Feynman integrals at large loop order and the $\log$-$\Gamma$ distribution

Abstract

We find empirically that the value of Feynman integrals follows a $\log$-$\Gamma$ distribution at large loop order. This result opens up a new avenue towards the large-order behavior in perturbative quantum field theory. Our study of the primitive contribution to the scalar ${\varphi }^4$ beta function in four dimensions up to 17 loops provides accompanying evidence. Guided by instanton considerations, we discuss the extrapolation of this contribution to all loop orders.

arXiv comment16 pages, v2: various comments and suggestions addressed; v3: version to appear in JHEP

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