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 experi- mentally 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 cap- ture, 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 experimen- tal interest already for decades. In particular, the neutrinoless double beta (0νββ) decay has become a popular subject since the emergence of the grand-unified theo- ries (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 num- ber 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 tech- niques 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) jh ij2 + + + 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 re- sonant 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 jhm ij 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).