Four-Loop Cusp Anomalous Dimension From Obstructions

TL;DR

This paper presents a novel method to extract the four-loop cusp anomalous dimension in N=4 Yang-Mills theory directly from four-gluon amplitudes, avoiding complex integral evaluations and confirming a conjectured value with high precision.

Contribution

The authors introduce a new technique that isolates the cusp anomalous dimension from obstructions in amplitudes, significantly improving computational efficiency and precision at four loops.

Findings

Analytical extraction of two- and three-loop anomalous dimensions.

Numerical confirmation of the four-loop anomalous dimension with high precision.

Strong evidence supporting the conjectured value of the four-loop cusp anomalous dimension.

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 = - (4\zeta^3_2+24\zeta_2\zeta_4+50\zeta_6- 4(1+r)\zeta_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.