[Paper Review] Chiral Phase Transitions

M. D. Scadron, P. Żenczykowski|arXiv (Cornell University)|Jun 18, 2001

High-Energy Particle Collisions Research4 citations

TL;DR

This paper proposes a self-consistent quark-level SU(2) linear sigma model (LσM) that uniquely describes chiral phase transitions in QCD with two light quarks. By thermally melting the quark mass, scalar σ meson mass, and quark condensate, it derives a critical temperature of $ T_c = 2f^{CL}_\pi \approx 180~\text{MeV} $ at zero chemical potential and a critical chemical potential of $ \mu_c \approx 325~\text{MeV} $ at zero temperature, with color superconductivity emerging at $ T_c^{(SC)} < 180~\text{MeV} $ due to diquark pairing.

ABSTRACT

We show that melting the quark mass, the scalar $σ$ mass and the quark condensate leads uniquely to the quark-level SU(2) linear $σ$ model field theory. Upon thermalization, the chiral phase transition curve requires $T_c =2f^{CL}_π\approx 180 MeV$ when $μ= 0$, while the critical chemical potential is $μ_c =m_q\approx 325 MeV$. Transition to the superconductive phase occurs at $T^{(SC)}_c=Δ/πe^{-γ_E}$. Coloured diquarks suggest $T_c^{(SC)}&lt;180 MeV$.

Motivation & Objective

• To establish a self-consistent field theory for chiral phase transitions in QCD with two light quarks.
• To derive the critical temperature $ T_c $ for chiral symmetry restoration from thermal melting of quark mass, σ meson mass, and quark condensate.
• To extend the model to finite chemical potential and determine $ \mu_c $ at zero temperature.
• To explore the onset of color superconductivity via diquark condensation at low temperatures.

Proposed method

• Thermalization of the fermion propagator using the Matsubara formalism with Fermi-Dirac statistics.
• Independent melting of the constituent quark mass, σ meson mass, and quark condensate to derive $ T_c $.
• Use of the quark-level linear sigma model (LσM) with SU(2) chiral symmetry and dynamical symmetry breaking.
• Application of the gap equation and BCS-like relation to estimate the color superconducting transition temperature $ T_c^{(SC)} $.
• Derivation of a chiral ellipse in the $ T-\mu $ plane via the generalized GTR condition $ \frac{T^2_c}{(2f_\pi)^2} + \frac{\mu^2_c}{m^2_q} = 1 $.
• Use of the relation $ m_\sigma = 2m_q $ and $ N_c = 3 $ to close the system of equations and fix model parameters.

Experimental results

Research questions

• RQ1 What is the critical temperature $ T_c $ for chiral symmetry restoration in the quark-level linear sigma model at $ \mu = 0 $?
• RQ2 What is the critical chemical potential $ \mu_c $ for chiral symmetry restoration at $ T = 0 $?
• RQ3 How does the inclusion of finite chemical potential modify the chiral phase transition structure in the $ T-\mu $ plane?
• RQ4 What is the predicted color superconducting transition temperature $ T_c^{(SC)} $, and how does it compare to $ T_c $?
• RQ5 Can the quark-level LσM model consistently describe both chiral and superconducting phases in dense, hot QCD matter?

Key findings

• The chiral phase transition temperature is derived as $ T_c = 2f^{CL}_\pi \approx 180~\text{MeV} $, consistent with lattice QCD results.
• The critical chemical potential at zero temperature is $ \mu_c = m_q \approx 325~\text{MeV} $, corresponding to the onset of chiral symmetry restoration.
• The phase transition line in the $ T-\mu $ plane forms a chiral ellipse described by $ \frac{T^2_c}{(2f_\pi)^2} + \frac{\mu^2_c}{m^2_q} = 1 $.
• The color superconducting transition temperature is estimated as $ T_c^{(SC)} \approx 0.567\Delta $, with $ \Delta \approx m_q $, implying $ T_c^{(SC)} < 180~\text{MeV} $.
• The model uniquely determines $ N_c = 3 $ and $ m_\sigma = 2m_q $ through consistency conditions on tadpole and combinatorial terms.
• The model reproduces the observed $ T_c \approx 180~\text{MeV} $ and $ \mu_c \approx 325~\text{MeV} $, with quantitative agreement to within 10% of lattice QCD estimates.