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We study tree-level scattering processes of arbitrary string states using the DDF formalism and suitable coherent vertex operators. We obtain new exact compact formulae for heavy-heavy-light-light scattering amplitudes in open or closed bosonic string theories, and derive explicit exact expressions for the absorption cross-sections, and corresponding emission rates, of highly excited string states using the optical theorem and time reversal symmetry. We show that these expressions are independent of the microscopic structure of the excited string states without averaging. For the absorption of massless modes in open string theory, in particular, we find a constant, frequency-independent cross-section. In contrast, the corresponding cross-section for the absorption of massless modes by excited closed strings depends linearly on the frequency, implying a non-trivial grey-body factor. In both cases, at energies below the scale set by the mass of the highly excited strings, we find emission rates with a Boltzmann factor at Hagedorn temperature., Comment: 54 pages, 3 figures

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 $\beta$- ensemble, writing an interpolation between the known results of the Gaussian orthogonal, unitary, and symplectic ensembles., Comment: 47 pages, 17 figures

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., Comment: 37 pages, 4 figures

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 $\beta$-ensemble of random matrix theory (RMT). We show how the continuous parameter $\beta$ 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.

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., Comment: 28 pages, 15 figures

We propose a novel measure of chaotic scattering amplitudes. It takes the form of a log-normal distribution function for the ratios $r_n={\delta_n}/{\delta_{n+1}}$ of (consecutive) spacings $\delta_n$ 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 $r_n$ obey the same distribution that governs the non-trivial zeros of Riemann zeta function., Comment: v2: small corrections, references added; v3: minor improvements to match published version

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., Comment: 32 pages, v2: typos fixed

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 (2001) 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 ${\cal A}\sim s^{2-\Delta/2}$ whereas in Witten's model ${\cal A} \sim s^{3-\Delta/2}$ and ${\cal A} \sim s^{7/3-2\Delta/3}$ for the 11D and 10D formulations, respectively. In the Regge regime ${\cal A}\sim {s^{2}}\,t^{-2+\alpha}\,(\log{s/ t})^{-1+\alpha}$ where $\alpha$ is the power found in the fixed angle regime. Using the pole expansion the result for each model is $Re[{\cal A}] \sim s^{-1}$, $Im[{\cal A}] \sim s^{\alpha}$. We compute the corresponding amplitudes for mesons using open strings and find qualitatively similar results as for closed strings., Comment: v1: 42 pages. v2: minor corrections, references added, 42 pages. v3: references added, and minor corrections made to text to match published version, 42 pages

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 $\alpha'$, slowly decaying (as $1/R$) at large distances and modulated in retarded time $u=t-R$. For spin $N$ states in the first Regge trajectory they entail a sequence of $N$ derivatives wrt $u$ on the `parent' $N=0$ 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., Comment: 24 pages, 5 figures

We identify string corrections to the EM memory effect. Though largely negligible in the low-energy limit, the effect become relevant in high-energy collisions and in extreme events. We illustrate our findings in a simple unoriented bosonic string model. Thanks to the coherent effect of the infinite tower of open string resonances, the corrections are non-perturbative in $\alpha'$, modulated in retarded time and slowly decaying even at large distances from the source. Remarkably compact expressions obtain for special choices of the kinematics in tree-level 4-point amplitudes. We discuss further corrections occurring at higher-points and the exponential damping resulting from broadening and shifting of the massive poles due to loops. Finally we estimate the range of the parameters and masses for detectability in semi-realistic (Type I) contexts and propose a rationale for this string memory effect., Comment: 16 pages, 5 figures

We discuss stringy ${\alpha'}$ 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 ${\alpha'}$ w.r.t. the General Relativity result, at tree level the string corrections seem to leave a footprint or a memory 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., Comment: 36 pages

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 $\alpha'$ expansion. We also discuss the unoriented Type IIB theory on $T^6/\mathbb{Z}_2\times\mathbb{Z}_2$ where two-point string scattering amplitudes of the complex K\"ahler 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 $\alpha'^2$-order with the (curvature)$^2$ terms in the low energy effective action of D-branes and $\Omega$-planes in both cases., Comment: 49 pages

We present a compact expression for a coherent vertex operator in superstring theory, in the Neveu-Schwarz sector, that can be easily extended to the Ramond sector by supersymmetric transformations in target space. We give also an explicit construction for the Ramond coherent vertex operator. These constructions provide the use of the supersymmetric version of the Del Giudice-Di Vecchia-Fubini operators and Operator Product Expansion techniques, displaying manifest BRST invariance and after normal ordering also manifest covariance. As a consistency check we compute tree level three-points scattering amplitudes involving Neveu-Schwarz coherent vertex operators in different superghost pictures. Extending these results to the closed superstring, we give the expressions of the coherent vertex operators in the case of type IIA and IIB theories., Comment: 29 pages

We study interactions of string coherent states in the DDF (after Di Vecchia, Del Giudice, Fubini) formalism. For simplicity we focus on open bosonic strings. After reviewing basic properties of DDF operators and of excited open strings, we present some classical profiles and show how they become more and more compact as the number of harmonics increases at fixed mass. We then compute various three- and four-point amplitudes with insertions of coherent states, tachyons and vector bosons on the boundary of the disk relying on a convenient choice of reference null momenta. We find that the amplitudes exponentiate in a rather subtle and interesting way. We then study the high-energy fixed-angle limit, dominated by a saddle-point when coherent states are present, and the soft behaviour as the momentum of a vector boson is taken to zero. We briefly comment on generalisation of our analysis to multiple intersecting and magnetised D-branes and to closed strings., Comment: 39 pages, 6 figures

We present our preliminary results of the non-perturbative determination of the valence mass dependent coefficients $b_\mathrm{A}-b_\mathrm{P}$ and $b_\mathrm{m}$ as well as the ratio $Z_\mathrm{P} Z_\mathrm{m}/ Z_\mathrm{A}$ entering the flavour non-singlet PCAC relation in lattice QCD with $N_f=3$ dynamical flavours. We apply the method proposed in the past for quenched approximation and $N_f=2$ cases, employing a set of finite-volume ALPHA configurations with Schr\"odinger functional boundary conditions, generated with $O(a)$ improved Wilson fermions and the tree-level Symanzik-improved gauge action for a range of couplings relevant for simulations at lattice spacings of about $0.09 \,$fm and below., Comment: 8 pages, 6 figures. Talk given at Lattice2017, the 35th International Symposium on Lattice Field Theory, 18-24 June 2017, Granada, Spain