Revisiting $O(N)$ $\sigma$ model at unphysical pion masses and high temperatures

Abstract

Roy-equation analyses on lattice data of $\pi \pi$ scattering phase shifts at $m_{\pi }=391$MeV reveals that the lowest $f_0$ meson becomes a bound state under this condition. In addition, there is a pair of complex poles below threshold generated by crossing symmetry [X.-H. Cao et al., Phys. Rev. D 108, 034009 (2023)]. We use the $N/D$ method to partially recover crossing symmetry of the $O(N)$ $\sigma$ model amplitude at leading order of $1/N$ expansion, and qualitatively reproduce the pole structure and pole trajectories with varying pion masses as revealed by Roy-equation analyses. The $\sigma$ pole trajectory with varying temperature is also discussed and found to be similar to its properties when varying $m_{\pi }$. As the temperature increases, the complex $\sigma$ poles firstly move from the second Riemann sheet to the real axis becoming two virtual state poles, and then one virtual state pole moves to the first sheet turning into a bound state pole and finally tends to the pion pole position at high temperature which is as expected from the chiral symmetry restoration. Our results provide further evidences that the lowest $f_0$ state extracted from experiments and lattice data plays the role of $\sigma$ meson in the spontaneous breaking of chiral symmetry. Finally, we also briefly discuss the problems of the effective potential in the situation when $m_{\pi }$ and temperature get large.

arXiv comment13 pages, 5 figures, 1 table; to match the published version

References in corpus (48)

• The chiral and deconfinement aspects of the QCD transition
• ππscattering
• Chiral crossover in QCD at zero and non-zero chemical potentials
• Mass and width of the lowest resonance in QCD
• The pion-pion scattering amplitude. IV: Improved analysis with once subtracted Roy-like equations up to 1100 MeV
• Roy equation analysis of pi pi scattering
• First results from 2+1 dynamical quark flavors on an anisotropic lattice: light-hadron spectroscopy and setting the strange-quark mass
• Meson-meson scattering within one loop Chiral Perturbation Theory and its unitarization
• Chiral phase transition temperature in (2+1)-Flavor QCD
• Isoscalar $\pi \pi$ scattering and the $\sigma$ meson resonance from QCD
• Two Particle States and the $S$-matrix Elements in Multi-channel Scattering
• Tetraquark Mesons in Large $N$ Quantum Chromodynamics
• Phase transitions in quantum chromodynamics
• Quark mass dependence of the rho and sigma from dispersion relations and Chiral Perturbation Theory
• Rummukainen-Gottlieb's formula on two-particle system with different mass
• Light scalars as tetraquarks or two-meson states from large Nc and unitarized Chiral Perturbation Theory
• The pi+ pi+ scattering length from maximally twisted mass lattice QCD
• Scalar Mesons a0(1450) and sigma(600) from Lattice QCD
• The $\kappa$ resonance in s wave $\pi K$ scatterings
• The Inverse Amplitude Method and Adler Zeros
• Chiral symmetry at finite temperature: linear vs nonlinear $\sigma$-models
• Remarks on pole trajectories for resonances
• Chiral extrapolation of light resonances from one and two-loop unitarized Chiral Perturbation Theory versus lattice results
• The pole structure of the unitary, crossing symmetric low energy $\pi \pi$ scattering amplitudes
• A Review on Partial-wave Dynamics with Chiral Effective Field Theory and Dispersion Relation
• Left Hand Singularities, Hadron Form Factors and the Properties of the Sigma Meson
• An improved study of the kappa resonance and the non-exotic $s$ wave $\pi K$ scatterings up to $\sqrt{s}=2.1$GeV of LASS data
• Glueball nature of the sigma/f_0(600) from pi-pi and gamma-gamma scatterings
• Pi-Pi scattering lengths at finite temperature
• Partial waves and large $N_C$ resonance sum rules
• The sigma and f_0(980) from Ke4 + pi-pi scatterings data
• Finite Temperature Pion Scattering to one-loop in Chiral Perturbation Theory
• Model dependent analyses on the $N_c$ dependence of the $\sigma$ pole trajectory
• Patterns and partners for chiral symmetry restoration
• Thermodynamics of O(N) sigma models: 1/N corrections
• Quark mass dependence of $\pi \pi$ scattering in isospin 0, 1, and 2 from lattice QCD
• $O(p^6)$ extension of the large--$N_C$ partial wave dispersion relations
• The [1,2] Padé Amplitudes for $\pi \pi$ Scatterings in Chiral Perturbation Theory
• Is the $f_0(600)$ meson a dynamically generated resonance? -- a lesson learned from the O(N) model and beyond
• Determination of crossing-symmetric $\pi \pi$ scattering amplitudes and the quark mass evolution of the $\sigma$ constrained by lattice QCD
• New Insights on Low Energy $\pi N$ Scattering Amplitudes
• Thermal behaviors of light scalar resonances at low temperatures
• A possible subthreshold pole in $S_{11}$ channel from $\pi N$ Roy-Steiner equation analyses
• Scrutinizing $\pi \pi$ scattering in light of recent lattice phase shifts
• Large-$N$ pion scattering at finite temperature: the $f_0(500)$ and chiral restoration
• A unitarity analysis on the I=0 $d$ wave $\pi \pi$ scattering amplitude
• The properties of the $\sigma$ and $\kappa$ resonances in a new unitarization approach
• Reply to "comment on `Scrutinizing $\pi \pi$ scattering in light of recent lattice phase shifts'"

Cited by in corpus (4)

• Revisiting $O(N)$ $\sigma$ model at unphysical pion masses and high temperatures. II. The vacuum structure and thermal $\sigma$ pole trajectory with cross-channel improvements
• On the pole trajectory of the subthreshold negative parity nucleon with varying pion masses
• The QCD scalar susceptibility and thermal scalar resonances in chiral symmetry restoration
• Review on recent progress in the study of the $N^{\ast }(920)$ subthreshold singularity and the $\sigma /f_0(500)$ meson