Accumulation-point amplitudes in string theory

Abstract

We point out some common qualitative features of the Coon amplitude — a family of deformations of the Veneziano amplitude with logarithmic Regge trajectories — and the open string scattering amplitude for strings ending on a D-brane in AdS. Both reduce to the Veneziano amplitude at relatively low energies. Both systems have an accumulation point in their spectrum, with an infinite number of states below a certain energy. The approach to this point is very similar. Both have the same high-energy fixed-angle behavior. Nevertheless, we find some differences in the spectrum of states with highest angular momentum. These similarities suggest that there may exist a string background that realizes the Coon amplitude.

Properties of infinite product amplitudes: Veneziano, Virasoro, and Coon

“…In any case, the Coon amplitude is a fruitful example for the study of general scattering amplitudes [5][6][7][8][9]. While there is yet no definitive field theory or string worldsheet realization of the Coon amplitude, accumulation point spectra like those exhibited by the Coon amplitude with q < 1 have been recently found in a stringy setup involving open strings ending on a D-brane [10]. Similar accumulation point spectra have also appeared in recent amplitude studies [6,7,[9][10][11][12].…”

“…While there is yet no definitive field theory or string worldsheet realization of the Coon amplitude, accumulation point spectra like those exhibited by the Coon amplitude with q < 1 have been recently found in a stringy setup involving open strings ending on a D-brane [10]. Similar accumulation point spectra have also appeared in recent amplitude studies [6,7,[9][10][11][12]. Famously, the hydrogen atom has energy levels E n = −13.6 eV/n 2 with an accumulation at E ∞ = 0.…”

On the positivity of Coon amplitude in D=4

“…However, it may still be possible that the underlying physical description suffers from problems resulting from negative norm states when loops are taken into account, and hence the physical description is inconsistent in the first place. In order to check such issues, one needs to understand the physics behind the amplitude, which may be similar to the prescription suggested by [7]. We aim to understand aspects of the underlying description in a forthcoming work.…”

“…In particular, this mysterious amplitude is the unique solution to duality constraints having an infinity of exchanges with logarithmic trajectories. Recently, it has been brought forward as an interesting case study on its own [4][5][6][7][8] while also serving as a useful reference to the study of accumulation point amplitudes with similar infinite towers of states [8][9][10][11][12].…”

“…Apart from certain aspects of asymptotic behaviour, not much about this amplitude is well understood. In particular, details regarding the underlying physical interpretation of the amplitude are lacking, although a proposal was recently put forward regarding the same [7]. Additionally, from previous works [15,16], it is not clear whether the amplitude (1.1) obeys unitarity constraints, though a recent numerical analysis indicates that it is so [5].…”