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 ${\int }_0^{\infty }{u}^{n+1}{K}_0(u{)}^p(u{K}_1(u){)}^{q}du$ 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.

9 pages, LaTeX 2e. A few sentences rephrased more exactly, misprints corrected