Improved Calculations of Quark Distributions in Hadrons: the case of pion

TL;DR

This paper presents an improved QCD sum rule method for calculating the valence quark distribution in pions, addressing previous limitations and providing results at a specific scale for use in evolution equations.

Contribution

The authors develop an enhanced sum rule approach using double dispersion and Borel transformation to accurately compute quark distributions in pions, including cases previously inaccessible.

Findings

Valence u-quark distribution in pion at intermediate x

Results obtained at Q^2=2 GeV^2

Method improves accuracy by eliminating background terms

Abstract

The earlier introduced method of calculation of quark distributions in hadrons, based on QCD sum rules, is improved. The imaginary part of the virtual photon forward scattering amplitude on some hadronic current is considered in the case, when initial and final virtualities of the current $p^2_1$, and $p^2_2$ are different, $p^2_1\not= p^2_2$. The operator product expansion (OPE) in $p^2_1$, $p^2_2$ is performed. The sum rule for quark distribution is obtained using double dispersion representation of the amplitude on one side in terms of calculated in QCD OPE and on the other side in terms of physical states contributions. Double Borel transformation in $p^2_1$, $p^2_2$ is applied to the sum rule, killing background non-diagonal transition terms, which deteriorated the accuracy in previous calculations. The case of valence quark distribution in pion is considered, which was impossible to treat by the previous method. OPE up to dimension 6 operators is performed and leading order perturbative corrections are accounted. Valence $u$-quark distribution in $\pi^+$ was found at intermediate $x$, $0.15 < x < 0.7$ and normalization point $Q^2=2 GeV^2$. These results may be used as input for evolution equations.