  In this manuscript, we study a class of linear-quadratic (LQ) mean field
control problems with a common noise and their corresponding $N$-particle
systems. The mean field control problems considered are not standard LQ mean
field control problems in the sense that their dependence on the mean field
terms can be non-linear and non-convex. Therefore, all the existing methods to
deal with LQ mean field control problems fail. The key idea to solve our LQ
mean field control problem is to utilize the common noise. We first prove the
global well-posedness of the corresponding Hamilton-Jacobi equations via the
non-degeneracy of the common noise. In contrast to the LQ mean field games
master equations, the Hamilton-Jacobi equations for the LQ mean field control
problems can not be reduced to finite-dimensional PDEs. We then globally solve
the Hamilton-Jacobi equations for $N$-particle systems. As byproducts, we
derive the optimal quantitative convergence results from the $N$-particle
systems to the mean field control problem and the propagation of chaos property
for the related optimal trajectories. This paper extends the results in [{\sc
M. Li, C. Mou, Z. Wu and C. Zhou}, \emph{Trans. Amer. Math. Soc.}, 376(06)
(2023), pp.~4105--4143] to the LQ mean field control problems.
