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This paper studies asymptotic solvability of a linear quadratic (LQ) meanfield social optimization problem with controlled diffusions and indefinitestate and control weights. Starting with an $N$-agent model, we employ arescaling approach to derive a low-dimensional Riccati ordinary differentialequation (ODE) system, which characterizes a necessary and sufficient conditionfor asymptotic solvability. The decentralized control obtained from the meanfield limit ensures a bounded optimality loss in minimizing the social costhaving magnitude $O(N)$, which implies an optimality loss of $O(1/N)$ peragent. We further quantify the efficiency gain of the social optimum withrespect to the solution of the mean field game.

Linear quadratic mean field social optimization: Asymptotic solvability and decentralized control