Lepton Numbers in the framework of Neutrino Mixing
S.M. Bilenky
Joint Institute for Nuclear Research, Dubna, Russia, and
INFN, Sez. di Torino, and Dip. di Fisica Teorica,
Univ. di Torino, I10125 Torino, Italy
and
C. Giunti
INFN, Sez. di Torino, and Dip. di Fisica Teorica,
Univ. di Torino, I10125 Torino, Italy
Abstract
In this short review we discuss the notion of lepton numbers.
The strong evidence in favor of neutrino oscillations
obtained
recently in the SuperKamiokande
atmospheric neutrino experiment
and in solar neutrino
experiments
imply that the law of conservation of family lepton numbers ,
and is strongly violated.
We consider the
states of flavor neutrinos , and
and we discuss the evolution
of these states in space and time
in the case of nonconservation of
family lepton numbers due to the mixing
of light neutrinos.
We discuss and compare different flavor neutrino discovery
experiments.
We stress that
experiments on the search for
and
oscillations
demonstrated that the
flavor neutrino
is a new type of neutrino, different from
and .
In the case of neutrino mixing, the lepton number (only one)
is connected with the nature of massive neutrinos.
Such conserved lepton number exist if massive neutrinos are Dirac
particles.
We review possibilities to check in future experiments whether
the conserved lepton number exists.

Journal:
Int. J. Mod. Phys. A 16, 3931 (2001).
Preprint:
DFTT 3/2001,
hepph/0102320.
Introduction
The strong evidence in favor of neutrino masses and mixing that was
obtained recently in the SuperKamiokande atmospheric neutrino
experiment
[1]
opened a new epoch in neutrino physics.
Evidence in favor of neutrino mixing was also obtained in all solar
neutrino experiments:
Homestake [2],
Kamiokande [3],
GALLEX [4],
SAGE [5],
SuperKamiokande [6],
GNO [7].
Indications in favor
of
oscillations were found in the
accelerator LSND experiment [8].
All these data can be explained
in terms of neutrino oscillations,
if the masses of neutrinos are different
from zero and the fields of massive neutrinos enter in the standard
CC and NC interaction Lagrangian
in the mixed form^{1}

(3) 
Here is the field of the neutrino with mass and
is the mixing matrix.
In the framework of theories with massless neutrinos it was customary to
introduce the family lepton numbers
, and , correspondingly, for the pairs
(
),
(
)
and
(
).
The observation of neutrino oscillations clearly demonstrates that
family lepton numbers are not conserved.
In Section 2
we consider the states of flavor neutrinos and the
evolution of these states in vacuum.
In Section 3
we discuss and compare different flavor neutrino
discovery experiments.
In Section 4
we review the
concept of lepton number in the framework of
the theory of neutrino masses and mixing.
Flavor neutrino states
In this Section we consider,
in the framework of neutrino mixing:
 Decays in which family lepton numbers are not conserved
(like
,
and others).
 Flavor neutrino states.
 Transitions between flavor neutrinos in vacuum (neutrino
oscillations).
Figure 1:
Diagrams contributing to the decay
at lowest order.

If there is neutrino mixing
(Eq. (3)),
processes
of transitions between leptons of different families,
as
,
and others,
become possible.
Let us consider, for example,
the decay
(see the diagrams in Fig.1).
In the simplest case of
mixing of two neutrinos with masses
and the ratio of the probability of
decay
to the probability of the decay
is given by [10,11,12,13,14]

(4) 
where is the mass of the boson,
and is the neutrino mixing angle.
The value of
in Eq. (4) is

(5) 
For
, for the ratio we have

(6) 
Thus, though the processes
,
and others
are in principle allowed
in the case of neutrino mixing,
it is practically
impossible to observe them^{2}.
The strong suppression of the probability of processes like
is due to the fact that the
coefficient in Eq. (5) is very small.
As we will see later,
in the case of neutrino oscillations
the corresponding
coefficient,
( is the neutrino energy and is the distance
between neutrino source and detector),
can be many orders of magnitude larger.
This is the main reason why effects of
violation of the law of conservation of lepton numbers can be revealed in
neutrino oscillation experiments^{3}.
Let us consider now,
in the framework of neutrino mixing,
the state
of a flavor neutrino produced in the CC
weak decay process

(7) 
For example,
in the case of neutrinos produced in nuclear
decay,
,
,
.
If there is neutrino mixing,
the flavor neutrino state
is a superposition of states of massive neutrinos :

(8) 
where
is the relevant element of the matrix.
From the data of laboratory experiments
and
astrophysical observations we know that
neutrino masses are very small
(see [20]),
few eV 
(9) 
Since in order to be detected
in presentday experiments
neutrinos must have energy
,
we have
.
Thus,
the kinematical dependence of the matrix element
on neutrino masses can be neglected with very good approximation,
leading to

(10) 
Here
is the matrix element evaluated in the Standard Model,
with zero neutrino masses.
From Eqs. (8) and (10),
the normalized state describing the flavor neutrino
produced in the decay process (7) is

(11) 
Neutrinos with mass are produced in
standard weak
decays in states with lefthanded as well as
righthanded helicities.
However, in the Standard Model the probability to produce
a neutrino in a state with
righthanded helicity is
negligibly small because it is proportional to
.
Thus, 's and, consequently, flavor neutrinos
are produced in standard weak interaction
processes
in almost pure lefthanded states.
The state of antineutrino
,
the particle that is produced in a CC
weak process together with a ,
is given by

(12) 
The state vector
describes
antineutrinos with righthanded helicity in the case of Dirac
or
neutrinos with righthanded helicity in the case of Majorana .
Hence,
the state vector
describes neutrinos with righthanded helicity.
In the general case of CP violation in the lepton sector,
there are phases in the neutrino
mixing matrix .
Therefore,
the states
and
differ not only by helicity, but also by the sign of the
CPviolating phases.
The violation of CP in the lepton sector can be
revealed through the investigation of neutrino oscillations
(see, for example, [14,9]).
In oscillation experiments neutrinos
are detected at some distance from
the source.
Neutrinos produced as flavor neutrinos
are described at the source by the state (11).
Taking into account the evolution
in space and time,
the neutrino beam
at the distance
from the source
and
at the time after production
is described by the state

(13) 
where
is the threemomentum of the massive neutrino
and
is its energy.
Let us consider
an experiment in which neutrinos
described by the state (13)
are detected through the observation of
the CC process

(14) 
Here is a target nucleon,
is the final lepton and
represents final hadrons.
The amplitude of the process (14) is proportional to

(15) 
As in the case of neutrino production,
for ultrarelativistic neutrinos
the contribution of neutrino masses
to the matrix element in Eq. (15)
can be neglected with very good approximation:

(16) 
where
is the matrix element evaluated in the Standard Model,
with massless neutrinos.
Therefore, the probability amplitude to observe a
flavor neutrino
at the distance
from the source
and
at the time after production of a flavor neutrino
is given by

(17) 
From this expression it is clear that
transitions between
different flavor neutrinos can take place only
if the following two conditions are satisfied:
 The matrix is nondiagonal.
 The phase factors
for different massive
neutrinos are different.
If neutrinos are massless,
and ,
leading to the transition probability

(18) 
In general,
using the unitarity of the neutrino mixing matrix,
we have

(19) 
Let us enumerate the neutrino masses in such a way that

(20) 
Taking into account the unitarity of the mixing matrix ,
the amplitude (17) of
transitions
can be written in the form

(21) 
It is obvious that the common phase
does not enter into the expression
for the transition probability.
Taking into account the fact that the neutrino is detected only
if its threemomentum is aligned along
(i.e.
),
the phase difference
in Eq. (21)
can be written as

(22) 
where
and
.
Furthermore,
we have

(23) 
where

(24) 
with
,
is the standard expression for the
phase difference
(see, for example, [21,14,22,9]).
Here
is the neutrino energy given
by the kinematics of the production process
neglecting neutrino masses.
The second term on the righthand side of Eq. (23)
is much smaller than
.
Indeed,
the kinematics of the production process
implies that

(25) 
Since the
velocity of the neutrino signal
is equal to the velocity of light
minus a correction of the order
,
we have

(26) 
Therefore,
the second term on the righthand side of Eq. (23)
is of the order

(27) 
and
can be neglected^{4}.
The probability of
transitions is given by

(28) 
where
is the distance between neutrino source and detector.
The transition probability depends on the quantity that
is determined by the experimental conditions.
If the quantity is so small that for all

(29) 
then
and violation of the law of conservation of family lepton numbers
cannot be observed.
A violation of this law
can be observed only if
the quantity is large enough so that for at least one
neutrino masssquared difference,
say
,

(30) 
This condition can be rewritten as

(31) 
where
is the sourcedetector distance in meters,
is neutrino energy in MeV,
and
is the neutrino masssquared difference in
.
In shortbaseline neutrino oscillation experiments
,
in longbaseline neutrino oscillation experiments
,
in atmospheric neutrino experiments
,
and
in solar neutrino experiments
,
leading to a sensitivity to
,
,
,
,
respectively.
In conclusion of this section,
let us stress that
in the case of
mixing of neutrinos with small masses,
flavor neutrinos and antineutrinos
are not quanta of the
,
and
fields
[28].
In other words,
,
and
are not fields of particles^{5}.
The neutrinos , ,
(antineutrinos
,
,
),
are produced
in CC weak decays together with, correspondingly,
,
,
(, , ),
and
produce, correspondingly,
,
(, , )
in CC processes of interaction with a nucleon, etc.
These neutrinos carry the flavor of corresponding leptons and
their appropriate names are flavor neutrinos^{6}.
The states of flavor neutrinos (antineutrinos) are superpositions of
states of neutrinos with definite masses and negative helicity
(positive helicity).
Thus, flavor neutrinos do not have definite mass.
The investigation of neutrino oscillations is the most sensitive method
to reveal the violation of the law of conservation of family lepton numbers
(see [21,14,22,9]).
Flavor neutrino discovery experiments
In this section
we discuss and compare different flavor neutrino
discovery experiments.
As it is well known,
the electron neutrino was
discovered by C.L. Cowan and F. Reines in the fifties
[29,30,31].
In 1962, in the Brookhaven experiment
of Lederman, Schwartz, Steinberger et al.
[32]
the second flavor neutrino was discovered.
In 2000
the tau neutrino
has been directly detected
in the DONUT experiment [33].
In the Cowan and Reines experiment,
electron (anti)neutrinos have been detected through
the observation of the process

(32) 
with antineutrinos from the powerful Savannah River reactor.
In a reactor
's are produced in a chain of
decays
of radioactive neutronrich nuclei, products
of the fission of U and U.
The energy of reactor antineutrinos
is less than about 10 MeV.
About
's are emitted per second per KW.
The power of the Savannah River
reactor was
(th).
Thus, about
's per second were emitted by the reactor.
The flux of
's in the Cowan and Reines experiment was
.
As it is well known,
the hypothesis of the existence of neutrino was
put forward in 1930
by W. Pauli in order to solve the problem of continuous
spectra and the problem of the spin and statistics
of some nuclei (like ).
In 1933 E. Fermi assumed that
an electron and an antineutrino are produced in the
process

(33) 
and proposed the first Hamiltonian of decay.
It is a direct consequence of quantum field theory
that an
that is produced in decay together
with an electron
must produce a positron in the process (32).
Moreover, if the interaction responsible for the decay of the neutron is
known, one can connect the cross
section of the process (32) at the small reactor energies
with the lifetime of the neutron.
Neglecting small corrections due to neutron recoil,
the total cross
section of the process (32) is given by
(see, for example, [34])

(34) 
Here
is the energy of the positron
( is the antineutrino energy),
is the lifetime of the neutron,
is the neutron
statistical factor that includes Coulomb interactions of the final proton
and electron,
, and are masses of
the neutron, proton and electron, respectively.
In the Cowan and Reines experiment
positrons and neutrons produced in the process
(32)
were detected
and
for the first time the corresponding very
small neutrino cross section
was measured.
This became possible because of the existence of an
intensive source
of antineutrinos (reactor) and because of the invention of
large scintillator counters.
The total cross section of the process (32) measured in the
Cowan and Reines experiment,

(35) 
was in an agreement with the expected cross section

(36) 
The Cowan and Reines experiment
was a crucial confirmation of the
PauliFermi hypothesis of existence of the neutrino.
This experiment also confirmed the correctness of the fieldtheoretical
relation (34)
between the lifetime the neutron and the cross section of the
crosssymmetrical process (32).
However,
since the energy of antineutrinos from a reactor is not
enough to produce muons, the Cowan and Reines experiment
could not reveal the existence of other flavor neutrinos, besides
.
The next flavor neutrino discovery experiment was the 1962 Brookhaven
experiment
of L.M. Lederman, M. Schwartz, J. Steinberger et al.
[32].
At that time there were some indications that the muon neutrino
(the neutrino that is produced in capture,
decay
and other weak processes in which the muon participates)
and the electron neutrino are different particles.
These indications were based
on the comparison of the results of calculations of the
probability
of
decay with
the experimental upper bound for the
probability of this decay.
If and are the same particle,
the decay
is allowed.
The probability of this decay was calculated in
Ref. [35]
in the framework of a nonrenormalizable theory with
intermediate boson
(diagrams are similar to the diagrams in Fig. 1),
assuming
that the boson has a normal magnetic
moment and that the cutoff mass
is equal to the mass of the .
The resulting value of
the ratio of the probability of the
decay
and
the total probability of muon decay
was
[35].
On the other hand,
the decay
was not observed experimentally.
At the time of the Brookhaven experiment,
the upper bound
was
[36,37]^{7}.
In spite of the indication of the existence of a muon neutrino
given by the nonobservation of
decays,
it was extremely important
to check whether and are the same
or different particles in a direct neutrino experiment.
The Brookhaven experiment was the ideal experiment
for this aim^{8}.
In this experiment the neutrino beam
was produced in the decays of pions
with a small admixture of neutrinos from the decays of kaons and muons.
The dominant decay mode of the meson is

(37) 
According to the universal theory of weak interactions of
Feynman and GellMann [39]
and
Marshak and Sudarshan [40],
the ratio of the probability of the decay

(38) 
and the probability of the decay in Eq. (37)
is about^{9}
.
Hence, the neutrino beam in the Brookhaven experiment was
practically a pure beam of muon neutrinos.
The pion beam in the experiment was produced by 15 GeV
protons striking a berillium target.
Neutrinos from the decays of pions had a
spectrum of energies
.
Neutrino interactions were observed in a 10 ton
aluminum spark chamber.
According to field theory,
a muon neutrinos produced in the decay
(37)
together with a must produce
in the process

(39) 
In order to investigate if
and
are the same or different particles,
one needs to check whether
's can produce also electrons in
the process

(40) 
If electron and muon neutrinos are the same particles,
according to the universal theory one must expect to
observe in the detector approximately
an equal number of electrons and muons.
In the Brookhaven experiment 34 single muon events have been observed,
with an expected background from cosmic rays of 5 events.
The measured cross
section was in agreement with the theory.
Six shower events were observed,
with a distribution of sparks totally different from that
expected for electrons.
If and are the same particles,
29 electron events with energy more than 400 MeV should have been
observed in the experiment^{10}.
Summarizing,
the Brookhaven experiment proved that muon neutrinos,
produced together with muons, cannot produce electrons in
the process (40).
Therefore, it was proved that
and are different flavor neutrinos.
The Brookhaven experiment also proved for the first time that
accelerator 's produced in the process (37)
can be detected.
Let us notice that the results of the Brookhaven experiment
and all other data existing at that time were interpreted in terms
of two
conserved family lepton numbers and that allowed to
distinguish
and
pairs
and to forbid processes of type (40).
We know now that in the framework of neutrino mixing family lepton
numbers are not conserved and muon neutrinos
at some distance can transform into electron neutrinos and produce
electrons (as in the case of LSND experiment [8]).
From the point of view of
neutrino mixing, flavor neutrino discovery experiments require
relatively small distances between neutrino sources and detector and
relatively large energies,
in order to satisfy the condition (29).
In 1975 the third lepton, , was discovered by M. Perl et al.
[44,45].
After this discovery many decay modes of have been investigated:
and others.
All experimental data on
decays are in good agreement with the Standard Model
[46,47,48].
It is a general a consequence of field theory that
the neutrino produced in decays
as those in Eq. (41)
can produce 's in process as

(42) 
and others.
Moreover,
universality of weak interactions
allows to predict the cross section of the
process (42).
Therefore,
the investigation of decays of the type (41)
and subsequent chargedcurrent processes as the one in Eq. (42)
do not allow to check if
is a new type of neutrino, different from
and .
However,
as in the case of and
this can be tested in a different type of neutrino experiment.
In order to prove that
is a new type of neutrino,
it is necessary to prove either that
's
cannot produce electrons or muons,
or that 's and 's
cannot produce 's.
The Brookhaven experiment proved that muon neutrinos produce
muons and do not produce
electrons with the predicted cross section.
However,
another type of experiment
could prove that
and
are different particles.
Imagine that it would be possible to create a pure beam of 's
with energies well above of the threshold of production.
If in an experiment with such a beam it were shown that
's
produce electrons and do not produce muons
with the predicted cross section
(under the assumption that and are the same particles)
it would be proven
that the flavor neutrinos and are different.
So far no experiment has proved that
's
cannot produce electrons or muons,
but several neutrino oscillation experiments
looking for
and
transitions have
proved that 's and 's
cannot produce 's.
These experiments are:
FNALE531
(
and
)
[49],
CHARM II
(
)
[50],
CCFR
(
)
[51],
CHORUS
(
and
)
[52]
NOMAD
(
and
)
[53].
For example,
the neutrino beam in the recent
CHORUS and NOMAD experiments,
produced with the CERN SPS accelerator,
was predominantly composed of 's,
with small
,
and
components.
The percentage of was about 0.9%
and the contamination
of
in the beam is negligible
(
).
The average
energies of and are 27 GeV
and
40 GeV, respectively. Notice that the threshold of production of
's
in the process (42) is 3.5 GeV.
No event of lepton production have been observed in the
CHORUS and NOMAD experiments
at a distance of about 600 m from the source,
leading to the following upper bounds
for the probabilities of
and
transitions:
These very stringent limits imply that
the flavor neutrinos
and are different from
.
If
and
were the same particle,
about 5014 one events
(events with
one reconstructed from the decay
)
would have been observed in the CHORUS experiment.
In reality no event of this type was observed.
If and
were the same particle,
about 23 events with a highly energetic from
the decay
would have been observed in the
NOMAD experiment.
No event of this type was observed above the expected background.
Let us notice that also
the experimental upper bounds for the
relative probabilities of the decays
and
,

(44) 
imply that
is different from
and .
This follows from
an argument that is similar to the one explained above for the decay
at the time of the Brookhaven
experiment,
based on the smallness of the upper limits (44)
with respect to
the value
expected if
and
or and
are the same particle [35].
The flavor neutrino
was directly detected for the first
time in the DONUT experiment [33].
The DONUT experiment is a beamdump experiment.
Neutrinos in this experiment
were produced in the decays of shortlived charm particles.
The neutrino beam was composed mainly
of 's and 's,
with about 5% of
's from the decay

(45) 
Neutrinos are detected in the DONUT experiment in emulsions
at a distance of 36 m from the source.
The important signature of production is the kink from
decay.
In a set of 203 neutrino interactions, four events with a kink,
which satisfy all requirements for the production and decay of ,
were found.
The estimated background is
events.
Up to now we considered only CC processes due to the intermediate
boson.
The investigation of NC processes due to the
intermediate boson, as

(46) 
and others,
have allowed to prove that
and interact with the boson
in accordance with the Standard Model.
The four LEP experiments
(ALEPH, DELPHI, L3, OPAL)
determined with high accuracy that the number
of light flavor neutrinos
(mass
)
produced in the decay of the boson is three
(see [20]):

(47) 
From the observation of the processes (46) it follows that
two flavor neutrinos that contribute to
in Eq. (47) are
and . The most plausible
candidate for
the third neutrino is
, discovered in CC reactions.
It is interesting that we still have no direct proof of that (for such a
proof the investigation of
induced NC processes is required).
Lepton number and neutrino mixing
In the case of neutrino mixing,
the possible existence of a conserved lepton number
can be connected only with neutrinos with definite masses.
The neutrino mass term has the form
h.c. 
(48) 
If the righthanded components and the lefthanded components
are independent,
the massive neutrinos are
Dirac particles.
Indeed, in this case the total Lagrangian is invariant
under the global gauge transformation

(49) 
where is an arbitrary constant.
This invariance implies that the lepton number ,
which has the same value
for , , and all 's,
is conserved.
In this case the
quanta of the fields are
neutrinos with and antineutrinos with .
On the other hand,
if the righthanded components and
the lefthanded components are not independent,
but connected by the relation

(50) 
(
is the matrix of charge conjugation),
the massive neutrinos are Majorana particles.
In this case there is no any gauge invariance of the total
Lagrangian^{11}
and the neutrino field
satisfies the Majorana condition

(51) 
This condition implies that the
quanta of the field
are truly neutral Majorana neutrinos (identical to
antineutrinos).
The problem of the nature of neutrino with definite masses is one of the
most fundamental problem of the physics of massive and mixed neutrinos
and is
connected with the origin of neutrino masses and neutrino mixing.
Dirac neutrino masses can be generated by the standard Higgs
mechanism.
Majorana neutrino masses require a new
mechanism of neutrino mass generation that is beyond the Standard Model.
One of the most popular mechanisms of neutrino mass generation
is the seesaw mechanism
[54,55,56].
This mechanism is based on the assumption that
the law of conservation of lepton number
is violated
at a scale that is much larger then the scale of violation of the
electroweak symmetry. The seesaw mechanism allows to connect
the smallness of neutrino masses with a large physical scale that
characterizes the violation of the lepton number conservation law.
In order to reveal the
Dirac or Majorana nature of neutrinos it is necessary to study neutrino
mass effects^{12}.
It is impossible to distinguish massive Dirac and Majorana neutrinos
through the investigation of neutrino oscillations
[58,59,60].
Indeed,
in the case of neutrino mixing the leptonic CC current has the form

(52) 
If are Dirac fields, the
mixing matrix is determined up to the transformation

(53) 
where
and are arbitrary parameters.
This is due to the fact that the phases of Dirac fields are arbitrary.
In the
case of Majorana neutrinos,
the Majorana condition (51) does not
allow to include arbitrary phases into the fields.
Thus, in the Majorana
case the mixing matrix is determined only up to the transformation

(54) 
From Eqs. (53) and (54) it follows that the number of physical
CPviolating phases in the Dirac and Majorana cases are different^{13}.
From Eq. (17) one can see that under both transformation
(53) and (54)
the amplitude of
transitions is transformed as

(55) 
and the probability of
transitions
is invariant under the transformation (55).
This means that the transition probability
is independent from the additional phases
in the Majorana case.
The most promising process that allows to investigate the nature
of massive neutrino (Dirac or Majorana?) is
neutrinoless double decay of eveneven nuclei:

(56) 
Figure 2:
Diagram of neutrinoless double decay.

The diagram of this process is depicted
in Fig. 2.
In the case of mixing of Majorana
neutrinos, the neutrino propagator in Fig. 2 is given by
From Eq. (57) it follows that in the case of small neutrino masses
the matrix element of neutrinoless double decay is proportional to

(58) 
Thus, the process (56) is allowed if
neutrinos are Majorana particles and massive. Notice that neutrino mixing is
not required for that^{14}.
Neutrinoless double decay
is a process of second order in the Fermi constant .
Its matrix element is proportional to small neutrino masses.
The expected lifetime of neutrinoless double decay
is much larger than the lifetime of usual decays.
However, because of the clear signature of the process
(two electrons with definite total energy in the final state),
several experiments have obtained very large lower bounds
for the lifetime of neutrinoless double decay
of different nuclei
(see [61]).
The lower limits for the lifetimes of
Ge and Xe
obtained in the HeidelbergMoscow experiment
[62]
and in the Gotthard experiment [63]
are
at 90% CL. 
(59) 
These limits imply upper bounds for the effective Majorana mass
,
with values that depend
on the calculation of nuclear matrix elements.
The results of different
calculations lead to the limits
at 90% CL. 
(60) 
Several new experiments searching for
neutrinoless double decay
are in preparation.
These future experiments are planned to be sensitive to values of
[64,65],
or even
[66,67,68].
The results of neutrino oscillation experiments give
information on neutrino masses
and on the elements of the neutrino mixing matrix.
The possible values of
,
depend, among others,
on the character of the neutrino mass spectrum,
on the real existence of the oscillations observed in the LSND experiment,
and
on the absolute values of neutrino masses
(see [69,70,71,72,73,74]).
If the results of the LSND experiment will not be confirmed by future
experiments,
all the other neutrino data can be explained by the existence
of only three massive and mixed neutrinos.
If there is a hierarchy of neutrino masses,

(61) 
there is a stringent upper bound for the effective Majorana mass
[70,72]:

(62) 
In the case of an ``inverted hierarchy'',

(63) 
the upper bound for the effective Majorana mass is less stringent
[72]:

(64) 
If the evidence in favor of
shortbaseline
oscillations
obtained in the LSND experiment will be confirmed by other experiments,
the effective Majorana mass could be as large as
or smaller than
,
depending on the neutrino mass spectrum
[69,70,71,72,73].
Conclusions
We have discussed the notion of lepton numbers
in the case of neutrino mixing.
We have stressed that the existence of
neutrino oscillations means that there are no conserved family
lepton numbers , , .
The flavor neutrinos , and
participate in weak
interactions
( is produced together with
a in decay, etc.).
In the case of small neutrino masses the states
of flavor neutrinos are superpositions of the states of
neutrinos with definite masses.
Flavor neutrinos are
not quanta of any field and they have no definite masses.
We have discussed the difference between different flavor neutrino
discovery experiments.
The results of
and
oscillation experiments clearly demonstrate that
is a
new type of flavor neutrino, different from and .
The
has been detected directly in the recent
DONUT experiment.
We have stressed that a conserved lepton number can exist only if
massive neutrinos are Dirac particles.
In this case the electron, muon, taulepton and
massive neutrinos
have the same value of .
The lepton number
distinguishes neutrinos from antineutrinos.
Different neutrinos
differ by the value of their masses.
If massive neutrinos are Majorana particles there are no conserved
lepton numbers.
The search for neutrinoless double decay
is the most promising method to
test the conservation of lepton number in the case of
neutrino mixing.
Acknowledgments
We would like to thank W. Grimus for useful comments
on a preliminary version of this paper.
We would also like to thank
W. Alberico and A. Bottino for useful discussions.
 1

S. Fukuda et al. (SuperKamiokande Coll.),
Phys. Rev. Lett. 85, 3999 (2000),
hepex/0009001; T. Kajita
(SuperKamiokande Coll.), Talk presented at NOW2000, Otranto, Italy,
September 2000 (http://www.ba.infn.it/~now2000).
 2

B. T. Cleveland et al.,
Astrophys. J. 496, 505 (1998).
 3

Kamiokande, Y. Fukuda et al.,
Phys. Rev. Lett. 77, 1683 (1996).
 4

GALLEX, W. Hampel et al.,
Phys. Lett. B447, 127 (1999).
 5

SAGE, J. N. Abdurashitov et al.,
Phys. Rev. C60, 055801 (1999), astroph/9907113.
 6

Y. Suzuki (SuperKamiokande Coll.), Talk presented at Neutrino 2000, Sudbury,
Canada, 1621 June 2000 (http://nu2000.sno.laurentian.ca);
Talk presented at NOW2000, Otranto, Italy, September 2000
(http://www.ba.infn.it/~now2000).
 7

GNO, M. Altmann et al.,
Phys. Lett. B490, 16 (2000), hepex/0006034.
 8

G. Mills (LSND Coll.), Talk presented at Neutrino 2000, Sudbury, Canada, 1621
June 2000 (http://nu2000.sno.laurentian.ca).
 9

S. M. Bilenky, C. Giunti, and W. Grimus,
Prog. Part. Nucl. Phys. 43, 1 (1999), hepph/9812360.
 10

S. T. Petcov,
Sov. J. Nucl. Phys. 25, 340 (1977),
Erratum, ibid. 25, 698 (1977).
 11

W. J. Marciano and A. I. Sanda,
Phys. Lett. B67, 303 (1977).
 12

B. W. Lee, S. Pakvasa, R. E. Shrock, and H. Sugawara,
Phys. Rev. Lett. 38, 937 (1977),
Erratum, ibid. 38, 1230 (1977).
 13

T. P. Cheng and L.F. Li,
Phys. Rev. Lett. 45, 1908 (1980).
 14

S. M. Bilenky and S. T. Petcov,
Rev. Mod. Phys. 59, 671 (1987).
 15

I.H. Lee,
Phys. Lett. B138, 121 (1984).
 16

R. Barbieri and L. J. Hall,
Phys. Lett. B338, 212 (1994), hepph/9408406.
 17

R. Barbieri, L. Hall, and A. Strumia,
Nucl. Phys. B445, 219 (1995), hepph/9501334.
 18

N. ArkaniHamed, H.C. Cheng, and L. J. Hall,
Phys. Rev. D53, 413 (1996), hepph/9508288.
 19

J. Hisano, T. Moroi, K. Tobe, and M. Yamaguchi,
Phys. Lett. B391, 341 (1997), hepph/9605296.
 20

D. E. Groom et al.,
Eur. Phys. J. C15, 1 (2000),
WWW page: http://pdg.lbl.gov.
 21

S. M. Bilenky and B. Pontecorvo,
Phys. Rept. 41, 225 (1978).
 22

C. W. Kim and A. Pevsner,
Neutrinos in physics and astrophysics (Harwood Academic Press,
Chur, Switzerland, 1993),
Contemporary Concepts in Physics, Vol. 8.
 23

C. Giunti and C. W. Kim,
(2000), hepph/0011074.
 24

L.B. Okun, Surveys High Energy Physics 15, 75 (2000).
 25

W. Grimus, S. Mohanty, and P. Stockinger,
(1999), hepph/9909341.
 26

C. Giunti, C. W. Kim, and U. W. Lee,
Phys. Rev. D44, 3635 (1991).
 27

C. Giunti and C. W. Kim,
Phys. Rev. D58, 017301 (1998), hepph/9711363.
 28

C. Giunti, C. W. Kim, and U. W. Lee,
Phys. Rev. D45, 2414 (1992).
 29

F. Reines and C. L. Cowan,
Phys. Rev. 92, 830 (1953).
 30

C. L. Cowan, F. Reines, F. B. Harrison, H. W. Kruse, and A. D. McGuire,
Science 124, 103 (1956).
 31

F. Reines and C. L. Cowan,
Phys. Rev. 113, 273 (1959).
 32

G. Danby et al.,
Phys. Rev. Lett. 9, 36 (1962).
 33

DONUT, K. Kodama et al.,
Phys. Lett. B504, 218 (2001), hepex/0012035.
 34

E.J. Konopinski, The Theory of Beta Radioactivity (Clarendon, Oxford,
1966).
 35

G. Feinberg,
Phys. Rev. 110, 1482 (1958).
 36

D. Bartlett et al.,
Phys. Rev. Lett. 8, 120 (1962).
 37

S. Frankel et al.,
Phys. Rev. Lett. 8, 123 (1962).
 38

B. Pontecorvo,
Sov. Phys. JETP 37, 1236 (1960).
 39

R. P. Feynman and M. GellMann,
Phys. Rev. 109, 193 (1958).
 40

E. C. G. Sudarshan and R. e. Marshak,
Phys. Rev. 109, 1860 (1958).
 41

T. Fazzini, G. Fidecaro, A. W. Merrison, H. Paul, and A. V. Tollestrup,
Phys. Rev. Lett. 1, 247 (1958).
 42

J. K. Bienlein et al.,
Phys. Lett. 13, 80 (1964).
 43

M. M. Block et al.,
Phys. Lett. 12, 281 (1964).
 44

M. L. Perl et al.,
Phys. Rev. Lett. 35, 1489 (1975).
 45

M. L. Perl,
Ann. Rev. Nucl. Part. Sci. 30, 299 (1980).
 46

A. Pich,
(2000), hepph/0012297.
 47

M. L. Perl,
(1998), hepph/9812400.
 48

A. Stahl,
Physics with tau leptons (Springer, Berlin, Germany, 2000).
 49

FERMILAB E531, N. Ushida et al.,
Phys. Rev. Lett. 57, 2897 (1986).
 50

CHARMII, M. Gruwe et al.,
Phys. Lett. B309, 463 (1993).
 51

CCFR/NuTeV, D. Naples et al.,
Phys. Rev. D59, 031101 (1999), hepex/9809023.
 52

CHORUS, E. Eskut et al.,
Phys. Lett. B497, 8 (2001).
 53

NOMAD, P. Astier et al.,
Phys. Lett. B483, 387 (2000).
 54

M. GellMann, P. Ramond and R. Slansky, in Supergravity, p. 315,
edited by F. van Nieuwenhuizen and D. Freedman, North Holland, Amsterdam,
1979.
 55

T. Yanagida, Proc. of the Workshop on Unified Theory and the Baryon
Number of the Universe, KEK, Japan, 1979.
 56

R. N. Mohapatra and G. Senjanovic,
Phys. Rev. Lett. 44, 912 (1980).
 57

R. E. Marshak, Riazuddin, and C. P. Ryan,
Theory of weak interactions in particle physics
(WileyInterscience, New York, 1969).
 58

S. M. Bilenky, J. Hosek, and S. T. Petcov,
Phys. Lett. B94, 495 (1980).
 59

M. Doi, T. Kotani, H. Nishiura, K. Okuda, and E. Takasugi,
Phys. Lett. B102, 323 (1981).
 60

P. Langacker, S. T. Petcov, G. Steigman, and S. Toshev,
Nucl. Phys. B282, 589 (1987).
 61

V. I. Tretyak and Y. G. Zdesenko,
Atomic Data and Nuclear Data Tables 61, 43 (1995).
 62

L. Baudis et al.,
Phys. Rev. Lett. 83, 41 (1999), hepex/9902014.
 63

R. Luescher et al.,
Phys. Lett. B434, 407 (1998).
 64

A. Alessandrello et al.,
Nucl. Phys. Proc. Suppl. 87, 78 (2000).
 65

NEMO, X. Sarazin and D. Lalanne,
(2000), hepex/0006031.
 66

H. V. KlapdorKleingrothaus, J. Hellmig, and M. Hirsch,
J. Phys. G24, 483 (1998).
 67

G. Bellini et al.,
Phys. Lett. B493, 216 (2000).
 68

M. Danilov et al.,
Phys. Lett. B480, 12 (2000), hepex/0002003.
 69

S. M. Bilenky, C. Giunti, C. W. Kim, and S. T. Petcov,
Phys. Rev. D54, 4432 (1996), hepph/9604364.
 70

S. M. Bilenky, C. Giunti, C. W. Kim, and M. Monteno,
Phys. Rev. D57, 6981 (1998), hepph/9711400.
 71

C. Giunti,
Phys. Rev. D61, 036002 (2000), hepph/9906275.
 72

S. M. Bilenky, C. Giunti, W. Grimus, B. Kayser, and S. T. Petcov,
Phys. Lett. B465, 193 (1999), hepph/9907234.
 73

H. V. KlapdorKleingrothaus, H. Pas, and A. Y. Smirnov,
Phys. Rev. D63, 073005 (2001), hepph/0003219.
 74

S. M. Bilenky, S. Pascoli, and S. T. Petcov,
(2001), hepph/0102265.
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Footnotes
 ... form^{1}

In order to describe all existing neutrino oscillation data,
including LSND data, it is necessary to assume that there are
transitions of flavor neutrinos
into sterile states
(see, for example, [9]).
We will not consider here this possibility.
 ... them^{2}

If the violation of the law of conservation
of lepton numbers is due to other mechanisms,
as supersymmetry, the probability of the decay
and other similar processes can be much larger than in the case
of neutrino mixing
(see [15,16,17,18,19]).
 ... experiments^{3}

In the case of neutrinoless double decay of nuclei
the suppression
of the decay probability
is less strong than
in the decay
and similar processes.
We consider neutrinoless double decay
in Section 4.
 ... neglected^{4}

Let us stress that we did not assume the
equality of momenta or equality of energies of massive neutrinos .
Such assumptions are often discussed in literature
(see [23,24,25]
and references therein).
The wave packet treatment
of neutrino transitions gives the same result
[26,27].
 ... particles^{5}

There is no difference of principle between neutrino mixing and quark
mixing. It is obvious that, for example, in the quark case
is not a particle field but
the combination of the lefthanded components
of the , and fields.
 ... neutrinos^{6}

Sometimes states of flavor neutrinos are called eigenstates of weak
interactions. We do not think that this name reflect the real content
of the notion of a flavor neutrino state.
 ...Bartlett62,Frankel62^{7}

Now the upper bound
is
[20].
 ... aim^{8}

The experiment was proposed by B. Pontecorvo in 1959
[38].
 ... about^{9}

This prediction of the VA
theory was beautifully confirmed in a CERN experiment in 1958
[41].
 ... experiment^{10}

In 1963 in CERN, with the invention of the magnetic horn, the intensity
and purity of neutrino beams was greatly improved.
The Brookhaven result was confirmed with good accuracy
in a large 45 tons
sparkchamber experiment
[42]
and in a large bubble chamber experiment
[43].
 ...
Lagrangian^{11}

In the Majorana case the transformation
requires
.
It is obvious that the
mass term (48) is not invariant under these transformations.
 ... effects^{12}

Dirac neutrinos and Majorana neutrinos are different only if
neutrino masses are different from zero. In the case of standard
electroweak interactions with lefthanded neutrino fields there is no
physical difference between massless Dirac and massless Majorana
neutrinos [57].
 ... different^{13}

For families the number of physical phases
in the case of Dirac neutrinos is
(in this case the number of phases is the same as in the quark case).
In the case of Majorana neutrinos the number of the physical phases is
larger: ,
i.e. there are additional phases.
 ... that^{14}

The probability of neutrinoless double decay is
suppressed because of the
smallness of neutrino masses.
If there is no mixing (i.e. is the unit matrix),
, where is the Majorana mass of .
In neutron decay both
righthanded and lefthanded Majorana electron neutrinos are emitted
together with 's.
The amplitude of the production of lefthanded neutrinos is
proportional to ( is the neutrino energy). The lefthanded
Majorana neutrino can be absorbed by
another neutron in a nucleus with the production of another .
The amplitude of absorption of righthanded neutrinos is
proportional to .
Hence, the amplitude of neutrinoless double
decay
is proportional to
for typical nuclear energies
.
Carlo Giunti
20031111