  This paper is devoted to solving a class of second order
Hamilton-Jacobi-Bellman (HJB) equations in the Wasserstein space, associated
with mean field control problems involving common noise. We provide the
well-posedness of viscosity solutions to the HJB equation in the sense of
Crandall-Lions' definition, under general assumptions on the coefficients. Our
approach adopts the smooth metric developed by Bayraktar, Ekren, and Zhang
[Proc. Amer. Math. Soc. (2023)] as our gauge function for the purpose of smooth
variational principle used in the proof of comparison theorem. Subsequently, we
derive further estimates and regularity of the metric, including a novel second
order derivative estimate with respect to the measure variable, in order to
ensure the uniqueness and existence.
