Do nonrelativistic neutrinos oscillate?
HJE
Do nonrelativistic neutrinos oscillate?
Evgeny Akhmedov 0 1
0 Also at the National Research Centre Kurchatov Institute , Moscow , Russia
1 Saupfercheckweg 1 , 69117 Heidelberg , Germany
We study the question of whether oscillations between nonrelativistic neutrinos or between relativistic and nonrelativistic neutrinos are possible. The issues of neutrino production and propagation coherence and their impact on the above question are discussed in detail. It is demonstrated that no neutrino oscillations can occur when neutrinos that are nonrelativistic in the laboratory frame are involved, except in a strongly We also discuss how this analysis depends on the choice of the Lorentz frame. Our results are for the most part in agreement with Hinchli e's rule.
Neutrino Physics; Beyond Standard Model

3
4
2.1
2.2
3.1
3.2
1 Introduction 2 Nonrelativistic neutrinos and coherence conditions
Nonrelativistic neutrinos and production coherence
2.2.1
2.2.2
General case
Nonrelativistic neutrinos
2.3
Wave packet separation and propagation coherence
Lorentz boosts
Usual neutrino oscillations in the rest frame of 2
Boosting nonrelativistic neutrinos
Summary and discussion
A Decay rates
B
Neutrino wave packet spreading
masseigenstate neutrino species have mass & 0:05 eV, or one of them has mass & 0:05 eV
and one other & 8
10 3 eV. The presentday temperature of the cosmic neutrino
background T
' 1:95 K ' 1:68 10 4 eV therefore means that at least two relic neutrino species
are currently nonrelativistic. Moreover, there are some indications of possible existence
of (predominantly) sterile neutrinos with an eVscale mass, and chie y sterile neutrinos
with keV  MeV  GeV scale masses are also being discussed [1{3]. If exist, such
neutrinos may well be nonrelativistic in some situations of practical interest. The question
of whether nonrelativistic neutrinos can oscillate is therefore not of completely academic
nature. In addition, as we shall see, it is related to some fundamental aspects of the theory
of neutrino oscillations and therefore is of considerable conceptual interest.
When asking whether nonrelativistic neutrinos can oscillate, one should obviously
specify the reference frame in which neutrinos are considered. Neutrinos that are
nonrelativistic in one Lorentz frame may be highly relativistic in another, and vice versa.
{ 1 {
If not otherwise speci ed, we shall be considering neutrinos in the laboratory frame, by
which we mean the frame in which the neutrino source is at rest or is slowly moving.1 For
neutrinos produced in decays, the source is just the parent particle; for neutrinos produced
in collisions, the de nition of the velocity of the source is more complicated and involves
a consideration of the velocities and wave packet lengths of all particles participating in
neutrino production [4].
The question of whether nonrelativistic neutrinos can take part in neutrino oscillations
has been discussed in refs. [4{10], with varying degree of detail and di ering conclusions.
In refs. [4{6] it was argued that nonrelativistic neutrinos cannot participate in neutrino
oscillations, whereas in refs. [7{10] the opposite conclusion has been reached (or the opposite
Nonrelativistic neutrinos and coherence conditions
It is well known that neutrino oscillations can only be observable if neutrino production
and detection processes cannot discriminate between di erent neutrino mass eigenstates.
This is because only
avour eigenstates undergo oscillations  mass eigenstates do not
oscillate.2 Flavour eigenstates are coherent linear superpositions of mass eigenstates, and
therefore the question of observability of neutrino oscillations is closely related to the
question of coherence of neutrino production, propagation and detection. If at any of
these stages coherence of di erent mass eigenstates is violated, oscillations will not be
observable. We therefore examine here if the coherence conditions are satis ed for
nonrelativistic neutrinos. The answer to this question depends on whether the neutrino mass
spectrum is hierarchical or quasidegenerate. We rst consider the latter case.
2.1
Quasidegenerate in mass nonrelativistic neutrinos
Conceptual issues of neutrino oscillation theory can only be consistently studied within
the quantummechanical (QM) wave packet framework [4, 6, 11{25]3 or in a formalism
based on the quantum
eld theoretic (QFT) approach [4, 26{37]. As has been mentioned
1This de nition may sometimes di er from what one would normally consider to be the laboratory frame
in a neutrino experiment. For instance, in accelerator neutrino experiments the conventional de nition
would likely be that this is the frame in which the neutrino detector is at rest, whereas neutrinos are
produced in decays of relativistic pions.
2In this paper we shall only deal with neutrino oscillations in vacuum. Mass eigenstates can oscillate in
also to the case of nonrelativistic but highly degenerate in mass neutrinos, i.e. when
j m2j
2E
E ;
where
m2
m2
i
m2k and E are the neutrino mass squared di erence and the average
energy of the neutrino mass eigenstates composing a produced neutrino
avour state,
respectively. As an example, in [22] it has been demonstrated that the conditions for neutrino
oscillations to occur and to be described by the standard oscillation probability are that
(i ) the energy di erence of the neutrino mass eigenstates composing the produced avour
eigenstate,
E
Eik =
qpi2 + m2
i
qp2k + m2 ;
k
HJEP07(21)
(2.1)
(2.2)
m2 and
(2.3)
is related to their momentum di erence
p
pi
pk, mass squared di erence
average group velocity vg by
E
2E
;
respectively: j Ej
which yields eq. (2.3).
E, j p
j
and (ii ) the coherence conditions for neutrino production, propagation and detection (to be
discussed in detail in sections 2.2 and 2.3) are satis ed. The relation in eq. (2.3) is satis ed
with very good accuracy for ultrarelativistic neutrinos; however, it is easy to see that for
its validity it is su cient that the neutrino mass eigenstates be quasidegenerate in mass,
i.e. condition (2.1) be satis ed. In this case di erent mass eigenstates are produced under
essentially the same kinematic conditions, i.e. their energy di erence
E and momentum
di erence
p are small compared to their average energy E and average momentum p,
p. This means that one can expand
E in
p and
m2,
The conditions of coherent neutrino production, propagation and detection put
upper limits on
m2=E2 and
m2=(2E) and actually do not require neutrinos to be
ultrarelativistic; they can also be satis ed if neutrinos are su ciently degenerate in mass.
Thus, the standard formalism of neutrino oscillations developed for ultrarelativistic
neutrinos applies without any modi cations also to nonrelativistic but highly degenerate
in mass neutrinos. Such neutrinos undergo the usual avour oscillations provided that the
standard coherence conditions are satis ed.
In the rest of this paper we shall be assuming that neutrinos are not quasidegenerate
in mass.
2.2
Nonrelativistic neutrinos and production coherence
Let us now consider neutrino production coherence in the case when nonrelativistic
neutrinos are produced. We start with a general discussion of neutrino production coherence.
{ 3 {
2.2.1
Di erent neutrino mass eigenstates can be emitted coherently and compose a avour state
only if the intrinsic QM uncertainties of their energies and momenta,
E and
p, are
su ciently large to accommodate their di ering energies in momenta. Assuming absence
of certain cancellations (as will be explained towards the end of this subsection), this
condition can be written as
j Ej
E
1 ;
j p
j
p
p are very small, then by measuring
4momenta of the other particles involved in neutrino production and using
energymomentum conservation, one could in principle determine the energy and momentum of
the produced neutrino state with high accuracy. This would allow one to accurately infer
the neutrino mass, which would mean that a mass (rather than avour) eigenstate has
been emitted, and neutrino oscillations would not take place.
Indeed, assume that by measuring the energies and momenta of all particles taking part
in neutrino production we determined neutrino energy and momentum with some accuracy.
According to QM uncertainty relations, the uncertainties of the determined neutrino energy
and momentum cannot be smaller than the intrinsic energy and momentum uncertainties
E and
p related to localization of the production process in
nite spacetime region.
Assuming E and p to be independent, from the onshell dispersion relation E2 = p2 + m2
one can then nd the minimum error in the determination of the squared neutrino mass [
12
]:
m2 = (2E E)2 + (2p p)2 1=2:
(2.5)
If m2 satis es
m2
j m2j = jmi2
m2kj, one cannot kinematically distinguish between
the mass eigenstates i and k in the production process, i.e. they can be emitted
coherently. Conversely, for m2 . j m2j one can nd out which neutrino eigenstate has actually
been produced; this means that i and k cannot be produced coherently. Since neutrino
oscillations are a result of interference of amplitudes corresponding to di erent mass
eigenstates, absence of their coherence means that no oscillations will take place. The avour
transition probabilities would then correspond to averaged neutrino oscillations.
This situation is quite similar to that with the electron interference in double slit
experiments. If there is no way to
nd out which slit the detected electron has passed
through, the detection probability will exhibit an interference pattern; however if such a
determination is possible, the interference pattern will be washed out.
In the general case when more than two neutrino species are involved, the mass
eigenstates i and k can be produced coherently if their mass di erence satis es j
even if the other pairs of mass eigenstates do not satisfy such a condition and therefore
m2 ,
mi2kj
cannot be produced coherently. In this case partial decoherence takes place.
As mentioned above, intrinsic energy and momentum uncertainties E and p
characterizing a produced neutrino state are related to spacetime localization of the production
process: the better the localization, the larger the E and p, and the easier it is to satisfy
{ 4 {
the production coherence condition. It is instructive to formulate this condition in con
guration space [
12, 20, 22
]; this will also allow us to nd out when eq. (2.4) represents the
coherent production condition. To this end, consider the oscillation phase acquired over
the distance x and the time interval t from the spacetime point at which neutrino was
produced:4
osc =
E t
p~ ~x :
The 4coordinate of the neutrino production point has an intrinsic uncertainty related to
the
nite spacetime extension of the production process; this leads to uctuations of the
oscillation phase
HJEP07(21)
osc =
E
t
p~
~x ;
where t and j ~xj are limited by the duration of the neutrino production process t and its
spatial extension X : t . t, j ~xj .
of the oscillation phase must satisfy j
oscj
X . For oscillations to be observable, the uctuations
1  otherwise oscillations will be washed
out upon averaging of the phase over the 4coordinate of neutrino production. That is,
observability of neutrino oscillations requires that the condition
(2.6)
(2.7)
(2.8)
j E
t
p~
~x
j
1
be satis ed. Barring accidental cancellations between the two terms in (2.8) and taking
into account that t
1
E , X
p 1, we arrive at eq. (2.4). Therefore,
di erent neutrino mass eigenstates are produced coherently and hence neutrino
oscillations may be observable only if the oscillation phase acquired over the
spacetime extension of the production region is much smaller than unity.
This condition essentially coincides with the obvious requirement that the size of the
neutrino production region be much smaller than the oscillation length (which corresponds to
osc = 2 ).
It should be noted that coherent neutrino production is necessary for observability of
neutrino oscillations, but it is not by itself su cient: for the oscillations to take place, also
the propagation and detection coherence conditions must be satis ed. Detection coherence
can be considered quite similarly to the production one; propagation coherence will be
discussed in section 2.3.
2.2.2
Nonrelativistic neutrinos
Let us now discuss the production coherence condition in the case when one or more
neutrino mass eigenstates are nonrelativistic in the frame where the neutrino source is
at rest or is slowly moving. In this case di erent neutrino mass eigenstates are produced
under very di erent kinematic conditions, and therefore have vastly di ering energies and
momenta. We shall demonstrate that large energy and momentum di erences will then
prevent coherent neutrino production.
4The phase
osc (as well as the energy di erence
E and the momentum di erence
p~) is actually
de ned for each pair of neutrino mass eigenstates i and k and should carry the indices ik. We suppress
them to simplify the notation.
{ 5 {
For illustration, we start with a concrete example. Consider neutrino production in
2body decays at rest X ! l i, where l denotes a charged lepton, i is the ith neutrino
mass eigenstate, and X is either a charged pseudoscalar meson ( , K, . . . ), or W boson,
or a charged scalar particle. The energies and momenta of the produced neutrino mass
eigenstates are
Ei =
m2X
2mX
ml2 + m2
i ;
pi =
[m2X
(ml2 + mi2)]2
distribution anomaly [38], which has been interpreted as a production in
!
decay of
a nonrelativistic neutrino with mass m ' 33:9 MeV and velocity v ' 0:02. Assuming that
the heaviest of the neutrino mass eigenstates produced in Xboson decay is nonrelativistic
and barring near degeneracy of the charged lepton and Xboson masses, from eq. (2.10)
we then nd
thus reduces to
Consider now the energy uncertainty E of the produced neutrino state. For neutrinos
born in decay of a free particle at rest, E is given by the energy uncertainty of the parent
particle, i.e. by its decay width
X . The rst of the two coherence conditions in eq. (2.4)
j Ej
mX :
j Ej
X :
X
mX :
{ 6 {
The decay rates (X ! l i) for the processes under discussion are given in appendix A.
Their common feature is that they can be written as X =
X mX , where mX is the mass of
the Xboson and X
1. The smallness of X is due to the fact that it contains a product
of small numerical and dynamical factors; in the cases when nonrelativistic neutrinos are
produced, the coe cients X are additionally suppressed by a small kinematic factor which
comes from the suppression of the phase space volume available to the nalstate particles.
We thus have
From eq. (2.11) it then follows that the coherent production condition (2.12) is strongly
violated, i.e. di erent neutrino mass eigenstates cannot be produced coherently.
Note that (2.13) is a general property of all unstable particles  their decay width is small compared to their mass. The only exception are decays of very broad resonances, for which
X
mX ; however, even in this case the coherent production condition (2.12) is
violated if a nonrelativistic neutrino is involved.
This result is actually quite general and holds also when neutrinos are produced in
more complicated decays or in reactions. To see this, recall that the produced neutrino
state is described by a wave packet, whose energy dispersion
E is determined by the
(2.11)
(2.12)
(2.13)
temporal localization of the production process. The mean energy of the neutrino state E
can be much larger than
E or of the order of
E, but can never be much smaller than
the energy dispersion
E. For processes with production of a nonrelativistic neutrino, the
di erences
E between its energy and the energies of the other neutrino mass eigenstates
are of the order of the corresponding mean energies. Therefore,
which means that the rst of the two coherent production conditions in eq. (2.4) is not met.
What about the second condition in eq. (2.4)?
When a nonrelativistic neutrino is
p between its momentum and momenta of the other mass
eigenstates satisfy j pj & p, where p is the mean momentum, similarly to what we found
for neutrino energy di erences and energy uncertainty. However, since momentum is a
vector whose projections on coordinate axes can be of either sign, one cannot in general
claim that the modulus of its mean value satis es p &
p
. In particular, for a wave
packet describing neutrino at rest, p =0 while p is nite. The momentum dispersion of
the produced neutrino state is determined by the momentum uncertainty inherent in the
production process, which in turn depends on the spatial localization of this process. The
latter depends on how the source particles were created and on other features of neutrino
production [4], and there are no simple and general arguments that would allow one to
tell if the condition
p
p is satis ed for nonrelativistic neutrinos, in contrast to the
situation with the requirement
E
E. However, for slow neutrino sources the temporal
duration and spatial localization of the neutrino production process are not directly related.
This means that in general no cancellations between the two terms in (2.8) occur, and for
neutrino production to be coherent both conditions in eq. (2.4) must be separately satis ed.
Hence, violation of the rst of these two conditions is su cient to prevent coherent neutrino
emission and thus neutrino oscillations.
One might naturally wonder what happens if we consider the usual neutrino oscillations
(such as e.g. oscillations of reactor, accelerator or atmospheric neutrinos)5 in a reference
frame where one of the neutrino mass eigenstates is slowly moving or at rest. Indeed,
in that case the relations in eq. (2.14) should also be valid, and yet neutrinos must be
oscillating: the answer to the question of whether neutrinos oscillate cannot depend on the
choice of the reference frame in which neutrinos are considered. We shall discuss this issue
in section 3.1.
2.3
Wave packet separation and propagation coherence
In addition to neutrino production and detection coherence, there is another important
coherence condition that has to be satis ed for neutrino oscillations to be observable:
propagation coherence. Coherence may be lost on the way between the neutrino source
and detector because the wave packets of di erent neutrino mass eigenstates propagate
with di erent group velocities. After long enough time (coherence time) they will separate
5In what follows by the `usual neutrino oscillations' we shall always mean oscillations of neutrinos which
are ultrarelativistic in the rest frame of their source.
{ 7 {
by a distance exceeding the spatial length
x of the wave packets, which then cease to
overlap. The coherence time can therefore be found from the relation
where
vg is the di erence of the group velocities of di erent neutrino mass eigenstates.
The corresponding coherence distance is given by
j vgj tcoh ' x ;
Lcoh ' vg tcoh '
vg
j vgj
x ;
(2.15)
(2.16)
where vg is the average group velocity of di erent neutrino mass eigenstates. Since in the
case of ultrarelativistic or quasidegenerate in mass neutrinos di erent mass eigenstates are
produced under essentially the same kinematic conditions, the lengths of their wave packets
xi are practically the same. For processes with emission of nonrelativistic neutrinos, the
lengths of the wave packets of di erent mass eigenstates may di er; the quantity
x in
eqs. (2.15) and (2.16) should then be understood as the largest among
xi. In all known
cases the lengths of the neutrino wave packets are tiny (microscopic);6 still, in the case of
the usual neutrino oscillations (with relativistic or highly degenerate in mass neutrinos)
the coherence distance Lcoh is macroscopic and very long because vg=j vgj ' 2E2=
m2 is
extremely large. The situation is quite di erent when one or more of the produced neutrinos
are nonrelativistic in the laboratory frame, and neutrinos are not quasidegenerate in mass.
The velocity di erences between di erent nonrelativistic neutrino mass eigenstates (or
between relativistic and nonrelativistic states) in that case are j vgj
1; the coherence
distance is therefore microscopic, Lcoh
x. This means that, even if nonrelativistic
neutrinos were produced coherently, they would have lost their coherence due to wave
packet separation practically immediately, before getting a chance of being detected.
3
Lorentz boosts
We have found that in the case when one or more of the produced neutrino mass eigenstates
are nonrelativistic in the reference frame where their source is at rest or is slowly moving,
the production coherence condition is violated and therefore neutrino oscillations cannot
take place. It is interesting to see how this analysis changes and what prevents neutrinos
from oscillating if we go to a frame where all neutrinos are ultrarelativistic.
A di erent but related question is this: how would the usual neutrino oscillations (such
as oscillations of reactor, accelerator or atmospheric neutrinos) look like in a reference frame
where one of the neutrino mass eigenstates is at rest? Neutrinos must obviously oscillate
in that frame as well, but it is very instructive to see where our previous arguments against
oscillations of nonrelativistic neutrinos fail in this case. We study this issue rst.
6A possible exception is the hypothetical recoilless neutrino emission from crystals in Mossbauertype
experiments, in which
x could actually be as long as a few meters [37]. It is not, however, clear if it will
ever be possible to realize such experiments.
{ 8 {
Consider for simplicity 2 avour neutrino oscillations in 1dimensional approach, i.e.
assuming that p~ k ~x. For de niteness, we shall assume that neutrinos are produced in pion
decays at rest. Extensions to the cases of more then two
avours and of moving
neutrino source are straightforward; extension to the full 3dimensional picture of neutrino
oscillations is somewhat more involved but does not pose any problems [4].
Consider rst the neutrino oscillation phase in the frame where the neutrino source
and detector are at rest. By making use of eq. (2.3) valid for ultrarelativistic neutrinos
one can rewrite eq. (2.6) as
1
vg
osc '
(x
vgt) E +
m2
2p
x :
The distance x and the time t between neutrino production and detection may both be
very large, but the di erence x
vgt is always small. It vanishes for pointlike neutrinos; in
the case when neutrinos are described by
nitesize wave packets, it is less than or of the
order of the spatial length of the neutrino wave packet
x: jx
vgtj .
x. The quantity
x is, in turn, determined by the spacetime extension of the neutrino production region
and is typically dominated by its temporal localization or, equivalently, by the energy
uncertainty
E inherent in the neutrino production process [4, 22, 34]. In particular, for
neutrinos produced by nonrelativistic sources x ' vg= E . The rst term on the right
hand side of eq. (3.1) is therefore .
xj Ej=vg ' j Ej= E . If the rst of the coherent
production conditions in eq. (2.4) is satis ed, this term can be neglected, and we obtain
the standard expression for the oscillation phase osc ' [ m2=(2p)]x.
Let us demonstrate that under very general assumptions the second condition in
eq. (2.4) actually follows from the rst one. From eq. (2.3) we nd that the condition
j Ej= E
1 is equivalent to
Barring accidental cancellations, this gives
vg j p
j
E
j m2j
2E E
'
j m2j x
2p
p; taking also into account that vg ' 1,
we
nd that the
rst strong inequality in (3.3) yields the second condition in (2.4), as
advertised. Note that the second condition in (3.3), [j
m2j=(2p)] x
meaning: the size of the neutrino wave packet x should be much smaller than the neutrino
1, has a simple
oscillation length losc = 4 p=
m2. In the example we consider (free pion decay at rest), we
have E '
' 2:5 10 8 eV, E ' p ' 29:8 MeV, and for
m2 =
m2atm ' 2:5 10 3 eV2
we nd that the coherent production conditions (2.4) are satis ed with a very large margin.
Let us now go to a frame where the heavier of the two neutrino mass eigenstates (which
we choose to be 2) is at rest. This is certainly not the best frame to consider neutrino
{ 9 {
oscillations, as the whole setup will look rather weird in it! Indeed, assume that in the
initial frame where the neutrino source and detector are at rest neutrinos are moving in
the positive direction of the xaxis. Then in the new frame 2 will be at rest, 1 will still
be moving in the positive direction of x (though with a smaller velocity), the parent pion
will be moving in the negative xdirection, and the detector will also be moving in the
negative direction of x towards the neutrinos. On top of that, the wave packet describing
the state of 2 will be fast spreading. Indeed, being in the rest frame of 2 means that
the mean momentum of its wave packet vanishes. Still, the neutrino wave packets are
characterized by a nite momentum spread, which means that in its rest frame 2 will have
both positive and negative momentum components along the xaxis, i.e. its wave packet
will quickly spread. Even though this will not a ect observability of neutrino oscillations
because the neutrino detector will \collide" with neutrinos before a signi cant spreading
occurs (see appendix B),7 this adds weirdness to the whole picture. Still, considering
neutrino oscillations in the rest frame of one of the mass eigenstates is very instructive for
understanding when and why nonrelativistic neutrinos can actually oscillate.
Let us go to a reference frame in which the whole neutrino source  detector setup is
boosted with velocity u along the xaxis. The standard Lorentz transformations read
x0 = u(x + ut) ;
Ei0 = u(Ei + upi) ;
t0 = u(t + ux) ;
p0i = u(pi + uEi) ;
where
u = (1
u2) 1=2 is the Lorentz factor of the boost and the prime refers to the
quantities in the new frame. To go to the rest frame of 2 we choose u =
vg2 =
(p2=E2),
which gives u = E2=m2. In the new frame we then have:8
E20 = m2 ;
p02 = 0 ;
E10 '
m22 + m21 ;
2m2
p01 '
m22
2m2
m21 :
For
E0
E0
2
E10 and
p0
p0
2
p01 this gives
E0 '
p0 '
m22
2m2
m21 :
where v0 = (1
transforms as
Next, we consider the transformation laws for neutrino energy and momentum
uncertainties. For neutrinos produced in pion decay at rest, the energy uncertainty is given by
the pion decay width:
E =
. In a moving frame in which the parent pion has velocity
v0 , the energy uncertainty is given by the pion decay width in that frame, 0 =
v02) 1=2 is the Lorentz factor of the boost from the pion's rest frame. That
is, upon going from the pion rest frame to a moving frame the neutrino energy uncertainty
E0 =
E =
v0
u
;
where we have taken into account that v0 coincides with the boost velocity u.
7The spreading of the wave packets of neutrinos that are ultrarelativistic in the rest frame of their
source has negligible e ect on their oscillations. Obviously, the same should also be true in any other frame,
including the rest frame of one of the neutrino mass eigenstates. We discuss these points in appendix B.
8Here and below we take into account that neutrinos are ultrarelativistic in the original frame, with
E1 ' E2 and p1 ' p2.
(3.4)
(3.5)
(3.6)
(3.7)
= v0 ,
(3.8)
Let us consider now the neutrino momentum uncertainty
p
. By the coordinate 
momentum uncertainty relation, it is the reciprocal of the neutrino coordinate uncertainty.
The latter essentially coincides with the length
x of the wave packet of the produced
neutrino. It has been demonstrated in [20, 22] that the quantity xj Ej is invariant under
Lorentz boosts, i.e.
where we have used eq. (3.5). For the momentum uncertainty pj ' 1= xj we therefore
have
i.e. the neutrino momentum uncertainty transforms in the same way as the neutrino energy.
To go from the rest frame of the parent pion to the 2 rest frame we choose u =
vg2,
and eqs. (3.9) and (3.10) give
x0j = xj E0
Ej =
j
xj
u(1 + uvgj )
;
p0j = pj u(1 + uvgj ) ;
are ultrarelativistic) all the neutrino mass eigenstates composing the produced
avour
eigenstate have essentially the same momentum uncertainty p and their wave packets have
the same length
x, this is no longer true in reference frames where some of the neutrino
mass eigenstates are nonrelativistic. In particular, in the rest frame of 2 its wave packet
is the longest one and therefore it is characterized by the smallest momentum uncertainty,
p0min = p02. Note that it is actually the smallest momentum uncertainty that is of interest
to us from the viewpoint of possible violation of the production coherence condition.
Combining eqs. (3.7), (3.8) and (3.11), we nd that in the rest frame of 2
j E0j
0
E
m2
u
' 2m2
' 2E
m2
2
u ;
j p0j
p0min
m2
' 2m2
vg2
u
' 2E
m2
2
u ;
(3.12)
where E '
l is the mean neutrino energy in the pion rest frame and we have taken
into account that u = E2=m2 ' E=m2. From eq. (3.12) it follows that both j E0j= E0 and
j p0j= p0min scale as u2. Therefore, even though conditions (2.4) are satis ed in the original
frame where the parent pion is at rest, they may be badly violated in the rest frame of 2
provided that the boost factor u is large enough, i.e. that the group velocity of the second
neutrino mass eigenstate in the pion rest frame vg2 is su ciently close to 1.
So, something went wrong here. To understand the root of the problem, let us note that
the primary condition of coherent neutrino production is the requirement (2.8) that the
variation of the oscillation phase with varying 4coordinate of the neutrino emission point
be small. Condition (2.4) is secondary and obtains from eq. (2.8) only under the assumption
that the two terms in (2.8) are uncorrelated and do not cancel (or approximately cancel)
each other. It is easy to see that it is actually this seemingly innocent assumption that led
to the above problem. To show this, let us note that the Lorentz transformation (3.4) with
u =
vg2 '
1 gives
t0 ' u( t
x) ;
x0 ' u( x
t) ;
(3.13)
(3.9)
(3.11)
(3.10)
i.e. t0 '
x0. Thus, even if in the original frame t and x are completely independent,
the corresponding quantities in the rest frame of 2 are highly correlated. In addition,
eq. (3.7) tells us that
E0 '
in (2.7) approximately cancel each other:
p0.9 Therefore, in the rest frame of 2 the two terms
0osc =
E0
t0
p0
x0 '
E0 ( t0 + x0) ' 0 :
(3.14)
This shows that (i) eq. (2.8) does not lead to the conditions in eq. (2.4) in this case and (ii)
no enhancement of
0osc actually occurs. More accurate calculation taking into account
the small deviation of u =
vg2 from
1 yields
0osc =
osc
neutrino production condition is satis ed in both frames.
1, so that the coherent
This is exactly as it must be: both the oscillation phase and its variation, being
products of two 4vectors, are Lorentz invariant. So must be the coherence conditions: the
answer to the question of whether di erent mass eigenstates are emitted coherently cannot
depend on the choice of the Lorentz frame in which we look at neutrinos. The conditions
in eq. (2.4), which are often used in the literature as the coherent production conditions,
are not Lorentz invariant; they follow from the Lorentz invariant condition (2.8) only in
reference frames where the neutrino source is nonrelativistic. Obviously, they cannot be
automatically extrapolated from one Lorentz frame to another.
So, we can now answer the question posed at the end of section 2.2.2. In the reference
frame in which neutrino source is at rest or is slowly moving the two terms in the expression
osc =
E t
p x do not in general cancel, and the coherent production condition (2.8)
reduces to (2.4). Since the usual neutrinos produced in pion decay at rest are highly
relativistic with very small energy and momentum di erences of their mass eigenstates,
the coherence conditions (2.4) are very well satis ed for them. In the frame where one of
the produced neutrino mass eigenstates is at rest, the energy and momentum di erences
of neutrino mass eigenstates become large, and conditions (2.4) are no longer satis ed, as
discussed in section 2.2.2. However, in this case eq. (2.4) does not represents the coherent
production condition and is actually irrelevant. This happens because the boost with a very
large Lorentz factor which is necessary to go to the new frame leads to near cancellation
of the two terms in eq. (2.8) in that frame. As a result, conditions (2.4) no longer follow
from the coherent production condition (2.8).
3.2
Boosting nonrelativistic neutrinos
the fact that in the new frame
transformation (3.5) with u ' 1.
E0 '
After we have studied in great detail coherence of the usual neutrino oscillations in the
rest frame of one of the neutrino mass eigenstates, it is easy to understand what happens
when neutrinos produced as nonrelativistic in the rest frame of their source are boosted
to become relativistic. In the original (laboratory) frame, the variations of temporal and
spatial coordinates of the neutrino production point within the production region are not
correlated, and neither are the energy and momentum di erences of the neutrino mass
9While eq. (3.7) is speci c to neutrinos produced in pion decays (or more generally in 2body decays),
p0 is actually quite general. It directly follows from the Lorentz
eigenstates. Under these circumstances the coherent production condition (2.8) leads to
eq. (2.4). Violation of the rst of the constraints in (2.4), j Ej
E, which, as discussed
in section 2.2.2, takes place in this case, therefore means that the coherent production
condition (2.8) is not met.
Assume now we go to a fast moving frame in which all neutrino mass eigenstates are
highly relativistic and have nearly the same energies and momenta. Because of Lorentz
invariance of
osc, the production coherence condition will be violated in the new frame as
well. Thus, boosting neutrinos that were nonrelativistic in the laboratory frame to make
them relativistic will not let them oscillate, as expected.
4
Summary and discussion
We have studied in detail the question of whether neutrinos that are nonrelativistic in
a reference frame in which their source is at rest or is slowly moving can oscillate. The
answer to this question depends on the neutrino mass spectrum. If neutrinos are highly
degenerate in mass, the standard formalism of neutrino oscillations applies to them without
any modi cations, and they do oscillate provided that the standard coherence conditions are
satis ed. This also answers the question why nonrelativistic neutral K, B and D mesons
oscillate: this is because their corresponding mass eigenstates are highly degenerate in mass.
If, however, nonrelativistic neutrinos are not quasidegenerate in mass, their large
energy and momentum di erences prevent di erent mass eigenstates from being produced
coherently. As a result, no oscillations with participation of nonrelativistic neutrinos are
possible. The avour transition probabilities would correspond to averagedout oscillations
in that case, and in particular survival probabilities would exhibit a constant suppression.10
We have also shown that even if nonrelativistic neutrinos were produced coherently,
they would have lost coherence due to their wave packet separation practically immediately,
at microscopic distances from their birthplace. Although propagation decoherence may in
principle be undone by a very coherent neutrino detection [14], in the case of nonrelativistic
neutrinos this would require a completely unrealistic degree of coherence of the detection
process. In addition, even though in general detection may restore neutrino coherence if
it was lost on the way between the source and the detector, the coherence can never be
restored if it was violated at neutrino production.
We have also considered in detail how the choice of the Lorentz frame in uences our
arguments and explicitly demonstrated that the coherence conditions are Lorentz invariant,
as they should be. In particular, since neutrinos which are nonrelativistic in the rest frame
of their source are produced incoherently and do not oscillate in that frame, they will also be
incoherent and will not oscillate upon a boost to a reference frame where they are all
ultrarelativistic. On the other hand, the usual neutrinos that are ultrarelativistic and oscillate
10When the production (or detection) coherence conditions are violated, the probability of the overall
neutrino production  propagation  detection process does not factorize into the production rate,
oscillation probability and detection cross section, so that the very notion of the oscillation probability loses its
sense. In that case one could still, in principle, study the oscillatory behaviour of the overall probability
(with or without lepton
avour change) as a function of the distance between the neutrino production and
detection points. Decoherence, however, means that no such oscillatory behaviour will take place.
in the frame where their source is at rest or is slowly moving will maintain their coherence
and will be oscillating also in the rest frame of any of the neutrino mass eigenstates.
Our discussion demonstrated that the conditions j Ej= E
1, j pj= p
1 that are
often employed as criteria of neutrino production coherence are not Lorentz invariant and
should be used with caution. They can only serve as the coherent production conditions
in the case of nonrelativistic neutrino sources, and in general should be replaced by the
Lorentzinvariant constraint on the variation of the oscillation phase over the neutrino
production region (2.8).
The main reason why neutrinos that are nonrelativistic in the frame where their source
is at rest or is slowly moving do not oscillate is their very large energy and momentum
di erences, which signi cantly exceed the corresponding energy and momentum
uncertainties inherent in the neutrino production process. This is very similar to the reason why
charged leptons do not oscillate [39].
The results of our study are for the most part in agreement with Hinchli e's
rule [40, 41].
A
Decay rates
We present here the rates of 2body decays X ! l i, where l denotes a charged lepton,
i stands for ith neutrino mass eigenstate, and X is either a charged pseudoscalar meson
( , K, . . . ), or W boson, or a charged scalar particle. All the rates are given in the rest
frame of the parent particle and are calculated to leading order in electroweak interaction,
initially without neglecting any masses of the involved particles.
We start with the rate of the charged pion decays
! l i. Direct calculation yields
m2
Here Uli is the element of the leptonic mixing matrix, g ' 0:65 is the SU(2)L gauge coupling
constant, mW is the W boson mass, f
' 130 MeV is the pion decay constant, the rest
of notation being selfexplanatory. Note that the pion decay rate is usually expressed
through the Fermi constant GF = p2g2=(8m2W ); we prefer to express it here through the
dimensionless gauge coupling constant g. For other charged pseudoscalar bosons (X = K,
B, . . . ) the decay rates (X ! l i) can be obtained from (A.1) by the obvious substitution
m
! mX , f
! fX .
Usually, the decay rates of charged pseudoscalar mesons are calculated under the
assumption that all neutrino masses are very small and can be neglected from the outset. The
X ! l l decay rates are then obtained by summing over all the neutrino mass eigenstates.
The resulting expressions are independent of Uli due to unitarity of the leptonic mixing
matrix. Such an approximation is not applicable if neutrinos with mass mi
mX exist.
For small lepton masses the factor ml2m+2mi2 in (A.1) describes chiral suppression of
the Xmeson decay; however, for decays with production of nonrelativistic neutrinos this
factor is not small, i.e. there is no chiral suppression. The factor in the curly brackets in
eq. (A.1) (and similar factors in eqs. (A.2) and (A.4) below) is of kinematic origin; it is
just the magnitude of the momentum of the produced neutrino (and of equal in magnitude
but opposite in sign momentum of the charged lepton) in units of the mass of the parent
particle. It vanishes when ml + mi approaches the parent particle's mass.
Consider next leptonic decays of W boson. The leading order W ! l i decay rate reads
(W ! l i) = g
2
48
mW jUlij2 1
ml2 + mi2
2m2W
(ml2
mi2)2
2m4W
1
Finally, we consider the rate of decay of a charged scalar caused by the Yukawatype
where y is the Yukawa coupling constant. Note that such charged scalars exist in many
extensions of the Standard Model, e.g. in 2 Higgs doublet models. Direct calculation to
the leading order in the Yukawa coupling y yields
( ! l i) = jyj2 m
8
1
(ml + mi)2
m2
1
The production coherence condition (2.12) is more easily satis ed for larger values of
X ; the latter are generally increased with decreasing mass of the produced charged lepton
ml (because this increases the phase space volume available to the nalstate particles). It
therefore may be useful to consider the decay widths of Xbosons also in an (unrealistic)
limit ml ! 0. The rates in eqs. (A.1), (A.2) and (A.4) then simplify to
m2
Neutrino wave packet spreading
Consider the spreading of the neutrino wave packets in the case of the usual neutrino
oscillations. We shall discuss how the e ects of this spreading change when going from
the rest frame of the neutrino source (where all neutrinos are ultrarelativistic) to the rest
frame of one of the neutrino mass eigenstates.
The wave packet spreading is caused by the velocity dispersion, i.e. by the dependence
of the group velocity ~vgi of the neutrino mass eigenstate i on its momentum. Indeed, from
Ei = (p~ 2 + m2)1=2 it follows that for mi 6= 0 the group velocity ~vgi = @Ei=@p~ = p~=Ei is
i
a function of p~. Therefore, the momentum spread within the neutrino wave packet means
that its di erent momentum components propagate with di erent velocities, leading with
time to its spreading. The spreading velocity is thus11
vj '
k
X 1
E
k
jk
vgj vgk
k
p ;
where pk is the neutrino momentum dispersion in the kth direction. For spreading of the
wave packet of the ith neutrino mass eigenstate in the direction of the neutrino propagation
(longitudinal spreading) we obtain
HJEP07(21)
vik =
m2
E 3i p ;
i
tspr i ' mi2 p2
i :
E3
tspr i
tosc
' 4
E2
m2
p2 m2 :
i
where p is the momentum dispersion in the longitudinal direction and we have taken into
account that for neutrinos that are ultrarelativistic in the rest frame of their source the
momentum dispersion of the di erent neutrino mass eigenstates is practically the same.
As follows from eq. (B.2), for ultrarelativistic neutrinos the longitudinal spreading
velocity is very small. As a result, the spreading of their wave packets is of no relevance
to neutrino oscillations. To show this, let us de ne the characteristic spreading time tspr i
as the time over which the wave packet of i spreads to about twice its initial length,
x
1= p: viktspr i ' 1= p. This gives
Let us compare this time with the oscillation time tosc (which for ultrarelativistic neutrinos
coincides with the oscillation length losc = 4 E= m2):
(B.1)
(B.2)
(B.3)
(B.4)
Barring quasidegeneracy of the neutrino masses, from the oscillation data it follows that
m2=mi2 &
m221=
m231
1=30. Next, we note that in realistic situations neutrino energy
is always very large compared to the energy uncertainty: E
!
decay at rest E ' (m2
m2 )=(2m ) ' 29:8 MeV and E '
that E= E ' 1:2
1015. Since for ultrarelativistic neutrinos p '
E. As an example, for
= 2:5
10 8 eV, so
E, the ratio in eq. (B.4)
is extremely large.12 Thus, it takes a much longer time for the neutrino wave packet to
spread by about a factor of two than for neutrino oscillation probability to reach its rst
maximum. This means that the e ects of wave packet spreading on neutrino oscillations
can be safely neglected in all realistic situations.
11We use superscripts to label the Cartesian components of the vectors, whereas lower indices are used
to mark the mass eigenstates. In eq. (B.1) the latter are omitted in order not to overload the notation.
12A possible exception are supernova neutrinos, for which E= E can be as small as
10. However, in
this case the spreading of the neutrino wave packets is not relevant to neutrino oscillations either [24].
Note that the requirement tspr i
losc as a condition for neglecting the wave packet
spreading e ects is actually a very conservative one. Indeed, for the usual neutrino
oscillations, the coherent production condition is satis ed with a large margin, which, in
particular, means that the initial length of the neutrino wave packets satis es x
losc
(see section 3.1). In these circumstances the value of x has no e ect on neutrino
oscillations,13 and neither will have its doubling.
One naturally expects that, if the wave packet spreading e ect on neutrino oscillations
is negligible in the rest frame of the neutrino source, the same will hold in all other Lorentz
frames.
We shall now demonstrate this explicitly. Let us go to the frame where the
neutrino source moves with a velocity u in the direction of neutrino emission. In the new
frame eq. (B.2) yields
HJEP07(21)
vik0 =
m2
E0i3 p0 ' vik u2(1 + uvgi)2
i
;
1
(B.5)
(B.6)
(B.7)
where we have used eqs. (3.5) and (3.10). Note that in the rest frame of the neutrino mass
eigenstate i (i.e. for u =
vgi) the longitudinal spreading velocity of its wave packet is a
factor u2 larger than it is in the rest frame of the parent pion.
Next, let us nd the characteristic spreading time t0spr i in the moving frame. Eqs. (3.9)
and (B.5) yield
0
vk0
i
t0spr i '
x = tspr i u(1 + uvgi) :
Let us now compare the spreading times tspr i and t0spr i with, respectively, the time intervals
between the i production and detection in the original frame and in the new frame,
ti and
t0 .14 Taking into account that
i
xi ' vgi ti, from the Lorentz transformation law (3.4)
one nds
t0i =
ti u(1 + uvgi). Together with eq. (B.6) this gives
Thus, if the wave packet spreading time is much larger than the neutrino ight time in the
rest frame of the neutrino source, the same will be true in any other Lorentz frame, including
the rest frame of one of the neutrino mass eigenstates. Therefore, the relative e ects of the
wave packet spreading on neutrino oscillations is frame independent, as expected.
13Except for neutrino propagation decoherence, for which the nite length of the neutrino wave packet
x is crucial, see section 2.3. However, as was shown in [24], the spreading of the neutrino wave packets
does not a ect the coherence length, which is therefore de ned by the initial value of x.
14Note that if we choose the new frame to be the rest frame of the neutrino mass eigenstate i, the
quantity
t0i will actually be the interval between the neutrino production time and the time when the
detector \collides" with the resting i.
t0
i
t0spr i = tspr i :
t
i
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1330019 [arXiv:1303.6912] [INSPIRE].
HJEP07(21)
[7] D.V. Ahluwalia, Notes on the kinematic structure of the three avor neutrino oscillation
framework, Int. J. Mod. Phys. A 12 (1997) 5081 [hepph/9612471] [INSPIRE].
[8] D.V. Ahluwalia and J.T. Goldman, Interplay of nonrelativistic and relativistic e ects in
neutrino oscillations, Phys. Rev. D 56 (1997) 1698 [hepph/9702308] [INSPIRE].
[9] J.M. Levy, Exercises with the neutrino oscillation length, hepph/0004221 [INSPIRE].
[10] H.T. He, Z.Y. Law, A.H. Chan and C.H. Oh, Nonrelativistic neutrino oscillation in dense
medium, in Proceedings of the Conference in Honour of Murray GellMann's 80th Birthday,
H. Fritzsch and K.K. Phua eds., World Scienti c (2012), pg. 480{487.
[INSPIRE].
[13] C. Giunti, C.W. Kim and U.W. Lee, When do neutrinos really oscillate?: Quantum
mechanics of neutrino oscillations, Phys. Rev. D 44 (1991) 3635 [INSPIRE].
[14] K. Kiers, S. Nussinov and N. Weiss, Coherence e ects in neutrino oscillations, Phys. Rev. D
53 (1996) 537 [hepph/9506271] [INSPIRE].
Phys. B 502 (1997) 3 [hepph/9703241] [INSPIRE].
[15] A.D. Dolgov, A.Yu. Morozov, L.B. Okun and M.G. Shchepkin, Do muons oscillate?, Nucl.
[16] C. Giunti and C.W. Kim, Coherence of neutrino oscillations in the wave packet approach,
Phys. Rev. D 58 (1998) 017301 [hepph/9711363] [INSPIRE].
[17] A.D. Dolgov, Neutrino oscillations and cosmology, hepph/0004032 [INSPIRE].
[18] A.D. Dolgov, Neutrinos in cosmology, Phys. Rept. 370 (2002) 333 [hepph/0202122]
B 805 (2008) 356 [arXiv:0803.0495] [INSPIRE].
[19] C. Giunti, Coherence and wave packets in neutrino oscillations, Found. Phys. Lett. 17 (2004)
[20] Y. Farzan and A.Yu. Smirnov, Coherence and oscillations of cosmic neutrinos, Nucl. Phys.
[arXiv:1309.1717] [INSPIRE].
avor mixing in quantum eld theory, Phys. Rev. D 61
(2000) 073006 [hepph/9909332] [INSPIRE].
(2002) 013003 [hepph/0202068] [INSPIRE].
[hepph/0205014] [INSPIRE].
[1] K.N. Abazajian et al., Light Sterile Neutrinos: A White Paper , arXiv: 1204 .5379 [INSPIRE].
[2] M. Drewes , The Phenomenology of Right Handed Neutrinos, Int. J. Mod. Phys. E 22 ( 2013 ) [3] M. Drewes et al., A White Paper on keV Sterile Neutrino Dark Matter, JCAP 01 ( 2017 ) 025 [4] M. Beuthe , Oscillations of neutrinos and mesons in quantum eld theory , Phys. Rept . 375 [5] C.W. Kim , C. Giunti and U.W. Lee , Oscillations of nonrelativistic neutrinos , Nucl. Phys.
[6] J. Rich , The quantum mechanics of neutrino oscillations , Phys. Rev. D 48 ( 1993 ) 4318 [11] S. Nussinov , Solar Neutrinos and Neutrino Mixing, Phys. Lett. B 63 ( 1976 ) 201 [INSPIRE].
[12] B. Kayser , On the Quantum Mechanics of Neutrino Oscillation, Phys. Rev. D 24 ( 1981 ) 110 [21] L. Visinelli and P. Gondolo , Neutrino Oscillations and Decoherence, arXiv: 0810 .4132 [22] E.K. Akhmedov and A. Yu . Smirnov, Paradoxes of neutrino oscillations , Phys. Atom . Nucl.
[23] D.V. Naumov , On the Theory of Wave Packets, Phys. Part. Nucl. Lett . 10 ( 2013 ) 642 [24] J. Kersten and A. Yu . Smirnov, Decoherence and oscillations of supernova neutrinos , Eur. Phys. J. C 76 ( 2016 ) 339 [arXiv: 1512 .09068] [INSPIRE]. Polon. B 29 ( 1998 ) 3925 [ hep ph/9810543] [INSPIRE].
[25] M. Zralek , From kaons to neutrinos: Quantum mechanics of particle oscillations , Acta Phys.
[26] I. Yu. Kobzarev , B.V. Martemyanov , L.B. Okun and M.G. Shchepkin, The phenomenology of neutrino oscillations , Sov. J. Nucl. Phys . 32 ( 1980 ) 823 [INSPIRE].
[27] I. Yu. Kobzarev , B.V. Martemyanov , L.B. Okun and M.G. Shchepkin , Sum Rules for Neutrino Oscillations, Sov . J. Nucl. Phys . 35 ( 1982 ) 708 [INSPIRE].
[28] C. Giunti , C.W. Kim , J.A. Lee and U.W. Lee , On the treatment of neutrino oscillations without resort to weak eigenstates , Phys. Rev. D 48 ( 1993 ) 4310 [ hep ph/9305276] [INSPIRE].
[29] W. Grimus and P. St ockinger, Real oscillations of virtual neutrinos , Phys. Rev. D 54 ( 1996 ) [40] S. Carroll , Guest Blogger: Joe Polchinski on the string debates , ( 2006 ), [41] B. Peon , Is Hinchli e's rule true?, ( 1988 ),