Analysis of the intermediate-state contributions to neutrinoless double beta-minus decays

A comprehensive analysis of the structure of the nuclear matrix elements (NMEs) of neutrinoless double beta-minus decays to the 0^+ ground and first excited states is performed in terms of the contributing multipole states in the intermediate nuclei of neutrinoless double beta-minus transitions. We concentrate on the transitions mediated by the light (l-NMEs) Majorana neutrinos. As nuclear model we use the proton-neutron quasiparticle random-phase approximation (pnQRPA) with a realistic two-nucleon interaction based on the Bonn one-boson-exchange G matrix. In the computations we include the appropriate short-range correlations, nucleon form factors, higher-order nucleonic weak currents and restore the isospin symmetry by the isoscalar-isovector decomposition of the particle-particle proton-neutron interaction parameter g_{pp}.


I. INTRODUCTION
Thanks to neutrino-oscillation experiments much is known about the basic properties of the neutrino concerning its mixing and squared mass differences.What is not known is the absolute mass scale, the related mass hierarchy, and the fundamental nature (Dirac or Majorana) of the neutrino.This can be studied by analyzing the neutrinoless double beta (0ν2β) decays of atomic nuclei [1][2][3][4] through analyses of the participating nuclear matrix elements (NMEs).The 0ν2β decays proceed by virtual transitions through states of all multipoles J π in the intermediate nucleus, J being the total angular momentum and π being the parity of the intermediate state.Most of the present interest is concentrated on the double beta-minus variant (0νβ − β − decay) of the 0ν2β decays due to their relatively large decay energies (Q values) and natural abundancies.
In this work we concentrate on analyses of the intermediate contributions to the 0νβ − β − decays for the 0 + → 0 + ground-state-to-ground-state and ground-stateto-excited-state transitions in nuclear systems of experimental interest.We focus on the light Majorana neutrino mediated transitions by taking into account the appropriate short-range nucleon-nucleon correlations [5], and contributions arising from the induced currents and the finite nucleon size [6].There are several nuclear models that have recently been used to compute the 0νβ − β − decay NMEs (see, e.g., the extensive discussions in [3,[7][8][9][10][11]).However, the only model that avoids the closure approximation and retains the contributions from individual intermediate states is the proton-neutron quasiparticle random-phase approximation (pnQRPA) [7,[12][13][14][15].Some analyses of the intermediate-state contributions within the pnQRPA approach have been performed in [12,14,16,17] and recently quite extensively in [18,19].In [18] an intermediate multipole J π decomposition was done for decays of 76 Ge, 82 Se, 96 Zr, 100 Mo, 110 Pd, 116 Cd, 124 Sn, 128,130 Te, 136 Xe to the ground state of the respective daughter nuclei, whereas in [19] both an intermediate multipole J π decomposition and a pair-angularmomentum decomposition were done for decays of 76 Ge, 82 Se, 96 Zr, 100 Mo, 110 Pd, 116 Cd, 124 Sn, 130 Te, 136 Xe to the lowest one or two excited 0 + states of the respective daughter nuclei.In this article we extend the analyses of [18,19] to a more detailed scrutiny of the intermediate contributions to the 0νβ − β − decay NMEs of the above-mentioned nuclei.

II. THEORY BACKGROUND
In this section a very brief introduction to the computational framework of the present calculations is given.The present analyses are based on the calculations done in Refs.[18,19] and more details on the used theory tools can be checked there and in [20].We assume here that the 0νβ − β − decay proceeds via the light Majorana neutrino so that the inverse half-life can be written as where G 0ν is a phase-space factor for the final-state leptons defined here without the axial-vector coupling constant g A .The quantity m ν denotes the neutrino effective mass and describes the physics beyond the standard model [18].The quantity M (0ν) is the light-neutrino nuclear matrix element (l-NME).The nuclear matrix element can be decomposed into Gamow-Teller (GT), Fermi (F) and tensor (T) contributions as arXiv:1604.01631v1 [nucl-th] 6 Apr 2016 where g V is the vector coupling constant.Each of the NMEs K = GT,F,T in (2) can be decomposed in terms of the intermediate multipole contributions J π as where each multipole contribution is, in turn, decomposed in terms of the two-particle transition matrix elements and one-body transition densities as where k 1 and k 2 label the different pnQRPA solutions for a given multipole J π and the indices p, p , n, n denote the proton and neutron single-particle quantum numbers.
The operators O K inside the two-particle matrix element contain the neutrino potentials for the light Majorana neutrinos, the characteristic two-particle operators for the different K = GT,F,T and a function taking into account the short-range correlations (SRC) between the two decaying neutrons in the mother nucleus of 0νβ − β − decay [18].The final 0 + state, 0 + f , can be either the ground state or an excited state of the 0νβ − β − daugter nucleus, and the overlap factor between the two one-body transition densities helps connect the corresponding intermediate J π states emerging from the pnQRPA calculations in the mother and daughter nuclei.The one-body transition densities are exposed in detail in the articles [18,19].
As mentioned before, our calculations contain the appropriate short-range correlators, nucleon form factors and higher-order nucleonic weak currents.In addition, we decompose the particle-particle proton-neutron interaction strength parameter g pp of the pnQRPA into its isoscalar (T = 0) and isovector (T = 1) components and adjust these components independently as described in [18]: The isovector component is fixed such that the NME of the two-neutrino double beta decay (2νβ − β − ) vanishes and the isospin symmetry is thus restored for both the 2νβ − β − and 0νβ − β − decays.The isoscalar component, in turn, is fixed such that the measured halflife of the 2νβ − β − decay is reproduced.The resulting values of both components of g pp are shown in Table I of Ref. [18].The details of the chosen valence spaces and the determination of the other hamiltonian parameters are presented in [18,19].

III. RESULTS AND DISCUSSION
In this section we discuss and present the results of our calculations.Presentation of the results follows top to bottom approach.First we analyze the multipole decompositions and total cumulative sums of the matrix elements.From these we can extract the most important multipole components and energy regions contibuting to the NMEs.After this we continue and dissect the most important multipole components into contributions coming from different individual states of the 0νββ intermediate nucleus.Throughout these computations we have used a conservatively quenched value of the axial vector coupling g A = 1.00.
A. Ground-state-to-ground-state transitions Let us begin by considering the ground-state-toground-state decays mediated by light neutrino exchange.In Fig. 1 (a)-(b) we have plotted the multipole decomposition (3) of the l-NMEs corresponding to the A = 96 and 136 nuclear systems.For most nuclei considered in this work, the leading multipole component is 1 − This is the case also for the nucleus 96 Zr shown in Fig. 1 (a).Most important contribution to the NMEs comes from the lowest multipole components 1 ± − 4 ± .It can also be observed that the shape of the overall multipole distribution is leveled when going towards heavier nuclei.This can be seen by comparing the distribution of 96 Zr with the distribution of 136 Xe displayed in Fig. 1

(b).
Nuclei can be grouped into different types according to the shapes of their cumulative NME distributions.For 0 + gs −→ 0 + gs transitions via light neutrino exchange, we can differentiate four types of nuclei.Type 1: Nuclei belonging to this type are 76 Ge, 82 Se, 96 Zr and 128 Te.Representative of this type, 76 Ge, is presented in Fig. 2, panel (a).Characteristic feature of the cumulative sum distribution belonging to type 1 is the strong drop in the value of the NME occuring between 12-17 MeV.Soon after this drop the NME saturates as can be seen from panel (a).Type 2: Nuclei belonging to this type are 100 Mo and 110 Pd.Representative of this type, 110 Pd, is presented in Fig. 2, panel (b).Characteristic feature of this type is the large enhancement and almost immediate cancellation of this enhancement around 10 MeV.This produces a spike like structure into the cumulative sum distribution as can be seen from panel (b).Type 3: Nuclei belonging to type 3 are 116 Cd, 124 Sn and 130 Te.Type 3 is represented by 124 Sn, shown in Fig. 2, panel (c).Characteristic features of this type are that there occurs neither sharp cancellation of the NME around 12-17 MeV, as in type 1, nor a spike like structure around 10 MeV, as in type 2. Value of the NME rather increases more or less smoothly to its highest value and then smoothly saturates to its final value around 20 MeV.Type 4: Type 4 is special in a sence that it icludes only one nucleus, 136 Xe.Cumulative sum of the NME for 136 Xe is shown in Fig. 2, panel (d).Characteristic feature of type 4 is that the lowest energy region, roughly between 0-1.5 MeV, contributes practically nothing to the value of the NME as can be noticed We notice a single-state dominance for the 2 − mode in nuclei 76 Ge, 82 Se and 96 Zr.In Ref. [21] an analysis of the unique first forbidden single β ± 2 − → 0 + ground-state-to-ground-state transitions in the mass region A = 72 − 132 was performed.It was found that a strong renormalization of the axial vector 2 − single β matrix elements is needed to be able to explain the experimental transition rates.It was then speculated that a same kind of an effect may also appear in the 0νββ NMEs.This may have a large effect on the 0νββ transition rates due to the important contribution of the 2 − multipole to the 0νββ NMEs.

B. Ground-state-to-excited-state decays
Let us then consider 0 + gs → 0 + 1 transitions mediated by the light neutrino exchange.In Fig. 3 (a)-(b) we have plotted the multipole decomposition of the l-NMEs corresponding to the A = 76 and 96 nuclear systems.The multipole distributions for the excited-state transitions are greatly different from those corresponding to the ground-state transitions.Usually there is only a couple of multipoles, 0 + and 1 + , which give by far the largest contribution to the NMEs.In this sense the excited-state transitions are more simple than the ground-state transitions.Typical example is the nucleus 76 Ge, displayed in Fig. 3 (a).One nucleus deviating from this trend is 96 Zr which is presented in Fig. 3 (b).Its multipole dis-tribution resembles somewhat more those shown for the ground-state decays in Fig. 1 (a)-(b).
Again we can divide nuclei into different groups by considering the shapes of their total cumulative sum distributions.For 0 + gs −→ 0 + 1 transitions via light neutrino exchange, we can differentiate two types of nuclei.Type 1: Nuclei belonging to type 1 are 76 Ge, 82 Se, 124 Sn, 130 Te and 136 Xe.Typical examples of this type, 76 Ge, 82 Se, 136 Xe, are shown in Fig. 4, panels (a), (b) and (d).Characteristic feature of this type is that there exist only few energy states which give most of the total matrix element producing a staircase like structure as seen in the panels.For example, for 76 Ge there seems to be only five such energy states.Type 2: Nuclei belonging to this type are 96 Zr, 100 Mo, 110 Pd and 116 Cd.Typical examples of this type are 96 Zr and 116 Cd shown in Fig. 4, panels (c) and (e).Characteristic feature of type 2 is that a large number of intermediate states give important contributions to the NMEs.In case of 116 Cd, panel (e), around 50% of the total NME comes from transitions through the ground state of the intermediate nucleus.The other 50% is distributed rather evenly on the interval 0-20 MeV.
Using the multipole decompositions, we extracted the most important multipole components contibuting to the light neutrino mediated 0 + gs → 0 + 1 decay transitions.These most important components were then again divided into contributions coming from different energy levels of the 0νββ intermediate nucleus.These contributions are collected into Table IV FIG. 3: Multipole decomposition of the l-NME for the nuclei 76 Ge and Zr corresponding to the 0 + gs → 0 + 1 decay transitions.

C. conclusions
In this article we have extended our previous works [18,19] on the ground-state-to-ground-state and ground-state-to-excited-state 0νββ decay transitions.In the present work we have concentrated our studies on the intermediate contributions to the NMEs involved in the light-neutrino mediated 0νββ decay.We have calculated We have done these computatations by using realistic two-body interactions and single-particle bases.All the appropriate short-range correlations, nucleon form factors and higher-order nucleonic weak currents are included in our present results.
We found in the calculations that often there exists only a few relevant intermediate states which collect most of the strength corresponding to a given multipole.We also found that there exists a single-state dominance in the important 2 − components related to the ground-state decays of nuclei 76 Ge, 82 Se and perhaps also for 96 Zr.

FIG. 2 :
FIG. 2: Cumulative values of the computed l-NMEs corresponding to the 0 + gs → 0 + gs decay transitions for the nuclear systems A = 76, 110, 124 and 136.The horizontal axis gives the excitation energies of the intermediate states contributing to the 0νββ transition.

FIG. 4 :
FIG. 4: Cumulative values of the computed l-NMEs corresponding to the 0 + gs → 0 + 1 decay transitions for the nuclear systems A = 76, 82, 96, 116 and 136.The horizontal axis gives the excitation energies of the intermediate states contributing to the 0νββ transition.
These most important components can be divided into contributions coming from different energy levels of the 0νββ intermediate nucleus.These contributions are collected into Table I for A = 76 − 100 systems, into Table II for A = 110−124 systems and into Table III for A = 128 − 136 systems.We see from the tables that often a very small set of states collects the largest part of a given multipole contribution to the NMEs.Also in some cases notable contributions are coming from high excitation energies, well above 10 MeV, like in the case of 1 − contributions for almost all nuclei, 1 + contributions for 76 Ge, 82 Se, 110 Pd, 116 Cd and 124 Sn, 2 + contributions for 130 Te and 136 Xe and a 3 − contribution for 124 Sn.
for A = 76−116 systems and into TableVfor A = 124 − 136 systems.Again we notice that often only a few intermediate states give the largest contibution to the dominant multipoles 1 + and 0 + .Extreme case is the nucleus 116 Cd for which the dominant intermediate ground state gives 81% of the total 1 + strength.Combining this with the fact that 1 + is by far the largest multipole component, we get a rather good approximation for the total NME by considering just a single virtual transition through the 1 + ground state of the intermediate nucleus116In.As for the ground-stateto-ground-state decays in some cases notable contributions are coming from high excitation energies, well above 10 MeV.There are high-energy contributions in case of 1 + multipole for all nuclei, and in the cases of 2 − and 2 + multipoles for 130 Te and 136 Xe.

TABLE I :
Most important multipoles and intermediate states contributing to the ground-state-to-ground-state 0νββ decays mediated by the light neutrino exchange.Columns E give the energies (in MeVs) and multipoles of the intermediate states.Multipoles are organized from left to right in terms of their importance, the most important being on the left.Columns labeled C give the corresponding NME contributions.Last two numbers in each C column give the summed contribution and the percentual part which the displayed states give to the total multipole strength.

TABLE II :
Most important multipoles and intermediate states contributing to the ground-state-to-ground-state 0νββ decays mediated by the light neutrino exchange.Columns E give the energies (in MeVs) and multipoles of the intermediate states.Multipoles are organized from left to right in terms of their importance, the most important being on the left.Columns labeled C give the corresponding NME contributions.Last two numbers in each C column give the summed contribution and the percentual part which the displayed states give to the total multipole strength.

TABLE III :
Most important multipoles and intermediate states contributing to the ground-state-to-ground-state 0νββ decays mediated by the light neutrino exchange.Columns E give the energies (in MeVs) and multipoles of the intermediate states.Multipoles are organized from left to right in terms of their importance, the most important being on the left.Columns labeled C give the corresponding NME contributions.Last two numbers in each C column give the summed contribution and the percentual part which the displayed states give to the total multipole strength.intermediate state multipole decompositions of the NMEs and extracted the most important multipole components.Cumulative sums of the NMEs were calculated to investigate the important energy regions contributing to the 0νββ transitions.Finally, the most important multipole components were divided into contributions coming from the virtual transitions through the individual states of the 0νββ intermediate nuclei.An extensive tabulation of these important intermediate states were given for all the nuclei considered in this paper. the