SUBROUTINE QIRDIS(LU,LL,X,WGT,N) - Generate $X$ as $dX/X^{N+1}$, with weight for $dX/X$ distribution returned

In the instanton-induced cross section,

$$\sigma_{eP}^{(I)}({\rm Cuts}) \simeq \frac{2\,\pi\,\alpha^2}{S}\,\sum\limits_{q^\prime} e_{q^\prime}^2... ...^\prime}\, \,\frac{\sigma^{(I)}_{q^\prime g}(x^\prime ,Q^{\prime 2})}{x^\prime}$$

$$\times \int\limits_{{\rm max}\left(\frac{Q^{\prime 2}}{Sx^\prime ... ...rm Bj\,max}-\frac{Q^{\prime 2}}{Sx^\prime z}}} \frac{dx_{\rm Bj}}{x_{\rm Bj}}\,$$ (1)

$$\times \int\limits_{{\rm max}\left( \frac{Q^{\prime 2}}{Sx^\prime... ..._{\rm min})\, \frac{1+(1-y_{\rm Bj})^2}{y_{\rm Bj}}\ P_{q^\prime}^{{ (I)}}\ ,$$

we encounter various integrals of the type

$$I(X_U,X_L)=\int\limits_{X_L}^{X_U} \frac{dx}{x}\,f(x) ,$$ (2)

where the function $f(x)$ steeply grows towards small $x$.

This makes the standard way of the Monte Carlo integration of Eq. (2) inefficient. The latter consists in generating $n$ random points $y_i=\ln x_i$ uniformly in the range from $\ln X_L$ to $\ln X_U$ and evaluating the sum

$$I(X_U,X_L)=\int\limits_{X_L}^{X_U} \frac{dx}{x}\,f(x)= \int\... ...f(x=\exp{y}) \ \approx\ \frac{1}{n}\,\sum\limits_{i=1}^n w_i,$$ (3)

with weights

$$w_i=(\ln X_U-\ln X_L)\,f(x_i=\exp{y_i}).$$ (4)

The efficiency is increased by introducing, instead of $y=\ln x$, the integration variable

$$z=x^{-N};\hspace{6ex} dz=-N\, x^{-(N+1)} dx,$$ (5)

with $N>1/2$, and generating $n$ random points $z_i=x_i^{-N}$ uniformly in the range from $X_U^{-N}$ to $X_L^{-N}$, such that

$$I(X_U,X_L)=\int\limits_{X_L}^{X_U} \frac{dx}{x}\,f(x)= \frac... ...=z^{-1/N})}{z}\ \approx\ \frac{1}{n}\,\sum\limits_{i=1}^n W_i,$$ (6)

with weights

$$W_i=\frac{1}{N}(X_L^{-N}-X_U^{-N})\,\frac{f(x_i=z_i^{-1/N})}{z_i}.$$ (7)

The in- and output variables of the subroutine QIRDIS(LU,LL,X,WGT,N) are described in Tables 1 and 2. The power $N$ used as default typically ranges between 1 and 5. For $N\leq 1/2$ the routine QIRDIF(LU,LL,X,WGT,-N) is called.

Table 1: Input variables of QIRDIS.

Name	Description
LU	Logarithm of upper integration limit, $\ln X_U$
LL	Logarithm of lower integration limit, $\ln X_L$
N	Power, $N>1/2$

Table 2: Output variables of QIRDIS.

Name	Description
X	$x_i$ value calculated from the generated $z_i$ value, $x_i=z_i^{-1/N}$
WGT	Monte Carlo weight according to Eq. (7), with $f\equiv 1$, $\frac{1}{N}(X_L^{-N}-X_U^{-N})\,\frac{1}{z_i}$