We consider turbulence of waves interacting weakly via four-wave scattering (sea waves, plasma waves, spin waves, etc.). In the first order in the interaction, a closed kinetic equation has stationary solutions describing turbulent cascades. We show that the higher-order terms generally diverge both at small (IR) and large (UV) wave numbers for direct cascades. The analysis up to the third order identifies the most UV-divergent terms. To gain qualitative analytic control, we sum a subset of the most UV divergent terms, to all orders, giving a perturbation theory free from UV divergence, showing that turbulence becomes independent of the dissipation scale when it goes to zero. On the contrary, the IR divergence (present in the majority of cases) makes the effective coupling parametrically larger than the naive estimate and growing with the pumping scale <math><mi>L</mi></math> (similar to anomalous scaling in fluid turbulence). In such cases, the kinetic equation does not describe wave turbulence even of arbitrarily small level at a given <math><mi>k</mi></math> if <math><mi>k</mi><mi>L</mi></math> is large enough that is the cascade is sufficiently long. We show that the character of strong turbulence is determined by whether the effective four-wave interaction is enhanced or suppressed by collective effects. The enhancement possibly signals that strong turbulence is dominated by multiwave bound states (solitons, shocks, cusps), similar to confinement in quantum chromodynamics.

United States

Vladimir Rosenhaus

Israel

Gregory Falkovich

Interaction Renormalization and Validity of Kinetic Equations for Turbulent States

Recent work has given a systematic way for studying the kinetics of classical weakly interacting waves beyond leading order, having analogies with renormalization in quantum field theory. An important context is weak wave turbulence, occurring for waves which are small in magnitude and weakly interacting, such as those on the surface of the ocean. Here we continue the work of perturbatively computing correlation functions and the kinetic equation in this far-from-equilibrium state. In particular, we obtain the next-to-leading-order kinetic equation for waves with a cubic interaction. Our main result is a simple graphical prescription for the terms in the kinetic equation, at any order in the nonlinearity.

Daniel Schubring

Md Shuvo

Michael Smolkin

Loop diagrams in the kinetic theory of waves

It has long been known that weakly nonlinear field theories can have a late-time stationary state that is not the thermal state, but a wave turbulent state with a far-from-equilibrium cascade of energy. We go beyond the existence of the wave turbulent state, studying fluctuations about the wave turbulent state. Specifically, we take a classical field theory with an arbitrary quartic interaction and add dissipation and Gaussian-random forcing. Employing the path integral relation between stochastic classical field theories and quantum field theories, we give a prescription, in terms of Feynman diagrams, for computing correlation functions in this system. We explicitly compute the two-point and four-point functions of the field to next-to-leading order in the coupling. Through an appropriate choice of forcing and dissipation, these correspond to correlation functions in the wave turbulent state. In particular, we derive the kinetic equation to next-to-leading order.

Feynman rules for forced wave turbulence

We compute the amplitude for an excited string in any precisely specified state to decay into another excited string in any precisely specified state, via emission of a tachyon or photon. For generic and highly excited string states, the amplitude is a complicated function of the outgoing kinematic angle, sensitive to the precise state. We compute the square of this amplitude, averaged over polarizations of the ingoing string and summed over polarizations of the outgoing string. The seeming intractability of these calculations is made possible by extracting amplitudes involving excited strings from amplitudes involving tachyons and a large number of photons; the number of photons grows with the complexity of the excited string state. Our work is in the spirit of the broad range of recent studies of statistical mechanics and chaos for quantum many-body systems. The number of different excited string states at a given mass is exponentially large, and our calculation gives the emission amplitude of a single photon from each of the microstates — which, through the Horowitz-Polchinski correspondence principle, are in correspondence with black hole microstates.

Italy

Maurizio Firrotta

Photon emission from an excited string

String theory provides a compact integral expression for the tree-level scattering amplitude of an arbitrary number of light strings. We focus on amplitudes involving a few tachyons and many photons, with a special choice of polarizations and kinematics. We pick out a particular pole in the amplitude—one corresponding to successive photon scatterings, which lead to an intermediate state with a highly excited string in a definite state. This provides a physical process that creates a highly excited string. The observed erratic behavior of the amplitude suggests that this may serve as a simple and explicit illustration of chaos in many-particle scattering.

Chaos in a Many-String Scattering Amplitude

A number of studies have shown that chaos occurs in scattering: the outgoing deflection angle is seen to be an erratic function of the impact parameter. We propose to extend this to quantum field theory and to use the erratic behavior of the many-particle <math><mi>S</mi></math>-matrix as a probe of chaos.

Chaos in the Quantum Field Theory <math><mrow><mi>S</mi></mrow></math>-Matrix

Motivated by the desire to understand chaos in the S-matrix of string theory, we study tree level scattering amplitudes involving highly excited strings. While the amplitudes for scattering of light strings have been a hallmark of string theory since its early days, scattering of excited strings has been far less studied. Recent results on black hole chaos, combined with the correspondence principle between black holes and strings, suggest that the amplitudes have a rich structure. We review the procedure by which an excited string is formed by repeatedly scattering photons off of an initial tachyon (the DDF formalism). We compute the scattering amplitude of one arbitrary excited string and any number of tachyons in bosonic string theory. At high energies and high mass excited state these amplitudes are determined by a saddle-point in the integration over the positions of the string vertex operators on the sphere (or the upper half plane), thus yielding a generalization of the “scattering equations”. We find a compact expression for the amplitude of an excited string decaying into two tachyons, and study its properties for a generic excited string. We find the amplitude is highly erratic as a function of both the precise excited string state and of the tachyon scattering angle relative to its polarization, a sign of chaos.

David Gross

Chaotic scattering of highly excited strings

We use the background field method to systematically derive CFT data for the critical ϕ 6 vector model in three dimensions, and the Gross-Neveu model in dimensions 2 ≤ d ≤ 4. Specifically, we calculate the OPE coefficients and anomalous dimensions of various operators, up to next-to-leading order in the 1/N expansion.

Mikhail Goykhman

The background field method and critical vector models

Smirnov and Zamolodchikov recently introduced a new class of two-dimensional quantum field theories, defined through a differential change of any existing theory by the determinant of the energy-momentum tensor. From this <math><mi>T</mi><mover><mi>T</mi><mo>¯</mo></mover></math> flow equation one can find a simple expression for both the energy spectrum and the <math><mi>S</mi></math>-matrix of the <math><mi>T</mi><mover><mi>T</mi><mo>¯</mo></mover></math> deformed theories. Our goal is to find the renormalized Lagrangian of the <math><mi>T</mi><mover><mi>T</mi><mo>¯</mo></mover></math> deformed theories. In the context of the <math><mi>T</mi><mover><mi>T</mi><mo>¯</mo></mover></math> deformation of an integrable theory, the deformed theory is also integrable and, correspondingly, the <math><mi>S</mi></math>-matrix factorizes into two-to-two <math><mi>S</mi></math>-matrices. One may thus hope to be able to extract the renormalized Lagrangian from the <math><mi>S</mi></math>-matrix. We do this explicitly for the <math><mi>T</mi><mover><mi>T</mi><mo>¯</mo></mover></math> deformation of a free massive scalar, to second order in the deformation parameter. Once one has the renormalized Lagrangian one can, in principle, compute all other observables, such as correlation functions. We briefly discuss this, as well as the relation between the renormalized Lagrangian, the <math><mi>T</mi><mover><mi>T</mi><mo>¯</mo></mover></math> flow equation, and the <math><mi>S</mi></math>-matrix. We also mention a more general class of integrability-preserving deformations of a free scalar field theory.

Integrability and renormalization under <math><mi>T</mi><mover><mi>T</mi><mo>¯</mo></mover></math>

Conformal blocks are the building blocks for correlation functions in conformal field theories. The four-point function is the most well-studied case. We consider conformal blocks for n-point correlation functions. For conformal field theories in dimensions d = 1 and d = 2, we use the shadow formalism to compute n-point conformal blocks, for arbitrary n, in a particular channel which we refer to as the comb channel. The result is expressed in terms of a multivariable hypergeometric function, for which we give series, differential, and integral representations. In general dimension d we derive the 5-point conformal block, for external and exchanged scalar operators.