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Temperature dependence of the $U_A(1)$ anomaly is investigated by taking intoaccount mesonic fluctuations in the $U(3)\times U(3)$ linear sigma model. Afield dependent anomaly coefficient function of the effective potential iscalculated within the finite temperature functional renormalization groupapproach. The applied approximation scheme is a generalization of the chiralinvariant expansion technique developed in [G. Fejos, Phys. Rev. D 90, 096011(2014)]. We provide an analytic expression and also numerical evidence thatdepending on the relationship between the two quartic couplings, mesonicfluctuations can either strengthen of weaken the anomaly as a function of thetemperature. The role of the six-point invariant of the $U(3)\times U(3)$group, and therefore the stability of the chiral expansion is also discussed indetail.

Functional dependence of axial anomaly via mesonic fluctuations in the three flavor linear sigma model