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RADIATIVE FOUR NEUTRINO MASSES and MIXINGS PROBIR ROY Tata Institute of Fundamental Research, Mumbai, India E-Mail

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RADIATIVE FOUR NEUTRINO MASSES and MIXINGS PROBIR ROY Tata Institute of Fundamental Research, Mumbai, India E-Mail View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by CERN Document Server 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). 3 0 < > 2 + + 1 1 l l jL iL jL jR 0 < > 1 0 < > 2 + + 1 2 l l iL sL jL jR 0 < > 1.
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