Kobe KaiseiCollege
KobeKaisei CollegeNew ansatz forneutrino

massmatrix

withUSY

KatsuichiHiguchia Masato Senamib
and KatsajiYamamotoc,

eDqpartmentofLiterature, Kbbe KkiiseiCotlage,Kbbe 657-080i eJopan
tllnstitute Cosmic Ray Research, Uhiversdy 7bkryo,Chiba 277-8582; eJopan
forof
9Department
ofAIitctearEngineering,K)rotoU)ziversity.
Iijroto606-8501,cJopan

Abstract
We propose anextensionof the Standard Model

new

Universal Strength of
Yukawa couplings (USY) forlepton sector. It describesthat the left-handed Majorana neutrino
mass
exists, the Dirac mass
for neutrino
is thesameas that for charged
lepton sector, and the strengthof vector-likechargedleptoncouplings with singlet scalar is the same as that ofvectorlike down-type quark. Under this ansatz
the masshierarchy between
ansatzonandneutrino
matrixthe structure

canchargedof Yukawa(SM) containing

couplings

withleptoncan be explained. Then the neutrino

mass

be reproduced.

1. Introduction

At present, phenomena of particle physics are almost understoodwithinthe SM[1].However there are many problems inthe SM model, particularlythe determinationof Yukawa couplings
is one of the most salient
In order to solve this problem, many suggestions
wereproposed inliteratures.
The USY hypothesis was proposed as an interestingsolution
in 1990[2].
Studiesof the USY hypothesiswere started from the formulaThetion of quark sector[2-9]. Then that of leptonsector was dohe[10-14].
extended
USY SM with extra fermionshas been also investigated[5,10ones.

12].
In this article we propose a new ansatz forIepton sector in USY model
vector-likefermions, charged
lepton( E,EC ),down-type
with some-117 -NII-Electronic Library ServiceKobe KaiseiCollege
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This article is organized
In Sec 2, the USY models fbr theas fo11ows.
lepton sector is reviewed.
In Sec 3, the model with new ansatz is presented. Sec.4is devoted to the summaryand discussion.
2. Reviewin lepton sector.

ofUSY

In literatures,the models

the USY hypothesis forthe lepton sec-with

tor have been studied[5,10-12].

These studies
It is shown
in Refs. [10-12]that the respective
to the lepton mass ratios.
plingsare connectedTABLE

1.The studies

onarein table.1.
phases of Yukawa cousummarized

the USY forthe leptonsector

DiracTrm--MajoranaExtraleton
Reference !t!S!gcmassmassoThisExplainedo
paper

FeatureThree

Possibilityof

veetor-like

Cv-Cr-C.-Cd

chargedleptons
andthree
vector-like,neutrlnos[10]

Noto0×Specialansatzforphasestructure=>sin220.,.=O,933,

sin22e.=o.olg(SIM},

andsrnallleptonm{xing

[5]

Noto×

Three

SU(4).LXSU(4).nXSU(2)LXSU(2)RXU{1)rvecter-likesymmetrybreaking.

chargedleptons
andthree

vector-likeneutrinos[11]Not×o×

Seesawmechanisminneutrinosector=>sin22e..=O.8g4,sin22e.,=1.0(LAdi4),

andlargeleptonmixingtt[121Noto×

FourthordinarySrnallphaseinYukawacouplings
neutrino-andlargeleptonmixing

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Kobe KaiseiCollege
KobeKaisei CollegeWe review the diagonalizationof a mass matrix
democraticordinary fermionmass matrix is given bywith

USY. A 3 × 3

11
M=c(2.1).

1
It isdiagonalized
asM-> F'MF

Mdi.

==coooooo

(2.2)

O03
whereF=11h-11V6

lfVii-lfE-11V61/Vi

1

21V6

lfVii.

(2.3)

Then two zero eigenvalues
andonenonzeroeigenvalue
are obtained.
Because of the two degenerate eigerrvalues, the firsttwo columns
in F canbe transformed by any 2 × 2 orthogonal
matrix,According to the experimental
data,all the fermionshave nonzero masses except for the lightest
neutrino,
has not been constrained
which
Hence
yet by any experiments.
the fiavordemocracy should be broken.In the USY scenario, this democracy is broken by the phases of the elementsof Yukawa
matrix
without
breaking the democracy of the magnitudeof Yukawacouplings.
Then, amassmatrix
in the USY scheme is given as

M=c-eiAedyeipteidaeide

eige'dueage@.

(2.4)

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Kaisei College

generated due tothe small phases A . By the use of USY mechanism,
it is explained
thatis much heavier
the third generation of fermionexcept for the neutrinos
than the firsttwo ones.
The massesof the firsttwo generations of fermions are

On the other hand,inthe neutrino
Super Kamiokandenonzero

sectoritisshownthat the three neutrinos

experiment

diffbrences.
An example

mass-squared

provides threedegenerate eigenvaluesofhave small

3 × 3 mass

matrix11l--nie3

1

1

1

which

is given in Refs.IIO-12]
as

g.ie3
1ve =gby the data of the(2.5)-SZie3

1

The ML is diagonalized
as(ZfMLq=gxdiag(e32zi-2ni-2ni+1,e3 +Le3 +1)

(2.6)

,

where

3nie3

1

1
1z.ie3
1

1

1-!ie4

q= ts･K,K=diag(-1,1,1).(2.7)

Zxie3

The specific features in this example
are cv <Kg
andthe large phases
which
break the flavordemocracy of Yukawa couplings.
Moreover, two specific cases of ansatz for the phase structureof theleptonmass matrices are proposed in Re £ [10,12].

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One[10] isas fo11ows,M)

whereaa ==e""

1

11ei"

1

11eib,

ci

3vi;i(ililillb=,large angle

ML

111eia111e'fi=cv

illil'f; I3 ia,n, and=for the atmosphericsolutioneia

This ansatz

n=(2.8)providesproblem and a small
angle
solution
forthe solar neutrino problem. However the small angle
solution
is not favared by the experimental
resultsof the Super
Kamiokande, Kamland and SNO.
The other[12] isas fbllows,Mi

whereangle

m=

ci

ei"

1

1

1eicr

1

11

neutrinoe-i<a+b),ML-g

9m

1

11ei"

1
1

1

e-i2a. This ansatzcan

a=3;Sr,b=li;X,anda=lln

forboththe atmospheric
the hierarchyfor gq

solutions

To explain
with USY isstudied[12].

ei"･

and'K,

solarThen, the neutrino
Otviv.T][.
..

mass

thelarge

explainproblems.

neutrino

6 × 6 neutrinoa(2.9)

mass

matrix

MLdw6

is given
asmatrix}][Ui](2.10)

where1MD:Z

111-,

M,

=P11.

(2.11)

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ML6x6can be diagonalizedas,[moofO(2.12)

M .']whereMofr "Hence, the existenceof111

11Z2'M',

MR

pt

=u1

largerighthanded neutrinopt]) A generates one small but nonzero
On the other hand, an alternative

Majorana mass

fbrthe ordinary

mass

mechanism

without

generates three extremely
es is proposed in Re £ [11].
It requires the existencemechanism

seesaw(2.13)

,

small

which

withneutrinos.

theresorting
neutrino

mass-of fourth generationof

near
zero andone large
Then, three quasi degenerate masses
In this model the structure of the phase is proposed as
are given.

neutrino.

mass

1

ein

ia(1+c)e1

1

IL4 K

11

111eiaia{1+fi)

11e1(2.14).

1If.thethree phases are
three neutrinos
realized

whethercanvery

small,

be explained.

the neutrinothen the very

It is notedmass

smallmasses

that this mechanismof thecanbeisgiven
by Dirac or Majorana massterm.

3.The model
3-1 Ansatz
A new vector-likefermion model with USY is presented in this section.
E6 inspiredvector-like charged
leptons(E,EC) , right-handed neutrino
and vector-like
down-typequarks (D,DC ) are ineludedin the model. The-122 -NII-Electronic Library Service

Kobe KaiseiCollege
KobeKaisei Collegequantum
shownSq(3)xSU,(2)xU(1)of these particles undernumbersareasE-(1,1;-1),EC -(1,1;1), v, -･ (1,1;O),D-(3,1;-g),
DC ･- (3,1;}). (3.1)

The reason why the top quark mass ismuch largerthan the bottom quark
one can be explainedby the existenceof heavy D andDe [9],On theother
hand, in the leptonsector the problem of mass hierarchy between
the neutrinos
andcharged
leptonsis more difificult
to explain. Quasi
degenerate neutrino
massesare generated by Majorana mass
All the democratic fermion mass matrices
areshownas1111
1111
rlll

111
Ml

Md =cd=gtemi[12].

kkkkxxx111xxx111ML=g(3.2)1111

xxx111111yyy111yyy11M} =cl,

1
1

11

1
1k' k' kt kt

111yyywherem.

=m.

=md=m,=m..

=m..

=m.

=mp

=O.Then,weassumethefo11owingconditions as ansatz.
1.The left-handedMajorana neutrino mass terms are present. (Forexample, these Majorana mass terms are generated by the Higgs triplet[14]).
2. The strength
3. The strength

g is the sameof k is the samethat of Ci .
as that of k'.

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3-2 The neutrino mass matrix
massmatrix.
We consider the model
Here, we focuson the neutrino
Majorana and Dirac mass
with both the leffhanded and right-handed
terms as the most general neutrinomass matrix. It isgiven
byX,Ytvzv{]q[Y(3.3), ,][:i]The model with x = O that is,the ordinary seesaw mass matrix was studwith
xtO
in
ied already[12].
In the present study we consider the modelgeneral.The Ml, ismade block-diagonalas,ML'-g[X6Y(3.4)

y9y]･

where,

y=(x+y)+ (x-y)2+4

Then the submatrices

2of,'MLarex=(x+y)- (x-y)2+4.(3.5)

2diagonalizedto obtainooooooooooooML ..

oo3x'ooo"goooooo
oooooo(3.6)'ooooo3yt
Here fbur neutrinos

remainmassless.

Aecording to the experimental

data

the Z decay at LEP, there are three lightneutrinos which couples tothe Z boson. If x<:1andy)>1,[MLding],, can be small and the threeneutrinos
eoupling
to the Z boson are lightenoughto satisfy the experimentalbound. The realistic neutrino masses
shouldbe generated in theof-124 -NII-Electronic Library ServiceKobe KaiseiCollege
KobeKaisei Collegeof available
USY phases will be enoughpresent USY scheme. The number
to reproduce
the realistic mass matrix. This issuewill be investigatedin afuturework
3-3 The mass charged leptonmatrix
The charged lepton mass matrix

isgiven by1111

1111

M} =Ct

(3.7)

1111

k' k' kt ktThis is diagonalizedas fo11ows,

jFrL
2L4},Fl.
cfiag (e,O,O,2c,mb)(3.8)=where11Vli-1lVii

E}oo-vh-11V6

21Vg

o-

k'1 3(kt2+3) k'l 3(kt2+3) kt/ 3(kt2+3) 31 3(k'2+3)lll,

1/V?T\511Vma5

11nt

11Vl!

-11V6

-11be

1!2o2iV6-11bell2-11j-lfJIi-11be112

31Vii

1/2k･IV?i-;i

(3.9)

=o
The firstthree column

oorthogonal

vectorsof ,F]LandER may

be changed

taransformation because the firstthree eigenvaluesby any 3 × 3
aredegen-

erate.

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4. Discussion and SummaryThere are a sufficient numberof degrees of freedom in the USY mechanism,
Eight USY phases in a 3 × 3 mass matrix, fifteenin a 4 × 4 mass's,matrix,
and eight scale factorsc
k,k',xand y exist, Moreover orthogonal transformations may
be made fbr the diagonalizationmatricesFof paramegenerating the large lepton mixings. To decrease the number
ters in USY models,
someconditions
are imposed as ansatz.
In the present model,
we have proposed the threecondition6
as fo11ows:
massterms are present.
1. The left-handedMajorana neutrino2. The strength of C" is the same as that of q.
3. The strength of k is the same as that of k'.
We hope that they are lead by a hidden symmetry.
In the future work, we should
specify
the rangeof the phases of
Yukawa couplings and that of k' ,x,andthe expery so as to reproduce
imental leptonmasses and mixings.
's,Acknowledgement
We would liketo thank Kobe Kaisei college for providing someence books.

-refer-

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