DOUBLE PRECISION FUNCTION GMULT(XPR,XI_MIN,XI_MAX,QLAM,KAPPA,NF,LOOPFL) - Computes average gluon multiplicity $\langle n_g\rangle$ via valley method

Computes average gluon multiplicity [1,2] via the saddle-point values $\xi_\ast$ and $v_\ast=Q^\prime \rho_\ast$ in 2-loop renormalization invariant formalism given ${\rm XPR}=x^\prime$, ${\rm QLAM}= Q^\prime/\Lambda^{(n_f)}_{\overline{\rm MS}}$, ${\rm KAPPA}=\mu_r/Q^\prime$ and $n_f$.

From an analysis via a generalized (Mueller [3]) optical theorem for the $q^\prime g \overline{g}$ forward scattering amplitude one infers [1,2] the mean gluon multiplicity produced in the instanton-induced subprocess as

$$\langle n_g\rangle^{(I)}\left(x^\prime ,\frac{Q^\prime} {\La... ... (\xi_\ast -2)\,\frac{dS^{(I\overline{I})}}{d\xi}(\xi_\ast ) ,$$ (1)

where the $I\overline{I}$-valley action is given by [4,5],

$$S^{(I\overline{I})}(\xi ) = 1-\frac{12}{f(\xi )} - \frac{96}{f(\xi )^2} +\frac{48}{f(\xi )^3}... ... 3f(\xi )+8\right] \ln\left[ \frac{1}{2\xi }\bigl( f(\xi ) +4\bigr)\right] \, ,$$ (2)

$$f(\xi ) = \xi^2+\sqrt{\xi^2-4}\xi-4 \, .$$ (3)

The stars $(\ast )$ in Eq. (1) indicate, that the $I$-size $\rho$, the ${\overline{I}}$-size $\overline{\rho}$ and their conformal distance $\xi$ are to be evaluated at the saddle-points $\rho_\ast=\overline{\rho}_\ast$ and $\xi_\ast$, satisfying [6]

$$\frac{1}{2}\frac{\sqrt{\frac{1-{x^\prime}}{{x^\prime}}}} {\sqrt{\... ...(\rho^\ast\mu_r\right)\right) \frac{dS^{(I\overline{I})}(\xi_\ast )}{d\xi^\ast} = 0 \,,$$ (4)

$$\left( \frac{1}{2}\,\sqrt{\frac{1-{x^\prime}}{{x^\prime}}} \sqrt{... ...ht) Q^\prime\rho^\ast+ \Delta_1\beta_0 S^{(I\overline{I})}(\xi^\ast ) -\Delta_2 = 0 \,,$$ (5)

with

$$\Delta_1\equiv 1+\frac{\beta_1}{\beta_0} \frac{\alpha_{\over... ...v 12\,\beta_0 \frac{\alpha_{\overline{\rm MS}}(\mu_r)}{4\pi} .$$ (6)

In this subroutine GMULT is calculated according to Eq. (1). An interpolation is used to find the saddle-point value $\xi_\ast$ (in a range between XI_MIN and XI_MAX) for a given $x^\prime$.