Quantum Aspects of Gauge Theories, Supersymmetry and Unification, Paris, 1999 PROCEEDINGS Three graviton scattering and recoil effects in Matrix theory A. Refollia ,N.Terzib and D. Zanonb a Instituut voor Theoretische Fysica, Katholieke Universiteit Leuven Celestijnenlaan 200D B-3001 Leuven, Belgium E-mail: [email protected] b Dipartimento di Fisica dell'Universit`adiMilanoand INFN, Sezione di Milano, Via Celoria 16,20133 Milano, Italy E-mail: [email protected] E-mail: [email protected] Abstract: We discuss recent results on three gravitons in M-atrix theory at finite N.Witha 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 . -theory seems to be a consistent quan- that they perfectly reproduce the three gravi- M tum theory which includes eleven dimen- ton scattering in supergravity, both in the di- sional supergravity as a low-energy approxima- rect three-body channel and in the two-body re- tion. Since the introduction of the matrix model coil exchange. The former, which corresponds to of M-theory [1] much progress has been made one-particle irreducible diagrams in supergravity, on the subject. According to the original conjec- has been computed in the first paper of ref. [6] ture, the degrees of freedom of M-theory in the and we confirm that result. We complete the infinite momentum frame are contained in the two–loop effective action calculation and evalu- dynamics of ND0-branes in the N →∞limit. ate two–body exchanges as well. We have found Subsequently it was argued that M-theory with a systematic way to separate the light and heavy one of the lightlike coordinate compactified is in matrix model degrees of freedom which allows to fact equivalent to the super Yang-Mills matrix obtain the full answer. We show that these con- model for finite N [2], [3]. tributions exactly match what expected from the A crucial test of the conjecture consists in the two-loop scattering of two D0-branes in M-atrix comparison of graviton scattering in supergravity theory [5]. with corresponding results from M(atrix) theory In the two body process there are only two ex- in the low energy limit. So far compelling tests of pansion parameters: the relative velocity v and this proposal have been the comparison of two- the inverse of the relative spatial separation r.In body [4, 5] and three-body [6] scattering. In the the low energy limit, from a supergravity compu- latter case there have been also some partial re- tation or directly from M(atrix)theory [4, 5], one sults [7, 8, 9, 10] and opposite claims [11, 12, 13]. obtains the following effective Lagrangian We focus on the M -atrix theory at finite N and explore further its correspondence with eleven dimensional supergravity. In particular we con- sider N = 3 and compute graphs up to two-loops v2 15 v4 225 v6 v8 L = + + +O in (0 + 1)-dimensional Yang-Mills [14]. We find 2R 16R3M 9 r7 64R5M 18 r14 r21 (1) Quantum Aspects of Gauge Theories, Supersymmetry and Unification, Paris, 1999 A. Refollia ,N.Terzib and D. Zanonb T √ T k 1. Three graviton scattering in +θ Dτθ− gθ γ [Xk,θ] (2.1) M(atrix) theory where we have denoted by g the Yang-Mills cou- In the following we will describe the full two loops pling constant and by γi nine real, symmetric effective potential calculation for the case of a gamma matrices satisfying {γi,γj} =2δij .The three body scattering as computed in Matrix the- covariant derivative is defined by ory. √ Our starting point is M-atrix theory at finite Dτ = ∂τ − i g[A, ] (2.2) N and we are interested in calculating the two- × loop effective action and to compare it to that The fields Xi, θ and A are N N hermitian coming from eleven dimensional supergravity. matrices of U(N), with i, j, k =1,2,...,9. We will consider super Yang Mills theory re- Being interested in quantum, perturbative duced to (0 + 1) dimensions with gauge group calculations it is convenient to use the background U(3). We will use a formalism that easily allows field method, which allows to maintain explicit to distinguish between the contributions from a the gauge invariance of the result. To this end direct three-body channel and the ones corre- one expands the action (2.1) around a classical sponding to a two-body recoil exchange. background field configuration Bi, setting Xi → The idea developed by [6] is that true three- Xi + Bi. body contributions are those which depend on After gauge fixing and background splitting all the three relative velocities of the gravitons; one finds the complete action: all the rest should sum up to give a two-body Z n 2 2 scattering. The calculation we present shows in S =Tr dτ (∂τ Xi) − [Bk,Xj] detail how this happens. √ g −2 g[Bk,Xj][Xk,Xj]− [Xk,Xj][Xk,Xj] In the next section we give the explicit form 2 2 2 of the gauged fixed action. The various fields are +∂τA −[A, Bk] − 4i∂τ Bk[A, Xk] − decomposed in terms of components on a U(3) √ √ 2i g∂τ Xk[A, Xk]+2 g[A, Bk][Xk,A] basis of hermitian matrices. The classical back- √ − 2 T − T ground is fixed with the three D-particles hav- g[A, Xk] + θ ∂τ θ i gθ [A, θ] √ T k T k ing relative velocities parallel to each others, or- − gθ γ [Xk,θ]−θ γ [Bk,θ] 2 √ thogonal to the corresponding relative displace- −2G∂˜ G − 2i g∂τ G˜[A, G] τ o ments. The result is analyzed keeping separate √ +2G˜ [Bk, [Bk,G]− g[G, Xk]] (2.3) the two types of contributions mentioned above, i.e. terms which depend on two distinct rela- Here Xi, A and θ are the quantum fluctuations, tive velocities (three-body interaction) and terms G and G˜ are the ghosts, while Bk is the external in which only one relative velocity appears (re- background. coil). Part of the calculations have been per- We want to extract results to be compared to formed with the help of Mathematica. the scattering of three gravitons in supergravity, thus the minimal choice for the Yang-Mills gauge 2. The action group that allows to describe the interaction of three D0-branes, is U(3). In fact the free motion The matrix model is simply obtained by reduc- of center of mass can be factored out; we thus ing (9 + 1)-dimensional U(N) super Yang-Mills deal only with a SU(3) gauge group. [16] to (0 + 1) dimensions. This theory describes asystemofND0-branes [17] in terms of nine 2.1 Cartan Lie algebra for SU(3) bosonic fields Xi and of sixteen fermionic super- partners θ, which are spinors under SO(9). The In order to clearly distinguish between light and Euclidean action is given by heavy Matrix model degrees of freedom it is con- Z n 2 g venient to use the Cartan basis for the Lie algebra S =Tr dτ (Dτ Xi) − [Xk,Xj][Xk,Xj] 2 of SU(3). With such a basis every matrix field is 2 Quantum Aspects of Gauge Theories, Supersymmetry and Unification, Paris, 1999 A. Refollia ,N.Terzib and D. Zanonb decomposed into components as one obtains a α ∗α α Xk ≡ Xk Ha + Xk Eα + Xk E−α α v τ if k =1 Rk = α a α ∗α bk if k>1 A ≡ A Ha + A Eα + A E−α a α ∗α θ ≡ θ Ha + θ Eα + θ E−α Now we can go back to the action in (2.3) and a α ∗α G ≡ G Ha + G Eα + G E−α perform the trace operation explicitly, using the ˜ ≡ ˜a ˜∗α ˜α notation G G Ha + G Eα + G E−α (2.4) X α 2 ≡ α α where H are the generator of Cartan subalge- (b ) bk bk k bra, E are the Cartan step operators and α = X α 2 ≡ α α ≡ 2 2 α 2 α1, α2, α3. (R ) Rk Rk vατ +(b ) k 2.1.1 Background choice For example the terms involving the X fields Now we make a specific choice of the background are found to be: Z configuration, i.e. straight line trajectories for a 2 a ∗α 2 α 2 α SX = dτ Xk (−@τ )Xk +2Xk (−@τ +(R ) )Xk the three particles. This amounts to have Bk in diagonal form with √ αβγ γ α β γ −2 g Rk (Xk Xj Xj α α ˜α ∗α ∗β ∗γ − α α ∗α a Bk =˜vkτ+bk α=1,2,3 (2.5) +Xk Xj Xj ) 2Rk Xj Xj Xk αa α α ∗α a α α ∗α a +Rk Xj Xk Xj αa + Rk Xk Xj Xj αa As mentioned above the free motion of the center − − α ∗α a 2 − a β α γ αβγ of mass can be factored out and ignored imposing g 2Xj Xj (Xk αa) 2Xk βaXj Xk Xj a b β ∗β − a ∗β ∗α ∗γ αβγ X3 X3 +2Xk βaXj βbXj Xk 2Xk βaXj Xk Xj α ˜α − α ∗η β ∗ρ αβγ ηργ α ∗α β ∗β · v˜k =0 bk =0 (2.6) Xk Xk Xj Xj + Xk Xj Xk Xj (α β) α=1 α=1 α ∗α β ∗β −Xk Xj Xj Xk (α · β) We simplify further our calculations considering ¿From the quadratic part of the action one can the case of parallel velocities for all three parti- easily read the mass matrix and obtain the fol- cles, e.g.