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Form factors of composite branch-point twist operators in the sinh-Gordon model on a multi-sheeted Riemann surface: semiclassical limit

Michael Lashkevich, Amir Nesturov

TL;DR

The paper addresses the problem of computing form factors for composite branch-point twist operators in the sinh-Gordon model on an $n$-sheeted Riemann surface in the semiclassical limit $b\to0$. It develops a radial-background semiclassical framework that linearizes fluctuations around a classical radial solution $\phi_\nu$ and leverages a Fredholm-determinant based machinery for the exponential CTOs while carefully renormalizing nonchiral descendants. The main contributions include explicit leading-order form factors for CTOs built from $V_\nu$ and their derivatives, as well as a detailed renormalization scheme for nonchiral descendants and explicit data for several resonance cases. The results illuminate the operator content of CTOs in the replica setting and are relevant for entanglement entropy calculations, providing a bridge between semiclassical methods and exact bootstrap form factors in integrable QFTs.

Abstract

Quantum sinh-Gordon model in 1+1 dimensions is one of the simplest and best-studied massive integrable relativistic quantum field theories. We consider this theory on a multi-sheeted Riemann surfaces with a flat metric, which can be seen as a pile of planes connected to each other along cut lines. The cut lines end at branch points, which are represented by a twist operator ${\cal T}_n.$ Operators of such kind are interesting in the framework of the problem of computing von Neumann and Renyi entanglement entropies in the original model on the plane. The composite branch-point twist operators (CTO) are a natural generalization of the twist operators, obtained by placing a local operator to a branch point by means of a certain limiting procedure. Correlation function in quantum field theory can be, in principle, found by means of the spectral decomposition. It allows one to express them in terms of form factors of local operators, i.e. their matrix elements in the basis of stationary states. In integrable models complete sets of exact form factors of all operators can be found exactly as solutions of a system of bootstrap equations. Nevertheless, identification of these solution to the operators in terms of the basic fields remains problematic. In this work, we develop a technique of computing form factors of a class of CTO determined in terms of the basic field in the semiclassical approximation.

Form factors of composite branch-point twist operators in the sinh-Gordon model on a multi-sheeted Riemann surface: semiclassical limit

TL;DR

The paper addresses the problem of computing form factors for composite branch-point twist operators in the sinh-Gordon model on an -sheeted Riemann surface in the semiclassical limit . It develops a radial-background semiclassical framework that linearizes fluctuations around a classical radial solution and leverages a Fredholm-determinant based machinery for the exponential CTOs while carefully renormalizing nonchiral descendants. The main contributions include explicit leading-order form factors for CTOs built from and their derivatives, as well as a detailed renormalization scheme for nonchiral descendants and explicit data for several resonance cases. The results illuminate the operator content of CTOs in the replica setting and are relevant for entanglement entropy calculations, providing a bridge between semiclassical methods and exact bootstrap form factors in integrable QFTs.

Abstract

Quantum sinh-Gordon model in 1+1 dimensions is one of the simplest and best-studied massive integrable relativistic quantum field theories. We consider this theory on a multi-sheeted Riemann surfaces with a flat metric, which can be seen as a pile of planes connected to each other along cut lines. The cut lines end at branch points, which are represented by a twist operator Operators of such kind are interesting in the framework of the problem of computing von Neumann and Renyi entanglement entropies in the original model on the plane. The composite branch-point twist operators (CTO) are a natural generalization of the twist operators, obtained by placing a local operator to a branch point by means of a certain limiting procedure. Correlation function in quantum field theory can be, in principle, found by means of the spectral decomposition. It allows one to express them in terms of form factors of local operators, i.e. their matrix elements in the basis of stationary states. In integrable models complete sets of exact form factors of all operators can be found exactly as solutions of a system of bootstrap equations. Nevertheless, identification of these solution to the operators in terms of the basic fields remains problematic. In this work, we develop a technique of computing form factors of a class of CTO determined in terms of the basic field in the semiclassical approximation.

Paper Structure

This paper contains 19 sections, 188 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Sinh-/Gordon model, its classical limit and saddle-/point approximation
  3. Properties of the sinh-/Bessel functions
  4. Definitions in terms of the Fredholm determinants
  5. Small $t$ expansions
  6. Vertex integrals
  7. Semiclassical form factors of the exponential operators and their descendants
  8. Operators ${\mathscr T}_nV_\nu$ and ${\mathscr T}_n(\partial^k\varphi\,V_\nu)$
  9. Operators ${\mathscr T}_n(\partial^k\varphi\,\bar{\partial}^k\varphi\,V_\nu)$
  10. The case $\sigma=\sigma'=1$, $k=2n$
  11. The cases $\sigma=1$, $\sigma'=0$ and $\sigma=0$, $\sigma'=1$
  12. Conclusion
  13. Consistency checks of the radial quantization scheme
  14. Integral representation for the connection coefficients
  15. General formulas
  16. ...and 4 more sections

Figures (2)

  • Figure 1: Schematic representation of a multi-/sheeted surface with two branch points $a$ and $b$.
  • Figure 2: Dragging an operator ${\mathscr O}$ around BPTO shifts it up one replica.