PACS: 04.62.+v; 14.20.Dh; 13.30.-a

It is known that the lifetime of a particle can be manipulated by exposing it to a large acceleration. For instance, among the various decays studied in the literature, the most interesting is the weak decay of a proton with acceleration

which is forbidden for an inertial proton. This problem was first considered by Müller in a toy model ^{1}, assuming all particles involved are scalars. Then, Matsas and Vanzella ^{2}, within a semiclassical approach in a two-dimensional spacetime, took the *p* and *n* as a classical current. This last assumption is suitable as far the nucleons are energetic enough to have a well defined trajectory. Besides doing this they computed the total decay rate in the coaccelerated proton frame, where according to the Unruh effect ^{3} the Minkowski vacuum corresponds to a thermal state of Rindler particles at the Unruh temperature *v*
^{
e
} and *e*
^{+} (or none), respectively. The decay rates in both reference frames were shown to agree by a numerical computation. Later, Suzuki and Yamada ^{4} did the same calculation analytically and in a 4-dimensional spacetime, confirming the result in ^{2}.

Another interesting decay for an accelerated proton are the strong processes

when the proton is in the presence of a very intense magnetic field ^{5}, or in circular motion under the influence of gravitational fields ^{5}. In these works the emphasis was the study of emission of cosmic and gamma rays from compact stellar objects associated with strong magnetic fields.

In the present work we revisited the strong decay ^{2}: (i) the nucleons constitute a two level quantum system described by a semiclassical current, (ii) the neutron velocity does not change with respect to the proton, the so-called no-recoil condition, and (iii) that proton acceleration *M*
_{
2
} where *M*
_{
1
} and *M*
_{
2
} are proton and neutron masses, respectively. We calculate the life-time of the proton in the inertial system of reference and in the non-inertial proton reference system, and we will show that the life-time is the same in both frames. The calculation turns out to be simpler than the one in ^{2}, since no Dirac spinors and ^{6}. We follow the procedure developed in ^{2} and ^{4}.

We consider motion in one spatial dimension and constant acceleration in the z-direction. In terms of the Rindler coordinates, the path of the proton is given by

The participation of a scalar particle requires to take as a semiclassical baryonic current the expression

where ^{7}, with *M*
_{
1
} and *M*
_{
2
} , the masses of the proton and neutron, respectively. The corresponding interaction is given by

where the scalar field

Then, the transition amplitude, from proton state

and the differential transition rate is given as

After substitution of (5) and (6) into (7) we obtain

where *z* is immediate, and using

with

we obtain

In terms of the other Rindler coordinate

where

Next, we perform a rotation

which gives

and

is the total proper time of proton. At this point the integral can be evaluated giving

Here

where *m* the mass of the pion. An identity for the Meijer function ^{8} gives

Now, from the point of view of a uniformly accelerating particle, empty space contains a gas of particles at a temperature proportional to acceleration. In the accelerated proton reference system, the decay process is seen as one in which the proton captures, from the particle bath he sees, a pion and then turning into a neutron: ^{12}
*K*, corresponding to an acceleration *m*
_{
p
} is equivalent to an acceleration of

The transition rate is computed in the following way. First, amplitude is given by

Here ^{9}),

where *m* the mass of the pion. Then,

The differential transition rate is given by

where ^{7}, bath temperature, in units where Boltzmann constant *k*
_{
B
} , reduced Planck constant *c* all take value one, is

Substitution of

Now, we integrate over pion energy to obtain finally the transition rate in the non-inertial reference frame

In summary, we have computed the transition width for the strong decay ^{2}, is that the Unruh effect is essential to obtain the proper decay rate in the uniformly accelerated frame.

The result in Eq. (19) can be compared to the calculation made by Ren and Weinberg ^{10} for emission from an accelerated scalar source. In this case

which is of the same form than their result (3.17) for the total emission probability, with the effective coupling *q* and recalling that we work in a two-dimensional spacetime.

Another interesting case is the same decay for protons in circular motion under the influence of an intense gravitational field, as the one considered by Fregolente *et al*^{11}. However, as was demonstrated by Letaw and Pfautsch ^{12} the spectrum of vacuum fluctuations in the non-inertial frame, is composed by a thermal energy plus a non-thermal contribution arising from the observer’s acceleration, which make the calculation quite complicated. This latter task is under current investigation.