The seesaw Lagrangian is $$-\mathcal{L}_{\rm mass}=\frac{Y^\ell_{ab}\langle H^0\rangle}{\sqrt{2}}\bar{\ell}_{aL}\ell_{bR}+\frac{Y^\nu_{ab}\langle H^0\rangle}{\sqrt{2}}\bar{\nu}_{aL}\nu_{bR}+\frac{1}{2}M_{ab}\bar{\nu}^C_{aR}\nu_{bR}+\text{h.c.}$$ For $3$ left(right)-handed charged leptons $\ell_{L(R)}$, $3$ left(right)-handed neutrinos $\nu_{L(R)}$, there are $18+18$ real parameters coming from the complex $3\times 3$ Yukawa matrices $Y^\ell, Y^\nu$ and $12$ more from complex symmetric $3\times 3$ Majorana matrix $M$. Therefore, the total real parameters in this Lagrangian is $48$.

  • What is wrong in this counting? Please help!

You have to take into account field redefinitions which remove unphysical parameters. The counting is therefore more delicate, however it is conceptually analogous to the counting in the quark sector. At the end you should find that the answer is 21 instead of 48.

You can follow these references for instance for detailed derivations:

  • $\begingroup$ I tried to read those. However, I could not completely follow. Can you mention what these 21 parameters would be? PMNS matrix has 3 angles and 3 physical phases. In addition, there are 3 charged lepton masses, 3 light neutrinos masses, 3 heavy neutrino masses. So I get a total of 15 real parameters. What are the other 6? Which I didn't count? $\endgroup$ – mithusengupta123 Mar 26 '20 at 5:36
  • $\begingroup$ The 21 physical parameters is the counting for the complete see saw theory. On the basis where the charged lepton Yukawa and the Majorana mass are diagonal, the counting is 3 charged lepton masses, 3 heavy neutrino masses, and 15 parameters in the neutrino Yukawa matrix (arxiv.org/abs/1207.5521). Those 15 parameters account for the light neutrino masses (3), PMNS mixing matrix parameters (6) and further interactions terms (6). See the Casas-Ibarra parametrization [arXiv:hep-ph/0103065] or see how dimension six effective operators are relevant [arxiv.org/abs/hep-ph/0210271] $\endgroup$ – Alejandro Celis Mar 26 '20 at 7:47
  • $\begingroup$ "and further interactions terms (6)" are they irrelevant at low energy? $\endgroup$ – mithusengupta123 Mar 26 '20 at 8:18
  • $\begingroup$ These are dimension six effects and come suppressed by 1/M^2 where M would be the scale of the heavy neutrinos in this case. You will need to do research about the experimental prospects to detect those effects if you are interested, but I can quote from [arxiv.org/abs/hep-ph/0210271] "The physical consequences of the low-energy dimension 6 operator are suppressed by two inverse powers of the large seesaw scale, and consequently, there is little practical hope to observe them, unless the seesaw scale turns out to be surprisingly small." $\endgroup$ – Alejandro Celis Mar 26 '20 at 8:37
  • $\begingroup$ Thanks. I will go more into this in detail. $\endgroup$ – mithusengupta123 Mar 26 '20 at 15:07

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