# "Splitting" magnetic catalysis effect prevents vacuum superconductivity in strong magnetic fields

## TL;DR

This paper investigates how splitting magnetic catalysis effects in multi-flavor NJL models prevent the expected vacuum superconductivity in strong magnetic fields, highlighting the complex interplay of quark flavor effects and meson mass behaviors.

## Contribution

It demonstrates that splitting magnetic catalysis effects inhibit vacuum superconductivity in strong magnetic fields using multi-flavor NJL models, providing insights into meson mass behaviors and flavor mixing.

## Key findings

- Vacuum superconductivity is disfavored due to splitting magnetic catalysis effects.
- Meson masses exhibit non-monotonic behavior with increasing magnetic field.
- Flavor mixing among neutral pseudoscalars is significantly affected by magnetic fields.

## Abstract

By comparing the two- and three-flavor Nambu--Jona-Lasinio (NJL) models, we demonstrate that the naively expected vacuum superconductivity (VSC) in constant magnetic field ${\bf B}=B\hat z$ is disfavored due to the splitting magnetic catalysis effect (MCE) to chiral condensates with different quark flavors. Based on the simple two-flavor NJL model, we illuminate, in the lowest Landau level approximation, the similar origins of $\pi^0$ and $\bar{\rho}^+_1$ (${\rho}^+$ meson with spin $S_z=1$) mass reductions with smaller $B$ and their different features at larger $B$. With the full Landau levels, the two-flavor NJL model is found to be invalid to study the magnetic field effect to $\bar{\rho}^+_1$ meson with physical vacuum mass $775~{\rm MeV}$. Then, restricted to $\rho$ meson mass below two-quark threshold in vacuum, that is $m_\rho^v<2m_q^v$, it is found that $\pi^0$ mass decreases and then increases with $B$ slowly, and $\bar{\rho}^+_1$ mass vanishing point is delayed to larger $B$ compared to the point particle result. In the more realistic three-flavor NJL model, all the quark masses split in strong magnetic field as a combinatorial result of their different current masses and electric charges. By choosing a vacuum mass closer to the physical one, $\bar{\rho}^+_1$ meson mass is found to be consistent with the LQCD results semi-quantitatively in smaller $B$ region but increase in larger $B$ region. These features are mainly outcomes of the interplay between the $S_z-B$ coupling effect and splitting MCE to the composite $u$ and $d$ quarks, which definitely disfavors VSC when the latter dominates. Furthermore, mesonic flavor mixing is modified by $B$ among the neutral pseudoscalars: $\pi^0,\eta_0$ and $\eta_8$, which is very important to suppress the mass enhancement of the effective mass eigenstates at large $B$.