Analytical solution to DGLAP integro-differential equation in a simple toy-model with a fixed gauge coupling

TL;DR

This paper presents an analytical solution to the DGLAP equation in a simplified gauge theory model with fixed coupling, and extends the method to more realistic splitting functions, revealing specific behaviors of gluon distributions.

Contribution

It introduces an analytical approach to solving the DGLAP equation in a fixed gauge coupling model and applies it to realistic splitting functions, providing new insights into gluon distribution behaviors.

Findings

Analytical solutions for simplified DGLAP in fixed coupling models.

Discovery of Bessel-like singular behavior near x=0.

Smooth behavior of gluon distribution near x=1.

Abstract

We consider a simple model for QCD dynamics in which DGLAP integro-differential equation may be solved analytically. This is a gauge model which possesses dominant evolution of gauge boson (gluon) distribution and in which the gauge coupling does not run. This may be ${\cal N} =4$ supersymmetric gauge theory with softly broken supersymmetry, other finite supersymmetric gauge theory with lower level of supersymmetry, or topological Chern-Simons field theories. We maintain only one term in the splitting function of unintegrated gluon distribution and solve DGLAP analytically for this simplified splitting function. The solution is found by use of the Cauchy integral formula. The solution restricts form of the unintegrated gluon distribution as function of momentum transfer and of Bjorken $x$. Then we consider an almost realistic splitting function of unintegrated gluon distribution as an input to DGLAP equation and solve it by the same method which we have developed to solve DGLAP equation for the toy-model. We study a result obtained for the realistic gluon distribution and find a singular Bessel-like behaviour in the vicinity of the point $x=0$ and a smooth behaviour in the vicinity of the point