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The restoration of spontaneous symmetry breaking for a scalar field theory for an accelerated observer is discussed by the one-loop effective potential calculation and by considering the effective potential for composite operators. Above a critical acceleration, corresponding to the critical restoration temperature,
T_{c}, for a Minkowski observer by Unruh relation,
i.e. a_{c}/2π=T_{c}, the symmetry is restored. This result confirms other recent calculations in effective field theories that symmetry restoration can occur for an observer with an acceleration larger than some critical value. From the physical point of view, a constant acceleration mimics a gravitational field and the critical acceleration to restore the spontaneous symmetry breaking corresponds to a huge gravitational effect which prevents boson condensation.

In quantum field theory in flat space-time a spontaneously broken symmetry can be restored above some critical temperature [1-3]. On the other hand, whenever the background geometry is endowed with a black-hole or an event horizon, the related vacuum physically behaves like a thermal bath of quanta with a temperature, , proportional to the surface gravity [

The previous discussion clearly indicates that a restoration of the symmetry can occur for an observer with an acceleration larger than some critical value independently on the specific dynamical mechanism that produces the acceleration.

Indeed in Reference [

The action for the scalar theory in Rindler spacetimes [

where, and the Rindler metric tensor is given by:

Since the two Rindler wedge are causally disconnected from each other [

Taking the Fourier transform with respect to:

the normalized solution turns out

where is the modified Bessel function of second kind and. The two-point Green’s function of the free scalar field in the right Rindler wedge, is defined by the equation:

The Fourier transform of with respect to is given by

where

By substituting back into Equation (6) one finds:

where

The world line in Rindler coordinates of a uniformly accelerated observer with proper constant acceleration is given as,

and it has been generally proved [

Since the Euclidean Rindler spacetime has a singularity at one requires that the period of the imaginary time is. With this particular choice the Euclidean formalism in Rindler coordinates coincides with the finite temperature Matsubara formalism and therefore the following substitutions will be necessary in evaluating the effective potential in the next sections:

Let us now study the effective potential for a Rindler observer when for a Minkowski observer the effective potential is assumed to possess a symmetry breaking solution. For classical constant field configuration, , the one loop effective action in Rindler coordinates turns out to be:

where and is given by

The effective potential (,) is defined as

and by the following relation,

where is defined by previous Equations (7)-(10) with the substitution, it turns out to be

In Euclidean space with the periodic boundary condition in Equations (12) and (13) and by introducing the Fourier transform of the Green’s function, Equation (7), the previous equation gives

where

and is now solution of the equation

which, according to Equation (5) can be written as

Since the world line in Rindler coordinates of a uniformly accelerated observer with proper constant acceleration is given by Equation (11), one sets and and by changing the integration variable in Equation (23) from to, it turns out

By performing the sum on the Matsubara frequencies the final result for the effective potential is

with and

In order to calculate the critical acceleration for symmetry restoration we will impose the following condition [

which gives

where we set for notation convenience. The computation of the integrals is straithforward and one gets

in complete analogy with the finite temperature case [

the critical acceleration is obtained by the equation

which, for large acceleration, gives

i.e.

in agreement with the one-loop calculation in Minkowski space-time at finite temperature [

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 [

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 [

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].

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

An efficient way to perform systematic selective summations is to use the method of the effective action for composite operators [

According [

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

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

As it was shown by in [

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.