_{1}

^{*}

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 (
and
, 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.

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

It is demonstrated in this paper how do the scalar fields with the weak and the hyper charge equal to

the hyper charge of the scalar gauge fields originate in the scalar index ^{1}, 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 [

The spin-charge-family theory [

The spin-charge-family theory predicts that there are at the low energy regime two decoupled groups of four families: The fourth [

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 [

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―mani- festing 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

Appendix A4 is a short review of the technique, taken from Ref. [

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.

This section follows a lot the similar one from Ref. [

Let me present the assumptions on which the theory is built, starting with the simple action in

A i. In the action [

kinds of the spin connection fields―

Here^{2}.

A ii. The manifold ^{3}.

A iii. There are additional breaks of symmetry: the manifold

A iv. There is a scalar condensate (

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

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

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

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

or two another (the second two triplets) of the two groups of four families - all are the superposition of

C i.e. The starting action contains also additional

ii., iii.: There are many ways of breaking symmetries from

Antiparticles are accessible from particles by the application of the operator

iv.: It is the condensate of two right handed neutrinos with the quantum numbers of the upper four families (

v.: At the electroweak break the scalar fields with the space index

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 [

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 [

where

all family members of all the

The first line of Equation (2) determines (in

fulfilling the commutation relations

and representing the colour, the weak and the hyper charge. The corresponding vector gauge fields

All vector gauge fields, appearing in the first line of Equation (2), except

The condensate,

The 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 (^{4} (with the sum of both equal to

Let us conclude this Subsection with the recognition that:

A. It is (only) one scalar condensate of two right handed neutrinos (

B. There are (only) the nonzero vacuum expectation values of the scalar gauge fields with the space index

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 (

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.

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 (

state | ||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

0 | 0 | 0 | 1 | −1 | 0 | 0 | 0 | 1 | −1 | 0 | 0 | 0 | 1 | |

0 | 0 | 0 | 0 | −1 | −1 | −1 | 0 | 1 | −1 | 0 | 0 | 0 | 1 | |

0 | 0 | 0 | −1 | −1 | −2 | −2 | 0 | 1 | −1 | 0 | 0 | 0 | 1 |

state | spin | |||||
---|---|---|---|---|---|---|

0 | 0 | 0 | ||||

0 | 0 | -1 | ||||

0 | 0 | 0 | ||||

0 | 0 | +1 |

Let us assume the infinitesimal coordinate transformations of the kind [

where we have made a choice of the symmetry

It follows for the vielbeins representing the background field

The background field in

where

From

Statement: These two vector gauge fields are just the superposition of

To prove this statement let us express the operators, appearing in Equation (5), as follows

Then we use the relation between the

Let us now put the vielbeins

Repeating equivalent calculations for the rest of components of

It is proven in this section that all the scalar gauge fields with the space index

It turnes out [

To see this one must take into account that the infinitesimal generators

determine spins of spinors, while

determine family charges of spinors (Equation (15)), while

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

All scalars carry correspondingly, besides the quantum numbers determined by the space index, also the quantum numbers

Statement: Scalar fields with the space index (7, 8) carry with respect to this space index the weak and the hyper charge (

To prove this statement let me introduce a common notation

Here

Let us make a choice of the superposition of the scalar fields so that they are eigenstates of

sition appears by itself if one rewrites the second line of Equation (2) as follows (the momentum ^{5})

with the summation over

The application of the operators

Since

These superpositions of

by

The operators

transform one member of a doublet from

This completes the proof of the above statement.

After the appearance of the condensate (^{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

The operator

Let me pay attention to the reader, that the term

handed ^{7}, which can, due to the properties of the scalar fields (Equation (19)), be interpreted also in the standard model way, namely, that

quark, while ^{th} line, which transforms under

the action of ^{th} line.

The operator ^{th} line into ^{th} line, where

The term

(35))) of the Higgs of the standard model. If

the same equation, then the operators

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 (

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.

It is not difficult to show that the scalar fields

Let us do this for

One finds

with

Similarly one finds properties with respect to the

The mass matrix of any family member, belonging to any of the two groups of the four families, manifests - due to the

Let us summarize this section: It is proven that all the scalar fields with the scalar index

by the standard model for the Higgs’s scalar (Equation (19)):

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 ([

where

All the other scalar fields:

The gauge fields with the space index

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.

This section discusses properties of vectors, tensors and spinors, appearing in the action in Equation (1), for

The presentation is based on Refs. [

where

The Clifford algebra objects have properties (Equation (49))

Either the coordinates

where

We see that

The linear vector space over the coordinate Grassmann space has the dimension

Grassmann coordinates in Equation (45) can be replaced by one of the Clifford algebra objects, let say by

provided that operation of

where

With this definition the relations from Equations ((47), (50)-(53)) remain valid. If

It is still true that the infinitesimal generators of the Lorentz transformations for vectors are

The two tangent spaces have the same metric tensors:

Let us transform any two vectors

Here

In Appendix A3 relations among the vielbeins

the spin connection fields,

rivative of the vielbeins is equal to zero (Equation (60)) relates the two affine connections,

Varying the action in Equation (1) with respect to

Variation of the action with respect to

One notices from Equations ((31), (32)) that if there are no spinor sources, then both spin connections―

The expressions for the two spin connection fields [

The condensate (

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 [

weak break, members of the two weak doublets (

These scalars (Equation (17)) interact besides through the weak and the hyper charge (determined by the space index

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 (

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

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

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 [

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 [

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 (some) of those questions remains so far the open question.

The author acknowledges funding of the Slovenian Research Agency, which terminated in December 2014.

Norma Susana MankočBorštnik, (2015) The Explanation for the Origin of the Higgs Scalar and for the Yukawa Couplings by the Spin-Charge-Family Theory. Journal of Modern Physics,06,2244-2274. doi: 10.4236/jmp.2015.615230

This section follows the similar section in Refs. [

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

For a chosen group the same coefficients

While

One finds [

where the generators

determine representations of the

determine representations of

One correspondingly finds the generators of the subgroups of

which determine representations of the two

determine representations of

The corresponding expressions for the generators of the above subgroups defining the representations of the corresponding gauge fields follow if replacing

One further defines the operators for the charges

The corresponding operators which apply on the corresponding gauge fields follow from the above relations, if either

The scalar fields, responsible [

of Equation (2). These scalar fields are included in the covariant derivatives as

One finds the scalar fields carrying the quantum numbers of the subgroups of the family groups, expressed in terms of

The expressions for the scalars, expressed in terms of

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 (

one finds the vector gauge fields in the “tilde” sector, or one just uses the expressions from Equations ((41), (40)), if replacing the scalar index s with the vector index m.

In this section the Lorentz transformations of

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 TransformationsI 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

is determined by a vector

leaves forms

invariant. While

A linear vector space over the coordinate Grassmann space has the dimension

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

The factors in front of the superposition of

and equivalently

An infinitesimal Lorentz transformation of the proper ortochronous Lorentz group is then

where

Let us write the operator of finite Lorentz transformations as follows

We see that the coordinates

Correspondingly one finds that compositions like

Also objects like

Relations among the vielbeins

The two kinds of vectors,

We express, after the parallel transport^{8} of each of these two kinds of vectors (belonging to two tangent spaces) from

where

The difference between the two vectors

We define the parallel transport also for the two kinds of vectors

The difference between the two vectors

The affine connection

When requiring that

Equation (60) relates the two affine connections,

Let us now vary the action Equation (1) with respect to

Multiplying both equations of Equation (32) by

The expression for the spin connection

Again one notices that if there are no spinor sources, carrying the spinor quantum numbers

Variation of the action Equation (1) with respect to

This appendix is a short review (taken from [

In this last stage we rewrite a spinor basis, written in Ref. [

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.

Appendix A2 briefly represents the starting point [

The objects

If B is a Clifford algebra object, let say a polynomial of

where

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

We assume the “Hermiticity” property for

in order that

One finds from Equation (66) that

Recognizing from Equation (65) that the two Clifford algebra objects

The choice for the Cartan subalgebra in

One proceeds equivalently for

algebra objects

To make the technique simple we introduce the graphic presentation as follows

where

One can easily check by taking into account the Clifford algebra relation (Equation (63)) and the definition of

which means that we get the same objects back multiplied by the constant

From Equation (72) it follows

From Equation (73) we conclude that

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

one recognizes that

Recognizing that

we define a vacuum state

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

All the states have the same handedness

spect to the group

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

While

Making a choice of the Cartan subalgebra set (Equation (67)) of the algebra

charge (2/3) and colour

This state is an eigenstate of all

The operators

charge (2/3) and the colour charge

Below some useful relations [

I present at the end one Weyl representation of

One Weyl representation of

The eight families of the first member of the eight-plet of quarks from

The eight-plets separate into two group of four families: One group contains doublets with respect to

The scalar fields which are the gauge scalars of

i | ||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

(Anti)octet, | ||||||||||||

1 | 1 | 1 | 0 | |||||||||

2 | 1 | 1 | 0 | |||||||||

3 | 1 | 1 | 0 | |||||||||

4 | 1 | 1 | 0 | |||||||||

5 | −1 | −1 | 0 | |||||||||

6 | −1 | −1 | 0 | |||||||||

7 | −1 | −1 | 0 | |||||||||

8 | −1 | −1 | 0 | |||||||||

9 | 1 | 1 | 0 | |||||||||

10 | 1 | 1 | 0 | |||||||||

∙∙∙ | ||||||||||||

17 | 1 | 1 | 0 | 0 | ||||||||

18 | 1 | 1 | 0 | 0 | ||||||||

∙∙∙ | ||||||||||||

25 | 1 | 1 | 0 | 0 | 0 | 0 | 0 | |||||

26 | 1 | 1 | 0 | 0 | 0 | 0 | 0 | |||||

27 | 1 | 1 | 0 | 0 | 0 | −1 | −1 |

28 | 1 | 1 | 0 | 0 | 0 | −1 | −1 | |||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

29 | −1 | −1 | 0 | 0 | 0 | −1 | ||||||

30 | −1 | −1 | 0 | 0 | 0 | −1 | ||||||

31 | −1 | −1 | 0 | 0 | 0 | 0 | ||||||

32 | −1 | −1 | 0 | 0 | 0 | 0 | ||||||

33 | −1 | 1 | 0 | |||||||||

34 | −1 | 1 | 0 | |||||||||

35 | −1 | 1 | 0 | |||||||||

36 | −1 | 1 | 0 | |||||||||

37 | 1 | −1 | 0 | |||||||||

38 | 1 | −1 | 0 | |||||||||

39 | 1 | −1 | 0 | |||||||||

40 | 1 | −1 | 0 | |||||||||

41 | −1 | 1 | 0 | |||||||||

∙∙∙ | ||||||||||||

49 | −1 | 1 | 0 | 0 | ||||||||

∙∙∙ | ||||||||||||

57 | −1 | 1 | 0 | 0 | 0 | 1 | 1 | |||||

58 | −1 | 1 | 0 | 0 | 0 | 1 | 1 | |||||

59 | −1 | 1 | 0 | 0 | 0 | 0 | 0 | |||||

60 | −1 | 1 | 0 | 0 | 0 | 0 | 0 | |||||

61 | 1 | −1 | 0 | 0 | 0 | 0 | ||||||

62 | 1 | −1 | 0 | 0 | 0 | 0 | ||||||

63 | 1 | −1 | 0 | 0 | 0 | 1 | ||||||

64 | 1 | −1 | 0 | 0 | 0 | 1 |

I | 0 | 0 | |||||||

I | 0 | 0 | |||||||

I | 0 | 0 | |||||||

I | 0 | 0 | |||||||

II | 0 | 0 | |||||||

II | 0 | 0 | |||||||

II | 0 | 0 | |||||||

II | 0 | 0 |

doublets with respect to these two groups. The scalar fields which are the gauge scalars of