Nội dung

Communications in Physics, Vol. 30, No. 3 (2020), pp. 231-244 DOI:10.15625/0868-3166/30/3/15071 ON A STANDARD MODEL EXTENSION WITH VECTOR-LIKE FERMIONS AND ABELIAN SYMMETRY TRAN MINH HIEU1† , DINH QUANG SANG2 AND TRIEU QUYNH TRANG3 1 Hanoi University of Science and Technology 1 Dai Co Viet Road, Hanoi, Vietnam 2 VNU University of Science, Vietnam National University - Hanoi 334 Nguyen Trai Road, Hanoi, Vietnam 3 Nam Dinh Teacher’s Training College 813 Truong Trinh Road, Nam Dinh, Vietnam † E-mail: hieu.tranminh@hust.edu.vn Received 16 May 2020 Accepted for publication 30 June 2020 Published 20 July 2020 Abstract. We investigate an extension of the standard model with vector-like fermions and an extra Abelian gauge symmetry. The particle mass spectrum is calculated explicitly. The Lagrangian terms for all the gauge interactions of leptons and quarks in the model are derived. We observe that while there is no new mixing in the lepton sector, the quark mixing plays an important role in the magnitudes of gauge interactions for quarks, particularly the interactions with massive W , Z and Z 0 bosons. We calculate the contributions of the new vector-like leptons and quarks to the Peskin-Takeuchi parameters as well as the ρ parameter of the electroweak precision tests, and show that the model is realistic. Keywords: Abelian symmetry, beyond the standard model, vector-like fermions. Classification numbers: 98.80.Cq, 12.60.Cn . I. INTRODUCTION The standard model (SM) has been continuously tested since it was born. Although many experiment results have shown good agreements with the SM predictions, there are evidences that new physics might exist. Examples of those include the neutrino oscillation, rare decay processes of B-mesons, the dark matter observation, and the muon anomalous magnetic moment. There are ©2020 Vietnam Academy of Science and Technology 232 T. M. HIEU, D. Q. SANG AND T. Q. TRANG various possibilities to extend the SM. New physics might come from additional symmetries, or new particles and interactions, see Refs. [1, 2] for examples. In this paper, we are interested in a class of models with vector-like fermions and an additional Abelian symmetry. In particular, we consider the model proposed in Refs. [3,4]. Vector-like fermions are particles whose left-handed and right-handed components transform in the same way under the symmetry group of the model [5]. Due to this property, vector-like fermions do not interact with the W and Z bosons as V − A currents like the SM chiral fermions, but as pure vector (V ) currents. These fermions can play an important role to realize the gauge coupling unification [6,7]. They also help to stabilize the electroweak vacuum [8], or explain observed discrepancies between experimental data and SM predictions [9]. Beside the SM gauge group SU(3)C × SU(2)L ×U(1)Y , the considered model include an additional Abelian symmetry U(1)X under which only new partices are charged. Such symmetry was also investigated in many other scenarios resulting in interesting phenomenology [10, 11]. Recently, Belle-II Collaboration has published new results in the search for the gauge boson Z 0 of this new Abelian symmetry [12]. In this context, we explicitly derive the analytic formulas for the new particle masses in the model. Refs. [3, 4] considered a simple version of the mixing for left-handed quarks. In particular, Ref. [3] only considered the mass mixing for the second and third generations of SM quarks in the calculation, and the mixing for first generation was neglected. In this paper, the full mixings between the SM fermions and the vector-like fermions are taken into account in our calculation leading to their modified gauge interactions. Structure of the paper is as follows. In Section II, we briefly describe all the ingredients of the model. In Section III, the formulas for new particle masses are derived. The modified gauge interactions for fermions are investigated in Section IV. In Section V, we briefly discuss a few phenomenological aspects of the model and show that the model is realistic. Finally, Section VI is devoted to conclusions. II. THE MODEL Beside the ordinary SM particles which have been observed experimentally, the model we consider consists of heavy vector-like leptons (LL , LR ) and quarks (QL , QR ) that transform as SU(2)L doublets:     NL,R UL,R LL,R = , QL,R = . (1) EL,R DL,R In this model, two complex scalars χ and φ are also introduced. They are singlets under the SM gauge groups. The SM symmetry is extended in this model by introducing an extra Abelian symmetry denoted as U(1)X . These above new particles are charged under U(1)X , while the SM particles are neutral under this symmetry. This is essential to ensure that the SM sector is consistent with experimental data. The properties of these new particles are given in Table 1. The Lagrangian of the model consists of two parts: L = LSM + LNP , (2) where the first part is the usual Lagrangian of the SM, and the second one describes new physics beyond the SM. Since the vector-like fermions transform in the same way as SM left-handed ON A STANDARD MODEL EXTENSION WITH VECTOR-LIKE FERMIONS AND ABELIAN SYMMETRY 233 Table 1. Properties of new particles introduced in the model [3]. Particles Spin SU(3)C LL , LR 1/2 1 QL , QR 1/2 3 χ 0 1 φ 0 1 SU(2)L U(1)Y 2 -1/2 2 1/6 1 0 1 0 U(1)X 1 -2 -1 2 fermion doublets, they can interact with the SM gauge bosons. Other interaction terms involving the new particles are given as   LNP ⊃ − λφ H |φ |2 |H|2 − λχH |χ|2 |H|2 − y`L LR χ + wqL QR φ + h.c. −V0 (φ , χ), (3) where H is the SM Higgs doublet, and lL and qL are the SM left-handed leptons and quarks:  e   νL u i i `L = , qL = L , (i = 1, 2, 3). eL i dL i (4) V0 is the scalar potential related to the new scalar fields φ and χ. Its explicit form is as follows
V0 (χ, φ ) = λφ |φ |4 + m2φ |φ |2 + λχ |χ|4 + m2χ |χ|2 + λφ χ |φ |2 |χ|2 + rφ χ 2 + h.c. . (5) The SM fermion mass terms are forbidden at the beginning due to the SU(2)L gauge symmetry. They obtain their masses only after the spontaneously breaking of the gauge group SU(2)L × U(1)Y . The situation for vector-like fermions is different because of the symmetry between their left-handed and right-handed components. Therefore, their mass terms can be introduced directly in the original Lagrangian: LNP ⊃ − (ML LL LR + MQ QL QR + h.c.) , (6) where ML and MQ are the vector-like fermion masses supposed to be large. In a model with two Abelian symmetries, the kinetic mixing term is allowed in general, LNP ⊃ kBµν X µν , (7) where k is the kinetic mixing coefficient. Here, we assume k = 0 for simplicity. For the treatment of the non-zero kinetic mixing, we refer the readers to Ref. [13] where it was studied in details. III. III.1. NEW PARTICLE MASSES Scalar bosons The SM Higgs’ vacuum expectation value (VEV), hHi = 174 GeV, plays a central role in generating the SM fermion and weak gauge boson masses. In our considered model, it induces two new quadratic terms in addition to the scalar potential (5): λφ H hHi2 |φ |2 + λχH hHi2 |χ|2 . (8) The new scalar potential for φ and χ can be written as
2 4 02 2 2 2 2 V (χ, φ ) = λφ |φ |4 + m02 φ |φ | + λχ |χ| + mχ |χ| + λφ χ |φ | |χ| + rφ χ + h.c. , (9) 234 T. M. HIEU, D. Q. SANG AND T. Q. TRANG where m02 = m2φ + λφ H hHi2 , φ (10) m02 = m2χ + λχH hHi2 . χ (11) 02 We assume that m02 φ < 0 and mχ > 0. Hence, only the scalar field φ can develops a VEV, s −m02 φ , hφ i = 2λφ (12) leading to the spontaneous breaking of the U(1)X group. Substituting1 1 = hφ i + √ (ϕr + iϕi ) (13) 2 into Eq. (9), where ϕr and ϕi are real scalar fields, we find the masses of these scalar fields as q mϕr = 2 λφ hφ i, (14) φ = 0. mϕi (15) While ϕr is a massive scalar boson, ϕi is a massless Nambu-Goldstone boson that can be absorbed by the U(1)X gauge field in the unitary gauge. Similarly, after the spontaneous breaking of the U(1)X group, the induced potential for the other scalar field χ now reads:
2 2 (16) V (χ) = λχ |χ|4 + m002 χ |χ| + rhφ iχ + h.c. , where 02 2 2 2 2 m002 χ = mχ + λφ χ hφ i = mχ + λχH hHi + λφ χ hφ i . (17) The coefficients of this potential are assumed such that they do not result in a non-zero VEV for χ. Substituting 1 √ (χr + iχi ) (18) 2 into Eq. (16), we obtain the mass terms relating to these field components as     
2 χr
m002 + (r + r∗ )hφ i 1 1 i(r − r∗ )hφ i χr χr χi Mχ χr χi = . (19) χi χi i(r − r∗ )hφ i m002 − (r + r∗ )hφ i 2 2 χ = The matrix Mχ2 is symmetric, and can be diagonalized by an orthogonal matrix. In the case where the coupling r is real, the squared mass matrix Mχ is diagonal. The masses of χr and χi are respectively m χr = m002 + 2rhφ i, (20) m χi = m002 − 2rhφ i. (21) We see that the mass splitting between these real scalar fields is proportional to the VEV of φ . 1The factor √1 is crucial for the canonical kinetic terms of the real scalar fields, ϕ and ϕ . r i 2 ON A STANDARD MODEL EXTENSION WITH VECTOR-LIKE FERMIONS AND ABELIAN SYMMETRY 235 III.2. U(1)X gauge boson Due to the U(1)X gauge symmetry, the mass term of the corresponding gauge field Z 0 is forbidden in the original Lagrangian. After the scalar field φ develops a non-zero VEV, hφ i, the group U(1)X is spontaneously broken. Because the field φ is invariant under the SM gauge symmetry, the covariant derivative of this scalar field is
Dµ φ = ∂ µ − igX Xφ Z 0µ φ , (22) where Xφ = 2 is the U(1)X charge of φ given in Table 1. Using Eq. (13) in the kinetic term with the above covariant derivative, we can extract the mass term for Z 0 :    

1 1 0 µ † µ 0µ (D φ ) Dµ φ = ∂ + 2igX Z hφ i + √ (ϕr − iϕi ) ∂µ − 2igX Zµ hφ i + √ (ϕr + iϕi ) 2 2 1 (23) ⊃ 4g2X hφ i2 Z 0µ Zµ0 ≡ m2Z 0 Z 0µ Zµ0 . 2 From the last identity, we obtain the mass of the Z 0 boson as √ mZ 0 = 2 2gX hφ i. (24) III.3. Vector-like fermions Since the scalar field χ does not develop a VEV, the vector-like lepton mass only comes from ML , and there is no mass mixing with the SM leptons. In general, ML is a 2 × 2 diagonal matrix:   mN 0 ML = , (25) 0 mE where mN and mE are the masses of the upper and lower components (N, E) of the vector-like lepton doublet L. The off-diagonal elements are forbidden by the charge conservation. The vector-like quark masses are more involved because the scalar field φ acquires a nonzero VEV, leading to their mixing with the SM quarks. The pure vector-like quark mass has a form similar to Eq. (25):   mU 0 MQ = , (26) 0 mD where mU and mD are the masses of the upper and lower components (U, D) of the vector-like quark doublet Q. The mass mixing between the vector-like quarks and the SM ones is controlled by the new Yukawa interaction shown in Eq. (3): −LYukawa ⊃ wqL QR φ = wuLUR φ + wdL DR φ . (27) After the gauge group U(1)X is spontaneously broken, the quark mass terms in the Lagrangian are given by quark −Lmass = Yiuj hHiuiL uRj +Yidj hHidLi dRj + wi hφ iuiLUR + wi hφ idLi DR + MU ULUR + MD DL DR  1  1 uR dR    2  2   u u d  dR  R+ 1 d2 d3 D = M (28) u1L u2L u3L UL M4×4  d L 4×4  d 3  , L L L  u3  R R UR DR 236 T. M. HIEU, D. Q. SANG AND T. Q. TRANG where, Y u and Y d are the up-type and down-type Yukawa coupling matrices in the SM. The two 4 × 4 mass matrices, M u and M d , are written in the basis of quark gauge eigenstates as follows   u u hHi Y u hHi w hφ i Y11 hHi Y12 1 13 Y u hHi Y u hHi Y u hHi w2 hφ i 23 22 21  (29) Mu =  u hHi Y u hHi Y u hHi w hφ i , Y31 3 32 33 0 0 0 mU   d d d Y11 hHi Y12 hHi Y13 hHi w1 hφ i Y d hHi Y d hHi Y d hHi w2 hφ i 21 22 23  (30) Md =  Y d hHi Y d hHi Y d hHi w3 hφ i . 33 32 31 0 0 0 mD We observe that there are three distinct scales exist in each mass matrices, i.e. (hHi, hφ i, mU ) for M u , and (hHi, hφ i, mD ) for M d . Each of these matrices can be diagonalized by a pair of unitary matrices: u Mdiag = VLu M u (VRu )† , (31) u Mdiag = VLd M d (VRd )† . (32) These unitary matrices act as rotations of the basis transforming the quark gauge eigenstates, (u1 , u2 , u3 ,U) and (d 1 , d 2 , d 3 , D), into the mass eigenstates, (u, c,t, U ) and (d, s, b, D):    1     1  uL,R dL,R uL,R dL,R   2 2   cL,R      
u d    uL,R   sL,R  = VL,R  dL,R  . (33) 3  tL,R  = VL,R 4×4  u3  ,  bL,R   dL,R  4×4 L,R UL,R DL,R UL,R DL,R IV. IV.1. GAUGE INTERACTIONS Gauge interactions for leptons Since the SM leptons do not mix with the vector-like leptons, their interactions with the gauge bosons (W ± , Z-bosons, and photon) remain the same as in the SM. Because the SM leptons have no charge under U(1)X , they do not interact with the new gauge boson Z 0 . The vector-like lepton interactions with gauge bosons can be derived from the kinetic terms: L ⊃ iLL γ µ Dµ LL + iLR γ µ Dµ LR , where the covariant derivatives of the vector-like leptons are given as 

ig2 ig2 Dµ LL,R = ∂µ − √ τ +Wµ+ + τ −Wµ− − I3 − sin2 θW Q Zµ − ieQAµ cos θW 2  0 − igX XZµ LL,R , where the 2 × 2 matrices τ ± are defined as   0 1 + τ = , 0 0   0 0 τ = , 1 0 − (34) (35) (36) ON A STANDARD MODEL EXTENSION WITH VECTOR-LIKE FERMIONS AND ABELIAN SYMMETRY 237 θW is the Weinberg angle, and the electric charge Q is determined by the Gell-Mann−Nishijima formula: Q = I3 +Y. (37) The U(1)X charges of the vector-like leptons are given in Table 1 as XLL,R = 1. As a result, the interaction terms between the vector-like leptons and the model’s gauge bosons are gauge Linteraction ⊃ LLLW + LLLZ + LLLA + LLLZ 0 , (38) where LLLW = LLLZ = g g √2 Nγ µ EWµ+ + √2 Eγ µ NWµ− , 2 2   g2 1 g2 µ 2 − + sin θW Eγ µ EZµ , Nγ NZµ + 2 cos θW cos θW 2 LLLA = − eEγ µ EAµ , (39) (40) (41) describe the interaction with the ordinary SM gauge bosons, and LLLZ 0 = gX Nγ µ NZµ0 + gX Eγ µ EZµ0 , (42) describes the interaction with the new massive gauge boson Z 0 . Here, we denote N = NL + NR , E = EL + ER , (43) as Dirac spinors for the upper and lower components of the vector-like lepton doublet L = LL + LR . IV.2. Gauge interactions for quarks Due to the mixing among the SM and the vector-like quarks (see Eq. (33)), the gauge interactions of the SM quarks are modified in comparison to those in the SM. Noting that the SM quarks are neutral under the U(1)X group, their covariant derivatives are  

λa a ig2 + + ig2 i − − 2 Dµ qL = ∂µ − ig3 Gµ − √ τ Wµ + τ Wµ − I3 − sin θW Q Zµ − ieQAµ qiL , 2 cos θW 2  
λa ig2 Dµ (u, d)iR = ∂µ − ig3 Gaµ − − sin2 θW Q Zµ − ieQAµ (u, d)iR . (44) 2 cos θW In the meanwhile, the vector-like quarks have non-zero U(1)X charges. Therefore, their covariant derivatives are 

λa ig2 ig2 Dµ QL,R = ∂µ − ig3 Gaµ − √ τ +Wµ+ + τ −Wµ− − I3 − sin2 θW Q Zµ − ieQAµ 2 cos θW 2  0 − igX XZµ QL,R . (45) Substituting these equations into the Dirac Lagrangian: L ⊃ iqiL γ µ Dµ qiL + iuiR γ µ Dµ uiR + idRi γ µ Dµ dRi + iQL γ µ Dµ QL + iQR γ µ Dµ QR , (46) we obtain the various gauge interaction terms for the model’s quarks: gauge Linteraction ⊃ LqqG + LqqW + LqqZ + LqqA + LqqZ 0 . (47) 238 T. M. HIEU, D. Q. SANG AND T. Q. TRANG Decomposing the quark doublets into different charged states, the interactions between these quarks and gluons are described by λa λa λa µ i a λa γ uL Gµ + g3 dLi γ µ dLi Gaµ + g3UL γ µ UL Gaµ + g3 DL γ µ DL Gaµ 2 2 2 2 λ λ λ λa a a a + g3 uiR γ µ uiR Gaµ + g3 dRi γ µ dRi Gaµ + g3UR γ µ UR Gaµ + g3 DR γ µ DR Gaµ 2 2 2 2 λ λ λ λ a a a a = g3 FLu γ µ FLu Gaµ + g3 FRu γ µ FRu Gaµ + g3 FLd γ µ FLu Gaµ + g3 FRd γ µ FRu Gaµ , (48) 2 2 2 2 LqqG = g3 uiL u,d where FL,R are used to denote the quark gauge eigenstates:  u1L,R  u2L,R  u  FL,R =  u3  , L,R UL,R  1  dL,R 2   dL,R d  = FL,R  d3  . L,R DL,R  (49) The interactions between quarks and W -bosons are found to be LqqW ig ig √2 uiL γ µ dLi Wµ+ + √2 dLi γ µ uiLWµ− 2 2 ig2 ig2 ig2 ig2 + √ UL γ µ DLWµ+ + √ DL γ µ ULWµ− + √ UR γ µ DRWµ+ + √ DR γ µ URWµ− 2 2 2 2
ig2 u µ W
ig 2 = √ FL γ CL 4×4 FLd Wµ+ + √ FRu γ µ CRW 4×4 FRd Wµ+ + h.c., (50) 2 2 = where CLW = Diag(1, 1, 1, 1), CRW = Diag(0, 0, 0, 1), (51) are 4 × 4 diagonal matrices acting on the generation space. The interaction terms of quarks and Z-bosons are     2 2 g2 g2 1 1 2 µ 1 i µ i i LqqZ = u γ − sin θW uL Zµ + d γ − + sin θW dLi Zµ cos θW L 2 3 cos θW L 2 3     g2 2 g 1 2 µ 2 i µ 2 i i + u γ − sin θW uR Zµ + d γ sin θW dRi Zµ cos θW R 3 cos θW R 3     g2 2 2 g2 1 1 2 µ 1 µ + UL γ − sin θW UL Zµ + DL γ − + sin θW DL Zµ cos θW 2 3 cos θW 2 3     g2 2 2 g2 1 1 2 µ 1 µ + UR γ − sin θW UR Zµ + DR γ − + sin θW DR Zµ cos θW 2 3 cos θW 2 3

g2 g 2 Z Z = F u γ µ CuL F uZ + F d γ µ CdL FdZ 4×4 L µ 4×4 L µ cos θW L cos θW L

g2 g2 Z u d γ µ CZ FRu γ µ CuR F Z + F FdZ , (52) + µ R dR R 4×4 4×4 R µ cos θW cos θW ON A STANDARD MODEL EXTENSION WITH VECTOR-LIKE FERMIONS AND ABELIAN SYMMETRY 239 where Z CuL  1 2 2 − sin θW · Diag(1, 1, 1, 1), 2 3   1 1 2 − + sin θW · Diag(1, 1, 1, 1), 2 3  2 2  0 0 0 − 3 sin θW   0 0 0 − 23 sin2 θW  , 2 2   0 0 − 3 sin θW 0 2 1 2 0 0 0 2 − 3 sin θW  1 2 0 0 0 3 sin θW 2 1   0 0 0 3 sin θW ,  2 1   0 0 0 3 sin θW 2 1 1 0 0 0 − 2 + 3 sin θW  = Z CdL = Z CuR = Z CdR = (53) (54) (55) (56) are 4 × 4 diagonal matrices acting on the generation space. From Eqs. (50) and (52), we can see that the SM quark weak currents are V − A type, while the vector-like quark weak currents are purely V type. The interaction terms between quarks and photons are written as 1 2 1 2 i µ i LqqA = eu γ uL Aµ − edLi γ µ dLi Aµ + euiR γ µ uiR Aµ − edRi γ µ dRi Aµ 3 L 3 3 3 1 2 1 2 + eUL γ µ UL Aµ − eDL γ µ DL Aµ + eUR γ µ UR Aµ − eDR γ µ DR Aµ 3 3 3 3 1 d µ d 2 u µ u 1 d µ d 2 u µ u eF γ FL Aµ − eFL γ FL Aµ + eFR γ FR Aµ − eFR γ FR Aµ . (57) = 3 L 3 3 3 Note that XQL,R = −2 as given in Table 1, the interaction terms between quarks and the Z 0 -boson are LqqZ 0 = − 2gX UL γ µ UL Zµ0 − 2gX DL γ µ DL Zµ0 − 2gX UR γ µ UR Zµ0 − 2gX DR γ µ DR Zµ0 = − 2gX FLu γ µ (CZ 0 )4×4 FLu Zµ0 − 2gX FLd γ µ (CZ 0 )4×4 FLd Zµ0 − 2gX FRu γ µ (CZ 0 )4×4 FRu Zµ0 − 2gX FRd γ µ (CZ 0 )4×4 FRd Zµ0 , (58) where CZ 0 = Diag(0, 0, 0, 1). (59) is a 4 × 4 diagonal matrix acting on the generation space. Next, we rewrite these above interaction terms in the basis of quark mass eigenstates (33),     uL,R dL,R  cL,R   sL,R  d d u u u d   (60) FL,R = FL,R =  bL,R  = VL,R FL,R ,  tL,R  = VL,R FL,R , DL,R UL,R that are physical states to be observed experimentally. In the calculation, we use the fact that the  † u,d u,d rotation matrices are unitary, namely VL,R VL,R = 14×4 = Diag(1, 1, 1, 1), in places where it can 240 T. M. HIEU, D. Q. SANG AND T. Q. TRANG be applied. To translate the Lagrangian from Weyl spinors for chiral states to Dirac spinor, we use the following relations: u,d FL,R = PL,R F u,d , PL,R = 1 ∓ γ5 , 2 F u,d = FLu,d + FRu,d . (61) (62) i. Quark−quark−gluon interaction λa µ u u† u a λa γ VL VL FL Gµ + g3 FRu γ µ VRuVRu† FRu Gaµ 2 2 λ λa a + g3 FLd γ µ VLd VLd† FLd Gaµ + g3 FRd γ µ VRd VRd† FRd Gaµ 2 2 λ λ λa λa a a = g3 FLu γ µ FLu Gaµ + g3 FRu γ µ FRu Gaµ + g3 FLd γ µ FLd Gaµ + g3 FRd γ µ FRd Gaµ 2 2 2 2 λ λ a a = g3 F u γ µ PL F u Gaµ + g3 F u γ µ PR F u Gaµ 2 2 λa λa + g3 F d γ µ PL F d Gaµ + g3 F d γ µ PR F d Gaµ 2 2 λ λ a a µ u a µ (63) = g3 F u γ F Gµ + g3 F d γ F d Gaµ . 2 2 LqqG = g3 FLu u,d From this equation, we see that, due to the unitarity of the rotation matrices VL,R , the strong interaction for quarks in this model is the same as that in the SM. ii. Quark−quark−W interaction LqqW ig ig √2 FLu γ µ VLuCLW VLd† FLd Wµ+ + √2 FRu γ µ VRuCRW VRd† FRd Wµ+ + h.c.. 2 2 = (64) Noting that CLW is the identity matrix (see Eq. (51)), the relevant term becomes simpler: LqqW (51) = = = ig ig √2 FLu γ µ VLuVLd† FLd Wµ+ + √2 FRu γ µ VRuCRW VRd† FRd Wµ+ + h.c. 2 2   ig √2 F u γ µ VLuVLd† PL +VRuCRW VRd† PR F d Wµ+ + h.c. 2 h  ig2 √ F u VLuVLd† +VRuCRW VRd† γµ 4×4 2 2   i − VLuVLd† −VRuCRW VRd† γ µ γ 5 F d Wµ+ + h.c. . 4×4 (65) In the SM, only left-handed quarks involve in the charged current. Due to the existence of the vector-like quarks and their mixing with SM quarks in this model, both left-handed and righthanded quarks take parts in the charged current.