Dedicated to Professor Apolodor Aristotel Răduţă’s 70th Anniversary
POSITRON-EMITTING AND DOUBLE-EC MODES
OF DOUBLE BETA DECAY
JOUNI SUHONEN
Department of Physics, University of Jyväskylä,
P. O. Box 35 (YFL), FI-40014 University of Jyväskylä, Finland
E-mail: [email protected]
Received April 22, 2013
This is a short review of the present status of the latest theoretical advances
on the positron-emitting and double-electron-capture (β + /EC) modes of double beta
decay. The double β − mode has been studied intensively for decades, both experimentally and theoretically, but the β + /EC modes have attracted little attention thus far.
Recently a boost to the β + /EC studies was given by the predicted enhancement of the
decay rates of the resonant neutrinoless double-electron capture. In order to verify the
fulfillment of the resonance condition a host of mass measurements have recently been
done by using Penning-type atom traps.
Key words: Double beta decay, positron-emitting modes, double electron capture, random-phase approximation, multiple-commutator model.
PACS: 21.60.Jz, 23.40.Bw, 23.40.Hc, 27.50.+e, 27.60.+j.
1. INTRODUCTION
The subject of double beta decay has attracted both theoretical and experimental interest already for decades. In particular, the neutrinoless double beta (0νββ)
decay has become a popular subject since the emergence of the grand-unified theories (GUT). These theories offered the possibility to lepton-number non-conservation
and to the existence of Majorana-type of massive neutrinos, potential mediators of
the decay. A further boost to the field was given by the discovery of the non-zero
neutrino mass by the neutrino-oscillation experiments during the last decade. This
decay mode yet awaits its (unambiguous) experimental discovery. On the contrary,
the two-neutrino mode (2νββ) of double beta decay has been discovered for a number of nuclei on the β − side of the stability line in the nuclear chart.
Concerning the β − type of 2νββ and 0νββ decays a huge theoretical effort
has been invested in calculation of the involved nuclear matrix elements (NMEs)
(see the reviews [1–4]). These NMEs are needed in order to extract information on
the neutrino masses and CP-violating phases of neutrino mixing from the measured
half-lives of 0νββ-decaying nuclei. In this context it is appropriate to mention the
important work done by Prof. A.A. Raduta and his various collaborators in the field.
Prof. Raduta has done pioneering work on applications of boson-expansion techniques to beta decays [5,6] and double beta decays [7,8]. Also renormalized versions
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of this expansion have been formulated [9, 10]. In addition, novel use of spherical
basis states for deformed nuclei has been accomplished in order to take into account
the effects of nuclear deformation on double beta decay [7, 8, 11–13]. Furthermore,
decays to excited states have also been considered [13] and recent refinements of the
theory have been described in Refs. [14–16].
On the positron-emitting/electron-capture (EC) decays there has been much
less work done, both experimentally and theoretically. On the experimental side
this is mostly due to the unfavorable decay energies (Q values) and less abundant
nuclear isotopes involved. On the theoretical side the work of Doi et al. [17, 18]
opened up the possibility to use the NMEs to quantitatively access the involved decay
modes: double positron emission (β + β + ), positron emission combined with electron
capture (β + EC) and double electron capture (ECEC). Some of the involved twoneutrino decay transitions to the ground state and excited states have been considered
in Refs. [19–30]. For neutrinoless modes of decays the phase-space mediated ECEC
decay is not possible since there are no final-state leptons involved to carry away
the released decay energy. Instead, the 0νβ + β + and 0νβ + EC modes have been
studied in Refs. [20, 29–34]. The neutrinoless double-electron capture (0νECEC) is
a special case and has to involve an additional mechanism to achieve energy balance
of the decay. In this context the idea of a resonant 0νECEC (R0νECEC) process is
lucrative due to its potential resonance enhancement. For this reason the Q value of
the decay has to be known accurately and work in this direction has been done in
Refs. [29, 34–42]. Recent experimental studies of R0νECEC processes have been
performed, e.g., in Refs. [43–50]. In the following a short review of the status of
these positron-emitting/electron-capture modes of double beta decays will be given.
2. OUTLINE OF THEORY
A lot of work has been done in experimental [51] and theoretical [1–4] investigations of the double β − decays of nuclei due to their favorable decay Q values. The
positron-emitting modes of decays, β + β + , β + EC and ECEC are much less studied.
Below some theoretical aspects of these decays are reviewed.
2.1. TWO-NEUTRINO DOUBLE BETA DECAYS
(2ν)
The 2νββ-decay half-life, t1/2 , for a transition from the initial ground state,
+
0i , to the final J + state, Jf+ (here either the ground state or some excited 0+ or 2+
state), can be compactly written in the form
h
i−1
(2ν) 2
(2ν)
+
(2ν)
t1/2 (0+
→
J
)
=
G
(J)
M
(J)
(1)
,
α
α
i
f α
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Jouni Suhonen
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(2ν)
where α = β + β + , β + EC, ECEC is the mode of double beta decay. Here Gα (J)
is the leptonic phase-space factor for the different double-beta channels: double
positron emission (β + β + ), positron emission combined with electron capture (β + EC)
and double electron capture (ECEC) [1, 17]. The largest available phase space and
decay energy are related to the ECEC mode since no positrons are emitted and the
rest mass (minus the binding energies) of the two captured electrons can be used
for the decay Q value. For the β + EC mode the situation is less favorable and the
least favored is the β + β + mode, where two positrons are emitted and no electon is
captured. The nuclear matrix elements of (1) are written explicitly in [29].
2.2. NEUTRINOLESS DOUBLE BETA DECAYS VIA PHASE SPACE
Along the lines described in Section 2.1 the 0νββ-decay half-life can be written
as [1, 29]
h
i−1
0 2
(0ν)
+
t1/2 (0+
= G(0ν)
M (0ν) |hmν i|2 , α = β + β + , β + EC , (2)
α
i → 0f )α
where hmν i is the effective neutrino mass [1], a linear combination of the products
of neutrino masses and matrix elements of the electron row of the neutrino mixing
matrix. The nuclear matrix element of (2) can be written as a linear combination
of the Gamow–Teller, Fermi and tensor terms as done e.g., in [29, 52, 53]. Here we
consider only the final ground state or excited 0+ states since 0νββ decays to 2+
(0ν)
final states are strongly suppressed [54]. Values for the phase-space factors Gα are
given in [1, 18, 31, 32].
An appropriate account of the nucleon-nucleon short-range correlations in the
neutrinoless decay is very important since the momentum of the virtual Majorana
neutrino, exchanged between the two decaying nucleons, is large enough to force
the nucleons to overlap. In the work [55] the traditionally used Jastrow short-range
correlations [56] were replaced by short-range correlations produced by the use of
the unitary correlation operator method (UCOM) [57]. This represented a definite
step forward and the UCOM short-range correlators were further studied and used in
Refs. [52, 58, 59]. In all the recent calculations the nucleon form factors of Ref. [60]
are used, instead of, e.g., the quark-model-derived ones in Refs. [61–63].
2.3. RESONANT NEUTRINOLESS ECEC DECAYS
The neutrinoless double electron capture has to run via a special mechanism
since there are no final-state leptons available. One possible mechanism is the resonant neutrinoless double electron capture (R0νECEC) which was studied in the
works [35,36] from the lepton aspects points of view. There it was suggested that the
fulfilment of a resonance condition in this decay could enhance the decay rates up to
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a factor of a million. The R0νECEC decay proceeds between two atomic states in
the form
e− + e− + (A, Z) → (A, Z − 2)∗ → (A, Z − 2) + γ + 2X,
(3)
where the capture of two atomic electrons leaves the final nucleus in an excited state
that decays by one or more gamma-rays and the atomic vacancies are filled by outer
electrons with emission of X-rays. The corresponding half-life can be written as
h
i−1
2
ECEC
|hmν i|2 Γ
R0νECEC
(Jf )
,
(4)
T1/2
(Jf )
= GECEC
(Jf ) M0ν
0ν
(Q − E)2 + Γ2 /4
where Jf is the angular momentum of the nuclear final state. The difference Q − E
is the degeneracy of the initial and final states, Q being the difference between the
masses of the initial and final atoms (decay Q value) and E is the total energy of
the excited state in the final atom (consisting of the nuclear excitation energy and
the excitation energy of the two holes in the electronic shells plus their Coulomb
repulsion). The quantity Γ is the decay width of the two holes in the atomic shells.
Details of the formalism related to R0vECEC processes are given in Ref. [42].
2.4. NUCLEAR MODELS
The calculations of the present review are based on the quasiparticle randomphase approximation (QRPA), and in particular on its proton-neutron variant (pnQRPA). Excited states of the daughter nucleus are described by starting from the
charge-conserving QRPA (ccQRPA). A thorough account of the formalism used in
the QRPA models is given in [1, 64, 65].
The multiple-commutator model (MCM) [66,67] is designed to connect excited
states of an even-even reference nucleus to states of the neighbouring odd-odd nucleus. Earlier the MCM has been used extensively in the calculations of double-betadecay rates e.g., in [21, 31, 32]. In this formalism the states of the odd-odd nucleus
are given by the pnQRPA and the excited states of the even-even nucleus are generated by the ccQRPA, and the transitions between them are handled in a higher-QRPA
framework. For more details concerning the positron-emission/electron-capture decays, see Refs. [29,42]. In fact, the MCM can be extended to a so-called microscopic
anharmonic-vibrator approach (MAVA) [68, 69] and it has already been used to describe two-neutrino double beta decays on the β − side of the line of stability [70,71].
3. RESULTS
In this section some examples of positron-emitting/electron-capture decays of
nuclei are presented. Furthermore, the present status of the R0νECEC decays is
reviewed.
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Jouni Suhonen
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3.1. DOUBLE BETA DECAY OF 124 Xe AS AN EXAMPLE
Fig. 1 summarizes the computed results for the two-neutrino double-beta halflives for all the possible positron-emitting/electron-capture modes of decay. The
calculations have been done by using the pnQRPA, combined with the MCM, as
described in section 2.4. The ranges of values of half-lives in the figure stem from the
different single-particle energy sets (Woods-Saxon or slightly adjusted Wood-Saxon
energies) adopted in the calculations, as also from the range gA = 1.00 − 1.25 used
for the axial-vector coupling constant. A complication in the present calculations
is that the location of the first 1+ state in the intermediate nucleus 124 I is unknown
and thus it is not possible to normalize the energy denominator of the related NME
experimentally. Instead, a reasonable range E(1+
1 ) = (0.15 − 1.00) MeV for the
excitation energies was assumed in the evaluation of the NMEs.
2−
gs
124
53 I71
0+
gs
124
54 Xe70
0+
1
1657.28 keV
ECEC: (1.7 − 580) × 1025 , β + EC: (4.4 − 38000) × 1032
2+
2
1325.51 keV
ECEC: (1.1 − 3700) × 1030 , β + EC: (2.0 − 13000) × 1031
2+
1
602.73 keV
ECEC: (2.3 − 11000) × 1028 , β + EC: (8.8 − 25000) × 1026
0+
gs
β + β + : (1.0 − 32) × 1043
ECEC: (4.0 − 88) × 1020 , β + EC: (9.4 − 97) × 1021
124
52 Te72
β + β + : (1.7 − 38) × 1026
Fig. 1 – Computed partial half-lives (in units of years) for two-neutrino double beta decays of 124 Xe.
From Fig. 1 one deduces that the decay mode β + β + has a positive Q value
(and thus can occur) only for the lowest two final states. Instead, the ECEC and
β + EC modes are possible for all the final states displayed in the figure. From the
figure it is obvious that decays to 0+ states are favoured over decays to 2+ states.
This is due to the extremely small NMEs for the latter decays, a feature first noticed
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in the early work of Ref. [21]. The available phase space for the ECEC decay is the
largest one, followed by the β + EC and β + β + phase spaces. In Fig. 1 the available
phase space plays the leading role in building the hierarchy of the half-lives of the
different decay modes. Concerning the detection possibilities of the two-neutrino
processes in 124 Xe, the best chances of detection in the near future offer the ECEC
and β + EC decays to the ground state with the computed half-lives in the range of
(0.4 − 97) × 1021 years.
XK
XK
(0+)
2854.87 keV
0+
1
1657.28 keV
2+
2
1325.51 keV
2+
1
602.73 keV
0+
gs
2−
gs
124
53 I71
R-ECEC: (1.9 − 5.6) × 1030
hmν i = 0.3 eV
124
52 Te72
0+
gs
124
54 Xe70
β +EC: ≥ 5.9 × 1032
hmν i = 0.3 eV
β +EC: (1.2 − 4.2) × 1027, β + β +: (2.3 − 7.7) × 1028
hmν i = 0.3 eV
Fig. 2 – Computed partial half-lives for neutrinoless double beta decays of 124 Xe. The UCOM shortrange correlations have been adopted. The half-lives are given in units of years for the effective neutrino
mass hmν i = 0.3 eV. The resonant double electron-capture transition is marked by ’R-ECEC’.
Let us next discuss the neutrinoless decays. In Fig. 2 the decays of 124 Xe to the
ground and excited 0+ states of 124 Te are displayed for the effective neutrino mass
hmν i = 0.3 eV (as extracted from the results of the famous Heidelberg-Moscow
0νββ experiment [72]). The ranges of values of half-lives in the figure stem from
the same sources as discussed for the two-neutrino decays in the context of Fig. 1.
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7
From Fig. 2 one observes that for the decays to the ground state the half-lives are in
the range of 1027 − 1028 years. For the decay to the 0+
1 state the half-lives are longer
than 1032 years and thus this transition is undetectable. For the resonant decay the
computed half-lives exceed 1030 years and thus this decay mode is also extremely
hard to detect.
3.2. STATUS OF THE RESONANT NEUTRINOLESS ECEC DECAYS
Table 1 lists the best known cases of R0νECEC transitions in various nuclei
where Q-value measurements have been conducted recently. These Q values have
been measured by exploiting the Penning-trap techniques. In the cases of 96 Ru,
106 Cd, 124 Xe and 130 Ba the assignment of 0+ spin-parity to the resonant state is
uncertain. In these cases further spectroscopic measurements are needed.
Table 1
R0νECEC decay transitions with the final-state spin-parity indicated in the second column and the
degeneracy parameters Q − E in the third column. Also the involved atomic orbitals have been given
in the fourth column. The second last column lists the currently available half-live estimates with the
references to the Q-value measurement and calculations indicated in the last column.
Transition
Se →
Ru →
74
74
96
96
102
106
Ge
Mo
Pd → 102 Ru
Cd → 106 Pd
Sn → 112 Cd
Xe → 124 Te
130
Ba → 130 Xe
136
Ce → 136 Ba
144
Sm → 144 Nd
152
Gd → 152 Sm
156
Dy → 156 Gd
112
124
Er → 162 Dy
Er → 164 Dy
168
Yb → 168 Er
180
W → 180 Hf
162
164
Jfπ
Q − E [keV]
Orbitals
C ECEC
Ref.
2+
2+
0+ ?
2+
0+ ?
(2, 3)−
0+
0+ ?
0+ ?
0+
2+
0+
gs
1−
0+
2+
2+
0+
gs
(2− )
0+
gs
2.23
8.92(13)
−3.90(13)
75.26(36)
8.39
−0.33(41)
−4.5
1.86(15)
10.18(30)
−11.67
171.89(87)
0.91(18)
0.75(10)
0.54(24)
0.04(10)
2.69(30)
6.81(13)
1.52(25)
11.24(27)
L2 L3
L1 L3
L1 L1
KL3
KK
KL3
KK
KK
KK
KK
KL3
KL1
KL1
L1 L1
M1 N3
KL3
L1 L1
M1 M3
KK
(0.2 − 100) × 1043
[38]
[73]
(2.1 − 5.7) × 1030
> 5.9 × 1029
(1.7 − 5.1) × 1029
(3 − 23) × 1032
(1.0 − 1.5) × 1027
(3.2 − 5.2) × 1031
(4.0 − 9.5) × 1029
[74]
[34]
[74]
[37]
[75]
[75]
[39]
[74]
[41, 76]
[77]
[77]
[77]
[73]
[41, 78]
[73]
[41, 79]
In the table we also list the estimated half-lives for the cases for which such
exist. The references of the last column indicates the origin of the Q-value measureRJP 58(Nos. 9-10), 1232–1241 (2013) (c) 2013-2013
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Positron-emitting and double-EC modes of double beta decay
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ment and the possible calculations of the related NME. In the table a quantity C ECEC
is given and it relates to the R0νECEC half-life through the expression
R0νECEC
T1/2
=
C ECEC
years ,
(hmν i[eV])2
(5)
where the effective neutrino mass should be given in units of eV. In all the listed
cases where C ECEC has been computed the decay rates are suppressed by the rather
sizeable magnitude of the degeneracy parameter. Decays to 0+ states are favoured
over the decays to 2+ or 1− , 2− , 3− etc. states due to the involved nuclear wave
functions and/or higher-order transitions. There are some favourable values of degeneracy parameters listed in Table 1, like 106 Cd → 106 Pd(2, 3)− and 156 Dy →
156 Gd(0+ , 1− , 2+ ) but the associated nuclear matrix elements are still waiting for
their evaluation. At the moment the most favourable case with half-life estimate is
the case 152 Gd → 152 Sm(0+
gs ) where the decay is to the ground state.
4. CONCLUSIONS
The nuclear double beta decay is of great current interest due to its strong impact on the physics of massive Majorana neutrinos. Much experimental and theoretical effort has been invested on double beta decays on the β − side of the nuclear
stability line. Contrariwise, on the positron-emitting/electron-capture side much less
experiments and theoretical work has been done. The two-neutrino ECEC, and possibly β + EC, decays to the ground states of the daughter nuclei might be within the
reach of (near-)future double-beta experiments for some of the candidates like 124 Xe,
taken as an example case in this article. The neutrinoless versions of these decays
are much harder to detect and are most likely too elusive for even the next-generation
double-beta experiments. The resonant neutrinoless double beta decay has attracted
a lot of experimental and theoretical interest lately. Unfortunately, based on the
presently available Penning-trap-measured Q values and nuclear-structure calculations, detection of this interesting decay mode is beyond the reach of any foreseeable
experiment.
Acknowledgements. This work was supported by the Academy of Finland under the Finnish
Center of Excellence Program 2012-2017 (Nuclear and Accelerator Based Program at JYFL).
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