# Neutrinoless Doubleβ+/EC Decays

Authors: Jukka Maalampi; Jouni Suhonen
Journal: Advances in High Energy Physics
 (2013)

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## Abstract

The relation of neutrino masses to neutrino oscillations and the nuclear double beta decay is highlighted. In particular, the neutrinolessβ+β+, β+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.


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## Body

## 1. Introduction

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 (0ν2β) 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 the0ν2β-decay experiments one needs to evaluate the NMEs in a reliable enough way. It has become customary to employ the neutrino-emitting correspondent of0ν2βdecay, the two-neutrino double beta (2ν2β) decay, to confine the nuclear-model degrees of freedom in the NME calculations. The2ν2βdecay is a second-order process in the standard model of the electroweak interactions and the associated half-lives have been measured for several nuclei [4].The neutrinoless doubleβ-(0νβ-β-) decays have been studied intensively over the years [2, 3] due to their favorable decayQvalues. The positron-emitting modes of decays,β+β+,β+EC, and ECEC, are much less studied. From here on we will denote all these decay modes as0νβ+/EC decays. The general, nuclear model independent frameworks of theory for these decays have been investigated in [7] for the0νβ+/EC-decay channelsβ+β+andβ+EC. The formalism for the resonant neutrinoless double electron capture (R0νECEC) was first developed in [8] and later discussed and extended to its radiative variant (0νγECEC) in [9]. Due to the resonant nature of the R0νECEC decay its studies have called for precise measurements of the mass differences of the atoms involved in the decays. The resonant mode of0νECEC 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 the0νβ+/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 fieldsνe,νμ, andντ, 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
(1)ℒCC=-g22∑ℓ‍ν-ℓγμ(1-γ5)ℓWμ+h.c.=-g2∑ℓ‍ν-ℓLγμℓLWμ+h.c.,
whereℓrefers to the three lepton flavors,ℓ=e,μ,τ,Wμis a vector field corresponding to the charged weak bosonW±,νℓLandℓLare the left-handed chiral components of the neutrino and charged lepton fields, andgis 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 fieldsνℓbut by fieldsνi(i=1,2,3) that have definite massesmi, 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:
(2)νℓL=∑ℓ=e,μ,τ‍UℓiνiL,
whereUis a unitary3×3matrix, 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(3)-12Mℓℓ′L(νℓL)C¯νℓ′L+h.c.,
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:-(Yℓℓ′/Λ)(νℓL)C¯(H0)2νℓ′L+h.c., whereYℓℓ′is the Yukawa coupling constant,Λis the energy scale of some new physics not present in the Standard Model, andH0is a neutral Higgs field. In the Standard Model, the vacuum expectation value of the neutral Higgs field is nonzero,〈H0〉=v/2≠0, giving rise to the following Majorana mass term for the left-handed neutrinos:
(4)-v2Yℓℓ′4Λ(νℓL)C¯νℓ′L+h.c.,
that is,Mℓℓ′L=v2Yℓℓ′/2Λ.One assigns leptons an additive quantum number called the lepton numberL, such thatL=+1for particles andL=-1for 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 fieldsνℓL, right-handed neutrino fieldsνℓR, the neutrino mass Lagrangian may contain also the Dirac mass terms-Mℓℓ′DνℓR¯νℓ′+h.c.and another type of Majorana mass terms-(1/2)Mℓℓ′R(νℓR)C¯νℓ′R+h.c. Unless the Majorana mass terms vanish, the fieldsνithat diagonalize the full mass Lagrangian are two-component Majorana fields obeying the condition (“Majorana condition”)
(5)νi=νiC.
There are in this case altogether six mass states. It is generally assumed thatMℓℓ′R≫Mℓℓ′D≫Mℓℓ′L(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 matrixU, and the neutrino masses has been obtained via solar, atmospheric, reactor, and accelerator neutrino oscillation experiments. The mixing matrixUcan be presented in terms of six measurable parameters, three rotation angles and three phases, as follows [26]: (6)U=(c12c13s12c13s13e-iδ-s12c23-c12s23s13eiδc12c23-s12s23s13eiδs23c13s12s23-c12c23s13eiδ-c12s23-s12c23s13eiδc23c13)P,whereP=diag(1,eiα,eiβ),sij=sinθij,cij=cosθij, and δ is called the Dirac phase and α and β the Majorana phases. The probability for the oscillatory transition from the neutrino flavorναto the flavorνβas a function of the distance of flightLand neutrino energyEis given by (see, e.g., [27])
(7)P(να⟶νβ)=δαβ-4∑i>j‍Re[Uαi*Uβj*UβiUαj]sin2Δmij2L4E+2∑i>j‍Im[Uαi*Uβj*UβiUαj]sinΔmij2L2E,
whereΔmij2=mi2-mj2. 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 differencesΔmij2. One can also easily show that neutrino oscillations are not sensitive to the Majorana phasesαandβ.A global fit to oscillation data yields the following values for the parameters [5]:
(8)Δm212=7.54-0.22+0.26×10-5eV2,Δm312≃Δm322=2.43-0.10+0.06×10-3eV2,sin2θ12=0.307-0.16+0.18,sin2θ23=0.386-0.21+0.24,sin2θ13=0.0241-0.0025+0.0025.
Here the normal mass hierarchym3>m1,m2is assumed; the values are slightly varied for the inverse hierarchym3<m1,m2(see [5]).The main goals of the forthcoming neutrino oscillation experiments are to measure the value of the CP phaseδand 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 massmβ=∑i‍|Uei|2mi2in beta decay experiments, as well as from the cosmological precision measurements of the sum of neutrino masses∑i‍mi. The current experimental upper limits formβare 2.3 eV [28] and 2.1 eV [29], and for the sum of neutrino masses∑i‍mithe recent Planck satellite data [6] imply the upper limit 0.66 eV.


## 3. Neutrino Masses and Double Beta Decay

In the standard picture the neutrinoless double beta decays(A,Z)→(A,Z+2)+2e-and(A,Z)→(A,Z-2)+2e+are 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(A,Z)→(A,Z-2)+2e+(see Figure 1). In the electroweak model the leptonic part of this process is described by a second-order perturbation given by
(9)(GF2)2e+¯γμ′(1+γ5)νecνe¯γμ(1-γ5)e-,
wheree-,e+,νe, andνecare the field operators of the electron, positron, electron neutrino, and electron antineutrino, respectively. The strength of the interaction is governed by the Fermi coupling constantGF/2=g2/(8mW2), wheregis the fundamental gauge coupling of the electroweak theory andmWis the mass ofW±. The propagator describing the internal neutrino is given by
(10)〈0|νec(x)ν¯e(y)|0〉=∑i‍(Uei*)2〈0|νi(x)ν¯i(y)|0〉=-i∑i‍(Uei*)2∫‍d4q(2π)4q+miq2-mi2exp(-iq·(x-y)),
where the condition (5) is used.Figure 1
Feynman diagram for the Majorana-neutrino-mediatedβ+β+decay.The amplitude of the process(A,Z)→(A,Z-2)+2e+is proportional to
(11)∑i‍GF2Uei*2γμγLq+miq2-mi2γμ′γL=∑i‍GF2Uei*2miq2-mi2γμγRγμ′,
whereqis the momentum of the exchanged neutrino andγL(R)are the chirality projection matrices γL(R)=(1(-/(+))γ5)/2. Note that theqpart of the neutrino propagator does not contribute due to chirality mismatch. Typicallyq≃100MeV, 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 byq2, leading to
(12)GF2(∑i‍Uei*2mi)1q2γμγRγμ′.The essential part of the amplitude from neutrino-physics point of view is the quantity:(13)〈mν〉≡∑i‍Uei*2mi,
whose absolute value is called the effective neutrino mass; that is,
(14)meff=|〈mν〉|.
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 neutrinoMeeL(see (3)).The modesβ+EC and ECEC (Figure 2) are described by the same operator (9) as theβ+β+mode, 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|〈mν〉|2.Feynman diagrams for the Majorana-neutrino-mediatedβ+EC (a) and ECEC (b) decays.
(a)
(b)Using the standard parametrization (6) of the mixing matrixU, one can cast〈mν〉in the following form:
(15)〈mν〉=c122c132m1+s122c132e-iφ12m2+s132e-iφ13m3,
whereφ12=αandφ13=β-δ. Depending on the phasesφ12andφ13, 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〈mν〉. Any values of the phases different from±1would mean violation of the CP symmetry.The amplitude of the electron-electron decay mode is proportional to the complex conjugate of〈mν〉. As the decay widths are proportional to|〈mν〉|2, the modesβ+β+andβ-β-, as well as of the modesβ+EC and ECEC, probe neutrino physics through the same quantity. Hence the CP is not manifestly broken in neutrinoless double beta decay, although the Majorana phasesφ12andφ13appear in〈mν〉. One can understand this also as a consequence of the fact that in the limitq2≫mi2the amplitudes depend on just one parameter of the mass Lagrangian, the elementMeeL, allowing for no measurable phases. To be sensitive to the Majorana CP phases, one should be able to distinguish between the mass statesνi.Apart from the CP phasesφ12andφ13, which are not observables of neutrino oscillations (the possible CP violation in oscillation phenomena is due to the Dirac phaseδ), there are two unknowns in the expression of the effective mass, namely, the absolute neutrino mass scale, say the massm0of the lightest neutrino, and the mass hierarchy, that is, whetherm3>m1,m2(normal hierarchy) orm3<m1,m2(inverted hierarchy). All three neutrino masses can be expressed in terms of the absolute massm0: in the case of the normal hierarchy
(16)m1=m0,m2=m02+Δm212,m3=m02+Δm312,
and in the case of inverted hierarchy
(17)m1=m02+|Δm312|,m2=m02+Δm212+|Δm312|,m3=m0.
The squared mass differenceΔm212and the absolute value of the mass differenceΔm312are 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,|〈mν〉|has lower limit of the order of10-2eV, as can be inferred from Figure 3, where the effective mass,meff=|〈mν〉|, is presented as a function of the massm0of the lightest neutrino for all possible values of the Majorana phasesφ12andφ13. 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|〈mν〉|<10-2 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.Figure 3
Absolute value of the effective neutrino massmeff=|〈mν〉|against the massm0of the lightest neutrino for both the normal and inverted mass hierarchy and for all possible values of the phasesφ12andφ13defined in (15). The best-fit values [5] are used for the oscillation parameters (see (8)). The cosmological upper limit form0, derived from the Planck satellite measurements [6], is also given.


## 4. Double Beta Decays on theβ+/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 the0νβ+/EC decays proceed exclusively via the exchange of massive Majorana neutrinos, as discussed in Section 3. The inverse half-lives for the neutrinolessβ+β+andβ+EC decays can be cast in the form
(18)[T0να(0+)]-1=G0να(0+)|M(0ν)′|2(meff[eV])2,α=β+β+,β+EC,
wheremeffis 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 integralsG0νβ+β+(0+)andG0νβ+EC(0+)are defined in [7]. 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
(19)M(0ν)′=(gA1.25)2[MGT(0ν)-(gVgA)2MF(0ν)+MT(0ν)],
wheregA=1.25corresponds to the bare-nucleon value of the axial-vector coupling constant andgV=1.00is 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 forgA=1.25independent of the value ofgAused in (19).In the case of the neutrinoless double electron capture,0νECEC, 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 radiative0νECEC decay [9] and the resonant decay, R0νECEC [8]. The resonance condition—close degeneracy of the initial and final (excited) atomic states—can enhance the decay rate by a factor as large as106. The R0νECEC process is of the form
(20)e-+e-+(A,Z)⟶(A,Z-2)*⟶(A,Z-2)+γ+2X,
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 parameterQ-E, whereEis the excitation energy of the final atomic state andQis 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-parity0+are the most favorable ones and the only ones discussed as examples in this review. The inverse half-life for transitions to0+states can be written as
(21)[T0νECEC(0+)]-1=g0νECEC(0+)|M(0ν)′|2meff2Γ(Q-E)2+Γ2/4,
where the daughter state(A,Z-2)*is a virtual state with energy
(22)E=E*+EH+EH′+EHH′,
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 quantityΓdenotes the combined nuclear and atomic radiative widths where the atomic widths dominate and are a few tens of electron volts [50]. The factorg0νECECcan 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
(23)g0νECEC(0+)=(GFcosθC2)4gA44π2ln2RA2me6𝒩n,κ𝒩n′,κ′,
where𝒩n,κis the normalization of the relativistic Dirac wave function for the atomic orbital specified by the quantum numbers (n,κ) [7] 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
(24)MK(0ν)=∑Jπ‍∑J′‍∑k1k2‍∑pp′nn′‍(-1)jp+jn′+J+J′×2J′+1{jnjpJjp′jn′J′}×mK(nn′,pp′;J′;k1,k2)×(0f+∥[cn′†c~p′]J∥Jk1π)〈Jk1π∣Jk2π〉(Jk2π∥[cn†c~p]J∥0i+),
whereK=F,GT and k1 and k2label the different nuclear-model solutions for a given multipoleJπ, the setk1stemming from the calculation based on the final nucleus and the setk2stemming from the calculation based on the initial nucleus. Here the one-body transition densities are(0f+∥[cn′†c~p′]J∥Jk1π)and(Jk2π∥[cn†c~p]J∥0i+), and they are given separately for the different types of0+final statesfin Section 4.3.The two-particle matrix element of (24) can be written as
(25)mK(nn′,pp′;J′;k1,k2)=J′^jp^jp′^jn^jn′^∑λS‍(2λ+1)(2S+1)FK×{lplp′λ1212Sjpjp′J′}{lnln′λ1212Sjnjn′J′}×∑n1n2lNL‍Mλ(n1lNL;nnlnnn′ln′)×Mλ(n2lNL;nplpnp′lp′)×∫‍d3rϕn1l(r)hK(r12,12(Ek1+Ek2))ϕn2l(r),
wherej^=2j+1andr12=|r1-r2|is the relative distance between the two decaying protons. The following auxiliary quantities have been defined
(26)FF=1,FGT=6(-1)S+1{1212S12121}.
The quantitiesMλare the Moshinsky brackets that mediate the transformation from the laboratory coordinatesr1andr2to the center-of-mass coordinateR=(1/2)(r1+r2)and the relative coordinater=(1/2)(r1-r2). In this way the short-range correlations of the two decaying protons are easily incorporated in the theory. The wave functionsϕnl(r)are taken to be the eigenfunctions of the isotropic harmonic oscillator.The neutrino potentialhK(r12,E),K=F,GT, in the integral of (26) is defined as
(27)hK(r12,E)=2πRA∫‍dqqhK(q2)q+E-(Ei+Ef)/2j0(qr12),
wherej0is the spherical Bessel function and the integration is performed over the exchanged momentumq. HereEi=Mic2is the ground-state mass energy of the initial nucleus andEfthe (ground-state or excited-state) mass energy of the final nucleus. In practice the lowest pnQRPA energies of the two setsk1andk2are normalized such that the energy difference of these energies and the mass energy of the initial nucleus match the corresponding experimental energy difference. The termhK(q2)in (27) includes the contributions arising from the short-range correlations, nucleon form factors, and higher-order terms of the nucleonic weak current [51]. For all the0νβ+/EC transitions of this work the NMEs have been computed by the use of both the Jastrow short-range correlations [52] and the UCOM correlations [53, 54]. Both short-range correlators have been recently used in many0νβ-β-calculations [48, 49, 55–58] and in some0νβ+/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. TheJπstates of the intermediate nucleus of theββdecays are generated by the usual proton-neutron QRPA (pnQRPA) [2, 62] in the form
(28)|JkπM〉=Q†(Jkπ,M)|QRPA〉=∑pn‍(XpnJkπ[ap†an†]JM-YpnJkπ[ap†an†]JM†)|QRPA〉,
whereXandYare the forward- and backward-going amplitudes of the pnQRPA, obtained by solving the pnQRPA equations of motion [62]. The excited statesIkπin the final even-even nuclei are described by the phonons of the charge-conserving QRPA (ccQRPA), expressed as
(29)|IkπM〉=Q†(Ikπ,M)|QRPA〉=∑ab‍(ZabIkπ[aa†ab†]IM-WabIkπ[aa†ab†]IM†)|QRPA〉,
where the symmetrized amplitudesZandWare obtained from the usual ccQRPA amplitudesXandY[62] through the transformation
(30)ZabIkπ={XabIkπ,ifa=b12XabIkπ,ifa<b12XbaIkπ,ifa>b
and similarly forWin terms ofY.Now one can take aIkπ=21+phonon of (29) and build an ideal two-phononI+state of the form
(31)|I2-ph+〉=12[Q†(21+)Q†(21+)]I|QRPA〉.
An ideal two-phonon state consists of partner statesIπ=0+,2+,4+that are degenerate in energy and exactly at an energy twice the excitation energy of the21+state. In practice this degeneracy is always lifted by the residual interaction between the one- and two-phonon states [63]. The one- and two-phonon states in the final even-even nucleus are connected to theJπstates of the intermediate nucleus by transition amplitudes obtained from a higher-QRPA framework called the multiple-commutator model (MCM), first introduced in [64] and further extended in [65].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 [66], 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 [62].As explained in detail in [45] the particle-hole and particle-particle parts of the proton-neutron two-body interaction are separately scaled by the particle-hole parametergphand particle-particle parametergpp. The value of the particle-hole parameter was fixed by the available systematics [62] on the location of the Gamow-Teller giant resonance (GTGR) state. The value of thegppparameter regulates theβ--decay amplitude of the first1+state in the intermediate nucleus [68] and hence also the decay rates of theββdecays. This value can be fixed by either the data onβ-decays [68] or by the data on2νβ-β--decay rates within the intervalgA=1.00–1.25of the axial-vector coupling constant [48, 54, 55, 57]. The experimental error and the uncertainty in the value ofgAthen induce an interval of physically acceptable values ofgpp, the minimum value ofgpprelated togA=1.00and the maximum value togA=1.25. This is because the magnitude of the calculated2νβ-β-NME,M(2ν), decreases with increasing value ofgppin a pnQRPA calculation [67, 69, 70] and this magnitude is compared with the magnitude of the experimental NME,M(2ν)(exp)∝(gA)-2, deduced from the experimental2νβ-β-half-life. The same correspondence betweengppandgAis adopted also here for the0νβ+/EC decays. In the absence of available half-life data on the0νβ+/EC side the ranges of the adoptedgppvalues are reasonable choices such that all the physically meaningful values of the0νβ+/EC NMEs are covered.For the ccQRPA the default valuegph=1.0was adopted and thegppparameter was fixed such that the experimental energy of the first2+state 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,
(32)(Jk2π∥[cn†c~p]J∥0i+)=J^(-1)jp+jn+J+1[vpunXpnJk2π+upvnYpnJk2π].
The transition density corresponding to the final ground state reads
(33)(0gs+∥[cn′†c~p′]J∥Jk1π)=J^(-1)jp′+jn′+J+1[u-p′v-n′X-p′n′Jk1π+v-p′u-n′Y-p′n′Jk1π],
wherev(v-) andu(u-) correspond to the BCS occupation and unoccupation amplitudes of the initial (final) even-even nucleus. The amplitudesXandY(X-andY-) come from the pnQRPA calculation starting from the initial (final) nucleus of the0νβ+/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 [71]. The ccQRPA phonon (29) defines a state in the final nucleus of the double beta decay. In particular, if this final state is thekthI+state, the related transition density, to be inserted in (24), becomes
(34)(Ik+∥[cn′†c~p′]L∥Jk1π)=2I^L^J^(-1)L+I+J×({{JILjn′jp′jn1}}∑p1‍[v-p′v-n′X-p1n′Jk1πZ-p′p1Ik+-u-p′u-n′Y-p1n′Jk1πW-p′p1Ik+]×{JILjp′jn′jp1}+∑n1‍(-1)I+jn′+jn1×[u-p′u-n′X-p′n1Jk1πZ-n′n1Ik+-v-p′v-n′Y-p′n1Jk1πW-n′n1Ik+]×{JILjn′jp′jn1}∑p1‍[v-p′v-n′X-p1n′Jk1πZ-p′p1Ik+-u-p′u-n′Y-p1n′Jk1πW-p′p1Ik+])
instead of the expression (33) for the ground-state transition. Againv(v-) andu(u-) correspond to the BCS occupation and unoccupation amplitudes of the initial (final) even-even nucleus. The amplitudesXandY(X-andY-) come from the pnQRPA calculation starting from the initial (final) nucleus of theββdecay. The amplitudesZ-andW-are the amplitudes of thekthI+state in the final nucleus. In the present applications we discuss onlyI+=0+final states.In the case of the two-phonon excitation the transition density to be inserted in (24) attains the form
(35)(I2-ph+∥[cn′†c~p′]L∥Jk1π)=402I^L^J^(-1)I+J+jp′+jn′×∑p1n1‍[v-p′u-n′X-p1n1Jk1πZ-p′p121+Z-n′n121++u-p′v-n′Y-p1n1Jk1πW-p′p121+W-n′n121+]×{jp′jp12jn′jn12LJI},
where, as usual, the barred quantities denote amplitudes obtained for the final nucleus of double beta decay. In the present work we use onlyI2-ph+=0+.


## 4.1. Half-Lives and Nuclear Matrix Elements

In this work it is assumed that the0νβ+/EC decays proceed exclusively via the exchange of massive Majorana neutrinos, as discussed in Section 3. The inverse half-lives for the neutrinolessβ+β+andβ+EC decays can be cast in the form
(18)[T0να(0+)]-1=G0να(0+)|M(0ν)′|2(meff[eV])2,α=β+β+,β+EC,
wheremeffis 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 integralsG0νβ+β+(0+)andG0νβ+EC(0+)are defined in [7]. 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
(19)M(0ν)′=(gA1.25)2[MGT(0ν)-(gVgA)2MF(0ν)+MT(0ν)],
wheregA=1.25corresponds to the bare-nucleon value of the axial-vector coupling constant andgV=1.00is 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 forgA=1.25independent of the value ofgAused in (19).In the case of the neutrinoless double electron capture,0νECEC, 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 radiative0νECEC decay [9] and the resonant decay, R0νECEC [8]. The resonance condition—close degeneracy of the initial and final (excited) atomic states—can enhance the decay rate by a factor as large as106. The R0νECEC process is of the form
(20)e-+e-+(A,Z)⟶(A,Z-2)*⟶(A,Z-2)+γ+2X,
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 parameterQ-E, whereEis the excitation energy of the final atomic state andQis 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-parity0+are the most favorable ones and the only ones discussed as examples in this review. The inverse half-life for transitions to0+states can be written as
(21)[T0νECEC(0+)]-1=g0νECEC(0+)|M(0ν)′|2meff2Γ(Q-E)2+Γ2/4,
where the daughter state(A,Z-2)*is a virtual state with energy
(22)E=E*+EH+EH′+EHH′,
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 quantityΓdenotes the combined nuclear and atomic radiative widths where the atomic widths dominate and are a few tens of electron volts [50]. The factorg0νECECcan 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
(23)g0νECEC(0+)=(GFcosθC2)4gA44π2ln2RA2me6𝒩n,κ𝒩n′,κ′,
where𝒩n,κis the normalization of the relativistic Dirac wave function for the atomic orbital specified by the quantum numbers (n,κ) [7] 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
(24)MK(0ν)=∑Jπ‍∑J′‍∑k1k2‍∑pp′nn′‍(-1)jp+jn′+J+J′×2J′+1{jnjpJjp′jn′J′}×mK(nn′,pp′;J′;k1,k2)×(0f+∥[cn′†c~p′]J∥Jk1π)〈Jk1π∣Jk2π〉(Jk2π∥[cn†c~p]J∥0i+),
whereK=F,GT and k1 and k2label the different nuclear-model solutions for a given multipoleJπ, the setk1stemming from the calculation based on the final nucleus and the setk2stemming from the calculation based on the initial nucleus. Here the one-body transition densities are(0f+∥[cn′†c~p′]J∥Jk1π)and(Jk2π∥[cn†c~p]J∥0i+), and they are given separately for the different types of0+final statesfin Section 4.3.The two-particle matrix element of (24) can be written as
(25)mK(nn′,pp′;J′;k1,k2)=J′^jp^jp′^jn^jn′^∑λS‍(2λ+1)(2S+1)FK×{lplp′λ1212Sjpjp′J′}{lnln′λ1212Sjnjn′J′}×∑n1n2lNL‍Mλ(n1lNL;nnlnnn′ln′)×Mλ(n2lNL;nplpnp′lp′)×∫‍d3rϕn1l(r)hK(r12,12(Ek1+Ek2))ϕn2l(r),
wherej^=2j+1andr12=|r1-r2|is the relative distance between the two decaying protons. The following auxiliary quantities have been defined
(26)FF=1,FGT=6(-1)S+1{1212S12121}.
The quantitiesMλare the Moshinsky brackets that mediate the transformation from the laboratory coordinatesr1andr2to the center-of-mass coordinateR=(1/2)(r1+r2)and the relative coordinater=(1/2)(r1-r2). In this way the short-range correlations of the two decaying protons are easily incorporated in the theory. The wave functionsϕnl(r)are taken to be the eigenfunctions of the isotropic harmonic oscillator.The neutrino potentialhK(r12,E),K=F,GT, in the integral of (26) is defined as
(27)hK(r12,E)=2πRA∫‍dqqhK(q2)q+E-(Ei+Ef)/2j0(qr12),
wherej0is the spherical Bessel function and the integration is performed over the exchanged momentumq. HereEi=Mic2is the ground-state mass energy of the initial nucleus andEfthe (ground-state or excited-state) mass energy of the final nucleus. In practice the lowest pnQRPA energies of the two setsk1andk2are normalized such that the energy difference of these energies and the mass energy of the initial nucleus match the corresponding experimental energy difference. The termhK(q2)in (27) includes the contributions arising from the short-range correlations, nucleon form factors, and higher-order terms of the nucleonic weak current [51]. For all the0νβ+/EC transitions of this work the NMEs have been computed by the use of both the Jastrow short-range correlations [52] and the UCOM correlations [53, 54]. Both short-range correlators have been recently used in many0νβ-β-calculations [48, 49, 55–58] and in some0νβ+/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. TheJπstates of the intermediate nucleus of theββdecays are generated by the usual proton-neutron QRPA (pnQRPA) [2, 62] in the form
(28)|JkπM〉=Q†(Jkπ,M)|QRPA〉=∑pn‍(XpnJkπ[ap†an†]JM-YpnJkπ[ap†an†]JM†)|QRPA〉,
whereXandYare the forward- and backward-going amplitudes of the pnQRPA, obtained by solving the pnQRPA equations of motion [62]. The excited statesIkπin the final even-even nuclei are described by the phonons of the charge-conserving QRPA (ccQRPA), expressed as
(29)|IkπM〉=Q†(Ikπ,M)|QRPA〉=∑ab‍(ZabIkπ[aa†ab†]IM-WabIkπ[aa†ab†]IM†)|QRPA〉,
where the symmetrized amplitudesZandWare obtained from the usual ccQRPA amplitudesXandY[62] through the transformation
(30)ZabIkπ={XabIkπ,ifa=b12XabIkπ,ifa<b12XbaIkπ,ifa>b
and similarly forWin terms ofY.Now one can take aIkπ=21+phonon of (29) and build an ideal two-phononI+state of the form
(31)|I2-ph+〉=12[Q†(21+)Q†(21+)]I|QRPA〉.
An ideal two-phonon state consists of partner statesIπ=0+,2+,4+that are degenerate in energy and exactly at an energy twice the excitation energy of the21+state. In practice this degeneracy is always lifted by the residual interaction between the one- and two-phonon states [63]. The one- and two-phonon states in the final even-even nucleus are connected to theJπstates of the intermediate nucleus by transition amplitudes obtained from a higher-QRPA framework called the multiple-commutator model (MCM), first introduced in [64] and further extended in [65].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 [66], 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 [62].As explained in detail in [45] the particle-hole and particle-particle parts of the proton-neutron two-body interaction are separately scaled by the particle-hole parametergphand particle-particle parametergpp. The value of the particle-hole parameter was fixed by the available systematics [62] on the location of the Gamow-Teller giant resonance (GTGR) state. The value of thegppparameter regulates theβ--decay amplitude of the first1+state in the intermediate nucleus [68] and hence also the decay rates of theββdecays. This value can be fixed by either the data onβ-decays [68] or by the data on2νβ-β--decay rates within the intervalgA=1.00–1.25of the axial-vector coupling constant [48, 54, 55, 57]. The experimental error and the uncertainty in the value ofgAthen induce an interval of physically acceptable values ofgpp, the minimum value ofgpprelated togA=1.00and the maximum value togA=1.25. This is because the magnitude of the calculated2νβ-β-NME,M(2ν), decreases with increasing value ofgppin a pnQRPA calculation [67, 69, 70] and this magnitude is compared with the magnitude of the experimental NME,M(2ν)(exp)∝(gA)-2, deduced from the experimental2νβ-β-half-life. The same correspondence betweengppandgAis adopted also here for the0νβ+/EC decays. In the absence of available half-life data on the0νβ+/EC side the ranges of the adoptedgppvalues are reasonable choices such that all the physically meaningful values of the0νβ+/EC NMEs are covered.For the ccQRPA the default valuegph=1.0was adopted and thegppparameter was fixed such that the experimental energy of the first2+state 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,
(32)(Jk2π∥[cn†c~p]J∥0i+)=J^(-1)jp+jn+J+1[vpunXpnJk2π+upvnYpnJk2π].
The transition density corresponding to the final ground state reads
(33)(0gs+∥[cn′†c~p′]J∥Jk1π)=J^(-1)jp′+jn′+J+1[u-p′v-n′X-p′n′Jk1π+v-p′u-n′Y-p′n′Jk1π],
wherev(v-) andu(u-) correspond to the BCS occupation and unoccupation amplitudes of the initial (final) even-even nucleus. The amplitudesXandY(X-andY-) come from the pnQRPA calculation starting from the initial (final) nucleus of the0νβ+/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 [71]. The ccQRPA phonon (29) defines a state in the final nucleus of the double beta decay. In particular, if this final state is thekthI+state, the related transition density, to be inserted in (24), becomes
(34)(Ik+∥[cn′†c~p′]L∥Jk1π)=2I^L^J^(-1)L+I+J×({{JILjn′jp′jn1}}∑p1‍[v-p′v-n′X-p1n′Jk1πZ-p′p1Ik+-u-p′u-n′Y-p1n′Jk1πW-p′p1Ik+]×{JILjp′jn′jp1}+∑n1‍(-1)I+jn′+jn1×[u-p′u-n′X-p′n1Jk1πZ-n′n1Ik+-v-p′v-n′Y-p′n1Jk1πW-n′n1Ik+]×{JILjn′jp′jn1}∑p1‍[v-p′v-n′X-p1n′Jk1πZ-p′p1Ik+-u-p′u-n′Y-p1n′Jk1πW-p′p1Ik+])
instead of the expression (33) for the ground-state transition. Againv(v-) andu(u-) correspond to the BCS occupation and unoccupation amplitudes of the initial (final) even-even nucleus. The amplitudesXandY(X-andY-) come from the pnQRPA calculation starting from the initial (final) nucleus of theββdecay. The amplitudesZ-andW-are the amplitudes of thekthI+state in the final nucleus. In the present applications we discuss onlyI+=0+final states.In the case of the two-phonon excitation the transition density to be inserted in (24) attains the form
(35)(I2-ph+∥[cn′†c~p′]L∥Jk1π)=402I^L^J^(-1)I+J+jp′+jn′×∑p1n1‍[v-p′u-n′X-p1n1Jk1πZ-p′p121+Z-n′n121++u-p′v-n′Y-p1n1Jk1πW-p′p121+W-n′n121+]×{jp′jp12jn′jn12LJI},
where, as usual, the barred quantities denote amplitudes obtained for the final nucleus of double beta decay. In the present work we use onlyI2-ph+=0+.


## 5. Typical Examples

In the present paper the neutrinolessβ+β+andβ+EC transitions in various nuclei are discussed. Considered are the transitions to the ground state,0gs+, and to the first0+state,01+. The0νβ+/EC decays to only the0+states are considered since large suppression of the mass mode for the decays to2+states is expected [72]. Furthermore, the R0νECEC transitions to the possible resonant states are considered. In the present work the analysis of the R0νECEC half-lives is performed by assuming a0+assignment 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 Figure4, where the decay of106Cd serves as paradigm. Both0νβ+β+and0νβ+EC transitions to the ground state are possible whereas only the0νβ+EC mode is possible for the decay to the01+state for phase-space reasons since the0νβ+β+decay has a negativeQvalue for this transition. The resonant0νECEC transition is also shown with the total energy (including the electron-hole contributions, see (22)) of the resonant atomic state.Figure 4
Schematic representation of the possible0νβ+/EC decay modes of106Cd. The atomic resonance at2766.26keV includes both the nuclear excitation energy and the energy related to the two electron holes in the atomic K shell as given in (22).The various0νβ+/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 intermediateJπstates 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 intermediateJπstates 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 rangegA=1.00–1.25of the axial-vector coupling constant and for the valuemeff=0.3eVof the effective neutrino mass (14).Figure 5
Computed partial0νβ+/EC decay half-lives of the various decay transitions from106Cd. The valuemeff=0.3eV is adopted for the effective neutrino mass and the UCOM short-range correlations have been assumed. The half-lives are given in units of years.As seen from Figure5, the fastest decay mode is0νβ+EC to the ground state of106Pd 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 of(2.3–6.3)×1031years, and is thus very hard to be detected in the foreseeable future.In Figure6 the half-lives of decay transitions in96Ru are shown for the rangegA=1.00–1.25and for the valuemeff=0.3eV. Again the fastest transition is0νβ+EC to the ground state of96Mo with a half-life of (5.5–6.3) × 1027  years, slightly slower than in the case of106Cd decay. Interestingly enough there are decays to two excited0+states at energies1148.13keVand1330keV. 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 twoL1electrons and emission of twoL1X-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.Figure 6
Computed partial0νβ+/EC decay half-lives of the various decay transitions from96Ru. The valuemeff=0.3eV is adopted for the effective neutrino mass and the UCOM short-range correlations have been assumed. The half-lives are given in units of years.The last example of this section pertains to the0νβ+/EC decays in124Xe, shown in Figure 7. The rangesgA= 1.00–1.25 andmeff=0.3eV were adopted in the calculations. The0νβ+EC decay to the ground state of124Te is the fastest with a half-life of(1.2–4.2)×1027years, being in the range of the corresponding decay transition in106Cd. The decay to the resonance state at2854.87keV proceeds with the capture of two K electrons and emission of two K X-rays. The computed half-life is(1.9–5.6)×1030years and is thus the fastest of the three discussed example cases, though hard to be detected in the near future.Figure 7
Computed partial0νβ+/EC decay half-lives of the various decay transitions from124Xe. The valuemeff=0.3eV is adopted for the effective neutrino mass and the UCOM short-range correlations have been assumed. The half-lives are given in units of years.The computed half-lives can be expressed by the use of the auxiliary quantitiesCβ+β+andCβ+ECin the following form:
(36)T1/2β+β+=Cβ+β+(meff[eV])-2,T1/2β+EC=Cβ+EC(meff[eV])-2,
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 rangegA=1.00–1.25of the axial-vector coupling constant.Table 1
Auxiliary factors of (36) computed for the nuclei under discussion by using gA = 1.00−1.25 and the UCOM short-range correlations. The results for 92Mo and 136Ce are taken from [30] with gA=1.261.
Nucleus
State
C
β
+
β
+
C
β
+
EC
78Kr
0
gs
+
(9.4−16) ×1026
(3.9−6.5) ×1026
0
1
+
—
(
1.7−4.8) × 1031
92Mo
0
gs
+
—
6.7
×
1
0
29
96Ru
0
gs
+
(5.9−6.7) ×1027
(5.0−5.7) ×1026
0
1
+
—
(1.0−7.6) ×1028
0
2
+
—
(3.4−7.3) ×1029
106Cd
0
gs
+
(1.9−2.2) ×1027
(
1.3−1.5) × 1026
0
1
+
—
(
9.8−12.2) × 1028
124Xe
0
gs
+
(2.1−6.9) ×1027
(
1.1−3.7) × 1026
0
1
+
—
>5.3× 1031
130Ba
0
gs
+
(5.3−15.9) ×1027
(5.6−16.9) ×1025
0
1
+
—
—
136Ce
0
gs
+
2.7
×
1
0
29
4.0
×
1
0
26From Table1 it can be evidenced that generally the fastest transitions are the0νβ+EC transitions to the ground state and transitions to the excited0+state(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 reasonableQvalues and decay by0νβ+/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. The0νβ+/EC NMEs

A systematics of the computed NMEs of the0νβ+/EC decays is shown in Figure 8. The values of NMEs for decays to the ground state (0gs+), first excited0+state (01+) and the resonant0+state (0res+) are shown. As mentioned before the assignment ofJπ=0+to the resonant states has to be taken in some cases, like for the106Cd decay, with a grain of salt. From the figure one notices that the ground-state NMEs are rather large (5.0or more), except for92Mo. This means that matrix-element-wise the0νβ+/EC decays are not suppressed relative to the0νβ-β-decays. This can be further evidenced in Figure 9 where these NMEs are shown together with those of0νβ-β-decays for nuclei with70≤A≤100(a) and for nuclei with104≤A≤136(b).Figure 8
Values of the computed nuclear matrix elements for0νβ+/EC decay transitions to the ground state (0gs+), the first excited0+state (01+) and the resonant0+state (0res+) for all the nuclei discussed in this paper.Values of the computed nuclear matrix elements for0νβ-β-and0νβ+/EC decays for masses70≤A≤100(a) and for104≤A≤136(b). Shown are the NMEs corresponding to the decay transitions to the ground state (0gs+) and the first excited0+state (01+).
(a)
(b)In Figure8 it is seen that the NMEs corresponding to the resonant0+states are larger than the NMEs corresponding to the decays to01+states. This stems from the fact that the resonant states are treated as one-ccQRPA-phonon states whereas the01+states are described as two-ccQRPA-phonon states. The MCM connects the two-phonon states weaker than the one-phonon states to theJπstates of the neighboring odd-odd nucleus due to theZZproducts and the 9jsymbol appearing in the associated transition density (35).From Figure9 it is seen that the0νβ+/EC NMEs show local maxima (96Ru,106Cd,124Xe,130Ba, and136Ce) for the ground-state transitions and even a global maximum:130Ba. On the other hand,92Mo shows a global minimum. For the decays to the01+states the0νβ+/EC NMEs are small relative to the0νβ-β-NMEs, except for96Ru which has a relatively large NME.


### 6.2. Fermi and Gamow-Teller Parts of the0νβ+/EC NMEs

One can also scrutinize the decomposition of the0νβ+/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,0gs+,01+, and0res+. In Figure 11 the same has been done in a global context by including also the ratios for the0νβ-β-emitters. In this figure one notices that the ratios have a rather universal value of roughly 2.0, except in the case of78Kr that has a ratio of about 6.0. The ratios for the01+transitions show pronounced peaks for the0νβ+/EC emitters78Kr and96Ru and for the0νβ-β-emitters110Pd and116Cd, whereas pronounced minima occur for106Cd and124Xe. Mostly these ratios for the01+are slightly above 2.0.Figure 10
Negative of the ratio between the Gamow-Teller and Fermi NMEs for decays to the ground state (0gs+), the first excited0+state (01+) and the resonant0+state (0res+) for all the nuclei discussed in this paper.Negative of the ratio between the Gamow-Teller and Fermi NMEs for0νβ-β-and0νβ+/EC decays for masses70≤A≤100(a) and for104≤A≤136(b). Shown are the ratios corresponding to the decay transitions to the ground state (0gs+) and the first excited0+state (01+).
(a)
(b)All in all, much more variation in the Gamow-Teller/Fermi ratio is seen for the01+states than for the ground states. From Figure 10 it is clear that the ratios for the resonant states are much higher than for the01+or0gs+states in the corresponding nuclei.


### 6.3. Decompositions of the0νβ+/EC NMEs

The0νβ+/EC NMEs can be decomposed into contributions of different intermediate multipoles as done in [59] for96Ru. Let us use here the decay of124Xe as an example. The decomposition of the0νβ/EC NMEsM(0ν)′(19) can be made in two ways, either through the different multipole statesJπof the intermediate nucleus (in this case the states of124I) or through different couplingsJ′of the two decaying nucleons [57, 73]. For the Gamow-Teller NME these decompositions can be schematically written as
(37)MGT(0ν)=∑Jπ‍∑J′‍MGT(0ν)(Jπ,J′),
whereMGT(0ν)(Jπ,J′)is given explicitly in (24). The decompositions (37) are shown for the Gamow-Teller NMEs of the decays of124Xe in Figures 12, 13, and 14. All the figures refer to calculations using the Jastrow short-range correlations and the valuegA=1.25for the axial-vector coupling constant.Decompositions of (37) inJπ(a) andJ′(b) for the ground-state Gamow-Teller NMEMGT(0ν)(0gs+)of124Xe. Results for the Jastrow short-range correlations are shown withgA=1.25.
(a)
(b)The same as in Figure12 for the two-phonon NMEMGT(0ν)(01+)corresponding to the01+state at1657keV in124Te.
(a)
(b)The same as in Figure12 for the resonance NMEMGT(0ν)(0res+)corresponding to the0+state at2790keV of nuclear excitation in124Te.
(a)
(b)From the decomposition figures one can make the following general observations. For the ground-state NME the decomposition in terms ofJπis the typical one of the pnQRPA calculations [45, 57, 59] and the decomposition in terms ofJ′is typical of the shell-model [73] and pnQRPA [36, 57, 59] calculations. Here typical for theJπdecomposition are the strong contributions of the high-multipole components2-,3+,4-, and5+. In this case the1+contribution is modest contrary to that of the96Ru decay [59]. This contribution depends strongly on the value of the strength parametergpp. For theJ′decomposition typical is the large positive monopole contribution and the much smaller, mostly negative, and higher-multipole contributions.For the lowest excited0+state,01+=02-ph+, the pattern is qualitatively different for theJπdecomposition since the other multipole components than2-are suppressed but sizable negative contributions from the higher multipole components appear. The relative contributions of the1±and2±states for the ground-state and01+NMEs are surprisingly similar. In the case of theJ′decomposition theJ′=0component is not the dominant one but, instead, theJ′=1component is. The higher-multipole components contribute sizably, with varying signs. The multipole decompositions of the0+resonant state,0res+, which is a one-phonon ccQRPA state, are rather blunt. They show both a strongJπ=1+component and a strong monopoleJ′=0component. The rest of the contributions play only a minor role.


### 6.4. Comparison of NMEs Produced by Different Models

In Table2 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 [31], and the projected Hartree-Fock-Bogoliubov (PHFB) results are taken from [32]. 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 constantgA.Table 2
Comparison of the ground-state NMEs for the QRPA, IBM-2 [31], and PHFB [32] models using the Jastrow short-range correlations. The results for 136Ce are taken from [30] with gA=1.261.
Nucleus
QRPA (gA=1.25)
IBM-2 (gA=1.269)
PHFB (gA=1.254)
M
GT
(
0
ν
)
M
F
(
0
ν
)
M
(
0
ν
)
′
M
GT
(
0
ν
)
M
F
(
0
ν
)
M
(
0
ν
)
′
M
(
0
ν
)
′
78Kr
3.271
-
0.331
3.482
3.384
-
2.146
4.478
—
96Ru
2.589
-
0.988
3.222
2.204
-
0.269
2.483
4.82
±
0.11
106Cd
4.920
-
1.586
5.935
2.757
-
0.255
3.106
7.97
±
0.72
124Xe
3.491
-
1.889
4.700
3.967
-
2.224
5.156
3.69
±
0.32
130Ba
5.412
-
2.528
7.031
3.911
-
2.108
5.043
2.75
±
0.82
136Ce
4.282
-
1.961
5.537
3.815
-
2.007
4.901
—From Table2 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 the0νβ-β-decaying nuclei as discussed extensively in [74].In Table3 the NMEs corresponding to the0νβ+/EC decays to the first excited0+state,01+, 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 for78Kr,106Cd, and124Xe and rather large NME for96Ru. For IBM-2 the opposite happens. This tension between the two calculations is more drastic than in the case of the0νβ-β-transitions, as analyzed in [74].Table 3
Comparison of the01+ NMEs for the QRPA and IBM-2 [31] models using the Jastrow short-range correlations.
Nucleus
QRPA (gA=1.25)
IBM-2 (gA=1.269)
M
GT
(
0
ν
)
M
F
(
0
ν
)
M
(
0
ν
)
′
M
GT
(
0
ν
)
M
F
(
0
ν
)
M
(
0
ν
)
′
78Kr
0.039
-
0.008
0.044
0.771
-
0.479
1.014
96Ru
2.004
-
0.396
2.258
0.036
-
0.012
0.045
106Cd
0.317
-
0.537
0.660
1.395
-
0.110
1.537
124Xe
-
0.005
-
0.050
0.028
0.647
-
0.359
0.839


### 6.5. Experimental Limits for the Half-Lives

Up to now only limits of half-lives have been extracted for the various0νβ+/EC processes. Measurements have been done, for example, for74Se [75], for96Ru [76, 77], for106Cd [78], for112Sn [79], for136Ce and138Ce [80], and for64Zn and180W [81]. The obtained lower limits are of the order of1020years for96Ru [77] and106Cd [78] and1015years for136Ce [80]. These limits are still very far from the theoretical estimates as implied by Table 1.


## 6.1. The0νβ+/EC NMEs

A systematics of the computed NMEs of the0νβ+/EC decays is shown in Figure 8. The values of NMEs for decays to the ground state (0gs+), first excited0+state (01+) and the resonant0+state (0res+) are shown. As mentioned before the assignment ofJπ=0+to the resonant states has to be taken in some cases, like for the106Cd decay, with a grain of salt. From the figure one notices that the ground-state NMEs are rather large (5.0or more), except for92Mo. This means that matrix-element-wise the0νβ+/EC decays are not suppressed relative to the0νβ-β-decays. This can be further evidenced in Figure 9 where these NMEs are shown together with those of0νβ-β-decays for nuclei with70≤A≤100(a) and for nuclei with104≤A≤136(b).Figure 8
Values of the computed nuclear matrix elements for0νβ+/EC decay transitions to the ground state (0gs+), the first excited0+state (01+) and the resonant0+state (0res+) for all the nuclei discussed in this paper.Values of the computed nuclear matrix elements for0νβ-β-and0νβ+/EC decays for masses70≤A≤100(a) and for104≤A≤136(b). Shown are the NMEs corresponding to the decay transitions to the ground state (0gs+) and the first excited0+state (01+).
(a)
(b)In Figure8 it is seen that the NMEs corresponding to the resonant0+states are larger than the NMEs corresponding to the decays to01+states. This stems from the fact that the resonant states are treated as one-ccQRPA-phonon states whereas the01+states are described as two-ccQRPA-phonon states. The MCM connects the two-phonon states weaker than the one-phonon states to theJπstates of the neighboring odd-odd nucleus due to theZZproducts and the 9jsymbol appearing in the associated transition density (35).From Figure9 it is seen that the0νβ+/EC NMEs show local maxima (96Ru,106Cd,124Xe,130Ba, and136Ce) for the ground-state transitions and even a global maximum:130Ba. On the other hand,92Mo shows a global minimum. For the decays to the01+states the0νβ+/EC NMEs are small relative to the0νβ-β-NMEs, except for96Ru which has a relatively large NME.


## 6.2. Fermi and Gamow-Teller Parts of the0νβ+/EC NMEs

One can also scrutinize the decomposition of the0νβ+/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,0gs+,01+, and0res+. In Figure 11 the same has been done in a global context by including also the ratios for the0νβ-β-emitters. In this figure one notices that the ratios have a rather universal value of roughly 2.0, except in the case of78Kr that has a ratio of about 6.0. The ratios for the01+transitions show pronounced peaks for the0νβ+/EC emitters78Kr and96Ru and for the0νβ-β-emitters110Pd and116Cd, whereas pronounced minima occur for106Cd and124Xe. Mostly these ratios for the01+are slightly above 2.0.Figure 10
Negative of the ratio between the Gamow-Teller and Fermi NMEs for decays to the ground state (0gs+), the first excited0+state (01+) and the resonant0+state (0res+) for all the nuclei discussed in this paper.Negative of the ratio between the Gamow-Teller and Fermi NMEs for0νβ-β-and0νβ+/EC decays for masses70≤A≤100(a) and for104≤A≤136(b). Shown are the ratios corresponding to the decay transitions to the ground state (0gs+) and the first excited0+state (01+).
(a)
(b)All in all, much more variation in the Gamow-Teller/Fermi ratio is seen for the01+states than for the ground states. From Figure 10 it is clear that the ratios for the resonant states are much higher than for the01+or0gs+states in the corresponding nuclei.


## 6.3. Decompositions of the0νβ+/EC NMEs

The0νβ+/EC NMEs can be decomposed into contributions of different intermediate multipoles as done in [59] for96Ru. Let us use here the decay of124Xe as an example. The decomposition of the0νβ/EC NMEsM(0ν)′(19) can be made in two ways, either through the different multipole statesJπof the intermediate nucleus (in this case the states of124I) or through different couplingsJ′of the two decaying nucleons [57, 73]. For the Gamow-Teller NME these decompositions can be schematically written as
(37)MGT(0ν)=∑Jπ‍∑J′‍MGT(0ν)(Jπ,J′),
whereMGT(0ν)(Jπ,J′)is given explicitly in (24). The decompositions (37) are shown for the Gamow-Teller NMEs of the decays of124Xe in Figures 12, 13, and 14. All the figures refer to calculations using the Jastrow short-range correlations and the valuegA=1.25for the axial-vector coupling constant.Decompositions of (37) inJπ(a) andJ′(b) for the ground-state Gamow-Teller NMEMGT(0ν)(0gs+)of124Xe. Results for the Jastrow short-range correlations are shown withgA=1.25.
(a)
(b)The same as in Figure12 for the two-phonon NMEMGT(0ν)(01+)corresponding to the01+state at1657keV in124Te.
(a)
(b)The same as in Figure12 for the resonance NMEMGT(0ν)(0res+)corresponding to the0+state at2790keV of nuclear excitation in124Te.
(a)
(b)From the decomposition figures one can make the following general observations. For the ground-state NME the decomposition in terms ofJπis the typical one of the pnQRPA calculations [45, 57, 59] and the decomposition in terms ofJ′is typical of the shell-model [73] and pnQRPA [36, 57, 59] calculations. Here typical for theJπdecomposition are the strong contributions of the high-multipole components2-,3+,4-, and5+. In this case the1+contribution is modest contrary to that of the96Ru decay [59]. This contribution depends strongly on the value of the strength parametergpp. For theJ′decomposition typical is the large positive monopole contribution and the much smaller, mostly negative, and higher-multipole contributions.For the lowest excited0+state,01+=02-ph+, the pattern is qualitatively different for theJπdecomposition since the other multipole components than2-are suppressed but sizable negative contributions from the higher multipole components appear. The relative contributions of the1±and2±states for the ground-state and01+NMEs are surprisingly similar. In the case of theJ′decomposition theJ′=0component is not the dominant one but, instead, theJ′=1component is. The higher-multipole components contribute sizably, with varying signs. The multipole decompositions of the0+resonant state,0res+, which is a one-phonon ccQRPA state, are rather blunt. They show both a strongJπ=1+component and a strong monopoleJ′=0component. The rest of the contributions play only a minor role.


## 6.4. Comparison of NMEs Produced by Different Models

In Table2 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 [31], and the projected Hartree-Fock-Bogoliubov (PHFB) results are taken from [32]. 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 constantgA.Table 2
Comparison of the ground-state NMEs for the QRPA, IBM-2 [31], and PHFB [32] models using the Jastrow short-range correlations. The results for 136Ce are taken from [30] with gA=1.261.
Nucleus
QRPA (gA=1.25)
IBM-2 (gA=1.269)
PHFB (gA=1.254)
M
GT
(
0
ν
)
M
F
(
0
ν
)
M
(
0
ν
)
′
M
GT
(
0
ν
)
M
F
(
0
ν
)
M
(
0
ν
)
′
M
(
0
ν
)
′
78Kr
3.271
-
0.331
3.482
3.384
-
2.146
4.478
—
96Ru
2.589
-
0.988
3.222
2.204
-
0.269
2.483
4.82
±
0.11
106Cd
4.920
-
1.586
5.935
2.757
-
0.255
3.106
7.97
±
0.72
124Xe
3.491
-
1.889
4.700
3.967
-
2.224
5.156
3.69
±
0.32
130Ba
5.412
-
2.528
7.031
3.911
-
2.108
5.043
2.75
±
0.82
136Ce
4.282
-
1.961
5.537
3.815
-
2.007
4.901
—From Table2 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 the0νβ-β-decaying nuclei as discussed extensively in [74].In Table3 the NMEs corresponding to the0νβ+/EC decays to the first excited0+state,01+, 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 for78Kr,106Cd, and124Xe and rather large NME for96Ru. For IBM-2 the opposite happens. This tension between the two calculations is more drastic than in the case of the0νβ-β-transitions, as analyzed in [74].Table 3
Comparison of the01+ NMEs for the QRPA and IBM-2 [31] models using the Jastrow short-range correlations.
Nucleus
QRPA (gA=1.25)
IBM-2 (gA=1.269)
M
GT
(
0
ν
)
M
F
(
0
ν
)
M
(
0
ν
)
′
M
GT
(
0
ν
)
M
F
(
0
ν
)
M
(
0
ν
)
′
78Kr
0.039
-
0.008
0.044
0.771
-
0.479
1.014
96Ru
2.004
-
0.396
2.258
0.036
-
0.012
0.045
106Cd
0.317
-
0.537
0.660
1.395
-
0.110
1.537
124Xe
-
0.005
-
0.050
0.028
0.647
-
0.359
0.839


## 6.5. Experimental Limits for the Half-Lives

Up to now only limits of half-lives have been extracted for the various0νβ+/EC processes. Measurements have been done, for example, for74Se [75], for96Ru [76, 77], for106Cd [78], for112Sn [79], for136Ce and138Ce [80], and for64Zn and180W [81]. The obtained lower limits are of the order of1020years for96Ru [77] and106Cd [78] and1015years for136Ce [80]. These limits are still very far from the theoretical estimates as implied by Table 1.


## 7. Present Status of the Resonant Processes

Table4 lists the known cases of R0νECEC transitions in various nuclei whereQ-value measurements have been conducted recently. TheseQvalues have been measured by using the Penning-trap techniques. In the cases of96Ru,106Cd,124Xe, and130Ba the assignment of 0+ spin-parity to the resonant state is uncertain. In these cases further experimental spectroscopy is needed.Table 4
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-life estimates with the references to the Q-value measurement and calculations indicated in the last column.
Transition
J
f
π
Q
-
E [keV]
Orbitals
C
ECEC
References
74
Se
→
74
Ge
2
+
2.23
L2L3
(0.2−100) ×1043
[33]
96
Ru
→
96
Mo
2
+
8.92 (13)
L1L3
[34]
0
+
?
−3.90 (13)
L1L1
102
Pd
→
102
Ru
2
+
75.26 (36)
KL3
[35]
106
Cd
→
106
Pd
0
+
?
8.39
KK
(2.1−5.7) ×1030
[36]
(
2
,
3
)
-
−0.33 (41)
KL3
[35]
112
Sn
→
112
Cd
0
+
−4.5
KK
>5.9× 1029
[37]
124
Xe
→
124
Te
0
+
?
1.86 (15)
KK
(1.7−5.1) ×1029
[38]
130
Ba
→
130
Xe
0
+
?
10.18 (30)
KK
[38]
136
Ce
→
136
Ba
0
+
−11.67
KK
(3−23) ×1032
[39]
144
Sm
→
144
Nd
2
+
171.89 (87)
KL3
[35]
152
Gd
→
152
Sm
0
gs
+
0.91 (18)
KL1
(1.0−1.5) ×1027
[40, 41]
156
Dy
→
156
Gd
1
-
0.75 (10)
KL1
[42]
0
+
0.54 (24)
L1L1
[42]
2
+
0.04 (10)
M1N3
[42]
162
Er
→
162
Dy
2
+
2.69 (30)
KL3
[34]
164
Er
→
164
Dy
0
gs
+
6.81 (13)
L1L1
(3.2−5.2) ×1031
[41, 43]
168
Yb
→
168
Er
(2-)
1.52 (25)
M1M3
[34]
180
W
→
180
Hf
0
gs
+
11.24 (27)
KK
(4.0−9.5) ×1029
[41, 44]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 theQ-value measurement and the possible calculations of the related NME. In the table an auxiliary quantityCECECis listed and its relation with the R0νECEC half-life stands as
(38)T1/2R0νECEC=CECEC(meff[eV])2years,
where the effective neutrino mass should be given in units of eV. In all the listed cases whereCECEChas been computed the decay rates are suppressed by the rather sizable magnitude of the degeneracy parameter. Decays to0+states are favored over the decays to2+or1-,2-,3-, and so forth states due to the involved nuclear wave functions and/or higher-order transitions. Also captures from atomic orbitals with orbital angular momentuml>0are suppressed [33].There are some favorable values of degeneracy parameters listed in Table4, like106Cd→106Pd(2,3)-and156Dy→156Gd(0+,1-,2+)but the associated nuclear matrix elements are still waiting for their evaluation. Strong suppression of the NMEs related to final states withJ>0is, however, expected. In case of the156Dy decay the deformation also plays an important role. At the moment the most favorable case with a half-life estimate is the case152Gd→152Sm(0gs+)which 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.



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*Source: 505874-2013-12-01.xml*