Field-Theoretical Analysis of Critical and Coexistence Singularities at Critical End Points

Continuum models with critical end points are considered whose Hamiltonian ${mathcal{H}}[phi,psi]$ depends on two densities $phi$ and $psi$. Field-theoretic methods are used to show the equivalence of the critical behavior on the critical line and at the critical end point and to give a systematic derivation of critical-end-point singularities like the thermal singularity $sim|{t}|^{2-alpha}$ of the spectator-phase boundary and the coexistence singularities $sim |{t}|^{1-alpha}$ or $sim|{t}|^{beta}$ of the secondary density $<psi>$. The appearance of a discontinuity eigenexponent associated with the critical end point is confirmed, and the mechanism by which it arises in field theory is clarified.