The Photon Content of the Proton

TL;DR

This paper provides a comprehensive derivation of the photon parton distribution function (PDF) of the proton, including all orders in QED and QCD, and introduces an explicit formula for its calculation from structure functions.

Contribution

It offers a novel, exact representation of the photon PDF in terms of structure functions, valid to all orders, and extends previous results to higher order and polarized cases.

Findings

Derived an exact photon PDF formula using structure functions.

Calculated the photon PDF to one order higher than previous work.

Connected the photon PDF with DGLAP splitting functions at higher orders.

Abstract

The photon PDF of the proton is needed for precision comparisons of LHC cross sections with theoretical predictions. In a recent paper, we showed how the photon PDF could be determined in terms of the electromagnetic proton structure functions $F_2$ and $F_L$ measured in electron-proton scattering experiments, and gave an explicit formula for the PDF including all terms up to next-to-leading order. In this paper we give details of the derivation. We obtain the photon PDF using the factorisation theorem and applying it to suitable BSM hard scattering processes. We also obtain the same PDF in a process-independent manner using the usual definition of PDFs in terms of light-cone Fourier transforms of products of operators. We show how our method gives an exact representation for the photon PDF in terms of $F_2$ and $F_L$, valid to all orders in QED and QCD, and including all non-perturbative corrections. This representation is then used to give an explicit formula for the photon PDF to one order higher than our previous result. We also generalise our results to obtain formul\ae\ for the polarised photon PDF, as well as the photon TMDPDF. Using our formula, we derive the $P_{\gamma i}$ subset of DGLAP splitting functions to order $\alpha \alpha_s$ and $\alpha^2$, which agree with known results. We give a detailed explanation of the approach that we follow to determine a photon PDF and its uncertainty within the above framework.