Abstract
For loop integrals the standard method is reduction. A well-known reduction method for one-loop integrals is the Passarino-Veltman (PV) reduction. Inspired by the recent paper [B. Feng, T. Li, and X. Li, J. High Energy Phys. 09 (2021) 081.] where the tadpole reduction coefficients have been solved, in this paper we show the same technique can be used to give a complete integral reduction for any one-loop integrals. The differential operator method is an alternative version of the PV-reduction method. Using this method, analytic expressions of all reduction coefficients of the master integrals can be given by algebraic recurrence relation easily. We demonstrate our method explicitly with several examples.
Highlights
The calculation of the scattering amplitude at higherloop level is always like a chronic disease to block the evolution of high energy physics
II we review the derivation of the differential equations of reduction coefficients and show how to obtain recursion relations of the expansion coefficients
In this paper we show how to use the differential operators to get the analytical expressions for the reduction coefficients of all master basis
Summary
The calculation of the scattering amplitude at higherloop level is always like a chronic disease to block the evolution of high energy physics. Since the loop integral is well defined using the dimensional regularization, the unitarity cut method in pure 4D needs to generalize to ð4 − 2εÞ-dimension, which has been done in [28,30]. Based on this generalization, the analytic expressions for reduction coefficients (except the tadpole coefficients) have been derived in a series of papers [31,32,33,34,35]. By comparing two sides of the expansion, we will achieve the recursion relations of the coefficients of the master integrals in differential form. In Appendix, we list all reduction coefficients for tensor triangles, boxes, and pentagons with rank 1 and rank 2
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