Can we predict the fourth family masses for quarks and leptons?
CCan we predict the fourth family masses for quarks and leptons?
G. Bregar, N.S. Mankoˇc BorˇstnikDepartment of Physics, FMF, University of Ljubljana,Jadranska 19, SI-1000 Ljubljana, Slovenia
In the ref. [1–4] four massless families of quarks and leptons before the electroweak breakare predicted. Mass matrices of all the family members demonstrate in this proposal the samesymmetry, determined by the family groups. There are scalar fields - two SU (2) triplets,the gauge fields of the family quantum numbers, and three singlets, the gauge fields of thethree charges ( Q, Q (cid:48) and Y (cid:48) )- all doublets with respect to the weak charge, which determinemass matrices on the tree level and, together with other contributions, also beyond the treelevel. The symmetry of mass matrices remains unchanged for all loop corrections. Thethree singlets are, in loop corrections also together with other contributors, responsible forthe differences in properties of the family members. Taking into account by the spin-charge-family theory proposed symmetry of mass matrices for all the family members and simplifyingstudy by assuming that mass matrices are Hermitian and real and mixing matrices real, wefit free parameters of mass matrices to experimental data within the experimental accuracy.Calculations are in progress. I. INTRODUCTION
There are several attempts in the literature to reconstruct mass matrices of quarks and leptonsout of the observed masses and mixing matrices and correspondingly to learn more about propertiesof fermion families [8]. The most popular is the n × n matrix, close to the democratic one, predictingthat ( n −
1) families must be very light in comparison with the n th one. Most of attempts treatneutrinos differently than the other family members, relying on the Majorana part, the Dirac partand the ”sea-saw” mechanism. Most often are the number of families taken to be equal to thenumber of the so far observed families, while symmetries of mass matrices are chosen in severaldifferent ways [9]. Also possibilities with four families are discussed [12].In this paper we follow the prediction of the spin-charge-family theory [1–4, 7] that there are fourmassless families above the electroweak break and that the scalar fields - the two triplets carryingthe family charges in the adjoint representations and the three singlets carrying the charges of thefamily members ( Q, Q (cid:48) and Y (cid:48) ) - all doublets with respect to the weak charge, cause (after gettingnonzero vacuum expectation values) the electroweak break. Assuming that the contributions of all a r X i v : . [ h e p - ph ] D ec the scalar (and in loop corrections also of other) fields to mass matrices of fermions are real andsymmetric, we are left with the following symmetry of mass matrices M α = − a − a e d be − a − a b dd b a − a eb d e a − a α , (1)the same for all the family members α ∈ { u, d, ν, e } . In appendix A 1 the evaluation of this massmatrix is presented and the symmetry commented. A change of phases of the left handed andthe right handed basis - there are (2 n −
1) free choices - manifests in a change of phases of massmatrices.The differences in the properties of the family members originate in the different charges ofthe family members and correspondingly in the different couplings to the corresponding scalar andgauge fields.We fit (sect. III B) the mass matrix Eq. (1) with 6 free parameters of any family member 6to the so far observed properties of quarks and leptons within the experimental accuracy. Thatis:
For a pair of either quarks or leptons, we fit twice free parameters of the two mass matricesto twice three so far measured masses and to the corresponding mixing matrix . Since we have thesame number of free parameters (two times 6 for each pair, since the mass matrices are assumedto be real) as there are measured quantities (two times 3 masses and 6 angles of the orthogonalmixing matrix under a simplification that the mixing matrix is real and Hermitian), we wouldpredict the fourth family masses uniquely, provided that the measured quantities are accurate.The n − n ≥
4. Theexperimental inaccuracy enable to determine only the interval for the fourth family masses.If the prediction of the spin-charge-family theory, that there are four families which manifest inthe massless basis the symmetry of Eq. (1), is correct, we expect that enough accurate experimentaldata for the properties of the so far observed three families will offer narrow enough intervals forthe fourth family masses.We treat all the family members, the quarks and the leptons, equivalently. We also estimatethe contributions of the fourth family members to the mesons decays in dependence of the fourthfamily masses, taking into account also the estimations of the refs. [15]. However, we must admitthat our estimations are so far pretty rough.In sect. III A we check on a toy model how accurate must be the experimental data that enablethe prediction of the fourth family masses: For two ”known” mass matrices, obeying the symmetryof Eq. (1), which lead approximately to the experimental data, we calculate masses and the mixingmatrix. Then, taking the mixing matrix and twice three lower masses as an input, we look back forthe starting two mass matrices with the required symmetry, allowing for the three lower families”experimental” inaccuracy. In the same section we then estimate the fourth family masses. So farthe results are preliminary. Although we spent quite a lot of efforts to make the results transparentand trustable, the numerical procedure to take into account the experimental inaccuracy of datais not yet good enough to allow us to determine the interval of the fourth family masses, even notfor quarks, so that all the results are very preliminary.Still we can say that the so far obtained support the prediction of the spin-charge-family theorythat there are four families of quarks and leptons, the mass matrices of which manifest the symmetrydetermined by the family groups – the same for all the family members, quarks and leptons. Themass matrices are quite close to the ”democratic” ones, in particular for leptons.Since the mass matrices offer an insight into the properties of the scalar fields, which determinemass matrices (together with other fields), manifesting effectively as the observed Higgs and theYukawa couplings, we hope to learn about the properties of these scalar fields also from the massmatrices of quarks and leptons.In appendix A we offer a very brief introduction into the spin-charge-family theory, which thereader, accepting the proposed symmetry of mass matrices without knowing the origin of thissymmetry, can skip.In sect. II the procedure to fit free parameters of mass matrices (Eq. (1) to the experimentaldata is discussed. We comment our studies in sect. IV.
II. PROCEDURE USED TO FIT FREE PARAMETERS OF MASS MATRICES TOEXPERIMENTAL DATA
Matrices, following from the spin-charge-family theory might not be Hermitian (appendix B).We, however, simplify our study, presented in this paper, by assuming that the mass matrix forany family member, that is for the quarks and the leptons, is real and symmetric. We take thesimplest phases up to signs, which depend on the choice of phases of the basic states, as discussedin appendices A 1 [19].The matrix elements of mass matrices, with the loop corrections in all orders taken into account,manifesting the symmetry of Eq. (1), are in this paper taken as free parameters.Let us first briefly overview properties of mixing matrices, a more detailed explanation of whichcan be found in subsection II A of this section.Let M α , α denotes the family member ( α = u, d, ν, e ), be the mass matrix in the massless basis(with all loop corrections taken into account). Let V αβ = S α S β † , where α represents either the u -quark and β the d -quark, or α represents the ν -lepton and β the e -lepton, denotes a (in generalunitary) mixing matrix of a particular pair.For n × n matrix ( n = 4 in our case) it follows:i. If a known submatrix ( n − × ( n −
1) of an unitary matrix n × n with n ≥ n × n , the n unitarity conditions determine (2(2( n −
1) + 1)) real unknownscompletely. If the submatrix ( n − × ( n −
1) of an unitary matrix is made unitary by itself, thenwe loose the information.ii. If the mixing matrix is assumed to be orthogonal, then the ( n − × ( n −
1) submatrix containsall the information about the n × n orthogonal matrix to which it belongs and the n ( n + 1) / n −
1) + 1 real unknowns completely for any n .If the submatrix of the orthogonal matrix is made orthogonal by itself, then we loose the informa-tion.We make in this paper, to simplify the present study, several assumptions [7], presented alreadyin the introduction. In what follows we present the procedure used in our study and repeat theassumptions.1. If the mass matrix M α is Hermitian, then the unitary matrices S α and T α , introduced inappendix B to diagonalize a non Hermitian mass matrix, differ only in phase factors de-pending on phases of basic vectors and manifesting in two diagonal matrices, F α S and F α T ,corresponding to the left handed and the right handed basis, respectively. For Hermitianmass matrices we therefore have: T α = S α F α S F α T † . By changing phases of basic vectorswe can change phases of (2 n −
1) matrix elements.2. We take the diagonal matrices M αd and the mixing matrices V αβ from the available experi-mental data. The mass matrices M α in Eq. (1) have, if they are Hermitian and real, 6 freereal parameters ( a α , a α , a α , b α , e α , d α ).3. We limit the number of free parameters of the mass matrix of each family member α bytaking into account n relations among free parameters, in our case n = 4, determined by theinvariants I α = − (cid:88) i =1 , m αi , I α = (cid:88) i>j =1 , m αi m αj ,I α = − (cid:88) i>j>k =1 , m αi m αj m αk , I α = m α m α m α m α , (2)which are expressions appearing at powers of λ α , λ α + λ α I + λ α I + λ α I + λ α I = 0, in theeigenvalue equation. The invariants are fixed, within the experimental accuracy of the data,by the observed masses of quarks and leptons and by the fourth family mass, if we make achoice of it. In appendix II B we present the relations among the reduced number of freeparameters for a chosen m α . There are (6 −
4) free parameters left for each mass matrix.4. The diagonalizing matrices S α and S β , each depending on the reduced number of free param-eters, are for real and symmetric mass matrices orthogonal. They follow from the procedure M α = S α M αd T α † , T α = S α F α S F α T † , M αd = ( m α , m α , m α , m α ) , (3)provided that S α and S β fit the experimentally observed mixing matrices V † αβ within theexperimental accuracy and that M α and M β manifest the symmetry presented in Eq. (1).We keep the symmetry of the mass matrices accurate. One can proceed in two ways. A. : S β = V † αβ S α , B. : S α = V αβ S β ,A. : V † αβ S α M βd S α † V αβ = M β , B. : V αβ S β M αd S β † V † αβ = M α . (4)In the case A. one obtains from Eq. (3), after requiring that the mass matrix M α has thedesired symmetry, the matrix S α and the mass matrix M α (= S α M αd S α † ), from where weget the mass matrix M β = V † αβ S α M βd S α † V αβ . In case B. one obtains equivalently thematrix S β , from where we get M α (= V αβ S β M αd S β † V † αβ ). We use both ways iterativelytaking into account the experimental accuracy of masses and mixing matrices.5. Under the assumption of the present study that the mass matrices are real and symmetric,the orthogonal diagonalizing matrices S α and S β form the orthogonal mixing matrix V αβ ,which depends on at most 6 (= n ( n − ) free real parameters (appendix B). Since, due towhat we have explained at the beginning of this section, the experimentally measured matrixelements of the 3 × × × × × Q α , Q (cid:48) α , Y (cid:48) α ), which in loop correctionstogether with other contributors in all orders contribute to all mass matrix elements andcause the difference among family members [20].Let us conclude. If the mass matrix of a family member obeys the symmetry required by the spin-charge-family theory, which in a simplified version (as it is taken in this study) is real andsymmetric, the matrix elements of the mixing matrices of quarks and leptons are correspondinglyreal, each of them with n ( n − free parameters. These six parameters of each mixing matrix are,within the experimental inaccuracy, determined by the three times three experimentally determinedsubmatrix. After taking into account three so far measured masses of each family member, thesix parameters of each mass matrix reduce to three. Twice three free parameters are within theexperimental accuracy correspondingly determined by the 3 × × × × × A. Submatrices and their extensions to unitary and orthogonal matrices
In this appendix well known properties of n × n matrices, extended from ( n − × ( n − n × n complex matrix has 2 n free parameters. The n + 2 n ( n − / n (= 2 n − ( n + 2 n ( n − / n − × ( n −
1) known submatrix of the unitary matrix. The submatrixcan be extended to the unitary matrix by (2 × [2( n −
1) + 1]) real parameters of the last columnand last line. The n unitarity conditions on the whole matrix reduce the number of unknowns to(2(2 n − − n ). For n = 4 and higher the ( n − × ( n −
1) submatrix contains all the informationabout the unitary n × n matrix. The ref. [6] proposes a possible extension of an ( n − × ( n − V ( n − n − into n × n unitary matrices V nn .The choice of phases of the left and the right basic states which determine the unitary matrix(like this is the case with the mixing matrices of quarks and leptons) reduces the number of freeparameters for (2 n − n − (2 n − n ( n −
1) real parameters and ( n − n − n − n ( n − − (2 n − n × n matrix has n free parameters which the n ( n + 1) orthogonality conditionsreduce to n ( n − n − × ( n −
1) submatrix of this orthogonal matrix can be extendedto this n × n orthogonal matrix with [2( n −
1) + 1] real parameters. The n ( n + 1) orthogonalityconditions reduce these [2( n −
1) + 1] free parameters to (2 n − − n ( n + 1)), which means thatthe ( n − × ( n −
1) submatrix of an n × n orthogonal matrix determine properties of its n × n orthogonal matrix completely. Any ( n − × ( n −
1) submatrix of an orthogonal matrix containsall the information about the whole matrix for any n . Making the submatrix of the orthogonalmatrix orthogonal by itself one looses the information about the n × n orthogonal matrix. B. Free parameters of mass matrices after taken into account invariants
It is useful for numerical evaluation purposes to take into account for each family member itsmass matrix invariants (sect. 2), expressible with three within the experimental accuracy knownmasses, while we keep the fourth one as a free parameter. We shall make a choice of a α instead ofthe fourth family mass.We shall skip in this section the family member index α and introduce new parameters as follows a, b , f = d + e , g = d − e , q = a + a √ , r = a − a √ . (5)After making a choice of a I , that is of the fourth family mass, four invariants of Eq. (2) reducethe number of free parameters to 2. The four invariants therefore relate six parameters leavingthree of them, the a included as a free parameter, undetermined. There are for each pair offamily members the measured mixing matrix elements, assumed in this paper to be orthogonaland correspondingly determined by six parameters, which then fixes these two times 3 parameters.The (accurately enough) measured 3 × × a, b, f, g, q, r ) from Eq. (5) we obtain a = I ,I (cid:48) = − I + 6 a − q − r − b = f + g ,I (cid:48) = − b ( I − aI + 4 a ) = f − g ,I (cid:48) = I − aI + a I − a = 14 ( q − r ) + ( q + r ) b + 12 ( q − r ) · ( ± ) · [ ± ] 2 gf + b ( f + g ) + 14 (2 gf ) . (6)We eliminate, using the first two equations, the parameters f and g , expressing them as functionsof I (cid:48) and I (cid:48) , which depend, for a particular family member, on the three known masses, theparameter a and the three parameters r , q and b . We are left with the four free parameters( a, b, q, r ) and the below relation among these parameters {−
12 ( q + r ) + ( − b + 12 ( − I + 6 a − b ))( q + r )+ ( I (cid:48) −
14 (( − I + 6 a − b ) + I (cid:48) ) + b ( − I + 6 a − b )) } = −
14 ( q − r ) (( − I + 6 a − b − ( q + r )) − I (cid:48) ) , (7)which reduces the number of free parameters to 3. These 3 free parameters must be determined,together with the corresponding three parameters of the partner, from the measured mixing matrix.We eliminate one of the 4 free parameters in Eq. (7) by solving the cubic equation for, let usmake a choice, q αq + βq + γq + δ = 0 . (8)Parameter ( α, β, γ, δ ) depend on the 3 free remaining parameters ( a, b, r ) and the three, withinexperimental accuracy, known masses.To reduce the number of free parameters from the starting 6 in Eq. (1) to the 3 left after takinginto account invariants of each mass matrix, we look for the solution of Eq (8) for all allowed valuesfor ( a, b, r ). We make a choice for a in the interval of ( a min , a max ), determined by the requirementthat a , which solves the equations, is a real number. Allowing only real values for parameters f and g we end up with the equation − I + 6 a − b − ( q + r ) > | I + 8 a − aI b | , (9)which determines the maximal positive b for q = 0 = r and also the minimal positive value for b .For each value of the parameter a the interval ( b min , b max ), as well as the interval ( r min = 0 , r max ),follow when taking into account experimental values for the three lower masses. III. NUMERICAL RESULTS
Taking into account the assumptions and the procedure explained in sect. II and in the ref. [7] weare looking for the 4 × × × × × × × × A. Checking on a toy model how much does the symmetry of mass matrices (Eq. (1)) limitthe fourth family properties
We check in this subsection on a toy model the reproducibility of the starting two mass matricesfrom the known two times three lower masses (say m u i , m d i , i = (1 , , × V ud ) i,j , i, j = (1 , , × m u i , m d i , i = (1 , ,
3) and ( V ud ) i,j , i, j = (1 , , M toy u = . . . . . . . . . . . . . . . . , M toy d = . . . . . . . . . . . . . . . . . (10)Diagonalizing these two mass matrices we find the following twice four masses M toy u d /M eV /c = (1 . , ., ., . ) , M toy d d /M eV /c = (2 . , ., ., . ) , (11)and the mixing matrix V toy ud = − . − . − . . . − . − . − . . . − . − . . . − . . . (12)In order to simulate experimental inaccuracies (intervals of values for twice three lower massesand for the matrix elements of the 3 × × m u in the interval ((300 − m u /GeV 300 500 600 650 700 800 1200”exp. inacc”/ σ σ of the mixing matrix elements of the 3 × m toy u -quark. m toy d massis kept equal to 700 GeV. × × σ ’s [21]. We keep in Table I the d mass equal to 700 GeV.Let us add that the accuracy, with which the 3 × × m toy d than it does on m toy u in this toy model case.We use this experience when evaluating intervals, within which the fourth family masses appearwhen taking into account the inaccuracies of the experimental data. B. Numerical results for the observed quarks and leptons with mass matrices obeyingEq. (1)
We take for the quark and lepton masses the experimental values [16], recalculated to the Z boson mass scale. We take from [16] also the experimentally declared inaccuracies for the so farmeasured 3 × × × spin-charge-family theory.Although the accurate enough mixing matrices and masses of quarks and leptons are essentialfor the prediction of the fourth family members masses, we still hope that even with the presentaccuracy of the experimental data the intervals for the fourth family masses shall not be too large,in particular not for quarks, for which the data are much more accurate than for leptons. Let uspoint out that from so far obtained results we are not yet able to predict the fourth family massintervals, which would be reliable enough.2We therefore present some preliminary results. Let us point out that all the mass matricesmanifest within a factor less then 2 the ”democratic” view. This is, as expected, more and morethe case, the higher might be the fourth family masses, and in particular is true for the leptons. • For quarks we take [16]:1. The quark mixing matrix [16] V ud = S u S d † | V ud | = . ± . . ± . . ± . | V u d | . ± .
011 1 . ± .
023 0 . ± . | V u d | . ± . . ± . . ± . | V u d || V u d | | V u d | | V u d | | V u d | , (13)determining for each assumed and experimentally allowed set of values for the mixingmatrix elements of the 3 × | V u i d | and | V u d j | ) from the unitarity condition for the 4 × M Z , while we take the fourthfamily masses as free parameters. We allow the values from 300 GeV up to more thanTeV to see the influence of the experimental inaccuracy on the fourth family masses. M ud / MeV / c = (1 .
27 + 0 . − . , ± ,
171 700 . ± ., m u >
335 000 . ) , M dd / MeV / c = (2 .
90 + 1 . − . ,
55 + 16 − , . ± ., m d >
300 000 . ) . (14) • For leptons we take [16]:1. We evaluate 3 × . · − ≤ ∆( m / MeV / c ) ≤ . · − , . · − ≤ ∆( m (31) , (32) / MeV / c ) ≤ . · − , . ≤ sin θ ≤ . , . ≤ sin θ ≤ . , sin θ < . . , sin θ = 0 . ± . , (15)which means that π − π ≤ θ ≤ π + π , π . − π ≤ θ ≤ π + π , θ < π .This reflects in the lepton mixing matrix V νe = S ν S e † | V νe | = . . . | V ν e | . . . | V ν e | . . . | V ν e || V ν e | | V ν e | | V ν e | | V ν e | , (16)3determining for each assumed value for any mixing matrix element within the exper-imentally allowed inaccuracy the corresponding fourth family mixing matrix elements( | V ν i e | and | V ν e j | ) from the unitarity condition for the 4 × M νd / MeV / c = (1 · − , · − , · − , m ν >
90 000 . ) , M ed / MeV / c = (0 .
486 570 161 ± .
000 000 042 , .
718 135 9 ± .
000 009 2 , . ± . , m e >
102 000 ) . (17)Following the procedure explained in sect. II we look for the mass matrices for the u -quarks andthe d -quarks and the ν -leptons and the e -leptons by requiring that the mass matrices reproduceexperimental data while manifesting symmetry of Eq. (1), predicted by the spin-charge-family theory.We look for several properties of the obtained mass matrices: i. We test the influence of theexperimentally declared inaccuracy of the 3 × × ii. We look forhow could different choices for the masses of the fourth family members limit the inaccuracy ofparticular matrix elements of the mixing matrices or the inaccuracy of the three lower massesof family members. iii.
We test how close to a democratic mass matrix are the obtained massmatrices in dependence of the fourth family masses.The numerical procedure, used in this contribution, is designed for quarks and leptons.In the two next subsections III B 1, III B 2 we present some preliminary results for 4 × spin-charge-family theory for quarks and leptons, respectively.
1. Mass matrices for quarks
Searching for mass matrices with the symmetries of Eq. (1) to determine the interval for thefourth family quark masses in dependence of the values of the mixing matrix elements within theexperimental inaccuracy, we have not yet found a trustable way to extract which experimentalinaccuracies of the mixing matrix elements should be taken more and which less ”seriously”. We4also need to evaluate more accurately the experimental limitations for the fourth family masses,originating in decay properties of mesons and other experiments. Although in the toy model casethe ”inaccuracy” of the matrix elements leads very clearly to the right fourth family masses, thisis not the case when the experimental data for the 3 × .
02% to 12%. The so far obtained results can not yet make the choice amongless or more trustable experimental values: We can not yet make more accurate choice for thosedata which have large experimental inaccuracies.We are still trying to improve our the procedure of searching for the masses of the fourth familyquarks.Let us still present two cases to demonstrate how do quark mass matrices change with respect tothe fourth family masses: The first two mass matrices lead to the fourth family masses m u = 300GeV and m d = 700 GeV, while the second two lead to the fourth family masses m u = 1 200 GeVand m d = 700 GeV. • M u = . . . . . . . . . . . . . . . . , M d = . . . . . . . . . . . . . . . . , (18) V ud = . . . − . . − . . − . . − . − . − . . . . . . (19)The corresponding masses are M ud / MeV / c = (1 . , . ,
172 000 .,
300 000 . ) , M dd / MeV / c = (2 . , . , . ,
700 000 . ) . (20) • M u = . . . . . . . . . . . . . . . . , M d = . . . . . . . . . . . . . . . . , (21)5 V ud = − . . − . . . . − . . − . − . − . − . . − . − . . . (22)The corresponding masses are M ud / MeV / c = (1 . , . ,
172 000 ., . ) , M dd / MeV / c = (2 . , . , . ,
700 000 . ) . (23)We notice: i. In both cases the required symmetry, Eq. (1), is (on purpose) kept very accurate. ii.
In both cases the mass matrices of quarks look quite close to the ”democratic” matrix, in thesecond case slightly more than in the first case. iii.
The mixing matrix elements are in the second case much closer (within the experimentalvalues are V , V , V and V , almost within the experimental values are V , V and V ) tothe experimentally allowed values, than in the first case (almost within the experimental allowedvalues are only V , V and V ).These results suggest that the fourth family masses m u = 1 200 GeV and m d = 700 GeV aremuch more trustable than m u = 300 GeV and m d = 700 GeV.
2. Mass matrices for leptons
We present here results for leptons, manifesting properties of the lepton mass matrices. Theseresults are less informative than those for quarks, since the experimental results are for leptonsmixing matrix much less accurate than in the case of quarks and also masses are known lessaccurately.We have • M ν =
14 021 .
14 968 .
14 968 . −
14 021 .
14 968 .
15 979 .
15 979 . −
14 968 .
14 968 .
15 979 .
15 979 −
14 968 . −
14 021 . −
14 968 . −
14 968 .
14 021 . , M e =
28 933 .
30 057 .
29 762 . −
27 207 .
30 057 .
32 009 .
31 958 . −
29 762 .
29 762 .
31 958 .
32 009 . −
30 057 . −
27 207 . −
29 762 . −
30 057 .
28 933 . , (24)6which leads to the mixing matrix V νe V νe = . . − . . − . . − . . − . . . . . . . . , (25)and the masses M νd / MeV / c = (5 · − , · − , . · − ,
60 000 . ) , M ed / MeV / c = (0 . , . , .
82 120 000) . (26)We did not adapt lepton masses to Z m mass scale. Zeros (0 . ) for the matrix elementsconcerning the fourth family members means that the values are less than 10 − .We notice: i. The required symmetry, Eq. (1), is kept very accurate. ii.
The mass matrices of leptons are very close to the ”democratic” matrix. iii.
The mixing matrix elements among the first three and the fourth family members are verysmall, what is due to our choice, since the matrix elements of the 3 × × spin-charge-family theory are very inaccurately known. IV. DISCUSSIONS AND CONCLUSIONS
One of the most important open questions in the elementary particle physics is: Where do thefamily originate? Explaining the origin of families would answer the question about the numberof families possibly observable at the low energy regime, about the origin of the scalar field(s) andYukawa couplings and would also explain differences in the fermions properties - the differences inmasses and mixing matrices among family members – quarks and leptons.Assuming that the prediction of the spin-charge-family theory that there are four rather thanso far observed three coupled families, the mass matrices of which demonstrate in the masslessbasis the SU (2) × SU (2) symmetry of Eq. (1), the same for all the family members - the quarksand the leptons - we look in this paper for: i. The origin of differences in the properties of the family members - quarks and leptons. ii.
The allowed intervals for the fourth family masses. iii.
The matrix elements in the mixing matrices among the fourth family members and the three7already measured ones.Our calculations presented here are preliminary and in progress.Let us tell that there are two kinds of the scalar fields in the spin-charge-family theory, re-sponsible for the masses and mixing matrices of quarks and leptons (and consequently also for themasses of the weak gauge fields): The ones which distinguish among the family members and theother ones which distinguish among the families. The differences between quarks and leptons andbetween u and d quarks and between ν and e leptons originate in the first kind of the scalar fields,which carry Q, Q (cid:48) (the two charges which, like in the standard model , originate in the weak andhyper charge) and Y (cid:48) (which originates in the hypercharge and in the fermion quantum number,similarly as in the SO (10) models).The existence of four coupled families seems almost unavoidable for the explanation of theproperties of the neutrino families if all the family members should start from the massless basisin an equivalent way: The 4 × × × spin-charge-family theory) of the two mass matrices withinthe experimental accuracy. The same procedure is used to study either quarks or leptons.
Expected results are not only themass matrices, but also the intervals within which masses of the fourth families should be observedand the corresponding mixing matrices.We developed a special procedure to extract the dependence of the fourth family masses onthe experimental inaccuracy of masses and mixing matrices. Our test of this procedure on a toymodel, in which we first postulate two mass matrices (leading to masses and mixing matrices veryclose to those of quarks), calculate the masses and the mixing matrix, and then from three lowestmasses and the 3 × × Yet the preliminaryresults presented here show, that the masses of the fourth family quarks with m u > lead tothe mixing matrix much closer to the experimental data than does m u ≈ × × Appendix A: A brief presentation of the spin-charge-family theory
We present in this section a very brief introduction into the spin-charge family theory [1–4].The reader can skip this appendix taking by the spin-charge family theory required symmetry ofmass matrices of Eq. (1) as an input to the study of properties of the 4 × spin-charge-family theory is offering a possible explanation for the origin of families and scalarfields, and in addition for the so far observed charges and the corresponding gauge fields.There are, namely, two (only two) kinds of the Clifford algebra objects: One kind, the Dirac γ a , takes care of the spin in d = (3 + 1), while the spin in d ≥ d = (3 + 1) in the low energy regime as the charges. In this part the spin-charge family theory is like the Kaluza-Klein theory, unifying spin (in the low energy regime,otherwise the total angular momentum) and charges, and offering a possible answer to the questionabout the origin of the so far observed charges and correspondingly also about the so far observedgauge fields. The second kind of the Clifford algebra objects, forming the equivalent representationswith respect to the Dirac kind, recognized by one of the authors (SNMB), is responsible for theappearance of families of fermions.There are correspondingly also two kinds of gauge fields, which appear to manifest in d = (3+1)as the so far observed vector gauge fields (the number of - obviously non yet observed - gauge fields9grows with the dimension) and as the scalar gauge fields. The scalar fields are responsible, aftergaining nonzero vacuum expectation values, for the appearance of masses of fermions and gaugebosons. They manifest as the so far observed Higgs [5] and the Yukawa couplings.All the properties of fermions and bosons in the low energy regime originate in the spin-charge-family theory in a simple starting action for massless fields in d = [1 + ( d − f αa and correspondingly with the two kinds of the spin connection fields: with ω abc = f αc ω abα which are the gauge fields of S ab = i ( γ a γ b − γ b γ a ) and with ˜ ω abc = f αc ˜ ω abα whichare the gauge fields of ˜ S ab = i (˜ γ a ˜ γ b − ˜ γ b ˜ γ a ). α, β, . . . is the Einstein index and a, b, . . . is the flatindex. The starting action is the simplest one S = (cid:90) d d x E L f + (cid:90) d d x E ( α R + ˜ α ˜ R ) , L f = 12 ( ¯ ψ γ a p a ψ ) + h.c.p a = f αa p α + 12 E { p α , Ef αa } − , p α = p α − S ab ω abα −
12 ˜ S ab ˜ ω abα , (A1) R = 12 { f α [ a f βb ] ( ω abα,β − ω caα ω cbβ ) } + h.c. , ˜ R = 12 f α [ a f βb ] (˜ ω abα,β − ˜ ω caα ˜ ω cbβ ) + h.c. . (A2)Fermions, coupled to the vielbeins and the two kinds of the spin connection fields, manifest (afterseveral breaks of the starting symmetries) before the electroweak break four massless families ofquarks and leptons , the left handed fermions are weak charged and the right handed ones are weakchargeless. The vielbeins and the two kinds of the spin connection fields manifest effectively as theobserved gauge fields and (those with the scalar indices in d = (1 + 3)) as several scalar fields. Themass matrices of the four family members (quarks and leptons) are after the electroweak breakexpressible on a tree level by the vacuum expectation values of the two kinds of the spin connectionfields and the corresponding vielbeins with the scalar indices ([4, 13]): i. One kind originates in the scalar fields ˜ ω abc , manifesting as the two SU (2) triplets –˜ A ˜ N L is , i = (1 , , , s = (7 , A ˜1 is , i = (1 , , , s = (7 , A ˜4 s , s = (7 ,
8) –contributing equally to all the family members. ii.
The second kind originates in the scalar fields ω abc , manifesting as three singlets – A Qs , A Q (cid:48) s , A Y (cid:48) , s = (7 ,
8) – contributing the same values to all the families and distinguishing amongfamily members. Q and Q (cid:48) are the quantum numbers from the standard model , Y (cid:48) originates inthe second SU (2) (a kind of a right handed ”weak”) charge.All the scalar fields manifest, transforming the right handed quarks and leptons into the corre-sponding left handed ones [23] and contributing also to the masses of the weak bosons, as doublets0with respect to the weak charge. Loop corrections, to which all the scalar and also gauge vectorfields contribute coherently, change contributions of the off-diagonal and diagonal elements on thetree level, keeping the tree level symmetry of mass matrices unchanged [24].
1. Mass matrices on the tree level and beyond which manifest SU (2) × SU (2) symmetry Let us make a choice of a massless basis ψ i , i = (1 , , , α .And let us take into account the two kinds of the operators, which transform the basis vectors intoone another ˜ N iL , i = (1 , , , ˜ τ iL , i = (1 , , , (A3)with the properties ˜ N L ( ψ , ψ , ψ , ψ ) = 12 ( − ψ , ψ , − ψ , ψ ) , ˜ N + L ( ψ , ψ , ψ , ψ ) = ( ψ , , ψ , , ˜ N − L ( ψ , ψ , ψ , ψ ) = (0 , ψ , , ψ ) , ˜ τ ( ψ , ψ , ψ , ψ ) = 12 ( − ψ , − ψ , ψ , ψ ) , ˜ τ + ( ψ , ψ , ψ , ψ ) = ( ψ , ψ , , , , ˜ τ − ( ψ , ψ , ψ , ψ ) = ( 0 , , ψ , ψ ) . (A4)This is indeed what the two SU (2) operators in the spin-charge-family theory do. The gaugescalar fields of these operators determine, together with the corresponding coupling constants, theoff diagonal and diagonal matrix elements on the tree level. In addition to these two kinds of SU (2)scalars there are three U (1) scalars, which distinguish among the family members, contributing onthe tree level the same diagonal matrix elements for all the families. In loop corrections in all ordersthe symmetry of mass matrices remains unchanged, while the three U (1) scalars, contributingcoherently with the two kinds of SU (2) scalars and all the massive fields to all the matrix elements,manifest in off diagonal elements as well. All the scalars are doublets with respect to the weakcharge, contributing to the weak and the hypercharge of the fermions so that they transform theright handed members into the left handed onces.With the above (Eq. (A4) presented choices of phases of the left and the right handed basicstates in the massless basis the mass matrices of all the family members manifest the symmetry,presented in Eq. (1). One easily checks that a change of the phases of the left and the right handedmembers, there are (2 n −
1) possibilities, causes changes in phases of matrix elements in Eq. (1).1
Appendix B: Properties of non Hermitian mass matrices
This pedagogic presentation of well known properties of non Hermitian matrices can be foundin many textbooks, for example [18]. We repeat this topic here only to make our discussionstransparent.Let us take a non Hermitian mass matrix M α as it follows from the spin-charge-family theory, α denotes a family member (index ± used in the main text is dropped).We always can diagonalize a non Hermitian M α with two unitary matrices, S α ( S α † S α = I )and T α ( T α † T α = I ) S α † M α T α = M αd = ( m α . . . m αi . . . m αn ) . (B1)The proof is added below.Changing phases of the basic states, those of the left handed one and those of the right handedone, the new unitary matrices S (cid:48) α = S α F αS and T (cid:48) α = T α F αT change the phase of the elementsof diagonalized mass matrices M αd S (cid:48) α † M α T (cid:48) α = F † αS M αd F αT = diag ( m α e i ( φ αS − φ αT ) . . . m αi e i ( φ αSi − φ αTi ) , . . . m αn e i ( φ αSn − φ αTn ) ) ,F αS = diag ( e − iφ αS , . . . , e − iφ αSi , . . . , e − iφ αSn ) ,F αT = diag ( e − iφ αT , . . . , e − iφ αTi , . . . , e − iφ αTn ) . (B2)In the case that the mass matrix is Hermitian T α can be replaced by S α , but only up to phasesoriginating in the phases of the two basis, the left handed one and the right handed one, since theyremain independent.One can diagonalize the non Hermitian mass matrices in two ways, that is either one diagonalizes M α M α † or M α † M α ( S α † M α T α )( S α † M α T α ) † = S α † M α M α † S α = M α dS , ( S α † M α T α ) † ( S α † M α T α ) = T α † M α † M α T α = M α dT , M α † dS = M αdS , M α † dT = M αdT . (B3)One can prove that M αdS = M αdT . The proof proceeds as follows. Let us define two Hermitian( H αS , H αT ) and two unitary matrices ( U αS , H αT ) H αS = S α M αdS S α † , H αT = T α M α † dT T α † ,U αS = H α − S M α , U αT = H α − T M α † , (B4)2It is easy to show that H α † S = H αS , H α † T = H αT , U αS U α † S = I and U αT U α † T = I . Then it follows S α † H αS S α = M αdS = M α † dS = S α † M α U α − S S α = S α † M α T α ,T α † H αT T α = M αdT = M α † dT = T α † M α † U α − T T α = T α † M α † S α , (B5)where we recognized U α − S S α = T α and U α − T T α = S α . Taking into account Eq. (B2) the startingbasis can be chosen so, that all diagonal masses are real and positive. [1] N.S. Mankoˇc Borˇstnik, Phys. Lett. B 292 (1992) 25; J. Math. Phys. (1993) 3731; Int. J. Theor. Phys.
315 (2001); Modern Phys. Lett.
A 10 (1995) 587, Proceedings of the 13 th Lomonosov conference onElementary Particle Physics in the EVE of LHC, World Scientific, (2009) p. 371-378, hep-ph/0711.4681p.94, arXiv:0912.4532 p.119;[2] A. Borˇstnik, N.S. Mankoˇc Borˇstnik, hep-ph/0401043, hep-ph/0401055, hep-ph/0301029; Phys. Rev.
D74 (2006) 073013, hep-ph/0512062.[3] N.S. Mankoˇc Borˇstnik, J. of Modern Phys. (2013) 823-847, doi:10.4236/jmp.2013.46113,http://arxiv.org/abs/1011.5765, http://arXiv:1012.0224, p. 105-130.[4] N.S. Mankoˇc Borˇstnik, ”Do we have the explanation for the Higgs and Yukawa couplings of the standardmodel ?”, http://arxiv.org/abs/1212.3184v2, (http://arxiv.org/abs/1207.6233).[5] CBC News, Mar 15, 2013 9:05.[6] C. Jarlskog, arxiv:math-ph/0504049, K. Fujii, arXiv:math-ph/0505047v3.[7] G. Bregar, N.S. Mankoˇc Borˇstnik, ”Masses and Mixing Matrices of Quarks Within the Spin-Charge-Family
Theory ”, http://arxiv.org/abs/1212.4055.[8] H. Fritzsch, Phys. Lett.
73 B , 317 (1978); Nucl. Phys.
B 155 (1979) 189, Phys. Lett.
B 184 (1987)391; C.D. Frogatt, H.B. Nielsen, Nucl. Phys.
B 147 (1979) 277. C. Jarlskog, Phys. Rev. Lett. (1985)1039; G.C. Branco and D.-D. Wu, ibid. (1988) 253; H. Harari, Y. Nir, Phys. Lett. B 195 (1987)586; E.A. Paschos, U. Turke, Phys. Rep. (1989) 173; C.H. Albright, Phys. Lett.
B 246 (1990)451; Zhi-Zhong Xing, Phys. Rev.
D 48 (1993) 2349; D.-D. Wu, Phys. Rev.
D 33 (1996) 860; E.J.Chun, A. Lukas, arxiv:9605377v2; B. Stech, Phys. Lett.
B 403 (1997) 114; E. Takasugi, M Yashimura,arxiv:9709367.[9] G.Altarelli, NJP 6 (2004) 106; S. Tatur, J. Bartelski, Phys. Rev.
D74 (2006) 013007, arXiv:0801.0095v3.[10] A. Kleppe, arXiv:1301.3812.[11] S. Rosati, INFN Roma, talk at Miami 2012, Atlass collaboration.[12] J. Erler, P. Langacker, arXiv:1003.3211; W.S. Hou,C.L. Ma, arXiv:1004.2186; Yu.A. Simonov,arXiv:1004.2672; A.N.Rozanov, M.I. Vysotsky, arXiv:1012.1483.[13] D. Lukman, N.S. Mankoˇc Borˇstnik, ”Families of spinors in d = (1 + 5) with zweibein and two kinds ofspin connection fields on an almost S ”, http://arxiv.org/abs/1212.2370. [14] A. Hernandez-Galeana, N.S. Mankoˇc Borˇstnik, ”Masses and Mixing matrices of families of quarks andleptons within the Spin-Charge-Family theory, Predictions beyond the tree level”, arXiv:1112.4368 p.105-130, arXiv:1012.0224 p. 166-176.[15] M.I. Vysotsky, arXiv:1312.0474; A. Lenz, Adv. High Energy Phys. (2013) 910275.[16] Z.Z. Xing, H. Zhang, S. Zhou, Phys. Rev. D 77 (2008) 113016, Beringer et al, Phys. Rev. D 86 (2012)010001, Particle Physics booklet, July 2012, PDG, APS physics.[17] G. Bregar, M. Breskvar, D. Lukman, N.S. Mankoˇc Borˇstnik, hep-ph/0711.4681, New J. of Phys. (2008) 093002, hep-ph/0606159, hep/ph-07082846, hep-ph/0612250, p.25-50.[18] Ta-Pei Cheng, Ling-Fong Li, Gauge theory of elementary particles , Clarendon Press Oxford, 1984.[19] In the ref. [17] we made a similar assumption, except that we allow that the symmetry on the treelevel of mass matrices might be changed in loop corrections. We got in that study dependance of massmatrices and correspondingly mixing matrices for quarks on masses of the fourth family.[20] There are also Majorana like terms contributing in higher order loop corrections [3] which might stronglyinfluence in particular the neutrino mass matrix.[21] We define σ as the difference of the reproduced mixing matrix elements and the exact matrix elements,following from the starting two mass matrices.[22] M.I.Vysotsky and A.Lenz comment in their very recent papers that the fourth family is excludedprovided that one assumes the standard model with one scalar field (the Higgs) and extends the numberof families from three to four while using loop corrections when evaluating the decay properties of theHiggs. We have, however, several scalar fields and first estimates show that the fourth family quarksmight have masses close to 1 TeV.[23] It is the term γ γ s φ Ais , where φ Ais , with s = (7 ,
8) denotes any of the scalar fields, which transforms theright handed fermions into the corresponding left handed partner [3, 4, 13]. This mass term originatesin ¯ ψ γ a p a ψ of the action Eq.(A1), with a = s = (7 ,
8) and p s = f σs ( p σ − ˜ S ab ˜ ω abσ − S st ω stσstσ