Analytical solution to DGLAP integro-differential equation via complex maps in domains of contour integrals

TL;DR

This paper presents an analytical solution to the simplified DGLAP integro-differential equation in QCD using complex maps and contour integrals, connecting it to Bessel functions and the Barnes integral.

Contribution

It introduces a novel analytical approach employing complex diffeomorphisms and contour integrals to solve the DGLAP equation with a single-term splitting function.

Findings

Solution expressed via Bessel functions for the simplified DGLAP equation.

Representation of inverse Laplace transform as Barnes contour integral.

Detailed derivation of the complex map and integral transformations.

Abstract

A simple model for QCD dynamics in which the DGLAP integro-differential equation may be solved analytically has been considered in our previous papers arXiv:1611.08787 [hep-ph] and arXiv:1906.07924 [hep-ph]. When such a model contains only one term in the splitting function of the dominant parton distribution, then Bessel function appears to be the solution to this simplified DGLAP equation. To our knowledge, this model with only one term in the splitting function for the first time has been proposed by Blumlein in hep-ph/9506403. In arXiv:1906.07924 [hep-ph] we have shown that a dual integro-differential equation obtained from the DGLAP equation by a complex map in the plane of the Mellin moment in this model may be considered as the BFKL equation. Then, in arXiv:1906.07924 we have applied a complex diffeomorphism to obtain a standard integral from Gradshteyn and Ryzhik tables starting from the contour integral for parton distribution functions that is usually taken by calculus of residues. This standard integral from these tables appears to be the Laplace transformation of Jacobian for this complex diffeomorphism. Here we write up all the formulae behind this trick in detail and find out certain important points for further development of this strategy. We verify that the inverse Laplace transformation of the Laplace image of the Bessel function may be represented in a form of Barnes contour integral.