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I wish to describe in simple but correct terms the analogy between the Cabibbo–Kobayashi–Maskawa (CMK) and Pontecorvo–Maki–Nakagawa–Sakata (PMNS) matrices. The CMK matrix describes the rotation between weak interaction eigenmodes and the flavour (mass?) eignstates of quarks. The PMNS matrix is the rotation between neutrino flavour states and the time-evolution (mass) eigenstates. Both are unitary and experimentally known.

Are the following two statements correct?

  1. Quark masses are large compared to amplitudes of the flavour-mixing weak interaction, thus the effect of flavour oscillations is negligible (and manifested mainly as decays of higher-mass quarks in the weak channel). On the other hand, the differences in neutrino eigenenergies are comparable to the mixing amplitudes thus the contrast of oscillations is high.

  2. The interaction responsible for the CMK matrix is the weak interaction, while the interaction responsible for PMNS and neutrino oscillations is unknown (?) due to non-observability or any other of its effects.

This answer sheds some very useful light on point no. 2, but I'm not sure whether the "non-renormalizable dimension 5 operators" mentioned there can be unambitious classified as different (in some well-defined sense) from the weak interaction or not.

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The dimension 5 operators are Higgs-Higgs-Lepton-Lepton direct coupling. These are in the Higgs sector, but no Ws and Z's, so it's up to you whether you call it a weak interaction. The CKM matrix is mathematically similar. I don't think it is fair to say that the interaction is unknown--- the effective description in terms of non-renormalizable operators is probably correct, but it depends on how the Higgs sector works for the details. –  Ron Maimon Mar 24 '12 at 7:14
@RonMaimon "Higgs-Higgs-Lepton-Lepton direct coupling" is what defines the quark mass eigenstates, it is their non-commutativity with the weak gauge interaction matrix that gives rise to to non-trivial CMK matrix. See Eqs. (40)-(41) of hep-ph/0304186 Thus in terms of admixture (oscillations of flavour) it the weak interaction which makes it possible. Thus this make sense? –  Slaviks Mar 26 '12 at 14:06
The CKM matrix comes from Higgs-Q-QR coupling, only one higgs, Q is the left-handed quark doublet and "QR" are the 6 right handed quark singlets. This interaction is the usual mass term. The Neutrino mass terms is H-H-L-L, two Higgs, two left-handed lepton fields. It is dimension 5 and requires two Higgs. –  Ron Maimon Mar 26 '12 at 16:09
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2 Answers

  • It's a little strong to hold that the PMNS matrix is known. It's mostly known (with $\theta_{1,3}$ non-zero at five sigma just this week! Congratulations, Daya Bay!{*}), but the CP violating phase ($\delta_{CP}$) is basically unconstrained as are the Majorana phases (if they apply). Nor is the "maximal" mixing angle known to high precision a fact which is of some concern going forward with $\delta_{CP}$ measurements.

  • The CKM matrix features small mixing angles, while the PMNS features a near maximal angle and another large one.

{*} Double Chooz was on paper sooner, but at lower significance; a fact I am compelled to mention because I had a hand in that measurement. Also, we can expect a better measurement from Double Chooz and something significant from Reno soon.

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Thanks for great references! CKM indeed is closer to unity than PMNS. I was just wondering why no one is taking about "quark oscillations" and sketched my reasons (which I'm note sure about) in the question. –  Slaviks Mar 10 '12 at 6:00
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Looking at the way CMK appears in the SM (I looked at hep-th/0304186, Sec. 3.1 in particular) it looks the most significant reason is simply the huge disparity of quark masses copared to their weak interaction. The admixture of higher mass quarks due to weak interaction is tiny because it the lattter is much weaker than the diffrences in Yuvawa couplings (masses) of quarks.

In contrast, PMNS is the only interaction of otherwise degenerate modes neutrino modes, and also, as nicely remarked in @dmckee answer, mass eigenstates are nearly orthogonal to flavour states, so the detected oscillation is profound.

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