Andreas Brandhuber

MHV amplitudes in N=4 super Yang-Mills and Wilson loops

Abstract It is a remarkable fact that MHV amplitudes in maximally supersymmetric Yang–Mills theory at arbitrary loop order can be written as the product of the tree amplitude with the same helicity configuration and a universal, helicity-blind function of the kinematic invariants. In this note we show how for one-loop MHV amplitudes with an arbitrary number of external legs this universal function can be derived using Wilson loops. Our result is in precise agreement with the known expression for the infinite sequence of MHV amplitudes in N = 4 super-Yang–Mills. In the four-point case, we are able to reproduce the expression of the amplitude to all orders in the dimensional regularisation parameter ϵ. This prescription disentangles cleanly infrared divergences and finite terms, and leads to an intriguing one-to-one mapping between certain Wilson loop diagrams and (finite) two-mass easy box functions.

Note on dual superconformal symmetry of the N=4 super Yang-Mills S matrix

We present a supersymmetric recursion relation for tree-level scattering amplitudes in N=4 super Yang-Mills. Using this recursion relation, we prove that the tree-level S matrix of the maximally supersymmetric theory is covariant under dual superconformal transformations. We further analyze the consequences that the transformation properties of the trees under this symmetry have on those of the loops. In particular, we show that the coefficients of the expansion of generic one-loop amplitudes in a basis of pseudoconformally invariant scalar box functions transform covariantly under dual superconformal symmetry, and in exactly the same way as the corresponding tree-level amplitudes.

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.

Wilson Loops in the Large N Limit at Finite Temperature

Using a proposal of Maldacena we compute in the framework of the supergravity description of N coincident D3 branes the energy of a quark anti-quark pair in the large N limit of U(N) N=4 SYM in four dimensions at finite temperature.

Wilson loops, confinement, and phase transitions in large N gauge theories from supergravity

We use the recently proposed supergravity approach to large N gauge theories to calculate ordinary and spatial Wilson loops of gauge theories in various dimensions. In this framework we observe an area law for spatial Wilson loops in four and five dimensional supersymmetric Yang-Mills at finite temperature. This can be interpreted as the area law of ordinary Wilson loops in three and four dimensional non-supersymmetric gauge theories at zero temperature which indicates confinement in these theories. Furthermore, we show that super Yang Mills theories with 16 supersymmetries at finite temperature do not admit phase transitions between the weakly coupled super Yang Mills and supergravity regimes. This result is derived by analyzing the entropy and specific heat of those systems as well as by computing ordinary Wilson loops at finite temperature. The calculation of the entropy was carried out in all different regimes and indicates that there is no first order phase transition in these systems. For the same theories at zero temperature we also compute the dependence of the quark anti-quark potential on the separating distance.

Gauge theory at large N and new G_2 holonomy metrics

We find a one-parameter family of new G_2 holonomy metrics and demonstrate that it can be extended to a two-parameter family. These metrics play an important role as the supergravity dual of the large N limit of four-dimensional supersymmetric Yang–Mills. We show that these G_2 holonomy metrics describe the M theory lift of the supergravity solution describing a collection of D6-branes wrapping the supersymmetric three-cycle of the deformed conifold geometry for any value of the string coupling constant.

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.

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.