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

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.