Article
Neutrino oscillations and uncertainty relations
Journal of Physics G Nuclear and Particle Physics (Impact Factor: 2.78). 02/2011; 38(11). DOI: 10.1088/09543899/38/11/115002
Source: arXiv
Fulltext
Available from: F. von Feilitzsch, Dec 20, 2013arXiv:1102.2770v1 [hepph] 14 Feb 2011
Neutrino oscillations and uncertainty relations
S. M. Bilenky
Joint Institute fo r Nuclear Research, Dubna , R14198 0, Russia
and
PhysikDepartment E15, Technische Universit¨at M¨unchen,
D85748 Garching, Germany
F. von Feilitzsch and W. Potzel
PhysikDepartment E15, Technische Universit¨at M¨unchen,
D85748 Garching, Germany
Abstract
We show that coherent ﬂavor neutrino states are produced (and
detected) due to the momentumcoordinate Heisenber g uncertainty
relation. The MandelstamTamm timeenergy uncertainty relation re
quires nonstationary neutrino states for oscillations to happen and de
termines the time interval (propagation length) which is necessary for
that. We compare diﬀerent approaches to neutrino oscillations which
are based on diﬀerent physical assumptions but lead to the same ex
pression for the neutrino transition probability in standard neutrino
oscillation experiments. We show that a M¨ossbauer neutrino experi
ment could allow to distinguish diﬀerent approaches an d we present
arguments in favor of the
163
Ho 
163
Dy system for such an experiment.
1 Introduction
The observation of neutrino oscillations in atmospheric [1], solar [2], reactor
[3] and accelerator experiments [4, 5] is one of the most important recent
discoveries in particle physics. Small neutrino masses can not be of Standard
Model orig in and are commonly considered as a signature of new physics
beyo nd the Standard Model.
All existing neutrinooscillation data with the exception o f the data of
the LSND [6] and MiniBooNE antineutrino experiments [7], which require
1
Page 1
conﬁrmation, are perfectly described under the assumption of threeneutrino
mixing
ν
lL
(x) =
3
X
i=1
U
li
ν
iL
(x). (1)
Here U is the PMNS [8, 9] 3 × 3 mixing matrix, which is characterized by
three mixing angles θ
12
, θ
23
, θ
13
and the CP phase δ, ν
i
(x) is the ﬁeld of
neutrinos (Dirac or Majorana) with mass m
i
, and the ”mixed ﬁeld” ν
lL
(x) is
the SM ﬁeld which enters into the standard charged current
j
α
(x) =
X
l=e,µ,τ
¯ν
lL
(x) γ
α
l
L
(x). (2)
Existing neutrinoo scillation data are analyzed under the assumption that
the t r ansition probabilities between diﬀerent ﬂavor neutrinos are given by
the following standard expression (see, for example, [10])
P(ν
l
→ ν
l
′
) = δ
l
′
l
− 2 Re
X
i>k
U
l
′
i
U
∗
li
U
∗
l
′
k
U
lk
(1 − e
−i
∆m
2
ki
L
2E
). (3)
Here, L is the distance between neutrino source and neutrino detector, E is
the neutrino energy, ∆m
2
ki
= m
2
i
− m
2
k
. Notice that it is also convenient to
use for the transition probability another expression
P(ν
l
→ ν
l
′
) = δ
l
′
l
− 2
X
i
U
li

2
(δ
l
′
l
− U
l
′
i

2
)(1 − cos
∆m
2
ji
L
2E
) (4)
+2 Re
X
i>k
U
l
′
i
U
∗
li
U
∗
l
′
k
U
lk
(e
−i
∆m
2
ji
L
2E
− 1 )(e
i
∆m
2
jk
L
2E
− 1),
where the index j is ﬁxed.
The character of neutrino oscillations is determined by the following two
observed features of the neutrinooscillation parameters:
• The solar KamLAND masssquared diﬀerence ∆m
2
S
is much smaller
than the atmosphericaccelerator masssquared diﬀerence ∆m
2
A
:
∆m
2
S
≃
1
30
∆m
2
A
. (5)
2
Page 2
• The mixing angle θ
13
is small [11]:
sin
2
θ
13
≤ 4 · 10
−2
. (6)
From (4), ( 5) and (6 ) follows (see, for example, [10]) that the leading
oscillations in the atmospheric and accelerator experiments are ν
µ
⇄ ν
τ
and
¯ν
µ
⇄ ¯ν
τ
and in solar and KamLAND experiments the leading oscillations
are ν
e
⇄ ν
µ,τ
and ¯ν
e
⇄ ¯ν
µ,τ
.
In the leading approximation it is impossible to distinguish two possible
neutrino mass spectra:
• Nor ma l spectrum
m
1
< m
2
< m
3
, ∆m
2
12
≪ ∆m
2
23
.
• Inverted spectrum
m
3
< m
1
< m
2
, ∆m
2
12
≪ ∆m
2
13
.
In the case of the normal spectrum ∆m
2
12
= ∆m
2
S
, ∆m
2
23
= ∆m
2
A
and in
the case of the inverted spectrum ∆m
2
12
= ∆m
2
S
, ∆m
2
13
= −∆m
2
A
.
From the recent threeneutrino analysis of the Super Kamiokande data
[1] the following 90% CL limits were found for the normal (inverted) neutrino
mass spectrum
1.9 (1 .7)·10
−3
≤ ∆m
2
A
≤ 2.6 (2.7)·10
−3
eV
2
, 0.407 ≤ sin
2
θ
23
≤ 0.583. (7)
For the parameter sin
2
θ
13
the following bounds were obtained
sin
2
θ
13
≤ 4 · 10
−2
(9 · 10
−2
). (8)
From the twoneutrino analysis of the MINOS data was found [5]
∆m
2
A
= (2.43 ± 0 .13) · 10
−3
eV
2
, sin
2
2θ
23
> 0.90 (9)
From the threeneutrino global analysis of the solar and reactor KamLAND
data was obtained [3]
∆m
2
S
= (7.50
+0.19
−0.20
) · 10
−5
eV
2
, tan
2
θ
12
= 0.452
+0.035
−0.032
(10)
3
Page 3
For the parameter sin
2
θ
13
was found
sin
2
θ
13
= 0.020
+0.016
−0.018
(11)
At present four neutrinooscillation parameters (∆m
2
S
, ∆m
2
A
, sin
2
2θ
23
and
tan
2
θ
12
) are known with accuracies within the (310 )% range. In the accel
erator neutrino oscillation experiment T2K [12] the parameter ∆m
2
A
will be
measured with an accuracy of δ∆m
2
A
< 10
−4
eV
2
and the parameter sin
2
2θ
23
will be measured with an accuracy of δ(sin
2
2θ
23
) ≃ 10
−2
. One o f the major
aims of this experiment and the reactor experiments DOUBLE CHOOZ[13],
RENO [14], and Daya Bay [15] is to determine the value (or to improve the
upper bound by one order of magnitude or better) of the parameter sin
2
θ
13
.
In case t hat this parameter is relatively lar ge, it is envisaged tha t in future
neutrino experiments the value of the CP phase δ will be determined and
the problem of the neutrino mass spectrum will be resolved ( see [12]).
Thus, we are entering into the era of high precision neutrino oscillation
exp eriments. Despite that the neutrino oscillation formalism, on which the
analysis of experimental data is based, has been developed and debated in
many papers starting fro m the 1970s (see reviews [16, 1 7]), these debates
and discussions are continuing (see recent papers [1 8]). From our point of
view the importa nce of uncertainty relations was not suﬃciently analyzed in
previous discussions. We will show here that the phenomenon of neutrino
oscillations is heavily based on the Heisenberg uncertainty relation and the
MandelstamTamm timeenergy uncertainty relat ion. We brieﬂy consider dif
ferent approaches to neutrino oscillations and discuss a M¨ossbauer neutrino
exp eriment which could allow to distinguish them.
2 Flavor neutrinos: production, evolution,
detection
Which neutrino states are pr oduced in CC weak processes together with
charg ed leptons in the case of neutrino mixing, eq.(1): Neutrino ﬂavor states,
coherent superpositions of plane waves, or superpositions of wave packets?
Here we will present arguments based on t he QFT, the Heisenberg uncer
tainty r elat ion and the knowledge of the neutrino masssquared diﬀerences
that ”mixed” ﬂavor states which describe the ﬂavor neutrinos ν
e
, ν
µ
and ν
τ
are physical states (fully analogous to the ”mixed” states which describe K
0
and
¯
K
0
, B
0
and
¯
B
0
, etc.).
4
Page 4
Let us consider (in the lab. system) the decay [19]
a → b + l
+
+ ν
i
, (i = 1, 2, 3) (12)
where a and b are some hadr ons.
The state of the ﬁnal particles is given by
fi =
X
i
ν
i
il
+
ib ihν
i
l
+
bSai, (13)
where hν
i
l
+
bSai is the matrix element of the transition a → b + l
+
+ ν
i
,
ν
i
i is the state of a neutrino with mass m
i
, momentum ~p
i
= p
i
~
k (
~
k is the
unit vector) and helicity equal to 1. We assume, as usual, that initial and
ﬁnal particles have deﬁnite momenta.
Because neutrino masses are small, we can use the expansion
p
i
=
q
E
2
i
− m
2
i
≃ E −
m
2
i
2E
, (14)
where E is the energy of neutrinos for m
2
i
→ 0. For the diﬀerence of neutrino
momenta we have
p
i
− p
k
 ≃
∆m
2
ki

2E
=
2π
L
r
osc
, (15)
where
L
r
osc
= 4π
E
∆m
2
r
≃ 2.48
(E/MeV)
(∆m
2
r
c
4
/eV
2
)
m, r = A, S (16)
is the oscillation length. For E ≃ 1 GeV and ∆m
2
A
≃ 2.4 · 10
−3
eV
2
(atmo
spheric and LBL accelerator neutrinos) we have L
A
osc
≃ 10
3
km. For E ≃ 3
MeV and ∆m
2
S
≃ 7.5 · 10
−5
eV
2
(reactor antineutrinos) we have L
S
osc
≃ 10
2
km.
On the other side, from the Heisenberg uncertainty relation we have
(∆p)
QM
≃
1
d
. (17)
Here d characterizes the quantummechanical size of the source. Taking into
account tha t
L
A,S
osc
≫ d (18)
we have
p
i
− p
k
 ≪ (∆p)
QM
. (19)
5
Page 5
Thus, we conclude that due to the uncertainty relat ion it is impossible to
resolve the emission of neutrinos with diﬀerent masses.
1
The operator
X
k
U
∗
lk
¯ν
kL
(x)γ
α
l
L
(x) (20)
determines the leptonic part of the matrix element of t he process (12). We
have
U
∗
li
¯u
L
(p
i
)γ
α
u
L
(−p
l
) ≃ U
∗
li
¯u
L
(p)γ
α
u
L
(−p
l
), (21)
where p
l
is the momentum o f l
+
, and p = E is t he momentum of the neutrino
for m
2
i
→ 0. For the total matrix element of the process (12) we have
hν
i
l
+
bSai ≃ U
∗
li
hν
l
l
+
bSai
SM
, (22)
where hν
l
l
+
bSai
SM
is the Standard Model matrix element of the emission
of the ﬂavor neutrino ν
l
with the momentum p in t he process
a → b + l
+
+ ν
l
. (23)
From (13) and ( 22) we ﬁnd
fi = ν
l
il
+
ib ihν
l
l
+
bSai
SM
, (24)
where the state of the ﬂavor neutrino ν
l
is given by the relation
ν
l
i =
X
i
U
∗
li
ν
i
i (l = e, µ, τ) (25)
and ν
i
i is the state of a neutrino with mass m
i
, negative helicity and mo
mentum p.
2
Let us stress that
• F lavor neutrino states do not depend on the production process.
1
For the energy of a neutrino with mass m
i
we have E
i
≃ E(1 +
m
2
i
2E
2
). In neutrino
oscillation experiments E & 1 MeV and
m
2
i
2E
2
. 10
−12
. Thus , it is impossible to re solve
diﬀerent neutrino energies in production (and de tection) processes.
2
Let us notice that the theory of the evolution of neutrinos in matter and the MSW
eﬀect [20, 21] are based on the assumption that a ﬂavor neutrino state is a state with
deﬁnite momentum.
6
Page 6
• F lavor states are characterized by the momentum (if there ar e no spe
cial conditions of neutrino production).
• F lavor states are orthogonal a nd no rmalized
hν
l
′
ν
l
i = δ
l
′
l
. (26)
The evolution of states in QFT is given by the Schr¨odinger equation
i
∂ Ψ(t)i
∂t
= H Ψ(t)i, (27)
where H is the total Hamiltonian and time t is a para meter bot h of which
chara cterize the evolution of the system.
If at t = 0 in a CC weak process ν
l
is produced, we have for the state of
the neutrino at the time t
ν
l
i
t
= e
−iHt
ν
l
i =
X
i
ν
i
ie
−iE
i
t
U
∗
li
, (28)
where
Hν
i
i = E
i
ν
i
i, E
i
≃ E +
m
2
i
2E
. (29)
Neutrinos are detected via the observation of weak CC and NC processes.
Let us consider the production of a lepton l
′
in the CC process
ν
i
+ N → l
′
+ X. (30)
Taking into account that eﬀects of neutrino masses can not be resolved in
neutrino processes we have
hl
′
XSν
i
Ni ≃ hl
′
XSν
l
′
Ni
SM
U
l
′
i
, (31)
where hl
′
XSν
l
′
Ni
SM
is the SM matrix element of the process
ν
l
′
+ N → l
′
+ X. (32)
From (24), (28) and (31) follows that the chain of processes a → b + l
+
+
ν
l
, ν
l
→ ν
l
′
, ν
l
′
+ N → l
′
+ X corresponds to the following factorized
product of amplitudes
hl
′
XSν
l
′
Ni
SM
X
i
U
l
′
i
e
−iE
i
t
U
∗
li
!
hb l
+
ν
l
Sai
SM
. (33)
7
Page 7
Only the amplitude of the transition ν
l
→ ν
l
′
A(ν
l
→ ν
l
′
) =
X
i
U
l
′
i
e
−iE
i
t
U
∗
li
(34)
depends on the properties of massive neutrinos (masssquared diﬀerences
and mixing angles). The matrix elements of the neutrino production and
detection are given by the Standard Model expressions in which eﬀects of
neutrino masses can safely be neglected. Let us stress that the property of
the fa cto rization (33) is based on the smallness of the neutrino masses and
on the Heisenberg uncertainty relation.
3 MandelstamTamm uncertainty relation
and neutrino oscillations
All uncertainty relations in Quantum Theory ar e based on the inequality
∆A ∆B ≥
1
2
ha[A, B]ai (35)
which follows from the Cauchy inequality. In (35) A and B are hermitian
operators, ai is any state, ∆A =
q
ha(A − A)
2
ai is the standard deviation
and A = haAai is the average value of the operator A. For example, for
operators of momentum p and coordinate q which satisfy the commutation
relation [p, q] =
1
i
we have the Heisenberg uncertainty relation ∆p ∆q ≥
1
2
.
The MandelstamTa mm timeenergy uncertainty relation [22] is based on
the inequality (35) and the equation
i
∂O(t)
∂t
= [O(t), H] (36)
for any operator O(t) in the Heisenberg representation (H is the to tal Hamil
tonian).
From (35) and (36) we have
∆E ∆O(t) ≥
1
2

d
dt
O(t) (37)
This inequality gives nontrivial constraints only in the case of nonstationary
states.
8
Page 8
Taking into account that ∆E does not depend on t we ﬁnd
∆E ∆t ≥
1
2
O(∆t) − O(0)
∆O(
¯
t)
(38)
For the time interval ∆t during which the state of the system is signiﬁcantly
changed (
O(t) is changed by the value which is characterized by the standard
deviation) t he righthand part of (38) is of the order of one. We obtain the
MandelstamTamm t imeenergy uncertainty relation
∆E ∆t & 1. (39)
From (34), f or the normalized probability of the transition ν
l
→ ν
l
′
we
obtain the expression
P (ν
l
→ ν
l
′
) = 
X
i6=j
U
l
′
i
(e
−i(E
i
−E
j
)t
− 1) U
∗
li
+ δ
l
′
l

2
, (40)
which obviously gives the standard transition probability (3).
From (40) follows that neutrino oscillations can be observed if the condi
tion
E
i
− E
j
 t & 1 (41)
is satisﬁed.
3
It is obvious that this inequality is the MandelstamTamm
timeenergy uncertainty relatio n. According to this relation a change of
the ﬂavo r neutrino state in time requires energy uncertainty (i.e., a non
stationary state). The time interval required for a signiﬁcant change of the
ﬂavor neutrino state is given by t ≃
1
E
i
−E
j

=
2E
∆m
2
ji

.
4
4 On plane wave and wave packet approaches
to neutrino oscillations
We will now brieﬂy discuss other approaches to neutrino oscillations. In the
approach based on the relativistic quantum mechanics, in CC pro cesses to
gether with charged leptons coherent superpositions of plane waves are
3
This is a necessary condition fo r the o bservation of oscillations. It is also necessary
that mixing angles would be relatively large.
4
Let us notice that the inequality (41) can be interpreted in a nother way: In order
to reveal a small energy diﬀerence E
i
− E
j
 ≃
∆m
2
ji

2E
we need a large time interval
t &
1
E
i
−E
j

. This corresponds to another interpretation of the timeenergy uncertainty
relation (see [23]).
9
Page 9
produced and absorbed. In this case, for the normalized ν
l
→ ν
l
′
transition
probability the fo llowing expression can be obtained (see, for example[24, 25])
P (ν
l
→ ν
l
′
) = 
X
i
U
l
′
i
e
−ip
i
·x
U
∗
li

2
= 
X
i6=j
U
l
′
i
(e
−i(p
i
−p
j
)·x
−1)U
∗
li
+ δ
l
′
l

2
. (42)
Here p
i
= (E
i
, ~p
i
) is the 4momentum of a neutrino with mass m
i
and x =
(t, ~x).
Let us assume that ~p
i
= p
i
~
k, where
~
k is the unit vector. For the phase
diﬀerence which is gained by a plain wave at the distance x = (~x
~
k) = L after
the time interval t we have
(p
i
− p
j
) · x = (E
i
− E
j
)t − (p
i
− p
j
)L. (43)
For ultrarelativistic neutrinos we have
t ≃ L. (44)
Taking into account that E
i
≃ p
i
+
m
2
i
2E
, from (43) and (44) we come to the
standard oscillation phase
(p
i
− p
j
) · x =
∆m
2
ji
2E
L (45)
and the standard expression (4) for the transition probability.
Let us stress that in the approach based on the QFT Schr¨odinger equation
the small oscillation phase diﬀerence is the result of the cancellation of large
terms in the expressions for the neutrino energies. The cancellation takes
place because neutrino states are characterized by deﬁnite momentum. In
the QM plane wave approach, small oscillation phases are the result of the
cancellation of large terms in the time and space parts of the phase diﬀerence.
The cancellation is due to the relation (44).
A direct generalization of the QM plane wave approach is the wave
packet approach (see [25] and references therein) in which the plane wave
transition probability (42) is changed to
P (ν
l
→ ν
l
′
) = 
X
i
U
l
′
i
Z
e
i(~p
i
′
~x−E
′
i
t)
f(~p
i
′
− ~p
i
) d
3
p
′
U
∗
li

2
, (46)
where E
′
i
=
p
(~p
i
′
)
2
+ m
2
i
and the function f(~p
i
′
−~p
i
) has a sharp maximum
at the point ~p
i
′
= ~p
i
.
10
Page 10
Expanding E
′
i
at the point ~p
i
′
= ~p
i
we ﬁnd
Z
e
i(~p
i
′
~x−E
′
i
t)
f(~p
i
′
− ~p
i
) d
3
p
′
= e
i(~p
i
~x−E
i
t)
g(~x −
~
v
i
t), (47)
where
g(~x −~v
i
t) =
Z
e
i~q (~x−~v
i
t)
f(~q) d
3
q (48)
and
~v
i
=
~p
i
E
i
, E
i
=
q
~p
i
2
+ m
2
i
. (49)
If we make the standard assumption that the function f(~q) has the Gaussian
form
f(~q) = N e
−
q
2
4σ
2
p
, (50)
(σ
p
is the width of the wave packet in the momentum space) we ﬁnd
g(~x −~v
i
t) = N(
π
σ
2
x
)
3/2
e
−
(~x−~v
i
t)
2
4σ
2
x
, (51)
where σ
x
=
1
2σ
p
chara cterizes the spacial width of the wave packet.
The probability o f the transition ν
l
→ ν
l
′
in the wave packet approach
is determined as a quantity integrated over time. From (47) we ﬁnd the
following expression for the integrated normalized transition probability
P(ν
l
→ ν
l
′
) =
X
i,k
U
l
′
i
U
∗
l
′
k
e
i[(p
i
−p
k
)−(E
i
−E
k
)]L
U
∗
li
U
lk
e
−(
L
L
ik
coh
)
2
e
−2π
2
ξ
2
(
σ
x
L
ik
osc
)
2
.
(52)
Here L is the distance between neutrino source and neutrino detector, L
ik
osc
is the oscillation length, ξ is a constant o f the order of one and
L
ik
coh
=
4
√
2σ
x
E
2
∆m
2
ik

. (53)
is the coherence length.
5
Taking into account that (p
i
− p
k
) − (E
i
− E
k
) =
−
∆m
2
ki
2E
we come to the conclusion that the ν
l
→ ν
l
′
transition probability in
5
We have v
i
− v
k
L
ik
coh
≃
∆m
2
ik

2E
2
L
ik
coh
∼ 2
√
2σ
x
. Thus, the coherence length is such
a distance between neutrino source and de tec tor at which ν
i
and ν
k
are separated by an
interval compa rable to the size of the wave packet.
11
Page 11
the wave packet approa ch is given by the standard expression (3) which is
multiplied by the decoherence factor e
−(
L
L
ik
coh
)
2
and the factor e
−2π
2
ξ
2
(
σ
x
L
ik
osc
)
2
.
Thus, the wave packet approach (after integration over t) assures the
equality t = L and the standard oscillation phase in the transition probability.
For usual neutrino oscillation experiments with L being a few times L
A,S
osc
, t he
two additional exponential factors are practically equal to one.
In many papers (see [1 8]), neutrinos pro pagating about 100 km (reactor
¯ν’s ) or about 1000 km (atmospheric and accelerator ν’s ), are considered
as virtual particles in a Feynman diagram like picture with the neutrino
production process at one vertex and the neutrino absorption process in
another vertex. This approach gives the wave packet picture of neutrino
oscillations with a transition probability which (befo r e integration over t)
depends on x and t.
The major diﬀerence between diﬀerent approaches to neutrino oscillations
can be summarized as follows:
1. The QF T approach with the Schr¨odinger evolution equation is based
on the assumption of the existence of ”mixed” ﬂavor neutrinos ν
e
, ν
µ
, ν
τ
which are described by coherent states  ν
l
i =
P
i
U
∗
li
ν
i
i. The import ant
chara cteristic feature of this approach is the MandelstamTamm time
energy uncertainty relation. Neutrino oscillations can take place only
in the case of nonstationary neutrino states with ∆E∆t & 1, where
∆t is the time int erval during which the oscillations happen. The QFT
approach is based o n the same general principles as the approach to
K
0
⇄
¯
K
0
, B
0
⇄
¯
B
0
, etc. oscillations studied in detail at B factories
and other facilities.
2. O t her approaches are based on the assumption that in weak processes,
mixed coherent superpositions of plane waves or wave packets describ
ing neutrinos with diﬀerent masses, are produced and detected. The
evolution o f mixed neutrino wave functions in space and time is de
termined by the Dirac equation. There is no notion of ﬂavor neutrino
states in these approaches. Neutrino o scillations are possible also in
the case of monochromatic neutrinos.
Diﬀerent approaches to neutrino oscillations lead to the same expression for
the neutrino transition probability P(ν
l
→ ν
l
′
) in the standard neutrino
oscillation experiments. In order to distinguish 1. and 2. special neutrino
12
Page 12
oscillation experiments are necessary. Such experiments could be M¨ossbauer
neutrino experiments which we will discuss in the next sections.
5 M¨ossbauer ¯ν
e
: Basic considerations
The basic concept is to use electron antineutrinos (¯ν
e
) which are emitted
without recoil in a boundstate βdecay and are resonantly captured again
without recoil in the reverse boundstate process. As an example, let us
consider the
3
H 
3
He system [26] with the transitions
3
H→
3
He +¯ν
e
(source) and ¯ν
e
+
3
He →
3
H (ta rget).
In the source, the electron (e
−
) is emitted directly into a boundstate
atomic orbit of
3
He. This decay is a twobody process, t hus the emitted ¯ν
e
has a ﬁxed energy (18.6 keV). In the targ et t he reverse process occurs, a
monochromatic ¯ν
e
with an energy of 18.6 keV and an e
−
in an atomic orbit
of
3
He are absorbed to form
3
H.
To suppress thermal motions of the
3
H and
3
He atoms, they have to be
imbedded in a solidstate lattice, e.g., in Nb metal [27]. In addition, for a
M¨ossbauer ¯ν
e
exp eriment it is mandatory that no phonons are excited in
the lattice when the ¯ν
e
is emitted or absorbed, because o nly then a highly
monochromatic ¯ν
e
radiation and the la rge cross section of the M¨ossbauer
resonance of typically 10
−19
to 10
−17
cm
2
can be achieved. However, it be
came apparent [28],[29],[30],[31] that there exist several basic diﬃculties to
observe M¨ossbauer ¯ν
e
with the system
3
H 
3
He in Nb metal. The main
problem originates from lattice expansion and contraction processes. They
occur when the nuclear transformations (from
3
H to
3
He and fro m
3
He to
3
H) take place during which the ¯ν
e
is emitted or absorbed and can cause
lattice excitations (pho nons) which change the ¯ν
e
energy and thus destroy
the M¨ossbauer resonance. It has been estimated that due t o these lattice ex
citations the probability for phononless emission and consecutive phononless
capture of ¯ν
e
is ∼ 7 · 10
−8
which ma kes a real experiment with the
3
H 
3
He
system extremely diﬃcult [28],[29],[30],[31]. Another basic problem is caused
by inhomogeneities in an imperfect lattice which directly inﬂuence the energy
of the ¯ν
e
[28].
A promising alternative is the ra reearth system
163
Ho 
163
Dy. It oﬀers
several advantages: Due to the highly similar chemical behaviour of the
rare earths also the lattice deformation energies for
163
Ho and
163
Dy can be
exp ected to be similar, thus leaving the ¯ν
e
energy practically unchanged.
13
Page 13
In addition, the ¯ν
e
energy is very low (2.6 keV), i.e., the recoil origina t ing
from the emitted (absorbed) ¯ν
e
is highly unlikely to generate phonons in the
lattice. Altogether, the probability of phononless emission and absorption
could be larger than for the
3
H 
3
He system by ∼ 7 orders of magnitude.
Furthermore, due to the similar chemical behaviour, the
163
Ho 
163
Dy system
can also be expected to be less sensitive to variat ions of the binding energies
in the lattice. For this reason, variations of the ¯ν
e
energy will also be reduced
improving the monochromaticity ( linewidth) of the ¯ν
e
M¨ossbauer resonance.
On the negative side, the magnetic moments of the 4f electrons of the
rareearth atoms are large and might cause broadening of the M¨ossbauer ¯ν
e
resonance [30],[31]. Fortunately, conventional M¨ossbauer spectroscopy (with
photons) gathered a wealth of informat ion on the behaviour of rareearth
systems in the past. Of particular interest is the 25.65 keV M¨ossbauer r es
onance in
161
Dy where an experimental linewidth of Γ
exp
≈ 5 · 10
−8
eV has
been r eached [32],[3 0]. We will show in the following section that the
163
Ho

163
Dy system might be suitable to investigate the question concerning t he
diﬀerent approaches to neutrino oscillations.
6 The
163
Ho 
163
Dy M¨ossbauer system and
the evolution of the ¯ν
e
state in time
If the evolution of the ¯ν
e
state occurs in t ime only, M¨ossbauer ¯ν
e
oscillations
with an oscillation length L
A
osc
determined by ∆m
2
A
will not be observed if
the relative energy uncertainty fulﬁlls the relation [33]
∆E
E
≪
1
4
∆m
2
A
c
4
E
2
(54)
where ∆m
2
A
≃ 2.4 · 10
−3
eV
2
is the atmospheric masssquared diﬀerence.
For the
163
Ho 
163
Dy system, eq. (54) requires
∆E
E
Ho−Dy
≪ 9.2 · 10
−11
or
∆E ≪ 2.4 · 10
−7
eV.
For the 25.65 keV γtransition in
161
Dy an experimental linewidth of
Γ
exp
≈ 5 ·1 0
−8
eV has been observed [32], which is ∼ 5 times below the limit
∆E . 2.4 · 10
−7
eV just mentioned. It might be expected that a similar
value f or Γ
exp
can be reached for the
163
Ho 
163
Dy system. In particular,
using the usual M¨ossbauer γtransition in
161
Dy, relevant physical properties,
e.g., the experimental linewidth in the Ho  Dy system can be investigated
14
Page 14
and improved if necessary. Thus it looks promising that the question if
M¨ossbauer ¯ν
e
oscillate can be answered experimentally. For Γ
exp
≈ 5 · 10
−8
eV, according to the MandelstamTa mm timeenergy uncertainty relation a
signiﬁcant change of the ¯ν
e
state in time can occur only very slowly leading
to a long oscillation pathlength L
change
since the ¯ν
e
is ultrarelativistic:
L
change
≃ c ·
~
Γ
exp
· 2π. (55)
For the
163
Ho 
163
Dy system, L
change
≈ 25 m for the ¯ν
e
state.
In compar ison, for an evolution of t he ¯ν
e
state in space and time, the
oscillation length is given by eq. (16). With E = 2.6 keV for the
163
Ho 
163
Dy system, and ∆m
2
A
≃ 2.4 · 10
−3
eV
2
, we obtain L
A
osc
≃ 2.6 m, about 10
times shorter tha n L
change
. If the evolution occurs in time only, in such a
M¨ossbauerneutrino experiment with Γ
exp
≈ 5 · 10
−8
eV, instead of L
A
osc
the
much longer L
change
would be observed.
If M¨ossbauer ¯ν
e
oscillate, an interesting application would be the search
for the conversion to sterile neutrinos ¯ν
e
→ ¯ν
sterile
[34] involving additional
mass eigenstates. Since ¯ν
sterile
does not show the weak interaction of the
Standard Model of elementa ry particle interactions, such a conversion would
have to be tested by the disappearance of ¯ν
e
. The results of the LSND
(Liquid Scintillator Neutrino Detector) experiment [6],[35] indicate a mass
splitting of ∆m
2
≈ 1 eV
2
[27]. Unfo r tunately, several experiments performed
by the MiniBooNE collaboration to check the LSND results have not been
conclusive, although the MiniBoo NE results are compatible with t he LSND
observation [36]. For M¨ossbauer ¯ν
e
of the
163
Ho 
163
Dy system (E= 2.6 keV)
the oscillation length L
A
osc
would be only ∼ 1 cm if ∆m
2
≈ 1 eV
2
.
7 Conclusions
After the golden years of the discovery of neutrino o scillations in atmospheric,
solar a nd reactor neutrino exp eriments we now enter into the era of detailed
studies of this phenomenon. Measurements of the small mixing a ngle θ
13
, of
the CP phase δ, and the establishment of the character of the neutrinomass
spectrum will require highprecision neutrinooscillation experiments which
are already ongoing now or are under preparation or in the R&D stage.
Is there a consensus in the treatment and understanding of the neutrino
oscillation phenomenon? Many recent papers on the theory of neutrino os
15
Page 15
cillations (see, for example, [18]) certify that such a consensus still does not
exist.
Is the notion of ﬂavor neutrinos ν
e
, ν
µ
and ν
τ
in the case of neutrino mix
ing a convenient terminology coming fr om ”the times of massless neutrinos”
or ar e they real physical states? From the momentumcoordinate Heisen
berg uncertainty relation follows that due to the small values of the neutrino
masssquared diﬀerences in weak processes ”mixed” ﬂavor neutrinos ν
l
(sim
ilar to the ”mixed” K
0
,
¯
K
0
; B
0
,
¯
B
0
; etc), which are described by coherent
superpositions of states of neutrinos with deﬁnite mass, are produced and
detected. We showed that in this approach for neutrino oscillations to be
observed the MandelstamTamm timeenergy uncertainty relation must be
satisﬁed. This means that neutrino oscillations can take place only in the
case of nonstationary neutrino states.
We compared diﬀerent approaches to neutrino oscillations. In approaches
in which ﬂavor neutrinos are described by coherent superpositions of plane
waves or wave packets and in the approach in which neutrinos are consid
ered as virtual particles in a Feynman diagr am with the neutrino production
process at one vertex a nd t he neutrino absorption process in another vertex
neutrino oscillations are possible also in the case of monochromatic neutrinos
(M¨ossbauer neutrinos).
Usual neutrino oscillation experiments do not allow to distinguish these
diﬀerent approaches. The realization of a n idea concerning the M¨ossbauer
resonance neutrino experiment with practically monoenergetic ¯ν
e
could be
the way of probing the real nature of mixed ﬂavor states, diﬀerent con
jectures on the evolution of such states and the universal applicability of
the timeenergy uncertainty relation. Such an experiment was discussed for
the
3
H −
3
He sourcedetector pair [27]. Recently, however, it was shown
that the performa nce of a M¨ossbauer neutrino experiment in the case of t he
3
H −
3
He system is most probably not possible in practice [28],[30],[31]. We
present here ar guments in favor of a M¨ossbauer neutrino experiment with
the
163
Ho −
163
Dy sourcedetector system [30]. The possibility to perform an
exp eriment in such a system looks promising but is still very challenging and
requires further investigations.
Acknowledgments
This work was supported by funds of the Deutsche Forschungsgemeinschaft
DFG (Transregio 27: Neutrinos and Beyond), the Munich Cluster of Ex
cellence (O rigin a nd Structure of the Universe), and the MaierLeibnitz
Laboratorium (Garching).
16
Page 16
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 "It follows from the Heisenberg uncertainty relation that this condition is satisfied in neutrino oscillation experiments with neutrino energies many orders of magnitude larger than neutrino masses. The possibility to resolve small neutrino masssquared differences is based on the timeenergy uncertainty relation (see [24]) ∆E ∆t 1. "
[Show abstract] [Hide abstract] ABSTRACT: The history of neutrino mixing and oscillations is briefly presented. Basics of neutrino mixing and oscillations and convenient formalism of neutrino oscillations in vacuum is given. The role of neutrino in the Standard Model and the Weinberg mechanism of the generation of the Majorana neutrino masses are discussed. 
 "To this, Raghavan brought in the bold proposal of using Niobium tritide as the actual source of recoilless emission and absorption of the neutrinos. Raghavan's proposal lead to a couple of intense controversies, one on whether the Mössbauer effect could truly be realized using Niobium tritide [5, 20,22232425, and another on the interpretation of the TimeEnergy uncertainty relation and its implications for the proposed experiments2627282930313233. At the same time, it generated a great deal of excitement and interest on the new kinds of experiments that would be made possible if Mössbauer neutrinos became a reality, and the types of physics it will allow us to probe34353637. "
[Show abstract] [Hide abstract] ABSTRACT: We discuss a direct test of the relation of time and energy in the very longlived decay of tritium (H3) (meanlife \tau ~ 18 yrs) with the width \Gamma ~ 10^{24} eV [set by the timeenergy uncertainty (TEU)], using the newfound possibility of resonance reactions H3 \leftrightarrow He3 with \Delta E/E ~ 5x10^{29}. The TEU is a keystone of quantum mechanics, but probed for the first time in this extreme timeenergy regime. Forestalling an apparent deviation from the TEU, we discuss the ramifications and a possible generalization of the TEU as \Delta E \Delta t > (\hbar/2)[1+(\Delta t/T)^n] where \Delta t = \tau is the time of measurement (the lifetime of the state), T=L/c the time for light to cross the Universe ~ 3x10^{18} s, and n a parameter subject to future measurements. (by R. S. Raghavan.) 
 "In particular, flavor oscillation provides the only means to measure the extremely small mass and decay rate splittings among the neutral mesons, and also provides convincing evidence for the existence of nonzero neutrino masses. The theoretical descriptions of flavor oscillation fall into several categories, including the basic plane wave Pontecorvo formalism [1, 2], intermediate3456789101112 and external1314151617181920 wavepacket approaches and quantum field theoretic results212223242526272829. Some detailed reviews of these approaches, their underlying assumptions, results and difficulties can be found in Refs. [2,303132 (and references therein). "
[Show abstract] [Hide abstract] ABSTRACT: We present a formalism for the flavor oscillation of unstable particles that relies only upon the structure of the time Fouriertransformed twopoint Green's function. We derive exact oscillation probability and integrated oscillation probability formulae, and verify that our results reproduce the known results for both neutrino and neutral meson oscillation in the expected regimes of parameter space. The generality of our approach permits us to investigate flavor oscillation in exotic parameter regimes, and present the corresponding oscillation formulae.