Communications in Physics, Vol. 26, No. 3 (2016), pp. 221-228 DOI:10.15625/0868-3166/26/3/8053 THE 3-3-1 MODEL WITH ARBITRARILY CHARGED LEPTONS HOANG NGOC LONG† Institute of Physics, Vietnam Academy of Science and Technology, 10 Dao Tan, Ba Dinh, Hanoi, Vietnam DUONG VAN LOI, NGUYEN CHI THAO Graduate University of Science and Technology, Vietnam Academy of Science and Technology, 18 Hoang Quoc Viet, Cau Giay, Hanoi, Vietnam THANH HUU HONG GIANG Graduate School, Department of Physics, Hanoi Pedagogical University 2, Phuc Yen, Vinh Phuc, Vietnam † E-mail: hnlong@iop.vast.ac.vn Received 27 December 2016 Accepted for publication 19 January 2017 Abstract. The gauge model based on SU(3)C ⊗ SU(3)L ⊗ U(1)X group with arbitrarily electric charged exotic leptons is presented. The mass eigenvalues and eigenstates for neutral gauge bosons are presented in the general form. We show that in the 3-3-1 models, there always exists triple Higgs self-coupling. The lepton number operator is also presented. Keywords: unified theories, models of strong, electroweak interactions. Classification numbers: 14.60.Pq, 12.60.Cn, 14.60.St, 14.70.Pw, 12.15.Ff. I. INTRODUCTION The current status of Particle Physics shows that the Standard Model (SM) must be extended. Among beyond the Standard Models, the models based on SU(3)C ⊗ SU(3)L ⊗ U(1)X (3-3-1) gauge group have some interesting properties due to the following reasons. The first reason concerns the generation number problem. In the SM, there exists the replica of the generations meaning the number of the latter is not constrained. However, in the 3-3-1 models [1, 2], one of quark generation behaves differently from the two other, and this fact leads to the consequence that the number of fermion generations is multiple of the quark color number (3), i.e., 3, 6, ... Combining with the Quantum Chromodynamics (QCD) asymptotic freedom requiring the quark generations is less than five, we get the answer why number of fermion generations is equal to c 2016 Vietnam Academy of Science and Technology 222 HOANG NGOC LONG, DUONG VAN LOI, NGUYEN CHI THAO AND THANH HUU HONG GIANG three. The second reason follows from the above mentioned fact that one of quark generations behaves differently from the two others, so we can explain why top quark is so heavy. The other issues such as neutrino and electric charge quantization [3] also get appropriate explanations. The different kind of the 3-3-1 models is defined by the beta parameter appearing in the electric charge operator (1) Q = T3 + β T8 + X . In Ref. [4], the 3-3-1 models with arbitrary beta has been presented. However, this approach gives unclear content of the Higgs components. Hence, in this paper, we present the 3-3-1 model with exotic leptons having arbitrarily electric charges. II. THE MODEL In (1), Ti , i = 3, 8 stand for the SU(3)L operators. For the fundamental representation triplet T3 = λ3 1 = diag(1, −1, 0), 2 2 1 λ8 = √ diag(1, 1, −2). 2 2 3 T8 = The lepton triplet is constituted by faL = (νa , la , Eaq )TL ∼ (1, 3, (q − 1)/3), a = e, µ , τ , (2) where q is electric charge of associated extra lepton. Other right-handed leptons are the singlets under SU(3)L q laR ∼ (1, 1, −1) , EaR ∼ (1, 1, q) . Applying Eq.(1) to Eq.(2) we obtain 2q + 1 q−1 . b = − √ , X faL = 3 3 Then, the electric charge operator, for triplet, has the form  1  3 (1 − q) + X . Q= − 13 (2 + q) + X 1 3 (1 + 2q) + X (3) (4) II.1. Yukawa couplings and fermion masses For the leptons faL = The mass of Eaq (νa , la , Eaq )TL laR ∼ (1, 1, −1) ,  q−1 , ∼ 1, 3, 3 q EaR ∼ (1, 1, q) .  is obtained from the Yukawa coupling q E − LYukawa = hEab faL Φ1 EbR + h.c. , where (5)   (−q)   Φ1 1 + 2q   Φ1 ∼ 1, 3, − =  Φ(−q−1)  . 1 3 Φ01 (6) (7) THE 3-3-1 MODEL WITH ARBITRARILY CHARGED LEPTONS Hence, if Φ01 has a vacuum expectation value (VEV) ω √ , 2 223 then Eaq gets mass from a mass matrix ω (mE )ab = hEab √ . 2 (8) Finally for the ordinary leptons, we have l − LYukawa = hlab faL Φ2 lbR + h.c. , (9) where If Φ02 has a VEV √v , 2   (+)   Φ2 2+q   =  Φ02  . Φ2 ∼ 1, 3, 3 (q+1) Φ2 (10) then the mass matrix related to masses of la is v (ml )ab = hlab √ . 2 (11) We turn now to the quark sector where     u3 1+q   d3 , ∼ 3, 3, Q3L = 3 T L u3R ∼ (3, 1, 2/3) , d3R ∼ (3, 1, −1/3) ,   2 + 3q . TR ∼ 3, 1, 3 (12) The u3 gets mass through the Yukawa part, t − LYukawa = ht Q3L Φ3 u3R + h.c. , (13) where If Φ03 has a VEV √u 2     Φ03 q−1 − Φ3 ∼ 1, 3, =  Φ3  . 3 (q) Φ3 (14) then the mass term of u3 is u mu3 = ht √ . 2 (15) The other Yukawa terms relating with Q3L are g3 = hb Q3L Φ2 d3R + hT Q3L Φ1 TR + h.c., − LYukawa (16) which give two mass terms: ω v md3 = hb √ , mT = hT √ . 2 2 (17) 224 HOANG NGOC LONG, DUONG VAN LOI, NGUYEN CHI THAO AND THANH HUU HONG GIANG Two other quark generations are   dα  q Qα L =  −uα  ∼ 3, 3∗ , − , 3 Dα L α = 1, 2, uα R ∼ (3, 1, 2/3) , dα R ∼ (3, 1, −1/3) ,   1 + 3q) . Dα R ∼ 3, 1, − 3 (18) The relevant Yukawa terms are † † 12 d2 D2 Qα L Φ†3 dβ R + hu2 −LYukawa = hαβ αβ Qα L Φ2 uβ R + hαβ Qα L Φ1 Dβ R + h.c., (19) from which it follows d2 u √ , (md2 )αβ = hαβ 2 v (mu2 )αβ = −hu2 αβ √ , 2 D2 ω √ . (mD2 )αβ = hαβ 2 (20) II.2. Gauge boson masses Gauge boson masses arise from the covariant kinetic term of Higgs, 3 LHiggs = ∑ (Dµ hΦi i)† Dµ hΦi i . (21) i=1 The covariant derivative is defined as 8 Dµ = ∂µ − ig ∑ Aaµ Ta − ig′ XB′µ T9 ≡ a=1 ∂µ − igPµNC − igPµCC , (22) where g, g′ and Aaµ , B′µ are gauge couplings and fields of the gauge groups SU(3)L and U(1)X , respectively. For the triplet, T9 = √16 diag(1, 1, 1), and the part relating with neutral currents is PµNC = r r 2 2 1 A8 A8 ′ diag A3 + √ + XtB , −A3 + √ + XtB′ , 2 3 3 3 3 ! r 2 2A8 ′ , −√ + XtB 3 3 µ (23) where the space-time indices of gauge fields, for compactness are omitted, and t ≡ g′ /g. The part associated with charged currents is PµCC = ∑ Ta Aaµ ; a = 1, 2, 4, 5, 6, 7 a =  0 W+ V −q 1 √  W− 0 Y −(1+q)  , 2 q (1+q) V Y 0 µ  (24) THE 3-3-1 MODEL WITH ARBITRARILY CHARGED LEPTONS 225 √ √ √ ±(1+q) where we have denoted 2Wµ± ≡ A1µ ∓iA2µ , 2Vµ±q ≡ A4µ ±iA5µ and 2Yµ ≡ A6µ ±iA7µ . The upper subscripts label the electric charges of gauge bosons. Remind that this part does not depend on the X-charges of triplets. To summary, with the following Higgs vacuum structure    T T  ω T u v √ ,0,0 , hΦ1 i = 0 , 0 , √ (25) , hΦ2 i = 0 , √ , 0 , hΦ3 i = 2 2 2 masses of non-Hermitian (charged) gauge bosons are given by g2 (v2 + u2 ) 2 g2 (u2 + ω 2 ) 2 g2 (v2 + ω 2 ) , mV = , mY = . (26) 4 4 4 By spontaneous symmetry breaking (SSB), the following relation should be in order: ω ≫ u, v; and from (26) ones get a consequence 2 = mW u2 + v2 = v2SM = 2462 GeV2 . (27) The diagonalization of the neutral gauge boson sector
is more complicated, because all the three gauge fields generally mix. In the basis A3µ , A8µ , B′µ , the respective squared mass matrix is given by   2 2t √ √1 (u2 − v2 ) [(q − 1)u2 − (q + 2)v2 ) u + v2 2 3 3 6 g  2t 1 2 2 2 2NG √ [(q − 1)u2 + (q + 2)v2 + 2(2q + 1)ω 2 ]  Mmass =   . (28) 3 (u + v + 4ω ) 9 2 4 2t 2 2 2 2 2 2 2 27 [(q − 1) u + (q + 2) v + (2q + 1) ω ] First of all, we can always obtain a zero eigenvalue (i.e. photon mass) with the corresponding eigenstate (i.e. photon field) as ! √ √ 3t 6 ′ B , Aµ = p A3µ + bA8µ + t µ 18 + 4(1 + q + q2 )t 2 which is independent of the VEVs as a consequence of the electric charge conservation [3]. Next, we can write electromagnetic interactions following the standard form given in [3], and thus the Weinberg’s angle (θW ) is defined as √ 3t , sW = p 18 + 4(1 + q + q2 )t 2 where note that sW = sin θW , cW = cos θW , and so forth. With this at hand, the photon field is rewritten in terms of ! √ 6tW ′ Aµ = sW A3µ + cW btW A8µ + Bµ . t The SM Z boson is orthogonal to the photon field as usual, ! √ 6tW ′ Bµ . Zµ = cW A3µ − sW btW A8µ + t 226 HOANG NGOC LONG, DUONG VAN LOI, NGUYEN CHI THAO AND THANH HUU HONG GIANG The model under consideration contains one new, neutral gauge bosons, called Z ′ , that are orthogonal to the field in the parentheses (as coupled to the weak hypercharge) appearing in the photon and Z fields, which are obtained by √  1 6A8µ − btB′µ , Zµ′ = √ 6 + b2 t 2 where note that t = √ √ 3 2sW 2 3−4(1+q+q2 )sW . Next, let us change to the new basis of (Aµ , Zµ , Zµ′ ). Correspondingly, the mass matrix 2NG is changed to Mmass   0 0 ′2 T 2NG , M = U MmassU = 0 M ′′2 where (A3µ , A8µ , B′µ )T = U(Aµ , Zµ , Zµ′ )T , and  sW cW  bt 2 cW2 bt 2 cW − 6+b 2t 2 ) 2 2 U = s (6+b W √  √ t 2 6tcW 6tcW − 6+b2t 2 sW (6+b2 t 2 ) 0 √ √ 6 6+b2 t 2 bt √ − 6+b 2t 2   .  We see that the photon field, Aµ , is decoupled, while the other states (Zµ , Zµ′ ) mix by themselves via a 2 × 2 mass matrix, M ′′2 , found to be   2 u2 −v2 −2[(1+q)u2 +qv2 ]sW u2 +v2 √ 2 2 2 2 g  cW  cW 3−4(1+q+q2 )sW M ′′2 =  u2 −v2 −2[(1+q)u 2 +v2 +4ω 2 −4s2 [(1+q)u2 −qv2 +2ω 2 −((1+q)2 u2 +q2 v2 +ω 2 )s2 ]  . 2 +qv2 ]s2 u W W W 4 √ 2 2 2 2 2 2 cW cW [3−4(1+q+q )sW ] 3−4(1+q+q )sW Diagonalizing their mass matrix, we obtain the corresponding physical states Z1µ = cε Zµ − sε Zµ′ , Z1′ µ = sε Zµ + cε Zµ′ , (29) where the Zµ -Zµ′ mixing angle (ξ ) is obtained by t2ξ = tan2ξ = ≃ q 2 [v2 − u2 + 2((1 + q)u2 + qv2 )s2 ] 3 − 4(1 + q + q2 )sW W 2 [v2 + q(qu2 + (2 + q)v2 ) − 2ω 2 + ((1 + q)2 u2 + q2 v2 + ω 2 )s2 ] u2 + v2 − 2ω 2 − 2sW W q 2 2 2 2 2 2 2 3 − 4(1 + q + q )sW [u − v − 2((1 + q)u + qv )sW ] , 4 2ω 2 cW (30) and the corresponding masses are m2Z1 ≃ g2 (u2 + v2 ) , 2 4cW m2Z ′ ≃ 1 2 2g2 ω 2 cW . 2 3 − 4(1 + q + q2 )sW Because of the condition u, v ≪ w, the Z1 boson has a small mass in the weak scales (u, v) which is identical to the standard model Z boson, whereas the Z1′ boson is a new, heavy charged gauge boson with mass proportional to ω scale. The mixing between these two fields is small since ξ → 0 due to the above condition. The spontaneous symmetry breaking follows the pattern ω u,v SU(3)L ⊗U(1)X −→ SU(2)L ⊗U(1)Y −→ U(1)Q . THE 3-3-1 MODEL WITH ARBITRARILY CHARGED LEPTONS 227 Corresponding to each step of the breaking, the neutral gauge boson states will be changed as follows: A3 ,A8 ,B′ A3 ,B,Z ′ SU(3)L ⊗U(1)X −→ SU(2)L ⊗U(1)Y −→ U(1)Q : A, Z, Z ′ . (31) Now we return to the Higgs content of the model. It is emphasized that if all fermions except neutrinos have the right-handed counterparts, then there need just three Higgs triplets. From the above presentation we explicitly see that: since sum of X-charges over three Higgs triplets vanishes, so in the Higgs potential, there always exists a triple Higgs self-coupling εi jk Φi1 Φ2j Φk3 . The lepton number operator is constructed from [5] L = α T3 + β T8 + L . (32) For the general case, let us assume that the new extra lepton E acquire lepton number l. Applying Eq.(32) to Eq.(2) we obtain 2+l 2(1 − l) √ , L faL = . 3 3 Hence, lepton number for triplet is in the form  1−l 3 +L 2(1 − l) 1−l  λ8 + L = L= √ 3 +L 3 2(l−1) α = 0,β = 3 The gauge bosons have the following lepton numbers (33) +L  . (34) L(W ) = 0 , L(V q ) = l − 1 , L(Y (1+q) ) = l − 1 . To finish this section, we notice that the model presented here is not applicable for the minimal 3-3-1 model [1], where right-handed charged lepton lies in the lepton triplet. Without right-handed lepton singlet, the scalar sextet has to be introduced to provide masses for all leptons. In the recent proposed model called the simplest 3-3-1 model [6], the exotic lepton has the electric charge ±1/2e. In this case the beta value vanishes. III. CONCLUSION In this paper, the 3-3-1 model in which exotic leptons has arbitrarily electric charge is presented. We have showed that the spontaneous symmetry breaking in the model requires just three Higgs triplets. 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