Support of Evidence for Neutrinoless Double Beta Decay

Indirect support for the evidence of neutrinoless double beta decay reported recently, is obtained by analysis of other Ge double beta experiments, which yield independent information on the background in the region of Q_(beta-beta). Some statistical features as well as background simulations with GEANT 4 of the HEIDELBERG-MOSCOW experiment are discussed which disprove recent criticism.

Recently first experimental evidence has been reported for neutrinoless double beta decay. Analysis of 55 kg y of data, taken by the HEIDELBERG-MOSCOW experiment in the GRAN SASSO over the years 1990 -2000, has led [1,2,3,4] to a half-life with best value of T 1/2 = 1.5×10 25 y, for the decay of the double beta emitter 76 Ge Assuming the decay amplitude to be dominated by exchange of a massive Majorana neutrino (see, e.g. [6]), this half-life results in a value of the effective neutrino mass < m >= U 2 ei m i = 0.05 − 0.84 eV (95% C.L.), with best value of 0.39 eV. Here a 50% uncertainty in the nuclear matrix elements has been taken into account (for details see [3]). Result of the peak-searching procedure performed on the HEIDELBERG-MOSCOW spectrum (taken with detectors 1,2,3,5) using the Maximum Likelihood approach (left) and the Bayesian method (right). On the y axis the probability of having a line at the corresponding energy in the spectrum is shown. This is for the first time that the absolute scale of the neutrino mass spectrum has been fixed, which cannot be achieved by neutrino oscillation experiments. This result restricts possible neutrino mass scenarios to degenerate or (still marginally allowed) inverse hierarchy [7,8,9]. In the degenerate case it leads to a common neutrino mass eigenvalue of This result is nicely consistent with later collected or analyzed experimental data, such as results from Large Scale Structure and CMB measurements [10, 11,12], or ultra-high cosmic rays [13]. The former yield an upper limit of i m i =1.0 eV (corresponding in the degenerate case to a common mass eigenvalue m 0 < 0.33 eV). The Z-burst scenario for ultra-high cosmic rays requires 0.1 -1.3 eV [13]. Tritium single beta decay cuts the upper range in eq.(4) down to 2.2 or 2.8 eV [14]. There is further theoretical support for a neutrino mass in the range fixed by the HEIDELBERG-MOSCOW experiment. A model basing on an A4 symmetry of the neutrino mass matrix requires the neutrinos to be degenerate and the common mass eigenvalue to be >0.2 eV [15]. Starting with the hypothesis that quark and lepton mixing are identical at or near the GUT scale, Mohapatra et. al. [16] show that the large solar and atmospheric neutrino mixing angles can be understood purely as result of renormalization group evolution, if neutrino masses are quasi-degenerate (with same CP parity). The common Majorana neutrino mass then must be larger than 0.1 eV. The fact that WMAP and less stringent tritium decay cuts away the upper part of the allowed range from double beta decay (eq. (4)), may indicate [5] that indeed the neutrino mass eigenvalues have the same sigen of CP phases -as required by [16].
In this Letter we report additional support of the double beta result of [1,2,3,4], by further discussion of the structure of the experimental background, by statistical considerations and by analysis of other double beta experiments investigating the decay of 76 Ge. Important points in the analysis of the measured spectrum are the structure of the background around Q ββ (=2039.006(50) keV [17]), and the energy range of analysis around Q ββ . Background lines in the vicinity of Q ββ : Fig. 1 shows the spectrum measured in the range 2000 -2100 keV in its original binning of 0.36 keV. By the peak search procedure developped [2,3] on basis of the Bayes and Maximum Likelihood Methods, exploiting as important input parameters the experimental knowledge on the shape and width of lines in the spectrum, weak lines of 214 Bi have been identified at the energies of 2010.78, 2016.7, 2021.6 and 2052.94 keV [1,2,3,4]. Fig. 2 shows the probability that there is a line of correct width and of Gaussian shape at a given energy, assuming all the rest of the spectrum as flat background (which is a highly conservative assumption). The intensities of these lines have been shown to be consistent with other, strong Bi lines in the measured spectrum according to the branching ratios given in the Table of Isotopes [18], and to Monte Carlo simulation of the experimental setup [3]. Note that the 2016 keV line, as an E0 transition, can be seen only by coincident summing of the two successive lines E = 1407.98 keV and E = 609.316 keV. Its observation proves that the 238 U impurity from which it is originating, is located in the Cu cap of the detectors. Recent measurements of the spectrum of a 214 Bi source as function of distance source-detector confirm this interpretation [19]. Premature estimates of the Bi intensities given in [25] thus are incorrect, because this long-known spectroscopic effect of true coincident summing [21] has not been taken into account, and also no simulation of the setup has been performed (for details see [3,20]). These 214 Bi lines occur also in other investigations of double beta decay with Ge -and -even more important -also the additional structures in Fig. 2, which cannot be attributed at present, are seen in these other investigations.
There are three other Ge experiments which have looked for double beta decay of 76 Ge. First there is the experiment by Caldwell et al. [22], using natural Germanium detectors (7.8% abundance of 76 Ge, compared to 86% in the HEIDELBERG-MOSCOW experiment). This was the most sensitive natural Ge experiment. With their background a factor of 9 higher than in the HEIDELBERG-MOSCOW experiment and their measuring time of 22.6 kg years, they had a statistics of the background by a factor of almost four l a r g e r than in the HEIDELBERG-MOSCOW experiment. This gives useful information on the composition of the background. Applying the same method of peak search as used in Fig. 2, yields indications for peaks essentially at the same energies as in Fig. 2 (see Fig. 3). This shows that these peaks are not fluctuations. In particular it sees the 2010.  Figure 6: Result of the peak-search procedure performed for the IGEX spectrum [24,26] using the ML approach (left) and the Bayesian statistics (right). On the y axis the probability of having a line at the corresponding energy in the spectrum is shown.
at higher energies. It finds, however, n o line at 2039 keV. This is consistent with the expectation from the rate found in the HEIDELBERG-MOSCOW experiment. About 16 observed events in the latter correspond to 0.6 expected events in the Caldwell experiment, because of the use of non-enriched material and the shorter measuring time. Fit of the Caldwell spectrum allowing for the 214 Bi lines and a 2039 keV line yields 0.4 events for the latter (see Fig. 4). The first experiment using enriched (but not high-purity) Germanium 76 detectors was that of Kirpichnikov and coworkers [23]. These authors show only the energy range between 2020 and 2064 keV of their measured spectrum. The peak search procedure finds also here indications of lines around 2028 keV and 2052 keV (see Fig. 5), but n o t any indication of a line at 2039 keV. This is consistent with the expectation, because for their low statistics of 2.95 kg y they would expect here (according to HEIDELBERG-MOSCOW) 0.9 counts. Another experiment (IGEX) used between 6 and 8.8 kg of enriched 76 Ge, but collected since beginning of the experiment in the early nineties till shutdown in 1999 only 8.8 kg years of statistics [24,26]. The authors of [24,26] unfortunately show only the range 2020 to 2060 keV of their measured spectrum in some detail. Fig. 6 shows the result of our peak scanning of this range. Clear indications are seen for the Bi lines at 2021 and 2052 keV, but also (as this is the case for the spectrum of [23], see Fig.5 ) of the unidentified structure around 2030 keV. Because of the conservative assumption on the background treatment in the scanning procedure (see above) there is no chance to see a signal at 2039 keV because of the 'hole' in the background of that spectrum (see Fig. 1 in [24]). With some good will one might see, however, an indication of 3 events here, consistent with the expectation of the HEIDELBERG-MOSCOW experiment of 2.6 counts. Jeffrey's prior) and the Maximum Likelihood Method. For each prior 1000 spectra have been generated with equal background and equal line intensity using random number generators available at CERN [28]. The average values of the best values agree (see Fig. 8) very well with the known intensities also for very low count rates (as in Fig. 7, left). A detailed simulation has been done to show the behavior of the position of the peak maximum of a line of 15 events, with width (standard deviation) of 4 channels, on a Poisson-distributed background with 0.5 events/channel. For that reason, a peak scan procedure was performed on 1000 randomly created spectra. For each spectrum, the deviation of the real position of the line to the position of the maximum probability was determined. The distribution of these deviations is shown in figure 9, which gives a RMS value of 4.00 channels, corresponding to ∼ 1.44 keV in the spectra from the HEIDELBERG MOSCOW experiment. The figure 9 describes the possible degree of deviation of the energy of the peak maximum from the transition energy, on the Influence of the choice of the energy range of analysis: The influence of the choice of the energy range of the analysis around Q ββ has been thoroughly discussed in [2,3]. Since erroneous ideas about this point are still around, a few further comments may be given here. In Fig. 10 we show the analysis of a simulated spectrum consisting of a Gaussian line of width (standard deviation) of 4 channels and intensity of 10 counts on a Poisson-distributed background of 4 events per channel (see fig. 7 left), as function of the width of the range of analysis. It is seen that a reliable result is obtained for a range of analysis of not smaller than 35 channels (i.e. ±18 channels)one channel corresponding to 0.36 keV in the HEIDELBERG-MOSCOW experiment. Fig. 11 shows this in a more general way. Every dot represents the mean deviation of the calculated line intensity to the real line intensity, obtained from 100 random spectra. The exact value for the deviation ∆ is calculated by seen that, if the evaluation width is more than 35 channels, the deviation is constant, so that an evaluation width of at least 35 channels is required for reliable results. This is an important result, since it is of course important to keep the range of analysis as s m a l l as possible, to avoid to include lines in the vicinity of the weak signal into the background. This unavoidably occurs when e.g. proceeding as suggested in [25]. The arguments given in those papers are therefore incorrect. Also Kirpichnikov, who states [29] that his analysis finds a 2039 keV signal in the HEIDELBERG-MOSCOW spectrum on a 4 sigma confidence level (as we also see it, when using the Feldman-Cousins method [27]), makes this mistake when analyzing the pulse-shape spectrum.
The background around Q ββ from GEANT4 simulation: Finally the background around Q ββ will be discussed from the side of simulation. A very careful new simulation of the different components of radioactive background in the HEIDELBERG-MOSCOW experiment has been performed recently by a new Monte Carlo program based on GEANT4 [30,31]. This simulation uses a new event generator for simulation of radioactive decays basing on ENSDF-data and describes the decay of arbitrary radioactive isotopes including alpha, beta and gamma emission as well as conversion electrons and X-ray emission. Also included in the simulation is the influence of neutrons in the energy range from thermal to high energies up to 100 MeV on the measured spectrum. Elastic and inelastic reactions, and capture have been taken into account, and the corresponding production of radioactive isotopes in the setup. The neutron fluxes and energy distributions were taken from published measurements performed in the Gran Sasso. Also simulated was the cosmic muon flux measured in the Gran Sasso, on the measured spectrum. To give a feeling for the quality of the simulation, Fig. 12 shows the simulated and the measured spectra for a 228 Th source spectrum for 4 of our five detectors. It should be noted, that this simulation is not a fit. It is a calculation of the expected spectra on the basis of the knowledge of the geometry of the setup, and its properties of absorption and scattering of gamma radiation, and of the known position and strength of the source. The agreement is excellent. The simulation of the background of the experiment reproduces a l l observed lines in the energy range between threshold (around 100 keV) and 2020 keV [27,30,31]. Also the potential contribution from decays of 77 Ge, 66 Ga, or 228 Ac, should not lead to signals visible in our measured spectrum near the signal at Q ββ . For details we refer to [31]. The structures around 2028 keV, 2066 keV and 2075 keV seen -as also the 214 Bi lines -in practically all Ge experiments (see above), cannot be identified at present.

Conclusion:
Concluding, additional support has been given for the evidence of a signal for neutrinoless double beta decay, by showing consistency of the result -for the signal, a n d for the background -with other double beta decay experiments using non-enriched or enriched Germanium detec- tors. In particular it has been shown that the lines seen in the vicinity of the signal (including those which at present cannot be attributed) are seen also in the other experiments. This is important for the correct treatment of the background. Furthermore, the sensitivity of the peak identification procedures has been demonstrated by extensive statistical simulations. It has been further shown by new extensive simulations of the expected background by GEANT4, that the background around Q ββ should be flat, and that no known gamma line is expected at the energy of Q ββ . The 2039 keV signal is seen o n l y in the HEIDELBERG-MOSCOW experiment, which has a factor of at least 10, but in general m u c h more, statistics than all other double beta experiments.