Kobe Kaisei CollegeCalculability of quark masses andwithin USY hypothesis

in THDM

Katsuichi Higuchia and Katsuji Yamamotob
aDepartment of Literature, Kobe Kaisei College, Kobe 657-0805, JapanbDepartment of Nuclear Engineering, Kyoto University, Kyoto 606-8501, JapanAbstract
We investigate the calculability of the quark masses and CabbiboKobayashi-Maskawa matrix () in a Two Higgs Doublet Model(THDM) within the Universality of Strength for Yukawa couplings(USY) hypothesis. It is assumed in this model that all the Yukawa
couplings have the same moduli. The two Higgs doublets are also
supposed to develop vacuum expectation values (vev’s) with almost
the same moduli.

In favor of this USY mechanism, the quark

mass hierarchical relations not only
also,

but

are naturally explained, as confirmed by a numericalcalculation.
1．Introduction
In a previous study we investigated a THDM restricted by the USY
hypothesis [1]. Both Higgs doublets couple to the u-type quarks andd-type quarks, and the moduli of the Yukawa couplings are the same.
Moreover the vev's of the two Higgs doublets take the same absolutevalues. As a result, the quark mass hierarchical relations not only,

but also

are naturally explained.

Furthermore, scalar mediated FCNC's are generated at the tree level,
while Z mediated FCNC's do not in this model. We showed roughlythat the six quark masses andare reproduced.

In this study, we investigate systematically the calculability of thequark masses andin the THDM within USY hypothesis.
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Kobe Kaisei CollegeThis article is organized as follows. In Sec 2, the model ispresented, and the quark mass matrices are diagonalized. In Sec 3, theformulation to analyze the quark masses andis explained, and anumerical result is presented. Sec 4 is devoted to summary.
2．The model and quark mass matrices
In the THDM within USY hypothesis, three specific ansatz areconsidered.
・ Both Higgs doublets couple to the u-type quarks and d-type quarks.
・ The moduli of the Yukawa couplings for the up-type quarks are thesame as those for the down-type quarks.
・ The vev's of the two Higgs doublets take almost the same absolutevalues.
2-1. Higgs potential
The two Higgs doublets are given as.

(2.1)

By the spontaneous symmetry breaking, they are expressed as,

where(2.2). The general gauge-invariant renomalizable Higgs

scalar potential is given by [2]− 74 −NII-Electronic Library Service

Kobe Kaisei College(2.3)

In this study, in view of GUT we take the relations,・・,・.

In order to provide the bottom quark mass within USY hypothesis, werequire the relations
・,・,

or for simplicity
・,・.

In view of the suppression of scalar mediated FCNC’s, it is also favoredthat.
Based on these ansaz, the Higgs potential is rewritten as− 75 −NII-Electronic Library ServiceKobe Kaisei College. (2.4)

2-2. Higgs masses
The scalar mass matrices can be transformed to diagonal matrices. Then,the charged scalar fields are mixed as→ N.G.bosons,→ physical charged bosons,

(2.5)

where

are the massless charged Nambu-Goldstone boson fields

andare the physical charged boson fields. Then, the masses of thephysical charged bosons are given as.
The value of(2.6)

can be taken large enough to satisfy the experimental

bound on the charged Higgs boson masses.
In the same way, the CP-even neutral scalar boson mass matrix isdiagonalized. Then, the two physical neutral fields are given as,.

(2.7)

The 2 × 2 mass matrix of the neutral scalar fields is given as(2.8)− 76 −NII-Electronic Library Service

Kobe Kaisei CollegeThere is no mixing between

and, i.e.,

,.

()

(2.9)

The CP-odd neutral scalar boson mass matrix is diagonalized with
→ N.G.boson,. → physical neutral boson.
For(2.10), there is no mixing among the neutral bosons. Then,the masses of the physical neutral bosons are given as,,.

(2.11)

2-3.Yukawa couplings and quark masses
Yukawa couplings are given by ,,where(2.12). Then, the neutral Yukawa couplings are,.

(2.13)− 77 −NII-Electronic Library Service

Kobe Kaisei CollegeThe quark mass matrices are provided as,.
Because of the hierarchy, andthat the quark mass matrices
democratic matrix as follows:and(2.14), we assume
are nearly equal to a,(2.15),(2.16)

whereandrepreset the center of pertubative calculation,
which will be determined in Sec.3.
The 3 × 3 democratic matrix is diagonalized as,⇒

whereof(2.17)

is an orthogonal matrix composed of the three eigenvectors(). Here, it is naturally seen that the quark masses of

the third generation are much larger than those of the first and secondgenerations. The eigenvalues and eigenvectors ofTable 1.

are shown in− 78 −NII-Electronic Library Service

Kobe Kaisei College

Table1.The eigenvalues and eigenvectors ofeigenvalues.

eigenvectors

Because of the two degenerate eigenvalues,eigenvectors may be arbitrary linear combinations of, theirand.

We start with the basis of the down-type quarks as.

The eigenvectors of,,,and(2.18), are given as,(2.19)

providing the basis of the up-type quarks,,

where,
is given byand(2.20)

are orthogonal each other. The CKM matrix,(2.21)− 79 −NII-Electronic Library Service

Kobe Kaisei College

which in the democratic limit becomes as.

By taking the input value(2.22), the experimental value of thequark mixing between the first and second generations is naturally
reproduced in the democratic limit.
The quark mass matrices, which are supposed to be Hermitian, arediagonalized by unitarity matrices's as,⇒where(2.23). Becausethe hierarchy of the up-type quark masses is larger than that of thedown-type ones, we suppose that,.

(2.24)

Then, we obtain,,(2.25)

where,− 80 −NII-Electronic Library Service

Kobe Kaisei College.

(2.26)

3．Determination of the USY phases
There are four conditions in this model as
1.

[GeV] from2.

from,,
from3.

4. Eq.(2.25)

and,for the d-type quark masses.
(3.1)

The mass matrix of the down-type quarks is expanded perturbatively as(3.2)− 81 −NII-Electronic Library Service

Kobe Kaisei Collegewith,(Theand).

(3.3)

for the center of perturbation are given as (see Fig. 1)
,,,.

(3.4)

Fig1. The center of perturbation.

From Eqs. (3.1) and (3.2), the USY phases are detemined as((),).

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We present a numerical result by taking the following values forthe input parameters.
・ Phases of the Yukawa couplings

・ Vacuum expectation values,・ Moduli of the Yukawa couplings.
The output values are given as,and,(3.6). They are all consistent with the experimental values [3].

4. Summary
The THDM within USY hypothesis provides a possible solution tothe flavor problem in the SM. In this study, three ansatz are imposed
as follows.
・ Both Higgs doublets couple to the u-type quarks and d-type quarks.
・ The moduli of the Yukawa couplings for the up-type quarks are thesame as those for the down-type quarks.
・ The vev's of the two Higgs doublets take almost the same absolutevalues.
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Based on these ansatz, the USY phases are determined systematically.
It is confirmed by a numerical calculation that the experimental values
are all reproduced for the quark masses,
invariant CP violation measure, and the rephrasing. A detailed investigation of thescalar mediated FCNC's is deserved for a future task.
Acknowledgement
We would like to thank Dr. Senami in Kyoto University for usefulsuggestion and Kobe Kaisei college for providing some businesssupplies.
References
[1]
[2]
[3]

K. Higuchi and K. Yamamoto, Kobe Kaisei Rev. 46, 119 (2008).
See for instance, S. Kraml, et al., Work shop on CP studies and Non-Standard Higgs
physics, hep-ph/0608079, August (2006).
Particle Data Group, http://pdg.lbl.gov/.

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