Index: trunk/src/physics/physics.nw =================================================================== --- trunk/src/physics/physics.nw (revision 8904) +++ trunk/src/physics/physics.nw (revision 8905) @@ -1,10074 +1,10074 @@ % -*- 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]]>>= <> module physics_defs <> <> use constants, only: one, two, three <> <> <> <> interface <> end interface end module physics_defs @ %def physics_defs @ <<[[physics_defs_sub.f90]]>>= <> submodule (physics_defs) physics_defs_s implicit none contains <> end submodule physics_defs_s @ %def physics_defs_s @ \subsection{Units} Conversion from energy units to cross-section units. <>= real(default), parameter, public :: & conv = 0.38937966e12_default @ Conversion from millimeter to nanoseconds for lifetimes. <>= real(default), parameter, public :: & ns_per_mm = 1.e6_default / 299792458._default @ Rescaling factor. <>= real(default), parameter, public :: & pb_per_fb = 1.e-3_default @ String for the default energy and cross-section units. <>= character(*), parameter, public :: & energy_unit = "GeV" character(*), parameter, public :: & cross_section_unit = "fb" @ \subsection{SM and QCD constants} <>= 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. <>= 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: <>= integer, parameter, public :: UNDEFINED = 0 @ %def UNDEFINED @ SM fermions: <>= 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: <>= 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: <>= integer, parameter, public :: PION = 111 integer, parameter, public :: PIPLUS = 211 integer, parameter, public :: PIMINUS = - PIPLUS @ %def PION PIPLUS PIMINUS @ Di-Quarks: <>= integer, parameter, public :: UD0 = 2101 integer, parameter, public :: UD1 = 2103 integer, parameter, public :: UU1 = 2203 @ %def UD0 UD1 UU1 @ Mesons: <>= 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: <>= 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: <>= integer, parameter, public :: SIGMAPLUS = 3222 integer, parameter, public :: SIGMA0 = 3212 integer, parameter, public :: SIGMAMINUS = 3112 @ %def SIGMAPLUS SIGMA0 SIGMAMINUS @ Charmed baryons: <>= integer, parameter, public :: SIGMACPLUSPLUS = 4222 integer, parameter, public :: SIGMACPLUS = 4212 integer, parameter, public :: SIGMAC0 = 4112 @ %def SIGMACPLUSPLUS SIGMACPLUS SIGMAC0 @ Bottom baryons: <>= integer, parameter, public :: SIGMAB0 = 5212 integer, parameter, public :: SIGMABPLUS = 5222 @ %def SIGMAB0 SIGMABPLUS @ 81-100 are reserved for internal codes. Hadron and beam remnants: <>= 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: <>= 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. <>= 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. <>= 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. <>= integer, parameter, public :: n_beams_rescaled = 2 @ %def n_beams_rescaled @ Orders of the electron PDFs. <>= integer, parameter, public :: EPDF_LL = 0 integer, parameter, public :: EPDF_NLL = 1 @ %def EPDF_LL EPDF_NLL @ <>= public :: component_status <>= interface component_status module procedure component_status_of_string module procedure component_status_to_string end interface <>= 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 <>= 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 @ <>= public :: is_nlo_component <>= elemental module function is_nlo_component (comp) result (is_nlo) logical :: is_nlo integer, intent(in) :: comp end function is_nlo_component <>= 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 @ <>= public :: is_subtraction_component <>= 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 <>= 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 <>= 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 @ @ <>= public :: thr_leg <>= module function thr_leg (emitter) result (leg) integer :: leg integer, intent(in) :: emitter end function thr_leg <>= 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]]>>= <> module c_particles use, intrinsic :: iso_c_binding !NODEP! <> <> <> interface <> end interface end module c_particles @ %def c_particles @ <<[[c_particles_sub.f90]]>>= <> submodule (c_particles) c_particles_s use io_units use format_defs, only: FMT_14, FMT_19 implicit none contains <> end submodule c_particles_s @ %def c_particles_s @ <>= public :: c_prt_t <>= 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]]. <>= public :: c_prt_write <>= module subroutine c_prt_write (prt, unit) type(c_prt_t), intent(in) :: prt integer, intent(in), optional :: unit end subroutine c_prt_write <>= 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]]>>= <> module lorentz <> use constants, only: zero, one use c_particles <> <> <> <> <> <> <> interface <> end interface end module lorentz @ %def lorentz @ <<[[lorentz_sub.f90]]>>= <> 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 <> 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. <>= public :: vector3_t <>= type :: vector3_t real(default), dimension(3) :: p end type vector3_t @ %def vector3_t @ Output a vector <>= public :: vector3_write <>= 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 <>= 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 <>= public :: vector3_null <>= type(vector3_t), parameter :: vector3_null = & vector3_t ([ zero, zero, zero ]) @ %def vector3_null @ Canonical three-vector: <>= public :: vector3_canonical <>= elemental module function vector3_canonical (k) result (p) type(vector3_t) :: p integer, intent(in) :: k end function vector3_canonical <>= 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. <>= public :: vector3_moving <>= interface vector3_moving module procedure vector3_moving_canonical module procedure vector3_moving_generic end interface <>= 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 <>= 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 <>= public :: operator(==), operator(/=) <>= interface operator(==) module procedure vector3_eq end interface interface operator(/=) module procedure vector3_neq end interface <>= 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 <>= 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 <>= public :: operator(+), operator(-) <>= interface operator(+) module procedure add_vector3 end interface interface operator(-) module procedure sub_vector3 end interface <>= 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 <>= 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: <>= public :: operator(*), operator(/) <>= 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 <>= 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 <>= 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: <>= interface operator(*) module procedure prod_vector3 end interface <>= elemental module function prod_vector3 (p, q) result (s) real(default) :: s type(vector3_t), intent(in) :: p,q end function prod_vector3 <>= 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 * <>= public :: cross_product <>= interface cross_product module procedure vector3_cross_product end interface <>= 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 <>= 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]]. <>= public :: operator(**) <>= interface operator(**) module procedure power_vector3 end interface <>= 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 <>= 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. <>= interface operator(-) module procedure negate_vector3 end interface <>= elemental module function negate_vector3 (p) result (q) type(vector3_t) :: q type(vector3_t), intent(in) :: p end function negate_vector3 <>= 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: <>= public :: sum <>= interface sum module procedure sum_vector3 end interface @ %def sum @ <>= public :: vector3_set_component <>= 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 <>= 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 @ <>= pure module function sum_vector3 (p) result (q) type(vector3_t) :: q type(vector3_t), dimension(:), intent(in) :: p end function sum_vector3 <>= 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: <>= public :: vector3_get_component @ %def component <>= 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 <>= 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. <>= public :: vector3_get_components <>= 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 <>= 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. <>= public :: direction <>= interface direction module procedure vector3_get_direction end interface <>= elemental module function vector3_get_direction (p) result (q) type(vector3_t) :: q type(vector3_t), intent(in) :: p end function vector3_get_direction <>= 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. <>= public :: vector4_t <>= type :: vector4_t real(default), dimension(0:3) :: p = & [zero, zero, zero, zero] contains <> end type vector4_t @ %def vector4_t @ Output a vector <>= public :: vector4_write <>= procedure :: write => vector4_write <>= 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 <>= 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 <>= public :: vector4_write_raw public :: vector4_read_raw <>= 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 <>= 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 <>= public :: vector4_null <>= type(vector4_t), parameter :: vector4_null = & vector4_t ([ zero, zero, zero, zero ]) @ %def vector4_null @ Canonical four-vector: <>= public :: vector4_canonical <>= elemental module function vector4_canonical (k) result (p) type(vector4_t) :: p integer, intent(in) :: k end function vector4_canonical <>= 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: <>= public :: vector4_at_rest <>= elemental module function vector4_at_rest (m) result (p) type(vector4_t) :: p real(default), intent(in) :: m end function vector4_at_rest <>= 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) <>= public :: vector4_moving <>= interface vector4_moving module procedure vector4_moving_canonical module procedure vector4_moving_generic end interface <>= 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 <>= 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 <>= interface operator(==) module procedure vector4_eq end interface interface operator(/=) module procedure vector4_neq end interface <>= 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 <>= 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: <>= interface operator(+) module procedure add_vector4 end interface interface operator(-) module procedure sub_vector4 end interface <>= 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 <>= 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: <>= 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 <>= 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 <>= 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: <>= interface operator(*) module procedure prod_vector4 end interface interface operator(**) module procedure power_vector4 end interface <>= elemental module function prod_vector4 (p, q) result (s) real(default) :: s type(vector4_t), intent(in) :: p,q end function prod_vector4 <>= 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]]. <>= 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 <>= 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 <>= interface operator(-) module procedure negate_vector4 end interface <>= elemental module function negate_vector4 (p) result (q) type(vector4_t) :: q type(vector4_t), intent(in) :: p end function negate_vector4 <>= 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: <>= interface sum module procedure sum_vector4, sum_vector4_mask end interface @ %def sum @ <>= 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 <>= 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: <>= public :: vector4_set_component <>= 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 <>= 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: <>= public :: vector4_get_component <>= 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 <>= 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. <>= public :: vector4_get_components <>= 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 <>= 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: <>= public :: space_part <>= interface space_part module procedure vector4_get_space_part end interface <>= 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 <>= 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. <>= interface direction module procedure vector4_get_direction end interface <>= elemental module function vector4_get_direction (p) result (q) type(vector3_t) :: q type(vector4_t), intent(in) :: p end function vector4_get_direction <>= 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 <>= public :: vector4_invert_direction <>= elemental module subroutine vector4_invert_direction (p) type(vector4_t), intent(inout) :: p end subroutine vector4_invert_direction <>= 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. <>= public :: assignment (=) <>= 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 <>= 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 <>= 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 @ <>= public :: vector4 <>= pure module function vector4 (a) result (p) type(vector4_t) :: p real(default), intent(in), dimension(4) :: a end function vector4 <>= 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 @ <>= procedure :: to_pythia6 => vector4_to_pythia6 <>= 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 <>= 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: <>= interface assignment (=) module procedure vector4_from_c_prt, c_prt_from_vector4 end interface <>= 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 <>= 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. <>= public :: vector4_to_c_prt <>= 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 <>= 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. <>= public :: azimuthal_angle <>= interface azimuthal_angle module procedure vector3_azimuthal_angle module procedure vector4_azimuthal_angle end interface <>= 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 <>= 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 <>= public :: azimuthal_angle_deg <>= interface azimuthal_angle_deg module procedure vector3_azimuthal_angle_deg module procedure vector4_azimuthal_angle_deg end interface <>= 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 <>= 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 $-\pi<\Delta\phi\leq\pi$. <>= public :: azimuthal_distance <>= interface azimuthal_distance module procedure vector3_azimuthal_distance module procedure vector4_azimuthal_distance end interface <>= 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 <>= 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: <>= public :: azimuthal_distance_deg <>= interface azimuthal_distance_deg module procedure vector3_azimuthal_distance_deg module procedure vector4_azimuthal_distance_deg end interface <>= 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 <>= 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. <>= public :: polar_angle <>= interface polar_angle module procedure polar_angle_vector3 module procedure polar_angle_vector4 end interface <>= 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 <>= 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: $-1\leq\cos\theta\leq 1$. <>= public :: polar_angle_ct <>= interface polar_angle_ct module procedure polar_angle_ct_vector3 module procedure polar_angle_ct_vector4 end interface <>= 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 <>= 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. <>= public :: polar_angle_deg <>= interface polar_angle_deg module procedure polar_angle_deg_vector3 module procedure polar_angle_deg_vector4 end interface <>= 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 <>= 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. <>= public :: enclosed_angle <>= interface enclosed_angle module procedure enclosed_angle_vector3 module procedure enclosed_angle_vector4 end interface <>= 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 <>= 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. <>= public :: enclosed_angle_ct <>= interface enclosed_angle_ct module procedure enclosed_angle_ct_vector3 module procedure enclosed_angle_ct_vector4 end interface <>= 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 <>= 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. <>= public :: enclosed_angle_deg <>= interface enclosed_angle_deg module procedure enclosed_angle_deg_vector3 module procedure enclosed_angle_deg_vector4 end interface <>= 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 <>= 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. <>= public :: enclosed_angle_rest_frame public :: enclosed_angle_ct_rest_frame public :: enclosed_angle_deg_rest_frame <>= 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 <>= 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 <>= 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) <>= public :: transverse_part <>= interface transverse_part module procedure transverse_part_vector4_beam_axis module procedure transverse_part_vector4_vector4 end interface <>= 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 <>= 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]]. <>= public :: longitudinal_part <>= interface longitudinal_part module procedure longitudinal_part_vector4 end interface <>= elemental module function longitudinal_part_vector4 (p) result (pL) real(default) :: pL type(vector4_t), intent(in) :: p end function longitudinal_part_vector4 <>= 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 <>= public :: space_part_norm <>= interface space_part_norm module procedure space_part_norm_vector4 end interface <>= 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 <>= 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) <>= public :: energy <>= interface energy module procedure energy_vector4 module procedure energy_vector3 module procedure energy_real end interface <>= 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 <>= 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 <>= 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. <>= public :: invariant_mass <>= interface invariant_mass module procedure invariant_mass_vector4 end interface <>= elemental module function invariant_mass_vector4 (p) result (m) real(default) :: m type(vector4_t), intent(in) :: p end function invariant_mass_vector4 <>= 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. <>= public :: invariant_mass_squared <>= interface invariant_mass_squared module procedure invariant_mass_squared_vector4 end interface <>= 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 <>= 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. <>= public :: transverse_mass <>= interface transverse_mass module procedure transverse_mass_vector4 end interface <>= elemental module function transverse_mass_vector4 (p) result (m) real(default) :: m type(vector4_t), intent(in) :: p end function transverse_mass_vector4 <>= 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 $p_\perp>0$) <>= public :: rapidity <>= interface rapidity module procedure rapidity_vector4 end interface <>= elemental module function rapidity_vector4 (p) result (y) real(default) :: y type(vector4_t), intent(in) :: p end function rapidity_vector4 <>= 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 $p_\perp>0$) <>= public :: pseudorapidity <>= interface pseudorapidity module procedure pseudorapidity_vector4 end interface <>= elemental module function pseudorapidity_vector4 (p) result (eta) real(default) :: eta type(vector4_t), intent(in) :: p end function pseudorapidity_vector4 <>= 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 $p_\perp>0$) <>= public :: rapidity_distance <>= interface rapidity_distance module procedure rapidity_distance_vector4 end interface <>= 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 <>= 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 $p_\perp>0$) <>= public :: pseudorapidity_distance <>= interface pseudorapidity_distance module procedure pseudorapidity_distance_vector4 end interface <>= 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 <>= 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: <>= public :: eta_phi_distance <>= interface eta_phi_distance module procedure eta_phi_distance_vector4 end interface <>= 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 <>= 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} <>= public :: lorentz_transformation_t <>= type :: lorentz_transformation_t private real(default), dimension(0:3, 0:3) :: L contains <> end type lorentz_transformation_t @ %def lorentz_transformation_t @ Output: <>= public :: lorentz_transformation_write <>= procedure :: write => lorentz_transformation_write <>= 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 <>= 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: <>= public :: lorentz_transformation_get_components <>= 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 <>= 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. <>= public :: inverse <>= interface inverse module procedure lorentz_transformation_inverse end interface <>= 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 <>= 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. <>= 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: <>= public :: identity <>= 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 <>= public :: space_reflection <>= 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. <>= public :: create_orthogonal <>= 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 <>= 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 @ <>= public :: create_unit_vector <>= 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 <>= 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 @ <>= public :: normalize <>= module function normalize(p) result (p_norm) type(vector3_t) :: p_norm type(vector3_t), intent(in) :: p end function normalize <>= 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]]. <>= public :: compute_resonance_mass <>= 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 <>= 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 @ <>= public :: get_resonance_momentum <>= 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 <>= 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 @ <>= public :: create_two_particle_decay <>= 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 <>= 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*} <>= public :: create_three_particle_decay <>= 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 <>= 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 @ <>= public :: evaluate_one_to_two_splitting_special <>= 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 @ <>= public :: generate_on_shell_decay <>= 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 <>= 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} <>= public :: boost <>= 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 $(m;\vec 0)$. <>= 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 <>= 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. <>= 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 <>= 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. <>= 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 <>= 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. <>= public :: rotation <>= 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. <>= 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 <>= 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. <>= 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. <>= public :: rotation_to_2nd <>= interface rotation_to_2nd module procedure rotation_to_2nd_generic module procedure rotation_to_2nd_canonical end interface <>= 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 <>= 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 $p$. <>= 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 $(p_1+p_2)^2 = m^2$. <>= public :: transformation <>= interface transformation module procedure transformation_rec_generic module procedure transformation_rec_canonical end interface @ %def transformation <>= 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 <>= 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. <>= interface operator(*) module procedure prod_LT_vector4 module procedure prod_LT_LT module procedure prod_vector4_LT end interface <>= 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 <>= 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. <>= public :: LT_compose_r3_r2_b3 <>= 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 <>= 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} <>= public :: LT_compose_r2_r3_b3 <>= 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 <>= 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 <>= public :: axis_from_p_r3_r2_b3, axis_from_p_b3 <>= 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 <>= 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 $\lambda(x,y,z)=x^2+y^2+z^2-2xy-2xz-2yz$. <>= public :: lambda <>= elemental module function lambda (m1sq, m2sq, m3sq) real(default) :: lambda real(default), intent(in) :: m1sq, m2sq, m3sq end function lambda <>= 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. <>= public :: colliding_momenta <>= 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 <>= 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. <>= public :: pacify <>= interface pacify module procedure pacify_vector3 module procedure pacify_vector4 module procedure pacify_LT end interface pacify <>= 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 <>= 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 @ <>= public :: vector_set_reshuffle <>= 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 <>= 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 @ <>= public :: vector_set_is_cms <>= 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 <>= 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 @ <>= public :: vector4_write_set <>= 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 <>= 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 @ <>= public :: vector4_check_momentum_conservation <>= 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 <>= 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. <>= public :: spinor_product <>= 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 <>= 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) !!! 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]]>>= <> module lorentz_ut use unit_tests use lorentz_uti <> <> contains <> end module lorentz_ut @ %def lorentz_ut @ <<[[lorentz_uti.f90]]>>= <> module lorentz_uti <> use constants, only: zero, Pi use format_defs, only: FMT_12 use lorentz <> <> contains <> end module lorentz_uti @ %def lorentz_ut @ API: driver for the unit tests below. <>= public :: lorentz_test <>= subroutine lorentz_test (u, results) integer, intent(in) :: u type(test_results_t), intent(inout) :: results <> end subroutine lorentz_test @ %def lorentz_test @ \subsubsection{Algebra with 3-vectors} <>= call test (lorentz_1, "lorentz_1", & "Test 3-vector functionality", & u, results) <>= public :: lorentz_1 <>= 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} <>= call test(lorentz_2, "lorentz_2", & "Test 4-vector functionality", u, results) <>= public :: lorentz_2 <>= 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} <>= call test(lorentz_3, "lorentz_3", & "Test 4-vector bilinear functions", u, results) <>= public :: lorentz_3 <>= 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} <>= call test(lorentz_4, "lorentz_4", & "Test Lorentz transformations", u, results) <>= public :: lorentz_4 <>= 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} <>= call test(lorentz_5, "lorentz_5", & "Test additional kinematics", u, results) <>= public :: lorentz_5 <>= 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]]>>= <> module kinematics_vars use kinds use lorentz use format_defs, only: FMT_19 <> <> <> <> interface <> end interface end module kinematics_vars @ %def kinematics_vars @ <<[[kinematics_vars_sub.f90]]>>= <> submodule (kinematics_vars) kinematics_vars_s implicit none contains <> end submodule kinematics_vars_s @ %def kinematics_vars_s @ <>= ! 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 @ <>= 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, & a2, b2, c2, d2, e2, f2 real(default) :: d20, d11, d21, e11, e20, e21 real(default) :: 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 <> <> 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, & a2, b2, c2, d2, e2, f2 real(default) :: d20, d21, e20, e21 real(default) :: 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 <> end subroutine mt2massless <>= public :: mt2_t @ <>= 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 @ <>= public :: mt2_init <>= 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 <>= 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 <>= integer function nsolutionsmassless (dsq) !!! calculates the number of solutions of quartic function in massless case - 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 <>= 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*16 :: z0, z1, z2, z3, z4, & + 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 <>= 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. <>= public :: write_mt2_input <>= module subroutine write_mt2_input (mt2, u) type(mt2_t), intent(in) :: mt2 integer, intent(in) :: u end subroutine write_mt2_input <>= 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 @ <>= public :: write_mt2_output <>= module subroutine write_mt2_output (mt2, scale, u) real(default), intent(in) :: mt2, scale integer, intent(in) :: u end subroutine write_mt2_output <>= 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 @ <>= function signchangepositive (t1, t2, t3, t4, t5) result (n_sign) !!! calculates the number of sign changes in the sturm sequence - 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 <>= integer function signchangenegative (t1, t2, t3, t4, t5) !!! calculates the number of times there is no sign !!! change in the sturm sequence - 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. <>= 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. <>= public :: mt2calc <>= 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 <>= 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. <>= public :: mt2_setevent <>= 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 <>= 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 real*16 :: 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]]>>= <> module kinematics_vars_ut use unit_tests use kinematics_vars_uti <> <> contains <> end module kinematics_vars_ut @ %def kinematics_vars_ut @ <<[[kinematics_vars_uti.f90]]>>= <> module kinematics_vars_uti <> use kinematics_vars <> <> contains <> 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. <>= public :: kinematics_vars_test <>= subroutine kinematics_vars_test (u, results) integer, intent(in) :: u type(test_results_t), intent(inout) :: results <> end subroutine kinematics_vars_test @ %def kinematics_vars_test @ \subsubsection{Testing kinematic variables like MT2} <>= call test (kinematics_vars_1, "kinematics_vars_1", "massive mT2 calculation", & u, results) <>= public :: kinematics_vars_1 <>= 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]]>>= <> module phs_points <> use lorentz, only: vector4_t use lorentz, only: lorentz_transformation_t use lorentz, only: sum <> <> <> <> interface <> end interface end module phs_points @ %def phs_points @ <<[[phs_points_sub.f90]]>>= <> 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 <> 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. <>= public :: phs_point_t <>= type :: phs_point_t private type(vector4_t), dimension(:), allocatable :: p contains <> 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. <>= procedure :: write => phs_point_write <>= 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 <>= 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 <>= public :: assignment(=) <>= 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 <>= 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 <>= 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 <>= 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 <>= 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). <>= public :: size <>= interface size module procedure phs_point_size end interface size <>= 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 <>= 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. <>= public :: operator(==) <>= interface operator(==) module procedure phs_point_eq end interface operator(==) <>= 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 <>= 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 <>= procedure :: get => phs_point_get <>= 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 <>= 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. <>= procedure :: select => phs_point_select <>= 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 <>= 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 <>= procedure :: get_msq => phs_point_get_msq <>= 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 <>= 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. <>= public :: operator(*) <>= interface operator(*) module procedure prod_LT_phs_point end interface operator(*) <>= 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 <>= 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. <>= public :: sum <>= interface sum module procedure phs_point_sum module procedure phs_point_sum_iarray end interface sum <>= 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 <>= 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. <>= procedure :: get_x => phs_point_get_x <>= 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 <>= 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]]>>= <> module phs_points_ut use unit_tests use phs_points_uti <> <> contains <> end module phs_points_ut @ %def phs_points_ut @ <<[[phs_points_uti.f90]]>>= <> module phs_points_uti <> use constants, only: zero use format_defs, only: FMT_12 use lorentz use phs_points <> <> contains <> end module phs_points_uti @ %def phs_points_ut @ API: driver for the unit tests below. <>= public :: phs_points_test <>= subroutine phs_points_test (u, results) integer, intent(in) :: u type(test_results_t), intent(inout) :: results <> end subroutine phs_points_test @ %def phs_points_test @ \subsubsection{PHS point unit test implementation} <>= call test (phs_points_1, "phs_points_1", & "Test PHS point functionality", & u, results) <>= public :: phs_points_1 <>= 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]]>>= <> module sm_physics <> use constants use physics_defs use lorentz <> <> <> interface <> end interface end module sm_physics @ %def sm_physics @ <<[[sm_physics_sub.f90]]>>= <> submodule (sm_physics) sm_physics_s use io_units use numeric_utils use diagnostics use permutations, only: factorial implicit none contains <> 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*} <>= public :: zeta2, zeta3, zeta4, zeta5 <>= 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*} <>= public :: eulerc <>= 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. <>= public :: beta0, beta1, beta2 public :: coeff_b0, coeff_b1, coeff_b2, coeffqed_b0, coeffqed_b1 <>= 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 <>= 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. <>= public :: running_as, running_as_lam, running_alpha, running_alpha_num <>= 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 <>= 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. <>= public :: lambda_qcd <>= 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 <>= 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} <>= real(default), parameter, public :: gamma_q = three/two * CF, & k_q = (7.0_default/two - pi**2/6.0_default) * CF @ %def gamma_q @ <>= public :: gamma_g, k_g <>= 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 <>= 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 $\frac{1.10^{19}}{19!}{\rm Bernoulli}_{19}=2.7\times 10^{-15}$. <>= public :: Li2 <>= elemental module function Li2 (x) real(default), intent(in) :: x real(default) :: Li2 end function Li2 <>= 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 @ <>= 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} <>= public :: psic public :: psir <>= 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 <>= 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} <>= public :: psim public :: psimr <>= 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 <>= 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. <>= public :: cnielsen public :: nielsen <>= 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 <>= 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 @ $\text{Li}_{n}(x) = S_{n-1,1}(x)$. <>= public :: polylog <>= module function polylog (n, x) result (plog) integer, intent(in) :: n real(default), intent(in) :: x real(default) :: plog end function polylog <>= 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 @ $\text{Li}_2(x)$. <>= public :: dilog <>= module function dilog (x) result (dlog) real(default), intent(in) :: x real(default) :: dlog end function dilog <>= module function dilog (x) result (dlog) real(default), intent(in) :: x real(default) :: dlog dlog = polylog (2,x) end function dilog @ %def dilog @ $\text{Li}_3(x)$. <>= public :: trilog <>= module function trilog (x) result (tlog) real(default), intent(in) :: x real(default) :: tlog end function trilog <>= 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. <>= public :: faux <>= elemental module function faux (x) result (y) real(default), intent(in) :: x complex(default) :: y end function faux <>= 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 @ <>= public :: fonehalf <>= elemental module function fonehalf (x) result (y) real(default), intent(in) :: x complex(default) :: y end function fonehalf <>= 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 @ <>= public :: fonehalf_pseudo <>= module function fonehalf_pseudo (x) result (y) real(default), intent(in) :: x complex(default) :: y end function fonehalf_pseudo <>= 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 @ <>= public :: fone <>= elemental module function fone (x) result (y) real(default), intent(in) :: x complex(default) :: y end function fone <>= 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 @ <>= public :: gaux <>= elemental module function gaux (x) result (y) real(default), intent(in) :: x complex(default) :: y end function gaux <>= 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 @ <>= public :: tri_i1 <>= elemental module function tri_i1 (a,b) result (y) real(default), intent(in) :: a,b complex(default) :: y end function tri_i1 <>= 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 @ <>= public :: tri_i2 <>= elemental module function tri_i2 (a,b) result (y) real(default), intent(in) :: a,b complex(default) :: y end function tri_i2 <>= 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, $\alpha_s$. <>= public :: run_b0 <>= elemental module function run_b0 (nf) result (bnull) integer, intent(in) :: nf real(default) :: bnull end function run_b0 <>= 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 @ <>= public :: run_b1 <>= elemental module function run_b1 (nf) result (bone) integer, intent(in) :: nf real(default) :: bone end function run_b1 <>= 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 @ <>= public :: run_aa <>= elemental module function run_aa (nf) result (aaa) integer, intent(in) :: nf real(default) :: aaa end function run_aa <>= 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 @ <>= public :: run_bb <>= 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. <>= public :: ff_dipole <>= 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 <>= 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 @ <>= public :: fi_dipole <>= 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 <>= 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 @ <>= public :: if_dipole <>= 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 <>= 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. <>= public :: ii_dipole <>= 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 <>= 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 $x=1-\frac{\epsilon}{2}$: <>= public :: delta <>= elemental module function delta (x,eps) result (z) real(default), intent(in) :: x, eps real(default) :: z end function delta <>= 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. <>= public :: plus_distr <>= elemental module function plus_distr (x,eps) result (plusd) real(default), intent(in) :: x, eps real(default) :: plusd end function plus_distr <>= 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. <>= public :: pqq <>= elemental module function pqq (x,eps) result (pqqx) real(default), intent(in) :: x, eps real(default) :: pqqx end function pqq <>= 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 @ <>= public :: pgq <>= elemental module function pgq (x) result (pgqx) real(default), intent(in) :: x real(default) :: pgqx end function pgq <>= 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 @ <>= public :: pqg <>= elemental module function pqg (x) result (pqgx) real(default), intent(in) :: x real(default) :: pqgx end function pqg <>= 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 @ <>= public :: pgg <>= elemental module function pgg (x, nf, eps) result (pggx) real(default), intent(in) :: x, nf, eps real(default) :: pggx end function pgg <>= 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} <>= public :: pqq_reg <>= elemental module function pqq_reg (x) result (pqqregx) real(default), intent(in) :: x real(default) :: pqqregx end function pqq_reg <>= 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 @ <>= public :: pgg_reg <>= elemental module function pgg_reg (x) result (pggregx) real(default), intent(in) :: x real(default) :: pggregx end function pgg_reg <>= 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} <>= public :: kbarqg <>= module function kbarqg (x) result (kbarqgx) real(default), intent(in) :: x real(default) :: kbarqgx end function kbarqg <>= 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 @ <>= public :: kbargq <>= module function kbargq (x) result (kbargqx) real(default), intent(in) :: x real(default) :: kbargqx end function kbargq <>= 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 @ <>= public :: kbarqq <>= module function kbarqq (x,eps) result (kbarqqx) real(default), intent(in) :: x, eps real(default) :: kbarqqx end function kbarqq <>= 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 @ <>= public :: kbargg <>= module function kbargg (x,eps,nf) result (kbarggx) real(default), intent(in) :: x, eps, nf real(default) :: kbarggx end function kbargg <>= 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} <>= public :: ktildeqq <>= module function ktildeqq (x,eps) result (ktildeqqx) real(default), intent(in) :: x, eps real(default) :: ktildeqqx end function ktildeqq <>= 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 @ <>= public :: ktildeqg <>= module function ktildeqg (x,eps) result (ktildeqgx) real(default), intent(in) :: x, eps real(default) :: ktildeqgx end function ktildeqg <>= 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 @ <>= public :: ktildegq <>= module function ktildegq (x,eps) result (ktildegqx) real(default), intent(in) :: x, eps real(default) :: ktildegqx end function ktildegq <>= 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 @ <>= public :: ktildegg <>= module function ktildegg (x,eps) result (ktildeggx) real(default), intent(in) :: x, eps real(default) :: ktildeggx end function ktildegg <>= 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, $\epsilon$. <>= public :: insert_q <>= pure module function insert_q () result (i_q) real(default), dimension(0:2) :: i_q end function insert_q <>= 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 @ <>= public :: insert_g <>= 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 <>= 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} <>= public :: k_q_al, k_g_al <>= 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 <>= 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. <>= public :: plus_distr_al <>= 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 <>= 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} <>= public :: kbarqg_al <>= module function kbarqg_al (x,alpha,eps) result (kbarqgx) real(default), intent(in) :: x, alpha, eps real(default) :: kbarqgx end function kbarqg_al <>= 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 @ <>= public :: kbargq_al <>= module function kbargq_al (x,alpha,eps) result (kbargqx) real(default), intent(in) :: x, alpha, eps real(default) :: kbargqx end function kbargq_al <>= 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 @ <>= public :: kbarqq_al <>= module function kbarqq_al (x,alpha,eps) result (kbarqqx) real(default), intent(in) :: x, alpha, eps real(default) :: kbarqqx end function kbarqq_al <>= 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 <>= public :: kbargg_al <>= 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 <>= 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} <>= public :: ktildeqq_al <>= module function ktildeqq_al (x,alpha,eps) result (ktildeqqx) real(default), intent(in) :: x, eps, alpha real(default) :: ktildeqqx end function ktildeqq_al <>= 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. <>= public :: log_plus_distr <>= module function log_plus_distr (x,eps) result (lpd) real(default), intent(in) :: x, eps real(default) :: lpd, eps2 end function log_plus_distr <>= 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, $2 \left( \frac{\log(1/(1-x))}{1-x} \right)_+$. <>= public :: log2_plus_distr <>= module function log2_plus_distr (x,eps) result (lpd) real(default), intent(in) :: x, eps real(default) :: lpd end function log2_plus_distr <>= 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, $2 \left( \frac{\log(1/(1-x))}{1-x} \right)_{1-\alpha}$. <>= public :: log2_plus_distr_al <>= 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 <>= 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: <>= public :: p_qqg public :: p_gqq public :: p_ggg @ $q\to q g$ <>= elemental module function p_qqg (z) result (P) real(default), intent(in) :: z real(default) :: P end function p_qqg <>= 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 @ $g\to q \bar{q}$ <>= elemental module function p_gqq (z) result (P) real(default), intent(in) :: z real(default) :: P end function p_gqq <>= 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 @ $g\to g g$ <>= elemental module function p_ggg (z) result (P) real(default), intent(in) :: z real(default) :: P end function p_ggg <>= 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: <>= public :: integral_over_p_qqg public :: integral_over_p_gqq public :: integral_over_p_ggg <>= 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 <>= 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: <>= 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 ($l_c=1$): <>= 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 <>= 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 $P_{qq}(N)$: <>= public :: pqqm <>= 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 <>= 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*} @ <>= public :: top_width_sm_lo <>= 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 <>= 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 @ <>= public :: g_mu_from_alpha <>= 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 <>= 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 @ <>= public :: alpha_from_g_mu <>= 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 <>= 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]]. <>= public :: top_width_sm_qcd_nlo_massless_b <>= 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 <>= 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 @ <>= public :: f0 <>= elemental module function f0 (w2) result (f) real(default) :: f real(default), intent(in) :: w2 end function f0 <>= 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 @ <>= public :: f1 <>= elemental module function f1 (w2) result (f) real(default) :: f real(default), intent(in) :: w2 end function f1 <>= 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. <>= public :: top_width_sm_qcd_nlo_jk <>= 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 <>= 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. <>= public :: top_width_sm_qcd_nlo_ce <>= 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 <>= 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 @ <>= public :: ff0 <>= elemental module function ff0 (eps2, w2) result (f) real(default) :: f real(default), intent(in) :: eps2, w2 end function ff0 <>= 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 @ <>= public :: ff_f0 <>= elemental module function ff_f0 (eps2, w2) result (f) real(default) :: f real(default), intent(in) :: eps2, w2 end function ff_f0 <>= 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 @ <>= public :: ff_lambda <>= elemental module function ff_lambda (eps2, w2) result (l) real(default) :: l real(default), intent(in) :: eps2, w2 end function ff_lambda <>= 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 @ <>= public :: ff1 <>= elemental module function ff1 (eps2, w2) result (f) real(default) :: f real(default), intent(in) :: eps2, w2 end function ff1 <>= 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]]>>= <> module sm_physics_ut use unit_tests use sm_physics_uti <> <> contains <> end module sm_physics_ut @ %def sm_physics_ut @ <<[[sm_physics_uti.f90]]>>= <> module sm_physics_uti <> use numeric_utils use format_defs, only: FMT_15 use constants use sm_physics <> <> contains <> end module sm_physics_uti @ %def sm_physics_ut @ API: driver for the unit tests below. <>= public :: sm_physics_test <>= subroutine sm_physics_test (u, results) integer, intent(in) :: u type(test_results_t), intent(inout) :: results <> end subroutine sm_physics_test @ %def sm_physics_test @ \subsubsection{Splitting functions} <>= call test (sm_physics_1, "sm_physics_1", & "Splitting functions", & u, results) <>= public :: sm_physics_1 <>= 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} <>= call test(sm_physics_2, "sm_physics_2", & "Top width", u, results) <>= public :: sm_physics_2 <>= 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} <>= call test (sm_physics_3, "sm_physics_3", & "Special functions", & u, results) <>= public :: sm_physics_3 <>= 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]]>>= <> module sm_qcd <> use physics_defs <> <> <> <> interface <> end interface end module sm_qcd @ %def sm_qcd @ <<[[sm_qcd_sub.f90]]>>= <> 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 <> end submodule sm_qcd_s @ %def sm_qcd_s @ \subsection{Coupling: Abstract Data Type} This is the abstract version of the QCD coupling implementation. <>= public :: alpha_qcd_t <>= type, abstract :: alpha_qcd_t contains <> end type alpha_qcd_t @ %def alpha_qcd_t @ There must be an output routine. <>= procedure (alpha_qcd_write), deferred :: write <>= 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. <>= procedure (alpha_qcd_get), deferred :: get <>= 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 $M_Z$. <>= public :: alpha_qcd_fixed_t <>= type, extends (alpha_qcd_t) :: alpha_qcd_fixed_t real(default) :: val = ALPHA_QCD_MZ_REF contains <> end type alpha_qcd_fixed_t @ %def alpha_qcd_fixed_t @ Output. <>= procedure :: write => alpha_qcd_fixed_write <>= 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 <>= 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. <>= procedure :: get => alpha_qcd_fixed_get <>= 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 <>= 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. <>= public :: alpha_qcd_from_scale_t <>= 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 <> end type alpha_qcd_from_scale_t @ %def alpha_qcd_from_scale_t @ Output. <>= procedure :: write => alpha_qcd_from_scale_write <>= 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 <>= 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. <>= procedure :: get => alpha_qcd_from_scale_get <>= 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 <>= 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. <>= public :: alpha_qcd_from_lambda_t <>= type, extends (alpha_qcd_t) :: alpha_qcd_from_lambda_t real(default) :: lambda = LAMBDA_QCD_REF integer :: order = 0 integer :: nf = 5 contains <> end type alpha_qcd_from_lambda_t @ %def alpha_qcd_from_lambda_t @ Output. <>= procedure :: write => alpha_qcd_from_lambda_write <>= 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 <>= 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. <>= procedure :: get => alpha_qcd_from_lambda_get <>= 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 <>= 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). <>= public :: qcd_t <>= type :: qcd_t class(alpha_qcd_t), allocatable :: alpha character(32) :: md5sum = "" integer :: n_f = -1 contains <> end type qcd_t @ %def qcd_t @ Output. We first print the polymorphic [[alpha]] which contains a headline, then any extra components. <>= procedure :: write => qcd_write <>= 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 <>= 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. <>= procedure :: compute_alphas_md5sum => qcd_compute_alphas_md5sum <>= module subroutine qcd_compute_alphas_md5sum (qcd) class(qcd_t), intent(inout) :: qcd integer :: unit end subroutine qcd_compute_alphas_md5sum <>= 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. <>= procedure :: get_md5sum => qcd_get_md5sum <>= module function qcd_get_md5sum (qcd) result (md5sum) character(32) :: md5sum class(qcd_t), intent(inout) :: qcd end function qcd_get_md5sum <>= 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]]>>= <> module sm_qcd_ut use unit_tests use sm_qcd_uti <> <> contains <> end module sm_qcd_ut @ %def sm_qcd_ut @ <<[[sm_qcd_uti.f90]]>>= <> module sm_qcd_uti <> use physics_defs, only: MZ_REF use sm_qcd <> <> contains <> end module sm_qcd_uti @ %def sm_qcd_ut @ API: driver for the unit tests below. <>= public :: sm_qcd_test <>= subroutine sm_qcd_test (u, results) integer, intent(in) :: u type(test_results_t), intent(inout) :: results <> end subroutine sm_qcd_test @ %def sm_qcd_test @ \subsubsection{QCD Coupling} We check two different implementations of the abstract QCD coupling. <>= call test (sm_qcd_1, "sm_qcd_1", & "running alpha_s", & u, results) <>= public :: sm_qcd_1 <>= 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]]>>= <> module sm_qed <> use physics_defs <> <> <> <> interface <> end interface end module sm_qed @ %def sm_qed @ <<[[sm_qed_sub.f90]]>>= <> submodule (sm_qed) sm_qed_s use io_units use format_defs, only: FMT_12 use md5 use sm_physics implicit none contains <> end submodule sm_qed_s @ %def sm_qed_s @ \subsection{Coupling: Abstract Data Type} This is the abstract version of the QCD coupling implementation. <>= public :: alpha_qed_t <>= type, abstract :: alpha_qed_t contains <> end type alpha_qed_t @ %def alpha_qed_t @ There must be an output routine. <>= procedure (alpha_qed_write), deferred :: write <>= 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. <>= procedure (alpha_qed_get), deferred :: get <>= 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. <>= public :: alpha_qed_fixed_t <>= type, extends (alpha_qed_t) :: alpha_qed_fixed_t real(default) :: val = ALPHA_QED_ME_REF contains <> end type alpha_qed_fixed_t @ %def alpha_qed_fixed_t @ Output. <>= procedure :: write => alpha_qed_fixed_write <>= 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 <>= 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. <>= procedure :: get => alpha_qed_fixed_get <>= 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 <>= 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. <>= public :: alpha_qed_from_scale_t <>= 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 <> end type alpha_qed_from_scale_t @ %def alpha_qed_from_scale_t @ Output. <>= procedure :: write => alpha_qed_from_scale_write <>= 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 <>= 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. <>= procedure :: get => alpha_qed_from_scale_get <>= 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 <>= 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. <>= public :: qed_t <>= type :: qed_t class(alpha_qed_t), allocatable :: alpha character(32) :: md5sum = "" integer :: n_f = -1 integer :: n_lep = -1 contains <> end type qed_t @ %def qed_t Output. We first print the polymorphic [[alpha]] which contains a headline, then any extra components. <>= procedure :: write => qed_write <>= 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 <>= 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. <>= procedure :: compute_alpha_md5sum => qed_compute_alpha_md5sum <>= module subroutine qed_compute_alpha_md5sum (qed) class(qed_t), intent(inout) :: qed integer :: unit end subroutine qed_compute_alpha_md5sum <>= 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. <>= procedure :: get_md5sum => qed_get_md5sum <>= module function qed_get_md5sum (qed) result (md5sum) character(32) :: md5sum class(qed_t), intent(inout) :: qed end function qed_get_md5sum <>= 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]]>>= <> module sm_qed_ut use unit_tests use sm_qed_uti <> <> contains <> end module sm_qed_ut @ %def sm_qed_ut @ <<[[sm_qed_uti.f90]]>>= <> module sm_qed_uti <> use physics_defs, only: ME_REF use sm_qed <> <> contains <> end module sm_qed_uti @ %def sm_qed_ut @ API: driver for the unit tests below. <>= public :: sm_qed_test <>= subroutine sm_qed_test (u, results) integer, intent(in) :: u type(test_results_t), intent(inout) :: results <> end subroutine sm_qed_test @ %def sm_qed_test @ \subsubsection{QED Coupling} We check two different implementations of the abstract QED coupling. <>= call test (sm_qed_1, "sm_qed_1", & "running alpha", & u, results) <>= public :: sm_qed_1 <>= 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]]>>= <> module shower_algorithms <> <> <> <> interface <> end interface end module shower_algorithms @ %def shower_algorithms <<[[shower_algorithms_sub.f90]]>>= <> submodule (shower_algorithms) shower_algorithms_s use diagnostics use constants implicit none contains <> <> 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. <>= 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 <>= 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 @ <>= 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 @ <>= 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.) <>= public :: shower_algorithms_test <>= subroutine shower_algorithms_test (u, results) integer, intent(in) :: u type(test_results_t), intent(inout) :: results <> end subroutine shower_algorithms_test @ %def shower_algorithms_test @ \subsubsection{Splitting functions} <>= call test (shower_algorithms_1, "shower_algorithms_1", & "veto technique", & u, results) <>= 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