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Durham University

Centre for Particle Theory

Research Staff

Publication details for Paul Heslop

Drummond, James, Duhr, Claude, Eden, Burkhard, Heslop, Paul, Pennington, Jeffrey & Smirnov, Vladimir A. (2013). Leading singularities and off-shell conformal integrals. Journal of High Energy Physics 2013(8): 133.

Author(s) from Durham


The three-loop four-point function of stress-tensor multiplets in N = 4 super
Yang-Mills theory contains two so far unknown, off-shell, conformal integrals, in addition
to the known, ladder-type integrals. In this paper we evaluate the unknown integrals, thus
obtaining the three-loop correlation function analytically. The integrals have the generic
structure of rational functions multiplied by (multiple) polylogarithms. We use the idea of
leading singularities to obtain the rational coefficients, the symbol — with an appropriate
ansatz for its structure — as a means of characterising multiple polylogarithms, and the
technique of asymptotic expansion of Feynman integrals to obtain the integrals in certain
limits. The limiting behaviour uniquely fixes the symbols of the integrals, which we then
lift to find the corresponding polylogarithmic functions. The final formulae are numerically
confirmed. The techniques we develop can be applied more generally, and we illustrate this
by analytically evaluating one of the integrals contributing to the same four-point function
at four loops. This example shows a connection between the leading singularities and the
entries of the symbol.