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RADIATIVE FOUR NEUTRINO MASSES AND MIXINGS
PROBIR ROY Tata Institute of Fundamental Research, Mumbai, India E-mail: [email protected]
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 2 2 2 1/2
successful relation (∆m )atm ' 2[(∆m )solar(∆m )LSND] as an added bonus.
0
< > We have summarized the constraints on neutrino 2
masses and mixings (within two-flavor oscillations at a +
time) from observed neutrino anomalies in Table 1. We +
1 2 2 2 1
have used the notation δmij ≡|mi −mj|for the mass
l l
jL
iL jR squared difference pertaining to flavor eigenstates i, j and jL
θij for their mixing angle. Here α, β can be any flavor
0
< >
other than e, µ since α 6= µ and β 6= e are implied by 1 other data. Let us first review the radiative Zee model 1 which Figure 1: One loop radiative νi − νj (i, j = e, µ, τ) mass due to charged Higgs exchange. was constructed for the νe − νµ − ντ system. The electroweak gauge group is SU(2)L × U(1)Y . In addi- tion to the SM leptons and two doublet Higgs bosons with φ+ Φ 1,2 ,withΦ being leptophobic, Zee also 1,2 ≡ 0 2 µv2 φ1,2 a = f (m2 − m2) F (m2 ,m2 ), (3) + [eµ] µ e v χ1 φ1 postulated a singlet charged Higgs χ carrying lepton 1 number L = −2. Thus the relevant part of the interac- tion is 2 2 µv2 2 2 b = f[eτ ](mτ − me) F (mχ1 ,mφ1), (4) v1 Zee + L = f[ij](νi``jL − `iLνjL)χ 1 µv i,j c = f (m2 m2 ) 2 F (m2 ,m2 ). (5) X [µτ ] τ − µ χ1 φ1 + ¯ 0 v1 + hi(¯νiLφ1 + `iLφ1)`iR i In (3) X+ 0 0 + − +µ(φ1 φ2 − φ1φ2 )χ + h.c. (1) 1 1 m2 F (m2 ,m2 )= `n χ , (6) 0 χ φ1 2 2 2 2 In (1) i, j are generation indices and hi ≡ m`i/hφ i are 16π m m m 1 χ − φ1 φ1 generation-hierarechical Yukawa coupling strengths of Φ1 + whereas f[ij], the Yukawa couplings for the charged Higgs with mχ,φ1 being the masses associated with χ , φ1.It + χ , are antisymmetric in i, j;furthermore,wehaveused follows by considering the νµ − ντ submatrix of (2) that ¯C the notation ψξ ≡ ψ ξ for any two fermionic spinor fields the corresponding mixing angle θµτ is given by Zee ψ and ξ.TheD=4termsinL1 above possess a 2 Glashow-Weinberg Z2 symmetry, softly broken by the 2a sin 2θµτ = . (7) Higgs self-interaction proportional to µ, protecting the (b − c)2 +4a2 system against unwanted FCNC effects. The first two terms in the RHS of (1) radiatively It may also be noted thatp experimental constraints 3 generate off diagonal neutrino Majorana masses through from electron-neutrino scattering and from the decay the one-loop diagram of Fig. 1 involving charged Higgs τ → µν¯µντ respectively imply exchange. The consequent Majorana mass-matrix is 3 |f |2 [eµ] < 0.036 G , (8) νeL νµL ντL 2 F Mχ ν C oab m = eL , (2) ν νC aoc |f |2 µL [µτ ] < 0.13 G . (9) C M 2 F ντL bco χ 1 Table 1: Experimental constraints on neutrino properties
Neutrino anomaly Differences of mass squared Mixing angle
2 −3 2 2 Atmospheric δmµβ ∼ (2 − 5) × 10 eV sin 2θµβ > 0.8 2 2 2 −3 LSND δmeµ ∼ (0.1 − 1) eV sin 2θeµ =(6±3) × 10 2 −6 2 2 −3 Solar Small angle δmeα =(1−10) × 10 eV sin 2θeα ' 5 × 10 MSW solution
2 −5 2 2 Solar Large angle δmeα =(0.1−10) × 10 eV sin 2θeα > 0.4 MSW solution
2 −10 2 2 Solar Vacuum solution δmeα ' 10 eV sin 2θeα > 0.75
Table 2: List of fermion and scalar fields in our model
0
< > 2 0
fermions L-parity SU(2)L × U(1)Y U(1)
+ +
1 2
(νi,li)L − (2, −1/2) 0
l l
iL sL jR liR − (1, −1) 0 jL
νsL − (1,0) 1
0
< > NR + (1,0) 1 1 SR + (1,0) 0 Figure 2: One loop radiative νi − νs (i = e, µ, τ) mass due to charged Higgs exchange. 0 scalars L-parity SU(2)L × U(1)Y U(1) in addition to Fig. 1, contributes to the neutrino Majo- + 0 (φ1,2,φ1,2) + (2,1/2) 0 rana mass matrix mν of (2). The latter is now extended + χ1 + (1,1) 0 to + νeL νµL ντL νsL χ2 + (1,1) 1 0 C χ2 + (1,0) 1 νµL oabd C mν = νeL aoce (11) C ντL bcof νC defo Evidently, the Zee model is unable to incorporate all the sL constraints of Table 1 since it does not have νs. Here the extra elements d, e, f are given by In our model 4 the gauge group is SU(2) U(1) L × Y × 0 0 0 0 2 2 0 d = f[eτ ]f mτ + f[eµ]f mµ µ hχ iF (m ,m ), (12) U (1) together with a Z2 discrete symmetry which we τ µ 2 χ1 χ2 call L-parity. The complete set of fields, along with their 0 0 0 0 2 2 e = f f mτ + f f me µ hχ iF (m ,m ), (13) transformation properties, are listed in Table 2. We have [µτ ] τ [µe] e 2 χ1 χ2 0 2 2 included a fourth sterile lefthanded neutrino νsL which 0 0 0 f = f[τµ]fµmµ +f[τe]feme µ hχ2iF(mχ1,mχ2). (14) is massless at the tree level. The righthanded neutral Neglecting m2 in comparison with m2 , (3) and (12) imply leptons NR and SR are taken to be heavy. Among the e τ + + that Higgs scalars {χ}, χ1 is the equivalent of Zee’s χ while 2 + 0 be mµ ff[eµ] χ and χ2 are extra. The doublets Φ1,2 are as in the Zee 2 d = 1 − 2 + . (15) scenario. The relevant part of the interaction now is c mτ ! f[τµ]
Zee 0 + 0 + − 0 We assume hierarchical couplings for the lepton non- LI = LI + fiν¯sL`iRχ2 + µ χ1 χ2 χ2 conserving Yukawa terms, i.e. i X ? 0 0 + h NRSRχ2 + h.c. (10) |f[eτ ]| |f[eµ]| |f[µτ ]|. (16) This means that |a| |b| |c|. Moreover, we assume 0 < Because of (10), an extra one-loop diagram (Fig. 2, that |fi | ∼ mi/v1. This leads us to conclude that c is 2 2 the dominant matrix element in (11) and, if |f[µτ ]| |f[eµ]f[eτ ]|, |ef| |ab|. Under these circumstances, the leading expressions for the mass eigenvalues of (11) are:
ab ef m '−2 ,m '−2 , 1 c 2 c ef ef m ' c + ,m '−c+ . (17) 3 c 4 c
Thus ν3 and ν4 make a pseudo-Dirac pair out of a max- imally mixed combination of νµ and ντ . On the other hand, ν1 and ν2 are predominantly νe, νs combinations with a small mixing angle. We now have 4e2f 2 (∆m2) ' , (∆m2) ' 4ef, solar c2 atmos 2 2 (∆m )LSND ' c . (18) and
2 2 2 1/2 (∆m )atm ' 2 (∆m )solar(∆m )LSND .
The mixings can be summarized as
2 2 2 2 sin 2θeµ ' 4b /c , sin 2θµτ ' 1, 2 f[eµ]c b mµ sin θes = + 2 . (19) 2[fµτ ]e 2f mτ
A suitable parametric choice is: a ' 3×10−5eV , c ' 1eV , e ' 0.12eV and f ' 0.01eV . These yield mνe ' −6 −3 2 −3 2×10 eV , mvs ' 2.4×10 eV and sin 2θes ∼ 6×10 to match with the small angle MSW solution of the so- lar neutrino anomaly in Table 1. In comparison, for the 2 −3 2 atmosphericanomaly,wehave∆mµτ ∼ 5 × 10 eV , 2 sin 2θµτ ' 1withνµ and ντ masses around 1eV .Fi- 2 2 nally, for the LSND anomaly, we have ∆meµ ∼ 1eV 2 −3 and sin 2θeµ ' 6 × 10 . This work has been done in collaboration with N. Gaur, A. Ghosal and E. Ma.
References
1. A. Zee, Phys. Lett. B 93, 389 (1980). 2. S.L. Glashow and S. Weinberg, Phys. Rev. D 15, 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. D 58, R7131 (1998).
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0
< >
2
+ +
1 1
l l
jL
iL
jL jR
0
< > 1
0
< >
2
+ +
1 2
l l
iL sL
jL jR
0
< > 1