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’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 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 ˜ ≡ ˜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. along the x1 axis, and relative displace- lowing particle content in the spectrum: ments transverse [7] • 8 complex bosons with mass (Rα)2 where v˜α =06 , v˜α =0 for k>1 1 k α =1,2,3; X9 α˜α α 2 v˜k bk =0 for α=1,2,3 • 1 complex boson with mass (R ) 2vα k=1 where α =1,2,3; α Settingv ˜1 =˜vαthe background matrices become • 20 real, massless bosons;   1  v˜1τ 00 ˜bk 00    2  • 2 complex ghost fields with mass (Rα)2 where B1 = 0˜v2τ0 ,Bk= 0˜bk 0 3 α =1,2,3; 00˜v3τ 00˜bk for k>1. • 2 real, massless ghosts;

We deﬁne relative velocities i α • 1 complex spinor with mass γ Ri where v1 =˜v2−v˜3 and cyclic α =1,2,3; and relative impact parameters • 2 Majorana, massless spinors.

1 2 3 bk = ˜bk − ˜bk and cyclic According to the conjecture strings stretching between D0-branes are associated to massive In terms of these quantities, setting degrees of freedom. X α a Correspondingly one obtains the propagators Rk = αaTr (H Bk ) (2.7) a=1,2 of the various ﬁelds [4].

3 Quantum Aspects of Gauge Theories, Supersymmetry and Uniﬁcation, Paris, 1999 A. Refollia ,N.Terzib and D. Zanonb

α In the next section we will present the two- loop calculation. ∆0

3. The two-loop eﬀective action Figure 1: Correspondence between M(atrix) d.o.f. and D-particle interactions The two-loop contributions are evaluated considering two types of diagrams, the ones involv- For example bosonic propagators are given ing one four-point vertex, the ﬁgure-eight graphs, by and the ones with two three-point vertices, the a b 0 1 ab 0 sunset-type graphs, respectively. In the present = δ δij ∆0(τ,τ ) 2 notation they correspond to (Fig2).

Z ∞ ∗α β 0 1 αβ α 0 α αβ = δ δij ds ∆ (τ,τ ,s) 2 0 β i, j =16 γ We have defined Figure 2: Typical diagrams: the Greek indices cor- 0 − 0 0 − ∆0(τ,τ )=θ(τ τ)(τ τ) (2.8) respond to massive degrees of freedom and r Given this setup, one can distinguish between α 0 −(bα)2s vα ∆ (τ,τ ,s)=e · the graphs that lead to genuine three-body ex- 2πSinh(2svα) 2 2 changes (one–particle irreducible diagrams for the −vαT Tanh(svα)−vαt Coth(svα) e three–graviton scattering in supergravity) and two- body recoil effects (one-particle reducible graphs in supergravity) in a very simple manner. We where we have introduced new time variables need collect terms that depend on two indepen- 1 1 T = (τ + τ 0) t = (τ − τ 0) dent relative velocities on one side, and terms 2 2 that depend only on one relative velocity on the In the fermionic sector the propagators are given other side. by Thus from the figure-eight diagrams one has aT b 0 1 ab 0 <θ (τ)θ (τ ) > = δ ∂τ ∆0(τ,τ ) a three-body exchange whenever the two prop- 2 agators come in with different masses, while a 1 recoil term is produced when the two masses are <θ∗αT (τ)θβ (τ 0) > = δαβ∆α (τ,τ0) 2 F equal. The connection between these graphs and We have defined supergravity diagrams is shown in (Fig 3). For example in the first diagram, since the graph de- 0 − 0 0 − ∆0(τ,τ )=θ(τ τ)(τ τ) pends on a single Eα, we have a string (graviton) stretching between only two D-particles. As we Z ∞ α 0 v α t 1 vαT have seen this corresponds to a string (fluctua- ∆F(τ,τ ) ≡ ds −II + γ 0 Sinh(vαs) Cosh(vαs) tion). The presence of a massless propagator would α α 1 α 0 +b/ Cosh(vαs)+b/ γ Sinh(vαs) ∆ (τ,τ ,s) lead to a vanishing contribution for this tadpole Once the propagators are known the one- kind of diagrams. loop contribution to the effective action is easily ¿From the sunset-type graphs we have direct computed [4] three-body contributions when the three propa- Z gators have three distinct masses (three differ- X3 4 (1) −15 (vα) ent relative velocities out of which two are in- Γ = dT α 2 2 2 7 16 α=1 [(b ) + T (vα) ] 2 dependent). Two-body forces are present when (2.9) two masses are equal, being the third one equal

4 Quantum Aspects of Gauge Theories, Supersymmetry and Uniﬁcation, Paris, 1999 A. Refollia ,N.Terzib and D. Zanonb

αα s2vβ s3vγ × 2Cosh( )Cosh( )− 2 2 s2vβ s3vγ β γ Sinh( )Sinh( ) ∆ (s2)∆ (s3)| 2 2 t=0 αβ and expanding at the leading order X Z 1 2 ΓY = − dT dtds1ds2ds3 (s1vα − s2vβ ) 18 α6=β6=γ 2 2 Figure 3: Correspondence between M(atrix) graphs (s1vα − s3vγ ) (s2vβ − s3vγ ) and supergravity tree diagrams 2 2 2 α β γ (s1vα + s2vβ + s3vγ )∆ (s1)∆ (s2)∆ (s3) to zero as required by momentum conservation. (3.1) exactly reproduces the corresponding term Having decomposed the fields as in (2.4) makes it obtainedinref.[6]. easy to identify the massless particles which are Now we turn to the calculation of the recoil given simply by the diagonal degrees of freedom. effects. Now we present the results obtained using 3.2 Two-body recoil contributions this procedure [14]. First let us consider all the contributions to the three-body forces1 Then we We are considering here all the diagrams not com- will concentrate on the two-loop contributions puted in the previous section, i.e. figure-eight that depend only on one relative velocity vα. graphs with the two propagators carrying the Their complete evaluation is conceptually sim- same mass, and sunset-type graphs with two prop- ple and algebraically manageable within our ap- agators of equal masses and the third one mass- proach. In [14] we have isolated the leading or- less. der term and proved its consistency with the re- α αα sult from two-graviton scattering obtained in [5]. ∆0 This shows without any further ambiguity that α we are dealing with recoil effects.

3.1 Three-body contributions Figure 4: All the remaining graphs As emphasized above one needs to compute all Looking at the spectrum of the various par- the two-loop terms which depend on two distinct 2 ticles it is clear that these contributions depend relative velocities , and this implies that only on a single relative velocity vα and are thus can- propagators associated to off-diagonal degrees of didates to represent supergravity two-body recoil freedom (see (2.4)) will enter this part of the cal- effects. In [14] we have determined them exactly culation. and shown that the above interpretation is in- In [14] we have presented the details of the deed correct. We find calculation, here we summarize the result. The Γrecoil =  final answer can be rearranged as a sum of two α 2 Z α 2 −(b ) (s1+s2) X (b ) V1 + V2 e contributions   − ds1ds2 √ √ 2πvα s1s2 s1 + s2 Γ=ΓV +ΓY (3.1) α 6 Explicit calculations lead to where (up to order vα) we find X Z 2 2 2 2 3 s vβ 3 s3vγ V1 = 75600 + 22680 s1vα + 2520 s1s2vα ΓV = − dT ds2ds3128Sinh ( )Sinh( ) 2 2 2 2 4 4 3 4 β6=γ +22680 s2vα + 630 s1vα − 2100 s1s2vα 1We will follow as much as possible the approach in 2 2 4 3 4 4 4 +10290 s1s2vα − 2100 s1s2vα + 630 s2vα ref. [6] so that a direct comparison can be made in a 6 6 5 6 4 2 6 straightforward manner. Complete agreement with their +139 s1vα + 761 s1s2vα + 1794 s1s2vα 3 3 6 2 4 6 5 6 findings is shown in the next subsection. −947 s1s2vα + 1794 s1s2vα + 761 s1s2vα 2 In sunrise diagrams there is a third relative velocity 6 6 s1s2 + 139 s2vα which is fixed due to momentum conservation 840 (s1 + s2)

5 Quantum Aspects of Gauge Theories, Supersymmetry and Uniﬁcation, Paris, 1999 A. Refollia ,N.Terzib and D. Zanonb

2 2 V2 = −75600 s1 + 302400 s1s2 − 75600 s2 irreducible tree diagrams in supergravity are in 4 2 3 2 2 2 2 −17640 s1vα − 27720 s1s2vα − 277200 s1s2vα agreement with two-loop contributions from M- 3 2 4 2 6 4 atrix theory. −27720 s1s2vα − 17640 s2vα − 6510 s1vα 5 4 4 2 4 In addition we have considered those two-loop +2940 s1s2vα + 68040 s1s2vα 3 3 4 2 4 4 contributions which depend only on one relative +211680 s1s2vα + 68040 s1s2vα velocity between the gravitons. We have shown 5 4 − 6 4 − 8 6 +2940 s1s2vα 6510 s2vα 97 s1vα how these terms give a total sum which is con- 7 6 6 2 6 5 3 6 +2065 s1s2vα − 7695 s1s2vα − 51738 s1s2vα sistent with results from a two-loop calculation 4 4 6 3 5 6 2 6 6 −103508 s1s2vα − 51738 s1s2vα − 7695 s1s2vα of two D-particle scattering [5]. This calcula- 7 6 − 8 6 1 tion gives additional support to the idea that +2065 s1s2vα 97 s2vα 2 3360 (s1 + s2) eleven-dimensional supergravity compactified in Finally the integrations on s1 and s2 can be a null direction is in direct correspondence with performed and summing the two contributions M-atrix theory at finite N. one obtains X 5 − 51975 vα Γrecoil = π α 13 (3.2) References α 65536 (b ) [1] T. Banks, W Fischler, S.H. Shenker, If we write (3.2) in the form Z L. Susskind, Phys. Rev. D55(1997) X 6 5112.[hep-th/9610043] − 225 vα Γrecoil = dT α 2 2 2 7 α 64 ((b ) + vαT ) [2] Leonard Susskind Another Conjecture about (3.3) M(atrix) Theory hep-th/9704080 then this expression can be directly compared [3] N. Seiberg, Why is the Matrix model with the result in (1) where the two graviton correct? Phys. Rev. Lett. 79 (1997) scattering was analyzed: we find that numbers 3577[hep-th/9710009] perfectly match. The expression in (3.3) with [4] K. Becker, M. Becker, A Two-Loop Test of ΓV and ΓY determines the full two–loop contri- M(atrix) Theory Nucl. Phys. B 506 (1997) 48 bution to the three D–particle effective action. [hep-th/9705091] [5] K. Becker, M. Becker, J. Polchinski, A. 4. Conclusions Tseytlin, Higher Order Graviton Scattering in M(atrix) Theory Phys. Rev. D56(1997) 3174 In this talk we have presented some checks of the [hep-th/9706072] finite N M(atrix) theory conjecture [2]. These [6] Y. Okawa, T. Yoneya, Multi-Body Inter- checks consist in comparing the M(atrix) theory actions of D-Particles in Supergravity and quantum low energy effective action in a scat- Matrix TheoryNucl. Phys. B 538 (1999) tering process between gravitons with analogous 67 67 [hep-th/9806108]; Equations of Mo- amplitudes from eleven dimensional supergravity tion and Galilei Invariance in D-Particle compactified on a lightlike circle. DynamicsNucl. Phys. B 541 (1999) 163 In particular we considered a three particle pro- [hep-th/9808188] cess for which it has been determined the full [7] M. Fabbrichesi, G. Ferretti, R. Iengo, Super- two-loop calculation in M-atrix theory. gravity and matrix theory do not disagree on Following [14] we have restricted our attention multi-graviton scattering J. High Energy Phys. to the case of three D-particles whose relative 06 (1998) 2 [hep-th/9806018] velocities are parallel and orthogonal to the cor- [8] W. Taylor, M. Raamsdonk, Three-graviton responding impact parameters. This restriction scattering in Matrix theory revisited Phys. Lett. has been forced on us by the necessity of keeping B 438 (1998) 248 [hep-th/9806066] calculations mageable. First we have found that [9] R. Echols, J. Grey, Comment on Multigraviton this calculation confirms what obtained in [6], Scattering in the Matrix Model Phys. Lett. B showing that the predictions from one-particle 449 (1999) 60 [hep-th/9806109]

6 Quantum Aspects of Gauge Theories, Supersymmetry and Uniﬁcation, Paris, 1999 A. Refollia ,N.Terzib and D. Zanonb

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