Violations of Lorentz invariance
Introduction to neutrinos
Lorentz-violating neutrinos

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Violations of Lorentz invariance

Lorentz invariance is a cornerstone of modern physics. As the symmetry that underlies Special Relativity, Lorentz invariance corresponds to a foundational constituent of General Relativity and the SM. Lorentz symmetry states that the laws of physics are independent of the orientation and speed of propagation of a system with respect to another used to make a measurement. This concept is usually used indistinguishably with the idea of invariance of the laws of physics under coordinate transformations; nonetheless, whether a physical system exhibits Lorentz symmetry is tied to the invariance under rotations and boosts of the studied system rather than the coordinates used to describe the measurement. The reason these two concepts are used indistinguishably is that in vacuum the two types of transformations are inversely related; nevertheless, this relation is no longer valid when Lorentz symmetry is broken. In other words, in the presence of Lorentz violation the laws of physics remain invariant under coordinate transformation (after all, Nature does not care about the coordinate system used by an observer), but measurable effects appear when the physical system is rotated or boosted with respect to another.

Take for instance a particle with magnetic moment $\color{blue}{\vec{\mu}}$; in vacuum, a rotation can be performed over the coordinates used to describe the system (observer transformation or coordinate transformation) or over the particle (particle transformation). In vacuum, these two different transformations (sometimes called passive and active, respectively) are inversely related

The invariance under coordinate transformations is unrelated to any physics; whereas, the invariance under particle transformations corresponds to the physical symmetry of the system. On the contrary, in the presence of a background field such as a magnetic field $\color{red}{\vec{B}}$, the system remains invariant under observer transformations (in fact, the interaction $U=-\color{blue}{\vec{\mu}}\cdot\color{red}{\vec{B}}$ is invariant under coordinate transformations);

nevertheless, under a particle transformation the system exhibits a particle invariance violation and correspondingly, the symmetry is broken. In other words, the system is physically distinguishable of its transformed version; therefore, the existence of the background field can be inferred in an experiment. Notice that the identification of passive vs. active as observer vs. particle transformations breaks down in the presence of a background field, which remains unchanged under a particle transformation. For this reason the names passive vs. active must be avoided in favor of observer vs. particle transformations. In this example, the broken symmetry is rotation invariance; nevertheless, the same idea can be extended to boosts; hence, to the full Lorentz group of transformations.

In 1989, mechanisms in string field theory were found that could produce the spontaneous breaking of Lorentz symmetry [Phys. Rev. D39, 683 (1989)]. Subsequently, it has been shown that a variety of candidates to quantum theories of gravity, such as loop-quantum gravity and noncommutative geometries, can also accommodate the breakdown of Lorentz invariance. Independent of the underlying theory, it is interesting to search for possible low-energy signatures of Lorentz invariance violation in current experiments

The Standard-Model Extension (SME) [Phys. Rev. D55, 6760 (1997); 58, 116002 (1998)] is an extended version of the SM that allows for minuscule violations of Lorentz invariance that preserve coordinate invariance and that reduces to the conventional SM in the absence of Lorentz violation. Based on effective field theory, the SME is a general framework that allows systematic studies of Lorentz violation in any possible sector of the SM. A subset of elements of the SME also breaks CPT invariance. The SME can also be constructed in a curved spacetime, so Lorentz-violating gravity can also be formally described [Phys. Rev. D69, 105009 (2004)].

In analogy to the $\color{blue}{\vec{\mu}}\cdot\color{red}{\vec{B}}$ interaction term in the example above that is rotationally invariant despite the fact that $\color{red}{\vec{B}}$ breaks the symmetry by setting a preferred direction, interaction terms that break Lorentz invariance while leaving the action invariant under observer transformations (coordinate invariance) can be constructed in the form
$$\mathcal{L}_\text{LV} \supset -\color{red}{a_\mu}\overline\psi\gamma^\mu\psi - \color{red}{b_\mu}\overline\psi\gamma_5\gamma^\mu\psi -\frac{1}{4}\color{red}{(k_F)_{\mu\nu\lambda\sigma}} \,F^{\mu\nu}F^{\lambda\sigma}.$$ The Lorentz-violating terms in the SME are constructed from SM fields properly contracted with coefficients to form observer scalars. These coefficients act like fixed background fields producing observable effects that can be studied in a variety of experiments.

One of the robust properties of the SME is that it allows performing generic experimental searches of Lorentz violation. This property has triggered a worldwide effort in many sectors including neutrino oscillations, oscillation and decay of neutral mesons, matter interferometry, high-energy astrophysical observations, collider experiments, laboratory and gravimetric tests of gravity, particle-antiparticle comparisons, and CMB polarization. To date, no compelling evidence of Lorentz violation exists. All the experimental results in the form of upper limits of coefficients for Lorentz violation are summarized in the Data tables for Lorentz and CPT violation, which are updated every year [Rev. Mod. Phys. 83, 11 (2011)].

Review articles:
- What do we know about Lorentz invariance?, J. D. Tasson, Rept. Prog.Phys. 77, 062901 (2014)
- Overview of the SME: Implications and phenomenology of Lorentz violation, R. Bluhm, Lect.Notes Phys. 702, 191 (2006)

Frequently asked questions on Lorentz and CPT violation
Articles on relativity violations for general public
Beyond Einstein: the search for relativity violations (video)
Relativity violations in neutrinos and photons
Relativity violations in gravity and matter
Relativity violations in gravity
Relativity violations in antihydrogen and Penning traps

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Neutrinos

Neutrinos are elementary particles that come in three different types (or flavors) named after the corresponding charged lepton they interact via weak interactions: electron neutrino $\color{blue}{\nu_e}$, muon neutrino $\color{green}{\nu_\mu}$, and tau neutrino $\color{Purple}{\nu_\tau}$. They are the only fundamental fermions that carry no electric charge and according to the Standard Model (SM) are massless. Invented by Pauli in 1930 to save the principle of conservation of energy in beta decays, they were first detected by Cowan and Reines in 1956. More than half of a century later, we are still learning that these little neutral particles are more misterious than we thought.

Neutrino oscillations

Neutrinos of a determined flavor are created in nuclear reactions accompanied by their corresponding charged lepton. For instance, fusion processes in the solar core produce electron neutrinos $\color{blue}{\nu_e}$ together with positrons when two protons fuse to form deuterium in the reaction $p + p \to\, ^{2}\text{H} + e^++ \color{blue}{\nu_e}$ and when an electron is captured by a berilium nucleus to form lithium in the reaction $^{7}\text{Be}+ e^- \to \,^{7}\text{Li}+\color{blue}{\nu_e}$. During the last decade, we have learned that neutrinos suffer of an identity crisis because they change from one flavor to another as they propagate.

For this reason, some of the electron neutrinos created at the solar core become muon neutrinos and tau neutrinos on their journey towards Earth. This fenomenon was first observed during the 60s and for a long time was known as the problem of the missing solar neutrinos. In 2001, the Sudbury Neutrino Observatory (SNO) verified the transformation of missing electron neutrinos from the Sun into muon and tau neutrinos. Similarly, the decay of energetic muons created by cosmic rays in the upper atmosphere produces muon neutrinos that can get transmuted into tau neutrinos if the travel long distances. In 1998, the Super-Kamiokande experiment confirmed this phenomenon. This change of a neutrino flavor into another is known as neutrino oscillation. The confirmation of these oscillations by several other experiments leads to two conclusions: 1) neutrinos mix just like quarks, meaning that the flavor states are linear superpositions of eigenstates with a well-defined energy and momentum; and 2) the three eigenstates cannot have the same energy-momentum relation. Independent of the underlying mechanism, neutrino oscillations constitute established evidence of physics beyond the SM. In general, the probability of measuring a neutrino flavor $\color{blue}{\nu_b}$ at a distance $L$ from a source of neutrinos of flavor $\color{blue}{\nu_a}$ has the form

where the amplitude of the probability is given by the elements of the mixing matrix and the oscillation phase corresponds to the energy difference of the three eigenstates.

Oscillation of massive neutrinos

Although the SM describes neutrinos as massless particles, a simple mechanism that can give rise to neutrino oscillations is the existence of tiny masses that modify the dispersion relations. The eigenenergies of the three neutrino states become different because of their masses in the form $E_{\color{red}{a'}}=|\pmb{p}|+m^2_{\color{red}{a'}}/2|\pmb{p}|$, which produces the well-know oscillation probability that oscillates with the phase $m^2_{\color{red}{a'b'}}/4|\pmb{p}|$. For instance, in a two-flavor approximation valid for atmospheric neutrinos, the probability of measuring muon neutrinos $\color{green}{\nu_\mu}$ after they travel a distance $L$ is

where $\theta$ is the mixing angle, $E$ is the neutrino energy, and $\Delta m^2$ is the mass-squared difference between the two relevant eigenstates. This description called the three-neutrino massive model leads to the important conclusion that neutrino oscillations are an indication of massive neutrinos. This description is parametrized by two mass-squared differences, three mixing angles, and a CP-violating phase. The mechanism to endow neutrino masses is subject of active research.
Notice that in principle any mechanism that modifies the neutrino dispersion relations in a different manner for each flavor can produce neutrino oscillations. Since oscillations constitute an interferometric phenomenon, tiny effects such as neutrino masses or other unconventional physics can be studied using neutrino oscillations.

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Lorentz-violating neutrinos

In the SME, neutrinos can have mass and also be affected by the breakdown of Lorentz invariance. Novel observable effects arise including direction dependence (due to the loss of rotation invariance), unconventional energy dependence, and neutrino-antineutrino oscillations. The basics of the theory of Lorentz-violating neutrinos in the SME appeared in 2004 [Phys. Rev. D69, 016005 (2004)]. The presence of Lorentz violation modifies neutrino dispersion relations, which opens the possibility that neutrino oscillations can be triggered not only by neutrino mass but also by Lorentz violation. One of the very interesting features of neutrinos in the SME, is that they can oscillate even in the absence of mass. In fact, a very elegant two-parameter model for massless neutrino oscillations was built using the SME framework that reproduced all the neutrino data known at that time [Phys. Rev. D70, 031902 (2004)]. This model known as the bicycle model, served as an example of the potential of the SME to describe experimental data in very simple way. One of the most remarkable features of the bicycle model is the fact that only requires two parameters to describe atmospheric and solar neutrinos. Part of my work in recent years consisted on the formulation of a modern model for neutrino oscillations that could describe all experimental data (see Alternative models for neutrino oscillations). Additionally, phenomena beyond neutrino oscillations can be studied including the effects of Lorentz violation in nuclear reactions involving neutrinos and astrophysical neutrinos.

Below the recording of a general overview of Lorentz and CPT violation in neutrinos.

For a detailed description of the theory and the experimental signatures of Lorentz violation in neutrinos see:

Neutrinos as probes of Lorentz invariance.
J. S. Díaz, Adv. High Energy Phys. 2014, 962410 (2014). [arXiv:1406.6838]

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