[Paper Review] Decay widths of resonances and pion scattering lengths in a globally invariant linear sigma model with vector mesons

Denis Parganlija, Francesco Giacosa+1 more|ArXiv.org|Dec 11, 2008

Quantum Chromodynamics and Particle Interactions15 references4 citations

TL;DR

This paper investigates low-energy meson decays and pion-pion scattering lengths in a globally invariant two-flavor linear sigma model including vector and axial-vector mesons. It finds that assigning the light scalar mesons $f_0(600)$ and $a_0(980)$ as $\bar{q}q$ states leads to significant discrepancies with experimental data—specifically, the $\sigma\to\pi\pi$ decay width is too small and the $a_1\to\rho\pi$ width too large—suggesting this assignment is inconsistent with observations.

ABSTRACT

We calculate low-energy meson decay processes and pion-pion scattering lengths in a two-flavour linear sigma model with global chiral symmetry, exploring the scenario in which the scalar mesons $f_0$(600) and $a_0$(980) are assumed to be $\bar q q$ states.

Motivation & Objective

• To assess the consistency of the $\bar{q}q$ assignment for light scalar mesons $f_0(600)$ and $a_0(980)$ within a globally invariant linear sigma model.
• To calculate decay widths of low-energy mesons and pion-pion scattering lengths using a model with global chiral symmetry and vector/axial-vector mesons.
• To test whether the inclusion of additional coupling constants in a globally invariant model improves agreement with experimental data compared to locally invariant models.
• To identify inconsistencies in the $\bar{q}q$ interpretation of light scalar mesons and explore alternative assignments in the 1–2 GeV region.
• To lay the groundwork for future studies including nucleon degrees of freedom and chiral symmetry restoration at finite temperature.

Proposed method

• Constructs a two-flavor linear sigma model with global $U(2)_R \times U(2)_L$ chiral symmetry, including scalar, pseudoscalar, vector, and axial-vector mesons.
• Introduces additional interaction terms in the Lagrangian to account for global symmetry, adding new coupling constants ($h_1, h_2, h_3, g_2$) not present in local models.
• Uses the $\chi^2$ minimization method to fit model parameters ($Z$, $g_2$, $h_1$, $h_2$, $m_\sigma$) to experimental data on decay widths and scattering lengths.
• Derives analytical expressions for decay widths ($\Gamma_{a_1\to\sigma\pi}$, $\Gamma_{\sigma\to\pi\pi}$, etc.) and scattering lengths ($a_0^0$, $a_0^2$) using the model's Lagrangian and Feynman rules.
• Fixes $m_\sigma = 330$ MeV based on fitting to $\pi\pi$ scattering lengths, and uses constraints from $a_0^0 = 0.233 \pm 0.023$ and $a_0^2 = -0.0471 \pm 0.015$.
• Evaluates the role of the bare $\rho$ meson mass ($m_1 \approx 758$ MeV) and the contribution of the quark condensate to the $\rho$ mass.

Experimental results

Research questions

• RQ1 Does the $\bar{q}q$ assignment of $f_0(600)$ and $a_0(980)$ as scalar mesons remain consistent with experimental decay widths and $\pi\pi$ scattering lengths in a globally invariant linear sigma model?
• RQ2 Can the inclusion of additional coupling constants in a globally invariant model improve agreement with experimental data compared to the locally invariant version?
• RQ3 What is the contribution of the bare $\rho$ meson mass to the physical $\rho$ mass in this model, and how does it affect the interpretation of the $\rho$ meson's structure?
• RQ4 Why do the calculated decay widths for $\sigma\to\pi\pi$ and $a_1\to\rho\pi$ deviate significantly from experimental values under the $\bar{q}q$ assumption?
• RQ5 Is an alternative assignment—identifying $\sigma$ and $a_0$ as $f_0(1370)$ and $a_0(1450}$—more consistent with experimental data?

Key findings

• The best-fit $\chi^2$ value is 0.7525 per degree of freedom, indicating a good fit to the data, with $Z = 1.5217$, $g_1 = 6.59$, $g_2 = 0.3365$, $h_1 = -100.7$, $h_2 = 106.045$, $h_3 = -2.63$, and $m_\sigma = 330$ MeV.
• The $\sigma \to \pi\pi$ decay width is calculated to be less than 10 MeV, which is significantly smaller than the experimental value, indicating a problem with the $\bar{q}q$ assignment.
• The $a_1 \to \rho\pi$ decay width is calculated at 1.4 GeV, which is far larger than the experimental value, further contradicting the $\bar{q}q$ interpretation.
• The bare $\rho$ meson mass ($m_1$) is found to be approximately 758 MeV, implying that the quark condensate contributes very little to the $\rho$ meson mass.
• The parameter $h_1$ is not suppressed by $1/N_C$, and its magnitude is about ten times larger than that of $2h_3$, indicating a strong coupling despite the trace structure.
• The model's results suggest that the $\bar{q}q$ assignment of $f_0(600)$ and $a_0(980)$ as light scalar mesons is inconsistent with experimental data, prompting a reconsideration of their structure in the 1–2 GeV region.