Abstract
The computational technique of $N$-fold Mellin-Barnes (MB) integrals, presented in a companion paper by the same authors, is used to derive sets of series representations of the massive one-loop conformal 3-point Feynman integral in various configurations. This shows the great simplicity and efficiency of the method in nonresonant cases (generic propagator powers) as well as some of its subtleties in the resonant ones (for unit propagator powers). We confirm certain results in the physics and mathematics literature and provide many new results, some of them dealing with the more general massive one-loop conformal $n$-point case. In particular, we prove two recent conjectures that give the massive one-loop conformal $n$-point integral (for generic propagator powers) in terms of multiple hypergeometric series. We show how these conjectures, that were deduced from a Yangian bootstrap analysis, are related by a tower of new quadratic transformations in Hypergeometric Functions Theory. Finally, we also use our MB method to identify spurious contributions that can arise in the Yangian approach.
Highlights
In Ref. [1], we have presented the first systematic computational method allowing us to derive sets of series representations of N-fold Mellin-Barnes (MB) integrals, in the case where N is a given positive integer
We prove two recent conjectures that give the massive one-loop conformal n-point integral in terms of multiple hypergeometric series. We show how these conjectures, that were deduced from a Yangian bootstrap analysis, are related by a tower of new quadratic transformations in hypergeometric functions theory
V, after having proved that for the three-point integral, region B is included in region A, we were naturally led to the discovery of a new quadratic transformation formula for HC in terms of the other well-known Srivastava HB triple hypergeometric series [5,9]. We inferred from this result that the conjectured expressions of [3] for the n-point integral in region A and B can be interpreted as the lhs and the rhs of a tower of quadratic transformation formulas for hypergeometric functions of nðn − 1Þ=2 variables which, apart from the lowest-order n 1⁄4 2 case, which involves the 2F1 Gauss hypergeometric series, seem to be unknown in hypergeometric functions theory. This comes from the fact that at one loop the conformal constraint, that fixes the sum of the generic propagator powers to be equal to the spacetime dimension, has the same form as the constraint satisfied by the parameters of hypergeometric functions obeying certain quadratic transformations
Summary
In Ref. [1], we have presented the first systematic computational method allowing us to derive sets of series representations of N-fold Mellin-Barnes (MB) integrals, in the case where N is a given positive integer. We inferred from this result that the conjectured expressions of [3] for the n-point integral in region A and B can be interpreted as the lhs and the rhs of a tower of quadratic transformation formulas for hypergeometric functions of nðn − 1Þ=2 variables which, apart from the lowest-order n 1⁄4 2 case, which involves the 2F1 Gauss hypergeometric series, seem to be unknown in hypergeometric functions theory This comes from the fact that at one loop the conformal constraint, that fixes the sum of the generic propagator powers to be equal to the spacetime dimension, has the same form as the constraint satisfied by the parameters of hypergeometric functions obeying certain quadratic transformations. An Appendix brings together all relevant results not explicitly stated in the main core of the text
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