  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 $|s_z|=1$ 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.
