected name; nl();
match tuples with
| [] -> ()
| _ ->
ignore (List.fold_left (fun i tuple ->
printf " @[<2>data table_flavor_%s(:,%4d) / %s / ! %s" name i
(String.concat ", "
(List.map (fun f -> Printf.sprintf "%3d" (M.pdg f)) tuple))
(String.concat " " (List.map M.flavor_to_string tuple));
nl (); succ i) 1 tuples)
let print_flavor_tables amplitudes =
let n = num_particles amplitudes in
(* [print_flavor_table_old n "f" "states_old"
(List.map (fun (fin, fout) -> fin @ fout) (CF.flavors amplitudes));] *)
print_flavor_table n "states"
(List.map (fun (fin, fout) -> fin @ fout) (CF.flavors amplitudes));
nl ()
let num_flavors amplitudes =
List.length (CF.flavors amplitudes)
let print_color_flows_table_old abbrev = function
| [] ->
printf " @[<2>integer, dimension(n_cindex, n_prt, n_cflow) ::";
printf "@ table_color_flows"; nl ()
| _ :: tuples' as tuples ->
ignore (List.fold_left (fun i tuple ->
printf
" @[<2>integer, dimension(n_cindex, n_prt), parameter, private ::";
printf "@ %s%04d = reshape ( (/ " abbrev i;
begin match CFlow.to_lists tuple with
| [] -> ()
| cf1 :: cfn ->
printf "@ %s" (String.concat "," (List.map string_of_int cf1));
List.iter (function cf ->
printf ",@ %s" (String.concat "," (List.map string_of_int cf))) cfn
end;
printf "@ /),@ (/ n_cindex, n_prt /) )";
nl (); succ i) 1 tuples);
printf
" @[<2>integer, dimension(n_cindex, n_prt, n_cflow), parameter ::";
printf "@ table_color_flows_old =@ reshape ( (/";
printf "@ %s%04d" abbrev 1;
ignore (List.fold_left (fun i tuple ->
printf ",@ %s%04d" abbrev i; succ i) 2 tuples');
printf "@ /),@ (/ n_cindex, n_prt, n_cflow /) )"; nl ()
let print_ghost_flags_table_old abbrev = function
| [] ->
printf " @[<2>logical, dimension(n_prt, n_cflow) ::";
printf "@ table_ghost_flags"; nl ()
| _ :: tuples' as tuples ->
ignore (List.fold_left (fun i tuple ->
printf
" @[<2>logical, dimension(n_prt), parameter, private ::";
printf "@ %s%04d = (/ " abbrev i;
begin match CFlow.ghost_flags tuple with
| [] -> ()
| gf1 :: gfn ->
printf "@ %s" (if gf1 then "T" else "F");
List.iter (function gf -> printf ",@ %s" (if gf then "T" else "F")) gfn
end;
printf "@ /)";
nl (); succ i) 1 tuples);
printf
" @[<2>logical, dimension(n_prt, n_cflow), parameter ::";
printf "@ table_ghost_flags_old =@ reshape ( (/";
printf "@ %s%04d" abbrev 1;
ignore (List.fold_left (fun i tuple ->
printf ",@ %s%04d" abbrev i; succ i) 2 tuples');
printf "@ /),@ (/ n_prt, n_cflow /) )"; nl ()
let print_color_flows_table tuples =
printf
" @[<2>integer, dimension(n_cindex,n_prt,n_cflow), save%s :: table_color_flows"
protected; nl ();
match tuples with
| [] -> ()
| _ :: tuples' as tuples ->
ignore (List.fold_left (fun i tuple ->
begin match CFlow.to_lists tuple with
| [] -> ()
| cf1 :: cfn ->
printf " @[<2>data table_color_flows(:,:,%4d) /" i;
printf "@ %s" (String.concat "," (List.map string_of_int cf1));
List.iter (function cf ->
printf ",@ %s" (String.concat "," (List.map string_of_int cf))) cfn;
printf "@ /"; nl ()
end;
succ i) 1 tuples)
let print_ghost_flags_table tuples =
printf
" @[<2>logical, dimension(n_prt,n_cflow), save%s :: table_ghost_flags"
protected; nl ();
match tuples with
| [] -> ()
| _ ->
ignore (List.fold_left (fun i tuple ->
begin match CFlow.ghost_flags tuple with
| [] -> ()
| gf1 :: gfn ->
printf " @[<2>data table_ghost_flags(:,%4d) /" i;
printf "@ %s" (if gf1 then "T" else "F");
List.iter (function gf -> printf ",@ %s" (if gf then "T" else "F")) gfn;
printf " /";
nl ()
end;
succ i) 1 tuples)
let format_power_of x
{ Color.Flow.num = num; Color.Flow.den = den; Color.Flow.power = pwr } =
match num, den, pwr with
| _, 0, _ -> invalid_arg "format_power_of: zero denominator"
| 0, _, _ -> "+zero"
| 1, 1, 0 | -1, -1, 0 -> "+one"
| -1, 1, 0 | 1, -1, 0 -> "-one"
| 1, 1, 1 | -1, -1, 1 -> "+" ^ x
| -1, 1, 1 | 1, -1, 1 -> "-" ^ x
| 1, 1, -1 | -1, -1, -1 -> "+1/" ^ x
| -1, 1, -1 | 1, -1, -1 -> "-1/" ^ x
| 1, 1, p | -1, -1, p ->
"+" ^ (if p > 0 then "" else "1/") ^ x ^ "**" ^ string_of_int (abs p)
| -1, 1, p | 1, -1, p ->
"-" ^ (if p > 0 then "" else "1/") ^ x ^ "**" ^ string_of_int (abs p)
| n, 1, 0 ->
(if n < 0 then "-" else "+") ^ string_of_int (abs n) ^ ".0_" ^ !kind
| n, d, 0 ->
(if n * d < 0 then "-" else "+") ^
string_of_int (abs n) ^ ".0_" ^ !kind ^ "/" ^
string_of_int (abs d)
| n, 1, 1 ->
(if n < 0 then "-" else "+") ^ string_of_int (abs n) ^ "*" ^ x
| n, 1, -1 ->
(if n < 0 then "-" else "+") ^ string_of_int (abs n) ^ "/" ^ x
| n, d, 1 ->
(if n * d < 0 then "-" else "+") ^
string_of_int (abs n) ^ ".0_" ^ !kind ^ "/" ^
string_of_int (abs d) ^ "*" ^ x
| n, d, -1 ->
(if n * d < 0 then "-" else "+") ^
string_of_int (abs n) ^ ".0_" ^ !kind ^ "/" ^
string_of_int (abs d) ^ "/" ^ x
| n, 1, p ->
(if n < 0 then "-" else "+") ^ string_of_int (abs n) ^
(if p > 0 then "*" else "/") ^ x ^ "**" ^ string_of_int (abs p)
| n, d, p ->
(if n * d < 0 then "-" else "+") ^
string_of_int (abs n) ^ ".0_" ^ !kind ^ "/" ^
string_of_int (abs d) ^
(if p > 0 then "*" else "/") ^ x ^ "**" ^ string_of_int (abs p)
let format_powers_of x = function
| [] -> "zero"
| powers -> String.concat "" (List.map (format_power_of x) powers)
let print_color_factor_table_old table =
let n_cflow = Array.length table in
let n_cfactors = ref 0 in
for c1 = 0 to pred n_cflow do
for c2 = 0 to pred n_cflow do
match table.(c1).(c2) with
| [] -> ()
| _ -> incr n_cfactors
done
done;
print_integer_parameter "n_cfactors" !n_cfactors;
if n_cflow <= 0 then begin
printf " @[<2>type(%s), dimension(n_cfactors) ::"
omega_color_factor_abbrev;
printf "@ table_color_factors"; nl ()
end else begin
printf
" @[<2>type(%s), dimension(n_cfactors), parameter ::"
omega_color_factor_abbrev;
printf "@ table_color_factors = (/@ ";
let comma = ref "" in
for c1 = 0 to pred n_cflow do
for c2 = 0 to pred n_cflow do
match table.(c1).(c2) with
| [] -> ()
| cf ->
printf "%s@ %s(%d,%d,%s)" !comma omega_color_factor_abbrev
(succ c1) (succ c2) (format_powers_of nc_parameter cf);
comma := ","
done
done;
printf "@ /)"; nl ()
end
(* \begin{dubious}
We can optimize the following slightly by reusing common color factor [parameter]s.
\end{dubious} *)
let print_color_factor_table table =
let n_cflow = Array.length table in
let n_cfactors = ref 0 in
for c1 = 0 to pred n_cflow do
for c2 = 0 to pred n_cflow do
match table.(c1).(c2) with
| [] -> ()
| _ -> incr n_cfactors
done
done;
print_integer_parameter "n_cfactors" !n_cfactors;
printf " @[<2>type(%s), dimension(n_cfactors), save%s ::"
omega_color_factor_abbrev protected;
printf "@ table_color_factors"; nl ();
let i = ref 1 in
if n_cflow > 0 then begin
for c1 = 0 to pred n_cflow do
for c2 = 0 to pred n_cflow do
match table.(c1).(c2) with
| [] -> ()
| cf ->
printf " @[<2>real(kind=%s), parameter, private :: color_factor_%06d = %s"
!kind !i (format_powers_of nc_parameter cf);
nl ();
printf " @[<2>data table_color_factors(%6d) / %s(%d,%d,color_factor_%06d) /"
!i omega_color_factor_abbrev (succ c1) (succ c2) !i;
incr i;
nl ();
done
done
end
let print_color_tables amplitudes =
let cflows = CF.color_flows amplitudes
and cfactors = CF.color_factors amplitudes in
(* [print_color_flows_table_old "c" cflows; nl ();] *)
print_color_flows_table cflows; nl ();
(* [print_ghost_flags_table_old "g" cflows; nl ();] *)
print_ghost_flags_table cflows; nl ();
(* [print_color_factor_table_old cfactors; nl ();] *)
print_color_factor_table cfactors; nl ()
let option_to_logical = function
| Some _ -> "T"
| None -> "F"
let print_flavor_color_table_old abbrev n_flv n_cflow table =
if n_flv <= 0 or n_cflow <= 0 then begin
printf " @[<2>logical, dimension(n_flv, n_cflow) ::";
printf "@ flv_col_is_allowed"; nl ()
end else begin
for c = 0 to pred n_cflow do
printf
" @[<2>logical, dimension(n_flv), parameter, private ::";
printf "@ %s%04d = (/@ %s" abbrev (succ c) (option_to_logical table.(0).(c));
for f = 1 to pred n_flv do
printf ",@ %s" (option_to_logical table.(f).(c))
done;
printf "@ /)"; nl ()
done;
printf
" @[<2>logical, dimension(n_flv, n_cflow), parameter ::";
printf "@ flv_col_is_allowed_old =@ reshape ( (/@ %s%04d" abbrev 1;
for c = 1 to pred n_cflow do
printf ",@ %s%04d" abbrev (succ c)
done;
printf "@ /),@ (/ n_flv, n_cflow /) )"; nl ()
end
let print_flavor_color_table n_flv n_cflow table =
printf
" @[<2>logical, dimension(n_flv, n_cflow), save%s :: @ flv_col_is_allowed"
protected; nl ();
if n_flv > 0 then begin
for c = 0 to pred n_cflow do
printf
" @[<2>data flv_col_is_allowed(:,%4d) /" (succ c);
printf "@ %s" (option_to_logical table.(0).(c));
for f = 1 to pred n_flv do
printf ",@ %s" (option_to_logical table.(f).(c))
done;
printf "@ /"; nl ()
done;
end
let print_amplitude_table a =
(* [print_flavor_color_table_old "a"
(num_flavors a) (List.length (CF.color_flows a)) (CF.process_table a);
nl ();] *)
print_flavor_color_table
(num_flavors a) (List.length (CF.color_flows a)) (CF.process_table a);
nl ();
printf
" @[<2>complex(kind=%s), dimension(n_flv, n_cflow, n_hel), save :: amp" !kind;
nl ();
nl ()
let print_helicity_selection_table () =
printf " @[<2>logical, dimension(n_hel), save :: ";
printf "hel_is_allowed = T"; nl();
printf " @[<2>real(kind=%s), dimension(n_hel), save :: " !kind;
printf "hel_max_abs = 0"; nl ();
printf " @[<2>real(kind=%s), save :: " !kind;
printf "hel_sum_abs = 0, ";
printf "hel_threshold = 1E10"; nl ();
printf " @[<2>integer, save :: ";
printf "hel_count = 0, ";
printf "hel_cutoff = 100"; nl ();
printf " @[<2>integer :: ";
printf "i"; nl ();
printf " @[<2>integer, save, dimension(n_hel) :: ";
printf "hel_map = (/(i, i = 1, n_hel)/)"; nl ();
printf " @[<2>integer, save :: hel_finite = n_hel"; nl ();
nl ()
(* \thocwmodulesubsection{Optional MD5 sum function} *)
let print_md5sum_functions = function
| Some s ->
printf " @[<5>"; if !fortran95 then printf "pure ";
printf "function md5sum ()"; nl ();
printf " character(len=32) :: md5sum"; nl ();
printf " ! DON'T EVEN THINK of modifying the following line!"; nl ();
printf " md5sum = \"%s\"" s; nl ();
printf " end function md5sum"; nl ();
nl ()
| None -> ()
(* \thocwmodulesubsection{Maintenance \&\ Inquiry Functions} *)
let print_maintenance_functions amplitudes =
if !whizard then begin
printf " subroutine init (par)"; nl ();
printf " real(kind=%s), dimension(*), intent(in) :: par" !kind; nl ();
printf " call import_from_whizard (par)"; nl ();
printf " end subroutine init"; nl ();
nl ();
printf " subroutine final ()"; nl ();
printf " end subroutine final"; nl ();
nl ();
printf " subroutine update_alpha_s (alpha_s)"; nl ();
printf " real(kind=%s), intent(in) :: alpha_s" !kind; nl ();
printf " call model_update_alpha_s (alpha_s)"; nl ();
printf " end subroutine update_alpha_s"; nl ();
nl ()
end
let print_inquiry_function_openmp () = begin
printf " pure function openmp_supported () result (status)"; nl ();
printf " logical :: status"; nl ();
printf " status = %s" (if !openmp then ".true." else ".false."); nl ();
printf " end function openmp_supported"; nl ();
nl ()
end
let print_inquiry_function_declarations name =
printf " @[<2>public :: number_%s,@ %s" name name;
nl ()
let print_numeric_inquiry_functions () =
printf " @[<5>"; if !fortran95 then printf "pure ";
printf "function number_particles_in () result (n)"; nl ();
printf " integer :: n"; nl ();
printf " n = n_in"; nl ();
printf " end function number_particles_in"; nl ();
nl ();
printf " @[<5>"; if !fortran95 then printf "pure ";
printf "function number_particles_out () result (n)"; nl ();
printf " integer :: n"; nl ();
printf " n = n_out"; nl ();
printf " end function number_particles_out"; nl ();
nl ()
let print_numeric_inquiry_functions (f, v) =
printf " @[<5>"; if !fortran95 then printf "pure ";
printf "function %s () result (n)" f; nl ();
printf " integer :: n"; nl ();
printf " n = %s" v; nl ();
printf " end function %s" f; nl ();
nl ()
let print_inquiry_functions name =
printf " @[<5>"; if !fortran95 then printf "pure ";
printf "function number_%s () result (n)" name; nl ();
printf " integer :: n"; nl ();
printf " n = size (table_%s, dim=2)" name; nl ();
printf " end function number_%s" name; nl ();
nl ();
printf " @[<5>"; if !fortran95 then printf "pure ";
printf "subroutine %s (a)" name; nl ();
printf " integer, dimension(:,:), intent(out) :: a"; nl ();
printf " a = table_%s" name; nl ();
printf " end subroutine %s" name; nl ();
nl ()
let print_color_flows () =
printf " @[<5>"; if !fortran95 then printf "pure ";
printf "function number_color_indices () result (n)"; nl ();
printf " integer :: n"; nl ();
printf " n = size (table_color_flows, dim=1)"; nl ();
printf " end function number_color_indices"; nl ();
nl ();
printf " @[<5>"; if !fortran95 then printf "pure ";
printf "function number_color_flows () result (n)"; nl ();
printf " integer :: n"; nl ();
printf " n = size (table_color_flows, dim=3)"; nl ();
printf " end function number_color_flows"; nl ();
nl ();
printf " @[<5>"; if !fortran95 then printf "pure ";
printf "subroutine color_flows (a, g)"; nl ();
printf " integer, dimension(:,:,:), intent(out) :: a"; nl ();
printf " logical, dimension(:,:), intent(out) :: g"; nl ();
printf " a = table_color_flows"; nl ();
printf " g = table_ghost_flags"; nl ();
printf " end subroutine color_flows"; nl ();
nl ()
let print_color_factors () =
printf " @[<5>"; if !fortran95 then printf "pure ";
printf "function number_color_factors () result (n)"; nl ();
printf " integer :: n"; nl ();
printf " n = size (table_color_factors)"; nl ();
printf " end function number_color_factors"; nl ();
nl ();
printf " @[<5>"; if !fortran95 then printf "pure ";
printf "subroutine color_factors (cf)"; nl ();
printf " type(%s), dimension(:), intent(out) :: cf"
omega_color_factor_abbrev; nl ();
printf " cf = table_color_factors"; nl ();
printf " end subroutine color_factors"; nl ();
nl ();
printf " @[<5>"; if !fortran95 then printf "pure ";
printf "function color_sum (flv, hel) result (amp2)"; nl ();
printf " integer, intent(in) :: flv, hel"; nl ();
printf " real(kind=%s) :: amp2" !kind; nl ();
printf " amp2 = real (omega_color_sum (flv, hel, amp, table_color_factors))"; nl ();
printf " end function color_sum"; nl ();
nl ()
let print_dispatch_functions () =
printf " @[<5>";
printf "subroutine new_event (p)"; nl ();
printf " real(kind=%s), dimension(0:3,*), intent(in) :: p" !kind; nl ();
printf " logical :: mask_dirty"; nl ();
printf " integer :: hel"; nl ();
printf " call calculate_amplitudes (amp, p, hel_is_allowed)"; nl ();
printf " if ((hel_threshold .gt. 0) .and. (hel_count .le. hel_cutoff)) then"; nl ();
printf " call @[<3>omega_update_helicity_selection@ (hel_count,@ amp,@ ";
printf "hel_max_abs,@ hel_sum_abs,@ hel_is_allowed,@ hel_threshold,@ hel_cutoff,@ mask_dirty)"; nl ();
printf " if (mask_dirty) then"; nl ();
printf " hel_finite = 0"; nl ();
printf " do hel = 1, n_hel"; nl ();
printf " if (hel_is_allowed(hel)) then"; nl ();
printf " hel_finite = hel_finite + 1"; nl ();
printf " hel_map(hel_finite) = hel"; nl ();
printf " end if"; nl ();
printf " end do"; nl ();
printf " end if"; nl ();
printf " end if"; nl();
printf " end subroutine new_event"; nl ();
nl ();
printf " @[<5>";
printf "subroutine reset_helicity_selection (threshold, cutoff)"; nl ();
printf " real(kind=%s), intent(in) :: threshold" !kind; nl ();
printf " integer, intent(in) :: cutoff"; nl ();
printf " integer :: i"; nl ();
printf " hel_is_allowed = T"; nl ();
printf " hel_max_abs = 0"; nl ();
printf " hel_sum_abs = 0"; nl ();
printf " hel_count = 0"; nl ();
printf " hel_threshold = threshold"; nl ();
printf " hel_cutoff = cutoff"; nl ();
printf " hel_map = (/(i, i = 1, n_hel)/)"; nl ();
printf " hel_finite = n_hel"; nl ();
printf " end subroutine reset_helicity_selection"; nl ();
nl ();
printf " @[<5>"; if !fortran95 then printf "pure ";
printf "function is_allowed (flv, hel, col) result (yorn)"; nl ();
printf " logical :: yorn"; nl ();
printf " integer, intent(in) :: flv, hel, col"; nl ();
printf " yorn = hel_is_allowed(hel) .and. ";
printf "flv_col_is_allowed(flv,col)"; nl ();
printf " end function is_allowed"; nl ();
nl ();
printf " @[<5>"; if !fortran95 then printf "pure ";
printf "function get_amplitude (flv, hel, col) result (amp_result)"; nl ();
printf " complex(kind=%s) :: amp_result" !kind; nl ();
printf " integer, intent(in) :: flv, hel, col"; nl ();
printf " amp_result = amp(flv, col, hel)"; nl ();
printf " end function get_amplitude"; nl ();
nl ()
(* \thocwmodulesubsection{Main Function} *)
let format_power_of_nc
{ Color.Flow.num = num; Color.Flow.den = den; Color.Flow.power = pwr } =
match num, den, pwr with
| _, 0, _ -> invalid_arg "format_power_of_nc: zero denominator"
| 0, _, _ -> ""
| 1, 1, 0 | -1, -1, 0 -> "+ 1"
| -1, 1, 0 | 1, -1, 0 -> "- 1"
| 1, 1, 1 | -1, -1, 1 -> "+ N"
| -1, 1, 1 | 1, -1, 1 -> "- N"
| 1, 1, -1 | -1, -1, -1 -> "+ 1/N"
| -1, 1, -1 | 1, -1, -1 -> "- 1/N"
| 1, 1, p | -1, -1, p ->
"+ " ^ (if p > 0 then "" else "1/") ^ "N^" ^ string_of_int (abs p)
| -1, 1, p | 1, -1, p ->
"- " ^ (if p > 0 then "" else "1/") ^ "N^" ^ string_of_int (abs p)
| n, 1, 0 ->
(if n < 0 then "- " else "+ ") ^ string_of_int (abs n)
| n, d, 0 ->
(if n * d < 0 then "- " else "+ ") ^
string_of_int (abs n) ^ "/" ^ string_of_int (abs d)
| n, 1, 1 ->
(if n < 0 then "- " else "+ ") ^ string_of_int (abs n) ^ "N"
| n, 1, -1 ->
(if n < 0 then "- " else "+ ") ^ string_of_int (abs n) ^ "/N"
| n, d, 1 ->
(if n * d < 0 then "- " else "+ ") ^
string_of_int (abs n) ^ "/" ^ string_of_int (abs d) ^ "N"
| n, d, -1 ->
(if n * d < 0 then "- " else "+ ") ^
string_of_int (abs n) ^ "/" ^ string_of_int (abs d) ^ "/N"
| n, 1, p ->
(if n < 0 then "- " else "+ ") ^ string_of_int (abs n) ^
(if p > 0 then "*" else "/") ^ "N^" ^ string_of_int (abs p)
| n, d, p ->
(if n * d < 0 then "- " else "+ ") ^ string_of_int (abs n) ^ "/" ^
string_of_int (abs d) ^ (if p > 0 then "*" else "/") ^ "N^" ^ string_of_int (abs p)
let format_powers_of_nc = function
| [] -> "0"
| powers -> String.concat " " (List.map format_power_of_nc powers)
let print_description cmdline amplitudes () =
printf "! File generated automatically by O'Mega"; nl();
printf "!"; nl();
printf "! %s" cmdline; nl();
printf "!"; nl();
printf "! with all scattering amplitudes for the process(es)"; nl ();
printf "!"; nl ();
printf "! flavor combinations:"; nl ();
printf "!"; nl ();
ThoList.iteri
(fun i process ->
printf "! %3d: %s" i (process_sans_color_to_string process); nl ())
1 (CF.flavors amplitudes);
printf "!"; nl ();
printf "! color flows:"; nl ();
printf "!"; nl ();
ThoList.iteri
(fun i cflow ->
printf "! %3d: %s" i (cflow_to_string cflow); nl ())
1 (CF.color_flows amplitudes);
printf "!"; nl ();
printf "! NB: i.g. not all color flows contribute to all flavor"; nl ();
printf "! combinations. Consult the array FLV_COL_IS_ALLOWED"; nl ();
printf "! below for the allowed combinations."; nl ();
printf "!"; nl ();
printf "! Color Factors:"; nl ();
printf "!"; nl ();
let cfactors = CF.color_factors amplitudes in
for c1 = 0 to pred (Array.length cfactors) do
for c2 = 0 to c1 do
match cfactors.(c1).(c2) with
| [] -> ()
| cfactor ->
printf "! (%3d,%3d): %s"
(succ c1) (succ c2) (format_powers_of_nc cfactor); nl ()
done
done;
printf "!"; nl ();
printf "! vanishing or redundant flavor combinations:"; nl ();
printf "!"; nl ();
List.iter (fun process ->
printf "! %s" (process_sans_color_to_string process); nl ())
(CF.vanishing_flavors amplitudes);
printf "!"; nl ();
begin
match CF.constraints amplitudes with
| None -> ()
| Some s ->
printf
"! diagram selection (MIGHT BREAK GAUGE INVARIANCE!!!):"; nl ();
printf "!"; nl ();
printf "! %s" s; nl ();
printf "!"; nl ()
end;
begin match RCS.description M.rcs with
| line1 :: lines ->
printf "! in %s" line1; nl ();
List.iter (fun s -> printf "! %s" s; nl ()) lines
| [] -> printf "! in %s" (RCS.name M.rcs); nl ()
end;
printf "!"; nl ()
let print_version () =
printf "! O'Mega revision control information:"; nl ();
List.iter (fun s -> printf "! %s" s; nl ())
(ThoList.flatmap RCS.summary (M.rcs :: rcs_list @ F.rcs_list))
(* \thocwmodulesubsection{Printing Modules} *)
type accessibility =
| Public
| Private
| Protected (* Fortran 2003 *)
let accessibility_to_string = function
| Public -> "public"
| Private -> "private"
| Protected -> "protected"
type used_symbol =
| As_Is of string
| Aliased of string * string
let print_used_symbol = function
| As_Is name -> printf "%s" name
| Aliased (orig, alias) -> printf "%s => %s" alias orig
type used_module =
| Full of string
| Full_Aliased of string * (string * string) list
| Subset of string * used_symbol list
let print_used_module = function
| Full name
| Full_Aliased (name, [])
| Subset (name, []) ->
printf " use %s" name;
nl ()
| Full_Aliased (name, aliases) ->
printf " @[<5>use %s" name;
List.iter
(fun (orig, alias) -> printf ", %s => %s" alias orig)
aliases;
nl ()
| Subset (name, used_symbol :: used_symbols) ->
printf " @[<5>use %s, only: " name;
print_used_symbol used_symbol;
List.iter (fun s -> printf ", "; print_used_symbol s) used_symbols;
nl ()
type fortran_module =
{ module_name : string;
default_accessibility : accessibility;
used_modules : used_module list;
public_symbols : string list;
print_declarations : (unit -> unit) list;
print_implementations : (unit -> unit) list }
let print_public = function
| name1 :: names ->
printf " @[<2>public :: %s" name1;
List.iter (fun n -> printf ",@ %s" n) names; nl ()
| [] -> ()
let print_public_interface generic procedures =
printf " public :: %s" generic; nl ();
begin match procedures with
| name1 :: names ->
printf " interface %s" generic; nl ();
printf " @[<2>module procedure %s" name1;
List.iter (fun n -> printf ",@ %s" n) names; nl ();
printf " end interface"; nl ();
print_public procedures
| [] -> ()
end
let print_module m =
printf "module %s" m.module_name; nl ();
List.iter print_used_module m.used_modules;
printf " implicit none"; nl ();
printf " %s" (accessibility_to_string m.default_accessibility); nl ();
print_public m.public_symbols; nl ();
begin match m.print_declarations with
| [] -> ()
| print_declarations ->
List.iter (fun f -> f ()) print_declarations; nl ()
end;
begin match m.print_implementations with
| [] -> ()
| print_implementations ->
printf "contains"; nl (); nl ();
List.iter (fun f -> f ()) print_implementations; nl ();
end;
printf "end module %s" m.module_name; nl ()
let print_modules modules =
List.iter print_module modules;
print_version ();
print_flush ()
let module_to_file line_length oc prelude m =
output_string oc (m.module_name ^ "\n");
let filename = m.module_name ^ ".f90" in
let channel = open_out filename in
setup_fortran_formatter line_length channel;
prelude ();
print_modules [m];
close_out channel
let modules_to_file line_length oc prelude = function
| [] -> ()
| m :: mlist ->
module_to_file line_length oc prelude m;
List.iter (module_to_file line_length oc (fun () -> ())) mlist
(* \thocwmodulesubsection{Chopping Up Amplitudes} *)
let num_fusions_brakets size amplitudes =
let num_fusions =
max 1 size in
let count_brakets =
List.fold_left
(fun sum process -> sum + List.length (F.brakets process))
0 (CF.processes amplitudes)
and count_processes =
List.length (CF.processes amplitudes) in
if count_brakets > 0 then
let num_brakets =
max 1 ((num_fusions * count_processes) / count_brakets) in
(num_fusions, num_brakets)
else
(num_fusions, 1)
let chop_amplitudes size amplitudes =
let num_fusions, num_brakets = num_fusions_brakets size amplitudes in
(ThoList.enumerate 1 (ThoList.chopn num_fusions (CF.fusions amplitudes)),
ThoList.enumerate 1 (ThoList.chopn num_brakets (CF.processes amplitudes)))
let print_compute_fusions1 dictionary (n, fusions) =
if !openmp then begin
printf " subroutine compute_fusions_%04d (%s)" n openmp_tld; nl ();
printf " @[<5>type(%s), intent(inout) :: %s" openmp_tld_type openmp_tld; nl ();
end else begin
printf " @[<5>subroutine compute_fusions_%04d ()" n; nl ();
end;
print_fusions dictionary fusions;
printf " end subroutine compute_fusions_%04d" n; nl ()
and print_compute_brakets1 dictionary (n, processes) =
if !openmp then begin
printf " subroutine compute_brakets_%04d (%s)" n openmp_tld; nl ();
printf " @[<5>type(%s), intent(inout) :: %s" openmp_tld_type openmp_tld; nl ();
end else begin
printf " @[<5>subroutine compute_brakets_%04d ()" n; nl ();
end;
List.iter (print_brakets dictionary) processes;
printf " end subroutine compute_brakets_%04d" n; nl ()
(* \thocwmodulesubsection{Common Stuff} *)
let omega_public_symbols =
["number_particles_in"; "number_particles_out";
"number_color_indices";
"reset_helicity_selection"; "new_event";
"is_allowed"; "get_amplitude"; "color_sum"; "openmp_supported"] @
ThoList.flatmap
(fun n -> ["number_" ^ n; n])
["spin_states"; "flavor_states"; "color_flows"; "color_factors"]
let whizard_public_symbols md5sum =
["init"; "final"; "update_alpha_s"] @
(match md5sum with Some _ -> ["md5sum"] | None -> [])
let used_modules () =
[Full "kinds";
Full Fermions.use_module;
Full_Aliased ("omega_color", ["omega_color_factor", omega_color_factor_abbrev])] @
List.map
(fun m -> Full m)
(match !parameter_module with "" -> !use_modules | pm -> pm :: !use_modules)
let public_symbols () =
if !whizard then
omega_public_symbols @ (whizard_public_symbols !md5sum)
else
omega_public_symbols
let print_constants amplitudes =
printf " ! DON'T EVEN THINK of removing the following!"; nl ();
printf " ! If the compiler complains about undeclared"; nl ();
printf " ! or undefined variables, you are compiling"; nl ();
printf " ! against an incompatible omega95 module!"; nl ();
printf " @[<2>integer, dimension(%d), parameter, private :: "
(List.length require_library);
printf "require =@ (/ @[";
print_list require_library;
printf " /)"; nl(); nl ();
(* Using these parameters makes sense for documentation, but in
practice, there is no need to ever change them. *)
List.iter
(function name, value -> print_integer_parameter name (value amplitudes))
[ ("n_prt", num_particles);
("n_in", num_particles_in);
("n_out", num_particles_out);
("n_cflow", num_color_flows); (* Number of different color amplitudes. *)
("n_cindex", num_color_indices); (* Maximum rank of color tensors. *)
("n_flv", num_flavors); (* Number of different flavor amplitudes. *)
("n_hel", num_helicities) (* Number of different helicty amplitudes. *) ];
nl ();
(* Abbreviations. *)
printf " ! NB: you MUST NOT change the value of %s here!!!" nc_parameter; nl();
printf " ! It is defined here for convenience only and must be"; nl ();
printf " ! compatible with hardcoded values in the amplitude!"; nl ();
print_real_parameter nc_parameter (CM.nc ()); (* $N_C$ *)
List.iter
(function name, value -> print_logical_parameter name value)
[ ("F", false); ("T", true) ]; nl ();
print_spin_tables amplitudes;
print_flavor_tables amplitudes;
print_color_tables amplitudes;
print_amplitude_table amplitudes;
print_helicity_selection_table ()
let print_interface amplitudes =
print_md5sum_functions !md5sum;
print_maintenance_functions amplitudes;
List.iter print_numeric_inquiry_functions
[("number_particles_in", "n_in");
("number_particles_out", "n_out")];
List.iter print_inquiry_functions
["spin_states"; "flavor_states"];
print_inquiry_function_openmp ();
print_color_flows ();
print_color_factors ();
print_dispatch_functions ();
nl();
current_continuation_line := 0;
if !km_write || !km_pure then (Targets_Kmatrix.Fortran.print !km_pure);
current_continuation_line := 1;
nl()
let print_calculate_amplitudes declarations computations amplitudes =
printf " @[<5>subroutine calculate_amplitudes (amp, k, mask)"; nl ();
printf " complex(kind=%s), dimension(:,:,:), intent(out) :: amp" !kind; nl ();
printf " real(kind=%s), dimension(0:3,*), intent(in) :: k" !kind; nl ();
printf " logical, dimension(:), intent(in) :: mask"; nl ();
printf " integer, dimension(n_prt) :: s"; nl ();
printf " integer :: h, hi"; nl ();
declarations ();
begin match CF.processes amplitudes with
| p :: _ -> print_external_momenta p
| _ -> ()
end;
ignore (List.fold_left print_momenta PSet.empty (CF.processes amplitudes));
printf " amp = 0"; nl ();
if num_helicities amplitudes > 0 then begin
printf " if (hel_finite == 0) return"; nl ();
if !openmp then begin
printf "!$OMP PARALLEL DO DEFAULT(SHARED) PRIVATE(s, h, %s) SCHEDULE(STATIC)" openmp_tld; nl();
end;
printf " do hi = 1, hel_finite"; nl ();
printf " h = hel_map(hi)"; nl ();
printf " s = table_spin_states(:,h)"; nl ();
ignore (List.fold_left print_externals WFSet.empty (CF.processes amplitudes));
computations ();
List.iter print_fudge_factor (CF.processes amplitudes);
(* This sorting should slightly improve cache locality. *)
let triple_snd = fun (_, x, _) -> x
in let triple_fst = fun (x, _, _) -> x
in let rec builder1 flvi flowi flows = match flows with
| (Some a) :: tl -> (flvi, flowi, flavors_symbol (flavors a)) :: (builder1 flvi (flowi + 1) tl)
| None :: tl -> builder1 flvi (flowi + 1) tl
| [] -> []
in let rec builder2 flvi flvs = match flvs with
| flv :: tl -> (builder1 flvi 1 flv) @ (builder2 (flvi + 1) tl)
| [] -> []
in let unsorted = builder2 1 (List.map Array.to_list (Array.to_list (CF.process_table amplitudes)))
in let sorted = List.sort (fun a b ->
if (triple_snd a != triple_snd b) then triple_snd a - triple_snd b else (triple_fst a - triple_fst b))
unsorted
in List.iter (fun (flvi, flowi, flv) ->
(printf " amp(%d,%d,h) = %s" flvi flowi flv; nl ();)) sorted;
(*i printf " else"; nl ();
printf " amp(:,h,:) = 0"; nl (); i*)
printf " end do"; nl ();
if !openmp then begin
printf "!$OMP END PARALLEL DO"; nl ();
end;
end;
printf " end subroutine calculate_amplitudes"; nl ()
let print_compute_chops chopped_fusions chopped_brakets () =
List.iter
(fun (i, _) -> printf " call compute_fusions_%04d (%s)" i
(if !openmp then openmp_tld else ""); nl ())
chopped_fusions;
List.iter
(fun (i, _) -> printf " call compute_brakets_%04d (%s)" i
(if !openmp then openmp_tld else ""); nl ())
chopped_brakets
(* \thocwmodulesubsection{Single Function} *)
let amplitudes_to_channel_single_function cmdline oc amplitudes =
let print_declarations () =
print_constants amplitudes
and print_implementations () =
print_interface amplitudes;
print_calculate_amplitudes
(fun () -> print_variable_declarations amplitudes)
(fun () ->
print_fusions (CF.dictionary amplitudes) (CF.fusions amplitudes);
List.iter
(print_brakets (CF.dictionary amplitudes))
(CF.processes amplitudes))
amplitudes in
let fortran_module =
{ module_name = !module_name;
used_modules = used_modules ();
default_accessibility = Private;
public_symbols = public_symbols ();
print_declarations = [print_declarations];
print_implementations = [print_implementations] } in
setup_fortran_formatter !line_length oc;
print_description cmdline amplitudes ();
print_modules [fortran_module]
(* \thocwmodulesubsection{Single Module} *)
let amplitudes_to_channel_single_module cmdline oc size amplitudes =
let print_declarations () =
print_constants amplitudes;
print_variable_declarations amplitudes
and print_implementations () =
print_interface amplitudes in
let chopped_fusions, chopped_brakets =
chop_amplitudes size amplitudes in
let dictionary = CF.dictionary amplitudes in
let print_compute_amplitudes () =
print_calculate_amplitudes
(fun () -> ())
(print_compute_chops chopped_fusions chopped_brakets)
amplitudes
and print_compute_fusions () =
List.iter (print_compute_fusions1 dictionary) chopped_fusions
and print_compute_brakets () =
List.iter (print_compute_brakets1 dictionary) chopped_brakets in
let fortran_module =
{ module_name = !module_name;
used_modules = used_modules ();
default_accessibility = Private;
public_symbols = public_symbols ();
print_declarations = [print_declarations];
print_implementations = [print_implementations;
print_compute_amplitudes;
print_compute_fusions;
print_compute_brakets] } in
setup_fortran_formatter !line_length oc;
print_description cmdline amplitudes ();
print_modules [fortran_module]
(* \thocwmodulesubsection{Multiple Modules} *)
let modules_of_amplitudes cmdline oc size amplitudes =
let name = !module_name in
let print_declarations () =
print_constants amplitudes
and print_variables () =
print_variable_declarations amplitudes in
let constants_module =
{ module_name = name ^ "_constants";
used_modules = used_modules ();
default_accessibility = Public;
public_symbols = [];
print_declarations = [print_declarations];
print_implementations = [] } in
let variables_module =
{ module_name = name ^ "_variables";
used_modules = used_modules ();
default_accessibility = Public;
public_symbols = [];
print_declarations = [print_variables];
print_implementations = [] } in
let dictionary = CF.dictionary amplitudes in
let print_compute_fusions (n, fusions) () =
if !openmp then begin
printf " subroutine compute_fusions_%04d (%s)" n openmp_tld; nl ();
printf " @[<5>type(%s), intent(inout) :: %s" openmp_tld_type openmp_tld; nl ();
end else begin
printf " @[<5>subroutine compute_fusions_%04d ()" n; nl ();
end;
print_fusions dictionary fusions;
printf " end subroutine compute_fusions_%04d" n; nl () in
let print_compute_brakets (n, processes) () =
if !openmp then begin
printf " subroutine compute_brakets_%04d (%s)" n openmp_tld; nl ();
printf " @[<5>type(%s), intent(inout) :: %s" openmp_tld_type openmp_tld; nl ();
end else begin
printf " @[<5>subroutine compute_brakets_%04d ()" n; nl ();
end;
List.iter (print_brakets dictionary) processes;
printf " end subroutine compute_brakets_%04d" n; nl () in
let fusions_module (n, _ as fusions) =
let tag = Printf.sprintf "_fusions_%04d" n in
{ module_name = name ^ tag;
used_modules = (used_modules () @
[Full constants_module.module_name;
Full variables_module.module_name]);
default_accessibility = Private;
public_symbols = ["compute" ^ tag];
print_declarations = [];
print_implementations = [print_compute_fusions fusions] } in
let brakets_module (n, _ as processes) =
let tag = Printf.sprintf "_brakets_%04d" n in
{ module_name = name ^ tag;
used_modules = (used_modules () @
[Full constants_module.module_name;
Full variables_module.module_name]);
default_accessibility = Private;
public_symbols = ["compute" ^ tag];
print_declarations = [];
print_implementations = [print_compute_brakets processes] } in
let chopped_fusions, chopped_brakets =
chop_amplitudes size amplitudes in
let fusions_modules =
List.map fusions_module chopped_fusions in
let brakets_modules =
List.map brakets_module chopped_brakets in
let print_implementations () =
print_interface amplitudes;
print_calculate_amplitudes
(fun () -> ())
(print_compute_chops chopped_fusions chopped_brakets)
amplitudes in
let public_module =
{ module_name = name;
used_modules = (used_modules () @
[Full constants_module.module_name;
Full variables_module.module_name ] @
List.map
(fun m -> Full m.module_name)
(fusions_modules @ brakets_modules));
default_accessibility = Private;
public_symbols = public_symbols ();
print_declarations = [];
print_implementations = [print_implementations] }
and private_modules =
[constants_module; variables_module] @ fusions_modules @ brakets_modules in
(public_module, private_modules)
let amplitudes_to_channel_single_file cmdline oc size amplitudes =
let public_module, private_modules =
modules_of_amplitudes cmdline oc size amplitudes in
setup_fortran_formatter !line_length oc;
print_description cmdline amplitudes ();
print_modules (private_modules @ [public_module])
let amplitudes_to_channel_multi_file cmdline oc size amplitudes =
let public_module, private_modules =
modules_of_amplitudes cmdline oc size amplitudes in
modules_to_file !line_length oc
(print_description cmdline amplitudes)
(public_module :: private_modules)
(* \thocwmodulesubsection{Dispatch} *)
let amplitudes_to_channel cmdline oc diagnostics amplitudes =
parse_diagnostics diagnostics;
match !output_mode with
| Single_Function ->
amplitudes_to_channel_single_function cmdline oc amplitudes
| Single_Module size ->
amplitudes_to_channel_single_module cmdline oc size amplitudes
| Single_File size ->
amplitudes_to_channel_single_file cmdline oc size amplitudes
| Multi_File size ->
amplitudes_to_channel_multi_file cmdline oc size amplitudes
let parameters_to_channel oc =
parameters_to_fortran oc (CM.parameters ())
end
module Fortran = Make_Fortran(Fortran_Fermions)
(* \thocwmodulesubsection{Majorana Fermions} *)
(* \begin{JR}
For this function we need a different approach due to our aim of
implementing the fermion vertices with the right line as ingoing (in a
calculational sense) and the left line in a fusion as outgoing. In
defining all external lines and the fermionic wavefunctions built out of
them as ingoing we have to invert the left lines to make them outgoing.
This happens by multiplying them with the inverse charge conjugation
matrix in an appropriate representation and then transposing it. We must
distinguish whether the direction of calculation and the physical direction
of the fermion number flow are parallel or antiparallel. In the first case
we can use the "normal" Feynman rules for Dirac particles, while in the
second, according to the paper of Denner et al., we have to reverse the
sign of the vector and antisymmetric bilinears of the Dirac spinors, cf.
the [Coupling] module.
Note the subtlety for the left- and righthanded couplings: Only the vector
part of these couplings changes in the appropriate cases its sign,
changing the chirality to the negative of the opposite.
\end{JR} *)
module Fortran_Majorana_Fermions : Fermions =
struct
let rcs = RCS.rename rcs_file "Targets.Fortran_Majorana_Fermions()"
[ "generates Fortran95 code for Dirac and Majorana fermions";
" using revision 2003_03_A of module omega95_bispinors" ]
open Coupling
open Format
let psi_type = "bispinor"
let psibar_type = "bispinor"
let chi_type = "bispinor"
let grav_type = "vectorspinor"
(* \begin{JR}
Because of our rules for fermions we are going to give all incoming fermions
a [u] spinor and all outgoing fermions a [v] spinor, no matter whether they
are Dirac fermions, antifermions or Majorana fermions.
\end{JR} *)
let psi_incoming = "u"
let brs_psi_incoming = "brs_u"
let psibar_incoming = "u"
let brs_psibar_incoming = "brs_u"
let chi_incoming = "u"
let brs_chi_incoming = "brs_u"
let grav_incoming = "ueps"
let psi_outgoing = "v"
let brs_psi_outgoing = "brs_v"
let psibar_outgoing = "v"
let brs_psibar_outgoing = "brs_v"
let chi_outgoing = "v"
let brs_chi_outgoing = "brs_v"
let grav_outgoing = "veps"
let psi_propagator = "pr_psi"
let psibar_propagator = "pr_psi"
let chi_propagator = "pr_psi"
let grav_propagator = "pr_grav"
let psi_projector = "pj_psi"
let psibar_projector = "pj_psi"
let chi_projector = "pj_psi"
let grav_projector = "pj_grav"
let psi_gauss = "pg_psi"
let psibar_gauss = "pg_psi"
let chi_gauss = "pg_psi"
let grav_gauss = "pg_grav"
let format_coupling coeff c =
match coeff with
| 1 -> c
| -1 -> "(-" ^ c ^")"
| coeff -> string_of_int coeff ^ "*" ^ c
let format_coupling_2 coeff c =
match coeff with
| 1 -> c
| -1 -> "-" ^ c
| coeff -> string_of_int coeff ^ "*" ^ c
(* \begin{dubious}
JR's coupling constant HACK, necessitated by tho's bad design descition.
\end{dubious} *)
let fastener s i =
try
let offset = (String.index s '(') in
if ((String.get s (String.length s - 1)) != ')') then
failwith "fastener: wrong usage of parentheses"
else
let func_name = (String.sub s 0 offset) and
tail =
(String.sub s (succ offset) (String.length s - offset - 2)) in
if (String.contains func_name ')') or
(String.contains tail '(') or
(String.contains tail ')') then
failwith "fastener: wrong usage of parentheses"
else
func_name ^ "(" ^ string_of_int i ^ "," ^ tail ^ ")"
with
| Not_found ->
if (String.contains s ')') then
failwith "fastener: wrong usage of parentheses"
else
s ^ "(" ^ string_of_int i ^ ")"
let print_fermion_current coeff f c wf1 wf2 fusion =
let c = format_coupling coeff c in
match fusion with
| F13 | F31 -> printf "%s_ff(%s,%s,%s)" f c wf1 wf2
| F23 | F21 -> printf "f_%sf(%s,%s,%s)" f c wf1 wf2
| F32 | F12 -> printf "f_%sf(%s,%s,%s)" f c wf2 wf1
let print_fermion_current2 coeff f c wf1 wf2 fusion =
let c = format_coupling_2 coeff c in
let c1 = fastener c 1 and
c2 = fastener c 2 in
match fusion with
| F13 | F31 -> printf "%s_ff(%s,%s,%s,%s)" f c1 c2 wf1 wf2
| F23 | F21 -> printf "f_%sf(%s,%s,%s,%s)" f c1 c2 wf1 wf2
| F32 | F12 -> printf "f_%sf(%s,%s,%s,%s)" f c1 c2 wf2 wf1
let print_fermion_current_vector coeff f c wf1 wf2 fusion =
let c = format_coupling coeff c in
match fusion with
| F13 -> printf "%s_ff(%s,%s,%s)" f c wf1 wf2
| F31 -> printf "%s_ff(-%s,%s,%s)" f c wf1 wf2
| F23 -> printf "f_%sf(%s,%s,%s)" f c wf1 wf2
| F32 -> printf "f_%sf(%s,%s,%s)" f c wf2 wf1
| F12 -> printf "f_%sf(-%s,%s,%s)" f c wf2 wf1
| F21 -> printf "f_%sf(-%s,%s,%s)" f c wf1 wf2
let print_fermion_current2_vector coeff f c wf1 wf2 fusion =
let c = format_coupling_2 coeff c in
let c1 = fastener c 1 and
c2 = fastener c 2 in
match fusion with
| F13 -> printf "%s_ff(%s,%s,%s,%s)" f c1 c2 wf1 wf2
| F31 -> printf "%s_ff(-(%s),%s,%s,%s)" f c1 c2 wf1 wf2
| F23 -> printf "f_%sf(%s,%s,%s,%s)" f c1 c2 wf1 wf2
| F32 -> printf "f_%sf(%s,%s,%s,%s)" f c1 c2 wf2 wf1
| F12 -> printf "f_%sf(-(%s),%s,%s,%s)" f c1 c2 wf2 wf1
| F21 -> printf "f_%sf(-(%s),%s,%s,%s)" f c1 c2 wf1 wf2
let print_fermion_current_chiral coeff f1 f2 c wf1 wf2 fusion =
let c = format_coupling coeff c in
match fusion with
| F13 -> printf "%s_ff(%s,%s,%s)" f1 c wf1 wf2
| F31 -> printf "%s_ff(-%s,%s,%s)" f2 c wf1 wf2
| F23 -> printf "f_%sf(%s,%s,%s)" f1 c wf1 wf2
| F32 -> printf "f_%sf(%s,%s,%s)" f1 c wf2 wf1
| F12 -> printf "f_%sf(-%s,%s,%s)" f2 c wf2 wf1
| F21 -> printf "f_%sf(-%s,%s,%s)" f2 c wf1 wf2
let print_fermion_current2_chiral coeff f c wf1 wf2 fusion =
let c = format_coupling_2 coeff c in
let c1 = fastener c 1 and
c2 = fastener c 2 in
match fusion with
| F13 -> printf "%s_ff(%s,%s,%s,%s)" f c1 c2 wf1 wf2
| F31 -> printf "%s_ff(-(%s),-(%s),%s,%s)" f c2 c1 wf1 wf2
| F23 -> printf "f_%sf(%s,%s,%s,%s)" f c1 c2 wf1 wf2
| F32 -> printf "f_%sf(%s,%s,%s,%s)" f c1 c2 wf2 wf1
| F12 -> printf "f_%sf(-(%s),-(%s),%s,%s)" f c2 c1 wf2 wf1
| F21 -> printf "f_%sf(-(%s),-(%s),%s,%s)" f c2 c1 wf1 wf2
let print_current = function
| coeff, _, VA, _ -> print_fermion_current2_vector coeff "va"
| coeff, _, V, _ -> print_fermion_current_vector coeff "v"
| coeff, _, A, _ -> print_fermion_current coeff "a"
| coeff, _, VL, _ -> print_fermion_current_chiral coeff "vl" "vr"
| coeff, _, VR, _ -> print_fermion_current_chiral coeff "vr" "vl"
| coeff, _, VLR, _ -> print_fermion_current2_chiral coeff "vlr"
| coeff, _, SP, _ -> print_fermion_current2 coeff "sp"
| coeff, _, S, _ -> print_fermion_current coeff "s"
| coeff, _, P, _ -> print_fermion_current coeff "p"
| coeff, _, SL, _ -> print_fermion_current coeff "sl"
| coeff, _, SR, _ -> print_fermion_current coeff "sr"
| coeff, _, SLR, _ -> print_fermion_current2 coeff "slr"
| coeff, _, POT, _ -> print_fermion_current_vector coeff "pot"
| coeff, _, _, _ -> invalid_arg
"Targets.Fortran_Majorana_Fermions: Not needed in the models"
let print_current_p = function
| coeff, Psi, SL, Psi -> print_fermion_current coeff "sl"
| coeff, Psi, SR, Psi -> print_fermion_current coeff "sr"
| coeff, Psi, SLR, Psi -> print_fermion_current2 coeff "slr"
| coeff, _, _, _ -> invalid_arg
"Targets.Fortran_Majorana_Fermions: Not needed in the used models"
let print_current_b = function
| coeff, Psibar, SL, Psibar -> print_fermion_current coeff "sl"
| coeff, Psibar, SR, Psibar -> print_fermion_current coeff "sr"
| coeff, Psibar, SLR, Psibar -> print_fermion_current2 coeff "slr"
| coeff, _, _, _ -> invalid_arg
"Targets.Fortran_Majorana_Fermions: Not needed in the used models"
(* This function is for the vertices with three particles including two
fermions but also a momentum, therefore with a dimensionful coupling
constant, e.g. the gravitino vertices. One has to dinstinguish between
the two kinds of canonical orders in the string of gamma matrices. Of
course, the direction of the string of gamma matrices is reversed if one
goes from the [Gravbar, _, Psi] to the [Psibar, _, Grav] vertices, and
the same is true for the couplings of the gravitino to the Majorana
fermions. For more details see the tables in the [coupling]
implementation. *)
(* We now have to fix the directions of the momenta. For making the compiler
happy and because we don't want to make constructions of infinite
complexity we list the momentum including vertices without gravitinos
here; the pattern matching says that's better. Perhaps we have to find a
better name now.
For the cases of $MOM$, $MOM5$, $MOML$ and $MOMR$ which arise only in
BRST transformations we take the mass as a coupling constant. For
$VMOM$ we don't need a mass either. These vertices are like kinetic terms
and so need not have a coupling constant. By this we avoid a strange and
awful construction with a new variable. But be careful with a
generalization if you want to use these vertices for other purposes.
*)
let format_coupling_mom coeff c =
match coeff with
| 1 -> c
| -1 -> "(-" ^ c ^")"
| coeff -> string_of_int coeff ^ "*" ^ c
let commute_proj f =
match f with
| "moml" -> "lmom"
| "momr" -> "rmom"
| "lmom" -> "moml"
| "rmom" -> "momr"
| "svl" -> "svr"
| "svr" -> "svl"
| "sl" -> "sr"
| "sr" -> "sl"
| "s" -> "s"
| "p" -> "p"
| _ -> invalid_arg "Targets:Fortran_Majorana_Fermions: wrong case"
let print_fermion_current_mom coeff f c wf1 wf2 p1 p2 p12 fusion =
let c = format_coupling_mom coeff c in
let c1 = fastener c 1 and
c2 = fastener c 2 in
match fusion with
| F13 -> printf "%s_ff(%s,%s,%s,%s,%s)" f c1 c2 wf1 wf2 p12
| F31 -> printf "%s_ff(%s,%s,%s,%s,%s)" f c1 c2 wf1 wf2 p12
| F23 -> printf "f_%sf(%s,%s,%s,%s,%s)" f c1 c2 wf1 wf2 p1
| F32 -> printf "f_%sf(%s,%s,%s,%s,%s)" f c1 c2 wf2 wf1 p2
| F12 -> printf "f_%sf(%s,%s,%s,%s,%s)" f c1 c2 wf2 wf1 p2
| F21 -> printf "f_%sf(%s,%s,%s,%s,%s)" f c1 c2 wf1 wf2 p1
let print_fermion_current_mom_vector coeff f c wf1 wf2 p1 p2 p12 fusion =
let c = format_coupling_mom coeff c in
let c1 = fastener c 1 and
c2 = fastener c 2 in
match fusion with
| F13 -> printf "%s_ff(%s,%s,%s,%s,%s)" f c1 c2 wf1 wf2 p12
| F31 -> printf "%s_ff(-%s,%s,%s,%s,%s)" f c1 c2 wf1 wf2 p12
| F23 -> printf "f_%sf(%s,%s,%s,%s,%s)" f c1 c2 wf1 wf2 p1
| F32 -> printf "f_%sf(%s,%s,%s,%s,%s)" f c1 c2 wf2 wf1 p2
| F12 -> printf "f_%sf(-%s,%s,%s,%s,%s)" f c1 c2 wf2 wf1 p2
| F21 -> printf "f_%sf(-%s,%s,%s,%s,%s)" f c1 c2 wf1 wf2 p1
let print_fermion_current_mom_sign coeff f c wf1 wf2 p1 p2 p12 fusion =
let c = format_coupling_mom coeff c in
let c1 = fastener c 1 and
c2 = fastener c 2 in
match fusion with
| F13 -> printf "%s_ff(%s,%s,%s,%s,%s)" f c1 c2 wf1 wf2 p12
| F31 -> printf "%s_ff(%s,%s,%s,%s,-(%s))" f c1 c2 wf1 wf2 p12
| F23 -> printf "f_%sf(%s,%s,%s,%s,%s)" f c1 c2 wf1 wf2 p1
| F32 -> printf "f_%sf(%s,%s,%s,%s,%s)" f c1 c2 wf2 wf1 p2
| F12 -> printf "f_%sf(%s,%s,%s,%s,-(%s))" f c1 c2 wf2 wf1 p2
| F21 -> printf "f_%sf(%s,%s,%s,%s,-(%s))" f c1 c2 wf1 wf2 p1
let print_fermion_current_mom_sign_1 coeff f c wf1 wf2 p1 p2 p12 fusion =
let c = format_coupling coeff c in
match fusion with
| F13 -> printf "%s_ff(%s,%s,%s,%s)" f c wf1 wf2 p12
| F31 -> printf "%s_ff(%s,%s,%s,-(%s))" f c wf1 wf2 p12
| F23 -> printf "f_%sf(%s,%s,%s,%s)" f c wf1 wf2 p1
| F32 -> printf "f_%sf(%s,%s,%s,%s)" f c wf2 wf1 p2
| F12 -> printf "f_%sf(%s,%s,%s,-(%s))" f c wf2 wf1 p2
| F21 -> printf "f_%sf(%s,%s,%s,-(%s))" f c wf1 wf2 p1
let print_fermion_current_mom_chiral coeff f c wf1 wf2 p1 p2 p12 fusion =
let c = format_coupling_mom coeff c and
cf = commute_proj f in
let c1 = fastener c 1 and
c2 = fastener c 2 in
match fusion with
| F13 -> printf "%s_ff(%s,%s,%s,%s,%s)" f c1 c2 wf1 wf2 p12
| F31 -> printf "%s_ff(%s,%s,%s, %s,-(%s))" cf c1 c2 wf1 wf2 p12
| F23 -> printf "f_%sf(%s,%s,%s,%s,%s)" f c1 c2 wf1 wf2 p1
| F32 -> printf "f_%sf(%s,%s,%s,%s,%s)" f c1 c2 wf2 wf1 p2
| F12 -> printf "f_%sf(%s,%s,%s,%s,-(%s))" cf c1 c2 wf2 wf1 p2
| F21 -> printf "f_%sf(%s,%s,%s,%s,-(%s))" cf c1 c2 wf1 wf2 p1
let print_fermion_g_current coeff f c wf1 wf2 p1 p2 p12 fusion =
let c = format_coupling coeff c in
match fusion with
| F13 -> printf "%s_grf(%s,%s,%s,%s)" f c wf1 wf2 p12
| F31 -> printf "%s_fgr(%s,%s,%s,%s)" f c wf1 wf2 p12
| F23 -> printf "gr_%sf(%s,%s,%s,%s)" f c wf1 wf2 p1
| F32 -> printf "gr_%sf(%s,%s,%s,%s)" f c wf2 wf1 p2
| F12 -> printf "f_%sgr(%s,%s,%s,%s)" f c wf2 wf1 p2
| F21 -> printf "f_%sgr(%s,%s,%s,%s)" f c wf1 wf2 p1
let print_fermion_g_2_current coeff f c wf1 wf2 p1 p2 p12 fusion =
let c = format_coupling coeff c in
match fusion with
| F13 -> printf "%s_grf(%s(1),%s(2),%s,%s,%s)" f c c wf1 wf2 p12
| F31 -> printf "%s_fgr(%s(1),%s(2),%s,%s,%s)" f c c wf1 wf2 p12
| F23 -> printf "gr_%sf(%s(1),%s(2),%s,%s,%s)" f c c wf1 wf2 p1
| F32 -> printf "gr_%sf(%s(1),%s(2),%s,%s,%s)" f c c wf2 wf1 p2
| F12 -> printf "f_%sgr(%s(1),%s(2),%s,%s,%s)" f c c wf2 wf1 p2
| F21 -> printf "f_%sgr(%s(1),%s(2),%s,%s,%s)" f c c wf1 wf2 p1
let print_fermion_g_current_rev coeff f c wf1 wf2 p1 p2 p12 fusion =
let c = format_coupling coeff c in
match fusion with
| F13 -> printf "%s_fgr(%s,%s,%s,%s)" f c wf1 wf2 p12
| F31 -> printf "%s_grf(%s,%s,%s,%s)" f c wf1 wf2 p12
| F23 -> printf "f_%sgr(%s,%s,%s,%s)" f c wf1 wf2 p1
| F32 -> printf "f_%sgr(%s,%s,%s,%s)" f c wf2 wf1 p2
| F12 -> printf "gr_%sf(%s,%s,%s,%s)" f c wf2 wf1 p2
| F21 -> printf "gr_%sf(%s,%s,%s,%s)" f c wf1 wf2 p1
let print_fermion_g_2_current_rev coeff f c wf1 wf2 p1 p2 p12 fusion =
let c = format_coupling coeff c in
match fusion with
| F13 -> printf "%s_fgr(%s(1),%s(2),%s,%s,%s)" f c c wf1 wf2 p12
| F31 -> printf "%s_grf(%s(1),%s(2),%s,%s,%s)" f c c wf1 wf2 p12
| F23 -> printf "f_%sgr(%s(1),%s(2),%s,%s,%s)" f c c wf1 wf2 p1
| F32 -> printf "f_%sgr(%s(1),%s(2),%s,%s,%s)" f c c wf2 wf1 p2
| F12 -> printf "gr_%sf(%s(1),%s(2),%s,%s,%s)" f c c wf2 wf1 p2
| F21 -> printf "gr_%sf(%s(1),%s(2),%s,%s,%s)" f c c wf1 wf2 p1
let print_fermion_g_current_vector coeff f c wf1 wf2 p1 p2 p12 fusion =
let c = format_coupling coeff c in
match fusion with
| F13 -> printf "%s_grf(%s,%s,%s)" f c wf1 wf2
| F31 -> printf "%s_fgr(-%s,%s,%s)" f c wf1 wf2
| F23 -> printf "gr_%sf(%s,%s,%s)" f c wf1 wf2
| F32 -> printf "gr_%sf(%s,%s,%s)" f c wf2 wf1
| F12 -> printf "f_%sgr(-%s,%s,%s)" f c wf2 wf1
| F21 -> printf "f_%sgr(-%s,%s,%s)" f c wf1 wf2
let print_fermion_g_current_vector_rev coeff f c wf1 wf2 p1 p2 p12 fusion =
let c = format_coupling coeff c in
match fusion with
| F13 -> printf "%s_fgr(%s,%s,%s)" f c wf1 wf2
| F31 -> printf "%s_grf(-%s,%s,%s)" f c wf1 wf2
| F23 -> printf "f_%sgr(%s,%s,%s)" f c wf1 wf2
| F32 -> printf "f_%sgr(%s,%s,%s)" f c wf2 wf1
| F12 -> printf "gr_%sf(-%s,%s,%s)" f c wf2 wf1
| F21 -> printf "gr_%sf(-%s,%s,%s)" f c wf1 wf2
let print_current_g = function
| coeff, _, MOM, _ -> print_fermion_current_mom_sign coeff "mom"
| coeff, _, MOM5, _ -> print_fermion_current_mom coeff "mom5"
| coeff, _, MOML, _ -> print_fermion_current_mom_chiral coeff "moml"
| coeff, _, MOMR, _ -> print_fermion_current_mom_chiral coeff "momr"
| coeff, _, LMOM, _ -> print_fermion_current_mom_chiral coeff "lmom"
| coeff, _, RMOM, _ -> print_fermion_current_mom_chiral coeff "rmom"
| coeff, _, VMOM, _ -> print_fermion_current_mom_sign_1 coeff "vmom"
| coeff, Gravbar, S, _ -> print_fermion_g_current coeff "s"
| coeff, Gravbar, SL, _ -> print_fermion_g_current coeff "sl"
| coeff, Gravbar, SR, _ -> print_fermion_g_current coeff "sr"
| coeff, Gravbar, SLR, _ -> print_fermion_g_2_current coeff "slr"
| coeff, Gravbar, P, _ -> print_fermion_g_current coeff "p"
| coeff, Gravbar, V, _ -> print_fermion_g_current coeff "v"
| coeff, Gravbar, VLR, _ -> print_fermion_g_2_current coeff "vlr"
| coeff, Gravbar, POT, _ -> print_fermion_g_current_vector coeff "pot"
| coeff, _, S, Grav -> print_fermion_g_current_rev coeff "s"
| coeff, _, SL, Grav -> print_fermion_g_current_rev coeff "sl"
| coeff, _, SR, Grav -> print_fermion_g_current_rev coeff "sr"
| coeff, _, SLR, Grav -> print_fermion_g_2_current_rev coeff "slr"
| coeff, _, P, Grav -> print_fermion_g_current_rev (-coeff) "p"
| coeff, _, V, Grav -> print_fermion_g_current_rev coeff "v"
| coeff, _, VLR, Grav -> print_fermion_g_2_current_rev coeff "vlr"
| coeff, _, POT, Grav -> print_fermion_g_current_vector_rev coeff "pot"
| coeff, _, _, _ -> invalid_arg
"Targets.Fortran_Majorana_Fermions: not used in the models"
let print_current_mom = function
| coeff, _, _, _ -> invalid_arg
"Targets.Fortran_Majorana_Fermions: Not needed in the models"
(* We need support for dimension-5 vertices with two fermions and two
bosons, appearing in theories of supergravity and also together with in
insertions of the supersymmetric current. There is a canonical order
[fermionbar], [boson_1], [boson_2], [fermion], so what one has to do is a
mapping from the fusions [F123] etc. to the order of the three wave
functions [wf1], [wf2] and [wf3]. *)
(* The function [d_p] (for distinct the particle) distinguishes which particle
(scalar or vector) must be fused to in the special functions. *)
let d_p = function
| 1, ("sv"|"pv"|"svl"|"svr"|"slrv") -> "1"
| 1, _ -> ""
| 2, ("sv"|"pv"|"svl"|"svr"|"slrv") -> "2"
| 2, _ -> ""
| _, _ -> invalid_arg "Targets.Fortran_Majorana_Fermions: not used"
let wf_of_f wf1 wf2 wf3 f =
match f with
| (F123|F423) -> [wf2; wf3; wf1]
| (F213|F243|F143|F142|F413|F412) -> [wf1; wf3; wf2]
| (F132|F432) -> [wf3; wf2; wf1]
| (F231|F234|F134|F124|F431|F421) -> [wf1; wf2; wf3]
| (F312|F342) -> [wf3; wf1; wf2]
| (F321|F324|F314|F214|F341|F241) -> [wf2; wf1; wf3]
let print_fermion_g4_brs_vector_current coeff f c wf1 wf2 wf3 fusion =
let cf = commute_proj f and
cp = format_coupling coeff c and
cm = if f = "pv" then
format_coupling coeff c
else
format_coupling (-coeff) c
and
d1 = d_p (1,f) and
d2 = d_p (2,f) and
f1 = (List.nth (wf_of_f wf1 wf2 wf3 fusion) 0) and
f2 = (List.nth (wf_of_f wf1 wf2 wf3 fusion) 1) and
f3 = (List.nth (wf_of_f wf1 wf2 wf3 fusion) 2) in
match fusion with
| (F123|F213|F132|F231|F312|F321) ->
printf "f_%sf(%s,%s,%s,%s)" cf cm f1 f2 f3
| (F423|F243|F432|F234|F342|F324) ->
printf "f_%sf(%s,%s,%s,%s)" f cp f1 f2 f3
| (F134|F143|F314) -> printf "%s%s_ff(%s,%s,%s,%s)" f d1 cp f1 f2 f3
| (F124|F142|F214) -> printf "%s%s_ff(%s,%s,%s,%s)" f d2 cp f1 f2 f3
| (F413|F431|F341) -> printf "%s%s_ff(%s,%s,%s,%s)" cf d1 cm f1 f2 f3
| (F241|F412|F421) -> printf "%s%s_ff(%s,%s,%s,%s)" cf d2 cm f1 f2 f3
let print_fermion_g4_svlr_current coeff f c wf1 wf2 wf3 fusion =
let c = format_coupling_2 coeff c and
f1 = (List.nth (wf_of_f wf1 wf2 wf3 fusion) 0) and
f2 = (List.nth (wf_of_f wf1 wf2 wf3 fusion) 1) and
f3 = (List.nth (wf_of_f wf1 wf2 wf3 fusion) 2) in
let c1 = fastener c 1 and
c2 = fastener c 2 in
match fusion with
| (F123|F213|F132|F231|F312|F321) ->
printf "f_svlrf(-(%s),-(%s),%s,%s,%s)" c2 c1 f1 f2 f3
| (F423|F243|F432|F234|F342|F324) ->
printf "f_svlrf(%s,%s,%s,%s,%s)" c1 c2 f1 f2 f3
| (F134|F143|F314) ->
printf "svlr2_ff(%s,%s,%s,%s,%s)" c1 c2 f1 f2 f3
| (F124|F142|F214) ->
printf "svlr1_ff(%s,%s,%s,%s,%s)" c1 c2 f1 f2 f3
| (F413|F431|F341) ->
printf "svlr2_ff(-(%s),-(%s),%s,%s,%s)" c2 c1 f1 f2 f3
| (F241|F412|F421) ->
printf "svlr1_ff(-(%s),-(%s),%s,%s,%s)" c2 c1 f1 f2 f3
let print_fermion_s2_current coeff f c wf1 wf2 wf3 fusion =
let cp = format_coupling coeff c and
cm = if f = "p" then
format_coupling (-coeff) c
else
format_coupling coeff c
and
cf = commute_proj f and
f1 = (List.nth (wf_of_f wf1 wf2 wf3 fusion) 0) and
f2 = (List.nth (wf_of_f wf1 wf2 wf3 fusion) 1) and
f3 = (List.nth (wf_of_f wf1 wf2 wf3 fusion) 2) in
match fusion with
| (F123|F213|F132|F231|F312|F321) ->
printf "%s * f_%sf(%s,%s,%s)" f1 cf cm f2 f3
| (F423|F243|F432|F234|F342|F324) ->
printf "%s * f_%sf(%s,%s,%s)" f1 f cp f2 f3
| (F134|F143|F314) ->
printf "%s * %s_ff(%s,%s,%s)" f2 f cp f1 f3
| (F124|F142|F214) ->
printf "%s * %s_ff(%s,%s,%s)" f2 f cp f1 f3
| (F413|F431|F341) ->
printf "%s * %s_ff(%s,%s,%s)" f2 cf cm f1 f3
| (F241|F412|F421) ->
printf "%s * %s_ff(%s,%s,%s)" f2 cf cm f1 f3
let print_fermion_s2p_current coeff f c wf1 wf2 wf3 fusion =
let c = format_coupling_2 coeff c and
f1 = (List.nth (wf_of_f wf1 wf2 wf3 fusion) 0) and
f2 = (List.nth (wf_of_f wf1 wf2 wf3 fusion) 1) and
f3 = (List.nth (wf_of_f wf1 wf2 wf3 fusion) 2) in
let c1 = fastener c 1 and
c2 = fastener c 2 in
match fusion with
| (F123|F213|F132|F231|F312|F321) ->
printf "%s * f_%sf(%s,-(%s),%s,%s)" f1 f c1 c2 f2 f3
| (F423|F243|F432|F234|F342|F324) ->
printf "%s * f_%sf(%s,%s,%s,%s)" f1 f c1 c2 f2 f3
| (F134|F143|F314) ->
printf "%s * %s_ff(%s,%s,%s,%s)" f2 f c1 c2 f1 f3
| (F124|F142|F214) ->
printf "%s * %s_ff(%s,%s,%s,%s)" f2 f c1 c2 f1 f3
| (F413|F431|F341) ->
printf "%s * %s_ff(%s,-(%s),%s,%s)" f2 f c1 c2 f1 f3
| (F241|F412|F421) ->
printf "%s * %s_ff(%s,-(%s),%s,%s)" f2 f c1 c2 f1 f3
let print_fermion_s2lr_current coeff f c wf1 wf2 wf3 fusion =
let c = format_coupling_2 coeff c and
f1 = (List.nth (wf_of_f wf1 wf2 wf3 fusion) 0) and
f2 = (List.nth (wf_of_f wf1 wf2 wf3 fusion) 1) and
f3 = (List.nth (wf_of_f wf1 wf2 wf3 fusion) 2) in
let c1 = fastener c 1 and
c2 = fastener c 2 in
match fusion with
| (F123|F213|F132|F231|F312|F321) ->
printf "%s * f_%sf(%s,%s,%s,%s)" f1 f c2 c1 f2 f3
| (F423|F243|F432|F234|F342|F324) ->
printf "%s * f_%sf(%s,%s,%s,%s)" f1 f c1 c2 f2 f3
| (F134|F143|F314) ->
printf "%s * %s_ff(%s,%s,%s,%s)" f2 f c1 c2 f1 f3
| (F124|F142|F214) ->
printf "%s * %s_ff(%s,%s,%s,%s)" f2 f c1 c2 f1 f3
| (F413|F431|F341) ->
printf "%s * %s_ff(%s,%s,%s,%s)" f2 f c2 c1 f1 f3
| (F241|F412|F421) ->
printf "%s * %s_ff(%s,%s,%s,%s)" f2 f c2 c1 f1 f3
let print_fermion_g4_current coeff f c wf1 wf2 wf3 fusion =
let c = format_coupling coeff c and
f1 = (List.nth (wf_of_f wf1 wf2 wf3 fusion) 0) and
f2 = (List.nth (wf_of_f wf1 wf2 wf3 fusion) 1) and
f3 = (List.nth (wf_of_f wf1 wf2 wf3 fusion) 2) in
match fusion with
| (F123|F213|F132|F231|F312|F321) ->
printf "f_%sgr(-%s,%s,%s,%s)" f c f1 f2 f3
| (F423|F243|F432|F234|F342|F324) ->
printf "gr_%sf(%s,%s,%s,%s)" f c f1 f2 f3
| (F134|F143|F314|F124|F142|F214) ->
printf "%s_grf(%s,%s,%s,%s)" f c f1 f2 f3
| (F413|F431|F341|F241|F412|F421) ->
printf "%s_fgr(-%s,%s,%s,%s)" f c f1 f2 f3
let print_fermion_2_g4_current coeff f c wf1 wf2 wf3 fusion =
let f1 = (List.nth (wf_of_f wf1 wf2 wf3 fusion) 0) and
f2 = (List.nth (wf_of_f wf1 wf2 wf3 fusion) 1) and
f3 = (List.nth (wf_of_f wf1 wf2 wf3 fusion) 2) in
let c = format_coupling_2 coeff c in
let c1 = fastener c 1 and
c2 = fastener c 2 in
match fusion with
| (F123|F213|F132|F231|F312|F321) ->
printf "f_%sgr(-(%s),-(%s),%s,%s,%s)" f c2 c1 f1 f2 f3
| (F423|F243|F432|F234|F342|F324) ->
printf "gr_%sf(%s,%s,%s,%s,%s)" f c1 c2 f1 f2 f3
| (F134|F143|F314|F124|F142|F214) ->
printf "%s_grf(%s,%s,%s,%s,%s)" f c1 c2 f1 f2 f3
| (F413|F431|F341|F241|F412|F421) ->
printf "%s_fgr(-(%s),-(%s),%s,%s,%s)" f c2 c1 f1 f2 f3
let print_fermion_2_g4_current coeff f c wf1 wf2 wf3 fusion =
let f1 = (List.nth (wf_of_f wf1 wf2 wf3 fusion) 0) and
f2 = (List.nth (wf_of_f wf1 wf2 wf3 fusion) 1) and
f3 = (List.nth (wf_of_f wf1 wf2 wf3 fusion) 2) in
let c = format_coupling_2 coeff c in
let c1 = fastener c 1 and
c2 = fastener c 2 in
match fusion with
| (F123|F213|F132|F231|F312|F321) ->
printf "f_%sgr(-(%s),-(%s),%s,%s,%s)" f c2 c1 f1 f2 f3
| (F423|F243|F432|F234|F342|F324) ->
printf "gr_%sf(%s,%s,%s,%s,%s)" f c1 c2 f1 f2 f3
| (F134|F143|F314|F124|F142|F214) ->
printf "%s_grf(%s,%s,%s,%s,%s)" f c1 c2 f1 f2 f3
| (F413|F431|F341|F241|F412|F421) ->
printf "%s_fgr(-(%s),-(%s),%s,%s,%s)" f c2 c1 f1 f2 f3
let print_fermion_g4_current_rev coeff f c wf1 wf2 wf3 fusion =
let c = format_coupling coeff c and
f1 = (List.nth (wf_of_f wf1 wf2 wf3 fusion) 0) and
f2 = (List.nth (wf_of_f wf1 wf2 wf3 fusion) 1) and
f3 = (List.nth (wf_of_f wf1 wf2 wf3 fusion) 2) in
match fusion with
| (F123|F213|F132|F231|F312|F321) ->
printf "f_%sgr(%s,%s,%s,%s)" f c f1 f2 f3
| (F423|F243|F432|F234|F342|F324) ->
printf "gr_%sf(-%s,%s,%s,%s)" f c f1 f2 f3
| (F134|F143|F314|F124|F142|F214) ->
printf "%s_grf(-%s,%s,%s,%s)" f c f1 f2 f3
| (F413|F431|F341|F241|F412|F421) ->
printf "%s_fgr(%s,%s,%s,%s)" f c f1 f2 f3
(* Here we have to distinguish which of the two bosons is produced in the
fusion of three particles which include both fermions. *)
let print_fermion_g4_vector_current coeff f c wf1 wf2 wf3 fusion =
let c = format_coupling coeff c and
d1 = d_p (1,f) and
d2 = d_p (2,f) and
f1 = (List.nth (wf_of_f wf1 wf2 wf3 fusion) 0) and
f2 = (List.nth (wf_of_f wf1 wf2 wf3 fusion) 1) and
f3 = (List.nth (wf_of_f wf1 wf2 wf3 fusion) 2) in
match fusion with
| (F123|F213|F132|F231|F312|F321) ->
printf "f_%sgr(%s,%s,%s,%s)" f c f1 f2 f3
| (F423|F243|F432|F234|F342|F324) ->
printf "gr_%sf(%s,%s,%s,%s)" f c f1 f2 f3
| (F134|F143|F314) -> printf "%s%s_grf(%s,%s,%s,%s)" f d1 c f1 f2 f3
| (F124|F142|F214) -> printf "%s%s_grf(%s,%s,%s,%s)" f d2 c f1 f2 f3
| (F413|F431|F341) -> printf "%s%s_fgr(%s,%s,%s,%s)" f d1 c f1 f2 f3
| (F241|F412|F421) -> printf "%s%s_fgr(%s,%s,%s,%s)" f d2 c f1 f2 f3
let print_fermion_2_g4_vector_current coeff f c wf1 wf2 wf3 fusion =
let d1 = d_p (1,f) and
d2 = d_p (2,f) and
f1 = (List.nth (wf_of_f wf1 wf2 wf3 fusion) 0) and
f2 = (List.nth (wf_of_f wf1 wf2 wf3 fusion) 1) and
f3 = (List.nth (wf_of_f wf1 wf2 wf3 fusion) 2) in
let c = format_coupling_2 coeff c in
let c1 = fastener c 1 and
c2 = fastener c 2 in
match fusion with
| (F123|F213|F132|F231|F312|F321) ->
printf "f_%sgr(%s,%s,%s,%s,%s)" f c1 c2 f1 f2 f3
| (F423|F243|F432|F234|F342|F324) ->
printf "gr_%sf(%s,%s,%s,%s,%s)" f c1 c2 f1 f2 f3
| (F134|F143|F314) -> printf "%s%s_grf(%s,%s,%s,%s,%s)" f d1 c1 c2 f1 f2 f3
| (F124|F142|F214) -> printf "%s%s_grf(%s,%s,%s,%s,%s)" f d2 c1 c2 f1 f2 f3
| (F413|F431|F341) -> printf "%s%s_fgr(%s,%s,%s,%s,%s)" f d1 c1 c2 f1 f2 f3
| (F241|F412|F421) -> printf "%s%s_fgr(%s,%s,%s,%s,%s)" f d2 c1 c2 f1 f2 f3
let print_fermion_g4_vector_current_rev coeff f c wf1 wf2 wf3 fusion =
let c = format_coupling coeff c and
d1 = d_p (1,f) and
d2 = d_p (2,f) and
f1 = (List.nth (wf_of_f wf1 wf2 wf3 fusion) 0) and
f2 = (List.nth (wf_of_f wf1 wf2 wf3 fusion) 1) and
f3 = (List.nth (wf_of_f wf1 wf2 wf3 fusion) 2) in
match fusion with
| (F123|F213|F132|F231|F312|F321) ->
printf "gr_%sf(%s,%s,%s,%s)" f c f1 f2 f3
| (F423|F243|F432|F234|F342|F324) ->
printf "f_%sgr(%s,%s,%s,%s)" f c f1 f2 f3
| (F134|F143|F314) -> printf "%s%s_fgr(%s,%s,%s,%s)" f d1 c f1 f2 f3
| (F124|F142|F214) -> printf "%s%s_fgr(%s,%s,%s,%s)" f d2 c f1 f2 f3
| (F413|F431|F341) -> printf "%s%s_grf(%s,%s,%s,%s)" f d1 c f1 f2 f3
| (F241|F412|F421) -> printf "%s%s_grf(%s,%s,%s,%s)" f d2 c f1 f2 f3
let print_fermion_2_g4_current_rev coeff f c wf1 wf2 wf3 fusion =
let c = format_coupling_2 coeff c in
let c1 = fastener c 1 and
c2 = fastener c 2 and
d1 = d_p (1,f) and
d2 = d_p (2,f) in
let f1 = (List.nth (wf_of_f wf1 wf2 wf3 fusion) 0) and
f2 = (List.nth (wf_of_f wf1 wf2 wf3 fusion) 1) and
f3 = (List.nth (wf_of_f wf1 wf2 wf3 fusion) 2) in
match fusion with
| (F123|F213|F132|F231|F312|F321) ->
printf "gr_%sf(%s,%s,%s,%s,%s)" f c1 c2 f1 f2 f3
| (F423|F243|F432|F234|F342|F324) ->
printf "f_%sgr(-(%s),-(%s),%s,%s,%s)" f c1 c2 f1 f2 f3
| (F134|F143|F314) ->
printf "%s%s_fgr(-(%s),-(%s),%s,%s,%s)" f d1 c1 c2 f1 f2 f3
| (F124|F142|F214) ->
printf "%s%s_fgr(-(%s),-(%s),%s,%s,%s)" f d2 c1 c2 f1 f2 f3
| (F413|F431|F341) ->
printf "%s%s_grf(%s,%s,%s,%s,%s)" f d1 c1 c2 f1 f2 f3
| (F241|F412|F421) ->
printf "%s%s_grf(%s,%s,%s,%s,%s)" f d2 c1 c2 f1 f2 f3
let print_fermion_2_g4_vector_current_rev coeff f c wf1 wf2 wf3 fusion =
(* Here we put in the extra minus sign from the coeff. *)
let c = format_coupling coeff c in
let c1 = fastener c 1 and
c2 = fastener c 2 in
let d1 = d_p (1,f) and
d2 = d_p (2,f) and
f1 = (List.nth (wf_of_f wf1 wf2 wf3 fusion) 0) and
f2 = (List.nth (wf_of_f wf1 wf2 wf3 fusion) 1) and
f3 = (List.nth (wf_of_f wf1 wf2 wf3 fusion) 2) in
match fusion with
| (F123|F213|F132|F231|F312|F321) ->
printf "gr_%sf(%s,%s,%s,%s,%s)" f c1 c2 f1 f2 f3
| (F423|F243|F432|F234|F342|F324) ->
printf "f_%sgr(%s,%s,%s,%s,%s)" f c1 c2 f1 f2 f3
| (F134|F143|F314) -> printf "%s%s_fgr(%s,%s,%s,%s,%s)" f d1 c1 c2 f1 f2 f3
| (F124|F142|F214) -> printf "%s%s_fgr(%s,%s,%s,%s,%s)" f d2 c1 c2 f1 f2 f3
| (F413|F431|F341) -> printf "%s%s_grf(%s,%s,%s,%s,%s)" f d1 c1 c2 f1 f2 f3
| (F241|F412|F421) -> printf "%s%s_grf(%s,%s,%s,%s,%s)" f d2 c1 c2 f1 f2 f3
let print_current_g4 = function
| coeff, Gravbar, S2, _ -> print_fermion_g4_current coeff "s2"
| coeff, Gravbar, SV, _ -> print_fermion_g4_vector_current coeff "sv"
| coeff, Gravbar, SLV, _ -> print_fermion_g4_vector_current coeff "slv"
| coeff, Gravbar, SRV, _ -> print_fermion_g4_vector_current coeff "srv"
| coeff, Gravbar, SLRV, _ -> print_fermion_2_g4_vector_current coeff "slrv"
| coeff, Gravbar, PV, _ -> print_fermion_g4_vector_current coeff "pv"
| coeff, Gravbar, V2, _ -> print_fermion_g4_current coeff "v2"
| coeff, Gravbar, V2LR, _ -> print_fermion_2_g4_current coeff "v2lr"
| coeff, Gravbar, _, _ -> invalid_arg "print_current_g4: not implemented"
| coeff, _, S2, Grav -> print_fermion_g4_current_rev coeff "s2"
| coeff, _, SV, Grav -> print_fermion_g4_vector_current_rev (-coeff) "sv"
| coeff, _, SLV, Grav -> print_fermion_g4_vector_current_rev (-coeff) "slv"
| coeff, _, SRV, Grav -> print_fermion_g4_vector_current_rev (-coeff) "srv"
| coeff, _, SLRV, Grav -> print_fermion_2_g4_vector_current_rev coeff "slrv"
| coeff, _, PV, Grav -> print_fermion_g4_vector_current_rev coeff "pv"
| coeff, _, V2, Grav -> print_fermion_g4_vector_current_rev coeff "v2"
| coeff, _, V2LR, Grav -> print_fermion_2_g4_current_rev coeff "v2lr"
| coeff, _, _, Grav -> invalid_arg "print_current_g4: not implemented"
| coeff, _, S2, _ -> print_fermion_s2_current coeff "s"
| coeff, _, P2, _ -> print_fermion_s2_current coeff "p"
| coeff, _, S2P, _ -> print_fermion_s2p_current coeff "sp"
| coeff, _, S2L, _ -> print_fermion_s2_current coeff "sl"
| coeff, _, S2R, _ -> print_fermion_s2_current coeff "sr"
| coeff, _, S2LR, _ -> print_fermion_s2lr_current coeff "slr"
| coeff, _, V2, _ -> print_fermion_g4_brs_vector_current coeff "v2"
| coeff, _, SV, _ -> print_fermion_g4_brs_vector_current coeff "sv"
| coeff, _, PV, _ -> print_fermion_g4_brs_vector_current coeff "pv"
| coeff, _, SLV, _ -> print_fermion_g4_brs_vector_current coeff "svl"
| coeff, _, SRV, _ -> print_fermion_g4_brs_vector_current coeff "svr"
| coeff, _, SLRV, _ -> print_fermion_g4_svlr_current coeff "svlr"
| coeff, _, V2LR, _ -> invalid_arg "Targets.print_current: not available"
let reverse_braket _ = false
let use_module = "omega95_bispinors"
let require_library =
["omega_bispinors_2010_01_A"; "omega_bispinor_cpls_2010_01_A"]
end
module Fortran_Majorana = Make_Fortran(Fortran_Majorana_Fermions)
(* \thocwmodulesubsection{\texttt{FORTRAN\,77}} *)
module Fortran77 = Dummy
(* \thocwmodulesection{O'Mega Virtual Machine} *)
module VM = Dummy
(* \thocwmodulesection{\texttt{C}} *)
module C = Dummy
(* \thocwmodulesubsection{\texttt{C++}} *)
module Cpp = Dummy
(* \thocwmodulesubsection{Java} *)
module Java = Dummy
(* \thocwmodulesection{O'Caml} *)
module Ocaml = Dummy
(* \thocwmodulesection{\LaTeX} *)
module LaTeX = Dummy
(*i
module VM_old (F : Fusion.T) (Make_MF : Fusion.MultiMaker)
(M : Model.T with type flavor = F.flavor and type constant = F.constant) =
struct
let rcs_list =
[ RCS.rename rcs_file "Targets.VM()"
[ "Bytecode for the O'Mega Virtual Machine" ] ]
module MF = Make_MF(F)
type amplitude = F.amplitude
type amplitudes = MF.amplitudes
type diagnostic = All | Arguments | Momenta | Gauge
let options = Options.empty
let flavors_to_string flavors =
String.concat " " (List.map M.flavor_to_string flavors)
let format_process amplitude =
flavors_to_string (F.incoming amplitude) ^ " -> " ^
flavors_to_string (F.outgoing amplitude)
open Format
open Coupling
let ovm_LOAD_SCALAR = 1
let ovm_LOAD_U = 2
let ovm_LOAD_UBAR = 3
let ovm_LOAD_V = 4
let ovm_LOAD_VBAR = 5
let ovm_LOAD_VECTOR = 6
let ovm_ADD_MOMENTA = 10
let ovm_PROPAGATE_SCALAR = 11
let ovm_PROPAGATE_SPINOR = 12
let ovm_PROPAGATE_CONJSPINOR = 13
let ovm_PROPAGATE_UNITARITY = 14
let ovm_PROPAGATE_FEYNMAN = 15
let ovm_PROPAGATE_TENSOR2 = 16
let ovm_FUSE_VECTOR_PSIBAR_PSI = 21
let ovm_FUSE_PSI_VECTOR_PSI = 22
let ovm_FUSE_PSIBAR_PSIBAR_VECTOR = 23
type instruction =
{ code : int; sign : int; coupl : int;
lhs : int; rhs1 : int; rhs2 : int }
let printi i =
printf "@\n%3d %3d %3d %3d %3d %3d"
i.code i.sign i.coupl i.lhs i.rhs1 i.rhs2
let load lhs f rhs =
let code =
match M.lorentz f with
| Scalar -> ovm_LOAD_SCALAR
| Spinor -> ovm_LOAD_U
| ConjSpinor -> ovm_LOAD_UBAR
| Majorana -> failwith "load: Majoranas not implemented yet"
| Maj_Ghost -> failwith "load: SUSY ghosts not implemented yet"
| Vector | Massive_Vector -> ovm_LOAD_VECTOR
| Vectorspinor -> invalid_arg "external spin must be <=1"
| Tensor_1 -> invalid_arg "Tensor_1 only internal"
| Tensor_2 -> invalid_arg "external spin must be <= 1"
| BRS _ -> invalid_arg "no BRST"
in
{ code = code; sign = 0; coupl = M.pdg f;
lhs = lhs; rhs1 = rhs; rhs2 = rhs }
let print_external count flavor =
printi (load count (F.flavor flavor) count);
succ count
let print_externals amplitude =
printf "@\n@[<2>BEGIN EXTERNALS";
ignore (List.fold_left print_external 1 (F.externals amplitude));
printf "@]@\nEND EXTERNALS"
let print_current rhs =
match F.coupling rhs with
| V3 (vertex, fusion, constant) -> printf "@\nV3"
| V4 (vertex, fusion, constant) -> printf "@\nV4"
| Vn (_, _, _) -> printf "@\nVn"
let p2s p =
if p >= 0 && p <= 9 then
string_of_int p
else if p <= 36 then
String.make 1 (Char.chr (Char.code 'A' + p - 10))
else
"_"
let format_p wf =
String.concat "" (List.map p2s (F.momentum_list wf))
let print_fusion fusion =
let lhs = F.lhs fusion in
let f = F.flavor lhs in
(*i let momentum = format_p lhs in i*)
List.iter print_current (F.rhs fusion);
let propagate code =
printi { code = code; sign = 0; coupl = 0;
lhs = int_of_string (format_p lhs);
rhs1 = abs (M.pdg f); rhs2 = abs (M.pdg f) } in
match M.propagator f with
| Prop_Scalar -> propagate ovm_PROPAGATE_SCALAR
| Prop_Col_Scalar ->
failwith "print_fusion: Prop_Col_Scalar not implemented yet!"
| Prop_Ghost ->
failwith "print_fusion: Prop_Ghost not implemented yet!"
| Prop_Spinor -> propagate ovm_PROPAGATE_SPINOR
| Prop_ConjSpinor -> propagate ovm_PROPAGATE_CONJSPINOR
| Prop_Majorana | Prop_Col_Majorana ->
failwith "print_fusion: Prop_Majorana not implemented yet!"
| Prop_Unitarity -> propagate ovm_PROPAGATE_UNITARITY
| Prop_Col_Unitarity ->
failwith "print_fusion: Prop_Col_Unitarity not implemented yet!"
| Prop_Feynman -> propagate ovm_PROPAGATE_FEYNMAN
| Prop_Col_Feynman ->
failwith "print_fusion: Prop_Col_Feynman not implemented yet!"
| Prop_Gauge xi ->
failwith "print_fusion: Prop_Gauge not implemented yet!"
| Prop_Rxi xi ->
failwith "print_fusion: Prop_Rxi not implemented yet!"
| Prop_Vectorspinor ->
failwith "print_fusion: Prop_Vectorspinor not implemented yet!"
| Prop_Tensor_2 -> propagate ovm_PROPAGATE_TENSOR2
| Aux_Scalar | Aux_Spinor | Aux_ConjSpinor | Aux_Majorana
| Aux_Vector | Aux_Tensor_1 -> ()
| Only_Insertion -> ()
module P = Set.Make (struct type t = int list let compare = compare end)
let rec add_momenta lhs = function
| [] | [_] -> invalid_arg "add_momenta"
| [rhs1; rhs2] ->
printi { code = ovm_ADD_MOMENTA; sign = 0; coupl = 0;
lhs = int_of_string (format_p lhs);
rhs1 = int_of_string (format_p rhs1);
rhs2 = int_of_string (format_p rhs2) }
| rhs1 :: rhs ->
add_momenta lhs rhs;
add_momenta lhs [lhs; rhs1]
let print_fusions amplitude =
printf "@\n@[<2>BEGIN FUSIONS";
let momenta =
List.fold_left (fun seen f ->
let wf = F.lhs f in
let p = F.momentum_list wf in
let momentum = format_p wf in
if not (P.mem p seen) then
add_momenta wf (F.children (List.hd (F.rhs f)));
print_fusion f;
P.add p seen) P.empty (F.fusions amplitude)
in
printf "@]@\nEND FUSIONS"
let print_brakets amplitude =
printf "@\n@[<2>BEGIN BRAKETS";
printf "@\n!!! not implemented yet !!!";
printf "@]@\nEND BRAKETS"
let print_fudge_factor amplitude =
printf "@\n@[<2>BEGIN FUDGE";
printf "@\n!!! not implemented yet !!!";
printf "@]@\nEND FUDGE"
let amplitude_to_channel oc diagnostics amplitude =
set_formatter_out_channel oc;
printf "@\n@[<2>BEGIN AMPLITUDE %s" (format_process amplitude);
print_externals amplitude;
print_fusions amplitude;
print_brakets amplitude;
print_fudge_factor amplitude;
printf "@]@\nEND AMPLITUDE"
let amplitudes_to_channel oc diagnostics amplitudes =
List.iter (amplitude_to_channel oc diagnostics) (MF.allowed amplitudes)
let parameters_to_channel oc =
set_formatter_out_channel oc;
(*i let params = M.parameters () in i*)
printf "@[<2>BEGIN PARAMETERS@\n";
printf "!!! not implemented yet !!!@]@\n";
printf "END PARAMETERS@\n"
end
i*)
(*i
* Local Variables:
* mode:caml
* indent-tabs-mode:nil
* page-delimiter:"^(\\* .*\n"
* End:
i*)
Index: trunk/share/models/SM_top_anom.mdl
===================================================================
--- trunk/share/models/SM_top_anom.mdl (revision 4047)
+++ trunk/share/models/SM_top_anom.mdl (revision 4048)
@@ -1,314 +1,325 @@
########################################################################
# Standard Model with trivial CKM matrix and anom. tt vector couplings
model "SM_top_anom"
# Independent parameters
### DO NOT CHANGE THE ORDER OF THESE PARAMETERS
parameter GF = 1.16639E-5 # Fermi constant
parameter mZ = 91.1882 # Z-boson mass
parameter mW = 80.419 # W-boson mass
parameter mH = 125 # Higgs mass
parameter alphas = 0.1178 # Strong coupling constant (Z point)
parameter me = 0.000510997 # electron mass
parameter mmu = 0.105658389 # muon mass
parameter mtau = 1.77705 # tau-lepton mass
parameter ms = 0.095 # s-quark mass
parameter mc = 1.2 # c-quark mass
parameter mb = 4.2 # b-quark mass
parameter mtop = 173.1 # t-quark mass
parameter wtop = 1.523 # t-quark width
parameter wZ = 2.443 # Z-boson width
parameter wW = 2.049 # W-boson width
parameter wH = 0.004143 # Higgs width
parameter khgaz = 1.000 # anomaly Higgs couplings K factors
parameter khgaga = 1.000 # anomaly Higgs couplings K factors
parameter khgg = 1.000 # anomaly Higgs couplings K factors
parameter xi0 = 0.000 # R_xi parameter for Z-boson
parameter xipm = 0.000 # R_xi parameter for W-boson
# General set of parameters for anomalous top-gauge and top-Higgs vertices
parameter tv_ttA = 0.000 # vector coeff. of ttA tensor coupling
parameter ta_ttA = 0.000 # axial coeff. (*i) of ttA tensor cpl.
parameter vl_ttZ = 0.000 # left-handed coeff. of ttZ vector cpl.
# FIXED by gauge inv. !
parameter vr_ttZ = 0.000 # right-handed coeff. of ttZ vector cpl.
parameter tv_ttZ = 0.000 # vector coeff. of ttZ tensor cpl.
# FIXED by gauge inv. !
parameter ta_ttZ = 0.000 # axial coeff. (*i) of ttZ tensor cpl.
# FIXED by gauge inv. !
parameter vl_tbW_Re = 0.000 # left-handed coeff. of tbW vector cpl.
parameter vl_tbW_Im = 0.000
parameter vr_tbW_Re = 0.000 # right-handed coeff. of tbW vector cpl.
parameter vr_tbW_Im = 0.000
parameter tl_tbW_Re = 0.000 # left-handed coeff. of bbartW- tensor cpl.
parameter tl_tbW_Im = 0.000
parameter tr_tbW_Re = 0.000 # right-handed coeff. of bbarW- tensor cpl.
parameter tr_tbW_Im = 0.000
parameter tv_ttG = 0.000 # vector coeff. of ttg tensor cpl.
parameter ta_ttG = 0.000 # axial coeff. (*i) of ttg tensor cpl.
parameter s_ttH = 0.000 # scalar coeff. of ttH cpl.
parameter p_ttH = 0.000 # pseudo scalar coeff. of ttH cpl.
parameter Lambda = 2000 # effective energy scale in [GeV]
parameter fun = 0 # ansatz for k^2-dependence of form factors
# (0) no dependence: ~ 1
# (1) Lorentzian: ~ 1 / (1 + k^2/Lambda^2)
# (2) dipole: ~ 1 / (1 + k^2/Lambda^2))^2
# (3) exponential: ~ exp(-k^2/Lambda^2)
# choose normalization convention
parameter norm_conv = 1
# Setting this switch to a positive value effects the usual normalization
# convention widely used in literature, where the ratio of fundamental scales
# is implicit inside the couplings (cf. e.g. 0811.3842 & 0904.2387).
# With a negative value, the ratio of scales appears explicitly in the
# normalization of the interaction terms, making the natural coupling sizes
# become of the order of the fundamental Wilson coefficients.
# Enforce gauge invariance
parameter gauge_inv = 1
# Setting this switch to a positive value enforces interrelations imposed by
# gauge invariance among some couplings introduced above and additional bottom
# couplings, i. e. any inconsistent values stated above (or zeros assumed for
# the bottom couplings) are OVERRIDEN such that the overall numerical factors
# N_i multiplying the interaction terms (i.e. independent of "norm_conv"!)
# respect the following constraints:
# N_vl_ttZ = N_vl_tbW_Re * sqrt(2) / cw (consequence of requiring vl_bbZ = 0)
# N_tv_bbA = N_tl_tbW_Re * sw / sqrt(2)
# N_ta_bbA = N_tl_tbW_Im * sw / sqrt(2)
# N_tv_bbZ = N_tl_tbW_Re * cw / sqrt(2)
# N_ta_bbZ = N_tl_tbW_Im * cw / sqrt(2)
# N_tr_tbW_Re = N_tv_ttZ * sqrt(2) * cw + N_tv_ttA * 2 * sw
# N_tr_tbW_Im = N_ta_ttZ * sqrt(2) * cw + N_ta_ttA * 2 * sw
# Partially redundant operator coefficients (turn on 4-fermion contact terms)
parameter Re_CqG = 0.000 # real part of C_qG
parameter Re_CuG = 0.000 # real part of C_uG
parameter Re_CqB = 0.000 # real part of C_qB
parameter Re_CuB = 0.000 # real part of C_uB
parameter Re_CqW = 0.000 # real part of C_qW
parameter Re_CDu = 0.000 # real part of C_Du
parameter Re_CDd = 0.000 # real part of C_Dd
parameter Im_CDd = 0.000 # imaginary part of C_Dd
+# Complete the list of anomalous dim6 charged-current top interactions
+parameter Re_Cquqd1_1 = 0.000 # real part of C_quqd8_1
+parameter Im_Cquqd1_1 = 0.000 # imaginary part of C_quqd8_1
+parameter Re_Cquqd1_2 = 0.000 # real part of C_quqd8_2
+parameter Im_Cquqd1_2 = 0.000 # imaginary part of C_quqd8_2
+parameter Re_Cquqd8_1 = 0.000 # real part of C_quqd8_1
+parameter Im_Cquqd8_1 = 0.000 # imaginary part of C_quqd8_1
+parameter Re_Cquqd8_2 = 0.000 # real part of C_quqd8_2
+parameter Im_Cquqd8_2 = 0.000 # imaginary part of C_quqd8_2
+
+
# Dependent SM parameters
derived v = 1 / sqrt (sqrt (2.) * GF) # v (Higgs vev)
derived cw = mW / mZ # cos(theta-W)
derived sw = sqrt (1-cw**2) # sin(theta-W)
derived ee = 2 * sw * mW / v # em-coupling (GF scheme)
derived alpha_em_i = 4 * pi / ee**2 # inverse fine structure constant
########################################################################
# Particle content
# The quarks
particle D_QUARK 1 parton
spin 1/2 charge -1/3 isospin -1/2 color 3
name d down
anti dbar D "d~"
tex_anti "\bar{d}"
particle U_QUARK 2 parton
spin 1/2 charge 2/3 isospin 1/2 color 3
name u up
anti ubar U "u~"
tex_anti "\bar{u}"
particle S_QUARK 3 like D_QUARK
name s strange
anti sbar S "s~"
tex_anti "\bar{s}"
mass ms
particle C_QUARK 4 like U_QUARK
name c charm
anti cbar C "c~"
tex_anti "\bar{c}"
mass mc
particle B_QUARK 5 like D_QUARK
name b bottom
anti bbar B "b~"
tex_anti "\bar{b}"
mass mb
particle T_QUARK 6 like U_QUARK
name t top
anti tbar T "t~"
tex_anti "\bar{t}"
mass mtop width wtop
# The leptons
particle E_LEPTON 11
spin 1/2 charge -1 isospin -1/2
name "e-" e1 electron e
anti "e+" E1 positron
tex_name "e^-"
tex_anti "e^+"
mass me
particle E_NEUTRINO 12 left
spin 1/2 isospin 1/2
name nue n1 "nu_e" ve "e-neutrino"
anti nuebar N1 "ve~"
tex_name "\nu_e"
tex_anti "\bar\nu_e"
particle MU_LEPTON 13 like E_LEPTON
name "mu-" e2 mu muon
anti "mu+" E2
tex_name "\mu^-"
tex_anti "\mu^+"
mass mmu
particle MU_NEUTRINO 14 like E_NEUTRINO
name numu "nu_mu" n2 vm "mu-neutrino"
anti numubar N2 "vm~"
tex_name "\nu_\mu"
tex_anti "\bar\nu_\mu"
particle TAU_LEPTON 15 like E_LEPTON
name "tau-" e3 tau "ta-" tauon
anti "tau+" E3 "ta+"
tex_name "\tau^-"
tex_anti "\tau^+"
mass mtau
particle TAU_NEUTRINO 16 like E_NEUTRINO
name nutau "nu_tau" n3 vt "tau_neutrino"
anti nutaubar N3 "vt~"
tex_name "\nu_\tau"
tex_anti "\bar\nu_\tau"
# The vector bosons
particle GLUON 21 parton gauge
spin 1 color 8
name gl g G gluon
particle PHOTON 22 gauge
spin 1
name A gamma photon
tex_name "\gamma"
particle Z_BOSON 23 gauge
spin 1
name Z
mass mZ width wZ
particle W_BOSON 24 gauge
spin 1 charge 1
name "W+" Wp
anti "W-" Wm
tex_name "W^+"
tex_anti "W^-"
mass mW width wW
# The Higgs
particle HIGGS 25
spin 0
name H h Higgs
mass mH width wH
# Hadrons
particle PROTON 2212
spin 1/2 charge 1
name p "p+"
anti pbar "p-"
particle HADRON_REMNANT 90
particle HADRON_REMNANT_SINGLET 91
particle HADRON_REMNANT_TRIPLET 92
color 3
particle HADRON_REMNANT_OCTET 93
color 8
########################################################################
# Vertices of the Standard model
# In graphs with identical structure the first vertex is kept for phase space
# therefore, lighter particles come before heavier ones.
# QED
vertex D d A
vertex U u A
vertex S s A
vertex C c A
vertex B b A
vertex T t A
vertex E1 e1 A
vertex E2 e2 A
vertex E3 e3 A
# QCD
vertex G G G
vertex G G G G
vertex D d G
vertex U u G
vertex S s G
vertex C c G
vertex B b G
vertex T t G
# Neutral currents
vertex D d Z
vertex U u Z
vertex S s Z
vertex C c Z
vertex B b Z
vertex T t Z
vertex E1 e1 Z
vertex E2 e2 Z
vertex E3 e3 Z
vertex N1 n1 Z
vertex N2 n2 Z
vertex N3 n3 Z
# Charged currents
vertex U d Wp
vertex C s Wp
vertex T b Wp
vertex D u Wm
vertex S c Wm
vertex B t Wm
vertex N1 e1 Wp
vertex N2 e2 Wp
vertex N3 e3 Wp
vertex E1 n1 Wm
vertex E2 n2 Wm
vertex E3 n3 Wm
# Yukawa
### keeping only 3rd generation for the moment
# vertex S s H
# vertex C c H
vertex B b H
vertex T t H
# vertex E2 e2 H
vertex E3 e3 H
# Vector-boson self-interactions
vertex Wp Wm A
vertex Wp Wm Z
vertex Wp Wm Z Z
vertex Wp Wp Wm Wm
vertex Wp Wm Z A
vertex Wp Wm A A
# Higgs - vector boson
vertex H Z A
vertex H A A
vertex H g g
vertex H Wp Wm
vertex H Z Z
vertex H H Wp Wm
vertex H H Z Z
# Higgs self-interactions
vertex H H H
vertex H H H H

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