In two related questions, it was asked how many free parameters the standard model has. Without neutrino oscillation, there are 19 free parameters. Now it turned out that neutrinos have a mass aswell, thus the Standard Model should/has been extended.

My question: How many parameters are required to take into account the oscillation? Does it work with only 3 mass-parameters for the 3 different known neutrinos, or are there additional parameters necessary (if so, what are they standing for)? Or is it possible to describe the effects with only 1 or 2 parameters (if so, what idea reduces the effect of the three masses to less free parameters)?

  • $\begingroup$ The answer didn't state it in so many words, although it is implied, but massive oscillating neutrinos add 7 parameters - three masses and four PMNS parameters. If neutrinos actually have Majorana rather than Dirac masses, you need a total of 9 additional parameters, since an analog to the PMNS matrix with Majorana neutrinos requires more parameters. $\endgroup$ – ohwilleke Jan 30 '17 at 1:56

The mass and weak eigenstates are related by the Pontecorvo-Maki–Nakagawa–Sakata matrix, which is a $3\times 3$ unitary matrix. Therefore, it can be parametrized by four parameters. The usual choice is the three mixing angles $\theta_{12}$, $\theta_{13}$ and $\theta_{23}$, and a CP-violating phase $\delta_{CP}$ (or, if neutrinos are Majorana fermions, 3 different CP-violating phases). Oscillation experiments provide information about the mixing angles, but not the phase.

Note that oscillation observation is not sensitive to neutrino masses, only to the square of the difference of neutrino masses. Therefore, it could be possible that only two of the three neutrino flavours have non-zero mass.

The consequences of the existence of the CP-violating phase are

  • In a difference between the oscillation $\nu_\alpha\to \nu_\beta$ and $\bar{\nu}_\alpha \to \bar{\nu}_\beta$ (and also $\nu_\beta\to \nu_\alpha$, if CPT symmetry is conserved).
  • A non-zero contribution to the electric dipole of charged leptons. Nevertheless, this effect is "neglibly small and unaccesible to experiments".
  • If neutrinos are Majorana fermions, the spectrum of the neutrinoless double beta decay might be sensitive to Majorana CP-violating phases.


  • K. Nakamura and S. T. Petcov: Neutrino mass, mixing and oscillations. Particle Data Group (Link to pdf)
  • G. C. Branco, R. Gonzalez Felipe and F. R. Joaquim: Leptonic CP violation. Rev.Mod.Phys. 84 (2012) 515-565. (arXiv preprint)
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  • $\begingroup$ Thank you, that is very interesting. I didn't know about the CP-violating phase. How do you measure that value, if you cannot see it in neutrino-oscillations, but it is not in the standard-model? That would mean, there is another beyond-SM effect related to neutrinos, or am I getting something wrong? $\endgroup$ – Mario Krenn Jul 29 '16 at 12:38

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