A power series identity and Bessel-type integrals over unitary groups

TL;DR

This paper proves a power series identity related to Bessel-type integrals over unitary groups, connecting supersymmetric extensions, determinantal formulas, and Schur functions with implications in quantum chromodynamics.

Contribution

It introduces a new power series identity that links Bessel-type integrals, supersymmetric extensions, and Schur functions, providing a novel proof via Cartan decomposition.

Findings

Proved a conjectured power series identity.

Derived a determinantal formula for Bessel-type integrals.

Established a probabilistic interpretation of the identity.

Abstract

In 2008, Lehner, Wettig, Guhr and Wei conjectured a power series identity and showed that it implied a determinantal formula for a Bessel-type integral over the unitary supergroup. The integral is the supersymmetric extension of Bessel-type integrals over the unitary group appearing as partition functions in quantum chromodynamics. The identity is proved by interpreting both sides as the same unitary integral, which can be computed using the Cartan decomposition. An equivalent identity of Schur functions is also given and interpreted probabilistically.