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 the ideal mode-coupling theory and mean-field discontinuous spin-glass models. Instead, 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 as a dynamical stochastic equation, called stochastic-beta relaxation (SBR) in Europhys. Lett. 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 exactly equivalent to a theory with quenched disorder. Qualitatively, the nonperturbative mechanism leading to the crossover is therefore the same as the mechanism of SBR. Quantitatively, SBR is equivalent to making the mean-field approximation once the quenched disorder has been generated.