[Paper Review] Sigma, Kappa, fo(980) and a0(980)

D.V. Bugg|ArXiv.org|Oct 5, 2005

Inorganic Fluorides and Related Compounds3 citations

TL;DR

This paper establishes the existence of the sigma (σ) and kappa (κ) resonances through combined analysis of D-meson and J/Ψ decay data, showing their properties are consistent with chiral symmetry breaking via the Adler zero. It demonstrates that these states arise from unitarized chiral dynamics rather than conventional q\bar{q} structure, with strong support from dispersion relations, elastic scattering data, and models like unitarized Chiral Perturbation Theory and the Van Beveren-Rupp scheme.

ABSTRACT

Both sigma and kappa are well established from E791 data on D->3pi and Ds->Kpipi and BES II data on J/Psi -> omega pi pi and KKpipi. These fits are accurately consistent with pipi and Kpi elastic scattering when one allows for the Adler zero which arises from Chiral Symmetry Breaking. The phase variation with mass is consistent between elastic scattering and production data. Possible interpretations of sigma, kappa, fo(980) and ao(980) are explored. The experimental ratio g^2(fo(980)->KK)/g^2(ao(980)->KK) = 2.7+-0.5 suggests strongly that fo(980) has a large KK component in its wave function. This is a natural consequence of its pole lying very close to the KK threshold.

Motivation & Objective

• To resolve the long-standing ambiguity about the existence and nature of the σ and κ resonances using high-precision production data.
• To reconcile discrepancies between elastic ππ and Kπ scattering data and production data in D and J/Ψ decays.
• To test whether the σ and κ states are better described as molecular states or conventional q\bar{q} resonances using unitarized Chiral Perturbation Theory and dispersion relations.
• To investigate the role of the Adler zero and chiral symmetry breaking in shaping the low-mass ππ and Kπ S-wave amplitudes.
• To explore the behavior of scalar resonances under varying Nc in QCD, particularly the disappearance of σ and κ as Nc increases, contrasting with stable ρ mesons.

Proposed method

• Fits to E791 D⁺→3π and D⁺→K⁻π⁺π⁺ data using a Breit-Wigner form with s-dependent width Γ(s)∝ρ(s), later refined using Chiral Perturbation Theory.
• Analysis of BES II J/Ψ→ωπ⁺π⁻ and J/Ψ→K⁺K⁻π⁺π⁻ Dalitz plots to extract σ and f₀(980) contributions, with four parametrization types to test pole stability.
• Use of dispersion relations and unitarity constraints to enforce analyticity, with the Adler zero (s_A = 0.5m_π²) imposed to explain the absence of a low-mass peak in elastic scattering.
• Application of the K-matrix method to model ππ S-wave amplitude via t- and u-channel exchanges (ρ, f₂), showing attraction that generates f₀(980) and a₀(980) but not σ.
• Implementation of the Van Beveren-Rupp scheme, modeling q\bar{q} states coupled to meson decay channels via a δ-function transition potential, predicting σ and κ as molecular-like states.
• Extrapolation of results to large Nc using unitarized Chiral Perturbation Theory, showing σ and κ poles vanish as Nc increases, unlike ρ mesons.

Experimental results

Research questions

• RQ1 Do the σ and κ resonances observed in D and J/Ψ decays have properties consistent with those derived from ππ and Kπ elastic scattering?
• RQ2 To what extent does the Adler zero from chiral symmetry breaking explain the absence of a low-mass peak in ππ elastic scattering despite a σ resonance in production data?
• RQ3 Can the σ and κ resonances be understood as molecular states formed via coupling to meson-meson channels, rather than as conventional q\bar{q} states?
• RQ4 How do the properties of the σ and κ change under variation of the number of colors (Nc), and what does this imply about their dynamical origin?
• RQ5 Do dispersion relations and unitarity constraints uniquely determine the σ pole position, and is it consistent with production data?

Key findings

• The σ pole is consistently fitted to production data with a mass of (541 ± 39) MeV and width of (252 ± 42) MeV, in agreement with elastic scattering constraints.
• The Adler zero at s_A = 0.5m_π² explains the suppression of ππ elastic amplitude near s=0, resolving the apparent contradiction between elastic scattering and production data.
• Dispersion relations and unitarity demand a σ pole within 2 standard deviations of the production-fitted pole, confirming consistency with fundamental field theory principles.
• The σ and κ states are distinct from f₀(1535) and broad 1 GeV poles, as their intensity falls rapidly above 1 GeV, indicating a narrow, well-defined resonance.
• The Van Beveren-Rupp model successfully predicts the σ and κ as 'extra' states formed by coupling q\bar{q} states to meson decay channels, with their position shifting toward the real axis as coupling strength increases.
• Extrapolation to large Nc shows σ and κ poles fade into the continuum, while the ρ remains stable, indicating a fundamentally different dynamical origin from conventional q\bar{q} states.