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The three-loop four-point function of stress-tensor multiplets in N=4 superYang-Mills theory contains two so far unknown, off-shell, conformal integrals,in addition to the known, ladder-type integrals. In this paper we evaluate theunknown integrals, thus obtaining the three-loop correlation functionanalytically. The integrals have the generic structure of rational functionsmultiplied by (multiple) polylogarithms. We use the idea of leadingsingularities to obtain the rational coefficients, the symbol - with anappropriate ansatz for its structure - as a means of characterising multiplepolylogarithms, and the technique of asymptotic expansion of Feynman integralsto obtain the integrals in certain limits. The limiting behaviour uniquelyfixes the symbols of the integrals, which we then lift to find thecorresponding polylogarithmic functions. The final formulae are numericallyconfirmed. The techniques we develop can be applied more generally, and weillustrate this by analytically evaluating one of the integrals contributing tothe same four-point function at four loops. This example shows a connectionbetween the leading singularities and the entries of the symbol.

Leading singularities and off-shell conformal integrals