  In this paper, we study the scattering theory for the defocusing
energy-critical Klein-Gordon equation with a cubic convolution $u_{tt}-\Delta
u+u+(|x|^{-4}\ast|u|^2)u=0$ in the spatial dimension $d \geq 5$. We utilize the
strategy in [S. Ibrahim, N. Masmoudi and K. Nakanishi, Scattering threshold for
the focusing nonlinear Klein-Gordon equation. Analysis and PDE., 4 (2011),
405-460.] derived from concentration compactness ideas to show that the proof
of the global well-posedness and scattering is reduced to disprove the
existence of the soliton-like solution. Employing technique from [B. Pausader,
Scattering for the Beam Equation in Low Dimensions. Indiana Univ. Math. J., 59
(2010), 791-822.], we consider a virial-type identity in the direction
orthogonal to the momentum vector so as to exclude such solution.
