Flavor structure in string models
Tatsuo Kobayashi
１．Introduction
2. Yukawa in heterotic orbifold models
3．Discrete flavor symmetry
4．Yukawa in intersecting D-brane models
5. Summary
based onKo, T.K. and Park, hep-ph/0406041, hep-ph/0503029
T.K., Raby and Zhang, hep-ph/0409098,
Higaki, Kitazawa, T.K. and Takahashi, hep-th/0504019

１．Introduction
The origin of fermion masses and mixing angles is
one of important issues in particle physics.
Fermion masses
Yukawa couplings between
fermions and Higgs fields
e.g. in the Standard Model
O(1) of Yukawa couplings are in a sense natural.
From this viewpoint, how to derive suppressed
Yukawa couplings is a key-point in understanding
fermion masses.

Quark masses and mixing angles
Mt

174

GeV,

Mb

4.3

GeV

Mc

1.2

GeV,

Ms

117

MeV

Mu

3

MeV,

Md

6.8

Vus

0.22,

Vcb

0.04,

Vub

MeV
0.004

RG effects are not drastic in usual cases except
PR-fixed points or superconformal fixed points.

Lepton masses and mixing angles
Me

0.5

MeV,

M

1.8

GeV,

2
M21
8 10 5sin2 12 0.3,

M

eV2 ,
sin2 23 0.5,

106

MeV

M312 2 10 3

eV2

sin2 13 0.00,

RG effects are not drastic in usual cases except
PR-fixed points or superconformal fixed points.

Superstring theory
Superstring theory :promising candidate for unified theory
including gravity
It is important to study flavor structure andYukawa structure from string theories.
Superstring theory

4D space-time+ 6D compact space

String models (String phenomenology)
Superstring ⇒ 10D space-time= 4D space-time+ 6D compact space

Origin of gauge groups SU(3), SU(2), U(1) ?
Origin of flavor structure
6/2 = 3 ?
Origin of parameters, g, Y, ...
?

6D compact space

Several types of string models have been constructed.
(Several 6D compact spaces consistent withsuperstring theory)
Heterotic models on Calabi-Yau manifold, Orbifolds,
fermionic construction,
Gepner, . . . . . . . . . . .
Intersecting D-brane models(magnetized D-branes)
These models lead to different flavor structures.

Flavor structure from 6D compact space
6D compact space consistent with string theory
How to count the family number is well-known.
Flavor (Yukawa) structure
selection rule for allowed (Yukawa) couplings
calculations of their magnitude
flavor symmetry
Classify possible patterns of Yukawa matrices forquarks and leptons
Heterotic orbifold models and Intersecting D-branemodels have similarlities.

Localized modes(in extra dimensional field theory)
Couplings among bulk modes→ O(1) Yukawa couplings( N=2,4 sector
g=Y)
Couplings among (quasi-) localized modes→ suppressed Yukawa couplings
String models with localized modes
Heterotic orbifold models
Intersecting D-brane models....

Localized modes
Heterotic orbifold models andIntersecting (magnetized) D-brane models
are interesting to derive suppressed Yukawa couplings,
because they have localized modes.
We would like to study somehow systematically.
e.g., intersecting D-brane configuration
Y = exp (-A) A: areau(1) su(2)
su(3)
H
Q
A
u

Flavor structure in string models
Ko, K.T., Park, ’04, ’05
Systematical analyses on Yukawa matrices inheterotic orbifold models
T.K., Raby, Zhang, ’04, (in preparation)
Nonabelian discrete flavor symmetry
controlling higher dimensional operators（Orbifold）
Higaki, Kitazawa, T.K.、Takahashi, ’05
Generic study on Yukawa structure inIntersecting D-brane models

2. Yukawa in hetero. orbifolds
The number of 6D orbifolds is finite,
Z3,Z4,Z6,……
(crystallographic) space group
For an orbifold all of fixed points, where
twisted string is localized, are known,
and its number is finite.
We know selection rules for allowed Yukawa
couplings, and their magnitudes.
In principle, systematic study is possible.

Orbifolds

Fixed points on 2D Z3 orbifold(0,0), (2/3,1/3), (1/3,2/3) in su(3) root basis
Twisted strings are associated with these fixed points.
The flavor structure and the selection rule aredetermined by geometrical aspect when we fix
orbifold.

2D Z6 orbifold
First twisted states T1
a single fixed point (0,0)
Second twisted states T2(0,0), (2/3,1/3), (1/3,2/3)

third ones T3(0,0), (1/2,0),(1/2,1/2)(0,1/2)

6D orbifold
6D Z3 orbifold = a product of 3 (2D Z3)
27 fixed points
6D Z6-I orbifold =a product of 2 (2D Z6) and (2D Z3)
T1 3 twisted states
T2 27 twisted states
T3 16 twisted states

Coupling selection rule
Yukawa couplings are allowed when three twisted
strings combine into a closed string, which shrink.
We have to consider equivalenceon the torus/orbifold.
selection rule← geometrical aspectsof compact space⇒ in general complicatedSpace group selection rule
X(
)
X(
0) e
( , e )( ' , e ' )( " , e" ) (1, 0 )Conjugacy class

X
(X
) e
( , e) ( , e (1 ) )In some case, only diagonal couplings areallowed, and in other case off-diagonal
couplings are also allowed.

Space group selection rule
2D Z3

Only T1 T1 T1 couplings are allowed.

fk
k=0,1,2i + j + k = 0 mod 3
diagonal
We can not get non-vanishing mixing angles.
This corresponds to Z3 symmetry, and its charge
is equal to k.
2D Z6 orbifold
Only T1 T2 T3 and T2 T2 T2 couplings are allowed.
Off-diagonal couplings are allowed for T1T2T3.
T2 T2 T2 couplings are diagonal.
For example, T1T1T1 couplings are not allowed.

Yukawa couplings on orbifold
CFT calculations

Dixon, et al ’87, Hamidi, Vafa, ’87,

Burwick, et al ’91, Erler, et al ’93, Casas, et al ’93……

……………. T.K., Lebedev ‘03Tˆ1 Tˆ 2 Tˆ 3

couplings

Y

3 /( 4exp[

Mf 3 )]( R1 ) 2

3( R1 ) 2 / 2

0

03( R1 ) 2 / 2
0

3( R1 ) 2000

(R2 )23(R 2 ) 2 / 2

003(R 2 ) 2 / 2

3(R 2 ) 2

Tˆ 2 Tˆ 2 Tˆ 2

couplings

Y

3 /( 16

exp[f3 )T M ( f2)( f 2)( f 2f3 )T M ( f2f 3 )]

Systematical study on Z6-I: Quark sector
Ko, T.K., Park ‘04
We study systematically all of possible assignmentsof 2nd and 3rd families to fixed points,
assume one pair of up and down Higgs fields,
examine allowed entries by the selection rule,
try to fit the mass ratios mc/mt, ms/mb andmixing angle Vcb by varying R1 and R2.

[mc / mt ]exp
[Vcb ]exp

0.0038,0.041[ms / mb ]exp

0.025

Assignment 1
Q 2 , Q 3,

u2,u3

d2,d3

H u,H dTˆ2( 2 ) , Tˆ2( 4 )Tˆ3( 3 ) , Tˆ3( 2 )Tˆ3( 1 ) , Tˆ3( 3 )Tˆ1 , Tˆ1

T1

T2

Yu

27 . 8 ,0 . 0416
0 . 0557

mc / mt

107

0 . 718
0 . 8480 . 0038 ,Yd

0 . 0313
0 . 0370

0 . 0416
0 . 0557

ms / mb

0 . 029 ,V cbThere are several examples leading to similar
results.

0 . 041

Assignment 2
Q 2 , Q 3,

u2 ,u3d2,d3

Hu,H dTˆ3( 2 ) , Tˆ3( 4 )Tˆ2( 3 ) , Tˆ2( 2 )Tˆ2(1 ) , Tˆ2( 3 )Tˆ1 , Tˆ1

T1

T2

Yu

24 . 0 ,0 . 0281
0 . 0371

mc / mt

150

0 . 439
0 . 665

0 . 0038 ,Yd

0 . 01990 . 0302

0 . 0281
0 . 0371

ms / mb

0 . 032 ,V cbThere are several examples leading to similar
results.

0 . 041

Assignment 5
Q 2 , Q 3,

u2,u3

d2,d3

Tˆ2( 2 ) , Tˆ2( 4 )Tˆ2( 2 ) , Tˆ2( 4 )Tˆ2( 3 ) , Tˆ2( 2 )

T1

T2

Yu

180 ,
0
0 . 0309

mc / mtTˆ2( 4 ) , Tˆ2( 4 )180

0 . 0309
0 . 500

0 . 0038 ,Hu,H dYd

ms / mb

0 . 00132
0 . 02140 . 029 ,0
0 . 0309
V cb

0 . 041

Results
We have found examples leading to a realistic
mixing angle as well as mass ratios between the
2nd and 3rd families of quarks.
Our results is the first examples to show thepossibilities for leading to realistic mixing angles
by use of stringy renomalizable couplings with
one pair of the up and down Higgs fields.
Results for three families of quarks are not
perfectly good.

Systematical study on Z6-I: Lepton sector
Ko, T.K., Park ‘05
We study systematically all of possible assignmentsof three families to fixed points,
assume one pair of up and down Higgs fields,
examine allowed entries by the selection rule,
try to fit the lepton mass ratios andmixing angles by varying R1 and R2.

[me / m ]exp2310.000288,2
21 exp[ m / m ][sin 2 23 ]exp

27,

0.50,

[m / m ]exp
[sin

20.0595]0.30[sin 2 13 ]exp

0.000

12 expDirac neutrino mass: Assignment 4
L1 , L2 , L3,

N1 , N2 , N3

Tˆ2( 2) , Tˆ2(3) , Tˆ2( 4, 1)

Tˆ2( 2 ) , Tˆ2( 4,1) , Tˆ2( 4, 1)

T1

26,

T2

me / m0.0003,sin 2 120.38,

e1 , e2 , e3

Hu , HdTˆ3(1) , Tˆ3( 2) , Tˆ3( 4,1)

Tˆ2( 2 ) , Tˆ1

21
m /m
sin 2

23

0.06,
0.70,

2
2
m31
/ m21
14,

sin 2 130.000,

There are several examples leading to similar
results、 but smaller ratios of neutrino massdifference.

Seesaw scenario: Assignment 4
We assume the right-handed Majorana neutrinomass to be proportional to identity matrix for
simplicity.
L1 , L2 , L3,

N1 , N2 , N3

Tˆ2( 2) , Tˆ2( 3) , Tˆ2( 4,1)
T1

23,

T2

me / m0.0003,sin 2 120.32,

e1 , e2 , e3Tˆ2( 2 ) , Tˆ2(3) , Tˆ2( 4,1)

Tˆ3(1) , Tˆ3( 2 ) , Tˆ3( 4,1)

Hu , HdTˆ2( 2 ) , Tˆ1

26
m /m
sin 2

23

0.06,
0.48,

2
2
m31
/ m21

sin 2 1329,

0.000,

There are several examples leading to similar
results.

Results
We have found examples leading to a realistic
values of charged lepton mass ratios, neutrinomass difference ratio and mixing angles forthree families.
It is quite non-trivial to fit six observables only by
two parameters.
Our results is the first examples to show thepossibilities for leading to realistic mixing angles
by use of stringy renomalizable couplings with
one pair of the up and down Higgs fields.

教訓
弦模型において、３点結合の選択則、その強さは、コンパクト空間の幾何学で決まっている。
→ ある程度現実的な湯川行列を導くコンパ
ト空間もあれば、そうでないものもある。

3. Discrete flavor symmetry
Effective Yukawa
higher dim. op. through symmetry
breaking
small Yukawa
e.g. Froggatt-Nielsen
Symmetries are important to control higher dim. Op.
e.g. U(2), S3, D4, A4, U(1), ZN,………
(Symmetrical approach)
What is the origin of these symmetries ?

Discrete Flavor Symmetry
What are their origins of discrete flavor symmetry likeS3, D4, A4 …… ?
They are symmetries of geometric solids.
So, they may be originated from geometry of extra
dimensional space.
Actually, we have found explicit string models, whose
three families = singlet + doublet under D4.
T.K., Raby, Zhang, ’04
This is the first explicit models leading to D4 flavor
structure in string models.

D4 flavor
Our model is from Z6-II orbifold, Z6=Z2*Z3 .
The essence that our model leads to D4 flavor
structure can be understood by the simplified
extra dim. space, S1/Z2.

There are two fixed points, on which two types ofstates exist.
In string models, these states are degenerate inmassless spectra. These correspond to D4
doublets and bulk modes correspond to singlets.

D4 Flavor Symmetry
Stringy symmetries require that
Lagrangian has the permutation symmetry between
1 and 2, and each coupling is controlled by Z2
symmetry.
0 1
1 0
1

1 0

,

3

0

1

D4 elements
1,

1

,

i

Geometry of compact space
origin of finite flavor symmetry

2

,

3

Z6 example
Gauge shift 6V=(22200000)(11000000)
Wilson lines 3W=(-11000000)(00200000)
Wilson line 2W=(10000111)(00000000)
Gauge group SU(4) SU(2) SU(2) (hidden group)
Matter3 [(4,2,1) + (4,1,2)] + (1,2,2) + exotics
Pati-Salam model with 3 families + exotics
3 families = (D4 doublets + D4 singlets)
Electroweak higgs = D4 singlet

Phenomenological aspects
D4 flavor structure has phenomenological aspects
on Yukawa couplings, SUSY breaking terms andso on.
Possibilities for realizing other non-abelian
discrete flavor symmetries ?
Work in progress

教訓
弦模型において、高次の結合の選択則、その強さは、コンパクト空間の幾何学で決まっている。
→ 場の理論の模型で仮定されてきたフレーバー対称性の起源を与える。

4. Flavor in intersecting D-branes
Dp-brane: (p+1) dimensional extended object,
where open string can end
gauge boson: open string, whose two end-points
are on the same (set of) D-brane(s)Intersecting D6 models
Setup of intersecting D6 models
6D compact space: a direct product of three
2D toriD6-branes wrap one cycle of each 2D torus

1st torus

2nd torus

3rd torus

Origin of flavor
Family number = intersection number
su(2)

Q1

Q2
Q3
1st plane

U(1)

su(3)

u1

u2
u3
2nd planesu(3)Yukawa in intersecting D-brane models
Flavor number = intersecting number

SU(3)

U(1)
U1
U2

U3
H Q3

Q2

SU(2)

Flavor structure

Higaki, Kitazawa, T.K.,Takahashi, ‘05
Analyses on flavor structure have been done
model by model so far.
We need to formulate model-independent analysis on
selection rules for allowed couplings like heterotic
orbifold models.
selection rules for 3-point couplings
Cremades, Ibanez, Marchesano, ’03
Our approach is different.
We would like to study somehow systematicallyflavor structure, which can be realized in intersecting
D-brane configuration.

Coupling selection rule
Yukawa couplings are allowed when three twisted
strings combine into a closed string, which shrink.
We have to consider equivalence on the torus.

selection rule← geometrical aspectsof compact space⇒ in general complicatedOpen string at intersecting D-brane

How can we describe open strings at different
intersecting points and open strings
equivalent up to torus ?

Intersecting points
Two D-brane Da, Db (winding numbers)

Wa(na , ma ),Wb (nb , mb ),I ab | na mb nb ma |Intersecting number
Equivalent open strings at intersecting points( 0 ,0 )

kW a

l Wb

Independent open strings at intersecting points
are described by shift vectors Vab corresponding tocoset representatives/ absub-lattice{Wa ,Wb },{(1,0), (0,1)}ab
I ab Vol(ab) / Vol( )Example (Coset representatives)
Two D-brane Da, Db (winding numbers)Wa(1,0),

Wb(1,3),

Intersecting number Iab=3
Intersecting points( k / 3 )W a ,k

0 ,1 , 2

Shift vectors represent threeintersecting points as coset rep..
V ab( 0 , 0 ), ( 0 ,1 ), ( 0 , 2 ),ab{( 1 , 0 ), ( 0 , 3 )},Open strings and selection rule
End points of open string

Xa

Xb

Vab

Three sets of intersecting D-branes

Wa (na , ma ),Wb (nb, mb ),Wc (nc , mc ),Condition for Yukawa couplings among three open
strings between Da-Db, Db-Dc and Dc-Da branes

Vab

Vbc

Vca

0

sub-lattices
only diagonal couplings or off-diagonal couplings ?

mod

ab

,

bc

,

ca

Flavor structure on 2D torus
Which types of flavor structure can be realized fromintersection D-brane configurations ?
infinite number of varieties ?
There is a rule for intersecting numbers.
Left-right (n,n) symmetric generation on torus
the number of Higgs scalars = n k
R n
for k=1
L
ab
bc
ca
n
Selection rule
Zn symmetryC
diagonal couplingsnk

Flavor structure
Winding vectors
WC (nC , mC ),WLIntersecting numbers(nL , mL ),| I CL | | I RC | n

WL

WR

WR

(nR , mR ),L

R
CkW C ,| I LR | nk

for k=1
CL
CR
LR
Selection rule Zn symmetry
for k > 1
Vol ( LR ) kVol ( CRLR )Selection rule still Zn
Higgs (LR) sector has Zk charges, but those areirrelevan to coupling selection rule.

Generic rule for intersecting numbers
Similarly, we can study generic case.
R mn
L
ln
C

Selection rule

Zn symmetrynkg.c.d.(k,l) = g.c.d.(l,m) = g.c.d.(m,k) = 1

Flavor structure
e.g. 3 families

3 Higgs

Left-handed Q
Right-handed Q
Higgs

Vab

( 0 , 0 ), ( 0 ,1), ( 0 , 2 )Vbc( 0 , 0 ), ( 0 ,1), ( 0 , 2 )Vca( 0 , 0 ), ( 0 ,1), ( 0 , 2 )Vab Vbc Vca (0,0)

mod

(0,3)

Selection rule for Yukawa couplingZ3 symmetry
diagonal couplings like Z3 heterotic orbifold
modelsYukawa matrix
Y

H0

H2

H1

H2

H1

H0

H1

H0

H2

Quark masses(m u , m c , m t )(m d , m s , m b )

(v

u
0

(v

d
0

,v

u
1

,v

d
1

u
2

)

,v

d
2

,v

)

Mixing angles

V12 : V13 : V23
Not realistic

mt / mc : m s / mb : md / mb

Asymmetic flavor structureIntersecting number
Left-handed Q
Right-handed Q
Higgs

Y ij
Rank one

I CL( 3 ,1 , 1 )I CR( 1 , 3 ,1 )I LR

( 1 ,1 , 1 )a ib jYukawa matrix not realistic

Result
We have formulated the selection rule for allowed Yukawa
couplings as well as generic n-point couplings.
There is a rule of intersecting numbers for genericconfigurations.
selection rule : Zn symmetry
Left-right symmetric 3 families
3 or more Higgs
Selection rule Z3 symmetry
Left-right asymmetric 3 families
Rank one matrix
Asymmetric structure with more families
non-trivial forms, but ...
We may need another type of flavor structures.

One of different points
Heterotic orbifold models seem better thanIntersecting D-brane models, in order to realize
realistic Yukawa matrices (by the present knowledge).
In heterotic orbifold models, discrete wilson lines
can break degenerate spectrum of fixed points.
If one can resolve degenerate spectrum ofintersecting points,
one may derive more realistic models.

Summary
We have studied flavor structure in string models.
Heterotic orbifold models have possibilities for realizing
realistic Yukawa matrices for quarks and leptons.
Other orbifolds ?
We have just started how to derive non-abelian
discrete flavor symmetries from string models.
We have formulated selection rule of Yukawa couplings
in intersecting D-brane models.
We have just started the classification of flavor
structure derived from D-brane configurations.

Future study
How to stabilize moduli VEVs at proper values is
an important issue to study.
Flavor structure(realistic) Yukawa matrices
Such flavor structure affects on SUSY breaking
terms, e.g. scalar masses and A-matrices.
Each model would have certain pattern of SUSY
breaking terms.
It is important to study such prediction on SUSY
breaking terms.