quantum electrodynamics - Lagrangian for electron radiating a photon

I know the electron-photon interaction Lagrangian from QED: ($-ie\bar{\Psi}\gamma^{\mu}A_{\mu}\Psi$). However, this interaction seems to describe the interaction of electron and positron with a photon (thus there is a $\bar{\Psi}$ for positron). This makes sense when an electron radiates a virtual photon (and becomes a positron) and then reabsorbs the photon again (and becomes an electron again). What about the case when an electron emits a real photon (and never becomes a positron) like the case of synchrotron radiation? What is the Lagrangian in that case since we wouldn't have any electron-positron vertex?