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Dynamical Landau Theory of the Glass Crossover

I introduce a dynamical field theory to describe the glassy behavior in supercooled liquids. The mean-field approximation of the theory predicts a dynamical arrest transition, as in ideal Mode-Coupling-Theory and mean-field discontinuous Spin-Glass Models. Instead {\it beyond} the mean-field approximation the theory predicts that the transition is avoided and transformed into a crossover, as observed in experiments and simulations. To go beyond mean-field a standard perturbative loop expansion is performed at first. Approaching the ideal critical point this expansion is divergent at all orders and I show that the leading divergent term at any given order is the same of a dynamical stochastic equation, called Stochastic-Beta-Relaxation (SBR) in {\it EPL 106, 56003 (2014)}. At variance with the original theory SBR can be studied beyond mean-field directly, without the need to resort to a perturbative expansion. Thus it provides a qualitative and quantitative description of the dynamical crossover. For consistency reasons it is important to establish the connection between the dynamical field theory and SBR beyond perturbation theory. This can be done with the help of a stronger result: the dynamical field theory is {\it exactly} equivalent to a theory with quenched disorder. Qualitatively the non-perturbative mechanism leading to the crossover is therefore the same of SBR. Quantitatively SBR corresponds to make the mean-field approximation once the quenched disorder has been generated.