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Iterated elliptic and hypergeometric integrals for Feynman diagrams. (English) Zbl 1394.81164

A central problem in calculating higher loop Feynman integrals in renormalizable quantum field theories consists in solving the differential equations obtained from the integration-by-parts identities (IBPs), which rule the master integrals. In the present paper, the authors have solved master integrals which correspond to irreducible differential equations of second order with more than three singularities fully analytically. They appear in the calculation of the QCD corrections to the rho-parameter at 3-loop order. They form typical examples for structures which appear in solving IBP-relations for Feynman diagrams beyond the well-understood case of singly factorizing integrals given as iterative integrals over a general alphabet. The latter case has been already algorithmized completely, even not needing any special choice of the basis. The second-order structures can be mapped to \({}_{2}F_{1}\) solutions under certain conditions presented in this paper. The authors have outlined the algorithmic analytic solution in this case in terms of iterative integrals over partly non-iterative letters. Indeed this holds even for much more general solutions than those of the \({}_{2}F_{1}\) type.

MSC:

81S40 Path integrals in quantum mechanics
81V05 Strong interaction, including quantum chromodynamics
81T15 Perturbative methods of renormalization applied to problems in quantum field theory
81T18 Feynman diagrams
33C75 Elliptic integrals as hypergeometric functions
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References:

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[188] As a convention, the modulus k2 = z is chosen in this paper and also used within the framework of Mathematica.
[189] For more involved physical problems, also irreducible higher-order differential equations may occur.
[190] For a simple earlier case, see, e.g., Refs. 16 and 62.
[191] Recent developments have been discussed also during the conference Elliptic Integrals, Elliptic Functions, and Modular Forms in Quantum Field Theory, DESY, Zeuthen, Oct. 23-26, 2017 .
[192] In the present case, only single poles appear; for Fuchsian differential equations, q(x) may have double poles.
[193] The sign can be adjusted by \(\psi_{1 b}^{(0)} \leftrightarrow \psi_{2 b}^{(0)}\).
[194] Iterative non-iterative integrals have been introduced by the 2nd 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. Ref. 96.
[195] This technique has also been used in Ref. 28.
[196] We thank P. Marquard for having provided all the necessary constants and a series of expansion parameters for the solutions given in Ref. 41 in a computer readable form.
[197] For q-expansions of the Weierstraß’ and σ functions, see, e.g., Ref. 109.
[198] In the literature, different definitions of the Jacobi ϑ functions are given, cf. Ref. 47, p. 305. We follow the one used by Mathematica.
[199] The ϑ and η functions, as well as their q-series, also play an important role in other branches of physics, as, e.g., in lattice models in statistical physics in the form of Rogers-Ramanujan identities; see, e.g., Refs. 45, 110 and 111, percolation theory,112 and other applications, e.g., in attempting to describe properties of deep-inelastic structure functions.113 In the latter case, the asymptotic behavior of Dedekind’s η function at x ∼ 1 seems to resemble the structure function for a wide range down to x ∼ 10−5. It has a surprisingly similar form as the small-x asymptotic wave equation solution,114 however, with a rising power of the soft pomeron.115
[200] It is usually desirable to work with η-functions depending on integer multiples of τ only, cf. Ref. 70, which can be achieved by rescaling the power of q.
[201] We introduced the symbols \(\overline{\eta}\) and \(\overline{\eta}^\prime\) here, instead of the original notation η, η′ to avoid confusion with Dedekind’s η function; here \(\overline{\eta}^\prime\) is a new function given by (6.47), but not the derivative of \(\overline{\eta}\).
[202] For its efficient evaluation, see, e.g., Ref. 131.
[203] The dimension of the corresponding vector space can also be calculated using the Sage program by Stein.134
[204] For a recent numerical representation of elliptic polylogarithms, using basic hypergeometric series,51 see Ref. 138. For |q| < 1, one obviously may always use suitable SL_{2} maps to obtain fast converging representations even using the above formulae. The problematic cases are the few isolated points |q| = 1.
[205] J.B. would like to thank S. Weinzierl for having pointed out these references to him.
[206] This is, besides the well-know Landen transformation,47,150 the next higher modular transformation. There exist even higher-order transformations, which were derived in Refs. 149 and 151-154. Also for the hypergeometric function \({}_2F^1 \left(\begin{matrix} \begin{matrix} \frac{1}{r}, \& 1 - \frac{1}{r} \end{matrix} \\ 1 \end{matrix} \left|\right. z(x)\right)\), there are rational modular transformations.155
[207] Representations of this kind are frequently used working first in a region which is free of singularities; see, e.g., Ref. 156.
[208] Eichler stated161 that there are five basic mathematical operations: addition, subtraction, multiplication, division, and modular forms.
[209] While the dispersive technique can be applied to usual Feynman integral calculations directly, this is not the case for diagrams containing local operator insertions.15-17,165,166 The latter short-distance representation would need to be re-derived after having performed the cut of the corresponding usual Feynman diagram.
[210] K3 stands for “Kummer, Kähler, and Kodaira”. The term has been introduced by Weil.169
[211] It is interesting to note that this q-series is closely related to the series used by Beukers179 in his series-proof of the irrationality of ζ_{2} and ζ_{3} related through an Eichler integral.180 Already his earlier proof based on integrals181 used functions playing a central role in the calculation of Feynman integrals.
[212] For similar investigations in the case of infinite sums, see Refs. 184 and 185.
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