Correlations among discontinuities in the QCD phase diagram

We show, in general, that when a discontinuity of either zeroth order or first order takes place in an order parameter such as the chiral condensate, discontinuities of the same order emerge in other order parameters such as the Polyakov loop. A condition for the coexistence theorem to be valid is clarified. Consequently, only when the condition breaks down, zeroth-order and first-order discontinuities can coexist on a phase boundary. We show with the Polyakov-loop extended Nambu-Jona-Lasinio model that such a type of coexistence is realized in the imaginary chemical potential region of the QCD phase diagram. We also present examples of coexistence of the same-order discontinuities in the real chemical potential region.