Elliptic integral evaluations of Bessel moments and applications

Abstract

We record what is known about the closed forms for various Bessel function moments arising in quantum field theory, condensed matter theory and other parts of mathematical physics. More generally, we develop formulae for integrals of products of six or fewer Bessel functions. In consequence, we are able to discover and prove closed forms for c n,k := ∞ 0 t k K n 0 (t) dt with integers n = 1, 2, 3, 4 and k ≥ 0, obtaining new results for the even moments c 3,2k and c 4,2k . We also derive new closed forms for the odd moments s n,2k+1 := ∞ 0 t 2k+1 I 0 (t) K n−1 0 (t) dt with n = 3, 4 and for t n,2k+1 := ∞ 0 t 2k+1 I 2 0 (t) K n−2 0

High-precision computation: Mathematical physics and dynamics

“…In yet a more recent study, co-authored with physicists David Broadhurst and Larry Glasser [5], we were able to analytically recognize many of these C n,k integrals-because, remarkably, these same integrals appear naturally in quantum field theory (for odd k). We also discovered, and then proved with considerable effort, that with c n,k normalized by C n,k = 2 n c n,k /(n!…”

The kite integral to all orders in terms of elliptic polylogarithms

“…More challenging are Feynman integrals, which cannot be expressed in terms of multiple polylogarithms. A prominent example is the two-loop sunrise integral with non-zero masses [17][18][19][20][21][22][23][24][25][26][27][28][29][30][31][32][33][34]. Evaluating this integral one encounters elliptic generalisations of (multiple) polylogarithms and every term of the Laurent expansion of the dimensionally regulated sunrise integral may be expressed in terms of functions from this class [32].…”

The iterated structure of the all-order result for the two-loop sunrise integral

“…The simplest Feynman integral which cannot be expressed in terms of multiple polylogarithms is the two-loop sunrise integral with non-vanishing masses. This Feynman integral has already received considerable attention in the literature [25][26][27][28][29][30][31][32][33][34][35][36][37][38][39][40][41][42][43]. In this paper we study the two-loop sunrise integral with equal non-zero masses in D = 2 − 2ε space-time dimensions.…”