On Laporta’s 4-loop sunrise formulae

  • Yajun ZhouEmail author


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


Watson integrals Bessel functions Feynman integrals Sunrise diagrams 

Mathematics Subject Classification

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



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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Authors and Affiliations

  1. 1.Program in Applied and Computational Mathematics (PACM)Princeton UniversityPrincetonUSA
  2. 2.Academy of Advanced Interdisciplinary Studies (AAIS)Peking UniversityBeijingPeople’s Republic of China

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