Four-point Amplitudes Research Articles

Symmetry properties of Wilson loops with a Lagrangian insertion

Null Wilson loops in mathcal{N} = 4 super Yang-Mills are dual to planar scattering amplitudes. This duality implies hidden symmetries for both objects. We consider closely related infrared finite observables, defined as the Wilson loop with a Lagrangian insertion, normalized by the Wilson loop itself. Unlike ratio and remainder functions studied in the literature, this observable is non-trivial already for four scattered particles and bears close resemblance to (finite parts of) scattering processes in non-supersymmetric Yang-Mills theory. Moreover, by integrating over the insertion point, one can recover information on the amplitude, as was recently done to compute the full four-loop cusp anomalous dimension. We study the general structure of the Wilson loop with a Lagrangian insertion, focusing in particular on its leading singularities and their (hidden) symmetry properties. Thanks to the close connection of the observable to integrands of MHV amplitudes, it is natural to expect that its leading singularities can be written as certain Grassmannian integrals. The latter are manifestly dual conformal. They also have a conformal symmetry, up to total derivatives. We find that, surprisingly, the conformal symmetry becomes an invariance in the frame where the Lagrangian insertion point is sent to infinity. Furthermore, we use integrability methods to study how higher Yangian charges act on the Grassmannian integral. We evaluate the n-particle observable both at tree- and at one-loop level, finding compact analytic formulas. These results are explicitly written in the form of conformal leading singularities, multiplied by transcendental functions. We then compare these formulas to known expressions for all-plus amplitudes in pure Yang-Mills theory. We find a remarkable new connection: the Wilson loop with Lagrangian insertion in mathcal{N} = 4 super Yang-Mills appears to predict the maximal weight terms of the planar pure Yang-Mills all-plus amplitude. We test this relationship for the two-loop n-point Yang-Mills amplitude, as well as for the three-loop four-point amplitude.

Construction of bulk solutions for towers of pole-skipping points

The pole-skipping phenomenon has been proposed as a connection between chaotic properties of black hole geometries and special points where regular solutions of linearized Einstein equations at horizons have extra free parameters. In this work, we pursue the special points in the near-horizon analysis of integer spin-$\ensuremath{\ell}$ fields on the Rindler-AdS black hole. We construct linear combinations of field components to simplify coupled equations of massive fields and investigate towers of the special points along with imaginary Matsubara frequencies $i\ensuremath{\omega}=2\ensuremath{\pi}(n+1\ensuremath{-}\ensuremath{\ell})T$ with a non-negative integer $n$ and the Hawking temperature $T$. We also propose that integrals of spin-$\ensuremath{\ell}$ bulk propagators over horizons of static black holes capture behaviors at the special points, which are generalizations of integrals of graviton propagators for shock wave geometries. Their interpretation is provided in terms of four-point amplitudes with the spin-$\ensuremath{\ell}$ exchange.

Perturbative higher-derivative terms in mathcal{N} = 6 asymmetric orbifolds

We analyse the perturbative four-point amplitudes in the simplest string theory examples of T-fold backgrounds, which enjoy mathcal{N} = 6 supersymmetries in four dimensions. There are two theories defined as asymmetric orbifolds of order 2 and 3, respectively. The perturbative spectrum and the one-loop four-point amplitudes are shown to be invariant under an arithmetic symplectic group defined over the Hurwitz (respectively Eisenstein) quaternions. The supersymmetry constraints on the low energy effective couplings are determined and we conjecture the U-duality group of the theory and the non-perturbative completion of the six-derivative coupling as a rank one theta series. We discuss the non-perturbative spectrum of BPS states in the light of our analysis.

Interactions of strings on a T-fold

We consider the interactions of strings on T-folds from the world-sheet point of view which are exact in α′. As a concrete example, we take a model where the internal torus at the SO(8) enhancement point is twisted by T-duality (T-folded), and compute the scattering amplitudes of a class of massless strings. The four-point amplitudes involving both twisted and untwisted strings are obtained in a closed form in terms of the hyper- geometric function. By their factorization, the three-point coupling of the twisted and untwisted strings is found to be suppressed by the chiral momenta along the internal torus, and quantized in integer powers of 1/4. The asymptotic forms of the four-point amplitudes in high-energy limits are also obtained. Our results rely only on general properties of the asymmetric orbifold by the T-duality twist and of the Lie algebra lattice from the symmetry enhancement, and thus may be extended qualitatively to more general T-folds.

Bulk locality from the celestial amplitude

In this paper, we study the implications of bulk locality on the celestial amplitude. In the context of the four-point amplitude, the fact that the bulk S-matrix factorizes locally in poles of Mandelstam variables is reflected in the imaginary part of the celestial amplitude. In particular, on the real axis in the complex plane of the boost weight, the imaginary part of the celestial amplitude can be given as a positive expansion on the Poincaré partial waves, which are nothing but the projection of flat-space spinning polynomials onto the celestial sphere. Furthermore, we derive the celestial dispersion relation, which relates the imaginary part to the residue of the celestial amplitude for negative even integer boost weight. The latter is precisely the projection of low energy EFT coefficients onto the celestial sphere. We demonstrate these properties explicitly on the open and closed string celestial amplitudes. Finally, we give an explicit expansion of the Poincaré partial waves in terms of 2D conformal partial waves.

Asymmetrically twisted strings

In this letter a new class of twisted strings is presented, with an asymmetry between the holomorphic and antiholomorphic sectors parametrized by an integer N. Their physical content is given by the massless resonances of the closed string plus the mass-level N spectrum of the open string. The appeal of this model is the singling out of the (higher spin) massive levels of string theory together with their self/gauge/gravity interactions. Motivated by the original tree level Kawai-Lewellen-Tye relation for closed strings, its asymmetrically twisted version at four points is conjectured and shown to naturally interpolate with conventional and twisted strings. The resulting four-point amplitudes have a generalized Virasoro-Shapiro dressing factor.

On-shell Higgsing for EFTs

We study the on-shell version of the Higgs mechanism in effective theories (EFTs) containing particles of different spins, focusing on contact terms as a simple starting point. We derive the massive contact terms and their coefficients from the massless amplitudes of the EFT above the symmetry breaking scale, by covariantizing the massless contact terms under the massive little group. In the little-group-covariant massive-spinor formalism, this notationally amounts to bolding spinor labels. Mass-suppressed contributions to the contact-term coefficients arise from higher-point contact terms with additional soft Higgs legs. We apply this procedure to obtain massive four-point amplitudes featuring scalars, spin 1/2 fermions and vectors, in the standard-model EFT. The subleading helicity-flipped components of each massive contact term, which are dictated by little-group covariance, are associated with the residues of factorizable massless amplitudes. Extra “frozen” Higgses emitted from each leg of a massless contact term supply the additional light-like momentum component, needed to form a massive leg of the same polarization. As another application, we derive various components of massive three-point amplitudes from massless amplitudes with up to three additional Higgses, in a standard-model-like toy model.

Scattering amplitudes of Kaluza-Klein strings and extended massive double-copy

We study the scattering amplitudes of massive Kaluza-Klein (KK) states of open and closed bosonic strings under toroidal compactification. We analyze the structure of vertex operators for the KK strings and derive an extended massive KLT-like relation which connects the N-point KK closed-string amplitude to the products of two KK open-string amplitudes at tree level. Taking the low energy field-theory limit of vanishing Regge slope, we derive double-copy construction formula of the N-point massive KK graviton amplitude from the sum of proper products of the corresponding KK gauge boson amplitudes. Then, using the string-based massive double-copy formula, we derive the exact tree-level four-point KK gauge boson amplitudes and KK graviton amplitudes, which fully agree with those given by the KK field-theory calculations. With these, we give an explicit prescription on constructing the exact four-point KK graviton amplitudes from the sum of proper products of the corresponding color-ordered KK gauge boson amplitudes. We further analyze the string-based double-copy construction of five-point and six-point scattering amplitudes of massive KK gauge bosons and KK gravitons.

Gravitational Equivalence Theorem and Double-Copy for Kaluza-Klein Graviton Scattering Amplitudes

We analyze the structure of scattering amplitudes of the Kaluza-Klein (KK) gravitons and of the KK gravitational Goldstone bosons in the compactified 5d general relativity (GR). Using a general R ξ gauge fixing, we study the geometric Higgs mechanism for the massive spin-2 KK gravitons. We newly propose and prove a gravitational equivalence theorem (GRET) to connect the scattering amplitudes of longitudinal KK gravitons to that of the KK gravitational Goldstone bosons, which formulates the geometric gravitational Higgs mechanism at the scattering S - matrix level. We demonstrate that the GRET provides a general energy cancellation mechanism guaranteeing the N -point longitudinal KK graviton scattering amplitudes to have their leading energy dependence cancelled down by a large power factor of E 2 N N ≥ 4 up to any loop level. We propose an improved double-copy approach to construct the massive KK graviton (Goldstone) amplitudes from the KK gauge boson (Goldstone) amplitudes. With these, we establish a new correspondence between the two types of energy cancellations in the four-point longitudinal KK amplitudes at tree level: E 4 ⟶ E 0 in the KK gauge theory and E 10 ⟶ E 2 in the KK GR theory.

Superstring Loop Amplitudes from the Field Theory Limit.

We propose a procedure to determine the moduli-space integrands of loop-level superstring amplitudes for massless external states in terms of the field theory limit. We focus on the type II superstring. The procedure is to (i)take a supergravity loop integrand written in a BCJ double-copy representation, (ii)use the loop-level scattering equations to translate that integrand into the ambitwistor string moduli-space integrand, localised on the nodal Riemann sphere, and (iii)uplift that formula to one on the higher-genus surface valid for the superstring, guided by modular invariance. We show how this works for the four-point amplitude at two loops, where we reproduce the known answer, and at three loops, where we present a conjecture that is consistent with a previous proposal for the chiral measure. Useful supergravity results are currently known up to five loops.

Double Copy Relation in AdS Space.

We present a double copy relation in AdS_{5} that relates tree-level four-point amplitudes of supergravity, super Yang-Mills, and bi-adjoint scalars.

Gluon scattering in AdS from CFT

We present a systematic study of holographic correlators in a vast array of SCFTs with non-maximal superconformal symmetry. These theories include 4d mathcal{N} = 2 SCFTs from D3-branes near F-theory singularities, 5d Seiberg exceptional theories and 6d E-string theory, as well as 3d and 4d phenomenological models with probe flavor branes. We consider current multiplets and their generalizations with higher weights, dual to massless and massive super gluons in the bulk. At leading order in the inverse central charge expansion, connected four-point functions of these operators correspond to tree-level gluon scattering amplitudes in AdS. We show that all such tree-level four-point amplitudes in all these theories are fully fixed by symmetries and consistency conditions and explicitly construct them. Our results encode a wealth of SCFT data and exhibit various interesting emergent structures. These include Parisi-Sourlas-like dimensional reductions, hidden conformal symmetry and an AdS version of the color-kinematic duality.

Towards the Virasoro-Shapiro amplitude in AdS5\xd7S5

We propose a systematic procedure for obtaining all single trace 1/2-BPS correlators in mathcal{N} = 4 super Yang-Mills corresponding to the four-point tree-level amplitude for type IIB string theory in AdS5 × S5. The underlying idea is to compute generalised contact Witten diagrams coming from a 10d effective field theory on AdS5 × S5 whose coefficients are fixed by the flat space Virasoro-Shapiro amplitude up to ambiguities related to commutators of the 10d covariant derivatives which require additional information such as localisation. We illustrate this procedure by computing stringy corrections to the supergravity prediction for all single trace 1/2-BPS correlators up to mathcal{O} (α′7), and spell out a general algorithm for extending this to any order in α′.

Carving out the space of open-string S-matrix

In this paper, we explore the open string amplitude’s dual role as a space-time S-matrix and a 2D holomorphic CFT correlation function. We pursue this correspondence in two directions. First, beginning with a general disk integrand dressed with a Koba-Nielsen factor, we demonstrate that exchange symmetry for the factorization residue of the amplitude forces the integrand to be expandable on SL(2,R) conformal blocks. Furthermore, positivity constraints associated with unitarity imply the SL(2,R) blocks must come in linear combinations for which the Virasoro block emerges at the “kink” in the space of solutions. In other words, Virasoro symmetry arises at the boundary of consistent factorization. Next, we consider the low energy EFT description, where unitarity manifests as the EFThedron in which the couplings must live. The existence of a worldsheet description implies, through the Koba-Nielsen factor, monodromy relations which impose algebraic identities amongst the EFT couplings. We demonstrate at finite derivative order that the intersection of the “monodromy plane” and the four-dimensional EFThedron carves out a tiny island for the couplings, which continues to shrink as the derivative order is increased. At the eighth derivative order, on a three-dimensional monodromy plane, the intersection fixes the width of this island to around 1.5% (of ζ(3)) and 0.2% (of ζ(5)) with respect to the toroidally compactified Type-I super string answer. This leads us to conjecture that the four-point open superstring amplitude can be completely determined by the geometry of the intersection of the monodromy plane and the EFThedron.

On genus-one string amplitudes on AdS5 \xd7 S5

We study non-planar correlators in mathcal{N} = 4 super-Yang-Mills in Mellin space. We focus in the stress tensor four-point correlator to order 1/N4 and in a strong coupling expansion. This can be regarded as the genus-one four-point graviton amplitude of type IIB string theory on AdS5× S5 in a low energy expansion. Both the loop supergravity result as well as the tower of stringy corrections have a remarkable simple structure in Mellin space, making manifest important properties such as the correct flat space limit and the structure of UV divergences.

A three-point form factor through five loops

We bootstrap the three-point form factor of the chiral part of the stress­tensor supermultiplet in planar mathcal{N} = 4 super-Yang-Mills theory, obtaining new results at three, four, and five loops. Our construction employs known conditions on the first, second, and final entries of the symbol, combined with new multiple-final-entry conditions, “extended-Steinmann-like” conditions, and near-collinear data from the recently-developed form factor operator product expansion. Our results are expected to give the maximally transcendental parts of the gg → Hg and H → ggg amplitudes in the heavy-top limit of QCD. At two loops, the extended-Steinmann-like space of functions we describe contains all transcendental functions required for four-point amplitudes with one massive and three massless external legs, and all massless internal lines, including processes such as gg → Hg and γ* → qoverline{q}g . We expect the extended-Steinmann-like space to contain these amplitudes at higher loops as well, although not to arbitrarily high loop order. We present evidence that the planar mathcal{N} = 4 three-point form factor can be placed in an even smaller space of functions, with no independent ζ values at weights two and three.

Effective field theory for closed strings near the Hagedorn temperature

We discuss interacting, closed, bosonic and superstrings in thermal equilibrium at temperatures close to the Hagedorn temperature in flat space. We calculate S-matrix elements of the strings at the Hagedorn temperature and use them to construct a low-energy effective action for interacting strings near the Hagedorn temperature. We show, in particular, that the four-point amplitude of massless winding modes leads to a positive quartic interaction. Furthermore, the effective field theory has a generalized conformal structure, namely, it is conformally invariant when the temperature is assigned an appropriate scaling dimension. Then, we show that the equations of motion resulting from the effective action possess a winding-mode-condensate background solution above the Hagedorn temperature and present a worldsheet conformal field theory, similar to a Sine-Gordon theory, that corresponds to this solution. We find that the Hagedorn phase transition in our setup is second order, in contrast to a first-order transition that was found previously in different setups.

All-loop-orders relation between Regge limits of mathcal{N} = 4 SYM and mathcal{N} = 8 supergravity four-point amplitudes

We examine in detail the structure of the Regge limit of the (nonplanar) mathcal{N} = 4 SYM four-point amplitude. We begin by developing a basis of color factors Cik suitable for the Regge limit of the amplitude at any loop order, and then calculate explicitly the coefficients of the amplitude in that basis through three-loop order using the Regge limit of the full amplitude previously calculated by Henn and Mistlberger. We compute these coefficients exactly at one loop, through mathcal{O}left({upepsilon}^2right) at two loops, and through mathcal{O}left({upepsilon}^0right) at three loops, verifying that the IR-divergent pieces are consistent with (the Regge limit of) the expected infrared divergence structure, including a contribution from the three-loop correction to the dipole formula. We also verify consistency with the IR-finite NLL and NNLL predictions of Caron-Huot et al. Finally we use these results to motivate the conjecture of an all-orders relation between one of the coefficients and the Regge limit of the mathcal{N} = 8 supergravity four-point amplitude.

Two-loop superstring five-point amplitudes. Part II. Low energy expansion and S-duality

In an earlier paper, we constructed the genus-two amplitudes for five external massless states in Type II and Heterotic string theory, and showed that the α′ expansion of the Type II amplitude reproduces the corresponding supergravity amplitude to leading order. In this paper, we analyze the effective interactions induced by Type IIB superstrings beyond supergravity, both for U(1)R-preserving amplitudes such as for five gravitons, and for U(1)R-violating amplitudes such as for one dilaton and four gravitons. At each order in α′, the coefficients of the effective interactions are given by integrals over moduli space of genus-two modular graph functions, generalizing those already encountered for four external massless states. To leading and sub-leading orders, the coefficients of the effective interactions D2ℛ5 and D4ℛ5 are found to match those of D4ℛ4 and D6ℛ4, respectively, as required by non-linear supersymmetry. To the next order, a D6ℛ5 effective interaction arises, which is independent of the supersymmetric completion of D8ℛ4, and already arose at genus one. A novel identity on genus-two modular graph functions, which we prove, ensures that up to order D6ℛ5, the five-point amplitudes require only a single new modular graph function in addition to those needed for the four-point amplitude. We check that the supergravity limit of U(1)R-violating amplitudes is free of UV divergences to this order, consistently with the known structure of divergences in Type IIB supergravity. Our results give strong consistency tests on the full five-point amplitude, and pave the way for understanding S-duality beyond the BPS-protected sector.

Closed strings and weak gravity from higher-spin causality

We combine old and new quantum field theoretic arguments to show that any theory of stable or metastable higher spin particles can be coupled to gravity only when the gravity sector has a stringy structure. Metastable higher spin particles, free or interacting, cannot couple to gravity while preserving causality unless there exist higher spin states in the gravitational sector much below the Planck scale Mpl. We obtain an upper bound on the mass Λgr of the lightest higher spin particle in the gravity sector in terms of quantities in the non-gravitational sector. We invoke the CKSZ uniqueness theorem to argue that any weakly coupled UV completion of such a theory must have a gravity sector containing infinite towers of asymptotically parallel, equispaced, and linear Regge trajectories. Consequently, gravitational four-point scattering amplitudes must coincide with the closed string four-point amplitude for s, t ≫ 1, identifying Λgr as the string scale. Our bound also implies that all metastable higher spin particles in 4d with masses m ≪ Λgr must satisfy a weak gravity condition.