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Geometrization of the ele tro-weak model bosoni omponent Fran es o Cianfrani*, Giovanni Montani* *ICRA(cid:22)International Center for Relativisti Astrophysi s 6 0 0 Dipartimento di Fisi a (G9), 2 Università di Roma, (cid:16)La Sapienza", n a Piazzale Aldo Moro 5, 00185 Rome, Italy. J 3 e-mail: montanii ra.it 1 fran es o. ianfranii ra.it 1 v 2 5 0 1 0 PACS: 11.15.-q, 04.50.+h 6 0 / c q - r g : v Abstra t i X r a In this work we develop a geometri al uni(cid:28) ation theory for gravity and the ele tro-weak model in a Kaluza-Klein approa h; in parti ular, from the urvature dimensional redu tion Einstein-Yang-Mills a tion is obtained. We onsider two possible spa e-time manifolds: V4 S1 S2 V4 S1 S3 1) ⊗ ⊗ where isospin doublets are identi(cid:28)ed with spinors; 2) ⊗ ⊗ in whi h both quarks and leptons doublets an be re ast into the same spinor, su h that the equal number of quark generations and leptoni families is explained. Finally a self-intera ting omplex s alar (cid:28)eld is introdu ed to reprodu e the spontaneous symmetry breaking me hanism; in this respe t, at the end we get an Higgs (cid:28)elds whose two omponents have got opposite hyper harges. 2 I. INTRODUCTION The best results of modern physi s are the two theories that des ribe the fundamental intera tions, i.e. Standard Model and General Relativity. However, at the present no sat- isfying way to unify them is known, espe ially be ause of the mathemati al and physi al di(cid:30) ulties in the development of a Quantum General Relativity. Even if it is a general hope that the quantization of gravity would lead to the uni(cid:28) ation of all intera tions, there is no indi ation that it will follow. For example Loop Quantum Gravity [1℄, one of the most promising andidate for a quantum General Relativity, does not ontain other intera tions, so ita hieves no uni(cid:28) ationyet. Moreover, be ause osmologi alsour es we observe are only gravitational ones, on osmologi al s ale we an treat gauge arriers (cid:28)elds as perturbations. Therefore, the oexisten e, in a uni(cid:28)ed pi ture, of a lassi al formulation for gravity and of a quantum theory for other intera tions an be interpreted as a low energy e(cid:27)e t. The great a hievement of General Relativity is the geometrization of the gravitational (cid:28)eld; following this approa h we an try to regard even others (cid:28)elds as geometri properties (geometri al uni(cid:28) ation models) and, in parti ular, as spa e-time metri omponents (Kaluza- Klein theories). This issue, obviously, implies extra geometri al degrees of freedom, that an be introdu ed by virtue of extra-dimensions. To make the multidimensional model onsistent with a four-dimensional phenomenology, we require the unobservability for this additional oordinates and, due to quantum un ertainty, it an be rea hed by taking a ompa ti(cid:28)ed extra-spa e. It is also possible to onsider non- ompa t Kaluza-Klein models, as in braneworld s enario [2℄, and they found appli ations in string theory; however, in those models a strange kind of uni(cid:28) ation arise, in fa t while the gravitational (cid:28)eld propagates in the bulk, the other intera tions are restri ted to the four-dimensional brane. The oldest model with non-gravitational(cid:28)elds in the metri is the originalKaluza-Kleinone [3℄ [4℄ [5℄, whi h deals with the uni(cid:28) ation of gravitational and ele tromagneti intera tions V4 S1 ina spa e-time manifold ⊗ . To extend this pro edure toa generi Yang-Millstheory, a geometri alimplementationof the gauge group and of its algebrahave to be performed. The latteriseasilya hievedbytheintrodu tionofanhomogeneousextra-spa e (forthede(cid:28)nition of homogeneous spa e see [6℄) and by onsidering its Killing ve tors algebra. Instead, the role of group transformations is played by extra-dimensional translations; however, unless the above mentioned unobservability is taken into a ount, the right gauge transformations 3 on (cid:28)elds are not reprodu ed [7℄. At the end, to geometrize in a Kaluza-Klein approa h a gauge intera tion, an extra-spa e, whose Killing ve tors algebra is the same as the gauge group one, is required (for a review see [8℄ [9℄). In our work we reprodu e in this way all the features of the ele tro-weak model, an issue failedbyothermultidimensionaltheories. In parti ular,we derive notonlyfree gaugebosons Lagrangian from the dimensional splitting of the multidimensional urvature, but also we show how their intera tion with spinor (cid:28)elds an be obtained by the splitting of free Dira Lagrangian. Therefore, we are able to give a ompletely geometri interpretation to the boson omponent and we have to introdu e just free spinors. Furthermore, the left- and right-handed four-dimensional (cid:28)elds, separately, as the lightest modes of an extra- oordinates expansion arise, while any their linear ombination a quires a mass term, having the order of the ompa ti(cid:28) ation s ale. So the need of a distin t treatment for the two hirality eigenstates be omes a low energy e(cid:27)e t. There are two spa e-times suitable for our approa h: V4 S1 S2 • a 7-dimensional manifold ⊗ ⊗ , V4 S1 S3 • a 8-dimensional manifold ⊗ ⊗ . In the (cid:28)rst ase, we deal with eight- omponents spinors, so we assign a geometri meaning to the isospin doublet. In the se ond ase, the number of dimensions of the gauge group is the same as that of the extra-spa e; from this ondition we get the identi(cid:28) ation of gauge harges with extra- omponents of the (cid:28)elds momentum and the equality of the number of leptoni families and quark generations. Moreover, as an extension respe t to anoni alKaluza-Kleintheories, a dependen e by some αm four-dimensional s alar (cid:28)elds ( ) for the extra-spa e metri is onsidered. At the end, the multidimensional analogous of the Higgs boson is introdu ed; in parti ular, we de(cid:28)ne it so that its two omponents have got opposite hyper harges. In this way it is U(1) possible to realize the spontaneous symmetry breaking to the ele tro-magneti group and to add, in the Lagrangian density, mass terms for all parti les. In parti ular, the work starts in se tion II with a short review of the ele tro-weak model, while in se tion III we de(cid:28)ne the spa e-time manifoldand show how the Einstein-Hilbert a - α tion redu tion leads to the Einstein-Yang-Millsone and to the terms des ribing dynami s; 4 in se tion IV, for matter (cid:28)elds we illustrate the form of the extra- oordinates dependen e able to give the right gauge transformations and the onservation of gauge harges. In se - V4 S1 S2 tion V we apply the results of the previous analysis to the spa e-time manifolds ⊗ ⊗ V4 S1 S3 and ⊗ ⊗ , where we re ast the observed four-dimensional(cid:28)elds into eight and sixteen omponents spinors, respe tively. Se tion VI deals with the introdu tion of the s alar (cid:28)eld SU(2) U(1) responsible of the ⊗ symmetry breaking, while in se tion VII brief on luding remarks follow. II. ELECTRO-WEAK MODEL SU(2) U(1) SU(2) The ele tro-weak model is a ⊗ gauge theory; transformations a t only on left-handed omponents in the following way i ψ = (I + gδωiT )ψ L′ 2 i L (1) δωi T i where arearbitraryin(cid:28)nitesimalfun tions, gisgroup's oupling onstants, aregroup's ψ L generators (Pauli matri es) and is onstituted by left-handed leptons (quarks) (cid:28)elds of the same family (generation) ν u lL qL ψ =   ψ =  . lL l qL d (2) L qL     SU(2) The onserved harges asso iated to the invarian e are the three omponents of the weak isospin Ii = ZE3d3xψl†Tiψl. (3) U(1) transformations a t also on right-handed states, in parti ular the in(cid:28)nitesimal transformation law for matter (cid:28)elds is the following one ψ = (I +ig y δω0I)ψ ′ ′ ψ (4) y ψ where g' is the U(1) oupling onstant and the hyper harge depends on the (cid:28)eld and of the hirality state. In fa t, we impose that the weak hyper harge, the onserved harge asso iated to these transformations Y = y d3xψ ψ, † Z (5) 5 is related to the ele tri harge and the third omponent of the weak isospin by the following relation Q = Y +I . e 3 (6) By substituting ordinary derivatives with the ones ontaining gauge onne tions and by SU(2) U(1) introdu ing gauge bosons free terms, the ⊗ invariant Lagrangian density from the Dira one is developed, i.e. 3 i~c Λ = D ψ¯ γµψ ψ¯ γµD ψ +D ν¯ γµν ν¯ γµD ν +D ¯l γµl X 2 h µ† lL lL − lL µ lL µ† lR lR − lR µ lR µ† R R − l,q=1 3 i~c ¯l γµD l + D ψ¯ γµψ ψ¯ γµD ψ +D u¯ γµu − R µ Ri X 2 h µ† qL qL − qL µ qL µ† qR qR − g=1 1 1 u¯ γµD u +D d¯ γµd d¯ γµD d B Bµν G Gµν − qR µ qR µ† qR qR − qR µ qRi− 4 µν − 4 iµν i where B = ∂ B ∂ B µν ν µ µ ν − Gi = ∂ Wi ∂ Wi +gci WjWk µν ν µ − µ ν jk µ ν 1 g D ψ = [∂ + igτ Wi i ′IB ]ψ µ lL µ 2 i µ − 2 µ lL D l = [∂ ig B ]l µ R µ ′ µ R − D ν = ∂ ν µ lR µ lR i g D ψ = [∂ + gτ Wi +i ′IB ]ψ µ qL µ 2 i µ 6 µ qL 2 D u = [∂ +i g B ]u µ qR µ ′ µ qR 3 1 D u = [∂ i g B ]u . µ qR µ ′ µ qR − 3 SU(2) U(1) Bosons we observe are related to ⊗ gauge ones by the transformations B = sinθ Z +cosθ A µ w µ w µ  −       W1 = 1 (W+ +W )  µ √2 −    .   (7) W2 = i (W+ W )  µ √2 − −         W3 = cosθ Z +sinθ A  µ w µ w µ    6 SU(2) U(1) In ordertoobtainaspontaneous ⊗ symmetrybreakingaHiggss alar(cid:28)eld, whi h 1 is an isospin doublet and has got 2 hyper harge, is introdu ed with a suitable Lagrangian density, i.e. 1 Λ = [D φ] [Dµφ] µ2φ φ λ[φ φ]2 g [ψ¯ l φ+φ ¯l ψ ] g [ψ¯ ν φ+φ ν¯ ψ ] φ 2 µ † − † − † − l lL R † R lL − νl lL lR † lR lL − X l e e g [ψ¯ u φ+φ u¯ ψ ] g [ψ¯ d φ+φ d¯ ψ ] uq qL qR † qR qL dq qL qR † qR qL − − (8) X q e e µ2 < 0 λ > 0 φ = i(φ T )T † 2 with , and − . The Higgs (cid:28)eld realizes a spontaneous symmetry e breaking, whi h dealstothe onservation ofonlytheele tri harge, by (cid:28)xingitsexpe tation value in the va uum as the following one 0 φ =  . 0 v (9)  √2  In this way, a uni(cid:28)ed pi ture for weak and ele tro-magneti intera tion is a hieved; furthermore, a me hanism by whi h massive parti les arise is obtained, even if the (cid:28)eld responsible of this pro ess have not been dete ted yet. III. GEOMETRIZATION OF THE ELECTRO-WEAK BOSONIC COMPONENT Let us onsider a spa e-time manifold Vn = V4 Bk ⊗ (10) V4 xµ(µ = 0,...,3) where is the ordinary 4-dimensional spa e-time, with oordinates , and Bk Bk stands for the extra-dimensional one. The spa e is introdu ed in su h a way that its SU(2) U(1) Killing ve tors reprodu e the Lie algebra of a ⊗ group, i.e. ξn ∂ ξm ξn ∂ ξm = CP ξm M n M − M n N¯ NM P (11) CP SU(2) U(1) where NM are ⊗ stru ture onstants. Bk In parti ular, we have the following two possible hoi es for i)S1 S2 ⊗ (12) ii)S1 S3 ; ⊗ (13) 7 S1 S2 S3 y0 yi(i = 1,..k 1) we will refer to oordinates on and or by and − , respe tively, ym (m = 0,..,k 1) while − will indi ate both of them. We develop a theory invariant under general four-dimensional oordinates transformations Bk and under translations along xµ = xµ(xν) ′ ′   ym = ym +ωM(xν)ξm(yn); (14) ′ M  for the metri we take the following ansantz g +γ ξmξnAMAN γ ξmAM  µν mn M N µ ν mn M µ  j =   AB   (15)        γmnξNnANν γmn    AM g where µ are gauge bosons (cid:28)elds and µν the four-dimensional metri [7℄. For the extra-dimensional metri , we assume γ (x;y) = γ (x)αm(x)αn(x) mn mn (16) α = α(x) (in the latter expression the indi es m and n are not summed) with some s alar (cid:28)elds that a ount for the dynami s of the extra-dimensional spa e. In order to forbid S2 S3 α1 = α2(= α3) = α anisotropies of the spa e or , we take , so allowingjust size hanges. Wenotethatinthe8-dimensional ase theextra-spa eis4-dimensional,soitsdimensionality is the same as that of the gauge group; thus, Killing ve tors an be hosen in su h a way they are dire tly related to extra-dimensional n-bein ve tors by the following relation ξ(n) = αne(n) n n (17) where in the latter the index n is not summed. We an arry on the dimensionalredu tion of the Einstein-Hilberta tion in 4+k-dimensions as c3 S = jnRd4xdky −16πGn ZV4 Bk p− (18) ⊗ and we get the geometrization of the gauge bosoni omponent, i.e. from the multidimensional urvature their Lagrangian density out omes c3 k−1 ∂ αn S = d4x√ g R+R 2gµν ∇µ ν −16πG ZV4 − h N − Xn=0 αn − 8 k−1 ∂ αm∂ αn 1 3 1 gµν µ ν (α)2GMGMµν (α)2BMBMµν − X αm αn − 4 X µν − 4 ′ µν i (19) n=m=0 M=1 6 R N where for Newton oupling onstant and for the urvature term we have, respe tively, (V is the volume of the extra-spa e) G G = n V 1 1 R = CM CN . N 4α2 NT TM For details, we remand to our pre edent work [7℄. Therefore in a Kaluza-Klein approa h we arry on the geometrization by identifying gauge bosons with mist metri omponents. Unless the introdu tion of super-symmetries this pro edure is not extensible to fermions, so to a ount for them we have to introdu e spinor matter (cid:28)elds. IV. MATTER FIELDS Any matter (cid:28)eld is a not-geometri and, thus, an undesired term in a geometri aluni(cid:28) a- tion model; however, a remarkable result a hieved in previous works [7℄ [10℄ is the possibility to deal simply with free spinor (cid:28)elds, on e we assume for them a suitable dependen e on extra- oordinates. In other words, we take the following matter (cid:28)elds 1 Ψ(x;y) = e−iTMλMNΘN(y)ψ(x), V (20) T ΘN 1 where M are gauge group's generators, s alar densities of weight 2 on the extra-spa e ψ λ and the four-dimensional (cid:28)elds, while we de(cid:28)ne the onstant matrix by the following relation 1 (λ 1)N = √ γ ξm∂ ΘN dKy. − M V Z − (cid:16) M m (cid:17) (21) ψ With su h a dependen e on extra- oordinates, the transformationof the (cid:28)eld under trans- ξm lations along Killing ve tors M, on e the role of the unobservability is taken into a ount [11℄, is like the a tion of a gauge group, i.e. ψ = ψ +iωMT ψ. ′ M (22) 9 Moreover, with the same assumption we demonstrate the onservation of gauge harges; in parti ular, in the 8-dimensional ase they an be regarded as extra- omponents of the (cid:28)elds momentum [10℄. The most remarkable result, that follows from the hypothesis (20), is the geometrization of spinor gauge onne tions. In fa t, starting from the free Dira Lagrangian, the intera tion with gauge bosons out omes √ γdKyΨ¯γ(µ)D Ψdky = ψ¯γ(µ)eµ (D +iT AM)ψ. Z − (µ) (µ) µ M µ (23) BK However, some additional terms, deriving from √ γdKyΨ¯γ(m)∂ Ψdky, Z − (m) (24) BK arise; they produ e the standard Kaluza-Klein ontribution to the spinor mass, of the same order as the ompa ti(cid:28) ations ale, and lead tovery heavy four-dimensionalparti les. These parti les annot be des ribed by a low energy theory. γ γ 5 Now, be ause the extra-dimensional matri es are developed by the four-dimensional one, we (cid:28)nd that for the left-handed and right-handed (cid:28)elds the expression (24) vanishes. Thelatterstatementexplainswhy, inamultidimensionaltheory,thefour-dimension hirality eigenstate still have to be onsidered: they orrespond to the lightest mode of an expansion in the extra- oordinates. In on lusion, to a ount for fermions intera ting with gauge bosons, we need just free left- handed and right handed spinors, where, again, we stress that the hirality we refer to is the four-dimensional one. V. SPINORS REPRESENTATION At this point, theintrodu tionof fermionsisbased on the analysisof the previousse tion. U(1) y0 = ϕ In parti ular, the a tion of the group is reprodu ed by adding a dependent phase in front of the four dimensional spinors 1 φ(xµ) Ψ = einϕφ ϕ = [0;2π) → √L (25) e L S1 φ where stands for length and n is related to the hyper harge of the (cid:28)eld ; in fa t, under ϕ ϕ+f(xµ) φ the in(cid:28)nitesimal translation → , the transformation law for is n φ φ+inf(xµ) = φ+i g(xµ). → 6 (26) 10 U(1) From the latter expression, by imposing it reprodu es the a tion of the hyper harge n group, the value of for ea h spinor is easily obtained n = 6y . ψ ψ (27) Ψ(ϕ) We observe that, in order to maintain the fundamental periodi ity of , n must be an e integer; this gives a justi(cid:28) ation for the hyper harge spe trum observed. SU(2) In an analogous but formally more omplex way, for the group we make use of Pauli S2 S3 matri es and, a ording with the equation (20), the form for the dependen e on or oordinates is the following 1 Ψ = e−iTMλMNΘN(yi)Ψ M,N = 1,2,3 √V (28) e T SU(2) V S2 S3 λ M where are generators, stands for or volume and obeys the relation(21). 2[2d] Now, be ause the number of spinor omponents in a d-dimensional spa e-time is [12℄, we perform a distin t treatment for a 7-dimensional and a 8-dimensional spa e-time. In the7-dimensional ase, we dealwitheight omponentsspinors, thusexplainingtheisospin doublet. In parti ular, we introdu e the following four spinors for any leptoni family and quark generation 1 einLϕν 1 ν ΨlL = e−iσIλIMΘM  lL  ΨlR =  lR  l = 1,2,3 √V√L einLϕl √V√L einlRϕl (29) L R     1 einLϕν 1 einuRϕu ΨqL = e−iσIλIMΘM  lL  ΨqR =  lR  q = 1,2,3 √V√L einLϕl √V√L eindRϕd (30) L lR     with obvious notation for four-dimensional (cid:28)elds. Thus, we in lude even the right handed (cid:28)elds of leptons (quarks) of the same family (generation) in the same spinor and this assumption is onsistent with their di(cid:27)erent intera tion properties, sin e we assume for them ϕ a di(cid:27)erent dependen e on the oordinate . In the 8-dimensional ase spinors have got 16 omponents and this allow us to reprodu e all Standard Model parti les from the following six ones einLϕν ν lL lR     1 einLϕl 1 einlRϕl ΨlL = e−iTIλIMΘM  L  ΨlR =  R  l = 1,2,3 √V√L einqLϕu  √V√L einuRϕu  (31)  lL   lR       einqLϕd   eindRϕd   lL   lR 