Notes on the scattering amplitude — Wilson loop duality

We consider the duality between the four-dimensional S-matrix of planar maximally supersymmetric Yang-Mills theory and the expectation value of polygonal shaped Wilson loops in the same theory. We extend the duality to amplitudes with arbitrary helicity states by introducing a suitable supersymmetric extension of the Wilson loop. We show that this object is determined by a host of recursion relations, which are valid at tree level and at loop level for a certain “loop integrand” defined within the Lagrangian insertion procedure. These recursion relations reproduce the BCFW ones obeyed by tree-level scattering amplitudes, as well as their extension to loop integrands which appeared recently in the literature, establishing the duality to all orders in perturbation theory. Finally, we propose that a certain set of finite correlation functions can be used to compute all first derivatives of the logarithm of MHV amplitudes.

Related papers:

title: The Bethe-ansatz for N = 4 super Yang-Mills
abstract: We derive the one loop mixing matrix for anomalous dimensions in N=4 Super Yang-Mills. We show that this matrix can be identified with the Hamiltonian of an integrable SO(6) spin chain with vector sites. We then use the Bethe ansatz to find a recipe for computing anomalous dimensions for a wide range of operators. We give exact results for BMN operators with two impurities and results up to and including first order 1/J corrections for BMN operators with many impurities. We then use a result of Reshetikhin's to find the exact one-loop anomalous dimension for an SO(6) singlet in the limit of large bare dimension. We also show that this last anomalous dimension is proportional to the square root of the string level in the weak coupling limit.

title: Transcendentality and crossing
abstract: We discuss possible phase factors for the S-matrix of planar gauge theory, leading to modifications at four-loop order as compared to an earlier proposal. While these result in a four-loop breakdown of perturbative BMN scaling, Kotikov?Lipatov transcendentality in the universal scaling function for large-spin twist operators may be preserved. One particularly natural choice, unique up to one constant, modifies the overall contribution of all terms containing odd-zeta functions in the earlier proposed scaling function based on a trivial phase. Excitingly, we present evidence that this choice is non-perturbatively related to a recently conjectured crossing-symmetric phase factor for perturbative string theory on AdS5 ? S5 once the constant is fixed to a particular value. Our proposal, if true, might therefore resolve the long-standing AdS/CFT discrepancies between gauge and string theory.

title: N=4 Yang-Mills and N=8 Supergravity as Limits of String Theories
abstract: The formulation of supersymmetric string theories in ten dimensions is generalized to incorporate compactified dimensions. Expressions for the one-loop four-particle S -matrix elements of N = 4 Yang-Mills and N = 8 supergravity in four dimensions are obtained by studying the string-theory loop amplitudes in the limit that the radii of the compactified dimensions and the Regge slope parameter simultaneously approach zero. If certain patterns that emerge should persist in the higher orders of perturbation theory, then N = 4 Yang-Mills in four dimensions would be ultraviolet finite to all orders, whereas N = 8 supergravity in four dimensions would have ultraviolet divergences starting at three loops.

title: The Four-Loop Planar Amplitude and Cusp Anomalous Dimension in Maximally Supersymmetric Yang-Mills Theory
abstract: We present an expression for the leading-color (planar) four-loop four-point amplitude of N = 4 supersymmetric Yang-Mills theory in 4-2{epsilon} dimensions, in terms of eight separate integrals. The expression is based on consistency of unitarity cuts and infrared divergences. We expand the integrals around {epsilon} = 0, and obtain analytic expressions for the poles from 1/{epsilon}{sup 8} through 1/{epsilon}{sup 4}. We give numerical results for the coefficients of the 1/{epsilon}{sup 3} and 1/e{sup 2} poles. These results all match the known exponentiated structure of the infrared divergences, at four separate kinematic points. The value of the 1/{epsilon}{sup 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}{sub 3}{sup 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.

title: 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 - 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}{sup 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.

title: On planar gluon amplitudes/Wilson loops duality
abstract: There is growing evidence that on-shell gluon scattering amplitudes in planar N=4 SYM theory are equivalent to Wilson loops evaluated over contours consisting of straight, light-like segments defined by the momenta of the external gluons. This equivalence was first suggested at strong coupling using the AdS/CFT correspondence and has since been verified at weak coupling to one loop in perturbation theory. Here we perform an explicit two-loop calculation of the Wilson loop dual to the four-gluon scattering amplitude and demonstrate that the relation holds beyond one loop. We also propose an anomalous conformal Ward identity which uniquely fixes the form of the finite part (up to an additive constant) of the Wilson loop dual to four- and five-gluon amplitudes, in complete agreement with the BDS conjecture for the multi-gluon MHV amplitudes

title: Gluon scattering amplitudes at strong coupling
abstract: We describe how to compute planar gluon scattering amplitudes at strong coupling in = 4 super Yang Mills by using the gauge/string duality. The computation boils down to finding a certain classical string configuration whose boundary conditions are determined by the gluon momenta. The results are infrared divergent. We introduce the gravity version of dimensional regularization to define finite quantities. The leading and subleading IR divergencies are characterized by two functions of the coupling that we compute at strong coupling. We compute also the full finite form for the four point amplitude and we find agreement with a recent ansatz by Bern, Dixon and Smirnov.

title: Comments on gluon scattering amplitudes via AdS/CFT
abstract: In this article we consider n gluon color ordered, planar amplitudes in = 4 super Yang Mills at strong 't Hooft coupling. These amplitudes are approximated by classical surfaces in AdS5 space. We compute the value of the amplitude for a particular kinematic configuration for a large number of gluons and find that the result disagrees with a recent guess for the exact value of the amplitude. Our results are still compatible with a possible relation between amplitudes and Wilson loops. In addition, we also give a prescription for computing processes involving local operators and asymptotic states with a fixed number of gluons. As a byproduct, we also obtain a string theory prescription for computing the dual of the ordinary Wilson loop, TrPeiA, with no couplings to the scalars. We also evaluate the quark-antiquark potential at two loops.

title: Fermionic T-Duality, Dual Superconformal Symmetry, and the Amplitude/Wilson Loop Connection
abstract: We show that tree level superstring theories on certain supersymmetric backgrounds admit a symmetry which we call ``fermionic T-duality''. This is a non-local redefinition of the fermionic worldsheet fields similar to the redefinition we perform on bosonic variables when we do an ordinary T-duality. This duality maps a supersymmetric background to another supersymmetric background with different RR fields and a different dilaton. We show that a certain combination of bosonic and fermionic T-dualities maps the full superstring theory on AdS5 × S5 back to itself in such a way that gluon scattering amplitudes in the original theory map to something very close to Wilson loops in the dual theory. This duality maps the ``dual superconformal symmetry'' of the original theory to the ordinary superconformal symmetry of the dual model. This explains the dual superconformal invariance of planar scattering amplitudes of N = 4 super Yang Mills and also sheds some light on the connection between amplitudes and Wilson loops. In the appendix, we propose a simple prescription for open superstring MHV tree amplitudes in a flat background.

title: Scattering amplitudes, Wilson loops and the string/gauge theory correspondence
abstract: We review, in a self-contained and pedagogical manner, recent developments and techniques for the evaluation of scattering amplitudes of planar N = 4 SYM theory at both weak and strong coupling. Special emphasis is placed on the newly discovered connection between a special class of amplitudes and the expectation values of particular cusped light-like Wilson loops.

title: Conformal properties of four-gluon planar amplitudes and Wilson loops
abstract: We present further evidence for a dual conformal symmetry in the four-gluon planar scattering amplitude in N=4 SYM. We show that all the momentum integrals appearing in the perturbative on-shell calculations up to four loops are dual to true conformal integrals, well defined off shell. Assuming that the complete off-shell amplitude has this dual conformal symmetry and using the basic properties of factorization of infrared divergences, we derive the special form of the finite remainder previously found at weak coupling and recently reproduced at strong coupling by AdS/CFT. We show that the same finite term appears in a weak coupling calculation of a Wilson loop whose contour consists of four light-like segments associated with the gluon momenta. We also demonstrate that, due to the special form of the finite remainder, the asymptotic Regge limit of the four-gluon amplitude coincides with the exact expression evaluated for arbitrary values of the Mandelstam variables.

title: The hexagon Wilson loop and the BDS ansatz for the six-gluon amplitude
abstract: As a test of the gluon scattering amplitude/Wilson loop duality, we evaluate the hexagonal light-like Wilson loop at two loops in N = 4 super-Yang–Mills theory. We compare its finite part to the Bern–Dixon–Smirnov (BDS) conjecture for the finite part of the six-gluon amplitude. We find that the two expressions have the same behavior in the collinear limit, but they differ by a non-trivial function of the three (dual) conformally invariant variables. This implies that either the BDS conjecture or the gluon amplitude/Wilson loop duality fails for the six-gluon amplitude, starting from two loops. Our results are in qualitative agreement with the analysis of Alday and Maldacena of scattering amplitudes with infinitely many external gluons.

title: Two-loop six-gluon maximally helicity violating amplitude in maximally supersymmetric Yang-Mills theory
abstract: We give a representation of the parity-even part of the planar two-loop six-gluon maximally helicity violating (MHV) amplitude of N=4 super-Yang-Mills theory, in terms of loop-momentum integrals with simple dual conformal properties. We evaluate the integrals numerically in order to test directly the Anastasiou-Bern-Dixon-Kosower/Bern-Dixon-Smirnov all-loop ansatz for planar MHV amplitudes. We find that the ansatz requires an additive remainder function, in accord with previous indications from strong-coupling and Regge limits. The planar six-gluon amplitude can also be compared with the hexagonal Wilson loop computed by Drummond, Henn, Korchemsky, and Sokatchev in . After accounting for differing singularities and other constants independent of the kinematics, we find that the Wilson loop and MHV-amplitude remainders are identical, to within our numerical precision. This result provides nontrivial confirmation of a proposed n-point equivalence between Wilson loops and planar MHV amplitudes, and suggests that an additional mechanism besides dual conformal symmetry fixes their form at six points and beyond.

title: Hexagon Wilson loop = six-gluon MHV amplitude
abstract: We compare the two-loop corrections to the finite part of the light-like hexagon Wilson loop with the recent numerical results for the finite part of the MHV six-gluon amplitude in N=4 SYM theory by Bern, Dixon, Kosower, Roiban, Spradlin, Vergu and Volovich () and demonstrate that they coincide within the error bars and, at the same time, they differ from the BDS ansatz by a non-trivial function of (dual) conformal kinematical invariants. This provides strong evidence that the Wilson loop/scattering amplitude duality holds in planar N=4 SYM theory to all loops for an arbitrary number of external particles.

title: Dual superconformal symmetry of scattering amplitudes in N=4 super-Yang-Mills theory
abstract: We argue that all scattering amplitudes in the maximally supersymmetric N=4 super-Yang–Mills theory possess a new, dual superconformal symmetry which extends the previously discovered dual conformal symmetry of MHV amplitudes. To reveal this property we formulate the scattering amplitudes as functions on the appropriate dual superspace. Rewritten in this form, all tree-level MHV and next-to-MHV amplitudes exhibit manifest dual superconformal symmetry. We propose a new, compact and Lorentz covariant formula for the tree-level NMHV amplitudes for arbitrary numbers and types of external particles. The dual superconformal symmetry is broken at loop level by infrared divergences. However, we provide evidence that the dual conformal anomaly of the MHV and NMHV superamplitudes is the same and, therefore, their ratio is dual conformally invariant. We show this explicitly for the six-particle amplitudes at one loop. We conjecture that these properties hold for all, MHV and non-MHV, superamplitudes in N=4 SYM both at weak and at strong coupling

title: A Note on dual superconformal symmetry of the N=4 super Yang-Mills S-matrix
abstract: 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.

title: The Grassmannian origin of dual superconformal invariance
abstract: A dual formulation of the S Matrix for $$ \mathcal {N} $$ = 4 SYM has recently been presented, where all leading singularities of n-particle Nk−2MHV amplitudes are given as an integral over the Grassmannian G(k, n), with cyclic symmetry, parity and superconformal invariance manifest. In this short note we show that the dual superconformal invariance of this object is also manifest. The geometry naturally suggests a partial integration and simple change of variable to an integral over G(k − 2, n). This change of variable precisely corresponds to the mapping between usual momentum variables and the “momentum twistors” introduced by Hodges, and yields an elementary derivation of the momentumtwistor space formula very recently presented by Mason and Skinner, which is manifestly dual superconformal invariant. Thus the G(k, n) Grassmannian formulation allows a direct understanding of all the important symmetries of $$ \mathcal {N} $$ = 4 SYM scattering amplitudes.

title: Dual superconformal invariance, momentum twistors and Grassmannians
abstract: Dual superconformal invariance has recently emerged as a hidden symmetry of planar scattering amplitudes in = 4 super Yang-Mills theory. This symmetry can be made manifest by expressing amplitudes in terms of `momentum twistors', as opposed to the usual twistors that make the ordinary superconformal properties manifest. The relation between momentum twistors and on-shell momenta is algebraic, so the translation procedure does not rely on any choice of space-time signature. We show that tree amplitudes and box coefficients are succinctly generated by integration of holomorphic δ-functions in momentum twistors over cycles in a Grassmannian. This is analogous to, although distinct from, recent results obtained by Arkani-Hamed et al. in ordinary twistor space. We also make contact with Hodges' polyhedral representation of NMHV amplitudes in momentum twistor space.

title: Eliminating spurious poles from gauge-theoretic amplitudes
abstract: A bstractThis note addresses the problem of spurious poles in gauge-theoretic scattering amplitudes. New twistor coordinates for the momenta are introduced, based on the concept of dual conformal invariance. The cancellation of spurious poles for a class of NMHV amplitudes is greatly simplified in these coordinates. The poles are eliminated altogether by defining a new type of twistor integral, dual to twistor diagrams as previously studied, and considerably simpler. The geometric features indicate a supersymmetric extension of the formalism at least to all NMHV amplitudes, allowing the dihedral symmetry of the super-amplitude to be made manifest. More generally, the definition of ‘momentum-twistor’ coordinates suggests a powerful new approach to the study of scattering amplitudes.

title: Symmetries and analytic properties of scattering amplitudes in N=4 SYM theory
abstract: In addition to the superconformal symmetry of the underlying Lagrangian, the scattering amplitudes in planar N = 4 super-Yang-Mills theory exhibit a new, dual superconformal symmetry. We address the question of how powerful these symmetries are to completely determine the scattering amplitudes. We use the example of the NMHV superamplitudes to show that the combined action of conventional and dual superconformal symmetries is not sufficient to fix all the freedom in the tree-level amplitudes. We argue that the additional information needed comes from the study of the analytic properties of the amplitudes. The requirement of absence of spurious singularities, together with the correct multi-particle singular behavior, determines the unique linear combination of superinvariants corresponding to the n−particle NMHV superamplitude. The same result can be obtained recursively, by relating the n− and (n − 1)−particle amplitudes in the singular collinear limit. We also formulate constraints on the loop corrections to the superamplitudes, following from the analytic behavior in the above limits. We then show that, at one-loop level, the holomorphic anomaly of the tree amplitudes leads to the breakdown of dual Poincaré supersymmetry (equivalent to ordinary special conformal supersymmetry) of the ratio of the NMHV and MHV superamplitudes, but this anomaly does not affect dual conformal symmetry. On leave from Laboratoire de Physique Théorique, Université de Paris XI, 91405 Orsay Cédex, France Unité de Recherche Associée au CNRS URA 2306 Laboratoire d’Annecy-le-Vieux de Physique Théorique, UMR 5108

title: More on the duality correlators/amplitudes
abstract: We continue the study of n-point correlation functions of half-BPS protected operators in N = 4 super-Yang–Mills theory, in the limit where the positions of the adjacent operators become light-like separated. We compute the l-loop corrections by making l Lagrangian insertions. We argue that there exists a simple relation between the ( n + l ) -point Born-level correlator with l Lagrangian insertions and the integrand of the n-particle l-loop MHV scattering amplitude, as obtained by the recent momentum twistor construction of Arkani-Hamed et al. We present several examples of this new duality, at one and two loops.

title: From correlation functions to scattering amplitudes
abstract: A bstractWe study the correlation functions of half-BPS protected operators in $ \mathcal{N} = {4} $ super-Yang-Mills theory, in the limit where the positions of adjacent operators become light-like separated. We compute the loop corrections by means of Lagrangian insertions. The divergences resulting from the light-cone limit are regularized by changing the dimension of the integration measure over the insertion points. Switching from coordinates to dual momenta, we show that the logarithm of the correlation function is identical with twice the logarithm of the matching MHV gluon scattering amplitude. We present a number of examples of this new relation, at one and two loops.

title: The all-loop integrand for scattering amplitudes in planar $ \mathcal{N} = 4 $ SYM
abstract: We give an explicit recursive formula for the all ℓ-loop integrand for scattering amplitudes in $ \mathcal{N} = 4 $ SYM in the planar limit, manifesting the full Yangian symmetry of the theory. This generalizes the BCFW recursion relation for tree amplitudes to all loop orders, and extends the Grassmannian duality for leading singularities to the full amplitude. It also provides a new physical picture for the meaning of loops, associated with canonical operations for removing particles in a Yangian-invariant way. Loop amplitudes arise from the “entangled” removal of pairs of particles, and are naturally presented as an integral over lines in momentum-twistor space. As expected from manifest Yangian invariance, the integrand is given as a sum over non-local terms, rather than the familiar decomposition in terms of local scalar integrals with rational coefficients. Knowing the integrands explicitly, it is straightforward to express them in local forms if desired; this turns out to be done most naturally using a novel basis of chiral, tensor integrals written in momentum-twistor space, each of which has unit leading singularities. As simple illustrative examples, we present a number of new multi-loop results written in local form, including the 6- and 7-point 2-loop NMHV amplitudes. Very concise expressions are presented for all 2-loop MHV amplitudes, as well as the 5-point 3-loop MHV amplitude. The structure of the loop integrand strongly suggests that the integrals yielding the physical amplitudes are “simple”, and determined by IR-anomalies. We briefly comment on extending these ideas to more general planar theories.

title: Loops and trees
abstract: We investigate relations between loop and tree amplitudes in quantum field theory that involve putting on-shell some loop propagators. This generalizes the so-called Feynman tree theorem which is satisfied at 1-loop. Exploiting retarded boundary conditions, we give a generalization to ℓ-loop expressing the loops as integrals over the on-shell phase space of exactly ℓ particles. We argue that the corresponding integrand for ℓ > 2 does not involve the forward limit of any physical tree amplitude, except in planar gauge theories. In that case we explicitly construct the relevant physical amplitude. Beyond the planar limit, abandoning direct integral representations, we propose that loops continue to be determined implicitly by the forward limit of physical connected trees, and we formulate a precise conjecture along this line. Finally, we set up technology to compute forward amplitudes in supersymmetric theories, in which specific simplifications occur.

title: The complete planar S-matrix of $ \mathcal{N} = 4 $ SYM as a Wilson loop in twistor space
abstract: We show that the complete planar S-matrix of $ \mathcal{N} = 4 $ super Yang-Mills — including all NkMHV partial amplitudes to all loops — is equivalent to the correlation function of a supersymmetric Wilson loop in twistor space. Remarkably, the entire classical S-matrix arises from evaluating the correlation function in the self-dual sector, while the expansion of the correlation function in powers of the Yang-Mills coupling constant provides the loop expansion of the amplitudes. We support our proposal with explicit computations of the n particle NMHV and N2MHV trees, the integrands of the 1-loop MHV and NMHV amplitudes, and the n particle 2-loop MHV amplitude. These calculations are performed using the twistor action in axial gauge. In this gauge, the Feynman diagrams of the correlation function are the planar duals of the usual MHV diagrams for the scattering amplitude. The results are presented in the form of a sum of products of dual superconformal invariants in (momentum) twistor space, and agree with the expressions derived in the companion paper  directly from the MHV rules. The twistor space Wilson loop is a natural supersymmetric generalization of the standard Wilson loop used to compute MHV amplitudes. We show how the Penrose-Ward transform can be used to determine a corresponding supersymmetrization on space-time and give the corresponding superconnection in the abelian case.

title: MHV diagrams in momentum twistor space
abstract: We show that there are remarkable simplifications when the MHV diagram formalism for $ \mathcal{N} = 4 $ super Yang-Mills is reformulated in momentum twistor space. The vertices are replaced by unity while each propagator becomes a dual superconformal ‘R-invariant’ whose arguments may be read off from the diagram, and include an arbitrarily chosen reference twistor. The momentum twistor MHV rules generate a formula for the full, all-loop planar integrand for the super Yang-Mills S-matrix that is manifestly dual superconformally invariant up to the choice of a reference twistor. We give a general proof of this reformulation and illustrate its use by computing the momentum twistor NMHV and N2MHV tree amplitudes and the integrands of the MHV and NMHV 1-loop and the MHV 2-loop planar amplitudes.

title: Self-dual sector of QCD amplitudes.
abstract: We provide an action for self-dual Yang-Mills theory which is a simple truncation of the usual Yang-Mills action. Only vertices that violate helicity conservation maximally are included. One-loop amplitudes in the self-dual theory then follow as a subset of the Yang-Mills ones. In light-cone gauges this action is almost identical to previously proposed actions, but in this formulation the vanishing of all higher-loop amplitudes is obvious; the explicit perturbative $S$-matrix is known. Similar results apply to gravity.

title: Dual conformal symmetry of 1-loop NMHV amplitudes in N = 4 SYM theory
abstract: We prove that 1-loop n-point NMHV superamplitudes in N = 4 SYM theory are dual conformal covariant for all numbers n of external particles (after regularization and subtraction of IR divergences). This property was previously established for n ≤ 9 in . We derive an explicit representation of these superamplitudes in terms of dual conformal cross-ratios. We also show that all the 1-loop ‘box coefficients’ obtained from maximal cuts of NMHV n-point functions are covariant under dual conformal transformations. ar X iv :0 90 5. 43 79 v1 [ he pth ] 2 7 M ay 2 00 9

title: Superconformal invariants for scattering amplitudes in N=4 SYM theory
abstract: Recent studies of scattering amplitudes in planar N = 4 SYM theory revealed the existence of a hidden dual superconformal symmetry. Together with the conventional superconformal symmetry it gives rise to powerful restrictions on the planar scattering amplitudes to all loops. We study the general form of the invariants of both symmetries. We first construct an integral representation for the most general dual superconformal invariants and show that it allows a considerable freedom in the choice of the integration measure. We then perform a half-Fourier transform to twistor space, where conventional conformal symmetry is realized locally, derive the resulting conformal Ward identity for the integration measure and show that it admits a unique solution. Thus, the combination of dual and conventional superconformal symmetries, together with invariance under helicity rescalings, completely fixes the form of the invariants. The expressions obtained generalize the known tree and one-loop superconformal invariants and coincide with the recently proposed coefficients of the leading singularities of the scattering amplitudes as contour integrals over Grassmannians.

title: From correlation functions to Wilson loops
abstract: We start with an n−point correlation function in a conformal gauge theory. We show that a special limit produces a polygonal Wilson loop with n sides. The limit takes the n points towards the vertices of a null polygonal Wilson loop such that successive distances $ x_{i,i + 1}^2 \to 0 $. This produces a fast moving particle that generates a “frame” for the Wilson loop. We explainin detail how the limit is approached, including some subtle effects from the propagation of a fast moving particle in the full interacting theory. We perform perturbative checks by doing explicit computations in $ \mathcal{N} = 4 $ super-Yang-Mills theory.

title: Direct proof of the tree-level scattering amplitude recursion relation in Yang-mills theory.
abstract: Recently, by using the known structure of one-loop scattering amplitudes for gluons in Yang-Mills theory, a recursion relation for tree-level scattering amplitudes has been deduced. Here, we give a short and direct proof of this recursion relation based on properties of tree-level amplitudes only.

title: Partial non-renormalisation of the stress-tensor four-point function in /N=4 SYM and AdS/CFT
abstract: We show that, although the correlator of four stress-tensor multiplets in N=4 SYM is known to have radiative corrections, certain linear combinations of its components are protected from perturbative renormalisation and remain at their free-field values. This result is valid for weak as well as for strong coupling and for any gauge group. Our argument uses Intriligator's insertion formula, and includes a proof that the possible contact term contributions cannot change the form of the amplitudes. Combining this new non-renormalisation theorem with Maldacena's conjecture allows us to make a prediction for the structure of the corresponding correlator in AdS supergravity. This is verified by first considerably simplifying the strong coupling expression obtained by recent supergravity calculations, and then showing that it does indeed exhibit the expected structure.

title: A non-renormalization theorem for conformal anomalies
abstract: We provide a non-renormalization theorem for the coefficients of the conformal anomaly associated with operators with vanishing anomalous dimensions. Such operators include conserved currents and chiral operators in superconformal field theories. We illustrate the theorem by computing the conformal anomaly of 2-point functions both by a computation in the conformal field theory and via the AdS/CFT correspondence. Our results imply that 2- and 3-point functions of chiral primary operators in N=4 SU(N) SYM will not renormalize provided that a “generalized Adler–Bardeen theorem” holds. We further show that recent arguments connecting the non-renormalizability of the above-mentioned correlation functions to a bonus U(1)Y symmetry are incomplete due to possible U(1)Y violating contact terms. The tree level contribution to the contact terms may be set to zero by considering appropriately normalized operators. Non-renormalizability of the above-mentioned correlation functions, however, will follow only if these contact terms saturate by free fields.

title: Classical polylogarithms for amplitudes and Wilson loops.
abstract: We present a compact analytic formula for the two-loop six-particle maximally helicity violating remainder function (equivalently, the two-loop lightlike hexagon Wilson loop) in N=4 supersymmetric Yang-Mills theory in terms of the classical polylogarithm functions Lik with cross ratios of momentum twistor invariants as their arguments. In deriving our formula we rely on results from the theory of motives.

title: Scattering into the fifth dimension of $$ \mathcal{N} $$ = 4 super Yang-Mills
abstract: We study an alternative to dimensional regularisation of planar scattering amplitudes in $$ \mathcal{N} $$ = 4 super Yang-Mills theory by going to the Coulomb phase of the theory. The infrared divergences are regulated by masses obtained from a Higgs mechanism, allowing us to work in four dimensions. The corresponding string theory set-up suggests that the amplitudes have an exact dual conformal symmetry. The latter acts on the kinematical variables of the amplitudes as well as on the Higgs masses in an effectively five dimensional space. We confirm this expectation by an explicit calculation in the gauge theory. A consequence of this exact dual conformal symmetry is a significantly reduced set of scalar basis integrals that are allowed to appear in an amplitude. For example, triangle sub-graphs are ruled out. We argue that the study of exponentiation of amplitudes is simpler in the Higgsed theory because evanescent terms in the mass regulator can be consistently dropped. We illustrate this by showing the exponentiation of a four-point amplitude to two loops. Finally, we also analytically compute the small mass expansion of a two-loop master integral with an internal mass.

title: More loops and legs in Higgs-regulated $$ \mathcal{N} = 4 $$ SYM amplitudes
abstract: We extend the analysis of Higgs-regulated planar amplitudes of $$ \mathcal{N} = 4 $$ supersymmetric Yang-Mills theory to four loops for the four-gluon amplitude and to two loops for the five-gluon amplitude. Our calculations are consistent with a proposed all-loop ansatz for planar MHV n-gluon amplitudes that is the analog of the BDS ansatz in dimensional regularization. In all cases considered, we have verified that the IR-finite parts of the logarithm of the amplitudes have the same dependence on kinematic variables as the corresponding functions in dimensionally-regulated amplitudes (up to overall additive constants, which we determine).We also study various Regge limits of $$ \mathcal{N} = 4 $$ SYM planar n-gluon amplitudes. Euclidean Regge limits of Higgs-regulated n ≥ 4 amplitudes yield results similar in form to those found using dimensional regularization, but with different expressions for the gluon trajectory and Regge vertices resulting from the different regulator scheme. We also show that the Regge limit of the four-gluon amplitude is dominated at next-to-leading-log order by vertical ladder diagrams together with the class of vertical ladder diagrams with a single H-shaped insertion.