Large $N_c$ Expansion and the Parity Violating $\pi, N, \Delta$ Couplings

TL;DR

This paper explores the large $N_c$ limit to analyze parity-violating couplings among pions, nucleons, and delta particles, deriving relations and consistency conditions that match known phenomenological values.

Contribution

It introduces a large $N_c$ framework to systematically determine relations among parity-violating couplings and reproduces known phenomenological ratios.

Findings

Derived the $N_c$ ordering of PV couplings.

Established relations among couplings consistent with chiral loop corrections.

Found $h_\Delta = -\frac{3}{\sqrt{5}} h_\pi$, matching phenomenological data.

Abstract

In the limit of large $N_c$ we first consider the $N_c$ ordering of the various parity violating $\pi, N, \Delta$ couplings. Then we derive the relations among these couplings and consistency relations from the stability of these couplings under the chiral loop corrections with and without the mass splitting between $N$ and $\Delta$. Especially we find that $h_\Delta =-{3\over \sqrt{5}}h_\pi$ in the large $N_c$ limit, which correctly reproduces the relative sign and magnitude of the "DDH" values for these PV couplings.