# Neutrino oscillations and uncertainty relations

**ABSTRACT** We show that coherent flavor neutrino states are produced (and detected) due

to the momentum-coordinate Heisenberg uncertainty relation. The Mandelstam-Tamm

time-energy uncertainty relation requires non-stationary neutrino states for

oscillations to happen and determines 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 expression for the neutrino transition probability in standard neutrino

oscillation experiments. We show that a Moessbauer neutrino experiment could

allow to distinguish different approaches and we present arguments in favor of

the 163Ho-163Dy system for such an experiment.

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arXiv:1102.2770v1 [hep-ph] 14 Feb 2011

Neutrino oscillations and uncertainty relations

S. M. Bilenky

Joint Institute for Nuclear Research, Dubna, R-141980, Russia

and

Physik-Department E15, Technische Universit¨ at M¨ unchen,

D-85748 Garching, Germany

F. von Feilitzsch and W. Potzel

Physik-Department E15, Technische Universit¨ at M¨ unchen,

D-85748 Garching, Germany

Abstract

We show that coherent flavor neutrino states are produced (and

detected) due to the momentum-coordinate Heisenberg uncertainty

relation. The Mandelstam-Tamm time-energy uncertainty relation re-

quires non-stationary 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 and we present

arguments in favor of the163Ho -163Dy system for such an experiment.

1Introduction

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 origin and are commonly considered as a signature of new physics

beyond the Standard Model.

All existing neutrino-oscillation data with the exception of the data of

the LSND [6] and MiniBooNE antineutrino experiments [7], which require

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confirmation, are perfectly described under the assumption of three-neutrino

mixing

3

?

Here U is the PMNS [8, 9] 3 × 3 mixing matrix, which is characterized by

three mixing angles θ12,θ23,θ13 and the CP phase δ, νi(x) is the field of

neutrinos (Dirac or Majorana) with mass mi, and the ”mixed field” νlL(x) is

the SM field which enters into the standard charged current

νlL(x) =

i=1

UliνiL(x).(1)

jα(x) =

?

l=e,µ,τ

¯ νlL(x)γαlL(x).(2)

Existing neutrino-oscillation data are analyzed under the assumption that

the transition probabilities between different flavor neutrinos are given by

the following standard expression (see, for example, [10])

P(νl→ νl′) = δl′l− 2 Re

?

i>k

Ul′iU∗

liU∗

l′kUlk(1 − e−i

∆m2

2E ).

kiL

(3)

Here, L is the distance between neutrino source and neutrino detector, E is

the neutrino energy, ∆m2

use for the transition probability another expression

ki= m2

i− m2

k. Notice that it is also convenient to

P(νl→ νl′) = δl′l− 2

?

i

|Uli|2(δl′l− |Ul′i|2)(1 − cos∆m2

jiL

2E

)(4)

+2 Re

?

i>k

Ul′iU∗

liU∗

l′kUlk(e−i

∆m2

jiL

2E

− 1)(ei

∆m2

jkL

2E

− 1),

where the index j is fixed.

The character of neutrino oscillations is determined by the following two

observed features of the neutrino-oscillation parameters:

• The solar-KamLAND mass-squared difference ∆m2

than the atmospheric-accelerator mass-squared difference ∆m2

Sis much smaller

A:

∆m2

S≃

1

30∆m2

A.(5)

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• The mixing angle θ13is small [11]:

sin2θ13≤ 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 νµ⇄ ντand

¯ νµ⇄ ¯ ντ and in solar and KamLAND experiments the leading oscillations

are νe⇄ νµ,τand ¯ νe⇄ ¯ νµ,τ.

In the leading approximation it is impossible to distinguish two possible

neutrino mass spectra:

• Normal spectrum

m1< m2< m3,∆m2

12≪ ∆m2

23.

• Inverted spectrum

m3< m1< m2,∆m2

12≪ |∆m2

13|.

In the case of the normal spectrum ∆m2

the case of the inverted spectrum ∆m2

From the recent three-neutrino analysis of the Super-Kamiokande data

[1] the following 90% CL limits were found for the normal (inverted) neutrino

mass spectrum

12= ∆m2

12= ∆m2

S,∆m2

13= −∆m2

23= ∆m2

Aand in

S, ∆m2

A.

1.9 (1.7)·10−3≤ ∆m2

For the parameter sin2θ13the following bounds were obtained

A≤ 2.6 (2.7)·10−3eV2,0.407 ≤ sin2θ23≤ 0.583. (7)

sin2θ13≤ 4 · 10−2(9 · 10−2).(8)

From the two-neutrino analysis of the MINOS data was found [5]

∆m2

A= (2.43 ± 0.13) · 10−3eV2,sin22θ23> 0.90(9)

From the three-neutrino global analysis of the solar and reactor KamLAND

data was obtained [3]

∆m2

S= (7.50+0.19

−0.20) · 10−5eV2,tan2θ12= 0.452+0.035

−0.032

(10)

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For the parameter sin2θ13was found

sin2θ13= 0.020+0.016

−0.018

(11)

At present four neutrino-oscillation parameters (∆m2

tan2θ12) are known with accuracies within the (3-10)% range. In the accel-

erator neutrino oscillation experiment T2K [12] the parameter ∆m2

measured with an accuracy of δ∆m2

will be measured with an accuracy of δ(sin22θ23) ≃ 10−2. One of 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 sin2θ13.

In case that this parameter is relatively large, it is envisaged that 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

experiments. Despite that the neutrino oscillation formalism, on which the

analysis of experimental data is based, has been developed and debated in

many papers starting from the 1970s (see reviews [16, 17]), these debates

and discussions are continuing (see recent papers [18]). From our point of

view the importance 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

Mandelstam-Tamm time-energy uncertainty relation. We briefly consider dif-

ferent approaches to neutrino oscillations and discuss a M¨ ossbauer neutrino

experiment which could allow to distinguish them.

S, ∆m2

A, sin22θ23and

Awill be

A< 10−4eV2and the parameter sin22θ23

2Flavor neutrinos: production, evolution,

detection

Which neutrino states are produced in CC weak processes together with

charged 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 the QFT, the Heisenberg uncer-

tainty relation and the knowledge of the neutrino mass-squared differences

that ”mixed” flavor states which describe the flavor neutrinos νe, νµand ντ

are physical states (fully analogous to the ”mixed” states which describe K0

and¯K0, B0and¯B0, etc.).

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Let us consider (in the lab. system) the decay [19]

a → b + l++ νi,(i = 1,2,3) (12)

where a and b are some hadrons.

The state of the final particles is given by

|f? =

?

i

|νi?|l+?|b ??νil+b|S|a?,(13)

where ?νil+b|S|a? is the matrix element of the transition a → b + l++ νi,

|νi? is the state of a neutrino with mass mi, momentum ? pi= pi?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

pi=

?

E2

i− m2

i≃ E −m2

i

2E,

(14)

where E is the energy of neutrinos for m2

momenta we have

i→ 0. For the difference of neutrino

|pi− pk| ≃|∆m2

2ELr

osc

ki|

=

2π

,(15)

where

Lr

osc= 4π

E

∆m2

r

≃ 2.48

(E/MeV)

(∆m2

rc4/eV2)m,r = A,S(16)

is the oscillation length. For E ≃ 1 GeV and ∆m2

spheric and LBL accelerator neutrinos) we have LA

MeV and ∆m2

km.

On the other side, from the Heisenberg uncertainty relation we have

A≃ 2.4 · 10−3eV2(atmo-

osc≃ 103km. For E ≃ 3

S≃ 7.5 · 10−5eV2(reactor antineutrinos) we have LS

osc≃ 102

(∆p)QM≃1

d.

(17)

Here d characterizes the quantum-mechanical size of the source. Taking into

account that

LA,S

osc≫ d(18)

we have

|pi− pk| ≪ (∆p)QM.(19)

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Thus, we conclude that due to the uncertainty relation it is impossible to

resolve the emission of neutrinos with different masses.1

The operator

?

determines the leptonic part of the matrix element of the process (12). We

have

U∗

k

U∗

lk¯ νkL(x)γαlL(x) (20)

li¯ uL(pi)γαuL(−pl) ≃ U∗

li¯ uL(p)γαuL(−pl), (21)

where plis the momentum of l+, and p = E is the momentum of the neutrino

for m2

i→ 0. For the total matrix element of the process (12) we have

?νil+b|S|a? ≃ U∗

where ?νll+b|S|a?SMis the Standard Model matrix element of the emission

of the flavor neutrino νlwith the momentum p in the process

li?νll+b|S|a?SM, (22)

a → b + l++ νl. (23)

From (13) and (22) we find

|f? = |νl?|l+?|b ??νll+b|S|a?SM,(24)

where the state of the flavor neutrino νlis given by the relation

|νl? =

?

i

U∗

li|νi?(l = e,µ,τ)(25)

and |νi? is the state of a neutrino with mass mi, negative helicity and mo-

mentum p.2

Let us stress that

• Flavor neutrino states do not depend on the production process.

1For the energy of a neutrino with mass mi we have Ei ≃ E(1 +

oscillation experiments E ? 1 MeV and

different neutrino energies in production (and detection) processes.

2Let 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.

m2

2E2). In neutrino

i

m2

2E2 ? 10−12. Thus, it is impossible to resolve

i

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• Flavor states are characterized by the momentum (if there are no spe-

cial conditions of neutrino production).

• Flavor states are orthogonal and normalized

?νl′|νl? = δl′l. (26)

The evolution of states in QFT is given by the Schr¨ odinger equation

i∂ |Ψ(t)?

∂t

= H |Ψ(t)?, (27)

where H is the total Hamiltonian and time t is a parameter both of which

characterize the evolution of the system.

If at t = 0 in a CC weak process νlis produced, we have for the state of

the neutrino at the time t

|νl?t= e−iHt|νl? =

?

i

|νi?e−iEitU∗

li, (28)

where

H|νi? = Ei|νi?,Ei≃ E +m2

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

νi+ N → l′+ X.(30)

Taking into account that effects of neutrino masses can not be resolved in

neutrino processes we have

?l′X|S|νiN? ≃ ?l′X|S|νl′N?SMUl′i,(31)

where ?l′X|S|νl′ N?SMis the SM matrix element of the process

νl′ + N → l′+ X.

From (24), (28) and (31) follows that the chain of processes a → b + l++

νl,νl → νl′,

product of amplitudes

(32)

νl′ + N → l′+ X corresponds to the following factorized

?l′X|S|νl′ N?SM

??

i

Ul′ie−iEitU∗

li

?

?b l+νl|S|a?SM.(33)

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Only the amplitude of the transition νl→ νl′

A(νl→ νl′) =

?

i

Ul′ie−iEitU∗

li

(34)

depends on the properties of massive neutrinos (mass-squared 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 factorization (33) is based on the smallness of the neutrino masses and

on the Heisenberg uncertainty relation.

3Mandelstam-Tamm uncertainty relation

and neutrino oscillations

All uncertainty relations in Quantum Theory are based on the inequality

∆A ∆B ≥1

2|?a|[A,B]|a?|(35)

which follows from the Cauchy inequality. In (35) A and B are hermitian

?

and A = ?a|A|a? 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

The Mandelstam-Tamm time-energy uncertainty relation [22] is based on

the inequality (35) and the equation

operators, |a? is any state, ∆A =

?a|(A − A)2|a? is the standard deviation

iwe have the Heisenberg uncertainty relation ∆p ∆q ≥1

2.

i∂O(t)

∂t

= [O(t),H](36)

for any operator O(t) in the Heisenberg representation (H is the total Hamil-

tonian).

From (35) and (36) we have

∆E ∆O(t) ≥1

2|d

dtO(t)|(37)

This inequality gives nontrivial constraints only in the case of non-stationary

states.

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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) the right-hand part of (38) is of the order of one. We obtain the

Mandelstam-Tamm time-energy uncertainty relation

∆E ∆t ? 1. (39)

From (34), for the normalized probability of the transition νl→ νl′ we

obtain the expression

P(νl→ νl′) = |

?

i?=j

Ul′i(e−i(Ei−Ej)t− 1) U∗

li+ δl′l|2,(40)

which obviously gives the standard transition probability (3).

From (40) follows that neutrino oscillations can be observed if the condi-

tion

|Ei− Ej| t ? 1

is satisfied.3

It is obvious that this inequality is the Mandelstam-Tamm

time-energy uncertainty relation.According to this relation a change of

the flavor 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 ≃

(41)

1

|Ei−Ej|=

2E

|∆m2

ji|.4

4On 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 processes to-

gether with charged leptons coherent superpositions of plane waves are

3This is a necessary condition for the observation of oscillations. It is also necessary

that mixing angles would be relatively large.

4Let us notice that the inequality (41) can be interpreted in another way: In order

to reveal a small energy difference |Ei− Ej| ≃

t ?

|Ei−Ej|. This corresponds to another interpretation of the time-energy uncertainty

relation (see [23]).

|∆m2

2E

ji|

we need a large time interval

1

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produced and absorbed. In this case, for the normalized νl→ νl′ transition

probability the following expression can be obtained (see, for example[24, 25])

P(νl→ νl′) = |

?

i

Ul′ie−ipi·xU∗

li|2= |

?

i?=j

Ul′i(e−i(pi−pj)·x−1)U∗

li+δl′l|2. (42)

Here pi= (Ei,? pi) is the 4-momentum of a neutrino with mass miand x =

(t,? x).

Let us assume that ? pi= pi?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

(pi− pj) · x = (Ei− Ej)t − (pi− pj)L.

For ultrarelativistic neutrinos we have

(43)

t ≃ L.

m2

2E, from (43) and (44) we come to the

(44)

Taking into account that Ei≃ pi+

standard oscillation phase

i

(pi− pj) · x =∆m2

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(νl→ νl′) = |

?

i

Ul′i

?

ei(? pi′? x−E′

it)f(? pi

′− ? pi) d3p′U∗

li|2,(46)

where E′

at the point ? pi

i=

?(? pi

′)2+ m2

′= ? pi.

iand the function f(? pi

′− ? pi) has a sharp maximum

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Expanding E′

iat the point ? pi

′= ? piwe find

?

ei(? pi′? x−E′

it)f(? pi

′− ? pi) d3p′= ei(? pi? x−Eit)g(? x −? vit),(47)

where

g(? x −? vit) =

?

ei? q (? x−? vit)f(? q) d3q(48)

and

? vi=? pi

Ei,Ei=

?

? pi

2+ m2

i.(49)

If we make the standard assumption that the function f(? q) has the Gaussian

form

f(? q) = N e

−q2

4σ2

p, (50)

(σpis the width of the wave packet in the momentum space) we find

g(? x −? vit) = N(π

σ2

x

)3/2e

−(? x−? vit)2

4σ2

x

, (51)

where σx=

The probability of the transition νl→ νl′ 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

1

2σpcharacterizes the spacial width of the wave packet.

P(νl→ νl′) =

?

i,k

Ul′iU∗

l′kei[(pi−pk)−(Ei−Ek)]LU∗

liUlke

−(

L

Lik

coh

)2

e

−2π2ξ2(

σx

Lik

osc)2.

(52)

Here L is the distance between neutrino source and neutrino detector, Lik

is the oscillation length, ξ is a constant of the order of one and

coh=4√2σxE2

osc

Lik

|∆m2

ik|

.(53)

is the coherence length.5Taking into account that (pi− pk) − (Ei− Ek) =

−∆m2

|∆m2

ik|

2E2 Lik

a distance between neutrino source and detector at which νiand νkare separated by an

interval comparable to the size of the wave packet.

ki

2E

we come to the conclusion that the νl→ νl′ transition probability in

coh∼ 2√2σx. Thus, the coherence length is such

5We have |vi− vk|Lik

coh≃

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the wave packet approach is given by the standard expression (3) which is

multiplied by the decoherence factor e

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 LA,S

two additional exponential factors are practically equal to one.

In many papers (see [18]), neutrinos propagating 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 (before integration over t)

depends on x and t.

The major difference between different approaches to neutrino oscillations

can be summarized as follows:

−(

L

Lik

coh

)2

and the factor e

−2π2ξ2(

σx

Lik

osc)2.

osc, the

1. The QFT approach with the Schr¨ odinger evolution equation is based

on the assumption of the existence of ”mixed” flavor neutrinos νe,νµ,ντ

which are described by coherent states |νl? =?

energy uncertainty relation. Neutrino oscillations can take place only

in the case of non-stationary neutrino states with ∆E∆t ? 1, where

∆t is the time interval during which the oscillations happen. The QFT

approach is based on the same general principles as the approach to

K0⇄¯K0, B0⇄¯B0, etc. oscillations studied in detail at B-factories

and other facilities.

iU∗

li|νi?. The important

characteristic feature of this approach is the Mandelstam-Tamm time-

2. Other 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 of 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 oscillations 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(νl → νl′) in the standard neutrino

oscillation experiments. In order to distinguish 1. and 2. special neutrino

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Page 13

oscillation experiments are necessary. Such experiments could be M¨ ossbauer

neutrino experiments which we will discuss in the next sections.

5 M¨ ossbauer ¯ νe: Basic considerations

The basic concept is to use electron antineutrinos (¯ νe) which are emitted

without recoil in a bound-state β-decay and are resonantly captured again

without recoil in the reverse bound-state process. As an example, let us

consider the3H -3He system [26] with the transitions

(source) and ¯ νe+3He →3H (target).

In the source, the electron (e−) is emitted directly into a bound-state

atomic orbit of3He. This decay is a two-body process, thus the emitted ¯ νe

has a fixed energy (18.6 keV). In the target the reverse process occurs, a

monochromatic ¯ νewith an energy of 18.6 keV and an e−in an atomic orbit

of3He are absorbed to form3H.

To suppress thermal motions of the3H and3He atoms, they have to be

imbedded in a solid-state lattice, e.g., in Nb metal [27]. In addition, for a

M¨ ossbauer ¯ νe experiment it is mandatory that no phonons are excited in

the lattice when the ¯ νeis emitted or absorbed, because only then a highly

monochromatic ¯ νe radiation and the large cross section of the M¨ ossbauer

resonance of typically 10−19to 10−17cm2can be achieved. However, it be-

came apparent [28],[29],[30],[31] that there exist several basic difficulties to

observe M¨ ossbauer ¯ νe with the system3H -3He in Nb metal. The main

problem originates from lattice expansion and contraction processes. They

occur when the nuclear transformations (from3H to3He and from3He to

3H) take place during which the ¯ νe is emitted or absorbed and can cause

lattice excitations (phonons) which change the ¯ νeenergy and thus destroy

the M¨ ossbauer resonance. It has been estimated that due to these lattice ex-

citations the probability for phononless emission and consecutive phononless

capture of ¯ νeis ∼ 7 · 10−8which makes a real experiment with the3H -3He

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 ¯ νe[28].

A promising alternative is the rare-earth system163Ho -163Dy. It offers

several advantages: Due to the highly similar chemical behaviour of the

rare earths also the lattice deformation energies for163Ho and163Dy can be

expected to be similar, thus leaving the ¯ νe energy practically unchanged.

3H→3He +¯ νe

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Page 14

In addition, the ¯ νeenergy is very low (2.6 keV), i.e., the recoil originating

from the emitted (absorbed) ¯ νeis highly unlikely to generate phonons in the

lattice. Altogether, the probability of phononless emission and absorption

could be larger than for the3H -3He system by ∼ 7 orders of magnitude.

Furthermore, due to the similar chemical behaviour, the163Ho -163Dy system

can also be expected to be less sensitive to variations of the binding energies

in the lattice. For this reason, variations of the ¯ νeenergy will also be reduced

improving the monochromaticity (linewidth) of the ¯ νeM¨ ossbauer resonance.

On the negative side, the magnetic moments of the 4f electrons of the

rare-earth atoms are large and might cause broadening of the M¨ ossbauer ¯ νe

resonance [30],[31]. Fortunately, conventional M¨ ossbauer spectroscopy (with

photons) gathered a wealth of information on the behaviour of rare-earth

systems in the past. Of particular interest is the 25.65 keV M¨ ossbauer res-

onance in161Dy where an experimental linewidth of Γexp≈ 5 · 10−8eV has

been reached [32],[30]. We will show in the following section that the163Ho

-163Dy system might be suitable to investigate the question concerning the

different approaches to neutrino oscillations.

6 The

the evolution of the ¯ νestate in time

163Ho -

163Dy M¨ ossbauer system and

If the evolution of the ¯ νestate occurs in time only, M¨ ossbauer ¯ νeoscillations

with an oscillation length LA

the relative energy uncertainty fulfills the relation [33]

oscdetermined by ∆m2

Awill not be observed if

∆E

E

≪1

4

∆m2

E2

Ac4

(54)

where ∆m2

For the163Ho -163Dy system, eq. (54) requires?∆E

For the 25.65 keV γ-transition in

Γexp≈ 5·10−8eV has been observed [32], which is ∼ 5 times below the limit

∆E ? 2.4 · 10−7eV just mentioned. It might be expected that a similar

value for Γexpcan be reached for the163Ho -163Dy system. In particular,

using the usual M¨ ossbauer γ-transition in161Dy, relevant physical properties,

e.g., the experimental linewidth in the Ho - Dy system can be investigated

A≃ 2.4 · 10−3eV2is the atmospheric mass-squared difference.

E

?

Ho−Dy≪ 9.2 · 10−11or

∆E ≪ 2.4 · 10−7eV.

161Dy an experimental linewidth of

14

Page 15

and improved if necessary.

M¨ ossbauer ¯ νeoscillate can be answered experimentally. For Γexp≈ 5 · 10−8

eV, according to the Mandelstam-Tamm time-energy uncertainty relation a

significant change of the ¯ νestate in time can occur only very slowly leading

to a long oscillation path-length Lchangesince the ¯ νeis ultrarelativistic:

Thus it looks promising that the question if

Lchange≃ c ·

?

Γexp

· 2π.(55)

For the163Ho -163Dy system, Lchange≈ 25 m for the ¯ νestate.

In comparison, for an evolution of the ¯ νe state in space and time, the

oscillation length is given by eq. (16). With E = 2.6 keV for the163Ho -

163Dy system, and ∆m2

times shorter than Lchange. If the evolution occurs in time only, in such a

M¨ ossbauer-neutrino experiment with Γexp≈ 5 · 10−8eV, instead of LA

much longer Lchangewould be observed.

If M¨ ossbauer ¯ νeoscillate, an interesting application would be the search

for the conversion to sterile neutrinos ¯ νe→ ¯ νsterile[34] involving additional

mass eigenstates. Since ¯ νsterile does not show the weak interaction of the

Standard Model of elementary particle interactions, such a conversion would

have to be tested by the disappearance of ¯ νe. The results of the LSND

(Liquid Scintillator Neutrino Detector) experiment [6],[35] indicate a mass

splitting of ∆m2≈ 1 eV2[27]. Unfortunately, several experiments performed

by the MiniBooNE collaboration to check the LSND results have not been

conclusive, although the MiniBooNE results are compatible with the LSND

observation [36]. For M¨ ossbauer ¯ νeof the163Ho -163Dy system (E=2.6 keV)

the oscillation length LA

A≃ 2.4 · 10−3eV2, we obtain LA

osc≃ 2.6 m, about 10

oscthe

oscwould be only ∼ 1 cm if ∆m2≈ 1 eV2.

7 Conclusions

After the golden years of the discovery of neutrino oscillations in atmospheric,

solar and reactor neutrino experiments we now enter into the era of detailed

studies of this phenomenon. Measurements of the small mixing angle θ13, of

the CP phase δ, and the establishment of the character of the neutrino-mass

spectrum will require high-precision neutrino-oscillation 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

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- Available from F. von Feilitzsch · May 15, 2014
- Available from ArXiv