Reconstructing Neutrino Mass Spectrum

32 Pages·0.32 MB·

by  A. Yu. Smirnov

RECONSTRUCTING NEUTRINO MASS SPECTRUMa
A. Yu. SMIRNOV
9 The Abdus Salam International Center of Theoretical Physics,
9 Strada Costiera 11, Trieste, Italy
9 Institute for Nuclear Research, RAS, Moscow, Russia
1 E-mail: [email protected]

n
a Reconstruction of the neutrino mass spectrum and lepton mixing is one of the
J fundamental problems of particle physics. In this connection we consider two
central topics: (i) the originof large lepton mixing, (ii)possible existence of new
3
(sterile) neutrino states. We discuss also possible relation between large mixing

andexistence ofsterileneutrinos.
1
v
8
1 Introduction
0
2
1 The experimental situation can be summarized in the following way:
0 1. Recent SuperKamiokande (SK) results on the atmospheric neutrinos
9 givestrongevidencefortheoscillationsofthemuonneutrinoswithlarge(max-
9 imal) depth1. An open question is to which extend the electron neutrinos are
/
h involved in the oscillations and whether an excess of the e-like events exists.
p
2. Thesituationwithsolarneutrinosisratheruncertain. Thedataindicate
-
p unexpected distortion of the recoil electrons energy spectrum2. It is unclear
e whether we deal with just statistical fluctuations, or distortion of the boron
h
neutrino spectrum or an excess of the events near the end point which is
:
v not related to boron neutrinos. No day-night asymmetry and no earth core
i enhancement of signal have been found.
X
3. LSNDcollaborationhasfurtherconfirmedtheoscillationinterpretation
r
a of their result3. At the same time, KARMEN4 does not see the oscillation
effect concluding that the data are approaching the situation when one can
speak on direct contradiction between the two experiments.
4. Recent cosmological observations (early galaxies, clusters evolution,
high redshift supernova type IA data) show that a contribution of neutrinos
to the energy density of the Universe should be smaller than it was thought
earlier, and the Hot Dark Matter (HDM) is not necessary for the fit of data
on the large scale structure5. At the same time, some amount of the HDM is
not excluded and may be needed for the further tuning of the data.
aTalkgivenatthe5thInternationalWEINSymposium: AConferenceonPhysicsBeyond
theStandardModel(WEIN 98),SantaFe,NM,June14–21,1998.
1
Keeping this in mind, we will concentrate on models which explain the
solar and the atmospheric neutrino data. We will consider main issues of the
present day discussions:
1. Origin of the large leptonic mixing.
2. Possible existence of new neutrino states (sterile neutrinos).
3. We also comment on possible relation of these two issues, addressing
the question: is large mixing the mixing with sterile neutrinos?
Accordingto the SK result,muonneutrinososcillateinto tauneutrinosor
probably into sterile neutrinos. The effective mixing angle which determines
the depth of oscillations should be large in both cases
sin22θ >0.8. (1)
The favouredmode is ν ν , although ν ν gives comparably good fit of
µ τ µ s
− −
thedata. Pureν ν channelisstronglydisfavoredbytheSuperKamiokande
µ e
−
data itself, and restricted by the CHOOZ result6. At the same time, a small
contributionof the ν ν channel is possible and probably desiredin view of
µ e
−
some excess of the e - like events.
In this connection the basic questions are
Why lepton mixing is large while quark mixing is small? Is this consis-
•
tent with quark-lepton symmetry (correspondence) and Grand Unifica-
tion? The question has more generalconceptual nature. The picture we
hadbefore is thatknownquarksandleptons formfamilies withweakin-
terfamily connection (characterized by mixing). Should we support this
conceptionin view of maximal mixing between the secondand the third
generations of leptons?
Is lepton mixing maximal between the second and the third generations
•
only, or probably all lepton mixings are large? In other words is the ob-
servedlargeleptonmixingthefeatureofthesecondandthirdgeneration
or it is the property of all leptons?
Theanswertothis questionwillcomefromstudies ofsolarneutrinos. In
thefirstcasethesmallmixingMSW-solutionisrealized,whereasinthe
second case the choice will be between the large mixing MSW solution
and long range vacuum oscillations (“just-so”).
Completelydifferentpossibilityisthatlargemixingisthemixingwithnew
(sterile) neutrino state. In this case the mixing between flavor states can be
small in analogy with quark mixing.
2
2 Patterns of neutrino mass and mixing
Before going into details of the theoretical analysis, we will describe possible
patternsoftheneutrinomassandmixingwhichareimpliedbyphenomenology.
Here weconsiderthree types ofneutrinoschemeswith single,double (bi-)and
triple maximal mixing.
2.1 Single maximal mixing
The scheme has the hierarchicalmass spectrum
m =(0.3 3) 10 1eV, m =(2 4) 10 3eV, m m (2)
3 − 2 − 1 2
− · − · ≪
with ν and ν mixed strongly in ν and ν (see fig. 1). The electron flavor
µ τ 2 3
is weakly mixed: it is mainly in ν with small admixtures in the heavy states.
1
The solar neutrino data are explained by ν ν resonance conversioninside
e 2
→
0
10
ν
(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1) e
10-1 ν
µ
ντ (cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)
-2
10 ATM
eV (cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)
, 10-3
m
ν
solar
-4
10 (cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)
-5
10 ν ν ν
1 2 3
Figure 1: Neutrino masses and mixing for the “solar and atmospheric” neutrinos. Boxes
correspondtothemasseigenstates. Thesizesofdifferentregionsintheboxesshowadmix-
turesofdifferentflavors. Weaklyhatchedregionscorrespondtotheelectronflavor,strongly
hatched regionsdepictthemuonflavor,blackregionspresentthetauflavor.
the Sun. Notice that ν converts to ν and ν in comparable portions. The
e µ τ
atmospheric neutrino problem is solved via ν ν oscillations. Small ν
µ τ e
↔
admixture in ν can lead to resonantly enhanced oscillations in matter of the
3
Earth.
There is no explanation of the LSND result, and the contribution to the
Hot Dark Matter component of the universe is small: Ω <0.01.
ν
The scheme can provide significant amount of the HDM without change
of the oscillation pattern if all three neutrinos have degenerate masses: m
i
≈
3
m 1 eV with small splitting:
0
∼
∆m2 ∆m2 ∆m2 ∆m2
∆m12 2 = ⊙, ∆m23 3 = atm, (3)
≈ 2m 2m ≈ 2m 2m
0 0 0 0
where m and m are defined in (2) and
2 3
∆m2⊙ ≈6·10−6eV2, ∆m2atm ≈(10−3−10−2)eV2. (4)
In this case an effective Majorana mass of the electron neutrino equals m
ee
≈
m and searches for the neutrinoless double beta decay give crucial check of
0
the scheme.
The scheme can be probed by the long baseline experiments.
2.2 Bi-large and bi-maximal mixings
Thepreviousscheme(fig.1)canbemodifiedinsuchawaythatsolarneutrino
dataareexplainedbylargeangleMSWconversionwithsin22θ 0.7 0.9and
∼ −
∆m2 =(2 20) 10 5 eV2.
−
−
In a version of the scheme with mass degeneracy, the cancellation in the
effective Majorana mass of the electron neutrino can occur, so that m
ee
∼
m 1 sin22θwillbesubstantiallylowerthanpresentboundevenform >1
0 0
−
eVp(see7 for recent discussion).
The solutionofthe atmospheric neutrino problemis basically the same as
in the previous case. There is a suggestion8 that mass splitting ∆m2 > 10 4
−
eV2betweenthetwolightstatescouldberelevantfortheatmosphericneutrino
problem. In particular, this mode can lead to the zenith angle dependence of
the detected events. In this case the 23-splitting could be much larger to
accommodate the LSND result. However, in8 the matter effect has not been
taken into account, and the latter, in fact, strongly suppresses the oscillation
depth.
In the bi-maximal scheme9 neutrinos have masses
m =(0.3 3) 10 1eV, m (0.7 2) 10 5eV, m m (5)
3 − 2 − 1 2
− · ∼ − · ≪
(see fig. 2); ν and ν mix maximally in ν = (ν +ν )/√2; the orthogonal
µ τ 3 µ τ
combination,ν (ν ν )/√2 strongly mixes with ν in ν andν . There is
2′ ≡ µ− τ e 1 2
no admixture of ν in the ν . The corresponding mixing matrix has the form:
e 3
1 1 0
√2 −√2
V = 1 1 1 , (6)
MNS 2 2 −√2
1 1 1
 
 2 2 √2 
4
m, eV
10-1
(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1) νe
ν
-2 µ
10
ν
τ
10-3
ATM
-4
10
ν
solar
-5
10 (cid:0)(cid:1)(cid:0)(cid:1) (cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1) (cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)
-6
10
ν ν ν
1 2 3
Figure2: Neutrinomassesandmixingpatternofthebi-maximalmixingscheme.
The solar neutrino problem can be solved via ν ν “Just-so” vacuum
e ↔ 2′
oscillations. Notice that ν converts equally to ν and ν . Larger values of
e µ τ
∆m2 lead to the averaged oscillation result which does not give a good fit of
the solar neutrino data.
The atmospheric neutrino anomaly is solved via ν ν maximal depth
µ τ
↔
oscillations.
Let us comment on the version of the bi-maximal scheme with inverted
mass hierarchy: m m m (3). Such a possibility can be realized in
1 2 3
≈ ≫
themodelwithapproximateL L L -symmetry. Thecorrespondingmass
e µ τ
− −
matrix has the form:
ǫ 1 1
′
mν =1 ǫ ǫ. (7)
1 ǫ ǫ
 
Two states with maximal (or large) ν mixing are heavy and degenerate,
e
whereas the third state with large ν ν mixing and small ν admixture
µ τ e
−
is light. In this scheme the ν ν level crossing occurs in the antineutrino
e − 3′
channel, so that in supernovae ν¯ will be strongly converted into combination
e
of ν¯ , ν¯ and vice versa. As the result the ν¯ ’s will have hard spectrum of the
µ τ e
original ν¯ .
µ
One can introduce a degeneracy of neutrinos (keeping the same ∆m2)
to get significant amount the HDM in the Universe without change of the
5
1
10
0
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ν
-4 τ
10 (cid:0)(cid:1)
ν ν ν
1 2 3
Figure 3: The neutrino mass and mixing in the bi-maximal mixing scheme with inverse
masshierarchy.
oscillation pattern. The effective Majorana mass of the electron neutrino is
zerointhe strictbi-maximalcase,sothatnoeffectinthe double beta decayis
expected due to light neutrinos. In this scheme one needs two mass splittings
of the order 10 3 eV and 10 10 eV respectively which looks very unnatural.
− −
2.3 Threefold maximal mixing
In such a scheme 10 all the elements of the mixing matrix are assumed to
be equal: U = 1/√3 (4). In all flavor channels All three frequencies of
αi
| |
oscillationscontributetoallflavorchannelsequally. Theatmosphericneutrino
problem is solved by ν ν and ν ν oscillations with equal depth:
µ e µ τ
sin22θ =4/9,sothattheν↔-disappeara↔nceis characterizedbysin22θ =8/9.
µ
The CHOOZ bound implies that ∆m2 <10 3 eV2.
−
ThesolarneutrinosurvivalprobabilityequalsP =4/9P +1/9,whereP is
2 2
thetwoneutrinooscillationprobabilitywithmaximaldepthandsmallestmass
splitting∆m2 . Itisassumedthat∆m2 <10 11 eV2,sothat1-2subsystem
12 12 −
of neutrinos is frozen and P = 1. As the result the solar neutrino flux will
2
haveenergyindependentsuppressionP =5/9. Thefitofboththeatmospheric
and the solar neutrino data is substantially worser than in previous schemes.
3 Large Lepton Mixing
6
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ν ν ν
1 2 3
Figure4: Neutrinomassesandmixingintheschemewiththreefoldmaximalmixing.
3.1 Is large lepton mixing the problem?
Let us first clarify whether the problem of large mixing exists at all. The
conception of families of fermions can be expressed in the following way. In
certain basis mass matrices of both upper and down fermions (from doublets)
have hierarchicalstructure with smalloff-diagonalelements. The matrices are
considered to be natural11 if the mixing angles θ satisfy inequality
ij
m
i
θ , (8)
ij
| |≤rmj
where m and m are the eigenvalues. (In this case no special arrangement of
i j
the matrix elements is needed).
Let us consider the secondand third generations of leptons and introduce
theanglesθ andθ whichdiagonalizethemassmatricesofthechargedleptons
cl ν
and the neutrinos correspondingly. Then the lepton mixing angle equals
θ =θ θ . (9)
l cl ν
−
Using (8) we get
m m
θ µ 130, θ 2 140 , (10)
cl ν
≈rm ∼ ≈rm ∼
τ 3
where for m 0.05 eV and m 0.003 eV are the values of masses required
3 2
∼ ∼
by solutions of the atmospheric and the solarneutrino problems. If the angles
7
θ and θ have opposite signs, so that θ = θ + θ , we find θ = 27 and
cl ν l cl ν l ◦
sin22θ = 0.67 0.72 - close to the desired v|alu|e13|. |Thus, the large lepton
−
mixing is consistent with the naturalness of the mass matrices.
Notice that if neutrino masses are due to the see-saw mechanism and the
mass matrix of the RH neutrinos has no hierarchy: M Iˆ, then m m2
∼ 0· i ∝ D
andthemixingangleisdeterminedbytheDiracmassmatrixm . Inthiscase
D
relation between the masses and mixing becomes θ 4 m /m 12 which
ν 2 3
leads to sin22θ=0.96. | |≈ p
For quarks the mixing is small if the corresponding angles θ and θ have
u d
the same signs and therefore cancel each other in the total mixing. The same
cancellationmayoccurforthemixing ofthefirstandsecondgenerations,thus
leading to a small mixing solutions of the solar neutrino problem.
So, the problem of the large mixing is reduced to explanation of signs
(phases in general) of contributions to mixing from the upper and the down
fermions. In fact, the change of the relative sign of the contributions in the
lepton sector can be related to the see-saw origin of the neutrino mass14.
Thus, the large lepton mixing can be well consistent with our “standard
notions”: quark - lepton symmetry (similarity of the Dirac mass matrices),
usual family structure and the see-saw mechanism.
Thealternativepossibilityisthatlargeleptonmixingisamanifestationof
newphysicsbeyondthe“standardnotion”. Inwhatfollowswewillconcentrate
on this interpretation.
3.2 Classifying possibilities
Trying to answerthe questionwhy the lepton mixing is large,while the quark
mixing is small one can think about the following possibilities:
Largeleptonmixingisthemixingofmuonneutrinowithsterileneutrino.
•
In this case the question does not exist: There is no analogue of ν ν
µ s
−
mixing in the quark sector.
If the atmospheric neutrino anomaly is due to ν ν mixing, there are
µ τ
• −
two options:
1). Mechanism of the neutrino mass generation differs from that of the
quarks. For instance, the mass matrix could be
m =mrad+msee saw , (11)
ν ν ν −
where msee saw is the see-saw contribution, whereas mrad is the contribution
ν − ν
from radiative mechanism. The radiative contribution can dominate and the
8
role of the see-saw is just to suppress the effect of the Dirac mass term. The
simplestversionofthe radiativemechanismwhichleadsto alargeleptonmix-
ing is the Zee-mechanism15. The key element is new chargedscalar boson S+
being singlet of SU(2). (Also second higgs doublet is introduced to have the
couplings with S+). In this case large lepton mixing is the consequence of
- SU(2)gaugesymmetry: thecouplingofthesingletwithleptondoublets
f LTiσ L S+ is antisymmetric in family index.
αβ α 2 β
- assumption that there is no strong inverse hierarchy of f ,
αβ
- mass hierarchy of charge leptons.
The model can be supplied by additional sterile neutrino to explain the
solar neutrino problem16.
2). Mechanismoftheneutrinomassandleptonmixinggenerationisclosely
relatedto generationofthe quarkand chargedlepton masses. This possibility
is realizedby the see-sawmechanism17. According to the see-sawmechanism:
m = mDM 1mD T +m , (12)
ν − ν − ν 0
where mD is the Dirac mass matrix of neutrinos, M is the Majorana matrix
ν
of the RH components and m is the direct majorana mass matrix of the left
0
componentswhichappearsifthescalar(SU )tripletexistswithno-zeroVEV.
2
In the quark sector the mixing is determined by two matrices: m for the
u
upper quarksandm forthe downquarks. The mixing(CKM-)matrixis the
d
product of matrices of the left component rotations:
VCKM =Vu†·Vd. (13)
In contrast, the lepton mixing is determined by three matrices mD, m and
ν l
M, and the lepton (MNS) mixing matrix18 can be written as:
V =V V V . (14)
MNS ss ν l
· ·
Here V is the see-saw matrix which specifies the see-saw mechanism itself
ss
19. It describes the influence of the matrix M structure on the lepton mixing.
Obviously, if M I, V = I and V = V V in analogy with the CKM
∝ ss MNS ν†· l
structure. According to (14), there are three possible sources of the large
mixing (of course, the interplay of several is possible):
V , thatis, the see-sawmechanismitself leadsto enhancement(the see-
ss
•
saw enhancement);
V which follows from Dirac neutrino mass matrix;
ν
•
9
V which follows from mass matrix of the charged leptons.
l
•
Here we have neglected possible effect of m , which in fact can also be
0
important. Notice, that precise origin of the enhancement (e.g. V or V )
ν l
depends on basis in which the mass matrices are introduced.
Recently, a number of models have been suggestedwhichrealize the three
above possibilities.
3.3 See-saw enhancement of lepton mixing
Notonlythesmallnessoftheneutrinomassbutalsolargeleptonmixingcanbe
related to Majorananature of neutrinos and both can follow from the see-saw
mechanism.
Itisnaturaltoassume(inaspiritofthegrandunification)thatthelepton
Diracmassmatricesaresimilartothequarkmassmatricesatsomeunification
scale: mD m and m m , and moreover, for the third generation one
ν ∼ u l ∼ d
may expect the equalities: m = m , m = m . Then the difference in the
ν3 t τ b
quark and lepton mixing can follow from specific structure of M .
If the influence of the first generation on the mixing of the second and
the third generations is small (and the problem is reduced to two generation
problem), one gets twodifferent conditionsof the strongsee-sawenhancement
20:
(i) Strong interfamily connection. In the basis where the neutrino Dirac
mass matrix is diagonal (Dirac basis), M should be off-diagonal:
a 0 0
M M00 0 b , (15)
∼
0 b 0
 
where a and b are some numbers. The off-diagonal form of M can in turn
be related to the Majorana nature of neutrinos. Prescribing the horizontal
charges (0, 1, -1) we reproduce (15).
(ii) Strong mass hierarchy. In the two generation case the matrix V can
ss
beparametrizedbythesee-sawangleθ whichcanberelatedtothehierarchies
ss
of the eigenvalues of the Dirac, mD, and Majorana, M , mass matrices: ǫ
i i D ≡
mD/mD, ǫ M /M . Introducing also ǫ M /M ǫ2 m /m (where
2 3 M ≡ 2 3 0 ≡ 02 03 ≡ D 3 2
M are the masses of the RH neutrinos which give in the absence of mixing
0i
in M the masses of the light neutrinos m and m ) we get20
2 3
ǫ2 ǫ
sinθ D 0 1 . (16)
ss
≈ ǫ (cid:18)rǫ − (cid:19)
0 M
10