Four-Loop Rapidity Anomalous Dimension and Event Shapes to Fourth Logarithmic Order

We obtain the quark and gluon rapidity anomalous dimension to fourth order in QCD. We calculate the ${\mathrm{N}}^3\mathrm{LO}$ rapidity anomalous dimensions to higher order in the dimensional regulator and make use of the soft and rapidity anomalous dimension correspondence in conjunction with the recent determination of the ${\mathrm{N}}^4\mathrm{LO}$ threshold anomalous dimensions to achieve our result. We show that the results for the quark and gluon rapidity anomalous dimensions at four loops are related by generalized Casimir scaling. Using the ${\mathrm{N}}^4\mathrm{LO}$ rapidity anomalous dimension, we perform the resummation of the energy-energy correlation in the back-to-back limit at ${\mathrm{N}}^4\mathrm{LL}$, achieving for the first time the resummation of an event shape at this logarithmic order. We present numerical results and observe a reduction of perturbative uncertainties on the resummed cross section to below 1%.

Figure 1

Boundary term of the rapidity quark anomalous dimension as a function of $b_T$ through four loops. The $b_T$ dependence enters only through the coupling constant. The diverging behavior at large $b_T$ is due to approaching the Landau pole. For recent work on extracting the anomalous dimension nonperturbatively at large $b_T$, see Refs. [102, 103, 104, 105, 106, 107, 108, 109, 110, 111, 112].

Figure 2

Resummed result for the EEC in the back-to-back region up to ${\mathrm{N}}^4\mathrm{LL}$ accuracy. Uncertainty bands reflect the residual perturbative uncertainty and are obtained with a 15-point scale variation of the resummation scales. See text for details.

Figure 3

Comparison of the central value for the EEC distribution between the resummed result computed with different values of the five-loop cusp anomalous dimension.