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The Four-Loop Dressing Phase of N=4 SYM

N. Beisert, T. McLoughlin, R. Roiban

TL;DR

The paper computes the four-loop dilatation operator for the su(2) sector of planar N=4 SYM by constraining the structure with the world-sheet S-matrix and determining the remaining coefficients from Feynman diagrams. It identifies a set of maximal-reshuffling scalar interactions that greatly simplify the calculation and uses a subtraction scheme together with Mellin-Barnes techniques to extract ultraviolet divergences, obtaining the leading dressing-phase coefficient $\beta_{2,3}=4\zeta(3)$ in agreement with known four-loop cusp results. This confirms the weak-coupling dressing phase and supports the integrability-based picture of the AdS/CFT correspondence, providing a pathway to higher-loop determinations via scalar diagrams and recursive master integrals.

Abstract

We compute the dilatation generator in the su(2) sector of planar N=4 super Yang-Mills theory at four-loops. We use the known world-sheet scattering matrix to constrain the structure of the generator. The remaining few coefficients can be computed directly from Feynman diagrams. This allows us to confirm previous conjectures for the leading contribution to the dressing phase which is proportional to zeta(3).

The Four-Loop Dressing Phase of N=4 SYM

TL;DR

The paper computes the four-loop dilatation operator for the su(2) sector of planar N=4 SYM by constraining the structure with the world-sheet S-matrix and determining the remaining coefficients from Feynman diagrams. It identifies a set of maximal-reshuffling scalar interactions that greatly simplify the calculation and uses a subtraction scheme together with Mellin-Barnes techniques to extract ultraviolet divergences, obtaining the leading dressing-phase coefficient in agreement with known four-loop cusp results. This confirms the weak-coupling dressing phase and supports the integrability-based picture of the AdS/CFT correspondence, providing a pathway to higher-loop determinations via scalar diagrams and recursive master integrals.

Abstract

We compute the dilatation generator in the su(2) sector of planar N=4 super Yang-Mills theory at four-loops. We use the known world-sheet scattering matrix to constrain the structure of the generator. The remaining few coefficients can be computed directly from Feynman diagrams. This allows us to confirm previous conjectures for the leading contribution to the dressing phase which is proportional to zeta(3).

Paper Structure

This paper contains 8 sections, 31 equations, 1 figure, 1 table.

Table of Contents

  1. Introduction and Overview
  2. Long-Range Heisenberg Hamiltonian
  3. Four-Loop Calculation
  4. Conclusions and Outlook
  5. Acknowledgments:
  6. The Triangle Rule
  7. Three-Loop Integrals
  8. Subtraction Scheme

Figures (1)

  • Figure 1: Maximal planar interactions up to four loops. Below the diagrams the permutation symbols are indicated. Solid and dashed lines correspond to two complex scalars in $\mathcal{N}=4$ SYM. Above the diagrams suitable momenta to remove IR singularities are indicated.