Abstract
We present the details of the analytic calculation of the three-loop angle-dependent cusp anomalous dimension in QCD and its supersymmetric extensions, including the maximally supersymmetric $\mathcal{N}=4$ super Yang-Mills theory. The three-loop result in the latter theory is new and confirms a conjecture made in our previous paper. We study various physical limits of the cusp anomalous dimension and discuss its relation to the quark-antiquark potential including the effects of broken conformal symmetry in QCD. We find that the cusp anomalous dimension viewed as a function of the cusp angle and the new effective coupling given by light-like cusp anomalous dimension reveals a remarkable universality property -- it takes the same form in QCD and its supersymmetric extensions, to three loops at least. We exploit this universality property and make use of the known result for the three-loop quark-antiquark potential to predict the special class of nonplanar corrections to the cusp anomalous dimensions at four loops. Finally, we also discuss in detail the computation of all necessary Wilson line integrals up to three loops using the method of leading singularities and differential equations.
Highlights
Introduction and summaryThe predictive power of QCD as a theory of strong interaction relies on the possibility to predict the scale dependence of various observables in terms of anomalous dimensions as a function of the strong coupling constant and various kinematical invariants
As was shown in [11, 12], the dependence of the scattering amplitude on both infrared divergences (IR) and UV cut-offs is controlled by the cusp anomalous dimension Γcusp(φ, αs), which depends on the Minkowskian recoil angle of the heavy quark, (v1v2) = cosh φM, where φ = iφM
It turned out that choosing such integrals as a basis rendered the physical answer much simpler already at the integrand level, before carrying out the integrations. (This is in part due to the close relationship between certain unitarity cuts and infrared divergences, whose appearance is clearer in the new basis choice.) The understanding of the relationship between Feynman integrals and uniform weight functions was put on a firmer footing in [29], by providing a way of proving the conjecture with the help of differential equations
Summary
The predictive power of QCD as a theory of strong interaction relies on the possibility to predict the scale dependence of various observables in terms of anomalous dimensions as a function of the strong coupling constant and various kinematical invariants. The latter can be readily evaluated numerically [35, 36], analytically continued, or expanded [35,36,37,38,39,40,41] around the above-mentioned interesting physical limits In this way, we reproduce the known result for Γcusp(φ, αs) in the light-like limit [42,43,44,45,46] and provide new insights into on the relation to the quark-antiquark potential [47,48,49,50] in the backtracking Wilson line limit. Appendix B discusses the heavy quark effective theory (HQET) approach to computing the cusped Wilson loop, appendices C and D contain a calculation of certain infinite classes of large nf terms of the cusp anomalous dimension and quark-antiquark potential
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