Local operators in the Sine-Gordon model: ${\partial }_{\mu }\varphi {\partial }_{\nu }\varphi$ and the stress tensor

Abstract

We consider the simplest non-trivial local composite operators in the massless Sine-Gordon model, which are ${\partial }_{\mu }\varphi {\partial }_{\nu }\varphi$ 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 $\hslash$) to the renormalised stress tensor to obtain a conserved quantity.

arXiv comment62 pages in AHP style, revised renormalisation (Lemmas 2 and 4), matches published version

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