CP violating experimental data and CKM model

CP violation was first discovered in Kaon mixing [7] in 1964. The CP eigenstates for the K₀ and K₀ system are

K₁ = 1

√2(K⁰− K⁰);

K₂ = 1

√2(K⁰+ K⁰). (1.45)

K₀ is the pesudoscalar particle with odd parity and in charge conjugatation trans-form CK₀ = K₀; CK₀ = K₀. It is obvious that K₁ is CP even eigenstate and K₂ is

Fig. 1.2: Box diagrams for neutral K-meson mixing

the eigenstate of CP odd. If the CP eigenstates are also the Hamiltonian eigenstates, it means that the CP is conserved under the system.

In general

H = M − i 2Γ =



 M₁₁ M₁₂ M₂₁^∗ M₂₂



 − i 2



 Γ₁₁ Γ₁₂ Γ^∗₂₁ Γ₂₂



 , (1.46)

where the M and Γ are 2 × 2 Hermitian matrices, so H is not Hermitian obvi-ously. This non-Hermitian Hamiltonian make the two state system decay during time evolution. The mass eigenstates are

K_S = 1

p1 + |˜²|²(K₁− ˜²K₂),

K_L = 1

p1 + |˜²|²(K₂+ ˜²K₁). (1.47)

Where |˜²| = (2.44 ± 0.04) × 10⁻³ [34] is the small value related to the mixing of two CP eigenstates. This formula indicates that the mass eigenstates(energy eigenstates) are not exactly identical to the CP eigenstates. This experimental data is explained by so-called box diagram shown in Fig.1.2 in the SM.

Direct CP violation in Kaon decay into ππ has also been discovered[35]. When treating K meson decay, we usually take the mass eigenstates K_S and K_L as the CP eigenstates instead of K₁ and K₂, because here we discuss only the CP violation from decay. It is convenient to define the quantities related to the decay amplitude

Fig. 1.3: Tree and penguin diagrams for K → ππ decays, where q can be u or d

for K meson to study direct CP violation,

η₀₀ = < π⁰π⁰|H|K_L>

< π⁰π⁰|H|K_S >; η+− = < π⁺π⁻|H|K_L>

< π⁺π⁻|H|K_S >. (1.48) CP violation in K → ππ is measured by ²⁰, which is defined as

η00= ˜² − 2²⁰;

η₊₋= ˜² + ²⁰. (1.49)

The Particle Data Group [12] gives the fitting for the value

Re(²⁰/˜²) = 1.65 ± 0.26. (1.50)

²⁰ is explained in the SM by the tree and penguin diagrams in Fig.1.3.

B decays can provide many tests for CKM model by measuring α, β, and γ in the unitary triangle in Fig 1.1.

β is the relative angle between V_tdV_tb^∗ and −V_cdV_cb^∗ on the complex plane. There are several ways people often take to determine this angle. The most popular process is the b → c¯cs process. This is because the amplitude of the tree level and loop diagram has approximately the same phase. One of the process often been used is B → J/ψK_s. The sin 2β is extracted from the relation [12]

S_f = −η_fsin 2β, (1.51)

where Sf is the quantity related to time-dependent CP asymmetry in B decays [36, 37], and η_f is the CP eigenvalue of f . The experimental result from average of the related decay by BaBar has the value [38]

sin 2β = 0.686 ± 0.039 ± 0.015. (1.52) From Fig 1.1 definition the α is the angle between V_udV_ub^∗ and −V_tdV_tb^∗. It can be extract from B → ππ process, via the measurement of S_π⁺_π⁻ and C_π⁺_π⁻. The measurement of BABAR gives that [40]

α = 96^◦+10₋₆◦ ^◦. (1.53)

The decay B → ρ₀π₀ are also applied to determined α, and the experimental result from Belle gives 68^◦ < α < 95^◦ at CL = 68%[41].

γ is the angle between V_cdV_cb^∗ and −V_udV_ub^∗. The measurement of γ determination uses the B decay process B → DK. BELLE [42] measured B⁻ → DK⁻, B⁻ → D^∗K⁻ and B⁻→ DK^∗− to obtain

(γ = 53⁺¹⁵₋₁₈± 3 ± 9)^◦ (1.54)

The process B → K⁺π⁻are usually applied to test the CP asymmetry, too. The experimental average is A_CP(B⁰ → K⁺π⁻) = −0.097 ± 0.012 by HFAG [43]. This asymmetry can be explained by SM [44, 45, 46].

The global fit for the unitary triangle is summarized in Fig.1.4 which is from CKMfitter [47]. The PDG review[12] provides the fitting values for Wolfenstein parameters, which are

λ = 0.2272 ± 0.0010; A = 0.818^+0.007_−0.017;

ρ = 0.221^+0.064_−0.028; η = 0.340^+0.017_−0.045. (1.55) Compare these values with the PDG parameters, we can derive s12 = 0.227 ± 0.001, s23= 0.0422±0.0004, s13= 0.00399±0.00007, and the phase sin δ13= 0.839±

0.006. From these values, one obtain[12]

J = (3.08^+0.16_−0.18) × 10⁻⁵. (1.56)

-1 -0.5 0 0.5

ρ

1 1.5 2 excluded area has CL > 0.95 excluded at CL > 0.95

Summer 2007

CKMf i t t e r

Fig. 1.4: The experimental fit for the ρ and η from CKMfitter Group [47]

From the above discussion for CKM matrix, we can see that the CKM matrix works very well in describing the meson decay, leptonic and semileptonic decay.

The phase in CKM matrix generates CP violation, and which is consistent with the experimental result for the CP violation phenomena like K meson mixing, K and B meson decay. However there are still some problems. One of them is the baryongenesis. That is, the amount of particles is more than that of antiparticles in our world, and one necessary condition for this phenomenon is the existence of CP violation. The quantity to estimate the asymmetry of universe is nB/nγ, with nB, nγ

denoting the baryon number density, and photon number density respectively. In high temperature the CKM model can produce about nB/nγ ≈ O(10⁻²⁰)[48], which is too small compared with observation nB/nγ ≈ 10⁻⁸[49]. There should be another source of CP violation beyond the CKM matrix in our world. Also, the CKM model does not provide the answer where CP violation is originated, but just put in by hand. It is desirable for some understanding of the origin of CP violation. In this

thesis we try to study how to connect the CKM matrix phase with the spontaneous CP violating phase for the explanation of the source of CP violation from CKM mechanism.

In chapter 2 we are going to discuss what the spontaneous CP violation is and treat some of the multi-Higgs models. In chapter 3 we will build a new model with the connection between spontaneous CP violating phase and CKM matrix phase. In chapter 4 we will use this model to discuss some phenomenology. In the last chapter we will summarize what we do in this thesis.

VIOLATION

Although the CKM matrix and its complex relation explain the CP violation of observation very well, it is still possible that CP is violated from other place. The multi-Higgs model is a popular topic in this area. Such models may also answer that CP violation comes from the so-called spontaneous CP violation(SCPV), a mechanics first proposed by T.D. Lee [50, 51].

When there are more than one Higgs, their vacuum expectation values might have the relative phases difference. If these phases are non-vanishing after symmetry breaking and irreducible in Higgs self-interaction or Yukawa terms with fermions, then they also produce CP violation. Because this kind of CP violation comes from the spontaneous symmetry breaking of Higgs, it is called spontaneous CP violation.

2.1 Two Higgs Doublet Model

In 1973, T. D. Lee proposed a model with two Higgs doublets [50, 51]. The most important property of this model is that if there is a phase difference between VEVs of two Higgs doublets, this phase can give the contribution to the CP violation. The two Higgs doublets are written in the following form [50]

φ₁ = e^iθ1H₁ = e^iθ1





√1

2(ρ1+ R1+ iA1) h⁻₁



 ;

φ₂ = e^iθ2H₂ = e^iθ2





√1

2(ρ₂+ R₂+ iA₂) h⁻₂



 . (2.1)

H₁ and H₂ are Higgs with real vacuum expectation values, and R₁, R₂, A₁, and A₂ are real parts and imaginary parts of them. The phase difference between two

Higgs doublets δ = θ2 − θ1 is the spontaneous CP violating phase. If this phase is non-zero and can’t be eliminated by fermion rotation, then this could produce the spontaneous CP violation. The Higgs potential can be built in the form [50]

V = − λ₁φ^†₁φ₁− λ₂φ^†₂φ₂+ A(φ^†₁φ₁)²+ B(φ^†₂φ₂)² + C(φ^†₁φ₁)(φ^†₂φ₂) + C(φ^†₁φ₂)(φ^†₂φ₁)

+ 1

2[(φ^†₁φ₂)(Dφ^†₁φ₂+ Eφ^†₁φ₁ + F φ^†₂φ₂) + h.c.], (2.2) where λ₁, λ₂, A − F , and C are all real numbers. The minimal condition by differ-entiating with respect to δ can give

cos δ = −(4Dρ₁ρ₂)⁻¹[Eρ²₁+ F ρ²₂]. (2.3)

If the right handed side is not required to be 1 or -1, CP is violated spontaneously.

There are eight real scalar fields in two Higgs doublets, and three of them are eaten by W^± and Z⁰. So there are five physical states in two Higgs doublet model after spontaneous symmetry breaking. The two Higgs model is different from SM because it has charged Higgs bosons. The interaction of charged Higgs and fermions is similar to charged weak interaction. So this model also have more contribution to flavor change process than SM.

Usually the two Higgs doublet models are classified into three types by different Yukawa interactions.

Type I Type I is that one Higgs couples with each fermions, like the Higgs in standard model, and another Higgs does not couple with fermions as below

Q_LΓuφ1UR+ Q_LΓdφe1DR+ LLΓeφe1ER+ h.c., (2.4)

where Γ_u, Γ_d, and Γ_e are real coupling matrices. We construct thise inter-actions of type I by introducing the discrete symmetry φ₂ → −φ₂ and other fields are unchanged, so that φ₂ only exists in the Higgs potential with even powers.

Type II Type II is that one Higgs doublet couples with up-type quarks and another one couples with down-type quarks. For example

Q_LΓ_uφ₁U_R+ Q_LΓ_dφe₂D_R. + L_LΓ_eφe₂E_R+ h.c.. (2.5)

We can construct type II model by introducing this discrete symmetry:

φ₁ → φ₁; φ₂ → −φ₂; U_R→ U_R; D_R→ −D_R; E_R→ −E_R;

QL → QL; LL→ LL. (2.6)

Type III The last model type III is the most general Yukawa interactions in which there are two Higgs coupling with each fermion,

Q_L(Γ_u1φ₁+ Γ_u2φ₂)U_R + Q_L(Γ_d1φe₁+ Γ_d2φe₂)D_R

+ L_L(Γ_e1φe₁+ Γ_e2φe₂)E_R+ h.c. (2.7)

so there could be the FCNC process because one can not diagonalize mass matrix v₁Γ₁+ v₂Γ₂ and coupling matrices Γ_1,2 simultaneously.

Type I and II can not have spontaneous CP violation, because (φ^†₁φ₂)(φ^†₁φ₁), and (φ^†₁φ₂)(φ^†₂φ₂) are not allowed, and this results in sin δ = 0. The spontaneous CP violating phase δ vanishes. So type I and type II with discrete symmetry have no spontaneous CP violation.

If we hope that the spontaneous CP violation exists, then only type III is allowed.

However, type III has the tree level FCNC contribution which is severely constrained by experimental data. Also, there are also too many unknown parameters in the model.

2.2 Weinberg Model

In 1976, Weinberg [52] proposed that by using some discrete symmetry, the model with three or more Higgs doublets gives the spontaneous CP violation without tree level flavor change neutral current. The three Higgs doublet model is called Weinberg

model. Three Higgs φ1, φ2, and φ3 are written as follows

φk = e^iθkHk = e^iθk





√1

2(v_k+ R_k+ iA_k) H_k⁻



 , k = 1 ∼ 3, (2.8)

where ^√1₂v_ke^iθk is the vacuum expectation value of neutral part in φ_k, and we let H_k⁰ = v_k+ R_k+ iA_k. R_k and A_k are the corresponding real part and imaginary part in H_k. Branco extended this idea to arbitrary number of generations with two sets of discrete symmetry [53]

D₁ : φ₁ → φ₁; φ₂ → −φ₂; φ₃ → φ₃; Q_L → Q_L; d_R→ d_R; u_R → −u_R D₂ : φ₁ → φ₁; φ₂ → φ₂; φ₃ → −φ₃; Q_L → Q_L; d_R→ d_R; u_R → u_R. (2.9)

The D₁ implies the constraint that φ₁ couples to the up-type quarks singlet u_R and φ₂ couples to the down-type ones d_R, which has the same propose as the discrete symmetry for type II of two Higgs doublet model. The D₂ can suppress φ₃ not to couple with quarks, but it can couple to leptons. Applying those discrete symmetry can inhibit the tree level FCNC process of Higgs exchange, and the CP violation can arise from the Higgs interaction themselves. The Yukawa terms of Weinberg model are written as

Q_LΓ_uφ₁U_R+ Q_LΓ_dφe₂D_R+ L_LΓ_eφe₃E_R+ h.c.. (2.10)

These interactions are similar to type II of two Higgs doublet model, and the spon-taneous CP violation will be produced in the Higgs potential.

After spontaneous symmetry breaking the Lagrangian is expanded as follows

L = −1

v₁ULMˆuURH₁⁰ − 1

v₂DLMˆdDRH₂⁰− 1

v₂LLMˆeERH₃⁰

−

√2

v₁ D_LV_CKM^† Mˆ_uU_RH₁⁻+

√2

v₂ U_LV_CKMMˆ_dD_RH₂⁺ +

√2 v3

L_LV_CKMMˆ_eE_RH₃⁺+ h.c., (2.11)

where M_u = −^√1₂Γ_uv₁, M_d = −^√1₂Γ_dv₂, and M_e = −^√1₂Γ_ev₃. V_CKM is assumed to be real matrix here. That is, CP violation does not come from the CKM matrix.

The CP should be arisen from the Higgs self-interaction. The parametrization for three Higgs doublet potential in discussion [54] is

V = − µ²₁φ^†₁φ₁− µ²₂φ^†₂φ₂− µ²₃φ^†₃φ₃+ h₁(φ^†₁φ₁)²+ h₂(φ^†₂φ₂)²+ h₃(φ^†₃φ₃)² + f₁₂(φ^†₁φ₁)(φ^†₂φ₂) + f₂₃(φ^†₂φ₂)(φ^†₃φ₃) + f₃₁(φ^†₃φ₃)(φ^†₁φ₁)

+ g₁₂(φ^†₁φ₂)(φ^†₂φ₁) + g₂₃(φ^†₂φ₃)(Dφ^†₃φ₂) + g₃₁(φ^†₃φ₁)(φ^†₁φ₃)

+ (k₁₂(φ^†₁φ₂)²+ k₂₃(φ^†₂φ₃)²+ k₃₁(φ^†₃φ₁)²+ h.c.). (2.12) It can be assumed that the coefficients in above formula are all real. In the potential only two phases δ12 = θ2 − θ1 and δ23 = θ3− θ2 exist. Differentiating with respect to the two phases, we get a condition below

k₁₂v₂²

v₃² sin 2δ₁₂= k₂₃v²₂

v²₁ sin 2δ₂₃ = −k₁₃sin 2(δ₁₂+ δ₂₃). (2.13) Eq.(2.13) reflects the fact that the phases δ₁₂ and δ₂₃ can be nonzero, and CP is violated here.

In this model, there are four charged and five neutral physical Higgs bosons. The mass matrix of Higgs can give CP properties of this model [54]. The resulting mass matrix of charged Higgs has the off diagonal complex elements. It means that the CP violation will arise from the exchanging of charge Higgs. Also, the neutral mass matrix has the mixing terms between scalar and pseudoscalar Higgs which lead to the CP violation under neutral Higgs exchange.

Although Weinberg model can provide the spontaneous CP violation without the tree level FCNC which is inhibited in Lee’s two Higgs doublet model, it still has some contradiction which had been provided by many authors[55, 56, 57]. The Weinberg model is decisively ruled out by data on sin 2β measurement in B → K_sJ/ψ. In Weinberg model, the upper bound for magnitude of sin 2β is | sin 2β| < 0.05[56, 57].

The present experimental data is shown in Eq.(1.52) that sin 2β = 0.686 ± 0.039 ± 0.015, which means that the Weinberg model has been ruled out. The neutron EDM calculation also rules out the Weinberg model, from which the estimation for neutron EDM has order 10⁻²³e cm [58], whereas the experimental upper bound is

|d_n| < 0.29 × 10⁻²⁵e cm [59].

2.3 The strong CP problem and Peccei-Quinn symmetry

For the Lagrangian in QCD, the term (θg_s²/32π²)G^µνGe_µν is allowed, where eG_µν =

1

2²_µνρσG^ρσ. This term also violates P and CP. This is a possible CP violation in strong interaction.

Because P and CP violation will cause the electric dipole moment(EDM) of particles, the measurement of EDM of particles is important to test the CP violation in standard model. The neutron EDM test is especially important, and in present the experimental upper bound is given [59]

|d_n| < 0.29 × 10⁻²⁵e cm. (2.14) SM theoretical calculation of CKM matrix CP violation gives the small value con-tribution about order less than 10⁻³¹e cm [61, 62, 63] without considering strong CP violation. With non-zero θ, d_n can be much larger. Experimental bound for neutron EDM constrains the θ very strongly for |θ| . 10⁻¹⁰ [12]. This is considered to be unnatural since other couplings with strong interaction are much larger. This is the problem.

For the multi-Higgs model with spontaneous CP violating phase, the strong phase θ would be large [60] at loop level. We need a mechanism to make this phase small.

In 1977, Peccei and Quinn proposed a mechanism to solve this problem [64, 65].

They introduced another global symmetry U(1)PQ in the standard model. This symmetry is generated by the chiral transformation defined as follows.

u → e^iαuγ5u; d → e^iαdγ5d;

φ₁ → e^i(αu+αd)φ₁; φ_i → e^{−i(αu+αd)}φ_i; i 6= 1, (2.15) where α_u and α_d are the chiral rotational phases for up-type quarks and down-type quarks respectively. The φ₁ and other φ_i are the multi-Higgs doublets. After the chiral rotation, the strong phase becomes

θ → θ − 4α_u− 4α_d. (2.16)

Since αu,d are arbitrary phases, one can choose these phase as θ = 4(αu + αd), therefore there is no strong CP phase and also without large contribution to neutron EDM.

Models with PQ symmetry have an axion resulting from spontaneous breaking down of PQ symmetry. No axion has been detected in experiments. One has to make the axion invisible, by extending the original PQ model[66, 67].

The multi-Higgs model can have the spontaneous CP violation(SCPV). This is a nice feature which provides a understanding of the origin of CP violation. But the two Higgs doublet model has tree level FCNC, leading to too many unknown parameters. To improve the situation, Weinberg proposed a three Higgs doublet model which has no tree level FCNC. However, the prediction of Weinberg model for sin 2β is not consistent with experimental data as mentioned before. Here we take the idea [1] that the CP violation is arisen from spontaneous symmetry breaking, but further make the spontaneous CP violating phase be identical to the CP violation in CKM matrix. In the following we build specific models to realize this.

3.1 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

Using the PDG parametrization with the same argument from Eq.(3.7), and

在文檔中 自發性CP破缺相角與CKM矩陣相角比較之研究 (頁 20-0)