The High Temperature Phase of QCD and {\boldmath $U(1)_A$} Symmetry
  Inequalities for QCD functional integrals are used to establish that up to certain technical assumptions. the high temperature chirally restored phase of QCD is effectively symmetric under $U(N_f) \times U(N_f)$ rather than $SU(N_f) \times SU(N_f)$. If these assumptions are correct, there are no effects due to anomalous breaking of $U(1)_A$ on correlation functions in this phase. 
Thomas. D. Cohen