Method of Analytic Evolution of Flat Distribution Amplitudes in QCD

TL;DR

This paper introduces an analytical method for evolving flat distribution amplitudes in QCD, especially effective when standard Gegenbauer expansion methods are inefficient, by deriving explicit evolution formulas for specific flat DAs.

Contribution

The paper develops a new analytical approach for ERBL evolution applicable to flat and antisymmetric flat distribution amplitudes, improving efficiency over traditional methods.

Findings

The method accurately describes evolution for flat DAs with good convergence for t<0.5.

Explicit formulas for evolution factors involving (x(1-x))^t and |1-2x|^{2t} are derived.

Standard Gegenbauer expansion remains useful for larger t values.

Abstract

A new analytical method of performing ERBL evolution is described. The main goal is to develop an approach that works for distribution amplitudes that do not vanish at the end points, for which the standard method of expansion in Gegenbauer polynomials is inefficient. Two cases of the initial DA are considered: a purely flat DA, given by the same constant for all x, and an antisymmetric DA given by opposite constants for x <1/2 and x>1/2. For a purely flat DA, the evolution is governed by an overall (x (1-x))^t dependence on the evolution parameter t times a factor that was calculated as an expansion in t. For an antisymmetric flat DA, an extra overall factor |1-2x|^{2t} appears due to a jump at x=1/2. A good convergence was observed in the t < 1/2 region. For larger t, one can use the standard method of the Gegenbauer expansion.