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We consider the simplest non-trivial local composite operators in themassless 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 additionalrenormalisation (beyond the free-field normal-ordering) at each order inperturbation theory. We further prove convergence of the renormalisedperturbative series for their expectation values, both in the Euclideansignature and in Minkowski space-time, and for the latter in an arbitraryHadamard state. Lastly, we show that one must add a quantum correction(proportional to $\hbar$) to the renormalised stress tensor to obtain aconserved quantity.

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