---
title: Double Beta Decay and Neutrino Mass
module: Beta Decay and the Weak Interaction
moduleNumber: 6
lessonNumber: 4
order: 604
summary: >
  For even-A isobars the pairing term splits the mass parabola into two curves,
  and a handful of even-even nuclides sit below their odd-odd neighbor yet above
  the next even-even one: single beta decay is forbidden but second-order double
  beta decay is allowed. The two-neutrino mode is a standard-model process with
  the longest measured lifetimes in nature; the neutrinoless mode would require
  the neutrino to be its own antiparticle and its rate measures the effective
  Majorana mass, the sharpest probe of the absolute neutrino mass scale.
topics: [Beta Decay and the Weak Interaction]
draft: false
sources:
  - book: Krane
    ref: "Ch. 9 — Beta Decay; §9.5 Double Beta Decay"
  - book: Wong
    ref: "Ch. 5 — Nuclear Reactions and Decays; even-even isobar systematics and second-order weak processes"
---

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.[^krane-double]

## The even-A double parabola

At fixed mass number the atomic mass is quadratic in $Z$, as in the
[semiempirical mass formula](/nuclear-physics/nuclear-properties/semi-empirical-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.

$$
% caption: For even A the pairing term splits the isobars into a lower even-even
% parabola and an upper odd-odd parabola; an even-even nuclide can sit below its
% odd-odd neighbor yet above the next even-even one, so single beta is forbidden
% but a double step to the lighter even-even isobar is allowed.
\begin{tikzpicture}[>=stealth, font=\footnotesize]
  \definecolor{acc}{HTML}{4A6FA5}
  \draw[->, black] (0,0) -- (8.4,0) node[right, black!70] {$Z$};
  \draw[->, black] (0,0) -- (0,5.0) node[above, black!70] {atomic mass};
  % lower parabola (even-even)
  \draw[very thick] (0.7,4.4) .. controls (2.8,0.1) and (5.6,0.1) .. (7.7,4.4);
  \node[black, anchor=west, font=\scriptsize] at (6.3,3.6) {even-even};
  % upper parabola (odd-odd), shifted up
  \draw[black, very thick, dashed] (1.1,4.8) .. controls (3.0,1.5) and (5.4,1.5) .. (7.3,4.8);
  \node[black, anchor=east, font=\scriptsize] at (2.4,3.6) {odd-odd};
  % three even-even points along lower curve: parent (left), skip, daughter (right)
  \fill[black!75] (2.6,2.05) circle (1.8pt);
  \node[black!75, anchor=south east, font=\scriptsize] at (2.6,2.15) {parent};
  \fill[black!75] (4.2,0.35) circle (1.8pt);
  \node[black!75, anchor=north, font=\scriptsize] at (4.2,0.25) {stable};
  \fill[black!75] (5.8,2.05) circle (1.8pt);
  % odd-odd intermediate (up on dashed curve, above parent level)
  \fill[black] (3.4,2.7) circle (1.8pt);
  \node[black, anchor=south, font=\scriptsize] at (3.4,2.8) {blocked};
  % single-step blocked (parent to intermediate is uphill)
  \draw[->, black, dashed] (2.75,2.15) -- (3.25,2.6);
  % double step allowed (parent to lighter even-even)
  \draw[->, acc, very thick] (2.8,1.85) .. controls (3.5,0.7) .. (4.05,0.5);
  \node[acc, anchor=west, font=\scriptsize] at (3.4,1.25) {double beta};
\end{tikzpicture}
$$

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**
($2\nu\beta\beta$),

$$
^{A}_{Z}\mathrm{X} \to\, ^{A}_{Z+2}\mathrm{X}' + 2e^- + 2\bar\nu_e,
$$

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.

> **Worked example (Counting rate from a huge half-life).** A source with
> $N = 10^{24}$ nuclei of $2\nu\beta\beta$ half-life $t_{1/2} = 2 \times
> 10^{21}\ \mathrm{yr}$ has decay constant $\lambda = \ln 2/t_{1/2} =
> 1.1 \times 10^{-29}\ \mathrm{s^{-1}}$ (using $1\ \mathrm{yr} = 3.16 \times
> 10^{7}\ \mathrm{s}$). The activity is $\lambda N \approx 1.1 \times
> 10^{-5}\ \mathrm{s^{-1}}$, about one decay every day. Detectable, but only with
> deep-underground shielding to suppress a background that would otherwise swamp
> it.

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** ($0\nu\beta\beta$),

$$
^{A}_{Z}\mathrm{X} \to\, ^{A}_{Z+2}\mathrm{X}' + 2e^-,
$$

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 $m_\nu/E$.

> **Definition (Majorana neutrino).** A Majorana fermion is identical to its own
> antiparticle, $\nu = \bar\nu$. A Dirac neutrino, by contrast, has a distinct
> antiparticle. Only the Majorana case permits the lepton-number-violating
> ($\Delta L = 2$) neutrinoless double beta decay.

$$
% caption: In the two-neutrino mode each vertex emits an electron and an
% antineutrino; in the neutrinoless mode a single Majorana neutrino exchanged
% between the two vertices absorbs both, so only two electrons leave and lepton
% number changes by two.
\begin{tikzpicture}[>=stealth, font=\footnotesize]
  \definecolor{acc}{HTML}{4A6FA5}
  % ---- two-neutrino ----
  \begin{scope}
    \node[black!70, anchor=south, font=\scriptsize] at (1.6,3.2) {two neutrino};
    \draw[black, thick] (0.2,2.7) -- (3.0,2.7);
    % two vertices
    \fill[black!70] (1.0,2.7) circle (1.6pt);
    \fill[black!70] (2.2,2.7) circle (1.6pt);
    % electrons up
    \draw[->, black!70] (1.0,2.7) -- (0.5,3.5);
    \draw[->, black!70] (2.2,2.7) -- (2.7,3.5);
    \node[black, anchor=east, font=\scriptsize] at (0.5,3.5) {e};
    \node[black, anchor=west, font=\scriptsize] at (2.7,3.5) {e};
    % antineutrinos down
    \draw[->, very thick] (1.0,2.7) -- (0.5,1.9);
    \draw[->, very thick] (2.2,2.7) -- (2.7,1.9);
    \node[black, anchor=east, font=\scriptsize] at (0.5,1.9) {nu};
    \node[black, anchor=west, font=\scriptsize] at (2.7,1.9) {nu};
  \end{scope}
  % ---- neutrinoless ----
  \begin{scope}[xshift=4.8cm]
    \node[black!70, anchor=south, font=\scriptsize] at (1.6,3.2) {neutrinoless};
    \draw[black, thick] (0.2,2.7) -- (3.0,2.7);
    \fill[black!70] (1.0,2.7) circle (1.6pt);
    \fill[black!70] (2.2,2.7) circle (1.6pt);
    % electrons up only
    \draw[->, black!70] (1.0,2.7) -- (0.5,3.5);
    \draw[->, black!70] (2.2,2.7) -- (2.7,3.5);
    \node[black, anchor=east, font=\scriptsize] at (0.5,3.5) {e};
    \node[black, anchor=west, font=\scriptsize] at (2.7,3.5) {e};
    % single exchanged Majorana neutrino between vertices
    \draw[acc, very thick] (1.0,2.7) .. controls (1.6,1.9) .. (2.2,2.7);
    \node[acc, anchor=north, font=\scriptsize] at (1.6,2.0) {Majorana nu};
  \end{scope}
\end{tikzpicture}
$$

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

$$
\bigl[T_{1/2}^{0\nu}\bigr]^{-1} = G^{0\nu}\,\lvert M^{0\nu}\rvert^2\,
\frac{\langle m_{\beta\beta}\rangle^2}{m_e^2},
$$

where $\langle m_{\beta\beta}\rangle$ is the **effective Majorana mass**. The rate
is proportional to the square of a mass, so a nonobservation sets an upper bound on
$\langle m_{\beta\beta}\rangle$, 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.

$$
% caption: The two-neutrino mode gives a continuous summed-electron spectrum up
% to the endpoint, while the neutrinoless mode deposits the full Q-value in the
% two electrons, appearing as a sharp peak at the endpoint of the continuum.
\begin{tikzpicture}[>=stealth, font=\footnotesize]
  \definecolor{acc}{HTML}{4A6FA5}
  \draw[->, black] (0,0) -- (7.0,0) node[right, black!70] {summed electron energy};
  \draw[->, black] (0,0) -- (0,4.0) node[above, black!70] {rate};
  % continuous 2-nu spectrum
  \draw[black, very thick]
    (0.15,0.1) .. controls (1.4,1.9) and (2.6,2.3) .. (3.4,2.2)
    .. controls (4.6,2.05) and (5.4,0.4) .. (5.8,0.03);
  \node[black, anchor=south, font=\scriptsize] at (2.6,2.3) {two neutrino};
  % sharp 0-nu peak at endpoint
  \draw[acc, very thick] (5.75,0.05) -- (5.8,3.4) -- (5.85,0.05);
  \node[acc, anchor=south, font=\scriptsize] at (5.8,3.4) {neutrinoless};
  \draw[black, dashed] (5.8,0) -- (5.8,0.05);
  \node[black!70, anchor=north, font=\scriptsize] at (5.8,-0.05) {endpoint};
\end{tikzpicture}
$$

## 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,

$$
\langle m_{\beta\beta}\rangle = \Bigl\lvert\sum_{i} U_{ei}^2\,m_i\Bigr\rvert,
$$

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 $\Delta m^2_{ij}$ and the mixing
angles but not the absolute masses or the overall scale, so $\langle m_{\beta\beta}
\rangle$ depends on the lightest mass $m_{\text{lightest}}$ and on the ordering of
the states.[^pdg-nu]

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

$$
% caption: The effective Majorana mass versus the lightest neutrino mass occupies
% two allowed bands, one for each mass ordering; the inverted band has a floor
% while the normal band can vanish, and current experiments probe from the top.
\begin{tikzpicture}[>=stealth, font=\footnotesize]
  \definecolor{acc}{HTML}{4A6FA5}
  \draw[->, black] (0,0) -- (7.0,0)
    node[right, black!70, font=\scriptsize] {lightest mass};
  \draw[->, black] (0,0) -- (0,4.6)
    node[above, black!70, align=center, font=\scriptsize] {Majorana\\mass};
  % inverted band (upper, has a floor at left)
  \draw[acc, thick] (0.3,3.1) -- (2.4,3.15) .. controls (4.0,3.3) .. (6.4,4.1);
  \draw[acc, thick] (0.3,2.4) -- (2.4,2.45) .. controls (4.0,2.7) .. (6.4,3.9);
  \node[acc, anchor=south, font=\scriptsize] at (1.4,3.2) {inverted};
  % normal band (lower, dips toward zero)
  \draw[black, thick, dashed] (0.3,1.7) .. controls (2.4,1.5) and (3.2,0.3) .. (3.9,0.15);
  \draw[black, thick, dashed] (0.3,1.0) .. controls (1.8,0.9) and (2.6,0.6) .. (4.2,1.3);
  \draw[black, thick, dashed] (4.2,1.3) .. controls (5.4,2.0) .. (6.4,3.2);
  \node[black, anchor=west, font=\scriptsize] at (0.4,0.65) {normal};
  % current limit from the top
  \draw[black, densely dotted] (0.2,4.2) -- (6.6,4.2);
  \node[black, anchor=east, font=\scriptsize] at (6.6,4.35) {current limit};
\end{tikzpicture}
$$

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 $\langle m_{\beta\beta}\rangle \lesssim 0.1\ \mathrm{eV}$.[^pdg-nu] 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.

[^krane-double]: **Krane**, _Introductory Nuclear Physics_, §9.5 (Double Beta Decay): the even-$A$ double parabola, the blocking of single beta decay, and the two-neutrino versus neutrinoless modes. The $^{76}\mathrm{Ge}$ Q-value and isobar masses are from the AME atomic-mass evaluation via NNDC, [nndc.bnl.gov](https://www.nndc.bnl.gov/).
[^pdg-nu]: **Particle Data Group**, _Review of Particle Physics_, "Neutrino Masses, Mixing, and Oscillations" and the neutrinoless double-beta-decay review: the effective Majorana mass, the normal and inverted ordering bands, and current half-life limits for $^{76}\mathrm{Ge}$ and $^{136}\mathrm{Xe}$, [pdg.lbl.gov](https://pdg.lbl.gov/).