The technology required for eikonal scattering amplitude calculations in
Matrix theory is developed. Using the entire supersymmetric completion of the
v^4/r^7 Matrix theory potential we compute the graviton-graviton scattering
amplitude and find agreement with eleven dimensional supergravity at tree
level.Comment: 10 pages, RevTeX, no figure

Andrew K. Waldron

B. de Wit

E. Witten

J. A. M. Vermaseren

J. F. Morales

J. Plefka

J. Polchinski

Jan C. Plefka

M. B. Green

M. Barrio

M. R. Douglas

Marco Serone

P. Kraus

S. Sannan

T. Banks

English

arXiv

Plefka, J.C.

Serone, M.

Waldron, A.K.

NARCIS 

The matrix theory S-matrix

International Migration, Integration and Social Cohesion online publications

UvA-DARE is a service provided by the library of the University of Amsterdam (https://dare.uva.nl)UvA-DARE (Digital Academic Repository)The matrix theory S matrixPlefka, J.C.; Serone, M.; Waldron, A.K.DOI10.1103/PhysRevLett.81.2866Publication date1998Published inPhysical Review LettersLink to publicationCitation for published version (APA):Plefka, J. C., Serone, M., & Waldron, A. K. (1998). The matrix theory S matrix. PhysicalReview Letters, 81, 2866. https://doi.org/10.1103/PhysRevLett.81.2866General rightsIt is not permitted to download or to forward/distribute the text or part of it without the consent of the author(s)and/or copyright holder(s), other than for strictly personal, individual use, unless the work is under an opencontent license (like Creative Commons).Disclaimer/Complaints regulationsIf you believe that digital publication of certain material infringes any of your rights or (privacy) interests, pleaselet the Library know, stating your reasons. In case of a legitimate complaint, the Library will make the materialinaccessible and/or remove it from the website. Please Ask the Library: https://uba.uva.nl/en/contact, or a letterto: Library of the University of Amsterdam, Secretariat, Singel 425, 1012 WP Amsterdam, The Netherlands. Youwill be contacted as soon as possible.Download date:27 Jul 2022VOLUME 81, NUMBER 14 P H Y S I C A L R E V I E W L E T T E R S 5 OCTOBER 1998erlands28Matrix Theory S MatrixJan C. Plefka,1 Marco Serone,2 and Andrew K. Waldron11NIKHEF, P.O. Box 41882, 1009 DB Amsterdam, The Netherlands2Department of Mathematics, University of Amsterdam, Plantage Muidergracht 24, 1018 TV Amsterdam, The Neth(Received 12 June 1998)The technology required for eikonal scattering amplitude calculations in matrix theory is developed.Using the entire supersymmetric completion of they4yr7 matrix theory potential we compute thegraviton-graviton scattering amplitude and find agreement with 11 dimensional supergravity at tree level.[S0031-9007(98)07213-5]PACS numbers: 11.25.Db, 04.60.–m, 04.65.+e, 11.55.–mofnsheryteerheofe-in-opu-vitytede-e-oftoeearveterntetseinM theory, the 11 dimensional quantum theory undelying perturbative strings, has in recent years headlindramatic changes in our understanding of string theoAt large distancesM theory reduces (by definition) to11 dimensional supergravity. According to the matrtheory conjecture of [1] the microscopic degrees of fredom of M theory are described by the largeN limit ofa quantum mechanicalsupersymmetric UsNd Yang-Millsmodel. The model itself arises, on the one hand, asregulating theory of the 11 dimensional supermembra[2] and on the other as the short distance descriptionD0-branes [3,4]. An essential feature of the model is texistence of asymptotic particle states carrying the quatum numbers of the 11 dimensional graviton supermulplet [1,5].A principal test of the matrix conjecture is the comparson of scattering amplitudes in the Yang-Mills quantumechanics with those of 11 dimensional supergraviTo date, typical matrix theory scattering experimeninvolve the comparison of classical gravity source-probactions with the background field effective action of supYang-Mills theory in s1 1 0d dimensions evaluated onstraight line configurations (see [6] for an exhaustive liof references). However, a matrix theory computatioyielding trueS-matrix elements, depending on momentand polarizations of the external particles, has remainelusive. In this Letter we carry out precisely suchcomputation.To this end we construct a matrix theory analog of thLehmann, Symanzik, and Zimmermann (LSZ) reductioformula which relates theS matrix to the background fieldexpansion of the matrix theory path integral. In essencwe have found that theS-matrix elements formed from theasymptotic supergraviton states of [5] induce exactly tboundary conditions in the matrix path integral satisfieby straight line diagonal background field configurationsIn fact, in order to obtain the polarization dependenof scattering amplitudes in matrix theory, it is necessato expand the effective potential in both bosonic anfermionic background fields. There exist, scattered in tliterature, some partial results for the fermionic part othe one loop matrix theory effective potential [7,8]. Her66 0031-9007y98y81(14)y2866(4)$15.00r-edry.ixe-theneofhen-ti-i-mty.tseerstnaedaene,hed.cerydhefewe present the full result which is based on the work[9,10]. Rather than a matrix theory Feynman diagramtourde force, all leading D-brane spin-dependent interactioare obtained by a string theory computation employing tGreen-Schwarz boundary state formalism [11].Combining the effective potential and our matrix theoLSZ reduction formula it is then possible to compueikonal S-matrix elements. As an example we consida graviton-graviton scattering process and find that tmatrix theory scattering amplitude agrees with that11 dimensional supergravity.Before presenting our results and formalism, a few rmarks are in order. Throughout this paper we worktheN ­ 2 sector of the matrix model. Since our computations are, for the time being, restricted to the one loleading terms of matrix theory which are protected by spersymmetry, there is no need to take the large-N limit.The demonstrated impressive agreement of supergraand matrix theory amplitudes at finiteN indeed confirmsthis claim. Despite the fact that this agreement is expecby supersymmetry, our results clearly show that matrix thory is aware of the tensorial structure ofLorentz invariant11 dimensional supergravity. Moreover, the formalism dveloped in the present Letter permits the computationmore general scattering amplitudes that will be crucialunderstand the range of validity of matrix theory. (A mordetailed analysis of the results presented here will appin a forthcoming paper.)LSZ for matrix theory.—The N ­ 2 matrix theoryHamiltonianH ­12 P0mP0m 1 f12$Pm ? $Pm 114 s $Xm 3 $Xnd21i2$Xm ? $ugm 3 $ug (1)is a sum of an interacting SU(2) part describing relatimotions and a free U(1) piece pertaining to the cenof mass. We use a vector notation for the adjoirepresentation of SU(2) ,$Xm ­ sYIm, xmd and $u ­ suI , u3d(with I ­ 1, 2 andm ­ 1, . . . , 9) and may choose a gaugin which YI9 ­ 0. The model has a potential with fladirections along a valley floor in the Cartan sectorxmand u3. The remaining degrees of freedom transverto the valley are supersymmetric harmonic oscillators© 1998 The American Physical SocietyVOLUME 81, NUMBER 14 P H Y S I C A L R E V I E W L E T T E R S 5 OCTOBER 1998nforav--l-d--fe-yn-et.ofe--the variablesYIm (m fi 9) and uI . Upon introducing alarge gauge invariant distancex ­ s $X9 ? $X9d1y2 ­ x9 asthe separation of a pair of particles, the Hamiltonian (was shown [5] to possess asymptotic two particle statesthe formjp1m, H1; p2m, H2l ­ j0B, 0Fl1x9eis p12p2dxeis p11p2dX03 jH 1lu01u3 jH 2lu02u3 . (2)Here p1,2m and H1,2 are, respectively, the momenta anpolarizations of the two particles. The statej0B, 0Fl isthe ground state of the superharmonic oscillators andpolarization states are the44 © 84 © 128 representationof the u0 6 u3 variables, corresponding to the gravitonthree-form tensor, and gravitino, respectively.For the computation of scattering amplitudes onmay now form the S matrix in the usual fashionSfi ­ koutj exph2iHT j jinl with the desired ingoingand outgoing quantum numbers according to (2). (Tasymptotic states above are constructed with respeca large separation in the same direction for both ingoiand outgoing particles, i.e., eikonal kinematics. Mogeneral kinematical situations are handled by introducia rotation operator into theS matrix [12].) The object ofinterest is then the vacuum to vacuum transition amplitueiGsx0m,xm,u3d ­ x0m k0B, 0F j exph2iHT j j0B, 0Flxm . (3)Note that the ground states actually depend on the Carvariablesxm and x0m through the oscillator mass. Alsoboth the left and right hand sides depend on the opetor u3.Our key observation is rather simple. In field theorone is accustomed to expand around a vanishing vacuexpectation value when computing the vacuum to vacuutransition amplitude for some field composed of oscillatmodes. In quantum mechanics the idea is of courexactly the same, and therefore if one is to represe(3) by a path integral one should expand the suposcillators transverse to the valley about a vanishivacuum expectation value. One may then write the mattheoryS matrix in terms of a path integral with the stateboundary conditionseiGsym,bm,u3d ­Z $Xm­s0,0,x0md, $u­s0,0,u3d$Xm­s0,0,xmd, $u­s0,0,u3dD s $Xm, $A, $b, $c, $ud3 exp√iZ Ty22Ty2LSYM!. (4)The LagrangianLSYM is that of the supersymmetricYang-Mills quantum mechanics with appropriate gaugfixing to which end we have introduced ghosts$b, $c andthe Lagrange multiplier gauge field$A. The effectiveaction Gsym, bm, u3d is most easily computed via anexpansion about classical trajectoriesX3mstd ; xclm std ­bm 1 ymt and constantu3std ­ u3 which yields the1)ofdthe,ehet tongrengdetan,ra-yummorsenterngrixdequoted boundary conditions through the identificatiobm ­ sx0m 1 xmdy2 andym ­ sx0m 2 xmdyT .Up to an overall normalizationN , our LSZ reductionformula for matrix theory is simplySfi ­ d9sk0m 2 kmde2ikmkmTy2Zd9x0d9x N exps2iwmx0m 1 iumxmd3 kH 3j kH 4jeiGsym,bm,u3djH 1l jH 2l . (5)The leading factor expresses momentum conservationthe center of mass where we have denotedkm ­ p1m 1p2m andk0m ­ p3m 1 p4m for the ingoing and outgoing par-ticles, respectively, and similarly for the relative momentum ­ sp1m 2 p2mdy2 andwm ­ sp4m 2 p3mdy2.In a loopwise expansion of the matrix theory pathintegral one findsGsym, bm, u3d ­ ymymTy2 1 Gs1d 1Gs2d 1 . . . of which we consider only the first two termsin order to compare our results with tree-level supergraity. Inserting this expansion into (5) and changing variablesd9x0d9x ! d9sTydd9b, the integral overTym maybe performed via stationary phase. Dropping the normaization and the overall center of mass piece theS matrixthen readsSfi ­ e2ifsu1wdy2g2Ty2Zd9b e2iqmbmkH 3j kH 4j3 eiGsloopsdfym­sum1wmdy2,bm,u3gjH 1l jH 2l , (6)where qm ­ wm 2 um. It is important to note that in(6) the variablesu3 are operatorshu3a , u3bj ­ dab whoseexpectation between polarization statesjH l yields thespin dependence of the scattering amplitude.The loopwise expansion of the effective action shoulbe valid for the eikonal regime, i.e., large impact parameter bm or small momentum transferqm. As we shall seebelow, this limit is dominated byt-channel physics on thesupergravity side.D0-brane computation of the matrix theory effective potential.—We must now determine the one-loop effectivematrix potentialGsy, b, u3d, namely, they4yr7 term andits supersymmetric completion. Fortunately the bulk othis computation has already been performed in string thory by [9,10] who applied the Green-Schwarz boundarstate formalism of [11] to a one-loop annulus computatiofor a pair of moving D0-branes. They found that the leading spin interactions are dictated by a simple zero modanalysis and their form is, in particular, scale independenThis observation allows one to extrapolate the results[9,10] to short distances and suggests a matrix theory dscription for tree-level supergravity interactions.Following [9,10], supersymmetric D0-brane interactions are computed from the correlatorV ­116Z `0dt kB, $x ­ 0je22pta0p1sP2i≠y≠x1d3 eshQ21h̃Q̃2deVB jB, $y ­ $bl (7)2867VOLUME 81, NUMBER 14 P H Y S I C A L R E V I E W L E T T E R S 5 OCTOBER 1998dfiedowingwith Q2, Q̃2 being the SO(8) supercharges broken by the presence of the D-brane,jBl the boundary state associatewith D0-branes, andVBHt­0ds sX f1≠sXig 112 Sg1iSd is the boost operator where the direction 1 has to be identiwith the time (see [9,10] for details). Expanding (7) and using the results in section four of [10], one finds the follcompact form for the leading one-loop matrix theory potential (normalizing to one they4 t rm and settinga0 ­ 1)V12loop ­∑y4 1 2iy2ymsugmnud≠n 2 2ypyqsugpmud sugqnud≠m≠n 24i9yqsugqmud sugnkud sugpkud≠m≠n≠p1263sugmlud sugnlud sugpkud sugqkud≠m≠n≠p≠q∏1r7, (8)gof-ofgsewhere u ­ sha, h̃ Ùad should be identified withu3y2 ofthe last section and all indices are nine dimensionThe general structure of this potential was noted in [1and its first, second, and last terms were calculated[14], [7], and [8], respectively. (It would be interestinto establish the supersymmetry transformations of tpotential. However, acceleration terms, set to zero insetup, should then be included [15].)et2868al.3]inhisurResults.—Our matrix computation is completed bytaking the quantum mechanical expectation of the efective potential (8) between the polarization states(6). Clearly one can now study any amplitude involvingravitons, three-form tensors, and gravitini. We chooto compute a h1 1 h2 ! h4 1 h3 graviton-gravitonprocess, and thus prepare statesjinl ­ 1256 h1mnsly1 gmly1 d sly1 gnly1 dh2pqsly2 gply2 d sly2 gqly2 d j2l ,koutj ­ 1256 k2jh4mnsl1gml1d sl1gnl1dh3pqsl2gpl2d sl2gql2d .(9)ant.-ndNote that (following [5]) we have complexified thMajorana center of mass and Cartan spinorsu0 and u3in terms of SO(7) spinorsl1,2 ­ su01 6 u31 1 iu02 6iu32dy2, where6 denotes projection with respect tog9.Actually the polarizations in (9) are seven dimensionbut may be generalized to the nine dimensional case aend of the calculation. We stress that these maneuverslthearepurely technical and our final results are SO(9) covariaThe creation and destruction operatorsly1,2 and l1,2annihilate the statesk2j andj2l, respectively.The resulting one-loop eikonal matrix theory gravitongraviton scattering amplitude is composed of 66 terms a[denoting, e.g.,sqh1h4yd ­ qmh1mnh4nryr and sh1h4d ­h1mnh4nm] is given byonly foragreeensionputationA ­1q2h 12 sh1h4d sh2h3dy4 1 2fsqh3h2yd sh1h4d 2 sqh2h3yd sh1h4dgy21 syh2yd sqh3qd sh1h4d 1 syh3yd sqh2qd sh1h4d 2 2sqh2yd sqh3yd sh1h4d2 2sqh1h4yd sqh3h2yd 1 sqh1h4yd sqh2h3yd 1 sqh4h1yd sqh3h2yd112 fsqh1h4h3h2qd 2 2sqh1h4h2h3qd 1 sqh4h1h2h3qd 2 2sqh2h3qd sh1h4dgy22 sqh2yd sqh3qd sh1h4d 1 sqh2qd sqh3yd sh1h4d 2 sqh1qd sqh2h3h4yd 1 sqh1qd sqh3h2h4yd2 sqh4qd sqh2h3h1yd 1 sqh4qd sqh3h2h1yd 2 sqh1yd sqh4h2h3qd 1 sqh1yd sqh4h3h2qd2 sqh4yd sqh1h2h3qd 1 sqh4yd sqh1h3h2qd 1 sqh1h4qd sqh2h3yd 2 sqh1h4qd sqh3h2yd118 fsqh1qd sqh2qd sh3h4d 1 2sqh1qd sqh4qd sh2h3d 1 2sqh1qd sqh3qd sh2h4d1 sqh3qd sqh4qd sh1h2dg 112 fsqh1qd sqh4h2h3qd 2 sqh1qd sqh2h4h3qd2 sqh1qd sqh4h3h2qd 2 sqh4qd sqh1h2h3qd 1 sqh4qd sqh1h3h2qd 2 sqh4qd sqh2h1h3qdg114 fsqh1h3qd sqh4h2qd 1 sqh1h2qd sqh4h3qd 1 sqh1h4qd sqh2h3qdgj1 fh1 √! h2, h3 √! h4g . (10)We have neglected all terms within the curly brackets proportional toq2 ; qmqm, i.e., those that cancel the1yq2pole. These correspond to contact interactions in the D0-brane computation, whereas this calculation is validnoncoincident branes.D ­ 11 supergravity.—The above leading order result for eikonal scattering in matrix theory is easily shown towith the corresponding 11 dimensional field theoretical amplitude. Tree-level graviton-graviton scattering is dimindependent and has been computed in [16]. We have double-checked that work by a type IIA string theory comVOLUME 81, NUMBER 14 P H Y S I C A L R E V I E W L E T T E R S 5 OCTOBER 1998r-r-e.,onms.Bh/r,n,,and will not display the explicit result here which dependon 11 momentapiM (with i ­ 1, . . . , 4) and polarizationshiMN subject to the de Donder gauge conditionpiNhi NM 2s1y2dpiMhi NN ­ 0 (no sum oni). Matrix theory, on theother hand, is formulated in terms of on shell degreesfreedom only, namely, transverse physical polarizationand euclidean nine momenta.Going to light-cone variables for the eleven momentpiM we take the case of vanishingp2 momentum ex-change, i.e., the scenario of our matrix computation,p1M ­ s212 sym 2 qmy2d2, 1, ym 2 qmy2d,p2M ­ s212 sym 2 qmy2d2, 1, 2ym 1 qmy2d ,p4M ­ s212 sym 1 qmy2d2, 1, ym 1 qmy2d,(11)p3M ­ s212 sym 1 qmy2d2, 1, 2ym 2 qmy2d .[We denotep6 ­ p7 ­ sp10 6 p0dyp2 and our metricconvention ishMN ­ diags2, 1 . . . , 1d.] By transverseGalilean invariance we have set to zero the nine dimesional center of mass momentum. We measure momenin units of p2 which we set to one. For this kinematicalsituation conservation ofp1 momentum clearly impliesymqm ­ 0. Note that the vectorsum and wm of (5) aresimply um ­ ym 2 qmy2 andwm ­ ym 1 qmy2.We reduce to physical polarizations by using thresidual gauge freedom to sethi1M ­ 0 and solve thede Donder gauge condition in terms of the transverstraceless polarizationshimn for which one findshi2M ­2pinhinM .Agreement with the matrix result (10) is then achieveby taking the eikonal limitym ¿ qm of the gravityamplitude in which thet-pole contributions dominate. (Inthe above parametrization, the Mandelstam variables at ­ q2m ­ 22p1MpM4 , s ­ 4y2m 1 q2m ­ 22p1MpM2 , andu ­ 4y2m ­ 22p1MpM3 ­ s 2 t.) One then reproducessofsan-taeedreexactly (10) as long as any pieces canceling thet pole(i.e., the aforementionedq2 terms) are neglected.Although we have presented here only a matrix scatteing amplitude restricted to the eikonal regime, we nevetheless believe the agreement found is rather impressivWe thank B. de Wit, R. Dijkgraaf, S. Moch, K. Peetersand J. Vermaseren for discussions. Our computatimade extensive use of the computer algebra systeFORM [17].[1] T. Banks, W. Fischler, S. Shenker, and L. Susskind, PhyRev. D55, 5112 (1997).[2] B. de Wit, J. Hoppe, and H. Nicolai, Nucl. Phys.B305,545 (1988).[3] J. Polchinski, Phys. Rev. Lett.75, 4724 (1995).[4] E. Witten, Nucl. Phys.B460, 3305 (1996).[5] J. Plefka and A. Waldron, Nucl. Phys.B512, 460 (1998).[6] T. Banks, Nucl. Phys. Proc. Suppl.B67, 180 (1998).[7] P. Kraus, Phys. Lett. B419, 73 (1998).[8] M. Barrio, R. Helling, and G. Polhemus, J. High EnergyPhys.5, 12 (1998).[9] J. F. Morales, C. A. Scrucca, and M. Serone, Phys. Lett.417, 233 (1998).[10] J. F. Morales, C. A. Scrucca, and M. Serone, hep-t9801183.[11] M. B. Green and M. Gutperle, Nucl. Phys.B476, 484(1996).[12] J. Plefka and A. Waldron, hep-th/9801093.[13] J. A. Harvey, hep-th/9706039.[14] M. R. Douglas, D. Kabat, P. Pouliot, and S. H. ShenkeNucl. Phys.B485, 85 (1997).[15] V. Periwal and R. von Unge, hep-th/9801121; S. PabaS. Sethi, and M. Stern, hep-th/9805018.[16] S. Sannan, Phys. Rev. D34, 1749 (1986).[17] J. A. M. Vermaseren,Symbolic Manipulation with FORM,version 2 (Computer Algebra Nederland, Amsterdam1991).2869

The matrix theory S matrix

UvA-DARE

UvA-DARE is a service provided by the library of the University of Amsterdam (https://dare.uva.nl)UvA-DARE (Digital Academic Repository)The matrix theory S-matrixPlefka, J.C.; Serone, M.; Waldron, A.K.DOI10.1103/PhysRevLett.81.2866Publication date1998Published inPhysical Review LettersLink to publicationCitation for published version (APA):Plefka, J. C., Serone, M., & Waldron, A. K. (1998). The matrix theory S-matrix. PhysicalReview Letters, 81, 2866-2869. https://doi.org/10.1103/PhysRevLett.81.2866General rightsIt is not permitted to download or to forward/distribute the text or part of it without the consent of the author(s)and/or copyright holder(s), other than for strictly personal, individual use, unless the work is under an opencontent license (like Creative Commons).Disclaimer/Complaints regulationsIf you believe that digital publication of certain material infringes any of your rights or (privacy) interests, pleaselet the Library know, stating your reasons. In case of a legitimate complaint, the Library will make the materialinaccessible and/or remove it from the website. Please Ask the Library: https://uba.uva.nl/en/contact, or a letterto: Library of the University of Amsterdam, Secretariat, Singel 425, 1012 WP Amsterdam, The Netherlands. Youwill be contacted as soon as possible.Download date:15 Apr 2023VOLUME 81, NUMBER 14 P H Y S I C A L R E V I E W L E T T E R S 5 OCTOBER 1998erlands28Matrix Theory S MatrixJan C. Plefka,1 Marco Serone,2 and Andrew K. Waldron11NIKHEF, P.O. Box 41882, 1009 DB Amsterdam, The Netherlands2Department of Mathematics, University of Amsterdam, Plantage Muidergracht 24, 1018 TV Amsterdam, The Neth(Received 12 June 1998)The technology required for eikonal scattering amplitude calculations in matrix theory is developed.Using the entire supersymmetric completion of they4yr7 matrix theory potential we compute thegraviton-graviton scattering amplitude and find agreement with 11 dimensional supergravity at tree level.[S0031-9007(98)07213-5]PACS numbers: 11.25.Db, 04.60.–m, 04.65.+e, 11.55.–mofnsheryteerheofe-in-opu-vitytede-e-oftoeearveterntetseinM theory, the 11 dimensional quantum theory undelying perturbative strings, has in recent years headlindramatic changes in our understanding of string theoAt large distancesM theory reduces (by definition) to11 dimensional supergravity. According to the matrtheory conjecture of [1] the microscopic degrees of fredom of M theory are described by the largeN limit ofa quantum mechanicalsupersymmetric UsNd Yang-Millsmodel. The model itself arises, on the one hand, asregulating theory of the 11 dimensional supermembra[2] and on the other as the short distance descriptionD0-branes [3,4]. An essential feature of the model is texistence of asymptotic particle states carrying the quatum numbers of the 11 dimensional graviton supermulplet [1,5].A principal test of the matrix conjecture is the comparson of scattering amplitudes in the Yang-Mills quantumechanics with those of 11 dimensional supergraviTo date, typical matrix theory scattering experimeninvolve the comparison of classical gravity source-probactions with the background field effective action of supYang-Mills theory in s1 1 0d dimensions evaluated onstraight line configurations (see [6] for an exhaustive liof references). However, a matrix theory computatioyielding trueS-matrix elements, depending on momentand polarizations of the external particles, has remainelusive. In this Letter we carry out precisely suchcomputation.To this end we construct a matrix theory analog of thLehmann, Symanzik, and Zimmermann (LSZ) reductioformula which relates theS matrix to the background fieldexpansion of the matrix theory path integral. In essencwe have found that theS-matrix elements formed from theasymptotic supergraviton states of [5] induce exactly tboundary conditions in the matrix path integral satisfieby straight line diagonal background field configurationsIn fact, in order to obtain the polarization dependenof scattering amplitudes in matrix theory, it is necessato expand the effective potential in both bosonic anfermionic background fields. There exist, scattered in tliterature, some partial results for the fermionic part othe one loop matrix theory effective potential [7,8]. Her66 0031-9007y98y81(14)y2866(4)$15.00r-edry.ixe-theneofhen-ti-i-mty.tseerstnaedaene,hed.cerydhefewe present the full result which is based on the work[9,10]. Rather than a matrix theory Feynman diagramtourde force, all leading D-brane spin-dependent interactioare obtained by a string theory computation employing tGreen-Schwarz boundary state formalism [11].Combining the effective potential and our matrix theoLSZ reduction formula it is then possible to compueikonal S-matrix elements. As an example we consida graviton-graviton scattering process and find that tmatrix theory scattering amplitude agrees with that11 dimensional supergravity.Before presenting our results and formalism, a few rmarks are in order. Throughout this paper we worktheN ­ 2 sector of the matrix model. Since our computations are, for the time being, restricted to the one loleading terms of matrix theory which are protected by spersymmetry, there is no need to take the large-N limit.The demonstrated impressive agreement of supergraand matrix theory amplitudes at finiteN indeed confirmsthis claim. Despite the fact that this agreement is expecby supersymmetry, our results clearly show that matrix thory is aware of the tensorial structure ofLorentz invariant11 dimensional supergravity. Moreover, the formalism dveloped in the present Letter permits the computationmore general scattering amplitudes that will be crucialunderstand the range of validity of matrix theory. (A mordetailed analysis of the results presented here will appin a forthcoming paper.)LSZ for matrix theory.—The N ­ 2 matrix theoryHamiltonianH ­12 P0mP0m 1 f12$Pm ? $Pm 114 s $Xm 3 $Xnd21i2$Xm ? $ugm 3 $ug (1)is a sum of an interacting SU(2) part describing relatimotions and a free U(1) piece pertaining to the cenof mass. We use a vector notation for the adjoirepresentation of SU(2) ,$Xm ­ sYIm, xmd and $u ­ suI , u3d(with I ­ 1, 2 andm ­ 1, . . . , 9) and may choose a gaugin which YI9 ­ 0. The model has a potential with fladirections along a valley floor in the Cartan sectorxmand u3. The remaining degrees of freedom transverto the valley are supersymmetric harmonic oscillators© 1998 The American Physical SocietyVOLUME 81, NUMBER 14 P H Y S I C A L R E V I E W L E T T E R S 5 OCTOBER 1998nforav--l-d--fe-yn-et.ofe--the variablesYIm (m fi 9) and uI . Upon introducing alarge gauge invariant distancex ­ s $X9 ? $X9d1y2 ­ x9 asthe separation of a pair of particles, the Hamiltonian (was shown [5] to possess asymptotic two particle statesthe formjp1m, H1; p2m, H2l ­ j0B, 0Fl1x9eis p12p2dxeis p11p2dX03 jH 1lu01u3 jH 2lu02u3 . (2)Here p1,2m and H1,2 are, respectively, the momenta anpolarizations of the two particles. The statej0B, 0Fl isthe ground state of the superharmonic oscillators andpolarization states are the44 © 84 © 128 representationof the u0 6 u3 variables, corresponding to the gravitonthree-form tensor, and gravitino, respectively.For the computation of scattering amplitudes onmay now form the S matrix in the usual fashionSfi ­ koutj exph2iHT j jinl with the desired ingoingand outgoing quantum numbers according to (2). (Tasymptotic states above are constructed with respeca large separation in the same direction for both ingoiand outgoing particles, i.e., eikonal kinematics. Mogeneral kinematical situations are handled by introducia rotation operator into theS matrix [12].) The object ofinterest is then the vacuum to vacuum transition amplitueiGsx0m,xm,u3d ­ x0m k0B, 0F j exph2iHT j j0B, 0Flxm . (3)Note that the ground states actually depend on the Carvariablesxm and x0m through the oscillator mass. Alsoboth the left and right hand sides depend on the opetor u3.Our key observation is rather simple. In field theorone is accustomed to expand around a vanishing vacuexpectation value when computing the vacuum to vacuutransition amplitude for some field composed of oscillatmodes. In quantum mechanics the idea is of courexactly the same, and therefore if one is to represe(3) by a path integral one should expand the suposcillators transverse to the valley about a vanishivacuum expectation value. One may then write the mattheoryS matrix in terms of a path integral with the stateboundary conditionseiGsym,bm,u3d ­Z $Xm­s0,0,x0md, $u­s0,0,u3d$Xm­s0,0,xmd, $u­s0,0,u3dD s $Xm, $A, $b, $c, $ud3 exp√iZ Ty22Ty2LSYM!. (4)The LagrangianLSYM is that of the supersymmetricYang-Mills quantum mechanics with appropriate gaugfixing to which end we have introduced ghosts$b, $c andthe Lagrange multiplier gauge field$A. The effectiveaction Gsym, bm, u3d is most easily computed via anexpansion about classical trajectoriesX3mstd ; xclm std ­bm 1 ymt and constantu3std ­ u3 which yields the1)ofdthe,ehet tongrengdetan,ra-yummorsenterngrixdequoted boundary conditions through the identificatiobm ­ sx0m 1 xmdy2 andym ­ sx0m 2 xmdyT .Up to an overall normalizationN , our LSZ reductionformula for matrix theory is simplySfi ­ d9sk0m 2 kmde2ikmkmTy2Zd9x0d9x N exps2iwmx0m 1 iumxmd3 kH 3j kH 4jeiGsym,bm,u3djH 1l jH 2l . (5)The leading factor expresses momentum conservationthe center of mass where we have denotedkm ­ p1m 1p2m andk0m ­ p3m 1 p4m for the ingoing and outgoing par-ticles, respectively, and similarly for the relative momentum ­ sp1m 2 p2mdy2 andwm ­ sp4m 2 p3mdy2.In a loopwise expansion of the matrix theory pathintegral one findsGsym, bm, u3d ­ ymymTy2 1 Gs1d 1Gs2d 1 . . . of which we consider only the first two termsin order to compare our results with tree-level supergraity. Inserting this expansion into (5) and changing variablesd9x0d9x ! d9sTydd9b, the integral overTym maybe performed via stationary phase. Dropping the normaization and the overall center of mass piece theS matrixthen readsSfi ­ e2ifsu1wdy2g2Ty2Zd9b e2iqmbmkH 3j kH 4j3 eiGsloopsdfym­sum1wmdy2,bm,u3gjH 1l jH 2l , (6)where qm ­ wm 2 um. It is important to note that in(6) the variablesu3 are operatorshu3a , u3bj ­ dab whoseexpectation between polarization statesjH l yields thespin dependence of the scattering amplitude.The loopwise expansion of the effective action shoulbe valid for the eikonal regime, i.e., large impact parameter bm or small momentum transferqm. As we shall seebelow, this limit is dominated byt-channel physics on thesupergravity side.D0-brane computation of the matrix theory effective potential.—We must now determine the one-loop effectivematrix potentialGsy, b, u3d, namely, they4yr7 term andits supersymmetric completion. Fortunately the bulk othis computation has already been performed in string thory by [9,10] who applied the Green-Schwarz boundarstate formalism of [11] to a one-loop annulus computatiofor a pair of moving D0-branes. They found that the leading spin interactions are dictated by a simple zero modanalysis and their form is, in particular, scale independenThis observation allows one to extrapolate the results[9,10] to short distances and suggests a matrix theory dscription for tree-level supergravity interactions.Following [9,10], supersymmetric D0-brane interactions are computed from the correlatorV ­116Z `0dt kB, $x ­ 0je22pta0p1sP2i≠y≠x1d3 eshQ21h̃Q̃2deVB jB, $y ­ $bl (7)2867VOLUME 81, NUMBER 14 P H Y S I C A L R E V I E W L E T T E R S 5 OCTOBER 1998dfiedowingwith Q2, Q̃2 being the SO(8) supercharges broken by the presence of the D-brane,jBl the boundary state associatewith D0-branes, andVBHt­0ds sX f1≠sXig 112 Sg1iSd is the boost operator where the direction 1 has to be identiwith the time (see [9,10] for details). Expanding (7) and using the results in section four of [10], one finds the follcompact form for the leading one-loop matrix theory potential (normalizing to one they4 t rm and settinga0 ­ 1)V12loop ­∑y4 1 2iy2ymsugmnud≠n 2 2ypyqsugpmud sugqnud≠m≠n 24i9yqsugqmud sugnkud sugpkud≠m≠n≠p1263sugmlud sugnlud sugpkud sugqkud≠m≠n≠p≠q∏1r7, (8)gof-ofgsewhere u ­ sha, h̃ Ùad should be identified withu3y2 ofthe last section and all indices are nine dimensionThe general structure of this potential was noted in [1and its first, second, and last terms were calculated[14], [7], and [8], respectively. (It would be interestinto establish the supersymmetry transformations of tpotential. However, acceleration terms, set to zero insetup, should then be included [15].)et2868al.3]inhisurResults.—Our matrix computation is completed bytaking the quantum mechanical expectation of the efective potential (8) between the polarization states(6). Clearly one can now study any amplitude involvingravitons, three-form tensors, and gravitini. We chooto compute a h1 1 h2 ! h4 1 h3 graviton-gravitonprocess, and thus prepare statesjinl ­ 1256 h1mnsly1 gmly1 d sly1 gnly1 dh2pqsly2 gply2 d sly2 gqly2 d j2l ,koutj ­ 1256 k2jh4mnsl1gml1d sl1gnl1dh3pqsl2gpl2d sl2gql2d .(9)ant.-ndNote that (following [5]) we have complexified thMajorana center of mass and Cartan spinorsu0 and u3in terms of SO(7) spinorsl1,2 ­ su01 6 u31 1 iu02 6iu32dy2, where6 denotes projection with respect tog9.Actually the polarizations in (9) are seven dimensionbut may be generalized to the nine dimensional case aend of the calculation. We stress that these maneuverslthearepurely technical and our final results are SO(9) covariaThe creation and destruction operatorsly1,2 and l1,2annihilate the statesk2j andj2l, respectively.The resulting one-loop eikonal matrix theory gravitongraviton scattering amplitude is composed of 66 terms a[denoting, e.g.,sqh1h4yd ­ qmh1mnh4nryr and sh1h4d ­h1mnh4nm] is given byonly foragreeensionputationA ­1q2h 12 sh1h4d sh2h3dy4 1 2fsqh3h2yd sh1h4d 2 sqh2h3yd sh1h4dgy21 syh2yd sqh3qd sh1h4d 1 syh3yd sqh2qd sh1h4d 2 2sqh2yd sqh3yd sh1h4d2 2sqh1h4yd sqh3h2yd 1 sqh1h4yd sqh2h3yd 1 sqh4h1yd sqh3h2yd112 fsqh1h4h3h2qd 2 2sqh1h4h2h3qd 1 sqh4h1h2h3qd 2 2sqh2h3qd sh1h4dgy22 sqh2yd sqh3qd sh1h4d 1 sqh2qd sqh3yd sh1h4d 2 sqh1qd sqh2h3h4yd 1 sqh1qd sqh3h2h4yd2 sqh4qd sqh2h3h1yd 1 sqh4qd sqh3h2h1yd 2 sqh1yd sqh4h2h3qd 1 sqh1yd sqh4h3h2qd2 sqh4yd sqh1h2h3qd 1 sqh4yd sqh1h3h2qd 1 sqh1h4qd sqh2h3yd 2 sqh1h4qd sqh3h2yd118 fsqh1qd sqh2qd sh3h4d 1 2sqh1qd sqh4qd sh2h3d 1 2sqh1qd sqh3qd sh2h4d1 sqh3qd sqh4qd sh1h2dg 112 fsqh1qd sqh4h2h3qd 2 sqh1qd sqh2h4h3qd2 sqh1qd sqh4h3h2qd 2 sqh4qd sqh1h2h3qd 1 sqh4qd sqh1h3h2qd 2 sqh4qd sqh2h1h3qdg114 fsqh1h3qd sqh4h2qd 1 sqh1h2qd sqh4h3qd 1 sqh1h4qd sqh2h3qdgj1 fh1 √! h2, h3 √! h4g . (10)We have neglected all terms within the curly brackets proportional toq2 ; qmqm, i.e., those that cancel the1yq2pole. These correspond to contact interactions in the D0-brane computation, whereas this calculation is validnoncoincident branes.D ­ 11 supergravity.—The above leading order result for eikonal scattering in matrix theory is easily shown towith the corresponding 11 dimensional field theoretical amplitude. Tree-level graviton-graviton scattering is dimindependent and has been computed in [16]. We have double-checked that work by a type IIA string theory comVOLUME 81, NUMBER 14 P H Y S I C A L R E V I E W L E T T E R S 5 OCTOBER 1998r-r-e.,onms.Bh/r,n,,and will not display the explicit result here which dependon 11 momentapiM (with i ­ 1, . . . , 4) and polarizationshiMN subject to the de Donder gauge conditionpiNhi NM 2s1y2dpiMhi NN ­ 0 (no sum oni). Matrix theory, on theother hand, is formulated in terms of on shell degreesfreedom only, namely, transverse physical polarizationand euclidean nine momenta.Going to light-cone variables for the eleven momentpiM we take the case of vanishingp2 momentum ex-change, i.e., the scenario of our matrix computation,p1M ­ s212 sym 2 qmy2d2, 1, ym 2 qmy2d,p2M ­ s212 sym 2 qmy2d2, 1, 2ym 1 qmy2d ,p4M ­ s212 sym 1 qmy2d2, 1, ym 1 qmy2d,(11)p3M ­ s212 sym 1 qmy2d2, 1, 2ym 2 qmy2d .[We denotep6 ­ p7 ­ sp10 6 p0dyp2 and our metricconvention ishMN ­ diags2, 1 . . . , 1d.] By transverseGalilean invariance we have set to zero the nine dimesional center of mass momentum. We measure momenin units of p2 which we set to one. For this kinematicalsituation conservation ofp1 momentum clearly impliesymqm ­ 0. Note that the vectorsum and wm of (5) aresimply um ­ ym 2 qmy2 andwm ­ ym 1 qmy2.We reduce to physical polarizations by using thresidual gauge freedom to sethi1M ­ 0 and solve thede Donder gauge condition in terms of the transverstraceless polarizationshimn for which one findshi2M ­2pinhinM .Agreement with the matrix result (10) is then achieveby taking the eikonal limitym ¿ qm of the gravityamplitude in which thet-pole contributions dominate. (Inthe above parametrization, the Mandelstam variables at ­ q2m ­ 22p1MpM4 , s ­ 4y2m 1 q2m ­ 22p1MpM2 , andu ­ 4y2m ­ 22p1MpM3 ­ s 2 t.) One then reproducessofsan-taeedreexactly (10) as long as any pieces canceling thet pole(i.e., the aforementionedq2 terms) are neglected.Although we have presented here only a matrix scatteing amplitude restricted to the eikonal regime, we nevetheless believe the agreement found is rather impressivWe thank B. de Wit, R. Dijkgraaf, S. Moch, K. Peetersand J. Vermaseren for discussions. Our computatimade extensive use of the computer algebra systeFORM [17].[1] T. Banks, W. Fischler, S. Shenker, and L. Susskind, PhyRev. D55, 5112 (1997).[2] B. de Wit, J. Hoppe, and H. Nicolai, Nucl. Phys.B305,545 (1988).[3] J. Polchinski, Phys. Rev. Lett.75, 4724 (1995).[4] E. Witten, Nucl. Phys.B460, 3305 (1996).[5] J. Plefka and A. Waldron, Nucl. Phys.B512, 460 (1998).[6] T. Banks, Nucl. Phys. Proc. Suppl.B67, 180 (1998).[7] P. Kraus, Phys. Lett. B419, 73 (1998).[8] M. Barrio, R. Helling, and G. Polhemus, J. High EnergyPhys.5, 12 (1998).[9] J. F. Morales, C. A. Scrucca, and M. Serone, Phys. Lett.417, 233 (1998).[10] J. F. Morales, C. A. Scrucca, and M. Serone, hep-t9801183.[11] M. B. Green and M. Gutperle, Nucl. Phys.B476, 484(1996).[12] J. Plefka and A. Waldron, hep-th/9801093.[13] J. A. Harvey, hep-th/9706039.[14] M. R. Douglas, D. Kabat, P. Pouliot, and S. H. ShenkeNucl. Phys.B485, 85 (1997).[15] V. Periwal and R. von Unge, hep-th/9801121; S. PabaS. Sethi, and M. Stern, hep-th/9805018.[16] S. Sannan, Phys. Rev. D34, 1749 (1986).[17] J. A. M. Vermaseren,Symbolic Manipulation with FORM,version 2 (Computer Algebra Nederland, Amsterdam1991).2869

Plefka J

Waldron A.

Serone, Marco

Sissa Digital Library

The Matrix theory S matrix

Crossref

Physical Review Letters

Matrix Theory<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mi mathvariant="italic">S</mml:mi></mml:math>Matrix

The Matrix Theory S-Matrix

The Matrix Theory S-Matrix

Abstract

Similar works

Full text

Available Versions

NARCIS

International Migration, Integration and Social Cohesion online publications

NARCIS

UvA-DARE

International Migration, Integration and Social Cohesion online publications

Sissa Digital Library

UvA-DARE

Crossref