γγ decay as a probe of neutrinoless ββ decay nuclear matrix elements
γγγ decay as a probe of neutrinoless ββ decay nuclear matrix elements B. Romeo,
1, 2
J. Men´endez, and C. Pe˜na Garay
2, 4 Donostia International Physics Center, 20018 San Sebasti´an, Spain Laboratorio Subterr´aneo de Canfranc, 22880 Canfranc-Estaci´on, Spain Department of Quantum Physics and Astrophysics and Institute of Cosmos Sciences,University of Barcelona, 08028 Barcelona, Spain Institute for Integrative Systems Biology (I SysBio) , Valencia, Spain.
We study double gamma ( γγ ) decay nuclear matrix elements (NMEs) for a wide range of nucleifrom titanium to xenon, and explore their relation to neutrinoless double-beta (0 νββ ) NMEs. Tofavor the comparison, we focus on double-magnetic dipole transitions in the final ββ nuclei, inparticular the γγ decay of the double isobaric analog of the initial ββ state into the ground state.For the most probable decay with equal-energy photons, our large-scale nuclear shell model resultsshow a good linear correlation between the γγ and 0 νββ NMEs. Our analysis reveals that thecorrelation holds for γγ transitions driven by the spin or orbital angular momentum due to thedominance of zero-coupled nucleon pairs, a feature common to 0 νββ decay. Our findings point outthe potential of future γγ decay measurements to constrain 0 νββ NMEs, which are key to answerfundamental physics questions based on 0 νββ experiments.
Introduction and main result.
The observation of thedecay of an atomic nucleus emitting only two electrons,neutrinoless double-beta (0 νββ ) decay, is the process ex-perimentally most feasible to demonstrate that neutrinosare their own antiparticles [1]. Moreover, 0 νββ decay isone of the most promising probes of physics beyond thestandard model (BSM) of particle physics [2]. For in-stance, the observation of change in lepton number in0 νββ decay could help to explain the prevalence of mat-ter in the universe [3, 4]. Because of this unique poten-tial, a very active program aims to detect 0 νββ decay.Currently the most stringent constraints reach half-liveslonger than 10 years [5–12], and next generation ton-scale experiments are being proposed, among others, for Ge,
Mo,
Te and
Xe nuclei.Since 0 νββ decay changes lepton number—no antineu-trinos are emitted to balance the two electrons—its decayrate depends on some unknown BSM parameter(s). Inthe standard scenario that 0 νββ is triggered by the ex-change of known neutrinos, this role is played by a com-bination of absolute neutrino masses and mixing matrixelements, m ββ . The decay rate also depends on a cal-culable phase-space factor [13, 14], and quadratically onthe nuclear matrix element (NME) that involves the ini-tial and final nuclear states [15]. Thus, 0 νββ NMEs areneeded to interpret current experimental half-life limitsand to anticipate the reach of future searches. However,typical NME calculations disagree up to a factor 3 [16–28], about an order of magnitude on the decay rate. Fur-thermore, first attempts to obtain more controlled 0 νββ
NMEs using ab initio techniques suggest smaller NMEvalues that most previous studies [29–31].A widely explored approach to reduce the uncertaintyin 0 νββ analyses is to study related nuclear observables.Nuclear structure [32] or muon capture [33] data are veryuseful to test nuclear models used to calculate NMEs, butthey are not directly related to 0 νββ decay. Likewise, ββ decay with neutrino emission shows no apparent corre-lation with 0 νββ , in spite of both being second-orderweak processes sharing initial and final states [34]. Use-ful insights could be gained from nuclear reactions, inthe same spirit of the β decay information obtained incharge-exchange experiments [35, 36]. Recent efforts in-clude the measurement of nucleon pair transfers [37, 38]and double charge-exchange reactions [39]. The good cor-relation found between 0 νββ and double Gamow-Tellertransitions [40] could in principle be exploited in dou-ble charge-exchange reactions, but the analyses are chal-lenged by tiny cross sections [41, 42] and involved reac-tion mechanisms [43, 44].In this Letter we study the correlation between theNMEs of 0 νββ and second-order electromagnetic (EM)transitions emitting two photons ( γγ ). In fact, the latterwere first studied in atoms by Goeppert-Mayer [45, 46],and it was the extension to the weak interaction whichled her to propose the ββ decay [47]. To ensure thatnuclear-structure aspects are as similar as possible in the γγ and ββ sectors, we focus on EM double-magneticdipole decays—which depend, like the 0 νββ operator,on the nuclear spin. In addition, isospin symmetry as-sures a good correspondence between the γγ and ββ nu-clear states if we consider the decay of the double isobaricanalogue state (DIAS) of the initial ββ state—an excitedstate of the final ββ nucleus—into the final ββ state—theground state (GS) of that nucleus. Our proposal expandsthe connections between first-order weak and EM transi-tions involving isobaric analogue states exploited in thepast [48–50].Figure 1 summarizes the main result of this work.We find a good linear correlation between γγ and 0 νββ NMEs obtained with the nuclear shell model, valid acrossthe nuclear chart. The upper panel presents results fordecays in nineteen nuclei comprising titanium, chromiumand iron isotopes with nucleon number 46 ≤ A ≤
60. The a r X i v : . [ nu c l - t h ] F e b α M γγ ( D I A S + ⟶ g s + ) [ μ N M e V - ] ●● ●● ●●●◆◆ ◆◆◆ ◆◆ ● ●● ●●◆ ◆◆ ◆◆● ● ●● ●◆ ◆ ◆◆ ◆ TiCrFe ≤ A ≤ ●●●● ◆◆◆◆ ▲▲▲▲ ●●●● ◆◆◆◆ ▲▲▲▲ ● ●●● ◆ ◆◆◆ ▲▲▲▲ ●● ◆◆ ▲▲ ●●●●● ◆◆◆◆◆ ●●●◆◆◆ ●●◆◆ ZnSeGeKr TeXeBa M νββ ( gs + ⟶ gs + ) ≤ A ≤ FIG. 1. Correlation between 0 νββ ( M νββ ) and double-magnetic dipole [ M γγ ( M M α is anisospin factor, see the text. Top panel: Results for − Ti, − Cr and − Fe obtained with the KB3G (circles) andGXPF1B (diamonds) effective interactions. Bottom panel:Results for − Zn, − Ge, − Se, , Kr obtained withGCN2850 (circles), JUN45 (diamonds) and JJ4BB (trian-gles); and for − Te, − Xe and , Ba calculatedwith the GCN5082 (circles) and QX (diamonds) interactions. lower panel covers twenty five nuclei comprising zinc, ger-manium, selenium, krypton, tellurium, xenon and bar-ium isotopes with 72 ≤ A ≤ γγ transitionshave been measured between 0 + first-excited states andGSs, where single- γ decay is forbidden [51–53], and, re-cently, among general nuclear states in competition with γ transitions [54, 55]. Future DIAS to GS γγ decay mea-surements, combined with the good linear correlation be-tween NMEs presented in this work, show as a promisingtool to give insights on 0 νββ NMEs.
Electromagnetic DIAS to GS transitions.
The γγ de-cay of a nuclear excited state is an EM process where twophotons are emitted simultaneously: N i ( p i ) −→ N f ( p f ) + γ λ ( k ) + γ λ (cid:48) ( k (cid:48) ) , (1)where N i , N f are the initial and final nuclear states withfour-momenta p i and p f , respectively, and photons have four-momenta k, k (cid:48) and helicities λ, λ (cid:48) .The theoretical framework of nuclear two-photon de-cay is presented in detail in Refs. [52, 56, 57]. The non-relativistic interaction Hamiltonian is given byˆ H I = (cid:90) d x ˆ J µ ( x ) A µ ( x ) (2)+ 12 (cid:90) d x d y ˆ B µν ( x, y ) A µ ( x ) A ν ( y ) , where A µ ( x ) denotes the EM field, ˆ J µ ( x ) the nuclear cur-rent, and ˆ B µν ( x, y ) is a contact (seagull) operator whichrepresents intermediate nuclear-state excitations not cap-tured by the nuclear model, such as nucleon-antinucleonpairs. Perturbation theory up to second order in the pho-ton field leads to the transition amplitudes M (1) = δ ( k + k (cid:48) + E f − E i ) (3) × (cid:88) n (cid:90) d x d y ε ∗ µλ ( k ) ε ∗ νλ (cid:48) ( k (cid:48) ) e − i ( k · x + k (cid:48) · y ) × (cid:34) (cid:104) f | ˆ J µ ( x ) | n (cid:105) (cid:104) n | ˆ J ν ( y ) | i (cid:105) E i − k (cid:48) − E n + i(cid:15) + (cid:104) f | ˆ J ν ( y ) | n (cid:105) (cid:104) n | ˆ J µ ( x ) | i (cid:105) E i − k − E n + i(cid:15) (cid:35) , M (2) = − (2 π ) δ ( k + k (cid:48) + E f − E i ) (4) × (cid:90) d x d y ε ∗ µλ ( k ) ε ∗ νλ (cid:48) ( k (cid:48) ) e − i ( k · x + k (cid:48) · y ) (cid:104) f | ˆ B µν ( x , y ) | i (cid:105) , where ε µλ ( k ) is the photon polarization vector. The ini-tial ( | i (cid:105) ), intermediate ( | n (cid:105) ) and final ( | f (cid:105) ) nuclear stateshave energies E i , E n and E f , respectively. The am-plitude M (2) can be neglected for DIAS to GS transi-tions, in the absence of subleading two-nucleon currents,because it involves a one-nucleon operator in isospinspace [52].It is very useful to perform a multipole decompositionof the γγ amplitude, because nuclear states have goodangular momentum. The expansion involves electric ( E )and magnetic ( M ) multipole operators with angular mo-mentum L , denoted as X . The transition amplitude sumsover multipoles, which factorize into a geometrical (phasespace) factor and the generalized nuclear polarizability, P J , containing all the information on the nuclear struc-ture and dynamics [52]: P J ( X (cid:48) X ; k , k (cid:48) ) = 2 π ( − J f + J i (cid:112) (2 L + 1)(2 L (cid:48) + 1) (5) × (cid:88) n,J n (cid:34)(cid:40) L L (cid:48) JJ i J f J n (cid:41) (cid:104) J f || (cid:101) O ( X ) || J n (cid:105)(cid:104) J n || (cid:101) O ( X (cid:48) ) || J i (cid:105) E n − E i + k (cid:48) + ( − Y (cid:40) L (cid:48) L JJ i J f J n (cid:41) (cid:104) J f || (cid:101) O ( X (cid:48) ) || J n (cid:105)(cid:104) J n || (cid:101) O ( X ) || J i (cid:105) E n − E i + k (cid:35) , where the 6 j -symbols depend on the total angular mo-menta of the initial, intermediate, and final states J i , J n , J f and Y = J − L − L (cid:48) . The reduced matrix ele-ments of the EM multipole operators involve the photonenergy: (cid:101) O ( X ) ∝ k L , as well as the nucleon radial r , an-gular spherical harmonics Y L , orbital angular momentum l , and spin s operators.Double EM and weak decays involve different nuclei: AZ Y ∗ N → AZ Y N + 2 γ vs AZ − X N +2 → AZ Y N + 2 e − , with N, Z the neutron and proton number. In order to study thecorrelation between 0 νββ and γγ NMEs, we focus on the γγ decay of the DIAS of the initial ββ state. This is anexcited state, with isospin T = T z +2, of the ββ daughternucleus with isospin third component T z = ( N − Z ) / γγ decay to the GS—the final ββ state—with T = T z , thus connects states with the same isospin struc-ture as ββ decay: an initial state with isospin T i = T f +2with a final one with T f = T z . Since isospin symmetryholds very well in nuclei, we expect the nuclear struc-ture aspects of DIAS to GS γγ and 0 νββ transitions tobe very similar. Altogether, the γγ decay involves thefollowing positive-parity J i = J f = 0 nuclear states: | + i (cid:105) γγ ≡ | + i (cid:105) ββ (DIAS) = T − T − K / | + i (cid:105) ββ , (6) | + f (cid:105) γγ ≡ | + f (cid:105) ββ , (7)with K a normalization constant and T − = (cid:80) Ai t − i thenucleus isospin lowering operator, which only changes T z .Angular momentum and parity conservation imposethat transitions between 0 + states just involve the zero-multipole polarizability P , with two EL or M L opera-tors. In the long wave approximation k | x | (cid:28)
1, satisfiedwhen Q = E i − E f ∼ −
10 MeV, dipole ( L = 1) decaysdominate. Since the nuclear spin is key for 0 νββ decay,we focus on double-magnetic dipole ( M M
1) processes,governed by the operator M = µ N (cid:114) π A (cid:88) i =1 g li l i + g si s i , (8)with µ N the nuclear magneton, and the neutron ( n ) andproton ( p ) spin and orbital g -factors: g sn = − . g sp =5 . g ln = 0, g lp = 1.We perform calculations for the most probable photonenergies k = k (cid:48) = Q/
2. For this case, Eq. (5) factorizesand the expression can be written in terms of a singleNME, M γγ ( M M P ( M M , Q/ , Q/
2) = 2 π √ Q M γγ ( M M , (9) M γγ ( M M
1) = (cid:88) n (cid:104) + f || M || + n (cid:105)(cid:104) + n || M || + i (cid:105) ε n , (10)where ε n = E n − ( E i + E f ) / M operator de-mands 1 + intermediate states. Furthermore Eq. (10) isalso valid for any values of the photon energies in all ββ nuclei explored in this work. In these cases the energydifference between the two photons is always much less Ti Se Te - E n ( MeV ) M γγ ( M M ) [ μ N M e V - ] FIG. 2. Contribution (solid lines) and cumulative (dashedlines) values of the M γγ ( M M
1) NME as a function of theexcitation energy of the intermediate states E n . The results,for the 0 +DIAS → +gs transition in Ti, Se and
Te, aresmoothed with a Lorentzian of width 0.1 MeV. than ε n , and M γγ ( M M
1) describes well the γγ transi-tion. Nuclear shell model calculations.
We calculate M γγ ( M M
1) for a broad range of 46 ≤ A ≤
136 nu-clei in the framework of the nuclear shell model [58–60].We cover three different configuration spaces spanningthe following harmonic oscillator single-particle orbitalsfor protons and neutrons: i) 0 f / , 1 p / , 0 f / and 1 p / ( pf shell) with the KB3G [61] and GXPF1B [62] effectiveinteractions; ii) 1 p / , 0 f / , 1 p / and 0 g / ( pfg space)with the GCN2850 [63], JUN45 [64] and JJ4BB [65] in-teractions; and iii) 1 d / , 0 g / , 2 s / , 1 d / and 0 h / ( sdgh space) with the GCN5082 [63] and QX [66] in-teractions. All the interactions are isospin symmetric.For our calculations we use the shell model codes AN-TOINE [58, 67] and NATHAN [58].First, we calculate the final γγ state and the initial ββ one, which we rotate in isospin to obtain its DIASas in Eq. (6). Next, we build a finite set of intermediatestates { + n } with the Lanczos strength function method,taking as doorway state the isospin T n = T f + 1 projec-tion of the isovector M operator applied to the finalstate: P T = T z +1 M IV | + f (cid:105) . This guarantees intermedi-ate states with correct angular momentum and isospin.We evaluate the energy denominator ε n using exper-imental energies when possible [68, 69]. For Ti, E f , E i (DIAS) and also the energy of a T = T z + 1 state (6 + )are known, and we use the latter to determine the energyof the lowest intermediate state E . With this experimen-tal input, M γγ ( M M
1) only varies the result obtainedwith calculated energies by 0 . ε n to a very good approximation [70].Therefore, in nuclei with unknown energy of the DIASor T = T z + 1 states, we use experimental data on statesof the same isospin multiplet in neighboring nuclei: the ββ parent to fix E i , and the ββ intermediate nucleus— ss ll ls T ss ll ls T ss ll ls T ss ll ls T ss ll ls T M xx ' γγ [ μ N ] Ti Zn Ge Xe Ba FIG. 3. Different contributions to the numerator NME ˆ M γγ for several nuclei: total (T), spin ˆ M γγss (ss), orbital ˆ M γγll (ll)and interference ˆ M γγls (ls) terms. when available—for E . Using these experimental ener-gies modifies M γγ ( M M
1) results by less than 5%.
Results.
With these ingredients we evaluate Eq. (10).Figure 2 shows M γγ ( M M
1) as a function of the exci-tation energy of the intermediate states, for nuclei cov-ering the three configuration spaces: Ti, Se and
Te. The Lanczos strength function gives convergedresults to ∼
1% after 50 −
100 iterations. Figure 2illustrates that, in general, intermediate states up to ∼
15 MeV can contribute to the double-magnetic dipoleNME, and that only a few states dominate each tran-sition. The comparison between weak and EM decaysneeds to take into account that while 0 νββ changes N and Z by two units, they are conserved in γγ decay.This is achieved by comparing isospin-reduced NMEs or,alternatively, by including the ratio of Clebsch-Gordancoefficients dictated by the Wigner-Eckart theorem [71]: α = (cid:113) C T f , ,T f +2 T f , ,T f +2 /C T f , ,T f +2 T f , , T f = (cid:112) (2 + T f )(3 + 2 T f ).Figure 1 shows the good linear correlation between0 νββ NMEs and double-magnetic dipole NMEs obtainedwith bare spin and orbital g -factors. We observe essen-tially the same correlation when using effective g -factorsthat give slightly better agreement with experimentalmagnetic dipole moments and transitions: g si (eff) =0 . g si , g lp (eff) = g lp + 0 . g ln (eff) = g ln − . pf shell [72]; and g si (eff) = 0 . g si for pfg nuclei [73].The slope of the linear correlation between γγ and0 νββ NMEs in Fig. 1 depends mildly on the mass num-ber, being larger in the pf shell than for pfg and sdgh nuclei. This distinct behaviour is due to the energy de-nominator in M γγ ( M M M γγ , the same linear correla-tion is common to all nuclei. This is consistent with thegeneral behaviour illustrated by Fig. 2: the intermediatestates that contribute more to M γγ ( M M
1) lie system-atically at lower energies in pf -shell nuclei, compared to A ≥
72 systems. In fact, the ratio of average energy ofthe dominant states contributing to M γγ ( M M
1) in the sslllsT - M xx ' γγ ( ) [ μ N ] FIG. 4. Decomposition of the
Ba numerator NME ˆ M γγ , interms of the two-nucleon angular momenta J : total (T), spinˆ M γγss (ss), orbital ˆ M γγll (ll) and interference ˆ M γγls (ls) parts. pf shell over the pfg − sdgh spaces matches very well theratio of the slopes in the top and bottom panels of Fig. 1.We can gain additional insights on the γγ − νββ corre-lation by decomposing the double-magnetic dipole NMEinto spin, orbital and interference parts. Since the en-ergy denominator plays a relatively minor role, we fo-cus on the changes in the numerator matrix element:ˆ M γγ = ˆ M γγss + ˆ M γγll + ˆ M γγls . Figure 3 shows the de-composition for the γγ decay of several nuclei. In somecases like Zn, the spin part dominates. Here, since ˆ M γγss is proportional to the double Gamow-Teller operator, avery good correlation with 0 νββ is expected [40]. In con-trast, the orbital ˆ M γγll part dominates in Xe or
Ba, sdgh nuclei with an l = 5 orbital. Remarkably, thesenuclei follow the common trend in Fig. 1, which meansthat the correlation with 0 νββ decay is not limited tooperators driven by the nuclear spin. The interferenceˆ M γγls is generally smaller, and can be of different sign tothe dominant terms. In fact, Fig. 3 also shows that thespin and orbital contributions to γγ decay always havethe same sign, preventing a cancellation that would blurthe correlation with 0 νββ decay.Figure 4 investigates further the relation between spinand orbital γγ contributions, decomposing the NMEs interms of the two-body angular momenta J of the twonucleons involved in the transition. Analogously to 0 νββ NMEs [17, 18], ˆ M γγ is dominated by the contributionof J = 0 pairs, partially canceled by that of J > M γγss and ˆ M γγll , witha more marked cancellation in the spin part, as expecteddue to the spin-isospin SU(4) symmetry of the isovectorspin operator [22, 74]. The J = 0 dominance suggeststhat spin and orbital S = L = 0 pairs are the mostrelevant in γγ DIAS to GS transitions, implying that s s = ( S − / / <
0, and likewise l l <
0. Since thespin and orbital isovector g -factors also share sign, thehierarchy in Fig. 4 explains the absence of cancellationsthat leads to the γγ correlation with 0 νββ decay.Future work includes evaluating two-nucleon cur-rent contributions to double-magnetic dipole [75] and0 νββ [76, 77] decays, but we do not expect these correc-tions to alter significantly the NME correlation. In con-trast, the recently-proposed leading-order contact con-tribution could modify sizeably 0 νββ NMEs [78], butnote that short-range NMEs can also be correlated tothe 0 νββ ones in Fig. 1 [79]. In addition, the correla-tion observed here can be tested with other many-bodyapproaches such as energy-density functional theory [26,27, 80], the interacting boson model [23, 81], the quasi-particle ramdom-phase approximation (QRPA) [21, 82]or ab initio methods [29–31, 83].
Summary.
We have observed a good linear correlationbetween γγ and 0 νββ NMEs. For our shell model cal-culations, the correlation holds across the nuclear chart,independently on the nuclear interaction used. This sug-gests a new avenue to reduce 0 νββ
NME uncertainties ifdouble-magnetic dipole DIAS to GS γγ transitions canbe measured, especially on the most relevant 0 νββ nu-clei. In fact, first steps in this direction are underway:Valiente-Dob´on et al. [84] recently proposed a flagshipexperiment to determine the conditions of a future pro-gram to measure the γγ decay of the Ca DIAS in Ti.Even though these experiments are challenging due tothe competition with single- γ , E E γγ and nucleon-emission channels, their potential cannot be diminished.Next generation 0 νββ experiments imply a significantinvestment with the promise to fully cover the invertedneutrino-mass hierarchy region [85], but current NMEuncertainties may limit the reach of the proposals underdiscussion. Acknowledgments.
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