Akio Tomiya

The decrease of the chiral pseudocritical temperature $T_{\mathrm{pc } }$ with an applied strong magnetic field has been extensively investigated by various QCD low-energy effective models and lattice QCD at physical point. We find that this decreasing feature may not hold in the case with a weak magnetic field and still depends on quark masses: when the quark masses get smaller, $T_{\mathrm{pc } }$ turns to increase with the weak magnetic field. This happens due to the significant electromagnetic-scale anomaly contribution in the thermomagnetic medium. We demonstrate this salient feature by employing the Polyakov Nambu-Jona-Lasinio model with 2 + 1 quark flavors including the electromagnetic-scale anomaly contribution. We observe a critical point in a sort of the Columbia plot, $(m_{0c}, m_{sc}) \simeq (3, 30) \mathrm{MeV}$ for the isospin symmetric mass for up and down quarks, $m_0$, and the strange quark mass, $m_s$, where $T_{\mathrm{pc } }$ decreases with the magnetic field if the quark masses exceed the critical values, and increases as the quark masses become smaller. Related cosmological implications, arising when the supercooled electroweak phase transition or dark QCD cosmological phase transition is considered along with a primordial magnetic field, are also briefly addressed.

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 $\sim \frac{|eB|^3}{f_\pi} \frac{|P|}{P^2 + m_0^2}$, where $eB$ denotes the magnetic field strength; $P$ the CP order parameter, and $m_0$ the isospin-symmetric current-quark mass.

We investigate the stability of topological charge under gradient flow taking the admissibility condition into account. For the $SU(2)$ Wilson gauge theory with $\beta=2.45$ and $L^4=12^4$, we numerically show that the gradient flows with the Iwasaki and DBW2 gauge actions stabilize the topological sectors significantly, and they have qualitatively different behaviors compared with the Wilson and tree-level Symanzik flows. By considering the classical continuum limit of the flow actions, we discuss that the coefficient of dimension-$6$ operators has to be positive for stabilizing the one-instanton configuration, and the Iwasaki and DBW2 actions satisfy this criterion while the Wilson and Symanzik actions do not. Moreover, we observe that the DBW2 flow stabilizes the topological sectors at the very early stage of the flow ($\hat{t}\approx 0.5$--$1$), suggesting that a further systematic investigation of the DBW2 flow is warranted to confirm its computational efficiency in determining the gauge topology.

We work on the functional renormalization group analysis on a four-fermion model with the CP and P violation in light of nonperturbative exploration of the infrared dynamics of quantum chromodynamics (QCD) arising from the spontaneous CP violation models in a view of the Wilsonian renormalization group. The fixed point structure reveals that in the large-$N_c$ limit, the CP $\bar{\theta}$ parameter is induced and approaches $\pi \cdot (N_f/2)$ (with the number of flavors $N_f$) toward the chiral broken phase due to the criticality and the large anomalous dimensions of the $U(1)$ axial violating four-fermion couplings. This trend seems to be intact even going beyond the large-$N_c$ leading, as long as the infrared dynamics of QCD is governed by the scalar condensate of the quark bilinear as desired. This gives an impact on modeling of the spontaneous CP violation scenarios: the perturbatively irrelevant four-fermion interactions nonperturbatively get relevant in the chiral broken phase, implying that the neutron electric dipole moment becomes too big, unless cancellations due to extra CP and P violating contributions outside of QCD are present at a certain intermediate infrared scale.