Proc Natl Conf Theor Phys 35 (2010), pp 255-259 NEUTRINO MASSES AND MIXING IN THE STANDARD MODEL WITH A4 -FLAVOR SYMMETRY NGO THI THU DINH University of Natural Science, 334 Nguyen Trai, Thanh Xuan, Hanoi PHUNG VAN DONG Institute of Physics, VAST, 10 Dao Tan, Ba Dinh, Hanoi Abstract Although the Standard Model is very successful, it also leaves many questions unanswered, one of which is massless of neutrinos In this talk we introduce A4 - flavour symmetry into the Standard Model with appropriate extension of scalar representations As a result, the neutrinos gain naturely small masses in agreement with experiment The neutrino mixing matrix in terms of tribimaximal form is obtained I INTRODUCTION The neutrino experiments imply: masses of neutrinos are small, and tribimaximal mixing neutrinos as proposed by Harrison-Perkins-Scott is given by:   UHPS =  √2 − √16 − √16 √1 √1 √1 √1 − √12    (1) The theories of neutrinos have recently been in trying to explain this form [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11] II THE MODEL II.1 Flavour symmetry A4 The finite group of the even permutation of four objects,A4 , has 12 elements and equivalence classes, with the number of elements 1, 4, 4, 3, respectively This means that there are irreducible representations, with dimensions ni, such that Σi n2i = 12 There is only one solution: n1 = n2 = n3 = and n4 = 3, and the character table of the representations is shown in Table below: 256 NGO THI THU DINH, PHUNG VAN DONG Table Character table of A4 Class χ1 χ1 C1 1 C2 ω C3 ω2 C4 1 χ1 ω2 ω χ3 0 −1 n 4 The complex number ω is the cube root of unity, i.e., e2πi/3 Hence + ω + ω = Calling the irreducible representations, and respectively, we have the decomposition: ⊗ = 1(11 + 22 + 33) ⊕ (11 + ω 22 + ω33) ⊕1 (11 + ω22 + ω 33) ⊕ 3(23, 31, 12) ⊕ 3(32, 13, 21) (2) The non-Abelian finite group A4 is also the symmetry group of the regular tetrahedron II.2 Lepton mass Under A4, the fermions and scalars of this model transform as follows: ψL = l1R ∼ (1, −2, 1), νL eL ∼ (2, −1, 3), l2R ∼ (1, −2, ), φ= φ+ φ0 l3R ∼ (1, −2, ), ∼ (2, 1, 3) (3) (4) (5) The Yukawa interactions are: = h1 (ψL φ)1 l1R + h2 (ψL φ)1 l2R + h3 (ψL φ)1 l3R + H.c Llep Y (6) The vacuum expectation value (VEV) of φ is (v1 , v2 , v3 ) under A4 It is now assumed that so that A4 is broken down to Z3 The mass Lagrangian for the changed leptons is: Llep mass = h1 v(l1L + l2L + l3L )l1R +h2 v(l1L + ωl2L + ω l3L )l2R +h3 v(l1L + ω l2L + ωl3L )l3R +H.c then the mass matrix is then diagonalized:  √    3h1 v √ 0 me 0  =  mµ  , UL−1 M lep UR =  3h2 v √ 0 mτ 0 3h3 v (7) (8) where   1 1 UR = 1, UL = √  ω ω  ≡ UlL ω2 ω (9) NEUTRINO MASSES AND MIXING IN THE STANDARD MODEL WITH 257 II.3 Neutrino sector To obtain arbitrary Majorana neutrino masses, four Higgs doublets are used: σ= + σ11 σ12 + ++ σ12 σ22 ∼ (3∗ , 2, 1), (10) s= s011 s+ 12 ++ s+ 12 s22 ∼ (3∗ , 2, 3) (11) The Yukawa interactions are: LνY = x(ψ¯Lc ψL )1 σ + y(ψ¯Lc ψL )3 s + H.c (12) The VEV of σ is: σ = u 0 (13) The VEV of s is put as: s = ( s1 , s2 , s3 ) (14) with s2 = s3 = 0, s1 = t 0 , (15) so that it is broken down to Z2 in neutrino sector The mass Lagrangian for the neutrinos is: LνY c ν c c c = x(ν1L 1L + ν2L ν2L + ν3L ν3L )u + yν2L ν3L t + H.c The mass matrices are then obtained by     xu 0 xu + 21 yt 0 T  UνL xu Mν =  xu 12 yt  = UνL  , 1 yt xu 0 xu − yt (16) (17) where   ULν =  √1 √1  0 √12   − √12 (18) The mismatch between Uν and UlL yields the tribimaximal mixing pattern as proposed by Harrison- Perkins- Scott: √   2/3 1/√3 0√ √ † UHP S = UlL Uν =  −1/√6 1/√3 −1/√  (19) −1/ 1/ 1/ This is a main result of the paper 258 NGO THI THU DINH, PHUNG VAN DONG II.4 Scalar Potential We can separate the general scalar potential into V = V φ + V σs , (20) where V φ = µ2φ (φ+ φ)1 + λφ1 [(φ+ φ)1 (φ+ φ)1 ] + λφ2 [(φ+ φ)1 (φ+ φ)1 ] +λφ3 [(φ+ φ)3s (φ+ φ)3a ] + λφ4 [(φ+ φ)3s (φ+ φ)3s + H.c.], (21) V σs = V (σ) + V (s) + V (σ, s) + V (φ, σ) + V (φ, s) + V (φ, σ, s) + V , (22) V (σ) = µ2σ T r(σ + σ) + λσ T r(σ + σ)2 + λ σ [T r(σ + σ)]2 , (23) with V (s) = T rV (φ → s) + λ1s T r(s+ s)1 T r(s+ s)1 + λ2s T r(s+ s)1 T r(s+ s)1 +λ3s T r(s+ s)3s T r(s+ s)3a + λ4s [T r(s+ s)3s T r(s+ s)3s + H.c.], (24) + + σs + + V (σ, s) = λσs T r[(σ s)3 (s σ)3 ] + λ1 T r(σ s)3 T r(s σ)3 + + σs + + +λσs T r[(σ σ)1 (s s)1 ] + λ2 T r(σ σ)1 T r(s s)1 + + σs + + +[λσs T r[(s s)3s (s σ)3 ] + λ3 T r(s s)3s T r(s σ)3 + + σs + + +λσs T r[(s s)3a (s σ)3 ] + λ4 T r(s s)3a T r(s σ)3 + H.c] + + σs + + +[λσs T r[(s σ)3 (s σ)3 ] + λ5 T r(s σ)3 T r(s σ)3 + H.c], (25) φσ + + + + V (φ, σ) = λφσ T r[(σ φ)3 (φ σ)3 ] + λ1 T r(σ φ)3 T r(φ σ)3 φσ + + + + +λφσ T r[(σ σ)1 (φ φ)1 ] + λ2 T r(σ σ)1 T r(φ φ)1 , (26) φs + + + + V (φ, s) = λφs 11 T r[(φ s)1 (s φ)1 ] + λ11 T r(φ s)1 T r(s φ)1 φs + + + + +λφs 12 T r[(φ s)1 (s φ)1 ] + λ12 T r(φ s)1 T r(s φ)1 φs + + + + +λφs 13 T r[(φ s)3s (s φ)3a ] + λ13 T r(φ s)3s T r(s φ)3a φs + + + + +[λφs 14 T r[(φ s)3s (s φ)3s ] + λ14 T r(φ s)3s T r(s φ)3s + H.c] φs + + + + +λφs 21 T r[(φ φ)1 (s s)1 ] + λ21 T r(φ φ)1 T r(s s)1 φs + + + + +λφs 22 T r[(φ φ)1 (s s)1 ] + λ22 T r(φ φ)1 T r(s s)1 φs + + + + +λφs 23 T r[(φ φ)3s (s s)3a ] + λ23 T r(φ φ)3s T r(s s)3a φs + + + + +[λφs 24 T r[(φ φ)3s (s s)3s ] + λ21 T r(φ φ)3s T r(s s)3s + H.c.], V (φ, σ, s) = µT r(φ+ σ + sφ) + H.c., (27) (28) NEUTRINO MASSES AND MIXING IN THE STANDARD MODEL WITH V = µ0 φT σφ + µ1 φT s1 φ 259 (29) III CONCLUSION We have shown the neutrinos gain naturely small masses in agreement with experiment The neutrino mixing matrix in terms of tribimaximal form is obtained Based on the flavour symmetry A4, we can understand neutrino experiments [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11] REFERENCES [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] P V Dong, L T Hue, H N Long, D V Soa, Phys Rev D 81 (2010) 053004 E Ma, arXiv:0908.3165 [hep-ph] E Ma, Phys Lett B 671 (2009) 366 G Altarelli, D Meloni, J Phys G 36 (2009) 085005 E Ma, Mod Phys Lett A 22 (2007) 101 C S Lam, Phys Lett B 656 (2007) 193 G Altarelli, F Feruglio, Nucl Phys B 741 (2006) 215 E Ma, Phys Rev D 73 (2006) 057304 E Ma, Mod Phys Lett A 21 (2006) 2931 P F Harrison, D H Perkins, W G Scott, Phys Lett B 530 (2002) 167 E Ma, G Rajasekaran, Phys Rev D 64 (2001) 113012 Received 30-09-2010  ... (28) NEUTRINO MASSES AND MIXING IN THE STANDARD MODEL WITH V = µ0 φT σφ + µ1 φT s1 φ 259 (29) III CONCLUSION We have shown the neutrinos gain naturely small masses in agreement with experiment The. .. masses in agreement with experiment The neutrino mixing matrix in terms of tribimaximal form is obtained Based on the flavour symmetry A4, we can understand neutrino experiments [1, 2, 3, 4, 5, 6,... ω  ≡ UlL ω2 ω (9) NEUTRINO MASSES AND MIXING IN THE STANDARD MODEL WITH 257 II.3 Neutrino sector To obtain arbitrary Majorana neutrino masses, four Higgs doublets are used: σ= + σ11 σ12 + ++