POSITRON-EMITTING and DOUBLE-EC MODES of DOUBLE BETA DECAY the Subject of Double Beta Decay Has Attracted Both Theoretical and E

Home , Double electron capture, Positron emission

Dedicated to Professor Apolodor Aristotel Radut¸˘ a’s˘ 70th Anniversary

POSITRON-EMITTING AND DOUBLE-EC MODES OF DOUBLE BETA DECAY

JOUNI SUHONEN Department of Physics, University of Jyvaskyl¨ a,¨ P. O. Box 35 (YFL), FI-40014 University of Jyvaskyl¨ a,¨ Finland E-mail: [email protected].ﬁ 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 RJPRom. 58(Nos. Journ. Phys., 9-10), Vol. 1232–1241 58, Nos. 9-10, (2013) P. 1232–1241, (c) 2013-2013 Bucharest, 2013 2 Positron-emitting and double-EC modes of double beta decay 1233 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 two- neutrino 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] investi- gations 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 2 (2ν) + → + (2ν) (2ν) t1/2 (0i Jf )α = Gα (J) Mα (J) , (1) RJP 58(Nos. 9-10), 1232–1241 (2013) (c) 2013-2013 1234 Jouni Suhonen 3

+ + + (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) + → + (0ν) (0ν) |h i|2 + + + t1/2 (0i 0f )α = Gα M mν , α = β β ,β EC , (2) where hmνi 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 RJP 58(Nos. 9-10), 1232–1241 (2013) (c) 2013-2013 4 Positron-emitting and double-EC modes of double beta decay 1235 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− 2 1 2 |hm i| Γ T R0νECEC(J ) = GECEC(J ) M ECEC(J ) ν , (4) 1/2 f 0ν f 0ν f (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 random- phase approximation (QRPA), and in particular on its proton-neutron variant (pn- QRPA). 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-beta- decay 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 genera- ted 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 de- scribe 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. RJP 58(Nos. 9-10), 1232–1241 (2013) (c) 2013-2013 1236 Jouni Suhonen 5

3.1. DOUBLE BETA DECAY OF 124Xe AS AN EXAMPLE

Fig. 1 summarizes the computed results for the two-neutrino double-beta half- lives 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 124I is unknown and thus it is not possible to normalize the energy denominator of the related NME + − experimentally. Instead, a reasonable range E(11 ) = (0.15 1.00) MeV for the excitation energies was assumed in the evaluation of the NMEs.

− 2gs 124 53I71 + 0gs 124 54Xe70 − × 25 + − × 32 + 1657.28 keV ECEC: (1.7 580) 10 , β EC: (4.4 38000) 10 01 − × 30 + − × 31 + 1325.51 keV ECEC: (1.1 3700) 10 , β EC: (2.0 13000) 10 22

− × 28 + − × 26 602.73 keV ECEC: (2.3 11000) 10 , β EC: (8.8 25000) 10 2+ 1 β+β+: (1.0 − 32) × 1043

− × 20 + − × 21 + ECEC: (4.0 88) 10 , β EC: (9.4 97) 10 0gs 124 β+β+: (1.7 − 38) × 1026 52Te72

Fig. 1 – Computed partial half-lives (in units of years) for two-neutrino double beta decays of 124Xe.

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 RJP 58(Nos. 9-10), 1232–1241 (2013) (c) 2013-2013 6 Positron-emitting and double-EC modes of double beta decay 1237 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 124Xe, 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.

− 2gs X X 124 K K 53I71

. . 30 + 2854.87 keV R-ECEC: (1 9 − 5 6) × 10 + (0 ) 0gs hmνi =0.3 eV 124 54Xe70

+ 32 + 1657.28 keV β EC: ≥ 5.9 × 10 01 hmν i =0.3 eV

+ 1325.51 keV 22

+ 602.73 keV 21

+ 27 + + 28 + β EC: (1.2 − 4.2) × 10 , β β : (2.3 − 7.7) × 10 0gs 124 hmν i =0.3 eV 52Te72

Fig. 2 – Computed partial half-lives for neutrinoless double beta decays of 124Xe. The UCOM short- range 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 124Xe to the ground and excited 0+ states of 124Te are displayed for the effective neutrino mass hmνi = 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. RJP 58(Nos. 9-10), 1232–1241 (2013) (c) 2013-2013 1238 Jouni Suhonen 7

From Fig. 2 one observes that for the decays to the ground state the half-lives are in 27 − 28 + the range of 10 10 years. For the decay to the 01 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 96Ru, 106Cd, 124Xe and 130Ba 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. π − ECEC Transition Jf Q E [keV] Orbitals C Ref. 74 74 + 43 Se → Ge 2 2.23 L2L3 (0.2 − 100) × 10 [38] 96 96 + Ru → Mo 2 8.92(13) L1L3 [73] + 0 ? −3.90(13) L1L1 102 102 + Pd → Ru 2 75.26(36) KL3 [74] 106Cd → 106Pd 0+? 8.39 KK (2.1 − 5.7) × 1030 [34] − (2,3) −0.33(41) KL3 [74] 112Sn → 112Cd 0+ −4.5 KK > 5.9 × 1029 [37] 124Xe → 124Te 0+? 1.86(15) KK (1.7 − 5.1) × 1029 [75] 130Ba → 130Xe 0+? 10.18(30) KK [75] 136Ce → 136Ba 0+ −11.67 KK (3 − 23) × 1032 [39] 144 144 + Sm → Nd 2 171.89(87) KL3 [74] 152 → 152 + − × 27 Gd Sm 0gs 0.91(18) KL1 (1.0 1.5) 10 [41,76] 156 156 − Dy → Gd 1 0.75(10) KL1 [77] + 0 0.54(24) L1L1 [77] + 2 0.04(10) M1N3 [77] 162 162 + Er → Dy 2 2.69(30) KL3 [73] 164 → 164 + − × 31 Er Dy 0gs 6.81(13) L1L1 (3.2 5.2) 10 [41,78] 168 168 − Yb → Er (2 ) 1.52(25) M1M3 [73] 180 → 180 + − × 29 W Hf 0gs 11.24(27) KK (4.0 9.5) 10 [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 measure- RJP 58(Nos. 9-10), 1232–1241 (2013) (c) 2013-2013 8 Positron-emitting and double-EC modes of double beta decay 1239 ment and the possible calculations of the related NME. In the table a quantity CECEC is given and it relates to the R0νECEC half-life through the expression ECEC R0νECEC C T1/2 = 2 years , (5) (hmνi[eV]) where the effective neutrino mass should be given in units of eV. In all the listed cases where CECEC 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 106Cd → 106Pd(2,3)− and 156Dy → 156Gd(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 152 → 152 + the case Gd Sm(0gs) where the decay is to the ground state.

4. CONCLUSIONS

The nuclear double beta decay is of great current interest due to its strong im- pact 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 pos- sibly 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 124Xe, 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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