  In this paper we study the large N limit of the $O(N)$-invariant linear sigma
model, which is a vector-valued generalization of the $\Phi^4$ quantum field
theory, on the three dimensional torus. We study the problem via its stochastic
quantization, which yields a coupled system of N interacting SPDEs. We prove
tightness of the invariant measures in the large N limit. For large enough mass
or small enough coupling constant, they converge to the (massive) Gaussian free
field at a rate of order $1/\sqrt N$ with respect to the Wasserstein distance.
We also obtain tightness results for certain $O(N)$ invariant observables.
These generalize some of the results in \cite{SSZZ20} from two dimensions to
three dimensions. The proof leverages the method recently developed by
\cite{GH18} and combines many new techniques such as uniform in $N$ estimates
on perturbative objects as well as the solutions.
