Double Beta Decay and Neutrino Mass

Most nuclides that can lower their energy by beta decay do so in a single step. A small set cannot: their immediate beta neighbor is heavier, blocking the one-step process, while the isobar two units of charge away is lighter. These nuclides decay by emitting two electrons at once — double beta decay — a second-order weak process so slow that it produces the longest lifetimes ever measured. Whether a rarer variant exists, in which no neutrinos are emitted, depends on the nature of the neutrino itself, making double beta decay the premier laboratory for the absolute neutrino mass.¹

The even-A double parabola

At fixed mass number the atomic mass is quadratic in $Z$, as in the semiempirical mass formula. For odd $A$ the pairing term $\delta$ vanishes and all isobars lie on one parabola with a single most-stable charge; single beta decay walks the chain down to it. For even $A$ the pairing term is nonzero and opposite in sign for the two classes of nuclide, splitting the parabola into two:

• even-even nuclides (both $Z$ and $N$ even) gain the pairing energy and lie on the lower parabola;
• odd-odd nuclides (both $Z$ and $N$ odd) pay it and lie on the upper parabola, displaced up by $2\delta$.

Beta decay steps $Z$ by one, so it always moves a nuclide from one parabola to the other. An even-even nuclide decays to an odd-odd daughter, and vice versa. This alternation creates the blocking configuration.

The classic case is ${}^{76}\mathrm{Ge}$. Its odd-odd neighbor ${}^{76}\mathrm{As}$ is slightly heavier, so ${}^{76}\mathrm{Ge}\to {}^{76}\mathrm{As}$ single ${\beta }^{-}$ is energetically forbidden. But ${}^{76}\mathrm{Se}$, two charges up and even-even, is lighter by $Q_{\beta \beta }=2.039\mathrm{MeV}$, so the two-step transition ${}^{76}\mathrm{Ge}\to {}^{76}\mathrm{Se}$ can proceed. Germanium-76 is therefore stable against single beta decay but unstable, extraordinarily slowly, against double beta decay.

The two-neutrino mode

The allowed, standard-model process is two-neutrino double beta decay ($2\nu \beta \beta$),

two ordinary beta decays occurring simultaneously through a virtual intermediate state. Because it is second order in the weak interaction, the rate carries two factors of $G_F^2$ and an intermediate-state energy denominator, so it is slower than a single allowed beta decay by roughly the square of the weak suppression. Measured half-lives run from $10^{18}$ to $10^{21}\mathrm{yr}$, longer than the age of the universe by up to eleven orders of magnitude, yet detectable because a kilogram of source contains of order $10^{24}$ nuclei.

The two neutrinos escape undetected, so the observable is the summed kinetic energy of the two electrons, $T_1+T_2$. Sharing the released energy among four light leptons gives a continuous spectrum from zero up to the endpoint $Q_{\beta \beta }$, peaked below the endpoint just as the ordinary beta spectrum is.

The neutrinoless mode

If lepton number is not exactly conserved, a second mode becomes possible: neutrinoless double beta decay ($0\nu \beta \beta$),

with no neutrinos emitted. The process requires the antineutrino emitted at one beta vertex to be reabsorbed as a neutrino at the other. That is possible only if the neutrino is its own antiparticle — a Majorana particle — and only if it has nonzero mass, because the reabsorption needs the wrong helicity component present in a massive neutrino at amplitude $m_{\nu }/E$.

The half-life of the neutrinoless mode factorizes into a phase-space factor, a nuclear matrix element, and the neutrino-mass term,

where $⟨m_{\beta \beta }⟩$ is the effective Majorana mass. The rate is proportional to the square of a mass, so a nonobservation sets an upper bound on $⟨m_{\beta \beta }⟩$, while a detection would measure it directly and prove lepton-number violation.

Distinguishing the modes

The two modes are separated by the summed-electron energy spectrum, which is the whole experimental signal because the neutrinos (if any) escape.

• Two-neutrino mode. Four light particles share $Q_{\beta \beta }$, so the summed electron energy $T_1+T_2$ is a continuous distribution from $0$ to $Q_{\beta \beta }$.
• Neutrinoless mode. The two electrons carry all of $Q_{\beta \beta }$ (the daughter recoil is negligible), so $T_1+T_2=Q_{\beta \beta }$ exactly, a sharp line at the endpoint.

Detecting $0\nu \beta \beta$ amounts to resolving a monoenergetic peak sitting on the tail of the continuous $2\nu \beta \beta$ spectrum, which demands excellent energy resolution and a background low enough that the peak region is nearly empty.

The neutrino mass scale

The effective Majorana mass is a coherent sum over the three neutrino mass eigenstates weighted by the squared mixing-matrix elements,

where the $U_{ei}$ are the elements of the leptonic mixing matrix and the phases in $U_{ei}^2$ include Majorana phases absent from oscillation physics. Oscillation experiments measure the mass-squared differences $\Delta m_{ij}^2$ and the mixing angles but not the absolute masses or the overall scale, so $⟨m_{\beta \beta }⟩$ depends on the lightest mass $m_{\text{lightest}}$ and on the ordering of the states.²

• Inverted ordering predicts $⟨m_{\beta \beta }⟩$ bounded below by about $15\mathrm{meV}$, a target within reach of next-generation experiments.
• Normal ordering allows $⟨m_{\beta \beta }⟩$ to dip toward zero through cancellations among the Majorana phases, so a null result there does not exclude Majorana neutrinos.

The leading experiments push the half-life bound above $10^{26}\mathrm{yr}$ for ${}^{76}\mathrm{Ge}$ and ${}^{136}\mathrm{Xe}$, translating (through the nuclear matrix elements) to $⟨m_{\beta \beta }⟩\lesssim 0.1\mathrm{eV}$.² A positive signal would establish three facts at once: that lepton number is violated, that the neutrino is a Majorana particle, and the absolute scale of the neutrino mass — three answers the beta-decay endpoint and neutrino oscillations cannot supply alone.