The system of partial differential equations for the $C_{_0}$ function

TL;DR

This paper introduces a novel method for analyzing scalar integrals in Feynman diagrams, recovering known results and deriving new insights into the $C_{0}$ function, applicable to epsilon-expansion coefficients.

Contribution

It presents a new approach that fully recovers existing results and generates new findings for the $C_{0}$ function in scalar Feynman integrals.

Findings

Recovered well-known results in the literature.

Produced new results on the $C_{0}$ function.

Applicable to epsilon-expansion coefficients in scalar integrals.

Abstract

We present an approach to analyze the scalar integrals of any Feynman diagrams in detail here. This method not only completely recovers some well-known results in the literature, but also produces some brand new results on the $C_{_0}$ function. The approach can be employed to evaluate the coefficient of arbitrary power of $\varepsilon$ in the expansion of a scalar integral, where $D=4-2\varepsilon$ denotes the time-space dimension.