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FortschrittederPhysik,17January2012 Exploration of the Tree-Level S-Matrix of Massless Particles PaoloBenincasa∗ Departamento de F´ısica de Part´ıculas, Universidade de Santiago de Compostela, E-157782, Santiago de Compostela,Spain Keywords ScatteringAmplitudes,PerturbationTheory 2 Inrecentyears,theBCFWconstructionprovidedaverypowerfultoolforcomputingscatteringamplitudes 1 aswellasitshedlightontheperturbationtheorystructure. Inthistalk,Idiscussthelong-standingissue 0 of the boundary term arising when the amplitudes do not vanish as some momenta are taken to infinity 2 alongsomecomplexdirection. Inparticular,weprovideanewsetofon-shellrecursionrelationsvalidfor n suchtheoriesanddiscussitsconsequencesonourunderstandingontheperturbationtheorystructureofthe a S-Matrix. J 6 Copyrightlinewillbeprovidedbythepublisher 1 ] h 1 Introduction t - p Interactingtheoriesinasymptoticallyflatspacetimecanbestudiedatweakcouplingthroughtheanalysis e h ofthescatteringamplitudes,whichprovidetheprobabilitythatacertainnumberofasymptoticstatesscatter [ toproduceotherasymptoticstates. ThebestwaythatscatteringamplitudeshavebeenunderstoodsofarisbymeanoftheFeynmandiagram 1 v representation,whicharediagrammaticrulescomingfromaLagrangianformulationofthetheories.Such 1 arepresentationmakesmanifesttwobasicpropertiesofthetheories:Poincareínvarianceandthelocalityof 9 theinteractions.Thereare,however,someside-effects.Firstofall,theindividualFeynmandiagramsmay 1 breakothersymmetriesofthetheories(e.g. itisindeedthecaseforgaugesymmetries). Furthermore,the 3 numberofdiagramsdramaticallyincreaseswiththeincreaseofthenumberofexternalstates,makingthe . 1 calculationmoreandmorecumbersome.Finally,thesimplicityofthescatteringamplitudecanbehidden, 0 asithappensforexampleinthecaseofscatteringofgluonsandgravitonsattreelevel.Intheformercase, 2 afterhavingsummedupahighnumberofFeynmandiagrams,then-gluonamplitudesturnsouttobezero 1 : ifalltheexternalgluonsareinthesamehelicitystate(oratmostonegluonisinadifferenthelicitystate v withrespecttotheothers),whileitacquirethesimpleParke-TaylorformforMHVamplitudes[1]. Inthe i X lattercase,instead,thecomputationofthen-gravitonamplitudewouldrequiretokeepintoaccountallthe r k-pointvertices(withk ≤ n). a Itisthereforefairtoaskwhetheritispossibletoformulatesomeotherrepresentationwhichcanpoint out the simplicity of the scattering amplitudes, and to which extent we really understand perturbation theory. Thequestionabouttheexistenceofotherdiagrammaticrepresentationscanbealreadyansweredposi- tively.AfirstexampleistheCSW-expansion[2],whichischaracterisedbyoff-shelldiagramsconstructed outofjustMHV-verticesandmakesmanifestlocalitywhiletheLorentzinvarianceofthetheoryisbroken by the individualdiagrams. This representation is however not general and it holds just for Yang-Mills theories. A second (and more general) example is provided by the BCFW-construction [3,4], which is instead anon-shelldiagrammaticexpansioninwhich thegaugeinvarianceofthe theoryis notbrokenat intermediatestages, Lorentzinvariancestaysmanifest. Thepriceonepaysisthattheindividualon-shell diagramsbreakslocality. ∗ e-mail:[email protected]. Copyrightlinewillbeprovidedbythepublisher 2 P.Benincasa:ExplorationoftheTree-LevelS-MatrixofMasslessParticles It is instead probablyneededa deeperunderstandingof the perturbativestructureof interactingtheo- ries, evenat tree level. First of all, one can try to use Occam’s razor and try to understandwhich is the minimalsetofassumptionstoformulateageneralS-matrixtheoryinflatspace. Inparticular,theBCFW- constructionseemstosuggestthatsucha minimalsetcanbeformedbyPoincare´ invariance,analyticity, Existence of one-particlestates, Locality of the wholeS-matrix [5]. Thisfact, togetherwith its intrinsic generalityasamethod,suggeststheBCFW-constructionasagoodstartingproblemtoconcretelyapproach thisproblem. 2 BCFW-construction A way to understand the structure of scattering amplitudes is through the analysis of their singularities. Thiscanhoweverbeareallydifficultproblemtoface,giventhatthescatteringamplitudescanbeseenas analytic functionsof Lorentzinvariantsand the numberof Lorentzinvariantsincreaseswith the number of externalstates. A drastic simplification can be obtainedby introducinga 1-parameterdeformationof the complexified momentum space, such that the both the massless on-shell condition p2 = 0 and the momentumconservationarepreserved[4]. Adeformationwithsuchcharacteristicsisindeednotunique, andthesimplestoneisobtainedbydeformingthemomentaoftwoparticles(namelylabelledbyiandj), leavingtheonesoftheothersunchanged: p(i)(z) = p(j)−zq, p(j)(z) = p(j)+zq, p(k)(z) = p(k), ∀k 6= {i, j}, (1) withtheon-shellconditionfixingthemomentumqtosatisfythefollowingrelations: q2 = 0, p(i)·q = 0 = p(j)·q. (2) With such a deformation, the amplitudes are mapped into a 1-parameter family of amplitudes M → n M(i,j)(z). One can thereforeanalyse the singularitystructure of the amplitudesas a functionof z only. n Generallyspeaking,thescatteringamplitudesarecharacterisedbybothpolesandbranchpoints.Focusing onthepolestructureishoweverequivalenttofocusingonthetreelevelandthepolesareprovidedbythe z-dependentinternalpropagators,i.e.theyarepresentinthosechannelswheretheinternalmomentumcan bewrittenasasumcontainingeitherp(i) orp(j) 1 1 P2 = ⇒ z = Ik . (3) [P2 (z)]2 P2 −2z P ·q Ik P ·q Ik Ik Ik Ik Asthelocationofapoleisapproac(cid:0)hed,the(cid:1)internalpropagatorgoeson-shell,andtheamplitudefactorises, withtheresidueofthepolewhichisgivenintermsoftheproductoftwoon-shellsub-amplitudes M(i,j)(z) z→∼zk ML(i,j)(zIk)MR(i,j)(zIk). (4) n P2 (z) Ik Thissuggeststhepossibilitytoconnectthewholeamplitudetotheon-shelllower-pointamplitudes 1 dz M(i,j)(z )M(i,j)(z ) 0 = M(i,j)(z) = M(i,j)(0)− L k R k −C(i,j), (5) 2πi z n n P2 n IR k∈XP(i,j) k wheretheintegrationisperformedalongthewholeRiemannsphereR,M(i,j)(0)coincideswiththephys- n icalamplitudeandtheboundarytermC(i,j) istheresidueofthesingularityatinfinity,whichiszeroifthe n amplitudevanishesasz → ∞(constructibilitycondition). Insuchacase,(5)impliesthattheamplitude canbewrittenassumofproductsofon-shelllower-pointsamplitudes M(i,j)(î,I ,−Pˆ )M(i,j)(Pˆ ,J ,ˆj) M = L k iIk R iIk k . (6) n P2 k∈XP(i,j) iIk ThisstructurehasbeenshowntoholdinYang-Millstheory[4], inGR[6], aswellasforN = 4 Super- symmetricYang-MillstheoryandN =8Supergravity[7]. Copyrightlinewillbeprovidedbythepublisher fdpheaderwillbeprovidedbythepublisher 3 3 BCFWconstruction: generalisedrecursive relations Therearehoweverseveraltheories(e.g.:Q.E.D.,Einstein-Maxwell)whichdonotsatisfytheconstructibil- ityconditionand,therefore,theboundarytermisnon-zero. Inthiscase,the1-parameterfamilyofampli- tudesM(i,j)(z)acquirestheform n M(i,j)(z )M(i,j)(z ) M(i,j)(z) = L k R k +C(i,j)(z). (7) n P2(z) n k∈XP(i,j) k Sincethefirsttermin(7)containsallthepolesatfinitelocation,thefunctionC(i,j)(z)isjustapolynomial n inzoforderν, ν C(i,j)(z) = C(i,j)+ a(i,j)zl, (8) n n l l=1 X whereν is aswelltheorderthe amplitudeM(i,j)(z)divergeswith, asz istakento infinity,andthe0-th n ordertermistheonlywhichsurvivesin(5)contributingtothephysicalamplitude. Thequestionwenow needtoanswerishowtoactuallycomputetheboundaryterm1. Theknowledgeofthepolesdoesnotseem to be enough to fully determine the amplitude when the constructibility condition is not satisfied. This means that we need to resort to some new quantity. Given that a tree-level amplitude is just a rational function,thenaturalquantitiestoconsidertogetherwithpolesarethezeroes[10]. Let{z(s)}besubsetof 0 zeroeswithmultiplicitym(s)andγ(s)beacontourincludingjustz(s)andnootherzeroorpoles. Fromthe 0 0 expressions(7)and(8),onegetsthefollowingsystemofequations Nfin 1 M(i,j)(z) P M(i,j)(z )M(i,j)(z ) 0 = dz n = (−1)r−1 L k R k +δ C(i,j)+ 2πiIγ0(s) z−z0(s) r Xk=1(−2Pk·q) z0(s)−zk r r,1 n (9) ν (cid:0) (cid:1) l! (cid:0) (cid:1) r = 1,...,m(s) + a(i,j)zl−r+1, with , (l−r+1)!(r−1)! l 0 s = 1,...,nz l=1 (cid:26) X whosesolutionrevealsaconnectionbetweenC(i,j)andasumofproductsofon-shellscatteringamplitudes n withfewerexternalstates. Theexplicitexpressionitself(see[10])isnotparticularlyilluminating. How- ever,onceitisreinsertedin(7),itallowstorewritethescatteringamplitudeinsuchawaythattheoverall structureoftheBCFWexpansionisstillpreserved[10] f(ν,n) M = M(i,j)(î,I ,−Pˆ ) iIk M(i,j)(Pˆ ,J ,ˆj), (10) n L k iIk P2 R iIk k k∈XP(i,j) iIk withthe“weights”f(ν,n)being iIk 1, ν < 0, f(ν,n) =  (11) iIk  νl=+11 1− P2Pi2Izk(l) , ν ≥ 0, (cid:18) iIk(cid:16) 0 (cid:17)(cid:19) Therecursionrelation(Q10)allowstostatethattheBCFW-structureisgeneralisedtoanyconsistenttheory at tree level, meaning that, for any consistent theory, the amplitudes can be expressed in terms of sum of products of lower-point on-shell amplitudes and propagators, now with a simple weight (11) which dependson the locationofa subsetof zeroesof the amplitudes. Iteratingthe recursiverelation, onecan 1 Somestepsinthisdirectionhasbeendoneinspecificcasesin[8,9] Copyrightlinewillbeprovidedbythepublisher 4 P.Benincasa:ExplorationoftheTree-LevelS-MatrixofMasslessParticles expressann-pointamplitudesjustintermsofpropagatorsandthesmallestamplitudepresentinthetheory. For theories with 3-particle interactions, this is indeed the 3-particle amplitude, which is determinedby momentum conservation and Lorentz invariance [5]. In the case instead the smallest interaction is an higherpointone,generallyspeakingonecanalwaysintroduceanauxiliarymassiveparticlestodefine3- particleinteractionsandthanintegrateitout,asithasbeendoneinthecaseofλφ4 [5]. Evenifthismight not be computationallyconvenient,it pointsout that also these theoriescan be determinedby 3-particle amplitudes. 4 Zeroes and collinear/multiparticlelimits If on one side (10) is a valid mathematical expression which reveals a general structure for tree-level amplitudes,ontheothersideitseemstobenotreallypracticalgiventhatageneralwaytodeterminethe zeroesisnotknown.However,therecursionrelation(10)providesarepresentationfortheamplitudesand, therefore,mustfactoriseproperlywhencollinear/multiparticlelimitsaretaken.Suchlimitscanbegrouped infourclasses lim P2 M = M(i, I , −P )M(P ,J ,j), Pi2Ik→0 iIk n k iIk iIk k lim P2M = M (K, −P )M (P ,Q,i,j), K n s+1 K n−s+1 K PK2 →0 (12) lim P2 M = M (k , k , −P )M (P ,K,i,j), Pk21k2→0 k1k2 n 3 1 2 k1k2 n−1 k1k2 lim P2M = M (i, j, −P )M (P ,K), Pi2j→0 ij n 3 ij n−1 ij andtheiranalysisleadstothefollowingconditionsofthezeroes P2(z(l)) = hi,kiα(l)[i,j], P2 (z(l)) = hi,jiα(l)[j,k], ik 0 ik jk 0 jk lim f(ν,n) = f(ν,n−s+1), lim f(ν,n) = 1, PK2→0 iIk iIk Pi2Ik→0 iIk lim f(ν,n) = f(ν,n−1), lim f(ν,n) = f(ν,n−1), [k1,k2]→0 ik¯ i(k1k2) hk1,k2i→0 jk¯ j(k1k2) (13) hi,ki δ−1 [i,j] 2hi+δ−ν H(k) lim (−1)2(hi+hj+hk)+δ+ν+1 n−1 = 1, [i,j]→0 k "(cid:18)hi,ji(cid:19) (cid:18)[i,k](cid:19) νl=+11αi(lk)# X [j,k] δ−1 hi,ji δ−2hj−ν HQ˜(k) lim (−1)2(hi+hk)+δ+ν+1 n−1 = 1, hi,ji→0 k "(cid:18)[i,j](cid:19) (cid:18)hj,ki(cid:19) νl=+11αj(lk)# X wherethenotationhasbeendetailedin[10]. Here,itisimportanttoknowjQustthatHisadimensionless helicityfactor,δisthenumberofderivativesofthe3-particleinteractions,andthattheBCFW-deformation (1) has been implemented by considering the bispinorial representation of the momenta p = λ λ˜ aa˙ a a˙ and shifting the spinorsas λ˜(i)(z) = λ˜(i) −zλ˜(j), λ(j) = λ(j) +zλ(i). The relations(13) suggeststhe possibilitytoconnectthe“weights”ofthen-particleamplitudeswiththeonesofthelowerpointones. At present,suchageneralconnectionisstillmissing. However,theconditions(13)aresolvableforanumber ofcases2. 5 Softlimitsof3-particleamplitudes andcomplex-UV behaviour The analysis in the previous section points out that the collinear limit P2 → 0 appears in the BCFW- ij representationasasoftsingularity[12],anditcanbetakenaspˆ(i) → 0orpˆ(j) → 0. Furthermore,inthis 2 Afurtheranalysisofthezeroeswasrecentlydonein[11]. Copyrightlinewillbeprovidedbythepublisher fdpheaderwillbeprovidedbythepublisher 5 limittheonlyon-shelldiagramswhichcontributearetheonesshowingathree-particleamplitudewithone ofthedeformedmomenta,thatisthemomentumwhichbecomessoft. Ifthesofteningofthismomentum isabletoproducethecorrectpolethantheamplitudefactorisesproperlyinthischannelandthestandard BCFW-representationisvalid,otherwisearetheweightswhichhavetocontributetotheformationofthe rightsingularity,as implied bythe first line in (13). The softbehaviourof the 3-particleamplitudescan therefore be taken as a criterion for the validity of the standard BCFW-representation. The analysis of the collinear limit P2 → 0 on the generalised BCFW-representation (10) allows to prove that such a ij soft-behaviourcoincideswiththelarge-zbehaviourofthewholeamplitude[13]. Arigorousproofofthis equalityinthecaseofthestandardBCFW-representationisnotknown,butithasbeencheckedforallthe knowncases. Suchabehaviourcanbegenerallywrittenjustintermsofthenumberofthederivativesof the3-particleinteractionandthehelicityofthedeformedparticles ν = δ+2h and/or ν = δ−2h , (14) i j which are respectively valid if the collinear limit P2 → 0 translates to the soft limit pˆ(i) → 0 or to ij pˆ(j) → 0,whileinthecaseboththesesoftlimitscanebetaken,thetworelationsin(14)coincide. This strikinglyimpliesthatalso theinformationaboutthecomplex-UVbehaviourofthe n-particleamplitude ofagiventheoryisencodedintheirbuildingblocksanditisindependentonthenumberofexternalstates. 6 Explorationofthespace oftheories Ithasbeennoticedin[13]thattheconditions(13)drasticallysimplifiesinthecaseofthe4-particleamplitudes Nfin P Pi2sr(z0(l)) = (−1)NPfin Pi2j NPfin, (15) r=1 Y (cid:0) (cid:1) whereNfinisthenumberofpolesatfinitelocation,whichcanbejustoneortwo.Suchaconditionallowsto P generalisethefour-particleconsistencytestproposedin[5].Theideaistocomputethe4-particleamplitude throughtwo differentBCFW-deformations. Imposing that the two results coincide bringson non-trivial constraintsontheS-matrix[5] M(i,j)(0) = M(i,k)(0). (16) 4 4 In[5]thistestallowedtorediscovertheJacobi-identityinYang-MillsandN = 1Supergravity,inwhich boththegaugesymmetryandsupersymmetryemergefromtheconsistencyofthetheory. Thegeneralised BCFW-representation, together with the condition on the zeroes (15), allows to rediscover the existent theories not satisfying the constructibility condition, which turn out to belong to the class of theories characterisedbythe3-particleamplitudeswith(∓s,∓s,±s)and(∓s′,±s′,∓s)(seetable1). Thetheoriescanbegenericallyclassifiedthroughthedimensionofthe3-particlecouplingconstant,or equivalentlythrough the numberof derivative of the 3-particle interaction, which implies that 3-particle amplitudesarecharacterisedbywell-definedhelicityconfigurations. Theconsistencytestseemstoreveal theappearanceofunexpectedinteractionsinwhichthespinoftheparticlesishigherthan2. Somecom- mentsarenowin order. The4-particleamplitudesinthese theoriesareallcharacterisedbythe presence of just one factorisation channel. This meansthat the representation(10) is still meaningfulif and only if the deformationchosenhassuch a channelas a BCFW-channel. Ifit wereotherwise, the 1-parameter familyofamplitudeswouldnothaveanypoleatfinitelocationandthewholeamplitudewouldbegiven bytheboundaryterm. Furthermore,intheformercase,theabsenceofasecondfactorisationchanneldoes not allow us to performthe analysis which broughtto the relation (15) which fixes the conditionon the zeroes. If one thinks aboutthe limits in which the amplitudesbecome trivial as a generic property,it is reasonabletoassumethatthecondition(15)canstillhold.Thisisindeedastrongassumptionwhichneeds tobechecked. Copyrightlinewillbeprovidedbythepublisher 6 P.Benincasa:ExplorationoftheTree-LevelS-MatrixofMasslessParticles s Conditions Interactions s=0 s′ = 1,κ=κ′ Yukawa 2 s′ =0,κ=0 scalarQEDandYM+scalars s=1 s′ = 1,κ=0 QEDandYM+fermions 2 s′ =1,κ=κ′ YM s′ =0,κ=κ′ scalarGR s=2 s′ = 1,κ=κ′ FermionGravity 2 s′ =1,κ=κ′ Einstein-Maxwell s′ = 3,κ=κ′ N =1supergravity 2 s′ =2,κ=κ′ GR Table1 Summaryofthetheoriescharacterisedbycouplingswiths-derivativeinteractions. Another characteristic of these theories is that the boundaryterm turns out to be a polynomialin the Lorentz invariants. A simple dimensionalanalysis revealsthat it has a structure of a contact interaction withL = (δ−2)n−2(δ−3)derivativeswhichincreaseasnincreases. Therefore,thesetheoriesseem n tobeendowedwithhigher-derivativeinteractiontermswiththeincreaseofthenumberofexternalstates. Thiscanbeinterpretedasasignatureofnon-localityforthesetheories. Furthermore,theconsistencyofthe4-particleamplitudesforthesetheoriesindeeddoesnotimplythe consistencyneitherofthewholetheoriesnoroftheirtree-level.Pathologiescan,forexample,alreadyarise at5-particlelevel: thefactthattheconsistencytesthasbeenpassedat4-particleleveldoesnotimplythat ithastobepassedgenerically.Furthermore,ghostsmightappearatloops. Finally,asageneralcomment,itisfairtopointoutthatouranalysisconcernsjustthosetheorieswhose propagatorsaregivenby1/P2. Acknowledgements IwouldliketothanktheorganisersforthestimulatingenvironmentcreatedattheXVIIth Eu- ropeanWorkshoponStringTheory. ItisalsoapleasuretothankEduardoCondewhocollaboratedwithmeonthe workswhichthistalkisbasedon.MyworkisfundedinpartbyMICINNundergrantFPA2008-01838,bytheSpanish Consolider-Ingenio2010ProgrammeCPAN(CSD2007-00042)andbyXuntadeGalicia(Conseller´ıadeEducac´ıon, grant INCITE09206 121 PR and grant PGIDIT10PXIB206075PR) and by FEDER.I am supported as well by the MInisteriodeCienciaeINNovaciońthroughtheJuandelaCiervaprogram. References [1] S.J.ParkeandT.R.Taylor,Phys.Rev.Lett.56,2459(1986). [2] F.Cachazo,P.Svrcek,andE.Witten,JHEP09,006(2004). [3] R.Britto,F.Cachazo,andB.Feng,Nucl.Phys.B715,499(2005). [4] R.Britto,F.Cachazo,B.Feng,andE.Witten,Phys.Rev.Lett.94(2005). [5] P.BenincasaandF.Cachazo(2007). [6] P.Benincasa,C.Boucher-Veronneau,andF.Cachazo,JHEP11,057(2007). [7] N.Arkani-Hamed,F.Cachazo,andJ.Kaplan,JHEP1009,016(2010). [8] B.Feng,J.Wang,Y.Wang,andZ.Zhang,JHEP01,019(2010). [9] B.FengandC.Y.Liu,JHEP07,093(2010). [10] P.BenincasaandE.Conde,JHEP1111,074(2011). [11] B.Feng,Y.Jia,H.Luo,andM.Luo(2011). [12] P.C.SchusterandN.Toro,JHEP0906,079(2009). [13] P.BenincasaandE.Conde(2011). Copyrightlinewillbeprovidedbythepublisher