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Revista mexicana de física

Print version ISSN 0035-001X

Rev. mex. fis. vol.63 n.5 México Sep./Oct. 2017



On the decay of an accelerated proton

A. Queijeiroa 

aDepartamento de Física, Escuela Superior de Física y Matemáticas, Edificio 9, Unidad Profesional Adolfo López Mateos, Instituto Politécnico Nacional Ciudad de México 07738, México. e-mail:


We compute the decay rate width of the strong decay pn+π+ for a linearly accelerated proton in both the inertial frame and in the coaccelerated proton frame. In this last reference system we use the Unruh effect, where the proton sees a bath of thermal particles at the temperature T=a/2π, where a is proton’s acceleration. Analytical results agree, thus giving a simpler example where the Unruh effect is necessary to keep the consistency between inertial and Rindler frame calculations of a physical observable.

Keywords: Proton decay; non-inertial reference system; unruh effect

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 a,

pn+e++νe, (1)

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 e± and ν(-)e as Dirac particles, and 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 T=a/2π, from where the particles e- and ν¯e (or both) are absorbed by the proton to transform into a neutron and emitting a 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

pp+π0,pn+π+ (2)

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 pn+π+ for a linearly uniformly accelerated proton, under the same set of assumptions as in 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 aM1, 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 γ-matrices in curved spacetime are involved. The only ingredient required is the solution of the Klein-Gordon equation in the accelerated frame. In what follows we present our calculations, first in the inertial frame and then in the non-inertial one, where the proton can absorb a pion present in the thermal bath 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

u=1/a (3)

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

j(x)=qˆ(τ)δ(u-a-1) (4)

where qˆ(τ) is an Hermitian operator 7, with qˆ(τ)=eiHˆ0τq0e-iHˆ0τ, and Hˆ0 the proper Hamiltonian of the proton-neutron system. This current is suitable for describing the strong conversion of a proton into a neutron, assuming that the back reaction on the neutron is negligible. In this sense the model consider the pair of nucleons as excited and unexcited states of the nucleon two-level system. They are eigenstates of the Hamiltonian Hˆ0 with eigenvalues M 1 and M 2 , the masses of the proton and neutron, respectively. The corresponding interaction is given by

SI=j(x)Φ(x)+Φ(x)d2x (5)

where the scalar field Φ(x) is described by annihilation aˆk and creation cˆk operators

Φ(x)=d3kaˆkϕk(+ω)(x)+cˆkϕ-k(-ω)(x) (6)

Then, the transition amplitude, from proton state p1 to neutron state p2, is

Ak=p2*gkSI0*p1 (7)

and the differential transition rate is given as

dPdk=Ak2 (8)

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


where ΔM=M1-M2, and τ is proper time of the proton, and gef=pqˆ(0)n is the effective strong coupling constant. The integral over z is immediate, and using dt=γdτ, where

γ=(1-v2)-1/2 (9)

with v being proton’s velocity

v=at1+a2t2, (10)

we obtain

Ak=gef(2π)1/22ω-dτeiΔMτ+ikx (11)

In terms of the other Rindler coordinate v=aτ, we write kx=ω˜sinhv-k˜coshv, where ω˜=ω/a and k˜=k/a; then (8) is given by

dPdk=|Ak|2=gef22π12ωa2-dv-dv'×eiΔM~(v-v')+iω~(sinhv-sinhv')-k~(coshv-coshv') (12)

where ΔM˜=(M1-M2)/a. A change in the integration variables gives

dPdk~=gef22π1ω~-dσ-ds×eiΔM~σ+2i(ω~coshs-k~sinhs')sinhσ (13)

Next, we perform a rotation

ω~'=ω~coshs-k~sinhs,k~=-ω~sinhs-k~coshs (14)

which gives

1TdPdk~'=gef22π1ω~'a2-dσ×e2i(ΔM~σ+ω~'sinhσ) (15)



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

1TdPdk˜'=gef22πa2e-πΔM˜1ω˜'K2iΔM˜(2ω˜'). (16)

Here K2iΔM˜(2ω˜') is the modified Bessel function of index 2iΔM˜. Total transition rate is obtained integrating, using the Mathematica program, over pion energy,

Γ=gef22πa2e-πΔM~m~ω~'2-m~2K2iΔM~(2ω~')dω~'=gef2m~8π1/2a2e-πΔM~G1,33,0×(m~20-12,-12+iΔM~,-12-iΔM~) (17)

where m˜=m/a, with m the mass of the pion. An identity for the Meijer function G1,33,0 in terms of the modified Bassel function 8 gives

Γ=gef24πa2e-πΔM~KiΔM~(ω~)2 (19)

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: p+π-n. The necessary temperature for having pions in the bath is greater than 1012 K, corresponding to an acceleration a of order 1031g, where g is the standard gravitational acceleration, and since proton mass m p is equivalent to an acceleration of 1032g, we still have a/mp<1.

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

Ak=gefd2xeiΔMτϕkω(x0,x3). (20)

Here ϕkω(x0,x3) is the pion function, solution of the Klein-Gordon equation in the accelerated frame of reference (by using the equivalence principle: a uniform acceleration is equivalent to a constant gravitational field, where the components of the metric tensor are given in terms of proton’s acceleration 9),

ϕkω(x0,x3)=12πa-1/2e-iωτ×sinh1/2(πω~)Kiω~(m~). (18)

where Kiω˜(m˜) is the modified Bessel function of index iω˜ and argument m˜=m/a, with m the mass of the pion. Then,

Ak=gef2πa3/2sinh1/2(πω˜)Kiω˜(m˜)δ(ω˜-ΔM˜) (19)

The differential transition rate is given by

1TdPdω˜=Ak2nB(ω). (20)

where nB(ω) is the bosonic thermal factor associated with the thermal bath and, as before, T is the proton proper time. According to Unruh 7, bath temperature, in units where Boltzmann constant k B , reduced Planck constant and speed of light c all take value one, is

T=a2π (21)

Substitution of nB(ω)=(eω/T-1)-1 with this temperature, gives

1TdPdω˜=gef24πa2e-πω˜Kiω˜(m˜)2δ(ω˜-ΔM˜) (22)

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

Γni=gef24πa2m~e-πω~|Kiω~(m~)|2δ(ω~-ΔM~)dω~=gef24πa2e-πΔM~|KiΔM~(m~)|2. (23)

In summary, we have computed the transition width for the strong decay pn+π+, for the uniformly accelerated proton, in an inertial frame and in proton’s frame. The analytical results Eq. (19) and Eq. (26) respectively, shows explicitly that they agree, providing a further theoretical check for the Unruh effect. The point of view in this work, following 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 ΔM˜=0, then Eq. (19) reduces to

Γ=gef2 4πa2K0(ω˜)2

which is of the same form than their result (3.17) for the total emission probability, with the effective coupling gef replaced by the corresponding coupling constant 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 al11. 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.


The author wishes to thank Drs. F. Angulo, A. Martínez and R. Cordero for stimulating discussions and encouragement. This work was partially supported by Comisión de Operación y Fomento de las Actividades Académicas and Estímulo al Desempeño de los Investigadores del Instituto Politécnico Nacional-México. He also acknowledges the anonymous referee for calling to his attention the works in 10.


1. R. Müller, Phys. Rev. D 36 (1997) 953. [ Links ]

2. G.E.A. Matsas and D.A.T. Vanzella, Phys. Rev. D 59 (1999) 094004. [ Links ]

3. W.G. Unruh, Phys. Rev. D 14 (1976) 870; S.A. Fulling, Phys. Rev. D 7 (1973) 2850; P.C.W. Davies, J. Phys. A 8 (1975) 609. For a didactic presentation of this result see Sean Carrol: Spacetime and Geometry, an introduction to General Relativity, (Addison Wesley 2004), pages 402-412, and J.F. Donoghue and Barry R. Holstein, Am. Jour. Phys. 52 (1984) 730, P.M. Alsing and P.W. Milonni, Am. Jour. Phys. 72 (2004) 1524. [ Links ]

4. H. Suzuki and K. Yamada, Phys. Rev. D 67 (2003) 065002. [ Links ]

5. V.L. Ginzburg, G.F. Zharkov, Sov. Phys. JETP 20 (1965) 1525; G.F. Zharkov G.F. Sov. Jour. Nucl. Phys. 1 (1965) 120. [ Links ]

7. B.S. de Witt, in General Relativity: An Einstein Centenary Survey, eds. S.W. Hawking and W. Israel (Cambridge University Press, Cambridge, England 1979) pp 680-745. [ Links ]

9. C.L.C.B. Higuchi, A. Matsas G.E.A, Rev. Mod. Phys. Rev. 80 (2008) 787, and references there in. [ Links ]

10. H. Ren and E.J. Weinberg, Phys. Rev. D 49 (1994) 6526. See also A. Higuchi, G.E.A. Matsas , and D. Sudarsky, Phys. Rev. D 45 (1992) R3308, where the emission of bremsstrahlung photons was computed in charge’s coaccelerated frame. [ Links ]

11. D. Fregolente, G.E.A. Matsas, and D.A.T. Vanzella, Phys. Rev. D 74 (2006) 045032. [ Links ]

12. J.R. Letaw and J.D. Pfautsch, Phys. Rev. D 22 (1980) 1345; see also U.H. Gerlach, Phys. Rev. D 27 (1983) 2310. [ Links ]

6We shall here consider the process of absorption of particles only, since from the point of view of the inertial observer, the Rindler particle emits as well as absorbs particles. The existence of Unruh radiation is not universally accepted, see for instance M. Visser, “Experimental Unruh radiation?” , arXiv: gr-qc/0102044, I. I. Smolyaninov, Phys. Lett. A 372 (2005) 7043; G.W. Ford and R.F. O’Connell, Phys. Lett. A 350 (2005) 7.

8See, for example, I.S. Gradshtein and I.M. Ryzhik, Table of Integrals, Series and Products sect. 9.3 (Academic, New York, 1980).

Received: April 20, 2017; Accepted: June 05, 2017

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