3.2 Massive Phase

The necessary remedy for the above inconsistency is the introduction of a mass term in the action (8). Paradoxically, the presence of mass is also necessary in order to restore gauge invariance, albeit in an extended form. As a matter of fact, the new action

$$S=\int d^2x \left[ {1\over 4} F^ {\mu\nu} F_ {\mu\n... ... + {m^2\over 2} A_\mu A^\mu -e J^\mu A_\mu \right]$$ (13)

reflects the fact that the original gauge invariance is not only topologically broken, i.e., implicitly broken by the boundary, but also explicitly broken by the presence of a mass term. However, we argue that there is a subtle interplay between those two mechanisms of symmetry breaking, so that manifest gauge invariance is actually restored. In order to further analyze the connection between the two mechanisms of symmetry breaking, it is convenient to separate the divergenceless, boundary free current from the non conserved boundary current. In two dimensions a generic vector can be decomposed into the sum of a ``hatted'', or divergence free component, and a ``tilded'', or curl free component: Thus, we write

$$A_\mu= \widehat A_\mu +\widetilde A_\mu ,\quad:\quad \pa... ...at A^\mu=0 ,\quad \partial_{[ \mu}\widetilde A_{\nu ]}=0$$ (14)

and a similar decomposition holds for the current,

$$J^\mu =\left( J^\mu- {\partial^\mu J\over \Box} \right) + {\pa... ...d:\quad \partial_\mu\widehat J^\mu =0 , \quad\partial_\mu\widetilde J^\mu=J .$$ (15)

In terms of this new set of fields and currents the action reads

$$S=\int d^2x \left[ {1\over 4} F^ {\mu\nu} F_ {\mu\nu}... ...ilde A^\mu -e \widetilde J^\mu \widetilde A_\mu \right]$$ (16)

and we find two systems of decoupled field equations: the divergence free vector field satisfies the Proca-Maxwell equation

$$\partial_\mu F^{\mu\nu} + m^2 \widehat A^\nu = e \widehat J^\nu$$ (17)

while the curl free part must satisfy the constraint

$$m^2 \partial_\nu \widetilde A^\nu = e J$$ (18)

or, equivalently,

$$\widetilde A_\mu = {e\over m^2} \partial_\mu {1\over\Box} J .$$ (19)

To the extent that the mass is linked to the divergence of the current, as shown by the above equations, it is also a measure of the ``symmetry leakage'' through the boundary. It is this connection between topological and explicit symmetry breaking that leads us to ask: Is there a way of restoring manifest gauge invariance in spite of the presence of a mass term in the action?