Chaotic scattering of highly excited strings

Abstract

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.

Scattering and Strebel graphs

“…In [16,17], Gross and Mende studied the high energy fixed angle regime of string scattering amplitudes of arbitrary loop, primarily motivated to discover the analogs of short distance physics like operator product expansion, renormalization group in string theory. See [33] for recent progress on the high energy behaviour of scattering amplitudes involving highly excited strings (in contrast to the light strings in Gross-Mende approach).…”

Chaotic instability in the BFSS matrix model

“…The complexity can be measured by positive Lyapunov exponents and non-quasiperiodic plots in Poincaré section. Recently, there has been an attempt to figure out chaotic behavior in the scattering process [1][2][3]. The S-matrix, which is one of the most important physical quantities, is a good target to tackle chaotic behavior in the context of field theory.…”

Maximally chaotic dynamical systems

“…Introduction.-The maximally chaotic dynamical systems continue to attract great attention of the researchers due to their rich physical properties and extended areas of application. In recent years the classical and quantum-mechanical concepts of maximally chaotic systems were developed in series of publications with application to the physics of black holes [1][2][3][4][5][6] , of non-Abelian gauge fields 7,8 and string theory 9 , to the fluid dynamics [13][14][15] and astrophysics [16][17][18][19] , foundation of statistical physics [20][21][22][23][24] and computer science [25][26][27] . There is a growing evidence that intrinsic properties of the Hawking black hole radiation can be understood in terms of classical and quantum theory of chaos [28][29][30][31][32][33][34] .…”

“…where c 1 , c 2 , a k are arbitrary constants. The spectrum belongs to the discrete values of spectral parameter a = 2πk in (9) and have a zero measure in the continuous spectrum of the limiting system (8). The inverse transformation ψ (−n) = G n ψ, n = 0, 1, 2, ... is therefore defined modulo kernel (23).…”

“…Its spectral characteristics are defined by the eigenvalue equation A x ψ(x) = λψ(x). Searching the eigenfunctions in the form of plane waves ψ a (x) = e iax(9)…”