The Principle of Maximal Transcendentality and the Four-Loop Collinear Anomalous Dimension

TL;DR

This work addresses the four-loop collinear anomalous dimension in planar $\mathcal{N}=4$ SYM and its connections to Regge, soft, and rapidity physics. The authors employ the principle of maximal transcendentality together with an eikonal bypass to map leading transcendental parts of large-$N_c$ QCD results to the planar theory, and then use the planar virtual anomalous dimension from integrability to return to the non-eikonal result. They obtain an analytic expression for the planar four-loop collinear anomalous dimension: $\mathcal{G}_0^{\text{planar N=4}} = -4 \zeta_3 g^4 + \bigl(32 \zeta_5 + \tfrac{80}{3}\zeta_2\zeta_3\bigr) g^6 - \bigl(300 \zeta_7 + 256 \zeta_2\zeta_5 + 384 \zeta_3\zeta_4\bigr) g^8$, which agrees with previous numerical results to roughly $0.2\%$. This analytic result also yields the four-loop Regge trajectory and the threshold soft and rapidity anomalous dimensions, and points to prospects for analytic access to subleading-color contributions once corresponding QCD data become available.

Abstract

We use the principle of maximal transcendentality and the universal nature of subleading infrared poles to extract the analytic value of the four-loop collinear anomalous dimension in planar ${\cal N}=4$ super-Yang-Mills theory from recent QCD results, obtaining $\hat{\cal G}_{0}^{(4)} = - 300 ζ_7 - 256 ζ_2 ζ_5 - 384 ζ_3 ζ_4$. This value agrees with a previous numerical result to within 0.2 percent. It also provides the Regge trajectory, threshold soft anomalous dimension and rapidity anomalous dimension through four loops.