First-order CP phase transition in two-flavor QCD at $\theta =\pi$ under electromagnetic scale anomaly via a Nambu-Jona-Lasinio description

We discuss the thermal CP phase transition in QCD at $\theta =\pi$ under a weak magnetic field background, where the electromagnetic scale anomaly gets significant. To explicitize, we work on a two-flavor Nambu-Jona-Lasinio model at $\theta =\pi$ in the mean field approximation, including the electromagnetic-scale anomaly term. We find that the thermal CP phase transition becomes first order and the strength of the first order gets more prominent as the magnetic field increases. The associated potential barrier is thermally created by the electromagnetic scale anomaly and gives rise to criticality due to the induced potential of a non-perturbative form $\sim \frac{\mathrm{\mid }eB{\mathrm{\mid }}^3}{f_{\pi }}\frac{\mathrm{\mid }P\mathrm{\mid }}{P^2+m_0^2}$, where $eB$ denotes the magnetic field strength; $P$ the CP order parameter, and $m_0$ the isospin-symmetric current-quark mass.