Neutral pseudoscalar and vector meson masses under strong magnetic fields in an extended NJL model: Mixing effects

Mixing effects on the mass spectrum of light neutral pseudoscalar and vector mesons in the presence of an external uniform magnetic field $\overrightarrow{B}$ are studied in the framework of a two-flavor Nambu-Jona-Lasinio (NJL)-like model. The model includes isoscalar and isovector couplings both in the scalar-pseudoscalar and vector sectors, and also incorporates flavor mixing through a ’t Hooft-like term. Numerical results for the $B$ dependence of meson masses are compared with present lattice QCD results. In particular, it is shown that the mixing between pseudoscalar and vector meson states leads to a significant reduction of the mass of the lightest state. The role of chiral symmetry and the effect of the alignment of quark magnetic moments in the presence of the magnetic field are discussed.

Figure 1

Effective quark masses $M_u$ (red upper band), $M_d$ (blue lower band) as functions of $eB$. The extremes of the bands correspond to $\alpha =0$ (dashed lines) and $\alpha =0.5$ (full line). The dotted lines correspond to $\alpha =0.1$.

Figure 2

Normalized average condensate (left) and normalized condensate difference (right) as functions of $eB$, for values of $\alpha$ from 0 to 1 (see text for definitions). LQCD results from Ref. [16] (gray bands) are added for comparison.

Figure 3

Masses of $S_z=0$ mesons as functions of $eB$, for $\alpha =0.1$. The dotted lines indicate $u\overline{u}$ and $d\overline{d}$ production thresholds.

Figure 4

Masses of the lightest $S_z=0$ mesons as functions of $eB$, for various values of $\alpha$. Dashed (dotted) lines correspond to the case in which the mixing between pseudoscalar and vector states is (is not) included. The dash-dotted line in the right panel is obtained from the approximate expression in Eq. (41).

Figure 5

Normalized mass of the $\tilde{\pi }$ meson (lightest state of the $S_z=0$ sector) as a function of $eB$, compared with LQCD results quoted in Ref. [54](quenched Wilson fermions), Ref. [55] (improved staggered quarks) and Refs. [15, 54, 80] (dynamical staggered quarks). Solid and dotted lines correspond to NJL results with and without pseudoscalar-vector meson mixing, respectively.

Figure 6

Masses of the $S_z=\pm 1$ vector meson states as functions of $eB$. Dotted and short-dotted lines indicate $m_d^{-}$ and $m_u^{-}$ quark-antiquark production thresholds, respectively.

Figure 7

Masses of $S_z=\pm 1$ vector mesons as functions of $eB$, for various values of $\alpha$. The full black lines correspond to the case in which the mixing between pseudoscalar and vector meson states is not considered.

Figure 8

Mass of the $\rho$ meson with $S_z=\pm 1$ for the case in which and $\alpha =0.5$ and there is no mixing between pseudoscalar and vector meson states. Results quoted in the literature using other regularization methods are also shown.

Figure 9

Masses of $S_z=\pm 1$ vector meson states for $\alpha =0.1$, compared with LQCD results given in Ref. [54].

Figure 10

Left (right) panel: masses of the $S_z=0$ ($S_z=\pm 1)$ meson states as functions of $eB$, for $B$-dependent couplings $g(eB)/g=g_v(eB)/g_v=\mathcal{F}(eB)$ [see Eq. (43)]. The results correspond to the case $\alpha =0.1$.