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Hypergeometric Functions and Feynman Diagrams

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Anti-Differentiation and the Calculation of Feynman Amplitudes

Abstract

The relationship between Feynman diagrams and hypergeometric functions is discussed. Special attention is devoted to existing techniques for the construction of the ε-expansion. As an example, we present a detailed discussion of the construction of the ε-expansion of the Appell function F 3 around rational values of parameters via an iterative solution of differential equations. As a by-product, we have found that the one-loop massless pentagon diagram in dimension d = 3 − 2𝜖 is not expressible in terms of multiple polylogarithms. Another interesting example is the Puiseux-type solution involving a differential operator generated by a hypergeometric function of three variables. The holonomic properties of the F N hypergeometric functions are briefly discussed.

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Notes

  1. 1.

    Several programs are available for the automatic generation of the Mellin–Barnes representation of Feynman diagrams [19,20,21].

  2. 2.

    The detailed discussion of A-functions and their properties is beyond our current consideration. There are many interesting papers on that subject, including (to mention only a few) Refs. [59,60,61].

  3. 3.

    See Koutschan’s paper in the present volume.

  4. 4.

    Rigorously speaking, this system of equations is correct when there is a contour in C n that is not changed under translations by an arbitrary unit vector, see Ref. [83], so that we treat the powers of propagator as parameters.

  5. 5.

    The monodromy group is the group of linear transformations of solutions of a system of hypergeometric differential equations under rotations around its singular locus. In the case when the monodromy is reducible, there is a finite-dimensional subspace of holomorphic solutions of the hypergeometric system on which the monodromy acts trivially.

  6. 6.

    It is interesting to note, that on-mass shell z = 1, this diagram has two Puiseux type solutions that do not have analytical continuations.

  7. 7.

    It was shown in [159, 160] that multiple Mellin–Barnes integrals related to Feynman diagrams could be evaluated analytically/numerically at each order in ε via multiple sums, without requiring a closed expression in terms of Horn-type hypergeometric functions.

  8. 8.

    See also Refs. [163, 164] for an alternative realization.

  9. 9.

    In general, it could be a more generic recurrence, \(\sum _{1=0}^k p_{k+j}(k+j)c(k+j) = r(k)\).

  10. 10.

    Recent results on the analytical evaluation of inverse binomial sums for particular values of the arguments have been presented in [175,176,177].

  11. 11.

    The appearance of \(1/\sqrt {3}\) in RG functions in seven loops was quite intriguing [122, 180].

  12. 12.

    The results have been written in terms of hyperlogarithms of primitive q-roots of unity.

  13. 13.

    The results of [201] were relevant for the reduction of multiple zeta values to the minimal basis.

  14. 14.

    An interesting construction of the coaction for the Feynman graph has been presented recently in Ref. [207].

  15. 15.

    Originally, such types of functions have been introduced in Ref. [197].

  16. 16.

    Unfortunately, the further prolongation of this project has not been supported by DFG, so that many interesting results remain unpublished.

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Acknowledgements

MYK is very indebted to Johannes Blümlein, Carsten Schneider and Peter Marquard for the invitation and for creating such a stimulating atmosphere in the workshop “Anti-Differentiation and the Calculation of Feynman Amplitudes” at DESY Zeuthen. BFLW thanks Carsten Schneider for kind the hospitality at RISC. MYK thanks the Hamburg University for hospitality while working on the manuscript. SAY acknowledges support from The Citadel Foundation.

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Kalmykov, M., Bytev, V., Kniehl, B.A., Moch, SO., Ward, B.F.L., Yost, S.A. (2021). Hypergeometric Functions and Feynman Diagrams. In: Blümlein, J., Schneider, C. (eds) Anti-Differentiation and the Calculation of Feynman Amplitudes. Texts & Monographs in Symbolic Computation. Springer, Cham. https://doi.org/10.1007/978-3-030-80219-6_9

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