We obtain concise analytic formulae for Wilson loops computed on special n-point polygonal contours through two-loops in weakly coupled N=4 supersymmetric gauge theory. The contours we consider can be embedded into a (1+1)-dimensional subspace of the 4-dimensional gauge theory, corresponding to the boundary of the AdS_3 on the string theory side. Our analytic results hold for any number of edges, thus generalising to arbitrary n the recently derived expressions for 2-dimensional octagons. These polygonal Wilson loops have been conjectured to be equivalent to MHV scattering amplitudes in planar N=4 SYM.
This paper has been withdrawn by the author due to an error made in the analysis of MHV amplitudes with external fundamental matter fields in the superconformal N=2 SQCD.
We investigate the appearance of straight infinite Wilson lines lying on the self-dual plane in the context of the Self Dual sector of the Yang Mills theory and in a connection to the Lagrangian implementing the MHV vertices (MHV Lagrangian) according to the Cachazo-Svrcek-Witten method. It was already recognized in the past by two of the authors, that such Wilson line functional provides the field transformation of positive helicity fields between the Yang-Mills theory on the light-cone and the MHV Lagrangian. Here we discuss in detail the connection to the Self Dual sector and we provide a n…
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 repr…
The MHV action is the Yang-Mills action quantized on the light-front, where the two explicit physical gluonic degrees of freedom have been canonically transformed to a new set of fields. This transformation leads to the action with vertices being off-shell continuations of the MHV amplitudes. We show that the solution to the field transformation expressing one of the new fields in terms of the Yang-Mills field is a certain type of the Wilson line. More precisely, it is a straight infinite gauge link with a slope extending to the light-cone minus and the transverse direction. One of the consequ…
We discuss the origin of the Wilson polygon - MHV amplitude duality at the perturbative level. It is shown that the duality for the MHV amplitudes at one-loop level can be proven upon the peculiar change of variables in Feynman parametrization and the use of the relation between Feynman integrals at the different space-time dimensions. Some generalization of the duality which implies the insertion of the particular vertex operator at the Wilson triangle is found for the 3-point function. We discuss analytical structure of Wilson loop diagrams and present the corresponding Landau equations. The…
Traditionally, the duality between Wilson loops and amplitudes beyond one loop in N=4 SYM is characterised by the remainder function. Because of the perturbative origins of the BDS expression, the remainder function is more natural at weak than at strong coupling. We advocate instead a more direct approach, based on considering ratios of Wilson loops. This allows us to define a manifestly finite, regularisation independent, conformally invariant quantity. It does not make a direct reference to the BDS expression and the definition is regularisation independent. It is a natural object at weak a…
We consider the (multi) Splitting function of Wilson loops and MHV gluon scattering S matrix elements in N=4 SYM. At strong coupling, one can utilize the methods of Alday and Maldacena and at weak coupling (one loop) the correspondence to light like Wilson loops is used. In both cases, the (multi) Splitting function corresponds to flattened cusps in the light like polygon, allowing for a clean disentanglement from the other gluons. We compute it in some cases and estimate some terms in other cases. We also prove the anomalous Ward identity of Drummond et al. in the strong coupling regime. Last…
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 (arXiv:0803.1465 [hep-th]) 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 ar…
The S-matrix for planar N = 4 super Yang-Mills theory can be computed as the correlation function for a holomorphic polygonal Wilson loop in twistor space. In an axial gauge, this leads to the construction of the all-loop integrand via MHV diagrams in twistor space. We show that at MHV, this formulation leads directly to expressions for loop integrands in d log form; i.e., the integrand is a product of exterior derivatives of logarithms of rational functions. For higher MHV degree, it is in d log form multiplied by delta functions. The parameters appearing in the d log form arise geometrically…
We develop a new classical action that in addition to $\mathrm{MHV}$ vertices contains also $\mathrm{N^kMHV}$ vertices, where $1\leq k \leq n-4$ with $n$ the number of external legs. The lowest order vertex is the four-point MHV vertex -- there is no three point vertex and thus the amplitude calculation involves fewer vertices than in the CSW method and, obviously, considerably fewer than in the standard Yang-Mills action. The action is obtained by a canonical transformation of the Yang-Mills action in the light-cone gauge, where the field transformations are based on the Wilson line functiona…
We perform a canonical transformation of fields that brings the Yang-Mills action in the light-cone gauge to a new classical action, which does not involve any triple-gluon vertices. The lowest order vertex is the four-point MHV vertex. Higher point vertices include the MHV and $\overline{\textrm{MHV}}$ vertices, that reduce to the corresponding amplitudes in the on-shell limit. In general, any $n$-leg vertex has $2\leq m \leq n-2$ negative helicity legs. The canonical transformation of fields can be compactly expressed in terms of path-ordered exponentials of fields and their functional deriv…
We give a representation of the parity-even part of the planar two-loop six-gluon 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 ABDK/BDS 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 i…
We propose that the complete planar S-matrix of N=4 super Yang-Mills - including all N^kMHV 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 NNMHV trees, the integrands of the 1-loop MHV an…
Wilson loops with lightlike polygonal contours have been conjectured to be equivalent to MHV scattering amplitudes in N=4 super Yang-Mills. We compute such Wilson loops for special polygonal contours at two loops in perturbation theory. Specifically, we concentrate on the remainder function R, obtained by subtracting the known ABDK/BDS ansatz from the Wilson loop. First, we consider a particular two-dimensional eight-point kinematics studied at strong coupling by Alday and Maldacena. We find numerical evidence that R is the same at weak and at strong coupling, up to an overall, coupling-depend…
A duality relation has been proposed between the planar gluon MHV amplitudes and light-like Wilson loops in N=4 super Yang-Mills. At six-point two-loop, the results for the planar gluon MHV amplitude and for the light-like Wilson loop agree, but they both differ from the Bern-Dixon-Smirnov ansatz by a finite remainder function. Recently Del Duca, Duhr and Smirnov presented an analytical result for the two-loop hexagon Wilson loop remainder function in general kinematics. Their result is rather lengthy, and the dependence on the conformal cross ratios appears in a complicated way. Here we prese…
We compute for the first time the two-loop corrections to arbitrary n-gon lightlike Wilson loops in N=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 i…
The MHV scattering amplitudes in planar N=4 SYM are dual to bosonic light-like Wilson loops. We explore various proposals for extending this duality to generic non-MHV amplitudes. The corresponding dual object should have the same symmetries as the scattering amplitudes and be invariant to all loops under the chiral half of the N=4 superconformal symmetry. We analyze the recently introduced supersymmetric extensions of the light-like Wilson loop (formulated in Minkowski space-time) and demonstrate that they have the required symmetry properties at the classical level only, up to terms proporti…
We study Feynman integrals and scattering amplitudes in ${\cal N}=4$ super-Yang-Mills by exploiting the duality with null polygonal Wilson loops. Certain Feynman integrals, including one-loop and two-loop chiral pentagons, are given by Feynman diagrams of a supersymmetric Wilson loop, where one can perform loop integrations and be left with simple integrals along edges. As the main application, we compute analytically for the first time, the symbol of the generic ($n\geq 12$) double pentagon, which gives two-loop MHV amplitudes and components of NMHV amplitudes to all multiplicities. We repres…
We perform an analytic continuation of the two-loop remainder function for the six-point planar MHV amplitude in N=4 SUSY, found by Goncharov, Spradlin, Vergu and Volovich from the light-like Wilson loop representation. The remainder function is continued into a physical region, where all but two energy invariants are negative. It turns out to be pure imaginary in the multi-Regge kinematics, which is in an agreement with the predictions based on the Steinmann relations for the Regge poles and Mandelstam cut contributions. The leading term reproduces correctly the expression calculated by one o…
We obtain a compact expression for the octagon MHV amplitude / Wilson loop at 3 loops in planar N=4 SYM and in special 2d kinematics in terms of 7 unfixed coefficients. We do this by making use of the cyclic and parity symmetry of the amplitude/Wilson loop and its behaviour in the soft/collinear limits as well as in the leading term in the expansion away from this limit. We also make a natural and quite general assumption about the functional form of the result, namely that it should consist of weight 6 polylogarithms whose symbol consists of basic cross-ratios only (and not functions thereof)…
A recent, integrability-based conjecture in the framework of the Wilson loop OPE for N=4 SYM theory, predicts the leading OPE contribution for the hexagon MHV remainder function and NMHV ratio function to all loops, in integral form. We prove that these integrals evaluate to a particular basis of harmonic polylogarithms, at any order in the weak coupling expansion. The proof constitutes an algorithm for the direct computation of the integrals, which we employ in order to obtain the full (N)MHV OPE contribution in question up to 6 loops, and certain parts of it up to 12 loops. We attach compute…
We report on the complete OPE series for the 6-gluon MHV and NMHV amplitudes in planar $\mathcal{N}=4$ SYM theory. Namely, we provide a finite coupling prediction for all the terms in the expansion of these amplitudes around the collinear limit. These furnish a non-perturbative representation of the full amplitudes.
In this paper we consider tree-level gauge invariant off-shell amplitudes (Wilson line form factors) in $\mathcal{N}=4$ SYM. For the off-shell amplitudes with one leg off-shell we present a conjecture for their Grassmannian integral representation in spinor helicity, twistor and momentum twistor parameterizations. The presented conjecture is successfully checked against BCFW results for MHV$_n$, NMHV$_4$ and NMHV$_5$ off-shell amplitudes. We have also verified that our Grassmannian integral representation correctly reproduces soft (on-shell) limit for the off-shell gluon momentum. It is shown …
This article reviews the recent progress in twistor approaches to Wilson loops, amplitudes and their duality for N=4 super Yang-Mills. Wilson loops and amplitudes are derived from first principles using the twistor action for maximally supersymmetric Yang-Mills theory. We start by deriving the MHV rules for gauge theory amplitudes from the twistor action in an axial gauge in twistor space, and show that this gives rise to the original momentum space version given by Cachazo, Svrcek and Witten. We then go on to obtain from these the construction of the momentum twistor space loop integrand usin…
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 ano…
We incorporate all gauge-invariant local composite operators into the twistor-space formulation of N=4 SYM theory, detailing and expanding on ideas we presented recently in arXiv:1603.04471. The vertices for these operators contain infinitely many terms and we show how they can be constructed by taking suitable derivatives of a light-like Wilson loop in twistor space and shrinking it down to a point. In particular, these vertices directly yield the tree-level MHV super form factors of all composite operators in N=4 SYM theory.
Loop-level scattering amplitudes for massless particles have singularities in regions where tree amplitudes are perfectly smooth. For example, a $2\to4$ gluon scattering process has a singularity in which each incoming gluon splits into a pair of gluons, followed by a pair of $2\to2$ collisions between the gluon pairs. This singularity mimics double parton scattering because it occurs when the transverse momentum of a pair of outgoing gluons vanishes. The singularity is logarithmic at fixed order in perturbation theory. We exploit the duality between scattering amplitudes and polygonal Wilson …
We construct a regularization for light-like polygonal Wilson loops in ${\cal N}=4$ SYM, which matches them to the off-shell MHV gluon scattering amplitudes. Explicit calculations are performed for the 1-loop four gluon case. The off light cone extrapolation has to be based on the local supersymmetric Wilson loop. The observed matching concerns Feynman gauge. Furthermore, the leading infrared divergent term is shown to be gauge parameter independent on 1-loop level.
The precise form of the multi-Regge asymptotics of the two-loop six-point MHV amplitude in N = 4 Super-Yang-Mills theory has been a subject of recent controversy. In this paper we utilize the amplitude/Wilson loop correspondence to obtain precise numerical results for the imaginary part of these asymptotics. The region of phase-space that we consider is interesting because it allowed Bartels, Lipatov, and Sabio Vera to determine that the two-loop six-point MHV amplitude is not fixed by the BDS ansatz. They proceeded by working in the framework of a high energy effective action, thus side-stepp…
We show that a certain class of light-like Wilson loops exhibits a Yangian symmetry at one loop, or equivalently, in an Abelian theory. The Wilson loops we discuss are equivalent to one-loop MHV amplitudes in N=4 super Yang-Mills theory in a certain kinematical regime. The fact that we find a Yangian symmetry constraining their functional form can be thought of as the effect of the original conformal symmetry associated to the scattering amplitudes in the N=4 theory.
We describe a general algorithm which builds on several pieces of data available in the literature to construct explicit analytic formulas for two-loop MHV amplitudes in N=4 super-Yang-Mills theory. The non-classical part of an amplitude is built from $A_3$ cluster polylogarithm functions; classical polylogarithms with (negative) cluster X-coordinate arguments are added to complete the symbol of the amplitude; beyond-the-symbol terms proportional to $π^2$ are determined by comparison with the differential of the amplitude; and the overall additive constant is fixed by the collinear limit. We p…
The MHV or CSW expansion of tree-level Yang-Mills amplitudes provides an elegant and simple way of obtaining analytic formulas for S-matrix elements. Inspired by the BCFW technique, a systematic approach to obtain the MHV expansion was introduced by Risager, using a particular complex deformation. Although it works for Yang-Mills amplitudes, Risager's technique fails to provide an MHV expansion already for Next-to-MHV gravity amplitudes with more than eleven particles, as shown by Bianchi, Elvang and Freedman in 2008. This fact implies that in this sector there is a contribution at infinity st…
Classically supersymmetric Wilson loop on a null polygonal contour possesses all symmetries required to match it onto non-MHV amplitudes in maximally supersymmetric Yang-Mills theory. However, to define it quantum mechanically, one is forced to regularize it since perturbative loop diagrams are not well-defined due to presence of ultraviolet divergences stemming from integration in the vicinity of the cusps. A regularization that is adopted by practitioners by allowing one to use spinor helicity formalism, on the one hand, and systematically go to higher orders of perturbation theory is based …
We discuss formulation of Wilson polygon - MHV amplitude duality at the perturbative level in various regularizations. For four gluons it is shown that at one loop one can formulate diagrammatic correspondence interpolating between the dimensional regularization and the off-shell one. We suggest new interpretation of all types of box diagrams in terms of the dual simplex in dimensional regularization and describe its degeneration to the Wilson polygon. The interesting nullification phenomena for the low-energy amplitudes in the Higgsed phase has been found.
Planar gluon amplitudes in N=4 SYM are remarkably similar to expectation values of Wilson loops made of light-like segments. We argue that the latter can be determined by making use of the conformal symmetry of the gauge theory, broken by cusp anomalies. We derive the corresponding anomalous conformal Ward identities valid to all loops and show that they uniquely fix the form of the finite part of a Wilson loop with n cusps (up to an additive constant) for n=4 and n=5 and reduce the freedom in it to a function of conformal invariants for n>=6. We also present an explicit two-loop calculation f…
We compute the one-loop expectation value of light-like polygonal Wilson loops in N=4 super-Yang-Mills theory in full superspace. When projecting to chiral superspace we recover the known results for tree-level next-to-maximally-helicity-violating (NMHV) scattering amplitude. The one-loop MHV amplitude is also included in our result but there are additional terms which do not immediately correspond to scattering amplitudes. We finally discuss different regularizations and their Yangian anomalies.
We consider generalization of the Cachazo-Svrcek-Witten (CSW) rules to one-loop amplitudes of N=4 super Yang-Mills theory in a recently developed holonomy formalism in twistor space. We first reconsider off-shell continuation of the Lorentz-invariant Nair measure for the incorporation of loop integrals. We then formulate an S-matrix functional for general amplitudes such that it implements the CSW rules at quantum level. For one-loop MHV amplitudes, the S-matrix functional correctly reproduces the analytic expressions obtained in the Brandhuber-Spence-Travaglini (BST) method. Motivated by this…
We present a compact analytic formula for the two-loop six-particle MHV remainder function (equivalently, the two-loop light-like hexagon Wilson loop) in N = 4 supersymmetric Yang-Mills theory in terms of the classical polylogarithm functions Li_k with cross-ratios of momentum twistor invariants as their arguments. In deriving our result we rely on results from the theory of motives.
We apply a bootstrap procedure to two-loop MHV amplitudes in planar N=4 super-Yang-Mills theory. We argue that the mathematically most complicated part (the $Λ^2 B_2$ coproduct component) of the n-particle amplitude is uniquely determined by a simple cluster algebra property together with a few physical constraints (dihedral symmetry, analytic structure, supersymmetry, and well-defined collinear limits). We present a concise, closed-form expression which manifests these properties for all n.
One of the many remarkable features of MHV scattering amplitudes is their conjectured equality to lightlike polygon Wilson loops, which apparently holds at all orders in perturbation theory as well as non-perturbatively. This duality is usually expressed in terms of purely four-dimensional quantities obtained by appropriate subtraction of the IR and UV divergences from amplitudes and Wilson loops respectively. In this paper we demonstrate, by explicit calculation, the completely unanticipated fact that the equality continues to hold at two loops through order epsilon in dimensional regularizat…
We calculate form factors of half-BPS operators in N=4 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 …
The Pentagon Operator Product Expansion represents polygonal Wilson loops in planar $\mathcal{N}=4$ super Yang-Mills in terms of a series of flux tube excitations for finite coupling. We demonstrate how to re-sum this series at the one loop level for the hexagonal Wilson loop dual to the six-point MHV amplitude. By summing over a series of effective excitations we find expressions which integrate to logarithms and polylogarithms, reproducing the known one-loop result.
One of the methods to calculate tree-level multi-gluon scattering amplitudes is to use the Berends-Giele recursion relation involving off-shell currents or off-shell amplitudes, if working in the light cone gauge. As shown in recent works using the light-front perturbation theory, solutions to these recursions naturally collapse into gauge invariant and gauge-dependent components, at least for some helicity configurations. In this work, we show that such structure is helicity independent and emerges from analytic properties of matrix elements of Wilson line operators, where the slope of the st…
This is a review of recent developments in the study of perturbative gauge theory and gravity using action functionals on twistor space. It is intended to provide a user-friendly introduction to twistor actions, geared towards researchers or graduate students interested in learning something about the utility, prospects, and shortcomings of this approach. For those already familiar with the twistor approach, it should provide a condensed overview of the literature as well as several novel results of potential interest. This work is based primarily upon the author's D.Phil. thesis. We first con…
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-…
We use momentum twistors to evaluate planar loop integrals. Infrared divergences are regulated by the recently proposed AdS-inspired mass regulator. We show that two-loop amplitudes in N=4 super Yang-Mills can be expanded in terms of basis integrals having twistor numerators. We argue that these integrals are considerably simpler compared to the ones conventionally used. Our case in point is the two-loop six-point MHV amplitude. We present analytical results for the remainder function in a kinematical limit, and find agreement with a recent Wilson loop computation. We also provide two-loop evi…
We present an explicit analytic calculation of the differential of the planar n-particle, two-loop MHV scattering amplitude in N=4 super Yang-Mills theory. The result is expressed only in terms of the polylogarithm functions Li_k(-x), for k=1,2,3, with arguments x belonging to the special class of dual conformal cross-ratios known as cluster X-coordinates. The surprising fact that these amplitudes may be expressed in this way provides a striking example of the manner in which the cluster structure on the kinematic configuration space underlies the structure of amplitudes in SYM theory.
Recent progress on scattering amplitudes has benefited from the mathematical technology of symbols for efficiently handling the types of polylogarithm functions which frequently appear in multi-loop computations. The symbol for all two-loop MHV amplitudes in planar SYM theory is known, but explicit analytic formulas for the amplitudes are hard to come by except in special limits where things simplify, such as multi-Regge kinematics. By applying symbology we obtain a formula for the leading behavior of the imaginary part (the Mandelstam cut contribution) of this amplitude in multi-Regge kinemat…
We use the leading singularity technique to determine the planar six-particle two-loop MHV amplitude in N=4 super Yang-Mills in terms of a simple basis of integrals. Our result for the parity even part of the amplitude agrees with the one recently presented in arXiv:0803.1465. The parity-odd part of the amplitude is a new result. The leading singularity technique reduces the determination of the amplitude to finding the solution to a system of linear equations. The system of equations is easily found by computing residues. We present the complete system of equations which determines the whole …
We consider the planar MHV amplitude in N=4 supersymmetric Yang-Mills theory for 2 -> 4 particle scattering at two and three loops in the Regge kinematics. We perform an analytic continuation of two-loop result for the remainder function found by Goncharov, Spradlin, Vergu and Volovich to the physical region, where the remainder function does not vanish in the Regge limit. After the continuation both the leading and the subleading in the logarithm of the energy terms are extracted and analyzed. Using this result we calculate the next-to-leading corrections to the impact factors required in the…
An intriguing new duality between planar MHV gluon amplitudes and light-like Wilson loops in N=4 super Yang-Mills is investigated. We extend previous checks of the duality by performing a two-loop calculation of the rectangular and pentagonal Wilson loop. Furthermore, we derive an all-order broken conformal Ward identity for the Wilson loops and analyse its consequences. Starting from six points, the Ward identity allows for an arbitrary function of conformal invariants to appear in the expression for the Wilson loop. We compute this function at six points and two loops and discuss its implica…
In this paper we define, independent of theories, two discriminant matrices involving a solution to the scattering equations in four dimensions, the ranks of which are used to divide the solution set into a disjoint union of subsets. We further demonstrate, {entirely within the Cachazo-He-Yuan formalism,} that each subset of solutions gives nonzero contribution to tree-level $\text{N}^{k}\text{MHV}$ gauge/gravity amplitudes only for a specific value of $k$. Thus the solutions can be characterized by the rank of their discriminant matrices, which in turn determines the value of $k$ of the $\tex…
We calculate corrections to the BDS formula for the six-particle planar MHV amplitude for the gluon transition 3->3 in the multi-Regge kinematics for the physical region, in which the Regge pole ansatz is not valid. The remainder function at two loops is obtained by an analytic continuation of the expression derived by Goncharov, Spradlin, Vergu and Volovich to the kinematic region described by the Mandelstam singularity exchange in the crossing channel. It contains both the imaginary and real contributions being in agreement with the BFKL predictions. The real part of the three loop expressio…
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 $AdS_5 \times S^5$ back to itself in such a way that gluon…
We show that scattering amplitudes in planar N = 4 Super Yang-Mills in multi-Regge kinematics can naturally be expressed in terms of single-valued iterated integrals on the moduli space of Riemann spheres with marked points. As a consequence, scattering amplitudes in this limit can be expressed as convolutions that can easily be computed using Stokes' theorem. We apply this framework to MHV amplitudes to leading-logarithmic accuracy (LLA), and we prove that at L loops all MHV amplitudes are determined by amplitudes with up to L + 4 external legs. We also investigate non-MHV amplitudes, and we …
We present a novel type of differential equations for on-shell loop integrals. The equations are second-order and importantly, they reduce the loop level by one, so that they can be solved iteratively in the loop order. We present several infinite series of integrals satisfying such iterative differential equations. The differential operators we use are best written using momentum twistor space. The use of the latter was advocated in recent papers discussing loop integrals in N=4 super Yang-Mills. One of our motivations is to provide a tool for deriving analytical results for scattering amplit…
