The $\sigma$-meson: four-quark vs. two-quark components and decay width in a Bethe-Salpeter approach

TL;DR

This paper investigates the internal structure and decay properties of the $\sigma$-meson by combining two- and four-quark components within a Bethe-Salpeter framework, revealing insights into its mass and decay width consistent with dispersive analyses.

Contribution

It extends a Bethe-Salpeter approach to include mixing of two- and four-quark states, providing a more comprehensive understanding of the $\sigma$-meson structure.

Findings

The four-quark component's $\pi\pi$ contribution lowers the meson's mass.

The analytic structure shows a branch cut at the two-pion threshold.

The decay width is indicated by a singularity in the second Riemann sheet.

Abstract

We study the dynamical generation of resonances in isospin singlet channels with mixing between two- and four-quark states. To this end we generalise a Bethe-Salpeter approach to four-quark states employed previously \cite{Heupel:2012ua} to accommodate for mixing diagrams. The $q\bar{q}q\bar{q}$ and $q\bar{q}$ components of the Bethe-Salpeter wave function (with light quarks $q\in\{u,d\}$) are determined consistently in a symmetry-preserving truncation of the underlying Dyson-Schwinger equations. As a prominent example we deal with the isospin-singlet $0^{++}$ meson with light quark content. We find that the $\pi\pi$ contribution of the four-quark component is mainly responsible for the low (real part of the) mass of the resulting state. We also study the analytic structure in the complex momentum plane and find a branch cut at the two-pion threshold and a singularity in the second Riemann sheet indicating a considerable decay width. Our findings are in excellent qualitative agreement with the general picture for the $\sigma/f_0(500)$ that emerged in the past two decades from dispersive approaches \cite{Pelaez:2015qba}.