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A new hypergeometric representation of one-loop scalar integrals in \(d\) dimensions. (English) Zbl 1058.81605
Summary: A difference equation w.r.t. space-time dimension \(d\) for \(n\)-point one-loop integrals with arbitrary momenta and masses is introduced and a solution presented. The result can in general be written as multiple hypergeometric series with ratios of different Gram determinants as expansion variables. Detailed considerations for 2-, 3- and 4-point functions are given. For the 2-point function we reproduce a known result in terms of the Gauss hypergeometric function \(_2F_1\). For the 3-point function an expression in terms of \(_2F_1\) and the Appell hypergeometric function \(F_1\) is given. For the 4-point function a new representation in terms of \(_2F_1\), \(F_1\) and the Lauricella-Saran functions \(F_S\) is obtained. For arbitrary \(d=4-2 \varepsilon\), momenta and masses the 2-, 3- and 4-point functions admit a simple one-fold integral representation. This representation will be useful for the calculation of contributions from the \(\varepsilon\)-expansion needed in higher orders of perturbation theory. Physically interesting examples of 3- and 4-point functions occurring in Bhabha scattering are investigated.

MSC:
81T18 Feynman diagrams
33C45 Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.)
Software:
Mincer; SHELL2; FORM
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References:
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