# Neutrino Physics

### The Discovery of Neutrinos and Flavors

There are three standard channels of decay in nuclear physics characterized by their decay products. Gamma decay emits a photon resulting in an energy change in the original atom, but no other changes in structure. Alpha decay induces a heavy change in the original atom due to the make up of the expelled alpha particle being 2 neutrons and 2 protons. In beta decay a proton in the original nucleus transforms into a neutron, with the emission of a beta particle (an electron). For a span of 19 years, from 1911 to 1930, there was an unexplained phenomenon in the kinetic energy of electrons emitted in beta decay. If beta decay were a two body problem as then hypothesized the kinetic energy of the electron should have a fixed value for all decays due to the conservation of energy and momentum in the decay. What was observed in experiment, and as pictured below, is a continuous spectrum. The spectrum goes to zero as the kinetic energy of the electron goes to the overall energy released, since a particle can’t acquire more energy than was released.

In 1930, Wolfgang Pauli solved this discrepancy between theory and observation by introducing a third body into the decay, with the third body being extremely light and neutrally charged. It wouldn’t be until 1956 that the neutrino (little neutron) was confirmed experimentally by the Cowan-Reines neutrino experiment, but many questions remained about the nature of the neutrino.

In the 1980s, the number of questions increased due to a newly discovered phenomenon called,neutrino oscillations. To understand oscillations, we first need to know that neutrinos have flavor types (such as electron neutrino and muon neutrino) which differ from the characteristic mass eigenstates. Neutrinos interact with other particles as a certain flavor type, for instance, in beta decay as an electron neutrino $\nu_e$ , but their time-evolution in a vacuum follows from  their mass eigenstates which are less illustratively called  $\nu_{1,2,3}$. The two bases are related by a unitary transformation, the neutrino mixing matrix below.

#### $\begin{pmatrix} \nu_e&\nu_\mu&\nu_\tau\end{pmatrix} = \begin{pmatrix} U_{e,1}&U_{e,2}&U_{e,3}\\U_{\mu,1}&U_{\mu,2}&U_{\mu,3}\\U_{\tau,1}&U_{\tau,2}&U_{\tau,3}\end{pmatrix}\begin{pmatrix}\nu_1 \\ \nu_2\\\nu_3\end{pmatrix}$

The difference in phase between the mass eigenstates as they propagate gives rise to neutrino oscillations. Neutrinos are created in the sun as electron type, but the probability to interact as any of the three flavor eigenstates, electron, muon and tau, changes as a function of distance travelled. The rate of oscillation is related to the difference in square in mass of the different mass eigenstates. This phenomenon was first observed in what is known as the solar neutrino problem, where too few electron type neutrinos were measured coming from the sun. The proof with that these electron neutrinos were oscillating to other flavor states was awarded the 2015 Nobel Prize in physics.

Rigorous measurement of these δm^2 values have resulted in a well known difference between the squared masses of neutrinos, but not the absolute mass scale, or actual ranking of the masses. This structure of neutrino masses is called the mass hierarchy with two well known but neither confirmed solutions.

In the investigation of the mass hierarchy, we know from experiments that two states are close to each other and that a third state is much further from either. This relative structure leads to two different possibilities, the normal and the inverted hierarchies. The normal hierarchy has the furthest mass eigenstate as the lightest with the two close eigenstates as heavier, whereas the inverted hierarchy has the two close masses as being lighter and the further mass as the heavier. A means of probing the absolute mass scale of neutrinos is the neutrino-less double beta decay discussed later.

### Ettore Majorana: His Equation and Fermions

We’ll leave the explicit discussion of neutri- nos now to briefly discuss Ettore Majorana and his relativistic equation of particles. An Italian physicist born in 1908, Ettore Majorana worked under Enrico Fermi and was a leading theoretician behind Fermi’s group’s seminal paper on beta decay. In 1932, while still working at the institute which Fermi chaired, Majorana published a paper on relativistic particle mechanics, in which he posited what would later be called the Majorana equation

$-i \eth \psi + m \psi_c = 0$

By all accounts a modest looking equation with it’s grand total of two terms, this equation implicates major violations in the current standard model. Particles whose wave functions satisfy this equation belong to a class of particle which we now call Majorana Particles. In particular such a particle has the curious characteristic of being its own anti-particle. One of the most accessible means of determining whether a particle is a Majorana particle or a Dirac particle (the particle upon which the standard model is built), is to measure the effective mass which differs whether the particle is Dirac or Majorana.

One candidate Majorana fermion is the neutrino, ν. Below are just a few characteristics of a Majorana Neutrino.

• Under charge conjugation, $\nu_m$ returns itself rather than it’s anti-particle like a dirac particle would have $C \vert \nu_m \rangle = \tilde{\eta}_c \vert \nu_m\rangle$
• In the field equation of the ν we no longer have terms for neutrino annihilation and anti-neutrino creation, but rather both annihilation and creation for the Majorana with an appropriate phase factor.
• The mechanics underlying the Majorana neutrino don’t conserve lepton number in the lagrangian.
• Proof of a neutrino behaving as a Majorana fermion, would be further proof that we need mechanics and physics beyond the standard model.

### The Double and Neutrinoless Double Beta Decays

Knowing what a single beta decay is, it wouldn’t be too hard to imagine that there are some isotopes which aren’t prone to single decay but rather a double beta decay. This would include two neutrons simultaneously decaying into two protons with two electrons and neutrinos as by products ( from here on we discuss only one flavor of neutrino, the electron neutrino). 35 isotopes are expected to exhibit this channel of decay, and it’s been well measured in a handful of these already, such as Xe-136,Ge-76 and Te-130. The half-lives of many 2νββ-decays are on the order of $10^[21]$ years. While these decays are important and can help refine models of heavy nucleus nuclear physics, they do little to elucidate whether neutrinos are Dirac or Majorana. For that, physicists are diving into the search for an even rarer decay type, neutrino-less double beta decay (0νββ). The rarity of this event means that for it to be discernible among a spectrum, all decays which could occur with similar energies must be heavily suppressed, for instance, we must choose a particle with heavily suppressed or no single beta decay mode.

The mechanics of the 0νββ-decay, depicted above,  are similar to those of the 2νββ-decay except for the neutrino emitted by one of the transitions being absorbed by the other transition as an anti-neutrino rather than traveling off as a free particle.  This feature is more readily explained by the neutrino being a “virtual” particle. This refers to the fact that under quantum field theory, this exchange of neutrinos is more of a facet of the mathematics involved, than the determination of real particles. Neglecting the subtleties of the derivation, the half-life of this decay is given by,

$(T^{0\nu}_{1/2})^{-1} = G^{0\nu}(E_0,Z) \vert M^{0\nu}\vert ^2 \langle m_{\beta \beta} \rangle^2,$

where the terms are the reciprocal of the half-life $T^{0\nu}_{1/2})^{-1}$, a phase factor $G^{0\nu}(E_0,Z)$, the norm squared of the nuclear matrix element $\vert M^{0\nu}\vert ^2$, which varies depending on the nucleus of the selected isotope, and finally the square of the effective neutrino mass $\langle m_{\beta \beta} \rangle^2$. The half-life of the decay is important, but also a means to a different end of finding the effective neutrino mass, which can also be defined as

$\langle m_{\beta \beta} \rangle = \biggl \vert \sum_j m_j U^2_{e j} \biggr \vert$

where the $m$ is some mass description for a flavor of neutrino and $U_{e,j}$ is the mixing matrix element for the electron flavor to $j^[th]$ flavor transition.

If the neutrino-less decay channel is confirmed then the Majorana nature of the neutrino will be shown true, but with a sensitive enough measurement we will also be able to determine bounds of the effective neutrino mass which will help determine the still elusive mass hierarchy of the neutrino. Pictured above is a graph of the regions of effective mass either disfavored or not yet explored, and the regions where mass would be expected to be found for a given hierarchy type. Also shown are the sensitivities of EXO-200, and the current expected sensitivity of nEXO. As we can see, the final operational limit of nEXO, will be below the entirety of the inverted hierarchy regime. This will let the nEXO project determine at least which hierarchy neutrinos fall into, if not succesfully detect neutrinoless double beta decay.

### Bibliography

[1] Lang, K. Future prospects for measurements of mass hierarchy and CP violation. Nuclear and Particle Physics Proceedings, 183-187. Ap. J. 295:305, 1985.

[2] Maneschg, W. Review of neutrinoless double beta decay experiments: Present status and near future. Nuclear and Particle Physics Proceedings, 260, 188-193, 2015.