Toward realistic string models: Long and Winding Roads Tatsuo Kobayashi Toward realistic string...
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models: models: Long and Winding Roads Long and Winding Roads Tatsuo Kobayashi Tatsuo Kobayashi １． １． Introduction Introduction 2. String models 2. String models 3 3 ． ． Parameters in the SM Parameters in the SM 4 4 ． ． Summary Summary
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Transcript of Toward realistic string models: Long and Winding Roads Tatsuo Kobayashi Toward realistic string...
Toward realistic string models: Toward realistic string models: Long and Winding Roads Long and Winding Roads Tatsuo Tatsuo KobayashiKobayashi
１．１． IntroductionIntroduction
2. String models2. String models
33 ． ． Parameters in the SMParameters in the SM
44 ． ． SummarySummary
1. Introduction1. Introduction
several (massless) modes of superstring :several (massless) modes of superstring :
gravitongraviton
higher dim. higher dim. ＳＹＭ （ＳＹＭ （ e.g. e.g. Ｅ８Ｅ８ , SO(32),SU(N), SO(32),SU(N) ）） including SM gauge bosons and matter including SM gauge bosons and matter fieldsfields
corresponding to quarks and leptons corresponding to quarks and leptons They include all of what we need.They include all of what we need.
A promising candidate for unified theory A promising candidate for unified theory
including gravityincluding gravity
Superstring theory = Theory of EverythingSuperstring theory = Theory of Everything
(anything ?)(anything ?)
String theory as fundamental String theory as fundamental theorytheory
Superstring theorySuperstring theory ： ： a candidate of unified a candidate of unified theory theory
including gravityincluding gravity
If it is really relevant to particle physics, If it is really relevant to particle physics,
Superstring Superstring the 4D Standard Model of the 4D Standard Model of
particle physics (at low energy)particle physics (at low energy)
including values of including values of parameters, parameters,
gauge couplings, Yukawa couplings, gauge couplings, Yukawa couplings,
Higgs sector (parameters), etc. Higgs sector (parameters), etc.
(Too) many string vacua (Too) many string vacua (models)(models) ProblemProblem ：： There are innumberable 4D string vacua There are innumberable 4D string vacua (models) (models)
Study on nonperturbative aspects to lift Study on nonperturbative aspects to lift up up
many vacua and/or to find out a principle many vacua and/or to find out a principle
leading to a unique vacuumleading to a unique vacuum
Study on particle phenomenological Study on particle phenomenological aspects of aspects of
already known 4D string vacuaalready known 4D string vacua
String phenomenologyString phenomenology
String phenomenologyString phenomenology Which class of string models lead to Which class of string models lead to
realistic aspects of particle physicsrealistic aspects of particle physics ？ ？ We do not need to care about string vacua We do not need to care about string vacua
without leading to e.g. without leading to e.g.
the top quark massthe top quark mass ＝ １７ ＝ １７ 22 GeVGeV
the electron massthe electron mass ＝ ０．５ ＝ ０．５ MeV,MeV,
no matter how many vacua exist.no matter how many vacua exist.
LetLet’’s study whether we can construct 4D string s study whether we can construct 4D string vacuavacua
really relevant to our Nature. really relevant to our Nature.
String phenomenologyString phenomenologySuperstringSuperstring ： ： theory around the Planck scaletheory around the Planck scale
SUSYSUSY ？ ？ GUTGUT ？ ？ ？？？？？ ？？？？？ Several scenarioSeveral scenario ？？？？？？ ？？？？？？Standard ModelStandard Model ： ： we know it up to 100 GeVwe know it up to 100 GeV
Long and winding roadsLong and winding roads
SuperstringSuperstring → → low energylow energy ？ （ ？ （ top-downtop-down ） ） Low energyLow energy → → underlying theoryunderlying theory ？ （ ？ （ bottom-ubottom-u
pp ）） We need more information from a bottom-up approach.We need more information from a bottom-up approach.
Both approaches are necessary to Both approaches are necessary to
connect between underlying theory and our Nature. connect between underlying theory and our Nature.
Cosmological aspectsCosmological aspects Cosmological constant (Dark energy)Cosmological constant (Dark energy)
Dark matter Dark matter
InflationInflation
Axion(s)Axion(s)
..................................................................
Several steps toward string Several steps toward string phenomenologyphenomenology
Massless modes Massless modes gauge bosons (gauge symmetry), gauge bosons (gauge symmetry), matter fermions, higgs bosons, matter fermions, higgs bosons, moduli fields, ............moduli fields, ............
Their effective action Their effective action gauge couplings, Yukawa couplings, ......... gauge couplings, Yukawa couplings, ......... Kahler potential (kinetic terms)Kahler potential (kinetic terms) (discrete/flavor) symmetry, ......(discrete/flavor) symmetry, ...... moduli stabilization, SUSY breaking, moduli stabilization, SUSY breaking, soft SUSY breaking terms, soft SUSY breaking terms, cosmology, ......cosmology, ......
2. String models and massless 2. String models and massless spectraspectra
Several string modelsSeveral string models KyaeKyae’’s talks talk
(before D-brane )(before D-brane ) 11stst string revolution string revolutionHeterotic models on Calabi-Yau manifold, Heterotic models on Calabi-Yau manifold, Orbifolds, Orbifolds,
fermionic construction, fermionic construction, Gepner, . . . . . . . . . . .Gepner, . . . . . . . . . . .( after D-brane ) 2( after D-brane ) 2ndnd string revolution string revolution type II models with D-branes type II models with D-branes Kitazawa Kitazawa’’s s talk talk
Intersecting D-brane models, Intersecting D-brane models, Magnetized D-branes, . . . . . . . . . . .Magnetized D-branes, . . . . . . . . . . . Nowadays Nowadays F-theory models F-theory models Hayashi Hayashi’’s and Chois and Choi’’s talks talk
OrbifoldsOrbifolds
T2/Z3 OrbifoldT2/Z3 Orbifold
There are three fixed points on Z3There are three fixed points on Z3 orbifoldorbifold
(0,0), (2/3,1/3), (1/3,2/3) su(3) root (0,0), (2/3,1/3), (1/3,2/3) su(3) root latticelattice
Orbifold = D-dim. Torus /twistOrbifold = D-dim. Torus /twist
Torus = D-dim flat space/ lattice Torus = D-dim flat space/ lattice
Intersecting D-brane modelsIntersecting D-brane models
gauge bosonsgauge bosons ： ： on brane on brane
quarks, leptons, higgs : quarks, leptons, higgs :
localized at intersecting localized at intersecting pointspoints
u(1) su(2)u(1) su(2)
su(3) su(3) HH
Q Q
LL
u,du,d
Massless spectraMassless spectra ⇒ ⇒ predict 6 extra dimensions predict 6 extra dimensions
in addition to our 4D space-timesin addition to our 4D space-times
Compact space (background)Compact space (background) 、、 D-brane configurationD-brane configuration 、、 ......
← ← constrained by string theoryconstrained by string theory (modular invariance, RR-charge cancellation,...)(modular invariance, RR-charge cancellation,...)
Once we choose background, all of modes can be Once we choose background, all of modes can be
investigated in principle (at the perturbative level).investigated in principle (at the perturbative level).
⇒ ⇒ Massless modes, which appear in low-energy Massless modes, which appear in low-energy
effective field theory, are completely determined.effective field theory, are completely determined.
Oscillations and momenta in compact space correspond Oscillations and momenta in compact space correspond
to quantum numbers of particles in 4d theory.to quantum numbers of particles in 4d theory.
It is not allowed to add/reduce some modes by hand.It is not allowed to add/reduce some modes by hand.
Massless spectraMassless spectra One can solve string in certain classes of One can solve string in certain classes of models, models,
heterotic string on orbifolds, Gepner heterotic string on orbifolds, Gepner manifolds, manifolds,
fermionic construction, ............................,fermionic construction, ............................,
type II intersecting D-brane models on type II intersecting D-brane models on torus, torus, …………
In this sense, these models are important In this sense, these models are important
(but these are special models).(but these are special models).
Other compactificationsOther compactifications (4D string (4D string models/vacua)models/vacua)Calabi-Yau and othersCalabi-Yau and others
Only topological aspects are known. Only topological aspects are known.
One can not solve string on such CY.One can not solve string on such CY.
First we take (10D) field-theory limit.First we take (10D) field-theory limit.
We try to solve the zero-mode equation,We try to solve the zero-mode equation,
0 mmDi
Realistic modelsRealistic modelsString model builders have done good jobs.String model builders have done good jobs.
We have constructed many (semi-)realistic We have constructed many (semi-)realistic models,models,
the SM gauge group (or its GUT extenstions) the SM gauge group (or its GUT extenstions)
three generations of quarks and leptons,three generations of quarks and leptons,
the minimal number of Higgs (or more)the minimal number of Higgs (or more)
++ hidden sector, singlets,hidden sector, singlets,
(extra matter fields charged under the SM (extra matter fields charged under the SM group)group)
For example, there are O(100)-O(1000) For example, there are O(100)-O(1000)
(semi-)realistic (heterotic orbifold) models.(semi-)realistic (heterotic orbifold) models.
Explicit Z6-II orbifold model: Pati-Explicit Z6-II orbifold model: Pati-SalamSalam T.K. Raby, Zhang T.K. Raby, Zhang ’’0404
4D massless spectrum4D massless spectrum
Gauge groupGauge group
Chiral fieldsChiral fields
Pati-Salam model with 3 generations + extra Pati-Salam model with 3 generations + extra fieldsfields
All of extra matter fields can become massiveAll of extra matter fields can become massive
)00000000)(10000111(2
)00200000)(10000001(3
)11000000)(22200000(6
5
3
W
W
V
5)1()'2()'10()2()2()4( USUSOSUSUSU
)1,1,6(2)2,1,4(:),2,2,1(6)1,1,6(6:),1,1,6()2,1,4(2:
)2,1;2,1,1(2)2,1,1(8)1,2,1(8)1,1,4(4)1,1,4(4)2,1,4(2)1,2,4(2:
)2,1,4()2,1,4(:)2,2,1(:),1,2,4(:
432
1
121
TTT
T
UUU
Explicit Z6-II orbifold model: Explicit Z6-II orbifold model: MSSMMSSM Buchmuller,et.al. Buchmuller,et.al. ’’06, Lebedev, et. al 06, Lebedev, et. al ‘‘0707
4D massless spectrum4D massless spectrum
Gauge groupGauge group
Chiral fieldsChiral fields
3 generations of MSSM + extra fields3 generations of MSSM + extra fields
All of extra matter fields can become massive All of extra matter fields can become massive
along flat directionsalong flat directions
There are O(100) models.There are O(100) models.
HY GUSUSU )1()2()3(
Explicit Z12-I orbifold model: Explicit Z12-I orbifold model: Kim, Kyae, Kim, Kyae, ‘‘0606
MSSM, Flipped SU(5), MSSM, Flipped SU(5), ……..
KyaeKyae’’s talks talk
Realistic modelsRealistic models We have just searched a narrow part of We have just searched a narrow part of string vacua.string vacua. For example, the number of certain heterotic For example, the number of certain heterotic orbifold models is finite, orbifold models is finite, Z3, Z4, Z6,Z7,Z8,Z12, ZNxZM with Wilson linesZ3, Z4, Z6,Z7,Z8,Z12, ZNxZM with Wilson lines We should analyze such string vacua We should analyze such string vacua systematicallysystematically
by computers and make their database of by computers and make their database of string vacua.string vacua. (See e.g. our telephone book, Katsuki, et.al. (See e.g. our telephone book, Katsuki, et.al. ’’90.)90.)
Flat directionsFlat directions realistic massless modes + extra modes realistic massless modes + extra modes with vector-like rep.with vector-like rep.Effective field theory has flat directions.Effective field theory has flat directions. VEVs of scalar fields along flat directions VEVs of scalar fields along flat directions ⇒⇒ vector-like rep. massive, no extra mattervector-like rep. massive, no extra matter
Such VEVs would correspond to deformation Such VEVs would correspond to deformation of orbifolds like blow-up of singular points.of orbifolds like blow-up of singular points. That is CY That is CY (as perturbation around the orbifold limit).(as perturbation around the orbifold limit).
Extra modesExtra modes Massless spectra of string models are not Massless spectra of string models are not mnimal, mnimal,
many singlets and several Higgs fields.many singlets and several Higgs fields. (Some of singlets correspond to flat directions.)(Some of singlets correspond to flat directions.)
What is their implicastion ?What is their implicastion ? (many Higgs)(many Higgs) Hetero E8xE8 Hetero E8xE8 the SM the SM many extra modesmany extra modes D-brane models have less extra modes.D-brane models have less extra modes.
Short summary on massless Short summary on massless spectraspectraOnce we choose a background( orbifold, gauge Once we choose a background( orbifold, gauge shift, wilson lines), a string model is fixed and shift, wilson lines), a string model is fixed and
its full massless spectrum can be analyzed its full massless spectrum can be analyzed
in principle.in principle.
We have 4D string models, whose massless We have 4D string models, whose massless
spectra realize spectra realize
the SM gauge group + 3 families the SM gauge group + 3 families （＋（＋ extra extra matter)matter)
and its extensions like the Pati-Salam model.and its extensions like the Pati-Salam model.
Similar situation for other compactificationsSimilar situation for other compactifications
Short summary on massless Short summary on massless spectraspectraMany models have been studied in Many models have been studied in
Z3, Z6-II, Z12 orbifolds.Z3, Z6-II, Z12 orbifolds.
the SM gauge group + 3 families the SM gauge group + 3 families （＋（＋ extra extra matter)matter)
and its extensions like the Pati-Salam model, and its extensions like the Pati-Salam model,
flipped SU(5) .flipped SU(5) .
Lots of extra matter fieldsLots of extra matter fields
3. Parameters in the Standard Model3. Parameters in the Standard ModelGauge bosonsGauge bosons SU(3)SU(3) 、 、 SU(2)SU(2) 、 、 U(1)U(1) parameters: threeparameters: three gauge couplingsgauge couplings Quarks, LeptonsQuarks, Leptons 3 families3 families hierarchical pattern of masses (mixing) hierarchical pattern of masses (mixing) Yukawa couplings to the Higgs field Yukawa couplings to the Higgs field Most of the parameters appear Most of the parameters appear from the Yukawa sector.from the Yukawa sector.Higgs fieldHiggs field the origin of massesthe origin of masses not discovered yet not discovered yet our purpose “derivation of the SM”our purpose “derivation of the SM” ⇒ ⇒ realize these massless modes and coupling realize these massless modes and coupling values values
Gauge couplingsGauge couplings experimental values experimental values RG flow ⇒RG flow ⇒ They approach each otherThey approach each other
and become similar valuesand become similar values
at high energyat high energy
In MSSM, they fit each In MSSM, they fit each other in a good accuracyother in a good accuracy
Gauge coupling unificationGauge coupling unification
Quark masses andQuark masses and mixing mixing anglesangles
These masses are obtained by Yukawa couplings These masses are obtained by Yukawa couplings
to the Higgs field with VEV, to the Higgs field with VEV, ｖ＝１７５ｖ＝１７５ GeV. GeV.
strong Yukawa coupling strong Yukawa coupling ⇒ ⇒ large masslarge mass
weakweak ⇒ ⇒ small masssmall mass
top Yukawa coupling =O(1)top Yukawa coupling =O(1)
other quarks ←other quarks ← suppressed Yukawa couplingssuppressed Yukawa couplings
004.0,04.0,22.0
8.6,3
117,2.1
3.4,172
ubcbus
du
sc
bt
VVV
MeVMMeVM
MeVMGeVM
GeVMGeVM
Lepton masses andLepton masses and mixing mixing anglesangles
mass squared differences and mixing angles mass squared differences and mixing angles
consistent with neutrino oscillationconsistent with neutrino oscillation
large mixing angleslarge mixing angles
,0.0sin,5.0sin,3.0sin
102,108
,8.1
106,5.0
132
232
122
23231
25221
eVMeVM
GeVM
MeVMMeVM e
Tri-bimaximal mixing angleTri-bimaximal mixing angle
large mixing angleslarge mixing angles
2
1
3
1
6
12
1
3
1
6
1
03
1
6
2
MNSV
Effective theoryEffective theory
Effective theory of massless modes is described Effective theory of massless modes is described
by supergravity- coupled gauge theory.by supergravity- coupled gauge theory.
In principle, we can compute gauge couplings,In principle, we can compute gauge couplings,
Yukawa couplings in string models, Yukawa couplings in string models, although although
such a computation is possible in certain such a computation is possible in certain models models
such as heterotic orbifold models, intersecting such as heterotic orbifold models, intersecting
D-brane models, etc.D-brane models, etc.
Those are functions of dilaton and moduli.Those are functions of dilaton and moduli.
3-2. Yukawa couplings 3-2. Yukawa couplings Certain modes are originated from 10D modes.Certain modes are originated from 10D modes. That is, they respect 4D N=4 (10D N=1) SUSYThat is, they respect 4D N=4 (10D N=1) SUSY vector multipletvector multiplet Yukawa couplingsYukawa couplings ← ← controlled by 4D N=4 controlled by 4D N=4 SUSY SUSY
Certain combinations are allowed: Selection Certain combinations are allowed: Selection rulerule
⇒ ⇒ Y = g = O(1) Y = g = O(1) That fits to the top Yukawa coupling That fits to the top Yukawa coupling (approximately)(approximately)
Yukawa couplingsYukawa couplings (good (good news)news)There are localized modes e.g. There are localized modes e.g. in heterotic orbifold models, in heterotic orbifold models, intersecting D-brane models, etc.intersecting D-brane models, etc. Couplings among Couplings among local fields are suppressed local fields are suppressed depending on their distance.depending on their distance. They can explain small Yukawa couplings They can explain small Yukawa couplings for light quark/leptonfor light quark/lepton ？ ？ LetLet’’s carry out stringy calculation s carry out stringy calculation (stringy selection rule)(stringy selection rule)
Coupling selection ruleCoupling selection ruleWhen three strings are localized far away When three strings are localized far away
each other, their couplings would be small.each other, their couplings would be small.
3-point coupling3-point couplingCalculate by inserting vertex op. Calculate by inserting vertex op.
corresponding to massless modescorresponding to massless modes
Yukawa couplings are suppressed by the Yukawa couplings are suppressed by the areaarea
that strings sweep to couple.that strings sweep to couple.
Those are favorable for light quarks/leptons.Those are favorable for light quarks/leptons.
AreaactionclassialScl :
qucl
cl
SS
Zqu
S
edZ
dZezzz
)()()( 321
n-point coplings n-point coplings Choi, T.K. Choi, T.K. ‘‘0808Selection ruleSelection rule ← ← gauge invariancegauge invariance
H-momentum conservationH-momentum conservation
space group selection rulespace group selection rule
Coupling strengthCoupling strength
calculated by inserting Vertex operatorscalculated by inserting Vertex operators
Calculations in intersecting D-brane models Calculations in intersecting D-brane models
are almost the same. are almost the same.
AreaactionclassialScl :
qucl
cl
SS
Zqun edZzzz
)()()( 21
Couplings among zero-Couplings among zero-modesmodes Extra dimensional effective field theoryExtra dimensional effective field theory
Non-trivial backgroundNon-trivial background → → non-trivial profile non-trivial profile of zero-mode wave functionof zero-mode wave function
4D couplings among quasi- localized modes4D couplings among quasi- localized modes = overlap integral along extra dimensions= overlap integral along extra dimensions → → suppressed Yukawa couplingssuppressed Yukawa couplings depending on their distancedepending on their distance
0 mmDi
Selection rule for allowed Selection rule for allowed couplingscouplings Allowed couplings : gauge invariantAllowed couplings : gauge invariant
conservation of quantum conservation of quantum numbersnumbers
e.g. momenta in 6D compact e.g. momenta in 6D compact space space
Some selection rules are not understood Some selection rules are not understood
by effective field theory. by effective field theory.
Stringy selection ruleStringy selection rule
Stringy coupling selection ruleStringy coupling selection rule
A string can be A string can be specified by specified by
its boundary its boundary condition.condition.
Two strings can be connected Two strings can be connected
to become a string if their to become a string if their
boundary conditions fit each boundary conditions fit each other.other.
coupling selection rulecoupling selection rule
symmetrysymmetry
)0( X
)( X
Coupling selection rule Coupling selection rule If three strings can be connected If three strings can be connected
and it becomes a shrinkable closed string, and it becomes a shrinkable closed string,
their coupling is allowed.their coupling is allowed.
selection selection rule rule
← ← geometrical structuregeometrical structure
of of compact spacecompact space
⇒⇒ in general complicatedin general complicated
Coupling selection rule (bad Coupling selection rule (bad news ?)news ?)In general, stringy coupling selection rule is In general, stringy coupling selection rule is tight. tight.
All of three generations can not couple with All of three generations can not couple with
a Higgs field as 3-point couplings.a Higgs field as 3-point couplings.
For example, only the top and bottom can For example, only the top and bottom can couple couple
One could not derive realistic quark/lepton One could not derive realistic quark/lepton
mass matrices.mass matrices.
Abelian discrete symmetriesAbelian discrete symmetries Stringy coupling selection ruleStringy coupling selection rule
Abelian discrete symmetries, Abelian discrete symmetries,
ZN x ZM x ZN x ZM x ………… symmetries symmetries
Boundary conditionsBoundary conditions
Three strings with the Three strings with the same same
gauge charges can be gauge charges can be
distinguished by distinguished by
boundary conditions, boundary conditions,
i.e. Zi.e. Z33 charges. charges.
Generic case ZGeneric case ZNN symmetries symmetries
)3(mod 2 1,,0
)()0( 22
XX
Heterotic orbifold modelsHeterotic orbifold modelsS1/Z2 OrbifoldS1/Z2 Orbifold
twisted stringtwisted string
untwisted string untwisted string
)0()( XX
2) (mod 1 ,0 , )0()( nenXX
2) (mod 1 ,0 ,
, )0()1()(
nm
enXX m
Closed strings on T2/Z3 Closed strings on T2/Z3 orbifoldorbifold
UntwistedUntwisted and and twistedtwisted strings strings
Twisted strings (first twisted sector)Twisted strings (first twisted sector)
second twisted sectorsecond twisted sector untwisted sector untwisted sector
)(e3 lattice toup twist,120
3) (mod 2 ,1 ,0 , )0()(
211
1
eenm
nenXX
3) (mod 2 ,1 ,0 , )0()( 12 nenXX
)0()( XX
Z3 x Z3 in Heterotic orbifold Z3 x Z3 in Heterotic orbifold modelsmodelsT2/Z3 OrbifoldT2/Z3 Orbifold
two Z3two Z3’’s s
twisted string (first twisted sector)twisted string (first twisted sector)
untwisted string untwisted string
vanishing Z3 charges for both Z3vanishing Z3 charges for both Z3
)3/2exp( ,
00
00
001
,
00
00
00
2
i
3) (mod ,2 1 ,0 ,
, )0()(
nm
enXX m
Higher Dimensional theory with flux
AbelianAbelian gauge field on magnetized torusgauge field on magnetized torus
Constant magnetic flux
Consistency requires Dirac’s quantization condition.
gauge fields of background
Torus with magnetic flux Torus with magnetic flux We solve the zero-mode Dirac equation,We solve the zero-mode Dirac equation,
e.g. for U(1) charge q=1.e.g. for U(1) charge q=1.
Torus background with magnetic flux Torus background with magnetic flux
leads to chiral spectra.leads to chiral spectra.
the number of zero-modes the number of zero-modes
= M (magnetic flux) = M (magnetic flux)
x q (charge) x q (charge)
0 mmDi
Zero-modesZero-modes
Wave-function = (gaussian) x (theta-function)Wave-function = (gaussian) x (theta-function)We have quantized momentum, We have quantized momentum,
The peaks of wave functions correspond to The peaks of wave functions correspond to
The momentum conservation The momentum conservation ZM discrete symmetry ZM discrete symmetry
e.g. M=3 Z3 symmetrye.g. M=3 Z3 symmetry
45445 2 ,0 , 2 yAAMF
M) (mod , 2 5 kP
/M 4 ky
Coupling selection rule (bad Coupling selection rule (bad news ?)news ?)In general, stringy coupling selection rule is tight. In general, stringy coupling selection rule is tight.
All of three generations can not couple with All of three generations can not couple with
a Higgs field as 3-point couplings.a Higgs field as 3-point couplings.
<- - controlled by discrete <- - controlled by discrete symmetriessymmetries
For example, only the top and bottom can couple For example, only the top and bottom can couple
One could not derive realistic quark/lepton One could not derive realistic quark/lepton
mass matrices.mass matrices.
Discrete symmetries are also important to Discrete symmetries are also important to
forbid dangerous couplings. forbid dangerous couplings.
Explicit modelsExplicit modelsExplicit models have flat directions and Explicit models have flat directions and
several scalar fields develop their VEVs. several scalar fields develop their VEVs.
Higher dim. Operators Higher dim. Operators
become effective Yukawa couplings become effective Yukawa couplings
after symmetry breaking.after symmetry breaking.
They would lead to suppressed Yukawa They would lead to suppressed Yukawa couplingscouplings
How to control n-point coupling is important.How to control n-point coupling is important.
So far, such analyses have been done So far, such analyses have been done
model by modelmodel by model. .
HQqM nn /...1
Explicit modelsExplicit modelsHigher dim. Operators with scalar VEVsHigher dim. Operators with scalar VEVs small number n small number n heavy quarks such as c heavy quarks such as c large number n large number n light quarks/leptons light quarks/leptons such as u,d,s, e,μsuch as u,d,s, e,μ quite model-dependent analysis quite model-dependent analysis results depend on moduli, flat directions and results depend on moduli, flat directions and scalar VEVsscalar VEVs Should we analyze model-by-model Should we analyze model-by-model quarks/lepton mass matrices quarks/lepton mass matrices for O(1000) or more models ? for O(1000) or more models ? Is it possible ? (Many model builders gave Is it possible ? (Many model builders gave up.) up.)
HQqM nn /...1
Short summary on effective Short summary on effective theory (coupling)theory (coupling)Several couplings are calculable.Several couplings are calculable.
All couplings are functions depending on All couplings are functions depending on
moduli (dilaton).moduli (dilaton).
We have to choose proper values of We have to choose proper values of
moduli VEVs. moduli VEVs.
Realization of quark/lepton masses and Realization of quark/lepton masses and mixing mixing
angles is still a challenging issue.angles is still a challenging issue.
We may need a new strategy.We may need a new strategy.
Quark and lepton mass Quark and lepton mass matricesmatricesWe have concentrated on (heavy) quarks.We have concentrated on (heavy) quarks.
Reasons:Reasons:
large couplings are important, large couplings are important,
we know better experimental data in we know better experimental data in
the quark sector than in the lepton sector, the quark sector than in the lepton sector,
neutrino masses and mixing angles.neutrino masses and mixing angles.
However, recent neutrino experiments However, recent neutrino experiments
show the typical pattern of mixing angles.show the typical pattern of mixing angles.
HQqM nn /...1
Tri-bimaximal mixing AnsatzTri-bimaximal mixing Ansatz
large mixing angleslarge mixing angles
2
1
3
1
6
12
1
3
1
6
1
03
1
6
2
MNSV
Non-Abelian discrete flavor Non-Abelian discrete flavor symm.symm.Recently, in field-theoretical model building, Recently, in field-theoretical model building,
several types of discrete flavor symmetries have several types of discrete flavor symmetries have
been proposed with showing interesting results, been proposed with showing interesting results,
e.g. S3, D4, A4, S4, Q6, Δ(27), ......e.g. S3, D4, A4, S4, Q6, Δ(27), ......
Review: e.g Review: e.g
Ishimori, T.K., Ohki, Okada, Shimizu, Tanimoto Ishimori, T.K., Ohki, Okada, Shimizu, Tanimoto ‘‘1010
⇒ ⇒ large mixing angles large mixing angles
one Ansatz: tri-bimaximalone Ansatz: tri-bimaximal
2/13/16/1
2/13/16/1
03/13/2
New viewpoint: New viewpoint: Non-Abelian symmetry Non-Abelian symmetry
String model builders have not cared about String model builders have not cared about
non-Abelian discrete symmetires.non-Abelian discrete symmetires.
Recently, we showed that certain non-Recently, we showed that certain non-Abelian Abelian
flavor symmetries appear in string models.flavor symmetries appear in string models.
Studies on discrete anomalies are also Studies on discrete anomalies are also important.important.
Non-Abelian discrete Non-Abelian discrete symmetriessymmetries Heterotic orbifold models Heterotic orbifold modelsS1/Z2 OrbifoldS1/Z2 Orbifold
Z2 even oddZ2 even odd
String theory has two Z2String theory has two Z2’’s.s.
In addition, the Z2 orbifold has the geometrical In addition, the Z2 orbifold has the geometrical
symmetry, i.e. Z2 permutation.symmetry, i.e. Z2 permutation.
2) (mod 1 ,0 ,
, )0()1()(
nm
enXX m
D4 Flavor SymmetryD4 Flavor SymmetryStringy symmetries require that Lagrangian has the Stringy symmetries require that Lagrangian has the
permutation symmetry between 1 and 2, and each permutation symmetry between 1 and 2, and each coupling is controlled by two Z2 symmetries. coupling is controlled by two Z2 symmetries.
Flavor symmeties: closed algebra S2 U(Z2xZ2) Flavor symmeties: closed algebra S2 U(Z2xZ2)
D4 elementsD4 elements
modes on two fixed points ⇒modes on two fixed points ⇒ doublet doublet untwisted (bulk) modes ⇒untwisted (bulk) modes ⇒ singletsingletGeometry of compact space Geometry of compact space origin of finite flavor symmetry origin of finite flavor symmetry Abelian part (Z2xZ2) : coupling selection ruleAbelian part (Z2xZ2) : coupling selection rule S2 permutation : one coupling is the same as S2 permutation : one coupling is the same as
another.another. T.K., Raby, Zhang, T.K., Raby, Zhang, ‘‘0505
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Heterotic orbifold as brane Heterotic orbifold as brane worldworld
Pati-Salam model Pati-Salam model T.K. Raby, Zhang, T.K. Raby, Zhang, ‘‘0505
2D Z2 orbifold2D Z2 orbifold
1 generation in bulk1 generation in bulk
two generations on two fixed pointstwo generations on two fixed points
unbrokenunbroken SU(4)SU(4) ＊＊ SU(2)SU(2) ＊＊ SU(2) D4SU(2) D4
bulk ⇒bulk ⇒ (4,2,1) + (4*,1,2)+... (4,2,1) + (4*,1,2)+... singletsinglet
localized modes ⇒localized modes ⇒ (4,2,1) + (4*,1,2) (4,2,1) + (4*,1,2) doubletdoublet
Heterotic orbifold modelsHeterotic orbifold modelsT2/Z3 OrbifoldT2/Z3 Orbifold
two Z3two Z3’’s s
Z3 orbifold has the S3 geometrical symmetry, Z3 orbifold has the S3 geometrical symmetry,
Their closed algebra is Δ(54).Their closed algebra is Δ(54).
T.K., Nilles, Ploger, Raby, Ratz, T.K., Nilles, Ploger, Raby, Ratz, ‘‘0707
)3/2exp( ,
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Heterotic orbifold modelsHeterotic orbifold models
T2/Z3 OrbifoldT2/Z3 Orbifold
has Δ(54) symmetry.has Δ(54) symmetry.
localized modes on three fixed points localized modes on three fixed points
Δ(54) tripletΔ(54) triplet
bulk modes Δ(54) singletbulk modes Δ(54) singlet
T.K., Nilles, Ploger, Raby, Ratz, T.K., Nilles, Ploger, Raby, Ratz, ‘‘0707
intersecting/magnetized intersecting/magnetized D-brane models D-brane models Abe, Choi, T.K. Ohki, Abe, Choi, T.K. Ohki, ’’09, 09, ‘‘1010
There is a Z2 permutation symmetry.There is a Z2 permutation symmetry.
The full symmetry is D4.The full symmetry is D4.
intersecting/magnetized intersecting/magnetized D-brane models D-brane models Abe, Choi, T.K. Ohki, Abe, Choi, T.K. Ohki, ’’09, 09, ‘‘1010
geometrical symm. Full symm. geometrical symm. Full symm. Z3 Δ(27) Z3 Δ(27)
S3 Δ(54)S3 Δ(54)
field theory: extension (Ohkifield theory: extension (Ohki’’s s talk)talk) Abe, Choi, T.K., Ohki, Sakai, Abe, Choi, T.K., Ohki, Sakai, ‘‘1010S1/Z2 Orbifold geometrical S1/Z2 Orbifold geometrical symm.symm.
String theory has two Z2String theory has two Z2’’s.s. We assign generic ZN charges to localized fields We assign generic ZN charges to localized fields on two fixed points, on two fixed points,
flavor symmetries flavor symmetries
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field theory: extension (Ohkifield theory: extension (Ohki’’s s talk)talk)T2/Z3 Orbifold geometrical T2/Z3 Orbifold geometrical symm.symm.
Z3, S3Z3, S3
String theory has two Z3String theory has two Z3’’s.s. We assign generic ZN charges to localized fields We assign generic ZN charges to localized fields on three fixed points, on three fixed points, flavor symmetries flavor symmetries Stringy derivation is not clear.Stringy derivation is not clear.
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New viewpoint: New viewpoint: Non-Abelian symmetry Non-Abelian symmetry
Indeed, this symmetry is the reason why we Indeed, this symmetry is the reason why we can not can not
derive realistic quark/lepton mass matrices derive realistic quark/lepton mass matrices
from 3-point couplings, from 3-point couplings,
where flavor symmetries do not break.where flavor symmetries do not break.
However, that would be interesting However, that would be interesting
from the viewpoint of large mixing angles from the viewpoint of large mixing angles
in the lepton sector.in the lepton sector.
2/13/16/1
2/13/16/1
03/13/2
Non-Abelian symmetryNon-Abelian symmetry We have just started this type of studies.We have just started this type of studies. We have not succeeded realization of We have not succeeded realization of realistic lepton mass matrices in purely realistic lepton mass matrices in purely stringy models stringy models (but field-theoretical model building). (but field-theoretical model building). Further phenomenological implicationsFurther phenomenological implications sfermion masses are controlled by non-Abelian sfermion masses are controlled by non-Abelian flavor symmetries.flavor symmetries. Higgs fields might be not singlets.Higgs fields might be not singlets. The number of Higgs fields might not be The number of Higgs fields might not be minimal.minimal.
sfermion mass sfermion mass SUSY breaking due to F-term of XSUSY breaking due to F-term of X
triplet triplet
1+2 1+2
1+ 11+ 1’’+1+1””
21
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22
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symmetry breakingsymmetry breaking Breaking of the flavor symmetries would Breaking of the flavor symmetries would
induce induce off-diagonal elements in the Kahler potential, off-diagonal elements in the Kahler potential, e.g.e.g.
and sfermion mass-squared matrix, and sfermion mass-squared matrix, e.g. e.g.
Large off-diagonal elements are not good from Large off-diagonal elements are not good from FCNC.FCNC. Large breaking is not good.Large breaking is not good.
))(( 2*1
*2112 qqqqXKK
21
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212
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Non-Abelian discrete Non-Abelian discrete symmetriessymmetries Symmetric background Symmetric background non-Abelian discrete non-Abelian discrete symmetriessymmetries
Generic CYGeneric CY’’s do not have such symmetries.s do not have such symmetries.
Lepton flavor model buildingLepton flavor model building in field theoryin field theory First, we assume a certain flavor symmetry.First, we assume a certain flavor symmetry. Then, we break it to a proper direction by flavon VEVs.Then, we break it to a proper direction by flavon VEVs. realistic MNS mixing matrix and lepton massesrealistic MNS mixing matrix and lepton masses We have achieved the first step for certain flavor We have achieved the first step for certain flavor
symmetries.symmetries. Flavon VEVs correspond to deformation of compact space, Flavon VEVs correspond to deformation of compact space, e.g. blow-up of orbifold singularity.e.g. blow-up of orbifold singularity. Which deformation is realistic ?Which deformation is realistic ?
Discrete symmetriesDiscrete symmetriesFormal viewpoint, Formal viewpoint,
it would be important to study anomalies it would be important to study anomalies
of discrete symmetires.of discrete symmetires.
Araki, T.K., Kubo, Ramos-Sanches, Ratz, Vaudrevange, Araki, T.K., Kubo, Ramos-Sanches, Ratz, Vaudrevange, ‘‘0808
Abe, et. al. work in progressAbe, et. al. work in progress
String theory String theory gravity/gauge anomalies gravity/gauge anomalies
Does string theory constrain discrete anomalies, Does string theory constrain discrete anomalies,
too ?too ?
Anomalies of discrete Anomalies of discrete symmetriessymmetries
Heterotic orbifold modelsHeterotic orbifold models Araki, T.K., Kubo, Ramos-Sanches, Ratz, Vaudrevange, Araki, T.K., Kubo, Ramos-Sanches, Ratz, Vaudrevange, ‘‘0808
Zn-G-G anomalies for G= non-Abelian gauge Zn-G-G anomalies for G= non-Abelian gauge groupsgroups
We have checked a number of models.We have checked a number of models.
universal anomalies of discrete symmetries universal anomalies of discrete symmetries
= universal = universal
for different gauge groupsfor different gauge groups
)( 2 RTq
Heterotic orbifold modelsHeterotic orbifold modelsU(1)-G-G anomalies U(1)-G-G anomalies are universal for different groups G.are universal for different groups G. 4D Green-Schwarz mechanism 4D Green-Schwarz mechanism due to a single axion (dilaton), due to a single axion (dilaton), which couples universally with gauge sectors.which couples universally with gauge sectors.ZN-G-G anomalies may also be cancelled ZN-G-G anomalies may also be cancelled by 4D GS mechanism.by 4D GS mechanism.There is a certain relations between There is a certain relations between U(1)-G-G and ZN-G-G anomalies,U(1)-G-G and ZN-G-G anomalies, anomalous U(1) generator is a linear combination anomalous U(1) generator is a linear combination of anomalous ZN generators.of anomalous ZN generators. Araki, T.K., Kubo, Ramos-Sanches, Ratz,Vaudrevange, Araki, T.K., Kubo, Ramos-Sanches, Ratz,Vaudrevange, ‘‘0808
)( 2 RTq
Flavor in string theoryFlavor in string theory
It is not so difficult to realize the generation It is not so difficult to realize the generation
number, i.e. the three generation.number, i.e. the three generation.
We have some explicit examples We have some explicit examples
to lead to semi-realistic patterns of to lead to semi-realistic patterns of
Yukawa matrices for quarks and leptons.Yukawa matrices for quarks and leptons.
However, realization of realistic Yukawa However, realization of realistic Yukawa
matrices is still a challenging issue.matrices is still a challenging issue.
(non-abelian flavor symmetries ?)(non-abelian flavor symmetries ?)
Summary Summary We have studied on particle We have studied on particle phenomenological phenomenological
aspects on string theory aspects on string theory to find out a scenario connecting to find out a scenario connecting string theory and the particle physics, string theory and the particle physics, in particular the Standard Model.in particular the Standard Model.
Several issues:Several issues: realistic spectra,realistic spectra, flavor structureflavor structure 、、 moduli stabilizationmoduli stabilization 、、SUSY breakingSUSY breaking 、、 cosmologycosmology 、 、 ..................
SummarySummaryRealistic massless spectraRealistic massless spectra all types of string theories are not badall types of string theories are not bad We have known already many string We have known already many string models, models,
which have the same content as which have the same content as the MSSM or its extensions.the MSSM or its extensions. Gauge couplings Gauge couplings Yukawa matricesYukawa matrices still a challenging issuestill a challenging issue Further studies: Moduli stabilizatinFurther studies: Moduli stabilizatin cosmological aspects, cosmological aspects, ……
