On Laporta’s 4-loop sunrise formulae

Abstract

We prove Laporta’s conjecture

which relates the 4-loop sunrise diagram in 2-dimensional quantum field theory to Watson’s integral for 4-dimensional hypercubic lattice. We also establish several related integral identities proposed by Laporta, including a reduction of the 4-loop sunrise diagram to special values of Euler’s gamma function and generalized hypergeometric series:

$$\begin{aligned}&\frac{4 \pi ^{5/2}}{\sqrt{3}}\left\{ \frac{\sqrt{3} }{2^6 }\left[ \frac{\Gamma \left( \frac{1}{3}\right) }{\sqrt{\pi }}\right] ^9\, _4F_3\left( \left. \begin{array}{c}\frac{1}{6},\frac{1}{3},\frac{1}{3},\frac{1}{2}\\ \frac{2}{3},\frac{5}{6},\frac{5}{6}\end{array} \right| 1\right) -\frac{2^{4}}{3}\left[ \frac{\sqrt{\pi }}{\Gamma \left( \frac{1}{3}\right) }\right] ^9\, _4F_3\left( \left. \begin{array}{c}\frac{1}{2},\frac{2}{3},\frac{2}{3},\frac{5}{6}\\ \frac{7}{6},\frac{7}{6},\frac{4}{3}\end{array} \right| 1\right) \right\} . \end{aligned}$$

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Notes

  1. 1.

    It is arguable whether \( L(f_{4,6},2)\) should be counted as a closed-form evaluation in its own right. As one may recall, Bloch–Kerr–Vanhove [7] and Samart [30] have expressed the 3-loop sunrise diagram \( 2^3\int _0^\infty I_0(t)[K_0(t)]^4t{{{\,\mathrm{d}\,}}}t\) as \( \frac{12\pi }{\sqrt{15}}L(f_{3,15},2)\), for a modular form \(f_{3,15}(z)=[\eta (3z)\eta (5z)]^3+[\eta (z)\eta (15z)]^3\) of weight 3 and level 15. Meanwhile, according to the work of Rogers–Wan–Zucker [28], such a special L-value can be reduced to a product of gamma values at rational arguments, thus leaving us a formula \( 2^3\int _0^\infty I_0(t)[K_0(t)]^4t{{{\,\mathrm{d}\,}}}t=\frac{1}{30\sqrt{5}}\Gamma \left( \frac{1}{15} \right) \Gamma \left( \frac{2}{15} \right) \Gamma \left( \frac{4}{15} \right) \Gamma \left( \frac{8}{15} \right) \) (see [39, Theorem 2.2.2] for a simplified proof of this integral identity). At the time of writing, it is not clear to us if the special L-value \( L(f_{4,6},2)\) admits a similar reduction.

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Acknowledgements

A large proportion of this work has been assembled from my research notes on hypergeometric series, which were prepared at Princeton in 2012. I thank Prof. Weinan E (Princeton University and Peking University) for running a seminar on mathematical problems in quantum fields at Princeton, covering both 2-dimensional and \((4-\varepsilon )\)-dimensional theories. I am grateful to Dr. David Broadhurst for many fruitful communications on recent progress in the arithmetic properties of Feynman diagrams. In particular, I thank him for suggesting the challenging integral identity in (1.11).

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The research was supported in part by the Applied Mathematics Program within the Department of Energy (DOE) Office of Advanced Scientific Computing Research (ASCR) as part of the Collaboratory on Mathematics for Mesoscopic Modeling of Materials (CM4).

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Zhou, Y. On Laporta’s 4-loop sunrise formulae. Ramanujan J 50, 465–503 (2019). https://doi.org/10.1007/s11139-018-0090-z

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Keywords

  • Watson integrals
  • Bessel functions
  • Feynman integrals
  • Sunrise diagrams

Mathematics Subject Classification

  • 33C05
  • 33C10
  • 33C20 (Primary)
  • 81T18
  • 81T40
  • 81Q30 (Secondary)