Note on the evaluation of one type scalar three-point integral extracted from the Higgs boson decay

TL;DR

This paper analytically evaluates a specific scalar three-point integral relevant to Higgs decay to gluons, providing formulas for different root cases and discussing applications in heavy quark decays.

Contribution

It presents complete analytic solutions for a scalar three-point integral with two internal masses, covering cases with and without real roots.

Findings

Analytic formulas derived for integrals with no real roots.

Formulas applicable to Higgs decay and heavy quark radiative decays.

Discussed applications in Standard Model and vector-like models.

Abstract

Motivated by the Higgs boson decaying to $gg$ at leading order approximation, the amplitude of scalar one loop three-point diagram with two different internal masses are evaluated and fully analytic results are obtained. The main ingredient of the evaluation is a integral in which the integrand is product of the reciprocal of the integral variable and a logarithm whose argument is a quadratic function of the general form. Complete analytic results for the two cases that there is no real root and there are two different real roots for the logarithm are presented. Applications of the results in this paper to radiative decays of the heavy quarks in the Standard Model and in the singlet vector-like model are discussed.