SUBROUTINE QIPSWT(PSWGT) - Generate kinematic weight for given phase space configuration

In this subroutine¹ an energy-weight factor $w$ corresponding to the leading-order matrix elements squared [1] is calculated: Each outgoing quark with four-momentum $p_q$ is weighted by its energy $p_q^t$, each outgoing gluon with four-momentum $p_k$ by its energy squared $p_k^{t\,2}$, such that

$$w = \prod_{q=1}^{2n_f-1} p_q^t\ \prod_{k=1}^{n_g} p_k^{t\,2} \, .$$ (1)

It is easy to show that the weight (1) takes is maximum for the following phase space configuration,

$$p_q^t =\frac{W_i}{2\,(n_g+n_f)-1}\ \forall\ q\in \{1,\ldots... ...c{W_i}{2\,(n_g+n_f)-1}\ \forall\ k\in \{ 1,\ldots ,n_g\} \, ,$$ (2)

where

$$W_i = \sum_{q=1}^{2n_f-1} p_q^t + \sum_{k=1}^{n_g} p_k^t$$ (3)

denotes the instanton CM energy. The maximum weight itself is given by

$$w_{\rm max}=2^{2\,n_g} \left[ \frac{W_i}{2\,(n_g+n_f)-1}\right]^{2\,(n_g+n_f)-1} ,$$ (4)

The output variable PSWGT of the subroutine QIPSWT is the relative weight

$$w_{\rm rel} = w/w_{\rm max} \, .$$ (5)