The massless sine-Gordon model with logarithmic correlations in arbitrary dimension
In this article, we study the sine-Gordon model with logarithmic correlations in arbitrary dimension $d \ge 1$. The model is defined as a Euclidean field theory whose interaction term in the classical action is \[ S_{\mathrm{int}}(\varphi) := 2z \int_{\Lambda} \cos(\sqrt{\beta}\,\varphi)\,\dx, \] where $\varphi$ is a logarithmically correlated Gaussian field on a compact domain $\Lambda \subset \mathbb{R}^d$ that coincides with the Gaussian free field when $d=2$, and where $z \in \mathbb{R}$ and $\beta>0$. We treat both the massive and massless cases, with the main emphasis on the massless regime. We construct the field without imposing boundary conditions, together with its charge and gradient correlation functions, and we show that the partition function is renormalizable. The results are non-perturbative, holding for all $z \in \mathbb{R}$. If $d=1$, our results are valid for the full subcritical range $\beta\in (0,4\pi)$ and if $d\geq 2$, the results are valid for $\beta\in (0,(d+1)2\pi)$. Our analysis is carried out directly in the continuum: we perform renormalization via a scale decomposition and control the partition function by controlling the renormalized potential that arises in this procedure, using the iterated Mayer expansion of Brydges and Kennedy~\cite{BrKe87a}.