Finding new relationships between hypergeometric functions by evaluating Feynman integrals

Abstract

Several new relationships between hypergeometric functions are found by comparing results for Feynman integrals calculated using different methods. A new expression for the one-loop propagator-type integral with arbitrary masses and arbitrary powers of propagators is derived in terms of only one Appell hypergeometric function F 1 . From the comparison of this expression with a previously known one, a new relation between the Appell functions F 1 and F 4 is found. By comparing this new expression for the case of equal masses with another known result, a new formula for reducing the F 1 function with particular arguments to the hypergeometric function F 2 3 is derived. By comparing results for a particular one-loop vertex integral obtained using different methods, a new relationship between F 1 functions corresponding to a quadratic transformation of the arguments is established. Another reduction formula for the F 1 function is found by analyzing the imaginary part of the two-loop self-energy integral on the cut. An explicit formula relating the F 1 function and the Gaussian hypergeometric function F 1 2 whose argument is the ratio of polynomials of degree six is presented.