Neutrino and antineutrino oscillations from decay of Z boson

0.14 MB·English

by  I. M. Pavlichenkov

Neutrino and antineutrino oscillations from decay of Z0 boson
I. M. Pavlichenkov1,∗
1Russian Research Center Kurchatov Institute, Moscow, 123182, Russia
(Dated: January 17, 2012)
The two-particle wave function of neutrino and antineutrino after the decay of a Z0 boson is
found as a solution of an initial value problem. The wave packetof thisstate involvesthe coherent
superposition of massive neutrinos and antineutrinos resulting in independent oscillations of their
flavor counterparts. Possible experiments to observe oscillations of a single lepton or both leptons
from thesame decay or the independentlyemitted pairs νeν¯e, νµν¯µ and ντν¯τ are considered.
PACSnumbers: 14.60.Pq,13.38.Dg,03.65.Ud
2
1
The standard theories of massive neutrino oscillations ing the time evolution of the boson and the neutrino-
0
consider the time-evolution of a neutrino independently antineutrinopairinthe courseofthedecayprocess. The
2
froma recoil. Inspite ofthe fact that almostallofthese wavefunctionwhichdescribesthespatiotemporalbehav-
n
phenomenologicaltheoriesprovidethe canonicalformula ior of the pair is obtained at times t > 1/Γ, where Γ is
a
for the probability of oscillations, some important ques- one half of the rate of the decay. In the far zone approx-
J
tions arise which are discussed in Ref. [1]. It has been imations this function is represented as a product of the
5
showninmyrecentwork[2]that,inatwo-particledecay, relativeandthecenterofmasswavefunctions,butitdoes
1
noninteracting particles, a neutrino and a recoil, do not not factorize in the neutrino, rν, and antineutrino, rν¯,

] evolve separately due to quantum correlation between spatial coordinates. Thus neutrino and antineutrino are
h
the space-like separated objects. Because of this the de- entangled. It will be shown that the coherent superpo-
p
cay energy converts to the kinetic energy of the recoil- sition of the neutrino and antineutrino mass eigenstates
-
p neutrino pair. Thus, I have rigorously proved that the has a fixed kinetic energy equal to the decay energy.
e neutrino mass eigenstates produced in the electron cap- The decaying system is described by the Hamiltonian
h
ture decay have the same energy. The question remains:
[ pˆ2
does this result still stand for other two-body weak de- H = +H0+Hw, (1)
1 cays? 2M
v
where pˆ is the bosonmomentum operator, H is the un-
3 InthiscommunicationIapplytheformalismdeveloped 0
0 in Ref. [2] to the description of the decay of a Z0 boson, perturbed Hamiltonian of the boson, neutrino and an-
1 Z0 ν +ν¯ ,whereν istheneutrinowithafixedflavor tineutrino fields, respectively, and the weak interaction
α α α
3 (α=→e,µ,τ). Inspiteoftheexoticsourceandthelongos- Hamiltonian is given by
.
1 cillation length compared with the Earth-Moondistance M (k,p)
20 tghleisd ppraorbtilcelmes,hnaesutprhinyosicaanldinatnetriensetutbreincaou,smeatywoosceinlltaatne-. Hw=i,j,p,k f√a2M Uα∗iUαjbpc+ip/2+kdjp/2−k+h.c., (2)
X
1
This is yet another peculiar realization of the Einstein-
: where U is the mixing matrix, b, c and d are the second
v Podolsky-Rosen thought experiment [3]. The question
quantizedoperatorsoftheboson,thethree(i,j =1,2,3)
i as to whether neutrino oscillations are observable in the
X massive neutrino and antineutrino, respectively. The
invisible Z0-decay has been considered by Bilenky and
r transition matrix element from the initial state a to the
Pontecorvo [4] and Smirnov and Zatsepin [5]. However,
a final one f has the form
the consensus among these authors on this subject does
i
not exist. Let us note also that Z0-decay differs notice- Mfa=−2gZεσu¯L(k+p/2)γσ21(1−γ5)vR(−k+p/2), (3)
ablyfromtheelectroncapturedecaybecauseofthesmall
parameter α = Q/M (where Q is the decay energy and where gZ is the coupling constant, uL and vR are the
M is the parent particle mass) used in Ref. [2] is equal Dirac spinors for the left-handed neutrino and the right-
to 1. handed antineutrino, and the Z0-boson transverse po-
larization ε = (0,1,i,0)/√2 corresponds to the state
σ
The analysis presented below is based on the solution −
in which its spin is directed along the axis z of the rest
of the time-dependent Schr¨odinger equation with initial
frame.
wavefunctionofadecayingZ0bosontakenintheformof
Thetime-dependentperturbationtheoryisusedtode-
afinite-sizewavepacketinitsrestframe. TheWeisskopf-
termine the temporal evolution of a decaying state. The
Wigner theory of spontaneous emission [6] is used to
solution of the Schr¨odinger equation with the Hamilto-
solvethecoupledsystemofdifferentialequationsdescrib-
nian (1) is sought by using the following ansatz for the
wave function
Ψ(t)= A(p,t)b+ 0 e−iEat
∗Electronicaddress: [email protected] p p| i
X
2
+ Bij(p,k,t)c+ip/2+kdjp/2−k|0ie−iEijt, (4) egxivpe(itkwro).teTrhmesrpemroapionrintigonfuanlcttoioenxpis(iekars)ilyanidnteegxrpa(tediktro)
i,Xj,p,k correspondingtooutgoingandincomingsphericalw−aves.
where 0 is the vacuum state and the eigenvalues of the Inthefarzone,theincomingwavegivesanexponentially
| i
non-perturbed Hamiltonian are small contribution which can be neglected [7].
For the outgoing wave, the vectors k and r are paral-
p2 lel and their direction is determined by the unit vector
E =M+ , E =ǫ (k+p/2)+ǫ (k p/2), (5)
a 2M ij i j − n(θ,ϕ) = r/r, which is the characteristic feature of a
far zone. Thus stated, the result of the integration of
where ǫ (k) = k2+m2 is the energy of the massive
i i Eq. (10) over dΩk is given by
neutrino or antineutrino with the mass m . The coeffi-
i
p
cients Bij(p,k,t) can be found in Weisskopf-Wigner ap- Ψ(R,r,t)= iMfa(n) A (p)eipRU∗ U ν ν¯
proximation just as they where in Ref. [2] (2π)2√2Mr 0 αi αj| i ji
i,j,p
X
A (p)M U∗ U
Bij(p,k,t)=√2M0 (E faEαi+αijΓ) 1−ei(Eij−Ea)t−Γt , ∞exp ikr i 2ǫij(k)+p2/4ǫij(k) t kdk
ij a − , (11)
− h i(6) × 2ǫ (k) M+p2/4ǫ (k) p2/2M +iΓ
Z ij(cid:8) − (cid:2) ij − (cid:3) (cid:9)
where the initial value of the coefficient A, 0
wherethe smallparameterp/M hasbeenused. The ma-
1
A (p)=(2√πd)3/2exp d2p2 , (7) trixelement(12)inthisapproximationtransformstothe
0
−2
(cid:18) (cid:19) form
isthe bosonwavefunctioninthe momentumrepresenta- g M
tionwhichdescribesbosonspatiallocalizationbeforede- Mfa(n)= Z (1 cosθ)eiϕ. (12)
− 2√2 −
cay. TheonehalfoftherateofthetransitionZ0 ν +ν¯
α α
→
is given by Theenergyǫ (k)dependsonthemassterm(m2+m2)/2.
ij i j
π The remaining integral in (11) is calculated using the
Γ= Uαi 2 Uαj 2 Mfa(k,p)2δ(Eij Ea), (8) residue method by changing the variable of integration
2M | | | | | | −
i,j,k from k to ǫ and expanding the former aroundthe half of
X
the decay energy M/2
The small dependence of Γ on the boson velocity is ig-
noredbecausep k,andultrarelativisticapproximation 1
≪ k(ǫ)=k + (ǫ M/2), (13)
reduces this expression to a conventionalvalue ij
v −
ij
g2M
Γ= Z =86MeV. (9) where kij and the group velocity of neutrinos, vij =
192π 2k /M, are taken at the energy M/2. The lower limit
ij
of integration in (11) can be extended to (since the
Now one can write the spatial part of the neutrino- −∞
contribution to the integral falls of sharply with increas-
antineutrino wave function in the coordinate representa-
ing ǫ owing to M Γ) and the integrant is continued
tion. With Bij taken from Eq. (6) at times t>1/Γ, it | | ≫
analytically in complex plane ǫ. In the lowest order of
has the form
the small parameter p/M the pole of the integrand is
1 A (p)M (k,p)U∗ U ν ν¯
Ψ= 0 fa αi αj| i ji
√2Mi,Xj,p,kǫi(k+p/2)+ǫj(k−p/2)−M−2pM2 +iΓ ǫp =(M −iΓ)/2, (14)
and the integral is evaluated straightforwardly
exp ikr+ipR i[ǫ (k+p/2)+ǫ (k p/2)]t , (10)
i j
× { − − } k (ǫ)eiP(ǫ)dǫ 2iπk
ij = ij eiP0Θ(2v t r). (15)
(wRhMer)earn=d Rrν=−(rrν¯ν i+s rthν¯e)/c2oiosrdthineatceenotferreolfatmivaessm(oCtiMon) CI vij(ǫ)(ǫ−ǫp) − vij ij −
one.
Here the contour C encloses the pole (14). The unit
This function is inconvenientto considerthe entangle-
step function Θ is due to the requirement of the integral
ment and neutrino oscillations, because it involves inte-
convergence on the contour C. The exponent P(ǫ) is
gration over k and p. The analytical integration over
defined as follows
k can be performed only approximately. The key ap-
proximationisreferredtoasthe farzoneapproximation, P(ǫ)=[k +(ǫ M/2)/v ]r 2ǫ+p2/4ǫ t, (16)
ij ij
which means that kr Mt > M/Γ 1. In this ap- − −
∼ ≫
proximation the main contribution to the integral over and its value in the pole can be tra(cid:0)nsformed to(cid:1)the form
dΩ is given to those k close to the direction of r. In
k
linewiththisassumption,onecanputkparallelrevery- p2t iΓ
P = Mt+k r + (2v t r). (17)
where in the integrand of Eq. (10) except in the factor 0 − ij − 2M 2v ij −
ij
3
Now, the wavefunction of relative motion can be con- wheret =Md2 isitsspreadingtime. Thistimeisless
CM
structed by using Eqs. (11), (15) and (17). In doing so, than the flight time if d<10−3 cm.
it is convenient to express the coupling constant g in Thetotalwavefunctiondescribingtheevolutionofthe
Z
terms ofΓ (9). As a resultthe RM wavefunction canbe neutrino-antineutrinopairafterdecayisaproductofthe
written as CM and RM parts
ψ(n,r,t)=F(n) Rij(r,t)Uα∗iUαj|νiν¯ji, (18) Ψ(R,r,t)=ΨCM(R,t)ψ(n,r,t)exp(−iMt). (25)
i,j
X It is seen that the energy of this state is equal to the
where its radial part involves the functions decayenergy,Q=M. Thisisageneralfeatureofthebi-
partite decay into noninteracting fragments constrained
√Γ 2v t r only by momentum and energy conservation. The wave
ij
Rij(r,t)= exp ikijr − Θ(2vijt r), (19) packet ψ(R,r,t) 2 increaseswith time in a longitudinal
r − D −
(cid:18) ij (cid:19) directio|n (along th|e vector r) with the velocity 2 and in
depending on r and t, and the angular one, a transverse one with the velocity 1/(2π3/2Md). Hence,
the spatiotemporal behavior of the joint quantum state
3 oftheneutrinoandtheantineutrinofollowingZ0-decayis
F(n)= (1 cosθ)eiϕ, (20)
in agreementwith the results obtained in the theoretical
16π −
r
studies of the electron capture decay [2]. The distinctive
depends on the orientation of the vector r. The func- featureofoursystemisthecoherentsuperpositionofthe
tion(18)isnormalizedintheultrarelativisticlimit,when two leptons with flavor mixing, which both oscillate. To
vij =1 and Rij=R0eikijr, where describethis phenomenon,itis necessarytoconsiderthe
two-particlewavefunction (25) in the observable coordi-
√Γ natesoftheneutrinoandtheantineutrino. This normal-
R (r)= exp[ Γ(t r/2)]Θ(2t r). (21)
0 r − − − ized function has the form
The function ψ describes a coherentsuperposition of ex- Ψ(r ,r ,t)=Ψ rν+rν¯,t ψ(n,r r ,t)e−iMt. (26)
ν ν¯ CM ν ν¯
ponential wave packets of massive neutrino-antineutrino 2 −
(cid:18) (cid:19)
pairs with different sharp edges at r = 2v t and widths
ij
D = 2v /Γ 10−13 cm. The later depend only on The function does not factorize in these variables – a
ij ij
∼ direct indication of the spatial entanglement of leptons.
the dynamics of the decay process and may change later
Each of the three massive neutrinos is entangled with
duetodispersion. Toestimatesuchspreadingeffect,itis
three massive antineutrinos.
necessary to extend the series expansion of the function
It is instructive to consider the recoilless decay of an
k(ǫ) in Eq. (13) up to the secondorderin ǫ M/2. This
− infinitelyheavymotherparticle(M ). Settingp=0
gives rise to an additional integral in Eq. (11) as shown
→∞
in Eq. (5) and performing exactly the same calculations
in Ref. [2]. The calculations described in this work gives
asabovewith A =1,onecanobtainthe functionofthe
the spreading time of the RM wave packet 0
pair
M3D2
tRM = 8vij(m2i +ijm2j) ∼100s. (22) Ψ∞(rν,rν¯,t)=δ rν+2rν¯ ψ(n,rν−rν¯,t)e−iMt. (27)
(cid:18) (cid:19)
The time required for a neutrino or an antineutrino This state is also entangled because massive neutrinos
to propagate the distance comparable with oscillation andantineutrinosarenotemittedinamomentumeigen-
length, ”flighttime”, is 10 times smaller thantRM. This stateduetothefactthattheinteractionHamiltonian(2)
allows to use the sharpedge wave packets for analysis of commutes only with the total momentum.
neutrino oscillations in Z0-decay. The spatial part of the neutrino-antineutrino wave
TheintegrationoverdpinEq.(11)canbereadilyper- packet (26) for fixed t is proportional to a product of
formed by using expressions (7) and (17). The resulting the Gaussian and exponential functions
CMwavefunctionhasthestandardformforthedecaying
bipartite system (r +r )2 r r
Ψ(r ,r )2 exp ν ν¯ +2| ν− ν¯| , (28)
1 R2 | ν ν¯ | ∼ − 4D2 D
Ψ (R,t)= exp . (23) (cid:26) R (cid:27)
CM π3/4 d+Mitd 3/2 "−2d d+Mitd # whereDij D =2/Γinthe ultrarelativisticapproxima-
≈
Thisisaspreading(cid:0)wavepa(cid:1)cketwiththe(cid:0)time-de(cid:1)pendent tion. The probability distribution of the two-particleco-
ordinatesdepends on r , r and the angle betweenthese
width ν ν¯
vectors. Therefore, the packet has an axially symmetri-
DR(t)="d2+(cid:18)Mtd(cid:19)2#1/2=(cid:26)t/Md,d, tt≪≫ttCCMM , (24) cCthaMaltsihpnarpothbeeawbdiitliirhteycrteriesopancehcotefsttoihtesthvpeeecaatkoxriwsnhp.eansIsttihniesgvetaehcsritoloyurgsthohatshveeee
4
equal lengths and are located on the symmetry axis an- inperfectanalogytoabovecalculations,andhastheform
tiparallel to each other. Hence, the daughter particles
aretravelingback-to-backintherestframeofZ0 aspro- dPα¯β¯ =W (2t)(1+cosϑ +cos2ϑ )
αβ ν¯ ν¯
vided by the classical picture. However,the decay is not dr dΩ
ν¯ ν¯
isotropic because of the spin polarization of the boson.
The preferred direction in which the particles travel is 3r2 r2
ν¯ exp ν¯ , (34)
determined by the factor F(n)2 (1 cosθ)2, which is × 4π3/2D3(t) −D2(t)
| | ∼ − R (cid:18) R (cid:19)
to say the antineutrino travelsmainly in the directionof
the boson spin. because Wα¯β¯=Wαβ.
Thestructureoftheneutrino-antineutrinowavepacket It is seen that the probability densities (33) and (34)
ofthe function(26)suggeststwokindsofexperimentsto involve the amplitude of oscillations depending on the
observe flavor oscillations. spatio-temporal localization of a neutrino. It owes its
Neutrino and antineutrino oscillations in one detec- origin to the evolution of an unstable localized system
tor experimen. The first is a well-establishedexperiment during its decay. Obviously, the amplitude can not be
for detecting neutrino or antineutrino oscillation. Sup- foundwithoutrigoroustreatmentofthedecayprocessas
pose that only a neutrino is detected regardless of the describedabove. Theprobabilityoftransitionisobtained
antineutrino position. In the detection of a flavor neu- by integration of (33) or (34) over a spatial coordinate,
trino, when the position of a detector is scanned, such and it is equal to Wαβ(2t).
measurementsallowstodeterminetheoscillationpicture. The detection time t is equal to distance between the
Theflavor-changingtransitionνα νβ isdeterminedby detector and the source, t = rν = r/2. Therefore, the
the probability density → oscillation length, associated with ∆m2 , is
ij
ddPrανβ =Zdrν¯(cid:12)Xl hνl|UβlΨ(rν,rν¯,t)(cid:12)2. (29) Lij=2π|∆Mm2ij|, (35)
(cid:12) (cid:12)
Changing the variable(cid:12)of integration from r(cid:12) to r, one where the neutrino energy is obviously equal to M/2.
ν¯
gets the integral Two conditions are necessary for the neutrino oscilla-
tions to be observed. The first condition is the coher-
dr (r+2r )2 2r
exp ν + F(θϕ)2W (r)Θ(2t r), ence of different mass eigenstates. Coherence is pre-
r2 − 4D2 D | | αβ − servedoverdistances notexceeding the coherencelength
Z (cid:20) R (cid:21)
(30) L . The latter is defined as the distance at which the
coh
where phase difference due to energy spreading of a neutrino,
δǫ = Γ, obeys the equation (∂φ/∂ǫ) δǫ = 2π, which
Wαβ(r)=δαβ 4 UαiUβiUαjUβjsin2(φij(r)) (31) gives L = L Q/Γ 103L . Themsecond condition
− coh osc osc
∼
Xi,j requires that at a distance, which is comparable with
the oscillation length, the spatial separation of the wave
is the probability of the neutrino flavor transition ν
α → packetscorrespondingtodifferentmasseigenstateswould
ν in the case of the CP invariance and
β beconsiderablylessthanthepacketwidth. Such,indeed,
m2 m2 is the case
φ (r)= i − jr (32)
ij
2M L δv ΓL ∂v Γ
osc = osc ∆m2 =π 1. (36)
vD 2v2 ∂m2 Q ≪
is the oscillation phase. The integrand of Eq. (30) has a (cid:12)(cid:18) (cid:19)ǫ(cid:12)
(cid:12) (cid:12)
sharp maximum at r = 2t, and integration can be done (cid:12) (cid:12)
Hence, the spatial s(cid:12)eparation(cid:12) of the massive neutrino
approximatelysincethewidthDoftheexponentialfunc-
wave packets does not affect the observation of neutrino
tion is much smaller than that of D (t) of the Gaussian
R oscillations in any two-particle decay.
one, if Γt 2d/λ , where λ is the Compton wave-
length of th≫e bosonZ(Γ 1023Zs−1). The integration is Allofthe aboveappliesto the decayofa singleZ0 bo-
∼ son. InthecaseofasteadyfluxofZ0’swhosedecayspro-
straightforward, albeit somewhat tedious, and its result
videtheincoherentfluxesI ofthe threetypesofneutri-
is α
nos(α=e,µ,τ),theintensityofneutrinoβatananydis-
dP tance from source is proportional to I W (r). Be-
αβ =W (2t)(1 cosϑ +cos2ϑ ) α α αβ
dr dΩ αβ − ν ν cause the probabilities of emission of different neutrino-
ν ν P
antineutrinopairsareequal,I =I ,thesumisequalto
α 0
3r2 r2 I according to Eq. (31). Hence, the neutrinos (as well
ν exp ν , (33) 0
× 4π3/2D3(t) −D2(t) as antineutrinos) do not oscillate, which is in agrement
R (cid:18) R (cid:19) with the result of Ref. [4].
whereϑ is theanglebetweenthe outgoingneutrinoand Neutrino and antineutrino oscillations in coincidence
ν
the direction of the Z0 boson spin. The probability den- measurements. The term ”coincidence” means that the
sity of the oscillationtransitionν¯ ν¯ may be derived flavors of both leptons from the same decay are fixed by
α β
→
5
twodetectors. The flavor-changingprocessν ν¯ ν ν¯ due to mixing of signals from two detectors. In order
α α β γ
→
is determined by the probability density to observe the independent oscillations of neutrinos and
antineutrinos, it is necessary to synchronize the time on
dPαα¯→βγ¯ = U U∗ ν ν¯ Ψ(r ,r ,t) 2. (37) two detectors to make sure that the leptons come from
drνdrν¯ βi γj i j| ν ν¯ one decay.
(cid:12)(cid:12)Xi,j (cid:10) (cid:12)(cid:12)
(cid:12) (cid:12)
By using Eq. (21) and (31), one finds
In summary, an accurate analytical solution for the
dPαα¯→βγ¯ rν+rν¯ 2 joint quantum state of flavor neutrino and antineutrino
= Ψ ,t
dr dr CM 2 following the decay of a Z0 boson has been found. The
ν ν¯
(cid:12) (cid:16) (cid:17)(cid:12)
(cid:12) (cid:12) evolutionofthe stateprovidesyetanotherexactlycalcu-
F(θϕ)2R2((cid:12)r,t)W (r)W ((cid:12)r). (38) lable illustration of the famous Einstein-Podolsky-Rosen
×| | 0 αβ αγ
thought experiment. The new effect is the entanglement
The highest output is obtained when the detectors are between two oscillating leptons. It is shown that the
locatedonthesymmetryaxisatequaldistancesfromthe oscillations of neutrino and antineutrino from the same
sourcer =r =t. Forthisconfiguration,theoscillation decay proceed independently and may be observed by
ν ν¯
amplitude is measuringtheflavorofbothparticles. Massiveneutrinos
andantineutrinosarenotemittedinastatewithdefinite
(1 cosθ)2Γr2 (1 cosθ)2Γ d 3 momentum, however the energy of the two-lepton wave
− ν = − . (39)
64π5/2D3(r ) 64π5/2r λ packet is fixed and is equal to the decay energy. Hence
R ν ν (cid:18) Z(cid:19)
it can be argued that the neutrino mass eigenstates pro-
This amplitude corresponds to the highest spatial corre- duced in bipartite decay have the same energy.
lation of the two leptons at a distance equal to 2t, that
is in complete agreement with the classical picture of a
bipartite decay. Despite its purely academic interest, the solved prob-
The probability of detecting the neutrino νβ and the lem is essential to the understanding of the physics of
antineutrinoν¯γ fromthe decayZ0 να+ν¯α is obtained neutrino oscillations. Besides, the sources of neutrinos
→
by the integration of Eq. (38) from the Z0-decay, as well as detectors recording neutri-
nos from a single decay may well be conceivable in the
Wαα¯→βγ¯ = dPαα¯→βγ¯ =Wαβ(2t)Wαγ(2t). (40) future experiments. Accordingly, I have considered the
experiments allowing to observe oscillations of one lep-
Z
ton as well as both leptons from the decay of a single
Itisseenthataneutrinoandanantineutrinooscillatein-
boson. If three neutrino-antineutrino pairs are emitted
dependently,andtheiroscillationlengthsaredetermined
independently, the oscillations can be observed also by
byEq.(36). Ifthe pairsν ν¯ , ν ν¯ andν ν¯ areemitted
e e µ µ τ τ the two detectors connected into the coincidence circuit
independently, and their fluxes are equal, then the prob-
withoneoutputorsynchronizedintime. Asimilarresult
ability of observingν in one detector and ν¯ in another
β γ was obtained in Ref. [5]. However, the results of that
one will be equal
study are based on the erroneous assumption that Z0
boson decays into a coherent superposition of neutrino-
W = W W , (41)
βγ¯ αβ αγ antineutrino pairs with different flavors [8].
α
X
which is different from Eq. (40) for the decay of a single
boson. For the detectors connected to a coincidence cir- The work was supported by the Grant NS-215.2012.2
cuit with one output, the oscillation pattern is distorted from the Russian Ministry of Education and Science.
[1] E. K.Akhmedovand A.Yu.Smirnov,Phys.Atom.Nucl. [6] V. Weisskopf and E. Wigner, Z.Phys. 63, 54 (1930).
72, 1363 (2009). [7] Therigoroussubstantiationofthefarzoneapproximation
[2] I. M. Pavlichenkov,Phys. Rev.D 84, 073005 (2011). has been given in thework of theauthor [2].
[3] A. Einstein, B. Podolsky and N. Rosen, Phys. Rev. 47, [8] Thisisthesamemistakethathasbeenmadeinthearticle
777 (1935). A.N.IvanovandP.Kienle,Phys.Rev.Lett.103,062502
[4] S. M. Bilenky and B. Pontecorvo, Lett. Nuovo Cimento (2009). According to quantum mechanics, a coherent su-
41, 531 (1984). perposition can not arise from the decay of an unstable
[5] A. Yu. Smirnov and G. T. Zatsepin, Mod. Phys. Lett. A state.
7, 1273 (1992).