Multiplicity distribution of produced gluons in deep inelastic scattering: main equations and their homotopy solutions for heavy nuclei

TL;DR

This paper derives new equations for gluon production in deep inelastic scattering within high energy QCD, introduces a homotopy solution method, and provides an analytical solution for large gluon multiplicities, analyzing entropy in heavy nuclei.

Contribution

It presents a new derivation of gluon production equations using the dipole approach and develops a homotopy method for solving these equations analytically and iteratively.

Findings

Derived equations match AGK rules within dipole QCD.

Developed a homotopy approach for solutions.

Obtained an analytical expression for large gluon multiplicities.

Abstract

In this paper we discuss the multiplicity distribution in the deep inelastic processes in the frame work of high energy QCD. We obtained three results. First, we get the new derivation of the equations for the cross sections of productions of $n$-cut Pomerons in the final states ($\sigma_n$). These equations coincide with the equations that have been derived using the Abramovsky, Gribov and Kancheli (AGK) cutting rules but based on the dipole approach to QCD. Second, we developed the homotopy approach for finding the solutions to these equations. It consists with the analytic solution for the first iteration and the converge procedure of calculating the next iterations using computing. Third, we found the analytical solution for $\sigma_n$ at large $n\,\gtrsim\,N(z) = 2 N_0 \,z\,\exp( z^2/(2 \kappa))$ with $z = \ln( r^2\,Q^2_s )$. Using this solution we calculate the entropy of the produced gluons at large $z$: $S_E = \ln \left( N(z)\right)$, where the saturation momentum $Q_s$ and all constants are discussed in the text.