Hadron masses in strong magnetic fields

Hadron masses under strong magnetic fields are studied. In the presence of strong magnetic fields exceeding the QCD energy scale $eB\gg {\mathrm{\Lambda }}_{\mathrm{QCD}}^2$, $\mathrm{SU}(3{)}_{\text{flavor}}\otimes \mathrm{SU}(2{)}_{\text{spin}}$ symmetry of hadrons is explicitly broken so that the quark components of hadrons differ from those with zero or weak magnetic fields $eB\lesssim {\mathrm{\Lambda }}_{\mathrm{QCD}}^2$. Also, squeezing of hadrons by strong magnetic fields affects the hadron mass spectrum. We develop a quark model which appropriately incorporates these features and analytically calculate various hadron masses including mesons, baryons, and those with strangeness.

Figure 1

The constituent quark mass $M$ (Eq. (2)) of $u\uparrow ,u\downarrow ,d\uparrow ,d\downarrow ,s\uparrow ,s\downarrow$ as a function of the strength of the magnetic field $eB$. Parameters are set as ${\sigma }_{\perp }={\sigma }_{\parallel }=(200\mathrm{MeV}{)}^2$ and $m_u=m_d=0\mathrm{MeV},m_s=350\mathrm{MeV}$.

Figure 2

Various meson masses as a function of the strength of the magnetic field $eB$. [Left] Light mesons. The thick lines are the model calculation and the symbols are the lattice results; filled symbols for Ref. [7] and open symbols for Ref. [8]. Different shapes for open symbols distinguish the lattice space and the volume of the simulations in Ref. [8]. The thin black dashed lines are the naive lowest Landau energies for charged $\pi$ mesons, $M_{{\pi }^{\pm }}=\sqrt{(M_{{\pi }^{\pm }}(B=0){)}^2+eB}$, and charged $\rho$ mesons, $M_{{\rho }^{\pm }}=\sqrt{(M_{{\rho }^{\pm }}(B=0){)}^2-eB}$. We note that only $|s_z|=1$ component of $\rho$ meson is considered here because $s_z=0$ component of $\rho$ meson is mixed up with $\pi$ meson under strong magnetic fields. [Right] Strange mesons. The thick lines are the model prediction. The thin black dashed lines are the naive lowest Landau energies for charged $K$ mesons, $M_{K^{\pm }}=\sqrt{(M_{K^{\pm }}(B=0){)}^2+eB}$, and charged $K^{*}$ mesons, $M_{K^{*\pm }}=\sqrt{(M_{K^{*\pm }}(B=0){)}^2-eB}$.

Figure 3

Various baryon masses as a function of the strength of the magnetic field $eB$. The thick lines are the model prediction. [Left] Light baryons. [Right] Strange baryons.