This paper proves that solutions of the mass‑critical Hartree equation exist for all time and behave like free waves at large times. In the defocusing case the result holds for any initial data with finite mass (that is, any function in L^2). In the focusing case the same conclusion holds provided the initial mass is strictly smaller than the mass of the ground state Q. The authors remove an earlier technical restriction that required solutions to be radially symmetric and thus establish the scattering conjecture at the scaling‑critical regularity for this equation.

The Hartree equation studied here is a nonlinear Schrödinger‑type equation with a nonlocal interaction. Physically it can be seen as a mean‑field model for many interacting bosons, where the nonlinearity involves a convolution with a long‑range potential proportional to |x|^{-2}. The equation is called mass‑critical because its natural scaling leaves the L^2 (mass) norm unchanged. Classical solutions conserve mass, momentum and energy. ‘‘Scattering’’ means that as time goes to plus or minus infinity the solution approaches a solution of the free (linear) Schrödinger equation.

To prove global existence and scattering the authors combine two modern tools for critical dispersive equations. They use the Kenig–Merle concentration‑compactness and rigidity framework and the long‑time Strichartz estimates developed by Dodson. Important ingredients in the analysis include interaction Morawetz type estimates and refined long‑time control of the nonlinear term. A key difficulty they address is the nonlocal nature of the Hartree nonlinearity, which makes several standard arguments for local nonlinearities harder to apply.

One concrete output of the work is a bound on the global spacetime norm of solutions (the norm that controls scattering), with that bound depending only on the mass of the initial data. In the defocusing case this yields global well‑posedness and scattering for every L^2 initial datum. In the focusing case the same conclusion holds under the explicit mass condition M(u_0)<M(Q), where Q is a positive radial ground state solving the associated elliptic equation.