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Phenomenological aspects of magnetized brane models Tatsuo Kobayashi １． Introduction 2. Magnetized extra dimensions 3. N-point couplings and flavor symmetries 4 ． Massive modes 5. Flavor and Higgs sector 6. Summary based on collaborations with based on collaborations with H.Abe, T.Abe, K.S.Choi, Y.Fujimoto, Y.Hamada,R.Maruyama, T.Miura, M.Murata, Y.Nakai, K.Nishiwaki, H.Ohki, A.Oikawa, M.Sakai, M.Sakamoto, K.Sumita, Y.Tatsuta A.Oikawa, M.Sakai, M.Sakamoto, K.Sumita, Y.Tatsuta

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１ Introduction Extra dimensional field theories, in particular in particular string-derived extra dimensional field theories, string-derived extra dimensional field theories, play important roles in particle physics play important roles in particle physics as well as cosmology. as well as cosmology.

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Extra dimensions 4 + n dimensions 4 + n dimensions 4D ⇒ our 4D space-time 4D ⇒ our 4D space-time nD ⇒ compact space nD ⇒ compact space Examples of compact space torus, orbifold, CY, etc. torus, orbifold, CY, etc.

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Field theory in higher dimensions 10D ⇒ 4D our space-time + 6D space 10D ⇒ 4D our space-time + 6D space 10D vector 10D vector 4D vector + 4D scalars 4D vector + 4D scalars SO(10) spinor ⇒ SO(4) spinor x SO(6) spinor x SO(6) spinor internal quantum number internal quantum number

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Several Fields in higher dimensions 4D (Dirac) spinor 4D (Dirac) spinor ⇒ (4D) Clifford algebra ⇒ (4D) Clifford algebra (4x4) gamma matrices (4x4) gamma matrices represention space ⇒ spinor representation represention space ⇒ spinor representation 6D Clifford algebra 6D Clifford algebra 6D spinor 6D spinor 6D spinor ⇒ 4D spinor x (internal spinor) 6D spinor ⇒ 4D spinor x (internal spinor) internal quantum internal quantum number number

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Field theory in higher dimensions Mode expansions Mode expansions KK decomposition KK decomposition

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KK docomposition on torus torus with vanishing gauge background Boundary conditions Boundary conditions We concentrate on zero-modes. We concentrate on zero-modes.

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Zero-modes Zero-mode equation ⇒ non-trival zero-mode profile the number of zero-modes the number of zero-modes

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4D effective theory Higher dimensional Lagrangian (e.g. 10D) integrate the compact space ⇒ 4D theory integrate the compact space ⇒ 4D theory Coupling is obtained by the overlap integral of wavefunctions integral of wavefunctions

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Couplings in 4D Zero-mode profiles are quasi-localized Zero-mode profiles are quasi-localized far away from each other in compact space far away from each other in compact space ⇒ suppressed couplings ⇒ suppressed couplings

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Chiral theory When we start with extra dimensional field theories, how to realize chiral theories is one of important issues from the viewpoint of particle physics. how to realize chiral theories is one of important issues from the viewpoint of particle physics. Zero-modes between chiral and anti-chiral fields are different from each other fields are different from each other on certain backgrounds, on certain backgrounds, e.g. CY, toroidal orbifold, warped orbifold, e.g. CY, toroidal orbifold, warped orbifold, magnetized extra dimension, etc. magnetized extra dimension, etc.

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Magnetic flux The limited number of solutions with non-trivial backgrounds are known. non-trivial backgrounds are known. Generic CY is difficult. Toroidal/Wapred orbifolds are well-known. Background with magnetic flux is one of interesting backgrounds. one of interesting backgrounds.

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Magnetic flux Indeed, several studies have been done in both extra dimensional field theories in both extra dimensional field theories and string theories with magnetic flux and string theories with magnetic flux background. background. In particular, magnetized D-brane models are T-duals of intersecting D-brane models. are T-duals of intersecting D-brane models. Several interesting models have been constructed in intersecting D-brane models, constructed in intersecting D-brane models, that is, the starting theory is U(N) SYM. that is, the starting theory is U(N) SYM.

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Phenomenology of magnetized brane models It is important to study phenomenological aspects of magnetized brane models such as aspects of magnetized brane models such as massless spectra from several gauge groups, massless spectra from several gauge groups, U(N), SO(N), E6, E7, E8,... U(N), SO(N), E6, E7, E8,... Yukawa couplings and higher order n-point Yukawa couplings and higher order n-point couplings in 4D effective theory, couplings in 4D effective theory, their symmetries like flavor symmetries, their symmetries like flavor symmetries, Kahler metric, etc. Kahler metric, etc. It is also important to extend such studies on torus background to other backgrounds on torus background to other backgrounds with magnetic fluxes, e.g. orbifold backgrounds. with magnetic fluxes, e.g. orbifold backgrounds.

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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 ｂ個の基底状態 k=0,1,2,…………,(b-1)

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2. Extra dimensions with magnetic fluxes: basic tools 2-1. Magnetized torus model 2-1. Magnetized torus model We start with N=1 super Yang-Mills theory in D = 4+2n dimensions. in D = 4+2n dimensions. For example, 10D super YM theory For example, 10D super YM theory consists of gauge bosons (10D vector) consists of gauge bosons (10D vector) and adjoint fermions (10D spinor). and adjoint fermions (10D spinor). We consider 2n-dimensional torus compactification with magnetic flux background. with magnetic flux background. We can start with 6D SYM (+ hyper multiplets), or non-SUSY models (+ matter fields ), similarly. or non-SUSY models (+ matter fields ), similarly.

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Higher Dimensional SYM theory with flux Cremades, Ibanez, Marchesano, ‘ 04 The wave functions eigenstates of corresponding internal Dirac/Laplace operator. 4D Effective theory <= dimensional reduction

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Higher Dimensional SYM theory with flux Abelian gauge field on magnetized torus Constant magnetic flux The boundary conditions on torus (transformation under torus translations) gauge fields of background

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Higher Dimensional SYM theory with flux We now consider a complex field with charge Q ( +/-1 ) Consistency of such transformations under a contractible loop in torus which implies Dirac ’ s quantization conditions.

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Dirac equation on 2D torus with twisted boundary conditions (Q=1) is the two component spinor. U(1) charge Q=1

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|M| independent zero mode solutions in Dirac equation. (Theta function) Dirac equation and chiral fermion Properties of theta functions :Normalizable mode :Non-normalizable mode By introducing magnetic flux, we can obtain chiral theory. chiral fermion

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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.

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

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Bi-fundamental Gaugino fields in off-diagonal entries correspond to bi-fundamental matter fields correspond to bi-fundamental matter fields and the difference M= m-m ’ of magnetic and the difference M= m-m ’ of magnetic fluxes appears in their Dirac equation. fluxes appears in their Dirac equation. F

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

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4D chiral theory 10D spinor 10D spinor light-cone 8s light-cone 8s even number of minus signs even number of minus signs 1 st ⇒ 4D, the other ⇒ 6D space 1 st ⇒ 4D, the other ⇒ 6D space If all of appear If all of appear in 4D theory, that is non-chiral theory. in 4D theory, that is non-chiral theory. If for all torus, If for all torus, only only appear for 4D helicity fixed. appear for 4D helicity fixed. ⇒ 4D chiral theory ⇒ 4D chiral theory

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U(8) SYM theory on T6 Pati-Salam group up to U(1) factors Three families of matter fields with many Higgs fields

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2-2. Wilson lines Cremades, Ibanez, Marchesano, ’ 04, Cremades, Ibanez, Marchesano, ’ 04, Abe, Choi, T.K. Ohki, ‘ 09 Abe, Choi, T.K. Ohki, ‘ 09 torus without magnetic flux torus without magnetic flux constant Ai mass shift constant Ai mass shift every modes massive every modes massive magnetic flux magnetic flux the number of zero-modes is the same. the number of zero-modes is the same. the profile: f(y) f(y +a/M) the profile: f(y) f(y +a/M) with proper b.c. with proper b.c.

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U(1)a*U(1)b theory magnetic flux, Fa=2πM, Fb=0 magnetic flux, Fa=2πM, Fb=0 Wilson line, Aa=0, Ab=C Wilson line, Aa=0, Ab=C matter fermions with U(1) charges, (Qa,Qb) matter fermions with U(1) charges, (Qa,Qb) chiral spectrum, chiral spectrum, for Qa=0, massive due to nonvanishing WL for Qa=0, massive due to nonvanishing WL when MQa >0, the number of zero-modes when MQa >0, the number of zero-modes is MQa. is MQa. zero-mode profile is shifted depending zero-mode profile is shifted depending on Qb, on Qb,

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

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

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Other models We can start with 10D SYM, 6D SYM (+ hyper multiplets), 6D SYM (+ hyper multiplets), or non-SUSY models (+ matter fields ) or non-SUSY models (+ matter fields ) with gauge groups, with gauge groups, U(N), SO(N), E6, E7,E8,...

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E6 SYM theory on T6 Choi, et. al. ‘ 09 Choi, et. al. ‘ 09 We introduce magnetix flux along U(1) direction, We introduce magnetix flux along U(1) direction, which breaks E6 -> SO(10)*U(1) which breaks E6 -> SO(10)*U(1) Three families of chiral matter fields 16 We introduce Wilson lines breaking We introduce Wilson lines breaking SO(10) -> SM group. SO(10) -> SM group. Three families of quarks and leptons matter fields with no Higgs fields with no Higgs fields

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Splitting zero-mode profiles Wilson lines do not change the (generation) number of zero-modes, but change localization point Q …… L Q …… L

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2.3 Orbifold with magnetic flux 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 T2/Z4, T2/Z6 T2/Z4, T2/Z6 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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Orbifold with magnetic flux Orbifold with magnetic flux Abe, T.K., Ohki, ‘ 08 Abe, T.K., Ohki, ‘ 08 The number of even and odd zero-modes The number of even and odd zero-modes We can also embed Z2 into the gauge space. We can also embed Z2 into the gauge space. => various models, various flavor structures => various models, various flavor structures

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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.

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Zero-modes on orbifold Adjoint matter fields are projected by orbifold projection. orbifold projection. We have degree of freedom to introduce localized modes on fixed points introduce localized modes on fixed points like quarks/leptons and higgs fields. like quarks/leptons and higgs fields.

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Zero-modes on ZN orbifold Similarly we can discuss 2D ZN orbifolds with magnetic fluxes 2D ZN orbifolds with magnetic fluxes for N=2,3,4 and 6. for N=2,3,4 and 6. Abe, Fujimoto, T.K., Miura, Nishiwaki, Abe, Fujimoto, T.K., Miura, Nishiwaki, Sakamoto, arXiv: Sakamoto, arXiv:

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3.N-point couplings and flavor symmetries The N-point couplings are obtained by The N-point couplings are obtained by overlap integral of their zero-mode w.f. ’ s. overlap integral of their zero-mode w.f. ’ s.

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Moduli Torus metric Torus metric Area Area We can repeat the previous analysis. Scalar and vector fields have the same wavefunctions. wavefunctions. Wilson moduli shift of w.f. shift of w.f.

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Zero-modes Cremades, Ibanez, Marchesano, ‘ 04 Cremades, Ibanez, Marchesano, ‘ 04 Zero-mode w.f. = gaussian x theta-function Zero-mode w.f. = gaussian x theta-function up to normalization factor up to normalization factor

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Products of wave functions products of zero-modes = zero-modes products of zero-modes = zero-modes

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3-point couplings Cremades, Ibanez, Marchesano, ‘ 04 Cremades, Ibanez, Marchesano, ‘ 04 The 3-point couplings are obtained by The 3-point couplings are obtained by overlap integral of three zero-mode w.f. ’ s. overlap integral of three zero-mode w.f. ’ s. up to normalization factor up to normalization factor

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Selection rule Each zero-mode has a Zg charge, Each zero-mode has a Zg charge, which is conserved in 3-point couplings. which is conserved in 3-point couplings. up to normalization factor up to normalization factor

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4-point couplings Abe, Choi, T.K., Ohki, ‘ 09 Abe, Choi, T.K., Ohki, ‘ 09 The 4-point couplings are obtained by The 4-point couplings are obtained by overlap integral of four zero-mode w.f. ’ s. overlap integral of four zero-mode w.f. ’ s. split insert a complete set insert a complete set up to normalization factor up to normalization factor for K=M+N for K=M+N

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4-point couplings: another splitting i k i k i k i k t j s l j l j s l j l

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N-point couplings Abe, Choi, T.K., Ohki, ‘ 09 Abe, Choi, T.K., Ohki, ‘ 09 We can extend this analysis to generic n-point couplings. We can extend this analysis to generic n-point couplings. N-point couplings = products of 3-point couplings N-point couplings = products of 3-point couplings = products of theta-functions = products of theta-functions This behavior is non-trivial. (It ’ s like CFT.) Such a behavior would be satisfied Such a behavior would be satisfied not for generic w.f. ’ s, but for specific w.f. ’ s. not for generic w.f. ’ s, but for specific w.f. ’ s. However, this behavior could be expected However, this behavior could be expected from T-duality between magnetized from T-duality between magnetized and intersecting D-brane models. and intersecting D-brane models.

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T-duality The 3-point couplings coincide between The 3-point couplings coincide between magnetized and intersecting D-brane models. magnetized and intersecting D-brane models. explicit calculation explicit calculation Cremades, Ibanez, Marchesano, ‘ 04 Cremades, Ibanez, Marchesano, ‘ 04 Such correspondence can be extended to Such correspondence can be extended to 4-point and higher order couplings because of 4-point and higher order couplings because of CFT-like behaviors, e.g., CFT-like behaviors, e.g., Abe, Choi, T.K., Ohki, ‘ 09 Abe, Choi, T.K., Ohki, ‘ 09

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Non-Abelian discrete flavor symmetry The coupling selection rule is controlled by Zg charges. Zg charges. For M=g, 1 2 g For M=g, 1 2 g Effective field theory also has a cyclic permutation symmetry of g zero-modes. Effective field theory also has a cyclic permutation symmetry of g zero-modes. These lead to non-Abelian flavor symmetires such as D4 and Δ(27) Abe, Choi, T.K, Ohki, ‘09 such as D4 and Δ(27) Abe, Choi, T.K, Ohki, ‘09 Cf. heterotic orbifolds, T.K. Raby, Zhang, ’ 04 Cf. heterotic orbifolds, T.K. Raby, Zhang, ’ 04 T.K. Nilles, Ploger, Raby, Ratz, ‘ 06 T.K. Nilles, Ploger, Raby, Ratz, ‘ 06

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Permutation symmetry D-brane models Permutation symmetry D-brane models Abe, Choi, T.K. Ohki, ’ 09, ‘ 10 Abe, Choi, T.K. Ohki, ’ 09, ‘ 10 There is a Z2 permutation symmetry. The full symmetry is D4.

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Permutation symmetry D-brane models Permutation symmetry D-brane models Abe, Choi, T.K. Ohki, ’ 09, ‘ 10 Abe, Choi, T.K. Ohki, ’ 09, ‘ 10 geometrical symm. Full symm. geometrical symm. Full symm. Z3 Δ(27) Z3 Δ(27) S3 Δ(54) S3 Δ(54)

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intersecting/magnetized D-brane models intersecting/magnetized D-brane models Abe, Choi, T.K. Ohki, ’ 09, ‘ 10 Abe, Choi, T.K. Ohki, ’ 09, ‘ 10 generic intersecting number g magnetic flux magnetic flux flavor symmetry is a closed algebra of flavor symmetry is a closed algebra of two Zg ’ s. two Zg ’ s. and Zg permutation and Zg permutation Certain case: Zg permutation larger symm. Like Dg Certain case: Zg permutation larger symm. Like Dg

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Magnetized brane-models Magnetic flux M D4 Magnetic flux M D ・・・ ・・・・・・ ・・・ ・・・ ・・・・・・ ・・・ Magnetic flux M Δ(27) (Δ(54)) x x ∑1 n n=1, …,9 9 ∑1 n n=1, …,9 (1 1 +∑2 n n=1, …,4) (1 1 +∑2 n n=1, …,4) ・・・ ・・・・・・・ ・・ ・・・ ・・・・・・・ ・・

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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), e.g. S3, D4, A4, S4, Q6, Δ(27), 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 one Ansatz: tri-bimaximal one Ansatz: tri-bimaximal

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3.2 Applications of couplings We can obtain quark/lepton masses and mixing angles. We can obtain quark/lepton masses and mixing angles. Yukawa couplings depend on volume moduli, Yukawa couplings depend on volume moduli, complex structure moduli and Wilson lines. complex structure moduli and Wilson lines. By tuning those values, we can obtain semi-realistic results. By tuning those values, we can obtain semi-realistic results. Ratios depend on complex structure moduli Ratios depend on complex structure moduli and Wilson lines. and Wilson lines.

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Quark/lepton masses matrices Abe, T.K., Ohki, Oikawa, Sumita, arXiv: Abe, T.K., Ohki, Oikawa, Sumita, arXiv: assumption on light Higgs scalar assumption on light Higgs scalar The overall gauge coupling is fixed through The overall gauge coupling is fixed through the gauge coupling unification. the gauge coupling unification. Vary other parameters, WLs and Vary other parameters, WLs and complex strucrure with 10% tuning complex strucrure with 10% tuning (5 free parameters) (5 free parameters)

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

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4. Massive modes Hamada, T.K. arXiv: Hamada, T.K. arXiv: Massive modes play an important role Massive modes play an important role in 4D LEEFT such as the proton decay, in 4D LEEFT such as the proton decay, FCNCs, etc. FCNCs, etc. It is important to compute mass spectra of It is important to compute mass spectra of massive modes and their wavefunctions. massive modes and their wavefunctions. Then, we can compute couplings among Then, we can compute couplings among massless and massive modes. massless and massive modes.

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Fermion massive modes Two components are mixed. Two components are mixed. 2D Laplace op. algebraic relations algebraic relations It looks like the quantum harmonic oscillator It looks like the quantum harmonic oscillator

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Fermion massive modes Creation and annhilation operators Creation and annhilation operators mass spectrum mass spectrum wavefunction wavefunction

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Fermion massive modes explicit wavefunction explicit wavefunction Hn: Hermite function Hn: Hermite function Orthonormal condition: Orthonormal condition:

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Scalar and vector modes The wavefunctions of scalar and vector fields The wavefunctions of scalar and vector fields are the same as those of spinor fields. are the same as those of spinor fields. Mass spectrum Mass spectrum scalar scalar vector vector Scalar modes are always massive on T2. Scalar modes are always massive on T2. The lightest vector mode along T2, The lightest vector mode along T2, i.e. the 4D scalar, is tachyonic on T2. i.e. the 4D scalar, is tachyonic on T2. Such a vector mode can be massless on T4 or T6.

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T-duality ? Mass spectrum Mass spectrum spinor spinor scalar scalar vector vector the same mass spectra as excited modes the same mass spectra as excited modes (with oscillator excitations ) (with oscillator excitations ) in intersecting D-brane models, i.e. “gonions” in intersecting D-brane models, i.e. “gonions” Aldazabal, Franco, Ibanez, Rabadan, Uranga, ‘01 Aldazabal, Franco, Ibanez, Rabadan, Uranga, ‘01

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Products of wavefunctions explicit wavefunction explicit wavefunction See also Berasatuce-Gonzalez, Camara, Marchesano, See also Berasatuce-Gonzalez, Camara, Marchesano, Regalado, Uranga, ‘12 Regalado, Uranga, ‘12Derivation: products of zero-mode wavefunctions products of zero-mode wavefunctions We operate creation operators on both LHS and RHS. and RHS.

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3-point couplings including higher modes The 3-point couplings are obtained by The 3-point couplings are obtained by overlap integral of three wavefunctions. overlap integral of three wavefunctions. (flavor) selection rule (flavor) selection rule is the same as one for the massless modes. is the same as one for the massless modes. (mode number) selection rule (mode number) selection rule

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3-point couplings: 2 zero-modes and one higher mode 3-point coupling 3-point coupling

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Higher order couplings including higher modes Similarly, we can compute higher order couplings Similarly, we can compute higher order couplings including zero-modes and higher modes. including zero-modes and higher modes. They can be written by the sum over products of 3-point couplings. products of 3-point couplings.

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3-point couplings including massive modes only due to Wilson lines Massive modes appear only due to Wilson lines Massive modes appear only due to Wilson lines without magnetic flux without magnetic flux We can compute the 3-point coupling e.g. e.g. Gaussian function for the Wilson line. Gaussian function for the Wilson line.

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3-point couplings including massive modes only due to Wilson lines For example, we have for for

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Several couplings Similarly, we can compute the 3-point couplings Similarly, we can compute the 3-point couplings including higher modes including higher modes Furthermore, we can compute higher order couplings including several modes, similarly. couplings including several modes, similarly.

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4.2 Phenomenological applications In 4D SU(5) GUT, The heavy X boson couples with quarks and leptons by the gauge coupling. by the gauge coupling. Their couplings do not change even after GUT breaking and it is the gauge coupling. and it is the gauge coupling. However, that changes in our models. However, that changes in our models.

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Phenomenological applications Phenomenological applications For example, we consider the SU(5)xU(1) GUT model and we put magnetic flux along extra U(1). and we put magnetic flux along extra U(1). The 5 matter field has the U(1) charge q, and the quark and lepton in 5 are quasi-localized and the quark and lepton in 5 are quasi-localized at the same place. at the same place. Their coupling with the X boson is given by Their coupling with the X boson is given by the gauge coupling before the GUT breaking. the gauge coupling before the GUT breaking.

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SU(5) => SM We break SU(5) by the WL along the U(1)Y direction. The X boson becomes massive. The quark and lepton in 5 remain massless, but their The quark and lepton in 5 remain massless, but their profiles split each other. profiles split each other. Their coupling with X is not equal to the gauge coupling, Their coupling with X is not equal to the gauge coupling, but includes the suppression factor but includes the suppression factor 5 Q L Q L

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Proton decay Similarly, the couplings of the X boson with quarks and leptons in the 10 matter fields can be suppressed. leptons in the 10 matter fields can be suppressed. That is important to avoid the fast proton decay. The proton life time would drastically change by the factor, change by the factor,

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Other aspects Other couplings including massless and massive modes can be suppressed and those would be important, can be suppressed and those would be important, such as right-handed neutrino masses and such as right-handed neutrino masses and off-diagonal terms of Kahler metric, etc. off-diagonal terms of Kahler metric, etc. Threshold corrections on the gauge couplings, Kahler potential after integrating out massive modes Kahler potential after integrating out massive modes

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5. Flavor and Higgs sector factorizable magnetic fluxes, factorizable magnetic fluxes, non-vanishing F45, F67, F89 non-vanishing F45, F67, F89 Three generations of both left-handed and Three generations of both left-handed and right-handed quarks originate only from right-handed quarks originate only from the same 2D torus. the same 2D torus. -> the number of higgs = 6. -> the number of higgs = 6. Δ(27) flavor symmetries Δ(27) flavor symmetries three families = triplet three families = triplet higgs fields = 2x (triplet) higgs fields = 2x (triplet)

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Flavor and Higgs sector Abe, T.K., Ohki, Sumita, Tatsuta ’ Abe, T.K., Ohki, Sumita, Tatsuta ’ non-factorizable magnetic fluxes, non-factorizable magnetic fluxes, non-vanishing F45, F67, F89 and F46, ……….. non-vanishing F45, F67, F89 and F46, ……….. Three generations of both left-handed and Three generations of both left-handed and right-handed quarks originate from 4D torus right-handed quarks originate from 4D torus -> several patterns of higgs sectors, -> several patterns of higgs sectors, one pair, two, ……. one pair, two, ……. Δ(27) flavor symmetries are vilolated. Δ(27) flavor symmetries are vilolated.

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Flavor and Higgs sector In most of cases, there are multi higgs fields. there are multi higgs fields. in a certain case, Δ(27) flavor symmetry in a certain case, Δ(27) flavor symmetry What is phenomenological aspects What is phenomenological aspects of multi higgs fields, e.g. in LHC ? of multi higgs fields, e.g. in LHC ?

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Summary We have studied phenomenological aspects of magnetized brane models. of magnetized brane models. Model building from U(N), E6, E7, E8 Model building from U(N), E6, E7, E8 N-point couplings are comupted. N-point couplings are comupted. 4D effective field theory has non-Abelian flavor 4D effective field theory has non-Abelian flavor symmetries, e.g. D4, Δ(27). symmetries, e.g. D4, Δ(27). Orbifold background with magnetic flux is Orbifold background with magnetic flux is also important. also important.

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Further studies Derivation of realistic values of quark/lepton masses and mixing angles. masses and mixing angles. Applications of flavor symmetries and studies on their anomalies and studies on their anomalies Phenomenological aspects of massive modes

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