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Breakdown of a renormalized perturbation expansion around mode-coupling theory of the glass transition

We analyze a renormalized perturbation expansion around the mode-coupling theory of the glass transition. We focus on the long-time limit of the irreducible memory function. We discuss a renormalized diagrammatic expansion for this function and re-sum two infinite classes of diagrams. We show that the resulting contributions to the irreducible memory function diverge at the mode-coupling transition. A further re-summation of ladder diagrams constructed by iterating these divergent contributions gives a finite result which cancels the mode-coupling theory's expression for the irreducible memory function.