non-MHV Amplitudes Research Articles

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.

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 overline{Q} -equation which imposes constraints on the derivatives of the transcendental functions appearing in loop amplitudes.

Tree-level gluon amplitudes on the celestial sphere

Pasterski, Shao and Strominger have recently proposed that massless scattering amplitudes can be mapped to correlators on the celestial sphere at infinity via a Mellin transform. We apply this prescription to arbitrary n-point tree-level gluon amplitudes. The Mellin transforms of MHV amplitudes are given by generalized hypergeometric functions on the Grassmannian Gr(4,n), while generic non-MHV amplitudes are given by more complicated Gelfand A-hypergeometric functions.

Resummed tree heptagon

The form factor program for the regularized space–time S-matrix in planar maximally supersymmetric gauge theory, known as the pentagon operator product expansion, is formulated in terms of flux-tube excitations propagating on a dual two-dimensional world-sheet, whose dynamics is known exactly as a function of 't Hooft coupling. Both MHV and non-MHV amplitudes are described in a uniform, systematic fashion within this framework, with the difference between the two encoded in coupling-dependent helicity form factors expressed via Zhukowski variables. The nontrivial SU(4) tensor structure of flux-tube transitions is coupling independent and is known for any number of charged excitations from solutions of a system of Watson and Mirror equations. This description allows one to resum the infinite series of form factors and recover the space–time S-matrix exactly in kinematical variables at a given order of perturbation series. Recently, this was done for the hexagon. Presently, we successfully perform resummation for the seven-leg tree NMHV amplitude. To this end, we construct the flux-tube integrands of the fifteen independent Grassmann component of the heptagon with an infinite number of small fermion–antifermion pairs accounted for in NMHV two-channel conformal blocks.

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.

OPE for all helicity amplitudes II. Form factors and data analysis

We present the general flux tube integrand for MHV and non-MHV amplitudes, in planar N = 4 SYM theory, up to a group theoretical rational factor. We find that the MHV and non-MHV cases only differ by simple form factors which we derive. This information allows us to run the operator product expansion program for all sorts of non-MHV amplitudes and to test the recently proposed map with the so called charged pentagons transitions. Perfect agreement is found, on a large sample of non-MHV amplitudes, with the perturbative data available in the literature.

Formula omitted] scattering amplitudes and the deformed Graßmannian

Some time ago the general tree-level scattering amplitudes of N=4 Super Yang–Mills theory were expressed as certain Graßmannian contour integrals. These remarkable formulas allow to clearly expose the super-conformal, dual super-conformal, and Yangian symmetries of the amplitudes. Using ideas from integrability it was recently shown that the building blocks of the amplitudes permit a natural multi-parameter deformation. However, this approach had been criticized by the observation that it seemed impossible to reassemble the building blocks into Yangian-invariant deformed non-MHV amplitudes. In this note we demonstrate that the deformations may be succinctly summarized by a simple modification of the measure of the Graßmannian integrals, leading to a Yangian-invariant deformation of the general tree-level amplitudes. Interestingly, the deformed building blocks appear as residues of poles in the spectral parameter planes. Given that the contour integrals also contain information on the amplitudes at loop-level, we expect the deformations to be useful there as well. In particular, applying meromorphicity arguments, they may be expected to regulate all notorious infrared divergences. We also point out relations to Gelfand hypergeometric functions and the quantum Knizhnik–Zamolodchikov equations.

Amplitudes of 3d Yang Mills theory

This paper studies various properties of amplitudes in 3d super Yang Mills theory. First we explain how to obtain the amplitudes of 3d super Yang Mills theories from 4d super Yang Mills theories and obtain their helicity structure. Next, we use a 3d BFCW recursion relation to show that the tree amplitudes and loop integrands of maximal 3d super Yang Mills have dual conformal covariance (although not invariance, so that the amplitudes themselves are not dual conformal). Finally, we argue that the one-loop amplitudes of maximal 3d super Yang-Mills can be reduced to scalar box diagrams and evaluate these diagrams using dimensional regularization. We find that the one-loop MHV amplitudes vanish and the one-loop non-MHV amplitudes are finite.

The super-correlator/super-amplitude duality: Part I

We extend the recently discovered duality between MHV amplitudes and the light-cone limit of correlation functions of a particular type of local scalar operators to generic non-MHV amplitudes in planar N=4 SYM theory. We consider the natural generalization of the bosonic correlators to super-correlators of stress-tensor multiplets and show, in a number of examples, that their light-cone limit exactly reproduces the square of the matching super-amplitudes. We show that the correlation function computed at Born level is dual to the tree-level amplitude if all of its points form a light-like polygon. If a subset of points are not on the light-like polygon but are integrated over, they become Lagrangian insertions generating the loop corrections to the correlator. In this case the duality with amplitudes holds at the level of the integrand. We build up the superspace formalism needed to formulate the duality and present the explicit example of the n-point NMHV tree amplitude as the dual of the lowest nilpotent level in the correlator.

Conformal anomaly of super Wilson loop

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 on a version of dimensional regularization, known as Four-Dimensional Helicity scheme. Recently it was demonstrated that its use for the super Wilson loop at one loop breaks both conformal symmetry and Poincaré supersymmetry. Presently, we exhibit the origin for these effects and demonstrate how one can undo this breaking. The phenomenon is alike the one emerging in renormalization group mixing of conformal operators in conformal theories when one uses dimensional regularization. The rotation matrix to the diagonal basis is found by means of computing the anomaly in the Ward identity for the conformal boost. Presently, we apply this ideology to the super Wilson loop. We compute the one-loop conformal anomaly for the super Wilson loop and find that the anomaly depends on its Grassmann coordinates. By subtracting this anomalous contribution from the super Wilson loop we restore its interpretation as a dual description for reduced non-MHV amplitudes which are expressed in terms of superconformal invariants.

Holonomies of gauge fields in twistor space 4: Functional MHV rules and one-loop amplitudes

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 result, we propose a novel regularization scheme by use of an iterated-integral representation of polylogarithms and obtain a set of new analytic expressions for one-loop NMHV and N 2MHV amplitudes in a conjectural form. We also briefly sketch how the extension to one-loop non-MHV amplitudes in general can be carried out.

New differential equations for on-shell loop integrals

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 $ \mathcal{N} = 4 $ super Yang-Mills. One of our motivations is to provide a tool for deriving analytical results for scattering amplitudes in this theory. We show that the integrals needed for planar MHV amplitudes up to two loops can be thought of as deriving from a single master topology. The master integral satisfies our differential equations, and so do most of the reduced integrals. A consequence of the differential equations is that the integrals we discuss are not arbitrarily complicated transcendental functions. For two specific two-loop integrals we give the full analytic solution. The simplicity of the integrals appearing in the scattering amplitudes in planar $ \mathcal{N} = 4 $ super Yang-Mills is strongly suggestive of a relation to the conjectured underlying integrability of the theory. We expect these differential equations to be relevant for all planar MHV and non-MHV amplitudes. We also discuss possible extensions of our method to more general classes of integrals.

From trees to loops and back

We argue that generic one-loop scattering amplitudes in supersymmetric Yang-Mills theories can be computed equivalently with MHV diagrams or with Feynman diagrams. We first present a general proof of the covariance of one-loop non-MHV amplitudes obtained from MHV diagrams. This proof relies only on the local character in Minkowski space of MHV vertices and on an application of the Feynman Tree Theorem. We then show that the discontinuities of one-loop scattering amplitudes computed with MHV diagrams are precisely the same as those computed with standard methods. Furthermore, we analyse collinear limits and soft limits of generic non-MHV amplitudes in supersymmetric Yang-Mills theories with one-loop MHV diagrams. In particular, we find a simple explicit derivation of the universal one-loop splitting functions in supersymmetric Yang-Mills theories to all orders in the dimensional regularisation parameter, which is in complete agreement with known results. Finally, we present concrete and illustrative applications of Feynman's Tree Theorem to one-loop MHV diagrams as well as to one-loop Feynman diagrams.

Compact formulas for tree amplitudes of six partons

Compact results are obtained for tree-level non-MHV amplitudes of six fermions and of four fermions and two gluons, by using extended BCF/BCFW rules. Combining with previous results, complete set of tree amplitudes of six partons are now available in compact forms.

MHV Rules for Higgs Plus Multi-Parton Amplitudes

We present MHV-rules for constructing perturbative amplitudes for a Higgs boson and an arbitrary number of partons. We give explicit expressions for amplitudes involving a Higgs and three negative helicity partons and any number of positive helicity partons — the NMHV amplitudes. We also present a recursive formulation of MHV rules that incorporates the Higgs, quarks and gluons. The recursion relations are valid for all non-MHV amplitudes. The general results agree numerically with all of the available Higgs + n-parton amplitudes and in some cases provide considerably shorter expressions.

Recursion Relations for Tree Amplitudes in Super Gauge Theories

Using newly proposed recursion relations in [25], [26], compact formulas are obtained for tree-level amplitudes of g1−g2−g3−g4+g5+...gn+. We then make an extension of these recursion relations to include fermions of multi-flavors, from which MHV and amplitudes are reproduced. We also calculate non-MHV amplitudes of processes with two fermions and four gluons. Results thus obtained are equivalent to those obtained by extended CSW prescriptions, and those by conventional field theory calculations.

An Early Route to MHV Tree Amplitudes

In the perturbative expansion of a gauge theory, large numbers of Feynman amplitudes combine to produce mathematically simple (and elegant) expressions. So many people had long suspected that deeper symmetry structures were involved, but those structures remained tantalizingly beyond reach until recently. The Search and Discovery story in the July 2004 issue of Physics Today (page 19) gives a lucid and detailed explanation of the evolving understanding of symmetry structures that underlie Feynman amplitudes. Identification of the connection between the maximally helicity-violating (MHV) amplitudes for the N = 4 super Yang–Mills theory and supertwistor space was a crucial ingredient in the recent developments.I would like to share the reasoning that led me, some years ago, to make an early proposal, 1 1. V. P. Nair, Phys. Lett. B 214, 215 (1988). https://doi.org/10.1016/0370-2693(88)91471-2 which unfortunately was not mentioned in the Physics Today story. I noticed that the Parke–Taylor formula for the MHV amplitudes involved the inverse of scalar products of spinor momenta that could be related to free fermion propagators (or current correlations of a Wess-Zumino-Witten theory) on the complex projective space CP 1. The space arises naturally (as the fiber) in twistor space. Earlier, Edward Witten had observed that the constraints defining the N = 4 super Yang–Mills theory could be nicely interpreted in supertwistor space. 2 2. E. Witten, Phys. Lett. B 77, 394 (1978). https://doi.org/10.1016/0370-2693(78)90585-3 Putting these observations together, I wrote a formula for MHV amplitudes in N = 4 Yang–Mills theory in twistor language. Conformal symmetry plays an important role. Witten’s recent work is more comprehensive, generalizing this complicated series of relationships to string theory and non-MHV amplitudes, and leading to many beautiful results that, for the special case of MHV amplitudes, are identical to mine. The work I’ve just described may be useful for practical calculations; but beyond that use, I hope there will emerge a new organizational principle for perturbation theory other than expansion in terms of Feynman amplitudes.REFERENCESSection:ChooseTop of pageREFERENCES <<1. V. P. Nair, Phys. Lett. B 214, 215 (1988). https://doi.org/10.1016/0370-2693(88)91471-2 , Google ScholarCrossref, CAS2. E. Witten, Phys. Lett. B 77, 394 (1978). https://doi.org/10.1016/0370-2693(78)90585-3 , Google ScholarCrossref© 2004 American Institute of Physics.

Tree-levelSmatrix of Yang-Mills theory

In this note we further investigate the procedure for computing tree-level amplitudes in Yang-Mills theory from connected instantons in the B-model on P^{3|4}, emphasizing that the problem of calculating Feynman diagrams is recast into the problem of finding solutions to a certain set of algebraic equations. We show that the B-model correctly reproduces all 6-particle amplitudes, including non-MHV amplitudes with three negative and three positive helicity gluons. As a further check, we also show that n-particle amplitudes obtained from the B-model obey a number of properties required of gauge theory, such as parity symmetry (which relates an integral over degree d curves to one over degree n-d-2 curves) and the soft and collinear gluon poles.