Random Aharonov-Bohm vortices and some exact families of integrals: Part II

TL;DR

This paper analyzes complex integrals involving Bessel functions arising from a perturbation theory problem in quantum physics, revealing their algebraic structure and relations to special constants like zeta values.

Contribution

It provides a detailed analysis of integrals involving Bessel functions, showing their linear relations to special constants and establishing recurrence relations for these integrals.

Findings

Integrals are expressible as linear combinations of zeta(5), specific Bessel integrals, and rational numbers.

A recurrence relation in parameter n is derived for these integrals.

The asymptotic behavior is linked to the smallest eigenvalue of a transition matrix.

Abstract

At 6th order in perturbation theory, the random magnetic impurity problem at second order in impurity density narrows down to the evaluation of a single Feynman diagram with maximal impurity line crossing. This diagram can be rewritten as a sum of ordinary integrals and nested double integrals of products of the modified Bessel functions $K_{\nu}$ and $I_{\nu}$, with $\nu=0,1$. That sum, in turn, is shown to be a linear combination with rational coefficients of $(2^5-1)\zeta(5)$, $\int_0^{\infty} u K_0(u)^6 du$ and $\int_0^{\infty} u^3 K_0(u)^6 du$. Unlike what happens at lower orders, these two integrals are not linear combinations with rational coefficients of Euler sums, even though they appear in combination with $\zeta(5)$. On the other hand, any integral $\int_0^{\infty} u^{n+1} K_0(u)^p (uK_1(u))^q du$ with weight $p+q=6$ and an even $n$ is shown to be a linear combination with rational coefficients of the above two integrals and 1, a result that can be easily generalized to any weight $p+q=k$. A matrix recurrence relation in $n$ is built for such integrals. The initial conditions are such that the asymptotic behavior is determined by the smallest eigenvalue of the transition matrix.