BCFW Recursion Research Articles

Double-soft limits of gluons and gravitons

The double-soft limit of gluon and graviton amplitudes is studied in four dimensions at tree level. In general this limit is ambiguous and we introduce two natural ways of taking it: a consecutive double-soft limit where one particle is taken soft before the other and a simultaneous limit where both particles are taken soft uniformly. All limits yield universal factorisation formulae which we establish by BCFW recursion relations down to the subleading order in the soft momentum expansion. These formulae generalise the recently discussed subleading single-soft theorems. While both types of limits yield identical results at the leading order, differences appear at the subleading order. Finally, we discuss double-scalar emission in $$ \mathcal{N}=4 $$ super Yang-Mills theory. These results should be of use in establishing the algebraic structure of potential hidden symmetries in the quantum gravity and Yang-Mills S-matrix.

Double soft theorems in gauge and string theories

We investigate the tree-level S-matrix in gauge theories and open superstring theory with several soft particles. We show that scattering amplitudes with two or three soft gluons of non-identical helicities behave universally in the limit, with multi-soft factors which are not the product of individual soft gluon factors. The results are obtained from the BCFW recursion relations in four dimensions, and further extended to arbitrary dimensions using the CHY formula. We also find new soft theorems for double soft limits of scalars and fermions in $$ \mathcal{N}=4 $$ and pure $$ \mathcal{N}=2 $$ SYM. Finally, we show that the double-soft-scalar theorems can be extended to open superstring theory without receiving any α′ corrections.

Recursion relation for boundary contribution

It is well known that under a BCFW-deformation, there is a boundary contribution when the amplitude scales as $$ \mathcal{O}\left({z}^0\right) $$ or worse. We show that boundary contributions have a similar recursion relation as scattering amplitude. Just like the BCFW recursion relation, where scattering amplitudes are expressed as the products of two on-shell subamplitudes (plus possible boundary contributions), our new recursion relation expresses boundary contributions as products of sub-amplitudes and boundary contributions with less legs, plus yet another possible boundary contribution. In other words, the complete scattering amplitude, including boundary contributions, can be obtained by multiple steps of recursions, unless the boundary contributions are still non-zero when all possible deformations are exploited. We demonstrate this algorithm by several examples. Especially, we show that for standard model like renormalizable theory in 4D, i.e., the theory including only gauge boson, fermions and scalars, the complete amplitude can always be computed by at most four recursive steps using our algorithm.

Recursion in multiplet bases for tree-level MHV gluon amplitudes

We investigate the construction of tree-level MHV gluon amplitudes in multiplet bases using BCFW recursion. The multiplet basis decomposition can either be obtained by decomposing results derived in (for example) the DDM basis or by formulating the recursion directly in the multiplet basis. We focus on the latter approach and show how to efficiently deal with the color structure appearing in the recursion. For illustration, we also explicitly calculate the four-, five- and six-gluon amplitudes.

General tree-level amplitudes by factorization limits

To find boundary contributions is a rather difficult problem when applying the BCFW recursion relation. In this paper, we propose an approach to bypass this problem by calculating general tree amplitudes that contain no polynomial using factorization limits. More explicitly, we construct an expression iteratively, which produces correct factorization limits for all physical poles, and does not contain other poles, then it should be the correct amplitude. To some extent, this approach can be considered as an alternative way to find boundary contributions. To demonstrate our approach, we present several examples: $\phi^4$ theory, pure gauge theory, Einstein-Maxwell theory, and Yukawa theory. While the amplitude allows the existence of polynomials which satisfy correct mass dimension and helicities, this approach is not applicable to determine the full amplitude.

The amplituhedron from momentum twistor diagrams

We propose a new diagrammatic formulation of the all-loop scattering amplitudes/Wilson loops in planar N=4 SYM, dubbed the "momentum-twistor diagrams". These are on-shell-diagrams obtained by gluing trivalent black and white vertices defined in momentum twistor space, which, in the reduced diagram case, are known to be related to diagrams in the original twistor space. The new diagrams are manifestly Yangian invariant, and they naturally represent factorization and forward-limit contributions in the all-loop BCFW recursion relations in momentum twistor space, in a fashion that is completely different from those in momentum space. We show how to construct and evaluate momentum-twistor diagrams, and how to use them to obtain tree-level amplitudes and loop-level integrands; in particular for the latter we identify an isolated bubble-structure for each loop variable, arising from a forward limit, or entangled removal of particles. From a given diagram one can directly read off the C, D matrices via a generalized "boundary measurement"; this in turn determines a cell in the amplituhedron associated with the amplitude, and our diagrammatic representations of the amplitude can provide triangulations of the amplituhedron with generally very intricate geometries. To demonstrate the computational power of the formalism, we give explicit results for general two-loop integrands, and the cells of the complete amplituhedron for two-loop MHV amplitudes.

Amplitudes for High Energy Factorization via BCFW Recursion

Recently, an extension of the BCFW on-shell recursion relation suitable to compute gauge invariant scattering amplitudes with off-shell particles has been presented for Yang-Mills theories with fermions. In particular, 4- and 5-point amplitudes have been completely worked out. The results are needed for the study of multi-parton scattering at hadron colliders in the framework of High Energy Factorization (HEF).

From six to four and more: massless and massive maximal super Yang-Mills amplitudes in 6d and 4d and their hidden symmetries

A self-consistent exposition of the theory of tree-level superamplitudes of the 4d N=4 and 6d N=(1,1) maximally supersymmetric Yang-Mills theories is provided. In 4d we work in non-chiral superspace and construct the superconformal and dual superconformal symmetry generators of the N=4 SYM theory using the non-chiral BCFW recursion to prove the latter. In 6d we provide a complete derivation of the standard and hidden symmetries of the tree-level superamplitudes of N=(1,1) SYM theory, again using the BCFW recursion to prove the dual conformal symmetry. Furthermore, we demonstrate that compact analytical formulae for tree-superamplitudes in N=(1,1) SYM can be obtained from a numerical implementation of the supersymmetric BCFW recursion relation. We derive compact manifestly dual conformal representations of the five- and six-point superamplitudes as well as arbitrary multiplicity formulae valid for certain classes of superamplitudes related to ultra-helicity-violating massive amplitudes in 4d. We study massive tree superamplitudes on the Coulomb branch of the N=4 SYM theory from dimensional reduction of the massless superamplitudes of the six-dimensional N=(1,1) SYM theory. We exploit this correspondence to construct the super-Poincare and enhanced dual conformal symmetries of massive tree superamplitudes in N=4 SYM theory which are shown to close into a finite dimensional algebra of Yangian type. Finally, we address the fascinating possibility of uplifting massless 4d superamplitudes to 6d massless superamplitudes proposed by Huang. We confirm the uplift for multiplicities up to eight but show that finding the uplift is highly non-trivial and in fact not of a practical use for multiplicities larger than five.

Loop corrections to soft theorems in gauge theories and gravity

In this paper, we study loop corrections to the recently proposed new soft theorem of Cachazo-Strominger, for both gravity and gauge theory amplitudes. We first review the proof of its tree-level validity based on BCFW recursion relations, which also establishes an infinite series of universals soft functions for MHV amplitudes, and a generalization to supersymmetric cases. For loop corrections, we focus on infrared finite, rational amplitudes at one loop, and apply recursion relations with boundary or double-pole contributions. For all-plus amplitudes, we prove that the subleading soft-theorems are exact to all multiplicities for both gauge and gravity amplitudes. For single-minus amplitudes, while the subleading soft-theorems are again exact for the minus-helicity soft leg, for plus-helicity loop corrections are required. Using recursion relations, we identify the source of such mismatch as stemming from the special contribution containing double poles, and obtain the all-multiplicity one-loop corrections to the subleading soft behavior in Yang-Mills theory. We also comment on the derivation of soft theorems using BCFW recursion in arbitrary dimensions.

The positive orthogonal Grassmannian and loop amplitudes of ABJM

In this paper we study the combinatorics associated with the positive orthogonal Grassmannian OGk and its connection to ABJM scattering amplitudes. We present a canonical embedding of OGk into the Grassmannian Gr(k, 2k), from which we deduce the canonical volume form that is invariant under equivalence moves. Remarkably the canonical forms of all reducible graphs can be converted into reduced ones with products of dlog forms. Unlike super Yang–Mills, here the Jacobian plays a crucial role to ensure the dlog form of the reduced representation. Furthermore in the positive region, we identify the functional map that arises from the triangle equivalence move as a 3-string scattering S-matrix which satisfies the tetrahedron equations by Zamolodchikov, implying (2 + 1)-dimensional integrability. We study the solution to the BCFW recursion relation for loop amplitudes, and demonstrate the presence of all physical singularities as well as the absence of all spurious ones. The on-shell diagram solution to the loop recursion relation exhibits manifest two-site cyclic symmetry and reveals that, to all loop, four and six-point amplitudes only have logarithmic singularities.

BCFW recursion for off-shell gluons

It is shown how tree-level multi-gluon helicity amplitudes with an arbitrary number of off-shell external gluons can be calculated via BCFW recursion. Compact expressions for helicity amplitudes for scattering processes of three and four gluons, with up to three of them off-shell, are presented. Also, maximum-helicity-violating configurations are identified for up to two off-shell gluons, and the expressions for their helicity amplitudes for an arbitrary number of on-shell gluons are derived.

Gravity in Twistor Space and its Grassmannian Formulation

We prove the formula for the complete tree-level $S$-matrix of $\mathcal{N}=8$ supergravity recently conjectured by two of the authors. The proof proceeds by showing that the new formula satisfies the same BCFW recursion relations that physical amplitudes are known to satisfy, with the same initial conditions. As part of the proof, the behavior of the new formula under large BCFW deformations is studied. An unexpected bonus of the analysis is a very straightforward proof of the enigmatic $1/z^2$ behavior of gravity. In addition, we provide a description of gravity amplitudes as a multidimensional contour integral over a Grassmannian. The Grassmannian formulation has a very simple structure; in the N$^{k-2}$MHV sector the integrand is essentially the product of that of an MHV and an $\overline{{\rm MHV}}$ amplitude, with $k+1$ and $n-k-1$ particles respectively.

Yang–Baxter operators and scattering amplitudes in [formula omitted] super-Yang–Mills theory

Yangian symmetry of amplitudes in N=4 super-Yang–Mills theory is formulated in terms of eigenvalue relations for monodromy matrix operators. The Quantum Inverse Scattering Method provides the appropriate tools to treat the extended symmetry and to recover as its consequences many known features like cyclic and inversion symmetry, BCFW recursion, Inverse Soft Limit construction, Grassmannian integral representation, R-invariants and on-shell diagram approach.

ABJM amplitudes and the positive orthogonal Grassmannian

A remarkable connection between perturbative scattering amplitudes of four-dimensional planar SYM, and the stratification of the positive grassmannian, was revealed in the seminal work of Arkani-Hamed et. al. Similar extension for three-dimensional ABJM theory was proposed. Here we establish a direct connection between planar scattering amplitudes of ABJM theory, and singularities there of, to the stratification of the positive orthogonal grassmannian. In particular, scattering processes are constructed through on-shell diagrams, which are simply iterative gluing of the fundamental four-point amplitude. Each diagram is then equivalent to the merging of fundamental OG_2 orthogonal grassmannian to form a larger OG_k, where 2k is the number of external particles. The invariant information that is encoded in each diagram is precisely this stratification. This information can be easily read off via permutation paths of the on-shell diagram, which also can be used to derive a canonical representation of OG_k that manifests the vanishing of consecutive minors as the singularity of all on-shell diagrams. Quite remarkably, for the BCFW recursion representation of the tree-level amplitudes, the on-shell diagram manifests the presence of all physical factorization poles, as well as the cancellation of the spurious poles. After analytically continuing the orthogonal grassmannian to split signature, we reveal that each on-shell diagram in fact resides in the positive cell of the orthogonal grassmannian, where all minors are positive. In this language, the amplitudes of ABJM theory is simply an integral of a product of dlog forms, over the positive orthogonal grassmannian.

Classical space–times from the S-matrix

We show that classical space–times can be derived directly from the S-matrix for a theory of massive particles coupled to a massless spin two particle. As an explicit example we derive the Schwarzchild space–time as a series in GN. At no point of the derivation is any use made of the Einstein–Hilbert action or the Einstein equations. The intermediate steps involve only on-shell S-matrix elements which are generated via BCFW recursion relations and unitarity sewing techniques. The notion of a space–time metric is only introduced at the end of the calculation where it is extracted by matching the potential determined by the S-matrix to the geodesic motion of a test particle. Other static space–times such as Kerr follow in a similar manner. Furthermore, given that the procedure is action independent and depends only upon the choice of the representation of the little group, solutions to Yang–Mills (YM) theory can be generated in the same fashion. Moreover, the squaring relation between the YM and gravity three point functions shows that the seeds that generate solutions in the two theories are algebraically related. From a technical standpoint our methodology can also be utilized to calculate quantities relevant for the binary inspiral problem more efficiently then the more traditional Feynman diagram approach.

Boundary behaviors for general off-shell amplitudes in Yang-Mills theory

The boundary behavior of amplitudes---the amplitudes' behavior under a large Britto-Cachazo-Feng-Witten (BCFW) momenta deformation for a pair of legs---in Yang-Mills theory is of great interest recently. In this article we analyze the boundary behavior of off-shell Yang-Mills amplitudes in Feynman gauge. The deformed legs can be either adjacent or nonadjacent. We find that a set of reduced vertices can be used to simplify the analysis and calculation of the boundary behavior of amplitudes. Boundary behavior for amplitudes with adjacent BCFW deformation is read off from the reduced vertices. Then we discover a relationship between a permutation sum with fixed color ordering of the legs and the improved boundary behavior for the off-shell amplitudes with a nonadjacent BCFW momenta deformation. Based on the boundary behavior, we generalize the BCFW recursion relation to calculate general tree-level off-shell amplitudes and analyze the relations between them.

On-shell recursion for massive fermion currents

Abstract We analyze the validity of BCFW recursion relations for currents of n − 2 gluons and two massive quarks, where one of the quarks is off shell and the remaining particles are on shell. These currents are gauge-dependent and can be used as ingredients in the unitarity-based approach to computing one-loop amplitudes. The validity of BCFW recursion relations is well known to depend on the so-called boundary behavior of the currents as the momentum shift parameter goes to infinity. With off-shell currents, a new potential problem arises, namely unphysical poles that depend on the choice of gauge. We identify conditions under which boundary terms are absent and unphysical poles are avoided, so that there is a natural recursion relation. In particular, we are able to choose a gauge in which we construct a valid shift for currents with at least n − 3 gluons of the same helicity. We derive an analytic formula in the case where all gluons have the same helicity. As by-products, we prove the vanishing boundary behavior of general off-shell objects in Feynman gauge, and we find a compact generalization of Berends-Giele gluon currents with a generic reference spinor.

Ward identity implies recursion relations in Yang-Mills theory

The Ward identity in gauge theory constrains the behavior of the amplitudes. We discuss the Ward identity for amplitudes with a pair of shifted lines with complex momenta. This will induce a recursion relation identical to BCFW recursion relations at the finite poles of the complexified amplitudes. Furthermore, according to the Ward identity, it is also possible to transform the boundary term into a simple form, which can be obtained by a new recursion relation. For the amplitude with one off-shell line in pure Yang-Mills theory, we find this technique is effective for obtaining the amplitude even when there are boundary contributions.

Gauge theory one-loop amplitudes and the Britto-Cachazo-Feng-Witten recursion relations

We calculate gauge theory one-loop amplitudes with the aid of the complex shift used in the Britto-Cachazo-Feng-Witten (BCFW) recursion relations of tree amplitudes. We apply the shift to the integrand and show that the contribution from the limit of infinite shift vanishes after integrating over the loop momentum, by a judicious choice of basis for polarization vectors. This enables us to write the one-loop amplitude in terms of on-shell tree and lower point one-loop amplitudes. Some of the tree amplitudes are forward amplitudes. We show that their potential singularities do not contribute and the BCFW recursion relations can be applied in such a way as to avoid these singularities altogether. We calculate in detail $n$-point one-loop amplitudes for $n=2,3,4$, and outline the generalization of our method to $n>4$.

Einstein supergravity amplitudes from twistor-string theory

This paper gives a twistor-string formulation for all tree amplitudes of Einstein (super-)gravities for and 4. Formulae are given with and without cosmological constant and with various possibilities for the gauging. The formulae are justified by use of Maldacena's observation that conformal gravity tree amplitudes with Einstein wavefunctions and non-zero cosmological constant will correctly give the Einstein tree amplitudes. This justifies the construction of Einstein gravity amplitudes at from twistor-string theory and is extended to by requiring the standard relation between the MHV-degree and the degree of the rational curve for Yang–Mills; this systematically excludes the spurious conformal supergravity gravity contributions. For comparison, BCFW recursion is used to obtain twistor-string-like formulae at degree 0 and 1 (anti-MHV and MHV) for amplitudes with supersymmetry with and without cosmological constant.