The Explanation for the Origin of the Higgs Scalar and for the Yukawa Couplings by the Spin-Charge-Family Theory

The spin-charge-family theory is a kind of the Kaluza-Klein theories, but with two kinds of the spin connection fields, which are the gauge fields of the two kinds of spins. The SO(13,1) representation of one kind of spins manifests in d = (3 + 1) all the properties of family members as assumed by the standard model; the second kind of spins explains the appearance of families. The gauge fields of the first kind, carrying the space index ( ) m = , , 0 3  , manifest in d = (3 + 1) all the vector gauge fields assumed by the standard model. The gauge fields of both kinds of spins, which carry the space index (7, 8) gaining at the electroweak break nonzero vacuum expectation values, manifest in d = (3 + 1) as scalar fields with the properties of the Higgs scalar of the standard model with respect to the weak and the hyper charge ( ± 1 2 and 1 2  , respectively), while they carry additional quantum numbers in adjoint representations, offering correspondingly the explanation for the scalar Higgs and the Yukawa couplings, predicting the fourth family and the existence of several scalar fields. The paper 1) explains why in this theory the gauge fields are with the scalar index ( ) s = , , , 5 6 7 8 doublets with respect to the weak and the hyper charge, while they are with respect to all the other charges in the adjoint representations; 2) demonstrates that the spin connection fields manifest as the Kaluza-Klein vector gauge fields, which arise from the vielbeins; and 3) explains the role of the vielbeins and of both kinds of the spin connection fields.

spect to the weak and the hyper charge ( ± 1 2 and 1 2  , respectively), while they carry additional quantum numbers in adjoint representations, offering correspondingly the explanation for the scalar Higgs and the Yukawa couplings, predicting the fourth family and the existence of several scalar fields.The paper 1) explains why in this theory the gauge fields are with the scalar index ( )

Introduction
The standard model assumed and the LHC confirmed the existence of the Higgs's scalar-the only so far observed boson with the fractional charges 1 2 ± .The question arises: where does the Higgs originate, why does it carry the "fermion" charges and where do the Yukawa couplings originate?
It is demonstrated in this paper how do the scalar fields with the weak and the hyper charge equal to 1 2 ± and 1 2  , respectively, appear from the simple starting action of the spin-charge-family theory.While the weak and the hyper charge of the scalar gauge fields originate in the scalar index ( ) , all the other charges of these scalar fields originate in the two kinds of the spin, carrying these additional charges in the adjoint representations.These scalars explain the appearance of families, of the Higgs scalar and the Yukawa couplings and their influence on the properties of the family members and on the families.
The relation between the vector gauge fields, when they are presented by the spin connections-this is the case in the spin-charge-family theory-and the vector gauge fields when they are expressed in terms of the vielbeins-which is usually used in the Kaluza-Klein theories-is discussed.
It was demonstrated in the paper [1] that all the scalars, that is all the gauge fields with the space index 5 s ≥ of the spin-charge-family theory, manifest in ( ) fractional charges with respect to the index s and the standard model charge groups: when carrying the space index ( ) ).When carrying the space index ( ) 9, ,14 s =  they are colour charge triplets (belonging to ( ) ( ) ). Doublets explain the weak and the hyper charge; triplets offer a possible explanation for the matter-antimatter asymmetry of the ordinary matter in the universe and for the proton decay.
The spin-charge-family theory [2]- [12] offers the explanation for all the assumptions of the standard model: for the properties of each family member-quarks and leptons, left and right handed (right handed neutrinos are in this theory regular members of each family)-for the appearance of the families, for the existence of the gauge vector fields of the family member charges and for the scalar field and the Yukawa couplings.It is offering the explanation also for the existence of phenomena, which are not included in the standard model, like there is the dark matter [11] and the (ordinary) matter-antimatter asymmetry [1].
The spin-charge-family theory predicts that there are at the low energy regime two decoupled groups of four families: The fourth [2] [4] [5] [10] to the already observed three families of quarks and leptons will be measured at the LHC [12], LHC will measure also some of the scalar fields (manifesting as the Higgs and the Yukawa couplings [4]).The lowest of the upper four families constitutes the dark matter [11].
In Subsection 1.1 a short introduction of the spin-charge-family theory is made: the simple starting action of the theory together with the assumptions made to achieve that the theory manifests at the low energies the observed phenomena are presented.
The main Section 3 discusses the properties of the scalar fields, offering the explanation for the appearance and properties of families of quarks and leptons, of the Higgs and the Yukawa coupling and correspondingly for the masses of the heavy bosons.
In Section 2 the relation between the vector gauge fields as appearing from the vielbeins (as one usually proceeds in the Kaluza-Klein theories [13]) and those expressible with the spin connections (as it is in the spin-charge-family theory) is discussed.I prove the statement that both gauge fields (those emerging from the vielbeins and those expressed by the spin connections) are equivalent for the ( ) symmetry of the space of coordinates ( ) and no fermion sources present.Section 5 presents a short summary of all the problems discussed in this paper.In the Sections 4, 7, and 8, properties of the vielbeins and both kinds of the spin connection fields-manifesting at the low energy regime the observed vector and scalar gauge fields-as well as properties of both kinds of the Clifford algebra objects-which determine either spins and charges or family quantum numbers of fermions, respectively-are discussed.
In Appendix A1 the infinitesimal generators of the subgroups of ( ) 13 1 SO + (determining spins and charges 1 In the spin-charge-family theory the weak and the hyper charge of the scalar fields originate in the ( ) ( ) ), having the analogue in the spin representation of the vector gauge fields in ( ) of fermions and of the corresponding gauge vector and scalar fields) and of  ( ) (determining family quantum numbers and the family charges of the scalar gauge fields), expressed by the generators ab S and ab S  , respectively, are presented, together with the corresponding gauge fields.
Appendix A4 is a short review of the technique, taken from Ref. [1].It is used in this paper to demonstrate properties of the spinor states, representing family members and families.
All appendices are added to make the paper easier to follow.Let me point out at the end of this part of the introduction that more I am working on the spin-charge-family theory (together with the collaborators) more answers to the open questions of the elementary particle physics and cosmology the theory is offering.In order that the reader will easier follow the achievements of this paper I repeat several topics which already have appeared in previous papers, cited in this one.The new achievements of this paper are presented and discussed in Sections 2 and 3 and supported by Appendix A2 and Appendix A3.

Spin-Charge-Family Theory, Action and Assumptions
This section follows a lot the similar one from Ref. [1].
Let me present the assumptions on which the theory is built, starting with the simple action in ( ) 13 1 d = + : A i. In the action [1] [2] [4] fermions ψ carry in ( ) as the internal degrees of freedom only two kinds of spins, no charges, determined by the two kinds of the Clifford objects (there exist no additional Clifford algebra objects) (Equations (( 14), (15), (47), ( 49), (63)))-a γ and a γ -and interact correspondingly with the two kinds of the spin connection fieldsabα ω and abα ω  , the gauge fields of ( )

M
+ ) fields than the right handed ones, the break can leave massless and mass protected α β γ  ), from the middle of both the alphabets the ob- served dimensions 0,1, 2,3 ( , , m n  and , , µ ν  ), indices from the bottom of the alphabets indicate the compactified dimensions ( , , s t  and , , σ τ  ).We assume the signature families remain massless and mass protected.Equivalent assumption, although not yet proved how does it really work, is made in the ( ) case.This study is in progress.
A iii.There are additional breaks of symmetry: the manifold ( ) There is a scalar condensate (Table 1) of two right handed neutrinos with the family quantum numbers of the upper four families, bringing masses of the scale above the unification scale ( 16 10

∝
) to all the vector and scalar gauge fields, which interact with the condensate [1].
A v. There are nonzero vacuum expectation values of the scalar fields with the space index (7, 8) conserving the electromagnetic and colour charge, which cause the electroweak break and bring masses to all the fermions and to the heavy bosons.
Comments on the assumptions: C i.This starting action enables to represent the standard model as an effective low energy manifestation of the spin-charge-family theory, offering an explanation for all the standard model assumptions, explaining also the appearance of the families, the Higgs and the Yukawa couplings: C i.a.One Weyl representation of ( ) , if analysed with respect to the subgroups ( ) (Equations ((33)-( 35)), Appendix A1), all the family members required by the standard model, with the right handed neutrinos in addition (Table 3): It contains the left handed weak ( ) ( ) ( ) chargeless colour triplet quarks and colourless leptons (neutrinos and electrons), and right handed weak chargeless and ( ) charged coloured quarks and colourless leptons, as well as the right handed weak charged and ( ) chargeless colour antitriplet antiquarks and (anti)colourless antileptons, and left handed weak chargeless and ( ) charged antiquarks and antileptons.The antifermion states are reachable from the fermion states by the application of the discrete symmetry operator     , pre- sented in Ref. [16].
C i.b.There are before the electroweak break all massless observable gauge fields: the gravity, the colour octet vector gauge fields (of the group ( ) , Equation ( 35)), the weak triplet vector gauge field (of the group ( ) , Equation (34)) and the hyper singlet vector gauge field (a superposition of ( ) and the third component of ( ) triplet, Equation ( 41)).All are the superposition of the c f α abα ω spinor gauge fields (Equation (41) represents the superposition for some scalar fields).C i.c.There are before the electroweak break all massless two decoupled groups of four families of quarks and leptons, in the fundamental representations of  ( ) L SO I SO SU SU × groups, respectively-the subgroups of  ( )  4).These eight families remain massless up to the electroweak break due to the "mass protection mechanism", that is due to the fact that the right handed members have no left handed partners with the same charges.C i.d.There are scalar fields, Section 3, with the space index (7,8) and with respect to the space index with the weak and the hyper charge of the Higgs's scalar (Equation ( 19)).They belong with respect to additional quantum numbers either to one of the two groups of two triplets, Equations ((36), (37)) (either to one of the two trip lets of the groups  ( ) 2 II SO SU , or to one of the two triplets of the groups  ( ) , respectively), which couple through the family quantum numbers to one (the first two triplets) or two another (the second two triplets) of the two groups of four families -all are the superposition of (Equation ( 40)), or they belong to three singlets, the scalar gauge fields of ( ) 39)), which couple to the family members of both groups of families-they are the superposition of s f σ abσ ω (Equation (41)).Both kinds of scalar fields determine the fermion masses (Equation ( 23)), offering the explanation for the Yukawa couplings and the heavy bosons masses (Equation ( 24)).
C i.e.The starting action contains also additional ( ) , Equation (34)) vector gauge fields (one of the components contributes to the hyper charge gauge fields as explained above), as well as the scalar fields with the space index ( )  ii., iii.: There are many ways of breaking symmetries from ( ) The assumed breaks explain why the weak and the hyper charge are connected with the handedness of spinors, manifesting correspondingly the observed properties of the family members-the quarks and the leptons, left and right handed (Table 3)-and of the observed vector gauge fields.
Antiparticles are accessible from particles by the application of the operator ⋅     , as explained in Refs.[16] [17].This discrete symmetry operator does not contain a γ 's degrees of freedom.To each family member there corresponds the antimember, with the same family quantum number.
iv.: It is the condensate of two right handed neutrinos with the quantum numbers of the upper four families (Table 1), which makes massive all the scalar gauge fields (with the index ( 5, 6, 7,8 ), as well as those with the index ( )  41), conserving the electromagnetic charge-change their mutual interaction, and gaining nonzero vacuum expectation values change correspondingly also their masses.They contribute to mass matrices of twice the four families, as well as to the masses of the heavy vector bosons (the two members of the weak triplet and the superposition of the third member of the triplet with the hyper vector field, Equation ( 24)).
All the rest scalar fields keep masses of the scale of the condensate and are correspondingly unobservable in the low energy regime.
The fourth family to the observed three ones is predicted to be observed at the LHC.Its properties are under consideration [12], the baryons of the stable family of the upper four families is offering the explanation for the dark matter [11].
Let us rewrite that part of the action of Equation ( 1), which determines the spinor degrees of freedom, in the way that we can clearly see how the action manifests under the above assumptions in the low energy regime by the standard model required degrees of freedom of fermions and bosons [2]- [12].
where ( ) , in agreement with the assumptions ii. and iii..In Subsection 2 the relation between the gauge fields, as obtain from the vielbeins in the ussual Kaluza-Klein procedure and those obtained from the spin connections as it is done in the spin-charge-family theory, is discussed.For a particular choice of the group ( ( )

SU
) when vielbeins have a particular symmetry the proof that both procedures lead to the same vector gauge field in ( ) is presented.The expressions of the vector gauge fields goes for all the charges in a similar way as in this particular case (Equation (10)).
All vector gauge fields, appearing in the first line of Equation ( 2), except A ′ get masses of the order of the condensate scale through the interaction with the condensate of the two right handed neutrinos with the quantum numbers of the upper four families (the assumption iv., Table 1).
The condensate, Table 1, gives masses of the order of the scale of its appearance also to all the scalar gauge fields, presented in the second and the third line of Equation (2).
The charges ) of the gauge fields are before the electroweak break the conserved charges, since the corresponding vector gauge fields don't interact with the condensate.After the electroweak break, when the scalar fields with the space index ( ) -those with the family quantum numbers and those with the quantum numbers ( , , 21)) gaining nonzero vacuum expectation values, the weak charge and the hyper charge are no longer conserved.The only conserved charges are then the colour and the electromagnetic charges.
In Equations ((41), ( 40)) the scalar fields with the space index (7,8), Equation (17), are presented as superpositions of the spin connection fields of both kinds.These scalar fields determine after the electroweak break the mass matrices of the two decoupled groups of four families (Equation ( 23)) and of the heavy bosons (Equation ( 24)).
Quarks and leptons have the "spinor" quantum number ( 4τ , originating in ( ) Let us conclude this Subsection with the recognition that: A. It is (only) one scalar condensate of two right handed neutrinos (Table 1), which gives masses to all the vector and the scalar gauge fields appearing in the spin-charge-family theory, except to those vector gauge fields which enter into the standard model as massless vector gauge fields (the gravity, the colour vector gauge fields, the weak vector gauge fields and the hyper ( ) gauge field, the last three gauge fields manifesting after the electroweak break as the heavy bosons and the massless colour and electromagnetic gauge fields).
B. There are (only) the nonzero vacuum expectation values of the scalar gauge fields with the space index ( ) (with the weak charge equal to 1 2 ± and the hyper charge correspondingly equal to 1 2  , both due to the space index), and with the family (twice two triplets) and family member quantum numbers (three singlets) in adjoint representations, which cause the electroweak break breaking the weak and the hyper charge symmetry.
The rest of the scalar fields, the members of the weak doublets (Table 2) with the space index ( ) , and the colour triplets and antitriplets with the space index ( ) 9,10, ,14 t =  [1] which contribute to transitions of antiparticles into particles and to proton decay, keep masses of the condensate scale.
Correspondingly the (only) two assumptions, iv. and v., make at the low energy regime observable the measured vector and scalar gauge fields, offering in addition the explanation also for the dark matter and the matter-antimatter asymmetry.

Relation between Spin Connections and Vielbeins When No Sources Are Present
It is demonstrated in this section for the case of spaces with no fermion sources present and with the symmetry of the vielbeins with the space indices ( ( ) , for any f, that both procedures-the ordinary Kaluza-Klein one with vielbeins and the procedure with spin connections used in the spin-charge-family theory-lead to the same gauge vector fields in ( ) 4 In the Pati-Salam model [18] this "spinor" quantum number is named 2 B L − quantum number and is twice the "spinor" quantum number, for quarks equal to 1 3 and for leptons to 1 − .
The two scalar weak doublets, one with 23  1 2 τ = − and the other with 23   1 2 τ = + , both with the "spinor" quantum number 4 0 τ = , are presented.In this table all the scalar fields carry besides the quantum numbers determined by the space index also the quantum numbers i  from Equation (17).Ai Ai A iA Let us assume the infinitesimal coordinate transformations of the kind [13] ( ) ( ) ( ) ( ) where we have made a choice of the symmetry ( ) ( ) and st S concern internal degrees of freedom of boson and fermion fields.The commutation relations for the generators of the ( ) The background field in ( ) is chosen to be flat, while the vielbeins m f σ represent the appearance of the vector gauge fields Both fields are functions of the coordinates in ( ) only.We make a choice [13] ( ) ( ) ( ) ( ) To prove this statement let us express the operators, appearing in Equation ( 5), as follows ( ) ( .) Then we use the relation between the stm ω fields and the vielbeins (Equation ( 62)), which in the case of no fermion sources present simplifies to .
Let us now put the vielbeins .
Repeating equivalent calculations for the rest of components of 1 m   and 2 m   one obtains that: , what completes the proof.

Scalar Fields Contributing to Electroweak Break Belong to Weak Charge Doublets
It is proven in this section that all the scalar gauge fields with the space index ( ) carry-with respect to the space index s-the weak and the hyper charge as does the Higgs's scalar of the standard model.These scalar fields, belonging either to (one of two times two ( ) ) triplets with respect to the family quantum numbers or to (one of the three) singlets with respect to the family members quantum numbers, offer the explanation for the origin of the Higgs's scalar and the Yukawa coupling of the standard model.
It turnes out [1] that all scalars (the gauge fields with the space index 5 ≥ ) of the action (Equation ( 1)) carry charges in the fundamental representations: They are either doublets (Table 2) or triplets [1] with respect to the space index 5 s ≥ .The scalars with the space indices ( ) 7,8 s ∈ and ( ) To see this one must take into account that the infinitesimal generators ab S , in accordance with the Equations (( 71)-( 73)).Expressions for the infinitesimal operators of the subgroups of the starting groups (presented in Equations ((33)-( 39))) are equivalent (have for the chosen Ai the same coefficients Ai ab c in Equation ( 3)) for all three kinds of degrees of freedom (Appendix A2.1, Appendix A1).All scalars carry correspondingly, besides the quantum numbers determined by the space index, also the quantum numbers Ai , the states of which belong to the adjoint representations.At the electroweak break all the scalar fields with the space index (7,8), those which belong to one of twice two triplets carrying the family quantum numbers ( Ai τ   , Equations ((36)-( 38))) and those which belong to one of the three singlets carrying the allowed family members quantum numbers ( , , 39), Section 1.1, the assumption v. and the corresponding comments), start to self interact, Equation (21).Gaining nonzero vacuum expectation values they break the weak, the hyper charge and the family charges.
Statement: Scalar fields with the space index (7, 8) carry with respect to this space index the weak and the

Ai s
A for all the scalar fields, independently of whether they originate in abs ω scalar fields-in this case ( ) -or in abs ω    scalar fields-in this case all the family quantum numbers of all eight families contribute.
Let us make a choice of the superposition of the scalar fields so that they are eigenstates of ( ) with the summation over Ai performed, since ).The application of the operators , Y Q (Equations (( 16), (39))) and 13   τ (Equation (34), if ab S is replaced by ab  ) on the fields ( ) Since 4   τ , Y , 13 τ and 1 1 τ τ , and since Y and 13 τ commute with the family quantum numbers, one sees that the scalar fields transform one member of a doublet from Table 2 into another member of the same doublet, keeping 23   τ un- changed.This completes the proof of the above statement.
After the appearance of the condensate (Table 1), which breaks the ( ) symmetry (bringing masses to all the scalar fields), the weak 1 τ  and the hyper charge Y remain the conserved charges 6 .At the electroweak break the scalar fields with the space index (7,8) start to interact among themselves so that the Lagrange density for these gauge fields changes from where  3 into the left handed 1 c L u quark from the seventh line of the same table 7 , which can, due to the properties of the scalar fields (Equation ( 19)), be interpreted also in the standard model way, namely, that    ( ) ).
The nonzero vacuum expectation values of the scalar fields of Equation ( 17) break the mass protection mechanism of quarks and leptons and determine correspondingly the mass matrices (Equation ( 23)) of the two groups of quarks and leptons.One group of four families carries the family quantum numbers ( , the other group of four families carries the family quantum numbers In loop corrections all the scalar and vector gauge fields which couple to fermions contribute.Correspondingly all the off diagonal matrix elements of the mass matrix (Equation ( 23)) depend on the family members quantum numbers.16), ( 36), (37))) or singlets as the gauge fields of 13

It is not difficult to show that the scalar fields
Let us do this for and Equation ( 16), and recognizing that  Similarly one finds properties with respect to the Ai quantum numbers for all the scalar fields The mass matrix of any family member, belonging to any of the two groups of the four families, manifests - Let us summarize this section: It is proven that all the scalar fields with the scalar index ( ) , which at the electroweak break start to mutual interact and gain nonzero vacuum expectation values (Equation ( 21)), keeping the electromagnetic charge conserved, carry the weak and the hyper charge quantum numbers as required by the standard model for the Higgs's scalar (Equation ( 19)): ( ) ( ) ( ) . These are the only scalar fields in this theory with the quantum numbers of the Higgs's field.These scalar fields carry additional quantum numbers: The triplet family quantum numbers and the singlet family members quantum numbers and form two groups of four families.They all contribute to masses of the heavy bosons ( [4], Equation ( 53)) ( reproducing on the tree level the expression for the mass term in the standard model where 2 I v are the contribution to the vacuum expectation value of all the scalar fields ( ) All the other scalar fields: A s∈ and A t t ′ ′ ∈  have masses of the order of the condensate scale and contribute to matter-antimatter asymmetry [1].

Triplets with Respect to Space Index s = (9, •••, 14)
The gauge fields with the space index ( ) 9, ,14 t ∈  form the triplets and antitriplets with respect to the space index ( ) 9, ,14 s =  . They are discussed in Ref. [1].The colour triplet scalars contribute to transition from antileptons into quarks and antiquarks into quarks and back, unless the scalar condensate of the two right handed neutrinos, presented in Table 1, breaks matter-antimatter symmetry [1], offering the explanation for the matter-antimatter asymmetry in our universe.This condensate leaves massless besides gravity only the colour, weak and the hyper charge vector gauge fields.Also all the scalar fields get masses through the interaction with the condensate.
There are no additional scalar indices and therefore no additional corresponding scalars with respect to the scalar indices in this theory.
Scalars, which do not get nonzero vacuum expectation values, keep masses on the condensate scale.

Vectors, Tensors and Spinors in Spin-Charge-Family Theory
This section discusses properties of vectors, tensors and spinors, appearing in the action in Equation ( 1), for ( ) -dimensional space-time, for any d in purpose to clarify the degrees of freedom of these fields, connected with the two kinds of the Clifford algebra objects: a γ and a γ (Equations ((47), (49))).The presentation is based on Refs.[7] [6] [19]- [21], where the two kinds of the Clifford objects a γ 's and a γ 's were introduced.In Ref. [7] these two kinds of the Clifford algebra objects were introduced in Grassmann space of anticommuting coordinates a θ .A part of this reference is briefly repeated in Appendix A2, where a γ 's and a γ 's are introduced as two superposition of a coordinate and its momentum (Equation (47)).
The Clifford algebra objects have properties (Equation (49 x or the corresponding momenta a p transform as vectors with respect to the Lo- rentz transformations in the tangent space (Equation (43)) and so do a θ and a p θ and correspondingly also a γ , a γ (Equation ( 53)).
where ab ω are parameters of transformations and ( ) is the operator of a finite Lorentz transformations.The Grassmann coordinate c θ transforms correspondingly as: 1 2 1 2 , which appear in Equation (1) ( a f α are the vielbeins), are also scalars.
The linear vector space over the coordinate Grassmann space has the dimension 2 d (Subsection 7.2).Any vector in Grassmann space can be presented as written in Equation (45).Any complex number  and Grassmann coordinates in Equation (45) can be replaced by one of the Clifford algebra objects, let say by a γ , and correspondingly the linear vector space can as well be described by the polynomials as follows ( ) provided that operation of a γ and a γ on such a vector space is understood as the left and the right multiplica- tion (Equation ( 64)) where 0 ψ is a vacuum state.
With this definition the relations from Equations ((47), (50)-( 53)) remain valid.If ab S  determine family quantum numbers, then ab S transform spinor states within one family (Table 3), keeping family quantum numbers unchanged, while ab S  transform a family member of one family into the same family member of another family (Table 4).
It is still true that the infinitesimal generators of the Lorentz transformations for vectors are ab ab ab S S = +   (Equation (47)), provided that we respect the rule of Equation (28).One correspondingly easily finds that any constant and the operator of handedness (Equation ( 68 Varying the action in Equation ( 1) with respect to a f α leads to the equations of motion .
Variation of the action with respect to abc ω and abs ω  , respectively, in the presence of the spinor fields leads [22] to the two equations One notices from Equations ((31), (32)) that if there are no spinor sources, then both spin connections- , and several others) different constants.

Conclusions
It is demonstrated in this paper (Section 3) that all the scalar gauge fields of the starting action (the second line in Equation ( 2)) of the spin-charge-family theory [1]- [11] with the space index ( ) 7,8 s = are, before the electroweak break, members of the two weak doublets (Table 2) with the hyper charge 1 2  , respectively.These scalars (Equation ( 17)) interact besides through the weak and the hyper charge (determined by the space index ( ) ) either through the family quantum numbers-they belong to twice two triplets (either to Correspondingly they either transform members of one group of four families of fermions among themselves, keeping the family member quantum number unchanged, or interact with each family member according to their eigenvalues of the family members charges ( , , Q Q Y ′ ′), keeping the family quantum numbers unchanged.When these scalars start to interact among themselves (Equation ( 21)), they gain nonzero vacuum expectation values, break the weak and the hyper charge, while preserving the electromagnetic charge, and cause the electroweak break.They determine mass matrices (Equation ( 23)) of two groups of four families as well as masses of the heavy bosons (Equation ( 24)).
These scalar fields with the space index ( ) and all the family charges in the adjoint representations offer an explanation for the appearance of the Higgs's scalar fields and the Yukawa couplings.
The paper discusses the relation between the Kaluza-Klein way through vielbeins and the spin-charge-family way through spin connections when explaining the appearance of the vector gauge fields in ( ) proven in Section 2 that, when there are no spinor sources present and the space exhibits in 5 d ≥ a large enough symmetry so that vielbeins in 5 d ≥ have the property The paper discusses also the Lorentz properties of the scalar and vector gauge fields of this theory-the vielbeins and the two kinds of the spin connection fields-showing up the difference among all three kinds of the gauge fields in the presence of the spinor sources, while in the absence of the spinor sources only one of these three kinds of gauge fields is the propagating field (Section 4, and Appendix A2, Appendix A3).
All the scalar and vector gauge fields, and all the family members and the families appearing in this theory have the interpretation in the observed fermion and boson fields.
The theory predicts two decoupled groups of four families [4] [5] [10] [11]: The fourth of the lower group of families will be measured at the LHC [12] and the lowest of the upper four families constitutes the dark matter [11].It also predicts that there will be several scalar fields observed sooner or later at the LHC, and that there is a new nuclear force among the fifth (and also the rest three of the upper group of four families) family baryons.The condensate contributes to the dark energy, as it does also the nonzero vacuum expectation values of the scalar fields with the space index (7,8).
Let me conclude with pointing out that the spin-charge-family theory is offering a possible next step beyond the standard model by offering the explanation for all the assumptions of the standard model and also so far to several phenomena of the cosmology, which are not yet understood: the dark matter [11], the matter/antimatter asymmetry [1].The spin-charge-family theory essentially differs from the unifying theories of Pati and Salam [18], Georgi and Glashow [23] and other ( ) ( ) SU n theories [24], and also from the Kaluza-Klein theories [25] [26], although all these unifying theories have many things in common-among themselves and with the spin-charge-family theory.
There are a lot of open questions in the elementary particle physics and cosmology which wait to be answered in addition to those presented in this paper.To see whether the spin-charge-family can offer answers also to ( ) ( ) ). Equivalently the generators of the family subgroups of the  ( ) determine generators of all three kinds of quantum numbers: Of those applying either on the family member or on the family quantum number of spinors, or on quantum numbers of bosons (of the vector and the scalar gauge fields).The difference among these three kinds of operators comes from the generators in d-dimensional space: ab S (for spins, Equation ( 14)), ab S  (for family quan- tum numbers, Equation ( 15)) and ab  (for quantum numbers of gauge fields, Equation ( 16)).
While ab S for spins of spinors is equal to ( ) One finds [2]- [9] [27] for the infinitesimal generators of the spin and the charge groups, which are the subgroups of ( ) determine representations of ( ) ( ) , originating in ( ) One correspondingly finds the generators of the subgroups of  ( ) (37) determine representations of  ( )  ( ) The corresponding operators which apply on the corresponding gauge fields follow from the above relations, if either ab S or ab S  are replaced by ab  from Equation ( 16).
The scalar fields, responsible [2]- [4]-after gaining nonzero vacuum expectation values and triggering the electroweak break-for masses of the family members and of the heavy bosons, are presented in the second line of Equation (2).These scalar fields are included in the covariant derivatives as ) ( ) , where notation a  is again used to point out that ( ) , a b belong in this case to the "tilde" space.
One finds the scalar fields carrying the quantum numbers of the subgroups of the family groups, expressed in terms of abs ω    (they contribute to mass matrices of quarks and leptons and masses of the heavy bosons), if taking into account Equations ((36), (37), (39)), ( ) The expressions for the scalars, expressed in terms of abc ω (contributing as well to the mass matrices of quarks and leptons and to masses of the heavy bosons) follow, if using Equations ((34), ( 35), (39)) ( ) Scalar fields from Equation (40) couple to the family quantum numbers, while those from Equation (41) couple to the family members quantum numbers.In Equation (41) the coupling constants are explicitly written in order to see the analogy with the gauge fields of the standard model.
Expressions for the vector gauge fields in terms of the spin connection fields and the vielbeins, which correspond to the colour charge ( 3 ), the ( ) ), the weak ( ) ) and the ( ) ), can be found by taking into account Equations ((34), ( 35)).Equivalently one finds the vector gauge fields in the "tilde" sector, or one just uses the expressions from ((41), (40)), if replacing the scalar index s with the vector index m.

Appendix A2. Symmetries of Vectors, Tensors and Spinors
In this section the Lorentz transformations of a γ 's, a γ 's and correspondingly of vector, tensor and spinor fields in ( ) -dimensional space-time are discussed.The presentation is based on the papers [6] [7], where the two kinds of the Clifford objects a γ 's, a γ 's were introduced (in those papers the notation a µ  and a µ   , respectively, was used, the present notation-a γ and a γ -was introduced in Refs.[19]- [21]).Ref [7] starts with the Grassmann space of anticommuting coordinates.I do the same in Subsection 7.1.The two kinds of the Clifford algebra objects (Equations ((63), (64))) in the Grassmann space are introduced in Subsection 7.2 as the two superposition of a Grassmann coordinate and its derivative and their properties are discussed, as well as the Lorentz transformations of a γ and a γ and of any vector or tensor.
One could start instead with the two kinds of the Clifford algebra objects, without using the Grassmannn space, as it is presented in Appendix A4 and explained in Section 4, Equation (27).

Appendix A2.1. Coordinate Space with Grassmann Character and Lorentz Transformations
I shall repeat here some properties of the anticommuting Grassmann coordinates, since the appearance of the two kinds of the Clifford algebra objects can in the Grassmann space easily be demonstrated.A point in d-dimensional Grassmann space of real anticommuting coordinates a θ , ( ) is determined by a vector { } ( ) . A Lorentz transformation on a vector component a θ and on a vector component a x , which is a real (ordinary) commuting coordinate, , , ( ) is a symmetric tensor made of complex numbers, is an antisymmetric tensor made of complex numbers.The requirements that ω ω + = .

Appendix A2.2. Linear Vector Space and Linear Operators over Coordinate Grassmann Space
A linear vector space over the coordinate Grassmann space has the dimension 2 d , due to the fact that ( ) Any vector in this space can be presented as a linear superposition of monomials ( ) The left derivative on vectors of the space of monomials is defined as follows The linear operators a p θ , a γ and a γ , ab  , ab S and ab S  are defined as .

{ }
, The factors in front of the superposition of a p θ and a θ in the second and the third line of Equation (47) are chosen so that both Clifford algebra objects, a γ and a γ , fulfil the same Clifford algebra relation (the third and the fourth line in Equation ( 49)).One finds and equivalently We see that the coordinates a θ and a x and the operators a γ and a γ transform as vectors Equation ( .
Correspondingly one finds that compositions like Also objects like 1 2 1 2 (1) transform with respect to the Lorentz transformations as scalars.

Appendix A3. Spin Connection Fields of Two Kinds and Vielbeins in Presence of Spinor Sources [22]
Relations among the vielbeins a f α and both kinds of the spin connection fields, abα ω and abα ω  , which are the gauge fields of ab S and ab S  , respectively, are studied under the assumption that ( ) − + space-time has a structure of a differentiable manifold [13].
The two kinds of vectors, a A and a A  , belonging to two different tangent spaces, transform with respect to the Lorentz transformations according to Equation (53).We express, after the parallel transport8 of each of these two kinds of vectors (belonging to two tangent spaces) from x to x dx + , in terms of the two kinds of the spin connections, , , .
We define the parallel transport also for the two kinds of vectors , .

A x dx A x
The difference between the two vectors A dA  , defines the covariant derivatives of the vector A α and of the vector ω  , respectively, in the presence of the spinor fields.One ends up [22] with the two expressions presented in Equation (32).Let me point out that if there are no spinor sources, then both spin connections- The expression for the spin connection c ab ω and c ab ω  can [22] then easily be found Again one notices that if there are no spinor sources, carrying the spinor quantum numbers ab S and ab S  for all ( ) , a b , then abα ω and abα ω  are completely determined by the vielbeins.In general both kinds of spinor sources affect the spin connections and vielbeins in their own way, making all three fields, vielbeins and both kinds of the spin connection fields, propagating fields.
Variation of the action Equation ( 1) with respect to a f α leads to the equations of motion Equation (31).
In this last stage we rewrite a spinor basis, written in Ref. [7] as products of polynomials of Grassmann coordinates of odd and even Grassmann character, chosen to be eigenstates of the Cartan subalgebra defined by the two kinds of the Clifford algebra objects, as products of nilpotents and projections, formed as odd and even objects of a γ 's, respectively, and chosen to be eigenstates of a Cartan subalgebra of the Lorentz groups defined by a γ 's and a γ '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 proper-ties of the states under any Clifford algebra object.
Appendix A2 briefly represents the starting point [7] of this technique in order to better understand the Lorentz transformation properties of both Clifford algebra objects, a γ 's and a γ 's, as well as of spinor, vector, tensor and scalar fields, appearing in the spin-charge-family theory, that is of the vielbeins and spin connections of both kinds, abα ω and abα ω  , and of spinor fields, family members and families.
The objects a γ and a γ have properties (49), , .
We assume the "Hermiticity" property for    To make the technique simple we introduce the graphic presentation as follows where which means that we get the same objects back multiplied by the constant From Equation (72) it follows From Equation (73) we conclude that ab S  generate the equivalent representations with respect to ab S and opposite.
Let us deduce some useful relations We recognize in Equation (74) the demonstration of the nilpotent and the projector character of the Clifford algebra objects ( ) Recognizing that we define a vacuum state 0 ψ so that one finds 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, only nilpotents ( ) ab k , one for each ab S of the Cartan subalgebra elements (Equation ( 67)), applying it on an vacuum state.For d odd the basic states are products of ( ) , 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.
The above graphic representation demonstrates that for 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 nilpotents ( ) We shall speak about left handedness when 1 Γ = − and about right handedness when 1 Γ = for either d even or odd.
While ab S which do not belong to the Cartan subalgebra (Equation ( 67)) generate all the states of one repre- sentation, ab S  which do not belong to the Cartan subalgebra (Equation (67)) generate the states of This state is an eigenstate of all ab S and ab S  which are members of the Cartan subalgebra (Equation ( 67)).The operators ab S  , which do not belong to the Cartan subalgebra (Equation (67)), generate families from the starting R u quark, transforming the R u quark from Equation (80) to the R u of another family, keeping all of the properties with respect to ab S unchanged.In particular, 01 S  applied on a right handed R u -quark from Equation (80) generates a state which is again a right handed R u -quark, weak chargeless, with spin up, hyper charge (2/3) and the colour charge ( ) ( )  charged quarks and leptons (electrons and neutrinos).It carries also the family quantum numbers, not mentioned in this table.The table is taken from Ref. [16].
The eight families of the first member of the eight-plet of quarks from Table 3, for example, that is of the right handed 1R u quark, are presented in the left column of Table 4 [4].In the right column of the same table the equivalent eight-plet of the right handed neutrinos 1R ν 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 d = (3 + 1) all the vector gauge fields assumed by the standard model.The gauge fields of both kinds of spins, which carry the space index (7, 8) gaining at the electroweak break nonzero vacuum expectation values, manifest in d = (3 + 1) as scalar fields with the properties of the Higgs scalar of the standard model with re- and R  are the two scalars ( R is a curvature), as it is presented in Section 4 and Appendix A3.A ii.The manifold both internal degrees of freedom-the one represented by a γ and the one represented by a γ .Since the left handed (with respect to ( ) space (a flat index), while Greek indices , , , , , , , , Einstein index (a curved index).Letters from the beginning of both the alphabets indicate a general index ( , , , a b c  and , , , of free fields (if no spinor source, carrying their quantum numbers, is present) both vielbeins (Subsection 2, Equations ((31), (62))), correspondingly only one kind of the three gauge fields are the propagating fields.

 2 τ
 are the gauge fields of the charges 1 τ  and , respectively, depending on the coordi- These two vector gauge fields are just the superposition of abe ω as used in the spin-charge-family theory:

m f σ from Equation ( 8 )
into the expressions for the gauge fields, let say hand side of Equation(12).Taking into account the symmetry of the space of the coordinates( ) of spinors (Equation (15)), while ab  (Equation (47)), which apply on the spin con- on either the space index e or any of the indices ( )


Let me pay attention to the reader, that the term ( ) quark from the first line of Table

uγ
giving it the weak and the hyper charge of the left handed 1 changes handedness.Equivalently happens to R ν from the 25 th line, which transforms under the action of from the third line of the

+
of the action (Equations ((1),(18))) takes care of the Yukawa couplings as well.All the scalar fields Equation(17), carry the weak and the hyper charge (Equations ((34), (35))) of the Higgs of the standard model.If Ai τ represents the first three operators in Equation (17) then it only multiplies the right handed family member with its eigenvalue.If Ai τ represents the last four operators of the same equation, then the operators respectively) trans- form the right handed family member of one family into the left handed partner of another family within the same group of four families, since these four operators manifest the symmetry twice (  ( ) ( )

±
are triplets as the gauge fields of the family quantum numbers account Equation (33) (where we replace ab S by ab  ) R I II SU SU × (either ( , L I ) or ( , R II )) structure of the scalar fields, which are the gauge fields of , symmetry presented in Equation (23).
of the infinitesimal Loorentz transformations fulfil the relations of Equation (50): tensor, are scalars with respect to the Lorentz transformations.
)) are the two scalars with respect to ab ab ab , for example, one finds[7] besides the two scalars (a constant and a product of all a γ ) al- so two three vectors and two four vectors.The two tangent spaces have the same metric tensors: any two vectors a A and a A  into the corresponding coordinate (curved) space with the vielbeins relations among the vielbeins a f α and the two kinds of the spin connection fields, abα ω and abα ω  , which are the gauge fields of ab S and ab S  , respectively, are studied under the assumption that space-time has a structure of a differentiable manifold [13].The relation among the two kinds of the spin connection fields, abα ω and abα ω  (Equations ((54), (55))), and the corresponding two kinds of the affine connections, α βγ Γ and α βγ Γ  (Equations ((56), (57))), is presented.The requirement that the covariant derivative of the vielbeins is equal to zero (Equation (60)) relates the two affine connections, are expressible with the vielbeins a f α in the same way and correspondingly equal to each other.Then also α βγ Γ and α βγ Γ  are equal (Equation (61).The only propagating fields are in this case the vielbeins a f α .The expressions for the two spin connection fields [22], c ab ω and c ab ω  , as functions of the vielbeins and the spinor sources are presented in Equation (62).If there are spinors present, then in general the two kinds of the spin connection fields are different.The condensate (Table1) of two right handed neutrinos, with the quantum numbers of the eighth family, contributes differently to abc ω than to abc ω  .It influences also a f α .In a flat space, that is with vielbeins equal to respectively-or through the family members quantum numbers-( , ,Q Q Y ′ ′)-as singlets.Triplets are the gauge scalar fields of the Clifford algebra objects of f; both ways lead to the same vector gauge fields.
and expressed.I present here also the gauge fields to the corresponding either the spins and charges or to the family quantum numbers in terms of either abc ab S  for families of spinors is equal to the space index e or the indices ( ) that the space index (e) of bde ω transforms according to the requirement of Equation (16), and so do , b d and , b d   .I used the notation , two independent groups.

ε
is the antisymmetric tensor, lead to a c ab ω are parameters of a transformation.Let us write the operator of finite Lorentz transformations as follows spin connection fields, related to tangent spaces.Requiring for the metric tensors, ab η and ab ab η η = , that they are the same at each point of each of the two tangent spaces, the relations parallel transport, defines the covariant derivative of the vector a A in the tangent space: introducing two kinds of the affine connections α βγ Γ and α βγ Γ  , respec- tively, (

The affine connection α ρβ Γ
(as well as the "tilde affine connection" α ρβ Γ  ) can be expressed by the torsion correspondingly relating two kinds of different coordinate systems for either a A or a A  , we find the related covariant derivatives vary the action Equation (1) with respect to abc ω and abs are expressible with the vielbeins a f α in the same way.Only one of these three fields are in this case the propagating field.Correspondingly also of Equation (32) by a e α and summing over both indices, the expressions for

ψ
is a vacuum state, defined in Equation (78) and ( ) B n − is equal to 1 for the term in the polynomial which has an even number of b γ 's, and to −1 for the term with an odd number of b γ 's, for any d, even or odd, and I is the unit element in the Clifford algebra.It follows from Equation (64) that the two kinds of the Clifford algebra objects are connected with the left and the right multiplication of any Clifford algebra objects B (Equation (28)).The Clifford algebra objects ab S and ab S  close the algebra of the Lorentz group (Equation (50)) the Cartan subalgebra in 4 d < is straightforward.It is useful to define one of the Casimirs of the Lorentz group-the handedness Γ {

(
by a γ 's.We understand the product of a γ 's in the ascending order with respect to the index a: 0 1 d γ γ γ  .It follows from Equation (66) for any choice of the signature aa We also find that for d even the handedness anticommutes with the Clifford algebra objects

1 2 k
in the case of ab S , while ab S  multiply ( ) ab k by k and [ ] ab k by ( ) k − rather than ( ) k .This also means that when ( ) ab k and [ ] ab k act from the left hand side on a vacuum state 0 ψ the obtained states are the eigenvectors of ab S .We further recognize that a ab 79)All the states have the same handedness Γ , since { } transforming all possible pairs of ( )( ) S , which do this.The procedure gives ( ) .A Clifford algebra object a γ 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.
and colourless leptons and right handed weak chargeless and the second ( )2 SU

τ
± on this particular member.The eight-plets separate into two group of four families: One group contains doublets with respect to R N group of families contains doublets with respect to L N   and 1 τ   , these families are singlets with respect to R N   and 2 τ   .The scalar fields which are the gauge scalars of R N   and 2 τ   couple only to the four families which are 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 A4.It contains the left handed ( ( ) leptons and the right handed weak chargeless leptons with the "spinor" contains also the of opposite charges, reachable from particle states by the application of the discrete symmetry operator     , presented in Refs.[16] [17].The vacuum state, on which the nilpotents and projectors operate, is not shown.The reader can find this Weyl representation also in Refs.[1] [4][27].

Table 1 )
with which they interact.They all are expressible with the superposition of

Table 1 .
[1]s table is taken from[1].The condensate of the two right handed neutrinos R , having the quantum numbers of the standard model Higgs' scalar.

Table 3 into
of this table, or R e from the 27 th line into L e from the 29 th line, where (16) Equation (39)) of these scalar fields, since4τ applied on the scalar index of these scalar fields gives zero, according to Equations ((34), (35)) in which ab S is replaced by ab  from Equation(16).line One further defines the operators for the charges Y and Q of the standard model, together with Q′ and Y ′ , and the corresponding operators of the family charges Y  , Y ′  , Q  , Q′  2 .