Table Of ContentIFT-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, Sa˜o Paulo, SP
9
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Brazil
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a
J
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Abstract
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We investigate the decay of accelerated protons and neutrons. Calcu-
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lations are carried out in the inertial and coaccelerated frames. Particle
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c interpretation of these processes are quite different in each frame but the
q
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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-
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clusions are not conceptually affected by this.
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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 differen-
tial 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 acceler-
ated 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 equa-
tion 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 rela-
tions
{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
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