Neutrinoless double beta-plus/EC decays

This is an electronic reprint of the original article. This reprint may differ from the original in pagination and typographic detail. Author(s): Title: Year: Version: Please cite the original version: All material supplied via JYX is protected by copyright and other intellectual property rights, and duplication or sale of all or part of any of the repository collections is not permitted, except that material may be duplicated by you for your research use or educational purposes in electronic or print form. You must obtain permission for any other use. Electronic or print copies may not be offered, whether for sale or otherwise to anyone who is not an authorised user. Neutrinoless double beta-plus/EC decays Maalampi, Jukka; Suhonen, Jouni Maalampi, J., & Suhonen, J. (2013). Neutrinoless double beta-plus/EC decays. Advances in High Energy Physics, 2013, Article 505874. https://doi.org/10.1155/2013/505874 2013 Hindawi Publishing Corporation Advances in High Energy Physics Volume 2013, Article ID 505874, 18 pages http://dx.doi.org/10.1155/2013/505874 Review Article Neutrinoless Double 𝛽+/EC Decays Jukka Maalampi and Jouni Suhonen Department of Physics, University of Jyv¨ askyl¨ a, P.O. Box 35 (YFL), Jyv¨ askyl¨ a40014,Finland Correspondence should be addressed to Jouni Suhonen; [email protected] Received 28 June 2013; Accepted 19 September 2013 Academic Editor: Srubabati Goswami Copyright © 2013 J. Maalampi and J. Suhonen. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. 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 (0]2𝛽) decays. The key issue here is how to cope with the involved nuclear-structure issues of the decays, crystallized in the form of the nuclear matrix elements (NMEs) [1– 3]. To be able to exploit the potential data extracted from the 0]2𝛽-decay experiments one needs to evaluate the NMEs in a reliable enough way. It has become customary to employ the neutrino-emitting correspondent of 0]2𝛽decay, the two-neutrino double beta (2]2𝛽)decay,toconfinethe nuclear-model degrees of freedom in the NME calculations. The 2]2𝛽decay is a second-order process in the standard modeloftheelectroweakinteractionsandtheassociatedhalfliveshavebeenmeasuredforseveralnuclei[4]. The neutrinoless double 𝛽−(0]𝛽−𝛽−) decays have been studied intensively over the years [2,3]duetotheir favorable decay 𝑄values. 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 as 0]𝛽+/EC decays. The general, nuclear model independent frameworks of theory for these decays have been investigated in [7]for the 0]𝛽+/EC-decay channels 𝛽+𝛽+and 𝛽+EC. The formalism for the resonant neutrinoless double electron capture (R0]ECEC) was first developed in [8]andlaterdiscussedand extended to its radiative variant (0]𝛾ECEC) in [9]. Due to the resonant nature of the R0]ECEC decay its studies have called for precise measurements of the mass differences of the atoms involved in the decays. The resonant mode of 0]ECEC decays is studied intensively for its potential enhanced sensitivity to discover the Majorana mass of the neutrino and that is why much experimental effort is being invested in observing this mode of decay. 2. Neutrino Masses and Oscillations In the calculations of transition rates of the 0]𝛽+/EC decays, the neutrino-physics part and nuclear-physics part factorize. We will start by considering the neutrino-physics part. The weak-interaction Lagrangian of leptons is diagonal in the neutrino fields ]𝑒,]𝜇,and]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 L𝐶𝐶 =−𝑔 2√2∑ ℓ ]ℓ𝛾𝜇(1−𝛾5)ℓ𝑊𝜇+h.c. =−𝑔 √2∑ ℓ ]ℓ𝐿𝛾𝜇ℓ𝐿𝑊𝜇+h.c., (1) 2Advances in High Energy Physics where ℓrefers to the three lepton flavors, ℓ=𝑒,𝜇,𝜏,𝑊𝜇is a vector field corresponding to the charged weak boson 𝑊±, ]ℓ𝐿 and ℓ𝐿are the left-handed chiral components of the neutrino and charged lepton fields, and 𝑔is 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 neutrinoflavor-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 fields ]ℓbut by fields ]𝑖(𝑖= 1,2,3) that have definite masses 𝑚𝑖, known as the mass eigenfields. The left-handed flavor eigenfields appearing in the interaction Lagrangian (1)aresuperpositionsofthelefthanded components of the mass eigenfields: ]ℓ𝐿=∑ ℓ=𝑒,𝜇,𝜏𝑈ℓ𝑖]𝑖𝐿,(2) where 𝑈is a unitary 3×3matrix, 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 −12𝑀𝐿 ℓℓ󸀠(]ℓ𝐿)𝐶]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: −(𝑌ℓℓ󸀠/Λ)(]ℓ𝐿)𝐶(𝐻0)2]ℓ󸀠𝐿+h.c., where 𝑌ℓℓ󸀠is the Yukawa coupling constant, Λis the energy scale of some new physics not present in the Standard Model, and 𝐻0is 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: −V2𝑌ℓℓ󸀠 4Λ (]ℓ𝐿)𝐶]ℓ󸀠𝐿+h.c., (4) that is, 𝑀𝐿 ℓℓ󸀠=V2𝑌ℓℓ󸀠/2Λ. One assigns leptons an additive quantum number called theleptonnumber𝐿,suchthat𝐿=+1for particles and 𝐿= −1forantiparticles.Theleptonnumberisconservedin the standard electroweak interactions, like in the chargedcurrent interactions described by the Lagrangian (1), but the Majorana mass term (3)breaksitbytwounits;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 lefthanded neutrino fields ]ℓ𝐿, right-handed neutrino fields ]ℓ𝑅, the neutrino mass Lagrangian may contain also the Dirac mass terms −𝑀𝐷 ℓℓ󸀠]ℓ𝑅]󸀠 ℓ+h.c.and another type of Majorana mass terms −(1/2)𝑀𝑅 ℓℓ󸀠(]ℓ𝑅)𝐶]ℓ󸀠𝑅+h.c. Unless the Majorana mass terms vanish, the fields ]𝑖that diagonalize the full mass Lagrangian are two-component Majorana fields obeying the condition (“Majorana condition”) ]𝑖=]𝐶 𝑖.(5) There are in this case altogether six mass states. It is generally assumed that 𝑀𝑅 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),whiletheotherthreeareveryheavyanddecouple from the low-energy physics. Even if the mixing between light and heavy sectors is neglected, the relation (2)isstill applicable. A lot of empirical information on the neutrino mixing, that is, the elements of the matrix 𝑈,andtheneutrino masses has been obtained via solar, atmospheric, reactor, and accelerator neutrino oscillation experiments. The mixing matrix 𝑈can be presented in terms of six measurable parameters, three rotation angles and three phases, as follows [26]: 𝑈=( 𝑐12𝑐13 𝑠12𝑐13 𝑠13𝑒−𝑖𝛿 −𝑠12𝑐23 −𝑐12𝑠23𝑠13𝑒𝑖𝛿 𝑐12𝑐23 −𝑠12𝑠23𝑠13𝑒𝑖𝛿 𝑠23𝑐13 𝑠12𝑠23 −𝑐12𝑐23𝑠13𝑒𝑖𝛿 −𝑐12𝑠23 −𝑠12𝑐23𝑠13𝑒𝑖𝛿 𝑐23𝑐13 )𝑃, (6) where 𝑃=diag(1,𝑒𝑖𝛼,𝑒𝑖𝛽),𝑠𝑖𝑗 =sin 𝜃𝑖𝑗,𝑐𝑖𝑗 =cos 𝜃𝑖𝑗,and 𝛿is called the Dirac phase and 𝛼and 𝛽the Majorana phases. The probability for the oscillatory transition from the neutrino flavor ]𝛼to the flavor ]𝛽as a function of the distance of flight 𝐿and neutrino energy 𝐸is given by (see, e.g., [27]) 𝑃(]𝛼󳨀→]𝛽) =𝛿𝛼𝛽 −4∑ 𝑖>𝑗 Re [𝑈∗ 𝛼𝑖 𝑈∗ 𝛽𝑗 𝑈𝛽𝑖 𝑈𝛼𝑗]sin2Δ𝑚2 𝑖𝑗𝐿 4𝐸 +2∑ 𝑖>𝑗 Im [𝑈∗ 𝛼𝑖 𝑈∗ 𝛽𝑗 𝑈𝛽𝑖 𝑈𝛼𝑗]sin Δ𝑚2 𝑖𝑗𝐿 2𝐸 , (7) where Δ𝑚2 𝑖𝑗 =𝑚2 𝑖−𝑚2 𝑗.Ascanbeseenfromthisformula, the neutrino oscillations do not bring us any information about the absolute neutrino mass scale, only about the squared mass differences Δ𝑚2 𝑖𝑗. One can also easily show that neutrino oscillations are not sensitive to the Majorana phases 𝛼and 𝛽. Advances in High Energy Physics 3 n0 n0 W+ W+ i e+ e+ p+ p+ Figure 1: Feynman diagram for the Majorana-neutrinomediated 𝛽+𝛽+decay. Aglobalfittooscillationdatayieldsthefollowingvalues for the parameters [5]: Δ𝑚2 21 =7.54+0.26 −0.22 ×10−5eV2, Δ𝑚2 31 ≃Δ𝑚2 32 =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 hierarchy 𝑚3>𝑚 1,𝑚2is assumed; the values are slightly varied for the inverse hierarchy 𝑚3< 𝑚1,𝑚2(see [5]). The main goals of the forthcoming neutrino oscillation experiments are to measure the value of the CP phase 𝛿and 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 mass 𝑚𝛽=√∑𝑖|𝑈𝑒𝑖|2𝑚2 𝑖in beta decay experiments, as well as from the cosmological precision measurements of the sum of neutrino masses ∑𝑖𝑚𝑖. The current experimental upper limits for 𝑚𝛽are 2.3 eV [28]and2.1eV[29],andforthesum of neutrino masses ∑𝑖𝑚𝑖the recent Planck satellite data [6] imply the upper limit 0.66 eV. n0 W+ W+ i e+ p+ n0 p+ e− (a) n0 W+ W+ i p+ n0 p+ e− e− (b) Figure 2: Feynman diagrams for the Majorana-neutrinomediated 𝛽+EC (a) and ECEC (b) decays. 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 positronemission 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 (𝐺F √2)2𝑒+𝛾𝜇󸀠(1+𝛾5)]𝑐 𝑒]𝑒𝛾𝜇(1−𝛾5)𝑒−,(9) 4Advances in High Energy Physics m0(eV2) 10−4 10−3 10−2 10−1 10−4 10−3 10−2 10−1 Inverted hierarchy Normal hierarchy Excluded by cosmology meff (eV2) Figure 3: Absolute value of the effective neutrino mass 𝑚eff = |⟨𝑚]⟩|against the mass 𝑚0of the lightest neutrino for both the normalandinvertedmasshierarchyandforallpossiblevaluesofthe phases 𝜑12 and 𝜑13 defined in (15). The best-fit values [5]areused for the oscillation parameters (see (8)). The cosmological upper limit for 𝑚0, derived from the Planck satellite measurements [6], is also given. where 𝑒−,𝑒+,]𝑒,and]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 constant 𝐺F/√2=𝑔2/(8𝑚2 𝑊),where 𝑔is the fundamental gauge coupling of the electroweak theory and 𝑚𝑊is the mass of 𝑊±. The propagator describing the internal neutrino is given by ⟨0󵄨󵄨󵄨󵄨]𝑐 𝑒(𝑥)]𝑒(𝑦)󵄨󵄨󵄨󵄨0⟩ =∑ 𝑖(𝑈∗ 𝑒𝑖)2⟨0󵄨󵄨󵄨󵄨]𝑖(𝑥)]𝑖(𝑦)󵄨󵄨󵄨󵄨0⟩ =−𝑖∑ 𝑖(𝑈∗ 𝑒𝑖)2∫d4𝑞 (2𝜋)4 𝑞+𝑚𝑖 𝑞2−𝑚2 𝑖 exp (−𝑖𝑞⋅(𝑥−𝑦)),(10) where the condition (5)isused. The amplitude of the process (𝐴,𝑍)→(𝐴,𝑍−2)+2𝑒+is proportional to ∑ 𝑖𝐺2 F𝑈∗2 𝑒𝑖 𝛾𝜇𝛾𝐿 𝑞+𝑚𝑖 𝑞2−𝑚2 𝑖𝛾𝜇󸀠𝛾𝐿=∑ 𝑖𝐺2 F𝑈∗2 𝑒𝑖 𝑚𝑖 𝑞2−𝑚2 𝑖𝛾𝜇𝛾𝑅𝛾𝜇󸀠,(11) where 𝑞is the momentum of the exchanged neutrino and 𝛾𝐿(𝑅) are the chirality projection matrices 𝛾𝐿(𝑅) = (1(−/(+))𝛾5)/2.Notethatthe 𝑞part of the neutrino propagator does not contribute due to chirality mismatch. Typically 𝑞≃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 by 𝑞2,leadingto 𝐺2 F(∑ 𝑖𝑈∗2 𝑒𝑖 𝑚𝑖)1 𝑞2𝛾𝜇𝛾𝑅𝛾𝜇󸀠.(12) The essential part of the amplitude from neutrino-physics pointofviewisthequantity: ⟨𝑚]⟩≡∑ 𝑖𝑈∗2 𝑒𝑖 𝑚𝑖,(13) whoseabsolutevalueiscalledtheeffectiveneutrinomass;that is, 𝑚eff =󵄨󵄨󵄨󵄨⟨𝑚]⟩󵄨󵄨󵄨󵄨.(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 lefthanded electron neutrino 𝑀𝐿 𝑒𝑒 (see (3)). The modes 𝛽+EC and ECEC (Figure 2) are described bythesameoperator(9)asthe𝛽+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]⟩|2. Using the standard parametrization (6)ofthemixing matrix 𝑈,onecancast⟨𝑚]⟩in the following form: ⟨𝑚]⟩=𝑐2 12𝑐2 13𝑚1+𝑠2 12𝑐2 13𝑒−𝑖𝜑12𝑚2+𝑠2 13𝑒−𝑖𝜑13𝑚3,(15) where 𝜑12 =𝛼and 𝜑13 =𝛽−𝛿. Depending on the phases 𝜑12 and 𝜑13, the contributions of the three neutrino mass states will add up constructively or destructively. In the case theCPsymmetryisconserved,thephasefactorsassume thevalues+1or−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]⟩. 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]⟩. As the decay widths are proportional to |⟨𝑚]⟩|2,the modes 𝛽+𝛽+and 𝛽−𝛽−,aswellasofthemodes𝛽+EC and ECEC, probe neutrino physics through the same quantity.HencetheCPisnotmanifestlybrokenin neutrinoless double beta decay, although the Majorana phases 𝜑12 and 𝜑13 appear in ⟨𝑚]⟩. One can understand this also as a consequence of the fact that in the limit 𝑞2≫𝑚2 𝑖the amplitudes depend on just one parameter of the mass Lagrangian, the element 𝑀𝐿 𝑒𝑒, allowing for no measurable phases. To be sensitive to the Majorana CP phases, one should be able to distinguish between the mass states ]𝑖. ApartfromtheCPphases𝜑12 and 𝜑13,whicharenot observables of neutrino oscillations (the possible CP violation in oscillation phenomena is due to the Dirac phase 𝛿), there are two unknowns in the expression of the effective mass,namely,theabsoluteneutrinomassscale,saythe mass 𝑚0of the lightest neutrino, and the mass hierarchy, that is, whether 𝑚3>𝑚 1,𝑚2(normal hierarchy) or 𝑚3< 𝑚1,𝑚2(inverted hierarchy). All three neutrino masses can be Advances in High Energy Physics 5 expressed in terms of the absolute mass 𝑚0:inthecaseofthe normal hierarchy 𝑚1=𝑚0, 𝑚2=√𝑚2 0+Δ𝑚2 21, 𝑚3=√𝑚2 0+Δ𝑚2 31, (16) andinthecaseofinvertedhierarchy 𝑚1=√𝑚2 0+󵄨󵄨󵄨󵄨Δ𝑚2 31󵄨󵄨󵄨󵄨, 𝑚2=√𝑚2 0+Δ𝑚2 21 +󵄨󵄨󵄨󵄨Δ𝑚2 31󵄨󵄨󵄨󵄨, 𝑚3=𝑚0. (17) The squared mass difference Δ𝑚2 21 and the absolute value of the mass difference Δ𝑚2 31 are 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]⟩|has lower limit of the order of 10−2 eV, as can be inferred from Figure 3,wherethe effective mass, 𝑚eff =|⟨𝑚 ]⟩|,ispresentedasafunctionof the mass 𝑚0of the lightest neutrino for all possible values of the Majorana phases 𝜑12 and 𝜑13.Ifnosignalofdoublebeta decayisobtainedabovethislimit,itwouldmeanthateither the hierarchy has to be the normal one or the neutrino is not a Majorana particle. An observation of double beta transition with |⟨𝑚]⟩|<10−2 eVwouldmeanthatthemasshierarchyis normal and the neutrino is a Majorana particle. On the other hand, nonobservation of the transition would not mean that theprocessdoesnotexist,sinceinthecaseofthenormalmass hierarchy the effective mass and hence the decay width can be arbitrarily small. 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 thereadercanhaveaunifiedpictureoftheformalismsused for various types of double beta transitions. 4.1. Half-Lives and Nuclear Matrix Elements. In this work it is assumed that the 0]𝛽+/EC decays proceed exclusively via the exchange of massive Majorana neutrinos, as discussed in Section 3. The inverse half-lives for the neutrinoless 𝛽+𝛽+and 𝛽+ECdecayscanbecastintheform [𝑇𝛼 0](0+)]−1 =𝐺𝛼 0](0+)󵄨󵄨󵄨󵄨󵄨󵄨𝑀(0])󸀠󵄨󵄨󵄨󵄨󵄨󵄨2(𝑚eff [eV])2, 𝛼=𝛽+𝛽+,𝛽+EC,(18) where 𝑚eff is the effective neutrino mass (14)thatshouldbe givenin(18) in units of eV. The decays described by (18) proceedviatheavailablephasespaceforthefinalstateleptons and the phase-space integrals 𝐺𝛽+𝛽+ 0](0+)and 𝐺𝛽+EC 0](0+)are defined in [7]. The involved nuclear matrix element (NME) canbewritten(see,e.g.,[45–47]) in terms of the GamowTeller (GT), Fermi (F), and tensor (T) matrix elements in the form 𝑀(0])󸀠=(𝑔𝐴 1.25)2[𝑀(0]) GT −(𝑔𝑉 𝑔𝐴)2𝑀(0]) F+𝑀(0]) T], (19) where 𝑔𝐴= 1.25corresponds to the bare-nucleon value of the axial-vector coupling constant and 𝑔𝑉= 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 for 𝑔𝐴=1.25independent of the value of 𝑔𝐴used in (19). In the case of the neutrinoless double electron capture, 0]ECEC,therearenoleptonsavailableinthe final state to carry away the decay energy. In this case one has to engage some additional mechanism to rid theinitialatomoftheexcessenergyofdecay.Thereare twoproposedmechanismstocopewiththissituation: the radiative 0]ECEC decay [9]andtheresonantdecay, R0]ECEC [8]. The resonance condition—close degeneracy of the initial and final (excited) atomic states—can enhance the decay rate by a factor as large as 106.TheR0]ECEC 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 socalled degeneracy parameter 𝑄−𝐸,where 𝐸is the excitation energy of the final atomic state and 𝑄is 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.Thefinalnuclearstateswith spin-parity 0+are the most favorable ones and the only ones discussed as examples in this review. The inverse half-life for transitions to 0+statescanbewrittenas [𝑇ECEC 0](0+)]−1 =𝑔ECEC 0](0+)󵄨󵄨󵄨󵄨󵄨󵄨𝑀(0])󸀠󵄨󵄨󵄨󵄨󵄨󵄨2𝑚2 effΓ (𝑄−𝐸)2+Γ2/4, (21) where the daughter state (𝐴,𝑍−2)∗is a virtual state with energy 𝐸=𝐸∗+𝐸𝐻+𝐸𝐻󸀠+𝐸𝐻𝐻󸀠,(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. 6Advances in High Energy Physics The quantity Γdenotes the combined nuclear and atomic radiative widths where the atomic widths dominate and are a fewtensofelectronvolts[50]. The factor 𝑔ECEC 0]can be called the atomic factor and it contains the information about the density distributions of the involved atomic orbitals at the nucleus.Itcanbewrittenas 𝑔ECEC 0](0+)=(𝐺Fcos 𝜃𝐶 √2)4𝑔4 𝐴 4𝜋2ln 2𝑅2 𝐴𝑚6 𝑒N𝑛,𝜅N𝑛󸀠,𝜅󸀠,(23) where N𝑛,𝜅 is 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(0]) 𝐾=∑ 𝐽𝜋∑ 𝐽󸀠∑ 𝑘1𝑘2∑ 𝑝𝑝󸀠𝑛𝑛󸀠(−1)𝑗𝑝+𝑗𝑛󸀠+𝐽+𝐽󸀠 ×√2𝐽󸀠+1{𝑗𝑛𝑗𝑝𝐽 𝑗𝑝󸀠𝑗𝑛󸀠𝐽󸀠} ×𝑚𝐾(𝑛𝑛󸀠,𝑝𝑝󸀠;𝐽󸀠;𝑘1,𝑘2) ×(0+ 𝑓󵄩󵄩󵄩󵄩󵄩󵄩[𝑐† 𝑛󸀠𝑐𝑝󸀠]𝐽󵄩󵄩󵄩󵄩󵄩󵄩𝐽𝜋 𝑘1)⟨𝐽𝜋 𝑘1|𝐽𝜋 𝑘2⟩(𝐽𝜋 𝑘2󵄩󵄩󵄩󵄩󵄩󵄩[𝑐† 𝑛𝑐𝑝]𝐽󵄩󵄩󵄩󵄩󵄩󵄩0+ 𝑖), (24) where 𝐾=F,GT and 𝑘1and 𝑘2label the different nuclear-model solutions for a given multipole 𝐽𝜋,the set 𝑘1stemming from the calculation based on the final nucleus and the set 𝑘2stemming from the calculation based on the initial nucleus. Here the one-body transition densities are (0+ 𝑓‖[𝑐† 𝑛󸀠𝑐𝑝󸀠]𝐽‖𝐽𝜋 𝑘1)and (𝐽𝜋 𝑘2‖[𝑐† 𝑛𝑐𝑝]𝐽‖0+ 𝑖),andtheyaregiven separately for the different types of 0+final states 𝑓in Section 4.3. The two-particle matrix element of (24)canbewrittenas 𝑚𝐾(𝑛𝑛󸀠,𝑝𝑝󸀠;𝐽󸀠;𝑘1,𝑘2) = 𝐽󸀠 𝑗𝑝 𝑗𝑝󸀠 𝑗𝑛 𝑗𝑛󸀠∑ 𝜆𝑆 (2𝜆+1)(2𝑆+1)𝐹𝐾 ×{ { { { { { { { { { { 𝑙𝑝𝑙𝑝󸀠𝜆 1212𝑆 𝑗𝑝𝑗𝑝󸀠𝐽󸀠 } } } } } } } } } } } { { { { { { { { { { { 𝑙𝑛𝑙𝑛󸀠𝜆 1212𝑆 𝑗𝑛𝑗𝑛󸀠𝐽󸀠 } } } } } } } } } } } ×∑ 𝑛1𝑛2𝑙𝑁𝐿𝑀𝜆(𝑛1𝑙𝑁𝐿;𝑛𝑛𝑙𝑛𝑛𝑛󸀠𝑙𝑛󸀠) ×𝑀𝜆(𝑛2𝑙𝑁𝐿;𝑛𝑝𝑙𝑝𝑛𝑝󸀠𝑙𝑝󸀠) ×∫𝑑3𝑟𝜙𝑛1𝑙(r)ℎ𝐾(𝑟12,12(𝐸𝑘1+𝐸𝑘2))𝜙𝑛2𝑙(r),(25) where  𝑗=√2𝑗+1and 𝑟12 =|r1−r2|is the relative distance between the two decaying protons. The following auxiliary quantities have been defined 𝐹F=1, 𝐹GT =6(−1)𝑆+1 { { { { { { { 1212𝑆 12121} } } } } } }.(26) The quantities 𝑀𝜆are the Moshinsky brackets that mediate the transformation from the laboratory coordinates r1and r2to the center-of-mass coordinate R=(1/√2)(r1+ r2)and the relative coordinate r=(1/ √2)(r1−r2).In this way the short-range correlations of the two decaying protons are easily incorporated in the theory. The wave functions 𝜙𝑛𝑙(r)aretakentobetheeigenfunctionsofthe isotropicharmonicoscillator. The neutrino potential ℎ𝐾(𝑟12,𝐸),𝐾=F,GT, in the integral of (26) is defined as ℎ𝐾(𝑟12,𝐸)=2 𝜋𝑅𝐴∫𝑑𝑞 𝑞ℎ𝐾(𝑞2) 𝑞+𝐸−(𝐸𝑖+𝐸𝑓)/2𝑗0(𝑞𝑟12), (27) where 𝑗0is the spherical Bessel function and the integration is performed over the exchanged momentum 𝑞.Here𝐸𝑖= 𝑀𝑖𝑐2is the ground-state mass energy of the initial nucleus and 𝐸𝑓the (ground-state or excited-state) mass energy of the final nucleus. In practice the lowest pnQRPA energies of the two sets 𝑘1and 𝑘2are normalized such that the energy difference of these energies and the mass energy of the initial nucleus match the corresponding experimental energy difference. The term ℎ𝐾(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 the 0]𝛽+/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 many 0]𝛽−𝛽−calculations [48,49,55–58]andin some 0]𝛽+/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. The 𝐽𝜋states of the intermediate nucleus of the 𝛽𝛽decays are generated by the usual proton-neutron QRPA (pnQRPA) [2,62]inthe form 󵄨󵄨󵄨󵄨𝐽𝜋 𝑘𝑀⟩=𝑄†(𝐽𝜋 𝑘,𝑀)|QRPA⟩ =∑ 𝑝𝑛 (𝑋𝐽𝜋 𝑘 𝑝𝑛[𝑎† 𝑝𝑎† 𝑛]𝐽𝑀 −𝑌𝐽𝜋 𝑘 𝑝𝑛[𝑎† 𝑝𝑎† 𝑛]† 𝐽𝑀)|QRPA⟩, (28) where 𝑋and 𝑌are the forwardand backward-going amplitudes of the pnQRPA, obtained by solving the pnQRPA Advances in High Energy Physics 7 511.86 keV 1128.0 keV 1133.7 keV 1229.2 keV 2766.26 keV = 2770(7) keV (0+) 0+ gs 4+ 1 2+ 2 2+ 1 1+ 1 0+ 1 0+ 1 Q R0ECEC 106 47 Ag59 106 46 Pd60 106 48 Cd58 XK XK 𝛽+EC0 ,𝛽+EC 𝛽+𝛽+ 00 = 202 keV Q𝛽− Figure 4: Schematic representation of the possible 0]𝛽+/EC decay modes of 106Cd. The atomic resonance at 2766.26keV includes both the nuclear excitation energy and the energy related to the two electron holes in the atomic K shell as given in (22). 511.86 keV 1128.0 keV 1133.7 keV 1229.2 keV 2766.26 keV (0+) 4+ 1 2+ 2 2+ 1 0+ 1 0+ gs = 2770(7) keV 1+ 1 Q 106 46 Pd60 106 48 Cd58 106 47 Ag59 0+ 1 XK XK R-ECEC: (2.3–6.3) ×10 31 𝛽+EC: (1.1–1.4) ×10 30 𝛽+EC: (1.5–1.7) ×10 27 𝛽+𝛽+: (2.1–2.4) ×10 28 = 202 keVQ𝛽− Figure 5: Computed partial 0]𝛽+/EC decay half-lives of the various decay transitions from 106Cd. The value 𝑚eff =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. equations of motion [62]. The excited states 𝐼𝜋 𝑘in the final even-even nuclei are described by the phonons of the chargeconserving QRPA (ccQRPA), expressed as 󵄨󵄨󵄨󵄨𝐼𝜋 𝑘𝑀⟩=𝑄†(𝐼𝜋 𝑘,𝑀)|QRPA⟩ =∑ 𝑎𝑏 (𝑍𝐼𝜋 𝑘 𝑎𝑏[𝑎† 𝑎𝑎† 𝑏]𝐼𝑀 −𝑊𝐼𝜋 𝑘 𝑎𝑏 [𝑎† 𝑎𝑎† 𝑏]† 𝐼𝑀)|QRPA⟩, (29) (0+) 7+ gs 2718.41 keV 96 43Tc53 0+ gs 0+ gs R0ECEC: (4.9–22) ×10 32 yr (|⟨m⟩| |⟨m⟩| |⟨m⟩| |⟨m⟩| |⟨m⟩| (|⟨m⟩| = 0.3 eV) 2+ 3 2+ 2 0+ 2 0+ 1 2+ 1 1625.90 keV 1497.79 keV 1330 keV 1148.13 keV 778.24 keV 𝛽+EC: (3.8–8.1) ×10 30 yr = 0.3 eV) 𝛽+EC: (1.2–8.4) ×10 29 yr ( = 0.3 eV) 𝛽+EC: (5.5–6.3) ×10 27 yr ( = 0.3 eV) 𝛽+𝛽+: (6.5–7.5) ×10 28 yr ( = 0.3 eV) 96 44Ru52 96 42Mo54 XL1 XL1 0 0 0 0 Figure 6: Computed partial 0]𝛽+/EC decay half-lives of the various decay transitions from 96Ru. The value 𝑚eff =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. where the symmetrized amplitudes 𝑍and 𝑊are obtained from the usual ccQRPA amplitudes 𝑋and 𝑌[62]through the transformation 𝑍𝐼𝜋 𝑘 𝑎𝑏 ={ { { { { { { { { { { { { { { 𝑋𝐼𝜋 𝑘 𝑎𝑏,if 𝑎=𝑏 12𝑋𝐼𝜋 𝑘 𝑎𝑏,if 𝑎<𝑏 12𝑋𝐼𝜋 𝑘 𝑏𝑎,if 𝑎>𝑏 (30) and similarly for 𝑊in terms of 𝑌. Now one can take a 𝐼𝜋 𝑘=2+ 1phonon of (29)andbuildan ideal two-phonon 𝐼+state of the form 󵄨󵄨󵄨󵄨󵄨𝐼+ 2−ph⟩=1 √2[𝑄†(2+ 1)𝑄†(2+ 1)]𝐼|QRPA⟩.(31) An ideal two-phonon state consists of partner states 𝐼𝜋= 0+,2+,4+that are degenerate in energy and exactly at an energy twice the excitation energy of the 2+ 1state. In practice this degeneracy is always lifted by the residual interaction between the oneand two-phonon states [63]. The oneandtwo-phononstatesinthefinaleven-evennucleusare connected to the 𝐽𝜋states 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 singleparticle spaces and the single-particle energies were generated by the use of a spherical Coulomb-corrected WoodsSaxon (WS) potential with a standard parametrization [66], 8Advances in High Energy Physics 2854.87 keV 602.73 keV 1325.51 keV 1657.28 keV 2+ 2 2+ 1 0+ 1 0+ gs 0+ gs (0+)R-ECEC: (1.9–5.6) ×10 30 |⟨m⟩| = 0.3 eV |⟨m⟩| = 0.3 eV |⟨m⟩| = 0.3 eV 124 54 Xe70 124 52 Te 72 𝛽+EC: ≥5.9 × 1032 ,𝛽+EC: (1.2–4.2) ×10 27 𝛽+𝛽+: (2.3–7.7) ×10 28 XK XK2gs − 124 53 I71 Figure 7: Computed partial 0]𝛽+/EC decay half-lives of the various decay transitions from 124Xe. The value 𝑚eff =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. 0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0 Nucleus 78Kr 92Mo 96Ru 106Cd 124Xe 130Ba 136Ce 0+ gs 0+ 1 0+ res M(0)󳰀 Figure 8: Values of the computed nuclear matrix elements for 0]𝛽+/EC decay transitions to the ground state (0+ gs), the first excited 0+state (0+ 1)andtheresonant 0+state (0+ res)forallthenuclei discussed in this paper. optimized for nuclei near the line of beta stability. Sometimes the Woods-Saxon based single-particle energies were slightly correctedneartheprotonand/orneutronFermisurfacesto 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, 0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 M(0)󳰀 70Zn 76Ge 78Kr 82Se 86Kr 92Mo 94Zr 96Zr 96Ru 100Mo 0+ gs 0+ 1 Nucleus (a) Nucleus 0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0 M(0)󳰀 104Ru 106Cd 110Pd 116Cd 124Sn 124Xe 136Xe 136Ce 128Te 130Ba 130Te 0+ gs 0+ 1 (b) Figure 9: Values of the computed nuclear matrix elements for 0]𝛽−𝛽−and 0]𝛽+/EC decays for masses 70 ≤ 𝐴 ≤ 100(a) and for 104≤𝐴≤136(b). Shown are the NMEs corresponding to the decay transitions to the ground state (0+ gs)andthefirst excited 0+state (0+ 1). 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]theparticle-holeand particle-particle parts of the proton-neutron two-body interaction are separately scaled by the particle-hole parameter 𝑔ph and particle-particle parameter 𝑔pp.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 the 𝑔pp parameter regulates the 𝛽−-decay amplitude of the first 1+state in the intermediate nucleus [68] and hence also the decay Advances in High Energy Physics 15 Table 4: R0]ECEC decay transitions with the final-state spin-parity indicated in the second column and the degeneracy parameters 𝑄−𝐸in 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 𝐽𝜋 𝑓𝑄−𝐸[keV] Orbitals 𝐶ECEC References 74Se →74Ge 2+2.23 L2L3(0.2−100) ×1043 [33] 96Ru →96Mo 2+8.92 (13) L1L3[34] 0+?−3.90 (13) L1L1 102Pd →102Ru 2+75.26 (36) KL3[35] 106Cd →106Pd 0+? 8.39 KK (2.1−5.7) ×1030 [36] (2,3)−−0.33 (41) KL3[35] 112Sn →112Cd 0+−4.5 KK >5.9 ×1029 [37] 124Xe →124Te 0+? 1.86 (15) KK (1.7−5.1) ×1029 [38] 130Ba →130Xe 0+? 10.18 (30) KK [38] 136Ce →136Ba 0+−11.67 KK (3−23) ×1032 [39] 144Sm →144Nd 2+171.89 (87) KL3[35] 152Gd →152Sm 0+ gs 0.91 (18) KL1(1.0−1.5) ×1027 [40,41] 156Dy →156Gd 1−0.75 (10) KL1[42] 0+0.54 (24) L1L1[42] 2+0.04 (10) M1N3[42] 162Er →162Dy 2+2.69 (30) KL3[34] 164Er →164Dy 0+ gs 6.81 (13) L1L1(3.2−5.2) ×1031 [41,43] 168Yb →168Er (2−)1.52(25)M 1M3[34] 180W→180Hf 0+ gs 11.24 (27) KK (4.0−9.5) ×1029 [41,44] ured by using the Penning-trap techniques. In the cases of 96Ru, 106Cd, 124Xe, and 130Ba the assignment of 0+spinparity to the resonant state is uncertain. In these cases further experimental spectroscopy is needed. In the table we also list the estimated half-lives for the cases for which such exist. The references of the last column indicate the origin of the 𝑄-value measurement and the possible calculations of the related NME. In the table an auxiliary quantity 𝐶ECEC is listed and its relation with the R0]ECEC half-life stands as 𝑇R0]ECEC 1/2 =𝐶ECEC (𝑚eff [eV])2years,(38) wheretheeffectiveneutrinomassshouldbegiveninunits of eV. In all the listed cases where 𝐶ECEC has been computed the decay rates are suppressed by the rather sizable magnitude of the degeneracy parameter. Decays to 0+states are favored over the decays to 2+or 1−,2−,3−,andsoforthstatesdue to the involved nuclear wave functions and/or higher-order transitions. Also captures from atomic orbitals with orbital angular momentum 𝑙>0are suppressed [33]. There are some favorable values of degeneracy parameters listed in Table 4,like106Cd →106Pd(2,3)−and 156Dy → 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 with 𝐽>0is, however, expected. In case of the 156Dy decay the deformation also plays an important role. At the moment the most favorable case with a half-life estimate is the case 152Gd →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 electroncapture modes of the neutrinoless double beta decays have been investigated for the associated nuclear matrix elements and decay half-lives. 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