Lee-Yang zero analysis for the study of QCD phase structure

We comment on the Lee-Yang zero analysis for the study of the phase structure of QCD at high temperature and baryon number density. We find that the sign problem for nonzero density QCD induces a serious problem in the finite volume scaling analysis of the Lee-Yang zeros. In the thermodynamic limit the Lee-Yang zeros will always approach the real axis of the complex parameter plane, except for $\mu_q=0$. A scaling behavior which would suggest a crossover transition thus is never obtained, which is in contrast with the usual expectation in the low density regime. This implies that the investigation of the order of phase transition by finite volume scaling does not provide an appropriate criterion for finding the endpoint of the first order phase transition line in the $(T, \mu_q)$ plane. To clarify this point, we discuss the Lee-Yang zero analysis for SU(3) pure gauge theory as a simple example without the sign problem, and consider possible other approaches that use the framework of the Lee-Yang zero analysis for the investigation of the critical endpoint