Matter vs. vacuum oscillations at long baseline accelerator neutrino experiments

The neutrino oscillation probabilities at the long-baseline accelerator neutrino experiments are expected to be modified by matter effects. We search for evidence of such modification in the data of T2K and NO$\nu$A, by fitting the data to the hypothesis of (a) matter modified oscillations and (b) vacuum oscillations. We find that vacuum oscillations provide as good a fit to the data as matter modified oscillations. Even extended runs of T2K and NO$\nu$A, with 5 years in neutrino mode $(5 \nu)$ and five years in anti-neutrino mode $(5 \bar{\nu})$, can {\bf not} make a $3~\sigma$ distinction between vacuum and matter modified oscillations. The proposed experiment DUNE, with neutrino and anti-neutrino runs of 5 years each $(5 \nu + 5 \bar{\nu})$, can rule out vacuum oscillations by itself at $5~\sigma$ if the hierarchy is normal. If the hierarchy is inverted, a $5~\sigma$ discrimination against vacuum oscillations requires the combination of $(5 \nu + 5 \bar{\nu})$ runs of T2K, \nova and DUNE.