Local operators in the Sine-Gordon model: $\partial_\mu \phi \, \partial_\nu \phi$ and the stress tensor

TL;DR

This paper analyzes the renormalization and conservation properties of local operators, specifically the stress tensor, in the massless Sine-Gordon model, demonstrating the necessity of additional renormalization and quantum corrections.

Contribution

It proves the need for extra renormalization of local operators in the Sine-Gordon model and establishes the convergence of their expectation values in various spacetime signatures.

Findings

Additional renormalization is required beyond normal-ordering.

Renormalized expectation values converge in Euclidean and Minkowski space.

Quantum correction is necessary for stress tensor conservation.

Abstract

We consider the simplest non-trivial local composite operators in the massless Sine-Gordon model, which are $\partial_\mu \phi \, \partial_\nu \phi$ and the stress tensor $T_{\mu\nu}$. We show that even in the finite regime $\beta^2 < 4 \pi$ 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.