We consider chiral perturbation theory for SU(2) at finite temperature T in
a constant magnetic background B. We compute the thermal mass of the pions
and the pion decay constant to leading order in chiral perturbation theory in
the presence of the magnetic field. The magnetic field gives rise to a
splitting between Mπ0 and Mπ± as well as between
Fπ0 and Fπ±. We also calculate the free energy and the
quark condensate to next-to-leading order in chiral perturbation theory. Both
the pion decay constants and the quark condensate are decreasing slower as a
function of temperature as compared to the case with vanishing magnetic field.
The latter result suggests that the critical temperature Tc for the chiral
transition is larger in the presence of a constant magnetic field. The increase
of Tc as a function of B is in agreement with most model calculations but
in disagreement with recent lattice calculations.Comment: 24 pages and 9 fig