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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
Full Text:
##### References:
 [1] ’t Hooft, G.; Veltman, M.J., Scalar one loop integrals, Nucl. phys. B, 153, 365, (1979) [2] Veltman, M.J., {\scformf}, a CDC program for numerical evaluation of the form factors, (1979), Utrecht, unpublished [3] van Oldenborgh, G.J.; Vermaseren, J.A., New algorithms for one loop integrals, Z. phys. C, 46, 425, (1990) [4] Fleischer, J.; Tarasov, O.V., SHELL2: package for the calculation of two loop on-shell Feynman diagrams in FORM, Comput. phys. commun., 71, 193, (1992) [5] S.A. Larin, F.V. Tkachov, J.A. Vermaseren, The FORM version of MINCER, NIKHEF-H-91-18 [6] Ferroglia, A.; Passera, M.; Passarino, G.; Uccirati, S., All-purpose numerical evaluation of one-loop multi-leg Feynman diagrams, Nucl. phys. B, 650, 162, (2003) · Zbl 1005.81059 [7] Fleischer, J.; Jegerlehner, F.; Tarasov, O.V., Algebraic reduction of one-loop Feynman graph amplitudes, Nucl. phys. B, 566, 423, (2000) · Zbl 0956.81054 [8] Tarasov, O.V., Connection between Feynman integrals having different values of the space – time dimension, Phys. rev. D, 54, 6479, (1996) · Zbl 0925.81121 [9] Miller, J.C.P., Linear difference equations, (1968), Benjamin New York · Zbl 0217.32802 [10] Milne-Thomson, L.M., The calculus of finite differences, (1960), Macmillan London · Zbl 0008.01801 [11] N. Bleistein, R.A. Handelsman, Asymptotic expansion of integrals, HBJ College and School Division, June 1975 · Zbl 0327.41027 [12] Scharf, R.; Tausk, J.B., Scalar two loop integrals for gauge boson selfenergy diagrams with a massless fermion loop, Nucl. phys. B, 412, 523, (1994) [13] () [14] Berends, F.; Davydychev, A.; Smirnov, V., Small-threshold behaviour of two-loop self-energy diagrams: two-particle thresholds, Nucl. phys. B, 478, 59, (1996) [15] Fleischer, J.; Jegerlehner, F.; Tarasov, O.V.; Veretin, O.L.; Fleischer, J.; Jegerlehner, F.; Tarasov, O.V.; Veretin, O.L., Two-loop QCD corrections of the massive fermion propagator, Nucl. phys. B, Nucl. phys. B, 571, 511, (2000), Erratum · Zbl 0956.81054 [16] Davydychev, A.I.; Kalmykov, M.Y.; Jegerlehner, F.; Kalmykov, M.Y.; Veretin, O.; Davydychev, A.I.; Kalmykov, M.Y., Massive Feynman diagrams and inverse binomial sums, Nucl. phys. B, 605, 266, (2001) · Zbl 0969.81598 [17] Tarasov, O.V., Application and explicit solution of recurrence relations with respect to space – time dimension, Nucl. phys. B (proc. suppl.), 89, 237, (2000) [18] Davydychev, A.I.; Delbourgo, R., A geometrical angle on Feynman integrals, J. math. phys., 39, 4299, (1998) · Zbl 0986.81082 [19] Olsson, Per O.M., Integration of the partial differential equations for the hypergeometric functions F1 and FD of two and more variables, J. math. phys., 5, 420, (1964) · Zbl 0122.31501 [20] Colavecchia, F.D.; Gasaneo, G.; Miraglia, J.E., Numerical evaluation of Appell’s F1 hypergeometric function, Comput. phys. commun., 138, 29-43, (2001) · Zbl 0984.65017 [21] Davydychev, A.I.; Osland, P.; Saks, L., Quark gluon vertex in arbitrary gauge and dimension, Phys. rev. D, 63, 014022, (2001) [22] Lauricella, G., Sulle funzioni ipergeometriche a pru variabili, Rend. circ. mat. Palermo, 7, 111-158, (1893) · JFM 25.0756.01 [23] Saran, S., Hypergeometric functions of three variables, Ganita, 5, 77-91, (1954) · Zbl 0058.29602 [24] Saran, S., Transformation of certain hypergeometric functions of three variables, Acta math., 93, 293-312, (1955) · Zbl 0064.30902 [25] Appell, P.; Kampé de Fériet, J., Fonctions hypergeometriques et hyperspheriques. polynomes d’Hermite, (1926), Gauthier-Villars Paris · JFM 52.0361.13
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