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1st or 2nd; the order of finite temperature phase transition of Nf=2 QCD from effective theory analysis

In the previous work, we have shown that the SU(2) chiral symmetry recovered above the critical temperature gives a strong constraint on the Dirac eigenvalue spectrum and this constraint is strong enough for a set of anomalous U(1) chiral symmetry breaking operators to vanish in the thermodynamical and chiral limits. We use this condition as an input and impose a constraint on the Landau low energy effective theory of QCD. The only constraint we can set is that the mass splitting term between the pion and eta meson should vanish. All the singlet/non-singlet scalar/pseudo-scalar mesons contribute to the effective theory. We evaluate the renormalization group $β$-function for the effective theory using the $ε$-expansion at one loop level, but find no stable infra-red fixed point except for the trivial Gaussian one. The chiral phase transition seems to be of first order.