Exploring the effect of Lorentz invariance violation with the currently running long-baseline experiments

Neutrinos are the fundamental particles, blind to all kind of interactions except the weak and gravitational. Hence, they can propagate very long distances without any deviation. This characteristic property can thus provide an ideal platform to investigate Planck suppressed physics through their long distance propagation. In this work, we intend to investigate CPT violation through Lorentz invariance violation (LIV) in the long-baseline accelerator based neutrino experiments. Considering the simplest four-dimensional Lorentz violating parameters, for the first time, we obtain the sensitivity limits on the LIV parameters from the currently running long-baseline experiments T2K and NO$\nu$A. In addition to this, we show their effects on mass hierarchy and CP violation sensitivities by considering NO$\nu$A as a case study. We find that the sensitivity limits on LIV parameters obtained from T2K are much weaker than that of NO$\nu$A and the synergy of T2K and NO$\nu$A can improve these sensitivities. All these limits are slightly weaker ($2 \sigma$ level) compared to the values extracted from Super-Kamiokande experiment with atmospheric neutrinos. Moreover, we observe that the mass hierarchy and CPV sensitivities are either enhanced or deteriorated significantly in the presence of LIV as these sensitivities crucially depend on the new CP-violating phases. We also present the correlation between $\sin^2 \theta_{23}$ and the LIV parameter $|a_{\alpha \beta}|$, as well as $\delta_{CP}$ and $|a_{\alpha \beta}|$.