Presentation on theme: " Symmetries and vanishing couplings in string-derived low-energy effective field theory Tatsuo Kobayashi １．Introduction."— Presentation transcript:
1 Symmetries and vanishing couplings in string-derived low-energy effective field theory Tatsuo Kobayashi１．Introduction2. Abelian Discrete Symmetries3． Non-Abelian Discrete Symmetries4． Anomalies5. Explicit stringy computations6. Summary
2 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. inheterotic 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 ?
3 4D low-energy effective field theory We have to control couplings in 4D LEEFT.realization of quark/lepton mass and mixingincluding the neutrino sector,avoiding the fast proton decay,stability of the LSP, suppressing FCNC, etcAbelian and non-Abelian discrete symmetries are useful in low-energy model buildingto control them.3
5 Non-Abelian discrete flavor symm. Recently, in field-theoretical model building,several types of discrete flavor symmetries havebeen proposed with showing interesting results,e.g. S3, D4, A4, S4, Q6, Δ(27),Δ(54),Review: e.gIshimori, T.K., Ohki, Okada, Shimizu, Tanimoto ‘10⇒ large mixing anglesin the lepton sectorone Ansatz: tri-bimaximal5
6 String-derived 4D LEEFT Can we derive these Abelian and non-Abeliandiscrete symmetries from string theory ?Which symmetires can appear in 4D LEEFTderived from string theory ?One can compute couplings of 4D LEEFTin 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
7 2. Abelian discrete symmetries coupling selection ruleA string can be specified byits boundary condition.Two strings can be connectedto become a string if theirboundary conditions fit each other.symmetry
8 Heterotic orbifold models S1/Z2 Orbifold There are two singular points,which are called fixed points.
9 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 latticeOrbifold = D-dim. Torus /twistTorus = D-dim flat space/ lattice
10 Closed strings on orbifold Untwisted and twisted stringsTwisted strings are associated with fixed points.“Brane-world” terminology:untwisted sector bulk modestwisted sector brane (localized) modes
13 Z2 x Z2 in Heterotic orbifold models S1/Z2 Orbifoldtwo Z2’stwisted stringuntwisted stringZ2 even for both Z2
14 Closed strings on orbifold Untwisted and twisted stringsTwisted strings (first twisted sector)second twisted sectoruntwisted sector
15 Z3 x Z3 in Heterotic orbifold models T2/Z3 Orbifoldtwo Z3’stwisted string (first twisted sector)untwisted stringvanishing Z3 charges for both Z3
16 3. Non-Abelian discrete symmetries Heterotic orbifold models S1/Z2 Orbifold String theory has two Z2’s.In addition, the Z2 orbifold has the geometricalsymmetry, i.e. Z2 permutation.
17 D4 Flavor SymmetryStringy 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 elementsmodes on two fixed points ⇒ doubletuntwisted (bulk) modes ⇒ singletGeometry of compact space origin of finite flavor symmetryAbelian part (Z2xZ2) : coupling selection ruleS2 permutation : one coupling is the same as another.T.K., Raby, Zhang, ‘05 T.K., Nilles, Ploger, Raby, Ratz, ‘07
22 Abelian discrete anomalies: path integral Zn transformationpath integral measureZN-G-G anomaliesanomaly free condition
23 Heterotic orbifold models There are two types of Abelian discrete symmetries.T2/Z3 Orbifoldtwo Z3’sOne is originated from twists,the other is originated from shifts.Both types of discrete anomaliesare universal for different groups G.Araki, T.K., Kubo, Ramos-Sanches, Ratz,Vaudrevange, ‘08
24 Heterotic orbifold models U(1)-G-G anomaliesare universal for different groups G.4D Green-Schwarz mechanismdue to a single axion (dilaton),which couples universally with gauge sectors.ZN-G-G anomalies may also be cancelledby 4D GS mechanism.There is a certain relations betweenU(1)-G-G and ZN-G-G anomalies,anomalous U(1) generator is a linear combinationof anomalous ZN generators.Araki, T.K., Kubo, Ramos-Sanches, Ratz,Vaudrevange, ‘08
25 4-2. Non-Abelian discrete anomalies Araki, T.K., Kubo, Ramos-Sanches, Ratz, Vaudrevange, ‘08Non-Abelian discrete groupfinite elementsEach 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.
26 Non-Abelian discrete anomalies matter fields = multiplets under non-Abeliandiscrete symmetryEach element is represented by a matrix on the multiplet.Such a multiplet does not contribute toZN-G-G anomalies.String models lead to certain combinations of multiplets. limited pattern of non-Abelian discreteanomalies
27 Heterotic string on Z2 orbifold: D4 Flavor Symmetry Flavor symmeties: closed algebra S2 U(Z2xZ2)modes on two fixed points ⇒ doubletuntwisted (bulk) modes ⇒ singletThe first Z2 is always anomaly-free, while the others can be anomalous.However, it is simple to arrange models such thatthe full D4 remains.e.g. left-handed and right-handed quarks/leptons1 + 2Such a pattern is realized in explicit models.
28 Heterotic models on Z3 orbifold two Z3’sZ3 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
29 5. Explicit stringy computation T.K., Parameswaran,Ramos-Sanchez, Zavala ‘11Explicit string computations tell something more.Heterotic string on orbifold:4D string, E8xE8 gauge part,r-moving fermionic string,6D string on orbifold, ghostA mode has a definite “quantum number”in each part.
31 3-pt correlation function Vertex operators:The correlation function vanishes unlessThat is the momentum conservation in the string ofthe gauge part, i.e. the gauge invariance.
32 String on 6D orbifold The 6D part is quite non-trivial. world-sheet instanton + quantum part 32
33 Classical solution The world-sheet instanton, which corresponds to string moving from a fixed pointto others.
34 symmetries of torus (sub)lattice Suppose that only holomorphic instanton can appear.Example: T2/Z3different fixed pointsZ3 symmetries (twist invariance)twist invariance+ H-momentum conservation=> discrete R-symmetry34
35 symmetries of torus (sub)lattice Suppose that only holomorphic instanton can appear.Example: T2/Z3three strings on the same fixed pointsZ6 symmetriesenhanced symmetriesfor certain couplings Rule 4Font, Ibanez, Nilles, Quevedo, ’88T.K., Parameswaran,Ramos-Sanchez, Zavala ‘1135
36 T2/Z3 orbifold Higher order couplings The situation is the same. Z3 symmetries among localized strings on different fixed pointsZ6 symmetries among localized strings on a fixed pointZ6 enhanced symmetries only for the couplingsof the same fixed points.=> Z3 twist invarince if the matter on the different fixed points is included36
37 Classical solution T2/Z2 Similarly, the classical solution corresponding tocouplings on the same fixed points has enhancedsymmetires, e.g. Z4 and Z6, depending the toruslattice, SO(5) torus and SU(3) torus.T.K., Parameswaran,Ramos-Sanchez, Zavala ‘1137
38 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 amongtwists.T.K., Parameswaran,Ramos-Sanchez, Zavala ‘1138
39 (anti-) Holomorphic instanton Condition for fine-valued actionVertex operators:39
40 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 symmetiresand explicit stringy computations would be important.
41 5. Something else T.K., Parameswaran,Ramos-Sanchez, Zavala ‘11 Explicit stringy calculations would tell us something more.Vertex operators:
42 3-pt correlation function Explicit stringy calculationsVertex operators:42