Bill Spence

One-loop gauge theory amplitudes in N=4 super Yang-Mills from MHV vertices

Abstract We propose a new, twistor string theory inspired formalism to calculate loop amplitudes in N = 4 super-Yang–Mills theory. In this approach, maximal helicity violating (MHV) tree amplitudes of N = 4 super-Yang–Mills are used as vertices, using an off-shell prescription introduced by Cachazo, Svrcek and Witten, and combined into effective diagrams that incorporate large numbers of conventional Feynman diagrams. As an example, we apply this formalism to the particular class of supersymmetric MHV one-loop scattering amplitudes with an arbitrary number of external legs in N = 4 super-Yang–Mills. Remarkably, our approach naturally leads to a representation of the amplitudes as dispersion integrals, which we evaluate exactly. This yields a new, simplified form for the MHV amplitudes, which is equivalent to the expressions obtained previously by Bern, Dixon, Dunbar and Kosower using the cut-constructibility approach.

Non-supersymmetric loop amplitudes and MHV vertices

Abstract We show how the MHV diagram description of Yang–Mills theories can be used to study non-supersymmetric loop amplitudes. In particular, we derive a compact expression for the cut-constructible part of the general one-loop MHV multi-gluon scattering amplitude in pure Yang–Mills theory. We show that in special cases this expression reduces to known amplitudes—the amplitude with adjacent negative-helicity gluons, and the five gluon non-adjacent amplitude. Finally, we briefly discuss the twistor space interpretation of our result.

The gauged nonlinear sigma model with Wess-Zumino term

Abstract The gauging of the isometry group of a general two-dimensional nonlinear sigma model with Wess-Zumino term is investigated. In general, there is a topological obstruction to gauging, but this is absent for models which are invariant under a Kac-Moody symmetry. When there is no topological obstruction, any “anomaly-free” subgroup of the isometry group can be gauged and we give a simple three-dimensional form of the gauged action. We show how the gauged Wess-Zumino-Novikov-Witten models arise as a particular case. Fermionization is discussed and supersymmetric extensions presented.

A twistor approach to one-loop amplitudes in N=1 supersymmetric Yang–Mills theory

Abstract We extend the twistor string theory inspired formalism introduced in hep-th/0407214 for calculating loop amplitudes in N = 4 super-Yang–Mills theory to the case of N = 1 (and N = 2 ) super-Yang–Mills. Our approach yields a novel representation of the gauge theory amplitudes as dispersion integrals, which are surprisingly simple to evaluate. As an application we calculate one-loop maximally helicity violating (MHV) scattering amplitudes with an arbitrary number of external legs. The result we obtain agrees precisely with the expressions for the N = 1 MHV amplitudes derived previously by Bern, Dixon, Dunbar and Kosower using the cut-constructibility approach.

Two-loop polygon Wilson loops in N = 4 SYM

We compute for the first time the two-loop corrections to arbitrary n-gon lightlike Wilson loops in = 4 supersymmetric Yang-Mills theory, using efficient numerical methods. The calculation is motivated by the remarkable agreement between the finite part of planar six-point MHV amplitudes and hexagon Wilson loops which has been observed at two loops. At n = 6 we confirm that the ABDK/BDS ansatz must be corrected by adding a remainder function, which depends only on conformally invariant ratios of kinematic variables. We numerically compute remainder functions for n = 7,8 and verify dual conformal invariance. Furthermore, we study simple and multiple collinear limits of the Wilson loop remainder functions and demonstrate that they have precisely the form required by the collinear factorisation of the corresponding two-loop n-point amplitudes. The number of distinct diagram topologies contributing to the n-gon Wilson loops does not increase with n, and there is a fixed number of ``master integrals, which we have computed. Thus we have essentially computed general polygon Wilson loops, and if the correspondence with amplitudes continues to hold, all planar n-point two-loop MHV amplitudes in the = 4 theory.

Loop amplitudes in pure Yang-Mills from generalised unitarity

We show how generalised unitarity cuts in D = 4?2 dimensions can be used to calculate efficiently complete one-loop scattering amplitudes in non-supersymmetric Yang-Mills theory. This approach naturally generates the rational terms in the amplitudes, as well as the cut-constructible parts. We test the validity of our method by re-deriving the one-loop ++++, -+++, --++, -+-+ and +++++ gluon scattering amplitudes using generalised quadruple cuts and triple cuts in D dimensions.

From trees to loops and back

We argue that generic one-loop scattering amplitudes in supersymmetric Yang-Mills theories can be computed equivalently with MHV diagrams or with Feynman diagrams. We first present a general proof of the covariance of one-loop non-MHV amplitudes obtained from MHV diagrams. This proof relies only on the local character in Minkowski space of MHV vertices and on an application of the Feynman Tree Theorem. We then show that the discontinuities of one-loop scattering amplitudes computed with MHV diagrams are precisely the same as those computed with standard methods. Furthermore, we analyse collinear limits and soft limits of generic non-MHV amplitudes in supersymmetric Yang-Mills theories with one-loop MHV diagrams. In particular, we find a simple explicit derivation of the universal one-loop splitting functions in supersymmetric Yang-Mills theories to all orders in the dimensional regularisation parameter, which is in complete agreement with known results. Finally, we present concrete and illustrative applications of Feynmans Tree Theorem to one-loop MHV diagrams as well as to one-loop Feynman diagrams.

Form Factors in N=4 Super Yang-Mills and Periodic Wilson Loops

We calculate form factors of half-BPS operators in

Higher-dimensional analogues of Donaldson-Witten theory

\mathcal{N} = 4

The Geometry of the gauged sigma model with Wess-Zumino term

super Yang-Mills theory at tree level and one loop using novel applications of recursion relations and unitarity. In particular, we determine the expression of the one-loop form factors with two scalars and an arbitrary number of positive-helicity gluons. These quantities resemble closely the MHV scattering amplitudes, including holomorphicity of the tree-level form factor, and the expansion in terms of two-mass easy box functions of the one-loop result. Next, we compare our result for these form factors to the calculation of a particular periodic Wilson loop at one loop, finding agreement. This suggests a novel duality relating form factors to periodic Wilson loops.