Texture of Yukawa coupling matrices in general two-Higgs doublet model

Yu-Feng ZhouJ. Phys. G: Nucl. Part. Phys.30 (2004) 783-792

Presented by Ardy

Introduction

• Unresolved problems within the SM: the origin of the fermion masses, mixing angles and CP violation, etc.

• Widely believed that the SM cannot be a fundamental theory of the basic interaction.

• Two-Higgs doublet model (2HDM) is the simplest extension of the SM.

The general 2HDM

• The Lagrangian for the Yukawa interaction is given by

• The two Higgs after spontaneous symmetry breaking have the following form:

where and are the absolute values of the VEV of the two Higgs fields, which satisfy

and define

• The relation to the Higgs mechanism in the SM is manifest if we apply a basis transformation of

• In this new basis, the two Higgs fields are

• We will focus on the case in which both VEVs are real, i.e. , and the effects of neutral scalar mixing are negligible.

• In this approximation, the fermion mass and the Yukawa coupling matrices are simply given by

• The main problem in the models with multi-Higgs doublet is that in general and cannot be diagonalized simultaneously, causing FCNC at tree level

• To forbid the tree level FCNC in 2HDM, the ad hoc discrete symmetries are often imposed on the Lagrangian

which defines two types of 2HDM without tree level FCNCs, referred to as models I and II of 2HDM.• However, since the most strict bound of tree

level FCNCs only comes from the light quark sector, there is a possibility that the tree level FCNC do occur, but are greatly suppressed.

• Abandoning the discrete symmetry (as type I and II), one arrives at the general type of 2HDM with small FCNCs.

• The small off-diagonal Yukawa matrix elements can be attributed to an approximate flavor symmetry which is slightly broken down.

• Another way to prevent large FCNCs in the general 2HDM is to adopt reasonable ansatz on the textures of the Yukawa interaction matrices .

• Without loss of generality, the mass matrices can be rotated to be Hermitian by suitable redefinition of the fermion fields in the flavor basis.

• In the case of both VEVs being real and arbitrary, and could also be rotated to be Hermitian.

2HDM with parallel texture

• Then, and can be diagonalized simultaneously if and only if .

• From Eq.(5), the commutator of and is given by

• Obviously, if have a parallel structure

the commutator will be close to zero.

• The simplest way to obtain the parallel texture is to impose an permutation symmetry between and

which leads to , and .• To obtain small off-diagonal elements such a

symmetry must be slightly broken down.• The breaking down of the exact parallel

texture can be parametrized as follows:

• In this parametrization, the commutator is given by

• The Hermitian mass matrix is diagonalized by a rotation matrix such that

• Applying the same transformation to , one arrives at the Yukawa matrix in the mass basis

Parallel textures with zero texture

• There have been extensive studies on the possible textures of mass matrices in the SM.

• The simplest among them are textures with texture-zeros.

• It is well known that a simple zero in the element of quark mass matrices leads to a correct prediction of the Cabibbo angle in the two-family case.

• In the three-family case, a similar texture with six texture-zeros is widely discussed and which is often referred to as ‘Fritzsch Matrix’, given by

where , and .• Motivated by this texture in the SM, an ansatz

of the Yukawa matrix was proposed by Cheng-and Sher, which suggest a similar structure for both Yukawa matrices in the flavor basis

• The free parameters ,, and are assumed to be of the same order of magnitude, i.e. of order 1.

• In order to reproduce the quark mass matrix they must satisfy the relation

• After being rotated into the mass basis, the Yukawa matrix takes the form

with being the functions of , and , and have the following values

• If there are no accidental cancelations, all are roughly of the same order of magnitude.

• To clearly see the hierarchy in the Yukawa matrix, it is convenient to introduce a common factor , with

• Thus the Yukawa coupling matrix can be approximately written as

• The FCNCs related to the light fermions are greatly suppressed by small fermion masses.

• For example, for transition, the Yukawa coupling is of the order of , which successfully explain the almost invisible FCNC in neutral Kaon system.

• However, later studies have already shown that the base of this ansatz is in severe problem in accounting for the current data of the CKM matrix elements from the known quark masses.

• The texture of Eq.(14) leads to the following predictions of the CKM matrix elements:

where and are two phase parameter.

• For a large value GeV, the six texture-zero-based mass matrix gives , which is too large compared with the current data .

Four texture-zero textures

• To accommodate all the current data of quark masses and mixing angles in the framework of texture-zeros, a natural choice is to give up the texture-zero in element of the mass matrices, the so-called four texture-zero textures.

• Currently, the most extensively studied and phenomenologically successful texture with four zeros is the one in which the zeros are located in and positions.

• This texture is given by

with .• In this four texture-zero texture the predicted

value of is , in good agreement with the experiment.

• Furthermore, in this texture there exist nontrivial complex phases which can directly result in .

• Using the four texture-zero texture, Eq.(15) is rewritten as

• One finds that the corresponding Yukawa coupling matrix in the mass basis is given by

with

• Similarly, all the couplings are roughly of the same order of magnitude, and can be approximately written as

• The Yukawa coupling matrix is then simplified to be

• Note that the complete Yukawa coupling matrix contains a global factor which has to be constrained or determined from the experiment.

• In the quark sector the most strict constraint comes from the light quark, especially mixing.

• The resultant bounds on the Yukawa coupling for down quarks and are the same.

• In the lepton sector, the strongest constraint comes from the radiative decay which is relevant to the and elements of the Yukawa coupling matrix .

• The branching ratio of at one-loop level is given by

• Taking the current upper bound of , one finds that

• The ratio obtained here can be directly used to make predictions.

• For a concrete illustration, we take the three-body lepton decays as examples, and calculate the branching ratios in the general 2HDM with two different ansatz in Eq.(26) and (19).

• For a concrete illustration, we take the three-body lepton decays as examples, and calculate the branching ratios in the general 2HDM with two different ansatz in Eq.(26) and (19).

• The predicted branching ratios for the various decay modes are given by

• Using the allowed range of and from Eq.(28), the predicted branching ratios are obtained and summarized in table 1

Summary

• We have discussed the texture of the Yukawa coupling matrices in the general 2HDM.

• In the framework of the parallel texture, suppression of the FCNC can be achieved.

• We have proposed a four texture-zero-based ansatz on the Yukawa coupling matrices which has several advantages over the six zero-texture-based ansatz proposed by Cheng and Sher.

• It is found that this new ansatz is more predictive in the lepton sector.

• As an example, the contribution from the general 2HDM to the lepton number violation decay modes are calculated in both ansatz.

• The result show that after considering the constraints from , the four texture-zero based ansatz can predict a decay rate two orders of magnitude higher than the six texture-zero based ansatz, and the branching ratios of and can reach and respectively.