Leading singularities and off-shell conformal integrals

Abstract

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.

60 pages, LaTeX, 2 figures, references added

References in corpus (16)

• Gluon scattering amplitudes at strong coupling
• Classical Polylogarithms for Amplitudes and Wilson Loops
• Conformal properties of four-gluon planar amplitudes and Wilson loops
• MHV Amplitudes in N=4 Super Yang-Mills and Wilson Loops
• Algorithm FIRE -- Feynman Integral REduction
• Hexagon Wilson loop = six-gluon MHV amplitude
• The Two-Loop Six-Gluon MHV Amplitude in Maximally Supersymmetric Yang-Mills Theory
• Presenting LiteRed: a tool for the Loop InTEgrals REDuction
• Feynman Integral Evaluation by a Sector decomposiTion Approach (FIESTA)
• FIRE4, LiteRed and accompanying tools to solve integration by parts relations
• Expansion by regions: revealing potential and Glauber regions automatically
• Geometric approach to asymptotic expansion of Feynman integrals
• Sharpening The Leading Singularity
• Group structure of the integration-by-part identities and its application to the reduction of multiloop integrals
• Three-loop universal structure constants in N=4 susy Yang-Mills theory
• On the decomposition of motivic multiple zeta values

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