  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-Landau 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.
