  We prove that the complex Euclidean field theory with local quartic
self-interaction in two dimensions arises as a limit of an interacting Bose gas
at positive temperature, when the density of the gas becomes large and the
range of the interaction becomes small. The field theory is supported on
distributions of negative regularity, which requires a renormalization by
divergent mass and energy counterterms. We obtain convergence of the relative
partition function and uniform convergence of the renormalized reduced density
matrices. The proof is based on three main ingredients: (a) a quantitative
analysis of the infinite-dimensional saddle point argument for the functional
integral introduced in [32] using continuity properties of Brownian paths, (b)
a Nelson-type estimate for a general nonlocal field theory in two dimensions,
and (c) repeated Gaussian integration by parts in field space to obtain uniform
control on the renormalized correlation functions. As a byproduct of our proof,
in two and three dimensions we also extend the results on the mean-field limit
from [32,56] to unbounded interaction potentials satisfying the optimal
integrability conditions proposed by Bourgain [13].
