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Symmetries and vanishing Symmetries and vanishing couplings couplings in string-derived low-energy in string-derived low-energy effective field theory effective field theory Tatsuo Tatsuo KobayashiKobayashi1.1. IntroductionIntroduction
2. Abelian Discrete Symmetries2. Abelian Discrete Symmetries
33 . . Non-Abelian Discrete SymmetriesNon-Abelian Discrete Symmetries
44 . . AnomaliesAnomalies
5. Explicit stringy computations5. Explicit stringy computations
6. Summary6. Summary
1. Introduction1. IntroductionNow, we have lots of 4D string models leading to Now, we have lots of 4D string models leading to (semi-)realistic massless spectra such as (semi-)realistic massless spectra such as
SU(3)xSU(2)xU(1) gauge groups,SU(3)xSU(2)xU(1) gauge groups, three chiral genenations, three chiral genenations, vector-like matter fields and lots of singlets vector-like matter fields and lots of singlets with and without chiral exotic fields, with and without chiral exotic fields,
e.g. in e.g. in heterotic models, heterotic models, type II intersecting D-brane models,type II intersecting D-brane models, type II magnetized D-brane models, type II magnetized D-brane models, etc.etc. What about their 4D low-energy effective theories ?What about their 4D low-energy effective theories ? Are they realistic ?Are they realistic ? What about the quark/lepton masses and mixing angles ?What about the quark/lepton masses and mixing angles ?
4D low-energy effective field 4D low-energy effective field theorytheory We have to control couplings in 4D LEEFT.We have to control couplings in 4D LEEFT.
realization of quark/lepton mass and mixingrealization of quark/lepton mass and mixing
including the neutrino sector, including the neutrino sector,
avoiding the fast proton decay,avoiding the fast proton decay,
stability of the LSP, suppressing FCNC, etc stability of the LSP, suppressing FCNC, etc
Abelian and non-Abelian discrete symmetries Abelian and non-Abelian discrete symmetries
are useful in low-energy model building are useful in low-energy model building
to control them.to control them.
Abelian discrete symmetriesAbelian discrete symmetries ZN symmetry ZN symmetry
R-symmetric and non-R-symmetric R-symmetric and non-R-symmetric
Flavor symmetryFlavor symmetry
R-parity, matter parity, R-parity, matter parity,
baryon triality, proton hexalitybaryon triality, proton hexality
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),Δ(54), ......e.g. S3, D4, A4, S4, Q6, Δ(27),Δ(54), ......
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
in the lepton sectorin the lepton sector
one Ansatz: tri-bimaximalone Ansatz: tri-bimaximal
2/13/16/1
2/13/16/1
03/13/2
String-derived 4D LEEFTString-derived 4D LEEFT Can we derive these Abelian and non-Abelian Can we derive these Abelian and non-Abelian
discrete symmetries from string theory ?discrete symmetries from string theory ?
Which symmetires can appear in 4D LEEFT Which symmetires can appear in 4D LEEFT
derived from string theory ?derived from string theory ?
One can compute couplings of 4D LEEFT One can compute couplings of 4D LEEFT
in string theory.in string theory.
(These are functions of moduli.)(These are functions of moduli.)
Control on anomalies is one of stringy features.Control on anomalies is one of stringy features.
What about anomalies of discrete symmetries ?What about anomalies of discrete symmetries ?
In this talk, we study heterotic orbifold models.In this talk, we study heterotic orbifold models.
2. Abelian discrete symmetries2. Abelian discrete symmetriescoupling selection rulecoupling 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
Heterotic orbifold modelsHeterotic orbifold models
S1/Z2 OrbifoldS1/Z2 Orbifold
There are two singular points, There are two singular points,
which are called fixed points.which are called fixed points.
)2/(~2/
~
eXeX
XX
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
Closed strings on orbifoldClosed strings on orbifold
UntwistedUntwisted and and twistedtwisted strings strings
Twisted strings are associated with fixed Twisted strings are associated with fixed points.points.
““Brane-worldBrane-world”” terminology: terminology: untwisted sector bulk modesuntwisted sector bulk modes twisted sector brane (localized) modestwisted sector brane (localized) modes
Heterotic orbifold modelsHeterotic orbifold models
S1/Z2 OrbifoldS1/Z2 Orbifold
)2/)0((2/)(
)0()(
eXeX
XX
2) (mod 1 ,0 , )0()( nenXX
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
Z2 x Z2 in Heterotic orbifold Z2 x Z2 in Heterotic orbifold modelsmodelsS1/Z2 OrbifoldS1/Z2 Orbifold
two Z2two Z2’’s s
twisted stringtwisted string
untwisted string untwisted string
Z2 even for both Z2Z2 even for both Z2
10
01 ,
10
01
2) (mod 1 ,0 ,
, )0()1()(
nm
enXX m
Closed strings on orbifoldClosed strings on orbifold
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
3. Non-Abelian discrete 3. Non-Abelian discrete symmetriessymmetries Heterotic orbifold models Heterotic orbifold modelsS1/Z2 OrbifoldS1/Z2 Orbifold
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 another.S2 permutation : one coupling is the same as another. T.K., Raby, Zhang, T.K., Raby, Zhang, ‘‘05 T.K., Nilles, Ploger, Raby, Ratz, 05 T.K., Nilles, Ploger, Raby, Ratz,
‘‘0707
10
01
10
011,
01
1031
321 ,,,1 i
Explicit Heterotic orbifold Explicit Heterotic orbifold modelsmodels T.K. Raby, Zhang ’05, Buchmuller, Hamaguchi, Lebedev, Ratz, T.K. Raby, Zhang ’05, Buchmuller, Hamaguchi, Lebedev, Ratz, ’06’06
Lebedev, Nilles, Raby, Ramos-Sanchez, Ratz, Lebedev, Nilles, Raby, Ramos-Sanchez, Ratz, Vaudrevange,Wingerter, Vaudrevange,Wingerter, ‘‘0707
2D Z2 orbifold2D Z2 orbifold
1 generation in bulk1 generation in bulk
two generations on two fixed pointstwo generations on two fixed points
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( ,
00
00
001
,
00
00
00
2
i
010
100
001
,
001
100
010
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
4. Discrete anomalies4. Discrete anomalies 4-1. Abelian discrete 4-1. Abelian discrete anomalies anomalies Symmetry violatedSymmetry violated quantum effectsquantum effects U(1)-G-G anomaliesU(1)-G-G anomalies anomaly free condition anomaly free condition ZN-G-G anomalies ZN-G-G anomalies anomaly free conditionanomaly free condition
0)( 2 RTq
) (mod 0)( 2 NRTq
Abelian discrete anomalies:Abelian discrete anomalies: path integral path integral Zn transformation Zn transformation path integral measurepath integral measure
ZN-G-G anomalies ZN-G-G anomalies anomaly free conditionanomaly free condition
'
) (mod 0)( 2 NRTq
integer )~
( tr
)( 1
)]~
( trexp[
4
321
2
4
32
2
2
FFxd
RTqN
A
FFxdAJ
DDJDD
i
Heterotic orbifold modelsHeterotic orbifold modelsThere are two types of Abelian discrete There are two types of Abelian discrete symmetries.symmetries.
T2/Z3 OrbifoldT2/Z3 Orbifold
two Z3two Z3’’s s
One is originated from twists, One is originated from twists,
the other is originated from shifts.the other is originated from shifts.
Both types of discrete anomalies Both types of discrete anomalies
are universal for different groups G.are universal for different groups G.
Araki, T.K., Kubo, Ramos-Sanches, Ratz,Vaudrevange, Araki, T.K., Kubo, Ramos-Sanches, Ratz,Vaudrevange, ‘‘0808
)( 2 RTq
3) (mod ,2 1 ,0 ,
, )0()(
nm
enXX m
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
4-2. Non-Abelian discrete 4-2. Non-Abelian discrete anomalies anomalies
Araki, T.K., Kubo, Ramos-Sanches, Ratz, Vaudrevange, Araki, T.K., Kubo, Ramos-Sanches, Ratz, Vaudrevange, ‘‘0808
Non-Abelian discrete groupNon-Abelian discrete group finite elementsfinite elements Each element generates an Abelian symmetry.Each element generates an Abelian symmetry.
We check ZN-G-G anomalies for each element. We check ZN-G-G anomalies for each element.
All elements are free from ZN-G-G anomalies.All elements are free from ZN-G-G anomalies. The full symmetry G is anomaly-free.The full symmetry G is anomaly-free. Some ZN symmetries for elements gSome ZN symmetries for elements gkk are are anomalous.anomalous.
The remaining symmetry corresponds to The remaining symmetry corresponds to the closed algebra without such the closed algebra without such elements.elements.
},,,{ 21 MgggG
) (mod 0)( 2 kk NRTq
1)( kNkg
Non-Abelian discrete anomalies Non-Abelian discrete anomalies matter fields = multiplets under non-Abelian matter fields = multiplets under non-Abelian discrete symmetry discrete symmetry Each element is represented by a matrix on Each element is represented by a matrix on the multiplet.the multiplet.
Such a multiplet does not contribute to Such a multiplet does not contribute to ZN-G-G anomalies.ZN-G-G anomalies. String models lead to certain combinations of String models lead to certain combinations of multiplets.multiplets.
limited pattern of non-Abelian limited pattern of non-Abelian discrete discrete
anomaliesanomalies
) (mod 0)( 2 kk NRTq 1)( det kg
Heterotic string on Z2 orbifold:Heterotic string on Z2 orbifold: D4 Flavor Symmetry D4 Flavor SymmetryFlavor symmeties: closed algebra S2 U(Z2xZ2) Flavor symmeties: closed algebra S2 U(Z2xZ2) modes on two fixed points ⇒modes on two fixed points ⇒ doublet doublet untwisted (bulk) modes ⇒untwisted (bulk) modes ⇒ singletsinglet
The first Z2 is always anomaly-free, while the The first Z2 is always anomaly-free, while the others can be anomalous.others can be anomalous.
However, it is simple to arrange models such However, it is simple to arrange models such that that
the full D4 remains.the full D4 remains.e.g. left-handed and right-handed e.g. left-handed and right-handed quarks/leptons quarks/leptons
1 + 21 + 2Such a pattern is realized in explicit models.Such a pattern is realized in explicit models.
10
01
10
011,
01
1031
Heterotic models on Z3 Heterotic models on Z3 orbifoldorbifold 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).
The full symmetry except Z2 is always anomaly-The full symmetry except Z2 is always anomaly-free.free.
That is, the Δ(27) is always anomaly-free.That is, the Δ(27) is always anomaly-free.
Abe, et. al. work in progressAbe, et. al. work in progress
)3/2exp( ,
00
00
001
,
00
00
00
2
i
010
100
001
,
001
100
010
5. Explicit stringy 5. Explicit stringy computationcomputation T.K., Parameswaran,Ramos-Sanchez, Zavala ‘11T.K., Parameswaran,Ramos-Sanchez, Zavala ‘11
Explicit string computations tell something more.Explicit string computations tell something more.
Heterotic string on orbifold:Heterotic string on orbifold: 4D string, E8xE8 gauge part, 4D string, E8xE8 gauge part, r-moving fermionic string,r-moving fermionic string, 6D string on orbifold, ghost 6D string on orbifold, ghost
A mode has a definite “quantum number” A mode has a definite “quantum number” in each part.in each part.
Vertex operatorsVertex operators Boson Boson
FermionFermion
IImsm
iL
iL
IImm
iL
iL
XiPHiqiN
iN
i
i
m
I
XiPHiqiN
iN
i
i
eeXXe
q
P
eeXXe
V
vector)(spinor, tion,representa Lorentz D10:
roots) (weights, groups gaugeunder numbers quantum :
gauge RNS string 6D ghost
V
)(3
1
2/1/2-
3
11-
3-pt correlation function3-pt correlation function Vertex operators: Vertex operators:
The correlation function vanishes unlessThe correlation function vanishes unless That is the momentum conservation in the string of That is the momentum conservation in the string of the gauge part, i.e. the gauge invariance.the gauge part, i.e. the gauge invariance.
iXiPHiqN
iN
i
i
iXiPHiqN
iN
i
i
IImsm
iL
iL
IImm
iL
iL
eeXXe
eeXXe
)(3
1
2/1/2-
3
11-
V
V
1/2-1/2-1- VVV
.0 IP
String on 6D orbifoldString on 6D orbifoldThe 6D part is quite non-trivial. The 6D part is quite non-trivial.
world-sheet instanton + quantum partworld-sheet instanton + quantum part
.0 with case heconsider t we,simplicityFor LN
1/2-1/2-1- VVV
SNNeXXDX
LL
SNeXDX
L
Classical solution The world-sheet instanton, which corresponds to string moving from a fixed point
to others.
symmetries of torus symmetries of torus (sub)lattice(sub)lattice Suppose that only holomorphic instanton can appear. Suppose that only holomorphic instanton can appear.
Example: T2/Z3Example: T2/Z3 different fixed pointsdifferent fixed points
Z3 symmetries (twist invariance)Z3 symmetries (twist invariance)
twist invariance twist invariance + H-momentum conservation + H-momentum conservation => discrete R-symmetry=> discrete R-symmetry
lattice (sub) torus points fixed of difference
)(
a
zahX cl
latticeadjoint weight SU(3) a
symmetries of torus symmetries of torus (sub)lattice(sub)lattice Suppose that only holomorphic instanton can appear. Suppose that only holomorphic instanton can appear.
Example: T2/Z3Example: T2/Z3 three strings on the same fixed pointsthree strings on the same fixed points
Z6 symmetries Z6 symmetries enhanced symmetries enhanced symmetries for certain couplingsfor certain couplings Rule 4Rule 4 Font, Ibanez, Nilles, Quevedo, ’88Font, Ibanez, Nilles, Quevedo, ’88 T.K., Parameswaran,Ramos-Sanchez, Zavala ‘11T.K., Parameswaran,Ramos-Sanchez, Zavala ‘11
ce(sub)latti torus points fixed of difference
)(
a
zahX cl
latticeadjoint a
T2/Z3 orbifoldT2/Z3 orbifold Higher order couplings Higher order couplings The situation is the same.The situation is the same.
Z3 symmetries among localized strings on different fixed Z3 symmetries among localized strings on different fixed pointspoints
Z6 symmetries among localized strings on a fixed pointZ6 symmetries among localized strings on a fixed point
Z6 enhanced symmetries only for the couplings Z6 enhanced symmetries only for the couplings of the same fixed points.of the same fixed points. => Z3 twist invarince if the matter on the different fixed => Z3 twist invarince if the matter on the different fixed points is includedpoints is included
SNeXDX
L
Classical solutionClassical solution T2/Z2 T2/Z2
Similarly, the classical solution corresponding to Similarly, the classical solution corresponding to couplings on the same fixed points has enhanced couplings on the same fixed points has enhanced symmetires, e.g. Z4 and Z6, depending the torus symmetires, e.g. Z4 and Z6, depending the torus lattice, SO(5) torus and SU(3) torus.lattice, SO(5) torus and SU(3) torus.
T.K., Parameswaran,Ramos-Sanchez, Zavala ‘11T.K., Parameswaran,Ramos-Sanchez, Zavala ‘11
)(zahX cl
String on 6D orbifoldString on 6D orbifoldThe 6D part is quite non-trivial. The 6D part is quite non-trivial.
world-sheet instanton + quantum partworld-sheet instanton + quantum part
Only instantons with fine-valued action contribute.Only instantons with fine-valued action contribute.
That leads to certain conditions on combinations That leads to certain conditions on combinations among among
twists.twists.
1/2-1/2-1- VVV
SNeXDX
L
T.K., Parameswaran,Ramos-Sanchez, Zavala ‘11
(anti-) Holomorphic (anti-) Holomorphic instantoninstanton Condition for fine-valued action Condition for fine-valued action Vertex operators: Vertex operators:
stringth theof twist : k
10for 01
:appearsinstanton cholomorphi-anti
10for 011
:appearsinstanton cholomorphi
kk
kk
k
Summary Summary We have studied discrete symmetries We have studied discrete symmetries and their anomalies.and their anomalies.Explicit stringy computations tell something Explicit stringy computations tell something more.more.
String-derived massless spectra are (semi-) String-derived massless spectra are (semi-) realistic, but their 4D LEEFT are not so, realistic, but their 4D LEEFT are not so,
e.g. derivation of quark/lepton masses and e.g. derivation of quark/lepton masses and mixing angles are still challenging.mixing angles are still challenging.
Applications of the above discrete symmetires Applications of the above discrete symmetires and explicit stringy computations would be and explicit stringy computations would be important.important.
5. Something else5. Something else T.K., Parameswaran,Ramos-Sanchez, T.K., Parameswaran,Ramos-Sanchez, Zavala ‘11Zavala ‘11
Explicit stringy calculations would tell us something Explicit stringy calculations would tell us something more. more.
Vertex operators: Vertex operators:
1-
3
10
3
1
2/1/2-
3
11-
V V
V
V
)(
j
jHiqjHiq
iXiPHiqN
iN
i
i
iXiPHiqN
iN
i
i
XeXee
eeXXe
eeXXe
mjvm
mjvm
IImsm
iL
iL
IImm
iL
iL
001/2-1/2-1- VVVVV
),1,0,0,0,0( ),0,1,0,0,0( ),0,0,1,0,0( 321 vm
vm
vm qqq
3-pt correlation function3-pt correlation function Explicit stringy calculations Explicit stringy calculations Vertex operators: Vertex operators:
321
00
s
qur
qu
sNcl
rNcl
S
X
N
s
LN
r
L
XX
XXes
N
r
N LLcl
cl
LL
1/2-1/2-1- VVV
321
321
F
quclLL
LL
SSN
qucl
N
qucl
SNN
eXXXXDX
eXXDX
Rule 5Rule 5 T.K., Parameswaran,Ramos-Sanchez, T.K., Parameswaran,Ramos-Sanchez, Zavala ‘11Zavala ‘11
Rule 5: Rule 5:
321
00
s
qur
qu
sNcl
rNcl
S
X
N
s
LN
r
L
XX
XXes
N
r
NF
LLcl
cl
LL
LL
LLL
LLL
NN
NNrN
NNsN
instanton No
instanton cholomorphi-antiOnly
instanton cholomorphiOnly
0 :instanton cholomorphi-anti No
0 :instanton cholomorphi No
cl
cl
X
X
srsqu
rqu XX