Burkhardt-Cottingham-type sum rules for light-cone and quasi-PDFs

TL;DR

This paper investigates Burkhardt-Cottingham-type sum rules for light-cone and quasi-PDFs, verifying their validity in different models and regularization schemes, and highlights the impact of rotational invariance breaking on these sum rules.

Contribution

It provides a perturbative check of BC-type sum rules for PDFs and quasi-PDFs in specific models, demonstrating their validity with dimensional regularization and violation with cut-off schemes.

Findings

Sum rules hold under dimensional regularization.

Sum rules are violated with cut-off regularization.

Breaking of rotational invariance causes sum rule violations.

Abstract

The Burkhardt-Cottingham (BC) sum rule connects the twist-3 light-cone parton distribution function (PDF) $g_{T}(x)$ to the twist-2 helicity PDF $g_{1}(x)$. The chiral-odd counterpart of the BC sum rule relates the twist-3 light-cone PDF $h_{L}(x)$ to the twist-2 transversity PDF $h_{1}(x)$. These BC-type sum rules can also be derived for the corresponding quasi-PDFs. We perform a perturbative check of the BC-type sum rules in the quark target model and the Yukawa model, by going beyond the ultra-violet (UV) divergent terms. We employ dimensional regularization (DR) and cut-off schemes to regulate UV divergences, and show that the BC-type sum rules hold for DR, while they are generally violated when using a cut-off. This violation can be traced back to the breaking of rotational invariance. We find corresponding results for the sum rule relating the mass of the target to the twist-3 PDF $e(x)$. Moreover, we supplement our analytical results with numerical calculations.