Effects of tau-neutrino detection on non-standard interactions at DUNE with a short discussion on the nature of neutrino mixing

In this paper, we investigate the effects of $\nu_\tau$ and $\bar{\nu}_\tau$ detection at the DUNE far detector on the experiment's sensitivity to Non-Standard Interactions (NSI) in neutrino propagation. We show that the strongest observable NSI effect in the $\nu_\tau$ and $\bar{\nu}_\tau$ appearance probabilities arises from $\epsilon_{\mu\tau}$. We have studied the hierarchy sensitivity, CP violation sensitivity and octant sensitivity of DUNE from $\nu_\tau$ and $\bar{\nu}_\tau$ appearance channels in presence of NSI. We have also studied the detection sensitivity of NSI phases and the future constaints on NSI parameters from the tau neutrino appearance channels in DUNE. Additionally, we examine the role of $\nu_\tau$ detection in constraining the unitary nature of the PMNS matrix. These studies emphasize the importance of incorporating $\nu_\tau$ detection in long-baseline neutrino experiments such as DUNE.