research article analysis of the intermediate-state...
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Research ArticleAnalysis of the Intermediate-StateContributions to Neutrinoless Double 120573minus Decays
Juhani Hyvaumlrinen and Jouni Suhonen
Department of Physics University of Jyvaskyla PO Box 35 40014 Jyvaskyla Finland
Correspondence should be addressed to Juhani Hyvarinen juhanithyvarinenstudentjyufi
Received 1 April 2016 Accepted 26 May 2016
Academic Editor Luca Stanco
Copyright copy 2016 J Hyvarinen and J SuhonenThis is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited The publication of this article was funded by SCOAP3
A comprehensive analysis of the structure of the nuclear matrix elements (NMEs) of neutrinoless double beta-minus (0]120573minus120573minus)decays to the 0+ ground and first excited states is performed in terms of the contributing multipole states in the intermediate nucleiof 0]120573minus120573minus transitions We concentrate on the transitions mediated by the light (l-NMEs)Majorana neutrinos As nuclear model weuse the proton-neutron quasiparticle random-phase approximation (pnQRPA) with a realistic two-nucleon interaction based onthe Bonn one-boson-exchange 119866 matrix In the computations we include the appropriate short-range correlations nucleon formfactors and higher-order nucleonic weak currents and restore the isospin symmetry by the isoscalar-isovector decomposition ofthe particle-particle proton-neutron interaction parameter 119892pp
1 Introduction
Thanks to neutrino-oscillation experiments much is knownabout the basic properties of the neutrino concerning itsmixing and squared mass differences What is not knownis the absolute mass scale the related mass hierarchy andthe fundamental nature (Dirac or Majorana) of the neutrinoThis can be studied by analyzing the neutrinoless doublebeta (0]2120573) decays of atomic nuclei [1ndash4] through analysesof the participating nuclear matrix elements (NMEs) The0]2120573 decays proceed by virtual transitions through states ofall multipoles 119869120587 in the intermediate nucleus 119869 being the totalangular momentum and 120587 being the parity of the interme-diate state Most of the present interest is concentrated onthe double beta-minus variant (0]120573minus120573minus decay) of the 0]2120573decays due to their relatively large decay energies (119876 values)and natural abundancies
In this work we concentrate on analyses of the interme-diate contributions to the 0]120573minus120573minus decays for the 0+ rarr 0
+
ground-state-to-ground-state and ground-state-to-excited-state transitions in nuclear systems of experimental interestWe focus on the lightMajorana neutrinomediated transitions
by taking into account the appropriate short-range nucleon-nucleon correlations [5] and contributions arising from theinduced currents and the finite nucleon size [6] There areseveral nuclear models that have recently been used tocompute the 0]120573minus120573minus decay NMEs (see eg the extensivediscussions in [3 7ndash11]) However the only model that avoidsthe closure approximation and retains the contributionsfrom individual intermediate states is the proton-neutronquasiparticle random-phase approximation (pnQRPA) [712ndash14]
Some analyses of the intermediate-state contributionswithin the pnQRPA approach have been performed in [1213 15 16] and recently quite extensively in [17] In [17]an intermediate multipole 119869120587 decomposition was done fordecays of 76Ge 82Se 96Zr 100Mo 110Pd 116Cd 124Sn 128130Teand 136Xe to the ground state of the respective daughternuclei In this paper we extend the analysis of [17] to a moredetailed scrutiny of the intermediate contributions to the0]120573minus120573minus decay NMEs of the above-mentioned nuclei We alsoextend the scope of [17] by considering transitions to the first0
+ excited states in addition to the ground-state-to-ground-state transitions
Hindawi Publishing CorporationAdvances in High Energy PhysicsVolume 2016 Article ID 4714829 13 pageshttpdxdoiorg10115520164714829
2 Advances in High Energy Physics
2 Theory Background
In this section a very brief introduction to the computationalframework of the present calculations is given The presentanalyses on ground-state-to-ground-state decays are basedon the calculations done in [17] Details considering theexcited-state decays are given in a future publication Weassume here that the 0]120573minus120573minus decay proceeds via the lightMajorana neutrino so that the inverse half-life can be writtenas
[119905
(0])12
(0
+
119894
997888rarr 0
+
119891
)]
minus1
= 119892
4
A1198660]1003816
1003816
1003816
1003816
1003816
119872
(0])10038161003816
1003816
1003816
1003816
2
1003816
1003816
1003816
1003816
â¨119898]âŠ1003816
1003816
1003816
1003816
2
(1)
where 1198660] is a phase-space factor for the final-state leptons
defined here without the axial vector coupling constant 119892AThe quantity â¨119898]⊠denotes the neutrino effective mass anddescribes the physics beyond the standard model [17] Thequantity119872(0]) is the light neutrino nuclearmatrix element (l-NME) The nuclear matrix element can be decomposed intoGamow-Teller (GT) Fermi (F) and tensor (T) contributionsas
119872
(0])= 119872
(0])GT minus (
119892V119892A
)
2
119872
(0])F +119872
(0])T
(2)
where 119892V is the vector coupling constantEach of the NMEs119870 = GT F and T in (2) can be decom-
posed in terms of the intermediate multipole contributions119869
120587 as
119872
(0])119870
= sum
119869
120587
119872
(0])119870
(119869
120587
) (3)
where each multipole contribution is in turn decomposedin terms of the two-particle transition matrix elements andone-body transition densities as
119872
(0])119870
(119869
120587
) = sum
119896
1119896
2119869
1015840
sum
119901119901
1015840119899119899
1015840
(minus1)
119895
119899+119895
1199011015840+119869+119869
1015840
radic
2119869
1015840
+ 1
times
119895
119901
119895
119899
119869
119895
119899
1015840 119895119901
1015840 119869
1015840
(119901119901
1015840
119869
10158401003817
1003817
1003817
1003817
O119870
1003817
1003817
1003817
1003817
119899119899
1015840
119869
1015840
)
times (0
+
119891
1003817
1003817
1003817
1003817
1003817
1003817
1003817
[119888
dagger
119901
1015840 119899
1015840]
119869
1003817
1003817
1003817
1003817
1003817
1003817
1003817
119869
120587
119896
1
) â¨119869
120587
119896
1
| 119869
120587
119896
2
⊠(119869
120587
119896
2
1003817
1003817
1003817
1003817
1003817
1003817
[119888
dagger
119901
119899
]
119869
1003817
1003817
1003817
1003817
1003817
1003817
0
+
119894
)
(4)
where 1198961
and 119896
2
label the different pnQRPA solutions fora given multipole 119869120587 and the indices 119901 1199011015840 119899 1198991015840 denote theproton and neutron single-particle quantum numbers Theoperators O
119870
inside the two-particle matrix element containthe neutrino potentials for the light Majorana neutrinosthe characteristic two-particle operators for the different 119870= GT F T and a function taking into account the short-range correlations (SRC) between the two decaying neutronsin the mother nucleus of 0]120573minus120573minus decay [17] The final 0+state 0+
119891
can be either the ground state or an excited stateof the 0]120573minus120573minus daughter nucleus and the overlap factorbetween the two one-body transition densities helps connectthe corresponding intermediate 119869120587 states emerging from thepnQRPA calculations in the mother and daughter nuclei
As mentioned before our calculations contain the appro-priate short-range correlators nucleon form factors andhigher-order nucleonic weak currents In addition wedecompose the particle-particle proton-neutron interactionstrength parameter 119892pp of the pnQRPA into its isoscalar(119879 = 0) and isovector (119879 = 1) components and adjustthese components independently as described in [17] theisovector component is fixed such that the NME of thetwo-neutrino double beta-decay (2]120573minus120573minus) vanishes and theisospin symmetry is thus restored for both the 2]120573minus120573minus and0]120573minus120573minus decays The isoscalar component in turn is fixedsuch that the measured half-life of the 2]120573minus120573minus decay isreproduced The resulting values of both components of 119892ppare shown in Table I of [17] The details of the chosen valencespaces and the determination of the other Hamiltonianparameters are presented in [17] We further note that in[17] two sets of NME computations related to the value ofthe axial vector coupling 119892A were performed first with thequenched value 119892A = 100 and then with the bare value 119892A= 126 In both computations the value of 119892A was fixed firstAfter this theHamiltonian parameters were adjusted by usingthe experimental data as briefly described above and morethoroughly in [17]
3 Results and Discussion
In this section we discuss and present the results of ourcalculations Presentation of the results follows top to bottomapproach First we analyze themultipole decompositions andtotal cumulative sums of the matrix elements From thesewe can extract the most important multipole componentsand energy regions contributing to the NMEs After this wecontinue and dissect the most important multipole compo-nents into contributions coming from different individualstates of the 0]120573120573 intermediate nucleus Throughout thesecomputations we have used a conservatively quenched valueof the axial vector coupling 119892A = 100 that is we usethe pnQRPA parameters which are related to the first set ofcomputations in [17] as was explained at the end of Section 2
There has been a lot of discussion about the correctvalue of 119892A in both the 2]120573120573 and 0]120573120573 decays lately Thisis so due to the fact that a large portion of the theoreticalhalf-life uncertainties are related to the present ambiguityin the value of 119892A In [9] the quenching of 119892A was studiedin the framework of IBM-2 and the interacting shell model(ISM) The effective 119892A values were parametrized as 119892effA =
1269119860
minus018 (IBM-2) and as 119892effA = 1269119860
minus012 (ISM) Theseparametrizations were obtained by comparing the modelcalculations with experimental data on 2]120573120573 decays Furtherstudieswere performedwithin the framework of the pnQRPAby using the available Gamow-Teller beta-decay and 2]120573120573decay data in several publications (see [18] and the referencestherein) A wide systematic study of the quenching of 119892Afor Gamow-Teller beta decays was performed in [18] Eventhe quenching related to spin-dipole 2minus states was studiedin [19] While the beta decays and 2]120573120573 decays are low-energy processes with small momentum transfers the 0]120573120573decay involves large momentum transfers and the thus
Advances in High Energy Physics 3
0 1 2 3 4 5 6 7 8 9 10 11 J120587
M(0)(J120587
)
minus05
0
05
1
+minus +minus +minus +minus +minus +minus +minus +minus +minus +minus +minus +minus
(a) 96Zr(0]120573120573 0+gs rarr 0+
gs)
0 1 2 3 4 5 6 7 8 9 10 11 J120587
minus05
0
05
1
M(0)(J120587
)
+minus +minus +minus +minus +minus +minus +minus +minus +minus +minus +minus +minus
(b) 136Xe(0]120573120573 0+gs rarr 0+
gs)
Figure 1 Multipole decomposition of the l-NME for the nuclei 96Zr and 136Xe corresponding to the 0+gs rarr 0
+
gs decay transitions
activated high-energy and high-multipolarity intermediatestates For higher momentum transfers the effective 119892A canbe momentum-dependent [20] and different multipoles canbe affected in different ways At present there exists noknown recipe on how to determine the value of 119892A forthe neutrinoless double beta decays and that is why wehave chosen in the present study to work with a moderatelyquenched value 119892A = 100 assumed to be the same for allintermediate multipoles We will study however the effectof changing the value of 119892A to the characteristics of theintermediate-state contributions in Section 33
31 Ground-State-to-Ground-State Transitions Let us beginby considering the ground-state-to-ground-state decaysmediated by light neutrino exchange In Figures 1(a) and 1(b)we have plotted the multipole decomposition (3) of the l-NMEs corresponding to the 119860 = 96 and 136 nuclear systemsFor most nuclei considered in this work the leading multi-pole component is 1minus This is the case also for the nucleus96Zr shown in Figure 1(a)Most important contribution to theNMEs comes from the lowest multipole components 1plusmn minus4plusmnIt can also be observed that the shape of the overall multipoledistribution is leveled when going towards heavier nucleiThis can be seen by comparing the distribution of 96Zr withthe distribution of 136Xe displayed in Figure 1(b)
Nuclei can be grouped into different types accordingto the shapes of their cumulative NME distributions For0
+
gs rarr 0
+
gs transitions via light neutrino exchange we candifferentiate four types of nuclei Type 1 nuclei belonging tothis type are 76Ge 82Se 96Zr and 128Te Representative of thistype 76Ge is presented in Figure 2(a) Characteristic featureof the cumulative sum distribution belonging to type 1 is thestrong drop in the value of the NME occurring between 12and 17MeV Soon after this drop the NME saturates as canbe seen from panel (a) Type 2 nuclei belonging to this typeare 100Mo and 110Pd Representative of this type 110Pd ispresented in Figure 2(b) Characteristic feature of this typeis the large enhancement and almost immediate cancellationof this enhancement around 10MeV This produces a spike-like structure into the cumulative sum distribution as canbe seen from panel (b) Type 3 nuclei belonging to type 3are 116Cd 124Sn and 130Te Type 3 is represented by 124Sn
shown in Figure 2(c) Characteristic features of this typeare that there occurs neither sharp cancellation of the NMEaround 12ndash17MeV as in type 1 nor a spike like structurearound 10MeV as in type 2 Value of the NME ratherincreases more or less smoothly to its highest value and thensmoothly saturates to its final value around 20MeV Type 4type 4 is special in a sense that it includes only one nucleus136Xe Cumulative sum of the NME for 136Xe is shown inFigure 2(d) Characteristic feature of type 4 is that the lowestenergy region roughly between 0 and 15MeV contributespractically nothing to the value of the NME as can be noticedfrom panel (d)
Using the multipole decompositions we have extractedthe most important multipole components contributing tothe light neutrino mediated ground-state-to-ground-statedecays These most important components can be dividedinto contributions coming from different energy levels of the0]120573120573 intermediate nucleusThese contributions are collectedinto Table 1 for 119860 = 76ndash100 systems into Table 2 for 119860 =
110ndash124 systems and into Table 3 for 119860 = 128ndash136 systemsWe see from the tables that often a very small set of statescollects the largest part of a given multipole contributionto the NMEs Also in some cases notable contributions arecoming from high excitation energies well above 10MeVlike in the case of 1minus contributions for almost all nuclei1
+ contributions for 76Ge 82Se 110Pd 116Cd and 124Sn 2+contributions for 130Te and 136Xe and a 3minus contribution for124Sn
We notice a single-state dominance for the 2
minus modein nuclei 76Ge 82Se and 96Zr In [19] an analysis of theunique first forbidden single 120573plusmn 2minus rarr 0
+ ground-state-to-ground-state transitions in the mass region 119860 = 72ndash132was performed It was found that a strong renormalizationof the axial vector 2minus single 120573matrix elements is needed to beable to explain the experimental transition rates It was thenspeculated that the same kind of an effect may also appearin the 0]120573120573 NMEs This may have a large effect on the 0]120573120573transition rates due to the important contribution of the 2minusmultipole to the 0]120573120573 NMEs
The energies of the intermediate states listed in Tables 12 and 3 (and also those in Tables 4 and 5 for the transitionsto the excited states) originate from pnQRPA calculationsUsually the pnQRPA cannot reproduce the fine details of
4 Advances in High Energy Physics
0
1
2
3
4
5
6
7
8
9
10Cu
mul
ativ
e NM
E co
ntrib
utio
n
0 2010
Energy (MeV)
(a) 76Ge
0
1
2
3
4
5
6
7
8
9
10
Cum
ulat
ive N
ME
cont
ribut
ion
0 2010
Energy (MeV)
(b) 110Pd
10 20 300
Energy (MeV)
0
1
2
3
4
5
6
7
8
9
10
Cum
ulat
ive N
ME
cont
ribut
ion
(c) 124Sn
0
1
2
3
4
5
6
7
8
9
10Cu
mul
ativ
e NM
E co
ntrib
utio
n
0 20 3010
Energy (MeV)
(d) 136Xe
Figure 2 Cumulative values of the computed l-NMEs corresponding to the 0+gs rarr 0
+
gs decay transitions for the nuclear systems 119860 =
76 110 124 and 136 The horizontal axis gives the excitation energies of the intermediate states contributing to the 0]120573120573 transition
0 1 2 3 4 5 6 7 8 9 10 11
0
1
2
3
M(0)(J120587
)
J120587+minus +minus +minus +minus +minus +minus +minus +minus +minus +minus +minus +minus
(a) 76Ge(0]120573120573 0+gs rarr 0+
1
)
+minus0
+minus1
+minus2
+minus3
+minus4
+minus5 6 7 8 9 10 11 J
120587
0
1
2
3
M(0)(J120587
)
+minus +minus +minus +minus +minus +minus
(b) 96Zr(0]120573120573 0+gs rarr 0+
1
)
Figure 3 Multipole decomposition of the l-NME for the nuclei 76Ge and 96Zr corresponding to the 0+gs rarr 0
+
1
decay transitions
Advances in High Energy Physics 5
0
1
2
3
4
5Cu
mul
ativ
e NM
E co
ntrib
utio
n
10 200
Energy (MeV)
(a) 76Ge
0 2010
Energy (MeV)
0
1
2
3
4
5
Cum
ulat
ive N
ME
cont
ribut
ion
(b) 82Se
10 20 300
Energy (MeV)
0
1
2
3
4
5
6
7
Cum
ulat
ive N
ME
cont
ribut
ion
(c) 96Zr
10 20 300
Energy (MeV)
0
1
2
3
4
5
6
7
Cum
ulat
ive N
ME
cont
ribut
ion
(d) 136Xe
10 20 300
Energy (MeV)
0
1
2
3
Cum
ulat
ive N
ME
cont
ribut
ion
(e) 116Cd
Figure 4 Cumulative values of the computed l-NMEs corresponding to the 0+gs rarr 0
+
1
decay transitions for the nuclear systems 119860 =
76 82 96 116 and 136 The horizontal axis gives the excitation energies of the intermediate states contributing to the 0]120573120573 transition
6 Advances in High Energy Physics
Table 1 Most important multipoles and intermediate states contributing to the ground-state-to-ground-state 0]120573120573 decays mediated by thelight neutrino exchange Columns 119864 give the energies (in MeVs) and multipoles of the intermediate states Multipoles are organized from leftto right in terms of their importance themost important being on the left Columns labeled119862 give the correspondingNMEcontributions Lasttwo numbers in each 119862 column give the summed contribution and the percentual part which the displayed states give to the total multipolestrength The percentage inside the parenthesis gives the fraction with which the displayed states contribute to the total NME
Nucleus 119864(2
minus
) 119862 119864(1
minus
) 119862 119864(2
+
) 119862 119864(1
+
) 119862 119864(3
minus
) 119862
76Ge
022 0748 587 0155 051 0166 000 0344 030 0189632 0058 187 0056 409 0199 092 0148700 0077 304 0094 448 0148 677 0056716 0064 362 0108 494 0578 685 0053827 0190 481 0058 1080 minus0053 1163 00621104 0084 773 0054 1175 minus0109 1207 minus00531203 0165 1352 minus05221670 0158
0748 0635 0556 0668 045479 (11) 83 (10) 77 (9) 88 (10) 70 (7)
Nucleus 119864(2
minus
) 119862 119864(2
+
) 119862 119864(1
minus
) 119862 119864(3
minus
) 119862 119864(1
+
) 119862
82Se
000 0510 065 0137 527 0116 007 0140 019 0264173 0065 685 0065 082 0138 316 0095222 0051 798 0134 407 0065356 0084 979 0083 455 0105405 0052 1225 0065 532 0559493 0054 1741 0070 701 minus0253
1453 minus03960510 0442 0393 0278 0439
81 (11) 81 (9) 76 (8) 57 (6) 92 (9)Nucleus 119864(1
minus
) 119862 119864(2
minus
) 119862 119864(3
minus
) 119862 119864(2
+
) 119862 119864(4
+
) 119862
96Zr
175 0056 092 0498 135 0151 064 0150 105 0099228 0063 221 0065 235 0051 163 0114 168 0071252 0150 375 0050 777 0064 569 0064446 0050 443 0052 1136 minus0091504 0077 853 0057527 0209 877 minus0056865 00601133 0061
0728 0666 0175 0265 023483 (16) 96 (15) 31 (4) 59 (6) 69 (5)
Nucleus 119864(1
minus
) 119862 119864(2
+
) 119862 119864(4
+
) 119862 119864(3
+
) 119862 119864(2
minus
) 119862
100Mo
312 0307 090 0192 132 0158 133 0202 176 0205441 0109 210 0154 226 0097 168 0076 285 0147668 0091 1168 0082 743 0152 552 minus00611115 0117 1197 minus0070 776 minus0173 1017 02191672 0058 1093 minus01942026 minus00602390 minus0060
0562 0359 0255 0257 031670 (11) 62 (7) 53 (5) 54 (5) 72 (6)
Advances in High Energy Physics 7
+minus0
+minus1
+minus2
+minus3
+minus4
+minus5
+minus6
+minus7
+minus8
+minus9
+minus10
+minus11
minus05
0
05
1
M(0)(J120587
)
J120587
= 100gA
(a) 76Ge(0]120573120573 0+gs rarr 0+
gs)
+minus0
+minus1
+minus2
+minus3
+minus4
+minus5
+minus6
+minus7
+minus8
+minus9
+minus10
+minus11
minus05
0
05
1
M(0)(J120587
)
J120587
= 126gA
(b) 76Ge(0]120573120573 0+gs rarr 0+
gs)
= 100
+minus0
+minus1
+minus2
+minus3
+minus4
+minus5
+minus6
+minus7
+minus8
+minus9
+minus10
+minus11
minus05
0
05
1
M(0)(J120587
)
J120587
gA
(c) 82Se(0]120573120573 0+gs rarr 0+
1
)
0 1 2 3 4 5 6 7 8 9 10 11
minus05
0
05
1
M(0)(J120587
)
J120587+minus +minus +minus +minus +minus +minus +minus +minus +minus +minus +minus +minus
= 126gA
(d) 82Se(0]120573120573 0+gs rarr 0+
1
)
Figure 5 Multipole decompositions for the ground-state-to-ground-state decay of the nucleus 76Ge ((a) and (b) panels) and for the ground-state-to-excited-state decay of the nucleus 82Se ((c) and (d) panels)The value119892A = 100was used for (a) and (c) panels and the value119892A = 126for (b) and (d) panels
the level structures found in all intermediate odd-odd nucleiconsidered in this work This is due to the general feature ofodd-odd nuclei the extremely high density of states even atlow energies This high density of nuclear states becomes aproblem not only for the pnQRPA but for any other nuclearmany-body approach including the nuclear shell model Thereason for this is that even small perturbations in the two-body interactionmatrix elements tend to change the orderingof the levels at random For this reason the spectra of the odd-odd intermediate nuclei are not a very good measure of thereliability of the calculations but instead a better way is toadjust the model parameters in such a way that the transitionrates of some other known processes for example single or2]120573120573 decays can be reproduced by the theory and this is thephilosophy which we have followed in this work
32 Ground-State-to-Excited-State Decays Let us then con-sider 0+gs rarr 0
+
1
transitions mediated by the light neutrinoexchange In Figures 3(a) and 3(b) we have plotted themultipole decomposition of the l-NMEs corresponding to the119860 = 76 and 96 nuclear systems The multipole distributionsfor the excited-state transitions are greatly different fromthose corresponding to the ground-state transitions Usuallythere is only a couple of multipoles 0+ and 1
+ whichgive by far the largest contribution to the NMEs In thissense the excited-state transitions are more simple than theground-state transitions Typical example is the nucleus 76Gedisplayed in Figure 3(a) One nucleus deviating from this
trend is 96Zr which is presented in Figure 3(b) Its multipoledistribution resembles somewhat more those shown for theground-state decays in Figures 1(a) and 1(b) Most of thisdiffering behaviour can be traced back to the one-phononstructure of the final 0+
1
excited state in the nucleus 96MoThe 0+
1
final states in this work are modeled as one-phononbasic QRPA excitations for the daughter nuclei 96Mo and116Sn Rest of the final states are modeled as two-quadrupole-phonon states Nucleus 96Zr is an exceptional case since the0
+
1
state in 96Mo has a relatively low excitation energy andthus boasts rather strong collective features This is why theexcited-state transition has a wide multipole distribution andis greatly enhanced
Again we can divide nuclei into different groups byconsidering the shapes of their total cumulative sum dis-tributions For 0+gs rarr 0
+
1
transitions via light neutrinoexchange we can differentiate two types of nuclei Type 1nuclei belonging to type 1 are 76Ge 82Se 124Sn 130Te and136Xe Typical examples of this type 76Ge 82Se and 136Xeare shown in Figures 4(a) 4(b) and 4(d) Characteristicfeature of this type is that there exist only few energy stateswhich give most of the total matrix element producing astaircase-like structure as seen in the panels For examplefor 76Ge there seems to be only five such energy statesType 2 nuclei belonging to this type are 96Zr 100Mo 110Pdand 116Cd Typical examples of this type are 96Zr and 116Cdshown in Figures 4(c) and 4(e) Characteristic feature oftype 2 is that a large number of intermediate states give
8 Advances in High Energy Physics
0 20 3010Energy (MeV)
0
1
2
3
4
5
6
7Cu
mul
ativ
e NM
E co
ntrib
utio
n
gA = 100
gA = 126
(a) 100Mo(0]120573120573 0+gs rarr 0+
gs)
0
1
2
3
4
5
6
7
Cum
ulat
ive N
ME
cont
ribut
ion
10 20 300Energy (MeV)
gA = 100
gA = 126
(b) 116Cd(0]120573120573 0+gs rarr 0+
gs)
0
1
2
Cum
ulat
ive N
ME
cont
ribut
ion
10 200Energy (MeV)
gA = 100
gA = 126
(c) 82Se(0]120573120573 0+gs rarr 0+
1
)
0
1
2
3
4
5
6
7
Cum
ulat
ive N
ME
cont
ribut
ion
10 20 300Energy (MeV)
gA = 100
gA = 126
(d) 96Zr(0]120573120573 0+gs rarr 0+
1
)
Figure 6 Cumulative values of the l-NMEs for ground-state-to-ground-state decays of the nuclei 100Mo and 116Cd ((a) and (b) panels) andfor the ground-state-to-excited-state decays of the nuclei 82Se and 96Zr ((c) and (d) panels) The horizontal axis gives the excitation energiesof the intermediate states contributing to the 0]120573120573 transition Two different values for the axial coupling were used as indicated in the panels
important contributions to the NMEs In case of 116Cd panel(e) around 50 of the total NME comes from transitionsthrough the ground state of the intermediate nucleus Theother 50 is distributed rather evenly on the interval 0ndash20MeV
Using the multipole decompositions we extracted themost important multipole components contributing to thelight neutrino mediated 0
+
gs rarr 0
+
1
decay transitionsThese most important components were then again dividedinto contributions coming from different energy levels of
the 0]120573120573 intermediate nucleus These contributions arecollected into Table 4 for119860 = 76ndash116 systems and into Table 5for 119860 = 124ndash136 systems Again we notice that often onlya few intermediate states give the largest contribution to thedominant multipoles 1+ and 0+ Extreme case is the nucleus116Cd forwhich the dominant intermediate ground state gives81 of the total 1+ strength Combining this with the factthat 1+ is by far the largest multipole component we get arather good approximation for the total NME by consideringjust a single virtual transition through the 1+ ground state
Advances in High Energy Physics 9
Table 2 Most important multipoles and intermediate states contributing to the ground-state-to-ground-state 0]120573120573 decays mediated by thelight neutrino exchange Columns 119864 give the energies (in MeVs) and multipoles of the intermediate states Multipoles are organized from leftto right in terms of their importance themost important being on the left Columns labeled119862 give the correspondingNMEcontributions Lasttwo numbers in each 119862 column give the summed contribution and the percentual part which the displayed states give to the total multipolestrength The percentage inside the parenthesis gives the fraction with which the displayed states contribute to the total NME
Nucleus 119864(1
minus
) 119862 119864(2
minus
) 119862 119864(1
+
) 119862 119864(3
minus
) 119862 119864(2
+
) 119862
110Pd
295 0130 082 0387 000 0938 114 0118 033 0244319 0106 247 0091 470 minus0062 164 0128 095 0087344 0221 263 0163 961 0153 191 0080 868 0061381 0426 594 0107 975 minus0146 307 0053 873 0075452 0126 888 0161 1022 0079 361 0075 900 0061911 0109 954 minus0256 1516 minus0316 533 0066 907 0053
1630 0256 808 0167 916 0199837 minus0051 930 minus0366845 0052868 0275911 minus0468
1118 0653 0903 0496 041477 (13) 68 (7) 96 (10) 60 (6) 55 (5)
Nucleus 119864(1
minus
) 119862 119864(3
minus
) 119862 119864(2
minus
) 119862 119864(1
+
) 119862 119864(3
+
) 119862
116Cd
361 0223 172 0081 184 0065 000 0378 090 0158447 0099 230 0053 293 0287 837 minus0102 140 0068537 0118 279 0051 756 0075 951 0087 389 0057587 0140 724 0104 831 0246 1074 minus0083861 0102 834 0103 1120 00812307 0071 845 minus0262 1375 0080
969 minus0051 1379 minus00871573 minus03631583 04401651 minus00821683 0092
0753 0289 0463 0439 028472 (15) 65 (6) 106 (9) 102 (9) 76 (6)
Nucleus 119864(1
minus
) 119862 119864(1
+
) 119862 119864(2
+
) 119862 119864(2
minus
) 119862 119864(3
minus
) 119862
124Sn
168 0522 000 0690 023 0157 052 0271 036 0050482 0089 100 minus0067 060 0083 182 0225 049 0110654 0082 256 0252 106 0056 455 0051 195 0150957 0059 331 0153 215 0066 760 0057 967 00861074 0159 672 minus0130 711 0055 766 minus0064 970 00621407 0065 951 0098 1020 0183 982 minus00871446 minus0099 1309 minus0112 1035 minus0094 1264 minus00531483 minus00581636 minus00871805 minus0067
0664 0884 0417 0629 031855 (8) 95 (11) 53 (5) 89 (8) 49 (4)
of the intermediate nucleus 116In As for the ground-state-to-ground-state decays in some cases notable contributionsare coming from high excitation energies well above 10MeVThere are high-energy contributions in case of 1+ multipolefor all nuclei and in the cases of 2minus and 2+multipoles for 130Teand 136Xe
33 Effects of 119892A on the Intermediate-State Contributions Asmentioned earlier we have used in this work the quenchedvalue for the axial vector coupling 119892A = 100 Next we shallbriefly examine how our results will change if we increasethe value of 119892A from the quenched value 100 to the barevalue 126The effect of this amplification of the axial coupling
10 Advances in High Energy Physics
Table 3 Most important multipoles and intermediate states contributing to the ground-state-to-ground-state 0]120573120573 decays mediated by thelight neutrino exchange Columns 119864 give the energies (in MeVs) and multipoles of the intermediate states Multipoles are organized from leftto right in terms of their importance themost important being on the left Columns labeled119862 give the correspondingNMEcontributions Lasttwo numbers in each 119862 column give the summed contribution and the percentual part which the displayed states give to the total multipolestrength The percentage inside the parenthesis gives the fraction with which the displayed states contribute to the total NME
Nucleus 119864(1
minus
) 119862 119864(2
+
) 119862 119864(3
minus
) 119862 119864(2
minus
) 119862 119864(3
+
) 119862
128Te
422 0200 004 0066 016 0055 061 0335 002 0074472 0060 051 0052 058 0140 402 0060 237 0063621 0078 293 0084 397 0052 455 0101 622 0078644 0059 397 0053 1004 0061 489 minus0070 677 minus0065807 minus0084 677 0050 1014 0136 982 minus0062830 0151 1057 minus0056 1027 minus0057898 0068 1155 00581069 minus00521112 01821748 minus00791919 minus0100
0484 0305 0308 0564 015469 (8) 52 (5) 58 (5) 120 (10) 33 (3)
Nucleus 119864(1
minus
) 119862 119864(2
+
) 119862 119864(3
minus
) 119862 119864(3
+
) 119862 119864(2
minus
) 119862
130Te
418 0184 013 0054 084 0113 010 0056 097 0277572 0059 316 0089 036 0052 1025 0061627 0059 470 0064 260 0056 1133 minus0100856 0096 1049 minus0065 635 00631126 0115 1057 0145 683 00541777 minus0062 1097 minus0088 1231 00591951 minus0079 1644 0150 1241 minus0065
1654 minus01320373 0216 0113 0166 0237
61 (7) 40 (4) 25 (2) 39 (3) 56 (5)Nucleus 119864(2
+
) 119862 119864(3
+
) 119862 119864(1
minus
) 119862 119864(2
minus
) 119862 119864(4
+
) 119862
136Xe
159 0033 135 0033 665 0141 486 0188 234 0032226 0038 239 0054 732 0044 742 0068 244 0061250 0037 481 0050 1031 0053 443 0031529 0067 827 0035 476 0049657 0033 999 0027 767 0034705 0027 1065 minus0027 910 00451411 0087 965 minus00381425 minus0085
0238 0172 0238 0256 021462 (7) 53 (5) 76 (7) 85 (7) 71 (6)
strength on the NMEs is demonstrated in Figure 5 where wehave plotted the multipole decompositions for nuclei 76Geand 82Se calculated with both values of the axial coupling119892A = 100 and 119892A = 126 In case of the ground-state-to-ground-state decays the 1+ multipole changes rather fastwhen the axial coupling is increased from 100 to 126 Thishappens mainly due to the changing of the 119892pp parameter(for each 119892A value the parameter 119892pp is adjusted in sucha way that the measured 2]120573120573 rate is reproduced) The 1+multipole contribution is very sensitive to the value of 119892ppWe can see from Figures 5(a) and 5(b) that for 119892A = 100 the
1
+ component is among the five most important multipoleswhile for 119892A = 126 it is not Some of the higher multipoleschange also somewhat but not so rapidly Ground-state-to-excited-state transitions proceed mainly through the 0+ and1
+multipole channelsWe see from Figures 5(c) and 5(d) thatincreasing the value of 119892A affects mostly the 0+ component
Figure 6 displays the total cumulative sum distributionsfor ground-state-to-ground-state decays of the nuclei 100Moand 116Cd (panels (a) and (b)) and for ground-state-to-excited-state decays of the nuclei 82Se and 96Zr (panels (c)and (d)) Axial coupling values 119892A = 100 and 119892A = 126
Advances in High Energy Physics 11
Table 4 Most important multipoles and intermediate states contributing to the ground-state-to-excited-state 0]120573120573 decays mediated by thelight neutrino exchange Columns 119864 give the energies (in MeVs) and multipoles of the intermediate states Multipoles are organized from leftto right in terms of their importance themost important being on the left Columns labeled119862 give the correspondingNMEcontributions Lasttwo numbers in each 119862 column give the summed contribution and the percentual part which the displayed states give to the total multipolestrength The percentage inside the parenthesis gives the fraction with which the displayed states contribute to the total NME
Nucleus 119864(1
+
) 119862 119864(0
+
) 119862
76Ge
000 0390 286 0053295 0062 428 1204494 0501787 04351352 0522
1909 125692 (50) 99 (33)
Nucleus 119864(1
+
) 119862 119864(0
+
) 119862
82Se
019 0149 545 0471532 0179701 01451453 0274
0746 047190 (41) 96 (26)
Nucleus 119864(1
+
) 119862 119864(2
minus
) 119862 119864(2
+
) 119862 119864(0
+
) 119862 119864(1
minus
) 119862
96Zr
000 0700 092 0283 064 0122 537 0033 228 0032781 0051 966 0057 163 0180 664 0025 252 0080899 0058 537 0035 732 0037 446 00261165 0065 692 0027 802 0036 527 00681617 0354 1012 0396 1133 0026
1200 00131228 0340 0364 0540 0232
91 (23) 60 (6) 65 (7) 98 (10) 59 (4)Nucleus 119864(1
+
) 119862 119864(0
+
) 119862
100Mo
942 0090 901 00761362 0079 955 01501431 00721512 0063
0303 022665 (40) 77 (30)
Nucleus 119864(1
+
) 119862 119864(0
+
) 119862
110Pd
000 0308 1026 01651022 00591373 00831456 00551516 0072
0577 016576 (53) 70 (15)
Nucleus 119864(1
+
) 119862
116Cd000 0735
073581 (48)
12 Advances in High Energy Physics
Table 5 Most important multipoles and intermediate states contributing to the ground-state-to-excited-state 0]120573120573 decays mediated by thelight neutrino exchange Columns 119864 give the energies (in MeVs) and multipoles of the intermediate states Multipoles are organized from leftto right in terms of their importance themost important being on the left Columns labeled119862 give the correspondingNMEcontributions Lasttwo numbers in each 119862 column give the summed contribution and the percentual part which the displayed states give to the total multipolestrength The percentage inside the parenthesis gives the fraction with which the displayed states contribute to the total NME
Nucleus 119864(1
+
) 119862 119864(0
+
) 119862
124Sn
000 0101 270 0667066 0433 460 0088100 0146 735 0567225 0051256 0227331 0664672 0289791 04241309 04731386 0161
2968 132294 (44) 94 (19)
Nucleus 119864(1
+
) 119862 119864(0
+
) 119862 119864(2
minus
) 119862 119864(2
+
) 119862
130Te
025 0390 740 0719 097 0340 041 0019150 0082 835 0631 507 0073 059 0047232 0097 1726 0106 183 0017276 0134 1895 0083 280 0043453 0064 470 0024559 0338 486 0027757 0592 518 00331466 0065 627 00161501 0259 675 00511507 0773 910 00321539 0089 1057 0045
1787 00262314 0024
2882 1351 0602 039095 (41) 996 (19) 79 (9) 65 (6)
Nucleus 119864(1
+
) 119862 119864(0
+
) 119862 119864(2
minus
) 119862 119864(2
+
) 119862
136Xe
230 0366 1224 0317 250 0060 486 0204306 0050 1254 0712 486 0056 1858 0083375 0066 529 0113 2034 0059446 0226 834 00531012 0232 1411 00541152 05651712 01371778 1031
2675 1029 0335 034693 (43) 101 (16) 53 (5) 67 (6)
Advances in High Energy Physics 13
were adopted We notice from the figures that increasing theaxial coupling strength shifts the distributions downwardsThis is especially true for the higher energy parts Despite thisfact the overall shapes of the cumulative sum distributionsdo not changemuch and the same classification of nuclei intodifferent categories according to their cumulative distributionshapes seems to hold also for larger values of 119892A
4 Conclusions
In this paper we have extended our previous work [17] onthe ground-state-to-ground-state 0]120573120573 decay transitions Inthe present work we have concentrated our studies on theintermediate contributions to the NMEs involved in thelight neutrino mediated 0]120573120573 decay We have calculated theintermediate state multipole decompositions of the NMEsand extracted the most important multipole componentsCumulative sums of the NMEs were calculated to investigatethe important energy regions contributing to the 0]120573120573 tran-sitions Finally the most important multipole componentswere divided into contributions coming from the virtualtransitions through the individual states of the 0]120573120573 inter-mediate nuclei An extensive tabulation of these importantintermediate states were given for all the nuclei considered inthis paper
We have done these computations by using realistic two-body interactions and single-particle bases All the appro-priate short-range correlations nucleon form factors andhigher-order nucleonic weak currents are included in ourpresent results
We found in the calculations that often there exists onlya few relevant intermediate states which collect most of thestrength corresponding to a given multipole We also foundthat there exists a single-state dominance in the important 2minuscomponents related to the ground-state decays of nuclei 76Ge82Se and perhaps also for 96Zr
Competing Interests
The authors declare that they have no competing interests
Acknowledgments
This work has been partially supported by the Academy ofFinland under the Finnish Centre of Excellence Programme2012ndash2017 (Nuclear and Accelerator Based Programme atJYFL)
References
[1] J Suhonen and O Civitarese ldquoWeak-interaction and nuclear-structure aspects of nuclear double beta decayrdquo Physics Reportvol 300 no 3-4 pp 123ndash214 1998
[2] F T Avignone S R Elliott and J Engel ldquoDouble beta decayMajorana neutrinos and neutrino massrdquo Reviews of ModernPhysics vol 80 no 2 pp 481ndash516 2008
[3] J D Vergados H Ejiri and F Simkovic ldquoTheory of neutrinolessdouble-beta decayrdquoReports on Progress in Physics vol 75 no 10p 106301 2012
[4] J Maalampi and J Suhonen ldquoNeutrinoless double 120573
+ECdecaysrdquo Advances in High Energy Physics vol 2013 Article ID505874 18 pages 2013
[5] M Kortelainen O Civitarese J Suhonen and J ToivanenldquoShort-range correlations and neutrinoless double beta decayrdquoPhysics Letters Section B Nuclear Elementary Particle andHigh-Energy Physics vol 647 no 2-3 pp 128ndash132 2007
[6] F Simkovic G Pantis J D Vergados and A Faessler ldquoAddi-tional nucleon current contributions to neutrinoless double 120573decayrdquo Physical Review C vol 60 no 5 Article ID 055502 14pages 1999
[7] J Suhonen and O Civitarese ldquoReview of the properties of the0]120573minus120573minus nuclear matrix elementsrdquo Journal of Physics G Nuclearand Particle Physics vol 39 no 12 Article ID 124005 2012
[8] P Vogel ldquoNuclear structure and double beta decayrdquo Journal ofPhysics G Nuclear and Particle Physics vol 39 no 12 p 1240022012
[9] J Barea J Kotila and F Iachello ldquoNuclear matrix elements fordouble-120573 decayrdquo Physical Review C vol 87 no 1 Article ID014315 2013
[10] A Neacsu and S Stoica ldquoStudy of nuclear effects in thecomputation of the 0]120573120573 decay matrix elementsrdquo Journal ofPhysics G Nuclear and Particle Physics vol 41 no 1 Article ID015201 2014
[11] J Engel ldquoUncertainties in nuclear matrix elements for neutri-noless double-beta decayrdquo Journal of Physics G Nuclear andParticle Physics vol 42 no 3 Article ID 034017 2015
[12] F Simkovic A Faessler V Rodin P Vogel and J EngelldquoAnatomyof the 0]120573120573nuclearmatrix elementsrdquoPhysical ReviewC Nuclear Physics vol 77 no 4 Article ID 045503 2008
[13] M T Mustonen and J Engel ldquoLarge-scale calculations ofthe double-120573 decay of 76Ge 130Te 136Xe and 150Nd in thedeformed self-consistent Skyrme quasiparticle random-phaseapproximationrdquo Physical Review C vol 87 Article ID 0643022013
[14] F Simkovic V Rodin A Faessler and P Vogel ldquo0]120573120573 and2]120573120573 nuclear matrix elements quasiparticle random-phaseapproximation and isospin symmetry restorationrdquo PhysicalReview C vol 87 no 4 Article ID 045501 9 pages 2013
[15] V A Rodin A Faessler F Simkovic and P Vogel ldquoAssessmentof uncertainties in QRPA 0]120573120573-decay nuclear matrix elementsrdquoNuclear Physics A vol 766 pp 107ndash131 2006
[16] D-L Fang A Faessler V Rodin and F Simkovic ldquoNeutri-noless double-120573 decay of deformed nuclei within quasiparticlerandom-phase approximationwith a realistic interactionrdquoPhys-ical Review C vol 83 no 3 Article ID 034320 8 pages 2011
[17] J Hyvarinen and J Suhonen ldquoNuclear matrix elements for0]120573120573 decays with light or heavy Majorana-neutrino exchangerdquoPhysical Review C vol 91 no 2 Article ID 024613 12 pages2015
[18] P Pirinen and J Suhonen ldquoSystematic approach to 120573 and 2]120573120573decays of mass119860 = 100minus136 nucleirdquo Physical Review C vol 91no 5 Article ID 054309 2015
[19] H Ejiri N Soukouti and J Suhonen ldquoSpin-dipole nuclearmatrix elements for double beta decays and astro-neutrinosrdquoPhysics Letters B vol 729 pp 27ndash32 2014
[20] J Menendez D Gazit and A Schwenk ldquoChiral two-bodycurrents in nuclei Gamow-Teller transitions and neutrinolessdouble-beta decayrdquo Physical Review Letters vol 107 no 6Article ID 062501 5 pages 2011
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AstronomyAdvances in
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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Superconductivity
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Statistical MechanicsInternational Journal of
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âComputationalââMethodsâinâPhysics
Journal of
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PhotonicsJournal of
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Journal of
Biophysics
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ThermodynamicsJournal of
2 Advances in High Energy Physics
2 Theory Background
In this section a very brief introduction to the computationalframework of the present calculations is given The presentanalyses on ground-state-to-ground-state decays are basedon the calculations done in [17] Details considering theexcited-state decays are given in a future publication Weassume here that the 0]120573minus120573minus decay proceeds via the lightMajorana neutrino so that the inverse half-life can be writtenas
[119905
(0])12
(0
+
119894
997888rarr 0
+
119891
)]
minus1
= 119892
4
A1198660]1003816
1003816
1003816
1003816
1003816
119872
(0])10038161003816
1003816
1003816
1003816
2
1003816
1003816
1003816
1003816
â¨119898]âŠ1003816
1003816
1003816
1003816
2
(1)
where 1198660] is a phase-space factor for the final-state leptons
defined here without the axial vector coupling constant 119892AThe quantity â¨119898]⊠denotes the neutrino effective mass anddescribes the physics beyond the standard model [17] Thequantity119872(0]) is the light neutrino nuclearmatrix element (l-NME) The nuclear matrix element can be decomposed intoGamow-Teller (GT) Fermi (F) and tensor (T) contributionsas
119872
(0])= 119872
(0])GT minus (
119892V119892A
)
2
119872
(0])F +119872
(0])T
(2)
where 119892V is the vector coupling constantEach of the NMEs119870 = GT F and T in (2) can be decom-
posed in terms of the intermediate multipole contributions119869
120587 as
119872
(0])119870
= sum
119869
120587
119872
(0])119870
(119869
120587
) (3)
where each multipole contribution is in turn decomposedin terms of the two-particle transition matrix elements andone-body transition densities as
119872
(0])119870
(119869
120587
) = sum
119896
1119896
2119869
1015840
sum
119901119901
1015840119899119899
1015840
(minus1)
119895
119899+119895
1199011015840+119869+119869
1015840
radic
2119869
1015840
+ 1
times
119895
119901
119895
119899
119869
119895
119899
1015840 119895119901
1015840 119869
1015840
(119901119901
1015840
119869
10158401003817
1003817
1003817
1003817
O119870
1003817
1003817
1003817
1003817
119899119899
1015840
119869
1015840
)
times (0
+
119891
1003817
1003817
1003817
1003817
1003817
1003817
1003817
[119888
dagger
119901
1015840 119899
1015840]
119869
1003817
1003817
1003817
1003817
1003817
1003817
1003817
119869
120587
119896
1
) â¨119869
120587
119896
1
| 119869
120587
119896
2
⊠(119869
120587
119896
2
1003817
1003817
1003817
1003817
1003817
1003817
[119888
dagger
119901
119899
]
119869
1003817
1003817
1003817
1003817
1003817
1003817
0
+
119894
)
(4)
where 1198961
and 119896
2
label the different pnQRPA solutions fora given multipole 119869120587 and the indices 119901 1199011015840 119899 1198991015840 denote theproton and neutron single-particle quantum numbers Theoperators O
119870
inside the two-particle matrix element containthe neutrino potentials for the light Majorana neutrinosthe characteristic two-particle operators for the different 119870= GT F T and a function taking into account the short-range correlations (SRC) between the two decaying neutronsin the mother nucleus of 0]120573minus120573minus decay [17] The final 0+state 0+
119891
can be either the ground state or an excited stateof the 0]120573minus120573minus daughter nucleus and the overlap factorbetween the two one-body transition densities helps connectthe corresponding intermediate 119869120587 states emerging from thepnQRPA calculations in the mother and daughter nuclei
As mentioned before our calculations contain the appro-priate short-range correlators nucleon form factors andhigher-order nucleonic weak currents In addition wedecompose the particle-particle proton-neutron interactionstrength parameter 119892pp of the pnQRPA into its isoscalar(119879 = 0) and isovector (119879 = 1) components and adjustthese components independently as described in [17] theisovector component is fixed such that the NME of thetwo-neutrino double beta-decay (2]120573minus120573minus) vanishes and theisospin symmetry is thus restored for both the 2]120573minus120573minus and0]120573minus120573minus decays The isoscalar component in turn is fixedsuch that the measured half-life of the 2]120573minus120573minus decay isreproduced The resulting values of both components of 119892ppare shown in Table I of [17] The details of the chosen valencespaces and the determination of the other Hamiltonianparameters are presented in [17] We further note that in[17] two sets of NME computations related to the value ofthe axial vector coupling 119892A were performed first with thequenched value 119892A = 100 and then with the bare value 119892A= 126 In both computations the value of 119892A was fixed firstAfter this theHamiltonian parameters were adjusted by usingthe experimental data as briefly described above and morethoroughly in [17]
3 Results and Discussion
In this section we discuss and present the results of ourcalculations Presentation of the results follows top to bottomapproach First we analyze themultipole decompositions andtotal cumulative sums of the matrix elements From thesewe can extract the most important multipole componentsand energy regions contributing to the NMEs After this wecontinue and dissect the most important multipole compo-nents into contributions coming from different individualstates of the 0]120573120573 intermediate nucleus Throughout thesecomputations we have used a conservatively quenched valueof the axial vector coupling 119892A = 100 that is we usethe pnQRPA parameters which are related to the first set ofcomputations in [17] as was explained at the end of Section 2
There has been a lot of discussion about the correctvalue of 119892A in both the 2]120573120573 and 0]120573120573 decays lately Thisis so due to the fact that a large portion of the theoreticalhalf-life uncertainties are related to the present ambiguityin the value of 119892A In [9] the quenching of 119892A was studiedin the framework of IBM-2 and the interacting shell model(ISM) The effective 119892A values were parametrized as 119892effA =
1269119860
minus018 (IBM-2) and as 119892effA = 1269119860
minus012 (ISM) Theseparametrizations were obtained by comparing the modelcalculations with experimental data on 2]120573120573 decays Furtherstudieswere performedwithin the framework of the pnQRPAby using the available Gamow-Teller beta-decay and 2]120573120573decay data in several publications (see [18] and the referencestherein) A wide systematic study of the quenching of 119892Afor Gamow-Teller beta decays was performed in [18] Eventhe quenching related to spin-dipole 2minus states was studiedin [19] While the beta decays and 2]120573120573 decays are low-energy processes with small momentum transfers the 0]120573120573decay involves large momentum transfers and the thus
Advances in High Energy Physics 3
0 1 2 3 4 5 6 7 8 9 10 11 J120587
M(0)(J120587
)
minus05
0
05
1
+minus +minus +minus +minus +minus +minus +minus +minus +minus +minus +minus +minus
(a) 96Zr(0]120573120573 0+gs rarr 0+
gs)
0 1 2 3 4 5 6 7 8 9 10 11 J120587
minus05
0
05
1
M(0)(J120587
)
+minus +minus +minus +minus +minus +minus +minus +minus +minus +minus +minus +minus
(b) 136Xe(0]120573120573 0+gs rarr 0+
gs)
Figure 1 Multipole decomposition of the l-NME for the nuclei 96Zr and 136Xe corresponding to the 0+gs rarr 0
+
gs decay transitions
activated high-energy and high-multipolarity intermediatestates For higher momentum transfers the effective 119892A canbe momentum-dependent [20] and different multipoles canbe affected in different ways At present there exists noknown recipe on how to determine the value of 119892A forthe neutrinoless double beta decays and that is why wehave chosen in the present study to work with a moderatelyquenched value 119892A = 100 assumed to be the same for allintermediate multipoles We will study however the effectof changing the value of 119892A to the characteristics of theintermediate-state contributions in Section 33
31 Ground-State-to-Ground-State Transitions Let us beginby considering the ground-state-to-ground-state decaysmediated by light neutrino exchange In Figures 1(a) and 1(b)we have plotted the multipole decomposition (3) of the l-NMEs corresponding to the 119860 = 96 and 136 nuclear systemsFor most nuclei considered in this work the leading multi-pole component is 1minus This is the case also for the nucleus96Zr shown in Figure 1(a)Most important contribution to theNMEs comes from the lowest multipole components 1plusmn minus4plusmnIt can also be observed that the shape of the overall multipoledistribution is leveled when going towards heavier nucleiThis can be seen by comparing the distribution of 96Zr withthe distribution of 136Xe displayed in Figure 1(b)
Nuclei can be grouped into different types accordingto the shapes of their cumulative NME distributions For0
+
gs rarr 0
+
gs transitions via light neutrino exchange we candifferentiate four types of nuclei Type 1 nuclei belonging tothis type are 76Ge 82Se 96Zr and 128Te Representative of thistype 76Ge is presented in Figure 2(a) Characteristic featureof the cumulative sum distribution belonging to type 1 is thestrong drop in the value of the NME occurring between 12and 17MeV Soon after this drop the NME saturates as canbe seen from panel (a) Type 2 nuclei belonging to this typeare 100Mo and 110Pd Representative of this type 110Pd ispresented in Figure 2(b) Characteristic feature of this typeis the large enhancement and almost immediate cancellationof this enhancement around 10MeV This produces a spike-like structure into the cumulative sum distribution as canbe seen from panel (b) Type 3 nuclei belonging to type 3are 116Cd 124Sn and 130Te Type 3 is represented by 124Sn
shown in Figure 2(c) Characteristic features of this typeare that there occurs neither sharp cancellation of the NMEaround 12ndash17MeV as in type 1 nor a spike like structurearound 10MeV as in type 2 Value of the NME ratherincreases more or less smoothly to its highest value and thensmoothly saturates to its final value around 20MeV Type 4type 4 is special in a sense that it includes only one nucleus136Xe Cumulative sum of the NME for 136Xe is shown inFigure 2(d) Characteristic feature of type 4 is that the lowestenergy region roughly between 0 and 15MeV contributespractically nothing to the value of the NME as can be noticedfrom panel (d)
Using the multipole decompositions we have extractedthe most important multipole components contributing tothe light neutrino mediated ground-state-to-ground-statedecays These most important components can be dividedinto contributions coming from different energy levels of the0]120573120573 intermediate nucleusThese contributions are collectedinto Table 1 for 119860 = 76ndash100 systems into Table 2 for 119860 =
110ndash124 systems and into Table 3 for 119860 = 128ndash136 systemsWe see from the tables that often a very small set of statescollects the largest part of a given multipole contributionto the NMEs Also in some cases notable contributions arecoming from high excitation energies well above 10MeVlike in the case of 1minus contributions for almost all nuclei1
+ contributions for 76Ge 82Se 110Pd 116Cd and 124Sn 2+contributions for 130Te and 136Xe and a 3minus contribution for124Sn
We notice a single-state dominance for the 2
minus modein nuclei 76Ge 82Se and 96Zr In [19] an analysis of theunique first forbidden single 120573plusmn 2minus rarr 0
+ ground-state-to-ground-state transitions in the mass region 119860 = 72ndash132was performed It was found that a strong renormalizationof the axial vector 2minus single 120573matrix elements is needed to beable to explain the experimental transition rates It was thenspeculated that the same kind of an effect may also appearin the 0]120573120573 NMEs This may have a large effect on the 0]120573120573transition rates due to the important contribution of the 2minusmultipole to the 0]120573120573 NMEs
The energies of the intermediate states listed in Tables 12 and 3 (and also those in Tables 4 and 5 for the transitionsto the excited states) originate from pnQRPA calculationsUsually the pnQRPA cannot reproduce the fine details of
4 Advances in High Energy Physics
0
1
2
3
4
5
6
7
8
9
10Cu
mul
ativ
e NM
E co
ntrib
utio
n
0 2010
Energy (MeV)
(a) 76Ge
0
1
2
3
4
5
6
7
8
9
10
Cum
ulat
ive N
ME
cont
ribut
ion
0 2010
Energy (MeV)
(b) 110Pd
10 20 300
Energy (MeV)
0
1
2
3
4
5
6
7
8
9
10
Cum
ulat
ive N
ME
cont
ribut
ion
(c) 124Sn
0
1
2
3
4
5
6
7
8
9
10Cu
mul
ativ
e NM
E co
ntrib
utio
n
0 20 3010
Energy (MeV)
(d) 136Xe
Figure 2 Cumulative values of the computed l-NMEs corresponding to the 0+gs rarr 0
+
gs decay transitions for the nuclear systems 119860 =
76 110 124 and 136 The horizontal axis gives the excitation energies of the intermediate states contributing to the 0]120573120573 transition
0 1 2 3 4 5 6 7 8 9 10 11
0
1
2
3
M(0)(J120587
)
J120587+minus +minus +minus +minus +minus +minus +minus +minus +minus +minus +minus +minus
(a) 76Ge(0]120573120573 0+gs rarr 0+
1
)
+minus0
+minus1
+minus2
+minus3
+minus4
+minus5 6 7 8 9 10 11 J
120587
0
1
2
3
M(0)(J120587
)
+minus +minus +minus +minus +minus +minus
(b) 96Zr(0]120573120573 0+gs rarr 0+
1
)
Figure 3 Multipole decomposition of the l-NME for the nuclei 76Ge and 96Zr corresponding to the 0+gs rarr 0
+
1
decay transitions
Advances in High Energy Physics 5
0
1
2
3
4
5Cu
mul
ativ
e NM
E co
ntrib
utio
n
10 200
Energy (MeV)
(a) 76Ge
0 2010
Energy (MeV)
0
1
2
3
4
5
Cum
ulat
ive N
ME
cont
ribut
ion
(b) 82Se
10 20 300
Energy (MeV)
0
1
2
3
4
5
6
7
Cum
ulat
ive N
ME
cont
ribut
ion
(c) 96Zr
10 20 300
Energy (MeV)
0
1
2
3
4
5
6
7
Cum
ulat
ive N
ME
cont
ribut
ion
(d) 136Xe
10 20 300
Energy (MeV)
0
1
2
3
Cum
ulat
ive N
ME
cont
ribut
ion
(e) 116Cd
Figure 4 Cumulative values of the computed l-NMEs corresponding to the 0+gs rarr 0
+
1
decay transitions for the nuclear systems 119860 =
76 82 96 116 and 136 The horizontal axis gives the excitation energies of the intermediate states contributing to the 0]120573120573 transition
6 Advances in High Energy Physics
Table 1 Most important multipoles and intermediate states contributing to the ground-state-to-ground-state 0]120573120573 decays mediated by thelight neutrino exchange Columns 119864 give the energies (in MeVs) and multipoles of the intermediate states Multipoles are organized from leftto right in terms of their importance themost important being on the left Columns labeled119862 give the correspondingNMEcontributions Lasttwo numbers in each 119862 column give the summed contribution and the percentual part which the displayed states give to the total multipolestrength The percentage inside the parenthesis gives the fraction with which the displayed states contribute to the total NME
Nucleus 119864(2
minus
) 119862 119864(1
minus
) 119862 119864(2
+
) 119862 119864(1
+
) 119862 119864(3
minus
) 119862
76Ge
022 0748 587 0155 051 0166 000 0344 030 0189632 0058 187 0056 409 0199 092 0148700 0077 304 0094 448 0148 677 0056716 0064 362 0108 494 0578 685 0053827 0190 481 0058 1080 minus0053 1163 00621104 0084 773 0054 1175 minus0109 1207 minus00531203 0165 1352 minus05221670 0158
0748 0635 0556 0668 045479 (11) 83 (10) 77 (9) 88 (10) 70 (7)
Nucleus 119864(2
minus
) 119862 119864(2
+
) 119862 119864(1
minus
) 119862 119864(3
minus
) 119862 119864(1
+
) 119862
82Se
000 0510 065 0137 527 0116 007 0140 019 0264173 0065 685 0065 082 0138 316 0095222 0051 798 0134 407 0065356 0084 979 0083 455 0105405 0052 1225 0065 532 0559493 0054 1741 0070 701 minus0253
1453 minus03960510 0442 0393 0278 0439
81 (11) 81 (9) 76 (8) 57 (6) 92 (9)Nucleus 119864(1
minus
) 119862 119864(2
minus
) 119862 119864(3
minus
) 119862 119864(2
+
) 119862 119864(4
+
) 119862
96Zr
175 0056 092 0498 135 0151 064 0150 105 0099228 0063 221 0065 235 0051 163 0114 168 0071252 0150 375 0050 777 0064 569 0064446 0050 443 0052 1136 minus0091504 0077 853 0057527 0209 877 minus0056865 00601133 0061
0728 0666 0175 0265 023483 (16) 96 (15) 31 (4) 59 (6) 69 (5)
Nucleus 119864(1
minus
) 119862 119864(2
+
) 119862 119864(4
+
) 119862 119864(3
+
) 119862 119864(2
minus
) 119862
100Mo
312 0307 090 0192 132 0158 133 0202 176 0205441 0109 210 0154 226 0097 168 0076 285 0147668 0091 1168 0082 743 0152 552 minus00611115 0117 1197 minus0070 776 minus0173 1017 02191672 0058 1093 minus01942026 minus00602390 minus0060
0562 0359 0255 0257 031670 (11) 62 (7) 53 (5) 54 (5) 72 (6)
Advances in High Energy Physics 7
+minus0
+minus1
+minus2
+minus3
+minus4
+minus5
+minus6
+minus7
+minus8
+minus9
+minus10
+minus11
minus05
0
05
1
M(0)(J120587
)
J120587
= 100gA
(a) 76Ge(0]120573120573 0+gs rarr 0+
gs)
+minus0
+minus1
+minus2
+minus3
+minus4
+minus5
+minus6
+minus7
+minus8
+minus9
+minus10
+minus11
minus05
0
05
1
M(0)(J120587
)
J120587
= 126gA
(b) 76Ge(0]120573120573 0+gs rarr 0+
gs)
= 100
+minus0
+minus1
+minus2
+minus3
+minus4
+minus5
+minus6
+minus7
+minus8
+minus9
+minus10
+minus11
minus05
0
05
1
M(0)(J120587
)
J120587
gA
(c) 82Se(0]120573120573 0+gs rarr 0+
1
)
0 1 2 3 4 5 6 7 8 9 10 11
minus05
0
05
1
M(0)(J120587
)
J120587+minus +minus +minus +minus +minus +minus +minus +minus +minus +minus +minus +minus
= 126gA
(d) 82Se(0]120573120573 0+gs rarr 0+
1
)
Figure 5 Multipole decompositions for the ground-state-to-ground-state decay of the nucleus 76Ge ((a) and (b) panels) and for the ground-state-to-excited-state decay of the nucleus 82Se ((c) and (d) panels)The value119892A = 100was used for (a) and (c) panels and the value119892A = 126for (b) and (d) panels
the level structures found in all intermediate odd-odd nucleiconsidered in this work This is due to the general feature ofodd-odd nuclei the extremely high density of states even atlow energies This high density of nuclear states becomes aproblem not only for the pnQRPA but for any other nuclearmany-body approach including the nuclear shell model Thereason for this is that even small perturbations in the two-body interactionmatrix elements tend to change the orderingof the levels at random For this reason the spectra of the odd-odd intermediate nuclei are not a very good measure of thereliability of the calculations but instead a better way is toadjust the model parameters in such a way that the transitionrates of some other known processes for example single or2]120573120573 decays can be reproduced by the theory and this is thephilosophy which we have followed in this work
32 Ground-State-to-Excited-State Decays Let us then con-sider 0+gs rarr 0
+
1
transitions mediated by the light neutrinoexchange In Figures 3(a) and 3(b) we have plotted themultipole decomposition of the l-NMEs corresponding to the119860 = 76 and 96 nuclear systems The multipole distributionsfor the excited-state transitions are greatly different fromthose corresponding to the ground-state transitions Usuallythere is only a couple of multipoles 0+ and 1
+ whichgive by far the largest contribution to the NMEs In thissense the excited-state transitions are more simple than theground-state transitions Typical example is the nucleus 76Gedisplayed in Figure 3(a) One nucleus deviating from this
trend is 96Zr which is presented in Figure 3(b) Its multipoledistribution resembles somewhat more those shown for theground-state decays in Figures 1(a) and 1(b) Most of thisdiffering behaviour can be traced back to the one-phononstructure of the final 0+
1
excited state in the nucleus 96MoThe 0+
1
final states in this work are modeled as one-phononbasic QRPA excitations for the daughter nuclei 96Mo and116Sn Rest of the final states are modeled as two-quadrupole-phonon states Nucleus 96Zr is an exceptional case since the0
+
1
state in 96Mo has a relatively low excitation energy andthus boasts rather strong collective features This is why theexcited-state transition has a wide multipole distribution andis greatly enhanced
Again we can divide nuclei into different groups byconsidering the shapes of their total cumulative sum dis-tributions For 0+gs rarr 0
+
1
transitions via light neutrinoexchange we can differentiate two types of nuclei Type 1nuclei belonging to type 1 are 76Ge 82Se 124Sn 130Te and136Xe Typical examples of this type 76Ge 82Se and 136Xeare shown in Figures 4(a) 4(b) and 4(d) Characteristicfeature of this type is that there exist only few energy stateswhich give most of the total matrix element producing astaircase-like structure as seen in the panels For examplefor 76Ge there seems to be only five such energy statesType 2 nuclei belonging to this type are 96Zr 100Mo 110Pdand 116Cd Typical examples of this type are 96Zr and 116Cdshown in Figures 4(c) and 4(e) Characteristic feature oftype 2 is that a large number of intermediate states give
8 Advances in High Energy Physics
0 20 3010Energy (MeV)
0
1
2
3
4
5
6
7Cu
mul
ativ
e NM
E co
ntrib
utio
n
gA = 100
gA = 126
(a) 100Mo(0]120573120573 0+gs rarr 0+
gs)
0
1
2
3
4
5
6
7
Cum
ulat
ive N
ME
cont
ribut
ion
10 20 300Energy (MeV)
gA = 100
gA = 126
(b) 116Cd(0]120573120573 0+gs rarr 0+
gs)
0
1
2
Cum
ulat
ive N
ME
cont
ribut
ion
10 200Energy (MeV)
gA = 100
gA = 126
(c) 82Se(0]120573120573 0+gs rarr 0+
1
)
0
1
2
3
4
5
6
7
Cum
ulat
ive N
ME
cont
ribut
ion
10 20 300Energy (MeV)
gA = 100
gA = 126
(d) 96Zr(0]120573120573 0+gs rarr 0+
1
)
Figure 6 Cumulative values of the l-NMEs for ground-state-to-ground-state decays of the nuclei 100Mo and 116Cd ((a) and (b) panels) andfor the ground-state-to-excited-state decays of the nuclei 82Se and 96Zr ((c) and (d) panels) The horizontal axis gives the excitation energiesof the intermediate states contributing to the 0]120573120573 transition Two different values for the axial coupling were used as indicated in the panels
important contributions to the NMEs In case of 116Cd panel(e) around 50 of the total NME comes from transitionsthrough the ground state of the intermediate nucleus Theother 50 is distributed rather evenly on the interval 0ndash20MeV
Using the multipole decompositions we extracted themost important multipole components contributing to thelight neutrino mediated 0
+
gs rarr 0
+
1
decay transitionsThese most important components were then again dividedinto contributions coming from different energy levels of
the 0]120573120573 intermediate nucleus These contributions arecollected into Table 4 for119860 = 76ndash116 systems and into Table 5for 119860 = 124ndash136 systems Again we notice that often onlya few intermediate states give the largest contribution to thedominant multipoles 1+ and 0+ Extreme case is the nucleus116Cd forwhich the dominant intermediate ground state gives81 of the total 1+ strength Combining this with the factthat 1+ is by far the largest multipole component we get arather good approximation for the total NME by consideringjust a single virtual transition through the 1+ ground state
Advances in High Energy Physics 9
Table 2 Most important multipoles and intermediate states contributing to the ground-state-to-ground-state 0]120573120573 decays mediated by thelight neutrino exchange Columns 119864 give the energies (in MeVs) and multipoles of the intermediate states Multipoles are organized from leftto right in terms of their importance themost important being on the left Columns labeled119862 give the correspondingNMEcontributions Lasttwo numbers in each 119862 column give the summed contribution and the percentual part which the displayed states give to the total multipolestrength The percentage inside the parenthesis gives the fraction with which the displayed states contribute to the total NME
Nucleus 119864(1
minus
) 119862 119864(2
minus
) 119862 119864(1
+
) 119862 119864(3
minus
) 119862 119864(2
+
) 119862
110Pd
295 0130 082 0387 000 0938 114 0118 033 0244319 0106 247 0091 470 minus0062 164 0128 095 0087344 0221 263 0163 961 0153 191 0080 868 0061381 0426 594 0107 975 minus0146 307 0053 873 0075452 0126 888 0161 1022 0079 361 0075 900 0061911 0109 954 minus0256 1516 minus0316 533 0066 907 0053
1630 0256 808 0167 916 0199837 minus0051 930 minus0366845 0052868 0275911 minus0468
1118 0653 0903 0496 041477 (13) 68 (7) 96 (10) 60 (6) 55 (5)
Nucleus 119864(1
minus
) 119862 119864(3
minus
) 119862 119864(2
minus
) 119862 119864(1
+
) 119862 119864(3
+
) 119862
116Cd
361 0223 172 0081 184 0065 000 0378 090 0158447 0099 230 0053 293 0287 837 minus0102 140 0068537 0118 279 0051 756 0075 951 0087 389 0057587 0140 724 0104 831 0246 1074 minus0083861 0102 834 0103 1120 00812307 0071 845 minus0262 1375 0080
969 minus0051 1379 minus00871573 minus03631583 04401651 minus00821683 0092
0753 0289 0463 0439 028472 (15) 65 (6) 106 (9) 102 (9) 76 (6)
Nucleus 119864(1
minus
) 119862 119864(1
+
) 119862 119864(2
+
) 119862 119864(2
minus
) 119862 119864(3
minus
) 119862
124Sn
168 0522 000 0690 023 0157 052 0271 036 0050482 0089 100 minus0067 060 0083 182 0225 049 0110654 0082 256 0252 106 0056 455 0051 195 0150957 0059 331 0153 215 0066 760 0057 967 00861074 0159 672 minus0130 711 0055 766 minus0064 970 00621407 0065 951 0098 1020 0183 982 minus00871446 minus0099 1309 minus0112 1035 minus0094 1264 minus00531483 minus00581636 minus00871805 minus0067
0664 0884 0417 0629 031855 (8) 95 (11) 53 (5) 89 (8) 49 (4)
of the intermediate nucleus 116In As for the ground-state-to-ground-state decays in some cases notable contributionsare coming from high excitation energies well above 10MeVThere are high-energy contributions in case of 1+ multipolefor all nuclei and in the cases of 2minus and 2+multipoles for 130Teand 136Xe
33 Effects of 119892A on the Intermediate-State Contributions Asmentioned earlier we have used in this work the quenchedvalue for the axial vector coupling 119892A = 100 Next we shallbriefly examine how our results will change if we increasethe value of 119892A from the quenched value 100 to the barevalue 126The effect of this amplification of the axial coupling
10 Advances in High Energy Physics
Table 3 Most important multipoles and intermediate states contributing to the ground-state-to-ground-state 0]120573120573 decays mediated by thelight neutrino exchange Columns 119864 give the energies (in MeVs) and multipoles of the intermediate states Multipoles are organized from leftto right in terms of their importance themost important being on the left Columns labeled119862 give the correspondingNMEcontributions Lasttwo numbers in each 119862 column give the summed contribution and the percentual part which the displayed states give to the total multipolestrength The percentage inside the parenthesis gives the fraction with which the displayed states contribute to the total NME
Nucleus 119864(1
minus
) 119862 119864(2
+
) 119862 119864(3
minus
) 119862 119864(2
minus
) 119862 119864(3
+
) 119862
128Te
422 0200 004 0066 016 0055 061 0335 002 0074472 0060 051 0052 058 0140 402 0060 237 0063621 0078 293 0084 397 0052 455 0101 622 0078644 0059 397 0053 1004 0061 489 minus0070 677 minus0065807 minus0084 677 0050 1014 0136 982 minus0062830 0151 1057 minus0056 1027 minus0057898 0068 1155 00581069 minus00521112 01821748 minus00791919 minus0100
0484 0305 0308 0564 015469 (8) 52 (5) 58 (5) 120 (10) 33 (3)
Nucleus 119864(1
minus
) 119862 119864(2
+
) 119862 119864(3
minus
) 119862 119864(3
+
) 119862 119864(2
minus
) 119862
130Te
418 0184 013 0054 084 0113 010 0056 097 0277572 0059 316 0089 036 0052 1025 0061627 0059 470 0064 260 0056 1133 minus0100856 0096 1049 minus0065 635 00631126 0115 1057 0145 683 00541777 minus0062 1097 minus0088 1231 00591951 minus0079 1644 0150 1241 minus0065
1654 minus01320373 0216 0113 0166 0237
61 (7) 40 (4) 25 (2) 39 (3) 56 (5)Nucleus 119864(2
+
) 119862 119864(3
+
) 119862 119864(1
minus
) 119862 119864(2
minus
) 119862 119864(4
+
) 119862
136Xe
159 0033 135 0033 665 0141 486 0188 234 0032226 0038 239 0054 732 0044 742 0068 244 0061250 0037 481 0050 1031 0053 443 0031529 0067 827 0035 476 0049657 0033 999 0027 767 0034705 0027 1065 minus0027 910 00451411 0087 965 minus00381425 minus0085
0238 0172 0238 0256 021462 (7) 53 (5) 76 (7) 85 (7) 71 (6)
strength on the NMEs is demonstrated in Figure 5 where wehave plotted the multipole decompositions for nuclei 76Geand 82Se calculated with both values of the axial coupling119892A = 100 and 119892A = 126 In case of the ground-state-to-ground-state decays the 1+ multipole changes rather fastwhen the axial coupling is increased from 100 to 126 Thishappens mainly due to the changing of the 119892pp parameter(for each 119892A value the parameter 119892pp is adjusted in sucha way that the measured 2]120573120573 rate is reproduced) The 1+multipole contribution is very sensitive to the value of 119892ppWe can see from Figures 5(a) and 5(b) that for 119892A = 100 the
1
+ component is among the five most important multipoleswhile for 119892A = 126 it is not Some of the higher multipoleschange also somewhat but not so rapidly Ground-state-to-excited-state transitions proceed mainly through the 0+ and1
+multipole channelsWe see from Figures 5(c) and 5(d) thatincreasing the value of 119892A affects mostly the 0+ component
Figure 6 displays the total cumulative sum distributionsfor ground-state-to-ground-state decays of the nuclei 100Moand 116Cd (panels (a) and (b)) and for ground-state-to-excited-state decays of the nuclei 82Se and 96Zr (panels (c)and (d)) Axial coupling values 119892A = 100 and 119892A = 126
Advances in High Energy Physics 11
Table 4 Most important multipoles and intermediate states contributing to the ground-state-to-excited-state 0]120573120573 decays mediated by thelight neutrino exchange Columns 119864 give the energies (in MeVs) and multipoles of the intermediate states Multipoles are organized from leftto right in terms of their importance themost important being on the left Columns labeled119862 give the correspondingNMEcontributions Lasttwo numbers in each 119862 column give the summed contribution and the percentual part which the displayed states give to the total multipolestrength The percentage inside the parenthesis gives the fraction with which the displayed states contribute to the total NME
Nucleus 119864(1
+
) 119862 119864(0
+
) 119862
76Ge
000 0390 286 0053295 0062 428 1204494 0501787 04351352 0522
1909 125692 (50) 99 (33)
Nucleus 119864(1
+
) 119862 119864(0
+
) 119862
82Se
019 0149 545 0471532 0179701 01451453 0274
0746 047190 (41) 96 (26)
Nucleus 119864(1
+
) 119862 119864(2
minus
) 119862 119864(2
+
) 119862 119864(0
+
) 119862 119864(1
minus
) 119862
96Zr
000 0700 092 0283 064 0122 537 0033 228 0032781 0051 966 0057 163 0180 664 0025 252 0080899 0058 537 0035 732 0037 446 00261165 0065 692 0027 802 0036 527 00681617 0354 1012 0396 1133 0026
1200 00131228 0340 0364 0540 0232
91 (23) 60 (6) 65 (7) 98 (10) 59 (4)Nucleus 119864(1
+
) 119862 119864(0
+
) 119862
100Mo
942 0090 901 00761362 0079 955 01501431 00721512 0063
0303 022665 (40) 77 (30)
Nucleus 119864(1
+
) 119862 119864(0
+
) 119862
110Pd
000 0308 1026 01651022 00591373 00831456 00551516 0072
0577 016576 (53) 70 (15)
Nucleus 119864(1
+
) 119862
116Cd000 0735
073581 (48)
12 Advances in High Energy Physics
Table 5 Most important multipoles and intermediate states contributing to the ground-state-to-excited-state 0]120573120573 decays mediated by thelight neutrino exchange Columns 119864 give the energies (in MeVs) and multipoles of the intermediate states Multipoles are organized from leftto right in terms of their importance themost important being on the left Columns labeled119862 give the correspondingNMEcontributions Lasttwo numbers in each 119862 column give the summed contribution and the percentual part which the displayed states give to the total multipolestrength The percentage inside the parenthesis gives the fraction with which the displayed states contribute to the total NME
Nucleus 119864(1
+
) 119862 119864(0
+
) 119862
124Sn
000 0101 270 0667066 0433 460 0088100 0146 735 0567225 0051256 0227331 0664672 0289791 04241309 04731386 0161
2968 132294 (44) 94 (19)
Nucleus 119864(1
+
) 119862 119864(0
+
) 119862 119864(2
minus
) 119862 119864(2
+
) 119862
130Te
025 0390 740 0719 097 0340 041 0019150 0082 835 0631 507 0073 059 0047232 0097 1726 0106 183 0017276 0134 1895 0083 280 0043453 0064 470 0024559 0338 486 0027757 0592 518 00331466 0065 627 00161501 0259 675 00511507 0773 910 00321539 0089 1057 0045
1787 00262314 0024
2882 1351 0602 039095 (41) 996 (19) 79 (9) 65 (6)
Nucleus 119864(1
+
) 119862 119864(0
+
) 119862 119864(2
minus
) 119862 119864(2
+
) 119862
136Xe
230 0366 1224 0317 250 0060 486 0204306 0050 1254 0712 486 0056 1858 0083375 0066 529 0113 2034 0059446 0226 834 00531012 0232 1411 00541152 05651712 01371778 1031
2675 1029 0335 034693 (43) 101 (16) 53 (5) 67 (6)
Advances in High Energy Physics 13
were adopted We notice from the figures that increasing theaxial coupling strength shifts the distributions downwardsThis is especially true for the higher energy parts Despite thisfact the overall shapes of the cumulative sum distributionsdo not changemuch and the same classification of nuclei intodifferent categories according to their cumulative distributionshapes seems to hold also for larger values of 119892A
4 Conclusions
In this paper we have extended our previous work [17] onthe ground-state-to-ground-state 0]120573120573 decay transitions Inthe present work we have concentrated our studies on theintermediate contributions to the NMEs involved in thelight neutrino mediated 0]120573120573 decay We have calculated theintermediate state multipole decompositions of the NMEsand extracted the most important multipole componentsCumulative sums of the NMEs were calculated to investigatethe important energy regions contributing to the 0]120573120573 tran-sitions Finally the most important multipole componentswere divided into contributions coming from the virtualtransitions through the individual states of the 0]120573120573 inter-mediate nuclei An extensive tabulation of these importantintermediate states were given for all the nuclei considered inthis paper
We have done these computations by using realistic two-body interactions and single-particle bases All the appro-priate short-range correlations nucleon form factors andhigher-order nucleonic weak currents are included in ourpresent results
We found in the calculations that often there exists onlya few relevant intermediate states which collect most of thestrength corresponding to a given multipole We also foundthat there exists a single-state dominance in the important 2minuscomponents related to the ground-state decays of nuclei 76Ge82Se and perhaps also for 96Zr
Competing Interests
The authors declare that they have no competing interests
Acknowledgments
This work has been partially supported by the Academy ofFinland under the Finnish Centre of Excellence Programme2012ndash2017 (Nuclear and Accelerator Based Programme atJYFL)
References
[1] J Suhonen and O Civitarese ldquoWeak-interaction and nuclear-structure aspects of nuclear double beta decayrdquo Physics Reportvol 300 no 3-4 pp 123ndash214 1998
[2] F T Avignone S R Elliott and J Engel ldquoDouble beta decayMajorana neutrinos and neutrino massrdquo Reviews of ModernPhysics vol 80 no 2 pp 481ndash516 2008
[3] J D Vergados H Ejiri and F Simkovic ldquoTheory of neutrinolessdouble-beta decayrdquoReports on Progress in Physics vol 75 no 10p 106301 2012
[4] J Maalampi and J Suhonen ldquoNeutrinoless double 120573
+ECdecaysrdquo Advances in High Energy Physics vol 2013 Article ID505874 18 pages 2013
[5] M Kortelainen O Civitarese J Suhonen and J ToivanenldquoShort-range correlations and neutrinoless double beta decayrdquoPhysics Letters Section B Nuclear Elementary Particle andHigh-Energy Physics vol 647 no 2-3 pp 128ndash132 2007
[6] F Simkovic G Pantis J D Vergados and A Faessler ldquoAddi-tional nucleon current contributions to neutrinoless double 120573decayrdquo Physical Review C vol 60 no 5 Article ID 055502 14pages 1999
[7] J Suhonen and O Civitarese ldquoReview of the properties of the0]120573minus120573minus nuclear matrix elementsrdquo Journal of Physics G Nuclearand Particle Physics vol 39 no 12 Article ID 124005 2012
[8] P Vogel ldquoNuclear structure and double beta decayrdquo Journal ofPhysics G Nuclear and Particle Physics vol 39 no 12 p 1240022012
[9] J Barea J Kotila and F Iachello ldquoNuclear matrix elements fordouble-120573 decayrdquo Physical Review C vol 87 no 1 Article ID014315 2013
[10] A Neacsu and S Stoica ldquoStudy of nuclear effects in thecomputation of the 0]120573120573 decay matrix elementsrdquo Journal ofPhysics G Nuclear and Particle Physics vol 41 no 1 Article ID015201 2014
[11] J Engel ldquoUncertainties in nuclear matrix elements for neutri-noless double-beta decayrdquo Journal of Physics G Nuclear andParticle Physics vol 42 no 3 Article ID 034017 2015
[12] F Simkovic A Faessler V Rodin P Vogel and J EngelldquoAnatomyof the 0]120573120573nuclearmatrix elementsrdquoPhysical ReviewC Nuclear Physics vol 77 no 4 Article ID 045503 2008
[13] M T Mustonen and J Engel ldquoLarge-scale calculations ofthe double-120573 decay of 76Ge 130Te 136Xe and 150Nd in thedeformed self-consistent Skyrme quasiparticle random-phaseapproximationrdquo Physical Review C vol 87 Article ID 0643022013
[14] F Simkovic V Rodin A Faessler and P Vogel ldquo0]120573120573 and2]120573120573 nuclear matrix elements quasiparticle random-phaseapproximation and isospin symmetry restorationrdquo PhysicalReview C vol 87 no 4 Article ID 045501 9 pages 2013
[15] V A Rodin A Faessler F Simkovic and P Vogel ldquoAssessmentof uncertainties in QRPA 0]120573120573-decay nuclear matrix elementsrdquoNuclear Physics A vol 766 pp 107ndash131 2006
[16] D-L Fang A Faessler V Rodin and F Simkovic ldquoNeutri-noless double-120573 decay of deformed nuclei within quasiparticlerandom-phase approximationwith a realistic interactionrdquoPhys-ical Review C vol 83 no 3 Article ID 034320 8 pages 2011
[17] J Hyvarinen and J Suhonen ldquoNuclear matrix elements for0]120573120573 decays with light or heavy Majorana-neutrino exchangerdquoPhysical Review C vol 91 no 2 Article ID 024613 12 pages2015
[18] P Pirinen and J Suhonen ldquoSystematic approach to 120573 and 2]120573120573decays of mass119860 = 100minus136 nucleirdquo Physical Review C vol 91no 5 Article ID 054309 2015
[19] H Ejiri N Soukouti and J Suhonen ldquoSpin-dipole nuclearmatrix elements for double beta decays and astro-neutrinosrdquoPhysics Letters B vol 729 pp 27ndash32 2014
[20] J Menendez D Gazit and A Schwenk ldquoChiral two-bodycurrents in nuclei Gamow-Teller transitions and neutrinolessdouble-beta decayrdquo Physical Review Letters vol 107 no 6Article ID 062501 5 pages 2011
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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FluidsJournal of
Atomic and Molecular Physics
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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Advances in Condensed Matter Physics
OpticsInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
AstronomyAdvances in
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Superconductivity
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Statistical MechanicsInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
GravityJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
AstrophysicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Physics Research International
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Solid State PhysicsJournal of
âComputationalââMethodsâinâPhysics
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Soft MatterJournal of
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AerodynamicsJournal of
Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
PhotonicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Biophysics
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
ThermodynamicsJournal of
Advances in High Energy Physics 3
0 1 2 3 4 5 6 7 8 9 10 11 J120587
M(0)(J120587
)
minus05
0
05
1
+minus +minus +minus +minus +minus +minus +minus +minus +minus +minus +minus +minus
(a) 96Zr(0]120573120573 0+gs rarr 0+
gs)
0 1 2 3 4 5 6 7 8 9 10 11 J120587
minus05
0
05
1
M(0)(J120587
)
+minus +minus +minus +minus +minus +minus +minus +minus +minus +minus +minus +minus
(b) 136Xe(0]120573120573 0+gs rarr 0+
gs)
Figure 1 Multipole decomposition of the l-NME for the nuclei 96Zr and 136Xe corresponding to the 0+gs rarr 0
+
gs decay transitions
activated high-energy and high-multipolarity intermediatestates For higher momentum transfers the effective 119892A canbe momentum-dependent [20] and different multipoles canbe affected in different ways At present there exists noknown recipe on how to determine the value of 119892A forthe neutrinoless double beta decays and that is why wehave chosen in the present study to work with a moderatelyquenched value 119892A = 100 assumed to be the same for allintermediate multipoles We will study however the effectof changing the value of 119892A to the characteristics of theintermediate-state contributions in Section 33
31 Ground-State-to-Ground-State Transitions Let us beginby considering the ground-state-to-ground-state decaysmediated by light neutrino exchange In Figures 1(a) and 1(b)we have plotted the multipole decomposition (3) of the l-NMEs corresponding to the 119860 = 96 and 136 nuclear systemsFor most nuclei considered in this work the leading multi-pole component is 1minus This is the case also for the nucleus96Zr shown in Figure 1(a)Most important contribution to theNMEs comes from the lowest multipole components 1plusmn minus4plusmnIt can also be observed that the shape of the overall multipoledistribution is leveled when going towards heavier nucleiThis can be seen by comparing the distribution of 96Zr withthe distribution of 136Xe displayed in Figure 1(b)
Nuclei can be grouped into different types accordingto the shapes of their cumulative NME distributions For0
+
gs rarr 0
+
gs transitions via light neutrino exchange we candifferentiate four types of nuclei Type 1 nuclei belonging tothis type are 76Ge 82Se 96Zr and 128Te Representative of thistype 76Ge is presented in Figure 2(a) Characteristic featureof the cumulative sum distribution belonging to type 1 is thestrong drop in the value of the NME occurring between 12and 17MeV Soon after this drop the NME saturates as canbe seen from panel (a) Type 2 nuclei belonging to this typeare 100Mo and 110Pd Representative of this type 110Pd ispresented in Figure 2(b) Characteristic feature of this typeis the large enhancement and almost immediate cancellationof this enhancement around 10MeV This produces a spike-like structure into the cumulative sum distribution as canbe seen from panel (b) Type 3 nuclei belonging to type 3are 116Cd 124Sn and 130Te Type 3 is represented by 124Sn
shown in Figure 2(c) Characteristic features of this typeare that there occurs neither sharp cancellation of the NMEaround 12ndash17MeV as in type 1 nor a spike like structurearound 10MeV as in type 2 Value of the NME ratherincreases more or less smoothly to its highest value and thensmoothly saturates to its final value around 20MeV Type 4type 4 is special in a sense that it includes only one nucleus136Xe Cumulative sum of the NME for 136Xe is shown inFigure 2(d) Characteristic feature of type 4 is that the lowestenergy region roughly between 0 and 15MeV contributespractically nothing to the value of the NME as can be noticedfrom panel (d)
Using the multipole decompositions we have extractedthe most important multipole components contributing tothe light neutrino mediated ground-state-to-ground-statedecays These most important components can be dividedinto contributions coming from different energy levels of the0]120573120573 intermediate nucleusThese contributions are collectedinto Table 1 for 119860 = 76ndash100 systems into Table 2 for 119860 =
110ndash124 systems and into Table 3 for 119860 = 128ndash136 systemsWe see from the tables that often a very small set of statescollects the largest part of a given multipole contributionto the NMEs Also in some cases notable contributions arecoming from high excitation energies well above 10MeVlike in the case of 1minus contributions for almost all nuclei1
+ contributions for 76Ge 82Se 110Pd 116Cd and 124Sn 2+contributions for 130Te and 136Xe and a 3minus contribution for124Sn
We notice a single-state dominance for the 2
minus modein nuclei 76Ge 82Se and 96Zr In [19] an analysis of theunique first forbidden single 120573plusmn 2minus rarr 0
+ ground-state-to-ground-state transitions in the mass region 119860 = 72ndash132was performed It was found that a strong renormalizationof the axial vector 2minus single 120573matrix elements is needed to beable to explain the experimental transition rates It was thenspeculated that the same kind of an effect may also appearin the 0]120573120573 NMEs This may have a large effect on the 0]120573120573transition rates due to the important contribution of the 2minusmultipole to the 0]120573120573 NMEs
The energies of the intermediate states listed in Tables 12 and 3 (and also those in Tables 4 and 5 for the transitionsto the excited states) originate from pnQRPA calculationsUsually the pnQRPA cannot reproduce the fine details of
4 Advances in High Energy Physics
0
1
2
3
4
5
6
7
8
9
10Cu
mul
ativ
e NM
E co
ntrib
utio
n
0 2010
Energy (MeV)
(a) 76Ge
0
1
2
3
4
5
6
7
8
9
10
Cum
ulat
ive N
ME
cont
ribut
ion
0 2010
Energy (MeV)
(b) 110Pd
10 20 300
Energy (MeV)
0
1
2
3
4
5
6
7
8
9
10
Cum
ulat
ive N
ME
cont
ribut
ion
(c) 124Sn
0
1
2
3
4
5
6
7
8
9
10Cu
mul
ativ
e NM
E co
ntrib
utio
n
0 20 3010
Energy (MeV)
(d) 136Xe
Figure 2 Cumulative values of the computed l-NMEs corresponding to the 0+gs rarr 0
+
gs decay transitions for the nuclear systems 119860 =
76 110 124 and 136 The horizontal axis gives the excitation energies of the intermediate states contributing to the 0]120573120573 transition
0 1 2 3 4 5 6 7 8 9 10 11
0
1
2
3
M(0)(J120587
)
J120587+minus +minus +minus +minus +minus +minus +minus +minus +minus +minus +minus +minus
(a) 76Ge(0]120573120573 0+gs rarr 0+
1
)
+minus0
+minus1
+minus2
+minus3
+minus4
+minus5 6 7 8 9 10 11 J
120587
0
1
2
3
M(0)(J120587
)
+minus +minus +minus +minus +minus +minus
(b) 96Zr(0]120573120573 0+gs rarr 0+
1
)
Figure 3 Multipole decomposition of the l-NME for the nuclei 76Ge and 96Zr corresponding to the 0+gs rarr 0
+
1
decay transitions
Advances in High Energy Physics 5
0
1
2
3
4
5Cu
mul
ativ
e NM
E co
ntrib
utio
n
10 200
Energy (MeV)
(a) 76Ge
0 2010
Energy (MeV)
0
1
2
3
4
5
Cum
ulat
ive N
ME
cont
ribut
ion
(b) 82Se
10 20 300
Energy (MeV)
0
1
2
3
4
5
6
7
Cum
ulat
ive N
ME
cont
ribut
ion
(c) 96Zr
10 20 300
Energy (MeV)
0
1
2
3
4
5
6
7
Cum
ulat
ive N
ME
cont
ribut
ion
(d) 136Xe
10 20 300
Energy (MeV)
0
1
2
3
Cum
ulat
ive N
ME
cont
ribut
ion
(e) 116Cd
Figure 4 Cumulative values of the computed l-NMEs corresponding to the 0+gs rarr 0
+
1
decay transitions for the nuclear systems 119860 =
76 82 96 116 and 136 The horizontal axis gives the excitation energies of the intermediate states contributing to the 0]120573120573 transition
6 Advances in High Energy Physics
Table 1 Most important multipoles and intermediate states contributing to the ground-state-to-ground-state 0]120573120573 decays mediated by thelight neutrino exchange Columns 119864 give the energies (in MeVs) and multipoles of the intermediate states Multipoles are organized from leftto right in terms of their importance themost important being on the left Columns labeled119862 give the correspondingNMEcontributions Lasttwo numbers in each 119862 column give the summed contribution and the percentual part which the displayed states give to the total multipolestrength The percentage inside the parenthesis gives the fraction with which the displayed states contribute to the total NME
Nucleus 119864(2
minus
) 119862 119864(1
minus
) 119862 119864(2
+
) 119862 119864(1
+
) 119862 119864(3
minus
) 119862
76Ge
022 0748 587 0155 051 0166 000 0344 030 0189632 0058 187 0056 409 0199 092 0148700 0077 304 0094 448 0148 677 0056716 0064 362 0108 494 0578 685 0053827 0190 481 0058 1080 minus0053 1163 00621104 0084 773 0054 1175 minus0109 1207 minus00531203 0165 1352 minus05221670 0158
0748 0635 0556 0668 045479 (11) 83 (10) 77 (9) 88 (10) 70 (7)
Nucleus 119864(2
minus
) 119862 119864(2
+
) 119862 119864(1
minus
) 119862 119864(3
minus
) 119862 119864(1
+
) 119862
82Se
000 0510 065 0137 527 0116 007 0140 019 0264173 0065 685 0065 082 0138 316 0095222 0051 798 0134 407 0065356 0084 979 0083 455 0105405 0052 1225 0065 532 0559493 0054 1741 0070 701 minus0253
1453 minus03960510 0442 0393 0278 0439
81 (11) 81 (9) 76 (8) 57 (6) 92 (9)Nucleus 119864(1
minus
) 119862 119864(2
minus
) 119862 119864(3
minus
) 119862 119864(2
+
) 119862 119864(4
+
) 119862
96Zr
175 0056 092 0498 135 0151 064 0150 105 0099228 0063 221 0065 235 0051 163 0114 168 0071252 0150 375 0050 777 0064 569 0064446 0050 443 0052 1136 minus0091504 0077 853 0057527 0209 877 minus0056865 00601133 0061
0728 0666 0175 0265 023483 (16) 96 (15) 31 (4) 59 (6) 69 (5)
Nucleus 119864(1
minus
) 119862 119864(2
+
) 119862 119864(4
+
) 119862 119864(3
+
) 119862 119864(2
minus
) 119862
100Mo
312 0307 090 0192 132 0158 133 0202 176 0205441 0109 210 0154 226 0097 168 0076 285 0147668 0091 1168 0082 743 0152 552 minus00611115 0117 1197 minus0070 776 minus0173 1017 02191672 0058 1093 minus01942026 minus00602390 minus0060
0562 0359 0255 0257 031670 (11) 62 (7) 53 (5) 54 (5) 72 (6)
Advances in High Energy Physics 7
+minus0
+minus1
+minus2
+minus3
+minus4
+minus5
+minus6
+minus7
+minus8
+minus9
+minus10
+minus11
minus05
0
05
1
M(0)(J120587
)
J120587
= 100gA
(a) 76Ge(0]120573120573 0+gs rarr 0+
gs)
+minus0
+minus1
+minus2
+minus3
+minus4
+minus5
+minus6
+minus7
+minus8
+minus9
+minus10
+minus11
minus05
0
05
1
M(0)(J120587
)
J120587
= 126gA
(b) 76Ge(0]120573120573 0+gs rarr 0+
gs)
= 100
+minus0
+minus1
+minus2
+minus3
+minus4
+minus5
+minus6
+minus7
+minus8
+minus9
+minus10
+minus11
minus05
0
05
1
M(0)(J120587
)
J120587
gA
(c) 82Se(0]120573120573 0+gs rarr 0+
1
)
0 1 2 3 4 5 6 7 8 9 10 11
minus05
0
05
1
M(0)(J120587
)
J120587+minus +minus +minus +minus +minus +minus +minus +minus +minus +minus +minus +minus
= 126gA
(d) 82Se(0]120573120573 0+gs rarr 0+
1
)
Figure 5 Multipole decompositions for the ground-state-to-ground-state decay of the nucleus 76Ge ((a) and (b) panels) and for the ground-state-to-excited-state decay of the nucleus 82Se ((c) and (d) panels)The value119892A = 100was used for (a) and (c) panels and the value119892A = 126for (b) and (d) panels
the level structures found in all intermediate odd-odd nucleiconsidered in this work This is due to the general feature ofodd-odd nuclei the extremely high density of states even atlow energies This high density of nuclear states becomes aproblem not only for the pnQRPA but for any other nuclearmany-body approach including the nuclear shell model Thereason for this is that even small perturbations in the two-body interactionmatrix elements tend to change the orderingof the levels at random For this reason the spectra of the odd-odd intermediate nuclei are not a very good measure of thereliability of the calculations but instead a better way is toadjust the model parameters in such a way that the transitionrates of some other known processes for example single or2]120573120573 decays can be reproduced by the theory and this is thephilosophy which we have followed in this work
32 Ground-State-to-Excited-State Decays Let us then con-sider 0+gs rarr 0
+
1
transitions mediated by the light neutrinoexchange In Figures 3(a) and 3(b) we have plotted themultipole decomposition of the l-NMEs corresponding to the119860 = 76 and 96 nuclear systems The multipole distributionsfor the excited-state transitions are greatly different fromthose corresponding to the ground-state transitions Usuallythere is only a couple of multipoles 0+ and 1
+ whichgive by far the largest contribution to the NMEs In thissense the excited-state transitions are more simple than theground-state transitions Typical example is the nucleus 76Gedisplayed in Figure 3(a) One nucleus deviating from this
trend is 96Zr which is presented in Figure 3(b) Its multipoledistribution resembles somewhat more those shown for theground-state decays in Figures 1(a) and 1(b) Most of thisdiffering behaviour can be traced back to the one-phononstructure of the final 0+
1
excited state in the nucleus 96MoThe 0+
1
final states in this work are modeled as one-phononbasic QRPA excitations for the daughter nuclei 96Mo and116Sn Rest of the final states are modeled as two-quadrupole-phonon states Nucleus 96Zr is an exceptional case since the0
+
1
state in 96Mo has a relatively low excitation energy andthus boasts rather strong collective features This is why theexcited-state transition has a wide multipole distribution andis greatly enhanced
Again we can divide nuclei into different groups byconsidering the shapes of their total cumulative sum dis-tributions For 0+gs rarr 0
+
1
transitions via light neutrinoexchange we can differentiate two types of nuclei Type 1nuclei belonging to type 1 are 76Ge 82Se 124Sn 130Te and136Xe Typical examples of this type 76Ge 82Se and 136Xeare shown in Figures 4(a) 4(b) and 4(d) Characteristicfeature of this type is that there exist only few energy stateswhich give most of the total matrix element producing astaircase-like structure as seen in the panels For examplefor 76Ge there seems to be only five such energy statesType 2 nuclei belonging to this type are 96Zr 100Mo 110Pdand 116Cd Typical examples of this type are 96Zr and 116Cdshown in Figures 4(c) and 4(e) Characteristic feature oftype 2 is that a large number of intermediate states give
8 Advances in High Energy Physics
0 20 3010Energy (MeV)
0
1
2
3
4
5
6
7Cu
mul
ativ
e NM
E co
ntrib
utio
n
gA = 100
gA = 126
(a) 100Mo(0]120573120573 0+gs rarr 0+
gs)
0
1
2
3
4
5
6
7
Cum
ulat
ive N
ME
cont
ribut
ion
10 20 300Energy (MeV)
gA = 100
gA = 126
(b) 116Cd(0]120573120573 0+gs rarr 0+
gs)
0
1
2
Cum
ulat
ive N
ME
cont
ribut
ion
10 200Energy (MeV)
gA = 100
gA = 126
(c) 82Se(0]120573120573 0+gs rarr 0+
1
)
0
1
2
3
4
5
6
7
Cum
ulat
ive N
ME
cont
ribut
ion
10 20 300Energy (MeV)
gA = 100
gA = 126
(d) 96Zr(0]120573120573 0+gs rarr 0+
1
)
Figure 6 Cumulative values of the l-NMEs for ground-state-to-ground-state decays of the nuclei 100Mo and 116Cd ((a) and (b) panels) andfor the ground-state-to-excited-state decays of the nuclei 82Se and 96Zr ((c) and (d) panels) The horizontal axis gives the excitation energiesof the intermediate states contributing to the 0]120573120573 transition Two different values for the axial coupling were used as indicated in the panels
important contributions to the NMEs In case of 116Cd panel(e) around 50 of the total NME comes from transitionsthrough the ground state of the intermediate nucleus Theother 50 is distributed rather evenly on the interval 0ndash20MeV
Using the multipole decompositions we extracted themost important multipole components contributing to thelight neutrino mediated 0
+
gs rarr 0
+
1
decay transitionsThese most important components were then again dividedinto contributions coming from different energy levels of
the 0]120573120573 intermediate nucleus These contributions arecollected into Table 4 for119860 = 76ndash116 systems and into Table 5for 119860 = 124ndash136 systems Again we notice that often onlya few intermediate states give the largest contribution to thedominant multipoles 1+ and 0+ Extreme case is the nucleus116Cd forwhich the dominant intermediate ground state gives81 of the total 1+ strength Combining this with the factthat 1+ is by far the largest multipole component we get arather good approximation for the total NME by consideringjust a single virtual transition through the 1+ ground state
Advances in High Energy Physics 9
Table 2 Most important multipoles and intermediate states contributing to the ground-state-to-ground-state 0]120573120573 decays mediated by thelight neutrino exchange Columns 119864 give the energies (in MeVs) and multipoles of the intermediate states Multipoles are organized from leftto right in terms of their importance themost important being on the left Columns labeled119862 give the correspondingNMEcontributions Lasttwo numbers in each 119862 column give the summed contribution and the percentual part which the displayed states give to the total multipolestrength The percentage inside the parenthesis gives the fraction with which the displayed states contribute to the total NME
Nucleus 119864(1
minus
) 119862 119864(2
minus
) 119862 119864(1
+
) 119862 119864(3
minus
) 119862 119864(2
+
) 119862
110Pd
295 0130 082 0387 000 0938 114 0118 033 0244319 0106 247 0091 470 minus0062 164 0128 095 0087344 0221 263 0163 961 0153 191 0080 868 0061381 0426 594 0107 975 minus0146 307 0053 873 0075452 0126 888 0161 1022 0079 361 0075 900 0061911 0109 954 minus0256 1516 minus0316 533 0066 907 0053
1630 0256 808 0167 916 0199837 minus0051 930 minus0366845 0052868 0275911 minus0468
1118 0653 0903 0496 041477 (13) 68 (7) 96 (10) 60 (6) 55 (5)
Nucleus 119864(1
minus
) 119862 119864(3
minus
) 119862 119864(2
minus
) 119862 119864(1
+
) 119862 119864(3
+
) 119862
116Cd
361 0223 172 0081 184 0065 000 0378 090 0158447 0099 230 0053 293 0287 837 minus0102 140 0068537 0118 279 0051 756 0075 951 0087 389 0057587 0140 724 0104 831 0246 1074 minus0083861 0102 834 0103 1120 00812307 0071 845 minus0262 1375 0080
969 minus0051 1379 minus00871573 minus03631583 04401651 minus00821683 0092
0753 0289 0463 0439 028472 (15) 65 (6) 106 (9) 102 (9) 76 (6)
Nucleus 119864(1
minus
) 119862 119864(1
+
) 119862 119864(2
+
) 119862 119864(2
minus
) 119862 119864(3
minus
) 119862
124Sn
168 0522 000 0690 023 0157 052 0271 036 0050482 0089 100 minus0067 060 0083 182 0225 049 0110654 0082 256 0252 106 0056 455 0051 195 0150957 0059 331 0153 215 0066 760 0057 967 00861074 0159 672 minus0130 711 0055 766 minus0064 970 00621407 0065 951 0098 1020 0183 982 minus00871446 minus0099 1309 minus0112 1035 minus0094 1264 minus00531483 minus00581636 minus00871805 minus0067
0664 0884 0417 0629 031855 (8) 95 (11) 53 (5) 89 (8) 49 (4)
of the intermediate nucleus 116In As for the ground-state-to-ground-state decays in some cases notable contributionsare coming from high excitation energies well above 10MeVThere are high-energy contributions in case of 1+ multipolefor all nuclei and in the cases of 2minus and 2+multipoles for 130Teand 136Xe
33 Effects of 119892A on the Intermediate-State Contributions Asmentioned earlier we have used in this work the quenchedvalue for the axial vector coupling 119892A = 100 Next we shallbriefly examine how our results will change if we increasethe value of 119892A from the quenched value 100 to the barevalue 126The effect of this amplification of the axial coupling
10 Advances in High Energy Physics
Table 3 Most important multipoles and intermediate states contributing to the ground-state-to-ground-state 0]120573120573 decays mediated by thelight neutrino exchange Columns 119864 give the energies (in MeVs) and multipoles of the intermediate states Multipoles are organized from leftto right in terms of their importance themost important being on the left Columns labeled119862 give the correspondingNMEcontributions Lasttwo numbers in each 119862 column give the summed contribution and the percentual part which the displayed states give to the total multipolestrength The percentage inside the parenthesis gives the fraction with which the displayed states contribute to the total NME
Nucleus 119864(1
minus
) 119862 119864(2
+
) 119862 119864(3
minus
) 119862 119864(2
minus
) 119862 119864(3
+
) 119862
128Te
422 0200 004 0066 016 0055 061 0335 002 0074472 0060 051 0052 058 0140 402 0060 237 0063621 0078 293 0084 397 0052 455 0101 622 0078644 0059 397 0053 1004 0061 489 minus0070 677 minus0065807 minus0084 677 0050 1014 0136 982 minus0062830 0151 1057 minus0056 1027 minus0057898 0068 1155 00581069 minus00521112 01821748 minus00791919 minus0100
0484 0305 0308 0564 015469 (8) 52 (5) 58 (5) 120 (10) 33 (3)
Nucleus 119864(1
minus
) 119862 119864(2
+
) 119862 119864(3
minus
) 119862 119864(3
+
) 119862 119864(2
minus
) 119862
130Te
418 0184 013 0054 084 0113 010 0056 097 0277572 0059 316 0089 036 0052 1025 0061627 0059 470 0064 260 0056 1133 minus0100856 0096 1049 minus0065 635 00631126 0115 1057 0145 683 00541777 minus0062 1097 minus0088 1231 00591951 minus0079 1644 0150 1241 minus0065
1654 minus01320373 0216 0113 0166 0237
61 (7) 40 (4) 25 (2) 39 (3) 56 (5)Nucleus 119864(2
+
) 119862 119864(3
+
) 119862 119864(1
minus
) 119862 119864(2
minus
) 119862 119864(4
+
) 119862
136Xe
159 0033 135 0033 665 0141 486 0188 234 0032226 0038 239 0054 732 0044 742 0068 244 0061250 0037 481 0050 1031 0053 443 0031529 0067 827 0035 476 0049657 0033 999 0027 767 0034705 0027 1065 minus0027 910 00451411 0087 965 minus00381425 minus0085
0238 0172 0238 0256 021462 (7) 53 (5) 76 (7) 85 (7) 71 (6)
strength on the NMEs is demonstrated in Figure 5 where wehave plotted the multipole decompositions for nuclei 76Geand 82Se calculated with both values of the axial coupling119892A = 100 and 119892A = 126 In case of the ground-state-to-ground-state decays the 1+ multipole changes rather fastwhen the axial coupling is increased from 100 to 126 Thishappens mainly due to the changing of the 119892pp parameter(for each 119892A value the parameter 119892pp is adjusted in sucha way that the measured 2]120573120573 rate is reproduced) The 1+multipole contribution is very sensitive to the value of 119892ppWe can see from Figures 5(a) and 5(b) that for 119892A = 100 the
1
+ component is among the five most important multipoleswhile for 119892A = 126 it is not Some of the higher multipoleschange also somewhat but not so rapidly Ground-state-to-excited-state transitions proceed mainly through the 0+ and1
+multipole channelsWe see from Figures 5(c) and 5(d) thatincreasing the value of 119892A affects mostly the 0+ component
Figure 6 displays the total cumulative sum distributionsfor ground-state-to-ground-state decays of the nuclei 100Moand 116Cd (panels (a) and (b)) and for ground-state-to-excited-state decays of the nuclei 82Se and 96Zr (panels (c)and (d)) Axial coupling values 119892A = 100 and 119892A = 126
Advances in High Energy Physics 11
Table 4 Most important multipoles and intermediate states contributing to the ground-state-to-excited-state 0]120573120573 decays mediated by thelight neutrino exchange Columns 119864 give the energies (in MeVs) and multipoles of the intermediate states Multipoles are organized from leftto right in terms of their importance themost important being on the left Columns labeled119862 give the correspondingNMEcontributions Lasttwo numbers in each 119862 column give the summed contribution and the percentual part which the displayed states give to the total multipolestrength The percentage inside the parenthesis gives the fraction with which the displayed states contribute to the total NME
Nucleus 119864(1
+
) 119862 119864(0
+
) 119862
76Ge
000 0390 286 0053295 0062 428 1204494 0501787 04351352 0522
1909 125692 (50) 99 (33)
Nucleus 119864(1
+
) 119862 119864(0
+
) 119862
82Se
019 0149 545 0471532 0179701 01451453 0274
0746 047190 (41) 96 (26)
Nucleus 119864(1
+
) 119862 119864(2
minus
) 119862 119864(2
+
) 119862 119864(0
+
) 119862 119864(1
minus
) 119862
96Zr
000 0700 092 0283 064 0122 537 0033 228 0032781 0051 966 0057 163 0180 664 0025 252 0080899 0058 537 0035 732 0037 446 00261165 0065 692 0027 802 0036 527 00681617 0354 1012 0396 1133 0026
1200 00131228 0340 0364 0540 0232
91 (23) 60 (6) 65 (7) 98 (10) 59 (4)Nucleus 119864(1
+
) 119862 119864(0
+
) 119862
100Mo
942 0090 901 00761362 0079 955 01501431 00721512 0063
0303 022665 (40) 77 (30)
Nucleus 119864(1
+
) 119862 119864(0
+
) 119862
110Pd
000 0308 1026 01651022 00591373 00831456 00551516 0072
0577 016576 (53) 70 (15)
Nucleus 119864(1
+
) 119862
116Cd000 0735
073581 (48)
12 Advances in High Energy Physics
Table 5 Most important multipoles and intermediate states contributing to the ground-state-to-excited-state 0]120573120573 decays mediated by thelight neutrino exchange Columns 119864 give the energies (in MeVs) and multipoles of the intermediate states Multipoles are organized from leftto right in terms of their importance themost important being on the left Columns labeled119862 give the correspondingNMEcontributions Lasttwo numbers in each 119862 column give the summed contribution and the percentual part which the displayed states give to the total multipolestrength The percentage inside the parenthesis gives the fraction with which the displayed states contribute to the total NME
Nucleus 119864(1
+
) 119862 119864(0
+
) 119862
124Sn
000 0101 270 0667066 0433 460 0088100 0146 735 0567225 0051256 0227331 0664672 0289791 04241309 04731386 0161
2968 132294 (44) 94 (19)
Nucleus 119864(1
+
) 119862 119864(0
+
) 119862 119864(2
minus
) 119862 119864(2
+
) 119862
130Te
025 0390 740 0719 097 0340 041 0019150 0082 835 0631 507 0073 059 0047232 0097 1726 0106 183 0017276 0134 1895 0083 280 0043453 0064 470 0024559 0338 486 0027757 0592 518 00331466 0065 627 00161501 0259 675 00511507 0773 910 00321539 0089 1057 0045
1787 00262314 0024
2882 1351 0602 039095 (41) 996 (19) 79 (9) 65 (6)
Nucleus 119864(1
+
) 119862 119864(0
+
) 119862 119864(2
minus
) 119862 119864(2
+
) 119862
136Xe
230 0366 1224 0317 250 0060 486 0204306 0050 1254 0712 486 0056 1858 0083375 0066 529 0113 2034 0059446 0226 834 00531012 0232 1411 00541152 05651712 01371778 1031
2675 1029 0335 034693 (43) 101 (16) 53 (5) 67 (6)
Advances in High Energy Physics 13
were adopted We notice from the figures that increasing theaxial coupling strength shifts the distributions downwardsThis is especially true for the higher energy parts Despite thisfact the overall shapes of the cumulative sum distributionsdo not changemuch and the same classification of nuclei intodifferent categories according to their cumulative distributionshapes seems to hold also for larger values of 119892A
4 Conclusions
In this paper we have extended our previous work [17] onthe ground-state-to-ground-state 0]120573120573 decay transitions Inthe present work we have concentrated our studies on theintermediate contributions to the NMEs involved in thelight neutrino mediated 0]120573120573 decay We have calculated theintermediate state multipole decompositions of the NMEsand extracted the most important multipole componentsCumulative sums of the NMEs were calculated to investigatethe important energy regions contributing to the 0]120573120573 tran-sitions Finally the most important multipole componentswere divided into contributions coming from the virtualtransitions through the individual states of the 0]120573120573 inter-mediate nuclei An extensive tabulation of these importantintermediate states were given for all the nuclei considered inthis paper
We have done these computations by using realistic two-body interactions and single-particle bases All the appro-priate short-range correlations nucleon form factors andhigher-order nucleonic weak currents are included in ourpresent results
We found in the calculations that often there exists onlya few relevant intermediate states which collect most of thestrength corresponding to a given multipole We also foundthat there exists a single-state dominance in the important 2minuscomponents related to the ground-state decays of nuclei 76Ge82Se and perhaps also for 96Zr
Competing Interests
The authors declare that they have no competing interests
Acknowledgments
This work has been partially supported by the Academy ofFinland under the Finnish Centre of Excellence Programme2012ndash2017 (Nuclear and Accelerator Based Programme atJYFL)
References
[1] J Suhonen and O Civitarese ldquoWeak-interaction and nuclear-structure aspects of nuclear double beta decayrdquo Physics Reportvol 300 no 3-4 pp 123ndash214 1998
[2] F T Avignone S R Elliott and J Engel ldquoDouble beta decayMajorana neutrinos and neutrino massrdquo Reviews of ModernPhysics vol 80 no 2 pp 481ndash516 2008
[3] J D Vergados H Ejiri and F Simkovic ldquoTheory of neutrinolessdouble-beta decayrdquoReports on Progress in Physics vol 75 no 10p 106301 2012
[4] J Maalampi and J Suhonen ldquoNeutrinoless double 120573
+ECdecaysrdquo Advances in High Energy Physics vol 2013 Article ID505874 18 pages 2013
[5] M Kortelainen O Civitarese J Suhonen and J ToivanenldquoShort-range correlations and neutrinoless double beta decayrdquoPhysics Letters Section B Nuclear Elementary Particle andHigh-Energy Physics vol 647 no 2-3 pp 128ndash132 2007
[6] F Simkovic G Pantis J D Vergados and A Faessler ldquoAddi-tional nucleon current contributions to neutrinoless double 120573decayrdquo Physical Review C vol 60 no 5 Article ID 055502 14pages 1999
[7] J Suhonen and O Civitarese ldquoReview of the properties of the0]120573minus120573minus nuclear matrix elementsrdquo Journal of Physics G Nuclearand Particle Physics vol 39 no 12 Article ID 124005 2012
[8] P Vogel ldquoNuclear structure and double beta decayrdquo Journal ofPhysics G Nuclear and Particle Physics vol 39 no 12 p 1240022012
[9] J Barea J Kotila and F Iachello ldquoNuclear matrix elements fordouble-120573 decayrdquo Physical Review C vol 87 no 1 Article ID014315 2013
[10] A Neacsu and S Stoica ldquoStudy of nuclear effects in thecomputation of the 0]120573120573 decay matrix elementsrdquo Journal ofPhysics G Nuclear and Particle Physics vol 41 no 1 Article ID015201 2014
[11] J Engel ldquoUncertainties in nuclear matrix elements for neutri-noless double-beta decayrdquo Journal of Physics G Nuclear andParticle Physics vol 42 no 3 Article ID 034017 2015
[12] F Simkovic A Faessler V Rodin P Vogel and J EngelldquoAnatomyof the 0]120573120573nuclearmatrix elementsrdquoPhysical ReviewC Nuclear Physics vol 77 no 4 Article ID 045503 2008
[13] M T Mustonen and J Engel ldquoLarge-scale calculations ofthe double-120573 decay of 76Ge 130Te 136Xe and 150Nd in thedeformed self-consistent Skyrme quasiparticle random-phaseapproximationrdquo Physical Review C vol 87 Article ID 0643022013
[14] F Simkovic V Rodin A Faessler and P Vogel ldquo0]120573120573 and2]120573120573 nuclear matrix elements quasiparticle random-phaseapproximation and isospin symmetry restorationrdquo PhysicalReview C vol 87 no 4 Article ID 045501 9 pages 2013
[15] V A Rodin A Faessler F Simkovic and P Vogel ldquoAssessmentof uncertainties in QRPA 0]120573120573-decay nuclear matrix elementsrdquoNuclear Physics A vol 766 pp 107ndash131 2006
[16] D-L Fang A Faessler V Rodin and F Simkovic ldquoNeutri-noless double-120573 decay of deformed nuclei within quasiparticlerandom-phase approximationwith a realistic interactionrdquoPhys-ical Review C vol 83 no 3 Article ID 034320 8 pages 2011
[17] J Hyvarinen and J Suhonen ldquoNuclear matrix elements for0]120573120573 decays with light or heavy Majorana-neutrino exchangerdquoPhysical Review C vol 91 no 2 Article ID 024613 12 pages2015
[18] P Pirinen and J Suhonen ldquoSystematic approach to 120573 and 2]120573120573decays of mass119860 = 100minus136 nucleirdquo Physical Review C vol 91no 5 Article ID 054309 2015
[19] H Ejiri N Soukouti and J Suhonen ldquoSpin-dipole nuclearmatrix elements for double beta decays and astro-neutrinosrdquoPhysics Letters B vol 729 pp 27ndash32 2014
[20] J Menendez D Gazit and A Schwenk ldquoChiral two-bodycurrents in nuclei Gamow-Teller transitions and neutrinolessdouble-beta decayrdquo Physical Review Letters vol 107 no 6Article ID 062501 5 pages 2011
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Advances in Condensed Matter Physics
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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
AstronomyAdvances in
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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Superconductivity
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Statistical MechanicsInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
GravityJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
AstrophysicsJournal of
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Physics Research International
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Solid State PhysicsJournal of
âComputationalââMethodsâinâPhysics
Journal of
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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Soft MatterJournal of
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AerodynamicsJournal of
Volume 2014
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PhotonicsJournal of
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Journal of
Biophysics
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
ThermodynamicsJournal of
4 Advances in High Energy Physics
0
1
2
3
4
5
6
7
8
9
10Cu
mul
ativ
e NM
E co
ntrib
utio
n
0 2010
Energy (MeV)
(a) 76Ge
0
1
2
3
4
5
6
7
8
9
10
Cum
ulat
ive N
ME
cont
ribut
ion
0 2010
Energy (MeV)
(b) 110Pd
10 20 300
Energy (MeV)
0
1
2
3
4
5
6
7
8
9
10
Cum
ulat
ive N
ME
cont
ribut
ion
(c) 124Sn
0
1
2
3
4
5
6
7
8
9
10Cu
mul
ativ
e NM
E co
ntrib
utio
n
0 20 3010
Energy (MeV)
(d) 136Xe
Figure 2 Cumulative values of the computed l-NMEs corresponding to the 0+gs rarr 0
+
gs decay transitions for the nuclear systems 119860 =
76 110 124 and 136 The horizontal axis gives the excitation energies of the intermediate states contributing to the 0]120573120573 transition
0 1 2 3 4 5 6 7 8 9 10 11
0
1
2
3
M(0)(J120587
)
J120587+minus +minus +minus +minus +minus +minus +minus +minus +minus +minus +minus +minus
(a) 76Ge(0]120573120573 0+gs rarr 0+
1
)
+minus0
+minus1
+minus2
+minus3
+minus4
+minus5 6 7 8 9 10 11 J
120587
0
1
2
3
M(0)(J120587
)
+minus +minus +minus +minus +minus +minus
(b) 96Zr(0]120573120573 0+gs rarr 0+
1
)
Figure 3 Multipole decomposition of the l-NME for the nuclei 76Ge and 96Zr corresponding to the 0+gs rarr 0
+
1
decay transitions
Advances in High Energy Physics 5
0
1
2
3
4
5Cu
mul
ativ
e NM
E co
ntrib
utio
n
10 200
Energy (MeV)
(a) 76Ge
0 2010
Energy (MeV)
0
1
2
3
4
5
Cum
ulat
ive N
ME
cont
ribut
ion
(b) 82Se
10 20 300
Energy (MeV)
0
1
2
3
4
5
6
7
Cum
ulat
ive N
ME
cont
ribut
ion
(c) 96Zr
10 20 300
Energy (MeV)
0
1
2
3
4
5
6
7
Cum
ulat
ive N
ME
cont
ribut
ion
(d) 136Xe
10 20 300
Energy (MeV)
0
1
2
3
Cum
ulat
ive N
ME
cont
ribut
ion
(e) 116Cd
Figure 4 Cumulative values of the computed l-NMEs corresponding to the 0+gs rarr 0
+
1
decay transitions for the nuclear systems 119860 =
76 82 96 116 and 136 The horizontal axis gives the excitation energies of the intermediate states contributing to the 0]120573120573 transition
6 Advances in High Energy Physics
Table 1 Most important multipoles and intermediate states contributing to the ground-state-to-ground-state 0]120573120573 decays mediated by thelight neutrino exchange Columns 119864 give the energies (in MeVs) and multipoles of the intermediate states Multipoles are organized from leftto right in terms of their importance themost important being on the left Columns labeled119862 give the correspondingNMEcontributions Lasttwo numbers in each 119862 column give the summed contribution and the percentual part which the displayed states give to the total multipolestrength The percentage inside the parenthesis gives the fraction with which the displayed states contribute to the total NME
Nucleus 119864(2
minus
) 119862 119864(1
minus
) 119862 119864(2
+
) 119862 119864(1
+
) 119862 119864(3
minus
) 119862
76Ge
022 0748 587 0155 051 0166 000 0344 030 0189632 0058 187 0056 409 0199 092 0148700 0077 304 0094 448 0148 677 0056716 0064 362 0108 494 0578 685 0053827 0190 481 0058 1080 minus0053 1163 00621104 0084 773 0054 1175 minus0109 1207 minus00531203 0165 1352 minus05221670 0158
0748 0635 0556 0668 045479 (11) 83 (10) 77 (9) 88 (10) 70 (7)
Nucleus 119864(2
minus
) 119862 119864(2
+
) 119862 119864(1
minus
) 119862 119864(3
minus
) 119862 119864(1
+
) 119862
82Se
000 0510 065 0137 527 0116 007 0140 019 0264173 0065 685 0065 082 0138 316 0095222 0051 798 0134 407 0065356 0084 979 0083 455 0105405 0052 1225 0065 532 0559493 0054 1741 0070 701 minus0253
1453 minus03960510 0442 0393 0278 0439
81 (11) 81 (9) 76 (8) 57 (6) 92 (9)Nucleus 119864(1
minus
) 119862 119864(2
minus
) 119862 119864(3
minus
) 119862 119864(2
+
) 119862 119864(4
+
) 119862
96Zr
175 0056 092 0498 135 0151 064 0150 105 0099228 0063 221 0065 235 0051 163 0114 168 0071252 0150 375 0050 777 0064 569 0064446 0050 443 0052 1136 minus0091504 0077 853 0057527 0209 877 minus0056865 00601133 0061
0728 0666 0175 0265 023483 (16) 96 (15) 31 (4) 59 (6) 69 (5)
Nucleus 119864(1
minus
) 119862 119864(2
+
) 119862 119864(4
+
) 119862 119864(3
+
) 119862 119864(2
minus
) 119862
100Mo
312 0307 090 0192 132 0158 133 0202 176 0205441 0109 210 0154 226 0097 168 0076 285 0147668 0091 1168 0082 743 0152 552 minus00611115 0117 1197 minus0070 776 minus0173 1017 02191672 0058 1093 minus01942026 minus00602390 minus0060
0562 0359 0255 0257 031670 (11) 62 (7) 53 (5) 54 (5) 72 (6)
Advances in High Energy Physics 7
+minus0
+minus1
+minus2
+minus3
+minus4
+minus5
+minus6
+minus7
+minus8
+minus9
+minus10
+minus11
minus05
0
05
1
M(0)(J120587
)
J120587
= 100gA
(a) 76Ge(0]120573120573 0+gs rarr 0+
gs)
+minus0
+minus1
+minus2
+minus3
+minus4
+minus5
+minus6
+minus7
+minus8
+minus9
+minus10
+minus11
minus05
0
05
1
M(0)(J120587
)
J120587
= 126gA
(b) 76Ge(0]120573120573 0+gs rarr 0+
gs)
= 100
+minus0
+minus1
+minus2
+minus3
+minus4
+minus5
+minus6
+minus7
+minus8
+minus9
+minus10
+minus11
minus05
0
05
1
M(0)(J120587
)
J120587
gA
(c) 82Se(0]120573120573 0+gs rarr 0+
1
)
0 1 2 3 4 5 6 7 8 9 10 11
minus05
0
05
1
M(0)(J120587
)
J120587+minus +minus +minus +minus +minus +minus +minus +minus +minus +minus +minus +minus
= 126gA
(d) 82Se(0]120573120573 0+gs rarr 0+
1
)
Figure 5 Multipole decompositions for the ground-state-to-ground-state decay of the nucleus 76Ge ((a) and (b) panels) and for the ground-state-to-excited-state decay of the nucleus 82Se ((c) and (d) panels)The value119892A = 100was used for (a) and (c) panels and the value119892A = 126for (b) and (d) panels
the level structures found in all intermediate odd-odd nucleiconsidered in this work This is due to the general feature ofodd-odd nuclei the extremely high density of states even atlow energies This high density of nuclear states becomes aproblem not only for the pnQRPA but for any other nuclearmany-body approach including the nuclear shell model Thereason for this is that even small perturbations in the two-body interactionmatrix elements tend to change the orderingof the levels at random For this reason the spectra of the odd-odd intermediate nuclei are not a very good measure of thereliability of the calculations but instead a better way is toadjust the model parameters in such a way that the transitionrates of some other known processes for example single or2]120573120573 decays can be reproduced by the theory and this is thephilosophy which we have followed in this work
32 Ground-State-to-Excited-State Decays Let us then con-sider 0+gs rarr 0
+
1
transitions mediated by the light neutrinoexchange In Figures 3(a) and 3(b) we have plotted themultipole decomposition of the l-NMEs corresponding to the119860 = 76 and 96 nuclear systems The multipole distributionsfor the excited-state transitions are greatly different fromthose corresponding to the ground-state transitions Usuallythere is only a couple of multipoles 0+ and 1
+ whichgive by far the largest contribution to the NMEs In thissense the excited-state transitions are more simple than theground-state transitions Typical example is the nucleus 76Gedisplayed in Figure 3(a) One nucleus deviating from this
trend is 96Zr which is presented in Figure 3(b) Its multipoledistribution resembles somewhat more those shown for theground-state decays in Figures 1(a) and 1(b) Most of thisdiffering behaviour can be traced back to the one-phononstructure of the final 0+
1
excited state in the nucleus 96MoThe 0+
1
final states in this work are modeled as one-phononbasic QRPA excitations for the daughter nuclei 96Mo and116Sn Rest of the final states are modeled as two-quadrupole-phonon states Nucleus 96Zr is an exceptional case since the0
+
1
state in 96Mo has a relatively low excitation energy andthus boasts rather strong collective features This is why theexcited-state transition has a wide multipole distribution andis greatly enhanced
Again we can divide nuclei into different groups byconsidering the shapes of their total cumulative sum dis-tributions For 0+gs rarr 0
+
1
transitions via light neutrinoexchange we can differentiate two types of nuclei Type 1nuclei belonging to type 1 are 76Ge 82Se 124Sn 130Te and136Xe Typical examples of this type 76Ge 82Se and 136Xeare shown in Figures 4(a) 4(b) and 4(d) Characteristicfeature of this type is that there exist only few energy stateswhich give most of the total matrix element producing astaircase-like structure as seen in the panels For examplefor 76Ge there seems to be only five such energy statesType 2 nuclei belonging to this type are 96Zr 100Mo 110Pdand 116Cd Typical examples of this type are 96Zr and 116Cdshown in Figures 4(c) and 4(e) Characteristic feature oftype 2 is that a large number of intermediate states give
8 Advances in High Energy Physics
0 20 3010Energy (MeV)
0
1
2
3
4
5
6
7Cu
mul
ativ
e NM
E co
ntrib
utio
n
gA = 100
gA = 126
(a) 100Mo(0]120573120573 0+gs rarr 0+
gs)
0
1
2
3
4
5
6
7
Cum
ulat
ive N
ME
cont
ribut
ion
10 20 300Energy (MeV)
gA = 100
gA = 126
(b) 116Cd(0]120573120573 0+gs rarr 0+
gs)
0
1
2
Cum
ulat
ive N
ME
cont
ribut
ion
10 200Energy (MeV)
gA = 100
gA = 126
(c) 82Se(0]120573120573 0+gs rarr 0+
1
)
0
1
2
3
4
5
6
7
Cum
ulat
ive N
ME
cont
ribut
ion
10 20 300Energy (MeV)
gA = 100
gA = 126
(d) 96Zr(0]120573120573 0+gs rarr 0+
1
)
Figure 6 Cumulative values of the l-NMEs for ground-state-to-ground-state decays of the nuclei 100Mo and 116Cd ((a) and (b) panels) andfor the ground-state-to-excited-state decays of the nuclei 82Se and 96Zr ((c) and (d) panels) The horizontal axis gives the excitation energiesof the intermediate states contributing to the 0]120573120573 transition Two different values for the axial coupling were used as indicated in the panels
important contributions to the NMEs In case of 116Cd panel(e) around 50 of the total NME comes from transitionsthrough the ground state of the intermediate nucleus Theother 50 is distributed rather evenly on the interval 0ndash20MeV
Using the multipole decompositions we extracted themost important multipole components contributing to thelight neutrino mediated 0
+
gs rarr 0
+
1
decay transitionsThese most important components were then again dividedinto contributions coming from different energy levels of
the 0]120573120573 intermediate nucleus These contributions arecollected into Table 4 for119860 = 76ndash116 systems and into Table 5for 119860 = 124ndash136 systems Again we notice that often onlya few intermediate states give the largest contribution to thedominant multipoles 1+ and 0+ Extreme case is the nucleus116Cd forwhich the dominant intermediate ground state gives81 of the total 1+ strength Combining this with the factthat 1+ is by far the largest multipole component we get arather good approximation for the total NME by consideringjust a single virtual transition through the 1+ ground state
Advances in High Energy Physics 9
Table 2 Most important multipoles and intermediate states contributing to the ground-state-to-ground-state 0]120573120573 decays mediated by thelight neutrino exchange Columns 119864 give the energies (in MeVs) and multipoles of the intermediate states Multipoles are organized from leftto right in terms of their importance themost important being on the left Columns labeled119862 give the correspondingNMEcontributions Lasttwo numbers in each 119862 column give the summed contribution and the percentual part which the displayed states give to the total multipolestrength The percentage inside the parenthesis gives the fraction with which the displayed states contribute to the total NME
Nucleus 119864(1
minus
) 119862 119864(2
minus
) 119862 119864(1
+
) 119862 119864(3
minus
) 119862 119864(2
+
) 119862
110Pd
295 0130 082 0387 000 0938 114 0118 033 0244319 0106 247 0091 470 minus0062 164 0128 095 0087344 0221 263 0163 961 0153 191 0080 868 0061381 0426 594 0107 975 minus0146 307 0053 873 0075452 0126 888 0161 1022 0079 361 0075 900 0061911 0109 954 minus0256 1516 minus0316 533 0066 907 0053
1630 0256 808 0167 916 0199837 minus0051 930 minus0366845 0052868 0275911 minus0468
1118 0653 0903 0496 041477 (13) 68 (7) 96 (10) 60 (6) 55 (5)
Nucleus 119864(1
minus
) 119862 119864(3
minus
) 119862 119864(2
minus
) 119862 119864(1
+
) 119862 119864(3
+
) 119862
116Cd
361 0223 172 0081 184 0065 000 0378 090 0158447 0099 230 0053 293 0287 837 minus0102 140 0068537 0118 279 0051 756 0075 951 0087 389 0057587 0140 724 0104 831 0246 1074 minus0083861 0102 834 0103 1120 00812307 0071 845 minus0262 1375 0080
969 minus0051 1379 minus00871573 minus03631583 04401651 minus00821683 0092
0753 0289 0463 0439 028472 (15) 65 (6) 106 (9) 102 (9) 76 (6)
Nucleus 119864(1
minus
) 119862 119864(1
+
) 119862 119864(2
+
) 119862 119864(2
minus
) 119862 119864(3
minus
) 119862
124Sn
168 0522 000 0690 023 0157 052 0271 036 0050482 0089 100 minus0067 060 0083 182 0225 049 0110654 0082 256 0252 106 0056 455 0051 195 0150957 0059 331 0153 215 0066 760 0057 967 00861074 0159 672 minus0130 711 0055 766 minus0064 970 00621407 0065 951 0098 1020 0183 982 minus00871446 minus0099 1309 minus0112 1035 minus0094 1264 minus00531483 minus00581636 minus00871805 minus0067
0664 0884 0417 0629 031855 (8) 95 (11) 53 (5) 89 (8) 49 (4)
of the intermediate nucleus 116In As for the ground-state-to-ground-state decays in some cases notable contributionsare coming from high excitation energies well above 10MeVThere are high-energy contributions in case of 1+ multipolefor all nuclei and in the cases of 2minus and 2+multipoles for 130Teand 136Xe
33 Effects of 119892A on the Intermediate-State Contributions Asmentioned earlier we have used in this work the quenchedvalue for the axial vector coupling 119892A = 100 Next we shallbriefly examine how our results will change if we increasethe value of 119892A from the quenched value 100 to the barevalue 126The effect of this amplification of the axial coupling
10 Advances in High Energy Physics
Table 3 Most important multipoles and intermediate states contributing to the ground-state-to-ground-state 0]120573120573 decays mediated by thelight neutrino exchange Columns 119864 give the energies (in MeVs) and multipoles of the intermediate states Multipoles are organized from leftto right in terms of their importance themost important being on the left Columns labeled119862 give the correspondingNMEcontributions Lasttwo numbers in each 119862 column give the summed contribution and the percentual part which the displayed states give to the total multipolestrength The percentage inside the parenthesis gives the fraction with which the displayed states contribute to the total NME
Nucleus 119864(1
minus
) 119862 119864(2
+
) 119862 119864(3
minus
) 119862 119864(2
minus
) 119862 119864(3
+
) 119862
128Te
422 0200 004 0066 016 0055 061 0335 002 0074472 0060 051 0052 058 0140 402 0060 237 0063621 0078 293 0084 397 0052 455 0101 622 0078644 0059 397 0053 1004 0061 489 minus0070 677 minus0065807 minus0084 677 0050 1014 0136 982 minus0062830 0151 1057 minus0056 1027 minus0057898 0068 1155 00581069 minus00521112 01821748 minus00791919 minus0100
0484 0305 0308 0564 015469 (8) 52 (5) 58 (5) 120 (10) 33 (3)
Nucleus 119864(1
minus
) 119862 119864(2
+
) 119862 119864(3
minus
) 119862 119864(3
+
) 119862 119864(2
minus
) 119862
130Te
418 0184 013 0054 084 0113 010 0056 097 0277572 0059 316 0089 036 0052 1025 0061627 0059 470 0064 260 0056 1133 minus0100856 0096 1049 minus0065 635 00631126 0115 1057 0145 683 00541777 minus0062 1097 minus0088 1231 00591951 minus0079 1644 0150 1241 minus0065
1654 minus01320373 0216 0113 0166 0237
61 (7) 40 (4) 25 (2) 39 (3) 56 (5)Nucleus 119864(2
+
) 119862 119864(3
+
) 119862 119864(1
minus
) 119862 119864(2
minus
) 119862 119864(4
+
) 119862
136Xe
159 0033 135 0033 665 0141 486 0188 234 0032226 0038 239 0054 732 0044 742 0068 244 0061250 0037 481 0050 1031 0053 443 0031529 0067 827 0035 476 0049657 0033 999 0027 767 0034705 0027 1065 minus0027 910 00451411 0087 965 minus00381425 minus0085
0238 0172 0238 0256 021462 (7) 53 (5) 76 (7) 85 (7) 71 (6)
strength on the NMEs is demonstrated in Figure 5 where wehave plotted the multipole decompositions for nuclei 76Geand 82Se calculated with both values of the axial coupling119892A = 100 and 119892A = 126 In case of the ground-state-to-ground-state decays the 1+ multipole changes rather fastwhen the axial coupling is increased from 100 to 126 Thishappens mainly due to the changing of the 119892pp parameter(for each 119892A value the parameter 119892pp is adjusted in sucha way that the measured 2]120573120573 rate is reproduced) The 1+multipole contribution is very sensitive to the value of 119892ppWe can see from Figures 5(a) and 5(b) that for 119892A = 100 the
1
+ component is among the five most important multipoleswhile for 119892A = 126 it is not Some of the higher multipoleschange also somewhat but not so rapidly Ground-state-to-excited-state transitions proceed mainly through the 0+ and1
+multipole channelsWe see from Figures 5(c) and 5(d) thatincreasing the value of 119892A affects mostly the 0+ component
Figure 6 displays the total cumulative sum distributionsfor ground-state-to-ground-state decays of the nuclei 100Moand 116Cd (panels (a) and (b)) and for ground-state-to-excited-state decays of the nuclei 82Se and 96Zr (panels (c)and (d)) Axial coupling values 119892A = 100 and 119892A = 126
Advances in High Energy Physics 11
Table 4 Most important multipoles and intermediate states contributing to the ground-state-to-excited-state 0]120573120573 decays mediated by thelight neutrino exchange Columns 119864 give the energies (in MeVs) and multipoles of the intermediate states Multipoles are organized from leftto right in terms of their importance themost important being on the left Columns labeled119862 give the correspondingNMEcontributions Lasttwo numbers in each 119862 column give the summed contribution and the percentual part which the displayed states give to the total multipolestrength The percentage inside the parenthesis gives the fraction with which the displayed states contribute to the total NME
Nucleus 119864(1
+
) 119862 119864(0
+
) 119862
76Ge
000 0390 286 0053295 0062 428 1204494 0501787 04351352 0522
1909 125692 (50) 99 (33)
Nucleus 119864(1
+
) 119862 119864(0
+
) 119862
82Se
019 0149 545 0471532 0179701 01451453 0274
0746 047190 (41) 96 (26)
Nucleus 119864(1
+
) 119862 119864(2
minus
) 119862 119864(2
+
) 119862 119864(0
+
) 119862 119864(1
minus
) 119862
96Zr
000 0700 092 0283 064 0122 537 0033 228 0032781 0051 966 0057 163 0180 664 0025 252 0080899 0058 537 0035 732 0037 446 00261165 0065 692 0027 802 0036 527 00681617 0354 1012 0396 1133 0026
1200 00131228 0340 0364 0540 0232
91 (23) 60 (6) 65 (7) 98 (10) 59 (4)Nucleus 119864(1
+
) 119862 119864(0
+
) 119862
100Mo
942 0090 901 00761362 0079 955 01501431 00721512 0063
0303 022665 (40) 77 (30)
Nucleus 119864(1
+
) 119862 119864(0
+
) 119862
110Pd
000 0308 1026 01651022 00591373 00831456 00551516 0072
0577 016576 (53) 70 (15)
Nucleus 119864(1
+
) 119862
116Cd000 0735
073581 (48)
12 Advances in High Energy Physics
Table 5 Most important multipoles and intermediate states contributing to the ground-state-to-excited-state 0]120573120573 decays mediated by thelight neutrino exchange Columns 119864 give the energies (in MeVs) and multipoles of the intermediate states Multipoles are organized from leftto right in terms of their importance themost important being on the left Columns labeled119862 give the correspondingNMEcontributions Lasttwo numbers in each 119862 column give the summed contribution and the percentual part which the displayed states give to the total multipolestrength The percentage inside the parenthesis gives the fraction with which the displayed states contribute to the total NME
Nucleus 119864(1
+
) 119862 119864(0
+
) 119862
124Sn
000 0101 270 0667066 0433 460 0088100 0146 735 0567225 0051256 0227331 0664672 0289791 04241309 04731386 0161
2968 132294 (44) 94 (19)
Nucleus 119864(1
+
) 119862 119864(0
+
) 119862 119864(2
minus
) 119862 119864(2
+
) 119862
130Te
025 0390 740 0719 097 0340 041 0019150 0082 835 0631 507 0073 059 0047232 0097 1726 0106 183 0017276 0134 1895 0083 280 0043453 0064 470 0024559 0338 486 0027757 0592 518 00331466 0065 627 00161501 0259 675 00511507 0773 910 00321539 0089 1057 0045
1787 00262314 0024
2882 1351 0602 039095 (41) 996 (19) 79 (9) 65 (6)
Nucleus 119864(1
+
) 119862 119864(0
+
) 119862 119864(2
minus
) 119862 119864(2
+
) 119862
136Xe
230 0366 1224 0317 250 0060 486 0204306 0050 1254 0712 486 0056 1858 0083375 0066 529 0113 2034 0059446 0226 834 00531012 0232 1411 00541152 05651712 01371778 1031
2675 1029 0335 034693 (43) 101 (16) 53 (5) 67 (6)
Advances in High Energy Physics 13
were adopted We notice from the figures that increasing theaxial coupling strength shifts the distributions downwardsThis is especially true for the higher energy parts Despite thisfact the overall shapes of the cumulative sum distributionsdo not changemuch and the same classification of nuclei intodifferent categories according to their cumulative distributionshapes seems to hold also for larger values of 119892A
4 Conclusions
In this paper we have extended our previous work [17] onthe ground-state-to-ground-state 0]120573120573 decay transitions Inthe present work we have concentrated our studies on theintermediate contributions to the NMEs involved in thelight neutrino mediated 0]120573120573 decay We have calculated theintermediate state multipole decompositions of the NMEsand extracted the most important multipole componentsCumulative sums of the NMEs were calculated to investigatethe important energy regions contributing to the 0]120573120573 tran-sitions Finally the most important multipole componentswere divided into contributions coming from the virtualtransitions through the individual states of the 0]120573120573 inter-mediate nuclei An extensive tabulation of these importantintermediate states were given for all the nuclei considered inthis paper
We have done these computations by using realistic two-body interactions and single-particle bases All the appro-priate short-range correlations nucleon form factors andhigher-order nucleonic weak currents are included in ourpresent results
We found in the calculations that often there exists onlya few relevant intermediate states which collect most of thestrength corresponding to a given multipole We also foundthat there exists a single-state dominance in the important 2minuscomponents related to the ground-state decays of nuclei 76Ge82Se and perhaps also for 96Zr
Competing Interests
The authors declare that they have no competing interests
Acknowledgments
This work has been partially supported by the Academy ofFinland under the Finnish Centre of Excellence Programme2012ndash2017 (Nuclear and Accelerator Based Programme atJYFL)
References
[1] J Suhonen and O Civitarese ldquoWeak-interaction and nuclear-structure aspects of nuclear double beta decayrdquo Physics Reportvol 300 no 3-4 pp 123ndash214 1998
[2] F T Avignone S R Elliott and J Engel ldquoDouble beta decayMajorana neutrinos and neutrino massrdquo Reviews of ModernPhysics vol 80 no 2 pp 481ndash516 2008
[3] J D Vergados H Ejiri and F Simkovic ldquoTheory of neutrinolessdouble-beta decayrdquoReports on Progress in Physics vol 75 no 10p 106301 2012
[4] J Maalampi and J Suhonen ldquoNeutrinoless double 120573
+ECdecaysrdquo Advances in High Energy Physics vol 2013 Article ID505874 18 pages 2013
[5] M Kortelainen O Civitarese J Suhonen and J ToivanenldquoShort-range correlations and neutrinoless double beta decayrdquoPhysics Letters Section B Nuclear Elementary Particle andHigh-Energy Physics vol 647 no 2-3 pp 128ndash132 2007
[6] F Simkovic G Pantis J D Vergados and A Faessler ldquoAddi-tional nucleon current contributions to neutrinoless double 120573decayrdquo Physical Review C vol 60 no 5 Article ID 055502 14pages 1999
[7] J Suhonen and O Civitarese ldquoReview of the properties of the0]120573minus120573minus nuclear matrix elementsrdquo Journal of Physics G Nuclearand Particle Physics vol 39 no 12 Article ID 124005 2012
[8] P Vogel ldquoNuclear structure and double beta decayrdquo Journal ofPhysics G Nuclear and Particle Physics vol 39 no 12 p 1240022012
[9] J Barea J Kotila and F Iachello ldquoNuclear matrix elements fordouble-120573 decayrdquo Physical Review C vol 87 no 1 Article ID014315 2013
[10] A Neacsu and S Stoica ldquoStudy of nuclear effects in thecomputation of the 0]120573120573 decay matrix elementsrdquo Journal ofPhysics G Nuclear and Particle Physics vol 41 no 1 Article ID015201 2014
[11] J Engel ldquoUncertainties in nuclear matrix elements for neutri-noless double-beta decayrdquo Journal of Physics G Nuclear andParticle Physics vol 42 no 3 Article ID 034017 2015
[12] F Simkovic A Faessler V Rodin P Vogel and J EngelldquoAnatomyof the 0]120573120573nuclearmatrix elementsrdquoPhysical ReviewC Nuclear Physics vol 77 no 4 Article ID 045503 2008
[13] M T Mustonen and J Engel ldquoLarge-scale calculations ofthe double-120573 decay of 76Ge 130Te 136Xe and 150Nd in thedeformed self-consistent Skyrme quasiparticle random-phaseapproximationrdquo Physical Review C vol 87 Article ID 0643022013
[14] F Simkovic V Rodin A Faessler and P Vogel ldquo0]120573120573 and2]120573120573 nuclear matrix elements quasiparticle random-phaseapproximation and isospin symmetry restorationrdquo PhysicalReview C vol 87 no 4 Article ID 045501 9 pages 2013
[15] V A Rodin A Faessler F Simkovic and P Vogel ldquoAssessmentof uncertainties in QRPA 0]120573120573-decay nuclear matrix elementsrdquoNuclear Physics A vol 766 pp 107ndash131 2006
[16] D-L Fang A Faessler V Rodin and F Simkovic ldquoNeutri-noless double-120573 decay of deformed nuclei within quasiparticlerandom-phase approximationwith a realistic interactionrdquoPhys-ical Review C vol 83 no 3 Article ID 034320 8 pages 2011
[17] J Hyvarinen and J Suhonen ldquoNuclear matrix elements for0]120573120573 decays with light or heavy Majorana-neutrino exchangerdquoPhysical Review C vol 91 no 2 Article ID 024613 12 pages2015
[18] P Pirinen and J Suhonen ldquoSystematic approach to 120573 and 2]120573120573decays of mass119860 = 100minus136 nucleirdquo Physical Review C vol 91no 5 Article ID 054309 2015
[19] H Ejiri N Soukouti and J Suhonen ldquoSpin-dipole nuclearmatrix elements for double beta decays and astro-neutrinosrdquoPhysics Letters B vol 729 pp 27ndash32 2014
[20] J Menendez D Gazit and A Schwenk ldquoChiral two-bodycurrents in nuclei Gamow-Teller transitions and neutrinolessdouble-beta decayrdquo Physical Review Letters vol 107 no 6Article ID 062501 5 pages 2011
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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FluidsJournal of
Atomic and Molecular Physics
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Advances in Condensed Matter Physics
OpticsInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
AstronomyAdvances in
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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Superconductivity
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Statistical MechanicsInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
GravityJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
AstrophysicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Physics Research International
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Solid State PhysicsJournal of
âComputationalââMethodsâinâPhysics
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Soft MatterJournal of
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AerodynamicsJournal of
Volume 2014
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PhotonicsJournal of
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Journal of
Biophysics
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
ThermodynamicsJournal of
Advances in High Energy Physics 5
0
1
2
3
4
5Cu
mul
ativ
e NM
E co
ntrib
utio
n
10 200
Energy (MeV)
(a) 76Ge
0 2010
Energy (MeV)
0
1
2
3
4
5
Cum
ulat
ive N
ME
cont
ribut
ion
(b) 82Se
10 20 300
Energy (MeV)
0
1
2
3
4
5
6
7
Cum
ulat
ive N
ME
cont
ribut
ion
(c) 96Zr
10 20 300
Energy (MeV)
0
1
2
3
4
5
6
7
Cum
ulat
ive N
ME
cont
ribut
ion
(d) 136Xe
10 20 300
Energy (MeV)
0
1
2
3
Cum
ulat
ive N
ME
cont
ribut
ion
(e) 116Cd
Figure 4 Cumulative values of the computed l-NMEs corresponding to the 0+gs rarr 0
+
1
decay transitions for the nuclear systems 119860 =
76 82 96 116 and 136 The horizontal axis gives the excitation energies of the intermediate states contributing to the 0]120573120573 transition
6 Advances in High Energy Physics
Table 1 Most important multipoles and intermediate states contributing to the ground-state-to-ground-state 0]120573120573 decays mediated by thelight neutrino exchange Columns 119864 give the energies (in MeVs) and multipoles of the intermediate states Multipoles are organized from leftto right in terms of their importance themost important being on the left Columns labeled119862 give the correspondingNMEcontributions Lasttwo numbers in each 119862 column give the summed contribution and the percentual part which the displayed states give to the total multipolestrength The percentage inside the parenthesis gives the fraction with which the displayed states contribute to the total NME
Nucleus 119864(2
minus
) 119862 119864(1
minus
) 119862 119864(2
+
) 119862 119864(1
+
) 119862 119864(3
minus
) 119862
76Ge
022 0748 587 0155 051 0166 000 0344 030 0189632 0058 187 0056 409 0199 092 0148700 0077 304 0094 448 0148 677 0056716 0064 362 0108 494 0578 685 0053827 0190 481 0058 1080 minus0053 1163 00621104 0084 773 0054 1175 minus0109 1207 minus00531203 0165 1352 minus05221670 0158
0748 0635 0556 0668 045479 (11) 83 (10) 77 (9) 88 (10) 70 (7)
Nucleus 119864(2
minus
) 119862 119864(2
+
) 119862 119864(1
minus
) 119862 119864(3
minus
) 119862 119864(1
+
) 119862
82Se
000 0510 065 0137 527 0116 007 0140 019 0264173 0065 685 0065 082 0138 316 0095222 0051 798 0134 407 0065356 0084 979 0083 455 0105405 0052 1225 0065 532 0559493 0054 1741 0070 701 minus0253
1453 minus03960510 0442 0393 0278 0439
81 (11) 81 (9) 76 (8) 57 (6) 92 (9)Nucleus 119864(1
minus
) 119862 119864(2
minus
) 119862 119864(3
minus
) 119862 119864(2
+
) 119862 119864(4
+
) 119862
96Zr
175 0056 092 0498 135 0151 064 0150 105 0099228 0063 221 0065 235 0051 163 0114 168 0071252 0150 375 0050 777 0064 569 0064446 0050 443 0052 1136 minus0091504 0077 853 0057527 0209 877 minus0056865 00601133 0061
0728 0666 0175 0265 023483 (16) 96 (15) 31 (4) 59 (6) 69 (5)
Nucleus 119864(1
minus
) 119862 119864(2
+
) 119862 119864(4
+
) 119862 119864(3
+
) 119862 119864(2
minus
) 119862
100Mo
312 0307 090 0192 132 0158 133 0202 176 0205441 0109 210 0154 226 0097 168 0076 285 0147668 0091 1168 0082 743 0152 552 minus00611115 0117 1197 minus0070 776 minus0173 1017 02191672 0058 1093 minus01942026 minus00602390 minus0060
0562 0359 0255 0257 031670 (11) 62 (7) 53 (5) 54 (5) 72 (6)
Advances in High Energy Physics 7
+minus0
+minus1
+minus2
+minus3
+minus4
+minus5
+minus6
+minus7
+minus8
+minus9
+minus10
+minus11
minus05
0
05
1
M(0)(J120587
)
J120587
= 100gA
(a) 76Ge(0]120573120573 0+gs rarr 0+
gs)
+minus0
+minus1
+minus2
+minus3
+minus4
+minus5
+minus6
+minus7
+minus8
+minus9
+minus10
+minus11
minus05
0
05
1
M(0)(J120587
)
J120587
= 126gA
(b) 76Ge(0]120573120573 0+gs rarr 0+
gs)
= 100
+minus0
+minus1
+minus2
+minus3
+minus4
+minus5
+minus6
+minus7
+minus8
+minus9
+minus10
+minus11
minus05
0
05
1
M(0)(J120587
)
J120587
gA
(c) 82Se(0]120573120573 0+gs rarr 0+
1
)
0 1 2 3 4 5 6 7 8 9 10 11
minus05
0
05
1
M(0)(J120587
)
J120587+minus +minus +minus +minus +minus +minus +minus +minus +minus +minus +minus +minus
= 126gA
(d) 82Se(0]120573120573 0+gs rarr 0+
1
)
Figure 5 Multipole decompositions for the ground-state-to-ground-state decay of the nucleus 76Ge ((a) and (b) panels) and for the ground-state-to-excited-state decay of the nucleus 82Se ((c) and (d) panels)The value119892A = 100was used for (a) and (c) panels and the value119892A = 126for (b) and (d) panels
the level structures found in all intermediate odd-odd nucleiconsidered in this work This is due to the general feature ofodd-odd nuclei the extremely high density of states even atlow energies This high density of nuclear states becomes aproblem not only for the pnQRPA but for any other nuclearmany-body approach including the nuclear shell model Thereason for this is that even small perturbations in the two-body interactionmatrix elements tend to change the orderingof the levels at random For this reason the spectra of the odd-odd intermediate nuclei are not a very good measure of thereliability of the calculations but instead a better way is toadjust the model parameters in such a way that the transitionrates of some other known processes for example single or2]120573120573 decays can be reproduced by the theory and this is thephilosophy which we have followed in this work
32 Ground-State-to-Excited-State Decays Let us then con-sider 0+gs rarr 0
+
1
transitions mediated by the light neutrinoexchange In Figures 3(a) and 3(b) we have plotted themultipole decomposition of the l-NMEs corresponding to the119860 = 76 and 96 nuclear systems The multipole distributionsfor the excited-state transitions are greatly different fromthose corresponding to the ground-state transitions Usuallythere is only a couple of multipoles 0+ and 1
+ whichgive by far the largest contribution to the NMEs In thissense the excited-state transitions are more simple than theground-state transitions Typical example is the nucleus 76Gedisplayed in Figure 3(a) One nucleus deviating from this
trend is 96Zr which is presented in Figure 3(b) Its multipoledistribution resembles somewhat more those shown for theground-state decays in Figures 1(a) and 1(b) Most of thisdiffering behaviour can be traced back to the one-phononstructure of the final 0+
1
excited state in the nucleus 96MoThe 0+
1
final states in this work are modeled as one-phononbasic QRPA excitations for the daughter nuclei 96Mo and116Sn Rest of the final states are modeled as two-quadrupole-phonon states Nucleus 96Zr is an exceptional case since the0
+
1
state in 96Mo has a relatively low excitation energy andthus boasts rather strong collective features This is why theexcited-state transition has a wide multipole distribution andis greatly enhanced
Again we can divide nuclei into different groups byconsidering the shapes of their total cumulative sum dis-tributions For 0+gs rarr 0
+
1
transitions via light neutrinoexchange we can differentiate two types of nuclei Type 1nuclei belonging to type 1 are 76Ge 82Se 124Sn 130Te and136Xe Typical examples of this type 76Ge 82Se and 136Xeare shown in Figures 4(a) 4(b) and 4(d) Characteristicfeature of this type is that there exist only few energy stateswhich give most of the total matrix element producing astaircase-like structure as seen in the panels For examplefor 76Ge there seems to be only five such energy statesType 2 nuclei belonging to this type are 96Zr 100Mo 110Pdand 116Cd Typical examples of this type are 96Zr and 116Cdshown in Figures 4(c) and 4(e) Characteristic feature oftype 2 is that a large number of intermediate states give
8 Advances in High Energy Physics
0 20 3010Energy (MeV)
0
1
2
3
4
5
6
7Cu
mul
ativ
e NM
E co
ntrib
utio
n
gA = 100
gA = 126
(a) 100Mo(0]120573120573 0+gs rarr 0+
gs)
0
1
2
3
4
5
6
7
Cum
ulat
ive N
ME
cont
ribut
ion
10 20 300Energy (MeV)
gA = 100
gA = 126
(b) 116Cd(0]120573120573 0+gs rarr 0+
gs)
0
1
2
Cum
ulat
ive N
ME
cont
ribut
ion
10 200Energy (MeV)
gA = 100
gA = 126
(c) 82Se(0]120573120573 0+gs rarr 0+
1
)
0
1
2
3
4
5
6
7
Cum
ulat
ive N
ME
cont
ribut
ion
10 20 300Energy (MeV)
gA = 100
gA = 126
(d) 96Zr(0]120573120573 0+gs rarr 0+
1
)
Figure 6 Cumulative values of the l-NMEs for ground-state-to-ground-state decays of the nuclei 100Mo and 116Cd ((a) and (b) panels) andfor the ground-state-to-excited-state decays of the nuclei 82Se and 96Zr ((c) and (d) panels) The horizontal axis gives the excitation energiesof the intermediate states contributing to the 0]120573120573 transition Two different values for the axial coupling were used as indicated in the panels
important contributions to the NMEs In case of 116Cd panel(e) around 50 of the total NME comes from transitionsthrough the ground state of the intermediate nucleus Theother 50 is distributed rather evenly on the interval 0ndash20MeV
Using the multipole decompositions we extracted themost important multipole components contributing to thelight neutrino mediated 0
+
gs rarr 0
+
1
decay transitionsThese most important components were then again dividedinto contributions coming from different energy levels of
the 0]120573120573 intermediate nucleus These contributions arecollected into Table 4 for119860 = 76ndash116 systems and into Table 5for 119860 = 124ndash136 systems Again we notice that often onlya few intermediate states give the largest contribution to thedominant multipoles 1+ and 0+ Extreme case is the nucleus116Cd forwhich the dominant intermediate ground state gives81 of the total 1+ strength Combining this with the factthat 1+ is by far the largest multipole component we get arather good approximation for the total NME by consideringjust a single virtual transition through the 1+ ground state
Advances in High Energy Physics 9
Table 2 Most important multipoles and intermediate states contributing to the ground-state-to-ground-state 0]120573120573 decays mediated by thelight neutrino exchange Columns 119864 give the energies (in MeVs) and multipoles of the intermediate states Multipoles are organized from leftto right in terms of their importance themost important being on the left Columns labeled119862 give the correspondingNMEcontributions Lasttwo numbers in each 119862 column give the summed contribution and the percentual part which the displayed states give to the total multipolestrength The percentage inside the parenthesis gives the fraction with which the displayed states contribute to the total NME
Nucleus 119864(1
minus
) 119862 119864(2
minus
) 119862 119864(1
+
) 119862 119864(3
minus
) 119862 119864(2
+
) 119862
110Pd
295 0130 082 0387 000 0938 114 0118 033 0244319 0106 247 0091 470 minus0062 164 0128 095 0087344 0221 263 0163 961 0153 191 0080 868 0061381 0426 594 0107 975 minus0146 307 0053 873 0075452 0126 888 0161 1022 0079 361 0075 900 0061911 0109 954 minus0256 1516 minus0316 533 0066 907 0053
1630 0256 808 0167 916 0199837 minus0051 930 minus0366845 0052868 0275911 minus0468
1118 0653 0903 0496 041477 (13) 68 (7) 96 (10) 60 (6) 55 (5)
Nucleus 119864(1
minus
) 119862 119864(3
minus
) 119862 119864(2
minus
) 119862 119864(1
+
) 119862 119864(3
+
) 119862
116Cd
361 0223 172 0081 184 0065 000 0378 090 0158447 0099 230 0053 293 0287 837 minus0102 140 0068537 0118 279 0051 756 0075 951 0087 389 0057587 0140 724 0104 831 0246 1074 minus0083861 0102 834 0103 1120 00812307 0071 845 minus0262 1375 0080
969 minus0051 1379 minus00871573 minus03631583 04401651 minus00821683 0092
0753 0289 0463 0439 028472 (15) 65 (6) 106 (9) 102 (9) 76 (6)
Nucleus 119864(1
minus
) 119862 119864(1
+
) 119862 119864(2
+
) 119862 119864(2
minus
) 119862 119864(3
minus
) 119862
124Sn
168 0522 000 0690 023 0157 052 0271 036 0050482 0089 100 minus0067 060 0083 182 0225 049 0110654 0082 256 0252 106 0056 455 0051 195 0150957 0059 331 0153 215 0066 760 0057 967 00861074 0159 672 minus0130 711 0055 766 minus0064 970 00621407 0065 951 0098 1020 0183 982 minus00871446 minus0099 1309 minus0112 1035 minus0094 1264 minus00531483 minus00581636 minus00871805 minus0067
0664 0884 0417 0629 031855 (8) 95 (11) 53 (5) 89 (8) 49 (4)
of the intermediate nucleus 116In As for the ground-state-to-ground-state decays in some cases notable contributionsare coming from high excitation energies well above 10MeVThere are high-energy contributions in case of 1+ multipolefor all nuclei and in the cases of 2minus and 2+multipoles for 130Teand 136Xe
33 Effects of 119892A on the Intermediate-State Contributions Asmentioned earlier we have used in this work the quenchedvalue for the axial vector coupling 119892A = 100 Next we shallbriefly examine how our results will change if we increasethe value of 119892A from the quenched value 100 to the barevalue 126The effect of this amplification of the axial coupling
10 Advances in High Energy Physics
Table 3 Most important multipoles and intermediate states contributing to the ground-state-to-ground-state 0]120573120573 decays mediated by thelight neutrino exchange Columns 119864 give the energies (in MeVs) and multipoles of the intermediate states Multipoles are organized from leftto right in terms of their importance themost important being on the left Columns labeled119862 give the correspondingNMEcontributions Lasttwo numbers in each 119862 column give the summed contribution and the percentual part which the displayed states give to the total multipolestrength The percentage inside the parenthesis gives the fraction with which the displayed states contribute to the total NME
Nucleus 119864(1
minus
) 119862 119864(2
+
) 119862 119864(3
minus
) 119862 119864(2
minus
) 119862 119864(3
+
) 119862
128Te
422 0200 004 0066 016 0055 061 0335 002 0074472 0060 051 0052 058 0140 402 0060 237 0063621 0078 293 0084 397 0052 455 0101 622 0078644 0059 397 0053 1004 0061 489 minus0070 677 minus0065807 minus0084 677 0050 1014 0136 982 minus0062830 0151 1057 minus0056 1027 minus0057898 0068 1155 00581069 minus00521112 01821748 minus00791919 minus0100
0484 0305 0308 0564 015469 (8) 52 (5) 58 (5) 120 (10) 33 (3)
Nucleus 119864(1
minus
) 119862 119864(2
+
) 119862 119864(3
minus
) 119862 119864(3
+
) 119862 119864(2
minus
) 119862
130Te
418 0184 013 0054 084 0113 010 0056 097 0277572 0059 316 0089 036 0052 1025 0061627 0059 470 0064 260 0056 1133 minus0100856 0096 1049 minus0065 635 00631126 0115 1057 0145 683 00541777 minus0062 1097 minus0088 1231 00591951 minus0079 1644 0150 1241 minus0065
1654 minus01320373 0216 0113 0166 0237
61 (7) 40 (4) 25 (2) 39 (3) 56 (5)Nucleus 119864(2
+
) 119862 119864(3
+
) 119862 119864(1
minus
) 119862 119864(2
minus
) 119862 119864(4
+
) 119862
136Xe
159 0033 135 0033 665 0141 486 0188 234 0032226 0038 239 0054 732 0044 742 0068 244 0061250 0037 481 0050 1031 0053 443 0031529 0067 827 0035 476 0049657 0033 999 0027 767 0034705 0027 1065 minus0027 910 00451411 0087 965 minus00381425 minus0085
0238 0172 0238 0256 021462 (7) 53 (5) 76 (7) 85 (7) 71 (6)
strength on the NMEs is demonstrated in Figure 5 where wehave plotted the multipole decompositions for nuclei 76Geand 82Se calculated with both values of the axial coupling119892A = 100 and 119892A = 126 In case of the ground-state-to-ground-state decays the 1+ multipole changes rather fastwhen the axial coupling is increased from 100 to 126 Thishappens mainly due to the changing of the 119892pp parameter(for each 119892A value the parameter 119892pp is adjusted in sucha way that the measured 2]120573120573 rate is reproduced) The 1+multipole contribution is very sensitive to the value of 119892ppWe can see from Figures 5(a) and 5(b) that for 119892A = 100 the
1
+ component is among the five most important multipoleswhile for 119892A = 126 it is not Some of the higher multipoleschange also somewhat but not so rapidly Ground-state-to-excited-state transitions proceed mainly through the 0+ and1
+multipole channelsWe see from Figures 5(c) and 5(d) thatincreasing the value of 119892A affects mostly the 0+ component
Figure 6 displays the total cumulative sum distributionsfor ground-state-to-ground-state decays of the nuclei 100Moand 116Cd (panels (a) and (b)) and for ground-state-to-excited-state decays of the nuclei 82Se and 96Zr (panels (c)and (d)) Axial coupling values 119892A = 100 and 119892A = 126
Advances in High Energy Physics 11
Table 4 Most important multipoles and intermediate states contributing to the ground-state-to-excited-state 0]120573120573 decays mediated by thelight neutrino exchange Columns 119864 give the energies (in MeVs) and multipoles of the intermediate states Multipoles are organized from leftto right in terms of their importance themost important being on the left Columns labeled119862 give the correspondingNMEcontributions Lasttwo numbers in each 119862 column give the summed contribution and the percentual part which the displayed states give to the total multipolestrength The percentage inside the parenthesis gives the fraction with which the displayed states contribute to the total NME
Nucleus 119864(1
+
) 119862 119864(0
+
) 119862
76Ge
000 0390 286 0053295 0062 428 1204494 0501787 04351352 0522
1909 125692 (50) 99 (33)
Nucleus 119864(1
+
) 119862 119864(0
+
) 119862
82Se
019 0149 545 0471532 0179701 01451453 0274
0746 047190 (41) 96 (26)
Nucleus 119864(1
+
) 119862 119864(2
minus
) 119862 119864(2
+
) 119862 119864(0
+
) 119862 119864(1
minus
) 119862
96Zr
000 0700 092 0283 064 0122 537 0033 228 0032781 0051 966 0057 163 0180 664 0025 252 0080899 0058 537 0035 732 0037 446 00261165 0065 692 0027 802 0036 527 00681617 0354 1012 0396 1133 0026
1200 00131228 0340 0364 0540 0232
91 (23) 60 (6) 65 (7) 98 (10) 59 (4)Nucleus 119864(1
+
) 119862 119864(0
+
) 119862
100Mo
942 0090 901 00761362 0079 955 01501431 00721512 0063
0303 022665 (40) 77 (30)
Nucleus 119864(1
+
) 119862 119864(0
+
) 119862
110Pd
000 0308 1026 01651022 00591373 00831456 00551516 0072
0577 016576 (53) 70 (15)
Nucleus 119864(1
+
) 119862
116Cd000 0735
073581 (48)
12 Advances in High Energy Physics
Table 5 Most important multipoles and intermediate states contributing to the ground-state-to-excited-state 0]120573120573 decays mediated by thelight neutrino exchange Columns 119864 give the energies (in MeVs) and multipoles of the intermediate states Multipoles are organized from leftto right in terms of their importance themost important being on the left Columns labeled119862 give the correspondingNMEcontributions Lasttwo numbers in each 119862 column give the summed contribution and the percentual part which the displayed states give to the total multipolestrength The percentage inside the parenthesis gives the fraction with which the displayed states contribute to the total NME
Nucleus 119864(1
+
) 119862 119864(0
+
) 119862
124Sn
000 0101 270 0667066 0433 460 0088100 0146 735 0567225 0051256 0227331 0664672 0289791 04241309 04731386 0161
2968 132294 (44) 94 (19)
Nucleus 119864(1
+
) 119862 119864(0
+
) 119862 119864(2
minus
) 119862 119864(2
+
) 119862
130Te
025 0390 740 0719 097 0340 041 0019150 0082 835 0631 507 0073 059 0047232 0097 1726 0106 183 0017276 0134 1895 0083 280 0043453 0064 470 0024559 0338 486 0027757 0592 518 00331466 0065 627 00161501 0259 675 00511507 0773 910 00321539 0089 1057 0045
1787 00262314 0024
2882 1351 0602 039095 (41) 996 (19) 79 (9) 65 (6)
Nucleus 119864(1
+
) 119862 119864(0
+
) 119862 119864(2
minus
) 119862 119864(2
+
) 119862
136Xe
230 0366 1224 0317 250 0060 486 0204306 0050 1254 0712 486 0056 1858 0083375 0066 529 0113 2034 0059446 0226 834 00531012 0232 1411 00541152 05651712 01371778 1031
2675 1029 0335 034693 (43) 101 (16) 53 (5) 67 (6)
Advances in High Energy Physics 13
were adopted We notice from the figures that increasing theaxial coupling strength shifts the distributions downwardsThis is especially true for the higher energy parts Despite thisfact the overall shapes of the cumulative sum distributionsdo not changemuch and the same classification of nuclei intodifferent categories according to their cumulative distributionshapes seems to hold also for larger values of 119892A
4 Conclusions
In this paper we have extended our previous work [17] onthe ground-state-to-ground-state 0]120573120573 decay transitions Inthe present work we have concentrated our studies on theintermediate contributions to the NMEs involved in thelight neutrino mediated 0]120573120573 decay We have calculated theintermediate state multipole decompositions of the NMEsand extracted the most important multipole componentsCumulative sums of the NMEs were calculated to investigatethe important energy regions contributing to the 0]120573120573 tran-sitions Finally the most important multipole componentswere divided into contributions coming from the virtualtransitions through the individual states of the 0]120573120573 inter-mediate nuclei An extensive tabulation of these importantintermediate states were given for all the nuclei considered inthis paper
We have done these computations by using realistic two-body interactions and single-particle bases All the appro-priate short-range correlations nucleon form factors andhigher-order nucleonic weak currents are included in ourpresent results
We found in the calculations that often there exists onlya few relevant intermediate states which collect most of thestrength corresponding to a given multipole We also foundthat there exists a single-state dominance in the important 2minuscomponents related to the ground-state decays of nuclei 76Ge82Se and perhaps also for 96Zr
Competing Interests
The authors declare that they have no competing interests
Acknowledgments
This work has been partially supported by the Academy ofFinland under the Finnish Centre of Excellence Programme2012ndash2017 (Nuclear and Accelerator Based Programme atJYFL)
References
[1] J Suhonen and O Civitarese ldquoWeak-interaction and nuclear-structure aspects of nuclear double beta decayrdquo Physics Reportvol 300 no 3-4 pp 123ndash214 1998
[2] F T Avignone S R Elliott and J Engel ldquoDouble beta decayMajorana neutrinos and neutrino massrdquo Reviews of ModernPhysics vol 80 no 2 pp 481ndash516 2008
[3] J D Vergados H Ejiri and F Simkovic ldquoTheory of neutrinolessdouble-beta decayrdquoReports on Progress in Physics vol 75 no 10p 106301 2012
[4] J Maalampi and J Suhonen ldquoNeutrinoless double 120573
+ECdecaysrdquo Advances in High Energy Physics vol 2013 Article ID505874 18 pages 2013
[5] M Kortelainen O Civitarese J Suhonen and J ToivanenldquoShort-range correlations and neutrinoless double beta decayrdquoPhysics Letters Section B Nuclear Elementary Particle andHigh-Energy Physics vol 647 no 2-3 pp 128ndash132 2007
[6] F Simkovic G Pantis J D Vergados and A Faessler ldquoAddi-tional nucleon current contributions to neutrinoless double 120573decayrdquo Physical Review C vol 60 no 5 Article ID 055502 14pages 1999
[7] J Suhonen and O Civitarese ldquoReview of the properties of the0]120573minus120573minus nuclear matrix elementsrdquo Journal of Physics G Nuclearand Particle Physics vol 39 no 12 Article ID 124005 2012
[8] P Vogel ldquoNuclear structure and double beta decayrdquo Journal ofPhysics G Nuclear and Particle Physics vol 39 no 12 p 1240022012
[9] J Barea J Kotila and F Iachello ldquoNuclear matrix elements fordouble-120573 decayrdquo Physical Review C vol 87 no 1 Article ID014315 2013
[10] A Neacsu and S Stoica ldquoStudy of nuclear effects in thecomputation of the 0]120573120573 decay matrix elementsrdquo Journal ofPhysics G Nuclear and Particle Physics vol 41 no 1 Article ID015201 2014
[11] J Engel ldquoUncertainties in nuclear matrix elements for neutri-noless double-beta decayrdquo Journal of Physics G Nuclear andParticle Physics vol 42 no 3 Article ID 034017 2015
[12] F Simkovic A Faessler V Rodin P Vogel and J EngelldquoAnatomyof the 0]120573120573nuclearmatrix elementsrdquoPhysical ReviewC Nuclear Physics vol 77 no 4 Article ID 045503 2008
[13] M T Mustonen and J Engel ldquoLarge-scale calculations ofthe double-120573 decay of 76Ge 130Te 136Xe and 150Nd in thedeformed self-consistent Skyrme quasiparticle random-phaseapproximationrdquo Physical Review C vol 87 Article ID 0643022013
[14] F Simkovic V Rodin A Faessler and P Vogel ldquo0]120573120573 and2]120573120573 nuclear matrix elements quasiparticle random-phaseapproximation and isospin symmetry restorationrdquo PhysicalReview C vol 87 no 4 Article ID 045501 9 pages 2013
[15] V A Rodin A Faessler F Simkovic and P Vogel ldquoAssessmentof uncertainties in QRPA 0]120573120573-decay nuclear matrix elementsrdquoNuclear Physics A vol 766 pp 107ndash131 2006
[16] D-L Fang A Faessler V Rodin and F Simkovic ldquoNeutri-noless double-120573 decay of deformed nuclei within quasiparticlerandom-phase approximationwith a realistic interactionrdquoPhys-ical Review C vol 83 no 3 Article ID 034320 8 pages 2011
[17] J Hyvarinen and J Suhonen ldquoNuclear matrix elements for0]120573120573 decays with light or heavy Majorana-neutrino exchangerdquoPhysical Review C vol 91 no 2 Article ID 024613 12 pages2015
[18] P Pirinen and J Suhonen ldquoSystematic approach to 120573 and 2]120573120573decays of mass119860 = 100minus136 nucleirdquo Physical Review C vol 91no 5 Article ID 054309 2015
[19] H Ejiri N Soukouti and J Suhonen ldquoSpin-dipole nuclearmatrix elements for double beta decays and astro-neutrinosrdquoPhysics Letters B vol 729 pp 27ndash32 2014
[20] J Menendez D Gazit and A Schwenk ldquoChiral two-bodycurrents in nuclei Gamow-Teller transitions and neutrinolessdouble-beta decayrdquo Physical Review Letters vol 107 no 6Article ID 062501 5 pages 2011
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Superconductivity
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âComputationalââMethodsâinâPhysics
Journal of
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Soft MatterJournal of
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Journal of
Biophysics
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
ThermodynamicsJournal of
6 Advances in High Energy Physics
Table 1 Most important multipoles and intermediate states contributing to the ground-state-to-ground-state 0]120573120573 decays mediated by thelight neutrino exchange Columns 119864 give the energies (in MeVs) and multipoles of the intermediate states Multipoles are organized from leftto right in terms of their importance themost important being on the left Columns labeled119862 give the correspondingNMEcontributions Lasttwo numbers in each 119862 column give the summed contribution and the percentual part which the displayed states give to the total multipolestrength The percentage inside the parenthesis gives the fraction with which the displayed states contribute to the total NME
Nucleus 119864(2
minus
) 119862 119864(1
minus
) 119862 119864(2
+
) 119862 119864(1
+
) 119862 119864(3
minus
) 119862
76Ge
022 0748 587 0155 051 0166 000 0344 030 0189632 0058 187 0056 409 0199 092 0148700 0077 304 0094 448 0148 677 0056716 0064 362 0108 494 0578 685 0053827 0190 481 0058 1080 minus0053 1163 00621104 0084 773 0054 1175 minus0109 1207 minus00531203 0165 1352 minus05221670 0158
0748 0635 0556 0668 045479 (11) 83 (10) 77 (9) 88 (10) 70 (7)
Nucleus 119864(2
minus
) 119862 119864(2
+
) 119862 119864(1
minus
) 119862 119864(3
minus
) 119862 119864(1
+
) 119862
82Se
000 0510 065 0137 527 0116 007 0140 019 0264173 0065 685 0065 082 0138 316 0095222 0051 798 0134 407 0065356 0084 979 0083 455 0105405 0052 1225 0065 532 0559493 0054 1741 0070 701 minus0253
1453 minus03960510 0442 0393 0278 0439
81 (11) 81 (9) 76 (8) 57 (6) 92 (9)Nucleus 119864(1
minus
) 119862 119864(2
minus
) 119862 119864(3
minus
) 119862 119864(2
+
) 119862 119864(4
+
) 119862
96Zr
175 0056 092 0498 135 0151 064 0150 105 0099228 0063 221 0065 235 0051 163 0114 168 0071252 0150 375 0050 777 0064 569 0064446 0050 443 0052 1136 minus0091504 0077 853 0057527 0209 877 minus0056865 00601133 0061
0728 0666 0175 0265 023483 (16) 96 (15) 31 (4) 59 (6) 69 (5)
Nucleus 119864(1
minus
) 119862 119864(2
+
) 119862 119864(4
+
) 119862 119864(3
+
) 119862 119864(2
minus
) 119862
100Mo
312 0307 090 0192 132 0158 133 0202 176 0205441 0109 210 0154 226 0097 168 0076 285 0147668 0091 1168 0082 743 0152 552 minus00611115 0117 1197 minus0070 776 minus0173 1017 02191672 0058 1093 minus01942026 minus00602390 minus0060
0562 0359 0255 0257 031670 (11) 62 (7) 53 (5) 54 (5) 72 (6)
Advances in High Energy Physics 7
+minus0
+minus1
+minus2
+minus3
+minus4
+minus5
+minus6
+minus7
+minus8
+minus9
+minus10
+minus11
minus05
0
05
1
M(0)(J120587
)
J120587
= 100gA
(a) 76Ge(0]120573120573 0+gs rarr 0+
gs)
+minus0
+minus1
+minus2
+minus3
+minus4
+minus5
+minus6
+minus7
+minus8
+minus9
+minus10
+minus11
minus05
0
05
1
M(0)(J120587
)
J120587
= 126gA
(b) 76Ge(0]120573120573 0+gs rarr 0+
gs)
= 100
+minus0
+minus1
+minus2
+minus3
+minus4
+minus5
+minus6
+minus7
+minus8
+minus9
+minus10
+minus11
minus05
0
05
1
M(0)(J120587
)
J120587
gA
(c) 82Se(0]120573120573 0+gs rarr 0+
1
)
0 1 2 3 4 5 6 7 8 9 10 11
minus05
0
05
1
M(0)(J120587
)
J120587+minus +minus +minus +minus +minus +minus +minus +minus +minus +minus +minus +minus
= 126gA
(d) 82Se(0]120573120573 0+gs rarr 0+
1
)
Figure 5 Multipole decompositions for the ground-state-to-ground-state decay of the nucleus 76Ge ((a) and (b) panels) and for the ground-state-to-excited-state decay of the nucleus 82Se ((c) and (d) panels)The value119892A = 100was used for (a) and (c) panels and the value119892A = 126for (b) and (d) panels
the level structures found in all intermediate odd-odd nucleiconsidered in this work This is due to the general feature ofodd-odd nuclei the extremely high density of states even atlow energies This high density of nuclear states becomes aproblem not only for the pnQRPA but for any other nuclearmany-body approach including the nuclear shell model Thereason for this is that even small perturbations in the two-body interactionmatrix elements tend to change the orderingof the levels at random For this reason the spectra of the odd-odd intermediate nuclei are not a very good measure of thereliability of the calculations but instead a better way is toadjust the model parameters in such a way that the transitionrates of some other known processes for example single or2]120573120573 decays can be reproduced by the theory and this is thephilosophy which we have followed in this work
32 Ground-State-to-Excited-State Decays Let us then con-sider 0+gs rarr 0
+
1
transitions mediated by the light neutrinoexchange In Figures 3(a) and 3(b) we have plotted themultipole decomposition of the l-NMEs corresponding to the119860 = 76 and 96 nuclear systems The multipole distributionsfor the excited-state transitions are greatly different fromthose corresponding to the ground-state transitions Usuallythere is only a couple of multipoles 0+ and 1
+ whichgive by far the largest contribution to the NMEs In thissense the excited-state transitions are more simple than theground-state transitions Typical example is the nucleus 76Gedisplayed in Figure 3(a) One nucleus deviating from this
trend is 96Zr which is presented in Figure 3(b) Its multipoledistribution resembles somewhat more those shown for theground-state decays in Figures 1(a) and 1(b) Most of thisdiffering behaviour can be traced back to the one-phononstructure of the final 0+
1
excited state in the nucleus 96MoThe 0+
1
final states in this work are modeled as one-phononbasic QRPA excitations for the daughter nuclei 96Mo and116Sn Rest of the final states are modeled as two-quadrupole-phonon states Nucleus 96Zr is an exceptional case since the0
+
1
state in 96Mo has a relatively low excitation energy andthus boasts rather strong collective features This is why theexcited-state transition has a wide multipole distribution andis greatly enhanced
Again we can divide nuclei into different groups byconsidering the shapes of their total cumulative sum dis-tributions For 0+gs rarr 0
+
1
transitions via light neutrinoexchange we can differentiate two types of nuclei Type 1nuclei belonging to type 1 are 76Ge 82Se 124Sn 130Te and136Xe Typical examples of this type 76Ge 82Se and 136Xeare shown in Figures 4(a) 4(b) and 4(d) Characteristicfeature of this type is that there exist only few energy stateswhich give most of the total matrix element producing astaircase-like structure as seen in the panels For examplefor 76Ge there seems to be only five such energy statesType 2 nuclei belonging to this type are 96Zr 100Mo 110Pdand 116Cd Typical examples of this type are 96Zr and 116Cdshown in Figures 4(c) and 4(e) Characteristic feature oftype 2 is that a large number of intermediate states give
8 Advances in High Energy Physics
0 20 3010Energy (MeV)
0
1
2
3
4
5
6
7Cu
mul
ativ
e NM
E co
ntrib
utio
n
gA = 100
gA = 126
(a) 100Mo(0]120573120573 0+gs rarr 0+
gs)
0
1
2
3
4
5
6
7
Cum
ulat
ive N
ME
cont
ribut
ion
10 20 300Energy (MeV)
gA = 100
gA = 126
(b) 116Cd(0]120573120573 0+gs rarr 0+
gs)
0
1
2
Cum
ulat
ive N
ME
cont
ribut
ion
10 200Energy (MeV)
gA = 100
gA = 126
(c) 82Se(0]120573120573 0+gs rarr 0+
1
)
0
1
2
3
4
5
6
7
Cum
ulat
ive N
ME
cont
ribut
ion
10 20 300Energy (MeV)
gA = 100
gA = 126
(d) 96Zr(0]120573120573 0+gs rarr 0+
1
)
Figure 6 Cumulative values of the l-NMEs for ground-state-to-ground-state decays of the nuclei 100Mo and 116Cd ((a) and (b) panels) andfor the ground-state-to-excited-state decays of the nuclei 82Se and 96Zr ((c) and (d) panels) The horizontal axis gives the excitation energiesof the intermediate states contributing to the 0]120573120573 transition Two different values for the axial coupling were used as indicated in the panels
important contributions to the NMEs In case of 116Cd panel(e) around 50 of the total NME comes from transitionsthrough the ground state of the intermediate nucleus Theother 50 is distributed rather evenly on the interval 0ndash20MeV
Using the multipole decompositions we extracted themost important multipole components contributing to thelight neutrino mediated 0
+
gs rarr 0
+
1
decay transitionsThese most important components were then again dividedinto contributions coming from different energy levels of
the 0]120573120573 intermediate nucleus These contributions arecollected into Table 4 for119860 = 76ndash116 systems and into Table 5for 119860 = 124ndash136 systems Again we notice that often onlya few intermediate states give the largest contribution to thedominant multipoles 1+ and 0+ Extreme case is the nucleus116Cd forwhich the dominant intermediate ground state gives81 of the total 1+ strength Combining this with the factthat 1+ is by far the largest multipole component we get arather good approximation for the total NME by consideringjust a single virtual transition through the 1+ ground state
Advances in High Energy Physics 9
Table 2 Most important multipoles and intermediate states contributing to the ground-state-to-ground-state 0]120573120573 decays mediated by thelight neutrino exchange Columns 119864 give the energies (in MeVs) and multipoles of the intermediate states Multipoles are organized from leftto right in terms of their importance themost important being on the left Columns labeled119862 give the correspondingNMEcontributions Lasttwo numbers in each 119862 column give the summed contribution and the percentual part which the displayed states give to the total multipolestrength The percentage inside the parenthesis gives the fraction with which the displayed states contribute to the total NME
Nucleus 119864(1
minus
) 119862 119864(2
minus
) 119862 119864(1
+
) 119862 119864(3
minus
) 119862 119864(2
+
) 119862
110Pd
295 0130 082 0387 000 0938 114 0118 033 0244319 0106 247 0091 470 minus0062 164 0128 095 0087344 0221 263 0163 961 0153 191 0080 868 0061381 0426 594 0107 975 minus0146 307 0053 873 0075452 0126 888 0161 1022 0079 361 0075 900 0061911 0109 954 minus0256 1516 minus0316 533 0066 907 0053
1630 0256 808 0167 916 0199837 minus0051 930 minus0366845 0052868 0275911 minus0468
1118 0653 0903 0496 041477 (13) 68 (7) 96 (10) 60 (6) 55 (5)
Nucleus 119864(1
minus
) 119862 119864(3
minus
) 119862 119864(2
minus
) 119862 119864(1
+
) 119862 119864(3
+
) 119862
116Cd
361 0223 172 0081 184 0065 000 0378 090 0158447 0099 230 0053 293 0287 837 minus0102 140 0068537 0118 279 0051 756 0075 951 0087 389 0057587 0140 724 0104 831 0246 1074 minus0083861 0102 834 0103 1120 00812307 0071 845 minus0262 1375 0080
969 minus0051 1379 minus00871573 minus03631583 04401651 minus00821683 0092
0753 0289 0463 0439 028472 (15) 65 (6) 106 (9) 102 (9) 76 (6)
Nucleus 119864(1
minus
) 119862 119864(1
+
) 119862 119864(2
+
) 119862 119864(2
minus
) 119862 119864(3
minus
) 119862
124Sn
168 0522 000 0690 023 0157 052 0271 036 0050482 0089 100 minus0067 060 0083 182 0225 049 0110654 0082 256 0252 106 0056 455 0051 195 0150957 0059 331 0153 215 0066 760 0057 967 00861074 0159 672 minus0130 711 0055 766 minus0064 970 00621407 0065 951 0098 1020 0183 982 minus00871446 minus0099 1309 minus0112 1035 minus0094 1264 minus00531483 minus00581636 minus00871805 minus0067
0664 0884 0417 0629 031855 (8) 95 (11) 53 (5) 89 (8) 49 (4)
of the intermediate nucleus 116In As for the ground-state-to-ground-state decays in some cases notable contributionsare coming from high excitation energies well above 10MeVThere are high-energy contributions in case of 1+ multipolefor all nuclei and in the cases of 2minus and 2+multipoles for 130Teand 136Xe
33 Effects of 119892A on the Intermediate-State Contributions Asmentioned earlier we have used in this work the quenchedvalue for the axial vector coupling 119892A = 100 Next we shallbriefly examine how our results will change if we increasethe value of 119892A from the quenched value 100 to the barevalue 126The effect of this amplification of the axial coupling
10 Advances in High Energy Physics
Table 3 Most important multipoles and intermediate states contributing to the ground-state-to-ground-state 0]120573120573 decays mediated by thelight neutrino exchange Columns 119864 give the energies (in MeVs) and multipoles of the intermediate states Multipoles are organized from leftto right in terms of their importance themost important being on the left Columns labeled119862 give the correspondingNMEcontributions Lasttwo numbers in each 119862 column give the summed contribution and the percentual part which the displayed states give to the total multipolestrength The percentage inside the parenthesis gives the fraction with which the displayed states contribute to the total NME
Nucleus 119864(1
minus
) 119862 119864(2
+
) 119862 119864(3
minus
) 119862 119864(2
minus
) 119862 119864(3
+
) 119862
128Te
422 0200 004 0066 016 0055 061 0335 002 0074472 0060 051 0052 058 0140 402 0060 237 0063621 0078 293 0084 397 0052 455 0101 622 0078644 0059 397 0053 1004 0061 489 minus0070 677 minus0065807 minus0084 677 0050 1014 0136 982 minus0062830 0151 1057 minus0056 1027 minus0057898 0068 1155 00581069 minus00521112 01821748 minus00791919 minus0100
0484 0305 0308 0564 015469 (8) 52 (5) 58 (5) 120 (10) 33 (3)
Nucleus 119864(1
minus
) 119862 119864(2
+
) 119862 119864(3
minus
) 119862 119864(3
+
) 119862 119864(2
minus
) 119862
130Te
418 0184 013 0054 084 0113 010 0056 097 0277572 0059 316 0089 036 0052 1025 0061627 0059 470 0064 260 0056 1133 minus0100856 0096 1049 minus0065 635 00631126 0115 1057 0145 683 00541777 minus0062 1097 minus0088 1231 00591951 minus0079 1644 0150 1241 minus0065
1654 minus01320373 0216 0113 0166 0237
61 (7) 40 (4) 25 (2) 39 (3) 56 (5)Nucleus 119864(2
+
) 119862 119864(3
+
) 119862 119864(1
minus
) 119862 119864(2
minus
) 119862 119864(4
+
) 119862
136Xe
159 0033 135 0033 665 0141 486 0188 234 0032226 0038 239 0054 732 0044 742 0068 244 0061250 0037 481 0050 1031 0053 443 0031529 0067 827 0035 476 0049657 0033 999 0027 767 0034705 0027 1065 minus0027 910 00451411 0087 965 minus00381425 minus0085
0238 0172 0238 0256 021462 (7) 53 (5) 76 (7) 85 (7) 71 (6)
strength on the NMEs is demonstrated in Figure 5 where wehave plotted the multipole decompositions for nuclei 76Geand 82Se calculated with both values of the axial coupling119892A = 100 and 119892A = 126 In case of the ground-state-to-ground-state decays the 1+ multipole changes rather fastwhen the axial coupling is increased from 100 to 126 Thishappens mainly due to the changing of the 119892pp parameter(for each 119892A value the parameter 119892pp is adjusted in sucha way that the measured 2]120573120573 rate is reproduced) The 1+multipole contribution is very sensitive to the value of 119892ppWe can see from Figures 5(a) and 5(b) that for 119892A = 100 the
1
+ component is among the five most important multipoleswhile for 119892A = 126 it is not Some of the higher multipoleschange also somewhat but not so rapidly Ground-state-to-excited-state transitions proceed mainly through the 0+ and1
+multipole channelsWe see from Figures 5(c) and 5(d) thatincreasing the value of 119892A affects mostly the 0+ component
Figure 6 displays the total cumulative sum distributionsfor ground-state-to-ground-state decays of the nuclei 100Moand 116Cd (panels (a) and (b)) and for ground-state-to-excited-state decays of the nuclei 82Se and 96Zr (panels (c)and (d)) Axial coupling values 119892A = 100 and 119892A = 126
Advances in High Energy Physics 11
Table 4 Most important multipoles and intermediate states contributing to the ground-state-to-excited-state 0]120573120573 decays mediated by thelight neutrino exchange Columns 119864 give the energies (in MeVs) and multipoles of the intermediate states Multipoles are organized from leftto right in terms of their importance themost important being on the left Columns labeled119862 give the correspondingNMEcontributions Lasttwo numbers in each 119862 column give the summed contribution and the percentual part which the displayed states give to the total multipolestrength The percentage inside the parenthesis gives the fraction with which the displayed states contribute to the total NME
Nucleus 119864(1
+
) 119862 119864(0
+
) 119862
76Ge
000 0390 286 0053295 0062 428 1204494 0501787 04351352 0522
1909 125692 (50) 99 (33)
Nucleus 119864(1
+
) 119862 119864(0
+
) 119862
82Se
019 0149 545 0471532 0179701 01451453 0274
0746 047190 (41) 96 (26)
Nucleus 119864(1
+
) 119862 119864(2
minus
) 119862 119864(2
+
) 119862 119864(0
+
) 119862 119864(1
minus
) 119862
96Zr
000 0700 092 0283 064 0122 537 0033 228 0032781 0051 966 0057 163 0180 664 0025 252 0080899 0058 537 0035 732 0037 446 00261165 0065 692 0027 802 0036 527 00681617 0354 1012 0396 1133 0026
1200 00131228 0340 0364 0540 0232
91 (23) 60 (6) 65 (7) 98 (10) 59 (4)Nucleus 119864(1
+
) 119862 119864(0
+
) 119862
100Mo
942 0090 901 00761362 0079 955 01501431 00721512 0063
0303 022665 (40) 77 (30)
Nucleus 119864(1
+
) 119862 119864(0
+
) 119862
110Pd
000 0308 1026 01651022 00591373 00831456 00551516 0072
0577 016576 (53) 70 (15)
Nucleus 119864(1
+
) 119862
116Cd000 0735
073581 (48)
12 Advances in High Energy Physics
Table 5 Most important multipoles and intermediate states contributing to the ground-state-to-excited-state 0]120573120573 decays mediated by thelight neutrino exchange Columns 119864 give the energies (in MeVs) and multipoles of the intermediate states Multipoles are organized from leftto right in terms of their importance themost important being on the left Columns labeled119862 give the correspondingNMEcontributions Lasttwo numbers in each 119862 column give the summed contribution and the percentual part which the displayed states give to the total multipolestrength The percentage inside the parenthesis gives the fraction with which the displayed states contribute to the total NME
Nucleus 119864(1
+
) 119862 119864(0
+
) 119862
124Sn
000 0101 270 0667066 0433 460 0088100 0146 735 0567225 0051256 0227331 0664672 0289791 04241309 04731386 0161
2968 132294 (44) 94 (19)
Nucleus 119864(1
+
) 119862 119864(0
+
) 119862 119864(2
minus
) 119862 119864(2
+
) 119862
130Te
025 0390 740 0719 097 0340 041 0019150 0082 835 0631 507 0073 059 0047232 0097 1726 0106 183 0017276 0134 1895 0083 280 0043453 0064 470 0024559 0338 486 0027757 0592 518 00331466 0065 627 00161501 0259 675 00511507 0773 910 00321539 0089 1057 0045
1787 00262314 0024
2882 1351 0602 039095 (41) 996 (19) 79 (9) 65 (6)
Nucleus 119864(1
+
) 119862 119864(0
+
) 119862 119864(2
minus
) 119862 119864(2
+
) 119862
136Xe
230 0366 1224 0317 250 0060 486 0204306 0050 1254 0712 486 0056 1858 0083375 0066 529 0113 2034 0059446 0226 834 00531012 0232 1411 00541152 05651712 01371778 1031
2675 1029 0335 034693 (43) 101 (16) 53 (5) 67 (6)
Advances in High Energy Physics 13
were adopted We notice from the figures that increasing theaxial coupling strength shifts the distributions downwardsThis is especially true for the higher energy parts Despite thisfact the overall shapes of the cumulative sum distributionsdo not changemuch and the same classification of nuclei intodifferent categories according to their cumulative distributionshapes seems to hold also for larger values of 119892A
4 Conclusions
In this paper we have extended our previous work [17] onthe ground-state-to-ground-state 0]120573120573 decay transitions Inthe present work we have concentrated our studies on theintermediate contributions to the NMEs involved in thelight neutrino mediated 0]120573120573 decay We have calculated theintermediate state multipole decompositions of the NMEsand extracted the most important multipole componentsCumulative sums of the NMEs were calculated to investigatethe important energy regions contributing to the 0]120573120573 tran-sitions Finally the most important multipole componentswere divided into contributions coming from the virtualtransitions through the individual states of the 0]120573120573 inter-mediate nuclei An extensive tabulation of these importantintermediate states were given for all the nuclei considered inthis paper
We have done these computations by using realistic two-body interactions and single-particle bases All the appro-priate short-range correlations nucleon form factors andhigher-order nucleonic weak currents are included in ourpresent results
We found in the calculations that often there exists onlya few relevant intermediate states which collect most of thestrength corresponding to a given multipole We also foundthat there exists a single-state dominance in the important 2minuscomponents related to the ground-state decays of nuclei 76Ge82Se and perhaps also for 96Zr
Competing Interests
The authors declare that they have no competing interests
Acknowledgments
This work has been partially supported by the Academy ofFinland under the Finnish Centre of Excellence Programme2012ndash2017 (Nuclear and Accelerator Based Programme atJYFL)
References
[1] J Suhonen and O Civitarese ldquoWeak-interaction and nuclear-structure aspects of nuclear double beta decayrdquo Physics Reportvol 300 no 3-4 pp 123ndash214 1998
[2] F T Avignone S R Elliott and J Engel ldquoDouble beta decayMajorana neutrinos and neutrino massrdquo Reviews of ModernPhysics vol 80 no 2 pp 481ndash516 2008
[3] J D Vergados H Ejiri and F Simkovic ldquoTheory of neutrinolessdouble-beta decayrdquoReports on Progress in Physics vol 75 no 10p 106301 2012
[4] J Maalampi and J Suhonen ldquoNeutrinoless double 120573
+ECdecaysrdquo Advances in High Energy Physics vol 2013 Article ID505874 18 pages 2013
[5] M Kortelainen O Civitarese J Suhonen and J ToivanenldquoShort-range correlations and neutrinoless double beta decayrdquoPhysics Letters Section B Nuclear Elementary Particle andHigh-Energy Physics vol 647 no 2-3 pp 128ndash132 2007
[6] F Simkovic G Pantis J D Vergados and A Faessler ldquoAddi-tional nucleon current contributions to neutrinoless double 120573decayrdquo Physical Review C vol 60 no 5 Article ID 055502 14pages 1999
[7] J Suhonen and O Civitarese ldquoReview of the properties of the0]120573minus120573minus nuclear matrix elementsrdquo Journal of Physics G Nuclearand Particle Physics vol 39 no 12 Article ID 124005 2012
[8] P Vogel ldquoNuclear structure and double beta decayrdquo Journal ofPhysics G Nuclear and Particle Physics vol 39 no 12 p 1240022012
[9] J Barea J Kotila and F Iachello ldquoNuclear matrix elements fordouble-120573 decayrdquo Physical Review C vol 87 no 1 Article ID014315 2013
[10] A Neacsu and S Stoica ldquoStudy of nuclear effects in thecomputation of the 0]120573120573 decay matrix elementsrdquo Journal ofPhysics G Nuclear and Particle Physics vol 41 no 1 Article ID015201 2014
[11] J Engel ldquoUncertainties in nuclear matrix elements for neutri-noless double-beta decayrdquo Journal of Physics G Nuclear andParticle Physics vol 42 no 3 Article ID 034017 2015
[12] F Simkovic A Faessler V Rodin P Vogel and J EngelldquoAnatomyof the 0]120573120573nuclearmatrix elementsrdquoPhysical ReviewC Nuclear Physics vol 77 no 4 Article ID 045503 2008
[13] M T Mustonen and J Engel ldquoLarge-scale calculations ofthe double-120573 decay of 76Ge 130Te 136Xe and 150Nd in thedeformed self-consistent Skyrme quasiparticle random-phaseapproximationrdquo Physical Review C vol 87 Article ID 0643022013
[14] F Simkovic V Rodin A Faessler and P Vogel ldquo0]120573120573 and2]120573120573 nuclear matrix elements quasiparticle random-phaseapproximation and isospin symmetry restorationrdquo PhysicalReview C vol 87 no 4 Article ID 045501 9 pages 2013
[15] V A Rodin A Faessler F Simkovic and P Vogel ldquoAssessmentof uncertainties in QRPA 0]120573120573-decay nuclear matrix elementsrdquoNuclear Physics A vol 766 pp 107ndash131 2006
[16] D-L Fang A Faessler V Rodin and F Simkovic ldquoNeutri-noless double-120573 decay of deformed nuclei within quasiparticlerandom-phase approximationwith a realistic interactionrdquoPhys-ical Review C vol 83 no 3 Article ID 034320 8 pages 2011
[17] J Hyvarinen and J Suhonen ldquoNuclear matrix elements for0]120573120573 decays with light or heavy Majorana-neutrino exchangerdquoPhysical Review C vol 91 no 2 Article ID 024613 12 pages2015
[18] P Pirinen and J Suhonen ldquoSystematic approach to 120573 and 2]120573120573decays of mass119860 = 100minus136 nucleirdquo Physical Review C vol 91no 5 Article ID 054309 2015
[19] H Ejiri N Soukouti and J Suhonen ldquoSpin-dipole nuclearmatrix elements for double beta decays and astro-neutrinosrdquoPhysics Letters B vol 729 pp 27ndash32 2014
[20] J Menendez D Gazit and A Schwenk ldquoChiral two-bodycurrents in nuclei Gamow-Teller transitions and neutrinolessdouble-beta decayrdquo Physical Review Letters vol 107 no 6Article ID 062501 5 pages 2011
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Superconductivity
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âComputationalââMethodsâinâPhysics
Journal of
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ThermodynamicsJournal of
Advances in High Energy Physics 7
+minus0
+minus1
+minus2
+minus3
+minus4
+minus5
+minus6
+minus7
+minus8
+minus9
+minus10
+minus11
minus05
0
05
1
M(0)(J120587
)
J120587
= 100gA
(a) 76Ge(0]120573120573 0+gs rarr 0+
gs)
+minus0
+minus1
+minus2
+minus3
+minus4
+minus5
+minus6
+minus7
+minus8
+minus9
+minus10
+minus11
minus05
0
05
1
M(0)(J120587
)
J120587
= 126gA
(b) 76Ge(0]120573120573 0+gs rarr 0+
gs)
= 100
+minus0
+minus1
+minus2
+minus3
+minus4
+minus5
+minus6
+minus7
+minus8
+minus9
+minus10
+minus11
minus05
0
05
1
M(0)(J120587
)
J120587
gA
(c) 82Se(0]120573120573 0+gs rarr 0+
1
)
0 1 2 3 4 5 6 7 8 9 10 11
minus05
0
05
1
M(0)(J120587
)
J120587+minus +minus +minus +minus +minus +minus +minus +minus +minus +minus +minus +minus
= 126gA
(d) 82Se(0]120573120573 0+gs rarr 0+
1
)
Figure 5 Multipole decompositions for the ground-state-to-ground-state decay of the nucleus 76Ge ((a) and (b) panels) and for the ground-state-to-excited-state decay of the nucleus 82Se ((c) and (d) panels)The value119892A = 100was used for (a) and (c) panels and the value119892A = 126for (b) and (d) panels
the level structures found in all intermediate odd-odd nucleiconsidered in this work This is due to the general feature ofodd-odd nuclei the extremely high density of states even atlow energies This high density of nuclear states becomes aproblem not only for the pnQRPA but for any other nuclearmany-body approach including the nuclear shell model Thereason for this is that even small perturbations in the two-body interactionmatrix elements tend to change the orderingof the levels at random For this reason the spectra of the odd-odd intermediate nuclei are not a very good measure of thereliability of the calculations but instead a better way is toadjust the model parameters in such a way that the transitionrates of some other known processes for example single or2]120573120573 decays can be reproduced by the theory and this is thephilosophy which we have followed in this work
32 Ground-State-to-Excited-State Decays Let us then con-sider 0+gs rarr 0
+
1
transitions mediated by the light neutrinoexchange In Figures 3(a) and 3(b) we have plotted themultipole decomposition of the l-NMEs corresponding to the119860 = 76 and 96 nuclear systems The multipole distributionsfor the excited-state transitions are greatly different fromthose corresponding to the ground-state transitions Usuallythere is only a couple of multipoles 0+ and 1
+ whichgive by far the largest contribution to the NMEs In thissense the excited-state transitions are more simple than theground-state transitions Typical example is the nucleus 76Gedisplayed in Figure 3(a) One nucleus deviating from this
trend is 96Zr which is presented in Figure 3(b) Its multipoledistribution resembles somewhat more those shown for theground-state decays in Figures 1(a) and 1(b) Most of thisdiffering behaviour can be traced back to the one-phononstructure of the final 0+
1
excited state in the nucleus 96MoThe 0+
1
final states in this work are modeled as one-phononbasic QRPA excitations for the daughter nuclei 96Mo and116Sn Rest of the final states are modeled as two-quadrupole-phonon states Nucleus 96Zr is an exceptional case since the0
+
1
state in 96Mo has a relatively low excitation energy andthus boasts rather strong collective features This is why theexcited-state transition has a wide multipole distribution andis greatly enhanced
Again we can divide nuclei into different groups byconsidering the shapes of their total cumulative sum dis-tributions For 0+gs rarr 0
+
1
transitions via light neutrinoexchange we can differentiate two types of nuclei Type 1nuclei belonging to type 1 are 76Ge 82Se 124Sn 130Te and136Xe Typical examples of this type 76Ge 82Se and 136Xeare shown in Figures 4(a) 4(b) and 4(d) Characteristicfeature of this type is that there exist only few energy stateswhich give most of the total matrix element producing astaircase-like structure as seen in the panels For examplefor 76Ge there seems to be only five such energy statesType 2 nuclei belonging to this type are 96Zr 100Mo 110Pdand 116Cd Typical examples of this type are 96Zr and 116Cdshown in Figures 4(c) and 4(e) Characteristic feature oftype 2 is that a large number of intermediate states give
8 Advances in High Energy Physics
0 20 3010Energy (MeV)
0
1
2
3
4
5
6
7Cu
mul
ativ
e NM
E co
ntrib
utio
n
gA = 100
gA = 126
(a) 100Mo(0]120573120573 0+gs rarr 0+
gs)
0
1
2
3
4
5
6
7
Cum
ulat
ive N
ME
cont
ribut
ion
10 20 300Energy (MeV)
gA = 100
gA = 126
(b) 116Cd(0]120573120573 0+gs rarr 0+
gs)
0
1
2
Cum
ulat
ive N
ME
cont
ribut
ion
10 200Energy (MeV)
gA = 100
gA = 126
(c) 82Se(0]120573120573 0+gs rarr 0+
1
)
0
1
2
3
4
5
6
7
Cum
ulat
ive N
ME
cont
ribut
ion
10 20 300Energy (MeV)
gA = 100
gA = 126
(d) 96Zr(0]120573120573 0+gs rarr 0+
1
)
Figure 6 Cumulative values of the l-NMEs for ground-state-to-ground-state decays of the nuclei 100Mo and 116Cd ((a) and (b) panels) andfor the ground-state-to-excited-state decays of the nuclei 82Se and 96Zr ((c) and (d) panels) The horizontal axis gives the excitation energiesof the intermediate states contributing to the 0]120573120573 transition Two different values for the axial coupling were used as indicated in the panels
important contributions to the NMEs In case of 116Cd panel(e) around 50 of the total NME comes from transitionsthrough the ground state of the intermediate nucleus Theother 50 is distributed rather evenly on the interval 0ndash20MeV
Using the multipole decompositions we extracted themost important multipole components contributing to thelight neutrino mediated 0
+
gs rarr 0
+
1
decay transitionsThese most important components were then again dividedinto contributions coming from different energy levels of
the 0]120573120573 intermediate nucleus These contributions arecollected into Table 4 for119860 = 76ndash116 systems and into Table 5for 119860 = 124ndash136 systems Again we notice that often onlya few intermediate states give the largest contribution to thedominant multipoles 1+ and 0+ Extreme case is the nucleus116Cd forwhich the dominant intermediate ground state gives81 of the total 1+ strength Combining this with the factthat 1+ is by far the largest multipole component we get arather good approximation for the total NME by consideringjust a single virtual transition through the 1+ ground state
Advances in High Energy Physics 9
Table 2 Most important multipoles and intermediate states contributing to the ground-state-to-ground-state 0]120573120573 decays mediated by thelight neutrino exchange Columns 119864 give the energies (in MeVs) and multipoles of the intermediate states Multipoles are organized from leftto right in terms of their importance themost important being on the left Columns labeled119862 give the correspondingNMEcontributions Lasttwo numbers in each 119862 column give the summed contribution and the percentual part which the displayed states give to the total multipolestrength The percentage inside the parenthesis gives the fraction with which the displayed states contribute to the total NME
Nucleus 119864(1
minus
) 119862 119864(2
minus
) 119862 119864(1
+
) 119862 119864(3
minus
) 119862 119864(2
+
) 119862
110Pd
295 0130 082 0387 000 0938 114 0118 033 0244319 0106 247 0091 470 minus0062 164 0128 095 0087344 0221 263 0163 961 0153 191 0080 868 0061381 0426 594 0107 975 minus0146 307 0053 873 0075452 0126 888 0161 1022 0079 361 0075 900 0061911 0109 954 minus0256 1516 minus0316 533 0066 907 0053
1630 0256 808 0167 916 0199837 minus0051 930 minus0366845 0052868 0275911 minus0468
1118 0653 0903 0496 041477 (13) 68 (7) 96 (10) 60 (6) 55 (5)
Nucleus 119864(1
minus
) 119862 119864(3
minus
) 119862 119864(2
minus
) 119862 119864(1
+
) 119862 119864(3
+
) 119862
116Cd
361 0223 172 0081 184 0065 000 0378 090 0158447 0099 230 0053 293 0287 837 minus0102 140 0068537 0118 279 0051 756 0075 951 0087 389 0057587 0140 724 0104 831 0246 1074 minus0083861 0102 834 0103 1120 00812307 0071 845 minus0262 1375 0080
969 minus0051 1379 minus00871573 minus03631583 04401651 minus00821683 0092
0753 0289 0463 0439 028472 (15) 65 (6) 106 (9) 102 (9) 76 (6)
Nucleus 119864(1
minus
) 119862 119864(1
+
) 119862 119864(2
+
) 119862 119864(2
minus
) 119862 119864(3
minus
) 119862
124Sn
168 0522 000 0690 023 0157 052 0271 036 0050482 0089 100 minus0067 060 0083 182 0225 049 0110654 0082 256 0252 106 0056 455 0051 195 0150957 0059 331 0153 215 0066 760 0057 967 00861074 0159 672 minus0130 711 0055 766 minus0064 970 00621407 0065 951 0098 1020 0183 982 minus00871446 minus0099 1309 minus0112 1035 minus0094 1264 minus00531483 minus00581636 minus00871805 minus0067
0664 0884 0417 0629 031855 (8) 95 (11) 53 (5) 89 (8) 49 (4)
of the intermediate nucleus 116In As for the ground-state-to-ground-state decays in some cases notable contributionsare coming from high excitation energies well above 10MeVThere are high-energy contributions in case of 1+ multipolefor all nuclei and in the cases of 2minus and 2+multipoles for 130Teand 136Xe
33 Effects of 119892A on the Intermediate-State Contributions Asmentioned earlier we have used in this work the quenchedvalue for the axial vector coupling 119892A = 100 Next we shallbriefly examine how our results will change if we increasethe value of 119892A from the quenched value 100 to the barevalue 126The effect of this amplification of the axial coupling
10 Advances in High Energy Physics
Table 3 Most important multipoles and intermediate states contributing to the ground-state-to-ground-state 0]120573120573 decays mediated by thelight neutrino exchange Columns 119864 give the energies (in MeVs) and multipoles of the intermediate states Multipoles are organized from leftto right in terms of their importance themost important being on the left Columns labeled119862 give the correspondingNMEcontributions Lasttwo numbers in each 119862 column give the summed contribution and the percentual part which the displayed states give to the total multipolestrength The percentage inside the parenthesis gives the fraction with which the displayed states contribute to the total NME
Nucleus 119864(1
minus
) 119862 119864(2
+
) 119862 119864(3
minus
) 119862 119864(2
minus
) 119862 119864(3
+
) 119862
128Te
422 0200 004 0066 016 0055 061 0335 002 0074472 0060 051 0052 058 0140 402 0060 237 0063621 0078 293 0084 397 0052 455 0101 622 0078644 0059 397 0053 1004 0061 489 minus0070 677 minus0065807 minus0084 677 0050 1014 0136 982 minus0062830 0151 1057 minus0056 1027 minus0057898 0068 1155 00581069 minus00521112 01821748 minus00791919 minus0100
0484 0305 0308 0564 015469 (8) 52 (5) 58 (5) 120 (10) 33 (3)
Nucleus 119864(1
minus
) 119862 119864(2
+
) 119862 119864(3
minus
) 119862 119864(3
+
) 119862 119864(2
minus
) 119862
130Te
418 0184 013 0054 084 0113 010 0056 097 0277572 0059 316 0089 036 0052 1025 0061627 0059 470 0064 260 0056 1133 minus0100856 0096 1049 minus0065 635 00631126 0115 1057 0145 683 00541777 minus0062 1097 minus0088 1231 00591951 minus0079 1644 0150 1241 minus0065
1654 minus01320373 0216 0113 0166 0237
61 (7) 40 (4) 25 (2) 39 (3) 56 (5)Nucleus 119864(2
+
) 119862 119864(3
+
) 119862 119864(1
minus
) 119862 119864(2
minus
) 119862 119864(4
+
) 119862
136Xe
159 0033 135 0033 665 0141 486 0188 234 0032226 0038 239 0054 732 0044 742 0068 244 0061250 0037 481 0050 1031 0053 443 0031529 0067 827 0035 476 0049657 0033 999 0027 767 0034705 0027 1065 minus0027 910 00451411 0087 965 minus00381425 minus0085
0238 0172 0238 0256 021462 (7) 53 (5) 76 (7) 85 (7) 71 (6)
strength on the NMEs is demonstrated in Figure 5 where wehave plotted the multipole decompositions for nuclei 76Geand 82Se calculated with both values of the axial coupling119892A = 100 and 119892A = 126 In case of the ground-state-to-ground-state decays the 1+ multipole changes rather fastwhen the axial coupling is increased from 100 to 126 Thishappens mainly due to the changing of the 119892pp parameter(for each 119892A value the parameter 119892pp is adjusted in sucha way that the measured 2]120573120573 rate is reproduced) The 1+multipole contribution is very sensitive to the value of 119892ppWe can see from Figures 5(a) and 5(b) that for 119892A = 100 the
1
+ component is among the five most important multipoleswhile for 119892A = 126 it is not Some of the higher multipoleschange also somewhat but not so rapidly Ground-state-to-excited-state transitions proceed mainly through the 0+ and1
+multipole channelsWe see from Figures 5(c) and 5(d) thatincreasing the value of 119892A affects mostly the 0+ component
Figure 6 displays the total cumulative sum distributionsfor ground-state-to-ground-state decays of the nuclei 100Moand 116Cd (panels (a) and (b)) and for ground-state-to-excited-state decays of the nuclei 82Se and 96Zr (panels (c)and (d)) Axial coupling values 119892A = 100 and 119892A = 126
Advances in High Energy Physics 11
Table 4 Most important multipoles and intermediate states contributing to the ground-state-to-excited-state 0]120573120573 decays mediated by thelight neutrino exchange Columns 119864 give the energies (in MeVs) and multipoles of the intermediate states Multipoles are organized from leftto right in terms of their importance themost important being on the left Columns labeled119862 give the correspondingNMEcontributions Lasttwo numbers in each 119862 column give the summed contribution and the percentual part which the displayed states give to the total multipolestrength The percentage inside the parenthesis gives the fraction with which the displayed states contribute to the total NME
Nucleus 119864(1
+
) 119862 119864(0
+
) 119862
76Ge
000 0390 286 0053295 0062 428 1204494 0501787 04351352 0522
1909 125692 (50) 99 (33)
Nucleus 119864(1
+
) 119862 119864(0
+
) 119862
82Se
019 0149 545 0471532 0179701 01451453 0274
0746 047190 (41) 96 (26)
Nucleus 119864(1
+
) 119862 119864(2
minus
) 119862 119864(2
+
) 119862 119864(0
+
) 119862 119864(1
minus
) 119862
96Zr
000 0700 092 0283 064 0122 537 0033 228 0032781 0051 966 0057 163 0180 664 0025 252 0080899 0058 537 0035 732 0037 446 00261165 0065 692 0027 802 0036 527 00681617 0354 1012 0396 1133 0026
1200 00131228 0340 0364 0540 0232
91 (23) 60 (6) 65 (7) 98 (10) 59 (4)Nucleus 119864(1
+
) 119862 119864(0
+
) 119862
100Mo
942 0090 901 00761362 0079 955 01501431 00721512 0063
0303 022665 (40) 77 (30)
Nucleus 119864(1
+
) 119862 119864(0
+
) 119862
110Pd
000 0308 1026 01651022 00591373 00831456 00551516 0072
0577 016576 (53) 70 (15)
Nucleus 119864(1
+
) 119862
116Cd000 0735
073581 (48)
12 Advances in High Energy Physics
Table 5 Most important multipoles and intermediate states contributing to the ground-state-to-excited-state 0]120573120573 decays mediated by thelight neutrino exchange Columns 119864 give the energies (in MeVs) and multipoles of the intermediate states Multipoles are organized from leftto right in terms of their importance themost important being on the left Columns labeled119862 give the correspondingNMEcontributions Lasttwo numbers in each 119862 column give the summed contribution and the percentual part which the displayed states give to the total multipolestrength The percentage inside the parenthesis gives the fraction with which the displayed states contribute to the total NME
Nucleus 119864(1
+
) 119862 119864(0
+
) 119862
124Sn
000 0101 270 0667066 0433 460 0088100 0146 735 0567225 0051256 0227331 0664672 0289791 04241309 04731386 0161
2968 132294 (44) 94 (19)
Nucleus 119864(1
+
) 119862 119864(0
+
) 119862 119864(2
minus
) 119862 119864(2
+
) 119862
130Te
025 0390 740 0719 097 0340 041 0019150 0082 835 0631 507 0073 059 0047232 0097 1726 0106 183 0017276 0134 1895 0083 280 0043453 0064 470 0024559 0338 486 0027757 0592 518 00331466 0065 627 00161501 0259 675 00511507 0773 910 00321539 0089 1057 0045
1787 00262314 0024
2882 1351 0602 039095 (41) 996 (19) 79 (9) 65 (6)
Nucleus 119864(1
+
) 119862 119864(0
+
) 119862 119864(2
minus
) 119862 119864(2
+
) 119862
136Xe
230 0366 1224 0317 250 0060 486 0204306 0050 1254 0712 486 0056 1858 0083375 0066 529 0113 2034 0059446 0226 834 00531012 0232 1411 00541152 05651712 01371778 1031
2675 1029 0335 034693 (43) 101 (16) 53 (5) 67 (6)
Advances in High Energy Physics 13
were adopted We notice from the figures that increasing theaxial coupling strength shifts the distributions downwardsThis is especially true for the higher energy parts Despite thisfact the overall shapes of the cumulative sum distributionsdo not changemuch and the same classification of nuclei intodifferent categories according to their cumulative distributionshapes seems to hold also for larger values of 119892A
4 Conclusions
In this paper we have extended our previous work [17] onthe ground-state-to-ground-state 0]120573120573 decay transitions Inthe present work we have concentrated our studies on theintermediate contributions to the NMEs involved in thelight neutrino mediated 0]120573120573 decay We have calculated theintermediate state multipole decompositions of the NMEsand extracted the most important multipole componentsCumulative sums of the NMEs were calculated to investigatethe important energy regions contributing to the 0]120573120573 tran-sitions Finally the most important multipole componentswere divided into contributions coming from the virtualtransitions through the individual states of the 0]120573120573 inter-mediate nuclei An extensive tabulation of these importantintermediate states were given for all the nuclei considered inthis paper
We have done these computations by using realistic two-body interactions and single-particle bases All the appro-priate short-range correlations nucleon form factors andhigher-order nucleonic weak currents are included in ourpresent results
We found in the calculations that often there exists onlya few relevant intermediate states which collect most of thestrength corresponding to a given multipole We also foundthat there exists a single-state dominance in the important 2minuscomponents related to the ground-state decays of nuclei 76Ge82Se and perhaps also for 96Zr
Competing Interests
The authors declare that they have no competing interests
Acknowledgments
This work has been partially supported by the Academy ofFinland under the Finnish Centre of Excellence Programme2012ndash2017 (Nuclear and Accelerator Based Programme atJYFL)
References
[1] J Suhonen and O Civitarese ldquoWeak-interaction and nuclear-structure aspects of nuclear double beta decayrdquo Physics Reportvol 300 no 3-4 pp 123ndash214 1998
[2] F T Avignone S R Elliott and J Engel ldquoDouble beta decayMajorana neutrinos and neutrino massrdquo Reviews of ModernPhysics vol 80 no 2 pp 481ndash516 2008
[3] J D Vergados H Ejiri and F Simkovic ldquoTheory of neutrinolessdouble-beta decayrdquoReports on Progress in Physics vol 75 no 10p 106301 2012
[4] J Maalampi and J Suhonen ldquoNeutrinoless double 120573
+ECdecaysrdquo Advances in High Energy Physics vol 2013 Article ID505874 18 pages 2013
[5] M Kortelainen O Civitarese J Suhonen and J ToivanenldquoShort-range correlations and neutrinoless double beta decayrdquoPhysics Letters Section B Nuclear Elementary Particle andHigh-Energy Physics vol 647 no 2-3 pp 128ndash132 2007
[6] F Simkovic G Pantis J D Vergados and A Faessler ldquoAddi-tional nucleon current contributions to neutrinoless double 120573decayrdquo Physical Review C vol 60 no 5 Article ID 055502 14pages 1999
[7] J Suhonen and O Civitarese ldquoReview of the properties of the0]120573minus120573minus nuclear matrix elementsrdquo Journal of Physics G Nuclearand Particle Physics vol 39 no 12 Article ID 124005 2012
[8] P Vogel ldquoNuclear structure and double beta decayrdquo Journal ofPhysics G Nuclear and Particle Physics vol 39 no 12 p 1240022012
[9] J Barea J Kotila and F Iachello ldquoNuclear matrix elements fordouble-120573 decayrdquo Physical Review C vol 87 no 1 Article ID014315 2013
[10] A Neacsu and S Stoica ldquoStudy of nuclear effects in thecomputation of the 0]120573120573 decay matrix elementsrdquo Journal ofPhysics G Nuclear and Particle Physics vol 41 no 1 Article ID015201 2014
[11] J Engel ldquoUncertainties in nuclear matrix elements for neutri-noless double-beta decayrdquo Journal of Physics G Nuclear andParticle Physics vol 42 no 3 Article ID 034017 2015
[12] F Simkovic A Faessler V Rodin P Vogel and J EngelldquoAnatomyof the 0]120573120573nuclearmatrix elementsrdquoPhysical ReviewC Nuclear Physics vol 77 no 4 Article ID 045503 2008
[13] M T Mustonen and J Engel ldquoLarge-scale calculations ofthe double-120573 decay of 76Ge 130Te 136Xe and 150Nd in thedeformed self-consistent Skyrme quasiparticle random-phaseapproximationrdquo Physical Review C vol 87 Article ID 0643022013
[14] F Simkovic V Rodin A Faessler and P Vogel ldquo0]120573120573 and2]120573120573 nuclear matrix elements quasiparticle random-phaseapproximation and isospin symmetry restorationrdquo PhysicalReview C vol 87 no 4 Article ID 045501 9 pages 2013
[15] V A Rodin A Faessler F Simkovic and P Vogel ldquoAssessmentof uncertainties in QRPA 0]120573120573-decay nuclear matrix elementsrdquoNuclear Physics A vol 766 pp 107ndash131 2006
[16] D-L Fang A Faessler V Rodin and F Simkovic ldquoNeutri-noless double-120573 decay of deformed nuclei within quasiparticlerandom-phase approximationwith a realistic interactionrdquoPhys-ical Review C vol 83 no 3 Article ID 034320 8 pages 2011
[17] J Hyvarinen and J Suhonen ldquoNuclear matrix elements for0]120573120573 decays with light or heavy Majorana-neutrino exchangerdquoPhysical Review C vol 91 no 2 Article ID 024613 12 pages2015
[18] P Pirinen and J Suhonen ldquoSystematic approach to 120573 and 2]120573120573decays of mass119860 = 100minus136 nucleirdquo Physical Review C vol 91no 5 Article ID 054309 2015
[19] H Ejiri N Soukouti and J Suhonen ldquoSpin-dipole nuclearmatrix elements for double beta decays and astro-neutrinosrdquoPhysics Letters B vol 729 pp 27ndash32 2014
[20] J Menendez D Gazit and A Schwenk ldquoChiral two-bodycurrents in nuclei Gamow-Teller transitions and neutrinolessdouble-beta decayrdquo Physical Review Letters vol 107 no 6Article ID 062501 5 pages 2011
Submit your manuscripts athttpwwwhindawicom
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Atomic and Molecular Physics
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AstronomyAdvances in
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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Superconductivity
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Statistical MechanicsInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
GravityJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
AstrophysicsJournal of
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Physics Research International
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Solid State PhysicsJournal of
âComputationalââMethodsâinâPhysics
Journal of
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Soft MatterJournal of
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AerodynamicsJournal of
Volume 2014
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PhotonicsJournal of
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Journal of
Biophysics
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ThermodynamicsJournal of
8 Advances in High Energy Physics
0 20 3010Energy (MeV)
0
1
2
3
4
5
6
7Cu
mul
ativ
e NM
E co
ntrib
utio
n
gA = 100
gA = 126
(a) 100Mo(0]120573120573 0+gs rarr 0+
gs)
0
1
2
3
4
5
6
7
Cum
ulat
ive N
ME
cont
ribut
ion
10 20 300Energy (MeV)
gA = 100
gA = 126
(b) 116Cd(0]120573120573 0+gs rarr 0+
gs)
0
1
2
Cum
ulat
ive N
ME
cont
ribut
ion
10 200Energy (MeV)
gA = 100
gA = 126
(c) 82Se(0]120573120573 0+gs rarr 0+
1
)
0
1
2
3
4
5
6
7
Cum
ulat
ive N
ME
cont
ribut
ion
10 20 300Energy (MeV)
gA = 100
gA = 126
(d) 96Zr(0]120573120573 0+gs rarr 0+
1
)
Figure 6 Cumulative values of the l-NMEs for ground-state-to-ground-state decays of the nuclei 100Mo and 116Cd ((a) and (b) panels) andfor the ground-state-to-excited-state decays of the nuclei 82Se and 96Zr ((c) and (d) panels) The horizontal axis gives the excitation energiesof the intermediate states contributing to the 0]120573120573 transition Two different values for the axial coupling were used as indicated in the panels
important contributions to the NMEs In case of 116Cd panel(e) around 50 of the total NME comes from transitionsthrough the ground state of the intermediate nucleus Theother 50 is distributed rather evenly on the interval 0ndash20MeV
Using the multipole decompositions we extracted themost important multipole components contributing to thelight neutrino mediated 0
+
gs rarr 0
+
1
decay transitionsThese most important components were then again dividedinto contributions coming from different energy levels of
the 0]120573120573 intermediate nucleus These contributions arecollected into Table 4 for119860 = 76ndash116 systems and into Table 5for 119860 = 124ndash136 systems Again we notice that often onlya few intermediate states give the largest contribution to thedominant multipoles 1+ and 0+ Extreme case is the nucleus116Cd forwhich the dominant intermediate ground state gives81 of the total 1+ strength Combining this with the factthat 1+ is by far the largest multipole component we get arather good approximation for the total NME by consideringjust a single virtual transition through the 1+ ground state
Advances in High Energy Physics 9
Table 2 Most important multipoles and intermediate states contributing to the ground-state-to-ground-state 0]120573120573 decays mediated by thelight neutrino exchange Columns 119864 give the energies (in MeVs) and multipoles of the intermediate states Multipoles are organized from leftto right in terms of their importance themost important being on the left Columns labeled119862 give the correspondingNMEcontributions Lasttwo numbers in each 119862 column give the summed contribution and the percentual part which the displayed states give to the total multipolestrength The percentage inside the parenthesis gives the fraction with which the displayed states contribute to the total NME
Nucleus 119864(1
minus
) 119862 119864(2
minus
) 119862 119864(1
+
) 119862 119864(3
minus
) 119862 119864(2
+
) 119862
110Pd
295 0130 082 0387 000 0938 114 0118 033 0244319 0106 247 0091 470 minus0062 164 0128 095 0087344 0221 263 0163 961 0153 191 0080 868 0061381 0426 594 0107 975 minus0146 307 0053 873 0075452 0126 888 0161 1022 0079 361 0075 900 0061911 0109 954 minus0256 1516 minus0316 533 0066 907 0053
1630 0256 808 0167 916 0199837 minus0051 930 minus0366845 0052868 0275911 minus0468
1118 0653 0903 0496 041477 (13) 68 (7) 96 (10) 60 (6) 55 (5)
Nucleus 119864(1
minus
) 119862 119864(3
minus
) 119862 119864(2
minus
) 119862 119864(1
+
) 119862 119864(3
+
) 119862
116Cd
361 0223 172 0081 184 0065 000 0378 090 0158447 0099 230 0053 293 0287 837 minus0102 140 0068537 0118 279 0051 756 0075 951 0087 389 0057587 0140 724 0104 831 0246 1074 minus0083861 0102 834 0103 1120 00812307 0071 845 minus0262 1375 0080
969 minus0051 1379 minus00871573 minus03631583 04401651 minus00821683 0092
0753 0289 0463 0439 028472 (15) 65 (6) 106 (9) 102 (9) 76 (6)
Nucleus 119864(1
minus
) 119862 119864(1
+
) 119862 119864(2
+
) 119862 119864(2
minus
) 119862 119864(3
minus
) 119862
124Sn
168 0522 000 0690 023 0157 052 0271 036 0050482 0089 100 minus0067 060 0083 182 0225 049 0110654 0082 256 0252 106 0056 455 0051 195 0150957 0059 331 0153 215 0066 760 0057 967 00861074 0159 672 minus0130 711 0055 766 minus0064 970 00621407 0065 951 0098 1020 0183 982 minus00871446 minus0099 1309 minus0112 1035 minus0094 1264 minus00531483 minus00581636 minus00871805 minus0067
0664 0884 0417 0629 031855 (8) 95 (11) 53 (5) 89 (8) 49 (4)
of the intermediate nucleus 116In As for the ground-state-to-ground-state decays in some cases notable contributionsare coming from high excitation energies well above 10MeVThere are high-energy contributions in case of 1+ multipolefor all nuclei and in the cases of 2minus and 2+multipoles for 130Teand 136Xe
33 Effects of 119892A on the Intermediate-State Contributions Asmentioned earlier we have used in this work the quenchedvalue for the axial vector coupling 119892A = 100 Next we shallbriefly examine how our results will change if we increasethe value of 119892A from the quenched value 100 to the barevalue 126The effect of this amplification of the axial coupling
10 Advances in High Energy Physics
Table 3 Most important multipoles and intermediate states contributing to the ground-state-to-ground-state 0]120573120573 decays mediated by thelight neutrino exchange Columns 119864 give the energies (in MeVs) and multipoles of the intermediate states Multipoles are organized from leftto right in terms of their importance themost important being on the left Columns labeled119862 give the correspondingNMEcontributions Lasttwo numbers in each 119862 column give the summed contribution and the percentual part which the displayed states give to the total multipolestrength The percentage inside the parenthesis gives the fraction with which the displayed states contribute to the total NME
Nucleus 119864(1
minus
) 119862 119864(2
+
) 119862 119864(3
minus
) 119862 119864(2
minus
) 119862 119864(3
+
) 119862
128Te
422 0200 004 0066 016 0055 061 0335 002 0074472 0060 051 0052 058 0140 402 0060 237 0063621 0078 293 0084 397 0052 455 0101 622 0078644 0059 397 0053 1004 0061 489 minus0070 677 minus0065807 minus0084 677 0050 1014 0136 982 minus0062830 0151 1057 minus0056 1027 minus0057898 0068 1155 00581069 minus00521112 01821748 minus00791919 minus0100
0484 0305 0308 0564 015469 (8) 52 (5) 58 (5) 120 (10) 33 (3)
Nucleus 119864(1
minus
) 119862 119864(2
+
) 119862 119864(3
minus
) 119862 119864(3
+
) 119862 119864(2
minus
) 119862
130Te
418 0184 013 0054 084 0113 010 0056 097 0277572 0059 316 0089 036 0052 1025 0061627 0059 470 0064 260 0056 1133 minus0100856 0096 1049 minus0065 635 00631126 0115 1057 0145 683 00541777 minus0062 1097 minus0088 1231 00591951 minus0079 1644 0150 1241 minus0065
1654 minus01320373 0216 0113 0166 0237
61 (7) 40 (4) 25 (2) 39 (3) 56 (5)Nucleus 119864(2
+
) 119862 119864(3
+
) 119862 119864(1
minus
) 119862 119864(2
minus
) 119862 119864(4
+
) 119862
136Xe
159 0033 135 0033 665 0141 486 0188 234 0032226 0038 239 0054 732 0044 742 0068 244 0061250 0037 481 0050 1031 0053 443 0031529 0067 827 0035 476 0049657 0033 999 0027 767 0034705 0027 1065 minus0027 910 00451411 0087 965 minus00381425 minus0085
0238 0172 0238 0256 021462 (7) 53 (5) 76 (7) 85 (7) 71 (6)
strength on the NMEs is demonstrated in Figure 5 where wehave plotted the multipole decompositions for nuclei 76Geand 82Se calculated with both values of the axial coupling119892A = 100 and 119892A = 126 In case of the ground-state-to-ground-state decays the 1+ multipole changes rather fastwhen the axial coupling is increased from 100 to 126 Thishappens mainly due to the changing of the 119892pp parameter(for each 119892A value the parameter 119892pp is adjusted in sucha way that the measured 2]120573120573 rate is reproduced) The 1+multipole contribution is very sensitive to the value of 119892ppWe can see from Figures 5(a) and 5(b) that for 119892A = 100 the
1
+ component is among the five most important multipoleswhile for 119892A = 126 it is not Some of the higher multipoleschange also somewhat but not so rapidly Ground-state-to-excited-state transitions proceed mainly through the 0+ and1
+multipole channelsWe see from Figures 5(c) and 5(d) thatincreasing the value of 119892A affects mostly the 0+ component
Figure 6 displays the total cumulative sum distributionsfor ground-state-to-ground-state decays of the nuclei 100Moand 116Cd (panels (a) and (b)) and for ground-state-to-excited-state decays of the nuclei 82Se and 96Zr (panels (c)and (d)) Axial coupling values 119892A = 100 and 119892A = 126
Advances in High Energy Physics 11
Table 4 Most important multipoles and intermediate states contributing to the ground-state-to-excited-state 0]120573120573 decays mediated by thelight neutrino exchange Columns 119864 give the energies (in MeVs) and multipoles of the intermediate states Multipoles are organized from leftto right in terms of their importance themost important being on the left Columns labeled119862 give the correspondingNMEcontributions Lasttwo numbers in each 119862 column give the summed contribution and the percentual part which the displayed states give to the total multipolestrength The percentage inside the parenthesis gives the fraction with which the displayed states contribute to the total NME
Nucleus 119864(1
+
) 119862 119864(0
+
) 119862
76Ge
000 0390 286 0053295 0062 428 1204494 0501787 04351352 0522
1909 125692 (50) 99 (33)
Nucleus 119864(1
+
) 119862 119864(0
+
) 119862
82Se
019 0149 545 0471532 0179701 01451453 0274
0746 047190 (41) 96 (26)
Nucleus 119864(1
+
) 119862 119864(2
minus
) 119862 119864(2
+
) 119862 119864(0
+
) 119862 119864(1
minus
) 119862
96Zr
000 0700 092 0283 064 0122 537 0033 228 0032781 0051 966 0057 163 0180 664 0025 252 0080899 0058 537 0035 732 0037 446 00261165 0065 692 0027 802 0036 527 00681617 0354 1012 0396 1133 0026
1200 00131228 0340 0364 0540 0232
91 (23) 60 (6) 65 (7) 98 (10) 59 (4)Nucleus 119864(1
+
) 119862 119864(0
+
) 119862
100Mo
942 0090 901 00761362 0079 955 01501431 00721512 0063
0303 022665 (40) 77 (30)
Nucleus 119864(1
+
) 119862 119864(0
+
) 119862
110Pd
000 0308 1026 01651022 00591373 00831456 00551516 0072
0577 016576 (53) 70 (15)
Nucleus 119864(1
+
) 119862
116Cd000 0735
073581 (48)
12 Advances in High Energy Physics
Table 5 Most important multipoles and intermediate states contributing to the ground-state-to-excited-state 0]120573120573 decays mediated by thelight neutrino exchange Columns 119864 give the energies (in MeVs) and multipoles of the intermediate states Multipoles are organized from leftto right in terms of their importance themost important being on the left Columns labeled119862 give the correspondingNMEcontributions Lasttwo numbers in each 119862 column give the summed contribution and the percentual part which the displayed states give to the total multipolestrength The percentage inside the parenthesis gives the fraction with which the displayed states contribute to the total NME
Nucleus 119864(1
+
) 119862 119864(0
+
) 119862
124Sn
000 0101 270 0667066 0433 460 0088100 0146 735 0567225 0051256 0227331 0664672 0289791 04241309 04731386 0161
2968 132294 (44) 94 (19)
Nucleus 119864(1
+
) 119862 119864(0
+
) 119862 119864(2
minus
) 119862 119864(2
+
) 119862
130Te
025 0390 740 0719 097 0340 041 0019150 0082 835 0631 507 0073 059 0047232 0097 1726 0106 183 0017276 0134 1895 0083 280 0043453 0064 470 0024559 0338 486 0027757 0592 518 00331466 0065 627 00161501 0259 675 00511507 0773 910 00321539 0089 1057 0045
1787 00262314 0024
2882 1351 0602 039095 (41) 996 (19) 79 (9) 65 (6)
Nucleus 119864(1
+
) 119862 119864(0
+
) 119862 119864(2
minus
) 119862 119864(2
+
) 119862
136Xe
230 0366 1224 0317 250 0060 486 0204306 0050 1254 0712 486 0056 1858 0083375 0066 529 0113 2034 0059446 0226 834 00531012 0232 1411 00541152 05651712 01371778 1031
2675 1029 0335 034693 (43) 101 (16) 53 (5) 67 (6)
Advances in High Energy Physics 13
were adopted We notice from the figures that increasing theaxial coupling strength shifts the distributions downwardsThis is especially true for the higher energy parts Despite thisfact the overall shapes of the cumulative sum distributionsdo not changemuch and the same classification of nuclei intodifferent categories according to their cumulative distributionshapes seems to hold also for larger values of 119892A
4 Conclusions
In this paper we have extended our previous work [17] onthe ground-state-to-ground-state 0]120573120573 decay transitions Inthe present work we have concentrated our studies on theintermediate contributions to the NMEs involved in thelight neutrino mediated 0]120573120573 decay We have calculated theintermediate state multipole decompositions of the NMEsand extracted the most important multipole componentsCumulative sums of the NMEs were calculated to investigatethe important energy regions contributing to the 0]120573120573 tran-sitions Finally the most important multipole componentswere divided into contributions coming from the virtualtransitions through the individual states of the 0]120573120573 inter-mediate nuclei An extensive tabulation of these importantintermediate states were given for all the nuclei considered inthis paper
We have done these computations by using realistic two-body interactions and single-particle bases All the appro-priate short-range correlations nucleon form factors andhigher-order nucleonic weak currents are included in ourpresent results
We found in the calculations that often there exists onlya few relevant intermediate states which collect most of thestrength corresponding to a given multipole We also foundthat there exists a single-state dominance in the important 2minuscomponents related to the ground-state decays of nuclei 76Ge82Se and perhaps also for 96Zr
Competing Interests
The authors declare that they have no competing interests
Acknowledgments
This work has been partially supported by the Academy ofFinland under the Finnish Centre of Excellence Programme2012ndash2017 (Nuclear and Accelerator Based Programme atJYFL)
References
[1] J Suhonen and O Civitarese ldquoWeak-interaction and nuclear-structure aspects of nuclear double beta decayrdquo Physics Reportvol 300 no 3-4 pp 123ndash214 1998
[2] F T Avignone S R Elliott and J Engel ldquoDouble beta decayMajorana neutrinos and neutrino massrdquo Reviews of ModernPhysics vol 80 no 2 pp 481ndash516 2008
[3] J D Vergados H Ejiri and F Simkovic ldquoTheory of neutrinolessdouble-beta decayrdquoReports on Progress in Physics vol 75 no 10p 106301 2012
[4] J Maalampi and J Suhonen ldquoNeutrinoless double 120573
+ECdecaysrdquo Advances in High Energy Physics vol 2013 Article ID505874 18 pages 2013
[5] M Kortelainen O Civitarese J Suhonen and J ToivanenldquoShort-range correlations and neutrinoless double beta decayrdquoPhysics Letters Section B Nuclear Elementary Particle andHigh-Energy Physics vol 647 no 2-3 pp 128ndash132 2007
[6] F Simkovic G Pantis J D Vergados and A Faessler ldquoAddi-tional nucleon current contributions to neutrinoless double 120573decayrdquo Physical Review C vol 60 no 5 Article ID 055502 14pages 1999
[7] J Suhonen and O Civitarese ldquoReview of the properties of the0]120573minus120573minus nuclear matrix elementsrdquo Journal of Physics G Nuclearand Particle Physics vol 39 no 12 Article ID 124005 2012
[8] P Vogel ldquoNuclear structure and double beta decayrdquo Journal ofPhysics G Nuclear and Particle Physics vol 39 no 12 p 1240022012
[9] J Barea J Kotila and F Iachello ldquoNuclear matrix elements fordouble-120573 decayrdquo Physical Review C vol 87 no 1 Article ID014315 2013
[10] A Neacsu and S Stoica ldquoStudy of nuclear effects in thecomputation of the 0]120573120573 decay matrix elementsrdquo Journal ofPhysics G Nuclear and Particle Physics vol 41 no 1 Article ID015201 2014
[11] J Engel ldquoUncertainties in nuclear matrix elements for neutri-noless double-beta decayrdquo Journal of Physics G Nuclear andParticle Physics vol 42 no 3 Article ID 034017 2015
[12] F Simkovic A Faessler V Rodin P Vogel and J EngelldquoAnatomyof the 0]120573120573nuclearmatrix elementsrdquoPhysical ReviewC Nuclear Physics vol 77 no 4 Article ID 045503 2008
[13] M T Mustonen and J Engel ldquoLarge-scale calculations ofthe double-120573 decay of 76Ge 130Te 136Xe and 150Nd in thedeformed self-consistent Skyrme quasiparticle random-phaseapproximationrdquo Physical Review C vol 87 Article ID 0643022013
[14] F Simkovic V Rodin A Faessler and P Vogel ldquo0]120573120573 and2]120573120573 nuclear matrix elements quasiparticle random-phaseapproximation and isospin symmetry restorationrdquo PhysicalReview C vol 87 no 4 Article ID 045501 9 pages 2013
[15] V A Rodin A Faessler F Simkovic and P Vogel ldquoAssessmentof uncertainties in QRPA 0]120573120573-decay nuclear matrix elementsrdquoNuclear Physics A vol 766 pp 107ndash131 2006
[16] D-L Fang A Faessler V Rodin and F Simkovic ldquoNeutri-noless double-120573 decay of deformed nuclei within quasiparticlerandom-phase approximationwith a realistic interactionrdquoPhys-ical Review C vol 83 no 3 Article ID 034320 8 pages 2011
[17] J Hyvarinen and J Suhonen ldquoNuclear matrix elements for0]120573120573 decays with light or heavy Majorana-neutrino exchangerdquoPhysical Review C vol 91 no 2 Article ID 024613 12 pages2015
[18] P Pirinen and J Suhonen ldquoSystematic approach to 120573 and 2]120573120573decays of mass119860 = 100minus136 nucleirdquo Physical Review C vol 91no 5 Article ID 054309 2015
[19] H Ejiri N Soukouti and J Suhonen ldquoSpin-dipole nuclearmatrix elements for double beta decays and astro-neutrinosrdquoPhysics Letters B vol 729 pp 27ndash32 2014
[20] J Menendez D Gazit and A Schwenk ldquoChiral two-bodycurrents in nuclei Gamow-Teller transitions and neutrinolessdouble-beta decayrdquo Physical Review Letters vol 107 no 6Article ID 062501 5 pages 2011
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
High Energy PhysicsAdvances in
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FluidsJournal of
Atomic and Molecular Physics
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Advances in Condensed Matter Physics
OpticsInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
AstronomyAdvances in
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Superconductivity
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Statistical MechanicsInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
GravityJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
AstrophysicsJournal of
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Physics Research International
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Solid State PhysicsJournal of
âComputationalââMethodsâinâPhysics
Journal of
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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Soft MatterJournal of
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AerodynamicsJournal of
Volume 2014
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PhotonicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Biophysics
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
ThermodynamicsJournal of
Advances in High Energy Physics 9
Table 2 Most important multipoles and intermediate states contributing to the ground-state-to-ground-state 0]120573120573 decays mediated by thelight neutrino exchange Columns 119864 give the energies (in MeVs) and multipoles of the intermediate states Multipoles are organized from leftto right in terms of their importance themost important being on the left Columns labeled119862 give the correspondingNMEcontributions Lasttwo numbers in each 119862 column give the summed contribution and the percentual part which the displayed states give to the total multipolestrength The percentage inside the parenthesis gives the fraction with which the displayed states contribute to the total NME
Nucleus 119864(1
minus
) 119862 119864(2
minus
) 119862 119864(1
+
) 119862 119864(3
minus
) 119862 119864(2
+
) 119862
110Pd
295 0130 082 0387 000 0938 114 0118 033 0244319 0106 247 0091 470 minus0062 164 0128 095 0087344 0221 263 0163 961 0153 191 0080 868 0061381 0426 594 0107 975 minus0146 307 0053 873 0075452 0126 888 0161 1022 0079 361 0075 900 0061911 0109 954 minus0256 1516 minus0316 533 0066 907 0053
1630 0256 808 0167 916 0199837 minus0051 930 minus0366845 0052868 0275911 minus0468
1118 0653 0903 0496 041477 (13) 68 (7) 96 (10) 60 (6) 55 (5)
Nucleus 119864(1
minus
) 119862 119864(3
minus
) 119862 119864(2
minus
) 119862 119864(1
+
) 119862 119864(3
+
) 119862
116Cd
361 0223 172 0081 184 0065 000 0378 090 0158447 0099 230 0053 293 0287 837 minus0102 140 0068537 0118 279 0051 756 0075 951 0087 389 0057587 0140 724 0104 831 0246 1074 minus0083861 0102 834 0103 1120 00812307 0071 845 minus0262 1375 0080
969 minus0051 1379 minus00871573 minus03631583 04401651 minus00821683 0092
0753 0289 0463 0439 028472 (15) 65 (6) 106 (9) 102 (9) 76 (6)
Nucleus 119864(1
minus
) 119862 119864(1
+
) 119862 119864(2
+
) 119862 119864(2
minus
) 119862 119864(3
minus
) 119862
124Sn
168 0522 000 0690 023 0157 052 0271 036 0050482 0089 100 minus0067 060 0083 182 0225 049 0110654 0082 256 0252 106 0056 455 0051 195 0150957 0059 331 0153 215 0066 760 0057 967 00861074 0159 672 minus0130 711 0055 766 minus0064 970 00621407 0065 951 0098 1020 0183 982 minus00871446 minus0099 1309 minus0112 1035 minus0094 1264 minus00531483 minus00581636 minus00871805 minus0067
0664 0884 0417 0629 031855 (8) 95 (11) 53 (5) 89 (8) 49 (4)
of the intermediate nucleus 116In As for the ground-state-to-ground-state decays in some cases notable contributionsare coming from high excitation energies well above 10MeVThere are high-energy contributions in case of 1+ multipolefor all nuclei and in the cases of 2minus and 2+multipoles for 130Teand 136Xe
33 Effects of 119892A on the Intermediate-State Contributions Asmentioned earlier we have used in this work the quenchedvalue for the axial vector coupling 119892A = 100 Next we shallbriefly examine how our results will change if we increasethe value of 119892A from the quenched value 100 to the barevalue 126The effect of this amplification of the axial coupling
10 Advances in High Energy Physics
Table 3 Most important multipoles and intermediate states contributing to the ground-state-to-ground-state 0]120573120573 decays mediated by thelight neutrino exchange Columns 119864 give the energies (in MeVs) and multipoles of the intermediate states Multipoles are organized from leftto right in terms of their importance themost important being on the left Columns labeled119862 give the correspondingNMEcontributions Lasttwo numbers in each 119862 column give the summed contribution and the percentual part which the displayed states give to the total multipolestrength The percentage inside the parenthesis gives the fraction with which the displayed states contribute to the total NME
Nucleus 119864(1
minus
) 119862 119864(2
+
) 119862 119864(3
minus
) 119862 119864(2
minus
) 119862 119864(3
+
) 119862
128Te
422 0200 004 0066 016 0055 061 0335 002 0074472 0060 051 0052 058 0140 402 0060 237 0063621 0078 293 0084 397 0052 455 0101 622 0078644 0059 397 0053 1004 0061 489 minus0070 677 minus0065807 minus0084 677 0050 1014 0136 982 minus0062830 0151 1057 minus0056 1027 minus0057898 0068 1155 00581069 minus00521112 01821748 minus00791919 minus0100
0484 0305 0308 0564 015469 (8) 52 (5) 58 (5) 120 (10) 33 (3)
Nucleus 119864(1
minus
) 119862 119864(2
+
) 119862 119864(3
minus
) 119862 119864(3
+
) 119862 119864(2
minus
) 119862
130Te
418 0184 013 0054 084 0113 010 0056 097 0277572 0059 316 0089 036 0052 1025 0061627 0059 470 0064 260 0056 1133 minus0100856 0096 1049 minus0065 635 00631126 0115 1057 0145 683 00541777 minus0062 1097 minus0088 1231 00591951 minus0079 1644 0150 1241 minus0065
1654 minus01320373 0216 0113 0166 0237
61 (7) 40 (4) 25 (2) 39 (3) 56 (5)Nucleus 119864(2
+
) 119862 119864(3
+
) 119862 119864(1
minus
) 119862 119864(2
minus
) 119862 119864(4
+
) 119862
136Xe
159 0033 135 0033 665 0141 486 0188 234 0032226 0038 239 0054 732 0044 742 0068 244 0061250 0037 481 0050 1031 0053 443 0031529 0067 827 0035 476 0049657 0033 999 0027 767 0034705 0027 1065 minus0027 910 00451411 0087 965 minus00381425 minus0085
0238 0172 0238 0256 021462 (7) 53 (5) 76 (7) 85 (7) 71 (6)
strength on the NMEs is demonstrated in Figure 5 where wehave plotted the multipole decompositions for nuclei 76Geand 82Se calculated with both values of the axial coupling119892A = 100 and 119892A = 126 In case of the ground-state-to-ground-state decays the 1+ multipole changes rather fastwhen the axial coupling is increased from 100 to 126 Thishappens mainly due to the changing of the 119892pp parameter(for each 119892A value the parameter 119892pp is adjusted in sucha way that the measured 2]120573120573 rate is reproduced) The 1+multipole contribution is very sensitive to the value of 119892ppWe can see from Figures 5(a) and 5(b) that for 119892A = 100 the
1
+ component is among the five most important multipoleswhile for 119892A = 126 it is not Some of the higher multipoleschange also somewhat but not so rapidly Ground-state-to-excited-state transitions proceed mainly through the 0+ and1
+multipole channelsWe see from Figures 5(c) and 5(d) thatincreasing the value of 119892A affects mostly the 0+ component
Figure 6 displays the total cumulative sum distributionsfor ground-state-to-ground-state decays of the nuclei 100Moand 116Cd (panels (a) and (b)) and for ground-state-to-excited-state decays of the nuclei 82Se and 96Zr (panels (c)and (d)) Axial coupling values 119892A = 100 and 119892A = 126
Advances in High Energy Physics 11
Table 4 Most important multipoles and intermediate states contributing to the ground-state-to-excited-state 0]120573120573 decays mediated by thelight neutrino exchange Columns 119864 give the energies (in MeVs) and multipoles of the intermediate states Multipoles are organized from leftto right in terms of their importance themost important being on the left Columns labeled119862 give the correspondingNMEcontributions Lasttwo numbers in each 119862 column give the summed contribution and the percentual part which the displayed states give to the total multipolestrength The percentage inside the parenthesis gives the fraction with which the displayed states contribute to the total NME
Nucleus 119864(1
+
) 119862 119864(0
+
) 119862
76Ge
000 0390 286 0053295 0062 428 1204494 0501787 04351352 0522
1909 125692 (50) 99 (33)
Nucleus 119864(1
+
) 119862 119864(0
+
) 119862
82Se
019 0149 545 0471532 0179701 01451453 0274
0746 047190 (41) 96 (26)
Nucleus 119864(1
+
) 119862 119864(2
minus
) 119862 119864(2
+
) 119862 119864(0
+
) 119862 119864(1
minus
) 119862
96Zr
000 0700 092 0283 064 0122 537 0033 228 0032781 0051 966 0057 163 0180 664 0025 252 0080899 0058 537 0035 732 0037 446 00261165 0065 692 0027 802 0036 527 00681617 0354 1012 0396 1133 0026
1200 00131228 0340 0364 0540 0232
91 (23) 60 (6) 65 (7) 98 (10) 59 (4)Nucleus 119864(1
+
) 119862 119864(0
+
) 119862
100Mo
942 0090 901 00761362 0079 955 01501431 00721512 0063
0303 022665 (40) 77 (30)
Nucleus 119864(1
+
) 119862 119864(0
+
) 119862
110Pd
000 0308 1026 01651022 00591373 00831456 00551516 0072
0577 016576 (53) 70 (15)
Nucleus 119864(1
+
) 119862
116Cd000 0735
073581 (48)
12 Advances in High Energy Physics
Table 5 Most important multipoles and intermediate states contributing to the ground-state-to-excited-state 0]120573120573 decays mediated by thelight neutrino exchange Columns 119864 give the energies (in MeVs) and multipoles of the intermediate states Multipoles are organized from leftto right in terms of their importance themost important being on the left Columns labeled119862 give the correspondingNMEcontributions Lasttwo numbers in each 119862 column give the summed contribution and the percentual part which the displayed states give to the total multipolestrength The percentage inside the parenthesis gives the fraction with which the displayed states contribute to the total NME
Nucleus 119864(1
+
) 119862 119864(0
+
) 119862
124Sn
000 0101 270 0667066 0433 460 0088100 0146 735 0567225 0051256 0227331 0664672 0289791 04241309 04731386 0161
2968 132294 (44) 94 (19)
Nucleus 119864(1
+
) 119862 119864(0
+
) 119862 119864(2
minus
) 119862 119864(2
+
) 119862
130Te
025 0390 740 0719 097 0340 041 0019150 0082 835 0631 507 0073 059 0047232 0097 1726 0106 183 0017276 0134 1895 0083 280 0043453 0064 470 0024559 0338 486 0027757 0592 518 00331466 0065 627 00161501 0259 675 00511507 0773 910 00321539 0089 1057 0045
1787 00262314 0024
2882 1351 0602 039095 (41) 996 (19) 79 (9) 65 (6)
Nucleus 119864(1
+
) 119862 119864(0
+
) 119862 119864(2
minus
) 119862 119864(2
+
) 119862
136Xe
230 0366 1224 0317 250 0060 486 0204306 0050 1254 0712 486 0056 1858 0083375 0066 529 0113 2034 0059446 0226 834 00531012 0232 1411 00541152 05651712 01371778 1031
2675 1029 0335 034693 (43) 101 (16) 53 (5) 67 (6)
Advances in High Energy Physics 13
were adopted We notice from the figures that increasing theaxial coupling strength shifts the distributions downwardsThis is especially true for the higher energy parts Despite thisfact the overall shapes of the cumulative sum distributionsdo not changemuch and the same classification of nuclei intodifferent categories according to their cumulative distributionshapes seems to hold also for larger values of 119892A
4 Conclusions
In this paper we have extended our previous work [17] onthe ground-state-to-ground-state 0]120573120573 decay transitions Inthe present work we have concentrated our studies on theintermediate contributions to the NMEs involved in thelight neutrino mediated 0]120573120573 decay We have calculated theintermediate state multipole decompositions of the NMEsand extracted the most important multipole componentsCumulative sums of the NMEs were calculated to investigatethe important energy regions contributing to the 0]120573120573 tran-sitions Finally the most important multipole componentswere divided into contributions coming from the virtualtransitions through the individual states of the 0]120573120573 inter-mediate nuclei An extensive tabulation of these importantintermediate states were given for all the nuclei considered inthis paper
We have done these computations by using realistic two-body interactions and single-particle bases All the appro-priate short-range correlations nucleon form factors andhigher-order nucleonic weak currents are included in ourpresent results
We found in the calculations that often there exists onlya few relevant intermediate states which collect most of thestrength corresponding to a given multipole We also foundthat there exists a single-state dominance in the important 2minuscomponents related to the ground-state decays of nuclei 76Ge82Se and perhaps also for 96Zr
Competing Interests
The authors declare that they have no competing interests
Acknowledgments
This work has been partially supported by the Academy ofFinland under the Finnish Centre of Excellence Programme2012ndash2017 (Nuclear and Accelerator Based Programme atJYFL)
References
[1] J Suhonen and O Civitarese ldquoWeak-interaction and nuclear-structure aspects of nuclear double beta decayrdquo Physics Reportvol 300 no 3-4 pp 123ndash214 1998
[2] F T Avignone S R Elliott and J Engel ldquoDouble beta decayMajorana neutrinos and neutrino massrdquo Reviews of ModernPhysics vol 80 no 2 pp 481ndash516 2008
[3] J D Vergados H Ejiri and F Simkovic ldquoTheory of neutrinolessdouble-beta decayrdquoReports on Progress in Physics vol 75 no 10p 106301 2012
[4] J Maalampi and J Suhonen ldquoNeutrinoless double 120573
+ECdecaysrdquo Advances in High Energy Physics vol 2013 Article ID505874 18 pages 2013
[5] M Kortelainen O Civitarese J Suhonen and J ToivanenldquoShort-range correlations and neutrinoless double beta decayrdquoPhysics Letters Section B Nuclear Elementary Particle andHigh-Energy Physics vol 647 no 2-3 pp 128ndash132 2007
[6] F Simkovic G Pantis J D Vergados and A Faessler ldquoAddi-tional nucleon current contributions to neutrinoless double 120573decayrdquo Physical Review C vol 60 no 5 Article ID 055502 14pages 1999
[7] J Suhonen and O Civitarese ldquoReview of the properties of the0]120573minus120573minus nuclear matrix elementsrdquo Journal of Physics G Nuclearand Particle Physics vol 39 no 12 Article ID 124005 2012
[8] P Vogel ldquoNuclear structure and double beta decayrdquo Journal ofPhysics G Nuclear and Particle Physics vol 39 no 12 p 1240022012
[9] J Barea J Kotila and F Iachello ldquoNuclear matrix elements fordouble-120573 decayrdquo Physical Review C vol 87 no 1 Article ID014315 2013
[10] A Neacsu and S Stoica ldquoStudy of nuclear effects in thecomputation of the 0]120573120573 decay matrix elementsrdquo Journal ofPhysics G Nuclear and Particle Physics vol 41 no 1 Article ID015201 2014
[11] J Engel ldquoUncertainties in nuclear matrix elements for neutri-noless double-beta decayrdquo Journal of Physics G Nuclear andParticle Physics vol 42 no 3 Article ID 034017 2015
[12] F Simkovic A Faessler V Rodin P Vogel and J EngelldquoAnatomyof the 0]120573120573nuclearmatrix elementsrdquoPhysical ReviewC Nuclear Physics vol 77 no 4 Article ID 045503 2008
[13] M T Mustonen and J Engel ldquoLarge-scale calculations ofthe double-120573 decay of 76Ge 130Te 136Xe and 150Nd in thedeformed self-consistent Skyrme quasiparticle random-phaseapproximationrdquo Physical Review C vol 87 Article ID 0643022013
[14] F Simkovic V Rodin A Faessler and P Vogel ldquo0]120573120573 and2]120573120573 nuclear matrix elements quasiparticle random-phaseapproximation and isospin symmetry restorationrdquo PhysicalReview C vol 87 no 4 Article ID 045501 9 pages 2013
[15] V A Rodin A Faessler F Simkovic and P Vogel ldquoAssessmentof uncertainties in QRPA 0]120573120573-decay nuclear matrix elementsrdquoNuclear Physics A vol 766 pp 107ndash131 2006
[16] D-L Fang A Faessler V Rodin and F Simkovic ldquoNeutri-noless double-120573 decay of deformed nuclei within quasiparticlerandom-phase approximationwith a realistic interactionrdquoPhys-ical Review C vol 83 no 3 Article ID 034320 8 pages 2011
[17] J Hyvarinen and J Suhonen ldquoNuclear matrix elements for0]120573120573 decays with light or heavy Majorana-neutrino exchangerdquoPhysical Review C vol 91 no 2 Article ID 024613 12 pages2015
[18] P Pirinen and J Suhonen ldquoSystematic approach to 120573 and 2]120573120573decays of mass119860 = 100minus136 nucleirdquo Physical Review C vol 91no 5 Article ID 054309 2015
[19] H Ejiri N Soukouti and J Suhonen ldquoSpin-dipole nuclearmatrix elements for double beta decays and astro-neutrinosrdquoPhysics Letters B vol 729 pp 27ndash32 2014
[20] J Menendez D Gazit and A Schwenk ldquoChiral two-bodycurrents in nuclei Gamow-Teller transitions and neutrinolessdouble-beta decayrdquo Physical Review Letters vol 107 no 6Article ID 062501 5 pages 2011
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
High Energy PhysicsAdvances in
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
FluidsJournal of
Atomic and Molecular Physics
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Advances in Condensed Matter Physics
OpticsInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
AstronomyAdvances in
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Superconductivity
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Statistical MechanicsInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
GravityJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
AstrophysicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Physics Research International
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Solid State PhysicsJournal of
âComputationalââMethodsâinâPhysics
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Soft MatterJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
AerodynamicsJournal of
Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
PhotonicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Biophysics
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
ThermodynamicsJournal of
10 Advances in High Energy Physics
Table 3 Most important multipoles and intermediate states contributing to the ground-state-to-ground-state 0]120573120573 decays mediated by thelight neutrino exchange Columns 119864 give the energies (in MeVs) and multipoles of the intermediate states Multipoles are organized from leftto right in terms of their importance themost important being on the left Columns labeled119862 give the correspondingNMEcontributions Lasttwo numbers in each 119862 column give the summed contribution and the percentual part which the displayed states give to the total multipolestrength The percentage inside the parenthesis gives the fraction with which the displayed states contribute to the total NME
Nucleus 119864(1
minus
) 119862 119864(2
+
) 119862 119864(3
minus
) 119862 119864(2
minus
) 119862 119864(3
+
) 119862
128Te
422 0200 004 0066 016 0055 061 0335 002 0074472 0060 051 0052 058 0140 402 0060 237 0063621 0078 293 0084 397 0052 455 0101 622 0078644 0059 397 0053 1004 0061 489 minus0070 677 minus0065807 minus0084 677 0050 1014 0136 982 minus0062830 0151 1057 minus0056 1027 minus0057898 0068 1155 00581069 minus00521112 01821748 minus00791919 minus0100
0484 0305 0308 0564 015469 (8) 52 (5) 58 (5) 120 (10) 33 (3)
Nucleus 119864(1
minus
) 119862 119864(2
+
) 119862 119864(3
minus
) 119862 119864(3
+
) 119862 119864(2
minus
) 119862
130Te
418 0184 013 0054 084 0113 010 0056 097 0277572 0059 316 0089 036 0052 1025 0061627 0059 470 0064 260 0056 1133 minus0100856 0096 1049 minus0065 635 00631126 0115 1057 0145 683 00541777 minus0062 1097 minus0088 1231 00591951 minus0079 1644 0150 1241 minus0065
1654 minus01320373 0216 0113 0166 0237
61 (7) 40 (4) 25 (2) 39 (3) 56 (5)Nucleus 119864(2
+
) 119862 119864(3
+
) 119862 119864(1
minus
) 119862 119864(2
minus
) 119862 119864(4
+
) 119862
136Xe
159 0033 135 0033 665 0141 486 0188 234 0032226 0038 239 0054 732 0044 742 0068 244 0061250 0037 481 0050 1031 0053 443 0031529 0067 827 0035 476 0049657 0033 999 0027 767 0034705 0027 1065 minus0027 910 00451411 0087 965 minus00381425 minus0085
0238 0172 0238 0256 021462 (7) 53 (5) 76 (7) 85 (7) 71 (6)
strength on the NMEs is demonstrated in Figure 5 where wehave plotted the multipole decompositions for nuclei 76Geand 82Se calculated with both values of the axial coupling119892A = 100 and 119892A = 126 In case of the ground-state-to-ground-state decays the 1+ multipole changes rather fastwhen the axial coupling is increased from 100 to 126 Thishappens mainly due to the changing of the 119892pp parameter(for each 119892A value the parameter 119892pp is adjusted in sucha way that the measured 2]120573120573 rate is reproduced) The 1+multipole contribution is very sensitive to the value of 119892ppWe can see from Figures 5(a) and 5(b) that for 119892A = 100 the
1
+ component is among the five most important multipoleswhile for 119892A = 126 it is not Some of the higher multipoleschange also somewhat but not so rapidly Ground-state-to-excited-state transitions proceed mainly through the 0+ and1
+multipole channelsWe see from Figures 5(c) and 5(d) thatincreasing the value of 119892A affects mostly the 0+ component
Figure 6 displays the total cumulative sum distributionsfor ground-state-to-ground-state decays of the nuclei 100Moand 116Cd (panels (a) and (b)) and for ground-state-to-excited-state decays of the nuclei 82Se and 96Zr (panels (c)and (d)) Axial coupling values 119892A = 100 and 119892A = 126
Advances in High Energy Physics 11
Table 4 Most important multipoles and intermediate states contributing to the ground-state-to-excited-state 0]120573120573 decays mediated by thelight neutrino exchange Columns 119864 give the energies (in MeVs) and multipoles of the intermediate states Multipoles are organized from leftto right in terms of their importance themost important being on the left Columns labeled119862 give the correspondingNMEcontributions Lasttwo numbers in each 119862 column give the summed contribution and the percentual part which the displayed states give to the total multipolestrength The percentage inside the parenthesis gives the fraction with which the displayed states contribute to the total NME
Nucleus 119864(1
+
) 119862 119864(0
+
) 119862
76Ge
000 0390 286 0053295 0062 428 1204494 0501787 04351352 0522
1909 125692 (50) 99 (33)
Nucleus 119864(1
+
) 119862 119864(0
+
) 119862
82Se
019 0149 545 0471532 0179701 01451453 0274
0746 047190 (41) 96 (26)
Nucleus 119864(1
+
) 119862 119864(2
minus
) 119862 119864(2
+
) 119862 119864(0
+
) 119862 119864(1
minus
) 119862
96Zr
000 0700 092 0283 064 0122 537 0033 228 0032781 0051 966 0057 163 0180 664 0025 252 0080899 0058 537 0035 732 0037 446 00261165 0065 692 0027 802 0036 527 00681617 0354 1012 0396 1133 0026
1200 00131228 0340 0364 0540 0232
91 (23) 60 (6) 65 (7) 98 (10) 59 (4)Nucleus 119864(1
+
) 119862 119864(0
+
) 119862
100Mo
942 0090 901 00761362 0079 955 01501431 00721512 0063
0303 022665 (40) 77 (30)
Nucleus 119864(1
+
) 119862 119864(0
+
) 119862
110Pd
000 0308 1026 01651022 00591373 00831456 00551516 0072
0577 016576 (53) 70 (15)
Nucleus 119864(1
+
) 119862
116Cd000 0735
073581 (48)
12 Advances in High Energy Physics
Table 5 Most important multipoles and intermediate states contributing to the ground-state-to-excited-state 0]120573120573 decays mediated by thelight neutrino exchange Columns 119864 give the energies (in MeVs) and multipoles of the intermediate states Multipoles are organized from leftto right in terms of their importance themost important being on the left Columns labeled119862 give the correspondingNMEcontributions Lasttwo numbers in each 119862 column give the summed contribution and the percentual part which the displayed states give to the total multipolestrength The percentage inside the parenthesis gives the fraction with which the displayed states contribute to the total NME
Nucleus 119864(1
+
) 119862 119864(0
+
) 119862
124Sn
000 0101 270 0667066 0433 460 0088100 0146 735 0567225 0051256 0227331 0664672 0289791 04241309 04731386 0161
2968 132294 (44) 94 (19)
Nucleus 119864(1
+
) 119862 119864(0
+
) 119862 119864(2
minus
) 119862 119864(2
+
) 119862
130Te
025 0390 740 0719 097 0340 041 0019150 0082 835 0631 507 0073 059 0047232 0097 1726 0106 183 0017276 0134 1895 0083 280 0043453 0064 470 0024559 0338 486 0027757 0592 518 00331466 0065 627 00161501 0259 675 00511507 0773 910 00321539 0089 1057 0045
1787 00262314 0024
2882 1351 0602 039095 (41) 996 (19) 79 (9) 65 (6)
Nucleus 119864(1
+
) 119862 119864(0
+
) 119862 119864(2
minus
) 119862 119864(2
+
) 119862
136Xe
230 0366 1224 0317 250 0060 486 0204306 0050 1254 0712 486 0056 1858 0083375 0066 529 0113 2034 0059446 0226 834 00531012 0232 1411 00541152 05651712 01371778 1031
2675 1029 0335 034693 (43) 101 (16) 53 (5) 67 (6)
Advances in High Energy Physics 13
were adopted We notice from the figures that increasing theaxial coupling strength shifts the distributions downwardsThis is especially true for the higher energy parts Despite thisfact the overall shapes of the cumulative sum distributionsdo not changemuch and the same classification of nuclei intodifferent categories according to their cumulative distributionshapes seems to hold also for larger values of 119892A
4 Conclusions
In this paper we have extended our previous work [17] onthe ground-state-to-ground-state 0]120573120573 decay transitions Inthe present work we have concentrated our studies on theintermediate contributions to the NMEs involved in thelight neutrino mediated 0]120573120573 decay We have calculated theintermediate state multipole decompositions of the NMEsand extracted the most important multipole componentsCumulative sums of the NMEs were calculated to investigatethe important energy regions contributing to the 0]120573120573 tran-sitions Finally the most important multipole componentswere divided into contributions coming from the virtualtransitions through the individual states of the 0]120573120573 inter-mediate nuclei An extensive tabulation of these importantintermediate states were given for all the nuclei considered inthis paper
We have done these computations by using realistic two-body interactions and single-particle bases All the appro-priate short-range correlations nucleon form factors andhigher-order nucleonic weak currents are included in ourpresent results
We found in the calculations that often there exists onlya few relevant intermediate states which collect most of thestrength corresponding to a given multipole We also foundthat there exists a single-state dominance in the important 2minuscomponents related to the ground-state decays of nuclei 76Ge82Se and perhaps also for 96Zr
Competing Interests
The authors declare that they have no competing interests
Acknowledgments
This work has been partially supported by the Academy ofFinland under the Finnish Centre of Excellence Programme2012ndash2017 (Nuclear and Accelerator Based Programme atJYFL)
References
[1] J Suhonen and O Civitarese ldquoWeak-interaction and nuclear-structure aspects of nuclear double beta decayrdquo Physics Reportvol 300 no 3-4 pp 123ndash214 1998
[2] F T Avignone S R Elliott and J Engel ldquoDouble beta decayMajorana neutrinos and neutrino massrdquo Reviews of ModernPhysics vol 80 no 2 pp 481ndash516 2008
[3] J D Vergados H Ejiri and F Simkovic ldquoTheory of neutrinolessdouble-beta decayrdquoReports on Progress in Physics vol 75 no 10p 106301 2012
[4] J Maalampi and J Suhonen ldquoNeutrinoless double 120573
+ECdecaysrdquo Advances in High Energy Physics vol 2013 Article ID505874 18 pages 2013
[5] M Kortelainen O Civitarese J Suhonen and J ToivanenldquoShort-range correlations and neutrinoless double beta decayrdquoPhysics Letters Section B Nuclear Elementary Particle andHigh-Energy Physics vol 647 no 2-3 pp 128ndash132 2007
[6] F Simkovic G Pantis J D Vergados and A Faessler ldquoAddi-tional nucleon current contributions to neutrinoless double 120573decayrdquo Physical Review C vol 60 no 5 Article ID 055502 14pages 1999
[7] J Suhonen and O Civitarese ldquoReview of the properties of the0]120573minus120573minus nuclear matrix elementsrdquo Journal of Physics G Nuclearand Particle Physics vol 39 no 12 Article ID 124005 2012
[8] P Vogel ldquoNuclear structure and double beta decayrdquo Journal ofPhysics G Nuclear and Particle Physics vol 39 no 12 p 1240022012
[9] J Barea J Kotila and F Iachello ldquoNuclear matrix elements fordouble-120573 decayrdquo Physical Review C vol 87 no 1 Article ID014315 2013
[10] A Neacsu and S Stoica ldquoStudy of nuclear effects in thecomputation of the 0]120573120573 decay matrix elementsrdquo Journal ofPhysics G Nuclear and Particle Physics vol 41 no 1 Article ID015201 2014
[11] J Engel ldquoUncertainties in nuclear matrix elements for neutri-noless double-beta decayrdquo Journal of Physics G Nuclear andParticle Physics vol 42 no 3 Article ID 034017 2015
[12] F Simkovic A Faessler V Rodin P Vogel and J EngelldquoAnatomyof the 0]120573120573nuclearmatrix elementsrdquoPhysical ReviewC Nuclear Physics vol 77 no 4 Article ID 045503 2008
[13] M T Mustonen and J Engel ldquoLarge-scale calculations ofthe double-120573 decay of 76Ge 130Te 136Xe and 150Nd in thedeformed self-consistent Skyrme quasiparticle random-phaseapproximationrdquo Physical Review C vol 87 Article ID 0643022013
[14] F Simkovic V Rodin A Faessler and P Vogel ldquo0]120573120573 and2]120573120573 nuclear matrix elements quasiparticle random-phaseapproximation and isospin symmetry restorationrdquo PhysicalReview C vol 87 no 4 Article ID 045501 9 pages 2013
[15] V A Rodin A Faessler F Simkovic and P Vogel ldquoAssessmentof uncertainties in QRPA 0]120573120573-decay nuclear matrix elementsrdquoNuclear Physics A vol 766 pp 107ndash131 2006
[16] D-L Fang A Faessler V Rodin and F Simkovic ldquoNeutri-noless double-120573 decay of deformed nuclei within quasiparticlerandom-phase approximationwith a realistic interactionrdquoPhys-ical Review C vol 83 no 3 Article ID 034320 8 pages 2011
[17] J Hyvarinen and J Suhonen ldquoNuclear matrix elements for0]120573120573 decays with light or heavy Majorana-neutrino exchangerdquoPhysical Review C vol 91 no 2 Article ID 024613 12 pages2015
[18] P Pirinen and J Suhonen ldquoSystematic approach to 120573 and 2]120573120573decays of mass119860 = 100minus136 nucleirdquo Physical Review C vol 91no 5 Article ID 054309 2015
[19] H Ejiri N Soukouti and J Suhonen ldquoSpin-dipole nuclearmatrix elements for double beta decays and astro-neutrinosrdquoPhysics Letters B vol 729 pp 27ndash32 2014
[20] J Menendez D Gazit and A Schwenk ldquoChiral two-bodycurrents in nuclei Gamow-Teller transitions and neutrinolessdouble-beta decayrdquo Physical Review Letters vol 107 no 6Article ID 062501 5 pages 2011
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
High Energy PhysicsAdvances in
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
FluidsJournal of
Atomic and Molecular Physics
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Advances in Condensed Matter Physics
OpticsInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
AstronomyAdvances in
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Superconductivity
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Statistical MechanicsInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
GravityJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
AstrophysicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Physics Research International
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Solid State PhysicsJournal of
âComputationalââMethodsâinâPhysics
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Soft MatterJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
AerodynamicsJournal of
Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
PhotonicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Biophysics
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
ThermodynamicsJournal of
Advances in High Energy Physics 11
Table 4 Most important multipoles and intermediate states contributing to the ground-state-to-excited-state 0]120573120573 decays mediated by thelight neutrino exchange Columns 119864 give the energies (in MeVs) and multipoles of the intermediate states Multipoles are organized from leftto right in terms of their importance themost important being on the left Columns labeled119862 give the correspondingNMEcontributions Lasttwo numbers in each 119862 column give the summed contribution and the percentual part which the displayed states give to the total multipolestrength The percentage inside the parenthesis gives the fraction with which the displayed states contribute to the total NME
Nucleus 119864(1
+
) 119862 119864(0
+
) 119862
76Ge
000 0390 286 0053295 0062 428 1204494 0501787 04351352 0522
1909 125692 (50) 99 (33)
Nucleus 119864(1
+
) 119862 119864(0
+
) 119862
82Se
019 0149 545 0471532 0179701 01451453 0274
0746 047190 (41) 96 (26)
Nucleus 119864(1
+
) 119862 119864(2
minus
) 119862 119864(2
+
) 119862 119864(0
+
) 119862 119864(1
minus
) 119862
96Zr
000 0700 092 0283 064 0122 537 0033 228 0032781 0051 966 0057 163 0180 664 0025 252 0080899 0058 537 0035 732 0037 446 00261165 0065 692 0027 802 0036 527 00681617 0354 1012 0396 1133 0026
1200 00131228 0340 0364 0540 0232
91 (23) 60 (6) 65 (7) 98 (10) 59 (4)Nucleus 119864(1
+
) 119862 119864(0
+
) 119862
100Mo
942 0090 901 00761362 0079 955 01501431 00721512 0063
0303 022665 (40) 77 (30)
Nucleus 119864(1
+
) 119862 119864(0
+
) 119862
110Pd
000 0308 1026 01651022 00591373 00831456 00551516 0072
0577 016576 (53) 70 (15)
Nucleus 119864(1
+
) 119862
116Cd000 0735
073581 (48)
12 Advances in High Energy Physics
Table 5 Most important multipoles and intermediate states contributing to the ground-state-to-excited-state 0]120573120573 decays mediated by thelight neutrino exchange Columns 119864 give the energies (in MeVs) and multipoles of the intermediate states Multipoles are organized from leftto right in terms of their importance themost important being on the left Columns labeled119862 give the correspondingNMEcontributions Lasttwo numbers in each 119862 column give the summed contribution and the percentual part which the displayed states give to the total multipolestrength The percentage inside the parenthesis gives the fraction with which the displayed states contribute to the total NME
Nucleus 119864(1
+
) 119862 119864(0
+
) 119862
124Sn
000 0101 270 0667066 0433 460 0088100 0146 735 0567225 0051256 0227331 0664672 0289791 04241309 04731386 0161
2968 132294 (44) 94 (19)
Nucleus 119864(1
+
) 119862 119864(0
+
) 119862 119864(2
minus
) 119862 119864(2
+
) 119862
130Te
025 0390 740 0719 097 0340 041 0019150 0082 835 0631 507 0073 059 0047232 0097 1726 0106 183 0017276 0134 1895 0083 280 0043453 0064 470 0024559 0338 486 0027757 0592 518 00331466 0065 627 00161501 0259 675 00511507 0773 910 00321539 0089 1057 0045
1787 00262314 0024
2882 1351 0602 039095 (41) 996 (19) 79 (9) 65 (6)
Nucleus 119864(1
+
) 119862 119864(0
+
) 119862 119864(2
minus
) 119862 119864(2
+
) 119862
136Xe
230 0366 1224 0317 250 0060 486 0204306 0050 1254 0712 486 0056 1858 0083375 0066 529 0113 2034 0059446 0226 834 00531012 0232 1411 00541152 05651712 01371778 1031
2675 1029 0335 034693 (43) 101 (16) 53 (5) 67 (6)
Advances in High Energy Physics 13
were adopted We notice from the figures that increasing theaxial coupling strength shifts the distributions downwardsThis is especially true for the higher energy parts Despite thisfact the overall shapes of the cumulative sum distributionsdo not changemuch and the same classification of nuclei intodifferent categories according to their cumulative distributionshapes seems to hold also for larger values of 119892A
4 Conclusions
In this paper we have extended our previous work [17] onthe ground-state-to-ground-state 0]120573120573 decay transitions Inthe present work we have concentrated our studies on theintermediate contributions to the NMEs involved in thelight neutrino mediated 0]120573120573 decay We have calculated theintermediate state multipole decompositions of the NMEsand extracted the most important multipole componentsCumulative sums of the NMEs were calculated to investigatethe important energy regions contributing to the 0]120573120573 tran-sitions Finally the most important multipole componentswere divided into contributions coming from the virtualtransitions through the individual states of the 0]120573120573 inter-mediate nuclei An extensive tabulation of these importantintermediate states were given for all the nuclei considered inthis paper
We have done these computations by using realistic two-body interactions and single-particle bases All the appro-priate short-range correlations nucleon form factors andhigher-order nucleonic weak currents are included in ourpresent results
We found in the calculations that often there exists onlya few relevant intermediate states which collect most of thestrength corresponding to a given multipole We also foundthat there exists a single-state dominance in the important 2minuscomponents related to the ground-state decays of nuclei 76Ge82Se and perhaps also for 96Zr
Competing Interests
The authors declare that they have no competing interests
Acknowledgments
This work has been partially supported by the Academy ofFinland under the Finnish Centre of Excellence Programme2012ndash2017 (Nuclear and Accelerator Based Programme atJYFL)
References
[1] J Suhonen and O Civitarese ldquoWeak-interaction and nuclear-structure aspects of nuclear double beta decayrdquo Physics Reportvol 300 no 3-4 pp 123ndash214 1998
[2] F T Avignone S R Elliott and J Engel ldquoDouble beta decayMajorana neutrinos and neutrino massrdquo Reviews of ModernPhysics vol 80 no 2 pp 481ndash516 2008
[3] J D Vergados H Ejiri and F Simkovic ldquoTheory of neutrinolessdouble-beta decayrdquoReports on Progress in Physics vol 75 no 10p 106301 2012
[4] J Maalampi and J Suhonen ldquoNeutrinoless double 120573
+ECdecaysrdquo Advances in High Energy Physics vol 2013 Article ID505874 18 pages 2013
[5] M Kortelainen O Civitarese J Suhonen and J ToivanenldquoShort-range correlations and neutrinoless double beta decayrdquoPhysics Letters Section B Nuclear Elementary Particle andHigh-Energy Physics vol 647 no 2-3 pp 128ndash132 2007
[6] F Simkovic G Pantis J D Vergados and A Faessler ldquoAddi-tional nucleon current contributions to neutrinoless double 120573decayrdquo Physical Review C vol 60 no 5 Article ID 055502 14pages 1999
[7] J Suhonen and O Civitarese ldquoReview of the properties of the0]120573minus120573minus nuclear matrix elementsrdquo Journal of Physics G Nuclearand Particle Physics vol 39 no 12 Article ID 124005 2012
[8] P Vogel ldquoNuclear structure and double beta decayrdquo Journal ofPhysics G Nuclear and Particle Physics vol 39 no 12 p 1240022012
[9] J Barea J Kotila and F Iachello ldquoNuclear matrix elements fordouble-120573 decayrdquo Physical Review C vol 87 no 1 Article ID014315 2013
[10] A Neacsu and S Stoica ldquoStudy of nuclear effects in thecomputation of the 0]120573120573 decay matrix elementsrdquo Journal ofPhysics G Nuclear and Particle Physics vol 41 no 1 Article ID015201 2014
[11] J Engel ldquoUncertainties in nuclear matrix elements for neutri-noless double-beta decayrdquo Journal of Physics G Nuclear andParticle Physics vol 42 no 3 Article ID 034017 2015
[12] F Simkovic A Faessler V Rodin P Vogel and J EngelldquoAnatomyof the 0]120573120573nuclearmatrix elementsrdquoPhysical ReviewC Nuclear Physics vol 77 no 4 Article ID 045503 2008
[13] M T Mustonen and J Engel ldquoLarge-scale calculations ofthe double-120573 decay of 76Ge 130Te 136Xe and 150Nd in thedeformed self-consistent Skyrme quasiparticle random-phaseapproximationrdquo Physical Review C vol 87 Article ID 0643022013
[14] F Simkovic V Rodin A Faessler and P Vogel ldquo0]120573120573 and2]120573120573 nuclear matrix elements quasiparticle random-phaseapproximation and isospin symmetry restorationrdquo PhysicalReview C vol 87 no 4 Article ID 045501 9 pages 2013
[15] V A Rodin A Faessler F Simkovic and P Vogel ldquoAssessmentof uncertainties in QRPA 0]120573120573-decay nuclear matrix elementsrdquoNuclear Physics A vol 766 pp 107ndash131 2006
[16] D-L Fang A Faessler V Rodin and F Simkovic ldquoNeutri-noless double-120573 decay of deformed nuclei within quasiparticlerandom-phase approximationwith a realistic interactionrdquoPhys-ical Review C vol 83 no 3 Article ID 034320 8 pages 2011
[17] J Hyvarinen and J Suhonen ldquoNuclear matrix elements for0]120573120573 decays with light or heavy Majorana-neutrino exchangerdquoPhysical Review C vol 91 no 2 Article ID 024613 12 pages2015
[18] P Pirinen and J Suhonen ldquoSystematic approach to 120573 and 2]120573120573decays of mass119860 = 100minus136 nucleirdquo Physical Review C vol 91no 5 Article ID 054309 2015
[19] H Ejiri N Soukouti and J Suhonen ldquoSpin-dipole nuclearmatrix elements for double beta decays and astro-neutrinosrdquoPhysics Letters B vol 729 pp 27ndash32 2014
[20] J Menendez D Gazit and A Schwenk ldquoChiral two-bodycurrents in nuclei Gamow-Teller transitions and neutrinolessdouble-beta decayrdquo Physical Review Letters vol 107 no 6Article ID 062501 5 pages 2011
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
High Energy PhysicsAdvances in
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
FluidsJournal of
Atomic and Molecular Physics
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Advances in Condensed Matter Physics
OpticsInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
AstronomyAdvances in
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Superconductivity
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Statistical MechanicsInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
GravityJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
AstrophysicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Physics Research International
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Solid State PhysicsJournal of
âComputationalââMethodsâinâPhysics
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Soft MatterJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
AerodynamicsJournal of
Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
PhotonicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Biophysics
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
ThermodynamicsJournal of
12 Advances in High Energy Physics
Table 5 Most important multipoles and intermediate states contributing to the ground-state-to-excited-state 0]120573120573 decays mediated by thelight neutrino exchange Columns 119864 give the energies (in MeVs) and multipoles of the intermediate states Multipoles are organized from leftto right in terms of their importance themost important being on the left Columns labeled119862 give the correspondingNMEcontributions Lasttwo numbers in each 119862 column give the summed contribution and the percentual part which the displayed states give to the total multipolestrength The percentage inside the parenthesis gives the fraction with which the displayed states contribute to the total NME
Nucleus 119864(1
+
) 119862 119864(0
+
) 119862
124Sn
000 0101 270 0667066 0433 460 0088100 0146 735 0567225 0051256 0227331 0664672 0289791 04241309 04731386 0161
2968 132294 (44) 94 (19)
Nucleus 119864(1
+
) 119862 119864(0
+
) 119862 119864(2
minus
) 119862 119864(2
+
) 119862
130Te
025 0390 740 0719 097 0340 041 0019150 0082 835 0631 507 0073 059 0047232 0097 1726 0106 183 0017276 0134 1895 0083 280 0043453 0064 470 0024559 0338 486 0027757 0592 518 00331466 0065 627 00161501 0259 675 00511507 0773 910 00321539 0089 1057 0045
1787 00262314 0024
2882 1351 0602 039095 (41) 996 (19) 79 (9) 65 (6)
Nucleus 119864(1
+
) 119862 119864(0
+
) 119862 119864(2
minus
) 119862 119864(2
+
) 119862
136Xe
230 0366 1224 0317 250 0060 486 0204306 0050 1254 0712 486 0056 1858 0083375 0066 529 0113 2034 0059446 0226 834 00531012 0232 1411 00541152 05651712 01371778 1031
2675 1029 0335 034693 (43) 101 (16) 53 (5) 67 (6)
Advances in High Energy Physics 13
were adopted We notice from the figures that increasing theaxial coupling strength shifts the distributions downwardsThis is especially true for the higher energy parts Despite thisfact the overall shapes of the cumulative sum distributionsdo not changemuch and the same classification of nuclei intodifferent categories according to their cumulative distributionshapes seems to hold also for larger values of 119892A
4 Conclusions
In this paper we have extended our previous work [17] onthe ground-state-to-ground-state 0]120573120573 decay transitions Inthe present work we have concentrated our studies on theintermediate contributions to the NMEs involved in thelight neutrino mediated 0]120573120573 decay We have calculated theintermediate state multipole decompositions of the NMEsand extracted the most important multipole componentsCumulative sums of the NMEs were calculated to investigatethe important energy regions contributing to the 0]120573120573 tran-sitions Finally the most important multipole componentswere divided into contributions coming from the virtualtransitions through the individual states of the 0]120573120573 inter-mediate nuclei An extensive tabulation of these importantintermediate states were given for all the nuclei considered inthis paper
We have done these computations by using realistic two-body interactions and single-particle bases All the appro-priate short-range correlations nucleon form factors andhigher-order nucleonic weak currents are included in ourpresent results
We found in the calculations that often there exists onlya few relevant intermediate states which collect most of thestrength corresponding to a given multipole We also foundthat there exists a single-state dominance in the important 2minuscomponents related to the ground-state decays of nuclei 76Ge82Se and perhaps also for 96Zr
Competing Interests
The authors declare that they have no competing interests
Acknowledgments
This work has been partially supported by the Academy ofFinland under the Finnish Centre of Excellence Programme2012ndash2017 (Nuclear and Accelerator Based Programme atJYFL)
References
[1] J Suhonen and O Civitarese ldquoWeak-interaction and nuclear-structure aspects of nuclear double beta decayrdquo Physics Reportvol 300 no 3-4 pp 123ndash214 1998
[2] F T Avignone S R Elliott and J Engel ldquoDouble beta decayMajorana neutrinos and neutrino massrdquo Reviews of ModernPhysics vol 80 no 2 pp 481ndash516 2008
[3] J D Vergados H Ejiri and F Simkovic ldquoTheory of neutrinolessdouble-beta decayrdquoReports on Progress in Physics vol 75 no 10p 106301 2012
[4] J Maalampi and J Suhonen ldquoNeutrinoless double 120573
+ECdecaysrdquo Advances in High Energy Physics vol 2013 Article ID505874 18 pages 2013
[5] M Kortelainen O Civitarese J Suhonen and J ToivanenldquoShort-range correlations and neutrinoless double beta decayrdquoPhysics Letters Section B Nuclear Elementary Particle andHigh-Energy Physics vol 647 no 2-3 pp 128ndash132 2007
[6] F Simkovic G Pantis J D Vergados and A Faessler ldquoAddi-tional nucleon current contributions to neutrinoless double 120573decayrdquo Physical Review C vol 60 no 5 Article ID 055502 14pages 1999
[7] J Suhonen and O Civitarese ldquoReview of the properties of the0]120573minus120573minus nuclear matrix elementsrdquo Journal of Physics G Nuclearand Particle Physics vol 39 no 12 Article ID 124005 2012
[8] P Vogel ldquoNuclear structure and double beta decayrdquo Journal ofPhysics G Nuclear and Particle Physics vol 39 no 12 p 1240022012
[9] J Barea J Kotila and F Iachello ldquoNuclear matrix elements fordouble-120573 decayrdquo Physical Review C vol 87 no 1 Article ID014315 2013
[10] A Neacsu and S Stoica ldquoStudy of nuclear effects in thecomputation of the 0]120573120573 decay matrix elementsrdquo Journal ofPhysics G Nuclear and Particle Physics vol 41 no 1 Article ID015201 2014
[11] J Engel ldquoUncertainties in nuclear matrix elements for neutri-noless double-beta decayrdquo Journal of Physics G Nuclear andParticle Physics vol 42 no 3 Article ID 034017 2015
[12] F Simkovic A Faessler V Rodin P Vogel and J EngelldquoAnatomyof the 0]120573120573nuclearmatrix elementsrdquoPhysical ReviewC Nuclear Physics vol 77 no 4 Article ID 045503 2008
[13] M T Mustonen and J Engel ldquoLarge-scale calculations ofthe double-120573 decay of 76Ge 130Te 136Xe and 150Nd in thedeformed self-consistent Skyrme quasiparticle random-phaseapproximationrdquo Physical Review C vol 87 Article ID 0643022013
[14] F Simkovic V Rodin A Faessler and P Vogel ldquo0]120573120573 and2]120573120573 nuclear matrix elements quasiparticle random-phaseapproximation and isospin symmetry restorationrdquo PhysicalReview C vol 87 no 4 Article ID 045501 9 pages 2013
[15] V A Rodin A Faessler F Simkovic and P Vogel ldquoAssessmentof uncertainties in QRPA 0]120573120573-decay nuclear matrix elementsrdquoNuclear Physics A vol 766 pp 107ndash131 2006
[16] D-L Fang A Faessler V Rodin and F Simkovic ldquoNeutri-noless double-120573 decay of deformed nuclei within quasiparticlerandom-phase approximationwith a realistic interactionrdquoPhys-ical Review C vol 83 no 3 Article ID 034320 8 pages 2011
[17] J Hyvarinen and J Suhonen ldquoNuclear matrix elements for0]120573120573 decays with light or heavy Majorana-neutrino exchangerdquoPhysical Review C vol 91 no 2 Article ID 024613 12 pages2015
[18] P Pirinen and J Suhonen ldquoSystematic approach to 120573 and 2]120573120573decays of mass119860 = 100minus136 nucleirdquo Physical Review C vol 91no 5 Article ID 054309 2015
[19] H Ejiri N Soukouti and J Suhonen ldquoSpin-dipole nuclearmatrix elements for double beta decays and astro-neutrinosrdquoPhysics Letters B vol 729 pp 27ndash32 2014
[20] J Menendez D Gazit and A Schwenk ldquoChiral two-bodycurrents in nuclei Gamow-Teller transitions and neutrinolessdouble-beta decayrdquo Physical Review Letters vol 107 no 6Article ID 062501 5 pages 2011
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
High Energy PhysicsAdvances in
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
FluidsJournal of
Atomic and Molecular Physics
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Advances in Condensed Matter Physics
OpticsInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
AstronomyAdvances in
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Superconductivity
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Statistical MechanicsInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
GravityJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
AstrophysicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Physics Research International
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Solid State PhysicsJournal of
âComputationalââMethodsâinâPhysics
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Soft MatterJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
AerodynamicsJournal of
Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
PhotonicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Biophysics
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
ThermodynamicsJournal of
Advances in High Energy Physics 13
were adopted We notice from the figures that increasing theaxial coupling strength shifts the distributions downwardsThis is especially true for the higher energy parts Despite thisfact the overall shapes of the cumulative sum distributionsdo not changemuch and the same classification of nuclei intodifferent categories according to their cumulative distributionshapes seems to hold also for larger values of 119892A
4 Conclusions
In this paper we have extended our previous work [17] onthe ground-state-to-ground-state 0]120573120573 decay transitions Inthe present work we have concentrated our studies on theintermediate contributions to the NMEs involved in thelight neutrino mediated 0]120573120573 decay We have calculated theintermediate state multipole decompositions of the NMEsand extracted the most important multipole componentsCumulative sums of the NMEs were calculated to investigatethe important energy regions contributing to the 0]120573120573 tran-sitions Finally the most important multipole componentswere divided into contributions coming from the virtualtransitions through the individual states of the 0]120573120573 inter-mediate nuclei An extensive tabulation of these importantintermediate states were given for all the nuclei considered inthis paper
We have done these computations by using realistic two-body interactions and single-particle bases All the appro-priate short-range correlations nucleon form factors andhigher-order nucleonic weak currents are included in ourpresent results
We found in the calculations that often there exists onlya few relevant intermediate states which collect most of thestrength corresponding to a given multipole We also foundthat there exists a single-state dominance in the important 2minuscomponents related to the ground-state decays of nuclei 76Ge82Se and perhaps also for 96Zr
Competing Interests
The authors declare that they have no competing interests
Acknowledgments
This work has been partially supported by the Academy ofFinland under the Finnish Centre of Excellence Programme2012ndash2017 (Nuclear and Accelerator Based Programme atJYFL)
References
[1] J Suhonen and O Civitarese ldquoWeak-interaction and nuclear-structure aspects of nuclear double beta decayrdquo Physics Reportvol 300 no 3-4 pp 123ndash214 1998
[2] F T Avignone S R Elliott and J Engel ldquoDouble beta decayMajorana neutrinos and neutrino massrdquo Reviews of ModernPhysics vol 80 no 2 pp 481ndash516 2008
[3] J D Vergados H Ejiri and F Simkovic ldquoTheory of neutrinolessdouble-beta decayrdquoReports on Progress in Physics vol 75 no 10p 106301 2012
[4] J Maalampi and J Suhonen ldquoNeutrinoless double 120573
+ECdecaysrdquo Advances in High Energy Physics vol 2013 Article ID505874 18 pages 2013
[5] M Kortelainen O Civitarese J Suhonen and J ToivanenldquoShort-range correlations and neutrinoless double beta decayrdquoPhysics Letters Section B Nuclear Elementary Particle andHigh-Energy Physics vol 647 no 2-3 pp 128ndash132 2007
[6] F Simkovic G Pantis J D Vergados and A Faessler ldquoAddi-tional nucleon current contributions to neutrinoless double 120573decayrdquo Physical Review C vol 60 no 5 Article ID 055502 14pages 1999
[7] J Suhonen and O Civitarese ldquoReview of the properties of the0]120573minus120573minus nuclear matrix elementsrdquo Journal of Physics G Nuclearand Particle Physics vol 39 no 12 Article ID 124005 2012
[8] P Vogel ldquoNuclear structure and double beta decayrdquo Journal ofPhysics G Nuclear and Particle Physics vol 39 no 12 p 1240022012
[9] J Barea J Kotila and F Iachello ldquoNuclear matrix elements fordouble-120573 decayrdquo Physical Review C vol 87 no 1 Article ID014315 2013
[10] A Neacsu and S Stoica ldquoStudy of nuclear effects in thecomputation of the 0]120573120573 decay matrix elementsrdquo Journal ofPhysics G Nuclear and Particle Physics vol 41 no 1 Article ID015201 2014
[11] J Engel ldquoUncertainties in nuclear matrix elements for neutri-noless double-beta decayrdquo Journal of Physics G Nuclear andParticle Physics vol 42 no 3 Article ID 034017 2015
[12] F Simkovic A Faessler V Rodin P Vogel and J EngelldquoAnatomyof the 0]120573120573nuclearmatrix elementsrdquoPhysical ReviewC Nuclear Physics vol 77 no 4 Article ID 045503 2008
[13] M T Mustonen and J Engel ldquoLarge-scale calculations ofthe double-120573 decay of 76Ge 130Te 136Xe and 150Nd in thedeformed self-consistent Skyrme quasiparticle random-phaseapproximationrdquo Physical Review C vol 87 Article ID 0643022013
[14] F Simkovic V Rodin A Faessler and P Vogel ldquo0]120573120573 and2]120573120573 nuclear matrix elements quasiparticle random-phaseapproximation and isospin symmetry restorationrdquo PhysicalReview C vol 87 no 4 Article ID 045501 9 pages 2013
[15] V A Rodin A Faessler F Simkovic and P Vogel ldquoAssessmentof uncertainties in QRPA 0]120573120573-decay nuclear matrix elementsrdquoNuclear Physics A vol 766 pp 107ndash131 2006
[16] D-L Fang A Faessler V Rodin and F Simkovic ldquoNeutri-noless double-120573 decay of deformed nuclei within quasiparticlerandom-phase approximationwith a realistic interactionrdquoPhys-ical Review C vol 83 no 3 Article ID 034320 8 pages 2011
[17] J Hyvarinen and J Suhonen ldquoNuclear matrix elements for0]120573120573 decays with light or heavy Majorana-neutrino exchangerdquoPhysical Review C vol 91 no 2 Article ID 024613 12 pages2015
[18] P Pirinen and J Suhonen ldquoSystematic approach to 120573 and 2]120573120573decays of mass119860 = 100minus136 nucleirdquo Physical Review C vol 91no 5 Article ID 054309 2015
[19] H Ejiri N Soukouti and J Suhonen ldquoSpin-dipole nuclearmatrix elements for double beta decays and astro-neutrinosrdquoPhysics Letters B vol 729 pp 27ndash32 2014
[20] J Menendez D Gazit and A Schwenk ldquoChiral two-bodycurrents in nuclei Gamow-Teller transitions and neutrinolessdouble-beta decayrdquo Physical Review Letters vol 107 no 6Article ID 062501 5 pages 2011
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
High Energy PhysicsAdvances in
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
FluidsJournal of
Atomic and Molecular Physics
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Advances in Condensed Matter Physics
OpticsInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
AstronomyAdvances in
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Superconductivity
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Statistical MechanicsInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
GravityJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
AstrophysicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Physics Research International
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Solid State PhysicsJournal of
âComputationalââMethodsâinâPhysics
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Soft MatterJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
AerodynamicsJournal of
Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
PhotonicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Biophysics
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
ThermodynamicsJournal of
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
High Energy PhysicsAdvances in
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
FluidsJournal of
Atomic and Molecular Physics
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Advances in Condensed Matter Physics
OpticsInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
AstronomyAdvances in
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Superconductivity
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Statistical MechanicsInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
GravityJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
AstrophysicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Physics Research International
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Solid State PhysicsJournal of
âComputationalââMethodsâinâPhysics
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Soft MatterJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
AerodynamicsJournal of
Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
PhotonicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Biophysics
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
ThermodynamicsJournal of