SUBROUTINE QIHINS - Generates instanton subprocess Bjorken variables $Q^{\prime 2}$ and $x^\prime$ and Monte Carlo weight

In this routine the Bjorken variables $Q^{\prime 2}$ and $x^\prime$ are generated and a Monte Carlo weight corresponding to the integral

$$\int\limits_{Q^{\prime 2}_{\rm min}}^{Q^{\prime 2}_{\rm max}... ...^{\prime 2}\ \sigma^{(I)}_{q^\prime g}(x^\prime, Q^{\prime 2})$$ (1)

is calculated. Here $\sigma^{(I)}_{q^\prime g}$ denotes the instanton-induced $q^\prime g$ total cross section from Ref. [1]. This proceeds in the following steps:

• Generation of $Q^{\prime 2}$ with weight corresponding to the integral
  

  $$\int\limits_{Q^{\prime 2}_{\rm min}}^{Q^{\prime 2}_{\rm max}} \frac{dQ^{\prime 2}}{Q^{\prime 2}} \, .$$ (2)

  
  
  For default settings of the contrôl flags in QIINIT, this is done as follows:

  $Q^{\prime 2}$ is generated by calling the subroutine QIRDIS(L2U,L2D,QIL2IN,AWGT,L2NV), which generates values of the variable
  

  $$z_{Q^{\prime 2}}=(Q^{\prime 2})^{-N_{Q^{\prime 2}}}; \hspace{6ex} {\rm with}\ N_{Q^{\prime 2}}>1/2,$$ (3)

  
  
  uniformly in the interval between $(Q^{\prime 2}_{\rm max})^{-N_{Q^{\prime 2}}}$ and $(Q^{\prime 2}_{\rm min})^{-N_{Q^{\prime 2}}}$. As an output of the subroutine QIRDIS we get the generated value of $Q^{\prime 2}$, calculated via
  

  $$Q^{\prime 2} = (z_{Q^{\prime 2}})^{-1/N_{Q^{\prime 2}}},$$ (4)

  
  
  from the generated value of $z_{Q^{\prime 2}}$, and the Monte Carlo weight associated with the integral (2),
  

  $$W_{Q^{\prime 2}} = \frac{1}{N_{Q^{\prime 2}}}\, \left[ (Q^... ... 2}}}\right]\, \frac{1}{(Q^{\prime 2})^{-N_{Q^{\prime 2}}}}.$$ (5)

  
  

  
  

  Table 1: Variables set in QIHINS.

  Name	Description
  L2U	Logarithm of upper integration limit of $Q^{\prime 2}$ integration, $\ln Q^{\prime 2}_{\rm max}$
  L2D	Logarithm of lower integration limit of $Q^{\prime 2}$ integration, $\ln Q^{\prime 2}_{\rm min}$
  L2NV	Power for generation of $Q^{\prime 2}$ with high efficiency, $N_{Q^{\prime 2}}=2.5$ (default)
  QIL2IN	Value of $Q^{\prime 2}$ inferred via Eq. (4) from the generated value of $z_{Q^{\prime 2}}$
  AWGT	Generated weight (5) corresponding to the integral (2)
  X1U	Logarithm of upper integration limit of $x^\prime$ integration, $\ln x^\prime_{\rm max}$
  X1D	Logarithm of lower integration limit of $x^\prime$ integration, $\ln x^\prime_{\rm min}$
  XPNV	Power for generation of $x^\prime$ with high efficiency, $N_{x^\prime}=5.0$ (default)
  QIX1HT	Value of $x^\prime$ inferred via Eq. (7) from the generated value of $z_{x^\prime}$
  BWGT	Generated weight (9) corresponding to the integral (6)
  GENWT	Generated weight corresponding to the product of the integrals (2) and (6)
  QIWGT	Generated weight (11) corresponding to the integral (1)

  
  

• Generation of $x^\prime$ with weight corresponding to the integral
  

  $$\int\limits_{x^\prime_{\rm min}}^{x^\prime_{\rm max}} \frac{dx^\prime}{x^\prime}\, .$$ (6)

  
  
  For default settings of the contrôl flags in QIINIT, this is done as follows:

  $x^\prime$ is generated by calling the subroutine QIRDIS(X1U,X1D,QIX1HT,BWGT,XPNV), which generates values of the variable
  

  $$z_{x^\prime}=(x^\prime)^{-N_{x^\prime}}; \hspace{6ex} {\rm with}\ N_{x^\prime}>1/2,$$ (7)

  
  
  uniformly in the interval between $(x^\prime_{\rm max})^{-N_{x^\prime}}$ and $(x^\prime_{\rm min})^{-N_{x^\prime}}$. As an output of the subroutine QIRDIS we get the generated value of $x^\prime$, calculated via
  

  $$x^\prime = (z_{x^\prime})^{-1/N_{x^\prime}},$$ (8)

  
  
  from the generated value of $z_{x^\prime}$, and the Monte Carlo weight associated with the integral (6),
  

  $$W_{x^\prime} = \frac{1}{N_{x^\prime}}\, \left[ (x^\prime_{... ...N_{x^\prime}}\right]\, \frac{1}{(x^\prime)^{-N_{x^\prime}}}.$$ (9)

  
  

• Computation of weight factor from the instanton-induced $q^\prime g$ total cross section,
  

  $$W_\sigma = \frac{2}{x^\prime}\ Q^{\prime 2}\ \sigma^{(I)}_{q^\prime g}(x^\prime, Q^{\prime 2}),$$ (10)

  
  
  evaluated at the generated values of $Q^{\prime 2}$ and $x^\prime$. The external function Q2SIG is called to calculate the corresponding weight factor.

• Multiplication of all weights,
  

  $$W_{Q^\prime x^\prime\sigma} = W_{Q^{\prime 2}}\, W_{x^\prime}\,W_\sigma .$$ (11)

  
  
  This weight corresponds to the integral (1), i.e. it is the relative contribution of the generated pair $(Q^{\prime 2},x^\prime )$ to the integral (1).