Diverging length scale and upper critical dimension in the Mode-Coupling Theory of the glass transition

We show that the glass transition predicted by the Mode-Coupling Theory (MCT) is a critical phenomenon with a diverging length and time scale associated to the cooperativity of the dynamics. We obtain the scaling exponents v and z that relate space and time scales to the distance from criticality, as well as the scaling form of the critical four-point correlation function. However, both these predictions and other well-known MCT results are mean. field in nature and are thus expected to change below the upper critical dimension d(c) = 6, as suggested by different forms of the Ginzburg criterion.