Revisiting SU(N) integrals

TL;DR

This paper explores integrals over SU(N) matrices, providing new explicit formulas for cases where the number of U and U† factors differ by N, and discusses their behavior in the large N limit.

Contribution

It introduces explicit expressions for SU(N) integrals with p=n+N, extending known results and analyzing their large N properties.

Findings

Derived explicit formulas for SU(N) integrals with p=n+N.

Compared the large N behavior of different integral cases.

Identified key differences and similarities in integral structures.

Abstract

In this note, I revisit integrals over $\SU(N)$ of the form $ \int DU\, U_{i_1j_1}\cdots U_{i_pj_p}\Ud_{k_1l_1}\cdots \Ud_{k_nl_n}$. While the case $p=n$ is well known, it seems that explicit expressions for $p=n+N$ had not appeared in the literature. Similarities and differences, in particular in the large $N$ limit, between the two cases are discussed