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On planar gluon amplitudes/Wilson loops duality

J. M. Drummond, J. Henn, G. P. Korchemsky, E. Sokatchev

TL;DR

This work provides a concrete two-loop verification of the proposed duality between planar N=4 SYM gluon scattering amplitudes and light-like Wilson loops by computing the Wilson loop dual to the four-gluon amplitude and matching its divergent and finite parts to the known amplitude result. Using nonabelian exponentiation, dimensional regularization, and dual coordinates, the authors demonstrate that the UV cusps-induced divergences of the Wilson loop reproduce the IR divergences of the amplitude, and that the finite part agrees with the BDS conjecture at two loops. They further derive an anomalous conformal Ward identity for Wilson loops, showing that it uniquely fixes the finite parts for four- and five-point cases (up to a constant) in agreement with BDS, while highlighting the role of conformal cross-ratios starting at six points. The results reinforce the amplitude/Wilson-loop duality at higher loops and motivate exploration of dual conformal symmetry and integrability in N=4 SYM for arbitrary numbers of external gluons.

Abstract

There is growing evidence that on-shell gluon scattering amplitudes in planar N=4 SYM theory are equivalent to Wilson loops evaluated over contours consisting of straight, light-like segments defined by the momenta of the external gluons. This equivalence was first suggested at strong coupling using the AdS/CFT correspondence and has since been verified at weak coupling to one loop in perturbation theory. Here we perform an explicit two-loop calculation of the Wilson loop dual to the four-gluon scattering amplitude and demonstrate that the relation holds beyond one loop. We also propose an anomalous conformal Ward identity which uniquely fixes the form of the finite part (up to an additive constant) of the Wilson loop dual to four- and five-gluon amplitudes, in complete agreement with the BDS conjecture for the multi-gluon MHV amplitudes.

On planar gluon amplitudes/Wilson loops duality

TL;DR

This work provides a concrete two-loop verification of the proposed duality between planar N=4 SYM gluon scattering amplitudes and light-like Wilson loops by computing the Wilson loop dual to the four-gluon amplitude and matching its divergent and finite parts to the known amplitude result. Using nonabelian exponentiation, dimensional regularization, and dual coordinates, the authors demonstrate that the UV cusps-induced divergences of the Wilson loop reproduce the IR divergences of the amplitude, and that the finite part agrees with the BDS conjecture at two loops. They further derive an anomalous conformal Ward identity for Wilson loops, showing that it uniquely fixes the finite parts for four- and five-point cases (up to a constant) in agreement with BDS, while highlighting the role of conformal cross-ratios starting at six points. The results reinforce the amplitude/Wilson-loop duality at higher loops and motivate exploration of dual conformal symmetry and integrability in N=4 SYM for arbitrary numbers of external gluons.

Abstract

There is growing evidence that on-shell gluon scattering amplitudes in planar N=4 SYM theory are equivalent to Wilson loops evaluated over contours consisting of straight, light-like segments defined by the momenta of the external gluons. This equivalence was first suggested at strong coupling using the AdS/CFT correspondence and has since been verified at weak coupling to one loop in perturbation theory. Here we perform an explicit two-loop calculation of the Wilson loop dual to the four-gluon scattering amplitude and demonstrate that the relation holds beyond one loop. We also propose an anomalous conformal Ward identity which uniquely fixes the form of the finite part (up to an additive constant) of the Wilson loop dual to four- and five-gluon amplitudes, in complete agreement with the BDS conjecture for the multi-gluon MHV amplitudes.

Paper Structure

This paper contains 9 sections, 45 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Light-like Wilson loops
  3. One-loop calculation
  4. Nonabelian exponentiation
  5. Two-loop calculation
  6. Duality relation to two loops
  7. A conformal Ward identity for the Wilson loop
  8. Conclusions
  9. Appendix

Figures (1)

  • Figure 1: The Feynman diagrams contributing to $\ln {W(C_4)}$ to two loops. The double line depicts the integration contour $C_4$, the wiggly line the gluon propagator and the blob the one-loop polarization operator.