5. The Coleman-Weinberg mechanism of mass generation

The scenario emerging from the last section is based on the assumption that the $\varphi (x)$-field can acquire a non vanishing vacuum expectation value. Note that there is no potential term for $\varphi (x)$ in the classical action (8). However, we wish to show that a self-consistent potential may originate from the quantum fluctuations of the $A _{\mu \nu \rho}(x)$ field leading to a non vanishing vacuum expectation value for the order parameter. On the technical side, this means to compute the one-loop effective potential for the $\varphi (x)$ field by integrating $A _{\mu \nu \rho}$ out of the functional integral

$$Z = \int \left[ D \Psi ^{*} [S] \right] \left[ D \Psi [S] \... ...\nu \rho}(x) ] \exp \left( i {\mathcal{S}} / \hbar \right) .$$ (31)

The quantization of higher rank gauge fields is a lengthy procedure involving a sequence of gauge fixing conditions together with various generations of ghosts [11]. In principle, these terms should be included in the functional measure in the action functional. However, in our case they are unnecessary since we know already that in the superconducting phase $A _{\mu \nu \rho}$ describes a massive scalar degree of freedom which is the only physical degree of freedom. Thus, the effective potential is

 

$$V ^{\mathrm{eff}} (\phi) \frac{1}{2} {\mathrm{Tr}} \, \ln \left[ \left( \partial ^{2} + \p... ...rho ( \Lambda )}{2} \phi ^{2} + \frac{\delta \lambda ( \Lambda )}{4!} \phi ^{4}$$ =

$$\frac{1}{32 \pi ^{2}} \left[ \phi ^{2} \Lambda ^{2} + \frac{\phi ... ...o ( \Lambda )}{2} \phi ^{2} + \frac{\delta \lambda ( \Lambda )}{4!} \phi ^{4} .$$ =

The ultraviolet divergences of the one-loop determinant have been regularized through the cutoff $\Lambda$, and the two counterterms $\delta \rho ( \Lambda )$ and $\delta \lambda ( \Lambda )$ are fixed by the renormalization conditions

  

$$\left( \frac{\partial ^{2} V ^{\mathrm{eff}} (\phi)}{\partial \phi ^{2}} \right) _{\phi = 0} = 0$$ (32)

$$\left( \frac{\partial ^{4} V ^{\mathrm{eff}} (\phi)}{\partial \phi ^{4}} \right) _{\phi = g / \mu} = 0$$ (33)

in which $\mu$ appears as an arbitrary renormalization scale. The scalar field $\varphi (x)$ has no classical dynamics of its own, i.e., it possesses no kinetic or potential term. This is the reason for imposing the two conditions (32),(33): equation (32) is the characteristic Coleman-Weinberg condition [3] ensuring that the mass of the gauge field is non vanishing only in the condensed phase $\phi \ne 0$; equation (33) follows from the absence of a classical quartic self-interaction. Of course, the physical properties of the system are insensitive to the choice of the renormalization condition. Then, with our choice, we find the Coleman-Weinberg potential for membranes

$$V _{CW} (\phi) = \frac{\phi ^{4}}{64 \pi ^{2}} \left[ \ln \frac{\phi ^{2} \mu ^{2}}{g ^{2}} - \frac{25}{6} \right] .$$ (34)

The absolute minimum of $V _{CW} (\phi)$ corresponds to a super-conducting phase characterized by a vacuum expectation value of the order parameter

$$\langle \phi \rangle ^{2} = \frac{g ^{2}}{\mu ^{2}} e ^{\displaystyle{\frac{11}{3}}}$$ (35)

and by a dynamical surface tension

$$\frac{\rho _{R} ^{2}}{g ^{2}} \equiv \left( \frac{\partial... ...2}}{8 \pi ^{2} \mu ^{2}} e ^{\displaystyle{\frac{11}{3}}} .$$ (36)

The factor $\langle \phi \rangle ^{2}$ as given by (35) is also the square of the dynamically generated mass for $A _{\mu \nu \rho}(x)$. Hence, the physical thickness of the membrane, is of the order of $\langle \phi \rangle ^{-1}$, and the dynamically generated surface tension is $\rho _{R} = g \langle \phi \rangle / \sqrt{8 \pi ^{2}}$. This quantity is positive, so that the bubbles of ordinary vacuum tend to collapse in the absence of a balancing internal pressure. In the case of ``hadronic bags'', this internal pressure is provided by the quark-gluon complex. In any event, the picture of the superconducting membrane vacuum is strongly reminiscent of the classical dynamics of a closed membrane coupled to its gauge partner i.e., $A _{\mu \nu \rho}(x)$ [4]. In both cases, vacuum bubbles created in one vacuum phase evolve and die in a different vacuum background. This suggests a new possibility of quantum vacuum polarization via the creation and annihilation of whole domains of spacetime in which the energy density is different from that of the ambient spacetime. As a matter of fact, the novelty of our field model is the onset of a new type of ``Higgs mechanism for membranes'' triggered solely by quantum fluctuations. The effect of such fluctuations can be accounted for by an effective potential. As in Umezawa's approach, this effective potential is consistent with the dynamical generation of a bag with surface tension out of the vacuum.