Kobe Kaisei College

Scalar mediated flavor changing neutral currents
in a THDM within USY hypothesis
Katsuichi Higuchia and Katsuji Yamamotob
aDepartment of Literature, Kobe Kaisei College, Kobe 657-0805, JapanbDepartment of Nuclear Engineering, Kyoto University, Kyoto 606-8501, Japan

Abstract
Scalar mediated flavor changing neutral currents (FCNC’s) areinvestigated in a two Higgs doublet model (THDM) within universalityof strength for Yukawa couplings (USY) hypothesis. They are inversely
proportional to the vacuum expectation values (vev’s) of the Higgs
doublets and proportional to the magnitude of the Yukawa couplings, and related to the Cabbibo-Kobayashi-Maskawa matrix. Inorder to satisfy the experimental constraints of the neutral meson
systems, some tuning is required for the model parameters to reduce
sufficiently the contributions of the scalar mediated FCNC’s. We
discuss that the scalar mediated FCNC’s may provide significant
effects in themixing as New Physics (NP).

1．Introduction
We have been studied a THDM within USY hypothesis[1,2]. In thismodel, three ansatz are imposed as follows.
・ Both Higgs doublets couple to the u-type quarks and the d-typequarks.
・ The moduli of the Yukawa couplings are the same for the up-typequarks and the down-type quarks.
・ The vev’s of the two Higgs doublets take almost the same absolutevalues.
These ansatz provide some favorable features:
・ The hierarchical relations for the quark masses,,− 81 −NII-Electronic Library Service

Kobe Kaisei Collegeand, are naturally explained.

・ The phase invariant CP violating parameter

isobtained sufficiently.
・ Scalar mediated flavor changing neutral currents (FCNC’s) arepresent at the tree level.
In this study, the scalar mediated FCNC’s for the d-type quarks
are investigated. They are described with several parameters.
Specifically, in order to satisfy the experimental constraints fromneutral meson systems, some tuning is required among the vev’s
and, the moduli of the Yukawa couplings

and the associatedphases of the Yukawa couplings.
This article is organized as follows. In Sec 2, the model and thescalar mediated FCNC’s are described. In Sec 3, the experimental
constraints are considered, and the possibility of NP is discussed. Sec
4 is devoted to summary.
2．Model and scalar mediated FCNC’s
2-1. The model
Three specific ansatz are imposed in the THDM within USYhypothesis.
・ Both Higgs doublets couple to the u-type quarks and the d-typequarks.
・ The moduli of the Yukawa couplings are the same for up-type
quarks and the down-type quarks.
・ The vev’s of the two Higgs doublets take almost the same absolutevalues.
Then the Yukawa couplings are given as(2.1)− 82 −NII-Electronic Library ServiceKobe Kaisei Collegewhere. The neutral Yukawa couplings are specifically given as(2.2)
The two Higgs doublets are given as(2.3)
After the spontaneous symmetry breaking, they are described as(2.4)
whereThe quark mass matrices are provided as(2.5)
Because of the hierarchical relations for the quark masses,
and ,we assume that

andare almost in a

democratic form as

(2.6)

(2.7)

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The center of perturbative calculation for the d-type quark masses isparticularly given with the angles

andas shown in Fig.1.

(2.8)

Fig. 1. The center of perturbative calculation.

Then the USY phases are determined as [2](2.9)

2-2. Scalar mediated FCNC’s
We concentrate on the Yukawa couplings of the d-type quarks in thefollowing. The mass matrix,, is diagonalized as− 84 −NII-Electronic Library Service

Kobe Kaisei College(2.10)
whereis unitary matrix as(2.11)

is another unitary matrix to diagonalize, andis an orthogonal

matrix composed of the three eigenvectors of the democratic matrix [2].
The Yukawa couplings of the d-type quarks with the scalar

are givenas(2.12)
The first term is diagonalized together with, without generating

FCNC’s. On the other hand, the second term generally provide scalar
mediated FCNC’s at the tree level. It is calculated as(2.13)− 85 −NII-Electronic Library Service

Kobe Kaisei College

wherein [2], andis a 3×3 Hermitian matrix, which is explicitly presented
are the elements ofas
(2.14)The Yukawa couplings with the scalar

are also calculated as(2.15)
It turns out that the scalar mediated FCNC’s are inversely
proportional toorand proportional torelated to the elements of. Moreover they areas(2.16)
where it follows from the experimental data onthat(2.17)

Then the generation hierarchy of the FCNC’s is automaticallyexplained. The scalar mediated FCNC’s between the first and− 86 −NII-Electronic Library Service

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second generations are strongly suppressed. On the other hand,those between the second and third generations are not suppressed
stringently. Hence New Physics may be discovered in theneutralmeson experiments.
3. Experimental constraints and possibility of NP
3-1. Experimental constraints
We discuss the experimental constraints on the scalar mediated
FCNC’s [3].
3-1-1.
We first consider the experimental constraints on themixing.
There is no FCNC at the tree level in the standard model (SM). Then theFCNC is dominantly induced by a box diagram with the intermediatecharged W boson (Fig.2). It is suppressed by the GIM mechanism.

Fig. 2. W boson mediated box diagram in themixing.

In this model there are also the neutral scalar mediated FCNC’s at thetree level (Fig.3).

Fig. 3. Scalar mediated tree diagram in themixing.

The state of mixture between
andis expressed as
Its time development is governed by the Schrodinger equation,.

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Kobe Kaisei College(3.1)
whereis the effective Hamiltonian given as(3.2)

In SM themixing is dominantly caused in the box diagram

mediated by the charged W boson. In this model, it may also be causedin the tree diagram mediated by the neutral scalars. The effective
Hamiltonian provided by the scalar mediated tree diagrams arewritten with the four-component Dirac fields as(3.3)
where(3.4)
Then, thetransition matrix element is calculated with theeffective Hamiltonian as(3.5)
where(3.6)
The mass differenceis given by the transition amplitude as(3.7)− 88 −NII-Electronic Library Service

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By using the vacuum insertion approximation, the neutral Higgs
scalar contribution to the neutralmeson mass difference is given as(3.8)
The CP asymmetry parameteris also given as(3.9)
The experimental values of these quantities [4] are given as(3.10)(3.11)
By using the coupling matricesin (2.13) and (2.15), thetransition amplitude is estimated perturbatively as(3.12)(3.13)

whereis a CP violating parameter from the neutral scalar mediated

FCNC’s. The contributions of the neutral scalar diagrams should notexceed the experimental values as(3.14)(3.15)
From the experimental values of,,, constraints for theparameters are placed as− 89 −NII-Electronic Library Service

Kobe Kaisei College(3.16)
Then, the scalar effect foris estimated with (3.12) as(3.17)

where possible cancellation between the contributions of the two scalar
diagrams is not considered. Since this estimate apparently exceeds
the experimental value (3.10), some tuning is required for the scalar
mediated FCNC’s. Furthermore, in order to reproduce the small CP
violation parameter as given in (3.11),

are assumed to be almost

Hermitian.
3-1-2.
We next consider the experimental constraint on theThe experimental value of the mass difference [4] is given asmixing.

(3.18)
Similarly to themixing, the transition amplitude generated
by a neutral scalar mediated tree diagram (Fig.4) is perturbatively
calculated as(3.19)
Then the scalar effect foris estimated naively as(3.20)

Since this exceeds slightly the experimental value (3.18), some tuning
is required for the scalar mediated FCNC’s.

Fig. 4. Scalar mediated tree diagram in themixing.

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3-1-3.
We further consider the experimental constraint on themixing. The experimental value of the mass difference [4] is given as(3.21)The transition amplitude generated by a neutral scalar mediated tree
diagram (Fig.5) is perturbatively calculated as(3.22)
Then the scalar effect foris estimated as(3.23)

Since this exceeds the experimental value (3.18), some tuning isrequired for the scalar mediated FCNC’s.

Fig. 5. Scalar mediated tree diagram in themixing.

3-1-4.
We here discus the direct CP violation parameter
decays. In order to calculate

in thewe introduce aparameter(3.24)In SM the decays are dominated by a penguin diagram mediated by
the charged W boson (Fig.6). In this model there is also a contribution− 91 −NII-Electronic Library ServiceKobe Kaisei College

by a penguin diagram mediated by the neutral scalar (Fig.7).

Fig. 6. W boson mediated penguin diagram in thedecays.

Fig. 7. Scalar mediated penguin diagram in thedecays.

Since the d-type quark masses are much smaller than that of theW boson, the FCNC’s by the neutral scalars are suppressed. TheW penguin mainly contributes to the real component of the decay
amplitude. On the other hand, the neutral scalar penguin contribution
can be large for the imaginary component, which is calculated as(3.25)

with(3.26)− 92 −NII-Electronic Library Service

Kobe Kaisei CollegeFor

GeV, we obtain. By using (2.13) forthe scalar mediated FCNC’sHermitian, we estimate, which are assumed to be almostas(3.27)The direct CP violation parameteris related toas
(3.28)

Then, the effect of scalar mediated diagram is given as(3.29)
while experimentally(3.30)
Hence there is a relatively large parameter space to reproduce thisexperimental value.
3-2. Possibility of New Physics
We now discuss the possibility of NP. In this paper we haverequired some tunings between the two scalar mediated tree diagramsfor the neutral meson mixings to reproduce the experimental dataof,,. We have also assumed that the scalar

mediated FCNC’s are almost Hermitian to reproduce the small direct
CP violation parameter. Under this realistic situation, there is agood chance for the scalar mediated FCNC’s to serve as NP especiallyin future experiments ofmeson.

4. Summary
We have investigated the scalar mediated FCNC’s in a THDM
within USY hypothesis. They are inversely proportional to thevev’s of the two Higgs doublets and proportional to the magnitude of− 93 −NII-Electronic Library ServiceKobe Kaisei Collegethe Yukawa couplings. Moreover they are related to the elements of

Cabbibo-Kobayashi-Maskawa matrix

so that the flavor hierarchyof the FCNC’s is automatically explained.
The effects of the scalar mediated FCNC’s for the neutral meson
mixings are perturbatively evaluated. The results appear to exceed
the experimental data, so that some tunings are required between thetwo diagrams mediated byand . Given these tunings reasonably,
there is a good chance for the scalar mediated FCNC’s to serve as NP
especially in future experiments ofmeson.

Acknowledgement
We would like to thank Dr. Senami at Kyoto University for usefulsuggestion and Kobe Kaisei College for providing some businesssupplies.

References
[1]
[2]
[3]

[4]

K. Higuchi and K. Yamamoto, Kobe Kaisei Rev. 46, 119 (2008).
K. Higuchi and K. Yamamoto, Kobe Kaisei Rev. 47, 73 (2009).
T. Matsushita, Master thesis at Department of Nuclear Engineering, Kyoto University(1996).
K. Higuchi, Master thesis at Department of Nuclear Engineering, Kyoto University(1998).
See http://pdg.lbl.gov/ (Particle Data Group);
HFAG, hep-ex 0704.3535 (2007).

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