extra dimensional models with magnetic fluxes tatsuo kobayashi
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Extra Dimensional Models Extra Dimensional Models with Magnetic Fluxes with Magnetic Fluxes Tatsuo KobayashiTatsuo Kobayashi1.1. IntroductionIntroduction2. Magnetized extra dimensions2. Magnetized extra dimensions3. Models3. Models44 .. N-point couplings and flavor symmetriesN-point couplings and flavor symmetries5. Summary5. Summary based on based on Abe, T.K., Ohki, arXiv: 0806.4748 Abe, T.K., Ohki, arXiv: 0806.4748 Abe, Choi, T.K., Ohki, 0812.3534, 0903.3800, 0904.26Abe, Choi, T.K., Ohki, 0812.3534, 0903.3800, 0904.26
31, 0907.5274,31, 0907.5274, Choi, T.K., Maruyama, Murata, Nakai, Ohki, Sakai, 0908.Choi, T.K., Maruyama, Murata, Nakai, Ohki, Sakai, 0908.
0395 0395
1 1 IntroductionIntroduction
Extra dimensional field theories, Extra dimensional field theories,
in particular in particular
string-derived extra dimensional field string-derived extra dimensional field theories, theories,
play important roles in particle physicsplay important roles in particle physics
as well as cosmology .as well as cosmology .
Chiral theoryChiral theoryWhen we start with extra dimensional field theories, When we start with extra dimensional field theories, how to realize chiral theories is one of important isshow to realize chiral theories is one of important iss
ues from the viewpoint of particle physics. ues from the viewpoint of particle physics.
Zero-modes between chiral and anti-chiral Zero-modes between chiral and anti-chiral fields are different from each other fields are different from each other on certain backgrounds, e.g. CY.on certain backgrounds, e.g. CY.
0 mmDi
Torus with magnetic flux Torus with magnetic flux
The limited number of solutions with The limited number of solutions with
non-trivial backgrounds are known.non-trivial backgrounds are known.
Torus background with magnetic flux Torus background with magnetic flux
is one of interesting backgrounds, is one of interesting backgrounds,
where one can solve zero-mode where one can solve zero-mode
Dirac equation.Dirac equation.
0 mmDi
Magnetic fluxMagnetic fluxIndeed, several studies have been done Indeed, several studies have been done in both extra dimensional field theories in both extra dimensional field theories and string theories with magnetic flux and string theories with magnetic flux background.background.In particular, magnetized D-brane models In particular, magnetized D-brane models are T-duals of intersecting D-brane models.are T-duals of intersecting D-brane models.Several interesting models have been Several interesting models have been constructed in intersecting D-brane models, constructed in intersecting D-brane models, that is, that is, the starting theory is U(N) SYM.the starting theory is U(N) SYM.
Magnetized D-brane modelsMagnetized D-brane models
The (generation) number of zero-modes The (generation) number of zero-modes
is determined by the size of magnetic is determined by the size of magnetic flux. flux.
Zero-mode profiles are quasi-localized.Zero-mode profiles are quasi-localized.
=> several interesting => several interesting phenomenologyphenomenology
Phenomenology of magnetized Phenomenology of magnetized brane modelsbrane models
It is important to study phenomenological It is important to study phenomenological aspects of magnetized brane models such as aspects of magnetized brane models such as massless spectra from several gauge groups, massless spectra from several gauge groups, U(N), SO(N), E6, E7, E8, ...U(N), SO(N), E6, E7, E8, ... Yukawa couplings and higher order n-pointYukawa couplings and higher order n-point couplings in 4D effective theory, couplings in 4D effective theory, their symmetries like flavor symmetries, their symmetries like flavor symmetries, Kahler metric, etc.Kahler metric, etc.It is also important to extend such studies It is also important to extend such studies on torus background to other backgrounds on torus background to other backgrounds with magnetic fluxes, e.g. orbifold backgrounds.with magnetic fluxes, e.g. orbifold backgrounds.
2. Extra dimensions with magnetic 2. Extra dimensions with magnetic fluxes: basic toolsfluxes: basic tools
2-1. Magnetized torus model2-1. Magnetized torus model
We start with N=1 super Yang-Mills theory We start with N=1 super Yang-Mills theory in D = 4+2n dimensions. in D = 4+2n dimensions. We consider 2n-dimensional torus compactificWe consider 2n-dimensional torus compactific
ation ation with magnetic flux background.with magnetic flux background.
Higher Dimensional SYM theory with flux Cremades, Ibanez, MarcCremades, Ibanez, Marchesano, ‘04hesano, ‘04
The wave functionsThe wave functions eigenstates of correspondinginternal Dirac/Laplace operator.
4D Effective theory <= dimensional reduction
Higher Dimensional SYM theory with flux
AbelianAbelian gauge field on magnetized torusgauge field on magnetized torus
Constant magnetic flux
The boundary conditions on torus (transformation under torus translations)
gauge fields of background
Higher Dimensional SYM theory with flux
We now consider a complex field with charge Q ( +/-1 )
Consistency of such transformations under a contractible loop in torus which implies Dirac’s quantization conditions.
Dirac equation
with twisted boundary conditions (Q=1)
is the two component spinor.
|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
(Ablian flux case )
Bi-fundamentalBi-fundamentalGaugino fields in off-diagonal entries Gaugino fields in off-diagonal entries correspond to bi-fundamental matter fields correspond to bi-fundamental matter fields and the difference M= m-m’ of magnetic and the difference M= m-m’ of magnetic fluxes appears in their Dirac equation.fluxes appears in their Dirac equation.
F F
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,
2-2. Wilson lines 2-2. Wilson lines Cremades, Ibanez, Marchesano, ’04, Cremades, Ibanez, Marchesano, ’04, Abe, Choi, T.K. Ohki, ‘09Abe, Choi, T.K. Ohki, ‘09 torus without magnetic fluxtorus without magnetic flux constant Ai constant Ai mass shift mass shift every modes massiveevery modes massive magnetic fluxmagnetic flux
the number of zero-modes is the same.the number of zero-modes is the same. the profile: f(y) the profile: f(y) f(y +a/M) f(y +a/M) with proper b.c.with proper b.c.
0 )(2 aMy
U(1)a*U(1)b theory U(1)a*U(1)b theory magnetic flux, Fa=2πM, Fb=0magnetic flux, Fa=2πM, Fb=0 Wilson line, Aa=0, Ab=CWilson line, Aa=0, Ab=C matter fermions with U(1) charges, (Qa,Qb)matter fermions with U(1) charges, (Qa,Qb) chiral spectrum, chiral spectrum, for Qa=0, massive due to nonvanishing WLfor Qa=0, massive due to nonvanishing WL when MQa >0, the number of zero-modeswhen MQa >0, the number of zero-modes is MQa.is MQa. zero-mode profile is shifted depending zero-mode profile is shifted depending on Qb, on Qb,
))/(( )( ab MQCQzfzf
2-3. Magnetized orbifold models2-3. Magnetized orbifold modelsWe consider orbifold compactification We consider orbifold compactification with magnetic flux.with magnetic flux.
Orbifolding is another way to obtain chiral theorOrbifolding is another way to obtain chiral theory.y.
Magnetic flux is invariant under the Z2 twist.Magnetic flux is invariant under the Z2 twist.
We consider the Z2 and Z2xZ2’ orbifolds.We consider the Z2 and Z2xZ2’ orbifolds.
Orbifold with magnetic fluxOrbifold with magnetic flux Abe, T.K., Ohki, ‘08Abe, T.K., Ohki, ‘08
Note that there is no odd massless modes Note that there is no odd massless modes on the orbifold without magnetic flux.on the orbifold without magnetic flux.
)()(:2 zzZ jMM
jM
jMM
jM
jMM
jM
:mode odd Z
:modeeven Z
2
2
ziyyzZ )(: 542
Zero-modesZero-modesEven and/or odd modes are allowed Even and/or odd modes are allowed as zero-modes on the orbifold with as zero-modes on the orbifold with magnetic flux.magnetic flux. On the usual orbifold without magnetic flux,On the usual orbifold without magnetic flux, odd zero-modes correspond only to odd zero-modes correspond only to massive modes.massive modes.
Adjoint matter fields are projected by Adjoint matter fields are projected by orbifold projection.orbifold projection.
Orbifold with magnetic fluxOrbifold with magnetic flux Abe, T.K., Ohki, ‘08Abe, T.K., Ohki, ‘08
The number of even and odd zero-modesThe number of even and odd zero-modes
We can also embed Z2 into the gauge We can also embed Z2 into the gauge space.space.
=> various models, various flavor => various models, various flavor structuresstructures
Localized modes on fixed Localized modes on fixed pointspoints
We have degree of freedom to We have degree of freedom to introduce localized modes on fixed points introduce localized modes on fixed points like quarks/leptons and higgs fields.like quarks/leptons and higgs fields.
That would lead to richer flavor structure.That would lead to richer flavor structure.
2-4. Orbifold with M.F. and W.L.2-4. Orbifold with M.F. and W.L. Abe, Choi, T.K., Ohki, ‘09Abe, Choi, T.K., Ohki, ‘09 Example: U(1)Example: U(1)aa x SU(2) theory x SU(2) theory SU(2) doublet with charge qSU(2) doublet with charge qaa
zero-modeszero-modes
the number of zero-modes = Mthe number of zero-modes = M
01
10 twist orbifold P
)()( ,2/1
,2/1 zz MjMMj
2/1
2/1
Another basisAnother basis
zero-modeszero-modes
the total number of zero-modes = Mthe total number of zero-modes = M
10
01' twist orbifold P
)(')('
)(')(',
2/1,
2/1
,2/1
,2/1
zz
zzMjMMj
MjMMj
2/1
2/1
'
'
Wilson linesWilson lines
zero-mode profileszero-mode profiles
10
01direction Cartan along lineWilson
))2/(())2/(( ,2/1
,2/1 MCzMCz bMjMbMj
SU(2) triplet SU(2) triplet
Wilson line along the Cartan directionWilson line along the Cartan directionzero-modeszero-modes
the number of zero-modes the number of zero-modes = M= M for the formerfor the former < M for the latter< M for the latter
001
010
100
twist orbifold P
)()(
)/()/(,
0,
0
,1
,1
zz
MCzMCzMjMMj
bMjMbMj
1
0
1
Orbifold, M.F. and W.L.Orbifold, M.F. and W.L. We can consider larger gauge groups We can consider larger gauge groups and several representations.and several representations. Non-trivial orbifold twists and Wilson linesNon-trivial orbifold twists and Wilson lines ⇒ ⇒ various models various models
Non-Abelian W.L. + fractional magnetic fluxesNon-Abelian W.L. + fractional magnetic fluxes (‘t Hooft toron background)(‘t Hooft toron background) ⇒ ⇒ interesting aspectsinteresting aspects Abe, Choi, T.K., Ohki, work in progressAbe, Choi, T.K., Ohki, work in progress
3. Models3. Models We can construct several models by using We can construct several models by using the above model building tools. the above model building tools. What is the starting theory ?What is the starting theory ? 10D SYM or 6D SYM (+ hyper multiplets), 10D SYM or 6D SYM (+ hyper multiplets), gauge groups, U(N), SO(N), E6, E7,E8,...gauge groups, U(N), SO(N), E6, E7,E8,... What is the gauge background ?What is the gauge background ? the form of magnetic fluxes, Wilson lines.the form of magnetic fluxes, Wilson lines. What is the geometrical background ?What is the geometrical background ? torus, orbifold, etc.torus, orbifold, etc.
U(N) theory on T6U(N) theory on T6
gauge group gauge group
kNk
N
zz
m
m
iF
0
0
211
matrixidentity)(: NNN
k
iiNUNU
1
)()(
54 iyyz
U(N) SYM theory on T6U(N) SYM theory on T6
Pati-Salam group up to U(1) factorsPati-Salam group up to U(1) factors
Three families of matter fields Three families of matter fields with many Higgs fieldswith many Higgs fields Orbifolding can lead to various 3-generation PS models.Orbifolding can lead to various 3-generation PS models. See See Abe, Choi, T.K., Ohki, ‘08Abe, Choi, T.K., Ohki, ‘08
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
21321
mmmm
Tmmmm
2,1,41,2,4
E6 SYM theory on T6E6 SYM theory on T6 Choi, et. al. ‘09Choi, et. al. ‘09 We introduce magnetix flux along U(1) direction, We introduce magnetix flux along U(1) direction,
which breaks E6 -> SO(10)*U(1)which breaks E6 -> SO(10)*U(1)
Three families of chiral matter fields 16Three families of chiral matter fields 16 We introduce Wilson lines breaking We introduce Wilson lines breaking SO(10) -> SM group.SO(10) -> SM group.Three families of quarks and leptons matter fields Three families of quarks and leptons matter fields with no Higgs fieldswith no Higgs fields
1100 161614578
1 ,1 ,3 321 mmm
Splitting zero-mode profilesSplitting zero-mode profilesWilson lines do not change the Wilson lines do not change the
(generation) number of zero-modes, (generation) number of zero-modes, but change localization point.but change localization point.
1616
QQ …… …… LL
E6 SYM theory on T6E6 SYM theory on T6 There is no electro-weak Higgs fieldsThere is no electro-weak Higgs fields By orbifolding, we can derive a similar model By orbifolding, we can derive a similar model with three generations of 16. with three generations of 16.
On the orbifold, there is singular points, i.e. On the orbifold, there is singular points, i.e. fixed points.fixed points.
We could assume consistently that We could assume consistently that electro-weak Higgs fields are localized modes electro-weak Higgs fields are localized modes on a fixed point.on a fixed point.
E7, E8 SYM theory on T6E7, E8 SYM theory on T6 Choi, et. al. ‘09Choi, et. al. ‘09 E7 and E8 have more ranks (U(1) factors) E7 and E8 have more ranks (U(1) factors) than E6 and SO(10). than E6 and SO(10). Those adjoint rep. include various matter fields.Those adjoint rep. include various matter fields.
Then, we can obtain various models including Then, we can obtain various models including MSSM + vector-like matter fieldsMSSM + vector-like matter fields See for its detail our coming paper.See for its detail our coming paper.
3.3. N-point couplings N-point couplings and flavor symmetries and flavor symmetries
The N-point couplings are obtained by The N-point couplings are obtained by overlap integral of their zero-mode w.f.’s.overlap integral of their zero-mode w.f.’s.
)()()(2 zzzzdgY kP
jN
iM
Zero-modes Zero-modes Cremades, Ibanez, Marchesano, ‘04Cremades, Ibanez, Marchesano, ‘04 Zero-mode w.f. = gaussian x theta-functionZero-mode w.f. = gaussian x theta-function
up to normalization factor up to normalization factor
),(0
/)]Im(exp[)( iMMz
MjzMziNz M
jM
,)()()(1
NM
m
MmjiNMijm
jN
iM zyzz
))(,0(0
))(/()(NMiMN
NMMNMNmMjNiyijm
MjNM ,,1 factor,ion normalizat:
3-point couplings3-point couplings Cremades, Ibanez, Marchesano, ‘04Cremades, Ibanez, Marchesano, ‘04
The 3-point couplings are obtained by The 3-point couplings are obtained by overlap integral of three zero-mode w.f.’s.overlap integral of three zero-mode w.f.’s.
up to normalization factor up to normalization factor
*2 )()()( zzzzdY k
NMjN
iMijk
NM
mijmkmMjiijk yY
1,
ikkM
iM zzzd *2 )()(
Selection rule Selection rule
Each zero-mode has a Zg charge, Each zero-mode has a Zg charge, which is conserved in 3-point couplings.which is conserved in 3-point couplings.
up to normalization factor up to normalization factor
)(, NMkmMjikmMji
))(,0(0
))(/()(NMiMN
NMMNMNmMjNiyijm
),gcd( when mod NMggkji
4-point couplings4-point couplings Abe, Choi, T.K., Ohki, ‘09Abe, Choi, T.K., Ohki, ‘09 The 4-point couplings are obtained by The 4-point couplings are obtained by overlap integral of four zero-mode w.f.’s.overlap integral of four zero-mode w.f.’s. splitsplit
insert a complete setinsert a complete set
up to normalization factor up to normalization factor for K=M+Nfor K=M+N
*2 )()()()( zzzzzdY l
PNMkP
jN
iMijkl
modes all
*)'()()'( zzzz n
KnK
*22 )'()'()'()()(' zzzzzzzzdd l
PNMkP
jN
iM
lsksijs
lijk yyY
4-point couplings: another 4-point couplings: another splittingsplitting
i k i ki k i k
t t
j s l j lj s l j l
*22 )'()'()'()()(' zzzzzzzzdd l
PNMjN
kP
iM
ltjtikt
lijk yyY
ltjtikt
lijk yyY lsksij
slijk yyY
N-point couplingsN-point couplings Abe, Choi, T.K., Ohki, ‘09 Abe, Choi, T.K., Ohki, ‘09 We can extend this analysis to generic n-point coupliWe can extend this analysis to generic n-point coupli
ngs.ngs. N-point couplings = products of 3-point couplingsN-point couplings = products of 3-point couplings = products of theta-functions= products of theta-functions
This behavior is non-trivial. (It’s like CFT.) This behavior is non-trivial. (It’s like CFT.) Such a behavior wouldSuch a behavior would be satisfied be satisfied not for generic w.f.’s, but for specific w.f.’s.not for generic w.f.’s, but for specific w.f.’s. However, this behavior could be expected However, this behavior could be expected from T-duality between magnetized from T-duality between magnetized and intersecting D-brane models.and intersecting D-brane models.
T-dualityT-duality The 3-point couplings coincide between The 3-point couplings coincide between magnetized and intersecting D-brane models. magnetized and intersecting D-brane models. explicit calculationexplicit calculation Cremades, Ibanez, Marchesano, ‘04Cremades, Ibanez, Marchesano, ‘04 Such correspondence can be extended to Such correspondence can be extended to 4-point and higher order couplings because of 4-point and higher order couplings because of CFT-like behaviors, e.g., CFT-like behaviors, e.g.,
Abe, Choi, T.K., Ohki, ‘09 Abe, Choi, T.K., Ohki, ‘09
lsksijs
lijk yyY
Heterotic orbifold modelsHeterotic orbifold models
Our results would be useful to n-point couplings Our results would be useful to n-point couplings of twsited sectors in heterotic orbifold models.of twsited sectors in heterotic orbifold models.
Twisted strings on fixed points might correspond Twisted strings on fixed points might correspond to quasi-localized modes with magnetic flux, to quasi-localized modes with magnetic flux, zero modes profile = gaussian x theta-function zero modes profile = gaussian x theta-function
amplitude string closedamplitude stringopen 2
orbifold heterotic
in coupling
brane ngintersecti
in couplings2
Non-Abelian discrete flavor symmetryNon-Abelian discrete flavor symmetry
The coupling selection rule is controlled by The coupling selection rule is controlled by Zg charges.Zg charges. For M=g, For M=g, 1 2 g 1 2 g
Effective field theory also has a cyclic permutation Effective field theory also has a cyclic permutation
symmetry of g zero-modes. symmetry of g zero-modes.
Non-Abelian discrete flavor symmetryNon-Abelian discrete flavor symmetryThe total flavor symmetry corresponds to The total flavor symmetry corresponds to the closed algebra of the closed algebra of
That is the semidirect product of Zg x Zg and Zg.That is the semidirect product of Zg x Zg and Zg. For example, For example, g=2 D4g=2 D4 g=3 Δ(27)g=3 Δ(27) Cf. heterotic orbifolds, Cf. heterotic orbifolds, T.K. Raby, Zhang, ’04T.K. Raby, Zhang, ’04 T.K. Nilles, Ploger, Raby, Ratz, ‘06T.K. Nilles, Ploger, Raby, Ratz, ‘06
]/2exp[ gi
0001
10
00
0010
,
1
1
g
SummarySummaryWe have studiedWe have studied phenomenological aspects phenomenological aspects of magnetized brane models.of magnetized brane models.
Model building from U(N), E6, E7, E8Model building from U(N), E6, E7, E8
N-point couplings are comupted.N-point couplings are comupted. 4D effective field theory has non-Abelian flavor 4D effective field theory has non-Abelian flavor symmetries, e.g. D4, Δ(27).symmetries, e.g. D4, Δ(27). Orbifold background with magnetic flux is Orbifold background with magnetic flux is also important. also important.
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