SO(3) Gauge Symmetry and Nearly Tri-bimaximal Neutrino Mixing

# SO(3) Gauge Symmetry and Nearly Tri-bimaximal Neutrino Mixing

Yue-Liang Wu
###### Abstract

In this note I mainly focus on the neutrino physics part in my talk and report the most recent progress made in . It is seen that the Majorana features of neutrinos and SO(3) gauge flavor symmetry can simultaneously explain the smallness of neutrino masses and nearly tri-bimaximal neutrino mixing when combining together with the mechanism of approximate global U(1) family symmetry. The mixing angle and CP-violating phase are in general nonzero and testable experimentally at the allowed sensitivity. The model also predicts the existence of vector-like Majorana neutrinos and charged leptons as well as new Higgs bosons, some of them can be light and explored at the LHC and ILC.

Keywords: Gauge Flavor Symmetry; Neutrino Mixing; Vector-like Fermions.

PACS numbers: 14.60.Pg, 12.60.-i, 11.30.Hv,14.60.St

The observed massive neutrinos and dark matter in our universe challenge the standard models of both particle physics and cosmology. So both flavor physics and cosmology may tell us fundamental physics. In this note I will concentrate on discussing the neutrino physics.

Various neutrino experiments provide more and more stringent constraints on the three mixing angles and mass squared differences for three neutrinos

 30∘<θ12<38∘,36∘<θ23<54∘,θ13<10∘ (1) 7.2×10−5 eV2<Δm221=m2νμ−m2νe<8.9×10−5 eV2, 2.1×10−3 eV2<Δm232=m2ντ−m2νμ<3.1×10−3 eV2 (2)

at the 99% confidence level. In comparison with the quark sector, it raises a puzzle that why neutrino masses are so tiny, but their mixing angles are so large. The only peculiar property for neutrinos is that they could be Majorana fermions, so a natural solution to the puzzle is most likely attributed to the Majorana features. Thus revealing the origin of large mixing angles and small masses of neutrinos is important not only for understanding neutrino physics, but also for exploring new physics beyond the standard model.

The current data on the neutrino mixing angles are consistent with the so-called tri-bimaximal mixing with , and , which was first proposed by Harrison, Perkins and Scott , and investigated by various groups. The mixing angle is the most unclear parameter and expected to be measured in near future. Such a tri-bimaximal mixing matrix has been found to be yielded by considering some interesting symmetries, especially the discrete symmetries. In general, it is shown that the discrete symmetries lead to . In such a case, it is hard to be directly tested experimentally. Alternatively, it is interesting to consider a non-abelian gauge family symmetry SO(3) instead of discrete symmetries discussed widely in literature. In this case, the tri-bimaximal neutrino mixing matrix is generally obtained as the lowest order approximation from diagonalizing a special symmetric mass matrix. In fact, the greatest success of the standard model (SM) is the gauge symmetry structure which has been tested by more and more precise experiments. SO(3) gauge family symmetry can be regarded as a simple extension of the standard model with three families and Majorana neutrinos. It is noted that only SO(3) rather than SU(3) is allowed due to the Majorana feature of neutrinos.

The invariant Lagrangian for Yukawa interactions of leptons with Majorana neutrinos can be constructed as follows

 LY = yν¯l~HνR+yN¯lHNN+12ξN¯NΦνN+12MR¯νRνcR (3) + ye¯lHE+ξeϕs¯EeR+12ξE¯EΦeE+H.c.

where , , , , and are all real Yukawa coupling constants and is the mass of right-handed Majorana neutrinos. All the fermions , , , , and belong to SO(3) triplets in family space. Where denote doublet leptons, and are doublet Higgs bosons with . are the right-handed neutrinos with the charge conjugated ones. are singlet vector-like charged leptons and the are singlet vector-like Majorana neutrinos with . is a singlet Higgs boson. The scalar fields and are SO(3) tri-triplets Higgs bosons satisfying , and which is required by the hermiticity condition of Lagrangian and the Majorana condition of vector-like neutrinos. The above Lagrangian is solely ensured by the following discrete symmetry ( and )

 N→iγ5 N,Φν→−Φν,HN→−iHN,ϕs→−ϕs,eR→−eR (4)

In terms of SO(3) representation, one can reexpress the real symmetric tri-triplet Higgs boson into the following general form

 Φν≡OνϕνOTν,Oν(x)=eiλiΘνi(x),ϕν(x)=⎛⎜ ⎜⎝ϕν1ϕν2ϕν3ϕν2ϕν3ϕν1ϕν3ϕν1ϕν2⎞⎟ ⎟⎠ (5)

with being the generators of SO(3). Where may be regarded as three rotational scalar fields of SO(3), and are three dilatation scalar fields.

SO(3) gauge invariance allows us to fix the gauge by making SO(3) gauge transformation , so that , we then arrive at the following Yukawa interactions

 LY = yν¯l~HνR+yN¯lHNN+12ξN¯NϕνN+12MR¯νRνcR (6) + ye¯lHE+ξeϕs¯EeR+12ξE¯E^ΦeE+H.c.

which is invariant under transformation. Where remains Hermitian and contains nine independent scalar fields, which can generally be reexpressed in terms of SO(3) representation as the following form

 ^Φe≡UeϕeU†e,Ue(x)≡PeOe,Oe(x)=eiλiχei(x), (7)

and

 (8)

where are regarded as three rotational scalar fields of SO(3), denote three phase scalar fields and are three dilation scalar fields.

We now consider the following general vacuum structure of scalar fields under the above gauge fixing condition

 =v,=vN <ϕs(x)>=vs,<ϕνi(x)>=vνi, (9) <ϕei(x)>=vei,<χei(x)>=θei,<ηei(x)>=δei

namely , and . Here (i=1,2,3) are CP phases arising from spontaneous symmetry breaking and are three rotational angles of SO(3).

Such a vacuum structure after spontaneous symmetry breaking leads to the following mass matrices for neutrinos with a type II like see-saw mechanism and for charged leptons with a generalized see-saw mechanism

 Mν=mDνM−1RmDν+mDNM−1NmDN, (10) Me=VemDEM−1EmDEV†e (11)

with , , , , and

 MN=ξN⎛⎜ ⎜⎝vν1vν2vν3vν2vν3vν1vν3vν1vν2⎞⎟ ⎟⎠,ME=ξE⎛⎜ ⎜⎝ve1000ve2000ve3⎞⎟ ⎟⎠ (12) Ve≡Peδ⎛⎜ ⎜⎝ce12ce13se12ce13se13−se12ce23−ce12se23se13ce12ce23−se12se23se13se23ce13se12se23−ce12ce23se13−ce12se23−se12ce23se13ce23ce13⎞⎟ ⎟⎠ (13)

with and , and are given as functions of . A similar special symmetric neutrino mass matrix like was also resulted for the Dirac-type neutrinos with a new symmetry and the Majorana-type neutrinos with the group.

When taking the Majorana neutrino masses and to be infinity large, the interactions with Majorana neutrinos decouple from the theory,

 y2νMR¯l~H~HTlc,  y2NMN¯lHNHTNlc→0,forMR,  MN→∞ (14)

which implies that the resulting Yukawa interactions in this limit generate additional global U(1) family symmetries for the charged lepton sector. Namely, once the Majorana neutrinos become very heavy, the Yukawa interactions possess approximate global U(1) family symmetries. Thus when applying the mechanism of approximate global U(1) family symmetries to the Yukawa interactions after SO(3) symmetry is broken down spontaneously, we have , and , namely

 Mν≪1,θeij≪1 (15)

which provides a possible explanation why the observed left-handed neutrinos are so light and meanwhile the charged lepton mixing angles must be small. Thus the neutrino mass matrix is given by a type II like see-saw mechanism and the charged lepton mass matrix is presented by a generalized see-saw mechanism. By diagonalizing the mass matrices, we obtain

 VTνMνVν=diag.(mνe,mνμ,mντ),V†eMeVe=diag.(me,mμ,mτ) (16)

where

 Vν=⎛⎜ ⎜ ⎜ ⎜ ⎜⎝2√6cν1√32√6sν−1√6cν−1√2sν1√31√2cν−1√6sν−1√6cν+1√2sν1√3−1√2cν−1√6sν⎞⎟ ⎟ ⎟ ⎟ ⎟⎠≡V0V1 (17)

with

 V0=⎛⎜ ⎜ ⎜ ⎜ ⎜⎝2√61√30−1√61√31√2−1√61√3−1√2⎞⎟ ⎟ ⎟ ⎟ ⎟⎠,V1=⎛⎜⎝cν0sν010−sν0cν⎞⎟⎠ (18)

Here is the so-called tri-bimaximal mixing matrix. For short, we have introduced the notations and with

 tan2θν=√3(vν21−vν31)vν21+vν31,vν21≡vν2−vν1,vν31≡vν3−vν1 (19)

As the smallness of charged lepton mixing angles can be attributed to the mechanism of approximate global U(1) family symmetries in the Yukawa sector, in a good approximation up to the first order of , the leptonic CKM-type mixing matrix in the mass eigenstates of leptons may be expressed as the following simplified form

 V=V†eVν≃ ⎛⎜ ⎜⎝1−se12−se13se121−se23se13se231⎞⎟ ⎟⎠Pe†δ⎛⎜ ⎜ ⎜ ⎜ ⎜⎝2√61√30−1√61√31√2−1√61√3−1√2⎞⎟ ⎟ ⎟ ⎟ ⎟⎠⎛⎜⎝cν0sν010−sν0cν⎞⎟⎠ (20)

The three vector-like heavy Majorana neutrino masses are obtained via diagonalizing the mass matrix , i.e.,

 mN1=−mN√1−Δ,mN2=mN,mN3=mN√1−Δ (21)

with

 mN≡ξN(vν1+vν2+vν3),Δ=3(vν1vν2+vν2vν3+vν3vν1)/(vν1+vν2+vν3)2 (22)

The masses of three left-handed light Majorana neutrinos are given in the physics basis as follows

 mνe=¯m0−m1(2+¯Δ),mνμ=¯m0,mντ=¯m0+m1¯Δ (23)

with , , and . It then enables us from the experimentally measured neutrino mass squire differences to extract two mass parameters and for a given value of parameter with .

The numerical results are presented in table 1 by taking the central values of the mass squire differences and .

Without losing generality, considering the case , we then have the following approximate relations

 Δ≃3r(1+r)2,tan2θν≃√3(1−r)1+r,r≡vν3/vν2 (24)

which shows that in this case both the mixing angle and the ratio can be determined for the given values of , an interesting solution is

 Δ=r=√3−1,tan2θν=2−√3,θν=7.5∘ (25)

For a numerical estimation, it is useful to investigate the following two interesting cases

 Case I:se13≃0, se12≃0 (26) Case II:se13≪se12∼√me/mμ≃0.07,δe1−δe2=π/2 (27)

which allows us to present a reasonable estimation for and (see table 2).

where (i.e., ) should be a more general and reasonable case when no symmetry is imposed, the resulting mixing angle can be large enough to be detected. For the case II, both mixing angle and CP-violating phase are in general testable by the future neutrino experiments. For the typical range , we are led to the most optimistic predictions for the mixing angle and CP-violating phase

 θ13≃7∘±4∘,δν≃35∘±20∘,r=vν3/vν2≃0.73±0.17 (28)

which can be tested in the future experiments.

When taking the Dirac type neutrino masses and to be at the order of MeV (i.e., at the same order of electron mass), and the mass parameter  GeV, we have the lightest vector-like Majorana neutrino masses and charged lepton mass to be

 mN1=mN3≃O(250) GeV∼O(25) TeV (29) mE3≃(127∼352) GeV (30)

which is at the electroweak scale and can be explored at LHC and ILC.

In conclusion, we have shown how the puzzles on the smallness of neutrino masses and the nearly tri-bimaximal neutrino mixing may simultaneously be understood in the flavor SO(3) gauge symmetry model with the mechanism of approximate global U(1) family symmetry. The vacuum structure of SO(3) symmetry breaking for the SO(3) tri-triplet Higgs bosons plays an important role. The mixing angle is in general nonzero and its typical values range from the experimentally allowed sensitivity to the current experimental bound. CP violation in the lepton sector is caused by a spontaneous symmetry breaking and can be significantly large for exploring via a long baseline neutrino experiment. A similar consideration can be extended to the quark sector, unlike the lepton sector with the features of Majorana neutrinos, the mechanism of approximate global U(1) family symmetry can be applied to understand the smallness of quark mixing angles.

## Acknowledgments

This work was supported in part by the National Science Foundation of China (NSFC) under the grant 10475105, 10491306, and the key Project of Chinese Academy of Sciences (CAS). The author is grateful to Chun Liu for his hard work on publishing the proceedings of ICFP2007.

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