1. Compared sensitivities of next generation DBD experiments IDEA - Zaragoza meeting – 7-8 November 2005 C. Augier presented by X. Sarazin LAL – Orsay – CNRS/IN2P3 and Université Paris-Sud XI

2. This work was realised and included in my HDR report in June 2005 (Section 4.3.7 « L’effet des éléments de matrice nucléaire », p.332-343) Presentation of this work Main goals of this work:  For GERDA and CUORE sensitivities, use of their published expected sensitivities for the  period.  Concerning SuperNEMO sensitivity, use of the preliminary calculations to give the two extremal values for the expected T 1/2 (0 ) period. The SuperNEMO period limit obtained from actual Monte-Carlo simulations is just below the best value used in this work. 1) study the effect of the large nuclear matrix element (NME) range on the experimental sensitivities for different isotopes, in case of the exchange of a light massive Majorana neutrino in  process 2) obtain a useful method to directly compare the different NME calculations in terms of  period sensitivity. 3) compare the predicted sensitivities on the effective neutrino mass for GERDA, CUORE and SuperNEMO projects, using both their predicted limits of  periods and this study of NME range.

3. where G  is the  phase space factor calculated for all nuclei by Doi, and then Vogel (see « F. Boehm and P. Vogel, Physics of massive neutrinos, Cambridge University Press, second edition, 1992 »). Presentation of this work  In case of light massive Majorana neutrino exchange in , period of the process is related to the effective neutrino mass by the relation [ T  ]  = G  |M  |   m     It is difficult to compare directly the NME values from the different publications. In fact, one can find in these publications the NME value |M  |, or the product of |M  | by the phase space factor G   C mm, or the effective mass corresponding to a given period value… Morevoer, some of the authors use their own calculations of the phase space factor. Others omit the electron mass in their calculation and one have to reintroduce it before the comparison…

4. Presentation of this work Using the values obtained in different publications, the results are presented in a Table which contains the T  values, where T  is defined as T  (y) =  T  (y)  m    (eV) From this table containing the T  values, it is useful to recalculate the effective neutrino mass (in eV) associated to any given  period (in y), using the relation In fact, the T  value corresponds to an effective neutrino mass  m  eV. T  (y)   m    (eV) =  TT /

5. Two NME calculation techniques Shell model Quasi Random Phase Approximation (QRPA) and extensions Choice of the NME publications and studied isotopes Criteria used for the publication choice - reproduction of relevant nuclear properties ( , , nuclear states,...) - publications with comparison of different isotopes - recent publications if authors explain why their new calculations are more credible Important note - ref [173] Staudt, Kuo, Klapdor-Kleingrothaus, Phys. Rev. C46 (1992) is NOT USED : it gives results only for 76 Ge, 130 Te and 136 Xe, with the most favored NME values for 76 Ge and 130 Te, which provide period sensitivities around one order most favored than for other calculations.

6. Shell model calculations Choice of the NME publications and studied isotopes Few publications, I decided to use Ref. [154] = E. Caurier, A. Gniady, F. Nowacki, « Beyond NEMO3 », Orsay, Dec. 2003, (NEMO meeting) in association with published results from the same authors + Ref. [163] = E. Caurier, G. Martinez-Pinedo, F. Nowacki, A. Poves and P. Zuber, Rev. Mod. Phys. 77 (2005) 427-488, also nucl-th/0402046 (2004) + Ref. [164] = E. Caurier, F. Nowacki, A. Poves and J. Retamosa, nucl-th/ 9601016 (1996) They give the NME values for 6 nuclei : 48 Ca, 76 Ge, 82 Se, 124 Sn, 130 Te and 136 Xe Arbitrary choice based on the fact that these authors calculate all parameters for a given nucleus, which are used to reproduce experimental nuclear levels with a good precision. Results are presented in the 1st line of the Table and in plots, refered as « Shell Model »

7. QRPA and extensions’ calculations Choice of the NME publications and studied isotopes 1)Ref. [155] = V.A. Rodin, A.Faessler, F. Simkovic, P. Vogel, « Systematic analysis of the uncertainty in the  decay nuclear matrix elements », nucl-th/0503063 - recent paper (2005) from authors issued from different theoretical groups - they give some arguments to explain their calculations ; - they use QRPA and RQRPA (renormalized) approach, both with two different values of the vector-axial coupling constant g A = 1.0 and 1.25, that means 4 results per isotope - they adjust the particle-particle coupling constant (g pp ) value to  experimental half-lives (which allow to have a slight dependance on the size of model space), with g ph = 1 (particle-hole interaction fixed to Gamow-Teller resonance), using « higher-order » terms of nucleon currents - they use their own phase space factor value, calculated with R = 1.1 A 1/3 They give the NME values for 9 nuclei : 76 Ge, 82 Se, 96 Zr, 100 Mo, 116 Cd, 128 Te, 130 Te, 136 Xe and 150 Nd (I do not present 128 Te results) Results are presented in lines 2 to 5 of the Table QRPA 1, QRPA 1.25, RQRPA 1., RQRPA 1.25, and the two extremal values are plotted, refered as RFSV 05 – RQRPA (avec g pp de  et) g A = 1 and RFSV 05 – QRPA (avec g pp de  et) g A = 1.25

8. QRPA and extensions’ calculations Choice of the NME publications and studied isotopes 2) Ref. [165] = F. Simkovic, G. Pantis, J.D. Vergados and A. Faessler, « Additional nucleon current contributions to neutrinoless double beta decay », Phys. Rev. C60 (1999) 055502 - paper with common authors than in the previous one, chosen for comparison with line 5 of the Table, - they use RQRPA (renormalized) approach, the vector-axial coupling constant g A = 1.25, - they use their own phase space factor value, calculated with R = 1.1 A 1/3 - the only difference is that they fix the particle-particle coupling constant (g pp ) value to 1, with g ph = 0.8 (particle-hole interaction) and using « higher- order » terms of nucleon currents They give the NME values for 9 nuclei : 76 Ge, 82 Se, 96 Zr, 100 Mo, 116 Cd, 128 Te, 130 Te, 136 Xe and 150 Nd (I do not present 128 Te results) Results are presented in line 6 of the Table, RQRPA 1.25, and plotted for all isotopes comparison, refered as SPVF 99 – RQRPA avec g pp = 1 et g A = 1.25

9. QRPA and extensions’ calculations Choice of the NME publications and studied isotopes 3) Ref. [166] = S. Stoica, H.V. Klapdor-Kleingrothaus, Nucl. Phys. A694 (2001) 269-294 - they use QRPA and 3 different extensions (RQRPA, f-RQRPA for fully renormalized, and SK-RQRPA for Stoica-Klapdor…) - For these 4 calculations, they use both small s and large l sizes of model space., with RQRPA approach, and the vector-axial coupling constant g A = 1.25, - they fix the particle-particle coupling constant (g pp ) value to the probability of  experimental transition, but only for J  = 1 + relevant state, and leave the strenght unrenormalized for the other states. They give the NME values for 8 nuclei : 76 Ge, 82 Se, 96 Zr, 100 Mo, 116 Cd, 128 Te, 130 Te, 136 Xe (I do not present 128 Te results) Results are presented in lines 7 to 14 of the Table (refered from QRPA s to SK- RQRPA l ). Also minimal and maximal values of T  from this publication are plotted and refered as SK 01 – min and SK 01 – max In this paper, NME values are different from one approximation to other, and one can find the most favored values of NME for numerous isotopes Also the needed phase space factor were corrected (for example for 100 Mo)

10. QRPA and extensions’ calculations Choice of the NME publications and studied isotopes 4) Ref. [167] = M. Aunola, J. Suhonen « Mean-field effects on neutrinoless double beta decay », Nucl. Phys. A643 (1998), and [168] J. Suhonen, M. Aunola, « Systematic study of neutrinoless double beta decay to excited 0+ states », Nucl. Phys. A723 (2003) - two review papers, with QRPA calculations. - the first one with AS1 (and AS2) for the use of standard (and adjusted) Woods-Saxon potential, the adjusted one used to obtain more realistic mean field ; the second paper refered AS3, is a compilation of different calculations of these authors. - for all the calculations, they use the vector-axial coupling constant g A = 1. - they adjust the particle-particle and particle-hole coupling constants (g pp and g ph ) values to the probability of  experimental transitions, They give the NME values for 8 nuclei : 76 Ge, 82 Se, 96 Zr, 124 Sn, 130 Te, 136 Xe, 100 Mo, 116 Cd Results are presented in lines 15 to 17 of the Table (refered from QRPA AS1 to QRPA AS3). With agreement of J. Suhonen, also minimal and maximal values of T  extracted from these two publications are plotted and refered as AS98 – AS03 – min and AS98 – AS03 – max

11. Results : T 0 values obtained from the studied publications Minimal and maximal values of T 0 used for the comparison plots Most favored valueLess favored value Model T 0 ( 76 Ge)T 0 ( 82 Se)T 0 ( 96 Zr)T 0 ( 100 Mo)T 0 ( 116 Cd)T 0 ( 130 Te)T 0 ( 136 Xe)

12. 76 Ge : T 0 = 1.77 x 10 25 y 82 Se : T 0 = 2.40 x 10 24 y 130 Te : T 0 = 9.0 x 10 23 y 136 Xe : T 0 = 1.3 x 10 24 y Most favored isotope Less favored isotope m (eV) =  T 1/2 (0 ) (yr) /T 0 (eV) (Caurier, Nowacki, publication 1996 + « beyond NEMO3 2003)S.M.

13. 76 Ge : T 0 = 4.60 x 10 24 y 82 Se : T 0 = 1.33 x 10 24 y 130 Te : T 0 = 1.96 x 10 24 y 136 Xe : T 0 = 4.17 x 10 24 y 96 Zr : T 0 = 2.18 x 10 27 y 100 Mo : T 0 = 2.79 x 10 24 y 116 Cd : T 0 = 1.72 x 10 24 y m (eV) =  T 1/2 (0 ) (yr) /T 0 (eV) (Rodin, Faessler, Simkovic, Vogel, 2005) Most favored isotope Less favored isotope RFSV 2005

14. 76 Ge : T 0 = 6.24 x 10 24 y 82 Se : T 0 = 2.03 x 10 24 y 130 Te : T 0 = 2.87 x 10 24 y 136 Xe : T 0 = 5.46 x 10 24 y 96 Zr : T 0 = 1.88 x 10 25 y 100 Mo : T 0 = 3.65 x 10 24 y 116 Cd : T 0 = 2.82 x 10 24 y m (eV) =  T 1/2 (0 ) (yr) /T 0 (eV) (Rodin, Faessler, Simkovic, Vogel, 2005) Most favored isotope Less favored isotope RFSV 2005

15. 76 Ge : T 0 = 4.23 x 10 24 y 82 Se : T 0 = 1.08 x 10 24 y 130 Te : T 0 = 1.46 x 10 24 y 136 Xe : T 0 = 1.04 x 10 25 y 96 Zr : T 0 = 1.61 x 10 24 y 100 Mo : T 0 = 4.6 x 10 23 y 116 Cd : T 0 = 9.99 x 10 23 y m (eV) =  T 1/2 (0 ) (yr) /T 0 (eV) (Simkovic, Pantis, Vogel, Faessler, 1999) Most favored isotope Less favored isotope SPVF 1999

16. The T 0 value, which corresponds to an effective mass m = 1 eV, has to be as low as possible to favor the possibility of  signal observation For 76 Ge : - the best sensitivity corresponds to the QRPA method with  adjustment, with T 0 = 1.96 x 10 24 y (Aunola, Suhonen, 1998), - the worst one corresponds to the QRPA- l method, with T 0 = 1.40 x 10 25 y (Stoica, Klapdor, 2001) For 82 Se : - the best sensitivity corresponds to the QRPA- s method, with T 0 = 2.96 x 10 23 y (Stoica, Klapdor, 2001), - the worst one corresponds to the Shell-Model calculations, with T 0 = 2.40 x 10 24 y (Caurier, Nowacki, 1996 and 2003) For 130 Te : - the best sensitivity corresponds to the QRPA- s method, with T 0 = 2.63 x 10 23 y (Stoica, Klapdor, 2001), - the worst one corresponds to the RQRPA method with  adjustment and g A =1, with T 0 = 3.60 x 10 24 y (Rodin, Faessler, Simkovic, Vogel, 2005) Study of the sensitivity range for 76 Ge, 82 Se and 130 Te

17. - Klapdor (best fit), T = 1.2 x 10 25 yr, 0.40 < 1.21 eV m (eV) =  T 1/2 (0 ) (y) /T 0 (eV) - IGEX best limit, T < 1.57 x 10 25 yr, 0.34 < 1.05 eV - HM best limit, T < 1.9 x 10 25 yr, 0.32 < 0.97 eV 76 Ge (Past experiments)

18. m (eV) =  T 1/2 (0 ) (y) /T 0 (eV) - GERDA phase I, T < 3 x 10 25 yr, 247 < 774 meV - GERDA phase II, T < 2 x 10 26 yr, 96 < 293meV - GERDA phase III, T < 3 x 10 27 yr, 25 < 77 meV 76 Ge (GERDA sensitivities)

19. m (eV) =  T 1/2 (0 ) (y) /T 0 - NEMO 3, T < 8 x 10 23 yr, 0.61 < 1.72 eV -SuperNEMO, « low » resolution T < 1 x 10 26 yr, 54 <155 meV -SuperNEMO, « high » resolution T < 2.2 x 10 26 yr, 36 < 105 meV (eV) 82 Se (SuperNEMO sensitivities)

20. m (eV) =  T 1/2 (0 ) (y) /T 0 - CUORICINO T < 4 x 10 24 yr, 0.26 < 0.84 eV - CUORE bkg = 0.001 with 130 TeO 2 T < 1.9 x 10 27 yr, 12 < 39 meV (eV) - CUORE bkg = 0.001 with nat TeO 2 T < 6.6 x 10 26 yr, 20 < 65 meV - CUORE bkg = 0.01 with nat TeO 2 T < 2.1 x 10 26 yr, 36 < 117 meV 130 Te (CUORE sensitivities)

21. T 0 values corresponding to an effective mass of 50 meV T range = 50 meV

22. T 0 values corresponding to an effective mass of 50 meV For 76 Ge : -  period between 7.8 x 10 26 y and 5.6 x 10 27 y  Possible with GERDA phase III (T 1/2  3 x 10 27 y) with 1000 kg.y and bkg = 0.001 cts.keV -1.kg -1.y -1 (same conclusions for MAJORANA) For 130 Te : -  period between 1.1 x 10 26 y and 1.4 x 10 27 y  No problem for CUORE with minimal value ; the maximal period could be reached for bkg = 0.001 cts.keV -1.kg -1.y -1 (T 1/2  6.6 x 10 26 y) or with enriched crystals (T 1/2  1.0 x 10 27 y) For 82 Se : -  period between 1.2 x 10 26 y and 9.6 x 10 27 y  No problem for SuperNEMO with minimal value, but it could be very difficult to measure if the NME value corresponds to the maximal period.

23. Shell Model: Caurier (2003) RQRPA Simkovic et al. (1999) Stoica et al. (2001) Suhonen et al. (1998 and 2003) Rodin, Simkovic (2005) Theoretical calculations of the NME Big theoretical uncertainties  Thus choice of the nucleus depends on: 1) detector technique 2) T 1/2 (  ) for  m  =50 meV In conclusion  enrichment possibility  high Q  value  high  period: T   10  y Goal  measure the highest possible experimental value of the  period... And wait for the good calculation

24. Used in the comparison plots Results : T 0 values obtained from the studied publications Model T 0 ( 76 Ge)T 0 ( 82 Se)T 0 ( 96 Zr)T 0 ( 100 Mo)T 0 ( 116 Cd)T 0 ( 130 Te)T 0 ( 136 Xe)

25. Study for other isotopes ( 48 Ca, 124 Sn, 150 Nd) Results for 48 Ca Ref. [163] = E. Caurier, G. Martinez-Pinedo, F. Nowacki, A. Poves and P. Zuber, Rev. Mod. Phys. 77 (2005) 427-488, for Shell Model calculation T 0 = 8.84 x 10 24 y Ref. [169] = C. Barbera et al., Nucl. Phys. A650 (1999) for QRPA calculations T 0 = 2.31 x 10 24 y Ref. [170] = Pantis, Simkovic,Vergados and Faessler, Phys. Rev C53 (1996), for QRPA calculations T 0 = 2.44 x 10 24 y  QRPA values are nearly the same, and three times more favorable than the value obtained from SM calculation 48 Ca (Z and N are magic numbers) 20 24

26. Study for other isotopes ( 48 Ca, 150 Nd, 124 Sn) Results for 124 Sn (Q  = 2.29 MeV, magic proton number Z = 50) Ref [167] Aunola, Suhonen, Nucl. Phys. A643 (1998) adjustment on  -decay transition AS1 :T 0 = 4.58 x 10 23 y(standard WS potential) AS2 : T 0 = 1.14 x 10 24 y(adjusted WS potential)  1) There is a factor 2.3 between the two QRPA calculations, 2) AS2-QRPA and SM calculations give nearly the same value Ref. [154] = E. Caurier, A. Gniady, F. Nowacki, « Beyond NEMO3 », Orsay, Dec. 2003, (NEMO meeting) T 0 = 1.60 x 10 24 y This nucleus is treated as a « core of 100 Sn » + 24 neutrons (stable) 50

27. Caurier, Nowacki, 1996 + « beyond NEMO3 » 2003 → Shell Model Rodin, Faessler, Simkovic, Vogel, 2005 QRPA g pp from  and g A = 1.25 → RFSV 05 – QRPA gA = 1.25 RQRPA g pp from  and g A = 1 → RFSV 05 – RQRPA gA = 1. Simkovic, Pantis, Vergados, Faessler, 1999, g pp =1 and g A = 1.25 → SPVF 99 – RQRPA gA = 1.25 Examples of isotope comparison for different publications Publications used  See the 4 comparison plots

28. Study for other isotopes ( 48 Ca, 150 Nd, 124 Sn) Results for 150 Nd (deformed nucleus, difficult to calculate) Ref. [155] = V.A. Rodin, A.Faessler, F. Simkovic, P. Vogel, « Systematic analysis of the uncertainty in the  decay nuclear matrix elements », nucl-th/0503063 from T 0 = 1.92 x 10 23 y (QRPA g A = 1.25) to T 0 = 3.03 x 10 23 y(RQRPA, g A = 1.0) Ref. [165] = F. Simkovic, G. Pantis, J.D. Vergados and A. Faessler, « Additional nucleon current contributions to neutrinoless double beta decay », Phys. Rev. C60 (1999) 055502 T 0 = 8.84 x 10 22 y (this value was more favorable)  All these QRPA values are nearly the same, even if the value from 1999 was more favorable.

29. QRPA and extensions’ calculations Choice of the NME publications and studied isotopes Other publications found as reference in previous papers. Results are put in the Table but not used in the plots because their T  values are included in the range obtained from previous publications (except for 100 Mo, where the T  value in QRPA (4) is only 6% higher than the maximal value used in the plots) Ref. [171] = Simkovic, Novak, Kaminski, Raduta, Faessler, Phys. Rev. C64 (2001) Ref. [172] = Muto, Bender and Klapdor-Kleingrothaus, Z. Phys. A334 (1989) Ref. [169] = C. Barbera et al., Nucl. Phys. A650 (1999) Ref. [170] = Pantis, Simkovic,Vergados and Faessler, Phys. Rev C53 (1996) Results are presented in lines 18 to 21 of the Table (refered from QRPA (1) to QRPA (4)).