Excited states by analytic continuation of TBA equations

Abstract

We suggest an approach to the problem of finding integral equations for the excited states of an integrable model, starting from the Thermodynamic Bethe Ansatz equations for its ground state. The idea relies on analytic continuation through complex values of the coupling constant, and an analysis of the monodromies that the equations and their solutions undergo. For the scaling Lee-Yang model, we find equations in this way for the one-and two-particle states in the spin-zero sector, and suggest various generalisations. Numerical results show excellent agreement with the truncated conformal space approach, and we also treat some of the ultraviolet and infrared asymptotics analytically.

Review of AdS/CFT Integrability, Chapter III.5: Lüscher Corrections

“…In order to describe excited states, one uses an analytical continuation trick due to Dorey and Tateo [14] that essentially introduces additional source terms into (2.14). These source terms are generated by singularities of the integrand 1 + e −ε(z i ) = 0, which, through integration by parts and an evaluation through residues give rise to additional source terms on the r.h.s.…”

“…the sinh-Gordon model). On the other hand, for a theory with μ-terms, like the SLYM, at least two source terms correspond to a single physical particle (see [14,15]). Now in order to obtain the Lüscher corrections, we have to perform a large volume expansion of these equations.…”

A next-to-leading Lüscher formula

“…We shall observe, at least in a diagonal case, that the insertion of such defects modifies the ground-state TBA equations in the same way as resulting from the standard analytical continuation of the ground-state TBA [15,16,38]. More importantly for our purposes, the twist matrices involved in the NLO Lüscher-like formula for the twisted vacuum [37] will be replaced by the S-matrices describing the scattering between physical and virtual particles.…”

“…of (2.18), with θ 1,2 being the rapidities of the bound-state's constituents. For example, in the Lee-Yang model, the leading order rapidities would be θ 1,2 = ±i/3 in our normalization, giving the same source term as in [38], once crossing symmetry and unitarity are used.…”

Quasi-local formulation of the mirror TBA

“…After integrability was discovered in the string worldsheet theory, the mirror Thermodynamic Bethe Ansatz (TBA) technique was proposed [3,4] to determine nonperturbatively the spectrum of the string theory. The TBA equations for AdS/CFT were first derived for the ground state [5,6,8,7,9] and then using an analytic continuation trick [10] TBA equations were conjectured for certain classes of states in the sl(2) sector of the theory [11,12,13].…”