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.
Highlights
To obtain the coefficients of tadpole integrals by the usual unitary cut method, an idea was proposed to add an auxiliary, unphysical propagator in the integrand [9]
For a general tensor one-loop integral, we will first introduce an auxiliary vector Rμ and assume its reduction to scalar master integrals,1 consider applying some differential operators with respect to R to the integral, so we obtain the differential equations of tadpole coefficients after comparing two sides of the equation
We have considered calculating the reduction tadpole coefficient of general one-loop integrals
Summary
We will consider the integral reduction of a general 1-loop tensor integral using differential operators with respect to an auxiliary vector R. In subsection 2.1 we will derive the differential equations for the coefficients of the master integrals. Using the general tensor structure of reduction coefficients, we transform the differential equations of tadpole coefficients into recurrence relations in subsection 2.2
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