We propose a novel indicator for chaotic quantum scattering processes, the scattering form factor (ScFF). It is based on mapping the locations of peaks in the scattering amplitude to random matrix eigenvalues, and computing the analog of the spectral form factor (SFF). We compute the spectral and scattering form factors of several non-chaotic systems. We determine the ScFF associated with the phase shifts of the leaky torus, closely related to the distribution of the zeros of Riemann zeta function. We compute the ScFF for the decay amplitude of a highly excited string states into two tachyons. We show that it displays the universal features expected from random matrix theory - a decline, a ramp and a plateau - and is in general agreement with the Gaussian unitary ensemble. It also shows some new features, owning to the special structure of the string amplitude, including a “bump” before the ramp associated with gaps in the average eigenvalue density. The “bump” is removed for highly excited string states with an appropriate state dependent unfolding. We also discuss the SFF for the Gaussian β-ensemble, writing an interpolation between the known results of the Gaussian orthogonal, unitary, and symplectic ensembles.

Italy

Massimo Bianchi

Greece

Maurizio Firrotta

Israel

Jacob Sonnenschein

South Korea

Dorin Weissman

From spectral to scattering form factor

We extend the Veneziano and Shapiro-Virasoro amplitudes to four arbitrarily excited states in bosonic string theory. We use the formalism of coherent string states based on the Di Vecchia-Del Giudice-Fubini construction. Within the same formalism, we also analyze the three string scattering finding the covariant version of the three reggeon interaction. Then studying the factorization properties of the extended four string scattering amplitudes we identify the covariant version of the three string interaction. Finally we obtain generalized Kawai-Lewellen-Tye relations connecting the scattering of four open and closed arbitrarily excited states.

Veneziano and Shapiro-Virasoro amplitudes of arbitrarily excited strings

We analyze the amplitudes of one highly excited string (HES) state with two or three tachyons in open bosonic string theory. We argue that these processes are chaotic by showing that the spacing ratios of successive peaks in the angular dependence of the amplitudes are distributed as predicted by the <math><mi>β</mi></math>-ensemble of random matrix theory (RMT). We show how the continuous parameter <math><mi>β</mi></math> depends on the level and helicity of the scattered HES state. We derive the scattering amplitude of an HES and three tachyons and show that it takes the form of the Veneziano amplitude times a dressing factor, and that the dressing is chaotic as a function of the scattering angle, in the sense that its spacing ratios match with RMT predictions.

Measuring chaos in string scattering processes

We analyse the most general process of a generic highly excited string that decays into a less excited, yet generic, highly excited string emitting a tachyon. We provide a simple and compact analytic description of the decay process which discriminates between and within the structure of every single microstate of the initial and final highly excited string. Taking into account the random nature of the decay process we extract the energy spectrum of highly excited strings, microstate by microstate, finding a behavior which corresponds to the greybody emission spectrum. In addition, by exploiting the analytic control of the decay process, we identify the origin of thermal effects which are triggered by the chaotic nature of the highly excited string interactions modeled by the microstates structure.

The chaotic emergence of thermalization in highly excited string decays

We propose a novel measure of chaotic scattering amplitudes. It takes the form of a log-normal distribution function for the ratios <math><msub><mi>r</mi><mi>n</mi></msub><mo>=</mo><msub><mi>δ</mi><mi>n</mi></msub><mo>/</mo><msub><mi>δ</mi><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msub></math> of (consecutive) spacings <math><msub><mi>δ</mi><mi>n</mi></msub></math> between two (consecutive) peaks of the scattering amplitude. We show that the same measure applies to the quantum mechanical scattering on a leaky torus as well as to the decay of highly excited string states into two tachyons. Quite remarkably, the <math><msub><mi>r</mi><mi>n</mi></msub></math> obey the same distribution that governs the nontrivial zeros of Riemann <math><mi>ζ</mi></math> function.

Japan

Measure for Chaotic Scattering Amplitudes

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.

United States

Vladimir Rosenhaus

Photon emission from an excited string

We study the emergence of partonic behavior in scattering processes at large Mandelstam’s variable s from string amplitudes in holographic backgrounds. We generalize the approach of Polchinski and Strassler [1] in two ways. (i) We analyze several holographic confining backgrounds, in particular the hard wall model, the soft wall model and Witten’s model. (ii) In addition to deriving the asymptotic behavior of the amplitudes at fixed angle and in the Regge limit, we also expand the amplitudes around their poles, integrate over the holographic direction and then re-sum the expansion. Due to dependence of the string tension on the holographic coordinate, the resulting singularities take the form of branch points rather than poles and the amplitudes display branch cuts and acquire a finite imaginary part. This may signal the failure of the PS prescription to reproduce the correct analytic structure at low energies. We also observe that the peaks are more pronounced in the region of small s but fade away for large s. In the fixed angle approximation we find in the hard and soft wall models that   <math> <mi>A</mi> </math>  $$ \mathcal{A} $$  ∼ s 2−∆/2 whereas in Witten’s model   <math> <mi>A</mi> </math>  $$ \mathcal{A} $$  ∼ s 3−∆/2 and   <math> <mi>A</mi> </math>  $$ \mathcal{A} $$  ∼ s 7/3−2∆/3 for the 11D and 10D formulations, respectively. In the Regge regime   <math> <mi>A</mi> </math>  $$ \mathcal{A} $$  ∼ s 2 t  −2+α  (log s/t) −1+α  where α is the power found in the fixed angle regime. Using the pole expansion the result for each model is Re [  <math> <mi>A</mi> </math>  $$ \mathcal{A} $$ ] ∼ s  −1, Im [  <math> <mi>A</mi> </math>  $$ \mathcal{A} $$ ] ∼ s  α . We compute the corresponding amplitudes for mesons using open strings and find qualitatively similar results as for closed strings.

Partonic behavior of string scattering amplitudes from holographic QCD models

We extend and generalise the string corrections to the EM memory to the Type I superstring including spin effects. Very much as in the simpler bosonic string context, the relevant corrections are non-perturbative in <math><msup><mrow><mi>α</mi></mrow><mrow><mo>′</mo></mrow></msup></math>, slowly decaying (as <math><mn>1</mn><mo>/</mo><mi>R</mi></math>) at large distances and modulated in retarded time <math><mi>u</mi><mo>=</mo><mi>t</mi><mo>−</mo><mi>R</mi></math>. For spin N states in the first Regge trajectory they entail a sequence of N derivatives wrt u on the ‘parent’ <math><mi>N</mi><mo>=</mo><mn>0</mn></math> amplitude. We also briefly discuss how to include loop effects, that broaden and shift the string resonances, and how to modify our analysis for macroscopic semi-classical quasi-BPS coherent states, whose collisions may lead to detectable string memory signals in viable Type I models.

Alice Aldi

Spinning-off stringy electro-magnetic memories

We discuss stringy <math><msup><mrow><mi>α</mi></mrow><mrow><mo>′</mo></mrow></msup></math> corrections to the gravitational wave signal generated in the merging of two black holes. We model the merging with two BPS compact massive objects in the heterotic string, described by standard vertex operators or coherent states. Despite the expected cubic suppression in <math><msup><mrow><mi>α</mi></mrow><mrow><mo>′</mo></mrow></msup></math> w.r.t. the General Relativity result in the low-energy expansion, at tree level the infinite tower of string resonances leaves a clear footprint on the gravitational wave signals within the sensitivity region of aLIGO/VIRGO and future interferometers. Including loop effects that broaden and destabilise the string resonances suggests a sort of lost stringy memory effect that can be regained through the analysis of the quasi normal modes in the ring-down phase.

China

Andrea Addazi

Antonino Marcianò

String memories ... lost and regained

We reconsider the two-point string scattering amplitudes of massless Neveu-Schwarz–Neveu-Schwarz states of Type IIB orientifold superstring theory on the disk and projective plane in ten dimensions and analyse its <math><msup><mrow><mi>α</mi></mrow><mrow><mo>′</mo></mrow></msup></math> expansion. We also discuss the unoriented Type IIB theory on <math><msup><mrow><mi>T</mi></mrow><mrow><mn>6</mn></mrow></msup><mo>/</mo><msub><mrow><mi>Z</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>×</mo><msub><mrow><mi>Z</mi></mrow><mrow><mn>2</mn></mrow></msub></math> where two-point string scattering amplitudes of the complex Kähler moduli and complex structures of the untwisted sector are computed on the disk and projective plane. New results are obtained together with known ones. Finally, we compare string scattering amplitudes results at <math><msup><mrow><msup><mrow><mi>α</mi></mrow><mrow><mo>′</mo></mrow></msup></mrow><mrow><mn>2</mn></mrow></msup></math>-order with the (curvature)$^{2}$ terms in the low energy effective action of D-branes and Ω-planes in both cases.