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Local operators in the Sine-Gordon model: $\partial_μφ\, \partial_νφ$ and the stress tensor

Markus B. Fröb, Daniela Cadamuro

TL;DR

The paper addresses renormalisation of simple local operators in the massless Sine-Gordon model, focusing on $O_{μν}=∂_μ φ ∂_ν φ$ and the stress tensor $T_{μν}$. It employs Gell-Mann–Low perturbation theory with IR/UV cutoffs in both Euclidean QFT and pAQFT for Minkowski space to define renormalised time-ordered products, proving the necessity of local counterterms beyond Wick ordering at every order. The main results establish the convergence of the renormalised perturbation series for the operators' expectation values in Euclidean space and in Minkowski space for Hadamard states, and show that a quantum correction proportional to $ħ$ yields a conserved quantum stress tensor, giving a modified tensor $\ ilde{T}_{μν}$. The work forges a bridge between Euclidean renormalisation and perturbative algebraic QFT while providing explicit local renormalisation structures and Ward-identity–related insights for local observables in the Sine-Gordon model.

Abstract

We consider the simplest non-trivial local composite operators in the massless Sine-Gordon model, which are $\partial_μφ\, \partial_νφ$ and the stress tensor $T_{μν}$. We show that even in the finite regime $β^2 < 4 π$ of the theory, these operators need additional renormalisation (beyond the free-field normal-ordering) at each order in perturbation theory. We further prove convergence of the renormalised perturbative series for their expectation values, both in the Euclidean signature and in Minkowski space-time, and for the latter in an arbitrary Hadamard state. Lastly, we show that one must add a quantum correction (proportional to $\hbar$) to the renormalised stress tensor to obtain a conserved quantity.

Local operators in the Sine-Gordon model: $\partial_μφ\, \partial_νφ$ and the stress tensor

TL;DR

The paper addresses renormalisation of simple local operators in the massless Sine-Gordon model, focusing on and the stress tensor . It employs Gell-Mann–Low perturbation theory with IR/UV cutoffs in both Euclidean QFT and pAQFT for Minkowski space to define renormalised time-ordered products, proving the necessity of local counterterms beyond Wick ordering at every order. The main results establish the convergence of the renormalised perturbation series for the operators' expectation values in Euclidean space and in Minkowski space for Hadamard states, and show that a quantum correction proportional to yields a conserved quantum stress tensor, giving a modified tensor . The work forges a bridge between Euclidean renormalisation and perturbative algebraic QFT while providing explicit local renormalisation structures and Ward-identity–related insights for local observables in the Sine-Gordon model.

Abstract

We consider the simplest non-trivial local composite operators in the massless Sine-Gordon model, which are and the stress tensor . We show that even in the finite regime of the theory, these operators need additional renormalisation (beyond the free-field normal-ordering) at each order in perturbation theory. We further prove convergence of the renormalised perturbative series for their expectation values, both in the Euclidean signature and in Minkowski space-time, and for the latter in an arbitrary Hadamard state. Lastly, we show that one must add a quantum correction (proportional to ) to the renormalised stress tensor to obtain a conserved quantity.
Paper Structure (1 section, 1 theorem, 3 equations)

This paper contains 1 section, 1 theorem, 3 equations.

Table of Contents

  1. Introduction

Key Result

theorem 1

Consider the massless Euclidean Sine-Gordon model in the finite regime $\beta^2 < 4\pi$ and with the free-field covariance $C^{\Lambda,\epsilon}$ with IR cutoff $\Lambda$ and UV cutoff $\epsilon$. There exists a choice of local counterterms (diverging logarithmically with the UV cutoff $\epsilon$) s in the free theory, with $\mathcal{N}_\mu$ denoting normal ordering with respect to the covariance

Theorems & Definitions (1)

  • theorem 1: Renormalisation in Euclidean signature