  We prove that the truncated correlation functions of the charge and gradient
fields associated with the massless sine-Gordon model on $\mathbb{R}^2$ with
$\beta=4\pi$ exist for all coupling constants and are equal to those of the
chiral densities and vector current of free massive Dirac fermions. This is an
instance of Coleman's prediction that the massless sine-Gordon model and the
massive Thirring model are equivalent (in the above sense of correlation
functions). Our main novelty is that we prove this correspondence starting from
the Euclidean path integral in the non-perturative regime of the infinite
volume models. We use this correspondence to show that the correlation
functions of the massless sine-Gordon model with $\beta=4\pi$ decay
exponentially and that the corresponding probabilistic field is localized.
