In TGD framework CKM mixing is induced by the mixing of the topologies of 2-D partonic surfaces characterized by genus g=0,1,2 (the number handles added to sphere to obtain topology of partonic 2-surface) assignable to quarks and also leptons (see this and this). The first three genera are special since they allow a global conformal symmetry always whereas higher genera allow it only for special values of conformal moduli. This suggests that handles behave like free particles in many particle state that for higher genera and for three lowest genera the analog of bound state is in question.

The mixing is in general different for different charge states of quark or lepton so that for quarks the unitary mixing matrices for U and type quarks - call them simply U and D - are different. Same applies in leptonic sector. CKM mixing matrix is determined by the topological mixing being of form CKM=UD^{†} for quarks and of similar form for charged leptons and neutrinos.

The usual time-dependent neutrino mixing would correspond to the topological mixing. The time constancy assumed for CKM matrix for quarks must be consistent with the time dependence of U and D. Therefore one should have U= U_{1}X(t) and D= D_{1}X(t), where U_{1} and D_{1} are time independent unitary matrices.

In the adelic approach to TGD (see this and this) fusing real and various p-adic physics (correlates for cognition) would have elements in some algebraic extension of rationals inducing extensions of various p-adic number fields. The number theoretical universality of U_{1} and D_{1} matrices is very powerful constraint. U_{1} and D_{1} would be expressible in terms of roots of unity and e (e^{p} is ordinary p-adic number so that p-adic extension is finite-dimensional) and would not allow exponential representation. These matrices would be constant for given algebraic extension of rationals.

It must be emphasized that the model for quark mixing developed for about 2 decades ago treats quarks as constituent quarks with rather larger masses determining hadron mass (constituent quark is identified as current valence quark plus its color magnetic body carrying most of the mass). The number theoretic assumptions about the mixing matrices are not consistent with the recent view: instead of roots of unity trigonometric functions reducing to rational numbers (Pythagorean triangles) were taken as the number theoretic ideal.

X(t) would be a matrix with real number/p-adic valued coefficients and in p-adic context it would be an imaginary exponential exp(itH) of a Hermitian generator H with the p-adic norm t < 1 to guarantee the existence of the p-adic exponential. CKM would be time independent for X_{U}=X_{D}. TGD view about what happens in state function reduction (see this, this, and this) implies that the time parameter t in time evolution operator is discretized and this would allow also X(t_{n}) to belong to the algebraic extension.

For quarks X_{U}= X_{D}=Id is consistent with what is known experimentally: of course, the time dependent topological mixing of U or D type quarks would be seen in the behavior of proton. One also expects that the time dependent mixing is very small for charged leptons whereas the non-triviality of X_{ν}(t) is suggested by neutrino mixing. Therefore the assumption X_{L}=X_{ν} is not consistent with the experimental facts and X_{L}(t)=Id seems to be true a good approximation so that only X_{ν}(t) would be non-trivial? Could the vanishing em charge of neutrinos and/or the vanishing weak couplings of right-handed neutrinos have something to do with this? If the μ-e anomaly in the decays of Higgs persists ( this), it could be seen as a direct evidence for CKM mixing in leptonic sector.

CP breaking is also possible. As a matter fact, one day after mentioning the CP breaking in leptonic sector I learned about indications for leptonic CP breaking emerging from T2K experiment performed in Japan: the rate for the muon-to-electron neutrino conversions is found to be higher than that for antineutrinos. Also the NOVA experiment in USA reports similar results. The statistical significance of the findings is rather low and the findings might suffer the usual fate. The topological breaking of CP symmetry would in turn induce the CP breaking the CKM matrix in both leptonic and quark sectors. Amusingly, it has never occurred to me whether topological mixing could provide the first principle explanation for CP breaking!

For background and details see the article Some comments about τ-μ anomaly of Higgs decays and anomalies of B meson decays.

For a summary of earlier postings see Latest progress in TGD.