Neutrinoless Double β⁺/EC Decays

Abstract

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

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 fields𝜈_𝑒,𝜈_𝜇, and𝜈_𝜏, 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 ℒ_𝐶𝐶𝑔=−2√2_ℓ𝜈_ℓ𝛾_𝜇1−𝛾₅ℓ𝑊^𝜇𝑔+h.c.=−√2_ℓ𝜈_ℓ𝐿𝛾_𝜇ℓ_𝐿𝑊^𝜇+h.c., (1) whereℓrefers to the three lepton flavors,ℓ=𝑒,𝜇,𝜏,𝑊^𝜇is a vector field corresponding to the charged weak boson𝑊^±,𝜈_ℓ𝐿andℓ_𝐿are 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 fields𝜈_ℓbut by fields𝜈_𝑖(𝑖=1,2,3) that have definite masses𝑚_𝑖, 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: 𝜈_ℓ𝐿=_{ℓ=𝑒,𝜇,𝜏}𝑈_ℓ𝑖𝜈_𝑖𝐿, (2) 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 −12𝑀^𝐿_ℓℓ′(𝜈_ℓ𝐿)^𝐶𝜈_ℓ′𝐿+h.c., (3) 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:−(𝑌_ℓℓ′/Λ)(𝜈_ℓ𝐿)^𝐶(𝐻⁰)²𝜈_ℓ′𝐿+h.c., where𝑌_ℓℓ′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: −𝑣²𝑌_ℓℓ′4Λ(𝜈_ℓ𝐿)^𝐶𝜈_ℓ′𝐿+h.c., (4) that is,𝑀^𝐿_ℓℓ′=𝑣²𝑌_ℓℓ′/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 fields𝜈_ℓ𝐿, right-handed neutrino fields𝜈_ℓ𝑅, the neutrino mass Lagrangian may contain also the Dirac mass terms−𝑀^𝐷_ℓℓ′𝜈_ℓ𝑅𝜈^′_ℓ+h.c.and another type of Majorana mass terms−(1/2)𝑀^𝑅_ℓℓ′(𝜈_ℓ𝑅)^𝐶𝜈_ℓ′𝑅+h.c. Unless the Majorana mass terms vanish, the fields𝜈_𝑖that diagonalize the full mass Lagrangian are two-component Majorana fields obeying the condition (“Majorana condition”) 𝜈_𝑖=𝜈^𝐶_𝑖. (5) There are in this case altogether six mass states. It is generally assumed that𝑀^𝑅_ℓℓ′≫𝑀^𝐷_ℓℓ′≫𝑀^𝐿_ℓℓ′(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]: 𝑐𝑈=¹²𝑐₁₃𝑠₁₂𝑐₁₃𝑠₁₃𝑒^−𝑖𝛿−𝑠₁₂𝑐₂₃−𝑐₁₂𝑠₂₃𝑠₁₃𝑒^𝑖𝛿𝑐₁₂𝑐₂₃−𝑠₁₂𝑠₂₃𝑠₁₃𝑒^𝑖𝛿𝑠₂₃𝑐₁₃𝑠₁₂𝑠₂₃−𝑐₁₂𝑐₂₃𝑠₁₃𝑒^𝑖𝛿−𝑐₁₂𝑠₂₃−𝑠₁₂𝑐₂₃𝑠₁₃𝑒^𝑖𝛿𝑐₂₃𝑐₁₃𝑃, (6)where𝑃=diag(1,𝑒^𝑖𝛼,𝑒^𝑖𝛽),𝑠_𝑖𝑗=sin𝜃_𝑖𝑗,𝑐_𝑖𝑗=cos𝜃_𝑖𝑗, and 𝛿 is called the Dirac phase and 𝛼 and 𝛽 the Majorana phases. The probability for the oscillatory transition from the neutrino flavor𝜈_𝛼to the flavor𝜈_𝛽as a function of the distance of flight𝐿and neutrino energy𝐸is given by (see, e.g., [27]) 𝑃𝜈_𝛼⟶𝜈_𝛽=𝛿_𝛼𝛽−4_𝑖>𝑗Re𝑈^∗_𝛼𝑖𝑈^∗_𝛽𝑗𝑈_𝛽𝑖𝑈_𝛼𝑗sin²Δ𝑚²_𝑖𝑗𝐿4𝐸+2_𝑖>𝑗Im𝑈^∗_𝛼𝑖𝑈^∗_𝛽𝑗𝑈_𝛽𝑖𝑈_𝛼𝑗sinΔ𝑚²_𝑖𝑗𝐿,2𝐸 (7) whereΔ𝑚²_𝑖𝑗=𝑚²_𝑖−𝑚²_𝑗. 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 differencesΔ𝑚²_𝑖𝑗. 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]: Δ𝑚²₂₁=7.54^+0.26_−0.22×10⁻⁵eV²,Δ𝑚²₃₁≃Δ𝑚²₃₂=2.43^+0.06_−0.10×10⁻³eV²,sin²𝜃₁₂=0.307^+0.18_−0.16,sin²𝜃₂₃=0.386^+0.24_−0.21,sin²𝜃₁₃=0.0241^+0.0025_−0.0025. (8) 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 mass𝑚_𝛽=∑_𝑖|𝑈_𝑒𝑖|²𝑚²_𝑖in beta decay experiments, as well as from the cosmological precision measurements of the sum of neutrino masses∑_𝑖𝑚_𝑖. The current experimental upper limits for𝑚_𝛽are 2.3 eV [28] and 2.1 eV [29], and for the sum of neutrino masses∑_𝑖𝑚_𝑖the 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 𝐺_F√2²𝑒⁺𝛾_𝜇′1+𝛾₅𝜈^𝑐_𝑒𝜈_𝑒𝛾_𝜇1−𝛾₅𝑒⁻, (9) where𝑒⁻,𝑒⁺,𝜈_𝑒, and𝜈^𝑐_𝑒are the field operators of the electron, positron, electron neutrino, and electron antineutrino, respectively. The strength of the interaction is governed by the Fermi coupling constant𝐺_𝐹/√2=𝑔²/(8𝑚²_𝑊), where𝑔is the fundamental gauge coupling of the electroweak theory and𝑚_𝑊is the mass of𝑊^±. The propagator describing the internal neutrino is given by ^{0𝜈𝑐𝑒(𝑥)𝜈𝑒𝑦0=𝑖𝑈∗𝑒𝑖20𝜈𝑖(𝑥)𝜈𝑖𝑦0}_{=−𝑖𝑖𝑈∗𝑒𝑖2}^𝑑4𝑞_()2𝜋/₄^𝑞+𝑚_{𝑖𝑞2−𝑚2𝑖,exp−𝑖𝑞⋅𝑥−𝑦} (10) 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 _{𝑖𝐺2F𝑈∗2𝑒𝑖𝛾𝜇𝛾𝐿}^𝑞/^+𝑚_{𝑖𝑞2−𝑚2𝑖𝛾𝜇′𝛾𝐿=𝑖𝐺2F𝑈∗2𝑒𝑖}^𝑚_{𝑖𝑞2−𝑚2𝑖𝛾𝜇𝛾𝑅𝛾𝜇′,} (11) where𝑞is the momentum of the exchanged neutrino and𝛾_𝐿(𝑅)are the chirality projection matrices 𝛾_𝐿(𝑅)=(1(−/(+))𝛾₅)/2. Note that the/^𝑞part 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 𝐺²_F𝑖𝑈^∗2_𝑒𝑖𝑚_𝑖1𝑞²𝛾_𝜇𝛾_𝑅𝛾_𝜇′. (12)

The essential part of the amplitude from neutrino-physics point of view is the quantity: ⟨𝑚_𝜈⟩≡_𝑖𝑈^∗2_𝑒𝑖𝑚_𝑖, (13) whose absolute value is called the effective neutrino mass; that is, 𝑚_eff=𝑚_𝜈. (14) 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 neutrino𝑀^𝐿_𝑒𝑒(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|⟨𝑚_𝜈⟩|².

(a)

(b)

Figure 2 

Feynman diagrams for the Majorana-neutrino-mediated𝛽+EC (a) and ECEC (b) decays.

Using the standard parametrization (6) of the mixing matrix𝑈, one can cast⟨𝑚_𝜈⟩in the following form: ⟨𝑚_𝜈⟩=𝑐²₁₂𝑐²₁₃𝑚₁+𝑠²₁₂𝑐²₁₃𝑒^{−𝑖𝜑12}𝑚₂+𝑠²₁₃𝑒^{−𝑖𝜑13}𝑚₃, (15) 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⟨𝑚_𝜈⟩. 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⟨𝑚_𝜈⟩. As the decay widths are proportional to|⟨𝑚_𝜈⟩|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⟨𝑚_𝜈⟩. One can understand this also as a consequence of the fact that in the limit𝑞²≫𝑚²_𝑖the amplitudes depend on just one parameter of the mass Lagrangian, the element𝑀^𝐿_𝑒𝑒, allowing for no measurable phases. To be sensitive to the Majorana CP phases, one should be able to distinguish between the mass states𝜈_𝑖.

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 𝑚₁=𝑚₀,𝑚₂=𝑚²₀+Δ𝑚²₂₁,𝑚₃=𝑚²₀+Δ𝑚²₃₁, (16) and in the case of inverted hierarchy 𝑚₁=𝑚²₀+Δ𝑚²₃₁,𝑚₂=𝑚²₀+Δ𝑚²₂₁+Δ𝑚²₃₁,𝑚₃=𝑚₀. (17) The squared mass differenceΔ𝑚²₂₁and the absolute value of the mass differenceΔ𝑚²₃₁are 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,|⟨𝑚_𝜈⟩|has lower limit of the order of10⁻²eV, as can be inferred from Figure 3, where the effective mass,𝑚_eff=|⟨𝑚_𝜈⟩|, 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|⟨𝑚_𝜈⟩|<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 the 𝜷⁺/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 𝑇^𝛼_0𝜈0⁺⁻¹=𝐺^𝛼_0𝜈0⁺𝑀^′(0𝜈)2𝑚_eff[eV]²,𝛼=𝛽⁺𝛽⁺,𝛽⁺EC, (18) where𝑚_effis 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 integrals𝐺^𝛽+𝛽+_0𝜈(0⁺)and𝐺^𝛽+EC_0𝜈(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 𝑀^′(0𝜈)=𝑔_𝐴1.25²𝑀^(0𝜈)_GT−𝑔_𝑉𝑔_𝐴²𝑀^(0𝜈)_F+𝑀^(0𝜈)_T, (19) where𝑔_𝐴=1.25corresponds to the bare-nucleon value of the axial-vector coupling constant and𝑔_𝑉=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 for𝑔_𝐴=1.25independent of the value of𝑔_𝐴used 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 𝑒⁻+𝑒⁻+(𝐴,𝑍)⟶(𝐴,𝑍−2)^∗⟶(𝐴,𝑍−2)+𝛾+2𝑋, (20) 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 𝑇^ECEC_0𝜈0⁺⁻¹=𝑔^ECEC_0𝜈0⁺𝑀^′(0𝜈)2𝑚²_effΓ(𝑄−𝐸)²+Γ²,/4 (21) where the daughter state(𝐴,𝑍−2)^∗is a virtual state with energy 𝐸=𝐸^∗+𝐸_𝐻+𝐸_𝐻′+𝐸_𝐻𝐻′, (22) 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 factor𝑔^ECEC_0𝜈can 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 𝑔^ECEC_0𝜈0⁺=𝐺_Fcos𝜃_𝐶√2⁴𝑔⁴_𝐴4𝜋²ln2𝑅²_𝐴𝑚⁶_𝑒𝒩_𝑛,𝜅𝒩_{𝑛′,𝜅′}, (23) where𝒩_𝑛,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 𝑀_𝐾^(0𝜈)=_{𝐽𝜋𝐽′𝑘1𝑘2𝑝𝑝′𝑛𝑛′}(−1)^{𝑗𝑝+𝑗𝑛′+𝐽+𝐽′}×2𝐽^′𝑗+1^𝑛𝑗_𝑝𝐽𝑗_𝑝′𝑗_𝑛′𝐽^′×𝑚_𝐾𝑛𝑛^′,𝑝𝑝^′;𝐽^′;𝑘₁,𝑘₂×0⁺_𝑓𝑐^†_𝑛′𝑐̃_𝑝′𝐽𝐽^𝜋_𝑘1𝐽^𝜋_𝑘1∣𝐽^𝜋_𝑘2𝐽^𝜋_𝑘2𝑐^†_𝑛𝑐̃_𝑝𝐽0⁺_𝑖, (24) where𝐾=F,GT and 𝑘₁ and 𝑘₂label the different nuclear-model solutions for a given multipole𝐽^𝜋, 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(0⁺_𝑓‖[𝑐^†_𝑛′𝑐̃_𝑝′]_𝐽‖𝐽^𝜋_𝑘1)and(𝐽^𝜋_𝑘2‖[𝑐^†_𝑛𝑐_𝑝̃]_𝐽‖0⁺_𝑖), 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 𝑚_𝐾𝑛𝑛^′,𝑝𝑝^′;𝐽^′;𝑘₁,𝑘₂=𝐽^′𝑗_𝑝𝑗_𝑝′𝑗_𝑛𝑗_{𝑛′𝜆𝑆}(2𝜆+1)(2𝑆+1)𝐹_𝐾×𝑙_𝑝𝑙_𝑝′𝜆1212𝑆𝑗_𝑝𝑗_𝑝′𝐽^′𝑙_𝑛𝑙_𝑛′𝜆1212𝑆𝑗_𝑛𝑗_𝑛′𝐽^′×_{𝑛1𝑛2𝑙𝑁𝐿}𝑀_𝜆𝑛₁𝑙𝑁𝐿;𝑛_𝑛𝑙_𝑛𝑛_𝑛′𝑙_𝑛′×𝑀_𝜆𝑛₂𝑙𝑁𝐿;𝑛_𝑝𝑙_𝑝𝑛_𝑝′𝑙_𝑝′×𝑑³𝑟𝜙_𝑛1𝑙(𝐫)ℎ_𝐾𝑟₁₂,12𝐸_𝑘1+𝐸_𝑘2𝜙_𝑛2𝑙(𝐫), (25) where𝑗̂=2𝑗+1and𝑟₁₂=|𝐫₁−𝐫₂|is the relative distance between the two decaying protons. The following auxiliary quantities have been defined 𝐹_F=1,𝐹_GT=6(−1)^𝑆+11212𝑆12121. (26) The quantities𝑀_𝜆are 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 functions𝜙_𝑛𝑙(𝐫)are taken to be the eigenfunctions of the isotropic harmonic oscillator.

The neutrino potentialℎ_𝐾(𝑟₁₂,𝐸),𝐾=F,GT, in the integral of (26) is defined as ℎ_𝐾𝑟₁₂=2,𝐸𝜋𝑅_𝐴𝑑𝑞𝑞ℎ_𝐾𝑞²𝑞+𝐸−𝐸_𝑖+𝐸_𝑓𝑗/2⁰𝑞𝑟₁₂, (27) where𝑗₀is the spherical Bessel function and the integration is performed over the exchanged momentum𝑞. Here𝐸_𝑖=𝑀_𝑖𝑐²is the ground-state mass energy of the initial nucleus and𝐸_𝑓the (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 termℎ_𝐾(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. The𝐽^𝜋states of the intermediate nucleus of the𝛽𝛽decays are generated by the usual proton-neutron QRPA (pnQRPA) [2, 62] in the form 𝐽^𝜋_𝑘𝑀=𝑄^†𝐽^𝜋_𝑘⟩=,𝑀|QRPA_𝑝𝑛𝑋^𝐽𝜋𝑘_𝑝𝑛𝑎^†_𝑝𝑎^†_𝑛𝐽𝑀−𝑌^𝐽𝜋𝑘_𝑝𝑛𝑎^†_𝑝𝑎^†_𝑛^†_𝐽𝑀|QRPA⟩, (28) where𝑋and𝑌are the forward- and backward-going amplitudes of the pnQRPA, obtained by solving the pnQRPA equations of motion [62]. The excited states𝐼^𝜋_𝑘in the final even-even nuclei are described by the phonons of the charge-conserving QRPA (ccQRPA), expressed as 𝐼^𝜋_𝑘𝑀=𝑄^†𝐼^𝜋_𝑘⟩=,𝑀|QRPA_𝑎𝑏𝑍^𝐼𝜋𝑘_𝑎𝑏𝑎^†_𝑎𝑎^†_𝑏𝐼𝑀−𝑊^𝐼𝜋𝑘_𝑎𝑏𝑎^†_𝑎𝑎^†_𝑏^†_𝐼𝑀|QRPA⟩, (29) where the symmetrized amplitudes𝑍and𝑊are obtained from the usual ccQRPA amplitudes𝑋and𝑌[62] through the transformation 𝑍^𝐼𝜋𝑘_𝑎𝑏=𝑋^𝐼𝜋𝑘_𝑎𝑏1,if𝑎=𝑏2𝑋^𝐼𝜋𝑘_𝑎𝑏1,if𝑎<𝑏2𝑋^𝐼𝜋𝑘_𝑏𝑎,if𝑎>𝑏 (30) and similarly for𝑊in terms of𝑌.

Now one can take a𝐼^𝜋_𝑘=2⁺₁phonon of (29) and build an ideal two-phonon𝐼⁺state of the form 𝐼⁺_2−ph=1√2𝑄^†(2⁺₁)𝑄^†(2⁺₁)_𝐼|QRPA⟩. (31) An ideal two-phonon state consists of partner states𝐼^𝜋=0⁺,2⁺,4⁺that are degenerate in energy and exactly at an energy twice the excitation energy of the2⁺₁state. 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 the𝐽^𝜋states 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 parameter𝑔_phand particle-particle parameter𝑔_pp. 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 the𝑔_ppparameter 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 interval𝑔_𝐴=1.00--1.25of the axial-vector coupling constant [48, 54, 55, 57]. The experimental error and the uncertainty in the value of𝑔_𝐴then induce an interval of physically acceptable values of𝑔_pp, the minimum value of𝑔_pprelated to𝑔_𝐴=1.00and the maximum value to𝑔_𝐴=1.25. This is because the magnitude of the calculated2𝜈𝛽⁻𝛽⁻NME,𝑀^(2𝜈), decreases with increasing value of𝑔_ppin a pnQRPA calculation [67, 69, 70] and this magnitude is compared with the magnitude of the experimental NME,𝑀^(2𝜈)(exp)∝(𝑔_𝐴)⁻², deduced from the experimental2𝜈𝛽⁻𝛽⁻half-life. The same correspondence between𝑔_ppand𝑔_𝐴is adopted also here for the0𝜈𝛽⁺/EC decays. In the absence of available half-life data on the0𝜈𝛽⁺/EC side the ranges of the adopted𝑔_ppvalues are reasonable choices such that all the physically meaningful values of the0𝜈𝛽⁺/EC NMEs are covered.

For the ccQRPA the default value𝑔_ph=1.0was adopted and the𝑔_ppparameter 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, 𝐽^𝜋_𝑘2𝑐^†_𝑛𝑐̃_𝑝𝐽0⁺_𝑖=𝐽(̂−1)^{𝑗𝑝+𝑗𝑛+𝐽+1}𝑣_𝑝𝑢_𝑛𝑋^{𝐽𝜋𝑘2}_𝑝𝑛+𝑢_𝑝𝑣_𝑛𝑌^{𝐽𝜋𝑘2}_𝑝𝑛. (32) The transition density corresponding to the final ground state reads 0⁺_gs𝑐^†_𝑛′𝑐̃_𝑝′𝐽𝐽^𝜋_𝑘1=𝐽(̂−1)^{𝑗𝑝′+𝑗𝑛′+𝐽+1}𝑢_𝑝′𝑣_𝑛′𝑋^{𝐽𝜋𝑘1}_{𝑝′𝑛′}+𝑣_𝑝′𝑢_𝑛′𝑌^{𝐽𝜋𝑘1}_{𝑝′𝑛′}, (33) where𝑣(𝑣) and𝑢(𝑢) correspond to the BCS occupation and unoccupation amplitudes of the initial (final) even-even nucleus. The amplitudes𝑋and𝑌(𝑋and𝑌) 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 𝐼⁺_𝑘𝑐^†_𝑛′𝑐̃_𝑝′𝐿𝐽^𝜋_𝑘1=2𝐼𝐿̂𝐽(̂−1)^{𝐿+𝐼+𝐽}×_𝑝1𝑣_𝑝′𝑣_𝑛′𝑋^{𝐽𝜋𝑘1}_𝑝1𝑛′𝑍^𝐼+𝑘_𝑝1′𝑝−𝑢_𝑝′𝑢_𝑛′𝑌^{𝐽𝜋𝑘1}_𝑝1𝑛′𝑊^𝐼+𝑘_𝑝1′𝑝×𝑗𝐽𝐼𝐿_𝑝′𝑗_𝑛′𝑗_𝑝1+_𝑛1(−1)^{𝐼+𝑗𝑛′+𝑗𝑛1}×𝑢_𝑝′𝑢_𝑛′𝑋^{𝐽𝜋𝑘1}_𝑝1′𝑛𝑍^𝐼+𝑘_𝑛1′𝑛−𝑣_𝑝′𝑣_𝑛′𝑌^{𝐽𝜋𝑘1}_𝑝1′𝑛𝑊^𝐼+𝑘_𝑛1′𝑛×𝑗𝐽𝐼𝐿_𝑛′𝑗_𝑝′𝑗_𝑛1 (34) 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(𝑋and𝑌) come from the pnQRPA calculation starting from the initial (final) nucleus of the𝛽𝛽decay. The amplitudes𝑍and𝑊are 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 𝐼⁺_2−ph𝑐^†_𝑛′𝑐̃_𝑝′𝐿𝐽^𝜋_𝑘1=40√2𝐼𝐿̂𝐽(̂−1)^{𝐼+𝐽+𝑗𝑝′+𝑗𝑛′}×_𝑝1𝑛1𝑣_𝑝′𝑢_𝑛′𝑋^{𝐽𝜋𝑘1}_𝑝1𝑛1𝑍²⁺¹_𝑝′𝑝1𝑍²⁺¹_𝑛′𝑛1+𝑢_𝑝′𝑣_𝑛′𝑌^{𝐽𝜋𝑘1}_𝑝1𝑛1𝑊²⁺¹_𝑝′𝑝1𝑊²⁺¹_𝑛′𝑛1×𝑗_𝑝′𝑗_𝑝12𝑗_𝑛′𝑗_𝑛12,𝐿𝐽𝐼 (35) where, as usual, the barred quantities denote amplitudes obtained for the final nucleus of double beta decay. In the present work we use only𝐼⁺_2−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,0⁺_gs, and to the first0⁺state,0⁺₁. 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 Figure 4, 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 the0⁺₁state 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 intermediate𝐽^𝜋states 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 intermediate𝐽^𝜋states 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 range𝑔_𝐴=1.00--1.25of the axial-vector coupling constant and for the value𝑚_eff=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 Figure 5, 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 Figure 6 the half-lives of decay transitions in96Ru are shown for the range𝑔_𝐴=1.00−−1.25and for the value𝑚_eff=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 ranges𝑔_A= 1.00–1.25 and𝑚_eff=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 quantities𝐶^𝛽+𝛽+and𝐶^𝛽+ECin the following form: 𝑇^𝛽+𝛽+_1/2=𝐶^𝛽+𝛽+𝑚_eff[eV]⁻²,𝑇^𝛽+EC_1/2=𝐶^𝛽+EC𝑚_eff[eV]⁻², (36) 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 range𝑔_𝐴=1.00--1.25of the axial-vector coupling constant.

Table 1 

Auxiliary factors of (36) computed for the nuclei under discussion by using 𝑔_𝐴 = 1.00−1.25 and the UCOM short-range correlations. The results for 92Mo and 136Ce are taken from [30] with 𝑔_𝐴=1.261.

From Table 1 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. The 0𝜈𝛽⁺/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 (0⁺_gs), first excited0⁺state (0⁺₁) and the resonant0⁺state (0⁺_res) are shown. As mentioned before the assignment of𝐽^𝜋=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 (0+gs), the first excited0+state (0+1) and the resonant0+state (0+res) for all the nuclei discussed in this paper.

(a)

(b)

Figure 9 

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 (0+gs) and the first excited0+state (0+1).

In Figure 8 it is seen that the NMEs corresponding to the resonant0⁺states are larger than the NMEs corresponding to the decays to0⁺₁states. This stems from the fact that the resonant states are treated as one-ccQRPA-phonon states whereas the0⁺₁states are described as two-ccQRPA-phonon states. The MCM connects the two-phonon states weaker than the one-phonon states to the𝐽^𝜋states of the neighboring odd-odd nucleus due to the𝑍𝑍products and the 9𝑗symbol appearing in the associated transition density (35).

From Figure 9 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 the0⁺₁states 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 the 0𝜈𝛽⁺/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,0⁺_gs,0⁺₁, and0⁺_res. 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 the0⁺₁transitions show pronounced peaks for the0𝜈𝛽⁺/EC emitters78Kr and96Ru and for the0𝜈𝛽⁻𝛽⁻emitters110Pd and116Cd, whereas pronounced minima occur for106Cd and124Xe. Mostly these ratios for the0⁺₁are slightly above 2.0.

Figure 10 

Negative of the ratio between the Gamow-Teller and Fermi NMEs for decays to the ground state (0+gs), the first excited0+state (0+1) and the resonant0+state (0+res) for all the nuclei discussed in this paper.

(a)

(b)

Figure 11 

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 (0+gs) and the first excited0+state (0+1).

All in all, much more variation in the Gamow-Teller/Fermi ratio is seen for the0⁺₁states than for the ground states. From Figure 10 it is clear that the ratios for the resonant states are much higher than for the0⁺₁or0⁺_gsstates in the corresponding nuclei.

6.3. Decompositions of the 0𝜈𝛽⁺/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 states𝐽^𝜋of 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 𝑀^(0𝜈)_GT=_{𝐽𝜋𝐽′}𝑀^(0𝜈)_GT𝐽^𝜋,𝐽^′, (37) where𝑀^(0𝜈)_GT(𝐽^𝜋,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 value𝑔_𝐴=1.25for the axial-vector coupling constant.

(a)

(b)

Figure 12 

Decompositions of (37) in𝐽𝜋(a) and𝐽′(b) for the ground-state Gamow-Teller NME𝑀(0𝜈)GT(0+gs)of124Xe. Results for the Jastrow short-range correlations are shown with𝑔𝐴=1.25.

(a)

(b)

Figure 13 

The same as in Figure 12 for the two-phonon NME𝑀(0𝜈)GT(0+1)corresponding to the0+1state at1657keV in124Te.

(a)

(b)

Figure 14 

The same as in Figure 12 for the resonance NME𝑀(0𝜈)GT(0+res)corresponding to the0+state at2790keV of nuclear excitation in124Te.

From the decomposition figures one can make the following general observations. For the ground-state NME the decomposition in terms of𝐽^𝜋is 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 the𝐽^𝜋decomposition 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 parameter𝑔_pp. 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,0⁺₁=0⁺_2−ph, the pattern is qualitatively different for the𝐽^𝜋decomposition 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 and0⁺₁NMEs 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,0⁺_res, which is a one-phonon ccQRPA state, are rather blunt. They show both a strong𝐽^𝜋=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 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 [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 constant𝑔_𝐴.

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 𝑔_𝐴=1.261.

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 the0𝜈𝛽⁻𝛽⁻decaying nuclei as discussed extensively in [74].

In Table 3 the NMEs corresponding to the0𝜈𝛽⁺/EC decays to the first excited0⁺state,0⁺₁, 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 the 0⁺₁ NMEs for the QRPA and IBM-2 [31] models using the Jastrow short-range correlations.

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

Table 4 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.

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 quantity𝐶^ECECis listed and its relation with the R0𝜈ECEC half-life stands as 𝑇^R0𝜈ECEC_1/2=𝐶^ECEC𝑚_eff[eV]²years, (38) where the effective neutrino mass should be given in units of eV. In all the listed cases where𝐶^ECEChas 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 Table 4, 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(0⁺_gs)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.

Acknowledgment

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).