# PMNS matrix versus CKM matrix -- how precise is the analogy?

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

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?

• Dec 27, 2020 at 15:27

$$\begin{bmatrix} V_{\nu_e e} & V_{\nu_e \mu} & V_{\nu_e \tau} \\ V_{\nu_{\mu} e} & V_{\nu_{\mu} \mu} & V_{\nu_{\mu} \tau} \\ V_{\nu_{\tau} e} & V_{\nu_{\tau} \mu} & V_{\nu_{\tau} \tau} \end{bmatrix} ?$$ 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.

• 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? Dec 27, 2020 at 15:06
• True. Not the PMNS matrix. The PMNS supplants the diagonal one here if you want to use mass eigenstates. Dec 27, 2020 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? Dec 27, 2020 at 17:29
• The starting point is PDG and references therein... Dec 27, 2020 at 17:52
• but that is a huge ref book... Dec 27, 2020 at 17:57