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

Continuum models with critical end points are considered whose Hamiltonian ${\mathcal{H}}[ϕ,ψ]$ depends on two densities $ϕ$ and $ψ$. 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 $\sim|{t}|^{2-α}$ of the spectator-phase boundary and the coexistence singularities $\sim |{t}|^{1-α}$ or $\sim|{t}|^β$ of the secondary density $<ψ>$. 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.