11institutetext: SISSA/ISAS and INFN Trieste, Italy

# Neutrino Physics

Andrea Romanino

###### Abstract

These lectures aim at providing a pedagogical overview of neutrino physics. We
will mostly deal with standard neutrinos, the ones that are part of the
Standard Model of particle physics, and with their standard dynamics, which is
enough to understand in a coherent picture most of the rich data available.
After introducing the basic theoretical framework, we will illustrate the
experimental determination of the neutrino parameters and their theoretical
implications, in particular for the origin of neutrino masses.

##  0.1 Introduction

Neutrino physics has played a crucial role in particle physics since the birth
of the theory of weak interactions, but the advances in a field requiring the
detection of such an elusive particle, have been characterized by long time
scales until 1998. After about 70 years of slow (but steady) progress, the
findings of the Super-Kamiokande (SK) experiment in 1998 triggered an
impressive acceleration and a renewed interest in the field.

There are various reasons for the interest in neutrino physics. First of all,
after decades in which the interpretation of neutrino experiments testing
neutrino transitions has been plagued by the uncertain knowledge of the
initial fluxes, the SK experiment started an era in which the data
interpretation has been relatively clean, with measurements either relatively
independent of the flux uncertainties or based on quite a precise knowledge of
the fluxes.

On the theoretical side, the evidence of small but non-vanishing neutrino
masses represents one of the few clear indications of physics beyond the
standard model of particle physics (SM). The latter predicts in fact vanishing
neutrino masses, unless supplemented by additional degrees of freedom or
effective interactions (which can be anyway considered as strong hints of new
degrees of freedom living at a higher energy scale).

Moreover, neutrinos play a crucial role in cosmology, another field which is
witnessing a fast expansion. They could be responsible for the origin of the
baryon asymmetry of the universe, i.e. our very existence, they enter the
determination of the spectrum of the cosmic microwave background (CMB), the
determination of the large scale structures (LSS) in the universe, the
delicate chemical equilibriums determining the light element abundances during
big bang nucleosynthesis (BBN). Not to mention astrophysics, where they
represent a powerful probe of the dynamics of the Sun, of core-collapse
supernovae, of high energy sources, etc.

Last but not least, neutrinos play an important role in many particle physics
models and phenomena, and may allow to indirectly access energy scales
otherwise largely out of the reach of (natural and laboratory) particle
accelerators. They are for example important ingredients in grand unified
theories (GUTs), flavour models, lepton flavour violation, lepton violation,
neutrinoless double beta decay.

In this lectures we will discuss many of the issues mentioned above. After
some historical remarks, we will discuss neutrino phenomenology, in particular
what do we know, and how, about the neutrino parameters. We will then discuss
the theoretical impact of the knowledge on neutrino parameters, in particular
the implications on the origin of neutrino masses and of the pattern of
neutrino masses and mixings. Useful general references are [1, 2]. We will use
natural units.

##  0.2 The birth of neutrino physics

(a) (b) (c)

Figure 1: Neutrino indirect discoveries: (a) electron neutrino from the beta
decay spectrum; (b) muon neutrino from the decay of pions from cosmic rays;
(c) tau neutrino from $e^{+}e^{-}\to e^{\pm}\mu^{\mp}X$ anomalous events at
SPEAR.

The existence of neutrinos has been first postulated by Pauli [3] in 1930 as a
“desperate remedy” to save energy conservation in the beta decay of nuclei. In
the absence of neutrinos, the two body decay of a nucleus made of $Z$ protons
and $A$ nucleons, $(A,Z)\to(A,Z+1)+e^{-}$, would produce a monochromatic
spectrum for the emitted electron. On the other hand, a continuous spectrum,
typical of a three body decay, was observed (see Fig. 1a). Hence Pauli’s
postulation of the existence of what we now call an electron anti-neutrino,
$\overline{\nu}_{e}$, in the final state of the decay:

$(A,Z)\to(A,Z+1)+e^{-}+\overline{\nu}_{e}.$ (1)

Fermi took Pauli seriously and in 1934 provided a quantitative description of
the phenomenon in terms of an effective interaction that would become the
basis of the theory of weak interactions [4, 5]. His success lead to a wide
acceptance of the neutrino hypothesis. The experimental progress in neutrino
physics has been quite slow in most of the neutrino history, and it was only
in 1956 that Reines and Cowan [6, 7] were able to establish experimentally the
existence of the neutrino emitted by nuclear reactors by detecting them
through the inverse beta reaction

$\overline{\nu}_{e}+p\to n+e^{+}.$ (2)

The muon neutrino was also introduced to account for missing momentum. Some
cosmic ray tracks observed in balloon experiments were in fact exhibiting
90-degrees kinks in correspondence to what would be interpreted as the decay
of a charged pion into a muon and a muon neutrino:
$\pi^{+}\to\mu^{+}\nu_{\mu}$ (see Fig. 1b). While such tracks were observed in
the late 40’s, when the pion was discovered, it was only in 1964 that
Lederman, Schwartz, and Steinberger [8] detected muon neutrinos by producing
the first, prototypical, artificial neutrino beam at Brookhaven. The latter
was obtained by sending a beam of protons on target to produce pions, mostly
decaying into muons and neutrinos as above. This is still a widely used method
to produce neutrinos, in the laboratory and in Nature. The neutrinos were then
detected through interactions with matter that were found to produce muons,
not electrons, thus indicating that the neutrino detected was not the same as
the electron neutrino detected by Reines and Cowan. Moreover, the separate
conservation of individual (electron, muon) lepton numbers holds in such
processes, once the neutrinos are given the same lepton number as their
corresponding lepton. As a consequence, when decaying into an electron, a muon
should produce both a muon neutrino and an electron antineutrino: $\mu\to
e\,\nu_{\mu}\overline{\nu}_{e}$. The fact that muon decay is mostly a three
body decay is confirmed by the analysis of the electron spectrum. Another
reason why the two neutrinos emitted in the muon decay should be different is
that otherwise the muon decay operator would induce a $\mu\to e\gamma$ decay
rate much larger than what experimentally allowed.

Tau neutrinos were associated in 1975 to the tau lepton discovery and detected
only recently, in 2000. Tau leptons were discovered at the SPEAR $e^{+}e^{-}$
accelerator, which observed anomalous events $e^{+}e^{-}\to e^{\pm}\mu^{\mp}X$
[9] (see Fig. 1c). $X$ represents here one or more invisible particles whose
presence in the final state was inferred by the measurement of the missing
momentum. The charged particles in the final state are produced by the decay
(too fast to be observed) of a $\tau^{+}\tau^{-}$ pair produced in the
collision. Analogously to the muon case, the $\tau$ lepton should decay into a
lighter lepton and a couple of neutrinos, to conserve the individual lepton
numbers. The tau neutrinos was thus introduced. Its detection at the DONUT
experiment [10] at FNAL was achieved by producing a tau neutrino flux through
the $D_{s}$ decay into $\tau\,\nu_{\tau}$ and through the challenging
observation of the tau produced by the $\nu_{\tau}$ interaction in matter.

The obvious question is now whether the story is over or there exists other
neutrino species, besides the electron, muon and tau ones. An important
constraint on the existence of such additional neutrinos is given by the
measurement of the $Z$-boson width at LEP. The decay width depends on the
number of (kinematically accessible) decays channels. The measurement agrees
with the prediction one obtains taking into account the known charged
particles with $m<M_{Z}/2$ and the three ($2.98\pm 0.01$) known neutrinos. The
existence of additional neutrinos with $m<M_{Z}/2$ is therefore excluded. Note
that by neutrino here we mean a particle with the same interactions as the
three known neutrinos. “Sterile neutrinos”, hypothetical particles not having
any SM gauge interaction, would not contribute to the $Z$ width and are
therefore allowed by the $Z$-width constraint. The latter, moreover,
translates into a constraint on the number of fermion families. The known
fermions are organised in three families with identical gauge quantum numbers,
each including a neutrino. The existence of additional families incorporating
light ($m<M_{Z}/2$) neutrinos is therefore also excluded.

Neutrino oscillation have also a long history. They were postulated in 1957 by
Pontecorvo [11, 12] who, in analogy with $K^{0}$-$\overline{K}^{0}$
oscillations, considered neutrino-antineutrino oscillations. The possibility
of mixing among electron and muon neutrinos was then considered by Maki,
Nakagawa, and Sakata [13] in 1962, in analogy to the Gell-Mann hypothesis of
quark mixing. The first experimental evidence of neutrino oscillations came in
1968, when Davis, using and experimental technique suggested by Pontecorvo
[14, 15], observed a deficit of about 50% in the measured solar neutrino flux
[16, 17] with respect to what predicted by Bahcall [18] a few years before.
Such an evidence was however plagued by the uncertainties on the theoretical
prediction of the solar neutrino flux. As we will see, it was only relatively
recently that the SNO experiment was able to get rid of such uncertainties and
confirm both the deficit and Bahcall’s prediction.

##  0.3 Neutrino parameters

Before discussing what we know about them and how, let us define the neutrino
parameters. In order to put the discussion in context, we start by describing
the theoretical framework and by illustrating the difference between Dirac and
Majorana neutrino.

Most neutrino experiments are characterized by energies much lower than the
electroweak scale, $v=174\,\mathrm{GeV}$. At such scales, the electroweak
symmetry is badly broken and it is convenient to describe the dynamics of the
particles light enough to be produced as initial or final states by means of
an effective hamiltonian that does not involve heavy fields and obeys the
unbroken QED and QCD symmetries (U(1)${}_{\text{em}}$ and SU(3)c
respectively).

Neutrinos do not couple to photons (QED) nor gluons (QCD). Their interactions
are described by an effective lagrangian generated by $W$ and $Z$ exchanges at
the electroweak scale. Such a four fermion interaction was first introduced by
Fermi and its detailed form was spelled out later [19, 20, 21]:

$\mathcal{L}_{E\ll
M_{Z}}^{\text{eff}}=\mathcal{L}_{\text{QED+QCD}}+4\,\frac{G_{F}}{\sqrt{2}}\,j_{c}^{\mu}j_{c\mu}^{\dagger}+\text{N.C.}+\ldots,$
(3)

where “N.C.” denotes the neutral current term, which is not relevant in the
following considerations, and the charged current is given by

$j^{\mu}_{c}=\overline{\nu_{e_{i}}}\,\gamma^{\mu}P_{L}\,e_{i}+\overline{u_{i}}\,\gamma^{\mu}P_{L}\,d_{i}$
(4)

in terms of the three charged lepton Dirac fields $e_{i}$ ($e_{1}\equiv e$,
$e_{2}\equiv\mu$, $e_{3}\equiv\tau$), the three neutrino “flavour eigenstates”
$\nu_{e_{i}}$, $i=1,2,3$, the three up quarks $u_{i}$ ($u_{1}\equiv u$,
$u_{2}\equiv c$, $u_{3}\equiv t$), the three down quarks $d_{i}$ ($d_{1}\equiv
d$, $d_{2}\equiv s$, $d_{3}\equiv b$). $P_{L}=(1-\gamma_{5})/2$ is the
projector on left-chirality fields, $\psi_{L}\equiv P_{L}\psi$, and
$P_{R}=(1+\gamma_{5})/2$ is the projector on right-chirality fields,
$\psi_{R}\equiv P_{R}\psi$. In the massless limit, left-chirality fields are
associated to left-handed helicity particles and right-handed helicity
antiparticles. In the massive limit this is not the case.

Neutrinos are allowed to have non-vanishing masses $m_{1}$, $m_{2}$, $m_{3}$.
We denote by $\nu_{i}$, $i=1,2,3$, the neutrino mass eigenstates fields, which
by definition diagonalize the mass matrix. The flavour eigenstates,
$\nu_{e_{i}}$, $i=1,2,3$, diagonalize the charged current. The neutrino
flavour eigenstates can be expressed in terms of the mass eigenstates through
a $3\times 3$ unitary matrix $U$ called Pontecorvo Maki Nakagawa Sakata (PMNS)
matrix. The matrix is unitary because it has to preserve the canonical form of
the kinetic term of the neutrino fields. As in the case of the CKM matrix
describing quark mixing, not all the 9 parameters parameterizing the matrix
$U$ are physical. The physical parameters are three mixing angles and,
depending on the Dirac or Majorana nature of the neutrinos (see below), one or
three CP-violating phases. The standard parameterization of the PMNS matrix in
the case of Majorana neutrinos is

$U=\left(\begin{array}[]{ccc}c_{12}c_{13}&s_{12}c_{13}&s_{13}e^{-i\delta}\\\
-s_{12}c_{23}-c_{12}s_{23}s_{13}e^{i\delta}&c_{12}c_{23}-s_{12}s_{23}s_{13}e^{i\delta}&s_{23}c_{13}\\\
s_{12}s_{23}-c_{12}c_{23}s_{13}s^{i\delta}&-c_{12}s_{23}-s_{12}c_{23}s_{13}s^{i\delta}&c_{23}c_{13}\end{array}\right)\begin{pmatrix}1&0&0\\\
0&e^{i\alpha}&0\\\ 0&0&e^{i\beta}\end{pmatrix},$ (5)

where $s_{ij}=\sin\theta_{ij}$, $c_{ij}=\cos\theta_{ij}$,
$\theta_{12},\theta_{23},\theta_{13}$ are three mixing angles, $\delta$ is a
CP-violating phases that is physical in both the Majorana and Dirac neutrino
cases, and $\alpha,\beta$ are two CP-violating phases that are physical only
in the case of Majorana neutrinos and are therefore sometimes called Majorana
phases. In the case of Dirac neutrinos, the standard parameterization of the
PMNS matrix is given by the first factor only in the RHS eq. (5).

From a qualitative point of view, Dirac and Majorana neutrinos differ as
follows. In the case of Dirac neutrinos, neutrino and antineutrino are two
different particles, each with two possible values of the helicity, for a
total of 4 degrees of freedom. Also, the neutrino and antineutrino fields are
independent and the neutrino field, as all Dirac fields, splits into two
independent components with definite chirality (value of $\gamma_{5}$):
$\nu=\nu_{L}+\nu_{R}$. A mass term for Dirac neutrinos does not break lepton
number.

In the case of Majorana neutrinos, the neutrino particle coincides with its
antiparticle111In the case of massless neutrinos, the two helicities do not
mix and can be associated to two massless fermions, the neutrino and the
antineutrino, with one degree of freedom each. In the case of massive
neutrinos, the two helicities are mixed by the mass term (they are both part
of the same irreducible representation of the Poincaré group) and represent a
single particle., which gives a total of 2 degrees of freedom. The neutrino
and antineutrino fields are not independent (they are related by a matrix
transformation) and only have one chirality: $\nu=\nu_{L}$. A mass term for a
single Majorana neutrino field necessarily breaks lepton number and any U(1)
charge associated to the neutrino.

In the massless limit, the distinction between Dirac and Majorana neutrinos is
irrelevant. Indeed, the $\nu_{R}$ component of the Dirac field, if it exists,
does not have in this case any interaction, gauge or Yukawa, nor it is mixed
by a mass term to the $\nu_{L}$ component. Therefore, it does not affect the
dynamics of the fields produced in the experiments. As a consequence, telling
Majorana from Dirac neutrinos in experiments in which the energy is much
larger than the neutrino mass (so that we approach the massless limit) is
difficult. In particular, the distinction between Majorana and Dirac neutrinos
is irrelevant in oscillation experiments and in most other neutrino
experiments except when lepton number violation plays a role, as in the case
of neutrinoless double beta decay (see Section 0.4.7).

From a pragmatic point of view, what above is what needed for the
comprehension of this and the next Section. In the next subsection we will
give additional theoretical details on the nature of neutrino masses that will
be mostly needed as a background to Section 0.5 only and can be omitted at a
first reading.

###  0.3.1 The neutrino mass term

In this interlude we would like to discuss in greater theoretical detail the
form of the neutrino mass term, the difference between Dirac and Majorana
neutrinos, and how the neutrino parameters arise. This requires a basic
knowledge of quantum field theory.

The charged fermions are described by Dirac spinors, four component complex
fields. From the point of view of Lorentz transformations, Dirac spinors are
not elementary, though. For example, the electron field, $e$, contains two
independent components that have different (inequivalent) Lorentz
transformations, characterized by their chirality: $e=e_{L}+e_{R}$. In order
to be able to write the most general Lorentz invariant mass term or
interaction, it is useful to list the fields with equivalent Lorentz
transformations. Each Standard Model charged fermion field, and each
conjugated field, decomposes into a left and a right component:
$u_{i}=u_{iL}+u_{iR}$, $\overline{u_{i}}=\overline{u_{iL}}+\overline{u_{iR}}$,
$d_{i}=d_{iL}+d_{iR}$, $\overline{d_{i}}=\overline{d_{iL}}+\overline{d_{iR}}$,
$e_{i}=e_{iL}+e_{iR}$, $\overline{e_{i}}=\overline{e_{iL}}+\overline{e_{iR}}$,
where $i=1,2,3$ is the family index and I have omitted the color index of
quarks. Only the left-handed component of the neutrino field has been observed
so far, therefore we do not include a right-handed component in the list for
the time being: $\nu_{i}=\nu_{iL}$, $\overline{\nu_{i}}=\overline{\nu_{iL}}$.
Note that the conjugated of a right-chirality field also has left-chirality.
The left-chirality fields are therefore $e_{iL}$, $u_{iL}$, $d_{iL}$,
$\overline{e_{i}}_{L}=\overline{e_{iR}}$,
$\overline{u_{i}}_{L}=\overline{u_{iR}}$,
$\overline{d_{i}}_{L}=\overline{d_{iR}}$, $\nu_{iL}$. The conjugated fields
have all right-chirality. The most general Lorentz invariant gauge
transformation can in principle mix all the left-chirality fields. Once the
gauge transformations have been defined, the most general mass term is given
by

$\frac{m_{ij}}{2}\psi_{iL}\psi_{jL}+\text{h.c.},$ (6)

where $\psi_{iL}$ denotes a generic left-chirality field (the 7 (per family)
fields listed above, in the case of the SM), a proper Lorentz invariant
contraction of the Lorentz indexes is understood, and the mass matrix $m_{ij}$
is symmetric and should be invariant under gauge transformation. It is then
easy to write the most general mass term for the fermions above. In the
effective theory we are considering, the relevant gauge symmetries are the QED
and QCD ones, U(1)${}_{\text{em}}$ and SU(3)c. Under SU(3)c transformations,
each left-chirality quark field $u_{iL}$, $d_{iL}$ transforms as a triplet and
each left-chirality antiquark field, $\overline{u_{iR}}$, $\overline{d_{iR}}$,
transforms as an anti-triplet (leptons are of course invariant). Under
U(1)${}_{\text{em}}$ transformations, each left chirality fermion transforms
according to its electric charge: $Q_{u_{L}}=2/3$, $Q_{d_{L}}=-1/3$,
$Q_{e_{L}}=-1$, $Q_{\nu_{L}}=0$, $Q_{\overline{u_{R}}}=-2/3$,
$Q_{\overline{d_{R}}}=1/3$, $Q_{\overline{e_{R}}}=1$. Note that the neutrino
field does not feel either QCD or QED interactions, hence its elusiveness. It
is then an easy exercise to write the most general mass term in the form in
eq. (6) and invariant under QED and QCD transformations:

$m^{U}_{ij}\overline{u_{iR}}u_{jL}+m^{D}_{ij}\overline{d_{iR}}d_{jL}+m^{U}_{ij}\overline{e_{iR}}e_{jL}+\frac{m^{L}_{ij}}{2}\nu_{iL}\nu_{jL}+\text{h.c.}.$
(7)

A few comments are in order. The charged fermions mass terms couple two
independent fields (that can be combined into a Dirac field). Such mass terms
are called “Dirac” mass terms. The factor 1/2 is missing there because, for
example,
$(m^{U}_{ij}/2)\overline{u_{iR}}u_{jL}+(m^{U}_{ji}/2)u_{iL}\overline{u_{jR}}=m^{U}_{ij}\overline{u_{iR}}u_{jL}$,
where we have used the fact that the mass term, when written in the form in
eq. (6), is symmetric. The neutrino mass term, on the other hand, involves the
same set of fields. Such a mass term is called a “Majorana” mass term. Any
charge carried by the $\nu_{iL}$ (such as total lepton number, for example) is
violated by such a mass term. That is why the neutrinos are the only fermions
in the above list for which a Majorana mass term is allowed: they are the only
neutral fields (under the gauge symmetries we are considering). Note that the
observed smallness of neutrino masses, compared to all the other fermion
masses, is not explained at this level: the QED and QCD gauge symmetries allow
a mass term for both the charged fermions and the neutrinos. We will see in
Section 0.5 that a natural explanation for the smallness of neutrino masses
arises once the whole SM gauge symmetry
$G_{\text{SM}}=\text{SU(3)}_{c}\times\text{SU(2)}_{L}\times\text{U(1)}_{Y}$ is
considered.

If neutrinos have a right-chirality component $\nu_{R}$ (also uncharged under
QED and QCD interactions), the most general mass term for neutrinos becomes

$\frac{m^{L}_{ij}}{2}\nu_{iL}\nu_{jL}\to\frac{m^{L}_{ij}}{2}\nu_{iL}\nu_{jL}+\frac{m^{R}_{ij}}{2}\nu_{iR}\nu_{jR}+m^{N}_{ij}\overline{\nu_{iR}}\nu_{jL}.$
(8)

The first two terms are Majorana, while the third one is a Dirac mass term.
The Dirac limit, in which lepton number is conserved if $\nu_{L}$ and
$\nu_{R}$ are given the same lepton number as $e_{L}$ and $e_{R}$, is obtained
for $m^{L}=m^{R}=0$. We will see that the right-chirality components of
neutrinos, if present, are allowed to get a mass term much heavier than the
electroweak scale. This may account for their absence from the effective
lagrangian we are considering and for the smallness of the light neutrino
masses. For the time being we stick to the economical and theoretically
appealing case in which only left-chirality neutrinos are present.

Fermion masses and mixings are obtained when writing the lagrangian in terms
of mass eigenstates. In order to do that, it suffices to determine new linear
combinations of the initial fields,

$\displaystyle d^{\prime}_{iL}$ $\displaystyle=U^{d_{L}}_{ij}d_{jL},$
$\displaystyle d^{\prime}_{iR}$ $\displaystyle=U^{d_{R}}_{ij}d_{jR},$
$\displaystyle u^{\prime}_{iL}$ $\displaystyle=U^{u_{L}}_{ij}u_{jL},$
$\displaystyle u^{\prime}_{iR}$ $\displaystyle=U^{u_{R}}_{ij}u_{jR},$ (9a)
$\displaystyle e^{\prime}_{iL}$ $\displaystyle=U^{e_{L}}_{ij}e_{jL},$
$\displaystyle e^{\prime}_{iR}$ $\displaystyle=U^{e_{R}}_{ij}e_{jR},$
$\displaystyle\nu^{\prime}_{iL}$ $\displaystyle=U^{\nu_{L}}_{ij}\nu_{jL},$
(9b)

such that: i) the kinetic term for the new field is still canonical and ii)
the mass terms can be written as

$\begin{split}m^{U}_{ij}\overline{u_{iR}}u_{jL}+m^{D}_{ij}\overline{d_{iR}}d_{jL}+m^{U}_{ij}\overline{e_{iR}}e_{jL}+\frac{m^{\nu}_{ij}}{2}\nu_{iL}\nu_{jL}+\text{h.c.}=\\\
m_{u_{i}}\overline{u^{\prime}_{iR}}u^{\prime}_{iL}+m_{d_{i}}\overline{d^{\prime}_{iR}}d^{\prime}_{iL}+m_{e_{i}}\overline{e^{\prime}_{iR}}e^{\prime}_{iL}+\frac{m_{i}}{2}\nu^{\prime}_{iL}\nu^{\prime}_{iL}+\text{h.c.}=\\\
m_{u_{i}}\overline{u^{\prime}_{i}}u^{\prime}_{i}+m_{d_{i}}\overline{d^{\prime}_{i}}d^{\prime}_{i}+m_{e_{i}}\overline{e^{\prime}_{i}}e^{\prime}_{i}+\left(\frac{m_{i}}{2}\nu^{\prime}_{iL}\nu^{\prime}_{iL}+\text{h.c.}\right).\end{split}$
(10)

Because of the first requirement, the mixing matrices $U$ must be unitary.
Because of the second one, they should satisfy

$m^{D}=U_{d_{R}}^{\dagger}m^{D}_{\text{diag}}U^{\phantom{\dagger}}_{d_{L}},\quad
m^{E}=U_{u_{R}}^{\dagger}m^{U}_{\text{diag}}U^{\phantom{\dagger}}_{u_{L}},\quad
m^{E}=U_{e_{R}}^{\dagger}m^{E}_{\text{diag}}U^{\phantom{\dagger}}_{e_{L}},\quad
m^{L}=U^{T}_{\nu}m^{\nu}_{\text{diag}}U^{\phantom{\dagger}}_{\nu},$ (11)

where the diagonal matrices have non-negative eigenvalues. It turns out that
given generic complex matrices $m^{U,D,E}$ and given a symmetric complex
matrix $m^{\nu}$, it is always possible to find unitary matrices
$U^{d_{L},d_{R},u_{L},u_{R},e_{L},e_{R},\nu_{L}}$ satisfying eqs. (11). We can
then express the whole lagrangian in terms of the primed fields with definite
mass, and drop the primes for convenience. The QED and QCD gauge lagrangian do
not change form, as the transformations in eqs. (0.3.1) conserve the gauge
quantum numbers. The only change arises in the charged current, which in terms
of the mass eigenstate fields, becomes

$j^{\mu}_{c}=V_{ij}\overline{u_{iL}}\gamma^{\mu}d_{jL}+U^{\dagger}_{ij}\overline{\nu_{iL}}\gamma^{\mu}e_{jL},$
(12)

where $V=U_{u_{L}}U^{\dagger}_{d_{L}}$ is the Cabibbo Kobayashi Maskawa (CKM)
quark mixing matrix and $U=U_{e_{L}}U^{\dagger}_{\nu_{L}}$ is the PMNS lepton
mixing matrix.

Not all the parameters in $V$, $U$ are physical. Let us consider the CKM
matrix first. It is possible to write
$V=\operatorname{Diag}(e^{i\gamma_{1}},e^{i\gamma_{2}},e^{i\gamma_{3}})V_{\text{standard}}\operatorname{Diag}(1,e^{i\alpha},e^{i\beta})$,
where $V_{\text{standard}}$ is in the form of the first factor in the RHS of
eq. (5). Moreover, it is possible to reabsorb the phases $\gamma_{i}$,
$i=1,2,3$, $\alpha,\beta$ through a redefinition of the left-handed fields
$u_{iL}\to u_{iL}e^{i\gamma_{i}}$, $d_{2L}\to d_{2L}e^{-i\alpha}$, $d_{3L}\to
d_{3L}e^{-i\beta}$. This allows to write $V$ in the standard form
$V_{\text{standard}}$. On the other hand, the field phase transformations used
to bring $V$ in the standard form, introduce phases in the mass terms in the
second line of eq. (10). In order to get rid of them once and for all, it is
possible to redefine the phases of the right-chirality fields $d_{iR},u_{iR}$.
This shows that the phases $\gamma_{i}$, $\alpha$, $\beta$ are not physical,
as they can be completely eliminated from the lagrangian. In the lepton sector
the story would be exactly the same if the neutrinos had a Dirac mass term
$m^{N}_{ij}\overline{\nu_{iR}}\nu_{iL}$ as the charged fermions. On the other
hand if, as we assume, the neutrino mass term is Majorana, the phases $\alpha$
and $\beta$ in
$U=\operatorname{Diag}(e^{i\gamma_{1}},e^{i\gamma_{2}},e^{i\gamma_{3}})U_{\text{standard}}\operatorname{Diag}(1,e^{i\alpha},e^{i\beta})$
end up being physical. Indeed, while it is possible to eliminate those phases
from $U$ by redefining $\nu_{2L}\to\nu_{2L}e^{-i\alpha}$,
$\nu_{3L}\to\nu_{3L}e^{-i\beta}$, this transformation would move those phases
in the neutrino Majorana mass term. As the latter does not involve an
independent field whose phase can be rotated to eliminate $\alpha,\beta$ once
and for all, the phases $\alpha,\beta$ turn out not to be unphysical. They are
called Majorana phases.

###  0.3.2 Physical lepton mass and mixing parameters (Majorana neutrinos)

Let us now go back to phenomenology. As we have seen, the physical mass and
mixing parameters in the lepton sector are the 6 charged lepton and neutrino
masses and the 6 mixing parameters

$m_{e},\;m_{\mu},\;m_{\tau},\quad
m_{1},\;m_{2},\;m_{3},\quad\theta_{23},\;\theta_{12},\;\theta_{13},\;\delta,\quad\alpha,\;\beta.$
(13)

The physical ranges of the above parameters are $m_{e,\mu,\tau,1,2,3}\geq 0$,
$0\leq\theta_{23,12,13}\leq\pi/2$, $0\leq\delta<2\pi$,
$0\leq\alpha,\beta<\pi$.

One important remark concerns the ordering of the neutrino mass eigenstates.
In the charged lepton sector (and in the quark sector), the mass eigenstates
are ordered with their masses: $m_{e_{1}}<m_{e_{2}}<m_{e_{3}}$. In the case of
neutrinos, the convention used is different. By definition, we call $\nu_{1}$
and $\nu_{2}$ the two neutrinos whose masses are closer in value, with
$m_{1}<m_{2}$. The third mass eigenstate has a larger separation in mass from
$\nu_{1}$ and $\nu_{2}$, but can be heavier or lighter. If $m_{3}>m_{1,2}$, we
say that the neutrinos have a “normal” hierarchy. If $m_{3}<m_{{1,2}}$, we say
they have an inverse hierarchy. Let us call

${\Delta m^{2}_{ij}}\equiv m^{2}_{j}-m^{2}_{i}.$ (14)

Then we have, by definition, ${\Delta m^{2}_{12}}>0$ and $0<{\Delta
m^{2}_{12}}<|{\Delta m^{2}_{23}}|$. Corresponding to the two possible
hierarchies, ${\Delta m^{2}_{23}}$ can have both signs: ${\Delta
m^{2}_{23}}>0$ in the case of normal hierarchy and ${\Delta m^{2}_{23}}<0$ in
the case of inverse hierarchy.

Neutrino oscillation phenomena do not depend on the absolute values of
neutrino masses but only on the squared mass differences in eq. (14). It is
then useful to use the following set of lepton mass and mixing parameters,
equivalent to the one in eq. (13):

$m_{e},\;m_{\mu},\;m_{\tau},\quad{\Delta m^{2}_{12}},\;|{\Delta
m^{2}_{23}}|,\;\operatorname{sign}({\Delta
m^{2}_{23}}),\;\theta_{23},\;\theta_{12},\;\theta_{13},\;\delta,\quad
m_{\text{lightest}},\;\alpha,\;\beta.$ (15)

The neutrino masses $m_{1}$, $m_{2}$, $m_{3}$ have been traded for the
equivalent set of parameters $m_{\text{lightest}}$, the lightest neutrino
mass, and ${\Delta m^{2}_{12}}$, $|{\Delta m^{2}_{23}}|$,
$\operatorname{sign}({\Delta m^{2}_{23}})$. The mass parameters ${\Delta
m^{2}_{12}}$, $|{\Delta m^{2}_{23}}|$, $\operatorname{sign}({\Delta
m^{2}_{23}})$, the mixing angles $\theta_{23}$, $\theta_{12}$, $\theta_{13}$
and the phase $\delta$ are accessible to neutrino oscillation experiments. The
absolute scale of neutrino masses, represented by $m_{\text{lightest}}$, and
the Majorana phases $\alpha,\beta$ (if physical) are not. The third squared
mass difference, ${\Delta m^{2}_{13}}$ is obviously not independent of the
first two, ${\Delta m^{2}_{13}}={\Delta m^{2}_{12}}+{\Delta m^{2}_{23}}$. The
experiment shows that ${\Delta m^{2}_{12}}\ll{\Delta m^{2}_{23}}$, so that
${\Delta m^{2}_{13}}\approx{\Delta m^{2}_{23}}$. We have then in first
approximation only two squared mass differences, which are sometimes called
“solar” and “atmospheric”: ${\Delta m^{2}_{\text{SUN}}}\equiv{\Delta
m^{2}_{12}}$, ${\Delta m^{2}_{\text{ATM}}}\equiv{\Delta
m^{2}_{23}}\approx{\Delta m^{2}_{13}}$. The name refers, as we will see, to
the neutrino source that was first used to measure those parameters. Analogous
names are sometimes used for the corresponding mixing angles:
$\theta_{\text{SUN}}\equiv\theta_{12}$,
$\theta_{\text{ATM}}\equiv\theta_{23}$.

The experimental situation is the following. The charged lepton masses are of
course well known. The solar and atmospheric squared mass differences,
together with the corresponding mixing angles, are also known. There are
bounds on $\theta_{13}$ and $m_{\text{lightest}}$. No information is available
at present on $\operatorname{sign}({\Delta m^{2}_{23}})$, $\alpha$, $\beta$.
Before discussing in detail the experimental determination of the neutrino
parameters, we summarize the most relevant information available at present:

$\begin{gathered}\begin{aligned} {\Delta m^{2}_{\text{ATM}}}&\sim 2.4\times
10^{-3}\,\mathrm{eV}^{2}&\theta_{23}&\sim 45^{\circ}&&\text{(ATM, K2K,
Minos)}\\\ {\Delta m^{2}_{\text{SUN}}}&\sim 0.76\times
10^{-4}\,\mathrm{eV}^{2}&\theta_{12}&\sim 35^{\circ}&&\text{(SUN, KamLAND)}\\\
&\qquad\theta_{13}<7^{\circ}\;\text{($2\sigma$)}&&&&\text{(CHOOZ, Minos + ATM,
SUN)}\end{aligned}\\\\[5.69054pt] \begin{aligned}
&|m_{ee}|=|U^{2}_{ei}m_{\nu_{i}}|<\mathcal{O}\left(1\right)\times
0.4\,\mathrm{eV}&\quad&\text{(Heidelberg-Moscow)}\\\
&(m^{\dagger}m)_{ee}=|U_{ei}|^{2}m^{2}_{\nu_{i}}<(2.2\,\mathrm{eV})^{2}&\quad&\text{(Mainz,
Troktsk)}\\\
&\sum_{i}m_{\nu_{i}}<\mathcal{O}\left(1\right)\,\mathrm{eV}\;\text{(priors)}&\quad&\text{(Cosmology)}.\end{aligned}\end{gathered}$
(16)

The experiments from which the information is obtained are also indicated.
“ATM” and “SUN” denote the atmospheric and solar neutrino experiments
respectively. $U_{ei}$ denotes the “1i” element of the PMNS matrix, which can
be expressed in terms of the parameters in eq. (15) through eq. (5). The bound
from cosmology is subject to uncertainties associated to the priors used in
the analysis.

##  0.4 The determination of the neutrino parameters

At present, most experimental information on the neutrino mass and mixing
parameters comes from experiments measuring neutrino transitions, which are by
now known to be due to neutrino oscillations. There are also beta decay
experiments aiming at a measurement of the absolute scale of neutrino masses,
$m_{\text{lightest}}$; neutrinoless double beta decay experiments, sensitive
to lepton number violation (Majorana vs Dirac neutrinos) and, in the case of
Majorana neutrinos, to both $m_{\text{lightest}}$ and the Majorana phases
$\alpha,\beta$; and experiments in astrophysics and cosmology, sensitive to
different neutrino properties.

###  0.4.1 The physics of neutrino oscillation experiments

Neutrino oscillations arise from the misalignment of the neutrino flavour
eigenstate fields, $\nu_{e,\mu\tau}$, coupled to the charged leptons in the
charged current interactions, and neutrino mass eigenstate fields,
$\nu_{1,2,3}$, eigenstates of the free hamiltonian and therefore associated to
the free propagation. Such a misalignment, as we have seen, is quantified by
the PMNS matrix: $\nu_{e_{i}}=U_{ih}\nu_{h}$,
$\overline{\nu}_{e_{i}}=U^{*}_{ih}\overline{\nu}_{h}$. The one-particle states
relations involve the conjugate matrix elements:
$|\nu_{e_{i}}\rangle=U^{*}_{ih}|\nu_{h}\rangle$,
$|\overline{\nu}_{e_{i}}\rangle=U_{ih}|\overline{\nu}_{h}\rangle$. Neutrinos
are produced by the charged current interactions of charged leptons, typically
electrons or muons. They are therefore in a flavour eigenstate, i.e. in a
coherent superposition of mass eigenstates. Suppose a neutrino
$|\nu_{e_{i}}\rangle=U^{*}_{ih}|\nu_{h}\rangle$ is produced by the interaction
with the lepton $e_{i}$ and it freely evolves. Let us compute the probability
that the neutrino is found after a time $t$ to be a $|\nu_{e_{j}}\rangle$
neutrino, for example by means of a charged current interaction with the
lepton $e_{j}$. The free evolution of the initial state gives
$e^{-iHt}|\nu_{e_{i}}\rangle=U^{*}_{ih}e^{-iE_{h}t}|\nu_{h}\rangle$, where
$E^{2}_{h}=(p^{2}+m^{2}_{h})$. The probability that the neutrino is found to
be a $|\nu_{e_{j}}\rangle$ neutrino is therefore

$P(\nu_{e_{i}}\rightarrow\nu_{e_{j}})=\left|\langle\nu_{e_{j}}|e^{-iHt}|\nu_{e_{i}}\rangle\right|^{2}=|U_{jh}e^{-iE_{h}t}U^{\dagger}_{hi}|^{2}\approx|U_{jh}e^{-i\frac{m^{2}_{h}}{2E}t}U^{\dagger}_{hi}|^{2},$
(17)

where we have approximated $E_{h}\approx p+m^{2}_{h}/(2E)$, as $E\gg m_{\nu}$
in all neutrino oscillation experiments, and the time $t$ can be replaced by
the length travelled $L$. We have not specified the helicity of the neutrino,
as it is not necessary. This is because in all neutrino oscillation
experiments the neutrino energy is way larger than its mass, $E\gg m_{\nu}$.
Since the neutrino interaction only involves the left-chirality component, the
neutrino produced will be mostly in an left-handed helicity state, whether it
is Majorana or Dirac, and helicity flips, either at production, detection, or
during propagation, are largely negligible. Moreover, in the $E\gg m_{\nu}$
limit, the oscillation probabilities do not depend on Majorana phases. The
oscillation probabilities satisfy
$P(\nu_{e_{i}}\to\nu_{e_{j}})=P(\overline{\nu}_{e_{j}}\to\overline{\nu}_{e_{i}})$,
because of CPT invariance. If CP is conserved,
$P(\nu_{e_{i}}\to\nu_{e_{j}})=P(\overline{\nu}_{e_{i}}\to\overline{\nu}_{e_{j}})$,
and equivalently $P(\nu_{e_{i}}\to\nu_{e_{j}})=P(\nu_{e_{j}}\to\nu_{e_{i}})$
if $T$ is conserved. The total oscillation probability is of course one,
$\sum_{j}P(\nu_{e_{i}}\to\nu_{e_{j}})=1$.

It is instructive to consider the simplest case of two neutrino oscillations.
Let us then consider the electron and muon neutrinos only. Up to phases
redefinitions, their mixing can be described by a real orthogonal $2\times 2$
matrix, i.e. a rotation by an angle $\theta$, which gives a simple expression
for the oscillation probability:

$\begin{aligned} \nu_{e}&=\nu_{1}\cos\theta+\nu_{2}\sin\theta\\\
\nu_{\mu}&=-\nu_{1}\sin\theta+\nu_{2}\cos\theta\end{aligned}\quad\Rightarrow\quad
P(\nu_{e}\rightarrow\nu_{\mu})=\sin^{2}2\theta\sin^{2}\frac{\Delta
m^{2}L}{4E},$ (18)

where ${\Delta m^{2}}=m^{2}_{2}-m^{2}_{1}$ can be taken positive by
definition. In order to obtain predictions for the outcome of realistic
experiments, the oscillation probability has to be convoluted with the source
energy spectrum, the distribution in the position of the neutrino emission and
detection, the scattering cross sections, the experimental resolution and
efficiency.

Let us comment on the form of the two neutrino oscillation formula. The
oscillation amplitude $A=\sin^{2}2\theta$ is determined by the mixing angle
$\theta$ and does not allow to distinguish (in vacuum) the physically
inequivalent $\theta$ and $\pi/2-\theta$ values. The squared mass difference
determines the oscillation length $\lambda=4\pi E/{\Delta m^{2}}\approx
2.48\,\text{km}(E(\text{GeV})/{\Delta m^{2}}(\text{eV}^{2}))$, or equivalently
the oscillation phase $\phi={\Delta m^{2}}L/(4E)\approx 1.27({\Delta
m^{2}}(\text{eV}^{2})L(\text{km})/E(\text{GeV}))$. In order to determine both
the oscillation parameters, it is best to consider an experiment in which the
neutrinos travel a distance comparable to their oscillation length,
$L\sim\lambda$. In the $L\ll\lambda$ limit, in fact,
$P(\nu_{e}\rightarrow\nu_{\mu})\approx\sin^{2}2\theta\,(\Delta
m^{2}L/(4E))^{2}$ and even a detailed measurement of the oscillation
probability as a function of $E$ and $L$ would determine the product
$\sin^{2}2\theta\cdot\Delta m^{2}$ only. In this limit, oscillations have not
enough time to occur and the expression for the probability can be obtained in
perturbation theory (which represents a check of the correctness of the
formula). The oscillation probability is proportional to $(L/E)^{2}$ and the
neutrino flux decreases with the geometrical factor $1/L^{2}$, therefore the
number of neutrino oscillation events measured in a detector is approximately
independent of the distance $L$, within this limit. In the $L\gg\lambda$
limit, on the other hand, the oscillations are so fast that they average out
and
$P(\nu_{e}\rightarrow\nu_{\mu})\approx\sin^{2}2\theta/2=\sin^{2}\theta\cos^{2}\theta+\cos^{2}\theta\sin^{2}\theta$.
Only the mixing angle can be measured in this limit. The oscillation
probability is independent of $E$, $L$ and the number of oscillation events
decreases with $1/L^{2}$. In this “classical” limit, the oscillation
probability is the sum (over $i$) of the probabilities that the initial
neutrino $\nu_{e}$ is a $\nu_{i}$ times the probability that the neutrino
$\nu_{i}$ is observed to be a $\nu_{\mu}$.

Figure 2: Typical sensitivity plot of a neutrino experiment.

The situation is illustrated in Fig. 2, where the typical sensitivity plot of
a neutrino experiment is plotted. Assuming that the experiment is sensitive to
a given (averaged) oscillation probability, the sensitivity in the
$\sin^{2}2\theta$–${\Delta m^{2}}$ plane is shown. The two limits considered
above can be recognized in the lower and upper part of the plot respectively.
In order to measure both $\sin^{2}2\theta$ and ${\Delta m^{2}}$, a measurement
of the averaged probability is not enough. The $E$ or $L$ dependence has also
to be measured, better if in the $L\sim\lambda$, or $({\Delta
m^{2}}L/(4E))\sim 1$, regime.

The above derivation of the oscillation formula is simplistic for a number of
reasons. First of all, it holds in vacuum only. The coherent (or incoherent)
effect of matter in neutrino propagation can be very important, as we will
show below. Moreover, the neutrino coherence assumed in the derivation of the
oscillation formula can be lost for a number of reasons, besides the necessary
averages mentioned above. Because the wave packets associated to the different
mass eigenstates making up a given flavour eigenstate travel at slightly
different velocities, for example (this is relevant when the distance
travelled is very large). Or because the neutrino production process typically
involves at least another particle in the final state. The quantum mechanics
reduction to the neutrino subsystem also induces a loss of coherence. Finally,
the derivation assumed that the neutrino mass eigenstates are all in a pure
eigenstate with same definite momentum. It is sometime argued that it is more
appropriate to assume that they have the same energy instead. A proper
derivation would take into account the precise form of the density matrix
describing the initial state and its momentum distribution, as obtained from
the dynamics of the production process. In this context, using the fixed
momentum or fixed energy description just amounts to a change of variable in
the integration over the momentum distribution of the initial neutrino state.

Let us consider now the three neutrino case. The exact three neutrino formulas
are

$\displaystyle\begin{aligned}
P(\nu_{e_{i}}\rightarrow\nu_{e_{j}})&=P(\overline{\nu}_{e_{j}}\rightarrow\overline{\nu}_{e_{i}})=P_{\text{CPC}}+P_{\text{CPV}}\\\
P(\overline{\nu}_{e_{i}}\rightarrow\overline{\nu}_{e_{j}})&=P(\nu_{e_{j}}\rightarrow\nu_{e_{i}})=P_{\text{CPC}}-P_{\text{CPV}}\end{aligned}$
$\displaystyle\begin{aligned}
P_{\text{CPC}}&=\delta_{ij}-4\operatorname{Re}(J^{ji}_{12})S^{2}_{12}-4\operatorname{Re}(J^{ji}_{23})S^{2}_{23}-4\operatorname{Re}(J^{ji}_{31})S^{2}_{31}\\\
P_{\text{CPV}}&=8\sigma_{ij}J_{\text{CP}}S_{12}S_{23}S_{31}\end{aligned}$ (19)
$\displaystyle S_{hk}=\sin\frac{{\Delta m^{2}_{hk}}L}{4E},\quad
J^{ji}_{hk}=U^{\phantom{\dagger}}_{jh}U^{\dagger}_{hi}U^{\phantom{\dagger}}_{ik}U^{\dagger}_{kj},\quad\operatorname{Im}(J^{ji}_{hk})=\sigma_{ji}\sigma_{hk}J_{\text{CP}},\quad\sigma_{ij}=\sum_{k}\epsilon_{ijk}=\pm
1,0,$

where $P_{\text{CPC}}$ and $P_{\text{CPV}}$ are the CP conserving and CP
violating parts of the oscillation probability respectively. Note again the
independence of the formulas above of Majorana phases and of the absolute
neutrino mass scale.

Despite the existence of three neutrinos, the results of neutrino oscillation
experiments are often shown in a two neutrino oscillation context and mainly
determine a single mixing angle and squared mass difference. This is because
the experimental values of the neutrino parameters are such that often, in
first approximation, the three neutrino oscillation formula reduces to a two
neutrino one. For example, the CHOOZ experiment, as we will see, measures the
probability of electron neutrino disappearance, $P(\nu_{e}\to\nu_{e})$, for
$L/E$ values such that the $S_{12}$ terms in eq. (19) are negligible (because
of the small ${\Delta m^{2}_{12}}$). For the same reason $S_{23}\approx
S_{12}$, so that we can approximate $P(\nu_{e}\rightarrow\nu_{e})\approx
1-\sin^{2}2\theta_{13}\sin^{2}({\Delta m^{2}_{23}}L/(4E))$, a two neutrino
oscillation formula with $\theta=\theta_{13}$ and ${\Delta m^{2}}={\Delta
m^{2}_{23}}$. This allows the CHOOZ experiment to set an upper bound on
$\theta_{13}$ [22] that makes the $\theta_{13}$ contribution negligible, in
first approximation, in the solar and atmospheric neutrino experiments. In
particular, in atmospheric neutrino experiments, $S^{2}_{12}\ll 1$,
$S^{2}_{23}\approx S^{2}_{13}$, $\theta_{13}\ll 1$, so that
$P(\nu_{\mu}\rightarrow\nu_{\tau})\approx\sin^{2}2\theta_{23}\sin^{2}({\Delta
m^{2}_{23}}L/(4E))$, $P(\nu_{e}\rightarrow\nu_{\mu,\tau})\ll 1$, and the
results can be interpreted in terms of $\nu_{\mu}\leftrightarrow\nu_{\tau}$
oscillations with $\theta=\theta_{23}$ and ${\Delta m^{2}}={\Delta
m^{2}_{23}}$. In solar neutrino experiments, detecting the disappearance of
the electron neutrinos produced by the Sun, the $S^{2}_{23}$ and $S^{2}_{13}$
terms are suppressed by $\theta_{13}$ and $P(\nu_{e}\rightarrow\nu_{e})\approx
1-\sin^{2}2\theta_{12}\sin^{2}({\Delta m^{2}_{12}}L/(4E))$, leading to a
determination of $\theta=\theta_{12}$ and ${\Delta m^{2}}={\Delta
m^{2}_{12}}$.

In the next subsections, we will discuss the experimental determination of
neutrino parameters by oscillation experiments.

###  0.4.2 Experimental determination of ${\Delta m^{2}_{23}}$ and
$\theta_{23}$

The experimental determination of ${\Delta m^{2}_{23}}$ and $\theta_{23}$ is
mostly due to the Super-Kamiokande (SK) measurement of atmospheric neutrinos,
and to the K2K, Minos, and Opera experiments. The result of a global fit is
shown in Fig. 3a [23]. Let us discuss the main ingredients entering the above
determination.

(a) (b)

Figure 3: Global fit of the ${\Delta m^{2}_{23}}$ and $\theta_{23}$ parameters
(a). Schematic representation of the distance travelled by atmospheric
neutrinos (b).

#### Atmospheric neutrinos

Atmospheric neutrinos arise from cosmic ray interactions in the atmosphere.
Charged pion produced by such interactions decay mostly through the decay
chain $\pi^{+}\to\mu^{+}\nu_{\mu}\to
e^{+}\nu_{e}\overline{\nu}_{\mu}\nu_{\mu}$ (analogously for negative pions, SK
does not tell neutrinos from antineutrinos), thus producing in first
approximation two muon neutrinos for each electron neutrino. The ratio of muon
to electron neutrinos reaching the Earth is actually slightly larger than two
because i) energetic muons have a long life-time and may not decay before
reaching the Earth and ii) Kaons are also produced by cosmic ray interactions.
Atmospheric neutrinos are detected by experiments placed underground (to
shield cosmic rays, but not neutrinos). The neutrinos travel a distance
ranging from $10\,\text{km}$ to more than $10^{4}\,\text{km}$, as shown in
Fig. 3b. Their energy ranges from 0.1 to 10 and more GeV. The oscillation
phase for ${\Delta m^{2}_{23}}$ oscillations is therefore typically ${\Delta
m^{2}_{23}}L/(4E)=10^{-2}$–$10^{2}$, centred around 1, the value we argued is
experimentally the best to reveal oscillation. As the neutrino flux produced
in the atmosphere is obtained by theoretical simulation characterized by
significant uncertainties, the measurement of the absolute muon or electron
neutrino flux does not allow to firmly establish the occurrence of neutrino
flavour transitions. On the other hand, the measurement of the muon to
electron neutrino ratio has a smaller theoretical uncertainty and is therefore
more reliable. Even more reliable is the variation of the muon and electron
fluxes (and their ratio) with the distance travelled, i.e. with the direction
(zenith angle) from which they reach the detector. The latter measurements by
SK provided in 1998 the first firm evidence of neutrino flavour transitions
and opened the modern era of neutrino physics.

Super-Kamiokande is a large water Cherenkov detector located in the Kamioka
mine, in Japan, $2.7\,\text{km}$ underground. It contains about 50 ktons of
water and is surrounded by about 13000 photomultipliers. In order to perform
the analysis above, a measurement of the neutrino flavour, direction, the
energy is needed. Let us see how such information is, at least partial,
obtained.

Neutrinos can be detected through their charged current interactions with the
nuclei: $\nu_{e_{i}}+N\to e_{i}+N^{\prime}$. The lepton $e_{i}=e,\mu$ produced
in the interaction is ultra-relativistic and produces a cone of Cherenkov
light while it travels through the water, which is detected by the
photomultipliers. When a lepton stops inside the detector the photomultipliers
detect a ring of Cherenkov light. The nature of the lepton can be told by the
shape of the ring: a muon produces a relatively clean ring, while the ring
produced by electrons is more fuzzy, as shown in Fig. 4. The position of the
ring allows to determine the lepton direction, which is correlated to the
neutrino direction if the neutrino energy is larger than about a GeV. If the
lepton is produced in the detector and stops inside the detector, its energy
can be measured by the amount of Cherenkov light collected by the
photomultipliers. The lepton energy is not strongly correlated to the neutrino
energy, but it cannot exceed it, which is enough to provide an handle on the
energy dependence of the neutrino flavour transition probability. The best
events are therefore the “fully contained multi-GeV” events. Neutrinos also
interact through neutral current interactions with nuclei such as
$\nu+N\to\nu+N+\pi^{0}\to\nu+N+\gamma\gamma$, which also produce a signal in
the detector. Tau leptons can also be produced if the neutrino is energetic
enough to exceed the kinematical threshold for production. Taus quickly decay
into hadrons, producing a signal similar to the neutral current one.

(a) (b)

Figure 4: Cherenkov rings produced by a muon (a) and an electron (b) in Super-
Kamiokande.

By now the statistics accumulated by Super-Kamiokande is impressive. Not only
it allows to establish neutrino transitions without any doubt, despite the
oscillation pattern is too smeared out by the poor neutrino-lepton energy
correlation to be observed explicitly, but it also allows to attribute the
transitions to $\nu_{\mu}\leftrightarrow\nu_{\tau}$ oscillations. In
particular, no depletion of the electron neutrino flux with the distance
travelled has been observed, which is compatible with the CHOOZ bound [22] on
$\nu_{e}$ transitions. Also, oscillations into sterile neutrinos, hypothetical
additional neutrinos not feeling any SM gauge interaction, are ruled out or
bound to have a marginal role. The same holds for exotic disappearance
mechanisms such as neutrino decay, or Lorentz or CPT violation.

#### Accelerator experiments

The Super-Kamiokande results have been confirmed by a number of experiments
using neutrinos produced at accelerators. Opera is a sophisticated detector at
the Gran Sasso laboratory in Italy designed to explicitly detect $\nu_{\tau}$
appearance from a $\nu_{\mu}$ neutrino beam produced at CERN. Such an
appearance would confirm the indirect, but solid, interpretation of the SK
results in terms of $\nu_{\mu}\leftrightarrow\nu_{\tau}$ oscillations. The tau
produced by the $\nu_{\tau}$ charged current interaction in the detector is
observed in emulsion films. Unfortunately, the expected statistics is not very
high, but a first candidate $\nu_{\tau}$ event has been recently reported
[24]. The K2K experiment in Japan used the SK detector to measure the
disappearance of $\nu_{\mu}$ from a pulsed beam produced at KEK. The initial
flux is measured by a detector placed near the neutrino source. The average
neutrino energy is slightly above $1\,\mathrm{GeV}$, and the distance
travelled by neutrinos is about $250\,\text{km}$, which gives an oscillation
phases of order one, as desired. The muon scattering angle in the detector can
be measured, together with its energy. The kinematics of the charged current
interaction then allows to reconstruct the neutrino energy. The experimental
results have been reported in [25]. Another important experiment is Minos, in
the Sudan mine, $735\,\text{km}$ north of Fermilab, where the (pulsed)
$\nu_{\mu}$ beam is produced. The average neutrino energy is higher than in
K2K, to give again an oscillation phase around one. As in the case of K2K, the
initial flux is measured by a near detector. Neutrino interactions in steel
are measured in this case by means of a magnetized tracking calorimeter. Minos
can observe $\nu_{\mu}$ charged current events (penetrating muons) and
therefore $\nu_{\mu}$ disappearance, which gives a determination of
$\theta_{23}$ and ${\Delta m^{2}_{23}}$ in agreement with the SK one [26]. It
can also see neutral current interactions of any neutrino (they produce a
diffuse hadron shower), which confirms that oscillations into sterile
neutrinos cannot account for the $\nu_{\mu}$ disappearance [27]. It can detect
$\nu_{e}$ charged current interactions (compact electromagnetic showers) and
therefore set a bound on $\nu_{\mu}\to\nu_{e}$ oscillations, which translates
into a bound on $\theta_{13}$ [28] compatible (although at present weaker)
with the CHOOZ one. The presence of a magnetic field allows Minos to tell
$\mu^{+}$ from $\mu^{-}$, which in turn allows to test CP-violation (although
with a poor sensitivity). The possibility to switch from a $\nu_{\mu}$ to a
$\overline{\nu}_{\mu}$ beam allows to test CPT violation. A mild, not very
significant, tension between the $\nu_{\mu}$ and $\overline{\nu}_{\mu}$
determinations of the oscillation parameters has been recently reported [29].

###  0.4.3 Experimental determination of ${\Delta m^{2}_{12}}$ and
$\theta_{12}$

The experimental determination of ${\Delta m^{2}_{12}}$ and $\theta_{12}$ is
mainly due to the SK, SNO, Borexino, KamLAND experiments. The result of a
global fit is shown in Fig. 5a [23]. Let us discuss the main ingredients
entering the above determination. In order to illustrate the physics of solar
neutrinos, it is necessary to discuss neutrino propagation in matter

(a) (b)

Figure 5: Global fit of the ${\Delta m^{2}_{12}}$ and $\theta_{12}$ parameters
(a). Dependence of the mixing angle in matter with the neutrino energy in the
case ${\Delta m^{2}}>0$ and for two values of the mixing angle in vacuum,
$\theta=0.6$ (dashed curve) and $\theta=0.06$ (solid curve) (b).

#### Matter effects in neutrino propagation

As neutrinos do not feel electromagnetic or strong interactions, they can
travel through ordinary matter without experiencing a single scattering
interaction. The mean free path of a neutrino in a medium as dense as the
Earth’s mantle is in fact $\lambda(E)\sim
10^{9}\,\text{km}\,(\text{GeV}/E)^{2}$ and even in the core of the Sun is
$\lambda(E)\sim 10^{10}\,\text{km}\,(10\,\mathrm{MeV}/E)^{2}$, both much
larger than the distance travelled in the medium. The energy normalization is
appropriate for atmospheric and solar neutrinos respectively. Only in
extraordinarily dense matter, such as a proto-neutron star core, neutrinos
have a mean free path, $\lambda(E)\sim
10\,\text{cm}\,(100\,\mathrm{MeV}/E)^{2}$, trapping them in a random walk
lasting about 10 seconds.

This does not mean, however, that matter does not affect neutrino propagation
in the Earth and in the Sun. While incoherent scattering is proportional to
the square of the weak interaction Fermi coupling suppressing the process,
forward coherent scattering [30], affecting the phase of the neutrino wave
function, is proportional to only one power of the Fermi coupling. Let us then
compare the rate of incoherent scattering, $dP_{\text{sc}}/dx$, where
$P_{\text{sc}}$ is the incoherent scattering probability, and the rate of
change of the neutrino phase due to coherent forward scattering,
$d\phi_{\text{co}}/dx$:

$\begin{aligned} &\text{incoherent:}&&dP_{\text{sc}}/dx\sim
G^{2}_{\text{F}}E^{2}n\\\ &\text{coherent:}&&d\phi_{\text{co}}/dx\sim
G_{\text{F}}n\end{aligned}\rightarrow\frac{dP_{\text{sc}}}{d\phi_{\text{co}}}\sim
G_{\text{F}}E^{2}\sim 10^{-5}\left(\frac{E}{\text{GeV}}\right)^{2},$ (20)

where $n$ is the matter number density and $E$ is the neutrino energy. We
therefore see that the coherent effect is largely dominant. While the effect
on the neutrino phase would be unobservable in the absence of neutrino
oscillations, the impact on oscillations may be significant, as the phases of
the three neutrino flavour eigenstates are affected in different ways.

Coherent scattering in the propagation can be accounted for by adding to the
free hamiltonian for the three neutrinos an effective “MSW” [30, 31, 32]
potential. In the flavour eigenstate basis, the Hamiltonian then reads

$H=\frac{1}{2E}U\begin{pmatrix}m^{2}_{1}&&\\\ &m^{2}_{2}&\\\
&&m^{2}_{3}\end{pmatrix}U^{\dagger}+\begin{pmatrix}V&&\\\ &0&\\\
&&0\end{pmatrix}+\text{universal terms,}$ (21)

where $U$ is the PMNS matrix and $V$ is the MSW potential. The three flavour
neutrinos feel different potentials in matter, because they have different
weak interactions. At the tree level, in neutral matter with no muon or tau
lepton number (or with $L_{\mu}=L_{\tau}$) and a negligible neutrino density,
such as the Earth or Sun matter, $V_{\mu}=V_{\tau}$ and
$V=V_{e}-V_{\mu}=\sqrt{2}G_{\text{F}}n_{e}$, where $n_{e}$ is the electron
neutrino number density. The difference between the electron neutrino
potential and the muon and tau one is due to the fact that the electron
neutrino can interact through charged current interactions with electrons,
while the muon and tau neutrinos cannot. The Hamiltonian in eq. (21)
determines neutrino propagation. In the antineutrino case, $U\to U^{*}$ and
$V\to-V$. Let us see what are the consequences of the presence of the MSW
potential are as far as neutrino propagation _in constant density_ is
concerned.

The case of constant density is relevant for the neutrino propagation in the
Earth, for example. The Earth has a density profile that is not constant but
can be in first approximated to be constant both in the mantle, where the mass
density is approximately $\rho_{m}\sim 3$–$5\,\text{g}/\text{cm}^{3}$, and in
the core, where $\rho_{c}\sim 10$–$15\,\text{g}/\text{cm}^{3}$. The effect of
the Earth in neutrino propagation is important for i) atmospheric neutrinos
(only through the subdominant $\nu_{e}\leftrightarrow\nu_{\mu,\tau}$
transitions, as $\nu_{\mu}\leftrightarrow\nu_{\tau}$ transitions are not
affected), ii) solar and supernova neutrinos, iii) terrestrial experiments (in
the case of a long baseline).

One of the most interesting consequences of the presence of the MSW term is
the possibility of a resonant enhancement of the oscillation amplitude. In
order to illustrate such an effect, let us consider a simple two neutrino
case, in which the Hamiltonian can be written as

$H=\begin{pmatrix}\displaystyle\sin^{2}\theta+\frac{2EV}{\Delta
m^{2}}&\sin\theta\cos\theta\\\
\sin\theta\cos\theta&\cos^{2}\theta\end{pmatrix}\frac{\Delta
m^{2}}{2E}+\text{universal terms}.$ (22)

It is then clear then a resonant enhancement of the mixing angle takes place
when the two diagonal elements coincide. In such a case, the mixing angle “in
matter” (i.e. obtained from the diagonalization of the matrix in eq. (22)),
$\theta_{m}$, becomes maximal, $\theta_{m}=45^{\circ}$, no matter how small is
the mixing angle in vacuum, and the squared mass difference in matter gets
correspondingly suppressed:

$\frac{2EV}{\Delta m^{2}}=\cos 2\theta\Rightarrow\left\\{\begin{aligned}
&(\sin 2\theta)_{m}=1,\;\\\ &(\Delta m^{2})_{m}=\Delta m^{2}\sin
2\theta\end{aligned}\right..$ (23)

Such a mixing enhancement takes place for an appropriate value of the neutrino
energy if $\theta<45^{\circ}$ and $V\cdot{\Delta m^{2}}>0$ or if
$\theta>45^{\circ}$ and $V\cdot{\Delta m^{2}}<0$. The dependence of the mixing
angle in matter with the neutrino energy is shown in Fig. 5b in the case
${\Delta m^{2}}>0$ and for two values of the mixing angle in vacuum,
$\theta=0.6$ (dashed curve) and $\theta=0.06$ (solid curve). We see that there
exists an energy for which the enhancement is maximal even for small mixing
angles, but the energy in which the angle is sizeable is correspondingly
small. It is also interesting to follow the dependence of the two Hamiltonian
neutrino eigenstates, $(\nu_{1,2})_{m}$, with the neutrino energy, in the case
$\theta\ll 1$, for example (still assuming $V\cdot{\Delta m^{2}}>0$, so that
the resonance does take place). When the neutrino energy is small,
$(\nu_{1,2})_{m}$ coincide with the mass eigenstates $\nu_{1,2}$. Since for
$\theta\ll 1$ the electron neutrino is close to $\nu_{1}$, we have
$\nu_{e}\approx(\nu_{1})_{m}$. In the opposite limit in which $(2EV/{\Delta
m^{2}})\gg 1$, the first diagonal element in eq. (22) becomes the heaviest and
dominates, which means that $\nu_{e}\approx(\nu_{2})_{m}$. By crossing the
resonance, the electron neutrino moves from the first to the second
Hamiltonian (propagation) eigenstate. The same effect takes place if the
neutrino energy is constant but the matter density, and therefore $V$, varies.
Which may play an important role for the neutrino propagation in matter with
varying density.

The precise relation between the mixing angle and squared mass difference in
vacuum and in matter is given, in the two neutrino case, by the following
formulas:

$\sin^{2}2\theta_{m}=\frac{\sin^{2}2\theta}{1+\displaystyle\left(\frac{2EV}{\Delta
m^{2}}\right)^{2}-2\cos 2\theta\frac{2EV}{\Delta m^{2}}},\quad(\Delta
m^{2})_{m}=\Delta m^{2}\left[1+\left(\frac{2EV}{\Delta m^{2}}\right)^{2}-2\cos
2\theta\frac{2EV}{\Delta m^{2}}\right]^{1/2}.$

We can define a resonant energy $E_{\text{res}}$ by

$\frac{2EV}{\Delta m^{2}}=\frac{E}{E_{\text{res}}}\cos 2\theta,\quad
E_{\text{res}}=\frac{\Delta m^{2}}{2V}\cos 2\theta\approx
8\,\mathrm{GeV}\left(\frac{\Delta m^{2}}{2\cdot
10^{-3}\,\mathrm{eV}^{2}}\frac{n_{e}}{1.65\,\text{gr/cm}^{3}}\right).$ (24)

Note also that

$\frac{(\sin^{2}2\theta)_{m}}{\sin^{2}2\theta}=\left[\frac{\Delta
m^{2}}{(\Delta m^{2})_{m}}\right]^{2}.$ (25)

The equations above show that matter effects are negligible when $E\ll
E_{\text{res}}$ or when $L\ll\lambda_{m}$, where $\lambda_{m}$ is the
oscillation length in matter. In the latter case, in fact, one can approximate
$\sin\phi\approx\phi$, where $\phi$ is the oscillation phase. Matter effects
then cancel because of eq. (25).

Let us now consider propagation in matter with varying density. Let us still
stick to the two neutrino case. The Hamiltonian is time dependent, as the MSW
potential varies during the propagation: $H(t)=H_{\text{free}}+V(t)$. The
exact solution for the evolution of the neutrino wave functions are non-
trivial and have usually to be obtained numerically. There is however one
important case in which the evolution is easy to follow: the adiabatic limit.
In such a limit, the variation of the Hamiltonian is much slower than the
variation of the oscillation phase. As a consequence, a neutrino which at a
certain time is in a given eigenstate of the full Hamiltonian $H(t)$ will
remain, in first approximation, in that eigenstate (which however will vary
together with $H(t)$). Such an adiabatic evolution takes place if

$\frac{d\theta_{m}}{dx}\ll\frac{(\Delta m^{2})_{m}}{2E}$ (26)

during the evolution, where $dx$ represents the variation in the neutrino
position.

An important consequence of the adiabatic evolution is the possibility of
large flavour swaps even for small mixing angles. This may happen if the
neutrino, while traveling, crosses a resonance because of the variation in the
matter density and therefore of $V$. The situation is illustrated in Fig. 6a.
There, the dependence of the two Hamiltonian eigenstates (in units of ${\Delta
m^{2}}/(2E)$) with $V$ ($2EV/{\Delta m^{2}}$) is shown for a small value of
the mixing angle, $\theta=0.06$. In the relevant case, solar neutrino
oscillations, the mixing angle will not be small, but this example better
shows how striking the effect can be. Consider the case in which an electron
neutrino is emitted in a medium with a density high enough that $2EV/{\Delta
m^{2}}\gg 1$ (the inner part of the Sun for example). As we have seen, in such
conditions, $\nu_{e}\approx(\nu_{2})_{m}$. Suppose now the density decreases
during the evolution until is vanishes and the evolution is adiabatic. Once
the neutrino is out of the medium, it will still be in the second eigenstate
of the Hamiltonian. Which in vacuum is
$\nu_{2}=\nu_{e}\sin\theta+\nu_{\mu}\cos\theta$. The probability that the
electron neutrino has become a muon neutrino is therefore
$P(\nu_{e}\to\nu_{\mu})\approx\cos^{2}\theta$. The transition probability
turns out to be close to one even if the mixing angle is small. Such an effect
cannot hold for arbitrarily small mixing angles of course (for $\theta=0$
there cannot be any effect), which means that the adiabatic approximation must
fail when $\theta$ is small enough. This can be seen from eq. (26). The
adiabatic condition is worse at the resonance, where eq. (26) becomes

$\gamma\equiv\frac{\Delta
m^{2}}{2E(V^{\prime}/V)_{\text{res}}}\frac{\sin^{2}2\theta}{\cos 2\theta}\gg
1,$ (27)

where $V^{\prime}$ is the derivative of the MSW potential with respect to the
position. If $\theta$ is small enough, the adiabatic condition at the
resonance is not fulfilled. If the adiabatic condition holds at production and
detection and it fails only in a small region around the resonance, the “level
crossing” probability is given in first approximation by the Landau-Zener
formula

$P(\nu_{1}\rightarrow\nu_{2})\equiv P_{c}\approx e^{-\gamma/2},$ (28)

where $\gamma$ is the adiabaticity parameter in eq. (27). The Landau-Zener
approximation fails in the extreme non-adiabatic regime, $\gamma\ll 1$.

(a) (b)

Figure 6: Dependence of the Hamiltonian eigenstates on the MSW potential for
$\theta=0.06$ (a). Contributions to the solar neutrino flux (b).

#### Solar neutrinos

The discussion of the neutrino evolution in varying density applies to solar
neutrinos. Solar neutrinos are electron neutrinos produced in the burning
process that produces the solar energy:
$4p+2e\to{}^{4}\text{He}+2\nu_{e}+26.7\,\mathrm{MeV}$. The process takes place
through different reactions. Correspondingly, we have different types of solar
neutrinos, characterized by different energy spectra. Among them, we have the
pp neutrinos, from the $pp\to de^{+}\nu_{e}$ reaction, that have by far the
largest flux (which is then well known because it can be derived from a
measurement of the total solar luminosity) but have quite a small energy,
$E<0.42\,\mathrm{MeV}$; the Be neutrinos, from
${}^{7}\text{Be}+e\to{}^{7}\text{Li}+\nu_{e}$, with a significant,
monochromatic flux with $E=0.863\,\mathrm{MeV}$; and B neutrinos, with a small
flux, but a more energetic spectrum, extending up to more than
$10\,\mathrm{MeV}$. The latter are the only ones that can be seen by the SK
and SNO experiments. The different contributions to the solar neutrino flux
are shown in Fig. 6b.

Several experiments have been devised to measure the solar neutrino flux,
starting from the historical Chlorine experiment in the Homestake mine in the
US, by Davis [17], which gave the first evidence of a neutrino deficit,
although with respect to an uncertain theoretical prediction. The latter was a
radiochemical experiment. The neutrino reaction
$\nu_{e}\,\mbox{}^{37}\text{Cl}\rightarrow e\,\mbox{}^{37}\text{Ar}$, with
energy threshold $E>0.814\,\mathrm{GeV}$, was detected by separating the few
tens of atoms of Argon produced by chemical methods and by counting them
through their beta decay back to the initial isotope. Analogous methods were
used in the Gallium experiments (SAGE [33], at the Baksan lake, Russia and
Gallex/GNO [34], at the Gran Sasso laboratories), exploiting the
$\nu_{e}{}^{71}\text{Ga}\to e\,{}^{71}\text{Ge}$ reaction, with threshold
$E>0.233\,\mathrm{MeV}$. Such experiments were not able to measure the time at
which the reaction happened nor the direction of the incoming neutrinos, but
they have the lowest energy thresholds, as shown in Fig. 6b. In particular,
the Gallium experiments are the only ones sensitive to pp neutrinos.

The Super-Kamiokande experiment detects solar neutrinos [35] through elastic
scattering with electrons in the water, $\nu_{e,x}\to\nu_{e,x}e$, where $x$
stands for $\mu$ or $\tau$, with an energy threshold $E>5.5\,\mathrm{MeV}$.
The electron and muon/tau neutrino cross sections are different, with the
latter smaller by a factor 6–7, because the charged current interactions do
not contribute.

The Sudbury Neutrino Observatory (SNO) experiment, near Sudbury, Canada, uses
heavy water, $D_{2}O$, and can detect neutrinos through three types of
processes. Elastic scattering (ES), $\nu_{e,x}e\to\nu_{e,x}e$, involves all
types of neutrinos. Electron and muon/tau neutrinos have different cross
sections, however, as in the SK case. If $\Phi_{e}$ and $\Phi_{\mu+\tau}$ are
the electron and muon/tau neutrino flux reaching the Earth, the ES measurement
determines $\Phi_{e}+0.155\Phi_{\mu+\tau}$. The neutrino direction can be
determined from the electron direction, which allows to tell the solar
neutrinos from the background by their direction. Charged current interactions
(CC), $\nu_{e}D\to ppe$, only involve electron neutrinos and therefore
determine $\Phi(\nu_{e})$. The neutrino energy spectrum can be reconstructed
from the electron one. Neutral current interactions (NC),
$\nu_{x}D\to\nu_{x}pn$, involve all types of neutrinos with equal cross
section. They therefore allow to determine the total neutrino flux
$\Phi_{e}+\Phi_{\mu+\tau}$. The SNO experiment has played for solar neutrino a
role similar to SK for atmospheric neutrinos, to the extent to which it
allowed to obtain a clear evidence of solar electron neutrino transitions,
independent of the theoretical uncertainties on the initial neutrino flux.
This is because the three reactions, ES, CC, and NC, measure three independent
linear combinations of the electron and muon/tau fluxes. It is then possible
to determine (and over-constrain) both fluxes, as shown in Fig. 7a [36]. In
particular, the total neutrino flux reaching the Earth (directly given by the
NC measurement), barring exotic phenomena, determines the initial electron
neutrino flux. From the experimental point of view, the use of heavy water is
necessary in order to obtain neutrino CC interactions (in water only
antineutrinos can interact with the proton in the Hydrogen and the neutrino
interaction with the neutrons in the Oxygen has a too high thresholds).
Chlorine was added in a second phase of the experiment to enhance the neutron
capture cross section, which, through the $\gamma$ produced, is an important
handle to detect the crucial NC processes [37]. Adding ${}^{3}\text{He}$
proportional chambers in the third phase of the experiment further improves
the NC detection, as it allows to see the single neutrons.

The Borexino experiment, at the Gran Sasso laboratories, also uses the elastic
scattering process, as SK and SNO, but is sensitive to lower energy neutrinos,
in particular to the ${}^{7}\text{Be}$ ones, as it uses a scintillator
detector. Such a measurement [38] is important as it constrains the electron
neutrino survival probability for values of the neutrino energy in which
matter effects in the sun are negligible. Moreover, such an experiment was
able to measure “geo-neutrinos” [39], $\overline{\nu}_{e}$ from natural
radioactivity with $E<3\,\mathrm{MeV}$.

#### KamLAND

The Kamioka Liquid-scintillator Anti-Neutrino Detector (KamLAND) experiment,
near the Super-Kamiokande experiment, also plays a crucial role in the
determination of the ${\Delta m^{2}_{12}}$ and $\theta_{12}$ parameters, as
the determination is not obtained by using solar neutrinos, but terrestrial
neutrinos ($\overline{\nu}_{e}$) emitted by several nuclear reactors in Japan.
The neutrino energy is of the order of a few MeV, the average distance
travelled is about 200 km, giving an order one oscillation phase for the
${\Delta m^{2}_{12}}$ oscillation frequency, ${\Delta m^{2}_{12}}L/(4E)\sim
1$. The detection is performed by means of the CC interaction
$\overline{\nu}_{e}p\to e^{+}n$ in the scintillator, with both the electron
and the delayed coincidence with the $\gamma$ signal from the neutron capture
used to observe it. The neutrino energy is directly related to the positron
energy, $E_{\nu_{e}}=E_{e^{+}}+m_{n}-m_{p}$, which allows to measure the
neutrino oscillation probability as a function of the energy and as a
consequence i) to obtain a good ${\Delta m^{2}_{12}}$ determination [40] and
ii) to observe the oscillation pattern, including an oscillation dip, in the
survival probability, as shown in Fig. 7b [40].

(a) (b)

Figure 7: Determination of the electron and muon/tau neutrino fluxes by SNO
(a). Electron antineutrino survival probability at KamLAND (b).

###  0.4.4 The unknown oscillation parameters

The mixing angle $\theta_{13}$ has not been measured yet, but both direct and
indirect bounds have been obtained from the CHOOZ and Minos experiments,
mentioned above, and from the analysis of subleading effects in the
atmospheric and solar neutrino experiments. The result of a global fit on
$\theta_{13}$ are shown in Fig. 8a [23].

The determination of $\theta_{13}$ is important for several reasons. It offers
an handle on the origin of the neutrino (and quark) masses and mixing angles.
In particular, it allows to discriminate among different flavour models. And
it is important for phenomenology, as it is crucial in the study of leptonic
CP-violation, supernova signals, and subleading effects, for example in
$\nu_{\mu}\leftrightarrow\nu_{\tau}$ transitions at the ${\Delta m^{2}_{23}}$
oscillation frequency. From the experimental point of view, a rich
experimental program is available. Several terrestrial experiments are running
or have been planned using different techniques: conventional beams obtained
from pion decays, so called “beta-beams”, obtained from the beta decay of
radio-active ions circulating in a storage ring with long straight sections,
and neutrino factory beams, obtained from the decay of muons also circulating
in a storage ring. A summary of the prospects on the $\theta_{13}$
determination are shown in Fig. 8b for different values of the experimental
parameters [41]. The figure uses the GLoBES package [42, 43]. References for
the single experiments are shown in Figure.

(a) (b)

Figure 8: Result of a global fit of $\theta_{13}$ (a). Future prospects on the
determination of $\theta_{13}$ (b).

Let us now discuss the determination of the sign of ${\Delta m^{2}_{23}}$. I
remind that this parameter determines the pattern of neutrino masses, enters
the analysis of supernova neutrino signals and of long baseline terrestrial
neutrino experiments, and determines the possibility to measure neutrinoless
double beta decay (see below). This parameter can be determined in the
presence of matter effects. Let us consider the three neutrino effective
Hamiltonian for propagation in matter and let us take the ${\Delta
m^{2}_{12}}=0$ limit for simplicity (${\Delta m^{2}_{12}}$ effects are
subleading in the experiments meant to measure $\operatorname{sign}({\Delta
m^{2}_{23}})$):

$H_{\text{eff}}=\frac{1}{2E}\left[U\begin{pmatrix}0&&\\\ &0&\\\ &&{\Delta
m^{2}_{23}}\end{pmatrix}U^{\dagger}\pm\begin{pmatrix}2EV&&\\\ &0&\\\
&&0\end{pmatrix}\right].$ (29)

The relative sign between the two terms on the RHS depends on whether neutrino
or antineutrino oscillations are considered. In the Earth, the MSW potential
is positive, $V>0$. We therefore easily see that if ${\Delta m^{2}_{23}}>0$
the resonant enhancement of oscillations can take place for neutrinos but not
antineutrinos, whereas if ${\Delta m^{2}_{23}}<0$ the enhancement takes place
for antineutrinos. This offers an handle to measure
$\operatorname{sign}({\Delta m^{2}_{23}})$. The neutrino energy should be of
the order of the resonant energy, say $10\,\mathrm{GeV}$, as determined by
$|{\Delta m^{2}_{23}}|$ and $V$. Moreover, a long baseline is needed, so that
the oscillation phase is not small. I remind in fact that for small
oscillation phases $\phi$, the oscillating factor in the probaility can be
approximated as $\sin\phi\approx\phi$ and matter effects cancel (see eq. (25)
and below). Finally, the effect shows up in the
$\nu_{e}\leftrightarrow\nu_{\mu,\tau}$ channel. This can be seen from eq. (29)
by observing that in the limit ${\Delta m^{2}_{12}}=0$ the $\theta_{12}$
rotation in the PMNS matrix is not physical and $U$ can be approximated with a
23 rotation by the angle $\theta_{23}$. In this limit, matter effects do not
affect the $\nu_{\mu}\leftrightarrow\nu_{\tau}$ oscillations. The prospects
for the measurement of the sign of ${\Delta m^{2}_{23}}$ are summarized in Fig
23 of [41].

Let us now discuss the possible determination of the CP-violating phase
$\delta$. We do not need to stress the importance of investigating whether CP-
violation is present not only in the quark sector, but also in the lepton
sector. On top of that, leptonic CP-violation could explain the origin of the
Baryon asymmetry in the universe through the leptogenesis mechanism. Neutrino
oscillation experiments offer the possibility to study leptonic CP-violation
associated to the CP phase $\delta$, which is certainly physical, whether
neutrinos are Dirac or Majorana. The Majorana phases $\alpha$, $\beta$, if
physical, cannot be accessed by oscillation experiments. The CP-violating
phase $\delta$ can determine a difference between the neutrino and
antineutrino oscillation probabilities:

$\displaystyle P(\nu_{e_{i}}\rightarrow\nu_{e_{j}})$
$\displaystyle=P(\overline{\nu}_{e_{j}}\rightarrow\overline{\nu}_{e_{i}})=P_{\text{CPC}}+P_{\text{CPV}}$
(30) $\displaystyle
P(\overline{\nu}_{e_{i}}\rightarrow\overline{\nu}_{e_{j}})$
$\displaystyle=P(\nu_{e_{j}}\rightarrow\nu_{e_{i}})=P_{\text{CPC}}-P_{\text{CPV}}$

(see also eq. (19)). At accelerators experiments aiming at a measurement of
such difference, due to the smallness of ${\Delta m^{2}_{12}}/|{\Delta
m^{2}_{23}}|$ and $\theta_{13}$, we can approximate

$\displaystyle\left.\begin{aligned}
P(\nu_{\mu}\leftrightarrow\nu_{\tau})_{\text{CPC}}&\approx\sin^{2}\theta_{23}\sin^{2}\frac{{\Delta
m^{2}_{23}}L}{4E}\\\
P(\nu_{e}\leftrightarrow\nu_{\mu})_{\text{CPC}}&\approx\sin^{2}\theta_{23}\sin^{2}2\theta_{13}\sin^{2}\frac{{\Delta
m^{2}_{23}}L}{4E}\\\
P(\nu_{e}\leftrightarrow\nu_{\tau})_{\text{CPC}}&\approx\cos^{2}\theta_{23}\sin^{2}2\theta_{13}\sin^{2}\frac{{\Delta
m^{2}_{23}}L}{4E}\end{aligned}\right\\}+{\Delta m^{2}_{\text{SUN}}}\text{
corr.}$ (31) $\displaystyle\;\;P_{\text{CPV}}=\pm\cos\theta_{13}\sin
2\theta_{12}\sin 2\theta_{23}\sin 2\theta_{13}\sin\delta\,\sin\frac{{\Delta
m^{2}_{12}}L}{4E}\sin^{2}\frac{{\Delta m^{2}_{23}}L}{4E}.$ (32)

The formulas above show that CP-violation has a chance to show up in
$\nu_{e}\leftrightarrow\nu_{\mu}$ oscillations [44]. First of all, two out of
the three angles entering the CP-violating part of the probability in eq. (32)
are large (unlike the quark mixing angles). If the baseline of the experiment
is large enough, the term oscillating with the atmospheric frequency is also
of order one. If the phase $\delta$ is not too small, the CP-violating part of
the probability is then only suppressed by $\sin 2\theta_{13}$ and the solar
phase $\sin({\Delta m^{2}_{12}}L/(4E))$, which are not necessarily too small.
On top of that, the CP-conserving part of the
$\nu_{e}\leftrightarrow\nu_{\mu}$ probability is suppressed by two powers of
$\sin 2\theta_{13}$, whereas the CP-violating part is suppressed by only one
power. This means that the smaller $\sin 2\theta_{13}$, the larger is the
asymmetry between the probabilities in the neutrino and antineutrino channel,

$a_{\text{CP}}=\frac{P(\nu_{e}\rightarrow\nu_{\mu})-P(\overline{\nu}_{e}\rightarrow\overline{\nu}_{\mu})}{P(\nu_{e}\rightarrow\nu_{\mu})+P(\overline{\nu}_{e}\rightarrow\overline{\nu}_{\mu})}\propto\frac{1}{\sin
2\theta_{13}+\text{ corr.}}.$ (33)

A smallish $\theta_{13}$ is therefore not necessarily a curse for CP-violation
[44]. On the one hand, the total number of events decreases with
$\sin^{2}2\theta_{13}$, and therefore the statistical error on the measurement
of the asymmetry increases as $\delta a\sim 1/\sqrt{N}\propto 1/\sin
2\theta_{13}$, where $N$ is the average number of events. On the other hand,
the asymmetry signal also increases with $1/\sin 2\theta_{13}$. The
statistical significance of the measurement, $\delta a/a$, is therefore
approximately constant [45]. Such a behaviour cannot hold for an arbitrarily
small value of $\theta_{13}$, of course. This is indeed the case for two
reasons: i) the corrections in eq. (33) become dominant compared to the $\sin
2\theta_{13}$ term and ii) the number of expected events may become smaller
than one.

Experimentally, the measurement of CP-violation is complicated by the fake
sources of neutrino-antineutrino asymmetry. In particular, one has to consider
the CP-asymmetry of the source, which typically does not emit the same number
of neutrinos and antineutrinos, the CP-asymmetry of the Earth, made of matter
and not antimatter, through which the neutrinos travel, and the CP-asymmetry
of the target. A measurement of CP-violation therefore requires a good
knowledge of the initial neutrino fluxes, of the Earth (electron) density
profile, and of the neutrino cross sections. To cope with such difficulties,
it would be useful to have a measurement of the energy spectrum, two
baselines, and to measure more than a single oscillation channel. Neutrino
factories are especially suited for measuring CP-violation, as the neutrino
flux is very high and quite pure. The prospects for the measurement of the
phase $\delta$ are summarized in Fig 22 of [41].

###  0.4.5 Supernova neutrinos

Supernova neutrinos are emitted during the core collapse of type-II
supernovas. Their study can i) provide further informations on the neutrino
parameters, modulo the uncertainties on the spectrum and intensity of the
source, ii) probe the physics of the collapse, and iii) constrain exotic
neutrino transitions, such as oscillations into sterile neutrinos.

Type-II supernovas originate from the collapse of large stars. The burning
process produces heavier and heavier elements in their core. If the star is
large enough, the gravitational pressure becomes too large to be stood by the
core, and leads to the collapse of the atomic structures. The core, which
before collapse has a radius $R\sim 8000\,$km, a density $\rho\sim
10^{9}\,\text{g/cm}^{3}$ and a temperature $T\sim 0.7\,\mathrm{MeV}$, shrinks
to a proto-neutron star formed by nuclear matter with $R\sim 30\,$km,
$\rho\sim 3\cdot 10^{14}\,\text{g/cm}^{3}$, $T\sim 30\,\mathrm{MeV}$. In the
process, an impressive amount of energy, $E\sim 3\cdot 10^{53}\,$erg,
corresponding essentially to the gravitational energy released, is emitted.
Only about 0.01% of this energy goes into light, about 1% goes into kinetic
energy, and the remaining 99% is emitted through neutrinos. The neutrino
emission is not instantaneous, however. The matter density in the proto-
neutron star is so high that the neutrino mean free path is of the order of 10
cm. The time it takes to the neutrinos to diffuse out is then
$t_{\text{diff}}\sim 3R^{2}/\lambda\sim 10\,$sec. A handful of nupernova
neutrinos where detected when the supernova SN 1987A exploded in the
Magellanic Cloud, 50 kpc away, in 1987. The time distribution of the neutrino
events confirmed the qualitative success of the picture above.

The observation of the neutrino emission constrains the possibility of
invisible, or faster escape channels for the energy to be released. One such
example is neutrino oscillations into sterile neutrinos. If the oscillation
rate was large enough, sterile neutrinos, which do not interact with matter
and would not be trapped inside the core, would immediately escape, carrying
away the neutrino energy. A strong bound on a possible active-sterile mixing
angle follows, $\sin^{2}2\theta_{s}\lesssim 10^{-8}$, which can be evaded if
the sterile neutrino mass is small enough [46, 47, 48, 49, 50]. Such limits
are particularly interesting [51, 52] in the case of neutrinos from extra-
dimension [53, 54, 55, 56, 57, 58, 59, 60]. Other invisible channels
constrained by the observation of supernova neutrinos are the conversion into
axions or into KK gravitons in large extra dimension scenarios.

Supernovas in our galaxy are expected to explode with an uncertain, but not
very exciting rate of about one every 30 years or more. However, if such an
event took place, the present neutrino detectors would gather an impressive
amount of data. A supernova 10 kpc away would produce about 8000 neutrino
events in SK, 800 in SNO, and 330 in KamLAND, thus allowing a detailed study
of the flavour, energy, and time spectrum of the neutrinos reaching us. The
distortions of such spectra compared to the expectations in the absence of
oscillations (which have a significant degree of uncertainty) might provide
information on $\theta_{13}$ and the sign of ${\Delta m^{2}_{23}}$ [61, 62].

###  0.4.6 Anomalous anomalies

While the three neutrino oscillation picture consistently and precisely
explains an impressive amount of experimental data, the results of the LSND
experiment [63, 64] do not fit in the picture. Using a neutrino beam from pion
decay detected in a scintillator, such an experiment found an evidence of
$\overline{\nu}_{\mu}\to\overline{\nu}_{e}$ transitions that, if interpreted
in terms of neutrino oscillations, would require a squared mass difference
larger than the atmospheric one, ${\Delta m^{2}_{\text{LSND}}}>{\Delta
m^{2}_{\text{ATM}}}$. Such a third squared mass difference would require the
introduction of a fourth light neutrino. As we have seen in the introduction,
the number of light “active” neutrinos (i.e. with the gauge interactions of
standard neutrinos) is bound by the measurement of the $Z$ boson width to be
three. The forth neutrino should then be sterile. Such an interpretation poses
a number of problems. From the theoretical point of view, in order to account
for a light sterile neutrino one should explain while an explicit mass term
for it, not forbidden by the electroweak symmetry (unlike the one for active
neutrinos), would be absent or extremely small. This can be however accounted
for by an appropriate symmetry. Moreover, even if the presence of a fourth
neutrino, it is not easy to fit the observed anomaly [65, 66, 67] because of
the bounds from the Karmen [68] and Bugey [69] experiments. The LSND anomaly
is being tested by MiniBOONE, which uses about the same value of $L/E$, but
with $\mathcal{O}\left(10\right)$ larger values of $L$, $E$. MiniBOONE can run
both in a neutrino and antineutrino mode. The present situation is the
following. The neutrino run excludes the LSND signal at more than 90%
confidence level (it observes an anomaly, but at the wrong value of $L/E$, and
in the low energy region that is more sensitive to the backgrounds) [70]. The
antineutrino run, on the other hand, seems to find an excess compatible with
LSND [71].

###  0.4.7 Beyond oscillations

We now discuss the bounds and prospects of determination of the neutrino
parameters that cannot be probed with oscillation experiments,
$m_{\text{lightest}}$, and the Majorana phases $\alpha$, $\beta$ (assuming
they are physical).

As said, the determination of the squared mass differences ${\Delta
m^{2}_{23}}$ and ${\Delta m^{2}_{12}}$ does not determine the absolute value
of neutrino masses. On the other hand, the latter can be obtained from the
additional knowledge of $m_{\text{lightest}}$. We have indeed
$m^{2}_{1}=m^{2}_{\text{lightest}}$, $m^{2}_{2}={\Delta
m^{2}_{12}}+m^{2}_{\text{lightest}}$,
$m^{2}_{3}=m^{2}_{\text{lightest}}+{\Delta m^{2}_{12}}+{\Delta m^{2}_{23}}$ in
the case of normal hierarchy and $m^{2}_{3}=m^{2}_{\text{lightest}}$,
$m^{2}_{2}=m^{2}_{\text{lightest}}-{\Delta m^{2}_{23}}$,
$m^{2}_{1}=m^{2}_{\text{lightest}}-{\Delta m^{2}_{12}}-{\Delta m^{2}_{23}}$ in
the case of inverse hierarchy (in which case ${\Delta m^{2}_{23}}<0$). In
principle, $m_{\text{lightest}}$ can have any value. If
$m_{\text{lightest}}\ll({\Delta m^{2}_{12}})^{1/2}\sim 0.01\,\mathrm{eV}$, the
three neutrinos have masses $m_{3}\approx|{\Delta m^{2}_{23}}|^{1/2}\sim
0.05\,\mathrm{eV}$, $m_{2}\approx({\Delta m^{2}_{12}})^{1/2}\sim
0.01\,\mathrm{eV}$, $m_{1}=m_{\text{lightest}}\ll m_{2}$ in the normal
hierarchy case and $m_{1}\approx m_{2}\approx|{\Delta m^{2}_{23}}|^{1/2}\sim
0.05\,\mathrm{eV}$, $m_{3}=m_{\text{lightest}}\ll m_{1,2}$ in the inverse
hierarchy case (in which case it actually suffices to assume
$m_{\text{lightest}}\ll|{\Delta m^{2}_{23}}|^{1/2}\sim 0.05\,\mathrm{eV}$). If
$m_{\text{lightest}}\gg|{\Delta m^{2}_{23}}|^{1/2}\sim 0.05\,\mathrm{eV}$, the
three neutrinos are approximately degenerate, $m_{1}\approx m_{2}\approx
m_{3}\approx m_{\text{lightest}}$.

Beta decay experiments exploit the fact that a non vanishing neutrino mass
modifies the endpoint of the electron spectrum in beta decays
$(A,Z)\to(A,Z+1)+e^{-}+\overline{\nu}_{e}$, where $A$ and $Z$ are the mass and
atomic number of the decaying atom. This is a purely kinematical effect
illustrated in Fig. 9a (taken from [72]). The tritium decay
${}^{3}\text{H}\to{}^{3}\text{He}+e^{-}+\overline{\nu}_{e}$ is often used for
this purpose. The decay spectrum depends in general on the composition of
$\nu_{e}$ in terms of the three mass eigenstates and on their masses. Given
that the present sensitivities are larger than $|{\Delta m^{2}_{23}}|^{1/2}$,
the spectrum only depends on the combination $(m^{\dagger}m)_{ee}$, where $m$
is the light neutrino mass matrix is the flavour basis:

$\displaystyle\frac{dN}{dE}\propto\sum|U_{eh}|^{2}\Gamma(m^{2}_{h},E)\approx\Gamma(m^{2}_{\nu_{e}},E),$
(34) $\displaystyle
m^{2}_{\nu_{e}}\equiv(m^{\dagger}m)_{ee}=|U_{eh}|^{2}m^{2}_{h}=c^{2}_{13}(m^{2}_{1}c^{2}_{12}+m^{2}_{2}s^{2}_{12})+m^{2}_{3}s^{2}_{13}.$
(35)

The present bound from the Mainz [73] and Troitsk [74] experiments is in the
degenerate neutrino regime, in which $m_{\nu_{e}}\approx m_{\text{lightest}}$,
and give $m_{\text{lightest}}<2.3\,\mathrm{eV}$. The Katrin experiment [75]
promises to lower the sensitivity down to $0.2\,\mathrm{eV}$.

(a) (b)

Figure 9: Modification of the beta decay spectrum in the presence of a non-
vanishing neutrino mass (a). Different beta decay processes and their
microscopic mechanisms (b).

Another handle on the absolute value of neutrino masses is provided by
neutrinoless double beta ($0\nu 2\beta$) decay, $(A,Z)\to(A,Z+2)+2e^{-}$. As
mentioned earlier, $0\nu 2\beta$ signals lepton number violation and is
induced by a Majorana neutrino mass term (see Fig. 9b) at a rate
$\Gamma\propto|m_{ee}|^{2}\langle Q\rangle^{2}$, where $\langle Q\rangle$ is
the matrix element of the hadronic part of the operator inducing the decay and

$m_{ee}=U_{eh}^{2}m_{h}=c^{2}_{13}(m_{1}c^{2}_{12}+m_{2}s^{2}_{12}e^{2i\alpha})+m_{3}s^{2}_{13}e^{2i\beta^{\prime}}$
(36)

is the 11 element of the light neutrino mass matrix in the flavour basis, with
$\beta^{\prime}=\beta-\delta$. The $0\nu 2\beta$ rate therefore probes both
the absolute scale of neutrino masses and the Majorana phases $\alpha$,
$\beta$.

In order to measure the $0\nu 2\beta$ decay, a nucleus $(A,Z)$ for which the
beta decay, but not the double beta one, is kinematically forbidden, is
needed. It is then possible to discriminate the neutrinoless decay from the
standard two neutrino decay $(A,Z)\to(A,Z+2)+2e^{-}+2\nu_{e}$. Indeed, the
latter has a continuous spectrum for the sum of the energies of the two
electrons, with endpoint at the $Q$-value of the decay, while in the former
the energy of the electrons must coincide with the $Q$ value. If the energy
resolution of the electron energy measurement is precise enough, it is then
possible to exclude most of the events due to the standard two neutrino decay.
The determination, or bound, one obtains is however plagued by the
$\mathcal{O}\left(50\%\right)$ uncertainty associated to the matrix element
$\langle Q\rangle$. The Heidelberg-Moscow collaboration, using the
$\mbox{}^{76}\text{Ge}\rightarrow\mbox{}^{76}\text{Se}+2e^{-}$ decay, sets a
limit $|m_{ee}|<\mathcal{O}\left(1\right)\times 0.4\,\mathrm{eV}$ [76]. The
claim of a signal has also been reported by a subgroup of the collaboration
[77]. A rich experimental program is available in this field, with prospects
of lowering the bound down to a few$\times 10^{-2}\,\mathrm{eV}$.

Neutrinos play a role in cosmology through their effect on the Cosmic
Microwave Background (CMB) and the formation of Large Scale Structures in the
universe (LSS). The effect on CMB is due to the fact that the anisotropies in
the photon radiation at decoupling (which takes place at a temperature of
about $0.3\,\mathrm{eV}$) are sensitive to the total radiation density, and in
particular to the energy fraction in neutrinos. In turn, the latter is
determined by the mere sum of the three neutrino masses,
$m_{\text{cosmo}}=m_{1}+m_{2}+m_{3}$, whose knowledge is of course equivalent
to the knowledge of $m_{\text{lightest}}$. The effect on LSS is due to the
fact that the free streaming of relativistic non-interacting particles
smoothes the density fluctuations leading to the large scale structures
observed today. The length scale of the effect depends again on the neutrino
masses.

By fitting the available data on CMB and LSS, it is possible to find an upper
bound on $m_{\text{lightest}}$. However, the latter depends on a number of
assumptions (although plausible and consistent) on the cosmological model. It
is assumed, for example, that the structures are generated by gaussian
adiabatic fluctuations, that the spectral index is constant, that the particle
spectrum is the SM one, that the dark matter is cold and the dark energy is
accounted for by a non-vanishing cosmological constant. Moreover, we note that
the LSS constraint is more powerful but less reliable, as the effect of
neutrino masses is larger at smaller scales, where the numerical simulations
are more difficult. The bound one obtains at 99% confidence level is
$m_{\text{cosmo}}<2.6\,\mathrm{eV}$ when conservatively using the CMB data
only and $m_{\text{cosmo}}<0.5\,\mathrm{eV}$ if the LSS data is also taken
into account [78].

Besides CMB and LSS, neutrinos also affect Big Bang nucleosynthesis (BBN) and
possibly Baryogenesis. The present relative abundance of protons, neutrons,
and light elements is determined during BBN by standard inverse beta reactions
involving neutrinos at their decoupling temperature $T\sim\,\mathrm{MeV}$. The
Baryon asymmetry in the universe is quantified by the number density of
Baryons (minus the negligible density of anti-Baryons), usually normalized to
the photon density, $n_{B}/n_{\gamma}\approx 6\cdot 10^{-10}$. It is believed
that the asymmetry between Baryons and anti-Baryons, $n_{B}>0$, originated
dynamically during the evolution of the universe. On the other hand, the SM of
particle physics cannot account for such a dynamical origin. However, it has
been proposed that the Baryon asymmetry could originate from a lepton
asymmetry generated by the simplest dynamics underlying the origin of neutrino
masses, the see-saw mechanism [79] (see next Section). More specifically, the
idea is that a lepton asymmetry is formed by the CP-asymmetric, out of
equilibrium decay of heavy right-handed neutrinos (transformed into a Baryon
asymmetry by sphalerons) [80, 81, 82]. Although by far not the only one, this
is an economical and successful Baryogenesis mechanism that allows, under
hypotheses, to relate the single number characterizing the Baryon asymmetry to
the neutrino parameters.

The summary of theoretical expectations and bounds on $m_{\text{lightest}}$ is
shown in Fig. 10 [2]. In Fig. 10a the parameter $m_{\nu_{e}}$ probed by beta
decay experiments is plotted as a function of the lightest neutrino mass,
taking into account the present uncertainties on ${\Delta m^{2}_{12}}$ and
$|{\Delta m^{2}_{23}}|$, for the two signs of ${\Delta m^{2}_{23}}$. The bound
from the Mainz and Troitsk experiments are shown together with the expected
bound from Katrin. Fig. 10b shows an analogous plot for the parameter probed
by $0\nu 2\beta$ decay. The darker regions correspond to the uncertainty
associated to the unknown Majorana phases, with the oscillation parameters
fixed at their present central values. The lighter region account for the
additional uncertainty on the oscillation parameters. Finally, Fig. 10c shows
the situation for the parameter probed by cosmology.

(a) (b) (c)

Figure 10: Summary of bounds on $m_{\text{lightest}}$.

##  0.5 Theoretical implications

After having illustrated the phenomenology associated to neutrino masses and
mixings and the determination of the neutrino parameters, we conclude by
discussing the theoretical impact of the information that has been gathered so
far.

From the theoretical point of view, the relevant information emerging from the
data in eq. (16) can be summarized as follows:

$\begin{gathered}m_{\nu_{i}}\ll 174\,\mathrm{GeV}\\\\[2.84526pt]
\theta_{23}\sim 45^{\circ}\text{(= $45^{\circ}$?)}\qquad\theta_{12}\sim
30^{\circ}\text{--}35^{\circ}\neq
45^{\circ}\qquad\theta_{13}<7^{\circ}\\\\[1.42262pt] |{\Delta
m^{2}_{12}}/{\Delta m^{2}_{23}}|\approx 0.035\ll 1.\end{gathered}$ (37)

The most important theoretical guideline is the smallness of neutrino masses.
We then have the surprising fact that two out of three mixing angles turn out
to be large, unlike what found in the quark sector. In particular,
$\theta_{23}$ is compatible with being maximal. While the present uncertainty
is too large to draw conclusions, it would be interesting to know whether
$\theta_{23}$ is indeed maximal (i.e. $45^{\circ}$ up to small corrections) or
just large (i.e. $\mathcal{O}\left(1\right)$). A maximal angle would in fact
be an indication of a non-trivial flavour structure [83]. As for the solar
angle, we know that it is definitely not maximal, although compatible with the
so called tri-bimaximal prediction [84]. The squared mass difference hierarchy
implies that the ratio $m_{2}/m_{3}$ is about 0.2 or larger, not as small as
the typical charged fermion mass hierarchy. In the following, we will
concentrate on the first guideline, the smallness of neutrino masses, and its
implications for the origin of neutrino masses.

There is no doubt that neutrino masses are indeed very small compared to the
natural scale of fermion masses, the electroweak scale, $v=174\,\mathrm{GeV}$:
$m_{\nu}/v\lesssim 10^{-12}$. On the other hand, some of the charged fermion
masses are also quite small compared to $v$, the smallest being the electron
mass, suppressed by a factor $m_{e}/v\approx 0.3\cdot 10^{-5}$. Still, the
smallness of neutrino masses seems to be peculiar. Not only because twelve
orders of magnitude are more than five. Also because all the three families of
neutrinos are bound to be that light. On the contrary, the suppression of the
lightest charged fermion masses seems to be related to the hierarchy among
different families, the heaviest families being suppressed compared to the
electroweak scale at most by a couple of orders of magnitude. Moreover, there
is a compelling explanation for the peculiar smallness of neutrino masses, as
we now see.

###  0.5.1 The origin of neutrino masses

We have seen in Section 0.3.1 that the observed smallness of neutrino masses
is not explained at the level of the effective theory below the electroweak
scale: the QED and QCD gauge symmetries allow a mass term for both the charged
fermions and the neutrinos. Things are different when considering the full SM
gauge symmetry
$G_{\text{SM}}=\text{SU(3)}_{c}\times\text{SU(2)}_{L}\times\text{U(1)}_{Y}$.
It is well known that such a symmetry forbids any fermion mass term, both for
charged fermions and neutrinos. In order to see that, it suffices to show that
no gauge invariant mass term in the form eq. (6) can be written for the left
handed fermion fields. The latter transform under the $\text{SU(2)}_{L}$ gauge
symmetry either as doublets, $L_{i}=(\nu_{iL},e_{iL})^{T}$,
$Q_{i}=(u_{iL},d_{iL})^{T}$, or as singlets, $\overline{e_{iR}}$,
$\overline{u_{iR}}$, $\overline{d_{iR}}$. Their hypercharges are -1/2, 1/6, 1,
-2/3, 1/3 respectively. It is then easy to see that it is not possible to
combine any two such left handed fermions in a gauge invariant combination.

The fact that no fermion mass term is allowed in the limit in which the
electroweak symmetry is unbroken can be rephrased by saying that the SM is a
“chiral” theory. This property might be the very reason why the fermions we
observe have a mass so much smaller than, say, the Planck scale: they are
protected by the electroweak symmetry. As a consequence, a SM fermion mass has
to be proportional to at least one power of $v$. Here is the crucial
difference between charged fermions and neutrinos: while the charged fermion
mass term arises proportional to one power of $v$, the neutrino mass term
needs at least two powers of $v$. Let us see why this is the case and what are
the consequences.

In the SM the electroweak symmetry is broken by the vacuum expectation value
(vev) of the Higgs field. The Higgs is a complex scalar field that transforms
as a doublet under SU(2)L, $H=(h^{+},h^{0})$, and has hypercharge 1/2. The
Higgs potential is such that the value of the (neutral component of the) field
in the ground state does not vanish. Such a vev is denoted by $\langle
h^{0}\rangle$ and provides the value of the electroweak scale, $\langle
h^{0}\rangle=v$. The electroweak symmetry is broken because the ground state
value of the Higgs is not invariant under electroweak gauge transformations.

While mass terms for the SM fermions are not allowed, Yukawa interactions with
the Higgs are. For example, the electron can interact with the Higgs through
the gauge invariant interaction
$\lambda_{E}\overline{e_{R}}L\,H^{\dagger}=\lambda_{E}\overline{e_{R}}(e_{L}h^{0}+\nu_{L}h^{+})$,
where family indexes have been understood. Once the Higgs field is expressed
in terms of the displacement from the ground state value, $h^{0}=v+\delta
h^{0}$, a mass term is generated for the electron in the form
$m_{E}\overline{e_{R}}e_{L}$, with $m_{E}=\lambda_{E}v$.

The electron Yukawa interaction above has the property of being
“renormalizable”. For our purposes, this means that the coupling in front has
a non negative dimension in mass (it is in fact dimensionless). Note that a
lagrangian (density) has dimension 4 in energy. Therefore, renormalizable
operators have dimension 4 or less. Non-renormalizable terms are instead
characterized by coefficients proportional to inverse powers of a mass scale,
or cut-off, $\Lambda$. Such terms are thought not to be fundamental but to
arise as remnant, effective terms from a more fundamental (possibly
renormalizable) theory living at a scale related to $\Lambda$. They are indeed
inconsistent (at least perturbatively) at energies larger than $\Lambda$. A
dimension $4+n$ non-renormalizable operator is suppressed by $n$ powers of
$\Lambda$.

One can then wonder what is the most general form of the renormalizable Yukawa
interactions of the SM fermions with the Higgs, and which is the most general
fermion mass term that can be generated at the renormalizable level. The
answer is provided by the SM flavour lagrangian

$\displaystyle\mathcal{L}_{\text{SM}}^{\text{flavor}}$
$\displaystyle=\lambda^{E}_{ij}\overline{e_{iR}}L_{j}H^{\dagger}$
$\displaystyle+\lambda^{D}_{ij}\overline{d_{iR}}Q_{j}H^{\dagger}$
$\displaystyle+\lambda^{U}_{ij}\overline{u_{iR}}Q_{j}H$
$\displaystyle+\text{h.c.}$ (38) $\displaystyle=m^{E}_{ij}e^{c}_{i}e_{j}$
$\displaystyle+m^{D}_{ij}d^{c}_{i}d_{j}$
$\displaystyle+m^{U}_{ij}u^{c}_{i}u_{j}$ $\displaystyle+\text{h.c.}+\ldots,$

with

$m^{E}_{ij}=\lambda^{E}_{ij}v\quad m^{D}_{ij}=\lambda^{D}_{ij}v\quad
m^{U}_{ij}=\lambda^{U}_{ij}v\quad m^{\nu}_{ij}=0.$ (39)

We therefore see that, unlike charged fermion masses, neutrino masses do not
arise in the SM even after electroweak symmetry breaking, if one sticks to
renormalizable interactions. This is a good starting point to understand the
smallness of neutrino masses. Of course, we have in the end to account for the
fact that neutrino masses are not vanishing. In order to do that, some
ingredient has to be added to the SM as a renormalizable theory. While there
are certainly several possibilities, we can distinguish two main options.
Either the new ingredients live at a scale $\Lambda\gg v$ (the standard
example being the addition of heavy right-handed neutrinos giving rise to the
see-saw mechanism) or the new ingredients live at a scale $\Lambda\lesssim v$
(the standard example being Dirac neutrinos). Let us consider the two
possibilities in turn.

###  0.5.2 $\Lambda\gg v$

This case is particularly interesting. It can be described in a model
independent way by making use of a central theorem of effective field theory.
At the electroweak scale and below, the effect of whatever are the additional
heavy degrees of freedom to be added in order to account for neutrino masses,
can be described in terms of effective interactions involving only light
degrees of freedom and symmetries. In particular, we do not need to know the
specific form of the high energy theory in order to parameterize its effect at
the electroweak scale in a model independent way. Such effective interactions
are non-renormalizable, i.e. suppressed by powers of the scale $\Lambda$ at
which they arise. We therefore have

$\mathcal{L}_{E\ll\Lambda}^{\text{eff}}=\mathcal{L}_{\text{SM}}^{\text{ren}}+\mathcal{L}_{\text{SM}}^{\text{NR}},$
(40)

where $\mathcal{L}_{\text{SM}}^{\text{ren}}$ is the renormalizable SM
lagrangian and $\mathcal{L}_{\text{SM}}^{\text{NR}}$ accounts for the
effective interactions.

The effect at energies $E\ll\Lambda$ of dimension $4+n$ effective interactions
arising at the scale $\Lambda$ is suppressed by $(E/\Lambda)^{n}$, as it can
be inferred from simple dimensional analysis. As a consequence, the most
relevant effective operators are those with lowest dimension: $4+1$. It turns
out that in the SM it is possible to write only one such dimension 5 operator:

$\mathcal{L}^{\text{NR}}_{\text{SM}}=\frac{h_{ij}}{2\Lambda}(HL_{i})(HL_{j})+\text{higher
dimension},$ (41)

where SU(2)L invariant contractions are understood. Note that separating the
coefficient of the operator in eq. (41) in a dimensionless numerator $h_{ij}$
and a dimensionful denominator $\Lambda$ is purely conventional. When doing
that, we are implicitly identifying with $\Lambda$ the scale at which the
degrees of freedom generating the operator live and with $h_{ij}$ the
combination of dimensionless couplings, loop factors, etc. entering the
determination of the operator.

Once the electroweak symmetry is broken and the neutral Higgs component
acquires a vev, a neutrino mass term is generated,

$m^{\nu}_{ij}=h_{ij}v\times\frac{v}{\Lambda}.$ (42)

We therefore see that, unlike charged fermions, neutrino masses turn out in
this context to be proportional to two powers of the electroweak symmetry
breaking scale, and therefore to be suppressed by a factor $v/\Lambda$
compared to the charged fermion masses, as a consequence of their
(unspecified) origin at the high scale $\Lambda$. The smallness of neutrino
masses is then understood in terms of the heaviness of the scale $\Lambda$ at
which they originate (and at which lepton number if broken). It is also
possible to invert the relation in eq. (42) to obtain an estimate of the scale
$\Lambda$ in terms of the measured value of the neutrino masses:

$\Lambda\sim 0.5\times
10^{15}\,\mathrm{GeV}h\left(\frac{0.05\,\mathrm{eV}}{m_{\nu}}\right).$ (43)

As the coupling $h$ cannot be much larger than 1, $\Lambda$ has to be of the
order or smaller than $10^{15}\,\mathrm{GeV}$. Still, $\Lambda$ might be not
too far from the GUT scale. Neutrino masses open in this context an indirect
window on scales that could never be probed directly.

Let us summarize the results of the discussion so far. The smallness of
neutrino masses can be economically understood in a model-independent way in
terms of the heaviness of the scale at which lepton number is violated. What
makes them special is the fact that they are the only fermions in the SM for
which a mass does not arise (after EWSB) from a renormalizable interaction
with the Higgs field. They turn out to be Majorana.

Such an understanding is very appealing, but is based on the fact that, unlike
the other charged fermions, neutrinos were not given a right-handed component.
In the presence of a right-handed component $\nu_{R}$, it would be possible to
write a renormalizable neutrino Yukawa interaction with the Higgs,
$\lambda^{N}_{ij}\overline{\nu_{iR}}L_{j}\,H$, providing neutrino masses
proportional to the electroweak scale, just as for all the other fermions. The
real question might then appear to be: why neutrinos should not have a right-
handed components, as all the other fermions? What makes them special from
this point of view?

The answer is simple. In order to make the neutrino Yukawa interaction gauge
invariant, the right-handed neutrinos should be neutral under all SM
interactions222To be precise, they could also be in a triplet of SU(2)L.. Then
they would be the only fermions for which an explicit mass term in the form in
eq. (6) would be allowed:

$\frac{M_{ij}}{2}\overline{\nu_{iR}}\,\overline{\nu_{jR}}.$ (44)

Unlike all other fermions, their mass would not be bound to vanish in the
limit in which the electroweak symmetry is unbroken, would not be bound to be
proportional to powers of the electroweak scale. Therefore, right-handed
neutrinos would be the only fermions for which a mass much larger than $v$
would be allowed. In which case, according to the effective theory theorem
mentioned above, their effect at the electroweak scale and below, can be
described in terms of the effective interaction in eq. (41). Indeed,
neglecting the momentum in the right-handed neutrino propagator in the “see-
saw” [85, 86, 87, 88, 89] diagram in Fig. 11a, one obtains the effective
interaction in eq. (41), with

$\frac{h}{\Lambda}=-\lambda_{N}^{T}\frac{1}{M}\lambda_{N}\quad\Rightarrow\quad
m_{\nu}=-m_{\text{D}}^{T}\frac{1}{M}m_{\text{D}},\quad\text{where}\quad
m_{D}=\lambda_{N}v.$ (45)

This is the celebrated see-saw formula. In this context, it turns out to be
just an example, probably the simplest, of heavy physics generating the
operator in eq. (41).

Figure 11: The see-saw diagram.

It turns out that there are only three possible types of heavy degrees of
freedom that can be exchanged at the tree level to generate the operator in
eq. (41): the exchange of SM-singlet fermions (type I see-saw, or just see-
saw), of a hypercharge -1 SU(2)L scalar triplet [90, 91, 92, 93] (type II see-
saw), or of a zero hypercharge SU(2)L fermion triplet [94] (type III see-saw).

###  0.5.3 $\Lambda\lesssim v$

While the $M\gg v$ option is very appealing, as it provides a solid,
economical, compelling understanding of the smallness of neutrino masses, only
based on the hypothesis that the new ingredients needed to account for
neutrino masses are heavier than the electroweak scale, the possibility that
such new ingredients are lighter than $v$ cannot be excluded. In this case,
the general effective description we used in the previous case does not hold
and each case should be considered separately.

A paradigmatic example is provided by Dirac neutrinos. As discussed above,
such neutrinos have a right-handed component, but their mass terms, which
could in principle be as heavy as the Planck scale, is assumed to be much
smaller than the neutrino masses themselves. The latter then arise from the
Yukawa interaction $\lambda^{N}_{ij}\overline{\nu_{iR}}L_{j}\,H$, as for the
other fermions, and turn out to be in the Dirac form (as in eq. (8) with
$m^{L}=m^{R}=0$), with $m_{N}=\lambda_{N}v$.

Such a possibility requires the Majorana mass term for the right-handed
neutrinos in eq. (44) to be smaller than the Planck scale by almost 30 orders
of magnitude and the $\lambda_{N}$ entries to be all smaller than about
$10^{-11}$. Lepton number conservation can force $M_{ij}=0$. We would then
only have to cope with very small Yukawas or to account for their smallness
with appropriate mechanisms [95, 96, 97]. In any case, it is fair to say that
additional structure must be added to explain what the simplest option,
$\Lambda\gg v$, gives for free.

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