Scalar one-loop 4-point integral with one massless vertex in loop regularization

TL;DR

This paper analytically evaluates the scalar one-loop 4-point function with a massless vertex using loop regularization, explicitly separating divergent and stable parts with detailed dilogarithm expressions.

Contribution

It introduces a loop regularization approach with a characteristic scale to evaluate the scalar one-loop 4-point function, explicitly handling divergences and providing detailed analytical results.

Findings

Explicit analytical expression with 44 dilogarithms

Clear separation of infrared divergent and stable parts

Regularization method applicable to similar loop integrals

Abstract

The scalar one-loop 4-point function with one massless vertex is evaluated analytically by employing the loop regularization method. According to the method a characteristic scale $\mu_{s}$ is introduced to regularize the divergent integrals. The infrared divergent parts, which take the form of $\ln^{2}(\lambda^{2}/\mu^{2}_{s})$ and $\ln(\lambda^{2}/\mu^{2}_{s})$ as $\mu_{s}\rightarrow 0$ where $\lambda$ is a constant and expressed in terms of masses and Mandelstam variables, and the infrared stable parts are well separated. The result is shown explicitly via $44$ dilogarithms in the kinematic sector in which our evaluation is valid.