Analytic tadpole coefficients of one-loop integrals

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.

Module Intersection and Uniform Formula for Iterative Reduction of One-loop Integrals

“…As pointed out in several papers [18,50,[54][55][56][57], arbitrary tensor structure can be compactly organized using an auxiliary vector R. Thus for TR of one-loop n-point integrals, we enlarge the set of propagators given in (2.1) by adding one new propagator, i.e.,…”

Reduction with degenerate Gram matrix for one-loop integrals

“…Recently, we proposed a new framework for general one-loop tensor reduction by employing an auxiliary vector R and two kinds of differential operators [1,2]. Similar to other reduction methods, our method also suffers from divergences for vanishing Gram determinant, which appears as the inverse of Gram matrix in the recursion constructions of reduction coefficients.…”

“…It is well known that the vanishing Gram determinant indicates external momenta are not completely independent of each other, thus the master integral of the highest topology will not be in the basis anymore, and it can be decomposed into combinations of master integrals of lower topologies. 1 With this decomposition, the reduction coefficients will be reorganized so that the divergences between different terms will cancel with each other. To see this picture clearly, calculating analytic coefficients will be important, which is exactly the merit of our new method.…”

“…To see this picture clearly, calculating analytic coefficients will be important, which is exactly the merit of our new method. In this paper, we will systematically study the degenerate case with vanishing Gram determinant using the results in [1,2]. It turns out that the decomposition coefficients can easily be solved as a series in the Gram determinant.…”

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An improved PV-reduction (Passarino-Veltman) method for one-loop integrals with auxiliary vector R has been proposed in [1,2]. It has also been shown that the new method is a self-completed method in [3].

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