Neutrino Masses and OscillationsView this Special Issue
Neutrinoless Double β+/EC Decays
The relation of neutrino masses to neutrino oscillations and the nuclear double beta decay is highlighted. In particular, the neutrinoless , EC, and resonant ECEC decays are investigated using microscopic nuclear models. Transitions to the ground state and excited 0+ states are analyzed. Systematics of the related nuclear matrix elements are studied and the present status of the resonant ECEC decays is reviewed.
The modern neutrino oscillation experiments have brought the study of neutrino properties to the era of precision measurements. At the same time the fundamental character (Majorana or Dirac) of the neutrino is still unknown, as is also its absolute mass scale. To gain information on these two unknowns the atomic nuclei can be engaged as the mediators of the Majorana-neutrino triggered neutrinoless double beta () decays. The key issue here is how to cope with the involved nuclear-structure issues of the decays, crystallized in the form of the nuclear matrix elements (NMEs) [1–3]. To be able to exploit the potential data extracted from the-decay experiments one needs to evaluate the NMEs in a reliable enough way. It has become customary to employ the neutrino-emitting correspondent ofdecay, the two-neutrino double beta () decay, to confine the nuclear-model degrees of freedom in the NME calculations. Thedecay is a second-order process in the standard model of the electroweak interactions and the associated half-lives have been measured for several nuclei .
The neutrinoless double() decays have been studied intensively over the years [2, 3] due to their favorable decayvalues. The positron-emitting modes of decays,,EC, and ECEC, are much less studied. From here on we will denote all these decay modes as/EC decays. The general, nuclear model independent frameworks of theory for these decays have been investigated in  for the/EC-decay channelsandEC. The formalism for the resonant neutrinoless double electron capture (RECEC) was first developed in  and later discussed and extended to its radiative variant (ECEC) in . Due to the resonant nature of the RECEC decay its studies have called for precise measurements of the mass differences of the atoms involved in the decays. The resonant mode ofECEC decays is studied intensively for its potential enhanced sensitivity to discover the Majorana mass of the neutrino and that is why much experimental effort is being invested in observing this mode of decay.
2. Neutrino Masses and Oscillations
In the calculations of transition rates of the/EC decays, the neutrino-physics part and nuclear-physics part factorize. We will start by considering the neutrino-physics part. The weak-interaction Lagrangian of leptons is diagonal in the neutrino fields,, and, called flavor eigenfields. The charged-current interaction part of the Lagrangian of the Standard Model of electroweak interactions, which is relevant to the considerations of this presentation, is given by whererefers to the three lepton flavors,,is a vector field corresponding to the charged weak boson,andare the left-handed chiral components of the neutrino and charged lepton fields, andis the gauge coupling constant. In all phenomena studied so far neutrinos appear as ultrarelativistic particles, but it is known that, albeit being extremely light compared with other fermions, neutrinos do have mass, evidenced by observations of many neutrino-flavor-oscillation phenomena (see, e.g., [10–18]). In neutrino oscillations transitions between neutrino flavors take place, indicating that neutrinos mix with each other. This mixing arises through the mechanism that gives neutrinos their mass. The mass part of the neutrino Lagrangian is hence not diagonalized by the flavor fieldsbut by fields() that have definite masses, known as the mass eigenfields. The left-handed flavor eigenfields appearing in the interaction Lagrangian (1) are superpositions of the left-handed components of the mass eigenfields: whereis a unitarymatrix, called the neutrino mixing matrix or Pontecorvo-Maki-Nakagawa-Sakata (PMNS) matrix [19, 20].
The mass of the left-handed neutrinos can arise from mass terms of the form the so-called Majorana mass terms. They can arise in the Standard Model of particle physics through nonrenormalizable interactions between neutrinos and neutral Higgs bosons:., whereis the Yukawa coupling constant,is the energy scale of some new physics not present in the Standard Model, andis a neutral Higgs field. In the Standard Model, the vacuum expectation value of the neutral Higgs field is nonzero,, giving rise to the following Majorana mass term for the left-handed neutrinos: that is,.
One assigns leptons an additive quantum number called the lepton number, such thatfor particles andfor antiparticles. The lepton number is conserved in the standard electroweak interactions, like in the charged-current interactions described by the Lagrangian (1), but the Majorana mass term (3) breaks it by two units; that is, Majorana mass terms are sources or sinks of the lepton number. No empirical evidence of nonconservation of the lepton number exists so far.
If one assumes that there exist, in addition to the left-handed neutrino fields, right-handed neutrino fields, the neutrino mass Lagrangian may contain also the Dirac mass termsand another type of Majorana mass terms. Unless the Majorana mass terms vanish, the fieldsthat diagonalize the full mass Lagrangian are two-component Majorana fields obeying the condition (“Majorana condition”) There are in this case altogether six mass states. It is generally assumed that(the so-called seesaw model [21–25]), implying that three of these six states are light, corresponding to the three ordinary neutrinos appearing in (2), while the other three are very heavy and decouple from the low-energy physics. Even if the mixing between light and heavy sectors is neglected, the relation (2) is still applicable.
A lot of empirical information on the neutrino mixing, that is, the elements of the matrix, and the neutrino masses has been obtained via solar, atmospheric, reactor, and accelerator neutrino oscillation experiments. The mixing matrixcan be presented in terms of six measurable parameters, three rotation angles and three phases, as follows : where,,, and is called the Dirac phase and and the Majorana phases. The probability for the oscillatory transition from the neutrino flavorto the flavoras a function of the distance of flightand neutrino energyis given by (see, e.g., ) where. As can be seen from this formula, the neutrino oscillations do not bring us any information about the absolute neutrino mass scale, only about the squared mass differences. One can also easily show that neutrino oscillations are not sensitive to the Majorana phasesand.
The main goals of the forthcoming neutrino oscillation experiments are to measure the value of the CP phaseand to determine the neutrino mass hierarchy, whether it is normal or inverted. The other important open questions of neutrino physics include determining the absolute mass scale of neutrinos and finding out whether neutrinos are Dirac particles or Majorana particles. These latter two questions could be at least partially solved by neutrinoless double beta decay and other lepton number violating processes. Information about the absolute neutrino mass can be also obtained by determining the effective electron neutrino massin beta decay experiments, as well as from the cosmological precision measurements of the sum of neutrino masses. The current experimental upper limits forare 2.3 eV  and 2.1 eV , and for the sum of neutrino massesthe recent Planck satellite data  imply the upper limit 0.66 eV.
3. Neutrino Masses and Double Beta Decay
In the standard picture the neutrinoless double beta decaysandare mediated by light neutrinos. These processes are of great importance from the particle-physics point of view, as they would indicate the violation of lepton number, which in turn would imply that light neutrinos are Majorana particles. This would be valuable information for understanding the origin of fermion masses.
We are considering in this work particularly the positron-emission mode(see Figure 1). In the electroweak model the leptonic part of this process is described by a second-order perturbation given by where,,, andare the field operators of the electron, positron, electron neutrino, and electron antineutrino, respectively. The strength of the interaction is governed by the Fermi coupling constant, whereis the fundamental gauge coupling of the electroweak theory andis the mass of. The propagator describing the internal neutrino is given by where the condition (5) is used.
The amplitude of the processis proportional to whereis the momentum of the exchanged neutrino andare the chirality projection matrices . Note that thepart of the neutrino propagator does not contribute due to chirality mismatch. TypicallyMeV, in accordance with a typical nuclear distance of 1 fm. Given that neutrinos are expected to be in the sub-eV mass scale, one can safely approximate the denominator of the neutrino propagator by, leading to
The essential part of the amplitude from neutrino-physics point of view is the quantity: whose absolute value is called the effective neutrino mass; that is, Although this quantity depends on a great number of observables, it is associated with just one single parameter of the fundamental Lagrangian, the Majorana mass term of the left-handed electron neutrino(see (3)).
The modesEC and ECEC (Figure 2) are described by the same operator (9) as themode, which is easily understandable since the antiparticle creation operator is always associated with the particle annihilation operator in the fermion fields. Hence all these processes probe the same effective neutrino mass. The decay rates of the processes are proportional to.
Using the standard parametrization (6) of the mixing matrix, one can castin the following form: whereand. Depending on the phasesand, the contributions of the three neutrino mass states will add up constructively or destructively. In the case the CP symmetry is conserved, the phase factors assume the values +1 or −1, depending on the intrinsic CP quantum numbers of the mass states, which in turn depend on the detailed structure of the mass matrix. There are four possible sign combinations which lead to different values for. Any values of the phases different fromwould mean violation of the CP symmetry.
The amplitude of the electron-electron decay mode is proportional to the complex conjugate of. As the decay widths are proportional to, the modesand, as well as of the modesEC and ECEC, probe neutrino physics through the same quantity. Hence the CP is not manifestly broken in neutrinoless double beta decay, although the Majorana phasesandappear in. One can understand this also as a consequence of the fact that in the limitthe amplitudes depend on just one parameter of the mass Lagrangian, the element, allowing for no measurable phases. To be sensitive to the Majorana CP phases, one should be able to distinguish between the mass states.
Apart from the CP phasesand, which are not observables of neutrino oscillations (the possible CP violation in oscillation phenomena is due to the Dirac phase), there are two unknowns in the expression of the effective mass, namely, the absolute neutrino mass scale, say the massof the lightest neutrino, and the mass hierarchy, that is, whether(normal hierarchy) or(inverted hierarchy). All three neutrino masses can be expressed in terms of the absolute mass: in the case of the normal hierarchy and in the case of inverted hierarchy The squared mass differenceand the absolute value of the mass differenceare known from neutrino oscillation experiments. The neutrino hierarchy will be determined in the forthcoming neutrino oscillation experiments. This information would be crucial for interpretation of the results of the double beta decay experiments. In the case of inverted hierarchy,has lower limit of the order ofeV, as can be inferred from Figure 3, where the effective mass,, is presented as a function of the massof the lightest neutrino for all possible values of the Majorana phasesand. If no signal of double beta decay is obtained above this limit, it would mean that either the hierarchy has to be the normal one or the neutrino is not a Majorana particle. An observation of double beta transition with eV would mean that the mass hierarchy is normal and the neutrino is a Majorana particle. On the other hand, nonobservation of the transition would not mean that the process does not exist, since in the case of the normal mass hierarchy the effective mass and hence the decay width can be arbitrarily small.
4. Double Beta Decays on the /Electron-Capture Side
In this section a rather detailed account of the basic theoretical ingredients of the half-life calculations is given. In this way the reader can have a unified picture of the formalisms used for various types of double beta transitions.
4.1. Half-Lives and Nuclear Matrix Elements
In this work it is assumed that the/EC decays proceed exclusively via the exchange of massive Majorana neutrinos, as discussed in Section 3. The inverse half-lives for the neutrinolessandEC decays can be cast in the form whereis the effective neutrino mass (14) that should be given in (18) in units of eV. The decays described by (18) proceed via the available phase space for the final state leptons and the phase-space integralsandare defined in . The involved nuclear matrix element (NME) can be written (see, e.g., [45–47]) in terms of the Gamow-Teller (GT), Fermi (F), and tensor (T) matrix elements in the form wherecorresponds to the bare-nucleon value of the axial-vector coupling constant andis the vector coupling constant. The tensor matrix element is neglected in the present calculations since its contribution is very small [48, 49]. The above defined NME is convenient since the phase-space factor to be used with it is always the one defined forindependent of the value ofused in (19).
In the case of the neutrinoless double electron capture,ECEC, there are no leptons available in the final state to carry away the decay energy. In this case one has to engage some additional mechanism to rid the initial atom of the excess energy of decay. There are two proposed mechanisms to cope with this situation: the radiativeECEC decay  and the resonant decay, RECEC . The resonance condition—close degeneracy of the initial and final (excited) atomic states—can enhance the decay rate by a factor as large as. The RECEC process is of the form where the capture of two atomic electrons leaves the final atom in an excited state, in most cases having the final nucleus in an excited state. The excited state of the nucleus decays by one or more gamma rays and the atomic vacancies is filled by outer electrons with emission of X-rays.
Fulfillment of the resonance condition depends on the so-called degeneracy parameter, whereis the excitation energy of the final atomic state andis the difference between the initial and final atomic masses. Possible candidates for such resonant decays are many and a representative list will be displayed in Section 7. The final nuclear states with spin-parityare the most favorable ones and the only ones discussed as examples in this review. The inverse half-life for transitions tostates can be written as where the daughter stateis a virtual state with energy including the possible nuclear excitation energy and the binding energies of the two captured electrons. The last term accounts for the Coulomb repulsion between the two holes. The quantitydenotes the combined nuclear and atomic radiative widths where the atomic widths dominate and are a few tens of electron volts . The factorcan be called the atomic factor and it contains the information about the density distributions of the involved atomic orbitals at the nucleus. It can be written as whereis the normalization of the relativistic Dirac wave function for the atomic orbital specified by the quantum numbers ()  in the presence of a uniformly charged spherical nucleus.
The Gamow-Teller and Fermi NMEs appearing in (19) can be written explicitly in the form where and and label the different nuclear-model solutions for a given multipole, the setstemming from the calculation based on the final nucleus and the setstemming from the calculation based on the initial nucleus. Here the one-body transition densities areand, and they are given separately for the different types offinal statesin Section 4.3.
The two-particle matrix element of (24) can be written as whereandis the relative distance between the two decaying protons. The following auxiliary quantities have been defined The quantitiesare the Moshinsky brackets that mediate the transformation from the laboratory coordinatesandto the center-of-mass coordinateand the relative coordinate. In this way the short-range correlations of the two decaying protons are easily incorporated in the theory. The wave functionsare taken to be the eigenfunctions of the isotropic harmonic oscillator.
The neutrino potential,, in the integral of (26) is defined as whereis the spherical Bessel function and the integration is performed over the exchanged momentum. Hereis the ground-state mass energy of the initial nucleus andthe (ground-state or excited-state) mass energy of the final nucleus. In practice the lowest pnQRPA energies of the two setsandare normalized such that the energy difference of these energies and the mass energy of the initial nucleus match the corresponding experimental energy difference. The termin (27) includes the contributions arising from the short-range correlations, nucleon form factors, and higher-order terms of the nucleonic weak current . For all the/EC transitions of this work the NMEs have been computed by the use of both the Jastrow short-range correlations  and the UCOM correlations [53, 54]. Both short-range correlators have been recently used in manycalculations [48, 49, 55–58] and in some/EC calculations [36, 59–61].
4.2. Nuclear Models and Model Parameters
In this work the wave functions of the nuclear states involved in the double beta decay transitions are calculated by the use of the quasiparticle random-phase approximation (QRPA) in realistically large single-particle model spaces. Thestates of the intermediate nucleus of thedecays are generated by the usual proton-neutron QRPA (pnQRPA) [2, 62] in the form whereandare the forward- and backward-going amplitudes of the pnQRPA, obtained by solving the pnQRPA equations of motion . The excited statesin the final even-even nuclei are described by the phonons of the charge-conserving QRPA (ccQRPA), expressed as where the symmetrized amplitudesandare obtained from the usual ccQRPA amplitudesand through the transformation and similarly forin terms of.
Now one can take aphonon of (29) and build an ideal two-phononstate of the form An ideal two-phonon state consists of partner statesthat are degenerate in energy and exactly at an energy twice the excitation energy of thestate. In practice this degeneracy is always lifted by the residual interaction between the one- and two-phonon states . The one- and two-phonon states in the final even-even nucleus are connected to thestates of the intermediate nucleus by transition amplitudes obtained from a higher-QRPA framework called the multiple-commutator model (MCM), first introduced in  and further extended in .
The calculations were done in sufficiently large single-particle spaces and the single-particle energies were generated by the use of a spherical Coulomb-corrected Woods-Saxon (WS) potential with a standard parametrization , optimized for nuclei near the line of beta stability. Sometimes the Woods-Saxon based single-particle energies were slightly corrected near the proton and/or neutron Fermi surfaces to better reproduce the low-energy spectra of the neighboring neutron-odd and/or proton-odd nuclei at the BCS level. The Bonn-A G-matrix has been used as the two-body interaction and it has been renormalized in the standard way [64, 67]. The quasiparticles are treated in the BCS formalism and the pairing matrix elements are scaled by a common factor, separately for protons and neutrons. In practice these factors are fitted such that the lowest quasiparticle energies obtained from the BCS match the experimental pairing gaps for protons and neutrons, respectively .
As explained in detail in  the particle-hole and particle-particle parts of the proton-neutron two-body interaction are separately scaled by the particle-hole parameterand particle-particle parameter. The value of the particle-hole parameter was fixed by the available systematics  on the location of the Gamow-Teller giant resonance (GTGR) state. The value of theparameter regulates the-decay amplitude of the firststate in the intermediate nucleus  and hence also the decay rates of thedecays. This value can be fixed by either the data ondecays  or by the data on-decay rates within the intervalof the axial-vector coupling constant [48, 54, 55, 57]. The experimental error and the uncertainty in the value ofthen induce an interval of physically acceptable values of, the minimum value ofrelated toand the maximum value to. This is because the magnitude of the calculatedNME,, decreases with increasing value ofin a pnQRPA calculation [67, 69, 70] and this magnitude is compared with the magnitude of the experimental NME,, deduced from the experimentalhalf-life. The same correspondence betweenandis adopted also here for the/EC decays. In the absence of available half-life data on the/EC side the ranges of the adoptedvalues are reasonable choices such that all the physically meaningful values of the/EC NMEs are covered.
For the ccQRPA the default valuewas adopted and theparameter was fixed such that the experimental energy of the firststate in the reference even-even nucleus was reproduced in the ccQRPA calculation.
4.3. Transition Densities
The various transition densities involved in the decay amplitudes (24) are addressed in this section. The initial-branch transition density remains always the same, namely, The transition density corresponding to the final ground state reads where() and() correspond to the BCS occupation and unoccupation amplitudes of the initial (final) even-even nucleus. The amplitudesand(and) come from the pnQRPA calculation starting from the initial (final) nucleus of the/EC decay.
For the excited states the multiple-commutator model (MCM) [64, 65] is used. It is designed to connect excited states of an even-even reference nucleus to states of the neighboring odd-odd nucleus. 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 . The ccQRPA phonon (29) defines a state in the final nucleus of the double beta decay. In particular, if this final state is thethstate, the related transition density, to be inserted in (24), becomes instead of the expression (33) for the ground-state transition. Again() and() correspond to the BCS occupation and unoccupation amplitudes of the initial (final) even-even nucleus. The amplitudesand(and) come from the pnQRPA calculation starting from the initial (final) nucleus of thedecay. The amplitudesandare the amplitudes of thethstate in the final nucleus. In the present applications we discuss onlyfinal states.
In the case of the two-phonon excitation the transition density to be inserted in (24) attains the form where, as usual, the barred quantities denote amplitudes obtained for the final nucleus of double beta decay. In the present work we use only.
5. Typical Examples
In the present paper the neutrinolessandEC transitions in various nuclei are discussed. Considered are the transitions to the ground state,, and to the firststate,. The/EC decays to only thestates are considered since large suppression of the mass mode for the decays tostates is expected . Furthermore, the RECEC transitions to the possible resonant states are considered. In the present work the analysis of the RECEC half-lives is performed by assuming aassignment for the resonant states. This assignment leads to a very likely enhancement in the decay rate. Since this assignment is in many cases only tentative or even unlikely, the calculated half-lives should be taken as optimistic estimates or as lower limits for the half-life.
All the discussed decay transitions are displayed in Figure 4, where the decay ofCd serves as paradigm. BothandEC transitions to the ground state are possible whereas only theEC mode is possible for the decay to thestate for phase-space reasons since thedecay has a negativevalue for this transition. The resonantECEC transition is also shown with the total energy (including the electron-hole contributions, see (22)) of the resonant atomic state.
The various/EC decay modes can now be treated by applying the formalisms outlined in Section 4. In particular, the BCS is used to create the quasiparticles in the chosen single-particle valence space and the pnQRPA is used to produce the intermediatestates involved in the NME (24). The excited states in the final nucleus are produced by the use of the ccQRPA and the final states are connected to the intermediatestates by the MCM prescriptions. After adjusting the parameters of the model Hamiltonian the rates related to the various decay transitions can be evaluated. The results are shown in our paradigm case in Figure 5 for the rangeof the axial-vector coupling constant and for the valueof the effective neutrino mass (14).
As seen from Figure 5, the fastest decay mode isEC to the ground state ofPd with a half-life of (1.5-1.7) × 1027 years. This could be in the range of detection sensitivity of the next generation of double beta experiments. The resonance transition proceeds by the capture of two K electrons and emission of two K X-rays, has a half-life ofyears, and is thus very hard to be detected in the foreseeable future.
In Figure 6 the half-lives of decay transitions inRu are shown for the rangeand for the value. Again the fastest transition isEC to the ground state ofMo with a half-life of (5.5–6.3) × 1027 years, slightly slower than in the case ofCd decay. Interestingly enough there are decays to two excitedstates at energiesand. The latter state is assumed to be a two-phonon state discussed in this paper, the former one being a one-ccQRPA-phonon state. In this case the resonant decay proceeds with the capture of twoelectrons and emission of twoX-rays. The computed half-life for the resonant decay is (4.9–22) × 1032 years which is impossible to be detected in the foreseeable future.
The last example of this section pertains to the/EC decays inXe, shown in Figure 7. The ranges 1.00–1.25 and were adopted in the calculations. TheEC decay to the ground state ofTe is the fastest with a half-life ofyears, being in the range of the corresponding decay transition inCd. The decay to the resonance state at proceeds with the capture of two K electrons and emission of two K X-rays. The computed half-life isyears and is thus the fastest of the three discussed example cases, though hard to be detected in the near future.
The computed half-lives can be expressed by the use of the auxiliary quantitiesandin the following form: where the effective neutrino mass should be inserted in units of eV. In Table 1 the auxiliary factors of the above equations are given for the nuclei and transitions under discussion. The UCOM short-range correlations have been used combining the results for the possible different basis sets used in the nuclear structure calculations and for the rangeof the axial-vector coupling constant.
From Table 1 it can be evidenced that generally the fastest transitions are theEC transitions to the ground state and transitions to the excitedstate(s) are quite much suppressed relative to the ground-state transitions.
6. Systematic Features of the Nuclear Matrix Elements
There are not too many nuclei that have reasonablevalues and decay by/EC decays, and only part of these can have a reasonable chance of decaying via the resonant neutrinoless double EC channel. It is nevertheless instructive to have a fresh view at the systematic features of the involved NMEs.
6.1. The /EC NMEs
A systematics of the computed NMEs of the/EC decays is shown in Figure 8. The values of NMEs for decays to the ground state (), first excitedstate () and the resonantstate () are shown. As mentioned before the assignment ofto the resonant states has to be taken in some cases, like for theCd decay, with a grain of salt. From the figure one notices that the ground-state NMEs are rather large (or more), except forMo. This means that matrix-element-wise the/EC decays are not suppressed relative to thedecays. This can be further evidenced in Figure 9 where these NMEs are shown together with those ofdecays for nuclei with(a) and for nuclei with(b).
In Figure 8 it is seen that the NMEs corresponding to the resonantstates are larger than the NMEs corresponding to the decays tostates. This stems from the fact that the resonant states are treated as one-ccQRPA-phonon states whereas thestates are described as two-ccQRPA-phonon states. The MCM connects the two-phonon states weaker than the one-phonon states to thestates of the neighboring odd-odd nucleus due to theproducts and the 9symbol appearing in the associated transition density (35).
From Figure 9 it is seen that the/EC NMEs show local maxima (Ru,Cd,Xe,Ba, andCe) for the ground-state transitions and even a global maximum:Ba. On the other hand,Mo shows a global minimum. For the decays to thestates the/EC NMEs are small relative to theNMEs, except forRu which has a relatively large NME.
6.2. Fermi and Gamow-Teller Parts of the /EC NMEs
One can also scrutinize the decomposition of the/EC NMEs to their Fermi and Gamow-Teller constituents. This decomposition is shown in Figure 10 where the negative of the ratio of these two constituents has been plotted for decays to the different final states,,, and. In Figure 11 the same has been done in a global context by including also the ratios for theemitters. In this figure one notices that the ratios have a rather universal value of roughly 2.0, except in the case ofKr that has a ratio of about 6.0. The ratios for thetransitions show pronounced peaks for the/EC emittersKr andRu and for theemittersPd andCd, whereas pronounced minima occur forCd andXe. Mostly these ratios for theare slightly above 2.0.
All in all, much more variation in the Gamow-Teller/Fermi ratio is seen for thestates than for the ground states. From Figure 10 it is clear that the ratios for the resonant states are much higher than for theorstates in the corresponding nuclei.
6.3. Decompositions of the /EC NMEs
The/EC NMEs can be decomposed into contributions of different intermediate multipoles as done in  forRu. Let us use here the decay ofXe as an example. The decomposition of the/EC NMEs(19) can be made in two ways, either through the different multipole statesof the intermediate nucleus (in this case the states ofI) or through different couplingsof the two decaying nucleons [57, 73]. For the Gamow-Teller NME these decompositions can be schematically written as whereis given explicitly in (24). The decompositions (37) are shown for the Gamow-Teller NMEs of the decays ofXe in Figures 12, 13, and 14. All the figures refer to calculations using the Jastrow short-range correlations and the valuefor the axial-vector coupling constant.
From the decomposition figures one can make the following general observations. For the ground-state NME the decomposition in terms ofis the typical one of the pnQRPA calculations [45, 57, 59] and the decomposition in terms ofis typical of the shell-model  and pnQRPA [36, 57, 59] calculations. Here typical for thedecomposition are the strong contributions of the high-multipole components, and. In this case thecontribution is modest contrary to that of theRu decay . This contribution depends strongly on the value of the strength parameter. For thedecomposition typical is the large positive monopole contribution and the much smaller, mostly negative, and higher-multipole contributions.
For the lowest excitedstate,, the pattern is qualitatively different for thedecomposition since the other multipole components thanare suppressed but sizable negative contributions from the higher multipole components appear. The relative contributions of theandstates for the ground-state andNMEs are surprisingly similar. In the case of thedecomposition thecomponent is not the dominant one but, instead, thecomponent is. The higher-multipole components contribute sizably, with varying signs. The multipole decompositions of theresonant state,, which is a one-phonon ccQRPA state, are rather blunt. They show both a strongcomponent and a strong monopolecomponent. The rest of the contributions play only a minor role.
6.4. Comparison of NMEs Produced by Different Models
In Table 2 we present the results of recent calculations for the NMEs of the discussed nuclei. The QRPA results are the ones of this work, the IBM-2 results are taken from , and the projected Hartree-Fock-Bogoliubov (PHFB) results are taken from . The IBA-2 model is based on a phenomenological Hamiltonian with connections to the underlying shell model via a mapping procedure. The PHFB is a mean-field model with phenomenological Hamiltonians. Both IBM-2 and PHFB can explicitly take into account deformation effects whereas the QRPA calculations assume spherical or nearly spherical shapes. Since IBM-2 and PHFB quote their results using the Jastrow short-range correlations, also the QRPA calculations have been done by using these correlations. All the quoted calculations in Table 2 use practically the same value of the axial-vector coupling constant.
From Table 2 one observes that the NMEs computed by the use of the QRPA and IBM-2 are rather similar whereas the PHFB NMEs deviate from them notably. These trends are similar to the ones for thedecaying nuclei as discussed extensively in .
In Table 3 the NMEs corresponding to the/EC decays to the first excitedstate,, are shown for the QRPA and the IBM-2. The PHFB model cannot access these NMEs since it is by definition a mean-field model describing only ground-state transitions. What is striking in Table 3 are the very different NMEs and their trends predicted by the two models. The QRPA produces small NMEs forKr,Cd, andXe and rather large NME forRu. For IBM-2 the opposite happens. This tension between the two calculations is more drastic than in the case of thetransitions, as analyzed in .
6.5. Experimental Limits for the Half-Lives
Up to now only limits of half-lives have been extracted for the various/EC processes. Measurements have been done, for example, forSe , forRu [76, 77], forCd , forSn , forCe andCe , and forZn andW . The obtained lower limits are of the order ofyears forRu  andCd  andyears forCe . These limits are still very far from the theoretical estimates as implied by Table 1.
7. Present Status of the Resonant Processes
Table 4 lists the known cases of RECEC transitions in various nuclei where-value measurements have been conducted recently. Thesevalues have been measured by using the Penning-trap techniques. In the cases ofRu,Cd,Xe, andBa the assignment of spin-parity to the resonant state is uncertain. In these cases further experimental spectroscopy is needed.
In the table we also list the estimated half-lives for the cases for which such exist. The references of the last column indicate the origin of the-value measurement and the possible calculations of the related NME. In the table an auxiliary quantityis listed and its relation with the RECEC half-life stands as where the effective neutrino mass should be given in units of eV. In all the listed cases wherehas been computed the decay rates are suppressed by the rather sizable magnitude of the degeneracy parameter. Decays tostates are favored over the decays toor, and so forth states due to the involved nuclear wave functions and/or higher-order transitions. Also captures from atomic orbitals with orbital angular momentumare suppressed .
There are some favorable values of degeneracy parameters listed in Table 4, likeandbut the associated nuclear matrix elements are still waiting for their evaluation. Strong suppression of the NMEs related to final states withis, however, expected. In case of theDy decay the deformation also plays an important role. At the moment the most favorable case with a half-life estimate is the casewhich describes a decay transition to the ground state.
8. Summary and Conclusions
Neutrino masses and their influence on neutrino oscillations and on the nuclear double beta decay have been addressed. The various positron-emitting and/or electron-capture modes of the neutrinoless double beta decays have been investigated for the associated nuclear matrix elements and decay half-lives. A QRPA-based theory framework with G-matrix-based two-body interactions and realistically large single-particle bases has been used in the calculations. The computed values of the nuclear matrix elements have been analyzed and contrasted globally with the double beta minus nuclear matrix elements. Special attention has been paid to the resonant neutrinoless double electron capture process to survey its potential for Majorana-mass detection in dedicated experiments. Generally, the resonance condition is poorly satisfied and the emerging half-lives are extremely hard to measure. Few exceptions occur but the associated nuclear matrix elements are not known. Further theoretical efforts in these cases are stringently called for.
This work was partly (Jouni Suhonen) 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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