Diagonal finite volume matrix elements in the sinh-Gordon model

Abstract

Using the fermionic basis we conjecture exact expressions for diagonal finite volume matrix elements of exponential operators and their descendants in the sinh-Gordon theory. Our expressions sum up the LeClair-Mussardo type infinite series generalized by Pozsgay for excited state expectation values. We checked our formulae against the Liouville three-point functions for small, while against Pozsgay's expansion for large volumes and found complete agreement. * Membre du CNRS the ground-state TBA densities with the excited state ones and contains additional factors, which can be interpreted as partial density of states [20,21].Alternatively, there is an other approach which focuses on specific theories and exploits their hidden (Grassmann) structure to provide compact expressions for finite volume matrix elements [22]. These specific continuum models arise as limits of integrable lattice models and the most studied examples are the sinh-Gordon and sine-Gordon models. There have been active work and relevant progress in deriving finite volume one-point functions for the exponential operators and their descendants in these theories [23,24]. These results were then extended for diagonal matrix elements in the sine-Gordon theory [25,26,27] and the aim of our paper is to provide similar expressions in the sinh-Gordon theory.The paper is organized as follows: Section 2 reviews the description of the finite size energy spectrum of the sinh-Gordon theory. A multi-particle state for large volumes can be labelled by momentum quantum numbers, which we relate at small volume to the spectrum of the Liouville conformal field theory by matching the eigenvalues of the conserved charges. In Section 3 we formulate our main conjecture for the finite volume exceptions values in the fermionic basis. The novelty compared to the vacuum expectation values is the discrete part of the convolutions, which carries information on the particles' rapidities. We check this conjecture for large volumes in Section 4. The discrete part of the convolution contains the polynomial, while the continuous part the exponentially small corrections in the volume. In Section 5 we compare our conjecture with Liouville three-point functions for low lying states including non-degenerate and degenerate L 0 subspaces. All the checks performed confirm our conjecture, thus we close the paper with conclusions in Section 6.

Exact Three-Point Functions of Determinant Operators in Planar N=4 Supersymmetric Yang-Mills Theory

“…These determinants can be viewed as the finite-volume version of the Gaudin determinants for the norm of the spin chain. They also resemble the finite-volume one-point functions in sin(h)-Gordon model [25][26][27].…”

Leading exponential finite size corrections for non-diagonal form factors

“…Although the expansion can be useful for practical applications, for obtaining exact results the series has to be summed up. In this respect the integral equation derived recently for diagonal form factors in the sinh-Gordon theory can be useful [44].…”

Effective Quantum Field Theory for the Thermodynamical Bethe Ansatz

“…More rigorous approach based on an integrable lattice regularization was developped in [27,36]. Some recent developments concern the computation of the one-point functions [37] and diagonal form factors [38]. As a scattering theory, the sinh-Gordon model involves a single massive relativistic particle whose mass m is determined by the coupling constant µ.…”