Abstract
A number of irreducible master integrals for L-loop sunrise and bubble Feynman diagrams with generic values of masses and external momenta are explicitly evaluated via the Mellin-Barnes representation.
Highlights
The aim of the present paper is to extend the approach described in refs. [27, 28] to the multivariable case with reducible monodromy
A number of irreducible master integrals for L-loop sunrise and bubble Feynman diagrams with generic values of masses and external momenta are explicitly evaluated via the Mellin-Barnes representation
Remark B : for sunrise and bubble diagrams, each term of the hypergeometric representation in eqs. (2.8) and (2.13) satisfies the conditions of Lemma F, so that the dimension of the space of nontrivial solutions of each term is defined by eq (3.5) and the differential reduction of each term is described by eq (3.6)
Summary
Any particular solution of the system defined by eq (2.3b) about the points zi = 0 for a generic set of parameters is a linear combination of solutions defined by eq (2.6) with undetermined coefficients To fix these coefficients, it is necessary to evaluate the Mellin-Barnes integral as a power series solution [33]. The main questions are how to find such linear relations between the parameters and how to define a minimal set of the additional PDEs. Our approach [27, 28] to these problems is based on studying the inverse differential operators: the exceptional case of parameters, where the dimension of the solution space is reduced, corresponds to the condition that the denominators of the functions Qi entering eq (2.5) are equal to zero for arbitrary values of zi [44,45,46].
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