Bound States in the Mirror TBA

TL;DR

This paper develops a mirror TBA framework for bound states with complex momenta in the AdS_5 imes S^5 superstring spectrum, revealing new analytic behaviors and energy formulas for these states.

Contribution

It introduces a novel TBA description for bound states with complex momenta, including new analytic features and quantization conditions, extending the understanding of the spectrum.

Findings

Y_Q-functions have poles in the analyticity strip.

New terms appear in the energy formula due to poles.

Quantization conditions vary for different three-particle states.

Abstract

The spectrum of the light-cone AdS_5 \times S^5 superstring contains states composed of particles with complex momenta including in particular those which turn into bound states in the decompactification limit. We propose the mirror TBA description for these states. We focus on a three-particle state which is a finite-size representative of a scattering state of a fundamental particle and a two-particle bound state and dual to an operator from the su(2) sector of N=4 SYM. We find that the analytic behavior of Y-functions differs drastically from the case of states with real momenta. Most importantly, Y_Q-functions exhibit poles in the analyticity strip which leads to the appearance of new terms in the formula for the energy of this state. In addition, the TBA equations are supplied by quantization conditions which involve Y_2. Considering yet another example of a three-particle state, we find that the corresponding quantization conditions do not even involve Y_1. Our treatment can be generalized to a wide class of states with complex momenta.