In one and two dimensions, transport coefficients may diverge in the thermodynamic limit due to long--time correlation of the corresponding currents. Indeed, power--law decay of correlations is expected to be a generic feature of one--dimensional systems in presence of conservation laws. A relevant example is the divergence of the thermal conductivity coefficient observed numerically in chains of anharmonic oscillators and 1D gases. To what extent are those observations universal? To answer this question we studied a 1d version of mode-coupling equations, which describe the relaxation of fluctuations of the displacement field. We show that the latter admit scaling solutions in the small-wavenumber limit which, at variance with usual linearized hydrodynamics, are not exponential. The theory predicts that the heat current autocorrelation should decay as t^{-2/3}, which is in agreement with recent simulation results on the 1D diatomic gas.