[Paper Review] S-wave meson scattering up to sqrt{s} < 2 GeV from chiral Lagrangians

M. Albaladejo, J. A. Oller+1 more|arXiv (Cornell University)|Nov 13, 2007

Quantum Chromodynamics and Particle Interactions3 citations

TL;DR

This paper develops a unitarized chiral perturbation theory framework to study S-wave meson scattering in I = 0 and I = 1/2 channels up to √s < 2 GeV, incorporating coupled channels including ππ, KK, ηη, σσ, ρρ, ωω, φφ, K*K*, a1π, and π⋆π. It identifies resonances such as σ(454 MeV, 475 MeV), f₀(980), f₀(1380), f₀(1500), f₀(1680), and f₀(1805) with good agreement to PDG data, demonstrating a parameter-efficient approach using chiral Lagrangians and unitarization with width effects via spectral functions.

ABSTRACT

The problem of scalar mesons still remains a challenging puzzle, for which we do not even know which are the right pieces to set up. The proliferation of resonances (some of them are very broad and appear on top of hadronic thresholds) and of coupled channels that interact strongly among each other makes the study of this sector a hard task. Our objective is the study of the strongly interacting mesons in coupled channels with quantum numbers J^{PC} = 0^{++} and I=0 and I=1/2, up to a center of mass energy sqrt{s} < 2 GeV. Our framework is based on Unitary Chiral Perturbation Theory. We include for I=0 the channels: \pi\pi, K\bar{K}, \eta\eta, \sigma\sigma, \eta\eta', ho ho, \omega\omega, \eta'\eta', \omega\phi, \phi\phi, K^\ast \bar{K}^\ast, a_1(1260)\pi and \pi^{\star}(1300)\pi. In addition, and in order to constrain our fits, we also study the I=1/2, 3/2 channels given by K\pi, K\eta and K\eta'. We finally present the resonant content of our fits with the $\sigma$, $f_0(980)$, $f_0(1310)$, $f_(1500)$, $f_0(1710)$ and $f_0(1790)$.

Motivation & Objective

• To address the unresolved puzzle of scalar meson spectroscopy, where multiple broad resonances and coupled channels complicate interpretation.
• To develop a predictive framework for I = 0 and I = 1/2 S-wave meson scattering up to √s ≲ 2 GeV using unitarized chiral Lagrangians.
• To constrain resonance parameters by fitting to experimental data across multiple coupled channels, including high-multiplicity final states.
• To incorporate width effects for broad resonances (e.g., σ, ρ, a₁(1260), π⋆(1300)) via spectral functions in loop integrals.
• To identify and characterize scalar resonances f₀(980), f₀(1310), f₀(1500), f₀(1710), f₀(1790), and the σ meson within a consistent chiral effective field theory approach.

Proposed method

• Uses lowest-order chiral Lagrangians and chirally invariant resonance Lagrangians to describe interactions among pseudoscalars and vector resonances.
• Applies unitarization via the Bethe-Salpeter equation in the K-matrix approximation, with the amplitude T = (I + N(s)g(s))⁻¹N(s), where g(s) is a once-subtracted dispersion relation.
• Incorporates width effects for unstable particles by replacing delta-function masses with spectral functions in loop integrals, replacing standard propagators.
• Derives the σσ amplitude via analytic continuation to the second Riemann sheet, using the limit lim_{s₁,s₂→sσ} T_{a→(ππ)₀(ππ)₀} / [D(s₁)D(s₂)] to extract N_{a→(σσ)₀} with D(s) = (1 + t²G(s))⁻¹.
• Treats the σ as a dynamically generated state from ππ I=0 S-wave interaction, with no additional free parameters beyond chiral Lagrangian couplings.
• Fits 13 parameters to ~373 experimental data points, including phase shifts and moduli from ππ, KK, ηη′, ηη, and Kπ scattering, using a global χ² minimization.

Experimental results

Research questions

• RQ1 What is the resonance content of S-wave meson scattering in I = 0 and I = 1/2 channels up to √s ≲ 2 GeV, when unitarized chiral Lagrangians are used?
• RQ2 How do broad resonances like the σ, ρ, a₁(1260), and π⋆(1300) affect the scattering amplitudes when their width effects are properly included?
• RQ3 Can a unified framework based on U(3) chiral symmetry and unitarization reproduce the masses and widths of known scalar resonances like f₀(980), f₀(1500), and f₀(1710)?
• RQ4 To what extent can the inclusion of coupled channels such as σσ, ρρ, φφ, K*K*, and a₁π improve the description of scalar meson dynamics?
• RQ5 How do the predicted resonance parameters compare with PDG and BES Collaboration data?

Key findings

• The σ meson is dynamically generated with a pole at 454 MeV in mass and 475 MeV in width, consistent with its broad, non-quark-antiquark nature.
• The f₀(980) resonance is well reproduced with a mass of 980 MeV and width of 44 MeV, in agreement with PDG values.
• The f₀(1380) resonance appears with a mass of 1380 MeV and width of 350 MeV, indicating a broad, broad state possibly involving K¯K or ππ components.
• The f₀(1500) resonance is predicted at approximately 1500 MeV with a width of 100–170 MeV, matching the PDG value of 1507 ± 5 MeV and 109 ± 7 MeV.
• The f₀(1710) resonance is found at 1680 MeV with a width of 160 MeV, consistent with the PDG value of 1718 ± 6 MeV and 137 ± 8 MeV.
• The f₀(1790) resonance is predicted at 1805 MeV with a width of 390 MeV, in good agreement with the PDG value of 1790+40−30 MeV and 270+60−30 MeV.