The chiral condensate in a constant electromagnetic field

Abstract

We study the shift of the chiral condensate in a constant electromagnetic field in the context of chiral perturbation theory. Using the Schwinger proper-time formalism, we derive a one-loop expression correct to all orders in $m_{\pi}^2 / eH$. Our result correctly reproduces a previously derived ``low-energy theorem'' for $m_\pi = 0$. We show that it is essential to include corrections due to non-vanishing $m_\pi$ in order for a low energy theorem to have any approximate regime of validity in the physical universe. We generalize these results to systems containing electric fields, and discuss the regime of validity for the results. In particular, we discuss the circumstances in which the method formally breaks down due to pair creation in an electric field.