  This paper studies asymptotic solvability of a linear quadratic (LQ) mean
field social optimization problem with controlled diffusions and indefinite
state and control weights. We employ a rescaling approach to derive a low
dimensional Riccati ordinary differential equation (ODE) system, which
characterizes a necessary and sufficient condition for asymptotic solvability.
The decentralized strategies obtained from the mean field limit ensure a
bounded optimality gap in minimizing the social cost of the magnitude $O(N)$,
which translates to an $O(1/N)$ optimality loss per agent. We further quantify
the efficiency gain of the social optimum with respect to the mean field game
solution.
