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IFT-P. 007/99 gr-qc/9901008 Decay of protons and neutrons induced by acceleration George E.A. Matsas and Daniel A.T. Vanzella Instituto de F´ısica Teo´rica Universidade Estadual Paulista Rua Pamplona 145 9 9 01405-900, Saõ Paulo, SP 9 1 Brazil n a J 5 Abstract 1 v 8 0 0 1 We investigate the decay of accelerated protons and neutrons. Calcu- 0 9 lations are carried out in the inertial and coaccelerated frames. Particle 9 / c interpretation of these processes are quite different in each frame but the q - r decay rates are verified to agree in both cases. For sake of simplicity our g : calculations are performed in a two-dimensional spacetime since our con- v i X clusions are not conceptually affected by this. r a 04.62.+v, 12.15.Ji, 13.30.-a, 14.20.Dh Typeset using REVTEX 1 I. INTRODUCTION It is well known that according to the Standard Model the mean proper lifetime of neutrons is about τ = 887s while protons are stable (τ > 1.6 1025 years) [1]. This is n p × only true, however, for inertial nucleons. There are a number of high-energy phenomena where acceleration plays a crucial role (see Refs. [2] and [3]- [4] for comprehensive discus- sions on electron depolarization in storage rings and bremsstrahlung respectively). The influence of acceleration in particle decay was only considered quite recently [5]. As it was pointed out, acceleration effects are not expected to play a significant role in most particle decays observed in laboratory. Notwithstanding, this might not be so under cer- tain astrophysical and cosmological conditions. Muller has estimated [5] the time decay of accelerated µ−, π− and p+ via the following processes: (i) µ− e−ν¯ ν , (ii) π− µ−ν¯ , (iii) p+ n e+ν , e µ µ e → → → as described in the laboratory frame. Here we analyze in more detail process (iii) and the related one (iv) n p+ e−ν¯ . e → Process (iii) is probably the most interesting one in the sense that the proton must be accelerated in order to give rise to a non-vanishing rate. In the remaining cases, non- vanishing rates are obtained even when the decaying particles (µ−, π−, n) are inertial. As a first approximation, Muller has considered that all particles involved are scalars. Here we shall treat e−, ν and the corresponding antiparticles as fermions while p+ and n e will be represented by a classical current. This is a suitable approximation as far as these nucleons areenergetic enoughtohavea welldefined trajectory. Moreover, we willanalyze β- and inverse β-decays in the coaccelerated frame in addition to in the inertial frame. This is interesting because the particle content of these decays will be quite different in each one of these frames. This is a consequence of the fact that the Minkowski vacuum corresponds to a thermal state of Rindler particles [6]- [7]. We have chosen to perform the calculations in a two-dimensional spacetime because there is no conceptual loss at all. A comprehensive (but restricted to the inertial frame) four-dimensional spectral analysis of the inverse β-decay for accelerated protons and a discussion of its possible importance to cosmology and astrophysics will be presented elsewhere. The paper is organized as follows: In Section II we introduce the classical current which suitably describes the decay of accelerated nucleons. Section III is devoted to calculate the β- and inverse β-decay rates in the inertial frame. In Section IV we review the quantization of the fermionic field in the coaccelerated frame. In Section V we compute the β- and inverse β-decay rates in the accelerated frame. For this purpose we must take into account the Fulling-Davies-Unruh thermal bath [6]- [7]. Finally, in Section VI we discuss our results and further perspectives. We will use natural units k = c = h¯ = 1 throughout this paper unless stated otherwise. B 2 II. DECAYING-NUCLEON CURRENT In order to describe the uniformly accelerated nucleon, it is convenient to introduce the Rindler wedge. The Rindler wedge is the portion of Minkowski spacetime defined by z > t where (t,z) are the usual Minkowski coordinates. It is convenient to cover the | | Rindler wedge with Rindler coordinates (v,u) which are related with (t,z) by t = usinhv , z = ucoshv, (2.1) where 0 < u < + and < v < + . As a result, the line element of the Rindler ∞ −∞ ∞ wedge is written as ds2 = u2dv2 du2. (2.2) − The worldline of a uniformly accelerated particle with proper acceleration a is given in these coordinates by u = a−1 = const. Particles following this worldline have proper time τ = v/a. Thus let us describe a uniformly accelerated nucleon through the vector current jµ = quµδ(u a−1), (2.3) − where q is a small coupling constant and uµ is the nucleon’s four-velocity: uµ = (a,0) and uµ = (√a2t2 +1,at) in Rindler and Minkowski coordinates respectively. Thecurrent aboveisfinetodescribestableacceleratednucleons butmustbeimproved to allow nucleon-decay processes. For this purpose, let us consider the nucleon as a two- level system [7]- [9]. In this scenario, neutrons n and protons p are going to be | i | i seen as excited and unexcited states of the nucleon respectively, and are assumed to be eigenstates of the nucleon Hamiltonian Hˆ: Hˆ n = m n , Hˆ p = m p , (2.4) n p | i | i | i | i where m and m are the neutron and proton mass respectively. Hence, in order to n p consider nucleon decay processes, we replace q in Eq. (2.3) by the Hermitian monopole qˆ(τ) eiHˆτqˆ e−iHˆτ . (2.5) 0 ≡ Here G m qˆ m will play the role of the Fermi constant in the two-dimensional F p 0 n ≡ |h | | i| Minkowski spacetime. As a result, current (2.3) will be replaced by ˆjµ = qˆ(τ)uµδ(u a−1) . (2.6) − III. INERTIAL FRAME CALCULATION OF THE β- AND INVERSE β-DECAY FOR ACCELERATED NUCLEONS Let us firstly analyze the decay of uniformly accelerated protons and neutrons in the inertial frame (see processes (iii) and (iv) in Sec. I). We shall describe electrons and neutrinos as fermionic fields: 3 +∞ Ψˆ(t,z) = dk ˆb ψ(+ω)(t,z)+dˆ† ψ(−ω) (t,z) , (3.1) kσ kσ kσ −k−σ σX=±Z−∞ (cid:16) (cid:17) ˆ ˆ† where b and d are annihilation and creation operators of fermions and antifermions, kσ kσ respectively, with momentum k and polarization σ. In the inertial frame, frequency, momentum and mass m are related as usually: ω = √k2 +m2 > 0. ψ(+ω) and ψ(−ω) are kσ kσ positive and negative frequency solutions of the Dirac equation iγµ∂ ψ(±ω) mψ(±ω) = 0. µ kσ − kσ By using the γµ matrices in the Dirac representation (see e.g. Ref. [4]), we find (ω m)/2ω ± ± ei(∓ωt+kz)  q 0  (±ω) ψ (t,z) = (3.2) k+ √2π  k/ 2ω(ω m)   ±   q 0      and 0 ei(∓ωt+kz)  (ω m)/2ω  (±ω) ψ (t,z) = ± ± . (3.3) k− √2π  q 0     k/ 2ω(ω m)    − ±   q  In order to keep a unified procedure for inertial and accelerated frame calculations, we have orthonormalizedmodes(3.2)-(3.3)accordingtothesameinner product definition[9] that will be used in Sec. IV: hψk(±σω),ψk(±′σω′′)i ≡ dΣµψ¯k(±σω)γµψk(±′σω′′) = δ(k −k′)δσσ′δ±ω±ω′ , (3.4) ZΣ where ψ¯ ψ†γ0, dΣ n dΣ with nµ being a unit vector orthogonal to Σ and pointing µ µ ≡ ≡ to the future, and Σ is an arbitrary spacelike hypersurface. (In this section, we have chosen t = const for the hypersurface Σ.) As a consequence canonical anticommutation relations for fields and conjugate momenta lead to the following simple anticommutation relations for creation and annihilation operators: {ˆbkσ,ˆb†k′σ′} = {dˆkσ,dˆ†k′σ′} = δ(k −k′) δσσ′ (3.5) and {ˆbkσ,ˆbk′σ′} = {dˆkσ,dˆk′σ′} = {ˆbkσ,dˆk′σ′} = {ˆbkσ,dˆ†k′σ′} = 0 . (3.6) Next we couple minimally electron Ψˆ and neutrino Ψˆ fields to the nucleon current e ν (2.6) according to the Fermi action Sˆ = d2x√ gˆj (Ψˆ¯ γµΨˆ +Ψˆ¯ γµΨˆ ) . (3.7) I µ ν e e ν − Z Note that the first and second terms inside the parenthesis at the r.h.s of Eq. (3.7) vanish for the β-decay (process (iv) in Sec. I) and inverse β-decay (process (iii) in Sec. I) respectively. 4 Let us consider firstly the inverse β-decay. The vacuum transition amplitude is given by p→n = n e+ ,ν Sˆ 0 p . (3.8) A(iii) h |⊗h keσe kνσν| I | i⊗| i ˆ By using current (2.6) in Eq. (3.7), and acting with S on the nucleon states in Eq. (3.8), I we obtain +∞ +∞ ei∆mτ p→n = G dt dz u δ(z √t2 +a−2) e+ ,ν Ψˆ¯ γµΨˆ 0 , A(iii) F −∞ −∞ √a2t2 +1 µ − h keσe kνσν| ν e | i Z Z (3.9) where ∆m m m , τ = a−1sinh−1(at) is the nucleon’s proper time and we recall that n p ≡ − in Minkowski coordinates the four-velocity is uµ = (√a2t2 +1,at) [see below Eq. (2.3)]. The numerical value of the two-dimensional Fermi constant G will be fixed further. By F using the fermionic field (3.1) in Eq. (3.9) and solving the integral in the z variable, we obtain (G /4π) δ +∞ p→n = − F σe,−σν dτei(∆mτ+a−1(ωe+ων)sinhaτ−a−1(ke+kν)coshaτ) A(iii) ω ω (ω +m )(ω m ) −∞ ν e ν ν e e Z − [(ω +mq)(ω m )+k k ]coshaτ [(ω +m )k +(ω m )k ]sinhaτ . ν ν e e ν e ν ν e e e ν ×{ − − − } The differential transition rate d2 p→n Pin = p→n 2 (3.10) dk dk |A(iii) | e ν σXe=±σXν=± calculated in the inertial frame will be, thus, d2 p→n G2 +∞ +∞ Pin = F dτ dτ exp[i∆m(τ τ )] dke dkν 8π2 Z−∞ 1Z−∞ 2 1 − 2 exp[i(ω +ω )(sinhaτ sinhaτ )/a i(k +k )(coshaτ coshaτ )/a)] e ν 1 2 e ν 1 2 × − − − (ω +m )(ω m )+k k (ω +m )k +(ω m )k ν ν e e ν e ν ν e e e ν − coshaτ − sinhaτ 1 1 × ω ω (ω +m )(ω m ) − ω ω (ω +m )(ω m )  ν e ν ν e e ν e ν ν e e − − q q  (ω +m )(ω m )+k k (ω +m )k +(ω m )k ν ν e e ν e ν ν e e e ν − coshaτ − sinhaτ . 2 2 × ω ω (ω +m )(ω m ) − ω ω (ω +m )(ω m )  ν e ν ν e e ν e ν ν e e − − q q  In order to decouple the integrals above, it is convenient to introduce first new variables s and ξ such that τ s+ξ/2 , τ s ξ/2 . After this, we write 1 2 ≡ ≡ − d2 p→n G2 +∞ +∞ Pin = F ds dξ ei(∆mξ+2a−1sinh(aξ/2)[(ων+ωe)coshas−(kν+ke)sinhas]) dk dk 4π2ω ω e ν ν e Z−∞ Z−∞ [(ω ω +k k )cosh2as (ω k +ω k )sinh2as m m coshaξ] . (3.11) ν e ν e e ν ν e ν e × − − Next, by defining a new change of variables: k k′ = ω sinh(as)+k cosh(as), e(ν) e(ν) e(ν) e(ν) → − 5 we are able to perform the integral in the s variable, and the differential transition rate (3.11) can be cast in the form 1 d2 p→n G2 +∞ Pin = F dξ exp i∆mξ +i2a−1(ω′ +ω′)sinh(aξ/2) T dk′ dk′ 4π2ω′ω′ e ν e ν e ν Z−∞ h i (ω′ω′ +k′ k′ m m coshaξ) , (3.12) × ν e ν e − ν e where T +∞ds is the total proper time and ω′ k′2 +m2 . ≡ −∞ e(ν) ≡ e(ν) e(ν) The totaRl transition rate Γpin→n = Pipn→n/T is obtaineqd after integrating Eq. (3.12) in both momentum variables. For this purpose it is useful to make the following change of variables: k′ k˜ k′ /a , ξ λ eaξ/2. e(ν) e(ν) e(ν) → ≡ → ≡ ˜ (Note that k is adimensional.) Hence we obtain e(ν) G2a +∞ dk˜ +∞ dk˜ +∞ dλ Γp→n = F e ν exp[i(ω˜ +ω˜ )(λ λ−1)] in 2π2 Z−∞ ω˜e Z−∞ ω˜ν Z−∞ λ1−i2∆m/a e ν − ω˜ ω˜ +k˜ k˜ m m (λ2 +λ−2)/(2a2) , (3.13) ν e ν e ν e × − (cid:16) (cid:17) where ω˜ k˜2 +m2 /a2. e(ν) ≡ e(ν) e(ν) Let us assuqme at this point m 0. In this case, using (3.871.3-4) of Ref. [10], ν → we perform the integration in λ and obtain the following final expression for the proton decay rate: 4G2a +∞ +∞ Γp→n = F dk˜ dk˜ K 2 k˜2 +m2/a2 +k˜ . (3.14) in π2eπ∆m/a e ν i2∆m/a e e ν Z0 Z0 (cid:20) (cid:18)q (cid:19)(cid:21) Performing analogous calculation for the β-decay, we obtain for the neutron differential and total decay rates the following expressions [see Eqs. (3.12) and (3.13)]: 1 d2 n→p G2 +∞ Pin = F dξ exp i∆mξ +i2a−1(ω′ +ω′)sinh(aξ/2) T dk′e dk′ν 4π2ωe′ων′ Z−∞ h− e ν i (ω′ω′ +k′ k′ m m coshaξ) , (3.15) × ν e ν e − ν e and G2a +∞ dk˜ +∞ dk˜ +∞ dλ Γn→p = F e ν exp[i(ω˜ +ω˜ )(λ λ−1)] in 2π2 Z−∞ ω˜e Z−∞ ω˜ν Z−∞ λ1+i2∆m/a e ν − ω˜ ω˜ +k˜ k˜ m m (λ2 +λ−2)/(2a2) . (3.16) ν e ν e ν e × − (cid:16) (cid:17) By making m 0 in the expression above, we end up with ν → 4G2a +∞ +∞ Γn→p = F dk˜ dk˜ K 2 k˜2 +m2/a2 +k˜ . (3.17) in π2e−π∆m/a e ν i2∆m/a e e ν Z0 Z0 (cid:20) (cid:18)q (cid:19)(cid:21) In order to determine the value of our two-dimensional Fermi constant G we will F impose the mean proper lifetime τ (a) = 1/Γn→p of an inertial neutron to be 887s [1]. n in 6 By taking a 0 and integrating both sides of Eq. (3.15) with respect to the momentum → variables, we obtain G2 +∞ dk′ +∞ dk′ +∞ Γn→p = F e ν dξ exp[i(ω′ +ω′)ξ] in |a→0 4π2 Z−∞ ωe′ Z−∞ ων′ Z−∞ e ν exp( i∆mξ)(ω′ω′ +k′ k′ m m ) . (3.18) × − ν e ν e − ν e Next, by performing the integral in ξ, we obtain 2G2 +∞ dω′ +∞ dω′ Γn→p = F e ν (ω′ω′ m m )δ(ω′ +ω′ ∆m) . in |a→0 π Zme ω′2 m2 Zmν ω′2 m2 ν e − ν e ν e − e − e ν − ν q q (3.19) Now it is easy to perform the integral in ω′ : ν 2G2 ∆m−mν dω′ ω′(∆m ω′) m m Γn→p = F e e − e − ν e . (3.20) in |a→0 π Zme ω′2e −m2e (∆m−ωe′)2 −m2ν q q By integrating the right-hand side of Eq. (3.20) with m 0 and imposing 1/ Γn→p ν → in |a→0 to be 887s, we obtain G = 9.918 10−13. Note that G 1 which corroborates our F F × ≪ perturbative approach. Now we are able to plot the neutron mean proper lifetime as a function of its proper acceleration a (see Fig. 1). Note that after an oscillatory regime it decays steadily. In Fig.2 we plot the proton mean proper lifetime. The necessary energy toallowprotonstodecay is provided bytheexternal accelerating agent. Foraccelerations such that the Fulling-Davies-Unruh (FDU) temperature (see discussion in Sec. V) is of order of m +m m , i.e. a/2π 1.8MeV, we have that τ τ (see Fig. 1 and Fig. 2). n e p p n − ≈ ≈ Such accelerations are considerably high. Just for sake of comparison, protons at LHC have a proper acceleration of about 10−8MeV. IV. FERMIONIC FIELD QUANTIZATION IN A TWO-DIMENSIONAL RINDLER WEDGE We shall briefly review [11] the quantization of the fermionic field in the accelerated frame since this will be crucial for our further purposes. Let us consider the two-dimensional Rindler wedge described by the line element (2.2). The Dirac equation in curved spacetime is (iγµ ˜ m)ψ = 0, where γµ (e )µγα are the Dirac matrices in curved spacetime, R˜∇µ −∂ +ωΓσ and Γ = 1[γRα,≡γβ](eα )λ˜ (e ) are the ∇µ ≡ µ µ µ 8 α ∇µ β λ Fock-Kondratenko connections. ( γµ are the usual flat-spacetime Dirac matrices.) In the Rindler wedge the relevant tetrads are (e )µ = u−1δµ, (e )µ = δµ. As a consequence, the 0 0 i i Dirac equation takes the form ∂ψ iα ∂ i ωσ = γ0mu 3 iuα ψ , (4.1) 3 ωσ ∂v − 2 − ∂u! 7 where α γ0γi. i ≡ We shall express the fermionic field as +∞ Ψˆ(v,u) = dω ˆb ψ (v,u)+dˆ† ψ (v,u) , (4.2) ωσ ωσ ωσ −ω−σ σX=±Z0 (cid:16) (cid:17) where ψ = f (u)e−iωv/a are positive (ω > 0) and negative (ω < 0) energy solutions ωσ ωσ with respect to the boost Killing field ∂/∂v with polarization σ = . From Eq. (4.1) we ± obtain Hˆ f = ωf , (4.3) u ωσ ωσ where iα ∂ Hˆ a muγ0 3 iuα . (4.4) u 3 ≡ " − 2 − ∂u# By “squaring” Eq. (4.3) and defining two-component spinors χ (j = 1,2) through j χ (u) f (u) 1 , (4.5) ωσ ≡ χ2(u) ! we obtain d d 1 ω2 iω u u χ = m2u2 + χ σ χ , (4.6) du du! 1 " 4 − a2# 1 − a 3 2 d d 1 ω2 iω u u χ = m2u2 + χ σ χ . (4.7) du du! 2 " 4 − a2# 2 − a 3 1 Next, by introducing φ± χ χ , we can define ξ± and ζ± through 1 2 ≡ ∓ ξ±(u) φ± . (4.8) ≡ ζ±(u)! In terms of these variables Eqs. (4.6)-(4.7) become d d u u ξ± = [m2u2 +(iω/a 1/2)2]ξ± , (4.9) du du! ± d d u u ζ± = [m2u2 +(iω/a 1/2)2]ζ± . (4.10) du du! ∓ The solutions of these differential equations can be written in terms of Hankel func- (j) tions H (imu), (j = 1,2), (see (8.491.6) of Ref. [10]) or modified Bessel functions iω/a±1/2 K (mu), I (mu) (see (8.494.1) of Ref. [10]). Hence, by using Eqs. (4.5) and iω/a±1/2 iω/a±1/2 (4.8), and imposing that the solutions satisfy the first-order Eq. (4.3), we obtain 8 K (mu)+iK (mu) iω/a+1/2 iω/a−1/2 0 f (u) = A   , (4.11) ω+ + K (mu)+iK (mu) iω/a+1/2 iω/a−1/2  −   0      0 K (mu)+iK (mu) f (u) = A  iω/a+1/2 iω/a−1/2  . (4.12) ω− − 0    K (mu) iK (mu)   iω/a+1/2 − iω/a−1/2    Note that solutions involving I turn out to be non-normalizable and thus must iω/a±1/2 be neglected. In order to find the normalization constants 1/2 mcosh(πω/a) A = A = , (4.13) + − " 2π2a # we have used [9] (see also Eq. (3.4)) hψωσ,ψω′σ′i ≡ dΣµψ¯ωσγRµψω′σ′ = δ(ω −ω′)δσσ′ , (4.14) ZΣ where ψ¯ ψ†γ0 and Σ is set to be v = const. Thus the normal modes of the fermionic ≡ field (4.2) are K (mu)+iK (mu) iω/a+1/2 iω/a−1/2 1/2 mcosh(πω/a) 0 ψ =  e−iωv/a , (4.15) ω+ " 2π2a # Kiω/a+1/2(mu)+iKiω/a−1/2(mu)  −   0      0 1/2 mcosh(πω/a) K (mu)+iK (mu) ψ =  iω/a+1/2 iω/a−1/2 e−iωv/a. (4.16) ω− " 2π2a # 0    K (mu) iK (mu)  iω/a+1/2 − iω/a−1/2    As a consequence, canonical anticommutation relations for fields and conjugate momenta lead annihilation and creation operators to satisfy the following anticommutation relations {ˆbωσ,ˆb†ω′σ′} = {dˆωσ,dˆ†ω′σ′} = δ(ω −ω′) δσσ′ (4.17) and {ˆbωσ,ˆbω′σ′} = {dˆωσ,dˆω′σ′} = {ˆbωσ,dˆω′σ′} = {ˆbωσ,dˆ†ω′σ′} = 0 . (4.18) 9 V. RINDLER FRAME CALCULATION OF THE β- AND INVERSE β-DECAY FOR ACCELERATED NUCLEONS Now we analyze the β- and inverse β-decay of accelerated nucleons from the point of view of the uniformly accelerated frame. Mean proper lifetimes must be the same of the ones obtained in Sec. III but particle interpretation changes significantly. This is so because uniformly accelerated particles in the Minkowski vacuum are immersed in the FDU thermal bath characterized by a temperature T = a/2π [6]- [7]. As it will be shown, the proton decay which is represented in the inertial frame, in terms of Minkowski particles, by process (iii) will be represented in the uniformly accelerated frame, in terms of Rindler particles, as the combination of the following processes: (v) p+ e− n ν , (vi) p+ ν¯ n e+ , (vii) p+ e−ν¯ n . → → → Processes (v) (vii) are characterized by the conversion of protons in neutrons due to − the absorption of e− and ν¯, and emission of e+ and ν from and to the FDU thermal bath. Note that process (iii) is forbidden in terms of Rindler particles because the proton is static in the Rindler frame. Let us calculate firstly the transition amplitude for process (v): p→n = n ν Sˆ e− p , (5.1) A(v) h |⊗h ωνσν| I | ωe−σe−i⊗| i whereSˆ isgivenbyEq.(3.7)withγµ replacedbyγµ andourcurrent isgivenbyEq.(2.6). I R Thus, we obtain [we recall that in Rindler coordinates uµ = (a,0)] G +∞ p→n = F dv ei∆mv/a ν Ψˆ†(v,a−1)Ψˆ (v,a−1) e− , (5.2) A(v) a −∞ h ωνσν| ν e | ωe−σe−i Z where we note that the second term in the parenthesis of Eq. (3.7) does not contribute. Next, by using Eq. (4.2), we obtain G +∞ p→n = Fδ dv ei∆mv/aψ† (v,a−1) ψ (v,a−1) . (5.3) A(v) a σe−,σν −∞ ωνσν ωe−σe− Z Using now Eq. (4.15) and Eq. (4.16) and performing the integral, we obtain 4G Ap(v→)n = πaF memν cosh(πωe−/a)cosh(πων/a) q × Re Kiων/a−1/2(mν/a) Kiωe−/a+1/2(me/a) δσe−,σνδ(ωe− −ων −∆m) . (5.4) h i Analogous calculations lead to the following amplitudes for processes (vi) and (vii): 4G p→n = F m m cosh(πω /a)cosh(πω /a) A(vi) πa e ν e+ ν¯ q Re K (m /a) K (m /a) δ δ(ω ω ∆m) , (5.5) × iωe+/a−1/2 e iων¯/a+1/2 ν σe+,σν¯ ν¯− e+ − h i 10