Analytic derivation of the leading-order gluon distribution function G(x,Q^2) = xg(x,Q^2) from the proton structure function F_2^p(x,Q^2)

TL;DR

This paper derives a direct method to compute the leading-order gluon distribution function from the proton structure function using a differential equation, providing a new analytical approach that aligns well with experimental data.

Contribution

It introduces a second-order differential equation linking the gluon distribution to the proton structure function, enabling direct determination without requiring separate quark or gluon evolution equations.

Findings

The derived G(x,Q^2) matches experimental data within the specified domain.

The new gluon distribution agrees with existing models for x>10^-3.

For x<10^-3, the new distribution is significantly smaller than traditional models.

Abstract

We derive a second-order linear differential equation for the leading order gluon distribution function G(x,Q^2) = xg(x,Q^2) which determines G(x,Q^2) directly from the proton structure function F_2^p(x,Q^2). This equation is derived from the leading order DGLAP evolution equation for F_2^p(x,Q^2), and does not require knowledge of either the individual quark distributions or the gluon evolution equation. Given an analytic expression that successfully reproduces the known experimental data for F_2^p(x,Q^2) in a domain x_min<=x<=x_max, Q_min^2<=Q^2<=Q_max^2 of the Bjorken variable x and the virtuality Q^2 in deep inelastic scattering, G(x,Q^2) is uniquely determined in the same domain. We give the general solution and illustrate the method using the recently proposed Froissart bound type parametrization of F_2^p(x,Q^2) of E. L. Berger, M. M. Block and C-I. Tan, PRL 98, 242001, (2007). Existing leading-order gluon distributions based on power-law description of individual parton distributions agree roughly with the new distributions for x>~10^-3 as they should, but are much larger for x<~10^-3.