[Paper Review] NLIE for the Sausage model

Changrim Ahn, János Balog+1 more|arXiv (Cornell University)|Jan 31, 2017

Black Holes and Theoretical Physics9 references3 citations

TL;DR

This paper proposes a finite set of nonlinear integral equations (NLIEs) for the sausage model, a deformed $O(3)$ sigma model preserving integrability, applicable to generic values of the deformation parameter $\lambda$, not just integer $1/\lambda$. Derived from truncated TBA equations via $T$-$Q$ relations, the NLIEs enable exact computation of the vacuum energy and effective central charge in both UV and IR limits, yielding a new exact relation between the $S$-matrix parameter $\lambda$ and the target space deformation $\nu$, differing from the original conjecture.

ABSTRACT

The sausage model, first proposed by Fateev, Onofri, and Zamolodchikov, is a deformation of the O(3) sigma model preserving integrability. The target space is deformed from the sphere to "sausage" shape by a deformation parameter ν. This model is defined by a factorizable S-matrix which is obtained by deforming that of the O(3) sigma model by a parameter λ. Clues for the deformed sigma model are provided by various UV and IR information through the thermodynamic Bethe ansatz (TBA) analysis based on the S-matrix. Application of TBA to the sausage model is, however, limited to the case of 1/λinteger where the coupled integral equations can be truncated to a finite number. In this paper, we propose a finite set of nonliear integral equations (NLIEs), which are applicable to generic value of λ. Our derivation is based on T-Q relations extracted from the truncated TBA equations. For consistency check, we compute next-leading order corrections of the vacuum energy and extract the S-matrix information in the IR limit. We also solved the NLIE both analytically and numerically in the UV limit to get the effective central charge and compared with that of the zero-mode dynamics to obtain exact relation between νand λ. This paper is a tribute to the memory of Prof. Petr Kulish.

Motivation & Objective

• To overcome the limitation of thermodynamic Bethe ansatz (TBA) in the sausage model, which is only tractable for integer $1/\lambda$, by deriving a finite set of NLIEs valid for generic $\lambda$.
• To establish a direct link between the $S$-matrix parameter $\lambda$ and the target space deformation $\nu$ via NLIEs, resolving discrepancies in the original conjecture.
• To verify the consistency of the NLIEs by computing next-leading order corrections to the vacuum energy in the IR limit and matching $S$-matrix elements.
• To solve the NLIEs analytically and numerically in the UV limit to extract the effective central charge and derive an exact relation between $\lambda$ and $\nu$.

Proposed method

• Derive NLIEs from the truncated TBA equations of the sausage model using $T$-$Q$ relations extracted from the $Y$-system.
• Apply analytic continuation to extend the TBA results from integer $1/\lambda$ to generic $\lambda$, with a posteriori consistency checks.
• Use the UV limit of the NLIEs to compute the effective central charge via analytical and numerical solutions, matching with zero-mode dynamics.
• Perform perturbative analysis in the UV using a coupling expansion to derive the central charge and relate it to the system size.
• Extract $S$-matrix elements from the NLIEs in the IR limit to verify consistency with the original $S$-matrix construction.
• Utilize symmetry properties and functional equations to reduce the NLIE system to a solvable form in the UV and IR regimes.

Experimental results

Research questions

• RQ1 Can a finite set of NLIEs be derived for the sausage model that are valid for generic $\lambda$, not restricted to integer $1/\lambda$?
• RQ2 What is the exact relation between the $S$-matrix parameter $\lambda$ and the target space deformation $\nu$ as determined by the NLIEs?
• RQ3 How do the NLIEs reproduce the correct $S$-matrix elements in the IR limit, confirming consistency with the original $S$-matrix?
• RQ4 What is the effective central charge in the UV limit as computed from the NLIEs, and how does it compare with the zero-mode dynamics?
• RQ5 Does the UV solution of the NLIEs yield a consistent perturbative expansion of the central charge that matches known results?

Key findings

• The NLIEs are successfully derived for generic $\lambda$ using $T$-$Q$ relations from truncated TBA equations, extending beyond the prior restriction to integer $1/\lambda$.
• In the IR limit, the NLIEs reproduce the correct $S$-matrix elements through next-leading order corrections to the vacuum energy, confirming consistency.
• In the UV limit, the effective central charge computed from the NLIEs matches exactly with the zero-mode dynamics, yielding a new exact relation between $\lambda$ and $\nu$.
• The derived relation between $\lambda$ and $\nu$ differs from the original conjecture in the literature, indicating a correction to the earlier ansatz.
• The UV perturbative expansion of the central charge is computed as $c(r) \approx 2 - \frac{3\pi^2}{2} \frac{N-2}{x^2} + \dots$, with $N=4$ corresponding to the sausage model, confirming the expected scaling.
• The UV solution shows no plateau in the $\varepsilon_a(0)$ functions, implying $\varepsilon_a(0) \to -\infty$, consistent with the sinh-Gordon model behavior.