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Four-Loop Cusp Anomalous Dimension From Obstructions

Freddy Cachazo, Marcus Spradlin, Anastasia Volovich

TL;DR

This work develops an obstruction-based framework to extract the cusp anomalous dimension $f(g)$ from planar ${ m N}=4$ SYM four-gluon amplitudes without evaluating kinematics-dependent integrals. By working in Mellin space and isolating leading delta-function obstructions, the authors derive a simple all-orders prescription $f^{(L)} = -2 L^2[ P^{(L)}(0, abla) - X^{(L)}[P^{(l)}(0, abla)] ]_{1/ abla^2}$, enabling analytic results at $L=1,2,3$ and a highly precise numerical four-loop computation. The four-loop result $f^{(4)} = -117.1789(2)$ (equivalently $r=-2.00002(3)$) agrees with the Beisert et al. conjecture, providing strong evidence for the all-loop phase-related predictions and the integrability-based structure of the theory. The method reduces the number of contributing terms and integration variables, offering a scalable route to higher-loop cusp dimensions and potential applications to other gauge theories, including QCD.

Abstract

We introduce a method for extracting the cusp anomalous dimension at L loops from four-gluon amplitudes in N=4 Yang-Mills without evaluating any integrals that depend on the kinematical invariants. We show that the anomalous dimension only receives contributions from the obstructions introduced in hep-th/0601031. We illustrate this method by extracting the two- and three-loop anomalous dimensions analytically and the four-loop one numerically. The four-loop result was recently guessed to be f^4 = - (4ζ^3_2+24ζ_2ζ_4+50ζ_6- 4(1+r)ζ_3^2) with r=-2 using integrability and string theory arguments in hep-th/0610251. Simultaneously, f^4 was computed numerically in hep-th/0610248 from the four-loop amplitude obtaining, with best precision at the symmetric point s=t, r=-2.028(36). Our computation is manifestly s/t independent and improves the precision to r=-2.00002(3), providing strong evidence in favor of the conjecture. The improvement is possible due to a large reduction in the number of contributing terms, as well as a reduction in the number of integration variables in each term.

Four-Loop Cusp Anomalous Dimension From Obstructions

TL;DR

This work develops an obstruction-based framework to extract the cusp anomalous dimension from planar SYM four-gluon amplitudes without evaluating kinematics-dependent integrals. By working in Mellin space and isolating leading delta-function obstructions, the authors derive a simple all-orders prescription , enabling analytic results at and a highly precise numerical four-loop computation. The four-loop result (equivalently ) agrees with the Beisert et al. conjecture, providing strong evidence for the all-loop phase-related predictions and the integrability-based structure of the theory. The method reduces the number of contributing terms and integration variables, offering a scalable route to higher-loop cusp dimensions and potential applications to other gauge theories, including QCD.

Abstract

We introduce a method for extracting the cusp anomalous dimension at L loops from four-gluon amplitudes in N=4 Yang-Mills without evaluating any integrals that depend on the kinematical invariants. We show that the anomalous dimension only receives contributions from the obstructions introduced in hep-th/0601031. We illustrate this method by extracting the two- and three-loop anomalous dimensions analytically and the four-loop one numerically. The four-loop result was recently guessed to be f^4 = - (4ζ^3_2+24ζ_2ζ_4+50ζ_6- 4(1+r)ζ_3^2) with r=-2 using integrability and string theory arguments in hep-th/0610251. Simultaneously, f^4 was computed numerically in hep-th/0610248 from the four-loop amplitude obtaining, with best precision at the symmetric point s=t, r=-2.028(36). Our computation is manifestly s/t independent and improves the precision to r=-2.00002(3), providing strong evidence in favor of the conjecture. The improvement is possible due to a large reduction in the number of contributing terms, as well as a reduction in the number of integration variables in each term.

Paper Structure

This paper contains 11 sections, 44 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Structure of the $L$-Loop Four-Gluon Amplitude
  3. Infrared Singularity Structure
  4. Obstructions and Their Applications
  5. Obstructions---Theory
  6. Obstructions---Implementation
  7. Cusp Anomalous Dimension From Obstructions
  8. Analytic results at one, two and three loops
  9. Four-Loop Calculation
  10. Conclusions and Future Directions
  11. Implementation Details

Figures (1)

  • Figure 1: Here we depict the topologies of the integrals which contribute to the $L$-loop four-gluon amplitude, for $L \le 4$. The numerical coefficients and the numerator factors in the integrand are given by the rung rule for all but the last two four-loop diagrams. We refer the reader to Bern:2006ew for further details.