Lorentz-violating models for neutrino oscillations

To date there is no compelling evidence of Lorentz violation; nonetheless, the observation of some anomalous results in neutrino oscillations led us to consider the breakdown of Lorentz invariance as a possible explanation of these anomalies. Many authors have proposed extensions of the neutrino massive model to accomodate the anomalies by adding more parameters. On the contrary, in this paper we address the problem using a more restricted approach: we use Lorentz violation to construct a model that describes all the well-established neutrino data and some of the anomalies. In other words, we construct a SME-based model that is independent of the massive neutrino model that can do a better job describing the experimental data. In order to make the model even more interesting, we included the restriction of constructing a model with fewer parameters than the coventional description (the coventional massive model uses six parameters). Since the SME is a general framework that contains all possible models based on effective field theory, it was stated in the paper that the model that works "was discovered by a systematic hunt through the jungle of possible SME-based models" and for this reason we called it the puma model. We included nonrenormalizable operators in the theory, which appear as terms proportional to powers of the neutrino energy greater than one in the hamiltonian.

For the construction of the puma model, we propose the texture of the hamiltonian, this is, the form of the $3\times3$ matrix in flavor space that describes the propagation and oscillations of three neutrino flavors. We began supposing at low energies the simplest and more symmetric $3\times3$ matrix that we can write. This is a well-known matrix called democratic because all its entries are equal
                                                          
where $A$ is a function of the energy $E$ of the form $A=m^2/2E$ (here, $m^2$ is a constant). This hamiltonian exhibits three important symmetries: it can be obtained from the Lorentz-invariant part of the SME; therefore, Lorentz and CPT invariance is preserved; aditionally, there is invariance under permutations of the three neutrino flavors, this means that the hamiltonian is left invariant under the action of the discrete group $S_3$.
We then break Lorentz invariance by incorporating a hamiltonian that dominates at high energies
                            

where $\color{blue}B$ and $\color{red}C$ are functions of positive powers of the energy. Both break Lorentz symmetry and $\color{blue}B$ also breaks CPT. It was found that by taking $\color{blue}B=\color{blue}aE^2$ and $\color{red}C=\color{red}cE^5$, then all the established experimental results (solar, atmospheric, long-baseline reactor, and long-baseline accelerator data) can be described at once after a careful choice of the three constant parameters $m^2$, $\color{blue}a$, and $\color{red}c$. For simplicity, the Lorentz-violating effects are taken as isotropic (no direction dependence).

Main results

• The simple form of the texture allows its analytical diagonalization with ease. One of the eigenvalues is null, which indicates that the eigenvalue equation reduces to a quadratic equation. This allows the direct construction of the mixing matrix and then the relevant oscillation probabilities.

• At low energies, the mass term $A$ dominates and the mixing becomes tri-bimaximal. This makes the puma model consistent with solar neutrinos and long-baseline reactor antineutrinos. Notice that the so-called solar mixing angle $\theta_{12}$ is not necessary as an extra parameter; on the contrary, it is derived from the texture.



• At high energies, the other two elements of the texture dominate breaking Lorentz and CPT invariance as well as the $S_3$ flavor symmetry.

• The breaking of the $S_3$ symmetry at high energies is implemented in such a way that the action of one of its $S_2$ subgroups remains as a symmetry. This $S_2$ symmetry is chosen to be in the $\mu$-$\tau$ sector of the hamiltonian, which leads to maximal mixing of atmospheric neutrinos. In other words, the mixing angle $\theta_{23}$ is not necessary as an extra parameter; on the contrary, it is derived from the texture.

• Even though at high energies the dominant terms in the hamiltonian contain positive powers of the energy, the phase of the oscillation probability becomes proportional to $\color{blue}B^2/\color{red}C\propto 1/E$, which allows the good description of the atmospheric-neutrino data. This Lorentz-violating seesaw mechanism was first implemented by Kostelecký and Mewes [Phys. Rev. D70, 031902 (2004)] and allows to mimic the effects of neutrino masses, despite the fact that in the puma model the mass parameter plays no role at high energies.



• The unconventional energy dependence in the hamiltonian makes the mixing angles to be energy dependent. The solar mixing angle at the relevant energies of a few MeV takes the value $\theta_{12}\approx34^\circ$. The remant $S_3$ symmetry and the Lorentz-violating seesaw mechanism push the value of $\theta_{23}\to45^\circ$ at high energies. The unknown (at that time) reactor angle $\theta_{13}$ is expected to take different values for searches using short-baseline reactor experiments (~MeV) and long-baseline accelerator experiments (~GeV). In particular, the puma model indicates that this angles should be larger for the MINOS experiment respect to the value for T2K experiment due to the different energies used. For short-baseline reactor experiments Daya Bay, Double Chooz, and RENO, the puma model predicts a null value.

• The KM plot shows how the puma model (solid lines) approaches the three-neutrino massive model (dashed line) for both low and high energies. This feature allows the puma model to describe neutrino oscillations in these two regimes.

• The Lorentz-violating seesaw mechanism appears in the KM plot as a rapid separation of the two independent eigenvalue differences.

• The KM plot shows when the seesaw mechanism is triggered, the eigenvalue difference that controls $\nu_\mu\to\nu_e$ oscillations (red line) drops rapidly crossing the MiniBooNE experiment, which indices oscillations in this experiment.

• The puma model shows that an oscillation signal should appear in the MiniBooNE experiment at the lowest part of the spectrum (200-400 MeV). This signal is consistent with the anomalous excess observed in this experiment, which cannot be explained by the three-neutrino massive model.

• We finally explore the necessary modifications to the puma model to include the description of the LSND experiment. It is found that a three-parameter enhancement of the hamiltonian with the same texture is consistent with the LSND data. Although this enhancement is somehow contrived, it shows that the description is possible.
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After the development of the puma model, it was found that the texture shown above can be used to build more than one model. This proved that the puma model is one example of a family of puma models. In this paper we explore the main features of the simplest models that can be constructed using the puma texture. The key difference between all these models is the mass dimension $d$ of the operators used to break Lorentz invariance, which lead to the energy dependence $E^{d-3}$. The texture of the hamiltonian is then preserved and the functions ${\color{blue}B}$ and ${\color{red}C}$ are now general functions of neutrino energy arising from operators of dimension $\color{blue}p$ and $\color{red}q$:
         

where $\color{blue}{k^{(p)}}$ and $\color{red}{k^{(q)}}$ are the coefficients for Lorentz violation. In the original puma model the dimension of the operators are $\color{blue}p=5$ and $\color{red}q=8$. At low energies we have preserved the hamiltonian; thus, no changes appear. The differences between different models are mostly noticeable at energies above 20 MeV.

Main results

• The Super-Kamiokande experiment observed a $L/E$ dependence of the oscillation probability [Phys. Rev. Lett. 93, 101801 (2004)]. It can be shown that all puma models can satisfy this condition via the Lorentz-violating seesaw mechanism by requiring the dimensions of the operators satisfy the relation $\color{red}q=2(\color{blue}p-1)$.

• The CPT transformation property of an operator of dimension $d$ is $(-1)^d$. This means that the result above implies that the dominant operator at high energies must be CPT even.

• The same condition can be used to classify the possible puma models by the dimension of the operators $(\color{blue}p,\color{red}q)$ that give raise to the two terms that break Lorentz invariance in the hamiltonian:
            $(\color{blue}3,\color{red}4)$: only renormalizable model (minimal SME)
            $(\color{blue}4,\color{red}6)$: simplest CPT-even model
            $(\color{blue}5,\color{red}8)$: original puma model
            ...
           $(\color{blue}p,\color{red}{2p-2})$

• The only puma model in the minimal SME $(\color{blue}3,\color{red}4)$ exhibits an interesting feature for solar neutrinos. The model is consistent with all the established experimental results and also includes a characteristic dip around 4-6 MeV. This is the only model that exhibits this feature. Although at low energies the puma model is contructed to behave just like the three-neutrino massive model, the operator of dimension $\color{blue}p=3$ leads to an energy-independent contribution in the hamiltonian, which appears at all energies, including the low-energy regime of the solar spectrum.
Interestingly, one month after this paper was released, the SNO experiment reported the study of 8B neutrinos that is consistent with the conventional MSW solution within the uncertainty; nevertheless, the curve predicted by the $(\color{blue}3,\color{red}4)$ puma model improves the fit to the data [arXiv:1109.0763].




• A model derived from the original puma model $(\color{blue}5,\color{red}8)$ was constructed by including a fourth parameter, producing a model $(\color{red}4,\color{blue}5,\color{red}8)$ that could easily explain the tension between neutrino and antineutrino data observed by the MINOS experiment [Phys.Rev.Lett.107, 021801 (2011)].
In September 2011, the MINOS experiment announced that the tension observed has been reduced after taking more data. This makes the necessity of a fourth parameter in the puma model unnecessary and therefore simpler.

A presentation of this work can be found as Three-Parameter Lorentz-Violating Model for Neutrino Oscillations at the 2011 Meeting of the Division of Particles and Fields of the American Physical Society DPF-2011, Providence, RI, August 09-13, 2011 [slides].

In 2012, the Daya Bay reactor experiment announced the discovery of the disappearance of electron antineutrinos over 2 km., corresponding to a nonzero value of the mixing angle $\theta_{13}$ [Phys.Rev.Lett. 108, 171803 (2012)]. This result shows that the puma model in its original form is excluded by the data. Some authors have tried to modify the low-energy structure of the model to make it compatible with the new data [Chin.Phys.Lett.29, 041402 (2012)].



Three-parameter Lorentz-violating texture for neutrino mixing.
J. S. Díaz and V. A. Kostelecký, Phys.Lett. B700, 25 (2011). [arXiv:1012.5985]

Lorentz- and CPT-violating models for neutrino oscillations.
J. S. Díaz and V. A. Kostelecký, Phys. Rev. D85, 016013 (2012). [arXiv:1108.1799]




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