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      Scattering amplitudes on the Coulomb branch of N=4 super Yang-Mills

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          Abstract

          We discuss planar scattering amplitudes on the Coulomb branch of N=4 super Yang-Mills. The vacuum expectation values on the Coulomb branch can be used to regulate infrared divergences. We argue that this has a number of conceptual as well as practical advantages over dimensional regularisation.

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          Iteration of Planar Amplitudes in Maximally Supersymmetric Yang-Mills Theory at Three Loops and Beyond

          , , (2005)
          We compute the leading-color (planar) three-loop four-point amplitude of N=4 supersymmetric Yang-Mills theory in 4 - 2 epsilon dimensions, as a Laurent expansion about epsilon = 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/epsilon^2 pole in the four-point amplitude determines the soft, or cusp, anomalous dimension at three loops in N = 4 supersymmetric Yang-Mills 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.
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            The Four-Loop Planar Amplitude and Cusp Anomalous Dimension in Maximally Supersymmetric Yang-Mills Theory

            Vladimir A. Smirnov, Zvi Bern, Michael Czakon … (2006)
            We present an expression for the leading-color (planar) four-loop four-point amplitude of N=4 supersymmetric Yang-Mills theory in 4-2 e dimensions, in terms of eight separate integrals. The expression is based on consistency of unitarity cuts and infrared divergences. We expand the integrals around e=0, and obtain analytic expressions for the poles from 1/e^8 through 1/e^4. We give numerical results for the coefficients of the 1/e^3 and 1/e^2 poles. These results all match the known exponentiated structure of the infrared divergences, at four separate kinematic points. The value of the 1/e^2 coefficient allows us to test a conjecture of Eden and Staudacher for the four-loop cusp (soft) anomalous dimension. We find that the conjecture is incorrect, although our numerical results suggest that a simple modification of the expression, flipping the sign of the term containing zeta_3^2, may yield the correct answer. Our numerical value can be used, in a scheme proposed by Kotikov, Lipatov and Velizhanin, to estimate the two constants in the strong-coupling expansion of the cusp anomalous dimension that are known from string theory. The estimate works to 2.6% and 5% accuracy, providing non-trivial evidence in support of the AdS/CFT correspondence. We also use the known constants in the strong-coupling expansion as additional input to provide approximations to the cusp anomalous dimension which should be accurate to under one percent for all values of the coupling. When the evaluations of the integrals are completed through the finite terms, it will be possible to test the iterative, exponentiated structure of the finite terms in the four-loop four-point amplitude, which was uncovered earlier at two and three loops.
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              Planar Amplitudes in Maximally Supersymmetric Yang-Mills Theory

              Z. Bern, L. Dixon, D. Kosower … (2003)
              The collinear factorization properties of two-loop scattering amplitudes in dimensionally-regulated N=4 super-Yang-Mills theory suggest that, in the planar ('t Hooft) limit, higher-loop contributions can be expressed entirely in terms of one-loop amplitudes. We demonstrate this relation explicitly for the two-loop four-point amplitude and, based on the collinear limits, conjecture an analogous relation for n-point amplitudes. The simplicity of the relation is consistent with intuition based on the AdS/CFT correspondence that the form of the large N_c L-loop amplitudes should be simple enough to allow a resummation to all orders.
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                Author and article information

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                Publication date Created: 17 May 2010
                Article
                DOI: 10.1016/j.nuclphysbps.2010.08.042
                ArXiV ID: 1005.2902
                SO-VID: 434ab8df-532c-4334-82b8-cd260a89f01c
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                http://arxiv.org/licenses/nonexclusive-distrib/1.0/

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                Report No HU-EP-10/22
                Journal reference Nucl.Phys.B, Proc.Suppl.205-206 2010:193-198,2010
                Comments 6 pages, 2 figures; talk given at 'Loops and Legs in Quantum Field Theory' (April 25-30, 2010
                Categories hep-ph hep-th

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