Masses of neutrinos

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As neutrino oscillations due to the square-mass difference Δmsol2 = (7.59 ± −5 2 atmospheric and accelerator neutrino 0.21) × 10 eV [22] and the observation of 2 −3 −3 2 oscillations due to the square-mass difference Δmatm = (2.43 ± 0.13 × 10 )×10 eV [61] give very robust evidence of three-neutrino mixing.
In this chapter, we consider the Gallium radioactive source experiments anomaly [73] which could be interpreted as indications of exotic neutrino physics beyond three-neutrino mixing. A description of these experiments is presented, as well as the analysis of the observed anomaly, under the hypothesis of the disappearance of electron neutri-nos due to neutrino oscillations [74, 75, 76] in the framework of two neutrino mixing. We also discuss the compatibility of this interpretation of the Gallium radioactive neu-trino experiments anomaly, with the results of the Bugey and Chooz nuclear reactor neutrino experiments, including a short description of each experiment.

Gallium radioactive source experiments

Radiochemical detection is one of the methods used for detecting neutrinos. In this method, a detector chemical interacts with neutrinos converting the initial element into a radioactive isotope of another element, νe + N (A, Z − 1) → e− + N (A, Z), (3.1) where Z is the atomic number and A the mass number. The atoms of the radioactive product are extracted and counted by using chemical techniques. This count gives a measure of the neutrino flux [77].
Radiochemical experiment using Gallium nuclei were proposed initially in [78] to detect pp solar neutrinos using the reaction νe + 71Ga −→ e− + 71Ge, (3.2) with a threshold energy Eνth = 0.233 MeV.
The experiment was carried out by the GALLEX [79, 80, 81] and SAGE [82, 83, 84, 73] groups, measuring a lower flux of electron neutrinos that that expected from the Solar Standard Model (SSM), result which is explained with the phenomenon of neutrino oscillations.
This results have important consequences in particle physics and astrophysics, so it is very important to test carefully the experimental techniques, to cancel any possible doubt on them and on the results. The most straightforward check is to expose these experiments to neutrino sources with known activity levels and appropriate energy, under conditions nearly identical to those used in solar exposures [79].
These tests were done by the two groups in their respective detectors, and the corre-sponding experimental characteristics are described in the following sections, together with the analysis of the final results.


The GALLium EXperiment detector was located at the Gran Sasso underground lab-oratory, in Italy, reducing the muon induced background by shielding the facility by about 3300 meters of water equivalent. The radiochemical source experiment was used to test the detector using two radioactive sources with small differences, as explained later.
For the test, it was necessary to fabricate an intense and portable neutrino source with the following specifications [79]:
• the source activity level must be such that the measurements reach the precision of the measurements of the solar neutrino flux after 4 years of data collection;
• the energy of the emitted neutrinos must be close to the mean energy of solar neutrinos detected by GALLEX;
• the source lifetime must be long enough to allow transport of the source to the underground detector and the posterior development of the experiment.
After all this considerations, the 51Cr nuclide was selected as the most suitable one. The radioactive nucleus 51Cr is produced by neutron capture on 50Cr, and has a lifetime of 27.7 days. It decays through electron capture to the ground state of 51V (see figure 3.1), e− + 2451Cr → 2351V + νe, (3.3) emitting νe lines with the energies and branching ratios listed in table 3.1, and a 320 keV γ.
The activity of the 51Cr source must be larger than 50 PBq1 in order to produce a signal about one order of magnitude grater than that of the Sun [79, 86]. To obtain the 51Cr source, the 50Cr isotope was transformed in 51Cr by neutron capture in a nuclear reactor: 50Cr(n, γ) 51Cr [87].
The activity of the final chromium source used as the neutrino sources in each of the two experiments, was measured using calorimetry, an ionization chamber, high-resolution gamma ray spectroscopy and neutron activation to measure the 51V pro-duction (more details can be found in references [79, 80]). The resulting mean activity of each 51Cr source at EOB2 were [80]:
ACr1 = 63.4 ± 0.5 PBq, for the first source; (3.4)
ACr2 = 69.1 ± 0.6 PBq, for the second source.
These sources were placed inside tanks which contained the GALLEX detector (see figure 3.2), composed of a GaCl3-HCl solution, with ∼ 30tons of Gallium. All the ex-perimental conditions were kept as close as possible to those for the solar experiments. The position of the radioactive sources and the size of the detectors are shown in table 3.2, where it should be noticed that for the second source experiment, the 51Cr source was located at a position 32 cm lower than the position of the first source.
As mentioned above, the 51Cr source decays through electron capture, equation (3.3), and the electron neutrinos generated in this process interact with the Gallium in the detector through the same process used for detecting solar neutrinos, equation (3.2). The neutrino flux can be measured precisely, by counting the amount Germanium produced in the process. The 71Ge atoms combine with the Cl forming the volatile molecule GeCl4, which is extracted from the tank by air circulation. Then, the Ge is transformed into germane GeH4 which is used as the gas of a proportional counter, used to observe the 71Ge decay (τ1 2 = 11.43 d) [87].
Then, the experimental production rate of 71Ge, QexpGe, is compared with the ex-pected one in absence of neutrino oscillations, QGe, using the ratio Qexp Rexp = Ge . (3.5) QGe
In table 3.3 the measured production rate of 71Ge and the corresponding ratio for the two runs are presented. The numbers are collected from reference [73], which has different R values to the reported by the GALLEX collaboration [79, 80], as the result of a later revision on the results (see reference [3] of [73]).
In absence of neutrino oscillation, it is expected to have R = 1, but the GALLEX Cr2 result in table 3.3 shows an almost 2σ deviation from one. Assuming that the Germanium counting process and results as well as the theoretical value of the cross section of the process (3.2) are correct, this anomalous result could be interpreted as a hint of the presence of electron neutrino disappearance, produced by neutrino oscillations, motivating the present analysis.
Here we do the analysis of these two individual experimental results to determine the neutrino oscillation parameters, (sin2(2θ), Δm2), adopting a Bayesian approach, as done in reference [88]. To do this calculation, the theoretical value of the ratio R of the predicted 71Ge production rates in the presence and absence of neutrino oscillations is defined as
dV L−2 i (B.R.) σ P νe→νe (L, E νi )
R = i i , (3.6)
i (B.R.)i σi dV L−2
where i is the index of the νe energy lines emitted in the 51Cr decay, as listed in table 3.1, (B.R.)i and σi are the branching ratio and the cross section at the corresponding energy (see table 3.1), and Pνe→νe (L, Eνi ) is the survival probability of electron neutrinos (in the effective framework of two neutrino oscillations) with energy Eν at a distance L from the source, given by (equation (2.49))
Pνe→νe (L, Eν ) = 1 − sin2(2θ) sin2 Δm2L , (3.7) 4Eν with θ the mixing angle and Δm2 the square-mass difference.
In equation (3.6), the integration over L accounts for the distance travelled by the neutrino from the production point at the source, to the detection point (capture in the detector). The integration is, then, performed approximating the GALLEX detector and source as having a cylindrical shape with the dimensions shown in table 3.2. We averaged the neutrino path length L with a Monte Carlo integration over the volume V of each cylindrical detector, taking into account the different positioning of the source.
In the Bayesian approach, R in equation (3.6) is considered as a random variable with a uniform (flat) prior probability distribution between zero and one. If Robs is the observed value of R, the normalized posterior probability distribution of R is given by
p(R R ) = p(Robs|R) . (3.8)
| obs 01 dR p(Robs|R)
Here, p(Robs|R) is the sampling distribution of Robs given R, which we assume to be a Gaussian with standard deviation equal to the experimental uncertainty,
1 R − Robs 2
p(R R) = exp . (3.9)

obs| 2π σexp − √
2 σexp
The allowed interval of R with a given Bayesian Confidence Level is given bay the Highest Posterior Density interval with integrated probability equal to the Confidence
Level (C.L.), dR p(R Robs) = N dR exp √− obs
1 α = Rup Rup 2 , (3.10)
1 R R
Rlow Rlow 2 σexp
where Rlow, up, are the limits such that p(R|Robs) is higher everywhere inside the inter-val [Rlow, Rup], than outside [63], and N is the value of the integral in the denominator of (3.8).
The resulting allowed regions are shown in Figure 3.3. One can note that the result for the first GALLEX source experiment (Cr1), for which the measured rate is within 1σ from unity, shows only upper limits for the mixing parameters. On the other hand, the second GALLEX experiment (Cr2), gives 2σ allowed bands, that shows a value of the square-mas difference of Δm2 > 1eV2, which is much larger than the known measurements from Solar and Atmospheric neutrino oscillation experiments.

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The other radioactive source experiment was developed to test the SAGE (Soviet-American Gallium Experiment) detector, which was used to measure the capture rate of solar neutrinos with a target of gallium metal in liquid state. The SAGE detector was located in the Baksan Neutrino Observatory, in Russia [82].
The test of the SAGE detector was performed using two different radioactive arti-ficial neutrino sources: the first one using 51Cr and the second one using 37Ar.
In the case of the 51Cr source experiment, similar considerations to the ones taken by the GALLEX collaboration were taken into account here. The chromium used in this experiment was enriched to 94.4% in 50Cr, which has the advantage of yielding a great specific activity and a small physical size, thus giving a high neutrino capture rate [82].
The 55 tons of Ga that SAGE used for solar neutrino measurements were contained in eight chemical reactors with approximately 7 tons in each. Figure 3.4 shows the layout of the ten reactors in the experimental area. In normal solar neutrino opera-tion, Ga is contained in reactors 2 − 5 and 7 − 10. All reactors except number 6 are equipped with the necessary mechanical equipment for the extraction process. Reactor 6 was modified for the Cr exposures by removing its stirring mechanism and replacing it with a reentrant Zr tube on its axis which extended to the reactor center. This modification increased the capacity of the reactor to 13 tons of Ga. To begin each irra-diation, a remote handling system (figure 3.5) was used to place the 51Cr source inside this reentrant tube at the reactor center. At the end of each irradiation, the source was moved to an adjacent calorimeter for activity measurement, and the gallium was pumped back to the two reactors where it was stored during solar neutrino runs [82]. For the Cr experiment, reactors 6 − 10 in figure 3.4 were used.
As for the GALLEX experiment, the source activity was determined by measuring its heat (energy deposited in its surroundings) with a calorimeter, and by the mea-surement of the 320-keV gamma rays emitted by the 51Cr. The average activity of the source was [82] A 51Cr = 19.114 PBq. (3.11)
The 51Cr source source was located in a central position inside the detector, as depicted in figure 3.5, which we approximated as cylindrical. Table 3.4 shows the dimensions of the detector and the height from the base of the detector at which the source was placed.
The source then, decays as depicted in equation (3.3), producing electron neutrinos with the energies and branching rations shown in table 3.1. The produced electron neutrinos interact with the Gallium in the detector via the process (3.2). The extracted Ge was synthesized into the counting gas GeH4, mixed with Xe, and inserted into a very low-background proportional counter [82].
For the second experiment to test the SAGE detector, the SAGE collaboration used an 37Ar source. Among the avantages of using 37Ar instead of 51Cr as the source, there are the following [73]:
• the desired active isotope must be chemically separated from the target following irradiation’s, allowing the remotion of almost all the impurities that are present in the target, so that the 37Ar source results to be practically free of radioactive impurities;
• 37Ar has a longer half-life, giving longer time to prepare the source end to make measurements;
• the energy of the produced neutrinos is greater than in the case of 51Cr, giving a higher cross section;
• there is no emission of γ rays, so the required shielding is less than for 51Cr and the source can be very compact.
37Ar decays (τ1 2 = 35.04 ± 0.04 d) to 37Cl by the electron capture process e− + 1837Ar −→ 1737Cl + νe. (3.12)
In figure 3.6 and table 3.5, the decay process and the neutrino energy lines, with the corresponding branching ratios are shown.
The method used to produce the 37Ar source was irradiation of calcium oxide, following the neutron capture reaction 40Ca(n, α) 37Ar, performed in the reactor BN-600 at Zarechny, Russia [73].
The experimental procedure and equipment were basically the same as those used by the SAGE Cr source experiment, described previously, using the same experimental area depicted in figure 3.4, and the same remote handling system shown in figure 3.5. In addition, the activity of the 37Ar source was measured using similar techniques to those used for the chromium source, obtaining an average activity of A 37Ar = 15.13 PBq. (3.13)
For the experiment, the source was located in the center of the cylindrical detector, considering the dimensions and position which are written in table 3.4. The electron neutrinos produced in the argon decay interact with the Gallium in the detector pro-ducing Ge as in equation (3.2), and the resulting Ge was synthesized into the counting gas GeH4, mixed with inactive Xe and inserted into a proportional counter with a carbon-film cathode.
As in the GALLEX experiment, the measured and predicted (in absence of os-cillation) production rates are compared using (3.5). The results for the two SAGE experiments are presented in table 3.6.
Also here there is an anomalous value of the ratio R coming from the SAGE 37Ar experiment. In this case, the result has a deviation larger than 2σ, so the possible hint of electron neutrino disappearance is also motivated in this case.
The analysis of these experimental result is performed in the same way that for the GALLEX results, considering the geometrical configuration shown in figure 3.5, taking the detector with cylindrical shape with the dimensions shown in table 3.4.
Following the highest posterior density procedure as described previously, and using the theoretical ratio as equation (3.6) with the information of tables 3.1 and 3.5 (for 51Cr and 37Ar, respectively), the SAGE 51Cr experimental datum results in the upper limits for the oscillation parameters shown in the left panel of figure 3.7, while in the right panel of the same figure an allowed band at 2σ is shown from the analysis of the SAGE 37Ar datum.
The value of the ratio of measured and expected events (without oscillations) from SAGE 37Ar, which presents a deviation from unity larger than 2σ, results in an allowed region for a Δm2 larger or equal to 1 eV2, and by comparison with the allowed region resulting from the GALLEX Cr2 (right panel of figure 3.3), one can see that there is a large overlap for the 2σ bands, for Δm2 & 1 eV.

Table of contents :

1 Introduction 
2 Neutrino Physics 
2.1 The Standard Model
2.2 Masses of neutrinos
2.2.1 Dirac masses
2.2.2 Majorana masses
2.2.3 Dirac-Majorana masses
2.3 Neutrino Oscillations and Mixing
2.3.1 Two neutrino mixing
2.3.2 Experimental evidence
2.4 Sterile Neutrinos
3 Analysis of the Gallium, Bugey and Chooz experiments 
3.1 Gallium radioactive source experiments
3.1.1 GALLEX
3.1.2 SAGE
3.1.3 Combined analysis
3.2 Nuclear Reactor experiments
3.2.1 Bugey
3.2.2 Chooz
3.3 Summary
4 Other nuclear reactor experiments 
4.1 I.L.L
4.2 S.R.S
4.3 G¨osgen
4.4 Summary
5 Neutrinos in Cosmology 
5.1 introduction
5.2 Basics on Cosmology
5.3 Brief history of the Universe
5.3.1 Neutrino Decoupling
5.4 Energy density of neutrinos
5.5 Relativistic particles in the Universe
5.6 Constraints on a light non-thermal sterile neutrino
5.6.1 Physical effects and parametrization
5.6.2 Data for the Analysis
5.6.3 General analysis
5.6.4 Mass/temperature bounds in the thermal case
5.6.5 Mass bounds in the DW case
5.6.6 Comparison with previous work
5.7 Summary
6 Conclusions 
List of Tables
List of Figures


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