Graviton Amplitude Research Articles

Differential equations for Carrollian amplitudes

Differential equations are powerful tools in the study of correlation functions in conformal field theories (CFTs). Carrollian amplitudes behave as correlation functions of Carrollian CFT that holographically describes asymptotically flat spacetime. We derive linear differential equations satisfied by Carrollian MHV gluon and graviton amplitudes. We obtain non-distributional solutions for both the gluon and graviton cases. We perform various consistency checks for these differential equations, including compatibility with conformal Carrollian symmetries.

On universal splittings of tree-level particle and string scattering amplitudes

In this paper, we study the newly discovered universal splitting behavior for tree-level scattering amplitudes of particles and strings [1]: when a set of Mandelstam variables (and Lorentz products involving polarizations for gluons/gravitons) vanish, the n-point amplitude factorizes as the product of two lower-point currents with n+3 external legs in total. We refer to any such subspace of the kinematic space of n massless momenta as “2-split kinematics”, where the scattering potential for string amplitudes and the corresponding scattering equations for particle amplitudes nicely split into two parts. Based on these, we provide a systematic and detailed study of the splitting behavior for essentially all ingredients which appear as integrands for open- and closed-string amplitudes as well as Cachazo-He-Yuan (CHY) formulas, including Parke-Taylor factors, correlators in superstring and bosonic string theories, and CHY integrands for a variety of amplitudes of scalars, gluons and gravitons. These results then immediately lead to the splitting behavior of string and particle amplitudes in a wide range of theories, including bi-adjoint ϕ3 (with string extension known as Z and J integrals), non-linear sigma model, Dirac-Born-Infeld, the special Galileon, etc., as well as Yang-Mills and Einstein gravity (with bosonic and superstring extensions). Our results imply and extend some other factorization behavior of tree amplitudes considered recently, including smooth splittings [2] and factorizations near zeros [3], to all these theories. A special case of splitting also yields soft theorems for gluons/gravitons as well as analogous soft behavior for Goldstone particles near their Adler zeros.

Computing NMHV gravity amplitudes at infinity

In this note we show how the solutions to the scattering equations in the NMHV sector fully decompose into subsectors in the z → ∞ limit of a Risager deformation. Each subsector is characterized by the punctures that coalesce in the limit. This naturally decomposes the E(n − 3, 1) solutions into sets characterized by partitions of n − 3 elements so that exactly one subset has more than one element. We present analytic expressions for the leading order of the solutions in an expansion around infinite z for any n. We also give a simple algorithm for numerically computing arbitrarily high orders in the same expansion. As a consequence, one has the ability to compute Yang-Mills and gravity amplitudes purely from this expansion around infinity. Moreover, we present a new analytic computation of the residue at infinity of the n = 12 NMHV tree-level gravity amplitude which agrees with the results of Conde and Rajabi. In fact, we present the analytic form of the leading order in 1/z of the Cachazo-Skinner-Mason/CHY formula for graviton amplitudes for each subsector and to all multiplicity. As a byproduct of the all-order algorithm, one has access to the numerical value of the residue at infinity for any n and hence to the corrected CSW (or MHV) expansion for NMHV gravity amplitudes.

On the anomaly interpretation of amplitudes in self-dual Yang-Mills and gravity

We investigate the integrability anomalies arising in the self-dual sectors of gravity and Yang-Mills theory, focusing on their connection to both the chiral anomaly and the trace anomaly. The anomalies in the self-dual sectors generate the one-loop all-plus amplitudes of gravitons and gluons, and have recently been studied via twistor constructions. On the one hand, we show how they can be interpreted as an anomaly of the chiral U(1) electric-magnetic-type duality in the self-dual sectors. We also note the similarity, for the usual fermionic chiral anomaly, between the 4D setting of self-dual Yang-Mills and the 2D setting of the Schwinger model. On the other hand, the anomalies in the self-dual theories also resemble the trace anomaly, sharing the same type of non-local effective action. We highlight the role of a Weyl-covariant fourth-order differential operator familiar from the trace anomaly literature, which (i) explains the conformal properties of the one-loop amplitudes, and (ii) indicates how this story may be extended to non-trivial spacetime backgrounds, e.g. with a cosmological constant. Moving beyond the self-dual sectors, and focusing on the gravity case, we comment on an intriguing connection to the two-loop ultraviolet divergence of pure gravity, whereby cancelling the anomaly at one-loop eliminates the two-loop divergence for the simplest helicity amplitudes.

Carrollian amplitudes and celestial symmetries

Carrollian holography aims to express gravity in four-dimensional asymptotically flat spacetime in terms of a dual three-dimensional Carrollian CFT living at null infinity. Carrollian amplitudes are massless scattering amplitudes written in terms of asymptotic or null data at I\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{I} $$\\end{document}. These position space amplitudes at I\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{I} $$\\end{document} are to be re-interpreted as correlation functions in the putative dual Carrollian CFT. We derive basic results concerning tree-level Carrollian amplitudes yielding dynamical constraints on the holographic dual. We obtain surprisingly compact expressions for n-point MHV gluon and graviton amplitudes in position space at I\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{I} $$\\end{document}. We discuss the UV/IR behaviours of Carrollian amplitudes and investigate their collinear limit, which allows us to define a notion of Carrollian OPE. By smearing the OPE along the generators of null infinity, we obtain the action of the celestial symmetries — namely, the S algebra for Yang-Mills theory and Lw1+∞ for gravity — on the Carrollian operators. As a consistency check, we systematically relate our results with celestial amplitudes using the link between the two approaches. Finally, we initiate a direct connection between twistor space and Carrollian amplitudes.

No U(1) ‘electric-magnetic’ duality in Einstein gravity

We revisit the question of whether classical general relativity obeys, beyond the linearised order, an analogue of the global U(1) electric-magnetic duality of Maxwell theory, with the Riemann tensor playing the role analogous to the field strength. Following contradictory claims in the literature, we present a simple gauge-invariant argument that the duality does not hold. The duality condition is the conservation of the helicity charge. Scattering amplitudes of gravitons in general relativity, and of gluons in Yang-Mills theory, violate this selection rule already at tree level. Indeed, the maximally-helicity-violating (MHV) amplitudes are famous for their simplicity. The duality in the linearised theories is, therefore, broken by the interactions. In contrast, the tree-level scattering amplitudes in duality-invariant theories of non-linear electromagnetism are known to obey helicity conservation. While the duality is not a symmetry of the full theory of general relativity, it does hold within a sector of the solution space, including vacuum type D solutions, where the duality is known to rotate between mass and NUT charge.

Three point amplitudes in matrix theory

We compute the three graviton amplitude in the Banks-Fischler-Shenker-Susskind matrix model for M-theory. Even though the three point amplitude is determined by super Poincare invariance in eleven dimensional M-theory, it requires a non-trivial computation in the matrix model. We consider a configuration where all three gravitons carry non-zero longitudinal momentum. To simplify the problem, we compactify one additional dimension and relate the amplitude to a supersymmetric index computation. We find agreement with the expected answer even at finite values of N.

Scalar-graviton amplitudes and celestial holography

We compute scattering amplitudes involving one massive scalar and two, three, or four gravitons. We show that when the conformal dimension of the massive scalar is set to zero, the resulting celestial correlators depend only on the coordinates of the gravitons. Such correlators of gravitons are well-defined and do not suffer from divergences associated with the Mellin transform of usual graviton amplitudes. Moreover, they are non-distributional and take the form of standard CFT correlators. We show that they are consistent with the usual OPEs but the statement of the soft theorem is modified.

Gravity Amplitudes from Double Bonus Relations.

In this Letter, we derive new expressions for tree-level graviton amplitudes in N=8 supergravity from Britto-Cachazo-Feng-Witten (BCFW) recursion relations combined with new types of bonus relations. These bonus relations go beyond the famous 1/z^{2} behavior under a large BCFW shift and use knowledge about certain zeros of graviton amplitudes in collinear kinematics. This extra knowledge can be used in the context of global residue theorems by writing the amplitude in a special form using canonical building blocks. In the next-to-maximally-helicity-violating case, these building blocks are dressed one-loop leading singularities, the same objects that appear in the expansion of Yang-Mills amplitudes, where each term corresponds to an R invariant. Unlike other approaches, our formula is not an expansion in terms of cyclic objects and does not manifest color-kinematics duality but rather preserves the permutational symmetry of its building blocks. We also comment on the possible connection to Grassmannian geometry and give some nontrivial evidence of such structure for graviton amplitudes.

Scattering on self-dual Taub-NUT

We derive exact solutions of massless free field equations and tree-level two-point amplitudes up to spin 2 on self-dual Taub-NUT (SDTN) spacetime, as well as on its single copy, the self-dual dyon (SDD). We use Killing spinors to build analogues of momentum eigenstates, finding that, in the spirit of color-kinematics duality, those for the SDD lift directly to provide states on the SDTN background if one replaces charge with energy. We discover that they are forced to have faster growth at infinity than in flat space due to the topological non-triviality of these backgrounds. The amplitudes for massless scalars and spinning particles in the and helicity configurations vanish for generic kinematics as a consequence of the integrability of the self-dual sector. The amplitudes are non-vanishing and we compute them exactly in the backgrounds, which are treated non-perturbatively. It is explained how spin is easily introduced via a Newman–Janis imaginary shift along the spin-vector leading directly to the well-known exponential factor in the dot product of the spin with the momenta. We also observe a double copy relation between the gluon amplitude on a SDD and graviton amplitude on a SDTN spacetime.

Scattering in black hole backgrounds and higher-spin amplitudes. Part II

We continue to investigate correspondences between, on the one hand, scattering amplitudes for massive higher-spin particles and gravitons in appropriate quantum-to-classical limits, and on the other hand, classical gravitational interactions of spinning black holes according to general relativity. We first construct an ansatz for a gravitational Compton amplitude, at tree level, constrained only by locality, crossing symmetry, unitarity and consistency with the linearized-Kerr 3-point amplitude, to all orders in the black hole’s spin. We then explore the extent to which a unique classical Compton amplitude can be identified by comparing with the results of the classical process of scattering long-wavelength gravitational waves off an exact Kerr black hole, determined by appropriate solutions of the Teukolsky equation. Up to fourth order in spin, we find complete agreement with a previously conjectured exponential form of the tree-level Compton amplitude. At higher orders, we extract tree-level contributions from the Teukolsky amplitude by an analytic continuation from a physical (a/GM < 1) to a particle-like (a/GM > 1) regime. Up to the sixth order in spin, we identify a unique conservative part of the amplitude which is insensitive both to the choice of boundary conditions at the black hole horizon and to branch choices in the analytic continuation. The remainder of the amplitude is determined modulo an overall sign from a branch choice, with the sign flipping under exchanging purely ingoing and purely outgoing boundary conditions at the horizon. Along the way, we make contact with novel applications of massive spinor-helicity variables pertaining to their relation to EFT operators and (spinning) partial amplitudes.

Constructing EYM amplitudes by inverse soft limit

It is well known that gravity amplitudes in four dimensions can be reconstructed by the inverse soft limit (ISL) method. According to ISL, a tree level n-graviton maximally-helicity-violating (MHV) amplitude is expressed in terms of deformed (n − 1)-graviton amplitudes accompanied by soft graviton factors. On another hand, single- and double-trace tree-level Einstein-Yang-Mills (EYM) MHV amplitudes have been proven to satisfy spanning forest formulas, where each edge in a forest has the same form with a term in soft graviton factor. It is not transparent that the formulas satisfied by EYM amplitudes can be constructed with ISL. In this paper, we construct the single- and double-trace MHV amplitudes in EYM, by the ISL and show that the known formulas can be precisely reproduced. Interesting identities which are based on Schouten identity and characterized by graphs are also introduced.

All plus four point (A)dS graviton function using generalized on-shell recursion relation

This paper presents a calculation of the four gravitons amplitude in (Anti)-de Sitter space, focusing specifically on external gravitons with positive helicity. To achieve this, we employ a generalized recursion method that involves complexifying all external momentum of the graviton function, which results in the factorization of AdS graviton amplitudes and eliminates the need for Feynman-Witten diagrams. Our calculations were conducted in three boundary dimensions, with a particular emphasis on exploring cosmology and aiding the cosmological bootstrap program. To compute the expression, we utilized the three-dimensional spinor helicity formalism. The final expression was obtained by summing over residues of physical poles, and we present both symbolic and numerical results. Additionally, we discuss the advantages and limitations of this approach, and highlight potential opportunities for future research.

Massive color-kinematics duality and double-copy for Kaluza-Klein scattering amplitudes

We study the structure of scattering amplitudes of massive Kaluza-Klein (KK) states under toroidal compactification. We present a shifting method to quantitatively derive the scattering amplitudes of massive KK gauge bosons and KK gravitons from the corresponding massless amplitudes in the noncompactified higher dimensional theories. With these we construct the massive KK scattering amplitudes by extending the double-copy relations of massless scattering amplitudes within the field theory framework, including both the BCJ and CHY methods, and build up their connections to the massive KK KLT relations. We present the massive BCJ-type double-copy construction of the N-point KK gauge boson/graviton scattering amplitudes, and as the applications we derive explicitly the four-point KK scattering amplitudes as well as the five-point KK scattering amplitudes. We further study the nonrelativistic limit of these massive scattering amplitudes with the heavy external KK states and discuss the impact of the compactified extra dimensions on the low energy gravitational potential. Finally, we analyze the four-point and N-point mass spectral conditions and newly propose a novel group theory approach to prove that only the KK theories under toroidal compactification can satisfy these conditions for directly realizing massive double-copy in the field theory framework.

On the associativity of 1-loop corrections to the celestial operator product in gravity

The question of whether the holomorphic collinear singularities of graviton amplitudes define a consistent chiral algebra has garnered much recent attention. We analyse a version of this question for infinitesimal perturbations around the self-dual sector of 4d Einstein gravity. The singularities of tree amplitudes in such perturbations do form a consistent chiral algebra, however at 1-loop its operator products are corrected by the effective graviton vertex. We argue that the chiral algebra can be interpreted as the universal holomorphic surface defect in the twistor uplift of self-dual gravity, and show that the same correction is induced by an anomalous diagram in the bulk-defect system. The 1-loop holomorphic collinear singularities do not form a consistent chiral algebra. The failure of associativity can be traced to the existence of a recently discovered gravitational anomaly on twistor space. It can be restored by coupling to an unusual 4th-order gravitational axion, which cancels the anomaly by a Green-Schwarz mechanism. Alternatively, the anomaly vanishes in certain theories of self-dual gravity coupled to matter, including in self-dual supergravity.

Topological Equivalence Theorem and Double-Copy for Chern-Simons Scattering Amplitudes.

We study the mechanism of topological mass generation for 3-dimensional Chern-Simons gauge theories and propose a brand-new topological equivalence theorem to connect scattering amplitudes of the physical gauge boson states to that of the transverse states under high-energy expansion. We prove a general energy cancelation mechanism for N-point physical gauge boson amplitudes, which predicts large cancelations of E4 - L → E(4 - L) - N at any L-loop level (L ⩾ 0). We extend the double-copy approach to construct massive graviton amplitudes and to study their structures. We newly uncovered a series of strikingly large energy cancelations E12 → E1 of the tree-level 4-graviton scattering amplitude under high-energy expansion and establish a new correspondence between the 2 energy cancelations in the topologically massive Yang-Mills gauge theory and the topologically massive gravity theory. We further study the scattering amplitudes of Chern-Simons gauge bosons and gravitons in the nonrelativistic limit.

Celestial holography meets twisted holography: 4d amplitudes from chiral correlators

We propose a new program for computing a certain integrand of scattering amplitudes of four-dimensional gauge theories which we call the form factor integrand, starting from 6d holomorphic theories on twistor space. We show that the form factor integrands can be expressed as sums of products of 1.) correlators of a 2d chiral algebra, related to the algebra of asymptotic symmetries uncovered recently in the celestial holography program, and 2.) OPE coefficients of a 4d non-unitary CFT. We prove that conformal blocks of the chiral algebras are in one-to-one correspondence with local operators in 4d. We use this bijection to recover the Parke-Taylor formula, the CSW formula, and certain one-loop scattering amplitudes. Along the way, we explain and derive various aspects of celestial holography, incorporating techniques from the twisted holography program such as Koszul duality. This perspective allows us to easily and efficiently recover the infinite-dimensional chiral algebras of asymptotic symmetries recently extracted from scattering amplitudes of massless gluons and gravitons in the celestial basis. We also compute some simple one-loop corrections to the chiral algebras and derive the three-dimensional bulk theories for which these 2d algebras furnish an algebra of boundary local operators.

Supersymmetric Massive Gravity

We initiate a systematic study of the self-interactions of a massive spin-2 “graviton” consistent with up to mathcal{N} = 4 supersymmetry. Using a recently developed massive on-shell superspace formalism, we construct the most general set of cubic massive graviton amplitudes in a form with all supersymmetry and Lorentz invariance manifest. We find that for mathcal{N} ≥ 3 supersymmetry, the family of consistent interactions coincide with those of the ghost-free dRGT model. For mathcal{N} = 4 (maximal) supersymmetry there is a single consistent cubic interaction which coincides with the unique structure required for the absence of asymptotic superluminality. Additionally, we discuss the structure of interactions in the high-energy limit, connections to supersymmetric Galileons and the possibility of a supersymmetric massive double copy.

Universal expansions of scattering amplitudes for gravitons, gluons, and Goldstone particles

Tree-level scattering amplitudes for gravitons, gluons and Goldstone particles in any dimensions are strongly constrained by basic principles, and they are intimately related to each other via various relations. We study two types of "universal expansions" with respect to gauge bosons and Goldstone bosons: the former express tree amplitudes in Einstein gravity (Yang-Mills) as linear combinations of single-trace Einstein-Yang-Mills (Yang-Mills-$\phi^3$) amplitudes with coefficients given by Lorentz products of polarizations and momenta; the latter express tree amplitudes in non-linear sigma model, (Dirac-)Born-Infeld and a special Galileon theory, as linear combinations of single-trace mixed amplitudes with particles of lower "degree of Adler's zero" and coefficients given by products of Mandelstam variables. We trace the origin of gauge-theory expansions to the powerful uniqueness theorem based on gauge invariance, and expansions in effective field theories can be derived from gauge-theory ones via a special dimension reduction.

Structure of Kaluza-Klein graviton scattering amplitudes from the gravitational equivalence theorem and double copy

We study the structure of scattering amplitudes of the Kaluza-Klein (KK) gravitons and of the gravitational KK Goldstone bosons in the compactified 5d General Relativity (GR). We analyze the geometric "Higgs" mechanism for mass-generation of KK gravitons under compactification with a general $R_\xi$ gauge-fixing, which is free from the vDVZ discontinuity. Then, we formulate the Gravitational Equivalence Theorem (GRET) to connect the longitudinal KK graviton amplitudes to the KK Goldstone amplitudes, which is a manifestation of the geometric Higgs mechanism at $S$-matrix level. We directly compute the tree-level KK Goldstone amplitudes which equal the longitudinal KK graviton amplitudes in the high energy limit. We further extend the double-copy method with color-kinematics duality to reconstruct the massive KK longitudinal graviton (Goldstone) amplitudes from the KK longitudinal gauge boson (Goldstone) amplitudes in the compactified 5d Yang-Mills (YM) theory under high energy expansion. From these, we reconstruct the GRET of the KK longitudinal graviton (Goldstone) amplitudes in the 5d GR from the KK longitudinal gauge boson (Goldstone) amplitudes in the 5d YM theory. Using either the GRET or the double-copy reconstruction, we provide a theoretical mechanism showing that the sum of all the energy-power terms [up to $O(E^{10})$] in the high-energy four longitudinal KK graviton amplitudes must cancel down to $O(E^2)$ as enforced by matching the energy-power dependence of the corresponding KK Goldstone amplitudes or by matching that of the double-copy amplitudes from the KK YM theory. With the double-copy approach, we establish a new correspondence between the two energy-cancellations: $E^4 \to E^0$ in the 5d KK YM theory and $E^{10} \to E^2$ in the 5d KK GR theory. We further analyze the structure of the residual terms in the GRET and uncover a new energy-cancellation mechanism therein.