Tests of Lorentz invariance using neutrino oscillations

Perturbative Lorentz violation for beam experiments
Neutrino-antineutrino oscillations in reactor experiments

Perturbative Lorentz and CPT violation for neutrino and antineutrino oscillations.
J. S. Díaz, V. A. Kostelecký, and M. Mewes, Phys.Rev. D80, 076007 (2009). [arXiv:0908.1401]

In this paper we develop techniques to search for generic Lorentz violation in neutrino experiments. The question we address is how can we test special relativity using modern neutrino oscillation experiments? Since neutrino oscillations constitute natural interferometers, the goal became to determine the observable signals of Lorentz violation in neutrino oscillation experiments.
In 2004, Kostelecký and Mewes developed this type of techniques for experiments in which the neutrino mass is irrelevant [Phys. Rev. D70, 076002 (2004)]. This is the case of short-baseline experiments, where the oscillation phase $\frac{m^2_{\color{red}{a'b'}}L}{4E}$ is too small to cause neutrino oscillations. In that case, oscillations (if observed) would be driven only by Lorentz violation. On the contrary, in experiments using long baselines such as MINOS, T2K, and NOvA, the oscillation phase driven by neutrino masses is dominant; therefore, Lorentz-violating effects can be treated as perturbative corrections to the mass-driven oscillations. We then use time-dependent perturbation theory and the SME to construct the oscillation probabilities in the presence of Lorentz violation. Conventionally, the propagation and oscillations of three neutrinos are described by a hamiltonian represented by a 3x3 matrix; similarly, the propagation and oscillations of three antineutrinos are described by a hamiltonian represented by another 3x3 matrix. These two matrices are related by a CP or T transformation (one matrix is the complex conjugate of the other) and there is no neutrino-antineutrino mixing. The fact that neutrinos and antineutrinos are decoupled allows to treat them independently. On the contrary, the presence of Lorentz violation taken into account by the SME leads to the mixing between neutrinos, between antineutrinos, and also between neutrinos and antineutrinos. This means that in general, neutrinos and antineutrinos cannot be treated independently and the complete 6x6 hamiltonian must be used. The 6x6 hamilatonian is written as $H=H_0+\delta H$, where the first term corresponds to the conventional hamiltonian and the second part contains all the new terms that break Lorentz invariance. Using standard time-dependent perturbation theory the oscillation probabilities can be constructed as power series of the form

where the zero-order term is the conventional oscillation probability, whereas the others are first and second order corrections in Lorentz violation. The indices $\color{blue}A$ and $\color{blue}B$ span the three flavors of neutrinos and antineutrinos.

Main results
In this paper, we study the effects of perturbative Lorentz and CPT violation on neutrino oscillations dominated by mass mixing. The primary focus is on corrections arising from renormalizable operators for Lorentz violation within effective field theory. The main results can be summarized as follows:

• The presence of Lorentz violation modifies the conventional neutrino and antineutrino oscillation probabilities. It also introduces neutrino-antineutrino oscillations, which violate lepton number.

• Modifications introduced by Lorentz violation include unconventional energy dependence and the sidereal variation of the oscillation probabilities.

• The sidereal variation of the oscillation probability arises as a consequence of the loss of invariance under rotations. This effect appears as a valuable tool to search for Lorentz violation in neutrino oscillation experiments:

• The effects of Lorentz violation grow with the baseline, making long-baseline experiments sensitive probes to test Lorentz invariance. Nonetheless, it is important to emphasize that the geographic location and orientation of the experiment also affects the sensitivity; hence, two experiments having different baselines can have comparative sensitivities because of their different orientations. This feature makes different experiments complementary in performing generic searches of Lorentz violation.

• Lorentz-violating effects that preserve CPT grow with baseline and energy, making long-baseline experiments studying the atmospheric $\Delta m^2$ very sensitive to these type of effects. Nevertheless, it is important to emphasize that experiments studying different oscillation channels are sensitive to different coefficients for Lorentz violation; hence, experiments with different energy and baseline (i.e. reactor vs. beam experiments) are complementary in the coverage of different Lorentz-violating effects that could arise.

• Comparisons between neutrinos and antineutrinos appear as a useful method to study CPT violation in the neutrino sector; however, some CPT-violating effects can be studied using neutrinos only.

• At first order in Lorentz violation there is no neutrino-antineutrino mixing. Neutrino-antineutrino oscillations appear as a second-order effect only.

Application to experiments

The application of the results in this work to long-baseline neutrino oscillation experiments in direct. In fact, illustrative examples are provided for several runing and future experiments including MINOS, T2K, and NOvA. Relevant oscillation probabilities are explicitly obtained for experiments searching for muon neutrino disappearace as well as electron neutrino appearance. For experiments with the capability of studying both neutrinos and antineutrinos, we construct generic quantities to quantify CPT violation. Finally, although neutrino-antineutrino oscillations appear as a second-order effect in Lorentz violation, the corresponding oscillation probabilitites are constructed to be applied to experiments. Its is also discussed that for short-baseline experiments, the formalism reduces to the one developed by Kostelecký and Mewes. In other words, the short-baseline approximation developed by these authors correspond to a limit of our formalism.

Experimental searches

During recent years, different experiments have searched for a sidereal variation in the neutrino oscillation data, one of the key signatures of Lorentz violation. For experiments such as LSND and MiniBooNE, the baseline is too short for our perturbative formalism to be valid thus the short-baseline approximation was used. The same approximation was valid for the study performed by MINOS in 2008 because data from the near detector was used and by IceCube because the very high energy of the atmospheric neutrinos used in the analysis made the mass term irrelevant, just like in short-baseline experiments. In 2010, the MINOS experiment performed the first search of sidereal variation in the neutrino data using the formalism described here. The long distance traveled from Fermilab to the far detector 735 km away allowed this collaboration to constrain coefficients for Lorentz violation at the 10-24 level.

The list of experimental searches for Lorentz violation in neutrinos is the following:

LSND experiment
Tests of Lorentz violation in $\bar{\nu}_\mu\to\bar{\nu}_e$ oscillations,
Phys.Rev.D72, 076004 (2005).

MINOS experiment
Testing Lorentz Invariance and CPT Conservation with NuMI Neutrinos in the MINOS Near Detector,
Phys.Rev.Lett.101, 151601 (2008).
Search for Lorentz Invariance and CPT Violation with the MINOS Far Detector, Phys.Rev.Lett.105, 151601 (2010).
Search for Lorentz invariance and CPT violation with muon antineutrinos in the MINOS Near Detector, Phys.Rev.D85, 031101R (2012).

IceCube experiment
Search for a Lorentz-violating sidereal signal with atmospheric neutrinos in IceCube, Phys.Rev.D82, 112003 (2010).

MiniBooNE experiment
Test of Lorentz and CPT violation with Short Baseline Neutrino Oscillation Excesses, Phys. Lett. B718, 1303 (2013).

Double Chooz experiment
First Test of Lorentz Violation with a Reactor-based Antineutrino Experiment, Phys. Rev. D 86, 112009 (2012).

Mufson & Rebel
The Search for Neutrino-Antineutrino Mixing Resulting from Lorentz Invariance Violation using neutrino interactions in MINOS, Astropart. Phys. 48 78 (2013).

Search for neutrino-antineutrino oscillations with a reactor experiment, Phys. Lett. B 727, 412 (2013).

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Search for neutrino-antineutrino oscillations with a reactor experiment.
J. S. Díaz, T. Katori, J. Spitz, and J. M. Conrad, Phys. Lett. B 727, 412 (2013). [arXiv:1307.5789]

Modern reactor experiments have reported the disappearance of electron antineutrinos $\color{blue}{\overline{\nu}_e}$ after propagating 1-2 kilometers from the source. This observation is interpreted as a nonzero value of the mixing angle $\theta_{13}$, which allows the oscillation of electron antineutrinos $\color{blue}{\overline{\nu}_e}$ into the other two antineutrino flavors $\color{blue}{\overline{\nu}_\mu}$ and $\color{blue}{\overline{\nu}_\tau}$.
One possible signal of Lorentz violation is the mixing between neutrinos and antineutrinos. The loss of invariance under rotations allows for violations of angular momentum conservation, which makes possible the oscillation of an active neutrino into an active antineutrino. This process is forbidden in the conventional model of massive neutrinos.
In this paper, we study the possibility that the disappearance of electron antineutrinos could be in part due to neutrino-antineutrino mixing via the oscillation into the three neutrino states:
$\color{blue}{\overline{\nu}_e}\to\color{red}{\nu_e}$, $\color{blue}{\overline{\nu}_e}\to\color{red}{\nu_\mu}$, and $\color{blue}{\overline{\nu}_e}\to\color{red}{\nu_\tau}$.

In a previous paper [Díaz et al., Phys.Rev.D80, 076007 (2009)], we proposed to use neutrino-oscillation experiments to search for neutrino-antineutrino mixing as a signal of Lorentz violation. In a recent study [Rebel & Mufson, Astropart.Phys. (2013)], our proposal was used to search for the sixty six coefficients producing sidereal variations of the neutrino event rate in the MINOS experiment as a consequence of neutrino-antineutrino oscillations. Fifteen coefficients remained unexplored because their time-independent effects are challenging to study without a precise measurement of the neutrino event energy spectrum.

In the present work, we use data from Double Chooz, which is the only reactor experiment that has made public the measured antineutrino energy spectrum. Electron antineutrinos are produced by two 4.25-MW reactor cores at the Chooz Nuclear Power Station (Ardennes, France), which travel 1.05 km to the detector. The observed disappearance of these antineutrinos is then interpreted as the combination of conventional oscillations into the other two antineutrino states $\color{blue}{\overline{\nu}_\mu}$ and $\color{blue}{\overline{\nu}_\tau}$ and a perturbative effect produced by the breakdown of Lorentz symmetry, which allows for oscillations into the neutrino flavors $\color{red}{\nu_e}$, $\color{red}{\nu_\mu}$, and $\color{red}{\nu_\tau}$. The fit to the antineutrino spectrum allows the study of the very distinctive energy dependence introduced by the Lorentz-violating modification. We find no evidence of neutrino-antineutrino oscillations in the Double Chooz data and set the first limits on the fifteen coefficients describing this unconventional mixing within the framework of the Standard-Model Extension.

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