EEffective double-beta-decay operator for Ge and Se Jason D. Holt
Institut f¨ur Kernphysik, Technische Universit¨at Darmstadt, 64289 Darmstadt, GermanyExtreMe Matter Institute EMMI, GSI Helmholtzzentrum f¨ur Schwerionenforschung GmbH, 64291 Darmstadt, GermanyDepartment of Physics and Astronomy, University of Tennessee, Knoxville, TN 37996, USA andPhysics Division, Oak Ridge National Laboratory, P.O. Box 2008, Oak Ridge, TN 37831, USA ∗ Jonathan Engel
Deptartment of Physics and Astronomy, University of North Carolina, Chapel Hill, NC, 27516-3255, USA † (Dated: November 13, 2018)We use diagrammatic many-body perturbation theory in combination with low-momentum inter-actions derived from chiral effective field theory to construct effective shell-model transition oper-ators for the neutrinoless double-beta decay of Ge and Se. We include all unfolded diagramsto first- and second-order in the interaction and all singly folded diagrams that can be constructedfrom them. The resulting effective operator, which accounts for physics outside the shell-modelspace, increases the nuclear matrix element by about 20% in Ge and 30% in Se.
PACS numbers: 23.40.-s, 21.60.Cs, 24.10.Cn, 27.50.+e
I. INTRODUCTION
The experimental discovery of neutrinoless double-beta (0 νββ ) decay, a nuclear-weak process that occursextremely slowly if at all, would have deep implicationsfor particle physics. Since 0 νββ decay can occur onlyif the neutrino is its own antiparticle, an observationwould at once establish the neutrino as a Majorana par-ticle. Furthermore, from a measured lifetime we could, inthe absence of exotic new physics, determine an averageneutrino mass m ν ≡ (cid:80) i U ei m i , where i labels the masseigenstates, and U is the neutrino mixing matrix [1]. Thismass cannot be extracted from a lifetime, however, with-out first knowing the value of a nuclear matrix elementthat also plays a role in the decay. The entanglementof nuclear and neutrino physics has led to a small butconcentrated effort within the nuclear structure commu-nity to calculate the nuclear matrix elements, which arenot themselves observable. While various theoretical ap-proaches agree to within factors of two or three, — arange many structure theorists might find not unreason-able — the uncertainty in the effective mass that canbe extracted from an observed lifetime is at least thatlarge as a result. Since large-scale experiments will bereporting results in the coming years, we need to workquickly to improve the accuracy of the matrix-elementcalculations.Of the theoretical methods currently employed, the nu-clear shell model is the only approach that offers an ex-act treatment of many-body correlations, albeit withina truncated single-particle (valence) space above someassumed inert core. Though most of the physics govern-ing double-beta ( ββ ) decay indeed resides in this valence ∗ [email protected] † [email protected] space, correlations involving neglected single-particle or-bitals may contribute non-negligibly to both the Hamil-tonian and the 0 νββ -decay transition operator, eachof which is a basic ingredient in any nuclear matrix-element calculation. Contributions to the Hamiltoniancan be, and have been, included in the construction ofan effective valence-space Hamiltonian, H eff , through dia-grammatic many-body perturbation theory (MBPT) [2].But the analogous contributions to an effective valence-space 0 νββ -decay operator, with the exception of a cruderenormalization of g A , have thus far been almost com-pletely ignored. The first and only work to apply MBPTto the 0 νββ -decay operator considered only diagramsthat were first order in the interaction (a G -matrix), plusa few selected higher-order contributions [3].In this article we carry out a much more compre-hensive computation, providing the first steps towardsa true first-principles calculation of nuclear matrix ele-ments based on chiral nuclear forces [4]. We first defineand compute an ˆ X -box consisting of all diagrams nowto second order in the interaction and in a much largerHilbert space than used in Ref. [3]. We then consider con-tributions of folded diagrams together with state norms,which must be explicitly computed for effective transitionoperators (see Eq. (7)). We finally apply the resultingtwo-body effective 0 νββ -decay operator, together withwavefunctions from existing shell-model calculations, toobtain corrected nuclear matrix elements in the pf -shell0 νββ -decay candidates Ge and Se.Assessing the accuracy of the perturbative expansionis a central challenge for MBPT. Although a nonpertur-bative treatment of core polarization found only mod-est changes in H eff [5], the analogous impact on effective ββ -decay operators is unclear, and ultimately a nonper-turbative method that goes beyond core polarization, al-lowing controlled approximations to both the effectiveHamiltonian and transition operators, will be preferable.Coupled-cluster theory [6, 7] and the in-medium similar- a r X i v : . [ nu c l - t h ] A p r ity renormalization group [8–10] are promising nonper-turbative methods, but neither has yet been applied to ββ decay. The situation may be different in a few years,but at present MBPT is still the best method to inves-tigate microscopic many-body corrections to the shell-model 0 νββ -decay operator. And even within MBPT,as we have noted, there is essentially no work, outsideof Ref. [3], on two-body transition operators, making thetopic almost completely unexplored.The remainder of this paper is structured as follows:Section II describes the ingredients of our calculation,including definitions of the matrix elements we compute,the framework for obtaining the nuclear interactions withwhich we begin, and the details of our many-body formal-ism for calculating effective ββ -decay operators. SectionIII presents our results for Ge and Se, updating thematrix element for Se first reported in Ref. [11]. Fi-nally, Section IV discusses the significance of the resultsand outlines steps that will improve their accuracy.
II. METHODSA. Decay Operator
In the closure approximation (which is good to at worst10% or so [12]), the nuclear matrix element governing0 νββ decay can be represented as the ground-state-to-ground-state matrix element of a two-body operator. Ne-glecting the so-called “tensor term,” the effect of whichis a few percent [13, 14], the matrix element is given by M ν = 2 Rπg A (cid:90) ∞ q dq (1) × (cid:104) f | (cid:88) a,b j ( qr ab ) [ h F ( q ) + h GT ( q ) (cid:126)σ a · (cid:126)σ b ] q + E − ( E i + E f ) / τ + a τ + b | i (cid:105) , where | i (cid:105) and | f (cid:105) are the ground states of the initial andfinal nuclei, r ab ≡ | (cid:126)r a − (cid:126)r b | is the distance between nu-cleons a and b , j is the usual spherical Bessel function,and the nuclear radius R is inserted to make the matrixelement dimensionless, with a compensating factor in thephase-space integral that multiplies the matrix element.The “form factors” h F and h GT are given by h F ( q ) ≡ − g V ( q ) , (2) h GT ( q ) ≡ g A ( q ) − g A ( q ) g P ( q ) q m p + g P ( q ) q m p + g M ( q ) q m p , where g V ( q ) = 1 (cid:0) q / (0 . ) (cid:1) , (3) g A ( q ) = 1 . (cid:0) q / (1 . ) (cid:1) , g P ( q ) = 2 m p g A ( q ) q + m π , g M ( q ) = 3 . g V ( q ) , and m p denotes the proton mass and m π the pion mass.The closure approximation is not good for two-neutrino double-beta (2 νββ ) decay, which we briefly dis-cuss later. The matrix element governing that processcontains a complete set of intermediate states, viz.: M ν ≈ (cid:88) n (cid:104) f | (cid:80) a (cid:126)σ a τ + a | n (cid:105) (cid:104) n | (cid:80) b (cid:126)σ b τ + b | i (cid:105) E n − ( M i + M f ) / , (4)where n denotes states in the intermediate nucleus withenergy E n , M i and M f are the masses of the initial andfinal nuclei, and the effects we neglect (e.g., forbiddencurrents, the Fermi matrix element, etc.) are small. Weare unable to obtain a complete set of intermediate states,so we can treat 2 νββ decay only in the closure approxi-mation, viz.: M cl2 ν = (cid:104) f | (cid:88) ab (cid:126)σ a · (cid:126)σ b τ + a τ + b | i (cid:105) . (5)Although the approximation is poor, and we cannot useit to deduce the real 2 νββ -decay matrix element, the clo-sure matrix element and the real one change in a similarway when correlations are added. B. Nuclear Interactions
Diagrammatic MBPT was reviewed extensively someyears ago [15, 16], but since then, driven by ad-vances in chiral effective field theory (EFT) [4] andrenormalization-group (RG) methods [17], it has seensomething of a revival [18]. Chiral EFT is a systematicexpansion of nuclear interactions and electroweak cur-rents in which three- (3N) and higher-body forces arisenaturally. Beginning from the chiral two-nucleon (NN)potential of Ref. [19], we construct a low-momentuminteraction ( V low k ), with cutoff Λ = 2 . − , via RGevolution [17, 20], explicitly decoupling high-momentumcomponents from those at the nuclear-structure scale. Incontrast, the G -matrix [16], often taken as a startingpoint in nuclear structure calculations, deals with high-momentum modes by particle-ladder resummation, anddoes not adequately decouple low- from high-momentumdegrees of freedom. As a result, many-body methodsbased on V low k tend to converge better than those us-ing a G -matrix [6]. Recent work with MBPT based onlow-momentum NN+3N interactions has led to the de-velopment of non-empirical valence-space Hamiltoniansfor proton- and neutron-rich systems [21–25]. While 3Nforces are neglected here, we plan to include them in ourfuture 0 νββ -decay nuclear-matrix-element calculations.The one drawback of using low-momentum interac-tions in calculations of effective operators is that high-momentum physics cannot be included explicitly. Theeffects of high-momentum (short-range) correlations onthe 0 νββ -decay operator are both small and now wellunderstood, however, and we include them via an effec-tive Jastrow function that has been fit to the results ofBrueckner-theory calculations [26]. C. Effective Two-Body Transition Operators
As we have noted, existing work on MBPT containslittle about effective two-body operators other than theHamiltonian, where Refs. [2, 27, 28] provide the mostcomprehensive discussion. No matter the two-body op-erator of interest, however, the starting point is alwaysthe construction of projection operators ˆ P and ˆ Q that di-vide the full many-body Hilbert space into a model space,in which subsequent exact diagonalization is carried out,and everything else. In our calculations in nuclei withmass near A = 80, the model space consists of the 0 f / ,1 p / , 1 p / , and 0 g / single-particle orbits, for bothprotons and neutrons, above a Ni core in a harmonic-oscillator basis of 13 major shells with (cid:126) ω = 10 . Q -box” (for the Hamiltonian) or an “ ˆ X -box” (for thetransition operator) and then writing the complete sum,including folded diagrams, in terms of the ˆ Q - and ˆ X -boxes and their derivatives with respect to unperturbedenergies. The first few terms in the ˆ Q - and ˆ X -boxes ap-pear in Figs. 1 and 2.Folding is significantly more complicated for a two-body transition operator, which combines ˆ X - and ˆ Q -boxes, than for the Hamiltonian, where only ˆ Q -boxes areneeded. Effective model-space operators in the basis ofenergy eigenstates are always defined (for a bare operator M ) via (cid:104) f eff | M eff | i eff (cid:105)(cid:104) f eff | f eff (cid:105) (cid:104) i eff | i eff (cid:105) = (cid:104) f | M | i (cid:105) , (6)where the states that lie in the model space, | i eff (cid:105) ≡ ˆ P | i (cid:105) and | f eff (cid:105) ≡ ˆ P | f (cid:105) , are not in general normalized. If M isthe Hamiltonian, then only diagonal matrix elements arenonzero, and the denominator is canceled by a similarfactor in the numerator. For two-body transition opera-tors, that is not the case, and state norms must be explic-itly computed. Prior authors have approached the issue ac bdˆ Q = ac bd V low-k + ac bd + . . .+ ac bd + ac bd+ a b dc + ac bd + . . . FIG. 1. The ˆ Q -box to second order in V low k (ellipses indicatehigher-order terms). The first line contains one-body con-tributions and the others two-body contributions. Exchangediagrams, though not shown, are included in our calculations. of norms in several ways. References [2] and [28], for in-stance, choose to expand the denominators and fold theminto the numerators, thus completely eliminating all dis-connected diagrams. The resulting expressions, however,become complicated as the number of folds increases, andthe approach requires the construction of a special basisas an intermediate step. For these reasons Ref. [27] advo- ac bdˆ X = ac bd M + ac bd+ ac bd + a b dc + a b dc+ ac bd + ac bd + . . . FIG. 2. (Color online) The ˆ X -box to first order in V low k .Solid (red) up- or down-going lines indicate neutrons and dot-ted (blue) lines protons. The wavy horizontal lines, as in Fig.1, represent V low k , and the dashed horizontal lines representthe 0 νββ -decay operator in Eq. (1). V low-k FIG. 3. Diagrams in the expansion of the effective interaction defining the two-body part of the second- and third-order ˆ Q -box.The wavy lines represent V low k . We obtain the first- and second-order ˆ X -box — the set of all unfolded first- and second-orderdiagrams for the two-body effective operator (not including norm diagrams) — by replacing one interaction in each of thesediagrams by a ββ -decay operator (in all possible ways) and restricting the sums over nucleons in the intermediate states toeither neutrons or protons, as in the first-order ˆ X -box diagrams in Fig. 2. cates keeping the denominator and numerator separate,at the price of introducing disconnected diagrams thatonly cancel when the sum is carried out completely. Here,though we evaluate the ˆ Q -box to third order and the ˆ X -box to second order in the interaction, we include onlyone fold in each of the three factors on the left hand sideof Eq. (6), and so opt to follow Refs. [2, 27] in expandingthe denominator and folding with the numerator. The re-sulting expression for the matrix elements of an operator M eff is approximately (cid:104) cd | M eff | ab (cid:105) = (7) d ˆ Q ( ε ) dε + 12 d ˆ Q ( ε ) d ε ˆ Q ( ε ) + 38 (cid:32) d ˆ Q ( ε ) dε (cid:33) . . . × (cid:34) ˆ X ( ε ) + ˆ Q ( ε ) ∂ ˆ X ( ε f , ε ) ∂ε f (cid:12)(cid:12)(cid:12)(cid:12) ε f = ε + ∂ ˆ X ( ε, ε i ) ∂ε i (cid:12)(cid:12)(cid:12)(cid:12) ε i = ε ˆ Q ( ε ) . . . (cid:35) × d ˆ Q ( ε ) dε + 12 d ˆ Q ( ε ) d ε ˆ Q ( ε ) + 38 (cid:32) d ˆ Q ( ε ) dε (cid:33) . . . cd,ab Because off the need for a special basis, this expression is onlystrictly correct when the terms in square brackets are diagonal.They are close to diagonal in the calculations presented here. where ε is the unperturbed energy of both the initial andfinal states (we take the energies to be the same). Bothˆ Q and ˆ X are matrices, with indices corresponding to thepossible two-body states in the valence space (e.g., a, b or c, d in Figs. 1 and 2). In this paper we report resultsof just the terms explicitly given above, which containbetween zero and five folds (there is a fold at every matrixmultiplication). The terms indicated by ellipses are morecomplicated and presumably less important; they awaitfuture investigation. D. Evaluation of ˆ Q - and ˆ X -Box Diagrams We turn now to the ˆ Q - and ˆ X -boxes themselves, con-structed from unfolded diagrams, that we use in Eq. (7).To construct the ˆ Q -box, we take all unfolded diagrams tothird order in V low k . The diagrams appear in AppendixA.2 of Ref. [16], and the two-body pieces are reproducedin Fig. 3. Our ˆ X -box has too many diagrams to dis-play here, so we characterize the set as follows: we takeall two-body ˆ Q -box diagrams in Fig. 3 and replace oneinteraction line in each diagram (in all possible ways)by a ββ -decay line. We then determine whether eachintermediate-state nucleon line should be a proton or aa a aa FIG. 4. (Color online) A Pauli-forbidden two-body diagramwith a spectator neutron and a three-body diagram, obtainedby exchanging two ingoing neutron lines, that cancels it ex-actly. neutron. The result is three times as many ˆ X -box di-agrams (at second order in V low k ) as ˆ Q -box diagrams inFig. 3.We make one nonstandard choice in evaluating the ˆ X -box: we restrict the particle lines in the intermediatestates to be essentially unoccupied. For example, in the ββ decay of Ge, we omit all contributions from inter-mediate protons in the 1 p / orbit and neutrons in 1 p / ,1 p / , or 0 f / orbits, ad we multiply the contributionsof graphs with intermediate neutrons in the 0 g / orbitby 0.4, its average occupancy. In the decay of Se, weomit the same contributions as in Ge and multiply thecontributions of graphs with intermediate neutrons in the0 g / orbit by 0.2 and those with intermediate protonsin the 0 f / orbit by 0.5. The reason for all this is thatin a nucleus with more than two valence nucleons, thediagram on the left of Fig. 4 — a two-body contribu-tion to the ββ -decay operator with a spectator neutron— would be canceled by the three-body diagram on theright if we were to include it. By omitting the two-bodydiagrams with intermediate particles in occupied orbitswe are effectively adding particular three-body diagrams(like those on the right of Fig. 4) to our calculation. Weare not including all three-body diagrams, just those thatcancel Pauli-forbidden two-body diagrams.We call this approach nonstandard because it is notusually followed in derivations of effective interactions.The reason is that in excluding some Pauli-forbidden di-agrams, one effectively includes unlinked one- and two-body diagrams (see, e.g., Fig. 10 of Ref. [29]) as wellas the exclusion-enforcing three-body diagrams we want.This problem, however, is more pronounced in the ˆ Q -boxthan the ˆ X -box since the latter has no one-body part andfar fewer ways to unlink diagrams by exchanging lines(the horizontal ββ -decay lines are restricted to have in-coming neutrons and outgoing protons). We thereforeeffectively include only very few unlinked diagrams byintroducing our restrictions in the ˆ X -box; the compen-sating benefit is a much better account of Pauli exclusion,an important physical effect. Diagrams such as the oneon the left of Fig. 4 result in large contributions thatshould not be present in a full calculation. We cancelthem with the implicit assumption that the cancelingcontribution from the figure on the right-hand side issignificantly greater than that of typical third-order dia-grams, which we omit. Eventually, though, this assump- Ge SeBare matrix element M ν X -box, without 3p-1h 5.44 4.86Full first-order ˆ X -box 2.20 2.40First order folded 3.11 2.79Full second-order ˆ X -box 4.14 3.92Final matrix element 3.77 3.62CD-Bonn G -matrix 3.62 3.45N LO G -matrix 3.48 3.33TABLE I. The 0 νββ -decay matrix elements M ν for Ge and Se at various approximations in our many-body framework. tion will have to be tested explicitly.
III. RESULTS
To obtain our final corrected shell-model 0 νββ -decaymatrix elements, we combine the individual two-bodymatrix elements of our effective operator with two-bodyshell-model transition densities. Since our aim is a consis-tent calculation without empirical adjustment, we reallyought to take two-body densities from the diagonaliza-tion of a valence-space interaction that is derived directlyfrom NN+3N forces. While work in this direction is inprogress, the computation is not yet possible in nucleithis heavy. Instead we use two-body densities from ex-isting shell-model calculations, the interactions for whichhave been tweaked to fit experimental data in nearby nu-clei. For Ge we use the calculation of Horoi [30] andfor Se that of Ref. [14]; the authors of both have kindlysupplied us with their transition densities.Table I presents our matrix elements at various levelsof ˆ X -box and folding approximations, using V low k andtaking intermediate-state excitations to 18 (cid:126) ω . Despitedifferences in the NN interaction and size of the basisspace, contributions from first-order diagrams in both Ge and Se largely agree with those first identified inRef. [3]: particle-particle and hole-hole ladders togetherenhance the matrix element, while the three-particle one-hole diagrams cause a dramatic reduction. When foldingis included, however, the net correction from first-orderˆ Q - and ˆ X -boxes essentially disappears. Taking the com-plete set of second-order diagrams into account, we find asignificant enhancement followed by a modest quenchingfrom folding. The final matrix element is approximately20% percent larger than the bare matrix element in Geand about 30% larger in Se. The primary reason forthe different effects in Se and Ge is the difference inthe omitted intermediate-state orbits discussed in Sec-tion II D. If we include those orbits, as is standard prac-tice in the construction of effective interactions, the ma-trix element is reduced by about 10% in Ge and 15% in Se. In Ref. [11], which contains a preliminary account (cid:126) ω (cid:126) ω (cid:126) ω (cid:126) ω (cid:126) ω (cid:126) ω Full 1st order 2 .
429 2 .
407 2 .
403 2 .
401 2 .
399 2 . .
908 3 .
932 3 .
940 3 .
931 3 .
925 3 . .
489 3 .
553 3 .
595 3 .
611 3 .
617 3 . νββ -decay matrix element in Sewith respect to allowed intermediate-state excitations. In allcases we work in a harmonic-oscillator basis of 13 major shells. of our calculations in Se, we obtained 3.56 instead of3.62. The small difference is due to the inclusion in Ref.[11] of ˆ Q -box restrictions and the addition here of a termin the expansion of the norm denominator. Though thetwo results are close, we believe that the one reportedhere is likely closer to the real matrix element.Several other aspects of the calculation are robust. Asseen in Table II, our V low k results at 18 (cid:126) ω are well con-verged to 3 or 4 digits. And as Table I shows, chang-ing the interaction to a G -matrix (in 11 major oscillatorshells) in place of V low k does not affect the results sub-stantially. Finally, although we emphasized our proce-dure of requiring intermediate-particle lines in ˆ X -box di-agrams to be unoccupied in the nucleus in question, otherprescriptions yield similar results once norms and foldingare included: in Se, for example, we obtain a final ma-trix element of 3.50 if we restrict particle lines in boththe ˆ Q and ˆ X boxes, and 3.03 if we impose no restrictionsat all. We should note, however, that at various interme-diate stages of the calculation, the procedures yield quitedifferent results. And other parts of the calculation leaveroom for change as more physics is included.We turn now to a discussion of 2 νββ decay. As notedabove, we use the closure matrix element M cl2 ν as a proxyfor the full matrix element, a step that limits how muchwe can say. Table III shows the matrix element for Gewith and without the intermediate-state restrictions weimpose on occupied or partially occupied orbits in calcu-lating M ν . Imposing the restrictions here increases thematrix element, as in 0 νββ decay; in this case, however,the result is probably undesirable, given that shell modelcalculations of 2 νββ decay in Ge overestimate M ν .On the other hand, omitting the restrictions increases Restricted UnrestrictedBare matrix element M cl2 ν X -box, without 3p-1h 0.99 0.89Full first-order ˆ X -box 0.37 -0.60Full second-order ˆ X -box 0.79 -0.37Final matrix element 0.96 0.70TABLE III. The 2 νββ -decay closure matrix M cl2 ν for Ge atseveral levels of approximation, with and without restrictionson occupied intermediate-particle lines. FIG. 5. (Color online) Schematic representation of diagramscontributing to renormalization of g A in 2 νββ decay. the negative contribution of the 3p-1h diagram to suchan extent that the matrix element changes sign. The signis eventually reversed by higher-order contributions andfolding, ultimately yielding a result that is approximatelyunchanged from the bare matrix element. It is difficultto be comfortable, however, with a low-order correctionthat changes the sign of the matrix element. The sensi-tivity of the numbers suggest that terms with more folds,of higher order, or involving more valence orbitals couldalso have a significant effect.Another reason (aside from the sign changes in theright-hand column of Table III) for preferring to restrictintermediate-sate orbits in the ˆ X -box is connected to thelong-standing problem of the apparent suppression of theaxial-vector coupling constant g A in the nuclear medium[31]. While the suppression probably has many sources,configurations outside the valence space are likely to playa key role. Though the bare operator governing weak de-cay is one-body, we can simulate the effect of g A suppres-sion in 2 νββ decay by including only closure diagramsthat have the form shown in Fig. 5. Such diagrams, inwhich only a single 2 νββ -decay line connects the twonucleons, incorporates only the renormalization of theone-body weak current.When we base our calculation of M cl2 ν on only thesediagrams and at the same time account for occupiedintermediate-state orbits, we find in Se that the fullresult is smaller than the bare result by 38%, implyingan effective g A of about 1.0, a reasonable value (in Gethe value is about 0.7). On the other hand, when wetake no account of occupied orbits we find (again, withonly diagrams of the form in Fig. 5 included) that theclosure matrix element changes sign, something that isimpossible through the renormalization of g A alone. Thesign change reflects the same strong effect of the 3p-1hdiagrams observed in Table III. Though these consider-ations are not conclusive, they do indicate that takingthe Pauli principle into account is a beneficial. Work iscurrently underway to investigate g A quenching more di-rectly. By focusing on the one-body operator, we can useanalogous many-body techniques, based again on chiralNN and 3N physics, with the effects of two-body cur-rents implemented consistently in the bare operator [32]to understand the origin of g A quenching.Whatever the outcome of that investigation, it is clearthat the 2 νββ matrix element is sensitive to many detailsin the wavefunctions, much more sensitive than its 0 νββ counterpart. Thus, although the increase of the 2 νββ closure matrix element does not bolster the case for our0 νββ calculation, neither, in our view, does it weaken itmuch. IV. DISCUSSION AND OUTLOOK
We have used chiral nuclear forces and many-bodyperturbation theory to calculate an effective shell-model0 νββ -decay operator, taking into account corrections tothe bare operator from configurations outside the valenceto second order in the interaction. The resulting nu-clear matrix element is approximately 20% larger thanthe bare matrix element in Ge and about 30% largerin Se. These new results represent our current bestestimates for the matrix elements but probably do nottell the whole story. We have omitted a number of ef-fects that could further alter the results. To do better,we must first establish consistency between the Hamil-tonian and our effective operator. This will require theconstruction of full non-empirical valence-space interac-tions in the pf shell from NN and 3N forces; work in thatdirection is in progress. A related improvement will beto include the effects of chiral 3N forces in the ˆ X -box, inaddition to the effects of chiral two-body currents in thebare operator [32].At the many-body level, the importance of third- andhigher-order terms in the ˆ X -box and additional foldingcontributions must be understood. Since we have foundthe effects of bubble diagrams to be the most importantin our perturbative expansion, it would be worthwhile to pursue a nonperturbative calculation of the effects of corepolarization (which these diagrams represent), like thatdone for effective interactions in Ref. [5]. Perhaps themost significant obstacle to a truly reliable result, how-ever, is the implementation of induced three-body oper-ators. Recent work [33, 34] indicates that such operatorsare not negligible, and even here we have shown thatthree-body diagrams of the form in Fig. 4 are important.Unfortunately, the number of induced three-body dia-grams is so large that nobody has computed them evenin the construction of effective interactions. We mustfind a way to at least estimate their size if we want topursue perturbation theory to its conclusion. Controllednonperturbative approaches [7, 9] are on the horizon, butthe inclusion of induced three-body terms is technicallydifficult there as well. In none of these approaches is theproblem impossible to overcome, but doing so will requirediligence and creativity. ACKNOWLEDGMENTS
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