Massless AdS 2 scattering and Bethe ansatz

Abstract

We first analyse the integrable scattering theory describing the massless excitations of AdS 2 × S 2 × T 6 superstrings in the relativistic limit. The matrix part of the S-matrix is obtained in the BMN limit from the conjectured exact expression, and compared to known S-matrices with N = 1 supersymmetry in 1 + 1 dimensions. A dressing factor, yet unknown for the complete theory, is here constructed based on relativistic crossing symmetry. We derive a Bethe-ansatz condition by employing a transfer-matrix technique based on the so-called free-fermion condition. This is known to overcome the problem of lack of a reference state. We then generalise the method to the massless non-relativistic case, and compare the resulting Bethe-ansatz condition with a simple massless limit of the one conjectured by Sorokin, Tseytlin, Wulff and Zarembo.

Geometry of massless scattering in integrable superstring

“…In [128], the dressing factor for the relativistic massless R-matrix, dubbed Solution 3 in that paper, has been found, and an integral representation is given in Appendix B. Supported by the check that our conjecture works also for the dressing factor in the AdS 3 case, we infer that the dressing factor for the massless non-relativistic AdS 2 case is…”

“…If one attempts to take a relativistic (BMN) limit, this turns out to be trivial for massive modes. Crucially, this is not the case in the massless sector: the relativistic limit is non-trivial between right-right and left-left movers [128]. The Lie-algebra reduces to N = 1 supersymmetry [129], but the S-matrix is rather different.…”

“…Integrable systems not admitting a reference state are an area of intense current investigation [135][136][137][138][139][140][141]. The approach of [128] relied on the so-called free-fermion condition [133,142] combined with the use of inversion relations [143].…”

“…The upper (lower) signs in (2.6) and (2.7) correspond to f → +1 ( f → −1) when considering the limit from the massive R-matrix, as explained in [128]. In order to understand the dispersion relation (2.5) as the vanishing of a quadratic Casimir, we extend the superalgebra su(1|1) c to the q-deformed Poincaré superalgebra in d = 2, E q (1|1).…”

“…The relativistic massless R-matrices to which (3.4) and (3.5) tend in the relativistic limit are encoded in Solution 3 and Solution 5 studied in [128], and they are respectively 7…”