Non-minimal Lorentz invariance violation in light of muon anomalous magnetic moment and long-baseline neutrino oscillation data

In light of the increasing hints of new physics at the muon $g-2$ and neutrino oscillation experiments, we consider the recently observed tension in the long-baseline neutrino oscillation experiments as a potential indication of Lorentz invariance violation. For this purpose, the latest data from T2K and NO$\nu$A is analysed in presence of non-minimal Lorentz invariance violation. Indeed, we find that isotropic violation in dimensions $D =$ 4, 5 and 6 can alleviate the tension in neutrino oscillation data by 0.4$-$2.4$\sigma$ CL significance, with the isotropic coefficient $\gamma^{(5)}_{\tau \tau} =$ 3.58$\times$10$^{-32}$GeV$^{-1}$ yielding the best fit. At the same time, the anomalous muon $g-2$ result can be reproduced with an additional non-isotropic violation of $d^{zt} =$ -1.7$\times$10$^{-25}$. The analysis highlights the possibility of simultaneous relaxation of experimental tensions with Lorentz invariance violation of mixed nature.