Magnetic polarizabilities of light mesons in $SU(3)$ lattice gauge theory

We investigate the masses (ground state energies) of neutral pseudoscalar and vector meson in $SU(3)$ lattice gauge theory in strong abelian magnetic field. The energy of ${\rho }^0$ meson with zero spin projection $s_z=0$ on the axis of the external magnetic field decreases, while the energies with non-zero spins $s_z=-1$ and $+1$ increase with the field. The energy of ${\pi }^0$ meson decrease as a function of the magnetic field. We calculated the magnetic polarizabilities of pseudoscalar and vector mesons for lattice volume ${18}^4$. For ${\rho }^0$ with spin $\mathrm{\mid }{s}_z\mathrm{\mid }=1$ and ${\pi }^0$ meson the extrapolations to zero lattice spacing have been done. We do not observe any evidence in favour of tachyonic mode existence.