Oscillations

The observation of the solar neutrino flux by the Homestake experiment in a deep mine in South Dakota was historically the first indication that neutrinos oscillate [4]. Raymond Davis designed an experiment consisting of a large tank of dry-cleaning fluid in which neutrinos emitted by nuclear fusion in the Sun could interact via the inverse beta decay process, i.e.

νe+³⁷Cl⁺→³⁷Ar⁺+e⁻. (2.14) The gaseous radioactive argon was regularly collected and its activity mea-sured to determine the number of neutrino interactions. The measurement was consistently very close to one-third of John Bahcall’s prediction. This gave rise to thesolar neutrino problem. Many subsequent radiochemical and water Cherenkov experiments confirmed the deficit [6,7,79–81].

Bruno Pontecorvo, a nuclear physicist who instigated several direct measurements for neutrinos, was the first to suggest that neutrinos may come in electron and muon flavours [82]. The existence of a second neutrino type was confirmed by the observation of the charged current production of muons at the Alternating Gradient Synchrotron [83]. In a 1957 breakthrough paper, he proposed that the two flavours may oscillate between each other in an analogy with the previously observed CP-violatingK⁰ oscillations [5,84].

For this to happen, the neutrinos cannot have zero-mass and therefore do not travel at the speed of light. In the two-neutrino model developed at the time, the flavour eigenstates are a rotation of the mass eigenstates through

νe

νµ

=

Ue1 Ue2

Uµ1 Uµ2

ν1

ν2

= cosθ sinθ

−sinθ cosθ ν1

ν2

, (2.15)

withν₁, ν₂ the mass eigenstates andθthe mixing angle.

The neutrino oscillation scheme is represented in figure2.5. When pro-duced in a weak process, the neutrino of flavourαis in a superimposition of the two mass eigenstates. Since theνi, i= 1,2 are mass eigenstates, their propagation can be described by plane wave solutions of the Schrödinger equation of the form

|νi(t)i=|νi(0)iexp (−i(Eit−pixi)/~), (2.16) witht the propagation time, Ei, pi and xi the energy, three-momentum and spacial position of the mass eigenstate, respectively, and~the reduced Planck constant. In the ultrarelativistic limit |pi| ≡pimic, the energy can be expanded at first order to

Ei=q

p²_ic²+m²_ic⁴'pic+m²_ic³ 2pi

'E+m²_ic⁴

2E , (2.17) withE the total energy of the neutrino, now approximatively identical in each eigenstate. In the limitv'c, the timetrelates to the distance through L/c. Dropping the phase factor, the wave function after a distanceLreads

|νi(L)i=|νi(0)iexp

−im²_ic³L 2~E

. (2.18)

Source

•

W να

`¯α

ν

• W

νβ

`¯β

L

Target

Figure 2.5:A neutrino of flavourαtravels a distanceLfrom its source to produces a charged leptonlβ in its interaction with the target.

As the flavour eigenstates are a linear combination of the mass eigenstates with evolving parameters, it is possible to observe a neutrino change its flavour during its propagation. The probability of a neutrino originally of flavour αto be later observed at target in the same flavour is

P_α→α=|hνα|να(L)i|²= 1−sin²(2θ) sin²

1.27∆m²L E

GeV eV²·km

, (2.19) with ∆m²=m²₂−m²₁. The effect of the oscillation is maximal whenL' E/∆m². The appropriate choice of baseline,L/E, allows for the observation

of the most significant disappearance of the emitted flavour or appearance of other neutrino flavours.

The survival probability of an electron neutrino is represented as a function ofL/E in the two-neutrino paradigm in figure 2.6 for θ = 33.5°

and ∆m² = 7.5×10⁻⁵eV². At a solar distance L ' 1.5×10⁸km, the oscillations of MeV neutrinos are in theLE/∆m² regime for which the probability effectively integrates to

P_α→α'1−1

2sin²(2θ)'0.575. (2.20) This is the ratio observed by solar neutrino experiments sensitive to sub-MeV neutrinos. At higher energies, the contribution of matter effects inside the sun become significant and reduces this ratio. The solar neutrino problem was definitely solved much later at Sudbury National Observatory. The detector consisted of heavy water sensitive to all neutrino flavours through neutral current interactions. It observed the expected total flux and showed that the multi-MeV electron neutrinos only make up a third of it [85].

10⁰ 10¹ 10²

0.2 0.4 0.6 0.8 1

L/E [km/MeV]

Pe→e

Figure 2.6:Survival probability of an electron neutrino in the two-neutrino model as a function of the baselineL/Eforθ= 33.5° and ∆m²= 7.5×10⁻5 eV².

The observation of CP-violating processes – not allowed by the 2×2 Cabbibo quark mixing matrix – lead Kobayashi and Maskawa to surmise the existence of a third generation of fermions [86]. The discovery of the τ lepton and its neutrino at the Stanford Linear Accelerator Centere⁺e⁻ colliding ring in 1975 confirmed its existence and prompted Maki, Nakagawa and Sakata to extend the dimensions of the neutrino mixing matrix to its current form [87]:

U_PMNS=





U_e1 U_e2 U_e3 U_µ1 U_µ2 U_µ3 Uτ1 Uτ2 Uτ3



, (2.21)

which is most commonly decomposed as number of neutrino generations was corroborated by the precise measurement of theZboson width by multiple experiments located on the Large Electron Positron collider at CERN [88]. Tau neutrino interactions were directly observed at the turn of the XXI^st century by a dedicated experiment at Fermilab, DONUT [89]. Figure2.7gives a schematic representation of the mixing angles between the two neutrino eigenbases.

ν1

Figure 2.7: Neutrino mixing angles represented as a product of Euler rotations:

(νe, νµ, ντ)^T =UPMNS(ν1, ν2, ν3)^T. Some representative values of the angles are used based on the best-fit values summarised in table2.1.

In the three-neutrino model, the probability of a neutrino of energyE produced in flavourαto be later observed in flavourβ after a propagation distanceL reads or, in terms of the mass splittings,

P_α→β=δαβ − 4X

with ∆m²_ij =m²_i −m²_j [90]. This formula now represents a superposition of two oscillations of frequencies driven by two mass splitting ∆m²₂₁ and

∆m²_3l ≡ ∆m²₃₂ ' ∆m²₃₁ ∆m²₁₂ and amplitudes determined by three mixing angles θ12, θ13 andθ23. The small splitting ∆m²₂₁ is defined to be positive but ∆²3l may be of either sign. Figure2.8 shows the two possible orderings. The mass eigenstate with the smallest fraction ofν_eis the heaviest in the normal ordering scenario and the lightest in the inverted ordering.

Normal Ordering (NO) Inverted Ordering (IO)

Figure 2.8:The two possible orderings of the neutrino mass eigenstates.

Equation 2.24 is usually well approximated by its two-neutrino form in equation 2.20, due to the relative small scale of ∆m²₂₁ and θ₁₃. The

∆m²₂₁ mass splitting and theθ₁₂ mixing angle are the leading factors in the long-baseline solar oscillationsν_e ↔ν_x measurements. The splitting was measured accurately by the KamLAND experiment, a kiloton scintillator tanker in Kamioka that observed the disappearance of MeV nuclear reactor

¯

νe at a mean distanceL0'180 km [91].

The large mass splitting ∆²3land θ23drive the atmospheric oscillations νµ↔νx. Multi-GeV muon neutrinos are produced in the upper atmosphere by pion and muon decays. The neutrinos do not oscillate before reaching the surface of the Earth but do when they cross it. The observation of the disappearance ofνµ coming from below the Super-Kamiokande 50 kT water Cherenkov detector demonstrated this second regime of oscillations [92].

Precision measurements of the mixing parameters were performed by the T2K and NOνA collaborations by observing the disappearance of accelerator ν_µ produced by dumping protons on targets at accelerator facilities and also measuring the appearance of the oscillatedν_e [93,94].

The small mixing angleθ₁₃ has its largest impact onν_edisappearance at short-baseline, before the solar oscillations dominate. Reactor neutrinos are an ideal candidate to measure this angle as they are produced in large quantities in commercial nuclear reactors. The oscillation maximum of an MeV neutrino occurs at a distance of order km from the source and constitutes

the optimal location to place a detector. Double Chooz was the first reactor experiment to report a non-zeroθ13at 1.9σconfidence level [95]. The initial measurement was soon confirmed and significantly improved since by the Daya Bay and RENO experiments [96,97].

The most recent summary of the mixing parameters produced from a combined fit of all the major experiments is summarised in table 2.1[12].

Normal Ordering (NO) Inverted Ordering (IO)

Best Fit 3σrange Best Fit 3σrange

sin²θ12 0.307^+0.013−0.012 0.272→0.346 0.307^+0.013−0.012 0.272→0.346 sin²θ₂₃ 0.538^+0.033−0.069 0.418→0.613 0.554^+0.022−0.033 0.435→0.616 sin²θ13 0.02206^+0.00075−0.00075 0.01981→0.02436 0.02277^+0.00074−0.00074 0.02006→0.02452

∆m²₂₁

10⁻⁵eV² 7.40^+0.21−0.20 6.80→8.02 7.40^+0.21−0.20 6.80→8.02

∆m²_3l

10⁻³eV² +2.494^+0.033−0.031 +2.399→+2.593 −2.465^+0.032−0.031 −2.562→ −2.369 Table 2.1: Three-flavour oscillation parameters from NuFIT’s fit to global data as of November 2017. The numbers in the first and second columns are obtained assuming normal ordering (NO) and inverted ordering (IO), respectively, i.e. relative to the respective local minimum. Note that ∆m²_3l= ∆m²₃₁>0 for NO and ∆m²_3l=

∆m²₃₂<0 for IO [12].