Smallness of matter effects in long-baseline muon neutrino disappearance

Current long -baseline accelerator experiments, NOvA and T2K, are making excellent measurements of neutrino oscillations and the next generation of experiments, DUNE and HK, will make measurements at the O(1%) level of precision. These measurements are a combination of the appearance channel which is more challenging experimentally but depends on many oscillation parameters, and the disappearance channel which is somewhat easier and allows for precision measurements of the atmospheric mass splitting and the atmospheric mixing angle. It is widely recognized that the matter effect plays a key role in the appearance probability, yet the effect on the disappearance probability is surprisingly small for these experiments. Here we investigate both exactly how small the effect is and show that it just begins to become relevant in the high statistics regime of DUNE.