Analytic Tadpole Coefficients of One-loop Integrals

TL;DR

This paper introduces a novel method using differential operators and recurrence relations to analytically compute tadpole coefficients in one-loop integrals, overcoming limitations of traditional unitarity cut techniques.

Contribution

It develops a new approach employing differential equations and tensor structures to derive analytic tadpole coefficients for one-loop integrals.

Findings

Derived differential equations for tadpole coefficients.

Transformed differential equations into recurrence relations.

Obtained explicit analytic expressions for tadpole coefficients.

Abstract

One remaining problem of unitarity cut method for one-loop integral reduction is that tadpole coefficients can not be straightforward obtained through this way. In this paper, we reconsider the problem by applying differential operators over an auxiliary vector $R$. Using differential operators, we establish the corresponding differential equations for tadpole coefficients at the first step. Then using the tensor structure of tadpole coefficients, we transform the differential equations to the recurrence relations for undetermined tensor coefficients. These recurrence relations can be solved easily by iteration, and we can obtain analytic expressions of tadpole coefficients for arbitrary one-loop integrals.