Topological Charge and U(1)a Symmetry in the High Temperature Phase of Qcd

We discuss the global symmetries of the high temperature phase of QCD with Nf massless quarks. We show that the U(Nf ) × U(Nf ) symmetries are only violated by operators of dimension ≥ 3Nf . For Nf > 2 this implies that the thermal two-point correlation functions of the η′ and πa’s are identical. We discuss the implications of this for the chiral phase transition at finite temperature. [email protected] [email protected] [email protected] Recently, Cohen [1] has used QCD inequalties to argue that the high temperature phase of massless QCD with Nf flavors of quarks is effectively symmetric under a global U(Nf )× U(Nf ) symmetry rather than just SU(Nf ) × SU(Nf ). Cohen examines the two-point correlation functions of different operators such as the η (q̄γ5q) and the π a (q̄τγ5q) which are in the same U(Nf )×U(Nf ) multiplet but not in the same SU(Nf )× SU(Nf ) multiplet. He argues that in the massless (or chiral) limit: mq → 0, the difference between the respective two-point correlation functions approaches zero. In this letter we show that the U(Nf ) × U(Nf ) symmetry is not completely restored in the high temperature phase although its breaking can only be manifested in operators of dimension ≥ 3Nf . Thus the U(Nf ) × U(Nf ) symmetry is only restored for the two-point correlators when Nf > 2. We concentrate on the relationship between (spontaneous) chiral symmetry breaking, the axial anomaly and topological charge and clarify some subtle points in the argument of [1]. It is well-known that the solution of the U(1)A problem [2] requires (A) the axial anomaly relation (mq = 0) ∂μJ 5 μ = g2Nf 16π tr[FF̃ ], (1)