Flavor structure in string theory

Transcription

1 Flavor structure in string theory Tatsuo Kobayashi.Introduction. Flavor Symmetries in string theories 3. D-brane instanton effects 4. Oblique magnetic fluxes 5. Numerical study 6. Summary based on collaborations with H.Abe, T.Abe, K.S.Choi, Y.Fujimoto, Y. Hamada, T.Miura, K.Nishiwaki, H.Ohki, A.Oikawa, M.Sakamoto, K.Sumita, Y.Tatsuta, S.Uemura,

2 . Introduction It is one of most important issues to study the flavor origin of quarks and leptons. Why 3 generations? What is the origin of mass hierarchy? What is the origin of mixing angles?

3 Quark masses and mixing angles , 0., 5, 00, 4, 73 ub cb us d u s c b t V V V MeV M MeV M MeV M GeV M GeV M GeV M

4 Charged lepton masses M e 0.5 MeV, M µ 06 MeV M τ.8 GeV

5 Mass hierarchy Quark mass ratios : : t c u 0.0 : : b s d Yukawa couplings with the Higgs field We need the Yukawa hierarchy. What is its origin? mass ratios among charged leptons : 0.06 :

6 Neutrino oscillations mass (squared) difference and mixing angles M ev, M ev sin θ 0.3, sin θ 3 0.5, sin θ 3 0.0, tiny masses and large mixing angles

7 Tri-bimaximal mixing Ansatz Harison, Perkins, Scott, 0 V PMNS some (flavor) symmetries? (approximation in the limit with θ3 0)

8 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, Δ(7), Δ(54),... Review: e.g Ishimori, T.K., Ohki, Okada, Shimizu, Tanimoto 0 large mixing angles one Ansatz: tri-bimaximal / 3 / 3 0 / 6 / 3 / / 6 / 3 /

9 Non-Abelian discrete flavor symm. First, assume a large discrete flavor symmetry. Then, break it to Zn, maybe different Zn s in the charged lepton masses and neutrino masses large mixing angles e.g. tri-bimaximal Ansatz / 3 / 3 0 / 6 / 3 / / 6 / 3 / θ3 0 is ruled out by recent experiments, but this approach is useful, still.

10 What we need Some ideas to lead to the generation number 3. Some mechanism for hierarchical Yukawa couplings. Mixing angles. Discrete flavor symmetries and their proper breaking may be useful for the lepton mixing angles. Symmetry is a good tool to build a bridge between the weak scale and a high scale like the Planck scale. cf. small quark mixing angles What s difference? One possibility is that the lepton sector has the right-handed Majorana neutrino masses, while there are only Dirac masses in the quark sector.

11 Superstring theory Superstring theory is a candidate for unified theory of all the interactions (including gravity) and quarks and leptons and higgs field(s). Superstring theory predicts 6D compact space in addition to our 4D spacetime.

12 String theory with compact space Compactification, geometry and gauge background, determines the generation number, e.g. 3 generations or others. One can compute Yukawa couplings in string theory. Some couplings are hierarchically suppressed. Compactification leads to some flavor symmetries, D4, Δ(7), Δ(54). What about neutrino masses?

13 . Non-Abelian discrete symmetries in string theories X ( σ π ) A string can be specified by its boundary condition. X ( σ 0) Two strings can be connected to become a string if their boundary conditions fit each other. coupling selection rule symmetry Geometry also leads to another symmetry.

14 D-brane models gauge boson: open string, whose two end-points are on the same (set of) D-brane(s) N parallel D-branes U(N) gauge group U() vector multiplet U()

15 . Intersecting D-branes Where are matter fields? U(N) (N,M) matter U(M) New modes appear between intersecting D-branes. They have charges under both gauge groups, i.e. bi-fundamental matter fields.

16 Toy model (in uncompact space) gauge bosons : on brane quarks, leptons, higgs : localized at intersecting points u() su() su(3) H Q L u,d

17 Generation number Torus compactification Family number intersection number su() U() Q Q Q3 su(3) u u u3 su(3)

18 -. Magnetized D-branes We consider torus compactification with magnetic flux background, F45, etc. U(M) U(N) F (N,M) matter

19 Fermions in bifundamentals Breaking the gauge group (Abelian flux case ) The gaugino fields gaugino of unbroken gauge bi-fundamental matter fields

20 Zero-mode Dirac equations No effect due to magnetic flux for adjoint matter fields, Total number of zero-modes of : Zero-modes : No zero-mode

21 Torus with magnetic flux We solve the zero-mode Dirac equation, iγ m Dmψ 0 e.g. for U() charge q. Torus background with magnetic flux leads to chiral spectra. the number of zero-modes M (magnetic flux) x q (charge)

22 Wave functions For the case of M3 Wave function profile on toroidal background Zero-modes wave functions are quasi-localized far away each other in extra dimensions. Therefore the hierarchirally small Yukawa couplings may be obtained.

23 Illustrating model: U(8) Pati-Salam SM m Ι N 0 F π Ι zz i m N N 4, N, N3 0 m3ι N3 Pati-Salam group U ( 4) U () U () L R ( m ( m m m ) ) ( m ( m 3 3 m m ) ) 3 for the first T for the ( other 4,,) + ( 4,, ) tori WLs along a U() in U(4) and a U() in U()R > Standard gauge group up to U() factors U ( 3) U () U C L U()Y is a linear combination. 3 ()

24 3-point couplings Cremades, Ibanez, Marchesano, 04 The 3-point couplings are obtained by overlap integral of three zero-mode w.f. s. Y ijk ( k ψ ( )) i j d zψ M ( z) ψ N ( z) M + N z * ( ψ ( z) ) i d z ψ ( z) k ik M M δ * If they are localized far away each other, the Yukawa coupling is suppressed. CP is violated. Gaussian suppression

25 Yukawa couplings in intersecting D-brane models stringy calculation -> results similar to ones in magnetized D-brane models u() su() su(3) H Q L Y~exp(-area) u,d

26 intersecting/magnetized D-brane models T-dual each other

27 ZN charge 0 Each zero-mode has Zn charge, 0,,,..

28 Flavor symmetries in D-brane models Abe, Choi, T.K. Ohki, 09, 0 There is a Z permutation symmetry. The full symmetry is D4.

29 Z x Z symmetries two Z s 0 They do not commute each other. Non-Abelian 0 0, 0 The full closed algebra is D4. Abe, Choi, T.K., Ohki, 09, 0

30 intersecting/magnetized D-brane models Abe, Choi, T.K. Ohki, 09, 0 geometrical symm. Z3 S3 Full symm. Δ(7) Δ(54)

31 Z3 x Z3 in Heterotic orbifold models Two Z3 s ω 0, 0 0, ω exp(πi 0 0 ω 0 0 / 3) They do not commute each other. Non-Abelian The full closed algebra is Δ(7). If there is Z reflection symmetry, the symmetry is enhanced into Δ(54). Abe, Choi, T.K., Ohki, 09, 0

32 Magnetized brane-models Magnetic flux M D Magnetic flux M Δ(7) (Δ(54)) x 3 9 n n,,9 (+ n n,,4)

33 Non-Abelian discrete flavor symmetry Similar non-abelian flavor symmetries are found in heterotic string theory on orbifolds. such as D4 and Δ(54) T.K. Raby, Zhang, 04 T.K. Nilles, Ploger, Raby, Ratz, 06

34 What we need Some ideas to lead to the generation number 3. Some mechanism for hierarchical Yukawa couplings. Mixing angles. Discrete flavor symmetries and their proper breaking may be useful for the lepton mixing angles. Each piece seems to be ready. What is the next? Nuetrino Majorana masses Certain breaking of flavor symmetries. Explicit studies on quark and lepton (Dirac) masses

35 3. D-brane instanton effects New terms are induced non-perturbatively in the gauge instanton background, depending on the numbers of zero-modes. INST

36 D-brane instanton: neutrino mass neutrino new zero-modes appears they couple with neutrinos Neutrino masses are induced. M n ( dα) ( dβ ) m e ναβ νν Blumenhage, Cvetic, Kachru, Weigand, 06 Ibanez, Uranga, 06 m

37 D-brane instanton: neutrino mass D-brane instantons break the symmetries of the D-brane system. No flavor symmetry?

38 D-brane instanton: neutrino mass We consider effects from all the instantons integration of the positions Cyclic permutation symmetry remains, Z3 Hamada, T.K., Uemura, arxiv:40:05

39 Induced right-handed neutrino mass matrix M A B B B A B B B A A and B are computed explicitly. Hamada, T.K., Uemura, arxiv:40:05

40 Lepton mixing matrix Hamada, T.K., Uemura, arxiv:40:05 We assume that Dirac mass matrices are (almost) diagonal. V PMNS c c c / / 3/ s s / / / / / s s / / s 6 6 / c + c 3 / / Maybe, we can compute similarly the neutrino masses in magnetized D-bran models because of T-duality.

41 4. Oblique magnetic fluxes i γ mψ m D Left-handed quarks (up or down) right-handed quarks F F 45 3 Δ(7) same Δ(7) triplet triplet Higgs sector F pairs x3

42 Oblique fluxes Abe, T.K. Ohki, Sumita, Tatsuta, arxiv: iγ m Dmψ Left-handed quarks (up or down) right-handed quarks F F 3 4 ' 5 ' Δ(7) different Δ(7) Δ(7) is broken. Higgs sector F pair of Higgs

43 Oblique fluxes Abe, T.K. Ohki, Sumita, Tatsuta, arxiv: Abe, T.K. Ohki, Sumita, Tatsuta, arxiv:403.xxxx i γ mψ m D 0 Left-handed quarks F, F F, right-handed quarks Δ(7), S3, Z3xZ3 different Δ(7), S3, Z3xZ3 They break in full Lagrangian. These lead to a rich flavor structure.

44 5. Numerical study Abe, T.K., Sumita, Tatsuta, arxiv:403.xxxx The 3-point couplings are obtained by overlap integral of three zero-mode w.f. s. Y ijk ( k ψ ( )) i j d zψ M ( z) ψ N ( z) M + N z * ( ψ ( z) ) i d z ψ ( z) k ik M M δ * We consider T/Z orbifold models with magnetic fluxes. Abe, T.K., Ohki, 08, Abe, Choi, T.K., Ohki, 09

45 Orbifold with magnetic flux S/Z Orbifold There are two singular points, which are called fixed points.

46 Orbifold with magnetic flux Abe, T.K., Ohki, 08 The number of even and odd zero-modes We can also embed Z into the gauge space. > various models, various flavor structures

47 Wave functions For the case of M3 Wave function profile on toroidal background Zero-modes wave functions are quasi-localized far away each other in extra dimensions. Therefore the hierarchirally small Yukawa couplings may be obtained.

48 Quark/lepton mass ratios and CKM Abe, T.K., Sumita, Tatsuta, arxiv:403.xxxx Left-handed Right-handed Quark M-5 (Z even) M-7 (Z odd) Lepton M-4(Z even) M-8 (Z odd) up sector (down sector) Higgs M (Z odd) 5 pairs of Higgs

49 Quark/lepton mass ratios and CKM Abe, T.K., Sumita, Tatsuta, arxiv:403.xxxx assumption on light Higgs scalar H u (, 0.9, 0, 0, 0) H d (, 0.38, 0., 0, 0) We have just one free parameter, the compex structure, for mass ratios and mixing angles.

50 Quark mass matrices twisted string (first twisted sector) /40) exp(, ) ( N v m N u u u u u u τπ ρ ρ ρ ρ ρ / ', /, / d d d d d d u u u H H H H H H ρ ρ ρ /40) exp(, ' ' ) ( N v m N d d d d d d τπ ρ ρ ρ ρ ρ

51 Quark/lepton masses and mixing angles Abe, T.K., Sumita, Tatsuta, arxiv:403.xxxx Example M M M t c u Im τ.5 GeV, GeV, MeV, M b 4 M s 70 M 8 d GeV MeV MeV V us 0.9, V cb 0.07, V ub 0.0 M M M τ µ e GeV, MeV, MeV, Flavor is still a challenging issue.

52 Orbifolds T/Z3 Orbifold There are three fixed points on Z3 orbifold T/Z4, T/Z6 ZN orbifolds with magnetic fluxes T.Abe, Y.Fujimoto, T.K., T.Miura, K,Nishiwaki, M.Sakamoto, arxiv: , 403.XXXX That leads to a rich structure for model building.

53 Summary We have studied the flavor structure in string theories. Certain string theories derive non-abelian discrete symmetries. Such flavor symmetries are violated by D-brane instanton effects or sophisticating flux configurations. That may be useful to understand the flavor sector, in particular mixing angles. Yukawa couplings are calculable and suppressed values can be obtained. It seems that we can fit fermion masses and mixing angles by a few parameters in rather simple models.

54 Summary It is still a challenging issues how to derive realistic quark/lepton masses and mixing angles.

55 F H 量子力学の復習 : 磁場中の粒子 (Landau) 45 π b, A4 0, A5 πby H m [ H, P5 ] 座標が k/b ずれた調和振動子 ( P + ( P by ) π 0 π m 4 5 4) P 5 4 π ( P + 4 b ( y k / b ) k 4 4 ) b 整数 b 個の基底状態 k0,,,,(b-)

56 Heterotic orbifold models S/Z Orbifold X ( σ π ) X ( σ 0) X ( σ π ) e / ( X ( σ 0) e / ) X ( σ π ) X ( σ 0) + n e, n 0, (mod )

57 Z x Z in Heterotic orbifold models S/Z Orbifold X ( σ π ) m, n 0, two Z s twisted string 0 m ( ) X ( (mod ) 0, untwisted string Z even for both Z 0 σ 0) 0 + n e,

58 Closed strings on orbifold Untwisted and twisted strings Twisted strings (first twisted sector) X ( σ π ) θx ( σ 0) + n e, n θ 0,, (mod 3) 0 twist, up to lattice Λ 3me + n( e e) second twisted sector X ( σ π ) θ X ( σ 0) + n e, n 0,, (mod 3) untwisted sector X ( σ π ) X ( σ 0)

59 Quark mass matrices twisted string (first twisted sector) /40) exp(, ) ( N v m N u u u u u u τπ ρ ρ ρ ρ ρ / ', /, / d d d d d d u u u H H H H H H ρ ρ ρ /40) exp(, ' ' ) ( N v m N d d d d d d τπ ρ ρ ρ ρ ρ

60 Heterotic orbifold models T/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 3) / exp(, , πi ω ω ω ω ω ω ,

61 Heterotic orbifold models T/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

62 Higher Dimensional theory with flux Abelian gauge field on magnetized torus Constant magnetic flux gauge fields of background Consistency requires Dirac s quantization condition.

63 Illustrating model: Pati-Salam SM model m Ι N 0 F π Ι zz i m N N 4, N, N3 0 m3ι N3 Pati-Salam group U ( 4) U () U () L R ( m ( m m m ) ) ( m ( m 3 3 m m ) ) 3 for the first T for the ( other 4,,) + ( 4,, ) tori WLs along a U() in U(4) and a U() in U()R > Standard gauge group up to U() factors U ( 3) U () U C L U()Y is a linear combination. 3 ()

64 Wilson lines Cremades, Ibanez, Marchesano, 04, Abe, Choi, T.K. Ohki, 09 torus without magnetic flux constant Ai mass shift every modes massive magnetic flux [ + ( + )] π My a [ π ( My + a) ] ψ 0 the number of zero-modes is the same. the profile: f(y) f(y +a/m) with proper b.c. ψ + 0

65 PS > SM Zero modes corresponding to three families of matter fields ( 4,,) + (4,,) remain after introducing WLs, but their profiles split (4,,) (3,,) + (,,) (4,,) (3,,) + (3,,) + (,,) + (,,) (4,,) Q L

66 Explicit magnetized brane models Abe, T.K. Ohki, Oikawa, Sumita, Starting point D9 brane model: 0D U(8) Super Yang-Mills theory Magenetic fluxes and Wilson lines 4D SU(3)xSU()xU()y three generations of quarks and leptons 6 pairs of Higgs Flavor symmetry: Δ(7) up-type, down-type of quarks and left-handed and right-handed leptons localize at different points