  This paper studies an asymptotic solvability problem for linear quadratic
(LQ) mean field games with controlled diffusions and indefinite weights for the
state and control in the costs. 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 rescaling technique is further used for performance estimates, establishing
an $O(1/N)$-Nash equilibrium for the obtained decentralized strategies.
