Neutrino and Lepton Physics

Neutrino observables answer different questions: oscillations determine mass-squared splittings and mixing, endpoint kinematics probes an incoherent mass norm, neutrinoless double-beta decay is sensitive to a coherent lepton-number-violating amplitude, cosmology constrains a model-dependent mass sum, and charged-lepton flavor violation tests other operators. This chapter supplies the mass-to-observable chain needed to keep those statements distinct, calculate each one, and combine them without turning an inference into a property of the Lagrangian.

Enter this chapter

The chapter uses natural units and the site’s global (+---) metric. Neutral-fermion mass matrices are diagonalized by a Takagi factorization, so the quoted masses are nonnegative even when an ordinary algebraic eigenvalue of a symmetric mass matrix is negative. Charged-current mixing is written

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

with $\Delta m_{ij}^{2}=m_i^2-m_j^2$ and $\Delta m_{21}^{2}>0$. “Normal” and “inverted ordering” describe the sign of the atmospheric splitting; neither phrase assumes a large hierarchy. Flavor states obey $|\nu_\alpha\rangle=\sum_iU_{\alpha i}^{*}|\nu_i\rangle$. For antineutrinos, $U\to U^*$ and the matter potential changes sign.

The vacuum phase is $\Delta m_{ij}^{2}L/(4E)$. If $\Delta m^2$ is in $\mathrm{eV}^2$, $L$ in km, and $E$ in GeV, its numerical value is $1.2669\,\Delta m^2L/E$. Every numerical application must state this unit conversion rather than silently mixing natural and laboratory units.

Choose a route

The six leaves below appear once, in their intended order. Routes 1–3 construct the mass, mixing, and propagation spine; routes 4 and 5 branch to nonoscillation observables; route 6 explains how a combined CP analysis is built and delimited.

Route	Use it when you need to	Result you should be able to produce
1. Neutrino Mass Mechanisms	Decide whether a proposed mass term is gauge invariant and what breaks lepton number	A Dirac, Majorana, Weinberg-operator, or Type-I-seesaw mass matrix with a controlled matching and diagonalization
2. Lepton Mixing, the PMNS Matrix, and Majorana Phases	Translate mass-basis rotations into charged-current observables	A parameter count, standard PMNS matrix, rephasing invariants, and a proof of which phases oscillations can see
3. Neutrino Oscillations, Coherence, and Matter Effects	Calculate a flavor-transition probability	Vacuum or matter evolution with ordering, antineutrino, coherence, averaging, and unit conventions explicit
4. Absolute Neutrino Mass and Majorana Probes	Compare endpoint, lepton-number-violating, oscillation, and cosmological information	The distinct combinations $m_\beta$, $m_{\beta\beta}$, $\sum_i m_i$, and $\Delta m^2_{ij}$ with their assumptions and cancellation structure
5. Charged-Lepton Flavor Violation	Connect a rare decay or conversion process to effective operators	A dipole, four-lepton, or semileptonic coefficient map evolved to the appropriate scale and matched to the target observable
6. Leptonic CP Observables and Combined Inference	Construct or scrutinize a joint CP analysis	A versioned likelihood with nuisance covariance, ordering and octant branches, degeneracy checks, and an explicit dated-evidence boundary

Hard dependencies and suggested order

Hard prerequisites are narrower than a productive reading sequence:

• Route 1 requires the electroweak representation content in Electroweak Gauge and Matter Structure, the neutral-spinor construction in Majorana Fields and Reality Conditions, and the matching logic in Integrating Out Heavy Fields.
• Route 2 requires route 1 and Yukawa Couplings and Fermion Masses.
• Route 3 requires route 2 and Linear ODEs, Evolution Operators, and Wronskians.
• Route 4 requires route 2. Route 5 instead begins from The Fermi Limit of Weak Interactions and Flavor, Hermiticity, and CP Bookkeeping; it does not require an oscillation calculation.
• Route 6 requires route 3 and Validation and Theory Uncertainties. Route 4 is helpful when external absolute-mass information enters the fit.

For a stable first pass, use 1 → 2 → 3 → 6. For absolute-mass or Majorana questions, use 1 → 2 → 4. For charged-lepton flavor violation, take the independent route 5 after its EFT prerequisites. Add routes 4 and 5 to the main spine only when the task genuinely compares oscillation and nonoscillation operators; neither branch is an optional way to measure the same parameter.

Readiness diagnostic

This diagnostic is informal and unscored. A missing answer points to a repair, not to a failed assessment.

Can you do this now?	Ready answer	Targeted repair
Test an electroweak mass operator	Contract the $SU(2)_L$ and hypercharge indices and state whether total lepton number is conserved	Review Electroweak Gauge and Matter Structure and Lorentz, Gauge, Flavor, and Discrete-Symmetry Constraints, then start at route 1
Diagonalize a complex symmetric mass matrix	Use $V^TMV=\operatorname{diag}(m_i)$ with $m_i\ge0$ and distinguish this from a Hermitian eigendecomposition	Repair with Majorana Fields and Reality Conditions, then use routes 1 and 2
Count physical phases	Identify which charged-lepton and neutrino field rephasings are allowed, then form a rephasing invariant	Compare the analogous construction in Quark Mixing and the CKM Matrix, then use route 2
Evolve a multicomponent state	Construct a unitary evolution operator for a constant Hamiltonian and an ordered exponential for a varying one	Repair with Linear ODEs, Evolution Operators, and Wronskians, then use route 3
Match a low-energy flavor observable	Declare a coefficient normalization, integrate out each threshold, run to the process scale, and identify hadronic or nuclear matrix elements	Review The Fermi Limit of Weak Interactions and Flavor, Hermiticity, and CP Bookkeeping, then use route 5
Combine correlated data	Write the likelihood with shared nuisance parameters or an equivalent joint covariance and preserve discrete branches	Repair with Validation and Theory Uncertainties, then use route 6

Synthesis: from an operator to an inference

The shortest gauge-invariant Majorana-mass route in the Standard Model EFT is the dimension-five Weinberg operator. After electroweak symmetry breaking it gives a complex symmetric light-neutrino mass matrix. In a Type-I seesaw with $\|m_D M_R^{-1}\|\ll1$,

$$m_\nu=-m_D M_R^{-1}m_D^T$$

at leading order, with matching-scale and sign conventions declared. This formula is a mechanism statement: it does not by itself determine the low-energy masses or mixing. Exact Takagi factors $V_{eL}$ and $V_\nu$ give the charged-current matrix $U=V_{eL}^\dagger V_\nu$; the Dirac-type invariant

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

can enter oscillations, while the extra Majorana phases cancel from every product $U_{\beta i}U_{\alpha i}^{*}$. The parameterization and oscillation conventions follow the durable review in Particle Data Group 2022, revised 2023, “Neutrino Masses, Mixing, and Oscillations,” §§14.2–14.3, PDF.

In vacuum the coherent amplitude is

$$\mathcal A_{\alpha\to\beta}(L) =\sum_i U_{\beta i} \exp\!\left(-i\frac{m_i^2L}{2E}\right) U_{\alpha i}^{*}.$$

In ordinary matter, the flavor-basis Hamiltonian acquires a potential,

$$H_f(x)=\frac{1}{2E} U\operatorname{diag}(m_1^2,m_2^2,m_3^2)U^\dagger +\operatorname{diag}\!\left(\sqrt2G_FN_e(x),0,0\right),$$

up to an irrelevant multiple of the identity. Production and detection localization must preserve interference between the relevant mass packets; loss of overlap or unresolved rapid phase variation produces an incoherent average, not a new mixing angle Beuthe 2003, §§4–6. Matter modifies propagation, not the vacuum mass operator.

Nonoscillation probes select different mass functions:

$$m_\beta^2=\sum_i|U_{ei}|^2m_i^2, \qquad m_{\beta\beta}=\left|\sum_iU_{ei}^2m_i\right|, \qquad \Sigma=\sum_i m_i.$$

$m_\beta$ is an incoherent kinematic norm. $m_{\beta\beta}$ is a coherent Majorana-sensitive combination that can cancel and whose conversion to a decay rate depends on a specified mechanism, phase-space convention, and nuclear matrix element. $\Sigma$ enters cosmological predictions only through a declared cosmological model and dataset likelihood. Thus an absolute-mass constraint cannot be substituted for a measured mass splitting, and observing a lepton-number-violating nuclear process would require mechanism discrimination before it could be interpreted as a value of $m_{\beta\beta}$ Dell’Oro et al. 2016, §§2–4.

Charged-lepton flavor violation is another operator problem. At a low scale its amplitude is schematically

$$\mathcal A_a=\sum_k C_k(\mu)\, \langle f_a|O_k(\mu)|i_a\rangle,$$

where dipole, four-lepton, and semileptonic coefficients mix under running and different processes project different coefficient combinations. A correlation such as dipole dominance is a hypothesis to test, not a definition of the theory Calibbi and Signorelli 2018, §§3–4.

Finally, a CP analysis maps parameters to predicted event bins and then to a likelihood,

$$\mu_b(\vartheta,\eta,o) =\int dE\,\Phi(E;\eta)\, \sigma(E;\eta)\,\epsilon_b(E;\eta)\, P_{\alpha\to\beta}(E;\vartheta,o),$$

where $o$ labels discrete ordering branches and $\eta$ contains shared nuisance parameters. The phase $\delta$ is a Lagrangian parameter; a preference, interval, or CP-violation statement is a dated property of specified datasets, likelihood versions, covariance assumptions, and test calibration. Discrete degeneracies and parameter boundaries can invalidate a naive chi-square interpretation Qian et al. 2012, §§II–III.

Informal synthesis review

These are work-product checks, not registered assessment. A satisfactory response states conventions and assumptions, shows enough algebra to reproduce the result, and names at least one failure mode.

1. Classify a mass proposal. Given a sterile singlet and the Standard Model fields, write every renormalizable neutral-fermion mass term and the leading light-field operator after the singlet is integrated out. Criteria: verify electroweak charges, lepton number, matrix symmetry, dimensions, and the seesaw expansion parameter. Repair: revisit route 1 and its exact $2\times2$ check.
2. Recover physical mixing. Starting from charged-lepton and neutrino mass matrices, construct $U$ and count its angles and phases for Dirac and Majorana neutrinos. Criteria: remove only allowed rephasings, reproduce a quartet invariant, and explain why Majorana phases cancel from oscillations. Repair: revisit route 2.
3. Derive a propagation limit. Reduce the three-flavor amplitude to a two-flavor probability, then add a constant matter potential. Criteria: recover $P_{\alpha\to\beta}(0)=0$, probability conservation, the vacuum limit, the antineutrino sign change, and the resonance condition. Repair: revisit route 3 and check the phase units independently.
4. Translate mass information. For specified masses and electron-row mixing, calculate $\Delta m^2$, $m_\beta$, the allowed $m_{\beta\beta}$ interval, and $\Sigma$. Criteria: keep coherent and incoherent sums distinct, exhibit a possible phase cancellation, and list the nuclear and cosmological assumptions required for observable predictions. Repair: revisit route 4.
5. Match a rare process. Starting from a declared high-scale coefficient, construct a low-scale prediction for a radiative decay or conversion process. Criteria: state basis and normalization, matching thresholds, running, matrix elements, interference, and a second process that tests the assumed operator direction. Repair: revisit route 5.
6. Build a combined CP fit. Write a minimal event likelihood with normalization and matter-density nuisances and both ordering branches. Criteria: profile or marginalize consistently, retain cross-sample covariance, locate intrinsic/order/octant degeneracies, calibrate the test statistic, and record dataset and likelihood identities. Repair: revisit route 6 and rerun the synthetic likelihood with both ordering branches.
7. Diagnose an overclaim. A report calls a current best-fit phase “the measured amount of leptonic CP violation.” Criteria: separate $\delta$, a CP-odd probability difference, $J_{\rm CP}$, and the dataset-dependent hypothesis test; state what additional evidence record is needed. Repair: combine routes 2, 3, and 6.

Purpose-keyed exits

• Test PMNS invariants, two-flavor matter evolution, and a profiled toy likelihood with declared conventions and tolerances.
• Compare the seesaw approximation with exact singular values and inspect the mixing-validity masks.
• For a structured particle-and-nuclear sequence, continue through QFT for particle and nuclear physics; its practice state should be read as guidance rather than formal assessment.
• For generic threshold matching and running before a neutrino or charged-lepton application, return to Matching, Decoupling, and Thresholds.
• For the nuclear currents and matrix elements that enter low-energy weak processes, continue to Electroweak Currents in Few-Body Systems.
• For versioned experimental or phenomenological status, continue to Effective Field Theory and Tests of the Standard Model; the timeless formulas here do not supply current intervals or limits.
• To choose another Standard Model or gauge-theory subject, return to Gauge Theories and the Standard Model.

References

• Beuthe, Mikael. “Oscillations of Neutrinos and Mesons in Quantum Field Theory.” Physics Reports 375 (2003): 105–218. DOI.
• Calibbi, Lorenzo, and Giovanni Signorelli. “Charged Lepton Flavour Violation: An Experimental and Theoretical Introduction.” Rivista del Nuovo Cimento 41 (2018): 71–174. DOI.
• Dell’Oro, Stefano, Simone Marcocci, Matteo Viel, and Francesco Vissani. “Neutrinoless Double Beta Decay: 2015 Review.” Advances in High Energy Physics 2016 (2016): 2162659. DOI.
• Particle Data Group. “Neutrino Masses, Mixing, and Oscillations.” In Review of Particle Physics, 2022 edition with 2023 update. PDF.
• Qian, X., A. Tan, W. Wang, J. J. Ling, R. D. McKeown, and C. Zhang. “Statistical Evaluation of Experimental Determinations of Neutrino Mass Hierarchy.” Physical Review D 86 (2012): 113011. DOI.