  Within the framework of Dyson-Schwinger equations (DSEs), we discuss the
chiral phase transition of QCD with a chiral chemical potential $\mu_5$ as an
additional scale. We focus especially on the issues related to the widely
accepted as well as interested critical end point (CEP). With the help of a
scalar susceptibility, we find that there might be no CEP$_5$ in the $T-\mu_5$
plane, and the phase transition in the $T-\mu_5$ plane might be totally
crossover when $\mu<50$ MeV, which has apparent consistency with the Lattice
QCD calculation. Our study may also provide some useful hints to some other
studies related to $\mu_5$.
