Absolute Neutrino Mass and Majorana-Sensitive Probes

Oscillations determine mass-squared differences, not the absolute mass scale. Endpoint kinematics measures $m_\beta^2=\sum_i|U_{ei}|^2m_i^2$ when individual endpoints are unresolved; light-Majorana exchange in neutrinoless double beta decay depends on the coherent sum $m_{\beta\beta}=|\sum_iU_{ei}^2m_i|$ ; standard cosmological analyses are primarily sensitive to $\Sigma=\sum_i m_i$ under a declared cosmological model. These are distinct combinations with distinct nuisance parameters, and none may be substituted for another without its mixing, mechanism, nuclear, and cosmological assumptions.

Required background. Lepton Mixing, PMNS Parameters, and Majorana Phases fixes mass labels, $U_{ei}$ , and the two Majorana phases.

Helpful background. Electroweak Currents in Few-Body Systems supplies the nuclear-current interface. Validation and Theory Uncertainties supplies the likelihood and correlated-systematic distinctions needed for cross-probe comparisons.

Four observable mass combinations

Use $m_i\ge0$ , $\Delta m_{ij}^2=m_i^2-m_j^2$ , and the standard PMNS convention. The main combinations are

Probe	Leading mass information	Coherence and model dependence
Flavor oscillation	$\Delta m_{ij}^2$ and mixing invariants	Coherent phases; independent of a common shift in every $m_i^2$
Beta-decay endpoint	$m_\beta^2=\sum_i\lvert U_{ei}\rvert^2m_i^2$ in the unresolved limit	Incoherent sum over final mass eigenstates; source and response model required
Neutrinoless double beta decay	$m_{\beta\beta}=\lvert\sum_iU_{ei}^2m_i\rvert$ for light-Majorana exchange	Coherent, phase-sensitive amplitude; nuclear matrix element and mechanism required
Cosmological structure and expansion	Usually $\Sigma=\sum_i m_i$ in a standard thermal scenario	Inference depends on cosmological model, datasets, nonlinear modeling, and priors

Oscillation splittings plus an ordering reduce the three masses to one continuous parameter. Define $\delta m^2=m_2^2-m_1^2>0$ and a positive atmospheric-scale splitting $\Delta m_A^2$ . With $m_0$ the lightest mass,

\begin{array}{c|ccc} &m_1&m_2&m_3\\ \hline \text{normal ordering} &m_0&\sqrt{m_0^2+\delta m^2} &\sqrt{m_0^2+\Delta m_A^2}\\[2pt] \text{inverted ordering} &\sqrt{m_0^2+\Delta m_A^2} &\sqrt{m_0^2+\Delta m_A^2+\delta m^2} &m_0 \end{array}

Here $\Delta m_A^2=m_3^2-m_1^2$ in the first row and $m_1^2-m_3^2$ in the second. Other atmospheric-splitting conventions differ by $O(\delta m^2)$ , so the definition must accompany any numerical input.

Endpoint kinematics and effective beta mass

Near an allowed beta endpoint $E_0$ , suppressing smooth nuclear and radiative factors, the electron spectrum is

\frac{d\Gamma}{dE_e} \propto F(Z,E_e)p_eE_e(E_0-E_e) \sum_i|U_{ei}|^2 \sqrt{(E_0-E_e)^2-m_i^2}\, \Theta(E_0-E_e-m_i).

Different $\nu_i$ are orthogonal final states, so their rates add incoherently and no Majorana phase appears. If the experimental response cannot resolve the individual kinks and the analysis interval is broad compared with their separations, expansion of the sum gives

m_\beta^2=\sum_i|U_{ei}|^2m_i^2.

This kinematic combination is the same for Dirac and Majorana neutrinos under the standard charged-current interaction. If an endpoint or an additional heavy state is resolved, use the full spectral sum rather than compressing it to $m_\beta$ . The derivation and response qualifications are reviewed by Weinheimer and Zuber 2013, §§ 2–3, pp. 567–572.

Majorana-sensitive decay

Assume standard left-handed currents and that exchange of the three light Majorana neutrinos dominates neutrinoless double beta decay. The helicity flip supplies $m_i$ , and the two electron-flavor vertices give

m_{\beta\beta} =\left|\sum_i m_iU_{ei}^2\right|.

In this chapter’s PMNS convention,

\begin{aligned} m_{\beta\beta}=\bigl|{}&m_1c_{12}^2c_{13}^2 +m_2s_{12}^2c_{13}^2e^{i\alpha_{21}}\\ &+m_3s_{13}^2e^{i(\alpha_{31}-2\delta)}\bigr|. \end{aligned}

The appearance of $\delta$ in the last convention-dependent phase does not make the result an ordinary oscillation observable; only the invariant relative phases of the three complex terms matter.

A common rate convention is

\left[T_{1/2}^{0\nu}\right]^{-1} =G^{0\nu}g_A^4|\mathcal M^{0\nu}|^2 \left|\frac{m_{\beta\beta}}{m_e}\right|^2.

Some nuclear calculations absorb powers of $g_A$ or radius factors into $G^{0\nu}$ or $\mathcal M^{0\nu}$ ; translate the complete product, not a quoted matrix element alone. The isotope, phase-space convention, nuclear Hamiltonian and current, many-body method, quenching prescription, and covariance are part of the inference Dell’Oro et al. 2016, §§ 3–5.

This formula is conditional on the light-neutrino mechanism. Other lepton-number-violating operators can contribute or interfere, so a half-life does not by itself determine $m_{\beta\beta}$ . Conversely, observation of neutrinoless double beta decay would imply a Majorana mass term through the black-box argument even if another short-distance mechanism dominated, but the induced mass need not set the observed rate Schechter and Valle 1982, pp. 2951–2954.

Phase cancellations and a synthetic spectrum

Let

a_i=m_i|U_{ei}|^2.

Varying the two independent relative phases gives the geometric bounds

\max\!\left(2\max_i a_i-\sum_i a_i,0\right) \le m_{\beta\beta}\le\sum_i a_i.

The lower endpoint follows from the triangle inequality: complete cancellation is possible only if the largest vector is no longer than the sum of the other two.

As a dimensionful but nonempirical fixture, take masses $(m_1,m_2,m_3)=(1,2,3)\mu$ and electron-row weights $(|U_{e1}|^2,|U_{e2}|^2,|U_{e3}|^2)=(1/2,1/3,1/6)$ . Then

m_\beta=\sqrt{\frac{10}{3}}\,\mu, \qquad \Sigma=6\mu, \qquad 0\le m_{\beta\beta}\le\frac53\mu.

The endpoint combination does not change as the two phases vary, while the Majorana combination can cancel. This example is algebra, not a fit or a claim about the physical spectrum.

Cosmological mass sum and its assumptions

For a standard thermal relic population within a specified cosmological model, neutrino energy density and free streaming make cosmological observables sensitive mainly to

\Sigma=m_1+m_2+m_3.

Unlike endpoint kinematics, this is not a direct laboratory observable. The mapping from spectra to $\Sigma$ depends on the thermal history, number and distribution of light species, background cosmology, primordial spectrum, nonlinear and baryonic modeling, scale calibration, dataset selection, likelihood, and priors. Extensions can broaden or shift the inferred mass distribution even while the particle-physics definition of $\Sigma$ is unchanged. The physical free-streaming mechanism and these model dependencies are developed in Lesgourgues and Pastor 2006, §§ 4–7, pp. 331–370.

Cross-probe consistency procedure

A defensible comparison proceeds in this order:

Freeze identities. Record the ordering and atmospheric-splitting convention, PMNS phase convention, isotope or beta source, cosmological model, datasets, and release versions.
Use a common parameter vector. Sample $m_0$ , the splittings, mixing parameters, Majorana phases, and every nuclear, response, and cosmological nuisance rather than translating independent one-dimensional intervals by eye.
Preserve likelihood shape and covariance. Physical boundaries at $m_i\ge0$ , phase cancellations, ordering branches, and non-Gaussian nuclear or cosmological nuisances make symmetric-error propagation unreliable.
Test mechanism assumptions. A tension in $m_{\beta\beta}$ may indicate a nuclear calculation, another lepton-number-violating operator, or incompatible data—not automatically the mass ordering.
Validate on held-out information. Do not use the same oscillation or calibration input twice and call the resulting agreement independent evidence.

Current limits, preferred orderings, cosmological combinations, and nuclear matrix-element rankings require dated evidence records; none is frozen here. Effective Field Theory and Tests of the Standard Model is the Research exit for versioned status, while the durable correlated-inference method is on Leptonic CP Observables and Combined Inference.

Common pitfalls

Inferring absolute mass from oscillations alone. Adding a common constant to every $m_i^2$ leaves all vacuum phases unchanged. An endpoint, cosmological, or other absolute-scale input is indispensable.

Calling a null decay result a model-independent Dirac verdict. The translation to $m_{\beta\beta}$ assumes a mechanism and nuclear matrix element, and destructive phases can suppress the light-neutrino amplitude. State exactly what parameter region and mechanism were tested.

Combining published limits as Gaussian measurements. Physical boundaries, ordering branches, shared oscillation inputs, phase cancellations, and prior-sensitive cosmological likelihoods invalidate a naive inverse-variance average.

References

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.
Lesgourgues, Julien, and Sergio Pastor. “Massive Neutrinos and Cosmology.” Physics Reports 429 (2006): 307–379. DOI.
Schechter, J., and J. W. F. Valle. “Neutrinoless Double-Beta Decay in $SU(2)\times U(1)$ Theories.” Physical Review D 25 (1982): 2951–2954. DOI.
Weinheimer, Christian, and Kai Zuber. “Neutrino Masses.” Annalen der Physik 525 (2013): 565–575. DOI.