Lee-Yang and Langer edge singularities from analytic continuation of scaling functions

We discuss the analytic continuation of scaling function in the three-dimensional $Z\left(2\right)$, $O\left(2\right)$ and $O\left(4\right)$ universality classes using the Schofield representation of the magnetic equation of state. We show that a determination of the location of Lee-Yang edge singularities and, in the case of $Z\left(2\right)$, also the Langer edge singularity yields stable results. Results for the former are in good agreement with functional renormalization group calculations. We also present results for the location of the Langer edge singularity in the $3d$, $Z\left(2\right)$ universality class. We find that in terms of the complex scaling variable $z$ the distance of the Langer edge singularity to the critical point agrees within errors with that of the Lee-Yang edge singularity. Furthermore the magnitude of the discontinuity along the Langer branch cut is an order of magnitude smaller than that along the Lee-Yang branch cut.