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Index: trunk/src/physics/physics.nw
===================================================================
--- trunk/src/physics/physics.nw (revision 8951)
+++ trunk/src/physics/physics.nw (revision 8952)
@@ -1,10082 +1,10082 @@
% -*- ess-noweb-default-code-mode: f90-mode; noweb-default-code-mode: f90-mode; -*-
% WHIZARD code as NOWEB source: physics and such
\chapter{Physics}
\includemodulegraph{physics}
Here we collect definitions and functions that we need for (particle)
physics in general, to make them available for the more specific needs
of WHIZARD.
\begin{description}
\item[physics\_defs]
Physical constants.
\item[c\_particles]
A simple data type for particles which is C compatible.
\item[lorentz]
Define three-vectors, four-vectors and Lorentz
transformations and common operations for them.
\item[phs\_point]
Collections of Lorentz vectors.
\item[sm\_physics]
Here, running functions are stored for special kinematical setup like
running coupling constants, Catani-Seymour dipoles, or Sudakov factors.
\item[sm\_qcd]
Definitions and methods for dealing with the running QCD coupling.
\item[shower\_algorithms]
Algorithms typically used in Parton Showers as well as in their
matching to NLO computations, e.g. with the POWHEG method.
\end{description}
\clearpage
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Physics Constants}
There is also the generic [[constants]] module. The constants listed
here are more specific for particle physics.
<<[[physics_defs.f90]]>>=
<<File header>>
module physics_defs
<<Use kinds>>
<<Use strings>>
use constants, only: one, two, three
<<Standard module head>>
<<Physics defs: public parameters>>
<<Physics defs: public>>
<<Physics defs: interfaces>>
interface
<<Physics defs: sub interfaces>>
end interface
end module physics_defs
@ %def physics_defs
@
<<[[physics_defs_sub.f90]]>>=
<<File header>>
submodule (physics_defs) physics_defs_s
implicit none
contains
<<Physics defs: procedures>>
end submodule physics_defs_s
@ %def physics_defs_s
@
\subsection{Units}
Conversion from energy units to cross-section units.
<<Physics defs: public parameters>>=
real(default), parameter, public :: &
conv = 0.38937966e12_default
@
Conversion from millimeter to nanoseconds for lifetimes.
<<Physics defs: public parameters>>=
real(default), parameter, public :: &
ns_per_mm = 1.e6_default / 299792458._default
@
Rescaling factor.
<<Physics defs: public parameters>>=
real(default), parameter, public :: &
pb_per_fb = 1.e-3_default
@
String for the default energy and cross-section units.
<<Physics defs: public parameters>>=
character(*), parameter, public :: &
energy_unit = "GeV"
character(*), parameter, public :: &
cross_section_unit = "fb"
@
\subsection{SM and QCD constants}
<<Physics defs: public parameters>>=
real(default), parameter, public :: &
NC = three, &
CF = (NC**2 - one) / two / NC, &
CA = NC, &
TR = one / two
@
\subsection{Parameter Reference values}
These are used exclusively in the context of
running QCD parameters. In other contexts, we rely on the uniform
parameter set as provided by the model definition, modifiable by the
user.
<<Physics defs: public parameters>>=
real(default), public, parameter :: MZ_REF = 91.188_default
real(default), public, parameter :: ME_REF = 0.000510998928_default
real(default), public, parameter :: ALPHA_QCD_MZ_REF = 0.1178_default
real(default), public, parameter :: ALPHA_QED_ME_REF = 0.0072973525693_default
real(default), public, parameter :: LAMBDA_QCD_REF = 200.e-3_default
@ %def alpha_s_mz_ref mz_ref lambda_qcd_ref
@
\subsection{Particle codes}
Let us define a few particle codes independent of the model.
We need an UNDEFINED value:
<<Physics defs: public parameters>>=
integer, parameter, public :: UNDEFINED = 0
@ %def UNDEFINED
@ SM fermions:
<<Physics defs: public parameters>>=
integer, parameter, public :: DOWN_Q = 1
integer, parameter, public :: UP_Q = 2
integer, parameter, public :: STRANGE_Q = 3
integer, parameter, public :: CHARM_Q = 4
integer, parameter, public :: BOTTOM_Q = 5
integer, parameter, public :: TOP_Q = 6
integer, parameter, public :: ELECTRON = 11
integer, parameter, public :: ELECTRON_NEUTRINO = 12
integer, parameter, public :: MUON = 13
integer, parameter, public :: MUON_NEUTRINO = 14
integer, parameter, public :: TAU = 15
integer, parameter, public :: TAU_NEUTRINO = 16
@ %def ELECTRON MUON TAU
@ Gauge bosons:
<<Physics defs: public parameters>>=
integer, parameter, public :: GLUON = 21
integer, parameter, public :: PHOTON = 22
integer, parameter, public :: PHOTON_OFFSHELL = -2002
integer, parameter, public :: PHOTON_ONSHELL = 2002
integer, parameter, public :: Z_BOSON = 23
integer, parameter, public :: W_BOSON = 24
@ %def GLUON PHOTON Z_BOSON W_BOSON
@ Light mesons:
<<Physics defs: public parameters>>=
integer, parameter, public :: PION = 111
integer, parameter, public :: PIPLUS = 211
integer, parameter, public :: PIMINUS = - PIPLUS
@ %def PION PIPLUS PIMINUS
@ Di-Quarks:
<<Physics defs: public parameters>>=
integer, parameter, public :: UD0 = 2101
integer, parameter, public :: UD1 = 2103
integer, parameter, public :: UU1 = 2203
@ %def UD0 UD1 UU1
@ Mesons:
<<Physics defs: public parameters>>=
integer, parameter, public :: K0L = 130
integer, parameter, public :: K0S = 310
integer, parameter, public :: K0 = 311
integer, parameter, public :: KPLUS = 321
integer, parameter, public :: DPLUS = 411
integer, parameter, public :: D0 = 421
integer, parameter, public :: B0 = 511
integer, parameter, public :: BPLUS = 521
@ %def K0L K0S K0 KPLUS DPLUS D0 B0 BPLUS
@ Light baryons:
<<Physics defs: public parameters>>=
integer, parameter, public :: PROTON = 2212
integer, parameter, public :: NEUTRON = 2112
integer, parameter, public :: DELTAPLUSPLUS = 2224
integer, parameter, public :: DELTAPLUS = 2214
integer, parameter, public :: DELTA0 = 2114
integer, parameter, public :: DELTAMINUS = 1114
@ %def PROTON NEUTRON DELTAPLUSPLUS DELTAPLUS DELTA0 DELTAMINUS
@ Strange baryons:
<<Physics defs: public parameters>>=
integer, parameter, public :: SIGMAPLUS = 3222
integer, parameter, public :: SIGMA0 = 3212
integer, parameter, public :: SIGMAMINUS = 3112
@ %def SIGMAPLUS SIGMA0 SIGMAMINUS
@ Charmed baryons:
<<Physics defs: public parameters>>=
integer, parameter, public :: SIGMACPLUSPLUS = 4222
integer, parameter, public :: SIGMACPLUS = 4212
integer, parameter, public :: SIGMAC0 = 4112
@ %def SIGMACPLUSPLUS SIGMACPLUS SIGMAC0
@ Bottom baryons:
<<Physics defs: public parameters>>=
integer, parameter, public :: SIGMAB0 = 5212
integer, parameter, public :: SIGMABPLUS = 5222
@ %def SIGMAB0 SIGMABPLUS
@ 81-100 are reserved for internal codes. Hadron and beam remnants:
<<Physics defs: public parameters>>=
integer, parameter, public :: BEAM_REMNANT = 9999
integer, parameter, public :: HADRON_REMNANT = 90
integer, parameter, public :: HADRON_REMNANT_SINGLET = 91
integer, parameter, public :: HADRON_REMNANT_TRIPLET = 92
integer, parameter, public :: HADRON_REMNANT_OCTET = 93
@ %def BEAM_REMNANT HADRON_REMNANT
@ %def HADRON_REMNANT_SINGLET HADRON_REMNANT_TRIPLET HADRON_REMNANT_OCTET
@
Further particle codes for internal use:
<<Physics defs: public parameters>>=
integer, parameter, public :: INTERNAL = 94
integer, parameter, public :: INVALID = 97
integer, parameter, public :: COMPOSITE = 99
@ %def INTERNAL INVALID COMPOSITE
@
\subsection{Spin codes}
Somewhat redundant, but for better readability we define named
constants for spin types. If the mass is nonzero, this is equal to
the number of degrees of freedom.
<<Physics defs: public parameters>>=
integer, parameter, public:: UNKNOWN = 0
integer, parameter, public :: SCALAR = 1, SPINOR = 2, VECTOR = 3, &
VECTORSPINOR = 4, TENSOR = 5
@ %def UNKNOWN SCALAR SPINOR VECTOR VECTORSPINOR TENSOR
@ Isospin types and charge types are counted in an analogous way,
where charge type 1 is charge 0, 2 is charge 1/3, and so on. Zero
always means unknown. Note that charge and isospin types have an
explicit sign.
Color types are defined as the dimension of the representation.
\subsection{NLO status codes}
Used to specify whether a [[term_instance_t]] of a
[[process_instance_t]] is associated with a Born, real-subtracted,
virtual-subtracted or subtraction-dummy matrix element.
<<Physics defs: public parameters>>=
integer, parameter, public :: BORN = 0
integer, parameter, public :: NLO_REAL = 1
integer, parameter, public :: NLO_VIRTUAL = 2
integer, parameter, public :: NLO_MISMATCH = 3
integer, parameter, public :: NLO_DGLAP = 4
integer, parameter, public :: NLO_SUBTRACTION = 5
integer, parameter, public :: NLO_FULL = 6
integer, parameter, public :: GKS = 7
integer, parameter, public :: COMPONENT_UNDEFINED = 99
@ % def BORN, NLO_REAL, NLO_VIRTUAL, NLO_SUBTRACTION, GKS
@ [[NLO_FULL]] is not strictly a component status code but having it is
convenient.
We define the number of additional subtractions for beam-involved NLO calculations.
Each subtraction refers to a rescaling of one of two beams.
Obviously, this approach is not flexible enough to support setups with just a
single beam described by a structure function.
<<Physics defs: public parameters>>=
integer, parameter, public :: n_beams_rescaled = 2
@ %def n_beams_rescaled
@
Orders of the electron PDFs.
<<Physics defs: public parameters>>=
integer, parameter, public :: EPDF_LL = 0
integer, parameter, public :: EPDF_NLL = 1
@ %def EPDF_LL EPDF_NLL
@
<<Physics defs: public>>=
public :: component_status
<<Physics defs: interfaces>>=
interface component_status
module procedure component_status_of_string
module procedure component_status_to_string
end interface
<<Physics defs: sub interfaces>>=
elemental module function component_status_of_string (string) result (i)
integer :: i
type(string_t), intent(in) :: string
end function component_status_of_string
elemental module function component_status_to_string (i) result (string)
type(string_t) :: string
integer, intent(in) :: i
end function component_status_to_string
<<Physics defs: procedures>>=
elemental module function component_status_of_string (string) result (i)
integer :: i
type(string_t), intent(in) :: string
select case (char(string))
case ("born")
i = BORN
case ("real")
i = NLO_REAL
case ("virtual")
i = NLO_VIRTUAL
case ("mismatch")
i = NLO_MISMATCH
case ("dglap")
i = NLO_DGLAP
case ("subtraction")
i = NLO_SUBTRACTION
case ("full")
i = NLO_FULL
case ("GKS")
i = GKS
case default
i = COMPONENT_UNDEFINED
end select
end function component_status_of_string
elemental module function component_status_to_string (i) result (string)
type(string_t) :: string
integer, intent(in) :: i
select case (i)
case (BORN)
string = "born"
case (NLO_REAL)
string = "real"
case (NLO_VIRTUAL)
string = "virtual"
case (NLO_MISMATCH)
string = "mismatch"
case (NLO_DGLAP)
string = "dglap"
case (NLO_SUBTRACTION)
string = "subtraction"
case (NLO_FULL)
string = "full"
case (GKS)
string = "GKS"
case default
string = "undefined"
end select
end function component_status_to_string
@ %def component_status
@
<<Physics defs: public>>=
public :: is_nlo_component
<<Physics defs: sub interfaces>>=
elemental module function is_nlo_component (comp) result (is_nlo)
logical :: is_nlo
integer, intent(in) :: comp
end function is_nlo_component
<<Physics defs: procedures>>=
elemental module function is_nlo_component (comp) result (is_nlo)
logical :: is_nlo
integer, intent(in) :: comp
select case (comp)
case (BORN : GKS)
is_nlo = .true.
case default
is_nlo = .false.
end select
end function is_nlo_component
@ %def is_nlo_component
@
<<Physics defs: public>>=
public :: is_subtraction_component
<<Physics defs: sub interfaces>>=
module function is_subtraction_component (emitter, nlo_type) result (is_subtraction)
logical :: is_subtraction
integer, intent(in) :: emitter, nlo_type
end function is_subtraction_component
<<Physics defs: procedures>>=
module function is_subtraction_component (emitter, nlo_type) result (is_subtraction)
logical :: is_subtraction
integer, intent(in) :: emitter, nlo_type
is_subtraction = nlo_type == NLO_REAL .and. emitter < 0
end function is_subtraction_component
@ %def is_subtraction_component
@
\subsection{Threshold}
Some commonly used variables for the threshold computation
<<Physics defs: public parameters>>=
integer, parameter, public :: THR_POS_WP = 3
integer, parameter, public :: THR_POS_WM = 4
integer, parameter, public :: THR_POS_B = 5
integer, parameter, public :: THR_POS_BBAR = 6
integer, parameter, public :: THR_POS_GLUON = 7
integer, parameter, public :: THR_EMITTER_OFFSET = 4
integer, parameter, public :: NO_FACTORIZATION = 0
integer, parameter, public :: FACTORIZATION_THRESHOLD = 1
integer, dimension(2), parameter, public :: ass_quark = [5, 6]
integer, dimension(2), parameter, public :: ass_boson = [3, 4]
integer, parameter, public :: PROC_MODE_UNDEFINED = 0
integer, parameter, public :: PROC_MODE_TT = 1
integer, parameter, public :: PROC_MODE_WBWB = 2
@
@
<<Physics defs: public>>=
public :: thr_leg
<<Physics defs: sub interfaces>>=
module function thr_leg (emitter) result (leg)
integer :: leg
integer, intent(in) :: emitter
end function thr_leg
<<Physics defs: procedures>>=
module function thr_leg (emitter) result (leg)
integer :: leg
integer, intent(in) :: emitter
leg = emitter - THR_EMITTER_OFFSET
end function thr_leg
@ %def thr_leg
@
\clearpage
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{C-compatible Particle Type}
For easy communication with C code, we introduce a simple C-compatible
type for particles. The components are either default C integers or
default C doubles.
The [[c_prt]] type is transparent, and its contents should be regarded
as part of the interface.
<<[[c_particles.f90]]>>=
<<File header>>
module c_particles
use, intrinsic :: iso_c_binding !NODEP!
<<Standard module head>>
<<C Particles: public>>
<<C Particles: types>>
interface
<<C Particles: sub interfaces>>
end interface
end module c_particles
@ %def c_particles
@
<<[[c_particles_sub.f90]]>>=
<<File header>>
submodule (c_particles) c_particles_s
use io_units
use format_defs, only: FMT_14, FMT_19
implicit none
contains
<<C Particles: procedures>>
end submodule c_particles_s
@ %def c_particles_s
@
<<C Particles: public>>=
public :: c_prt_t
<<C Particles: types>>=
type, bind(C) :: c_prt_t
integer(c_int) :: type = 0
integer(c_int) :: pdg = 0
integer(c_int) :: polarized = 0
integer(c_int) :: h = 0
real(c_double) :: pe = 0
real(c_double) :: px = 0
real(c_double) :: py = 0
real(c_double) :: pz = 0
real(c_double) :: p2 = 0
end type c_prt_t
@ %def c_prt_t
@ This is for debugging only, there is no C binding. It is a
simplified version of [[prt_write]].
<<C Particles: public>>=
public :: c_prt_write
<<C Particles: sub interfaces>>=
module subroutine c_prt_write (prt, unit)
type(c_prt_t), intent(in) :: prt
integer, intent(in), optional :: unit
end subroutine c_prt_write
<<C Particles: procedures>>=
module subroutine c_prt_write (prt, unit)
type(c_prt_t), intent(in) :: prt
integer, intent(in), optional :: unit
integer :: u
u = given_output_unit (unit); if (u < 0) return
write (u, "(1x,A)", advance="no") "prt("
write (u, "(I0,':')", advance="no") prt%type
if (prt%polarized /= 0) then
write (u, "(I0,'/',I0,'|')", advance="no") prt%pdg, prt%h
else
write (u, "(I0,'|')", advance="no") prt%pdg
end if
write (u, "(" // FMT_14 // ",';'," // FMT_14 // ",','," // &
FMT_14 // ",','," // FMT_14 // ")", advance="no") &
prt%pe, prt%px, prt%py, prt%pz
write (u, "('|'," // FMT_19 // ")", advance="no") prt%p2
write (u, "(A)") ")"
end subroutine c_prt_write
@ %def c_prt_write
@
\clearpage
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Lorentz algebra}
Define Lorentz vectors, three-vectors, boosts, and some functions to
manipulate them.
To make maximum use of this, all functions, if possible, are declared
elemental (or pure, if this is not possible).
<<[[lorentz.f90]]>>=
<<File header>>
module lorentz
<<Use kinds with double>>
use constants, only: zero, one
use c_particles
<<Standard module head>>
<<Lorentz: public>>
<<Lorentz: public operators>>
<<Lorentz: public functions>>
<<Lorentz: types>>
<<Lorentz: parameters>>
<<Lorentz: interfaces>>
interface
<<Lorentz: sub interfaces>>
end interface
end module lorentz
@ %def lorentz
@
<<[[lorentz_sub.f90]]>>=
<<File header>>
submodule (lorentz) lorentz_s
use constants, only: pi, twopi, degree, two, tiny_07, eps0
use numeric_utils
use io_units
use format_defs, only: FMT_11, FMT_13, FMT_15, FMT_19
use format_utils, only: pac_fmt
use diagnostics
implicit none
contains
<<Lorentz: procedures>>
end submodule lorentz_s
@ %def lorentz_s
@
\subsection{Three-vectors}
First of all, let us introduce three-vectors in a trivial way. The
functions and overloaded elementary operations clearly are too much
overhead, but we like to keep the interface for three-vectors and
four-vectors exactly parallel. By the way, we might attach a label to
a vector by extending the type definition later.
<<Lorentz: public>>=
public :: vector3_t
<<Lorentz: types>>=
type :: vector3_t
real(default), dimension(3) :: p
end type vector3_t
@ %def vector3_t
@ Output a vector
<<Lorentz: public>>=
public :: vector3_write
<<Lorentz: sub interfaces>>=
module subroutine vector3_write (p, unit, testflag)
type(vector3_t), intent(in) :: p
integer, intent(in), optional :: unit
logical, intent(in), optional :: testflag
end subroutine vector3_write
<<Lorentz: procedures>>=
module subroutine vector3_write (p, unit, testflag)
type(vector3_t), intent(in) :: p
integer, intent(in), optional :: unit
logical, intent(in), optional :: testflag
character(len=7) :: fmt
integer :: u
u = given_output_unit (unit); if (u < 0) return
call pac_fmt (fmt, FMT_19, FMT_15, testflag)
write(u, "(1x,A,3(1x," // fmt // "))") 'P = ', p%p
end subroutine vector3_write
@ %def vector3_write
@ This is a three-vector with zero components
<<Lorentz: public>>=
public :: vector3_null
<<Lorentz: parameters>>=
type(vector3_t), parameter :: vector3_null = &
vector3_t ([ zero, zero, zero ])
@ %def vector3_null
@ Canonical three-vector:
<<Lorentz: public>>=
public :: vector3_canonical
<<Lorentz: sub interfaces>>=
elemental module function vector3_canonical (k) result (p)
type(vector3_t) :: p
integer, intent(in) :: k
end function vector3_canonical
<<Lorentz: procedures>>=
elemental module function vector3_canonical (k) result (p)
type(vector3_t) :: p
integer, intent(in) :: k
p = vector3_null
p%p(k) = 1
end function vector3_canonical
@ %def vector3_canonical
@ A moving particle ($k$-axis, or arbitrary axis). Note that the
function for the generic momentum cannot be elemental.
<<Lorentz: public>>=
public :: vector3_moving
<<Lorentz: interfaces>>=
interface vector3_moving
module procedure vector3_moving_canonical
module procedure vector3_moving_generic
end interface
<<Lorentz: sub interfaces>>=
elemental module function vector3_moving_canonical (p, k) result(q)
type(vector3_t) :: q
real(default), intent(in) :: p
integer, intent(in) :: k
end function vector3_moving_canonical
pure module function vector3_moving_generic (p) result(q)
real(default), dimension(3), intent(in) :: p
type(vector3_t) :: q
end function vector3_moving_generic
<<Lorentz: procedures>>=
elemental module function vector3_moving_canonical (p, k) result(q)
type(vector3_t) :: q
real(default), intent(in) :: p
integer, intent(in) :: k
q = vector3_null
q%p(k) = p
end function vector3_moving_canonical
pure module function vector3_moving_generic (p) result(q)
real(default), dimension(3), intent(in) :: p
type(vector3_t) :: q
q%p = p
end function vector3_moving_generic
@ %def vector3_moving
@ Equality and inequality
<<Lorentz: public operators>>=
public :: operator(==), operator(/=)
<<Lorentz: interfaces>>=
interface operator(==)
module procedure vector3_eq
end interface
interface operator(/=)
module procedure vector3_neq
end interface
<<Lorentz: sub interfaces>>=
elemental module function vector3_eq (p, q) result (r)
logical :: r
type(vector3_t), intent(in) :: p,q
end function vector3_eq
elemental module function vector3_neq (p, q) result (r)
logical :: r
type(vector3_t), intent(in) :: p,q
end function vector3_neq
<<Lorentz: procedures>>=
elemental module function vector3_eq (p, q) result (r)
logical :: r
type(vector3_t), intent(in) :: p,q
r = all (abs (p%p - q%p) < eps0)
end function vector3_eq
elemental module function vector3_neq (p, q) result (r)
logical :: r
type(vector3_t), intent(in) :: p,q
r = any (abs(p%p - q%p) > eps0)
end function vector3_neq
@ %def == /=
@ Define addition and subtraction
<<Lorentz: public operators>>=
public :: operator(+), operator(-)
<<Lorentz: interfaces>>=
interface operator(+)
module procedure add_vector3
end interface
interface operator(-)
module procedure sub_vector3
end interface
<<Lorentz: sub interfaces>>=
elemental module function add_vector3 (p, q) result (r)
type(vector3_t) :: r
type(vector3_t), intent(in) :: p,q
end function add_vector3
elemental module function sub_vector3 (p, q) result (r)
type(vector3_t) :: r
type(vector3_t), intent(in) :: p,q
end function sub_vector3
<<Lorentz: procedures>>=
elemental module function add_vector3 (p, q) result (r)
type(vector3_t) :: r
type(vector3_t), intent(in) :: p,q
r%p = p%p + q%p
end function add_vector3
elemental module function sub_vector3 (p, q) result (r)
type(vector3_t) :: r
type(vector3_t), intent(in) :: p,q
r%p = p%p - q%p
end function sub_vector3
@ %def + -
@ The multiplication sign is overloaded with scalar multiplication;
similarly division:
<<Lorentz: public operators>>=
public :: operator(*), operator(/)
<<Lorentz: interfaces>>=
interface operator(*)
module procedure prod_integer_vector3, prod_vector3_integer
module procedure prod_real_vector3, prod_vector3_real
end interface
interface operator(/)
module procedure div_vector3_real, div_vector3_integer
end interface
<<Lorentz: sub interfaces>>=
elemental module function prod_real_vector3 (s, p) result (q)
type(vector3_t) :: q
real(default), intent(in) :: s
type(vector3_t), intent(in) :: p
end function prod_real_vector3
elemental module function prod_vector3_real (p, s) result (q)
type(vector3_t) :: q
real(default), intent(in) :: s
type(vector3_t), intent(in) :: p
end function prod_vector3_real
elemental module function div_vector3_real (p, s) result (q)
type(vector3_t) :: q
real(default), intent(in) :: s
type(vector3_t), intent(in) :: p
end function div_vector3_real
elemental module function prod_integer_vector3 (s, p) result (q)
type(vector3_t) :: q
integer, intent(in) :: s
type(vector3_t), intent(in) :: p
end function prod_integer_vector3
elemental module function prod_vector3_integer (p, s) result (q)
type(vector3_t) :: q
integer, intent(in) :: s
type(vector3_t), intent(in) :: p
end function prod_vector3_integer
elemental module function div_vector3_integer (p, s) result (q)
type(vector3_t) :: q
integer, intent(in) :: s
type(vector3_t), intent(in) :: p
end function div_vector3_integer
<<Lorentz: procedures>>=
elemental module function prod_real_vector3 (s, p) result (q)
type(vector3_t) :: q
real(default), intent(in) :: s
type(vector3_t), intent(in) :: p
q%p = s * p%p
end function prod_real_vector3
elemental module function prod_vector3_real (p, s) result (q)
type(vector3_t) :: q
real(default), intent(in) :: s
type(vector3_t), intent(in) :: p
q%p = s * p%p
end function prod_vector3_real
elemental module function div_vector3_real (p, s) result (q)
type(vector3_t) :: q
real(default), intent(in) :: s
type(vector3_t), intent(in) :: p
q%p = p%p/s
end function div_vector3_real
elemental module function prod_integer_vector3 (s, p) result (q)
type(vector3_t) :: q
integer, intent(in) :: s
type(vector3_t), intent(in) :: p
q%p = s * p%p
end function prod_integer_vector3
elemental module function prod_vector3_integer (p, s) result (q)
type(vector3_t) :: q
integer, intent(in) :: s
type(vector3_t), intent(in) :: p
q%p = s * p%p
end function prod_vector3_integer
elemental module function div_vector3_integer (p, s) result (q)
type(vector3_t) :: q
integer, intent(in) :: s
type(vector3_t), intent(in) :: p
q%p = p%p/s
end function div_vector3_integer
@ %def * /
@ The multiplication sign can also indicate scalar products:
<<Lorentz: interfaces>>=
interface operator(*)
module procedure prod_vector3
end interface
<<Lorentz: sub interfaces>>=
elemental module function prod_vector3 (p, q) result (s)
real(default) :: s
type(vector3_t), intent(in) :: p,q
end function prod_vector3
<<Lorentz: procedures>>=
elemental module function prod_vector3 (p, q) result (s)
real(default) :: s
type(vector3_t), intent(in) :: p,q
s = dot_product (p%p, q%p)
end function prod_vector3
@ %def *
<<Lorentz: public functions>>=
public :: cross_product
<<Lorentz: interfaces>>=
interface cross_product
module procedure vector3_cross_product
end interface
<<Lorentz: sub interfaces>>=
elemental module function vector3_cross_product (p, q) result (r)
type(vector3_t) :: r
type(vector3_t), intent(in) :: p,q
end function vector3_cross_product
<<Lorentz: procedures>>=
elemental module function vector3_cross_product (p, q) result (r)
type(vector3_t) :: r
type(vector3_t), intent(in) :: p,q
integer :: i
do i=1,3
r%p(i) = dot_product (p%p, matmul(epsilon_three(i,:,:), q%p))
end do
end function vector3_cross_product
@ %def cross_product
@ Exponentiation is defined only for integer powers. Odd powers mean
take the square root; so [[p**1]] is the length of [[p]].
<<Lorentz: public operators>>=
public :: operator(**)
<<Lorentz: interfaces>>=
interface operator(**)
module procedure power_vector3
end interface
<<Lorentz: sub interfaces>>=
elemental module function power_vector3 (p, e) result (s)
real(default) :: s
type(vector3_t), intent(in) :: p
integer, intent(in) :: e
end function power_vector3
<<Lorentz: procedures>>=
elemental module function power_vector3 (p, e) result (s)
real(default) :: s
type(vector3_t), intent(in) :: p
integer, intent(in) :: e
s = dot_product (p%p, p%p)
if (e/=2) then
if (mod(e,2)==0) then
s = s**(e/2)
else
s = sqrt(s)**e
end if
end if
end function power_vector3
@ %def **
@ Finally, we need a negation.
<<Lorentz: interfaces>>=
interface operator(-)
module procedure negate_vector3
end interface
<<Lorentz: sub interfaces>>=
elemental module function negate_vector3 (p) result (q)
type(vector3_t) :: q
type(vector3_t), intent(in) :: p
end function negate_vector3
<<Lorentz: procedures>>=
elemental module function negate_vector3 (p) result (q)
type(vector3_t) :: q
type(vector3_t), intent(in) :: p
integer :: i
do i = 1, 3
if (abs (p%p(i)) < eps0) then
q%p(i) = 0
else
q%p(i) = -p%p(i)
end if
end do
end function negate_vector3
@ %def -
@ The sum function can be useful:
<<Lorentz: public functions>>=
public :: sum
<<Lorentz: interfaces>>=
interface sum
module procedure sum_vector3
end interface
@ %def sum
@
<<Lorentz: public>>=
public :: vector3_set_component
<<Lorentz: sub interfaces>>=
module subroutine vector3_set_component (p, i, value)
type(vector3_t), intent(inout) :: p
integer, intent(in) :: i
real(default), intent(in) :: value
end subroutine vector3_set_component
<<Lorentz: procedures>>=
module subroutine vector3_set_component (p, i, value)
type(vector3_t), intent(inout) :: p
integer, intent(in) :: i
real(default), intent(in) :: value
p%p(i) = value
end subroutine vector3_set_component
@ %def vector3_set_component
@
<<Lorentz: sub interfaces>>=
pure module function sum_vector3 (p) result (q)
type(vector3_t) :: q
type(vector3_t), dimension(:), intent(in) :: p
end function sum_vector3
<<Lorentz: procedures>>=
pure module function sum_vector3 (p) result (q)
type(vector3_t) :: q
type(vector3_t), dimension(:), intent(in) :: p
integer :: i
do i=1, 3
q%p(i) = sum (p%p(i))
end do
end function sum_vector3
@ %def sum
@ Any component:
<<Lorentz: public>>=
public :: vector3_get_component
@ %def component
<<Lorentz: sub interfaces>>=
elemental module function vector3_get_component (p, k) result (c)
type(vector3_t), intent(in) :: p
integer, intent(in) :: k
real(default) :: c
end function vector3_get_component
<<Lorentz: procedures>>=
elemental module function vector3_get_component (p, k) result (c)
type(vector3_t), intent(in) :: p
integer, intent(in) :: k
real(default) :: c
c = p%p(k)
end function vector3_get_component
@ %def vector3_get_component
@ Extract all components. This is not elemental.
<<Lorentz: public>>=
public :: vector3_get_components
<<Lorentz: sub interfaces>>=
pure module function vector3_get_components (p) result (a)
type(vector3_t), intent(in) :: p
real(default), dimension(3) :: a
end function vector3_get_components
<<Lorentz: procedures>>=
pure module function vector3_get_components (p) result (a)
type(vector3_t), intent(in) :: p
real(default), dimension(3) :: a
a = p%p
end function vector3_get_components
@ %def vector3_get_components
@ This function returns the direction of a three-vector, i.e., a
normalized three-vector. If the vector is null, we return a null vector.
<<Lorentz: public functions>>=
public :: direction
<<Lorentz: interfaces>>=
interface direction
module procedure vector3_get_direction
end interface
<<Lorentz: sub interfaces>>=
elemental module function vector3_get_direction (p) result (q)
type(vector3_t) :: q
type(vector3_t), intent(in) :: p
end function vector3_get_direction
<<Lorentz: procedures>>=
elemental module function vector3_get_direction (p) result (q)
type(vector3_t) :: q
type(vector3_t), intent(in) :: p
real(default) :: pp
pp = p**1
if (pp > eps0) then
q%p = p%p / pp
else
q%p = 0
end if
end function vector3_get_direction
@ %def direction
@
\subsection{Four-vectors}
In four-vectors the zero-component needs special treatment, therefore
we do not use the standard operations. Sure, we pay for the extra
layer of abstraction by losing efficiency; so we have to assume that
the time-critical applications do not involve four-vector operations.
<<Lorentz: public>>=
public :: vector4_t
<<Lorentz: types>>=
type :: vector4_t
real(default), dimension(0:3) :: p = &
[zero, zero, zero, zero]
contains
<<Lorentz: vector4: TBP>>
end type vector4_t
@ %def vector4_t
@ Output a vector
<<Lorentz: public>>=
public :: vector4_write
<<Lorentz: vector4: TBP>>=
procedure :: write => vector4_write
<<Lorentz: sub interfaces>>=
module subroutine vector4_write &
(p, unit, show_mass, testflag, compressed, ultra)
class(vector4_t), intent(in) :: p
integer, intent(in), optional :: unit
logical, intent(in), optional :: show_mass, testflag, compressed, ultra
end subroutine vector4_write
<<Lorentz: procedures>>=
module subroutine vector4_write &
(p, unit, show_mass, testflag, compressed, ultra)
class(vector4_t), intent(in) :: p
integer, intent(in), optional :: unit
logical, intent(in), optional :: show_mass, testflag, compressed, ultra
logical :: comp, sm, tf, extreme
integer :: u
character(len=7) :: fmt
real(default) :: m
comp = .false.; if (present (compressed)) comp = compressed
sm = .false.; if (present (show_mass)) sm = show_mass
tf = .false.; if (present (testflag)) tf = testflag
extreme = .false.; if (present (ultra)) extreme = ultra
if (extreme) then
call pac_fmt (fmt, FMT_19, FMT_11, testflag)
else
call pac_fmt (fmt, FMT_19, FMT_13, testflag)
end if
u = given_output_unit (unit); if (u < 0) return
if (comp) then
write (u, "(4(F12.3,1X))", advance="no") p%p(0:3)
else
write (u, "(1x,A,1x," // fmt // ")") 'E = ', p%p(0)
write (u, "(1x,A,3(1x," // fmt // "))") 'P = ', p%p(1:)
if (sm) then
m = p**1
if (tf) call pacify (m, tolerance = 1E-6_default)
write (u, "(1x,A,1x," // fmt // ")") 'M = ', m
end if
end if
end subroutine vector4_write
@ %def vector4_write
@ Binary I/O
<<Lorentz: public>>=
public :: vector4_write_raw
public :: vector4_read_raw
<<Lorentz: sub interfaces>>=
module subroutine vector4_write_raw (p, u)
type(vector4_t), intent(in) :: p
integer, intent(in) :: u
end subroutine vector4_write_raw
module subroutine vector4_read_raw (p, u, iostat)
type(vector4_t), intent(out) :: p
integer, intent(in) :: u
integer, intent(out), optional :: iostat
end subroutine vector4_read_raw
<<Lorentz: procedures>>=
module subroutine vector4_write_raw (p, u)
type(vector4_t), intent(in) :: p
integer, intent(in) :: u
write (u) p%p
end subroutine vector4_write_raw
module subroutine vector4_read_raw (p, u, iostat)
type(vector4_t), intent(out) :: p
integer, intent(in) :: u
integer, intent(out), optional :: iostat
read (u, iostat=iostat) p%p
end subroutine vector4_read_raw
@ %def vector4_write_raw vector4_read_raw
@ This is a four-vector with zero components
<<Lorentz: public>>=
public :: vector4_null
<<Lorentz: parameters>>=
type(vector4_t), parameter :: vector4_null = &
vector4_t ([ zero, zero, zero, zero ])
@ %def vector4_null
@ Canonical four-vector:
<<Lorentz: public>>=
public :: vector4_canonical
<<Lorentz: sub interfaces>>=
elemental module function vector4_canonical (k) result (p)
type(vector4_t) :: p
integer, intent(in) :: k
end function vector4_canonical
<<Lorentz: procedures>>=
elemental module function vector4_canonical (k) result (p)
type(vector4_t) :: p
integer, intent(in) :: k
p = vector4_null
p%p(k) = 1
end function vector4_canonical
@ %def vector4_canonical
@ A particle at rest:
<<Lorentz: public>>=
public :: vector4_at_rest
<<Lorentz: sub interfaces>>=
elemental module function vector4_at_rest (m) result (p)
type(vector4_t) :: p
real(default), intent(in) :: m
end function vector4_at_rest
<<Lorentz: procedures>>=
elemental module function vector4_at_rest (m) result (p)
type(vector4_t) :: p
real(default), intent(in) :: m
p = vector4_t ([ m, zero, zero, zero ])
end function vector4_at_rest
@ %def vector4_at_rest
@ A moving particle ($k$-axis, or arbitrary axis)
<<Lorentz: public>>=
public :: vector4_moving
<<Lorentz: interfaces>>=
interface vector4_moving
module procedure vector4_moving_canonical
module procedure vector4_moving_generic
end interface
<<Lorentz: sub interfaces>>=
elemental module function vector4_moving_canonical (E, p, k) result (q)
type(vector4_t) :: q
real(default), intent(in) :: E, p
integer, intent(in) :: k
end function vector4_moving_canonical
elemental module function vector4_moving_generic (E, p) result (q)
type(vector4_t) :: q
real(default), intent(in) :: E
type(vector3_t), intent(in) :: p
end function vector4_moving_generic
<<Lorentz: procedures>>=
elemental module function vector4_moving_canonical (E, p, k) result (q)
type(vector4_t) :: q
real(default), intent(in) :: E, p
integer, intent(in) :: k
q = vector4_at_rest(E)
q%p(k) = p
end function vector4_moving_canonical
elemental module function vector4_moving_generic (E, p) result (q)
type(vector4_t) :: q
real(default), intent(in) :: E
type(vector3_t), intent(in) :: p
q%p(0) = E
q%p(1:) = p%p
end function vector4_moving_generic
@ %def vector4_moving
@ Equality and inequality
<<Lorentz: interfaces>>=
interface operator(==)
module procedure vector4_eq
end interface
interface operator(/=)
module procedure vector4_neq
end interface
<<Lorentz: sub interfaces>>=
elemental module function vector4_eq (p, q) result (r)
logical :: r
type(vector4_t), intent(in) :: p,q
end function vector4_eq
elemental module function vector4_neq (p, q) result (r)
logical :: r
type(vector4_t), intent(in) :: p,q
end function vector4_neq
<<Lorentz: procedures>>=
elemental module function vector4_eq (p, q) result (r)
logical :: r
type(vector4_t), intent(in) :: p,q
r = all (abs (p%p - q%p) < eps0)
end function vector4_eq
elemental module function vector4_neq (p, q) result (r)
logical :: r
type(vector4_t), intent(in) :: p,q
r = any (abs (p%p - q%p) > eps0)
end function vector4_neq
@ %def == /=
@ Addition and subtraction:
<<Lorentz: interfaces>>=
interface operator(+)
module procedure add_vector4
end interface
interface operator(-)
module procedure sub_vector4
end interface
<<Lorentz: sub interfaces>>=
elemental module function add_vector4 (p,q) result (r)
type(vector4_t) :: r
type(vector4_t), intent(in) :: p,q
end function add_vector4
elemental module function sub_vector4 (p,q) result (r)
type(vector4_t) :: r
type(vector4_t), intent(in) :: p,q
end function sub_vector4
<<Lorentz: procedures>>=
elemental module function add_vector4 (p,q) result (r)
type(vector4_t) :: r
type(vector4_t), intent(in) :: p,q
r%p = p%p + q%p
end function add_vector4
elemental module function sub_vector4 (p,q) result (r)
type(vector4_t) :: r
type(vector4_t), intent(in) :: p,q
r%p = p%p - q%p
end function sub_vector4
@ %def + -
@ We also need scalar multiplication and division:
<<Lorentz: interfaces>>=
interface operator(*)
module procedure prod_real_vector4, prod_vector4_real
module procedure prod_integer_vector4, prod_vector4_integer
end interface
interface operator(/)
module procedure div_vector4_real
module procedure div_vector4_integer
end interface
<<Lorentz: sub interfaces>>=
elemental module function prod_real_vector4 (s, p) result (q)
type(vector4_t) :: q
real(default), intent(in) :: s
type(vector4_t), intent(in) :: p
end function prod_real_vector4
elemental module function prod_vector4_real (p, s) result (q)
type(vector4_t) :: q
real(default), intent(in) :: s
type(vector4_t), intent(in) :: p
end function prod_vector4_real
elemental module function div_vector4_real (p, s) result (q)
type(vector4_t) :: q
real(default), intent(in) :: s
type(vector4_t), intent(in) :: p
end function div_vector4_real
elemental module function prod_integer_vector4 (s, p) result (q)
type(vector4_t) :: q
integer, intent(in) :: s
type(vector4_t), intent(in) :: p
end function prod_integer_vector4
elemental module function prod_vector4_integer (p, s) result (q)
type(vector4_t) :: q
integer, intent(in) :: s
type(vector4_t), intent(in) :: p
end function prod_vector4_integer
elemental module function div_vector4_integer (p, s) result (q)
type(vector4_t) :: q
integer, intent(in) :: s
type(vector4_t), intent(in) :: p
end function div_vector4_integer
<<Lorentz: procedures>>=
elemental module function prod_real_vector4 (s, p) result (q)
type(vector4_t) :: q
real(default), intent(in) :: s
type(vector4_t), intent(in) :: p
q%p = s * p%p
end function prod_real_vector4
elemental module function prod_vector4_real (p, s) result (q)
type(vector4_t) :: q
real(default), intent(in) :: s
type(vector4_t), intent(in) :: p
q%p = s * p%p
end function prod_vector4_real
elemental module function div_vector4_real (p, s) result (q)
type(vector4_t) :: q
real(default), intent(in) :: s
type(vector4_t), intent(in) :: p
q%p = p%p/s
end function div_vector4_real
elemental module function prod_integer_vector4 (s, p) result (q)
type(vector4_t) :: q
integer, intent(in) :: s
type(vector4_t), intent(in) :: p
q%p = s * p%p
end function prod_integer_vector4
elemental module function prod_vector4_integer (p, s) result (q)
type(vector4_t) :: q
integer, intent(in) :: s
type(vector4_t), intent(in) :: p
q%p = s * p%p
end function prod_vector4_integer
elemental module function div_vector4_integer (p, s) result (q)
type(vector4_t) :: q
integer, intent(in) :: s
type(vector4_t), intent(in) :: p
q%p = p%p/s
end function div_vector4_integer
@ %def * /
@ Scalar products and squares in the Minkowski sense:
<<Lorentz: interfaces>>=
interface operator(*)
module procedure prod_vector4
end interface
interface operator(**)
module procedure power_vector4
end interface
<<Lorentz: sub interfaces>>=
elemental module function prod_vector4 (p, q) result (s)
real(default) :: s
type(vector4_t), intent(in) :: p,q
end function prod_vector4
<<Lorentz: procedures>>=
elemental module function prod_vector4 (p, q) result (s)
real(default) :: s
type(vector4_t), intent(in) :: p,q
s = p%p(0)*q%p(0) - dot_product(p%p(1:), q%p(1:))
end function prod_vector4
@ %def *
@ The power operation for four-vectors is signed, i.e., [[p**1]] is
positive for timelike and negative for spacelike vectors. Note that
[[(p**1)**2]] is not necessarily equal to [[p**2]].
<<Lorentz: sub interfaces>>=
elemental module function power_vector4 (p, e) result (s)
real(default) :: s
type(vector4_t), intent(in) :: p
integer, intent(in) :: e
end function power_vector4
<<Lorentz: procedures>>=
elemental module function power_vector4 (p, e) result (s)
real(default) :: s
type(vector4_t), intent(in) :: p
integer, intent(in) :: e
s = p * p
if (e /= 2) then
if (mod(e, 2) == 0) then
s = s**(e / 2)
else if (s >= 0) then
s = sqrt(s)**e
else
s = -(sqrt(abs(s))**e)
end if
end if
end function power_vector4
@ %def **
@ Finally, we introduce a negation
<<Lorentz: interfaces>>=
interface operator(-)
module procedure negate_vector4
end interface
<<Lorentz: sub interfaces>>=
elemental module function negate_vector4 (p) result (q)
type(vector4_t) :: q
type(vector4_t), intent(in) :: p
end function negate_vector4
<<Lorentz: procedures>>=
elemental module function negate_vector4 (p) result (q)
type(vector4_t) :: q
type(vector4_t), intent(in) :: p
integer :: i
do i = 0, 3
if (abs (p%p(i)) < eps0) then
q%p(i) = 0
else
q%p(i) = -p%p(i)
end if
end do
end function negate_vector4
@ %def -
@ The sum function can be useful:
<<Lorentz: interfaces>>=
interface sum
module procedure sum_vector4, sum_vector4_mask
end interface
@ %def sum
@
<<Lorentz: sub interfaces>>=
pure module function sum_vector4 (p) result (q)
type(vector4_t) :: q
type(vector4_t), dimension(:), intent(in) :: p
end function sum_vector4
pure module function sum_vector4_mask (p, mask) result (q)
type(vector4_t) :: q
type(vector4_t), dimension(:), intent(in) :: p
logical, dimension(:), intent(in) :: mask
end function sum_vector4_mask
<<Lorentz: procedures>>=
pure module function sum_vector4 (p) result (q)
type(vector4_t) :: q
type(vector4_t), dimension(:), intent(in) :: p
integer :: i
do i = 0, 3
q%p(i) = sum (p%p(i))
end do
end function sum_vector4
pure module function sum_vector4_mask (p, mask) result (q)
type(vector4_t) :: q
type(vector4_t), dimension(:), intent(in) :: p
logical, dimension(:), intent(in) :: mask
integer :: i
do i = 0, 3
q%p(i) = sum (p%p(i), mask=mask)
end do
end function sum_vector4_mask
@ %def sum
@
\subsection{Conversions}
Manually set a component of the four-vector:
<<Lorentz: public>>=
public :: vector4_set_component
<<Lorentz: sub interfaces>>=
module subroutine vector4_set_component (p, k, c)
type(vector4_t), intent(inout) :: p
integer, intent(in) :: k
real(default), intent(in) :: c
end subroutine vector4_set_component
<<Lorentz: procedures>>=
module subroutine vector4_set_component (p, k, c)
type(vector4_t), intent(inout) :: p
integer, intent(in) :: k
real(default), intent(in) :: c
p%p(k) = c
end subroutine vector4_set_component
@ %def vector4_get_component
Any component:
<<Lorentz: public>>=
public :: vector4_get_component
<<Lorentz: sub interfaces>>=
elemental module function vector4_get_component (p, k) result (c)
real(default) :: c
type(vector4_t), intent(in) :: p
integer, intent(in) :: k
end function vector4_get_component
<<Lorentz: procedures>>=
elemental module function vector4_get_component (p, k) result (c)
real(default) :: c
type(vector4_t), intent(in) :: p
integer, intent(in) :: k
c = p%p(k)
end function vector4_get_component
@ %def vector4_get_component
@ Extract all components. This is not elemental.
<<Lorentz: public>>=
public :: vector4_get_components
<<Lorentz: sub interfaces>>=
pure module function vector4_get_components (p) result (a)
real(default), dimension(0:3) :: a
type(vector4_t), intent(in) :: p
end function vector4_get_components
<<Lorentz: procedures>>=
pure module function vector4_get_components (p) result (a)
real(default), dimension(0:3) :: a
type(vector4_t), intent(in) :: p
a = p%p
end function vector4_get_components
@ %def vector4_get_components
@ This function returns the space part of a four-vector, such that we
can apply three-vector operations on it:
<<Lorentz: public functions>>=
public :: space_part
<<Lorentz: interfaces>>=
interface space_part
module procedure vector4_get_space_part
end interface
<<Lorentz: sub interfaces>>=
elemental module function vector4_get_space_part (p) result (q)
type(vector3_t) :: q
type(vector4_t), intent(in) :: p
end function vector4_get_space_part
<<Lorentz: procedures>>=
elemental module function vector4_get_space_part (p) result (q)
type(vector3_t) :: q
type(vector4_t), intent(in) :: p
q%p = p%p(1:)
end function vector4_get_space_part
@ %def space_part
@ This function returns the direction of a four-vector, i.e., a
normalized three-vector. If the four-vector has zero space part, we
return a null vector.
<<Lorentz: interfaces>>=
interface direction
module procedure vector4_get_direction
end interface
<<Lorentz: sub interfaces>>=
elemental module function vector4_get_direction (p) result (q)
type(vector3_t) :: q
type(vector4_t), intent(in) :: p
end function vector4_get_direction
<<Lorentz: procedures>>=
elemental module function vector4_get_direction (p) result (q)
type(vector3_t) :: q
type(vector4_t), intent(in) :: p
real(default) :: qq
q%p = p%p(1:)
qq = q**1
if (abs(qq) > eps0) then
q%p = q%p / qq
else
q%p = 0
end if
end function vector4_get_direction
@ %def direction
@ Change the sign of the spatial part of a four-vector
<<Lorentz: public>>=
public :: vector4_invert_direction
<<Lorentz: sub interfaces>>=
elemental module subroutine vector4_invert_direction (p)
type(vector4_t), intent(inout) :: p
end subroutine vector4_invert_direction
<<Lorentz: procedures>>=
elemental module subroutine vector4_invert_direction (p)
type(vector4_t), intent(inout) :: p
p%p(1:3) = -p%p(1:3)
end subroutine vector4_invert_direction
@ %def vector4_invert_direction
@ This function returns the four-vector as an ordinary array. A
second version for an array of four-vectors.
<<Lorentz: public>>=
public :: assignment (=)
<<Lorentz: interfaces>>=
interface assignment (=)
module procedure array_from_vector4_1, array_from_vector4_2, &
array_from_vector3_1, array_from_vector3_2, &
vector4_from_array, vector3_from_array
end interface
<<Lorentz: sub interfaces>>=
pure module subroutine array_from_vector4_1 (a, p)
real(default), dimension(:), intent(out) :: a
type(vector4_t), intent(in) :: p
end subroutine array_from_vector4_1
pure module subroutine array_from_vector4_2 (a, p)
type(vector4_t), dimension(:), intent(in) :: p
real(default), dimension(:,:), intent(out) :: a
end subroutine array_from_vector4_2
pure module subroutine array_from_vector3_1 (a, p)
real(default), dimension(:), intent(out) :: a
type(vector3_t), intent(in) :: p
end subroutine array_from_vector3_1
pure module subroutine array_from_vector3_2 (a, p)
type(vector3_t), dimension(:), intent(in) :: p
real(default), dimension(:,:), intent(out) :: a
end subroutine array_from_vector3_2
pure module subroutine vector4_from_array (p, a)
type(vector4_t), intent(out) :: p
real(default), dimension(:), intent(in) :: a
end subroutine vector4_from_array
pure module subroutine vector3_from_array (p, a)
type(vector3_t), intent(out) :: p
real(default), dimension(:), intent(in) :: a
end subroutine vector3_from_array
<<Lorentz: procedures>>=
pure module subroutine array_from_vector4_1 (a, p)
real(default), dimension(:), intent(out) :: a
type(vector4_t), intent(in) :: p
a = p%p
end subroutine array_from_vector4_1
pure module subroutine array_from_vector4_2 (a, p)
type(vector4_t), dimension(:), intent(in) :: p
real(default), dimension(:,:), intent(out) :: a
integer :: i
forall (i=1:size(p))
a(:,i) = p(i)%p
end forall
end subroutine array_from_vector4_2
pure module subroutine array_from_vector3_1 (a, p)
real(default), dimension(:), intent(out) :: a
type(vector3_t), intent(in) :: p
a = p%p
end subroutine array_from_vector3_1
pure module subroutine array_from_vector3_2 (a, p)
type(vector3_t), dimension(:), intent(in) :: p
real(default), dimension(:,:), intent(out) :: a
integer :: i
forall (i=1:size(p))
a(:,i) = p(i)%p
end forall
end subroutine array_from_vector3_2
pure module subroutine vector4_from_array (p, a)
type(vector4_t), intent(out) :: p
real(default), dimension(:), intent(in) :: a
p%p(0:3) = a
end subroutine vector4_from_array
pure module subroutine vector3_from_array (p, a)
type(vector3_t), intent(out) :: p
real(default), dimension(:), intent(in) :: a
p%p(1:3) = a
end subroutine vector3_from_array
@ %def array_from_vector4 array_from_vector3
@
<<Lorentz: public>>=
public :: vector4
<<Lorentz: sub interfaces>>=
pure module function vector4 (a) result (p)
type(vector4_t) :: p
real(default), intent(in), dimension(4) :: a
end function vector4
<<Lorentz: procedures>>=
pure module function vector4 (a) result (p)
type(vector4_t) :: p
real(default), intent(in), dimension(4) :: a
p%p = a
end function vector4
@ %def vector4
@
<<Lorentz: vector4: TBP>>=
procedure :: to_pythia6 => vector4_to_pythia6
<<Lorentz: sub interfaces>>=
pure module function vector4_to_pythia6 (vector4, m) result (p)
real(double), dimension(1:5) :: p
class(vector4_t), intent(in) :: vector4
real(default), intent(in), optional :: m
end function vector4_to_pythia6
<<Lorentz: procedures>>=
pure module function vector4_to_pythia6 (vector4, m) result (p)
real(double), dimension(1:5) :: p
class(vector4_t), intent(in) :: vector4
real(default), intent(in), optional :: m
p(1:3) = vector4%p(1:3)
p(4) = vector4%p(0)
if (present (m)) then
p(5) = m
else
p(5) = vector4 ** 1
end if
end function vector4_to_pythia6
@ %def vector4_to_pythia6
@
\subsection{Interface to [[c_prt]]}
Transform the momentum of a [[c_prt]] object into a four-vector and
vice versa:
<<Lorentz: interfaces>>=
interface assignment (=)
module procedure vector4_from_c_prt, c_prt_from_vector4
end interface
<<Lorentz: sub interfaces>>=
pure module subroutine vector4_from_c_prt (p, c_prt)
type(vector4_t), intent(out) :: p
type(c_prt_t), intent(in) :: c_prt
end subroutine vector4_from_c_prt
pure module subroutine c_prt_from_vector4 (c_prt, p)
type(c_prt_t), intent(out) :: c_prt
type(vector4_t), intent(in) :: p
end subroutine c_prt_from_vector4
<<Lorentz: procedures>>=
pure module subroutine vector4_from_c_prt (p, c_prt)
type(vector4_t), intent(out) :: p
type(c_prt_t), intent(in) :: c_prt
p%p(0) = c_prt%pe
p%p(1) = c_prt%px
p%p(2) = c_prt%py
p%p(3) = c_prt%pz
end subroutine vector4_from_c_prt
pure module subroutine c_prt_from_vector4 (c_prt, p)
type(c_prt_t), intent(out) :: c_prt
type(vector4_t), intent(in) :: p
c_prt%pe = p%p(0)
c_prt%px = p%p(1)
c_prt%py = p%p(2)
c_prt%pz = p%p(3)
c_prt%p2 = p ** 2
end subroutine c_prt_from_vector4
@ %def vector4_from_c_prt c_prt_from_vector4
@ Initialize a [[c_prt_t]] object with the components of a four-vector
as its kinematical entries. Compute the invariant mass, or use the
optional mass-squared value instead.
<<Lorentz: public>>=
public :: vector4_to_c_prt
<<Lorentz: sub interfaces>>=
elemental module function vector4_to_c_prt (p, p2) result (c_prt)
type(c_prt_t) :: c_prt
type(vector4_t), intent(in) :: p
real(default), intent(in), optional :: p2
end function vector4_to_c_prt
<<Lorentz: procedures>>=
elemental module function vector4_to_c_prt (p, p2) result (c_prt)
type(c_prt_t) :: c_prt
type(vector4_t), intent(in) :: p
real(default), intent(in), optional :: p2
c_prt%pe = p%p(0)
c_prt%px = p%p(1)
c_prt%py = p%p(2)
c_prt%pz = p%p(3)
if (present (p2)) then
c_prt%p2 = p2
else
c_prt%p2 = p ** 2
end if
end function vector4_to_c_prt
@ %def vector4_to_c_prt
@
\subsection{Angles}
Return the angles in a canonical system. The angle $\phi$ is defined
between $0\leq\phi<2\pi$. In degenerate cases, return zero.
<<Lorentz: public functions>>=
public :: azimuthal_angle
<<Lorentz: interfaces>>=
interface azimuthal_angle
module procedure vector3_azimuthal_angle
module procedure vector4_azimuthal_angle
end interface
<<Lorentz: sub interfaces>>=
elemental module function vector3_azimuthal_angle (p) result (phi)
real(default) :: phi
type(vector3_t), intent(in) :: p
end function vector3_azimuthal_angle
elemental module function vector4_azimuthal_angle (p) result (phi)
real(default) :: phi
type(vector4_t), intent(in) :: p
end function vector4_azimuthal_angle
<<Lorentz: procedures>>=
elemental module function vector3_azimuthal_angle (p) result (phi)
real(default) :: phi
type(vector3_t), intent(in) :: p
if (any (abs (p%p(1:2)) > 0)) then
phi = atan2(p%p(2), p%p(1))
if (phi < 0) phi = phi + twopi
else
phi = 0
end if
end function vector3_azimuthal_angle
elemental module function vector4_azimuthal_angle (p) result (phi)
real(default) :: phi
type(vector4_t), intent(in) :: p
phi = vector3_azimuthal_angle (space_part (p))
end function vector4_azimuthal_angle
@ %def azimuthal_angle
@ Azimuthal angle in degrees
<<Lorentz: public functions>>=
public :: azimuthal_angle_deg
<<Lorentz: interfaces>>=
interface azimuthal_angle_deg
module procedure vector3_azimuthal_angle_deg
module procedure vector4_azimuthal_angle_deg
end interface
<<Lorentz: sub interfaces>>=
elemental module function vector3_azimuthal_angle_deg (p) result (phi)
real(default) :: phi
type(vector3_t), intent(in) :: p
end function vector3_azimuthal_angle_deg
elemental module function vector4_azimuthal_angle_deg (p) result (phi)
real(default) :: phi
type(vector4_t), intent(in) :: p
end function vector4_azimuthal_angle_deg
<<Lorentz: procedures>>=
elemental module function vector3_azimuthal_angle_deg (p) result (phi)
real(default) :: phi
type(vector3_t), intent(in) :: p
phi = vector3_azimuthal_angle (p) / degree
end function vector3_azimuthal_angle_deg
elemental module function vector4_azimuthal_angle_deg (p) result (phi)
real(default) :: phi
type(vector4_t), intent(in) :: p
phi = vector4_azimuthal_angle (p) / degree
end function vector4_azimuthal_angle_deg
@ %def azimuthal_angle_deg
@ The azimuthal distance of two vectors. This is the difference of
the azimuthal angles, but cannot be larger than $\pi$: The result is
between $-\pi<\Delta\phi\leq\pi$.
<<Lorentz: public functions>>=
public :: azimuthal_distance
<<Lorentz: interfaces>>=
interface azimuthal_distance
module procedure vector3_azimuthal_distance
module procedure vector4_azimuthal_distance
end interface
<<Lorentz: sub interfaces>>=
elemental module function vector3_azimuthal_distance (p, q) result (dphi)
real(default) :: dphi
type(vector3_t), intent(in) :: p,q
end function vector3_azimuthal_distance
elemental module function vector4_azimuthal_distance (p, q) result (dphi)
real(default) :: dphi
type(vector4_t), intent(in) :: p,q
end function vector4_azimuthal_distance
<<Lorentz: procedures>>=
elemental module function vector3_azimuthal_distance (p, q) result (dphi)
real(default) :: dphi
type(vector3_t), intent(in) :: p,q
dphi = vector3_azimuthal_angle (q) - vector3_azimuthal_angle (p)
if (dphi <= -pi) then
dphi = dphi + twopi
else if (dphi > pi) then
dphi = dphi - twopi
end if
end function vector3_azimuthal_distance
elemental module function vector4_azimuthal_distance (p, q) result (dphi)
real(default) :: dphi
type(vector4_t), intent(in) :: p,q
dphi = vector3_azimuthal_distance &
(space_part (p), space_part (q))
end function vector4_azimuthal_distance
@ %def azimuthal_distance
@ The same in degrees:
<<Lorentz: public functions>>=
public :: azimuthal_distance_deg
<<Lorentz: interfaces>>=
interface azimuthal_distance_deg
module procedure vector3_azimuthal_distance_deg
module procedure vector4_azimuthal_distance_deg
end interface
<<Lorentz: sub interfaces>>=
elemental module function vector3_azimuthal_distance_deg (p, q) result (dphi)
real(default) :: dphi
type(vector3_t), intent(in) :: p,q
end function vector3_azimuthal_distance_deg
elemental module function vector4_azimuthal_distance_deg (p, q) result (dphi)
real(default) :: dphi
type(vector4_t), intent(in) :: p,q
end function vector4_azimuthal_distance_deg
<<Lorentz: procedures>>=
elemental module function vector3_azimuthal_distance_deg (p, q) result (dphi)
real(default) :: dphi
type(vector3_t), intent(in) :: p,q
dphi = vector3_azimuthal_distance (p, q) / degree
end function vector3_azimuthal_distance_deg
elemental module function vector4_azimuthal_distance_deg (p, q) result (dphi)
real(default) :: dphi
type(vector4_t), intent(in) :: p,q
dphi = vector4_azimuthal_distance (p, q) / degree
end function vector4_azimuthal_distance_deg
@ %def azimuthal_distance_deg
@ The polar angle is defined $0\leq\theta\leq\pi$. Note that
[[ATAN2]] has the reversed order of arguments: [[ATAN2(Y,X)]]. Here,
$x$ is the 3-component while $y$ is the transverse momentum which is
always nonnegative. Therefore, the result is nonnegative as well.
<<Lorentz: public functions>>=
public :: polar_angle
<<Lorentz: interfaces>>=
interface polar_angle
module procedure polar_angle_vector3
module procedure polar_angle_vector4
end interface
<<Lorentz: sub interfaces>>=
elemental module function polar_angle_vector3 (p) result (theta)
real(default) :: theta
type(vector3_t), intent(in) :: p
end function polar_angle_vector3
elemental module function polar_angle_vector4 (p) result (theta)
real(default) :: theta
type(vector4_t), intent(in) :: p
end function polar_angle_vector4
<<Lorentz: procedures>>=
elemental module function polar_angle_vector3 (p) result (theta)
real(default) :: theta
type(vector3_t), intent(in) :: p
if (any (abs (p%p) > 0)) then
theta = atan2 (sqrt(p%p(1)**2 + p%p(2)**2), p%p(3))
else
theta = 0
end if
end function polar_angle_vector3
elemental module function polar_angle_vector4 (p) result (theta)
real(default) :: theta
type(vector4_t), intent(in) :: p
theta = polar_angle (space_part (p))
end function polar_angle_vector4
@ %def polar_angle
@ This is the cosine of the polar angle: $-1\leq\cos\theta\leq 1$.
<<Lorentz: public functions>>=
public :: polar_angle_ct
<<Lorentz: interfaces>>=
interface polar_angle_ct
module procedure polar_angle_ct_vector3
module procedure polar_angle_ct_vector4
end interface
<<Lorentz: sub interfaces>>=
elemental module function polar_angle_ct_vector3 (p) result (ct)
real(default) :: ct
type(vector3_t), intent(in) :: p
end function polar_angle_ct_vector3
elemental module function polar_angle_ct_vector4 (p) result (ct)
real(default) :: ct
type(vector4_t), intent(in) :: p
end function polar_angle_ct_vector4
<<Lorentz: procedures>>=
elemental module function polar_angle_ct_vector3 (p) result (ct)
real(default) :: ct
type(vector3_t), intent(in) :: p
if (any (abs (p%p) > 0)) then
ct = p%p(3) / p**1
else
ct = 1
end if
end function polar_angle_ct_vector3
elemental module function polar_angle_ct_vector4 (p) result (ct)
real(default) :: ct
type(vector4_t), intent(in) :: p
ct = polar_angle_ct (space_part (p))
end function polar_angle_ct_vector4
@ %def polar_angle_ct
@ The polar angle in degrees.
<<Lorentz: public functions>>=
public :: polar_angle_deg
<<Lorentz: interfaces>>=
interface polar_angle_deg
module procedure polar_angle_deg_vector3
module procedure polar_angle_deg_vector4
end interface
<<Lorentz: sub interfaces>>=
elemental module function polar_angle_deg_vector3 (p) result (theta)
real(default) :: theta
type(vector3_t), intent(in) :: p
end function polar_angle_deg_vector3
elemental module function polar_angle_deg_vector4 (p) result (theta)
real(default) :: theta
type(vector4_t), intent(in) :: p
end function polar_angle_deg_vector4
<<Lorentz: procedures>>=
elemental module function polar_angle_deg_vector3 (p) result (theta)
real(default) :: theta
type(vector3_t), intent(in) :: p
theta = polar_angle (p) / degree
end function polar_angle_deg_vector3
elemental module function polar_angle_deg_vector4 (p) result (theta)
real(default) :: theta
type(vector4_t), intent(in) :: p
theta = polar_angle (p) / degree
end function polar_angle_deg_vector4
@ %def polar_angle_deg
@ This is the angle enclosed between two three-momenta. If one of the
momenta is zero, we return an angle of zero. The range of the result
is $0\leq\theta\leq\pi$. If there is only one argument, take the
positive $z$ axis as reference.
<<Lorentz: public functions>>=
public :: enclosed_angle
<<Lorentz: interfaces>>=
interface enclosed_angle
module procedure enclosed_angle_vector3
module procedure enclosed_angle_vector4
end interface
<<Lorentz: sub interfaces>>=
elemental module function enclosed_angle_vector3 (p, q) result (theta)
real(default) :: theta
type(vector3_t), intent(in) :: p, q
end function enclosed_angle_vector3
elemental module function enclosed_angle_vector4 (p, q) result (theta)
real(default) :: theta
type(vector4_t), intent(in) :: p, q
end function enclosed_angle_vector4
<<Lorentz: procedures>>=
elemental module function enclosed_angle_vector3 (p, q) result (theta)
real(default) :: theta
type(vector3_t), intent(in) :: p, q
theta = acos (enclosed_angle_ct (p, q))
end function enclosed_angle_vector3
elemental module function enclosed_angle_vector4 (p, q) result (theta)
real(default) :: theta
type(vector4_t), intent(in) :: p, q
theta = enclosed_angle (space_part (p), space_part (q))
end function enclosed_angle_vector4
@ %def enclosed_angle
@ The cosine of the enclosed angle.
<<Lorentz: public functions>>=
public :: enclosed_angle_ct
<<Lorentz: interfaces>>=
interface enclosed_angle_ct
module procedure enclosed_angle_ct_vector3
module procedure enclosed_angle_ct_vector4
end interface
<<Lorentz: sub interfaces>>=
elemental module function enclosed_angle_ct_vector3 (p, q) result (ct)
real(default) :: ct
type(vector3_t), intent(in) :: p, q
end function enclosed_angle_ct_vector3
elemental module function enclosed_angle_ct_vector4 (p, q) result (ct)
real(default) :: ct
type(vector4_t), intent(in) :: p, q
end function enclosed_angle_ct_vector4
<<Lorentz: procedures>>=
elemental module function enclosed_angle_ct_vector3 (p, q) result (ct)
real(default) :: ct
type(vector3_t), intent(in) :: p, q
if (any (abs (p%p) > 0) .and. any (abs (q%p) > 0)) then
ct = p*q / (p**1 * q**1)
if (ct>1) then
ct = 1
else if (ct<-1) then
ct = -1
end if
else
ct = 1
end if
end function enclosed_angle_ct_vector3
elemental module function enclosed_angle_ct_vector4 (p, q) result (ct)
real(default) :: ct
type(vector4_t), intent(in) :: p, q
ct = enclosed_angle_ct (space_part (p), space_part (q))
end function enclosed_angle_ct_vector4
@ %def enclosed_angle_ct
@ The enclosed angle in degrees.
<<Lorentz: public functions>>=
public :: enclosed_angle_deg
<<Lorentz: interfaces>>=
interface enclosed_angle_deg
module procedure enclosed_angle_deg_vector3
module procedure enclosed_angle_deg_vector4
end interface
<<Lorentz: sub interfaces>>=
elemental module function enclosed_angle_deg_vector3 (p, q) result (theta)
real(default) :: theta
type(vector3_t), intent(in) :: p, q
end function enclosed_angle_deg_vector3
elemental module function enclosed_angle_deg_vector4 (p, q) result (theta)
real(default) :: theta
type(vector4_t), intent(in) :: p, q
end function enclosed_angle_deg_vector4
<<Lorentz: procedures>>=
elemental module function enclosed_angle_deg_vector3 (p, q) result (theta)
real(default) :: theta
type(vector3_t), intent(in) :: p, q
theta = enclosed_angle (p, q) / degree
end function enclosed_angle_deg_vector3
elemental module function enclosed_angle_deg_vector4 (p, q) result (theta)
real(default) :: theta
type(vector4_t), intent(in) :: p, q
theta = enclosed_angle (p, q) / degree
end function enclosed_angle_deg_vector4
@ %def enclosed_angle
@ The polar angle of the first momentum w.r.t.\ the second momentum,
evaluated in the rest frame of the second momentum. If the second
four-momentum is not timelike, return zero.
<<Lorentz: public functions>>=
public :: enclosed_angle_rest_frame
public :: enclosed_angle_ct_rest_frame
public :: enclosed_angle_deg_rest_frame
<<Lorentz: interfaces>>=
interface enclosed_angle_rest_frame
module procedure enclosed_angle_rest_frame_vector4
end interface
interface enclosed_angle_ct_rest_frame
module procedure enclosed_angle_ct_rest_frame_vector4
end interface
interface enclosed_angle_deg_rest_frame
module procedure enclosed_angle_deg_rest_frame_vector4
end interface
<<Lorentz: sub interfaces>>=
elemental module function enclosed_angle_rest_frame_vector4 (p, q) result (theta)
type(vector4_t), intent(in) :: p, q
real(default) :: theta
end function enclosed_angle_rest_frame_vector4
elemental module function enclosed_angle_ct_rest_frame_vector4 (p, q) result (ct)
type(vector4_t), intent(in) :: p, q
real(default) :: ct
end function enclosed_angle_ct_rest_frame_vector4
elemental module function enclosed_angle_deg_rest_frame_vector4 (p, q) &
result (theta)
type(vector4_t), intent(in) :: p, q
real(default) :: theta
end function enclosed_angle_deg_rest_frame_vector4
<<Lorentz: procedures>>=
elemental module function enclosed_angle_rest_frame_vector4 (p, q) result (theta)
type(vector4_t), intent(in) :: p, q
real(default) :: theta
theta = acos (enclosed_angle_ct_rest_frame (p, q))
end function enclosed_angle_rest_frame_vector4
elemental module function enclosed_angle_ct_rest_frame_vector4 (p, q) result (ct)
type(vector4_t), intent(in) :: p, q
real(default) :: ct
if (invariant_mass(q) > 0) then
ct = enclosed_angle_ct ( &
space_part (boost(-q, invariant_mass (q)) * p), &
space_part (q))
else
ct = 1
end if
end function enclosed_angle_ct_rest_frame_vector4
elemental module function enclosed_angle_deg_rest_frame_vector4 (p, q) &
result (theta)
type(vector4_t), intent(in) :: p, q
real(default) :: theta
theta = enclosed_angle_rest_frame (p, q) / degree
end function enclosed_angle_deg_rest_frame_vector4
@ %def enclosed_angle_rest_frame
@ %def enclosed_angle_ct_rest_frame
@ %def enclosed_angle_deg_rest_frame
@
\subsection{More kinematical functions (some redundant)}
The scalar transverse momentum (assuming the $z$ axis is longitudinal)
<<Lorentz: public functions>>=
public :: transverse_part
<<Lorentz: interfaces>>=
interface transverse_part
module procedure transverse_part_vector4_beam_axis
module procedure transverse_part_vector4_vector4
end interface
<<Lorentz: sub interfaces>>=
elemental module function transverse_part_vector4_beam_axis (p) result (pT)
real(default) :: pT
type(vector4_t), intent(in) :: p
end function transverse_part_vector4_beam_axis
elemental module function transverse_part_vector4_vector4 (p1, p2) result (pT)
real(default) :: pT
type(vector4_t), intent(in) :: p1, p2
end function transverse_part_vector4_vector4
<<Lorentz: procedures>>=
elemental module function transverse_part_vector4_beam_axis (p) result (pT)
real(default) :: pT
type(vector4_t), intent(in) :: p
pT = sqrt(p%p(1)**2 + p%p(2)**2)
end function transverse_part_vector4_beam_axis
elemental module function transverse_part_vector4_vector4 (p1, p2) result (pT)
real(default) :: pT
type(vector4_t), intent(in) :: p1, p2
real(default) :: p1_norm, p2_norm, p1p2, pT2
p1_norm = space_part_norm(p1)**2
p2_norm = space_part_norm(p2)**2
! p1p2 = p1%p(1:3)*p2%p(1:3)
p1p2 = vector4_get_space_part(p1) * vector4_get_space_part(p2)
pT2 = (p1_norm*p2_norm - p1p2)/p1_norm
pT = sqrt (pT2)
end function transverse_part_vector4_vector4
@ %def transverse_part
@ The scalar longitudinal momentum (assuming the $z$ axis is
longitudinal). Identical to [[momentum_z_component]].
<<Lorentz: public functions>>=
public :: longitudinal_part
<<Lorentz: interfaces>>=
interface longitudinal_part
module procedure longitudinal_part_vector4
end interface
<<Lorentz: sub interfaces>>=
elemental module function longitudinal_part_vector4 (p) result (pL)
real(default) :: pL
type(vector4_t), intent(in) :: p
end function longitudinal_part_vector4
<<Lorentz: procedures>>=
elemental module function longitudinal_part_vector4 (p) result (pL)
real(default) :: pL
type(vector4_t), intent(in) :: p
pL = p%p(3)
end function longitudinal_part_vector4
@ %def longitudinal_part
@ Absolute value of three-momentum
<<Lorentz: public functions>>=
public :: space_part_norm
<<Lorentz: interfaces>>=
interface space_part_norm
module procedure space_part_norm_vector4
end interface
<<Lorentz: sub interfaces>>=
elemental module function space_part_norm_vector4 (p) result (p3)
real(default) :: p3
type(vector4_t), intent(in) :: p
end function space_part_norm_vector4
<<Lorentz: procedures>>=
elemental module function space_part_norm_vector4 (p) result (p3)
real(default) :: p3
type(vector4_t), intent(in) :: p
p3 = sqrt (p%p(1)**2 + p%p(2)**2 + p%p(3)**2)
end function space_part_norm_vector4
@ %def momentum
@ The energy (the zeroth component)
<<Lorentz: public functions>>=
public :: energy
<<Lorentz: interfaces>>=
interface energy
module procedure energy_vector4
module procedure energy_vector3
module procedure energy_real
end interface
<<Lorentz: sub interfaces>>=
elemental module function energy_vector4 (p) result (E)
real(default) :: E
type(vector4_t), intent(in) :: p
end function energy_vector4
elemental module function energy_vector3 (p, mass) result (E)
real(default) :: E
type(vector3_t), intent(in) :: p
real(default), intent(in), optional :: mass
end function energy_vector3
elemental module function energy_real (p, mass) result (E)
real(default) :: E
real(default), intent(in) :: p
real(default), intent(in), optional :: mass
end function energy_real
<<Lorentz: procedures>>=
elemental module function energy_vector4 (p) result (E)
real(default) :: E
type(vector4_t), intent(in) :: p
E = p%p(0)
end function energy_vector4
@ Alternative: The energy corresponding to a given momentum and mass.
If the mass is omitted, it is zero
<<Lorentz: procedures>>=
elemental module function energy_vector3 (p, mass) result (E)
real(default) :: E
type(vector3_t), intent(in) :: p
real(default), intent(in), optional :: mass
if (present (mass)) then
E = sqrt (p**2 + mass**2)
else
E = p**1
end if
end function energy_vector3
elemental module function energy_real (p, mass) result (E)
real(default) :: E
real(default), intent(in) :: p
real(default), intent(in), optional :: mass
if (present (mass)) then
E = sqrt (p**2 + mass**2)
else
E = abs (p)
end if
end function energy_real
@ %def energy
@ The invariant mass of four-momenta. Zero for lightlike, negative for
spacelike momenta.
<<Lorentz: public functions>>=
public :: invariant_mass
<<Lorentz: interfaces>>=
interface invariant_mass
module procedure invariant_mass_vector4
end interface
<<Lorentz: sub interfaces>>=
elemental module function invariant_mass_vector4 (p) result (m)
real(default) :: m
type(vector4_t), intent(in) :: p
end function invariant_mass_vector4
<<Lorentz: procedures>>=
elemental module function invariant_mass_vector4 (p) result (m)
real(default) :: m
type(vector4_t), intent(in) :: p
real(default) :: msq
msq = p*p
if (msq >= 0) then
m = sqrt (msq)
else
m = - sqrt (abs (msq))
end if
end function invariant_mass_vector4
@ %def invariant_mass
@ The invariant mass squared. Zero for lightlike, negative for
spacelike momenta.
<<Lorentz: public functions>>=
public :: invariant_mass_squared
<<Lorentz: interfaces>>=
interface invariant_mass_squared
module procedure invariant_mass_squared_vector4
end interface
<<Lorentz: sub interfaces>>=
elemental module function invariant_mass_squared_vector4 (p) result (msq)
real(default) :: msq
type(vector4_t), intent(in) :: p
end function invariant_mass_squared_vector4
<<Lorentz: procedures>>=
elemental module function invariant_mass_squared_vector4 (p) result (msq)
real(default) :: msq
type(vector4_t), intent(in) :: p
msq = p*p
end function invariant_mass_squared_vector4
@ %def invariant_mass_squared
@ The transverse mass. If the mass squared is negative, this value
also is negative.
<<Lorentz: public functions>>=
public :: transverse_mass
<<Lorentz: interfaces>>=
interface transverse_mass
module procedure transverse_mass_vector4
end interface
<<Lorentz: sub interfaces>>=
elemental module function transverse_mass_vector4 (p) result (m)
real(default) :: m
type(vector4_t), intent(in) :: p
end function transverse_mass_vector4
<<Lorentz: procedures>>=
elemental module function transverse_mass_vector4 (p) result (m)
real(default) :: m
type(vector4_t), intent(in) :: p
real(default) :: msq
msq = p%p(0)**2 - p%p(1)**2 - p%p(2)**2
if (msq >= 0) then
m = sqrt (msq)
else
m = - sqrt (abs (msq))
end if
end function transverse_mass_vector4
@ %def transverse_mass
@ The rapidity (defined if particle is massive or $p_\perp>0$)
<<Lorentz: public functions>>=
public :: rapidity
<<Lorentz: interfaces>>=
interface rapidity
module procedure rapidity_vector4
end interface
<<Lorentz: sub interfaces>>=
elemental module function rapidity_vector4 (p) result (y)
real(default) :: y
type(vector4_t), intent(in) :: p
end function rapidity_vector4
<<Lorentz: procedures>>=
elemental module function rapidity_vector4 (p) result (y)
real(default) :: y
type(vector4_t), intent(in) :: p
y = .5 * log( (energy (p) + longitudinal_part (p)) &
& /(energy (p) - longitudinal_part (p)))
end function rapidity_vector4
@ %def rapidity
@ The pseudorapidity (defined if $p_\perp>0$)
<<Lorentz: public functions>>=
public :: pseudorapidity
<<Lorentz: interfaces>>=
interface pseudorapidity
module procedure pseudorapidity_vector4
end interface
<<Lorentz: sub interfaces>>=
elemental module function pseudorapidity_vector4 (p) result (eta)
real(default) :: eta
type(vector4_t), intent(in) :: p
end function pseudorapidity_vector4
<<Lorentz: procedures>>=
elemental module function pseudorapidity_vector4 (p) result (eta)
real(default) :: eta
type(vector4_t), intent(in) :: p
eta = -log( tan (.5 * polar_angle (p)))
end function pseudorapidity_vector4
@ %def pseudorapidity
@ The rapidity distance (defined if both $p_\perp>0$)
<<Lorentz: public functions>>=
public :: rapidity_distance
<<Lorentz: interfaces>>=
interface rapidity_distance
module procedure rapidity_distance_vector4
end interface
<<Lorentz: sub interfaces>>=
elemental module function rapidity_distance_vector4 (p, q) result (dy)
type(vector4_t), intent(in) :: p, q
real(default) :: dy
end function rapidity_distance_vector4
<<Lorentz: procedures>>=
elemental module function rapidity_distance_vector4 (p, q) result (dy)
type(vector4_t), intent(in) :: p, q
real(default) :: dy
dy = rapidity (q) - rapidity (p)
end function rapidity_distance_vector4
@ %def rapidity_distance
@ The pseudorapidity distance (defined if both $p_\perp>0$)
<<Lorentz: public functions>>=
public :: pseudorapidity_distance
<<Lorentz: interfaces>>=
interface pseudorapidity_distance
module procedure pseudorapidity_distance_vector4
end interface
<<Lorentz: sub interfaces>>=
elemental module function pseudorapidity_distance_vector4 (p, q) result (deta)
real(default) :: deta
type(vector4_t), intent(in) :: p, q
end function pseudorapidity_distance_vector4
<<Lorentz: procedures>>=
elemental module function pseudorapidity_distance_vector4 (p, q) result (deta)
real(default) :: deta
type(vector4_t), intent(in) :: p, q
deta = pseudorapidity (q) - pseudorapidity (p)
end function pseudorapidity_distance_vector4
@ %def pseudorapidity_distance
@ The distance on the $\eta-\phi$ cylinder:
<<Lorentz: public functions>>=
public :: eta_phi_distance
<<Lorentz: interfaces>>=
interface eta_phi_distance
module procedure eta_phi_distance_vector4
end interface
<<Lorentz: sub interfaces>>=
elemental module function eta_phi_distance_vector4 (p, q) result (dr)
type(vector4_t), intent(in) :: p, q
real(default) :: dr
end function eta_phi_distance_vector4
<<Lorentz: procedures>>=
elemental module function eta_phi_distance_vector4 (p, q) result (dr)
type(vector4_t), intent(in) :: p, q
real(default) :: dr
dr = sqrt ( &
pseudorapidity_distance (p, q)**2 &
+ azimuthal_distance (p, q)**2)
end function eta_phi_distance_vector4
@ %def eta_phi_distance
@
\subsection{Lorentz transformations}
<<Lorentz: public>>=
public :: lorentz_transformation_t
<<Lorentz: types>>=
type :: lorentz_transformation_t
private
real(default), dimension(0:3, 0:3) :: L
contains
<<Lorentz: lorentz transformation: TBP>>
end type lorentz_transformation_t
@ %def lorentz_transformation_t
@ Output:
<<Lorentz: public>>=
public :: lorentz_transformation_write
<<Lorentz: lorentz transformation: TBP>>=
procedure :: write => lorentz_transformation_write
<<Lorentz: sub interfaces>>=
module subroutine lorentz_transformation_write (L, unit, testflag, ultra)
class(lorentz_transformation_t), intent(in) :: L
integer, intent(in), optional :: unit
logical, intent(in), optional :: testflag, ultra
end subroutine lorentz_transformation_write
<<Lorentz: procedures>>=
module subroutine lorentz_transformation_write (L, unit, testflag, ultra)
class(lorentz_transformation_t), intent(in) :: L
integer, intent(in), optional :: unit
logical, intent(in), optional :: testflag, ultra
integer :: u, i
logical :: ult
character(len=7) :: fmt
ult = .false.; if (present (ultra)) ult = ultra
if (ult) then
call pac_fmt (fmt, FMT_19, FMT_11, ultra)
else
call pac_fmt (fmt, FMT_19, FMT_13, testflag)
end if
u = given_output_unit (unit); if (u < 0) return
write (u, "(1x,A,3(1x," // fmt // "))") "L00 = ", L%L(0,0)
write (u, "(1x,A,3(1x," // fmt // "))") "L0j = ", L%L(0,1:3)
do i = 1, 3
write (u, "(1x,A,I0,A,3(1x," // fmt // "))") &
"L", i, "0 = ", L%L(i,0)
write (u, "(1x,A,I0,A,3(1x," // fmt // "))") &
"L", i, "j = ", L%L(i,1:3)
end do
end subroutine lorentz_transformation_write
@ %def lorentz_transformation_write
@ Extract all components:
<<Lorentz: public>>=
public :: lorentz_transformation_get_components
<<Lorentz: sub interfaces>>=
pure module function lorentz_transformation_get_components (L) result (a)
type(lorentz_transformation_t), intent(in) :: L
real(default), dimension(0:3,0:3) :: a
end function lorentz_transformation_get_components
<<Lorentz: procedures>>=
pure module function lorentz_transformation_get_components (L) result (a)
type(lorentz_transformation_t), intent(in) :: L
real(default), dimension(0:3,0:3) :: a
a = L%L
end function lorentz_transformation_get_components
@ %def lorentz_transformation_get_components
@
\subsection{Functions of Lorentz transformations}
For the inverse, we make use of the fact that
$\Lambda^{\mu\nu}\Lambda_{\mu\rho}=\delta^\nu_\rho$. So, lowering the
indices and transposing is sufficient.
<<Lorentz: public functions>>=
public :: inverse
<<Lorentz: interfaces>>=
interface inverse
module procedure lorentz_transformation_inverse
end interface
<<Lorentz: sub interfaces>>=
elemental module function lorentz_transformation_inverse (L) result (IL)
type(lorentz_transformation_t) :: IL
type(lorentz_transformation_t), intent(in) :: L
end function lorentz_transformation_inverse
<<Lorentz: procedures>>=
elemental module function lorentz_transformation_inverse (L) result (IL)
type(lorentz_transformation_t) :: IL
type(lorentz_transformation_t), intent(in) :: L
IL%L(0,0) = L%L(0,0)
IL%L(0,1:) = -L%L(1:,0)
IL%L(1:,0) = -L%L(0,1:)
IL%L(1:,1:) = transpose(L%L(1:,1:))
end function lorentz_transformation_inverse
@ %def lorentz_transformation_inverse
@ %def inverse
@
\subsection{Invariants}
These are used below. The first array index is varying fastest in
[[FORTRAN]]; therefore the extra minus in the odd-rank tensor
epsilon.
<<Lorentz: parameters>>=
integer, dimension(3,3), parameter :: delta_three = &
& reshape( source = [ 1,0,0, 0,1,0, 0,0,1 ], &
& shape = [3,3] )
integer, dimension(3,3,3), parameter :: epsilon_three = &
& reshape( source = [ 0, 0,0, 0,0,-1, 0,1,0, &
& 0, 0,1, 0,0, 0, -1,0,0, &
& 0,-1,0, 1,0, 0, 0,0,0 ],&
& shape = [3,3,3] )
@ %def delta_three epsilon_three
@ This could be of some use:
<<Lorentz: public>>=
public :: identity
<<Lorentz: parameters>>=
type(lorentz_transformation_t), parameter :: &
& identity = &
& lorentz_transformation_t ( &
& reshape( source = [ one, zero, zero, zero, &
& zero, one, zero, zero, &
& zero, zero, one, zero, &
& zero, zero, zero, one ],&
& shape = [4,4] ) )
@ %def identity
<<Lorentz: public>>=
public :: space_reflection
<<Lorentz: parameters>>=
type(lorentz_transformation_t), parameter :: &
& space_reflection = &
& lorentz_transformation_t ( &
& reshape( source = [ one, zero, zero, zero, &
& zero,-one, zero, zero, &
& zero, zero,-one, zero, &
& zero, zero, zero,-one ],&
& shape = [4,4] ) )
@ %def space_reflection
@ Builds a unit vector orthogal to the input vector in the xy-plane.
<<Lorentz: public functions>>=
public :: create_orthogonal
<<Lorentz: sub interfaces>>=
module function create_orthogonal (p_in) result (p_out)
type(vector3_t), intent(in) :: p_in
type(vector3_t) :: p_out
end function create_orthogonal
<<Lorentz: procedures>>=
module function create_orthogonal (p_in) result (p_out)
type(vector3_t), intent(in) :: p_in
type(vector3_t) :: p_out
real(default) :: ab
ab = sqrt (p_in%p(1)**2 + p_in%p(2)**2)
if (abs (ab) < eps0) then
p_out%p(1) = 1
p_out%p(2) = 0
p_out%p(3) = 0
else
p_out%p(1) = p_in%p(2)
p_out%p(2) = -p_in%p(1)
p_out%p(3) = 0
p_out = p_out / ab
end if
end function create_orthogonal
@ %def create_orthogonal
@
<<Lorentz: public functions>>=
public :: create_unit_vector
<<Lorentz: sub interfaces>>=
module function create_unit_vector (p_in) result (p_out)
type(vector4_t), intent(in) :: p_in
type(vector3_t) :: p_out
end function create_unit_vector
<<Lorentz: procedures>>=
module function create_unit_vector (p_in) result (p_out)
type(vector4_t), intent(in) :: p_in
type(vector3_t) :: p_out
p_out%p = p_in%p(1:3) / space_part_norm (p_in)
end function create_unit_vector
@ %def create_unit_vector
@
<<Lorentz: public functions>>=
public :: normalize
<<Lorentz: sub interfaces>>=
module function normalize(p) result (p_norm)
type(vector3_t) :: p_norm
type(vector3_t), intent(in) :: p
end function normalize
<<Lorentz: procedures>>=
module function normalize(p) result (p_norm)
type(vector3_t) :: p_norm
type(vector3_t), intent(in) :: p
real(default) :: abs
abs = sqrt (p%p(1)**2 + p%p(2)**2 + p%p(3)**2)
p_norm = p / abs
end function normalize
@ %def normalize
@ Computes the invariant mass of the momenta sum given by the indices in
[[i_res_born]] and the optional argument [[i_emitter]].
<<Lorentz: public>>=
public :: compute_resonance_mass
<<Lorentz: sub interfaces>>=
pure module function compute_resonance_mass (p, i_res_born, i_gluon) result (m)
real(default) :: m
type(vector4_t), intent(in), dimension(:) :: p
integer, intent(in), dimension(:) :: i_res_born
integer, intent(in), optional :: i_gluon
end function compute_resonance_mass
<<Lorentz: procedures>>=
pure module function compute_resonance_mass (p, i_res_born, i_gluon) result (m)
real(default) :: m
type(vector4_t), intent(in), dimension(:) :: p
integer, intent(in), dimension(:) :: i_res_born
integer, intent(in), optional :: i_gluon
type(vector4_t) :: p_res
p_res = get_resonance_momentum (p, i_res_born, i_gluon)
m = p_res**1
end function compute_resonance_mass
@ %def compute_resonance_mass
@
<<Lorentz: public>>=
public :: get_resonance_momentum
<<Lorentz: sub interfaces>>=
pure module function get_resonance_momentum &
(p, i_res_born, i_gluon) result (p_res)
type(vector4_t) :: p_res
type(vector4_t), intent(in), dimension(:) :: p
integer, intent(in), dimension(:) :: i_res_born
integer, intent(in), optional :: i_gluon
end function get_resonance_momentum
<<Lorentz: procedures>>=
pure module function get_resonance_momentum &
(p, i_res_born, i_gluon) result (p_res)
type(vector4_t) :: p_res
type(vector4_t), intent(in), dimension(:) :: p
integer, intent(in), dimension(:) :: i_res_born
integer, intent(in), optional :: i_gluon
integer :: i
p_res = vector4_null
do i = 1, size (i_res_born)
p_res = p_res + p (i_res_born(i))
end do
if (present (i_gluon)) p_res = p_res + p (i_gluon)
end function get_resonance_momentum
@ %def get_resonance_momentum
@
<<Lorentz: public>>=
public :: create_two_particle_decay
<<Lorentz: sub interfaces>>=
module function create_two_particle_decay (s, p1, p2) result (p_rest)
type(vector4_t), dimension(3) :: p_rest
real(default), intent(in) :: s
type(vector4_t), intent(in) :: p1, p2
end function create_two_particle_decay
<<Lorentz: procedures>>=
module function create_two_particle_decay (s, p1, p2) result (p_rest)
type(vector4_t), dimension(3) :: p_rest
real(default), intent(in) :: s
type(vector4_t), intent(in) :: p1, p2
real(default) :: m1_sq, m2_sq
real(default) :: E1, E2, p
m1_sq = p1**2; m2_sq = p2**2
p = sqrt (lambda (s, m1_sq, m2_sq)) / (two * sqrt (s))
E1 = sqrt (m1_sq + p**2); E2 = sqrt (m2_sq + p**2)
p_rest(1)%p = [sqrt (s), zero, zero, zero]
p_rest(2)%p(0) = E1
p_rest(2)%p(1:3) = p * p1%p(1:3) / space_part_norm (p1)
p_rest(3)%p(0) = E2; p_rest(3)%p(1:3) = -p_rest(2)%p(1:3)
end function create_two_particle_decay
@ %def create_two_particle_decay
@ This function creates a phase-space point for a $1 \to 3$ decay in
the decaying particle's rest frame. There are three rest frames for
this system, corresponding to $s$-, $t$,- and $u$-channel momentum
exchange, also referred to as Gottfried-Jackson frames. Below, we choose
the momentum with index 1 to be aligned along the $z$-axis. We then
have
\begin{align*}
s_1 &= \left(p_1 + p_2\right)^2, \\
s_2 &= \left(p_2 + p_3\right)^2, \\
s_3 &= \left(p_1 + p_3\right)^2, \\
s_1 + s_2 + s_3 &= s + m_1^2 + m_2^2 + m_3^2.
\end{align*}
From these we can construct
\begin{align*}
E_1^{R23} = \frac{s - s_2 - m_1^2}{2\sqrt{s_2}} &\quad P_1^{R23} = \frac{\lambda^{1/2}(s, s_2, m_1^2)}{2\sqrt{s_2}},\\
E_2^{R23} = \frac{s_2 + m_2^2 - m_3^2}{2\sqrt{s_2}} &\quad P_2^{R23} = \frac{\lambda^{1/2}(s_2, m_2^2, m_3^2)}{2\sqrt{s_2}},\\
E_3^{R23} = \frac{s_2 + m_3^2 - m_2^2}{2\sqrt{s_2}} &\quad P_3^{R23} = P_2^{R23},
\end{align*}
where $R23$ denotes the Gottfried-Jackson frame of our choice. Finally, the scattering angle $\theta_{12}^{R23}$ between
momentum $1$ and $2$ can be determined to be
\begin{equation*}
\cos\theta_{12}^{R23} = \frac{(s - s_2 - m_1^2)(s_2 + m_2^2 - m_3^2) + 2s_2 (m_1^2 + m_2^2 - s_1)}
{\lambda^{1/2}(s, s_2, m_1^2) \lambda^{1/2}(s_2, m_2^2, m_3^2)}
\end{equation*}
<<Lorentz: public>>=
public :: create_three_particle_decay
<<Lorentz: sub interfaces>>=
module function create_three_particle_decay (p1, p2, p3) result (p_rest)
type(vector4_t), dimension(4) :: p_rest
type(vector4_t), intent(in) :: p1, p2, p3
end function create_three_particle_decay
<<Lorentz: procedures>>=
module function create_three_particle_decay (p1, p2, p3) result (p_rest)
type(vector4_t), dimension(4) :: p_rest
type(vector4_t), intent(in) :: p1, p2, p3
real(default) :: E1, E2, E3
real(default) :: pr1, pr2, pr3
real(default) :: s, s1, s2, s3
real(default) :: m1_sq, m2_sq, m3_sq
real(default) :: cos_theta_12
type(vector3_t) :: v3_unit
type(lorentz_transformation_t) :: rot
m1_sq = p1**2
m2_sq = p2**2
m3_sq = p3**2
s1 = (p1 + p2)**2
s2 = (p2 + p3)**2
s3 = (p3 + p1)**2
s = s1 + s2 + s3 - m1_sq - m2_sq - m3_sq
E1 = (s - s2 - m1_sq) / (two * sqrt (s2))
E2 = (s2 + m2_sq - m3_sq) / (two * sqrt (s2))
E3 = (s2 + m3_sq - m2_sq) / (two * sqrt (s2))
pr1 = sqrt (lambda (s, s2, m1_sq)) / (two * sqrt (s2))
pr2 = sqrt (lambda (s2, m2_sq, m3_sq)) / (two * sqrt(s2))
pr3 = pr2
cos_theta_12 = ((s - s2 - m1_sq) * (s2 + m2_sq - m3_sq) + two * s2 * (m1_sq + m2_sq - s1)) / &
sqrt (lambda (s, s2, m1_sq) * lambda (s2, m2_sq, m3_sq))
v3_unit%p = [zero, zero, one]
p_rest(1)%p(0) = E1
p_rest(1)%p(1:3) = v3_unit%p * pr1
p_rest(2)%p(0) = E2
p_rest(2)%p(1:3) = v3_unit%p * pr2
p_rest(3)%p(0) = E3
p_rest(3)%p(1:3) = v3_unit%p * pr3
p_rest(4)%p(0) = (s + s2 - m1_sq) / (2 * sqrt (s2))
p_rest(4)%p(1:3) = - p_rest(1)%p(1:3)
rot = rotation (cos_theta_12, sqrt (one - cos_theta_12**2), 2)
p_rest(2) = rot * p_rest(2)
p_rest(3)%p(1:3) = - p_rest(2)%p(1:3)
end function create_three_particle_decay
@ %def create_three_particle_decay
@
<<Lorentz: public>>=
public :: evaluate_one_to_two_splitting_special
<<Lorentz: interfaces>>=
abstract interface
subroutine evaluate_one_to_two_splitting_special (p_origin, &
p1_in, p2_in, p1_out, p2_out, msq_in, jac)
import
type(vector4_t), intent(in) :: p_origin
type(vector4_t), intent(in) :: p1_in, p2_in
type(vector4_t), intent(inout) :: p1_out, p2_out
real(default), intent(in), optional :: msq_in
real(default), intent(inout), optional :: jac
end subroutine evaluate_one_to_two_splitting_special
end interface
@ %def evaluate_one_to_two_splitting_special
@
<<Lorentz: public>>=
public :: generate_on_shell_decay
<<Lorentz: sub interfaces>>=
recursive module subroutine generate_on_shell_decay (p_dec, &
p_in, p_out, i_real, msq_in, jac, evaluate_special)
type(vector4_t), intent(in) :: p_dec
type(vector4_t), intent(in), dimension(:) :: p_in
type(vector4_t), intent(inout), dimension(:) :: p_out
integer, intent(in) :: i_real
real(default), intent(in), optional :: msq_in
real(default), intent(inout), optional :: jac
procedure(evaluate_one_to_two_splitting_special), intent(in), &
pointer, optional :: evaluate_special
end subroutine generate_on_shell_decay
<<Lorentz: procedures>>=
recursive module subroutine generate_on_shell_decay (p_dec, &
p_in, p_out, i_real, msq_in, jac, evaluate_special)
type(vector4_t), intent(in) :: p_dec
type(vector4_t), intent(in), dimension(:) :: p_in
type(vector4_t), intent(inout), dimension(:) :: p_out
integer, intent(in) :: i_real
real(default), intent(in), optional :: msq_in
real(default), intent(inout), optional :: jac
procedure(evaluate_one_to_two_splitting_special), intent(in), &
pointer, optional :: evaluate_special
type(vector4_t) :: p_dec_new
integer :: n_recoil
n_recoil = size (p_in) - 1
if (n_recoil > 1) then
if (present (evaluate_special)) then
call evaluate_special (p_dec, p_in(1), sum (p_in (2 : n_recoil + 1)), &
p_out(i_real), p_dec_new)
call generate_on_shell_decay (p_dec_new, p_in (2 : ), p_out, &
i_real + 1, msq_in, jac, evaluate_special)
else
call evaluate_one_to_two_splitting (p_dec, p_in(1), &
sum (p_in (2 : n_recoil + 1)), p_out(i_real), p_dec_new, msq_in, jac)
call generate_on_shell_decay (p_dec_new, p_in (2 : ), p_out, &
i_real + 1, msq_in, jac)
end if
else
call evaluate_one_to_two_splitting (p_dec, p_in(1), p_in(2), &
p_out(i_real), p_out(i_real + 1), msq_in, jac)
end if
end subroutine generate_on_shell_decay
subroutine evaluate_one_to_two_splitting (p_origin, &
p1_in, p2_in, p1_out, p2_out, msq_in, jac)
type(vector4_t), intent(in) :: p_origin
type(vector4_t), intent(in) :: p1_in, p2_in
type(vector4_t), intent(inout) :: p1_out, p2_out
real(default), intent(in), optional :: msq_in
real(default), intent(inout), optional :: jac
type(lorentz_transformation_t) :: L
type(vector4_t) :: p1_rest, p2_rest
real(default) :: m, msq, msq1, msq2
real(default) :: E1, E2, p
real(default) :: lda, rlda_soft
call get_rest_frame (p1_in, p2_in, p1_rest, p2_rest)
msq = p_origin**2; m = sqrt(msq)
msq1 = p1_in**2; msq2 = p2_in**2
lda = lambda (msq, msq1, msq2)
if (lda < zero) then
print *, 'Encountered lambda < 0 in 1 -> 2 splitting! '
print *, 'lda: ', lda
print *, 'm: ', m, 'msq: ', msq
print *, 'm1: ', sqrt (msq1), 'msq1: ', msq1
print *, 'm2: ', sqrt (msq2), 'msq2: ', msq2
stop
end if
p = sqrt (lda) / (two * m)
E1 = sqrt (msq1 + p**2)
E2 = sqrt (msq2 + p**2)
p1_out = shift_momentum (p1_rest, E1, p)
p2_out = shift_momentum (p2_rest, E2, p)
L = boost (p_origin, p_origin**1)
p1_out = L * p1_out
p2_out = L * p2_out
if (present (jac) .and. present (msq_in)) then
jac = jac * sqrt(lda) / msq
rlda_soft = sqrt (lambda (msq_in, msq1, msq2))
!!! We have to undo the Jacobian which has already been
!!! supplied by the Born phase space.
jac = jac * msq_in / rlda_soft
end if
contains
subroutine get_rest_frame (p1_in, p2_in, p1_out, p2_out)
type(vector4_t), intent(in) :: p1_in, p2_in
type(vector4_t), intent(out) :: p1_out, p2_out
type(lorentz_transformation_t) :: L
L = inverse (boost (p1_in + p2_in, (p1_in + p2_in)**1))
p1_out = L * p1_in; p2_out = L * p2_in
end subroutine get_rest_frame
function shift_momentum (p_in, E, p) result (p_out)
type(vector4_t) :: p_out
type(vector4_t), intent(in) :: p_in
real(default), intent(in) :: E, p
type(vector3_t) :: vec
vec%p(1:3) = p_in%p(1:3) / space_part_norm (p_in)
p_out = vector4_moving (E, p * vec)
end function shift_momentum
end subroutine evaluate_one_to_two_splitting
@ %def generate_on_shell_decay
@
\subsection{Boosts}
We build Lorentz transformations from boosts and rotations. In both
cases we can supply a three-vector which defines the axis and (hyperbolic)
angle. For a boost, this is the vector $\vec\beta=\vec p/E$,
such that a particle at rest with mass $m$ is boosted to a particle
with three-vector $\vec p$. Here, we have
\begin{equation}
\beta = \tanh\chi = p/E, \qquad
\gamma = \cosh\chi = E/m, \qquad
\beta\gamma = \sinh\chi = p/m
\end{equation}
<<Lorentz: public functions>>=
public :: boost
<<Lorentz: interfaces>>=
interface boost
module procedure boost_from_rest_frame
module procedure boost_from_rest_frame_vector3
module procedure boost_generic
module procedure boost_canonical
end interface
@ %def boost
@ In the first form, the argument is some four-momentum, the space
part of which determines a direction, and the associated mass (which
is not checked against the four-momentum). The boost vector
$\gamma\vec\beta$ is then given by $\vec p/m$. This boosts from the
rest frame of a particle to the current frame. To be explicit, if
$\vec p$ is the momentum of a particle and $m$ its mass, $L(\vec p/m)$
is the transformation that turns $(m;\vec 0)$ into $(E;\vec p)$.
Conversely, the inverse transformation boosts a vector \emph{into} the
rest frame of a particle, in particular $(E;\vec p)$ into $(m;\vec
0)$.
<<Lorentz: sub interfaces>>=
elemental module function boost_from_rest_frame (p, m) result (L)
type(lorentz_transformation_t) :: L
type(vector4_t), intent(in) :: p
real(default), intent(in) :: m
end function boost_from_rest_frame
elemental module function boost_from_rest_frame_vector3 (p, m) result (L)
type(lorentz_transformation_t) :: L
type(vector3_t), intent(in) :: p
real(default), intent(in) :: m
end function boost_from_rest_frame_vector3
<<Lorentz: procedures>>=
elemental module function boost_from_rest_frame (p, m) result (L)
type(lorentz_transformation_t) :: L
type(vector4_t), intent(in) :: p
real(default), intent(in) :: m
L = boost_from_rest_frame_vector3 (space_part (p), m)
end function boost_from_rest_frame
elemental module function boost_from_rest_frame_vector3 (p, m) result (L)
type(lorentz_transformation_t) :: L
type(vector3_t), intent(in) :: p
real(default), intent(in) :: m
type(vector3_t) :: beta_gamma
real(default) :: bg2, g, c
integer :: i,j
if (m > eps0) then
beta_gamma = p / m
bg2 = beta_gamma**2
else
bg2 = 0
L = identity
return
end if
if (bg2 > eps0) then
g = sqrt(1 + bg2); c = (g-1)/bg2
else
g = one + bg2 / two
c = one / two
end if
L%L(0,0) = g
L%L(0,1:) = beta_gamma%p
L%L(1:,0) = L%L(0,1:)
do i=1,3
do j=1,3
L%L(i,j) = delta_three(i,j) + c*beta_gamma%p(i)*beta_gamma%p(j)
end do
end do
end function boost_from_rest_frame_vector3
@ %def boost_from_rest_frame
@ A canonical boost is a boost along one of the coordinate axes, which
we may supply as an integer argument. Here, $\gamma\beta$ is scalar.
<<Lorentz: sub interfaces>>=
elemental module function boost_canonical (beta_gamma, k) result (L)
type(lorentz_transformation_t) :: L
real(default), intent(in) :: beta_gamma
integer, intent(in) :: k
end function boost_canonical
<<Lorentz: procedures>>=
elemental module function boost_canonical (beta_gamma, k) result (L)
type(lorentz_transformation_t) :: L
real(default), intent(in) :: beta_gamma
integer, intent(in) :: k
real(default) :: g
g = sqrt(1 + beta_gamma**2)
L = identity
L%L(0,0) = g
L%L(0,k) = beta_gamma
L%L(k,0) = L%L(0,k)
L%L(k,k) = L%L(0,0)
end function boost_canonical
@ %def boost_canonical
@ Instead of a canonical axis, we can supply an arbitrary axis which
need not be normalized. If it is zero, return the unit matrix.
<<Lorentz: sub interfaces>>=
elemental module function boost_generic (beta_gamma, axis) result (L)
type(lorentz_transformation_t) :: L
real(default), intent(in) :: beta_gamma
type(vector3_t), intent(in) :: axis
end function boost_generic
<<Lorentz: procedures>>=
elemental module function boost_generic (beta_gamma, axis) result (L)
type(lorentz_transformation_t) :: L
real(default), intent(in) :: beta_gamma
type(vector3_t), intent(in) :: axis
if (any (abs (axis%p) > 0)) then
L = boost_from_rest_frame_vector3 (beta_gamma * axis, axis**1)
else
L = identity
end if
end function boost_generic
@ %def boost_generic
@
\subsection{Rotations}
For a rotation, the vector defines the rotation axis, and its length
the rotation angle. All of these rotations rotate counterclockwise
in a right-handed coordinate system.
<<Lorentz: public functions>>=
public :: rotation
<<Lorentz: interfaces>>=
interface rotation
module procedure rotation_generic
module procedure rotation_canonical
module procedure rotation_generic_cs
module procedure rotation_canonical_cs
end interface
@ %def rotation
@ If $\cos\phi$ and $\sin\phi$ is already known, we do not have to
calculate them. Of course, the user has to ensure that
$\cos^2\phi+\sin^2\phi=1$, and that the given axis [[n]] is normalized to
one. In the second form, the length of [[axis]] is the rotation
angle.
<<Lorentz: sub interfaces>>=
elemental module function rotation_generic_cs (cp, sp, axis) result (R)
type(lorentz_transformation_t) :: R
real(default), intent(in) :: cp, sp
type(vector3_t), intent(in) :: axis
end function rotation_generic_cs
elemental module function rotation_generic (axis) result (R)
type(lorentz_transformation_t) :: R
type(vector3_t), intent(in) :: axis
end function rotation_generic
elemental module function rotation_canonical_cs (cp, sp, k) result (R)
type(lorentz_transformation_t) :: R
real(default), intent(in) :: cp, sp
integer, intent(in) :: k
end function rotation_canonical_cs
elemental module function rotation_canonical (phi, k) result (R)
type(lorentz_transformation_t) :: R
real(default), intent(in) :: phi
integer, intent(in) :: k
end function rotation_canonical
<<Lorentz: procedures>>=
elemental module function rotation_generic_cs (cp, sp, axis) result (R)
type(lorentz_transformation_t) :: R
real(default), intent(in) :: cp, sp
type(vector3_t), intent(in) :: axis
integer :: i,j
R = identity
do i=1,3
do j=1,3
R%L(i,j) = cp*delta_three(i,j) + (1-cp)*axis%p(i)*axis%p(j) &
& - sp*dot_product(epsilon_three(i,j,:), axis%p)
end do
end do
end function rotation_generic_cs
elemental module function rotation_generic (axis) result (R)
type(lorentz_transformation_t) :: R
type(vector3_t), intent(in) :: axis
real(default) :: phi
if (any (abs(axis%p) > 0)) then
phi = abs(axis**1)
R = rotation_generic_cs (cos(phi), sin(phi), axis/phi)
else
R = identity
end if
end function rotation_generic
@ %def rotation_generic_cs rotation_generic
@ Alternatively, give just the angle and label the coordinate axis by
an integer.
<<Lorentz: procedures>>=
elemental module function rotation_canonical_cs (cp, sp, k) result (R)
type(lorentz_transformation_t) :: R
real(default), intent(in) :: cp, sp
integer, intent(in) :: k
integer :: i,j
R = identity
do i=1,3
do j=1,3
R%L(i,j) = -sp*epsilon_three(i,j,k)
end do
R%L(i,i) = cp
end do
R%L(k,k) = 1
end function rotation_canonical_cs
elemental module function rotation_canonical (phi, k) result (R)
type(lorentz_transformation_t) :: R
real(default), intent(in) :: phi
integer, intent(in) :: k
R = rotation_canonical_cs(cos(phi), sin(phi), k)
end function rotation_canonical
@ %def rotation_canonical_cs rotation_canonical
@
This is viewed as a method for the first argument (three-vector):
Reconstruct the rotation that rotates it into the second three-vector.
<<Lorentz: public functions>>=
public :: rotation_to_2nd
<<Lorentz: interfaces>>=
interface rotation_to_2nd
module procedure rotation_to_2nd_generic
module procedure rotation_to_2nd_canonical
end interface
<<Lorentz: sub interfaces>>=
elemental module function rotation_to_2nd_generic (p, q) result (R)
type(lorentz_transformation_t) :: R
type(vector3_t), intent(in) :: p, q
end function rotation_to_2nd_generic
elemental module function rotation_to_2nd_canonical (k, p) result (R)
type(lorentz_transformation_t) :: R
integer, intent(in) :: k
type(vector3_t), intent(in) :: p
end function rotation_to_2nd_canonical
<<Lorentz: procedures>>=
elemental module function rotation_to_2nd_generic (p, q) result (R)
type(lorentz_transformation_t) :: R
type(vector3_t), intent(in) :: p, q
type(vector3_t) :: a, b, ab
real(default) :: ct, st
if (any (abs (p%p) > 0) .and. any (abs (q%p) > 0)) then
a = direction (p)
b = direction (q)
ab = cross_product(a,b)
ct = a * b; st = ab**1
if (abs(st) > eps0) then
R = rotation_generic_cs (ct, st, ab / st)
else if (ct < 0) then
R = space_reflection
else
R = identity
end if
else
R = identity
end if
end function rotation_to_2nd_generic
@ %def rotation_to_2nd_generic
@
The same for a canonical axis: The function returns the transformation that
rotates the $k$-axis into the direction of $p$.
<<Lorentz: procedures>>=
elemental module function rotation_to_2nd_canonical (k, p) result (R)
type(lorentz_transformation_t) :: R
integer, intent(in) :: k
type(vector3_t), intent(in) :: p
type(vector3_t) :: b, ab
real(default) :: ct, st
integer :: i, j
if (any (abs (p%p) > 0)) then
b = direction (p)
ab%p = 0
do i = 1, 3
do j = 1, 3
ab%p(j) = ab%p(j) + b%p(i) * epsilon_three(i,j,k)
end do
end do
ct = b%p(k); st = ab**1
if (abs(st) > eps0) then
R = rotation_generic_cs (ct, st, ab / st)
else if (ct < 0) then
R = space_reflection
else
R = identity
end if
else
R = identity
end if
end function rotation_to_2nd_canonical
@ %def rotation_to_2nd_canonical
@
\subsection{Composite Lorentz transformations}
This function returns the transformation that, given a pair of vectors
$p_{1,2}$, (a) boosts from the rest frame of the c.m. system (with
invariant mass $m$) into the lab frame where $p_i$ are defined, and
(b) turns the given axis (or the canonical vectors $\pm
e_k$) in the rest frame into the directions of $p_{1,2}$ in the lab frame.
Note that the energy components are not used; for a
consistent result one should have $(p_1+p_2)^2 = m^2$.
<<Lorentz: public functions>>=
public :: transformation
<<Lorentz: interfaces>>=
interface transformation
module procedure transformation_rec_generic
module procedure transformation_rec_canonical
end interface
@ %def transformation
<<Lorentz: sub interfaces>>=
elemental module function transformation_rec_generic (axis, p1, p2, m) result (L)
type(vector3_t), intent(in) :: axis
type(vector4_t), intent(in) :: p1, p2
real(default), intent(in) :: m
type(lorentz_transformation_t) :: L
end function transformation_rec_generic
elemental module function transformation_rec_canonical (k, p1, p2, m) result (L)
integer, intent(in) :: k
type(vector4_t), intent(in) :: p1, p2
real(default), intent(in) :: m
type(lorentz_transformation_t) :: L
end function transformation_rec_canonical
<<Lorentz: procedures>>=
elemental module function transformation_rec_generic (axis, p1, p2, m) result (L)
type(vector3_t), intent(in) :: axis
type(vector4_t), intent(in) :: p1, p2
real(default), intent(in) :: m
type(lorentz_transformation_t) :: L
L = boost (p1 + p2, m)
L = L * rotation_to_2nd (axis, space_part (inverse (L) * p1))
end function transformation_rec_generic
elemental module function transformation_rec_canonical (k, p1, p2, m) result (L)
integer, intent(in) :: k
type(vector4_t), intent(in) :: p1, p2
real(default), intent(in) :: m
type(lorentz_transformation_t) :: L
L = boost (p1 + p2, m)
L = L * rotation_to_2nd (k, space_part (inverse (L) * p1))
end function transformation_rec_canonical
@ %def transformation_rec_generic transformation_rec_canonical
@
\subsection{Applying Lorentz transformations}
Multiplying vectors and Lorentz transformations is straightforward.
<<Lorentz: interfaces>>=
interface operator(*)
module procedure prod_LT_vector4
module procedure prod_LT_LT
module procedure prod_vector4_LT
end interface
<<Lorentz: sub interfaces>>=
elemental module function prod_LT_vector4 (L, p) result (np)
type(vector4_t) :: np
type(lorentz_transformation_t), intent(in) :: L
type(vector4_t), intent(in) :: p
end function prod_LT_vector4
elemental module function prod_LT_LT (L1, L2) result (NL)
type(lorentz_transformation_t) :: NL
type(lorentz_transformation_t), intent(in) :: L1,L2
end function prod_LT_LT
elemental module function prod_vector4_LT (p, L) result (np)
type(vector4_t) :: np
type(vector4_t), intent(in) :: p
type(lorentz_transformation_t), intent(in) :: L
end function prod_vector4_LT
<<Lorentz: procedures>>=
elemental module function prod_LT_vector4 (L, p) result (np)
type(vector4_t) :: np
type(lorentz_transformation_t), intent(in) :: L
type(vector4_t), intent(in) :: p
np%p = matmul (L%L, p%p)
end function prod_LT_vector4
elemental module function prod_LT_LT (L1, L2) result (NL)
type(lorentz_transformation_t) :: NL
type(lorentz_transformation_t), intent(in) :: L1,L2
NL%L = matmul (L1%L, L2%L)
end function prod_LT_LT
elemental module function prod_vector4_LT (p, L) result (np)
type(vector4_t) :: np
type(vector4_t), intent(in) :: p
type(lorentz_transformation_t), intent(in) :: L
np%p = matmul (p%p, L%L)
end function prod_vector4_LT
@ %def *
@
\subsection{Special Lorentz transformations}
These routines have their application in the generation and extraction
of angles in the phase-space sampling routine. Since this part of the
program is time-critical, we calculate the composition of
transformations directly instead of multiplying rotations and boosts.
This Lorentz transformation is the composition of a rotation by $\phi$
around the $3$ axis, a rotation by $\theta$ around the $2$ axis, and a
boost along the $3$ axis:
\begin{equation}
L = B_3(\beta\gamma)\,R_2(\theta)\,R_3(\phi)
\end{equation}
Instead of the angles we provide sine and cosine.
<<Lorentz: public functions>>=
public :: LT_compose_r3_r2_b3
<<Lorentz: sub interfaces>>=
elemental module function LT_compose_r3_r2_b3 &
(cp, sp, ct, st, beta_gamma) result (L)
type(lorentz_transformation_t) :: L
real(default), intent(in) :: cp, sp, ct, st, beta_gamma
end function LT_compose_r3_r2_b3
<<Lorentz: procedures>>=
elemental module function LT_compose_r3_r2_b3 &
(cp, sp, ct, st, beta_gamma) result (L)
type(lorentz_transformation_t) :: L
real(default), intent(in) :: cp, sp, ct, st, beta_gamma
real(default) :: gamma
if (abs(beta_gamma) < eps0) then
L%L(0,0) = 1
L%L(1:,0) = 0
L%L(0,1:) = 0
L%L(1,1:) = [ ct*cp, -ct*sp, st ]
L%L(2,1:) = [ sp, cp, zero ]
L%L(3,1:) = [ -st*cp, st*sp, ct ]
else
gamma = sqrt(1 + beta_gamma**2)
L%L(0,0) = gamma
L%L(1,0) = 0
L%L(2,0) = 0
L%L(3,0) = beta_gamma
L%L(0,1:) = beta_gamma * [ -st*cp, st*sp, ct ]
L%L(1,1:) = [ ct*cp, -ct*sp, st ]
L%L(2,1:) = [ sp, cp, zero ]
L%L(3,1:) = gamma * [ -st*cp, st*sp, ct ]
end if
end function LT_compose_r3_r2_b3
@ %def LT_compose_r3_r2_b3
@ Different ordering:
\begin{equation}
L = B_3(\beta\gamma)\,R_3(\phi)\,R_2(\theta)
\end{equation}
<<Lorentz: public functions>>=
public :: LT_compose_r2_r3_b3
<<Lorentz: sub interfaces>>=
elemental module function LT_compose_r2_r3_b3 &
(ct, st, cp, sp, beta_gamma) result (L)
type(lorentz_transformation_t) :: L
real(default), intent(in) :: ct, st, cp, sp, beta_gamma
end function LT_compose_r2_r3_b3
<<Lorentz: procedures>>=
elemental module function LT_compose_r2_r3_b3 &
(ct, st, cp, sp, beta_gamma) result (L)
type(lorentz_transformation_t) :: L
real(default), intent(in) :: ct, st, cp, sp, beta_gamma
real(default) :: gamma
if (abs(beta_gamma) < eps0) then
L%L(0,0) = 1
L%L(1:,0) = 0
L%L(0,1:) = 0
L%L(1,1:) = [ ct*cp, -sp, st*cp ]
L%L(2,1:) = [ ct*sp, cp, st*sp ]
L%L(3,1:) = [ -st , zero, ct ]
else
gamma = sqrt(1 + beta_gamma**2)
L%L(0,0) = gamma
L%L(1,0) = 0
L%L(2,0) = 0
L%L(3,0) = beta_gamma
L%L(0,1:) = beta_gamma * [ -st , zero, ct ]
L%L(1,1:) = [ ct*cp, -sp, st*cp ]
L%L(2,1:) = [ ct*sp, cp, st*sp ]
L%L(3,1:) = gamma * [ -st , zero, ct ]
end if
end function LT_compose_r2_r3_b3
@ %def LT_compose_r2_r3_b3
@ This function returns the previous Lorentz transformation applied to
an arbitrary four-momentum and extracts the space part of the result:
\begin{equation}
\vec n = [B_3(\beta\gamma)\,R_2(\theta)\,R_3(\phi)\,p]_{\rm space\ part}
\end{equation}
The second variant applies if there is no rotation
<<Lorentz: public functions>>=
public :: axis_from_p_r3_r2_b3, axis_from_p_b3
<<Lorentz: sub interfaces>>=
elemental module function axis_from_p_r3_r2_b3 &
(p, cp, sp, ct, st, beta_gamma) result (n)
type(vector3_t) :: n
type(vector4_t), intent(in) :: p
real(default), intent(in) :: cp, sp, ct, st, beta_gamma
end function axis_from_p_r3_r2_b3
elemental module function axis_from_p_b3 (p, beta_gamma) result (n)
type(vector3_t) :: n
type(vector4_t), intent(in) :: p
real(default), intent(in) :: beta_gamma
end function axis_from_p_b3
<<Lorentz: procedures>>=
elemental module function axis_from_p_r3_r2_b3 &
(p, cp, sp, ct, st, beta_gamma) result (n)
type(vector3_t) :: n
type(vector4_t), intent(in) :: p
real(default), intent(in) :: cp, sp, ct, st, beta_gamma
real(default) :: gamma, px, py
px = cp * p%p(1) - sp * p%p(2)
py = sp * p%p(1) + cp * p%p(2)
n%p(1) = ct * px + st * p%p(3)
n%p(2) = py
n%p(3) = -st * px + ct * p%p(3)
if (abs(beta_gamma) > eps0) then
gamma = sqrt(1 + beta_gamma**2)
n%p(3) = n%p(3) * gamma + p%p(0) * beta_gamma
end if
end function axis_from_p_r3_r2_b3
elemental module function axis_from_p_b3 (p, beta_gamma) result (n)
type(vector3_t) :: n
type(vector4_t), intent(in) :: p
real(default), intent(in) :: beta_gamma
real(default) :: gamma
n%p = p%p(1:3)
if (abs(beta_gamma) > eps0) then
gamma = sqrt(1 + beta_gamma**2)
n%p(3) = n%p(3) * gamma + p%p(0) * beta_gamma
end if
end function axis_from_p_b3
@ %def axis_from_p_r3_r2_b3 axis_from_p_b3
@
\subsection{Special functions}
The K\"all\'en function, mostly used for the phase space.
This is equivalent to $\lambda(x,y,z)=x^2+y^2+z^2-2xy-2xz-2yz$.
<<Lorentz: public functions>>=
public :: lambda
<<Lorentz: sub interfaces>>=
elemental module function lambda (m1sq, m2sq, m3sq)
real(default) :: lambda
real(default), intent(in) :: m1sq, m2sq, m3sq
end function lambda
<<Lorentz: procedures>>=
elemental module function lambda (m1sq, m2sq, m3sq)
real(default) :: lambda
real(default), intent(in) :: m1sq, m2sq, m3sq
lambda = (m1sq - m2sq - m3sq)**2 - 4*m2sq*m3sq
end function lambda
@ %def lambda
@ Return a pair of head-to-head colliding momenta, given the collider
energy, particle masses, and optionally the momentum of the
c.m. system.
<<Lorentz: public functions>>=
public :: colliding_momenta
<<Lorentz: sub interfaces>>=
module function colliding_momenta (sqrts, m, p_cm) result (p)
type(vector4_t), dimension(2) :: p
real(default), intent(in) :: sqrts
real(default), dimension(2), intent(in), optional :: m
real(default), intent(in), optional :: p_cm
end function colliding_momenta
<<Lorentz: procedures>>=
module function colliding_momenta (sqrts, m, p_cm) result (p)
type(vector4_t), dimension(2) :: p
real(default), intent(in) :: sqrts
real(default), dimension(2), intent(in), optional :: m
real(default), intent(in), optional :: p_cm
real(default), dimension(2) :: dmsq
real(default) :: ch, sh
real(default), dimension(2) :: E0, p0
integer, dimension(2), parameter :: sgn = [1, -1]
if (abs(sqrts) < eps0) then
call msg_fatal (" Colliding beams: sqrts is zero (please set sqrts)")
p = vector4_null; return
else if (sqrts <= 0) then
call msg_fatal (" Colliding beams: sqrts is negative")
p = vector4_null; return
end if
if (present (m)) then
dmsq = sgn * (m(1)**2-m(2)**2)
E0 = (sqrts + dmsq/sqrts) / 2
if (any (E0 < m)) then
call msg_fatal &
(" Colliding beams: beam energy is less than particle mass")
p = vector4_null; return
end if
p0 = sgn * sqrt (E0**2 - m**2)
else
E0 = sqrts / 2
p0 = sgn * E0
end if
if (present (p_cm)) then
sh = p_cm / sqrts
ch = sqrt (1 + sh**2)
p = vector4_moving (E0 * ch + p0 * sh, E0 * sh + p0 * ch, 3)
else
p = vector4_moving (E0, p0, 3)
end if
end function colliding_momenta
@ %def colliding_momenta
@ This subroutine is for the purpose of numerical checks and
comparisons. The idea is to set a number to zero if it is numerically
equivalent with zero. The equivalence is established by comparing
with a [[tolerance]] argument. We implement this for vectors and
transformations.
<<Lorentz: public functions>>=
public :: pacify
<<Lorentz: interfaces>>=
interface pacify
module procedure pacify_vector3
module procedure pacify_vector4
module procedure pacify_LT
end interface pacify
<<Lorentz: sub interfaces>>=
elemental module subroutine pacify_vector3 (p, tolerance)
type(vector3_t), intent(inout) :: p
real(default), intent(in) :: tolerance
end subroutine pacify_vector3
elemental module subroutine pacify_vector4 (p, tolerance)
type(vector4_t), intent(inout) :: p
real(default), intent(in) :: tolerance
end subroutine pacify_vector4
elemental module subroutine pacify_LT (LT, tolerance)
type(lorentz_transformation_t), intent(inout) :: LT
real(default), intent(in) :: tolerance
end subroutine pacify_LT
<<Lorentz: procedures>>=
elemental module subroutine pacify_vector3 (p, tolerance)
type(vector3_t), intent(inout) :: p
real(default), intent(in) :: tolerance
where (abs (p%p) < tolerance) p%p = zero
end subroutine pacify_vector3
elemental module subroutine pacify_vector4 (p, tolerance)
type(vector4_t), intent(inout) :: p
real(default), intent(in) :: tolerance
where (abs (p%p) < tolerance) p%p = zero
end subroutine pacify_vector4
elemental module subroutine pacify_LT (LT, tolerance)
type(lorentz_transformation_t), intent(inout) :: LT
real(default), intent(in) :: tolerance
where (abs (LT%L) < tolerance) LT%L = zero
end subroutine pacify_LT
@ %def pacify
@
<<Lorentz: public>>=
public :: vector_set_reshuffle
<<Lorentz: sub interfaces>>=
module subroutine vector_set_reshuffle (p1, list, p2)
type(vector4_t), intent(in), dimension(:), allocatable :: p1
integer, intent(in), dimension(:), allocatable :: list
type(vector4_t), intent(out), dimension(:), allocatable :: p2
end subroutine vector_set_reshuffle
<<Lorentz: procedures>>=
module subroutine vector_set_reshuffle (p1, list, p2)
type(vector4_t), intent(in), dimension(:), allocatable :: p1
integer, intent(in), dimension(:), allocatable :: list
type(vector4_t), intent(out), dimension(:), allocatable :: p2
integer :: n, n_p
n_p = size (p1)
if (size (list) /= n_p) return
allocate (p2 (n_p))
do n = 1, n_p
p2(n) = p1(list(n))
end do
end subroutine vector_set_reshuffle
@ %def vector_set_reshuffle
@
<<Lorentz: public>>=
public :: vector_set_is_cms
<<Lorentz: sub interfaces>>=
module function vector_set_is_cms (p, n_in) result (is_cms)
logical :: is_cms
type(vector4_t), intent(in), dimension(:) :: p
integer, intent(in) :: n_in
end function vector_set_is_cms
<<Lorentz: procedures>>=
module function vector_set_is_cms (p, n_in) result (is_cms)
logical :: is_cms
type(vector4_t), intent(in), dimension(:) :: p
integer, intent(in) :: n_in
integer :: i
type(vector4_t) :: p_sum
p_sum%p = 0._default
do i = 1, n_in
p_sum = p_sum + p(i)
end do
is_cms = all (abs (p_sum%p(1:3)) < tiny_07)
end function vector_set_is_cms
@ %def vector_set_is_cms
@
<<Lorentz: public>>=
public :: vector4_write_set
<<Lorentz: sub interfaces>>=
module subroutine vector4_write_set (p, unit, show_mass, testflag, &
check_conservation, ultra, n_in)
type(vector4_t), intent(in), dimension(:) :: p
integer, intent(in), optional :: unit
logical, intent(in), optional :: show_mass
logical, intent(in), optional :: testflag, ultra
logical, intent(in), optional :: check_conservation
integer, intent(in), optional :: n_in
end subroutine vector4_write_set
<<Lorentz: procedures>>=
module subroutine vector4_write_set (p, unit, show_mass, testflag, &
check_conservation, ultra, n_in)
type(vector4_t), intent(in), dimension(:) :: p
integer, intent(in), optional :: unit
logical, intent(in), optional :: show_mass
logical, intent(in), optional :: testflag, ultra
logical, intent(in), optional :: check_conservation
integer, intent(in), optional :: n_in
logical :: extreme
integer :: i, j
real(default), dimension(0:3) :: p_tot
character(len=7) :: fmt
integer :: u
logical :: yorn, is_test
integer :: n
extreme = .false.; if (present (ultra)) extreme = ultra
is_test = .false.; if (present (testflag)) is_test = testflag
u = given_output_unit (unit); if (u < 0) return
n = 2; if (present (n_in)) n = n_in
p_tot = 0
yorn = .false.; if (present (check_conservation)) yorn = check_conservation
do i = 1, size (p)
if (yorn .and. i > n) then
forall (j=0:3) p_tot(j) = p_tot(j) - p(i)%p(j)
else
forall (j=0:3) p_tot(j) = p_tot(j) + p(i)%p(j)
end if
call vector4_write (p(i), u, show_mass=show_mass, &
testflag=testflag, ultra=ultra)
end do
if (extreme) then
call pac_fmt (fmt, FMT_19, FMT_11, testflag)
else
call pac_fmt (fmt, FMT_19, FMT_15, testflag)
end if
if (is_test) call pacify (p_tot, 1.E-9_default)
if (.not. is_test) then
write (u, "(A5)") 'Total: '
write (u, "(1x,A,1x," // fmt // ")") "E = ", p_tot(0)
write (u, "(1x,A,3(1x," // fmt // "))") "P = ", p_tot(1:)
end if
end subroutine vector4_write_set
@ %def vector4_write_set
@
<<Lorentz: public>>=
public :: vector4_check_momentum_conservation
<<Lorentz: sub interfaces>>=
module subroutine vector4_check_momentum_conservation (p, n_in, unit, &
abs_smallness, rel_smallness, verbose)
type(vector4_t), dimension(:), intent(in) :: p
integer, intent(in) :: n_in
integer, intent(in), optional :: unit
real(default), intent(in), optional :: abs_smallness, rel_smallness
logical, intent(in), optional :: verbose
end subroutine vector4_check_momentum_conservation
<<Lorentz: procedures>>=
module subroutine vector4_check_momentum_conservation (p, n_in, unit, &
abs_smallness, rel_smallness, verbose)
type(vector4_t), dimension(:), intent(in) :: p
integer, intent(in) :: n_in
integer, intent(in), optional :: unit
real(default), intent(in), optional :: abs_smallness, rel_smallness
logical, intent(in), optional :: verbose
integer :: u, i
type(vector4_t) :: psum_in, psum_out
logical, dimension(0:3) :: p_diff
logical :: verb
u = given_output_unit (unit); if (u < 0) return
verb = .false.; if (present (verbose)) verb = verbose
psum_in = vector4_null
do i = 1, n_in
psum_in = psum_in + p(i)
end do
psum_out = vector4_null
do i = n_in + 1, size (p)
psum_out = psum_out + p(i)
end do
p_diff = vanishes (psum_in%p - psum_out%p, &
abs_smallness = abs_smallness, rel_smallness = rel_smallness)
if (.not. all (p_diff)) then
call msg_warning ("Momentum conservation: FAIL", unit = u)
if (verb) then
write (u, "(A)") "Incoming:"
call vector4_write (psum_in, u)
write (u, "(A)") "Outgoing:"
call vector4_write (psum_out, u)
end if
else
if (verb) then
write (u, "(A)") "Momentum conservation: CHECK"
end if
end if
end subroutine vector4_check_momentum_conservation
@ %def vector4_check_momentum_conservation
@ This computes the quantities
\begin{align*}
\langle ij \rangle &= \sqrt{|S_{ij}|} e^{i\phi_{ij}},
[ij] &= \sqrt{|S_{ij}|} e^{\i\tilde{\phi}_{ij}},
\end{align*}
with $S_{ij} = \left(p_i + p_j\right)^2$. The phase space factor
$\phi_{ij}$ is determined by
\begin{align*}
\cos\phi_{ij} &= \frac{p_i^1p_j^+ - p_j^1p_i^+}{\sqrt{p_i^+p_j^+S_{ij}}},
\sin\phi_{ij} &= \frac{p_i^2p_j^+ - p_j^2p_i^+}{\sqrt{p_i^+p_j^+S_{ij}}}.
\end{align*}
After $\langle ij \rangle$ has been computed according to these
formulae, $[ij]$ can be obtained by using the relation $S_{ij} =
\langle ij \rangle [ji]$ and taking into account that $[ij] =
-[ji]$. Thus, a minus-sign has to be applied.
<<Lorentz: public>>=
public :: spinor_product
<<Lorentz: sub interfaces>>=
module subroutine spinor_product (p1, p2, prod1, prod2)
type(vector4_t), intent(in) :: p1, p2
complex(default), intent(out) :: prod1, prod2
end subroutine spinor_product
<<Lorentz: procedures>>=
module subroutine spinor_product (p1, p2, prod1, prod2)
type(vector4_t), intent(in) :: p1, p2
complex(default), intent(out) :: prod1, prod2
real(default) :: sij
complex(default) :: phase
real(default) :: pp_1, pp_2
pp_1 = p1%p(0) + p1%p(3)
pp_2 = p2%p(0) + p2%p(3)
sij = (p1+p2)**2
phase = cmplx ((p1%p(1)*pp_2 - p2%p(1)*pp_1)/sqrt (sij*pp_1*pp_2), &
(p1%p(2)*pp_2 - p2%p(2)*pp_1)/sqrt (sij*pp_1*pp_2), &
default)
!!! <ij>
prod1 = sqrt (sij) * phase
!!! [ij]
if (abs(prod1) > 0) then
prod2 = - sij / prod1
else
prod2 = 0
end if
end subroutine spinor_product
@ %def spinor_product
@
\subsection{Unit tests}
Test module, followed by the corresponding implementation module.
<<[[lorentz_ut.f90]]>>=
<<File header>>
module lorentz_ut
use unit_tests
use lorentz_uti
<<Standard module head>>
<<Lorentz: public test>>
contains
<<Lorentz: test driver>>
end module lorentz_ut
@ %def lorentz_ut
@
<<[[lorentz_uti.f90]]>>=
<<File header>>
module lorentz_uti
<<Use kinds>>
use constants, only: zero, Pi
use format_defs, only: FMT_12
use lorentz
<<Standard module head>>
<<Lorentz: test declarations>>
contains
<<Lorentz: tests>>
end module lorentz_uti
@ %def lorentz_ut
@ API: driver for the unit tests below.
<<Lorentz: public test>>=
public :: lorentz_test
<<Lorentz: test driver>>=
subroutine lorentz_test (u, results)
integer, intent(in) :: u
type(test_results_t), intent(inout) :: results
<<Lorentz: execute tests>>
end subroutine lorentz_test
@ %def lorentz_test
@
\subsubsection{Algebra with 3-vectors}
<<Lorentz: execute tests>>=
call test (lorentz_1, "lorentz_1", &
"Test 3-vector functionality", &
u, results)
<<Lorentz: test declarations>>=
public :: lorentz_1
<<Lorentz: tests>>=
subroutine lorentz_1 (u)
integer, intent(in) :: u
type(vector3_t) :: v3_1, v3_2
write (u, "(A)") "* Test output: lorentz_1"
write (u, "(A)") "* Purpose: testing vector3_t"
write (u, "(A)")
write (u, "(A)")
write (u, "(A)") "* Null 3-vector"
write (u, "(A)")
call vector3_write (vector3_null, u, testflag = .true.)
write (u, "(A)")
write (u, "(A)") "* Canonical 3-vector"
write (u, "(A)")
call vector3_write (vector3_canonical (1), u, testflag = .true.)
call vector3_write (vector3_canonical (2), u, testflag = .true.)
call vector3_write (vector3_canonical (3), u, testflag = .true.)
write (u, "(A)")
write (u, "(A)") "* Canonical moving 3-vector"
write (u, "(A)")
call vector3_write (vector3_moving (42._default, 1), u, testflag = .true.)
call vector3_write (vector3_moving (42._default, 2), u, testflag = .true.)
call vector3_write (vector3_moving (42._default, 3), u, testflag = .true.)
write (u, "(A)")
write (u, "(A)") "* Generic moving 3-vector"
write (u, "(A)")
call vector3_write (vector3_moving ([3._default, 4._default, 5._default]), &
u, testflag = .true.)
write (u, "(A)")
write (u, "(A)") "* Simple algebra with 3-vectors"
write (u, "(A)")
v3_1 = vector3_moving ([3._default, 4._default, 5._default])
v3_2 = vector3_moving ([-2._default, 5._default, -1._default])
write (u, "(1x,A)") "v3_1:"
call vector3_write (v3_1, u, testflag=.true.)
write (u, "(1x,A)") "v3_2:"
call vector3_write (v3_2, u, testflag=.true.)
write (u, "(1x,A)") "-v3_1:"
call vector3_write (-v3_1, u, testflag=.true.)
write (u, "(1x,A)") "v3_1 / |v3_1|:"
call vector3_write (direction (v3_1), u, testflag=.true.)
write (u, "(1x,A," // FMT_12 // ")") "v3_1(x): ", vector3_get_component (v3_1, 1)
write (u, "(1x,A," // FMT_12 // ")") "v3_1(y): ", vector3_get_component (v3_1, 2)
write (u, "(1x,A," // FMT_12 // ")") "v3_1(z): ", vector3_get_component (v3_1, 3)
write (u, "(1x,A)") "v3_1 + v3_2:"
call vector3_write (v3_1 + v3_2, u, testflag=.true.)
write (u, "(1x,A)") "v3_1 - v3_2:"
call vector3_write (v3_1 - v3_2, u, testflag=.true.)
write (u, "(1x,A,L1)") "v3_1 == v3_2: ", v3_1 == v3_2
write (u, "(1x,A,L1)") "v3_1 /= v3_2: ", v3_1 /= v3_2
write (u, "(1x,A)") "2 * v3_1:"
call vector3_write (2._default * v3_1, u, testflag=.true.)
write (u, "(1x,A)") "v3_2 / 4:"
call vector3_write (v3_2 / 4, u, testflag=.true.)
write (u, "(1x,A," // FMT_12 // ")") "v3_1, azimuth (radians):", azimuthal_angle (v3_1)
write (u, "(1x,A," // FMT_12 // ")") "v3_1, azimuth (degrees):", &
azimuthal_angle_deg (v3_1)
write (u, "(1x,A," // FMT_12 // ")") "v3_1, polar (radians) :", polar_angle (v3_1)
write (u, "(1x,A," // FMT_12 // ")") "v3_1, polar (degrees) :", &
polar_angle_deg (v3_1)
write (u, "(1x,A," // FMT_12 // ")") "v3_1, cosine polar :", &
polar_angle_ct (v3_1)
write (u, "(1x,A," // FMT_12 // ")") "v3_1, energy w. mass=1 :", &
energy (v3_1, 1._default)
write (u, "(1x,A)") "3-vector orthogonal to v3_1:"
call vector3_write (create_orthogonal (v3_1), u, testflag=.true.)
write (u, "(1x,A)") "unit 3-vector from v3_1:"
write (u, "(A)")
write (u, "(A)") "* Dot and cross product"
write (u, "(A)")
write (u, "(1x,A," // FMT_12 // ")") "v3_1 * v3_2: ", v3_1 * v3_2
write (u, "(1x,A," // FMT_12 // ")") "v3_1**3 : ", v3_1**3
write (u, "(1x,A)") "v3_1 x v3_2:"
call vector3_write (cross_product (v3_1, v3_2), u, testflag=.true.)
write (u, "(1x,A," // FMT_12 // ")") "enclosed angle (radians):", &
enclosed_angle (v3_1, v3_2)
write (u, "(1x,A," // FMT_12 // ")") "enclosed angle (degrees):", &
enclosed_angle_deg (v3_1, v3_2)
write (u, "(1x,A," // FMT_12 // ")") "cosine (enclosed angle) :", &
enclosed_angle_ct (v3_1, v3_2)
write (u, "(A)")
write (u, "(A)") "* Test output end: lorentz_1"
end subroutine lorentz_1
@ %def lorentz_1
@
\subsubsection{Algebra with 4-vectors}
<<Lorentz: execute tests>>=
call test(lorentz_2, "lorentz_2", &
"Test 4-vector functionality", u, results)
<<Lorentz: test declarations>>=
public :: lorentz_2
<<Lorentz: tests>>=
subroutine lorentz_2 (u)
integer, intent(in) :: u
type(vector3_t) :: v3_1, v3_2
type(vector4_t) :: v4_1, v4_2, v4_1_inv
write (u, "(A)") "* Test output: lorentz_2"
write (u, "(A)") "* Purpose: testing vector4_t"
write (u, "(A)")
write (u, "(A)")
write (u, "(A)") "* Null 4-vector"
write (u, "(A)")
call vector4_write (vector4_null, u, testflag = .true.)
write (u, "(A)")
write (u, "(A)") "* Canonical 4-vector"
write (u, "(A)")
call vector4_write (vector4_canonical (0), u, testflag = .true.)
call vector4_write (vector4_canonical (1), u, testflag = .true.)
call vector4_write (vector4_canonical (2), u, testflag = .true.)
call vector4_write (vector4_canonical (3), u, testflag = .true.)
write (u, "(A)")
write (u, "(A)") "* 4-vector at rest with mass m = 17"
write (u, "(A)")
call vector4_write (vector4_at_rest (17._default), u, testflag = .true.)
write (u, "(A)")
write (u, "(A)") "* Canonical moving 4-vector"
write (u, "(A)")
call vector4_write (vector4_moving (17._default, 42._default, 1), u, testflag = .true.)
call vector4_write (vector4_moving (17._default, 42._default, 2), u, testflag = .true.)
call vector4_write (vector4_moving (17._default, 42._default, 3), u, testflag = .true.)
write (u, "(A)")
write (u, "(A)") "* Generic moving 4-vector"
write (u, "(A)")
v3_1 = [3._default, 4._default, 5._default]
call vector4_write (vector4_moving (17._default, v3_1), u, testflag = .true.)
write (u, "(A)")
write (u, "(A)") "* Simple algebra with 4-vectors"
write (u, "(A)")
v3_2 = [-2._default, 5._default, -1._default]
v4_1 = vector4_moving (8._default, v3_1)
v4_2 = vector4_moving (zero, v3_2)
write (u, "(1x,A)") "v4_1:"
call vector4_write (v4_1, u, testflag=.true.)
write (u, "(1x,A)") "v4_2:"
call vector4_write (v4_2, u, testflag=.true.)
write (u, "(1x,A)") "-v4_1:"
call vector4_write (-v4_1, u, testflag=.true.)
v4_1_inv = v4_1
call vector4_invert_direction (v4_1_inv)
write (u, "(1x,A)") "v4_1, inverted direction:"
call vector4_write (v4_1_inv, u, testflag=.true.)
write (u, "(1x,A)") "(v4_1)_spatial / |(v4_1)_spatial|:"
call vector3_write (direction (v4_1), u, testflag=.true.)
write (u, "(1x,A," // FMT_12 // ")") "v4_1(E): ", energy (v4_1)
write (u, "(1x,A," // FMT_12 // ")") "v4_1(x): ", vector4_get_component (v4_1, 1)
write (u, "(1x,A," // FMT_12 // ")") "v4_1(y): ", vector4_get_component (v4_1, 2)
write (u, "(1x,A," // FMT_12 // ")") "v4_1(z): ", vector4_get_component (v4_1, 3)
write (u, "(1x,A)") "space_part (v4_1):"
call vector3_write (space_part (v4_1), u, testflag=.true.)
write (u, "(1x,A," // FMT_12 // ")") "norm space_part (v4_1): ", &
space_part_norm (v4_1)
write (u, "(1x,A)") "unit vector from v4_1:"
call vector3_write (create_unit_vector (v4_1), u, testflag = .true.)
write (u, "(1x,A)") "v4_1 + v4_2:"
call vector4_write (v4_1 + v4_2, u, testflag=.true.)
write (u, "(1x,A)") "v4_1 - v4_2:"
call vector4_write (v4_1 - v4_2, u, testflag=.true.)
write (u, "(1x,A,L1)") "v4_1 == v4_2: ", v4_1 == v4_2
write (u, "(1x,A,L1)") "v4_1 /= v4_2: ", v4_1 /= v4_2
write (u, "(1x,A)") "2 * v4_1:"
call vector4_write (2._default * v4_1, u, testflag=.true.)
write (u, "(1x,A)") "v4_2 / 4:"
call vector4_write (v4_2 / 4, u, testflag=.true.)
write (u, "(A)")
write (u, "(A)") "* Angles and kinematic properties of 4-vectors"
write (u, "(A)")
write (u, "(1x,A," // FMT_12 // ")") "v4_1, azimuth (radians):", azimuthal_angle (v4_1)
write (u, "(1x,A," // FMT_12 // ")") "v4_1, azimuth (degrees):", &
azimuthal_angle_deg (v4_1)
write (u, "(1x,A," // FMT_12 // ")") "v4_1, polar (radians) :", polar_angle (v4_1)
write (u, "(1x,A," // FMT_12 // ")") "v4_1, polar (degrees) :", &
polar_angle_deg (v4_1)
write (u, "(1x,A," // FMT_12 // ")") "v4_1, cosine polar :", &
polar_angle_ct (v4_1)
write (u, "(1x,A," // FMT_12 // ")") "v4_1, invariant mass :", &
invariant_mass (v4_1)
write (u, "(1x,A," // FMT_12 // ")") "v4_1, invariant mass sq:", &
invariant_mass_squared (v4_1)
write (u, "(1x,A," // FMT_12 // ")") "v4_2, invariant mass :", &
invariant_mass (v4_2)
write (u, "(1x,A," // FMT_12 // ")") "v4_2, invariant mass sq:", &
invariant_mass_squared (v4_2)
write (u, "(1x,A," // FMT_12 // ")") "v4_1, transverse mass :", &
transverse_mass (v4_1)
write (u, "(1x,A," // FMT_12 // ")") "v4_1, rapidity :", &
rapidity (v4_1)
write (u, "(1x,A," // FMT_12 // ")") "v4_1, pseudorapidity :", &
pseudorapidity (v4_1)
write (u, "(1x,A," // FMT_12 // ")") "v4_1, pT :", &
transverse_part (v4_1)
write (u, "(1x,A," // FMT_12 // ")") "v4_1, pL :", &
longitudinal_part (v4_1)
write (u, "(A)")
write (u, "(A)") "* Test output end: lorentz_2"
end subroutine lorentz_2
@ %def lorentz_2
@
\subsubsection{Bilinear functions of 4-vectors}
<<Lorentz: execute tests>>=
call test(lorentz_3, "lorentz_3", &
"Test 4-vector bilinear functions", u, results)
<<Lorentz: test declarations>>=
public :: lorentz_3
<<Lorentz: tests>>=
subroutine lorentz_3 (u)
integer, intent(in) :: u
type(vector3_t) :: v3_1, v3_2
type(vector4_t) :: v4_1, v4_2
write (u, "(A)") "* Test output: lorentz_3"
write (u, "(A)") "* Purpose: testing bilinear functions of 4-vectors"
write (u, "(A)")
write (u, "(A)")
write (u, "(A)") "* Products and distances of 4-vectors"
write (u, "(A)")
v3_1 = [3._default, 4._default, 5._default]
v3_2 = [-2._default, 5._default, -1._default]
v4_1 = vector4_moving (8._default, v3_1)
v4_2 = vector4_moving (6._default, v3_2)
write (u, "(1x,A," // FMT_12 // ")") "v4_1 * v4_2: ", v4_1 * v4_2
write (u, "(1x,A," // FMT_12 // ")") "rapidity distance :", &
rapidity_distance (v4_1, v4_2)
write (u, "(1x,A," // FMT_12 // ")") "pseudorapidity distance :", &
pseudorapidity_distance (v4_1, v4_2)
write (u, "(1x,A," // FMT_12 // ")") "eta phi distance :", &
eta_phi_distance (v4_1, v4_2)
write (u, "(1x,A," // FMT_12 // ")") "enclosed angle (radians):", &
enclosed_angle (v4_1, v4_2)
write (u, "(1x,A," // FMT_12 // ")") "enclosed angle (degrees):", &
enclosed_angle_deg (v4_1, v4_2)
write (u, "(1x,A," // FMT_12 // ")") "cosine (enclosed angle) :", &
enclosed_angle_ct (v4_1, v4_2)
write (u, "(1x,A," // FMT_12 // ")") "rest frame theta (rad) :", &
enclosed_angle_rest_frame (v4_1, v4_2)
write (u, "(1x,A," // FMT_12 // ")") "rest frame theta (deg) :", &
enclosed_angle_deg_rest_frame (v4_1, v4_2)
write (u, "(1x,A," // FMT_12 // ")") "rest frame cosine(theta):", &
enclosed_angle_ct_rest_frame (v4_1, v4_2)
write (u, "(1x,A," // FMT_12 // ")") "v4_1_T w.r.t. v4_2 :", &
transverse_part (v4_1, v4_2)
write (u, "(A)")
write (u, "(A)") "* Test output end: lorentz_3"
end subroutine lorentz_3
@ %def lorentz_3
@
\subsubsection{Tests for Lorentz transformations}
<<Lorentz: execute tests>>=
call test(lorentz_4, "lorentz_4", &
"Test Lorentz transformations", u, results)
<<Lorentz: test declarations>>=
public :: lorentz_4
<<Lorentz: tests>>=
subroutine lorentz_4 (u)
integer, intent(in) :: u
type(vector3_t) :: v3_1, v3_2
type(vector4_t) :: v4
type(lorentz_transformation_t) :: LT
real(default) :: tol
write (u, "(A)") "* Test output: lorentz_4"
write (u, "(A)") "* Purpose: testing Lorentz transformations"
write (u, "(A)")
write (u, "(A)")
write (u, "(A)") "* Basic Lorentz transformatios"
write (u, "(A)")
write (u, "(1x,A)") "LT = 1:"
call lorentz_transformation_write (identity, u, testflag=.true.)
write (u, "(A)")
write (u, "(1x,A)") "LT = space reflection:"
call lorentz_transformation_write (space_reflection, u, testflag=.true.)
write (u, "(A)")
write (u, "(A)") "* Lorentz transformations: rotations"
write (u, "(A)")
v3_1 = [1._default, 2._default, 3._default]
v3_2 = [-2._default, 1._default, -5._default]
tol = 1.e-12_default
write (u, "(1x,A)") "Rotation of Pi/4 around 1-axis, def. by cos and sin:"
LT = rotation (0.707107_default, 0.707107_default, 1)
call pacify (LT, tol)
call lorentz_transformation_write (LT, u, testflag=.true.)
write (u, "(1x,A)") "Rotation of Pi/4 around 2-axis, def. by cos and sin:"
LT = rotation (0.707107_default, 0.707107_default, 2)
call pacify (LT, tol)
call lorentz_transformation_write (LT, u, testflag=.true.)
write (u, "(1x,A)") "Rotation of Pi/4 around 3-axis, def. by cos and sin:"
LT = rotation (0.707107_default, 0.707107_default, 3)
call pacify (LT, tol)
call lorentz_transformation_write (LT, u, testflag=.true.)
write (u, "(1x,A)") "Rotation of Pi/4 around 1-axis, def. by angle:"
LT = rotation (Pi/4._default, 1)
call pacify (LT, tol)
call lorentz_transformation_write (LT, u, testflag=.true.)
write (u, "(1x,A)") "Rotation of Pi/4 around 2-axis, def. by angle:"
LT = rotation (Pi/4._default, 2)
call pacify (LT, tol)
call lorentz_transformation_write (LT, u, testflag=.true.)
write (u, "(1x,A)") "Rotation of Pi/4 around 3-axis, def. by angle:"
LT = rotation (Pi/4._default, 3)
call pacify (LT, tol)
call lorentz_transformation_write (LT, u, testflag=.true.)
write (u, "(1x,A)") "Rotation of Pi/4 around axis = (1,2,3):"
call lorentz_transformation_write (rotation (0.707107_default, 0.707107_default, &
normalize (v3_1)), u, testflag=.true.)
write (u, "(1x,A)") "Rotation in plane to axis = (1,2,3), angle given by length of axis:"
call lorentz_transformation_write (rotation (v3_1), u, testflag=.true.)
write (u, "(1x,A)") "Rotation from v3_1=(1,2,3) to v3_2=(-2,1,-5):"
call lorentz_transformation_write (rotation_to_2nd (v3_1,v3_2), u, testflag=.true.)
write (u, "(1x,A)") "Rotation from 1-axis to v3_2=(-2,1,-5):"
call lorentz_transformation_write (rotation_to_2nd (1,v3_2), u, testflag=.true.)
write (u, "(1x,A)") "Rotation from 2-axis to v3_2=(-2,1,-5):"
call lorentz_transformation_write (rotation_to_2nd (2,v3_2), u, testflag=.true.)
write (u, "(1x,A)") "Rotation from 3-axis to v3_2=(-2,1,-5):"
call lorentz_transformation_write (rotation_to_2nd (3,v3_2), u, testflag=.true.)
write (u, "(A)")
write (u, "(A)") "* Lorentz transformations: boosts"
write (u, "(A)")
write (u, "(1x,A)") "Boost from rest frame to 3-vector, mass m=10:"
call lorentz_transformation_write (boost (v3_1, 10._default), u, testflag=.true.)
write (u, "(1x,A)") "Boost from rest frame to 4-vector, mass m=10:"
v4 = vector4_moving (42._default, v3_1)
call lorentz_transformation_write (boost (v4, 10._default), u, testflag=.true.)
write (u, "(1x,A)") "Boost along 1-axis, beta*gamma = 12"
call lorentz_transformation_write (boost (12._default, 1), u, testflag=.true.)
write (u, "(1x,A)") "Boost along 2-axis, beta*gamma = 12"
call lorentz_transformation_write (boost (12._default, 2), u, testflag=.true.)
write (u, "(1x,A)") "Boost along 3-axis, beta*gamma = 12"
call lorentz_transformation_write (boost (12._default, 3), u, testflag=.true.)
write (u, "(1x,A)") "Boost along axis=(1,2,3), beta*gamma = 12"
call lorentz_transformation_write (boost (12._default, v3_1), u, testflag=.true.)
write (u, "(A)")
write (u, "(A)") "* Test output end: lorentz_4"
end subroutine lorentz_4
@ %def lorentz_4
@
\subsubsection{Tests for additional kinematic functions and sets of 4-vectors}
<<Lorentz: execute tests>>=
call test(lorentz_5, "lorentz_5", &
"Test additional kinematics", u, results)
<<Lorentz: test declarations>>=
public :: lorentz_5
<<Lorentz: tests>>=
subroutine lorentz_5 (u)
integer, intent(in) :: u
type(vector4_t), dimension(2) :: p
real(default), dimension(2) :: m
real(default) :: sqrts
type(vector4_t), dimension(8) :: tt_mom
type(vector4_t), dimension(:), allocatable :: tin, tout
integer, dimension(:), allocatable :: shuffle
write (u, "(A)") "* Test output: lorentz_5"
write (u, "(A)") "* Purpose: testing additional kinematics and sets of 4-vectors"
write (u, "(A)")
write (u, "(A)")
write (u, "(A)") "* Colliding momenta, 13 TeV, massless"
write (u, "(A)")
sqrts = 13000._default
p = colliding_momenta (sqrts)
call vector4_write (p(1), u, testflag=.true.)
call vector4_write (p(2), u, testflag=.true.)
write (u, "(A)")
write (u, "(A)") "* Colliding momenta, 10 GeV, massive muons"
write (u, "(A)")
sqrts = 10._default
m = [0.1057_default, 0.1057_default]
p = colliding_momenta (sqrts, m)
call vector4_write (p(1), u, testflag=.true.)
call vector4_write (p(2), u, testflag=.true.)
write (u, "(A)")
write (u, "(A)") "* Kinematical function lambda"
write (u, "(A)")
write (u, "(1x,A," // FMT_12 // ")") "s = 172.3**2, m1 = 4.2, m2 = 80.418:", &
lambda (172.3_default**2, 4.2_default**2, 80.418_default**2)
write (u, "(A)")
write (u, "(A)") "* Test vector_set"
write (u, "(A)")
tt_mom(1) = [2.5000000000000000e+02_default, zero, zero, 2.4999999999947777e+02_default]
tt_mom(2) = [2.5000000000000000e+02_default, zero, zero, -2.4999999999947777e+02_default]
tt_mom(3) = [1.1557492413664579e+02_default, 3.9011599241011098e+01_default, &
-6.4278142734963140e+01_default, 8.7671766153043137e+01_default]
tt_mom(4) = [1.4617918132729235e+02_default, -1.0947970597860679e+02_default, &
1.5484441802571380e+01_default, -9.5525593923398418e+01_default]
tt_mom(5) = [5.2637589215119526e+01_default, -4.7413198564695762e+01_default, &
1.0087885417286579e+01_default, 2.0516525153079229e+01_default]
tt_mom(6) = [5.4760292922264796e+01_default, 1.5197406985690520e+01_default, &
5.1527071739328015e+01_default, -1.0615525413924287e+01_default]
tt_mom(7) = [3.2415057664609684e+01_default, 7.5539389341684711e+00_default, &
-1.5935831743946720e+01_default, -2.7139737100881156e+01_default]
tt_mom(8) = [9.8432954734067863E+01_default, 9.5129959382432399e+01_default, &
3.1145755197238966e+00_default, 2.5092565132081496e+01_default]
write (u, "(1x,A)") "Write routine for vector sets, maximal compression:"
call vector4_write_set (tt_mom, u, show_mass=.true., testflag=.true., &
check_conservation=.true., ultra=.true.)
write (u, "(1x,A,L1)") "Vector set is CMS frame: ", vector_set_is_cms (tt_mom, 2)
write (u, "(1x,A)") "Reshuffle vector set, final state inverted:"
allocate (tin (8))
tin = tt_mom
allocate (shuffle (8), source = [1,2,8,7,6,5,4,3])
call vector_set_reshuffle (tin, shuffle, tout)
call vector4_write_set (tout, u, show_mass=.true., testflag=.true., &
check_conservation=.true., ultra=.true.)
write (u, "(1x,A)") "Vector set, check momentum conservation:"
call vector4_check_momentum_conservation (tt_mom, 2, u, &
abs_smallness = 1.e-12_default, verbose=.true.)
write (u, "(A)")
write (u, "(A)") "* Test output end: lorentz_5"
end subroutine lorentz_5
@ %def lorentz_5
@
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
This is the module file for the $M_{T2}$ calculator based on
algorithm by Han and Cheng by Tom James (DESY) August 2013.
<<[[kinematics_vars.f90]]>>=
<<File header>>
module kinematics_vars
use kinds
use lorentz
use format_defs, only: FMT_19
<<Standard module head>>
<<Kinematics vars: public>>
<<Kinematics vars: parameters>>
<<Kinematics vars: types>>
interface
<<Kinematics vars: sub interfaces>>
end interface
end module kinematics_vars
@ %def kinematics_vars
@
<<[[kinematics_vars_sub.f90]]>>=
<<File header>>
submodule (kinematics_vars) kinematics_vars_s
implicit none
contains
<<Kinematics vars: procedures>>
end submodule kinematics_vars_s
@ %def kinematics_vars_s
@
<<Kinematics vars: parameters>>=
! Calculation parameters
real(kind=default), parameter, public :: relative_precision = 0.00001, &
absolute_precision=0.0, min_mass=0.1, step=0.1
real(kind=default), public :: prec
! scale and precision
! Define precision=relative_precision*scale where scale=max{Ea, Eb}
! Don't set relative precision <0.00001
! If mass a < min_mass and mass b < min_mass use massless code
! Gives massless particles a very small mass
! scanning step size
@ %def relative_precision prec
@
<<Kinematics vars: procedures>>=
subroutine mt2calcmassive (mt2, mn, mt2_b)
!!! Code implementation of the massive algorithm for calculating mt2
type(mt2_t), intent(inout) :: mt2
real(default), intent(in) :: mn
real(default) :: Ea, Eb
real(default) :: masq, mbsq, mnsq
real(default), intent(out) :: mt2_b
real(default):: deltasq0, x00, y00, dis, p2x0, p2y0, deltasqmid, &
deltasqhigh, deltasqhigh1, deltasqhigh2, deltasqhigh21, &
deltasqhigh22, deltasqlow
! Calculation coefficients for the quadratic and quartic curves
- real(default) :: a1, b1, c1, d1, e1, f1, &
+ real*16 :: a1, b1, c1, d1, e1, f1, &
a2, b2, c2, d2, e2, f2
- real(default) :: d20, d11, d21, e11, e20, e21
- real(default) :: f10, f12, f20, f21, f22
+ real*16 :: d20, d11, d21, e11, e20, e21
+ real*16 :: f10, f12, f20, f21, f22
integer:: nsolshigh, nsolslow, findhigh, nsolsmid
print *, 'using calculation for massive visible particles.'
!!! Allowed transverse 2-momenta for each invisible partice incloses
!!! an elliptical region on the px, py plane.
!!! To safisfy kinematic constraints, the two elliptical regions must
!!! overlap. MT2 is therefore given by the minimum initial decay mass
!!! in each branch (mt2_b) which gives overlapping elipses
!!! We therefore increase the initial decay mass in each branch (mt2_b)
!!! until the ellipses overlap - the "balanced configuration".
!!! In the balanced configuration, the two quadtratic equations
!!! can be combined into one quartic equation.
!!! We search for the point at which the two ellipses are tangent,
!!! using the Sturm sequence method and bisection
!!! When the smaller ellipse is already inside the larger ellipse, this
!!! is called an "unbalanced configuration", and mt2 is easy to calculate.
deltasq0 = mt2%ma*(mt2%ma + 2*mt2%mn)
!!! Find co-efficients at deltasq = deltasq0 for the quadratic
!!! equations that describe each ellipse
masq = mt2%ma**2
mbsq = mt2%mb**2
mnsq = mt2%mn**2
Ea = sqrt(mt2%Ea2)
Eb = sqrt(mt2%Eb2)
a1 = 1 - mt2%pax**2 / mt2%Ea2
b1 = - mt2%pax * mt2%pay / mt2%Ea2
c1 = 1 - mt2%pay**2 / mt2%Ea2
d1 = - (mt2%pax*(deltasq0 - masq))/(2.d0*mt2%Ea2)
e1 = - (mt2%pay * (deltasq0 - masq))/(2.d0*mt2%Ea2)
a2 = 1 - (mt2%pbx * mt2%pbx)/(mt2%Eb2)
b2 = - (mt2%pbx * mt2%pby) / (mt2%Eb2)
c2 = 1 - (mt2%pby * mt2%pby / (mt2%Eb2))
d2 = -mt2%pmx + ((mt2%pbx * (deltasq0 - mbsq))/(2.d0*mt2%Eb2)) + &
(mt2%pbx*((mt2%pbx*mt2%pmx) + (mt2%pby * mt2%pmy)))/(mt2%Eb2)
e2 = -mt2%pmy + (mt2%pby * (deltasq0 - mbsq))/(2.d0*mt2%Eb2) + &
(mt2%pby*((mt2%pbx*mt2%pmx) + (mt2%pby * mt2%pmy)))/(mt2%Eb2)
f2 = mt2%pmx*mt2%pmx + mt2%pmy * mt2%pmy-((deltasq0-mbsq)/(2*Eb) + &
(mt2%pbx*mt2%pmx + mt2%pby * mt2%pmy)/Eb)*((deltasq0-mbsq)/(2*Eb)+ &
((mt2%pbx*mt2%pmx + mt2%pby * mt2%pmy)/Eb))+ mnsq
!!! find the centre of the small ellipse
x00 = (c1*d1 - b1*e1)/(b1*b1 - a1*c1)
y00 = (a1*e1 - b1*d1)/(b1*b1 - a1*c1)
!!! check if the large ellipse contains the small one -
!!! "unbalanced configuration"
dis = a2*x00*x00 + 2.d0*b2*x00*y00 + c2*y00*y00 + &
2.d0*d2*x00 + 2.d0*e2*y00 + f2
mt2_b = 0
if (dis<0.01) then
mt2_b = sqrt(mnsq + deltasq0)
return
end if
!!! find linear and constant terms in quadratic equations for
!!! delta = (deltasq-msq)/(2Esq) as quadratic terms do not change as we adjust delta
d11 = -mt2%pax
e11 = -mt2%pay
f10 = mnsq
f12 = -mt2%Ea2
d21 = (mt2%Ea2 * mt2%pbx)/(mt2%Eb2)
d20 = ((masq - mbsq)*mt2%pbx)/(2.d0*mt2%Eb2) - mt2%pmx + &
(mt2%pbx*(mt2%pbx*mt2%pmx + mt2%pby * mt2%pmy)) / mt2%Eb2
e21 = (mt2%Ea2 * mt2%pby) / mt2%Eb2
e20 = ((masq - mbsq) * mt2%pby)/(2.d0*mt2%Eb2) - mt2%pmy + &
(mt2%pby * (mt2%pbx*mt2%pmx + mt2%pby * mt2%pmy)) / mt2%Eb2
f22 = -mt2%Ea2*mt2%Ea2 / mt2%Eb2
f21 = (-2.d0*mt2%Ea2*((masq - mbsq)/(2.d0*Eb) + &
(mt2%pbx*mt2%pmx + mt2%pby * mt2%pmy)/Eb))/Eb
f20 = mnsq + mt2%pmx*mt2%pmx + mt2%pmy**2 &
- ((masq - mbsq)/(2.d0*Eb) + (mt2%pbx*mt2%pmx + &
mt2%pby * mt2%pmy)/Eb)*((masq - mbsq)/(2.d0*Eb) &
+ (mt2%pbx*mt2%pmx + mt2%pby * mt2%pmy)/Eb)
!!! Estimate upper bound of mT2 if large ellipse contains centre of small ellipse
p2x0 = mt2%pmx - x00
p2y0 = mt2%pmy - y00
deltasqhigh1 = 2*Eb*sqrt(p2x0*p2x0+p2y0*p2y0+mnsq) - &
2 * mt2%pbx * p2x0 - 2 * mt2%pby * p2y0 + mbsq
!!! Estimate upper bound of mT2 if both ellipses enclose the origin
deltasqhigh21 = 2.d0*Eb*sqrt(mt2%pmx*mt2%pmx + mt2%pmy * mt2%pmy+mnsq) &
- 2.d0 * mt2%pbx * mt2%pmx - 2.d0 * mt2%pby * mt2%pmy + mbsq
deltasqhigh22 = 2.d0*Ea*mn + masq
if ( deltasqhigh21 < deltasqhigh22 ) then
deltasqhigh2 = deltasqhigh22
else
deltasqhigh2 = deltasqhigh21
end if
! Upper bound on mt2 is the smallest of these two estimates
if (deltasqhigh1 < deltasqhigh2) then
deltasqhigh = deltasqhigh1
else
deltasqhigh = deltasqhigh2
end if
deltasqlow=deltasq0
if (nsolutionsmassive(deltasqlow) > 0) then
mt2_b = sqrt(mnsq + deltasqlow)
return
end if
nsolslow = nsolutionsmassive (deltasqlow)
nsolshigh = nsolutionsmassive (deltasqhigh)
if (nsolshigh==nsolslow) then
if (nsolshigh==4) then
findhigh = findhighfunc(deltasqhigh)
print *, 'findhigh=', findhigh
if(findhigh==0) then
mt2_b = sqrt(deltasqlow + mnsq)
return
end if
end if
end if
do while ((sqrt(deltasqhigh + mnsq) - sqrt(deltasqlow + mnsq)) > prec)
deltasqmid = (deltasqhigh + deltasqlow)/(2.)
nsolsmid = nsolutionsmassive(deltasqmid)
if (nsolsmid==4) then
deltasqhigh=deltasqmid
findhigh = findhighfunc(deltasqhigh)
cycle
end if
if (nsolsmid == nsolslow) then
deltasqlow=deltasqmid
else
deltasqhigh = deltasqmid
end if
end do
mt2_b = sqrt(mnsq + deltasqhigh)
contains
<<Kinematics vars: findhighfunc>>
<<Kinematics vars: nsolutionsmassive>>
end subroutine mt2calcmassive
subroutine mt2massless (mt2, mn, mt2_b)
!!! Code implementation of the massless algorithm for calculating mt2
type(mt2_t), intent(inout) :: mt2
real(default), intent(in) :: mn
real(default) :: Ea, Eb
real(default), intent(out) :: mt2_b
real(default) :: mnsq, Easq, Ebsq
real(default) :: pmissxsq, pmissysq
real(kind=default):: theta, s, c, pxtemp, pytemp, &
deltasq0, deltasqlow, deltasqhigh, deltasqlow1, deltasqlow2, &
deltasqhigh1, deltasqhigh2, &
minmass, maxmass, mass, &
deltamid, midmass, nsolsmid
- real(default) :: a1, b1, c1, d1, e1, f1, &
+ real*16 :: a1, b1, c1, d1, e1, f1, &
a2, b2, c2, d2, e2, f2
- real(default) :: d20, d21, e20, e21
- real(default) :: f20, f21, f22
+ real*16 :: d20, d21, e20, e21
+ real*16 :: f20, f21, f22
integer :: nsolshigh, nsolslow, findhigh
print *, 'using calculation for massless visible particles.'
! first rotate system so that pay=0
theta = atan(mt2%pay / mt2%pax)
s = sin(theta)
c = cos(theta)
mnsq = mn*mn
pmissxsq = mt2%pmx**2
pmissysq = mt2%pmy**2
Easq = mt2%pax**2 + mt2%pay**2
Ebsq = (mt2%pbx * mt2%pbx) + (mt2%pby * mt2%pby)
Ea = sqrt(Easq)
Eb = sqrt(Ebsq)
pxtemp = mt2%pax * c + mt2%pay * s
mt2%pax = pxtemp
mt2%pay = 0
pxtemp = mt2%pbx *c + mt2%pby * s
pytemp = -s * mt2%pbx + c * mt2%pby
mt2%pbx= pxtemp
mt2%pby= pytemp
pxtemp = mt2%pmx*c + mt2%pmy*s
pytemp = -s*mt2%pmx + c * mt2%pmy
mt2%pmx = pxtemp
mt2%pmy = pytemp
a2 = 1 - mt2%pbx * mt2%pbx / Ebsq
b2 = - mt2%pbx * mt2%pby / Ebsq
c2 = 1 - mt2%pby * mt2%pby / Ebsq
d21 = (Easq * mt2%pbx)/Ebsq
d20 = - mt2%pmx + (mt2%pbx * (mt2%pbx * mt2%pmx + mt2%pby * mt2%pmy)) / Ebsq
e21 = (Easq * mt2%pby) / Ebsq
e20 = - mt2%pmy + (mt2%pby * (mt2%pbx * mt2%pmx + mt2%pby * mt2%pmy)) / Ebsq
f22 = -(Easq*Easq/Ebsq)
f21 = -2*Easq*(mt2%pbx * mt2%pmx + mt2%pby * mt2%pmy) / Ebsq
f20 = mnsq + pmissxsq + pmissysq - (mt2%pbx * mt2%pmx + &
mt2%pby * mt2%pmy)*(mt2%pbx * mt2%pmx + mt2%pby * mt2%pmy) / Ebsq
deltasq0 = 0
deltasqlow = deltasq0 + prec
nsolslow = nsolutionsmassless (deltasqlow)
mt2_b = 0
if (nsolslow > 1) then
mt2_b = sqrt(deltasq0 + mnsq)
return
end if
deltasqhigh1 = (2*Eb*sqrt((mt2%pmx**2 * mt2%pmy**2) + mnsq)) &
- (2 * mt2%pbx *mt2%pmx) - (2 * mt2%pby * mt2%pmy)
deltasqhigh2 = 2*Ea*mn
if (deltasqhigh1 < deltasqhigh2) then
deltasqhigh = deltasqhigh2
else
deltasqhigh = deltasqhigh1
end if
nsolshigh = nsolutionsmassless(deltasqhigh)
if (nsolshigh==nsolslow) then
findhigh=0
minmass = mn
maxmass = sqrt(mnsq + deltasqhigh)
mass = minmass + step
do while (mass < maxmass)
deltasqhigh = (mass*mass) - mnsq
nsolshigh = nsolutionsmassless(deltasqhigh)
if (nsolshigh>0) then
findhigh=1
deltasqlow = (mass-step)*(mass-step) - mnsq
end if
if(findhigh==0) then
mt2_b = sqrt (deltasqlow + mnsq)
return
end if
end do
end if
if (nsolslow == nsolshigh) then
mt2_b = sqrt(mnsq + deltasqlow)
return
end if
minmass = sqrt (deltasqlow + mnsq)
maxmass = sqrt (deltasqhigh + mnsq)
do while ((maxmass - minmass) > prec)
midmass = (minmass+maxmass)/2
nsolsmid = nsolutionsmassless(deltamid)
if (nsolsmid == nsolslow) then
minmass = midmass
else
maxmass = midmass
end if
end do
mt2_b = minmass
contains
<<Kinematics vars: nsolutionsmassless>>
end subroutine mt2massless
<<Kinematics vars: public>>=
public :: mt2_t
@
<<Kinematics vars: types>>=
type :: mt2_t
private
real(default) :: mt2 = 0._default
real(default) :: mn = 0._default
real(default) :: ma = 0._default
real(default) :: mb = 0._default
real(default) :: Ea2 = 0._default
real(default) :: Eb2 = 0._default
real(default) :: pax = 0._default
real(default) :: pay = 0._default
real(default) :: pbx = 0._default
real(default) :: pby = 0._default
real(default) :: pmx = 0._default
real(default) :: pmy = 0._default
real(default) :: scale = 0._default
end type mt2_t
@ %def mt2_t
@
<<Kinematics vars: public>>=
public :: mt2_init
<<Kinematics vars: sub interfaces>>=
module subroutine mt2_init (mt2, mn, ma, mb, Easq, Ebsq, &
pax, pay, pbx, pby, pmissx, pmissy, scale)
type(mt2_t), intent(out) :: mt2
real(default), intent(in) :: mn, ma, mb
real(default), intent(in) :: Easq, Ebsq
real(default), intent(in) :: pax, pay, pbx, pby
real(default), intent(in) :: pmissx, pmissy, scale
end subroutine mt2_init
<<Kinematics vars: procedures>>=
module subroutine mt2_init (mt2, mn, ma, mb, Easq, Ebsq, &
pax, pay, pbx, pby, pmissx, pmissy, scale)
type(mt2_t), intent(out) :: mt2
real(default), intent(in) :: mn, ma, mb
real(default), intent(in) :: Easq, Ebsq
real(default), intent(in) :: pax, pay, pbx, pby
real(default), intent(in) :: pmissx, pmissy, scale
mt2%mn = mn
mt2%ma = ma
mt2%mb = mb
mt2%Ea2 = Easq
mt2%Eb2 = Ebsq
mt2%pax = pax
mt2%pay = pay
mt2%pbx = pbx
mt2%pby = pby
mt2%pmx = pmissx
mt2%pmy = pmissy
mt2%scale = scale
end subroutine mt2_init
@ %def mt2_init
<<Kinematics vars: nsolutionsmassless>>=
integer function nsolutionsmassless (dsq)
!!! calculates the number of solutions of quartic function in massless case
!!! Problem with macOS XCode 14/15
! real*16 :: delta, a, b, &
real(kind=default) :: delta, a, b, &
z0, z1, z2, z3, z4, &
z0sq, z1sq, z2sq, z3sq, z4sq, &
y0, y1, y2, y3, &
x0, x1, x2, &
w0, w1, &
v0, &
t1, t2, t3, t4, t5
real(kind=default)::deltasqlow, tempmass, maxmass, dsq
integer:: findhigh, nsolshigh, nsol
delta = dsq / (2 * mt2%Ea2)
!!! d1 = d11*delta seems not to be used
!!! e1 = e11*delta seems not to be used
!!! f1 = f12*delta*delta+f10 seems not to be used
d2 = d21*delta+d20
e2 = e21*delta+e20
f2 = f22*delta*delta+f21*delta+f20
if (mt2%pax > 0) then
a = Ea/Dsq
else
a = -Ea/Dsq
end if
if (mt2%pax > 0) then
b = ((mnsq*Ea)/Dsq)-(Dsq/(4*Ea))
else
b = Dsq/(4*Ea) - ((mnsq*Ea)/Dsq)
end if
z4 = a*a*a2
z3 = 2*a*b2/Ea
z2 = (2*a*a2*b+c2+2*a*d2) / mt2%Ea2
z1 = (2*b*b2+2*e2) / (mt2%Ea2*Ea)
z0 = (a2*b*b+2*b*d2+f2)/(mt2%Ea2*mt2%Ea2)
z0sq = z0*z0
z1sq = z1*z1
z2sq = z2*z2
z3sq = z3*z3
z4sq = z4*z4
y3 = 4*z4
y2 = 3*z3
y1 = 2*z2
y0 = z1
x2 = -(z2/2 - 3*z3sq/(16*z4))
x1 = -(3*z1/4. -z2*z3/(8*z4))
x0 = -z0 + z1*z3/(16*z4)
w1 = -y1 - (y3*x0*x1/x2 - y3*x0 -y2*x1)/x2
w0 = -y0 - y3 *x0 *x1/(x2*x2)+ y2*x0/x2
v0 = -x0 - x2*w0*w0/(w1*w1) + x1*w0/w1
t1 = z4
t2 = z4
t3 = x2
t4 = w1
t5 = v0
!!! number of solutions depends on the number of sign changes
nsol = signchangenegative(t1, t2, t3, t4, t5) - signchangepositive(t1, t2, t3, t4, t5)
if (nsol<0) then
nsol=0
end if
nsolutionsmassless=nsol
end function nsolutionsmassless
@ %def nsolutionsmassless
<<Kinematics vars: nsolutionsmassive>>=
integer function nsolutionsmassive ( dsq )
!!! calculates the number of solutions of quartic function in massive case
integer:: nsol, nsolsmid
real(kind=default):: deltasqmid, dsq, delta
real(kind=default) :: z0, z1, z2, z3, z4, &
z0sq, z1sq, z2sq, z3sq, z4sq, &
y0, y1, y2, y3, &
x0, x1, x2, &
w0, w1, &
v0, &
t1, t2, t3, t4, t5
delta = (dsq-masq) / (2*mt2%Ea2)
!!! calculate quadratic co-efficients
d1 = d11*delta
e1 = e11*delta
f1 = (f12*delta*delta)+f10
d2 = (d21*delta)+d20
e2 = (e21*delta)+e20
f2 = (f22*delta*delta)+f21*delta+f20
!!! Calculate quartic co-efficients, divided by Ea^n to keep dimensionless
z0 = (-4.d0*a2*d1*d2*f1 + 4.d0*a1*d2*d2*f1 + a2*a2*f1*f1 &
+ 4.d0*a2*d1*d1*f2 - 4.d0*a1*d1*d2*f2 &
- 2.d0*a1*a2*f1*f2+a1*a1*f2*f2) / (mt2%Ea2 * mt2%Ea2)
z1 = (-8.d0*a2*d1*d2*e1 + 8.d0*a1*d2*d2*e1 + 8.d0*a2*d1*d1*e2 &
- 8.d0*a1*d1*d2*e2 - 4.d0*a2*b2*d1*f1 - 4.d0*a2*b1*d2*f1 + &
8.d0*a1*b2*d2*f1 &
+ 4.d0*a2*a2*e1*f1 - 4.d0*a1*a2*e2*f1 + 8.d0*a2*b1*d1*f2 - &
4.d0*a1*b2*d1*f2 - 4.d0*a1*b1*d2*f2 - 4.d0*a1*a2*e1*f2 + &
4.d0*a1*a1*e2*f2) / (mt2%Ea2*Ea)
z2 = (4.d0*a2*c2*d1*d1 - 4.d0*a2*c1*d1*d2 - 4.d0*a1*c2*d1*d2 + &
4.d0*a1*c1*d2*d2 - 8.d0*a2*b2*d1*e1 - 8.d0*a2*b1*d2*e1 + &
16.d0*a1*b2*d2*e1 &
+ 4.d0*a2*a2*e1*e1 + 16.d0*a2*b1*d1*e2 - 8.d0*a1*b2*d1*e2 - &
8.d0*a1*b1*d2*e2 - 8.d0*a1*a2*e1*e2 + 4.d0*a1*a1*e2*e2 - &
4.d0*a2*b1*b2*f1 &
+ 4.d0*a1*b2*b2*f1 + 2.d0*a2*a2*c1*f1 - 2.d0*a1*a2*c2*f1 + &
4.d0*a2*b1*b1*f2 - 4.d0*a1*b1*b2*f2 - 2.d0*a1*a2*c1*f2 + &
2.d0*a1*a1*c2*f2) / mt2%Ea2
z3 = (-4*a2*b2*c1*d1 + 8*a2*b1*c2*d1 - 4*a1*b2*c2*d1 - 4*a2*b1*c1*d2 + &
8*a1*b2*c1*d2 - 4*a1*b1*c2*d2 - 8*a2*b1*b2*e1 + 8*a1*b2*b2*e1 + &
4*a2*a2*c1*e1 - 4*a1*a2*c2*e1 + 8*a2*b1*b1*e2 - 8*a1*b1*b2*e2 - &
4*a1*a2*c1*e2 + 4*a1*a1*c2*e2)/Ea
z4 = - (4.d0*a2*b1*b2*c1) + (4.d0*a1*b2*b2*c1) + (a2*a2*c1*c1) + &
(4.d0*a2*b1*b1*c2) - (4.d0*a1*b1*b2*c2) - (2.d0*a1*a2*c1*c2) + &
(a1*a1*c2*c2)
if (z4==0) then
z4=1e-33
end if
z0sq = z0*z0
z1sq = z1*z1
z2sq = z2*z2
z3sq = z3*z3
z4sq = z4*z4
y0 = z1
y1 = 2.d0*z2
y2 = 3.d0*z3
y3 = 4.d0*z4
x0 = - z0 + (z1*z3)/(16.d0*z4)
x1 = - (3.d0*z1/4.d0) + (z2*z3)/(8.d0*z4)
x2 = - (z2/2.d0) + (3.d0*z3sq)/(16.d0*z4)
w0 = - y0 - ((y3*x0 *x1)/(x2*x2))+ ((y2*x0)/x2)
w1 = - y1 - ((y3*x1*x1/x2) - y3*x0 - y2*x1)/x2
v0 = - x0 - (x2*w0*w0)/(w1*w1) + (x1*w0/w1)
!!! Sturm sequence coefficients of leading order
t1 = z4
t2 = z4
t3 = x2
t4 = w1
t5 = v0
nsol = signchangenegative (t1, t2, t3, t4, t5) - &
signchangepositive (t1, t2, t3, t4, t5)
!!! correction for roundoff error if it gives a negative result
if (nsol<0) then
nsol = 0
endif
nsolutionsmassive = nsol
end function nsolutionsmassive
@ %def nsolutionsmassive
<<Kinematics vars: findhighfunc>>=
function findhighfunc (dsq)
real(kind=default):: x00, y00, deltasqlow, dis, &
deltasqmid, deltasqhigh, dsq
integer:: findhighfunc, nsolsmid !!! , nsolutionsmassive
x00 = (c1*d1 - b1*e1)/(b1*b1 - a1*c1)
y00 = (a1*e1 - b1*d1)/(b1*b1 - a1*c1)
deltasqlow = (mn + mt2%ma)*(mn + mt2%ma) - mnsq
do while ((deltasqhigh - deltasqlow) > 0.001)
deltasqmid = (deltasqhigh + deltasqlow)/2
nsolsmid = nsolutionsmassive(deltasqmid)
if (nsolsmid==2) then
deltasqhigh = deltasqmid
end if
if ( nsolsmid==4) then
deltasqhigh = deltasqmid
end if
if (nsolsmid==0) then
d1 = -mt2%pax * (deltasqmid - masq) / (2 * mt2%Ea2)
e1 = -mt2%pay * (deltasqmid - masq) / (2 * mt2%Ea2)
d2 = -mt2%pmx + mt2%pbx * (deltasqmid - mbsq)/(2*mt2%Eb2) + &
mt2%pbx * (mt2%pbx *mt2%pmx + mt2%pby * mt2%pmy) / mt2%Eb2
e2 = - mt2%pmy + mt2%pby * (deltasqmid - mbsq)/(2*mt2%Eb2) &
+ mt2%pby * (mt2%pbx * mt2%pmx + mt2%pby * mt2%pmy) / mt2%Eb2
f2 = mt2%pmx*mt2%pmx + mt2%pmy**2 - ((deltasqmid - mbsq)/(2*Eb) &
+ (mt2%pbx * mt2%pmx + mt2%pby * mt2%pmy)/Eb) * &
((deltasqmid - mbsq)/(2*Eb) + (mt2%pbx * mt2%pmx + &
mt2%pby * mt2%pmy)/Eb) + mnsq
dis = a2*x00*x00 + 2*b2*x00*y00 + c2*y00*y00 + 2*d2*x00 + &
2*e2*y00 + f2
!!! Check if large ellipse contains small one
if (dis < 0 ) then
deltasqhigh = deltasqmid
else
deltasqlow = deltasqmid
end if
end if
end do
findhighfunc=0
end function findhighfunc
@ %def findhighfunc
Print input values.
<<Kinematics vars: public>>=
public :: write_mt2_input
<<Kinematics vars: sub interfaces>>=
module subroutine write_mt2_input (mt2, u)
type(mt2_t), intent(in) :: mt2
integer, intent(in) :: u
end subroutine write_mt2_input
<<Kinematics vars: procedures>>=
module subroutine write_mt2_input (mt2, u)
type(mt2_t), intent(in) :: mt2
integer, intent(in) :: u
!!! prints the input variables as a check
write (u, "(1x,A," // FMT_19 // ")") 'pax =', mt2%pax * mt2%scale
write (u, "(1x,A," // FMT_19 // ")") 'pay =', mt2%pay * mt2%scale
write (u, "(1x,A," // FMT_19 // ")") 'ma =', mt2%ma * mt2%scale
write (u, "(1x,A," // FMT_19 // ")") 'pbx =', mt2%pbx * mt2%scale
write (u, "(1x,A," // FMT_19 // ")") 'pby =', mt2%pby * mt2%scale
write (u, "(1x,A," // FMT_19 // ")") 'mb =', mt2%mb * mt2%scale
write (u, "(1x,A," // FMT_19 // ")") 'pmissx =', mt2%pmx * mt2%scale
write (u, "(1x,A," // FMT_19 // ")") 'pmissy =', mt2%pmy * mt2%scale
write (u, "(1x,A," // FMT_19 // ")") 'mn =', mt2%mn * mt2%scale
!!! write (u, *) 'precision = ', prec
end subroutine write_mt2_input
@ %def write_mt2_input
@
<<Kinematics vars: public>>=
public :: write_mt2_output
<<Kinematics vars: sub interfaces>>=
module subroutine write_mt2_output (mt2, scale, u)
real(default), intent(in) :: mt2, scale
integer, intent(in) :: u
end subroutine write_mt2_output
<<Kinematics vars: procedures>>=
module subroutine write_mt2_output (mt2, scale, u)
real(default), intent(in) :: mt2, scale
integer, intent(in) :: u
!!! Prints the output (MT2)
! mt2=mt2_b*scale
write (u, "(1x,A,ES10.4)") 'mt2=', mt2 * scale
end subroutine write_mt2_output
@ %def write_mt2_output
@
<<Kinematics vars: procedures>>=
function signchangepositive (t1, t2, t3, t4, t5) result (n_sign)
!!! calculates the number of sign changes in the sturm sequence
!!! Problem with macOS XCode 14/15
! real*16 :: t1, t2, t3, t4, t5
real(kind=default) :: t1, t2, t3, t4, t5
integer :: n_sign
integer :: n
n = 0
if (t1*t2 < 0) then
n = n+1
end if
if (t2*t3 < 0) then
n = n+1
end if
if (t3*t4 < 0) then
n = n+1
end if
if (t4*t5 < 0) then
n = n+1
end if
n_sign = n
end function signchangepositive
@ %def signchangepositive
<<Kinematics vars: procedures>>=
integer function signchangenegative (t1, t2, t3, t4, t5)
!!! calculates the number of times there is no sign
!!! change in the sturm sequence
!!! Problem with macOS XCode 14/15
! real*16 :: t1, t2, t3, t4, t5
real(kind=default) :: t1, t2, t3, t4, t5
integer:: n
n=0
if((t1*t2)>0) then
n=n+1
endif
if((t2*t3)>0) then
n=n+1
endif
if((t3*t4)>0) then
n=n+1
endif
if((t4*t5)>0) then
n=n+1
endif
signchangenegative = n
end function signchangenegative
@ %def signchangenegative
It looks like as if this function is not used in the code.
<<XXX Kinematics vars: procedures>>=
integer function scanhigh (mn, deltasqhigh)
real(default), intent(in) :: mn
real(kind=default):: tempmass, maxmass, deltasqlow, deltasqhigh, mass
integer:: nsolshigh, findhigh
mnsq = mn*mn
tempmass = mn+ma
findhigh = 0
maxmass = sqrt(mnsq + deltasqhigh)
mass = tempmass + step
do while ( mass < maxmass)
deltasqhigh = mass*mass - mnsq
nsolshigh = nsolutionsmassive(deltasqhigh)
if (nsolshigh > 0) then
deltasqlow = (mass - step)*(mass - step) - mnsq
findhigh = 1
exit
end if
end do
scanhigh=findhigh
end function scanhigh
@ %def scanhight
All this subroutine does is to decide whether to use the algorithm
for massless visible particles, or the algorithm for massive
visible particles.
<<Kinematics vars: public>>=
public :: mt2calc
<<Kinematics vars: sub interfaces>>=
module subroutine mt2calc (mt2, mn, mt2_b)
type(mt2_t), intent(inout) :: mt2
real(default), intent(in) :: mn
real(default), intent(out) :: mt2_b
end subroutine mt2calc
<<Kinematics vars: procedures>>=
module subroutine mt2calc (mt2, mn, mt2_b)
type(mt2_t), intent(inout) :: mt2
real(default), intent(in) :: mn
real(default), intent(out) :: mt2_b
if (mt2%ma**2 < min_mass .and. mt2%mb**2 < min_mass) then
call mt2massless (mt2, mn, mt2_b)
else
call mt2calcmassive (mt2, mn, mt2_b)
end if
end subroutine mt2calc
@ %def mt2calc
@ This subroutine sets up the event for calculation by defining the
input variables, and setting them to a scale that aids calculation.
The output variables have been rescaled back.
<<Kinematics vars: public>>=
public :: mt2_setevent
<<Kinematics vars: sub interfaces>>=
module subroutine mt2_setevent (mn, ma, mb, Easq, Ebsq, pax, pay, pbx, pby, &
pmissx, pmissy, scale, u)
real(default), intent(out) :: ma, mb ! mass of particles a and b
real(default), intent(out) :: mn ! invisible particle mass scaled and unscaled
real(default), intent(out) :: scale
real(default), intent(out) :: pax, pay, pbx, pby
real(default), intent(out) :: pmissx, pmissy
real(default), intent(out) :: Easq, Ebsq
integer, intent(in) :: u
end subroutine mt2_setevent
<<Kinematics vars: procedures>>=
module subroutine mt2_setevent (mn, ma, mb, Easq, Ebsq, pax, pay, pbx, pby, &
pmissx, pmissy, scale, u)
real(default), intent(out) :: ma, mb ! mass of particles a and b
real(default), intent(out) :: mn ! invisible particle mass scaled and unscaled
real(default), intent(out) :: scale
real(default), intent(out) :: pax, pay, pbx, pby
! x and y momentum of particles a and b
! missing x and y momentum
real(default) :: Ea, Eb
type(vector4_t) :: pmissa, pa, pb
real(default), intent(out) :: pmissx, pmissy
real(default) :: masq, mbsq, mnsq, Emiss, pmissz
real(default), intent(out) :: Easq, Ebsq
real(default) :: pmissxsq, pmissysq, mn_unscaled
!!! Problem with macOS XCode 14/15
! real*16 :: temp, scalesq, rt, relative_precision100
real(default) :: temp, scalesq, rt, relative_precision100
integer, intent(in) :: u
!!! Here the 8 input variables are entered. This should be read in from
!!! the syndarin file - let me know if I can help accomplish this
!!! The units of input are equal to the units of the output e.g GeV
!!! in --> GeV out, MeV in --> MeV out
ma=100 ! total mass of visible particles in branch a
mb=100 ! total mass of visible particles in branch b
!!! 0810.5178
!!! mT2 is invariant under longitudinal boosts => set paz = pbz = 0
! mass = transverse mass here!???
pax=40 ! total x momentum of visible partivles in branch a
pay=40 ! total y momentum of visible partivles in branch a
pbx=40 ! total x momentum of visible partivles in branch b
pby=40 ! total y momentum of visible partivles in branch b
Emiss=0
pmissz=0
pmissx=20 ! Total missing x momentum
pmissy=20 ! Total missing y momentum
pmissa = [ Emiss, pmissx, pmissy, pmissz]
mn=0 ! starting guess for mass of invisible particle
mn_unscaled = 0 ! starting guess for mass of invisible particle
scale=1.d0 ! scale is originally defined as 1
masq = ma*ma
Easq = masq + pax*pax + pay*pay
Ea = sqrt(Easq)
pa = [Ea, pax, pay, 0._default]
mbsq = mb*mb
Ebsq = mbsq + pbx*pbx + pby*pby
Eb = sqrt(Ebsq)
pmissxsq = pmissx*pmissx
pmissysq = pmissy*pmissy
! set ma>=mb to keep consistency
if (masq < mbsq) then
temp = pax
pax = pbx
pbx = temp
temp = pay
pay = pby
pby = temp
temp = Ea
Ea = Eb
Eb = temp
temp = Easq
Easq = Ebsq
Ebsq = temp
temp = masq
masq = mbsq
mbsq = temp
temp = ma
ma = mb
mb = temp
end if
! normalise to the scale
if (Ea > Eb) then
scale = Ea/100.d0
else
scale = Eb/100.d0
end if
rt=(sqrt(pmissxsq+pmissysq))/100.d0
if (rt > scale) then
scale = (sqrt(pmissxsq+pmissysq))/100.d0
end if
!!! write (u, *) 'scale=', scale
scalesq = scale*scale
ma = ma/scale
mb = mb/scale
masq = masq/scalesq
mbsq = mbsq/scalesq
pax = pax/scale
pay = pay/scale
pbx = pbx/scale
pby = pby/scale
Ea = Ea/scale
Eb = Eb/scale
Easq = Easq/scalesq
Ebsq = Ebsq/scalesq
pmissx = pmissx/scale
pmissy = pmissy/scale
pmissxsq = pmissxsq/scalesq
pmissysq = pmissysq/scalesq
mn = mn_unscaled/scale
mnsq = mn*mn
! calculating precision
relative_precision100 = relative_precision*100.d0
if (absolute_precision > relative_precision100) then
prec = absolute_precision
else
prec = 100*relative_precision
end if
end subroutine mt2_setevent
@ %def mt2_setevent
@
\subsection{Unit tests}
Test module, followed by the corresponding implementation module.
<<[[kinematics_vars_ut.f90]]>>=
<<File header>>
module kinematics_vars_ut
use unit_tests
use kinematics_vars_uti
<<Standard module head>>
<<Kinematics vars: public test>>
contains
<<Kinematics vars: test driver>>
end module kinematics_vars_ut
@ %def kinematics_vars_ut
@
<<[[kinematics_vars_uti.f90]]>>=
<<File header>>
module kinematics_vars_uti
<<Use kinds>>
use kinematics_vars
<<Standard module head>>
<<Kinematics vars: test declarations>>
contains
<<Kinematics vars: tests>>
end module kinematics_vars_uti
@ %def kinematics_vars_ut
This has been written by Tom James as a DESY summer student in August
2013. It is a self-contained Fortran90 program for calculating
$M_{T2}$ using the Cheng-Han algorithm.
<<Kinematics vars: public test>>=
public :: kinematics_vars_test
<<Kinematics vars: test driver>>=
subroutine kinematics_vars_test (u, results)
integer, intent(in) :: u
type(test_results_t), intent(inout) :: results
<<Kinematics vars: execute tests>>
end subroutine kinematics_vars_test
@ %def kinematics_vars_test
@
\subsubsection{Testing kinematic variables like MT2}
<<Kinematics vars: execute tests>>=
call test (kinematics_vars_1, "kinematics_vars_1", "massive mT2 calculation", &
u, results)
<<Kinematics vars: test declarations>>=
public :: kinematics_vars_1
<<Kinematics vars: tests>>=
subroutine kinematics_vars_1 (u)
integer, intent(in) :: u
real(default) :: Easq, Ebsq
real(default) :: ma, mb, mn
real(default) :: pmissx, pmissy
real(default) :: mt2_b, scale
real(default) :: pax, pay, pbx, pby
type(mt2_t) :: mt2
write (u, "(A)") "* Test output: mt2_1"
write (u, "(A)") "* Purpose: calculate MT2 for the massive case"
write (u, "(A)")
call mt2_setevent (mn, ma, mb, Easq, Ebsq, pax, pay, pbx, pby, &
pmissx, pmissy, scale, u)
call mt2_init (mt2, mn, ma, mb, Easq, Ebsq, pax, pay, pbx, pby, &
pmissx, pmissy, scale)
call write_mt2_input (mt2, u)
call mt2calc (mt2, mn, mt2_b)
call write_mt2_output (mt2_b, scale, u)
end subroutine kinematics_vars_1
@ %def kinematics_vars_1
@
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Collections of Lorentz Vectors}
The [[phs_point]] type is a container for an array of Lorentz vectors. This
allows us to transfer Lorentz-vector arrays more freely, and to collect vector
arrays of non-uniform size.
<<[[phs_points.f90]]>>=
<<File header>>
module phs_points
<<Use kinds>>
use lorentz, only: vector4_t
use lorentz, only: lorentz_transformation_t
use lorentz, only: sum
<<Standard module head>>
<<PHS points: public>>
<<PHS points: types>>
<<PHS points: interfaces>>
interface
<<PHS points: sub interfaces>>
end interface
end module phs_points
@ %def phs_points
@
<<[[phs_points_sub.f90]]>>=
<<File header>>
submodule (phs_points) phs_points_s
use lorentz, only: vector4_null
use lorentz, only: vector4_write_set
use lorentz, only: operator(==)
use lorentz, only: operator(*)
use lorentz, only: operator(**)
implicit none
contains
<<PHS points: procedures>>
end submodule phs_points_s
@ %def phs_points_s
@
\subsection{PHS point definition}
This is a trivial container for an array of momenta. The main
application is to store a non-uniform array of phase-space points.
<<PHS points: public>>=
public :: phs_point_t
<<PHS points: types>>=
type :: phs_point_t
private
type(vector4_t), dimension(:), allocatable :: p
contains
<<PHS points: phs point: TBP>>
end type phs_point_t
@ %def phs_point_t
@
\subsection{PHS point: basic tools}
Output. This is instrumented with options, which have to be
provided by the caller.
<<PHS points: phs point: TBP>>=
procedure :: write => phs_point_write
<<PHS points: sub interfaces>>=
module subroutine phs_point_write (phs_point, unit, show_mass, testflag, &
check_conservation, ultra, n_in)
class(phs_point_t), intent(in) :: phs_point
integer, intent(in), optional :: unit
logical, intent(in), optional :: show_mass
logical, intent(in), optional :: testflag, ultra
logical, intent(in), optional :: check_conservation
integer, intent(in), optional :: n_in
end subroutine phs_point_write
<<PHS points: procedures>>=
module subroutine phs_point_write (phs_point, unit, show_mass, testflag, &
check_conservation, ultra, n_in)
class(phs_point_t), intent(in) :: phs_point
integer, intent(in), optional :: unit
logical, intent(in), optional :: show_mass
logical, intent(in), optional :: testflag, ultra
logical, intent(in), optional :: check_conservation
integer, intent(in), optional :: n_in
if (allocated (phs_point%p)) then
call vector4_write_set (phs_point%p, &
unit = unit, &
show_mass = show_mass, &
testflag = testflag, &
check_conservation = check_conservation, &
ultra = ultra, &
n_in = n_in)
end if
end subroutine phs_point_write
@ %def phs_point_write
@ Non-intrinsic assignment
<<PHS points: public>>=
public :: assignment(=)
<<PHS points: interfaces>>=
interface assignment(=)
module procedure phs_point_from_n
module procedure phs_point_from_vector4
module procedure vector4_from_phs_point
end interface
@ Initialize with zero momenta but fixed size
<<PHS points: sub interfaces>>=
pure module subroutine phs_point_from_n (phs_point, n_particles)
type(phs_point_t), intent(out) :: phs_point
integer, intent(in) :: n_particles
end subroutine phs_point_from_n
<<PHS points: procedures>>=
pure module subroutine phs_point_from_n (phs_point, n_particles)
type(phs_point_t), intent(out) :: phs_point
integer, intent(in) :: n_particles
allocate (phs_point%p (n_particles), source = vector4_null)
end subroutine phs_point_from_n
@ %def phs_point_init_from_n
@ Transform from/to plain vector array
<<PHS points: sub interfaces>>=
pure module subroutine phs_point_from_vector4 (phs_point, p)
type(phs_point_t), intent(out) :: phs_point
type(vector4_t), dimension(:), intent(in) :: p
end subroutine phs_point_from_vector4
pure module subroutine vector4_from_phs_point (p, phs_point)
class(phs_point_t), intent(in) :: phs_point
type(vector4_t), dimension(:), allocatable, intent(out) :: p
end subroutine vector4_from_phs_point
<<PHS points: procedures>>=
pure module subroutine phs_point_from_vector4 (phs_point, p)
type(phs_point_t), intent(out) :: phs_point
type(vector4_t), dimension(:), intent(in) :: p
phs_point%p = p
end subroutine phs_point_from_vector4
pure module subroutine vector4_from_phs_point (p, phs_point)
class(phs_point_t), intent(in) :: phs_point
type(vector4_t), dimension(:), allocatable, intent(out) :: p
if (allocated (phs_point%p)) p = phs_point%p
end subroutine vector4_from_phs_point
@ %def phs_point_from_vector4
@ %def vector4_from_phs_point
@ Query the size of the momentum array (assuming it is allocated).
<<PHS points: public>>=
public :: size
<<PHS points: interfaces>>=
interface size
module procedure phs_point_size
end interface size
<<PHS points: sub interfaces>>=
pure module function phs_point_size (phs_point) result (s)
class(phs_point_t), intent(in) :: phs_point
integer :: s
end function phs_point_size
<<PHS points: procedures>>=
pure module function phs_point_size (phs_point) result (s)
class(phs_point_t), intent(in) :: phs_point
integer :: s
if (allocated (phs_point%p)) then
s = size (phs_point%p)
else
s = 0
end if
end function phs_point_size
@ %def phs_point_size
@ Equality, implemented only for valid points.
<<PHS points: public>>=
public :: operator(==)
<<PHS points: interfaces>>=
interface operator(==)
module procedure phs_point_eq
end interface operator(==)
<<PHS points: sub interfaces>>=
elemental module function phs_point_eq &
(phs_point_1, phs_point_2) result (flag)
class(phs_point_t), intent(in) :: phs_point_1, phs_point_2
logical :: flag
end function phs_point_eq
<<PHS points: procedures>>=
elemental module function phs_point_eq &
(phs_point_1, phs_point_2) result (flag)
class(phs_point_t), intent(in) :: phs_point_1, phs_point_2
logical :: flag
if (allocated (phs_point_1%p) .and. (allocated (phs_point_2%p))) then
flag = all (phs_point_1%p == phs_point_2%p)
else
flag = .false.
end if
end function phs_point_eq
@ %def phs_point_eq
@ Extract all momenta, as a method
<<PHS points: phs point: TBP>>=
procedure :: get => phs_point_get
<<PHS points: sub interfaces>>=
pure module function phs_point_get (phs_point) result (p)
class(phs_point_t), intent(in) :: phs_point
type(vector4_t), dimension(:), allocatable :: p
end function phs_point_get
<<PHS points: procedures>>=
pure module function phs_point_get (phs_point) result (p)
class(phs_point_t), intent(in) :: phs_point
type(vector4_t), dimension(:), allocatable :: p
if (allocated (phs_point%p)) then
p = phs_point%p
else
allocate (p (0))
end if
end function phs_point_get
@ %def phs_point_select
@ Extract a subset of all momenta.
<<PHS points: phs point: TBP>>=
procedure :: select => phs_point_select
<<PHS points: sub interfaces>>=
elemental module function phs_point_select (phs_point, i) result (p)
class(phs_point_t), intent(in) :: phs_point
integer, intent(in) :: i
type(vector4_t) :: p
end function phs_point_select
<<PHS points: procedures>>=
elemental module function phs_point_select (phs_point, i) result (p)
class(phs_point_t), intent(in) :: phs_point
integer, intent(in) :: i
type(vector4_t) :: p
if (allocated (phs_point%p)) then
p = phs_point%p(i)
else
p = vector4_null
end if
end function phs_point_select
@ %def phs_point_select
@ Return the invariant mass squared for a subset of momenta
<<PHS points: phs point: TBP>>=
procedure :: get_msq => phs_point_get_msq
<<PHS points: sub interfaces>>=
pure module function phs_point_get_msq (phs_point, iarray) result (msq)
class(phs_point_t), intent(in) :: phs_point
integer, dimension(:), intent(in) :: iarray
real(default) :: msq
end function phs_point_get_msq
<<PHS points: procedures>>=
pure module function phs_point_get_msq (phs_point, iarray) result (msq)
class(phs_point_t), intent(in) :: phs_point
integer, dimension(:), intent(in) :: iarray
real(default) :: msq
if (allocated (phs_point%p)) then
msq = (sum (phs_point%p(iarray)))**2
else
msq = 0
end if
end function phs_point_get_msq
@ %def phs_point_get_msq
@
\subsection{Lorentz algebra pieces}
Lorentz transformation.
<<PHS points: public>>=
public :: operator(*)
<<PHS points: interfaces>>=
interface operator(*)
module procedure prod_LT_phs_point
end interface operator(*)
<<PHS points: sub interfaces>>=
elemental module function prod_LT_phs_point (L, phs_point) result (phs_point_LT)
type(lorentz_transformation_t), intent(in) :: L
type(phs_point_t), intent(in) :: phs_point
type(phs_point_t) :: phs_point_LT
end function prod_LT_phs_point
<<PHS points: procedures>>=
elemental module function prod_LT_phs_point (L, phs_point) result (phs_point_LT)
type(lorentz_transformation_t), intent(in) :: L
type(phs_point_t), intent(in) :: phs_point
type(phs_point_t) :: phs_point_LT
if (allocated (phs_point%p)) phs_point_LT%p = L * phs_point%p
end function prod_LT_phs_point
@ %def prod_LT_phs_point
@ Compute momentum sum, analogous to the standard [[sum]] function
(mask), and additionally using an index array.
<<PHS points: public>>=
public :: sum
<<PHS points: interfaces>>=
interface sum
module procedure phs_point_sum
module procedure phs_point_sum_iarray
end interface sum
<<PHS points: sub interfaces>>=
pure module function phs_point_sum (phs_point, mask) result (p)
class(phs_point_t), intent(in) :: phs_point
logical, dimension(:), intent(in), optional :: mask
type(vector4_t) :: p
end function phs_point_sum
pure module function phs_point_sum_iarray (phs_point, iarray) result (p)
class(phs_point_t), intent(in) :: phs_point
integer, dimension(:), intent(in) :: iarray
type(vector4_t) :: p
end function phs_point_sum_iarray
<<PHS points: procedures>>=
pure module function phs_point_sum (phs_point, mask) result (p)
class(phs_point_t), intent(in) :: phs_point
logical, dimension(:), intent(in), optional :: mask
type(vector4_t) :: p
if (allocated (phs_point%p)) then
if (present (mask)) then
p = sum (phs_point%p, mask)
else
p = sum (phs_point%p)
end if
else
p = vector4_null
end if
end function phs_point_sum
pure module function phs_point_sum_iarray (phs_point, iarray) result (p)
class(phs_point_t), intent(in) :: phs_point
integer, dimension(:), intent(in) :: iarray
type(vector4_t) :: p
logical, dimension(:), allocatable :: mask
integer :: i
allocate (mask (size (phs_point)), source = .false.)
mask(iarray) = .true.
p = sum (phs_point, mask)
end function phs_point_sum_iarray
@ %def phs_point_sum
@
\subsection{Methods for specific applications}
Convenience method: compute the pair of energy fractions w.r.t.\ the
specified beam energy. We assume that the momenta represent a
scattering process (two incoming particles) in the c.m.\ frame.
<<PHS points: phs point: TBP>>=
procedure :: get_x => phs_point_get_x
<<PHS points: sub interfaces>>=
pure module function phs_point_get_x (phs_point, E_beam) result (x)
class(phs_point_t), intent(in) :: phs_point
real(default), dimension(2) :: x
real(default), intent(in) :: E_beam
end function phs_point_get_x
<<PHS points: procedures>>=
pure module function phs_point_get_x (phs_point, E_beam) result (x)
class(phs_point_t), intent(in) :: phs_point
real(default), dimension(2) :: x
real(default), intent(in) :: E_beam
x = phs_point%p(1:2)%p(0) / E_beam
end function phs_point_get_x
@ %def phs_point_get_x
@
\subsection{Unit tests}
Test module, followed by the corresponding implementation module.
<<[[phs_points_ut.f90]]>>=
<<File header>>
module phs_points_ut
use unit_tests
use phs_points_uti
<<Standard module head>>
<<PHS points: public test>>
contains
<<PHS points: test driver>>
end module phs_points_ut
@ %def phs_points_ut
@
<<[[phs_points_uti.f90]]>>=
<<File header>>
module phs_points_uti
<<Use kinds>>
use constants, only: zero
use format_defs, only: FMT_12
use lorentz
use phs_points
<<Standard module head>>
<<PHS points: test declarations>>
contains
<<PHS points: tests>>
end module phs_points_uti
@ %def phs_points_ut
@ API: driver for the unit tests below.
<<PHS points: public test>>=
public :: phs_points_test
<<PHS points: test driver>>=
subroutine phs_points_test (u, results)
integer, intent(in) :: u
type(test_results_t), intent(inout) :: results
<<PHS points: execute tests>>
end subroutine phs_points_test
@ %def phs_points_test
@
\subsubsection{PHS point unit test implementation}
<<PHS points: execute tests>>=
call test (phs_points_1, "phs_points_1", &
"Test PHS point functionality", &
u, results)
<<PHS points: test declarations>>=
public :: phs_points_1
<<PHS points: tests>>=
subroutine phs_points_1 (u)
integer, intent(in) :: u
type(vector4_t), dimension(8) :: tt_mom
type(phs_point_t) :: phs_p
type(vector4_t) :: p_sum
type(vector4_t), dimension(:), allocatable :: p_tau, p_out
write (u, "(A)") "* Test output: phs_points_1"
write (u, "(A)") "* Purpose: handling a 2->6 PSP"
write (u, "(A)")
write (u, "(A)")
write (u, "(A)") "* Setting up a 2->6 off-shell top PSP"
write (u, "(A)")
tt_mom(1) = [2.5000000000000000e+02_default, zero, zero, 2.4999999999947775e+02_default]
tt_mom(2) = [2.5000000000000000e+02_default, zero, zero, -2.4999999999947775e+02_default]
tt_mom(3) = [1.1557492413664579e+02_default, 3.9011599241011098e+01_default, &
-6.4278142734963140e+01_default, 8.7671766153043137e+01_default]
tt_mom(4) = [1.4617918132729235e+02_default, -1.0947970597860679e+02_default, &
1.5484441802571380e+01_default, -9.5525593923398418e+01_default]
tt_mom(5) = [5.2637589215119526e+01_default, -4.7413198564695762e+01_default, &
1.0087885417286579e+01_default, 2.0516525153079229e+01_default]
tt_mom(6) = [5.4760292922264796e+01_default, 1.5197406985690520e+01_default, &
5.1527071739328015e+01_default, -1.0615525413924287e+01_default]
tt_mom(7) = [3.2415057664609684e+01_default, 7.5539389341684711e+00_default, &
-1.5935831743946720e+01_default, -2.7139737100881156e+01_default]
tt_mom(8) = [9.8432954734067863e+01_default, 9.5129959382432389e+01_default, &
3.1145755197238953e+00_default, 2.5092565132081493e+01_default]
phs_p = tt_mom
write (u, "(A)")
write (u, "(A)") "* Retrieving the size of PSP"
write (u, "(A)")
write (u, "(3x,A,I0)") "Size PSP = ", size (phs_p)
write (u, "(A)")
write (u, "(A)") "* Returning the set of 4-momenta from PSP"
write (u, "(A)")
p_out = phs_p%get ()
write (u, "(3x,A)") "set 4-mom. = "
call vector4_write_set (p_out, u, testflag = .true., ultra = .true.)
write (u, "(A)")
write (u, "(A)") "* Sum of momenta of PSP"
write (u, "(A)")
p_sum = sum (phs_p)
call pacify (p_sum, tolerance = 1.e-12_default)
write (u, "(3x,A)") "Sum:"
call p_sum%write (u)
write (u, "(A)")
write (u, "(A)") "* Reconstructing top/antitop candidate invariant masses from PSP"
write (u, "(A)")
write (u, "(3x,A," // FMT_12 // ")") "m2(top) = ", sqrt (phs_p%get_msq ([3,6,8]))
write (u, "(3x,A," // FMT_12 // ")") "m2(a-top) = ", sqrt (phs_p%get_msq ([4,5,7]))
write (u, "(A)")
write (u, "(A)") "* Select a specific 4-vector from PSP, here for a tau"
write (u, "(A)")
p_tau = phs_p%select ([7])
write (u, "(3x,A)") "p(tau):"
call p_tau(1)%write (u, show_mass = .true., testflag = .true.)
write (u, "(A)")
write (u, "(A)") "* Test output end: phs_points_1"
end subroutine phs_points_1
@ %def phs_points_1
@
\clearpage
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Special Physics functions}
Here, we declare functions that are specific for the Standard Model,
including QCD: fixed and running $\alpha_s$, Catani-Seymour
dipole terms, loop functions, etc.
To make maximum use of this, all functions, if possible, are declared
elemental (or pure, if this is not possible).
<<[[sm_physics.f90]]>>=
<<File header>>
module sm_physics
<<Use kinds with double>>
use constants
use physics_defs
use lorentz
<<Standard module head>>
<<SM physics: public>>
<<SM physics: parameters>>
interface
<<SM physics: sub interfaces>>
end interface
end module sm_physics
@ %def sm_physics
@
<<[[sm_physics_sub.f90]]>>=
<<File header>>
submodule (sm_physics) sm_physics_s
use io_units
use numeric_utils
use diagnostics
use permutations, only: factorial
implicit none
contains
<<SM physics: procedures>>
end submodule sm_physics_s
@ %def sm_physics_s
@
\subsection{Constants for Quantum Field Theory calculations}
For loop calculations in quantum field theories, one needs the
numerical values of the Riemann zeta function:
\begin{align*}
\zeta(2) &=\; 1.64493406684822643647241516665\ldots \; \\
\zeta(3) &=\; 1.20205690315959428539973816151\ldots \; \\
\zeta(4) &=\; 1.08232323371113819151600369654\ldots \; \\
\zeta(5) &=\; 1.03692775514336992633136548646\ldots \;
\end{align*}
<<SM physics: public>>=
public :: zeta2, zeta3, zeta4, zeta5
<<SM physics: parameters>>=
real(default), parameter :: &
zeta2 = 1.64493406684822643647241516665_default, &
zeta3 = 1.20205690315959428539973816151_default, &
zeta4 = 1.08232323371113819151600369654_default, &
zeta5 = 1.03692775514336992633136548646_default
@ %def zeta2 zeta3 zeta4
@ The Euler-Mascheroni constant is
\begin{equation*}
\gamma_E =
\end{equation*}
<<SM physics: public>>=
public :: eulerc
<<SM physics: parameters>>=
real(default), parameter :: &
eulerc =0.5772156649015328606065120900824024310422_default
@ %def eulerc
@
\subsection{Running $\alpha_s$}
Then we define the coefficients of the beta function of QCD (as a
reference cf. the Particle Data Group), where $n_f$ is the number of
active flavors in two different schemes:
\begin{align}
\beta_0 &=\; 11 - \frac23 n_f \\
\beta_1 &=\; 51 - \frac{19}{3} n_f \\
\beta_2 &=\; 2857 - \frac{5033}{9} n_f + \frac{325}{27} n_f^2
\end{align}
\begin{align}
b_0 &=\; \frac{1}{12 \pi} \left( 11 C_A - 2 n_f \right) \\
b_1 &=\; \frac{1}{24 \pi^2} \left( 17 C_A^2 - 5 C_A n_f - 3 C_F n_f \right) \\
b_2 &=\; \frac{1}{(4\pi)^3} \biggl( \frac{2857}{54} C_A^3 -
\frac{1415}{54} * C_A^2 n_f - \frac{205}{18} C_A C_F n_f + C_F^2 n_f
+ \frac{79}{54} C_A n_f**2 + \frac{11}{9} C_F n_f**2 \biggr)
\end{align}
The functions [[sumQ2q]] and [[sumQ4q]] provide the sum of the squared
and quartic electric charges of a number [[nf]] of active quark flavors.
<<SM physics: public>>=
public :: beta0, beta1, beta2
public :: coeff_b0, coeff_b1, coeff_b2, coeffqed_b0, coeffqed_b1
<<SM physics: sub interfaces>>=
pure module function beta0 (nf)
real(default), intent(in) :: nf
real(default) :: beta0
end function beta0
pure module function beta1 (nf)
real(default), intent(in) :: nf
real(default) :: beta1
end function beta1
pure module function beta2 (nf)
real(default), intent(in) :: nf
real(default) :: beta2
end function beta2
pure module function coeff_b0 (nf)
real(default), intent(in) :: nf
real(default) :: coeff_b0
end function coeff_b0
pure module function coeff_b1 (nf)
real(default), intent(in) :: nf
real(default) :: coeff_b1
end function coeff_b1
pure module function coeff_b2 (nf)
real(default), intent(in) :: nf
real(default) :: coeff_b2
end function coeff_b2
pure module function coeffqed_b0 (nf, nlep)
integer, intent(in) :: nf, nlep
real(default) :: n_lep, coeffqed_b0
end function coeffqed_b0
pure module function coeffqed_b1 (nf, nlep)
integer, intent(in) :: nf, nlep
real(default) :: n_lep, coeffqed_b1
end function coeffqed_b1
<<SM physics: procedures>>=
pure module function beta0 (nf)
real(default), intent(in) :: nf
real(default) :: beta0
beta0 = 11.0_default - two/three * nf
end function beta0
pure module function beta1 (nf)
real(default), intent(in) :: nf
real(default) :: beta1
beta1 = 51.0_default - 19.0_default/three * nf
end function beta1
pure module function beta2 (nf)
real(default), intent(in) :: nf
real(default) :: beta2
beta2 = 2857.0_default - 5033.0_default / 9.0_default * &
nf + 325.0_default/27.0_default * nf**2
end function beta2
pure module function coeff_b0 (nf)
real(default), intent(in) :: nf
real(default) :: coeff_b0
coeff_b0 = (11.0_default * CA - two * nf) / (12.0_default * pi)
end function coeff_b0
pure module function coeff_b1 (nf)
real(default), intent(in) :: nf
real(default) :: coeff_b1
coeff_b1 = (17.0_default * CA**2 - five * CA * nf - three * CF * nf) / &
(24.0_default * pi**2)
end function coeff_b1
pure module function coeff_b2 (nf)
real(default), intent(in) :: nf
real(default) :: coeff_b2
coeff_b2 = (2857.0_default/54.0_default * CA**3 - &
1415.0_default/54.0_default * &
CA**2 * nf - 205.0_default/18.0_default * CA*CF*nf &
+ 79.0_default/54.0_default * CA*nf**2 + &
11.0_default/9.0_default * CF * nf**2) / (four*pi)**3
end function coeff_b2
pure module function coeffqed_b0 (nf, nlep)
integer, intent(in) :: nf, nlep
real(default) :: n_lep, coeffqed_b0
n_lep = real(nlep, kind=default)
coeffqed_b0 = - (three * sumQ2q (nf) + n_lep) / (three*pi)
end function coeffqed_b0
pure module function coeffqed_b1 (nf, nlep)
integer, intent(in) :: nf, nlep
real(default) :: n_lep, coeffqed_b1
n_lep = real(nlep, kind=default)
coeffqed_b1 = - (three * sumQ4q (nf) + n_lep) / (four*pi**2)
end function coeffqed_b1
pure function sumQ2q (nf)
integer, intent(in) :: nf
real(default) :: sumQ2q
select case (nf)
case (0)
sumQ2q = zero
case (1)
sumQ2q = 1.0_default/9.0_default
case (2)
sumQ2q = 5.0_default/9.0_default
case (3)
sumQ2q = 2.0_default/3.0_default
case (4)
sumQ2q = 10.0_default/9.0_default
case (5)
sumQ2q = 11.0_default/9.0_default
case (6:)
sumQ2q = 5.0_default/3.0_default
end select
end function sumQ2q
pure function sumQ4q (nf)
integer, intent(in) :: nf
real(default) :: sumQ4q
select case (nf)
case (0)
sumQ4q = zero
case (1)
sumQ4q = 1.0_default/81.0_default
case (2)
sumQ4q = 17.0_default/81.0_default
case (3)
sumQ4q = 2.0_default/9.0_default
case (4)
sumQ4q = 34.0_default/81.0_default
case (5)
sumQ4q = 35.0_default/81.0_default
case (6:)
sumQ4q = 17.0_default/27.0_default
end select
end function sumQ4q
@ %def beta0 beta1 beta2
@ %def coeff_b0 coeff_b1 coeff_b2 coeffqed_b0 coeffqed_b1
@ %def sumQ2q sumQ4q
@ There should be two versions of running $\alpha_s$, one which takes
the scale and $\Lambda_{\text{QCD}}$ as input, and one which takes the
scale and e.g. $\alpha_s(m_Z)$ as input. Here, we take the one which
takes the QCD scale and scale as inputs from the PDG book.
<<SM physics: public>>=
public :: running_as, running_as_lam, running_alpha, running_alpha_num
<<SM physics: sub interfaces>>=
pure module function running_as (scale, al_mz, mz, order, nf) result (ascale)
real(default), intent(in) :: scale
real(default), intent(in), optional :: al_mz, nf, mz
integer, intent(in), optional :: order
real(default) :: ascale
end function running_as
pure module function running_as_lam (nf, scale, lambda, order) result (ascale)
real(default), intent(in) :: nf, scale
real(default), intent(in), optional :: lambda
integer, intent(in), optional :: order
real(default) :: ascale
end function running_as_lam
pure module function running_alpha &
(scale, al_me, me, order, nf, nlep) result (ascale)
real(default), intent(in) :: scale
real(default), intent(in), optional :: al_me, me
integer, intent(in), optional :: order, nf, nlep
real(default) :: ascale
end function running_alpha
pure module function running_alpha_num &
(scale, al_me, me, order, nf, nlep) result (ascale)
real(default), intent(in) :: scale
real(default), intent(in), optional :: al_me, me
integer, intent(in), optional :: order, nf, nlep
real(default) :: ascale
end function running_alpha_num
<<SM physics: procedures>>=
pure module function running_as (scale, al_mz, mz, order, nf) result (ascale)
real(default), intent(in) :: scale
real(default), intent(in), optional :: al_mz, nf, mz
integer, intent(in), optional :: order
integer :: ord
real(default) :: az, m_z, as_log, n_f, b0, b1, b2, ascale
real(default) :: as0, as1
if (present (mz)) then
m_z = mz
else
m_z = MZ_REF
end if
if (present (order)) then
ord = order
else
ord = 0
end if
if (present (al_mz)) then
az = al_mz
else
az = ALPHA_QCD_MZ_REF
end if
if (present (nf)) then
n_f = nf
else
n_f = 5
end if
b0 = coeff_b0 (n_f)
b1 = coeff_b1 (n_f)
b2 = coeff_b2 (n_f)
as_log = one + b0 * az * log(scale**2/m_z**2)
as0 = az / as_log
as1 = as0 - as0**2 * b1/b0 * log(as_log)
select case (ord)
case (0)
ascale = as0
case (1)
ascale = as1
case (2)
ascale = as1 + as0**3 * (b1**2/b0**2 * ((log(as_log))**2 - &
log(as_log) + as_log - one) - b2/b0 * (as_log - one))
case default
ascale = as0
end select
end function running_as
pure module function running_as_lam (nf, scale, lambda, order) result (ascale)
real(default), intent(in) :: nf, scale
real(default), intent(in), optional :: lambda
integer, intent(in), optional :: order
real(default) :: lambda_qcd
real(default) :: as0, as1, logmul, b0, b1, b2, ascale
integer :: ord
if (present (lambda)) then
lambda_qcd = lambda
else
lambda_qcd = LAMBDA_QCD_REF
end if
if (present (order)) then
ord = order
else
ord = 0
end if
b0 = beta0(nf)
logmul = log(scale**2/lambda_qcd**2)
as0 = four*pi / b0 / logmul
if (ord > 0) then
b1 = beta1(nf)
as1 = as0 * (one - two* b1 / b0**2 * log(logmul) / logmul)
end if
select case (ord)
case (0)
ascale = as0
case (1)
ascale = as1
case (2)
b2 = beta2(nf)
ascale = as1 + as0 * four * b1**2/b0**4/logmul**2 * &
((log(logmul) - 0.5_default)**2 + &
b2*b0/8.0_default/b1**2 - five/four)
case default
ascale = as0
end select
end function running_as_lam
pure module function running_alpha &
(scale, al_me, me, order, nf, nlep) result (ascale)
real(default), intent(in) :: scale
real(default), intent(in), optional :: al_me, me
integer, intent(in), optional :: order, nf, nlep
integer :: ord, n_f, n_lep
real(default) :: ae, m_e, a_log, b0, b1, ascale
real(default) :: a0, a1
if (present (me)) then
m_e = me
else
m_e = ME_REF
end if
if (present (order)) then
ord = order
else
ord = 0
end if
if (present (al_me)) then
ae = al_me
else
ae = ALPHA_QED_ME_REF
end if
if (present (nf)) then
n_f = nf
else
n_f = 5
end if
if (present (nlep)) then
n_lep = nlep
else
n_lep = 1
end if
b0 = coeffqed_b0 (n_f, n_lep)
b1 = coeffqed_b1 (n_f, n_lep)
a_log = one + b0 * ae * log(scale**2/m_e**2)
a0 = ae / a_log
a1 = ae / (a_log + ae * b1/b0 * &
log((a_log + ae * b1/b0)/(one + ae * b1/b0)))
select case (ord)
case (0)
ascale = a0
case (1)
ascale = a1
case default
ascale = a0
end select
end function running_alpha
pure module function running_alpha_num &
(scale, al_me, me, order, nf, nlep) result (ascale)
real(default), intent(in) :: scale
real(default), intent(in), optional :: al_me, me
integer, intent(in), optional :: order, nf, nlep
integer, parameter :: n_steps = 20
integer :: ord, n_f, n_lep, k1
real(default), parameter :: sxth = 1._default/6._default
real(default) :: ae, ascale, m_e, log_q, dlr, &
b0, b1, xk0, xk1, xk2, xk3
if (present (order)) then
ord = order
else
ord = 0
end if
if (present (al_me)) then
ae = al_me
else
ae = ALPHA_QED_ME_REF
end if
if (present (me)) then
m_e = me
else
m_e = ME_REF
end if
if (present (nf)) then
n_f = nf
else
n_f = 5
end if
if (present (nlep)) then
n_lep = nlep
else
n_lep = 1
end if
ascale = ae
log_q = log (scale**2/m_e**2)
dlr = log_q / n_steps
b0 = coeffqed_b0 (n_f, n_lep)
b1 = coeffqed_b1 (n_f, n_lep)
! ..Solution of the evolution equation depending on ORD
! (fourth-order Runge-Kutta beyond the leading order)
select case (ord)
case (0)
ascale = ae / (one + b0 * ae * log_q)
case (1:)
do k1 = 1, n_steps
xk0 = dlr * beta_qed (ascale)
xk1 = dlr * beta_qed (ascale + 0.5 * xk0)
xk2 = dlr * beta_qed (ascale + 0.5 * xk1)
xk3 = dlr * beta_qed (ascale + xk2)
ascale = ascale + sxth * (xk0 + 2._default * xk1 + &
2._default * xk2 + xk3)
end do
end select
contains
pure function beta_qed (alpha)
real(default), intent(in) :: alpha
real(default) :: beta_qed
beta_qed = - alpha**2 * (b0 + alpha * b1)
end function beta_qed
end function running_alpha_num
@ %def running_as
@ %def running_as_lam
@ %def running_alpha running_alpha_num
@ This routine determines the Landau pole $\Lambda^{(n_f)}_{\overline{MS}}$
for given $\alpha_s$, scale, number of flavors and order.
<<SM physics: public>>=
public :: lambda_qcd
<<SM physics: sub interfaces>>=
module function lambda_qcd (as_q, q, nf, order) result (lambda)
real(default), intent(in) :: as_q, q
integer, intent(in) :: order, nf
real(default) :: lambda
end function lambda_qcd
<<SM physics: procedures>>=
module function lambda_qcd (as_q, q, nf, order) result (lambda)
real(default), intent(in) :: as_q, q
integer, intent(in) :: nf, order
real(default) :: lambda
real(default), parameter :: acc = 1e-8_default
if (order == 0) then
lambda = lambda_qcd_lo (as_q, q, nf)
else if (order == 1) then
lambda = lambda_qcd_nlo (as_q, q, nf)
else if (order == 2) then
lambda = lambda_qcd_nnlo (as_q, q, nf)
else
call msg_error ("lambda_qcd: order unknown")
end if
contains
function lambda_qcd_lo (as_q, q, nf) result (lambda)
real(default) :: lambda
real(default), intent(in) :: as_q, q
integer, intent(in) :: nf
real(default) :: b0, t0, t1, as0, as1
b0 = coeff_b0(real(nf, default))
t1 = one/b0/as_q
FIND_ROOT: do
if (signal_is_pending ()) return
t0 = t1
as0 = one/b0/t1
as1 = - one/b0/t1**2
t1 = (as_q-as0)/as1 + t1
if (abs(t0-t1)/t0 < acc) exit FIND_ROOT
end do FIND_ROOT
lambda = q * exp(-t1/two)
end function lambda_qcd_lo
function lambda_qcd_nlo (as_q, q, nf) result (lambda)
real(default) :: lambda
real(default), intent(in) :: as_q, q
integer, intent(in) :: nf
real(default) :: b0, b1, t0, t1, as0, as1, logt
b0 = coeff_b0(real(nf, default))
b1 = coeff_b1(real(nf, default))
t1 = one/b0/as_q
FIND_ROOT: do
if (signal_is_pending ()) return
logt = log(t1)
t0 = t1
as0 = one/b0/t1 - b1/b0 * logt/(b0 * t1)**2
as1 = - one/b0/t1**2 - b1/b0**3 * (one - two*logt)/t1**3
t1 = (as_q-as0)/as1 + t1
if (abs(t0-t1)/t0 < acc) exit FIND_ROOT
end do FIND_ROOT
lambda = q * exp(-t1/two)
end function lambda_qcd_nlo
function lambda_qcd_nnlo (as_q, q, nf) result (lambda)
real(default) :: lambda
real(default), intent(in) :: as_q, q
integer, intent(in) :: nf
real(default) :: b0, b1, b2, t0, t1, as0, as1, logt
b0 = coeff_b0(real(nf, default))
b1 = coeff_b1(real(nf, default))
b2 = coeff_b2(real(nf, default))
t1 = one/b0/as_q
FIND_ROOT: do
if (signal_is_pending ()) return
logt = log(t1)
t0 = t1
as0 = one/b0/t1 * (one - b1/b0**2 * logt/t1 + (b1/b0**2 * logt/t1)**2 &
- (b1**2 * (logt + one) - b0*b2)/b0**4/t1**2)
as1 = one/b0/t1 * (-two*b1**2 * logt**2/(b0**4 * t1**3) &
+ two*(b1**2 * (logt + one) - b0*b2)/(b0**4 * t1**3) &
+ b1 * logt/(b0**2 * t1**2) + two*b1**2 * logt/(b0**4 * t1**3) &
- b1/(b0**2 * t1**2) - b1**2/(b0**4 * t1**3)) &
- (b1**2 * logt**2/(b0**4 * t1**2) - (b1**2 * (logt + one) &
- b0*b2)/(b0**4 * t1**2) - b1 * logt/(b0**2 * t1) + one)/(b0 * t1**2)
t1 = (as_q-as0)/as1 + t1
if (abs(t0-t1)/t0 < acc) exit FIND_ROOT
end do FIND_ROOT
lambda = q * exp(-t1/two)
end function lambda_qcd_nnlo
end function lambda_qcd
@ %def lambda_qcd
@
\subsection{Catani-Seymour Parameters}
These are fundamental constants of the Catani-Seymour dipole formalism.
Since the corresponding parameters for the gluon case depend on the
number of flavors which is treated as an argument, there we do have
functions and not parameters.
\begin{equation}
\gamma_q = \gamma_{\bar q} = \frac{3}{2} C_F \qquad \gamma_g =
\frac{11}{6} C_A - \frac{2}{3} T_R N_f
\end{equation}
\begin{equation}
K_q = K_{\bar q} = \left( \frac{7}{2} - \frac{\pi^2}{6} \right) C_F \qquad
K_g = \left( \frac{67}{18} - \frac{\pi^2}{6} \right) C_A -
\frac{10}{9} T_R N_f
\end{equation}
<<SM physics: parameters>>=
real(default), parameter, public :: gamma_q = three/two * CF, &
k_q = (7.0_default/two - pi**2/6.0_default) * CF
@ %def gamma_q
@
<<SM physics: public>>=
public :: gamma_g, k_g
<<SM physics: sub interfaces>>=
elemental module function gamma_g (nf) result (gg)
real(default), intent(in) :: nf
real(default) :: gg
end function gamma_g
elemental module function k_g (nf) result (kg)
real(default), intent(in) :: nf
real(default) :: kg
end function k_g
<<SM physics: procedures>>=
elemental module function gamma_g (nf) result (gg)
real(default), intent(in) :: nf
real(default) :: gg
gg = 11.0_default/6.0_default * CA - two/three * TR * nf
end function gamma_g
elemental module function k_g (nf) result (kg)
real(default), intent(in) :: nf
real(default) :: kg
kg = (67.0_default/18.0_default - pi**2/6.0_default) * CA - &
10.0_default/9.0_default * TR * nf
end function k_g
@ %def gamma_g
@ %def k_g
@
\subsection{Mathematical Functions}
The dilogarithm. This simplified version is bound to double
precision, and restricted to argument values less or equal to unity,
so we do not need complex algebra. The wrapper converts it to default
precision (which is, of course, a no-op if double=default).
The routine calculates the dilogarithm through mapping on the area
where there is a quickly convergent series (adapted from an F77
routine by Hans Kuijf, 1988): Map $x$ such that $x$ is not in the
neighbourhood of $1$. Note that $|z|=-\ln(1-x)$ is always smaller
than $1.10$, but $\frac{1.10^{19}}{19!}{\rm Bernoulli}_{19}=2.7\times
10^{-15}$.
<<SM physics: public>>=
public :: Li2
<<SM physics: sub interfaces>>=
elemental module function Li2 (x)
real(default), intent(in) :: x
real(default) :: Li2
end function Li2
<<SM physics: procedures>>=
elemental module function Li2 (x)
real(default), intent(in) :: x
real(default) :: Li2
Li2 = real( Li2_double (real(x, kind=double)), kind=default)
end function Li2
@ %def: Li2
@
<<SM physics: procedures>>=
elemental function Li2_double (x) result (Li2)
real(double), intent(in) :: x
real(double) :: Li2
real(double), parameter :: pi2_6 = pi**2/6
if (abs(1-x) < tiny_07) then
Li2 = pi2_6
else if (abs(1-x) < 0.5_double) then
Li2 = pi2_6 - log(1-x) * log(x) - Li2_restricted (1-x)
else if (abs(x) > 1.d0) then
! Li2 = 0
! call msg_bug (" Dilogarithm called outside of defined range.")
!!! Reactivate Dilogarithm identity
Li2 = -pi2_6 - 0.5_default * log(-x) * log(-x) - Li2_restricted (1/x)
else
Li2 = Li2_restricted (x)
end if
contains
elemental function Li2_restricted (x) result (Li2)
real(double), intent(in) :: x
real(double) :: Li2
real(double) :: tmp, z, z2
z = - log (1-x)
z2 = z**2
! Horner's rule for the powers z^3 through z^19
tmp = 43867._double/798._double
tmp = tmp * z2 /342._double - 3617._double/510._double
tmp = tmp * z2 /272._double + 7._double/6._double
tmp = tmp * z2 /210._double - 691._double/2730._double
tmp = tmp * z2 /156._double + 5._double/66._double
tmp = tmp * z2 /110._double - 1._double/30._double
tmp = tmp * z2 / 72._double + 1._double/42._double
tmp = tmp * z2 / 42._double - 1._double/30._double
tmp = tmp * z2 / 20._double + 1._double/6._double
! The first three terms of the power series
Li2 = z2 * z * tmp / 6._double - 0.25_double * z2 + z
end function Li2_restricted
end function Li2_double
@ %def Li2_double
@ Complex digamma function. For this we use the asymptotic formula in
Abramoqicz/Stegun, Eq. (6.3.18), and the recurrence formula
Eq. (6.3.6):
\begin{equation}
\psi^{(0})(z) := \psi(z) = \frac{\Gamma'(z)}{\Gamma(z)}
\end{equation}
<<SM physics: public>>=
public :: psic
public :: psir
<<SM physics: sub interfaces>>=
elemental module function psic (z) result (psi)
complex(default), intent(in) :: z
complex(default) :: psi
end function psic
elemental module function psir (x) result (psi)
real(default), intent(in) :: x
real(default) :: psi
end function psir
<<SM physics: procedures>>=
elemental module function psic (z) result (psi)
complex(default), intent(in) :: z
complex(default) :: psi
complex(default) :: shift, zz, zi, zi2
shift = 0
zz = z
if (abs (aimag(zz)) < 10._default) then
do while (abs (zz) < 10._default)
shift = shift - 1 / zz
zz = zz + 1
end do
end if
zi = 1/zz
zi2 = zi*zi
psi = shift + log(zz) - zi/2 - zi2 / 5040._default * ( 420._default + &
zi2 * ( -42._default + zi2 * (20._default - 21._default * zi2)))
end function psic
elemental module function psir (x) result (psi)
real(default), intent(in) :: x
real(default) :: psi
psi = real (psic (cmplx (x,0,kind=default)), kind=default)
end function psir
@ %def psic psir
@ Complex polygamma function. For this we use the asymptotic formula in
Abramoqicz/Stegun, Eq. (6.4.11), and the recurrence formula
Eq. (6.4.11):
\begin{equation}
\psi^{(m})(z) := \frac{d^m}{dz^m} \psi(z) = \frac{d^{m+1}}{dz^{m+1}}
\ln \Gamma(z)
\end{equation}
<<SM physics: public>>=
public :: psim
public :: psimr
<<SM physics: sub interfaces>>=
elemental module function psim (z, m) result (psi)
complex(default), intent(in) :: z
integer, intent(in) :: m
complex(default) :: psi
end function psim
elemental module function psimr (x, m) result (psi)
real(default), intent(in) :: x
integer, intent(in) :: m
real(default) :: psi
end function psimr
<<SM physics: procedures>>=
elemental module function psim (z, m) result (psi)
complex(default), intent(in) :: z
integer, intent(in) :: m
complex(default) :: psi
complex(default) :: shift, rec, zz, zi, zi2
real(default) :: c1, c2, c3, c4, c5, c6, c7
integer :: i
if (m < 1) then
psi = psic(z)
else
shift = 0
zz = z
if (abs (aimag (zz)) < 10._default) then
CHECK_ABS: do
rec = (-1)**m * factorial (m) / zz**(m+1)
shift = shift - rec
zz = zz + 1
if (abs (zz) > 10._default) exit CHECK_ABS
end do CHECK_ABS
end if
c1 = 1._default
c2 = 1._default / 2._default
c3 = 1._default / 6._default
c4 = - 1._default / 30._default
c5 = 1._default / 42._default
c6 = - 1._default / 30._default
c7 = 5._default / 66._default
do i = 2, m
c1 = c1 * (i-1)
c2 = c2 * i
c3 = c3 * (i+1)
c4 = c4 * (i+3)
c5 = c5 * (i+5)
c6 = c6 * (i+7)
c7 = c7 * (i+9)
end do
zi = 1/zz
zi2 = zi*zi
psi = shift + (-1)**(m-1) * zi**m * ( c1 + zi * ( c2 + zi * ( &
c3 + zi2 * ( c4 + zi2 * ( c5 + zi2 * ( c6 + ( c7 * zi2)))))))
end if
end function psim
elemental module function psimr (x, m) result (psi)
real(default), intent(in) :: x
integer, intent(in) :: m
real(default) :: psi
psi = real (psim (cmplx (x,0,kind=default), m), kind=default)
end function psimr
@ %def psim psimr
@ Nielsen's generalized polylogarithms,
\begin{equation*}
S_{n,m}(x) = \frac{(-1)^{n+m-1}}{(n-1)!m!} \int_0^1 t^{-1}
\; \ln^{n-1} t \; \ln^m (1-xt) \; dt \; ,
\end{equation*}
adapted from the CERNLIB function [[wgplg]] for real arguments [[x]]
and integer $n$ and $m$ satisfying $1 \leq n \leq 4$, $1 \leq m \leq 4$,
$n+m \leq 5$, i.e. one of the functions $S_{1,1}$, $S_{1,2}$,
$S_{2,1}$, $S_{1,3}$, $S_{2,2}$, $S_{3,1}$, $S_{1,4}$, $S_{2,3}$,
$S_{3,2}$, $S_{4,1}$. If $x\leq1$, $S_{n,m}(x)$ is real, and the
imaginary part is set to zero.
<<SM physics: public>>=
public :: cnielsen
public :: nielsen
<<SM physics: sub interfaces>>=
module function cnielsen (n, m, x) result (nplog)
integer, intent(in) :: n, m
real(default), intent(in) :: x
complex(default) :: nplog
end function cnielsen
module function nielsen (n, m, x) result (nplog)
integer, intent(in) :: n, m
real(default), intent(in) :: x
real(default) :: nplog
end function nielsen
<<SM physics: procedures>>=
module function cnielsen (n, m, x) result (nplog)
integer, intent(in) :: n, m
real(default), intent(in) :: x
complex(default) :: nplog
real(default), parameter :: c1 = 4._default/3._default, &
c2 = 1._default/3._default
real(default), dimension(0:4), parameter :: &
fct = [1.0_default,1.0_default,2.0_default,6.0_default,24.0_default]
real(default), dimension(4,4) :: s1, cc
real(default), dimension(0:30,10) :: aa
complex(default), dimension(0:5) :: vv
real(default), dimension(0:5) :: uu
real(default) :: x1, h, alfa, b0, b1, b2, qq, rr
complex(default) :: sj, sk
integer, dimension(10), parameter :: &
nc = [24,26,28,30,22,24,26,19,22,17]
integer, dimension(31), parameter :: &
index = [1,2,3,4,0,0,0,0,0,0,5,6,7,0,0,0,0,0,0,0, &
8,9,0,0,0,0,0,0,0,0,10]
real(default), dimension(0:4), parameter :: &
sgn = [1._default, -1._default, 1._default, -1._default, 1._default]
integer :: it, j, k, l, m1, n1
if ((n<1) .or. (n>4) .or. (m<1) .or. (m>4) .or. (n+m > 5)) then
call msg_fatal &
("The Nielsen dilogarithms cannot be applied for these values.")
end if
s1 = 0._default
s1(1,1) = 1.6449340668482_default
s1(1,2) = 1.2020569031596_default
s1(1,3) = 1.0823232337111_default
s1(1,4) = 1.0369277551434_default
s1(2,1) = 1.2020569031596_default
s1(2,2) = 2.7058080842778e-1_default
s1(2,3) = 9.6551159989444e-2_default
s1(3,1) = 1.0823232337111_default
s1(3,2) = 9.6551159989444e-2_default
s1(4,1) = 1.0369277551434_default
cc = 0._default
cc(1,1) = 1.6449340668482_default
cc(1,2) = 1.2020569031596_default
cc(1,3) = 1.0823232337111_default
cc(1,4) = 1.0369277551434_default
cc(2,2) =-1.8940656589945_default
cc(2,3) =-3.0142321054407_default
cc(3,1) = 1.8940656589945_default
cc(3,2) = 3.0142321054407_default
aa = 0._default
aa( 0,1) = 0.96753215043498_default
aa( 1,1) = 0.16607303292785_default
aa( 2,1) = 0.02487932292423_default
aa( 3,1) = 0.00468636195945_default
aa( 4,1) = 0.00100162749616_default
aa( 5,1) = 0.00023200219609_default
aa( 6,1) = 0.00005681782272_default
aa( 7,1) = 0.00001449630056_default
aa( 8,1) = 0.00000381632946_default
aa( 9,1) = 0.00000102990426_default
aa(10,1) = 0.00000028357538_default
aa(11,1) = 0.00000007938705_default
aa(12,1) = 0.00000002253670_default
aa(13,1) = 0.00000000647434_default
aa(14,1) = 0.00000000187912_default
aa(15,1) = 0.00000000055029_default
aa(16,1) = 0.00000000016242_default
aa(17,1) = 0.00000000004827_default
aa(18,1) = 0.00000000001444_default
aa(19,1) = 0.00000000000434_default
aa(20,1) = 0.00000000000131_default
aa(21,1) = 0.00000000000040_default
aa(22,1) = 0.00000000000012_default
aa(23,1) = 0.00000000000004_default
aa(24,1) = 0.00000000000001_default
aa( 0,2) = 0.95180889127832_default
aa( 1,2) = 0.43131131846532_default
aa( 2,2) = 0.10002250714905_default
aa( 3,2) = 0.02442415595220_default
aa( 4,2) = 0.00622512463724_default
aa( 5,2) = 0.00164078831235_default
aa( 6,2) = 0.00044407920265_default
aa( 7,2) = 0.00012277494168_default
aa( 8,2) = 0.00003453981284_default
aa( 9,2) = 0.00000985869565_default
aa(10,2) = 0.00000284856995_default
aa(11,2) = 0.00000083170847_default
aa(12,2) = 0.00000024503950_default
aa(13,2) = 0.00000007276496_default
aa(14,2) = 0.00000002175802_default
aa(15,2) = 0.00000000654616_default
aa(16,2) = 0.00000000198033_default
aa(17,2) = 0.00000000060204_default
aa(18,2) = 0.00000000018385_default
aa(19,2) = 0.00000000005637_default
aa(20,2) = 0.00000000001735_default
aa(21,2) = 0.00000000000536_default
aa(22,2) = 0.00000000000166_default
aa(23,2) = 0.00000000000052_default
aa(24,2) = 0.00000000000016_default
aa(25,2) = 0.00000000000005_default
aa(26,2) = 0.00000000000002_default
aa( 0,3) = 0.98161027991365_default
aa( 1,3) = 0.72926806320726_default
aa( 2,3) = 0.22774714909321_default
aa( 3,3) = 0.06809083296197_default
aa( 4,3) = 0.02013701183064_default
aa( 5,3) = 0.00595478480197_default
aa( 6,3) = 0.00176769013959_default
aa( 7,3) = 0.00052748218502_default
aa( 8,3) = 0.00015827461460_default
aa( 9,3) = 0.00004774922076_default
aa(10,3) = 0.00001447920408_default
aa(11,3) = 0.00000441154886_default
aa(12,3) = 0.00000135003870_default
aa(13,3) = 0.00000041481779_default
aa(14,3) = 0.00000012793307_default
aa(15,3) = 0.00000003959070_default
aa(16,3) = 0.00000001229055_default
aa(17,3) = 0.00000000382658_default
aa(18,3) = 0.00000000119459_default
aa(19,3) = 0.00000000037386_default
aa(20,3) = 0.00000000011727_default
aa(21,3) = 0.00000000003687_default
aa(22,3) = 0.00000000001161_default
aa(23,3) = 0.00000000000366_default
aa(24,3) = 0.00000000000116_default
aa(25,3) = 0.00000000000037_default
aa(26,3) = 0.00000000000012_default
aa(27,3) = 0.00000000000004_default
aa(28,3) = 0.00000000000001_default
aa( 0,4) = 1.0640521184614_default
aa( 1,4) = 1.0691720744981_default
aa( 2,4) = 0.41527193251768_default
aa( 3,4) = 0.14610332936222_default
aa( 4,4) = 0.04904732648784_default
aa( 5,4) = 0.01606340860396_default
aa( 6,4) = 0.00518889350790_default
aa( 7,4) = 0.00166298717324_default
aa( 8,4) = 0.00053058279969_default
aa( 9,4) = 0.00016887029251_default
aa(10,4) = 0.00005368328059_default
aa(11,4) = 0.00001705923313_default
aa(12,4) = 0.00000542174374_default
aa(13,4) = 0.00000172394082_default
aa(14,4) = 0.00000054853275_default
aa(15,4) = 0.00000017467795_default
aa(16,4) = 0.00000005567550_default
aa(17,4) = 0.00000001776234_default
aa(18,4) = 0.00000000567224_default
aa(19,4) = 0.00000000181313_default
aa(20,4) = 0.00000000058012_default
aa(21,4) = 0.00000000018579_default
aa(22,4) = 0.00000000005955_default
aa(23,4) = 0.00000000001911_default
aa(24,4) = 0.00000000000614_default
aa(25,4) = 0.00000000000197_default
aa(26,4) = 0.00000000000063_default
aa(27,4) = 0.00000000000020_default
aa(28,4) = 0.00000000000007_default
aa(29,4) = 0.00000000000002_default
aa(30,4) = 0.00000000000001_default
aa( 0,5) = 0.97920860669175_default
aa( 1,5) = 0.08518813148683_default
aa( 2,5) = 0.00855985222013_default
aa( 3,5) = 0.00121177214413_default
aa( 4,5) = 0.00020722768531_default
aa( 5,5) = 0.00003996958691_default
aa( 6,5) = 0.00000838064065_default
aa( 7,5) = 0.00000186848945_default
aa( 8,5) = 0.00000043666087_default
aa( 9,5) = 0.00000010591733_default
aa(10,5) = 0.00000002647892_default
aa(11,5) = 0.00000000678700_default
aa(12,5) = 0.00000000177654_default
aa(13,5) = 0.00000000047342_default
aa(14,5) = 0.00000000012812_default
aa(15,5) = 0.00000000003514_default
aa(16,5) = 0.00000000000975_default
aa(17,5) = 0.00000000000274_default
aa(18,5) = 0.00000000000077_default
aa(19,5) = 0.00000000000022_default
aa(20,5) = 0.00000000000006_default
aa(21,5) = 0.00000000000002_default
aa(22,5) = 0.00000000000001_default
aa( 0,6) = 0.95021851963952_default
aa( 1,6) = 0.29052529161433_default
aa( 2,6) = 0.05081774061716_default
aa( 3,6) = 0.00995543767280_default
aa( 4,6) = 0.00211733895031_default
aa( 5,6) = 0.00047859470550_default
aa( 6,6) = 0.00011334321308_default
aa( 7,6) = 0.00002784733104_default
aa( 8,6) = 0.00000704788108_default
aa( 9,6) = 0.00000182788740_default
aa(10,6) = 0.00000048387492_default
aa(11,6) = 0.00000013033842_default
aa(12,6) = 0.00000003563769_default
aa(13,6) = 0.00000000987174_default
aa(14,6) = 0.00000000276586_default
aa(15,6) = 0.00000000078279_default
aa(16,6) = 0.00000000022354_default
aa(17,6) = 0.00000000006435_default
aa(18,6) = 0.00000000001866_default
aa(19,6) = 0.00000000000545_default
aa(20,6) = 0.00000000000160_default
aa(21,6) = 0.00000000000047_default
aa(22,6) = 0.00000000000014_default
aa(23,6) = 0.00000000000004_default
aa(24,6) = 0.00000000000001_default
aa( 0,7) = 0.95064032186777_default
aa( 1,7) = 0.54138285465171_default
aa( 2,7) = 0.13649979590321_default
aa( 3,7) = 0.03417942328207_default
aa( 4,7) = 0.00869027883583_default
aa( 5,7) = 0.00225284084155_default
aa( 6,7) = 0.00059516089806_default
aa( 7,7) = 0.00015995617766_default
aa( 8,7) = 0.00004365213096_default
aa( 9,7) = 0.00001207474688_default
aa(10,7) = 0.00000338018176_default
aa(11,7) = 0.00000095632476_default
aa(12,7) = 0.00000027313129_default
aa(13,7) = 0.00000007866968_default
aa(14,7) = 0.00000002283195_default
aa(15,7) = 0.00000000667205_default
aa(16,7) = 0.00000000196191_default
aa(17,7) = 0.00000000058018_default
aa(18,7) = 0.00000000017246_default
aa(19,7) = 0.00000000005151_default
aa(20,7) = 0.00000000001545_default
aa(21,7) = 0.00000000000465_default
aa(22,7) = 0.00000000000141_default
aa(23,7) = 0.00000000000043_default
aa(24,7) = 0.00000000000013_default
aa(25,7) = 0.00000000000004_default
aa(26,7) = 0.00000000000001_default
aa( 0,8) = 0.98800011672229_default
aa( 1,8) = 0.04364067609601_default
aa( 2,8) = 0.00295091178278_default
aa( 3,8) = 0.00031477809720_default
aa( 4,8) = 0.00004314846029_default
aa( 5,8) = 0.00000693818230_default
aa( 6,8) = 0.00000124640350_default
aa( 7,8) = 0.00000024293628_default
aa( 8,8) = 0.00000005040827_default
aa( 9,8) = 0.00000001099075_default
aa(10,8) = 0.00000000249467_default
aa(11,8) = 0.00000000058540_default
aa(12,8) = 0.00000000014127_default
aa(13,8) = 0.00000000003492_default
aa(14,8) = 0.00000000000881_default
aa(15,8) = 0.00000000000226_default
aa(16,8) = 0.00000000000059_default
aa(17,8) = 0.00000000000016_default
aa(18,8) = 0.00000000000004_default
aa(19,8) = 0.00000000000001_default
aa( 0,9) = 0.95768506546350_default
aa( 1,9) = 0.19725249679534_default
aa( 2,9) = 0.02603370313918_default
aa( 3,9) = 0.00409382168261_default
aa( 4,9) = 0.00072681707110_default
aa( 5,9) = 0.00014091879261_default
aa( 6,9) = 0.00002920458914_default
aa( 7,9) = 0.00000637631144_default
aa( 8,9) = 0.00000145167850_default
aa( 9,9) = 0.00000034205281_default
aa(10,9) = 0.00000008294302_default
aa(11,9) = 0.00000002060784_default
aa(12,9) = 0.00000000522823_default
aa(13,9) = 0.00000000135066_default
aa(14,9) = 0.00000000035451_default
aa(15,9) = 0.00000000009436_default
aa(16,9) = 0.00000000002543_default
aa(17,9) = 0.00000000000693_default
aa(18,9) = 0.00000000000191_default
aa(19,9) = 0.00000000000053_default
aa(20,9) = 0.00000000000015_default
aa(21,9) = 0.00000000000004_default
aa(22,9) = 0.00000000000001_default
aa( 0,10) = 0.99343651671347_default
aa( 1,10) = 0.02225770126826_default
aa( 2,10) = 0.00101475574703_default
aa( 3,10) = 0.00008175156250_default
aa( 4,10) = 0.00000899973547_default
aa( 5,10) = 0.00000120823987_default
aa( 6,10) = 0.00000018616913_default
aa( 7,10) = 0.00000003174723_default
aa( 8,10) = 0.00000000585215_default
aa( 9,10) = 0.00000000114739_default
aa(10,10) = 0.00000000023652_default
aa(11,10) = 0.00000000005082_default
aa(12,10) = 0.00000000001131_default
aa(13,10) = 0.00000000000259_default
aa(14,10) = 0.00000000000061_default
aa(15,10) = 0.00000000000015_default
aa(16,10) = 0.00000000000004_default
aa(17,10) = 0.00000000000001_default
if (x == 1._default) then
nplog = s1(n,m)
else if (x > 2._default .or. x < -1.0_default) then
x1 = 1._default / x
h = c1 * x1 + c2
alfa = h + h
vv(0) = 1._default
if (x < -1.0_default) then
vv(1) = log(-x)
else if (x > 2._default) then
vv(1) = log(cmplx(-x,0._default,kind=default))
end if
do l = 2, n+m
vv(l) = vv(1) * vv(l-1)/l
end do
sk = 0._default
do k = 0, m-1
m1 = m-k
rr = x1**m1 / (fct(m1) * fct(n-1))
sj = 0._default
do j = 0, k
n1 = n+k-j
l = index(10*n1+m1-10)
b1 = 0._default
b2 = 0._default
do it = nc(l), 0, -1
b0 = aa(it,l) + alfa*b1 - b2
b2 = b1
b1 = b0
end do
qq = (fct(n1-1) / fct(k-j)) * (b0 - h*b2) * rr / m1**n1
sj = sj + vv(j) * qq
end do
sk = sk + sgn(k) * sj
end do
sj = 0._default
do j = 0, n-1
sj = sj + vv(j) * cc(n-j,m)
end do
nplog = sgn(n) * sk + sgn(m) * (sj + vv(n+m))
else if (x > 0.5_default) then
x1 = 1._default - x
h = c1 * x1 + c2
alfa = h + h
vv(0) = 1._default
uu(0) = 1._default
vv(1) = log(cmplx(x1,0._default,kind=default))
uu(1) = log(x)
do l = 2, m
vv(l) = vv(1) * vv(l-1) / l
end do
do l = 2, n
uu(l) = uu(1) * uu(l-1) / l
end do
sk = 0._default
do k = 0, n-1
m1 = n-k
rr = x1**m1 / fct(m1)
sj = 0._default
do j = 0, m-1
n1 = m-j
l = index(10*n1 + m1 - 10)
b1 = 0._default
b2 = 0._default
do it = nc(l), 0, -1
b0 = aa(it,l) + alfa*b1 - b2
b2 = b1
b1 = b0
end do
qq = sgn(j) * (b0 - h*b2) * rr / m1**n1
sj = sj + vv(j) * qq
end do
sk = sk + uu(k) * (s1(m1,m) - sj)
end do
nplog = sk + sgn(m) * uu(n) * vv(m)
else
l = index(10*n + m - 10)
h = c1 * x + c2
alfa = h + h
b1 = 0._default
b2 = 0._default
do it = nc(l), 0, -1
b0 = aa(it,l) + alfa*b1 - b2
b2 = b1
b1 = b0
end do
nplog = (b0 - h*b2) * x**m / (fct(m) * m**n)
end if
end function cnielsen
module function nielsen (n, m, x) result (nplog)
integer, intent(in) :: n, m
real(default), intent(in) :: x
real(default) :: nplog
nplog = real (cnielsen (n, m, x))
end function nielsen
@ %def cnielsen nielsen
@ $\text{Li}_{n}(x) = S_{n-1,1}(x)$.
<<SM physics: public>>=
public :: polylog
<<SM physics: sub interfaces>>=
module function polylog (n, x) result (plog)
integer, intent(in) :: n
real(default), intent(in) :: x
real(default) :: plog
end function polylog
<<SM physics: procedures>>=
module function polylog (n, x) result (plog)
integer, intent(in) :: n
real(default), intent(in) :: x
real(default) :: plog
plog = nielsen (n-1,1,x)
end function polylog
@ %def polylog
@ $\text{Li}_2(x)$.
<<SM physics: public>>=
public :: dilog
<<SM physics: sub interfaces>>=
module function dilog (x) result (dlog)
real(default), intent(in) :: x
real(default) :: dlog
end function dilog
<<SM physics: procedures>>=
module function dilog (x) result (dlog)
real(default), intent(in) :: x
real(default) :: dlog
dlog = polylog (2,x)
end function dilog
@ %def dilog
@ $\text{Li}_3(x)$.
<<SM physics: public>>=
public :: trilog
<<SM physics: sub interfaces>>=
module function trilog (x) result (tlog)
real(default), intent(in) :: x
real(default) :: tlog
end function trilog
<<SM physics: procedures>>=
module function trilog (x) result (tlog)
real(default), intent(in) :: x
real(default) :: tlog
tlog = polylog (3,x)
end function trilog
@ %def trilog
@
\subsection{Loop Integrals}
These functions appear in the calculation of the effective one-loop coupling of
a (pseudo)scalar to a vector boson pair.
<<SM physics: public>>=
public :: faux
<<SM physics: sub interfaces>>=
elemental module function faux (x) result (y)
real(default), intent(in) :: x
complex(default) :: y
end function faux
<<SM physics: procedures>>=
elemental module function faux (x) result (y)
real(default), intent(in) :: x
complex(default) :: y
if (1 <= x) then
y = asin(sqrt(1/x))**2
else
y = - 1/4.0_default * (log((1 + sqrt(1 - x))/ &
(1 - sqrt(1 - x))) - cmplx (0.0_default, pi, kind=default))**2
end if
end function faux
@ %def faux
@
<<SM physics: public>>=
public :: fonehalf
<<SM physics: sub interfaces>>=
elemental module function fonehalf (x) result (y)
real(default), intent(in) :: x
complex(default) :: y
end function fonehalf
<<SM physics: procedures>>=
elemental module function fonehalf (x) result (y)
real(default), intent(in) :: x
complex(default) :: y
if (abs(x) < eps0) then
y = 0
else
y = - 2.0_default * x * (1 + (1 - x) * faux(x))
end if
end function fonehalf
@ %def fonehalf
@
<<SM physics: public>>=
public :: fonehalf_pseudo
<<SM physics: sub interfaces>>=
module function fonehalf_pseudo (x) result (y)
real(default), intent(in) :: x
complex(default) :: y
end function fonehalf_pseudo
<<SM physics: procedures>>=
module function fonehalf_pseudo (x) result (y)
real(default), intent(in) :: x
complex(default) :: y
if (abs(x) < eps0) then
y = 0
else
y = - 2.0_default * x * faux(x)
end if
end function fonehalf_pseudo
@ %def fonehalf_pseudo
@
<<SM physics: public>>=
public :: fone
<<SM physics: sub interfaces>>=
elemental module function fone (x) result (y)
real(default), intent(in) :: x
complex(default) :: y
end function fone
<<SM physics: procedures>>=
elemental module function fone (x) result (y)
real(default), intent(in) :: x
complex(default) :: y
if (abs(x) < eps0) then
y = 2.0_default
else
y = 2.0_default + 3.0_default * x + &
3.0_default * x * (2.0_default - x) * &
faux(x)
end if
end function fone
@ %def fone
@
<<SM physics: public>>=
public :: gaux
<<SM physics: sub interfaces>>=
elemental module function gaux (x) result (y)
real(default), intent(in) :: x
complex(default) :: y
end function gaux
<<SM physics: procedures>>=
elemental module function gaux (x) result (y)
real(default), intent(in) :: x
complex(default) :: y
if (1 <= x) then
y = sqrt(x - 1) * asin(sqrt(1/x))
else
y = sqrt(1 - x) * (log((1 + sqrt(1 - x)) / &
(1 - sqrt(1 - x))) - &
cmplx (0.0_default, pi, kind=default)) / 2.0_default
end if
end function gaux
@ %def gaux
@
<<SM physics: public>>=
public :: tri_i1
<<SM physics: sub interfaces>>=
elemental module function tri_i1 (a,b) result (y)
real(default), intent(in) :: a,b
complex(default) :: y
end function tri_i1
<<SM physics: procedures>>=
elemental module function tri_i1 (a,b) result (y)
real(default), intent(in) :: a,b
complex(default) :: y
if (a < eps0 .or. b < eps0) then
y = 0
else
y = a*b/2.0_default/(a-b) + a**2 * b**2/2.0_default/(a-b)**2 * &
(faux(a) - faux(b)) + &
a**2 * b/(a-b)**2 * (gaux(a) - gaux(b))
end if
end function tri_i1
@ %def tri_i1
@
<<SM physics: public>>=
public :: tri_i2
<<SM physics: sub interfaces>>=
elemental module function tri_i2 (a,b) result (y)
real(default), intent(in) :: a,b
complex(default) :: y
end function tri_i2
<<SM physics: procedures>>=
elemental module function tri_i2 (a,b) result (y)
real(default), intent(in) :: a,b
complex(default) :: y
if (a < eps0 .or. b < eps0) then
y = 0
else
y = - a * b / 2.0_default / (a-b) * (faux(a) - faux(b))
end if
end function tri_i2
@ %def tri_i2
@
\subsection{More on $\alpha_s$}
These functions are for the running of the strong coupling constants,
$\alpha_s$.
<<SM physics: public>>=
public :: run_b0
<<SM physics: sub interfaces>>=
elemental module function run_b0 (nf) result (bnull)
integer, intent(in) :: nf
real(default) :: bnull
end function run_b0
<<SM physics: procedures>>=
elemental module function run_b0 (nf) result (bnull)
integer, intent(in) :: nf
real(default) :: bnull
bnull = 33.0_default - 2.0_default * nf
end function run_b0
@ %def run_b0
@
<<SM physics: public>>=
public :: run_b1
<<SM physics: sub interfaces>>=
elemental module function run_b1 (nf) result (bone)
integer, intent(in) :: nf
real(default) :: bone
end function run_b1
<<SM physics: procedures>>=
elemental module function run_b1 (nf) result (bone)
integer, intent(in) :: nf
real(default) :: bone
bone = 6.0_default * (153.0_default - 19.0_default * nf)/run_b0(nf)**2
end function run_b1
@ %def run_b1
@
<<SM physics: public>>=
public :: run_aa
<<SM physics: sub interfaces>>=
elemental module function run_aa (nf) result (aaa)
integer, intent(in) :: nf
real(default) :: aaa
end function run_aa
<<SM physics: procedures>>=
elemental module function run_aa (nf) result (aaa)
integer, intent(in) :: nf
real(default) :: aaa
aaa = 12.0_default * PI / run_b0(nf)
end function run_aa
@ %def run_aa
@
<<SM physics: pubic functions>>=
public :: run_bb
<<SM physics: procedures>>=
elemental function run_bb (nf) result (bbb)
integer, intent(in) :: nf
real(default) :: bbb
bbb = run_b1(nf) / run_aa(nf)
end function run_bb
@ %def run_bb
@
\subsection{Functions for Catani-Seymour dipoles}
For the automated Catani-Seymour dipole subtraction, we need the
following functions.
<<SM physics: public>>=
public :: ff_dipole
<<SM physics: sub interfaces>>=
pure module subroutine ff_dipole (v_ijk, y_ijk, p_ij, pp_k, p_i, p_j, p_k)
type(vector4_t), intent(in) :: p_i, p_j, p_k
type(vector4_t), intent(out) :: p_ij, pp_k
real(default), intent(out) :: y_ijk
real(default), intent(out) :: v_ijk
end subroutine ff_dipole
<<SM physics: procedures>>=
pure module subroutine ff_dipole (v_ijk, y_ijk, p_ij, pp_k, p_i, p_j, p_k)
type(vector4_t), intent(in) :: p_i, p_j, p_k
type(vector4_t), intent(out) :: p_ij, pp_k
real(default), intent(out) :: y_ijk
real(default) :: z_i
real(default), intent(out) :: v_ijk
z_i = (p_i*p_k) / ((p_k*p_j) + (p_k*p_i))
y_ijk = (p_i*p_j) / ((p_i*p_j) + (p_i*p_k) + (p_j*p_k))
p_ij = p_i + p_j - y_ijk/(1.0_default - y_ijk) * p_k
pp_k = (1.0/(1.0_default - y_ijk)) * p_k
!!! We don't multiply by alpha_s right here:
v_ijk = 8.0_default * PI * CF * &
(2.0 / (1.0 - z_i*(1.0 - y_ijk)) - (1.0 + z_i))
end subroutine ff_dipole
@ %def ff_dipole
@
<<SM physics: public>>=
public :: fi_dipole
<<SM physics: sub interfaces>>=
pure module subroutine fi_dipole (v_ija, x_ija, p_ij, pp_a, p_i, p_j, p_a)
type(vector4_t), intent(in) :: p_i, p_j, p_a
type(vector4_t), intent(out) :: p_ij, pp_a
real(default), intent(out) :: x_ija
real(default), intent(out) :: v_ija
end subroutine fi_dipole
<<SM physics: procedures>>=
pure module subroutine fi_dipole (v_ija, x_ija, p_ij, pp_a, p_i, p_j, p_a)
type(vector4_t), intent(in) :: p_i, p_j, p_a
type(vector4_t), intent(out) :: p_ij, pp_a
real(default), intent(out) :: x_ija
real(default) :: z_i
real(default), intent(out) :: v_ija
z_i = (p_i*p_a) / ((p_a*p_j) + (p_a*p_i))
x_ija = ((p_i*p_a) + (p_j*p_a) - (p_i*p_j)) &
/ ((p_i*p_a) + (p_j*p_a))
p_ij = p_i + p_j - (1.0_default - x_ija) * p_a
pp_a = x_ija * p_a
!!! We don't not multiply by alpha_s right here:
v_ija = 8.0_default * PI * CF * &
(2.0 / (1.0 - z_i + (1.0 - x_ija)) - (1.0 + z_i)) / x_ija
end subroutine fi_dipole
@ %def fi_dipole
@
<<SM physics: public>>=
public :: if_dipole
<<SM physics: sub interfaces>>=
pure module subroutine if_dipole (v_kja, u_j, p_aj, pp_k, p_k, p_j, p_a)
type(vector4_t), intent(in) :: p_k, p_j, p_a
type(vector4_t), intent(out) :: p_aj, pp_k
real(default), intent(out) :: u_j
real(default), intent(out) :: v_kja
end subroutine if_dipole
<<SM physics: procedures>>=
pure module subroutine if_dipole (v_kja, u_j, p_aj, pp_k, p_k, p_j, p_a)
type(vector4_t), intent(in) :: p_k, p_j, p_a
type(vector4_t), intent(out) :: p_aj, pp_k
real(default), intent(out) :: u_j
real(default) :: x_kja
real(default), intent(out) :: v_kja
u_j = (p_a*p_j) / ((p_a*p_j) + (p_a*p_k))
x_kja = ((p_a*p_k) + (p_a*p_j) - (p_j*p_k)) &
/ ((p_a*p_j) + (p_a*p_k))
p_aj = x_kja * p_a
pp_k = p_k + p_j - (1.0_default - x_kja) * p_a
v_kja = 8.0_default * PI * CF * &
(2.0 / (1.0 - x_kja + u_j) - (1.0 + x_kja)) / x_kja
end subroutine if_dipole
@ %def if_dipole
@ This function depends on a variable number of final state particles
whose kinematics all get changed by the initial-initial dipole insertion.
<<SM physics: public>>=
public :: ii_dipole
<<SM physics: sub interfaces>>=
pure module subroutine ii_dipole (v_jab, v_j, p_in, p_out, flag_1or2)
type(vector4_t), dimension(:), intent(in) :: p_in
type(vector4_t), dimension(size(p_in)-1), intent(out) :: p_out
logical, intent(in) :: flag_1or2
real(default), intent(out) :: v_j
real(default), intent(out) :: v_jab
end subroutine ii_dipole
<<SM physics: procedures>>=
pure module subroutine ii_dipole (v_jab, v_j, p_in, p_out, flag_1or2)
type(vector4_t), dimension(:), intent(in) :: p_in
type(vector4_t), dimension(size(p_in)-1), intent(out) :: p_out
logical, intent(in) :: flag_1or2
real(default), intent(out) :: v_j
real(default), intent(out) :: v_jab
type(vector4_t) :: p_a, p_b, p_j
type(vector4_t) :: k, kk
type(vector4_t) :: p_aj
real(default) :: x_jab
integer :: i
!!! flag_1or2 decides whether this a 12 or 21 dipole
if (flag_1or2) then
p_a = p_in(1)
p_b = p_in(2)
else
p_b = p_in(1)
p_a = p_in(2)
end if
!!! We assume that the unresolved particle has always the last
!!! momentum
p_j = p_in(size(p_in))
x_jab = ((p_a*p_b) - (p_a*p_j) - (p_b*p_j)) / (p_a*p_b)
v_j = (p_a*p_j) / (p_a * p_b)
p_aj = x_jab * p_a
k = p_a + p_b - p_j
kk = p_aj + p_b
do i = 3, size(p_in)-1
p_out(i) = p_in(i) - 2.0*((k+kk)*p_in(i))/((k+kk)*(k+kk)) * (k+kk) + &
(2.0 * (k*p_in(i)) / (k*k)) * kk
end do
if (flag_1or2) then
p_out(1) = p_aj
p_out(2) = p_b
else
p_out(1) = p_b
p_out(2) = p_aj
end if
v_jab = 8.0_default * PI * CF * &
(2.0 / (1.0 - x_jab) - (1.0 + x_jab)) / x_jab
end subroutine ii_dipole
@ %def ii_dipole
@
\subsection{Distributions for integrated dipoles and such}
Note that the following formulae are only meaningful for
$0 \leq x \leq 1$.
The Dirac delta distribution, modified for Monte-Carlo sampling,
centered at $x=1-\frac{\epsilon}{2}$:
<<SM physics: public>>=
public :: delta
<<SM physics: sub interfaces>>=
elemental module function delta (x,eps) result (z)
real(default), intent(in) :: x, eps
real(default) :: z
end function delta
<<SM physics: procedures>>=
elemental module function delta (x,eps) result (z)
real(default), intent(in) :: x, eps
real(default) :: z
if (x > one - eps) then
z = one / eps
else
z = 0
end if
end function delta
@ %def delta
@ The $+$-distribution, $P_+(x) = \left( \frac{1}{1-x}\right)_+$, for
the regularization of soft-collinear singularities. The constant part
for the Monte-Carlo sampling is the integral over the splitting
function divided by the weight for the WHIZARD numerical integration
over the interval.
<<SM physics: public>>=
public :: plus_distr
<<SM physics: sub interfaces>>=
elemental module function plus_distr (x,eps) result (plusd)
real(default), intent(in) :: x, eps
real(default) :: plusd
end function plus_distr
<<SM physics: procedures>>=
elemental module function plus_distr (x,eps) result (plusd)
real(default), intent(in) :: x, eps
real(default) :: plusd
if (x > one - eps) then
plusd = log(eps) / eps
else
plusd = one / (one - x)
end if
end function plus_distr
@ %def plus_distr
@ The splitting function in $D=4$ dimensions, regularized as
$+$-distributions if necessary:
\begin{align}
P^{qq} (x) = P^{\bar q\bar q} (x) &= \; C_F \cdot \left( \frac{1 +
x^2}{1-x} \right)_+ \\
P^{qg} (x) = P^{\bar q g} (x) &= \; C_F \cdot \frac{1 + (1-x)^2}{x}\\
P^{gq} (x) = P^{g \bar q} (x) &= \; T_R \cdot \left[ x^2 + (1-x)^2
\right] \\
P^{gg} (x) &= \; 2 C_A \biggl[ \left( \frac{1}{1-x} \right)_+ +
\frac{1-x}{x} - 1 + x(1-x) \biggl] \notag{}\\
&\quad + \delta(1-x) \left( \frac{11}{6} C_A -
\frac{2}{3} N_f T_R \right)
\end{align}
Since the number of flavors summed over in the gluon splitting
function might depend on the physics case under consideration, it is
implemented as an input variable.
<<SM physics: public>>=
public :: pqq
<<SM physics: sub interfaces>>=
elemental module function pqq (x,eps) result (pqqx)
real(default), intent(in) :: x, eps
real(default) :: pqqx
end function pqq
<<SM physics: procedures>>=
elemental module function pqq (x,eps) result (pqqx)
real(default), intent(in) :: x, eps
real(default) :: pqqx
if (x > (1.0_default - eps)) then
pqqx = (eps - one) / two + two * log(eps) / eps - &
three * (eps - one) / eps / two
else
pqqx = (one + x**2) / (one - x)
end if
pqqx = CF * pqqx
end function pqq
@ %def pqq
@
<<SM physics: public>>=
public :: pgq
<<SM physics: sub interfaces>>=
elemental module function pgq (x) result (pgqx)
real(default), intent(in) :: x
real(default) :: pgqx
end function pgq
<<SM physics: procedures>>=
elemental module function pgq (x) result (pgqx)
real(default), intent(in) :: x
real(default) :: pgqx
pgqx = TR * (x**2 + (one - x)**2)
end function pgq
@ %def pgq
@
<<SM physics: public>>=
public :: pqg
<<SM physics: sub interfaces>>=
elemental module function pqg (x) result (pqgx)
real(default), intent(in) :: x
real(default) :: pqgx
end function pqg
<<SM physics: procedures>>=
elemental module function pqg (x) result (pqgx)
real(default), intent(in) :: x
real(default) :: pqgx
pqgx = CF * (one + (one - x)**2) / x
end function pqg
@ %def pqg
@
<<SM physics: public>>=
public :: pgg
<<SM physics: sub interfaces>>=
elemental module function pgg (x, nf, eps) result (pggx)
real(default), intent(in) :: x, nf, eps
real(default) :: pggx
end function pgg
<<SM physics: procedures>>=
elemental module function pgg (x, nf, eps) result (pggx)
real(default), intent(in) :: x, nf, eps
real(default) :: pggx
pggx = two * CA * ( plus_distr (x, eps) + (one-x)/x - one + &
x*(one-x)) + delta (x, eps) * gamma_g(nf)
end function pgg
@ %def pgg
@ For the $qq$ and $gg$ cases, there exist ``regularized'' versions of
the splitting functions:
\begin{align}
P^{qq}_{\text{reg}} (x) &= - C_F \cdot (1 + x) \\
P^{gg}_{\text{reg}} (x) &= 2 C_A \left[ \frac{1-x}{x} - 1 + x(1-x) \right]
\end{align}
<<SM physics: public>>=
public :: pqq_reg
<<SM physics: sub interfaces>>=
elemental module function pqq_reg (x) result (pqqregx)
real(default), intent(in) :: x
real(default) :: pqqregx
end function pqq_reg
<<SM physics: procedures>>=
elemental module function pqq_reg (x) result (pqqregx)
real(default), intent(in) :: x
real(default) :: pqqregx
pqqregx = - CF * (one + x)
end function pqq_reg
@ %def pqq_reg
@
<<SM physics: public>>=
public :: pgg_reg
<<SM physics: sub interfaces>>=
elemental module function pgg_reg (x) result (pggregx)
real(default), intent(in) :: x
real(default) :: pggregx
end function pgg_reg
<<SM physics: procedures>>=
elemental module function pgg_reg (x) result (pggregx)
real(default), intent(in) :: x
real(default) :: pggregx
pggregx = two * CA * ((one - x)/x - one + x*(one - x))
end function pgg_reg
@ %def pgg_reg
@ Here, we collect the expressions needed for integrated
Catani-Seymour dipoles, and the so-called flavor kernels. We always
distinguish between the ``ordinary'' Catani-Seymour version, and the
one including a phase-space slicing parameter, $\alpha$.
The standard flavor kernels $\overline{K}^{ab}$ are:
\begin{align}
\overline{K}^{qg} (x) = \overline{K}^{\bar q g} (x) & = \;
P^{qg} (x) \log ((1-x)/x) + CF \times x \\
%%%
\overline{K}^{gq} (x) = \overline{K}^{g \bar q} (x) & = \;
P^{gq} (x) \log ((1-x)/x) + TR \times 2x(1-x) \\
%%%
\overline{K}^{qq} &=\; C_F \biggl[ \left( \frac{2}{1-x} \log
\frac{1-x}{x} \right)_+ - (1+x) \log ((1-x)/x) +
(1-x) \biggr] \notag{}\\
&\quad - (5 - \pi^2) \cdot C_F \cdot \delta(1-x) \\
%%%
\overline{K}^{gg} &=\; 2 C_A \biggl[ \left( \frac{1}{1-x} \log
\frac{1-x}{x} \right)_+ + \left( \frac{1-x}{x} - 1 + x(1-x)
\right) \log((1-x)/x) \biggr] \notag{}\\
&\quad - \delta(1-x) \biggl[ \left(
\frac{50}{9} - \pi^2 \right) C_A - \frac{16}{9} T_R N_f \biggr]
\end{align}
<<SM physics: public>>=
public :: kbarqg
<<SM physics: sub interfaces>>=
module function kbarqg (x) result (kbarqgx)
real(default), intent(in) :: x
real(default) :: kbarqgx
end function kbarqg
<<SM physics: procedures>>=
module function kbarqg (x) result (kbarqgx)
real(default), intent(in) :: x
real(default) :: kbarqgx
kbarqgx = pqg(x) * log((one-x)/x) + CF * x
end function kbarqg
@ %def kbarqg
@
<<SM physics: public>>=
public :: kbargq
<<SM physics: sub interfaces>>=
module function kbargq (x) result (kbargqx)
real(default), intent(in) :: x
real(default) :: kbargqx
end function kbargq
<<SM physics: procedures>>=
module function kbargq (x) result (kbargqx)
real(default), intent(in) :: x
real(default) :: kbargqx
kbargqx = pgq(x) * log((one-x)/x) + two * TR * x * (one - x)
end function kbargq
@ %def kbarqg
@
<<SM physics: public>>=
public :: kbarqq
<<SM physics: sub interfaces>>=
module function kbarqq (x,eps) result (kbarqqx)
real(default), intent(in) :: x, eps
real(default) :: kbarqqx
end function kbarqq
<<SM physics: procedures>>=
module function kbarqq (x,eps) result (kbarqqx)
real(default), intent(in) :: x, eps
real(default) :: kbarqqx
kbarqqx = CF*(log_plus_distr(x,eps) - (one+x) * log((one-x)/x) + (one - &
x) - (five - pi**2) * delta(x,eps))
end function kbarqq
@ %def kbarqq
@
<<SM physics: public>>=
public :: kbargg
<<SM physics: sub interfaces>>=
module function kbargg (x,eps,nf) result (kbarggx)
real(default), intent(in) :: x, eps, nf
real(default) :: kbarggx
end function kbargg
<<SM physics: procedures>>=
module function kbargg (x,eps,nf) result (kbarggx)
real(default), intent(in) :: x, eps, nf
real(default) :: kbarggx
kbarggx = CA * (log_plus_distr(x,eps) + two * ((one-x)/x - one + &
x*(one-x) * log((1-x)/x))) - delta(x,eps) * &
((50.0_default/9.0_default - pi**2) * CA - &
16.0_default/9.0_default * TR * nf)
end function kbargg
@ %def kbargg
@ The $\tilde{K}$ are used when two identified hadrons participate:
\begin{equation}
\tilde{K}^{ab} (x) = P^{ab}_{\text{reg}} (x) \cdot \log (1-x) +
\delta^{ab} \mathbf{T}_a^2 \biggl[ \left( \frac{2}{1-x} \log (1-x)
\right)_+ - \frac{\pi^2}{3} \delta(1-x) \biggr]
\end{equation}
<<SM physics: public>>=
public :: ktildeqq
<<SM physics: sub interfaces>>=
module function ktildeqq (x,eps) result (ktildeqqx)
real(default), intent(in) :: x, eps
real(default) :: ktildeqqx
end function ktildeqq
<<SM physics: procedures>>=
module function ktildeqq (x,eps) result (ktildeqqx)
real(default), intent(in) :: x, eps
real(default) :: ktildeqqx
ktildeqqx = pqq_reg (x) * log(one-x) + CF * ( - log2_plus_distr (x,eps) &
- pi**2/three * delta(x,eps))
end function ktildeqq
@ %def ktildeqq
@
<<SM physics: public>>=
public :: ktildeqg
<<SM physics: sub interfaces>>=
module function ktildeqg (x,eps) result (ktildeqgx)
real(default), intent(in) :: x, eps
real(default) :: ktildeqgx
end function ktildeqg
<<SM physics: procedures>>=
module function ktildeqg (x,eps) result (ktildeqgx)
real(default), intent(in) :: x, eps
real(default) :: ktildeqgx
ktildeqgx = pqg (x) * log(one-x)
end function ktildeqg
@ %def ktildeqg
@
<<SM physics: public>>=
public :: ktildegq
<<SM physics: sub interfaces>>=
module function ktildegq (x,eps) result (ktildegqx)
real(default), intent(in) :: x, eps
real(default) :: ktildegqx
end function ktildegq
<<SM physics: procedures>>=
module function ktildegq (x,eps) result (ktildegqx)
real(default), intent(in) :: x, eps
real(default) :: ktildegqx
ktildegqx = pgq (x) * log(one-x)
end function ktildegq
@ %def ktildeqg
@
<<SM physics: public>>=
public :: ktildegg
<<SM physics: sub interfaces>>=
module function ktildegg (x,eps) result (ktildeggx)
real(default), intent(in) :: x, eps
real(default) :: ktildeggx
end function ktildegg
<<SM physics: procedures>>=
module function ktildegg (x,eps) result (ktildeggx)
real(default), intent(in) :: x, eps
real(default) :: ktildeggx
ktildeggx = pgg_reg (x) * log(one-x) + CA * ( - &
log2_plus_distr (x,eps) - pi**2/three * delta(x,eps))
end function ktildegg
@ %def ktildegg
@ The insertion operator might not be necessary for a GOLEM interface
but is demanded by the Les Houches NLO accord. It is a
three-dimensional array, where the index always gives the inverse
power of the DREG expansion parameter, $\epsilon$.
<<SM physics: public>>=
public :: insert_q
<<SM physics: sub interfaces>>=
pure module function insert_q () result (i_q)
real(default), dimension(0:2) :: i_q
end function insert_q
<<SM physics: procedures>>=
pure module function insert_q () result (i_q)
real(default), dimension(0:2) :: i_q
i_q(0) = gamma_q + k_q - pi**2/three * CF
i_q(1) = gamma_q
i_q(2) = CF
end function insert_q
@ %def insert_q
@
<<SM physics: public>>=
public :: insert_g
<<SM physics: sub interfaces>>=
pure module function insert_g (nf) result (i_g)
real(default), intent(in) :: nf
real(default), dimension(0:2) :: i_g
end function insert_g
<<SM physics: procedures>>=
pure module function insert_g (nf) result (i_g)
real(default), intent(in) :: nf
real(default), dimension(0:2) :: i_g
i_g(0) = gamma_g (nf) + k_g (nf) - pi**2/three * CA
i_g(1) = gamma_g (nf)
i_g(2) = CA
end function insert_g
@ %def insert_g
@ For better convergence, one can exclude regions of phase space with
a slicing parameter from the dipole subtraction procedure. First of
all, the $K$ functions get modified:
\begin{equation}
K_i (\alpha) = K_i - \mathbf{T}_i^2 \log^2 \alpha + \gamma_i (
\alpha - 1 - \log\alpha)
\end{equation}
<<SM physics: public>>=
public :: k_q_al, k_g_al
<<SM physics: sub interfaces>>=
pure module function k_q_al (alpha)
real(default), intent(in) :: alpha
real(default) :: k_q_al
end function k_q_al
pure module function k_g_al (alpha, nf)
real(default), intent(in) :: alpha, nf
real(default) :: k_g_al
end function k_g_al
<<SM physics: procedures>>=
pure module function k_q_al (alpha)
real(default), intent(in) :: alpha
real(default) :: k_q_al
k_q_al = k_q - CF * (log(alpha))**2 + gamma_q * &
(alpha - one - log(alpha))
end function k_q_al
pure module function k_g_al (alpha, nf)
real(default), intent(in) :: alpha, nf
real(default) :: k_g_al
k_g_al = k_g (nf) - CA * (log(alpha))**2 + gamma_g (nf) * &
(alpha - one - log(alpha))
end function k_g_al
@ %def k_q_al
@ %def k_g_al
@ The $+$-distribution, but with a phase-space slicing parameter,
$\alpha$, $P_{1-\alpha}(x) = \left( \frac{1}{1-x}
\right)_{1-x}$. Since we need the fatal error message here, this
function cannot be elemental.
<<SM physics: public>>=
public :: plus_distr_al
<<SM physics: sub interfaces>>=
module function plus_distr_al (x,alpha,eps) result (plusd_al)
real(default), intent(in) :: x, eps, alpha
real(default) :: plusd_al
end function plus_distr_al
<<SM physics: procedures>>=
module function plus_distr_al (x,alpha,eps) result (plusd_al)
real(default), intent(in) :: x, eps, alpha
real(default) :: plusd_al
if ((one - alpha) >= (one - eps)) then
plusd_al = zero
call msg_fatal ('sm_physics, plus_distr_al: alpha and epsilon chosen wrongly')
elseif (x < (1.0_default - alpha)) then
plusd_al = 0
else if (x > (1.0_default - eps)) then
plusd_al = log(eps/alpha)/eps
else
plusd_al = one/(one-x)
end if
end function plus_distr_al
@ %def plus_distr_al
@ Introducing phase-space slicing parameters, these standard flavor
kernels $\overline{K}^{ab}$ become:
\begin{align}
\overline{K}^{qg}_\alpha (x) = \overline{K}^{\bar q g}_\alpha (x) & = \;
P^{qg} (x) \log (\alpha (1-x)/x) + C_F \times x \\
%%%
\overline{K}^{gq}_\alpha (x) = \overline{K}^{g \bar q}_\alpha (x) & = \;
P^{gq} (x) \log (\alpha (1-x)/x) + T_R \times 2x(1-x) \\
%%%
\overline{K}^{qq}_\alpha &=
C_F (1 - x) + P^{qq}_{\text{reg}} (x) \log \frac{\alpha(1-x)}{x}
\notag{}\\ &\quad
+ C_F \delta (1 - x) \log^2 \alpha
+ C_F \left( \frac{2}{1-x} \log \frac{1-x}{x} \right)_+ \notag{}\\
&\quad
- \left( \gamma_q + K_q(\alpha) - \frac56 \pi^2 C_F \right) \cdot
\delta(1-x) \; C_F \Bigl[ + \frac{2}{1-x} \log \left(
\frac{\alpha (2-x)}{1+\alpha-x} \right)
- \theta(1 - \alpha - x) \cdot \left( \frac{2}{1-x} \log
\frac{2-x}{1-x} \right) \Bigr] \\
%%%
\overline{K}^{gg}_\alpha &=\;
P^{gg}_{\text{reg}} (x) \log \frac{\alpha(1-x)}{x}
+ C_A \delta (1 - x) \log^2 \alpha \notag{}\\
&\quad
+ C_A \left( \frac{2}{1-x} \log \frac{1-x}{x} \right)_+
- \left( \gamma_g + K_g(\alpha) - \frac56 \pi^2 C_A \right) \cdot
\delta(1-x) \; C_A \Bigl[ + \frac{2}{1-x} \log \left(
\frac{\alpha (2-x)}{1+\alpha-x} \right)
- \theta(1 - \alpha - x) \cdot \left( \frac{2}{1-x} \log
\frac{2-x}{1-x} \right) \Bigr]
\end{align}
<<SM physics: public>>=
public :: kbarqg_al
<<SM physics: sub interfaces>>=
module function kbarqg_al (x,alpha,eps) result (kbarqgx)
real(default), intent(in) :: x, alpha, eps
real(default) :: kbarqgx
end function kbarqg_al
<<SM physics: procedures>>=
module function kbarqg_al (x,alpha,eps) result (kbarqgx)
real(default), intent(in) :: x, alpha, eps
real(default) :: kbarqgx
kbarqgx = pqg (x) * log(alpha*(one-x)/x) + CF * x
end function kbarqg_al
@ %def kbarqg_al
@
<<SM physics: public>>=
public :: kbargq_al
<<SM physics: sub interfaces>>=
module function kbargq_al (x,alpha,eps) result (kbargqx)
real(default), intent(in) :: x, alpha, eps
real(default) :: kbargqx
end function kbargq_al
<<SM physics: procedures>>=
module function kbargq_al (x,alpha,eps) result (kbargqx)
real(default), intent(in) :: x, alpha, eps
real(default) :: kbargqx
kbargqx = pgq (x) * log(alpha*(one-x)/x) + two * TR * x * (one-x)
end function kbargq_al
@ %def kbargq_al
@
<<SM physics: public>>=
public :: kbarqq_al
<<SM physics: sub interfaces>>=
module function kbarqq_al (x,alpha,eps) result (kbarqqx)
real(default), intent(in) :: x, alpha, eps
real(default) :: kbarqqx
end function kbarqq_al
<<SM physics: procedures>>=
module function kbarqq_al (x,alpha,eps) result (kbarqqx)
real(default), intent(in) :: x, alpha, eps
real(default) :: kbarqqx
kbarqqx = CF * (one - x) + pqq_reg(x) * log(alpha*(one-x)/x) &
+ CF * log_plus_distr(x,eps) &
- (gamma_q + k_q_al(alpha) - CF * &
five/6.0_default * pi**2 - CF * (log(alpha))**2) * &
delta(x,eps) + &
CF * two/(one -x)*log(alpha*(two-x)/(one+alpha-x))
if (x < (one-alpha)) then
kbarqqx = kbarqqx - CF * two/(one-x) * log((two-x)/(one-x))
end if
end function kbarqq_al
@ %def kbarqq_al
<<SM physics: public>>=
public :: kbargg_al
<<SM physics: sub interfaces>>=
module function kbargg_al (x,alpha,eps,nf) result (kbarggx)
real(default), intent(in) :: x, alpha, eps, nf
real(default) :: kbarggx
end function kbargg_al
<<SM physics: procedures>>=
module function kbargg_al (x,alpha,eps,nf) result (kbarggx)
real(default), intent(in) :: x, alpha, eps, nf
real(default) :: kbarggx
kbarggx = pgg_reg(x) * log(alpha*(one-x)/x) &
+ CA * log_plus_distr(x,eps) &
- (gamma_g(nf) + k_g_al(alpha,nf) - CA * &
five/6.0_default * pi**2 - CA * (log(alpha))**2) * &
delta(x,eps) + &
CA * two/(one -x)*log(alpha*(two-x)/(one+alpha-x))
if (x < (one-alpha)) then
kbarggx = kbarggx - CA * two/(one-x) * log((two-x)/(one-x))
end if
end function kbargg_al
@ %def kbargg_al
@ The $\tilde{K}$ flavor kernels in the presence of a phase-space slicing
parameter, are:
\begin{equation}
\tilde{K}^{ab} (x,\alpha) = P^{qq, \text{reg}} (x)
\log\frac{1-x}{\alpha} + ..........
\end{equation}
<<SM physics: public>>=
public :: ktildeqq_al
<<SM physics: sub interfaces>>=
module function ktildeqq_al (x,alpha,eps) result (ktildeqqx)
real(default), intent(in) :: x, eps, alpha
real(default) :: ktildeqqx
end function ktildeqq_al
<<SM physics: procedures>>=
module function ktildeqq_al (x,alpha,eps) result (ktildeqqx)
real(default), intent(in) :: x, eps, alpha
real(default) :: ktildeqqx
ktildeqqx = pqq_reg(x) * log((one-x)/alpha) + CF*( &
- log2_plus_distr_al(x,alpha,eps) - Pi**2/three * delta(x,eps) &
+ (one+x**2)/(one-x) * log(min(one,(alpha/(one-x)))) &
+ two/(one-x) * log((one+alpha-x)/alpha))
if (x > (one-alpha)) then
ktildeqqx = ktildeqqx - CF*two/(one-x)*log(two-x)
end if
end function ktildeqq_al
@ %def ktildeqq_al
@ This is a logarithmic $+$-distribution, $\left(
\frac{\log((1-x)/x)}{1-x} \right)_+$. For the sampling, we need the
integral over this function over the incomplete sampling interval
$[0,1-\epsilon]$, which is $\log^2(x) + 2 Li_2(x) -
\frac{\pi^2}{3}$. As this function is negative definite for $\epsilon
> 0.1816$, we take a hard upper limit for that sampling parameter,
irrespective of the fact what the user chooses.
<<SM physics: public>>=
public :: log_plus_distr
<<SM physics: sub interfaces>>=
module function log_plus_distr (x,eps) result (lpd)
real(default), intent(in) :: x, eps
real(default) :: lpd, eps2
end function log_plus_distr
<<SM physics: procedures>>=
module function log_plus_distr (x,eps) result (lpd)
real(default), intent(in) :: x, eps
real(default) :: lpd, eps2
eps2 = min (eps, 0.1816_default)
if (x > (1.0_default - eps2)) then
lpd = ((log(eps2))**2 + two*Li2(eps2) - pi**2/three)/eps2
else
lpd = two*log((one-x)/x)/(one-x)
end if
end function log_plus_distr
@ %def log_plus_distr
@ Logarithmic $+$-distribution, $2 \left( \frac{\log(1/(1-x))}{1-x} \right)_+$.
<<SM physics: public>>=
public :: log2_plus_distr
<<SM physics: sub interfaces>>=
module function log2_plus_distr (x,eps) result (lpd)
real(default), intent(in) :: x, eps
real(default) :: lpd
end function log2_plus_distr
<<SM physics: procedures>>=
module function log2_plus_distr (x,eps) result (lpd)
real(default), intent(in) :: x, eps
real(default) :: lpd
if (x > (1.0_default - eps)) then
lpd = - (log(eps))**2/eps
else
lpd = two*log(one/(one-x))/(one-x)
end if
end function log2_plus_distr
@ %def log2_plus_distr
@ Logarithmic $+$-distribution with phase-space slicing parameter, $2
\left( \frac{\log(1/(1-x))}{1-x} \right)_{1-\alpha}$.
<<SM physics: public>>=
public :: log2_plus_distr_al
<<SM physics: sub interfaces>>=
module function log2_plus_distr_al (x,alpha,eps) result (lpd_al)
real(default), intent(in) :: x, eps, alpha
real(default) :: lpd_al
end function log2_plus_distr_al
<<SM physics: procedures>>=
module function log2_plus_distr_al (x,alpha,eps) result (lpd_al)
real(default), intent(in) :: x, eps, alpha
real(default) :: lpd_al
if ((one - alpha) >= (one - eps)) then
lpd_al = zero
call msg_fatal ('alpha and epsilon chosen wrongly')
elseif (x < (one - alpha)) then
lpd_al = 0
elseif (x > (1.0_default - eps)) then
lpd_al = - ((log(eps))**2 - (log(alpha))**2)/eps
else
lpd_al = two*log(one/(one-x))/(one-x)
end if
end function log2_plus_distr_al
@ %def log2_plus_distr_al
@
\subsection{Splitting Functions}
@ Analogue to the regularized distributions of the last subsection, we
give here the unregularized splitting functions, relevant for the parton
shower algorithm. We can use this unregularized version since there will
be a cut-off $\epsilon$ that ensures that $\{z,1-z\}>\epsilon(t)$. This
cut-off seperates resolvable from unresolvable emissions.
[[p_xxx]] are the kernels that are summed over helicity:
<<SM physics: public>>=
public :: p_qqg
public :: p_gqq
public :: p_ggg
@ $q\to q g$
<<SM physics: sub interfaces>>=
elemental module function p_qqg (z) result (P)
real(default), intent(in) :: z
real(default) :: P
end function p_qqg
<<SM physics: procedures>>=
elemental module function p_qqg (z) result (P)
real(default), intent(in) :: z
real(default) :: P
P = CF * (one + z**2) / (one - z)
end function p_qqg
@ $g\to q \bar{q}$
<<SM physics: sub interfaces>>=
elemental module function p_gqq (z) result (P)
real(default), intent(in) :: z
real(default) :: P
end function p_gqq
<<SM physics: procedures>>=
elemental module function p_gqq (z) result (P)
real(default), intent(in) :: z
real(default) :: P
P = TR * (z**2 + (one - z)**2)
end function p_gqq
@ $g\to g g$
<<SM physics: sub interfaces>>=
elemental module function p_ggg (z) result (P)
real(default), intent(in) :: z
real(default) :: P
end function p_ggg
<<SM physics: procedures>>=
elemental module function p_ggg (z) result (P)
real(default), intent(in) :: z
real(default) :: P
P = NC * ((one - z) / z + z / (one - z) + z * (one - z))
end function p_ggg
@ %def p_qqg p_gqq p_ggg
@ Analytically integrated splitting kernels:
<<SM physics: public>>=
public :: integral_over_p_qqg
public :: integral_over_p_gqq
public :: integral_over_p_ggg
<<SM physics: sub interfaces>>=
pure module function integral_over_p_qqg (zmin, zmax) result (integral)
real(default), intent(in) :: zmin, zmax
real(default) :: integral
end function integral_over_p_qqg
pure module function integral_over_p_gqq (zmin, zmax) result (integral)
real(default), intent(in) :: zmin, zmax
real(default) :: integral
end function integral_over_p_gqq
pure module function integral_over_p_ggg (zmin, zmax) result (integral)
real(default), intent(in) :: zmin, zmax
real(default) :: integral
end function integral_over_p_ggg
<<SM physics: procedures>>=
pure module function integral_over_p_qqg (zmin, zmax) result (integral)
real(default), intent(in) :: zmin, zmax
real(default) :: integral
integral = (two / three) * (- zmax**2 + zmin**2 - &
two * (zmax - zmin) + four * log((one - zmin) / (one - zmax)))
end function integral_over_p_qqg
pure module function integral_over_p_gqq (zmin, zmax) result (integral)
real(default), intent(in) :: zmin, zmax
real(default) :: integral
integral = 0.5_default * ((two / three) * &
(zmax**3 - zmin**3) - (zmax**2 - zmin**2) + (zmax - zmin))
end function integral_over_p_gqq
pure module function integral_over_p_ggg (zmin, zmax) result (integral)
real(default), intent(in) :: zmin, zmax
real(default) :: integral
integral = three * ((log(zmax) - two * zmax - &
log(one - zmax) + zmax**2 / two - zmax**3 / three) - &
(log(zmin) - zmin - zmin - log(one - zmin) + zmin**2 &
/ two - zmin**3 / three) )
end function integral_over_p_ggg
@ %def integral_over_p_gqq integral_over_p_ggg integral_over_p_qqg
@ We can also use (massless) helicity dependent splitting functions:
<<SM physics: public>>=
public :: p_qqg_pol
@ $q_a\to q_b g_c$, the helicity of the quark is not changed by gluon
emission and the gluon is preferably polarized in the branching plane
($l_c=1$):
<<SM physics: sub interfaces>>=
elemental module function p_qqg_pol (z, l_a, l_b, l_c) result (P)
real(default), intent(in) :: z
integer, intent(in) :: l_a, l_b, l_c
real(default) :: P
end function p_qqg_pol
<<SM physics: procedures>>=
elemental module function p_qqg_pol (z, l_a, l_b, l_c) result (P)
real(default), intent(in) :: z
integer, intent(in) :: l_a, l_b, l_c
real(default) :: P
if (l_a /= l_b) then
P = zero
return
end if
if (l_c == -1) then
P = one - z
else
P = (one + z)**2 / (one - z)
end if
P = P * CF
end function p_qqg_pol
@
\subsubsection{Mellin transforms of splitting functions}
As Mellin transforms necessarily live in the complex plane, all
functions are defined as complex functions:
@ Splitting function $P_{qq}(N)$:
<<SM physics: public>>=
public :: pqqm
<<SM physics: sub interfaces>>=
module function pqqm (n, c_f) result (pqq_m)
integer, intent(in) :: n
real(default), intent(in) :: c_f
complex(default) :: pqq_m
end function pqqm
<<SM physics: procedures>>=
module function pqqm (n, c_f) result (pqq_m)
integer, intent(in) :: n
real(default), intent(in) :: c_f
complex(default) :: pqq_m
pqq_m = three - four * (eulerc + &
psic(cmplx(N+1,zero,kind=default))) + two/N/(N+1)
end function pqqm
@ %def pqqm
@
\subsection{Top width}
In order to produce sensible results, the widths have to be recomputed
for each parameter and order.
We start with the LO-expression for the top width given by the decay
$t\,\to\,W^+,b$, cf. [[doi:10.1016/0550-3213(91)90530-B]]:\\
The analytic formula given there is
\begin{equation*}
\Gamma = \frac{G_F m_t^2}{16\sqrt{2}\pi}
\left[\mathcal{F}_0(\varepsilon, \xi^{-1/2}) -
\frac{2\alpha_s}{3\pi} \mathcal{F}_1 (\varepsilon, \xi^{-1/2})\right],
\end{equation*}
with
\begin{align*}
\mathcal{F}_0 &= \frac{\sqrt{\lambda}}{2} f_0, \\
f_0 &= 4\left[(1-\varepsilon^2)^2 + w^2(1+\varepsilon^2) - 2w^4\right], \\
\lambda = 1 + w^4 + \varepsilon^4 - 2(w^2 + \varepsilon^2 + w^2\varepsilon^2).
\end{align*}
Defining
\begin{equation*}
u_q = \frac{1 + \varepsilon^2 - w^2 - \lambda^{1/2}}{1 +
\varepsilon^2 - w^2 + \lambda^{1/2}}
\end{equation*}
and
\begin{equation*}
u_w = \frac{1 - \varepsilon^2 + w^2 - \lambda^{1/2}}{1 -
\varepsilon^2 + w^2 + \lambda^{1/2}}
\end{equation*}
the factor $\mathcal{F}_1$ can be expressed as
\begin{align*}
\mathcal{F}_1 = \frac{1}{2}f_0(1+\varepsilon^2-w^2)
& \left[\pi^2 + 2Li_2(u_w) - 2Li_2(1-u_w) - 4Li_2(u_q) \right. \\
& -4Li_2(u_q u_w) + \log\left(\frac{1-u_q}{w^2}\right)\log(1-u_q)
- \log^2(1-u_q u_w) \\
& \left.+\frac{1}{4}\log^2\left(\frac{w^2}{u_w}\right) - \log(u_w)
\log\left[\frac{(1-u_q u_w)^2}{1-u_q}\right]
-2\log(u_q)\log\left[(1-u_q)(1-u_q u_w)\right]\right] \\
& -\sqrt{\lambda}f_0(2\log(w) + 3\log(\varepsilon) - 2\log{\lambda}) \\
& +4(1-\varepsilon^2)\left[(1-\varepsilon^2)^2 +
w^2(1+\varepsilon^2) - 4w^4\right]\log(u_w) \\
& \left[(3 - \varepsilon^2 + 11\varepsilon^4 - \varepsilon^6)
+ w^2(6 - 12\varepsilon^2 +2\varepsilon^4) - w^4(21 +
5\varepsilon^2) + 12w^6\right] \log(u_q) \\
& 6\sqrt{\lambda} (1-\varepsilon^2) (1 + \varepsilon^2 - w^2)
\log(\varepsilon)
+ \sqrt{\lambda}\left[-5 + 22\varepsilon^2 - 5\varepsilon^4 -
9w^2(1+\varepsilon^2) + 6w^4\right].
\end{align*}
@
<<SM physics: public>>=
public :: top_width_sm_lo
<<SM physics: sub interfaces>>=
elemental module function top_width_sm_lo (alpha, sinthw, vtb, mtop, mw, mb) &
result (gamma)
real(default) :: gamma
real(default), intent(in) :: alpha, sinthw, vtb, mtop, mw, mb
end function top_width_sm_lo
<<SM physics: procedures>>=
elemental module function top_width_sm_lo (alpha, sinthw, vtb, mtop, mw, mb) &
result (gamma)
real(default) :: gamma
real(default), intent(in) :: alpha, sinthw, vtb, mtop, mw, mb
real(default) :: kappa
kappa = sqrt ((mtop**2 - (mw + mb)**2) * (mtop**2 - (mw - mb)**2))
gamma = alpha / four * mtop / (two * sinthw**2) * &
vtb**2 * kappa / mtop**2 * &
((mtop**2 + mb**2) / (two * mtop**2) + &
(mtop**2 - mb**2)**2 / (two * mtop**2 * mw**2) - &
mw**2 / mtop**2)
end function top_width_sm_lo
@ %def top_width_sm_lo
@
<<SM physics: public>>=
public :: g_mu_from_alpha
<<SM physics: sub interfaces>>=
elemental module function g_mu_from_alpha (alpha, mw, sinthw) result (g_mu)
real(default) :: g_mu
real(default), intent(in) :: alpha, mw, sinthw
end function g_mu_from_alpha
<<SM physics: procedures>>=
elemental module function g_mu_from_alpha (alpha, mw, sinthw) result (g_mu)
real(default) :: g_mu
real(default), intent(in) :: alpha, mw, sinthw
g_mu = pi * alpha / sqrt(two) / mw**2 / sinthw**2
end function g_mu_from_alpha
@ %def g_mu_from_alpha
@
<<SM physics: public>>=
public :: alpha_from_g_mu
<<SM physics: sub interfaces>>=
elemental module function alpha_from_g_mu (g_mu, mw, sinthw) result (alpha)
real(default) :: alpha
real(default), intent(in) :: g_mu, mw, sinthw
end function alpha_from_g_mu
<<SM physics: procedures>>=
elemental module function alpha_from_g_mu (g_mu, mw, sinthw) result (alpha)
real(default) :: alpha
real(default), intent(in) :: g_mu, mw, sinthw
alpha = g_mu * sqrt(two) / pi * mw**2 * sinthw**2
end function alpha_from_g_mu
@ %def alpha_from_g_mu
@ Cf. (3.3)-(3.7) in [[1207.5018]].
<<SM physics: public>>=
public :: top_width_sm_qcd_nlo_massless_b
<<SM physics: sub interfaces>>=
elemental module function top_width_sm_qcd_nlo_massless_b &
(alpha, sinthw, vtb, mtop, mw, alphas) result (gamma)
real(default) :: gamma
real(default), intent(in) :: alpha, sinthw, vtb, mtop, mw, alphas
end function top_width_sm_qcd_nlo_massless_b
<<SM physics: procedures>>=
elemental module function top_width_sm_qcd_nlo_massless_b &
(alpha, sinthw, vtb, mtop, mw, alphas) result (gamma)
real(default) :: gamma
real(default), intent(in) :: alpha, sinthw, vtb, mtop, mw, alphas
real(default) :: prefac, g_mu, w2
g_mu = g_mu_from_alpha (alpha, mw, sinthw)
prefac = g_mu * mtop**3 * vtb**2 / (16 * sqrt(two) * pi)
w2 = mw**2 / mtop**2
gamma = prefac * (f0 (w2) - (two * alphas) / (3 * Pi) * f1 (w2))
end function top_width_sm_qcd_nlo_massless_b
@ %def top_width_sm_qcd_nlo_massless_b
@
<<SM physics: public>>=
public :: f0
<<SM physics: sub interfaces>>=
elemental module function f0 (w2) result (f)
real(default) :: f
real(default), intent(in) :: w2
end function f0
<<SM physics: procedures>>=
elemental module function f0 (w2) result (f)
real(default) :: f
real(default), intent(in) :: w2
f = two * (one - w2)**2 * (1 + 2 * w2)
end function f0
@ %def f0
@
<<SM physics: public>>=
public :: f1
<<SM physics: sub interfaces>>=
elemental module function f1 (w2) result (f)
real(default) :: f
real(default), intent(in) :: w2
end function f1
<<SM physics: procedures>>=
elemental module function f1 (w2) result (f)
real(default) :: f
real(default), intent(in) :: w2
f = f0 (w2) * (pi**2 + two * Li2 (w2) - two * Li2 (one - w2)) &
+ four * w2 * (one - w2 - two * w2**2) * log (w2) &
+ two * (one - w2)**2 * (five + four * w2) * log (one - w2) &
- (one - w2) * (five + 9 * w2 - 6 * w2**2)
end function f1
@ %def f1
@ Basically, the same as above but with $m_b$ dependence,
cf. Jezabek / Kuehn 1989.
<<SM physics: public>>=
public :: top_width_sm_qcd_nlo_jk
<<SM physics: sub interfaces>>=
elemental module function top_width_sm_qcd_nlo_jk &
(alpha, sinthw, vtb, mtop, mw, mb, alphas) result (gamma)
real(default) :: gamma
real(default), intent(in) :: alpha, sinthw, vtb, mtop, mw, mb, alphas
end function top_width_sm_qcd_nlo_jk
<<SM physics: procedures>>=
elemental module function top_width_sm_qcd_nlo_jk &
(alpha, sinthw, vtb, mtop, mw, mb, alphas) result (gamma)
real(default) :: gamma
real(default), intent(in) :: alpha, sinthw, vtb, mtop, mw, mb, alphas
real(default) :: prefac, g_mu, eps2, i_xi
g_mu = g_mu_from_alpha (alpha, mw, sinthw)
prefac = g_mu * mtop**3 * vtb**2 / (16 * sqrt(two) * pi)
eps2 = (mb / mtop)**2
i_xi = (mw / mtop)**2
gamma = prefac * (ff0 (eps2, i_xi) - &
(two * alphas) / (3 * Pi) * ff1 (eps2, i_xi))
end function top_width_sm_qcd_nlo_jk
@ %def top_width_sm_qcd_nlo_jk
@ Same as above, $m_b > 0$, with the slightly different implementation
(2.6) of arXiv:1204.1513v1 by Campbell and Ellis.
<<SM physics: public>>=
public :: top_width_sm_qcd_nlo_ce
<<SM physics: sub interfaces>>=
elemental module function top_width_sm_qcd_nlo_ce &
(alpha, sinthw, vtb, mtop, mw, mb, alpha_s) result (gamma)
real(default) :: gamma
real(default), intent(in) :: alpha, sinthw, vtb, mtop, mw, mb, alpha_s
end function top_width_sm_qcd_nlo_ce
<<SM physics: procedures>>=
elemental module function top_width_sm_qcd_nlo_ce &
(alpha, sinthw, vtb, mtop, mw, mb, alpha_s) result (gamma)
real(default) :: gamma
real(default), intent(in) :: alpha, sinthw, vtb, mtop, mw, mb, alpha_s
real(default) :: pm, pp, p0, p3
real(default) :: yw, yp
real(default) :: W0, Wp, Wm, w2
real(default) :: beta2
real(default) :: f
real(default) :: g_mu, gamma0
beta2 = (mb / mtop)**2
w2 = (mw / mtop)**2
p0 = (one - w2 + beta2) / two
p3 = sqrt (lambda (one, w2, beta2)) / two
pp = p0 + p3
pm = p0 - p3
W0 = (one + w2 - beta2) / two
Wp = W0 + p3
Wm = W0 - p3
yp = log (pp / pm) / two
yw = log (Wp / Wm) / two
f = (one - beta2)**2 + w2 * (one + beta2) - two * w2**2
g_mu = g_mu_from_alpha (alpha, mw, sinthw)
gamma0 = g_mu * mtop**3 * vtb**2 / (8 * pi * sqrt(two))
gamma = gamma0 * alpha_s / twopi * CF * &
(8 * f * p0 * (Li2(one - pm) - Li2(one - pp) - two * Li2(one - pm / pp) &
+ yp * log((four * p3**2) / (pp**2 * Wp)) + yw * log (pp)) &
+ four * (one - beta2) * ((one - beta2)**2 + w2 * (one + beta2) - four * w2**2) * yw &
+ (3 - beta2 + 11 * beta2**2 - beta2**3 + w2 * (6 - 12 * beta2 + two * beta2**2) &
- w2**2 * (21 + 5 * beta2) + 12 * w2**3) * yp &
+ 8 * f * p3 * log (sqrt(w2) / (four * p3**2)) &
+ 6 * (one - four * beta2 + 3 * beta2**2 + w2 * (3 + beta2) - four * w2**2) * p3 * log(sqrt(beta2)) &
+ (5 - 22 * beta2 + 5 * beta2**2 + 9 * w2 * (one + beta2) - 6 * w2**2) * p3)
end function top_width_sm_qcd_nlo_ce
@ %def top_width_sm_qcd_nlo_ce
@
<<SM physics: public>>=
public :: ff0
<<SM physics: sub interfaces>>=
elemental module function ff0 (eps2, w2) result (f)
real(default) :: f
real(default), intent(in) :: eps2, w2
end function ff0
<<SM physics: procedures>>=
elemental module function ff0 (eps2, w2) result (f)
real(default) :: f
real(default), intent(in) :: eps2, w2
f = one / two * sqrt(ff_lambda (eps2, w2)) * ff_f0 (eps2, w2)
end function ff0
@ %def ff0
@
<<SM physics: public>>=
public :: ff_f0
<<SM physics: sub interfaces>>=
elemental module function ff_f0 (eps2, w2) result (f)
real(default) :: f
real(default), intent(in) :: eps2, w2
end function ff_f0
<<SM physics: procedures>>=
elemental module function ff_f0 (eps2, w2) result (f)
real(default) :: f
real(default), intent(in) :: eps2, w2
f = four * ((1 - eps2)**2 + w2 * (1 + eps2) - 2 * w2**2)
end function ff_f0
@ %def ff_f0
@
<<SM physics: public>>=
public :: ff_lambda
<<SM physics: sub interfaces>>=
elemental module function ff_lambda (eps2, w2) result (l)
real(default) :: l
real(default), intent(in) :: eps2, w2
end function ff_lambda
<<SM physics: procedures>>=
elemental module function ff_lambda (eps2, w2) result (l)
real(default) :: l
real(default), intent(in) :: eps2, w2
l = one + w2**2 + eps2**2 - two * (w2 + eps2 + w2 * eps2)
end function ff_lambda
@ %def ff_lambda
@
<<SM physics: public>>=
public :: ff1
<<SM physics: sub interfaces>>=
elemental module function ff1 (eps2, w2) result (f)
real(default) :: f
real(default), intent(in) :: eps2, w2
end function ff1
<<SM physics: procedures>>=
elemental module function ff1 (eps2, w2) result (f)
real(default) :: f
real(default), intent(in) :: eps2, w2
real(default) :: uq, uw, sq_lam, fff
sq_lam = sqrt (ff_lambda (eps2, w2))
fff = ff_f0 (eps2, w2)
uw = (one - eps2 + w2 - sq_lam) / &
(one - eps2 + w2 + sq_lam)
uq = (one + eps2 - w2 - sq_lam) / &
(one + eps2 - w2 + sq_lam)
f = one / two * fff * (one + eps2 - w2) * &
(pi**2 + two * Li2 (uw) - two * Li2 (one - uw) - four * Li2 (uq) &
- four * Li2 (uq * uw) + log ((one - uq) / w2) * log (one - uq) &
- log (one - uq * uw)**2 + one / four * log (w2 / uw)**2 &
- log (uw) * log ((one - uq * uw)**2 / (one - uq)) &
- two * log (uq) * log ((one - uq) * (one - uq * uw))) &
- sq_lam * fff * (two * log (sqrt (w2)) &
+ three * log (sqrt (eps2)) - two * log (sq_lam**2)) &
+ four * (one - eps2) * ((one - eps2)**2 + w2 * (one + eps2) &
- four * w2**2) * log (uw) &
+ (three - eps2 + 11 * eps2**2 - eps2**3 + w2 * &
(6 - 12 * eps2 + 2 * eps2**2) - w2**2 * (21 + five * eps2) &
+ 12 * w2**3) * log (uq) &
+ 6 * sq_lam * (one - eps2) * &
(one + eps2 - w2) * log (sqrt (eps2)) &
+ sq_lam * (- five + 22 * eps2 - five * eps2**2 - 9 * w2 * &
(one + eps2) + 6 * w2**2)
end function ff1
@ %def ff1
@
\subsection{Unit tests}
Test module, followed by the corresponding implementation module.
<<[[sm_physics_ut.f90]]>>=
<<File header>>
module sm_physics_ut
use unit_tests
use sm_physics_uti
<<Standard module head>>
<<SM physics: public test>>
contains
<<SM physics: test driver>>
end module sm_physics_ut
@ %def sm_physics_ut
@
<<[[sm_physics_uti.f90]]>>=
<<File header>>
module sm_physics_uti
<<Use kinds>>
use numeric_utils
use format_defs, only: FMT_15
use constants
use sm_physics
<<Standard module head>>
<<SM physics: test declarations>>
contains
<<SM physics: tests>>
end module sm_physics_uti
@ %def sm_physics_ut
@ API: driver for the unit tests below.
<<SM physics: public test>>=
public :: sm_physics_test
<<SM physics: test driver>>=
subroutine sm_physics_test (u, results)
integer, intent(in) :: u
type(test_results_t), intent(inout) :: results
<<SM physics: execute tests>>
end subroutine sm_physics_test
@ %def sm_physics_test
@
\subsubsection{Splitting functions}
<<SM physics: execute tests>>=
call test (sm_physics_1, "sm_physics_1", &
"Splitting functions", &
u, results)
<<SM physics: test declarations>>=
public :: sm_physics_1
<<SM physics: tests>>=
subroutine sm_physics_1 (u)
integer, intent(in) :: u
real(default) :: z = 0.75_default
write (u, "(A)") "* Test output: sm_physics_1"
write (u, "(A)") "* Purpose: check analytic properties"
write (u, "(A)")
write (u, "(A)") "* Splitting functions:"
write (u, "(A)")
call assert (u, vanishes (p_qqg_pol (z, +1, -1, +1)), "+-+")
call assert (u, vanishes (p_qqg_pol (z, +1, -1, -1)), "+--")
call assert (u, vanishes (p_qqg_pol (z, -1, +1, +1)), "-++")
call assert (u, vanishes (p_qqg_pol (z, -1, +1, -1)), "-+-")
!call assert (u, nearly_equal ( &
!p_qqg_pol (z, +1, +1, -1) + p_qqg_pol (z, +1, +1, +1), &
!p_qqg (z)), "pol sum")
write (u, "(A)")
write (u, "(A)") "* Test output end: sm_physics_1"
end subroutine sm_physics_1
@ %def sm_physics_1
@
\subsubsection{Top width}
<<SM physics: execute tests>>=
call test(sm_physics_2, "sm_physics_2", &
"Top width", u, results)
<<SM physics: test declarations>>=
public :: sm_physics_2
<<SM physics: tests>>=
subroutine sm_physics_2 (u)
integer, intent(in) :: u
real(default) :: mtop, mw, mz, mb, g_mu, sinthw, alpha, vtb, gamma0
real(default) :: w2, alphas, alphas_mz, gamma1
write (u, "(A)") "* Test output: sm_physics_2"
write (u, "(A)") "* Purpose: Check different top width computations"
write (u, "(A)")
write (u, "(A)") "* Values from [[1207.5018]] (massless b)"
mtop = 172.0
mw = 80.399
mz = 91.1876
mb = zero
mb = 0.00001
g_mu = 1.16637E-5
sinthw = sqrt(one - mw**2 / mz**2)
alpha = alpha_from_g_mu (g_mu, mw, sinthw)
vtb = one
w2 = mw**2 / mtop**2
write (u, "(A)") "* Check Li2 implementation"
call assert_equal (u, Li2(w2), 0.2317566263959552_default, &
"Li2(w2)", rel_smallness=1.0E-6_default)
call assert_equal (u, Li2(one - w2), 1.038200378935867_default, &
"Li2(one - w2)", rel_smallness=1.0E-6_default)
write (u, "(A)") "* Check LO Width"
gamma0 = top_width_sm_lo (alpha, sinthw, vtb, mtop, mw, mb)
call assert_equal (u, gamma0, 1.4655_default, &
"top_width_sm_lo", rel_smallness=1.0E-5_default)
alphas = zero
gamma0 = top_width_sm_qcd_nlo_massless_b &
(alpha, sinthw, vtb, mtop, mw, alphas)
call assert_equal (u, gamma0, 1.4655_default, &
"top_width_sm_qcd_nlo_massless_b", rel_smallness=1.0E-5_default)
gamma0 = top_width_sm_qcd_nlo_jk &
(alpha, sinthw, vtb, mtop, mw, mb, alphas)
call assert_equal (u, gamma0, 1.4655_default, &
"top_width_sm_qcd_nlo", rel_smallness=1.0E-5_default)
write (u, "(A)") "* Check NLO Width"
alphas_mz = 0.1202 ! MSTW2008 NLO fit
alphas = running_as (mtop, alphas_mz, mz, 1, 5.0_default)
gamma1 = top_width_sm_qcd_nlo_massless_b &
(alpha, sinthw, vtb, mtop, mw, alphas)
call assert_equal (u, gamma1, 1.3376_default, rel_smallness=1.0E-4_default)
gamma1 = top_width_sm_qcd_nlo_jk &
(alpha, sinthw, vtb, mtop, mw, mb, alphas)
! It would be nice to get one more significant digit but the
! expression is numerically rather unstable for mb -> 0
call assert_equal (u, gamma1, 1.3376_default, rel_smallness=1.0E-3_default)
write (u, "(A)") "* Values from threshold validation (massive b)"
alpha = one / 125.924
! ee = 0.315901
! cw = 0.881903
! v = 240.024
mtop = 172.0 ! This is the value for M1S !!!
mb = 4.2
sinthw = 0.47143
mz = 91.188
mw = 80.419
call assert_equal (u, sqrt(one - mw**2 / mz**2), sinthw, &
"sinthw", rel_smallness=1.0E-6_default)
write (u, "(A)") "* Check LO Width"
gamma0 = top_width_sm_lo (alpha, sinthw, vtb, mtop, mw, mb)
call assert_equal (u, gamma0, 1.5386446_default, &
"gamma0", rel_smallness=1.0E-7_default)
alphas = zero
gamma0 = top_width_sm_qcd_nlo_jk &
(alpha, sinthw, vtb, mtop, mw, mb, alphas)
call assert_equal (u, gamma0, 1.5386446_default, &
"gamma0", rel_smallness=1.0E-7_default)
write (u, "(A)") "* Check NLO Width"
alphas_mz = 0.118 !(Z pole, NLL running to mu_h)
alphas = running_as (mtop, alphas_mz, mz, 1, 5.0_default)
write (u, "(A," // FMT_15 // ")") "* alphas = ", alphas
gamma1 = top_width_sm_qcd_nlo_jk &
(alpha, sinthw, vtb, mtop, mw, mb, alphas)
write (u, "(A," // FMT_15 // ")") "* Gamma1 = ", gamma1
mb = zero
gamma1 = top_width_sm_qcd_nlo_massless_b &
(alpha, sinthw, vtb, mtop, mw, alphas)
alphas = running_as (mtop, alphas_mz, mz, 1, 5.0_default)
write (u, "(A," // FMT_15 // ")") "* Gamma1(mb=0) = ", gamma1
write (u, "(A)")
write (u, "(A)") "* Test output end: sm_physics_2"
end subroutine sm_physics_2
@ %def sm_physics_2
@
\subsubsection{Special functions}
<<SM physics: execute tests>>=
call test (sm_physics_3, "sm_physics_3", &
"Special functions", &
u, results)
<<SM physics: test declarations>>=
public :: sm_physics_3
<<SM physics: tests>>=
subroutine sm_physics_3 (u)
integer, intent(in) :: u
complex(default) :: z1 = (0.75_default, 1.25_default)
complex(default) :: z2 = (1.33_default, 11.25_default)
complex(default) :: psiz
real(default) :: x1 = 0.045847700_default
real(default) :: psir
write (u, "(A)") "* Test output: sm_physics_3"
write (u, "(A)") "* Purpose: check special functions"
write (u, "(A)")
write (u, "(A)") "* Complex digamma function:"
write (u, "(A)")
psiz = psic (z1)
write (u, "(1x,A,'(',F5.2,',',F5.2,')')") " z1 = ", &
real(z1), aimag(z1)
write (u, "(1x,A,'(',F7.5,',',F7.5,')')") " psi(z1) = ", &
real(psiz), aimag(psiz)
psiz = psic (z2)
write (u, "(1x,A,'(',F5.2,',',F5.2,')')") " z2 = ", &
real(z2), aimag(z2)
write (u, "(1x,A,'(',F7.5,',',F7.5,')')") " psi(z2) = ", &
real(psiz), aimag(psiz)
write (u, "(A)")
write (u, "(A)") "* Complex polygamma function:"
write (u, "(A)")
psiz = psim (z1,1)
write (u, "(1x,A,'(',F5.2,',',F5.2,')')") " z1 = ", &
real(z1), aimag(z1)
write (u, "(1x,A,'(',F8.5,',',F8.5,')')") " psi(z1,1) = ", &
real(psiz), aimag(psiz)
psiz = psim (z2,1)
write (u, "(1x,A,'(',F5.2,',',F5.2,')')") " z2 = ", &
real(z2), aimag(z2)
write (u, "(1x,A,'(',F8.5,',',F8.5,')')") " psi(z2,1) = ", &
real(psiz), aimag(psiz)
write (u, "(A)")
psiz = psim (z1,2)
write (u, "(1x,A,'(',F5.2,',',F5.2,')')") " z1 = ", &
real(z1), aimag(z1)
write (u, "(1x,A,'(',F8.5,',',F8.5,')')") " psi(z1,2) = ", &
real(psiz), aimag(psiz)
psiz = psim (z2,2)
write (u, "(1x,A,'(',F5.2,',',F5.2,')')") " z2 = ", &
real(z2), aimag(z2)
write (u, "(1x,A,'(',F8.5,',',F8.5,')')") " psi(z2,2) = ", &
real(psiz), aimag(psiz)
write (u, "(A)")
psiz = psim (z1,3)
write (u, "(1x,A,'(',F5.2,',',F5.2,')')") " z1 = ", &
real(z1), aimag(z1)
write (u, "(1x,A,'(',F8.5,',',F8.5,')')") " psi(z1,3) = ", &
real(psiz), aimag(psiz)
psiz = psim (z2,3)
write (u, "(1x,A,'(',F5.2,',',F5.2,')')") " z2 = ", &
real(z2), aimag(z2)
write (u, "(1x,A,'(',F8.5,',',F8.5,')')") " psi(z2,3) = ", &
real(psiz), aimag(psiz)
write (u, "(A)")
psiz = psim (z1,4)
write (u, "(1x,A,'(',F5.2,',',F5.2,')')") " z1 = ", &
real(z1), aimag(z1)
write (u, "(1x,A,'(',F8.5,',',F8.5,')')") " psi(z1,4) = ", &
real(psiz), aimag(psiz)
psiz = psim (z2,4)
write (u, "(1x,A,'(',F5.2,',',F5.2,')')") " z2 = ", &
real(z2), aimag(z2)
write (u, "(1x,A,'(',F8.5,',',F8.5,')')") " psi(z2,4) = ", &
real(psiz), aimag(psiz)
write (u, "(A)")
psiz = psim (z1,5)
write (u, "(1x,A,'(',F5.2,',',F5.2,')')") " z1 = ", &
real(z1), aimag(z1)
write (u, "(1x,A,'(',F8.5,',',F8.5,')')") " psi(z1,5) = ", &
real(psiz), aimag(psiz)
psiz = psim (z2,5)
write (u, "(1x,A,'(',F5.2,',',F5.2,')')") " z2 = ", &
real(z2), aimag(z2)
write (u, "(1x,A,'(',F8.5,',',F8.5,')')") " psi(z2,5) = ", &
real(psiz), aimag(psiz)
write (u, "(A)")
write (u, "(A)") "* Real polygamma function:"
write (u, "(A)")
psir = psimr (x1,1)
write (u, "(1x,A,'(',F8.5,')')") " x1 = ", x1
write (u, "(1x,A,'(',F8.4,')')") " psir = ", psir
write (u, "(A)")
write (u, "(A)") "* Generalized Nielsen polylogarithm:"
write (u, "(A)")
write (u, "(1x,A,F8.5)") " S(1,1,0) = ", &
nielsen(1,1,0._default)
write (u, "(1x,A,F8.5)") " S(1,1,-1) = ", &
nielsen(1,1,-1._default)
write (u, "(1x,A,F8.5)") " S(1,2,-1) = ", &
nielsen(1,2,-1._default)
write (u, "(1x,A,F8.5)") " S(2,1,-1) = ", &
nielsen(2,1,-1._default)
write (u, "(1x,A,F8.5)") " S(1,3,-1) = ", &
nielsen(1,3,-1._default)
write (u, "(1x,A,F8.5)") " S(2,2,-1) = ", &
nielsen(2,2,-1._default)
write (u, "(1x,A,F8.5)") " S(3,1,-1) = ", &
nielsen(3,1,-1._default)
write (u, "(1x,A,F8.5)") " S(1,4,-1) = ", &
nielsen(1,4,-1._default)
write (u, "(1x,A,F8.5)") " S(2,3,-1) = ", &
nielsen(2,3,-1._default)
write (u, "(1x,A,F8.5)") " S(3,2,-1) = ", &
nielsen(3,2,-1._default)
write (u, "(1x,A,F8.5)") " S(4,1,-1) = ", &
nielsen(4,1,-1._default)
write (u, "(1x,A,F8.5)") " S(1,1,0.2) = ", &
nielsen(1,1,0.2_default)
write (u, "(1x,A,F8.5)") " S(1,2,0.2) = ", &
nielsen(1,2,0.2_default)
write (u, "(1x,A,F8.5)") " S(2,1,0.2) = ", &
nielsen(2,1,0.2_default)
write (u, "(1x,A,F8.5)") " S(1,3,0.2) = ", &
nielsen(1,3,0.2_default)
write (u, "(1x,A,F8.5)") " S(2,2,0.2) = ", &
nielsen(2,2,0.2_default)
write (u, "(1x,A,F8.5)") " S(3,1,0.2) = ", &
nielsen(3,1,0.2_default)
write (u, "(1x,A,F8.5)") " S(1,4,0.2) = ", &
nielsen(1,4,0.2_default)
write (u, "(1x,A,F8.5)") " S(2,3,0.2) = ", &
nielsen(2,3,0.2_default)
write (u, "(1x,A,F8.5)") " S(3,2,0.2) = ", &
nielsen(3,2,0.2_default)
write (u, "(1x,A,F8.5)") " S(4,1,0.2) = ", &
nielsen(4,1,0.2_default)
write (u, "(1x,A,F8.5)") " S(1,1,1) = ", &
nielsen(1,1,1._default)
write (u, "(1x,A,F8.5)") " S(1,2,1) = ", &
nielsen(1,2,1._default)
write (u, "(1x,A,F8.5)") " S(2,1,1) = ", &
nielsen(2,1,1._default)
write (u, "(1x,A,F8.5)") " S(1,3,1) = ", &
nielsen(1,3,1._default)
write (u, "(1x,A,F8.5)") " S(2,2,1) = ", &
nielsen(2,2,1._default)
write (u, "(1x,A,F8.5)") " S(3,1,1) = ", &
nielsen(3,1,1._default)
write (u, "(1x,A,F8.5)") " S(1,4,1) = ", &
nielsen(1,4,1._default)
write (u, "(1x,A,F8.5)") " S(2,3,1) = ", &
nielsen(2,3,1._default)
write (u, "(1x,A,F8.5)") " S(3,2,1) = ", &
nielsen(3,2,1._default)
write (u, "(1x,A,F8.5)") " S(4,1,1) = ", &
nielsen(4,1,1._default)
write (u, "(1x,A,F8.5)") " S(1,1,0.75) = ", &
nielsen(1,1,0.75_default)
write (u, "(1x,A,F8.5)") " S(1,3,0.75) = ", &
nielsen(1,3,0.75_default)
write (u, "(1x,A,F8.5)") " S(1,4,0.75) = ", &
nielsen(1,4,0.75_default)
write (u, "(1x,A,F8.5)") " S(2,2,0.75) = ", &
nielsen(2,2,0.75_default)
write (u, "(1x,A,'(',F8.5,',',F8.5,')')") " S(1,1,2) = ", &
real(cnielsen(1,1,3._default)), &
aimag(cnielsen(1,1,3._default))
write (u, "(A)")
write (u, "(A)") "* Dilog, trilog, polylog:"
write (u, "(A)")
write (u, "(1x,A,F8.5)") " Li2(0.66) = ", &
dilog(0.66_default)
write (u, "(1x,A,F8.5)") " Li3(0.66) = ", &
trilog(0.66_default)
write (u, "(1x,A,F8.5)") " Poly(4,0.66) = ", &
polylog(4,0.66_default)
write (u, "(A)")
write (u, "(A)") "* Test output end: sm_physics_3"
end subroutine sm_physics_3
@ %def sm_physics_3
@
\clearpage
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{QCD Coupling}
We provide various distinct implementations of the QCD coupling. In
this module, we define an abstract data type and three
implementations: fixed, running with $\alpha_s(M_Z)$ as input, and
running with $\Lambda_{\text{QCD}}$ as input. We use the functions
defined above in the module [[sm_physics]] but provide a common
interface. Later modules may define additional implementations.
<<[[sm_qcd.f90]]>>=
<<File header>>
module sm_qcd
<<Use kinds>>
use physics_defs
<<Standard module head>>
<<SM qcd: public>>
<<SM qcd: types>>
<<SM qcd: interfaces>>
interface
<<SM qcd: sub interfaces>>
end interface
end module sm_qcd
@ %def sm_qcd
@
<<[[sm_qcd_sub.f90]]>>=
<<File header>>
submodule (sm_qcd) sm_qcd_s
use io_units
use format_defs, only: FMT_12
use numeric_utils
use diagnostics
use md5
use sm_physics
implicit none
contains
<<SM qcd: procedures>>
end submodule sm_qcd_s
@ %def sm_qcd_s
@
\subsection{Coupling: Abstract Data Type}
This is the abstract version of the QCD coupling implementation.
<<SM qcd: public>>=
public :: alpha_qcd_t
<<SM qcd: types>>=
type, abstract :: alpha_qcd_t
contains
<<SM qcd: alpha qcd: TBP>>
end type alpha_qcd_t
@ %def alpha_qcd_t
@ There must be an output routine.
<<SM qcd: alpha qcd: TBP>>=
procedure (alpha_qcd_write), deferred :: write
<<SM qcd: interfaces>>=
abstract interface
subroutine alpha_qcd_write (object, unit)
import
class(alpha_qcd_t), intent(in) :: object
integer, intent(in), optional :: unit
end subroutine alpha_qcd_write
end interface
@ %def alpha_qcd_write
@ This method computes the running coupling, given a certain scale. All
parameters (reference value, order of the approximation, etc.) must be
set before calling this.
<<SM qcd: alpha qcd: TBP>>=
procedure (alpha_qcd_get), deferred :: get
<<SM qcd: interfaces>>=
abstract interface
function alpha_qcd_get (alpha_qcd, scale) result (alpha)
import
class(alpha_qcd_t), intent(in) :: alpha_qcd
real(default), intent(in) :: scale
real(default) :: alpha
end function alpha_qcd_get
end interface
@ %def alpha_qcd_get
@
\subsection{Fixed Coupling}
In this version, the $\alpha_s$ value is fixed, the [[scale]] argument
of the [[get]] method is ignored. There is only one parameter, the
value. By default, this is the value at $M_Z$.
<<SM qcd: public>>=
public :: alpha_qcd_fixed_t
<<SM qcd: types>>=
type, extends (alpha_qcd_t) :: alpha_qcd_fixed_t
real(default) :: val = ALPHA_QCD_MZ_REF
contains
<<SM qcd: alpha qcd fixed: TBP>>
end type alpha_qcd_fixed_t
@ %def alpha_qcd_fixed_t
@ Output.
<<SM qcd: alpha qcd fixed: TBP>>=
procedure :: write => alpha_qcd_fixed_write
<<SM qcd: sub interfaces>>=
module subroutine alpha_qcd_fixed_write (object, unit)
class(alpha_qcd_fixed_t), intent(in) :: object
integer, intent(in), optional :: unit
end subroutine alpha_qcd_fixed_write
<<SM qcd: procedures>>=
module subroutine alpha_qcd_fixed_write (object, unit)
class(alpha_qcd_fixed_t), intent(in) :: object
integer, intent(in), optional :: unit
integer :: u
u = given_output_unit (unit); if (u < 0) return
write (u, "(3x,A)") "QCD parameters (fixed coupling):"
write (u, "(5x,A," // FMT_12 // ")") "alpha = ", object%val
end subroutine alpha_qcd_fixed_write
@ %def alpha_qcd_fixed_write
@ Calculation: the scale is ignored in this case.
<<SM qcd: alpha qcd fixed: TBP>>=
procedure :: get => alpha_qcd_fixed_get
<<SM qcd: sub interfaces>>=
module function alpha_qcd_fixed_get (alpha_qcd, scale) result (alpha)
class(alpha_qcd_fixed_t), intent(in) :: alpha_qcd
real(default), intent(in) :: scale
real(default) :: alpha
end function alpha_qcd_fixed_get
<<SM qcd: procedures>>=
module function alpha_qcd_fixed_get (alpha_qcd, scale) result (alpha)
class(alpha_qcd_fixed_t), intent(in) :: alpha_qcd
real(default), intent(in) :: scale
real(default) :: alpha
alpha = alpha_qcd%val
end function alpha_qcd_fixed_get
@ %def alpha_qcd_fixed_get
@
\subsection{Running Coupling}
In this version, the $\alpha_s$ value runs relative to the value at a
given reference scale. There are two parameters: the value of this
scale (default: $M_Z$), the value of $\alpha_s$ at this scale, and the
number of effective flavors. Furthermore, we have the order of the
approximation.
<<SM qcd: public>>=
public :: alpha_qcd_from_scale_t
<<SM qcd: types>>=
type, extends (alpha_qcd_t) :: alpha_qcd_from_scale_t
real(default) :: mu_ref = MZ_REF
real(default) :: ref = ALPHA_QCD_MZ_REF
integer :: order = 0
integer :: nf = 5
contains
<<SM qcd: alpha qcd from scale: TBP>>
end type alpha_qcd_from_scale_t
@ %def alpha_qcd_from_scale_t
@ Output.
<<SM qcd: alpha qcd from scale: TBP>>=
procedure :: write => alpha_qcd_from_scale_write
<<SM qcd: sub interfaces>>=
module subroutine alpha_qcd_from_scale_write (object, unit)
class(alpha_qcd_from_scale_t), intent(in) :: object
integer, intent(in), optional :: unit
end subroutine alpha_qcd_from_scale_write
<<SM qcd: procedures>>=
module subroutine alpha_qcd_from_scale_write (object, unit)
class(alpha_qcd_from_scale_t), intent(in) :: object
integer, intent(in), optional :: unit
integer :: u
u = given_output_unit (unit); if (u < 0) return
write (u, "(3x,A)") "QCD parameters (running coupling):"
write (u, "(5x,A," // FMT_12 // ")") "Scale mu = ", object%mu_ref
write (u, "(5x,A," // FMT_12 // ")") "alpha(mu) = ", object%ref
write (u, "(5x,A,I0)") "LL order = ", object%order
write (u, "(5x,A,I0)") "N(flv) = ", object%nf
end subroutine alpha_qcd_from_scale_write
@ %def alpha_qcd_from_scale_write
@ Calculation: here, we call the function for running $\alpha_s$ that
was defined in [[sm_physics]] above. The function does not take into
account thresholds, so the number of flavors should be the correct one
for the chosen scale. Normally, this should be the $Z$ boson mass.
<<SM qcd: alpha qcd from scale: TBP>>=
procedure :: get => alpha_qcd_from_scale_get
<<SM qcd: sub interfaces>>=
module function alpha_qcd_from_scale_get (alpha_qcd, scale) result (alpha)
class(alpha_qcd_from_scale_t), intent(in) :: alpha_qcd
real(default), intent(in) :: scale
real(default) :: alpha
end function alpha_qcd_from_scale_get
<<SM qcd: procedures>>=
module function alpha_qcd_from_scale_get (alpha_qcd, scale) result (alpha)
class(alpha_qcd_from_scale_t), intent(in) :: alpha_qcd
real(default), intent(in) :: scale
real(default) :: alpha
alpha = running_as (scale, alpha_qcd%ref, alpha_qcd%mu_ref, &
alpha_qcd%order, real (alpha_qcd%nf, kind=default))
end function alpha_qcd_from_scale_get
@ %def alpha_qcd_from_scale_get
@
\subsection{Running Coupling, determined by $\Lambda_{\text{QCD}}$}
In this version, the inputs are the value $\Lambda_{\text{QCD}}$ and
the order of the approximation.
<<SM qcd: public>>=
public :: alpha_qcd_from_lambda_t
<<SM qcd: types>>=
type, extends (alpha_qcd_t) :: alpha_qcd_from_lambda_t
real(default) :: lambda = LAMBDA_QCD_REF
integer :: order = 0
integer :: nf = 5
contains
<<SM qcd: alpha qcd from lambda: TBP>>
end type alpha_qcd_from_lambda_t
@ %def alpha_qcd_from_lambda_t
@ Output.
<<SM qcd: alpha qcd from lambda: TBP>>=
procedure :: write => alpha_qcd_from_lambda_write
<<SM qcd: sub interfaces>>=
module subroutine alpha_qcd_from_lambda_write (object, unit)
class(alpha_qcd_from_lambda_t), intent(in) :: object
integer, intent(in), optional :: unit
end subroutine alpha_qcd_from_lambda_write
<<SM qcd: procedures>>=
module subroutine alpha_qcd_from_lambda_write (object, unit)
class(alpha_qcd_from_lambda_t), intent(in) :: object
integer, intent(in), optional :: unit
integer :: u
u = given_output_unit (unit); if (u < 0) return
write (u, "(3x,A)") "QCD parameters (Lambda_QCD as input):"
write (u, "(5x,A," // FMT_12 // ")") "Lambda_QCD = ", object%lambda
write (u, "(5x,A,I0)") "LL order = ", object%order
write (u, "(5x,A,I0)") "N(flv) = ", object%nf
end subroutine alpha_qcd_from_lambda_write
@ %def alpha_qcd_from_lambda_write
@ Calculation: here, we call the second function for running $\alpha_s$ that
was defined in [[sm_physics]] above. The $\Lambda$ value should be
the one that is appropriate for the chosen number of effective
flavors. Again, thresholds are not incorporated.
<<SM qcd: alpha qcd from lambda: TBP>>=
procedure :: get => alpha_qcd_from_lambda_get
<<SM qcd: sub interfaces>>=
module function alpha_qcd_from_lambda_get (alpha_qcd, scale) result (alpha)
class(alpha_qcd_from_lambda_t), intent(in) :: alpha_qcd
real(default), intent(in) :: scale
real(default) :: alpha
end function alpha_qcd_from_lambda_get
<<SM qcd: procedures>>=
module function alpha_qcd_from_lambda_get (alpha_qcd, scale) result (alpha)
class(alpha_qcd_from_lambda_t), intent(in) :: alpha_qcd
real(default), intent(in) :: scale
real(default) :: alpha
alpha = running_as_lam (real (alpha_qcd%nf, kind=default), scale, &
alpha_qcd%lambda, alpha_qcd%order)
end function alpha_qcd_from_lambda_get
@ %def alpha_qcd_from_lambda_get
@
\subsection{QCD Wrapper type}
We could get along with a polymorphic QCD type, but a monomorphic wrapper type
with a polymorphic component is easier to handle and probably safer
(w.r.t.\ compiler bugs). However, we keep the object transparent, so we can
set the type-specific parameters directly (by a [[dispatch]] routine).
<<SM qcd: public>>=
public :: qcd_t
<<SM qcd: types>>=
type :: qcd_t
class(alpha_qcd_t), allocatable :: alpha
character(32) :: md5sum = ""
integer :: n_f = -1
contains
<<SM qcd: qcd: TBP>>
end type qcd_t
@ %def qcd_t
@ Output. We first print the polymorphic [[alpha]] which contains a headline,
then any extra components.
<<SM qcd: qcd: TBP>>=
procedure :: write => qcd_write
<<SM qcd: sub interfaces>>=
module subroutine qcd_write (qcd, unit, show_md5sum)
class(qcd_t), intent(in) :: qcd
integer, intent(in), optional :: unit
logical, intent(in), optional :: show_md5sum
end subroutine qcd_write
<<SM qcd: procedures>>=
module subroutine qcd_write (qcd, unit, show_md5sum)
class(qcd_t), intent(in) :: qcd
integer, intent(in), optional :: unit
logical, intent(in), optional :: show_md5sum
logical :: show_md5
integer :: u
u = given_output_unit (unit); if (u < 0) return
show_md5 = .true.; if (present (show_md5sum)) show_md5 = show_md5sum
if (allocated (qcd%alpha)) then
call qcd%alpha%write (u)
else
write (u, "(3x,A)") "QCD parameters (coupling undefined)"
end if
if (show_md5 .and. qcd%md5sum /= "") &
write (u, "(5x,A,A,A)") "md5sum = '", qcd%md5sum, "'"
end subroutine qcd_write
@ %def qcd_write
@ Compute an MD5 sum for the [[alpha_s]] setup. This is
done by writing them to a temporary file, using a standard format.
<<SM qcd: qcd: TBP>>=
procedure :: compute_alphas_md5sum => qcd_compute_alphas_md5sum
<<SM qcd: sub interfaces>>=
module subroutine qcd_compute_alphas_md5sum (qcd)
class(qcd_t), intent(inout) :: qcd
integer :: unit
end subroutine qcd_compute_alphas_md5sum
<<SM qcd: procedures>>=
module subroutine qcd_compute_alphas_md5sum (qcd)
class(qcd_t), intent(inout) :: qcd
integer :: unit
if (allocated (qcd%alpha)) then
unit = free_unit ()
open (unit, status="scratch", action="readwrite")
call qcd%alpha%write (unit)
rewind (unit)
qcd%md5sum = md5sum (unit)
close (unit)
end if
end subroutine qcd_compute_alphas_md5sum
@ %def qcd_compute_alphas_md5sum
@
@ Retrieve the MD5 sum of the qcd setup.
<<SM qcd: qcd: TBP>>=
procedure :: get_md5sum => qcd_get_md5sum
<<SM qcd: sub interfaces>>=
module function qcd_get_md5sum (qcd) result (md5sum)
character(32) :: md5sum
class(qcd_t), intent(inout) :: qcd
end function qcd_get_md5sum
<<SM qcd: procedures>>=
module function qcd_get_md5sum (qcd) result (md5sum)
character(32) :: md5sum
class(qcd_t), intent(inout) :: qcd
md5sum = qcd%md5sum
end function qcd_get_md5sum
@ %def qcd_get_md5sum
@
\subsection{Unit tests}
Test module, followed by the corresponding implementation module.
<<[[sm_qcd_ut.f90]]>>=
<<File header>>
module sm_qcd_ut
use unit_tests
use sm_qcd_uti
<<Standard module head>>
<<SM qcd: public test>>
contains
<<SM qcd: test driver>>
end module sm_qcd_ut
@ %def sm_qcd_ut
@
<<[[sm_qcd_uti.f90]]>>=
<<File header>>
module sm_qcd_uti
<<Use kinds>>
use physics_defs, only: MZ_REF
use sm_qcd
<<Standard module head>>
<<SM qcd: test declarations>>
contains
<<SM qcd: tests>>
end module sm_qcd_uti
@ %def sm_qcd_ut
@ API: driver for the unit tests below.
<<SM qcd: public test>>=
public :: sm_qcd_test
<<SM qcd: test driver>>=
subroutine sm_qcd_test (u, results)
integer, intent(in) :: u
type(test_results_t), intent(inout) :: results
<<SM qcd: execute tests>>
end subroutine sm_qcd_test
@ %def sm_qcd_test
@
\subsubsection{QCD Coupling}
We check two different implementations of the abstract QCD coupling.
<<SM qcd: execute tests>>=
call test (sm_qcd_1, "sm_qcd_1", &
"running alpha_s", &
u, results)
<<SM qcd: test declarations>>=
public :: sm_qcd_1
<<SM qcd: tests>>=
subroutine sm_qcd_1 (u)
integer, intent(in) :: u
type(qcd_t) :: qcd
write (u, "(A)") "* Test output: sm_qcd_1"
write (u, "(A)") "* Purpose: compute running alpha_s"
write (u, "(A)")
write (u, "(A)") "* Fixed:"
write (u, "(A)")
allocate (alpha_qcd_fixed_t :: qcd%alpha)
call qcd%compute_alphas_md5sum ()
call qcd%write (u)
write (u, *)
write (u, "(1x,A,F10.7)") "alpha_s (mz) =", &
qcd%alpha%get (MZ_REF)
write (u, "(1x,A,F10.7)") "alpha_s (1 TeV) =", &
qcd%alpha%get (1000._default)
write (u, *)
deallocate (qcd%alpha)
write (u, "(A)") "* Running from MZ (LO):"
write (u, "(A)")
allocate (alpha_qcd_from_scale_t :: qcd%alpha)
call qcd%compute_alphas_md5sum ()
call qcd%write (u)
write (u, *)
write (u, "(1x,A,F10.7)") "alpha_s (mz) =", &
qcd%alpha%get (MZ_REF)
write (u, "(1x,A,F10.7)") "alpha_s (1 TeV) =", &
qcd%alpha%get (1000._default)
write (u, *)
write (u, "(A)") "* Running from MZ (NLO):"
write (u, "(A)")
select type (alpha => qcd%alpha)
type is (alpha_qcd_from_scale_t)
alpha%order = 1
end select
call qcd%compute_alphas_md5sum ()
call qcd%write (u)
write (u, *)
write (u, "(1x,A,F10.7)") "alpha_s (mz) =", &
qcd%alpha%get (MZ_REF)
write (u, "(1x,A,F10.7)") "alpha_s (1 TeV) =", &
qcd%alpha%get (1000._default)
write (u, *)
write (u, "(A)") "* Running from MZ (NNLO):"
write (u, "(A)")
select type (alpha => qcd%alpha)
type is (alpha_qcd_from_scale_t)
alpha%order = 2
end select
call qcd%compute_alphas_md5sum ()
call qcd%write (u)
write (u, *)
write (u, "(1x,A,F10.7)") "alpha_s (mz) =", &
qcd%alpha%get (MZ_REF)
write (u, "(1x,A,F10.7)") "alpha_s (1 TeV) =", &
qcd%alpha%get (1000._default)
write (u, *)
deallocate (qcd%alpha)
write (u, "(A)") "* Running from Lambda_QCD (LO):"
write (u, "(A)")
allocate (alpha_qcd_from_lambda_t :: qcd%alpha)
call qcd%compute_alphas_md5sum ()
call qcd%write (u)
write (u, *)
write (u, "(1x,A,F10.7)") "alpha_s (mz) =", &
qcd%alpha%get (MZ_REF)
write (u, "(1x,A,F10.7)") "alpha_s (1 TeV) =", &
qcd%alpha%get (1000._default)
write (u, *)
write (u, "(A)") "* Running from Lambda_QCD (NLO):"
write (u, "(A)")
select type (alpha => qcd%alpha)
type is (alpha_qcd_from_lambda_t)
alpha%order = 1
end select
call qcd%compute_alphas_md5sum ()
call qcd%write (u)
write (u, *)
write (u, "(1x,A,F10.7)") "alpha_s (mz) =", &
qcd%alpha%get (MZ_REF)
write (u, "(1x,A,F10.7)") "alpha_s (1 TeV) =", &
qcd%alpha%get (1000._default)
write (u, *)
write (u, "(A)") "* Running from Lambda_QCD (NNLO):"
write (u, "(A)")
select type (alpha => qcd%alpha)
type is (alpha_qcd_from_lambda_t)
alpha%order = 2
end select
call qcd%compute_alphas_md5sum ()
call qcd%write (u)
write (u, *)
write (u, "(1x,A,F10.7)") "alpha_s (mz) =", &
qcd%alpha%get (MZ_REF)
write (u, "(1x,A,F10.7)") "alpha_s (1 TeV) =", &
qcd%alpha%get (1000._default)
write (u, "(A)")
write (u, "(A)") "* Test output end: sm_qcd_1"
end subroutine sm_qcd_1
@ %def sm_qcd_1
@
\clearpage
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{QED Coupling}
On the surface similar to the QCD coupling module but much simpler.
Only a fixed QED couping $\alpha_\text{em}$ is allowed.
Can be extended later if we want to enable a running of
$\alpha_\text{em}$ as well.
<<[[sm_qed.f90]]>>=
<<File header>>
module sm_qed
<<Use kinds>>
use physics_defs
<<Standard module head>>
<<SM qed: public>>
<<SM qed: types>>
<<SM qed: interfaces>>
interface
<<SM qed: sub interfaces>>
end interface
end module sm_qed
@ %def sm_qed
@
<<[[sm_qed_sub.f90]]>>=
<<File header>>
submodule (sm_qed) sm_qed_s
use io_units
use format_defs, only: FMT_12
use md5
use sm_physics
implicit none
contains
<<SM qed: procedures>>
end submodule sm_qed_s
@ %def sm_qed_s
@
\subsection{Coupling: Abstract Data Type}
This is the abstract version of the QCD coupling implementation.
<<SM qed: public>>=
public :: alpha_qed_t
<<SM qed: types>>=
type, abstract :: alpha_qed_t
contains
<<SM qed: alpha qed: TBP>>
end type alpha_qed_t
@ %def alpha_qed_t
@ There must be an output routine.
<<SM qed: alpha qed: TBP>>=
procedure (alpha_qed_write), deferred :: write
<<SM qed: interfaces>>=
abstract interface
subroutine alpha_qed_write (object, unit)
import
class(alpha_qed_t), intent(in) :: object
integer, intent(in), optional :: unit
end subroutine alpha_qed_write
end interface
@ %def alpha_qed_write
@ This method computes the running coupling, given a certain scale. All
parameters (reference value, order of the approximation, etc.) must be
set before calling this.
<<SM qed: alpha qed: TBP>>=
procedure (alpha_qed_get), deferred :: get
<<SM qed: interfaces>>=
abstract interface
function alpha_qed_get (alpha_qed, scale) result (alpha)
import
class(alpha_qed_t), intent(in) :: alpha_qed
real(default), intent(in) :: scale
real(default) :: alpha
end function alpha_qed_get
end interface
@ %def alpha_qed_get
@
\subsection{Fixed Coupling}
In this version, the $\alpha$ value is fixed, the [[scale]] argument
of the [[get]] method is ignored. There is only one parameter, the
value. The default depends on the electroweak scheme chosen in the
model.
<<SM qed: public>>=
public :: alpha_qed_fixed_t
<<SM qed: types>>=
type, extends (alpha_qed_t) :: alpha_qed_fixed_t
real(default) :: val = ALPHA_QED_ME_REF
contains
<<SM qed: alpha qed fixed: TBP>>
end type alpha_qed_fixed_t
@ %def alpha_qed_fixed_t
@ Output.
<<SM qed: alpha qed fixed: TBP>>=
procedure :: write => alpha_qed_fixed_write
<<SM qed: sub interfaces>>=
module subroutine alpha_qed_fixed_write (object, unit)
class(alpha_qed_fixed_t), intent(in) :: object
integer, intent(in), optional :: unit
end subroutine alpha_qed_fixed_write
<<SM qed: procedures>>=
module subroutine alpha_qed_fixed_write (object, unit)
class(alpha_qed_fixed_t), intent(in) :: object
integer, intent(in), optional :: unit
integer :: u
u = given_output_unit (unit); if (u < 0) return
write (u, "(3x,A)") "QED parameters (fixed coupling):"
write (u, "(5x,A," // FMT_12 // ")") "alpha = ", object%val
end subroutine alpha_qed_fixed_write
@ %def alpha_qed_fixed_write
@ Calculation: the scale is ignored in this case.
<<SM qed: alpha qed fixed: TBP>>=
procedure :: get => alpha_qed_fixed_get
<<SM qed: sub interfaces>>=
module function alpha_qed_fixed_get (alpha_qed, scale) result (alpha)
class(alpha_qed_fixed_t), intent(in) :: alpha_qed
real(default), intent(in) :: scale
real(default) :: alpha
end function alpha_qed_fixed_get
<<SM qed: procedures>>=
module function alpha_qed_fixed_get (alpha_qed, scale) result (alpha)
class(alpha_qed_fixed_t), intent(in) :: alpha_qed
real(default), intent(in) :: scale
real(default) :: alpha
alpha = alpha_qed%val
end function alpha_qed_fixed_get
@ %def alpha_qed_fixed_get
@
\subsection{Running Coupling}
In this version, the $\alpha$ value runs relative to the value at a
given reference scale. There are two parameters: the value of this
scale (default: $M_Z$), the value of $\alpha$ at this scale, and the
number of effective flavors. Furthermore, we have the order of the
approximation.
<<SM qed: public>>=
public :: alpha_qed_from_scale_t
<<SM qed: types>>=
type, extends (alpha_qed_t) :: alpha_qed_from_scale_t
real(default) :: mu_ref = ME_REF
real(default) :: ref = ALPHA_QED_ME_REF
integer :: order = 0
integer :: nf = 5
integer :: nlep = 1
logical :: analytic = .true.
contains
<<SM qed: alpha qed from scale: TBP>>
end type alpha_qed_from_scale_t
@ %def alpha_qed_from_scale_t
@ Output.
<<SM qed: alpha qed from scale: TBP>>=
procedure :: write => alpha_qed_from_scale_write
<<SM qed: sub interfaces>>=
module subroutine alpha_qed_from_scale_write (object, unit)
class(alpha_qed_from_scale_t), intent(in) :: object
integer, intent(in), optional :: unit
end subroutine alpha_qed_from_scale_write
<<SM qed: procedures>>=
module subroutine alpha_qed_from_scale_write (object, unit)
class(alpha_qed_from_scale_t), intent(in) :: object
integer, intent(in), optional :: unit
integer :: u
u = given_output_unit (unit); if (u < 0) return
write (u, "(3x,A)") "QED parameters (running coupling):"
write (u, "(5x,A," // FMT_12 // ")") "Scale mu = ", object%mu_ref
write (u, "(5x,A," // FMT_12 // ")") "alpha(mu) = ", object%ref
write (u, "(5x,A,I0)") "LL order = ", object%order
write (u, "(5x,A,I0)") "N(flv) = ", object%nf
write (u, "(5x,A,I0)") "N(lep) = ", object%nlep
write (u, "(5x,A,L1)") "analytic = ", object%analytic
end subroutine alpha_qed_from_scale_write
@ %def alpha_qed_from_scale_write
@ Calculation: here, we call the function for running $\alpha_s$ that
was defined in [[sm_physics]] above. The function does not take into
account thresholds, so the number of flavors should be the correct one
for the chosen scale. Normally, this should be the $Z$ boson mass.
<<SM qed: alpha qed from scale: TBP>>=
procedure :: get => alpha_qed_from_scale_get
<<SM qed: sub interfaces>>=
module function alpha_qed_from_scale_get (alpha_qed, scale) result (alpha)
class(alpha_qed_from_scale_t), intent(in) :: alpha_qed
real(default), intent(in) :: scale
real(default) :: alpha
end function alpha_qed_from_scale_get
<<SM qed: procedures>>=
module function alpha_qed_from_scale_get (alpha_qed, scale) result (alpha)
class(alpha_qed_from_scale_t), intent(in) :: alpha_qed
real(default), intent(in) :: scale
real(default) :: alpha
if (alpha_qed%analytic) then
alpha = running_alpha (scale, alpha_qed%ref, alpha_qed%mu_ref, &
alpha_qed%order, alpha_qed%nf, alpha_qed%nlep)
else
alpha = running_alpha_num (scale, alpha_qed%ref, alpha_qed%mu_ref, &
alpha_qed%order, alpha_qed%nf, alpha_qed%nlep)
end if
end function alpha_qed_from_scale_get
@ %def alpha_qed_from_scale_get
@
\subsection{QED type}
This module is similar to [[qcd_t]], defining the type [[qed_t]].
It stores the [[alpha_qed_t]] type which is either constant or a running $\alpha$
with different options.
<<SM qed: public>>=
public :: qed_t
<<SM qed: types>>=
type :: qed_t
class(alpha_qed_t), allocatable :: alpha
character(32) :: md5sum = ""
integer :: n_f = -1
integer :: n_lep = -1
contains
<<SM qed: qed: TBP>>
end type qed_t
@ %def qed_t
Output. We first print the polymorphic [[alpha]] which contains a headline,
then any extra components.
<<SM qed: qed: TBP>>=
procedure :: write => qed_write
<<SM qed: sub interfaces>>=
module subroutine qed_write (qed, unit, show_md5sum)
class(qed_t), intent(in) :: qed
integer, intent(in), optional :: unit
logical, intent(in), optional :: show_md5sum
end subroutine qed_write
<<SM qed: procedures>>=
module subroutine qed_write (qed, unit, show_md5sum)
class(qed_t), intent(in) :: qed
integer, intent(in), optional :: unit
logical, intent(in), optional :: show_md5sum
logical :: show_md5
integer :: u
u = given_output_unit (unit); if (u < 0) return
show_md5 = .true.; if (present (show_md5sum)) show_md5 = show_md5sum
if (allocated (qed%alpha)) then
call qed%alpha%write (u)
else
write (u, "(3x,A)") "QED parameters (coupling undefined)"
end if
if (show_md5 .and. qed%md5sum /= "") &
write (u, "(5x,A,A,A)") "md5sum = '", qed%md5sum, "'"
end subroutine qed_write
@ % def qed_write
@ Compute an MD5 sum for the [[alpha]] setup. This is
done by writing them to a temporary file, using a standard format.
<<SM qed: qed: TBP>>=
procedure :: compute_alpha_md5sum => qed_compute_alpha_md5sum
<<SM qed: sub interfaces>>=
module subroutine qed_compute_alpha_md5sum (qed)
class(qed_t), intent(inout) :: qed
integer :: unit
end subroutine qed_compute_alpha_md5sum
<<SM qed: procedures>>=
module subroutine qed_compute_alpha_md5sum (qed)
class(qed_t), intent(inout) :: qed
integer :: unit
if (allocated (qed%alpha)) then
unit = free_unit ()
open (unit, status="scratch", action="readwrite")
call qed%alpha%write (unit)
rewind (unit)
qed%md5sum = md5sum (unit)
close (unit)
end if
end subroutine qed_compute_alpha_md5sum
@ %def qed_compute_alphas_md5sum
@
@ Retrieve the MD5 sum of the qed setup.
<<SM qed: qed: TBP>>=
procedure :: get_md5sum => qed_get_md5sum
<<SM qed: sub interfaces>>=
module function qed_get_md5sum (qed) result (md5sum)
character(32) :: md5sum
class(qed_t), intent(inout) :: qed
end function qed_get_md5sum
<<SM qed: procedures>>=
module function qed_get_md5sum (qed) result (md5sum)
character(32) :: md5sum
class(qed_t), intent(inout) :: qed
md5sum = qed%md5sum
end function qed_get_md5sum
@ %def qed_get_md5sum
@
\subsection{Unit tests}
Test module, followed by the corresponding implementation module.
<<[[sm_qed_ut.f90]]>>=
<<File header>>
module sm_qed_ut
use unit_tests
use sm_qed_uti
<<Standard module head>>
<<SM qed: public test>>
contains
<<SM qed: test driver>>
end module sm_qed_ut
@ %def sm_qed_ut
@
<<[[sm_qed_uti.f90]]>>=
<<File header>>
module sm_qed_uti
<<Use kinds>>
use physics_defs, only: ME_REF
use sm_qed
<<Standard module head>>
<<SM qed: test declarations>>
contains
<<SM qed: tests>>
end module sm_qed_uti
@ %def sm_qed_ut
@ API: driver for the unit tests below.
<<SM qed: public test>>=
public :: sm_qed_test
<<SM qed: test driver>>=
subroutine sm_qed_test (u, results)
integer, intent(in) :: u
type(test_results_t), intent(inout) :: results
<<SM qed: execute tests>>
end subroutine sm_qed_test
@ %def sm_qed_test
@
\subsubsection{QED Coupling}
We check two different implementations of the abstract QED coupling.
<<SM qed: execute tests>>=
call test (sm_qed_1, "sm_qed_1", &
"running alpha", &
u, results)
<<SM qed: test declarations>>=
public :: sm_qed_1
<<SM qed: tests>>=
subroutine sm_qed_1 (u)
integer, intent(in) :: u
type(qed_t) :: qed
write (u, "(A)") "* Test output: sm_qed_1"
write (u, "(A)") "* Purpose: compute running alpha"
write (u, "(A)")
write (u, "(A)") "* Fixed:"
write (u, "(A)")
allocate (alpha_qed_fixed_t :: qed%alpha)
call qed%compute_alpha_md5sum ()
call qed%write (u)
write (u, *)
write (u, "(1x,A,F10.7)") "alpha (me) =", &
qed%alpha%get (ME_REF)
write (u, "(1x,A,F10.7)") "alpha (10 GeV) =", &
qed%alpha%get (10._default)
write (u, "(1x,A,F10.7)") "alpha (1 TeV) =", &
qed%alpha%get (1000._default)
write (u, *)
deallocate (qed%alpha)
write (u, "(A)") "* Running from me (LO):"
write (u, "(A)")
allocate (alpha_qed_from_scale_t :: qed%alpha)
call qed%compute_alpha_md5sum ()
call qed%write (u)
write (u, *)
write (u, "(1x,A,F10.7)") "alpha (me) =", &
qed%alpha%get (ME_REF)
write (u, "(1x,A,F10.7)") "alpha (10 GeV) =", &
qed%alpha%get (10._default)
write (u, "(1x,A,F10.7)") "alpha (1 TeV) =", &
qed%alpha%get (1000._default)
write (u, *)
write (u, "(A)") "* Running from me (NLO, analytic):"
write (u, "(A)")
select type (alpha => qed%alpha)
type is (alpha_qed_from_scale_t)
alpha%order = 1
end select
call qed%compute_alpha_md5sum ()
call qed%write (u)
write (u, *)
write (u, "(1x,A,F10.7)") "alpha (me) =", &
qed%alpha%get (ME_REF)
write (u, "(1x,A,F10.7)") "alpha (10 GeV) =", &
qed%alpha%get (10._default)
write (u, "(1x,A,F10.7)") "alpha (1 TeV) =", &
qed%alpha%get (1000._default)
write (u, *)
write (u, "(A)") "* Running from me (NLO, numeric):"
write (u, "(A)")
select type (alpha => qed%alpha)
type is (alpha_qed_from_scale_t)
alpha%order = 1
alpha%analytic = .false.
end select
call qed%compute_alpha_md5sum ()
call qed%write (u)
write (u, *)
write (u, "(1x,A,F10.7)") "alpha (me) =", &
qed%alpha%get (ME_REF)
write (u, "(1x,A,F10.7)") "alpha (10 GeV) =", &
qed%alpha%get (10._default)
write (u, "(1x,A,F10.7)") "alpha (1 TeV) =", &
qed%alpha%get (1000._default)
write (u, *)
deallocate (qed%alpha)
write (u, "(A)")
write (u, "(A)") "* Test output end: sm_qed_1"
end subroutine sm_qed_1
@ %def sm_qed_1
@
\clearpage
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Shower algorithms}
<<[[shower_algorithms.f90]]>>=
<<File header>>
module shower_algorithms
<<Use kinds>>
<<Standard module head>>
<<Shower algorithms: public>>
<<Shower algorithms: interfaces>>
interface
<<Shower algorithms: sub interfaces>>
end interface
end module shower_algorithms
@ %def shower_algorithms
<<[[shower_algorithms_sub.f90]]>>=
<<File header>>
submodule (shower_algorithms) shower_algorithms_s
use diagnostics
use constants
implicit none
contains
<<Shower algorithms: procedures>>
<<Shower algorithms: tests>>
end submodule shower_algorithms_s
@ %def shower_algorithms_s
@
@ We want to generate emission variables [[x]]$\in\mathds{R}^d$
proportional to
\begin{align}
&\quad f(x)\; \Delta(f, h(x)) \quad\text{with}\\
\Delta(f, H) &= \exp\left\{-\int\text{d}^d x'f(x') \Theta(h(x') -
H)\right\}
\end{align}
The [[true_function]] $f$ is however too complicated and we are only
able to generate [[x]] according to the [[overestimator]] $F$. This
algorithm is described in Appendix B of 0709.2092 and is proven e.g.~in
1211.7204 and hep-ph/0606275. Intuitively speaking, we overestimate the
emission probability and can therefore set [[scale_max = scale]] if the
emission is rejected.
<<Shower algorithms: sub interfaces>>=
module subroutine generate_vetoed (x, overestimator, true_function, &
sudakov, inverse_sudakov, scale_min)
real(default), dimension(:), intent(out) :: x
!class(rng_t), intent(inout) :: rng
procedure(XXX_function), pointer, intent(in) :: overestimator, true_function
procedure(sudakov_p), pointer, intent(in) :: sudakov, inverse_sudakov
real(default), intent(in) :: scale_min
end subroutine generate_vetoed
<<Shower algorithms: procedures>>=
module subroutine generate_vetoed (x, overestimator, true_function, &
sudakov, inverse_sudakov, scale_min)
real(default), dimension(:), intent(out) :: x
!class(rng_t), intent(inout) :: rng
procedure(XXX_function), pointer, intent(in) :: overestimator, true_function
procedure(sudakov_p), pointer, intent(in) :: sudakov, inverse_sudakov
real(default), intent(in) :: scale_min
real(default) :: random, scale_max, scale
scale_max = inverse_sudakov (one)
do while (scale_max > scale_min)
!call rng%generate (random)
scale = inverse_sudakov (random * sudakov (scale_max))
call generate_on_hypersphere (x, overestimator, scale)
!call rng%generate (random)
if (random < true_function (x) / overestimator (x)) then
return !!! accept x
end if
scale_max = scale
end do
end subroutine generate_vetoed
@ %def generate_vetoed
@
<<Shower algorithms: procedures>>=
subroutine generate_on_hypersphere (x, overestimator, scale)
real(default), dimension(:), intent(out) :: x
procedure(XXX_function), pointer, intent(in) :: overestimator
real(default), intent(in) :: scale
call msg_bug ("generate_on_hypersphere: not implemented")
end subroutine generate_on_hypersphere
@ %def generate_on_hypersphere
@
<<Shower algorithms: interfaces>>=
interface
pure function XXX_function (x)
import
real(default) :: XXX_function
real(default), dimension(:), intent(in) :: x
end function XXX_function
end interface
interface
pure function sudakov_p (x)
import
real(default) :: sudakov_p
real(default), intent(in) :: x
end function sudakov_p
end interface
@
\subsection{Unit tests}
(Currently unused.)
<<XXX Shower algorithms: public>>=
public :: shower_algorithms_test
<<XXX Shower algorithms: tests>>=
subroutine shower_algorithms_test (u, results)
integer, intent(in) :: u
type(test_results_t), intent(inout) :: results
<<Shower algorithms: execute tests>>
end subroutine shower_algorithms_test
@ %def shower_algorithms_test
@
\subsubsection{Splitting functions}
<<XXX shower algorithms: execute tests>>=
call test (shower_algorithms_1, "shower_algorithms_1", &
"veto technique", &
u, results)
<<XXX shower algorithms: tests>>=
subroutine shower_algorithms_1 (u)
integer, intent(in) :: u
write (u, "(A)") "* Test output: shower_algorithms_1"
write (u, "(A)") "* Purpose: check veto technique"
write (u, "(A)")
write (u, "(A)") "* Splitting functions:"
write (u, "(A)")
!call assert (u, vanishes (p_qqg_pol (z, +1, -1, +1)))
!call assert (u, nearly_equal ( &
!p_qqg_pol (z, +1, +1, -1) + p_qqg_pol (z, +1, +1, +1),
!p_qqg (z))
write (u, "(A)")
write (u, "(A)") "* Test output end: shower_algorithms_1"
end subroutine shower_algorithms_1
@ %def shower_algorithms_1

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