Analytic tadpole coefficients of one-loop integrals

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.

Automated summary

By applying differential operators on an auxiliary vector, we established differential equations for tadpole coefficients and transformed them into recurrence relations for undetermined tensor coefficients. Solving these recurrence relations iteratively allows us to obtain analytic expressions of tadpole coefficients for arbitrary one-loop integrals.