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Symmetries and vanishing couplings in string-derived low-energy effective field theory Tatsuo Kobayashi １．Introduction 2. Abelian Discrete Symmetries 3． Non-Abelian Discrete Symmetries 4． Anomalies 5. Explicit stringy computations 6. Summary

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**1. Introduction Now, we have lots of 4D string models leading to**

(semi-)realistic massless spectra such as SU(3)xSU(2)xU(1) gauge groups, three chiral genenations, vector-like matter fields and lots of singlets with and without chiral exotic fields, e.g. in heterotic models, type II intersecting D-brane models, type II magnetized D-brane models, etc. What about their 4D low-energy effective theories ? Are they realistic ? What about the quark/lepton masses and mixing angles ?

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**4D low-energy effective field theory**

We have to control couplings in 4D LEEFT. realization of quark/lepton mass and mixing including the neutrino sector, avoiding the fast proton decay, stability of the LSP, suppressing FCNC, etc Abelian and non-Abelian discrete symmetries are useful in low-energy model building to control them. 3

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**Abelian discrete symmetries**

ZN symmetry R-symmetric and non-R-symmetric Flavor symmetry R-parity, matter parity, baryon triality, proton hexality 4

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**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, Δ(27),Δ(54), Review: e.g Ishimori, T.K., Ohki, Okada, Shimizu, Tanimoto ‘10 ⇒ large mixing angles in the lepton sector one Ansatz: tri-bimaximal 5

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**String-derived 4D LEEFT**

Can we derive these Abelian and non-Abelian discrete symmetries from string theory ? Which symmetires can appear in 4D LEEFT derived from string theory ? One can compute couplings of 4D LEEFT in string theory. (These are functions of moduli.) Control on anomalies is one of stringy features. What about anomalies of discrete symmetries ? In this talk, we study heterotic orbifold models. 6

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**2. Abelian discrete symmetries**

coupling selection rule A string can be specified by its boundary condition. Two strings can be connected to become a string if their boundary conditions fit each other. symmetry

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**Heterotic orbifold models**

S1/Z2 Orbifold There are two singular points, which are called fixed points.

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**Orbifolds T2/Z3 Orbifold There are three fixed points on Z3 orbifold**

(0,0), (2/3,1/3), (1/3,2/3) su(3) root lattice Orbifold = D-dim. Torus /twist Torus = D-dim flat space/ lattice

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**Closed strings on orbifold**

Untwisted and twisted strings Twisted strings are associated with fixed points. “Brane-world” terminology: untwisted sector bulk modes twisted sector brane (localized) modes

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**Heterotic orbifold models**

S1/Z2 Orbifold

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**Heterotic orbifold models**

S1/Z2 Orbifold twisted string untwisted string

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**Z2 x Z2 in Heterotic orbifold models**

S1/Z2 Orbifold two Z2’s twisted string untwisted string Z2 even for both Z2

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**Closed strings on orbifold**

Untwisted and twisted strings Twisted strings (first twisted sector) second twisted sector untwisted sector

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**Z3 x Z3 in Heterotic orbifold models**

T2/Z3 Orbifold two Z3’s twisted string (first twisted sector) untwisted string vanishing Z3 charges for both Z3

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**3. Non-Abelian discrete symmetries Heterotic orbifold models**

S1/Z2 Orbifold String theory has two Z2’s. In addition, the Z2 orbifold has the geometrical symmetry, i.e. Z2 permutation.

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D4 Flavor Symmetry Stringy symmetries require that Lagrangian has the permutation symmetry between 1 and 2, and each coupling is controlled by two Z2 symmetries. Flavor symmeties: closed algebra S2 U(Z2xZ2) D4 elements modes on two fixed points ⇒ doublet untwisted (bulk) modes ⇒ singlet Geometry of compact space origin of finite flavor symmetry Abelian part (Z2xZ2) : coupling selection rule S2 permutation : one coupling is the same as another. T.K., Raby, Zhang, ‘05 T.K., Nilles, Ploger, Raby, Ratz, ‘07

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**Explicit Heterotic orbifold models**

T.K. Raby, Zhang ’05, Buchmuller, Hamaguchi, Lebedev, Ratz, ’06 Lebedev, Nilles, Raby, Ramos-Sanchez, Ratz, Vaudrevange,Wingerter, ‘07 2D Z2 orbifold 1 generation in bulk two generations on two fixed points

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**Heterotic orbifold models**

T2/Z3 Orbifold two Z3’s Z3 orbifold has the S3 geometrical symmetry, Their closed algebra is Δ(54). T.K., Nilles, Ploger, Raby, Ratz, ‘07

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**Heterotic orbifold models**

T2/Z3 Orbifold has Δ(54) symmetry. localized modes on three fixed points Δ(54) triplet bulk modes Δ(54) singlet T.K., Nilles, Ploger, Raby, Ratz, ‘07

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**4. Discrete anomalies 4-1. Abelian discrete anomalies**

Symmetry violated quantum effects U(1)-G-G anomalies anomaly free condition ZN-G-G anomalies anomaly free condition

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**Abelian discrete anomalies: path integral**

Zn transformation path integral measure ZN-G-G anomalies anomaly free condition

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**Heterotic orbifold models**

There are two types of Abelian discrete symmetries. T2/Z3 Orbifold two Z3’s One is originated from twists, the other is originated from shifts. Both types of discrete anomalies are universal for different groups G. Araki, T.K., Kubo, Ramos-Sanches, Ratz,Vaudrevange, ‘08

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**Heterotic orbifold models**

U(1)-G-G anomalies are universal for different groups G. 4D Green-Schwarz mechanism due to a single axion (dilaton), which couples universally with gauge sectors. ZN-G-G anomalies may also be cancelled by 4D GS mechanism. There is a certain relations between U(1)-G-G and ZN-G-G anomalies, anomalous U(1) generator is a linear combination of anomalous ZN generators. Araki, T.K., Kubo, Ramos-Sanches, Ratz,Vaudrevange, ‘08

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**4-2. Non-Abelian discrete anomalies**

Araki, T.K., Kubo, Ramos-Sanches, Ratz, Vaudrevange, ‘08 Non-Abelian discrete group finite elements Each element generates an Abelian symmetry. We check ZN-G-G anomalies for each element. All elements are free from ZN-G-G anomalies. The full symmetry G is anomaly-free. Some ZN symmetries for elements gk are anomalous. The remaining symmetry corresponds to the closed algebra without such elements.

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**Non-Abelian discrete anomalies**

matter fields = multiplets under non-Abelian discrete symmetry Each element is represented by a matrix on the multiplet. Such a multiplet does not contribute to ZN-G-G anomalies. String models lead to certain combinations of multiplets. limited pattern of non-Abelian discrete anomalies

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**Heterotic string on Z2 orbifold: D4 Flavor Symmetry**

Flavor symmeties: closed algebra S2 U(Z2xZ2) modes on two fixed points ⇒ doublet untwisted (bulk) modes ⇒ singlet The first Z2 is always anomaly-free, while the others can be anomalous. However, it is simple to arrange models such that the full D4 remains. e.g. left-handed and right-handed quarks/leptons 1 + 2 Such a pattern is realized in explicit models.

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**Heterotic models on Z3 orbifold**

two Z3’s Z3 orbifold has the S3 geometrical symmetry, Their closed algebra is Δ(54). The full symmetry except Z2 is always anomaly-free. That is, the Δ(27) is always anomaly-free. Abe, et. al. work in progress

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**5. Explicit stringy computation**

T.K., Parameswaran,Ramos-Sanchez, Zavala ‘11 Explicit string computations tell something more. Heterotic string on orbifold: 4D string, E8xE8 gauge part, r-moving fermionic string, 6D string on orbifold, ghost A mode has a definite “quantum number” in each part.

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Vertex operators Boson Fermion

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**3-pt correlation function**

Vertex operators: The correlation function vanishes unless That is the momentum conservation in the string of the gauge part, i.e. the gauge invariance.

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**String on 6D orbifold The 6D part is quite non-trivial.**

world-sheet instanton + quantum part 32

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**Classical solution The world-sheet instanton, which**

corresponds to string moving from a fixed point to others.

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**symmetries of torus (sub)lattice**

Suppose that only holomorphic instanton can appear. Example: T2/Z3 different fixed points Z3 symmetries (twist invariance) twist invariance + H-momentum conservation => discrete R-symmetry 34

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**symmetries of torus (sub)lattice**

Suppose that only holomorphic instanton can appear. Example: T2/Z3 three strings on the same fixed points Z6 symmetries enhanced symmetries for certain couplings Rule 4 Font, Ibanez, Nilles, Quevedo, ’88 T.K., Parameswaran,Ramos-Sanchez, Zavala ‘11 35

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**T2/Z3 orbifold Higher order couplings The situation is the same.**

Z3 symmetries among localized strings on different fixed points Z6 symmetries among localized strings on a fixed point Z6 enhanced symmetries only for the couplings of the same fixed points. => Z3 twist invarince if the matter on the different fixed points is included 36

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**Classical solution T2/Z2**

Similarly, the classical solution corresponding to couplings on the same fixed points has enhanced symmetires, e.g. Z4 and Z6, depending the torus lattice, SO(5) torus and SU(3) torus. T.K., Parameswaran,Ramos-Sanchez, Zavala ‘11 37

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**String on 6D orbifold The 6D part is quite non-trivial.**

world-sheet instanton + quantum part Only instantons with fine-valued action contribute. That leads to certain conditions on combinations among twists. T.K., Parameswaran,Ramos-Sanchez, Zavala ‘11 38

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**(anti-) Holomorphic instanton**

Condition for fine-valued action Vertex operators: 39

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**Summary We have studied discrete symmetries and their anomalies.**

Explicit stringy computations tell something more. String-derived massless spectra are (semi-) realistic, but their 4D LEEFT are not so, e.g. derivation of quark/lepton masses and mixing angles are still challenging. Applications of the above discrete symmetires and explicit stringy computations would be important.

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**5. Something else T.K., Parameswaran,Ramos-Sanchez, Zavala ‘11**

Explicit stringy calculations would tell us something more. Vertex operators:

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**3-pt correlation function**

Explicit stringy calculations Vertex operators: 42

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Rule 5 T.K., Parameswaran,Ramos-Sanchez, Zavala ‘11 Rule 5: 43

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Z3 twist invariance Z3 twist invariance 44

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