In this work, we develop machine learning techniques to study nonperturbative scattering amplitudes. We focus on the two-to-two scattering amplitude of identical scalar particles and study a toy model where we set the double discontinuity to zero, thereby simplifying the problem numerically. Neural networks provide an efficient parameterization for scattering amplitudes, offering a flexible toolkit to describe the details of their nonperturbative structure. Combined with the bootstrap approach based on the dispersive representation of the amplitude and machine learning’s gradient descent algorithms, they offer a new method to explore the space of consistent S-matrices. We derive bounds on the values of the first two low-energy Taylor coefficients of the amplitude and characterize the resulting amplitudes that populate the allowed region. Crucially, we parallel our neural network analysis with the standard S-matrix bootstrap, both primal and dual, and observe perfect agreement across all approaches.

France

Mehmet Gümüş

Damien Leflot

Piotr Tourkine

Switzerland

Alexander Zhiboedov

The S-matrix bootstrap with neural optimizers. Part I. Zero double discontinuity

We explore the space of meromorphic amplitudes with extra constraints coming from the shape of the leading Regge trajectory. This information comes in two guises: it bounds the maximal spin of exchanged particles of a given mass; it leads to sum rules obeyed by the discontinuity of the amplitude, which express the softness of scattering at high energies. We assume that the leading Regge trajectory is linear, and we derive bounds on the low-energy Wilson coefficients using the dual and primal approaches. For the graviton-graviton scattering in four dimensions, the maximal spin constraint leads to slightly more stringent bounds than those that follow from general constraints of analyticity, crossing, and unitarity. The exponential softness at high energies is manifest in our primal approach and is not used in our implementation of the dual approach. Nevertheless, we observe the agreement between the bounds obtained from both. We conclude that high-energy superpolynomial softness does not leave an obvious imprint on the low-energy observables. We exhibit a unitary three-parameter deformation of the Veneziano amplitude for the open string case. It has a novel, exponentially soft behavior at high energies and fixed angles. We generalize the previous analysis of this regime and present a stringy version of the lower bound on high-energy, fixed-angle scattering by Cerulus and Martin.

CERN

Kelian Häring

The stringy S-matrix bootstrap: maximal spin and superpolynomial softness

Lorentzian correlators of local operators exhibit surprising singularities in theories with gravity duals. These are associated with null geodesics in an emergent bulk geometry. We analyze singularities of the thermal response function dual to propagation of waves on the AdS Schwarzschild black hole background. We derive the analytic form of the leading singularity dual to a bulk geodesic that winds around the black hole. Remarkably, it exhibits a boundary group velocity larger than the speed of light, whose dual is the angular velocity of null geodesics at the photon sphere. The strength of the singularity is controlled by the classical Lyapunov exponent associated with the instability of nearly bound photon orbits. In this sense, the bulk-cone singularity can be identified as the universal feature that encodes the ubiquitous black hole photon sphere in a dual holographic CFT. To perform the computation analytically, we express the two-point correlator as an infinite sum over Regge poles, and then evaluate this sum using WKB methods. We also compute the smeared correlator numerically, which in particular allows us to check and support our analytic predictions. We comment on the resolution of black hole bulk-cone singularities by stringy and gravitational effects into black hole bulk-cone “bumps”. We conclude that these bumps are robust, and could serve as a target for simulations of black hole-like geometries in table-top experiments.

Matthew Dodelson

Italy

Cristoforo Iossa

Robin Karlsson

United States

Alexandru Lupsasca

Black hole bulk-cone singularities

An important element of the S-matrix bootstrap program is the relationship between the modulus of an S-matrix element and its phase. Unitarity relates them by an integral equation. Even in the simplest case of elastic scattering, this integral equation cannot be solved analytically and numerical approaches are required. We apply modern machine learning techniques to studying the unitarity constraint. We find that for a given modulus, when a phase exists it can generally be reconstructed to good accuracy with machine learning. Moreover, the loss of the reconstruction algorithm provides a good proxy for whether a given modulus can be consistent with unitarity at all. In addition, we study the question of whether multiple phases can be consistent with a single modulus, finding novel phase-ambiguous solutions. In particular, we find a new phase-ambiguous solution which pushes the known limit on such solutions significantly beyond the previous bound.

Aurélien Dersy

Matthew Schwartz

Reconstructing S-matrix Phases with Machine Learning

We show that holographic thermal two-sided two-point correlators take the form of a product over quasi-normal modes (QNMs). Due to this fact, the two-point function admits a natural dispersive representation with a positive discontinuity at the location of QNMs. We explore the general constraints on the structure of QNMs that follow from the operator product expansion, the presence of the singularity inside the black hole, and the hydrodynamic expansion of the correlator. We illustrate these constraints through concrete examples. We suggest that the product formula for thermal correlators may hold for more general large N chaotic systems, and we check this hypothesis in several models.

A thermal product formula

We study energy correlations in states created by a heavy operator acting on the vacuum in a conformal field theory. We argue that the energy correlations in such states exhibit two characteristic regimes as functions of the angular separations between the calorimeters: power-like growth at small angles described by the light-ray OPE and slowly varying, or “flat”, function at larger angles. The transition between the two regimes is controlled by the scaling dimension of the heavy operator and the dynamics of the theory. We analyze this phenomenon in detail in the planar   <math> <mi>N</mi> </math>  $$ \mathcal{N} $$  = 4 SYM theory both at weak and strong coupling. An analogous transition was previously observed in QCD in the measurement of the angular energy distribution of particles belonging to the same energetic jet. In that case it corresponds to the transition from the light-ray OPE, perturbative regime described in terms of correlations between quarks and gluons to the flat, non-perturbative regime described in terms of correlations between hadrons.

Dmitry Chicherin

Gregory Korchemsky

Emery Sokatchev

Energy correlations in heavy states

We present and numerically implement a computational method to construct relativistic scattering amplitudes that obey analyticity, crossing, elastic and inelastic unitarity in three and four spacetime dimensions. The algorithm is based on the Mandelstam representation of the amplitude and iterations of unitarity. The input for the iterative procedure is given by the multi-particle double spectral density, the S-wave inelasticity, and the value of the amplitude at the crossing-symmetric point. The output, obtained at the fixed point of the iteration of unitarity, is a nonperturbative scattering amplitude. The amplitudes we obtain exhibit interesting features, such as non-zero particle production, intricate high-energy and near the two-particle threshold behavior. Scattering amplitudes obtained by initializing the iteration process with zero (or small) multi-particle input end up close to saturating the S-matrix bounds derived by other methods. There is a version of the iterative algorithm that is directly related to Feynman diagrams: it effectively re-sums infinitely many two-particle reducible planar Feynman graphs in the ϕ 4 theory, which remarkably produces a unitary nonperturbative scattering amplitude function. Finally, we discuss how the algorithm can be further refined by including multi-particle unitarity.

Scattering amplitudes from dispersive iterations of unitarity

We explore the implications of stable gravitational orbits around an AdS black hole for the boundary conformal field theory. The orbits are long-lived states that eventually decay due to gravitational radiation and tunneling. They appear as narrow resonances in the heavy-light OPE when the spectrum becomes effectively continuous due to the presence of the black hole horizon. Alternatively, they can be identified with quasi-normal modes with small imaginary part in the thermal two-point function. The two pictures are related via the eigenstate thermalisation hypothesis. When the decay effects can be neglected the orbits appear as a discrete family of double-twist operators. We investigate the connection between orbits, quasi-normal modes, and double-twist operators in detail. Using the corrected Bohr-Sommerfeld formula for quasi-normal modes, we compute the anomalous dimension of double-twist operators. We compare our results to the prediction of the light-cone bootstrap, finding perfect agreement where the results overlap. We also compute the orbit decay time due to scalar radiation and compare it to the tunneling rate. Perturbatively in spin, in the light-cone bootstrap framework double-twist operators appear as a small fraction of the Hilbert space which violate the eigenstate thermalization hypothesis, a phenomenon known as many-body scars. Nonperturbatively in spin, the double-twist operators become long-lived states that eventually thermalize. We briefly discuss the connection between perturbative scars in holographic theories and known examples of scars in the condensed matter literature.

Gravitational orbits, double-twist mirage, and many-body scars

Defect characterizes the depth of factorization of terms in differential (cyclotomic) expansions of knot polynomials, i.e. of the non-perturbative Wilson averages in the Chern-Simons theory. We prove the conjecture that the defect can be alternatively described as the degree in  $$q^{\pm 2}$$  <math> <msup> <mi>q</mi> <mrow> <mo>±</mo> <mn>2</mn> </mrow> </msup> </math>   of the fundamental Alexander polynomial, which formally corresponds to the case of no colors. We also pose a question if these Alexander polynomials can be arbitrary integer polynomials of a given degree. A first attempt to answer the latter question is a preliminary analysis of antiparallel descendants of the 2-strand torus knots, which provide a nice set of examples for all values of the defect. The answer turns out to be positive in the case of defect zero knots, what can be observed already in the case of twist knots. This proved conjecture also allows us to provide a complete set of C-polynomials for the symmetrically colored Alexander polynomials for defect zero. In this case, we achieve a complete separation of representation and knot variables.

Russia

E. Lanina

A. Morozov

Defect and degree of the Alexander polynomial

We introduce a new class of collider-type observables in conformal field theories which we call generalized event shapes. They are defined as matrix elements of light-ray operators that are sensitive to the longitudinal, or time-dependent, structure of the state produced in the collision. Generalized event shapes can be studied using both correlation functions and scattering amplitudes. They are infrared finite and smoothly transit over to the familiar event shapes. We compute them in planar   <math> <mi>N</mi> </math>  $$ \mathcal{N} $$  = 4 super-Yang-Mills theory at weak and strong coupling, and study their physical properties. We show that at strong coupling both the stringy and quantum-gravitational corrections to the energy-energy correlation exhibit longitudinal broadening that manifests itself through the presence of long-time tails in the energy flux measured by the detectors.