1. Nuclear Matrix Elements in Double Beta Decay S.J.Freeman University of Manchester What they are and why they are important How they can be calculated What experiment can do to help

2. Nuclear Matrix Elements Double beta decay with neutrinos: Double beta decay without neutrinos: In both cases, the nuclear matrix element connects initial and final nuclear states. d uu d W–W– W–W– e–e– e–e– udd uu u uud d d d W–W– W–W– e–e– e–e– pp nn udd uu u uud d d d W–W– W–W– e–e– e–e– pp nn d uu d W–W– W–W– e–e– e–e– A-2

3. Double beta decay with neutrinos, 2νββ Rate of the process: Nuclear matrix element: Phase space factors ~Q 11, weak coupling constants, Coulomb effects GT part:Fermi part: Amplitude for   decay or (p,n) reaction Amplitude for   decay or (n,p) reaction Sum over states in the intermediate nucleus: e.g. 76 As for 76 Ge decay to 76 Se

4. Double beta decay with neutrinos, 2νββ Process goes through states in the intermediate nucleus: GT part:  J=±1,0, except no J=0 to J=0 Fermi part:  J=0; super allowed if T i =T f Essentially GT transitions via 1 + states in the intermediate nucleus 76 Ge 0 + gs T=6 T=5 T=4 gs 1+1+ T=5 0 + ias 0 + das T=6 76 Se 76 As Effects of nuclear structure high, depends on specific locations of 1+ states i.e. GT strength function. BUT 0νββ is very different…… Neglect to good approx.

5. Neutrinoless double beta decay, 0νββ Mediation by a virtual neutrino gives different features: A: Energy of intermediate excited states can be large up to tens of MeV, and range of states excited is larger than their spacing, c.f. few MeV for 2νββ. B: Angular momentum transfer is also large, up to 5-6 hbar, c.f. 1 for 2νββ. Sum over pairs of nucleons “Neutrino potential”: depends on where nucleons are and energy of intermediate state, due to (A) can replace by average, ( CLOSURE ) Both F and GT transitions. When expanding H into multipoles expect contributions up to 6 or more.

6. Neutrinoless double beta decay, 0νββ 76 Ge 0 + gs T=6 T=4 76 Se 76 As ~ tens MeV

7. Neutrino Masses For neutrinoless double beta decay mediated by a light massive neutrino: The effective neutrino mass extracted for a half life of 4x10 27 years 76 Ge. NO other experimentally accessible process can directly determine the nuclear matrix element. “The uncertainty in the calculated nuclear matrix elements for neutrinoless double beta decay will constitute the principal obstacle to answering some basic questions about neutrinos” John Bahcall 2004

8. Calculating Nuclear Matrix Elements Need to know the wave functions of the intermediate states, and of initial and final ground states. For the foreseeable future, it will not be possible to calculate the properties of large nuclei using QCD. Current “best” nuclear-structure calculations take empirical NN interactions, corrected for many-body forces, and solve the many-body Schrödinger equation directly using Monte-Carlo techniques, giving accurate wave functions as heavy as 12 C. Mean-field theory: independently moving nucleons » Slater determinant of independent particles Add correlations between nucleons, using QRPA or shell model. For candidate double beta decay nuclei, have to resort to:

9. Quasiparticle Random-Phase Approximation (and many variants) PLUS! Treats many nucleons as “active” and allows them to move in a large single-particle space MINUS! Can only cope with specific correlations of simple type (actually best suited for describing collective excitations). METHOD: small deviations from the unperturbed ground state by bosonizing Hamiltonian and transition operators (RPA). Pairing-like correlations taken into account using BCS theory (QRPA). ISSUES: 2νββ decay rate is VERY sensitive to particle-particle coupling strength; small change results in large fluctuations and the RPA approx collapses. VERY many variants to try to get around this issue (‘second’ QRPA, RQRPA, BCS-RQRPA….), but none clearly superior to the others. ALL leave out (non-RPA!) correlations. ALL have some truncation of model space so effective operators are needed.

10. Shell Models PLUS! Allows arbitrary correlations between nucleons. MINUS! Can only cope with a small number of “active” nucleons METHOD: Diagonalize empirically deduced effective interactions within a restricted model space. Could get arbitrary accuracy, provided a large enough model space is used; current computer technology allows bases of millions of states, but this barely sufficient in heavy systems. ISSUES: Nucleon-nucleon interaction in a restricted basis needs modification, and effective interactions are usually based on fitting to “simple” configurations in “simple” nuclei. Any truncation of model space requires effective operators to account for effects of nucleons outside of the model space.

11. Theoretical Prospects (as viewed by an experimentalist!) Despite everything, most “good” calculations agree to within a factor of a few. Attempts are made to judge uncertainty from the range of calculations. Increases in computing power guarantee that shell-model calculations will get better and better; although issues with effective operators for double beta decay have not been fully dealt with. Various avenues still need to be taken to understand the virtues and deficiencies of QRPA. Can help by measuring related observables: The more observables a calculation can reproduce, in general, the more trust you can put in it. If there are free parameters, other observables could be used to fix them. If renormalization is needed, other observables could be used to do it.

12. GT Distributions and Charge exchange reactions (A,Z) (A,Z+1) (A,Z+2) (p,n) ( 3 He, t) (n,p) (t, 3 He) (d, 2 He) (π ,π   etc. Forward-angle, single-charge exchange reactions (L=0) are sensitive to GT distributions populating 1 + states in intermediate system. Very useful for checking calculations of 2νββ For L≠0 transitions important for 0νββ are complex in hadronic reactions; no simple relationship exists to convert cross sections into the relevant strength distributions. Also relative phases of amplitudes not directly measurable! Double charge exchange, e.g. (π +,π – ), ( 11 Be, 11 Li), convert two neutrons into protons but different operators involved AND complex reaction mechanisms.

13. Beta decay and 2νββ (A,Z) (A,Z+1) (A,Z+2) β+β+ ββ Some authors claim that fitting to 2νββ and measured masses, allows all QRPA variants to agree to within 30% independent of model space, interactions, quenching…. Others dispute the paramount importance of 2νββ rates beyond other info…. In particular, calculations fitted to 2νββ rates don’t seem to be able to reproduce single β decay of intermediate nucleus. Precision mass measurements (at tens of eV level) using trapped ions have an important role to play, esp given Q value dependencies. 2νββ

14. Neutrinoless double beta decay: 76 Ge 0 + gs T=6 T=4 76 Se 76 As ~ tens MeV Given the shear range and numbers of states populated in the intermediate system, calculations MAY be more sensitive to our knowledge of the ground states of the parent and daughter!

15. Pair Transfer Check on the pairing properties of the initial and final states… Reactions involving transfer of a pair of nucleons to a simple BCS wave function are enhanced. Se/Ge nuclei are noted for several low-lying 0 + excited states; which in some cases carrying a significant fragment of the BCS wave function. Spectra from (p,t) reactions: For 76 Ge and 76 Se (p,t) strength is predominately to the ground states, indicating they are nice simple BCS paired states. They have quantitatively similar pair correlations, significantly simplifying calculations.

16. Single-Nucleon Transfer Independent nucleons will fill single- particle levels up to Fermi surface… …correlations and interactions scattered particles leading to a “smeared” Fermi surface and partial occupancy of single-particle levels. Ground-state wave functions are characterized by particular occupancies of single-particle orbitals…. In single-nucleon transfer, the cross section for adding a nucleon to a particular single-particle orbital depends on how empty it is. Conversely, the cross section for removing a nucleon depends on orbital occupancy. Under certain circumstances, and with particular care and attention, orbital occupancies and vacancies can be obtained from experiment. Occupancy

17. Neutron Occupancies Series of neutron-adding and neutron- moving reactions…done with particular attention to consistent experimental technique and analysis… …(α, 3 He), ( 3 He, α), (d,p) and (p,d) on 76 Ge and 76 Se targets. Measured neutron occupancies ARE different to those used in QRPA…but consequences for matrix elements are not so clear at present. Proton occupancies are also important, and will be measured next week!

18. Conclusions The calculation of nuclear matrix elements is an important but difficult topic. Theoretical consensus is really yet to form on what aspects of nuclear wave functions are most important (probably small and ill defined) AND what methods are best used to calculate them. BUT there are reasons to be hopeful about significant improvements in calculations and their convergence. Although there are NO direct measurements, other than 0νββ itself, experiments have much to offer in constraining the calculations. Beyond the ones mentioned, others have been proposed: neutrino-nucleus collisions, muon capture… Probably needs more effort on both sides to crack the issue in the short term….

19. Thanks to my collaborators Argonne National Laboratory: J. Schiffer, K.E. Rehm, S. Gros, C.L. Jiang, X.D. Tang Yale University: J.A. Clark, C. Deibel, A. Parikh, P.D. Parker, C. Wrede, A. Heinz, V. Werner, J. Qian Manchester University: B.P. Kay and C.R. Fitzpatrick GANIL: A.C.C. Villari Open University: D. Hirata