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We discuss recent results on three gravitons in M -atrix theory at finite N . With a specific choice of the background we obtain the complete result up to two loops. The contributions from three-body forces agree with the ones presented in recent papers. We evaluate the two–body exchanges as well. We show that the result we have obtained from M -atrix theory precisely matches the result from one-particle reducible tree diagrams in eleven-dimensional supergravity . M -theory seems to be a consistent quantum theory which includes eleven dimensional supergravity as a low-energy approximation. Since the introduction of the matrix model of M -theory [1] much progress has been made on the subject. According to the original conjecture, the degrees of freedom of M -theory in the infinite momentum frame are contained in the dynamics of N D0-branes in the N → ∞ limit. Subsequently it was argued that M -theory with one of the lightlike coordinate compactified is in fact equivalent to the super Yang-Mills matrix model for finite N [2], [3]. A crucial test of the conjecture consists in the comparison of graviton scattering in supergravity with corresponding results from M(atrix) theory in the low energy limit. So far compelling tests of this proposal have been the comparison of twobody [4, 5] and three-body [6] scattering. In the latter case there have been also some partial results [7, 8, 9, 10] and opposite claims [11, 12, 13]. We focus on the M -atrix theory at finite N and explore further its correspondence with eleven dimensional supergravity. In particular we consider N = 3 and compute graphs up to two-loops in (0 + 1)-dimensional Yang-Mills [14]. We find that they perfectly reproduce the three graviton scattering in supergravity, both in the direct three-body channel and in the two-body recoil exchange. The former, which corresponds to one-particle irreducible diagrams in supergravity, has been computed in the first paper of ref. [6] and we confirm that result. We complete the two–loop effective action calculation and evaluate two–body exchanges as well. We have found a systematic way to separate the light and heavy matrix model degrees of freedom which allows to obtain the full answer. We show that these contributions exactly match what expected from the two-loop scattering of two D0-branes in M -atrix theory [5]. In the two body process there are only two expansion parameters: the relative velocity v and the inverse of the relative spatial separation r. In the low energy limit, from a supergravity computation or directly from M(atrix)theory [4, 5], one obtains the following effective Lagrangian L = v 2R + 15 16R3M9 v r7 + 225 64R5M18 v r14 +O ( v r21 )

Three graviton scattering and recoil effects in Matrix theory