NMHV Amplitudes Research Articles

Non-planar BCFW Grassmannian geometries

In this paper, we study non-adjacent BCFW recursion relations and their connection to positive geometry. For an adjacent BCFW shift, the n-point NkMHV tree-level amplitude in mathcal{N} = 4 SYM theory is expressed as a sum over planar on-shell diagrams, corresponding to canonical “dlog” forms on the cells in the positive Grassmannian G+(k, n). Non-adjacent BCFW shifts naturally lead to an expansion of the amplitude in terms of a different set of objects, which do not manifest the cyclic ordering and the hidden Yangian symmetry of the amplitude. We show that these terms can be interpreted as dlog forms on the non-planar Grassmannian geometries, generalizing the cells of the positive Grassmannian G+(k, n) to a larger class of objects which live in G(k, n). We focus mainly on the case of NMHV amplitudes and discuss in detail the Grassmannian geometries. We also propose an alternative way to calculate the associated on-shell functions and dlog forms using an intriguing connection between Grassmannian configurations and the geometry in the kinematical space.

Recursion relations for scattering amplitudes with massive particles II: Massive vector bosons

Using the recently introduced recursion relations with covariant massive-massless shift, we study tree-level scattering amplitudes involving a pair of massive vector bosons and an arbitrary number of gluons in the massive spinor-helicity formalism. In particular, we derive compact expressions for cases in which i) all gluons are of the same helicity and ii) one gluon has flipped helicity and is colour adjacent to one of the massive particles. We provide numerous consistency checks of our results including the exact match of high energy limits with well known MHV and NMHV amplitudes in pure Yang-Mills theory. As a corollary, we obtain an alternative novel representation of the NMHV amplitude.

Comments on all-loop constraints for scattering amplitudes and Feynman integrals

We comment on the status of “Steinmann-like” constraints, i.e. all-loop constraints on consecutive entries of the symbol of scattering amplitudes and Feynman integrals in planar mathcal{N} = 4 super-Yang-Mills, which have been crucial for the recent progress of the bootstrap program. Based on physical discontinuities and Steinmann relations, we first summarize all possible double discontinuities (or first-two-entries) for (the symbol of) amplitudes and integrals in terms of dilogarithms, generalizing well-known results for n = 6, 7 to all multiplicities. As our main result, we find that extended-Steinmann relations hold for all finite integrals that we have checked, including various ladder integrals, generic double-pentagon integrals, as well as finite components of two-loop NMHV amplitudes for any n; with suitable normalization such as minimal subtraction, they hold for n = 8 MHV amplitudes at three loops. We find interesting cancellation between contributions from rational and algebraic letters, and for the former we have also tested cluster-adjacency conditions using the so-called Sklyanin brackets. Finally, we propose a list of possible last-two-entries for MHV amplitudes up to 9 points derived from overline{Q} equations, which can be used to reduce the space of functions for higher-point MHV amplitudes.

Next-to-MHV Yang-Mills kinematic algebra

Kinematic numerators of Yang-Mills scattering amplitudes possess a rich Lie algebraic structure that suggest the existence of a hidden infinite-dimensional kinematic algebra. Explicitly realizing such a kinematic algebra is a longstanding open problem that only has had partial success for simple helicity sectors. In past work, we introduced a framework using tensor currents and fusion rules to generate BCJ numerators of a special subsector of NMHV amplitudes in Yang-Mills theory. Here we enlarge the scope and explicitly realize a kinematic algebra for all NMHV amplitudes. Master numerators are obtained directly from the algebraic rules and through commutators and kinematic Jacobi identities other numerators can be generated. Inspecting the output of the algebra, we conjecture a closed-form expression for the master BCJ numerator up to any multiplicity. We also introduce a new method, based on group algebra of the permutation group, to solve for the generalized gauge freedom of BCJ numerators. It uses the recently introduced binary BCJ relations to provide a complete set of NMHV kinematic numerators that consist of pure gauge.

The symbol and alphabet of two-loop NMHV amplitudes from overline{Q} equations

We study the symbol and the alphabet for two-loop NMHV amplitudes in planar mathcal{N} = 4 super-Yang-Mills from the overline{Q} equations, which provide a first-principle method for computing multi-loop amplitudes. Starting from one-loop N2MHV ratio functions, we explain in detail how to use overline{Q} equations to obtain the total differential of two-loop n-point NMHV amplitudes, whose symbol contains letters that are algebraic functions of kinematics for n ≥ 8. We present explicit formula with nice patterns for the part of the symbol involving algebraic letters for all multiplicities, and we find 17 − 2m multiplicative-independent letters for a given square root of Gram determinant, with 0 ≤ m ≤ 4 depending on the number of particles involved in the square root. We also observe that these algebraic letters can be found as poles of one-loop four-mass leading singularities with MHV or NMHV trees. As a byproduct of our algebraic results, we find a large class of components of two-loop NMHV, which can be written as differences of two double-pentagon integrals, particularly simple and free of square roots. As an example, we present the complete symbol for n = 9 whose alphabet contains 59 × 9 rational letters, in addition to the 11 × 9 independent algebraic ones. We also give all-loop NMHV last-entry conditions for all multiplicities.

Mathcal{N} = 7 On-shell diagrams and supergravity amplitudes in momentum twistor space

We derive an on-shell diagram recursion for tree-level scattering amplitudes in mathcal{N} = 7 supergravity. The diagrams are evaluated in terms of Grassmannian integrals and momentum twistors, generalising previous results of Hodges in momentum twistor space to non-MHV amplitudes. In particular, we recast five and six-point NMHV amplitudes in terms of mathcal{N} = 7 R-invariants analogous to those of mathcal{N} = 4 super-Yang-Mills, which makes cancellation of spurious poles more transparent. Above 5-points, this requires defining momentum twistors with respect to different orderings of the external momenta.

Rooting out letters: octagonal symbol alphabets and algebraic number theory

It is widely expected that NMHV amplitudes in planar, maximally supersymmetric Yang-Mills theory require symbol letters that are not rationally expressible in terms of momentum-twistor (or cluster) variables starting at two loops for eight particles. Re- cent advances in loop integration technology have made this an ‘experimentally testable’ hypothesis: compute the amplitude at some kinematic point, and see if algebraic symbol letters arise. We demonstrate the feasibility of such a test by directly integrating the most difficult of the two-loop topologies required. This integral, together with its rotated image, suffices to determine the simplest NMHV component amplitude: the unique component finite at this order. Although each of these integrals involve algebraic symbol alphabets, the combination contributing to this amplitude is — surprisingly — rational. We describe the steps involved in this analysis, which requires several novel tricks of loop integration and also a considerable degree of algebraic number theory. We find dramatic and unusual simplifications, in which the two symbols initially expressed as almost ten million terms in over two thousand letters combine in a form that can be written in five thousand terms and twenty-five letters.

The twistor Wilson loop and the amplituhedron

The amplituhedron provides a beautiful description of perturbative superamplitude integrands in mathcal{N}=4 SYM in terms of purely geometric objects, generalisations of polytopes. On the other hand the Wilson loop in supertwistor space also gives an explicit description of these superamplitudes as a sum of planar Feynman diagrams. Each Feynman diagram can be naturally associated with a geometrical object in the same space as the amplituhedron (although not uniquely). This suggests that these geometric images of the Feynman diagrams give a tessellation of the amplituhedron. This turns out to be the case for NMHV amplitudes. We argue however that beyond NMHV this is not true. Specifically, each Feynman diagram leads to an image with a physical boundary and spurious boundaries. The spurious ones should be “internal”, matching with neighbouring diagrams. We however show that there is no choice of geometric image of the Wilson loop Feynman diagrams which yields a geometric object without leaving unmatched spurious boundaries.

Boundaries of amplituhedra and NMHV symbol alphabets at two loops

In this sequel to [3] we classify the boundaries of amplituhedra relevant for determining the branch points of general two-loop amplitudes in planar mathcal{N}=4 super-Yang-Mills theory. We explain the connection to on-shell diagrams, which serves as a useful cross-check. We determine the branch points of all two-loop NMHV amplitudes by solving the Landau equations for the relevant configurations and are led thereby to a conjecture for the symbol alphabets of all such amplitudes.

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 left(overline{Q}right) 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.

Hexagonal Wilson loops in planar SYM theory at finite coupling

We report on the complete OPE series for the 6-gluon MHV and NMHV amplitudes in planar 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.

Towards NMHV amplitudes at strong coupling

Pentagon Operator Product Expansion provides a non-perturbative framework for analysis of scattering amplitudes in planar maximally supersymmetric gauge theory building up on their duality to null polygonal superWilson loop and integrability. In this paper, we construct a systematic expansion for the main ingredients of the formalism, i.e., pentagons, at large 't Hooft coupling as a power series in its inverse value. The calculations are tested against relations provided by the so-called Descent Equation which mixes transitions at different perturbative orders. We use leading order results to have a first glimpse into the structure of scattering amplitude at NMHV level at strong coupling.

Towards the amplituhedron volume

It has been recently conjectured that scattering amplitudes in planar N=4 super Yang-Mills are given by the volume of the (dual) amplituhedron. In this paper we show some interesting connections between the tree-level amplituhedron and a special class of differential equations. In particular we demonstrate how the amplituhedron volume for NMHV amplitudes is determined by these differential equations. The new formulation allows for a straightforward geometric description, without any reference to triangulations. Finally we discuss possible implications for volumes related to generic N^kMHV amplitudes.

On factorization of multiparticle pentagons

We address the near-collinear expansion of multiparticle NMHV amplitudes, namely, the heptagon and octagons in the dual language of null polygonal super Wilson loops. In particular, we verify multiparticle factorization of charged pentagon transitions in terms of pentagons for single flux-tube excitations within the framework of refined operator product expansion. We find a perfect agreement with available tree and one-loop data.

Nonsinglet pentagons and NMHV amplitudes

Scattering amplitudes in maximally supersymmetric gauge theory receive a dual description in terms of the expectation value of the super Wilson loop stretched on a null polygonal contour. This makes the analysis amenable to nonperturbative techniques. Presently, we elaborate on a refined form of the operator product expansion in terms of pentagon transitions to compute twist-two contributions to NMHV amplitudes. To start with, we provide a novel derivation of scattering matrices starting from Baxter equations for flux-tube excitations propagating on magnon background. We propose bootstrap equations obeyed by pentagon form factors with nonsinglet quantum numbers with respect to the R-symmetry group and provide solutions to them to all orders in 't Hooft coupling. These are then successfully confronted against available perturbative calculations for NMHV amplitudes to four-loop order.

Eliminating spurious poles from gauge-theoretic amplitudes

This 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.

The Multi-Regge limit of NMHV amplitudes in N=4 SYM theory

We consider the multi-Regge limit for N=4 SYM NMHV leading color amplitudes in two different formulations: the BFKL formalism for multi-Regge amplitudes in leading logarithm approximation, and superconformal N=4 SYM amplitudes. It is shown that the two approaches agree to two-loops for the 2->4 and 3->3 six-point amplitudes. Predictions are made for the multi-Regge limit of three loop 2->4 and 3->3 NMHV amplitudes, as well as a particular sub-set of two loop 2 ->2 +n N^kMHV amplitudes in the multi-Regge limit in the leading logarithm approximation from the BFKL point of view.

Superpropagator and superconformal invariants

We construct a superpropagator in maximally supersymmetric Yang–Mills theory which is invariant off-shell under a chiral half of supersymmetries. Motivated by the duality with scattering amplitudes in this theory, we apply this superpropagator to supersymmetric Wilson loop on polygonal contours. By performing explicit one-loop calculations we confirm the absence of anomalies and verify the duality between the object under study and NMHV amplitudes.

Uplifting amplitudes in special kinematics

We consider scattering amplitudes in planar N = 4 supersymmetric Yang-Mills theory in special kinematics where all external four-dimensional momenta are restricted to a (1+1)-dimensional subspace. The amplitudes are known to satisfy non-trivial factorisation properties arising from multi-collinear limits, which we further study here. We are able to find a general solution to these multi-collinear limits. This results in a simple formula which represents an n-point superamplitude in terms of a linear combination of functions S_m which are constrained to vanish in all appropriate multi-collinear limits. These collinear-vanishing building blocks, S_m, are dual-conformally-invariant functions which depend on the reduced m-point kinematics with 8 \leq m \leq 4l. For MHV amplitudes they can be constructed directly using, for example, the approach in Ref. [1]. This procedure provides a universal uplift of lower-point collinearly vanishing building blocks S_m to all higher-point amplitudes. It works at any loop-level l \geq 1 and for any MHV or N^kMHV amplitude. We compare this with explicit examples involving n-point MHV amplitudes at 2-loops and 10-point MHV amplitudes at 3-loops. Tree-level superamplitudes have different properties and are treated separately from loop-level amplitudes in our approach. To illustrate this we derive an expression for n-point tree-level NMHV amplitudes in special kinematics.

Double parton scattering singularity in one-loop integrals

We present a detailed study of the double parton scattering (DPS) singularity, which is a specific type of Landau singularity that can occur in certain one-loop graphs in theories with massless particles. A simple formula for the DPS singular part of a four-point diagram with arbitrary internal/external particles is derived in terms of the transverse momentum integral of a product of light cone wavefunctions with tree-level matrix elements. This is used to reproduce and explain some results for DPS singularities in box integrals that have been obtained using traditional loop integration techniques. The formula can be straightforwardly generalised to calculate the DPS singularity in loops with an arbitrary number of external particles. We use the generalised version to explain why the specific MHV and NMHV six-photon amplitudes often studied by the NLO multileg community are not divergent at the DPS singular point, and point out that whilst all NMHV amplitudes are always finite, certain MHV amplitudes do contain a DPS divergence. It is shown that our framework for calculating DPS divergences in loop diagrams is entirely consistent with the `two-parton GPD' framework of Diehl and Schafer for calculating proton-proton DPS cross sections, but is inconsistent with the `double PDF' framework of Snigirev.