C\'alculo de la integral escalar a un lazo $A(m^2)$ mediante el teorema del residuo

TL;DR

This paper calculates the one-loop scalar integral in quantum field theory using dimensional regularization and the Residue Theorem, demonstrating their equivalence through Gamma function properties.

Contribution

It introduces a novel application of the Residue Theorem to evaluate scalar integrals in quantum field theory, comparing it with dimensional regularization methods.

Findings

Successful calculation of the scalar integral using the Residue Theorem

Demonstration of equivalence between the Residue Theorem and dimensional regularization

Enhanced pedagogical understanding of radiative correction calculations

Abstract

Radiative corrections in the phenomenology of particle physics lead to great predictions on the observables of the Standard Model (SM) which are in good agreement with different measurements on Particle Accelerators and Detectors and in the case of numeric predictions on Quantum Chromodynamics (QCD) and its study in the lattice for the low energy regime. This is a pedagogical paper in which we calculate the one loop scalar integral $A(m^2)$, that arises in the simplest radiative correction calculation, by means of the dimensional regularization. As main result, we perform also the evaluation of this integral by using the Residue Theorem on complex variable. In addition, the equivalence between the two procedures is shown through Gamma function properties.