The Yukawa couplings

Before discussion the phenomenology, we show the Lagrangian for the quarks couplings with Higgs as The above formula shows that the FCNC process is produced from the H₁ and a₁ exchange, because for H₁,a₁ couplings there is a non-diagonal coupling matrix proportional to V_CKMIm(V_CKM^† ) ˆM_ue^iδ/(sin δ) for up-type quarks in model(a). In model(b) has the same situation with −V_CKM^† Im(V_CKM) ˆM_de^−iδ/(sin δ) for down-type quarks. Also note that the flavor conserving interaction with H₄⁰ and a can be neglected because of the small factor 1/v_s.

The FCNC coupling matrices for PDG parametrization are expressed as follows

where Eq.(3.56) is FCNC related coupling matrix for up-type quarks in model(a) and Eq.(3.57) is for down-type quarks in model(b). From Eq. (3.56) we find that there exist all the mixing contributions between u − c, u − t, c − t. However, when considering the meson mixing, there are no meson which is constructed by t quark, we only apply the u − c couplings to neutral meson mixing later. The model(b) FCNC coupling matrices shown in Eq.(3.57) produce only the d − s couplings.

The FCNC coupling matrices for KM parametrization are

V_CKMIm(V_CKM^† ) ˆM_u e^iδ

These formulae show that there are only s−b and c−t couplings in KM parametriza-tion.

After building this model, we try to connect it to some experimental result and find whether this model is consistent with experimental data. We concentrate on effects for neutral meson mixing and neutron electric dipole moment. The multi-Higgs Yukawa coupling model we built leads to the FCNC phenomena, which makes contribution to the neutral meson mixing.

4.1 Neutral meson mixing

The FCNC from Higgs provides the tree level contribution for neutral meson oscillation. It occurs by the exchange of scalar Higgs, pseudoscalar Higgs, or both scalar and pseudoscalar Higgs with the cross terms between them. We write down the Yukawa interaction and the quadratic Higgs interaction for H₁⁰ and a₁ as follow-ing form

L = ¯q_i(a_ij + b_ijγ₅)q_jH₁⁰+ i ¯q_i(b_ij + a_ijγ₅)q_ja₁+ λ_H1⁰_a1H₁⁰a₁, (4.1)

where aij, bij are coupling constants of H₁⁰, a1, and i, j quarks, and λ_H1⁰_a1 is the mixing term between H₁⁰ and a1, with λ_H1⁰_a1 = 2m²_H0

1a1.

The total amplitude for mixing from the sum of three contribution which are shown in Fig 4.1 can be written as follows

M = − 1

m²_H0 1

¯

q_i(a_ij + b_ijγ₅)q_jq¯_i(a_ij + b_ijγ₅)q_j

+ 1

m²_a1 q¯_i(b_ij + a_ijγ₅)q_jq¯_i(b_ij + a_ijγ₅)q_j

− i λ_H1⁰_a1

m²_a1m²_H1 q¯_i(a_ij + b_ijγ₅)q_jq¯_i(b_ij + a_ijγ₅)q_j. (4.2)

Fig. 4.1: Mixing due to the exchange by (a) H₁⁰, (b) a₁, and (c) both H₁⁰ and a₁

This amplitude is the sum of the contribution of the s-channel diagrams. The t-channel contribution also needs to be considered. So the total amplitude is the s-channel contribution in Eq.(4.2) with that of Fierz transformation for fermionic fields together. The fermionic Fierz transformation for scalar interaction is S →

−¹₄(S + V + T − A + P ), with S, V, T, A, P which are the interaction of scalar, vector, tensor, axial vector, and pseudoscalar, respectively. In other words, the

¯

qi(aij + bijγ5)qjq¯i(aij+ bijγ5)qj is seen as scalar interaction and can be transformed into

1

3 × (−1

4)[( ¯qi(aij + bijγ5)qjq¯i(aij + bijγ5)qj) + ( ¯q_iγ_µ(a_ij+ b_ijγ₅)q_jq¯_iγ^µ(a_ij + b_ijγ₅)q_j)

− ( ¯q_iγ_µγ₅(a_ij + b_ijγ₅)q_jq¯_iγ^µγ₅(a_ij + b_ijγ₅)q_j)

+ ( ¯q_iγ₅(a_ij + b_ijγ₅)q_jq¯_iγ₅(a_ij + b_ijγ₅)q_j)], (4.3)

where the factor 1/3 comes from the constraint of the colorless meson, and note that there are no tensor contribution in this formula. Then we put the amplitude in Eq.(4.2) with the Fierz transformation in above discussion into the scattering amplitude matrix element M12= h ¯P |M|P i. We get the matrix element for neutral

meson mixing terms from our tree level contribution as follows,

where mP and fP are the mass and decay constant of meson. Note that the term with m²_H0

1a1 is the imaginary part in M12. So it has no contribution to ∆m, and also it will cause CP violation in meson mixing. We will mention this later in K⁰− K⁰. The quantity x = ∆m/Γ = 2M₁₂/Γ is useful when we discuss the meson mixing, where ∆m is the mass difference in neutral meson, and Γ is the decay width of the meson.

Without considering t quark interaction, the non-zero off diagonal matrix ele-ments of a_ij and b_ij are shown as follows

PDG model(a)

In numerical analysis the quark masses we are using [74] m_u(1GeV) = 5MeV, m_d(1GeV) = 10MeV, m_s(1GeV) = 187MeV, m_c(m_c) = 1.30GeV, m_b(m_b) = 4.34GeV and m_t = 174GeV. The meson decay constants which we take are [27] f_K = 156MeV, f_D = 201MeV, f_Bs = 260MeV.

D⁰− D⁰ mixing

Using the PDG parametrization, for model (a) we only discuss D⁰ − D⁰ mix-ing because the mesons with t quark have not been found yet. Here we define

tan β = v1/v2. BABAR [75] and BELLE [76, 77] experimental results give x = The effective Higgs mass, which has the relation with the scalar and pseudoscalar Higgs 1/m²_Heff = 1/m²_H0

1−1/m²_a1, can be with order 100GeV if one choose tan β = 40.

K⁰− K⁰ mixing

In model(a) there is no contribution to this meson mixing, so we consider the model(b). The only nonzero off-diagonal element is a₂₁ and b₂₁ which is related to K⁰− K⁰ mixing. The contribution to the this mixing is

∆mK effective Higgs mass should be at the scale of order TeV.

From Eq.(4.4), the ratio of ImM12 to ReM12 is written as

¯¯

Using the experimental value for ² in neutral K meson mixing [79] we derive the bound

This bounds will constrain the neutron electric dipole moment from exchange of H₁⁰ and a⁰₁. We will discuss in next section.

B_s⁰− B⁰_s mixing

Here we discuss the KM parametrization with the model(b) because there is no down-type quark mixing by neutral Higgs in other models. The non-zero elements are s and b quark mixing, which is corresponding to the B_s⁰ − B⁰_s mixing. The couplings a32 and b32 are from Eq.(4.5). Then we have

∆m_Bs

If we choose v1 = v2 = v3 and using the experimental value for ∆mBs = 17.77ps⁻¹ [12], with the estimation that the new physics is allowed to give contribution about 10% of them [80, 81, 82]. Then the mass of Higgs can be about 300GeV.

The above discussion for neutral meson anti-meson mixing provides the bounds for neutral Higgs mass. For model(a) with PDG parametrization the Higgs mass with hundred GeV is allowed in D⁰ − D⁰, and for model(b) the Higgs mass can be with the order TeV from K⁰ − K⁰ mixing. In KM parametrization, experimental data for ∆m_Bs makes the Higgs mass with lower bound of 300 GeV.

4.2 Electric dipole moment of neutron

The experimental upper bound for neutron EDM we mentioned in previous is 0.29 × 10⁻²⁵e cm (CL = 90%), which is larger for comparing with the standard model prediction. In our model we will use the parameters like VEVs and Higgs mass in previous neutral meson mixing discussion as input to examine whether the neutral EDM we calculated can be close to the experimental upper bound.

At first we consider the one loop contribution to quark EDM. Note that in our model, the exchange of only one Higgs we obtained can not produce the quark EDM because all the couplings with H’s and a’s are real and pure imaginary, respectively.

So we discuss the contribution shown as Fig.4.2(a), in which the EDM contribution comes from the cross terms between H₀ⁱ and a_k. For example, with the flavor con-serving interaction, the u quark EDM is generated by exchange of a u quark in the loop. Using H₁⁰ and a₁ as the example, the contribution is shown as

d^H_u^10a1 = e(2/3) 32π²

m_um²_H0 1a1

m²_H0

1 − m²_a1(−2(v₁m_u v₁₂v₂

− s²₁₃v12mu

v₁v₂ )²)[f (m²_H1, m²_u) − f (m²_a1, m²_u)], (4.11) where f (x, y) = 2

Z ₁

0

dz z²

x(1 − z) + z²y.

This formula shows that the one loop contribution is small because it is proportional to m³_u which is small.

Fig. 4.2: The neutron EDM contribution from (a) quark EDM for q with one loop diagram.

The cross sign means the interaction between H_i⁰ and a_k (b) quark EDM at two loop diagram, and (c) gluon color EDM

We therefore consider the three dominant two-loop contribution as shown in Fig.4.2(b), the electromagnetic operator O^γ [83, 84], the color EDM O^C [83, 84], and the gluon color EDM operator O^g in Fig.4.2(c) proposed by Weinberg [85, 86], which is often called Weinberg operator. These operators are written as

O^γ = −d_q

2i¯qσ_µνγ₅F^µνq, O^C = −f_q

2ig_sqσ¯ _µνγ₅G^µνq, O^g = −1

6Cf_abcG^a_µνG^b_µαGe^c_να (4.12) The corresponding electric dipole moment contributions from Eq.(4.12) are written in the form [61, 62, 63]

d^γ_n= η_d[4 3d_d−1

3d_u]_Λ ; d^C_n = eη_f[4 9f_d+2

9f_u]_Λ; (4.13) d^g_n ≈ eM

4π ξC, (4.14)

where d^γ_nis the radiative contribution from O^γ; d^C_n is the gluon emitted contribution from O^C. dq, fq are the contribution to neutron EDM from photon and gluon radiative contribution to quark q respectively, and the subscript Λ indicates that the hadronic energy scale. Eq.(4.14) is the approximation contribution for the color EDM of gluon operator O^C_g, and M = 1.190GeV indicates the scale related to the chiral symmetry breaking. The factor C will be defined later. The ηdand ηf [87, 88]

are

where g(Λ) = 4π/6 [85] is the strong coupling constant at hadronic scale.

The quark EDM q_i, quark color EDM f_i and the factor C in gluon color EDM formula Eq.(4.14) are written as follows

d_q = eαemQq

24π³ m_qG_q; f_q = αs

64π³m_qG_q; C = 1

8πH_g, (4.17) where mq is the mass of quark and Qq is the charge of quark, and αem, αs are electromagnetic coupling constant and strong coupling constant respectively. The factor Gq and Hg are defined as

The functions f, g,and h are

f (z) = z

Summation of Eq.(4.13) and Eq.(4.14) is the totally contribution to the neutron EDM. That is

d_n= d^γ_n+ d^C_n + d^g_n. (4.20) and we only consider the flavor conserving interaction because the flavor violating contribution is suppressed by s₁₂,s₂₃,s₁₃ for PDG parametrization, or s₁,s₂,s₃ for KM parametrization.

From Eq.(3.40) to Eq.(3.54), we find that for mass mixing terms of scalar and pseudoscalar, there are m²_H0

1a1, m²_H0

1a2, m²_H0

2a1, m²_H0

3a1, and m²_H0

4a1 which are nonzero.

We will not consider the H₄⁰ − a₁ contribution because the the factor 1/v_s in Yukawa couplings suppresses the H₄⁰ and a contribution. For model(a) with PDG parametrization, we use H₃⁰, a₁ as an example. Writing down all Y_ij³¹ in following with i, j indicating quarks,

Using the input from previous D⁰−D⁰ discussion for this model with tan β = 40, v₁₂ = 240GeV, and v₃ = 10GeV. When the neutral Higgs mass is about order 100GeV, we substitute the functions f, g, h difference between input by m²_t/m²_Hl and m²_t/m²_ak by (∆f, ∆g, ∆h) = (1, 2, 0.1).

Substituting Eq.(4.21) into Eq.(4.17, 4.18, 4.13, 4.14), we obtain the relation dn(H₃⁰− a1) ≈ −3 × 10⁻²⁵ m²_H0

3a1

m²_H0

3 − m²_a1 e cm. (4.22) For the other three kinds of Higgs pair exchange

d_n(H₂⁰− a₁) ≈ −2 × 10⁻²⁶ m²_H0

So neutron electric dipole moment is dominated by the contribution of H₃⁰ − a1

exchange. At this moment λ₃₁ = m²_H0

3a1/(m²_H0

3 − m²_a1) . 0.1 is required.

In model(b) with PDG parametrization, we note that there is no up-type quarks coupling with H₁⁰ and a₁. If we take the neutral Higgs mass to be with order TeV, we choose (∆f, ∆g, ∆h) = (0.2, 0.2, 0.03), and then treat the H₂⁰− a₁ contribution as

dn≈ −1 × 10⁻²⁶ m²_H0 1a2

m²_H0

1 − m²_a2 e cm (4.24)

In KM parametrization of model(a), we consider the H₁⁰ − a₂ process. If the choice for VEVs is v₁ = v₂ = v₃, with the Higgs mass to be 100GeV, then the contribution to neutron EDM for H₁⁰ − a₂ exchange is

dn≈ 7 × 10⁻²⁶ m²_H0 1a2

m²_H0

1 − m²_a2 e cm (4.25)

For small λ₁₂ . 0.4 this contribution can saturated the upper bound of neutron EDM. Also note that from CP phenomenon in K⁰− K⁰ mixing H₁⁰− a₁ contribution is small.

For model(b), H₁⁰ − a₁ interaction are also not including the interaction with top quarks, so the this interaction will not give the dominate contribution. Taking Higgs mass 100GeV and v₁ = v₂ = v₃. The H₁⁰− a₂ gives

d_n ≈ 1 × 10⁻²⁵ m²_H0 1a2

m²_H0

1 − m²_a2 e cm. (4.26)

If we choose Higgs mass about 300 GeV, which is the same condition as that for B_s⁰− B_s⁰ mixing discussion. The contribution to neutron EDM will be small

dn ≈ 4 × 10⁻²⁶ m²_H0 1a2

m²_H0

1 − m²_a2 e cm. (4.27)

Using λ₁₂ . 0.7 the result can be close to the upper bound.

In above discussion, we treat the two loop contribution to neutron electric dipole moment. Using PDG parametrization in model(a), with effective neutral Higgs mass about 100GeV and λ₃₁ . 0.1, the result can be close to the experimental bounds.

In model(b), the effective Higgs mass we choose is 1TeV, which is the same as that

in K⁰− K⁰ mixing. For KM parametrization, we take Higgs mass about 100GeV in model(a) with λ₁₂. 0.4 and 300GeV in model(b) with λ₁₂. 0.7 to get close results to experimental bound of neutron EDM.

The CKM matrix can not deal with problems from the baryogenesis, and also it can not deal with the question where CP violation come from. So another source for CP violation is required. With more than one Higgs doublets, these problems may be answered. CP violation can be a result of spontaneous symmetry breaking.

That is, spontaneous CP violation. With two Higgs doublets, these are three types model with two Higgs doublets, which is so-called Lee model. Type I and type II introduce the discrete symmetry, and it lead to the vanishing of spontaneous CP violating phase. Type III can make the spontaneous CP violation, and it has tree level FCNC with too many parameters arbitrary. The Weinberg model solves the problem for Lee model. It introduces three Higgs doublets which is the minimal model to have the spontaneous CP violating phase but without tree level FCNC process. However, the Weinberg model has been ruled out by the experimental data for sin 2β. This motivation makes us to study new models with the spontaneous CP violation. We summary our work in the following

• We introduce an idea that make the spontaneous CP violating phase be identi-cal to the CKM matrix phase. Two kinds of Yukawa interactions are discussed.

One is called model(a) where two Higgs doublet couple to the up-type quarks and one Higgs couples to the down-type quarks. Another one is model(b) with two Higgs doublets couple to the down-type quarks and one Higgs doublets couple to the up-type quarks.

• For model(a) using the PDG parametrization a phase is absorbed into the up-type quarks to make the CKM matrix with uniform phase δ₁₃. We obtain

δ = −δ₁₃,

and all coupling matrices are determined. Here δ13 is the phase causing spon-taneous CP violation.

The same process can be apply to KM parametrization. The phase relation is similar to that of PDG,

δ = −δ_KM.

The model(b) has the same phase relation as that of the model(a).

• We construct a model with three Higgs doublets and one Higgs singlet, with the Pessei-Quinn symmetry to make small enough neutron electric dipole mo-ment. The minimal condition of the Higgs potential makes the spontaneous CP violating phase δ be the only one phase in the Higgs potential, and the spontaneous CP violating phase is the source of CP violation.

• We extract the Goldstone boson eaten by W^± and Z⁰, also the axion by appropriate rotation, and then we derive the corresponding Yukawa couplings.

From the couplings we find that the H₄⁰ and a interaction are neglected by the factor 1/v_s. Tree level FCNC only occurs in the interaction by exchanging Higgs H₁⁰ and a₁. The coupling matrices are related to the V_CKM. When we choose an explicit parametrization for CKM matrix, all couplings can be written in terms of CKM parameters and quark masses.

• Using experimental data on meson and anti-meson mixing, the mass of effective neutral Higgs with the relation 1/m²_eff = 1/m²_H0

1 − 1/m²_a1 are constrained.

• We use the result from the previous discussion of meson and anti-meson mix-ing to discuss the neutron electric dipole moment. It is well-known that the one loop contribution for quarks EDM with exchanging Higgs is small and negligible, so we calculate the two loop contribution from quark electric dipole moment, quark color electric dipole moment, and the gluon color electric dipole moment. The result is shown that the EDM could be close to the present upper bound for neutron electric dipole moment.

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在文檔中 自發性CP破缺相角與CKM矩陣相角比較之研究 (頁 47-0)