11institutetext: Universitat de València – CSIC, València, Spain

# Neutrino Physics

G. Barenboim

###### Abstract

The Standard Model has been incredibly successful in predicting the outcome of
almost all the experiments done up so far. In it, neutrinos are mass-less.
However, in recent years we have accumulated evidence pointing to tiny masses
for the neutrinos (as compared to the charged leptons). These masses allow
neutrinos to change their flavour and oscillate. In these lectures I review
the properties of neutrinos in and beyond the Standard Model.

##  0.1 Introduction

The last decade witnessed a revolution in neutrino physics. It has been
observed that neutrinos have nonzero masses, and that leptons mix. This fact
was proven by the observation that neutrinos can change from one type, or
“flavour”, to another. Almost all the knowledge we have gathered about
neutrinos, is only fifteen years old. But before diving into the recent "news"
about neutrinos, lets find out how neutrinos were born.

The ’20s witnessed the assassination of many sacred cows, and physics was no
exception, one of physic’s most holly principles, the conservation of energy,
appeared not to hold within the subatomic world. For some radioactive nuclei,
it seemed that part of its energy just disappear, leaving no footprint of its
existence.

In 1920, in a letter to a conference, Pauli wrote, "Dear radioactive Ladies
and Gentlemen, … as a desperate remedy to save the principle of energy
conservation in beta decay, … I propose the idea of a neutral particle of spin
half". Pauli postulated that the energy loss was taken off by a new particle,
whose properties were such that it would not yet be seen: it had no electric
charge and rarely interacted with matter at all. This way, the neutrino was
born into the world of particle physics.

Soon afterwards, Fermi wrote the four-Fermi Hamiltonian for beta decay using
the neutrino, electron, neutron and proton. A new field came to existence: the
field of weak interactions. And two decades after Pauli’s letter, Cowan and
Reines finally observed anti-neutrinos emitted by a nuclear reactor. As more
and more particles were discovered in the following years and observed to
participate in weak processes, weak interactions got legitimacy as a new force
of nature and the neutrino became a key ingredient of this interactions.

Further experiments over the course of the next 30 years showed us that there
were three kinds, or “flavours” of neutrinos (electron neutrinos ($\nu_{e}$),
muon neutrinos ($\nu_{\mu}$) and tau neutrinos ($\nu_{\tau}$)) and that, as
far as we could tell, had no mass at all. The neutrino saga might have stop
there, but new experiments in solar physics taught us that the neutrino story
was just beginning….

Within the Standard Model, neutrinos have zero mass and therefore interact
diagonally in flavour space,

$\displaystyle W^{+}\longrightarrow
e^{+}\;+\;\nu_{e}\;\;\;\;\;\;;\;\;\;\;\;\;Z\longrightarrow\nu_{e}\;+\;\bar{\nu}_{e}$
$\displaystyle
W^{+}\longrightarrow\mu^{+}\;+\;\nu_{\mu}\;\;\;\;\;\;;\;\;\;\;\;\;Z\longrightarrow\nu_{\mu}\;+\;\bar{\nu}_{\mu}$
(1) $\displaystyle
W^{+}\longrightarrow\tau^{+}\;+\;\nu_{\tau}\;\;\;\;\;\;;\;\;\;\;\;\;Z\longrightarrow\nu_{\tau}\;+\;\bar{\nu}_{\tau}$

Since they are mass-less, they move at the speed of light and therefore their
flavour remains the same from production up to detection. It is obvious then,
that at least as flavour is concerned, zero mas neutrinos are almost not
interesting as compared to quarks.

On the other hand, neutrinos masses different from zero, mean that there are
three neutrino mass eigenstates $\nu_{i},i=1,2,\ldots$, each with a mass
$m_{i}$. The meaning of leptonic mixing can be understood by analysing the
leptonic decays, $W^{+}\longrightarrow\nu_{i}+\overline{\ell_{\alpha}}$ of the
charged $W$ boson. Where, $\alpha=e,\mu$, or $\tau$, and $\ell_{e}$ represents
the electron, $\ell_{\mu}$ the muon, or $\ell_{\tau}$ the tau. We refer to
particle $\ell_{\alpha}$ as the charged lepton of flavour $\alpha$. Mixing
essentially means that when the $W^{+}$ decays to a given flavour of charged
lepton $\overline{\ell_{\alpha}}$, the neutrino that comes along is not always
the same mass eigenstate $\nu_{i}$. Any of the different $\nu_{i}$ can show
up. The amplitude for the decay of a $W^{+}$ to a specific combination
$\overline{\ell_{\alpha}}+\nu_{i}$ is designated by $U^{*}_{\alpha i}$. The
neutrino that is emitted in $W^{+}$ decay along with the given charged lepton
$\overline{\ell_{\alpha}}$ is then

$|\nu_{\alpha}>=\sum_{i}U_{\alpha i}^{*}\;|\nu_{i}>\leavevmode\nobreak\
\leavevmode\nobreak\ .$ (2)

This particular combination of mass eigenstates is the neutrino of flavour
$\alpha$.

The quantities $U_{\alpha i}$ can be collected in a unitary matrix (analogue
to the CKM matrix of the quark sector) known as the leptonic mixing matrix
[1]. The unitarity of $U$ guarantees that every time a neutrino of flavour
$\alpha$ interacts in a detector and produces a charged lepton, such a charged
lepton will be always $\ell_{\alpha}$, the charged lepton with flavour
$\alpha$. That is, a $\nu_{e}$ indefectibly creates an $e$, a $\nu_{\mu}$ a
$\mu$, and a $\nu_{\tau}$ a $\tau$.

The relation (2), describing a neutrino of definite flavour as a linear
combination of mass eigenstates, may be inverted to describe each mass
eigenstate $\nu_{i}$ as a linear combination of flavours:

$|\nu_{i}>=\sum_{\alpha}U_{\alpha i}\;|\nu_{\alpha}>\leavevmode\nobreak\
\leavevmode\nobreak\ .$ (3)

The $\alpha$-flavour "content" (or fraction) of $\nu_{i}$ is clearly
$|U_{\alpha i}|^{2}$. When a $\nu_{i}$ interacts and generates a charged
lepton, this $\alpha$-flavour fraction becomes the probability that the
emerging charged lepton be of flavour $\alpha$.

##  0.2 Neutrino oscillations in vacuum

A standard neutrino flavour transition, or "oscillation", can be understood as
follows. A neutrino is produced by a source together with a charged lepton
$\overline{\ell_{\alpha}}$ of flavour $\alpha$. Therefore, at the production
point, the neutrino is a $\nu_{\alpha}$. Then, after birth, the neutrino
travels a distance $L$ until it is detected. There, it is where it reaches a
target with which it interacts and produces another charged lepton
$\ell_{\beta}$ of flavour $\beta$. Thus, at the interaction point, the
neutrino is a $\nu_{\beta}$. If $\beta\neq\alpha$ (for example, if
$\ell_{\alpha}$ is a $\mu$ but $\ell_{\beta}$ is a $\tau$), then, during its
trip from the source to the detection point, the neutrino has transitioned
from a $\nu_{\alpha}$ into a $\nu_{\beta}$.

This morphing of neutrino flavour, $\nu_{\alpha}\longrightarrow\nu_{\beta}$,
is a text-book example of a quantum-mechanical effect.

Because, as described by Eq. (2), a $\nu_{\alpha}$ is really a coherent
superposition of mass eigenstates $\nu_{i}$, the neutrino that propagates
since it is created until it interacts, can be any one of the $\nu_{i}$’s,
therefore we must add the contributions of all the different $\nu_{i}$
coherently. Then, the transition amplitude,
Amp($\nu_{\alpha}\longrightarrow\nu_{\beta}$) contains a share of each
$\nu_{i}$ and it is a product of three factors. The first one is the amplitude
for the neutrino born at the production point in combination with an
$\overline{\ell_{\alpha}}$ to be, specifically, a $\nu_{i}$. As we have
mentioned already , this amplitude is given by $U_{\alpha i}^{*}$. The second
factor is the amplitude for the $\nu_{i}$ created by the source to propagate
until it reaches the detector. We will call this factor Prop($\nu_{i}$) and
will find out its value later. The third factor is the amplitude for the
charged lepton produced by the interaction of the $\nu_{i}$ with the detector
to be, specifically, an $\ell_{\beta}$. As the Hamiltonian that describes the
interaction of neutrinos, charged leptons and $W$ bosons is hermitian , it
ensues that if Amp($W\longrightarrow\overline{\ell_{\alpha}}\nu_{i})=U_{\alpha
i}^{*}$, then Amp$(\nu_{i}\longrightarrow\ell_{\beta}W)=U_{\beta i}$.
Therefore, the third and last factor in the $\nu_{i}$ contribution is
$U_{\beta i}$, and

$\mathrm{Amp}(\nu_{\alpha}\longrightarrow\nu_{\beta})=\sum_{i}U_{\alpha
i}^{*}\;\;\mathrm{Prop}(\nu_{i})\;\;U_{\beta i}\leavevmode\nobreak\
\leavevmode\nobreak\ .$ (4)

Figure 1: Neutrino flavour change (oscillation) in vacuum

It still remains to be established the value of Prop($\nu_{i}$). To determine
it, we’d better study the $\nu_{i}$ in its rest frame. We will label the time
in that system $\tau_{i}$. If $\nu_{i}$ does have a rest mass $m_{i}$, then in
this frame its state vector satisfies the good old Schrödinger equation

$i\frac{\partial}{\partial\tau_{i}}|\nu_{i}(\tau_{i})>\;=m_{i}|\nu_{i}(\tau_{i})>\leavevmode\nobreak\
\leavevmode\nobreak\ .$ (5)

whose solution is given clearly by

$|\nu_{i}(\tau_{i})>\;=e^{-im_{i}\tau_{i}}|\nu_{i}(0)>\leavevmode\nobreak\
\leavevmode\nobreak\ .$ (6)

Then, the amplitude for the mass eigenstate $\nu_{i}$ to propagate for a time
$\tau_{i}$, is simply the amplitude $<\nu_{i}(0)|\nu_{i}(\tau_{i})>$ for
observing the original $\nu_{i}$ $|\nu_{i}(0)>$ after some time as the
evoluted state $|\nu_{i}(\tau_{i})>$, i.e. $\exp[-im_{i}\tau_{i}]$. Thus
Prop($\nu_{i}$) is only this amplitude where we have used that the time taken
by $\nu_{i}$ to travel from the neutrino source to the detector is $\tau_{i}$,
the proper time.

Nevertheless, if we want Prop($\nu_{i}$) to be of any use to us, we must write
it first in terms of variables in the laboratory system. The natural choice is
obviously the laboratory-frame distance, $L$, that the neutrino covers between
the source and the detector, and the laboratory-frame time, $t$, that slips
away during the journey. The distance $L$ is set by the experimentalists
through the selection of the place of settlement of the source and that of the
detector. Likewise, the value of the time $t$ is selected by the
experimentalists through their election for the time at which the neutrino is
created and that when it is detected. Thus, $L$ and $t$ are chosen (hopefully
carefully) by the experiment design, and are the same for all the $\nu_{i}$
components of the beam. Different $\nu_{i}$ do travel through an identical
distance $L$, in an identical time $t$.

We still need two other laboratory-frame variables, they are the laboratory-
frame energy $E_{i}$ and momentum $p_{i}$ of the neutrino mass eigenstate
$\nu_{i}$. With the four lab-frame variable and using Lorentz invariance, we
can obtain the phase $m_{i}\tau_{i}$ in the $\nu_{i}$ propagator
Prop($\nu_{i}$) we have been looking for, which (expressed in terms of
laboratory frame variables) is given by

$m_{i}\tau_{i}=E_{i}t-p_{i}L\leavevmode\nobreak\ \leavevmode\nobreak\ .$ (7)

At this point however one may argue that, in real life, neutrino sources are
basically constant in time, and that the time $t$ that slips away since the
neutrino is produced till it dies in the detector is actually not measured.
This argument is right. In real life, an experiment averages over the time $t$
used by the neutrino to complete its journey. However, lets consider that two
constituents of the neutrino beam, the first one with energy $E_{1}$ and the
second one with energy $E_{2}$ (both measured in the lab frame), contribute
coherently to the neutrino signal produced in the detector. Now, if we call
$t$ to the the time used by the neutrino to cover the distance separating the
production and detection points, then by the time the constituent whose energy
is $E_{j}\;(j=1,2)$ arrives to the detector, it has raised a phase factor
$\exp[-iE_{j}t]$. Therefore, we will have an interference between the $E_{1}$
and $E_{2}$ beam participants that will include a phase factor
$\exp[-i(E_{1}-E_{2})t]$. When averaged over the non-observed travel time $t$,
this factor goes away, except when $E_{2}=E_{1}$. Therefore, only components
of the neutrino beam that share the same energy contribute coherently to the
neutrino oscillation signal [2, 3]. Specifically, only the different mass
eigenstate constituents of a beam that have the same energy contribute
coherently to the oscillation signal.

A mass eigenstate $\nu_{i}$, with mass $m_{i}$, and energy $E$, has a momentum
$p_{i}$ given by

$p_{i}=\sqrt{E^{2}-m_{i}^{2}}\cong E-\frac{m_{i}^{2}}{2E}\leavevmode\nobreak\
\leavevmode\nobreak\ .$ (8)

Where, we have used that as the masses of the neutrinos are miserably small,
$m_{i}^{2}\ll E^{2}$ for any energy $E$ attainable at a realistic experiment.
From Eqs. (7) and (8), we see that at energy $E$ the phase $m_{i}\tau_{i}$
appearing in Prop($\nu_{i}$) takes the value

$m_{i}\tau_{i}\cong E(t-L)+\frac{m_{i}^{2}}{2E}L\leavevmode\nobreak\
\leavevmode\nobreak\ .$ (9)

As the phase $E(t-L)$ appears in all the interfering terms it will eventually
disappear when calculating the transition amplitude. Thus, we can get rid of
it already now and use

$\mathrm{Prop}(\nu_{i})=\exp[-im_{i}^{2}\frac{L}{2E}]\leavevmode\nobreak\
\leavevmode\nobreak\ .$ (10)

Applying this result, we can obtain from Eq. (4) that the amplitude for a
neutrino born as a $\nu_{\alpha}$ to be detected as a $\nu_{\beta}$ after
covering a distance $L$ through vacuum with energy $E$ yields

$\mathrm{Amp}(\nu_{\alpha}\longrightarrow\nu_{\beta})=\sum_{i}U_{\alpha
i}^{*}\,e^{-im_{i}^{2}\frac{L}{2E}}U_{\beta i}\leavevmode\nobreak\
\leavevmode\nobreak\ .$ (11)

The expression above is valid for an arbitrary number of neutrino flavours and
mass eigenstates. The probability P($\nu_{\alpha}\longrightarrow\nu_{\beta}$)
for $\nu_{\alpha}\longrightarrow\nu_{\beta}$ can be found by squaring it,
giving

$\displaystyle\mathrm{P}(\nu_{\alpha}\longrightarrow\nu_{\beta})$
$\displaystyle=$
$\displaystyle|\mathrm{Amp}(\nu_{\alpha}\longrightarrow\nu_{\beta})|^{2}$ (12)
$\displaystyle=$
$\displaystyle\delta_{\alpha\beta}-4\sum_{i>j}\Re(U^{*}_{\alpha i}U_{\beta
i}U_{\alpha j}U^{*}_{\beta j})\sin^{2}\left(\Delta
m^{2}_{ij}\frac{L}{4E}\right)$
$\displaystyle\phantom{\delta_{\alpha\beta}}+2\sum_{i>j}\Im(U^{*}_{\alpha
i}U_{\beta i}U_{\alpha j}U^{*}_{\beta j})\sin\,\left(\Delta
m^{2}_{ij}\frac{L}{2E}\right)\leavevmode\nobreak\ \leavevmode\nobreak\ ,$

with

$\Delta m_{ij}^{2}\equiv m_{i}^{2}-m_{j}^{2}\leavevmode\nobreak\
\leavevmode\nobreak\ .$ (13)

In order to get Eq. (12) we have used that the mixing matrix $U$ is unitary.

The oscillation probability P($\nu_{\alpha}\longrightarrow\nu_{\beta}$) we
have just obtained corresponds to that of a neutrino, and not to an
antineutrino, as we have used that the oscillating neutrino was produced along
with a charged antilepton $\bar{\ell}$, and gives birth to a charged lepton
$\ell$ once it reaches the detector. The corresponding probability
P($\overline{\nu_{\alpha}}\longrightarrow\overline{\nu_{\beta}}$) for an
antineutrino oscillation can be obtained from
P($\nu_{\alpha}\longrightarrow\nu_{\beta}$) taking advantage of the fact that
the two transitions
$\overline{\nu_{\alpha}}\longrightarrow\overline{\nu_{\beta}}$ and
$\nu_{\beta}\longrightarrow\nu_{\alpha}$ are CPT conjugated processes. Thus,
assuming that neutrino interactions respect CPT[4],

$\mathrm{P}(\overline{\nu_{\alpha}}\longrightarrow\overline{\nu_{\beta}})=\mathrm{P}(\nu_{\beta}\longrightarrow\nu_{\alpha})\leavevmode\nobreak\
\leavevmode\nobreak\ .$ (14)

Then, from Eq. (12) we obtain that

$\mathrm{P}(\nu_{\beta}\longrightarrow\nu_{\alpha};\>U)=\mathrm{P}(\nu_{\alpha}\longrightarrow\nu_{\beta};\>U^{*})\leavevmode\nobreak\
\leavevmode\nobreak\ .$ (15)

Therefore, if CPT is a good symmetry (with respect to neutrino interactions),
Eq. (12) tells us that

$\displaystyle\mathrm{P}(\shortstack{{\tiny(\rule[0.86108pt]{5.0pt}{0.28453pt})}
\\\ [-.7ex]
$\nu_{\alpha}$}\longrightarrow\shortstack{{\tiny(\rule[0.86108pt]{5.0pt}{0.28453pt})}
\\\ [-.7ex] $\nu_{\beta}$})$ $\displaystyle=$
$\displaystyle\delta_{\alpha\beta}-4\sum_{i>j}\Re(U^{*}_{\alpha i}U_{\beta
i}U_{\alpha j}U^{*}_{\beta j})\sin^{2}\left(\Delta
m^{2}_{ij}\frac{L}{4E}\right)$ (16)
$\displaystyle\phantom{\delta_{\alpha\beta}}{\rm\raise
3.44444pt\hbox{+}\kern-8.99994pt\lower
2.58334pt\hbox{{\tiny(}--{\tiny)}}}\;2\sum_{i>j}\Im(U^{*}_{\alpha i}U_{\beta
i}U_{\alpha j}U^{*}_{\beta j})\sin\,\left(\Delta
m^{2}_{ij}\frac{L}{2E}\right)\leavevmode\nobreak\ \leavevmode\nobreak\ .$

These expressions make it clear that if the mixing matrix $U$ is complex,
P($\overline{\nu_{\alpha}}\longrightarrow\overline{\nu_{\beta}}$) and
P($\nu_{\alpha}\longrightarrow\nu_{\beta}$) will not be identical, in general.
As $\overline{\nu_{\alpha}}\longrightarrow\overline{\nu_{\beta}}$ and
$\nu_{\alpha}\longrightarrow\nu_{\beta}$ are CP conjugated processes,
P$(\overline{\nu_{\alpha}}\longrightarrow\overline{\nu_{\beta}})\neq\mathrm{P}(\nu_{\alpha}\longrightarrow\nu_{\beta})$
would be an evidence of CP violation in neutrino oscillations. So far, CP
violation has been observed only in the quark sector, so its measurement in
neutrino oscillations would be quite exciting.

So far, we have been working in natural units, if we return now the $\hbar$’s
and $c$ factor (we have happily left out) into the oscillation probability we
find that

$\sin^{2}\left(\Delta
m^{2}_{ij}\frac{L}{4E}\right)\;\;\longrightarrow\;\;\sin^{2}\left(\Delta
m^{2}_{ij}c^{4}\frac{L}{4\hbar cE}\right)$ (17)

Having done that, it is easy and instructive to explore the semi-classical
limit, $\hbar\longrightarrow 0$. In this limit the oscillation length goes to
zero (the oscillation phase goes to infinity) and the oscillations are
averaged out. The same happens if we let the mass difference $\Delta m^{2}$
become large. This is exactly what happens in the quark sector (and the reason
why we never study quark oscillations despite knowing that mass eigenstates do
not coincide with flavour eigenstates).

In terms of real life units (which are not "natural" units), the oscillation
phase is given by

$\Delta m^{2}_{ij}\frac{L}{4E}=1.27\,\Delta
m^{2}_{ij}(\mathrm{eV}^{2})\frac{L\,(\mathrm{km})}{E\,(\mathrm{GeV})}\leavevmode\nobreak\
\leavevmode\nobreak\ .$ (18)

then, since $\sin^{2}[1.27\,\Delta
m^{2}_{ij}(\mathrm{eV}^{2})L\,(\mathrm{km})/E\,(\mathrm{GeV})]$ can be
experimentally observed only if its argument is of order unity or larger, an
experimental set-up with a baseline $L$ (km) and an energy $E$ (GeV) is
sensitive to neutrino mass squared differences $\Delta
m^{2}_{ij}(\mathrm{eV}^{2})$ larger that or equal to
$\sim[L\,(\mathrm{km})/E\,(\mathrm{GeV}]^{-1}$. For example, an experiment
with $L\sim 10^{4}$ km, roughly the size of Earth’s diameter, and $E\sim 1$
GeV is sensitive to $\Delta m^{2}_{ij}$ down to $\sim\\!10^{-4}$ eV2. This
fact makes it clear that neutrino oscillation experiments can test super tiny
neutrino masses. It does so by exhibiting quantum mechanical interferences
between amplitudes whose relative phases are proportional to these super tiny
neutrino mass squared differences, which can be transformed into sizeable
effects by choosing an $L/E$ large enough.

But let’s keep analysing the oscillation probability and see whether we can
learn more about neutrino oscillations by studying its expression.

It is clear from
$\mathrm{P}(\shortstack{{\tiny(\rule[0.86108pt]{5.0pt}{0.28453pt})} \\\
[-.7ex]
$\nu_{\alpha}$}\longrightarrow\shortstack{{\tiny(\rule[0.86108pt]{5.0pt}{0.28453pt})}
\\\ [-.7ex] $\nu_{\beta}$})$ that if neutrinos have zero mass, in such a way
that all $\Delta m^{2}_{ij}=0$, then,
$\mathrm{P}(\shortstack{{\tiny(\rule[0.86108pt]{5.0pt}{0.28453pt})} \\\
[-.7ex]
$\nu_{\alpha}$}\longrightarrow\shortstack{{\tiny(\rule[0.86108pt]{5.0pt}{0.28453pt})}
\\\ [-.7ex] $\nu_{\beta}$})=\delta_{\alpha\beta}$. Therefore, the experimental
observation that neutrinos can morph from one flavour to a different one
indicates that neutrinos are not only massive but also that their masses are
not degenerate. Actually, it was precisely this evidence the one that led to
the conclusion that neutrinos are massive.

However, every neutrino oscillation seen so far has involved at some point
neutrinos that travel through matter. But the expression we derived is valid
only for flavour change in vacuum, and does not take into account any
interaction between the neutrinos and the matter traversed between their
source and their detector. Thus, one might ask whether flavour-changing
interactions between neutrinos and matter are indeed responsible of the
observed flavour changes, and not neutrino masses. Regarding this question,
two points can be made. First, although it is true that the Standard Model of
elementary particle physics contains only mass-less neutrinos, it provides an
amazingly well corroborated description of neutrino interactions, and this
description clearly establishes that neutrino interactions with matter do not
change flavour. Second, for at least some of the observed flavour changes,
matter effects are expected to be miserably small, and there is solid evidence
that in these cases, the flavour transition probability depends on $L$ and $E$
through the combination $L/E$, as anticipated by the oscillation hypothesis.
Modulo a constant, $L/E$ is precisely the proper time that goes by in the rest
frame of the neutrino as it covers a distance $L$ possessing an energy $E$.
Thus, these flavour transitions behave as if they were a progression of the
neutrino itself over time, rather than a result of interaction with matter.

Now, suppose the leptonic mixings were trivial. This would mean that in the
decay $W^{+}\longrightarrow\overline{\ell_{\alpha}}+\nu_{i}$, which as we
established has an amplitude $U_{\alpha i}^{*}$, the emerging charged
antilepton $\overline{\ell_{\alpha}}$ of flavour $\alpha$ comes along always
with the same neutrino mass eigenstate $\nu_{i}$. That is, if $U_{\alpha
i}^{*}\neq 0$, then $U_{\alpha j}$ becomes zero for all $j\neq i$. Therefore,
from Eq. (16) it is clear that,
$\mathrm{P}(\shortstack{{\tiny(\rule[0.86108pt]{5.0pt}{0.28453pt})} \\\
[-.7ex]
$\nu_{\alpha}$}\longrightarrow\shortstack{{\tiny(\rule[0.86108pt]{5.0pt}{0.28453pt})}
\\\ [-.7ex] $\nu_{\beta}$})=\delta_{\alpha\beta}$. Thus, the observation that
neutrinos can change flavour indicates mixing.

Then, there are basically two ways to detect neutrino flavour change. The
first one is to observe, in a beam of neutrinos which are all created of the
same flavour, say $\alpha$, some appearance of neutrinos of a new flavour
$\beta$ that is different from the flavour $\alpha$ we started with. This is
what is called an appearance experiment. The second way is to start with a
beam of identical $\nu_{\alpha}$s, whose flux is known, and observe that this
known $\nu_{\alpha}$ flux is depleted. This is called a disappearance
experiment.

As Eq. (16) shows, the transition probability in vacuum does not only depend
on $L/E$ but also oscillates with it. It is because of this fact that neutrino
flavour transitions are named “neutrino oscillations”. Now notice also that
neutrino transition probabilities depend only on neutrino squared-mass
splittings, and not on the individual squared neutrino masses themselves.
Thus, oscillation experiments can only measure the neutrino squared-mass
spectral pattern, but not its absolute scale, i.e. the distance above zero the
entire pattern lies.

It is clear that neutrino transitions cannot modify the total flux in a
neutrino beam, but simply alter its distribution between the different
flavours. Actually, from Eq. (16) and the unitarity of the $U$ matrix, it
follows that

$\sum_{\beta}\mathrm{P}(\shortstack{{\tiny(\rule[0.86108pt]{5.0pt}{0.28453pt})}
\\\ [-.7ex]
$\nu_{\alpha}$}\longrightarrow\shortstack{{\tiny(\rule[0.86108pt]{5.0pt}{0.28453pt})}
\\\ [-.7ex] $\nu_{\beta}$})=1\leavevmode\nobreak\ \leavevmode\nobreak\ ,$ (19)

where the sum runs over all flavours $\beta$, including the original flavour
$\alpha$. Eq. (19) makes it transparent that the probability that a neutrino
morphs its flavour, added to the probability that it does not do so, is one.
Ergo, flavour transitions do not change the total flux. Nevertheless, some of
the flavours $\beta\neq\alpha$ into which a neutrino can oscillate into may be
sterile flavours; that is, flavours that do not take part in weak interactions
and therefore escape detection. If any of the original (active) neutrino flux
turns into sterile, then an experiment measuring the total active neutrino
flux—that is, the sum of the $\nu_{e},\;\nu_{\mu}$, and $\nu_{\tau}$
fluxes—will find it to be less than the original flux. In the experiments
performed up today, no flux was ever missed.

In the literature, description of neutrino oscillation normally assume that
the different mass eigenstates $\nu_{i}$ that contribute coherently to a beam
share the same momentum, rather than the same energy as we have argued they
must have. While the supposition of equal momentum is technically wrong, it is
an inoffensive mistake, since, as can easily be shown [5], it conveys to the
same oscillation probabilities as we have found.

A relevant and interesting case of the (not that simple) formula for
P$(\overline{\nu_{\alpha}}\longrightarrow\overline{\nu_{\beta}})$ is the case
where only two flavours participate in the oscillation. The only-two-neutrino
scenario is a rather rigorous description of a vast number of experiments.
Lets assume then, that only two mass eigenstates, which we will name $\nu_{1}$
and $\nu_{2}$, and two corresponding flavour states, which we will name
$\nu_{\mu}$ and $\nu_{\tau}$, are relevant. There is then only one squared-
mass splitting, $m^{2}_{2}-m^{2}_{1}\equiv\Delta m^{2}$. Even more, neglecting
phase factors that can be proven to have no effect on oscillation, the mixing
matrix $U$ takes the simple form

$\left(\begin{array}[]{c}\nu_{\mu}\\\
\nu_{\tau}\end{array}\right)=\left(\begin{array}[]{cc}\phantom{-}\cos\theta&\sin\theta\\\
-\sin\theta&\cos\theta\end{array}\right)\left(\begin{array}[]{c}\nu_{1}\\\
\nu_{2}\end{array}\right)$ (20)

The $U$ of Eq. (20) is just a 2$\times$2 rotation matrix, and the rotation
angle $\theta$ within it is referred to as the mixing angle. Inserting the $U$
of Eq. (20) and the unique $\Delta m^{2}$ into the general expression for
P$(\overline{\nu_{\alpha}}\longrightarrow\overline{\nu_{\beta}})$, Eq. (16),
we immediately find that, for $\beta\neq\alpha$, when only two neutrinos are
relevant,

$\mathrm{P}(\overline{\nu_{\alpha}}\longrightarrow\overline{\nu_{\beta}})=\sin^{2}2\theta\sin^{2}\left(\frac{\Delta
m^{2}\;L}{4E}\right)\leavevmode\nobreak\ \leavevmode\nobreak\ .$ (21)

Moreover, the survival probability, i.e. the probability that the neutrino
remains with the same flavour its was created with is, as usual, unity minus
the probability that it changes flavour.

##  0.3 Neutrino oscillations in matter

When we create a beam of neutrinos on earth through an accelerator and send it
up to thousand kilometres away to a meet detector, the beam does not move
through vacuum, but through matter, earth matter. The beam of neutrinos then
scatters (coherently forward) from particles it meets along the way. Such a
scattering can have a large effect on the transition probabilities. We will
assume that neutrino interactions with matter are flavour conserving, as
described by the Standard Model. Then a neutrino in matter have two
possibilities to enjoy coherent forward scattering from matter particles .
First, if it is an electron neutrino, $\nu_{e}$—and only in this case—can
exchange a $W$ boson with an electron. Neutrino-electron coherent forward
scattering via $W$ exchange opens up an extra interaction potential energy
$V_{W}$ suffered exclusively by electron neutrinos. Obviously, this additional
weak interaction energy has to be proportional to $G_{F}$, the Fermi coupling
constant. In addition, the interaction energy coming from $\nu_{e}-e$
scattering grows with $N_{e}$, the number of electrons per unit volume. From
the Standard Model, we find that

$V_{W}=+\sqrt{2}\,G_{F}\,N_{e}\leavevmode\nobreak\ \leavevmode\nobreak\ ,$
(22)

clearly, this interaction energy affects also antineutrinos (in a opposite way
though), it changes sign if we replace the $\nu_{e}$ by $\overline{\nu_{e}}$.

The second interaction corresponds to the case where a neutrino in matter
exchanges a $Z$ boson with an matter electron, proton, or neutron. The
Standard Model teaches us that weak interactions are flavour blind. Every
flavour of neutrino enjoys them, and the amplitude for this $Z$ exchange is
always the same. It also teaches us that, at zero momentum transfer, the $Z$
couplings to electrons and protons have equal strength and opposite sign.
Therefore, counting on the fact that the matter through which our neutrino
moves is electrically neutral (it contains equal number of electrons and
protons), the electron and proton contribution to coherent forward neutrino
scattering through $Z$ exchange will add up to zero. Then, the effect of the
$Z$ exchange contribution to the interaction potential energy $V_{Z}$ will be
equal to all flavors and will depends exclusively on $N_{n}$, the number
density of neutrons. From the Standard Model, we find that

$V_{Z}=-\frac{\sqrt{2}}{2}\,G_{F}\,N_{n}\leavevmode\nobreak\
\leavevmode\nobreak\ ,$ (23)

as was the case before, for $V_{W}$, this contribution will flip sign if we
replace the neutrinos by anti-neutrinos.

But we already learnt that Standard Model interactions do not change neutrino
flavour. Therefore, unless non-Standard-Model flavour changing interactions
play a role, neutrino flavour transitions or neutrino oscillations points also
to neutrino mass and mixing even when neutrinos are propagating through
matter.

Neutrino propagation in matter is easy to understand when analyzed through a
time dependent Schrödinger equation in the laboratory frame

$i\frac{\partial}{\partial
t}|\nu(t)>\;=\mathcal{H}|\nu(t)>\leavevmode\nobreak\ \leavevmode\nobreak\ .$
(24)

where, $|\nu(t)\\!>$ is a multicomponent neutrino vector state, in which each
neutrino flavour corresponds to one component. In the same way, the
Hamiltonian $\mathcal{H}$ is a matrix in flavour space. To make our lives
easy, lets analyze the case where only two neutrino flavours are relevant, say
$\nu_{e}$ and $\nu_{\mu}$. Then

$|\nu(t)>\;=\left(\begin{array}[]{c}f_{e}(t)\\\
f_{\mu}(t)\end{array}\right)\leavevmode\nobreak\ \leavevmode\nobreak\ ,$ (25)

where $f_{e}(t)^{2}$ is the fraction of the neutrino that is a $\nu_{e}$ at
time $t$, and similarly for $f_{\mu}(t)$. Analogously, $\mathcal{H}$ is a
2$\times$2 matrix in $\nu_{e}-\nu_{\mu}$ space.

It will prove to be clarifying to work out the two flavour case in vacuum
first, and add matter effects afterwards. Using Eq. (2) for $|\nu_{\alpha}>$
as a linear combination of mass eigenstates, we can see that the
$\nu_{\alpha}-\nu_{\beta}$ matrix element of the Hamiltonian in vacuum,
$\mathcal{H}_{\mathrm{Vac}}$, can be written as

$\displaystyle<\nu_{\alpha}|\mathcal{H}_{\mathrm{Vac}}|\nu_{\beta}>$
$\displaystyle=$ $\displaystyle<\sum_{i}U^{*}_{\alpha
i}\nu_{i}|\mathcal{H}_{\mathrm{Vac}}|\sum_{j}U^{*}_{\beta j}\nu_{j}>$ (26)
$\displaystyle=$ $\displaystyle\sum_{j}U_{\alpha j}U^{*}_{\beta
j}\sqrt{p^{2}+m_{j}^{2}}\leavevmode\nobreak\ \leavevmode\nobreak\ .$

where we are supposing that neutrino belongs to a beam where all its mass
components (the mass eigenstates) share the same definite momentum $p$. (As we
have already mentioned, this supposition is technically wrong, however it
leads anyway to the right oscillation probability.) In the second line of Eq.
(26), we have used that

$\displaystyle\mathcal{H}_{\mathrm{Vac}}|\nu_{j}>=E_{j}|\nu_{j}>$ (27)

with $E_{j}=\sqrt{p^{2}+m_{j}^{2}}$ the energy of the mass eigenstate
$\nu_{j}$ with momentum $p$, and the fact that the mass eigenstates of the
Hermitian Hamiltonian $\mathcal{H}_{\mathrm{Vac}}$ constitute a basis and
therefore are orthogonal.

As we have already mentioned, neutrino oscillations are the archetype quantum
interference phenomenon, where only the relative phases of the interfering
states play a role. Therefore, only the relative energies of these states,
which set their relative phases, are relevant. As a consequence, if it proves
to be convenient, we can feel free to happily remove from the Hamiltonian
$\mathcal{H}$ any contribution proportional to the identity matrix $I$. As we
have said, this substraction will leave unaffected the differences between the
eigenvalues of $\mathcal{H}$, and therefore will leave unaffected the
prediction of $\mathcal{H}$ for flavour transitions.

It goes without saying that as in this case only two neutrinos are relevant,
there are only two mass eigenstates, $\nu_{1}$ and $\nu_{2}$, and only one
mass splitting $\Delta m^{2}\equiv m^{2}_{2}-m^{2}_{1}$, and therefore the $U$
matrix is given by Eq. (20). Inserting this matrix into Eq. (26), and applying
the high momentum approximation $(p^{2}+m^{2}_{j})^{1/2}\cong p+m^{2}_{j}/2p$,
and removing from $\mathcal{H}_{\mathrm{Vac}}$ a term proportional to the the
identity matrix (a removal we know is going to be harmless), we get

$\mathcal{H}_{\mathrm{Vac}}=\frac{\Delta
m^{2}}{4E}\left(\begin{array}[]{cc}-\cos 2\theta&\sin 2\theta\\\
\phantom{-}\sin 2\theta&\cos 2\theta\end{array}\right)\leavevmode\nobreak\
\leavevmode\nobreak\ .$ (28)

To write this expression, we have used that $p\cong E$, where $E$ is the
average energy of the neutrino mass eigenstates in our neutrino beam of ultra
high momentum $p$.

It is not difficult to corroborate that the Hamiltonian
$\mathcal{H}_{\mathrm{Vac}}$ of Eq. (28) for the two neutrino scenario would
give an identical oscillation probability , Eq. (21), as the one we have
already obtained in a different way. For example, lets have a look at the
oscillation probability for the process $\nu_{e}\longrightarrow\nu_{\mu}$.
From Eq. (20) it is clear that in terms of the mixing angle, the electron
neutrino state composition is

$|\nu_{e}>\;=\phantom{-}|\nu_{1}>\cos\theta+|\nu_{2}>\sin\theta\leavevmode\nobreak\
\leavevmode\nobreak\ ,$ (29)

while that of the muon neutrino is given by

$|\nu_{\mu}>\;=-|\nu_{1}>\sin\theta+|\nu_{2}>\cos\theta\leavevmode\nobreak\
\leavevmode\nobreak\ .$ (30)

Now, the eigenvalues of $\mathcal{H}_{\mathrm{Vac}}$, 25, read

$\lambda_{1}=-\frac{\Delta m^{2}}{4E}\leavevmode\nobreak\ \leavevmode\nobreak\
,\;\lambda_{2}=+\frac{\Delta m^{2}}{4E}\leavevmode\nobreak\
\leavevmode\nobreak\ .$ (31)

The eigenvectors of this Hamiltonian, $|\nu_{1}>$ and $|\nu_{2}>$, can also be
written in terms of $|\nu_{e}>$ and $|\nu_{\mu}>$ by means of Eqs. (29) and
(30). Therefore, with $\mathcal{H}$, its vacuum expression,
$\mathcal{H}_{\mathrm{Vac}}$ of Eq. (28), the Schrödinger equation of Eq. (24)
tells us that if at time $t=0$ we begin from a $|\nu_{e}>$, then after some
time $t$ this $|\nu_{e}>$ will progress into the state

$|\nu(t)>\;=|\nu_{1}>e^{+i\frac{\Delta
m^{2}}{4E}t}\cos\theta+|\nu_{2}>e^{-i\frac{\Delta
m^{2}}{4E}t}\sin\theta\leavevmode\nobreak\ \leavevmode\nobreak\ .$ (32)

Thus, the probability P$(\nu_{e}\longrightarrow\nu_{\mu})$ that this evoluted
neutrino be detected as a different flavour $\nu_{\mu}$, from Eqs. (30) and
(32), is given by,

$\displaystyle\mathrm{P}(\nu_{e}\longrightarrow\nu_{\mu})$ $\displaystyle=$
$\displaystyle|<\nu_{\mu}|\nu(t)>|^{2}$ (33) $\displaystyle=$
$\displaystyle|\sin\theta\cos\theta(-e^{i\frac{\Delta
m^{2}}{4E}t}+e^{-i\frac{\Delta m^{2}}{4E}t})|^{2}$ $\displaystyle=$
$\displaystyle\sin^{2}2\theta\sin^{2}\left(\Delta
m^{2}\frac{L}{4E}\right)\leavevmode\nobreak\ \leavevmode\nobreak\ .$

Where we have substituted the time $t$ travelled by our highly relativistic
state by the distance $L$ it has covered. The flavour transition or
oscillation probability of Eq. (33), as expected, is exactly the same we have
found before, Eq. (21).

We can now move on to analyze neutrino propagation in matter. In this case,
the 2$\times$2 vacuum Hamiltonian $\mathcal{H}_{\mathrm{Vac}}$ receives two
additional contributions and becomes $\mathcal{H}_{M}$, which is given by

$\mathcal{H}_{M}=\mathcal{H}_{\mathrm{Vac}}+V_{W}\left(\begin{array}[]{cc}1&0\\\
0&0\end{array}\right)+V_{Z}\left(\begin{array}[]{cc}1&0\\\
0&1\end{array}\right)\leavevmode\nobreak\ \leavevmode\nobreak\ .$ (34)

In the new Hamiltonian, the first additional contribution corresponds to the
interaction potential due to $W$ exchange, Eq. (22). As this interaction is
suffered only by $\nu_{e}$, this contribution is different from zero only in
the upper left, $\nu_{e}-\nu_{e}$, element of $\mathcal{H}_{M}$. The second
additional contribution, the last term of Eq. (34) comes from the interaction
potential due to $Z$ exchange, Eq. (23). Since this interaction is flavour
blind, it affects every neutrino flavour in the same way, its contribution to
$\mathcal{H}_{M}$ is proportional to the identity matrix, and can be safely
neglected. Then

$\mathcal{H}_{M}=\mathcal{H}_{\mathrm{Vac}}+\frac{V_{W}}{2}+\frac{V_{W}}{2}\left(\begin{array}[]{cc}1&0\\\
0&-1\end{array}\right)\leavevmode\nobreak\ \leavevmode\nobreak\ ,$ (35)

where (for reasons that are going to become clear later) we have divided the
$W$-exchange contribution into two pieces, one a multiple of the identity
matrix (that we will disregard in the next step) and, a piece that it is not a
multiple of the identity matrix. Disregarding the first piece as promised, we
have from Eqs. (28) and (35)

$\mathcal{H}_{M}=\frac{\Delta m^{2}}{4E}\left(\begin{array}[]{cc}-(\cos
2\theta-A)&\sin 2\theta\\\ \sin 2\theta&(\cos
2\theta-A)\end{array}\right)\leavevmode\nobreak\ \leavevmode\nobreak\ ,$ (36)

where

$A\equiv\frac{V_{W}/2}{\Delta m^{2}/4E}=\frac{2\sqrt{2}G_{F}N_{e}E}{\Delta
m^{2}}\leavevmode\nobreak\ \leavevmode\nobreak\ .$ (37)

Clearly, $A$ shows the size (the importance) of the matter effects as compared
to the neutrino squared-mass splitting and signal the situations when they
become important.

Now, if we define

$\Delta m^{2}_{M}\equiv\Delta m^{2}\sqrt{\sin^{2}2\theta+(\cos
2\theta-A)^{2}}$ (38)

and

$\sin^{2}2\theta^{M}\equiv\frac{\sin^{2}2\theta}{\sin^{2}2\theta+(\cos
2\theta-A)^{2}}\leavevmode\nobreak\ \leavevmode\nobreak\ ,$ (39)

$\mathcal{H}_{M}$ can be given as

$\mathcal{H}_{M}=\frac{\Delta m^{2}_{M}}{4E}\left(\begin{array}[]{cc}-\cos
2\theta^{M}&\sin 2\theta^{M}\\\ \phantom{-}\sin 2\theta^{M}&\cos
2\theta^{M}\end{array}\right)\leavevmode\nobreak\ \leavevmode\nobreak\ .$ (40)

That is, the Hamiltonian in matter, $\mathcal{H}_{M}$, becomes formally
identical to its vacuum counterpart, $\mathcal{H}_{\mathrm{Vac}}$, Eq. (28),
except that the vacuum parameters $\Delta m^{2}$ and $\theta$ are now given by
the matter ones, $\Delta m^{2}_{M}$ and $\theta^{M}$, respectively.

Obviously, the eigenstates of $\mathcal{H}_{M}$ are not identical to their
vacuum counterparts. The splitting between the squared masses of the matter
eigenstates is not the same as the vacuum splitting $\Delta m^{2}$, and the
same happens with the mixing angle, the mixing in matter—the angle that
rotates from the $\nu_{e},\nu_{\mu}$ basis, to the mass basis—is different
from the vacuum mixing angle $\theta$. Clearly however, all the physics of
neutrino propagation in matter is controlled by the matter Hamiltonian
$\mathcal{H}_{M}$. However, according to Eq. (40), at least at the formal
level, $\mathcal{H}_{M}$ has the same functional dependence on the matter
parameters $\Delta m^{2}_{M}$ and $\theta^{M}$ in as the vacuum Hamiltonian
$\mathcal{H}_{\mathrm{Vac}}$, Eq. (28), on the vacuum ones, $\Delta m^{2}$ and
$\theta$. Therefore, $\Delta m^{2}_{M}$ corresponds to the effective splitting
between the squared masses of the eigenstates in matter, and $\theta^{M}$
corresponds to the effective mixing angle in matter.

In a typical experimental set-up where the neutrino beam is generated by an
accelerator and sent away to a detector that is, say, several hundred, or even
thousand kilometers away, it traverses through earth matter, but only
superficially , it does not get deep into the earth. The matter density met by
such a beam en voyage can be taken to be approximately constant. Therefore,
the electron density $N_{e}$ is also constant, and the same happens with the
parameter $A$, and the matter Hamiltonian $\mathcal{H}_{M}$. They all become
approximately position independent, and therefore quite analogue to the vacuum
Hamiltonian $\mathcal{H}_{\mathrm{Vac}}$, which was absolutely position
independent. Comparing Eqs. (40) and (28), we can immediately conclude that
since $\mathcal{H}_{\mathrm{Vac}}$ gives rise to the vacuum oscillation
probability P$(\nu_{e}\longrightarrow\nu_{\mu})$ of Eq. (33),
$\mathcal{H}_{M}$ must give rise to a matter oscillation probability of the
form

$\mathrm{P}_{M}(\nu_{e}\longrightarrow\nu_{\mu})=\sin^{2}2\theta^{M}\sin^{2}\left(\Delta
m^{2}_{M}\frac{L}{4E}\right)\leavevmode\nobreak\ \leavevmode\nobreak\ .$ (41)

That is, the oscillation probability in matter (formally) is the same as in
vacuum, except that the vacuum parameters $\theta$ and $\Delta m^{2}$ are
replaced by their matter counterparts.

In theory, judging simply its potential, matter effects can have very drastic
effects. From Eq. (39) for the effective mixing angle in matter, $\theta^{M}$,
we can appreciate that even when the vacuum mixing angle $\theta$ is
incredible small, say, $\sin^{2}2\theta=10^{-4}$, if we get to have
$A\cong\cos 2\theta$, then $\sin^{2}2\theta^{M}$ can be brutally enhanced as
compared to its vacuum value and can even reach its maximum possible value,
one. This brutal enhancement of a tiny mixing angle in vacuum up to a sizeable
one in matter is the “resonant” version of the Mikheyev-Smirnov-Wolfenstein
effect [6, 7, 8, 9]. In the beginning of solar neutrino experiments, people
entertained the idea that this brutal enhancement was actually taking place
inside the sun. Nonetheless, as we will see soon the solar neutrino mixing
angle is quite sizeable ($\sim 34^{\circ}$) already in vacuum [10]. Then,
although matter effects on the sun are important and they do enhance the solar
mixing angle, unfortunately they are not as drastic as we once dreamt.

##  0.4 Evidence for neutrino oscillations

###  0.4.1 Atmospheric and accelerator neutrinos

Almost fifteen year have elapsed since we were presented convincing evidence
of neutrino masses and mixings, and since then, the evidence has only grown.
SuperKamiokande (SK) was the first experiment to present compelling evidence
of $\nu_{\mu}$ disappearance in their atmospheric neutrino fluxes, see [12] .
In Fig. 2 the zenith angle (the angle subtended with the horizontal)
dependence of the multi-GeV $\nu_{\mu}$ sample is shown together with the
disappearance as a function of $L/E$ plot. These data fit amazingly well the
naive two component neutrino hypothesis with

$\Delta m^{2}_{\mbox{atm}}=2-3\times
10^{-3}{\mbox{eV}}^{2}\;\;\;{\mbox{and}}\;\;\;\sin^{2}\theta_{\mbox{atm}}=0.50\pm
0.15$ (42)

Roughly speaking SK corresponds to an $L/E$ for oscillations of 500 km/GeV and
almost maximal mixing (the mass eigenstates are nearly even admixtures of muon
and tau neutrinos). No signal of an involvement of the third flavour,
$\nu_{e}$ is found so the assumption is that atmospheric neutrino
disappearance is basically $\nu_{\mu}\longrightarrow\nu_{\tau}$.

Figure 2: Superkamiokande’s evidence for neutrino oscillations both in the
zenith angle and L/E plots

After atmospheric neutrino oscillations were established, two new experiments
were built, sending (man-made) beams of $\nu_{\mu}$ neutrinos to detectors
located at large distances: the K2K experiment [13, 14], sends neutrinos from
the KEK accelerator complex to the old SK mine, with a baseline of 120 km
while the MINOS experiment [15], sends its beam from Fermilab, near Chicago,
to the Soudan mine in Minnesota, a baseline of 735 km. Both experiments have
seen evidence for $\nu_{\mu}$ disappearance consistent with the one found by
SK. The results of both are summarised in Fig. 3.

Figure 3: Allowed regions in the $\Delta m^{2}_{\mbox{atm}}$ vs
$\sin^{2}\theta_{\mbox{atm}}$ plane for MINOS data as well as for K2K data and
two of the SK analyses. MINOS’s best fit point is at
$\sin^{2}\theta_{\mbox{atm}}=1$ and $\Delta m^{2}_{\mbox{atm}}=2.7\times
10^{-3}{\mbox{eV}}^{2}$.

###  0.4.2 Reactor and solar neutrinos

The KamLAND reactor experiment, an antineutrino disappearance experiment,
receiving neutrinos from sixteen different reactors, at distances ranging from
hundred to thousand kilometers, with an average baseline of 180 km and
neutrinos of a few ev, [16, 17], has seen evidence of neutrino oscillations .
Such evidence was collected not only at a different $L/E$ than the atmospheric
and accelerator experiments but also consists on oscillations involving
electron neutrinos, $\nu_{e}$, the ones which were not involved before. These
oscillations have also been seen for neutrinos coming from the sun (the sun
produces only electron neutrinos). However,in order to compare the two
experiments we should assume that neutrinos (solar) and antineutrinos
(reactor) behave in the same way, i.e. assume CPT conservation. The best fit
values in the two neutrino scenario for the KamLAND experiment are

$\Delta m^{2}_{\odot}=8.0\pm 0.4\times
10^{-5}\mbox{eV}^{2}\;\;\;\mbox{and}\;\;\;\sin^{2}\theta_{\odot}=0.31\pm 0.03$
(43)

In this case, the $L/E$ involved is 15 km/MeV which is more than an order of
magnitude larger than the atmospheric scale and the mixing angle, although
large, is clearly not maximal.

Fig. 4 shows the disappearance probability for the $\bar{\nu}_{e}$ for KamLAND
as well as several older reactor experiments with shorter baselines.The second
panel depicts the flavour content of the 8Boron solar neutrino flux (with GeV
energies) measured by SNO, [18], and SK, [19]. The reactor outcome can be
explained in terms of two flavour oscillations in vacuum, given that the fit
to the disappearance probability, is appropriately averaged over $E$ and $L$.

Figure 4: Disappearance of the $\bar{\nu}_{e}$ observed by reactor experiments
as a function of distance from the reactor. The flavour content of the 8Boron
solar neutrinos for the various reactions for SNO and SK. CC:
$\nu_{e}+d\longrightarrow e^{-}+p+p$, NC:$\nu_{x}+d\longrightarrow\nu_{x}+p+n$
and ES: $\nu_{\alpha}+e^{-}\longrightarrow\nu_{\alpha}+e^{-}$.

The analysis of neutrinos coming from the sun is slightly more sophisticated
because it should include the matter effects that the neutrinos suffer since
they are born (at the centre of the sun) until they leave it, which are
important at least for the 8Boron neutrinos. The pp and 7Be neutrinos are less
energetic and therefore are not significantly altered by the presence of
matter and leave the sun as if it where ethereal. 8Boron neutrinos on the
other hand, leave the sun strongly affected by the presence of matter and this
is evidenced by the fact that they leave the sun as the $\nu_{2}$ mass
eigenstate and therefore do not undergo oscillations. This difference is, as
mentioned, due mainly to their differences at birth. While pp (7Be) neutrinos
are created with an average energy of 0.2 MeV (0.9 MeV), 8B are born with 10
MeV and as we have seen the impact of matter effects grows with the energy of
the neutrino.

However, we should stress that we do not really see solar neutrino
oscillations. To trace the oscillation pattern, we need a kinematic phase of
order one. In the case of neutrinos coming from the sun the kinematic phase is

$\displaystyle\Delta_{\odot}=\frac{\Delta m^{2}_{\odot}L}{4E}=10^{7\pm 1}.$
(44)

Therefore, solar neutrinos behave as "effectively incoherent" mass eigenstates
once they leave the sun, and remains being so once they reach the earth.
Consequently the $\nu_{e}$ survival probability is given by

$\displaystyle\langle
P_{ee}\rangle=f_{1}\cos^{2}\theta_{\odot}+f_{2}\sin^{2}\theta_{\odot}$ (45)

where $f_{1}$ is the $\nu_{1}$ content or fraction of $\nu_{\mu}$ and $f_{2}$
is the $\nu_{2}$ content of $\nu_{\mu}$ and therefore both fractions satisfy

$\displaystyle f_{1}+f_{2}=1.$ (46)

However, as we have mentioned, pp and 7Be solar neutrinos are not affected by
the solar matter and oscillate as in vacuum and thus, in their case
$f_{1}\approx\cos^{2}\theta_{\odot}=0.69$ and
$f_{2}\approx\sin^{2}\theta_{\odot}=0.31$. In the 8B a neutrino case, however,
matter effects are important and the corresponding fractions are substantially
altered, see Fig. 5.

Figure 5: The sun produces $\nu_{e}$ in the core but once they exit the sun
thinking about them in the mass eigenstate basis is useful. The fraction of
$\nu_{1}$ and $\nu_{2}$ is energy dependent above 1 MeV and has a dramatic
effect on the 8Boron solar neutrinos, as first observed by Davis.

In a two neutrino scenario, the day-time CC/NC measured by SNO, which is
roughly identical to the day-time average $\nu_{e}$ survival probability,
$\langle P_{ee}\rangle$, reads

$\displaystyle\left.\frac{CC}{NC}\right|_{\mbox{day}}=\langle
P_{ee}\rangle=f_{1}\cos^{2}\theta_{\odot}+f_{2}\sin^{2}\theta_{\odot},$ (47)

where $f_{1}$ and $f_{2}=1-f_{1}$ are the $\nu_{1}$ and $\nu_{2}$ contents of
the muon neutrino, respectively, averaged over the 8B neutrino energy spectrum
appropriately weighted with the charged current current cross section.
Therefore, the $\nu_{1}$ fraction (or how much $f_{2}$ differs from 100% ) is
given by

$\displaystyle
f_{1}=\frac{\left(\left.\frac{CC}{NC}\right|_{\mbox{day}}-\sin^{2}\theta_{\odot}\right)}{\cos
2\theta_{\odot}}=\frac{\left(0.347-0.311\right)}{0.378}\approx 10\%$ (48)

where the central values of the last SNO analysis, [18], were used. As there
are strong correlations between the uncertainties of the CC/NC ratio and
$\sin^{2}\theta_{\odot}$ it is not obvious how to estimate the uncertainty on
$f_{1}$ from their analysis. Note, that if the fraction of $\nu_{2}$ were
100%, then $\left.\frac{CC}{NC}\right|_{\mbox{day}}=\sin^{2}\theta_{\odot}$.

Using the analytical analysis of the Mikheyev-Smirnov-Wolfenstein (MSW)
effect, provided in [20], one can obtain the mass eigenstate fractions, which
are given by

$\displaystyle f_{2}=1-f_{1}=\langle\sin^{2}\theta_{\odot}^{M}+P_{x}\cos
2\theta_{\odot}^{M}\rangle_{{}^{8}{\mbox{B}}},$ (49)

with $\theta_{\odot}^{M}$ being the mixing angle as given at the $\nu_{e}$
production point and $P_{x}$ is the probability of the neutrino to hop from
one mass eigenstate to the second one during the Mikheyev-Smirnov resonance
crossing. The average $\langle...\rangle_{{}^{8}{\mbox{B}}}$ is over the
electron density of the 8B $\nu_{e}$ production region in the centre of the
Sun as given by the Solar Standard Model and the energy spectrum of 8B
neutrinos appropriately weighted with SNO’s charged current cross section. All
in all, the 8B energy weighted average content of $\nu_{2}$’s measured by SNO
is

$\displaystyle f_{2}=91\pm 2\%{\mbox{ at the 95 \% C.L.}}$ (50)

Therefore, it is obvious that the 8B solar neutrinos are the purest mass
eigenstate neutrino beam known so far and SK super famous picture of the sun
taken (underground) with neutrinos is made with approximately 90% of $\nu_{2}$
.

On March 8, 2012 a newly built reactor neutrino experiment, the Daya Bay
experiment, located in China, announced the measurement of the third mixing
angle [11], the only one which was still missing and found it to be

$\displaystyle\sin^{2}(2\theta_{12})=0.092\pm 0.017\;$ (51)

The fact that this angle, although smaller that the other two, is still
sizeable opens the door to a new generation of neutrino experiments aiming to
answer the open questions in the field.

##  0.5  $\nu$ Standard Model

Now that we have understood the physics behind neutrinos oscillations and have
learnt the experimental evidence about the parameters driving this
oscillations, we can move ahead and construct the Neutrino Standard Model:

  * •

it consists of three light ($m_{i}$ $<$ 1 eV) neutrinos, i.e. it involves only
two mass differences

$\Delta m^{2}_{\mbox{atm}}\approx 2.5\times 10^{-3}\mbox{eV}^{2}$ and $\Delta
m^{2}_{\mbox{solar}}\approx 8.0\times 10^{-5}{\mbox{eV}}^{2}$ .

  * •

so far we have not seen any experimental indication (or need) for additional
neutrinos. As we have measured long time ago the invisible width of the $Z$
boson and found it to be 3, if new neutrinos are going to be incorporated into
the model, they cannot couple to the $Z$ boson, they cannot enjoy weak
interactions, so we call them sterile. However, as sterile neutrinos have not
been seen, and are not needed, our Neutrino Standard Model will contain only
the three active flavours: $e$, $\mu$ and $\tau$.

Figure 6: Flavour content of the three neutrino mass eigenstates (not
including the dependence on the cosine of the CP violating phase $\delta$).If
CPT is conserved, the flavour content must be the same for neutrinos and anti-
neutrinos. Notice that oscillation experiments cannot tell us how far above
zero the entire spectrum lies.

  * •

the unitary mixing matrix, called the PMNS matrix, which describes the
relation between flavour eigenstates and mass eigenstates, comprises three
mixing angles (the so called solar mixing angle:$\theta_{12}$, the atmospheric
mixing angle $\theta_{23}$, and the recently measured reactor mixing
angle$\theta_{13}$) , one Dirac phase ($\delta$) and possibly two Majorana
phases ($\alpha$, $\beta$) and is given by

$\displaystyle\mid\nu_{\alpha}\rangle=U_{\alpha i}\mid\nu_{i}\rangle$
$\displaystyle U_{\alpha i}=\left(\begin{array}[]{ccc}1&&\\\ &c_{23}&s_{23}\\\
&-s_{23}&c_{23}\end{array}\right)\left(\begin{array}[]{ccc}c_{13}&&s_{13}e^{-i\delta}\\\
&1&\\\
-s_{13}e^{i\delta}&&c_{13}\end{array}\right)\left(\begin{array}[]{ccc}c_{12}&s_{12}&\\\
-s_{12}&c_{12}&\\\ &&1\end{array}\right)\left(\begin{array}[]{ccc}1&&\\\
&e^{i\alpha}&\\\ &&e^{i\beta}\end{array}\right)$ (64)

where $s_{ij}=\sin\theta_{ij}$ and $c_{ij}=\cos\theta_{ij}$. Thanks to the
hierarchy in mass differences (and to a less extent the smallness of the
reactor mixing angle) we are allowed to identify the (23) label in the three
neutrino scenario with the atmospheric $\delta m^{2}_{\mbox{atm}}$ we obtained
in the two neutrino scenario, identically the (12) label can be assimilated to
the solar $\delta m^{2}_{\odot}$. The (13) sector drives the $\nu_{e}$ flavour
oscillations at the atmospheric scale, and the depletion in reactor neutrino
fluxes see [21]. Therefore,

$\displaystyle\sin^{2}\theta_{12}=0.31\pm
0.03\;\;\;;\;\;\;\sin^{2}\theta_{23})=0.50\pm
0.15\;\;\;;\;\;\;\sin^{2}\theta_{13}=0.15\pm 0.03$

and the mass splittings are

$\displaystyle\mid\Delta m^{2}_{32}\mid=2.7\pm 0.4\times
10^{-3}{\mbox{eV}}^{2}\;\;\;{\mbox{and}}\;\;\;\Delta m^{2}_{21}=+8.0\pm
0.4\times 10^{-5}{\mbox{eV}}^{2}.$

These mixing angles and mass splittings are summarised in Fig. 6.

  * •

The absolute mass scale of the neutrinos, or the mass of the lightest neutrino
is not know yet, but the heaviest one must be lighter than about .5 eV.

  * •

As transition or survival probabilities depend on the combination
$U^{*}_{\alpha i}U_{\beta i}$ no trace of the Majorana phases could appear on
oscillation phenomena, however they can have observable effects in those
processes where the Majorana character of the neutrino is essential for the
process to happen, like neutrino-less double beta decay.

##  0.6 Neutrino mass and character

###  0.6.1 Absolute neutrino mass

The absolute mass scale of the neutrino cannot be obtained in oscillation
experiments, however this does not mean we cannot have it. Direct experiments
like tritium beta decay, or neutrinoless double beta decay and indirect ones,
like cosmological observations, have potential to feed us the information on
the absolute scale of neutrino mass, we so desperately need. The Katrin
tritium beta decay experiment, [22], has sensitivity down to 200 meV for the
"mass" of $\nu_{e}$ defined as

$\displaystyle m_{\nu_{e}}=\mid U_{e1}\mid^{2}m_{1}+\mid
U_{e2}\mid^{2}m_{2}+\mid U_{e3}\mid^{2}m_{3}.$ (65)

Figure 7: The effective mass measured in double $\beta$ decay, in cosmology
and in Tritium $\beta$ decay versus the mass of the lightest neutrino. Below
the dashed lines, only the normal hierarchy is allowed.

Neutrino-less double beta decay experiments, see [23] for a review, do not
measure the absolute mass of the neutrino directly but a combination of
neutrino masses and mixings,

$\displaystyle m_{\beta\beta}=\mid\sum m_{i}U_{ei}^{2}\mid=\mid
m_{a}c_{13}^{2}c_{12}^{2}+m_{2}c_{13}^{2}s_{12}^{2}e^{2i\alpha}+m_{3}s_{13}^{2}e^{2i\beta}\mid,$
(66)

where it is understood that neutrinos are taken to be Majorana particles. The
new generation of experiments seeks to reach below 10 meV for $m_{\beta\beta}$
in double beta decay.

Cosmological probes measure the sum of the neutrino masses,

$\displaystyle m_{\mbox{cosmo}}=\sum_{i}m_{i}.$ (67)

If $\sum m_{i}\approx 50$ eV, the energy balance of the universe saturates the
bound coming from its critical density. The current limit, [24], is a few % of
this number, $\sim 1$ eV. These bounds are model dependent but they do all
give numbers of the same order of magnitude. However, given the systematic
uncertainties characteristic of cosmology, a solid limit of less that 100 meV
seems way too aggressive.

Fig. 7 shows the allowed parameter space for the neutrino masses (as a
function of the absolute scale) for both the normal and inverted hierarchy.

###  0.6.2 Majorana vs Dirac

A fermion mass is nothing but a coupling between a left handed state and a
right handed one. Thus, if we examine a massive fermion at rest, then one can
regards this state as a linear combination of two massless particles, one
right handed and one left handed particle. If the particle we are examining is
electrically charged, like an electron, both particles, the left handed as
well as the right handed must have the same charge (we want the mass term to
be electrically neutral). This is a Dirac mass term. However, for a neutral
particle, like a sterile neutrino, a new possibility opens up, the left handed
particle can be coupled to the right handed anti-particle, (a term which would
have a net charge, if the fields are not absolutely and totally neutral) this
is a Majorana mass term.

Thus a neutral particle does have two ways of getting a mass term, a la Dirac
or a la Majorana, and in principle can have them both, as shown :

In the case of a neutrino, the left chiral field couples to $SU(2)\times U(1)$
implying that a Majorana mass term is forbidden by gauge symmetry. However,
the right chiral field carries no quantum numbers, is totally and absolutely
neutral. Then, the Majorana mass term is unprotected by any symmetry and it is
expected to be very large, of the order of the largest scale in the theory. On
the other hand, Dirac mass terms are expected to be of the order of the
electroweak scale times a Yukawa coupling, giving a mass of the order of
magnitude of the charged lepton or quark masses. Putting all the pieces
together, the mass matrix for the neutrinos results as in Fig. 8.

Figure 8: The neutrino mass matrix with the various right to left couplings,
$M_{D}$ is the Dirac mass terms while 0 and $M$ are Majorana masses for the
charged and uncharged (under $SU(2)\times U(1)$) chiral components.

To get the mass eigenstates we need to diagonalise the neutrino mass matrix.
By doing so, one is left with two Majorana neutrinos, one super-heavy Majorana
neutrino with mass $\simeq M$ and one light Majorana neutrino with mass
$m_{D}^{2}/M$, i.e. one mass goes up while the other sinks, this is what we
call the seesaw mechanism, [25, 26, 27]. The light neutrino(s) is(are) the
one(s) observed in current experiments (its mass differences) while the heavy
neutrino(s) are not accessible to current experiments and could be responsible
for explaining the baryon asymmetry of the universe through the generation of
a lepton asymmetry at very high energy scales since its decays can in
principle be CP violating (they depend on the two Majorana phases on the PNMS
matrix which are invisible for oscillations).

If neutrinos are Majorana particles lepton number is no longer a good quantum
number and a plethora of new processes forbidden by lepton number conservation
can take place, it is not only neutrino-less double beta decay. For example, a
muon neutrino can produce a positively charged muon. However, this process and
any processes of this kind, would be suppressed by $(m_{\nu}/E)^{2}$ which is
tiny, $10^{-20}$, and therefore, although they are technically allowed, are
experimentally unobservable.

##  0.7 Conclusions

The experimental observations of neutrino oscillations, meaning that neutrinos
have mass and mix, answered questions that had endured since the establishment
of the Standard Model. As those veils have disappeared, new questions open up
and challenge our understanding:

  * •

what is the nature of the neutrino? are they Majorana or Dirac? are neutrinos
totally neutral?

  * •

is the spectrum normal or inverted?

  * •

is CP violated (is $\sin\delta\neq 0$)?

  * •

which is the absolute mass scale of the neutrinos?

  * •

are there new interactions?

  * •

can neutrinos violate CPT [28]?

  * •

are these intriguing signals in short baseline reactor neutrino experiments
(the missing fluxes) a real effect? Do they imply the existence of sterile
neutrinos?

We would like to answer these questions. For doing it, we are doing right now,
and we plan to do new experiments. These experiments will, for sure bring some
answers and clearly open new, pressing questions. Only one thing is clear. Our
journey into the neutrino world is just beginning.

## Acknowledgements

I would like to thank the students and the organisers of the European School
on HEP for giving me the opportunity to present these lectures in such a
wonderful atmosphere. I did enjoy each day of the school enormously.

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