libmcmule reference guide

The following types, variables, and routines are unlikely to be needed by the typical user and are instead aimed at McMule’s developers.

The particle framework

integer maxparticles: The maximal number of particles allowed

type mlm

Type fields:

% momentum (4) [real(kind=prec)] :: the momentum

type particle

Type fields:

% momentum (4) [real(kind=prec)] :: the momentum
% effcharge [integer] :: the effective charge, corresponding to the +charge for incoming and -charge for outgoing particles.
% charge [integer] :: the actual charge
% incoming [logical] :: .true. for incoming particles
% lepcharge [integer] :: the lepton family (1 for electrons, 2 for muons, 3 for taus), defaults to zero

type particles

Type fields:

% vec (maxparticles) [type(particle)] :: the constituent partciles
% n [integer] :: the number of particles actually used
% combo [character(len=1)] :: the flavour combination used, allowed values are * (any combination), x (only mixed), e (only electronic), m (only muonic), t (only tauonic)

function make_mlm(qq)

Construct a mlm, i.e. a massless momentum

Parameters:: qq (4) [real(kind=prec),in] :: the momentum

function part(qq, charge, inc[, lepcharge])

Construct a particle.

Parameters:

qq (4) [real(kind=prec),in] :: the momentum
charge [integer,in] :: the charge of the particle
inc [integer,in] :: +1 for incoming, -1 for outgoing

Options:

lepcharge [integer,1] :: the lepton family number

function parts(ps[, combo])

Construct particles from a list of particles

Parameters:: ps (*) [type(particle),in] :: a list of particle
Options:: combo [character(len=1)] :: the flavour combination used, allowed values are * (any combination), x (only mixed), e (only electronic), m (only muonic), t (only tauonic)

function eik()

An interface to construct the eikonal factor. eik can be called with

(kg, pp), using the type particles. The optional flavour combination combo restricts the emission to the desired set of fermion lines. If combo is set to x, all contributions but the self-eikonal are included.
({q1,k1}, kg, {q2,k2}), with an explicit call to the momenta of the {massive, massless} emitter, before (1) and after (2) the emission.

Parameters:

pp [type(particles),in] :: the fermions involved in the photon emission
qi (4) [real(kind=prec),in] :: the momenta of the massive emitter
ki [type(mlm),in] :: the momenta of the massless emitter
kg [type(mlm),in] :: the momentum of the photon

Return:

eik :: the eikonal factor

function ieik()

An interface to construct the integrated eikonal factor [28]. ieik can be called with

(xicut, epcmf, pp[, pole]), using the type particles. The optional flavour combination combo restricts the emission to the desired set of fermion lines. If combo is set to x, all contributions but the self-eikonal are included.
(xicut, epcmf, q1, q2[, pole]), with an explicit call to the momenta of the massive emitter, before (1) and after (2) the emission.

Parameters:

xicut [real(kind=prec),in] :: \(\xi_c\) (cf. Section Running at NLO and beyond)
epcmf [real(kind=prec),in] :: square root of scms
pp [type(particles),in] :: the fermions involved in the photon emission
qi (4) [real(kind=prec),in] :: the momenta of the massive emitter

Options:

pole [real(kind=prec),out] :: the singular part of the integrated eikonal, as a coefficient of \(1/\epsilon\)

Return:

ieik :: the finite part of the integrated eikonal factor

function ntssoft(pp, kk, pole)

The (universal) soft contribution to the LBK theorem at 1 loop [9], i.e. the NTS soft function. The optional flavour combination for the particles pp restricts the emission to the desired set of fermion lines. [1]

Parameters:

pp [type(particles),in] :: the fermions involved in the photon emission
kk (4) [real(kind=prec),in] :: the momentum of the photon

Options:

pole [real(kind=prec),out] :: the singular part of the NTS soft function, as a coefficient of \(1/\epsilon\)

Return:

ieik :: the finite part of the NTS soft function

Matrix element interface

function partInterface(q1, q2, q3, q4, q5, q6, q7)

an abstract interface to construct particles for a given process.

Parameters:: qi (4) [real(kind=prec)] :: the momenta
Return:: partInterface [particles] :: the constructed particle string

Package-X function

Note

This section needs to be completed, link to issue

function DiscB()

function DiscB_cplx()

function ScalarC0IR6()

function ScalarC0IR6_cplx()

function ScalarC0()

function ScalarC0_cplx()

function ScalarD0IR16()

function ScalarD0IR16_cplx()

VP functions

Note

This section needs to be completed, link to issue

Phase spaces

McMule has implemented a number of phase routines that map from the hypercube to the physical momenta. Here is a list of currently used ones

subroutine PSD3(ra, q1, m1, q2, m2, q3, m3, weight)

Generic phase space routine for \(1\to2\) decays

Parameters:

ra (2) [real(kind=prec),in] :: the random numbers
qi (4) [real(kind=prec),out] :: the momenta
mi [real(kind=prec),in] :: the masses
weight [real(kind=prec),out] :: the Jacobian

subroutine PSD4(ra, q1, m1, q2, m2, q3, m3, q4, m4, weight)

Generic phase space routine for \(1\to3\) decays

Parameters:

ra (5) [real(kind=prec),in] :: the random numbers
qi (4) [real(kind=prec),out] :: the momenta
mi [real(kind=prec),in] :: the masses
weight [real(kind=prec),out] :: the Jacobian

subroutine PSD4_FKS(ra, q1, m1, q2, m2, q3, m3, q4, weight)

FKS phase space routine for \(1\to3\) decays, requires \(m_4=0\). Tuned for \(\sphericalangle(p_2,q_4)\) and \(E_4\)

Parameters:

ra (5) [real(kind=prec),in] :: the random numbers
qi (4) [real(kind=prec),out] :: the momenta
mi [real(kind=prec),in] :: the masses
weight [real(kind=prec),out] :: the Jacobian

subroutine PSD5(ra, q1, m1, q2, m2, q3, m3, q4, m4, q5, m5, weight)

Generic phase space routine for \(1\to4\) decays

Parameters:

ra (8) [real(kind=prec),in] :: the random numbers
qi (4) [real(kind=prec),out] :: the momenta
mi [real(kind=prec),in] :: the masses
weight [real(kind=prec),out] :: the Jacobian

subroutine PSD5_25(ra, q1, m1, q2, m2, q3, m3, q4, m4, q5, m5, weight)

Phase space routine for \(1\to4\) decays, tuned for \(\sphericalangle(p_2,q_5)\) and \(E_5\), collinear limit is ra(2) -> 0

Parameters:

ra (8) [real(kind=prec),in] :: the random numbers
qi (4) [real(kind=prec),out] :: the momenta
mi [real(kind=prec),in] :: the masses
weight [real(kind=prec),out] :: the Jacobian

subroutine PSD5_FKS(ra, q1, m1, q2, m2, q3, m3, q4, m4, q5, weight)

FKS phase space routine for \(1\to4\) decays, requires \(m_5=0\). Tuned for \(\sphericalangle(p_2,q_5)\) and \(E_5\)

Parameters:

ra (8) [real(kind=prec),in] :: the random numbers
qi (4) [real(kind=prec),out] :: the momenta
mi [real(kind=prec),in] :: the masses
weight [real(kind=prec),out] :: the Jacobian

subroutine PSD6(ra, q1, m1, q2, m2, q3, m3, q4, m4, q5, m5, q6, m6, weight)

Generic phase space routine for \(1\to5\) decays

Parameters:

ra (11) [real(kind=prec),in] :: the random numbers
qi (4) [real(kind=prec),out] :: the momenta
mi [real(kind=prec),in] :: the masses
weight [real(kind=prec),out] :: the Jacobian

subroutine PSD6_23_24_34(ra, q1, m1, q2, m2, q5, m5, q6, m6, q3, m3, q4, m4, weight)

Phase space routine for \(1\to5\) decays with FKS-ish tuning. This is designed for the decay \(\mu^+\to e^+\nu\bar\nu e^+e^-\). q2 should be the unique particle (electron) and q3 and q4 are the identical particles (postirons):

                   _/  q
                   /|   2
---<---*~~~~~~~~~~*
       |          |
       ^          ^  q
       | q        |   3
          4

The ‘spectator’ neutrinos are q5 and q6. Start by generating p2 and p3 at an angle * = arccos(y2):

              ^ p2
             |||
             |||
p3         __|||
 --__     /  |||
     --__/  *|||
         --__|||
             |||

Generate p4 at an angle * = arccos(y3) and rotating by an angle phi w.r.t. to p3:

             |||      / p4
          _--|||--_  /
         -_  |||  _-/
p3         --___-- /
 --__        |||__/
     --__    |||*/
         --__|||/
             |||

Parameters:

ra (11) [real(kind=prec),in] :: the random numbers
qi (4) [real(kind=prec),out] :: the momenta
mi [real(kind=prec),in] :: the masses
weight [real(kind=prec),out] :: the Jacobian

subroutine PSD6_23_24_34_E56(ra, q1, m1, q2, m2, q5, m5, q6, m6, q3, m3, q4, m4, weight)

Phase space routine for \(1\to5\) decays with FKS-ish tuning, similar to PSD6_23_24_34() but with special tuning on the \(E_5+E_6\) tail.

Parameters:

ra (11) [real(kind=prec),in] :: the random numbers
qi (4) [real(kind=prec),out] :: the momenta
mi [real(kind=prec),in] :: the masses
weight [real(kind=prec),out] :: the Jacobian

subroutine PSD6_FKS(ra, q1, m1, q2, m2, q3, m3, q4, m4, q5, m5, q6, weight)

FKS phase space routine for \(1\to5\) decays, requires \(m_6=0\). Tuned for \(\sphericalangle(p_2,q_6)\) and \(E_6\)

Parameters:

ra (11) [real(kind=prec),in] :: the random numbers
qi (4) [real(kind=prec),out] :: the momenta
mi [real(kind=prec),in] :: the masses
weight [real(kind=prec),out] :: the Jacobian

subroutine PSD6_25_26_m50_FKS(ra, q1, m1, q2, m2, q3, m3, q4, m4, q5, m5, q6, weight)

FKS phase space routine for \(1\to5\) decays, requires \(m_5=m_6=0\). Tuned for \(\sphericalangle(p_2,q_{5,6})\) and \(E_{5,6}\)

Parameters:

ra (11) [real(kind=prec),in] :: the random numbers
qi (4) [real(kind=prec),out] :: the momenta
mi [real(kind=prec),in] :: the masses
weight [real(kind=prec),out] :: the Jacobian

subroutine PSD6_FKSS(ra, q1, m1, q2, m2, q3, m3, q4, m4, q5, q6, weight)

Double-FKS phase space routine for \(1\to5\) decays, requires \(m_5=m_6=0\). Tuned for \(\sphericalangle(p_2,q_{5,6})\) and \(E_{5,6}\)

Parameters:

ra (11) [real(kind=prec),in] :: the random numbers
qi (4) [real(kind=prec),out] :: the momenta
mi [real(kind=prec),in] :: the masses
weight [real(kind=prec),out] :: the Jacobian

subroutine PSD7(ra, q1, m1, q2, m2, q3, m3, q4, m4, q5, m5, q6, m6, q7, m7, weight)

Generic phase space routine for \(1\to6\) decays

Parameters:

ra (14) [real(kind=prec),in] :: the random numbers
qi (4) [real(kind=prec),out] :: the momenta
mi [real(kind=prec),in] :: the masses
weight [real(kind=prec),out] :: the Jacobian

subroutine PSD7_27_37_47_FKS(ra, q1, m1, q2, m2, q3, m3, q4, m4, q5, m5, q6, m6, q7, weight)

FKS phase space routine for \(1\to6\) decays, tuned for \(\sphericalangle(p_{2,3,4},q_7)\) and \(E_7\)

Parameters:

ra (14) [real(kind=prec),in] :: the random numbers
qi (4) [real(kind=prec),out] :: the momenta
mi [real(kind=prec),in] :: the masses
weight [real(kind=prec),out] :: the Jacobian

subroutine PSD7_27_37_47_E56_FKS(ra, q1, m1, q2, m2, q3, m3, q4, m4, q5, m5, q6, m6, q7, weight)

FKS phase space routine for \(1\to6\) decays, tuned for \(\sphericalangle(p_{2,3,4},q_7)\) and \(E_7\) and tuned for the \(E_5+E_6\) tail

Parameters:

ra (14) [real(kind=prec),in] :: the random numbers
qi (4) [real(kind=prec),out] :: the momenta
mi [real(kind=prec),in] :: the masses
weight [real(kind=prec),out] :: the Jacobian

subroutine PSX2(ra, q1, m1, q2, m2, q3, m3, q4, m4, weight)

Generic phase space routine for \(2\to2\) cross sections

Parameters:

ra (2) [real(kind=prec),in] :: the random numbers
qi (4) [real(kind=prec),out] :: the momenta
mi [real(kind=prec),in] :: the masses
weight [real(kind=prec),out] :: the Jacobian

subroutine PSX3_FKS(ra, q1, m1, q2, m2, q3, m3, q4, m4, q5, weight)

FKS phase space routine for \(2\to3\) cross sections, requires \(m_5=0\). Tuned for ISR and not FSR

Parameters:

ra (5) [real(kind=prec),in] :: the random numbers
qi (4) [real(kind=prec),out] :: the momenta
mi [real(kind=prec),in] :: the masses
weight [real(kind=prec),out] :: the Jacobian

subroutine PSX3_35_FKS(ra, q1, m1, q2, m2, q3, m3, q4, m4, q5, weight[, sol])

FKS phase space routine for \(2\to3\) cross sections, requires \(m_5=0\). Tuned for \(\sphericalangle(q_3,q_5)\) and \(E_5\)

Parameters:

ra (5) [real(kind=prec),in] :: the random numbers
qi (4) [real(kind=prec),out] :: the momenta
mi [real(kind=prec),in] :: the masses
weight [real(kind=prec),out] :: the Jacobian

Options:

sol [integer,in] :: which solution to pick

subroutine PSX4(ra, q1, m1, q2, m2, q3, m3, q4, m4, q5, m5, q6, m6, weight)

Generic phase space routine for \(2\to4\) cross sections

Parameters:

ra (8) [real(kind=prec),in] :: the random numbers
qi (4) [real(kind=prec),out] :: the momenta
mi [real(kind=prec),in] :: the masses
weight [real(kind=prec),out] :: the Jacobian

subroutine PSX4_FKSS(ra, q1, m1, q2, m2, q3, m3, q4, m4, q5, q6, weight)

Double-FKS phase space routine for \(2\to4\) cross sections, requires \(m_5=m_6=0\). Tuned for ISR and not :term`FSR`

Parameters:

ra (8) [real(kind=prec),in] :: the random numbers
qi (4) [real(kind=prec),out] :: the momenta
mi [real(kind=prec),in] :: the masses
weight [real(kind=prec),out] :: the Jacobian

subroutine PSX4_35_36_FKSS(ra, q1, m1, q2, m2, q3, m3, q4, m4, q5, q6, weight[, sol])

Double-FKS phase space routine for \(2\to4\) cross sections, requires \(m_5=m_6=0\). Tuned for \(\sphericalangle(q_3,q_{5,6})\) and \(E_{5,6}\)

Parameters:

ra (8) [real(kind=prec),in] :: the random numbers
qi (4) [real(kind=prec),out] :: the momenta
mi [real(kind=prec),in] :: the masses
weight [real(kind=prec),out] :: the Jacobian

Options:

sol [integer,in] :: which solution to pick

subroutine PSD6_P_25_26_m50_FKS(ra, q1, m1, q2, m2, q3, m3, q4, m4, q5, m5, q6, weight)

FKS phase space routine for \(1\to5\) decays, requires \(m_5=m_6=0\). Tuned for \(\sphericalangle(p_2,q_{5,6})\) and \(E_{5,6}\) Partioning of PSD6_25_26_m50_FKS() with \(s_{26}<s_{25}\).

Parameters:

ra (11) [real(kind=prec),in] :: the random numbers
qi (4) [real(kind=prec),out] :: the momenta
mi [real(kind=prec),in] :: the masses
weight [real(kind=prec),out] :: the Jacobian

subroutine PSD6_26_2x5(ra, q1, m1, q2, m2, q3, m3, q4, m4, q5, m5, q6, m6, weight)

Phase space routine for \(1\to5\) decays with FKS-ish tuning. Modification of PSD6_23_24_34() with \(2\leftrightarrow 5\). This is designed for the decay \(\mu^+\to e^+\nu\bar\nu e^+e^-\).

Parameters:

ra (11) [real(kind=prec),in] :: the random numbers
qi (4) [real(kind=prec),out] :: the momenta
mi [real(kind=prec),in] :: the masses
weight [real(kind=prec),out] :: the Jacobian

subroutine PSD7_27_37_47_2x5_FKS(ra, q1, m1, q2, m2, q3, m3, q4, m4, q5, m5, q6, m6, q7, weight)

FKS phase space routine for \(1\to6\) decays, tuned for \(\sphericalangle(p_{2,3,4},q_7)\) and \(E_7\) Modification of PSD7_27_37_47_FKS() with \(2\leftrightarrow 5\).

Parameters:

ra (14) [real(kind=prec),in] :: the random numbers
qi (4) [real(kind=prec),out] :: the momenta
mi [real(kind=prec),in] :: the masses
weight [real(kind=prec),out] :: the Jacobian

subroutine PSX3_P_15_FKS(ra, q1, m1, q2, m2, q3, m3, q4, m4, q5, weight)

FKS phase space routine for \(2\to3\) cross sections, requires \(m_5=0\). Tuned for ISR and not FSR. Partioning of PSX3_FKS() with \(s_{15}<s_{35}\).

Parameters:

ra (5) [real(kind=prec),in] :: the random numbers
qi (4) [real(kind=prec),out] :: the momenta
mi [real(kind=prec),in] :: the masses
weight [real(kind=prec),out] :: the Jacobian

subroutine PSX3_P13_35_FKS(ra, q1, m1, q2, m2, q3, m3, q4, m4, q5, weight)

FKS phase space routine for \(2\to3\) cross sections, requires \(m_5=0\). Tuned for \(\sphericalangle(q_3,q_5)\) and \(E_5\) Partioning of PSX3_35_FKS() with \(s_{15}>s_{35}\).

Parameters:

ra (5) [real(kind=prec),in] :: the random numbers
qi (4) [real(kind=prec),out] :: the momenta
mi [real(kind=prec),in] :: the masses
weight [real(kind=prec),out] :: the Jacobian

subroutine PSX3_coP13_35_FKS(ra, q1, m1, q2, m2, q3, m3, q4, m4, q5, weight)

The corner piece to PSX3_P13_35_FKS()

Parameters:

ra (5) [real(kind=prec),in] :: the random numbers
qi (4) [real(kind=prec),out] :: the momenta
mi [real(kind=prec),in] :: the masses
weight [real(kind=prec),out] :: the Jacobian

subroutine PSX3_P_15_25_FKS(ra, q1, m1, q2, m2, q3, m3, q4, m4, q5, weight)

FKS phase space routine for \(2\to3\) cross sections, requires \(m_5=0\). Tuned for ISR and not FSR. Partioning of PSX3_FKS() with \({\rm min}\big(s_{15},s_{25}\big)<{\rm min}\big(s_{35},s_{45}\big)\).

Parameters:

ra (5) [real(kind=prec),in] :: the random numbers
qi (4) [real(kind=prec),out] :: the momenta
mi [real(kind=prec),in] :: the masses
weight [real(kind=prec),out] :: the Jacobian

subroutine PSX3_P_35_FKS(ra, q1, m1, q2, m2, q3, m3, q4, m4, q5, weight)

FKS phase space routine for \(2\to3\) cross sections, requires \(m_5=0\). Tuned for \(\sphericalangle(q_3,q_5)\) and \(E_5\) Partioning of PSX3_35_FKS() with \(s_{35}<{\rm min}\big(s_{15},s_{25},s_{45}\big)\).

Parameters:

ra (5) [real(kind=prec),in] :: the random numbers
qi (4) [real(kind=prec),out] :: the momenta
mi [real(kind=prec),in] :: the masses
weight [real(kind=prec),out] :: the Jacobian

subroutine PSX3_P_45_FKS(ra, q1, m1, q2, m2, q3, m3, q4, m4, q5, weight)

FKS phase space routine for \(2\to3\) cross sections, requires \(m_5=0\). Tuned for \(\sphericalangle(q_3,q_5)\) and \(E_5\) Partioning of PSX3_35_FKS() with \(s_{45}<{\rm min}\big(s_{15},s_{25},s_{35}\big)\) and \(3\leftrightarrow4\)

Parameters:

ra (5) [real(kind=prec),in] :: the random numbers
qi (4) [real(kind=prec),out] :: the momenta
mi [real(kind=prec),in] :: the masses
weight [real(kind=prec),out] :: the Jacobian

subroutine PSX3_coP_35_FKS(ra, q1, m1, q2, m2, q3, m3, q4, m4, q5, weight)

The corner piece to PSX3_P_35_FKS()

Parameters:

ra (5) [real(kind=prec),in] :: the random numbers
qi (4) [real(kind=prec),out] :: the momenta
mi [real(kind=prec),in] :: the masses
weight [real(kind=prec),out] :: the Jacobian

subroutine PSX3_coP_45_FKS(ra, q1, m1, q2, m2, q3, m3, q4, m4, q5, weight)

The corner piece to PSX3_P_45_FKS()

Parameters:

ra (5) [real(kind=prec),in] :: the random numbers
qi (4) [real(kind=prec),out] :: the momenta
mi [real(kind=prec),in] :: the masses
weight [real(kind=prec),out] :: the Jacobian

subroutine PSX4_P_15_16_FKSS(ra, q1, m1, q2, m2, q3, m3, q4, m4, q5, q6, weight)

Double-FKS phase space routine for \(2\to4\) cross sections, requires \(m_5=m_6=0\). Tuned for ISR and not :term`FSR` Partioning of PSX4_FKSS() with \({\rm min}\big(s_{15}, s_{16}\big)<{\rm min}\big(s_{35},s_{36}\big)\).

Parameters:

ra (8) [real(kind=prec),in] :: the random numbers
qi (4) [real(kind=prec),out] :: the momenta
mi [real(kind=prec),in] :: the masses
weight [real(kind=prec),out] :: the Jacobian

subroutine PSX4_P_35_36_FKSS(ra, q1, m1, q2, m2, q3, m3, q4, m4, q5, q6, weight)

Double-FKS phase space routine for \(2\to4\) cross sections, requires \(m_5=m_6=0\). Tuned for \(\sphericalangle(q_3,q_{5,6})\) and \(E_{5,6}\) Partioning of PSX4_35_36_FKSS() with \({\rm min}\big(s_{15}, s_{36}\big)<{\rm min}\big(s_{15},s_{25},s_{45},s_{16},s_{26},s_{46}\big)\).

Parameters:

ra (8) [real(kind=prec),in] :: the random numbers
qi (4) [real(kind=prec),out] :: the momenta
mi [real(kind=prec),in] :: the masses
weight [real(kind=prec),out] :: the Jacobian

subroutine PSX4_coP_35_36_FKSS(ra, q1, m1, q2, m2, q3, m3, q4, m4, q5, q6, weight)

The corner piece to PSX4_P_35_36_FKSS()

Parameters:

ra (8) [real(kind=prec),in] :: the random numbers
qi (4) [real(kind=prec),out] :: the momenta
mi [real(kind=prec),in] :: the masses
weight [real(kind=prec),out] :: the Jacobian

subroutine PSX4_P13_35_36_FKSS(ra, q1, m1, q2, m2, q3, m3, q4, m4, q5, q6, weight)

Double-FKS phase space routine for \(2\to4\) cross sections, requires \(m_5=m_6=0\). Tuned for \(\sphericalangle(q_3,q_{5,6})\) and \(E_{5,6}\) Partioning of PSX4_35_36_FKSS() with \({\rm min}\big(s_{15}, s_{16}\big)>{\rm min}\big(s_{35},s_{36}\big)\).

Parameters:

ra (8) [real(kind=prec),in] :: the random numbers
qi (4) [real(kind=prec),out] :: the momenta
mi [real(kind=prec),in] :: the masses
weight [real(kind=prec),out] :: the Jacobian

subroutine PSX4_coP13_35_36_FKSS(ra, q1, m1, q2, m2, q3, m3, q4, m4, q5, q6, weight)

The corner piece to PSX4_P13_35_36_FKSS()

Parameters:

ra (8) [real(kind=prec),in] :: the random numbers
qi (4) [real(kind=prec),out] :: the momenta
mi [real(kind=prec),in] :: the masses
weight [real(kind=prec),out] :: the Jacobian

subroutine PSX4_P_15_16_25_26_FKSS(ra, q1, m1, q2, m2, q3, m3, q4, m4, q5, q6, weight)

Double-FKS phase space routine for \(2\to4\) cross sections, requires \(m_5=m_6=0\). Tuned for ISR and not :term`FSR` Partioning of PSX4_FKSS() with \({\rm min}\big(s_{15}, s_{16},s_{25}, s_{26}\big)<{\rm min}\big(s_{35},s_{36},s_{54},s_{46}\big)\).

Parameters:

ra (8) [real(kind=prec),in] :: the random numbers
qi (4) [real(kind=prec),out] :: the momenta
mi [real(kind=prec),in] :: the masses
weight [real(kind=prec),out] :: the Jacobian

subroutine PSX4_P_45_46_FKSS(ra, q1, m1, q2, m2, q3, m3, q4, m4, q5, q6, weight)

Double-FKS phase space routine for \(2\to4\) cross sections, requires \(m_5=m_6=0\). Tuned for \(\sphericalangle(q_3,q_{5,6})\) and \(E_{5,6}\) Partioning of PSX4_35_36_FKSS() with \({\rm min}\big(s_{45}, s_{46}\big)>{\rm min}\big(s_{15},s_{25},s_{35},s_{16},s_{26},s_{36}\big)\).

Parameters:

ra (8) [real(kind=prec),in] :: the random numbers
qi (4) [real(kind=prec),out] :: the momenta
mi [real(kind=prec),in] :: the masses
weight [real(kind=prec),out] :: the Jacobian

subroutine PSX4_coP_45_46_FKSS(ra, q1, m1, q2, m2, q3, m3, q4, m4, q5, q6, weight)

The corner piece to PSX4_P_45_46_FKSS()

Parameters:

ra (5) [real(kind=prec),in] :: the random numbers
qi (4) [real(kind=prec),out] :: the momenta
mi [real(kind=prec),in] :: the masses
weight [real(kind=prec),out] :: the Jacobian