Scalar Spin of Elementary Fermions

We show that, using the experimentally observed values of CKM and PMNS mixing matrices, all known elementary fermions can be assigned a new quantum number, the scalar spin, in a unique way. This is achieved without introduction of new degrees of freedom. The assignment implies that tau-neutrino should be an anti-Dirac spinor, while mu-tau leptons and charm-top, strange-bottom quarks form Dirac-anti-Dirac scalar spin doublets. The electron and its neutrino remain as originally described by Dirac.


Introduction
The origin of the quark and lepton mixing matrices and the difference in their textures has been a long-standing puzzle in elementary particle physics. Its resolution remains elusive and it is one of the top three mysteries of the Standard Model [1,2,3,4]. Explanations of the textures of mixing matrices usually employ introduction of extra dynamics using either extra gauge or discrete degrees of freedom or additional space-time dimensions. Most common approaches use extra degrees of freedom with a symmetry that is broken down to some discrete subgroup. By exhaustive search of all discrete symmetry groups it is possible to find a reasonably good fit with the experimentally observed values of the lepto-quark mixing matrices [5]. However, the many parameters that are brought along make such fits less satisfactory. A number of attempts have been also made to derive a unified framework for lepto-quark mixing. Some recent work explores quark-lepton complementarity [6], TBM-Cabibbo mixing [7], the use of  
In this letter we derive a common representation for the quark CKM and the lepton PMNS mixing matrices without introduction of additional degrees of freedom. The difference in textures of the two matrices appears as a result of assignment of lepton and quark pairs to different multiplets of a two element discrete symmetry. The symmetry is not present in the Standard Model (SM). It appears if, instead of the Dirac spinors, we use a bi-spinor 1 representation of fermions discovered by Ivanenko and Landau [10] and further developed in [11,12].
When Dirac degrees of freedom are extracted from bi-spinors the symmetry, called scalar spin, appears automatically as the remnant of the second Lorentz transformation invariance of bi-spinors. However, it acts in the generation space. We show that within the context of bispinor gauge theory [13,14], mixing matrix textures for both quarks and leptons are -2 -essentially unique. Alternatively, the results can be viewed as derivation from known leptoquark mixing matrix textures of unique scalar spin multiplet assignments to all elementary fermions. This paper expands the results in [13,14,15] about the common form of mixing for leptons and quarks.

Scalar Spin and Lepto-Quark Flavor Mix
For convenience, we will work with both three and four generations of elementary fermions, assuming massive Dirac neutrinos. The interplay between three and generations will be made clear below.
After spontaneous symmetry breakdown the free-field lepto-quark part of the SM or SM4 Lagrangian with massive Dirac neutrinos is given by where Since of the SM are arbitrary. One of the most enduring puzzles of the SM is that both its spectrum and mixing seem to exhibit patterns. Within a single generation of lepto-quarks there is a clear exponential dependence of mass on the "size" of the gauge group of symmetry with quarks being the heaviest and neutrinos the lightest. Also the mass splitting between the members of   L SU 2 doublets generally depends on the size of the group. At the same time there is a pronounced difference in texture of mixing matrices for quarks and leptons: while for quarks mixing of the first two generations dominates , for leptons, after the recent discovery of large 13 sin , values of PMNS U are of the same order of magnitude. In this paper we explore a possible solution to the mixing puzzle within the context of bispinor gauge theory, where fermionic degrees of freedom are described by bi-spinors instead of the standard Dirac spinors [13]. Bi-spinor gauge theory has a number of interesting features not present in the SM or any of its extensions. For example it allows explicit mass terms for -3 -fermions in bi-fundamental representations and a realization of supersymmetry that places the observed fermions and bosons in supersymmetry multiplets, thus possibly explaining nonobservation of supersymmetric partners of the SM particles. The current status of quantum field theory of bi-spinors and its perturbation theory is described in [16]. Leaving calculation of mass hierarchy of lepto-quarks till a follow-up publication, here we will concentrate on the mixing puzzle.
In bi-spinor gauge theory the free-field Lagrangian for lepto-quarks comes in three possible forms, involving either a single generation or a generation pair. It is given by  for a Dirac-anti-Dirac doublet spinors. Dirac and anti-Dirac spinors describe a single generation each, while Dirac-anti-Dirac doublet describes two generations, but a single elementary fermion. Any other free-field fermionic Lagrangian can be formed by a adding together arbitrary number of generations the three types above.
Like in the SM, the interacting bi-spinor gauge theory is obtained by minimally gauging the free-field theory. Thus, the only formal difference between Lagrangian (1-3) and Lagrangian (4)(5)(6) and between the corresponding interacting theories is that some generations in bi-spinor theory contribute to the Lagrangians with the negative sign. We will now show that this modification is sufficient to explain the difference in the textures of mixing of quarks and leptons in a unique way.
We will begin with the four-generation case, eventually reducing the number of generations to three. The key observation is that in bi-spinor gauge theory explicit mass matrices u,d M ,  , M l are not arbitrary but have a specific form [16]. Generically, where m is a parameter with dimension of mass, Dimensionless matrix M is also a direct sum of two two-dimensional matrices but now the first summand mixes generations 1 and 3 only, while the second summand mixes generations 2 and 4

Mass degeneracy induced by
reflects the fact that this case describes a single 8component Dirac-anti-Dirac particle, consisting of generation doublet of two algebraic Dirac spinors labeled by an additional quantum number, called scalar spin [16]. The degeneracy is lifted at one-loop level by interactions. The computation of the lifting at one loop will be presented elsewhere.
Having listed all possible mass terms, we turn to the diagonalization procedure. In the SM the diagonalization procedure is a linear transformation in the field generation space. Diagonalization is always possible, since after representing mass matric in its polar decomposition form one can always redefine away the unitary factors, which is possible because the free field kinetic term bilinear form corresponds to the unit matrix that commutes with the unitary matrices used in the field redefinitions.
Mixing matrices in bi-spinor gauge theory are defined exactly the same as in the SM. The quark mixing matrix is defined as where L L D U , are mass basis transforms for the fields in (5) The lepton mixing matrix is defined analogously as where L L N E , define lepton mass basis transforms for the fields in (6) In the SM transformation to mass basis is defined as a transformation that decouples the fields of different generations. After the transformation free field fermionic SM Lagrangian becomes a sum of four Lagrangians each containing the fields for one of the four generations. In bi-spinor theory the situation is somewhat different. While for diagonal mass matrix summand in (9) the diagonalization means exactly the same as in SM, for the   2 R M summand in (9) such diagonalization is impossible, because the corresponding kinetic term bilinear matrix (it would be the unit matrix for the SM with two generations) does not commute with the transformations that diagonalize mass matrices.
Therefore, the definition of diagonalization in the present case has to be modified. Instead of insisting on separating the fermionic free field Lagrangian into four independent terms, for case we will define diagonalization as the transformation that diagonalizes the equations of motion. This definition is sufficient for definition of mass eigenstates [16].
The transformation that decouples equations of motion for where W mixes either indexes 1 and 3 or 2 and 4. Note that the order in which stripping of the unitary factors and application of (16,18) is applied is fixed. First comes stripping of the unitary factors in (7) and then transformation (19). For Therefore, we obtain the complete generic diagonalizing transformation that corresponds to four possible mass matrices (11)(12)(13)(14) is given by A convenient expression for   After such generation swap   q p W , becomes block-diagonal Since is arbitrary we can write down Q T , the generic quark, or L T , the generic lepton mixing matrix as where the upper-left block of L Q W , mixes generations 1 and 2, while its lower-right block mixes generations 3 and 4.
. However, in (23) the first block in   q p W , mixes generations 1 and 3, while the second generations 2 and 4. Explicitly, the four possible matrices   q p W , are given by It follows from (23) that altogether there are 16 possible types of mixing matrices in 4generation bi-spinor theory that differ in texture. They are all parameterized by arbitrary block-diagonal L Q W , For convenience, their explicit forms are listed in the Appendix. We can now compare the 3 3 sub-matrices of the 16 matrices (23) with the experimentally observed SM 3 3 mixing textures and try to find whether any of them provide a reasonable fit. For elementary particles describable by Dirac spinors the generic form of 3 3 unitary mixing matrix has a single CP violating phase . In the most commonly used Chau-Keung parameterization both for quarks and leptons it can be written as  13  23  13  23  12  23  12  13  23  12  23  12   13  23  13  23  12  23  12  13  23  12  23  12   13  13  12  13  12   c  c  e  s  c  s  s  c  e  s  c  c  s  s   c  s  e  s  s  s  c  c  e  s  s  c  c  s   e  s  c  s  c  c The entries of quark CKM and lepton PMNS matrices in the SM are denoted as Determined from the four experimentally measured Volfenstein parameters  , where td V , ts V in (33) are not directly measurable and hence marked by (X). They can be calculated via box diagrams assuming 3 3 unitarity and . For leptons we have from direct measurements of absolute values The PMNS U is close to absolute values of the tri-maximal matrix from its values. We can now examine the 16 possible 4 4 unitary matrices (23) that are listed in the Appendix. We seek two 4 4 matrices such that after cutoff of one of the generations the resulting (non-unitary) 3 3 matrices approximate the experimental data in the best way. It is not to difficult to find that there is an unequivocal best fit choice for both quarks and leptons given by 2  1  2  1  2  1  2  1   2  1  2  1  2  1  2  1   2  1  2  1  2  1  2  1   2  1  2  1  2  1  2  1   4   2   1 (1,1)(1,1) where the "hatted" matrix elements denote a different choice of parameters of the unitary block diagonal matrices in (23) and where we multiplied the last row of 4 matrix with one real parameter and three phases, which can be represented as From (36, 37) we obtain two 3 3 mixing matrices rows and columns via spinbein cutoff. The reason why specifically the third rows and columns are eliminated is simple. It provides the best fit between and experimental data. The elimination is in fact done in a generally covariant way by imposing a generally covariant constraint of the type 0 det   , where  is a quark or lepton bi-spinor field [16,22,23]. We arrive at the final form of 3 3 mixing matrices for the bi-spinor gauge theory with Dirac neutrinos Unlike the general SM mixing matrix in (25), quark bi-spinor mixing matrix has two real parameters and three phases, while lepton bi-spinor mixing matrix has two real parameters and no phases, that is, by adjustment of phases of fermion fields it can be chosen to be real. As we will see below, additional phases may appear as a result of renormalization. The parameter count can be obtained as follows.
Since we have three generations of quarks, there are six re-scaling phases available, which however may be taken to sum to zero, since the Lagrangian is invariant with regard to rescaling with the same phase. Thus, we are left with five phases available for rescaling. For quarks, since we want to preserve the form of (39) we have to fix one of the phases to ensure that after rescaling tb cs has seven independent entries, the -10 -phases of which depend on six phases, three from each of the two   2 U that enter (23). We can define six new phases that are the phases of the last two entries in the first row plus three entries in the second row plus the first entry in the third row. The phase of 2 1 x x  in (39)  blocks in (23) that describe the experimental data the best. We obtain the experimentally observed textures in (25,26) if we take for quarks and leptons Under the assumption (42) the resulting quark mixing matrix (39) correctly predicts that cb ts V V  , while leaving ub td V V , independent. It also predicts that mixing of the third generation with the first two is suppressed. If we assume (42) with the equality sign for leptons then b-PMNS matrix reduces to the TBM matrix (35).
We now notice that the choice of (36, 37) uniquely specifies scalar spin assignments of quarks and leptons in 4-generation bi-spinor gauge theory. We conclude that ' t u  , t c  , ' b d  , and b s  are scalar spin 1/2 DaD doublets. For leptons, e and 4 e have scalar spin zero, where e is a Dirac spinor, while 4 e is an anti-Dirac spinor. At the same time,    -11 -form a scalar spin 1/2 Dirac-anti-Dirac doublet. Note, that DaD doublets q p  in fact must be considered as manifestations of a single particle, where p is a state of q p  with scalar spin up, while q is a state of q p  with scalar spin down. (The direction in the space of scalar spin is determined by interaction with gauge fields.) Therefore, we can assign index 4 , , 1   A to the four generations of b-gauge theory according to  .
Note that in this assignment the conventional numbering of the fourth and the third generations are switched. However, masses of ' , ' b t and 4 , 4 e e ν should not be assumed to be smaller than masses of b t, and τ ν τ, , respectively. Their values are in any case irrelevant, because the dynamics of the fourth generation is cut off from the Lagrangian. The cutoff should not be confused with the well-known effect of decoupling of dynamics due to very large mass of a particle. The cut-off generation four leaves no traces in the dynamics, e.g., in loop calculations. Its presence can be detected only kinematically, through its influence on the form of mixing matrices. After the cutoff we obtain the final assignment in 3-generation bi-SM    [17]. The accuracy of this measurement is not sufficient to distinguish between the exact 3 3 unitarity SM and the approximate 3 3 unitarity of bi-spinor theory. However, increasing accuracy in determination of tb V by two orders of magnitude and that of cd V by one order of magnitude would make it possible.
Despite the encouraging hints from the experiment, there are two problems that stand in the way of treating scalar spin as physical quantity. The first problem is that we have obtained the tree level relation . The second problem is that members of Dirac-anti-Dirac doublets are degenerate in mass. Both mass degeneracy and 0 3  e U contradict observations. Let us consider the two problems in turn.
We begin with . So far we provided a possible explanation of the textures for CKM and PMNS mixing matrices by assuming approximations (42). From the analysis of renormalization of the propagator and inter-generation mixing matrices in the SM [26] it follows the renormalization effects could be as high as one percent. OF course, since in the SM all Yukawa couplings are arbitrary, in the SM this result is of little significance.
-13 - The situation changes in bi-spinor SM. There the tree-level mixing matrix entries are no longer arbitrary but can be grouped according to their order of magnitude: for quarks there are four entries in CKM matrix that are on order or less of a percent in absolute value, while in PMNS matrix there is only one, the 3 e U , which is approximately fifteen percent in value. Therefore, in the bi-spinor gauge theory it is reasonable to try to explain the small observed values in mixing matrices as originating from radiative correction to tree-level values, assuming that at tree level relations (42) are exact. If radiative corrections can modify mixing of the second and third lepton generation, for example, if a   from the right is in fact a slight generalization of the proposal in [27], where one begins with the TBM PMNS matrix and then corrects it by multiplication from the right by a two parameter matrix that is a member of    blocks in (23), if instead of (42) we begin with tree-level for quarks and -14 -   . As far as mass degeneracy of DaD doublets is concerned, it is lifted by the interactions of the doublets with gauge fields, because the interaction term does not commute with the piece of the free Lagrangian that is proportional to  sinh parameter in (10). As a result, there will be corrections to the fermion self-energy that are proportional to  sinh that depend on whether scalar spin is up or down. Hence mass degeneracy is be broken by radiative corrections. Notably, the 1-loop corrections are proportional to coupling constant squared, which implies that mass splitting for quarks should be larger than that for leptons, which is indeed the case. Similar effect takes place at one loop for corrections to the Despite restrictions on mixing matrices, b-SM does not predict masses. Bare masses of Dirac, anti-Dirac, DaD particles and  sinh are free parameters of bi-spinor gauge theory, hence absolute values of masses cannot be predicted. However, the ratio of mass difference to the average mass of a scalar spin doublet would be calculable. Such detailed calculations are beyond the scope of the paper and will be presented elsewhere.

Summary
We have shown that within the framework of bi-spinor gauge theory the measured textures of the lepto-quark mixing matrices lead to unique assignment of scalar spin multiplets to all experimentally observed elementary fermions. The result raises a number of questions, the answers to which at present are mostly lacking. The questions illuminate, -15 -however, further research that needs to be carried out in order to make scalar spin not a experimentally hypothetical but physical quantity that originates from a well defined quantum field theory. Of course, the answers to the questions can also rule out both scalar spin and bispinor gauge theories as alternatives to the standard gauge theories. Let us consider the most obvious questions in turn.
First we summarize. We began with free bi-spinor dynamics. It has been known for some time that, unlike in the SM, bi-spinor gauge theories with left-right asymmetry admit explicit mass terms [13]. This happens, because generically bi-spinors transform in bi-fundamental representations of the gauge group. Subsequently, it was established that the explicit mass terms are severely restricted in their form [16]. The restrictions appear when one extracts from bi-spinors the Dirac degrees of freedom via spinbein decomposition, because the free-field Lagrangian expressed in terms of algebraic Dirac spinors retains remnants of bi-spinor transformation property of bi-spinors. Thus, scalar spin appears as a residue of symmetry with regard to the Lorentz transformation applied to the second bi-spinor Dirac index.
The results presented in this paper are based on the tree-level analysis of this free-field fermionic bi-spinor Lagrangian, which differs from free-field fermionic Lagrangian of the SM by the fact that the massless part of the Lagrangian for some of generations enters the Lagrangian with the negative sign. They indicate that, using the experimental data on quark and lepton mixing, scalar spin value can be assigned to all known fermions in a unique way. Naturally, a question arises whether one can take the gauge group of the SM and construct its bi-spinor analog using minimal gauging of the bi-spinor free field Lagrangian as is done with the SM? This of course can be easily done at tree level. Assuming that full quantum field theory for bi-spinor gauge theories is constructed, what would be relation of such theory, let us call it b-SM, to the SM and what would be its phenomenological consequences? Would it be consistent with the experimental data, which the SM fits so well?
Unfortunately, at this stage in the development of quantum bi-spinor gauge theory it is not possible to answer questions, answers to which rely on loop calculations. The reason for this is that quantum field theory of bi-spinor and gauge fields differs from that of spinor and gauge fields. Therefore, a careful analysis and construction of general bi-spinor gauge theory is needed before detailed loop calculations can be carried out. Hence, here we will restrict ourselves to very general arguments. For progress on bi-spinor QFT we refer to [16].
As far as general arguments are concerned, massless or explicitly massive bi-spinor gauge theory obtained from massless or massive free-field bi-spinor theory by minimal gauging is renormalizable by power count. All of its coupling constants are dimensionless. Clearly minimal gauging would affect not only electroweak but the QCD interactions as well: the massless part of the interacting fermionic action of some generations of quarks and, separately, of leptons could enter the total action with the negative sign. Minimal gauging for such generations would add interaction terms with coupling constants negative of those in the analogous SM. Nevertheless, coupling in b-SM is as universal as in the SM: gauge fields couple to fermions with coupling constants that have the same sign relative to free-field Lagrangian.
The next natural question is about the role of Higgs field in the theory. In the SM all masses both for gauge and fermionic fields are generated as a result of the existence of nonzero vacuum expectation value of Higgs field. Bi-spinor SM seems to offer an alternative for fermions. There fermionic masses can appear as the result of normalization of spinbeins, in a process that is purely kinematical in origin, where a surrogate Higgs field doublet appears from spinbeins. How to combine the standard Higgs effect with the kinematic Higgs effect in b-SM is not clear at the moment. It is also not clear, whether the Higgs doublet should be complex or real. Supersymmetry in bi-spinor gauge theory requires that spin zero counterpart of fermionic members of a chiral supermultiplet is real [22,23].
We emphasize, that although we formally started with four generations and in the derivation of the results the fourth generation is essential for fitting the experimental data, the fourth generation of quarks and leptons is excluded from the effective dynamics of Dirac degrees of freedom in b-SM not because of standard decoupling argument, which is based heavy mass of the fourth generation, but because the fourth generation is cut off from the dynamics by the choice of degenerate spinbein. As a result, the fourth generation does not enter the Lagrangian, except in CKM and PMNS mixing matrices. Because the fourth generation is present in the theory only kinematically and not dynamically, it does not contribute to loop integrals at all. Hence it does not have to be decoupled.
We saw above that electron and electron neutrino are the only particles in bi-spinor SM that are described by the standard Dirac spinor action. In b-SM the Dirac's quantum theory of electron would still hold. At the same time    and their neutrinos are not described by Dirac theory but form two DaD doublets of scalar spin ½. As for quarks, all of them are members of DaD doublets, where by  we symbolically denoted the fourth generation states cut off by the spinbein. Apparently, there exists a difference between the SM and bi-spinor SM. Is the difference physical? Can it be detected?
The difference is physical, because anti-Dirac (DaD) particles could couple differently to fields of integer spin than Dirac particles, but detection of the difference would not be straightforward. This is because the most pronounced differences between Dirac and anti-Dirac (DaD) particles would appear in the amplitudes where contributions of Dirac and non-Dirac particles would create an interference effect. This could be difficult to detect because of large differences in particle masses. The differences in mass would lead to suppression of the contribution to the amplitude of the lighter particle by the ration of two masses. It follows then that the interference terms would at most contribute some percentage points to the amplitudes and even less to the scattering probabilities.
But what about the precision electroweak measurements, and the S,T,U parameters [28] that are designed to detect in electroweak vacuum polarization contributions of yet unseen heavy particles? Even with the S, T, U parameters the situation presently is not so clear. This is because the hypothetical bi-spinor SM is not an extension of the SM: the propagators of anti-Dirac and DaD doublets differ from the standard Feynman propagators of Dirac particles. As a result, the S, T, U parameters for bi-spinor SM are not compatible with those of the SM. This means that to enforce the EW constraints one has to re-derive the whole machinery of oblique corrections and then compare the experimental values of new S, T, U bi-SM parameters with their theoretical predictions for bi-spinor SM. This work is in progress.
Before one carries out the construction of full b-SM and computes S, T, U parameters, however, there are two urgent problems to solve. The first problem is that we have obtained . The second problem is that members of Dirac-anti-Dirac doublets are degenerate in mass. Both contradict observations. As was outlined at the end of the preceding section both 0 3  e U and mass degeneracy should be cured by one loop corrections, the detailed calculations, however, are beyond the scope of the present work.
In summary, the results presented here can be best be considered as providing motivation for carrying out further work in rewriting the spinor part of the standard QFT in terms of bispinors and constructing a well-defined b-SM that includes Higgs sector. Of course, this work is conditioned on the satisfactory resolution of 0 3  e U and DaD mass degeneracy problems.