Making SCPV phase identical to CKM phase

Model(a)

In our new model, we try to build a model with the spontaneous CP violating phase from Higgs identical to the the phase in CKM matrix. We couple two independent Higgs doublets to the up-type quarks and one Higgs doublet to the down type quarks as below

L = Q_L(Γu1φ1+ Γu2φ2)UR+ Q_LΓdφeddR+ h.c., (3.1) where Γ_u1 and Γ_u2 are real 3 × 3 coupling matrices, and the tilde sign on Higgs means eφ_k= −iσ₂φ^∗_k. The φ₁, φ₂ and φ_d are Higgs doublets, which are written as

φ_k = e^iθk





√1

2(v_k+ R_k+ iA_k) h⁻_k



 , (3.2)

where k can be 1, 2, and d. It is convenient to redefine these Higgs doublets so that they have real vacuum expectation values. That is, φ_k = e^iθkH_k. Here we call

Eq.(3.1) as model(a).

After spontaneous symmetry breaking, the mass terms of model(a) Lagrangian appear as

L_m = −U_L(M_u1e^iθ1 + M_u2e^iθ2)U_R− D_L(M_de^−iθd)D_R+ h.c., (3.3)

where M_u1,u2 = −Γ_u1,u2v_1,2/√

2 and M_d = −Γ_dv_d/√

2. The phase θ₁ and −θ_d can be absorbed into the U_R and D_R respectively.

From previous section, Eq.(1.20) shows the relations between flavor eigenstates and mass eigenstates of quarks

U_L= V_L^uU_L^m; U_R= V_R^uU_R^m; DL = V_L^dD_L^m; DR= V_R^dD_R^m.

We make the M_d to be diagonal without loss of generality. That is, D_L and D_R are already the mass eigenstates D_L^m and D^m_R, and the mixing matrix V_L^d and V_R^d are unit matrices. The mass terms become

L_m = −U_L(M_u1+ M_u2e^iδ)U_R− D_LMˆ_dD_R+ h.c., (3.4)

where the relative phase δ = θ₂− θ₁ is the spontaneous CP violating phase in the Yukawa terms, and ˆM_d is the diagonal mass matrix. If δ is non-zero, it could cause the spontaneous CP violation in the model. The total mass matrix of up-type quarks can be diagonalized by left and right handed matrices V_R^u and V_L^u. That is

Mˆ_u = V_L^u†M_uV_R^u, (3.5)

where ˆM_uis the diagonal up-quark mass matrix and M_u = M_u1+e^iδM_u2. To simplify the discussion we assume that V_R^u is a unit matrix. This simplification can help us to make the identity relation between δ and the phase in CKM matrix. Because V_L^d is a unit matrix, from V_CKM = V_L^u†V_L^d it is obvious that V_L^u† is equal to V_CKM. Eq(3.5) becomes ˆM_u = V_CKM(M_u1+ e^iδM_u2) , and we obtain the relation

V_CKM^† = (Mu1+ e^iδMu2) ˆMu

−1. (3.6)

At this step we need the explicit CKM parametrization with a phase. First we use the Particle Data Group parametrization which is shown in Eq.(1.30)

V_CKM =

This parametrization makes more than one phases in the V_CKM elements. That is, one phase δ₁₃ in V₂₁, V₂₂, V₃₁, V₃₂ and another phase −δ₁₃ in V₁₃. We solve the problem by pulling the phase −δ₁₃ out and then decomposing the V_CKM into two matrices as below,

Absorbing the left handed diagonal matrix by redefining the quark sector UL. The remaining matrix has the uniform phase in each element. By comparing two side of Eq.(3.6), we introduce the identical relation

δ = −δ13. (3.8)

This relation implies that the CKM phase comes from the spontaneous CP violating phase. Also, this relation is related to the phase δ₁₃ which has been measured in experiments, so if the spontaneous CP violating phase δ can be nonzero after solving the minimal condition in Higgs potential, it is independent of the masses of Higgs, and the CP phenomena always exists.

By substituting Eq.(3.8) into Eq.(3.6) we determine the coupling matrices M_u1

and Mu2

Note that these matrices M_u1 and M_u2 depend on quark masses and the angles in CKM parametrization. This is not true for other multi-Higgs models, with which the spontaneous CP violating phase and CKM phase are concerned.

We apply the same idea to another CKM parametrization, the original Kobayashi Maskawa parametrization in Eq.(1.29)

Then the step of previous discussion for PDG parametrization makes the same relation as Eq.(3.8),

δ = −δKM. (3.10)

This relation gives the expression for the mass matrix with respect to KM parametriza-tion,

In general we can express the two coupling matrices M_u1 and M_u2 in terms of the CKM matrix, quark mass matrices, and the spontaneous CP violating phase as

follows,

M_u1 = V_CKM^† Mˆ_u− e^iδ

sin δIm(V_CKM^† ) ˆM_u; M_u2 = 1

sin δIm(V_CKM^† ) ˆM_u. (3.12) This relation is useful when we treat the Yukawa couplings of Higgs, and it is inde-pendent of the parametrization of V_CKM. Choosing a specific CKM representation implies a choice of a model.

Model(b)

Now we treat another kind of Yukawa interactions which is called model(b),

L = ¯Q_LΓ_uφ_uU_R+ ¯Q_L(Γ₁φe₁+ Γ₂φe₂)d_R. + h.c.. (3.13) In this Lagrangian two Higgs doublets φ₁ and φ₂ couple with down-type quarks and one Higgs-doublet φ_u is coupled with up-type quarks. Γ_d(1,2) and Γ_u are 3 × 3 real matrices. Here the φ₁ and φ₂ are defined as those in model(a), and φ_u is with the same definition as that in Eq.(3.2) and in which k is replaced by u. After symmetry breaking the mass terms is written below

Lm = −UL(Mue^iθu)UR− DL(Md1e^−iθ1 + Md2e^−iθ2)DR+ h.c., (3.14) where M_d(1,2) = −Γ_d(1,2)v_1,2/√

2 and M_u = −Γ_uv_u/√

2. Following the same treat-ment for model(a), we absorb the phases −θ1 and θu into DR and UR respectively, and we also treat the Mu as diagonal mass matrix. So V_L^u = V_R^u = 1 and the mass terms become

L_m = −U_LMˆ_uU_R− D_L(M_d1+ M_d2e^−iδ)D_R+ h.c.. (3.15) The ˆM_u indicates that it is diagonal, and δ is the spontaneous CP violating phase with δ = θ₂ − θ₁. The diagonal down-type mass matrix ˆM_d has the relation to M_d= M_d1+ e^−iδM_d2,

Mˆ_d= V_L^†dM_dV_R^d, (3.16) We make V_R^d= 1 and the V_L^d† is equal to V_CKM^† . That makes us to express the mass matrices Md1 and Md2 in terms of VCKM as

V_CKM= (M_d1+ e^−iδM_d2) ˆM_d⁻¹. (3.17)

Using the PDG parametrization with the same argument from Eq.(3.7), and com-paring phases in two sides of Eq.(3.17), we can write down the phase relation as follows

δ = −δ₁₃. (3.18)

Note that this relation is the same as Eq.(3.8) discussed in model(a). Also, this relation makes coupling matrices M_d1 and M_d2 as

Md1 =

Like the model(a), the mass matrices are determined by down-type quark masses and three angles of CKM matrix parametrization.

Using the KM parametrization we assume the same relation as Eq.(3.10) in model(a), which leads to determine the coupling matrices M_d1 and M_d2 as

M_d1 =

In this section we have discussed how to make the spontaneous CP violating phase identical to the CKM matrix phase, this identical relation leads to the deter-mination of couplings matrices, which depends on three angles in CKM parametriza-tion and the quark masses. For different CKM parametrizaparametriza-tion, the related coupling matrix are also different. In next section we will study a particular multi-Higgs model, and then apply it to our Yukawa coupling models built in this section.