The Dichotomous Nature of the $σ$ Meson and the Nucleon D-Term

TL;DR

The paper investigates how mass generation in QCD relates to the gravitational structure of hadrons by analyzing D-terms for the pion, the $\sigma$ meson, and the nucleon within a symmetry-preserving Dyson–Schwinger framework using a momentum-independent (contact) interaction. It identifies the $\sigma$ as a dual object—both a $q\bar q$ bound state and a (pseudo) NG boson of spontaneous scale-symmetry breaking (dilaton)—and shows that above a critical coupling $\alpha_c$ the D-terms converge to universal values $D_q=-1/3$, $D_{\pi}=-1$, $D_{\sigma}=-7/3$, with a discontinuity at $\alpha_c$. The work yields a nucleon D-term prediction of $D_N\simeq -3$, in agreement with lattice QCD, continuum, and dispersive analyses, when the dilaton perspective and scalar-meson dominance are incorporated. Overall, the results link chiral and scale symmetries to the gravitational structure of hadrons and propose experimental tests of the nucleon D-term at the Electron–Ion Collider to validate the framework.

Abstract

Employing a symmetry-preserving contact-interaction formulation of the Dyson-Schwinger equations in quantum chromodynamics (QCD), we examine the identity of the $σ$ meson and its implications for the gravitational structure of hadrons. In this framework, the scalar meson emerges as the chiral partner of the pion, with both states' properties tightly connected to the mechanisms of mass generation in QCD. We find that, above a critical coupling that triggers dynamical chiral symmetry breaking, the D-terms of the constituent quark, the pion, and $σ$ saturate at fixed values $D_{q,π,σ}=-1/3,-1,-7/3$. By examining the coupling strength evolution of the D-terms, this pattern follows naturally once a dual nature for the $σ$ meson is recognized: it behaves both as a quark-antiquark composite and as a dilaton arising from spontaneous scale symmetry breaking. This unified picture yields the prediction $D_N\sim -3$ for the nucleon D-term, consistent with contemporary lattice-QCD, continuum, and dispersive studies.

Figures (2)

• Figure 1: Dependence of the dressed-quark, pion, and $\sigma$ meson masses on the effective coupling $\alpha_{eff}$. The vertical dotted line marks the critical coupling $\alpha_c$. Below this threshold, pion and $\sigma$ are degenerate states.
• Figure 2: Dependence of the dressed-quark, pion, and $\sigma$ meson D-terms on the effective coupling $\alpha_{eff}$. At the critical point, corresponding to the vertical dotted line, one finds: $D_q=0$, and $D_\pi=D_\sigma=-1/3$. Below $\alpha_c$, pion and $\sigma$ become degenerate states.