Radiative four neutrino masses and mixings

a r X i v :h e p -p h /9810448v 1 22 O c t 1998

RADIATIVE FOUR NEUTRINO MASSES AND MIXINGS

PROBIR ROY

Tata Institute of Fundamental Research,Mumbai,India

E-mail:probir@theory.tifr.res.in

The radiative model,proposed by Zee,is extended from the three neutrino (νe,μ,τ)sector to include a fourth sterile neutrino νs .The oscillations νe ?νs (solar),νμ?ντ(atmospheric)and νe ?νμ(LSND)are invoked to explain all oscillation data with the successful relation (?m 2)atm ?2[(?m 2)solar (?m 2)LSND ]1/2as an added bonus.

We have summarized the constraints on neutrino masses and mixings (within two-?avor oscillations at a time)from observed neutrino anomalies in Table 1.We

have used the notation δm 2ij ≡|m 2i ?m 2

j |for the mass squared di?erence pertaining to ?avor eigenstates i,j and θij for their mixing angle.Here α,βcan be any ?avor other than e,μsince α=μand β=e are implied by other data.

Let us ?rst review the radiative Zee model 1which was constructed for the νe ?νμ?ντsystem.The electroweak gauge group is SU (2)L ×U (1)Y .In addi-tion to the SM leptons and two doublet Higgs bosons

Φ1,2≡ φ+1,2

φ01,2

,with Φ2being leptophobic,Zee also

postulated a singlet charged Higgs χ+carrying lepton number L =?2.Thus the relevant part of the interac-tion is

L Zee

1=

i,j

f [ij ](νi??jL ??iL νjL )χ++

i

h i (ˉνiL φ+1+ˉ?iL φ01)?iR

+μ(φ+1φ0

2

?

φ01φ+2)χ

?

+h.c.(1)

In (1)i,j are generation indices and h i ≡m ?i / φ01 are

generation-hierarechical Yukawa coupling strengths of Φ1whereas f [ij ],the Yukawa couplings for the charged Higgs χ+,are antisymmetric in i,j ;furthermore,we have used

the notation ψξ≡ˉψ

C ξfor any two fermionic spinor ?elds ψand ξ.The

D =4terms in L Zee 1

above possess a Glashow-Weinberg 2

Z 2symmetry,softly broken by the Higgs self-interaction proportional to μ,protecting the system against unwanted FCNC e?ects.

The ?rst two terms in the RHS of (1)radiatively generate o?diagonal neutrino Majorana masses through the one-loop diagram of Fig.1involving charged Higgs exchange.The consequent Majorana mass-matrix is 3

m ν=

νeL

νμL ντL νC μL

a

o

c

v 1

F (m 2χ1,m 2

φ1),

(3)

b =

f [eτ](m 2τ

?m 2e )

μv 2

v 1

F (m 2χ1,m 2

φ1).

(5)

In (3)

F (m 2χ,m 2

φ1)

=

1

m 2χ?m 2φ1

?n

m 2χ

M 2χ

<0.036G F ,(8)

|f [μτ]|2

Table 1:Experimental constraints on neutrino properties

Neutrino anomaly

Mixing angle δm 2μβ～(2?5)×10

?3

eV 2LSND

sin 22θeμ=(6±3)×10?3

δm 2eα=(1?10)×10

?6eV 2Solar Large angle sin 22θeα>0.4

MSW solution

δm 2eα?10

?10

eV 2L-parity

U (1)′

(νi ,l i )L

(2,?1/2)

?

νsL

(1,0)

+

1

S R

(1,0)

L-parity

U (1)′

(φ+1,2,φ0

1,2)

(2,1/2)

+

0χ+2

(1,1)

+

1

νC μL o a b d νC τL

b

c

o

f

c

1?

m 2μ

f [τμ]

.(15)

We assume hierarchical couplings for the lepton non-conserving Yukawa terms,i.e.

|f [eτ]|?|f [eμ]|?|f [μτ]|.

(16)

This means that |a |?|b |?|c |.Moreover,we assume

that |f ′i |<～m i /v 1.This leads us to conclude that c is

the dominant matrix element in(11)and,if|f[μτ]|2?|f[eμ]f[eτ]|,|ef|?|ab|.Under these circumstances,the leading expressions for the mass eigenvalues of(11)are:

m1??2ab

c

,

m3?c+ef

c

.(17)

Thusν3andν4make a pseudo-Dirac pair out of a max-imally mixed combination ofνμandντ.On the other hand,ν1andν2are predominantlyνe,νs combinations with a small mixing angle.

We now have

(?m2)solar?

4e2f2

2[fμτ]e +

b

m2τ

.(19)

A suitable parametric choice is:a?3×10?5eV,c?1eV, e?0.12eV and f?0.01eV.These yield mνe?2×10?6eV,m v s?2.4×10?3eV and sin22θes～6×10?3 to match with the small angle MSW solution of the so-lar neutrino anomaly in Table1.In comparison,for the atmospheric anomaly,we have?m2μτ～5×10?3eV2, sin22θμτ?1withνμandντmasses around1eV.Fi-nally,for the LSND anomaly,we have?m2eμ～1eV2 and sin22θeμ?6×10?3.

This work has been done in collaboration with N. Gaur,A.Ghosal and E.Ma.

References

1.A.Zee,Phys.Lett.B93,389(1980).

2.S.L.Glashow and S.Weinberg,Phys.Rev.D15,

1958(1977).

3.A.Yu.Smirnov and M.Tanimoto,Phys.Rev.D

55,1665(1997).

4.N.Gaur,A.Ghosal,E.Ma and P.Roy,Phys.Rev.

D58,R7131(1998).