CCIPANP2015-EngelOctober 14, 2018
Nuclear Matrix Elements for Double- β Decay
Jonathan Engel Department of Physics and AstronomyCB 3255, University of North Carolina, Chapel Hill, NC 27599-3255 USA
Recent progress in nuclear-structure theory has been dramatic. I de-scribe recent and future applications of ab initio calculations and thegenerator coordinate method to double-beta decay. I also briefly discussthe old and vexing problem of the renormalization of the weak nuclearaxial-vector coupling constant “in medium” and plans to resolve it.PRESENTED AT
Twelfth Conference on the Intersections of Particle andNuclear PhysicsVail, Colorado, May 1924, 2015 Work supported in part by the U.S. Department of Energy, Office of Science, Office of NuclearPhysics, under Contract Nos. DE-FG02-97ER41019 and de-sc0008641, ER41896 (NUCLEI SciDACcollaboration). a r X i v : . [ nu c l - t h ] O c t eutrinoless double-beta (0 νββ ) decay occurs if neutrinos are Majorana particles,at a rate that depends on a weighted average of neutrino masses (see Refs. [1, 2]for reviews). New experiments to search for 0 νββ decay are planned or underway.Extracting a mass from the results, however, or setting a reliable upper limit, willrequire accurate values of the nuclear matrix elements governing the decay. Thesecannot be measured and so must be calculated.The matrix elements have been computed in venerable and sophisticated models,but vary by factors of two or three. All the models can be improved, however.Here I focus on two of them: the shell model and the generator coordinate method(GCM). I first discuss effective interactions and decay operators for the shell modelthat will connect that method to ab initio nuclear-structure calculations, which havemade rapid progress recently. I then show how the GCM avoids problems of thequasiparticle random phase approximation (QRPA) and, moreover, can be extendedso that it incorporates the QRPA’s ability to capture proton-neutron pairing. Finally,I briefly examine the currently unsettling “renormalization” of the nuclear weak axialcoupling constant g A , and argue that the cause will be identified soon through andinvestigation of many-body currents and the effective enlargement of model spaces.The lifetime for 0 νββ decay, if the exchange of the familiar light neutrinos isresponsible, is given by the product of a phase space factor (recently recomputed inRef. [3]) an effective mass m ν = (cid:80) i U ei m i , where m i is the mass of the i th eigenstateand U ei weights each mass by the mixing angle of the associated eigenstate with theelectron neutrino, and M ν is the nuclear matrix element. The matrix element iscomplicated but can be simplified without significantly altering its value through the“closure approximation.” In this approximation, and neglecting two-body currents(which I take up briefly later), one can write the matrix element as 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 ] + 3 j ( qr ab ) h T ( q ) (cid:126)σ a · (cid:126)r ab (cid:126)σ b · (cid:126)r ab q + E − ( E i + E f ) / τ + a τ + b | i (cid:105) , where r ab ≡ | (cid:126)r a − (cid:126)r b | is the distance between nucleons a and b , j and j are theusual spherical Bessel functions, E is an average excitation energy to which the ma-trix element is insensitive, and the nuclear radius R ≡ . A / fm is inserted with acompensating factor in the phase-space function to make the matrix element dimen-sionless. The “form factors” h F , h GT , and h T are given in Refs. [4] and [5].As already indicated, researchers have applied a variety of nuclear models to ββ decay. At the moment, it is difficult to assess the uncertainty in any one of thematrix-element calculations. Quantifying uncertainty is an important task for thenext few years, but just as important is reducing the uncertainty by improving thecalculations. We can do both by linking model Hamiltonians and decay operators1o data through ab initio calculations. The lighter ββ nuclei ( Ge, Se), or thosesuch as
Xe that are near closed shells, will be the easiest to connect to ab initiowork. The heavier nuclei will generally require a different treatment. I discuss thosesuitable for an ab initio treatment first.The shell model is a complete diagonalization in a subspace of the full many-bodyHilbert space that consists of all possible configurations of valence particles withina few valence single-particle orbitals, outside a core that is forced to remain inert.The model thus neglects excitations of the particles in the core into the valence levelsor higher-lying levels, as well as excitations of the valence nucleons into higher-lyinglevels. At present, practitioners usually deal with this problem by constructing a phe-nomenological Hamiltonian for use in the shell model space. Nuclear-structure theoryis reaching the point where we can do better, however. A variety of many-body meth-ods now yield accurate solutions to the Schr¨odinger equation for nuclei with closedshells and nuclei with one or two nucleons outside closed shells. Among the methodsare the coupled clusters approach [6] and the in-medium similarity renormalizationgroup (IMSRG) [7]. These approaches can be used to construct, for up to two orthree nucleons in the valence shell, a unitary transformation that transforms the fullmany-body Hamiltonian into block diagonal form, with a piece H eff in the shell-modelspace that reproduces the lowest-lying energies exactly.The procedure for obtaining the 0 νββ matrix element in Ge via a coupled-clusters-based shell-model calculation would go something like this:1. Derive a two- and three-nucleon Hamiltonian from chiral effective field theory[8] or phenomenology in few-nucleon systems.2. Do ab initio coupled-clusters calculations of the ground state of the closed shellnucleus Ni, of the low-lying eigenstates states of the closed-shell-plus-one nu-clei Ni and Cu, and of the low-lying states of the closed-shell-plus-two nuclei Ni, Cu, and Zn. Eventually, when it becomes possible, do the same inclosed-shell+three nuclei as well.3. Perform a “Lee-Suzuki” mapping [10] of the low-lying states in these nucleionto states in the valence shell containing one and two (and eventually, three)nucleons. The mapping is designed to maximize the overlap of the full ab initioeigenstates with their shell-model images, while preserving orthogonality of theimages [11].4. Use the mapping of states to construct the shell-model interaction H eff thatgives the image states the same energies as their parents. Construct an effectivedouble-beta operator that gives the same matrix elements between image statesas the bare operator does between the associated parents.2 E ne r g y [ M e V ] / + / + / + / + / + / + / + / + / + / + / + / + / + / + / + / + / + / + / + / + / + * / + * / + * / + * / + * / + * / + / + * / + / + * O + + + + + + + + + + + + + + + + + * + * + * + * + + + + + O 0123456 / + / + / + / + / + / + / + / + / + ? / + / + / + / + * / + / + / + / + / + / + / + O CC E I E x p . U S D E ne r g y [ M e V ] + + + + + + + + + + + + + + + + + O CC E I E x p . U S D / + / + / + / + / + / + / + / + / + / + / + / + / + / + / + / + / + / + O CC E I E x p . U S D + + + + + + + + + + ? + + + + + + + O FIG. 2. (Color online) Excitation spectra of neutron-rich oxygen isotopes. The left columns (red lines) contain the CCEIresults, the middle columns (black lines) the known experimental data, and the right columns (blue lines) the spectra obtainedwith the USD shell-model Hamiltonian [7, 8]. A star next to the excitation levels in the right columns indicates that the levelwas included in the fit of the USD Hamiltonian. The gray bands indicate states above the neutron decay threshold.
Λ-CCSD(T) ground-state energies in , , , O. Our Λ-CCSD(T) calculations use the model space mentionedearlier, while the calculations that determine our CCEIuse N max = 12 and N + N + N = 12. We be-lieve that our CCEI results are converged to within ∼
100 keV. Both our Λ-CCSD(T) and CCEI results arein good agreement with experimental binding energies.Our CCEI and Λ-CCSD(T) calculations also agree wellwith a variety of recent calculations in the oxygen iso-topes that start with the same Hamiltonian [54, 55].If we look more closely, we see that the reference Λ-CCSD(T) results in , O are in excellent agreementwith our CCEI results. In , O the CCEI resultsstart to deviate from the Λ-CCSD(T) reference values.In O the CCEI ground-state is less bound by about3 . O with an EOM coupled-cluster calculation that includes singles and doubles excitations [56]. EOM-CCSD can accurately describe low-lying states that aredominated by one-particle-one-hole excitations [48], andwe therefore choose those states for comparison. In Owe obtain low-lying 2 + and 3 + states with 2 . . . . + state in O is not yet converged; itmoves down by ∼
150 keV when we increase the modelspace size from N = 10 to N = 12 oscillator shells. The2 + state changes only by ∼ + state in O, which isdominated by two-particle-two-hole excitations from theground-state.We turn now to carbon. The Λ-CCSD(T) ground-state energies of , , C are − . − . − . C the resultagrees well with the experimental ground-state energy
Figure 1: Spectra of neutron-rich oxygen isotopes. The left column contains theresults of sd -shell-model calculations with an effective Hamiltonian derived form abinitio chiral two- and three-body forces [8] and coupled-cluster calculations in , , O.The middle column contains experimental data and the right column contains thepredictions with the phenomenological USD interaction [9] that was fit to data in thesame shell.5. Put 4 protons and 16 neutrons (for Ge) and 6 protons and 14 neutrons (for Se) in the valence shell and use the effective interaction and decay operatorderived in the previous step to calculate the ground-state-to-ground-state decaymatrix element.We have just begun to carry out this program [12], starting in lighter nuclei. Usingcoupled cluster calculations in , , O, we predicted the spectra of oxygen isotopeswith more neutrons. Fig. 1 shows the results. The left column for each isotopecontains our predictions, the middle column the experimental data, and the rightcolumn the “predictions” of the USD shell-model interaction [9] that was fit longago to lots of data in the sd shell itself . Our interaction, which uses only data in3 M ν g pn pn-GCMQRPAOrdinary GCM Figure 2: 0 νββ matrix element for decay of Ge in three models: the pn-QRPA(black dashed curve), the ordinary GCM without explicit proton-neutron correlations(flat blue solid line), and the GCM with proton-neutron pairing correlations (solidred curve). The quantity g pn is the strength of the proton-neutron pairing in theHamiltonian in Eq. (2). The quadrupole interaction in that Hamiltonian is turned offfor this illustration. The calculations use two full oscillator shells as the single-particlemodel space.two- and three-nucleon systems produces results which are at least as good. Thoughwe have yet to investigate matrix elements of the double-beta operator, these initialresults for energy levels are extremely promising. Including an effective three-nucleoninteraction, from still extremely difficult ab initio calculations in O and C (ensuringthat our predictions exactly match the ab initio results in those isotopes) shouldimprove the spectra further. A larger shell model space would do the same. A similarprogram is being undertaken within the IMSRG [13].For heavier complicated nuclei such as
Te or
Nd, fully ab initio calculationsin the region are still a ways off. In those, we may have to use a more restricted wavefunction. Fortunately, recent work suggest that collective correlations may be most ofwhat you need for an accurate matrix element [14]. The phenomenological methods,e.g., density functional theory, are built for collective correlations.Such methods, particularly the GCM, have already been applied to ββ decay [15,16], but not all collective correlations have been included. Neutron-proton pairing,in particular, is omitted because its effects are hard to see in nuclear spectra andtransitions. It does, however, play a significant role in ββ decay. To see this, wehave carried out calculation of the decay of Ge in a Hilbert space consisting of 364ucleons in two full oscillator shells with a semi-realistic interaction of the form H = h − (cid:88) µ = − g T =1 µ S † µ S µ − χ (cid:88) K = − Q † K Q K − g pn (cid:88) ν = − P † ν P ν + g ph (cid:88) µ,ν = − F µ † ν F µν , (2)where h contains single particle energies, the Q K are components of the quadrupoleoperator, and S † µ = 1 √ (cid:88) l √ l + 1[ c † l c † l ] µ , P † µ = 1 √ (cid:88) l √ l + 1[ c † l c † l ] µ , F µν = 12 (cid:88) i σ µ ( i ) τ ν ( i ) . (3)In this last line c † l is a creation operator, l labels single-particle multiplets with goodorbital angular momentum, S † µ creates a correlated pair with total orbital angularmomentum L = 0, spin S = 0, and isospin T = 1 (with µ labeling the isospincomponent τ = T z ), P † µ creates an isoscalar proton-neutron pair with L = 0 and S = 1 ( S z = µ ), and the F µν are the components of the Gamow-Teller operator.The Hamiltonian incorporates like-particle and proton-neutron pairing, a quadrupole-quadrupole interaction, and a repulsive “spin-isospin” interaction. Ref. [14] showsthat in the f p shell, anyway, this kind of interaction reproduces full shell-modelresults accurately.Fig. 2 shows the GCM results for the decay of Ge; the quadrupole interactionis temporarily turned off. The dashed curve is from the QRPA; it blows up whenthe proton-neutron pairing strength is near 1.5 (a realistic value). The reason isthat the mean field on which the QRPA is based undergoes a phase transition froma condensate of like-particle pairs to a condensate of proton-neutron pairs at thatpoint. The QRPA is unable to accommodate more than one mean field; it breaksdown at the transition point. The GCM on the other hand, is explicitly designed tomix many mean fields. The technique is usually applied to nuclei that don’t have adefinite shape (many ββ -decay candidates are in this class), with wave functions thatare superpositions of states with a range of deformation. The GCM generates meanfields with that range by minimizing the mean-field energy under the constraint thatthe quadrupole moment take a particular value, then repeating the minimization forlots of other values for the quadrupole moment. The interaction is then diagonalizedin the space of constrained mean-fields, usually after each has been projected ontostates with well defined angular momentum and particle number.The method as just described was applied together with the phenomenologicaldensity-dependent Gogny interaction to 0 νββ decay in Refs. [15, 16]. The resultingmatrix elements are usually larger than those of the shell model or QRPA. One reasonis the absence of the proton-neutron correlations that in the QRPA shrink the matrixelement as in Fig. 2 (before ruining it completely for very strong proton-neutronpairing). But one can add the physics of proton-neutron pairing by mixing togethermean-fields (quasiparticle vacua) with different degrees of that pairing. One does5 .10.2 Ge00.10.2 0 2 4 6 8 10 | Ψ ( φ ) | φ = pn pairing amplitude Se φ for Ge φ f o r S e -15-10-50510 Figure 3: Left: Squares of collective wave functions, as a function of the proton-neutron pairing amplitude φ in Ge (top) and Se (bottom), the initial and finalnuclei in the decay of Ge, for the same Hamiltonian (2) as used for Fig. 2. Right:0 νββ matrix element as a function of the pairing amplitudes in the projected mean-field states making up the ground states of the initial and final nuclei. The matrixelement has a maximum for no proton-neutron pairing and is reduced at the point atwhich the collective wave functions peak.so by imposing constraints on both the quadrupole moment and the proton-neutronpairing amplitude, that is by minimizing H (cid:48) = H − λ Z N Z − λ N N N − λ Q Q − λ P (cid:16) P + P † (cid:17) , (4)where the Lagrange multipliers λ Z and λ N fix the expectation values of the proton andneutron number operators N Z and N N — this is part of the usual HFB minimization— and the other multipliers fix the quadrupole moment (cid:104) Q (cid:105) and the proton-neutronpairing amplitude φ ≡ (cid:104) P + P † (cid:105) . Such a minimization requires generalizing the usualBCS-like wave functions to include proton-neutron, leading to quasiparticles that arepart proton and part neutron as well part particle and part hole (which they are evenin the usual treatment).Ref. [17] carries out this calculation. With the quadrupole moment turned off,it produces “collective wave functions” of the proton-neutron pairing amplitude, thesquares of which appear on the left side of Fig. 3. These represent the probabilitythat the final diagonalized ground states in Ge and Se contain a given proton-neutronpairing amplitude φ = (cid:104) P + P † (cid:105) . One can see that the wave functions are peakedaround φ = 4 or 5. The right side of the figure shows the 0 νββ matrix element as afunction of the two pairing amplitudes. At the point representing the peak of the twowave functions, the matrix element is noticeably smaller than the point at which the6airing amplitudes are zero. Finally, the part of Fig. 2 I haven’t focused on shows the0 νββ matrix element as a function of the proton-neutron pairing strength. The GCMcurve mirrors that produced by the QRPA until a point close to the mean-field phasetransition, around and after which it behaves smoothly (as it should; there’s no realphase transition beyond mean-field theory). The matrix element is indeed smallerthan that of the ordinary GCM, which captures no proton-neutron correlations ofthis type and thus produces a result that is independent of g pn .The next step in the development of this approach is to move beyond the semi-realistic calculation discussed here and marry this enlarged GCM with sophisticatedSkyrme or Gogny density functionals, which work in complete single-particle spaces,with all the nucleons active. The result will almost certainly be matrix elements thatare closer to those of the shell model.I turn finally to the renormalization of the axial-vector coupling g A . It has beenknow for some time (see, e.g., Ref. [9]) that matrix elements for β and 2 νββ decayare smaller in reality than in our calculations. If 0 νββ matrix elements are as smallcompared to our calculations as 2 νββ matrix elements, experiments are in trouble.Fortunately, the issue can now be investigated systematically. There can only betwo sources of the quenching: many-body weak currents, which would alter the pre-dictions of calculations with the one-body Gamow-Teller operator, and model spacetruncation, i.e. the omission of important configurations. Work is now beginning toexamine both these sources. The effects of many-body currents have traditionallybeen thought to be small [18], but the construction of those currents in chiral effec-tive field theory — currents that should go along with the interactions used by abinitio calculations — may lead to larger effects [19, 20]. Crucially, however, thoseeffects should be smaller for 0 νββ decay than for 2 νββ decay. The issue should becleared up by careful EFT parameter fits in the near future. The other source ofquenching, model-space truncation, can be investigated in the ab initio shell-modelcalculations described earlier. Those implicitly include many configurations from out-side the model space in the effective interactions and operators. We should soon beable to see whether 0 νββ decay is quenched, and if so, by how much. ACKNOWLEDGEMENTS
I am grateful to my collaborators G.R. Jansen, G. Hagen, N. Hinohara, M. Mustonen,P. Navratil, A. Signoracci, F. ˇSimkovic, and P. Vogel. This work was supported inpart by the U.S. Department of Energy,
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