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Theoretical Physics
Abstract:
Through the IBP reduction procedure, any Feynman integral can be obtained as a linear combi- nation of a finite number of basis integrals. The latter can be shown to satisfy a system of linear, first-order, partial differential equations and in the case where the integrals evaluate to multiple polylogarithms, these differential equations are particularly easy to solve, when they are cast into canonical form. However, it is also well-known that from two loops onwards more complicated special functions can appear in the solutions. The simplest extension includes elliptic generaliza- tions of multiple polylogarithms. For this case, we propose an alternative "pre-canonical" form for the differential equations. In addition to ε-factorized terms, it allows also terms independent of ε in the differential equation matrix, while still making the Fuchsian property manifest. We conjecture that we can find such a basis always by the means of basis transformations rational in the kinematics and ε when starting from a generic set of basis integrals fulfilling rational dif- ferential equations. Consequently, more complicated functions are avoided completely at the differential equations level. Furthermore, we analyze how established techniques to identify canonical basis integrals can be adapted to find integral bases for our new form. The ideas pre- sented can be applied successfully to multiple elliptic examples, including the famous sunrise graph with three internal masses, as we will illustrate explicitly.
