In 1935, Maria Goeppert-Mayer predicted the existence of the two neutrino double
beta decay process (DBD).

This decay mode is theoretically allowed in the SM framework; it
is labelled ββ2ν.
This process (the β^{-}β^{-} mode) consists
in the simultaneous transmutation of two neutrons into
protons inside a nucleus thanks to
β decay like processes (figure 10).

Note: β^{+}β^{+} decay exists too...

Figure 10: Double beta decay; two neutrons are changed into protons
inside the nucleus, emitting two electrons and two antineutrinos.

The double beta decay process can be observed
in a few isotopes
for which all other decay channels are forbidden for energetic
reasons (figure 11).

Figure 11: Energy scheme for the double β^{-} decay from parent nucleus ^{A}X to
daughter nucleus ^{A}Y.
The single β^{-} decay to the intermediate isotope ^{A}T is forbidden by
the energy conservation rule.

Calculations predict that this process is very rare,
with mean lifetimes of the order of 10^{18-24} years
depending of the emitter isotope.
The half life time is given by the following formula:

a_{2ν} ~ 2 10^{-22} y^{-1} is a dimensional factor

F_{2ν} is a known phase space factor proportionnal to
Q_{ββ}^{11}

M_{2ν} is the nuclear matrix element (n.m.e.).

The main uncertainty in the estimation of ββ2ν half lives
comes from the n.m.e. calculations that are not very reliable
(one gets results in a nearly one order of magnitude range
between different calculations).

Only a few ββ emitting isotopes are
good candidates for experimental studies. They are shown in table 4.
The ββ2ν process have been experimentally observed for most of them.

Isotope

Q_{ββ} (MeV)

Isotopic abundance (%)

F_{2ν}

Half life T_{1/2,2ν} (y) exp.

^{48}Ca

4.271

0.0035

~ 140 10^{3}

~ 4.0 10^{19}

^{76}Ge

2.039

7.8

~ 0.5 10^{3}

~ 1.4 10^{21}

^{82}Se

2.995

9.2

~ 15 10^{3}

~ 0.9 10^{20}

^{96}Zr

3.350

2.8

~ 70 10^{3}

~ 2.1 10^{19}

^{100}Mo

3.034

9.6

~ 33 10^{3}

~ 8.0 10^{18}

^{116}Cd

2.802

7.5

~ 28 10^{3}

~ 3.3 10^{19}

^{128}Te

0.868

31.7

~ 2.8

~ 2.5 10^{24}

^{130}Te

2.533

34.5

~ 16 10^{3}

~ 0.9 10^{21}

^{136}Xe

2.479

8.9

~ 16 10^{3}

not observed yet

^{150}Nd

3.367

5.6

~ 400 10^{3}

~ 7.0 10^{18}

Table 4: ββ emitters of experimental interest
(space factor data from F. Mauger,
half life data from A.S. Barabash in nucl-ex/0203001).

Suppose we use 1 kg of ^{100}Mo (10 moles), how many
ββ2ν decays do we get in one year?
Answer:

N_{2ν} = 10 N_{A} log(2) / T_{1/2,2ν}
~ 5 10^{5}

where N_{A} is the Avogadro constant.

Assuming that our ^{100}Mo sample is contaminated
by some natural ^{214}Bi radioactivity
at the level of 1 Bq/kg, we get about 3 10^{7}
beta decays due to natural ^{214}Bi radioactivity for the same period:
this is about two orders of magnitude higher than the expected
number of ββ2ν events.
That means that natural radioactivity is likely to act as a background
provider, preventing an
efficient direct observation of the very rare double beta process.

In 1939, Wolfgang Furry proposed that a double beta decay without
emission of neutrino (labelled ββ0ν)
could occur in ββ emitting nuclei if new physics exist beyond the standard model.
In ββ0ν decay, two neutrons in a nucleus are simultaneously
changed in protons emitting two electrons but without emitting any anti-neutrinos (figure 12).

Figure 12: The ββ0&nu decay; only two electrons are emitted while two neutrons
simultaneously transmute into two protons.

The existence of such a process violates the lepton number
conservation rule, thus it is forbidden by the SM.
Nevertheless, ββ0ν decay could exist if (figure 13):

(anti-)neutrinos have a non zero mass,

neutrinos are a Majorana particles, i.e. neutrinos and anti-neutrinos
are the same particle.

Figure 13: Feynman diagram for the neutrinoless double beta decay
(ββ0&nu).
This process could be observed
thanks to the exchange of a Majorana massive neutrino
between the two W^{-} bosons.

Note: other mechanisms have also been invented to explain such a process.

Neutrinoless double beta decay is the only known process
that enables to test experimentally the Majorana nature of neutrino together
with its absolute mass scale. That makes this topics
very attractive.

The half life time is given by the following formula:

a_{0ν} ~ 5 10^{-17} y^{-1} is a dimensional factor

F_{0ν} is a known phase space factor proportionnal to
Q_{ββ}^{5}

M_{0ν} is the nuclear matrix element (n.m.e.).

η = <m_{ν}>/m_{e} with
<m_{ν}> being
the effective mass of the exchanged neutrino and m_{e}
the mass of the electron (0.511 MeV/c^{2}).

From the formula above, we see that the observation of the
ββ0ν process enables the estimation of the (Majorana)
neutrino mass scale <m_{ν}>.
Here again the calculation of the nuclear matrix element
is an important source of uncertainty.