Presentation on theme: "Phenomenological aspects of magnetized brane models Tatsuo Kobayashi"— Presentation transcript:
1 Phenomenological aspects of magnetized brane models Tatsuo Kobayashi １．Introduction2. Magnetized extra dimensions3. N-point couplings and flavor symmetries4．Massive modes5. Flavor and Higgs sector6. Summarybased on collaborations withH.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
2 １ IntroductionExtra dimensional field theories, in particular string-derived extra dimensional field theories, play important roles in particle physics as well as cosmology .
3 Extra dimensions 4 + n dimensions 4D ⇒ our 4D space-time nD ⇒ compact space Examples of compact spacetorus, orbifold, CY, etc.
4 Field theory in higher dimensions 10D ⇒ 4D our space-time + 6D space10D vector4D vector + 4D scalarsSO(10) spinor ⇒ SO(4) spinorx SO(6) spinorinternal quantum number
5 Several Fields in higher dimensions 4D (Dirac) spinor⇒ (4D) Clifford algebra (4x4) gamma matricesrepresention space ⇒ spinor representation6D Clifford algebra6D spinor6D spinor ⇒ 4D spinor x (internal spinor)internal quantumnumber
6 Field theory in higher dimensions Mode expansionsKK decomposition
7 KK docomposition on torus torus with vanishing gauge backgroundBoundary conditions We concentrate on zero-modes.
8 Zero-modes ⇒ non-trival zero-mode profile the number of zero-modes Zero-mode equation⇒ non-trival zero-mode profilethe number of zero-modes
9 4D effective theory integrate the compact space ⇒ 4D theory Higher dimensional Lagrangian (e.g. 10D)integrate the compact space ⇒ 4D theoryCoupling is obtained by the overlapintegral of wavefunctions
10 Couplings in 4D ⇒ suppressed couplings Zero-mode profiles are quasi-localizedfar away from each other in compact space⇒ suppressed couplings
11 Chiral theory Zero-modes between chiral and anti-chiral When we start with extra dimensional field theories,how to realize chiral theories is one of important issues from the viewpoint of particle physics.Zero-modes between chiral and anti-chiralfields are different from each otheron certain backgrounds, e.g. CY, toroidal orbifold, warped orbifold,magnetized extra dimension, etc.
12 Magnetic fluxThe 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.
13 Magnetic fluxIndeed, 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.
14 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.
16 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.
17 Higher Dimensional SYM theory with flux Cremades, Ibanez, Marchesano, ‘044D Effective theory <= dimensional reductioneigenstates of correspondinginternal Dirac/Laplace operator.The wave functions
18 Higher Dimensional SYM theory with flux Abelian gauge field on magnetized torusConstant magnetic fluxgauge fields of backgroundThe boundary conditions on torus (transformation under torus translations)
19 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.
20 is the two component spinor. Dirac equation on 2D torusis the two component spinor. U(1) charge Q=1with twisted boundary conditions (Q=1)
21 Dirac equation and chiral fermion |M| independent zero mode solutions in Dirac equation.(Theta function)Properties of theta functionschiral fermionBy introducing magnetic flux, we can obtain chiral theory.:Normalizable mode:Non-normalizable mode
22 Wave functionsFor the case of M=3Wave function profile on toroidal backgroundZero-modes wave functions are quasi-localized far away each other in extra dimensions. Therefore the hierarchirally small Yukawa couplings may be obtained.
23 Breaking the gauge group Fermions in bifundamentalsBreaking the gauge group(Abelian flux case )The gaugino fieldsgaugino of unbroken gaugebi-fundamental matter fields
24 Bi-fundamental Gaugino fields in off-diagonal entries correspond to bi-fundamental matter fieldsand the difference M= m-m’ of magneticfluxes appears in their Dirac equation.F
25 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
26 4D chiral theory10D 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 for all torus, only appear for 4D helicity fixed. ⇒ 4D chiral theory
27 U(8) SYM theory on T6 Pati-Salam group up to U(1) factors Three families of matter fieldswith many Higgs fields
28 2-2. Wilson lines torus without magnetic flux constant Ai mass shift Cremades, Ibanez, Marchesano, ’04,Abe, Choi, T.K. Ohki, ‘09torus without magnetic fluxconstant Ai mass shiftevery modes massivemagnetic flux the number of zero-modes is the same.the profile: f(y) f(y +a/M)with proper b.c.
29 U(1)a*U(1)b theory Wilson line, Aa=0, Ab=C magnetic flux, Fa=2πM, Fb=0Wilson line, Aa=0, Ab=Cmatter fermions with U(1) charges, (Qa,Qb)chiral spectrum,for Qa=0, massive due to nonvanishing WLwhen MQa >0, the number of zero-modesis MQa.zero-mode profile is shifted dependingon Qb,
30 Pati-Salam 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 (the others are massive.)U(1)Y is a linear combination.
31 PS => SM Zero modes corresponding to three families of matter fieldsremain after introducing WLs, but their profiles split(4,2,1)Q L
32 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,...
33 E6 SYM theory on T6 Choi, et. al. ‘09 We introduce magnetix flux along U(1) direction,which breaks E6 -> SO(10)*U(1)Three families of chiral matter fields 16We introduce Wilson lines breakingSO(10) -> SM group.Three families of quarks and leptons matter fieldswith no Higgs fields
34 Splitting zero-mode profiles Wilson lines do not change the (generation) number of zero-modes, but change localization point.16Q …… L
35 2.3 Orbifold with magnetic flux S1/Z2 Orbifold There are two singular points,which are called fixed points.
36 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 latticeT2/Z4, T2/Z6Orbifold = D-dim. Torus /twistTorus = D-dim flat space/ lattice
37 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
38 Wave functionsFor the case of M=3Wave function profile on toroidal backgroundZero-modes wave functions are quasi-localized far away each other in extra dimensions. Therefore the hierarchirally small Yukawa couplings may be obtained.
39 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.
40 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:
41 N-point couplings and flavor symmetries The N-point couplings are obtained byoverlap integral of their zero-mode w.f.’s.
42 ModuliTorus metric Area We can repeat the previous analysis. Scalar and vector fields have the same wavefunctions. Wilson moduli shift of w.f.
43 Zero-modesCremades, Ibanez, Marchesano, ‘04 Zero-mode w.f. = gaussian x theta-function up to normalization factor
44 Products of wave functions products of zero-modes = zero-modes
45 3-point couplings Cremades, Ibanez, Marchesano, ‘04 The 3-point couplings are obtained byoverlap integral of three zero-mode w.f.’s.up to normalization factor
46 Selection ruleEach zero-mode has a Zg charge, which is conserved in 3-point couplings. up to normalization factor
47 4-point couplings Abe, Choi, T.K., Ohki, ‘09 The 4-point couplings are obtained byoverlap integral of four zero-mode w.f.’s.splitinsert a complete setup to normalization factorfor K=M+N
48 4-point couplings: another splitting i k i k t j s l j l
49 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-functionsThis behavior is non-trivial. (It’s like CFT.)Such a behavior would be satisfiednot for generic w.f.’s, but for specific w.f.’s.However, this behavior could be expectedfrom T-duality between magnetizedand intersecting D-brane models.
50 T-duality magnetized and intersecting D-brane models. The 3-point couplings coincide betweenmagnetized and intersecting D-brane models.explicit calculationCremades, Ibanez, Marchesano, ‘04Such correspondence can be extended to4-point and higher order couplings because ofCFT-like behaviors, e.g.,Abe, Choi, T.K., Ohki, ‘09
51 Non-Abelian discrete flavor symmetry The coupling selection rule is controlled byZg charges.For M=g, gEffective field theory also has a cyclic permutation symmetry of g zero-modes.These lead to non-Abelian flavor symmetiressuch as D4 and Δ(27) Abe, Choi, T.K, Ohki, ‘09 Cf. heterotic orbifolds, T.K. Raby, Zhang, ’04T.K. Nilles, Ploger, Raby, Ratz, ‘06
52 Permutation symmetry D-brane models Abe, Choi, T.K. Ohki, ’09, ‘10There is a Z2 permutation symmetry.The full symmetry is D4.
54 intersecting/magnetized D-brane models Abe, Choi, T.K. Ohki, ’09, ‘10generic intersecting number gmagnetic fluxflavor symmetry is a closed algebra of two Zg’s.and Zg permutationCertain case: Zg permutation larger symm. Like Dg
55 Magnetized brane-models Magnetic flux M D4 ・・・ ・・・・・・・・・Magnetic flux M Δ(27) (Δ(54))x 31∑1n n=1,…,9(11+∑2n n=1,…,4)
56 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),Review: e.gIshimori, T.K., Ohki, Okada, Shimizu, Tanimoto ‘10⇒ large mixing anglesone Ansatz: tri-bimaximal
57 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 moduliand Wilson lines.
58 Quark/lepton masses matrices Abe, T.K., Ohki, Oikawa, Sumita, arXiv:assumption on light Higgs scalarThe overall gauge coupling is fixed throughthe gauge coupling unification.Vary other parameters, WLs andcomplex strucrure with 10% tuning (5 free parameters)
59 Quark/lepton masses and mixing angles Abe, T.K., Ohki, Oikawa, Sumita, arXiv:ExampleFlavor is still a challenging issue.
60 4. Massive modes Hamada, T.K. arXiv:1207.6867 Massive modes play an important rolein 4D LEEFT such as the proton decay,FCNCs, etc.It is important to compute mass spectra ofmassive modes and their wavefunctions.Then, we can compute couplings amongmassless and massive modes.
61 Fermion massive modesTwo components are mixed. 2D Laplace op. algebraic relations It looks like the quantum harmonic oscillator
62 Fermion massive modesCreation and annhilation operators mass spectrum wavefunction
63 Fermion massive modesexplicit wavefunction Hn: Hermite function Orthonormal condition:
64 Scalar and vector modes The wavefunctions of scalar and vector fieldsare the same as those of spinor fields.Mass spectrumscalarvectorScalar 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.
65 T-duality ? spinor scalar vector Mass spectrumspinorscalarvector the same mass spectra as excited modes(with oscillator excitations )in intersecting D-brane models, i.e. “gonions”Aldazabal, Franco, Ibanez, Rabadan, Uranga, ‘01
66 Products of wavefunctions explicit wavefunctionSee also Berasatuce-Gonzalez, Camara, Marchesano,Regalado, Uranga, ‘12Derivation:products of zero-mode wavefunctionsWe operate creation operators on both LHSand RHS.
67 3-point couplings including higher modes The 3-point couplings are obtained byoverlap integral of three wavefunctions.(flavor) selection ruleis the same as one for the massless modes.(mode number) selection rule
68 3-point couplings: 2 zero-modes and one higher mode
69 Higher order couplings including higher modes Similarly, we can compute higher order couplingsincluding zero-modes and higher modes.They can be written by the sum overproducts of 3-point couplings.
70 3-point couplings including massive modes only due to Wilson lines Massive modes appear only due to Wilson lineswithout magnetic fluxWe can compute the 3-point couplinge.g.Gaussian function for the Wilson line.
71 3-point couplings including massive modes only due to Wilson lines For example, we havefor
72 Several couplings Similarly, we can compute the 3-point couplings including higher modesFurthermore, we can compute higher ordercouplings including several modes, similarly.
73 4.2 Phenomenological applications In 4D SU(5) GUT,The heavy X boson couples with quarks and leptonsby the gauge coupling.Their couplings do not change even after GUT breakingand it is the gauge coupling.However, that changes in our models.
74 Phenomenological applications For example,we consider the SU(5)xU(1) GUT modeland 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-localizedat the same place.Their coupling with the X boson is given bythe gauge coupling before the GUT breaking.
75 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 theirprofiles split each other.Their coupling with X is not equal to the gauge coupling,but includes the suppression factor5Q L
76 Proton decay The proton life time would drastically Similarly, the couplings of the X boson with quarks andleptons in the 10 matter fields can be suppressed.That is important to avoid the fast proton decay.The proton life time would drasticallychange by the factor,
77 Other aspects Other couplings including massless and massive modes can be suppressed and those would be important ,such as right-handed neutrino masses andoff-diagonal terms of Kahler metric, etc.Threshold corrections on the gauge couplings,Kahler potential after integrating out massive modes
78 5. Flavor and Higgs sector factorizable magnetic fluxes,non-vanishing F45, F67, F89Three generations of both left-handed andright-handed quarks originate only fromthe same 2D torus.-> the number of higgs = 6. Δ(27) flavor symmetriesthree families = triplet higgs fields = 2x (triplet)
79 Flavor and Higgs sector Abe, T.K. , Ohki, Sumita, Tatsuta ’ non-factorizable magnetic fluxes,non-vanishing F45, F67, F89 and F46, ………..Three generations of both left-handed andright-handed quarks originate from 4D torus-> several patterns of higgs sectors,one pair, two , ……. Δ(27) flavor symmetries are vilolated.
80 Flavor and Higgs sector In most of cases,there are multi higgs fields.in a certain case, Δ(27) flavor symmetry What is phenomenological aspectsof multi higgs fields, e.g. in LHC ?
81 Summary We have studied phenomenological aspects of magnetized brane models.Model building from U(N), E6, E7, E8N-point couplings are comupted.4D effective field theory has non-Abelian flavorsymmetries, e.g. D4, Δ(27).Orbifold background with magnetic flux isalso important.
82 Further studies Derivation of realistic values of quark/lepton masses and mixing angles.Applications of flavor symmetriesand studies on their anomaliesPhenomenological aspects of massive modes