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PMNS matrix is said to be the matrix for the neutrinos as the CKM matrix for the quarks.

See https://en.wikipedia.org/wiki/Pontecorvo–Maki–Nakagawa–Sakata_matrix#The_PMNS_matrix

However, I am confused why this is true.

  1. The PMNS matrix $M_{PMNS}$ is the matrix changing between the neutrino flavor eigenstate and the neutrino mass eigenstate $$ \begin{bmatrix} {\nu_e} \\ {\nu_\mu} \\ {\nu_\tau} \end{bmatrix} = \begin{bmatrix} U_{e 1} & U_{e 2} & U_{e 3} \\ U_{\mu 1} & U_{\mu 2} & U_{\mu 3} \\ U_{\tau 1} & U_{\tau 2} & U_{\tau 3} \end{bmatrix} \begin{bmatrix} \nu_1 \\ \nu_2 \\ \nu_3 \end{bmatrix} = M_{PMNS} \begin{bmatrix} \nu_1 \\ \nu_2 \\ \nu_3 \end{bmatrix} $$

  2. However, the CKM matrix is obtained from (see p.723 of Peskin QFT) $$ V_{CKM} =U_u^\dagger U_d $$ where $U_u$ is a matrix of $u,c,t$ flavor to flavor matrix. $U_d$ is a matrix of $d,s,b$ flavor to flavor matrix. The $U_u$ and $U_d$ are obtained in an attempt to diagonalizing the Higgs Yukawa term to a diagonalized form as the mass eigenstates. The $ V_{CKM}$ is the weak charge current coupling to the $W$ bosons with flavor changing process.

So $$ V_{CKM} = \begin{bmatrix} V_{ud} & V_{us} & V_{ub} \\V_{cd} & V_{cs} & V_{cb} \\ V_{td} & V_{ts} & V_{tb} \end{bmatrix} $$

Question

So how can $M_{PMNS}$ for neutrinos is an analogy of $V_{CKM}$ for quarks?

I thought the correct analogy for lepton sectors (as $V_{CKM}$ for quarks) would be a matrix of the form like $$ \begin{bmatrix} V_{\nu_e e} & V_{\nu_e \mu} & V_{\nu_e \tau} \\ V_{\nu_{\mu} e} & V_{\nu_{\mu} \mu} & V_{\nu_{\mu} \tau} \\ V_{\nu_{\tau} e} & V_{\nu_{\tau} \mu} & V_{\nu_{\tau} \tau} \end{bmatrix} ? $$ Not the $M_{PMNS}$. True or false?

Qmechanic
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ann marie cœur
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1 Answers1

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$$ \begin{bmatrix} V_{\nu_e e} & V_{\nu_e \mu} & V_{\nu_e \tau} \\ V_{\nu_{\mu} e} & V_{\nu_{\mu} \mu} & V_{\nu_{\mu} \tau} \\ V_{\nu_{\tau} e} & V_{\nu_{\tau} \mu} & V_{\nu_{\tau} \tau} \end{bmatrix} ? $$ Not the $M_{PMNS}$. True or false?

True. Not the $M_{PMNS}$. Indeed, by definition, the oxymoronic off-diagonal elements of the matrix you wrote vanish: we define the e-neutrino to be precisely that linear combination of the three neutrino mass eigenstates which weak-couples to the electron; and analogously for the other two leptons.

Analogously, for quarks, the up quark couples to a linear combination of downlike quark eigenstates, namely $V_{ud} d+V_{us} s+V_{ub} b$. If we utilized the same aggressively confusing "flavor eigenstate" language, we'd call this combination something like $D_u$, clearly stupid, since, for quarks, flavor is defined by mass eigenstates.

So the "real", mass eigenstate particles appearing in the SM lepton sector are $e,\mu,\tau; \nu_L, \nu_M, \nu_H $, (Lightest, Middle, Heaviest; most probably 1,2,3, in the most likely, normal, hierarchy alternative). Mercifully, the SM wall charts confusing generations have now been corrected to reflect this.

The PMNS matrix is a very close analog to the CKM matrix, indeed, and it is only historical usage (ritual misuse) of the slippery term "flavor" that victimizes students with devilish glee. For quarks, flavor indicates the mass of the particle, hence aligns with generation; whereas for neutrinos, it indicates the charged lepton the state couples to, and straddles generations, except in the older, misguided charts.

Cosmas Zachos
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  • But the $ \begin{bmatrix} V_{\nu_e e} & V_{\nu_e \mu} & V_{\nu_e \tau} \ V_{\nu_{\mu} e} & V_{\nu_{\mu} \mu} & V_{\nu_{\mu} \tau} \ V_{\nu_{\tau} e} & V_{\nu_{\tau} \mu} & V_{\nu_{\tau} \tau} \end{bmatrix} $ is Not the $M_{PMNS}$, correct? – ann marie cœur Dec 27 '20 at 15:06
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    True. Not the PMNS matrix. The PMNS supplants the diagonal one here if you want to use mass eigenstates. – Cosmas Zachos Dec 27 '20 at 15:17
  • Do we know in the literature how many CP violation phase there are in a generic matrix with e, mu. tau left-handed neutrinos, and N right-handed neutrinos? for some positive N =1,2,3? Please suggest any ref or book? – ann marie cœur Dec 27 '20 at 17:29
  • The starting point is PDG and references therein... – Cosmas Zachos Dec 27 '20 at 17:52
  • but that is a huge ref book... – ann marie cœur Dec 27 '20 at 17:57
  • This is what I had in mind https://physics.stackexchange.com/q/603426/42982 ;) – ann marie cœur Dec 27 '20 at 18:14
  • Excuse me, 1) how many CP violation phase in the weak lepton sector in the PMNS matrix? – ann marie cœur Dec 28 '20 at 19:12
  • If I use the CKM matrix way to look at the weak current for leptonic sectors $\begin{bmatrix} V_{\nu_e e} & V_{\nu_e \mu} & V_{\nu_e \tau} \ V_{\nu_{\mu} e} & V_{\nu_{\mu} \mu} & V_{\nu_{\mu} \tau} \ V_{\nu_{\tau} e} & V_{\nu_{\tau} \mu} & V_{\nu_{\tau} \tau} \end{bmatrix}$, there are again 9 d.o.f. but it can be rotated as the same argument as CKM, so it only has 1 complex CP violating phase left – ann marie cœur Dec 28 '20 at 19:14
  • The oxymoronic matrix for leptons you wrote is meaningless, as I explained. It must be diagonal. The proper PMNS one has only one phase, like CKM, if neutrinos are Dirac, and two more if they are Majorana; for Majorana, the total is N(N-1)/2, as explained in the PDG review. – Cosmas Zachos Dec 28 '20 at 19:24
  • thanks that helps = which page or section of PDG? "The oxymoronic matrix for leptons you wrote is meaningless" --> why is that? Is that just a convention to choose the PMNS matrix instead of my $V$ matrix? or is there a physical reason why "The oxymoronic matrix is meaningless" and only PMNS matrix is physical? – ann marie cœur Dec 28 '20 at 19:44
  • I thought PMNS matrix vs my matrix is just a choice of convention? – ann marie cœur Dec 28 '20 at 19:44

  • which page or section of PDG? -> sec 14, p 252 I found - thanks

     – ann marie cœur Dec 28 '20 at 19:49
  • (14.33). Your V matrix is the identity matrix, provided you kill the impossible, meaningless, oxymoronic off-diagonal terms. It basically defines "flavor basis neutrinos", in themself of the PMNS matrix. – Cosmas Zachos Dec 28 '20 at 19:51
  • Thanks PDG really helps! It has an answer - I need to think thru – ann marie cœur Dec 28 '20 at 23:06