# Asymptotics of the TBA equation

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).

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 $R$ , and the small-$R$ asymptotics has been studied in that connection . The $R\rightarrow 0$ limit (called the "massless" or "ultraviolet" limit in the physics literature) has the form of an Airy function .
 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].
• The first and third papers deal with a slightly different version they call the $N=2$ TBA equation. Do their results generalize to other TBA-type equations (in particular, the one I mentioned)? – Pavel Safronov Oct 30 '11 at 22:08