4D low-energy effective field theory from magnetized D-brane models Tatsuo Kobayashi

1
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

2
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.
• .

3
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.

4
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

5
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)

6
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.Cvetics talk
• Lets discuss the merge between the two sides.

7
Plan

• ?1.Introduction
• 2. Intersecting/magnetized D-brane models
• 3. N-point couplings and flavor symmetries
• 4.Massive modes
• 5. Moduli
• 6. Summary

8
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)xU(1)
• U(1) ? U(2)
  

9
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.
  

10
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
  
• 
  These are
• 
  localized modes
• Twisted boundary condition

11
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

12
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

13
Generation number

• Torus compactification
• Family number intersection number
• su(2)
  
• 
  
• 
  
• 
  
• Q1 Q2 Q3
  su(3)
• on T2
  

14
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

15
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.

16
Toy model on T2xT2xT2

• Intersecting number family number
• u(1)xu(1) su(2)
• su(3) H
• u(1) Q 3
• L 3 3
• 3 u,d
• e,
  neutrino

17
Hidden sector

• u(1) su(2)
• su(3) H
• Q 3
• L 3 3
• 3 u,d

18
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
• L 3 3
• 3 3 u,d

19
Other configurations

• Other types of D-brane configurations
• are also interesting.
• ,
  ,
• etc

20
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.

21
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.

22
Some extensions

• orbifold, Calabi-Yau, etc.

23
2-2. Magnetized D-branes

• We consider torus compactification
• with magnetic flux background.
• F

24
Boundary conditions on magnetized D-branes

• similar to the boundary condition of
• open string between intersecting D-branes
• T-dual to (type IIA) intersecting D-brane
  models

25
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

26
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

27
2-3-1 Field theory in higher dimensions
generic aspects

• 10D ? 4D our space-time 6D space
• 10D vector
• 4D vector 4D scalars
• SO(10) spinor ? SO(4) spinor
• x SO(6) spinor
• internal quantum
  number

28
Several Fields in higher dimensions

• 4D (Dirac) spinor
• ? (4D) Clifford algebra
• (4x4) gamma matrices
• represention space ? spinor representation
• 6D Clifford algebra
• 6D spinor
• 6D spinor ? 4D spinor x (internal spinor)
• 
  internal quantum
• 
  number

29
Field theory in higher dimensions

• Mode expansions
• KK decomposition

30
KK docomposition on torus

• torus with vanishing gauge background
• Boundary conditions
• First, we concentrate on zero-modes.

31
Zero-modes

• Zero-mode equation
• ? non-trival zero-mode profile
• the number of zero-modes

32
4D effective theory

• Higher dimensional Lagrangian (e.g. 10D)
• integrate the compact space ? 4D theory
• Coupling is obtained by the overlap
• integral of wavefunctions

33
Couplings in 4D

• Zero-mode profiles are quasi-localized
• far away from each other in compact space
• ? suppressed couplings

34
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.

35
2-3-2 Higher Dimensional SYM theory with flux
Cremades, Ibanez, Marchesano, 04
4D Effective theory lt dimensional reduction
eigenstates of corresponding internal
Dirac/Laplace operator.
The wave functions
36
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)
37
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
Diracs quantization conditions.
38
Dirac equation on 2D torus
is the two component spinor.
U(1) charge Q1
with twisted boundary conditions (Q1)
39
Dirac equation and chiral fermion
M independent zero mode solutions in Dirac
equation.
(Theta function)
Properties of theta functions
chiral fermion
By introducing magnetic flux, we can obtain
chiral theory.
zero-modes
no zero-mode
40
Wave functions
For the case of M3
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.
41
Fermions in bifundamentals
Breaking the gauge group
(Abelian flux case )
The gaugino fields
gaugino of unbroken gauge
bi-fundamental matter fields
42
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

43
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
44
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

45
(Simple) U(8) SYM theory on T6

• Pati-Salam group up to U(1) factors
• Three families of matter fields
• with many Higgs fields

46
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.

47
U(1)aU(1)b theory

• magnetic flux, Fa2pM, Fb0
• Wilson line, Aa0, AbC
• matter fermions with U(1) charges, (Qa,Qb)
• chiral spectrum,
• for Qa0, massive due to nonvanishing WL
• when MQa gt0, the number of zero-modes
• is MQa.
• zero-mode profile is shifted depending
• on Qb,

48
Pati-Salam model

• Pati-Salam group
• WLs along a U(1) in U(4) and a U(1) in U(2)R
• gt Standard gauge group up to U(1) factors
• U(1)Y is a linear combination.

49
PS gt SM

• Zero modes corresponding to
• three families of matter fields
• remain after introducing WLs, but their profiles
  split
• (4,2,1)
• 
  Q L

50
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.
• gt various models, various flavor structures

51
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.

52
S2 with magnetic flux

• Conlon, Maharana, Quevedo, 08
• Fubuni-Study metric
• Zero-mode eq.
• with spin
• connection
• M gt 0
• M0

53
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.

54
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

55
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.

56
Moduli

• Torus metric
• Area
• We can repeat the previous analysis.
• Scalar and vector fields have the same
• wavefunctions.
• Wilson moduli
• shift of w.f.

57
Zero-modes

• Cremades, Ibanez, Marchesano, 04
• Zero-mode w.f. gaussian x theta-function
• Product of zero-mode wavefunctions

58
Products of wave functionsHint to understand

• 
  
• products of zero-modes zero-modes

59
3-point couplings

• Cremades, Ibanez, Marchesano,
  04
• The 3-point couplings are obtained by
• overlap integral of three zero-mode w.f.s.

60
Selection rule

• Each zero-mode has a Zg charge,
• which is conserved in 3-point couplings.
• up to normalization factor

61
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
• 
  for KMN

62
4-point couplings another splitting

• i k i
  k
• t
  
• j s l j
  l

63
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. (Its 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.

64
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
  

65
3.2 Non-Abelian discrete flavor
symmetry

• The coupling selection rule is controlled by Zg
  charges.
• For Mg, 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 ?(27) Abe, Choi, T.K, Ohki, 09
  
• Cf. heterotic orbifolds, T.K. Raby, Zhang, 04
• T.K. Nilles,
  Ploger, Raby, Ratz, 06

66
Permutation symmetry D-brane models

• Abe, Choi,
  T.K. Ohki, 09, 10
• 
  
• 
  
• 
  
• There is a Z2 permutation symmetry.
• The full symmetry is D4.
  

67
Permutation symmetry D-brane models

• Abe, Choi, T.K. Ohki, 09,
  10
• 
  
• 
  
• 
  
• geometrical symm. Full symm.
• Z3 ?(27)
  
  
• S3 ?(54)

68
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 Zgs.
• and Zg permutation
• Certain case Zg permutation larger symm.
  Like Dg

69
Magnetized brane-models

• Magnetic flux M D4
• 2 2
• 4 1 1- 1- 1--
  
• Magnetic flux M ?(27) (?(54))
• 3 31
• 6 2 x 31
  
• 9 ?1n
  n1,,9
• (11?2n
  n1,,4)

70
Discrete flavor symmetry

• ZN symmetry is originated from
• anomalous U(1) symmetries.
• Berasatuce-Gonzalez, Camara, Marchesano,
• Regalado, Uranga, 12

71
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, ?(27), ......
• Review e.g
• Ishimori, T.K., Ohki, Okada, Shimizu, Tanimoto
  10
• ? large mixing angles
• one Ansatz tri-bimaximal

72
Quark masses and mixing angles

• These masses are obtained by Yukawa couplings
• to the Higgs field with VEV, v175GeV.
• strong Yukawa coupling ? large mass
• weak ? small mass
• top Yukawa coupling O(1)
• other quarks ? suppressed Yukawa couplings

73
Lepton masses and mixing angles

• mass squared differences and mixing angles
• consistent with neutrino oscillation
• large mixing angles

74
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.

75
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.

76
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.

77
Fermion massive modes

• Two components are mixed.
• 2D Laplace op.
• algebraic relations
• It looks like the quantum harmonic oscillator

78
Fermion massive modes

• Creation and annhilation operators
• mass spectrum
• wavefunction

79
Fermion massive modes

• explicit wavefunction
• Hn Hermite function
• Orthonormal condition

80
Scalar and vector modes

• The wavefunctions of scalar and vector fields
• are the same as those of spinor fields.
• Mass spectrum
• scalar
• vector
• 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.
  

81
Products of wavefunctions

• explicit wavefunction
• See also Berasatuce-Gonzalez, Camara,
  Marchesano,
• Regalado, Uranga, 12
• Derivation
• products of zero-mode wavefunctions
• We operate creation operators on both LHS
• and RHS.

82
3-point couplings including higher modes

• The 3-point couplings are obtained by
• overlap integral of three wavefunctions.
• (flavor) selection rule
• is the same as one for the massless
  modes.
• (mode number) selection rule

83
3-point couplings2 zero-modes and one higher
mode

• 3-point coupling

84
Higher order couplings including higher modes

• Similarly, we can compute higher order
  couplings
• including zero-modes and higher modes.
• They can be written by the sum over
• products of 3-point couplings.

85
3-point couplings including massive modes only
due to Wilson lines

• Massive modes appear only due to Wilson lines
• without magnetic flux
• We can compute the 3-point coupling
• e.g.
• Gaussian function for the Wilson line.
  

86
3-point couplings including massive modes only
due to Wilson lines

• For example, we have
• for

87
Several couplings

• Similarly, we can compute the 3-point couplings
• including higher modes
• Furthermore, we can compute higher order
• couplings including several modes, similarly.

88
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.

89
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.

90
SU(5) gt 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
• 
  Q L

91
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,
• ?

92
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

93
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.

94
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.

95
Gauge kinetic function

• For simplicity, we consider the factorizable
  torus,
• T2xT2xT2.
• SUSY condition
  
• 
  Berkooz, Douglas, Leigh, 96
• DBI ?
• Lust, Mayr, Richter,
  Stieberger, 04

96
Non-perturbative terms

• Non-perturbative effects such as gaugino
  condensation
• would induce terms like
• D-brane instanton effects
• 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.

97
3-form flux compactification

• The 3-form flux may stabilize the dilaton and
• complex structure moduli.

98
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.

99
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 ?

100
Discussions

• The moduli stabilization ?
• Inflation ? Axions ?
• Lets kick off to merge two approaches.

101
(2-form) magnetic fluxes

• SUSY condition
  
• may stabilize some of Kahler moduli ?
  
• Antoniadis, Maillar, 04
• Antoniadis, Kumar,
  Maillard, 06