Analytic structure of one-loop coefficients

TL;DR

This paper reformulates one-loop coefficients in Lorentz-invariant forms, revealing their analytic structure, singularity types, and physical behavior under various limits, enhancing understanding of quantum field theory calculations.

Contribution

It introduces Lorentz-invariant contraction forms for one-loop coefficients, clarifying their analytic structure and singularity behavior, which was not explicitly shown in previous spinor form expressions.

Findings

Coefficients contain only second-type singularities.

Highest degree of singularity correlates with numerator momentum degree.

Same singularities appear across different coefficients, aiding physical interpretation.

Abstract

By the unitarity cut method, analytic expressions of one-loop coefficients have been given in spinor forms. In this paper, we present one-loop coefficients of various bases in Lorentz-invariant contraction forms of external momenta. Using these forms, the analytic structure of these coefficients becomes manifest. Firstly, coefficients of bases contain only second-type singularities while the first-type singularities are included inside scalar bases. Secondly, the highest degree of each singularity is correlated with the degree of the inner momentum in the numerator. Thirdly, the same singularities will appear in different coefficients, thus our explicit results could be used to provide a clear physical picture under various limits (such as soft or collinear limits) when combining contributions from all bases.