Study on neutron-proton effective mass splitting in the transport models

The study of neutron-proton effective mass splitting in neutron-rich nuclear matter holds significant importance for understanding the equation of state of nuclear matter, as well as addressing numerous issues in nuclear and astrophysics. Constraints on effective mass splitting are primarily reflected in the sign of the splitting: whether the neutron effective mass exceeds the proton's (positive splitting) or vice versa (negative splitting). Due to the complexity of nuclear forces in a medium, a consensus on neutron-proton effective mass splitting has not yet been reached.This study reviews three forms of momentum-dependent symmetry potentials within transport models that lead to neutron-proton effective mass splitting, including the logarithmic, Skyrme, and Gogny-type momentum-dependent potentials, along with their origins and development. Observables in heavy-ion collisions that can be used to study neutron-proton effective mass splitting are also discussed. These include isotopic yield ratios (such as neutron-proton ratios and double ratios), isospin diffusion (isospin transport ratio), isospin mixing, collective flow (e.g., neutron-proton direct flow difference and elliptic flow difference), differential collective flow (e.g., differential direct flow and elliptic flow differences), and meson yield ratios (such as π^-/π^+, K^0/K^+, and the double ratio of π^-/π^+). In transport model calculations, adopting a negative nucleon effective mass splitting produces larger neutron-proton yield ratios, more negative elliptic flows, greater neutron-proton differential direct flows, and lower meson yield ratios than adopting a positive nucleon effective mass splitting. Current heavy-ion collision experimental data are unable to provide a consistent determination of nucleon effective mass splitting.