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CKM and PMNS mixing matrices from discrete subgroups of SU(2).

1 Introduction

The very successful Standard Model (SM) local gauge group SU[(2).sub.L] x U[(1).sub.[gamma]] x SU[(3).sub.C] defines an electroweak (EW) interaction part and a color interaction part. Experiments have determined that the left-handed EW isospin flavor states are linear superpositions of mass eigenstates. One of the greatest challenges in particle physics is to determine the first principles origin of the quark and lepton mixing matrices CKM and PMNS that relate the flavor states to the mass states.

In a recent article [1] I derived the lepton PMNS mixing matrix by using the quaternion (i.e., spinor) generators of three specific discrete (i.e., finite) binary rotational subgroups of the EW gauge group SU[(2).sub.L] x U[(1).sub.[gamma]], one group for each lepton family, while remaining within the realm of the SM lagrangian. All the derived PMNS matrix element values are within the 1[sigma] range of the empirically determined absolute values.

The three lepton family groups, binary rotational groups called [3,3,2], [4,3,2], and [5,3,2], (or 2T, 2O, and 2I), have discrete rotational symmetries in [R.sup.3]. Each group has two degenerate basis states which must be taken in linear superposition to form the two orthogonal fermion flavor states in each family, i.e., ([v.sub.e], e), ([v.sub.[mu]], [mu]), and ([v.sub.[tau]], [tau]).

In order to have a consistent geometrical approach toward understanding the SM, I have proposed in a series of articles [2-4] over several years that the quark flavor states represent discrete binary rotational groups also. However, one must move up one spatial dimension from [R.sup.3] to [R.sup.4] and use the related four discrete binary rotational subgroups [3,3,3], [4,3,3], [3,4,3], and [5,3,3], (or 5-cell, 16-cell, 24-cell, and 600-cell), for the quarks, thereby dictating four quark families. Recall that both [R.sup.3] and [R.sup.4] are subspaces of the unitary space [C.sup.2].

Therefore, following up the success I had deriving the neutrino PMNS matrix, the CKM mixing matrix should be derivable by using the same geometrical method, i.e., based upon the quaternion generators of the four groups of specific discrete rotational symmetries. In this quark case, however, first one determines a 4x4 mixing matrix called CKM4 and then extracts the appropriate 3x3 submatrix as the traditional CKM matrix.

These seven closely-related groups representing specific discrete rotational symmetries dictate the three known lepton families in R3 and four related quark families in [R.sup.4], the fourth quark family still to be discovered. That is, neither leptons nor quarks are to be considered as point objects at the fundamental Planck scale of about [10.sup.-35] meters. If this geometrical derivation of both the PMNS and CKM mixing matrices is based upon the correct reason for the mixing of flavor states to make the mass states, then one must reconcile the empirical data with the prediction of a fourth quark family.

My proposal that leptons are 3-D entities and that quarks are 4-D entities has several advantages. There is a clear distinction between leptons and quarks determined by inherent geometrical properties such as explaining that leptons do not experience the color interaction via SU[(3).sub.c] because gluons and quarks would involve 4-D rotations associated with the three color charges defined in [R.sup.4]. Also, one now has a geometrical reason for there being more than one family of leptons and of quarks. In addition, the mass ratios of the fundamental fermions are determined by the group relationships to the j-invariant of the Monster Group. These physical properties and many other physical consequences are discussed in my previous papers.

2 Review of the PMNS matrix derivation

This section reviews the mathematical procedure used in my 2013 derivation [1] of the PMNS matrix from first principles. One constructs the three SU(2) generators, the [U.sub.1] = j, [U.sub.2] = k, and the [U.sub.3] = i, (i.e., the Pauli matrices in quaternion form), from the three quaternion generators from each of the discrete subgroups [3,3,2], [4,3,2], and [5,3,2]. As you know, the three Pauli matrices, i.e., the quaternions i, j, and k, can generate all rotations in [R.sup.3] about a chosen axis or, equivalently, all rotations in the plane perpendicular to this axis. For example, the quaternion k is a binary rotation by 180[degrees] in the i-j plane.

The complete mathematical description [5] for the generators operating on the unit vector x in R3 extending from the origin to the surface of the unit sphere S2 is given by [R.sub.s] = i x [U.sub.s] where s = 1, 2, 3 and

[U.sub.1] = j, [U.sub.2] = -icos [[pi]/q] -jcos [[pi]/p] + k sin [[pi]/h], [U.sub.3] = i, (1)

with h = 4, 6, 10 for the three lepton flavor groups [p, q, 2], respectively. Their U2 generators are listed in Table 1.

My three lepton family binary rotational groups, [3, 3, 2], [4,3,2], and [5,3,2], all have generators [U.sub.1] = j and [U.sub.3] = i, but each [U.sub.2] is a different quaternion generator operating in [R.sup.3]. One obtains the correct neutrino PMNS mixing angles from the linear superposition of their [U.sub.2]'s by making the total [U.sub.2] = k, agreeing with SU(2). This particular combination of three discrete angle rotations is now equivalent to a rotation in the i-j plane by the quaternion k.

The sum of all three [U.sub.2] generators should be k, so there are three equations for the three unknown factors, which are determined to be: -5.537, 16.773, and -11.236. Let the quantity [phi] = ([square root of 5] + 1)/2, the golden ratio. The resulting angles in Table 1 are the arccosines of these factors (normalized), i.e., their projections to the k-axis, but they are twice the rotation angles required in [R.sup.3], a property of quaternion rotations.

Using one-half of these angles produces

[[theta].sub.1] = 52.67[degrees], [[theta].sub.2] = 18.38[degrees], [[theta].sub.3] = 61.23[degrees], (2)

resulting in mixing angles

[[theta].sub.12] = 34.29[degrees], [[theta].sub.13] = -8.56[degrees], [[theta].sub.23] = -42.85[degrees]. (3)

The absolute values of these mixing angles are all within the 1[sigma] range of their values for the normal mass hierarchy [6-11] as determined from several experiments:

[[theta].sub.12] = [+ or -] 34.47[degrees], [[theta].sub.13] = [+ or -] 8.73[degrees], [[theta].sub.23] = [+ or -] (38.39[degrees]-45.81[degrees]). (4)

The experimental 1[sigma] uncertainty in 612 is about 6%, in [[theta].sub.13] about 14%, and [[theta].sub.23] has the range given. The [+ or -] signs arise from the squares of the sines of the angles determined by the experiments.

For three lepton families, one has the neutrino flavor states [v.sub.e], [v.sub.[mu]], [v.sub.T], and the mass states [v.sub.1], [v.sub.2], [v.sub.3], related by the PMNS matrix [V.sub.ij]


The PMNS entries are the products of the sines and cosines of the derived angles (3) using the standard parametrization of the matrix, producing:


For direct comparison, the empirically estimated PMNS matrix for the normal hierarchy of neutrino masses is


Comparing the [V.sub.e3] elements from each, the phase angle 6 is confined to be 0[degrees] [less than or equal to] [delta] [less than or equal to] [+ or -]14.8[degrees], an angle in agreement with the T2K collaboration value of [delta] [approximately equal to] 0 but quite different from other proposed [delta] [approximately equal to] [pi] values.

3 The CKM4 matrix derivation

The success of the above geometrical procedure for deriving the lepton PMNS matrix by using the quaternion generators from the 3 discrete binary rotation groups demands that the same approach should work for the quark families in R4 using the 4 discrete binary rotation groups [3, 3, 3], [4,3,3], [3,4,3], and [5, 3, 3]. If this procedure succeeds in deriving the CKM matrix elements as a 3 x 3 submatrix of CKM4, then a fourth sequential quark family, call its quark states b' and t', exists in Nature.

These 4 binary rotational groups for the quark family flavors each have rotation subgroups of SO(4) = SO(3) x SO(3), and they also have the double covering SU(2) x SU(2). The SO(4) is the rotation group of the unit hypersphere [S.sup.3] in [R.sup.4], with every 4-D rotation being simultaneous rotations in two orthogonal planes.

The only finite (i.e., discrete) quaternion groups are [12]

2I, 2O, 2T, [2D.sub.2n], 2Cn, 1Cn (n odd) (5)

with the 2 in front meaning binary (double) group, the double cover of the normal 3-D rotation group by SU(2) over SO(3). Mathematically, the 4 discrete binary groups for the quark families each can be identified as (L/[L.sub.K]; R/[R.sub.K]) with the homomorphism L/[L.sub.K] = R/[R.sub.K]. Here L and R are specific discrete groups of quaternions and LK and RK are their kernels.

P. DuVal [13] established that one only needs the cyclic groups 2Cn and 1Cn when considering the four discrete rotational symmetry groups, i.e., the ones I am using for the quark families. Essentially, vertices on the 4-D regular polytope can be projected to be a regular polygon on each of the two orthogonal planes in R4.

There will be 6 quaternion generators for each of the 4 groups, producing simultaneous rotations in two orthogonal planes. The two sets of Pauli matrices for producing continuous rotations can be identified as i, j, k, and another i, j, k, but they act on the two different S2 spheres, i.e, in the two orthogonal planes. One can consider this 4-D rotational transformation as the result of a bi-quaternion operation [14], or equivalently, a bi-spinor or Ivanenko-Landau-Kahler spinor or Dirac-Kahler spinor operation.

For three quark families, one has the "down" flavor states d', s', b', and their mass states d, s, b, related by the CKM matrix. This quark mixing matrix for the left-handed components is defined in the standard way as

V = [U.sub.L][D.sup.[dagger].sub.L], (6)

but for four quark families the mathematics is a little different, for one must consider the bi-quaternion case in which there will be Bogoliubov mixing [14], producing two subfactors for each component, i.e.,

[U.sub.L] = [W.sup.u.sub.14,23][W.sup.u.sub.12,34], [D.sub.L] = [W.sup.d.sub.14,23][W.sup.d.sub.12,34] (7)

with the [W.sup.u] and [W.sup.d] factor on the right mixing the 1st and 2nd generations and, separately, mixing the 3rd and 4th generations. The Bogoliubov mixing in the factor on the left mixes the 1st and 4th generations and, separately, the 2nd and 3rd generations. Therefore, the CKM4 matrix derives from

[V.sub.CKM4] = [U.sub.L][D.sup.[dagger].sub.L] = [W.sup.u.sub.14,23] [W.sup.u.sub.12,34], ([W.sup.d.sub.14,23] [W.sup.d.sub.12,34]). (8)

The product [W.sup.u.sub.12,34] [W.sup.d[dagger].sub.12,34] is given by


The upper left block is an SU(2) matrix that mixes generations 1 and 2 while the lower right block is an SU(2) matrix that mixes generations 3 and 4. Each 2x2 block relates the rotation angles and the phases via


The 4x4 matrix that achieves the Bogoliubov mixing has four possible forms for the four possible isospin cases obeying SU(2) x SU(2): (0, 0), (1/2, 0), (0, 1/2), and (1/2, 1/2). The (1/2, 1/2) is the one for equal, simultaneous, isospin 1/2 rotations in the two orthogonal planes for CKM4:


Multiplying out these three 4x4 bi-quaternion mixing matrices, one determines that


in which the phases [alpha], [beta], [gamma], [delta] have been ignored.

One determines the angles 61 and 62 from the quaternion generators of the 4 discrete binary rotation groups for the quark families. Projections of each of the four discrete symmetry 4-D entities onto the two orthogonal planes produces a regular polygon [5,13] with the generator iexp[2[pi]j/h], as given in Table 2, where the h values are 5, 8, 12, 30, for the [3,3,3], [4,3,3], [3,4,3], and [5,3,3], respectively.

Again, we need to determine the contribution from each group generator that will make the sum add to 180[degrees], i.e., make their collective action produce the rotation [U.sub.2] = k. Expanding out the exponentials in terms of sines and cosines reveals four unknowns but only two equations. Alternately, because the four rotation angles sum to only 159[degrees], we can use the same factor for each group, i.e., the ratio 180[degrees]/159[degrees] = 1.132.

In the last column of Table 2 are the normalized angles which are twice the angle required. Therefore, taking the appropriate half-angle differences produces the mixing angles

[[theta].sub.1] = 15.282[degrees], [[theta].sub.2] = 10.188[degrees]. (9)

Substituting the cosines and sines of these two derived angles into the CKM4 matrix form above produces a mixing matrix symmetrical about the diagonal. Remember that I have ignored up to eight possible phases in the 2x2 blocks.


One can compare the upper left 3x3 submatrix to the most recent estimated absolute values [7]


Note that most of these estimated [V.sub.CKM] values are probably good to within a few percent but some could have uncertainties as large as 10% or more.

Of concern are my low values of 0.2203 for [] and []. However, according to the Particle Data Group (2013) there are two possible values [7]: 0.2253 and 0.2204, the latter from tau decays. Also, my derived symmetric CKM4 matrix [V.sub.ub] value is high while the [] value is reasonable, i.e., [] at 0.0098 compares well with the estimated value of 0.0088.

The [V.sub.tb] element of CKM4 is 0.9744, quite a bit smaller than the suggested 0.9991 [V.sub.tb] value for the 3x3 CKM matrix. However, if one imposes the unitarity condition on the rows and columns of the extracted CKM, the new value for this [V.sub.tb] matrix element would be 0.999, in agreement.

My final comment is that if one calculates CKM using only the first three quark groups [3,3,3], [4,3,3], and [3,4,3], the resulting 3x3 CKM matrix will disagree significantly with the known CKM matrix. Therefore, one cannot eliminate a fourth quark family when discrete rotational subgroups of SU(2) are considered.

4 Discussion

In the SM the EW symmetry group is the Lie group SU(2)L x U(1)y. This local gauge group operating on the lepton and quark states works extremely well, meaning that all its predictions agree with experiments so far. However, in this context there is no reason for Nature to have more than one fermion family, and certainly no reason for having 3 lepton families and at least 3 quark families. As far as I know, the normal interpretation of the SM provides no answer that dictates the actual number of families, although the upper limit of 3 lepton families with low mass neutrinos is well established via [Z.sup.0] decays and via analysis of the CMB background.

My geometrical approach with discrete symmetries alters the default reliance upon SU(2) and its continuous symmetry transformations, for I utilize discrete binary rotational subgroups of SU(2) for the fundamental fermion states, a different subgroup for each lepton family and for each quark family. In this scenario one can surmise that the enormous success of the SM occurs because SU[(2).sub.L] x U[(1).sub.[gamma]] is acting like a mathematical "cover group" for the actual underlying discrete rotations operating on the lepton states and quark states.

Assuming that the above matrix derivations are correct, the important question is: Where is the b' quark of the predicted 4th quark family? In 1992 I predicted a top quark mass of about 160 GeV, a b' quark mass of 65-80 GeV, and a t' quark at a whopping 2600 GeV. These mass predictions were based upon the mass ratios being determined by the j-invariant function of elliptic modular functions and of fractional linear transformations, i.e., Mobius transformations. Note that all seven discrete groups I have for the fermions are related to the j-invariant and Mobius transformations, which have direct connections to numerous areas of fundamental mathematics.

With a predicted b' mass that is much smaller than the top quark mass of 173.3 GeV and even smaller than the W mass at 80.4 GeV, one would have expected some production of the b' at LEP, Fermilab, and the LHC. Yet, no clear indication of the b' quark has appeared.

Perhaps the b' quark has escaped detection at the LHC and lies hidden in the stored data from the runs at 7 TeV and 8 TeV. With a mass value below the W and Z masses, the b' quark must decay via flavor changing neutral current (FCNC) decay channels [16] such as b' [right arrow] b + [gamma] and b' [right arrow] b + gluon. The b' could have an average lifetime too long for the colliders to have detected a reasonable number of its decays within the detector volumes and/or the energy and angle cuts. However, the b' quark and t' quark would affect certain other decays that depend upon the heaviest "top" quark in a box diagram or penguin diagram.

Another possibility is that a long lifetime might allow the formation of the quarkonium bound state b'-anti-b', which has its own specific decay modes, to bb-bar, gg, [gamma][gamma], and WW* [right arrow] vvll. Depending upon the actual quarkonium bound state, the spin and parity [J.sup.PC] = [0.sup.++] or [0.sup.-+].

And finally, there is an important theoretical problem associated with the mismatch of three lepton families to four quark families, e.g., the famous triangle anomalies do not cancel in the normal manner. Perhaps my fundamental leptons and quarks, being extended particles into 3 and 4 dimensions, respectively, can avoid this problem which occurs for point particles. Someone would need to work on this possibility.

5 The bigger picture!

We know that the SM is an excellent approximation for understanding the behavior of leptons, quarks, and the interaction bosons in the lower energy region when the spatial resolution is less than [10.sup.-24] meters. At smaller distance scales, perhaps one needs to consider a discrete space-time, for which the discrete binary rotation groups that I have suggested for the fundamental particles would be appropriate. Quite possibly, with this slight change in emphasis to discrete subgroups of the local gauge group, the SM lagrangian will hold true all the way down to the Planck scale.

If indeed the SM applies at the Planck scale, then one can show [2] that the Monster group dictates all of physics! The surprising consequence: The Universe is mathematics and is unique. Indeed, we humans are mathematics!

This connection to the Monster Group is present already in determining the lepton and quark mass ratios, which are proportional to the j-invariant of elliptic modular functions, the same j-invariant that is the partition function for the Monster Group in a quantum field theory [17].

The mathematics of these discrete groups does even more for us, for there is a direct connection [2] from the lepton groups [3, 3, 2], [4, 3, 2], [5, 3, 2], and the quark groups [3, 3, 3], [4, 3, 3], [3, 4, 3], [5, 3, 3], in R3 and R4, respectively, via special quaternions called icosians to the discrete space R8. One then brings in another R8 for relativistic space-time transformations. The two spaces combine into a 10-D discrete space-time obeying the discrete symmetry transformations of "Weyl" SO(9,1) = Weyl [E.sub.8] x Weyl [E.sub.8]. This proposed unique connection to "Weyl" SO(9,1) was a surprise to me because one has two 8-D spaces combining to make a 10-D spacetime! Its direct and unique relationship to the SM certainly is a welcome replacement to the [10.sup.500] ways for M-theory.

Finally, among the advantages to having a fourth family of quarks is a possible explanation of the baryon asymmetry of the Universe (BAU). From the CKM and the PMNS matrices, one learns that the predicted CP violation (CPV) is at least 10 orders of magnitude too small to explain the BAU. That is, the important quantity called the Jarlskog value is much too small. But a 4th quark family resolves this issue [18] because substituting the fourth quark family mass values into the Jarlskog expression increases the CPV value by more than 1013! Voila. One now has penguin diagrams distinguishing the particle and antiparticle decays with sufficient difference to have the particle Universe we experience.

6 Conclusion

The quark mixing matrix CKM4 has been derived using four quark families. Using quaternion generators from four specific related discrete binary rotational groups [3,3,3], [4,3,3], [3,4,3], and [5,3,3], I have derived the quark CKM4 and its CKM submatrix. However, neither quark of the 4th quark family has been detected at the colliders. Their appearance could mean that the Standard Model lagrangian might be a good approximation to the ultimate lagrangian all the way down to the Planck scale if space-time is discrete.


The author wishes to thank for financial support and encouragement.

Submitted on April 2, 2014/Accepted on April 10, 2014


[1.] Potter, F. Geometrical Derivation of the Lepton PMNS Matrix Values. Progress in Physics, 2013, v. 9 (3), 29-30.

[2.] Potter, F. Our Mathematical Universe: I. How the Monster Group Dictates All of Physics. Progress in Physics, 2011, v. 7 (4), 47-54.

[3.] Potter, F. Unification of Interactions in Discrete Spacetime. Progress in Physics, 2006, v. 2 (1), 3-9.

[4.] Potter, F. Geometrical Basis for the Standard Model. International Journal of Theoretical Physics, 1994, v. 33, 279-305.

[5.] Coxeter, H. S. M. Regular Complex Polytopes. Cambridge University Press, Cambridge, 1974.

[6.] An, F. P. et al. (Daya Bay Collaboration). Spectral Measurement of Electron Antineutrino Oscillation Amplitude and Frequency at Daya Bay. Physical Review Letters, 2014, v. 112, 061801. arXiv:1310.6732.

[7.] Beringer, J. et al. (Particle Data Group). The Review of Particle Physics: Vud, Vus, the Cabbibo Angle, and CKM Unitarity. Physical Review, 2012 and 2013 partial update, v. D86, 010001, 6-7.

[8.] Capozzi, F., Fogli, G. J., et al. Status of three-neutrino oscillation parameters, circa 2013. arXiv:1312.2878v1.

[9.] Fogli, G.I. Global analysis of neutrino masses, mixings and phases: Entering the era of leptonic CP violation searches. Physical Review, 2012, v. D86, 013012. arXiv:1205.5254v3.

[10.] Forero, D. V., Tortola, M., Valle, J. W. F. Global status of neutrino oscillation parameters after Neutrino-2012. Physical Review, 2012, v. D86, 073012. arXiv:1205.4018.

[11.] T2K Collaboration, Abe, K. et al. Indication of Electron Neutrino Appearance from an Accelerator-produced Off-axis Muon Neutrino Beam. Physical Review Letters, 2011, v. 107, 041801. arXiv:1106.2822.

[12.] Conway, J. H., Smith, D. A. On Quaternions and Octonions: Their Geometry, Arithmetic, and Symmetry. A.K. Peters, Wellesley, Massachusetts, 2003.

[13.] Du Val, P. Homographies, Quaternions, and Rotations. Oxford University Press, Oxford, 1964.

[14.] Jourjine, A. Scalar Spin of Elementary Fermions. Physics Letters, 2014, v. B728, 347-357. arXiv:1307.2694.

[15.] Beringer, J. et al. (Particle Data Group). The Review of Particle Physics: Neutrino Mass, Mixing and Oscillations. Physical Review, 2012 and 2013 partial update, v. D86, 010001, 46-48.

[16.] Arhrib, A., Hou, W.S. CP Violation in Fourth Generation Quark Decays. Physical Review, 2009, v. D80, 076005. arXiv:0908.0901v1.

[17.] Witten, E. Three-Dimensional Gravity Reconsidered. arXiv:0706.3359.

[18.] Hou, W. S. Source of CP Violation for the Baryon Asymmetry of the Universe. International Journal of Modern Physics, 2011, v. D20, 1521-1532. arXiv:1101.2161v1.

Franklin Potter, 8642 Marvale Drive, Huntington Beach, CA 92646 USA E-mail:
Table 1: Lepton Family Quaternion Generators [U.sub.2]

Fam. Grp. Generator Factor Angle

[v.sub.e], e 332 -[1/2]i - [1/2]j -0.2645 105.337
 + [1/[square root
 of 2]]k

[v.sub.[mu]], 432 -[1/2]i - [1/[square 0.8012 36.755
[mu] root of 2]]k + [1/2]j

[v.sub.[tau]], 532 -[1/2]i - [[phi]/2]j -0.5367 122.459
[tau] + [[[phi].sup.1]/2]k

Table 2: Quark Family Discrete Group Assignments for [U.sub.2]

Fam. Grp. Generator Angle Factor Angle
 [degrees] [degrees]

u, d 333 exp[2[pi]i/5] 72 1.132 81.504
c, s 433 exp[2[pi]i/8] 45 1.132 50.940
t, b 343 exp[2[pi]i/12] 30 1.132 33.960
t', b' 533 exp[2[pi]i/30] 12 1.132 13.584
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Date:Jul 1, 2014
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