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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

http://neutrinos.uv.es

ABSTRACT

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.

1. Introduction

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

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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

Page 3

by diagonalizing the following mass matrix in the basis ν,νc

?ML

DT

D

MR

?

(1)

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

Ris the

13.

Mν eff= ML− DM−1

RDT.(2)

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

some.

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

0MT

0D

0

0

DT

M

µ

(3)

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The Majorana masses for the neutrinos are determined from

ML= DM−1µMT−1DT

(4)

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

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νν

~d

dd

d

c

c

~

Figure 1: One-loop-induced Neutrino Mass.

+

h+

k++

lR

c

lc

LLl

h

σ

νν

l

L

R

Rc

xx

x

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.

(1)

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.

34.The

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

35, which

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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

−1

2gv3

0

M1

0

2g′vd

1

2g′vu

2g′v3

0−1

1

2gvd

2g′vd

1

2g′vu

−1

−µ

0

ǫ3

−1

1

2gv3

2g′v3

M2

1

2gvd

−1

1

2gvu

−1

00

ǫ3

0

2gvu

−µ

(5)

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′

given as

v′

mZ4

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

3

3≈

ǫ3µ

?

v′

d∆M2+ µ′vu∆B

?

(6)

∆M2

≈

3h2

8π2m2

b

ZlnMGUT

mZ

(7)

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

3are cal-

3

3, which is induced radiatively as the parameters

3

2

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10

-6

10

-5

10

-4

10

-3

10

-2

10

-1

1

10

10

-9

10

-7

10-5

10-3

ξ / (100 GeV)4

10

-11

mντ (MeV)

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

physics43.

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

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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.

?

? νc?

, with

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

implies49

?mνi<∼92 Ωνh2eV ,(8)

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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

laboratory experiments.

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

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2.5

3

3.5

4

4.5

5

5.5

6

6.5

7

110

m(ντ) (MeV)

Neq

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.

From ref.55

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.

ν

, a wide range of tau neutrino masses

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10

-5

10

-4

10

-3

110

m(ντ) (MeV)

g

Figure 5: The region above each curve is allowed for the corresponding Nmax

ref.55

ν

. From

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

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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

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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

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Figure 8: Presently allowed MSW solar neutrino parameters for 2-flavour active neu-

trino conversions with an enhanced hep flux, from ref.66

Figure 9:

fluctuations, ref.69

Solar neutrino survival probability in the presence of random density

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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-day data sample reported by the SK collaboration at Neutrino 983is that

it exhibits a zenith-angle-dependent deficit of muon neutrinos which is inconsistent