Relationship between Neutrinoless ββ-Decay and Double Charge-Exchange Resonances
C. De Conti, V. dos S. Ferreira, A.R. Samana, C.A. Barbero, F. Krmpoti?
aa r X i v : . [ nu c l - t h ] J a n Relationship between Neutrinoless ββ -Decay and Double Charge-ExchangeResonances C. De Conti , V. dos S. Ferreira , A.R. Samana , C.A. Barbero , and F. Krmpoti´c Campus Experimental de Rosana, Universidade Estadual Paulista, CEP 19274-000, Rosana, SP, Brazil Instituto de F´ısica, Universidade do Estado do Rio de Janeiro, CEP 20550-900, Rio de Janeiro-RJ, Brazil Departamento de Ciˆencias Exactas e Tecnol´ogicas,Universidade Estadual de Santa Cruz, CEP 45662-000 Ilh´eus, Bahia-BA, Brazil and Instituto de F´ısica La Plata, CONICET, Universidad Nacional de La Plata, CP 1900 La Plata, Argentina,
To describe the double-charge-exchange (DCE) processes, we have designed recently the( pn, p n )-QTDA model which fully includes the pairing correlations and four quasiparticle ex-citations. It has been applied in 2 ν double beta decays (DBDs), and the double charge-exchangeresonances (DCERs). Here we extend it to 0 ν DBD and discuss the relationship between the nuclearmatrix elements (NMEs), and the DCE reaction matrix elements (RMEs) with the same spin-isospinstructure. We do it for all final 0 + states, even in the region of DCERs, where the DBD is energet-ically forbidden. As an example, we evaluate the DBD Ge → Se, both for 2 ν and 0 ν modes, aswell as the associated DCE sum rules, excitation energies within the Q -value window for DBD, andthe Q -value itself. We find that the 0 ν NMEs are correlated with the RMEs, both at low energy, andin the DCER region where most of the transition strength is concentrated. These findings occur inother nuclei as well and suggest that measurements of 0 + DCERs could provide useful informationregarding the 0 ν DBD. An analogous comparison and conclusion cannot be made for the 2 + states,since the 0 ν NMEs and RMEs transition operators are not similar to each other in this case.
Introduction:
The single charge-exchange (SCE) op-erators O = τ ∓ , and O = τ ∓ σ , as well as the DCEoperators D ∓ J J = [ O ∓ J × O ∓ J ] J , with J = 0 ,
1, and J = 0 ,
2, play a fundamental role in single β ∓ -decays(SBD ∓ s): ( A, Z ) → ( A, Z ± β ∓ -decays(DBD ∓ s): ( A, Z ) → ( A, Z ± h m ν i .The SCE QRPA was developed in 1967 [1] to describethe SBDs. In the 1990s, by averaging over two singleSBDs, this model was adapted for calculations of 0 ν and2 ν DBDs to ground states of final nuclei [2–7]. This av-eraging procedure is known as the pn -QRPA [8]. Themain reason for the popularity of pn -QRPA is its pro-nounced sensitivity to pairing correlations, allowing it inthis way to account for extremely large DBD half lives(see, for instance, [3, Fig. 2]). Unfavorable aspects ofthe pn -QRPA are: 1) it is unable to describe the DBDsto excited states, and 2) it does not allow the extrac-tion of information on 0 ν NMEs from the DCE reactionmeasurements.In recent years a lot of attention is being paid to DBDsto the final excited 0 + states, and there are several largeunderground experiments operating for the detection of h m ν i , such as the Majorana Demonstrator’s search in Ge [9].On the other hand, we still have not learned any-thing about 0 ν DBDs from heavy ion reactions. But, theNUMEN heavy ion multidetector, designed for a comple- mentary approach to the 0 ν NMEs, is taking data at thepresent time [10–14]. Theoretical comparisons betweenthe 0 ν NMEs and the RMEs were made, but only for theground states [15–17], where a very tiny portion of thetotal reaction strength is located.We have recently constructed a new model, based onthe BCS mean field, which appropriately describes thenuclear (
A, Z ) → ( A, Z ±
2) phenomena, by fully takinginto account both the pairing correlations, and the four-quasiparticle excitations. It is called ( pn, p n )-QTDA,or DCE QTDA [18], and it is a natural extension of theSCE QRPA model, with several advantages over the pn -QRPA, such as: i) it allows us to study the NMEs of allfinal 0 + and 2 + states, accounting at the same time fortheir excitation energies and the corresponding Q -values,ii) together with the DBD NMEs, and the RMEs, it alsoallows the evaluation of the energy spectra, as well as thepositions of Double Isobaric Analog State (DIAS), andmonopole and quadrupole Double Gamow-Teller Reso-nances (DGTRs), and their sum rules values.There is a close correlation between the 0 ν NMEs andthe RMEs, since, except for the pseudoscalar and weakmagnetism terms (see [19, Eqs.(15)]), the underlying nu-clear spin-isospin structure is the same. Here we incor-porate the evaluation of 0 ν DBD into the DCE QTDA,drawing largely on our recent works [8, 18], and thencomparing the energy spectra of 0 ν NMEs and of RMEs.The transition operators for the 0 ν decays to the fi-nal 0 + and 2 + states are completely different from eachother, since they are spawned from different parts of thevector (V), and axial-vector (A) weak-hadronic-currents J µ = (cid:0) ρ, j (cid:1) ; see, for instance [20, Eqs.(7), (8)]. In fact,while 0 ν NMEs for the 0 + states come from ρ V , and j A , which are akin to D , and D , respectively, the 2 + states come from the velocity dependent parts of j V , and j A , that does not resemble D ; see, [21, Eqs.(5), (6)].Then, the comparison between 0 ν NMEs and RMEs onlymakes sense for the 0 + states. Formalism:
The DCE QTDA expressions for the 2 ν DBD are given in [18, Eq. (2.9)]. To derive the formu-las for 0 ν DDB, it is convenient to deal with the two-body density matrix, defined by [18, Eq.(2.10)], whichfor DBD − reads (see [18, Eqs. (2.44)-(2.46)]), ̺ − ( pnp ′ n ′ ; J πα , J + f ) = ˆ J ˆ J u p ′ v n ′ X p ′ n ′ J πα × X p ′′ n ′′ J p J n ( − ) J p + J n ˆ J p ˆ J n N ( nn ′′ ) N ( pp ′′ ) Y pp ′′ J p , nn ′′ J n ; J + f × ¯ P ( nn ′′ J n ) ¯ P ( pp ′′ J p ) p p ′′ J p n n ′′ J n J J J X p ′′ n ′′ J πα u p v n , (1)where ˆ J = √ J + 1, and the operator¯ P ( p p J ) = 1 + ( − ) p − p + J P ( p ↔ p ) , (2)takes into account the Pauli principle by exchanging theparticles p and p , and acts only on the right handside. X pnJ πα , and Y pp ′ J p , nn ′ J n ; J + f , are, respectively, the pn , and 2 p n QTDA amplitudes for excitations from ini-tial state | + i i going to intermediate states | J πα i , with J π = 0 ± , ± , · · · , and energies ω J πα , and final states |J πf i ,with J π = 0 + , + , and energies Ω J f . The correspond-ing DBD + density matrix ̺ + ( pnp ′ n ′ ; J πα , J + f ) is obtainedfrom (1) after making u p v n → u n v p , and u p ′ v n ′ → u n ′ v p ′ .The 0 ν NMs arise not only from V and A weak currents,but also from weak magnetism (M), and pseudoscalar (P)currents [8, Eq. (1.1)]. Thus M ν (0 + f ) = X X M νX (0 + f ) , (3)where X = V, A, M, P . The corresponding weak cou-plings are fixed as follows: (i) g V = 1, and g M = 3 . g A = 1 . g P =2 M N g A / ( k + m π ) from the assumption of Partially Con-served Axial Current.Finite Nucleon Size effects are introduced through theusual dipole form factors g X → g X ( k ) ≡ g X (1 + k / Λ X ) − , (4)where g X = g V , g A , ( g M + g V ) / (2 M N ), and g ′ P = g P / (2 M N ). Λ V = Λ M = 0 .
85 GeV, and Λ A = Λ P =1 .
086 GeV are cut-off parameters as found in [19]. TheShort Range Correlations are included in the way indi-cated in [8, Eqs.(2.29)-(2.31)].The individual pieces M νX (0 + f ) are obtained from Eqs.(20) in [8], after performing the substitutions ̺ ( pnp ′ n ′ ; J πα ) → ̺ ∓ ( pnp ′ n ′ ; J πα , + f ) ,ω J πα → D J πα , + f = ω J πα − Ω + f / . (5) For instance, the axial-vector 0 ν ∓ NME reads M ν ∓ A (0 + f ) = X LJ πα ( − ) L +1 X pp ′ nn ′ ̺ ± ( pnp ′ n ′ ; J πα , + f ) × W L J ( pn ) W L J ( p ′ n ′ ) R ALL ( pnp ′ n ′ ; D J πα , + f ) , (6)where W LSJ ( pn ) is a purely angular momenta factor, asdefined in [8, Eq. (21)]. The two-body radial integralsare defined as (see [20]) R XLL ′ ( pnp ′ n ′ ; D ) = r N Z dkk v X ( k ; D ) × R L ( pn ; k ) R L ′ ( p ′ n ′ ; k ) , (7)with R L ( pn ; k ) = Z ∞ u n p l p ( r ) u n n l n ( r ) j L ( kr ) r dr, (8)and v X ( k, D ) = 2 π G X ( k ) k ( k + D ) , (9)where G X ( k ) = g V ( k ) , g A ( k ) , k f M ( k ) , k g P ( k )[2 g A ( k ) − k g ′ P ( k )], and r N = 1 . A / fm is introduced to makethe 0 ν NMEs dimensionless.The 0 ν NMEs are usually evaluated in closure approx-imation (CA) where the denominators D ≡ D J πα , + f areapproximated by a constant value of the order of 10MeV [23]. Here, we go a step beyond and make D J πα , + f → ¯ ω J π − Ω + f / ≡ D ¯ J π , + f (10)where ¯ ω J π are centroid energies for the axial-vector op-erators O ∓ J π = τ ∓ ( σY L r L ) J π , with J = 1 , , · · · , and (i) L = J for: J even, and π = +, or J odd, and π = − ,and (ii) L = J − J even, and π = − , or J odd, and π = +. For J π = 0 ∓ these operators are O ∓ + = τ ∓ , and O ∓ − = τ ∓ ( σr ) − . This procedure will be called average-energy closure approximation (AECA).The AECA allows us to simplify greatly the numericalevaluation of the 0 ν NME within the SCE QTDA, sincewe make use of the closure relation for the intermediatestates | J πα i in (1), by summing over α . This leads to X α ̺ ∓ ( pnp ′ n ′ ; J πα , J + f ) ≡ ˜ ̺ ∓ ( pnp ′ n ′ ; J π , J + f )= ˆ J ˆ J X pp ′ nn ′ J p J n ( − ) J p + J n N ( nn ′ ) N ( pp ′ ) Y pp ′ J p , nn ′ J n ; J + f × ˆ J p ˆ J n ¯ P ( nn ′ J n ) ¯ P ( pp ′ J p ) p p ′ J p n n ′ J n J J J (cid:26) u p v n u p ′ v n ′ v p u n v p ′ u n ′ , (11)where the dependence of ˜ ̺ ∓ on parity π is implicit in thecoupling ( pn ) J π . Thus, instead of (6), we have now M ν ∓ A (0 + f ) = X LJ π ( − ) L +1 X pp ′ nn ′ ˜ ̺ ∓ ( pnp ′ n ′ ; J π , + f ) × W L J ( pn ) W L J ( p ′ n ′ ) R ALL ( pnp ′ n ′ ; D ¯ J π , + f ) , (12)and similarly for V, M , and P ν NMEs in (3).The corresponding half-lives are evaluated from[ τ ( J + f )] − = (cid:12)(cid:12)(cid:12) F M ( J + f ) (cid:12)(cid:12)(cid:12) G ( J + f ) , (13)where the DBD mode factors are F ν = 1, and F ν = h m ν i , with h m ν i given in natural units (¯ h = m e = c = 1).All leptonic kinematics factors G ν ( J + f ), and G ν ( J + f )are taken from [24], except for G ν (2 +2 ) that is found in[25].The RMEs hJ + f || D ∓ J J || + i i , with J = 0 ,
1, and J =0 ,
2, are evaluated in the same way, and one gets M ∓ J ( J + f ) ≡ hJ + f || D ∓ J J || + i i = ˆ J − X pnp ′ n ′ × ˜ ̺ ∓ ( pnp ′ n ′ ; J , J + f ) W JJ ( pn ) W JJ ( p ′ n ′ ) . (14)Comparisons of M ν − V with M − , and of M ν − A with M − , for ground states 0 +1 in several nuclei have beenrecently done [15–17]. In doing this, one should keep inmind that, while for M − J only the intermediate state J + contributes, in the construction of M ν − V , and M ν − A allintermediate states J π participate, as seen, for instance,from (12). Therefore, it might be more appropriate toconfront the RMEs with the 0 ν NMEs that arise fromthese unique states, which are J π = 0 + , and J π = 1 + for V, and A NMEs, respectively, that will be labeled as M ν − V , and M ν − A .The charge-exchange transition strengths going to theintermediate and final states | J + α i and |J + f i are S {∓ } J ≡ X α |h J + α ||O ∓ J || + i i| , (15)and S {∓ } J J = X f |hJ f || D ∓ J J || + i i| ≡ X f S {∓ } J J (0 + f ) . (16)When both | J + α i and |J + f i are a complete set of excitedstates that can be reached by operating with O ± J , and D ± J J on | + i i , their differences S { } J = S {− } J − S { +1 } J S { } J J = S {− } J J − S { +2 } J J , (17)obey the following SCE Ikeda sum rule S { } J = N − Z, (18)and the DCE sum rules (DCESR)[26–28] S { } = 2( N − Z )( N − Z − , (19) S { } = 2( N − Z ) (cid:16) N − Z + 1 + 2 S {− } (cid:17) − C, S { } = 10( N − Z ) (cid:16) N − Z − S {− } (cid:17) + 53 C. where C is a relatively small quantity given by [27, (4)].The corresponding centroid energies are defined as¯ E {∓ } J J = P f E J + f | M ∓ J ( J + f ) | S {∓ } J J , (20)where E f = E {∓ } + f − E {∓ } +1 , (21)are the excitation energies in the ( A, Z ±
2) nuclei whichhave the same value in both nuclei in the present model.We will also discuss the total 0 ν strength S ν − = X f | M ν − (0 + f ) | ≡ X f S ν − (0 + f ) , (22)and its spectral distribution S ν − (0 + f ), which will beconfronted, respectively, with S {− } J J , and S {− } J J (0 + f ).A similar comparison will be done for M ν − V (0 + f ), M ν − A (0 + f ), M ν − V (0 + f ), and M ν − A (0 + f ) with M − J (0 + f ).The Q -values for the 2 β − -decay, and for the 2 e -captureare defined as Q β − = M ( Z, A ) − M ( Z + 2 , A ) ,Q e = M ( Z, A ) − M ( Z − , A ) , (23)where the M ’s are the atomic masses. In our model, theyread Q β − = E { } + − E { +2 } +1 = − Ω +1 − λ p − λ n ) ,Q e = E { } + − E {− } +1 = − Ω +1 + 2( λ p − λ n ) . (24)These are the windows of excitation energies where DBDscan be observed. Numerical Results:
We show some numerical resultsfor the decay Ge → Se, which is being experimentallystudied at the moment by Alvis et al. [9].To satisfy thesum rules (18) and (19), which is a necessary condition tohave control over numerical calculations, we use 9 singleparticle levels within the 2¯ hω − hω shells. The single-particle energies (spes) were obtained as described in [30].Hence, we get 2045 0 + and 8456 2 + states in Se.The δ -force V = − π ( v s P s + v t P t ) δ ( r ) MeV · fm , (25)is used as the residual interaction, with the pairingstrengths for protons and neutrons, v s pair ( p ) and v s pair ( n )obtained from the fitting of the corresponding experi-mental pairing gaps. The isovector ( v s ) and isoscalar ( v t )parameters within the particle-particle (pp) and particle-hole (ph) channels, as well as the ratios s = v s pp /v s pair , and t = v t pp /v s pair , with v s pair = ( v s pair ( p ) + v s pair ( n )) /
2, werefixed in the same way as in the pn -QRPA calculations.In detail, TABLE I. Calculated and experimental Q -values (in MeV).Par/Exp Q β − Q e ∆ Q (a) 2.420 -9.403 -11.82(b) 2.424 -9.400 -11.82Exp. 2.039 -10.910 -12.95 (a) by invoking the Partial Restoration of the SU4Symmetry, as done in [8, 18]: s = 1, t = 1 . v s ph = 27, and v t ph = 64, and(b) by choosing the parameter t to reproduce the mea-sured value of M ν (0 +1 ), as it is usually done [19]: s = 1, t = 2 . v s ph = 27, and v t ph = 64. TABLE II. Calculated and experimental excitation energies E ( J + f ) in Se, and the 2 ν NMEs M ν ( J + f ) for the decays of Ge to J + f = 0 +1 , , and 2 +1 , states in Se. The measuredhalf-live τ ν (0 +1 ) is from [31], and the corresponding NME M ν (0 +1 ) was evaluated from Eq.(13).Par/Exp 0 +1 +2 +1 +2 E ( J + f )(MeV) (a) 0 . .
25 0 .
50 2 . . .
26 0 .
54 2 . . .
12 0 .
56 1 . M ν ( J + f )(naturalunits × ) (a) 28 . − . .
11 0 . − . − .
61 7 . τ ν ( J + f )(yr) × × × × (a) 27 . . . .
92 2 .
04 0 .
02 0 . . ± . M ν ( J + f ), andthe RMEs M J ( J + f ), besides the half-life τ ν for the J + f =0 +1 , states in Se, within the parameterizations (a) and (b). M ν ( J + f ) and M J ( J + f ) are dimensionless, and multiplied by10 . τ ν are in units of yr × , and were evaluated from(13) for h m ν i = 1 . + f M νV M νV M M νA M νA M M νP M νM M ν τ ν (a)0 +1 -158 -92.3 144 -634 -285 242 64.3 -34.2 -762 1 . +2 . +1 -245 -166 266 -994 -591 518 84.5 -40.4 -1200 12 . +2 . Although our main goal is to discuss the relationshipbetween the 0 ν − DBD and DCE resonances, we also show
TABLE IV. Results for the transition strengths S {∓ } J J , theenergy centroids ¯ E {− } J J , and the 0 ν − DBD rates S ν − V,A , and S ν − V ,A , defined in (22). J J S {− } J J S { +2 } J J S { } J J S { } J J ¯ E {− } J J S ν − V,A S ν − V ,A (a)00 292 0 .
26 292 264 22 .
39 296 28610 315 0 .
44 315 ≤
334 16 .
70 365 31512 1439 2 .
10 1437 ≥ . .
26 292 264 22 .
25 296 28610 315 0 .
44 315 ≤
332 12 .
10 364 31512 1439 2 .
10 1437 ≥ . the results for other observables that are directly related,and have been measured. They are:1) The Q -values in Ge, given in Table I. Note thattheir difference ∆ Q ≡ Q e − Q β − = 4( λ p − λ n ), whichdepends only on the mean field, is correctly reproducedby the calculations. On the contrary, with the spes fromRef. [29] we obtain ∆ Q = 1 .
52 MeV.2) The calculated excitation energies E ( J + f ) in Se,and the 2 ν NMEs M ν ( J + f ) for the decays of Ge to J + f = 0 +1 , , and 2 +1 , states in Se, are listed in Ta-ble II. The corresponding half-lives τ ν ( J + f ) are alsoshown. The NMEs M ν (0 + f ), and the RMEs M J (0 + f )for the J + f = 0 +1 , states in Se are compared in TableIII. It makes sense to only compare the absolute values.But, we display also their signs to indicate the interfer-ence between different components. As expected from(3) and (12), the NME M νA , and even more the M νA ,agrees better with the RME M − than M ν − . The con-tributions of M νP , and M νM are small, but not negligible.We observe that both current values of 0 ν NME for theground state are notably less than that obtained in our pn -QRPA calculation [8], which was M ν − (0 +1 ) = 3 . J = 0 , J = 0), or DIAS, andmonopole ( J = 1 , J = 0), and quadrupole ( J = 1 , J =2) GT transition strengths S {∓ } J J , evaluated from (16),ii) their differences S { } J J , together with the correspondingpredicted sum rules S { } J J given by (19), and iii) the energycentroids ¯ E {− } J J given by (20) are shown in Table IV. Thefact that S {− } J J ≫ S { +2 } J J is due to a large neutron excess.The sum rules for the DIAS and DGTRs, S { } J J ∼ = S { } J J ,are fairly well fulfilled. The inequalities are due to theomission of the C term in (19).Also given in Table IV are the V and A parts of total 0 ν decay rates S ν , defined in (22). We compare S {− } with S ( E ) ( M e V - ) E (MeV) S S S /2 S V + S A + S V S A S S ( E ) S ( f + ) E FIG. 1. DCE reaction strength distributions S {− } J J ( E ) (for J = 0 ,
1, and J = 0 , ν − DBD strength distributions S ν − V ( E ), S ν − V ( E ), S ν − A ( E ), S ν − A ( E ), and S ν − ( E ) in Se, as a function of excitation energy E . The results for thedimensionless state-dependent strengths S {− } (0 +1 , ), and S νA (0 +1 , ) are shown in the insert plot. S νV and S νV , where only M νV or M νV contribute, and S {− } with S ν − A and S ν − A , where only M νA or M νA contribute. The axial-vector strengths were evaluatedfor g A = 1, since g A does not appear in the RMEs. Theagreement of S {− } with S νV and S νV , as well as of S {− } with S ν − A and S ν − A is a surprise. There is noreason, in principle, for such an agreement! The justmentioned 0 ν strengths are not, of course, measurable,but the reaction strengths, S {− } J J , are. Thus, the agree-ment between calculated and observed values for the lat-ter might be a very valuable information on the 0 ν DBD.Moreover, by examining Tables III and IV, it can be seenthat all total strengths are less sensitive to the variationof the model parameters than the individual 0 ν NMEs.Possibly these quantities are also independent of the nu-clear structure model used, as long as it is capable ofevaluating them correctly. The different strength distri-butions S {− } J J (0 + f ) and S ν (0 + f ), evaluated within theparametrization (b) and folded as S ( E ) = ∆ π X f S (0 + f )( E − E f ) + ∆ , (26)with the energy interval ∆ = 1 MeV, are represented inFigure 1. Following the same reasoning as in Table IV,the axial-vector strengths were evaluated by assuming g A = 1. It can be seen that also here S νV , and S νV are close to S {− } , just as S νA and S νA are close to S {− } .Furthermore, it is noted from this figure that the agree-ment manifested in the region of the DCERs also existsin the low energy region, where the 0 ν DBDs are en-ergetically allowed. As seen from Table IV, identicalresults are obtained for transition strengths within theparametrization (a), except that the GT resonances areshifted upwards by about 5 MeV. Therefore, the abovestatement on the agreement of calculated and measuredvalues is also relevant here.We present in Table IV and in Figure 1 also the resultsfor the 2 + RMEs, although the comparison with the 2 + ν NMEs cannot be made. This is done because the2 + strengths are much larger than the 0 + strengths, andthey can be experimentally confused with each other.In summary, we suggest that the experimental studyof the DCERs, which are mainly in the continuum ofthe energy spectrum, could provide useful informationregarding the neutrinoless double beta decay.We sincerely thank Wayne Seale and Tom T. S. Kuo fortheir careful and enlightening reading of the manuscript.This work was financed in part by the Coordena¸c˜ao deAperfei¸coamento de Pessoal de N´ıvel Superior Brasil Fi-nance Code 001. A.R.S. acknowledges the financial sup-port of Funda¸c˜ao de Amparo `a Pesquisa do Estado daBahia (T.O. PIE0013/2016) and the partial support ofUESC (PROPP 00220.1300.1832). [1] J. A. Halbleib and R. A. Sorensen, Nucl. Phys. A98 , 542(1967).[2] P. Vogel and M. R. Zirnbauer, Phys. Rev. Lett , 3148(1986).[3] O. Civitarese, A. Faessler and T. Tomoda, Phys. Lett. B194 , 11 (1987).[4] T. Tomoda and A. Faessler, Phys. Lett.
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