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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 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 １ Introduction Extra dimensional field theories, in particular string-derived extra dimensional field theories, play important roles in particle physics as well as cosmology . Extra dimensions 4 + n dimensions 4D ⇒ our 4D space-time nD ⇒ compact space Examples of compact space torus, orbifold, CY, etc. Field theory in higher dimensions 10D ⇒ 4D our space-time + 6D space 10D vector AM A , Am 4D vector + 4D scalars SO(10) spinor ⇒ SO(4) spinor x SO(6) spinor internal quantum number Several Fields in higher dimensions 4D (Dirac) spinor i D 0 ⇒ (4D) Clifford algebra { , } 2 (4x4) gamma matrices represention space ⇒ spinor representation 6D Clifford algebra { M , N } 2 MN M 4 ( M 0,1,2,3) , I 44 5 6D spinor 6D spinor I 44 1 , 3 2 4 D 2 D ⇒ 4D spinor x (internal spinor) internal quantum number Field theory in higher dimensions Mode expansions M M Am ( 6 ) Am 0 i ( D m Dm ) 0 KK decomposition KK docomposition on torus torus with vanishing gauge background Boundary conditions ( y4 1, y5 ) ( y4 , y5 ) ( y4 , y5 1) ( y4 , y5 ) 0 : constantmode n : exp(ikny) k 2 / R We concentrate on zero-modes. mn 0 mn kn Zero-modes Zero-mode equation i Dm 0 m ⇒ non-trival zero-mode profile the number of zero-modes 4D effective theory Higher dimensional Lagrangian (e.g. 10D) L10 g d xd y ( x, y ) A( x, y ) ( x, y ) 4 6 integrate the compact space ⇒ 4D theory L4 Y d x ( x) ( x) ( x) 4 Y g d y ( y ) ( y ) ( y ) 6 Coupling is obtained by the overlap integral of wavefunctions Couplings in 4D Zero-mode profiles are quasi-localized far away from each other in compact space ⇒ suppressed couplings 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. i Dm 0 m Zero-modes between chiral and anti-chiral fields are different from each other on certain backgrounds, e.g. CY, toroidal orbifold, warped orbifold, magnetized extra dimension, etc. Magnetic flux i Dm 0 m The limited number of solutions with non-trivial backgrounds are known. Generic CY is difficult. Toroidal/Wapred orbifolds are well-known. Background with magnetic flux is one of interesting backgrounds. Magnetic flux Indeed, several studies have been done in both extra dimensional field theories and string theories with magnetic flux background. In particular, magnetized D-brane models are T-duals of intersecting D-brane models. Several interesting models have been constructed in intersecting D-brane models, that is, the starting theory is U(N) SYM. Phenomenology of magnetized brane models It is important to study phenomenological aspects of magnetized brane models such as massless spectra from several gauge groups, U(N), SO(N), E6, E7, E8, ... Yukawa couplings and higher order n-point couplings in 4D effective theory, their symmetries like flavor symmetries, Kahler metric, etc. It is also important to extend such studies on torus background to other backgrounds with magnetic fluxes, e.g. orbifold backgrounds. 量子力学の復習：磁場中の粒子(Landau) F45 2b, A4 0, A5 2by4 1 2 2 H P4 ( P5 2by 4 ) 2m [ H , P5 ] 0 P5 2k 1 2 2 2 2 H P4 4 b ( y4 k / b) 2m 座標がｋ/bずれた調和振動子 ｂ個の基底状態 b=整数 k=0,1,2,…………,(b-1) 2. Extra dimensions with magnetic fluxes: basic tools 2-1. Magnetized torus model We start with N=1 super Yang-Mills theory in D = 4+2n dimensions. For example, 10D super YM theory consists of gauge bosons (10D vector) and adjoint fermions (10D spinor). We consider 2n-dimensional torus compactification with magnetic flux background. We can start with 6D SYM (+ hyper multiplets), or non-SUSY models (+ matter fields ), similarly. Higher Dimensional SYM theory with flux Cremades, Ibanez, Marchesano, ‘04 4D Effective theory <= dimensional reduction The wave functions eigenstates of corresponding internal Dirac/Laplace operator. Higher Dimensional SYM theory with flux Abelian gauge field on magnetized torus Constant magnetic flux gauge fields of background The boundary conditions on torus (transformation under torus translations) Higher Dimensional SYM theory with flux We now consider a complex field Consistency of such transformations under a contractible loop in torus which implies Dirac’s quantization conditions. with charge Q ( +/-1 ) Dirac equation on 2D torus is the two component spinor. U(1) charge Q=1 4 i 5 , 4 i 5 with twisted boundary conditions (Q=1) Dirac equation and chiral fermion |M| independent zero mode solutions in Dirac equation. (Theta function) Properties of theta functions chiral fermion :Normalizable mode :Non-normalizable mode By introducing magnetic flux, we can obtain chiral theory. Wave functions For the case of M=3 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. Fermions in bifundamentals Breaking the gauge group (Abelian flux case The gaugino fields gaugino of unbroken gauge bi-fundamental matter fields ) Bi-fundamental Gaugino fields in off-diagonal entries correspond to bi-fundamental matter fields and the difference M= m-m’ of magnetic fluxes appears in their Dirac equation. F Zero-modes Dirac equations No effect due to magnetic flux for adjoint matter fields, Total number of zero-modes of :Normalizable mode :Non-Normalizable mode 4D chiral theory 10D spinor ( , ) light-cone 8s even number of minus signs 1st ⇒ 4D, the other ⇒ 6D space If all of appear in 4D theory, that is non-chiral theory. If only for all torus, ab N a , Nb , appear for 4D helicity fixed. ⇒ 4D chiral theory U(8) SYM theory on T6 Fzz m1 N1 2i 0 0 m2 N 2 N1 4, N2 2, N3 2 m3 N 3 U (4) U (2) L U (2) R Pati-Salam group up to U(1) factors T 2 (m1 m2 ) (m3 m1 ) 3 for t hefirst 4,2,1 4,1,2 (m1 m2 ) (m3 m1 ) 1 for t heot her t ori Three families of matter fields with many Higgs fields (4,2,1) ( 4,1,2) (1,2,2) 2-2. 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 2 (My a) 2 ( My a) the number of zero-modes is the same. the profile: f(y) f(y +a/M) with proper b.c. 0 0 U(1)a*U(1)b theory magnetic flux, Fa=2πM, Fb=0 Wilson line, Aa=0, Ab=C matter fermions with U(1) charges, (Qa,Qb) chiral spectrum, for Qa=0, massive due to nonvanishing WL when MQa >0, the number of zero-modes is MQa. zero-mode profile is shifted depending on Qb, f ( z) f ( z CQ /(MQ )) b a Pati-Salam model Fzz m1 N1 2i 0 Pati-Salam group 0 m2 N 2 m3 N 3 N1 4, N2 2, N3 2 U (4) U (2) L U (2) R (m1 m2 ) (m3 m1 ) 3 for t hefirst T 2 t ori (m1 m2 ) (m3 m1 ) 1 for t heot her 4,2,1 4,1,2 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 (3)C U (2) L U (1)3 U(1)Y is a linear combination. (the others are massive.) PS => SM Zero modes corresponding to (4,2,1) ( 4,1,2) three families of matter fields remain after introducing WLs, but their profiles split (4,2,1) (3,2,1) (1,2,1) ( 4,1,2) ( 3,1,1) ( 3,1,1) (1,1,1) (1,1,1) (4,2,1) Q L Other models We can start with 10D SYM, 6D SYM (+ hyper multiplets), or non-SUSY models (+ matter fields ) with gauge groups, U(N), SO(N), E6, E7,E8,... E6 SYM theory on T6 Choi, et. al. ‘09 We introduce magnetix flux along U(1) direction, which breaks E6 -> SO(10)*U(1) 78 450 10 161 161 m1 3, m2 1, m3 1 Three families of chiral matter fields 16 We introduce Wilson lines breaking SO(10) -> SM group. Three families of quarks and leptons matter fields with no Higgs fields Splitting zero-mode profiles Wilson lines do not change the (generation) number of zero-modes, but change localization point. 16 Q …… L 2.3 Orbifold with magnetic flux S1/Z2 Orbifold There are two singular points, which are called fixed points. 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 T2/Z4, T2/Z6 Orbifold = D-dim. Torus /twist Torus = D-dim flat space/ lattice 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 For the case of M=3 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. Zero-modes on orbifold Adjoint matter fields are projected by orbifold projection. We have degree of freedom to introduce localized modes on fixed points like quarks/leptons and higgs fields. Zero-modes on ZN orbifold Similarly we can discuss 2D ZN orbifolds with magnetic fluxes for N=2,3,4 and 6. Abe, Fujimoto, T.K., Miura, Nishiwaki, Sakamoto, arXiv:1309.4925 3. N-point couplings and flavor symmetries The N-point couplings are obtained by overlap integral of their zero-mode w.f.’s. Y g d z ( z ) ( z ) ( z ) 2 i M z y4 iy5 j N k P Moduli Torus metric ds 2(2R) dzdz z x y 2 2 Area A 4 R Im We can repeat the previous analysis. Scalar and vector fields have the same wavefunctions. M Wilson moduli ( z) ( z ) shift of w.f. 2 2 Zero-modes Cremades, Ibanez, Marchesano, ‘04 ( z) N M j M j/M exp[iMz Im(z )] ( Mz, iM ) 0 j 1,, M N M : normalization factor, Zero-mode w.f. = gaussian x theta-function ( z ) ( z ) i M j N M N y m 1 ijm i j Mm M N ( z ), up to normalization factor yijm ( Ni Mj MNm) /(MN ( M N )) (0, iMN ( M N )) 0 Products of wave functions 2My 0, 2Ny 0, 2 ( M N ) y M N ( M N ) ( M N ) M N products of zero-modes = zero-modes 0, 3-point couplings Cremades, Ibanez, Marchesano, ‘04 The 3-point couplings are obtained by overlap integral of three zero-mode w.f.’s. Yijk d z ( z ) ( z ) 2 i M j N k M N ( z) * 2 i k ik d z ( z ) ( z ) M M Yijk * M N m 1 i j mM , k yijm up to normalization factor Selection rule i j mM ,k i j mM k (M N ) i j k mod g when g gcd( M , N ) Each zero-mode has a Zg charge, which is conserved in 3-point couplings. yijm ( Ni Mj MNm) /(MN ( M N )) (0, iMN ( M N )) 0 up to normalization factor 4-point couplings Abe, Choi, T.K., Ohki, ‘09 The 4-point couplings are obtained by overlap integral of four zero-mode w.f.’s. split Yijkl d z ( z ) ( z ) ( z ) 2 d 2 i M j N k P l M N P ( z) zd z ' ( z ) ( z ) ( z z ' ) ( z ' ) 2 i M j N k P insert a complete set ( z z' ) n K s l M N P n ( z) K ( z' ) all modes Yijkl yijs yskl * * up to normalization factor for K=M+N ( z' ) * 4-point couplings: another splitting d 2 zd z ' ( z ) ( z ) ( z z ' ) ( z ' ) 2 i M k P j N l M N P ( z' ) Yijkl yikt ytjl t Yijkl yijs yskl Yijkl yikt ytjl t s i k i k t j s l j l * N-point couplings Abe, Choi, T.K., Ohki, ‘09 We can extend this analysis to generic n-point couplings. N-point couplings = products of 3-point couplings = products of theta-functions This behavior is non-trivial. (It’s like CFT.) Such a behavior would be satisfied not for generic w.f.’s, but for specific w.f.’s. However, this behavior could be expected from T-duality between magnetized and intersecting D-brane models. T-duality The 3-point couplings coincide between magnetized and intersecting D-brane models. explicit calculation Cremades, Ibanez, Marchesano, ‘04 Such correspondence can be extended to 4-point and higher order couplings because of CFT-like behaviors, e.g., Yijkl yijs yskl s Abe, Choi, T.K., Ohki, ‘09 Non-Abelian discrete flavor symmetry The coupling selection rule is controlled by Zg charges. For M=g, 1 2 g 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 Cf. heterotic orbifolds, T.K. Raby, Zhang, ’04 T.K. Nilles, Ploger, Raby, Ratz, ‘06 Permutation symmetry D-brane models Abe, Choi, T.K. Ohki, ’09, ‘10 There is a Z2 permutation symmetry. The full symmetry is D4. Permutation symmetry D-brane models Abe, Choi, T.K. Ohki, ’09, ‘10 geometrical symm. Z3 S3 Full symm. Δ(27) Δ(54) intersecting/magnetized D-brane models Abe, Choi, T.K. Ohki, ’09, ‘10 generic intersecting number g magnetic flux flavor symmetry is a closed algebra of two Zg’s. 1 , g 1 and Zg permutation Certain case: Zg permutation Dg 0 0 1 0 0 1 , e 2i / g 0 0 larger symm. Like Magnetized brane-models Magnetic flux M 2 4 ・・・ Magnetic flux M 3 6 9 ・・・ D4 2 1++ + 1+- +1-+ + 1-・・・・・・・・・ Δ(27) (Δ(54)) 31 2 x 31 ∑1n n=1,…,9 (11+∑2n n=1,…,4) ・・・・・・・・・ 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, Q6, Δ(27), ...... Review: e.g Ishimori, T.K., Ohki, Okada, Shimizu, Tanimoto ‘10 ⇒ large mixing angles 2/3 1/ 3 0 one Ansatz: tri-bimaximal 1 / 6 1 / 3 1/ 2 1/ 6 1/ 3 1 / 2 3.2 Applications of couplings We can obtain quark/lepton masses and mixing angles. Yukawa couplings depend on volume moduli, complex structure moduli and Wilson lines. By tuning those values, we can obtain semi-realistic results. Ratios depend on complex structure moduli and Wilson lines. Quark/lepton masses matrices Abe, T.K., Ohki, Oikawa, Sumita, arXiv:1211.437 assumption on light Higgs scalar H u ( 2.7, 1.3, 0, 0, 0, 0) H d (5.8, 5.8, 0.1, 0.1, 0, 0) tan 25 The overall gauge coupling is fixed through the gauge coupling unification. Vary other parameters, WLs and complex strucrure with 10% tuning (5 free parameters) Quark/lepton masses and mixing angles Abe, T.K., Ohki, Oikawa, Sumita, arXiv:1211.437 Example M t 170 GeV , Mb 6 M c 1.0 GeV , M s 150 Mu 3 Vus 0.21, M 3 MeV , Md 3 Vcb 0.04, GeV , M 60 MeV , M e 0.5 MeV , GeV Flavor is still a challenging issue. Vub 0.002 MeV MeV 4. Massive modes Hamada, T.K. arXiv:1207.6867 Massive modes play an important role in 4D LEEFT such as the proton decay, FCNCs, etc. It is important to compute mass spectra of massive modes and their wavefunctions. Then, we can compute couplings among massless and massive modes. Fermion massive modes Two components are mixed. DD 0 2D Laplace op. ,n 0 , n 2 mn DD ,n ,n {D , D} / 2 algebraic relations [ D, D ] 4M / A [, D ] 4MD / A, [, D] 4MD / A It looks like the quantum harmonic oscillator Fermion massive modes Creation and annhilation operators a D A / 4M , a D A / 4M , [a, a ] 1 mass spectrum m 4Mn / A 2 n wavefunction j ,M n (1/ n!)(a ) n j ,M 0 Fermion massive modes explicit wavefunction nj , M ( 2 M Im )1/ 4 ( 2 n n! A)1/ 2 Hn j ,M k ( z , ) k 2M Im k j / M Im(z ) / Im kj ,M ( z, ) exp[M Im (k j / M Im z / Im ) 2 iM Re z(2k 2 j / M Im z / Im ) iM Re (k j / M ) Hn: Hermite function Orthonormal condition: 2 j ,M k ,M * d z ( ) jk n n Scalar and vector modes The wavefunctions of scalar and vector fields are the same as those of spinor fields. Mass spectrum 2 m n 2M (2n 1) / A scalar vector mn2 2M (2n 1) / A Scalar modes are always massive on T2. The lightest vector mode along T2, i.e. the 4D scalar, is tachyonic on T2. Such a vector mode can be massless on T4 or T6. m2 2 (M1 / A1 M 2 / A2 M3 / A3 ) T-duality ? Mass spectrum spinor scalar vector mn2 2M ( 2 n) / A mn2 2M ( 2 n 1) / A 2 mn 2M (2n 1) / A the same mass spectra as excited modes (with oscillator excitations ) in intersecting D-brane models, i.e. “gonions” Aldazabal, Franco, Ibanez, Rabadan, Uranga, ‘01 Products of wavefunctions explicit wavefunction ni , M ( z ) nj , N ( z ) 1 2 yijm s n1 C M N n1 n2 m 1 0 s 0 i j Mm , N M yijm ( z ), s s n2 s ( n2 s ) / 2 ( n1 s ) / 2 ( n1 n2 1) / 2 C ( 1 ) N M ( N M ) s n2 ( s )!( n1 n2 s )!/( n1! n2 !) Nj NMm , nMi 1 n2 s NM ( N M ) (0, ) See also Berasatuce-Gonzalez, Camara, Marchesano, Regalado, Uranga, ‘12 i,M ( z ) j,N ( z) M N y i j Mm , M N ( z ), Derivation: m 1 products of zero-mode wavefunctions We operate creation operators on both LHS and RHS. ijm 3-point couplings including higher modes The 3-point couplings are obtained by overlap integral of three wavefunctions. ijk n1 n2 n3 Y d z 2 i,N n1 d z 2 ijk n1n2 n3 Y M N i,M n1 ( z ) ( z) n2 m 1 0 s 0 j ,M n2 j ,M s ( z) k ,M N n3 ( z) * ( z ) ik s * i j mM ,k s ,n yijm s 3 i j k mod M (flavor) selection rule is the same as one for the massless modes. (mode number) selection rule n n n 3 1 2 3-point couplings: 2 zero-modes and one higher mode 3-point coupling n1 n3 0 n3 n1 n2 ( M N ) j , NM ( N M ) Y N n2 / 2 ( N M )( n2 1) / 2 nMk (0, ) 2 Higher order couplings including higher modes Similarly, we can compute higher order couplings including zero-modes and higher modes. Y d z 2 i,N n1 ( z ) j ,M n2 ( z) k ,P nm They can be written by the sum over products of 3-point couplings. ( z) * 3-point couplings including massive modes only due to Wilson lines Massive modes appear only due to Wilson lines without magnetic flux n(Wn) A1/ 2 exp[i (2nR Im / Im ) Re z i ( Re 2(nI nR Re )) Im z / Im ] R I We can compute the 3-point coupling jk (W ) nR nI Y d z 2 (W ) nR nI ( z ) j ,M 0 (z 1) k ,M 0 (z 2 ) * | A exp[ | 2 1 | /(2 Im )] e.g. | Y Gaussian function for the Wilson line. jk (W ) nR 0 nI 0 1/ 2 M 1 M 2 2 3-point couplings including massive modes only due to Wilson lines jk (W ) nR 0 nI 0 |Y 1/ 2 | A exp[ | 2 1 | /(2 Im )] For example, we have | Y(Wjk ) nR 0nI 0 | exp[ ] 0.04 for | 2 1 |2 /(2 Im ) 1 2 Several couplings Similarly, we can compute the 3-point couplings including higher modes jk n1 n2 (W ) nR nI Y d z 2 (W ) nR nI ( z ) j ,M n1 ( z) k ,M n2 ( z) * Furthermore, we can compute higher order couplings including several modes, similarly. Y d z 2 (W ) nR nI ( z ) j ,M n1 ( z ) k ,M n2 ( z) * 4.2 Phenomenological applications In 4D SU(5) GUT, The heavy X boson couples with quarks and leptons by the gauge coupling. Their couplings do not change even after GUT breaking and it is the gauge coupling. However, that changes in our models. Phenomenological applications For example, we consider the SU(5)xU(1) GUT model 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 at the same place. Their coupling with the X boson is given by the gauge coupling before the GUT breaking. 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 profiles split each other. Their coupling with X is not equal to the gauge coupling, but includes the suppression factor 5 | Y(Wjk ) nR 0nI 0 | exp[ ] 0.04 Q L Proton decay Similarly, the couplings of the X boson with quarks and 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, O(10 10 ) 4 5 | Y(Wjk ) nR 0nI 0 | exp[ ] 0.04 Other aspects Other couplings including massless and massive modes can be suppressed and those would be important , such as right-handed neutrino masses and off-diagonal terms of Kahler metric, etc. Threshold corrections on the gauge couplings, Kahler potential after integrating out massive modes 5. Flavor and Higgs sector factorizable magnetic fluxes, non-vanishing F45, F67, F89 Three generations of both left-handed and right-handed quarks originate only from the same 2D torus. -> the number of higgs = 6. Δ(27) flavor symmetries three families = triplet higgs fields = 2x (triplet) Flavor and Higgs sector Abe, T.K. , Ohki, Sumita, Tatsuta ’ 1307.1831 non-factorizable magnetic fluxes, non-vanishing F45, F67, F89 and F46, ……….. Three generations of both left-handed and right-handed quarks originate from 4D torus -> several patterns of higgs sectors, one pair, two , ……. Δ(27) flavor symmetries are vilolated. Flavor and Higgs sector In most of cases, there are multi higgs fields. in a certain case, Δ(27) flavor symmetry What is phenomenological aspects of multi higgs fields, e.g. in LHC ? Summary We have studied phenomenological aspects of magnetized brane models. Model building from U(N), E6, E7, E8 N-point couplings are comupted. 4D effective field theory has non-Abelian flavor symmetries, e.g. D4, Δ(27). Orbifold background with magnetic flux is also important. Further studies Derivation of realistic values of quark/lepton masses and mixing angles. Applications of flavor symmetries and studies on their anomalies Phenomenological aspects of massive modes