$B \wedge F$ Term by Spontaneous Symmetry Breaking in a generalized Abelian Higgs Model

We show that the topological $B \wedge F$ term in $3+1$ dimensions can be generated via spontaneous symmetry breaking in a generalized Abelian Higgs model. Further, we also show that even in $D$ dimensions $( D \geq 3 )$, a $B \wedge F$ term gives rise to the topological massive excitations of the Abelian gauge field and that such a $B \wedge F$ term can also be generated via Higgs mechanism.