Lepton Mixing, PMNS Parameters, and Majorana Phases

The PMNS matrix is the mismatch between the left-handed rotations that diagonalize the charged-lepton and neutrino mass sectors. For three nondegenerate massive neutrinos it contains three angles and one Dirac phase; if the neutrinos are Majorana particles, two additional phases survive because their fields cannot be continuously rephased while their masses remain real and positive. Ordinary oscillations depend on quartic rephasing invariants and lose the Majorana phases exactly, whereas lepton-number-violating amplitudes contain products such as $m_iU_{ei}^2$ and retain them.

Required background. Dirac, Majorana, and Seesaw Neutrino Masses supplies the Dirac singular-value and Majorana Takagi diagonalizations. Yukawa Couplings and Fermion Masses supplies the charged-lepton mass-basis rotation.

Helpful background. Quark Mixing and the CKM Matrix supplies the analogous Dirac mixing construction and Jarlskog invariant.

Mass-basis misalignment

Choose weak-basis fields and diagonalize the charged-lepton matrix by

$$V_{eL}^\dagger M_eV_{eR}=D_e, \qquad D_e=\operatorname{diag}(m_e,m_\mu,m_\tau)>0.$$

For Dirac neutrinos use $V_{\nu L}^\dagger M_\nu V_{\nu R}=D_\nu$; for Majorana neutrinos use the Takagi factorization $V_\nu^{\mathsf T}M_\nu V_\nu=D_\nu$ with $D_\nu\ge0$. In either case, the charged current becomes

$$\mathcal L_{\rm CC}= -\frac{g}{\sqrt2}\, \overline{\ell_{L\alpha}}\gamma^\mu U_{\alpha i}\nu_{Li}W^-_\mu+\text{h.c.}, \qquad U=V_{eL}^\dagger V_\nu .$$

The label “flavor state” therefore refers to the charged-current production or detection channel, while $i$ labels propagation mass eigenstates. Changing either weak-basis rotation alone changes $U$; changing both by the same weak-basis transformation does not.

Counting physical parameters

An $n\times n$ unitary matrix contains $n(n-1)/2$ real angles and $n(n+1)/2$ phases. Field rephasings remove unphysical phases:

Neutrino type	Removable continuous phases	Physical phases in $U$
Dirac	$2n-1$ from charged leptons and neutrinos, after one common phase is counted once	$(n-1)(n-2)/2$ Dirac phases
Majorana	$n$ charged-lepton phases; neutrino rephasings would complexify the positive Majorana masses	$n(n-1)/2$ phases

The Majorana count splits into $(n-1)(n-2)/2$ Dirac-type phases plus $n-1$ Majorana phases. Hence at $n=3$,

$$3\ \text{angles} +1\ \text{Dirac phase} +2\ \text{Majorana phases}.$$

This generic count assumes nondegenerate nonzero masses and nonvanishing mixing needed to expose each phase. If a neutrino is exactly massless, one additional continuous rephasing is available and one Majorana phase disappears. Exact degeneracies permit further rotations, and if $s_{13}=0$ the standard Dirac phase is unobservable. These are reductions of the physical parameter space, not failures of the generic count.

Standard three-neutrino convention

This chapter fixes

$$U=R_{23}U_{13}(\delta)R_{12}P_M, \qquad P_M=\operatorname{diag} \left(1,e^{i\alpha_{21}/2},e^{i\alpha_{31}/2}\right),$$

with $s_{ij}=\sin\theta_{ij}$, $c_{ij}=\cos\theta_{ij}$, $0\le\theta_{ij}\le\pi/2$, and phases defined modulo $2\pi$. Explicitly,

$$\begin{aligned} R_{23}U_{13}R_{12} =\begin{pmatrix} c_{12}c_{13}&s_{12}c_{13}&s_{13}e^{-i\delta}\\ -s_{12}c_{23}-c_{12}s_{23}s_{13}e^{i\delta} &c_{12}c_{23}-s_{12}s_{23}s_{13}e^{i\delta} &s_{23}c_{13}\\ s_{12}s_{23}-c_{12}c_{23}s_{13}e^{i\delta} &-c_{12}s_{23}-s_{12}c_{23}s_{13}e^{i\delta} &c_{23}c_{13} \end{pmatrix}. \end{aligned}$$

Some sources put two phases on the first two columns or reverse the sign of $\delta$. Translate them with diagonal row phases, allowed Majorana signs, and—if necessary—a column permutation. A complete convention translation records the mass ordering as well as the matrix: relabeling $\nu_i$ sends $U\to UP$ and $D_\nu\to P^{\mathsf T}D_\nu P$. The invariant checks below must survive. The Particle Data Group gives this standard parameterization and its phase ranges in Workman et al. 2022 with 2023 update, § 14.2.2, pp. 7–8 (PDF).

Rephasing invariants

For distinct flavors and masses define

$$J_{\alpha\beta}^{ij} =\operatorname{Im}\!\left( U_{\alpha i}^*U_{\beta i} U_{\alpha j}U_{\beta j}^* \right).$$

For three generations every nonzero $J_{\alpha\beta}^{ij}$ equals $\pm J$ according to index orientation, with

$$J=c_{12}s_{12}c_{23}s_{23}c_{13}^2s_{13}\sin\delta.$$

$J$ is unchanged by arbitrary charged-lepton and Dirac-neutrino row or column rephasings, and it changes sign under complex conjugation. It vanishes if any required mixing factor or $\sin\delta$ vanishes. This is the lepton analogue of Jarlskog’s basis-independent CP criterion Jarlskog 1985, pp. 1039–1042.

Majorana-sensitive rephasing information can be carried by quantities such as

$$I_{\alpha;ij} =\operatorname{Im}\!\left[ (U_{\alpha i}U_{\alpha j}^*)^2 \right],$$

or, directly, by the phase of a lepton-number-violating sum $\sum_i m_iU_{\alpha i}U_{\beta i}$. These are invariant under charged-lepton rephasings and allowed Majorana sign flips. They need not be “pure” functions of one named Majorana phase; the standard phases and mixing angles can enter the same invariant.

Why oscillations lose Majorana phases

With the production convention

$$|\nu_\alpha\rangle=\sum_iU_{\alpha i}^*|\nu_i\rangle,$$

the coherent vacuum amplitude is

$$\mathcal A_{\alpha\to\beta}(L) =\sum_iU_{\beta i} e^{-im_i^2L/(2E)}U_{\alpha i}^*.$$

Writing $U=VP_M$, each term contains

$$U_{\beta i}U_{\alpha i}^* =V_{\beta i}e^{i\alpha_i/2} V_{\alpha i}^*e^{-i\alpha_i/2} =V_{\beta i}V_{\alpha i}^*.$$

Thus both Majorana phases cancel before squaring the amplitude. The cancellation relies on lepton-number-conserving production, propagation, and detection; it does not say the phases are unphysical. For light-Majorana exchange in a lepton-number-violating process, the propagator supplies a mass insertion and the combination is instead $\sum_i m_iU_{\alpha i}U_{\beta i}$, so no conjugate removes $P_M$. This separation was established for Dirac and Majorana oscillations by Bilenky, Hošek, and Petcov 1980, pp. 495–498.

Exact fixture and convention checks

A reproducible calculation uses

$$(s_{12},c_{12})=(3/5,4/5),\quad (s_{23},c_{23})=(5/13,12/13),\quad (s_{13},c_{13})=(7/25,24/25),$$

with $\delta=\pi/2$, $\alpha_{21}=\pi/3$, and $\alpha_{31}=\pi/2$. Direct multiplication must give $U^\dagger U=I$, while

$$J=\frac{580608}{13203125}.$$

Varying either Majorana phase must leave every coherent oscillation probability unchanged but change generic terms in $\sum_i m_iU_{ei}^2$. Additional checks are:

• each row and column has unit norm and distinct rows and columns are orthogonal;
• $J$ computed from any oriented choice of two rows and columns agrees up to the expected sign;
• a row rephasing changes displayed entries but no charged-current rate or invariant;
• a column permutation accompanied by the same mass relabeling leaves every amplitude unchanged;
• setting all angles to zero gives $U=P_M$, whose phases still cancel from oscillations.

Neutrino Oscillations, Coherence, and Matter Effects uses this convention for propagation. Absolute Neutrino Mass and Majorana-Sensitive Probes uses the same columns and phases in $m_\beta$ and $m_{\beta\beta}$.

Common pitfalls

Counting phases before fixing the mass type. A Dirac neutrino can be continuously rephased; a positive-mass Majorana field generally can only change sign. Applying the Dirac subtraction to a Majorana matrix removes physical phases incorrectly.

Changing PMNS convention without relabeling masses. A column permutation changes which splitting is called $\Delta m_{ij}^2$. Transform $U$, the mass vector, and every phase definition together, then check an amplitude.

Saying oscillations prove Majorana phases vanish. They cancel from ordinary oscillation products because one PMNS element is conjugated. Lepton-number-violating products contain two unconjugated elements and probe different invariants.

References

• Bilenky, S. M., J. Hošek, and S. T. Petcov. “On Oscillations of Neutrinos with Dirac and Majorana Masses.” Physics Letters B 94 (1980): 495–498. DOI.
• Jarlskog, Cecilia. “Commutator of the Quark Mass Matrices in the Standard Electroweak Model and a Measure of Maximal CP Nonconservation.” Physical Review Letters 55 (1985): 1039–1042. DOI.
• Workman, R. L., et al. (Particle Data Group). “Review of Particle Physics.” Progress of Theoretical and Experimental Physics 2022 (2022): 083C01, with 2023 update, review 14, “Neutrino Masses, Mixing, and Oscillations.” DOI.