arXiv subjects

J. M. Drummond

Publications and source records attributed to J. M. Drummond.

At least 37 records · Page 2Linked to original sources

arXiv2012

Generalised ladders and single-valued polylogarithms

We introduce and solve an infinite class of loop integrals which generalises the well-known ladder series. The integrals are described in terms of single-valued polylogarithmic functions which satisfy certain differential equations. The combination of the differential equations and single-valued behaviour allow us to explicitly construct the polylogarithms recursively. For this class of integrals the symbol may be read off from the integrand in a particularly simple way. We give an explicit formula for the simplest generalisation of the ladder series. We also relate the generalised ladder integrals to a class of vacuum diagrams which includes both the wheels and the zigzags.

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arXiv2011

Tree-level amplitudes and dual superconformal symmetry

We review the structure of gauge theory scattering amplitudes at tree level and describe how a compact expression can be found which encodes all the tree-level amplitudes in the maximally supersymmetric N=4 theory. The expressions for the amplitudes reveal a dual superconformal symmetry. We describe how these ideas can be extended to leading singularities and the loop integrand in the planar theory and discuss the appearance of dual conformal symmetry in higher-dimensional gauge theories. This article is an invited review for a special issue of Journal of Physics A devoted to "Scattering Amplitudes in Gauge Theories".

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arXiv2011

The Yangian origin of the Grassmannian integral

In this paper we analyse formulas which reproduce different contributions to scattering amplitudes in N=4 super Yang-Mills theory through a Grassmannian integral. Recently their Yangian invariance has been proved directly by using the explicit expression of the Yangian level-one generators. The specific cyclic structure of the form integrated over the Grassmannian enters in a crucial way in demonstrating the symmetry. Here we show that the Yangian symmetry fixes this structure uniquely.

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arXiv2011

Review of AdS/CFT Integrability, Chapter V.2: Dual Superconformal Symmetry

Scattering amplitudes in planar N=4 super Yang-Mills theory reveal a remarkable symmetry structure. In addition to the superconformal symmetry of the Lagrangian of the theory, the planar amplitudes exhibit a dual superconformal symmetry. The presence of this additional symmetry imposes strong restrictions on the amplitudes and is connected to a duality relating scattering amplitudes to Wilson loops defined on polygonal light-like contours. The combination of the superconformal and dual superconformal symmetries gives rise to a Yangian, an algebraic structure which is known to be related to the appearance of integrability in other regimes of the theory. We discuss two dual formulations of the symmetry and address the classification of its invariants.

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arXiv2011

Hidden Simplicity of Gauge Theory Amplitudes

These notes were given as lectures at the CERN Winter School on Supergravity, Strings and Gauge Theory 2010. We describe the structure of scattering amplitudes in gauge theories, focussing on the maximally supersymmetric theory to highlight the hidden symmetries which appear. Using the BCFW recursion relations we solve for the tree-level S-matrix in N=4 super Yang-Mills theory, and describe how it produces a sum of invariants of a large symmetry algebra. We review amplitudes in the planar theory beyond tree-level, describing the connection between amplitudes and Wilson loops, and discuss the implications of the hidden symmetries.

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arXiv2010

Yangian symmetry of light-like Wilson loops

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.

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arXiv2010

Yangians, Grassmannians and T-duality

We investigate the Yangian symmetry of scattering amplitudes in N=4 super Yang-Mills theory and show that its formulations in twistor and momentum twistor space can be interchanged. In particular we show that the full symmetry can be thought of as the Yangian of the dual superconformal algebra, annihilating the amplitude with the MHV part factored out. The equivalence of this picture with the one where the ordinary superconformal symmetry is thought of as fundamental is an algebraic expression of T-duality. Motivated by this, we analyse some recently proposed formulas, which reproduce different contributions to amplitudes through a Grassmannian integral. We prove their Yangian invariance by directly applying the generators.

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arXiv2010

A note on N=8 counterterms

The most difficult counterterms to construct in any supersymmetric theory are those that cannot be written as full superspace integrals of gauge-invariant integrands. In $D=4$ maximal supergravity it has been known for some time that there are just three of these at the linearised level. In this article we discuss these counterterms again from the point of view of representations of the superconformal group. In particular, we show that the only independent invariants constructed from shortened superconformal multiplets in $D=4$ are BPS.

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arXiv2010

Simple loop integrals and amplitudes in N=4 SYM

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 evidence that the logarithm of MHV amplitudes can be written in terms of simple twistor space integrals.

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arXiv2009

Conformal Ward identities for Wilson loops and a test of the duality with gluon amplitudes

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 for n=5. The result confirms the form predicted by the Ward identities and exactly matches the finite part of the two-loop five-gluon planar MHV amplitude. This constitutes another non-trivial test of the Wilson loop/gluon amplitude duality.

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arXiv2009

Yangian symmetry of scattering amplitudes in N=4 super Yang-Mills theory

Tree-level scattering amplitudes in N=4 super Yang-Mills theory have recently been shown to transform covariantly with respect to a 'dual' superconformal symmetry algebra, thus extending the conventional superconformal symmetry algebra psu(2,2|4) of the theory. In this paper we derive the action of the dual superconformal generators in on-shell superspace and extend the dual generators suitably to leave scattering amplitudes invariant. We then study the algebra of standard and dual symmetry generators and show that the inclusion of the dual superconformal generators lifts the psu(2,2|4) symmetry algebra to a Yangian. The non-local Yangian generators acting on amplitudes turn out to be cyclically invariant due to special properties of psu(2,2|4). The representation of the Yangian generators takes the same form as in the case of local operators, suggesting that the Yangian symmetry is an intrinsic property of planar N=4 super Yang-Mills, at least at tree level.

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arXiv2009

Tree-Level Amplitudes in N=8 Supergravity

We present an algorithm for writing down explicit formulas for all tree amplitudes in N=8 supergravity, obtained from solving the supersymmetric on-shell recursion relations. The formula is patterned after one recently obtained for all tree amplitudes in N=4 super Yang-Mills which involves nested sums of dual superconformal invariants. We find that all graviton amplitudes can be written in terms of exactly the same structure of nested sums with two modifications: the dual superconformal invariants are promoted from N=4 to N=8 superspace in the simplest manner possible--by squaring them--and certain additional non-dual conformal gravity dressing factors (independent of the superspace coordinates) are inserted into the nested sums. To illustrate the procedure we give explicit closed-form formulas for all NMHV, NNMHV and NNNMV gravity superamplitudes.

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arXiv2009

All tree-level amplitudes in N=4 SYM

We give an explicit formula for all tree amplitudes in N=4 SYM, derived by solving the recently presented supersymmetric tree-level recursion relations. The result is given in a compact, manifestly supersymmetric form and we show how to extract from it all possible component amplitudes for an arbitrary number of external particles and any arrangement of external particles and helicities. We focus particularly on extracting gluon amplitudes which are valid for any gauge theory. The formula for all tree-level amplitudes is given in terms of nested sums of dual superconformal invariants and it therefore manifestly respects both conventional and dual superconformal symmetry.

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arXiv2009

Hexagon Wilson loop = six-gluon MHV amplitude

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 arbitrary number of external particles.

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arXiv2008

Generalized unitarity for N=4 super-amplitudes

We develop a manifestly supersymmetric version of the generalized unitarity cut method for calculating scattering amplitudes in N=4 SYM theory. We illustrate the power of this method by computing the one-loop n-point NMHV super-amplitudes. The result confirms two conjectures which we made in arXiv:0807.1095 [hep-th]. Firstly, we derive the compact, manifestly dual superconformally covariant form of the NMHV tree amplitudes for arbitrary number and types of external particles. Secondly, we show that the ratio of the one-loop NMHV to the MHV amplitude is dual conformal invariant.

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arXiv2008

Dual superconformal symmetry of scattering amplitudes in N=4 super-Yang-Mills theory

We argue that the scattering amplitudes in the maximally supersymmetric N=4 super-Yang-Mills theory possess a new symmetry which extends the previously discovered dual conformal symmetry. To reveal this property we formulate the scattering amplitudes as functions in 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 conformal symmetry is broken at loop level by infrared divergences. However, we provide evidence that the anomalous contribution to the MHV and NMHV superamplitudes is the same and, therefore, their ratio is a dual conformal invariant function. We identify this function by an explicit calculation of 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.

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arXiv2008

The hexagon Wilson loop and the BDS ansatz for the six-gluon amplitude

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.

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arXiv2008

Conformal properties of four-gluon planar amplitudes and Wilson loops

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 five 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.

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