aa r X i v : . [ h e p - ph ] A ug UCRHEP-T472August 2009
Neutrino Tribimaximal Mixing from A Alone
Ernest Ma
Department of Physics and Astronomy, University of California,Riverside, California 92521, USA
Abstract
Neutrino tribimaximal mixing is obtained from the breaking of A to Z in thecharged-lepton sector and to Z in the neutrino sector. To enforce this conflictingpattern, extra particles and symmetries are usually invoked, often accompanied bynonrenormalizable interactions and even extra dimensions. It is shown here in a specificrenormalizable model how A alone will accomplish this, with only the help of leptonnumber. ntroduction : The observed neutrino mixing matrix is very close to the tribimaximalform [1] and is best understood in terms of the tetrahedral symmetry A [2, 3, 4, 5]. Thekey to its success is the pattern of symmetry breaking with preserved subgroups [6, 7] suchthat A → Z and A → Z in two different sectors. This misalignment is technicallychallenging to achieve and many auxiliary symmetries and particles have been invoked tomake it happen [5, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17], often with nonrenormalizable operatorsand even in the context of extra dimensions, thereby complicating the simple original ideaof using just A .On closer examination, it turns out that the breaking of Z by Z through a judiciouschoice of soft terms in the Higgs potential would work because there is a hidden symmetrywhich prevents the appearance of the other unwanted terms. This is accomplished in arenormalizable model using Higgs doublets and triplets, both transforming as 1 and 3 of A alone. Lepton and Higgs Assignments : The three families of leptons are considered [18] astriplets under A : L i = ( ν i , l i ) ∼ , l ci ∼ . (1)To obtain arbitrary lepton masses m e,µ,τ , four Higgs doublets are used:Φ = ( φ , φ − ) ∼ , Φ i = ( φ i , φ − i ) ∼ . (2)The Yukawa interactions are then given by L Y = h ( ν φ − − l φ ) l c + h ( ν φ − − l φ ) l c + h ( ν φ − − l φ ) l c + (1 → , → , →
1) + (1 → , → , →
1) +
H.c., (3)resulting in M l = h v h v h v h v h v h v h v h v h v . (4)2or v = v = v = v , this is exactly diagonalized: M l = U L m e m µ
00 0 m τ U † R , (5)where U L = U R = 1 √ ω ω ω ω , (6)with ω = exp(2 πi/
3) = − / i √ / m e m µ m τ = U L √ h v h vh v . (7)Before discussing the Higgs potential, note that the above is invariant under the trans-formation Φ → Φ , Φ , → − Φ , , v → v , v , → − v , , (8) L → L , L , → − L , , l c → l c , l c , → − l c , . (9)This hidden symmetry will prove to be crucial in obtaining tribimaximal mixing with A alone. Scalar Potential of Higgs Doublets : Consider the scalar potential V of Φ by itself. Ithas just 2 terms: V = µ (Φ † Φ ) + 12 λ (Φ † Φ ) . (10)The corresponding potential V of Φ , , was shown to be [2] V = µ X i (Φ † i Φ i ) + 12 λ X i (Φ † i Φ i ) + λ | Φ † Φ + ω Φ † Φ + ω Φ † Φ | + λ h | Φ † Φ | + | Φ † Φ | + | Φ † Φ | i + (cid:26) λ h (Φ † Φ ) + (Φ † Φ ) + (Φ † Φ ) i + H.c. (cid:27) . (11)3he terms connecting the two are given by V = λ (Φ † Φ ) X i (Φ † i Φ i ) + ( λ X i (Φ † Φ i )(Φ † Φ i ) + H.c. ) + h λ (Φ † Φ )(Φ † Φ ) + (Φ † Φ )(Φ † Φ ) + (Φ † Φ )(Φ † Φ ) + H.c. i + h λ (Φ † Φ )(Φ † Φ ) + (Φ † Φ )(Φ † Φ ) + (Φ † Φ )(Φ † Φ ) + H.c. i . (12)It is clear from the above that the breaking of A to Z is maintained with v = v = v .The new observation of here is that this solution is physically indistinguishable from v = − v = − v . Neutrino Sector : To obtain arbitrary Majorana neutrino masses, four Higgs triplets areused: ξ = ( ξ ++0 , ξ +0 , ξ ) ∼ , ξ i = ( ξ ++ i , ξ + i , ξ i ) ∼ . (13)The Yukawa interactions are then of the form ξ P i ν i ν i and ξ ν ν + ξ ν ν + ξ ν ν . It isnow assumed that u = u = 0, where u i = h ξ i i , so that A breaks to Z , resulting in [4, 5] M ν = a a d d a = U ν a + d a
00 0 − a + d U Tν , (14)where U ν = / √ − / √
20 1 / √ / √ i . (15)The mismatch between U L and U ν yields [4] U † L U ν = q / / √ − / √ / √ − / √ − / √ / √ / √ , (16)i.e. tribimaximal mixing. This is the simplest such realization, which is conistent with onlythe normal hierarchy of neutrino masses ( m < m < m ), with the prediction [19] | m ν e | ≃ | m ee | + ∆ m atm / , (17)4here m ν e is the kinematical mass of ν e , and m ee is its mass measured in neutrinoless doublebeta decay. Note that ξ may be assigned the global lepton number L = − L → ( − L . Addition of Higgs Triplets : It is assumed that ξ and ξ , , are all very heavy [20] withmasses M and M respectively, so they do not appear as physical particles at or below theTeV scale. They have quartic interactions among themselves similar to those of V , , . It iseasy to establish that the analog of Eq. (8) holds here as well, i.e. ξ → ξ , ξ , → − ξ , . (18)To obtain the desirable solution u = 0, u = 0, and u = u = 0, the global lepton number L as well as A must be broken. The new observation of this note is that the following choiceof soft scalar trilinear terms V ′ = µ ′ ξ Φ Φ + µ ′ ξ X i Φ i Φ i + µ ′′ ξ Φ Φ + µ ′′ ξ X i Φ i Φ i (19)works. From V ′ , very small values of u and u are induced [20] by v and v = v = v = v ,i.e. u ≃ − µ ′ v − µ ′ v M , u ≃ − µ ′′ v − µ ′′ v M , (20)but u , cannot appear. To understand this, note first that in order for ξ or ξ , , to havea vacuum expectation value, L must be broken and that can only be achieved through theterms of V ′ . However, ξ or ξ are only connected to these terms through their quartic orYukawa couplings, which always preserve the symmetry ψ , → − ψ , , where ψ may be ( ν, l )or l c or Φ or ξ . Hence they always appear together and protect each other from getting avacuum expectation value if neither has one to begin with. Phenomenology of Higgs Doublets : Since the Higgs triplets are very heavy, they may beintegrated away, and from the explicit choice of V ′ , the effective scalar potential of Higgsdoublets at the electroweak scale is still given by V + V + V . Therefore, A symmetry is5aintained and its breaking ( v = v = v = v ) results in a residual Z symmetry, underwhich Φ and Φ ′ = (Φ + Φ + Φ ) / √ ′ = 1 √ + ω Φ + ω Φ ) ∼ ω, Φ ′ = 1 √ + ω Φ + ω Φ ) ∼ ω . (21)The charged leptons are similarly redefined: e = 1 √ l + l + l ) ∼ , e c = 1 √ l c + l c + l c ) ∼ , (22) µ = 1 √ l + ωl + ω l ) ∼ ω , µ c = 1 √ l c + ω l c + ωl c ) ∼ ω, (23) τ = 1 √ l + ω l + ωl ) ∼ ω, τ c = 1 √ l c + ωl c + ω l c ) ∼ ω . (24)As a result, φ couples to the charged leptons according to13 v ( m e + m µ + m τ )( ee c + µµ c + τ τ c ) , (25)and φ ′ according to13 √ v [(2 m e − m µ − m τ ) ee c + (2 m µ − m τ − m e ) µµ c + (2 m τ − m e − m µ ) τ τ c ] (26)The linear combination ( v φ + √ vφ ′ ) / q v + 3 v corresponds to the Standard-Model Higgsboson which couples to ( m e ee c + m µ µµ c + m τ τ τ c ) / q v + 3 v as expected.The Yukawa interactions of φ ′ and φ ′ are given by13 √ v [(2 m µ − m τ − m e ) eτ c + (2 m τ − m e − m µ ) µe c + (2 m e − m µ − m τ ) τ µ c ] , (27)13 √ v [(2 m τ − m e − m µ ) eµ c + (2 m e − m µ − m τ ) µτ c + (2 m µ − m τ − m e ) τ e c ] , (28)respectively. Note that the above Yukawa couplings are different from those of the original A model [2], where l c , , ∼ , ′ , ′′ . Note also that the residual Z symmetry is a discretelepton flavor symmetry which forbids [16] transitions such as µ → eγ and µ → eee . However,6here are processes forbidden by the usual lepton flavor ( L e , L µ , L τ ) conservation, but allowedby Z . From Eqs. (27) and (28), it is clear that τ + → e + e + µ − , τ + → µ + µ + e − , (29)are possible. Their branching fractions are easily calculated to be B ( τ + → e + e + µ − ) ≃ B ( τ + → µ + µ + e − ) ≃ m τ m eff ! v + 3 v v ! B ( τ → µνν ) ≃ . × −
100 GeV m eff ! v + 3 v v ! , (30)where m eff is the effective contribution of the two scalar fields φ ′ , , as compared to theexperimental upper bound of 1 . × − . Of course, if φ ′ , are produced at the LargeHadron Collider (LHC), then their decays into two different charged leptons would be aunique signature of this model.The contribution of φ ′ , to the muon anomalous magnetic moment is given by∆ a µ = 5 G F m τ √ π m µ m eff ! v + 3 v v ! ≃ . × −
100 GeV m eff ! v + 3 v v ! , (31)well below the current experimental sensitivity. Conclusion : It has been shown how a renormalizable model of Higgs doublets andtriplets based on A alone, with only the help of lepton number, is able to sustain neutrinotribimaximal mixing. The key is the judicious choice of the soft terms of Eq. (19), whichbreak lepton number L to ( − L , as well as A , but preserving the symmetry whereby allfields with indices 2 and 3 are reflected, i.e. ψ , → − ψ , . As a result, ξ , are protected byeach other so that u , = 0 is a consistent solution, which then leads to tribimaximal mixing.For the choice of Higgs doublets, there are also specific predictions of lepton flavor structure,which are testable at the LHC. 7 cknowledgement : This work was supported in part by the U. S. Department of Energyunder Grant No. DE-FG03-94ER40837. References [1] P. F. Harrison, D. H. Perkins, and W. G. Scott, Phys. Lett.
B530 , 167 (2002).[2] E. Ma and G. Rajasekaran, Phys. Rev.
D64 , 113012 (2001).[3] K. S. Babu, E. Ma, and J. W. F. Valle, Phys. Lett. , 207 (2003).[4] E. Ma, Phys. Rev.
D70 , 031901 (2004).[5] G. Altarelli and F. Feruglio, Nucl. Phys.
B720 , 64 (2005).[6] C. S. Lam, Phys. Lett.
B656 , 193 (2007).[7] A. Blum, C. Hagedorn, and M. Lindner, Phys. Rev.
D77 , 076004 (2008).[8] K. S. Babu and X.-G. He, arXiv:0507217 [hep-ph].[9] G. Altarelli and F. Feruglio, Nucl. Phys.
B741 , 215 (2006).[10] E. Ma, Phys. Rev.
D73 , 057304 (2006).[11] X.-G. He, Nucl. Phys. Proc. Suppl. , 350 (2007).[12] S. Morisi, M. Picariello, and E. Torrente-Lujan, Phys. Rev.
D75 , 075015 (2007).[13] E. Ma, Mod. Phys. Lett.
A22 , 101 (2007).[14] G. Altarelli, F. Feruglio, and C. Hagedorn, JHEP , 052 (2008).[15] F. Bazzochi, M. Frigerio, and S. Morisi, Phys. Rev.
D78 , 116018 (2008).[16] E. Ma, Phys. Lett.
B671 , 366 (2009).[17] G. Altarelli and D. Meloni, J. Phys.
G36 , 085005 (2009).[18] E. Ma, Mod. Phys. Lett.
A21 , 2931 (2006).[19] E. Ma, Phys. Rev.
D72 , 037301 (2005).[20] E. Ma and U. Sarkar, Phys. Rev. Lett.80