3 We can define two correlations functions favor of the expectation value of the overlap (i.e. q ) and all other physical variables can be expressed as function of q ; this is the usual Legendre transformation of statis- tical mechanics. Our assumption is that in the critical region ( ϵ → 0 and C → C$_{P}$ )the quantities where the overline denotes the average over the σ ’s and 〈·〉 the average over the τ ’s. In the case of a fluid with only one type of particles, q$_{σ,τ}$ ( x ) can be written as σ ( x ) τ ( x ) (where σ ( x ) and τ ( x ) are smeared densities around the point x ) and the previ- ous equations become computed in perturbation theory in the replica approach are the same as those computed in the mode-coupling theory (as function of C ). This relation holds only in the region of q where the forcing field h ≈ 0. One finds that near the critical point (neglecting loops) G$_{0}$ ( x ) = 〈 q$_{σ,τ}$ ( x ) 〉〈 q$_{σ,τ}$ (0) 〉 − 〈 q$_{σ,τ}$ (0) 〉 2 , (11) G$_{1}$ ( x ) = 〈 q$_{σ,τ}$ ( x ) q$_{σ,τ}$ (0) 〉 − 〈 q$_{σ,τ}$ ( x ) 〉〈 q$_{σ,τ}$ (0) 〉 , χ$_{4}$ ( ϵ | q ) , Γ$_{4}$( x − y, ϵ | q ) , (8) G$_{0}$ ( x ) = 〈 τ ( x ) 〉〈 τ (0) 〉 σ ( x ) σ (0) − 〈 τ (0) 〉 σ (0) 2 , (12) G$_{1}$ ( x ) = 〈 τ ( x ) τ (0) 〉$_{c}$ σ ( x ) σ (0) . ˜ G$_{0}$ ( k, ϵ ) = 1 ( ϵ 1 / 2 + k $^{2}$)2 , ˜ G$_{1}$ ( k, ϵ ) = 1 ϵ 1 / 2 + k 2 . (13) Let us start the perturbative computation in the replica approach. The computations in the mode-coupling ap- proach will be rather similar. It is quite clear from Fig. 1 that changing the temperature there is a critical point. Our task it to compute the critical properties using the standard renormalization group tools. We want to de- termine the critical exponents γ and ν and the scaling function g$_{4}$ ( x/ξ ), that according to the previous assump- tion are in the same perturbative universality class in the mode-coupling theory and in the replica approach. Let us discuss for the moment only the region below the dy- namical temperature, i.e. ϵ < 0, since this region is very well defined in the perturbative expansion. In order to compute the loop corrections one must use a precise replica setting. In the approach presented here, there is an annoying asymmetry among the replica σ and the replica τ . In the replica approach it corresponds to take one privileged replica σ and n replicas of type τ and sending the value of n to zero. It is known that a partially equivalent formalism consists in using m replicas that are constrained to be at overlap q and take the limit where m goes to one. In this different formulation we have a symmetry Z$_{m}$ in the limit m → 1 and it may be more convenient to use [22]. We checked that the two formulations are equivalent near the stationary point and for simplicity we present the analysis in this second formalism. A detailed discussion of this point will be presented elsewhere [23]. In high dimensions the correlation function is just given by the mean field result [12, 19, 20] thus leading to the following mean field exponents The diagrammatics is the same of an usual φ 3 theory with two differences [12, 21]: A) some of the propaga- tors have a single pole, others have double poles; B) the multiplicity of the diagrams have to be computed in the limit where m goes to one and some diagrams give zero contribution in this limit. As we will see, there is a crucial difference among the double pole behaviour that we find to be valid in the general case and the single pole behavior. Notice that the double pole form in Eq.(9) is at variance with the single pole behavior predicted in [19, 20], if no local conservation laws are present. The reasons for this discrepancy will be discussed later, when we confirm the presence of a double pole by a dynamical analysis. ˜ G$_{4}$ ( k, ϵ ) = 1 ( | ϵ | 1 / 2 + k $^{2}$)2 , (9) ν = 1 4 γ = 1 . (10) One should do a careful analysis: at the end of a long analysis [23] we find that many diagrams give zero contri- bution and the dimension where the perturbative correc- tions are divergent is 8 [33]. Moreover the diagrams are the same of those for lattice animals, with the difference that here the effective coupling constant is positive (for lattice animals is negative) [24–26]. The value 8 for the upper critical dimension may provide an explanation for many of the anomalies found in [27]. In order to compute the critical exponents of the dy- namical transition we will start from the perturbation theory around the mean field theory. We will consider the most divergent diagrams near the critical point and we will identify the universality class of the problem. In this context we do not see that the critical point is in the metastable phase because this would be a non- perturbative effect. This computation can be done within the replica method of [5] assuming a universal cubic effective replica action close to the inflection point of W ( q ) for ϵ → 0. For disordered systems like Potts model, however, this model should be in the same universality class of struc- tural glasses as far the dynamical transition is concerned. What happens below (and near) 8 dimensions? The situation is quite puzzling: the renormalization group pushes the coupling constant g 2 toward a large value and there is no perturbative fixed point that we can analyze. Moreover the terms in the perturbation theory have all positive sign and therefore it not easy to estimate the sum. This result is not so disturbing. Metastable states have finite life time, the free-energy acquires an imaginary