Prospect of the Zee model

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by  Yoshio Koide

1
Prospects of the Zee model
Yoshio Koide a
∗
a Department of Physics, University of Shizuoka, 52-1 Yada, Shizuoka, Japan 422-8526
2 The Zee model is one of promising models of neutrino mass generation mechanism. However, the original Zee
0 model is not on the framework of the ground unification scenario, and moreover, it is recently pointed out that
0 the predicted value of sin22θsolar must be satisfied the relation sin22θsolar >0.99. We discuss whether possible
2 GUTversionsoftheZeemodelcanbefreefromthesevereconstraintsin22θsolar >0.99ornot. Wewillconclude
thatthefollowingtwomodelsarepromising: anR-parityviolatingSUSYGUTmodelandanSO(10)modelwith
n
a a 126-plet scalar.
J

7
2
1. INTRODUCTION that they depend on the Yukawa coupling con-
1
stants y , while only neutrino masses are gener-
v TheZeemodel[1]isoneoftheattractivemod- ij
ated radiatively, so that they depend on the Zee
0
els for neutrino mass matrix. The mass matrix
5 coupling constants fij. In general, fij are inde-
form is given by
2 pendentofy . Wemustseekforaprinciplewhich
ij
1 relates f to y . (For such an attempt, for ex-
0 0 a c ij ij
ample, see Ref.[4],)
2 Mν = a 0 b  , (1.1) (2)TheZeemodelisnotembeddedinaGrand
0 c b 0
/   Unification Theory (GUT). We must seek for a
h room for the Zee scalar h+ in a GUT scenario.
p where
- a=f (m2 m2)K , (3) The Zee model leads to a severe constraint
p eµ µ− e [6] sin22θ > 0.99 under the observed ratio
solar
:he b=fµτ(m2τ −m2µ)K , (1.2) d∆amta2so[l5a]rs/h∆omw2astimn2≪2θs1o.laTrhep0r.8e.seWntesmoluarstnienuvtersintio-
Xiv c=fτe(m2e−m2τ)K , gthaetesaevmeroedcifioendstZraeienmt soidne2l∼2wθhichc>an0.b99e.freefrom
solar
and K is a common factor. The model has only
r The problem (3) has recently pointed out by
a 3 free parameters and it can naturally lead to a the author[6]. A parameter independent investi-
largeneutrino mixing [2]. Especially,forthe case
gationleadstoasevereconstraintonthevalueof
a=c≫b, it leads to a bi-maximal mixing [3] sin22θsolar
U ≃ √1122 −21√12 −0√12  , (1.3) sin22θsolar ≥1− 116(cid:18)∆∆mm2s2aotlmar(cid:19)2. (1.4)
1 1 1
 2 2 √2  The conclusion cannot be loosened even if we
takeRGEeffectsintoconsideration: Also,wecan
with ∆m2 /∆m2 √2b/a.
12 23 ≃ show that the two-loop effect is negligibly small.
However,fromthestandpointoftheunification
The simple ways to escape from the constraint
ofquarksandleptons,therearesomeproblemsto
(1.8)willbyasfollows: Oneistoconsider[7]that
be overcome.
the Yukawa vertices of the charged leptons can
(1) The masses of the quarks and charged lep-
couple to both scalars φ1 and φ2. Another one
tons are generated by the Higgs mechanism, so
[8] is to introduce a single right-handed neutrino
∗E-mailaddress: [email protected] νR and a second singlet Zee scalar S+. Also, a
2
model with a new doubly charged scalar k++ is from the candidates of the extended Zee model
interesting because the two loop effects in such based on a GUT scenario.
a model can give non-negligible contributions to For example, in the Haba-Matsuda-Tanimoto
the neutrino masses [9]. As another attractive mode [12], there is no SU(5) 5-plet scalar (ex-
model,thereisanidea[10,11]thatinanR-parity cept for the conventional Higgs scalars), so that
violating SUSY model we identify Zee scalar h+ there is no proton decay due to A5 and B5
as slepton e˜ . Then, we can obtain additional terms. In their model, in principle, the down-
R
contributions from d-quark loops to the neutrino quark loop diagrams can contribute to the neu-
masses. However,ifwewanttoextendthemodel trino masses in addition to the charged lepton
to a GUT scenario, we will meet a new trouble loop diagrams. Such the down-quark loop di-
“proton decay” as stated in the next section. agrams can contribute to the non-diagonal ele-
Anyhow, these models are not connected to ments of the neutrino mass matrix. However,
GUTscenarios. WewanttoembedtheZeemodel the down-quark loop diagrams include a colored
into aGUT scenario. ForanextendedZeemodel Higgs scalar (the triplet component of the SU(5)
based on a GUT scenario, there is, for example, 5-pletHiggsscalar). Wesupposethatthecolored
the Haba-Matsuda-Tanimoto model [12]. They Higgs scalarhas a mass of the order of the grand
have regarded the Zee scalar h+ as a member of unification scale in order to suppress the proton
the messengerfieldM10+M10 ofSUSY-breaking decay due to the coloredHiggs scalar. Therefore,
on the basis of an SU(5) SUSY GUT. However, thecontributionsarenegligiblysmall,sothatthe
their model cannot escape from the constraint model cannot still be free from the severe con-
(1.8) as we discuss in the next section. straint sin22θ >0.99.
solar
2.2. R-parity violating model
2. EXTENDED ZEE MODEL WITH
Note that if we assume that R-parity violat-
GUT
ing interactions are allowed only for special gen-
We identify the Zee scalar h+ as a member of erations (families), then we can choose such the
SU(5) 10-plet scalar (including a case of 5+10 A5 andB5 interactionsas they do notcontribute
sfermion). Then, we must pay attention to the to the proton decay without breaking the SU(5)
following items: GUT.
(a) Proton decay is safely forbidden.
For example, we assume
(b)Themodelisfreefromthesevereconstraint
2si.n12.2θHsoolwar >to0a.9v9oiidn tphreoZtoeenmdoedcealy. AA150≡≡λλ3ij3ij(ψ(ψ5c5c)Ai)Ai(ψ(ψ105))Bj3A(Bψ(1ψ0e)53)ABjB,, (2.4)
Generally,the5-pletand10-pletscalarsφ5 and i.e., only the interactions witeh the SU(5) 10-plet
φ10 can couple to the 5+10-plet fermions as fol- superfield of the third generation violate the R-
lows. parity. Besides, we must assume this expression
BAA5150≡≡≡(εψ(Aψc5Bc5)CA)AD(ψ(Eψ1(05ψ))c1BA0B()φA(1φB05())ψAB1B0,),CD(φ5)E . (2.1) (mtm5hai.ex4si)spnrigmsotVatort1∗nru3i.exdIoeinncsaodtyrhidaesgetrioblnlatasooilsc,fcooburnebrcsiwdatuhhtsihrecoehuifutgh-ihttemtiushipexn-iqoCntugKasMraokt,
energy scale, we must assume additional flavor
The terms A10 can contribute to the radiative symmetries (a discrete symmetry, and so on).
neutrinomass,whilethetermsA5 andB5 induce Althoughtheassumption(2.4)issomewhatun-
theprotondecay. WewantthetermsA10,butdo natural,anyhow,themodelcangivenonvanishing
not want terms A5 and B5. diagonal elements in the neutrino mass matrix.
The conventional R-parity violating SUSY
The contributions give the form
model [11] contains terms A10, but also contains
terms A5 and B5, so that the model is ruled out Miej,d loop =m0fifj with fi =λ3i3 . (2.5)
3
This matrix form (2.5)correspondsto the Drees- SUSY GUT model with restricted R-parity vio-
Pakvasa-Tata-Veldhuismodel [10]. lating which was discussed in 2.2 is also interest-
ing. The details will be given elsewhere.
2.3. Model with a SU(2) triplet scalar
Except for the “restricted” R-parity violating
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ij ij i j
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Phys. Rev. D64 (2001) 053015 .
the proton decay, is ruled out if the solar neu-
trino data establish sin22θ 0.8. On the
solar
∼
other hand, an SO(10) 126 model is promising
if neutrino data show sin22θ 0.8. How-
solar
∼
ever,suchamodelwithatripletscalarwillcause
another problems. More study is required. An