Transcript KKLT type models with moduli-mixing superpotential Tatsuo

4D low-energy effective field theory
from magnetized D-brane models
Tatsuo Kobayashi
1．Introduction
2. Intersecting/magnetized D-brane models
3. N-point couplings and flavor symmetries
4． Massive modes
5. Moduli
6. Summary
１ Introduction
The workshop website says
In the last decade, there have been exciting developments on
two sides of the string phenomenology.
First, realistic low energy particle models that include the
MSSM and GUTs after appropriate moduli parameter fixing
were constructed by various superstring theories,
…………………………………………………...
Secondly, the KKLT scenario that can realise complete moduli
stabilization by flux compactification of type IIB superstring
theory and may lead to a realistic inflationary universe model
was proposed.
……………………………………………………..
This argument, however, has been based on simple models
that neglect low energy particle physics.
Objectives: purpose
The workshop website says
Thus, it is a good time now to merge these two approaches
and look for compactifications of string theory
that are fully satisfactory both from the low energy particle
physics and from cosmology/the moduli problem.
The main purpose of this workshop is to kick off this
challenging program by overviewing the
present status of the above two approaches to the string
phenomenology and discussing promising models to be
pursued.
Various string models
(before D-brane ) 1st string revolution
Heterotic models on Calabi-Yau manifold, Orbifolds,
fermionic construction,
Gepner, . . . . . . . . . . .
( after D-brane ) 2nd string revolution
Intersecting D-brane models
Magnetized D-branes, . . . . . . . . . . .
⇒ (semi-)realistic low-energy particle models
SU(3)xSU(2)xU(1)Y (or GUT)
three families of quarks and leptons
and extra matter
This talk
chose one type of model constructions,
(Type IIB) magnetized D-brane models,
T-dual to (type IIA) intersecting D-brane models.
(Hetoric models, in particular heterotic orbifold models,
are quite interesting.)
For the first side (particle models),
explain their properties as low-energy particle models,
i.e.
massless spectrum (realistic property)
gauge bosons (gauge symmetries)
matter fermions, higgs fields, moduli, ……………
their action (low-energy effective theory)
This talk
action of massless modes
gauge couplings, Yukawa couplings, …..
Kahler potential (kinetic terms)
(discrete/flavor) symmetries
(phenomenological aspects)
Towards the second side (moduli pheno./cosmology),
we study moduli-dependence of 4D LEEFT such as
gauge couplings, Yukawa couplings, higher order couplings,
D-terms, etc.
(perturbative terms).
non-perturbative terms ?
⇒ M.Cvetic’s talk
Let’s discuss the merge between the two sides.
Plan
✔1．Introduction
2. Intersecting/magnetized D-brane models
3. N-point couplings and flavor symmetries
4．Massive modes
5. Moduli
6. Summary
2. Intersecting/magnetized
D-brane models
gauge boson: open string, whose two end-points
are on the same (set of) D-brane(s)
N parallel D-branes ⇒ U(N) gauge group
gauge bosons, gauginos
adjoint fermions
U(1)
U(1)xU(1)
⇒ U(2)
Intersecting/magnetized
D-brane models
See for a review Ibanez and Uranga texbook and references
therein.
Intersecting D-brane models:
geometrical picture is simple.
Magnetized D-brane models
are also interesting.
(We mainly study this types of models.)
Generic models are their
mixture.
2.1 Intersecting D-branes
Berkooz, Douglas, Leigh, ‘96
Where are the matter fields ?
New modes appear between intersecting D-branes.
They have charges under both gauge groups, i.e.
bi-fundamental matter fields
under U(N)xU(M) gauge group.
boundary condition
X 2 (  0)  0　,  X 1 (  0)  0
X (   ) tan   X (   )  0　,　1
2
 X 1 (   )    X 2 (   ) tan   0
Twisted boundary condition
These are
localized modes
Toy model (in uncompact space)
gauge bosons ： on brane
quarks, leptons, higgs :
localized at intersecting points
u(1)xu(1) su(2)
su(3)
H
u(1)
Q
L
u,d
e, neutrino
Toy model (in uncompact space)
gauge group can be enhanced
from U(3)xU(1)xU(2)xU(1)xU(1)
⇒ U(4)xU(2)xU(2) (Pati-Salam)
u(1)xu(1) su(2)
su(3)
H Split of branes
u(1)
Q
 Wilson lines
L
u,d
e,neutrino
Generation number
Torus compactification
Family number = intersection number
su(2)
Q1
Q2
on T2
Q3
su(3)
Type IIA D6-brane models
T2xT2xT2 compactification
D6 branes wrap a factorizable three-cycle
(one-cycle of each T2).
1st plane
2nd plane
3rd plane
Intersecting modes
Always massless fermions appear at intersecting points.
Scalar modes are sometimes tachyonic.
D-brane configuration is unstable ⇒
symmetry breaking (recombination of D-branes)
(local) SUSY guarantees that the lightest scalar is not tachyonic,
but massless.
1   2  3  0
Toy model on T2xT2xT2
Intersecting number = family number
u(1)xu(1) su(2)
su(3)
H
u(1)
Q 3
L3
3
3 u,d
e, neutrino
Hidden sector
u(1) su(2)
H
su(3)
Q 3
L3
3
3 u,d
RR-charge cancellation
D6 brane has a RR-charge.
The total charge should vanish
along compact extra dimensions.
u(1) su(2)
su(3)
H
Q 3
L3
3
3
3 u,d
Other configurations
Other types of D-brane configurations
are also interesting.
,
etc
,
Orientifold
Orientifold also has a RR charge.
U(N)
U(N)’
(N,N)
⇒ symm.
and anti-symm. reps.
after identifying U(N) and U(N)’
These modes may also correspond to
SM matter fields.
Orientifold
U(N)
U(N)’
(N,N)
⇒ symm.
and anti-symm. reps.
These modes may also correspond to SM matte fields.
The three generations of quarks and leptons are not
just originated from the intersections of one type of
bi-fundamental matter fields, but the flavor structure
becomes rich.
Some extensions
orbifold, Calabi-Yau, etc.
2-2. Magnetized D-branes
We consider torus compactification
with magnetic flux background.
F
Boundary conditions
on magnetized D-branes
~
 F45 X
5
 0,
  X
5
 0,
 X
~
F45 X
4
4
similar to the boundary condition of
open string between intersecting D-branes
T-dual to (type IIA) intersecting D-brane models
~
cot   F45
Type IIB magnetized D-brane models
D9, D7, D5, D3
D9: wrapping on T2xT2xT2 with magnetic fluxes
D7: wrapping on T2xT2 with magnetic fluxes
D5: wrapping on T2 with magnetic fluxes
2.3 LEEFT of magnetized D-branes
Low-energy effective field theory
of D-brane models
= higher dimensional super Yang-Mills theory
e.g.
D9-brane models
⇒ 10D SYM (gauge bosons, gauginos)
KK decomposition
4D LEEFT
2-3-1 Field theory in higher
dimensions: generic aspects
10D ⇒ 4D our space-time + 6D space
10D vector
AM  A , Am
4D vector + 4D scalars
SO(10) spinor ⇒ SO(4) spinor
x SO(6) spinor
internal quantum number
Several Fields in higher dimensions
4D (Dirac) spinor
i

D  0
⇒ (4D) Clifford algebra {  ,   }  2 
(4x4) gamma matrices
represention space ⇒ spinor representation
6D Clifford algebra
{ M ,  N }  2 MN


M
4
( M  0,1,2,3)

, 
I 44  
5
6D spinor
6D spinor
I 44   1 ,


3
2
 4 D  2 D
⇒ 4D spinor x (internal spinor)
internal quantum
number
Field theory in higher dimensions
Mode expansions

M

 M Am  (     6 ) Am  0
i (  D   m Dm )  0
KK decomposition
KK docomposition on torus
torus with vanishing gauge background
Boundary conditions
 ( y4  1, y5 )   ( y4 , y5 )
 ( y4 , y5  1)   ( y4 , y5 )
0 : constantmode
n : exp(ikny) k  2 / R
mn  0
mn  kn
First, we concentrate on zero-modes.
Zero-modes
Zero-mode equation
i Dm  0
m
⇒ non-trival zero-mode profile
the number of zero-modes
4D effective theory
Higher dimensional Lagrangian (e.g. 10D)
L10  g  d xd y  ( x, y ) A( x, y ) ( x, y )
4
6
integrate the compact space ⇒ 4D theory
L4  Y  d x ( x) ( x)  ( x)
4
Y  g  d y ( y ) ( y ) ( y )
6
Coupling is obtained by the overlap
integral of wavefunctions
Couplings in 4D
Zero-mode profiles are quasi-localized
far away from each other in compact space
⇒ suppressed couplings
Chiral theory
When we start with extra dimensional field theories,
how to realize chiral theories is one of important
issues from the viewpoint of particle physics.
i Dm  0
m
Zero-modes between chiral and anti-chiral
fields are different from each other
on certain backgrounds,
e.g. CY, toroidal orbifold, warped orbifold,
magnetized extra dimension, etc.
2-3-2
Higher Dimensional SYM theory with flux
Cremades, Ibanez, Marchesano, ‘04
4D Effective theory <= dimensional reduction
The wave functions
eigenstates of corresponding
internal Dirac/Laplace operator.
Higher Dimensional SYM theory with flux
Abelian gauge field on magnetized torus
Constant magnetic flux
gauge fields of background
The boundary conditions on torus (transformation under torus translations)
Higher Dimensional SYM theory with flux
We now consider a complex field
Consistency of such transformations under
a contractible loop in torus which implies
Dirac’s quantization conditions.
with charge Q ( +/-1 )
Dirac equation on 2D torus
is the two component spinor.
U(1) charge Q=1
   4  i 5 ,
   4  i 5
with twisted boundary conditions (Q=1)
Dirac equation and chiral fermion
|M| independent zero mode solutions in Dirac equation.
(Theta function)
Properties of
theta functions
chiral fermion
: zero-modes
: no zero-mode
By introducing magnetic
flux, we can obtain chiral
theory.
Wave functions
For the case of M=3
Wave function profile on toroidal background
Zero-modes wave functions are quasi-localized far away each
other in extra dimensions. Therefore the hierarchirally small
Yukawa couplings may be obtained.
Fermions in bifundamentals
Breaking the gauge group
(Abelian flux case
The gaugino fields
gaugino of unbroken gauge
bi-fundamental matter fields
)
Bi-fundamental
Gaugino fields in off-diagonal entries
correspond to bi-fundamental matter fields
and the difference M= m-m’ of magnetic
fluxes appears in their Dirac equation.
F
Zero-mode Dirac equations
No effect due to magnetic flux for adjoint matter fields,
Total number of zero-modes of
: Zero-modes
: No zero-mode
4D chiral theory
10D spinor
( ,    )
light-cone 8s
even number of minus signs
1st ⇒ 4D, the other ⇒ 6D space
If all of
appear
in 4D theory, that is non-chiral theory.
If
only
for all torus,

ab
N
a
, Nb  ,  
appear for 4D helicity fixed.
⇒ 4D chiral theory
(Simple) U(8) SYM theory on T6
Fzz
 m1 N1

 2i

 0
0
m2  N 2
N1  4, N2  2, N3  2
m3 N 3





U (4) U (2) L U (2) R
Pati-Salam group up to U(1) factors
  T 2
(m1  m2 )  (m3  m1 )  3 for t hefirst
4,2,1  4,1,2
(m1  m2 )  (m3  m1 )  1 for t heot her t ori
Three families of matter fields
with many Higgs fields
(4,2,1)  ( 4,1,2)
(1,2,2)
Wilson lines
Cremades, Ibanez, Marchesano, ’04,
Abe, Choi, T.K. Ohki, ‘09
torus without magnetic flux
constant Ai  mass shift
every modes massive
magnetic flux
  2 (My  a)

  2 ( My  a) 
the number of zero-modes is the same.
the profile: f(y)  f(y +a/M)
with proper b.c.
0
0
U(1)a*U(1)b theory
magnetic flux, Fa=2πM, Fb=0
Wilson line, Aa=0, Ab=C
matter fermions with U(1) charges, (Qa,Qb)
chiral spectrum,
for Qa=0, massive due to nonvanishing WL
when MQa >0, the number of zero-modes
is MQa.
zero-mode profile is shifted depending
on Qb, f ( z)  f ( z  CQ /(MQ ))
b
a
Pati-Salam model
Fzz
 m1 N1

 2i

 0
Pati-Salam group
0
m2  N 2
m3 N 3





N1  4, N2  2, N3  2
U (4) U (2) L U (2) R
(m1  m2 )  (m3  m1 )  3 for t hefirst T
2
  t ori
(m1  m2 )  (m3  m1 )  1 for t heot her
4,2,1  4,1,2
WLs along a U(1) in U(4) and a U(1) in U(2)R
=> Standard gauge group up to U(1) factors
U (3)C U (2) L U (1)
U(1)Y is a linear combination.
3
PS => SM
(4,2,1)  ( 4,1,2)
Zero modes corresponding to
three families of matter fields
remain after introducing WLs, but their profiles split
(4,2,1)  (3,2,1)  (1,2,1)
( 4,1,2)  ( 3,1,1)  ( 3,1,1)  (1,1,1)  (1,1,1)
(4,2,1)
Q
L
2.4 Other backgrounds:
Orbifold with magnetic flux
Abe, T.K., Ohki, ‘08
The number of even and odd zero-modes
We can also embed Z2 into the gauge space.
=> various models, various flavor structures
Zero-modes on orbifold
Adjoint matter fields are projected by
orbifold projection.
We have degree of freedom to
introduce localized modes on fixed points
like quarks/leptons and higgs fields.
S2 with magnetic flux
Conlon, Maharana, Quevedo, ‘08
Fubuni-Study metric
2 2
2
2
ds  4 R (1  z ) dzd z



Zero-mode eq. (1  z 2 )  (M  1) z / 2 ( z )  0

with spin
2
(1  z )  (M  1) z / 2  ( z )  0
connection
2
M>0
  ( z )  Nz m /(1  z ) ( M 1) / 2 , 0  m  M  1
  ( z)
M=0
  ( z)
  ( z)
no solut ion
no solut ion
no solut ion

Short summary
In magnetized D-brane models,
Zero-modes are quasi-localized and
the number of zero-modes,
i.e., the family number, is
determined by the size of magnetic flux.
2.5 Generic models
Generic model would be a mixture of intersecting and
magnetized D-brane models.
for example,
IIB intersecting D7-branes with magnetic fluxes,
IIA intersecting D8-branes with magnetic fluxes
………………………………………………………………
on CY
3. N-point couplings
and flavor symmetries
3.1 N-point couplings of zero-modes
The N-point couplings are obtained by
overlap integral of their zero-mode w.f.’s.
Y  g  d z ( z ) ( z )  ( z )
2
i
M
z  y4  iy5
j
N
k
P
Moduli
Torus metric
ds  2(2R) dzdz
z  x  y
2
2
Area A  4 R Im
We can repeat the previous analysis.
Scalar and vector fields have the same
wavefunctions.
  M
Wilson moduli
 ( z)   ( z   )
shift of w.f.
2
2
Zero-modes
Cremades, Ibanez, Marchesano, ‘04
 ( z)  N M
j
M
 j/M
exp[iMz Im(z ) / Im ]  
( Mz,M )

 0 

N M  2M Im / A

2 1/4
,
j  1,, M
Zero-mode w.f. = gaussian x theta-function
Product of zero-mode wavefunctions
 ( z )  ( z ) 
i
M
yijm
j
N
M N
y
m 1

ijm
i  j  Mm
M N
( z ),
N N N M ( Ni  Mj  MNm) /(MN (M  N ))


(0,MN ( M  N ))

0
N N M 

Products of wave functions:
Hint to understand
  2My  0,
  2Ny   0,
  2 ( M  N ) y 
M
N
( M  N )
 ( M  N )    M   N 
products of zero-modes = zero-modes
 0,
3-point couplings
Cremades, Ibanez, Marchesano, ‘04
The 3-point couplings are obtained by
overlap integral of three zero-mode w.f.’s.

Yijk   d z ( z ) ( z ) 
2
i
M
j
N


k
M N
ik
d
z

(
z
)

(
z
)



2
Yijk 
i
M
k
M
*
M N

m 1
i  j  mM , k
yijm
( z)

*
Selection rule
i  j mM ,k  i  j  mM  k (M  N )
i j k
mod g 
when g  gcd( M , N )
Each zero-mode has a Zg charge,
which is conserved in 3-point couplings.
yijm
( Ni  Mj  MNm) /(MN ( M  N ))
 
(0,MN ( M  N ))

0


up to normalization factor
4-point couplings
Abe, Choi, T.K., Ohki, ‘09
The 4-point couplings are obtained by
overlap integral of four zero-mode w.f.’s.
split

Yijkl   d z ( z ) ( z ) ( z ) 
2
d
2
i
M
j
N
k
P
l
M N P

( z)
zd z ' ( z ) ( z ) ( z  z ' ) ( z ' ) 
2
i
M
j
N
k
P
insert a complete set
 
 ( z  z' ) 
n
K
l
M N P

n
( z)  K
( z' )
*
all modes
Yijkl 

s only zero-modes
yijs yskl

*
for K=M+N
( z' )

*
4-point couplings: another splitting
d
2

zd z ' ( z ) ( z ) ( z  z ' ) ( z ' ) 
2
i
M
k
P
j
N
l
M N P
( z' )
Yijkl   yikt ytjl
t
Yijkl   yijs yskl
Yijkl   yikt ytjl
t
s
i
k
i
k
t
j
s
l
j
l

*
N-point couplings
Abe, Choi, T.K., Ohki, ‘09
We can extend this analysis to generic n-point
couplings.
N-point couplings = products of 3-point couplings
= products of theta-functions
This behavior is non-trivial. (It’s like CFT.)
Such a behavior would be satisfied
not for generic w.f.’s, but for specific w.f.’s.
However, this behavior could be expected
from T-duality between magnetized
and intersecting D-brane models.
T-duality
The 3-point couplings coincide between
magnetized and intersecting D-brane models.
explicit calculation
Cremades, Ibanez, Marchesano, ‘04
Such correspondence can be extended to
4-point and higher order couplings because of
CFT-like behaviors, e.g.,
Yijkl   yijs yskl
s
Abe, Choi, T.K., Ohki, ‘09
3.2 Non-Abelian discrete flavor
symmetry
The coupling selection rule is controlled by Zg charges.
For M=g,
1
2
g
Effective field theory also has a cyclic permutation
symmetry of g zero-modes.
These lead to non-Abelian discrete flavor symmetires
such as D4 and Δ(27) Abe, Choi, T.K, Ohki, ‘09
Cf. heterotic orbifolds, T.K. Raby, Zhang, ’04
T.K. Nilles, Ploger, Raby, Ratz, ‘06
Permutation symmetry
D-brane models
Abe, Choi, T.K. Ohki, ’09, ‘10
There is a Z2 permutation symmetry.
The full symmetry is D4.
Permutation symmetry
D-brane models
Abe, Choi, T.K. Ohki, ’09, ‘10
geometrical symm.
Z3
S3
Full symm.
Δ(27)
Δ(54)
intersecting/magnetized
D-brane models Abe, Choi, T.K. Ohki, ’09, ‘10
generic intersecting number g
magnetic flux
flavor symmetry is a closed algebra of
two Zg’s.
1










,


g 1 
 
and Zg permutation
Certain case: Zg permutation
Dg
0

0




1
0







0
1




,   e 2i / g



 
0

0




larger symm. Like
Magnetized brane-models
Magnetic flux M
2
4
・・・
Magnetic flux M
3
6
9
・・・
D4
2
1++ + 1+- +1-+ + 1-・・・・・・・・・
Δ(27)
(Δ(54))
31
2 x 31
∑1n
n=1,…,9
(11+∑2n n=1,…,4)
・・・・・・・・・
Discrete flavor symmetry
ZN symmetry is originated from
anomalous U(1) symmetries.
Berasatuce-Gonzalez, Camara, Marchesano,
Regalado, Uranga, ‘12
Non-Abelian discrete flavor symm.
Recently, in field-theoretical model building,
several types of discrete flavor symmetries have
been proposed with showing interesting results,
e.g. S3, D4, A4, S4, Q6, Δ(27), ......
Review: e.g
Ishimori, T.K., Ohki, Okada, Shimizu, Tanimoto ‘10
⇒ large mixing angles
 2/3
1/ 3
0 


one Ansatz: tri-bimaximal   1 / 6 1 / 3
1/ 2 

  1/ 6
1/ 3

 1 / 2 
Quark masses and mixing angles
M t  174
GeV ,
M b  4.3
GeV
M c  1.2
GeV ,
M s  117
MeV
Mu  3
Vus  0.22,
MeV ,
Vcb  0.04,
M d  6.8
MeV
Vub  0.004
These masses are obtained by Yukawa couplings
to the Higgs field with VEV, ｖ＝１７５GeV.
strong Yukawa coupling ⇒ large mass
weak
⇒ small mass
top Yukawa coupling =O(1)
other quarks ← suppressed Yukawa couplings
Lepton masses and mixing angles
M e  0.5
MeV ,
M   1.8
GeV ,
M   106
MeV
mass squared differences and mixing angles
consistent with neutrino oscillation
M 212  8 10 5
sin 2 12  0.3,
eV 2 ,
sin 2  23  0.5,
large mixing angles
M 312  2 10 3
sin 2 13  0.02,
eV 2
Applications of couplings
We can obtain quark/lepton masses and mixing angles.
Yukawa couplings depend on volume moduli,
complex structure moduli and Wilson lines.
By tuning those values, we can obtain semi-realistic results.
Abe, Choi, T.K., Ohki ‘08
Abe, et. al. work in progress
Ratios depend on complex structure moduli
and Wilson lines.
Flavor is still a challenging issue.
Short summary
We have studied 3-point couplings
and higher order couplings among massless modes.
They may be useful to realize hierarchical
quark/lepton masses and mixing angles
and other aspects.
The discrete flavor symmetry would be useful.
4. Massive modes
Hamada, T.K. ‘12
Massive modes play an important role
in 4D LEEFT such as the proton decay,
FCNCs, etc.
It is important to compute mass spectra of
massive modes and their wavefunctions.
Then, we can compute couplings among
massless and massive modes.
Fermion massive modes
Two components are mixed.
 DD

 0

2D Laplace op.
  ,n 
0   , n 
2




 mn 






DD   ,n 
 ,n 
  {D , D} / 2
algebraic relations
[ D, D ]  4M / A
[, D ]  4MD / A,
[, D]  4MD / A
It looks like the quantum harmonic oscillator
Fermion massive modes
Creation and annhilation operators
a  D A / 4M ,
a  D
A / 4M ,

[a, a ]  1
mass spectrum
m  4Mn / A
2
n
wavefunction

j ,M
n
 (1/ n!)(a ) 
 n
j ,M
0
Fermion massive modes
explicit wavefunction
 nj , M
( 2 M Im )1/ 4

( 2 n n! A)1/ 2
 Hn

j ,M

 k ( z   , )
k

2M Im k  j / M  Im(z   ) / Im 
kj ,M ( z, )  exp[M Im (k  j / M  Im z / Im ) 2
 iM Re z(2k  2 j / M  Im z / Im )  iM Re (k  j / M )
Hn: Hermite function
Orthonormal condition:
2
j ,M
k ,M *
d
z

(

)   jk n
n


Scalar and vector modes
The wavefunctions of scalar and vector fields
are the same as those of spinor fields.
Mass spectrum
2
m
n  2M (2n  1) / A
scalar
vector
mn2  2M (2n 1) / A
Scalar modes are always massive on T2.
The lightest vector mode along T2,
i.e. the 4D scalar, is tachyonic on T2.
Such a vector mode can be massless on T4 or T6.
m2  2 (M1 / A1  M 2 / A2  M3 / A3 )
Products of wavefunctions
explicit wavefunction
 ni , M ( z )  nj , N ( z ) 
1
2
yijm
s 

n1 C
M  N n1
n2
 
m 1  0 s 0
i  j  Mm , N  M
yijm

( z ),
 s  s
n2  s
( n2    s ) / 2
( n1    s ) / 2
 ( n1  n2 1) / 2
C
(

1
)
N
M
(
N

M
)
s
n2
(  s )!( n1  n2    s )!/( n1! n2 !)
 Nj  NMm ,
 nMi

1 n2    s
NM ( N  M )
(0, )
See also Berasatuce-Gonzalez, Camara, Marchesano,
Regalado, Uranga, ‘12

i,M
( z ) 
j,N
( z) 
M N
y
 i  j  Mm , M  N ( z ),
Derivation:
m 1
products of zero-mode wavefunctions
We operate creation operators on both LHS
and RHS.
ijm
3-point couplings including higher
modes
The 3-point couplings are obtained by
overlap integral of three wavefunctions.
ijk
n1 n2 n3
Y
  d z
2
i,N
n1
d z 
2
ijk
n1n2 n3
Y

M N
i,M

n1
( z )

( z) 
n2
 
m 1
 0 s 0
j ,M
n2
j ,M
s

( z) 
k ,M  N
n3
( z)

*

( z )   ik s
*
 i  j  mM ,k    s ,n yijm
s
3
i  j  k mod M
(flavor) selection rule
is the same as one for the massless modes.
(mode number) selection rule
n  n n
3
1
2
3-point couplings:
2 zero-modes and one higher mode
3-point coupling
n1  n3  0
n3  n1  n2
( M  N ) j , NM ( N  M )
Y  N n2 / 2 ( N  M )( n2 1) / 2 nMk
(0, )
2
Higher order couplings including
higher modes
Similarly, we can compute higher order couplings
including zero-modes and higher modes.
Y   d z
2
i,N
n1
( z )
j ,M
n2

( z) 
k ,P
nm
They can be written by the sum over
products of 3-point couplings.
( z)

*
3-point couplings including massive
modes only due to Wilson lines
Massive modes appear only due to Wilson lines
without magnetic flux
 n(Wn)  A1/ 2 exp[i (2nR  Im / Im ) Re z
 i ( Re   2(nI  nR Re  )) Im z / Im ]
R
I
We can compute the 3-point coupling
jk
(W ) nR nI
Y
  d z
2
(W )
nR nI
( z )
j ,M
0

(z  1) 
k ,M
0
(z   2 )

*
| A exp[ |  2   1 | /(2 Im )]
e.g. | Y
Gaussian function for the Wilson line.
jk
(W ) nR 0 nI 0
1/ 2
M 1    M 2
2
3-point couplings including massive
modes only due to Wilson lines
jk
(W ) nR 0 nI 0
|Y
1/ 2
| A
exp[ |  2   1 | /(2 Im )]
For example, we have
| Y(Wjk ) nR 0nI 0 | exp[ ]  0.04
for
|  2   1 |2 /(2 Im )  1
2
Several couplings
Similarly, we can compute the 3-point couplings
including higher modes
jk
n1 n2 (W ) nR nI
Y
  d z
2
(W )
nR nI
( z )
j ,M
n1

( z) 
k ,M
n2
( z)

*
Furthermore, we can compute higher order
couplings including several modes, similarly.
Y   d z
2
(W )
nR nI
( z )
j ,M
n1

( z ) 
k ,M
n2
( z)

*
4.2 Phenomenological applications
In 4D SU(5) GUT,
The heavy X boson couples with quarks and leptons
by the gauge coupling.
Their couplings do not change even after GUT breaking
and it is the gauge coupling.
However, that changes in our models.
Phenomenological applications
For example,
we consider the SU(5)xU(1) GUT model
and we put magnetic flux along extra U(1).
The 5 matter field has the U(1) charge q,
and the quark and lepton in 5 are quasi-localized
at the same place.
Their coupling with the X boson is given by
the gauge coupling before the GUT breaking.
SU(5) => SM
We break SU(5) by the WL along the U(1)Y direction.
The X boson becomes massive.
The quark and lepton in 5 remain massless, but their
profiles split each other.
Their coupling with X is not equal to the gauge coupling,
but includes the suppression factor
5
| Y(Wjk ) nR 0nI 0 | exp[ ]  0.04
Q
L
Proton decay
Similarly, the couplings of the X boson with quarks and
leptons in the 10 matter fields can be suppressed.
That is important to avoid the fast proton decay.
The proton decay life time would drastically
change by the factor,
O(10  10 )
4
5

| Y(Wjk ) nR 0nI 0 | exp[ ]  0.04
Other aspects
Other couplings including massless and massive modes
can be suppressed and those would be important ,
such as right-handed neutrino masses and
off-diagonal terms of Kahler metric, etc.
Threshold corrections on the gauge couplings,
Kahler potential after integrating out massive modes
Short summary
We have studied mass spectra and wavefunctions
of higher modes.
We have computed couplings including higher modes.
We can write the LEEFT with the full modes.
These results have important implications.
We know that couplings among zero-modes coincide
between the magnetized and intersecting D-brane models.
What about couplings including higher modes ?
Anyway, the mass spectra coincide each other.
5. Moduli (discussions)
We have used the basis that the kinetic term is canonical.
The holomorphic parts of the couplings depend only on
the complex structure moduli as well as
the Wilson line moduli.
The holomorphic part of couplings ⇒ superpotential
the non-holomorphic part
⇒ Kahler metric
Cremades, Ibanez, Marchesano ’04
Di Vecchia, et. al. ‘09
Abe, T.K., Ohki, Sumita, ‘12
Kahler moduli appear only in the Kahler metric.
Gauge kinetic function
For simplicity, we consider the factorizable torus,
T2xT2xT2.
SUSY condition
F / A   F / A
i
i
i
i
Berkooz, Douglas, Leigh, ‘96
DBI ⇒
g
2
D9
 Re(s  F1F2t3  F1F3t2  F2 F3t1 )
Lust, Mayr, Richter, Stieberger, ‘04
Re( s )  e
10
A1 A2 A3
Re( ti )  e 10 Ai
Non-perturbative terms
Non-perturbative effects such as gaugino condensation
would induce terms like
exp[bs  m1t1  m2t2  m3t3 ]
D-brane instanton effects
Q1 Qn f ( , ) exp[b' s  m1 ' t1  m2 ' t2  m3 ' t3 ]
This form is determied by (anomalous) U(1) symmetries
and discrete (flavor) symmetreis.
On the other hand, holomorphic perturbative couplings
depend on complex structure moduli
as well as open string (WL) moduli.
3-form flux compactification
The 3-form flux may stabilize the dilaton and
complex structure moduli.
Summary
We have studied phenomenological aspects
of magnetized D-brane models.
We can construct models with realistic
massless spectrum, SM gauge group
(and GUT extensions) and
three generations of quarks and leptons.
We can write the 4D LEEFT of massless modes,
perturbative coupling terms and their moduli
dependence.
Summary
We can also write the perturbative coupling terms
of the full modes.
The 4D LEEFT has certain discrete (flavor)
symmetries.
What about their anomalies ?
Discussions
The moduli stabilization ?
Inflation ? Axions ?
Let’s kick off to merge two approaches.
(2-form) magnetic fluxes
SUSY condition
F / A   F / A
i
i
i
i
may stabilize some of Kahler moduli ?
Antoniadis, Maillar, ’04
Antoniadis, Kumar, Maillard, ‘06