arXiv:hep-ph/9809234v2 4 Sep 1998
NEUTRINO MASSES, WHERE DO WE STAND?∗
JOS´E W. F. VALLE
Inst. de F´ ısica Corpuscular - C.S.I.C. - Dept. de F´ ısica Te` orica, Univ. de Val` encia
46100 Burjassot, Val` encia, Spain
I review the status of neutrino physics post-Neutrino 98, including the implica-
tions of solar and atmospheric neutrino data, which strongly indicate nonzero
neutrino masses. LSND and the possible role of neutrinos as hot dark matter
(HDM) are also mentioned. The simplest schemes proposed to reconcile these
requirements invoke a light sterile neutrino in addition to the three active ones,
two of them at the MSW scale and the other two maximally-mixed neutrinos
at the HDM/LSND scale. In the simplest theory the latter scale arises at one-
loop, while the solar and atmospheric parameters ∆m2⊙& ∆m2atmappear at
the two-loop level. The lightness of the sterile neutrino, the nearly maximal
atmospheric neutrino mixing, and the generation of ∆m2⊙ & ∆m2atm follow
naturally from the assumed lepton-number symmetry and its breaking. These
two basic schemes can be distinguished at future solar & atmospheric neutrino
experiments and have different cosmological implications.
Neutrinos are the only fermions which the Standard Model (SM) predicts to be
massless. This ansatz was justified due to the apparently masslessness of neutrinos in
most experiments. However, the situation has changed due to the important impact
of underground experiments, since the pioneer geochemical experiments of Davis and
collaborators, to the more recent Gallex, Sage, Kamiokande and SuperKamiokande
experiments1,2,3,4,5. Altogether they provide solid evidence for the solar and the at-
mospheric neutrino problems, two milestones in the search for physics beyond the SM.
Of particular importance has been the recent confirmation by the SuperKamiokande
collaboration3of the atmospheric neutrino zenith-angle-dependent deficit, which has
marked a turning point in our understanding of neutrinos, providing a strong evidence
for νµconversions. In addition to the neutrino data from underground experiments
there is also some possible indication for neutrino oscillations from the LSND ex-
periment6. To this we may add the possible rˆ ole of neutrinos in the dark matter
problem and structure formation7,8,9. If one boldly insists in including also the last
two requirements, together with the data on solar and atmospheric neutrinos, then we
have three mass scales involved in neutrino oscillations. The simplest way to reconcile
these requirements invokes the existence of a light sterile neutrino10,11,12. The pro-
∗Proceedings of New Trends in Neutrino Physics, May 1998, Ringberg Castle, Tegernsee, Germany.
Presented also as a set of three lectures at the V Gleb Wataghin School on High Energy Phenomenol-
ogy, Campinas, Brazil, July 1998
totype models proposed in10,11enlarge the SU(2) ⊗U(1) Higgs sector in such a way
that neutrinos acquire mass radiatively, without unification nor seesaw. Out of the
four neutrinos, two of them lie at the MSW scale and the other two maximally-mixed
neutrinos are at the HDM/LSND scale. The latter scale arises at one-loop, while the
solar and atmospheric scales come in at the two-loop level. The lightness of the sterile
neutrino, the nearly maximal atmospheric neutrino mixing, and the generation of the
solar and atmospheric neutrino scales all result naturally from the assumed lepton-
number symmetry and its breaking. Either νe- ντconversions explain the solar data
with νµ- νsoscillations accounting for the atmospheric deficit10, or else the rˆ oles
of ντand νsare reversed
future solar & atmospheric neutrino experiments, as well as cosmology.
11. These two basic schemes have distinct implications at
2. Theories of Neutrino Mass
One of the most unpleasant features of the SM is that the masslessness of neutrinos
is not dictated by an underlying principle, such as that of gauge invariance in the case
of the photon: the SM simply postulates that neutrinos are massless by choosing a
restricted multiplet content. Why are neutrinos so special when compared with the
other fundamental fermions? If massive, neutrinos would present another puzzle:
Why are their masses so small compared to those of the charged fermions? The fact
that neutrinos are the only electrically neutral elementary fermions may hold the key
to the answer, namely neutrinos could be Majorana fermions, the most fundamental
kind of fermion. In this case the suppression of their mass could be associated to the
breaking of lepton number symmetry at a very large energy scale within a unification
approach, which can be implemented in many extensions of the SM. Alternatively,
neutrino masses could arise from garden-variety weak-scale physics characterized by
a scale ?σ? = O (mZ) where ?σ? denotes a SU(2) ⊗U(1) singlet vacuum expectation
value which owes its smallness to the symmetry enhancement which would result if
?σ? and mν→ 0.
One should realize however that, although the physics of neutrinos can be rather
different in various gauge theories of neutrino mass, there is hardly any predictive
power on masses and mixings, this is one of the aspects of the so-called flavour
problem which is probably the toughest open problem in physics.
2.1. Unification Approach
An attractive possibility is to ascribe the origin of parity violation in the weak
interaction to the spontaneous breaking of B-L symmetry in the context of left-right
symmetric extensions such as the SU(2)L⊗SU(2)R⊗U(1)13, SU(4)⊗SU(2)⊗SU(2)14
or SO(10) gauge groups15. In this case the masses of the light neutrinos are obtained
by diagonalizing the following mass matrix in the basis ν,νc
where D is the standard SU(2)⊗U(1) breaking Dirac mass term and MR= MT
isosinglet Majorana mass that may arise from a 126 vacuum expectation value (vev)
in SO(10). The magnitude of the MLνν term16is also suppressed by the left-right
breaking scale, ML∝ 1/MR
In the seesaw approximation, one finds
Mν eff= ML− DM−1
As a result one is able to explain naturally the relative smallness of neutrino masses
since mν∝ 1/MR. Although MRis expected to be large, its magnitude heavily de-
pends on the model and it may have different possible structures in flavour space
(so-called textures)17. As a result it is hard to make firm predictions for the corre-
sponding light neutrino masses and mixings that are generated through the seesaw
mechanism. In fact this freedom has been exploited in model building in order to
account for an almost degenerate seesaw-induced neutrino mass spectrum18.
One virtue of the unification approach is that it may allow one to gain a deeper
insight into the flavour problem. There have been interesting attempts at formulating
supersymmetric unified schemes with flavour symmetries and texture zeros in the
Yukawa couplings. In this context a challenge is to obtain the large lepton mixing
now indicated by the atmospheric neutrino data.
2.2. Weak-Scale Approach
Although very attractive, the unification approach is by no means the only way
to generate neutrino masses. There are many schemes which do not require any large
mass scale. The extra particles employed to generate the neutrino masses have masses
O (mZ) accessible to present experiments. There is a variety of such mechanisms, in
which neutrinos acquire mass either at the tree level or radiatively. Let us look at
2.2.1. Tree Level
For example, it is possible to extend the lepton sector of the SU(2)⊗U(1) theory
by adding a set of two 2-component isosinglet neutral fermions, denoted νciand Si,
i = e, µ or τ in each generation. In this case one can consider the mass matrix (in
the basis ν,νc,S)19
The Majorana masses for the neutrinos are determined from
In the limit µ → 0 the exact lepton number symmetry is recovered and will keep
neutrinos strictly massless to all orders in perturbation theory, as in the SM. The
corresponding texture of the mass matrix has been suggested in various theoretical
models20, such as superstring inspired models21. In the latter the zeros arise due to
the lack of Higgs fields to provide the usual Majorana mass terms. The smallness of
neutrino mass then follows from the smallness of µ. The scale characterizing M, unlike
MRin the seesaw scheme, can be low. As a result, in contrast to the heavy neutral
leptons of the seesaw scheme, those of the present model can be light enough as to be
produced at high energy colliders such as LEP22or at a future Linear Collider. The
smallness of µ is in turn natural, in t’Hooft’s sense, as the symmetry increases when
µ → 0, i.e. total lepton number is restored. This scheme is a good alternative to the
smallness of neutrino mass, as it bypasses the need for a large mass scale, present in the
seesaw unification approach. One can show that, since the matrices D and M are not
simultaneously diagonal, the leptonic charged current exhibits a non-trivial structure
that cannot be rotated away, even if we set µ ≡ 0. The phenomenological implication
of this, otherwise innocuous twist on the SM, is that there is neutrino mixing despite
the fact that light neutrinos are strictly massless. It follows that flavour and CP are
violated in the leptonic currents, despite the masslessness of neutrinos. The loop-
induced lepton flavour and CP non-conservation effects, such as µ → e + γ23,24, or
CP asymmetries in lepton-flavour-violating processes such as Z → e¯ τ or Z → τ¯ e25
are precisely calculable. The resulting rates may be of experimental interest26,27,28,
since they are not constrained by the bounds on neutrino mass, only by those on
universality, which are relatively poor. In short, this is a conceptually simple and
phenomenologically rich scheme.
Another remarkable implication of this model is a new type of resonant neutrino
conversion mechanism29, which was the first resonant mechanism to be proposed
after the MSW effect30, in an unsuccessful attempt to bypass the need for neutrino
mass in the resolution of the solar neutrino problem. According to the mechanism,
massless neutrinos and anti-neutrinos may undergo resonant flavour conversion, under
certain conditions. Though these do not occur in the Sun, they can be realized in the
chemical environment of supernovae31. Recently it has been pointed out how they
may provide an elegant approach for explaining the observed velocity of pulsars32.
2.2.2. Radiative Level
There is also a large variety of radiative models, where the SU(2)⊗U(1) multiplet
content is extended in order to generate neutrino masses. The prototype one-loop
scheme is the one proposed by Zee33. Supersymmetry with explicitly broken R-parity
Figure 1: One-loop-induced Neutrino Mass.
Figure 2: Two-loop-induced Neutrino Mass
also provides an alternative one-loop mechanism to generate neutrino mass. These
arise, for example, from scalar quark or scalar lepton contributions, as shown in Fig.
A two-loop scheme to induce neutrino mass was suggested by Babu
relevant diagram is shown in Fig. (2)a.
In the above examples active neutrinos acquire radiative mass. One can also
employ the radiative approach to construct models including sterile neutrinos, such as
those in ref.10,11. In this case some new Feynman diagram topologies are encountered.
2.3. A Hybrid Approach
I now describe an interesting mechanism of neutrino mass generation that com-
bines seesaw and radiative mechanisms. It invokes supersymmetry with broken R-
parity, as the origin of neutrino mass and mixings36. The simplest model is a unified
minimal supergravity model with universal soft breaking parameters (MSUGRA) and
bilinear breaking of R–parity36,37. Contrary to a popular misconception, the bilinear
violation of R–parity implied by the ǫ3term in the superpotential is physical, and
can not be rotated away38. It leads also by a minimization condition, to a non-zero
aNote here that I have used the slight variant of the Babu model suggested in ref.
incorporates the idea of spontaneous, rather than explicit lepton number violation
sneutrino vev, v3. It is well-known39that in such models of broken R–parity the tau
neutrino ντacquires a mass, due to the mixing between neutrinos and neutralinos. It
comes from the matrix
where the first two rows are gauginos, the next two Higgsinos, and the last one denotes
the tau neutrino. The vu and vd are the standard vevs, g′s are gauge couplings
and M1,2are the gaugino mass parameters. Since the ǫ3and the v3are related, the
simplest (one-generation) version of this model contains only one extra free parameter
in addition to those of the MSUGRA model. The universal soft supersymmetry-
breaking parameters at the unification scale mXare evolved via renormalization group
equations down to the weak scale O (mZ). This induces an effective non-universality
of the soft terms at the weak scale which in turn implies a non-zero sneutrino vev v′
where the primed quantities refer to a basis in which we eliminate the ǫ3term from
the superpotential (but reintroduce it, of course, in other sectors of the theory).
The scalar soft masses and bilinear mass parameters obey ∆M2= 0 and ∆B = 0
at mX. However at the weak scale they are calculable from radiative corrections as
Note that eq. (6) implies that the R–parity-violating effects induced by v′
culable in terms of the primordial R–parity-violating parameter ǫ3. It is clear that
the universality of the soft terms plays a crucial rˆ ole in the calculability of the v′
and hence of the resulting neutrino mass36. Thus eq. (5) represents a new kind of
see-saw scheme in which the MRof eq. (1) is the neutralino mass, while the rˆ ole of
the Dirac entry D is played by the v′
evolve from mXto the weak scale. Thus we have a hybrid see-saw mechanism, with
naturally suppressed Majorana ντ mass induced by the mixing between the weak
eigenstate tau neutrino and the zino.
Let me now turn to estimate the expected ντmass. For this purpose let me
first determine the tau neutrino mass in the most general supersymmetric model
with bilinear breaking of R-parity, without imposing soft universality. The ντmass
depends quadratically on an effective parameter ξ defined as ξ ≡ (ǫ3vd+ µv3)2∝ v′
characterizing the violation of R–parity. The expected mντvalues are illustrated
in Fig. (3). The band shown in the figure is obtained through a scan over the
3, which is induced radiatively as the parameters
ξ / (100 GeV)4
Figure 3: Tau neutrino mass versus ξ ≡ (ǫ3vd+ µv3)2, from ref.36
parameter space requiring that the supersymmetric particles are not too light. Let
us now compare this with the cosmologically allowed values of the tau neutrino mass.
The cosmological critical density bound mντ<∼92Ωh2eV only holds if neutrinos are
stable. In the present model (with 3-generations) the ντcan decay into 3 neutrinos,
via the neutral current16,40, or by slepton exchanges. This decay will reduce the relic
ντabundance to the required level, as long as ντis heavier than about 100 KeV or
so. On the other hand primordial Big-Bang nucleosynthesis implies that ντis lighter
than about an MeV or so41.
However, if one adopts a SUGRA scheme where universality of the soft supersym-
metry breaking terms at mXis assumed, then the ντmass is theoretically predicted
in terms of hband can be small in this case due to a natural cancellation between
the two terms in the parameter ξ, which follows from the assumed universality of the
softs at mX. One can verify that mντmay easily lie in the electron-volt range, in
which case ντcould be a component of the hot dark matter of the Universe.
Notice that νeand νµremain massless in this approximation. They get masses
either from scalar loop contributions in Fig. (1) or by mixing with singlets in models
with spontaneous breaking of R-parity42. It is important to notice that even when
mντis small, many of the corresponding R-parity violating effects can be sizeable.
An obvious example is the fact that the lightest neutralino decay will typically decay
inside the detector, unlike standard R-parity-conserving supersymmetry. This leads
to a vastly unexplored plethora of phenomenological possibilities in supersymmetric
In conclusion I can say that, other than the seesaw scheme, none of the above
models requires a large mass scale. As a result they lead to a potentially rich phe-
nomenology, since the extra particles required have masses at scales that could be ac-
cessible to present experiments. In the simplest versions of these models the neutrino
mass arises from the explicit violation of lepton number. Their phenomenological
potential gets richer if one generalizes the models so as to implement a spontaneous
violation scheme. This brings me to the next section.
2.4. Weak-scale majoron
If lepton number (or B-L) is an ungauged symmetry and if it is arranged to break
spontaneously, the generation of neutrino masses will be accompanied by the existence
of a physical Goldstone boson that we generically call majoron. Except for the left-
right symmetric unification approach, in which B-L is a gauge symmetry, in all of the
above schemes one can implement the spontaneous violation of lepton number. One
can also introduce it in an SU(2)⊗U(1) seesaw framework44, as originally proposed,
but I do not consider this case here, see ref.45for a review. Here I will mainly
concentrate on weak-scale physics. In all models I consider the lepton-number breaks
at a scale given by a vacuum expectation value ?σ? ∼ mweak. Such scale arises as the
most natural one since in all of these models, as already mentioned, we have that the
neutrino masses vanish as the lepton-breaking scale ?σ? → 046.
It is also clear that in any acceptable model one must arrange for the majoron
to be mainly an SU(2) ⊗ U(1) singlet, ensuring that it does not affect the invisible
Z decay width, well-measured at LEP. In models where the majoron has L=2 the
neutrino mass is proportional to an insertion of ?σ?, as indicated in Fig. (2). In
the supersymmetric model with broken R-parity the majoron is mainly a singlet
sneutrino, which has lepton number L=1, so that mν∝ ?σ?2, where ?σ? ≡
? νcdenoting the singlet sneutrino. The presence of the square, just as in the parameter
ξ in Fig. (3), reflects the fact that the neutrino gets a Majorana mass which has
lepton number L=2. The sneutrino gets a vev at the effective supersymmetry breaking
scale msusy= mweak.
The weak-scale majorons may have remarkable phenomenological implications,
such as the possibility of invisibly decaying Higgs bosons46. Unfortunately I have no
time to discuss it here (see, for instance43).
If the majoron acquires a KeV mass (natural in weak-scale models) from gravita-
tional effects at the Planck scale47it may play a rˆ ole in cosmology as dark matter48.
In what follows I will just focus on two examples of how the underlying physics of
weak-scale majoron models can affect neutrino cosmology in an important way.
2.4.1. Heavy neutrinos and the Universe Mass
Neutrinos of mass less than O (100 KeV) or so, are cosmologically stable if they
have only SM interactions. Their contribution to the present density of the universe
?mνi<∼92 Ωνh2eV , (8)
where the sum is over all isodoublet neutrino species with mass less than O (1 MeV).
The parameter Ωνh2≤ 1, where h2measures the uncertainty in the present value of
the Hubble parameter, 0.4<∼h<∼1, while Ων= ρν/ρc, measures the fraction of the
critical density ρcin neutrinos. For the νµand ντthis bound is much more stringent
than the laboratory limits.
In weak-scale majoron models the generation of neutrino mass is accompanied
by the existence of a physical majoron, with potentially fast majoron-emitting decay
channels such as45,50
ν′→ ν + J .(9)
as well as new annihilations to majorons,
ν′+ ν′→ J + J .(10)
These could eliminate relic neutrinos and therefore allow neutrinos of higher mass,
as long as the rates are large enough to allow for an adequate red-shift of the heavy
neutrino decay and/or annihilation products. While the annihilation involves a diag-
onal majoron-neutrino coupling g, the decays proceed only via the non-diagonal part
of the coupling, in the physical mass basis. A careful diagonalization of both mass
matrix and coupling matrix is essential in order to avoid wild over-estimates of the
heavy neutrino decay rates, such as that in ref.44. The point is that, once the neu-
trino mass matrix is diagonalized, there is a danger of simultaneously diagonalizing
the majoron couplings to neutrinos. That would be analogous to the GIM mecha-
nism present in the SM for the couplings of the Higgs to fermions. Models that avoid
this GIM mechanism in the majoron-neutrino couplings have been proposed, e.g. in
ref.50. Many of them are weak-scale majoron models19,46,42. A general method to
determine the majoron couplings to neutrinos and hence the neutrino decay rates in
any majoron model was first given in ref.
spontaneously broken R-parity51see ref.42.
In short one may say that neutrino lifetimes can be shorter than required by the
cosmological mass bound, for all values of the masses which are presently allowed by
40. For an estimate in the model with
2.4.2. Heavy neutrinos and Cosmological Nucleosynthesis
Similarly, the number of light neutrino species is also restricted by cosmological
Big Bang Nucleosynthesis (BBN). Due to its large mass, an MeV stable (lifetime
longer than ∼ 100 sec) tau neutrino would be equivalent to several SM massless neu-
trino species and would therefore substantially increase the abundance of primordially
produced elements, such as4He and deuterium52,53,54. This can be converted into
restrictions on the ντmass. If the bound on the effective number of massless neutrino
species is taken as Nν< 3.4−3.6, one can rule out ντmasses above 0.5 MeV41. If we
Figure 4: The dashed line shows the effective number of massless SM neutrinos equiv-
alent to the heavy ντ(g = 0). Depending on the value of g (in units of 10−5) one can
lower Nνbelow the canonical SM value Nν= 3 due to the effect of ντannihilations.
take Nν< 4.554the mντlimit loosens accordingly, as seen from Fig. (4), and allows
a ντof about an MeV or so.
In the presence of ντannihilations the BBN mντbound is substantially weakened
or eliminated55. In Fig. (4) we also give the expected Nνvalue for different values of
the coupling g between ντ’s and J’s, expressed in units of 10−5. Comparing with the
SM g = 0 case one sees that for a fixed Nmax
is allowed for large enough values of g. No ντmasses below the LEP limit can be
ruled out, as long as g exceeds a few times 10−4. One can also see from the figure
that Nνcan also be lowered below the canonical SM value Nν= 3 due to the effect
of the heavy ντannihilations to majorons. These results may be re-expressed in the
mντ− g plane, as shown in figure 5. We note that the required values of g(mντ) fit
well with the theoretical expectations of many weak-scale majoron models.
The above discussion has been on the effect of ντannihilations to majorons in
BBN. In some weak-scale majoron models decays in eq. (9) may lead to short enough
ντlifetimes that they may also play an important rˆ ole in BBN56.
Before concluding the discussion on majorons, let me comment that the majoron
may be realized even in the context of models where B-L is a gauge symmetry, such as
left-right-symmetric models, by suitably implementing a spontaneously broken global
U(1) symmetry similar to lepton number. It plays an interesting rˆ ole in such models
as it allows the left-right scale to be relatively low57.
, a wide range of tau neutrino masses
Figure 5: The region above each curve is allowed for the corresponding Nmax
3. Indications for Neutrino Mass
The most solid indications in favour of nonzero neutrino masses come from under-
ground experiments on solar and atmospheric neutrinos. I will provide a theorist’s
sketch of the present experimental situation.
3.1. Solar Neutrinos
The puzzle posed by the data collected by the Homestake, Kamiokande, and the
radiochemical Gallex and Sage experiments still defy an explanation in terms of the
Standard Model. The most recent data on rates are summarized as: 2.56 ± 0.23
SNU (chlorine), 72.2 ± 5.6 SNU (Gallex and Sage gallium experiments sensitive to
the pp neutrinos), and (2.44±0.10)×106cm−2s−1(8B flux from SuperKamiokande)1.
This has been re-confirmed by the 504 days data sample now collected by the Su-
perKamiokande (SK) collaboration and reported at Neutrino 98
can see the predictions of various standard solar models in the plane defined by the
7Be and8B neutrino fluxes, normalized to the predictions of the BP98 solar model59.
Abbreviations such as BP95, identify different solar models, as given in ref.60. The
rectangular error box gives the 3σ error range of the BP98 fluxes. The values of these
fluxes indicated by present data on neutrino event rates are also shown by the contours
in the figure. The best-fit7Be neutrino flux is negative! Possible non-standard astro-
physical solutions are strongly constrained by helioseismology studies58 61. Within
the standard solar model approach, the theoretical predictions clearly lie far from the
best-fit solution, and even far from the 3σ contour, leading us to conclude that new
5. In Fig. (6) one
Figure 6: Recent SSM predictions, from ref.58
particle physics is the only way to account for the data62.
The most likely possibility is to assume the existence of neutrino conversions
involving very small neutrino masses. The most attractive theoretical schemes are
the MSW effect30, vacuum neutrino oscillations or just-so solution and, possibly,
the Spin-Flavour Precession mechanism proposed in ref.63, aided by the Resonant
enhancement due to matter effects in the Sun found in ref.64. The resulting RSFP
mechanism still provides a viable solution to the solar neutrino problem65.
The recent SK data updates the 300 days situation we had before Neutrino 984
without major surprises, except that the SK collaboration has now given the first de-
tailed report of the recoil energy spectrum produced by solar neutrino interactions5.
The measured spectrum they reported at Neutrino 98 shows more events at the high-
est bins than would have been expected from the most popular neutrino oscillation
parameters discussed previously. At first sight this might seem bad news for the os-
cillation scenarios. However, Bahcall and Krastev have noted that if the low energy
cross section for3He + p →
times larger than the best (but uncertain) theoretical estimates, then this reaction
could significantly influence the electron energy spectrum produced by solar neutrino
interactions in the high recoil region. This would hardly have any effect at lower ener-
gies. They compare the predicted energy spectra for different assumed hep fluxes and
different neutrino oscillation scenarios with the one measured at SuperKamiokande.
Fig. 7 shows the ratio of the measured5to the calculated number of events with
electron recoil energy E. The crosses are the recent SK measurements5, while the
calculated curves are global fits to all of the data. The horizontal line at Ratio = 0.37
represents the ratio of the total event rate measured by SuperKamiokande to the
predicted event rate59with no oscillations and only8B neutrinos. One sees how the
4He + e++ νe, the so-called hep reaction, is >∼20
Figure 7: Combined8B plus hep energy spectrum from ref.66. The total flux of hep
neutrinos was varied so as to obtain the best-fit for each scenario.
spectra with enhanced hep contributions provide better fits to the SK data, suggesting
that these neutrinos may be playing a rˆ ole.
One can determine the required solar neutrino parameters ∆m2and sin22θ through
a χ2fit of the experimental data. In Fig. (8) we show the allowed two-flavour regions
obtained in an updated MSW global fit analysis of the solar neutrino data for the case
of active neutrino conversions. The data include the chlorine, Gallex, Sage1and SK
total event rates5, the SK energy spectrum5, as well as the SK day-night asymme-
try5, which would be expected in the MSW scheme due to regeneration effects at the
Earth. The data also includes the recent SK 504 days sample. The analysis uses the
BP98 model but with an arbitrary hep neutrino flux67. One notices from the analysis
that rate-independent observables, such as the electron recoil energy spectrum and
the day-night asymmetry (zenith angle distribution), play an important rˆ ole in ruling
out large regions of MSW parameters.
A theoretical issue which has raised some interest recently is the study of the
possible effect of random fluctuations in the solar matter density68,69,70. The possible
existence of noise fluctuations at a few percent level is not excluded by present helio-
seismology studies. In Fig. (9) we show averaged solar neutrino survival probability
as a function of E/∆m2, for sin22θ = 0.01. This figure was obtained via a numerical
integration of the MSW evolution equation in the presence of noise, using the density
profile in the Sun from BP95 in ref.60, and assuming that the correlation length L0
(which corresponds to the scale of the fluctuation) is L0= 0.1λm, where λmis the
neutrino oscillation length in matter. An important assumption in the analysis is that
lfree≪ L0≪ λm, where lfree∼ 10 cm is the mean free path of the electrons in the
solar medium. The fluctuations may strongly affect the7Be neutrino component of
the solar neutrino spectrum so that the Borexino experiment should provide an ideal
Figure 8: Presently allowed MSW solar neutrino parameters for 2-flavour active neu-
trino conversions with an enhanced hep flux, from ref.66
Solar neutrino survival probability in the presence of random density
Figure 10: Presently allowed vacuum oscillation parameters, from ref.67
test, if sufficiently small errors can be achieved. The potential of Borexino in probing
the level of solar matter density fluctuations provides an additional motivation for
the experiment71. This is discussed in more detail in ref.69.
The most popular alternative solution to the solar neutrino problem is the vacuum
oscillation solution which clearly requires large neutrino mixing and just-so adjust-
ment of the oscillation length so as to coincide roughly with the Earth-Sun distance.
This solution fits with simplistic see-saw inspired-numerology and has attractive fea-
tures, as recently advocated in ref.72. Fig. 10 shows the regions of just-so oscillation
parameters obtained in a recent global fit of the data including the 504 days SK data
sample, both the rates and the recoil energy spectrum. Seasonal effects are expected
in this scenario and could potentially be used to further constrain the parameters, as
described in ref.73, and also to help discriminating it from the MSW scenario.
3.2. Atmospheric Neutrinos
Showers initiated when primary cosmic rays hit the Earth’s atmosphere origi-
nate secondary mesons, mostly pions and kaons, which decay producing νe’s, νµ’s
as well as ¯ νe’s and ¯ νµ’s74. There has been a long-standing discrepancy between
the predicted and measured νµ/νeratio of the atmospheric neutrino fluxes2. The
anomaly was found both in water Cerenkov experiments, such as Kamiokande, Su-
perKamiokande and IMB4, as well as in the iron calorimeter Soudan2 experiment.
Negative experiments, such as Frejus and Nusex have much larger errors.
Although individual νµor νefluxes are only known to within 30% accuracy, the
νµ/νeratio is known to 5%. The most important feature of the atmospheric neutrino
535-daydata sample reported by the SK collaboration at Neutrino 983is that
it exhibits a zenith-angle-dependent deficit of muon neutrinos which is inconsistent
Figure 11: Theoretically expected zenith angle distributions for SK electron and
muon-like sub-GeV and multi-GeV events in the SM (no-oscillation) and for the best-
fit points of the various oscillation channels, from ref.75,76. The crosses correspond
to the SK observations reported at Neutrino 98.
with expectations based on calculations of the atmospheric neutrino fluxes. For recent
analyses see ref.75,76,77. Experimental biases and uncertainties in the prediction of
neutrino fluxes and cross sections are unable to explain the data.
In Fig. (11) I show the measured zenith angle distribution of electron-like and
muon-like sub-GeV and multi-GeV events, as well as the one predicted in the absence
of oscillation. I also give the expected distribution in various neutrino oscillation
schemes. The thick-solid histogram is the theoretically expected distribution in the
absence of oscillation, while the predictions for the best-fit points of the various
oscillation channels is indicated as follows: for νµ→ νs(solid line), νµ→ νe(dashed
line) and νµ→ ντ (dotted line). The error displayed in the experimental points is
In the theoretical analysis we have used the latest improved calculations of the
atmospheric neutrino fluxes as a function of zenith angle, including the muon polar-
ization effect and took into account a variable neutrino production point78. Clearly
the data are not reproduced by the no-oscillation hypothesis, adding substantially to
our confidence that the atmospheric neutrino anomaly is real.
In Fig. (12) I show the allowed neutrino oscillation parameters obtained in a recent
global fit of the sub-GeV and multi-GeV (vertex-contained) atmospheric neutrino
data75,76including the recent data reported at Neutrino 98, as well as all other
Fig. 12. Allowed atmospheric oscillation parameters for all experiments including the SK data
reported at Neutrino 98, combined at 90 (thick solid line) and 99 % CL (thin solid line) for all
possible oscillation channels, from ref.75,76. In each case the best-fit point is denoted by a star
and always corresponds to maximal mixing, a feature which is well-reproduced by the theoretical
predictions of the models proposed in ref.10,11. The sensitivity of the present accelerator and reactor
experiments as well as the expectations of upcoming long-baseline experiments is also displayed.
experiments combined at 90 (thick solid line) and 99 % CL (thin solid line) for each
oscillation channel considered. The two lower panels Fig. (12) differ in the sign
of the ∆m2which was assumed in the analysis of the matter effects in the Earth
for the νµ → νs oscillations. We found that νµ → ντ oscillations give a slightly
better fit than νµ→ νsoscillations. At present the atmospheric neutrino data cannot
distinguish between the νµto ντand νµto νschannels.
neutral-to-charged current ratios are important observables in neutrino oscillation
phenomenology, which are especially sensitive to the existence of singlet neutrinos,
light or heavy16. The atmospheric neutrinos produce isolated neutral pions (π0-
events) mainly in neutral current interactions. One may therefore study the ratios
of π0-events and the events induced mainly by the charged currents, as recently
advocated in ref.79. This has the virtue of minimizing uncertainties related to the
It is well-know that the
original atmospheric neutrino fluxes. In fact the SK collaboration has already tried
to do this by estimating the double ratio of π0over e-like events in their sample3
and found R = 0.93 ± 0.07 ± 0.19. This is consistent both with νµto ντor νµto
νschannels, with a slight preference for the former. The situation should improve in
We also display in Fig. (12) the sensitivity of present accelerator and reactor
experiments, as well as that expected at future long-baseline (LBL) experiments in
each channel. The first point to note is that the Chooz reactor80data already excludes
the region indicated for the νµ→ νechannel when all experiments are combined at
From the upper-left panel in Fig. (12) one sees that the regions of νµ→ ντ os-
cillation parameters obtained from the atmospheric neutrino data analysis cannot be
fully tested by the LBL experiments, as presently designed. One might expect that,
due to the upward shift of the ∆m2indicated by the fit for the sterile case (due
to the effects of matter in the Earth) it would be possible to completely cover the
corresponding region of oscillation parameters. Although this is the case for the MI-
NOS disappearance test, in general most of the LBL experiments can not completely
probe the region of oscillation parameters allowed by the νµ→ νsatmospheric neu-
trino analysis. This is so irrespective of the sign of ∆m2assumed. For a discussion
of the various potential tests that can be performed at the future LBL experiments
in order to unravel the presence of oscillations into sterile channels see ref.76.
3.3. LSND, Dark Matter & Pulsars
A search for ¯ νµ→ ¯ νeoscillations has been conducted at the Los Alamos Meson
Physics Facility by using ¯ νµ from µ+decay at rest6. The ¯ νe’s are detected via
the reaction ¯ νep → e+n, correlated with a γ from np → dγ (2.2MeV). The use
of tight cuts to identify e+events with correlated γ rays yields 22 events with e+
energy between 36 and 60MeV and only 4.6 ± 0.6 background events. A fit to the
e+events between 20 and 60MeV yields a total excess of 51.8+18.7
attributed to ¯ νµ→ ¯ νeoscillations, this corresponds to an oscillation probability of
shaded regions are the favoured likelihood regions given in ref.6. The curves show
the 90 % and 99 % likelihood allowed ranges from LSND, and compares them to
limits from BNL776, KARMEN1, Bugey, CCFR, and NOMAD. A search for νµ→
νeoscillations has also been conducted by the LSND collaboration. Using νµfrom
π+decay in flight, the νeappearance is detected via the charged-current reaction
C(νe,e−)X. Two independent analyses are consistent with the above signature,
after taking into account the events expected from the νecontamination in the beam
and the beam-off background. If interpreted as an oscillation signal, the observed
−16.9± 8.0 events. If
−0.10± 0.05)% and leads to the oscillation parameters shown in Fig. (13). The
90% (L/Lmax > 0.1)
99% (L/Lmax > 0.01)
LSND 93-97 Preliminary
Fig. 13. Allowed LSND oscillation parameters versus competing experiments81
oscillation probability of 2.6±1.0±0.5×10−3is consistent with the ¯ νµ→ ¯ νeoscillation
evidence described above. Fig. 14 compares the LSND region with the expected
sensitivity from MiniBooNE, which was recently approved to run at Fermilab81.
A possible confirmation of the LSND anomaly would be a discovery of far-reaching
The research on the nature of the cosmological dark matter and the origin of
galaxies and large scale structure in the Universe within the standard theoretical
framework of gravitational collapse of fluctuations as the origin of structure in the
expanding universe has undergone tremendous progress recently. Indeed the obser-
vations of cosmic background temperature anisotropies on large scales performed by
the COBE satellite7combined with cluster-cluster correlation data e.g. from IRAS9
can not be reconciled with the simplest cold dark matter (CDM) model. Barring a
non-zero cosmological constant and high value of the Hubble parameter (h>∼0.7) the
simplest model that have a chance to work is Cold + Hot Dark Matter (MDM, for
mixed dark matter), if the Hubble parameter and age parameter allow for an Ω = 1
cosmology8, suggested by inflation. Electron-volt mass neutrinos are the most well-
motivated HDM candidate. This mass scale is similar to that indicated by the hints
reported by the LSND experiment6.
However it is too early to be confident on the MDM scenario, and one should for
the moment keep an open mind. For example, I note that an MeV range (unstable) tau
neutrino is an interesting possibility to consider from the point of view of dark matter.
If such neutrino decays before the matter dominance epoch, its decay products would
add energy to the radiation, thereby delaying the time at which the matter and
Fig. 14. Expected sensitivity of the proposed MiniBooNE experiment81
radiation contributions to the energy density of the universe become equal. Such
delay would allow one to reduce the density fluctuations on the smaller scales purely
within the standard cold dark matter scenario82.
Future sky maps of the cosmic microwave background radiation (CMBR) with
high precision at the upcoming MAP and PLANCK missions should bring more light
into the nature of the dark matter and the possible rˆ ole of neutrinos83.
One of the most challenging problems in modern astrophysics is to find a con-
sistent explanation for the high velocity of pulsars. Observations84show that these
velocities range from zero up to 900 km/s with a mean value of 450 ± 50 km/s. An
attractive possibility is that pulsar motion arises from an asymmetric neutrino emis-
sion during the supernova explosion. In fact, neutrinos carry more than 99% of the
new-born proto-neutron star’s gravitational binding energy so that even a 1% asym-
metry in the neutrino emission could generate the observed pulsar velocities. One
possible explanation to this puzzle may reside in the interplay between the parity
non-conservation present in weak interactions and the strong magnetic fields which
are expected during a SN explosion. Possible realizations of this idea in the frame-
work of the Standard Model (SM) have been proposed85,86However, it has recently
been noted87that no asymmetry in neutrino emission can be generated in thermal
equilibrium, even in the presence of parity violation. This suggests that alternative
mechanism is at work. Several neutrino conversion mechanisms in matter have been
invoked as a possible engine for powering pulsar motion. They all share in common
the feature that neutrino propagation properties are affected by the polarization88of
the SN medium which is provided by the strong magnetic fields 1015Gauss present
during a SN explosion. This would give rise to some angular dependence of the
matter-induced neutrino potentials leading to a deformation of the ”neutrino-sphere”
for, say, tau neutrinos and hence to an anisotropic neutrino emission. As a conse-
quence, in the presence of non-vanishing ντmass and mixing the resonance sphere for
the νe− ντconversions is distorted. If the resonance surface lies between the ντand
νeneutrino spheres, such a distortion would induce a temperature anisotropy in the
flux of the escaping tau-neutrinos produced by the conversions, hence a recoil kick of
the proto-neutron star. This mechanism was realized in ref.89invoking MSW conver-
sions30with mντ>∼100 eV or so, assuming a negligible νemass. This is necessary
in order for the resonance surface to be located between the two neutrino-spheres.
It should be noted, however, that such requirement is at odds with cosmological
bounds on neutrinos masses unless the τ-neutrino is unstable. On the other hand
in ref.90a realization was proposed in the resonant spin-flavour precession scheme
(RSFP)64. Here the magnetic field not only affects the medium properties, but also
induces the spin-flavour precession through its coupling to the neutrino transition
magnetic moment63. Perhaps the simplest suggestion was proposed in ref.32where
the required pulsar velocities would arise from anisotropic neutrino emission induced
by resonant conversions of massless neutrinos (hence no magnetic moment)29. This
mechanism arises in the model described in eq. (3) and has been shown to be of
potential relevance for SN physics31.
Very recently, however, Raffelt and Janka91have claimed that the asymmet-
ric neutrino emission effect was vastly overestimated, because the variation of the
temperature over the deformed neutrino-sphere is not an adequate measure for the
anisotropy of the neutrino emission. This would invalidate the oscillation mecha-
nisms, leaving the pulsar velocity problem without any known viable solution. The
only potential way out of their criticism would invoke conversions into sterile neu-
trinos, since the conversions would take place deeper in the star. However, it is too
early to tell whether or not it works92.
4. Reconciling the neutrino puzzles
It is easy to accommodate the solar and atmospheric neutrino data by themselves
in a general gauge theory of neutrino mass, since it lacks predictivity. One could even
have a situation where three-neutrino mixing could be bi-maximal, i.e. maximal in
both the atmospheric as well as solar neutrino transitions, if the solution chosen by
nature is just-so72. The challenge to reconcile these two requirements arise mainly
if one wishes to do that in a predictive quark-lepton unification scheme that relates
lepton and quark mixing angles. This especially so since the latter are small, in
contrast to the lepton mixing indicated by the SK atmospheric data. The story gets
more complicated if one wishes to account also for the LSND anomaly and for the
hot dark matter. There has been a lot of effort to solve the bigger puzzle posed by
the inclusion of any of these additional hints10,11,12. As we have seen the atmospheric
neutrino data requires ∆m2
indicated by the solar neutrino data, either in the context of the MSW mechanism
or the just-so solution. These two experiments fix two different scales for neutrino
mass differences, so that with just the three known neutrinos and without discarding
any experimental data, there is no room to include the LSND scale indicated in Fig.
(13), nor the HDM scale which is roughly similarb.
Reconciling the neutrino puzzles may be attempted within the unification ap-
proach or the weak-scale approach to the theory of neutrino mass. I will concentrate
mostly on the latter, because it is an interesting and simpler alternative to the former.
atmwhich is much larger than the scale ∆m2⊙which is
4.1. Almost Degenerate Neutrinos
The only possibility to fit solar, atmospheric and HDM scales in a world with
just the three known neutrinos is if all of them have nearly the same mass12, of
about ∼ 1.5 eV or so in order to provide the right amount of HDM8(all three
active neutrinos contribute to HDM). There is no room in this case to accommodate
the LSND anomaly. This can be arranged in the unification approach discussed in
sec. 2 using the MLterm present in general in seesaw models. With this in mind
one can construct, e.g. unified SO(10) seesaw models where all neutrinos lie at the
above HDM mass scale (∼ 1.5 eV), due to a suitable horizontal symmetry, while the
parameters ∆m2⊙& ∆m2atmappear as symmetry breaking effects. An interesting
fact is that the ratio ∆m2⊙/ ∆m2atmappears as mc2/mt2 18.
4.2. Four-Neutrino Models
The simplest way to open the possibility of incorporating the LSND scale is to
invoke a sterile neutrino, i.e. one whose interaction with standard model particles
(such as the W and the Z) is much weaker than the SM weak interaction. It must
come in as an SU(2) ⊗ U(1) singlet ensuring that it does not affect the invisible Z
decay width, well-measured at LEP. The sterile neutrino νsmust also be light enough
in order to participate in the oscillations involving the three active neutrinos. The
theoretical challenges we have are:
• to understand why the sterile neutrino is so light (it is clear that if a sterile
bI will ignore the pulsar velocity problem since there is no clear working-model at the moment.
neutrino is introduced into the SM, the SU(2) ⊗ U(1) gauge symmetry allows
it to have a bare mass, which could be large)
• to account for the maximal neutrino mixing indicated by the atmospheric data
• to account for the three scales ∆m2
atm, ∆m2⊙ and ∆m2
With this in mind we have formulated the simplest and first schemes10,11which pro-
vide an answer to the above points. I will denote them, (eτ)(µ s)10and (es)(µτ)
respectively. One should realize that a given phenomenological scheme (mainly deter-
mined by the structure of the leptonic charged current) may be realized in more than
one theoretical model. For example, an alternative to the model in
in ref.12. There have been many attempts to reproduce the above phenomenological
scenarios from different theoretical assumptions, as has been discussed here93,94,95.
These two basic schemes are characterized by a very symmetric mass spectrum in
which there are two ultra-light neutrinos at the solar neutrino scale and two maxi-
mally mixed almost degenerate eV-mass neutrinos (LSND/HDM scale), split by the
atmospheric neutrino scale10,11. The HDM problem requires the heaviest neutrinos
at about 2 eV mass96. These scales are generated radiatively due to the additional
Higgs bosons which are postulated, as follows: ∆m2
sults, it naturally focussed on accounting for the HDM problem, rather than LSND.
However, it has been realized that the LSND oscillation effects may be accounted for
in its framework. These are the simplest theories based only on weak-scale physics,
in which one explains the lightness of the sterile neutrino, the large lepton mixing
required by the atmospheric neutrino data, as well as the generation of the mass
splittings responsible for solar and atmospheric neutrino conversions. These follow
naturally from the underlying lepton-number-like symmetry and its breaking10,11.
These models are minimal in the sense that they add a single SU(2)⊗U(1) singlet
lepton to the SM. Before breaking the symmetry the heaviest neutrinos are exactly
degenerate, while the other two which will be responsible for the explanation of the
solar neutrino problem are still massless97. After the global U(1) lepton symmetry
breaks the massive ones split and the light ones get mass. The models differ according
to whether the νslies at the dark matter scale or at the solar neutrino scale. In the
(eτ)(µ s) scheme the νslies at the LSND/HDM scale, as illustrated in Fig. (15) while
in the alternative (es)(µτ) model, νsis at the solar neutrino scale as shown in Fig.
(16)11 93. In the (eτ)(µ s) case the atmospheric neutrino puzzle is explained by νµto
νsoscillations, while in (es)(µτ) it is explained by νµto ντoscillations. Correspond-
ingly, the deficit of solar neutrinos is explained in the first case by νeto ντconversions,
while in the second the relevant channel is νeto νs. The two models are therefore
clearly inequivalent. In both cases it is possible to fit all present observations together.
LSND/HDMarises at one-loop, while
atmand ∆m2⊙are two-loop effects. Since this proposal pre-dated the LSND re-
Figure 15: (eτ)(µ s) scheme: νe- ντconversions explain the solar neutrino data and
νµ- νsoscillations account for the atmospheric deficit, ref.10.
Figure 16: (es)(µτ) scheme: νe- νsconversions explain the solar neutrino data and
νµ- ντoscillations account for the atmospheric deficit, ref.11.
I now turn to the consistency of the models with BBN. The presence of additional
weakly interacting light particles, such as our light sterile neutrino νs, is constrained
by BBN since the νswould enter into equilibrium with the active neutrinos in the early
Universe (and therefore would contribute to Nmax
∆m2sin42θ<∼3 × 10−6
eV2where ∆m2denotes the mass-square difference of the
active and sterile species and θ is the vacuum mixing angle. However, systematical
uncertainties in the derivation of BBN bounds still caution us not to take them
too literally. For example, it has been argued in54that present observations of
primordial Helium and deuterium abundances can allow up to Nν = 4.5 neutrino
species if the baryon to photon ratio is small. Adopting this as a limit, clearly both
models described above are consistent. Should the BBN constraints get tighter, e.g.
< 3.5 they could rule out the (eτ)(µ s) model, and leave out only the competing
scheme as a viable alternative. For recent work on this see ref.99.
The two models would be distinguishable both from the analysis of future solar
as well as atmospheric neutrino data. For example they may be tested in the SNO
current data and compare it with the corresponding charged current value (ΦCC
the solar neutrinos convert to active neutrinos, as in the (eτ)(µ s) model, then one
≃ .5, whereas in the (es)(µτ) scheme (νeconversion to νs), the
) via neutrino oscillations98, unless
100once they measure the solar neutrino flux (ΦNC
) in their neutral
ν ). If
above ratio would be nearly ≃ 1. Looking at pion production via the neutral current
reaction ντ+N → ντ+π0+N in atmospheric data might also help in distinguishing
between these two possibilities79, since this reaction is absent in the case of sterile
neutrinos, but would exist in the (es)(µτ) scheme.
If light sterile neutrinos indeed exist, as suggested by the current solar and at-
mospheric neutrino data, together with the LSND experiment, one can show that in
some four-neutrino scenarios, neutrinos would contribute to a cosmic hot dark matter
component and to an increased radiation content at the epoch of matter-radiation
equality. These effects leave their imprint in sky maps of the cosmic microwave
background radiation (CMBR) and may thus be detectable with the precision mea-
surements of the upcoming MAP and PLANCK missions as noted recently in ref.83.
4.3. MeV Tau Neutrino
In ref.101a model was presented where an unstable MeV Majorana tau neu-
trino naturally reconciles the cosmological observations of large and small-scale den-
sity fluctuations with the cold dark matter picture (CDM). The model assumes the
spontaneous violation of a global lepton number symmetry at the weak scale. The
breaking of this symmetry generates the cosmologically required decay of the ντwith
lifetime τντ∼ 102− 104sec, as well as the masses and oscillations of the three light
neutrinos νe, νµand νs. One can also verify that the BBN constraints can be satis-
fied. The cosmological attractiveness of this scheme should encourage one to check
whether one can indeed account for the present solar and atmospheric data through
oscillations among the three light neutrinos, after taking into account the recent SK-
5. In conclusion
A major news has been the re-confirmation of an angle-dependent atmospheric
neutrino deficit by the SK collaboration, providing a strong evidence for neutrino
masses, similar to that offered by the solar neutrino data. Unfortunately future LBL
experiments do not all probe the full region indicated by the atmospheric data. If
the LSND result stands the test of time, this would be a puzzling indication for
the existence of a light sterile neutrino. Who ordered it? The two most attractive
schemes to reconcile these observations invoke either νe- ντconversions to explain
the solar data, with νµ- νsoscillations accounting for the atmospheric deficit, or the
other way around. These two basic schemes have distinct implications at future solar
& atmospheric neutrino experiments. SNO and SuperKamiokande have the potential
to distinguish them due to their neutral current sensitivity.
How about heavy neutrinos? Although cosmological bounds are a fundamental
tool to restrict neutrino masses, in many theories heavy neutrinos will either decay or
annihilate very fast, thereby loosening or evading the cosmological bounds. From this
point of view, neutrinos can have any mass presently allowed by laboratory experi-
ments, and it is therefore important to search for manifestations of heavy neutrinos
at the laboratory.
Last but not least, though most of the recent excitement comes from underground
experiments, one should note that models of neutrino mass may lead to a plethora of
new signatures which may be accessible also at accelerators, thus illustrating the com-
plementarity between the two approaches in unravelling the properties of neutrinos
and probing for signals beyond the SM.
I am grateful to Bernd Kniehl and Georg Raffelt for the kind hospitality at the
Ringberg castle. My thanks to John Bahcall, Plamen Krastev and Bill Louis, for
making their postcript figures available to me, and to Thomas Janka and Eligio Lisi
for correspondence. I thank all my collaborators, especially Hiroshi Nunokawa for
going over the first draft of this manuscript critically. This work was supported by
DGICYT grant PB95-1077 and by the EEC under the TMR contract ERBFMRX-
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