Four-loop cusp anomalous dimension from obstructions

Abstract

We introduce a method for extracting the cusp anomalous dimension at $L$ loops from four-gluon amplitudes in $\mathcal{N}=4$ 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 [F. Cachazo, M. Spradlin, and A. Volovich, J. High Energy Phys. 07 (2006) 007]. 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 ${f}^{(4)}=\ensuremath{-}(4{\ensuremath{\zeta}}_{2}^{3}+24{\ensuremath{\zeta}}_{2}{\ensuremath{\zeta}}_{4}+50{\ensuremath{\zeta}}_{6}\ensuremath{-}4(1+r){\ensuremath{\zeta}}_{3}^{2})$ with $r=\ensuremath{-}2$ using integrability and string theory arguments in [N. Beisert, B. Eden, and M. Staudacher, J. Stat. Mech. (2007) P021]. Simultaneously, ${f}^{(4)}$ was computed numerically in [Z. Bern, M. Czakon, L. J. Dixon, D. A. Kosower, and V. A. Smirnov, Phys. Rev. D 75, 085010 (2007)] from the four-loop amplitude obtaining, with best precision at the symmetric point $s=t$, $r=\ensuremath{-}2.028(36)$. Our computation is manifestly $s/t$ independent and improves the precision to $r=\ensuremath{-}2.000\text{ }02(3)$, 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.