The chiral condensate in a constant electromagnetic field

TL;DR

This paper investigates how a constant electromagnetic field affects the chiral condensate using chiral perturbation theory, deriving a comprehensive one-loop expression that accounts for finite pion mass effects and exploring the implications for electric fields.

Contribution

It provides a novel all-orders one-loop calculation of the chiral condensate shift in electromagnetic fields, including finite pion mass corrections and conditions for the validity of the low-energy theorem.

Findings

Derived a one-loop expression valid to all orders in $m_{C6}^2 / eH$

Showed the importance of pion mass corrections for the low-energy theorem

Discussed the breakdown of the method due to pair creation in electric fields

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.