Abstract
We present an efficient method to shorten the analytic integration-by-parts (IBP) reduction coefficients of multi-loop Feynman integrals. For our approach, we develop an improved version of Leinartas’ multivariate partial fraction algorithm, and provide a modern implementation based on the computer algebra system Singular. Furthermore, we observe that for an integral basis with uniform transcendental (UT) weights, the denominators of IBP reduction coefficients with respect to the UT basis are either symbol letters or polynomials purely in the spacetime dimension D. With a UT basis, the partial fraction algorithm is more efficient both with respect to its performance and the size reduction. We show that in complicated examples with existence of a UT basis, the IBP reduction coefficients size can be reduced by a factor of as large as ∼ 100. We observe that our algorithm also works well for settings without a UT basis.
Highlights
IBP reduction of complicated multi-loop Feynman integrals, we obtain reduction coefficients with a huge size, as rational functions of the spacetime parameter D and kinematic variables
We develop an improved version of Leinartas’ multivariate partial fraction algorithm, and provide a modern implementation based on the computer algebra system Singular
As mentioned in ref. [33], we suggest that when a uniform transcendental (UT) master integral basis for the integral family under consideration exists, it is advantageous to first reduce Feynman integrals to the UT basis, and run our partial fraction algorithm to shorten the size of the IBP
Summary
2.1 Integration-by-parts identities and master integrals There are many algebraic relations between different Feynman integrals and it is very efficient to use these relations to obtain further Feynman integrals from the ones we already know. A very useful set of relations can be obtained via the integration-by-parts (IBP) identities, which relate different integrals of a given integral family. Where L is the number of loops, αi are integer indisces and the denominators are given by. I.e are quadratic or linear functions of the external momenta pi and the loop momenta li. L with qk a linear combination of loop momenta and external momenta. With the IBP identities, we can find the basis of a given integral family, which are called master integrals (MIs). A Feynman integral can be written as a linear combination of master integrals, I[α1, . There are many public IBP reduction codes, like AIR, FIRE, Kira, Reduze, LiteRed [5,6,7,8,9,10,11,12,13,14,15]
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