Neutrino Oscillation Experiments

  • Samoil Bilenky
Chapter
Part of the Lecture Notes in Physics book series (LNP, volume 947)

Abstract

A long period of the searching for neutrino oscillations started in 1970 with the Homestake solar neutrino radiochemical experiment by Davis et al. In this experiment, the observed rate of solar ν e was found to be 2-3 times smaller that the rate, predicted by the Standard Solar Model (SSM). This discrepancy was called the solar neutrino problem.

11.1 Introduction

A long period of the searching for neutrino oscillations started in 1970 with the Homestake solar neutrino radiochemical experiment by Davis et al. In this experiment, the observed rate of solar ν e was found to be two to three times smaller that the rate, predicted by the Standard Solar Model (SSM). This discrepancy was called the solar neutrino problem.

Before the Homestake experiment started, B. Pontecorvo suggested that because of neutrino oscillations the observed flux of the solar neutrinos might be two times smaller than the predicted flux.1 After the Davis results were obtained the idea of neutrino oscillations as a possible reason for the solar neutrino deficit became more and more popular.

In the eighties, the second solar neutrino experiment Kamiokande was performed. In this direct-counting experiment a large water-Cherenkov detector was used. The solar neutrino rate measured by the Kamiokande experiment was also smaller than the rate predicted by the SSM.

In the Homestake and Kamiokande experiments high-energy solar neutrinos, produced mainly in the decay of8B, were detected. The flux of these neutrinos is about 10−4 of the total solar neutrino flux and the predicted value of the flux depends on the model.

In the nineties new radiochemical solar neutrino experiments SAGE and GALLEX were performed. In these experiments neutrinos from all reactions of the solar pp and CNO cycles, including low-energy neutrinos from the reaction pp → de+ν e , were detected. This reaction gives the largest contribution to the flux of the solar neutrinos. The flux of the pp neutrinos can be predicted in a model independent way. The event rates measured in the SAGE and GALLEX experiments were approximately two times smaller than the predicted rates. Thus, in these experiments important evidence was obtained in favor of the disappearance of solar ν e on the way from the central region of the sun, where solar neutrinos are produced, to the earth.

Another indications in favor of neutrino oscillations were obtained in the nineties in the Kamiokande and IMB neutrino experiments in which atmospheric muon and electron neutrinos were detected. These neutrinos are produced in decays of pions and kaons, created in interactions of cosmic rays with nuclei in the atmosphere, and in decays of muons, which are produced in the decays of pions and kaons. It was found in these experiments that the ratio of the numbers of ν μ and ν e events is significantly smaller than the predicted (practically model independent) ratio.

On the other side, no indications in favor of neutrino oscillations were found in the eighties and nineties in numerous reactor and accelerator short baseline experiments.2

A first model independent evidence in favor of neutrino oscillations was obtained in 1998 in the water-Cherenkov Super-Kamiokande experiment. In this experiment a significant up-down asymmetry of the high-energy atmospheric neutrino muon events was observed. It was discovered that the number of up-going high-energy muon neutrinos, passing through the earth, is about two times smaller than the number of the down-going muon neutrinos coming directly from the atmosphere.

In 2002 in the SNO solar neutrino experiment evidence in the favor of the disappearance of solar ν e was obtained. In this experiment solar neutrinos were detected through the observation of CC and NC reactions. A Model independent evidence of the disappearance of solar ν e was obtained. It was shown that the flux of the solar ν e is approximately three times smaller than the flux of ν e , ν μ and ν τ .

In 2002 in the KamLAND reactor neutrino experiment a model independent evidence in favor of oscillations of reactor antineutrinos was obtained. In this experiment was found that the number of reactor \(\bar \nu _{e}\) events at the average distance of ∼180 km from the reactors is about 0.6 of the number of the expected events. In 2004 a significant distortion of the \(\bar \nu _{e}\) spectrum was observed in the KamLAND experiment.

All these experiments complete the first period of the brilliant discovery of neutrino oscillations. It was proven that neutrinos have small masses and that the flavor neutrinos ν e , ν μ , ν τ are “mixed particles”. All observed data can be described if we assume the three-neutrino mixing. The values of four neutrino oscillation parameters (two-mass squared differences and two mixing angles) were determined.

The muon neutrino disappearance were observed in the accelerator long-baseline K2K, MINOS and later T2K and NOvA experiments. These experiments confirm the results obtained in the pioneer atmospheric Super-Kamiokande experiment.

Oscillations of atmospheric and accelerator neutrinos (solar and reactor KamLAND neutrinos) are determined mainly by the large mass-squared difference \(\varDelta m_{A}^{2}\) and large mixing angle θ23 (small mass-squared difference \(\varDelta m_{S}^{2}\) and large mixing angle θ12). During many years from the reactor CHOOZ experiment only upper bound for the small mixing angle θ13 was known.

The angle θ13 determines subdominant \(\nu _{\mu }\leftrightarrows \nu _{e}\) oscillations of accelerator and atmospheric neutrinos and disappearance of reactor antineutrinos driven by the atmospheric mass-squared difference. First evidence in favor of \(\nu _{\mu }\leftrightarrows \nu _{e}\) oscillations was obtained in the accelerator T2K experiment. In 2012–2016 the angle θ13 was measured with high precision in the reactor Daya Bay, RENO and Double Chooz experiments. This was very important development in the investigation of a new phenomenon, neutrino oscillations. The way to the determination of the character of the neutrino mass spectrum (normal or inverted mass ordering?) and to the measurement of the CP phase δ was open.

In this chapter we will briefly discuss the major neutrino oscillation experiments.

11.2 Solar Neutrino Experiments

11.2.1 Introduction

Solar ν e ’s are produced in reactions of the thermonuclear pp and CNO cycles in which the energy of the sun is generated. The thermonuclear reactions are going on in the central, most hot region of the sun. In this region the temperature is about 15 ⋅ 106 K. At such a temperature the major contribution to the energy production is given by the pp cycle. The estimated contribution of the CNO cycle to the sun energy production is about 1%.3

The pp cycle starts with the pp and pep reactions
$$\displaystyle \begin{aligned} p+p\to d +e^{+}+ \nu_{e}\quad \mathrm{and}\quad p+e^{-}+p\to d + \nu_{e}.\end{aligned} $$
(11.1)
The pp reaction gives the dominant contribution to the deuterium production (99.77%). The contribution of the pep reaction is 0.23%.
Deuterium and proton produce 3He in the reaction
$$\displaystyle \begin{aligned} p+d\to ^{3}\mathrm{He}+\gamma.\end{aligned} $$
(11.2)
Nuclei 3He disappear due to the following three reactions
$$\displaystyle \begin{aligned} ^{3}\mathrm{He}+^{3}\mathrm{He}\to ^{4}\mathrm{He}+p+p \quad (84.92\%). \end{aligned} $$
(11.3)
$$\displaystyle \begin{aligned} ^{3}\mathrm{He}+p\to ^{4}\mathrm{He}+e^{+}+ \nu_{e} \quad (\mathrm{about} ~~10^{-5}\%) \end{aligned} $$
(11.4)
$$\displaystyle \begin{aligned} ^{3}\mathrm{He}+^{4}\mathrm{He}\to ^{7}\mathrm{Be}+\gamma\quad (15.08\%) \end{aligned} $$
(11.5)
In the first two reactions 4He is produced. Nuclei 7Be, produced in the third reaction, take part in two chains of reactions terminated by the production of 4He nuclei
$$\displaystyle \begin{aligned} ^{7}\mathrm{Be}+e^{-}\to {}^{7}\mathrm{Li}+\nu_{e},\quad ^{7}\mathrm{Li} +p\to ^{4}\mathrm{He}+^{4}\mathrm{He}. \end{aligned} $$
(11.6)
and
$$\displaystyle \begin{aligned} p+^{7}\mathrm{Be}\to ^{8}\mathrm{B}+\gamma,\quad ^{8}\mathrm{B}\to ^{8}\mathrm{Be}^{*}+e^{+}+ \nu_{e},\quad ^{8}\mathrm{Be}^{*}\to^{4}\mathrm{He}+^{4}\mathrm{He}. \end{aligned} $$
(11.7)
Positrons annihilate with electrons and produce photons. Thus, the energy of the sun is generated in the transition4
$$\displaystyle \begin{aligned} 4p+2 e^{-}\to ^{4}\mathrm{He}+2 \nu_{e}+ Q~, \end{aligned} $$
(11.8)
where
$$\displaystyle \begin{aligned} Q=4m_{p}+2m_{e}-m_{{}^{4}\mathrm{He}}\simeq 26.73~\mathrm{MeV} \end{aligned} $$
(11.9)
is the energy produced in the transition (11.8).5 From (11.8) follows that the production of \(\frac {1}{2} Q\simeq 13.36\) MeV is accompanied by the emission of one neutrino. Let us consider a neutrino with energy E. The production of such neutrino is accompanied by the emission of luminous energy equal to \(\frac {1}{2} Q-E\). If ϕ r (E) is the flux of neutrinos from the source r (r = pp,7Be,8B, …) on the earth, we have the following relation
$$\displaystyle \begin{aligned} \sum_{r}\int (\frac{1}{2} Q-E)~\phi_{r}(E)~dE=\frac{\mathscr{L}_\odot}{4\pi R^{2}}, \end{aligned} $$
(11.10)
where \(\mathscr {L}_\odot \) is the luminosity of the sun and R is the sun-earth distance.
The relation (11.10) is called luminosity relation. It is a general constraint on the fluxes of solar neutrinos. The luminosity relation is based on the following assumptions
  1. 1.

    The solar energy is of thermonuclear origin.

     
  2. 2.

    The sun is in a stationary state.

     
The last assumption is connected with the fact that neutrinos observed in a detector were produced about 8 min before the detection. On the other side it takes about 105 years for photons produced in the central region of the sun to reach its surface.
We can rewrite the luminosity relation in the form
$$\displaystyle \begin{aligned} \sum_r \left( \frac{ Q }{ 2 } - \overline{ E }_r \right) \varPhi_r = \frac{ \mathscr{L}_\odot }{ 4 \pi R^2 }. \end{aligned} $$
(11.11)
Here
$$\displaystyle \begin{aligned} \overline{ E }_r=\frac{1}{\varPhi_r}~ \int E~\phi_{r}(E)~dE \end{aligned} $$
(11.12)
is the average neutrino energy from the source r and \(\varPhi _r= \int \phi _{r}(E)~dE\) is the total flux of neutrinos from the source r.

The calculation of neutrino fluxes from different reactions can be done only in the framework of a solar model. Usually the results of the Standard Solar Model (SSM) calculations are used.6

In Table 11.1 we present SSM fluxes of ν e from different reactions. In this Table we included also SSM fluxes from the following reactions of the CNO cycle: 13N →13C + e+ + ν e , 15O →15N + e+ + ν e and 17F →17O + e+ + ν e . In the last column of Table 11.1 neutrino energies are given.
Table 11.1

Solar neutrino-producing reactions and SSM neutrino fluxes

Abbreviation

Reaction

SSM flux (cm−2 s−1)

Neutrino energy (MeV)

pp

p + p → d + e+ + ν e

5.97 (1 ± 0.006) ⋅ 1010

≤0.42

pep

p + e + p → d + ν e

1.41 (1 ± 0.011) ⋅ 108

1.44

7Be

e +7Be →7Li + ν e

5.07 (1 ± 0.06) ⋅ 109

0.86

8B

8B →8Be + e+ + ν e

5.94 (1 ± 0.11) ⋅ 106

\( \lesssim \)15

hep

3He + p →4He + e+ + ν e

7.90 (1 ± 0.15) ⋅ 103

≤ 18.8

13N

13N →13C + e+ + ν e

2.88 (1 ± 0.15) ⋅ 108

≤1.20

15O

15O →15N + e+ + ν e

2.15 (1 ± 0.17) ⋅ 108

≤ 1.73

17F

17F →17O + e+ + ν e

5.82 (1 ± 0.19) ⋅ 106

≤1.74

It is evident from Table 11.1 that the second term of the luminosity relation (11.10) is much smaller than the first one. If we neglect this term, we find the following estimate for the total flux of neutrinos
$$\displaystyle \begin{aligned} \varPhi =\sum_{r}\varPhi_r\simeq \frac{ \mathscr{L}_\odot }{ 2 \pi R^{2} Q }.\end{aligned} $$
(11.13)
Taking into account that \( \mathscr {L}_\odot = 2.40 \cdot 10^{39} \, \mathrm {MeV} \, \mathrm {s}^{-1} \) and R = 1.496 ⋅ 1013 cm we find
$$\displaystyle \begin{aligned} \varPhi\simeq 6 \cdot 10^{10} \, \mathrm{cm}^{-2} \, \mathrm{s}^{-1}~.\end{aligned} $$
(11.14)
In Fig. 11.1 predicted by the SSM spectra of neutrinos from different reactions are presented.
Fig. 11.1

Predicted by the Standard Solar Model spectra of solar neutrinos from different reactions

11.2.2 Homestake Chlorine Solar Neutrino Experiment

The pioneer experiment, in which solar electron neutrinos were detected, was the Homestake experiment by R. Davis et al.7 The experiment continued from 1968 till 1994.

In the Davis experiment radiochemical chlorine-argon method, proposed by B. Pontecorvo in 1946, was used. Solar electron neutrinos were detected through the observation of the reaction
$$\displaystyle \begin{aligned} \nu_e + {}^{37}\mathrm{Cl} \to e^- + {}^{37}\mathrm{Ar}~.\end{aligned} $$
(11.15)
The 37Ar atoms are radioactive. They decay via electron-capture with emission of Auger electrons. The half-life of the decay is 34.8 days.

A tank filled with 615 tons of liquid tetrachloroethylene (C2Cl4) was a part of the detector in the Davis experiment. In order to decrease the cosmic ray background, the experiment was performed in the Homestake mine (USA) at depth of about 1480 m (4100 m water equivalent). The radioactive 37Ar atoms, produced by solar ν e via the reaction (11.15) during the exposure time (about 2 months), were extracted from the tank by purging with 4He gas. About 16 atoms of 37Ar were extracted during one exposure run. The gas with radioactive 37Ar atoms was placed into a low-background proportional counter in which the signal (Auger electrons) was detected. An important feature of the experiment was the measurement of the rise time of the signal. This allowed to suppress the background.

The energy threshold of the Cl-Ar reaction is equal to 0.814 MeV, i.e. it is larger than the maximal energy of pp neutrinos, constituting the major part of the solar neutrinos flux (see Table 11.1). At high 8B energies the transition to an excited state of 37Ar significantly increase the cross section of the process (11.15). As a result, the dominant contribution to the counting rate give the high energy 8B neutrinos. The SSM contribution of the 8B neutrinos to the event rate is approximately equal to 5.8 SNU.8 The SSM contribution to the event rate of 7Be neutrinos is approximately equal to 1.2 SNU. Other much smaller contributions come from pep and CNO neutrinos.

The averaged over 108 runs (between 1970 and 1994) event rate, measured in the Homestake experiment, is equal to
$$\displaystyle \begin{aligned} R_{\mathrm{Cl}}= (2.56\pm 0.16 \pm 0.16)~~\mathrm{SNU}\end{aligned} $$
(11.16)
The measured event rate is significantly smaller than the rate predicted by the SSM (under the assumption that there are no neutrino oscillations):
$$\displaystyle \begin{aligned} (R_{\mathrm{Cl}})_{SSM}=8.6\pm 1.2~~\mathrm{SNU} \end{aligned} $$
(11.17)

11.2.3 Radiochemical GALLEX-GNO and SAGE Experiments

Neutrinos from all solar neutrino reactions, including low-energy neutrinos from the pp reaction, were detected in the radiochemical gallium GALLEX-GNO and SAGE experiments. In these experiments neutrinos were detected by the radiochemical method through the observation of the reaction
$$\displaystyle \begin{aligned} \nu_e + {}^{71}\mathrm{Ga} \to e^- + {}^{71}\mathrm{Ge}, \end{aligned} $$
(11.18)
in which radioactive 71Ge was produced. The threshold of this reaction is equal to 0.233 MeV. The half-life of 71Ge is equal to 11.43 days.

The detector in the GALLEX-GNO experiment was a tank containing 100 tons of a water solution of gallium chloride (30.3 tons of 71Ga). The experiment was done in the underground Gran Sasso Laboratory (Italy). During 1991–2003 there were 123 GALLEX and GNO exposure runs. The duration of one run was about 4 weeks. About 10 atoms of 71Ge were produced during one run. Radioactive 71Ge atoms were extracted from the detector by a chemical procedure and introduced into a small proportional counter in which Auger electrons, produced in the capture e +71Ge →71Ga + ν e , were detected.

The measured event rate averaged over 123 runs is equal to
$$\displaystyle \begin{aligned} R_{\mathrm{Ga}}=(67.5 \pm 5.1) ~\mathrm{SNU}. \end{aligned} $$
(11.19)
The SSM event rate
$$\displaystyle \begin{aligned} (R_{\mathrm{Ga}})_{\mathrm{SSM}}=(128~^{+9}_{-7})~\mathrm{SNU}. \end{aligned} $$
(11.20)
is about two times larger than the measured rate.

The major contribution to the SSM predicted event rate comes from the pp neutrinos (69.7 SNU). Contributions of 7Be and 8B neutrinos to the SSM event rate are equal to 34.2 SNU and 12.1 SNU, respectively.

In SAGE gallium experiment about 50 tons of 71Ga in the liquid metal form were used. The experiment was done in the Baksan Neutrino Observatory (Caucasus mountains, Russia) in a hall with an overburden of 4700 m of water equivalent. Neutrinos were detected through the observation of the reaction (11.18). An exposure time in this experiment was about 4 weeks. The 71Ge atoms, produced by the solar neutrinos, are chemically extracted from the target and are converted to GeH4. Auger electrons, produced in decay of germanium, were detected in a small proportional counter.

The germanium production rate, measured in the SAGE experiment, averaged over 92 runs (1990–2001) was equal to
$$\displaystyle \begin{aligned} R_{\mathrm{Ga}}=(70.8 ^{+5.3}_{-5.2}(\mathrm{stat})^{+3.7}_{-3.2}(\mathrm{syst})) ~\mathrm{SNU}. \end{aligned} $$
(11.21)
As it is seen from (11.19) and (11.21), the rates measured in the SAGE and in the GALLEX-GNO experiments are in a good agreement.

11.2.4 Kamiokande and Super-Kamiokande Solar Neutrino Experiments

In radiochemical experiments the neutrino direction can not be determined. The first experiment in which the neutrino direction was measured was Kamiokande. It was proved that the detected neutrinos were coming from the sun.

In the Kamiokande experiment a 3000 ton water-Cherenkov detector was used. The experiment was done in the Kamioka mine (Japan) at a depth of about 1000 m (2700 m water equivalent).

In the Kamiokande experiment the solar neutrinos were detected through the observation of recoil electrons in the elastic neutrino-electron scattering
$$\displaystyle \begin{aligned} \nu_{x} + e \to \nu_{x} + e. \quad (x=e,\mu,\tau) \end{aligned} $$
(11.22)
All types of flavor neutrinos could be detected via observation of the process (11.22). However, the cross section of νμ,τ − e scattering is significantly smaller than the cross section of ν e  − e scattering (σ(νμ,τe → νμ,τe) ≃ 0.16 σ(ν e e → ν e e)). Thus, mainly the flux of solar ν e was measured in the Kamiokande experiment.

Solar neutrinos were detected via the observation of the Cherenkov radiation of electrons in water. About 1000 large (50 cm in diameter) photomultipliers, which covered about 20% of the surface of the detector, were utilized in the experiment. Because of the contamination of Rn in the water it was necessary to apply a 7.5 MeV energy threshold for the recoil electrons.

At high energies recoil electrons are emitted in a narrow (about 15) cone around the initial neutrino direction. In the experiment a strong correlation between the direction of recoil electrons and the direction to the sun was observed This correlation was an important signature which allowed to suppress background and to prove that the observed events were due to solar neutrinos.

Because of the high threshold mainly 8B neutrinos were detected in the Kamiokande experiment. The total flux of high energy 8B neutrinos obtained from the data of the Kamiokande experiment was equal to
$$\displaystyle \begin{aligned} \varPhi_{\nu}^{\mathrm{K}} = ( 2.80\pm 0.19 \pm 0.33) \cdot 10^6 \, \mathrm{cm}^{-2} \, \mathrm{s}^{-1}. {} \end{aligned} $$
(11.23)
The ratio of the measured solar neutrino flux to the flux predicted by the SSM (under the assumption that there are no neutrino oscillations) was equal to RK = 0.51 ± 0.04 ± 0.06.

The Kamiokande result was an important confirmation of the existence of the solar neutrino problem, discovered in Davis et al. in the Homestake experiment.9

The Kamiokande experiment was running during 9 years from 1987 till 1995. In 1996 the Super-Kamiokande, experiment of the next generation, started. In this experiment a huge 50 kton water-Cherenkov detector (fiducial volume 22.5 kton) was used.

There were four phases of the Super-Kamiokande experiment. The SK-I phase started in 1996 and finished in 2001. In this phase 11,146 photomultipliers (PMT) were used. In 2001 an accident happened in which about 60% of PMTs were destroyed. After about a year, the data-taking started with 5182 photomultipliers (SK-II). This phase finished in 2005. In 2006 SK-III started with 11,129 PMTs. The fourth phase of the experiment started in 2006 and finished in 2014.

During the SK-I phase the threshold for the kinetic energy of the recoil electrons was 6 MeV (first 280 days) and 4.5 MeV for the remaining days. In the SK-IV phase the recoil electron threshold was 3.49 MeV. Due to high threshold only 8B and hep neutrinos where detected in the Super-Kamiokande experiment.

Improvement in the electronics, in the water circulation system, in calibration and in methods of analysis allowed to reach in the SK-IV much smaller systematic uncertainty than during other phases of the experiment. The measured in the SK-IV flux of the solar neutrinos is equal to
$$\displaystyle \begin{aligned} \varPhi_{\nu}^{\mathrm{SK-IV}} = ( 2.308\pm 0.020~(\mathrm{stat.})\pm 0.039~(\mathrm{syst.}) \cdot 10^6 \, \mathrm{cm}^{-2} \, \mathrm{s}^{-1}. \end{aligned} $$
(11.24)
Combining the results of all Super-Kamiokande phases it was found
$$\displaystyle \begin{aligned} \varPhi_{\nu}^{\mathrm{SK}} = ( 2.345\pm 0.014~(\mathrm{stat.})\pm 0.036~(\mathrm{syst.}) \cdot 10^6 \, \mathrm{cm}^{-2} \, \mathrm{s}^{-1}. \end{aligned} $$
(11.25)
Due to the earth matter effect the fluxes of solar neutrinos during day and during night must be different (in the night flux it must be more ν e than in the day flux). The high statistics of the events allowed the Super-Kamiokande Collaboration to measure the day-night asymmetry. In the SK-IV it was found
$$\displaystyle \begin{aligned} A_{\mathrm{D}-\mathrm{N}}=(-3.6 \pm 1.6 \pm 0.6) \% \end{aligned} $$
(11.26)
No distortion of the spectrum of recoil electrons with respect to the expected spectrum was observed.10 From analysis of the all SK data for the solar neutrino mass-squared difference the following value was obtained
$$\displaystyle \begin{aligned} \varDelta m^{2}_{S}=(4.8^{+1.5}_{-0.8})\cdot 10^{-5}~\mathrm{eV}^{2}. \end{aligned} $$
(11.27)
From the SK-IV data for the parameter \(\sin ^{2}\theta _{12}\) it was found
$$\displaystyle \begin{aligned} \sin^{2}\theta_{12}=0.327^{+0.026}_{-0.031}. \end{aligned} $$
(11.28)
From the analysis of all SK data it was obtained
$$\displaystyle \begin{aligned} \sin^{2}\theta_{12}=0.334^{+0.027}_{-0.023}. \end{aligned} $$
(11.29)
Finally, from the fit of all solar and KamLAND data (assuming that the parameter \(\sin ^{2}\theta _{13}\) is given by the reactor value (see later)) it was found
$$\displaystyle \begin{aligned} \varDelta m^{2}_{S}=(7.49^{+0.19}_{-0.18})\cdot 10^{-5}~\mathrm{eV}^{2},\quad \sin^{2}\theta_{12}=0.307^{+0.013}_{-0.012}. \end{aligned} $$
(11.30)

11.2.5 SNO Solar Neutrino Experiment

The fluxes of solar neutrinos, measured in the Homestake, GALLEX-GNO, SAGE, Kamiokande and Super-Kamiokande experiments, were significantly smaller than the fluxes, predicted by the Standard Solar model. From the analysis of the data of these experiments, strong indications in favor of neutrino transitions in matter, driven by neutrino masses and mixing, were obtained.

The first model-independent evidence for transitions of solar ν e into ν μ and ν τ was obtained in the SNO solar neutrino experiment. The SNO detector was located in the Creighton mine (Sudbury, Canada) at a depth of 2092 m (5890 ± 94 m water equivalent). The detector consisted of the transparent acrylic vessel (a sphere, 12 m in diameter) containing 1 kton of pure heavy water D2O. About 7 kton of H2O shielded the vessel from external radioactive background. An array of 9456 PMTs detected Cherenkov radiation produced in the D2O and H2O.

A crucial feature of the SNO experiment was the detection of the solar neutrinos via three different processes.
  1. 1.
    The CC process
    $$\displaystyle \begin{aligned} \nu_e + d \to e^- + p + p ~. \end{aligned} $$
    (11.31)
     
  2. 2.
    The NC process
    $$\displaystyle \begin{aligned} \nu_x + d \to \nu_x + p + n \quad (x=e,\mu,\tau) \end{aligned} $$
    (11.32)
     
  3. 3.
    Elastic neutrino-electron scattering (ES)
    $$\displaystyle \begin{aligned} \nu_x + e \to \nu_x + e~. \end{aligned} $$
    (11.33)
     
The CC and ES processes were observed through the detection of the Cherenkov light produced by electrons in the heavy water. The NC process was observed via the detection of neutrons. There were three phases of the SNO experiment in which different methods of the detection of neutrons were used.
  • The NC neutrons were captured in D2O and produced 6.25 MeV γ-quanta in the reaction n + d →3H + γ. During Phase I the Cherenkov light of secondary Compton electrons and e+ − e pairs was detected.

  • In Phase II of the SNO experiment about two tons of NaCl were dissolved in the heavy water. Neutrons were detected through the observation of γ-quanta from the capture of neutrons by 35Cl nuclei. For thermal neutrons the cross section of this process is equal to 44 barn while the cross section of the process nd →3Hγ is equal to 0.5 mb. Thus, the addition of the salt significantly enhanced the NC signal.

  • In Phase III an array of proportional counters was deployed in the heavy water. Neutrons were detected through the observation of the reaction n +3He → p +3H in which proton and 3H had a total kinetic energy 0.76 MeV. Charged particles in the proportional counters produced ionization electrons and the induced by them voltage was recorded as a function of time. This technic allowed to reduce background significantly.

The SNO Collaboration started to collect data in 1999. The last phase was finished in 2006. The SNO threshold for the detection of the electrons from the CC and the ES processes was equal to Tthr = 5.5 MeV. The neutrino energy threshold for NC process is 2.2 MeV (the deuterium bounding energy). Thus, in the SNO experiments mostly high energy solar 8B neutrinos were detected.

The initial spectrum of ν e from the 8B decay is known. It was obtained from the measurement of α-spectrum from the 8B decay. The SNO and other solar neutrino data are compatible with the assumption that in the high-energy 8B region ν e  → ν e survival probability is a constant.

From the observation of the CC events for the flux ν e the following value was obtained
$$\displaystyle \begin{aligned} \varPhi^{CC}_{\nu_{e}} = ( 1.68\pm 0.06~ (\mathrm{stat.}) {}^{+0.08}_{-0.09}~(\mathrm{syst.}) ) \cdot 10^6 \, \mathrm{cm}^{-2} \, \mathrm{s}^{-1}. {} \end{aligned} $$
(11.34)
Because of the ν e  − ν μ  − ν τ universality of the NC neutrino-hadron interaction the observation of NC events allows to determine the total flux of all flavor neutrinos. In the SNO experiment was found that the total flux of all flavor neutrinos is equal to
$$\displaystyle \begin{aligned} \varPhi^{NC}_{\nu_{e,\mu,\tau}}=( 5.25\pm 0.16~ (\mathrm{stat.}) {}^{+0.11}_{-0.13}~(\mathrm{syst.}) ) \cdot 10^6 \, \mathrm{cm}^{-2} \, \mathrm{s}^{-1} ~. \end{aligned} $$
(11.35)
The value (11.35) of the total flux of all flavor neutrinos is in agreement with the flux of the 8B neutrinos predicted by the Standard Solar Model (see Table 11.1).

The SNO experiment solved the solar neutrino problem. If we compare the flux of ν e with the total flux of ν e , ν μ and ν τ , we come to the model independent conclusion that solar ν e on the way from the sun to the earth are transformed into ν μ and ν τ .

From the three-neutrino analysis of the SNO and other solar neutrino data and also data of the reactor KamLAND experiment for the neutrino oscillation parameters the following values were obtained
$$\displaystyle \begin{aligned} \varDelta m^{2}_{S}=(7.41^{+0.21}_{-0.19})\cdot 10^{-5}~\mathrm{eV}^{2},~~ \tan^{2}\theta_{12}=0.446^{+0.030}_{-0.029},~~ \sin^{2}\theta_{13}=2.5^{+1.8}_{-1.5}\cdot 10^{-2}. \end{aligned} $$
(11.36)

11.2.6 Borexino Solar Neutrino Experiment

Due to very low background in the Borexino solar neutrino experiment low energy pp, 7Be and pep neutrinos and also high energy 8B neutrinos were detected.

The Borexino detector is located in the Gran Sasso underground laboratory (Italy). It consists of concentric shells of increasing radio-purity. The Inner Detector is a nylon vessel which contains 280 tons of liquid scintillator. It is surrounded by layers of buffer liquid and highly purified water which allows to suppress background of cosmic muons.

The Borexino collaboration are taking data since 2007. In the Phase I of the experiment (2207–2010) rates of solar pp, 7Be and pep neutrinos were measured and a bound on the flux of CNO neutrinos was obtained. In the Phase II, started in 2011, the rate of pp neutrinos was determined.

The solar neutrinos are observed in the Borexino experiment through the detection of recoil electrons from the elastic neutrino-electron scattering
$$\displaystyle \begin{aligned} \nu_x +e \to \nu_x +e,\quad x=e,\mu,\tau.\end{aligned} $$
(11.37)
The scintillation light is detected by 2212 PMTs uniformly distributed on the inner surface of the detector. The measurement of the scintillation light allows to determine the energy of the electrons. There is no information about the direction of the electrons. Because the energy threshold in the Borexino experiment must be low, the major requirement is an extremely low radioactive contamination of the scintillator (9–10 order of magnitude lower than the natural radioactivity).
In the reaction 7Be + e7Li + ν e neutrino with the energy 0.86 MeV is produced. The signature of 0.86 MeV ν e is a characteristic Compton-like spectrum with a shoulder at about 660 keV. The 7Be rate was obtained from the fit of the data in which contributions of decays of 85Kr, 210Bi, 11C and 210Po were taken into account. For the interaction rate of the 7Be neutrinos it was found the value
$$\displaystyle \begin{aligned} R_{{}^{7}\mathrm{Be}}=(46\pm 1.5~(\mathrm{stat.})\pm 1.5 (\mathrm{syst.}))~\mathrm{cpd}/(100 ~\mathrm{tons}),\end{aligned} $$
(11.38)
where cpd ≡counts/(day). Assuming that the probability of 7Be electron neutrino to survive is given by the standard MSW value for the flux of 7Be neutrinos from (11.38) we find
$$\displaystyle \begin{aligned} \varPhi^{{}^{7}\mathrm{Be}}_{\nu_{e}}=(4.43\pm 0.22)\cdot 10^{9}~\mathrm{cm}^{-2}\,\mathrm{s}^{-1}. \end{aligned} $$
(11.39)
The pep neutrinos have an energy 1.44 MeV. Taking into account the major background from the 11C decay in the Borexino experiment for the pep rate was obtained
$$\displaystyle \begin{aligned} R_{\mathrm{pep}}=(3.1\pm 0.6~(\mathrm{stat.})\pm 0.3 (\mathrm{syst.}))~\mathrm{cpd}/(100~ \mathrm{tons}), \end{aligned} $$
(11.40)
For the flux of pep neutrinos it was found
$$\displaystyle \begin{aligned} \varPhi^{\mathrm{pep}}_{\nu_{e}}=(1.63\pm 0.35)\cdot 10^{8}~\mathrm{cm}^{-2}\,\mathrm{s}^{-1}. \end{aligned} $$
(11.41)
From the fit of the data (in which the pep rate was fixed at the value (11.40)) for the rate of the CNO neutrinos the following upper bound was obtained
$$\displaystyle \begin{aligned} R_{\mathrm{CNO}}<7.4~\mathrm{cpd}/(100~ \mathrm{tons}). \end{aligned} $$
(11.42)
This bound implies the following bound for the CNO neutrinos
$$\displaystyle \begin{aligned} \varPhi^{\mathrm{CNO}}_{\nu_{e}}<7.7 \cdot 10^{8}~\mathrm{cm}^{-2}\,\mathrm{s}^{-1}. \end{aligned} $$
(11.43)
The flux of the low energy pp neutrinos (the maximal neutrino energy is 420 keV) is the major flux of the solar neutrinos. In the Phase II of the Borexino experiment the rate of the pp neutrinos was determined. The main problem was a background from the 14C β decay. Taking into account in the fit of the data also decays of other background nuclei (210Po, 210Bi and others) the following rate of the pp neutrinos was obtained
$$\displaystyle \begin{aligned} R_{\mathrm{pp}}=(144\pm 13~(\mathrm{stat.})\pm 10 (\mathrm{syst.}))~\mathrm{cpd}/(100~ \mathrm{tons}). \end{aligned} $$
(11.44)
From (11.44) the following flux of the pp neutrinos was inferred
$$\displaystyle \begin{aligned} \varPhi^{\mathrm{pep}}_{\nu_{e}}=(6.6\pm 0.7)\cdot 10^{10}~\mathrm{cm}^{-2}\,\mathrm{s}^{-1}. \end{aligned} $$
(11.45)
This flux is in an agreement with the flux predicted by SSM.
If we use the SSM flux, from (11.44) for the probability of the low energy pp neutrinos to survive we obtain the value
$$\displaystyle \begin{aligned} P(\nu_{e}\to \nu_{e})=0.64\pm 0.12. \end{aligned} $$
(11.46)
In the Borexino experiment the rate of the high-energy 8B neutrinos was also determined. The threshold for recoil electron energy in this experiment was equal to 3 MeV. For the rate 8B neutrinos it was found
$$\displaystyle \begin{aligned} R_{{}^{8}\mathrm{B}}=(0.22\pm 0.04~(\mathrm{stat.})\pm 0.01 (\mathrm{syst.}))~\mathrm{cpd}/(100~ \mathrm{tons}). \end{aligned} $$
(11.47)
The flux of 8B neutrinos, determined from (11.47)
$$\displaystyle \begin{aligned} \varPhi^{{}^{8}\mathrm{B}}_{\nu_{e}}=(5.2\pm 0.3)\cdot 10^{6}~\mathrm{cm}^{-2}\,\mathrm{s}^{-1}. \end{aligned} $$
(11.48)
is in agreement with Super-Kamiokande and SNO data.
The results obtained in the Borexino experiment allow to obtain the ν e survival probability of pp, 7Be, pep and 8B solar neutrinos (see Fig. 11.2). As it is seen from Fig. 11.2 the Borexino data are in agreement with MSW prediction. Up to now no indications in favor of a non-standard physics were obtained.
Fig. 11.2

Borexino experiment: ν e survival probability of pp, 7Be, pep and 8B solar neutrinos. Curve is MSW prediction (arXiv:1707.9279)

11.3 Super-Kamiokande Atmospheric Neutrino Experiment

The Super-Kamiokande is a multi-purpose detector. In the previous section we considered the Super-Kamiokande solar neutrino experiment. In this section we will consider the Super-Kamiokande atmospheric neutrino experiment. In the Super-Kamiokande atmospheric neutrino experiment the first model independent evidence in favor of neutrino oscillations was obtained (1998). This discovery opened a new era in the study of the problem of neutrino masses, mixing and oscillations.

The Super-Kamiokande detector is a 50 kton water Cerenkov detector which is optically separated into inner detector ID (32 kton, fiducial volume 22.5 kton) viewed by 11,146 inward-facing 50 cm PMTs and outer detector OD with 1885 20 cm PMTs which is used as a veto for events induced by the cosmic rays. Cherenkov radiation produced by charged particles, traveling through detector, is collected by PMTs.

The Super-Kamiokande atmospheric neutrino experiment started in April 1966. There were four phases of the experiment. The SK-I phase started in April 1996 and finished in November 2001 when the accident with photo-tubes happened. The SK-II phase, with only half of PMTs operating, continued from 2002 till 2005. In 2006 after the total number of PMTs was restored, the SK-III phase started. This phase finished in 2008 when a new phase SK-IV with upgraded electronics began.

In the Super-Kamiokande atmospheric neutrino experiment neutrinos (and antineutrinos) are detected in a wide range of energies from about 100 MeV to about 10 TeV and distances from about 10 km to about 13,000 km. Atmospheric neutrinos originate from the decays of pions and kaons, produced in the processes of interaction of cosmic rays with nuclei of the atmosphere, and consequent decays of muons. Neutrinos with energies \(\lesssim \)5 GeV are produced mainly in the decays of pions and muons
$$\displaystyle \begin{aligned} \pi^{\pm}\to \mu^{\pm}+\nu_{\mu}(\bar\nu_{\mu}),\quad \mu^{\pm}\to e^{\pm}+ \bar\nu_{\mu}(\nu_{\mu})+\nu_{e}(\bar\nu_{e}) \end{aligned} $$
(11.49)
At higher energies the contribution of kaons becomes also important.
Neutrinos and antineutrinos are detected through the observation of electrons and muons produced in the CC processes
$$\displaystyle \begin{aligned} \nu_{l}(\bar \nu_{l}) +N\to l^{-}(l^{+}) +X \quad (l=e,\mu). \end{aligned} $$
(11.50)
Atmospheric neutrino events are divided into three categories
  • Fully contained events (FC). Events are called FC if initial vertexes are in the ID fiducial volume and all energies are deposited in the inner detector. Such events are separated into two samples: sub-GeV (E ≤ 1.33 GeV) and multi-GeV (E > 1.33 GeV).

  • Partially contained events (PC). If a high energy muon escapes the inner detector and deposits part of its energy in the outer veto detector such an event is called a PC event.

  • Upward going muons (Upμ). Upward going muon events are due to interaction of muon neutrinos in the rock outside of the Super-Kamiokande detector which produce muons entering into the detector from below. There are two categories of such events. Upward stopping muons are those muons which come to rest in the detector. Upward through-going muons are those muons which pass the whole detector.

FC events are produced by neutrinos with energies of a few GeV. PC events are produced by neutrinos with energies about an order of magnitude higher. The energies of neutrinos which produce upward stopping muons is about 10 GeV. Upward through-going muons are produced by neutrinos with an average energy of about 100 GeV.

During four phases of the Super-Kamiokande experiment it was observed 10,386 (10,493) sub-GeV μ (e)-like FC events, 4370 (4076) multi-GeV μ (e)-like FC events and 3003 PC events.

A model-independent evidence in favor of neutrino oscillations was obtained by the Super-Kamiokande Collaboration through the investigation of the zenith-angle dependence of the atmospheric electron and muon events. The zenith angle θ is determined in such a way that neutrinos going vertically downward have θ = 0 and neutrinos coming vertically upward through the earth have θ = π. Because of the geomagnetic cutoff at small energies (0.3–0.5 GeV) the flux of downward going neutrinos is lower than the flux of upward going neutrinos. At neutrino energies E ≥ 0.9 GeV the fluxes of muon and electron neutrinos are symmetric under the change θ → π − θ. Thus, if there are no neutrino oscillations at high energies the numbers of electron and muon events must satisfy the relation
$$\displaystyle \begin{aligned} N_{l}(\cos\theta)=N_{l}(-\cos\theta)\quad l=e,\mu. \end{aligned} $$
(11.51)
A significant violation of this relation was found in the Super-Kamikande experiment.

For the study of flavor neutrino oscillations it is crucial to distinguish electrons and muons produced in the processes (11.50). In the Super-Kamiokande experiment leptons are observed through the detection of the Cherenkov radiation. The shapes of the Cherenkov rings of electrons and muons are completely different. In the case of electrons the Cherenkov rings exhibit a more diffuse light than in the muon case. The probability of a misidentification of electrons and muons is below 2%.

First indication in favor of neutrino oscillations came from the measurement of the ratio r of the \((\nu _{\mu }+ \bar \nu _{\mu })\) and \((\nu _{e}+ \bar \nu _{e})\) fluxes. This ratio can be predicted with an accuracy of about 3%. In the SK-I phase, for the double ratio \(R=\frac {r_{\mathrm {meas}}}{r_{\mathrm {MC}}}\) (rmeas is the measured and rMC is the predicted ratios) following value was obtained in the sub-GeV region
$$\displaystyle \begin{aligned} R_{\mathrm{sub-GeV}}= 0.658\pm 0.016\pm 0.035~. \end{aligned} $$
(11.52)
In the multi-GeV region was found
$$\displaystyle \begin{aligned} R_{\mathrm{multi-GeV}}= 0.702\pm 0.032\pm 0.101~. \end{aligned} $$
(11.53)
If there are no neutrino oscillations the double ratio R must be equal to one.
The most important Super-Kamiokande result was obtained from the measurement of the zenith-angle distribution of the electron and muon events. The latest results of the measurement of these distributions are presented in Fig. 11.3. As is seen from Fig. 11.3 the distributions of sub-GeV and multi-GeV electron events are in agreement with the expected distributions. In the distributions of the sub-GeV and multi-GeV muon events and upward stopping muon events a significant deficit of upward-going muons is observed.
Fig. 11.3

Super-Kamiokande atmospheric neutrino experiment: zenith angle dependence of the numbers of electron and muon events. The MC prediction assuming that there are no oscillations and distribution of events obtained with best-fit values of the oscillation parameters are shown (arXiv:1412.5234v1 [hep-ex])

This result can be explained by the disappearance of muon neutrinos due to neutrino oscillations. As we have seen before, in the case of neutrino oscillations the probability of ν μ to survive depends on the distance between neutrino source and neutrino detector. Downward going neutrinos (θ ≃ 0) pass a distance of about 10–20 km. On the other side upward going neutrinos (θ ≃ π) pass a distance of about 13,000 km (earth diameter). The measurement of the dependence of the numbers of the electron and muon events on the zenith angle θ allows to span distances from about 10 km to about 13,000 km. The energies of the atmospheric neutrinos are in the range 100 MeV–100 GeV Such wide ranges of energies and distances allow the Super-Kamikande Collaboration to study neutrino oscillations in details.

The Super-Kamiokande data can be explained by the ν μ (and \(\bar \nu _{\mu }\)) disappearance due to dominant \(\nu _{\mu }\leftrightarrows \nu _{\tau }\) oscillations. Taking into account that \(\varDelta m_{S}^{2}\ll \varDelta m_{S}^{2}\) and neglecting small contribution of \(\sin ^{2}\theta _{13}\) we have
$$\displaystyle \begin{aligned} \mathrm{P}(\nu_{\mu}\to \nu_{\mu})=\mathrm{P}(\bar\nu_{\mu}\to \bar\nu_{\mu})=1 -\sin^{2}2\theta_{23}~\sin^{2}\frac{\varDelta m_{A}~L}{4E}. \end{aligned} $$
(11.54)
From analysis of the Super-Kamiokande data, obtained during SK-I phase, the following 90% CL ranges of the neutrino oscillation parameters were obtained
$$\displaystyle \begin{aligned} 1.5\cdot 10^{-3}<\varDelta m_{A}^{2}< 3.4\cdot 10^{-3}~\mathrm{eV}^{2},\quad \sin^{2}2\theta_{23}>0.92. \end{aligned} $$
(11.55)
In the standard Super-Kamiokande analysis of the data the dependence of the probability on \(\frac {L}{E}\) is practically washed out because of the poor resolution. In order to reveal the oscillatory behavior of the probability, the Super-Kamiokande Collaboration performed a special analysis. A subset of events with high resolution in the variables L and E was chosen for the analysis. This allowed to determine the ν μ survival probability as a function of \(\frac {L}{E}\) and to reveal the first minimum of the survival probability (see Fig. 11.4). It is seen from Fig. 11.4 that the minimum of the survival probability is reached at
$$\displaystyle \begin{aligned} \left(\frac{L}{E}\right)_{\mathrm{min}}=\frac{2\pi\hbar c}{\varDelta m_{A}^{2}c^{4}}\simeq 500 ~\frac{\mathrm{km}}{\mathrm{GeV}} \end{aligned} $$
(11.56)
From this number we can estimate the neutrino mass-squared difference:
$$\displaystyle \begin{aligned} \varDelta m_{A}^{2}\simeq 2.5 \cdot 10^{-3}~\mathrm{eV}^{2}. \end{aligned} $$
(11.57)
This value is in agreement with (11.55).
Fig. 11.4

Values of the probability P(ν μ  → ν μ ) as a function of the parameter \( \frac {L}{E}\), determined from the data of the Super-Kamiokande atmospheric neutrino experiment. The best-fit two-neutrino oscillation curve is also plotted (arXiv:hep-ex/0404034)

The detection in the Super-Kamiokande detector of ν τ , produced in \(\nu _{\mu }\leftrightarrows \nu _{\tau }\) oscillations, is a difficult problem.11 This is connected with the fact the threshold for production of τ in CC ν τ  − N processes is about 3.5 GeV and the majority of the atmospheric neutrinos have energies which are below of this threshold. Nevertheless the Super-Kamiokande Collaboration by special analysis of the data, obtained during first three phases, found \(180.1\pm 44.3~(\mathrm {stat})^{+17.8}_{-15.2}~(\mathrm {syst})\) ν τ -interactions. This result confirm ν τ appearance at 3.8 σ level.

Finally from the three-neutrino analysis of the SK-I, SK-II and SK-III data for the NO (IO) neutrino mass spectrum the following 90% CL intervals were found
$$\displaystyle \begin{aligned} \begin{array}{rcl}{} (1.9 ~(1.7)&\displaystyle <&\displaystyle \varDelta m_{A}^{2}< 2.6 ~(2.7))\cdot 10^{-3}~\mathrm{eV}^{2}, ~~0.407< \sin^{2}\theta_{23}< 0.583,\\ \sin^{2}2\theta_{13}&\displaystyle <&\displaystyle 0.04~ (0.09). \end{array} \end{aligned} $$
(11.58)

11.4 KamLAND Reactor Neutrino Experiment

In the KamLAND reactor experiment oscillations of reactor \(\bar \nu _{e}\), driven by the solar mass-squared difference \(\varDelta m_{12}^{2}\equiv \varDelta m_{S}^{2}\) were observed. The KamLAND detector is located in the Kamioka mine (Japan) at a depth of about 1 km (2700 m water equivalent). It contains a 13 m-diameter transparent spherical nylon balloon filled with a 1 kton liquid scintillator. The balloon is suspended in 1800 m3 non-scintillating purified mineral buffer oil. The internal detector (balloon and buffer oil) is contained in a 18 m-diameter stainless steel spherical vessel. On the inner surface of the vessel there are 1879 50-cm diameter PMTs (the PMT coverage is 34%). The internal detector is surrounded by 3.2 kton water with 225 PMTs (outer detector). This Cherenkov detector serves as a veto which provides shielding from cosmic-ray muons and external radioactivity.

In the KamLAND experiment \(\bar \nu _{e}\) from about 50 power reactors were detected. A flux-averaged distance between reactors and the Kamioka mine was ∼180 km. About 80% of antineutrinos came from 26 reactors at distances 138–214 km.

Reactor \(\bar \nu _{e}\)s are produced in decays of nuclei, which are products of fission of 235U (57%), 238U (7.8%), 239Pu (29.5%) and 241Pu (5.7%). Each fission, in which about 200 MeV is produced, is accompanied by the emission of 6 \(\bar \nu _{e}\). A reactor with power about 3 GW th emits about \(6\cdot 10^{20}~\bar \nu _{e}/\mathrm {s}\).

Reactor antineutrinos are detected through the observation of the inverse β-decay
$$\displaystyle \begin{aligned} \bar\nu _{e}+p \to e^{+}+n. \end{aligned} $$
(11.59)
The threshold of this process is 1.8 MeV. Two γ-quanta from the annihilation of e+ (prompt signal) and γ-quanta produced in a neutron capture by a proton or a 12C nuclei (delayed signal with a mean delay time (207.5 ± 2.8) ⋅μs) are detected in the experiment. The signature of the event in the KamLAND experiment (and in other reactor neutrino experiments) is a coincidence between the prompt and delayed signals. It provides a strong suppression of a radioactive background.
The prompt energy E p is connected with the neutrino energy E by the relation
$$\displaystyle \begin{aligned} E\simeq E_{p}+\bar E_{n} +0.8~\mathrm{MeV}. \end{aligned} $$
(11.60)
where \(\bar E_{n}\) is average neutron recoil energy (≃10 keV). The prompt energy is the sum of the positron kinetic energy and the annihilation energy (2 m e ).

In the KamLAND experiment not only \(\bar \nu _{e}\) from reactors but also \(\bar \nu _{e}\), which are produced in decay chains of 238U and 232Th in the earth (geo-neutrinos), are detected. The prompt energy released in the interaction of geo-neutrinos with protons is less than 2.6 MeV. In order to avoid geo-neutrino background in the study of neutrino oscillations the KamLAND Collaboration imposed 2.6 MeV E p -threshold.

As we have seen before, the neutrino oscillation length is given by the expression
$$\displaystyle \begin{aligned} L_{12}\simeq 2.5 ~\frac{E}{\varDelta m_{12}^{2}}~\mathrm{m}, \end{aligned} $$
(11.61)
where E is the neutrino energy in MeV and \(\varDelta m_{12}^{2}\) is the neutrino mass-squared difference in eV2. The average energy of the reactor antineutrinos is 3.6 MeV. For the solar neutrino mass-squared difference \(\varDelta m_{12}^{2}\simeq 8\cdot 10^{-5}~\mathrm {eV}^{2}\) at E = 3.6 MeV we have L12 ≃ 120 km. From this estimate we conclude that distances between the KamLAND detector and Japanese reactors are appropriate to study neutrino oscillations driven by the solar neutrino mass-squared difference.

The KamLAND experiment started in 2002 and continued up to 2012. During this period 2611 \(\bar \nu _{e}\) events were detected with an estimated background (excluding geo-neutrinos) 364.1 ± 30.5 events. Expected number of the events (assuming that there are no neutrino oscillations) is 3564 ± 145.

From the latest 3-neutrino analysis of all KamLAND data for the neutrino oscillation parameters \(\varDelta m_{S}^{2}\), \(\tan ^{2}\theta _{12}\) and \(\sin ^{2}\theta _{13}\) the following values were obtained
$$\displaystyle \begin{aligned} \varDelta m_{S}^{2}=(7.54^{+0.19}_{-0.18})\cdot 10^{-5}~\mathrm{eV}^{2},~ \tan^{2}\theta_{12}=0.481^{+0.092}_{-0.080},~\sin^{2}\theta_{13}=0.010^{+0.033}_{-0.034}. \end{aligned} $$
(11.62)
From a joint analysis of the KamLAND data and the data of all solar neutrino experiments a better accuracy for the parameter \(\tan ^{2}\theta _{12}\) can be inferred12:
$$\displaystyle \begin{aligned} \varDelta m_{S}^{2}=(7.53^{+0.19}_{-0.18})\cdot 10^{-5}~\mathrm{eV}^{2},~ \tan^{2}\theta_{12}=0.437^{+0.029}_{-0.026},~\sin^{2}\theta_{13}=0.023^{+0.015}_{-0.015}. \end{aligned} $$
(11.63)
In Fig. 11.5 the ratio of the numbers of the observed and expected events (the \(\bar \nu _{e}\) survival probability) as a function of L0/E is plotted (L0 = 180 km is the flux-averaged distance between reactors and the KamLAND detector, E is the neutrino energy). The curve is the calculated survival probability with best-fit parameters obtained from the three-neutrino analysis of the KamLAND data. Figure 11.5 illustrates oscillatory behavior of the \(\bar \nu _{e}\) survival probability determined from the data of the KamLAND experiment.
Fig. 11.5

Ratio of \(\bar \nu _{e}\) spectrum measured in the KamLAND experiment to the spectrum, expected in the case of no oscillations as a function of \( \frac {L_{0}}{E}\) (L0 = 180 km is flux-weighted average distance from reactors to the detector). The expected ratio calculated with the values of the oscillation parameters obtained by the KamLAND collaboration is also shown (arXiv:0801.4589)

11.5 Measurement of the Angle θ13 in Reactor Experiments

11.5.1 Introduction: CHOOZ Reactor Experiment

From the data of first neutrino oscillation experiments an information about two mass-squared differences (atmospheric \(\varDelta m_{A}^{2}\) and solar \(\varDelta m_{S}^{2}\)) and two mixing angles (θ23 and θ12) was obtained. During many years only an upper bound for the parameter \(\sin ^{2}\theta _{13}\) was known. This bound was obtained from the data of the reactor CHOOZ experiment.

The \(\bar \nu _{e}\)-survival probability, driven by the atmospheric mass-squared difference, in the major two-neutrino approximation is given by the expression
$$\displaystyle \begin{aligned} P(\bar\nu_{e}\to \bar\nu_{e})=1 -\sin^{2}2 \theta_{13}~\sin^{2}(\frac{\varDelta m_{A}^{2}L}{4E})\end{aligned} $$
(11.64)
Thus the study of the disappearance of the reactor \(\bar \nu _{e}\) allows to determine the parameter \(\sin ^{2}2\theta _{13}\). For the average energy of the reactor antineutrinos (3.6 MeV) the corresponding oscillation length is equal to
$$\displaystyle \begin{aligned} L_{\mathrm{osc}}\simeq 2.5 \frac{E}{\varDelta m_{A}^{2}}~m\simeq 3.6\,\mathrm{km}. \end{aligned} $$
(11.65)
In the CHOOZ experiment one antineutrino detector was exposed to two reactors of the CHOOZ power station (8.5 GW th ). The distance between the detector and reactors was about 1 km. The CHOOZ detector comprised 5 tons of Gd-loaded liquid scintillator contained in an acrylic vessel. The antineutrinos were detected through the observation of the classical reaction
$$\displaystyle \begin{aligned} \bar\nu_e + p \to e^+ + n. \end{aligned} $$
(11.66)
From April 1997 till July 1998, in the CHOOZ experiment 3600 antineutrino events were recorded. For the ratio R of the total number of detected and the expected events it was found
$$\displaystyle \begin{aligned} R =1.01 \pm 2.8\%\,(\mathrm{stat.})\pm\pm 2.7\%\,(\mathrm{syst.}). \end{aligned} $$
(11.67)
From the data of the experiment the following upper bound
$$\displaystyle \begin{aligned} \sin^{2}2\,\theta_{13} \leq 0.16. \end{aligned} $$
(11.68)
was obtained.

In 2012 three new reactor experiments Daya Bay, RENO and Double Chooz started. The aim of these experiments was to measure the angle θ13 (or to improve the CHOOZ bound). The parameter \(\sin ^{2}2\,\theta _{13}\) was successfully measured (it occurred that its value is close to the CHOOZ upper bound (11.68)). This finding is extremely important for the future investigation of the problem of the neutrino mixing. As we have seen before, the CP phase δ enter into the PMNS mixing matrix the form \(\sin \theta _{13}~e^{-i\delta }\). Thus such fundamental effect of the three-neutrino mixing as CP violation in the lepton sector can be studied only if the mixing angle θ13 is not equal to zero (and relatively large). Another problem, the solution of which requires nonzero θ13, is the problem of the character of the neutrino mass spectrum (normal or inverted ordering?). In the next subsections we will briefly discuss Daya Bay, RENO and Double Chooz experiments.

11.5.2 Daya Bay Experiment

In the Daya Bay experiment antineutrinos from six commercial nuclear reactors, located at Daya Bay and Ling Ao nuclear power stations (China), are detected. Each reactor has a thermal power 2.9 GWth. Antineutrino detectors are disposed in two near underground halls and one far underground hall (correspondingly, at the distances 350–600 m and 1500–1950 m from the reactors). In each near hall there are two antineutrino detectors. In the first phase of the experiment (December 2011–July 2012) there were two antineutrino detectors in the far hall. In October 2012 two additional antineutrino detectors were disposed in the far hall.

All eight antineutrino detectors have identical three-zone structure: 20 tons of Gd-loaded liquid scintillator in the inner zone (\(\bar \nu _{e}\) detector), 22 tons of liquid scintillator in the middle zone and 37 tons of mineral oil in external zone. Scintillation light is detected by 192 8-in. PMTs.

Reactor \(\bar \nu _{e}\)’s are detected via the observation of the inverse β-decay
$$\displaystyle \begin{aligned} \bar\nu_e + p \to e^+ + n. \end{aligned} $$
(11.69)
Detection of photons produced in e+ − e annihilation (prompt signal) allows to determine the positron energy (Epr = T + 2m e , T being positron kinetic energy). The neutron, produced in (11.69), thermalizes and is captured by a Gd nucleus, producing γ-rays with total energy 8 MeV, or a proton, producing γ-quantum with the energy 2.2 MeV (delayed signal).
During 217 days of the data-taking with six antineutrino detectors and 1013 days of the data-taking with eight antineutrino detectors in the Daya Bay experiment the total number of 2.5 ⋅ 105 \(\bar \nu _{e}\)-events were observed. Such large statistics allowed the Daya Bay Collaboration to obtain a very precise value of the parameter \(\sin ^{2}\theta _{13}\). For the ratio of the \(\bar \nu _{e}\) total rates in the far and near detectors it was found
$$\displaystyle \begin{aligned} R=0.949 \pm 0.002~(\mathrm{stat.})\pm 0.002~(\mathrm{syst.}) \end{aligned} $$
(11.70)
From the three-neutrino analysis of observed rate and energy spectrum the following values of the parameters \(\sin ^{2}2\theta _{13}\) and \(|\varDelta m^{2}_{ee}|\) were obtained
$$\displaystyle \begin{aligned} \sin^{2}2\theta_{13}=0.0841 \pm 0.0027~(\mathrm{stat.})\pm 0.0019~(\mathrm{syst.}) \end{aligned} $$
(11.71)
and
$$\displaystyle \begin{aligned} |\varDelta m^{2}_{ee}|=(2.50 \pm 0.06~(\mathrm{stat.})\pm 0.06~(\mathrm{syst.}))\cdot 10^{-3}~\mathrm{eV}^{2}, \end{aligned} $$
(11.72)
where
$$\displaystyle \begin{aligned} |\varDelta m^{2}_{ee}|=\cos^{2}\theta_{12}|\varDelta m^{2}_{13}|+\sin^{2}\theta_{12}|\varDelta m^{2}_{23}|. \end{aligned} $$
(11.73)
is the effective (“average”) reactor mass-squared difference.
In the case of the normal ordering of the neutrino masses it was found
$$\displaystyle \begin{aligned} \mathrm{NO} ~~~\varDelta m^{2}_{A}=(2.45 \pm 0.06~(\mathrm{stat.})\pm 0.06~(\mathrm{syst.}))\cdot 10^{-3}~\mathrm{eV}^{2}. \end{aligned} $$
(11.74)
For the inverted ordering it was obtained
$$\displaystyle \begin{aligned} \mathrm{IO} ~~~\varDelta m^{2}_{A}=(2.56 \pm 0.06~(\mathrm{stat.})\pm 0.06~(\mathrm{syst.}))\cdot 10^{-3}~\mathrm{eV}^{2}. \end{aligned} $$
(11.75)
These values for the atmospheric mass-squared difference are in a good agreement with the values which were found from the data of the accelerator neutrino experiments which we will discuss later.
In Fig. 11.6 the reactor \(\bar \nu _{e}\) survival probability as function of the parameter Leff/E is presented. The points are the ratios of the observed and expected events. The curve was calculated using the best-fit values of the parameters \(\sin ^{2}2\theta _{13}\) and \(|\varDelta m^{2}_{ee}|\). The curve demonstrates oscillation behavior of the event rates observed in the Daya Bay experiment.
Fig. 11.6

Daya Bay experiment: the reactor \(\bar \nu _{e}\) survival probability as function of the parameter Leff/E. The points are the ratios of the observed and expected events. The solid line was calculated using the best-fit values of the parameters \(\sin ^{2}2\theta _{13}\) and \(|\varDelta m^{2}_{ee}|\) (arXiv:1610.04802)

11.5.2.1 RENO Experiment

The reactor neutrino experiment RENO started in August 2011. Antineutrinos from six reactors at Hanbit nuclear power plant (Korea) are detected by two underground detectors, located at the distances 294 and 1383 m from the center of reactor array. The thermal power of each reactor is about 2.8 GWth.

Near and far detectors in the RENO experiment are identical. The innermost part of the detector is an acrylic vessel filled with 16 tons of Gd-doped liquid scintillator (target). It is contained in another acrylic vessel filled with liquid scintillator (γ-catcher in which γ-quanta, escaping from the target region, are detected). Outside the γ-catcher there is a 70-cm thick layer of mineral buffer oil which provides shielding from external radioactivity. Produced light is detected by 354 10-in. PMTs which are mounted on the inner wall of a stainless steel container. The container is surrounded by a veto water-Cherenkov detector which provides shielding against γ-quanta and neutrons from surrounding rocks.

During 500 days of the data-taking it was detected 290,775 (31,514) \(\bar \nu _{e}\) events in the near (far) detector. Two identical detectors allow to perform a relative measurement of antineutrino rates and spectra. This relative measurement allowed to reduce systematic errors coming from uncertainties of the reactor neutrino flux and detection efficiency. From analysis of the RENO data for the neutrino oscillation parameters it was found the values
$$\displaystyle \begin{aligned} \sin^{2}2\theta_{13}=0.082 \pm 0.009~(\mathrm{stat.})\pm 0.006~(\mathrm{syst.}) \end{aligned} $$
(11.76)
and
$$\displaystyle \begin{aligned} |\varDelta m^{2}_{ee}|=(2.62^{+0.21}_{-0.23} ~(\mathrm{stat.})^{+0.12}_{-0.13}~(\mathrm{syst.}))\cdot 10^{-3}~\mathrm{eV}^{2}, \end{aligned} $$
(11.77)
which are in agreement with Daya Bay values (11.71) and (11.72).
In Fig. 11.7 (top) the dependence of the number of the \(\bar \nu _{e}\) events, observed in the far RENO detector, on the prompt energy is plotted. The shaded histogram was found from near detector data. On the bottom the ratio of the number of \(\bar \nu _{e}\) far detector events to the number of predicted events (assuming that there is no neutrino oscillations) is depicted (points). The shaded band is the ratio of the number of the far detector events to the number of MC predicted best-fit events.
Fig. 11.7

Top: the dependence of the number of the \(\bar \nu _{e}\) events, observed in the far RENO detector, on the prompt energy. The shaded histogram was found from near detector data. Bottom: points are the ratio of the number of \(\bar \nu _{e}\) far detector events to the number of predicted events (assuming that there is no neutrino oscillations). The shaded band is the ratio of the number of the far detector events to the number of MC predicted best-fit events (arXiv:1610.04326v3)

11.5.2.2 Double Chooz Experiment

In the Double Chooz experiment \(\bar \nu _{e}\)’s from two reactors (the thermal power of each reactor is 4.25 GWth) are detected. The experiment is performed at CHOOZ-B power plant, Chooz, France. The Double Chooz experiment started in 2011 with one detector at the average distance 1050 m from reactors. In 2015 the near detector at the average distance 400 m was constructed. Both detectors have identical structure. The inner target detector has 10.3 m3 of Gd-doped liquid scintillator. It is surrounded by 22.4 m3 liquid scintillator (γ-catcher) and 100 m3 non-scintillating mineral buffer oil. Photons, produced in the target and γ-catcher, are detected by 390 10-in. diameter PMTs. Optically separated from three inter volumes there is an external veto detector.

Reactor antineutrinos are detected via classical reaction
$$\displaystyle \begin{aligned} \bar\nu_e + p \to e^+ + n. \end{aligned} $$
(11.78)
Coincidence of a prompt signal from annihilation of e+ with e in the liquid scintillator (energies from 1 to 11 MeV) and a delayed signal from capture of neutron by a Gd nucleus (photons with total energy 8 MeV) is a good signature of the event.
Disappearance of the reactor \(\bar \nu _{e}\) was observed in the experiment. In the case of the three-neutrino mixing the probability of \(\bar \nu _{e}\) to survive can be presented in the form
$$\displaystyle \begin{aligned} \mathrm{P}(\bar\nu_{e}\to \bar\nu_{e})\simeq 1-\sin^{2}2\theta_{13}\sin^{2}\left(\frac{\varDelta m^{2}_{ee}L}{4E}\right) -\cos^{4}\theta_{13}\sin^{2}2\theta_{12}\sin^{2}\left(\frac{\varDelta m^{2}_{S}L}{4E}\right), \end{aligned} $$
(11.79)
where the effective reactor mass-squared difference \(\varDelta m^{2}_{ee}\) is given by (11.73).
From analysis of the data of 673 days of far detector and 151 days of near detector, using the constraint
$$\displaystyle \begin{aligned} |\varDelta m^{2}_{ee}|=(2.44\pm 0.09)\cdot 10^{-3}~\mathrm{eV}^{2}, \end{aligned} $$
(11.80)
obtained from the data of the accelerator MINOS experiment, in the Double Chooz experiment it was found
$$\displaystyle \begin{aligned} \sin^{2}2\theta_{13}=0.111 \pm 0.018~(\mathrm{stat.}+\mathrm{syst.}). \end{aligned} $$
(11.81)

11.6 Long-Baseline Accelerator Neutrino Experiments

11.6.1 K2K Accelerator Neutrino Experiment

In long baseline accelerator neutrino experiments there is a possibility to use beams of neutrinos and antineutrinos, to work with narrow band neutrino beams (off-axis neutrino beams) etc. This allow to study CP violation in the lepton sector, to reveal the character of the neutrino mass spectrum and to perform a high precision measurement of neutrino oscillation parameters.

Oscillations in the long baseline accelerator neutrino experiments are driven predominantly by the atmospheric mass-squared difference \(\varDelta m_{A}^{2}\). For a neutrino energy E ≃ 1 GeV and \(\varDelta m_{A}^{2}\simeq 2.5 \cdot 10^{-3}~\mathrm {eV}^{2}\) the oscillation length Losc is given by
$$\displaystyle \begin{aligned} L_{\mathrm{osc}}\simeq 2.5 ~\frac{E}{\varDelta m_{A}^{2}}~\mathrm{m}\simeq 10^{3}~\mathrm{km}~.\end{aligned} $$
(11.82)
In the first long baseline K2K experiment the distance between the neutrino source (KEK accelerator, Japan) and the neutrino detector (Super-Kamiokande) was about 250 km.

Protons with an energy of 12 GeV from the KEK-PS accelerator bombard an aluminum target in which secondary particles were produced. Positively charged particles (mainly π+) were focused in horns and decayed in a 200 m-long decay pipe. After a beam dump in which all hadrons and muons were absorbed a neutrino beam was produced (there were 97.3% of ν μ , 1.3% of ν e and 1.4% of \(\bar \nu _{\mu }\) in the beam). The neutrinos had energies in the range (0.5–1.5) GeV.

In the K2K experiment the disappearance of muon neutrinos was searched for. The two-neutrino probability of ν μ to survive has the form
$$\displaystyle \begin{aligned} \mathrm{P}(\nu_{\mu}\to \nu_{\mu})\simeq 1 -\sin^{2}2 \theta_{23}~\sin^{2}(1.27~\varDelta m_{A}^{2}\frac{L}{E}),\end{aligned} $$
(11.83)
where E is the neutrino energy in GeV, L is the source-detector distance in km and \(\varDelta m_{A}^{2}\) is the atmospheric neutrino mass-squared difference in eV2.

From 1999 till 2004 in the K2K experiment 112 neutrino events were detected. For the number of the expected events (in the case if there were no neutrino oscillations) was found the value \(158^{+9.2}_{-8.6}\). In the low energy region the distortion of the neutrino spectrum was observed.

From the two-neutrino analysis of the K2K data under the assumption that \(\sin ^{2}2\theta _{23}=1\) it was found the following 90% CL range for the parameter \(\varDelta m_{A}^{2}\):
$$\displaystyle \begin{aligned} 1.9\cdot 10^{-3}~ \mathrm{eV}^2 <\Delta m_{A}^{2}< 3.5 \cdot 10^{-3}~ \mathrm{eV}^2. \end{aligned} $$
(11.84)
The K2K experiment was the first experiment with artificially produced neutrinos which confirmed the existence of neutrino oscillations discovered in the atmospheric Super-Kamiokande neutrino experiment.

11.6.2 MINOS Accelerator Neutrino Experiment

In the long baseline MINOS experiment muon neutrinos produced at the Fermilab Main Injector facility were detected in the Sudan mine (Minnesota, USA) at a distance of 735 km. Data-taking started in the MINOS experiment in 2005 and finished in 2012. In 2013 the experiment MINOS+, successor of the MINOS, started.

Protons with an energy of 120 GeV, extracted from the Main Injector proton accelerator, bombarded a graphite target and produced (predominantly) pions and kaons. Positively (or negatively) charged particles were focused by two magnetic horns and directed into a 675 m long decay pipe. After the pipe there was an absorber for hadrons and 240 m of rock in which muons were stopped.

Muon neutrinos were produced in the decays π+(K+) → μ+ + ν μ . Electron neutrinos were produced in the decays \(\mu ^{+}\to e^{+}+ \nu _{e}+\bar \nu _{\mu }\) and \(K^{+}(K^{0}_{L})\to e^{+}+ \nu _{e} +\pi ^{0}(\pi ^{-})\). Antineutrinos were created in charge conjugated processes. The ν μ -dominated beam consisted of ν μ (91.7%), \(\bar \nu _{\mu }\) (7%), ν e and \(\bar \nu _{e}\) (1.3%). The \(\bar \nu _{\mu }\)-enhanced beam consisted of ν μ (58.1%), \(\bar \nu _{\mu }\) (39.9%), ν e and \(\bar \nu _{e}\) (2.0%). For the MINOS experiment the Main Injector supplied 10.71 ⋅ 1020 protons-on-target (POT) in order to produce the ν μ -dominated beam and 3.36 ⋅ 1020 POT in order to produce the \(\bar \nu _{\mu }\)-enhanced beam.

The majority of the MINOS data was obtained with the low-energy neutrino beam (1 ≤ E ≤ 5 GeV) which has a peak at 3 GeV. There were two identical neutrino detectors in the MINOS experiment. The near detector (ND) with a mass of about 1 kton was at a distance of about 1.04 km from the target and about 100 m underground. The far detector (FD) with a mass of 5.4 kton was at a distance of 735 km from the target and 705 m underground (2070 m water equivalent). The detectors were steel (2.54 cm thick)-scintillator (1 cm thick) calorimeters magnetized to 1.3 T (ND) and 1.4 T (FD). The measurement of the curvature of the muon tracks allows to distinguish ν μ from \(\bar \nu _{\mu }\) and to measure energy of muons which leave the detector. The energies of the muons which are stopped in the detector are determined by their ranges.

Muon neutrinos and antineutrinos were detected in the MINOS experiment via the observation of the CC process
$$\displaystyle \begin{aligned} \nu_{\mu}(\bar\nu_{\mu})+\mathrm{Fe}\to \mu^{-}(\mu^{+})+X. \end{aligned} $$
(11.85)
Such events are characterized by tracks caused by muons and a hadronic showers. The neutrino energy is given by the sum of the muon energy and the energy of the hadronic shower. Electron neutrinos were detected via the observation of the CC process
$$\displaystyle \begin{aligned} \nu_{e}+\mathrm{Fe}\to e^{-}+X. \end{aligned} $$
(11.86)
The signature of such events is an electromagnetic shower. Because electrons do not have track-like topology, ν e and \(\bar \nu _{e}\) events can not be separated.

In the near detector the initial neutrino spectrum was measured. This measurement allowed to predict the expected spectrum of the muon neutrinos in the far detector in the case if there were no neutrino oscillations and to determine the ν μ (\(\bar \nu _{\mu }\)) survival probability as a function of the neutrino energy. The search for ν μ  → ν e and \(\bar \nu _{\mu }\to \bar \nu _{e}\) appearance was also performed.

In the MINOS experiment not only Main Injector beam generated neutrino events but also atmospheric neutrino events were observed (starting from 2003). In the final three-neutrino oscillation analysis all detected events were taken into account. In this analysis the average reactor neutrino value \(\sin ^{2}\theta _{13}=0.0242\pm 0.0025\) was used. The solar-KamLAND values \(\varDelta m_{S}^{2}=7.54\cdot 10^{-5}~\mathrm {eV}^{2}\) and \(\sin ^{2}\theta _{12}=0.307\) were also kept fixed in the fit. In the case of the normal mass ordering from analysis of the MINOS data it was found
$$\displaystyle \begin{aligned} \varDelta m_{A}^{2}=(2.28-2.46)\cdot 10^{-3}~\mathrm{eV}^{2}~(68\%),\quad \sin^{2}\theta_{23}=(0.35-0.65) ~(90\%) \end{aligned} $$
(11.87)
In the case of the inverted mass ordering it was obtained
$$\displaystyle \begin{aligned} \varDelta m_{A}^{2}=(2.32-2.53)\cdot 10^{-3}~\mathrm{eV}^{2}~(68\%),\quad \sin^{2}\theta_{23}=(0.34-0.67) ~(90\%) \end{aligned} $$
(11.88)
The MINOS+ experiment is running in the high-energy region 3–10 GeV. During the first year of the operation, starting from September 2013, it was collected 2.99 ⋅ 1020 POT. Oscillation parameters found from analysis of the MINOS+ data are in agreement with the oscillation parameters (11.87) and (11.88) obtained from the analysis of the MINOS data.
In Fig. 11.8 neutrino energy spectrum, obtained from the results of the MINOS and MINOS+ experiments, is presented. The curve is the prediction calculated with best-fit MINOS oscillation parameters. The histogram is the expected spectrum in the case if there is no neutrino oscillations.
Fig. 11.8

Neutrino energy spectrum, obtained from the results of the MINOS and MINOS+ experiments. The curve is the prediction calculated with best-fit MINOS oscillation parameters. The histogram is the expected spectrum in the case of no neutrino oscillations (arXiv:1601.05233v3)

11.6.3 T2K Experiment

The T2K experiment is performed at J-PARC accelerator facility in Tokai (Japan). In order to produce a neutrino beam a 30 GeV protons hit a graphite target and produce charged pions and kaons which are focused by three magnetic horns. Either positive or negative mesons are focused resulting in a beam of predominantly ν μ or \(\bar \nu _{\mu }\) produced in a 96 m long decay tube. The decay volume is followed by the beam dump and muon monitors. Neutrinos are detected by an on axis near detector and of axis, at 2.5 relative to the beam direction, near and far detectors. The off axis narrow band neutrino energy spectrum has a peak at 0.6 GeV.

Two T2K near detectors INGRID (on axes) and ND280 (off axis) are located at the distance 280 m from the target. They measure the beam direction, composition, neutrino spectrum and the event rate. The 50 kton water-Cherenkov Super-Kamiokande detector is used as a far detector in the T2K experiment. It is located off axes at the distance 295 km from the target.13

From January 2010 till May 2013 in the T2K experiment only neutrino data were collected. From May 2014 till May 2016 predominantly antineutrino events were observed. This corresponds to a neutrino and antineutrino beam exposures on the far detector, correspondingly, 7.48 ⋅ 1020 POT and 7.47 ⋅ 1020 POT.

From the three-neutrino analysis of ν μ and \(\bar \nu _{\mu }\) disappearance data it was found
$$\displaystyle \begin{aligned} \sin^{2}\theta_{23}=0.514^{+0.055}_{-0.056},\quad \varDelta m_{A}^{2}=(2.51\pm 0.10)\cdot 10^{-3}~\mathrm{eV}^{2} \end{aligned} $$
(11.89)
in the case of the normal mass ordering and
$$\displaystyle \begin{aligned} \sin^{2}\theta_{23}=0.511\pm 0.055,\quad \varDelta m_{A}^{2}=(2.48\pm 0.10)\cdot 10^{-3}~\mathrm{eV}^{2} \end{aligned} $$
(11.90)
for the inverted mass ordering.
In 2011 in the T2K experiment the first 2.5 σ evidence in favor of ν μ  → ν e transition was obtained. This was the first indication in favor θ13 ≠ 0. During the exposure which finished in 2014 twenty eight electron neutrino events were observed. The energy distribution of these events was consistent with ν e appearance due to neutrino oscillations. Assuming \(\varDelta m_{A}^{2}=2.45 \cdot 10^{-3}~\mathrm {eV}^{2}\), \(\sin ^{2}\theta _{23}=0.5\), δ = 0, it was found from analysis of the ν e appearance data
$$\displaystyle \begin{aligned} \sin^{2}2\theta_{13}=0.140^{+0.038}_{-0.032}~~(\mathrm{NO}),\quad \sin^{2}2\theta_{13}=0.170^{+0.045}_{-0.037} ~~(\mathrm{IO}). \end{aligned} $$
(11.91)

11.6.4 NOvA Experiment

A new long baseline accelerator neutrino oscillation experiment NOvA with near and far identical detectors (ND and FD) started to collect data in 2014. The energy spectrum of neutrinos, produced at FermiLab Main Injector facility, is measured by the 290-ton ND, located 1 km away from the Main Injector target, 100 m underground. The 14-kton FD is located on the surface, 14.6 mrad off axis, at the distance 810 km from the FermiLab (Ash River, Minnesota, USA). A narrow band neutrino beam with peak energy about 2 GeV (the first oscillation maximum) is utilized in the NOvA experiment. The flavor composition of the neutrino beam at the FD is estimated to be 97.8% ν μ , 1.6% \(\bar \nu _{\mu }\) and 0.6% (\(\nu _{e}+\bar \nu _{e}\)) assuming that there are no oscillations.

Both ND and FD are segmented tracking calorimeters. Reflective cells of length 15.5 m (3.9 m) in the FD (ND) with a 3.9 × 6.6 cm2 cross section are filled with liquid scintillator. Light, produced by charged particles, is collected in each cell by optical fiber and measured with an avalanche photodiode.

The study of ν μ disappearance requires identification in FD of the reactionν μ  + N → μ + X and the measurement of the neutrino energy. The neutrino energy is given by a sum of the reconstructed muon energy and recoil hadronic energy. The investigation of ν e appearance requires identification in FD of the process ν e  + N → e + X and understanding background processes. The signature of this CC process in the NOvA detectors is an electromagnetic shower and associated hadronic recoil energy.

From the analysis of the NOvA ν μ disappearance data, obtained from 6.05 ⋅ 1020 POT exposure, in the case of the normal mass ordering it was found
$$\displaystyle \begin{aligned} \varDelta m_{A}^{2}=(2.67\pm 0.11)\cdot 10^{-3}~\mathrm{eV}^{2} \end{aligned} $$
(11.92)
For the parameter \(\sin ^{2}\theta _{23}\) two statistically-degenerate values were obtained
$$\displaystyle \begin{aligned} \sin^{2}\theta_{23}=0.404^{+0.030}_{-0.022},\quad \sin^{2}\theta_{23}=0.624^{+0.022}_{-0.030} \end{aligned} $$
(11.93)
In the case of the inverted mass ordering it was found
$$\displaystyle \begin{aligned} \varDelta m_{A}^{2}=(2.72\pm 0.11)\cdot 10^{-3}~\mathrm{eV}^{2} \end{aligned} $$
(11.94)
and
$$\displaystyle \begin{aligned} \sin^{2}\theta_{23}=0.398^{+0.030}_{-0.022},~~\mathrm{or}~~ \sin^{2}\theta_{23}=0.618^{+0.022}_{-0.030} \end{aligned} $$
(11.95)
In Fig. 11.9 NOvA 90% CL allowed region in the plane of the parameters (\(\sin ^{2}\theta _{23}, \varDelta m_{A}^{2}\)) are shown. T2K and MINOS allowed regions are also presented.
Fig. 11.9

NOvA experiment: 90% CL allowed region in the plane of the parameters \(\sin ^{2}\theta _{23} ~\mathrm {and}~ \varDelta m_{A}^{2}\). T2K and MINOS allowed regions are also shown (arXiv:1701.05891v1)

With the same exposure 6.05 ⋅ 1020 POT 33 ν e candidate events with a background 8.2 ± 0.8 events were observed in the NOvA experiment. Combing these data with NOvA ν μ disappearance data and with the reactor value of the parameter \(\sin ^{2}\theta _{13}\) the NOvA Collaboration concluded that inverted neutrino mass spectrum with \(\theta _{23}<\frac {\pi }{4}\) is disfavored at 93% CL for all values of the CP phase δ.

Footnotes

  1. 1.

    At that time only ν e and ν μ were known.

  2. 2.

    Notice that the spectrum of the reactor \(\bar \nu _{e}\) was recently recalculated. As a result, old reactor data are considered at present as an indication in favor of active-sterile neutrino oscillations (see later).

  3. 3.

    In stars significantly heavier than the sun the central temperatures are higher and the CNO cycle gives important contribution to the energy production.

  4. 4.

    The CNO cycle is the following chain of reactions: p +12C →13N + γ, 13N →13C + e+ + ν e , p +13C →14N + γ, p +14N →15O + γ, 15O →15N + e+ + ν e . There are two branches of reactions with nuclei 15N, terminated with the production of 4He: p +15N →12C +4He or p +15N →16O + γ, p +16O →17F + γ, 17F →17O + e+ + ν e , p +17O →14N +4He.

  5. 5.

    The energy, produced by the sun, is emitted in the form of photons (about 98%) and neutrinos (about 2%).

  6. 6.

    The Standard Solar Model is based on the assumption that the sun is a spherically symmetric plasma sphere in hydrostatic equilibrium. The effects of rotation and of the magnetic field are neglected.

  7. 7.

    For this experiment R. Davis was awarded with the Nobel Prize in 2002.

  8. 8.

    The solar neutrino unit (SNU) is determined as follows:1 SNU = 10−36events atom−1 s−1.

  9. 9.

    In 1987 the Kamiokande Collaboration (and also the IMB and Baksan Collaborations) observed neutrinos from the explosion of the supernova SN1987A. This was the first observation of supernova neutrinos. The experiment confirmed the general theory of the gravitational collapse.

  10. 10.

    The initial 8B solar neutrino spectrum is determined by the weak decay 8B → e+ + ν e  + 2α. This spectrum can be obtained from the laboratory measurement of the α-spectrum. The fact that the electron spectrum, measured in the Super-Kamiokande experiment, is in an agreement with the expected spectrum means that in the high-energy 8B region the probability of the solar ν e to survive is a constant.

  11. 11.

    ν τ produced in \(\nu _{\mu }\leftrightarrows \nu _{\tau }\) oscillations were observed in the long baseline experiment OPERA. The distance between the source of ν μ (CERN) and the detector (Gran Sasso Laboratory) in this experiment was about 730 km. The production of τ in ν τ -nucleon CC processes was detected in an emulsion. Five ν τ events were observed.

  12. 12.

    This analysis is based on the assumption of the CPT invariance.

  13. 13.

    For E ≃ 0.6 GeV and L=295 km we have \(1.27\frac {\varDelta m^{2}_{A}L}{E}\simeq \frac {\pi }{2}\). Thus, neutrino energy and the source-detector distance in the T2K experiment correspond to the first maximum of oscillations driven by the atmospheric mass-squared difference.

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Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  • Samoil Bilenky
    • 1
  1. 1.TRIUMFVancouverCanada

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