Asymptotics of the TBA equation

Question

The Thermodynamic Bethe Ansatz equation is an integral equation that was derived by Yang and Yang to study some interacting systems. In the simplest case, it is $$\epsilon(\beta)=R\cosh\beta-\int\frac{d\beta'}{\pi\cosh(\beta-\beta')}\log(1+\exp(-\epsilon(\beta'))).$$

A reference for this is Al.B. Zamolodchikov, Thermodynamic Bethe Ansatz in relativistic models, Nucl Phys B342 (1990) 695-720.

I am interested in the asymptotics of the solutions as $R\rightarrow 0$. Zamolodchikov gives a heuristic argument that for small $R\cosh\beta$ we can neglect the first term. Therefore, the $\beta\rightarrow\beta+const$ invariance is restored and the solution $\epsilon(\beta)$ becomes independent of $\beta$ for small $R\cosh\beta$. Can one deduce a more precise asymptotic behavior? In particular, I would like to write down the small $R$ corrections.

One can easily show that this integral operator acting on $\exp(-\epsilon)$ maps the ball of radius $(e^R-1)^{-1}$ in $C^0(\mathbf{R})$ to itself. Furthermore, it is a contraction for large $R$ (see e.g. http://arxiv.org/abs/0807.4723, appendix C).

Was this equation studied anywhere in the mathematical literature?

Answer 1

Some pointers to the literature that may help: The small-$R$ asymptotics of the Thermodynamic Bethe Ansatz equation can be expressed in terms of a Painleve III function with independent variable $R$ [1], and the small-$R$ asymptotics has been studied in that connection [2]. The $R\rightarrow 0$ limit (called the "massless" or "ultraviolet" limit in the physics literature) has the form of an Airy function [3].

[1] C.A. Tracy and H. Widom, Proofs of Two Conjectures Related to the Thermodynamic Bethe Ansatz, Commun.Math.Phys. 179 (1996) 667-680 [arXiv:solv-int/9509003].

[2] P. Fendley and H. Saleur, $N=2$ Supersymmetry, Painleve III and Exact Scaling Functions in 2D Polymers, Nucl.Phys.B 388 (1992) 609-626 [arXiv:hep-th/9204094].

[3] P. Fendley, Airy functions in the thermodynamic Bethe ansatz, Lett.Math.Phys. 49 (1999) 229-233 [arXiv:hep-th/9906114].