Weak decay of uniformly accelerated protons and related processes

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Weak decay of uniformly accelerated protons and related processes

Daniel A. T. Vanzella and George E. A. Matsas

Instituto de Fı´sica Teo´rica, Universidade Estadual Paulista, Rua Pamplona 145, 01405-900, Sa˜o Paulo, SP, Brazil 共Received 31 January 2000; published 6 December 2000兲

We investigate the weak interaction emission of spin-1/2 fermions from accelerated currents. As particular applications, we analyze the decay of uniformly accelerated protons and neutrons, and the neutrino-antineutrino emission from uniformly accelerated electrons. The possible relevance of our results to astrophys-ics is also discussed.

DOI: 10.1103/PhysRevD.63.014010 PACS number共s兲: 13.30.⫺a, 04.62.⫹v, 14.20.Dh, 95.30.Cq I. INTRODUCTION

We investigate the weak interaction emission of spin-1/2 fermions from classical and semiclassical currents. We de-note by semiclassical those currents which possess classical trajectories and are endowed with quantized inner energy levels. Our results can be used to investigate a broad class of processes involving accelerated particles provided that they have a well defined world line, as indeed verified in many situations of interest. In the case where the particle is accel-erated by a background electromagnetic field, such processes could be fully analyzed quantum mechanically. As a conse-quence, any recoil effects due to the fermion emission would be automatically taken into account 关1,2兴. For instance, in Ref.关1兴 a quasiclassical approach to quantum electrodynam-ics was developed to consider ␥-synchrotron radiation from an electron in a classical background magnetic field. This approach is basically characterized by assuming that the electron motion is quasiclassical. This is always possible as long as the magnetic field is not very strong, namely, H ⰆH0and␥Ⰷ1, where H0⫽4.4⫻1013G and␥is the Lorentz factor for the electron. This quasiclassical approach applied to neutrino-antineutrino emission is analyzed in detail in Sec. 6.1 of Ref. 关2兴.

Although it is hard to take into account the current recoil in our semiclassical approach, the relations here obtained, which agree with the full quantum mechanical treatment in the proper limit (␹Ⰶ1, see, e.g., Sec. VI兲, are easily appli-cable when the process involves particle decay and the tra-jectory itself 共rather than the underlying dynamical process which generates it兲 is inferred from the observational data. Explicit results for uniformly accelerated currents are exhib-ited.

As far as we know, the first ones to call attention to the possibility that noninertial protons may decay were Ginzburg and Syrovatskii关3兴 but only recently Muller 关4兴 obtained the first estimation for the decay rate associated with the process p→ne⫹␯e by assuming that all the involved particles are scalars. Here, as a particular application of modeling accel-erated particles by semiclassical currents, we perform a com-prehensive 共inertial-frame兲 analysis of the inverse ␤ decay for uniformly accelerated protons. We show that under cer-tain astrophysical conditions, high-energy protons in strong background magnetic fields should rapidly decay.

The observation of noninertial neutrons is less trivial. Notwithstanding the calculation of the ␤ decay rate for

ac-celerated neutrons may be of some relevance in situations where they are under the influence of ‘‘relatively’’ strong background gravitational fields and, thus, will also be per-formed.

Some features of the ␤ and inverse ␤ decays for uni-formly accelerated nucleons will be discussed in terms of the Fulling-Davies-Unruh 共FDU兲 effect. The FDU effect asserts that the Minkowski vacuum corresponds to a thermal bath with respect to uniformly accelerated observers 关5,6兴. It is perhaps remarkable that although inertial observers associate the inverse␤decay to the channel p→ne⫹␯e, coaccelerated observers associate the same proton decay event to one of the following channels: pe⫺→n␯e, p¯␯e→ne⫹ or pe⫺¯␯e→n, where the absorbed e⫺and¯␯eare Rindler particles present in the FDU thermal bath ‘‘attached’’ to the proton’s frame关7兴. The corresponding branching ratios can be also calculated as a function of the proton acceleration.

Under a certain restriction, we can make our semiclassical current to behave as a classical one. This is suitable to inves-tigate neutrino-antineutrino emission from accelerated elec-trons. This process is of relevance in some astrophysical situ-ations as, e.g., in the cooling of neutron stars. Eventually we compare our results in the proper limit with the ones in the literature obtained by quantizing electrons in a background magnetic field 关8–10兴.

The paper is organized as follows. In Sec. II we introduce the semiclassical currents and discuss how they model par-ticle decays. In Sec. III, we introduce the weak-interaction action and couple our current to a spin-1/2 fermion-antifermion field. Afterwards we calculate the differential transition probability for currents following arbitrary world lines. In Sec. IV we use the results obtained in the previous section to explicitly evaluate the fermion emission rate and radiated power for the particular case of a uniformly accel-erated current. The next two sections, Secs. V and VI, are dedicated to analyze in detail the decay of uniformly accel-erated protons and neutrons, and the neutrino-antineutrino emission from uniformly accelerated electrons, respectively. We also comment on the possible astrophysical relevance of our results. We dedicate Sec. VII for our final discussions. We will use natural units c⫽ប⫽kB⫽1 throughout this paper unless stated otherwise.

II. SEMICLASSICAL VECTOR CURRENT

Let us consider a particle in a four-dimensional Minkowski spacetime covered by the usual inertial

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coordi-nates (t,x)苸R4. Let x␮(␶) be the particle’s world line and␶ its proper time. The classical vector current associated with this particle is given by

j␮共x兲⫽qu ␮_共_␶_兲 u0共␶兲 ␦

3_关x⫺x共_␶_兲兴, _共2.1兲 where q is a ‘‘small’’ coupling constant and u␮(␶) ⬅dx␮_/d_␶_{. The current above is suitable to describe a} point-like classical 共i.e., with noinner structure兲 particle. Eventu-ally it can be used to describe a fermion f1 and antifermion f¯2 emission from an accelerated particle p1:

p1→p1f1f¯2

共e.g., a noninertial electron emitting a neutrino-antineutrino pair兲. Notwithstanding, this current must be improved in or-der to allow more general processes of the form

p₁→p₂f₁f¯₂, 共2.2兲 where particle p1 turns into particle p2 with a fermion-antifermion pair emission共e.g., decay of an accelerated pro-ton into a neutron with a positron-neutrino emission兲. This is attained by replacing the real coupling constant q by an operator-valued function共see, e.g., Ref. 关11兴兲

qˆ共␶兲⫽eiHˆ0␶_qˆ

0e⫺iHˆ0␶. 共2.3兲 This can be regarded as the usual first-quantization proce-dure, where a classical observable q is replaced by a self-adjoint operator qˆ0 evolved by the one-parameter group of unitary operators Uˆ (␶)⫽e⫺iHˆ0␶_{. Here H}ˆ

0 is the proper Hamiltonian of the system, i.e.,

Hˆ₀兩p_j

典

兩 is the effective coupling constant. Note that we can recover current 共2.1兲 from Eq. 共2.6兲 by making M2⫽M1 and Geff⫽q.

We will assume that the fermion emission does not change appreciably the four-velocity of p2 with respect to p1. We will denominate this assumption ‘‘no-recoil condi-tion.’’ This is verified as far as the momentum of the emitted

fermions 共with respect to the inertial frame instantaneously at rest with the current兲 satisfies 兩k˜兩ⰆM1, M2. In order to be conservative we will impose 兩k˜兩⬍␻˜ⰆM1, M2. It will be-come clear further that the typical energy of the emitted fer-mions␻˜ is of the order of the current’s proper acceleration a. Hence our condition above can be recast in the suitable form aⰆM1, M2. Our results should be accurate as far as this condition is verified.

III. FERMION-ANTIFERMION EMISSION FROM A SEMICLASSICAL CURRENT

We shall describe the emitted fermions by spinorial fields ⌿ˆ共x兲⫽

兺

␴⫽⫾

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d 3_k_关bˆk ␴␺k␴ (⫹␻)_{共x兲⫹dˆk} ␴ † _␺ ⫺k⫺␴ (⫺␻) _共x兲兴, 共3.1兲 where bˆk␴and dˆk␴ †

are annihilation and creation operators of fermions and antifermions, respectively, with three-momentum k⫽(kx,ky,kz) and polarization␴. We will adopt the notation used in Ref. 关7兴. Energy␻, momentum k, and mass m are related as usual: ␻⫽

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k2⫹m2⬎0. ␺_k(⫹␻)_␴ and ␺k␴

(⫺␻)

are positive and negative frequency solutions of the Dirac equation i␥␮⳵_␮␺(_k⫾␻)_␴ ⫺m␺_k(⫾␻)_␴ ⫽0. By using the␥␮ matrices in the Dirac representation共see, e.g., Ref. 关12兴兲, we find ␺k⫹ (⫾␻)_共x兲⫽ e i(⫿␻t⫹k•x)

冑

16␲3␻共␻⫾m兲

冉

m⫾␻ 0 kz kx_⫹iky

冊

共3.2兲 and ␺k_⫺ (⫾␻)_共x兲⫽ e i(⫿␻t⫹k•x)

冑

16␲3␻共␻⫾m兲

冉

0 m⫾␻ kx⫺iky ⫺kz

冊

. 共3.3兲

We have orthonormalized modes共3.2兲, 共3.3兲 according to the inner product 关11兴

具␺

k␴ (⫾␻) ,␺_k(⫾␻_⬘_␴_⬘⬘)

典

⬅

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⌺d⌺␮␺ ¯ k␴ (⫾␻)_␥_␮_␺ k⬘␴⬘ (⫾␻⬘) ⫽␦3_共k⫺k

_⬘

_兲_␦ ␴␴_⬘␦⫾␻⫾␻_⬘, 共3.4兲 where d⌺_␮⬅n_␮d⌺ with n␮being a unit vector orthogonal to ⌺ and pointing to the future, and ⌺ is an arbitrary spacelike hypersurface. We have chosen t⫽const for the hypersurface ⌺. As a consequence, canonical anticommutation relations for fields and conjugate momenta lead to the following simple anticommutation relations for creation and annihila-tion operators: 兵bˆk␴,bˆk⬘␴⬘ † 其⫽兵dˆk␴,dˆk⬘␴⬘ † 其⫽␦3_共k⫺k

_⬘

_兲_␦ ␴␴_⬘ 共3.5兲

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and 兵bˆ_k_␴,bˆ_k_⬘_␴_⬘其⫽兵dˆ_k_␴,dˆ_k_⬘_␴_⬘其⫽兵bˆ_k_␴,dˆ_k_⬘_␴_⬘其⫽兵bˆ_k_␴,dˆ_k ⬘␴_⬘ † 其⫽0. 共3.6兲 Next we minimally couple the spinorial fields⌿ˆ1and⌿ˆ2, associated with the emitted fermions f₁ and f¯₂, respectively, to our general current jˆ␮ according to the weak-interaction action

SˆI⫽

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d4x jˆ_␮兵⌿R₁␥␮共cV⫺c_A␥5兲⌿ˆ₂⫹⌿R₂␥␮共cV⫺c_A␥5兲⌿ˆ₁其, 共3.7兲 where c_V and c_A will be settled further.

The vacuum transition amplitude for process 共2.2兲 at the tree level is given by

A_k 1k2

␴1␴2⫽

具

_p

2兩丢

具

f1k₁␴₁, f¯2k₂␴₂兩SˆI兩0

典

丢兩p1

典

. 共3.8兲 Note that the second term inside the parenthesis at the right hand side of Eq. 共3.7兲 vanishes in this case. By using the field decomposition 共3.1兲 in Eq. 共3.7兲, and acting SˆI in Eq. 共3.8兲, we obtain A_k 1k2 ␴1␴2⫽

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_d4_{x j} ␮ ( p1→p2) ␺ ¯ k₁␴₁ (⫹␻₁) ␥␮_共cV⫺c A␥ 5_兲_␺ ⫺k2⫺␴2 (⫺␻₂) , 共3.9兲 where j_␮( pi→pj) _and␺ k_j␴_j (⫾␻_j)

兲其. 共3.13兲 Equation共3.12兲 can be cast in the form

J_␮␯( p1→p2)共x,x

⬘

兲⫽Geff

2 u␮共␶兲u␯共␶⬘兲 u0共␶兲u0共␶⬘兲e

i⌬M(␶⫺␶⬘)_␦3_关x⫺x共_␶_兲兴

⫻␦3_关x

_⬘

_⬘

_兲⫽ 共k”⫾m兲 2共2␲兲3␻e ⫾ik␭(x⫺x⬘)_␭_, 共3.16兲 where k␭⫽(␻,k) is the emitted fermion’s four momentum and k”⫽k”␭␥_␭. Applying the above expression in Eq. 共3.15兲, and using␥-matrix trace identities, we obtain

G_k 1k2 ␮␯ _共x,x

_⬘

_兲⫽ei(k1⫹k2) ␭_(x_⫺x_⬘₎ ␭ 4共2␲兲6␻1␻2 兵共cV2_⫹cA2_兲tr关_␥␮_k_” 2␥␯k”1兴 ⫹2cVc_Atr关␥5_␥␮_k_” 2␥␯k”1兴 ⫺m1m2共cV 2_⫺cA2_兲tr关_␥_␮_␥_␯_兴_其 ⫽e i(k₁_⫹k₂)␭(x_⫺x⬘)_␭ 共2␲兲6_␻ 1␻2 兵共cV2_⫹cA2_兲 ⫻关2k1 (␮ k2␯)⫺␩␮␯k␣1k2␣兴⫺m1m2共cV 2_⫺cA2_兲_␩_␮␯ ⫹2icVcA⑀␮␯␣␤k1␣k2␤其, 共3.17兲 where ⑀␮␣␯␤ is the totally skew-symmetric Levi-Civita pseudotensor 共with ⑀0123⫽⫺1) and k₁(␮k₂␯)⬅(k₁␮k₂␯ ⫹k1␯k2␮)/2. Letting Eqs. 共3.14兲 and 共3.17兲 into 共3.11兲, we obtain the differential transition probability

dPp1→p2 d3k1d3k2 ⫽ Geff 2 共2␲兲6_␻ 1␻2

冕

⫺⬁ ⫹⬁ d␶

冕

⫺⬁ ⫹⬁ d␶⬘ei⌬M(␶⫺␶⬘) ⫻ei(k₁⫹k₂)␭[x(␶)⫺x(␶⬘)]_␭_兵₂_关共cV2_⫹c A 2_兲k 1 (␮ k₂␯) ⫹icVcA⑀␮␯␣␤k1␣k2␤兴u␮共␶兲u␯共␶⬘兲⫺关共cV

2_⫺cA2_兲 ⫻m1m2⫹共cV 2_⫹c A 2_兲k 1 ␣_k 2␣兴u␮共␶兲u␮共␶⬘兲其, 共3.18兲 where we have used that d␶⫽dt/u0

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IV. UNIFORMLY ACCELERATED CURRENTS

The world line of a uniformly accelerated particle with proper acceleration a can be given in the usual Minkowski coordinates (t,x)苸R4 by

x␮共␶兲⫽共a⫺1sinh a␶, 0, 0, a⫺1cosh a␶兲. 共4.1兲 The corresponding four-velocity is

u␮共␶兲⫽共cosh a␶, 0, 0, sinha␶兲. 共4.2兲 Let us now define new coordinates

␰⬅共␶⫺␶⬘兲/2 and s⬅共␶⫹␶⬘兲/2, 共4.3兲 which allows us to rewrite Eq. 共3.18兲 as

dPp1→p2 d3k1d3k2 ⫽ 2Geff 2 共2␲兲6_␻ 1␻2

冕

⫺⬁ ⫹⬁ ds

冕

⫺⬁ ⫹⬁ d␰exp兵2i关⌬M␰ ⫹共k1⫹k2兲␭u␭共s兲sinh共a␰兲/a兴其 ⫻兵2共cV2⫹cA 2_兲k 1 ␮_k 2 ␯_关u

␮共s兲u␯共s兲cosh2共a␰兲 ⫺a2_x

␮共s兲x␯共s兲sinh2共a␰兲兴⫺cosh共2a␰兲 ⫻关共cV2_⫺cA2_兲m 1m2⫹共cV 2_⫹cA2_兲共k 1 ␣_k 2␣兲兴 ⫹2iacVcAsinh共2a␰兲⑀␮␯␣␤x␮共s兲u␯共s兲k1␣k2␤其,

共4.4兲 where we have used 关x(␶)⫺x(␶⬘)兴␮⫽2a⫺1sinh(a␰)u␮(s), u␮(␶)⫽cosh(a␰)u␮(s)⫹a sinh(a␰)x␮(s), u␮(␶⬘) ⫽cosh(a␰)u␮(s)⫺a sinh(a␰)x␮(s), and u␮(␶)u_␮(␶⬘) ⫽cosh(2a␰).

In order to decouple the integrals in Eq.共4.4兲, let us make the following change in the momentum variable:

k␮→k˜␮⫽共␻˜ ,k˜兲⬅关k␭u_␭共s兲, kx, ky, ⫺ak␭x_␭共s兲兴. 共4.5兲 Using Eqs.共4.1兲 and 共4.2兲 we can verify explicitly that the transformation共4.5兲 corresponds to a boost in the z direction. Indeed, k˜␮are the components of the emitted fermion’s four-momentum in the inertial frame instantaneously at rest with the current at the proper time s. Hence the transition prob-ability per proper time ⌫p1→p2⬅dPp1→p2/ds for process 共2.2兲 can be written from Eq. 共4.4兲 as

d⌫p1→p2 d3k˜1d3k˜2 ⫽ 2Geff 2 共2␲兲6_␻_˜ 1␻˜2

冕

⫺⬁ ⫹⬁ d␰exp兵2i关⌬M␰ ⫹a⫺1_sinh_共a_␰_兲共_␻˜₁_⫹_␻˜₂_兲兴_其 ⫻兵共cV2_⫹cA2_兲共 ␻ ˜₁_␻˜₂_⫹k˜ 1 z k ˜ 2 z_兲 ⫺2icVcAsinh共2a␰兲共k˜1⫻k˜2兲

z_⫹关共cV2_⫹cA2_兲 ⫻共k˜⬜1•k˜⬜2兲⫺共cV 2_⫺c A 2_兲m 1m2兴cosh共2a␰兲其, 共4.6兲

where k˜1⫻k˜2 is the usual three-vector product and k˜⬜1•k˜⬜2 ⬅k˜1 x_˜_k 2 x_⫹k˜ 1 y_˜_k 2

y_{. In order to integrate Eq.} _{共4.6兲, it is} conve-nient to use spherical coordinates in the momenta space (k˜ 苸R⫹_,˜_␪_苸关0,_␲_兴,_␾˜_苸关0,2_␲_)), _where ˜_kx_⫽k˜sin_␪_{˜ cos}_␾_{˜ , k˜}y ⫽k˜sin␪˜ sin␾˜ , k˜z_⫽k˜cos_{˜, and the following change of inte-}_␪ gration variable:␰→␭⬅ea␰_{. By using expression}_{共3.471.10兲} of Ref.关13兴, we obtain d⌫p1→p2 d3k˜1d3˜k2 ⫽4Geff 2 e⫺␲⌬M/a 共2␲兲6_␻_˜ 1␻˜2a 兵共cV 2_⫹cA2_兲共 ␻ ˜₁_␻˜₂_⫹k˜₁˜_k₂ ⫻cos␪˜

1cos˜␪2兲K2i⌬M/a关2共␻˜1⫹␻˜2兲/a兴 ⫹2cVc_A˜k₁˜k₂sin˜␪₁sin˜␪₂sin共␾˜₁⫺␾˜₂兲 ⫻Im兵K2_⫹2i⌬M/a关2共␻˜1⫹␻˜2兲/a兴其 ⫹关共cV2_⫺cA2_兲m

1m2⫺共cV 2_⫹c

A 2_兲 ⫻k˜1˜k2sin˜␪1sin˜␪2cos共␾˜1⫺␾˜2兲兴

⫻Re兵K2⫹2i⌬M/a关2共␻˜1⫹␻˜2兲/a兴其其, 共4.7兲 where Re兵z其 and Im兵z其are the real and imaginary parts of a complex number z, respectively, and K_␯(z) is the modified Bessel function.

We note that the uncorrelated emission of f1 and f¯2 is spherically symmetric in the instantaneously comoving frame. This can be seen by tracing out共i.e., integrating兲 one of the momentum variables in Eq.共4.7兲,

␻˜ 1 2_⫺m 1 2

冑

_␻_˜ 2 2_⫺m 2 2_共共c V 2_⫹cA2_兲 ⫻␻˜ 1␻˜2K2i⌬M/a关2共␻˜1⫹␻˜2兲/a兴⫹共cV 2_⫺c A 2_兲 ⫻m1m2Re兵K2_⫹2i⌬M/a关2共␻˜1⫹␻˜2兲/a兴其兲. 共4.11兲 Assuming that f1 or f¯2 is a massless particle, we can perform explicitly the integrals that appear in Eqs.共4.10兲 and 共4.11兲. For this purpose, we make the change of variables (␻˜1,␻˜2)→(␳,␨), where

␳⬅␻˜

l/␻˜i⫹1 and ␨⬅␻˜i 2

/m2, 共4.12兲

and here we label the massless and massive 共with mass m) particles with l and i indices, respectively. Applying Eq. 共4.12兲 in Eqs. 共4.10兲 and 共4.11兲 with ml⫽0, we have

冉

4a4⌬M⫹5a2⌬M3⫹⌬M5 e2␲⌬M/a⫺1

冊

共4.26兲 and Wp1→p2 ⫽Geff 2 _共cV2_⫹c A 2_兲 3840␲3 ⫻

冉

225a 6_⫹1036a4_⌬M2_⫹560a2_⌬M4_⫹64⌬M6 e2␲⌬M/a⫹1

冊

. 共4.27兲 In the next sections we use these formulas to investigate some selected reactions.

V. ACCELERATED PROTON AND NEUTRON DECAY

Let us now consider the processes

p→ne⫹␯e 共5.1兲

and

n→pe⫺¯␯e 共5.2兲

for uniformly accelerated protons and neutrons, respectively. We will assume the neutrino mass to vanish because even if this is not so, it would be neglectable in comparison with any other energy scale involved in the problem. The effective coupling constant G_eff⫽G_pn for processes共5.1兲, 共5.2兲 is ob-tained by imposing that the mean proper lifetime of inertial neutrons is 887 s关14兴, i.e.,

⌫inn_→p_⬅⌫n→p_{共a→0兲⫽1/887s}⫺1_. _共5.3兲 This phenomenological procedure has the advantage of by passing any uncertainties on the influence of the nucleon inner structure. For sake of convenience, we take the a→0 limit in Eq. 共4.6兲 rather than in Eq. 共4.19兲, obtaining

d⌫inn→p d3k˜_ed3˜k_␯⫽ 4Gpn 2 共2␲兲6_␻_˜ e␻˜␯

冕

⫺⬁ ⫹⬁ d␰e2i␰(⌬M⫹␻˜e⫹␻˜␯) ⫻共␻˜_e_␻˜_␯_⫹k˜e_•k˜␯兲 ⫽2Gpn 2 共2␲兲5

冉

1⫹ k ˜_e•k˜␯ ␻˜_e_␻˜_␯

冊

␦共␻˜e⫹␻˜␯⫺⌬M 兲, 共5.4兲 where we have used cV⫽cA⫽1 关15兴 since only left-handed massless neutrinos are known to exist. After integrating Eq. 共5.4兲 in angular coordinates and in ␻˜

e, we find ⌫in n→p_⫽Gpn 2 ␲3

冕

₀ ⌬M⫺me d␻˜_␯␻˜_␯2共⌬M⫺␻˜␯兲

冑

共⌬M⫺␻˜␯兲2⫺m_e2. 共5.5兲 Evaluating numerically Eq.共5.5兲 with me⫽0.511 MeV, and ⌬M⫽(mn⫺mp)⫽1.29 MeV, we end up with ⌫in

n→p_⫽1.81 ⫻10⫺3_G

pn

2 _MeV5_{. Hence by imposing condition} _{共5.3兲, we} obtain Gpn⫽1.74GF, where G_F⬅1.166⫻10⫺5 GeV⫺2 is the Fermi coupling constant关14兴. Now we are able to use Eq. 共4.19兲 to plot in Fig. 1 the proton and neutron mean proper lifetimes ␶_p(a)⫽(⌫p→n)⫺1 and ␶_n(a)⫽(⌫n→p)⫺1, respec-tively. Let us note that

␶n共a兲⫽e⫺2␲兩⌬M兩/a_␶_p共a兲. _共5.6兲 We have only considered accelerations aⰆmp⫽938 MeV in order to respect our no-recoil condition共see Sec. II兲. We call attention to the fact that for accelerations aⰇac⬅2␲兩⌬M兩 ⬇8 MeV, we have ␶p(a)⬇␶n(a). This is easier to under-stand in the coaccelerated frame with the current, where 共ac-cording to the FDU effect 关5,6兴兲 a thermal bath of Rindler particles with temperature TFDU⫽a/2␲ is ‘‘attached’’ to the

(7)

current. Thus, for aⰇac we have TFDUⰇ兩⌬M兩, which leads both nucleons to behave similarly. 共See Ref. 关7兴 for a more comprehensive discussion on this issue.兲

In order to estimate how much energy is carried out in form of leptons, we may use Eqs.共4.20兲 and 共4.21兲 to obtain

Wj p_→n

and Wn_j→p⫽e2␲兩⌬M兩/aWp_j→n for j⫽e,␯. Although

We p→n

and W_␯p→n 共as well as W_en→p and W_␯n→p) are not manifestly identical, they seem to be according to Fig. 2.

In order to investigate the energy distribution of the emit-ted leptons, let us define the normalized energy distribution

Np_j₁→p2 ⬅ 1 ⌫p1→p2 d⌫p1→p2 d␻˜j 共5.7兲

with j⫽e,␯, where d⌫p1→p2/d␻˜j is defined in Eq. 共4.9兲. Note that Np_j→n⫽Nn_j→p. In Fig. 3 we plot the distributions

Nj p→n

for two values of acceleration: a⫽1.0 and 2.0 MeV. We see that the typical energy共in the inertial frame instan-taneously at rest with the nucleon兲 of the emitted electrons and neutrinos is ␻˜⬇a, which justifies our no-recoil condi-tion.

In order to roughly estimate how small is the proper life-time of circularly moving protons at the CERN Large Had-ron Collider 共LHC兲 we use directly Eq. 共5.6兲 with a⫽a

LHC

⬇10⫺8 _{MeV for the proton’s proper acceleration, obtaining} ␶p(a_LHC)⬇103⫻10

8

yr, where we have used that ␶_n(a Ⰶme,兩⌬M兩)⬇103s. Although Eq.共5.6兲 was derived assum-ing uniformly accelerated motion, this should not be seen as a major problem: Because of the huge proper lifetime ob-tained for the proton, our estimation turns out to be non-sensitive up to an inaccuracy of hundreds of thousands of orders of magnitude共which should not be the case兲.

Astrophysics seems to provide much more suitable condi-tions for the observation of the decay of accelerated protons. Although our decay rate 共4.19兲 was obtained considering uniformly accelerated protons, let us assume that this is ap-proximately valid for circularly moving protons with proper acceleration aⰇ⌬M,1/R, where R is the local curvature ra-dius of the proton trajectory. Indeed we can test this assump-tion, e.g., for two-level scalar systems, whose excitation rates, at the tree level, are given by关16兴

⌫lin⫽ c₀2 2␲ ⌬E e2␲⌬E/a⫺1 共5.8兲 and

FIG. 1. The mean proper lifetime of protons,␶p共full line兲, and

neutrons,␶n 共dashed line兲, are plotted as functions of their proper

acceleration a. Note that ␶p→⫹⬁ and ␶n→887 s as a→0. For

accelerations aⰇac⬅2␲⌬M⬇8 MeV we have that␶p⬇␶n.

FIG. 2. Wj p→n

andWj n→p

are plotted in full and dashed lines, respectively, for j⫽e,␯, as functions of the nucleon proper accel-erations. Our numerical results suggest that We

p→n_⫽W ␯p→n and We n→p_⫽W ␯ n→p .

FIG. 3. The normalized energy distribution of emitted positrons,

Ne

p→n_{, and neutrinos,} _N

␯

p→n_{, are plotted for two values of the}

proton’s proper acceleration: a⫽1.0 MeV 共full line兲 and 2.0 MeV

共dashed line兲. Note that the typical energy of the emitted particles 共in the inertial frame instantaneously at rest with the proton兲 is

(8)

⌫cir⫽ c₀2 2␲

ae⫺冑12⌬E/a

2

冑

12 共5.9兲

for uniformly accelerated and circularly moving relativistic sources, respectively, where c0 is a small coupling constant and⌬E is the two-level system energy gap. Note that in the limit aⰇ⌬E, Eqs. 共5.8兲 and 共5.9兲 give us ⌫_lin/⌫_cir⫽1.103.

In order to illustrate an astrophysical situation where pro-cess 共5.1兲 may be important, let us consider a cosmic ray proton with energy Ep⫽␥mp⬇1.6⫻1014eV under the influ-ence of a magnetic field B⬇1014G of a typical pulsar. Pro-tons under these conditions have proper accelerations of aB ⫽␥eB/mp⬇110 MeVⰇ兩⌬M兩. For practical purposes the acceleration of the proton will be assumed as constant along the process. For the chosen values of E_p and B, the proton is confined in a cylinder with typical radius R⬇␥2/aB⬇5 ⫻10⫺3_cm_ⰆlB_{, where l}

B is the typical size of the magnetic field region. According to Eq.共4.19兲 we obtain ␶p⬇10⫺7 s. As a result, protons would have a ‘‘laboratory’’ mean life-time of tp⫽␥␶p⬇10⫺1 s. For lB⬇107 _{cm, we obtain that} less than 兩⌬Np/Np兩⫽(1⫺e⫺lB/tp₎⬇lB_/tp⬇1% of the

pro-tons would decay via process共5.1兲. We note that we did not take into account the influence of the magnetic field on the emitted positron. Clearly a more precise estimation should take into account this effect as well as other ones as, e.g., the nonuniformity of the magnetic field and energy losses through electromagnetic sinchrotron radiation. The last one in particular may not be a problem since energy may be furnished to the proton from dynamo processes. A more careful analysis of such astrophysical issues would be wel-come but this is beyond the scope of the present field-theoretical investigation.

VI. NEUTRINO EMISSION FROM UNIFORMLY ACCELERATED ELECTRONS

In this section, we will consider the emission of neutrinos from accelerated electrons:

e⫺→e⫺␯e␯¯e. 共6.1兲 The description of the creation of neutrino-antineutrino pairs by electrons in an external electromagnetic field in the con-text of the standard model is contained in Sec. 6.1 of Ref. 关2兴. Here we analyze this process for uniformly accelerated electrons by using the formulas derived in Sec. IV where both emitted fermions are massless. From Eqs. 共4.26兲 and 共4.27兲 we get for the emission rate of␯e¯␯epairs

⌫␯␯¯⫽ G_e2_␯a5

15␲4 , 共6.2兲

and for the total radiated power

W␯␯¯⫽

15G_e2_␯a6

256␲3 , 共6.3兲

where we have used ⌬M⫽0, cV⫽cA⫽1 and Ge_␯ is the corresponding effective coupling constant.

In order to determine the value of Ge_␯, we assume that Eq. 共6.3兲 describes the instantaneous emitted power from an electron with arbitrary world line at the point where it has proper acceleration a. This is indeed verified for photon共see Larmor formula in Ref. 关17兴兲 and scalar particle 关18兴 emis-sion from accelerated sources. 关We emphasize that this equivalence is not fully 共although it is approximately兲 veri-fied for Eq. 共6.2兲, which depends in general on the source’s world line.兴 Thus we will impose that Eq. 共6.3兲 gives the radiated power for the neutrino emission from circularly moving relativistic electrons in a uniform magnetic field B provided that a⫽␥eB/meⰆme共no-recoil condition兲. Here␥ is the usual Lorentz factor for the electron and e is its electric charge. The differential emission rate of␯e¯␯epairs in a back-ground magnetic field was calculated in detail 关2兴共see Ref. 关8兴 for the form used below兲:

d⌫_␯␯¯L P ds ⫽ G_F2m_e4 16共2␲兲3 m_e ␥ ␹5_s3_⫹1/2 共1⫹␹s3/2_兲4⫻

再

共CV 2_⫹CA2_兲 ␹ 2_s3 共1⫹␹s3/2_兲 ⫻

冕

s ⬁

冋

2⫹1 3共2s⫹y兲共y⫺s兲 2

册

_Ai_{共y兲dy⫹共C} V 2_⫹CA2_兲 ⫻

冋

冕

s ⬁

关6⫹共y⫺s兲共s2_{⫹共s⫺y兲}2_{兲兴Ai共y兲dy⫺sAi共s兲}

册

⫹8sCA

2

冋

3 4

冉

冕

s

⬁

共s⫺y兲2_Ai_共y兲dy

冊

_⫹Ai共s兲

册

冎

_, _共6.4兲 where ␹⬅a/me, Ai(z) is the Airy function, and s 苸关0,␥/␹兴 is defined such that

␻␯⫹␻␯¯⬅

me␥␹s3/2

共1⫹␹s3/2兲. 共6.5兲

The parameters CV and CA give the vector and axial contri-butions to the electric current, respectively. Using Eqs.共6.4兲 and共6.5兲 we have, in the limit␹Ⰶ1,

W_␯␯¯L P ⫽

冕

0 ␥/␹ ds共␻_␯⫹␻_␯¯兲 d⌫_␯␯¯L P ds ⫽ 5共2CV2⫹23CA2兲 108␲3 GF 2 m_e6␹6. 共6.6兲 Letting C_V2⫽0.93 and C_A2⫽0.25 关9兴, we have W_␯␯¯L P⫽1.14 ⫻10⫺2_G

F

2_a6_{. By comparing this expression with our Eq.} 共6.3兲 we obtain Ge␯⫽2.45GF. In Figs. 4 and 5 we plot Eqs. 共6.2兲 and 共6.3兲, respectively, for uniformly accelerated elec-trons with a⭐me.

The normalized energy distribution of emitted neutrino-antineutrino N␯␯¯⬅ 1 ⌫_␯␯¯ d⌫_␯␯¯ d␻˜_␯ 共6.7兲

(9)

is plotted in Fig. 6 for electrons with proper acceleration a ⫽0.1 MeV and 0.2 MeV, where 关see Eq. 共4.9兲兴

d⌫_␯␯¯ d␻˜_␯ ⫽ 2G_e2_␯ ␲4_a␻˜␯ 2

冕

0 ⬁ d␻˜_␯¯␻˜_␯¯2K0关2共␻˜␯⫹␻˜␯¯兲/a兴. 共6.8兲 共Neutrinos and antineutrinos have identical emission energy distribution.兲 Note again that a defines the typical energy of the emitted neutrinos.

VII. DISCUSSIONS

We have investigated the weak interaction emission of spin-1/2 fermions from classical and semiclassical currents. As a particular application of modeling the accelerated par-ticle by a semiclassical current, we have analyzed the inverse ␤ decay of uniformly accelerated protons. We have shown that although protons in laboratory storage rings are not

likely to decay in this way, under some astrophysical condi-tions high-energy protons in background magnetic fields may have a considerably short lifetime. Moreover, we have ana-lyzed the modification of the usual ␤ decay for uniformly accelerated neutrons. This may be of some relevance when neutrons are under the influence of strong background gravi-tational fields. Although a full curved spacetime calculation is desirable to treat these situations, our calculation should be a good approximation when the gravitational field is ‘‘mod-erate’’ 关18兴. In this case, neutrons can be treated as being accelerated in Minkowski space.

By restricting our semiclassical current to behave classi-cally, we were able to use our formalism to investigate the neutrino-antineutrino pair emission from uniformly acceler-ated electrons and compare our results with the ones in the literature obtained by quantizing the electron field in a back-ground magnetic field. Our formalism allows the utilization of currents associated with more general world lines. De-pending on the accuracy level required, however, one can use directly the formulas derived for uniformly accelerated currents. This may be particularly useful in some astrophysi-cal situations.

ACKNOWLEDGMENTS

The authors are thankful to J. C. Montero, V. Pleitez, and A. A. Natale for discussions. D. V. was fully supported by Fundaçaõ de Amparo a` Pesquisa do Estado de Saõ Paulo while G.M. was partially supported by Conselho Nacional de Desenvolvimento Cientı´fico e Tecnolo´gico.

APPENDIX: INTEGRATION OF EQS.„4.24…, „4.25…

In order to solve Eqs.共4.24兲 and 共4.25兲 let us consider the integral

FIG. 4. The emission probability per proper time of␯e␯¯epairs is

plotted for a⭐me.

FIG. 5. The total radiated power in form of neutrinos is plotted for a⭐me.

FIG. 6. The normalized energy distribution of the emitted neu-trinos共and antineutrinos兲 is plotted for two values of the electron’s proper acceleration: a⫽0.1 MeV 共full line兲 and 0.2 MeV 共dashed line兲. Note that the typical energy of the emitted particles 共in the inertial frame instantaneously at rest with the electron兲 is given by

(10)

In⫹⬅

冕

0 ⬁ dw w n⫹2i⌬M/a 共w2_⫺1⫹2i_⑀_w_兲n_⫹1, n苸N. 共A1兲 Note that the analytic extension of the integrand above has poles of order (n⫹1) at w⫾⫽⫾1⫺i⑀⫹O(⑀2) . This im-plies that we can make ⑀⫽0 in Eq. 共A1兲 provided we con-tour the pole at w⫹⫽1 by the upper half-plane, i.e.,

In⫹⫽

冕

␥⫹ dww n⫹2i⌬M/a 共w2_⫺1兲n⫹1, 共A2兲 where ␥⫾⬅关0,1⫺⑀⬘兴艛兵1⫾⑀⬘ei␪; ␪苸关0,␲兴其艛关1⫹⑀⬘,⬁) with⑀⬘→0_⫹. Using the residue theorem we see that

In⫺⫺In⫹⫽2␲i Res共 fn兲w⫽1, 共A3兲 whereI_n⫺is obtained substituting␥_⫹by␥_⫺in Eq.共A2兲, and we denote the residue value of the function

fn共w兲⬅ w

n_⫹2i⌬M/a

共w2_⫺1兲n⫹1 共A4兲

at the point w⫽w⫾ by Res( fn)w_⫽w⫾. Now, let us define

In⬅

冕

⫺⬁ ⫹⬁ dw w n⫹2i⌬M/a 共w2_⫺1⫹2i_⑀_w_兲n⫹1, 共A5兲 which can be written for ⑀→0 as

In⫽共⫺1兲n_e⫺2␲⌬M/a_I n ⫺_⫹I

n

⫹_. _共A6兲

Since the integrand of In is analytic in the upper half-plane and goes to zero as兩w兩⫺(n⫹2) as兩w兩→⬁, it follows that In ⫽0. As a consequence Eqs. 共A6兲 and 共A3兲 imply

In⫹⫽ ⫺2␲i Res共 fn兲w_⫽1 1⫹共⫺1兲n_e2␲⌬M/a, 共A7兲 with Res共 fn兲w_⫽w⫾⫽ 1 n! dn dwn兵共w⫺w ⫾_兲n_⫹1_f_n共w兲_其_兩w ⫽w⫾. 共A8兲 Using function 共A4兲 to explicitly evaluate Eq. 共A8兲 for n ⫽5, 6, we obtain I5⫹⫽ ⫺␲ 120a5

冉

4a4⌬M⫹5a2⌬M3⫹⌬M5 e2␲⌬M/a⫺1

冊

, 共A9兲 and I6⫹⫽ i␲ 46080a6 ⫻

冉

225a 6_⫹1036a4_⌬M2_⫹560a2_⌬M4_⫹64⌬M6 e2␲⌬M/a_⫹1

冊

. 共A10兲

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