Topological transition in a parallel electromagnetic field

Gaoqing Cao

Phys. Rev. D 109, 094020 – Published 10 May 2024

Abstract

In this work, we attack the problem of “chiral phase instability” ( $χ PI$ ) in a quantum chromodynamics (QCD) system under a parallel and constant electromagnetic field. The $χ PI$ refers to the fact that, when $I_{2} \equiv E \cdot B$ is larger than the threshold $I_{2}^{c}$ , no homogeneous solution can be found for $σ$ or $π^{0}$ condensate, and the chiral phase $θ$ becomes unstable. Within the two-flavor chiral perturbation theory, we obtain an effective Lagrangian density for $θ (x)$ where the chiral anomalous Wess-Zumino-Witten term is found to play a role of “source” to the “potential field” $θ (x)$ . The Euler-Lagrangian equation is applied to derive the equation of motion for $θ (x)$ , and physical solutions are worked out for several shapes of systems. In the case $I_{2} > I_{2}^{c}$ , it is found that the $χ PI$ actually implies an inhomogeneous QCD phase with $θ (x)$ spatially dependent. By its very nature, the homogeneous-inhomogeneous phase transition is of pure topological and second order at $I_{2}^{c}$ . Finally, the work is extended to the three-flavor case, where an inhomogeneous $η$ condensation is also found to be developed for $I_{2} > I_{2}^{c}$ . Correspondingly, there is a second critical point, $I_{2}^{c^{'}} = 24.3 I_{2}^{c}$ , across which the transition is also of topological and second order by its very nature.