Benchmarking 136 Xe Neutrinoless ββ Decay Matrix Element Calculations with the 138 Ba(p,t) Reaction
B. M. Rebeiro, S. Triambak, P. E. Garrett, B. A. Brown, G. C. Ball, R. Lindsay, P. Adsley, V. Bildstein, C. Burbadge, A. Diaz Varela, T. Faestermann, D. L. Fang, R. Hertenberger, M. Horoi, B. Jigmeddorj, M. Kamil, K. G. Leach, P. Z. Mabika, J. C. Nzobadila Ondze, J. N. Orce, H. -F. Wirth
aa r X i v : . [ nu c l - e x ] F e b Benchmarking
Xe Neutrinoless ββ Decay Matrix Element Calculations with the
Ba( p, t ) Reaction
B. M. Rebeiro, ∗ S. Triambak, † P. E. Garrett,
2, 1
B. A. Brown, G. C. Ball, R. Lindsay, P. Adsley,
5, 6
V. Bildstein, C. Burbadge, A. Diaz Varela, T. Faestermann, D. L. Fang,
8, 9
R. Hertenberger, M. Horoi, B. Jigmeddorj, M. Kamil, K. G. Leach, P. Z. Mabika,
1, 13
J. C. Nzobadila Ondze, J. N. Orce, and H. -F. Wirth Department of Physics and Astronomy, University of the Western Cape, P/B X17, Bellville 7535, South Africa. Department of Physics, University of Guelph, Guelph, Ontario N1G 2W1, Canada. Department of Physics and Astronomy and National Superconducting Cyclotron Laboratory,Michigan State University, East Lansing, Michigan 48824-1321, USA TRIUMF, 4004 Wesbrook Mall, Vancouver, British Columbia V6T 2A3, Canada. School of Physics, University of the Witwatersrand, Johannesburg 2050, South Africa iThemba LABS, P.O. Box 722, Somerset West 7129, South Africa Physik Department, Technische Universit¨at M¨unchen, D-85748 Garching, Germany. Institute of Modern Physics, Chinese Academy of Sciences, Lanzhou, 730000, China School of Nuclear Science and Technology, University of Chinese Academy of Sciences, Beijing 100049, China Fakult¨at f¨ur Physik, Ludwig-Maximilians-Universit¨at M¨unchen, D-85748 Garching, Germany. Department of Physics, Central Michigan University, Mount Pleasant, Michigan 48859, USA Department of Physics, Colorado School of Mines, Golden, Colorado 80401, USA Department of Physics and Engineering, University of Zululand,Private Bag X1001, KwaDlangezwa 3886, South Africa. (Dated: February 11, 2020)We used a high-resolution magnetic spectrograph to study neutron pair-correlated 0 + states in Ba, produced via the
Ba( p, t ) reaction. In conjunction with state-of-the-art shell model calcu-lations, these data benchmark part of the dominant Gamow-Teller component of the nuclear matrixelement (NME) for
Xe neutrinoless double beta (0 νββ ) decay. We demonstrate for the first timean evaluation of part of a 0 νββ decay NME by use of an experimental observable, presenting a newavenue of approach for more accurate calculations of 0 νββ decay matrix elements.
The massive nature of neutrinos leads to a violation ofthe γ invariance [1] for weak interactions. Consequently,there is substantial interest worldwide [2–4] to searchfor standard-model-forbidden neutrinoless double beta(0 νββ ) decays, that violate lepton number conservationby 2 units. The observation of such decays would provethat the electron neutrino ( ν e ) is a Majorana fermion,and therefore indistinguishable from its antiparticle (¯ ν e ).This is consistent with most theories beyond the stan-dard model [5], that attribute the smallness of neutrinomasses to a violation of total lepton number at an energyscale of ∼ GeV [3, 5].If the mechanism driving a 0 νββ decay is via the ex-change of a light left-handed Majorana neutrino, thenthe decay amplitude is proportional to m ee M ν = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)X j | U ej | e iα j m j (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) M ν , (1)where m ee is the effective Majorana mass of the elec-tron neutrino and M ν is the nuclear matrix element(NME) for the decay. The NME is expressed as the sum ∗ [email protected] † [email protected] of Gamow-Teller (GT), Fermi (F) and Tensor (T) com-ponents M ν = M νGT − (cid:18) g V g A (cid:19) M νF + M νT , (2)where the Gamow-Teller contribution is the dominantterm. In Eq. (1), the U ej are elements of the Pontecorvo-Maki-Nakagawa-Sakata (PMNS) neutrino mixing ma-trix [6, 7], the m j ’s are the light neutrino masses andthe α j ’s are phases in the mixing matrix. For the specialcase of three-neutrino mixing, the PMNS matrix is pa-rameterized in terms of three mixing angles and one Diracand two Majorana CP-violating phases [2]. It is evidentfrom Eq. (1) that in addition to the observation of a 0 νββ decay process, it is also equally important to determineits half-life, which would establish the absolute neutrinomass scale. Furthermore, such a measurement also hasthe potential to identify the correct neutrino mass spec-trum [8] and find extra sources of CP-violation in the lep-tonic sector [9]. However, achieving the above (or plac-ing stringent constraints on any new physics) requires anaccurate evaluation of the NME for the decay. This hasbeen at the forefront of nuclear physics research in recenttimes, with several approaches being used to calculate theNMEs for 0 νββ decay candidates [10, 11]. Depending onthe method used, the calculations for specific isotopesdisagree with one another, differing by factors of three ormore in many cases [10, 11]. These discrepancies result TABLE I. Some recent evaluations of the NME for
Xe 0 νββ decay using different theoretical approaches. The calculationsassume light Majorana neutrino exchange.Method M ν Deformed WS-QRPA (Jilin-T¨ubingen-Bratislava) [20] 1 . . . . . . . . . . in large uncertainties for the NMEs, which not only limitthe physics that can be addressed, but also the planningand execution of future 0 νββ decay experiments [11]. Incontrast, the NMEs for the rare (yet standard-model-allowed) two-neutrino double beta (2 νββ ) decays can beextracted directly from measured half-lives. These andother experimentally derived spectroscopic informationhave played a critical role in constraining various NMEcalculations [12–17].One of the most promising candidates for observing0 νββ decays is Xe ββ → Ba. Its 2 νββ decay half-lifeis much longer than most other cases [18]. As a result,the ratio of the 0 νββ decay signal to the irreducible 2 νββ decay background in the vicinity of the decay endpointenergy is expected to be larger for this particular case.In fact, a highly sensitive search for 0 νββ decays was re-cently reported for
Xe by the KamLAND-Zen collabo-ration [19], who placed the most stringent upper limits todate on the effective neutrino mass m ee < −
165 meV,depending on the choice of NME used.We list some recent evaluations of M ν for Xe ββ decay in Table I. While some of these results are in rea-sonable agreement with each other, there still exist largediscrepancies in the calculated values, depending on themethod used. Needless to say, this is a pressing issueas future Xe 0 νββ decay experiments aim to improvetheir sensitivity by at least one order of magnitude [29].Additionally, next generation experiments also intend touse the method of barium ion tagging [30] in xenon timeprojection chambers (TPCs). This technique has the po-tential to reduce room background contributions to in-significant levels, which undoubtedly would further en-hance the sensitivity of
Xe 0 νββ decay experimentsover other experimental searches.Finally, it is expected that due to these advantages,future Xe ββ decay experiments would also place thetightest constraints on possible CP-violating Majoranaphases in the PMNS matrix [31]. The phases in the neu-trino mixing matrix are potential leptogenesis parameters that may explain the observed baryon asymmetry in theuniverse [32–34].Quite naturally, due to the several reasons listed above,the 0 νββ decay of Xe presents a compelling case toaddress the accuracy in its calculated NME . Two meth-ods that have been traditionally used to calculate 0 νββ decay NMEs are the interacting shell model (ISM) andthe quasiparticle random phase approximation (QRPA).Unlike the latter, the shell model calculations use a lim-ited configuration space that is comprised of relativelyfewer single-particle states in the vicinity of the Fermisurface. Despite this restriction, large-scale ISM calcula-tions allow arbitrarily complex correlations between thevalence nucleons. On the other hand, the QRPA calcula-tions make use of a much larger model space with com-paratively simpler configurations. In general, the ISMcalculations are known to yield smaller values of M ν ,compared to the QRPA [2, 11]. This discrepancy hasbeen attributed to different approaches in treating thepairing (or seniority) structure of the nuclear wavefunc-tions [35, 36]. For the most part, previous QRPA calcula-tions assumed spherical ground states for the parent anddaughter nuclei, wherein the pairing correlations betweenlike nucleons were taken into account using the Bardeen-Cooper-Schrieffer (BCS) approximation [2, 11]. It wasonly recently that deformed QRPA NME calculationswere performed for 0 νββ decays [20, 22], whose resultsfor Xe are listed in Table I. Compared to the sphericalQRPA [21], the deformed calculations yield smaller val-ues for the NME, and are in reasonable agreement withthe ISM results. The authors of Refs. [20, 22] point outthat the suppression of the NME in their calculations ismainly due to differences in the pairing content of theinitial and final mean fields. Unlike the spherical QRPA,the deformed calculations accounted for the sharp neu-tron Fermi surface in
Xe due to the neutron number N = 82 shell closure. This curtails the overlap betweenthe BCS wavefunctions and leads to a significant reduc-tion in the calculated NMEs [20, 22]. The calculationsalso suggest that the NME can be even more suppressedif the parent and daughter nuclei have different deforma-tions. Such a scenario will either further reduce [20, 22]the QRPA overlap factors mentioned above or result ina similar seniority mismatch between the ISM wavefunc-tions, due to high-seniority [37] components introducedby the deformation. In comparison, the NME calcula-tions using other many-body approaches such as the non-relativistic energy density functional (NREDF) theory,covariant density functional theory (CDFT), the inter-acting boson model (IBM-2) or the generator coordinatemethod (GCM), predict higher values for the NME (Ta-ble I). It has been suggested that these values are mostlikely overestimated, because of the omission of both col-lective as well as non-collective correlations, dependingon the calculation [11, 28, 38, 39].In light of the above, precise experimental informa- C oun t s p e r k e V Excitation energy (keV) ✽ ✽▲ ▲▲▲▲ ▲ ▲ ▲✽ ✽ FIG. 1. Excitation energy spectrum in
Ba obtained at θ lab = 25 ◦ . Previously known 0 + states are marked withasterisks, while the newly identified ones from this work areshown with filled triangles. tion elucidating the properties of Xe and
Ba nu-clei are crucial to benchmark the NME calculations andfurther reduce their model dependence. Indeed, differ-ences in the valence nucleon occupancies for these nu-clei were recently determined using one nucleon trans-fer reactions [40, 41]. Furthermore, the ground state of
Ba is not expected to have a nearly spherical struc-ture as
Xe or
Ba. The even barium isotopes in the N ≤
82 region are known to be transitional, displayinga structural evolution from spherical to γ -soft behaviorwith decreasing neutron number [42–44]. In this Letterwe discuss neutron pairing correlations in Ba, stud-
15 25 35 45 55 65 75 85 95 105 115
Scattering Angle ( θ CM ) σ ( θ ) / σ ( θ ) R Ref. [47]Ref. [48]Ref. [49]Ref. [50]Ref. [51]
FIG. 2. Measured
Ba( p, p ) angular distribution from thiswork (expressed in terms of ratio to the Rutherford cross sec-tions) compared to various DWBA predictions based on dif-ferent global OMP parameters. ied with the
Ba( p, t ) reaction. The experiment wasperformed at the Maier-Leibnitz-Laboratorium (MLL)in Garching, Germany, where a 1 . µ A, 23 MeV pro-ton beam from the MLL tandem accelerator was inci-dent onto a 40 µ g/cm thick, 99.9% isotopically enriched BaO target, that was evaporated on a 30 µ g/cm car-bon backing. The light reaction ejectiles were momentumanalyzed using the high-resolution Q3D magnetic spec-trograph [45, 46], whose solid angle acceptance rangedfrom 2 . − . ∼ . ◦ to thebeam, and connected to a Brookhaven Instruments Cor-poration (BIC) current integrator.For our measurements, we obtained triton angular dis-tributions using four magnetic field settings and at tenspectrograph angle settings, ranging from θ lab = 5 ◦ to50 ◦ . Fig. 1 shows sample triton spectra obtained fromthis experiment, where we observe states in Ba up to ∼ . .
10 keV. We alsotook additional
Ba( p, p ) elastic scattering data over anangular range of θ lab = 15 ◦ to 115 ◦ , in 5 ◦ steps. Thesedata were used to determine both the effective areal den-sity of the Ba target nuclei, as well as the appropriateglobal optical model potential (OMP) parameters for theincoming
Ba + p reaction channel [47–51]. The latterwere used in a zero-range distorted wave Born approxi-mation (DWBA) analysis of our data, for which we usedthe DWUCK4 code [52] with Woods-Saxon potentials.As shown in Fig. 2, based on a comparison of variousDWBA calculations with our elastic scattering measure-ments, we chose the global proton OMP parameters rec-ommended by Varner et al. [51] for the incoming (proton)channel. For the outgoing Ba + t channel we used theOMP parameters provided by Li et al. [53], as they gavethe best agreement with our measured triton angular dis-tribution for the ground state in Ba (c.f. Fig. 3). Thetransfer form factor was calculated assuming a single-step pick up of a di-neutron in a singlet relative s -state.For the core-2 n coupling we used the global OMP fromRef. [54], whose well depth was adjusted to reproduce thebinding energy of each neutron [55].The above approach was used to perform a comprehen-sive analysis of the angular distributions for all the tritonpeaks shown in Fig. 1. We defer a detailed discussion onthe analysis to a forthcoming article [56]. In this Letterwe only focus on the L = 0 angular distributions, whichare critical for studying pair-correlated states in even- -2 -1 -4 -3 -2 -3 -2 -1 -4 -3 -2 -3 -2 -1 -3 -2 -3 -2 -1 -4 -3 -2 -4 -3 -2 -3 -2 Scattering angle ( θ CM ) -4 -3 -2 -1 d σ / d Ω ( m b / s r) Scattering angle ( θ CM ) -4 -3 d σ / d Ω ( m b / s r) Ex = 0.0 keVEx = 1579 keVEx = 2315 keVEx = 2784 keVEx = 2977 keVEx = 3279 keV Ex = 3427 keVEx = 3921 keVEx = 4147 keVEx = 4344 keVEx = 4444 keVEx = 4534 keV
FIG. 3. Angular distributions of 0 + states identified in thiswork. The solid curves are normalized DWUCK4 DWBApredictions for L = 0 transfer, assuming a (0 h / ) configu-ration [41] for the form factor. The newly identified 0 + statesobserved in this work are labeled in red. even nuclei. Our measured angular distributions iden-tify eight new 0 + states in Ba [57, 58]. These resultsare shown in Fig. 3, together with normalized DWBAcross sections. The data show reasonable agreement withDWUCK4 predictions, except for the well-established 0 +2 state at 1579 keV [57], where the first minimum occurs atapproximately twice the predicted value. This is due toan inherent shortcoming of the DWUCK4 calculations.For example, the calculations ignore multi-step processesand interference from different configurations to the pairtransfer amplitude, which can alter the shape of the angu-lar distribution. In Table II we list the measured absolutecross sections for these states at θ lab = 5 ◦ , in addition tothe L = 0 transfer strengths to the excited 0 + states rel-ative to the ground state, denoted by ǫ i . The latter weredetermined for each excited state by the product ǫ i = " (cid:0) dσd Ω (cid:1) data0 + ex (cid:0) dσd Ω (cid:1) DWBA0 + ex i " (cid:0) dσd Ω (cid:1) DWBAG . S . (cid:0) dσd Ω (cid:1) dataG . S . , (3)which was obtained by normalizing the DWBA calcula-tions to the data at forward angles (where the DWBA TABLE II. Measured cross sections at θ lab = 5 ◦ and relative( p, t ) strengths obtained from the data shown in Fig. 3. E x ( dσ/d Ω) ◦ ǫ i [keV] [mb/sr] [%]0 2 . . . . . . . . . . . . . . . . . . . . . . a ... . L = 0 strengthrelative to the ground state P ǫ i = 53(3)% a We could not determine the cross section for the 4534 keVstate at low angles due to the presence of a kinematicallybroadened light-ion contaminant peak in the region. is best satisfied) for each plot in Fig. 3 and then takingthe ratio of the normalization factor for the 0 + i state tothe ground state normalization factor. This effectivelyremoves the Q -value dependence and other kinematic ef-fects in determining the relative ( p, t ) strengths.Our results in Fig. 3 and Table II show that in additionto a number of hitherto unknown 0 + states in Ba, weobserve a large fragmentation of the L = 0 , ( p, t ) strengthto these states, with ∼
30% of the ground state strengthconcentrated at 2.3 and 2.8 MeV. This manifestly indi-cates a breakdown of the BCS approximation for neu-trons in
Ba [13]. A similarly large breakdown was notobserved in the − Ba isotopes [44, 59, 60]. Never-theless, such a departure from superfluid behavior is notunexpected in a shape transitional region, particularlyaround closed shell nuclei [13, 61]. Therefore, our resultsindicate that the ground state wavefunctions of , Bacould be largely dissimilar due to the ‘non-spherical’ na-ture of the latter. Additionally, the data also presentevidence of a small pairing gap in
Ba, that could oc-cur due to vibrational modes in the pairing field [61, 62].While both possibilities cannot be ruled out, the formerwill have important ramifications for
Xe 0 νββ decayNME calculations. Previous work showed that the staticquadrupole moment of the first 2 + state in Ba couldbe as large as − .
19 or +0 . e b [63–65], which does notrule out a significantly deformed Ba ground state.The results from this experiment also allow testsof the nuclear structure models that are used to cal-culate the NME for
Xe 0 νββ decay. We per-formed one such test using configuration interaction shellmodel calculations with the NuShellX code [66]. Un-like the DWUCK4 calculations which merely served to
Excitation energy (MeV) R unn i ng s u m o f d σ / d Ω ( m b / s r) Experimentsn100pnsn100tGCN50:82
Excitation energy (MeV)
Experimentsn100pn-CPsn100t-CPGCN50:82-CP
FIG. 4. A comparison of the running sum of experimental( p, t ) cross sections at θ lab = 5 ◦ with the calculations de-scribed in the text. Left panel: The calculated values withoutcore-polarization corrections to the two-neutron transfer am-plitude. Right panel: Results after core-polarization effectsare taken into account. identify the observed 0 + states and determine relativestrengths, the shell model calculations were used to as-sess the absolute ( p, t ) cross sections, distributed over 0 + states in Ba. The calculations used the five-orbital(0 g / , d / , d / , s / , h / ) valence space for pro-tons and neutrons to determine the wavefunctions for the0 + ground state of Ba and the lowest fifty 0 + states in Ba. NuShellX was then used to calculate two-neutrontransfer amplitudes between these states, that served asinput for the Fresco [67] coupled-reaction channels codeto generate
Ba( p, t ) angular distribution predictions.In the Fresco calculations we used the same OMP pa-rameters as the DWUCK4 calculations (for the protonand triton channels) and took into account the coherentsum of both direct and sequential two-step transfer. Thesingle nucleon transfer amplitudes for the two-step partwere obtained assuming one intermediate state in
Bafor each of the transferred ( n, ℓ, j ) values. For the
Ba-deuteron coupling, we used the global OMP parametersrecommended by An and Cai [68], based on a compari-son with
Ba( d, d ) angular distribution data (similar toFig. 2) that we obtained independently. We used threedifferent Hamiltonians for the calculations, which werecorrected for core-polarization due to configuration mix-ing with orbitals outside the model space [69]. The firstHamiltonian is from Ref. [70] and is called sn100pn in theNuShellX interaction library [66]. The second Hamilto-nian (that we call sn100t) is very similar to sn100pn, ex-cept with minor modifications and was used in Ref. [12]to calculate M ν for Xe ββ decay, while the thirdGCN50:82 [71] Hamiltonian was used by the authors ofRef. [36] to calculate the NME for the decay.In the left panel of Fig. 4 we compare the running sumof our measured ( p, t ) cross sections at the most forwardangle (where the L = 0 strength is concentrated) to theshell model/Fresco calculations. Similar to our experi-mental results, the calculations predict the cross sectionto be dominated by the transition to the ground state in Ba, with smaller contributions from excited 0 + states.However, the theory predictions are found to be about afactor of two smaller than the experimental values. Suchan underestimation should not be surprising, given thatthe model space for neutrons is limited to only five or-bitals near the Fermi surface. Coherent contributionsfrom all orbitals outside the valence space are known toenhance the calculated L = 0 two-neutron transfer crosssection [72]. We next considered the effects of such core-polarization by calculating ladder-diagram corrections tothe two-nucleon transfer amplitudes (TNA), as describedin Ref. [72], assuming the scattering of pairs of neutronsto twenty three orbitals beyond the model space (up to i / ). As shown in the right panel of Fig. 4, the revisedcalculations that incorporated the core-polarization ef-fects show enhancements in the predicted cross sectionsby about a factor of 1 .
5, and agree with experiment to ∼
22% for the GCN50:82 Hamiltonian and ∼
14% forthe others. It is worthy of note that the relative dis-tribution of the predicted cross sections over 0 + statesalso agree reasonably well with experiment, particularlyfor the GCN50:82 Hamiltonian. The agreement did notsignificantly improve on making small adjustments ofthe single-particle energies and pairing strengths of theHamiltonians.How does the above relate to NME calculations for Xe 0 νββ decay? The connection is discussed inRef. [73], where it was shown that the 0 νββ decay NMEfor a parent nucleus with mass number A can be ex-panded as a sum over states in an intermediate nucleuswith mass number ( A − Xe, wecan similarly evaluate the NME by summing over theproducts of the TNA for two-neutron removal to
Xe,the TNA for two-proton addition to
Ba, and the two-body matrix element for the double-beta decay operators(c.f. Eq. 9 in Ref. [73]). The most significant contribu-tion to the NME is through the 0 + ground state in the Xe, while
J > J = 0 term [73]. This is similar to other calcula-tions [36, 74] that separate the NME in terms of nucleonpairs coupled to angular momentum and parity J π = 0 + and J π = 0 + , where the J > J π = 0 + term (see Fig. 1 inRef. [36]).Since the Xe → Xe transition is similar to Ba → Ba (they both correspond to a transforma-tion from N = 82 to N = 80), we can benchmark thedominant J π = 0 + Gamow-Teller (GT) component ofthe NME, by calculating it using the expanded set ofTNA that better reproduces our measured
Ba( p, t )cross section. To do so, we first performed a five-orbitalvalence space ISM calculation of this part of the NME(for light neutrino exchange) with the sn100t Hamilto-nian. On using the CD-Bonn potential [75] for two-nucleon short range correlations (SRC) and further in-cluding higher-order contributions (HOC) due to inducednucleon currents [76], we determine the matrix elementto be M νGT ( J π = 0 + ) = 5 .
67. Next we evaluated theNME through the J π = 0 + ground state in Xe, bothwith and without the core-polarization corrections to theTNA described above. These calculations showed thatthe five-orbital valence space ISM result for the NMErequired an enhancement factor of ∼ .
58, due to therequired core-polarization corrections. Such an enhance-ment significantly increases the magnitude of the NME,so that its revised value is M νGT ( J π = 0 + ) = 8 .
96. Tomake further comparisons, we also performed a large-scale spherical QRPA calculation of the NME (with HOCand the CD-Bonn SRC), using the model parameters ofRef. [23] and 28 orbitals for major oscillator shells with N ≤
6. This resulted in a value M νGT ( J π = 0 + ) = 9 . J = 0 component of the Gamow-TellerNME. We recommend improved calculations of this partof NME as well as the canceling M νGT ( J >
0) term, per-haps along the lines of a many-body perturbation theorytreatment [77], that takes into account physics contri-butions from beyond the model space. Since the de-tails of the cancellation between the J = 0 and J > M νGT ( J π = 0 + ) along these lines. We note that in or-der to make the connection with two-nucleon transferreaction data, it is important that the results of thesecalculations be expressed in terms of their J π decompo-sition.In summary, this work demonstrates for the first timea direct evaluation of part of a 0 νββ decay NME usingexperimental data. In addition to providing a benchmarkfor future calculations, it also presents a new avenue ofapproach for evaluating 0 νββ decay NMEs more accu-rately, motivating similar investigations in other candi-dates. We also report for the first time a large break-down of the neutron BCS approximation in an even bar-ium nucleus with N ≤
82. Our observations motivatea reassessment of the neutron pairing approximation in
Ba, used in some NME calculations for
Xe 0 νββ decay and invite further investigations of the shape of
Ba.We are grateful to Ian Thompson for his assistancewith the Fresco calculations. This work was partiallysupported by the National Research Foundation (NRF)of South Africa, the Natural Sciences and EngineeringResearch Council of Canada (NSERC), the U.S. NationalScience Foundation under Grant No. PHY-1811855 andthe U.S. Department of Energy, Office of Science un-der Grant No. de-sc0017649. P.A. acknowledges fund-ing from the Claude Leon Foundation in the form of a postdoctoral fellowship. [1] J. J. Sakurai,
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