On the origin of families of quarks and leptons - predictions for four families

The approach unifying all the internal degrees of freedom--proposed by one of us--is offering a new way of understanding families of quarks and leptons: A part of the starting Lagrange density in d(=1+13), which includes two kinds of spin connection fields--the gauge fields of two types of Clifford algebra objects--transforms the right handed quarks and leptons into the left handed ones manifesting in d=1+3 the Yukawa couplings of the Standard model. We study the influence of the way of breaking symmetries on the Yukawa couplings and estimate properties of the fourth family--the quark masses and the mixing matrix, investigating the possibility that the fourth family of quarks and leptons appears at low enough energies to be observable with the new generation of accelerators.


I. INTRODUCTION
The Standard model of the electroweak and strong interactions (extended by assuming nonzero masses of the neutrinos) fits with around 25 parameters and constraints all the existing experimental data. It leaves, however, unanswered many open questions, among which are also the questions about the origin of families, the Yukawa couplings of quarks and leptons and the corresponding Higgs mechanism. Understanding the mechanism for generating families, their masses and mixing matrices might be one of the most promising ways to physics beyond the Standard model.
The approach unifying spins and charges [1,2,3,4,5,6,7,8,9,10] might-by offering a new way of describing families-give an explanation about the origin of the Yukawa couplings. It was demonstrated in [5,7,8,9] that a left handed SO (1,13) Weyl spinor multiplet includes, if the representation is analyzed in terms of the subgroups SO(1, 3), SU(2), SU (3) and the sum of the two U(1)'s, all the spinors of the Standard model-that is the left handed SU(2) doublets and the right handed SU(2) singlets of quarks and leptons. There are the (two kinds of) spin connection fields and the vielbein fields in d = (1 + 13)−dimensional space, which might manifest-after some appropriate compactifications (or some other kind of making the rest of d − 4 space unobservable at low energies)-in the four dimensional space as all the gauge fields of the known charges, as well as the Yukawa couplings.
The paper [10] analyzes, how do terms, which lead to masses of quarks and leptons, appear in the approach unifying spins and charges as a part of the spin connection and vielbein fields. No Higgs is needed in this approach to "dress" the right handed spinors with the weak charge, since the terms of the starting Lagrangean, which include γ 0 γ s , with s = 7, 8, do the job of a Higgs field. The approach predicts more than three families.
In this paper we study properties of mass matrices (Yukawa couplings) following from the approach unifying spins and charges after assuming some (two) possible breaks of symmetries. To calculate from the starting Lagrangean with only one (or at most two parameters) all the properties of the observed quarks and leptons is a too ambitious project at this moment (even in the limit when gravity can be treated-as it can be in this particular case, since breaks are supposed to occut far bellow the Planck scale-as ordinary gauge fields) because of all possible perturbative and nonperturbative effects. Instead we study the influence of particular breaks of symmetries from the symmetry of the Lagrange density on properties of mass matrices. We make calculations on a tree level and leave the fields determining mass matrices after the assumed breaks as free parameters to be determined by the experimental data. In this way we try to understand how might the right way of breaking symmetries go if our approach has some meaning, and whether the fourth family of quarks and leptons might at all appear at energies observable with new accelerators. We present masses and mixing matrices for the four families of quarks.
There are several attempts in the literature to explain the origin of families. In ref. [14], for example, the authors investigate the possibility that the unification of the charges (described by SO (10)) and the family quantum number (described by SO (8)) within the group SO (18) might be the right way to understand the replication of the family of quarks and leptons at low energies. In ref. [15] the authors assume the Standard model group to the third power to reproduce families. In the approach unifying spins and charges spinors (due to two types of γ a operators) carry two indices, one index takes care of the ordinary spin, the other of the family. There are the ordinary S ab which transform one state of a spinor representation into another state, whileS ab transform the family index.
The approach unifying spins and charges shares with the Kaluza-Klein-like theories the difficulty how to ensure masslessness of spinors in d = 1 + 3 as well as their chiral coupling to the corresponding gauge fields ( [16]). The Kaluza-Klein-like theories are also in danger to manifest in d = 1 + 3 charges of both signs, in disagreement with the experimental data, since in the second quantization procedure spinors of the opposite charges (antiparticles) appear anyhow. We proposed in the ref. [12,13] the boundary condition for spinors in d = 1 + 5 compactified on a finite disk that ensures masslessness of spinors in d = 1 + 3 allowing at the same time charges of only one sign. We hope that such a toy model might be extended to the case of d = 1 + 13.
Although we are far from being able to calculate from the simple starting action in d = 1 + 13 the properties of the families of quarks and leptons as manifested at measurable energies directly-each break causes perturbative and nonperturbative effects, which are by themselves hard problems (not yet solved even in the hadron physics)-the approach manifests several nice features: i) In one Weyl representation in d = 1 + 13 all the quarks and the leptons of one family appear, but only the left handed quarks and leptons are weak charged while the right handed ones are weak chargeless.
ii) The starting Lagrange density offers the mechanism for generating families by including two kinds of the Clifford algebra objects.
iii) It is a part of the starting Lagrange density in d = 1 + 13 which transforms the right handed weak chargeless spinors into the left handed weak charged spinors manifesting the Yukawa couplings of the Standard model.
The assumed breaks of symmetries relate mass matrix elements quite strongly and make accordingly possible predictions about properties of the fourth family of quarks and leptons in dependence of a way of breaking symmetries-after connecting free parameters of mass matrices with the known experimental data.

ANCE OF FAMILIES OF QUARKS AND LEPTONS
This section repeats briefly the approach unifying spins and charges as presented in ref. [10]. We assume that only a left handed Weyl spinor in (1 + 13)-dimensional space exists. A spinor carries only the spin (no charges) and interacts accordingly with only the gauge gravitational fields-with the spin connections and the vielbeins. We assume two kinds of the Clifford algebra objects and allow accordingly two kinds of gauge fields [1,2,3,4,5,6,7,8,9,10]. One kind is the ordinary gauge field (gauging the Poincaré symmetry in d = 1 + 13). The corresponding spin connection fields appear for spinors as gauge fields of S ab (Eq.2) defined in terms of γ a , which are the ordinary Dirac operators These gauge fields manifest at "physical energies" as all the gauge fields of the Standard model, and they also contribute-by connecting the right handed weak chargeless quarks or leptons to the left handed weak charged partners within one family of spinors-to the diagonal terms of mass matrices.
The second kind of gauge fields is in our approach responsible for the appearance of families of spinors and accordingly also for couplings among families, contributing to diagonal matrix elements as well. It might explain, together with the first kind of the spin connection fields, the Yukawa couplings of the Standard model of the electroweak and colour interactions. The corresponding spin connection fields appear for spinors as gauge fields of withγ a as the second kind of the Clifford algebra objects [2,18].
γ s p 0s ψ + the rest. Here , while A Ai m , m = 0, 1, 2, 3, denote the gauge fields (expressible in terms of ω stm ) corresponding to the charges defined by the generators τ Ai . One easily sees from Table I  We are not yet able to answer all these questions. Assuming that particular two ways of breaking symmetries could occur, we are in this paper trying to find out possible connections between breaks of symmetries and the symmetries of the corresponding Yukawa couplings and to predict accordingly what are properties of the fourth family in each of these two ways of breaking symmetries. The larger are the symmetries of the Yukawa couplings after the assumed breaks of symmetries the smaller is the number of free parameters in the Yukawa couplings following from our approach and the more predictive is our approach unifying spins and charges in this simple application of it.
The terms responsible for the Yukawa couplings in our approach can be rearranged to be written in terms of nilpotents 78 (±) as follows with s = 7, 8 and p 0st± = p 0s ∓ ip 0t and that we can writẽ for any c. We can accordingly rewrite − (a,b) Having the spinor basis written in terms of projectors and nilpotents (Table I) and knowing the relations of Eq. (14) it turns out that it is convenient to rewrite the mass term L Y = s=7,8ψ γ s p 0s ψ in Eq.(7) as follows terms of the gauge fields ω abc andω abc . The diagonal matrix elements are expressed as the gauge fields of the operators τ 21 , τ 41 as well as the operatorsÑ 3 ± := 1 2 (S 12 ± iS 03 ),τ 13 := 1 2 (S 56 −S 78 ),τ 21 := 1 2 (S 56 +S 78 ),τ 41 := − 1 3 (S 9 10 +S 11 12 +S 13 14 ). Taking into account with the pairs (m, n) = (0, 3), (1, 2); (s, t) = (5, 6), (7,8), belonging to the Cartan subalgebra and Ω ± = Ω 7 ∓ iΩ 8 , where Ω 7 , Ω 8 mean any of the above fieldsω ab7 ,ω ab8 , . Let us point out that this is true only before any break of the symmetries occur. We repeat that ω abc = f α c ω abα and ω abc = f α cωabα . We have for the non diagonal mass matrix the elements with r = i, if (ab) = (03) and r = 1 otherwise. We simplify the index kl in the exponent of fieldsÃ kl ± ((ac), (bd)) to ±, omitting i. A way of breaking any of the two symmetries -the Poincaré one and the symmetry determined by the generatorsS ab in d = 1 + 13 -strongly influences the Yukawa couplings of Eq.(10), relating the parametersω abc and influencing the coupling constants.
In this paper we assume two ways of breaking symmetries and investigate under which conditions each of these two ways of breaking symmetries leads to up to now measured properties of fermions.

A. Properties of Clifford algebra objects
Since S ab = i 2 γ a γ b , for a = b (for a = b S ab = 0), it is useful to know the following properties of γ a 's, if they are applied on nilpotents and projectors Accordingly, for example, S ac The operators, which are an even product of nilpotents appear to be the raising and lowering operators for a particular pair (ab, cd) belonging to the Cartan subalgebra of the group SO(q, d − q), with q = 1 in our case. There are always four possibilities for products of nilpotents with respect to the sign of (k 1 ) and (k 2 ), since It is useful to have in mind [18,19] the following properties of the nilpotents ab (k): which the reader can easily check if taking into account Eq. (12).

B. Families of spinors
Commuting with S ab ({S ab , S ab } − = 0), the generatorsS ab generate equivalent representations, which we recognize as families. To evaluate the application ofS ab on the starting family, presented in Table I, we take into account the Clifford algebra properties ofγ a . We Accordingly it follows The operators, which are an even product of nilpotents in theγ a sector appear (equivalently as τ ± (ab,cd),k 1 ,k 2 in the S ac sector) as the raising and lowering operators, when a pair (ab), (cd) belongs to the Cartan subalgebra of the algebraS ac , transforming a member of one family into the same member of another family.  [−] 11 12 [+] 13 14 (−), which has all the properties with respect to the operators S ab the same as u c R from Table I.

III. FROM EIGHT TO FOUR FAMILIES OF QUARKS AND LEPTONS
Assuming that the break of the symmetry from SO(1,  Table I (the right handed weak chargeless u c R -quark with spin 1/2, for example, as well as the right handed weak chargeless neutrino with spin 1/2-both differ only in the part which concerns the SU(3) and the U(1) charge (U(1) from SO (6)) and which stay unchanged under the application ofS ab , with (a, b) ∈ {0, 1, 2, 3, 5, 6, 7, 8},) appears in the following 8  [+] || · · · .
The rest seven members of each of the above eight families can be obtained, as in Table I, by the application of the operators S ab on the above particular member (or with the help of the raising and lowering operators τ ± (ab,cd),k 1 ,k 2 ). One easily checks (by checking the quantum numbers represented in Table I)  be justified [12,13].) The break of symmetries influences both, the (Poincaré) symmetry described by S ab and the symmetries described byS ab .
ii. Further breaks lead to two (almost) decoupled massive four families, well separated in masses.
iii. We make estimates on a "tree level". iv. We assume the mass matrices to be real and symmetric expecting that the complexity and the nonsymmetric properties of the mass matrices do not influence considerably masses and the real part of the mixing matrices of quarks and leptons. In this paper we do not study the CP breaking.
The following two ways of breaking symmetries leading to four "low lying" families of We extend this requirement also to the operators ab (k) cd (l). This means that all the fields of the typeÃ kl ± ((ab), (cd)), with either k or l equal to ± and with either (ab) or (cd) equal to (56), are put to zero. Then the eight families split into decoupled two times four families.
One easily sees that the diagonal matrix elements can be chosen in such a way that one of the two four families has much larger diagonal elements then the other (which guarantees correspondingly also much higher masses of the corresponding fermions). Accordingly we are left to study the properties of one four family, decoupled from the other four family.
We present this study in subsection III A.
b. In the second way of breaking symmetries from SO(1, 7) × U(1) × SU(3) to the observed "low energy" sector we assume that no matrix elements of the type S ms ω msc orS smω smc , with m = 0, 1, 2, 3, and s − 5, 6, 7, 8, are allowed. This means that all the matrix elements of the typeÃ kl ± ((ab), (cd)), with either k or l equal to ± and with (ab) equal to (03) or (12) and (cd) equal to (56) or (78), are put to zero. This means that the symmetry SO(1, 7) × U(1) breaks into SO(1, 3) × SO(4) × U(1) and further into SO(1, 3) × U(1). Again the mass matrix of eight families splits into two times decoupled four families. We recognize that in this way of breaking symmetries the diagonal matrix elements of the higher four families are again much larger than the diagonal matrix elements of the lower four families. We study the properties of the four families with the lower diagonal matrix elements in subsection III B.
To simplify the problem we assume in both cases, in a. and in b., that the mass matrices are real and symmetric. To determine free parameters of mass matrices by fitting masses and mixing matrices of four families to the measured values for the three known families within the known accuracy, is by itself quite a demanding task. And we hope that after analyzing two possible breaks of symmetries even such a simplified study can help to understand the origin of families and to predict properties of the fourth family.

A. Four families of quarks in proposal no. I
The assumption that there are no matrix elements of the typeÃ kl ± ((ab), (cd)), with k = ± and l = ± (in all four combinations) and with either (ab) or (cd) equal to (56)

03
(+i) 12 [+] | 56 (+) 78 [+] ||.... (19) and to the corresponding mass matrices presented in Table II and Table III. It is easy to see that the parameters can be chosen so that the second four families, decoupled from the first four, have much higher diagonal matrix elements and determine accordingly fermions of much higher masses.
Let us assume that the mass matrices are real and symmetric (assumption iv. in section III) and in addition that the break of symmetries leads to two heavy and two light families and that the mass matrices are diagonalizable in a two steps process [23,24] so that the first diagonalization transforms the mass matrices into block-diagonal matrices with two 2 × 2 sub-matrices. We follow the ref. [24] (where the reader can find all the details). It is easy to prove that a 4 × 4 matrix is diagonalizable in two steps only if it has a structure Since A and C are assumed to be symmetric 2×2 matrices, so must be B. The parameter k, which is an unknown parameter, has the property that k = k u = −k d , where the index u or d denotes the u and the d quarks, respectively. The above assumption requires that with a, which determines the lower two times two matrices and b the higher two times two matrices after the first step diagonalization. Then the angles of rotations in the u and the d quark case are related: i. For the angle of the first rotation (which leads to two by diagonal matrices) we find tan ϕ u = tan −1 ϕ d , with ϕ u = π 4 − ϕ 2 . ii. For the angles of the second rotations in the sector a and b we correspondingly find for the u-quark a,b ϕ u = π 4 − a,b ϕ 2 and for the d-quark a,b ϕ d = π 4 + a,b ϕ 2 . It is now easy to express all the fieldsω abc in terms of the masses and the parameters k and a,b η u,dω with a u = A I u − 1 2ω 038u + 1 2 ( k 2 − 1 + ( k 2 ) 2 )(ω 078u +ω 387δ ), and equivalently for the d-quarks, where a,b η u stays unchanged (Eq. (20)).
The experimental data offer the masses of six quarks and the corresponding mixing matrix for the three families (within the measured accuracy and the corresponding calculational errors). Due to our assumptions the mixing matrix is real and antisymmetric where with the angles described by the three parameters k, a η u , b η u .
We present numerical results in the next section. The assumptions which we made left us with the problem of fitting twelve parameters for both types of quarks with the experimental data. Since the parameter k, which determines the first step of diagonalization of mass matrices, turns out (experimentally) to be very small, the ratios of the fieldsω abc for uquarks and d-quarks (ω abc ũ ω abc d ) are almost determined with the values a,b η u (Eq. (20)) and we are left with seven parameters, which should be fitted to twice three masses of quarks and (in our simplified case) three angles within the known accuracy.

B. Four families of quarks in proposal no. II
The assumption made in the previous subsection (III A) takes care-in the S ab sectorthat the mass term conserves the electromagnetic charge. The same assumption was made also in theS ab sector.
In this subsection we study the break of the symmetries from SO(1, 7) × U(1) × SU (3) down to SO(1, 3) × U(1) × SU(3) which occurs in the following steps. First we assume that all the matrix elementsÃ kl ± ((ab), (cd)), which have (ab) equal to either (03) or (12) and (cd) equal to either (56) or (78) are equal to zero, which means that the symmetry

03
[+i] We shall see that the parameters of the second four families lead accordingly to much higher masses.
In Eq. (10) The fieldsÃ kl ± ((ab)(cd)) in Eq. (11)  (+)Ã −+ ± . We assume that at the break of SO(4) × U(1) into SU(2) × U(1) appearing at some large scale new fieldsÃ Y ± andÃ Y ′ ± manifest (in a similar way in the Standard model new fields occur when the weak charge breaks) and accordingly also new operators It then follows for theS abω ab± sector of the mass matrix 1 2S Hereg Y =g 4 cosθ 2 ,g Y ′ =g 2 cosθ 2 and tanθ 2 =g 4 g 2 . Let at the weak scale the SU(2) × U(1) break further into U(1) leading again to new fieldsÃ 13 ± =Ã ± sinθ 1 +Z ± cosθ 1 , withẽ =g Y cosθ 1 ,g ′ =g 1 cosθ 1 and tanθ 1 =g Ỹ g 1 . Ifθ 2 appears to be very small andg 2Ã2± ± andg Y ′Ã Y ′ ±Ỹ ′ very large, the second four families (decoupled from the first one) appear to be very heavy in comparison with the first four families. The first four families mass matrix (evaluated on a tree level) for the u-quarks (−) and the d-quarks (+) is presented in Table IV.
In Table IV a ± stands for the contribution to the mass matrices from the S ab ω ab± part (which distinguishes among the members of each particular family) and from the diagonal terms of theS abω ab± part. The mass matrix in Table IV is in general complex. To be able to make an estimate of the properties of the four families of quarks we assume (as in subsection III A) that the mass matrices are real and symmetric. We then treat the elements as they appear in Table IV as free parameters and fit them to the experimental data. Accordingly we rewrite the mass matrix in Table IV in the form presented in Table V. The parameters b ± , c ± , d i± , i = 1, 2, 3 are expressible in terms of the real and symmetric part of the matrix elements of Table IV. We present the way of adjusting parameters to the experimental data for the three known families in the next section.  Table IV, taken in this case to be real and parameterized in a transparent way. −, + denote the u-quarks and the d-quarks, respectively.

IV. NUMERICAL RESULTS
The two types of mass matrices in section III followed from the two assumed ways of Since the problem of deriving the Yukawa couplings explicitly from the starting Lagrange density of the approach unifying spins and charges is very complex, we make in this paper a rough estimation for each of the two proposed breaks of symmetries in order to see whether the approach can be the right way to go beyond the Standard model of the electroweak and colour interactions and what does the approach teach us about the families. We hope that the perturbative and nonperturbative effects manifest at least to some extent in the parameters of the mass matrices, which we leave to be adjusted so that the masses and the mixing matrix for the three known families of quarks agree (within the declaired accuracy) with the experimental data. We also investigate a possibility of making predictions about the properties of the fourth family.

A. Experimental data for quarks
We take the experimental data for the known three families of quarks from ref. [27,28].
We use for masses the data   [27,28] for the parameters k, a η and b η determining the mixing matrices for the four families of quarks is presented.
Predicting four families of quarks and leptons at "physical" energies, we require the unitarity condition for the mixing matrices for four rather than three measured families of quarks [27]     There are six free parameters in each of the two mass matrices. The two off diagonal elements together with three out of four diagonal elements determine the orthogonal transformation, which diagonalizes the mass matrix (subtraction of a constant times the unit matrix does not change the orthogonal transformation). The four times four matrix is diagonizable with the orthogonal transformation depending on six angles (in general with n(n − 1)/2).
We use the Monte-Carlo method to fit the free parameters of each of the two mass matrices to the elements of the quark mixing matrix Eqs.(34) and the quark masses Eqs.(33) of the known three families. One notices that the matrix in Table V splits into two times two matrices, if we put parameters c ± equal to zero. Due to experimental data we expect that c ± must be small. The quark mixing matrix is assumed to be real (but not also symmetric as it was in IV B). Since there are more free parameters than the experimental data to be fitted, we look for the best fit in dependence on the quark masses of the fourth family.

V. DISCUSSIONS AND CONCLUSIONS
We study in this paper whether the approach of one of us [1,2,3,4,5,6,7,8,9]  In one of the two ways of breaking symmetries we assume that there are no matrix elements of the typeÃ kl ± ((ab), (cd)), with k = ± and l = ± (in all four combinations) and with either (ab) or (cd) equal to (56). (In the Poincaré sector such a choice guarantees the conservation of the electromagnetic charge.) We also assume that the mass matrices are symmetric and real (hoping that this assumption does not influence considerably the real part of the mixing matrices and the masses) and diagonalizable in two steps.
In the second choice of breaking the starting symmetries we instead assume that all the matrix elementsÃ kl ± ((ab), (cd)), which have (ab) equal to either (03) or (12) and (cd) equal to (56) or (78) are equal to zero, which means that the symmetry SO(1, 7) × U(1) breaks into SO(1, 3) × SO(4) × U(1). We then break SO(4) × U(1) in the sectorω abs , s = 7, 8, so that at some high scale one of SU(2) in SO(4) × U(1) breaks together with U(1) into SU(2) × U(1) and then-at much lower scale-the break of the symmetry of SU(2) × U (1) to U(1) appears. We again end up with the four families decoupled from the much heavier four families with the quark mass matrices differing strongly from the mass matrices in the first choice. We assume again that the mass matrices are real.
We make the calculations on the tree level, obtaining mass matrices for quarks in both chosen ways parameterized with the fields, whose strengths depend on the way and the scale of breaking symmetries. We let the perturbative and nonperturbative effects to be (at least to some extent) included in the fields, for which we assume that they are free parameters to be determined by fitting the masses and the mixing matrix to the known experimental data within the known accuracy.
The symmetries of the mass matrices in the first chosen way of breaking the starting symmetries locate (after assuming the real and symmetric mass matrices, diagonalizable in two steps) the masses of the four families to be in the region for which the analyzes in refs. [20,21,22] show that it is experimentally allowed. The second choice of breaking the symmetries (although each of the mass matrices having only two off diagonal elements) does not determine the masses of the fourth quark family, leaving these masses as free parameters.
Both choices predict rather strong coupling among the observed three and the fourth family.
The fourth family decouples in the second choice of breaking symmetries from the first three for pretty high values for the fourth family quark masses. The calculated ratio |V td |/|V ts | differs for both assumed ways of breaking the symmetries from the measured one (we obtain |V td |/|V ts | equal to 0.128 − 0.149 in the second case), which is expected since the measured value is obtained with the inclusion of the calculations made under the assumption that there are only three families.
Both chosen ways are very approximate. To say which of the two ways is more trustable, further (more demanding) calculations have to be made. However, it seems quite acceptable that breaks of symmetries go in both sectors-the Poincaré one and the one connected withS ab -through two steps of breaking the symmetries SO(1, 7) × U(1) (as suggested by the second way of breaking symmetries) resembling in both steps the Standard model way of breaking the symmetry in the spinor sector, suggesting that the second way might be the right one, although one can not at all expect that the break of symmetries in both sectors manifests in the same way. This paper is to be understood as a first step to further calculations, which should at the end tell whether our way of describing charges and families is the right way beyond the Standard model of the electroweak and colour interactions.
Let us make a note that the decoupled four families might be candidates for forming the dark matter [29].