Investigating Lorentz Violation with the long baseline experiment P2O

One of the basic propositions of quantum field theory is Lorentz invariance. The spontaneous breaking of Lorentz symmetry at a high energy scale can be studied at low energy extensions like the Standard model in a model-independent way through effective field theory (EFT). The present and future Long-baseline neutrino experiments can give a scope to observe such a Planck-suppressed physics of Lorentz invariance violation (LIV). A proposed long baseline experiment, Protvino to ORCA (dubbed "P2O") with a baseline of 2595 km, is expected to provide good sensitivities to unresolved issues, especially neutrino mass ordering. P2O can offer good statistics even with a moderate beam power and runtime, owing to the very large ($\sim 6$ Mt) detector volume at KM3NeT/ ORCA. Here we discuss in detail, how the individual LIV parameters affect neutrino oscillations at P2O and DUNE baselines at the level of probability and derive analytical expressions to understand interesting degeneracies and other features. We estimate $\Delta \chi^{2}$ sensitivities to the LIV parameters, analyzing their correlations among each other, and also with the standard oscillation parameters. We calculate these results for P2O alone and also carry out a combined analysis of P2O with DUNE. We point out crucial features in the sensitivity contours and explain them qualitatively with the help of the relevant probability expressions derived here. Finally, we estimate constraints on the individual LIV parameters at $95\%$ confidence level (C.L.) intervals stemming from the combined analysis of P2O and DUNE datasets and highlight the improvement over the existing constraints. We also find out that the additional degeneracy induced by the LIV parameter $a_{ee}$ around $-22 \times 10^{-23}$ GeV is lifted by the combined analysis at $95\%$ C.L.