I wish to describe in simple but correct terms the analogy between the Cabibbo–Kobayashi–Maskawa (CKM) and Pontecorvo–Maki–Nakagawa–Sakata (PMNS) matrices. The CKM 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 CKM 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 unambigously classified as different (in some well-defined sense) from the weak interaction or not.

  • 1
    $\begingroup$ 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. $\endgroup$
    – Ron Maimon
    Mar 24, 2012 at 7:14
  • $\begingroup$ @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? $\endgroup$
    – Slaviks
    Mar 26, 2012 at 14:06
  • $\begingroup$ 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. $\endgroup$
    – Ron Maimon
    Mar 26, 2012 at 16:09

2 Answers 2


My understanding of this question is really two different questions. Let me answer each of these in turn.

1) What is the relation between the CKM and PMNS matrices?

To see how this works consider the relevant quark interaction terms without any choice of basis, \begin{equation} - m _d \bar{d} d - m _u \bar{u} u - i W _\mu \bar{d} \gamma ^\mu P _L \bar{u} \end{equation} Here $ m _d $ and $ m _u $ are completely arbitrary $ 3 \times 3 $ matrices.

We can redefine the down type quarks such that $ m _d $ is diagonal, $ d \rightarrow U _d d $. This matrix can then be reabsorbed into $ u $ (by a choice of basis for $u$) keeping the charged current diagonal. However, after this second redefinition we can't redefine the up type quarks again since we lost that freedom.

Therefore to have mass eigenstates we must introduce a mixing matrix which we call the CKM (this is often referred to as a product of the transformations on the down-type and up-type quarks but this is a bit unnecessary since we can always redefine one of either the down-type or up-type quarks to be in the diagonal basis). The CKM appears in the charged current interaction, \begin{equation} W _\mu \bar{d} \gamma ^\mu P _L \bar{u} = W _\mu \bar{d}' \gamma ^\mu P _L V _{ CKM}\bar{u} ' \end{equation} Then we define a quark to be the mass eigenstates. The "cost" of this is that then we have to deal with uncertainty about which particle is produced in the charged current interaction since now particles of different generations can interact with the charged current. Its important here to note that this would not have been true if we called our ``quarks'' the fields that had a diagonal charged current.

That being said lets contrast with the charge lepton sector. Here we have, \begin{equation} - m _\ell \bar{\ell } \ell - m _\nu \bar{\nu } \nu - i W _\mu \bar{\ell } \gamma ^\mu P _L \bar{\nu } \end{equation} If the neutrinos were massless ($ m _\nu = 0 $) then we can just redefine the charged lepton basis such that their mass matrix is diagonal and we don't introduce any mixings into the charged current. However, if neutrinos do get a small mass then we have a choice we can diagonalize the neutrino matrix or leave the charged current diagonal.

On the other hand, unlike for the quarks, the mass eigenstates of the neutrino are almost impossible to produce. We have very little control over the neutrinos and they are typically made in one of the interaction eigenstates (in the basis in which $ m _\nu $ is nondiagonal), due to some charged current interaction. Thus the neutrinos are going to oscillate between the different mass eigenstates due to the state being in a superposition of energy eigenstates. Since we can't produce these mass eigenstates it is more convenient to call our "neutrinos" the states which we produce and let them oscillate.

Finally note that we often do diagonalize the neutrino matrix and define the analogue to the CKM known as the PMNS matrix, however this is more of a convenient way to parametrize the neutrino mass matrix then anything else.

2) Do quarks experience particle oscillations?

In general whenever the interaction eigenstates are not equal to the mass eigenstates particles can experience oscillations. In practice whether or not these oscillations are observable will depend on the interactions of the outgoing particles. Quarks interact significantly with their environment making their oscillations not observable in a physical experiment. To see how this plays out consider some collider producing down-type quarks (this can be say from top decays). The outgoing states will take the form,

$$|\rm outgoing\rangle = \#_1 |d\rangle +\#_2 |s \rangle+\#_3 |b \rangle$$

with the different coefficients determined by the CKM angle. When acted on by the time evolution operator, this state will mix into the other interaction eigenstates and hence when $|\rm outgoing \rangle$ propagates, it oscillates.

However, once these states are produced they are quickly "measured" by the environment through the subsequent processes such as showering and hadronization. The timescale for hadronization is $\Lambda_{QCD}^{-1} $ or a length scale of about a femtometer. This is way shorter than where we could place our detectors to see such oscillations. Once hadronization takes place the states decohere and quantum effects are no longer observable. Hence the linear combination is destroyed well before these particles are allowed to reach our detectors.

  • 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.

  • $\begingroup$ 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. $\endgroup$
    – Slaviks
    Mar 10, 2012 at 6:00
  • 1
    $\begingroup$ FWIW, and with no criticism intended towards @dmckee (who can't predict the future many years in advance) the precision of the PMNS matrix element measurements has increased dramatically since March 2012 (six and a half years ago). One parameter of the PMNS matrix is known to less than 3% precision and two others are known to about 5% precision (the CP violation parameter has only about 15% precision but excludes zero at two sigma), and the PMNS model has weathered the test of time. The neutrino mass difference parameters for neutrino oscillation are known to roughly 1% precision. $\endgroup$
    – ohwilleke
    Sep 27, 2018 at 20:22
  • $\begingroup$ Yes. This is very much a outdated historical perspective. But I'm out of that business now, and haven't been keeping up in detail. @ohwilleke, if you wanted to write up a more current response I'd be please to lend it my support. $\endgroup$ Sep 27, 2018 at 21:26

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