Consider the following theory in which a $2$-form field $B_{\mu\nu}$ with field strength $P_{\alpha\mu\nu}=\partial_{[\alpha}B_{\mu\nu]}$ is coupled to a $3$-form gauge field $C_{\alpha\beta\gamma}$ with field strength $F_{\mu\alpha\beta\gamma} = \partial_{[\mu}C_{\alpha\beta\gamma]}$.
$$ \mathcal L = 12 m^2(P_{\alpha\beta\gamma} - C_{\alpha\beta\gamma})^2+\frac12 F_{\mu\alpha\beta\gamma}F^{\mu\alpha\beta\gamma}\,.\tag{12} $$
Note that the field $B_{\mu\nu}$ has only $1$ propagating degree of freedom and is, in fact, physically equivalent to a scalar field. The gauge field $C_{\alpha\beta\gamma}$ has no propagating degrees of freedom at all. Therefore, the above theory describes only $1$ propagating degree of freedom, namely a scalar field which has been gauged and put into a Higgs phase of the gauge theory.
Usually, in the Higgs phase, the gauge field acquires mass. I would think that in the above theory the gauge field eats the $2$-form and acquires one massive degree of freedom. However, to the contrary, the scalar field somehow survives and, in fact, acquires mass [cf. G. Dvali (2005)].
I want to know the following. How do we find out for which field there are no massless excitations in the theory? In light of the above example, how do we “read off” from the Lagrangian that it is $B_{\mu\nu}$ who eats the gauge field and gets massive? Or, is it that both fields get massive and in order to see it explicitly, we have to integrate out the other field?
