Sum rules of four-neutrino mixing in matter
  Assuming the existence of one light sterile neutrino, we investigate the neutrino flavor mixing matrix in matter. Sum rules between the mixing parameters in vacuum and their counterparts in matter are derived. By using these new sum rules, we obtain the simple but exact expressions of the effective flavor mixing matrix in matter in terms of neutrino masses and the mixing parameters in vacuum. The rephasing invariants, sides of unitarity quadrangles and oscillation probabilities in matter are also achieved. Our model-independent results will be very helpful for analyzing flavor mixing and CP violation in the future long-baseline neutrino oscillation experiments. 
He Zhang