Firewall boundaries and mixed phases of rotating quark matter in linear sigma model

TL;DR

This work analyzes rotating quark matter in the linear sigma model with quarks (LSM$_q$) under three mean-field condensate treatments: a uniform, a slowly varying, and a fully inhomogeneous condensate. By coupling rotation and temperature through a local thermal equilibrium framework and introducing firewall boundary conditions at the light cylinder, the authors uncover how rotation drives chiral symmetry restoration and yields mixed-phase structures with spatially varying condensates, including a Klein-Gordon-type equation for Model 3 and a Rankine vortex extension for inhomogeneous rotation. The key result is that rotation lowers the chiral transition threshold and, beyond a critical angular velocity $\Omega_c \approx 175$ MeV, restores chirality across the system; the analysis also demonstrates how boundary conditions and gradients qualitatively alter the phase structure compared to uniform- or gradientless approximations. These insights provide a controlled setting to study rotating strongly interacting matter and have potential implications for modeling rotating quark-gluon plasma boundaries, with suggested extensions to include quantum corrections and Polyakov-loop dynamics.

Abstract

A rigidly-rotating body in unbounded space is usually considered a pathological system since it leads to faster-than-light velocities and associated breaches of causality. However, numerical results on chiral symmetry breaking in rotating plasmas of interacting fermions reveal surprisingly close correspondence in predictions between the rigorous bounded and formal unbounded approaches. To provide insight into this correlation, we consider the linear sigma model coupled to quarks, undergoing rigid rotation in unbounded Minkowski space-time. Within the mean-field approach, we adopt three consecutive levels of approximation to the ground state of the system that feature uniform (model 1), weakly inhomogeneous (model 2) and fully inhomogeneous (model 3) condensates. Models 1 and 2 that do not take into account spatial gradients of the condensate show agreement with the Tolman-Ehrenfest law. Model 3 exhibits a deviation from the Tolman-Ehrenfest prediction due to the appearance of a new energy scale set by the inhomogeneity of the ground state. Its boundary conditions are fixed by imposing regularity at the rotation axis and by demanding the global minimization of the grand potential. We dub the latter as ``firewall boundary conditions,'' translating into the requirement of vanishing condensate on the light cylinder, which follows from the fact that the system state formally diverges at the light cylinder. In all models, we present the phase diagram of the system and point out that in models 2 and 3, the system resides either in a chirally-restored phase, or in a mixed phase that possesses spatially-separated chirally-restored and chirally-broken phases. Finally, we discuss the properties of the system under inhomogeneous rotation using the relativistic version of the Rankine vortex model.

Figures (23)

• Figure 1: Uniform ground state: Normalized value of the condensate $\sigma$, obtained as a solution of the mass gap equation \ref{['eq_sbar_saddle']}, for a set of on-axis temperatures $T_0$ at two values of the on-axis chemical potential $\mu_0$. At small chemical potential, the transition is crossover, whereas at large chemical potential, it is of first order.
• Figure 2: Uniform ground state: The same as in Fig. \ref{['fig_Averaged-sofR-mu']} for fixed temperature and varying chemical potential. At small temperatures, the transition goes from crossover to first order as the chemical potential increases. At larger temperatures, the transition is always crossover.
• Figure 3: Uniform ground state: Numerical confirmation of the asymptotic behaviour of the condensate plotted as the logarithmic function of the Lorentz factor at the boundary. Dashed lines correspond to the analytical expression \ref{['eq_averaged-near-firewall']} while the points are the numerical solution of the mass gap equation \ref{['eq_sbar_saddle']}.
• Figure 4: Uniform ground state: Dimensionless critical radius $\Omega R_c$ of the system in a cylinder of radius $R$ that rotates with the angular velocity $\Omega$. The critical radius is shown as a function of the ratio of the critical temperature in the absence of rotation, $T_c^{\Omega = 0}$, and the on-axis temperature $T_0$ in a rotating system (this ratio of temperatures is equal to the ratio of the corresponding chemical potentials). The system resides in the chirally-broken (-restored) phase at $R < R_c$ ($R > R_c$). Dashed lines correspond to the analytical formula \ref{['eq_effective_thermo']}, while data points correspond to the critical value obtained by solving the mass gap equation \ref{['eq_sbar_saddle']}.
• Figure 5: Slowly varying condensate: Normalized value of the inhomogeneous condensate $\sigma$, shown as a function of the rescaled radial distance $\rho \Omega$. The condensate is a solution of the mass gap equation \ref{['eq_massgap_slow']}, shown for a set of on-axis temperatures $T_0$ at two values of the on-axis chemical potential $\mu_0$. The colored (black) points represent the thermodynamically (un)favored solutions.