Abstract
Unitarity cut method has been proved to be very useful in the computation of one-loop integrals. In this paper, we generalize the method to the situation where the powers of propagators in the denominator are larger than one in general. We show how to use the trick of differentiation over masses to translate the problem to the integrals where all powers are just one. Then by using the unitarity cut method, we can find the wanted reduction coefficients of all basis except the tadpole. Using this method, we calculate the reduction of scalar bubble, scalar triangle, scalar box and scalar pentagon with general power of propagators.
Highlights
In this paper, we consider the reduction of one-loop integrals with general pole structures, i.e., propagators could have general power
Different master integrals have different analytic structures for the imaginary part, if we can analytically computer the left hand side, we can do the spliting according to the analytic signature of each master integral and find expansion coefficients ct at the right hand side
We have considered the reduction of one-loop integrals with higher poles using the unitarity cut method
Summary
To be the general scalar bubble integral with higher power of propagators. The is well defined for n ≥ 0 and it is easy to derive a recursion relation by integration-by-part. We find that when doing the reduction for triangles and boxes, we will meet the form Bub(n) with the negative integer n For this case, we can use (2.7) to analytically continue from positive n to negative n. For the general bubble coefficients of the reduction of I2(n+1, m+1), using the relation. When doing reduction at both sides, we must have the bubble coefficient to be same, i.e, c2→2(n + 1, m + 1)[K; M1, M2] = c2→2(m + 1, n + 1)[−K; M2, M1]. We have used the Mathematica to check the massless case of our general result (2.18). The third check is that we have used the LiteRed [23] to explicitly calculate some examples with nonzero M1, M2 and we do find the match
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