Chiral Perturbation Theory, the ${1/N_c}$ expansion and Regge behaviour determine the structure of the lightest scalar meson

TL;DR

This paper investigates the nature of the lightest scalar meson using Chiral Perturbation Theory and the $1/N_c$ expansion, revealing a transition from a di-pion dominated state to a sub-dominant quark-antiquark component as $N_c$ increases.

Contribution

It demonstrates how unitarised Chiral Perturbation Theory beyond one loop order uncovers the evolving quark structure of the $\sigma$ meson with increasing $N_c$, connecting resonance behavior with Regge theory.

Findings

At one loop, the $ ho$ is a ${ar q}q$ meson, while the $\sigma$ is not.

Beyond one loop, the $\sigma$ develops a sub-dominant ${ar q}q$ component at higher $N_c$.

Semi-local duality is maintained for $N_c o 15$ due to this quark component evolution.

Abstract

The leading $1/N_c$ behaviour of Unitarised Chiral Perturbation Theory distinguishes the nature of the $\rho$ and the $\sigma$. At one loop order the $\rho$ is a ${\bar q}q$ meson, while the $\sigma$ is not. However, semi-local duality between resonances and Regge behaviour cannot be satisfied for larger $N_c$, if such a distinction holds. While the $\sigma$ at $N_c=3$ is inevitably dominated by its di-pion component, Unitarised Chiral Perturbation Theory beyond one loop order reveals that as $N_c$ increases above 6-8, the $\sigma$ has a sub-dominant ${\bar q}q$ fraction up at 1.2 GeV. Remarkably this ensures semi-local duality is fulfilled for the range of $N_c \lesssim 15$, where the unitarisation procedure adopted applies.