  In this paper, we study the global well-posedness and scattering for the wave
equation with a cubic convolution $\partial_{t}^2u-\Delta
u=\pm(|x|^{-3}\ast|u|^2)u$ in dimensions $d\geq4$. We prove that if the radial
solution $u$ with life-span $I$ obeys $(u,u_t)\in L_t^\infty(I;\dot
H^{1/2}_x(\mathbb R^d)\times\dot H^{-1/2}_x(\mathbb R^d))$, then $u$ is global
and scatters. By the strategy derived from concentration compactness, we show
that the proof of the global well-posedness and scattering is reduced to
disprove the existence of two scenarios: soliton-like solution and high to low
frequency cascade. Making use of the No-waste Duhamel formula and double
Duhamel trick, we deduce that these two scenarios enjoy the additional
regularity by the bootstrap argument of [Dodson,Lawrie, Anal.PDE, 8(2015),
467-497]. This together with virial analysis implies the energy of such two
scenarios is zero and so we get a contradiction.
