3.1 Electric Stückelberg Mechanism

In this subsection, as already anticipated above, we wish to discuss in more detail the case of open $p$-branes endowed with electric charge only, in order to emphasize in a simpler case the peculiarities of the open case with respect to the closed one that we have summarized in section 2. In particular it is important to observe the physical (extended!) origin of the field content in the action of the starting theory. Honoring the procedure we employed so far, we associate to each extended object a source current, coupled to a suitable gauge potential, which in turn ``generates'' a field strength in the $(D+1)$-dimensional spacetime. The source current is divergenceless, i.e. the corresponding charge is conserved, only if the associated object is closed; moreover under this condition the theory is gauge invariant, since ``going to the boundary'' in the action integral yields no ``breaking terms'': indeed for closed objects there is no boundary at all. After this remarks it is thus clear the way in which the open case differs from the closed one. The source current of the open $p$-brane is not divergenceless now, due to the presence of the boundary (which is a $(p-1)$-brane). Hence there is a leakage of current through the boundary, and this breaks the gauge symmetry of the action

$$S [A , J] = \int \left [ - \frac{1}{2} F ^{\, (p+2)} (... ...) + e _{p}\, A _{(p+1)}\, J ^{\, (p+1)} \right ] \quad ,$$ (31)

which was the starting point for the treatment of the electric closed $p$-brane, since now we have

$$\delta S [A , J] _{A \to A + d \Lambda} = e _{p} \int d \Lambda _{(p)} J ^{\, (p+1)}$$

$$= e _{p} \int \Lambda _{(p)}\, \partial\, J ^{\, (p+1)}$$

$$= e _{p} \int \Lambda _{(p)}\, j ^{(p)} \neq 0 \quad .$$ (32)

But the gauge invariance of the theory can be restored if we couple also to the boundary of the extended objects an antisimmetric tensor field, mimicking what we did for the higher dimensional object. We also note that the boundary of the $p$-brane can also be described in terms of a current, $j ^{(p)}$ which takes into account how much the source of the $p$-brane fails to be conserved, i.e., mathematically,

$$j ^{(p)} = \partial J _{\mathrm{e}} ^{\, (p+1)} \quad .$$

Thus the gauge invariant form of the closed action (31) in the open case is

$$Z [ \, \overline{N} _{\mathrm{e}} \, ] = {1 \over Z [ \, 0 \, ]} \int [ {\mathcal{D}}A ] [ {\mathcal{D}}C ] \, e ^{ - S [ \, A , C , J _{\mathrm{e}} \, ]}$$

$$S [ \, A , C , \overline{N} _{\mathrm{e}} \, ] = \int \left[ \, - { 1\over 2 } F _{(p+2)} ( A )\, F ^{\, (p+2)} ( A ) \right. +$$

$$+ e _{p}\left. \left(\, A _{(p+1)} - d C _{(p)}\, \right)\, J _{\ma... ...appa\over 2 }\left(\, A _{(p+1)}- d C _{(p)} \, \right) ^{2} \, \right] \quad .$$ (33)

The new field $C$ is a Stückelberg compensating field, shifting under the transformation (31) as

$$\delta A _{(p+1)}= d\Lambda _{(p)}$$ (34)

$$\delta C_{(p)}= \Lambda _{(p)}$$ (35)

in order to prevent the leakage of symmetry through the boundary. It is coupled to the $(p-1)$-brane and thus its origin is physically related to the presence of the boundary. Note that this is a natural way to write the interactions and the kinetic term for $C$, preserving gauge invariance.

Now we will see how, the presence of a boundary, and thus of a further interaction on it mediated by the Stückelberg field, can be traded off with a massive interaction within the $p$-brane elements: in some sense, we can forget about the presence of the boundary, when we concentrate on the gauge invariance properties of the theory, and simply reinterpret it as the propagation of massive degrees of freedom on the $p$-dimensional extended object. This is seen as follows.
First, we derive from the action (33) the field equations, by means of variations with respect to the two gauge potentials, $A$ and $C$, respectively:

$$\partial\, F ^{\, (p+2)} + \kappa \left( A ^{\, (p+1)} - d C ^{(p)} \right) = e _{p}\, J ^{\, (p+1)}$$ (36)

$$\kappa \, \partial \left( A ^{\, (p+1)} - d C ^{(p)} \right) = e _{p}\, j ^{(p)} \quad .$$ (37)

Second, we solve the two equations above in terms of the currents and of the propagators; we split $A$ and $J$ into the sum of divergenceless, hatted part, and a curl-free tilded piece:

$$J ^{\, (p+1)}= \hat{J} ^{\, (p+1)}+\widetilde{J} ^{\, (p+1)}$$ (38)

$$\partial\, \hat{J} ^{\, (p+1)}=0\ ,\quad d\,\widetilde{J} ^{\, (p+1)}=0$$ (39)

$$\partial\, J ^{\, (p+1)}= j^ { (p) }\qquad\longrightarrow \widetilde{J} ^{\, (p+1)}= d\,\frac{1}{\Box}\, j ^{(p)}$$ (40)

$$A ^{\, (p+1)}= \hat{A} ^{\, (p+1)}+\widetilde{A} ^{\, (p+1)}$$ (41)

$$\partial\, \hat{A} ^{\, (p+1)}=0\ ,\quad d\,\widetilde{A} ^{\, (p+1)}=0$$ (42)

$$F _{(p+2)}=d\hat{A} _{(p+1)}\qquad\longrightarrow \hat{A} _{(p+1)}= \partial {1\over\Box } F _{(p+2)}$$ (43)

Then, we get from eq.(37)

$$\widetilde{A}^{\, (p+1)}- d C ^{(p)} = \frac{e _{p}}{\kappa} d\, { 1\over \Box }\, j ^{(p)}$$ (44)

while, the Maxwell-like (36) equation can be written in terms of the field strength as

$$\partial \left(\, { \Box + \kappa\over \Box }\, F ^{\, (p+2... ...)}= e _{p}\, d\, { 1\over \Box +\kappa }\, \hat{J}^{\, (p+1)}$$ (45)

Third, we substitute the solutions into the action (33) and we get,

$$S [\, J\, , j\,] = \int \left [ - { 1\over 2 } F _{(p+2)} F ^{\, (p+2)} - { \kappa\o... ... (p+2)} - e _{p} \, F _{(p+2)}\, d{ 1 \over \Box }\hat{J}^{\, (p+1)}\, \right ]$$

$$= \int \left [\, -{ e _{p}^2\over 2 } \hat{J} ^{\, (p+1)} \frac{1}{... ... + { e _{p}^2 \over 2\kappa } j ^{(p)} \frac{1}{\Box} j ^{(p)} \right ] \quad .$$ (46)

As a consequence of the gauge invariance (35) introduced by the Stückelberg conpensator the action (46) is written in terms of divergence free currents. The first term in (46) represents a short range, bulk interaction mediated by a massive vector boson, as it could be expected since the very beginning. However, this is conclusion less trivial than it could appear at first sight: we have no Higgs Model for $p$-branes and no way to break gauge symmetry spontaneously. At present, the only, gauge invariant way to provide a mass tensor gauge field with a mass term is through the Stückelberg mechanism discussed above. The cosmological implications of this new conversion mechanism, transforming the vacuum energy ``stored'' by a massless tensor gauge potential into massive particles is currently under investigation[12].
Finally, the last term in (46) describe a long range interaction confined on the boundary. With hindsight we can recognize this term as the memory of the interaction mediated by the massless Stückelberg field $C_{(p)}$. In a sort of Meissner-like effect the conpensator is `` expelled '' form the bulk and trapped over the boundary of the extended object. It has been suggested that in the limiting case $D=p+1$ this ``secret long range force'' can produce confinement, in $D=2$, and glueball formation in $D=4$ [13]. Accordingly, we expect similar effects in higher dimensions [6].
The above effects represent the physical output of the Stückelberg Mechanism for electric $p$-branes. To investigate the strong coupling dynamics of this mechanism we have to switch to the `` magnetic phase '' of the model (31). The condition (7) guarantees that a strong electric coupling regime can be equivalently described in terms of a dual weak magnetic coupling phase. Thus, duality procedure can now be developed, but we defer it to the next subsection, since the same set-up can be now applied in the more general case in which electric as well as magnetic charges are present.