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Leading singularities and off-shell conformal integrals

James Drummond, Claude Duhr, Burkhard Eden, Paul Heslop, Jeffrey Pennington, Vladimir A. Smirnov

TL;DR

The paper advances analytic evaluation of the three-loop four-point stress-tensor correlator in planar N=4 SYM by isolating two unknown off-shell conformal integrals (Easy and Hard) and expressing them as sums of leading singularities times pure polylogarithmic functions. It develops and deploys a cohesive framework—leading singularities to fix rational prefactors, symbol analysis to constrain polylogarithm structure, and asymptotic expansions to determine symbols and uplift to full functions—yielding compact analytic results for both Easy and Hard integrals, verified numerically. Extending the method to a four-loop example via a differential equation approach demonstrates the robustness and generality of the approach, including the emergence of generalized polylogarithms beyond SVHPLs. The work clarifies the link between leading singularities and symbol entries and lays groundwork for applying these techniques to higher-loop conformal integrals and related correlators.

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.

Leading singularities and off-shell conformal integrals

TL;DR

The paper advances analytic evaluation of the three-loop four-point stress-tensor correlator in planar N=4 SYM by isolating two unknown off-shell conformal integrals (Easy and Hard) and expressing them as sums of leading singularities times pure polylogarithmic functions. It develops and deploys a cohesive framework—leading singularities to fix rational prefactors, symbol analysis to constrain polylogarithm structure, and asymptotic expansions to determine symbols and uplift to full functions—yielding compact analytic results for both Easy and Hard integrals, verified numerically. Extending the method to a four-loop example via a differential equation approach demonstrates the robustness and generality of the approach, including the emergence of generalized polylogarithms beyond SVHPLs. The work clarifies the link between leading singularities and symbol entries and lays groundwork for applying these techniques to higher-loop conformal integrals and related correlators.

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.

Paper Structure

This paper contains 32 sections, 198 equations, 2 figures, 4 tables.

Table of Contents

  1. Introduction
  2. Conformal four-point integrals and single-valued polylogarithms
  3. The symbol
  4. Single-Valued Harmonic Polylogarithms (SVHPLs)
  5. The $\bar{x} \rightarrow 0$ limit of SVHPLs
  6. The short-distance limit
  7. The Easy integral
  8. Residues of the Easy integral
  9. The symbol of $E(x,\bar{x})$
  10. The analytic result for $E(x,\bar{x})$: uplifting from the symbol
  11. The analytic result for $E(x,\bar{x})$: the direct approach
  12. Numerical consistency tests for $E$
  13. The Hard integral
  14. Residues of the Hard integral
  15. The symbols of $H^{(a)}(x,\bar{x})$ and $H^{(b)}(x,\bar{x})$
  16. ...and 17 more sections

Figures (2)

  • Figure 1: The Easy and Hard integrals contributing to the correlator of stress tensor multiplets at three loops.
  • Figure 2: The four-loop integral $I^{(4)}_{14;23}$ defined in eq. \ref{['I4int']}.