Implications of electromagnetic scale anomaly to QCD chiral phase transition in smaller quark mass regime: $T_{\mathrm{pc}}$ does not drop with eB

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 Nambu-Jona-Lasinio model with 2 + 1 quark flavors including the electromagnetic-scale anomaly contribution. We observe that at $(m_{0c}, m_{sc}) \simeq (2, 20) \mathrm{MeV}$ for the isospin symmetric mass for up and down quarks, $m_0$, and the strange quark mass, $m_s$, $T_{\mathrm{pc}}$ 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.