We discuss the improvement of the sensitivity of ESSνSB to the discovery of CP violation and to new neutrino physics which can be obtained with a two-detector fit of the data of the near and far detectors. In particular, we consider neutrino non-standard interactions generated by very heavy vector mediators, nonunitary neutrino mixing, and neutrino oscillations due to the mixing of the ordinary active neutrinos with a light sterile neutrino. Contents I. Introduction 1 II. Simulation details 2 III. CP-sensitivity 2 IV. Non-standard interactions 3 V. Nonunitary neutrino mixing 4 VI. Light sterile neutrinos 9 VII. Summary and conclusions 11 References 12