arXiv subjects

James Drummond

Publications and source records attributed to James Drummond.

At least 19 recordsLinked to original sources

arXiv2026

BCFW recursion for light-like Wilson loop correlators

We apply the idea of holomorphic linking (arXiv:1101.1329) to correlators of multiple light-like loop operators in $\mathcal{N}=4$ super Yang-Mills theory. This allows us to extend the BCFW recursion relations for planar scattering amplitude integrands or single Wilson loops in the planar theory to correlators of multiple loop operators. We discover a novel term which feeds the recursion for single loops into the relation for multiple loop operators. We illustrate the recursion relations with several examples and highlight its applicability in combination with the $\bar{Q}$-equation for multiple loop operators. Finally, we again employ holomorphic linking to derive a recursion relation for colour-exact Wilson loop correlators.

hep-th↗

arXiv2026

Tropical symmetries of cluster algebras

We study tropicalisations of quasi-automorphisms of cluster algebras and show that their induced action on the g-vectors can be realized by tropicalising their action on the homogeneous $\hat{y}$ (or $\mathcal{X}$) variables of a chosen initial cluster. This perspective allows us to interpret the action on g-vectors as a change of coordinates in the tropical setting. Focusing on Grassmannian cluster algebras, we analyse tropicalisations of quasi-automorphisms in detail. We derive tropical analogues of the braid group action and the twist map on both g-vectors and tableaux. We introduce the notions of unstable and stable fixed points for quasi-automorphisms, which prove useful for constructing cluster monomials and non-real modules, respectively. As an application, we demonstrate that the counts of prime non-real tableaux with a fixed number of columns in $\mathrm{SSYT}(3, [9])$ and $\mathrm{SSYT}(4, [8])$, arising from the braid group action on stable fixed points, are governed by Euler's totient function. Furthermore, we apply our findings to scattering amplitudes in physics, providing a novel interpretation of the square root associated with the four-mass box integral via stable fixed points of quasi-automorphisms of the Grassmannian cluster algebra $\CC[\Gr(4,8)]$.

math.RT↗

arXiv2026

Leading singularities of Wilson loop correlators from twistor Wilson loop diagrams

The leading singularities of one-loop scattering amplitudes in planar $\mathcal{N}=4$ super Yang-Mills theory are known to factorise into products of tree-level amplitudes, and this can be seen from a number of different perspectives e.g. generalised unitarity or on-shell diagrams. Here we investigate the leading singularities from the perspective of the Wilson loop expectation values to which these amplitudes are dual, in particular making use of the twistor Wilson loop formalism. We show that the factorisation of one-loop leading singularities of a null Wilson loop's expectation value into a product of tree-level objects is manifest at the level of twistor Wilson loop diagrams, and is a simple consequence of planarity, without appeal to e.g. unitarity on the amplitude side of the duality. We then use the same approach to derive compact formulae for the one-loop leading singularities of correlators of multiple light-like Wilson loop operators in terms of tree-level objects. Via the chiral box expansion, these formulae provide a simple route to writing down the $O(g^2)$ correlation function of any number of Wilson loops at any MHV degree.

hep-th↗

arXiv2026

Light-like Wilson loops and the $\bar{Q}$-equation

In recent work we began a study of the correlators of multiple light-like Wilson loops in $\mathcal{N}=4$ super Yang-Mills theory, focussing primarily on tree-level calculations and, beyond tree-level, to the Abelian theory. Here we calculate $O(g^2)$ correlators of multiple light-like Wilson loops in the $SU(N)$ theory. We use the chiral box expansion and a study of the leading singularities of the loop integrand to arrive at integrated expressions for these objects. We then use the results of these calculations to verify that a natural generalisation of the $\bar{Q}$-equation, familiar from the study of single Wilson loops, holds in the $SU(N)$ theory. This $\bar{Q}$-equation should provide a valuable tool for the computation of multiple Wilson loop correlators at higher order in the coupling.

hep-th↗

arXiv2025

Light-like Wilson loop correlators

It is well-known that the expectation values of null polygonal Wilson loops computed in planar $\mathcal{N}=4$ super Yang-Mills theory are dual to MHV amplitudes in that theory, and moreover that the duality can be extended to higher helicity sectors through the introduction of super Wilson loops. In this first of a series of papers, we investigate the natural generalisation posed by correlation functions of multiple light-like loop operators, both in the bosonic case and in the case of super Wilson loops. Explicit calculations are presented in several cases and we verify that, in the Abelian theory, these objects obey a natural generalisation of the $\bar{Q}$-equation which relates different loop orders, kinematic configurations and Grassmann sectors.

hep-th↗

arXiv2025

Singularities of massless scattering and cluster algebras

Partial flag varieties arise in the context of massless scattering kinematics. They can be associated to both spinor-helicity variables and momentum twistor variables in two separate yet natural ways. Here we report on evidence at five and six points that the cluster algebras associated to these partial flag varieties contain information relevant to non-dual conformal massless scattering amplitudes and related observables. At five points both spinor-helicity and momentum twistor cluster structures capture similar information about symbol alphabets. At six points we demonstrate that the momentum twistor structure captures a larger subset of the alphabet compared to the spinor-helicity one. We also observe that the associated cluster structures correctly predict the appearance of certain triples of symbol letters related to cluster mutation relations.

hep-th↗

arXiv2021

Tropical Grassmannians, cluster algebras and scattering amplitudes

We provide a cluster-algebraic approach to the computation of the recently introduced generalised biadjoint scalar amplitudes related to Grassmannians ${\rm Gr}(k,n)$. A finite cluster algebra provides a natural triangulation for the tropical Grassmannian whose volume computes the scattering amplitudes. Using this method one can construct the entire colour-ordered amplitude via mutations starting from a single term.

hep-th↗

arXiv2020

Cluster adjacency and the four-loop NMHV heptagon

We exploit the recently described property of cluster adjacency for scattering amplitudes in planar $\mathcal{N}=4$ super Yang-Mills theory to construct the symbol of the four-loop NMHV heptagon amplitude. We use a manifestly cluster adjacent ansatz and describe how the parameters of this ansatz are determined using simple physical consistency requirements. We then specialise our answer for the amplitude to the multi-Regge limit, finding agreement with previously available results up to the next-to-leading logarithm, and obtaining new predictions up to (next-to)$^3$-leading-logarithmic accuracy.

hep-th↗

arXiv2020

Tropical fans, scattering equations and amplitudes

We describe a family of tropical fans related to Grassmannian cluster algebras. These fans are related to the kinematic space of massless scattering processes in a number of ways. For each fan associated to the Grassmannian ${\rm Gr}(k,n)$ there is a notion of a generalised $ϕ^3$ amplitude and an associated set of scattering equations which further generalise the ${\rm Gr}(k,n)$ scattering equations that have been recently introduced. Here we focus mostly on the cases related to finite Grassmannian cluster algebras and we explain how face variables for the cluster polytopes are simply related to the scattering equations. For the Grassmannians ${\rm Gr}(4,n)$ the tropical fans we describe are related to the singularities (or symbol letters) of loop amplitudes in planar $\mathcal{N}=4$ super Yang-Mills theory. We show how each choice of tropical fan leads to a natural class of polylogarithms, generalising the notion of cluster adjacency and we describe how the currently known loop data fit into this classification.

hep-th↗

arXiv2019

Algebraic singularities of scattering amplitudes from tropical geometry

We address the appearance of algebraic singularities in the symbol alphabet of scattering amplitudes in the context of planar $\mathcal{N}=4$ super Yang-Mills theory. We argue that connections between cluster algebras and tropical geometry provide a natural language for postulating a finite alphabet for scattering amplitudes beyond six and seven points where the corresponding Grassmannian cluster algebras are finite. As well as generating natural finite sets of letters, the tropical fans we discuss provide letters containing square roots. Remarkably, the minimal fan we consider provides all the square root letters recently discovered in an explicit two-loop eight-point NMHV calculation.

hep-th↗

arXiv2019

One-loop amplitudes in $AdS_5\times S^5$ supergravity from $\mathcal{N}=4$ SYM at strong coupling

We explore the structure of maximally supersymmetric Yang-Mills correlators in the supergravity regime. We develop an algorithm to construct one-loop supergravity amplitudes of four arbitrary Kaluza-Klein supergravity states, properly dualised into single-particle operators. We illustrate this algorithm by constructing new explicit results for multi-channel correlation functions, and we show that correlators which are degenerate at tree level become distinguishable at one-loop. The algorithm contains a number of subtle features which have not appeared until now. In particular, we address the presence of non-trivial low twist protected operators in the OPE that are crucial for obtaining the correct one-loop results. Finally, we outline how the differential operators $\widehat{\mathcal{D}}_{pqrs}$ and $Δ^{(8)}$, which play a role in the context of the hidden 10d conformal symmetry at tree level, can be used to reorganise our one-loop correlators.

hep-th↗

arXiv2018

Amplitudes in the Multi-Regge Limit of N=4 SYM

A novel way of computing high-order amplitudes in the multi-Regge limit of planar maximally supersymmetric Yang-Mills theory is presented. In this framework, we are able to obtain high-loop and high-leg results by an easy operation on known amplitudes with fewer loops and lower multiplicity. This mechanism will be reviewed, along with an ensuing factorisation which allows us to determine leading logarithmic MHV results for any number of legs at a fixed loop order.

hep-th↗

arXiv2018

The seven-gluon amplitude in multi-Regge kinematics beyond leading logarithmic accuracy

We present an all-loop dispersion integral, well-defined to arbitrary logarithmic accuracy, describing the multi-Regge limit of the 2->5 amplitude in planar N=4 super Yang-Mills theory. It follows from factorization, dual conformal symmetry and consistency with soft limits, and specifically holds in the region where the energies of all produced particles have been analytically continued. After promoting the known symbol of the 2-loop N-particle MHV amplitude in this region to a function, we specialize to N=7, and extract from it the next-to-leading order (NLO) correction to the BFKL central emission vertex, namely the building block of the dispersion integral that had not yet appeared in the well-studied six-gluon case. As an application of our results, we explicitly compute the seven-gluon amplitude at next-to-leading logarithmic accuracy through 5 loops for the MHV case, and through 3 and 4 loops for the two independent NMHV helicity configurations, respectively.

hep-th↗

arXiv2018

Cluster adjacency beyond MHV

We explore further the notion of cluster adjacency, focussing on non-MHV amplitudes. We extend the notion of adjacency to the BCFW decomposition of tree-level amplitudes. Adjacency controls the appearance of poles, both physical and spurious, in individual BCFW terms. We then discuss how this notion of adjacency is connected to the adjacency already observed at the level of symbols of scattering amplitudes which controls the appearance of branch cut singularities. Poles and symbols become intertwined by cluster adjacency and we discuss the relation of this property to the $\bar{Q}$-equation which imposes constraints on the derivatives of the transcendental functions appearing in loop amplitudes.

hep-th↗

arXiv2018

Multi-Loop Amplitudes in the High-Energy Limit in N=4 SYM

We introduce a novel way to perform high-order computations in multi-Regge-kinematics in planar N=4 supersymmetric Yang-Mills theory and generalize the existing factorization into building blocks at two loops to all loop orders. Afterwards, we will explain how this framework can be used to easily obtain higher-loop amplitudes from existing amplitudes and how to relate them to amplitudes with higher number of legs.

hep-th↗

arXiv2017

Cluster adjacency properties of scattering amplitudes

We conjecture a new set of analytic relations for scattering amplitudes in planar N=4 super Yang-Mills theory. They generalise the Steinmann relations and are expressed in terms of the cluster algebras associated to Gr(4,n). In terms of the symbol, they dictate which letters can appear consecutively. We study heptagon amplitudes and integrals in detail and present symbols for previously unknown integrals at two and three loops which support our conjecture.

hep-th↗

arXiv2017

Heptagons from the Steinmann Cluster Bootstrap

We reformulate the heptagon cluster bootstrap to take advantage of the Steinmann relations, which require certain double discontinuities of any amplitude to vanish. These constraints vastly reduce the number of functions needed to bootstrap seven-point amplitudes in planar $\mathcal{N} = 4$ supersymmetric Yang-Mills theory, making higher-loop contributions to these amplitudes more computationally accessible. In particular, dual superconformal symmetry and well-defined collinear limits suffice to determine uniquely the symbols of the three-loop NMHV and four-loop MHV seven-point amplitudes. We also show that at three loops, relaxing the dual superconformal ($\bar{Q}$) relations and imposing dihedral symmetry (and for NMHV the absence of spurious poles) leaves only a single ambiguity in the heptagon amplitudes. These results point to a strong tension between the collinear properties of the amplitudes and the Steinmann relations.

hep-th↗

arXiv2016

Multi-Regge kinematics and the moduli space of Riemann spheres with marked points

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 show that they can be obtained by convoluting the MHV results with a certain helicity flip kernel. We classify all leading singularities that appear at LLA in the Regge limit for arbitrary helicity configurations and any number of external legs. Finally, we use our new framework to obtain explicit analytic results at LLA for all MHV amplitudes up to five loops and all non-MHV amplitudes with up to eight external legs and four loops.

hep-th↗