Kobe KaiseiCollege
KobeKaisei CollegeTwo Higgsdoubletmodel

within

USY hypohtesis

KatsuichiHiguchiaand KatsuJ'iYamamotobEtz)apartment

tlDepartment

Kbbe KixiseiCollage,Kbbe 657-080a Jopan
ofLiterature,Engineering,K5,otoU)tiversity,
1fyoto606-8501,
ofNuclearcJdpanAbstract
We investigatephenomenology in a Two Higgs Doublet Model (THDM)
withinthe UniversalityStrengthforYukawa couplings(USY) hypothesis.
It is assumed
in this model that all the Yukawa couplings have the samemoduli.
The two Higgs doublets are also supposed
to developvacuum
In favor of this mechawith the same moduli.
expectation values(vev's)m.,m, `K mt, mdnism,
not only
the quark mass hierarchical relations
mb but also n;b <K m, are naturally
explained.
Furthermore, the suf, m,large phase invariantCP violationparameter J is obtained. In
ficiently
addition,
the scalar mediated
FCNC's are generated at the tree level.
(K1. Introduction
withAt present,phenomena of particlephysicsare almost understoodin the Standard Model (SM) [1].
However, there are many problems in thecannotbeSM. Particularly,
the flavorstructure of the Yukarvvacouplings
explanation
forthe quark mass
predicted, without
providinga reasonable
mb andmb <{ mt.
In order to solve thisflahierarchy,m.,m. <K mt, md,m,vor problem, manysuggestions
have been proposed in literature.The
UniversalityStrength fbrYukawa eouplings(USY)hypothesis was proandsince then it has been
posed in 1990 [2]as an interestingsolution,<Kstudiedextensively[2-12].

In the SM within USY hypothesis,while the six ordinary quark masses andthe absolute values of the Cabbibo-Kobayashi-Maskawa matrixelements
are reproduced
correctly,
the sufficiently
largephasebeinvariant CP violation
J=im(PC.P(].V].'U.')cannot
parameter

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obtained,asexperimental

singlets andshownvalue

by Branco, et al. [4].Then, in order to reproduce theof J, an extended
USY model including extra quark

Higgs singlets wasstudied[12].

It this article, a Two Higgs Doublet Model (THDM) within
hypothesis is proposed as an alternative USY model. Two specific

USYansatz

are considered.
'.

''

In view of the grand unified theory (GUT) the moduli of the Yukawa
couplings
for the up-type
as those forthe downquarks are the sametype quarks.
The vev's of the two Higgs donhlets
take the same absolute value.
Then, the followingresults wi11be obtained.
The scalar mediated
FCNC's appear
at the tree level,
while
the Z mediated ones do not.
The top-botom mass hierarcy mb `K m, is naturally
In addiexplained.
tion, the CKM quark mixing
matrix7bicitfandrephrasing
invariantCPviolation
measureJ can be reproduced.
This article is organizedasfbllows.In Sec 2, the SM within USY
hypothesis is reviewed.
In See 3, the THDM withinUSY is presented,anditsphenomenology isinvestigated.See 4 is devoted to summary.
2. Review of the SM within USY hypothesis
2-1 The democratic limit
It wi11be shown that the quark mixing between the firstandin the democratic limit.
generations in PbKA4is obtained
A 3 × 3 democratic ordinary quark mass matrix isgivenby.

Ma=wherea=uor d.

second11%v

11E-(2.1)

11Itis diagonalizedas-120 -NII-Electronic Library Service

Kobe KaiseiCollege
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A2XVooooooM.,d,.

=j,(2.2)

O 0 3of the three eigenvectors
ofisan orthogonal matrix composed
M.. Here,itisnaturally seen that the quark masses of the thirdgeneraand
secondtion are much largerthan those of the firstgenerations. The

where,FL

eigenvalues

and eigenvectorsof M. are shown

Tablel.Theeigenvalues

andof

Xl=j'j'1-1o11o2-

X2=ts'V6'V63Ava

111

X3=ts,vi'vii

Because of the two degenerateeigenvalues, m.
of xi andtorsmay be arbitrary linearcombinations
We start] with the basisofthe up-type quarks as=m.

XlX2X34-Ml,.

eigenvectors

miomt=j.

eigenvectors

eigenvalues

Mc=in tahle 1

11E
11V6

=O,theireigenvec-x2.

o-11j

11V6

-21V6

(2.3)

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Kobe KaiseiCollege
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ofMdtt,xitand,x2'xlx3

gtven asarecose+x2sine=xl,x2=xl

sine-x2cose(2.4),'X3=X3

providing the basisof the down-type quarks,
Xl4'

wherex i ,x2t

X2e + x2 sin e
xl sin O - x, cos e

X3

X3xl cos

tandx3andareorthogonaleach(2.5)'other.

The CKM

matrixis given
by%sw-4ny'which(2.6)

indemocratic limitbeeomes as
cose-sinsineoecoseoo1,%sw=o(2.7)

By taking the inputvalue

e=O.227 [rad],theexperimental
result of thequark mixing between the firstand second generations are naturally
reproducedin the democraticlimit.It is,however, clear that the quark
mixinginvolvingthe third generation cannot appear in Eq. (2.7)
of thedemocratic limit.

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2-2.Reproducingthe experimental
In the democraticlimit,either quark masses

ofthe firstandsecondgenerations or the quark mixing involvingthe third generation are notof flavor democracy, somegenerated. Then, keeping the advantage
schemeshouldbe considered
to generate them. The USY is aextended
in such an extendedflavordemocratic model.
powerful mechanism
these can be generated by the aid of adding phasWithin USY mechanism
es to the Yukawa couplings.
The quark mass matrices are provided within USY aseiwhil

Ma=eiwh12

eiwh13%v ewh21 eiwhn

jei%31(2.8)

eiwh23

eiG32

ei%33.

In 1997, itwas shown by Branco et al. that ifsuitable values are taken fbr
matrixvaluesof the CKM
all the quark masses
and the absolute{A,ij,
arecorrectlyreproduced[4].The part of itsresult is givenasm,
174GeV, md6.5MeV,m, = 165MeV, mb 5.4GeV ,m. = 1.0MeV,m, 1.35GeV,fi?bicnfl====andO.2213 O.O032
O.2210 O.9745 O.0391
O.Ol17 O.0375 O.9992.

I%..1-O.9752
Then, the rephrasingasinvariantCP

violationmeasure(2.9)

J is estimated

J=1.8 × 10-'Thisis much.

smallerItJ,xp.=3.08
× 10-5[13].
to CP violationbeyond the SM
contribution
as a problem why
the top quark is muchthan the experimental

value

is hence clear that some new
It also remains
is necessary.
heavierthan the bottom quark.

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3. THDM within USY hypothesis
3-1,The model
There remain two problems in the SM withinUSY.
1 The rephrasing
invariant CP violation
measureJ is muchsmaller
than the experimental
value.
2 It is not naturally explained why the top quark is much heavierthan
the bottom quark.
To solve these problems, we investigate
the THDM
within
USY hypothesis, by taking the fo11owingtwo ansatz.
The moduli
ofthe
Yukawa couplings are the same as c .
[[hevev's of the two Higgs doublets
have the sarne absolute value as v
The Yukawa couplings are givenas
.

.

･'.

LJi,k...Ai,ij(i)qiLt
O(i)u.m+ Ai,ij(2)q,,t¢ (2)u.il?
=A,ay(i)q,,top(i)4il,+ A,,ij(2)g,,'¢ (2)c(pt(!;tlllJl.e,rke..A,t/"(:l. Z),.
) -'' e.X,P(iipaij(X))a"dqiL=

t.C.,e.lg[lg.i`.Sal

M.ij

=

Mdij

･Then, the,)exp(iqij(i))+ exp(i6.(2)) exp(iAij(2))][exp(i6.(i))･Iiili(3.2)=[exp(ioj(i))
Cili(3.3)･･exp(i{A,ij(i))+ exp(i6,(2)) dexp(i%(2))]

here that the deviationfrom the flavordemocracy is small

It is supposed
as(3.1)+ h.c.

Lij(X)1 .
<KFor simplieity we begin to study
the democratic mass matrices,in the limit M. =md

1Micv･,/li'exp(itSL,('))

1

=M,

with

1+exp(i6h(2))

1-=M.

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Md=cv

exp(i6,(i))

E1111+ exp(i6,(2) )

(35)

1

1

1

d only when 6.(i)In thislimitthe top-bottom hierarchy mb <K m, is deriveand6d(2)isandthe difference
between 6d(i)is almost the same as q(2),nearlyn.

6.(2))
3cv

exp(i6.(i))

j3cvexp(i6,(2)),h]Mb

3cvexp(i6,('))

jFigl.Representationof mb<Km, .

to reproduce
fbund the phases whij(X)invariantCP viovalues
oftheP?JKni
and the rephrasing
themaybe represented
withJ. The CKM matrix

By perturbativecalculation
the experimental

welation measure
Wolfenstein parameterizationas

1-A2 12%Kni=A
1-A2/2-z
AZ3 (1p - iny)--AA2

AA3 (p- in)AA21(3.6)

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For simplicity it is supposed that 6.(i)= 6,{2),andv= 29
Then, the relations for the phases 6d(i)are found in terms of the{fiuij(X)'s
Wolfenstein parameters as.

,6d(2>,and,Sd(1) -6,(2)m,(1-2A)="-o Aij(2)oQ,ij(')=3v,=,,2m Zwhn<i){adii{2)O wh22(i)whn(2)3vb 4hi33(1)"==,=m,A[2-AnA2(1+A)]=={il12(1)
ipd21(2)6v===m,A[3+AnA2(1+A)]=t{El23(1)
ipti32(2):==mbZ[2+Anz2(1+A)]

ipd3t(1){zai13{2)6v,

wt2t(1)wh12(2),m,Z[1-Ana3],

ipd13{1}{hd31(2):{Zb.A･=eZ,,,`2),=MbA[1+Anl3]

6v

6v

''

nzbZ[3-AnA2(1+A)]

6v,

wh32(i)g4i23(2)
=

=6v

.

(3.7)
3-2 Scalarmediated
FCNC's[l]hisTHDM
provides FCNC's･Scalarmediated･ Gauge

FCNC'sboson Z mediatedasfbllows.

present at the tree level.
FCNC's are absent at the tree level.
areOn the perturbativecalculation, the scalar mediated
FCNC's appear to be
too largeto satisfy experimental
value
in the case that the nondiagonal
elementsof theYukawa

couplings

areof the sameorderasthe diagonal

In order to suppress the FCNC's sufficiently, it wi11be necessarythat the firstand the second terms in Eq. (3.1)
are almost simultaneously
diagonalizedby the unitary transfomiations ULL and Urk to diagonalize
ones.

,M.

the thirdand the fourth terms in Eq,
are almost
simultaneously
diagonalizedby the unitary transforma(3.1)
tions Udi , and U,m to diagonalize
the down-type quark mass
matrixMdHence, the phases
to break the democracy should be much
-6d(2) . It is a futuretask to findsmallerthan the phase difference
6d(i)the allowed
range
of {a,ij(")
for estimating
the possiblemagnitudeof thethe up-type

quark matrix, while.

.

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mediated

FCNC's.

4. Summaryis fascinatingas a possible solution
The THDM withinUSY hypothesis
to the flavor,
problem in the SM. It is shown that the quark masses, Vbicaf
measureand rephrasing
invariantCP violation
J can al1be reproduced.
while
theMoreover, the scalar mediated
FCNC's appear at the tree-level,'ssuitablevaluesof Aij{X)are not
Z boson mediatedones
do not. So fair,
FCNC's satfound yet so as to suppresses
sufficiently the scalar mediatedisfyingthe experimental
data. This will be deservedfora futureinvestigation.

Acknowledgement
We would liketo than k Kobe Kaisei college
ence books.

for providing somerefer-References[1]

S.L.Glashow,Nucl.Phys.22(1961)579;

S.Weinberg,Phys.Rev.19(1967)1264;
ParticlePhysics,ed. By N.Svartholm
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W.Marciano and[2]
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H.Pagels,Phys.Rep.36(1978)137.

G.C.BrancoandJ.I,Silva-Mareos,Phys.Lett,B237(1990)446,
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707.
I.S.Sogami,H.Tanaka and T.Shinohara, Prog.TheerPhys.101 (1999)
T,Shinohara,H.TanakaandI.S.Sogami,Prog,Theor.Phys.100(1998)615.

J.I.Silva-Marcos,Phys.Lett.B443(1998)276.
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[11] J.I.Silva-Marcos,Phys.Rev.D62(2000>073004.
[12] KHiguchi,M.SellamiandK,YamamotoKobeKaisei.Rev,41,139(2oo3);
K. Higuchi,M. Senami and K Yamamoto, Phys. Lett.B638, 492 (2006).
[13] Seehttp:Z/pdg.lbLgev/(ParticleDataGroup).

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