Flavor structure in string theory Tatsuo Kobayashi １．Introduction 2. 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,

1. 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 ?

Quark masses and mixing angles

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Charged lepton masses

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Mass hierarchy Quark mass ratios 1 : 0.007 : 0.00002 t c u 0.02 : 0.0007: 0.00004 b s d Yukawa couplings with the Higgs field We need the Yukawa hierarchy. What is its origin ? mass ratios among charged leptons 1 : 0.06 : 0.0003

Neutrino oscillations

mass (squared) difference and mixing angles tiny masses and large mixing angles

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102,108

132

232

122

23231

25221

===

×=∆×=∆ −−

θθθ

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Tri-bimaximal mixing Ansatz Harison, Perkins, Scott, ‘02 ⇒ some (flavor) symmetries ? (approximation in the limit with θ13 =0)

−

−−≈

21

31

61

21

31

61

03

16

2

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

−−−

2/13/16/12/13/16/1

03/13/2

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 θ13 = 0 is ruled out by recent experiments, but this approach is useful, still.

−−−

2/13/16/12/13/16/1

03/13/2

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.

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.

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(27), Δ（54）. What about neutrino masses ?

2. Non-Abelian discrete symmetries in string theories 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. coupling selection rule symmetry Geometry also leads to another symmetry.

)0( =σX)( πσ =X

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(1) vector U(2) multiplet

2.1 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.

Toy model (in uncompact space)

gauge bosons ： on brane quarks, leptons, higgs : localized at intersecting points u(1) su(2) su(3) H Q L u,d

Generation number Torus compactification Family number = intersection number su(2) U(1) Q1 Q2 Q3 su(3) u1 u2 u3 su(3)

2-2. Magnetized D-branes

We consider torus compactification with magnetic flux background, F45, etc. U(M) U(N) (N,M) F matter

Fermions in bifundamentals

The gaugino fields

Breaking the gauge group

bi-fundamental matter fields

gaugino of unbroken gauge

(Abelian flux case )

Zero-mode Dirac equations

Total number of zero-modes of

: Zero-modes

: No zero-mode

No effect due to magnetic flux for adjoint matter fields,

Torus with magnetic flux We solve the zero-mode Dirac equation, e.g. for U(1) charge q=1. Torus background with magnetic flux leads to chiral spectra. the number of zero-modes = M (magnetic flux) x q (charge)

0=ψγ mmDi

Wave functions

Wave function profile on toroidal background

For the case of M=3

Zero-modes wave functions are quasi-localized far away each other in extra dimensions. Therefore the hierarchirally small Yukawa couplings may be obtained.

Illustrating model: U(8) Pati-Salam SM

Pati-Salam group WLs along a U(1) in U(4) and a U(1) in U(2)R => Standard gauge group up to U(1) factors U(1)Y is a linear combination.

( ) ( )2,1,41,2,4 +

ΙΙ

Ι=

3

2

1

3

2

1

0

02

N

N

N

zz

mm

miF π 2 ,2 ,4 321 === NNN

RL UUU )2()2()4( ××

other tori for the 1)()(first for the 3)()(

1321

21321

=−=−=−=−

mmmmTmmmm

3)1()2()3( UUU LC ××

3-point couplings Cremades, Ibanez, Marchesano, ‘04

The 3-point couplings are obtained by overlap integral of three zero-mode w.f.’s.

If they are localized far away each other, the Yukawa coupling is suppressed. CP is violated. Gaussian suppression

( )∫ = ikkM

iM zzzd δψψ

*2 )()(

( )∫ +=*2 )()()( zzzzdY k

NMj

NiMijk ψψψ

Yukawa couplings in intersecting D-brane models stringy calculation -> results similar to ones in magnetized D-brane models u(1) su(2) su(3) H Q L Y~exp(-area) u,d

intersecting/magnetized D-brane models

⇔ T-dual each other

ZN charge

0 1 2

Each zero-mode has Zn charge, 0, 1, 2, ………..

Flavor symmetries in D-brane models Abe, Choi, T.K. Ohki, ’09, ‘10

There is a Z2 permutation symmetry. The full symmetry is D4.

Z2 x Z2 symmetries two Z2’s They do not commute each other. Non-Abelian The full closed algebra is D4. Abe, Choi, T.K., Ohki, ‘09, ‘10

− 01

10 ,

1001

intersecting/magnetized D-brane models Abe, Choi, T.K. Ohki, ’09, ‘10

geometrical symm. Full symm. Z3 Δ(27)

S3 Δ(54)

Z3 x Z3 in Heterotic orbifold models Two Z3’s They do not commute each other. Non-Abelian The full closed algebra is Δ(27). If there is Z2 reflection symmetry, the symmetry is enhanced into Δ(54). Abe, Choi, T.K., Ohki, ‘09, ‘10

)3/2exp( ,001100010

,00

00001

2

iπωω

ω =

Magnetized brane-models

Magnetic flux M D4 2 2 4 1++ + 1+- +1-+ + 1-- ・・・ ・・・・・・・・・ Magnetic flux M Δ(27) (Δ(54)) 3 31

6 2 x 31

9 ∑1n n=1,…,9 (11+∑2n n=1,…,4) ・・・ ・・・・・・・・・

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

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

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

INST

D-brane instanton: neutrino mass

neutrino new zero-modes appears they couple with neutrinos Neutrino masses are induced. Blumenhage, Cvetic, Kachru, Weigand, ’06 Ibanez, Uranga, ‘06

ννβα ναβ meddM mn ⇒−∫ )()(

D-brane instanton: neutrino mass

D-brane instantons break the symmetries of the D-brane system. No flavor symmetry ?

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:1402:2052

Induced right-handed neutrino mass matrix

A and B are computed explicitly.

Hamada, T.K., Uemura, arXiv:1402:2052

=

ABBBABBBA

M

Lepton mixing matrix Hamada, T.K., Uemura, arXiv:1402:2052 We assume that Dirac mass matrices are (almost) diagonal. Maybe, we can compute similarly the neutrino masses in magnetized D-bran models because of T-duality.

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4. Oblique magnetic fluxes (up or down) Left-handed quarks right-handed quarks Δ(27) same Δ(27) triplet triplet Higgs sector 6 pairs 2x3

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Oblique fluxes Abe, T.K. Ohki, Sumita, Tatsuta, arXiv:1307.1831 (up or down) Left-handed quarks right-handed quarks Δ(27) different Δ(27) Δ(27) is broken. Higgs sector 1 pair of Higgs

345 =F 3''54 =F

1=F

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Oblique fluxes Abe, T.K. Ohki, Sumita, Tatsuta, arXiv:1307.1831 Abe, T.K. Ohki, Sumita, Tatsuta, arXiv:1403.XXXX Left-handed quarks right-handed quarks Δ(27) , S3, Z3xZ3 different Δ(27), S3, Z3xZ3 They break in full Lagrangian. These lead to a rich flavor structure.

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5. Numerical study Abe, T.K., Sumita, Tatsuta, arXiv:1403.XXXX

The 3-point couplings are obtained by overlap integral of three zero-mode w.f.’s. We consider T2/Z2 orbifold models with magnetic fluxes. Abe, T.K., Ohki, ‘08, Abe, Choi, T.K., Ohki, ‘09

( )∫ = ikkM

iM zzzd δψψ

*2 )()(

( )∫ +=*2 )()()( zzzzdY k

NMj

NiMijk ψψψ

Orbifold with magnetic flux

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

Orbifold with magnetic flux Abe, T.K., Ohki, ‘08 The number of even and odd zero-modes We can also embed Z2 into the gauge space. => various models, various flavor structures

Wave functions

Wave function profile on toroidal background

For the case of M=3

Zero-modes wave functions are quasi-localized far away each other in extra dimensions. Therefore the hierarchirally small Yukawa couplings may be obtained.

Quark/lepton mass ratios and CKM Abe, T.K., Sumita, Tatsuta, arXiv:1403.XXXX Left-handed Right-handed Quark M=-5 (Z2 even) M=-7 (Z2 odd) Lepton M=-4(Z2 even) M=-8 (Z2 odd) up sector (down sector) Higgs M=12 (Z2 odd) 5 pairs of Higgs

Quark/lepton mass ratios and CKM Abe, T.K., Sumita, Tatsuta, arXiv:1403.XXXX assumption on light Higgs scalar We have just one free parameter, the compex structure, for mass ratios and mixing angles.

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∝∝

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u

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Quark mass matrices twisted string (first twisted sector)

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1411

83223

2025100)( Nvm N

u

uu

uuu τπη

ηρηηρηρηηρηηρη

−=

−−−

≈

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/420)exp( ,''

2

1411

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Quark/lepton masses and mixing angles Abe, T.K., Sumita, Tatsuta, arXiv:1403.XXXX Example Flavor is still a challenging issue.

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Orbifolds

T2/Z3 Orbifold There are three fixed points on Z3 orbifold T2/Z4, T2/Z6 ZN orbifolds with magnetic fluxes T.Abe, Y.Fujimoto, T.K., T.Miura, K,Nishiwaki, M.Sakamoto, arXiv: 1309.4925, 1403.XXXX That leads to a rich structure for model building.

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.

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

量子力学の復習：磁場中の粒子(Landau)

座標がｋ/bずれた調和振動子 b=整数

( )245

24 )2(

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H π−+=

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0],[ 5 =PH kP π25 =

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2224 )/(4

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H −+= π

ｂ個の基底状態 k=0,1,2,…………,(b-1)

Heterotic orbifold models S1/Z2 Orbifold

)2/)0((2/)()0()(

eXeXXX

−=−=−==−==

σπσσπσ

2) (mod 1 ,0 , )0()( =+=−== nenXX σπσ

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

−

−

−10

01 ,

1001

2) (mod 1 ,0 , , )0()1()(

=+=−==

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Closed strings on orbifold

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

)(e3 lattice toup twist,120

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)0()( === σπσ XX

Quark mass matrices twisted string (first twisted sector)

/420)exp( , 2

1411

83223

2025100)( Nvm N

u

uu

uuu τπη

ηρηηρηρηηρηηρη

−=

−−−

≈

424343 /',/,/ dddddduuu HHHHHH === ρρρ

/420)exp( ,''

2

1411

8323

1525100)( Nvm N

d

dd

ddd τπη

ηρηηρηρηηρηηρη

−=

−−−

≈

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

)3/2exp( ,00

00001

,00

0000

2

iπωω

ωω

ωω

=

010100001

,001100010

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

Higher Dimensional theory with flux

Abelian gauge field on magnetized torus

Constant magnetic flux

Consistency requires Dirac’s quantization condition.

gauge fields of background

Illustrating model: Pati-Salam SM model

Pati-Salam group WLs along a U(1) in U(4) and a U(1) in U(2)R => Standard gauge group up to U(1) factors U(1)Y is a linear combination.

( ) ( )2,1,41,2,4 +

ΙΙ

Ι=

3

2

1

3

2

1

0

02

N

N

N

zz

mm

miF π 2 ,2 ,4 321 === NNN

RL UUU )2()2()4( ××

other tori for the 1)()(first for the 3)()(

1321

21321

=−=−=−=−

mmmmTmmmm

3)1()2()3( UUU LC ××

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 the number of zero-modes is the same. the profile: f(y) f(y +a/M) with proper b.c.

[ ][ ] 0 )(2

0 )(2=+−∂=++∂

−

+

ψπψπ

aMyaMy

PS => SM Zero modes corresponding to three families of matter fields remain after introducing WLs, but their profiles split (4,2,1) Q L

)2,1,4()1,2,4( +

)1,1,1()1,1,1()1,1,3()1,1,3()2,1,4()1,2,1()1,2,3()1,2,4(

+++=

+=

Explicit magnetized brane models Abe, T.K. Ohki, Oikawa, Sumita,’12 Starting point D9 brane model: 10D U(8) Super Yang-Mills theory Magenetic fluxes and Wilson lines 4D SU(3)xSU(2)xU(1)y three generations of quarks and leptons 6 pairs of Higgs Flavor symmetry: Δ(27) up-type, down-type of quarks and left-handed and right-handed leptons localize at different points