Phenomenology of Neutrino Oscillations

S.M. Bilenky

Joint Institute for Nuclear Research, Dubna, Russia, and
Institut für Theoretische Physik, Technische Universität München, D-85748 Garching, Germany

C. Giunti

INFN, Sezione di Torino, and Dipartimento di Fisica Teorica,
Università di Torino, Via P. Giuria 1, I-10125 Torino, Italy, and
School of Physics, Korea Institute for Advanced Study, Seoul 130-012, Korea

W. Grimus

Institute for Theoretical Physics, University of Vienna,
Boltzmanngasse 5, A-1090 Vienna, Austria

Abstract

This review is focused on neutrino mixing and neutrino oscillations in the light of the recent experimental developments. After discussing possible types of neutrino mixing for Dirac and Majorana neutrinos and considering in detail the phenomenology of neutrino oscillations in vacuum and matter, we review all existing evidence and indications in favour of neutrino oscillations that have been obtained in the atmospheric, solar and LSND experiments. We present the results of the analyses of the neutrino oscillation data in the framework of mixing of three and four massive neutrinos and investigate possibilities to test the different neutrino mass and mixing schemes obtained in this way. We also discuss briefly future neutrino oscillation experiments.

Journal: Prog. Part. Nucl. Phys. 43, 1 (1999).
Preprint: UWThPh-1998-61, DFTT 69/98, KIAS-P98045, SFB 375-310, TUM-HEP 340/98, hep-ph/9812360.

Contents

• Introduction
• Neutrino Mixing
  • Dirac mass term
  • Dirac-Majorana mass term
  • Majorana mass term
  • The one-generation case
  • The see-saw mechanism
  • Effective Lagrangians
  • Maximal mixing
  
  
• Neutrino oscillations in vacuum
  • The general formalism
  • Oscillations in the two-neutrino case
  
  
• Neutrino oscillations and transitions in matter
  • The effective Hamiltonian for neutrinos in matter
  • The two-neutrino case and adiabatic transitions
  • The resonance
  • Non-adiabatic neutrino oscillations in matter and crossing probabilities
  
  
• Indications of neutrino oscillations
  • Atmospheric neutrino experiments
  • Solar neutrino experiments
  • The LSND experiment
  
  
• Analysis of neutrino oscillation data
  • Mixing of three massive neutrinos
  • Mixing of four massive neutrinos
  
  
• Conclusions
• Note added
  • Atmospheric neutrinos
  • Solar neutrinos
  
  
• Acknowledgements
• Properties of Majorana neutrinos and fields
• Bibliography
• About this document ...

Introduction

The strong evidence in favour of oscillations of atmospheric neutrinos found by the Super-Kamiokande Collaboration [1,2] opened a new era in particle physics. There is no doubt that new experiments are necessary to understand the nature of neutrino masses and mixing which are intimately connected with neutrino oscillations, but the first decisive step has been done: massive and mixed neutrinos can now be considered as real physical objects.

The problem of neutrino mass has a long history. Originally, Pauli considered the neutrino as a particle with a small but non-zero mass (smaller than the electron mass) [3] and the method for the measurement of the neutrino mass through the investigation of the $\beta$-spectrum near the end point was proposed in the first theoretical papers on $\beta$-decay of Fermi [4,5] and Perrin [6].

The first experiments on the measurement of the neutrino mass, based on the Fermi-Perrin method, yielded the upper bound $m_{\nu} \lesssim 500 \, \mathrm{eV}$ [7] which was improved in the fifties to $m_{\nu} \lesssim 250 \, \mathrm{eV}$ [8]. Therefore, it became evident that the neutrino mass (if non-zero at all) is much smaller than the electron mass. This was the main reason that in 1957, after the discovery of parity violation in $\beta$-decay, the authors of the two-component theory of the neutrino (Landau [9], Lee and Yang [10], Salam [11]) assumed that the neutrino is a massless particle, the field of which is either a left-handed field $\nu_{L}$ or a right-handed field $\nu_{R}$.

In 1958, Goldhaber et al. [12] measured the helicity of the neutrino. The result of this experiment was in agreement with the two-component neutrino theory and it was established that the neutrino field is $\nu_{L}$.¹The results of the experiment of Goldhaber et al. could not exclude, however, the possibility of a small neutrino mass. In the V$-$A theory (Feynman and Gell-Mann [15], Sudarshan and Marshak [16]) the Hamiltonian of weak interactions contains the left-handed component of the neutrino field $\nu_{L}$, and also the left-handed components of all massive fields. Therefore the possibility for the neutrino to be nevertheless a massive particle became more natural [17] after the confirmation of the V$-$A theory.

In 1957, B. Pontecorvo [18,19] proposed the idea that the state of neutrinos produced in weak interaction processes is a superposition of states of two Majorana neutrinos [20] with definite masses (analogous to the states $\vert K^0\rangle$ and $\vert\bar{K}^0\rangle$ which are the superposition of $\vert K_1\rangle$ and $\vert K_2\rangle$, the states of particles with definite masses and widths). In this way, B. Pontecorvo arrived at the hypothesis of neutrino oscillations (analogous to $K^0 \leftrightarrows \bar{K}^0$ oscillations). At that time only one type of neutrino was known. The possibility of mixing of the two species of neutrinos $\nu_e$ and $\nu_{\mu}$ was considered in Ref. [21]. All possible types of neutrino oscillations for this case were investigated by Pontecorvo in 1967 [22].

Gribov and Pontecorvo proposed in 1969 [23] the first phenomenological theory of neutrino mixing and oscillations. In this theory, the two left-handed neutrino fields $\nu_{{e}L}$ and $\nu_{{\mu}L}$ are linear combinations of the left-handed components of the fields of Majorana neutrinos with definite masses and the neutrino mass term contains only the left-handed fields $\nu_{{e}L}$ and $\nu_{{\mu}L}$.

In 1976, neutrino oscillations were considered in the scheme of mixing of two Dirac neutrinos based on the analogy between quarks and leptons [24,25] and in the same year in the general Dirac-Majorana scheme [26] (for later works see Refs. [27,28,29,30]).

The theoretical arguments in favour of non-zero neutrino masses and mixing are based on the models beyond the Standard Model (see, for example, Ref. [31]). In such models the fields of quarks, charged leptons and neutrinos are grouped in the same multiplets and the generation of the masses of quarks and charged leptons with the Higgs mechanism as a rule provides also non-zero neutrino masses.

In 1979, the see-saw mechanism for the generation of neutrino masses was proposed [32,33,34]. This mechanism connects the smallness of neutrino masses with the possible violation of lepton number conservation at a very large energy scale.

At present the effects of neutrino masses and mixing are investigated in many different experiments. There are three types of experiments in which the effects of small neutrino masses (say, of the order of 1 eV or smaller) and mixing can be revealed (for a review and references see Ref. [35]):

1. Neutrino oscillations experiments.

2. Experiments on the search for neutrinoless double $\beta$-decay.

3. Experiments on the measurement of the electron neutrino mass with the precise investigation of the high energy part of the $\beta$-spectrum of $^3$H.

Three indications in favour of neutrino masses and mixing have been found so far. These indications were obtained in the following experiments:

1. Solar neutrino experiments (Homestake [36,37,38], Kamiokande [39,40,41], GALLEX [42,43], SAGE [44,45], Super-Kamiokande [46,47,48]).

2. Atmospheric neutrino experiments (Super-Kamiokande [1,2], Kamiokande [49], IMB [50], Soudan [51], MACRO [52]).

3. The accelerator LSND experiment [53,54].

Many other neutrino oscillation experiments with neutrinos from reactor and accelerators did not find any evidence for neutrino oscillations. In the experiments on the search for neutrinoless double $\beta$-decay no indications for non-zero neutrino masses were found (see Section 6.2). The present upper bound for the electron neutrino mass obtained in the Troitsk experiment [55] is 2.7 eV (see also the Mainz experiment [56]). The upper limits on the masses of $\nu_\mu$ and $\nu_\tau$ are 170 keV (90% CL) and 18.2 MeV (95% CL), respectively [35]. Neutrinos play an important role in cosmology and astrophysics and many bounds on neutrino properties can be derived in this context. For reviews see, e.g., Refs. [31,57].

In this review we discuss the phenomenological theory of neutrino mixing (Section 2), neutrino oscillations in vacuum (Section 3), neutrino oscillations and transitions in matter (Section 4) and experimental data and results of analyses of the data (Sections 5 and 6). We also consider the implications of the existing experimental results on neutrino oscillations for experiments in preparation. After the conclusions (Section 7) we discuss some properties of Majorana neutrinos and fields in Appendix A.

We hope that this review will be useful not only for the physicists that are working in the field but also for those who are interested in this exciting field of physics. In many cases we present not only results but also derivations of the results. For those who start to study the subject we refer to the books Refs. [58,59,60,31,61] and the reviews Refs. [62,63,64,65,66,67,68,69,70,71,72,73].

Neutrino Mixing

All the numerous data on weak interaction processes are perfectly well described by the Standard Model [74,75,76]. The standard weak interactions are due to the coupling of quarks and leptons with the gauge $W$ and $Z$ vector bosons, described by the charged-current (CC) and neutral-current (NC) interaction Lagrangians

$$\mathcal{L}_{I}^{\mathrm{CC}} = - \frac{g}{2\sqrt{2}} \, j^{\mathrm{CC}}_{\rho} \, W^{\rho} + \mathrm{h.c.} \,,$$ (2.1)

$$\mathcal{L}_{I}^{\mathrm{NC}} = - \frac{g}{2\cos\theta_{W}} \, j^{\mathrm{NC}}_{\rho} \, Z^{\rho} \,.\setlength{\arraycolsep}{\templength}$$ (2.2)

Here $g$ is the SU(2)$_L$ gauge coupling constant, $\theta_{W}$ is the weak angle and the charged and neutral currents $j^{\mathrm{CC}}_{\rho}$ and $j^{\mathrm{NC}}_{\rho}$ are given by the expressions

$$j^{\mathrm{CC}}_{\rho} = 2 \sum_{\ell=e,\mu,\tau} \overline{\nu_{{\ell}L}} \, \gamma_{\rho} \, \ell_L + \ldots \,,$$ (2.3)

$$j^{\mathrm{NC}}_{\rho} = \sum_{\ell=e,\mu,\tau} \overline{\nu_{{\... ...gamma_{\rho} \, \nu_{{\ell}L} + \ldots \,,\setlength{\arraycolsep}{\templength}$$ (2.4)

where the $\ell$ are the physical charged lepton fields with masses $m_\ell$ and we have written explicitly only the terms containing the neutrino fields. The flavour neutrinos $\nu_e$, $\nu_{\mu}, \ldots$ are determined by CC weak interactions: for example, the $\nu_{\mu}$ is the particle produced in the decay $\pi^+ \to \mu^+ + \nu_\mu$ and so on. The number of light flavour neutrinos is given by the invisible width of the $Z$ boson [77] in the Standard Model. It was shown in the famous LEP experiments on the measurement of the invisible width of the $Z$ boson that the number of the light flavour neutrinos is equal to 3. The most recent experimental value of the number of neutrino flavours is $2.994 \pm 0.012$ [35], showing that are no other neutrino flavours than the well-known $\nu_e$, $\nu_\mu$, $\nu_\tau$.

The CC and NC interactions conserve the electron $L_e$, muon $L_{\mu}$ and tau $L_{\tau}$ lepton numbers, which are assigned as shown in Table 2.1.

	$L_{e}$	$L_{\mu}$	$L_{\tau}$
$\left( \nu_{e} \, , \, e^{-} \right)$	$+1$	0	0
$\left( \nu_{\mu} \, , \, \mu^{-} \right)$	0	$+1$	0
$\left( \nu_{\tau} \, , \, \tau^{-} \right)$	0	0	$+1$

Table 2.1. Assignment of lepton numbers. The corresponding antiparticles have opposite lepton numbers.

There are no indications in favour of violation of the law of conservation of lepton numbers in weak processes and very strong bounds on the probabilities of the lepton number violating processes have been obtained from the experimental data. The most stringent limits (90% CL) are (see Ref. [35]):

$$\Gamma( \mu \to e \, \gamma ) / \Gamma( \mu \to \mathrm{all} ) < 4.9 \times 10^{-11} \,,$$ (2.5)

$$\Gamma( \mu \to 3 \, e ) / \Gamma( \mu \to \mathrm{all} ) < 1.0 \times 10^{-12} \,,$$ (2.6)

$$\sigma( \mu^- \, \mathrm{Ti} \to e^- \, \mathrm{Ti} ) / \sigma( \mu^- \, \mathrm{Ti} \to \mathrm{capture} ) < 4.3 \times 10^{-12} \,,$$ (2.7)

$$\Gamma( K_L \to e \, \mu ) / \Gamma( K_L \to \mathrm{all} ) < 3.3 \times 10^{-11} \,,$$ (2.8)

$$\Gamma( K^+ \to \pi^+ \, e^- \, \mu^+ ) / \Gamma( K^+ \to \mathrm{all} ) < 2.1 \times 10^{-10} \,.\setlength{\arraycolsep}{\templength}$$ (2.9)

According to the neutrino mixing hypothesis [18,19,21], the conservation of the lepton numbers is only approximate. It is violated because of non-zero neutrino masses and neutrino mixing. In the case of neutrino mixing, the left-handed flavour neutrino fields $\nu_{{\ell}L}$ are superpositions of the left-handed components $\nu_{kL}$ of the fields of neutrinos with definite masses $m_k$:

$$\nu_{{\ell}L} = \sum_{k=1}^{n} U_{{\ell}k} \, \nu_{kL} \qquad (\ell=e,\mu,\tau) \,,$$ (2.10)

where $U$ is a unitary mixing matrix. The number of massive neutrino fields $n$ could be equal or more than 3. If $n$ is larger than 3 there are sterile neutrinos that do not take part in the standard weak interactions (2.3) and (2.4).

It is well-established that quarks take part in CC weak interactions in mixed form with the V$-$A current

$$\sum_{q'=u,c,t} \, \sum_{q=d,s,b} \, \bar{q}'_L\, \gamma_\rho V_{q'q} \, q_L \,,$$ (2.11)

where $V$ is the Cabibbo-Kobayashi-Maskawa mixing matrix [78,79]. The relation (2.10) is analogous to the mixing in Eq.(2.11). However, between the mixing of neutrinos and quarks there can be a fundamental difference. In fact, quarks are four-component Dirac particles: quarks and antiquarks have opposite electric and baryonic charges. Instead, neutrinos are electrically neutral particles. If the total lepton charge

$$L = L_e + L_\mu + L_\tau$$ (2.12)

is conserved, neutrinos with definite masses are four-component Dirac particles like quarks (in this case a neutrino differs from an antineutrino by the opposite value of $L$). If the total lepton number (2.12) is not conserved, massive neutrinos are truly neutral two-component Majorana particles. These possibilities are realized in different models and correspond to different neutrino mass terms.

Dirac mass term

A Dirac neutrino mass term can be generated by the Higgs mechanism with the standard Higgs doublet which is responsible for the generation of the masses of quarks and charged leptons.²In this case the neutrino mass term is given by³

$$\mathcal{L}^{\mathrm{D}} = - \sum_{\ell,\ell'} \overline{\nu... ...\, \nu_{{\ell'}L} + \mathrm{h.c.} \qquad (\ell=e,\mu,\tau) \,,$$ (2.13)

and $M^{\mathrm{D}}$ is a complex $3 \times 3$ matrix.

The Dirac mass term (2.13) can be diagonalized taking into account that the complex matrix $M^{\mathrm{D}}$ can be written as

$$M^{\mathrm{D}} = V \, \hat{m} \, U^\dagger \,,$$ (2.14)

where $V$ and $U$ are unitary matrices and $\hat{m}$ is a positive definite diagonal matrix, $\hat{m}_{kj} = m_k \delta_{kj}$, with $m_k \geq 0$. Hence, the Dirac mass term (2.13) can be written in the diagonal form

$$\mathcal{L}^{\mathrm{D}} = - \sum_{k=1}^{3} m_k \, \overline{\nu_k} \, \nu_k \,,$$ (2.15)

with

$$\nu_{{\ell}L} = \sum_{k=1}^{3} U_{{\ell}k} \, \nu_{kL} \qquad (\ell=e,\mu,\tau) \,.$$ (2.16)

Therefore, in the case of the Dirac mass term (2.13) the three flavour fields $\nu_{{\ell}L}$ ( $\ell=e,\mu,\tau$) are linear unitary combinations of the left-handed components $\nu_{kL}$ of three fields of neutrinos with masses $m_k$ ($k=1,2,3$). On the other hand, we also have

$$\nu_{\ell R} = \sum_{k=1}^{3} V_{\ell k} \, \nu_{kR} \qquad (\ell=e,\mu,\tau) \,,$$ (2.17)

but these right-handed fields do not occur in the standard weak interaction Lagrangian. Therefore, right-handed singlets are sterile and are not mixed in this scheme with the active neutrinos.⁴

The field $\nu_k$ is a Dirac field if not only the mass term (2.13) but also the total Lagrangian is invariant under the global U(1) transformation

$$\nu_\ell \to e^{i\varphi} \, \nu_\ell \,, \qquad \ell \to e^{i\varphi} \, \ell \qquad (\ell=e,\mu,\tau) \,,$$ (2.18)

where the phase $\varphi$ is the same for all neutrino and charged lepton fields. Then, using Noether's theorem, one can see that the invariance of the Lagrangian under the transformation (2.18) implies that the total lepton charge $L$ is conserved. Therefore, $L$ is the quantum number that distinguishes a neutrino from an antineutrino.

The Dirac mass term (2.13) allows processes like $\mu \to e + \gamma$, $\mu^- \to e^- + e^+ + e^-$. However, the contribution of neutrino mixing to the probabilities of such processes is negligibly small [81,82,83].

The unitary $3 \times 3$ mixing matrix $U$ can be written in terms of 3 mixing angles and 6 phases. However, only one phase is measurable [79]. This is due to the fact that in the Standard Model the only term in the Lagrangian where the mixing matrix $U$ enters is the CC interaction Lagrangian (2.1). With neutrino mixing the lepton charged-current is given by

$${j_{\rho}^{\mathrm{CC}}}^\dagger = 2 \sum_{\ell=e,\mu,\tau} ... ...erline{\ell_L} \, \gamma_{\rho} \, U_{{\ell}k} \, \nu_{kL} \,.$$ (2.19)

Because the phases of Dirac fields are arbitrary one can eliminate from the mixing matrix $U$ five phases by a redefinition of the phases of the charged lepton and neutrino fields, with only one physical phase remaining in $U$. The presence of this phase causes the violation of CP invariance in the lepton sector. A convenient parameterization of the mixing matrix $U$ is the one proposed in Ref. [84]:

$$U = \left( \begin{array}{ccc} c_{12} c_{13} & s_{12} c_{13} ... ...s_{13} e^{i\delta_{13}} & c_{23} c_{13} \end{array}\right) \,,$$ (2.20)

where $c_{ij} \equiv \cos\vartheta_{ij}$ and $s_{ij} \equiv \sin\vartheta_{ij}$ and $\delta_{13}$ is the CP-violating phase.

Since in the parameterization (2.20) of the mixing matrix the CP-violating phase $\delta_{13}$ is associated with $s_{13}$, it is clear that CP violation is negligible in the lepton sector if the mixing angle $\vartheta_{13}$ is small. More generally, it is possible to show that if any of the elements of the mixing matrix is zero, the CP-violating phase can be rotated away by a suitable rephasing of the charged lepton and neutrino fields.⁵

Dirac-Majorana mass term

If none of the lepton numbers is conserved and both, left-handed flavour fields $\nu_{\ell L}$ ( $\ell=e,\mu,\tau$) and sterile right-handed gauge singlet fields $\nu_{sR}$ ( $s=s_1,s_2,\ldots$) enter into the mass term we have the so-called Dirac-Majorana mass term

$$\mathcal{L}^{\mathrm{D+M}} = \mathcal{L}^{\mathrm{M}}_L + \mathcal{L}^{\mathrm{D}} + \mathcal{L}^{\mathrm{M}}_R \,,$$ (2.21)

with

$$\mathcal{L}^{\mathrm{D}} = - \sum_{s,\ell} \overline{\nu_{sR}} \, M^{\mathrm{D}}_{s\ell} \, \nu_{{\ell}L} + \mathrm{h.c.} \,,$$ (2.22)

$$\mathcal{L}^{\mathrm{M}}_L = - \frac{1}{2} \sum_{\ell,\ell'} \ove... ...e{(\nu_{{\ell}L})^c} \, M^{L}_{\ell\ell'} \, \nu_{{\ell'}L} + \mathrm{h.c.} \,,$$ (2.23)

$$\mathcal{L}^{\mathrm{M}}_R = - \frac{1}{2} \sum_{s,s'} \overline{... ...{R}_{ss'} \, (\nu_{s'R})^c + \mathrm{h.c.}\setlength{\arraycolsep}{\templength}$$ (2.24)

Here $M^{\mathrm{D}}$, $M^L$ and $M^R$ are complex matrices and the indices $s,s'$ run over $n_R$ values. Let us stress that the number of right-handed singlet fields could be different from the number of neutrino flavours (see [88,30]).

The charge-conjugate fields are defined by

$$(\nu_{{\ell}L})^c \equiv \mathcal{C} \, \overline{\nu_{{\ell... ...d (\nu_{sR})^c \equiv \mathcal{C} \, \overline{\nu_{sR}}^T \,,$$ (2.25)

where $\mathcal{C}$ is the charge conjugation matrix (see Appendix A). Notice that $(\nu_{{\ell}L})^c$ and $(\nu_{sR})^c$ are right-handed and left-handed components, respectively. Indeed, with $\mathcal{C}^{-1} \gamma_5 \mathcal{C} = \gamma_5^T$ one finds

$$\frac{1+\gamma_5}{2} (\nu_{{\ell}L})^c = \mathcal{C} \left(... ...T = \mathcal{C} \overline{\nu_{{\ell}L}}^T = (\nu_{{\ell}L})^c$$ (2.26)

and the analogous reasoning holds for $(\nu_{sR})^c$. Furthermore, we have

$$\overline{(\nu_{{\ell}L})^c} = - \nu_{{\ell}L}^T \, \mathcal... ...\overline{(\nu_{sR})^c} = - \nu_{sR}^T \, \mathcal{C}^{-1} \,.$$ (2.27)

The matrices $M^{L}$ and $M^{R}$ are symmetric. This can be shown with the help of the relation $\nu_{\ell L}^T \mathcal{C}^{-1} \nu_{\ell' L} = \nu_{\ell' L}^T \mathcal{C}^{-1} \nu_{\ell L}$, which follows from the fact that $\mathcal{C}$ is an antisymmetric matrix and from the anticommutation property of fermion fields. An analogous relation holds for the right-handed fields. Then, using Eq.(2.27), one obtains

$$\sum_{\ell,\ell'} \overline{(\nu_{{\ell}L})^c} \, M^{L}_{\el... ...{(\nu_{{\ell}L})^c} \, M^{L}_{\ell'\ell} \, \nu_{{\ell'}L} \,.$$ (2.28)

Let us introduce the left-handed column vector

$$n_L \equiv \left( \begin{array}{c} \displaystyle \nu_L \\ \d... ...{{\mu}L} \\ \displaystyle \nu_{{\tau}L} \end{array}\right) \,,$$ (2.29)

where $\nu_R$ is the column vector of right-handed fields. With the relation

$$\sum_{s,\ell} \overline{\nu_{sR}} \, M^{\mathrm{D}}_{s\ell} ... ...\ell}L})^c} \, (M^{\mathrm{D}})^T_{\ell s} \, (\nu_{sR})^c \,,$$ (2.30)

the Dirac-Majorana mass term $\mathcal{L}^{\mathrm{D+M}}$ can be written in the form

$$\mathcal{L}^{\mathrm{D+M}} = - \frac{1}{2} \, \overline{(n_L)^c} \, M^{\mathrm{D+M}} \, n_L + \mathrm{h.c.} \,,$$ (2.31)

with the symmetric $(3+n_R) \times (3+n_R)$ matrix

$$M^{\mathrm{D+M}} \equiv \left( \begin{array}{cc} \displaysty... ...T \\ \displaystyle M^{\mathrm{D}} & M^R \end{array}\right) \,.$$ (2.32)

The complex symmetric matrix $M^{\mathrm{D+M}}$ can be diagonalized with the help of a unitary matrix $U$ [89,90]:

$$M^{\mathrm{D+M}} = (U^\dagger)^T \, \hat{m} \, U^\dagger \,,$$ (2.33)

with $\hat{m}_{kj} = m_{k} \delta_{kj}$ and $m_{k} \geq 0$. Using this relation the mass term (2.31) can be written as

$$\mathcal{L}^{\mathrm{D+M}} = - \frac{1}{2} \, \overline{N} \... ...{1}{2} \sum_{k=1}^{3+n_R} m_k \, \overline{\nu_k} \, \nu_k \,,$$ (2.34)

with

$$N \equiv \left( \begin{array}{c} \displaystyle \nu_1 \\ \dis... ... \end{array}\right) = U^\dagger \, n_L + (U^\dagger n_L)^c \,.$$ (2.35)

The fields $\nu_k$ satisfy the Majorana condition

$$(\nu_k)^c = \nu_k \qquad (k=1,\ldots,3+n_R) \,.$$ (2.36)

Therefore, in the general case of a Dirac-Majorana mass term the fields of particles with definite masses are Majorana fields. Majorana particles are particles with spin $1/2$ and all charges equal to zero (particle $=$ antiparticle). Some properties of Majorana fields are discussed in Appendix A.

We want to emphasize that it is natural that the diagonalization of the mass term (2.31) leads to fields of Majorana particles with definite masses: the mass term (2.31) for the case of a general matrix $M^{\mathrm{D+M}}$ is not invariant under any global phase transformation. In other words, in the general case of the Dirac and Majorana mass term there are no conserved quantum numbers that allow to distinguish a particle from its antiparticle.

From Eq.(2.35), for the left-handed components of the neutrino fields we have the mixing relations

$$\nu_{{\ell}L} = \sum_{k=1}^{3+n_R} U_{{\ell}k} \, \nu _{kL} \qquad (\ell=e,\mu,\tau) \,,$$ (2.37)

$$(\nu_{sR})^c = \sum_{k=1}^{3+n_R} U_{sk} \, \nu _{kL} \,.\setlength{\arraycolsep}{\templength}$$ (2.38)

Thus, in the case of a Dirac-Majorana mass term the flavour fields $\nu_{\ell L}$ are linear unitary combinations of the left-handed components of the Majorana fields $\nu_k$ of neutrinos with definite masses. These components are also connected with the sterile fields $(\nu_{sR})^c$ through the relation (2.38). If the masses $m_k$ are small, the mixing relations (2.37) and (2.38) imply that flavour neutrinos $\nu_e$, $\nu_{\mu}$ and $\nu_{\tau}$ can oscillate into sterile states that are quanta of the right-handed fields $\nu_{sR}$. We will discuss such transitions in Section 6.

Let us stress that in order to have all three terms in the Dirac-Majorana mass term (2.31) not only a Higgs doublet but also a Higgs triplet [91,92,93] and a Higgs singlet are necessary. Thus, it can be generated only in the framework of models beyond the Standard Model. A typical example is the SO(10) model (see, for example, [31]). For a discussion of radiative corrections to the Dirac-Majorana mass term see Ref. [94].

Up to now we did not make any special assumption about the Dirac-Majorana mass term (2.31). Let us now consider the possibility that CP invariance in the lepton sector holds. In this case we have

$$U_{\mathrm{CP}} \, \mathcal{L}^{\mathrm{D+M}}(x) \, U_{\mathrm{CP}}^{-1} = \mathcal{L}^{\mathrm{D+M}}(x_{\mathrm{P}}) \,.$$ (2.39)

where $U_{CP}$ is the operator of CP conjugation, $x\equiv(x^0,\vec{x})$ and $x_{\mathrm{P}}\equiv(x^0,-\vec{x})$. For the fields $n_{L}(x)$ we take the usual CP transformation

$$U_{\mathrm{CP}} \, n_{L}(x) \, U_{\mathrm{CP}}^{-1} = \eta \... ...mma^0 \, \mathcal{C} \, \overline{n_{L}}^T(x_{\mathrm{P}}) \,,$$ (2.40)

where $\eta$ is a diagonal matrix of phase factors. The mass term $\mathcal{L}^{\mathrm{D+M}}$ can be written in the form

$$\mathcal{L}^{\mathrm{D+M}}(x) = \frac{1}{2} n_L^T(x) \, \mat... ...athrm{D+M}})^\dagger \, \mathcal{C} \, \overline{n_L}^T(x) \,.$$ (2.41)

Using Eq.(2.40) we obtain

$$U_{\mathrm{CP}} \, n_L^T(x) \, \mathcal{C}^{-1} \, M^{\mathr... ...}} \eta \, \mathcal{C} \, \overline{n_L}^T(x_{\mathrm{P}}) \,.$$ (2.42)

Hence, the CP invariance condition (2.39) is satisfied if

$$\eta M^{\mathrm{D+M}} \eta = - (M^{\mathrm{D+M}})^\dagger \,.$$ (2.43)

Up to now $\eta$ was an arbitrary diagonal matrix of phase factors. If we choose $\eta = i$, in this case CP invariance in the lepton sector implies that

$$M^{\mathrm{D+M}} = (M^{\mathrm{D+M}})^\dagger \,.$$ (2.44)

Taking into account that $M^{\mathrm{D+M}}$ is a symmetric matrix, the condition (2.44) is equivalent to

$$M^{\mathrm{D+M}} = (M^{\mathrm{D+M}})^* \,.$$ (2.45)

The real symmetric matrix $M^{\mathrm{D+M}}$ can be diagonalized with the transformation

$$M^{\mathrm{D+M}} = \mathcal{O} \, m' \, \mathcal{O}^T \,,$$ (2.46)

where $\mathcal{O}$ is an orthogonal matrix ( $\mathcal{O}^T=\mathcal{O}^{-1}$) and $m'$ is a diagonal matrix, $m'_{kj} = m'_k \delta_{kj}$. The eigenvalues $m'_k$ can be positive or negative and the neutrino masses are thus given by $m_k = \vert m'_k\vert$. Therefore, we write the eigenvalues as

$$m'_k = m_k \, \rho_k \,,$$ (2.47)

where $\rho_k = \pm 1$ is the sign of the $k^{\mathrm{th}}$ eigenvalue of the matrix $M^{\mathrm{D+M}}$. Then the relation (2.46) can be rewritten in the form

$$M^{\mathrm{D+M}} = (U^\dagger)^T \, \hat{m} \, U^\dagger$$ (2.48)

with

$$U^\dagger = \sqrt{\rho} \, \mathcal{O}^T$$ (2.49)

and $\hat{m}_{kj} = m_k \, \delta_{kj}$. Here $\rho$ is the diagonal matrix with elements $\rho_{kj} = \rho_k \delta_{kj}$. Therefore, if CP is conserved in the lepton sector, the neutrino mixing matrix has the simple structure shown in Eq.(2.49), which implies that

$$U^* = U \, \rho \,.$$ (2.50)

The CP transformation of the Majorana field $\nu_k$ is given by (see Appendix A)

$$U_{\mathrm{CP}} \, \nu _k(x) \, U_{\mathrm{CP}}^{-1} = \eta^{\mathrm{CP}}_k \, \gamma_0 \, \nu _k(x_\mathrm{P}) \,,$$ (2.51)

where $\eta^{\mathrm{CP}}_k$ is the CP parity of the Majorana field $\nu_k$ which is $\eta^{\mathrm{CP}}_k = \pm i$ (see Ref. [65]). From Eq.(2.51) we have

$$U_{\mathrm{CP}} \, \nu _{kL}(x) \, U_{\mathrm{CP}}^{-1} = \eta^{\mathrm{CP}}_k \, \gamma_0 \, \nu _{kR}(x_\mathrm{P}) \,.$$ (2.52)

This relation and Eqs.(2.35), (2.40) and (2.50) imply that the CP parity of the Majorana field $\nu_k$ is [95,96,97]

$$\eta^{\mathrm{CP}}_k = i \, \rho_k \,.$$ (2.53)

Indeed, we have

$$U_{\mathrm{CP}} \, N_{L}(x) \, U_{\mathrm{CP}}^{-1} = U^\dag... ...mathrm{P}}) = i \, \rho \, \gamma_0 \, N_R(x_{\mathrm{P}}) \,.$$ (2.54)

Majorana mass term

If only left-handed neutrino flavour fields $\nu_{{\ell}L}$ ( $\ell=e,\mu,\tau$) enter into the Lagrangian we can write down the mass term [23,98]

$$\mathcal{L}^{\mathrm{M}} = - \frac{1}{2} \sum_{\ell,\ell'} \... ...{(\nu_{{\ell}L})^c} \, M^{L}_{\ell\ell'} \, \nu_{{\ell'}L} \,,$$ (2.55)

where $M^{L}$ is a symmetric complex matrix (see previous section). Then we have for the mixing

$$\nu_{{\ell}L} = \sum_{k=1}^{3} U_{{\ell}k} \, \nu_{kL} \,,$$ (2.56)

where $\nu_k$ is the field of Majorana neutrinos with mass $m_k$. In this case the number of massive Majorana neutrinos is equal to the number of lepton flavours (three). Note that the generation of the Majorana mass term (2.55) requires an enlargement of the scalar sector of the Minimal Standard Model [99]: with a Higgs triplet like in the Majoron models Ref. [91,92,93] Majorana masses are obtained at the tree level, whereas radiative generation is possible at the 1-loop level with a singly charged Higgs singlet (plus an additional Higgs doublet) [100,101,102,103,104,31]) or at the 2-loop level with a doubly charged Higgs singlet (plus an additional singly charged scalar) [105].

Since the Majorana condition (2.36) does not allow the rephasing of the neutrino fields, only three of the six phases in the $3 \times 3$ mixing matrix $U$ can be absorbed into the charged lepton fields in the charged current (2.19). Therefore, in the Majorana case the mixing matrix contains three CP-violating phases [28,29,106] in contrast to the single CP-violating phase of the Dirac case discussed in Section 2.1. However, the additional CP-violating phases in the Majorana case have no effect on neutrino oscillations in vacuum [28,29,106] (see Section 3) as well as in matter [107].

The one-generation case

Let us consider the Dirac-Majorana mass term in the simplest case of one generation. We have

$$\mathcal{L}^{\mathrm{D+M}} \null \null = \null \null - \frac{1}{2} \, m_{L} \, \overline{(\nu_L)^c} \, \nu_L - m... ..., \nu_L - \frac{1}{2} \, m_{R} \, \overline{\nu_R} \, (\nu_R)^c + \mathrm{h.c.}$$

$$\null \null = \null \null - \frac{1}{2} \, m_{R} \, \overline{(n_L)^c} \, M \, n_L + \mathrm{h.c.} \,,\setlength{\arraycolsep}{\templength}$$ (2.57)

with

$$n_L \equiv \left( \begin{array}{c} \displaystyle \nu_L \\ \d... ...m{D}} \null & \null \displaystyle m_{R} \end{array}\right) \,.$$ (2.58)

For simplicity we assume CP invariance in the lepton sector (see Subsection 2.2). In this case $m_{L}$, $m_{\mathrm{D}}$ and $m_{R}$ are real parameters. In order to diagonalize the matrix $M$, let us write it in the form

$$M = \frac{1}{2} \, \mathrm{Tr}M + M^0 = \frac{1}{2} \, ( m_L + m_R ) + M^0 \,,$$ (2.59)

with

$$M^0 = \left( \begin{array}{cc} \displaystyle - \frac{1}{2} \... ...splaystyle \frac{1}{2} \, ( m_R - m_L ) \end{array}\right) \,.$$ (2.60)

The eigenvalues of the matrix $M^0$ are

$$m^0_{1,2} = \mp \frac{1}{2} \, \sqrt{ ( m_R - m_L )^2 + 4 \, m_{\mathrm{D}}^2 } \,.$$ (2.61)

Furthermore, we have

$$M^0 = \mathcal{O} \, m^0 \mathcal{O}^T \,,$$ (2.62)

where $m^0 = \mathrm{diag}(m^0_1,m^0_2)$ and $\mathcal{O}$ is the orthogonal matrix

$$\mathcal{O} = \left( \begin{array}{rr} \displaystyle \cos\va... ...a \null & \null \displaystyle \cos\vartheta \end{array}\right)$$ (2.63)

with the mixing angle $\vartheta$ given by

$$\cos 2\vartheta = \frac{m_R-m_L}{\sqrt{ ( m_R - m_L )^2 + 4... ...ac{2m_D}{\sqrt{ ( m_R - m_L )^2 + 4 \, m_{\mathrm{D}}^2 }} \,.$$ (2.64)

Hence, the matrix $M$ can be written as

$$M = \mathcal{O} \, m' \, \mathcal{O}^T \,,$$ (2.65)

where $m' = \mathrm{diag}(m'_1,m'_2)$ and

$$m'_{1,2} = \frac{1}{2} \left[ ( m_R + m_L ) \mp \sqrt{ ( m_R - m_L )^2 + 4 \, m_{\mathrm{D}}^2 } \right] \,.$$ (2.66)

The eigenvalues of the matrix $M$ are real but can have positive or negative sign. Let us write them as

$$m'_k = m_k \, \rho_k \,,$$ (2.67)

where $m_k = \vert m'_k\vert$ and $\rho_k$ is the sign of the $k^{\mathrm{th}}$ eigenvalue of the matrix $M$. As shown in Eq.(2.53), the CP parity of the Majorana field $\nu_k$ with definite mass $m_k$ is $\eta^{\mathrm{CP}}_k = i \rho_k$. The relation (2.65) can be written in the form

$$M = (U^\dagger)^T \, \hat{m} \, U^\dagger \,,$$ (2.68)

with $\hat{m} = \mathrm{diag}(m_1,m_2)$ and

$$U^\dagger \equiv \sqrt{\rho} \, \mathcal{O}^T \,,$$ (2.69)

with $\rho = \mathrm{diag}(\rho_1,\rho_2)$. Using now the general formulas obtained in Section 2.2, we have the mixing relation

$$\left( \begin{array}{c} \nu_L \\ (\nu_R)^c \end{array} \righ... ... \begin{array}{c} \nu_{1L} \\ \nu_{2L} \end{array} \right) \,,$$ (2.70)

with

$$U = \left( \begin{array}{rr} \displaystyle (\sqrt{\rho_1})^*... ...tyle (\sqrt{\rho_2})^* \, \cos\vartheta \end{array}\right) \,.$$ (2.71)

Therefore, the three parameters $m_{L}$, $m_{\mathrm{D}}$, $m_{R}$ are related with the mixing angle $\vartheta$ and the neutrino masses $m_k$ by the relations (2.64) and (2.66), (2.67). The signs of the eigenvalues of $M$ determine the CP parities of the massive Majorana fields $\nu_k$.

In the framework of CP conservation, the relations obtained so far are general. In the following part of this section we consider some particular cases with special physical significance.

The see-saw mechanism

Let us consider the Dirac-Majorana mass term (2.57) and assume [32,33,34] that $m_{L} = 0$, $m_{\mathrm{D}} \simeq m^f$, where $m^{f}$ is the mass of a quark or a charged lepton of the same generation, and $m_R \simeq \mathcal{M} \gg m^f$. In this case, from the relations (2.64), (2.66) and (2.67) we have

$$\vartheta \simeq \frac{m^f}{\mathcal{M}} \ll 1 \,,$$ (2.72)

$$m_1 \simeq \frac{(m^f)^2}{\mathcal{M}} \ll m^f \,, \qquad \rho_1 = -1 \,,$$ (2.73)

$$m_2 \simeq \mathcal{M} \,, \qquad \qquad \rho_2 = 1 \,.\setlength{\arraycolsep}{\templength}$$ (2.74)

These relations imply that approximately

$$\nu_L = - i \, \nu_{1L} \,, \qquad (\nu_R)^c = \nu_{2L} \,,$$ (2.75)

and the Majorana fields $\nu_1$, $\nu_2$ are connected with the fields $\nu_{L}$ and $\nu_{R}$ by

$$\nu_1 = i \, [ \nu_L - (\nu_L)^c ] \,, \qquad \nu_2 = \nu_R + (\nu_R)^c \,.$$ (2.76)

The mechanism which we consider here is called see-saw mechanism [32,33,34]. It is based on the assumption that the conservation of the total lepton number $L$ is violated by the right-handed Majorana mass term at the scale $\mathcal{M}$ that is much larger than the scale of the electroweak symmetry breaking. Several models which implement the see-saw mechanism are possible (see, for example, [31,66,68] and references therein) and the scale $\mathcal{M}$ depends on the model. This scale could be as low as the TeV scale (for example, in left-right symmetric models [108,109,31]) or an intermediate scale, or as high as the grand unification scale $\sim 10^{15} \, \mathrm{GeV}$ or even the Plank scale $\sim 10^{19} \, \mathrm{GeV}$. The great attractiveness of the see-saw model lies in the fact that, through the relation (2.73), it gives an explanation of the smallness of neutrino masses with respect to the masses of other fundamental fermions.⁶

In the case of three generations the see-saw mechanism leads to a spectrum of masses of Majorana particles with three light neutrino masses $m_k$ and three very heavy masses $M_k$ ($k=1,2,3$) of the order of the scale of violation of the lepton numbers. This is realized if the mass matrix (2.32) has $M^L=0$,

$$M^{\mathrm{D+M}} \equiv \left( \begin{array}{cc} \displaysty... ...tyle M^{\mathrm{D}} & \displaystyle M^R \end{array}\right) \,,$$ (2.77)

and $M^R$ is such that all its eigenvalues are much bigger than the elements of $M^{\mathrm{D}}$. In this case the mass matrix is block-diagonalized (up to corrections of order $(M^R)^{-1}M^{\mathrm{D}}$) by the unitary transformation

$$W^T \, M^{\mathrm{D+M}} \, W \simeq \left( \begin{array}{cc}... ...le 0 & \displaystyle M_{\mathrm{heavy}} \end{array}\right) \,,$$ (2.78)

with

$$W \simeq \left( \begin{array}{cc} \displaystyle 1 - \frac{1}... ...m{D}})^\dagger \, {(M^R)^\dagger}^{-1} \end{array}\right) \,.$$ (2.79)

The matrices for the light and heavy masses are given by [112,113]

$$M_{\mathrm{light}} \simeq -(M^{\mathrm{D}})^T \, (M^R)^{-1} \, M^{\mathrm{D}} \,, \qquad M_{\mathrm{heavy}} \simeq M^R \,.$$ (2.80)

The mass eigenvalues of the light neutrinos are determined by the specific form of $M^{\mathrm{D}}$ and $M^R$. Note that in left-right symmetric models and in SO(10) models the matrix $M^L$ in the big matrix $M^\mathrm{D+M}$ (2.32) can be important (see, e.g., Ref. [114]).

Two simple possibilities are discussed in the literature (see Refs. [115,116]):

1. If $M^R = \mathcal{M} \, I$, where $I$ is the identity matrix, one obtains the quadratic see-saw,
   

   $$M_{\mathrm{light}} \simeq - \frac{ (M^{\mathrm{D}})^T \, M^{\mathrm{D}} }{ \mathcal{M} } \,,$$ (2.81)

   
   
   and the light neutrino masses are given by
   

   $$m_k = \frac{ (m^f_k)^2 }{ \mathcal{M} } \qquad (k=1,2,3) \,,$$ (2.82)

   
   
   where $m^f_k$ is the mass of a quark or a charged lepton of the $k^{\mathrm{th}}$ generation. In this case the neutrino masses $m_k$ scale as the squares of the masses $m^f_k$:
   

   $$m_1 : m_2 : m_3 = (m^f_1)^2 : (m^f_2)^2 : (m^f_3)^2 \,.$$ (2.83)

   
   

2. If $M^R = \frac{ \mathcal{M} }{ \mathcal{M}_{\mathrm{D}} } \, M_{\mathrm{D}}$, where $\mathcal{M}_{\mathrm{D}}$ characterizes the scale of $M_{\mathrm{D}}$, one obtains the linear see-saw (see, for example, Ref. [117]),
   

   $$M_{\mathrm{light}} \simeq - \frac{ \mathcal{M}_{\mathrm{D}} }{ \mathcal{M} } \, M^{\mathrm{D}} \,,$$ (2.84)

   
   
   and the light neutrino masses are given by
   

   $$m_k = \frac{ \mathcal{M}_{\mathrm{D}} }{ \mathcal{M} } \, m^f_k \qquad (k=1,2,3) \,.$$ (2.85)

   
   
   In this case the neutrino masses $m_k$ scale as the masses $m^f_k$:
   

   $$m_1 : m_2 : m_3 = m^f_1 : m^f_2 : m^f_3 \,.$$ (2.86)

   
   

Let us stress that in any case the see-saw mechanism implies the hierarchical relation

$$m_1 \ll m_2 \ll m_3$$ (2.87)

for the three light Majorana neutrino masses.

Effective Lagrangians

In the Standard Model without right-handed singlet neutrino fields there are no renormalizable interactions that give masses to the neutrinos after the spontaneous breaking of the $\mathrm{SU(2)}_L \times \mathrm{U(1)}_Y$ symmetry with the Higgs doublet mechanism. However, there is a general belief that the Standard Model is the low-energy manifestation of a more complete theory [118,119] (for reviews see Refs. [120,31]). The effect of this new theory is to induce in the Lagrangian of the Standard Model non-renormalizable interactions which preserve the $\mathrm{SU(2)}_L \times \mathrm{U(1)}_Y$ symmetry above the electroweak symmetry breaking scale, but violate the conservation of lepton and baryon numbers (see Ref. [121] and references therein). These non-renormalizable interactions are operators of dimension $d > 4$ and must be multiplied by coupling constants that have dimension $\mathcal{M}^{4-d}$, where $\mathcal{M}$ is a mass scale characteristic of the new theory. It is clear that the dominant effects at low energies are produced by the operators with lowest dimension.

In the Standard Model the lepton number non-conserving operator with minimum dimension that can generate a neutrino mass is⁷[122,123,124,125,126,127]

$$\frac{ 1 }{ \mathcal{M} } \sum_{\ell,\ell'} \frac{ g_{\ell\e... ...{\hskip-1pt\vec{\hskip1pt\sigma}}\, \phi ) + \mathrm{h.c.} \,,$$ (2.88)

where $g$ is a $3 \times 3$ matrix of coupling constants, $\ensuremath{\hskip-1pt\vec{\hskip1pt\sigma}}$ are the Pauli matrices, $L_\ell$ are the standard leptonic doublets and $\phi$ is the standard Higgs doublet:

$$L_\ell \equiv \left( \begin{array}{c} \displaystyle \nu_{{\e... ...le \varphi^+ \\ \displaystyle \varphi^0 \end{array}\right) \,.$$ (2.89)

The effective operator in Eq.(2.88) has dimension five and the coupling constant is proportional to $\mathcal{M}^{-1}$.

When $\mathrm{SU(2)}_L \times \mathrm{U(1)}_Y$ is broken by the vacuum expectation value $v / \sqrt{2}$ of $\varphi^0$, the effective interaction (2.88) generates the Majorana mass term

$$\mathcal{L}^{\mathrm{M}} = \frac{1}{2} \, \frac{ v^2 }{ \mat... ...}L})^c} \, g_{\ell\ell'} \, \nu_{{\ell'}L} + \mathrm{h.c.} \,,$$ (2.90)

where $v \simeq 246$ GeV. In this mass term there is a suppression factor $v/\mathcal{M}$ which is responsible for the smallness of the neutrino masses [122,123].

Maximal mixing

The expression (2.64) for the mixing angle $\vartheta$ implies that the mixing is maximal, i.e., $\vartheta = \pi/4$, if $m_{R}=m_{L}$. In this case, assuming that $m_{L} \geq \vert m_{\mathrm{D}}\vert$, the Majorana neutrino masses are given by

$$m_{1,2} = m_L \mp m_\mathrm{D} \,.$$ (2.91)

The fields $\nu_{L}$ and $(\nu_{R})^c$ are connected with the fields $\nu_1$ and $\nu_2$ by the relations

$$\nu_L = \frac{1}{\sqrt{2}} \left( \nu_{1L} + \nu_{2L} \right... ... = \frac{1}{\sqrt{2}} \left( - \nu_{1L} + \nu_{2L} \right) \,.$$ (2.92)

The massive Majorana fields are given by

$$\nu_{1,2} = \frac{1}{\sqrt{2}} \left\{ \left[ \nu_L + (\nu_L)^c \right] \mp \left[ \nu_R + (\nu_R)^c \right] \right\} \,.$$ (2.93)

If $m_{R}=m_{L}=0$ and $m_{\mathrm{D}}>0$, the mass term (2.57) is simply a Dirac mass term. Applying Eqs.(2.64) and (2.66), we obtain

$$\vartheta = \frac{\pi}{4} \,, \qquad m'_{1,2} = \mp m_{\mathrm{D}} \,.$$ (2.94)

Using Eq.(2.71) for the mixing matrix $U$, one can see that the fields $\nu_{L}$ and $(\nu_{R})^c$ are connected to the Majorana fields $\nu_1$ and $\nu_2$ by the relations

$$\nu_L = \frac{1}{\sqrt{2}} \left( - i \nu_{1L} + \nu_{2L} \r... ... = \frac{1}{\sqrt{2}} \left( i \nu_{1L} + \nu_{2L} \right) \,,$$ (2.95)

where $\nu_{1,2}$ are Majorana fields with the same mass $m_1 = m_2 = m_{\mathrm{D}}$ and with CP parities $\eta^{\mathrm{CP}}_1 = - i$ and $\eta^{\mathrm{CP}}_2 = + i$ (see Eq.(2.53)). Eq.(2.93) implies for a Dirac field

$$\nu = \frac{1}{\sqrt{2}} \left( - i \nu_{1} + \nu_{2} \right) \,.$$ (2.96)

Thus we arrive at the well-known result that a Dirac field can always be represented as an equal mixture of two Majorana fields with the same mass and opposite CP parities.

One can see this result also directly:

$$\nu = - \frac{i}{\sqrt{2}} \left( i \, \frac{ \nu - \nu^c }{... ... \frac{i}{\sqrt{2}} \, \nu_1 + \frac{1}{\sqrt{2}} \, \nu_2 \,,$$ (2.97)

where

$$\nu_1 = i \, \frac{ \nu - \nu^c }{ \sqrt{2} } \,, \qquad \nu_2 = \frac{ \nu + \nu^c }{ \sqrt{2} }$$ (2.98)

are Majorana fields.

Finally, there is the possibility that $\vert m_L\vert , \vert m_R\vert \ll m_{\mathrm{D}}$ but at least one of the parameters $m_{L,R}$ is non-zero. In this case Eqs.(2.95) and (2.96) are approximately valid and $\nu_{1,2}$ are two Majorana neutrinos with opposite CP parities and almost degenerate masses given by

$$m_{1,2} \simeq m_{\mathrm{D}} \mp \frac{1}{2} \left( m_L + m_R \right) \,.$$ (2.99)

The field $\nu$ is called in this case a pseudo-Dirac neutrino field [128,62,65,129,130].

Neutrino oscillations in vacuum

The general formalism

If there is neutrino mixing, the left-handed components of the neutrino fields $\nu_{{\alpha}L}$ ( $\alpha=e,\mu,\tau,s_1,s_2,\ldots$) are unitary linear combinations of the left-handed components of the $n$ (Dirac or Majorana) neutrino fields $\nu_k$ ($k=1,\ldots,n$) with masses $m_k$:

$$\nu_{{\alpha}L} = \sum_{k=1}^{n} U_{{\alpha}k} \, \nu_{kL} \,.$$ (3.1)

The number $n$ of massive neutrinos is 3 for the Dirac mass term discussed in Section 2.1 and for the Majorana mass term discussed in Section 2.3, in which cases there are only the three active flavour neutrinos. The number $n$ of massive neutrinos is more than three in the case of a Dirac-Majorana mass term discussed in Section 2.2 with a mixing of both, active and sterile neutrinos. In general, the number of light massive neutrinos can be more than three. We enumerate the neutrino masses in such a way that

$$m_1 \leq m_2 \leq m_3 \leq \ldots \leq m_n \,.$$ (3.2)

In this section we will consider in detail the phenomenon of neutrino oscillations in vacuum which is implied by the mixing relation (3.1).

If all neutrino mass differences are small, a state of a flavour neutrino $\nu_\alpha$ produced in a weak process (as the $\pi^+ \to \mu^+ \nu_\mu$ decay, nuclear beta-decays, etc.) with momentum $p \gg m_k$ is described by the coherent superposition of mass eigenstates (for a discussion of the quantum mechanical problems of neutrino oscillations see Refs. [131,62,132,133,134,135,136,137,61,138,139,140,141,142,143,144,145,146])

$$\vert\nu_\alpha\rangle = \sum_{k=1}^{n} U_{{\alpha}k}^* \, \vert\nu_k\rangle \,.$$ (3.3)

Here $\vert\nu_k\rangle$ is the state of a neutrino with negative helicity, mass $m_k$ and energy

$$E_k = \sqrt{ p^2 + m_k^2 } \simeq p + \frac{ m_k^2 }{ 2 p } \,.$$ (3.4)

Let us assume that at the production point and at time $t=0$ the state of a neutrino is described by Eq.(3.3). According to the Schrödinger equation the mass eigenstates $\vert\nu_k\rangle$ evolve in time with the phase factors $\exp (-i E_k t)$ and at the time $t$ at the detection point we have

$$\vert\nu_\alpha\rangle_t = \sum_{k=1}^{n} U_{{\alpha}k}^* \, e^{ - i E_k t } \, \vert\nu_k\rangle \,.$$ (3.5)

Neutrinos are detected by observing weak interaction processes. Expanding the state (3.5) in the basis of flavour neutrino states $\vert\nu_\beta\rangle$, we obtain

$$\vert\nu_\alpha\rangle_t = \sum_\beta \mathcal{A}_{\nu_\alpha\to\nu_\beta}(t) \, \vert\nu_\beta\rangle \,,$$ (3.6)

where

$$\mathcal{A}_{\nu_\alpha\to\nu_\beta}(t) = \sum_{k=1}^{n} U_{{\beta}k} \, e^{ - i E_k t } \, U_{{\alpha}k}^*$$ (3.7)

is the amplitude of $\nu_\alpha\to\nu_\beta$ transitions at the time $t$ at a distance $L \simeq t$. Consequently, the probability of this transition is given by

$$P_{\nu_\alpha\to\nu_\beta} = \vert\mathcal{A}_{\nu_\alpha\to... ...eta}k} \, e^{ - i E_k t } \, U_{{\alpha}k}^* \right\vert^2 \,.$$ (3.8)

This formula has a very simple interpretation. $U^*_{\alpha k}$ is the amplitude to find the neutrino mass eigenstate $\vert\nu_k\rangle$ with energy $E_k$ in the state of the flavour neutrino $\vert \nu_\alpha \rangle$, the factor $\exp (-i E_k t)$ gives the time evolution of the mass eigenstate and, finally, the term $U_{{\beta}k}$ gives the amplitude to find the flavour neutrino state $\vert\nu_\beta\rangle$ in the mass eigenstate $\vert\nu_k\rangle$. We want to remark that weak interaction processes responsible for neutrino production and detection involve active neutrinos. Therefore, strictly speaking, in the derivation of the Eqs.(3.7) and (3.8) the indices $\alpha$ and $\beta$ take only active flavour indices. However, transitions into sterile neutrinos can be revealed through neutral current neutrino experiments (disappearance of active neutrinos). In this sense, the formulas (3.7) and (3.8) have a meaning also for transitions between active and sterile states.

Notice that in order to have a non-negligible active-sterile transition probability the sterile fields must have a mixing with the light neutrino mass eigenfields the number of which must be more than three. Such a possibility is phenomenologically given by the Dirac-Majorana mass term (2.21), but it is not realized in the simple see-saw scheme discussed in Section 2.5, where the scale of the right-handed Majorana mass term is large. However, the see-saw scenario can be modified by additional assumptions to include light sterile neutrinos (``singular see-saw'' [147], ``universal see-saw [148,149]).

At this point a remark concerning the unitarity of the mixing matrix $U$ is at order. If some of the mass eigenstates are so heavy that they are not produced in the standard weak processes then these mass eigenstates will not occur in the flavour state $\vert \nu_\alpha \rangle$ (3.3). Let us assume that the first $n'$ mass eigenstates are light ($n' < n$). Consequently, only that part of $U$ plays a role in neutrino oscillations where $k \leq n'$. In the following we will always assume that in the situation described here we can confine ourselves to an $n' \times n'$ submatrix of $U$ which is unitary to a good approximation (see, e.g., Ref. [150]). This is realized in the see-saw mechanism with a sufficiently large right-handed scale. In the further discussion we will drop the distinction between $n$ and $n'$.

From the relation (3.1) it follows that the state describing a flavour antineutrino $\bar\nu_\alpha$ is given by

$$\vert\bar\nu_\alpha\rangle = \sum_{k=1}^{n} U_{{\alpha}k} \, \vert\bar\nu_k\rangle \,.$$ (3.9)

Thus the amplitude of $\bar\nu_\alpha\to\bar\nu_\beta$ transitions is given by

$$\mathcal{A}_{\bar\nu_\alpha\to\bar\nu_\beta}(t) = \sum_{k=1}^{n} U_{{\beta}k}^* \, e^{ - i E_k t } \, U_{{\alpha}k} \,.$$ (3.10)

Notice that the amplitude for antineutrino transitions differs from the corresponding amplitude (3.7) for neutrinos only by the exchange $U \to U^*$.

Using the unitarity relation

$$\sum_{k=1}^{n} U_{{\beta}k} \, U_{{\alpha}k}^* = \delta_{\alpha\beta}\,,$$ (3.11)

the probability (3.8) can be written as

$$P_{\nu_\alpha\to\nu_\beta} = \left\vert \delta_{\alpha\beta}... ...lta{m}^2_{k1} L }{ 2 E } \right) - 1 \right] \right\vert^2 \,,$$ (3.12)

where $\Delta m^2_{kj} \equiv m^2_k - m^2_j$ and the ultrarelativistic approximation (3.4) has been used. Thus the probability of $\nu_\alpha\to\nu_\beta$ transitions depends on the elements of the mixing matrix, on $n-1$ independent mass-squared differences and on the parameter $L/E$, whose range is determined by the experimental setup.

If there is no mixing ($U=I$) or/and $\Delta{m}^2_{k1} L / E \ll 1$ for all $k=2,\ldots,n$, there are no transitions ( $P_{\nu_\alpha\to\nu_\beta} = \delta_{\alpha\beta}$). Neutrino transitions can be observed only if neutrino mixing takes place and at least one⁸$\Delta{m}^2$ satisfies the condition

$$\Delta{m}^2 \gtrsim \frac{E}{L} \,.$$ (3.13)

In this inequality, $\Delta{m}^2$ is the neutrino mass-squared in eV$^2$, $L$ is the distance between neutrino source and detector in m (km) and $E$ is neutrino energy in MeV (GeV). Thus, the larger the value of the parameter $L/E$, the smaller are values of $\Delta{m}^2$ which can be probed in an experiment. The inequality (3.13) allows to estimate (for large mixing angles) the sensitivity to the parameter $\Delta m^2$ of different types of neutrino oscillation experiments. These estimates are presented in Table 3.1. Let us stress that this table gives only a very rough idea of the sensitivity to $\Delta m^2$. For example, in the LSND short-baseline (SBL) experiment antineutrino energies are between 20 and 60 MeV, the distance is approximately 30 m and the minimal value of $\Delta m^2$ probed in this experiment is $4 \times 10^{-2}$ eV$^2$.

Experiment	$L$ (m)	$E$ (MeV)	$\Delta{m}^2$ (eV$^2$)
Reactor SBL	$10^2$	$1$	$10^{-2}$
Reactor LBL	$10^3$	$1$	$10^{-3}$
Accelerator SBL	$10^3$	$10^3$	$1$
Accelerator LBL	$10^6$	$10^3$	$10^{-3}$
Atmospheric	$10^7$	$10^3$	$10^{-4}$
Solar	$10^{11}$	$1$	$10^{-11}$

Table 3.1. Order of magnitude estimates of the values of $\Delta{m}^2$ which can be probed in reactor short-baseline (SBL) and long-baseline (LBL), accelerator SBL and LBL, atmospheric and solar neutrino oscillation experiments. Note, however, that the energies and distances of the various types of experiments can vary in a wide range and only some representative values are given in this table.

The probability (3.8) and the corresponding one for antineutrinos are invariant under the phase transformation

$$U_{{\alpha}k} \to e^{-i\varphi_\alpha} \, U_{{\alpha}k} \, e^{i\psi_k}$$ (3.14)

Therefore, it is clear that the probabilities of $\nu_\alpha\to\nu_\beta$ and $\bar\nu_\alpha\to\bar\nu_\beta$ transitions do not depend [28,29,106] on the Majorana CP-violating phases discussed at the end of Section 2.3 and it is not possible to distinguish the Dirac and Majorana cases by the observation of neutrino oscillations.

Comparing the expressions (3.7) and (3.10) for the transition amplitudes of neutrinos and antineutrinos we see that $\mathcal{A}_{\nu_\alpha\to\nu_\beta}(t) = \mathcal{A}_{\bar\nu_\beta\to\bar\nu_\alpha}(t)$. Therefore, for the transition probabilities we have

$$P_{\nu_\alpha\to\nu_\beta} = P_{\bar\nu_\beta\to\bar\nu_\alpha} \,.$$ (3.15)

This relation is a consequence of CPT invariance inherent in any local field theory [151]. From the equality (3.15) it follows that the neutrino and antineutrino survival probabilities are equal:

$$P_{\nu_\alpha\to\nu_\alpha} = P_{\bar\nu_\alpha\to\bar\nu_\alpha} \,.$$ (3.16)

On the other hand, the transition probabilities of neutrinos and antineutrinos in general are different for $\alpha \neq \beta$. They are equal only if there is CP invariance in the lepton sector. In fact, in the case of massive Dirac neutrinos the phases of the neutrino fields and of the charged lepton fields can be chosen in such a way that the mixing matrix $U$ is real. In the case of Majorana neutrinos, CP invariance implies that (see Eq.(2.50))

$$U_{{\alpha}k}^* = U_{{\alpha}k} \, \rho_k \,,$$ (3.17)

with $\rho_k = - i \, \eta^{\mathrm{CP}}_k = \pm 1$, where $\eta^{\mathrm{CP}}_k$ is the CP parity of the Majorana neutrinos with mass $m_k$ (see Eq.(2.53)). Thus, in the Dirac as well as in the Majorana case, CP invariance implies that

$$P_{\nu_\alpha\to\nu_\beta} = P_{\bar\nu_\alpha\to\bar\nu_\beta} \,.$$ (3.18)

Oscillations in the two-neutrino case

The results of neutrino oscillation experiments are usually analysed under the simplest assumption of oscillations between two neutrino types. In this case, for the transition probability (3.12) we get

$$P_{\nu_\alpha\to\nu_\beta} = \left\vert \delta_{\alpha\beta}... ...{ \Delta{m}^2 L }{ 2 E } \right) - 1 \right] \right\vert^2 \,,$$ (3.19)

where $\Delta m^2 = m_2^2 - m_1^2$ and $\alpha$, $\beta$ are $e$, $\mu$, or $\mu$, $\tau$, etc$\ldots$ Thus in the simplest case of transitions between two neutrino types the probability is determined only by the elements of $U$ which connect flavour neutrinos with $\nu_2$ (or $\nu_1$). It is obvious that phases drop out in the expression (3.19). This is an illustration of Eq.(3.14) and of the fact that no information about CP violation can be obtained in the case of transitions between only two neutrino types. If we put $U_{\alpha 2} = \sin \vartheta$ then we have $U_{\beta 2} = \cos \vartheta$ and the transition and survival probabilities are given by the following standard expressions valid for both, neutrinos and antineutrinos:

$$P_{\stackrel{\makebox[0pt][l] {$\hskip-3pt\scriptscriptstyle(-)$}... ...ft( 1 - \cos \frac{ \Delta{m}^2 L }{ 2 E } \right) \qquad (\alpha\neq\beta) \,,$$ (3.20)

$$P_{\stackrel{\makebox[0pt][l] {$\hskip-3pt\scriptscriptstyle(-)$}... ...\scriptscriptstyle(-)$}}{\nu_{\beta}}} \,.\setlength{\arraycolsep}{\templength}$$ (3.21)

The transition probability (3.20) can be written in the form

$$P_{\stackrel{\makebox[0pt][l] {$\hskip-3pt\scriptscriptstyle... ...rac{ \Delta{m}^2 L }{ E } \right) \qquad (\alpha\neq\beta) \,.$$ (3.22)

where $L$ is the source - detector distance expressed in m (km), $E$ is the neutrino energy in MeV (GeV) and $\Delta{m}^2$ is the neutrino mass-squared difference in eV$^2$. Thus, the transition probability is a periodic function of $L/E$. This phenomenon is called neutrino oscillations. The amplitude of oscillations is $\sin^22\vartheta$ and the oscillation length is given by

$$L^{\mathrm{osc}} = \frac{ 4 \, \pi \, E }{ \Delta{m}^2 } \si... ...athrm{MeV}) }{ \Delta{m}^2 \ (\mathrm{eV}^2) } \: \mbox{m} \,.$$ (3.23)

The condition (3.13) can be rewritten in the form

$$L^{\mathrm{osc}} \lesssim L \,.$$ (3.24)

Therefore, neutrino oscillations can be observed if the oscillation length is not much larger than the source - detector distance $L$.

The oscillatory behaviour of the transition probability (3.20) with $\sin^22\vartheta=1$ is shown in Fig. 3.1, where we have plotted it as a function of $\Delta{m}^2 L / 4 \pi E = L / L^{\mathrm{osc}}$. The grey line represents the transition probability (3.20), whereas, in order to demonstrate the effect of energy averaging, the black line represents the transition probability (3.20) averaged over a Gaussian energy distribution with mean value $E$ and standard deviation $\sigma = E/10$. The averaged probability is the measurable quantity in neutrino oscillation experiments. One can see that the averaging over the energy spectrum practically reduces the probability to the constant $1 - \sin^22\vartheta / 2$ for $L \gg L^{\mathrm{osc}}$.

The expressions (3.20) and (3.21) are usually employed in analyses of the data of neutrino oscillation experiments. In many SBL experiments with neutrinos from reactors and accelerators, no indication in favour of neutrino oscillations was found. The data of these experiments give an upper bound for the transition probability which implies an excluded region in the space of the parameters $\Delta{m}^2$ and $\sin^2{2\vartheta}$. A typical exclusion plot is presented in Fig. 3.2 [152]. This plot shows the exclusion curves in the $\nu_\mu\to\nu_\tau$ channel obtained in the CDHS [153], FNAL E531 [154], CHARM II [155], CCFR [156], CHORUS [157] and NOMAD [158] experiments. The excluded region lies on the right of the curves.

The two most stringent exclusion curves in Fig. 3.2 have been obtained in the CHORUS [157] and NOMAD [158] experiments, which are operating at CERN using the neutrino beam from the SPS (with an average energy of about 30 GeV). 800 kg of emulsions are used in the CHORUS experiment as target. The production and decay of $\tau$'s in the emulsion is searched for. In the NOMAD experiment a magnetic detector is used and the production of $\tau$'s is identified with kinematical criteria.

	Figure 3.2. Exclusion curves (90% CL) in the $\nu_\mu\to\nu_\tau$ channel obtained in the CDHS [153], FNAL E531 [154], CHARM II [155], CCFR [156], CHORUS [157] and NOMAD [158] experiments.	Figure 3.3. Exclusion curves (90% CL) in the $\bar\nu_e\to\bar\nu_e$ channel obtained in the CHOOZ [159], Gösgen [160], Krasnoyarsk [161] and Bugey [162] experiments. The shadowed region is allowed by the results of the Kamiokande atmospheric neutrino experiment [49].

Figure 3.3 shows the exclusion curves obtained in the CHOOZ [159], Gösgen [160], Krasnoyarsk [161] and Bugey [162] reactor $\bar\nu_e\to\bar\nu_e$ experiments. The region allowed by the results of the Kamiokande atmospheric neutrino experiment [49] (see Section 5.1) is also depicted in this figure. The Gösgen, Krasnoyarsk and Bugey experiments are SBL reactor experiments, whereas the recent CHOOZ experiment is the first LBL reactor experiment. In this experiment the detector (5 tons of liquid scintillator loaded with Gd) is at the distance of about 1 km from the CHOOZ power station, which has two water reactors with a total thermal power of 8.5 GW. The antineutrinos are detected through the observation of the reaction

$$\bar\nu_e + p \to e^+ + n$$ (3.25)

(the photons from annihilation of the positron and the delayed photons from the capture of neutron by Gd are detected). No indications in favour of neutrino oscillations were found in the CHOOZ experiment. The ratio $R$ of the numbers of measured antineutrino events and of expected antineutrino events in the case of absence of neutrino oscillations in the CHOOZ detector is [159]

$$R = 0.98 \pm 0.04 \pm 0.04 \,.$$ (3.26)

Since the average value of $L/E$ in the CHOOZ experiment is approximately 300 ( $\langle{E}\rangle \simeq 3 \, \mathrm{MeV}$, $L \simeq 1000 \, \mathrm{m}$), in this experiment it is possible to probe the value of the relevant neutrino mass-squared difference down to $10^{-3} \, \mathrm{eV}^2$.

With the help of the expression (3.22), it is possible to understand qualitatively the general features of exclusion curves. In the region of large $\Delta{m}^2$ such that the oscillation length is much smaller than the source - detector distance $L$, the cosine in the expression (3.22) oscillates very rapidly as a function of the neutrino energy $E$. Since in practice all neutrino beams have an energy spectrum and the neutrino sources and detectors are extended in space, only the average transition probability

$$\langle P_{\stackrel{\makebox[0pt][l] {$\hskip-3pt\scriptscr... ...(-)$}}{\nu_{\beta}}} \rangle = \frac{1}{2} \, \sin^22\vartheta$$ (3.27)

can be determined in the region of large $\Delta m^2$. The average probability is independent from $\Delta{m}^2$ and, therefore, from an experimental upper bound $\langle P_{\stackrel{\makebox[0pt][l] {$\hskip-3pt\scriptscriptstyle(-)$}}{\nu... ...{\makebox[0pt][l] {$\hskip-3pt\scriptscriptstyle(-)$}}{\nu_{\beta}}} \rangle_0$ on $\langle P_{\stackrel{\makebox[0pt][l] {$\hskip-3pt\scriptscriptstyle(-)$}}{\nu... ...el{\makebox[0pt][l] {$\hskip-3pt\scriptscriptstyle(-)$}}{\nu_{\beta}}} \rangle$ one obtains the vertical-line part of the exclusion curve.

At

$$\Delta{m}^2 \simeq \frac{\pi}{2} \, \frac{\langle E \rangle}{1.27 \, \langle L \rangle} \,,$$ (3.28)

where $\langle E \rangle$ is the average energy and $\langle L \rangle$ is the average distance, the parameter $\sin^22\vartheta$ has the minimal value

$$(\sin^22\vartheta)_{\mathrm{min}} = \langle P_{\stackrel{\m... ...] {$\hskip-3pt\scriptscriptstyle(-)$}}{\nu_{\beta}}} \rangle_0$$ (3.29)

on the boundary curve. Note that we are using here and in the rest of the section the same units as in Eq.(3.22).

Typically, the upper bound $\langle P_{\stackrel{\makebox[0pt][l] {$\hskip-3pt\scriptscriptstyle(-)$}}{\nu... ...{\makebox[0pt][l] {$\hskip-3pt\scriptscriptstyle(-)$}}{\nu_{\beta}}} \rangle_0$ is much less than one. Then, in the region where $\sin^22\vartheta$ is large the expression (3.22) for the transition probability can be approximated by

$$P_{\stackrel{\makebox[0pt][l] {$\hskip-3pt\scriptscriptstyle... ...2\vartheta \left( 1.27\, \frac{ \Delta{m}^2 L }{ E } \right)^2$$ (3.30)

and in this region the boundary curve in the exclusion plot is given by

$$\Delta{m}^2 \simeq \frac { \sqrt{ \langle P_{\stackrel{\make... ...vartheta \, \langle L^2 \rangle \langle E^{-2} \rangle }} \,.$$ (3.31)

Therefore, this part of the exclusion curve is a straight line in the $\log\sin^22\vartheta$ - $\log\Delta{m}^2$ plot as can be seen from Fig. 3.2. From Eq.(3.31) it follows that the minimal value of the parameter $\Delta{m}^2$ that can be probed by an experiment is

$$\Delta{m}^2 \simeq \frac { \sqrt{ \langle P_{\stackrel{\make... ...27 \, \sqrt{\langle L^2 \rangle \langle E^{-2} \rangle }} \,.$$ (3.32)

It corresponds to $\sin^22\vartheta=1$.

Neutrino oscillations and transitions in matter

The effective Hamiltonian for neutrinos in matter

It has been pointed out by Wolfenstein [163] and by Mikheyev and Smirnov [164] that the neutrino oscillation pattern in vacuum can get significantly modified by the passage of neutrinos through matter because of the effect of coherent forward scattering. This effect can be described by an effective Hamiltonian. Starting with neutrino oscillations in vacuum, one can easily check that the transition probability (3.12) can be obtained by considering the evolution of the state vector $\psi$ of the neutrino types with the ``Schrödinger equation''

$$i \frac{\mathrm{d}\psi(x)}{\mathrm{d}x} = H^{\mathrm{vac}}_\nu \psi(x) = \frac{1}{2E} \, U \hat{m}^2 U^\dagger \psi(x) \,,$$ (4.1)

where $H^{\mathrm{vac}}_\nu$ is the effective Hamiltonian for neutrino oscillations in vacuum with $U$ being the mixing matrix, $E$ the neutrino energy and $\hat{m}$ the diagonal neutrino mass matrix. The presence of matter will give a correction to Eq.(4.1). Note that the variable in Eq.(4.1) is not time but space and the components of $\psi(x)$ are given by the amplitudes denoted by $a_\alpha(x)$ for the neutrino types $\alpha = e,\mu,\tau,s$. The corresponding effective Hamiltonian for antineutrinos $H^{\mathrm{vac}}_{\bar \nu}$ is obtained from $H^{\mathrm{vac}}_\nu$ by the exchange $U \to U^*$. After the seminal work of Wolfenstein about neutrino oscillations in matter (see also Ref. [165]) and the discovery of the importance of $H^{\mathrm{mat}}_\nu$, the analogue of $H^{\mathrm{vac}}_\nu$ in matter, for the solar neutrino problem [164] many rederivations of the effective matter Hamiltonian [166,167] have been presented using a coupled system of Dirac equations [168,169], field theory [170] or studying coherent forward scattering in more detail [171].

We want to sketch a derivation using the Dirac equation and following Ref. [169]. The starting point in most derivations is the expectation value of the currents for isotropic non-relativistic matter given by [163,164,172]

$$\langle \bar f_L \gamma_\mu f_L \rangle_{\mathrm{matter}} = \frac{1}{2} N_f \delta_{\mu 0} \,,$$ (4.2)

where $N_f$ is the number density of the particles represented by the field $f$. Note that the $\gamma_5$ term does not contribute for non-aligned spins [173]. Eq.(4.2) is a good approximation even for electrons in the core of the sun with a temperature $T \simeq 16 \times 10^6$ K [174,175] and thus an average velocity of the electrons of around 10% of the velocity of light and corrections of order $kT/m_e \simeq 2.6 \times 10^{-3}$. Starting from the weak interaction Lagrangians (2.1) and (2.2) one gets for low-energy neutrino interactions of flavour $\ell$ with the background matter

$$- \mathcal{L}^{\mathrm{mat}}_{\nu_\ell} = \frac{G_F}{\sqrt{... ..._f N_f(\delta_{\ell f} + T_{3f_L} - 2 \sin^2 \theta_W Q_f) \,,$$ (4.3)

where $G_F$ is the Fermi coupling constant, $\theta_W$ the weak mixing angle, $T_{3f_L}$ the eigenvalue of the field $f_L$ of the third component of the weak isospin ($T_{3f_R} = 0$ in the Standard Model) and $Q_f$ is the charge of $f$. In the matter Lagrangian (4.3) the charged current interaction is represented by the Kronecker symbol $\delta_{\ell f}$ saying that for neutrinos of flavour $\ell$ the charged current only contributes when background matter containing charged leptons of the same flavour is present. Concentrating now on realistic matter with electrons, protons and neutrons which is electrically neutral, i.e., $N_e = N_p$, and taking into account that we have $T_{3e_L} = -T_{3p_L} = T_{3n_L} = -1/2$ and $Q_e = -Q_p = -1$, $Q_n = 0$ for electrons, protons and neutrons, respectively,⁹we get an effective Hamiltonian

$$H^D_\nu = -i \vec{\alpha} \cdot \vec{\nabla} + \beta (M P_L... ...ac{1}{2} N_n, - \frac{1}{2} N_n, - \frac{1}{2} N_n \right) P_L$$ (4.4)

acting on the vector $\Psi(t,x)$ of the flavour neutrino wave functions, where we have defined $P_{L,R} = ( \mathbf{1} \mp \gamma_5 )/2$, $\alpha_j = \gamma^0 \gamma^j$ and $\beta = \gamma^0$. $M$ is the non-diagonal mass matrix. The last term in Eq.(4.4) is called the matter potential term. Several remarks concerning the Hamiltonian (4.4), which is the point of departure for deriving the effective matter Hamiltonian in Refs. [168,169], are at order:

1. $H^D_\nu$ has spinor and flavour indices, it leads thus to a system of Dirac equations coupled via the mass term.
2. The neutral current contributions of electrons and protons cancel for realistic matter because of opposite $T_{3f_L}$ quantum numbers and electric charges (see discussion above before Eq.(4.4)).
3. The Hamiltonian (4.4) is valid for Dirac neutrinos. For antineutrinos, $M$ is replaced by $M^T$ and the matter potential term of $H^D_{\bar \nu}$ has the opposite sign and the projector $P_R$ instead of $P_L$. This is a consequence of proceeding along the same lines as before but using the charge-conjugate fields to describe antineutrinos ( $\bar \nu_\ell \gamma_\rho P_L \nu_\ell = - \overline{\nu^c_\ell} \gamma_\rho P_R \nu^c_\ell$).
4. In the Majorana case, in order to obtain the Hamiltonian $H^M_\nu$ the antineutrino matter potential has to be added to $H^D_\nu$ and the neutrino field vector $\Psi$ is subject to the Majorana condition $\Psi = \mathcal{C} \gamma_0^T \Psi^*$. This condition is compatible with the time evolution governed by $H^M_\nu$ [169].
5. Sterile neutrinos can easily be incorporated into $H^{D}_{\stackrel{\scriptscriptstyle (-)}{\nu}}$ and $H^M_\nu$ by simply adding the entry 0 to the matter potential term. Note that the probability of active - active transitions does not depend on the neutron density $N_n$, which is, however, important for active - sterile transitions (see Table 4.1).

The essence of deriving an effective Hamiltonian for neutrino oscillations in matter is to get rid of the spinor indices in $H^D_\nu$ and to obtain an equation involving only the indices $\alpha$ for the different neutrino types as in the vacuum case (4.1). To this end, let us now assume that the neutrino propagation proceeds along the $x^3 \equiv x$ axis, that the neutrino momentum $p>0$ corresponding to propagation in vacuum is much larger than the matter potentials and that the neutrinos are ultrarelativistic. Thus we consider a one-dimensional problem from now on. Defining a wave function $\phi$ via

$$\Psi(t,x) = e^{ipx} \phi(x,t) \quad \mbox{with} \quad i \fra... ...i}{\partial t} = \left( p \alpha_3 + H^D_\nu \right) \phi \,,$$ (4.5)

this wave function changes little over distances of the order of the de Broglie wave length of the neutrino. With respect to $\alpha_3$ we can decompose the Hamiltonian (4.4) into $H^D_\nu = H_{\mathrm{even}} + H_{\mathrm{odd}}$ which are the parts commuting or anticommuting with $\alpha_3$. $H_{\mathrm{odd}}$ consists solely of the mass term and $H_{\mathrm{even}}$ contains the rest of $H^D_\nu$. With $p$ being a large parameter we can perform a Foldy - Wouthuysen transformation where we truncate the series at $1/p$ leading to the Hamiltonian [176,169]

$$H^{D}_{\mathrm{FW},\nu} = p \alpha_3 + H_{\mathrm{even}} + \frac{1}{2p} \alpha_3 H_{\mathrm{odd}}^2$$

$$= \alpha_3 \left( p -i \frac{\partial}{\partial x} \right) + \sqrt{... ... \left( N_e - \frac{1}{2} N_n, - \frac{1}{2} N_n, - \frac{1}{2} N_n \right) P_L$$

$$+ \frac{1}{2p}\, \alpha_3 ( M^\dagger M P_L + M M^\dagger P_R ) \,,$$ (4.6)

such that all terms in $H^D_{\mathrm{FW},\nu}$ commute with $\alpha_3$ and, therefore, all matrices in $H^D_{\mathrm{FW},\nu}$ with spinor indices can be diagonalized at the same time in order to separate positive and negative energy states. Denoting this unitary diagonalization matrix by $U_0$ we can achieve $U_0 \alpha_3 U_0^\dagger = \mbox{diag} (1,1,-1,-1)$, $U_0 \frac{\mathbf{1}-\gamma_5}{2} U_0^\dagger = \mbox{diag} (0,1,1,0)$ and $U_0 \frac{\mathbf{1}+\gamma_5}{2} U_0^\dagger = \mbox{diag} (1,0,0,1)$. Note that there are no transitions between left and right-handed states through $H^{D}_{\mathrm{FW},\nu}$. For neutrinos with magnetic moments (electric dipole moments) the method described above would also yield the appropriate left-right transitions [169].

Going back to the Foldy - Wouthuysen transform of the wave function $\Psi$ instead of $\phi$ removes the $p$ from the Hamiltonian (4.6). The final step in deriving the effective matter Hamiltonian consists of considering stationary states and splitting off the plane wave part by

$$\Psi_{\mathrm{FW}}(t,x) = e^{-iE(t-x)} \psi(x) \,.$$ (4.7)

Then the wave function $\psi(x)$, taken for positive energies and left-handed neutrinos, fulfills the first order differential equation (4.1) with $H^{\mathrm{vac}}_\nu$ replaced by [163]

$$H^{\mathrm{mat}}_\nu = \frac{1}{2E} \left( M^\dagger M + 2\... ... N_n, - \frac{1}{2} N_n, - \frac{1}{2} N_n \right) \right) \,,$$ (4.8)

correct up to order $1/E$. This effective Hamiltonian is valid for left-handed Dirac neutrinos or Majorana neutrinos, whereas in the case of right-handed (Dirac) antineutrinos or right-handed Majorana neutrinos we have

$$H^{\mathrm{mat}}_{\bar \nu} = \frac{1}{2E} \left( M^T M^* - ... ... N_n, - \frac{1}{2} N_n, - \frac{1}{2} N_n \right) \right) \,,$$ (4.9)

because for (Dirac) antineutrinos one has to replace $M$ by $M^T$ in the Hamiltonian (4.6) and for Majorana neutrinos $M=M^T$ holds (see also points 3 and 4 after Eq.(4.4)). With

$$M = U_R \hat{m} U_L^\dagger \quad \mbox{and} \quad U_L \equiv U$$ (4.10)

only the left-handed mixing matrix $U$ enters and connection between Eqs.(4.8) and (4.1) is made. Thus in the final results (4.8) and (4.9) there is no distinction between the Dirac and Majorana case for ultrarelativistic neutrinos. This is an illustration of the fact that with neutrino oscillation experiments the Dirac or Majorana nature cannot be distinguished [98,177].

The Hamiltonians (4.8) and (4.9) have been used to investigate neutrino oscillations in the sun, in the earth and in supernovae. In the following we will only be concerned with the first two subjects. Application limits of the neutrino evolution equations in matter have been discussed in Ref. [178]. Elastic and inelastic neutrino scattering introduces quantum damping into the evolution equations which is proportional to the neutrino interaction rate [179]. In the sun and the earth this effect is negligible, in particular, for low energies, whereas in the early universe it is of crucial importance [179]. Density fluctuations have been found to influence considerably neutrino propagation in the sun [180], however, more realistic considerations with helioseismic waves as density fluctuations show no observable effect on the solar neutrino problem discussed in terms of neutrino oscillations [181]. Solar neutrinos are also influenced by their passage through the earth [182].

The two-neutrino case and adiabatic transitions

Let us now concentrate on left-handed neutrinos and specify the effective Hamiltonian (4.8) to two neutrino types. Thus for two-neutrino oscillations in matter with the definitions

$$N(\nu_\alpha) \equiv \delta_{\alpha e} N_e - \frac{1}{2} N_n \quad (\alpha = e, \mu, \tau) \,, \quad N(\nu_s) \equiv 0$$ (4.11)

and

$$A \equiv 2\sqrt{2} G_F E \left( N(\nu_\alpha) - N(\nu_\beta) \right) \,,$$ (4.12)

the differential equation

$$i \frac{\mathrm{d}}{\mathrm{d}x} \left( \begin{array}{c} a_\alpha \\ a_\beta \end{array} \right) = H^{\mathrm{mat}}_\nu \left( \begin{array}{c} a_\alpha \\ a_\beta \end{array} \right)$$

$$= \frac{1}{4E} \left\{ \left[ m_1^2+m_2^2 + 2\sqrt{2} G_F \left( N(... ...t) \right] \left( \begin{array}{cc} 1 & 0 \\ 0 & 1 \end{array} \right) \right.$$

$$+ \left. \left( \begin{array}{cc} A - \Delta m^2 \cos 2\vartheta ... ...} a_\alpha \\ a_\beta \end{array} \right)\setlength{\arraycolsep}{\templength}$$ (4.13)

has to be studied [163,164,178,183,172,184], where $a_\alpha$, $a_\beta$ are the amplitudes for the neutrino types $\alpha$, $\beta$ ( $\alpha, \beta = e,\mu,\tau,s$), respectively. The evolution equation for antineutrinos is obtained from Eq.(4.13) by $A \to -A$. In the two-neutrino case there is one $\Delta m^2 = m_2^2 - m_1^2$ and the mixing matrix

$$U = \left( \begin{array}{rr} \cos \vartheta & \sin \vartheta \\ -\sin \vartheta & \cos \vartheta \end{array} \right)$$ (4.14)

is real without loss of generality.

	$\nu_e\to\nu_{\mu,\tau}$	$\nu_e\to\nu_s$	$\nu_\mu\to\nu_\tau$	$\nu_{\mu,\tau}\to\nu_s$
$\frac{A}{2\sqrt{2}EG_F}$	$N_e$	$N_e-\frac{1}{2}N_n$	$0$	$-\frac{1}{2}N_n$

Table 4.1. The list of matter densities relevant for two-neutrino oscillations (for the definition of $A$ see Eq.(4.12)).

The list of all possible matter densities which determine $A$ and occur in the different oscillation channels is given in Table 4.1. Evidently, $\nu_\mu\leftrightarrow\nu_\tau$ oscillations proceed as in vacuum because $A=0$ and the term proportional to the unit matrix in Eq.(4.13) has no effect on transitions. As mentioned before, in the rest of this section we will have in mind that Eq.(4.13) is applied to neutrino propagation in the sun and the earth.

Let us first define the eigenfunctions of the effective Hamiltonian as

$$H^{\mathrm{mat}}_\nu \psi_{mj} = E_j \psi_{mj} \quad \mbox{w... ... \sin \vartheta_m \\ \cos \vartheta_m \end{array} \right) \,,$$ (4.15)

where the matter angle $\vartheta_m$ and the energy eigenvalues $E_j$ are functions of $x$. In the limit of vanishing matter density $\vartheta_m$ is identical with the vacuum angle $\vartheta$ (4.14). The eigenvalues $E_j$ and the matter angle are given by

$$E_{1,2} \null \null = \null \null \frac{1}{4E} \left\{ m_1^2 + m_2^2 + 2\sqrt{2} G_F \left( N(\nu_\alpha) + N(\nu_\beta) \right) \right.$$

$$\null \null \null \left. \hspace{2cm} \mp \sqrt{(A-\Delta m^2 \cos 2 \varthet... ...+ (\Delta m^2 \sin 2\vartheta)^2} \right\}\setlength{\arraycolsep}{\templength}$$ (4.16)

and

$$\tan 2\vartheta_m = \frac{\tan 2\vartheta}{\displaystyle 1 - \frac{A}{\Delta m^2 \cos 2\vartheta}} \,,$$ (4.17)

respectively. The general solution of the differential equation (4.13) can be represented as

$$\psi(x) = \sum_{j=1}^2 a_j(x) \psi_{mj}(x) \exp \left( -i \int_{x_0}^x \mathrm{d}x' E_j(x') \right)$$ (4.18)

with the coefficients $a_{1,2}$ fulfilling

$$\frac{\mathrm{d}}{\mathrm{d}x} \left( \begin{array}{c} a_1 ... ...\right) \left( \begin{array}{c} a_1 \\ a_2 \end{array} \right)$$ (4.19)

and $\Delta E \equiv E_2-E_1$.

The adiabatic solution is defined as an approximate solution where $\dot{a_j} \simeq 0$ ($j=1,2$).¹⁰Postponing the discussion of the question under which condition adiabaticity is fulfilled, we consider temporarily the case of an arbitrary number $n$ of neutrino flavours or types and define the mixing matrix in matter $U_m(x)$ via

$$U_m(x)^\dagger H^{\mathrm{mat}}_\nu U_m(x) = \mbox{diag} (E... ...\mbox{with} \quad U_m(x) = (\psi_{m1}, \ldots, \psi_{m n}) \,,$$ (4.20)

where the eigenvectors $\psi_{mj}$ generalize Eq.(4.15). One readily obtains the generalization of the vacuum oscillation amplitude Eq.(3.12) for neutrino oscillations in matter if the evolution of the neutrino wave function is adiabatic:

$$\mathcal{A}^{\mathrm{adiab}}_{\nu_\alpha\to\nu_\beta} = \su... ...mathrm{d}x' E_j(x') \right\} \right) U^*_m(x_0)_{\alpha j} \,.$$ (4.21)

In this formula neutrino production and detection happen at $x_0$ and $x_1$, respectively. The adiabatic phases $\delta_j$, which are defined by

$$\delta_j \equiv -i \int_{x_0}^{x_1} \mathrm{d}x' \psi_{mj}(x')^\dagger \dot{\psi}_{mj}(x') \,,$$ (4.22)

are necessary for the correct evolution of $\psi(x)$ in the adiabatic limit. Usually, they can be absorbed into the eigenvectors $\psi_{mj}$ [185], except for special matter densities, where these phases acquire a topological meaning, together with the presence of CP violation [186]. Note that for real vectors $\psi_{mj}$ one has $\psi_{mj}^\dagger \dot{\psi}_{mj} = 0$ and therefore $\delta_j = 0$.¹¹ This is so in the two-dimensional case Eq.(4.15).

Evaluating Eq.(4.21) for two neutrino types and assuming that an averaging over neutrino energies takes place such that

$$\left\langle \exp \left( -i \int_{x_0}^{x_1} \mathrm{d}x' \Delta E(x') \right) \right\rangle_{\mathrm{av}} = 0 \,,$$ (4.23)

it is easy to show that the averaged two-neutrino survival probability can be written in the adiabatic case as

$$\bar P_{\nu_\alpha\to\nu_\alpha} = \frac{1}{2} (1 + \cos 2\vartheta_m(x_0) \cos 2\vartheta_m(x_1)) \,.$$ (4.24)

Let us now derive a condition for adiabaticity in the two-neutrino case. The evolution of the neutrino state in matter is adiabatic if the right-hand side of Eq.(4.19) can be neglected. A formal solution of Eq.(4.19) is given by

$$\left( \begin{array}{c} a_1(x) \\ a_2(x) \end{array} \right)... ... \begin{array}{c} a_1(x_0) \\ a_2(x_0) \end{array} \right) \,,$$ (4.25)

where $J$ denotes the matrix on the right-hand side of Eq.(4.19). Defining the variable

$$\alpha \equiv \frac{1}{2} \int_{x_0}^x \mathrm{d}x' \Delta E(x')$$ (4.26)

and the adiabaticity parameter (as a function of $x$) [187,188,189,184]

$$\gamma (x) \equiv \frac{\Delta E}{2 \vert\dot{\vartheta}_m\vert}$$ (4.27)

one can determine an upper bound to the typical integral which occurs in all the terms of the sum (4.25) on the right end. In an interval $[y_0,y_1]$ where $\gamma$ is monotonous there is a value $y_c$ such that [190]

$$\pm \int_{y_0}^{y_1} dy\, \dot{\vartheta}_m \cos 2\alpha = \... ... + \frac{1}{\gamma(y_1)} \int_{y_c}^{y_1} d\alpha \cos 2\alpha$$ (4.28)

according to the second mean value theorem of integral calculus ( $y_0 < y_c < y_1$). This consideration allows to put the bound

$$\left\vert \int_{y_0}^{y_1} dy\, \dot{\vartheta}_m e^{2i\alp... ...gamma_\mathrm{min} \equiv \min_{y \in [y_0,y_1]} \gamma(y) \,.$$ (4.29)

Having found this bound, in the rest of the integrals in the terms in Eq.(4.25) one can simply take the absolute values of the integrands and one is left with integrations of the type $\mathrm{d}x \vert\dot{\vartheta}_m\vert = \vert\mathrm{d} \vartheta_m\vert$. With the refinement that the interval $[x_0,x]$ possibly has to be divided into several parts labelled by $a$ such that in each part interval $\gamma(y)$ is monotonous we can define

$${\bar\gamma}^{-1} \equiv \sum_a \gamma_{\mathrm{min},a}^{-1}$$ (4.30)

and obtain the exact bound

$$\vert a_j(x) - a_j(x_0) \vert \leq \frac{2\sqrt{2}}{\bar\gam... ...t \\ Vert a_2(x_0)\vert \end{array} \right) \right\vert _j \,,$$ (4.31)

where the symbol $\vert _j$ denotes the $j$-th element of the vector on the right-hand side of this inequality and $\Delta \Theta$ has a contribution $\vert\Delta \vartheta_m\vert$ for every interval where $\dot{\vartheta}_m$ has a definite sign. ( $\Delta \vartheta_m$ is the difference between the angle $\vartheta_m$ in the initial and the final point of such an interval.) Thus with the exact bound (4.31) we have found the following sufficient condition:

$$\bar\gamma \gg 1 \quad \Rightarrow \quad \mbox{\textbf{adiabatic evolution}} \,.$$ (4.32)

It is fulfilled if on the scale of the oscillation length in matter given by $1/\Delta E$ the matter angle $\vartheta_m$ changes very little, i.e., $\vert\dot{\vartheta}_m\vert \ll \Delta E$. Eq.(4.31) allows to get an upper bound on the crossing or jumping probability from $\psi_{m1}$ to $\psi_{m2}$. Assuming the initial conditions $a_1(x_0)=1$ and $a_2(x_0)=0$ we get

$$\vert a_2(x)\vert \leq \frac{2\sqrt{2}}{\bar\gamma} \sinh \Delta \Theta$$ (4.33)

which illustrates once more the importance of $\bar\gamma$ for adiabaticity.

The resonance

In the context of the solar neutrino problem the discovery that a resonance in the passage of $\nu_e$ through the sun is possible [164] gave a major boost to the investigation of the propagation of neutrinos in matter. The possibility of a resonance is most easily understood in the adiabatic approximation by looking at Eqs.(4.17) and (4.24). For different neutrino masses we can always label them in such a way that $\Delta m^2 > 0$. If on the way from the creation point $x_0$ in matter of $\nu_e$ to a point $x_1$ in vacuum the neutrino passes through a point $x_{\mathrm{res}}$ where the resonance condition

$$A(x_{\mathrm{res}}) = \Delta m^2 \cos 2\vartheta$$ (4.34)

is fulfilled then $\vartheta_m(x_1) \equiv \vartheta$ has to be between $0^\circ$ and $45^\circ$ (for $\nu_e$ the quantity $A$ is positive!) whereas according to Eq.(4.17) the matter angle $\vartheta_m(x_0)$ is found between $45^\circ$ and $90^\circ$.¹²Consequently, the product of cosines in the $\nu_e$ survival probability (4.24) is negative and the probability to find a $\nu_e$ after the passage through the sun is less than 1/2. The interesting phenomenon is that this can happen even for relatively small mixing angles, provided the matter potential at the production point is large enough such that there is a point $x_{\mathrm{res}}$ where the condition (4.34) is fulfilled. Indeed, for $\vartheta$ close to $0^\circ$, one can even have $\vartheta_m(x_1)$ close to $90^\circ$ and thus $\bar P_{\nu_e\to\nu_e} \simeq 0$ (4.24). It has been shown that the resonance is also effective in a certain area in the $\Delta m^2$- $\sin^22\vartheta$ plane where the evolution of the neutrino state is non-adiabatic. For antineutrinos a resonance is possible if $45^\circ < \vartheta < 90^\circ$.

If there is a resonance, then for reasonable matter densities the adiabaticity parameter (4.27) is smallest at the resonance and thus adiabaticity is most likely violated there. Therefore, considering $\gamma$ at the resonance, from Eq.(4.17) one easily computes $\left. \dot{\vartheta}_m \right\vert _{\mathrm{res}} = \dot{A}_{\mathrm{res}}/(2 \sin 2\vartheta\, \Delta m^2)$. A suitable measure of adiabaticity in the case of a resonance is thus given by [187,188]

$$\gamma_\mathrm{res} \equiv \left. \frac{\Delta E}{2 \vert\d... ...2E \cos 2\vartheta\, (\vert\dot{A}\vert/A)_{\mathrm{res}}} \,.$$ (4.35)

In the solar interior the electron density is maximal in the center with $\left. N_e \right\vert _\mathrm{max} \simeq 100 \times N_\mathrm{avo}$ where $N_\mathrm{avo} = 6.022 \times 10^{23} \; \mbox{cm}^{-3}$ and $\left. N_n \right\vert _\mathrm{max} \simeq \frac{1}{2} \left. N_e \right\vert _\mathrm{max}$ [174]. This leads to

$$2\sqrt{2} G_F E \left. N_e \right\vert _\mathrm{max} \simeq ... ...5} \left( \frac{E}{1 \; \mbox{MeV}} \right) \: \mbox{eV}^2 \,.$$ (4.36)

Considering the resonance condition (4.34) one can read off from this equation for which $\Delta m^2$ a resonance is possible. Whereas the density variation of $N_e$ and $N_n$ in the sun is smooth, in the earth due to its layered structure (lithosphere - mantle - core) the densities can be approximated as step functions [191,192]. Furthermore, approximately $N_n \simeq N_e$ is valid in all layers. In the lithosphere, which is relevant for LBL neutrino oscillation experiments, the average mass density $\rho$ is around 3 g/cm$^3$. It increases to around 13 g/cm$^3$ in the core. Thus we can conveniently write

$$2\sqrt{2} G_F E\, N_e \simeq 2.3 \times 10^{-4} \: \mbox{eV}... ...m{cm}^{-3}} \right) \left( \frac{E}{1 \; \mbox{GeV}} \right) .$$ (4.37)

Non-adiabatic neutrino oscillations in matter and crossing probabilities

The analogue of the adiabatic formula (4.21) in terms of probabilities is given by

$$P_{\nu_\alpha\to\nu_\beta} = \left\vert \sum_{j,k} U_m(x_1)_{\beta j} B(x_1,x_0)_{jk} U^*_m(x_0)_{\alpha k} \right\vert^2$$ (4.38)

with

$$B(x_1,x_0) = U_m^\dagger(x_1)\, P \exp \left\{ -i \int_{x_0}^{x_1} \mathrm{d}x\, H_\nu^\mathrm{mat}(x) \right\} U_m(x_0)$$

$$= P \exp \left\{ -i \int_{x_0}^{x_1} \mathrm{d}x \left( \hat{E} -i\, U_m^\dagger(x) \dot{U}_m(x) \right) \right\} \,,$$ (4.39)

where $\hat{E}$ is the diagonal matrix of energy eigenvalues present in Eq.(4.20) (see Ref. [183]). $B(x_1,x_0)$ is a unitary $n \times n$ matrix which is diagonal in the adiabatic limit corresponding to $-i\, U_m^\dagger(x) \dot{U}_m(x) \simeq \mathrm{diag}\, (\delta_1, \ldots, \delta_n)$ (see Eq.(4.22)).

Confining ourselves now to two neutrino types and having in mind neutrino production in matter and detection in vacuum we use the notation $\vartheta_m(x_0) \equiv \vartheta_m^0$ and $\vartheta_m(x_1) = \vartheta$. With the crucial assumption that averaging over neutrino energies and the neutrino production region all terms other than probabilities can be dropped we get

$$\langle B_{jk} B^*_{j'k'} \rangle_\mathrm{av} = \delta_{jj'}... ..._c \,,\; \vert B_{11}\vert^2 = \vert B_{22}\vert^2 = 1-P_c \,,$$ (4.40)

where $P_c$ is the crossing probability from $\psi_{m1}$ to $\psi_{m2}$ and in the second part of Eq.(4.40) unitarity has been used. Such averaging procedures have been shown to be effective in the context of solar neutrinos, e.g., in Refs. [193,194]. Inserting the relations (4.40) into Eq.(4.38) specialized to the survival probability of $\nu_e$ one obtains after some algebra with trigonometric functions the Parke formula

$$\bar P_{\nu_e\to\nu_e} = \frac{1}{2} + \left( \frac{1}{2} - P_c \right) \cos 2\vartheta \cos 2\vartheta_m^0 \,,$$ (4.41)

which is the generalization of Eq.(4.24) [189,183]. This formula was first derived in the context of an exact solution of Eq.(4.13) for matter densities linear in $x$ [188,195] (see also Refs. [196,187]).

The probability $P_c$ can be estimated with the Landau - Zener method [197,196]. Following the derivation of Landau, the idea is to make an analytic continuation of the matter densities and therefore of the effective matter Hamiltonian (4.8) into the complex plane by $x \to z$ where $z$ is a complex variable. Then also the neutrino state $\psi$ (4.18) is analytic. Considering the energy eigenvalues $E_{1,2}(x)$ (4.16) as functions of $z$ we find two branching points $z_0$ and $z^*_0$ ( $\mbox{Im}\, z_0 > 0$) defined by the equation

$$E_1(z_0) = E_2(z_0) \quad \Leftrightarrow A(z_0) = \Delta m^2 e^{\pm 2i\vartheta} \,.$$ (4.42)

We consider $E_1(z)$ and $\psi_{m1}(z)$ for definiteness and follow its evolution along a curve $C(z)$ which starts at a value $z$ in the vicinity of the real axis and goes along a negatively oriented loop around $z_0$ such that we end at the same point $z$ but now located on the second sheet.¹³Along this path $E_1(z)$ changes into $E_2(z)$ [197] and $\psi_{m1}(z)$ into $\psi_{m2}(z)$.¹⁴ The crucial observation is that, though $E_j(z)$ and $\psi_{mj}(z)$ ($j=1,2$) are two-valued analytic objects, the analytic continuation $\psi(z)$ of the solution (4.18) of the differential equation (4.13) is single-valued [198]. From this fact it can easily be shown with Eq.(4.18) that [198]

$$\tilde{a}_1(z) = \exp \left( i\int_{C(z)} E_1(z)dz \right) a_2(z) \,,$$ (4.43)

where $\tilde{a}_1(z)$ is the analytic continuation of $a_1(x)$ in the vicinity of the real axis and then along $C(z)$. The Landau - Zener crossing probability is derived from the following consideration. Starting on the real axis at $x_- \ll x_\mathrm{res}$, where the evolution of $\psi$ is adiabatic and where the boundary conditions $a_1(x_-) = 1$, $a_2(x_-) = 0$ hold, and going along a path in the complex plane which passes above $z_0$ and returning to the real axis at a point $x_+ \gg x_\mathrm{res}$ on the second sheet this path can be chosen such that the evolution in the complex variable $z$ is adiabatic because we have passed the resonance in safe distance and we have thus $\vert\tilde{a}_1(x_+)\vert \simeq 1$. On the other hand, we can go from $x_+$ on the first sheet to the point $x_+$ on the second sheet via $C(x_+)$ and with Eq.(4.43) we thus obtain

$$P_c = \vert a_2(x_+)\vert^2 = \left\vert \exp \left( -i\int_{C(x_+)} E_1(z)dz \right) \right\vert^2 \,,$$ (4.44)

neglecting deviations of $\vert\tilde{a}_1(x_+)\vert$ from 1. From this formula we can read off that integrations along the real axis do not contribute to $P_c$. Because of analyticity, the path $C(x_+)$ can be deformed such that it goes along the real axis from $x_+$ to $x_\mathrm{res}$ from where it leads to $z_0$, circles around the branching point with infinitely small radius and then goes back to $x_\mathrm{res}$ and $x_+$ on the second sheet. In this way, we obtain from Eq.(4.44) the final result for the Landau - Zener crossing probability

$$\ln P_c = -\frac{1}{E}\, \mbox{Im} \int_{x_\mathrm{res}}^{z_... ...cos 2\vartheta)^2 + (\Delta m^2 \sin 2\vartheta)^2 ]^{1/2} \,.$$ (4.45)

To get this formula we have used the explicit form of the energies of the adiabatic states (4.16).

With the variable transformation $A = A(z)$ and $dz = dA/ \frac{dA}{dz}$ it is easy to evaluate Eq.(4.45) for a linear density [187,188,189,172] and is also possible for an exponential density [199]. These calculations can be summarized by [189,183]

$$P_c = \exp \left(-\frac{\pi}{2} \gamma_\mathrm{res}F \right)$$ (4.46)

with $F=1$ for the linear and $F=1-\tan^2 \vartheta$ for the exponential case. For an exponentially varying matter density the dependence of $P_c$ on $\vartheta$ is particularly simple because $\gamma_\mathrm{res}F = 4\delta \sin^2 \vartheta$ with $\delta \equiv \gamma_\mathrm{res} \cos 2\vartheta/\sin^2 2\vartheta$ which is independent of the vacuum mixing angle. For a more general discussion of the crossing probability (4.45) see Ref. [200].

Exact solutions of the differential equation (4.13) for neutrino oscillations in matter exist not only for the linear case [196,187,188,195] in terms of Weber functions but also for the exponentially varying matter density in terms of Whittaker functions [201,193]. This case is of particular importance for solar neutrinos because it approximates the real density variation in the sun. Further exact solutions are known for $A$ varying with $\tanh x$ [202] and $1/x$ [189]. In Refs. [189,183] a list of the factors $F$ for all these cases is given, calculated with the Landau - Zener formula (4.45).

The survival probability (4.41) with $P_c$ in the Landau - Zener approximation does not reproduce well the exact survival probability in the extremely non-adiabatic region [195,193] (see also Refs. [189,183,172]). In these references the example of an extremely dense medium ($A \to \infty$ and thus $\vartheta^0_m = \pi/2$) with a sharp boundary to the vacuum is given. In this case the neutrino does not oscillate in the medium and therefore $\bar P_{\nu_e\to\nu_e} = 1 - \frac{1}{2} \sin^2 2\vartheta$ stems purely from vacuum oscillations. This has to be compared with $P_c = 1$, because of the jump in density one gets $\gamma_\mathrm{res}=0$, inserted into Eq.(4.41). Obviously, the resulting expression $\bar P_{\nu_e\to\nu_e} = \cos^2 \vartheta$ does not agree with the previous one.

A remedy of this deficiency was found in the framework of the exact solution for an exponentially varying matter density leading to the following modification [193] of the Landau - Zener crossing probability $P_c$ (4.45):

$$P_c = \frac{\exp \left(-{\displaystyle \frac{\pi}{2}} \gamma... ...m{res} {\displaystyle \frac{F}{\sin^2 \vartheta}} \right)} \,.$$ (4.47)

Numerical calculations for solar neutrinos with an exponential electron density using this formula agree very well with numerical integrations of the differential equation (4.13), typically the agreement is in the percent range for relevant mixing parameters [203]. It has been conjectured in Ref. [189] that the form (4.47) of $P_c$ holds for a wide class of matter density profiles. In any case, in the limit $\gamma_\mathrm{res} \to 0$ one gets $P_c \to \cos^2 \vartheta$ which, when inserted into formula (4.41), correctly describes the survival probability in the above example of an extremely non-adiabatic evolution.

Concluding this section, we consider again an arbitrary number of neutrino flavours or types and envisage the interesting case where one of the eigenfunctions (4.20) of $H^\mathrm{mat}_\nu$ labelled by the index $\ell_0$ has an adiabatic evolution whereas the other part of $\psi$ has an arbitrary evolution. We assume exact adiabaticity for $\psi_{m \ell_0}$ which amounts to $B_{j\ell_0} = B_{\ell_0 k} = 0$ for $j,k \neq \ell_0$. This allows to write

$$P_{\nu_\alpha\to\nu_\beta} = (1-\vert U_m(x_1)_{\beta \ell_... ...\beta \ell_0}\vert^2 \vert U_m(x_0)_{\alpha \ell_0}\vert^2 \,,$$ (4.48)

where we have defined

$$P^{(\ell_0)}_{\nu_\alpha\to\nu_\beta} \null \null = \null \null [(1-\vert U_m(x_1)_{\beta \ell_0}\vert^2) (1-\vert U_m(x_0)... ... \ell_0} U_m(x_1)_{\beta j} B(x_1,x_0)_{jk} U^*_m(x_0)_{\alpha k} \right\vert^2$$

$$\null \null \leq \null \null 1 \,.\setlength{\arraycolsep}{\templength}$$ (4.49)

That $P^{(\ell_0)}_{\nu_\alpha\to\nu_\beta}$ is smaller than 1 follows from the Cauchy - Schwarz inequality and from the above assumption for $B$ because after dropping the row and the column labelled by $\ell_0$ in $B$ the remaining matrix is again unitary. Eq.(4.49) and similar cases are useful if neutrino masses differing by orders of magnitude occur.

Indications of neutrino oscillations

Atmospheric neutrino experiments

The atmospheric neutrino flux

In 1912 it was discovered by V.F. Hess [204] in a manned balloon flight that the intensity of the ionizing radiation in the atmosphere as a function of the altitude did not conform with the idea that this ionization was caused by radioactive elements in the surface of the earth but rather pointed to an extraterrestrial origin. In the following decades, before the advent of accelerator physics, this radiation, which was called first ``ultraradiation'' and later baptized ``cosmic rays'' by R.A. Millikan, proved to be one of the most fruitful means for doing particle physics experiments. At the end of the first half of the 20th century such experiments had lead to the discovery of the positron, the pion and the muon and also the first particles with strangeness were found with cosmic rays [205]. Eventually, in the beginning of the fifties, proton beams from accelerators replaced the cosmic proton flux as an experimental tool. However, after many years where accelerator physics was dominating in particle physics, at the end of the 20th century cosmic rays play again a major role through atmospheric neutrinos which allow to use the whole globe as a neutrino physics laboratory and to probe neutrino mass-squared differences down to a few $10^{-4}$ eV$^2$. In this way convincing evidence for the existence of neutrino oscillations and thus for non-zero neutrino masses has been obtained [2].

In a simplified picture, the production of atmospheric neutrinos [206,207,208,209,210,211,212,213,214,215,216,217,218] proceeds in three steps [219]. In the first step the primary cosmic rays [219,220] hit the nuclei in the atmosphere, thereby producing charged pions and kaons, either directly or via intermediate particles. In the second step, the decay of these particles gives rise to part of the atmospheric $\nu_\mu$ and $\bar{\nu}_\mu$ neutrino fluxes:

$$\pi^+ \to \mu^+ + \nu_\mu \,, \; \pi^- \to \mu^- + \bar{\nu}... ...+ \to \mu^+ + \nu_\mu \,, \; K^- \to \mu^- + \bar{\nu}_\mu \,.$$ (5.1)

In the third step, the $\nu_e$ and $\bar{\nu}_e$ fluxes and further $\nu_\mu$ and $\bar{\nu}_\mu$ fluxes are produced by

$$\mu^+ \to e^+ + \nu_e + \bar{\nu}_\mu \quad \mbox{and} \quad \mu^- \to e^- + \bar{\nu}_e + \nu_\mu \,.$$ (5.2)

There is also a contribution to the neutrino fluxes from the decays

$$K_L \to \pi^+ + \ell^- + \bar{\nu}_\ell \quad \mbox{and} \quad K^+ \to \pi^0 + \ell^+ + \nu_\ell$$ (5.3)

with $\ell = e, \mu$ and the charge conjugate processes [219,221,215] which does not conform with the simple 3-step picture. At low energies the most important process in Eq.(5.1) is the pion decay. The contribution of $K_{\ell 3}$ decays (5.3) to the neutrino fluxes is small [222,223]. Let us now describe some useful details of the production of the atmospheric neutrino flux and its properties.

Cosmic rays:

Galactic cosmic rays enter the solar system as an isotropic flux of particles. There is convincing evidence that the bulk of the radiation with energies less than $10^6$ GeV comes from our galaxy. Though there is no conclusive proof of the origin of cosmic rays yet, plausible mechanisms range from material ejected by supernovae to interstellar medium accelerated in supernova shock waves [224]. Solar cosmic rays are emitted irregularly by major solar flares on the sun. Apart from electrons, primary cosmic rays consist of protons and bare nuclei (mostly He). Roughly speaking, the chemical composition of the galactic cosmic ray particles is given approximately by 90 % H, 9 % He and less than 1 % heavier nuclei. However, the chemical composition varies with energy. At energies of around 100 MeV per nucleon the particle number ratio H/He is less than 5, it increases to 10 at 1 GeV and is around 30 at 100 GeV [225,226]. The absolute flux of cosmic protons is not very large: it is of the order of $1000 / \mbox{m}^2 \times \mbox{sec} \times \mbox{sr}$ for energies of a few GeV and above the atmosphere. As a function of the energy the proton flux above a few GeV is well described by the power law $E^{-2.7}$ until $10^6$ GeV [225]. Above an energy of 100 GeV per nucleon the cosmic ray fluxes are less precisely known [225,226].

Solar modulation:

The solar cosmic rays (solar wind) are important in so far as they weaken and modulate the flux of galactic cosmic ray particles with the solar activity. The stronger the solar wind the more difficult it is for the low energy galactic cosmic rays to enter the solar sphere of influence. This modulation is noticeable for kinetic energies of around 10 GeV per nucleon or less [224,215]. This variation of the primary cosmic ray flux with the approximate 11-year cycle of solar activity can be parameterized as a function of the neutron monitor at Mt. Washington [215]. It induces a corresponding modulation of the low energy atmospheric neutrino flux but its influence is small. The effect of the solar wind on cosmic ray particles depends on the rigidity of the nuclei which is defined by momentum/charge. It is strongest for cosmic ray particles with small rigidity. For neutrino energies $E_\nu \sim 2$ GeV it is practically negligible, for energies around 1 GeV only appreciable at high geomagnetic latitudes because the geomagnetic cut-off admits cosmic ray particles with smaller rigidity there.

Geomagnetic cut-off:

The geomagnetic field prevents primary cosmic particles with low rigidity from entering the atmosphere [227,215,228,218]. In the atmospheric neutrino flux calculations it is assumed that the momenta of the neutrinos have the same direction as the primary cosmic ray particles responsible for their production [215,217]. This is a good approximation for $E_\nu > 200$ MeV as shown in Ref. [229]. In a first approximation, for a given direction, there is a cut-off in rigidity for the primary cosmic ray particles below which such a particle cannot reach the top of the atmosphere in a vertical altitude of around 20 km. The value of the geomagnetic cut-off can be obtained by a computer simulation with the help of the ``backtracking technique''. In this technique, to establish if a particle with a given charge and momentum coming from a certain direction can reach a point on top of the atmosphere, one integrates the equations of motion for a particle with opposite charge and reflected momentum starting from its final position. If the backtracked particle reaches infinity it is assumed that the trajectory is allowed. As infinity one can take a distance of around 10 times the radius of the earth where the geomagnetic field has decreased to the level of the interstellar magnetic field with a strength $\sim 3 \times 10^{-8}$ Tesla [215]. More refined calculations replace the rigidity cut-off by a probability distribution [227,228,218]. In the rigidity cut-off approximation one can give for every experimental location a contour map depicting curves with constant cut-off in a plot with the coordinate axes representing azimuth and zenith angles of the primary cosmic ray particle direction. For Kamioka such a plot is found in Ref. [215]. At the geomagnetic latitude of $90^\circ$ the geomagnetic cut-off disappears, therefore experiments situated at high geomagnetic latitudes have a higher flux of low energy atmospheric neutrinos. The magnetic field of the earth is not exactly of dipole form but also has a considerable multipole contribution which affects the local rigidity cut-off maps and thus the atmospheric neutrino flux in particular for down-going neutrinos [215].

Hadronic interactions:

After entering the atmosphere the cosmic ray particles collide with the nuclei in the air. Though the fractions of other particles than protons in the cosmic rays are small, their importance for the atmospheric neutrino flux gets enhanced because the predecessors of the neutrinos, pions and kaons, are created through the hadronic interactions of the cosmic rays with the air nuclei. These processes depend rather on the number of nucleons than on the number of nuclei. The ratio of the charged kaon to the charged pion flux depends sensitively on the description of hadronic interactions. In this context the quantities

$$Z_{p \mathcal{M}} \equiv \int_0^1 \mathrm{d}x \, x^\gamma \frac{\mathrm{d}N_{p \mathcal{M}}}{\mathrm{d}x} \,,$$ (5.4)

where $\mathcal{M} = \pi, K$ and $\gamma \simeq 1.7$ comes from the power law of the primary cosmic protons, $x$ is the fraction of the proton momentum carried by the meson and $\mathrm{d}N/\mathrm{d}x$ is the distribution of the charged mesons produced by collisions of protons with nuclei in the atmosphere, are important indicators for the fraction of the muon neutrino flux originating from the $\mathcal{M}$ meson flux. The ratio $Z_{pK^\pm}/Z_{p\pi^\pm}$ ranges from 0.10 to 0.15 in different calculations [217]. As a consequence, at neutrino energies below 100 GeV the pions dominate as neutrino sources [221,216].

Characteristics of the atmospheric neutrino fluxes:

Some of the characteristic properties of the atmospheric neutrino fluxes are simple consequences of the production mechanisms (5.1) and (5.2), whereas others follow from a Monte Carlo or analytic calculation [221] of the neutrino fluxes. As mentioned above, the flux calculations are one-dimensional and energy losses in the air, in particular, for the muons, have to be taken into account [219].

a..

Approximately, the $\stackrel{\scriptscriptstyle (-)}{\nu}_{\hskip-3pt \mu}$ fluxes have a power law energy spectrum $E_\nu^{-3}$ for $1 \lesssim E_\nu \lesssim 10^3$ GeV whereas the $\stackrel{\scriptscriptstyle (-)}{\nu}_{\hskip-3pt e}$ fluxes decrease like $E_\nu^{-3.5}$ [215].

b..

Fixing a neutrino energy $E_\nu$, the largest contribution to the flux of this energy originates from cosmic ray particles with energy $E_{\mathrm{cr}} \sim 10 \times E_\nu$ [215,216].

c..

Denoting neutrino fluxes by $(\nu)$ then it follows immediately from Eqs.(5.1) and (5.2) that

$$\frac{(\nu_\mu) + (\bar{\nu}_\mu)}{(\nu_e) + (\bar{\nu}_e)} \simeq 2 \,.$$ (5.5)

However, it can easily be estimated that, at neutrino energies larger than 1 GeV, muons from the reactions (5.1) start to reach the surface of the earth before they decay. Therefore the neutrino fluxes from muon decay decrease, in particular, the $\stackrel{\scriptscriptstyle (-)}{\nu}_{\hskip-3pt e}$ flux, and the ratio (5.5) begins to rise [219,215].

d..

For the same reason muons cease to be the dominant source of $(\nu_e) + (\bar{\nu}_e)$ above $E_\nu \simeq 100$ GeV and the reactions (5.3) take over [219,221].

e..

Pions cease to be the dominant source of the $\stackrel{\scriptscriptstyle (-)}{\nu}_{\hskip-3pt \mu}$ flux for neutrino energies above 100 GeV [221]. The reason is given by the different masses and lifetimes of the parent mesons $\mathcal{M} = \pi, K$ in the decays $\mathcal{M} \to \mu \nu$. For relativistic parent mesons and for a given energy $E_\nu$, the lower bound on the energy of the parent meson in such a decay is given by $E_{\mathcal{M}} \geq E_\nu / (1-m_\mu^2/m^2_{\mathcal{M}})$. Thus, neutrinos with energy $E_\nu$ from the decay of the heavier parent meson need in average a lower parent meson energy where the corresponding meson flux is higher. This together with the shorter kaon lifetime leads to the prevailing of kaons as the source of muon neutrinos at sufficiently high neutrino energies [230,221,215,216].

f..

The ratio $(\bar{\nu}_e) / (\nu_e)$ is smaller than 1 because among the primary cosmic ray nucleons the protons by far dominate which leads to an enhancement of positively charged pions over negative ones in the hadronic interactions and, consequently,¹⁵ $(\pi^+)/(\pi^-) \simeq (\mu^+)/(\mu^-) > 1$ [221,215].

g..

On the other hand, for $E_\nu \lesssim 1$ GeV, where the muons have time to decay in the atmosphere, one has $(\bar{\nu}_\mu)/(\nu_\mu) \simeq 1$ because the muons supply the antineutrinos (neutrinos) to the neutrinos (antineutrinos) from pions and kaons. However, for neutrino energies above a few GeV, when part of the muons reaches the surface of the earth before they decay, this ratio becomes smaller than one because the $\pi ^+$ are more numerous than the $\pi^-$ (see above).

h..

For $E_\nu \gtrsim 5$ GeV the geomagnetic effects become negligible.

The largest error in the atmospheric neutrino flux calculations arises from an overall uncertainty of primary cosmic ray measurements of the order of $\pm 15 \%$ [215,217]. Therefore, it is mandatory to use quantities which are ratios such that this uncertainty cancels or, in the case of a fit to the data, to include the overall normalization of the primary cosmic ray flux in the set of parameters to be determined by the fit. Among the flux calculations of the different groups the major source of differences comes from the treatment of pion production in collisions of protons with light nuclei [217,231]. Thus, the calculated absolute fluxes have uncertainties of the order of $\pm 20 \%$ and even larger uncertainties at neutrino energies below 1 GeV, whereas flux ratios differ in general only by a few percent (see the comparison between different flux calculations in Refs. [217,231]).

The angular dependence of the atmospheric neutrino flux:

The direction of the atmospheric neutrino flux at the location of an experiment is described by the zenith angle $\theta$ and the azimuth angle $\phi$. Neutrinos going vertically downward have $\theta = 0$ whereas those coming vertically upward through the earth have $\theta = \pi$. The azimuth angle indicates the direction of the flux in the horizontal plane. There are two natural causes for an angular dependence [218] of the atmospheric neutrino flux:

a..

The development of cosmic ray showers in the atmosphere depends on the density of the air. At zenith angles $\theta \sim \pi/2$, where along the line of flight the increase in density is less steep, pion and kaon decay is enhanced compared to vertical directions. It is easily checked that the situation is symmetric with respect to $\theta \to \pi - \theta$. Therefore, the generation of pions and kaons in the atmosphere caused by primary cosmic rays induces a dependence of the atmospheric neutrino flux on $\vert\cos \theta\vert$. In other words, this mechanism does not introduce an up-down asymmetry in the flux.

b..

The geomagnetic field, which acts on the primary cosmic ray flux, is the cause for a $\phi$ and $\theta$ dependence of the atmospheric neutrino flux. Because of the positive charge of the primary cosmic rays the neutrino flux is highest (lowest) for directions coming from the west (east). The nature of the geomagnetic effects also generates an up-down asymmetry for low energy neutrinos [215,218].

A third cause for a dependence on the zenith angle is possibly given by neutrino oscillations which arises because varying $\theta$ from 0 to $\pi$ the neutrino path length varies from around 10 km [232] to around 13000 km. Clearly, this dependence is not up-down symmetric. If it is disentangled from the up-down asymmetry caused by the geomagnetic effects, e.g., by using the high energy component of the neutrino flux, it provides us with valuable information on neutrino masses and mixing.

Experiments with atmospheric neutrinos

Early efforts for detecting atmospheric neutrinos (see Ref. [233] for a summary) concentrated on neutrino-induced upward muons, i.e., muons with zenith angles $90^\circ \leq \theta \leq 180^\circ$, or horizontal muons, i.e., muons with a zenith angle around $90^\circ$, using the process

$$\stackrel{\scriptscriptstyle (-)}{\nu}_{\hskip-3pt \mu} + N \to \mu^\pm + X \,,$$ (5.6)

where $N$ is a nucleon in the rock surrounding the detector located underground [234,235]. This method was proposed to distinguish muons generated by atmospheric neutrinos from muons originating in the meson decays (5.1) in the atmosphere. There were two experiments, the Kolar Gold Field experiment in India [236] and an experiment in South Africa [237], reporting the first evidence for atmospheric muon neutrinos. Since these experiments could not distinguish up and down directions they used horizontal muons crossing the detector which lay, due to the great depth of the experimental sites, in the large zenith angle intervals $60^\circ \leq \theta \leq 120^\circ$ and $45^\circ \leq \theta \leq 135^\circ$ for the Indian and the South African experiment, respectively. These early experiments were accompanied already by atmospheric neutrino flux calculations [238,239,230,240].

Recent detectors [241] are divided into two classes: water Cherenkov detectors where the neutrino target is a large volume of water surveyed by a huge array of photomultiplier tubes sitting on the surface of the volume (the Kamiokande [242,243,49], Super-Kamiokande [2] and IMB [244,50,245] collaborations) and iron plate calorimeters where neutrino-induced charged particles ionize the gas between the plates and the particle paths are reconstructed electronically (the Fréjus [246], NUSEX [247] and Soudan-2 [51] collaborations). In contrast to the early detectors the recent detectors are sensitive to the direction of tracks and can thus distinguish between up and down through-going tracks. However, they cannot measure the charge of the leptons $\ell^\pm$ and thus cannot distinguish between $\nu_\ell$ and $\bar \nu_\ell$. Therefore, at low energies they are approximately sensitive the flux combination $(\nu_\ell)+\frac{1}{3}(\bar{\nu}_\ell)$ ($\ell = e, \mu$) because at low energies quasi-elastic scattering is predominant and the ratio of quasi-elastic antineutrino to neutrino cross sections is approximately 1/3 [233,248,217]. In atmospheric neutrino physics it is important to distinguish $e$-like and $\mu$-like events which is accomplished by distinguishing between showers or diffuse Cherenkov rings ( $e^+, e^-, \gamma$) and tracks or sharp Cherenkov rings ($\mu^+, \mu^-$ and charged pions, kaons, protons etc.). The separation between $e$-like and $\mu$-like events is very good, e.g., for Super-Kamiokande its efficiency is estimated to be 98% or better [2].

For the deep underground detectors two different event classes are defined [241]. Events in which the neutrino interacts with the material inside the detector and where all particles from the neutrino interaction deposit their energies inside the detector are called contained events. The second class refers to events where muon neutrinos interact with the material surrounding the detector via charged current interactions such that the high energy muons enter the detector [249]. In this way one distinguishes through-going muons and stopping muons. Recent measurements of the upward muon flux were performed by the Baksan [250], Kamiokande [251,252], IMB [50], Fréjus [246], MACRO [52] and Super-Kamiokande [1] collaborations.

Super-Kamiokande - and before also Kamiokande - has an inner detector surrounded by an outer detector. This allows to further subdivided the contained events into fully contained events (FC) with all energy of an event deposited in the inner detector and partially contained events (PC) which have exiting tracks detected also in the outer detector. In Kamiokande and Super-Kamiokande, FC events are separated into those having a visible energy $E_{\mathrm{vis}} < 1.33$ GeV, the sub-GeV events, and those with $E_{\mathrm{vis}} > 1.33$ GeV, the multi-GeV events. Among the contained events in Cherenkov detectors the single-ring events are well understood, they are predominantly produced by quasi-elastic scattering of electron and muon neutrinos. In Kamiokande and Super-Kamiokande for the analyses of FC events only single-ring events are used with the additional criteria $p_e \geq 100$ MeV and $p_\mu \geq 200$ MeV for electron and muon momenta, respectively, in the case of sub-GeV events. To quote the numbers of Super-Kamiokande [2], a Monte Carlo simulation has shown that 88% (96%) of the sub-GeV $e$-like ($\mu$-like) events are charged current interactions whereas for multi-GeV events the number is 84% (99%). The remainder is given by neutral current events. The PC events were estimated to be 98% $\stackrel{\scriptscriptstyle (-)}{\nu}_{\hskip-3pt \mu}$-induced events for single and multi-ring configurations and, therefore, all PC events are used for the analyses.

For the analyses of atmospheric neutrino data it is very important to have a good understanding of the neutrino interactions in the detector. The deep-inelastic scattering (DIS) formulas are only valid for sufficiently high momentum transfer $Q^2$ from the leptons to the hadrons. For neutrinos with a few GeV interacting with the detector material, the lowest multiplicity exclusive channels represent an important fraction of the cross section. Therefore the following decomposition of the neutrino cross section [253] has been proposed:

$$\sigma_\nu^{\mathrm{CC}} = \sigma_{\mathrm{QEL}} + \sigma_{1\pi} + \sigma_{\mathrm{DIS}} \,,$$ (5.7)

where QEL indicates the quasi-elastic $\nu_\ell + n \to \ell^- + p$ and $\bar{\nu}_\ell + p \to \ell^+ + n$ cross sections and $1\pi$ single pion production. In the case of the latter it is assumed that $\Delta(1232)$ dominates. To avoid double counting in $\sigma_{1\pi}+\sigma_{\mathrm{DIS}}$, the maximal mass of the pion - nucleon system in the final state is taken to be $W_c = 1.4$ GeV for single pion production whereas the kinematical region for DIS is bounded by $W \geq W_c$, where $W$ is the mass of the hadronic final state.

The atmospheric neutrino anomaly

The first observable to be measured in recent atmospheric neutrino experiments was the ratio of $\mu$-like to $e$-like events denoted by $(\mu/e)_{\mathrm{data}}$. As discussed earlier, in flux ratios the large uncertainty in the overall normalization of the primary cosmic ray flux cancels and there is also some cancellation of errors in the theoretical calculation. However, the above ratio is only a limited reflection of the corresponding ratio of atmospheric neutrino fluxes because of detector efficiencies and event selection criteria. Thus for the expected ratio one has to fold the theoretical flux calculations with the cross sections of the neutrino interactions in the detector and the detection efficiencies and apply the event selection criteria. Quoting the result of the Monte Carlo calculation of Super-Kamiokande as an example, this collaboration obtains $(\mu/e)_{\mathrm{MC}} = 1.50$ and $2.83$ for the sub-GeV ratio of FC events and the ratio considering multi-GeV FC and PC events, respectively [2], using the neutrino flux calculations of Ref. [215]. Note that these numbers significantly deviate from the naive expectation 2. Therefore, the actual physically relevant quantity is given by the double ratio

$$R = \frac{(\mu/e)_{\mathrm{data}}}{(\mu/e)_{\mathrm{MC}}} \,.$$ (5.8)

The first indication that this ratio is smaller than 1 was reported more than ten years ago [254]. In the meantime the most impressive measurements of $R$ are represented by the Kamiokande and Super-Kamiokande results:¹⁶

$$R = \left\{ \begin{array}{rr} \left. \begin{array}{crr} 0.6... ... \right\} & \quad \mbox{Super-Kamiokande} \end{array}\right.$$ (5.9)

The PC events have been added to the multi-GeV data. The results $R = 0.54 \pm 0.05 \pm 0.11$ of the IMB Collaboration [50,255] and $R = 0.61 \pm 0.15 \pm 0.05$ of Soudan-2 [256] are in agreement with those of the experiments in Kamioka. However, the early iron calorimeter experiments found values of $R$ compatible with 1, namely $R = 1.00 \pm 0.15 \pm 0.08$ (Fréjus Coll. [246]) and $R = 0.96 \begin{array}{l} +0.32 \\ [-3pt] -0.28 \end{array}$ (NUSEX Coll. [247]). Apart from the latter two experiments, all others hint at a reduction of $R$ compared to the expectation $R=1$.

Kamiokande and, in particular, Super-Kamiokande have enough statistics to study the zenith angle dependence of the measured $\stackrel{\scriptscriptstyle (-)}{\nu}_{\hskip-3pt e}$ and $\stackrel{\scriptscriptstyle (-)}{\nu}_{\hskip-3pt \mu}$ fluxes. To this end, the $\cos \theta$ interval $[-1, 1]$ is divided into five bins of length 0.4. Kamiokande [49] has observed a zenith angle variation of $R$ for the FC multi-GeV + PC events with indications that the zenith angle variation rather comes from $\mu$-like events than from the $e$-like events. All this is amply confirmed by Super-Kamiokande with much more statistics and with a significant zenith angle variation in the $\mu$-like events for both, sub-GeV and multi-GeV.

It is important to study the zenith angle variation independent of the double ratio $R$ in order to disentangle the zenith angle dependencies of the electron and muon neutrino fluxes. To this end, in addition to the oscillation parameters, also the normalization of primary cosmic ray flux has to be fitted. Super-Kamiokande has performed a statistical analysis of the data under the assumption of $\stackrel{\scriptscriptstyle (-)}{\nu}_{\hskip-3pt \mu} \to \stackrel{\scriptscriptstyle (-)}{\nu}_{\hskip-3pt \tau}$ oscillations. With $e$-like and $\mu$-like events in five $\cos \theta$ bins and seven momentum bins there are altogether 70 data points and three quantities to be determined: the mixing angle $\vartheta$, the neutrino mass-squared difference $\Delta m^2$ and the overall neutrino flux normalization. The best fit gives $\sin^22\vartheta=1$ and $\Delta m^2 = 2.2 \times 10^{-3}$ eV$^2$ with $\chi^2_{\mathrm{min}} = 65.2$ for 67 DOF. The regions in the $\sin^22\vartheta$-$\Delta{m}^2$ plane allowed at 68%, 90% and 99% confidence level in the case of $\stackrel{\scriptscriptstyle (-)}{\nu}_{\hskip-3pt \mu} \to \stackrel{\scriptscriptstyle (-)}{\nu}_{\hskip-3pt \tau}$ oscillations are shown in Fig. 5.1 [2]. At 90% CL the mass-squared difference lies in the interval $5 \times 10^{-4} \: \mbox{eV}^2 < \Delta m^2 < 6 \times 10^{-3} \: \mbox{eV}^2$. For the simulation of neutrino oscillations the profiles for the neutrino production heights of Ref. [232] were used. An analogous procedure with the hypothesis of $\stackrel{\scriptscriptstyle (-)}{\nu}_{\hskip-3pt \mu} \leftrightarrow \stackrel{\scriptscriptstyle (-)}{\nu}_{\hskip-3pt e}$ oscillations, taking into account matter effects in the earth, gives a poor fit with $\chi^2_{\mathrm{min}} = 87.8$ for 67 DOF.

	Figure 5.1. The 68%, 90% and 99% confidence regions in the $\sin^22\vartheta$-$\Delta{m}^2$ plane for $\nu_\mu\to\nu_\tau$ oscillations obtained from 33.0 kton year of Super-Kamiokande data [2]. The 90% confidence region obtained by the Kamiokande experiment is also shown.	Figure 5.2. Up-down asymmetry measured in the Super-Kamiokande atmospheric neutrino experiment as a function of the momentum of $e$-like and $\mu$-like events [2]. The hatched region shows the theoretical expectation without neutrino oscillations, with statistical and systematic errors added in quadrature. The dashed line for $\mu$-like events is the theoretical expectation in the case of two-generation $\nu_\mu\to\nu_\tau$ oscillations with $\Delta{m}^2 = 2.2 \times 10^{-3} \, \mathrm{eV}^2$ and $\sin^22\vartheta = 1.0$.

It is interesting to note that, as shown in Fig. 5.1, a fit for Kamiokande with $\stackrel{\scriptscriptstyle (-)}{\nu}_{\hskip-3pt \mu} \to \stackrel{\scriptscriptstyle (-)}{\nu}_{\hskip-3pt \tau}$ oscillations gives $5 \times 10^{-3} \: \mbox{eV}^2 < \Delta m^2 < 3 \times 10^{-2} \: \mbox{eV}^2$ with 90% CL [49] for the multi-GeV data whereas the sub-GeV data show no indication for a zenith angle dependence of the number of events. However, Kamiokande has a lower statistics than Super-Kamiokande. In Refs. [257,258,259,260,261] analyses of all available experiments were performed using the three possible oscillation hypotheses $\stackrel{\scriptscriptstyle (-)}{\nu}_{\hskip-3pt \mu} \leftrightarrow \stackrel{\scriptscriptstyle (-)}{\nu}_{\hskip-3pt e,\tau,s}$ with similar results as described above and showing in addition that also the hypothesis of oscillations into sterile neutrinos gives a satisfactory fit. For analyses with three neutrinos and including all neutrino oscillation experiments see [262,263,264,265,266,261]. In Ref. [245] (IMB) no zenith angle variation of $R$ was seen though with rather small statistics.

In the Super-Kamiokande experiment a significant up-down asymmetry for the $\mu$-like events was found. The measured value of the asymmetry $(U-D)/(U+D)$ as a function of momentum for $e$-like and $\mu$-like events is shown in Fig. 5.2 [2]. Here $U$ is the number of upward-going events with zenith angles in the range $-1 < \cos \theta < -0.2$ and $D$ the number of downward-going events with $0.2 < \cos \theta < 1$. The value of the asymmetry for FC and PC multi-GeV $\mu$-like events is [2]

$$A_\mu \equiv \left(\frac{U-D}{U+D}\right)_\mu = -0.296 \pm 0.048 \pm 0.01 \,.$$ (5.10)

The geomagnetic effect which is one of the causes of an up-down asymmetry contributes less than $\pm 0.01$ for multi-GeV events [2]. For the PC events with a mean neutrino energy of 15 GeV [2] this effect is even less important. Note that for multi-GeV events the average angle between the incoming neutrino direction and the charged lepton seen in the single-ring events is of the order of $20^\circ$ or less. Since $\cos \theta = \pm 0.2$ corresponds to $90^\circ \pm 11.5^\circ$, the asymmetry $A$ of the $\mu$-like events is in effect an up-down asymmetry of the $\stackrel{\scriptscriptstyle (-)}{\nu}_{\hskip-3pt \mu}$ flux. The result (5.10) constitutes the best indication in favour of neutrino oscillations found so far with $A_\mu$ being 6 standard deviations away from 0. This is corroborated by the plots of $A_{e,\mu}(p)$ as a function of the lepton momentum $p$ in the case of the single-ring events shown in Fig. 5.2 [2]. For $e$-like events, $A_e(p)$ is a flat function consistent with no $\stackrel{\scriptscriptstyle (-)}{\nu}_{\hskip-3pt e} \leftrightarrow \stackrel{\scriptscriptstyle (-)}{\nu}_{\hskip-3pt \mu}$ oscillations. The asymmetry for $e$-like events analogous to Eq.(5.10) is compatible with zero: $A_e = -0.036 \pm 0.067 \pm 0.02$. For the $\mu$-like events, $A_\mu(p)$ starts with zero and decreases, thus indicating that at small energies both up and down-going neutrinos oscillate with averaged $\stackrel{\scriptscriptstyle (-)}{\nu}_{\hskip-3pt \mu} \to \stackrel{\scriptscriptstyle (-)}{\nu}_{\hskip-3pt \mu}$ probabilities 1/2 whereas at muon energies in the GeV range the probability of $\stackrel{\scriptscriptstyle (-)}{\nu}_{\hskip-3pt \mu} \to \stackrel{\scriptscriptstyle (-)}{\nu}_{\hskip-3pt \mu}$ survival for the down-going neutrinos approaches 1. This behaviour is in agreement with the $L/E_\nu$ dependence of the oscillation probabilities. It is interesting to note that the experimental values of the asymmetries $A_{e,\mu}$ can be used to discriminate between different oscillation scenarios [267,268,269,270].

The importance of the results of the atmospheric neutrino oscillation experiments requires further scrutiny to test the interpretation in terms of neutrino oscillations. It has been proposed for Super-Kamiokande to use ratios of charged current events (CC) to neutral current (NC) events [271,272] in the spirit of the SNO experiment [273] in the context of solar neutrinos. The basic idea is that in Super-Kamiokande NC events could be seen through

$$\stackrel{\scriptscriptstyle (-)}{\nu}_{\hskip-3pt \ell} + N... ...uad \mbox{with} \quad N = p,n \, \quad \ell = e, \mu, \tau \,,$$ (5.11)

whereas CC reactions are likely to produce a single charged pion via

$$\stackrel{\scriptscriptstyle (-)}{\nu}_{\hskip-3pt \ell} + N... ...ll^\mp + \pi^\pm + N \quad \mbox{with} \quad \ell = e, \mu \,.$$ (5.12)

To produce a test with the reactions (5.11) and (5.12) it is necessary to discuss their experimental signatures and the contaminations of these signature with other processes. Neutral pions are selected by taking events with two diffuse rings from the two decay photons with an invariant mass between 90 and 180 MeV and momenta smaller 400 MeV in order to separate the two rings. Such events were already observed in Kamiokande [274]. Because of the selection criteria for $\pi^0$'s, single pion production is important only for neutrino energies of order 1 GeV, therefore it involves the sub-GeV events and no tau lepton will produced in the CC reaction. Defining $N_{\pi^0}$ via this procedure then the two-ring ratios [272]

$$\mathcal{R}_e \equiv \frac{N_{DS}}{N_{\pi^0}} \quad \mbox{and} \quad \mathcal{R}_\mu \equiv \frac{N_{SS}}{N_{\pi^0}}$$ (5.13)

act as measures of the CC to NC event ratios because in the ideal situation one can make the identification $N_{\pi^0} = N^{NC}_{\pi^0}$, $N_{DS} = N_{e^\mp \pi^\pm}^{CC}$ and $N_{SS} = N_{\mu^\mp \pi^\pm}^{CC}$ since electrons produce diffuse and muons and charged pions sharp rings. This picture is blurred [272,275] by detector efficiencies (e.g., it is 0.77 for the identification of neutral pions via two diffuse rings), misidentifications (like misidentifying a $\pi^0$ whose two photons cannot be resolved with an electron) and contaminations (most notably, NC events with $\pi^+ \pi^-$ in the final state contribute to $N_{SS}$, electronic CC events with a $\pi^0$ whose photons cannot be resolved add to $N_{\pi^0}$ and muonic CC events with the same $\pi^0$ configuration add to $N_{DS}$). Taking all this into account [272,275], one has nevertheless reason the expect that with increasing statistics in Super-Kamiokande the two-ring events can be used to obtain information on distinguishing between oscillation of the muon neutrino into tau or sterile neutrinos and to discriminate between different regions of the parameter space of neutrino mixing [272,275,276]. Furthermore, it was suggested to use the asymmetry $A_N \equiv (U_{\pi^0}-D_{\pi^0})/(U_{\pi^0}+D_{\pi^0})$ for up and down-going $\pi^0$'s coming from the NC reaction (5.11) as an observable to distinguish muon neutrino oscillations into active neutrinos from those into sterile neutrinos [277]. One has to assume in this case that up and down $\pi^0$'s originate from up and down neutrinos, respectively, with high probability. This is not so obvious, however, since for the identification of the $\pi^0$ the events should rather have low energy as explained above.

The further tests discussed here concern the stopping and through-going muon events where special efforts have been made to calculate the fluxes [249,278,253]. Whereas the FC events have neutrino energies of around 1 GeV the stopping muon events correspond to a mean neutrino energy of 10 GeV and the through-going muons to 100 GeV [249,231]. Thus we are now discussing a different range of energy compared to the discussion above (with the exception of the PC sample).

Obviously, also the zenith angle distribution of upward stopping or through-going muons can be used to test the neutrino oscillation hypothesis [279]. Among other experiments (see above) upward through-going muons have been studied by Kamiokande [252] and upward muons by MACRO [52]. Kamiokande has 372 such events above an energy threshold of 1.6 GeV. Fitting the data to the $\stackrel{\scriptscriptstyle (-)}{\nu}_{\hskip-3pt \mu} \to \stackrel{\scriptscriptstyle (-)}{\nu}_{\hskip-3pt \tau}$ oscillation hypothesis yields a best fit with $\Delta m^2 = 3.2 \times 10^{-3}$ eV$^2$ agreeing rather well with the Super-Kamiokande result. The analysis of the MACRO Coll. based on 479 events gives a similar result for $\Delta m^2$, however, the zenith angle distribution does not fit very well with the oscillation hypothesis into tau neutrinos. An attempt has been made to explain the zenith angle distribution of the MACRO experiment with $\stackrel{\scriptscriptstyle (-)}{\nu}_{\hskip-3pt \mu} \to \stackrel{\scriptscriptstyle (-)}{\nu}_{\hskip-3pt s}$ oscillations where matter effects in the earth play a crucial role [280].

In the earth, the density profile can approximately be represented by constant densities in the mantle and the core, respectively (see end of Section 4.3). Such a profile can lead to an enhancement of neutrino transitions due to the matter effect¹⁷in the earth if atmospheric neutrinos cross the core¹⁸ such that the phase picked up by a neutrino wave function traversing the mantle for the first time and the phase acquired by traversing the core are each approximately equal to $\pi$. Such an effect for atmospheric neutrinos was recently considered in detail in Refs. [280,282,283,284,285]. It has also been proposed to exploit this effect to discriminate between $\stackrel{\scriptscriptstyle (-)}{\nu}_{\hskip-3pt \mu} \to \stackrel{\scriptscriptstyle (-)}{\nu}_{\hskip-3pt \tau}$ (no matter effects) and $\stackrel{\scriptscriptstyle (-)}{\nu}_{\hskip-3pt \mu} \to \stackrel{\scriptscriptstyle (-)}{\nu}_{\hskip-3pt s}$ transitions of atmospheric neutrinos [280] and to explain an excess of $e$-like events [286] possibly seen in the Super-Kamiokande experiment [2].

It was suggested in Ref. [244] to use the observable

$$r = \frac{N_{\mathrm{stop}}}{N_{\mathrm{thru}}}$$ (5.14)

where $N_{\mathrm{stop}}$ and $N_{\mathrm{thru}}$ are the numbers of stopping and though-going muons, respectively, as an indicator for neutrino oscillations because this ratio is reduced for neutrino oscillations with respect to the no-oscillation hypothesis [253,287]. Also the ratios of ``horizontal'' to ``vertical'' muon events for stopping and through-going muons defined by

$$S_{\mathrm{stop,thru}} \equiv \left. \frac{N_{\mathrm{hor}}}{N_{\mathrm{vert}}} \right\vert _{\mathrm{stop, thru}} \,,$$ (5.15)

with

$$N_{\mathrm{hor}} = \int^0_{\cos \theta_c} d \cos \theta \fra... ...nt^{\cos \theta_c}_{-1} d \cos \theta \frac{dN}{d \cos \theta}$$ (5.16)

are useful observables as shown in Ref. [287]. In this work a study was made for Super-Kamiokande, taking $\cos \theta_c = -0.5$ as the boundary between ``horizontal'' and ``vertical'', showing that the prospects are good for confirming atmospheric neutrino oscillations found with the FC single-ring and PC events by using stopping and through-going muons and the observables $r$ (5.14), $S_{\mathrm{stop}}$ and $S_{\mathrm{thru}}$ (5.15). However, for a precise determination of $\Delta m^2$ these variables are not suitable.

Long-baseline experiments and tests of the atmospheric neutrino oscillation parameters

The explanation of the atmospheric neutrino anomaly in terms of neutrino oscillations can be checked with long-baseline neutrino oscillation experiments. The first long-baseline reactor experiment CHOOZ [159] (see Section 3.2 and Fig. 3.3) has already excluded atmospheric $\nu_\mu\leftrightarrows\nu_e$ oscillations with a large mixing angle for $\Delta{m}^2_{\mathrm{atm}} \gtrsim 10^{-3} \, \mathrm{eV}^2$. Two other long-baseline reactor experiments are under construction: Palo Verde [288,289] and Kam-Land [290,291]. The Palo Verde experiment has a setup and a sensitivity similar to the CHOOZ experiment, whereas the Kam-Land experiment, which is the result of the conversion of the old Kamiokande detector to a liquid scintillator detector, will detect $\bar\nu_e$'s produced by Japanese reactors 150-200 km away and will be sensitive to $\Delta{m}^2 \gtrsim 10^{-5} \, \mathrm{eV}^2$ and a large mixing angle. The Borexino experiment (see the end of the Section 5.2 and [292,291]) will allow to perform a similar measurement.

Accelerator long-baseline experiments will study the oscillation channels $\nu_\mu\to\nu_{e,\mu,\tau}$. The K2K [271] experiment, with a baseline of about 235 km from KEK to Super-Kamiokande and a neutrino beam with 1.4 GeV average energy, will be sensitive to $\nu_\mu$ disappearance and $\nu_\mu\to\nu_e$ transitions with $\Delta{m}^2 \gtrsim 2 \times 10^{-3} \, \mathrm{eV}^2$. A near 1 kton water-Cherenkov detector will be placed at a distance of about 1 km from the beam dump and will allow to measure the initial flux and energy spectrum of $\nu_\mu$'s. This experiment is under construction and is planned to begin taking data in the year 1999.

Also the MINOS [293] experiment is under construction. This experiment will have a near detector at Fermilab and a baseline of about 730 km from Fermilab to the Soudan mine, where the far detector will be placed. The neutrino beam will be produced by protons from the new Main Injector at Fermilab and will have an average energy of about 10 GeV. The far detector is an 8 kton sampling calorimeter made of magnetized iron and scintillators. This experiment will be sensitive to $\nu_\mu$ disappearance and $\nu_\mu\to\nu_e$, $\nu_\mu\to\nu_\tau$, $\nu_\mu\to\nu_s$ transitions, with the possibility to distinguish the different channels, for $\Delta{m}^2 \gtrsim 10^{-3} \, \mathrm{eV}^2$ (the possibility to extend the sensitivity to $\Delta{m}^2 \gtrsim 5 \times 10^{-5} \, \mathrm{eV}^2$ lowering the neutrino energy is under study). In particular $\nu_\mu\to\nu_s$ transitions can be revealed through the measurement of a deficit in the NC/CC ratio. The MINOS experiment is scheduled to start data-taking around the year 2003.

The ICARUS experiment [294] in Gran Sasso, constituted of a 0.6 kton liquid argon detector is scheduled to start in the year 2000. In the future three new modules with a total mass of 2.4 kton will be installed. This detector will be sensitive to atmospheric and solar neutrinos and will allow to reveal long-baseline $\nu_\mu\to\nu_\tau$ oscillations using a neutrino beam produced at CERN about 730 km away. Since the average energy of the neutrino beam is rather high, about 25 GeV, in order to allow the detection of $\nu_\tau$ through the CC production of a $\tau$, this experiment will be sensitive to $\Delta{m}^2 \gtrsim 10^{-3} \, \mathrm{eV}^2$. Four other detectors for future LBL CERN-Gran Sasso experiments, OPERA [295], NOE [296], AQUA-RICH [297] and NICE [298], have been proposed and are under consideration (see Ref. [299]), together with the possibility of a new atmospheric neutrino detector consisting of a large-mass and high-density tracking calorimeter [300].

In Ref. [301] a comparison is made between the possibilities using atmospheric neutrinos and LBL neutrino experiments for the determination of the oscillation parameters. The feasibility to distinguish between atmospheric $\nu_\mu\to\nu_\tau$ and $\nu_\mu\to\nu_s$ oscillations using a combination of the results of future atmospheric, LBL and SBL experiments is discussed in Ref. [302].

Solar neutrino experiments

The earliest indication in favour of neutrino oscillations was obtained about 30 years ago in the radiochemical solar neutrino experiment by R. Davies et al. [36]. The flux of solar electron neutrinos measured in this experiment was significantly less than the predicted one. This phenomenon was called solar neutrino problem. The existence of this problem was confirmed in all five solar neutrino experiments (Homestake [36,37,38], Kamiokande [39,40,41], GALLEX [42,43], SAGE [44,45] and Super-Kamiokande [46,47,48]) which measure a flux of electron neutrinos significantly smaller than the one predicted by the Standard Solar Model (SSM) [174,60,303,175,304,305,306,307,308,309,310]. The solar neutrino problem (see, for example, [60,311,312,313,314,315,316]) arose in the Homestake experiment by the low counting rate showing that the flux of the high-energy $^8\mathrm{B}$ neutrinos ( $E_{^8\mathrm{B}} \lesssim 15 \, \mathrm{MeV}$) and of the medium-energy $^7\mathrm{Be}$ neutrinos ( $E_{^7\mathrm{Be}} = 0.862 \, \mathrm{MeV}$) is suppressed by a factor of about 1/3 with respect to the SSM prediction. In 1988 the solar neutrino problem was confirmed by the results of the real-time water-Cherenkov Kamiokande experiment [39] which measured a flux of $^8\mathrm{B}$ neutrinos of about half of the SSM flux. The measurements of the Kamiokande experiment proved that the observed neutrinos arrive at the detector from the direction of the sun. In 1992 the radiochemical GALLEX [42] and SAGE experiments [45] succeeded in measuring the neutrino flux with a low energy threshold $E_{\mathrm{th}} = 233 \, \mathrm{keV}$, which allowed to detect low-energy $pp$ neutrinos produced by the fundamental reaction of the $pp$ cycle. Also these experiments measured a neutrino flux of about half of the one predicted by the SSM. Finally, the Super-Kamiokande experiment has recently confirmed [46,47,48] with high statistics the suppression of the $^8\mathrm{B}$ neutrino flux with respect to the SSM one by a factor of about 1/2.

[l]

	Figure 5.3. The $pp$ cycle.	Figure 5.4. The CNO cycle.

The energy of the sun is produced in the reactions of the thermonuclear $pp$ and CNO cycles shown in Figs. 5.3 and 5.4 (see, e.g., Ref. [60]). The overall result of both cycles is the transition

$$4 \, p + 2 \, e^- \to {}^4\mathrm{He} + 2 \, \nu_e + Q \,,$$ (5.17)

where $Q = 4 \, m_p + 2 \, m_e - m_{^4\mathrm{He}} = 26.73 \, \mathrm{MeV}$ is the energy release.¹⁹Hence, the production of energy in the sun is accompanied by the emission of electron neutrinos. The main part of the solar energy is radiated through photons and a small part (about 2%) is emitted through neutrinos. The sources of solar neutrinos are listed in Table 5.1. The $pp$, $pep$, $^7$Be, $^8$B and $hep$ reactions belong to the $pp$ cycle (see Fig. 5.3), whereas the $^{13}$N, $^{15}$O and $^{17}$F reactions belong to the CNO cycle (see Fig. 5.4), which produces only about 2% of the solar energy. The average and maximum neutrino energies listed in Table 5.1 are taken from Refs. [317,318]. The neutrino fluxes and predictions for the neutrino capture rates in the chlorine Homestake experiment and in the gallium GALLEX and SAGE experiments given by the Bahcall-Pinsonneault 1998 (BP98) [304] SSM are listed in Table 5.2.

Source $r$	Reaction	Average Neutrino Energy $\langle{E}\rangle_r$ (MeV)	Maximum Neutrino Energy (MeV)
$pp$	$p + p \to d + e^+ + \nu_e$	$0.2668$	$0.423 \pm 0.03$ |
$pep$	$p + e^- + p \to d + \nu_e$	$1.445$	$1.445$ |
$^7$Be	$e^- + {}^7\mathrm{Be} \to {}^7\mathrm{Li} + \nu_e$	$0.3855$ $0.8631$	$0.3855$ $0.8631$ |
$^8$B	${}^8\mathrm{B} \to {}^8\mathrm{Be}^* + e^+ + \nu_e$	$6.735 \pm 0.036$	$\sim 15$ |
$hep$	${}^3\mathrm{He} + p \to {}^4\mathrm{He} + e^+ + \nu_e$	$9.628$	$18.778$ |
$^{13}$N	${}^{13}\mathrm{N} \to {}^{13}\mathrm{C} + e^+ + \nu_e$	$0.7063$	$1.1982 \pm 0.0003$ |
$^{15}$O	${}^{15}\mathrm{O} \to {}^{15}\mathrm{N} + e^+ + \nu_e$	$0.9964$	$1.7317 \pm 0.0005$ |
$^{17}$F	${}^{17}\mathrm{F} \to {}^{17}\mathrm{O} + e^+ + \nu_e$	$0.9977$	$1.7364 \pm 0.0003$ |

Table 5.1. Sources of solar neutrinos [317,318,319].

Source $r$	Flux $\Phi_r$ ( $\mathrm{cm}^{-2} \, \mathrm{s}^{-1}$)	$\langle\sigma_{\mathrm{Cl}}\rangle_r$ ( $10^{-44} \, \mathrm{cm}^2$)	$S_{\mathrm{Cl}}^{(r)}$ (SNU)	$\langle\sigma_{\mathrm{Ga}}\rangle_r$ ( $10^{-44} \, \mathrm{cm}^2$)	$S_{\mathrm{Ga}}^{(r)}$ (SNU)
$pp$	$( 5.94 \pm 0.06 ) \times 10^{10}$	-	-	$0.117 \pm 0.003$	$69.6 \pm 0.7$ |
$pep$	$( 1.39 \pm 0.01 ) \times 10^{8}$	$0.16$	$0.2$	$2.04 \, {}^{+0.35}_{-0.14}$	$2.8$ |
$^7$Be	$( 4.80 \pm 0.43 ) \times 10^{9}$	$0.024$	$1.15 \pm 0.1$	$0.717 \, {}^{+0.050}_{-0.0.021}$	$34.4 \pm 3.1$ |
$^8$B	$( 5.15 \, {}^{+0.98}_{-0.72} ) \times 10^{6}$	$114 \pm 11$	$5.9 \, {}^{+1.1}_{-0.8}$	$240 \, {}^{+77}_{-36}$	$12.4 \, {}^{+2.4}_{-1.7}$ |
$hep$	$2.10 \times 10^{3}$	$390$	$0.0$	$714 \, {}^{+228}_{-114}$	$0.0$ |
$^{13}$N	$( 6.05 \, {}^{+1.15}_{-0.77} ) \times 10^{8}$	$0.017$	$0.1$	$0.604 \, {}^{+0.036}_{-0.018}$	$3.7 \, {}^{+0.7}_{-0.5}$ |
$^{15}$O	$( 5.32 \, {}^{+1.17}_{-0.80} ) \times 10^{8}$	$0.068 \pm 0.001$	$0.4 \pm 0.1$	$1.137 \, {}^{+0.136}_{-0.057}$	$6.0 \, {}^{+1.3}_{-0.9}$ |
$^{17}$F	$( 6.33 \, {}^{+0.76}_{-0.70} ) \times 10^{6}$	$0.069$	$0.0$	$1.139 \, {}^{+0.137}_{-0.057}$	$0.1$ |
Total			$7.7 \, {}^{+1.2}_{-1.0}$		$129 \, {}^{+8}_{-6}$ |

Table 5.2. Standard Solar Model [304] neutrino fluxes, average neutrino cross sections [60,317,318] and SSM predictions for the neutrino capture rates [304] in the chlorine (Cl) Homestake experiment and in the gallium (Ga) GALLEX and SAGE experiments.

As it is seen from the Tables 5.1 and 5.2, the major part of solar neutrinos are low energy neutrinos coming from the $pp$ reaction. Monoenergetic neutrinos with intermediate energy are produced in the capture of electrons by $^7$Be and in the $pep$ reaction. High energy neutrinos are produced in the decay of $^8$B (the flux of $hep$ neutrinos is so small that its contribution to the event rates of solar neutrino experiments is negligible). The flux of $^8$B neutrinos is much smaller than the fluxes of $pp$, $^7$Be and $pep$ neutrinos. However, as we will see later, these neutrinos give the major contribution to the event rates of experiments with a high energy detection threshold. The CNO $^{13}\mathrm{N}$, $^{15}\mathrm{O}$, $^{17}\mathrm{F}$ reactions are sources of intermediate energy neutrinos with a spectrum that extends up to about $1.7 \, \mathrm{MeV}$. Their contribution to the event rates of solar neutrino experiment is small but not negligible.

The neutrino flux coming from each source as a function of the neutrino energy $E$ can be written as

$$\phi_r(E)=\Phi_r\,X_r(E) \qquad \qquad (r=pp,pep,{}^7\mathrm... ...ep, {}^{13}\mathrm{N},{}^{15}\mathrm{O},{}^{17}\mathrm{F}) \,,$$ (5.18)

where $\Phi_r$ is the total flux and $X_r(E)$ is the energy spectrum ( $\int \mathrm{d}E \, X_r(E) = 1$). The energy spectrum $X_r(E)$ for each source $r$ is known with negligible uncertainties [320,317] because it is determined by the weak interactions and it is practically independent from solar physics. On the other hand, the total flux $\Phi_r$ of each source $r$ must be calculated with a solar model and the resulting uncertainties represent one of the main problem for the interpretation of the experimental results. However, there are some relations that allow to extract model-independent information on the neutrino fluxes from the experimental data. The main one is the luminosity constraint, which is based on the assumption that the sun is in a stable state (the energy is produced in the central region of the sun and for its electromagnetic part it takes more than $10^4$ years to reach the surface, whereas neutrinos escape the sun in about two seconds). Let us consider a solar neutrino with energy $E$. The luminous energy released together with this neutrino is $Q/2 - E$. Multiplying this quantity with the total flux of neutrinos $\sum_r \phi_r(E)$ and integrating over the neutrino energy one obtains the luminosity constraint²⁰

$$\sum_r \left( \frac{ Q }{ 2 } - \langle{E}\rangle_r \right) \Phi_r = \frac{ \mathcal{L}_\odot }{ 4 \pi R^2 } \,,$$ (5.19)

where $\langle{E}\rangle_r = \int E \, X_r(E) \, \mathrm{d}E$ is the average energy of the neutrinos from the source $r$ (see Table 5.1), $\mathcal{L}_\odot = 2.40 \times 10^{39} \, \mathrm{MeV} \, \mathrm{s}^{-1}$ [35] is the luminosity of the sun, $R = 1.496 \times 10^{13} \, \mathrm{cm}$ [35] is the sun - earth distance. The luminosity constraint can be rewritten in the compact form

$$\sum_r Q_r \, \Phi_r = K_\odot \,,$$ (5.20)

where $Q_r \equiv Q/2 - \langle{E}\rangle_r$ and $K_\odot \equiv \mathcal{L}_\odot / 4 \pi R^2 = 8.54 \times 10^{11} \, \mathrm{MeV} \, \mathrm{cm}^{-2} \, \mathrm{s}^{-1}$ is the solar constant. Let us emphasize that the luminosity relation is valid under the assumption that solar $\nu_e$'s on their way to the earth do not transform into other states. Neglecting the small fraction of energy carried away by neutrinos, the luminosity constraint gives the approximate value of the total solar neutrino flux $\Phi = \sum_r \Phi_r$:

$$\Phi \simeq \frac{ 2 \, K_\odot }{ Q } = 6.4 \times 10^{10} \, \mathrm{cm}^{-2} \, \mathrm{s}^{-1} \,.$$ (5.21)

Since the $^3\mathrm{He}$ nuclei necessary for the formation of $^7\mathrm{Be}$ and $^8\mathrm{B}$ are created by the $pp$ or $pep$ reactions, there is another model-independent constraint for the solar neutrino fluxes of the $pp$ cycle (see [321]):

$$\phi_{^7\mathrm{Be}} + \phi_{^8\mathrm{B}} \leq \phi_{pp} + \phi_{pep} \,.$$ (5.22)

Let us now consider the experimental data. The results of five solar neutrino experiments are available at present and are listed in Table 5.3.

Experiment	Result	Theory	$\frac{\mbox{Result}}{\mbox{Theory\rule[-0.2cm]{0pt}{0.5cm}}}$ |
Homestake [38]	$2.56 \pm 0.16 \pm 0.16$ ($2.56 \pm 0.23$)	$7.7 \, {}^{+1.2}_{-1.0}$	$0.33 \, {}^{+0.06}_{-0.05}$
GALLEX [322]	$77.5 \pm 6.2 \, {}^{+4.3}_{-4.7}$ ($78 \pm 8$)	$129 \, {}^{+8}_{-6}$	$0.60 \pm 0.07$
SAGE [323]	$66.6 \, {}^{+6.8}_{-7.1} \, {}^{+3.8}_{-4.0}$ ($67 \pm 8$)	$129 \, {}^{+8}_{-6}$	$0.52 \pm 0.07$
Kamiokande [41]	$2.80 \pm 0.19 \pm 0.33$ ($2.80 \pm 0.38$)	$5.15 \, {}^{+1.0}_{-0.7}$	$0.54 \pm 0.07$
Super-Kamiokande [48]	$2.44 \pm 0.05 \, {}^{+0.09}_{-0.07}$ ( $2.44 \, {}^{+0.10}_{-0.09}$)	$5.15 \, {}^{+1.0}_{-0.7}$	$0.47 \, {}^{+0.07}_{-0.09}$

Table 5.3. The results of solar neutrino experiments confronted with the corresponding theoretical predictions [304]. The results of the Homestake, GALLEX and SAGE experiments are expressed in terms of event rates in SNU units ( $1 \:\mathrm{SNU} \equiv 10^{-36} \: \mathrm{events} \: \mathrm{atom}^{-1} \, \mathrm{s}^{-1}$), whereas the results of the Kamiokande and Super-Kamiokande experiments are expressed in terms of the $^8\mathrm{B}$ neutrino flux in units of $10^6 \, \mathrm{cm}^{-2} \mathrm{s}^{-1}$. The first experimental error is statistical and the second is systematic. The experimental values in parenthesis have the statistical and systematic errors added in quadrature.

Homestake [36,37,38], GALLEX [42,43] and SAGE [44,45] are radiochemical experiments. In the pioneering chlorine Homestake experiment of R. Davis et. al., which started in 1967, the detector is a tank with a volume of $6 \times 10^5$ liters filled with C$_2$Cl$_4$. Radioactive atoms of $^{37}$Ar are produced by solar electron neutrinos through the reaction [324,325]

$$\nu_e + {}^{37}\mathrm{Cl} \to e^- + {}^{37}\mathrm{Ar} \,,$$ (5.23)

which has an energy threshold $E_{\mathrm{th}} = 0.81 \, \mathrm{MeV}$. The radioactive $^{37}$Ar atoms that are created during the time of exposition of each run (about two months) are extracted from the detector by purging it with $^4$He and counted in small proportional counters which detect the Auger electron produced in the electron-capture of the $^{37}$Ar nuclei. About 0.5 atoms of $^{37}$Ar are produced every day by solar neutrinos and about 16 atoms are extracted in each run (this number is smaller than 30 because of the $^{37}$Ar lifetime of about 35 days and because of the extraction efficiency of about 90%). Since the energy threshold is above the end-point of the $pp$ neutrino spectrum and the cross section of the detecting process (5.23) grows with the neutrino energy, the main contribution to the counting rate in the Homestake experiment comes from $^8$B and $^7$Be neutrinos. According to the BP98 SSM [304] the event rate in the Homestake experiment should be²¹ $7.7 \, {}^{+1.2}_{-1.0} \, \mathrm{SNU}$, of which 5.9 SNU come from $^8$B neutrinos, 1.15 SNU are produced by $^7$Be neutrinos and 0.7 SNU are due to $pep$ and CNO neutrinos (see Table 5.2). The Homestake event rate presented in Table 5.3 is the event rate averaged over 108 runs [38].

In the radiochemical gallium experiments GALLEX and SAGE electron neutrinos from the sun are detected through the observation of radioactive $^{71}$Ge that is produced in the process

$$\nu_e + {}^{71}\mathrm{Ga} \to e^- + {}^{71}\mathrm{Ge} \,.$$ (5.24)

The GALLEX detector is a tank containing 30.3 tons of $^{71}$Ga (100 tons of a water solution of gallium chloride), whereas the SAGE experiment uses about 57 tons of $^{71}$Ga in metallic form. The event rate measured in the GALLEX experiment is $0.699 \pm 0.069$ events per day. (Since the $^{71}$Ge lifetime is about 16.5 days, only around 7 atoms of $^{71}$Ge are extracted in each 3-weeks run.)

Since the threshold of the process (5.24) is $E_{\mathrm{th}}^{^{71}\mathrm{Ga}} = 0.233 \, \mathrm{MeV}$, neutrinos from all sources are detected in gallium experiments. According to the SSM, the contributions to the total predicted event rate from $pp$, $^7$Be and $^8$B neutrinos are [304] 54%, 27% and 10%, respectively.

As can be seen from Table 5.3, both gallium detectors measure an event rate that is about one half of the SSM prediction. The weighted average of the GALLEX and SAGE results yields the event rate

$$S_{\mathrm{Ga}}^{\mathrm{exp}} = 72.5 \pm 6 \: \mathrm{SNU} \,.$$ (5.25)

Taking into account the experimental and theoretical errors, this rate differs from the SSM prediction by about seven standard deviations!

Both the GALLEX and SAGE detectors have been calibrated using an intense $^{51}\mathrm{Cr}$ neutrino source. The ratio of observed and expected events is $0.93 \pm 0.08$ for GALLEX [326] and $0.95 \pm 0.12$ for SAGE [45]. These results demonstrate the absence of unexpected systematic errors at the 10% level in both experiments. In addition, the GALLEX Collaboration calibrated the detector by introducing a known number of radioactive $^{71}\mathrm{As}$ atoms in the target solution [326,327]. The atoms of $^{71}\mathrm{Ge}$ resulting from $^{71}\mathrm{As}$ decay have been extracted in the usual way, and the As tests prove, at the 1% level, the reliability of the technique [328] (the number of $^{71}\mathrm{Ge}$ atoms produced in these tests is of the order of $10^5$, whereas the number produced in the $^{51}\mathrm{Cr}$ tests is of the order of ten per day). As emphasized by the GALLEX Collaboration [326,327], these results rule out the presence of unexpected radiochemical effects that could explain the deficit of solar $\nu_e$'s measured by the gallium experiments.

It is necessary to emphasize that the gallium experiments are not only very important for the assessment of the solar neutrino problem but also for the theory of thermonuclear energy production in the sun: They have provided the first observation of low-energy solar neutrinos produced in the $pp$ reaction that is the basic reaction of the $pp$ cycle and, therefore, the first direct experimental confirmation of the theory of thermonuclear origin of solar energy production.

In the Kamiokande [39,40,41] and Super-Kamiokande [46,47,48] experiments water-Cherenkov detectors are used for the detection of solar neutrinos through the observation of the Cherenkov light emitted by the recoil electrons in the elastic-scattering process

$$\nu + e^- \to \nu + e^- \,.$$ (5.26)

Since the direction of the recoil electron is peaked in the direction of the incoming neutrino, water-Cherenkov detectors measure the direction of the neutrino flux and the results of Kamiokande and Super-Kamiokande have proven that there is a flux of high-energy neutrinos coming from the sun with an intensity about half of that predicted by the SSM.

The energy threshold of water-Cherenkov detectors is given by the threshold for the detection of the recoil electron in the reaction (5.26). It is higher than in other solar neutrino experiments because of the large background at low energies: $E_{\mathrm{th}}^{\mathrm{Kam}} \simeq 7 \, \mathrm{MeV}$ in the Kamiokande experiment and $E_{\mathrm{th}}^{\mathrm{SK}} \simeq 6.5 \, \mathrm{MeV}$ in the Super-Kamiokande experiment. Therefore, only $^8\mathrm{B}$ neutrinos can be detected in water-Cherenkov experiments. If nothing happens to electron neutrinos during their trip from the core of the sun to the earth (i.e., no neutrino oscillations or other transitions), the results of the Kamiokande and Super-Kamiokande experiments provide a measurement of the total flux $\Phi_{^8\mathrm{B}}$ of $^8\mathrm{B}$ neutrinos. Therefore, the results of the Kamiokande and Super-Kamiokande experiments are usually presented in terms of the measured flux of $^8\mathrm{B}$ neutrinos. In the Super-Kamiokande experiment it was found

$$\Phi_{^8\mathrm{B}}^{\mathrm{SK}} = ( 2.44 \, {}^{+0.10}_{-0.09} ) \times 10^6 \, \mathrm{cm}^{-2} \, \mathrm{s}^{-1} \,,$$ (5.27)

which is about one half of the flux predicted by the SSM (see Table 5.3).²²

From the experimental results and the most updated theoretical predictions [304] listed in Table 5.3 one can see that the observed event rates in all solar neutrino experiments are significantly smaller than the predicted rates (see also [329] where the comparison with other theoretical predictions is discussed). This discrepancy constitutes the solar neutrino problem. The SSM is robust and it has been recently tested in a convincing way by comparing its predicted value for the sound speed in the interior of the sun with precise helioseismological measurements (see [330,331,304,315,332,333]). However, it is clear that a model-independent proof of the existence of the solar neutrino problem would be more convincing. Such a model-independent approach has been discussed in several papers [334,308,335,336,337,338,339,340,341,321,312,342,329,343] using the model-independent luminosity constraint (5.20) and the fact that the energy spectra $X_r(E)$ Eq.(5.18) of the various neutrino sources are practically independent from solar physics [320,317].

From the luminosity constraint it is possible to obtain a model-independent lower bound on the gallium event rate. Indeed, since $pp$ neutrinos have the smallest average energy and the $\nu_e$- $^{71}\mathrm{Ga}$ cross section $\sigma_{\mathrm{Ga}}(E)$ increases with the neutrino energy $E$, we have

$$S_{\mathrm{Ga}} = \int \sigma_{\mathrm{Ga}}(E) \sum_r \phi_r... ..., \Phi_r \geq \langle\sigma_{\mathrm{Ga}}\rangle_{pp} \Phi \,,$$ (5.28)

where $\langle\sigma_{\mathrm{Ga}}\rangle_r = \int \sigma_{\mathrm{Ga}}(E) \, X_r(E) \, \mathrm{d}E$ is the average cross section of neutrinos from the source $r$ (see Table 5.2). Since the luminosity constraint (5.20) implies that

$$Q_{pp} \, \Phi \geq K_\odot \,,$$ (5.29)

we obtain

$$S_{\mathrm{Ga}} \geq \frac{ \langle\sigma_{\mathrm{Ga}}\rangle_{pp} \, K_\odot }{ Q_{pp} } = 76 \pm 2 \: \mathrm{SNU} \,,$$ (5.30)

where we used the value of $\langle\sigma_{\mathrm{Ga}}\rangle_{pp} = 11.7 \times 10^{-46} \, \mathrm{cm}^2$ given in Table 5.2 and $Q_{pp} = 13.1 \, \mathrm{MeV}$.

The lower bound (5.30) is just compatible with the combined result (5.25) of the gallium experiments. This means that the results of the GALLEX and SAGE experiments can be explained if practically only $pp$ neutrinos are emitted by the sun. This possibility is incompatible with any solar model constrained by the helioseismological data [304,315].

More stringent model-independent conclusions on the existence of the solar neutrino problem can be obtained by comparing the results of different solar neutrino experiments. If the survival probability of solar $\nu_e$'s is equal to one, the result of the Super-Kamiokande experiment gives the value (5.27) for the flux of $^8\mathrm{B}$ neutrinos, whose contributions to the event rates of the chlorine and gallium experiments are

$$S_{\mathrm{Cl}}^{^8\mathrm{B},\mathrm{SK}} = \langle\sigma_{\math... ...rm{B}} \, \Phi_{^8\mathrm{B}}^{\mathrm{SK}} = 2.78 \pm 0.27 \, \mathrm{SNU} \,,$$ (5.31)

$$S_{\mathrm{Ga}}^{^8\mathrm{B},\mathrm{SK}} = \langle\sigma_{\math... ....9 \, {}^{+1.9}_{-0.9} \, \mathrm{SNU} \,.\setlength{\arraycolsep}{\templength}$$ (5.32)

Subtracting the contribution (5.31) of $^8\mathrm{B}$ $\nu_e$'s from the event rate measured in the Homestake experiment (see Table 5.3), one obtains

$$S_{\mathrm{Cl}}^{\mathrm{exp}} - S_{\mathrm{Cl}}^{^8\mathrm{B},\mathrm{SK}} = - 0.22 \pm 0.35 \,.$$ (5.33)

Since the quantity $S_{\mathrm{Cl}} - S_{\mathrm{Cl}}^{^8\mathrm{B}}$ represents the contribution to the chlorine event rate due to $pep$, $^7\mathrm{Be}$, $hep$ and CNO neutrinos, it must be positive (or zero) but the result in Eq.(5.33) is only marginally compatible with a positive value (the probability is less than 26%). Furthermore, the result Eq.(5.33) shows that the fluxes of intermediate energy neutrinos ($pep$, $^7\mathrm{Be}$ and CNO neutrinos) are strongly suppressed with respect to the SSM prediction (see Table 5.2). This is incompatible with any solar model constrained by the helioseismological data [304,315] and with the fact that, since both the reactions that produce $^7\mathrm{Be}$ and $^8\mathrm{B}$ neutrinos originate from $^7\mathrm{Be}$ nuclei (see Fig. 5.3), it is very difficult to have a suppression of the $^7\mathrm{Be}$ neutrino flux (with respect to the SSM value) that is stronger than the suppression of the $^8\mathrm{B}$ neutrino flux.²³

Let us consider now the gallium experiments. Subtracting the contribution of $^8\mathrm{B}$ neutrinos from the luminosity constraint (5.20), we have

$$K_\odot - Q_{^8\mathrm{B}} \, \Phi_{^8\mathrm{B}} = \sum_{r\... ..._r \, \Phi_r \leq Q_{pp} \sum_{r\neq{}^8\mathrm{B}} \Phi_r \,.$$ (5.34)

Hence, following the same reasoning as in Eq.(5.28) we obtain

$$S_{\mathrm{Ga}}^{\mathrm{min}} = S_{\mathrm{Ga}}^{^8\mathrm{... ...hrm{SK}} } { Q_{pp} } = 82 \, {}^{+3}_{-2} \, \mathrm{SNU} \,,$$ (5.35)

and the contribution of $pep$, $^7\mathrm{Be}$, $hep$ and CNO neutrinos to the gallium event rate cannot be bigger than

$$S_{\mathrm{Ga}}^{\mathrm{exp}} - S_{\mathrm{Ga}}^{\mathrm{min}} = - 9.5 \pm 6 \,.$$ (5.36)

This result is compatible with a positive value with a probability smaller than 6%. Again, the result (5.36) implies that the fluxes of intermediate energy neutrinos ($pep$, $^7\mathrm{Be}$ and CNO neutrinos) are strongly suppressed with respect to the SSM prediction, in contradiction with any solar model constrained by the helioseismological data and with the moderate suppression of the $^8\mathrm{B}$ neutrino flux.

Further model-independent methods for proving the existence of the solar neutrino problem on the basis of the data of solar neutrino experiments are nicely discussed in Ref. [312]. In the following we will assume that there is a solar neutrino problem that is caused by neutrino oscillations.

The solar neutrino data have been analysed in many papers under the assumption of two-neutrino mixing (see [342,344,329] and references therein) and in a few papers under the assumption of three-neutrino mixing (see [345,346,347,348] and references therein). Here we will briefly review the results of the most updated two-generation analysis [329] of all the solar neutrino data, including those obtained in the first 504 days of operation of the Super-Kamiokande experiment (see Table 5.3).

The deficit of solar $\nu_e$'s can be explained in terms of two-generation neutrino mixing either through vacuum oscillations or through MSW resonant transitions in matter [164,163]. Furthermore, the two mixed neutrinos can be the electron neutrino and another active ($\nu_\mu$ or $\nu_\tau$) neutrino or the electron neutrino and a sterile neutrino. There are two differences between transitions of solar $\nu_e$'s into active and sterile neutrinos:

1. The probability of MSW transitions is different in the two cases, because sterile neutrinos do not have the weak-interaction potential due to elastic forward scattering that is present for active neutrinos.

2. The active neutrinos ($\nu_\mu$'s or $\nu_\tau$'s) that are produced by the oscillation mechanism (either in vacuum or in matter) contribute to the event rates measured in the Kamiokande and Super-Kamiokande water-Cherenkov experiments (notice, however, that the cross section of $\nu_\mu$-$e^-$ and $\nu_\tau$-$e^-$ scattering is about six times smaller than the $\nu_e$-$e^-$ cross section).

	Figure 5.5. MSW $\nu_e\to\nu_\mu$ or $\nu_e\to\nu_\tau$ transitions: result of the fit of the experimental event rates listed in Table 5.3. The shadowed regions are allowed at 99% CL [329]. The dots indicate the best-fit points in each allowed region.	Figure 5.6. Vacuum $\nu_e\to\nu_\mu$ or $\nu_e\to\nu_\tau$ oscillations: result of the fit of the experimental event rates listed in Table 5.3. The shadowed regions are allowed at 99% CL [329]. The dot indicates the best-fit point.

The formulas for the survival probability of solar $\nu_e$'s in the case of vacuum oscillations and MSW transitions are given in Eq.(3.21) (with $\alpha=e$) and in Eq.(4.41) together with Eq.(4.47) [193], respectively. These formulas depend on the two mixing parameters $\Delta{m}^2$ and $\sin^22\vartheta$. The allowed regions in the $\sin^22\vartheta$-$\Delta{m}^2$ plane, obtained in Ref. [329] from the fit of the measured event rates listed in Table 5.3 by using the BP98 SSM [304] and the analytic formulas in Ref. [193] for the MSW case, are shown in Figs. 5.5-5.6.

Figure 5.5 [329] shows the three allowed regions in the case of MSW $\nu_e\to\nu_\mu$ or $\nu_e\to\nu_\tau$ transitions. They are the small mixing angle (SMA-active) region at

$$\Delta{m}^2 \simeq 5 \times 10^{-6} \, \mathrm{eV}^2 \,, \qq... ... \simeq 6 \times 10^{-3} \qquad \qquad \mbox{(SMA-active)} \,,$$ (5.37)

and the large mixing angle (LMA) region at

$$\Delta{m}^2 \simeq 2 \times 10^{-5} \, \mathrm{eV}^2 \,, \qquad \sin^22\vartheta \simeq 0.76 \qquad \qquad \mbox{(LMA)} \,.$$ (5.38)

The best fit, with a confidence level of 19%, is obtained in the SMA region, whereas the LMA region has a confidence level of 4%. In Ref. [329] there is also the so-called low mass (LOW) region at $\Delta{m}^2 \simeq 8 \times 10^{-8} \: \mathrm{eV}^2$ and $\sin^22\vartheta \simeq 0.96$, however, the LOW region is only marginally acceptable, with a confidence level of 0.7%.

The allowed regions in the $\sin^22\vartheta$-$\Delta{m}^2$ plane in the case of $\nu_e\to\nu_\mu$ or $\nu_e\to\nu_\tau$ vacuum oscillations are shown in Fig. 5.5 [329]. These regions extend over large ranges of $\sin^22\vartheta$ and $\Delta{m}^2$ around the best fit values

$$\Delta{m}^2 \simeq 8 \times 10^{-11} \, \mathrm{eV}^2 \,, \q... ...n^22\vartheta \simeq 0.75 \qquad \qquad \mbox{(Vac. Osc.)} \,.$$ (5.39)

The confidence level of the fit is 3.8%. There is no allowed region in the case of $\nu_e\to\nu_s$ vacuum oscillations (the fit has a confidence level of 0.05%) [329,349].

As shown in Fig. 5.7 [329], only the small mixing angle (SMA-sterile) region at

$$\Delta{m}^2 \simeq 4 \times 10^{-6} \, \mathrm{eV}^2 \,, \qq... ...\simeq 7 \times 10^{-3} \qquad \qquad \mbox{(SMA-sterile)} \,,$$ (5.40)

is allowed in the case of MSW $\nu_e\to\nu_s$ transitions, with a confidence level of 19% (the large mixing angle and low mass solutions have a confidence level of 0.001% and 0.003%, respectively) [329,350].

	Figure 5.7. MSW $\nu_e\to\nu_s$ transitions: result of the fit of the experimental event rates listed in Table 5.3. The shadowed region is allowed at 99% CL [329]. The dot indicates the best-fit point.	Figure 5.8. MSW $\nu_e\to\nu_\mu$ or $\nu_e\to\nu_\tau$ transitions: result of the global fit of the experimental event rates listed in Table 5.3 and of the energy spectrum and zenith-angle distribution measured in the Super-Kamiokande experiment [48]. The shadowed region is allowed at 99% CL [329]. The dot indicates the best-fit point.

The authors of Ref. [329] performed also global fits of the solar neutrino data including the energy spectrum and zenith-angle distribution of the recoil electrons measured in the Super-Kamiokande experiment [48]. As noted in Ref. [329], since these data are still preliminary, the results of this analysis are less robust than those obtained fitting only the global rates. However, it is interesting to note that in the case of MSW $\nu_e\to\nu_\mu$ or $\nu_e\to\nu_\tau$ transitions only the SMA-active region remains allowed, as shown in Fig. 5.8 [329], with a confidence level of 7%.

During the night, solar neutrinos pass through the earth and the matter effect can cause a regeneration of $\nu_e$'s (see Ref. [316] and references therein) and, therefore, a zenith angle dependence of the solar neutrino flux. The size of this effect depends on the values of $\Delta{m}^2$ and $\sin^22\vartheta$ and is sizable only for large values of the mixing angle. The preliminary value of the day-night asymmetry of solar neutrino events measured in the Super-Kamiokande experiment is [48]

$$\frac{D-N}{D+N} = - 0.023 \pm 0.020 \pm 0.014 \,,$$ (5.41)

compatible with zero. Hence this experimental result tends to exclude solutions of the solar neutrino problem with a large mixing angle.

As was shown in Refs. [283,351] the step-like profile of the matter density of the earth could lead to an enhancement of neutrino transitions not only for atmospheric neutrinos (see Section 5.1.3) but also for solar neutrinos.

In conclusion of this section, we would like to emphasize that the results of solar neutrino experiments provide a rather strong indication on favour of neutrino mixing and several experiments are under construction [273,352,322] or in project (see [353,354] and references therein). In particular, the measurement of the electron neutrino spectrum at SNO may allow to obtain model-independent information on the neutrino oscillation probability and on the flux $\Phi_{^8\mathrm{B}}$ of $^8\mathrm{B}$ neutrinos [355,356,357,358,359], the measurement of the flux of $^7\mathrm{Be}$ neutrinos on the earth in the Borexino experiment and the results of Super-Kamiokande, SNO, Borexino and GNO will allow to distinguish the different possible solutions of the solar neutrino problem (see Refs. [350,360,361,362,363] and references therein).

The Sudbury Neutrino Observatory (SNO) [273,364] is located 6800 feet below ground in the Creighton mine, near Sudbury in Ontario (Canada). In the SNO experiment a Cherenkov detector with 1 kton of heavy water (D$_2$O) contained in an acrylic vessel of 12 m diameter will be used. The Cherenkov light is detected with a geodesic array of $10^4$ photomultiplier tubes surrounding the heavy water vessel. The heavy water detector is immersed in normal water in order to reduce the background. Solar neutrinos will be observed in real-time through the charged-current (CC), neutral-current (NC) and elastic scattering (ES) reactions

$$\nu_e + d \to e^- + p + p \null \null \qquad \qquad \null \null \mbox{(CC)} \,,$$ (5.42)

$$\nu_\ell + d \to \nu_\ell + p + n \null \null \,, \qquad \ell=e,\mu,\tau \,, \qquad \null \null \mbox{(NC)} \,,$$ (5.43)

$$\nu_\ell + e^- \to \nu_\ell + e^- \null \null \,, \qquad \ell=e,\mu,\tau \,, \qquad \null \null \mbox{(ES)} \,.$$ (5.44)

Since the energy threshold for the observation of the recoil electrons in the CC and ES processes is about 5 MeV and the neutrino energy threshold for the NC reaction is 2.2 MeV, only $^8$B neutrinos can be observed. The event rates predicted by the SSM are around 23 per day for the CC reaction, 7 per day for the NC reaction and 3 per day for the ES reaction. From the hit pattern of the photomultipliers, the direction and energy of the neutrino in the CC reaction can be reconstructed, allowing a direct measurement of the spectrum of the solar electron neutrino flux on the earth. The observation of a distortion of this spectrum with respect to the one calculated without neutrino oscillations will represent a model-independent proof of the occurrence of neutrino oscillations. The NC reaction will allow to detect all active neutrinos with the same cross section (because of the $e$-$\mu$-$\tau$ universality of weak interactions), whereas the cross sections of $\nu_\mu$ and $\nu_\tau$ scattering in the ES reaction is about six times smaller than that of $\nu_e$. The measurement of the neutral current reaction will allow to determine the total flux of active neutrinos from the sun [