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Probing Lorentz violation in $2νββ$ using single electron spectra and angular correlations

We show that the current search for Lorentz invariance violation (LIV) in the summed energy spectra of electrons in $2νββ$ decay can be extended by investigating the single electron spectra and the angular correlation between the emitted electrons. We derive and calculate the LIV contributions to these spectra associated with the anisotropic part of the countershaded operator and controlled through the coefficient $\mathring{a}_{\text{of}}^{(3)}$ and discuss possible signatures that may be probed in experiments. First, we show that some distortion occurs in the single electron spectrum, maximal at small electron energies. Then, we show that other LIV effects may be highlighted by analysing the angular correlation spectra and the ratio between the Standard Model Extension (SME) electron spectra and their Standard Model (SM) forms. We found that these LIV signatures depend on the magnitude of $\mathring{a}_{\text{of}}^{(3)}$, manifest differently for positive and negative values of this coefficient, and become more pronounced as the electron energy approaches the $Q$-value. Finally, we propose an alternative, new method to constrain $\mathring{a}_{\text{of}}^{(3)}$ through the measurement of the angular correlation coefficient. Using this method, and considering only statistical uncertainties, we obtain bounds of $\mathring{a}_{\text{of}}^{(3)}$ at the level of present ones, obtained from summed energy spectra. We show that future experiments can improve these limits significantly. Our study is performed for $^{100}$Mo, but the results hold qualitatively for other nuclei that undergo a double-beta decay. We hope our results will provide additional motivation for the LIV analyses performed in DBD experiments.