Physical aspects of the CKM matrix

Fig. 1: the CKM matrix gives branching path amplitudes for beta decay (or other weak interaction) transitions of quarks, since quarks can decay in different ways in a probability tree, by changing flavours (changing between generations in the standard model). The sum of all possible branches for a given interaction is shown by a probability tree (amplitudes must be squared to determine relative path probabilities). A muon always decays “in an electron” by beta decay, so it has a relative CKM matrix amplitude of 1. The CKM matrix is therefore a statement of the relative amplitudes for various different decay paths in beta decay for quarks. The standard model offers no physical explanation for it; it’s just a matrix of numbers. Obviously the total probability for all different possible decay paths is 1, e.g. for upquarks the CKM matrix gives the sum of probabilities as 0.974282 + 0.22532 + 0.03472 = 1, so knowing that there are 3 possibilities (3 generations) permits you to represent one number by subtracting the squares of the other two from 1: 0.97 = (1 – 0.22532 – 0.03472)1/2. This nursery skill with adding and subtracting numbers is used (with plentiful obfuscating symbolism) to “cleverly compress” the CKM matrix values mathematically in the standard model, but the ability to subtract is not equivalent to “doing physics”. We want to be able to predict all the values in the matrix, and to understand the mechanism for inter-generation decays!

The conventional viewpoint in the standard model is that all lepton to lepton weak interactions have transition amplitudes of 1, so that there is no branching for lepton decays at all. However, as noted in the previous post, we know that over long distances (e.g. the sun to earth distance of 150 million km) neutrinos oscillate between all three generations, and although this mixing is not observable in the laboratory, it is nevertheless physical evidence for one mechanism that constitutes a physical process for inter-generation mixing effects which we need to consider for the case of the CKM matrix weak interaction inter-generation mixing. The standard model is set up as if leptons have an effectively non-mixing CKM matrix with within-generation transition amplitudes of 1 in all cases, and across-generation transition amplitudes of 0 in all cases. However, that’s an assumption based on a lack of evidence, due to the lack of neutrino mixing over short distances.

The only reason for this lack of mixing between lepton flavours in beta decay is due to the short range of the weak force in laboratory experiments, since if weak interactions extended over longer distances, neutrinos would oscillate between flavours appreciably and we would then observe for lepton-lepton weak transitions cross-generation mixing (driven by neutrino flavour oscillations), just as we observe for quarks in the CKM matrix.

Applying this argument back to explain the quark CKM matrix, it follows that the cross-generation transition amplitudes arise from something akin to the neutrino oscillations which are observed over long distances.

Notice that as the mass of the quarks increases, the branching amplitudes for cross-generation mixing become smaller in the CKM matrix. E.g., transition amplitudes within the lightest generation (up and down quarks) are about 0.97, compared to about 0.999 for the heaviest generation (top and bottom quarks).

So moving to heavier quarks makes inter-generation mixing less likely. Why is this? Answer: heavier masses involve shorter-range interactions, and a shorter-range provides less physical spacetime for the “oscillation” of particles (not just) neutrinos between generations! Therefore, some particle that oscillates in the beta decay is able to oscillate more in the lower mass vacuum field of light quarks than that of heavy quarks, and this extra amount of oscillation for light quarks increases the probability of inter-generation interactions.

So we have a physical mechanism for the CKM matrix, explaining the relationship between the masses of the particles to the transition amplitudes. Lepton to lepton transitions show no detectable flavour change amplitude, but have very low masses! Why don’t they change flavour under this mechanism? It is not proved that neutrinos are the only particles to oscillate, so we need to keep all other options open until we have a reason to rule them out. So what is oscillating between flavours in weak interactions of quarks, to produce the observed CKM matrix values?

Weak boson flavour oscillations

Does the weak boson oscillate in flavour? In the standard model, it’s not supposed to have any flavour, but there is an analogy of interest. Photons are supposed to be electrically neutral, but they contain a superposition of positive and negative electromagnetic fields, which does couple with the fields in a block of glass through which a photon moves, thus slowing it down. Therefore, it is not true to say that something that is “neutral” (through balance of fields) has no interaction with electromagnetic fields. A “neutral” photon can and does interact with electromagnetic fields, as observed in the refraction of light by glass.

The weak boson in beta interactions is off-shell and can have various effective masses, so although it can be created in observable, on-shell form using its “rest mass” (80 GeV if charged, 91 GeV if not). A muon simply doesn’t have enough energy to create an on-shell weak boson during decay; it utilizes an off-shell weak boson created briefly through the annihilation of virtual fermion pairs in the vacuum at short ranges, in the strong electric field very close to the muon.

Weak interactions are “weak” precisely because such heavy (weak) bosons are not produced very abundantly from pair annihilations in the vacuum; the annihilation of pairs produces more electromagnetic photons than weak bosons, so the electromagnetic interaction has a much higher coupling than the weak interaction.

To be continued.


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