Three Graviton Scattering and Recoil Effects in Matrix Theory

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'UniversitàdiMilanoand 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 consider 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 ˜ ≡ ã ˜∗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.

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