Iteration of planar amplitudes in maximally supersymmetric Yang-Mills theory at three loops and beyond

Abstract

We compute the leading-color (planar) three-loop four-point amplitude of N = 4 supersymmetric Yang-Mills theory in 4 − 2ǫ dimensions, as a Laurent expansion about ǫ = 0 including the finite terms. The amplitude was constructed previously via the unitarity method, in terms of two Feynman loop integrals, one of which has been evaluated already. Here we use the Mellin-Barnes integration technique to evaluate the Laurent expansion of the second integral. Strikingly, the amplitude is expressible, through the finite terms, in terms of the corresponding one-and two-loop amplitudes, which provides strong evidence for a previous conjecture that higher-loop planar N = 4 amplitudes have an iterative structure. The infrared singularities of the amplitude agree with the predictions of Sterman and Tejeda-Yeomans based on resummation. Based on the four-point result and the exponentiation of infrared singularities, we give an exponentiated ansatz for the maximally helicity-violating n-point amplitudes to all loop orders. The 1/ǫ 2 pole in the four-point amplitude determines the soft, or cusp, anomalous dimension at three loops in N = 4 supersymmetric YangMills theory. The result confirms a prediction by Kotikov, Lipatov, Onishchenko and Velizhanin, which utilizes the leading-twist anomalous dimensions in QCD computed by Moch, Vermaseren and Vogt. Following similar logic, we are able to predict a term in the three-loop quark and gluon form factors in QCD.2

Amplitudes in the β-deformed conformal Yang-Mills

“…It also has a dual string theory description [19,20] provided by the AdS/CFT correspondence. From the perspective of the on-shell amplitudes it is not clear at present which of these features (if any) are responsible for the iterative structure uncovered in [1]. This makes it particularly interesting to investigate the amplitudes in theories related to the N = 4 SYM but with lower degree of symmetry.…”

“…The proposal made in [1] gives an exponentiated ansatz which predicts the finite part of n-point planar MHV amplitudes in the superconformal N = 4 Yang-Mills to all orders…”

Refining threshold resummations

“…At the level of amplitudes, a generalization of the work of [6] to multiparton configurations has been provided in [9], proving an earlier statement [10] on the structure of single poles in QCD amplitudes at two loops. This was in turn an important ingredient in the formulation of a striking conjecture on the all-order structure of multileg amplitudes in N = 4 supersymmetric Yang-Mills theory, for which strong evidence was provided in [11,12]. A two-loop analysis of the soft functions appearing in the exponentiation of multiparton amplitudes was very recently started along these lines in [13].…”

Star product and the general Leigh–Strassler deformation

“…Organizing the Feynman diagrams by decomposed momentum and helicity, instead of momentum and polarized spin, has shown to dramatically reduce their complexity. These MHV diagrams share an iterative structure for computing higher loops [8].…”

“…In [8] it was found that maximally helicity violating (MHV) planar amplitudes in N = 4 SYM have an iterative structure for all n-point amplitudes. These results were then transferred to the β-deformed theory in [7] by placing the deformation into the so-called star product.…”