Dynamical field theory for glass-forming liquids, self-consistent resummations and time-reversal symmetry

We analyze the symmetries and the self-consistent perturbative approaches of dynamical field theories for glass-forming liquids. In particular we focus on the time-reversal symmetry which is crucial to obtain fluctuation dissipation relations (FDR). Previous field theoretical treatment violated this symmetry, whereas others pointed out that constructing symmetry preserving perturbation theories is a crucial and open issue. In this work we solve this problem and then apply our results to the Mode Coupling Theory of the glass transition (MCT). We show that in the context of dynamical field theories for glassforming liquids time-reversal symmetry is expressed as a non-linear field transformation that leaves the action invariant. Because of this non-linearity standard perturbation theories generically do not preserve time-reversal symmetry and in particular fluctuation-dissipation relations. We show how one can cure this problem and set up symmetry preserving perturbation theories by introducing some auxiliary fields. As an outcome we obtain Schwinger-Dyson dynamical equations that preserve automatically FDR and that serve as a basis for carrying out symmetry preserving approximations. We apply our results to the Mode-Coupling Theory of the glass transition revisiting previous field theory derivations of MCT equations and showing that they generically violate FDR. We obtain symmetry preserving modecoupling equations and discuss their advantages and drawbacks. Furthermore we show, contrary to previous works, that the structure of the dynamic equations is such that the ideal glass transition is not cut-off at any finite order of perturbation theory, even in the presence of coupling between current and density. The opposite results found in previous field theoretical works, as the ones based on non-linear fluctuating hydrodynamics, were only due to an incorrect treatment of time-reversal symmetry.