19 July 2001

Physics Letters B 512 (2001) 247–251www.elsevier.com/locate/npe

A model for neutrino warm dark matter and neutrino oscillations

Chun Liua, Jeonghyeon Songb

a Institute of Theoretical Physics, Chinese Academy of Sciences, P.O. Box 2735, Beijing 100080, PR Chinab Korea Institute for Advanced Study, 207-43 Cheongryangri-dong, Dongdaemun-gu, Seoul 130-012, South Korea

Received 20 February 2001; received in revised form 21 May 2001; accepted 4 June 2001Editor:T. Yanagida

Abstract

The muon- and tau-neutrinos with the mass in the keV range, which are allowed in a low reheating temperature cosmology,can compose the warm dark matter of the universe. A model of four light neutrinos including the keV scaleνµ andντ is studied,which combines the seesaw mechanism and the Abelian flavor symmetry. The atmospheric neutrino anomaly is due to theνµ–ντ oscillation. The solar neutrino problem is answered by the oscillation into the light sterile neutrino, where the SMA, LMA,and LOW–QVO solutions can be accommodated in our scenario. 2001 Published by Elsevier Science B.V.

PACS: 14.60.Pq; 11.30.Hv; 95.35+d; 14.60.St

Cosmological studies show increasing evidencesthat the dark matter is in fact warm, neither coldnor hot, so as to explain the observed structure andbehavior of the expanding universe [1]. In particlephysics, as the candidates of the warm dark matter(WDM), gravitinos or sterile neutrinos with massesin the keV range have been proposed. In this Letter,we consider the possibility that the ordinary activeneutrinos are the WDM.

Experiments have provided various constraints onthe neutrino mass spectrum. From the direct experi-mental search, we havemν1 2.5 eV,mν2 170 keVand mν3 15.5 MeV [2]. More information comesfrom the neutrino oscillation experiments. The Super-Kamiokande (Super-K) data for the atmospheric neu-trino anomaly suggest that theνµ is maximally mixedwith νx (x = e) with m2

µx 2.2×10−3 eV2 [3]. Andthex = τ case is strongly favored [4]. The solar neu-

E-mail address: jhsong@kias.re.kr (J. Song).

trino deficit [5] may imply the oscillation ofνe intoνy . There are several allowed parameter regions. Forexample, if theνy is a sterile neutrino, the currently fa-vored solution is the small mixing angle (SMA) withm2

ey 5 × 10−6 eV2 and tan2 θey 10−3. If oneexperiment among Super-K, Ga and Cl is removedfrom the data analysis, the large mixing angle (LMA)solution with m2 10−5–10−4 eV2 and the low-mass and quasi-vacuum oscillation (LOW–QVO) so-lution with m2 10−10–10−7 eV2 are also allowed.The LSND experiment has reported positive appear-ance results ofνµ → νe oscillations, which impliesm2

eµ 1 eV2 and sin2(2θeµ) 10−2 [6]. However,a large part of its parameter space is excluded by thenull results of the KARMEN data [7].

One of the most stringent constraints on the neu-trino mass comes from the cosmological considera-tion to avoid over-closing the universe. In the stan-dard cosmology [8], the stable neutrinos should be noheavier than 20 eV. Recently it has been shown that ifthe reheating temperatureTRH is low, the densities of

0370-2693/01/$ – see front matter 2001 Published by Elsevier Science B.V.PII: S0370-2693(01)00730-4

248 C. Liu, J. Song / Physics Letters B 512 (2001) 247–251

neutrinos can become much less than usually assumed[9,10]. The cosmological constraint is very much re-laxed. Indeed, for 1 MeV TRH 3 MeV, the abun-dance of tau- and muon-neutrinos is [11]

(1)Ωντ h2 = Ωνµh

2 =(

mν

4 keV

)(TRH

1 MeV

)3

.

This leads to Ref. [11] takingνµ andντ as the WDM.The above WDM consideration makes the neutrino

mass pattern quite unique. First the Super-K resultsconstrain the mass of bothνµ and ντ around keVscale [4]. Their mass-squared difference of order 10−3

eV2 implies that the masses ofνµ andντ are highlydegenerate. In this case, the solar neutrino problemcan only be understood by introducing a light sterileneutrino,νs . The νe (and in the large mixing casethe νs ) should be lighter than eV order, for theconsistence with the laboratory experiments ofmνe .The LSND result, however, cannot be compatible withthe presence of keV scale muon-neutrinos. Instead themixing between theνµ andνe should be very small,as expected from the keV scale mass ofνµ and thesub-eV scale mass ofνe.

In this Letter, a four light neutrino model is built togive the above-mentioned neutrino mass pattern. Howto construct the mass spectrum of four light neutrinoswith hierarchies is a theoretically challenging prob-lem. We extend a method proposed in Ref. [12]. Itsimply incorporates the seesaw mechanism [13] withflavor symmetry [14]. Introducing three right-handedneutrinos and assuming singularity in both the Diracand Majorana mass matrices, the neutrino mass ma-trix has approximately the following form:

(2)M =

0 0 0 0 0 00 0 0 0 m22 m230 0 0 0 m32 m330 0 0 0 0 00 m22 m32 0 M22 M230 m23 m33 0 M23 M33

.

In this mass spectrum, there are two heavy neutrinos ofmasses∼ M, two light neutrinos of masses∼ m2/M,and two massless neutrinos; three mass scales,M,m2/M, and 0, characterizes this model. Thus the fourlight neutrinos are naturally divided into two pairs witha mass gap of∼ m2/M, and each pair consists of twodegenerate mass eigenstates.

Abelian flavor symmetry [14] can be used to gen-erate such a neutrino mass matrix. Furthermore thissymmetry should also provide the maximal mixing forthe atmosphere neutrino anomaly which is not guaran-teed in the form ofM in Eq. (2). A softly breaking ofthe symmetry is necessary to generate small massesfor two massless neutrinos and to lift the degener-acy in each pair. In the following discussion, super-symmetry is implied. The flavor symmetry is sponta-neously broken by a vacuum expectation value (VEV)of an electroweak singlet fieldX. As long as the fla-vor charges balance under the symmetry, the followinginteractions are allowed:

LαHNβ

( 〈X〉VEV

Λ

)mαβ

,

(3)MNαNβ

( 〈X〉VEV

Λ

)nαβ

,

where Lα (α = e, µ, τ ), H and Nα denote thelepton doublets, a Higgs field, and the right-handedneutrino fields, respectively. TheΛ is the flavorsymmetry breaking scale, and themαβ and nαβ arenon-negative integers, required for the holomorphy ofthe superpotential. The order parameter for this newsymmetry is defined by

(4)λ ≡ 〈X〉VEV

Λ 1,

which is a typical order of Cabbibo angle∼ 0.1The assignment of the Abelian flavor charges rele-

vant to Eq. (2) and to maximalνµ–ντ mixing is as-sumed to be

Le(z + 1/2), Lµ(a), Lτ (−a),

Ece(−z + 9/2), Ec

µ(−a + 3), Ecτ (a + 2),

(5)Ne(x − 1/2), Nµ(−a), Nτ (a), X(−1),

with the positive integersa and x(> a > 1). Thesign of the integerz is to be set later. TheEc

α ’sin Eq. (5) are the antiparticle fields of the SU(2)singlet charged leptons. In order to obtain the physicalmixing angles of neutrinos, the charged lepton massmatrix should be simultaneously taken into account.The gauge and Higgs bosons possess vanishing flavorcharges. Note that the flavor charges for the firstgeneration are half-integers while those for the secondand third generations are integers expressed by a singleparametera. Compared to analogous analysis for three

C. Liu, J. Song / Physics Letters B 512 (2001) 247–251 249

light neutrino scenario, the choice of the flavor chargeshere is more limited. One tricky point is that oneof the right-handed neutrino masses is made to bevanishingly small ( m2/M).

The flavor charge assignment in Eq. (5) producesthe Dirac and Majorana mass matrices of neutrinos as

MD = m

(Y11λ

x+z 0 00 1 Y23λ

2a

0 0 1

),

(6)MM = M

(ζ1λ

2x−1 0 00 0 10 1 ζ4λ

2a

),

and the mass matrix of charged leptons as

(7)Ml = m

(η11λ

5 0 00 η22λ

3 η23λ2a+2

0 0 η33λ2

),

where Y ’s, ζ ’s and η’s are order one coefficients.The Ml and MD are almost diagonal, while theMM is mainly off-diagonal. Therefore to leadingorder, the mass matrix of four light neutrinos in the(νs, νe, νµ, ντ ) basis is obtained as,

(8)M(0)ν m2

M

0 0 0 00 0 0 00 0 0 −10 0 −1 0

.

The neutrino mass spectrum due toM(0)ν is

mνs = mνe = 0, mνµ = mντ = m2

M,

(9)sinθµτ = 1√2.

It is to be noted that the maximal mixing betweenthe νµ andντ results from the flavor symmetry. Therequirement of keV scaleνµ andντ , i.e.,

(10)m2

M∼ 1 keV,

is achieved via the seesaw mechanism withm 300 GeV andM 1011 GeV. Theε, defined by theratiom/M, is then of order 10−8.5.

First let us examine the mass spectrum in thecharged lepton sector. The eigenvalues of theMl areof orderλ5m, λ3m, andλ2m, which yields appropriatemass scales of the charged leptons. TheMl is diago-nalized by

(11)RLl MlR

R†l = Diag(me,mµ,mτ ),

where theRLl diagonalizes the hermitian mass-squared

matrixMlM†l .

In the neutrino sector the mass matrix of fourlight neutrinos can be obtained by the method de-scribed in Ref. [15]. We finally have the followingsymmetric mass matrix of four light neutrinos in the(νs, νe, νµ, ντ ) basis withλx/ε ≡ λβ :

(12)Mν = m2

M

λ2β−1 λβ+z 0 0λβ+z 0 0 0

0 0 λ2a −10 0 −1 0

,

where the charged lepton mixing effects are incorpo-rated so that the charged lepton fields have been ro-tated into mass eigenstates. In Eq. (12) the combina-tions of order one parameters such asYij , ηi , andζiare omitted, for simplicity.

In our scenario, there is no mixing between the twopairs due to our assignment of half-integer chargesfor the first generation, and integer charges for thesecond and third generations. The absence of theνe–νµ mixing is compatible with the KARMEN data,predicts null results in future laboratory searchingfor the νe–νµ or νe–ντ mixing, and has no influenceon astrophysical processes [16]. Then theMν isautomatically factorized out intoMνs–νe andMνµ–ντ .

The requirement of Super-K data determines thevalue ofa as

(13)

2m23 λ2a(m2

M

)2

3× 10−3 eV2, for a = 4.

The effective mass matrix ofνe andνs in the unit ofeV can be written by

(14)Mνs–νe eV

(λ2β−4 λβ+z−3

λβ+z−3 0

).

The case ofβ − 1 < z (β − 1 > z) corresponds tothe small (large) mixing angle solution for the solarneutrino problem. Some possible solutions are listedin Table 1. Note that in the SMA case, themνe is thesmaller mass since theMνs−νe in Eq. (14) is in the(νs, νe) basis.

It is to be checked in the SNO [17] and KamLANDexperiments. For one of the LOW–QVO solution,the mass of the electron neutrino is just an order ofmagnitude lower than the current experiment limit. Itis possible to probe this value in future experiments

250 C. Liu, J. Song / Physics Letters B 512 (2001) 247–251

Table 1

Solution z β (x) mνe (eV) m2 (eV2) sin2 2θSMA 4 3.5 (12) 10−6 10−6 10−3

LMA 1 3.5 (12) 3× 10−2 3× 10−5 1

LOW–QVO 1 4.5 (13) 3× 10−3 3× 10−8 1

2 4.5 (13) 3× 10−4 3× 10−9 1

3 4.5 (13) 3× 10−5 3× 10−10 1

0 5.5 (14) 3× 10−3 3× 10−10 1

−1 5.5 (14) 3× 10−2 3× 10−9 1

−2 5.5 (14) 0.3 3× 10−8 1

directly. In addition, it may have observable effect inthe cosmic microwave background anisotropies. Theνe and νs compose a Dirac neutrino which does notresult in any observable neutrinoless doubleβ decayof relevant next generation experiments.

In summary, it would be simple if the muon- andtau-neutrinos are just the WDM. We have presented aneutrino model which generates keV scaleνµ andντand meanwhile provides the neutrino oscillation so-lutions for the solar and atmospheric neutrino exper-iments. It combines the seesaw mechanism and theFrogatt–Nielsen mechanism. The key point is that theneutrino mass matrix has a singular form. A light ster-ile neutrino is obtained due to the flavor symmetry.

Finally some remarks should be mentioned.

• Compared to the model in Ref. [12], the lightneutrino spectrum is similar to the ordinary 2+ 2light neutrino scheme [18]. But the two neutrinopairs are more widely separated (by keV). Forthe solar neutrino problem, the SMA, LMA orLOW–QVO oscillation solutions into light sterileneutrinos can be accommodated in our scenario.This is achieved by taking the U(1) flavor chargesto be half-integers for the first generation and tobe integers for the second and third generations.Of course, the LSND results cannot be relevant inthis model. But our results are compatible with theKARMEN data.

• The LMA and LOW–QVO solutions have been ob-tained in fact due to the singular seesaw mecha-nism. The singular seesaw mechanism was appliedto the atmospheric neutrino anomaly in Ref. [15].However, the current Super-K data do not favor the

νµ–νs oscillation. What we have done in Ref. [12]and in this Letter for LMA and LOW–QVO scenar-ios are to obtain a singular seesaw mechanism forthe solar neutrinos.

• It seems a drawback to introduce both the seesawand the Frogatt–Nielsen mechanism to make neu-trinos light. However the seesaw mechanism itselfcannot predict the neutrino flavor structure, espe-cially the maximalνµ–ντ mixing. Another under-lying motivation is to make the four light neutrinoscenario easier to be understood in the frameworkof grand unification theories, which will be studiedfurther.

• Other interesting aspects of keV neutrinos were dis-cussed before. Such neutrinos may be responsiblefor the large velocity of pulsars [19]. Because theneutrino densities in the early universe are muchsmaller than that assumed in the standard cosmol-ogy, the cosmological constraints for neutrinos fromthe big bang nucleosynthesis is different from thatin the standard one [20]. This needs more detailedstudy. In addition to the keV active neutrinos, cer-tain amount of some other keV sterile neutrinos [21]may also contribute to the WDM.

Acknowledgements

C.L. was supported in part by the National NaturalScience Foundation of China with grant no. 10047005.

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