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Communications in Physics, Vol. 25, No. 3 (2015), pp. 227-237 DOI:10.15625/0868-3166/25/3/6068 GAUGE BOSONS IN THE 3-3-1 MODEL WITH THREE NEUTRINO SINGLETS HOANG NGOC LONG Institute of Physics, Vietnam Academy of Science and Technology, 10 Dao Tan, Ba Dinh, Hanoi, Vietnam LAM PHU AN HUY, TRAN THANH THUY, LY THI TU TRAN, VU THI HOANG YEN Graduate School, Can Tho University, Can Tho, Vietnam Received 12 April 2015 Accepted for publication 30 September 2015 E-mail: hnlong@iop.vast.ac.vn Abstract. We show that the mass matrix of electrically neutral gauge bosons in the recent proposed model based on SU(3)C ⊗ SU(3)L ⊗ U(1)X group with three neutrino singlets [9] has two exact eigenvalues and corresponding eigenvectors. Hence the neutral non-Hermitian gauge boson Xµ0 is properly determined. With extra vacuum expectation values of the Higgs fields, there are mixings among charged gauge bosons W ± and Y ± as well as among neutral gauge bosons Z, Z 0 and X 0 . Keywords: neutrino mass, beyond the standard model. I. INTRODUCTION At present, it is well known that neutrinos are massive that contradicts the Standard Model (SM). The experimental data [1] show that masses of neutrinos are tiny small and neutrinos mix with special pattern in approximately tribimaximal form [2]. The neutrino masses, dark matter and the baryon asymmetry of Universe (BAU) are the facts requiring extension of the SM. Among the extensions beyond the SM, the models based on SU(3)C ⊗ SU(3)L ⊗ U(1)X gauge group (called 3-3-1 for short) [3, 4] have some interesting features including the ability to explain the generation problem [3, 4] and the electric charge quantization [5]. Concerning the content in lepton triplet, the exist two main versions of 3-3-1 models: the minimal version [3] without extra lepton and the model with right-handed neutrinos [4] without exotic charged particles. Due to the fact that particles with different lepton numbers lie in the same triplet, the lepton number is violated and it is better to deal with a new conserved charge L commuting with the gauge symmetry [6] 4 (1) L = √ T8 + L . 3 c 2015 Vietnam Academy of Science and Technology 228 HOANG NGOC LONG et al In the framework of the 3-3-1 models, almost issues concerning neutrino physics are solvable. In the framework of the minimal 3-3-1 model where perturbative regime is trustable until 4-5 TeV, to realize idea of seesaw, the effective dimension-5 operator is used [7]. In regard to the 3-3-1 model with right-handed neutrinos, effective-5 operators are sufficient to generate light neutrino masses. The effective dimension-5 operator may be realized through a kind of type-II seesaw mechanism implemented by a sextet of scalars belonging to the GUT scale [8]. There are two ways to explain smallness of neutrino masses: the radiative mechanism and the seesaw one. The seesaw mechanism is the most easy and elegant way of generating small neutrino masses by using the Majorana neutrinos with mass belonging to GUT scale. With such high scale, the Majorana neutrinos are unavailable for laboratory searches. There are attempts to improve the situation. In the recently proposed model [9], the authors have introduced three neutrino singlets and used radiative mechanism to get model, where the seesaw mechanism is realized at quite low scale of few TeV scale. We remind that in the 3-3-1 model with right-handed neutrinos, there are two scalar triplets η, χ containing two neutral components lying at top and bottom of triplets: η10 , χ10 and η30 , χ30 . In the previous version [4], only η10 and χ30 have VEVs, however, in new version, the η30 carrying lepton number 2 has larger VEV of new physics scale. This leads to the mixing in both charged and neutral gauge bosons sectors. In the neutral gauge boson sector, the mass mixing matrix is 4 × 4. In general, the diagonalizing process for 4 × 4 matrix is approximate only. However, in this paper, we show that two exact eigenvalues and eigenstates, consequently the diagonalization is exact! In this paper we study in details the above mentioned model. In Sect. II, we briefly give particle content of the model. II. THE MODEL As usual [4], the left-handed leptons are assigned to the triplet representation of SU(3)L  
1 1 ` − c T , `R ∼ (1, 1, −1, 1) (2) fL = ν` , ` , N` L ∼ 1, 3, − , 3 3 where ` = 1, 2, 3 ≡ e, µ, τ. The numbers in bracket are assignment in SU(3)C , SU(3)L ,U(1)X and L. The third quark generation is in triplet     2 1 2 T 3 , TR ∼ 3, 1, , −2 , QL = (t , b , T )L ∼ 3, 3, , − 3 3 3     2 1 tR ∼ 3, 1, , 0 , bR ∼ 3, 1, − , 0 (3) 3 3 Two first quark generations are in anti triplet   2 T i ∗ QL = (di , −ui , Di )L ∼ 3, 3 , 0, − , i = 1, 2, 3       1 2 1 DiR ∼ 3, 1, − , 2 , uiR ∼ 3, 1, , 0 , diR ∼ 3, 1, − , 0 (4) 3 3 3 In addition to the new two-component neutral fermions present in the lepton triplet NLc ≡ (N c )L ≡ (νR )c where ψ c = −Cψ T and C is the charge conjugation matrix, ones introduce new sequential THE 3-3-1 MODEL WITH THREE NEUTRINO SINGLETS 229 lepton-number-carrying gauge singlets S = {S1 , S2 , S3 } with the following number [9] Si ∼ (1, 1, 0, −1). With the above L assignment the electric charge operator is given in terms of the U(1)X generator X and the diagonal generators of the SU(3)L as 1 Q = T3 − √ T8 + X . (5) 3 In order to spontaneously break the weak gauge symmetry, we introduce three scalar triplets   0 T  1 4 0 − 0 χ = χ ,χ ,χ ∼ 1, 3, − , ; hχi = (0 , 0 , n1 )T (6) 3 3  T   0 1 2 ; hηi = (k2 , 0 , n2 )T (7) ∼ 1, 3, − , − η = η 0 ,η− ,η0 3 3     0+ T 2 2 + 0 ; hρi = (0 , k1 , 0)T . (8) ∼ 1, 3, , − ρ = ρ ,ρ ,ρ 3 3 Note that the third component of η carries two units of lepton number, and the k1 and k2 VEVs are at the electroweak scale and correspond to the VEV of the SU(2)L ⊂ SU(3)L doublets. The VEVs n1 and n2 are isosinglet VEVs that characterize the SU(3)L breaking scale. Note that while η takes VEV in both electrically neutral directions, the second VEV of χ is neglected, so that lepton number is broken only by SU(2)L singlets. This pattern gives the simplest consistent neutrino mass spectrum, avoiding the linear seesaw contribution. As we discuss below, this structure of VEVs leads to mixing not only among charged gauge bosons, but also among electrically neutral gauge bosons. The spontaneous symmetry breaking follows the pattern n1,2 k1,2 SU(3)L ⊗U(1)X −−→ SU(2)L ⊗U(1)Y −−→ U(1)Q . III. GAUGE BOSON SECTOR The kinetic term for the scalar fields is LKin = (Dµ H)† (Dµ H) , (9) Dµ = ∂µ − ig Aµa Ta − ig0 XBµ T9 , (10) ∑ H=χ,η,ρ where covariant derivative is defined as here X is the U(1)X charge of the field, Aµa and Bµ are the gauge bosons of SU(3)L and U(1)X , respectively. The above equation applies for triplet: Ta → λa /2 , T9 → λ9 /2 where λi are the Gellq Mann matrices, and λ9 = 2 3 diag (1, 1, 1). The matrix Aµ ≡ ∑a Aaµ λa is √ µ+  µ µ µ µ 2W12 A4 − iA5 A3 + √13 A8 √ µ−  µ µ  √ µ− −A3 + √13 A8 2W67  . Aµ =  2W12 √ µ+ µ µ µ A4 + iA5 2W67 − √23 A8 230 HOANG NGOC LONG et al The charged states are defined as 1 1 µ µ µ µ± µ µ± W12 = √ (A1 ∓ iA2 ) , W67 = √ (A6 ± iA7 ) . 2 2 The mass Lagrangian for gauge fields is given by Lmass = ∑ (11) (Dµ hHi)† (Dµ hHi) . (12) H=χ,η,ρ µ In charged gauge boson sector, the mass Lagrangian in (12) gives one decoupled A5 with mass g2 2 (n + n22 + k22 ) , 4 1 µ µ and two others with the mass matrix given in the basis of (W12 ,W67 ) as   g2 k12 + k22 n2 k2 . Mcharged = n2 k2 n21 + n22 + k12 2 m2A5 = (13) (14) The matrix in Eq. (14) has two eigenvalues √  1 2 2 2 2 n + n2 + k2 ± ∆ , λ1,2 − k1 = 2 1 where (15) ∆ = (n21 + n22 − k22 )2 + 4n22 k22 . (16) In the limit n1 ∼ n2  k1 ∼ k2 , one has  √ 2 2 ∆ ' (n1 + n2 ) 1 − 2 = n21 + n22 + k22 − k22 4n22 k22 + (n21 + n22 ) (n21 + n22 )2  12 2n21 k22 . (n21 + n22 ) (17) We will identify the light eigenvalue with square mass of the SM W boson, while the heavy one with that of the new charged gauge boson Y :   g2 g2 2 n21 k22 2 mW = λ1 ' k + 2 2 1 (n11 + n22 )   g2 2 k22 k1 + , (18) ' 2 2   g2 g2 2 n21 k22 2 2 2 2 mY = λ2 ' n + n2 + k1 + k2 − 1 2 2 1 (n1 + n22 ) g2 2 (n + n22 ). 2 1 Note that in the limit n1 ∼ n2  k1 ∼ k2 , our result is consistent with that in [9]. Two physical bosons are determined as [10] ' (19) − − Wµ− = cos θ Wµ12 − sin θ Wµ67 , − − Yµ− = sin θ Wµ12 + cos θ Wµ67 , (20) THE 3-3-1 MODEL WITH THREE NEUTRINO SINGLETS 231 where the W −Y mixing angle θ charaterizing lepton number violation is given by tan 2θ ≡ ε ∼ 2n2 k2 2 n1 + n22 − k22 . (21) Now we turn to the electrically neutral gauge boson sector. Five neutral fields, namely µ µ µ A3 , A8 , Bµ , A4 mix q   2 2 + 2k2 ) √1 (k2 − k2 ) −t (k n k k12 + k22 2 2 1 1 27 2 3 2   1 2 + n2 ) + (k2 + k2 )] 2 [4(n M − √13 n2 k2  g 23 2 1 2 1 2 3 ,  q M =  4  2  M33 −2t 27 n2 k2  n21 + n22 + k22 (22) √ 2 where M23 ≡ 92 t[2(n21 + n22 ) + 2k12 − k22 ], M33 ≡ 2t27 (n21 + n22 + 4k12 + k22 ) and t is given (see the last paper in Ref. [4]) √ g0 3 2 sin θW (m0Z ) t= =q . g 2 0 3 − 4 sin θW (mZ ) The matrix (22) has one massless state √  √ 1 Aµ = √ 3tA3µ − tA8µ + 3 2Bµ , 18 + 4t 2 (23) which is identified to the photon. The second eigenvalue of (22) is defined m2A0 = 4 g2 2 (n + n22 + k22 ), 2 1 (24) with eigenstate A04µ = = √ n2 k2 3n2 k2 A3µ + 2 A8µ + A4µ n21 + n22 − k22 n1 + n22 − k22 √ 3t2θ 1 t2θ q A3µ + q A8µ + q A4µ , 2 2 2 2 1 + 4t2θ 2 1 + 4t2θ 1 + 4t2θ (25) where t2θ ≡ tan 2θ is determined the same as in (21). Comparing (13) with (24) we see that two components of W45 have, as expected, the same mass. Hence we can identify 1 Xµ0 = √ (A04µ − iA5µ ) (26) 2 as physical electrically neutral non-Hermitian gauge boson. It is easy to see that this gauge boson Xµ0 carries lepton number two, hence it is called bilepton gauge boson. With this exact eigenvalues, we return to the diagonalization of the matrix in (22). It is difficult to find two states which are orthogonal simultaneously to two states Aµ and A4µ . Thus we 232 HOANG NGOC LONG et al follow the method in [10]. The diagonalization separates into three steps. Firstly, let us take   s 2 t tW Aµ = sW W3µ + cW − √ W8µ + 1 − W Bµ  , 3 3   s 2 t tW Zµ = cW W3µ − sW − √ W8µ + 1 − W Bµ  , 3 3 s t2 tW 1 − W W8µ + √ Bµ Zµ0 = 3 3 A4µ = A4µ . (27) or in matrix form    q − √sw3 s2w √ c qw 3 3−4s2w 1 cw 3 3−4s2w q3 2 w −tw 3−4s 3 0 0   A3µ  A8µ   T    Bµ  = S  A4µ  Aµ Zµ  . Zµ0  A4µ sw Aµ    Zµ  c  0  =   w  Zµ    0 A4µ 0 0     0    0  1 tw √ 3  A3µ A8µ  . Bµ  A4µ (28) Then  (29) Let us denote  sw cw s2w √ cq w 3 2 w −tw 3−4s 3   − √sw3 q U1 = S =   3−4s2w  3 T 0 1 cw q0 3−4s2w 3 tw √ 3 0 0 0   0  ,  0  1 then  M 02 = U1T MU1 0 0 k12 +k22 c2w 2 k2 (1−2s2w )−k12 c2w 3−4s2w n2 k2 cw  0   =  √ 2  0  0   0 0 = , 0 M 33 g2 0 √ − k22 (1−2s2w )−k12 c2w 3−4s2w 4 (n2 +n2 ) k12 +c22w k22 +4cW 1 2 c2w (3−4s2w ) √n2 k2 cw 3−4s2w 0 n2 k2 cw √n2 k2 cw 3−4s2w n21 + n22 + k22        (30) THE 3-3-1 MODEL WITH THREE NEUTRINO SINGLETS 233 where we have denoted m2Z M 33 ≡  m2ZZ 0 m2ZA4  m2ZZ 0 m2Z 0 m2Z 0 A0 4  m2ZA4 m2Z 0 A4  . m2A4 (31) We turn to the second step. To see explicitly that the following basis is orthogonal and normalized, comparing with (25), let us put t2θ sθ 0 ≡ q , (32) 2 cw 1 + 4t2θ which leads to A04µ  q  q 0 2 2 2 = sθ 0 Zµ + cθ 0 tθ 0 3 − 4sw Zµ + 1 − tθ 0 (3 − 4sw )A4µ . (33) Note that the mixing angle in this step θ 0 is the same order as the mixing angle in the charged 2 ' 0.231, from (32) we get s 0 ' 2.28s . It is now gauge boson sector. Taking into account [1] sW θ θ easy to choose two remaining gauge vectors  q  q 0 2 2 2 Zµ = cθ 0 Zµ − sθ 0 tθ 0 3 − 4sw Zµ + 1 − tθ 0 (3 − 4sw )A4µ , q q Zµ0 = 1 − tθ2 0 (3 − 4s2w )Zµ0 − tθ 0 3 − 4s2w A4µ . (34) Therefore, in the base of (Aµ , Zµ , Zµ0 ,A04µ ) the matrix is quasi-diagonal by matrix    Aµ  Zµ       Zµ0  = S2  A04µ  Aµ Zµ  , Zµ0  A04µ (35) where  1 0   0 cθ 0  S2 =   0 0  0 sθ 0 0 q 0 p 2 −sθ 0 tθ 0 3 − 4sw −sθ 0 1 − tθ2 0 (3 − 4s2w ) q p 1 − tθ2 0 (3 − 4s2w ) −tθ 0 3 − 4s2w q p cθ 0 tθ 0 3 − 4s2w cθ 0 1 − tθ2 0 (3 − 4s2w )        Let us denote  1 0 0 0  0 0 c 0 s θ θ0  q p p 2 U2 = S2T =  2 2 −sθ 0 tθ 0 3 − 4sw 1 − tθ 0 (3 − 4sw ) cθ 0 tθ 0 3 − 4s2w  0  q q p 0 −sθ 0 1 − tθ2 0 (3 − 4s2w ) −tθ 0 3 − 4s2w cθ 0 1 − tθ2 0 (3 − 4s2w ) (36)    ,   234 HOANG NGOC LONG et al then the mass matrix M 02 has a quasi-diagonal form  0 0 0 2 2 2 g m 0 m Z ZZ0 M 002 = U2T M 02U2 =  2  0 m2Z Z 0 m2Z 0 0 0 0  0  0 ,  0 2 2 2 (n1 + n2 + k2 ) (37) where m2Z = m2Z Z 0 = m2Z 0 1 [(4 − s22θ )k22 + 4k12 − s22θ (n21 + n22 )] , − s22θ ) 1 q {[4c2θ c2W + s2θ t2θ (2c2W + 1)]k22 − 4c2θ k12 2 2 2 4(4cW − s2θ ) 4cW − 1 2 4(4cW −s2θ t2θ (n21 + n22 )} , 1 2 4 = [(4cW − s22θ )k22 + 4c22θ k12 + (16cW − s22θ )(n21 + n22 )] . 2 − s2 )(4c2 − 1) 4(4cW W 2θ (38) As expected, we get mass of A4µ as given in (24). The final step is determined as     Aµ Aµ  Z1µ   Z1µ       Zµ0  = U3  Z2µ  , A04µ A4µ (39) where  1 0  0 cζ U3 =   0 −sζ 0 0 0 sζ cζ 0  0 0   0  1 (40) and M 000 2 = U3T M”2 U3 = diag(0, m2Z1 , m2Z2 , m2A0 ) 4 (41) The mixing angle is defined as tan 2ζ = 2m2ZZ 0 . m2Z 0 − m2Z (42) Note that in the limit n1 ∼ n2  k1 ∼ k2 , two angles are in the same order, i.e., θ ∝ ζ . Masses of heavy physical bosons are given by m2Z1 = m2Z2 = √ i g2 1 h 2 (n1 + n22 )(18 + t 2 ) + k22 (18 + 4t 2 ) + k12 (18 + t 2 ) − ∆0 2 27 √ i g2 1 h 2 (n1 + n22 )(18 + t 2 ) + k22 (18 + 4t 2 ) + k12 (18 + t 2 ) + ∆0 , 2 27 (43) (44) THE 3-3-1 MODEL WITH THREE NEUTRINO SINGLETS 235 where   ∆0 = [(n21 + n22 + k22 )(18 + t 2 ) + 2k12 (9 + 2t 2 )]2 − 108(9 + 2t 2 ) n21 k22 + (n21 + n22 + k22 )k12  4(9 + 2t 2 ) k2 2 2 2 2 2 ' (n1 + n2 ) (18 + t ) 1 + 2 2 2 2 + k12 2 (n1 + n2 ) (n1 + n22 )(18 + t 2 )   (9 + 2t 2 ) n21 k22 2 − 108 2 k + . (45) (n1 + n22 )(18 + t 2 )2 1 (n21 + n22 ) Then √ ∆0 ' (n21 + n22 )(18 + t 2 ) + k22 (18 + t 2 ) + 2(9 + 2t 2 )k12   (9 + 2t 2 ) 2 n21 k22 k + − 54 (18 + t 2 ) 1 (n21 + n22 )    (3 − 4s2w ) 2 n21 k22 54 2 2 2 2 2 (n1 + n2 + k2 )cw + k1 − k1 + 2 . = (3 − 4s2w ) 2c2w (n1 + n22 ) (46) Substituting (46) into (43) yields m2Z1    54(9 + 2t 2 ) 2 g2 n21 k22 2 2 2 ' 3t (k2 − k1 ) + k1 + 2 54 18 + t 2 (n1 + n22 )    1 n21 k22 s2w 2 2 2 2 (k − k1 ) + 2 k1 + 2 = g . (3 − 4s2w ) 2 2cw (n1 + n22 ) (47) Similarly, for the heavy extra neutral gauge boson Z2 , one obtains m2Z2 ' − = − g2  2 2(n1 + n22 + k12 + k22 )(18 + t 2 ) + 3t 2 (k22 + k12 ) 54   54(9 + 2t 2 ) 2 n21 k22 k1 + 2 (18 + t 2 ) (n1 + n22 )  2 g2 2(n1 + n22 + k12 + k22 )c2w + (k12 + k22 )s2w 2 (3 − 4sw )   n21 k22 (3 − 4s2w ) 2 k1 + 2 2c2w (n1 + n22 ) (48) In summary, the physical vector fields relate to the gauge states as     Aµ A3µ  A8µ   Z1µ       Bµ  = U  Z2µ  , A04µ A4µ (49) 236 HOANG NGOC LONG et al where U = U1 U2 U3  sw  s  √ 3w  2 =  3−4s  √3 w  0 cw cθ 0 cζ M22 cw cθ 0 sζ M23 M32 M33 M42 M43 cw sθ 0 3cw sθ 0 √ 3 0 cθ 0 q 1 − tθ2 0 (3 − 4s2w )        (50) with M22 = M23 = M32 = M33 = M42 = M43 = " # q q cζ (s2w − 3c2w s2θ 0 ) 1 √ − sζ 3 − 4s2w 1 − tθ2 0 (3 − 4s2w ) , cθ 0 3cw # " q q sζ (s2w − 3c2w s2θ 0 ) 1 √ + cζ 3 − 4s2w 1 − tθ2 0 (3 − 4s2w ) , cθ 0 3cw   q q tw cζ 3 − 4s2w − sζ 1 − tθ2 0 (3 − 4s2w ) , −√ 3 cθ 0   q q sζ tw 2 2 2 −√ 3 − 4sw − cζ 1 − tθ 0 (3 − 4sw ) , 3 cθ 0 q q −cζ sθ 0 1 − tθ2 0 (3 − 4s2w ) + sζ tθ 0 3 − 4s2w , q q −sζ sθ 0 1 − tθ2 0 (3 − 4s2w ) − cζ tθ 0 3 − 4s2w . Note that sθ 0 ∼ sζ  1, then the matrix in (50) becomes Up ' U(sθ 0 ∼ sζ  1)  0 sw   2 cw cθ p c s  ζ w sw 2 √1 − s 3 − 4s  ζ w c 3cw √3  θ0 p  =   3−4s2w c t w − √3 c ζ0 3 − 4s2w − sζ  √3 θ 0 −cζ sθ 0 cw sζ cw sθ 0  p 2 √1 + c 3 − 4s ζ w 3cw  p  s − √tw3 c ζ0 3 − 4s2w − cζ θ p −cζ tθ 0 3 − 4s2w 3cw sθ 0 √ 3  sζ s2w cθ 0 0       cθ 0 (51) The matrix Up in (51) is very useful for practical calculation in consideration of phenomenology. IV. CONCLUSION In this paper, we have showed that the mass matrix of electrically neutral gauge bosons in the recent proposed 3-3-1 model with three neutrino singlets [9] has two exact eigenvalues and corresponding eigenvectors. Hence the 4 × 4 mass matrix is diagonalized exactly. Two components of neutral bilepton boson Xµ0 have the same mass, hence the neutral non-Hermitian gauge boson Xµ0 is properly determined. This contradicts to previous analysis in Ref. [9]. With extra vacuum expectation values of the Higgs fields, there are mixings among charged gauge bosons W ± and Y ± as well as among neutral gauge bosons Z, Z 0 and X 0 . Due to these mixings, the lepton