  This manuscript constructs global in time solutions to the $master\
equations$ for potential Mean Field Games. The study concerns a class of
Lagrangians and initial data functions, which are $displacement\ convex$ and
so, it may be in dichotomy with the class of so--called $monotone$ functions,
widely considered in the literature. We construct solutions to both the scalar
and vectorial master equations in potential Mean Field Games, when the
underlying space is the whole space $\mathbb{R}^d$ and so, it is not compact.
