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Overview of Extra Dimensional Models
with Magnetic Fluxes Tatsuo Kobayashi
1.Introduction
2. Magnetized extra dimensions
3. N-point couplings and flavor symmetries
4.Summary
1 Introduction
Extra dimensional field theories,
in particular
string-derived extra dimensional field theories,
play important roles in particle physics
as well as cosmology .
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.
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.
0 m
mDi
Magnetic flux
The limited number of solutions with
non-trivial backgrounds are known.
Generic CY is difficult.
Toroidal/Wapred orbifolds are well-known.
Background with magnetic flux is
one of interesting backgrounds.
0 m
mDi
Magnetic flux
Indeed, several studies have been done
in both extra dimensional field theories
and string theories with magnetic flux
background.
In particular, magnetized D-brane models
are T-duals of intersecting D-brane models.
Several interesting models have been
constructed in intersecting D-brane models,
that is, the starting theory is U(N) SYM.
Phenomenology of magnetized brane models
It is important to study phenomenological
aspects of magnetized brane models such as
massless spectra from several gauge groups,
U(N), SO(N), E6, E7, E8, ...
Yukawa couplings and higher order n-point
couplings in 4D effective theory,
their symmetries like flavor symmetries,
Kahler metric, etc.
It is also important to extend such studies
on torus background to other backgrounds
with magnetic fluxes, e.g. orbifold backgrounds.
2. Extra dimensions with magnetic fluxes: basic tools
2-1. Magnetized torus model
We start with N=1 super Yang-Mills theory
in D = 4+2n dimensions.
For example, 10D super YM theory
consists of gauge bosons (10D vector)
and adjoint fermions (10D spinor).
We consider 2n-dimensional torus compactification
with magnetic flux background.
We can start with 6D SYM (+ hyper multiplets),
or non-SUSY models (+ matter fields ), similarly.
Higher Dimensional SYM theory with flux
Cremades, Ibanez, Marchesano, ‘04
The wave functions eigenstates of corresponding
internal Dirac/Laplace operator.
4D Effective theory <= dimensional reduction
Higher Dimensional SYM theory with flux
Abelian gauge field on magnetized torus
Constant magnetic flux
The boundary conditions on torus (transformation under torus translations)
gauge fields of background
Dirac equation on 2D torus
with twisted boundary conditions (Q=1)
is the two component spinor.
5454 , ii
U(1) charge Q=1
|M| independent zero mode solutions in Dirac equation.
(Theta function)
Dirac equation and chiral fermion
Properties of
theta functions
:Normalizable mode
:Non-normalizable
mode
By introducing magnetic
flux, we can obtain chiral
theory.
chiral fermion
Wave functions
Wave function profile on toroidal background
For the case of M=3
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
The gaugino fields
Breaking the gauge group
bi-fundamental matter fields
gaugino of unbroken gauge
(Abelian flux case )
Zero-modes Dirac equations
Total number of zero-modes of
:Normalizable mode
:Non-Normalizable mode
No effect due to magnetic flux for adjoint matter fields,
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 for all torus,
only
appear for 4D helicity fixed.
⇒ 4D chiral theory
),(
, , ba
ab NN
U(8) SYM theory on T6
Pati-Salam group up to U(1) factors
Three families of matter fields
with many Higgs fields
2,1,41,2,4
3
2
1
3
2
1
0
0
2
N
N
N
zz
m
m
m
iF
2 ,2 ,4 321 NNNRL UUU )2()2()4(
other tori for the 1)()(
first for the 3)()(
1321
2
1321
mmmm
Tmmmm
)2,1,4()1,2,4(
)2,2,1(
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
the number of zero-modes is the same.
the profile: f(y) f(y +a/M)
with proper b.c.
0 )(2
0 )(2
aMy
aMy
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,
))/(( )( ab MQCQzfzf
Pati-Salam model
Pati-Salam group
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(1)Y is a linear combination.
2,1,41,2,4
3
2
1
3
2
1
0
0
2
N
N
N
zz
m
m
m
iF 2 ,2 ,4 321 NNN
RL UUU )2()2()4(
other tori for the 1)()(
first for the 3)()(
1321
2
1321
mmmm
Tmmmm
3)1()2()3( UUU LC
PS => SM Zero modes corresponding to
three families of matter fields
remian after introducing WLs, but their profiles split
(4,2,1)
Q L
)2,1,4()1,2,4(
)1,1,1()1,1,1()1,1,3()1,1,3()2,1,4(
)1,2,1()1,2,3()1,2,4(
Othere models
We can start with 10D SYM,
6D SYM (+ hyper multiplets),
or non-SUSY models (+ matter fields )
with gauge groups,
U(N), SO(N), E6, E7,E8,...
E6 SYM theory on T6 Choi, et. al. ‘09
We introduce magnetix flux along U(1) direction,
which breaks E6 -> SO(10)*U(1)
Three families of chiral matter fields 16
We introduce Wilson lines breaking
SO(10) -> SM group.
Three families of quarks and leptons matter fields
with no Higgs fields
1100 161614578
1 ,1 ,3 321 mmm
Splitting zero-mode profiles
Wilson lines do not change the (generation) number of zero-modes, but change localization point.
16
Q …… L
E8 SYM theory on T6 T.K., et. al. ‘10
E8 -> SU(3)*SU(2)*U(1)Y*U(1)1*U(1)2*U(1)3*U(1)4
We introduce magnetic fluxes and Wilson lines
along five U(1)’s.
E8 248 adjoint rep. include various matter fields.
248 = (3,2)(1,1,0,1,-1) + (3,2)(1,0,1,1,-1) + (3,2)(1,-1,-1,1,-1)
+ (3,2)(1,1,0,1,-1) + (3,2)(1,0,1,1,-1) + ......
We have studied systematically the possibilities
for realizing the MSSM.
Results: E8 SYM theory on T6 1.
We can get exactly 3 generations of
quarks and leptons,
but there are tachyonic modes and no top Yukawa.
2.
We allow MSSM + vector-like generations
and require the top Yukawa couplings and
no exotic fields
as well as no vector-like quark doublets.
three classes
Results: E8 SYM theory on T6 A.
B.
C.
charges U(1)Yno with singlets
])1,1()1,1[(15])2,1()2,1[(12
])1,3()1,3[(21])1,3()1,3[(10
])1,1()2,1()1,3()1,3()2,3[(3
6633
2244
63241
charges U(1)Yno with singlets
])1,1()1,1[()156(])2,1()2,1[()44(
])1,3()1,3[()36(])1,3()1,3[()96(
])1,1()2,1()1,3()1,3()2,3[(3
66
2
33
2
22
2
44
2
63241
nnnn
nnnn
charges U(1)Yno with singlets
])1,1()1,1[(180])2,1()2,1[(45
])1,3()1,3[(18])1,3()1,3[(66
])1,1()2,1()1,3()1,3()2,3[(3
6633
2244
63241
Results: E8 SYM theory on T6 There are many vector-like generations.
They have 3- and 4-point couplings with
singlets (with no hypercharges).
VEVs of singlets mass terms of vector-like
generations
Light modes depend on such mass terms.
That is, low-energy phenomenologies
depends on VEVs of singlets (moduli).
2.3 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.
2.4 S2 with magnetic flux
solution no )(
10 ,)1/()( 2/)1(2
z
MmzNzz Mm
0)(2/)1()1(
0)(2/)1()1(
2
2
zzMz
zzMz
zdzdzRds 2222 )1(4
solution no )(
solution no )(
z
z
Conlon, Maharana, Quevedo, ‘08
Fubuni-Study metric
Zero-mode eq.
with spin
connection
M > 0
M=0
3. N-point couplings and flavor symmetries
The N-point couplings are obtained by
overlap integral of their zero-mode w.f.’s.
)()()(2 zzzzdgY k
P
j
N
i
M
54 iyyz
Zero-modes Cremades, Ibanez, Marchesano, ‘04
Zero-mode w.f. = gaussian x theta-function
up to normalization factor
),(0
/)]Im(exp[)( iMMz
MjzMziNz M
j
M
,)()()(1
NM
m
Mmji
NMijm
j
N
i
M zyzz
))(,0(0
))(/()(NMiMN
NMMNMNmMjNiyijm
MjNM ,,1 factor,ion normalizat:
Products of wave functions
products of zero-modes = zero-modes
NMNM
NM
N
M
yNM
Ny
My
)(
)( ,0 )(2
,0 2
,0 2
3-point couplings
Cremades, Ibanez, Marchesano, ‘04
The 3-point couplings are obtained by
overlap integral of three zero-mode w.f.’s.
up to normalization factor
ikk
M
i
M zzzd *2 )()(
*2 )()()( zzzzdY k
NM
j
N
i
Mijk
NM
m
ijmkmMjiijk yY1
,
Selection rule
Each zero-mode has a Zg charge,
which is conserved in 3-point couplings.
up to normalization factor
)(, NMkmMjikmMji
))(,0(0
))(/()(NMiMN
NMMNMNmMjNiyijm
),gcd( when mod NMggkji
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 insert a complete set up to normalization factor for K=M+N
*2 )()()()( zzzzzdY l
PNM
k
P
j
N
i
Mijkl
modes all
*)'()()'( zzzz n
K
n
K
*22 )'()'()'()()(' zzzzzzzzdd l
PNM
k
P
j
N
i
M
lsksij
slijk
yyY
4-point couplings: another splitting
i k i k
t
j s l j l
*22 )'()'()'()()(' zzzzzzzzdd l
PNM
j
N
k
P
i
M
ltjtik
tlijk
yyY
ltjtik
tlijk
yyY lsksij
slijk
yyY
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.,
Abe, Choi, T.K., Ohki, ‘09
lsksij
slijk
yyY
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
T.K., et. al. ‘10,
Abe, et. al. work in progress
Other applications
up to normalization factor
Summary
We have studied phenomenological aspects
of magnetized brane models.
Model building from U(N), E6, E7, E8
N-point couplings are comupted.
4D effective field theory has non-Abelian flavor
symmetries, e.g. D4, Δ(27).
Orbifold background with magnetic flux is
also important. S2 and others are also interesting.
4D effective theory Higher dimensional Lagrangian (e.g. 10D)
integrate the compact space ⇒ 4D theory
Coupling is obtained by the overlap
integral of wavefunctions
)()()(6 yyyydgY
),(),(),( 64
10 yxyxAyxyxddgL
)()()(4
4 xxxxdYL
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 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
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