Anomalous mesonic interactions near a chiral phase transition.

The axial anomaly is the observation that for fermions coupled to a gauge field, the divergence of the current for axial fermion number is not just the standard contribution from the classical equations of motion. In addition, at one loop order the divergence also contains a new term from quantum effects [1]. Due to deep geometrical reasons, there are no further corrections to this new term beyond one loop order [2]. For similar reasons, the axial anomaly is not altered by the presence of a medium, such as a thermal bath, in either two [3] or four [4] spacetime dimensions. Besides the axial anomaly for fermions, there are also “anomalous” mesonic interactions [1,5]. These are like the fermion anomaly, in that they arise from fermion loop graphs involving an odd number of the Dirac matrix γ, and so in the end are proportional to the antisymmetric tensor αβδγ . Despite this superficial similarity, in this Letter I show that while the axial anomaly for fermions is completely unaffected by the presence of a medium, near a chiral phase transition of second order, anomalous mesonic interactions change dramatically. As a medium I consider a thermal bath [6], which may be produced in the central region of heavy ion collisions at ultrarelativistic energies. I begin by considering the prototypical anomalous mesonic interaction, the decay of π → γγ [1,5]. Because the lifetime of the π is electromagnetic, and so much longer than hadronic timescales, this decay is not of experimental interest for heavy ion collisions. The essential physics, however, applies to the (anomalous) decay of the ω meson, which does decay over hadronic timescales. I work in a constituent quark model, with two flavors and Nc = 3 colors of a quark field ψ. I couple quarks couple to photons, Aα, and to mesons, Φ, ignoring their coupling to gluons. I assume that the (quantum mechanical) breaking of the axial U(1) chiral symmetry is large at all temperatures [7], so that the relevant chiral symmetry is SU(2)`×SU(2)r . Then the only [8] meson fields required are Φ = σt0 + i~π · ~t, with σ a J = 0 meson, and ~π the 0− pions; the flavor matrices are t0 = 1/2 and tr(tt) = δ/2. I take a positive definite euclidean metric, with (γ) = +1. Left and right handed quark fields are constructed by using the projectors P`,r = (1 ∓ γ)/2, ψ`,r = P`,rψ. The lagrangian density for quarks is