Generalized Covariant Derivative on Extra Dimension and Weinberg-Salam Model Yoshitaka Okumura Department of Natural Science, Chubu University, Kasugai, 487-8501, Japan Abstract [5] into that on the product space of M and ex- 6 4 0 tra continuous compact space. This compact space 0 The generalized covariant derivative on 5-dimen- doesnotneedtobespecified,butwecallitS1 with 2 sionalspaceincluding 1-dimensionalextracompact argument y. We first define the generalized covari- n space is defined, and, by use of it, the Weinberg- ant derivatives on M4 S1 and then, we obtain × a Salam model is reconstructed. The spontaneous the generalized field strength following the usual J breakdown of symmetry takes place owing to the procedure of gauge theory, from which the Yang- 1 extra dimension under the settings that the Higgs Mills-HiggsLagrangianisderived. ThisLagrangian 3 field exists in the extra dimensional space depend- contains the term which provokesthe correctspon- ing on the argument y of this extra space, whereas taneoussymmetrybreakdown. Inordertoyieldthe 1 v the gauge and fermion fields do not depend on y. Lagrangian in Weinberg-Salam model, we have to 0 Both Yang-Mills-Higgs and fermion Lagrangiansin assume that the Higgs field intrudes into the extra 6 Weinberg-Salam model are correctly reproduced. dimensional space S so as to depend on the argu- 1 2 ment y whereas the gauge field as well as leptonic 1 fields do not contain y, and exist uniformly in S . 0 1 Introduction 1 6 We do not consider the Kaluza-Klein mode in this 0 paper. The leptonic Lagrangian is also obtained / Thesupersymmetricstringtheory[1]isconsistently by use of the generalized covariantderivatives, and h formulated in 10 dimensional space. The extra 6- p wecansuccessfullyreconstructtheWeinberg-Salam dimensional space over M has to be compactified - 4 model. p in order not to be observed by the present experi- This paper consists of four sections. The next e mentalfacilities. However,thisextra6-dimensional h section presents the basic formulation of the gen- spaceaftercompactifiedhasthe profoundpossibili- : eralized covariant derivatives in order to yield the v ties to explicate many unresolvedproblems such as i Yang-Mills Higgs Lagrangian as well as fermionic X particle generations, internal flavor symmetry and Lagrangian. In the third section, the Weinberg- its spontaneous breakdown, CKM matrix, particle r Salam model is reconstructed. The last section is a masses and so on. For these purposes, the extra 6- devoted to concluding remarks. dimensional space has recently attracted much at- tentions. 2 Generalized covariant deri- Manton [2] initiated efforts to derive the Weinberg-Salam model from the Yang-Mills the- vative ory in 6-dimensional space containing extra com- pact two dimensional space. He elucidated in his Sogami[5]reconstructedthe spontaneousbroken work that the Higgs field is a part of gauge fields. gauge theories such as standard model and grand Meanwhile, Connes [3] proposed non-commutative unified theory by use of the generalized covariant geometry and applied it to construct the sponta- derivative smartly defined by him. Let us explain neous broken gauge theory on two-sheeted discrete hismethodintheversionofWeinberg-Salammodel. space followed by M , which provided a geometri- 4 He divided the space of fermion fields into two cal understanding of the Higgs mechanism without sectors which consist of the left-handed and right- extra physical degrees of freedom as in the Kaluza- handed fermions, respectively. Klein theory. There are several versions of this approach[4], however, in any case gauge and Higgs ψ =ψ L>+ψ R>, (2.1) L R fieldsarewrittentogetherandyieldtheYang-Mills- | | Higgs Lagrangian. where L>and R>arethebaseofleft-andright- | | In this paper, we try to extend the generalized handedfermionspaces,respectivelyandψ andψ L R covariant derivative method proposed by Sogami aretheleft-andright-handedfermionfieldsdenoted by InordertoobtaintheYang-Mills-HiggsLagrangian with correct signs, we define the counter covariant ν ψ = L , ψ =e . (2.2) derivatives to (2.5). L e R R L (cid:18) (cid:19) D¯µ =∂µ+Aµ L><L +Aµ R><R, Then, he defined the generalized covariant deriva- L| | R| | tive D¯y =∂y ΦL><R, (2.8) − | | Dµ =∂µ−igALµ|L><L|−ig′ARµ|R><R| D¯y¯=∂y−Φ†|R><L|, 1 γµ ΦL><R +Φ† R><L +c+c5γ5, from whichthe counterfield strengthes to (2.7) are −4 | | | | derived. (2.3) (cid:0) (cid:1) from which the generalizedfield strength is yielded ¯µν = ∂µAν ∂νAµ +[Aµ, Aν] L><L F L− L L L | | by the equation + ∂(cid:0)µAνR−∂νAµR+[AµR, AνR](cid:1)|R><R| [Dµ, Dν]=−igFLµν−ig′FRµν − 4iFµ(0ν). (2.4) F¯µy =(cid:0)−∂∂yµAΦµ+LA>µL<ΦL−ΦA∂µRyA|µL(cid:1)R><><R|R −(cid:0) L| |− (cid:1)R| | (2.9) Hsyemmsuectcreyedberdokiennrgecaoungsetrtuhcetoinrgy tbhye usspeonotfantehoeuses F¯µy¯=− ∂µΦ†+AµRΦ†−Φ†AµL |R><L| ∂yAµ R><R ∂yAµ L><L items. −(cid:0) R| |− L(cid:1)| | Inthispaper,weapplyhisideatoreconstructthe ¯yy¯= ∂yΦ R><L +∂yΦL><R † F − | | | | Weinberg-Salam model on the 5-dimensional space +Φ†ΦR><R ΦΦ† L><L including one dimensional extra compact space. | |− | | Though the extra space is not necessary to be S , Then, we define the Lagrangian by use of field 1 we write this 5-dimensional space to be M S strengthes in (2.7) and (2.9). 4 1 × with the argument x and y. We define the gener- µ 1 alized covariant derivative on 5-dimensional space (x,y)= Tr< ¯µν (x,y), (x,y)> L − 2g2 F † Fµν as 1 1 Dµ =∂µ+ALµ|L><L|+ARµ|R><R|, − g2Tr<F¯µy†(x,y), Fµy(x,y)> 2 Dy =∂y+Φ|L><R|, (2.5) 1 Tr< ¯µy¯ (x,y), (x,y)> Dy¯=∂y+Φ†|R><L|. − g32 F † Fµy¯ 1 According to (2.1) and (2.5), we can describe the Tr< ¯yy¯ (x,y), (x,y)> − g2 F † Fyy¯ leptonic Lagrangian 4 (2.10) D(x,y)=ψ¯(iγµDµ+gYDy+gYDy¯)ψ 1 1 L =ψ¯Liγµ(∂µ+ALµ)ψL+ψ¯Riγµ(∂µ+ARµ)ψR =− 2g12TrFLµν†FLµν − 2g12TrFRµν†FRµν +gY′ ψ¯LΦψR+gY′ ψ¯RΦ†ψL. (2.6) + g12 + g12 Tr DµΦ † DµΦ (cid:18) 2 3(cid:19) (cid:16)(cid:0) (cid:1) (cid:0) (cid:1) Then, field strengthes are derived in usual way − ∂yAµL † ∂yALµ − ∂yAµR † ∂yARµ Fµν =[Dµ, Dν] + g(cid:0)22Tr (cid:1)∂(cid:0)yΦ † ∂(cid:1)yΦ (cid:0)− Φ†Φ(cid:1) (cid:0)† Φ†Φ (cid:1)(cid:17), = ∂µALν −∂νALµ+[ALµ, ALν] |L><L| 4 (cid:16)(cid:0) (cid:1) (cid:0) (cid:1) (cid:0) (cid:1) (cid:0) (2.1(cid:1)(cid:17)1) +(cid:0)∂µARν ∂νARµ+[ARµ, ARν](cid:1) R><R − | | where =[D , D ] µy(cid:0) µ y (cid:1) F Fµν =∂µAν ∂νAµ +[Aµ, Aν], = ∂µΦ+ALµΦ ΦARµ L><R L L− L L L − | | Fµν =∂µAν ∂νAµ +[Aµ, Aν], (2.12) (cid:0) ∂yALµ L><L ∂(cid:1)yARµ R><R R R− R R R − | |− | | µΦ=∂µΦ+AµΦ ΦAµ. µy¯=[Dµ, Dy¯] D L − R F = ∂ Φ +A Φ Φ A R><L Here,weaddressthegaugetransformationofthe µ † Rµ † † Lµ − | | present formulation. The transformation function ∂ A R><R ∂ A L><L y(cid:0) Rµ y Lµ (cid:1) − | |− | | is denoted by =[D , D ] yy¯ y y¯ F g(x)=g (x)L><L +g (x)R><R, (2.13) =∂ Φ R><L ∂ ΦL><R L R y † y | | | | | |− | | +Φ†ΦR><R ΦΦ† L><L in which we should note that g(x) does not depend | |− | | (2.7) on the argument y of S . It is evident that the 1 2 covariant derivatives are gauge covariant as they calculation should be. 3 1 1 (x,y)= FiµνFi BµνB L − 4 L Lµν − 4 µν g(x)Dµg−1(x)=∂µ+AgLµ|L><L| Xi=1 +AgRµ|R><R|=Dµg, (2.14) + DµΦ † DµΦ −λ′ Φ†Φ 2+α2 ∂yΦ † ∂yΦ g(x)Dyg−1(x)=∂y +Φg|L><R|=Dyg, β(cid:0)2 3(cid:1) ∂(cid:0)yAiµ(cid:1) ∂ A(cid:0)i (cid:1) β2 ∂y(cid:0)Bµ (cid:1)∂(cid:0)B (cid:1), g(x) g 1(x)=∂ +Φg R><L = g. − L L y Lµ − R y µ Dy¯ − y †| | Dy¯ Xi=1(cid:0) (cid:1)(cid:0) (cid:1) (cid:0) (cid:1)(cid:0) (3(cid:1).3) where Similarly, we can prove the covariant derivatives givenin(2.8)arealsocovariantforthegaugetrans- Fi =∂ Ai ∂ Ai +gf Aj Ak , Lµν µ Lν − ν Lµ ijk Lµ Lν formation. According to (2.14), field strengthes ex- B =∂ B ∂ B , µν µ ν ν µ pressedin(2.7)and(2.9)aregaugecovariantwhich − 3 yieldsthattheLagrangianin(2.10)isgaugeinvari- i Φ= ∂ σkgAk +σ0g B Φ, ant. It is also evident that the leptonic Lagrangian Dµ ( µ− 2 Lµ ′ µ!) k=1 in (2.6) is gauge invariant. X (3.4) with a= 1 and b= 2 for left- and right-handed − − leptons in (2.2), respectively. In this paper, we consider the case that only 3 Reconstruction of Weinberg Higgs field is infiltrated into the extra dimensional space such as model Φ(x,y)=φ(x)f(y), (3.5) Let us specify gauge fields in W-S model as wheref′(y)=0andwenormalizethefunctionf(y) 6 such as 1 2πR 3 f2(y)dy =1, (3.6) i 2πR ALµ =−2( σkgAkLµ+aσ0g′Bµ), whereas gauge fieldZs0penetrate the extra dimension k=1 X i uniformly and therefore don’t depend on the argu- A = bg B , (3.1) Rµ −2 ′ µ ment y of extra dimensional space. φ+ Ai (x,y)=Ai (x), B (x,y)=B (x). (3.7) Φ= , L L µ µ φ0 (cid:18) (cid:19) From these settings, we find the 4-dimensional Yang-Mills-Higgs Lagrangian whereAk andB areSU(2)andU(1)gaugefields with couLpµling conµstants g and g , respectively and, 1 2πR ′ L = (x,y)dy YMH aandbaretheU(1)hyperchargesofleft-andright- 2πR L Z0 handed leptons, respectively. From (2.7), we can 3 1 1 form the fields strengthes =− 4 FLiµν(x)FLiµν(x)− 4Bµν(x)Bµν(x) i=1 X +( φ(x)) ( µφ(x)) i 3 Dµ † D Fµν = g σi ∂µAiν ∂νAiµ+gfijkAjµAkν α2 2πR L − 2 L − L L L + f 2(y)dy φ (x)φ(x) ′ † i=1 2πR i X (cid:0) (cid:1) Z0 − 2ag′σ0 ∂µBν −∂νBµ , λ′ 2πRf4(y)dy (cid:0)φ†(x)φ(x)(cid:1)2, − 2πR FRµν =− 2ibg′σ0(cid:0)∂µBν −∂νBµ(cid:1), Z0 (cid:0) (cid:1) (3.8) i (cid:0) 3 (cid:1) where we shouldnotice that the metric structure is DµΦ=(∂µ− 2 σkgAkLµ−σ0g′Bµ!)Φ. Xk=1 (3.2) ∂y =∂y. (3.9) After insertion of (3.2) into (2.11) and rescaling of The effective potential in tree level is known to be fields,theLagrangiantakestheform,withconstant parameters α, β , β and λ resulting from proper V(φ)=λ φ (x)φ(x) 2 µ2 φ (x)φ(x) (3.10) L R ′ † † − (cid:0) (cid:1) (cid:0) (cid:1) 3 with the parameters λ = λ′ 2πRf4(y)dy and 4 Conclusions 2πR 0 µ2 = α2 2πRf 2(y)dy. Here, we adopt the uni- 2πR 0 ′ R We have reconstructed the Weinberg-Salam model tary gauge and then Higgs field is expressed as R based on the generalized covariant derivative method on the product space M S , where the 0 4 1 × φ= ϕ+v , (3.11) gauge symmetry is spontaneously broken owing to  √2  the penetration of the Higgs field into the extra compact space S . This breakdown of symmetry   1 where φ0 = 0 v/√2 t gives the minimal point to isrealizedwithoutconsideringthequantumeffects. This is favorable point of our model. theeffectivepotentialV(φ),andsov =µ/√λ. The (cid:0) (cid:1) It is assumed that the gauge and fermion fields field ϕ is the neutral Higgs boson. The effective do not depend on the argumenty in S , and there- potential as a function of ϕ is 1 fore the Kaluza-Klein modes of those fields do not λ appear on the stage. This assures the renormaliz- V(ϕ)= ϕ4+λvϕ3+µ2ϕ2 (3.12) abilityandstabilityofourmodelformulatedinthis 4 paper since the y derivative terms of gauge fields exceptforthe constantterm. Thecovariantderiva- in (2.11) give the imaginary masses to the Kaluza- tive of φ is written as Klein modes. The existence of the Kaluza-Klein modes of Higgs field is also undesirable since it in- √2gW+ 1 0 i µ ϕ+v creasesthegaugebosonmass. Thisnon-existenceof φ= , Dµ √2(cid:18)∂µϕ(cid:19)− 2 g2+g′2Zµ √2 Kaluza-Kleinmodes is acleardifference fromother model [6]. LHC or more powerful machine will de-   (3.13) p cide whether Kaluza-Klein modes exists or not. whereW+andZ arethechargedandneutralweak µ µ boson fields, respectively. Finally, we obtain the Yang-Mills-Higgs Lagrangian Acknowledgement 2 1 L = Fi (x)Fiµν(x) The author wouldlike to expresshis sincerethanks YMH − 4 Lµν L to Professor H. Kase and Professor K. Morita for i=1 1X 1 useful suggestion and invaluable discussions. F (x)Fµν(x) F (x)Fµν(x) − 4 Zµν Z − 4 µν + 1(∂ ϕ)2 V(ϕ) References µ 2 − 1 (g2+g′2) [1] M.B.Green,J.H.Schwartz,andE.Witten,Su- +4(ϕ+v)2 g2Wµ+W−µ+ 2 Zµ2 , perstring Theory I, II, Cambridge Univ. Press ! (1987), (3.14) whereFi , Fµν andF (x)arethefieldstrengths J. Polchinski, Superstring Theory I, II, Cam- Lµν Z µν bridge Univ. Press (1998). of charged, neutral weak gauge fields and photon field, respectively. From (3.14), the famous mass [2] N.S. Manton, Nucl. Phys. B158 (1979) 141, relation mW =mZcosθW follows. ibit. B193 (1981) 502. Under the assumption that leptons also stay at S uniformly, the 4-dimensional Dirac Lagrangian [3] A. Connes, p.9 in The Interface of Mathe- 1 obtained by integrating (2.6) takes the form maticsandParticlePhysics,ed.D.G.Quillen, G.B.Segal,andTsou.S.T.,ClarendonPress, 1 2πR Oxford, 1990. L = dy D 2πR LD A. Connes and J. Lott, Nucl. Phys. B18B Z0 3 (Proc. Suppl.) (1990), 57. i =ψ¯ iγµ ∂ σkgAk σ0g B ψ A. Connes, Non-commutative differential Ge- L ( µ− 2 Lµ− ′ µ!) L ometry, Cambridge Univ. Press, (1993). k=1 X +e¯ iγ (∂µ+ig B )e +g e¯eϕ+m e¯e, R µ ′ µ R Y e [4] M. Dubois-Violette, R.Kernerand J.Madore, (3.15) J . Math. Phys. 31 (1990), 316. where gY = 2g√Y′2√απλ′R 02πRf(y)dy and the electron R. Coquereaux, G. Esposito-Farese and G. massm =g v. Theequation(3.15)isequaltothe Vaillant, Nucl. Phys., B353 (1991), 689. e Y R lepton part of Lagrangian in the Weinberg-Salam R. Coquereaux, G. Esposito-Farese and model. F. Scheck, Int. Journ. Mod. Phys., A7 (1992), 4 6555. A. Sitarz, Phys. Lett., B308 (1993), 311 ; Jour. Geom. Phys. 15 (1995), 123. S. Naka and E. Umezawa, Prog. Theor. Phys. 92 (1994), 189. A. H. Chamseddine, G. Felder and J. Fr¨olich, Phys. Letters, B296 (1992), 109; Nucl. Phys. B395 (1993), 672. A. H. Chamseddine and J. Fr¨olich, Phys. Let- ters, B314 (1993), 308 Phys. Rev. D 50 (1994), 2893. C. P. Mart´in, J. M. Gracia-Bondia, J. S. Var- illy, Phys. Rep. 294 (1998), 363. K. Morita and Y. Okumura, Prog. Theor. Phys.91,959(1994),Phys.Rev.D,50(1994), 1016, Y. Okumura, Phys. Rev. D, 50 (1994), 1026, Prog.Theor.Phys. 95 (1996), 969,Eur. Phys. J. C4 (1998), 711. [5] I.S. Sogami, Prog. Theor. Phys. 94 (1995), 117; ibid. 95 (1996), 637. [6] Y. Hosotani, Phys. Lett. B 126 (1983) 309, Annals of Phys. 190 (1989) 233. N. Haba, M. Harada, Y. Hosotani and Y. Kawamura,Nucl. Phys. B657 (2003) 169, N. Haba, Y. Hosotani, Y. Kawamura and T. Yamashita, Phys. Rev. D70 (2004) 015010, N.HabaandT.Yamashita,JHEP0404(2004) 016, and references therein. 5