Neutrinos are particles that have evoked the curiosity of physicists and non-physicists alike since their discovery almost 60 years ago. They were born out of theory in an effort to protect the sanctity of the conservation of momentum. They are the most abundant lepton in the universe, and perhaps the most difficult to detect. Most recently the topic of the 2015 Nobel Prize in Physics, their dynamic properties have pointed towards them having mass, a property not accounted for explicitly in electroweak theory. What implications do massive neutrinos have for the Standard Model of particle physics?, and could they hold the answer to other open questions in high energy physics? Which theoretical models look most promising? What are the current experimental bounds on the neutrino mass? We explore the origins of neutrinos, their place in the Standard Model, and look at the models and experimental status of the determination of neutrino masses and their dynamics.

Neutrinos were first proposed by Wolfgang Pauli in 1930 as a solution to missing momentum in the process of beta decay. Consider the analogous nuclear process in Figure \ref{fig:bubble} \[p^{+} + \nu_{\mu} \rightarrow p^{+} + \pi^{+} + \mu^{-}. \label{decay_process}\] As neutral particles do not condense inside the chamber, the decay track looks like a “vee” in the chamber which appears to violate the conservation of momentum. A second problem surfaces when examining the emitted electron energy spectrum in a beta decay process. Comparing the energy spectrum of the beta particle to that of the well-understood energy spectrum of an alpha particle, the beta particle energy spectrum allows for a range of energies, while the alpha particle spectrum is discrete (which is expected given the restricted decay modes of a given nucleus). This continuous energy spectrum suggests that either energy is not conserved in this process, or that the process is more complicated than what is observed. Indeed, these were two of the solutions proposed at the time by two notable physicists; Niels Bohr thought that the conservation of energy may be a statistical phenomena and was not necessarily explicitly conserved, while Enrico Fermi felt that the conservation of energy was too fundamental of a concept to compromise. Fermi instead proposed that the missing energy was being carried away by a neutral particle unobserved in the bubble chamber. Fermi’s explanation not only correctly account for the missing energy, but also resolved discrepancies in the conservation of angular momentum; decaying mother atoms with integer/fractional spin were observed to produce daughter particles with a corresponding integer/fractional spin. This particle proposed by Fermi would eventually be found to be the electron neutrino we know of today.

In 1979, the Nobel prize was awarded to Sheldon Glashow, Abdus Salam, and Steven Weinberg for their work unifying the electromagnetic and weak forces under a single theory. In formulating a unified electroweak theory, several unique experimentally observed properties of the weak and electromagnetic interactions needed to be accounted for:

CP violation was observed by Chien-Shiung Wu in 1957 through experiments of \(\beta\) decay in Cobalt-60 (Wu 1957). This showed an asymmetry in the interaction of leptons in a chiral basis; left-handed leptons only interacted with right-handed anti-leptons, and right-handed leptons did not participate in the weak interaction. To date, there has yet to be observed a right-handed neutrino.

Descriptions of forces in nature are well described by gauge theories, with forces being mediated through gauge bosons resulting from some gauge symmetry (required for a renormalizable theory). The universal coupling strength of the weak interaction is a strong indication of it being described by a gauge theory.

For a unification of electromagnetism and the weak interaction, a description including four gauge bosons was required for charge conservation in weak processes: a positively-charged boson, negatively charged boson, a neutral boson, and a massless boson for the electromagnetic force (the photon).

Given that the weak interaction was a short-range force, it implied the gauge bosons involved were

**massive**.The interactions between left-handed leptons and right-handed anti-leptons seemed to suggest a \(SU(2)\) non-abelian symmetry. Mathematically, this implies the coupling of the theory will be asymptotically-free (similar to in quantum chromodynamics). As no weak bound states had been observed, this seemed to contradict observation. For a non-abelian theory to have the appropriate coupling behaviour, it implied the presence of a scalar particle.

One of the difficulties accomplishing this was in the formulation of lepton and gauge boson masses under \(SU(2)\) gauge invariance. The Dirac equation is commonly used to describe fermionic particles in a quantum field theory \[\bar{\psi}(i \gamma^{\mu}\partial_{\mu}-m)\psi = 0. \label{dirac}\] When promoting this into our gauge-invariant theory however, we run into a problem. As the electroweak theory is a CP-violating theory, we express our proposed Lagrangian in terms of the right and left handed chiral fields \(\phi_{R}\) and \(\phi_{L}\). We define Dirac spinor in the chiral representation as \[\psi = \begin{pmatrix} \psi_{R}\\ \psi_{L} \end{pmatrix}\] In terms of chiral fields, (\ref{dirac}) becomes \[\bar{\psi}(i \gamma^{\mu}\partial_{\mu}-m)\psi = i (\bar{\psi}_{R} \bar{\sigma}^{\mu}\partial_{\mu}\psi_{R}+\bar{\psi}_{L}\sigma^{\mu}\partial_{\mu}\psi_{L}) -m (\bar{\psi}_{L}\psi_{R}+\bar{\psi}_{R}\psi_{L}), \label{dirac_chiral}\] where the gamma matrices in the chiral basis are defined as \[\gamma^{\mu} = \begin{pmatrix} 0 & \sigma^{\mu}\\ \bar{\sigma}^{\mu} & 0 \\ \end{pmatrix}. \label{chiralgamma}\] We see that our Dirac mass term in the chiral basis contains both left and right-handed fields. In forming the electroweak theory, this becomes problematic. We have previously touched on how the electroweak interaction is CP-violating, and the left and right-handed components of the spinors interact distinctly. In forming electroweak theory, the left-handed spinors live in a \(U(1)\times SU(2)\) space, while the right-handed spinors (which do not participate in the weak interaction) live in the \(U(1)\) space. Thus terms such as \(\bar{\psi}_{L}\psi_{R}\) are not gauge-invariant, and cannot be included in our Lagrangian. In forming the electroweak Lagrangian, lepton masses and our gauge boson masses must be generated in an alternate way. In 1964, inspired by work done by Anderson (Anderson 1963), Brout, Englert, and Higgs published a model for the production of mass in the electroweak theory, now called the Brout-Englert-Higgs Mechanism (Englert 1964, Higgs 1964). Through a spontaneous local symmetry breaking, the masses for both the \(W^{\pm}\) and \(Z^{0}\) gauge bosons are generated, alongside the lepton masses. We will not go into detail about the mechanism here, but instead consider the final Lagrangian after applying the Higgs mechanism. In terms of the Higgs vacuum expectation value \(v\), the fermion mass terms that appear in the electroweak Lagrangian resulting from the spontaneous symmetry breaking proposed by (Englert 1964, Higgs 1964) are \[%\left(\frac{g_{2} v }{2}\right)^{2} W^{+}_{\mu}{W^{-}}^{\mu} + %\frac{v^2}{8} \begin{pmatrix} W_{3} & B_{\mu} \end{pmatrix} %\begin{pmatrix} g_{1}^{2} & -g_{1} g_{2}\\ -g_{1} g_{2} & g_{2}^{2} \end{pmatrix%} \begin{pmatrix} W_{3}\\B_{\mu} \end{pmatrix} \begin{split} \ldots &-f_{u} \frac{v}{\sqrt{2}}\begin{pmatrix} \bar{u}_{L} & \bar{c}_{L} &\bar{t}_{L}\end{pmatrix}\begin{pmatrix} u_{R}\\c_{R}\\t_{R} \end{pmatrix}\\ &-f_{d} \frac{v}{\sqrt{2}}\begin{pmatrix} \bar{d}_{L}&\bar{s}_{L}&\bar{b}_{L} \end{pmatrix}\begin{pmatrix} d_{R} \\s_{R} \\ b_{R} \end{pmatrix}\\ &-f_{e} \frac{v}{\sqrt{2}}\begin{pmatrix} \bar{e}_{L} & \bar{\mu}_{L} & \bar{\tau}_{L} \end{pmatrix}\begin{pmatrix} e_{R} \\ \mu_{R} \\ \tau_{R} \end{pmatrix} + \ldots, \end{split} \label{higgsgeneratedmasses}\] where \(f_{u}\), \(f_{d}\), and \(f_{e}\) are the Yukawa couplings from the Higgs-fermion interactions. We notice that the neutrino mass terms are absent; this is a direct result of the CP-violation of the electroweak interaction. There has yet to be substantive experimental observation to warrant the inclusion of right-handed neutrinos. Without these right-handed neutrinos, a Dirac mass term in the electroweak Lagrangian remains absent. Is this the only formulation of neutrinos in electroweak theory, and is it physically consistent? Are neutrinos massless?

The discovery of the electron antineutrino by Cowan and Reines in 1956 (Cowan 1956) uncovered a new tool for experimental physics. Though neutrinos were difficult to detect they provided a method of investigating nuclear reactions from within stars (as neutral particles, neutrinos are unaffected by any electric or magnetic fields). Through the neutrino, astrophysicists Raymond Davis and John Bahcall hoped to understand more about the nuclear processes taking place inside our Sun, and designed the Homestake experiment to investigate solar neutrinos originating from these processes. What Davis and Bahcall found was only a third of the neutrinos predicted from the solar models were observed (Cleveland 1998). This discrepancy was to go unsolved for over thirty years, and was known as the “Solar Neutrino Problem”. Since the discovery of the electron antineutrino and the Homestake experiment, physicists have definitively observed two other neutrino “flavours” corresponding with their leptonic partners: the muon neutrino was discovered in 1962 by Lederman, Schwartz, and Steinberger (Danby 1962), while the tau neutrino was not observed until 2000 at Fermilab (Kodama 2001). In the meantime, physicists questioned the assumption in the electroweak theory that neutrinos were massless.

By 1964, it was known that the kaon (a quark-antiquark bound state of a down and a strange quark) could “oscillate” between its particle and antiparticle state through the weak interaction Figure \ref{fig:kaon}. It was proposed by Bruno Pontecorvo (Pontecorvo 1968) that, perhaps in some similar fashion, the neutrino oscillated between its own flavour states. However, in order for neutrino oscillations to take place, neutrinos must carry a non-zero mass, as we will see. Pontecorvo described this as a rotation from a neutrino flavour eigenstate basis to a mass eigenstate basis in perfect analogy to the CKM matrix and the rotation of quarks between their flavour and mass eigenstates. Much like with the CKM Matrix, we may write the relationship between the neutrino flavour eigenstates and mass eigenstates as \[\begin{pmatrix} \nu_{e}\\ \nu_{\mu}\\ \nu_{\tau} \end{pmatrix} =
\begin{pmatrix} U_{e_1} & U_{e_2} & U_{e_3}\\ U_{\mu_1} & U_{\mu_2} & U_{\mu_3}\\ U_{\tau_1} & U_{\tau_2} & U_{\tau_3} \end{pmatrix}
\begin{pmatrix} \nu_{1}\\ \nu_{2}\\ \nu_{3} \end{pmatrix}.
\label{neutrino_osc}\] This mixing matrix with elements \(U_{l_{i}}\) is known as the Pontecorvo–Maki–Nakagawa–Sakata matrix (PMNS matrix). If the off-diagonal elements are zero, the PMNS matrix indicates the mass eigenstates are identical to the flavour eigenstates. We can calculate the probability that a neutrino will change flavour (“oscillate”) (Boehm 1992). Recalling that we can express our time-dependent states using the usual time-evolution operator, consider the neutrino mass eigenstates in (\ref{neutrino_osc}) described in bra-ket notation, \[\Ket{\nu_{\alpha}(t)}= \sum_{i} U_{\alpha i}^{*}e^{-i E_{i} t}\Ket{\nu_{i}},
\label{braketneutrino_osc1}\] implying that (as \(U\) is unitary) \[\Ket{\nu_{i}}= \sum_{\alpha} U_{\alpha i}\Ket{\nu_{\alpha}},
\label{braketneutrino_osc2}\] where \(\nu_{\alpha}\) represents the flavour eigenstates and \(\nu_{i}\) the mass eigenstates. \(U_{\alpha i}\) is the PMNS matrix discussed above. To examine the conditions on flavour oscillations, we look at the probability of a neutrino transitioning from one flavour state to another using our expressions (\ref{braketneutrino_osc1}) and (\ref{braketneutrino_osc2}) ^{1} \[P(\alpha\rightarrow\beta)= |\Braket{\nu_{\beta}(t)|\nu_{\alpha}(t)}|^{2}.
\label{prob_flavourosc1}\] From (\ref{braketneutrino_osc1}) and (\ref{braketneutrino_osc2}), \[\Ket{\nu_{\alpha}(t)}= \sum_{\beta = e,\mu,\tau}\left(\sum_{k} U_{\alpha k}^{*}e^{-i E_{k} t}U_{\beta k}\right)\Ket{\nu_{\beta}}.
\label{braket_neutrino_state}\] For \(p\gg m\), we have \(E \approx p+\frac{m^2}{2p} \approx E+ \frac{m^2}{2E}\). Thus, utilizing (\ref{braket_neutrino_state}), we obtain \[P(\alpha\rightarrow\beta)=\sum_{k,j}U^{*}_{\alpha k}U_{\beta k}U_{\alpha j} U^{*}_{\beta j} e^{-\frac{i t}{2 E}\left( m^{2}_{k} - m^{2}_{j}\right)}
\label{prob_osc}\] This implies that in order for neutrinos to be able to oscillate from one flavour to another (i.e. \(\alpha \neq \beta\)) a non-zero mass difference is required. Though it was theoretically postulated over forty years ago, it was not until 1998 that neutrino oscillations were definitively observed. Super -Kamiokande (Super-K), a Japanese neutrino detector, observed neutrinos changing flavours as they interacted with the Earth’s atmosphere. In 2001 the Sudbury Neutrino Observatory (SNOLAB), a Canadian laboratory with the capacity to distinguish between electron neutrinos and muon or tau neutrinos, observed that neutrinos originating from the Sun were composed of roughly one-third electron neutrinos, with the rest comprising muon and tau neutrinos. With this discovery, the Solar Neutrino Problem was put to rest. Both Super-K and SNOLAB shared the 2015 Nobel prize for their measurement of neutrino oscillation (Media).

Given the problems with incorporating masses into the electroweak theory, how do we introduce neutrino masses into the Standard Model? We saw previously that a Dirac mass term did not occur in the electroweak Lagrangian due to CP-violation. In order for a Dirac mass term to appear in the theory, a right-handed neutrino must exist. However, it must exist in a singlet representation in order to preserve gauge-invariance, and thus not interact weakly or strongly, and (due to their neutral charge) electromagnetically. Though this would make them extremely difficult to detect, they would make interesting candidate for dark matter (Canetti 2013). Another option is if the neutrinos are their own antiparticles. In 1937, Ettore Majorana suggested that it might be possible for a neutral fermion to be its own antiparticle (Majorana 2006); he proposed what is now called the Majorana equation \[i \gamma_{\mu} \partial^{\mu} \psi = m \psi_{c}, \label{majorana}\] where \(\psi_{c}\) is the charge conjugate of the spinor \(\psi\). If this is the case, then we can write down a term in our Lagrangian resembling \[m \bar{\nu_{L}}\nu_{L}^{c} \label{majorana_term}\] This would happen in a very similar fashion as the kaon oscillations described earlier (Kayser 2009); neutrinos could once again “oscillate”, but this time between their particle and anti-particle states. It is worth noting that these solutions to adding massive neutrinos are not mutually exclusive options. It is possible that neutrinos could carry both Dirac and Majorana mass expressions, in which case a combination of the above approaches would be applied.

Though we have not been able to d

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