On Laporta's 4-loop sunrise formulae
arXiv:1801.02182 · doi:10.1007/s11139-018-0090-z
Abstract
We prove Laporta's conjecture\begin{align*}&\int_0^\infty\frac{\mathrm d\, x_1}{x_1}\int_0^\infty\frac{\mathrm d\, x_2}{x_2}\int_0^\infty\frac{\mathrm d\, x_3}{x_3}\int_0^\infty\frac{\mathrm d\, x_4}{x_4}\frac{1}{\left(1+\sum^4_{k=1}x_k\right)\left(1+\sum^4_{k=1}\frac{1}{x_{k}} \right)-1}\\={}&\frac43 \int_{0}^π\mathrm d\, ϕ_1 \int_{0}^π\mathrm d\, ϕ_2\int_{0}^π\mathrm d\, ϕ_3 \int_{0}^π\mathrm d\, ϕ_4\frac{1}{4-\sum_{k=1}^4\cos ϕ_k}, \end{align*} 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{align*} \frac{4 π^{5/2}}{\sqrt{3}}\left\{ \frac{\sqrt{3} }{2^6 }\left[\frac{Γ\left(\frac{1}{3}\right)}{\sqrtπ}\right]^9\, _4F_3\left(\left. \begin{array}{c}\frac{1}{6},\frac{1}{3},\frac{1}{3},\frac{1}{2}\\[4pt]\frac{2}{3},\frac{5}{6},\frac{5}{6}\end{array} \right|1\right)-\frac{2^{4}}{3}\left[\frac{\sqrtπ}{Γ\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}\\[4pt]\frac{7}{6},\frac{7}{6},\frac{4}{3}\end{array} \right|1\right) \right\}. \end{align*}
i+28 pages
References in corpus (6)
- High-precision calculation of the 4-loop contribution to the electron g-2 in QED
- Elliptic integral evaluations of Bessel moments
- Analytical expressions of 3 and 4-loop sunrise Feynman integrals and 4-dimensional lattice integrals
- Wrońskian factorizations and Broadhurst-Mellit determinant formulae
- Wick rotations, Eichler integrals, and multi-loop Feynman diagrams
- Hilbert Transforms and Sum Rules of Bessel Moments