. (33)

If the symmetry is unbroken then is a minimum for and the second derivative is positive. However in our case the symmetry is spontaneously broken and the effective potential develops two symmetric non-vanishing minima while becomes a maximum. Then the fact that does not mean that the mass square is negative but simply tells us that we are computing the mass square in the wrong vacuum state i.e. in a field configuration which, because of symmetry breaking, is not the true vacuum state (the minimum) anymore (indeed it is a maximum and hence the second derivative is negative).

Although the final result is as expected, the calculation in not entirely trivial. Moreover the determination of the critical acceleration from one loop effective potential suffers the same infrared problem of the finite temperature calculation related with the mode. As well known, in finite temperature field theory a reliable evaluation of the critical temperature requires the resummation of an infinite subset of diagrams [3]. This can be more easily done by considering the effective potential for composite operators (CJT) [11], extensively applied at finite temperature [13], since the relevant, infinite, subset of diagrams is automatically resummed by the gap equations corresponding to the minimum conditions of the effective potential with respect to the relevant physical parameters in the theory. In the analysis of the spontaneous symmetry breaking and its restoration for theory by CJT method in the Hartree-Fock approximation (i.e. by considering the lowest order contribution to the gap equation) the relevant operators are and and the corresponding parameters are the vacuum expectation value of the field and the mass in the two-point function. Calculations at finite temperature have been carried out in [14]. Since the gap equation in the Hartree-Fock approximation correspond to one-loop selfconsistent calculation of the self-energy (see [14] for details) from our previous, explicit, oneloop calculation and from the complete analogy of the Green’s functions between a Minkowski observer at finite temperature and an accelerated observer with [7], it follows that a more reliable evaluation of the critical acceleration with respect to the one-loop result in Equation (32) can be obtained by following the same analysis of [14]. However the most interesting aspect is not the exact value of the critical acceleration but the restoration of the symmetry for an accelerated observer (see [7], Section 4, for a different point of view).

4. Comments and Conclusions

The restoration of chiral and color symmetries in the Nambu Jona-Lasinio model for an observer with a costant acceleration above a critical value [8,9] and the calculation performed in the previous section clearly indicate that one can restore broken symmetries by acceleration. Although the technical aspects of the previous calculations are sound, the physical mechanism of the restoration is unclear if one does not recall that a constant acceleration is locally equivalent to a gravitational field. The critical acceleration to restore the spontaneous symmetry breaking corresponds to a huge gravitational effect which prevents boson condensation as in the case of a non relativistic, ideal Bose gas [15].

More generally, the acceleration associated with the Hawking-Unruh temperature (and radiation) due to the observed gravitational fields is too small to produce measurable effects. There are very interesting attempts to find gravity-analogue of the Hawking-Unruh radiation [16,17] and, in our opinion, high energy particle physics seems the more promising sector to observe this effect. Indeed, a temperatute MeV, corresponding to an acceleration can be reached in relativistic heavy ion collisions and the hadronic production can be understood as Hawking-Unruh radiation in Quantum Chromodynamics [18-20].

5. Acknowledgements

The authors thank H. Satz and D. Zappalà for useful comments.

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Appendix: The Effective Action for Composite Operators

An efficient way to perform systematic selective summations is to use the method of the effective action for composite operators [11]. In this case, the effective action is the generating functional of the two-particle ireducible (2PI) vacuum graphs (a graph is called two-particle irreducible if it does not become disconnected upon opening two lines). Now, the effective action, depends not only on, but on, as well. These two quantities are to be realized as the possible expectation values of a quantum field and as the time ordered product of the field operator respectively. There is an advantage in using the CJT method to calculate the effective potential in certain approximations as is, for example, the Hartree-Fock approximation of the theory. Indeed, if we use an ansatz for a dressed propagator, we need to evaluate only the double bubble graph in Figure 1 (with lines representing dressed propagators), instead of summing the infinite class of daisy and super-daisy graphs. In order to define the effective action for composite operators, we can follow a path analogous to the one leading to the ordinary effective action. The essential difference is that the partition function depends also on a bilocal source, in addition to the local source. As an example, we consider the theory with Lagrangian:

(34)

According [11], the generating functional for the Green functions in the presence of sources and is given by (we set)

Figure 1. The double bubble graph contributing to the effective potential in the CJT method. The lines represent dressed propagators G(x, y).

(35)

where, is the generating functional for the connected Green functions, while is the classical action. The effective action for composite operators, is obtained through a double Legendre transformation of

(36)

Physical processes correspond to vanishing sources, so the stationarity conditions which determine the expectation value of the field and the (dressed) propagator are given by

(37)

(38)

As it was shown by in [11], the effective action is given by

(39)

where in this last equation is the sum of all two-particle irreducible (2PI) diagrams in which all lines represent full (dressed) propagators, while is the inverse of the tree-level propagator. When translation invariance is not broken it is sufficient to consider a classical constant field configuration. Under this assumption we can factorize an overall four-dimensional volume factor and define the effective potential for composite operators exactly as we did for the standard effective potential. Then the stationary condition are given by Equations (37) and (38) with replaced by.