We study analytically the chiral phase transition for hot quark matter in presence of a strong magnetic background, focusing on the existence of a critical point at zero baryon chemical potential and nonzero magnetic field. We build up a Ginzburg-Lan
dau effective potential for the chiral condensate at finite temperature, computing the coefficients of the expansion within a chiral quark-meson model. Our conclusion is that the existence of the critical point at finite $bm B$ is very sensitive to the way the ultraviolet divergences of the model are treated. In particular, we find that after renormalization, no chiral critical point is present in the phase diagram. On the other hand, such a critical point there exists when the ultraviolet divergences are not removed by a proper renormalization of the thermodynamic potential.
