Nakanishi integral representation for the quark-photon vertex

TL;DR

This paper develops a nonperturbative integral representation framework for the quark-photon vertex using Dyson-Schwinger equations, enabling solutions in Minkowski space and advancing understanding of quark-photon interactions.

Contribution

It introduces Nakanishi-like integral representations for the quark-photon vertex and related functions, transforming Dyson-Schwinger equations into coupled integro-differential equations for nonperturbative solutions.

Findings

Derived integral representations for the vertex.

Transformed Dyson-Schwinger equations into coupled equations.

Provided a self-consistent solution framework in Minkowski space.

Abstract

Using a nonperturbative framework of Dyson-Schwinger equations a class of Nakanishi's like integral representations for the transverse part of the quark-photon vertex is derived. For this but also for its own purpose the two and single variable integral representations for untruncated quark-antiquark-photon vertex is proposed as well. To exhibit the adequacy of proposed representation, the Dyson-Schwinger equation for the vertex is transformed into the equivalent set of coupled integro-differential equations for Nakanishi weight functions: functions that appear linearly in the numerator of given integral representation. Their knowledge then provide self-consistent nonperturbative solution for the vertex in the entire Minkowski space.