Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

On multipoint correlation functions in the Sinh-Gordon 1+1 dimensional quantum field theory

Karol K. Kozlowski, Alex Simon

TL;DR

The paper addresses the problem of constructing exact, per se truncated $k$-point functions in the 1+1D Sinh-Gordon quantum field theory within a rigorous bootstrap framework, tackling longstanding gaps for $k\ge3$.It develops a comprehensive methodology based on the factorised $S$-matrix, form-factor axioms, and a $K$-transform approach to generate explicit, structured representations for multi-particle kernels and their action on smeared multipoint operators.Key contributions include explicit direct and dual representations of multi-particle kernels, a master smeared integral representation for $r$-truncated $k$-point functions, and several closed, combinatorial expressions that organize form factors, $S$-matrices, and delta-distributions with precise boundary-values.Although convergence of the full series remains an open question, the results provide rigorous, implementable formulas and bounds that pave the way toward verifying the Wightman axioms and extending the framework to other integrable models such as sine-Gordon.

Abstract

This work provides a closed, explicit and rigorous expression for the appropriately truncated $k$-point function of the integrable 1+1 dimensional Sinh-Gordon quantum field theory. The results are obtained within the bootstrap program setting.

On multipoint correlation functions in the Sinh-Gordon 1+1 dimensional quantum field theory

TL;DR

The paper addresses the problem of constructing exact, per se truncated $k$-point functions in the 1+1D Sinh-Gordon quantum field theory within a rigorous bootstrap framework, tackling longstanding gaps for $k\ge3$.It develops a comprehensive methodology based on the factorised $S$-matrix, form-factor axioms, and a $K$-transform approach to generate explicit, structured representations for multi-particle kernels and their action on smeared multipoint operators.Key contributions include explicit direct and dual representations of multi-particle kernels, a master smeared integral representation for $r$-truncated $k$-point functions, and several closed, combinatorial expressions that organize form factors, $S$-matrices, and delta-distributions with precise boundary-values.Although convergence of the full series remains an open question, the results provide rigorous, implementable formulas and bounds that pave the way toward verifying the Wightman axioms and extending the framework to other integrable models such as sine-Gordon.

Abstract

This work provides a closed, explicit and rigorous expression for the appropriately truncated -point function of the integrable 1+1 dimensional Sinh-Gordon quantum field theory. The results are obtained within the bootstrap program setting.

Paper Structure

This paper contains 23 sections, 24 theorems, 427 equations.

Table of Contents

  1. Introduction
  2. The operator content & the Bootstrap program for the Sinh-Gordon model
  3. The basic operators
  4. The bootstrap program for the zero particle sector
  5. The bootstrap program for the multi-particle sector
  6. The direct representation for multi-particle matrix elements
  7. Dual representation and its equivalence with the direct one
  8. Toward truncated multi-point correlation functions: the smeared integral representation
  9. A premilinary expression
  10. An auxiliary representation
  11. The smeared integral representation
  12. The per se correlation function
  13. Various auxiliary bounds
  14. Closed representation for the $\boldsymbol{r}$-truncated multipoint functions
  15. A mixed representation
  16. ...and 8 more sections

Key Result

Proposition 2.1

BabujianKarowskiBreatherFFSineGordon Let $\boldsymbol{\ell}_n\in \{0,1\}^n$ and $p_{n}^{( \boldsymbol{ \texttt{O} } )}(\boldsymbol{\beta}_n\mid \boldsymbol{\ell}_n)$ be a solution to the below constraints Then, the sequence of meromorphic functions defined by means of the $\mathcal{K}$-transform definition K transformee fct p solves the bootstrap axioms ${\bf I)} -{\bf IV)}$.

Theorems & Definitions (27)

  • Proposition 2.1
  • Proposition 2.2
  • Proposition 2.3
  • Lemma 2.4
  • Lemma 2.5
  • Lemma 2.6
  • Proposition 2.7
  • Lemma 2.8
  • Lemma 2.9
  • Proposition 2.10
  • ...and 17 more