# How do we determine who eats who (or which field gets massive) in a Higgs phase transition?

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?

• I suspect you are misreading the message of your reference. Reviewing its Abelian Higgs mechanism, the gauge field allows the original field a to get a mass maintaining its shift symmetry and removes all massless excitations from the spectrum (indeed, Lorentz invariance connects the engorged original photon degrees of freedom to the massive a, but one not care about that). Likewise, GD demonstrates massive modes, so there is no massless d.o.f. left. Whom it belongs to is a discretionary "academic" matter, as it is in the SM--but rarely discussed. – Cosmas Zachos Jan 29 at 17:11
• The upshot in his actual system is that there is only one massive degree of freedom, and it does not matter what fields or combinations thereof one chooses to give it residence in. It does not even Lorentz-transform into anything else, as there is nothing else. – Cosmas Zachos Jan 29 at 18:09
• @CosmasZachos Wow, that changes everything. Are you saying that the concept of mass is dissociated from the concept of what exactly possesses the mass, and that the latter is an “academic” matter only? If what you are saying is true, then it's an amazing way of thinking about the matter. – Nanashi No Gombe Jan 29 at 20:23
• Nono... There is a massive excitation. But the field description thereof, especially with gauge freedom, is not fixed. In the SM, you real one uses the unitary gauge, or other more computationally suited ones. In Susskind's complementarity description, the massive states are all "bound-state" composite gauge singlets... – Cosmas Zachos Jan 29 at 21:17