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Iterative Non-iterative Integrals in Quantum Field Theory

Part of the Texts & Monographs in Symbolic Computation book series (TEXTSMONOGR)


Single scale Feynman integrals in quantum field theories obey difference or differential equations with respect to their discrete parameter N or continuous parameter x. The analysis of these equations reveals to which order they factorize, which can be different in both cases. The simplest systems decouple to linear differential equations which factorize to first-order. For them complete solution algorithms exist. The next interesting level is formed by those cases that decouple to linear differential equations in which also irreducible second-order factors emerge. We give a survey on the latter case. The solutions can be obtained as general \(_2F_1\) solutions. The corresponding solutions of the associated inhomogeneous differential equations form so-called iterative non-iterative integrals. There are known conditions under which one may represent the solutions by complete elliptic integrals. In this case one may find representations in terms of meromorphic modular forms, out of which special cases allow representations in the framework of elliptic polylogarithms with generalized parameters. These are in general weighted by a power of \(1/\eta (\tau )\), where \(\eta (\tau )\) is Dedekind’s \(\eta \)-function. Single scale elliptic solutions emerge in the \(\rho \)-parameter, which we use as an illustrative example. They also occur in the 3-loop QCD corrections to massive operator matrix elements and the massive 3-loop form factors.

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  1. 1.

    Iterative non-iterative integrals have been introduced by the author in a talk on the 5th International Congress on Mathematical Software, held at FU Berlin, July 11–14, 2016, with a series of colleagues present, cf. [41].

  2. 2.

    In the present case only single poles appear; for Fuchsian differential equations q(x) may have double poles.

  3. 3.

    Here we use the notation applied by Mathematica. In some part of the literature one defines: etc.

  4. 4.

    This will not apply to simpler cases like or , however.

  5. 5.

    The dimension of the corresponding vector space can be also calculated using the Sage program by W. Stein [113].

  6. 6.

    This is, besides the well-know Landen transformation [78, 123], the next higher modular transformation; for a survey cf. [124]. Also for the hypergeometric function there are rational modular transformations [125].


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I would like to thank J. Ablinger, A. De Freitas, M. van Hoeij, E. Imamoglu, P. Marquard, C.G. Raab, C.-S. Radu, and C. Schneider for collaboration in two projects and A. Behring, D. Broadhurst, H. Cohen, G. Köhler, P. Paule, E. Remiddi, M. Steinhauser, J.-A. Weil, S. Weinzierl and D. Zagier for discussions.

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Blümlein, J. (2019). Iterative Non-iterative Integrals in Quantum Field Theory. In: Blümlein, J., Schneider, C., Paule, P. (eds) Elliptic Integrals, Elliptic Functions and Modular Forms in Quantum Field Theory. Texts & Monographs in Symbolic Computation. Springer, Cham.

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