Bled Workshops in Physics Vol. 15, No. 2 A Proceedings to the 17th Workshop What Comes Beyond ... (p. 123) Bled, Slovenia, July 20-28, 2014 9 Can Spin-charge-family Theory Explain Baryon Number Non-conservation? N.S. Mankoc Borštnik Department of Physics, FMF, University of Ljubljana, Jadranska 19, SI-1000 Ljubljana, Slovenia Abstract. The spin-charge-family theory [1-12,14], in which spinors carry besides the Dirac spin also the second kind of the Clifford object, no charges, is a kind of the Kaluza-Klein theories [13]. The Dirac spinors of one Weyl representation in d = (13 + 1) manifest [1,4,3,10,14,15] in d = (3 + 1) at low energies all the properties of quarks and leptons assumed by the standard model. The second kind of spins explains the origin of families. Spinors interact with the vielbeins and the two kinds of the spin connection fields, the gauge fields of the two kinds of the Clifford objects, which manifest in d = (3 + 1) besides the gravity and the known gauge vector fields also several scalar gauge fields. Scalars with the space index s € (7, 8) carry the weak charge and the hyper charge j, ± -, respectively), explaining the origin of the Higgs and the Yukawa couplings. It is demonstrated in this paper that the scalar fields with the space index t € (9,10,..., 14) carry the triplet colour charges, causing transitions of antileptons and antiquarks into quarks and back, enabling the appearance and the decay of baryons. These scalar fields are offering in the presence of the right handed neutrino condensate, which breaks the CP symmetry, the answer to the question about the matter-antimatter asymmetry. Povzetek. V teoriji spinov-nabojev-družin [1-12,14] nosijo spinorji dve vrsti kvantnih stevil, ki jih določata dve vrsti operatorjev ya: Diracovi operatorji ya in avtoričini Ya, obe sta povezani z mnoZenjem Cliffordovih objektov, ena vrsta z leve, druga z desne. Obe vrsti spina sta neodvisni in tvorita druga drugi ekvivalentne upodobitve. Analiza Lorentzove grupe SO(13,1) s podgrupami te grupe pokaze, da vsebuje ena Weylova upodobitev Dira-covih spinorjev v d = (13 + 1) vse kvarke in leptone (ter antikvarke in antileptone) s kvantnimi stevili kot jih predpise standardni model pred elektrosibko zlomitvijo, le da so desnorocni nevtrini enakopravni partnerji elektronom [1,4,3,10,14,15]. Druga vrsta spina pojasni izvor druzin. Spinorji interagirajo s tetradami in s polji dveh vrst spinskih povezav, ki so umeritvena polja obeh vrst operatorjev gamma. Po zlomitvi simetrij, tedaj pri opa-zljivih nizkih energijah, dolocajo ta polja, skupaj z vektorskimi sveznji, gravitacijo in vsa znana umeritvena vektorska polja. Dolocajo pa tudi skalarna polja. Skalarna polja s prostorskim indexom s = (7, 8) so sibki dubleti (t13 = ^ 1 ,Y = ± -), kar pojasni izvor Higg-sovega skalarnega polja in Yukawinih sklopitev. Skalarna polja s prostorskim indeksom t € (9,10,..., 14) pa so barvni tripleti, ki povzrocajo prehode antileptonov in antikvarkov v kvarke in obratno, kar omogoci nastanek in razpad barionov. Vsa skalarna polja nosijo glede na kvantna stevila, ki jih dolocajo Diracovi ya in druzinski Ya, tudi druzinska in Diracova kvantna stevila v adjungirani upodobitvi grup. Lepota te teorije je, da en sam kon-denzat iz dveh desnorocnih nevtrinov z druzinskimi kvantnimi stevili, ki niso druzinska kvantna stevila spodnjih stirih druzin, zlomi diskretno simetrijo CP in poskrbi za maso 124 N.S. Mankoč Borštnik vseh skalarnih polj, ter še neopaženega vektorskega polja. Ta skalarna polja ponujajo v prisotnosti kondenžata desnoročnih nevtrinov, ki žlomi simetrijo CP, odgovor na vprasanje kako je v nasem vesolju nastala opažljiva asimetrija med snovjo in antisnovjo. Skalarna polja s prostorskim indexom s = (7, 8) žlomijo ž neničelno vakuumsko pričakovano vrednostjo se sibki in hipernaboj, in spremenijo tudi lastno maso, ter tako pojasnijo vse privžetke standardnega modela. Ker teorija napoveduje dve ločeni gruči po stiri družine kvarkov in leptonov, pojasni stabilna od žgornjih sirih družin ižvor temne snovi. Teorija pa napoveduje tudi, da bodo na LHC ižmerili četrto k trem že opaženim družinam, ižmerili pa bodo tudi več skalarnih polj. 9.1 Introduction The spin-charge-family [1-12,14] theory is offering, as a kind of the Kaluža-Klein like theories, the explanation for the čharges of quarks and leptons (right handed neutrinos are in this theory the regular members of a family) and antiquarks and antileptons [15,16], and for the existenče of the čorresponding gauge večtor fields. The theory explains, by using besides the Dirač kind of the Clifford algebra obječts also the sečond kind of the Clifford algebra obječts (there are only two kinds [57,3,17,18,20,19], assočiated with the left and the right multipličation of any Clifford obječt), the origin of families of quarks and leptons and čorrespondingly the origin of the sčalar gauge fields čausing the elečtroweak break. These sčalar fields are responsible, after gaining nonžero vačuum expečtation values, for the masses and mixing matričes of quarks and leptons [9-11] and for the masses of the weak večtor gauge fields. They manifest, čarrying the weak čharge and the hyper čharge equal to (± 2, T 2, respečtively) [14], as the Higgs field and the Yukawa čouplings of the standard model. The spin-charge-family theory predičts two dečoupled groups of four families [3,4,9-11]: The fourth of the lower group will be measured at the LHC [10], while the lowest of the upper four families čonstitutes the dark matter [12]. This theory also predičts the existenče of the sčalar fields whičh čarry the triplet čolour čharges. All the sčalars fields čarry the fračtional quantum numbers with respečt to the sčalar index s > 5, either the ones of SU(2) or the ones of SU(3), while they are with respečt to other groups in the adjoint representations. Neither these sčalar fields nor the sčalars čausing the elečtroweak break are the supersymmetrič sčalar partners of the quarks and leptons, sinče they do not čarry all the čharges of a family member. These sčalar fields with the triplet čolour čharges čause transitions of antileptons into quarks and antiquarks into quarks and bačk, offering, in the presenče of the čondensate of the two right handed neutrinos with the family quantum numbers belonging to the upper four families whičh breaks the CP symmetry, the explanation for the matter-antimatter asymmetry. This is the topič of the present paper. Let me point out that the spin-charge-family theory overlaps in many points with other unifying theories [26-31], sinče all the unifying groups čan be seen as the subgroups of the large enough orthogonal groups, with family groups inčluded. But there are also many differenčes. While the theories built on čhosen 9 Can Spin-charge-family Theory Explain Baryon Number Non-conservation? 125 groups must for their choice propose the Lagrange densities designed for these groups and representations (which means that there must be a theory behind this effective Lagrange densities), the spin-charge-family theory starts with a very simple action, from where all the properties of spinors and the gauge vector and scalar fields follow, provided that the breaks of symmetries occur. Consequently this theory differs from other unifying theories in the degrees of freedom of spinors and scalar and vector gauge fields which show up on different levels of the break of symmetries, in the unification scheme, in the family degrees of freedom and correspondingly also in the evolution of our universe. It will be demonstrated in this paper that one condensate of two right handed neutrinos makes all the scalar gauge fields and all the vector gauge fields massive on the scale of the appearance of the condensate, except the vector gauge fields which appear in the standard model action before the electroweak break as massless fields. The scalar gauge fields, which cause the electroweak break while gaining nonzero vacuum expectation values and changing their masses, then explain masses of quarks and leptons and of the weak bosons. It is an extremely encouraging fact, that one scalar condensate and the nonzero vacuum expectation values of some scalar fields, those with the weak and the hyper charge equal to by the standard model required charges for the Higgs's scalar, can bring the simple starting action in d =(13 +1) to manifest in d = (3 +1) in the low energy regime the observed phenomena of fermions and bosons, explaining the assumptions of the standard model and can possibly answer also the open questions, like the ones of the appearance of family members, of families, of the dark matter and of the matter-antimatter asymmetry. The paper leaves, however, many a question connected with the break of symmetries open. Although the scales of breaks of symmetries can roughly be estimated, for the trustworthy predictions a careful study of the properties of fermions and bosons in the expanding universe is needed. It stays to be checked under which conditions in the expanding universe, the starting fields (fermions with the two kinds of spins and the corresponding vielbeins and the two kind of the spin connection fields) after the spontaneous breaks manifest in the low energy regime the observed phenomena. This is a very demanding study, a first simple step of which was done in the refs. [12,22]. The present paper is a step towards understanding the matter-antimatter asymmetry within the spin-charge-family theory. In the subsection 9.1.1 I present the action and the assumptions of the spin-charge-family theory, with the comments added. In sections (9.2,9.4,9.5,9.3) the properties of the scalar and vector gauge fields and of the condensate are discussed. In appendices the discrete symmetries of the spin-charge-family theory and the technique used for representing spinors, with the one Weyl representation of SO(13,1) and the families in SO (7,1) included, is briefly presented. The final discussions are presented in sect. 9.7. 9.1.1 The action of the spin-charge-family theory and the assumptions In this subsection all the assumptions of the spin-charge-family theory are presented and commented. This subsection follows to some extend a similar subsection of 126 N.S. Mankoč Borštnik the ref. [14]. i. The space-time is d(= (13 +1)) dimensional. Spinors carry besides the internal degrees of freedom, determined by the Dirac Ya's operators, also the second kind of the Clifford algebra operators [5-7,4], called Ya's. ii. In the simple action [3,1] fermions ^ carry in d = (13 + 1) only two kinds of spins, no charges, and interact correspondingly with only the two kinds of the spin connection gauge fields, daba and d> aba, and the vielbeins, faa. S = | ddx E Lf + ddxE (aR + aR), Lf = 2 (^ YaPoa^) + h.c., poa = faapoa + 2^ {Pa,Efaa}-, P0a = Pa — 2Sabdaba — ^ daba, R = 2 (fa[afpb] (daba,p - dcaa dCbP)} + h.C. , R = 2 fa[af«b] (ddaba,P - dcaadd%p) + h.C. . (9.1) Here 1 fa[af«b] = faafPb - fabfPa. sab and Sab are generators (Eqs.(9.5, 9.37, 9.37) of the groups SO(13,1) and SO(13,1), respectively, expressible by Ya and Y a. iii. The manifold M(13+1) breaks first into M(7+1) times M(6) (which manifests as SU(3) xU(1)), affecting both internal degrees of freedom, SO(13 + 1) and SO(13 + 1). After this break there are 2((7+1 )/2-1) massless families, all the rest families get heavy masses 2. Both internal degrees of freedom, the ordinary SO(13 + 1) one (where Ya determine spins and charges of spinors) and the SO (13 +1) (where Ya determine family quantum numbers), break simultaneously with the manifolds. iv. There are additional breaks of symmetry: The manifold M(7+1' breaks further 1 faa are inverted vielbeins to eaa with the properties eaafab = 5%, eaafPa = 6«, E = det(eaa). Latin indices a, b,.., m, n,.., s, t,.. denote a tangent space (a flat index), while Greek indices a, |3,.., |a, v, ..a, T,.. denote an Einstein index (a curved index). Letters from the beginning of both the alphabets indicate a general index (a, b, c,.. and a, Y,.. ), from the middle of both the alphabets the observed dimensions 0,1,2,3 (m, n,.. and v,..), indices from the bottom of the alphabets indicate the compactified dimensions (s, t,.. and a, T,..). We assume the signature r|ab = diag{1, —1, —1, • • • , —1}. 2 A toy model [22,23,15] was studied in d = (5 + 1) with the same action as in Eq.'(9.1). For a particular choice of vielbeins and for a class of spin connection fields the manifold Ms+1 breaks into M(3+1' times an almost S2, while 2((3+1 )/2-1' families stay massless and mass protected. Equivalent assumption, although not yet proved that it really works, is made also in the case that M(13+1) breaks first into M(7+1) x M(6). The study is in progress. 9 Can Spin-charge-family Theory Explain Baryon Number Non-conservation? 127 into M<3+1)x M(4). v. There is a scalar condensate of two right handed neutrinos with the family quantum numbers of the upper four families, bringing masses of the scale above the unification scale, to all the vector and scalar gauge fields, which interact with the condensate. vi. There are nonzero vacuum expectation values of the scalar fields with the scalar indices (7,8), which cause the electroweak break and bring masses to the fermions and weak gauge bosons, conserving the electromagnetic and colour charge. Comments on the assumptions: i.: There are, as already written above, two (only two) kinds of the Clifford algebra objects. The Dirac one (Eq.(9.35)) (ya) will be used to describe spins of spinors (fermions) in d = (13 + 1), manifesting in d = (3 + 1) the spin and all the fermion charges, the second one (Eq.(9.35)) (Ya) will describe families of spinors. The representations of Ya's and Ya's are orthogonal to one another 3. There are correspondingly two groups determining internal degrees of freedom of spinors in d = (13 + 1): The Lorentz group SO(13,1) and the group SO(13,1). One Weyl representation of SO (13,1) contains, if analysed [1,3,4,15] with respect to the standard model groups, all the family members, assumed by the standard model, with the right handed neutrinos included (the family members are presented in table 9.3). It contains the left handed weak (SU(2)i) charged and SU(2)n chargeless colour triplet quarks and colourless leptons (neutrinos and electrons), the right handed weakless and SU(2)II charged quarks and leptons, as well as the right handed weak charged and SU(2)II chargeless colour antitriplet antiquarks and (anti)colourless antileptons, and the left handed weakless and SU(2)II charged antiquarks and antileptons. The reader can easily check the properties of the representations of spinors (table 9.3), presented in the "technique" (appendix 9.9) way, if using Eqs. (9.5, 9.8,9.9, 9.11,9.14). Each family member carries the family quantum numbers, originating in Ya's degrees of freedom. Correspondingly Sab changes the family quantum numbers, leaving the family member quantum number unchanged. ii.: This starting action enables to represent the standard model as an effective low energy manifestation of the spin-charge-family theory, which explains all the standard model assumptions, with the families included. There are gauge vector fields, massless before the electroweak break: gravity, colour SU(3) octet vector gauge fields, weak SU(2) (it will be named SU(2)I) triplet vector gauge field and "hyper" U(1) (it will be named U(1 )I) singlet vector gauge fields. All are superposition of fac ojaba. There are (eight rather than the observed three) families of quarks and leptons, massless before the electroweak break. These eight families are indeed two decoupled groups of four families, in the fundamental representations with respect to twice sU(2)x SU(2) groups, the 3 One can learn in Eq. (9.44)of appendix (9.9) that S ab transforms one state of the representation into another state of the same representation, while S ab transforms the state into the state belonging to another representation. 128 N.S. Mankoč Borštnik subgroups of SO(3,1) x SO(4) G SO(7,1). The scalar gauge fields, determining the mass matrices of quarks and leptons, carry with respect to the scalar index s G (7,8) the weak and the hyper charge of the scalar Higgs, while they carry if they are the superposition of fas & aba two kinds of the family quantum numbers in the adjoint representations, representing two (orthogonal) groups, each of the group contains two triplets (with respect to SU(2)gO(3 1) x SU(2)so(4)). The scalar fields with the quantum numbers (Q, Q', Y', which are the superposition of fas &aba) are the three singlets, again carrying the weak and the hyper charge of the scalar Higgs. One group of two triplets determine, together with the three singlets, after gaining nonzero expectation values, the Higgs's scalar and the Yukawa couplings of the standard model. The starting action contains also the additional SU(2)n (from SO(4)) vector gauge field and the scalar fields with the space index s G (5,6) and t G (9,10,11,12), as well as the auxiliary vector gauge fields expressible (Eqs. (9.56, 9.55) in the appendix 9.10) with vielbeins. They all remain either auxiliary (if there are no spinor sources manifesting their quantum numbers) or become massive after the appearance of the condensate. iii., iv.: The assumed break from M(13+1' first into M(7+1' times M(6) (manifesting the symmetry SU(3) xU(1 )n) explains why the weak and the hyper charge are connected with the handedness of spinors. In the spinor representation of SO (7,1) there are left handed weak charged quarks and leptons with the hyper charges (6, — 2, respectively) and the right handed weak chargeless quarks with the hyper charge either | or — 3, while the right handed weak chargeless leptons carry the hyper charge either 0 or (—1). A further break from M(7+1' into M(3+1' x M4), manifesting the symmetry SO(3,1) xSU(2)i x SU(2)n xU(1)„x SU(3), explains the observed properties of the family members - the coloured quarks, left handed weak charged and SU(2)II chargeless and right handed weak chargeless and SU(2)ii charged and colourless leptons, again left handed weak charged and SU(2)II chargeless and right handed weak chargeless and SU(2)II charged, quarks with the "spinor" charge 6 and leptons with the "spinor" charge — 2 - and of the observed vector gauge fields and the scalar fields (through Higgs's scalar and Yukawa couplings). Since the left handed members distinguish from the right handed partners in the weak and the hyper charges, the family members of all the families stay massless and mass protected up to the electroweak break 4. Antiparticles are accessible from particles by the Cn and pn, as explained in refs. [15,16] and briefly also in the appendix (9.8). This discrete symmetry operator does not contain ya's degrees of freedom. To each family member there corresponds the antimember, with the same family quantum number. v.: It is a condensate of the two right handed neutrinos with the quantum numbers of the upper four families (table 9.2), appearing in the energy region 4 As long as the left handed family members and their right handed partners carry different conserved charges, they can not behave as massive particles, they are mass protected. It is the appearance of nonzero vacuum expectation values of the scalar fields, carrying the weak and the hyper charge, which cause non conservation of these two charges, which makes the superposition of the left and the right handed family members possible, and breaks the mass protection. 9 Can Spin-charge-family Theory Explain Baryon Number Non-conservation? 129 above the unification scale, which makes all the scalar gauge fields (those with the space index (5,6,7,8), as well as those with the space index (9,..., 14)) and the vector gauge fields, manifesting nonzero quantum numbers t4, t23, Q ,Y, t4, t23, Q ,Y,NR (Eqs. (9.8,9.9, 9.11,9.12, 9.13,9.14)) massive. vi.: At the electroweak break the scalar fields with the space index s = (7,8), triplets with respect to the family index (originating in d>abs, Eq. (9.16)) and the three singlets carrying the charges (Q, Q', Y') (originating in dts's, Eq. (9.15)), all with the weak and the hyper charge equal to (^ 2, ± 1, respectively), get nonzero vacuum expectation values, changing also their masses and breaking the weak and the hyper charge symmetry. These scalars determine mass matrices of twice the four families, as well as the masses of the weak bosons. All the rest scalar fields keep masses of the condensate scale and are correspondingly unobservable in the low energy regime 5. The fourth family to the observed three ones will (sooner or later) be observed at the LHC. Its properties are under the consideration [10], while the stable of the upper four families is the candidate for the dark matter constituents. The above assumptions enable that the starting action (Eq. (9.1)) manifests effectively in d = (3 + 1) in the low energy regime fermion and boson fields as assumed by the standard model. To see this [3,1,4-8,2,9,10,12,14], let us formally rewrite the Lagrange density for a Weyl spinor of (Eq.(9.1)), as follows Lf = (pm - X 9ATAiAmi)^ + Ai {Y_ ^YsPos + s = 7,8 { Y_ ^YVot , t=5,6,9,...,14 P0s = Ps — s ds's"s — abdabs , P0t = Pt — 2stVdt't"t — abdabt , (9.2) where m e (0,1,2,3), s e 7,8, (s', s") e (5,6,7,8), (a,b) (appearing in Sab) run within e (0,1,2,3) and e (5,6,7,8), t e (5,6,9,..., 13,14), (t',t") e (5,6,7,8) and e (9,10,..., 14). ^ represents all family members of all the families. The generators of the charge groups TAi (expressed in Eqs. (9.3), (9.9), (9.11) in terms of Sab) fulfil the commutation relations „-Ai V" „Ai cab T = 2_ c ab S , a,b {TAi, TBj}_ = i6ABfAijkTAk . (9.3) 5 Correspondingly d = (13 + 1) manifests in d = (3 + 1) spins and charges as if there would be d = (9 + 1), since the plane (5, 6) and the plane in which the vector t4 lies, are unobservable at low energies. 130 N.S. Mankoč Borštnik The spin generators are defined in Eq. (9.8). These group generators determine all the internal degrees of freedom of one family members as seen from the point of view of d = (3 + 1): The colour charge (SU(3) with the generators t3) and the "spinor charge" (U(1 )n) with the generator t4 originating in SO(6), the weak charge SU(2)i with the generators t1 and the second SU(2)n charge with the generators t2 originating in SO(4) (SU(2)II breaks in the presence of the condensate into U(1 )l7 defining together with t4 the hyper charge Y (= t23 + t4) and the spin determined by SO(3,1). The condensate of two right handed neutrinos with the family quantum numbers of the upper four families bring masses (of the unifying scale > 1016 GeV or above) to all the scalar and those vector gauge fields which are not observed at so far measurable energies. The scalar fields causing, when getting nonzero vacuum expectation values, the electroweak phase transitions changing at the transition also their own masses, bring masses to the eight families and to the weak bosons. We shall comment all these fields in what follows. The first line of Eq. (9.2) describes [1,3] before the electroweak break the dynamics of eight families of massless fermions in interaction with the massless colour Am, weak Am and hyper A^ (= sin A^5 + cos A^J gauge fields, all are the superposition of cabm 6. The second line of the same equation (Eq. (9.2)) determines the mass term, which after the electroweak break brings masses to all the family members of the eight families and to the weak bosons. The scalar fields responsible - after getting nonzero vacuum expectation values - for masses of the family members and of the weak bosons are namely included in the second line of Eq. (9.2) as (— 2Ss s"cs <-■ - — Js's s 1S a b cD a b s, s =€ (7,8), (s ',s") € (5,6,7,8), (a ,1b) G (0 ,T ,...,8)) 7. The properties of these scalar fields are discussed in sect. (9.4), where the proof is presented that they all carry the weak charge and the hyper charge as the standard model Higgs's scalar, while they are either triplets with respect to the family quantum numbers or singlets with respect to the charges Q, Q' and Y'. While the two triplets (A], AJN1 L) interact with the lower four families, interact (/A:?, A J"1 R) with the upper four families. These twice two triplets are superposition of ] Sab cDabs, s =€ (7,8), Eq. (9.16). The three singlets (AQ, AQ and Am') are superposition of cds 's"s, Eq.(9.15). They interact with the family members of all the families, "seeing" family members charges. The third line of Eq. (9.2) represents fermions in interaction with all the rest scalar fields. Scalar fields become massive after interacting with the condensate. Those which do not gain nonzero vacuum expectation values, keep the heavy masses of the order of the scale of the condensate up to low energies. The massive 6 These superposition can easily be found by using Eqs. (9.11, 9.9). They are explicitly written in the ref. [3]. The interaction with the condensate makes the fields Am, Eq. (9.14), Am and Am2 very massive (at the scale of the condensate). 7 To point out that Sab and S ab belong to two different kinds of the Clifford algebra objects are the indices (a, b) are in Sab in this paragraph written as (a, lb). Normally only (a, b) will be used for S ab and ¡5ab. 9 Can Spin-charge-family Theory Explain Baryon Number Non-conservation? 131 scalars with the space index t e (5,6) transform (table 9.3) uR-quarks into dL-quarks and vR-leptons into eL-leptons and back, as well as UR-antiquarks into dL-antiquarks and back and VR-antileptons into eL-antileptons and back, breaking in the presence of the condensate the Q global symmetry. Those scalar fields with the space index t = (9,10, ••• ,14) transform antileptons into quarks and antiquarks into quarks and back. They are offering in the presence of the scalar condensate breaking the CP symmetry the explanation for the observed matterantimatter asymmetry, as we shall show in sect. 9.2. Let us write down the part of the fermion action which in the presence of the condensate offers the explanation for the observed matter/antimatter asymmetry. Lf = V Y0 y' { X [Pt-( 2SS V D 'st + 1st " c/ft t=(9,10,...14) + 1sab cDabt)]} (9.4) where (s',s") e (5,6,7,8), (t,t',t") e (9,10,..., 14) and (a,b) e (0,1,2,3) and e (5,6,7,8), in agreement with the assumed breaks in sect. (9.1). Again operators Sab determine family quantum numbers and Sab determine family members quantum numbers. Correspondingly the superposition of the scalar fields cDabt and the superposition of the scalar fields cabt carry the quantum numbers determined by either the superposition of Sab or by the superposition Sab in the adjoint representations, while they carry the colour charge, determined by the space index t e (9,10,..., 14), in the triplet representation of the SU(3) group, as we shall see. Similarly the scalars with the space index s e (7,8) carry the weak and the hyper charge in the doublets representations. The condensate of two right handed neutrinos with the family quantum numbers of the upper four families carries (table 9.2) t4 = 1, t23 = — 1, t13 = 0, Y = 0, Q = 0, and the family quantum numbers of the upper four families and gives masses to scalar and vector gauge fields with the nonzero corresponding quantum numbers. The only vector gauge fields which stay massless up to the electroweak break are the hyper charge field (A^), the weak charge field (Am) and the colour charge field (Am). The standard model subgroups of the SO (13 + 1) and SO (13 + 1) groups and the corresponding gauge fields This section follows to large extend the refs. cite-JMP,NscalarsweakY2014. To calculate quantum numbers of one Weyl representation presented in table 9.3 in terms of the generators of the standard model charge groups tai (= ^ ab cAlab Sab) one must look for the coefficients cAlab (Eq. (9.3)). Similarly also the spin and the family degrees of freedom can be expressed. The same coefficients cAlab determine operators which apply on spinors and on vectors. The difference among the three kinds of operators - vector and two kinds of spinor - lies in the difference among Sab, Sab and Sab. While Sab for spins of spinors is equal to Sab = 7(Ya Yb - Yb Ya), (Ya ,Yb}+ = 2^ab , (9.5) 132 N.S. Mankoč Borštnik and Sab for families of spinors is equal to Sab = 4(YaYb - YbYa), {Ya ,Yb}+ = 2nab, {Ya ,?b}+ = 0, (9.6) one must take, when Sab apply on the spin connections ^bde (= fae ^bda) and •bde (= fae •bda), on either the space index e or the indices (b, d, 13, d), the operator (S ab)ceAd'.'e'.,g = i(nacsb - nbc6a) Ad...e...g. (9.7) This means that the space index (e) of ^bde transforms according to the requirement of Eq. (9.7), and so do b, d and 13, d. Here I used again the notation 13, d to point out that Sab and Sab (= Sa ) are the generators of two independent groups[14]. One finds [1,3-8,2] for the generators of the spin and the charge groups, which are the subgroups of SO(13,1), the expressions: N±(= N(L,R)):= 2(S23 ± iS01,S31 ± iS02,S12 ± iS03), (9.8) where the generators N± determine representations of the two SU(2) invariant subgroups of SO(3,1), the generators T1 and T2, t1 : = 1 (S58 - S67, S57 + S68, S56 - S78), (9.9) t2 : = 1 (S58 + S67, S57 - S68, S56 + S78), (9.10) determine representations of the SU(2)jX SU(2)n invariant subgroups of the group SO(4), which is further the subgroup of SO(7,1) (SO(4), SO(3,1) are subgroups of SO(7,1)), and the generators T3, t4 and T4 t3 := 1 {S912 - S1011 ,S911 + S1012, S910 - S11 12, S9 14 - S10 13 S913 + s10 14 Shm - §12 13 S11 13 + S1214, -L(S910 + S11 12 - 2S1314)}, V3 t4 := -1 (S910 + S11 12 + S1314), T4 := -3(S910 + S1112 + S13 14), (9.11) determine representations of SU(3) x U(1), originating in SO (6), and of T4 originating in SO(6). One correspondingly finds the generators of the subgroups of SO(7,1), Nl,r : = 2(S23 ± iS01, S31 ± iS02, S12 ± iS03), (9.12) 9 Can Spin-charge-family Theory Explain Baryon Number Non-conservation? 133 which determine representations of the two SU(2) invariant subgroups of SO (3,1 ), while T : = 1 (S58 - S67, S57 + S68, S56 - S78), T2 : = 2(S58 + S67, S57 - S68, S56 + S78), (9.13) determine representations of SU(2)ix SU(2)n of SO(4). Both, SO(3,1) and SO(4), are the subgroups of SO(7,1 ). One further finds [3] Y = t4 + t23 , Y' = —t4 tan2 £2 + t23 , Q = t13 + Y, Q' =-Ytan2^ + t13 , Y = t4 + t23 , Y' = -t4 tan2 $2 + t23 , Q = Y + t13 , Q ' =-Ytan2 d 1 + t13 . (9.14) The scalar fields, responsible [1-3] - after getting in the electroweak break nonzero vacuum expectation values - for the masses of the family members and of the weak bosons, and presented in the second line of Eq. (9.2), can be expressed in terms of ^abc fields and &abc fields as presented in Eq. (9.15), 9.16). One can find the below expressions by taking into account Eqs. (9.9,9.11,9.12, 9.13) and Eq. (9.14). - 1ss 's" Vs = -(g23 t23 A23 + g13 t13 A]3 + g4 t4 a4) , g13 t13 A]3 + g23 t23 A23 + g4 t4 A4 = gQ QAQ + gQ' Q'aq' + gY' Y' Aj', A4 = -(^9 10 s + ^11 12 s + ^13 14 s) , A]3 = (^56s - ^78s) , A;:3 = (^56s + ^78s) , aq = sin^ a]3 + cos^ Aj , aq ' = cos^ a]3 - sin^ Aj , Aj = sin A23 + cos A4 , Aj' = cos A23 - sin A4 , (s G (7,8)). (9.15) In Eq. (9.15) the coupling constants were explicitly written to see the analogy with the gauge fields in the standard model. - 2sab &abs = -(T1 A ] + N L A + T2 A 2 + N R A ), A1 = (%8s - %7s + &68s, %6s - %8s) , AN = (^23s + ^31s + i&02s, %2s + i^03s) , A2 = (%8s + %7s - &68s, %6s + %8s) , AN = (^23s - ^31s - i&02s, %2s - i^03s) , (s G (7,8)). (9.16) 134 N.S. Mankoč Borštnik Scalar fields from Eq. (9.16) couple to the family quantum numbers, while those from Eq. (9.15) distinguish among family members. Expressions for the vector gauge fields in terms of the spin connection fields and the vielbeins, which correspond to the colour charge (Am), the SU(2)n charge (Amm), the weak charge (SU(2)i) (Am) and the U(1 )n charge originating in SO(6) (Am), can be found by taking into account Eqs. (9.9, 9.11). Equivalently one finds the vector gauge fields in the "tilde" sector. One really can use just the expressions from Eqs. (9.15,9.16), if replacing the scalar index s with the vector index m. 9.2 Properties of scalar and vector gauge fields, contributing to transitions of antileptons into quarks In this - the main - part of the present paper the properties, quantum numbers and discrete symmetries of those scalar and vector gauge fields appearing in the action (Eqs.(9.1,9.2), 9.4) of the spin-charge-family theory [1-9,12] are studied, which cause transitions of antileptons into quarks and back, and antiquarks into quarks and back. These scalar gauge fields carry the triplet or antitriplet colour charge (see table 9.1) and the fractional hyper and electromagnetic charge. The Lagrange densities from Eqs. (9.1, 9.2, 9.4) manifest Cn • pn invariance (appendix (9.8)). All the vector and the spinor gauge fields are before the appearance of the condensate massless and reactions creating particles from antiparticles and back goes in both directions equivalently Correspondingly there is no matterantimatter asymmetry. The spin-charge-family theory breaks the matter-antimatter symmetry by the appearance of the condensate (sect. 9.3) and also by nonzero vacuum expectation values of the scalar fields causing the electroweak phase transition (sect. 9.4). I shall show that there is the condensate of two right handed neutrinos which breaks this symmetry, giving masses to all the scalar gauge fields and to all those vector gauge fields which would be in contradiction with the observations. Let us start by analysing the Lagrange density presented in Eq. (9.4) before the appearance of the condensate. The term Yt 2 Ss s" ws s"t in Eq. (9.4) can be rewritten, if taking into account Eq. (9.42), as follows yt 's" Ws vt = XL (©) > 's" w tt, , 2 f-fc) 2 s"s"(e) w tt' : = w tt' = (Ws 's"t T /S"t'), s"s"(© ) s"s"(±) tt' tt' 1 :=(±)= 2 (Yt ± Yt ), (tt') e ((910), (11 12), (1314)). (9.17) tt ' I introduced the notations (©) and w tt' to distinguish among different super- s"s"(© ) position of states in equations below. 9 Can Spin-charge-family Theory Explain Baryon Number Non-conservation? 135 tt' ; Using Eqs. (9.9, 9.11) the expression (©) i Ss s" ^ tt' can be further s"s" (©) rewritten as follows «' 1 , . -Ss s w s"s" (®) 2- ~ tt' tt' (@) {t2+ A2+ + T2- A2- + T23 A^, + t1+ A1+, + T1- A1- + T13 a!3, } , J L tt ' tt ' tt ' 1 tt ' tt ' tt ' J * (® a20 tt , (® A2t, (® a'® tt , (® A't3, (® (®) (®) (®) (©) (®) ) Ttt, - w tt,), ^ 57(®) 68(®) = (d tt' + W tt ' 58(®) 67(®) = (d tt' + W tt ' 56(®) 78 (®) = d tt' - W tt ' ) 58(®) 67(®) = (d tt' - W tt ' 56(®) 78 (®) ) , ) T tt' + w tt,) , ^ 57(®) 68(®) ). (9.18) Equivalently one expresses the term Yt 2 Sab wabt in Eq. (9.4), by using Eqs. (9.12, 9.13), as 1 tt' i Yt ,Sab ddabt = (@) ,Sab dd tt, = 2 2 ab (®) tt' (ffi) { t2+ A2+ + T2- A2- + T23 A2t' + t' + A'+ + T1- A+ T13 A13' + L tt' tt ' tt' tt' tt' tt' (®) (®) (®) (®) (®) (®) NN + Anr+ + NN- Anr- + NNR Anr3 + NN + Anl,+ + NN- Anl,- + NNL Anl,3}, R tt ' R tt ' R tt ' L tt ' L tt ' L tt ' J > (®) (®) (®) (®) (±) (®) - nRs □ Att, =(dd tt' -idd tt') Ti(w tt, -iw tt,), t®) 2 3 (®) 01(®) ^ 31(®) 02(®) /ANtR,3 = (id tt' - id tt'), t®) 12 (®) 03 (®) ANtL,± = (w tt' + i id tt') T i(w tt' + i id tt'), t®) 2 3 (®) 01 (®) ^ 31 (®) 02(®) /ANtR,3 = (d tt' + idd tt'), (9.19) (® ) 12 (®) 03 (® ) with aA , aA"tt', A and aA'3, following from expressions for A , a23,, A (t®®t ') (t®®t ') (t®®t ') (t®®t ') (t®®t ') (t®®t ') (t®®t ') and A13,, respečtively, in (Eq.(9.18)), if replačing w tt, by w tt,. (te®) s"s"(®) s"s" (®) There is the additional term in Eq. (9.4): Yt 2 St t" dt t"t. This term čan be written with respečt to the generators St t" as one čolour očtet sčalar field and one U(1)n sčalar field (Eq. 9.11) Yt 1 Stt'" dt"t '"t = XL (©) {T3 • A 3tt' + t4 • A4tt, }, 2 +,- (tt') (®) (®) (tt') G ((910),11 12),1314)). (9.20) 136 N.S. Mankoč Borštnik Taking all above equations (9.17,9.18, 9.19, 9.20) into account Eq. (9.4) can be rewritten, if we leave out p tt / since in the low energy limit the momentum does c®) not play any role, as follows ++' Lf" = V Y0(-) X +>- (tt') [ t2+ A2+' + t2- A2-' + t23 A2tt' tt ' tt ' tt ' c®) (®) c®) + t1+ A1+' + t1-A1-' + t13 A13' tt' tt' tt' c®) c®) c®) + t2+ A 2+' + t2- A2-' + t23 A23' tt' tt' tt' c®) c®) c®) + t1 + A1+' + t1-A1- + t13 A13' tt' tt' tt' c®) c®) c®) + N + Anr+ + N- Anr- + NR Anr3 R tt ' R tt ' R tt ' c®) c®) c®) + N + Anl'+ + N- Anl'- + N3 Anl'3 L tt' L tt ' 3 tt' c®) c®) c®) + T3i A3ti' + t4 A4tt' ]}(9.21) ct®®t ) ct®®t ) where (t,t') run in pairs over [(9,10),... (13,14)] and the summation must go over + and — of tt'. ct®®t ) Let us calculate now quantum numbers of the scalar and vector gauge fields appearing in Eq. (9.21) by taking into account that the spin of gauge fields is determined according to Eq. (9.7) ((Sab)cdAd."e".9 = i(nacSb — nbc^a) Ad."e".9, for each index (e (d... g)) of a bosonic field Ad".g separately). We must take into account also the relation among S ab and the charges (the relations are, of course, the same for bosons and fermions) (Eqs. (9.8, 9.9, 9.11)). On table 9.1 properties of the scalar gauge fields appearing in Eq. (9.21) are presented. The scalar fields with the scalar index s = (9,10, • • • ,14), presented in table 9.1, carry one of the triplet colour charges and the "spinor" charge equal to twice the quark "spinor" charge, or the antitriplet colour charges and the anti "spinor" charge. They carry in addition the quantum numbers of the adjoint representations originating in Sab or in Sab. Although carrying the colour charge in one of the triplet or antitriplet states, these fields can not be interpreted as superpartners of the quarks as required by, let say, the N = 1 supersymmetry. The hyper charges and the electromagnetic charges are namely not those required by the supersymmetric partners to the family members. Let us have a look what do the scalar fields, appearing in Eq. (9.21) and in table 9.1, do when being applied on the left handed members of the Weyl representation presented on table 9.3, containing quarks and leptons and antiquarks and antileptons [4,21,15]. Let us choose the 57th line of table 9.3, which represents in the spinor technique the left handed positron, e+. If we make, let say, the choice 910 of the term (y0 (+) t2B ) A^ (the scalar field A^ is presented in the 7th line in c®) c®) table 9.1 and in the second line of Eq. (9.21)), the family quantum numbers will 9 Can Spin-charge-family Theory Explain Baryon Number Non-conservation? 137 field prop. T4 t'3 T23 (t33,t38 ) Y Q T 4 T '3 T 23 N L N R A 1 H A9 10 (® ) A 13 A9 1 0 (© ) scalar scalar a 3 a 3 H 1 0 0 0 (a H 2VJ) (a H 213) © 3 © 3 © 3+©1 a 3 0 0 0 0 0 0 0 0 0 0 A'H A11 12 (© ) A 13 A 11 12 (© ) scalar scalar a 3 a 3 11 0 0 0 (a H 213) (a H 213) © 3 © 3 © 3+©1 a 3 0 0 0 0 0 0 0 0 0 0 A'H A 13 14 (© ) A 13 A 1 3 14 (© ) scalar scalar a 3 a 3 11 0 0 0 (0, a13) (0, a13) © 3 © 3 © 3+©1 a 3 0 0 0 0 0 0 0 0 0 0 ?H A9 10 (® ) A23 A 9 1 0 (® ) scalar scalar a 3 a 3 0 0 H1 0 (a H 213) (a H ) © 3+©1 © 3 © 3+©1 a 3 0 0 0 0 0 0 0 0 0 0 A 1± A 910 (© ) A 13 A 9 10 (© ) scalar scalar a 3 a 3 0 0 0 0 (a H 2ÜJ) (a H ) © 3 © 3 a 3 a 3 0 0 H1 0 0 0 0 0 0 0 2 H A Q A 910 () scalar a 3 0 0 (a 2, H ) © 3 a 3 0 0 H1 0 0 A 2a) A 910 (© ) scalar a 3 0 0 (a 2, H ) © 3 a 3 0 0 0 0 0 A nlH A 910 (H ) A NL3 A 910 (H) scalar a 3 0 0 (a H 213) © 3 a 3 0 0 0 H1 0 scalar a 3 0 0 (a H ^) © 3 a 3 0 0 0 0 0 A nr± A 910 (® ) A NR3 A 910 (© ) scalar a 3 0 0 (a H 213) © 3 a 3 0 0 0 0 H1 scalar a 3 0 0 (a H ^) © 3 a 3 0 0 0 0 0 a31 A 9 1 0 (a ) scalar a 3 0 0 (H 1+ H 2, H 213 ) © 3 a 3 0 0 0 0 0 A 4 A 9 1 0 (a ) scalar a 3 0 0 (a I, a ) (a 2 , H 213 ) © 3 a 3 0 0 0 0 0 A 3 A m vector 0 0 0 octet 0 0 0 0 0 0 0 A4 A m vector 0 0 0 0 0 0 0 0 0 0 0 Table 9.1. Quantum numbers of the scalar gauge fields carrying the space index t = (9,10, • • • ,14), appearing in Eq. (9.21), are presented. To the colour charge of all these scalar fields the space degrees of freedom contribute one of the triplets values. These scalars are with respect to the two SU(2) charges, (T1 and T2), and the two SU(2) charges, (T1 and T2), triplets (that is in the adjoint representations of the corresponding groups), and they all carry twice the "spinor" number (t4) of the quarks. The quantum numbers of the two vector gauge fields, the colour and the U(1 )ii ones, are added. 138 N.S. Mankoč Borštnik not be affected and they can be any. The state carries the "spinor" (lepton) number t4 = 1, the weak charge t13 = 0, the second SU(2)n charge t23 = 1 and the colour charge (t33, t38) = (0,0). Correspondingly, its hyper charge (Y(= t4 + t23)) is 1 and the electromagnetic charge Q(= Y + t13) is 1. 910 So, what does the term y0 (+) t2B A2B0 make on this spinor? Making use (e) 910 of Eqs. (9.44, 9.46, 9.54) of appendix 9.9 one easily finds that operator y° (+) t2- 03 12 56 78 9 10 11 12 13 14 transforms the left handed positron into (+i) (+) | [-] [-] || (+) (-) (-), which 03 03 910 is dR1, presented on line 3 of table 9.3. Namely, y0 transforms [-i] into (+i), (+) 9 10 9 10 56 78 56 78 56 78 transforms [-] into (+), while t2- (= - (-) (-)) transforms (+) (+) into [-] [-]. The state dR1 carries the "spinor" (quark) number t4 = 1, the weak charge t13 = 0, the second SU(2)n charge t23 = -§ and the colour charge (t33,t38) = (§, §73). Correspondingly its hyper charge is (Y = t4 + t23 =) -1 and the electromagnetic charge (Q = Y + t13 =) -3. The scalar field A2B0 carries just the needed quantum (e) numbers as we can see in the 7th line of table 9.1. If the antiquark uR2, from the line 43 (it is not presented, but one can very easily construct it) in table 9.3, with the "spinor" charge t4 = -6, the weak charge t13 = 0, the second SU(2)n charge t23 = -1, the colour charge (t33,t38) = (1, -273), the hyper charge Y(= t4 + t23 =) -§ and the electromagnetic charge Q( = Y + t13 =) -§ submits the A§B0 scalar field, it transforms into uR3 from 3 (e) the line 17 of table 9.3, carrying the quantum numbers t4 = 1, t13 = 0, t23 = 1, (t33, t38) = (0, -73), Y = § and Q = § . These two quarks, dR1 and uR3 can bind together with uR2 from the 9th line of the same table (at low enough energy, after the electroweak transition, and if they belong in a superposition with the left handed partners to the first family) into the colour chargeless baryon - a proton. This transition is presented in figure 9.1. The opposite transition at low energies would make the proton decay. Let us look at one more example. The 63th line of table 9.3 represents in the spinor technique the right handed positron, e+. Since we shall again not have a look on a transition, in which scalar fields with the nonzero family quantum numbers are involved, the family quantum number of this positron is not important. The state carries the "spinor" (lepton) number t4 = 1, the weak charge t13 = §, the second SU(2)n charge t23 = 0 and the colour charge (t33, t38) = (0,0). Correspondingly, its hyper charge (Y = t4 + t23) is § and the electromagnetic charge Q = Y + t13 is 1. 910 What does, let say, the term y0 (+) t1b A1b0 (the scalar field A1B0 is pre- (e) (e0 sented in the first line of table 9.1) make on e+? Making use of Eqs. (9.44, 9.46, 9.54) of appendix 9.9 one easily finds that the right handed positron transforms 910 03 12 56 78 9 10 1 1 12 13 14 under the application of y0 t1- (+) into [-i] (+) | [-] (+) || (+) (-) (-), which 03 03 910 is dR1 presented on line 5 of table 9.3. Namely, y0 transforms (+i) into [-i], (+) 9 Can Spin-charge-family Theory Explain Baryon Number Non-conservation? 139 t 4 =1 ,t 13=0,r 23 = (t 23,t 38) = (0,0) y=1,q=1 t4 = 1 T13 = ' 6 0,t 23 = - 2 (t 33,t 38) = ( ) Y=-3 ,Q=-3 dR A2B L a9 10, ( + ) t 4=2x(-6 ),t 13=0,t 23=-1 (t 33 ,-38)=( 2, 2- ) Y=-3 ,q=-3 c2 Ul t 4=-1 ,t13=0,t 23=-2 (t 33,t 38) = ( 22- ) Y=-3,Q=-2 U t 4 = 1 ,t 13=0,t 23 = § (t 33 ,t 38) = (0,--1= ) Y = 6 ,Q= 3 L c2 c2 UR -*- UR T4 = 1 T 13 = 0 T23 = 1 ' 6 '' 0'' 2 (T 33,T 38) = (- 2 , Y=§ >Q= 3 Fig. 9.1. The birth of a proton out of an positron e+, antiquark uf2 and quark (spectator) uR2. The family quantum number can be any. 9 10 9 10 56 78 56 56 56 56 transforms [-] into (+), while t1B (=(-) (+)) transforms (+) [-] into [-] (+). The state dL1 carries the "spinor" (quark) number t4 = 1, the weak charge t13 = -1, the second SU(2)n charge t23 = 0 and the colour charge (t33,t38) = (1, 2773). Correspondingly its hyper charge is (Y = t4 + t23 =) 6 and the electromagnetic charge (Q = Y + t13 =) -1. The scalar field AjB0 carries all the needed quantum numbers, as one can see in figure 9.1. If the antiquark uR2, from the line 47 in table 9.3 (the reader can easily find 03 12 56 78 9 10 11 12 13 14 the expression (+i) (+) | [-] (+) || (+) (-) [+] ), with the "spinor" charge t4 = -1, the weak charge t13 = -J, the second SU(2)n charge t23 = 0, the colour charge (t33, t38) = (1, - j^j), the hypercharge (Y = t4 + t23 =) -1 and the electromagnetic charge (Q = Y + t13 =) - J, submits the A1B0 scalar field, (®) 03 12 56 78 9 10 11 12 it transforms into uR3 from the line 23 of table 9.3 ([-i] (+) | (+) [-] || [-] (-) 13 14 [+] ), carrying the quantum numbers t4 = 1, t13 = J, t23 = 0, (t33,t38) = (0, -73), Y = 6 and Q = J. These two quarks, dR1 and uR3, can bind (at low enough energy, when making after the electroweak transition the superposition with the right handed partners) together with uR2 from the 15th line of the same 140 N.S. Mankoč Borštnik table, into the colour chargeless baryon - a proton. This transition is presented in figure 9.2. The opposite transition would make the proton decay. ~2' 2 _33 _38 ) = (0,0) Y=2 ,Q=1 T4 = 6 ,T 13 = - 2 ,T23=0 (T 33,T 38) = ( 2 ^ ) Y=1 ,Q=—3 5+ -R df 1B 9 10 (+) t 4 = 2x( —3 ),T13 = — 1 ,T 23=0 (T 33 _38\_ / 1 ) = ( 2 , 213 ) Y=—3 ,Q=—3 u T 4 = — 1 T13 = — 1 t23=0 i 3 ,i 2 ,' 0 (T33 T38) — ( 1 1 ) (T ,T )=( 2 ,—273 ) Y=—1 Q=—2 1 = 3 ,Q= 3 u c3 (T 33,t 38) = (0, — ^ ) Y = 6 ,Q= 2 L T 4= 1 ,T13 1 T 23 3 2 c2 c2 Ut -►- Uc/ T 4— 1 T13= 1 T 23 = 1 = 3 = 2 = 33 38 1 1 0 5) = (-1 ) )=( 2,2V3) Y —1 Q= 2 Y — 6 ,Q— 3 Fig. 9.2. The birth of a proton out of a positron e+, antiquark uR2 and quark (spectator) uL2 The family quantum number can be any. Similar transitions go also with other scalars from Eq. (9.21) and table 9.1. The A1, t., A2, t., ANand Afields cause transitions among the family members, (+) (+) (+) (+) changing a particular member into the antimember of another colour and of 910 N - another family. The term y0 (+) N- A91R0 transforms e + into uf1, changing the c®) family quantum numbers. The action from Eqs. (9.1,9.2,9.4) manifests Cn •Pn invariance. All the vector and the spinor gauge fields are massless. Since no one of the scalar fields from table 9.1 have been observed and also no vector gauge fields like Am, A^ and other scalar and vector fields, we shall discuss this topic in sect. 9.5, it must exist a mechanism, which makes the non observed scalar and vector gauge fields massive enough. Scalar fields from table 9.1 carry the colour and the electromagnetic charge. Therefore their nonzero vacuum expectation values would not be in agreement with the observed phenomena. One, however, notices that all the scalar gauge 9 Can Spin-charge-family Theory Explain Baryon Number Non-conservation? 141 fields from table 9.1 and several other scalar and vector gauge fields (see sect. 9.5) couple to the condensate with the nonzero quantum number t4 and t23 and nonzero family quantum numbers. It is not difficult to recognize that the desired condensate must have spin zero, Y = t4+ t23 = 0, Q = Y + t13 = 0 and t1 = 0 in order that in the low energy limit the spin-charge-family theory would manifest effectively as the standard model. I make a choice of the two right handed neutrinos of the VIIIth family coupled into a scalar, with t4 = -1, t23 = 1, correspondingly Y = 0, Q = 0 and t1 = 0, and with family quantum numbers (Eqs. (9.13, 9.12)) T4 = — 1, t23 = 1, NR = 1, and correspondingly with Y = T4 + T23 = 0, Q = Y + T13 = 0, and T = 0. The condensate carries the family quantum numbers of the upper four families. The condensate made out of spinors couples to spinors differently than to antispinors - "anticondensate" would namely carry t4 = 1, and t23 = — 1 - breaking correspondingly the Cn• Pn symmetry: The reactions creating particles from antiparticles are not any longer symmetric to those creating antiparticle from particles. Such a condensate leaves the hyper field A^ (= sin Am3 + cos A^J (for the choice that sin$2 = cos$2 and g4 = g2, there is no justification for such a choice, Am = (Am + Am)) massless, while it gives masses to A^^ and Am (= (Am — Amm3) for sin = cos $2) and it gives masses also to all the scalar gauge fields from table 9.1, since they all couple to the condensate through t4. The weak vector gauge fields, A m, the hyper charge vector gauge fields, Am, and the colour vector gauge fields, A m, stay massless. The scalar fields with the scalar space index s = (7,8) - those which couple to all eight families, those which couple only to the upper and those which couple only to the lower four families - carrying the weak and the hyper charges of the Higgs's scalar - wait for getting nonzero vacuum expectation values to change their masses while causing the electroweak break. The condensate does what is needed so that in the low energy regime the spin-charge-family manifests as an effective theory which agrees with the standard model to the extend that it is in agreement with the observed phenomena, explaining the standard model assumptions and predicting new fermion and boson fields. It also may hopefully explain also the observed matter-antimatter asymmetry if the conditions in the expanding universe would be appropriate (9.6). The work needed to check these conditions in the expanding universe within the spin-charge-family theory is very demanding. Although we do have some experience with following the history of the expanding universe [12], this study needs much more efforts, not only in the calculations, but also in understanding the mechanism of appearing the condensate, relations among the velocity of the expansion, the temperature and the dimension of space-time in the period of the appearance of the condensate. This study has not yet been really started. 9.3 Properties of the condensate In table 9.2 the properties of the condensate of the two right handed neutrinos (|vVI11 >-| |vVI11 >2 ), one with spin up and another with spin down (table 9.3, line 142 N.S. Mankoč Borštnik 25 and 26), carrying the family quantum numbers of the VIIIth family (table 9.4 ), are presented. The condensate carries the quantum numbers of SU(2)n, t23 = 1 (Eq. (9.9)), of U(1)n originating in SO(6), t4 = —1 (Eq.9.11), correspondingly Y = 0, Q = 0, and the family quantum numbers (table 9.4) T4 = —1 (Eq. (9.11)), T23 = 1 (Eq. (9.13)), and NR = 1 (Eq. (9.12)). Each of the two neutrinos could belong to a different family of the upper four families. In this case the family quantum numbers of the condensate change. The condensate is presented in the first line of table 9.2 as a member of a triplet of the group SU(2)n with the generators t2\ Correspondingly the condensate couples to all the vector gauge fields which carry nonzero t2t, t4, T2t, NR and T4. The fields A^, Am and A m stay massless. The condensate couples also to all the scalar gauge fields with the scalar indices s G (5,6,7,8,9,..., 14), since they all carry nonzero either t4 or t23. state S03 si2 t13 t23 t4 Y Q T13 T23 T4 Y Q N L N R (|v1R >1 |v2R >2) 0 0 0 1 —1 0 0 0 1 —1 0 0 0 1 (|v1R >1 |e2R >2) r^VIII . „VIII . (|e1R >1 |e2R >2) 0 0 0 0 —1 —1 —1 0 0 0 —1 —1 —2 —2 0 1 —1 0 0 0 1 0 1 —1 0 0 0 1 Table 9.2. The condensate of the two right handed neutrinos vR, with the VIIItH family quantum number, coupled to spin zero and belonging to a triplet with respect to the generators T2i, is presented, together with its two partners. The condensate carries T1 = 0, t23 = 1, t4 = -1 and Q = 0 = Y. The triplet carries T4 = -1, T23 = 1 and NR = 1, NL = 0, Y = 0, Q = 0. The family quantum numbers are presented in table 9.4. Coupling of the scalar gauge fields to the condensate is proportional to tt' tt' (1 IvVR11 >) c®) c®) tt > tt' > tt (© ) (© ) (© ) (© ) A 3 (© ) .2A A (© ) , aa23 ', .1 (© ) A , , A tt ' ' (© ) 13 tt' (© ) Anlq anl3 Anr± Anr3, tt' ? tt' ? tt' ? tt' (© ) (©) (© ) (© ) (tt') G [(910), (11 12), (1314)], A 2B 1 23 AY , = ^ (A2 ss ' ' ss P) ^ ss (© ) (© ) V2 (© ) (ss') G [(56), (78)], A2@ AY' _ 1 (a23 a4 ) Am , Am _ (Am — Am) , A2 A4 A nr Am , Am , Am , m G (0,1,2,3). A4 ' ) ss ' (© ) (9.23) In expression for A^ s $2 = f is chosen, just for simplicity, with no justification so far. It stays as an open question what does make the right handed neutrinos to form such a condensate in the history of the universe. Since A^1, s G (5,6) couple to the condensate and get masses, while (by assumption) they do not get nonzero vacuum expectation values during the electroweak break (what changes the masses of the scalar fields A^1, s G (7,8)) the restriction in the sum in Eq. (9.2) is justified. The scalar fields, causing the birth of baryons, have the triplet colour charges. They resemble the supersymmetric partners of the quarks, but since they do not carry all the quantum numbers of the quarks, they are not. 9.4 Properties of scalar fields which determine mass matrices of fermions This section is a short overview of the ref. [14]. There are two kinds of the scalar gauge fields, which gain at the electroweak break nonzero vacuum expectation values and determine correspondingly the masses of the families of quarks and leptons and to the masses of gauge weak bosons: The kind originating in d>a5s and the kind originating in s, ^56s and d78s, both kinds have the space index s = (7,8) and both carry the weak and the hyper charge as the Higgs's scalar. These scalar fields are presented in the Lagrange density for fermions (Eq. (9.2)) in the second line. The "tilde" kind influences the family quantum numbers of fermions, the "Dirac" kind influences the family members quantum numbers. 144 N.S. Mankoč Borštnik The two triplets (A], AJ^1- ) influence the lower four families (the lowest three already observed), while (AAN) influence the upper four families, the stable of which constitute the dark matter. Recognizing that f1 A] +N L A+T2 A2 +NR AN = ] Sab cDabs, s = (7,8), one easily finds, taking into account Eqs. (9.12, 9.13), the expressions A1 = (cD58s - cD67s, CD57s + CD68s, CD56s - CD78s) , AN = (cD 23s + i CD ois, CD 31s + iCD02s,CD 12s + iCD03s) , /A-2 = (cD58s + CD67s, CD57s - CD68s, CD56s + CD78s) , A nr = (cD 23s - iDD ois, DD 31s - i CD 02s,CD 12s - i CD 03s) , s = (7,8), (9.24) presented already in Eq. (9.16). Similarly one finds, taking into account Eqs. (9.8, 9.9, 9.11, 9.14), the expressions for AQ, Aj and Aj', presented in Eqs. (9.15). The scalar fields AQ, Aj and Aj' distinguish among the family members, coupling to the family members quantum numbers through Q (= t13 + Y), Y (= t23 + t4) and Y' = t23 - tan£2 t4, t4 = -3(S910 + S1112 + S1314). The scalars originating in o>abs and distinguishing among families, couple the family quantum numbers through (t1 and NL), or through (T2 and NR), all in the adjoint representations of the corresponding groups. Let us now prove that all the scalar fields with the space (scalar with respect to d = (3 + 1)) index s = (7,8) carry the weak and the hyper charge (t13, Y) equal to either (- 2,1) or to (1, -1). Let us first simplify the notation, using a common name AA for all the scalar fields with the scalar index s = (7,8) aa1 = (ASQ, aq ', Aj', A4, A 2, A1, A , A NL), (9.25) and let us rewrite the term Y.s=7 8 1 Ys p0s 1 in Eq. (9.2) as follows Y 1 Ys P0s 1, s = 7,8 78 78 = 1{ (+) P0 + + (-) P0-} 1, P0± = (P07 T ip08) , (P07 T ip08) = (P7 T ip8) - ta1 (AA1 t iA^1) 78 1 (±)= 1 (Y7 ± iY8). (9.26) Let us now apply the operators Y, Q, Eq. (9.14), and t13 = 1 (S56 - S78), Eq. (9.9), on the fields AA1 = (AA1 T iAA1). One finds (±) 1 2 1 2 Q (AA1 T iAA1)= 0, (9.27) T13 (AA1 T iAA1) = ±1 (AA1 t iAA1), Y (AA1 T iAA1) = T 1 (AA1 T iAA1), 9 Can Spin-charge-family Theory Explain Baryon Number Non-conservation? 145 This is, with respect to the weak, the hyper and the electromagnetic charge, just what the standard model assumes for the Higgs' scalars. The proof is complete. 78 One can check also, using Eq. (9.44), that y0 (-) transforms the uR1 from the first line of table 9.3 into u^1 from the seventh line of the same table, or vR from the 25th line into the vL from the 31th line of the same table. The scalars AA7i obviously bring the weak charge 2 and the hyper charge — 2 to the right handed family members (uR, vR), and the scalars Aa1 bring the weak charge — 2 and the hyper charge 2 to (dR, eR). Let us now prove that the scalar fields of Eq. (9.25) are either triplets with respect to the family quantum numbers (NR, NL, T2, T1; Eqs. (9.12,9.13)) or singlets as the gauge fields of Q = t13 + Y, Q' = t13 — Y tan2 $1 and Y' = t23 — tan2 $2 t4. One can prove this by applying T2, T1, NR, NL and Q, Q', Y' on their eigenstates. Let us calculate, as an example, N L and Q on An78l3 and on Aq8 , taking into (±) (±) account Eqs. (9.12, 9.11,9.9, 9.7) NL AN8LS = ± A, INL Anl3 = 0, (±) (±) (±) QAQ8 = 0, (±) - NL± A 78n = {(cD 78 + icD 78 ) ffl i (cD 78 + icD 78 )}, ■ 78 (±) 78 2 3 (±) 01(±) 1—1 31(±) 02 (± ) AN7L3 = (cD 78 + icD 78 ) (±) 12(±) 03(±) Aq8 = sin$1 A1738 + sin$1 (—)(c 78 + c 78 + c 78 ), (9.28) (±) (±) 910(±) 11 12 (±) 13 14 (±) with Q = S56 + t4 = S56 — 3(S910 + S11 12 + S1314), and with t4 defined in Eq. (9.11)). Nonzero vacuum expectation values of the scalar fields (Eq. (9.25)), which carry the scalar index s = (7,8), and correspondingly the weak and the hyper charges, break the mass protection mechanism of quarks and leptons of the lower and the upper four families. In the loop corrections contribute to all the matrix elements of mass matrices of any family members besides A^1 and the scalar fields which are the gauge fields of Q, Q', Y' also the vector gauge fields. The gauge fields of N R and i2 contribute only to masses of the upper four families, while the gauge fields of N L and t1 contribute only to masses of the lower four families. The triplet scalar fields with the scalar index s = (7,8) and the family charges N R and i2 transform any family member belonging to the group of the upper four families into the same family member belonging to another family of the same group of four families, changing the right handed member into the left handed partner, while those triplets with the family charges NL and t1 transform any family member of particular handedness and belonging to the lower four families into its partner of opposite handedness, belonging to another family of the lower four families. 146 N.S. Mankoč Borštnik The scalars aQ (Eq. (9.28)), aQ (= cos $1 A1738 - sin$1A478 ) and A^g (±) (±) (±) (±) (±) (Eq. (9.23)) contribute to all eight families, distinguishing among the family members and not among the families. The mass matrix of any family member, belonging to any of the two groups of the four families, manifests - due to the SU(2)(R,L) x SU(2) (H,i) (either (R, II) or (L, I)) structure of the scalar fields, which are the gauge fields of the NR,L and T2,1 - the symmetry presented in Eq. (9.29) Ma = /-a1 - a e d b \ e -a2 - a b d d b a2 - a e \ b d e a1 - a/ (9.29) In the ref. gn2014 the mass matrices for quarks, which are in the agreement with the experimental data, are presented and predictions made. pc 9.5 Condensate and nonzero vacuum expectation values of scalar fields make spinors and most of scalar and vector gauge fields massive Let us shortly overview properties of the scalar and the vector gauge fields after a. two right handed neutrinos (coupled to spin zero and with the family quantum numbers (table 9.4) of the upper four families) make a condensate (table 9.2) at the scale > 1016 GeV and after b. the electroweak break, when the scalar fields with the space index s = (7,8) get nonzero vacuum expectation values. All the scalar gauge fields A^1, t G (5,6,7,8,9,..., 14) (Eqs. (9.2, 9.21, 9.23), table 9.1) interact with the condensate through the quantum numbers t4 and t23, those with the family quantum numbers of the upper four families interact also through the family quantum numbers T2 or N R, getting masses of the order of the condensate scale (Eq.(9.23)). At the electroweak break the scalar fields A^1, s G (7,8), from Eqs. (9.25,9.25) get nonzero vacuum expectation values, changing correspondingly their own masses and determining masses of quarks and leptons, as well as of the weak vector gauge fields. The vector gauge fields A^P, Am', A ^P, A and A £ (Eq. (9.23)) get masses due to the interaction with the condensate through t23 and t4, the first two, and/or also due to the family quantum numbers of the upper four families, the last three, respectively. The vector gauge fields AAand A^ stay massless up to the electroweak break when the scalar gauge fields, which are weak doublets with the hypercharge making electromagnetic charge Q equal to zero, give masses to the weak bosons 1± ' (Am± = ^ (A£ T iA£) and a£ = cos $1 A^5 - sin $1 A^J, while the electromagnetic vector field (a£ = sin $1 A^ + cos $1 A^J and the colour vector gauge field stay massless. 9 Can Spin-charge-family Theory Explain Baryon Number Non-conservation? 147 At the electroweak break, when the nonzero vacuum expectation values of the scalar fields break the weak and the hypercharge global symmetry, also all the eight families of quarks and leptons get masses. Until the electroweak break the families were mass protected, since the right handed partners distinguished from the left handed ones in the weak and hyper charges, what disabled them to make the superposition manifesting masses. 9.6 Sakharov conditions as seen in view of the spin-charge-family theory The condensate of the right handed neutrinos, as well as the nonzero vacuum expectation values of the scalar fields AA7i - if leading to the complex matrix (±) elements of the mixing matrices - cause the Cn Pn violation terms, which generate the matter-antimatter asymmetry. It is the question whether both generators of the matter-antimatter asymmetry - the condensate and the complex phases of the mixing matrices of quarks and leptons (this last alone can not with one complex phase and also very probably not with the three complex phases of the lower four families) - can explain at all the observed matter-antimatter asymmetry of the "ordinary" matter, that is the matter of mostly the first family of quarks and leptons. The lowest of the upper four families determine the dark matter. For the dark matter any relation among matter and antimatter is so far experimentally allowed. Both origins of the matter-antimatter asymmetry - the condensate and the nonzero vacuum expectation values of the scalar fields carrying the weak and the hyper charge - (are assumed to) appear spontaneously. Sakharov [24] states that for the matter-antimatter asymmetry three conditions must be fulfilled: a. (Cn and) Cn Pn must not be conserved. b. Baryon number non conserving processes must take place. c. Thermal non equilibrium must be present not to equilibrate the number of baryons and antibaryons. Sakharov uses for c. the requirement that CPT must be conserved and that {CPT, H}_ = 0. In a thermal equilibrium the average number of baryons < nB >= Tr(e-PHnB) = Tr(e-PHCPTnB(CPT)-1) = < nB >. Therefore < nB > - < nB >= 0 at the thermal equilibrium and there is no excess of baryons with respect to antibaryons. In the expanding universe, however, the temperature is changing with time. It is needed that the discrete symmetry Cn Pn is broken to break the symmetry between matter and antimatter, if the universe starts with no matterantimatter asymmetry. The spin-charge-family theory starting action (Eq.(9.1)) is invariant under the Cn Pn symmetry. The scalar fields (Eq.(9.21)) of this theory cause transitions, in which a quark is born out of a positron (figures (9.1, 9.2)) and a quark is born out of antiquark, and back. These reactions go in both directions with the same probability, until the spontaneous break of the Cn Pn symmetry is caused by the appearance of the condensate of the two right handed neutrinos (table 9.2). 148 N.S. Mankoč Borštnik But after the appearance of the condensate (and in addition of the appearance of the non zero vacuum expectation values of the scalar fields with the space index s e (7,8)), family members "see" the vacuum differently than the antimembers. And this might explain the matter-antimatter asymmetry. It is also predicting the proton decay. It is, of course, the question whether both phenomena can at all explain the observed matter-antimatter asymmetry. I agree completely with the referee of this paper that before answering the question whether or not the spin-charge-family theory explains this observed phenomena, one must do a lot of additional work to find out: i. Which is the order of phase transition, which leads to the appearance of the condensate. ii. How strong is the thermal nonequilibrium, which leads to the matter-antimatter asymmetry during the phase transition. iii. How rapid is the appearance of the matter-antimatter asymmetry in comparison with the expansion of the universe. iv. Does the later history of the expanding universe enable that the produced asymmetry survives up to today. Although we do have some experience with solving the Boltzmann equations for fermions and antifermions [12] to follow the history of the dark matter within the spin-charge-family theory, the study of the history of the universe from the very high temperature to the baryon production within the same theory in order to see the matter-antimatter asymmetry in the present time is much more demanding task. These is under consideration, but still at a very starting point since a lot of things must be understood before starting with the calculations. What I can conclude is that the spin-charge-family theory does offer the opportunity also for the explanation for the observed matter-antimatter asymmetry. 9.7 Conclusions The spin-charge-family [1,3-8,2,9,12,14,15] theory is a kind of the Kaluza-Klein theories in d = (13 + 1) but with the families introduced by the second kind of gamma operators - the Ya operators in addition to the Dirac Ya in d = (13 + 1). The theory assumes a simple starting action (Eq. (9.1)) in d = (13 +1). This simple action manifests in the low energy regime, after the breaks of symmetries (subsection 9.1.1), all the degrees of freedom assumed in the standard model, offering the explanation for all the properties of quarks and leptons (right handed neutrinos are in this theory the regular members of each family) and antiquarks and antileptons. The theory explains the existence of the observed gauge vector fields. It explains the origin of the scalar fields (the Higgs and the Yukawa couplings) responsible for the quark and lepton masses and the masses of the weak bosons and carrying the weak and the hyper charge of the standard model Higgs ([14]). The theory is offering the explanation also for the matter-antimatter asymmetry and for the appearance of the dark matter. The spin-charge-family theory predicts two decoupled groups of four families [3,4,9,12]: The fourth of the lower group of four families will be measured at the LHC [10] and the lowest of the upper four families constitutes the dark matter [12] and was already seen. It also predicts that there might be several scalar 9 Can Spin-charge-family Theory Explain Baryon Number Non-conservation? 149 fields observable at the LHC. The upper four families manifest, due to their high masses, a new "nuclear force" among their baryons. All these degrees of freedom are already contained in the simple starting action. The scalar fields with the weak and the hyper charges equal to (t 1, ± 1), respectively (section 9.4), have the space index s = (7,8), while they carry in addition to the weak and the hyper charges also the family quantum numbers, originating in Sab (they form two groups of twice SU(2) triplets), or the family members quantum numbers, originating in Sab (they form three singlets with the quantum numbers (Q, Q', Y'))• These scalar fields cause the transitions of the right handed family members into the left handed partners and back. Those with the family quantum numbers cause at the same time transitions among families within each of the two family groups of four families. They all gain in the electroweak break nonzero vacuum expectation values, giving masses to both groups of four families of quarks and leptons and to weak bosons (changing also their own masses). There are in this theory also the scalar fields with the space index s = (5,6); They carry with respect to this degree of freedom they the weak charge equal to the hyper charge (t 1, T1, respectively). They carry also additional quantum numbers Eq.(9.23) like all the scalar fields: The family quantum numbers, originating in S ab and the family members quantum numbers originating in Sab. And there are also the scalar fields with the scalar index s = (9,10, • • • ,14). These scalars carry the triplet colour charge with respect to the space index and the additional quantum numbers (table 9.1), originating in family quantum numbers S ab and in family members quantum numbers Sab. There are no additional scalar gauge fields. There are the vector gauge fields with respect to d = (3 + 1): A^1, with Ai staying for the groups SU(3) and U(1) (both originating in SO (6) of SO (13,1)), for the groups SU(2)„ and SU(2)i (both originating in SO(4) of SO(7,1)) and for the groups SU(2) x SU(2) (e So(3, 1)), in both sectors, the Sab and Sab ones. The condensate of the two right handed neutrinos with the family charges of the upper four families (table 9.2) gives masses to all the scalar and vector gauge fields, except to the colour octet vector, the hyper singlet vector and the weak triplet vector gauge fields, to which the condensate does not couple. It gives masses also to all the vector gauge fields to which the condensate couples. Those vector gauge fields of either Sab or S ab origin, which do not couple to the condensate, are expressible with the corresponding vielbeins (appendices 9.55, 9.56). The condensate breaks the Cn Pn symmetry (sections (9.3, 9.8)). There are no additional vector gauge fields in this theory. Nonzero vacuum expectation values of the scalar gauge fields with the space index s = (7,8) and the quantum numbers as explained in the fourth paragraph of this section change in the electroweak break their masses, while all the other scalars or vectors either stay massless (the colour octet, the electromagnetic field), or keep the masses of the scale of the condensate. The only before the electroweak massless vector fields, which become at the electroweak break massive, are the heavy bosons. 150 N.S. Mankoč Borštnik It is extremely encouraging that the simple starting action of the spin-charge-family offers at low energies the explanations for so many observed phenomena, although the starting assumptions (section 9.1.1) wait to be derived from the initial and boundary conditions of the expanding universe. This paper is a step towards understanding the matter-antimatter asymmetry within the spin-charge-family theory, predicting also the proton decay. The theory obviously offers the possibility that the scalar gauge fields with the space index s = (9,10, ••• ,14) explain, after the appearance of the condensate, the matterantimatter asymmetry. To prove, however, that this indeed happen, requires the additional study: Following the universe through the phase transitions which breaks the Cn Pn symmetry at the level of the condensate and further through the electroweak phase transition up to today, to check how much of the matterantimatter asymmetry is left. The experience when following the history of the expanding universe to see whether the spin-charge-family theory can explain the dark matter content [12] is of some help. However, answering the question to which extend this theory can explain the observed matter-antimatter asymmetry requires a lot of additional understanding and a lot of work. Let me conclude with the recognition, pointed out already in the introduction, that the spin-charge-family theory overlaps in many points with other unifying theories [26-31], since all the unifying groups can be recognized as the subgroups of the large enough orthogonal groups, with family groups included. But there are also many differences: The spin-charge-family theory starts with a very simple action, from where all the properties of spinors and the gauge vector and scalar fields follow, provided that the breaks of symmetries occur in the desired way. Consequently it differs from other unifying theories in the degrees of freedom of spinors and scalar and vector gauge fields which show up on different levels of the break of symmetries, in the unification scheme, in the family degrees of freedom and correspondingly also in the evolution of our universe. 9.8 APPENDIX: Discrete symmetry operators [15] I present here the discrete symmetry operators in the second quantized picture, for the description of which the Dirac sea is used. I follow the reference [15]. The discrete symmetry operators of this reference are designed for the Kaluza-Klein like theories, in which the total angular momentum in higher than (3 + 1) dimensions manifest as charges in d = (3 + 1). The dimension of space-time is even, as it is in the case of the spin-charge-family theory. 3 cn = n Ym r(3+1) k Ix6,xs,...,xd, 3ym,m=0 3 Tn = n Ym r(3+1) KIx0 Ixs)X7,...)Xd-1 , pn = y0 r(3+1) r(d) ix3. (9.30) 9 Can Spin-charge-family Theory Explain Baryon Number Non-conservation? 151 The operator of handedness in even d dimensional spaces is defined as r(d) := (i)d/2 n (Vn^Ya), (9.31) a with products of Ya in ascending order. We choose y0, y1 real, y2 imaginary, y3 real, y5 imaginary, y6 real, alternating imaginary and real up to Yd real. Operators I operate as follows: Ix0 x0 = —x0; Ixxa = —xa; Ix0 xa = (—x0, x); I«x = —x; II3 xa = (x0, —x1, —x2, —x3,x5,x6,...,xd); Ix5,x7,...,xd-1 (x0,x1 ,x2,x3,x5,x6,x7,x8,...,xd-1,xd) = , , , (x0,x1 ,x2,x3, —x5 ,x6, —x7,..., —xd-1,xd); Ix6,x8,...,xd (x0,x1 ,x2,x3,x5,x6,x7,x8,...,xd-1,xd) = (x0, x1, x2, x3, x5, —x6, x7, —x8,..., xd-1, —xd), d = 2n. Cjn transforms the state, put on the top of the Dirac sea, into the corresponding negative energy state in the Dirac sea. The operator, it is named [1,16,15] Cj, is needed, which transforms the starting single particle state on the top of the Dirac sea into the negative energy state and then empties this negative energy state. This hole in the Dirac sea is the an-tiparticle state put on the top of the Dirac sea. Both, a particle and its antiparticle state (both put on the top of the Dirac sea), must solve the Weyl equations of motion. This Cj is defined as a product of the operator [1,16] "emptying", (making transformations into a completely different Fock space) "emptying" = n Ya K = (—)d+1 n Ya r(d)K, (9.32) mya 3ya and Cj d 3 Cn = n Ya K n Ym r(3+1) KIx6,x8,...,xd OTya,a=0 3ym,m=0 d = n Ys Ix6,x8,...,xd . (9.33) OTys,s=5 We shall need inded only the product of operators CnPn, TN and CnPn TN, since either Cn or Pn have in even dimensional spaces with d = 2(2n +1) an odd number of Ya operators, transforming accordingly states from the representation 152 N.S. Mankoč Borštnik of one handedness in d = 2(2n + 1) into the Weyl of another handedness. d CnPN = Y0 n YS Ixs Ix6,x8,...,xd , 3ys,s=5 d CnPnTn = n Ya KIx . (9.34) 3ya,a=0 9.9 APPENDIX: Short presentation of technique [6,18,20] I make in this appendix a short review of the technique [18,20], initiated and developed [5-8] when proposing the spin-charge-family theory [5,6,8,4,1,2,12,9] assuming that all the internal degrees of freedom of spinors, with family quantum number included, are describable in the space of d-anticommuting (Grassmann) coordinates [6], if the dimension of ordinary space is d. There are two kinds of operators in the Grassmann space, fulfilling the Clifford algebra, which anticom-mute with one another. The technique was further developed in the present shape together with H.B. Nielsen [18,20] by identifying one kind of the Clifford objects with Ys's and another kind with Ya's. The objects Ya and Ya have properties {Ya,Yb}+ = 2nab, {Ya,Yb}+ = 2nab, , {Ya,Yb}+ = 0, YaB := i(-)nB BYa |^0 >, B = a.0 + aaYa + aabYaYb + • • • + aa, -adYai ... Yad)l^0 > (9.35) for any d, even or odd. I is the unit element in the Clifford algebra. The two kinds of the Clifford algebra objects are connected with the left and the right multiplication of any Clifford algebra objects B. In Eq. (9.35) B is expressed as a polynomial of Ya, (-)nB = +1, -1, when the object B has a Clifford even (+1) or odd (+1) character, respectively. |^0 > is a vacuum state on which the operators Ya apply. If B is a Clifford algebra object, let say a polynomial of Ya, then one finds (YaB : = i(-)nB BYa ) |^0 >, B = a0 + aa0Ya0 + aa,a2YaiYa2 + ••• + aa,...adYai ••• Yad , (9.36) where |^0 > is a vacuum state, defined in Eq. (9.50) and (-)nB is equal to 1 for the term in the polynomial which has an even number of Yb's, and to -1 for the term with an odd number of Yb's. In this last stage we constructed a spinor basis as products of nilpotents and projections formed as odd and even objects of Ya's, respectively, and chosen to be eigenstates of a Cartan subalgebra of the Lorentz groups defined by Ya's and Ya's. The technique can be used to construct a spinor basis for any dimension d and any signature in an easy and transparent way. Equipped with the graphic presentation of basic states, the technique offers an elegant way to see all the quantum numbers of states with respect to the two Lorentz groups, as well as transformation properties of the states under any Clifford algebra object. 9 Can Spin-charge-family Theory Explain Baryon Number Non-conservation? 153 The Clifford algebra objects Sab and Sab close the algebra of the Lorentz group Sab := (i/4)(yayb - ybya), Sab := (i/4)(YaYb - YbYa), {Sab,Scd}_ = 0, {Sab,Scd}- = i(nadSbc + nbcSad - nacSbd - nbdSac), {S ab,S cd}_ = i(nad5Sbc + nbcSad - nacSbd - nbdS ac), (9.37) We assume the "Hermiticity" property for ya's and ya's yat = naaYa , Yat = naaYa , (9.38) in order that Ya and Ya are compatible with (9.35) and formally unitary, i.e. Ya t Ya = I and Ya ^a = I. One finds from Eq.(9.38) that (Sab)t = naanbbSab. Recognizing from Eq.(9.37) that two Clifford algebra objects Sab, Scd with all indices different commute, and equivalently for Sab, Scd, we select the Cartan subalgebra of the algebra of the two groups, which form equivalent representations with respect to one another S03,S12,S56, ••• ,Sd-1d, if d = 2n > 4, S03,S12, ••• ,Sd-2d-1, if d =(2n + 1) >4, S 03,S 12,S56, ••• ,S d-1 d, if d = 2n > 4, S03,S12, ••• ,Sd-2d-1, if d =(2n + 1) >4. (9.39) The choice for the Cartan subalgebra in d < 4 is straightforward. It is useful to define one of the Casimirs of the Lorentz group - the handedness r ({r, Sab}_ = 0) in any d r(d) :=(i)d/2 ^ (VnaaYa), if d = 2n, a r(d) :=(i)(d-1)/2 ^ (V'naaYa), if d = 2n + 1. (9.40) a One proceeds equivalently for f(d), subtituting Ya's by Ya's. We understand the product of Ya's in the ascending order with respect to the index a: y0y1 • • • Yd .It follows from Eq.(9.38) for any choice of the signature naa that rt = r, r2 = I. We also find that for d even the handedness anticommutes with the Clifford algebra objects Ya ({Ya, H+ = 0), while for d odd it commutes with Ya ({Ya, H- = 0). To make the technique simple we introduce the graphic presentation as follows ab 1 naa ab 1 i (k): = ^(Ya + n^Yb), [k]:= ^(1 + kYaYb), +: = 1 (1 + r), -:= 1 (1 - r), (9.41) 154 N.S. Mankoc Borstnik where k2 = naanbb. It follows then ab ab ab ab Ya = (k) + (-k), Yb = iknaa ((k) - (-k)), k ab ab Sab = k ([k] - [-k]) (9.42) One can easily check by taking into account the Clifford algebra relation (Eq.9.35) and the definition of Sab and Sab (Eq.9.37) that if one multiplies from the left hand ab ab side by Sab or Sab the Clifford algebra objects (k) and [k], it follows that ab 1 ab ab 1 ab Sab (k)= 2 k (k), Sab [k]= ^k [k], ab 1 ab ab 1 ab Sab (k)= 2 k (k), Sab [k]= -^ k [k], (9.43) which means that we get the same objects back multiplied by the constant 2k in ab ab the case of Sab, while Sab multiply (k) by k and [k] by (-k) rather than (k). This ab ab also means that when (k) and [k] act from the left hand side on a vacuum state } the obtained states are the eigenvectors of Sab. We further recognize that Ya ab ab ab ab ab ab transform (k) into [-k], never to [k], while Ya transform (k) into [k], never to [-k] ab ab ab ab ab ab ab ab Ya (k)= naa [-k], Yb (k)= -ik [-k], Ya [k]=(-k), Yb [k]= -iknaa (-k), ab ab ab ab ab ab ab ab Y~a (k) = -inaa [k], Y~b (k)= -k [k], Y~a [k]= i (k), Y~b [k] = -knaa (k) (9.44) From Eq.(9.44) it follows ab cd i ab cd ab cd i abcd Sac (k)(k) = -2-naVc [-k][-k], Sac (k)(k)= ^naancc [k][k], ab cd i ab cd ab cd i ab cd Sac [k][k] = 2 (-k)(-k), Sac [k][k]= -2 (k)(k), ab cd i ab cd ab cd i ab cd Sac (k)[k] = -^naa [-k](-k), Sac (k)[k]= -2-naa [k](k), ab cd i ab cd ab cd i ab cd Sac [k](k) = ^ncc (-k) [-k], Sac [k](k)= ^ncc (k)[k] . (9.45) From Eqs. (9.45) we conclude that Sab generate the equivalent representations with respect to Sab and opposite. Let us deduce some useful relations ab ab (k)(k) = 0, ab ab ab [k] [k] = [k], ab ab (k)[k] = 0, ab ab ab (k)[-k] = (k), ab ab ab (k)(-k)= naa [k]. ab ab [k][-k]= 0, ab ab ab [k](k)=(k), ab ab [k](-k)= 0, ab ab ab ab ab (-k)(k) = naa [-k], (-k)(-k)= 0, ab ab ab ab ab [-k][k]= 0, ab ab ab (-k)[k]=(-k), ab ab [-k](k)= 0, [-k][-k]=[-k], ab ab (-k)[-k]= 0, ab ab ab [-k](-k)=(-k) . (9.46) 9 Can Spin-charge-family Theory Explain Baryon Number Non-conservation? 155 We recognize in the first equation of the first line and the first and the second equation of the second line the demonstration of the nilpotent and the projector ab ab character of the Clifford algebra objects (k) and [k], respectively. Defining ab 1 ab (±i)= 2 (Ya T Yb), 1 (±1)= 2 (Ya ± iYb), one recognizes that ab ab (k) (k) = 0, Recognizing that ab ab —k) (k) = in ab [k], ab ab ab (k) [k]= i (k), (9.47) ab ab (k) [—k]= 0. (9.48) ab ^ ab ab ^ ab (k) = naa (-k), [k]=[k]: we define a vacuum state |^0 > so that one finds ab ^ ab < (k) (k) >= 1 , ab^ ab < [k] [k] >= 1 . (9.49) (9.50) Taking into account the above equations it is easy to find a Weyl spinor irreducible representation for d-dimensional space, with d even or odd. For d even we simply make a starting state as a product of d/2, let us say, ab only nilpotents (k), one for each Sab of the Cartan subalgebra elements (Eq.(9.39)), applying it on an (unimportant) vacuum state. For d odd the basic states are products of (d — 1)/2 nilpotents and a factor (1 ± r). Then the generators Sab, which do not belong to the Cartan subalgebra, being applied on the starting state from the left, generate all the members of one Weyl spinor. 0d 12 35 (kûd)(ki2)(k35) 0d 12 35 -k0d][—k12](k35) 0d 12 35 -k0d](k12)[—k35] d-1 d-2 (kd-1 d-2) ^0 d-1 d-2 (kd-1 d-2) ^0 d-1 d-2 (kd- 1 d-2) ^0 0d 12 35 d-1 d-2 [—k0d](ki2)(k35) • • • [—kd-1 d-2] ^0 od 12 35 d-1 d-2 (k0d)[—k12][—k35] • • • (kd-1 d-2) ^0 . (9.51) All the states have the handedness r, since {r, Sab} = 0. States, belonging to one multiplet with respect to the group SO(q, d — q), that is to one irreducible representation of spinors (one Weyl spinor), can have any phase. We made a choice of the simplest one, taking all phases equal to one. 156 N.S. Mankoč Borštnik The above graphic representation demonstrate that for d even all the states of one irreducible Weyl representation of a definite handedness follow from a ab starting state, which is, for example, a product of nilpotents (kab), by transforming ab mn ab mn all possible pairs of (kabHW) into [-kab][-kmn]. There are Sam, San, Sbm, Sbn, which do this. The procedure gives 2(d/2-1' states. A Clifford algebra object Ya being applied from the left hand side, transforms a Weyl spinor of one handedness into a Weyl spinor of the opposite handedness. Both Weyl spinors form a Dirac spinor. For d odd a Weyl spinor has besides a product of (d — 1)/2 nilpotents or projectors also either the factor o:= 1 (1 + r) or the factor •:= 1 (1 — r). As in the case of d even, all the states of one irreducible Weyl representation of a definite handedness follow from a starting state, which is, for example, a product of (1 + r) ab ab mn and (d — 1)/2 nilpotents (kab), by transforming all possible pairs of (kab)(kmn) ab mn into [—kab][—kmn]. But Ya's, being applied from the left hand side, do not change the handedness of the Weyl spinor, since {r,Ya}_ = 0 for d odd. A Dirac and a Weyl spinor are for d odd identical and a "family" has accordingly 2(d-1)/2 members of basic states of a definite handedness. We shall speak about left handedness when r = — 1 and about right handedness when r = 1 for either d even or odd. While Sab which do not belong to the Cartan subalgebra (Eq. (9.39)) generate all the states of one representation, generate S ab which do not belong to the Cartan subalgebra(Eq. (9.39)) the states of 2d/2-1 equivalent representations. Making a choice of the Cartan subalgebra set (Eq.(9.39)) of the algebra Sab and Sab a left handed (r(13,1' = — 1) eigen state of all the members of the Cartan subalgebra, representing a weak chargeless uR-quark with spin up, hyper charge (2/3) and colour (1/2,1/(2%/3)), for example, can be written as 03 12 56 78 9 101 1 1213 14 (+i)(+)l ( + )( + ) II (+) ( — ) ( — ) IW> = (Y0 — Y3)(Y1 + iY2)I(Y5 + iY6)(Y7 + iY8)II (Y9 + iY10)(Y11 — iY12)(Y13 — iY14)I^>. (9.52) This state is an eigen state of all Sab and Sab which are members of the Cartan subalgebra (Eq. (9.39)). The operators Sab, which do not belong to the Cartan subalgebra (Eq. (9.39)), generate families from the starting uR quark, transforming uR quark from Eq. (9.52) to the uR of another family, keeping all the properties with respect to Sab unchanged. In particular S01 applied on a right handed uR-quark, weak charge-less, with spin up, hyper charge (2/3) and the colour charge (1/2,1/(2%/3)) from Eq. (9.52) generates a state which is again a right handed uR-quark, weak charge-less, with spin up, hyper charge (2/3) and the colour charge (1/2,1/(2%/3)) 03 12 56 78 91011121314 i 03 12 56 78 91011 121314 S01 (+iH+)I(+)(+) II (+)(—)(—) = — 2 [+i][ + ] I (+)(+) II (+) (—)(—) . 2 (9.53) 9 Can Spin-charge-family Theory Explain Baryon Number Non-conservation? Below some useful relations [4] are presented 157 03 12 03 12 N± = N+ ± iN+ = - (Ti)(±), N± _ N- ± iN- _(±i)(±), "+ ± iN+ 03 12 03 12 :i± _ f^wii Ki± IN + = — C=Fi)C=±), IN± =(±i)(±), 56 78 56 78 1± - _ T'± _(f)(±)(f) , T 56 78 _(T (±)(T) , =1± (T (t)(t), 56 78 2T _(T) (FF)(T) . (9.54) I present at the end one Weyl representation of SO (13 +1) and the family quantum numbers of the two groups of four families. One Weyl representation of SO(13 + 1) contains left handed weak charged and the second SU(2) chargeless coloured quarks and colourless leptons and right handed weak chargeless and the second SU(2) charged quarks and leptons (electrons and neutrinos). It carries also the family quantum numbers, not mentioned in this table. The table is taken from the reference [15]. ^t3TTt r (1,7) = of quarks ^i >_ = 1, r(6) = -1, and leptons 03 12 +i) ( + ) I 10 1112 13 14 + ) ( -) ( -) 2 V3 5 ( + 10 1112 13 14 + ) ( -) ( -) 2 V3 12 ( + ) 10 1112 13 14 + ) ( -) ( -) 2 V3 0 1112 13 14 ( -) ( -) 2 V3 12 ( + ) I 10 1112 13 14 + ) ( -) ( -) 2 V3 +i) [ -] I [ 10 1112 13 14 + ) ( -) ( -) 2 V3 12 (+) I 10 1112 13 14 + ) ( -) ( -) 2 V3 +i) [ -] 5 ( + 10 1112 13 14 + ) (-) (-) 2 V3 12 ( + ) I 10 1112 13 14 -] [+] (-) 2 V3 5 ( + 10 1112 13 14 -] [+] (-) 2 V3 1 2 0 1 V3 1 1 2 0 1 V3 1 1 2 0 0 1 1 2 0 0 1 2 1 - 2 0 0 1 2 1 - 2 0 0 1 2 0 0 0 1 2 0 0 0 1 2 0 0 0 1 2 0 0 0 1 2 1 2 1 2 1 2 V3 1 6 1 2 1 2 1 2 V3 1 6 1 2 1 2 1 2 V3 1 6 12 5 ( + ) I ( + 10 1112 13 14 -] (-) [+] 5 ( + 1112 13 14 (-) [+] 12 ( + ) I 10 + ) 11 12 [+] 13 14 [+] 5 ( + 11 12 [+] 13 14 [+] 12 ( + ) 11 12 [+] 13 14 [+] 11 12 [+] 13 14 [+] 12 ( + ) I 11 12 [+] 13 14 [+] 11 12 [+] 13 14 [+] 12 ( + ) I 11 12 [+] 13 14 [+] 5 ( + 11 12 [+] 13 14 [+] 12 5 ( + ) I ( + 10 1112 13 14 -] [+] [+] 5 I ( + 78 ( + ) I 11 12 13 14 [+] [+] 03 12 5 -i] ( + ) I [- 78 910 1112 1314 [-] II [-] [+] [+] - Continued on nexi page 2 3 23 33 8 4 a S Octet 9 1 2 1 2 1 2 1 6 2 3 2 3 c R 0 03 —i 2 1 2 2 3 R 2 I 0 u 2 2 6 3 5 78 1 2 1 2 1 2 1 6 1 3 1 3 c R d I 0 03 2 5 8 9 1 2 1 2 R 4 d I 0 2 6 3 3 03 5 1 2 1 2 1 2 1 6 1 6 1 3 dL I 0 03 2 5 1 2 1 2 c L d I 0 2 6 6 3 03 78 1 2 1 2 1 2 1 6 1 6 2 3 L I 0 03 2 8 1 2 1 2 1 2 1 1 ~G 2 3 c L 8 I 0 u c2 R 1 2 2 3 9 0 u 2 2 6 3 03 2 9 c2 R 1 2 1 2 1 2 1 6 2 3 2 3 10 I 0 c3 R 1 2 2 3 2 3 17 0 u 3 2 0 c3 R 1 2 2 3 2 3 18 I 0 0 0 25 0 R 03 -i 2 9 26 I 0 0 0 v R 5 78 27 I 0 1 R 03 2 5 8 9 28 I 0 1 R 5 29 I 1 e L 2 2 5 30 I 1 e L 2 8 31 I 0 v L 2 2 8 32 I 0 v L 2 9 33 0 L 2 3 2 9 0 c 1 L 1 2 1 3 34 0 3 2 3 35 0 L 2 3 158 N.S. Mankoč Borštnik i > r(3,1) S12 r (4) x13 T23 T33 T38 t4II y| Q Octet, r(1,7) = 1, r(6) = —1, of quarks and leptons 36 uj1 03 12 56 78 910 1112 1314 (+i) [ —] I [ —] [ —] ll[ —] [+] [+] -1 1 — 2 1 0 1 —2 1 — 2 1 2 V3 1 — 6 2 — 2 2 — 3 37 d c 1 aR 03 12 56 78 9 10 11 12 13 14 (+i) ( + ) I (+) [ —] II [ —] [+] [+] 1 1 2 -1 1 2 0 1 — 2 1 2 V3 1 — 6 1 — 6 1 3 38 03 12 56 78 9 10 11 12 13 14 [ —i] [ —] I ( + ) [ —] II [ —] [+] [+] 1 1 2 -1 1 2 0 1 2 1 2 V3 1 6 1 6 1 3 39 uR1 03 12 56 78 9 10 11 12 13 14 (+i) ( + ) I [ —] ( + ) I I [ —] [+] [+] 1 1 2 -1 1 2 0 1 2 1 2 V3 1 6 1 6 2 3 40 uR1 03 12 56 78 9 10 11 12 13 14 [ —i] [ —] I [ —] ( + ) 11 [ —] [+] [+] 1 1 2 -1 1 2 0 1 2 1 2 V3 1 6 1 6 2 3 41 dp 03 12 56 78 9 10 11 12 13 14 [ —i] (+) I ( + ) ( + ) II ( + ) ( —) [+] -1 1 2 1 0 1 2 1 2 1 2 a/3 1 6 1 3 1 3 49 aj3 03 12 56 78 9 10 11 12 13 14 [ —i] (+) I ( + ) ( + ) II ( + ) [+] ( — ) -1 1 2 1 0 1 2 0 1 A3 1 — 6 1 3 1 3 57 eL 03 12 56 78 9 10 11 12 13 14 [ —i] (+) I ( + ) ( + ) I I [ —] ( —) ( — ) -1 1 2 1 0 1 2 0 0 1 2 1 1 58 eL 03 12 56 78 9 10 11 12 13 14 (+i) [ —] I ( + ) ( + ) I I [ —] ( —) ( — ) -1 1 — 2 1 0 1 2 0 0 1 2 1 1 59 VL 03 12 56 78 9 10 11 12 13 14 [ —i] ( + ) I [ —] [ —] 11 [ —] ( —) ( —) -1 1 2 1 0 1 — 2 0 0 1 2 0 0 60 VL 03 12 56 78 9 10 11 12 13 14 (+i) [ —] I [ —] [ —] 11 [ —] ( —) ( —) -1 1 2 1 0 1 2 0 0 1 2 0 0 61 VR 03 12 56 78 9 10 11 12 13 14 (+i) ( + ) I [ —] ( + ) I I [ —] ( —) ( — ) 1 1 2 -1 1 2 0 0 0 1 2 1 2 0 62 VR 03 12 56 78 9 10 11 12 13 14 [ —i] [ —] I [ —] ( + ) 11 [ —] ( —) ( —) 1 1 2 -1 1 2 0 0 0 1 2 1 2 0 63 eR 03 12 56 78 9 10 11 12 13 14 (+i) ( + ) I (+) [ —] I I [ —] ( —) ( — ) 1 1 7 -1 1 2 0 0 0 1 2 1 2 1 64 eR 03 12 56 78 9 10 11 12 13 14 [ —i] [ —] I ( + ) [ —] 11 [ —] ( —) ( —) 1 1 2 -1 1 2 0 0 0 1 2 1 2 1 Table 9.3. The left handed (F(13,11 = -1) multiplet of spinors - the members of the SO(13,1) group, manifesting the subgroup SO(7,1) - of the colour charged quarks and anti-quarks and the colourless leptons and anti-leptons, is presented in the massless basis using the technique presented in Appendix 9.9. It contains the left handed (F(3,11 = — 1) weak charged (t13 = ±2) and SU(2)n chargeless (t23 = 0) quarks and the right handed weak chargeless and SU(2)n charged (t23 = ± 1) quarks of three colours (cl = (t33,t38)) with the "spinor" charge (t4 = 1) and the colourless left handed weak charged leptons and the right handed weak chargeless leptons with the "spinor" charge (t4 = — 2). S12 defines the ordinary spin ±2. The vacuum state |vac >fQm, on which the nilpotents and projectors operate, is not shown. The reader can find this Weyl representation also in the refs. [21,3]. Left handed antiquarks and anti leptons are weak chargeless and carry opposite charges. The eight families of the first member of the eight-plet of quarks from Table 9.3, for example, that is of the right handed u1R quark, are presented in the left column of Table 9.4 [3]. In the right column of the same table the equivalent eight-plet of the right handed neutrinos v1R are presented. All the other members of any of the eight families of quarks or leptons follow from any member of a particular family by the application of the operators N± L and t<2'1)± on this particular member. The eight-plets separate into two group of four families: One group contains doublets with respect to N R and T2, these families are singlets with respect to NL and T1. Another group of families contains doublets with respect to N L and t1, these families are singlets with respect to N R and T2. The scalar fields which are the gauge scalars of N R and T2 couple only to the four families which are doublets with respect to these two groups. The scalar fields which are the gauge scalars of N L and t1 couple only to the four families which are doublets with respect to these last two groups. ?2J Njr t 03 12 56 78 9 10 1 1 12 13 14 03 12 56 78 9 10 1112 13 14 I (+i) [+1 W ( + ) II (+) [- -] H ^r2 (+i) [+] W (+) (+) (+) (+) 1 2 0 1 2 0 1 3 03 12 56 78 9 10 1 1 12 13 14 03 12 56 78 9 10 1112 13 14 I [+i] (+) W ( + ) II (+) [- -] H [+i] (+) W (+) (+) (+) (+) 1 2 0 1 2 0 1 3 U& 03 12 56 78 9 10 1 1 12 13 14 03 12 56 78 9 10 1112 13 14 I (+i) [+] (+) W II (+) [- -] h] vr3 (+i) [+] (+) W (+) (+) (+) 1 2 0 1 2 0 1 3 ur4 03 12 56 78 9 10 1 1 12 13 14 03 12 56 78 9 10 1112 13 14 I [+i] (+) (+) W II (+) [- -] h] vr4 [+i] (+) (+) W (+) (+) (+) 1 2 0 1 2 0 1 3 Ur5 03 12 56 78 9 10 11 12 13 14 03 12 56 78 9 10 1112 13 14 II [+i] [+] [+] W II (+) h -] H vr5 [+i] [+] [+] [+] (+) ( + ) (+) 0 1 2 0 1 2 1 3 03 12 56 78 9 10 1 1 12 13 14 03 12 56 78 9 10 1112 13 14 II (+i) (+) 1 [+] [+] II (+) [- -] h] vr6 (+i) (+) 1 [+] [+] (+) (+) (+) 0 1 2 0 1 2 1 3 UcrV 03 12 56 78 9 10 1 1 12 13 14 03 12 56 78 9 10 1112 13 14 II [+i] [+] 1 (+) ( + ) II (+) [- -] h] [+i] [+] 1 (+) (+) (+) (+) (+) 0 1 2 0 1 2 1 3 03 12 56 78 9 10 1 12 13 14 03 12 56 78 9 10 11 12 13 14 II (+i) (+) 1 (+) (+) II (+) [ -] h] vr8 (+i) (+) 1 (+) (+) II (+) (+) (+) 0 1 2 0 1 2 1 3 Table 9.4. Eight families of the right handed Ur1 (9.3) quark with spin j, the colour charge (t33 = 1 /2, t38 = 1 /(2-^/3), and of the colourless right handed neutrino vr of spin j (9.2) are presented in the left and in the right column, respectively. They belong to two groups of four families, one (I) is a doublet with respect to (Nl and t'1') and a singlet with respect to (Nr and t'2'), the other (II) is a singlet with respect to (Nl and t'1') and a doublet with with respect to (Nr and f'2'). All the families follow from the starting one by the application of the operators (N^ L, t1-2'1^), Eq. (9.54). The generators (N^ L, t'2,1 (Eq. (9.54)) transform ui r to all the members of one family of the same colour. The same generators transform equivalently the right handed neutrino vi r to all the colourless members of the same family. 160 N.S. Mankoč Borštnik 9.10 APPENDIX: Expressions for the spin connection fields in terms of vielbeins and the spinor sources [14] The expressions for the spin connection of both kind, caba and cDaba in terms of the vielbeins and the spinor sources of both kinds are presented, obtained by the variation of the action Eq.(9.1). The expression for the spin connection caba is taken from the ref. [32]. One notices that if there are no spinor sources, carrying the spinor quantum numbers Sab, then caba is completely determined by the vielbeins. Equivalently one obtains expressions for the spin connection fields carryin family quantum numbers Acknowledgments The author acknowledges funding of the Slovenian Research Agency. References 1. N.S. Mankoc Borstnik, "Spin-charge-family theory is explaining appearance of families of quarks and leptons, of Higgs and Yukawa couplings", in Proceedings to the 16th Workshop "What comes beyond the standard models", Bled, 14-21 of July, 2013, eds. N.S. Mankoc Borstnik, H.B. Nielsen and D. Lukman (DMFA ZaloZnistvo, Ljubljana, December 2013) p.113 [arxiv:1312.1542]. (9.55) (9.56) 9 Can Spin-charge-family Theory Explain Baryon Number Non-conservation? 161 2. N.S. Mankoc Borštnik, "Do we have the explanation for the Higgs and Yukawa couplings of the standard model", http://arxiv.org/abs/1212.3184v2, (http://arxiv.org/abs/1207.6233), in Proceedings to the 15 th Workshop "What comes beyond the standard models", Bled, 9-19 of July, 2012, Ed. N.S. Mankoc Borstnik, H.B. Nielsen, D. Lukman, DMFA Založnistvo, Ljubljana, December 2012, p.56-71, [arxiv.1302.4305]. 3. N.S. Mankoc Borstnik, J. of Modern Phys. 4 (2013) 823 [arxiv:1312.1542]. 4. A. Borstnik Bracic and N.S. Mankoc Borstnik, Phys. Rev. D 74 (2006) 073013 [hep-ph/0512062, hep-ph/9905357, p. 52-57]. 5. N.S. Mankoc Borstnik, Phys. Lett. B 292 (1992) 25. 6. N.S. Mankoc Borstnik, J. Math. Phys. 34 (1993) 3731. 7. N.S. Mankoc Borstnik, Int. J. Theor. Phys. 40 (2001) 315. 8. N.S. Mankoc Borstnik, Modern Phys. Lett. A 10 (1995) 587. 9. G. Bregar, M. Breskvar, D. Lukman and N.S. Mankoc Borstnik, New J. of Phys. 10 (2008) 093002. 10. G. Bregar and N.S. Mankoc Borstnik, "Can we predict the fourth family masses for quarks and leptons?", Proceedings (arxiv:1403.4441) to the 16th Workshop "What comes beyond the standard models", Bled, 14-21 of July, 2013, Ed. N.S. Mankoc Borštnik, H.B. Nielsen, D. Lukman, DMFA Založnistvo, Ljubljana December 2013, p. 31-51, http://arxiv.org/abs/1212.4055. 11. G. Bregar and N.S. Mankoc Borstnik, "The new experimental data for the quarks mixing matrix are in better agreement with the spin-charge-family theory predictions", Proceedings to the 17th Workshop "What comes beyond the standard models", Bled, 2028 of July, 2014, Ed. N.S. Mankoc Borstnik, H.B. Nielsen, D. Lukman, DMFA Založnistvo, Ljubljana December 2014. 12. G. Bregar and N.S. Mankoc Borstnik, Phys. Rev. D 80 (2009) 083534. 13. The authors of the works presented in An introduction to Kaluza-Klein theories, Ed. by H. C. Lee, World Scientific, Singapore 1983, T. Appelquist, A. Chodos, P.G.O. Freund (Eds.), Modern Kaluza-Klein Theories, Reading, USA: Addison Wesley, 1987. 14. N.S. Mankoc Borstnik,"The spin-charge-family theory explains why the scalar Higgs carries the weak charge ^ 1 and the hyper charge ± j", to appear in Proceedings to the 17th Workshop "What comes beyond the standard models", Bled, 20-28 of July, 2014, eds. N.S. Mankocš Borsštnik, H.B. Nielsen and D. Lukman (DMFA Založšnisštvo, Ljubljana, December 2014). 15. N.S. Mankoc Borstnik, H.B. Nielsen, "Discrete symmetries in the Kaluza-Klein-like theories", JHEP 04 (2014) 165 http://arxiv.org/abs/1212.2362v2. 16. T. Troha, D. Lukman, N.S. Mankoc Borstnik, Int. J. of Mod. Phys. A 29 (2014) 1450124, arxiv:1312.1541. 17. N.S. Mankoc Borstnik, [arxiv:1212.3184, arxiv:1011.5765]. 18. N.S. Mankoc Borstnik and H.B. Nielsen, J. of Math. Phys. 43 (2002) 5782 [hep-th/0111257]. 19. N.S. Mankoc Borštnik and H.B. Nielsen, Phys. Rev. D 62 (2000) 044010 [hep-th/9911032]. 20. N.S. Mankoc Borstnik and H.B. Nielsen, J. of Math. Phys. 44 (2003) 4817-4827, hep-th/0303224. 21. A. Borstnik, N.S. Mankoc Borstnik, Proceedings to the Euroconference on Symmetries Beyond the Standard Model, Portorož, July 12 -17, 2003, Ed. Norma Mankoc Borstnik, Holger Bech Nielsen, Colin Froggatt, Dragan Lukman, DMFA, Založnistvo, Ljubljana, December 2003, p. 31-57, hep-ph/0401043, hep-ph/0401055. 22. D. Lukman, N.S. Mankoc Borstnik and H.B. Nielsen, New J. Phys. 13 (2011) 103027. 23. D. Lukman and N.S. Mankoc Borstnik, J. Phys. A: Math. Theor. 45 (2012) 465401 [arxiv:1205.1714, hep-ph/0412208 p.64-84]. 24. A. D. Sakharov, JETP 5 (1967) 24. 162 N.S. Mankoč Borštnik 25. C. Itzykson adn J. Zuber, Quantum Field Theory (McGraw-Hill 1980). 26. H. Georgi, S. Glashow, Phys. Rev. Lett. 32 (1974) 438; Phys. Rev. Lett. 32 (1974) 438. 27. H. Georgi, in Proceedings of the American Institue of Physics, ed. Carlson (1974). 28. H. Fritzsch and P. Minkowski, Ann. Phys. 93 (1975) 193. 29. M. Gell-Mann, P. Ramond and R. Slansky, Rev. Mod. Phys. 50 (1978) 721. 30. A. Buras, J. Ellis, M. Gaillard and D.Nanopoulos, Nucl. Phys. B 135 (1978) 66. 31. Unity of Forces in the Universe, ed. A. Zee (World Scientific,1982). 32. N.S. Mankoc Borstnik, H.B. Nielsen, D. Lukman, "An example of Kaluza-Klein-like theories leading after compactification to massless spinors coupled to a gauge field-derivations and proofs", Proceedings to the 7th Workshop "What Comes Beyond the Standard Models", Bled, July 19 - 31, 2004, Ed. by Norma Mankoc Borstnik, Holger Bech Nielsen, Colin Froggatt, Dragan Lukman, DMFA ZaloZnistvo, Ljubljana December 2004, p.64-84, [hep-ph/0412208].