the Standard Model
1.2 Neutrino Oscillations
128
The Super-Kamiokande experiment in 1998 [16] and the SNO experiment in 2001 [17,
129
18] firmly established that neutrinos oscillate and have mass, and for this discovery
130
the Nobel Prize in Physics was awarded to Takaaki Kajita and Arthur B. McDonald
131
in 2015. These measurements eventually solved long-standing problems in neutrino
132
physics, namely the solar neutrino puzzle and the up-down asymmetry in atmospheric
133
neutrinos: for a long time several experiments measured neutrinofluxes in disagreement
134
with the predictions, which were based on the non-oscillating neutrino hypothesis.
135
Neutrino masses and cross sections are so tiny that it took several decades to prove
136
that these disagreements were due to neutrino oscillations.
137
1.2.1 Solar Neutrinos
138
Neutrinos are constantly produced in the Sun (mainly from the p−p chain and the CNO cycle) with a net reaction of:
4p→4 He + 2e++ 2νe+ 26.731 MeV.
In 1968 the Homestake experiment [19, 20] measured the solar electron neutrinoflux
139
and compared the result to the prediction of the Standard Solar Model. They concluded
140
that the flux was between one third and one half of what was expected [21]. This was
141
the first measurement of the solar neutrino problem [22]. More recent experiments,
142
SAGE [23], GALLEX [24] and Kamiokande [25] published results consistent with the
143
Homestake experiment.
144
This suppression in the electron neutrinoflux from the Sun is now explained by the
145
neutrino oscillations and it wasfinally confirmed in 2001 by the SNO experiment [17,
146
18]: SNO measured the flux of all the 3 flavours of neutrinos and found consistent
147
results with the prediction of the Standard Solar Model. Fig. 1.2 shows the solar flux
148
result from SNO.
149
1.2.2 Atmospheric Neutrinos
150
Primary cosmic rays entering the Earth’s atmosphere are mostly made of protons (up
151
to 90 %). When they enter the atmosphere, hadronic interactions produce hadrons
152
such as pions and kaons which decay producing neutrinos. The pions decay into µ
153
Figure 1.2: Solar neutrino result. Thexaxis is theflux ofνe and they axis is theflux of νµ and ντ from SNO measurements. The red band is from the SNO CC result. The blue one is from the SNO NC result and the light green is from the SNO elastic scattering result. The dark green band is from the Super-Kamiokande elastic scattering result. The bands represent the 1σ error. The sum of the neutrinofluxes is consistent with the SSM expectation (dashed line) [17, 18],
and νµ, then subsequently the µ decay into e,νe, and νµ. These neutrinos are called
154
atmospheric neutrinos.
155
Since the dominant interaction producing neutrinos is the decay chainπ+→µ++νµ
156
and subsequentlyµ+→e++νµ+νe[26], the amount ofνµ,νµis approximately twice as
157
much asνe, and the ratio ofνµflux toνeflux should be approximately isotropic, at least
158
for sub-GeV muons (high-energy muons from the zenith might not decay before reaching
159
the detector). Nevertheless in 1998 the Super-Kamiokande experiment measured an
160
anisotropic flux ratio, revealing a zenith angle dependence in the neutrino direction,
161
and proving that the flux depends on how far neutrinos travel. This was the first
162
robust proof of the neutrino oscillations. Fig. 1.3 [16] shows the best-fit with neutrino
163
oscillation is well consistent with the atmospheric neutrino measurements.
164
1.2.3 Neutrino Oscillations in Matter: the MSW Effect
165
While the atmospheric neutrinoflux is explained by the neutrino oscillation in vacuum,
166
the solar neutrinoflux anomaly is mainly due to the Mikheyev, Smirnov and Wolfenstein
167
(MSW) effect [27, 28], or simply matter effect.
168
When the electron neutrinos travel in matter, they can interact with electrons in
169
matter through both the neutral weak current and the charged weak current, while
170
other flavours of neutrinos only interact through the neutral weak current. Therefore,
171
there will be a phase difference between the components of the different flavours. An
172
effective potential term is added to the Hamiltonian to take into account the MSW
173
effect: this term is proportional to the neutrino energy and the density of electrons in
174
the material it traverses, and it has opposite sign forνeandνe, which will be important
175
to determine the mass hierarchy and to test the lepton CP violation (Section 1.3.2.2).
176
Because of the high density in the core of the Sun, the MSW effect turns a significant
177
amount of electron neutrinos into muon and tau neutrinos.
178
1.2.4 Neutrino Oscillation Framework
179
The very first proposal of a neutrino oscillation framework was formulated in 1957 by
180
Bruno Pontecorvo, imaging the oscillations between neutrinos and anti-neutrinos [29,
181
30]. Few years later, as soon as the muon neutrino was discovered, Pontecorvo also
182
proposed the oscillations between νe and νµ, further extended into a three neutrino
183
Figure 1.3: Zenith angle distributions: non-oscillated Monte Carlo predictions are in the dotted histograms and the best-fit expectations for νµ to ντ oscillations are in the solid histograms.
oscillation scheme by Z. Maki, M. Nakagawa and S. Sakata in 1962 [31]. Even if
184
the early neutrino experiments discussed in the previous sections were not motivated
185
by this theoretical proposal (or not even aware thereof), the oscillation framework of
186
Pontecorvo, Maki, Nakagawa and Sakata turned out to be the right starting point.
187
As discussed in Section 1.1, neutrino oscillations are a consequence of the
non-188
zero neutrino masses. The fundamental principle of neutrino oscillations is that the
189
neutrino flavour eigenstates, states involved in neutrino interactions (cf. Section 1.4),
190
are not identical to the neutrino mass eigenstates, i.e. the stable energy eigenstates.
191
Theflavour eigenstatesνe,νµand ντ are superpositions of the mass eigenstatesν1,ν2
192
andν3. This implies that neutrinos cannot be massless and that they can change their
193
flavour during propagation (hence the name “neutrino oscillations”) [32]. Therefore,
194
the picture of the particles in the Standard Model of particle physics has changed from
195
Fig. 1.1 to Fig. 1.4.
196
Figure 1.4: Particles in the Standard Model of particle physics
The neutrino flavour eigenstates,|να�, can be expressed as a linear combination of
197
mass eigenstates,|νi�, i= 1,2,3, using the unitary matrixU, known as the
Pontecorvo-198
Maki-Nakagawa-Sakata (PMNS) neutrino mixing matrix [33, 34]:
199
|να�=
dimensional space as in Fig. 1.5.
201
Figure 1.5
The PMNS matrix is a 3× 3 complex unitary matrix similar to the
Cabibbo-202
Kobayashi-Maskawa (CKM) matrix in the quark sector [32, 35]. There are many ways
203
to parametrise the PMNS matrix. Usually, it is written in the following extended form
204
which also include the Majorana term (cf. Section 1.3.2.4):
205
and α andβ are Majorana phases that have no effect on neutrino oscillations.
207
If a neutrino is produced at a time t = 0 in the state να, the time evolution of
208
neutrinoflavour states is given by:
209
|να(t)�=
�3 i=1
Uαi∗e−iEit|νi�. (1.10)
Similarly, a mass eigenstate can be expressed as a superposition of the flavour
210
eigenstates
Then, the probability of να→νβ transitions is given by:
213
P(να→νβ) =|Aνα→νβ(t)|2 =|
�3 i=1
Uβie−iEitUαi∗ |2 . (1.13)
As neutrinos are highly relativistic (E ≈|p|), this approximation can be made:
214
Ei =
�
p2+m2i ≈E+m2i
2E (1.14)
and the oscillation probability can be re-written as:
215
whereΔm2ij =m2j−m2i is the mass squared difference of neutrino mass eigenstates.
216
The T2K neutrino oscillation experiment probes oscillations coming from a muon
217
neutrino beam. The oscillation and survival probabilities are:
218
whereLis the distance travelled by neutrinos in km,Eν is neutrino energy in GeV,
219
Δm2 in eV2, and the factor 1.27 comes from 1/�cin the conversion of units to km and
220
GeV. These equations show that the magnitude of theνµ→νeoscillations is governed
221
by the mixing angles θ13 and θ23, whereas the frequency of oscillations depends on
222
Δm232.
223
When we measure θ23 with the survival probability P(νµ → νµ) which is
propor-224
tional to sin22θ23 to first order, there is an octant ambiguity: either θ23 ≤ 45◦ (in
225
the first octant) or θ23 > 45◦ (in the second octant). By combining the
measure-226
ments of P(νµ → νµ) and P(νµ → νe), future long baseline experiments, like
Hyper-227
Kamiokande [36] and DUNE [37], can reach the necessary sensitivity to determine the
228
θ23 octant.
229
The formal 3-flavour probability for neutrino oscillations in vacuum can be written
230
where the symbols are those previously defined, andU it the PMNS matrix.
232
There are 9 oscillation parameters: three mixing angles, three neutrino masses, and
233
one single CP phase and two possible Majorana phases (cf. Section 1.3.2.4). Most
234
of parameters have been measured with certain precision; the latest measurements
235
are discussed in Section 1.3.1. However, there are still missing pieces, described in
236
Section 1.3.2.
237