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Symmetries and vanishing couplings in string-derived low-energy effective field theory Tatsuo Kobayashi 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 (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 ?

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4D low-energy effective field theory We have to control couplings in 4D LEEFT. We have to control couplings in 4D LEEFT. realization of quark/lepton mass and mixing realization 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.

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Abelian discrete symmetries ZN symmetry ZN symmetry R-symmetric and non-R-symmetric Flavor symmetry Flavor symmetry R-parity, matter parity, baryon triality, proton hexality baryon triality, proton hexality

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Non-Abelian discrete flavor symm. 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 ‘ 10 Ishimori, T.K., Ohki, Okada, Shimizu, Tanimoto ‘ 10 ⇒ large mixing angles ⇒ large mixing angles in the lepton sector in the lepton sector one Ansatz: tri-bimaximal one Ansatz: tri-bimaximal

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String-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. 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.

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2. Abelian discrete symmetries coupling selection rule A string can be specified by A string can be specified by its boundary condition. its boundary 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 other. boundary conditions fit each other. coupling selection rule coupling selection rule symmetry symmetry

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Heterotic orbifold models Heterotic orbifold models S1/Z2 Orbifold There are two singular points, There are two singular points, which are called fixed 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 (0,0), (2/3,1/3), (1/3,2/3) su(3) root lattice Orbifold = D-dim. Torus /twist Orbifold = D-dim. Torus /twist Torus = D-dim flat space/ lattice 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 untwisted sector bulk modes twisted sector brane (localized) 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 untwisted string

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Z2 x Z2 in Heterotic orbifold models S1/Z2 Orbifold two Z2 ’ s two Z2 ’ s twisted string twisted string untwisted string Z2 even for both Z2 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 second twisted sector untwisted sector

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Z3 x Z3 in Heterotic orbifold models T2/Z3 Orbifold two Z3 ’ s two Z3 ’ s twisted string (first twisted sector) twisted string (first twisted sector) untwisted string vanishing Z3 charges for both Z3 vanishing Z3 charges for both Z3

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3. Non-Abelian discrete symmetries Heterotic orbifold models 3. Non-Abelian discrete symmetries Heterotic orbifold models S1/Z2 Orbifold String theory has two Z2 ’ s. String theory has two Z2 ’ 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.

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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 modes on two fixed points ⇒ doublet untwisted (bulk) modes ⇒ singlet untwisted (bulk) modes ⇒ singlet Geometry of compact space origin of finite flavor symmetry origin of finite flavor symmetry Abelian part (Z2xZ2) : coupling selection rule Abelian 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, ‘ 05 T.K., Nilles, Ploger, Raby, Ratz, ‘ 07 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 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 1 generation in bulk two generations on two fixed points two generations on two fixed points

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Heterotic orbifold models T2/Z3 Orbifold two Z3 ’ s two Z3 ’ 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, ‘ 07 T.K., Nilles, Ploger, Raby, Ratz, ‘ 07

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

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4. Discrete anomalies 4-1. Abelian discrete anomalies 4. Discrete anomalies 4-1. Abelian discrete anomalies Symmetry violated quantum effects quantum effects U(1)-G-G anomalies U(1)-G-G anomalies anomaly free condition anomaly free condition ZN-G-G anomalies ZN-G-G anomalies anomaly free condition anomaly free condition

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

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Heterotic orbifold models There are two types of Abelian discrete symmetries. T2/Z3 Orbifold T2/Z3 Orbifold two Z3 ’ s two Z3 ’ 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 are universal for different groups G. are universal for different groups G. Araki, T.K., Kubo, Ramos-Sanches, Ratz,Vaudrevange, ‘ 08 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. 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 by 4D GS mechanism. by 4D GS mechanism. 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, ‘ 08 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 Araki, T.K., Kubo, Ramos-Sanches, Ratz, Vaudrevange, ‘ 08 Non-Abelian discrete group finite elements finite elements Each element generates an Abelian symmetry. Each element generates an Abelian symmetry. 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 g k are anomalous. Some ZN symmetries for elements g k are anomalous. The remaining symmetry corresponds to The remaining symmetry corresponds to the closed algebra without such elements. the closed algebra without such elements.

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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 the multiplet. Each element is represented by a matrix on 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 multiplets. String models lead to certain combinations of multiplets. limited pattern of non-Abelian discrete limited pattern of non-Abelian discrete anomalies 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 untwisted (bulk) modes ⇒ singlet The first Z2 is always anomaly-free, while the others can be anomalous. 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. the full D4 remains. e.g. left-handed and right-handed quarks/leptons Such a pattern is realized in explicit models.

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Heterotic models on Z3 orbifold two Z3 ’ s two Z3 ’ 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-free. That is, the Δ(27) is always anomaly-free. Abe, et. al. work in progress Abe, et. al. work in progress

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5. Explicit stringy computation 5. Explicit stringy computation T.K., Parameswaran,Ramos-Sanchez, Zavala ‘11 T.K., Parameswaran,Ramos-Sanchez, Zavala ‘11 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.

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Vertex operators Vertex operators Boson Boson Fermion

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3-pt correlation function 3-pt correlation function Vertex operators: Vertex operators: The 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.

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String on 6D orbifold String on 6D orbifold The 6D part is quite non-trivial. world-sheet instanton + quantum part world-sheet instanton + quantum part

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

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

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T2/Z3 orbifold T2/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 points Z3 symmetries among localized strings on different fixed points Z6 symmetries among localized strings on a fixed point Z6 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 points is included => Z3 twist invarince if the matter on the different fixed points is included

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Classical solution Classical solution 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 ‘11 T.K., Parameswaran,Ramos-Sanchez, Zavala ‘11

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String on 6D orbifold String on 6D orbifold The 6D part is quite non-trivial. world-sheet instanton + quantum part world-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 among That leads to certain conditions on combinations among twists. twists. T.K., Parameswaran,Ramos-Sanchez, Zavala ‘11

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(anti-) Holomorphic instanton (anti-) Holomorphic instanton Condition for fine-valued action Condition for fine-valued action Vertex operators: Vertex operators:

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Summary We have studied discrete symmetries and their anomalies. and their anomalies. Explicit stringy computations tell something more. String-derived massless spectra are (semi-) realistic, but their 4D LEEFT are not so, 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. and explicit stringy computations would be important.

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5. Something else 5. Something else T.K., Parameswaran,Ramos-Sanchez, Zavala ‘11 T.K., Parameswaran,Ramos-Sanchez, Zavala ‘11 Explicit stringy calculations would tell us something more. Explicit stringy calculations would tell us something more. Vertex operators: Vertex operators:

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3-pt correlation function 3-pt correlation function Explicit stringy calculations Explicit stringy calculations Vertex operators: Vertex operators:

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

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Z3 twist invariance Z3 twist invariance

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