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High Energy Scattering and Renormalization in the NS+ String and Large N QCD

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Title:
High Energy Scattering and Renormalization in the NS+ String and Large N QCD
Creator:
Rojas, Francisco J
Place of Publication:
[Gainesville, Fla.]
Florida
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University of Florida
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Language:
english
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1 online resource (126 p.)

Thesis/Dissertation Information

Degree:
Doctorate ( Ph.D.)
Degree Grantor:
University of Florida
Degree Disciplines:
Physics
Committee Chair:
Thorn, Charles B
Committee Members:
Acosta, Darin E
Sikivie, Pierre
Ramond, Pierre
Groisser, David J
Graduation Date:
5/5/2012

Subjects

Subjects / Keywords:
Analytics ( jstor )
Approximation ( jstor )
D branes ( jstor )
Gluons ( jstor )
Integrands ( jstor )
Natural logarithms ( jstor )
Region of integration ( jstor )
Sine function ( jstor )
Trajectories ( jstor )
Vertices ( jstor )
Physics -- Dissertations, Academic -- UF
analytic-continuation -- hard-scattering -- neveu-schwarz -- open-string -- regge -- renormalization
Genre:
bibliography ( marcgt )
theses ( marcgt )
government publication (state, provincial, terriorial, dependent) ( marcgt )
born-digital ( sobekcm )
Electronic Thesis or Dissertation
Physics thesis, Ph.D.

Notes

Abstract:
In this work we study a model of open strings and its low energy limit as a device to understand certain aspects of gauge theory. More specifically, we use the Neveu-Schwarz model with the open strings attached to a stack of N coincident Dp-branes as our starting point. After various projections, the low energy limit of this string theory is pure Yang-Mills and the field theory of sum of the planar open string diagrams corresponds to the sum of the leading diagrams of large N QCD. The initial expression for the amplitude is given in an integral form and diverges in various corners of the integration region as usual. However, we regularized all these divergences by suspending total momentum conservation by an amount p, as first suggested by Goddard and by Neveu and Scherk [2, 3]. Using this regulator, we construct a completely finite expression for the one looop amplitude of open string massless vectors which become the “gluons” of large N QCD in the low energy limit. We show that all the counterterms we need to introduce, after analytic continuation to p = 0, are proportional to the tree amplitude which then simply amounts to a renormalization the open string coupling constant. Using this renormalized expression, we also study its high-energy behavior. The open string Regge trajectory a(t) through one loop is obtained also as a function of the dimensionality of the Dp-branes. We also study the field theory limit of a(t) which is simply the small t behavior and we obtain the exact same answer that conventional calculations in gauge theory provide. We think that this is a useful insight that open string theory provides into the understanding of gauge theory. We also work out the high-energy regime at fixed scattering angle (hard scattering) for the renormalized amplitude through one-loop. We obtain the well known exponential falloff expected for stringy amplitudes. By studing the limit where s is much larger than t of this expression, which involves taking various approximations to the integral representation found for the amplitude, we recover the Regge limit at high t as is indeed expected. ( en )
General Note:
In the series University of Florida Digital Collections.
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Includes vita.
Bibliography:
Includes bibliographical references.
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Description based on online resource; title from PDF title page.
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This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Thesis:
Thesis (Ph.D.)--University of Florida, 2012.
Local:
Adviser: Thorn, Charles B.
Statement of Responsibility:
by Francisco J Rojas.

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UFRGP
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Copyright Rojas, Francisco J. Permission granted to the University of Florida to digitize, archive and distribute this item for non-profit research and educational purposes. Any reuse of this item in excess of fair use or other copyright exemptions requires permission of the copyright holder.
Resource Identifier:
867650424 ( OCLC )
Classification:
LD1780 2012 ( lcc )

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HIGHENERGYSCATTERINGANDRENORMALIZATIONINTHENS+STRINGANDLARGENQCDByFRANCISCOJAVIERROJASFERNANDEZADISSERTATIONPRESENTEDTOTHEGRADUATESCHOOLOFTHEUNIVERSITYOFFLORIDAINPARTIALFULFILLMENTOFTHEREQUIREMENTSFORTHEDEGREEOFDOCTOROFPHILOSOPHYUNIVERSITYOFFLORIDA2012

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c2012FranciscoJavierRojasFernandez 2

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IdedicatethistomywifeBarbara,mybabygirlValentina,andtoallmyfamily,fortheirendlessandunconditionallove. 3

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ACKNOWLEDGMENTS Iamdeeplyindebtedtomyadvisor,Prof.CharlesThornforhisconstantguidance,patienceandforthenearlyinnitenumberofhourshespentteachingmephysicsandguidingmethroughmyprojects.IamgratefultoProf.RichardWoodardformanydiscussions,adviceandfriendship.IalsothankProf.PierreRamondformanydiscussionsandadvice.IwanttothankmybeautifulwifeBarbaraforallherconstantandunconditionalloveandsupport.IalsothankmybabygirlValentinathat,withoutknowingityet,haslitmylifeforeverwithherpresence.IthankmyparentsLauraandEsteban,mybrotherRodrigo,andallofmyfamilyfortheirendlessloveandforalwaysbeenthereforme.BeingfarfromthemisoneofthehardestthingIhaveeverhadtogothroughinmylifeandsomehowtheyalwaysmanagedtomakemefeeltheywereonlyacouplestepsaway.IalsowouldliketothankmyfriendshereattheDepartmentofPhysicsfortheirfriendshipandforhelpingmeinsomanyways:DanArenas,NilanjanBanik,EvanDonoghue,JesusEscobar,Sung-SooKim,KatieLeonard,PedroMora,ChrisPankow,GeorgiosPapathanasiou,MyeonghunPark,SohyunPark,JayPerez,PabloPerez,GaurabSarangi,HeywoodTam,JonathanThompsonandJueZhang. 4

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TABLEOFCONTENTS page ACKNOWLEDGMENTS .................................. 4 LISTOFFIGURES ..................................... 7 ABSTRACT ......................................... 9 CHAPTER 1INTRODUCTION ................................... 11 1.1LargeNQCD .................................. 14 1.1.1TheBuildingBlocks .......................... 15 1.1.1.1GluonPropagator ...................... 15 1.1.1.2QuarkPropagator ...................... 17 1.1.1.3ThreeGluonVertex ..................... 17 1.1.1.4FourGluonVertex ...................... 17 1.1.1.5Gluon-TwoQuarkVertex .................. 18 1.1.2Examples ................................ 18 1.1.2.1AFour-GluonTreeDiagram ................. 18 1.1.2.2GluonLoopvs.FermionLoop ............... 18 1.1.2.3PlanarvsNon-Planar .................... 19 1.1.3TheTopologicalExpansion ...................... 20 1.2PlanarOpenStringAmplitudesandLargeNQCD ............. 22 2ANALYTICCONTINUATIONANDRENORMALIZATIONOFTHEONE-LOOPM-GLUONAMPLITUDE ............................... 26 2.1TheGNSRegulatorandtheRemovalofSpuriousDivergences ...... 26 2.1.1LinearDivergences ........................... 35 2.1.2LogarithmicDivergences ........................ 42 2.2RenormalizedM-gluonAmplitude ...................... 50 2.3D8andD9BranesandReggeSlopeRenormalization ........... 52 3REGGETRAJECTORYANDITSFIELDTHEORYLIMIT ............ 55 3.1TheOne-LoopOpenStringReggeTrajectory ................ 55 3.2SmalltBehaviorof(t),FieldTheoryLimitandGluonReggeization ... 63 3.3LargetBehaviorof(t) ............................ 68 3.4NumericalAnalysisandGraphics ....................... 70 3.5OrbifoldProjection ............................... 75 4HARDSCATTERING ................................ 79 4.1TheOneLoopAmplitude ........................... 80 4.2TheHardScatteringLimit ........................... 83 5

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4.2.1HardScatteringLimitThroughOneLoop ............... 83 4.2.2ComparisonwithTreeAmplitude ................... 102 4.2.3RecoveryoftheReggeBehavioratHight .............. 102 4.3HighEnergyScatteringintheTypeISuperstring .............. 105 4.3.1ReggeBehavioratOne-Loop ..................... 106 4.3.2HardScatteringatOneLoop ..................... 111 4.3.3RecoveryoftheReggeLimit ...................... 113 5CONCLUSIONS ................................... 117 APPENDIX ALOGARITHMICCOUNTERTERMS ........................ 121 BANIDENTITYUSINGTHEGNSREGULATOR .................. 123 REFERENCES ....................................... 124 BIOGRAPHICALSKETCH ................................ 126 6

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LISTOFFIGURES Figure page 1-1EachlinecorrespondstoaKroneckerdeltawherethearrowpointsinthedirectionfromanupperindextoalowerone. ........................ 16 1-2Fromthisdiagramwecanimmediatelywrite(Ta)ij(Tb)jk(Tc)ki=tr(TaTbTc). .. 17 1-3Fromthisdiagramwecanimmediatelywrite(Ta)ij(Tb)jk(Tc)kl(Td)li=tr(TaTbTcTd). 18 1-4Gluon-twoquarkvertexinthedoublelinenotation. ................ 18 1-5Fourgluontree .................................... 18 1-6Thegluonloopatthetopgivesafactorofg2kk=g2Nwhereasthefermionlooponlygivesg2=(g2N)1=N,thusitissubleadingbyafactorof1=Nwithrespecttothegluonloop. .............................. 19 1-7(a)A3-loopgluonplanardiagram.Thenumberofcolorloopsiseasilyvisualizedontheleft.Thestrengthofthisdiagramisthereforeg6N3=(g2N)3.(b)ThisdiagramisalsoofthesameorderintheYang-Millscouplingconstant,i.e.g6,butthereisonlyonecolorloop,thusthisdiagramcontributeswithg6N=(g2N)31=N2.Thisisanexampleofanon-planardiagramwhichissuppressedbyafactorof1=N2comparedtotheplanaroneonpart(a). .................. 20 1-8Vacuumbubbles(a)Threefaces,sixedges,zeroholes,andzerohandles(b)Oneface,sixedges,onehandle,andzeroholes(c)twofaces,fouredges,onehole,andzerohandles. ............................. 21 1-9Afreelypropagatingopenstringdiagram.Thetopologyofthisdiagramisthatofadisk.Theindicesiandjrepresentthetwoend-pointsoftheopenstringattachedtoD-branesiandjrespectively. ..................... 23 1-10Openstringplanarone-loopdiagram.Sincethereisoneinternalboundaryandtwointeractionverticesthisdiagramisg2oN. .................. 24 1-11Openstringplanartwo-loopdiagram.Therearetwointernalboundariesandfourinteractionvertices,thusthisdiagramisg4oN2=(g2oN)2. ........... 24 1-12Openstringnon-planarloopdiagram.Duetonon-planarity,thisdiagramhasonlyoneinternalboundary.Countingthenumberofinteractionverticesandinternalboundariesthisdiagramgivesg6oN=(g2oN)3N)]TJ /F4 7.97 Tf 6.58 0 Td[(2. ............. 25 2-1The3-simplexaboveshowstheregionofintegrationatxedqforthe4-pointamplitude.Edgesandverticescorrespondtotheplaceswherespuriousandrealdivergencescanoccur. ............................. 36 7

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2-2Theedgecorrespondingto320isshownasthehighlightedlineinthegure.Thisregioncorrespondstoaloopinsertioninoneoftheexternalstateswhichforcesthepropagatorforstatenumber4tobeevaluatedon-shellproducingadivergence ............................... 44 3-1Integrationregionoverthekvariables.TheedgehighlightedinyellowistheregionthatdominatesintheReggelimit,whiletheonesinorangecorrespondtotheplaceswheretheamplitudeisdivergent.ThefactthedominantregionintheReggelimit(yellowline)overlapswiththelogarithmicallydivergentregions(orangelines)onlyattheverticesimpliesthatweonlyneedtheleadingcountertems(whicharisefromthevertices)tocomputetheone-loopcorrectiontotheleadingReggetrajectory. ................................... 61 3-2Thedotscorrespondtothedirectnumericalintegrationof(t)=(t)whilethesolidlineisthepredictedbehavioratlargejtjbothasafunctionofln()]TJ /F5 11.955 Tf 9.29 0 Td[(0t).Atlargevaluesof0jtjthenumericalintegrationapproachesthepredictedbehaviorfrombelow. ................................. 71 3-3Thetoftheleadingandthreesubleadingcorrectionstothedatapointsispresentedasthesolidline.Hereweseethattheagreementbetweenthedatapointsandthetincreasesforlargervaluesof0tasexpected,anditisalreadygoodstartingatln(0t)6sincewehaveincludedsubleadingcorrections. .. 72 3-4ThesmalltbehaviorisshownforD=5.Weexpect(t)togotozerowithinniteslopewhichcanbeappreciatedfromthisgure. ............. 73 3-5Zoominsmallt.Theexpected()]TJ /F5 11.955 Tf 9.3 0 Td[(0t)1=2behaviorneart=0canbeappreciatedmoreclearlyinthisgure.Thesolidlineisthepredictedasymptoticbehavior(t)=)]TJ /F2 11.955 Tf 24.6 0 Td[(23=24()]TJ /F5 11.955 Tf 9.3 0 Td[(0t)1=2ast!0whichmatcheswellwiththedatapointsinthislimit. ................................ 74 3-6Alargerrangethatincludesbothlargeandsmalltbehaviorisshown.Inthisplotitispossibletoseethetwoasymptoticregionswithsomeaccuracy.Thelargetregiongrowsast=(lnt)3=2asdescribedinSection5.Althoughitisnotcompletelyevidentfromthisgure,(t)isgoingtozerowithinniteslopeas()]TJ /F5 11.955 Tf 9.3 0 Td[(0t)1=2intheregionneart=0(seeFigure 3-5 ) ............... 75 4-1(a)Anopenstringone-loopamplitudeisobtainedbyintegratingovertheradiusoftheannulusdiagram.(b)Theregionofintegrationwheretheradiusqgetsclosetozerobecomesindistinguishablefromtheopenstringtreeamplitudewhichhasthetopologyofadisk. .......................... 86 8

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AbstractofDissertationPresentedtotheGraduateSchooloftheUniversityofFloridainPartialFulllmentoftheRequirementsfortheDegreeofDoctorofPhilosophyHIGHENERGYSCATTERINGANDRENORMALIZATIONINTHENS+STRINGANDLARGENQCDByFranciscoJavierRojasFernandezMay2012Chair:CharlesB.ThornMajor:PhysicsInthisworkwestudyamodelofopenstringsanditslowenergylimitasadevicetounderstandcertainaspectsofgaugetheory.Morespecically,weusetheevenG-paritysectoroftheNeveu-SchwarzmodelwiththeopenstringsendpointsattachedtoastackofNcoincidentDp-branesasourstartingpoint.Aftervariousprojections,thelowenergylimitofthisstringtheoryispureYang-Millsandtheeldtheorylimitofthesumoftheplanaropenstringdiagramsbecomestothesumoftheleadingdiagramsof'tHooft'slargeNexpansionofQCD.Thetypicalexpressionfortheamplitudewestudyhere,namelyforthescatteringofopenstringmasslessvectorbosons,comesinanintegralformthatshowsmultiple(spurious)divergencesinvariouscornersoftheintegrationregion.However,weregularizeallofthesedivergencesbymomentarilysuspendingtotalmomentumconservationbyanamountp,asrstsuggestedbyGoddardandbyNeveuandScherk[ 1 2 ],andconstructtheappropriatecounterterms.Weshowthatallthecountertermsweneedtointroduce,afteranalyticcontinuationtop=0,areallproportionaltothetreeamplitudewhichthenamountstoarenormalizationthestringcouplingconstant.Usingthisrenormalizedexpression,wealsostudyitshigh-energybehavior.Weobtaintheone-loop(O(g2))correctiontotheopenstringReggetrajectory(t)alsoasafunctionofthespacetimedimensionalityoftheDp-branes.Wealsostudy 9

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theeldtheorylimitofthenewtrajectorywhichamountstoanalyzingthesmalltbehaviorof(t).WeobtaintheexactsameanswergivenbyconventionalcalculationsindimensionallyregularizedgaugetheorywhichputsthegluononaReggetrajectoryoforderg2.Wethinkthatthisisausefulinsightthatopenstringtheoryprovidesintotheunderstandingofgaugetheory.Additionallywestudythebehaviorofthenewtrajectory(t)fortawayfromzerowhichisrelevantforopenstringphysics.Finallyweworkoutthehigh-energyregimeatxedscatteringangle(hardscattering)fortherenormalizedamplitudethroughone-loop.Weobtainthewellknownexponentialfalloffexpectedforstringyamplitudes,butwealsoprovidethecompletefactorthatmultipliesthisexponentialintermsofthekinematicalinvariants.Bystudyingthestlimitofthisnovelexpression,whichinvolvestakingvariousapproximationsintheintegralrepresentationfoundfortheamplitude,werecovertheReggelimitathightasisindeedexpected. 10

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CHAPTER1INTRODUCTIONAtthecoreofeveryatom,protonsandneutronsareboundtogetherthroughthestrongnuclearforce.Atevensmallerdistances,protons,neutronsandalloftheotherparticlesthatinteractwitheachotherthroughthestrongnuclearforce(hadrons)aremadeupofsmallerparticleswhicharethequarksandgluons.Ourcurrentunderstandingisthateverythinginthenuclearworldismadeofcompositesofquarksandgluons,aconclusionthathasbeenexperimentallytestedandconrmedwithastonishingprecisionforseveraldecadesalready.ThetheoreticalframeworkthatmakesthesepredictionsiscalledQuantumChromodynamicsorQCDforshort.However,evenafteralltheseyears,therearemanyquestionsaboutQCDthatarestillunanswered.Amongthese,explainingconnementandchiralsymmetrybreakingfromrstprinciplesremainassomeofthemostimportantunansweredquestionsintheoreticalparticlephysicsuntilthisday.Onepossibleavenuefordealingwiththeseissuesistousethemathematicalmachineryofstringtheory,whichisthecentraltopicofthiswork.Aswithalltheotherthreeforcesofnature(Gravity,ElectromagnetismandtheWeakNuclearforce),thecouplingconstantkeepstrackofthestrengthoftheinteractionsamongtheparticles.Whenthisnumberissmall,itispossibletouseperturbativeQCDtomakepredictionsabouttheoutcomeofexperimentswherewecollidetheseparticles.However,whenperformingtheseexperiments,manyprocessescanhappenandaconsiderableamountoftheeventsthatoccurhavetodowithprocesseswherethecouplingconstantisalargenumber.Inthiscase,theperturbativetoolsofQCDarenolongerreliableandwecannotuseittoexplainorpredictresults.Asoftoday,theonlyknownwaytogetpredictionsfromQCDinthenonperturbativeregimeisthroughnumericalcomputations.Oneofthemostremarkableoutcomesofcolliderexperimentsregardingthestrongnuclearforcerealmisthatquarksandgluonsseemtobeconnedinsidethenucleus. 11

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Thisisthewellknownconnementhypothesiswhichdoesnothaveacompletelysatisfactorytheoreticalexplanationyet.CertainstringtheoriescanbeadjustedtogiveexactlyQCDintheirlowenergylimit.Inaddition,thefactthatstringsareone-dimensionalobjects(insteadofpointparticlesasinQCD)allowsustohaveabettermathematicalframeworkinwhichconnementcouldbeexplainedusingstringsbycomputingscatteringamplitudesinthesocalledplanarlimit.Thereasonforthisisthatthenumberofstringtheorydiagramsissignicantlylessthanthenumberofeldtheorydiagrams.Thus,thesumoverstringtheorydiagramsmightbemoretractablethanthesumovereldtheoryones.Inthepresentwork,westudythesestringyscatteringamplitudesthroughonelooporderandcomputetheirlowenergylimitinordertocomparethemwithQCD.Thisthesisisorganizedasfollows:inthefollowingsectionwegiveabriefintroductiontothelargeNexpansioninQCDanditsconnectionwithperturbativeopenstringtheory.Weshowhowthesumofplanarmulti-loopopenstringdiagramsisanextensionofthelargeNexpansioningaugetheoriesandhowitsusecouldhelpintheunderstandingofnonperturbativeissuesingaugetheoriessuchasconnement.Theninchapter 2 weshowtherenormalizationofone-loopplanaramplitudesintheNS+modelusingaregularizationtechniquebasedonsuspendingtotalmomentumconservationbyaniteamountpandanalyticallycontinuingtheanswertop=0attheveryend.Wealsoshowthatthisprocedureiscompatiblewithgaugeinvariance.Inchapter 3 westudytheReggelimitofthecompleteplanaramplitudethroughoneloopandshowhowthelowenergylimitofthenewReggetrajectory,i.e.includingtheone-loopcorrection,correctlyreproducestheknownresultsfromconventionalgaugetheorycomputations.TheuseofthedimensionalityoftheDp-branes(D=p+1)asaregulatingparameterforinfrareddivergencesofthestringamplitudeswillalsobeshowntoreproducetheIRdivergenttermsindimensionallyregularizedgaugeeldtheory.Wealsoprovidenumericalanalysisthatsupportsouranalyticresults.Inchapter 4 we 12

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studytheothertypeofhigh-energyregime,namelythehardscatteringorxed-anglehighenergylimit.Weshowthattheone-loopamplitudeisexponentiallysuppressedathighenergiesasexpected,butwealsoprovidethefullkinematicfactormultiplyingthisexponentialintermstheratiot=swhichisanovelfeatureandallowstocomparethehardandReggeregimestoeachother.Asaby-productwealsoperformthisanalysisforthetypeIsuperstringandndthattheuseofanoldidentityhelpscomputingtheprefactorinanexactmay.Finally,weclosethisthesiswithconclusionsandadiscussionaboutpossiblefuturedirectionsofthiswork. 13

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1.1LargeNQCDBecauseoftheasymptoticfreedomofQCD,atveryhighenergies(smalldistances)wecanuseperturbativemethodstodescribetheinteractionsofquarksandgluonswithremarkableaccuracy.Atlargedistanceswealsoexpectthedynamicsofquarksandgluonstodescribeandexplainphenomenasuchasconnementandchiralsymmetrybreaking.Howeverinthisregime,thecouplingconstantceasestobesmallandweloseallhopeofusingperturbationtheoryforthesepurposes.Thismeansthatweneedtoappealtononperturbativemethodstotackletheseproblems.However,becauseofdimensionaltransmutation,thereisnotanalternativedimensionlessexpansionparameter.Inspiteofthis,'tHooftin1974[ 3 ]inventedanewremarkablewayinwhichtoexpandandobtainnon-perturbativeinformationfromtheQCDdynamics.QCDisagaugetheorybasedonthecolorSU(3)group,but'tHooftfoundthatifwebaseQCDonthegroupSU(N)instead,remarkablesimplicationsoccurintheN!1limit.Thus,thenewdimensionlessexpansionparameteris1=N.IntherealworldN=3,andatrstsight1=N=0.33doesnotseemtobeagoodexpansionparameter.Still,considerthefollowing[ 4 ]:inQEDtheexpansionparameteristypicallye2=4'1=137whichmeansthate'0.30.TheelectricchargeeistheonlyfreeparameterinQED,andifewouldhavebeenregardedastheexpansionparameterofthetheory,perturbativemethodswouldhavebeendiscardedasusefulinQEDalso!HistoricallythisiscertainlynottruesincemostoftheexperimentallysuccessfulpredictionsofQEDreliedonperturbationtheory.Thus,thereasonwhytheperturbativeexpansionofQEDissosuccessfulisbecausethetypicalexpansionparameterisnotebutrathere2=4'0.007.Therefore,untilwecomputetheactualphysicalimplicationsthatstemfromthe1=Nexpansion,wehavenoapriorireasontobelievethisexpansionisnotausefulone.Muchonthecontrary,thelargeNexpansionmotivatessomenicephenomenologicalpredictionssuchastheratioofmassoftherhomesontothatoftheprotonm=mPto 14

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anaccuracyofabout10%,poledominanceinlowenergyscatteringofmesons,Zweig'sruleandsuppressionofexotics.WewillnowbrieyreviewthemainideasaboutthelargeNexpansionforQCD[ 3 5 ]startingfrom'tHooft'sdoublelinenotation.Forthissectionwealsofollow[ 6 ]. 1.1.1TheBuildingBlocksThestartingpointistheQCDlagrangiandensity LQCD=)]TJ /F2 11.955 Tf 10.49 8.09 Td[(1 4tr(FF)+i @ +g A )]TJ /F2 11.955 Tf 11.96 0 Td[(m (1) wherethematrixeldsA(x)arewrittenintermsoftheSU(N)generatorsintheadjointrepresentationTathroughA(x)ij=Aa(x)(Ta)ij,thustheaindexrunsfrom1toN2)]TJ /F2 11.955 Tf 12.38 0 Td[(1.WearenotwritingthecompleteQCDlargrangian,butthispartsufcesforthediscussionbelow. 1.1.1.1GluonPropagatorToderivethegluonpropagatorallweneedisthekineticpartfortheAa(x)eldwhichalsoincludesagaugexingterm.IntheRgaugethistermis Lgf=)]TJ /F2 11.955 Tf 13.34 8.09 Td[(1 2@Aa@Aa (1) Aftersomeintegrationsbyparts,thequadraticpartofthepureYang-Millslagrangianplusthegaugexingtermread 1 2Aa(@2)]TJ /F5 11.955 Tf 11.96 0 Td[(@@)Aa)]TJ /F2 11.955 Tf 16 8.08 Td[(1 2@Aa@Aa (1) fromwherewecanreadoffthepropagatorintheRgauge ab(k)=ab k2)]TJ /F2 11.955 Tf 11.96 0 Td[(i)]TJ /F2 11.955 Tf 13.15 8.09 Td[(kk k2+kk k2 (1) Thecentralideaof'tHooft'sdoublelinenotationistousethecolorindicesi,jtodrawFeynmandiagramsinsteadofthecustomarya,bindices.Considerforexampletheterm 15

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inthelagrangian )]TJ /F2 11.955 Tf 13.15 8.09 Td[(1 2@Aa@Aa (1) Usingthenormalizationtr(TaTb)=ab,wehaveAa(x)=tr(A(x)Ta),thuswecanwrite )]TJ /F2 11.955 Tf 13.15 8.08 Td[(1 2@Aa@Aa=)]TJ /F2 11.955 Tf 10.49 8.08 Td[(1 2@(A)ij@(A)kl(Ta)ji(Ta)lk (1) Thus,insteadofusingtheaindicestoconstructthepropagatorbutusingthei,jindicesinstead,wenowreadoffthepropagatoras jilk()=(Ta)ji(Ta)lk k2)]TJ /F2 11.955 Tf 11.95 0 Td[(i)]TJ /F2 11.955 Tf 13.15 8.09 Td[(kk k2+kk k2 (1) ForSU(N)thegeneratorsTaaretraceless,whichyields (Ta)lk(Ta)ij=iklj)]TJ /F2 11.955 Tf 14.61 8.08 Td[(1 Nijlk (1) therefore,thepropagatoris jilk()=iklj)]TJ /F4 7.97 Tf 14.2 4.7 Td[(1 Nijlk k2)]TJ /F2 11.955 Tf 11.95 0 Td[(i)]TJ /F2 11.955 Tf 13.15 8.09 Td[(kk k2+kk k2 (1) SinceweareinterestedintheN!1limitwecandropthelastterminprefactorofthenumeratorabove.Thus,theasymptotiaoftheSU(N)gaugetheoryisthesameasthatofaU(N)gaugetheoryinthelargeNlimit.Therefore,thenalpropagatorofinterestis jilk()=iklj k2)]TJ /F2 11.955 Tf 11.96 0 Td[(i)]TJ /F2 11.955 Tf 13.15 8.09 Td[(kk k2+kk k2 (1) Weseethatitisthennaturaltoassignadoublelinetothediagramthatrepresentsthispropagatorwhichisshowningure 1-1 .EachlinerepresentsaKroneckerdeltawith Figure1-1. EachlinecorrespondstoaKroneckerdeltawherethearrowpointsinthedirectionfromanupperindextoalowerone. thearrowpointingfromanupperindextoalowerone. 16

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1.1.1.2QuarkPropagatorBythesametoken,thekineticpartforthefermioneld @ yieldsthepropagator S(k)ij=)]TJ /F5 11.955 Tf 9.3 0 Td[(k+m k2+m2)]TJ /F2 11.955 Tf 11.96 0 Td[(iij (1) whichhasonlyoneKroneckerdelta,thusitisrepresentedbyasingleline. 1.1.1.3ThreeGluonVertexThisvertexcomesfromtheterm)]TJ /F2 11.955 Tf 9.3 0 Td[(gtr(AA@A)inthelagrangianwhich,usingthecolorindices,canbewrittenas )]TJ /F2 11.955 Tf 11.96 0 Td[(gAaAb@Ac(Ta)ij(Tb)jk(Tc)ki (1) Fromhereweseethatthestrengthoftheinteractionisproportionalto)]TJ /F2 11.955 Tf 9.3 0 Td[(g(Ta)ij(Tb)jk(Tc)kiwhichisnaturallydrawnasthevertexshowningure 1-2 ,wherenowwealsoassigna(Ta)ijmatrixtoeachtipofthevertex.Thisisschematicallyshownbyspecifyingjustthepairofindicesijoneachtip. Figure1-2. Fromthisdiagramwecanimmediatelywrite(Ta)ij(Tb)jk(Tc)ki=tr(TaTbTc). 1.1.1.4FourGluonVertexThisvertexcontainsthefactortr(TaTbTcTd)andgure 1-3 showsitsdoublelinerepresentation. 17

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Figure1-3. Fromthisdiagramwecanimmediatelywrite(Ta)ij(Tb)jk(Tc)kl(Td)li=tr(TaTbTcTd). 1.1.1.5Gluon-TwoQuarkVertexTheterminthelagrangianresponsibleforthisvertexis )]TJ /F2 11.955 Tf 11.95 0 Td[(ig i(A)ji j=)]TJ /F2 11.955 Tf 9.3 0 Td[(ig i(Aa)(Ta)ji j (1) thusitisrepresentedaspicturedingure 1-4 Figure1-4. Gluon-twoquarkvertexinthedoublelinenotation. 1.1.2Examples 1.1.2.1AFour-GluonTreeDiagramThisisshowningure 1-5 .fromwhichweobtaing2tr(TaTdTcTb). Figure1-5. Fourgluontree 1.1.2.2GluonLoopvs.FermionLoopAgluonloopandafermionloopareshowningure 1-6 .Readingoffthedouble-linenotationwehavethatthegluonloopdiagram,depictedatthetopofgure 1-6 18

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contributeswith g2(Ta)ij(Tb)jikk=g2tr(TaTb)N=g2abN (1) fromwhichweseethateveryclosedloopcontributeswithkk=N.Thefermionloophoweveris g2(Ta)ij(Tb)ji=g2tr(TaTb)=g2ab (1) whichissuppressedbyafactorof1=Nwithrespecttothegluonloop.ThisfactliesattheheartoftheusefulnessofthelargeNexpansionsincethisisageneralresultandtheleadingdiagramsaretheonesthatcontaingluonicloopsonly.ThebasicreasonforthisisthatthegluonsareintheadjointrepresentationofSU(N)andthereforethereareN2)]TJ /F2 11.955 Tf 12.59 0 Td[(1ofthemcirculatingtheloop,whereassincethequarksareinthefundamentalrepresentation,thereareonlyNofthenavailabletocirculateintheloops. Figure1-6. Thegluonloopatthetopgivesafactorofg2kk=g2Nwhereasthefermionlooponlygivesg2=(g2N)1=N,thusitissubleadingbyafactorof1=Nwithrespecttothegluonloop. 1.1.2.3PlanarvsNon-PlanarThisexampleembodiesoneofthecentralpointsofthelargeNexpansion.Ingure 1-7 considerthetwodiagramsontheleftwiththeircorrespondingdouble-linenotationsontheright.Whilediagrams(a)and(b)arebothoforderg6,therstoneisproportionaltoN3whilethesecondoneisproportionaltoNonly.Thus,inthelargeN 19

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Figure1-7. (a)A3-loopgluonplanardiagram.Thenumberofcolorloopsiseasilyvisualizedontheleft.Thestrengthofthisdiagramisthereforeg6N3=(g2N)3.(b)ThisdiagramisalsoofthesameorderintheYang-Millscouplingconstant,i.e.g6,butthereisonlyonecolorloop,thusthisdiagramcontributeswithg6N=(g2N)31=N2.Thisisanexampleofanon-planardiagramwhichissuppressedbyafactorof1=N2comparedtotheplanaroneonpart(a). limittherstonedominates.Thisisahallmarkexampleofthemoregeneralresultthatplanardiagramsdominateovernon-planaronesinthelargeNexpansion. 1.1.3TheTopologicalExpansionThepreviousexamplesshowedtwofeaturesthatarethecentralideasofthelargeNexpansion: Gluonloopsdominateoverfermionloops Planardiagramsdominateovernon-planaronesThestartingpointisagaintheQCDlagrangian( 1 ).Therstobservationisthatsincethenumberofeldsinthelagrangiangetsincreasinglylarge(N2)]TJ /F2 11.955 Tf 12.16 0 Td[(1forthegluoneldsandNforthequarkelds)wecanredeneeldsinsuchawaythatNscalesoutinfrontofthelagrangian: L!N g2)]TJ /F2 11.955 Tf 10.49 8.09 Td[(1 4F2+i @ + A )]TJ /F2 11.955 Tf 11.96 0 Td[(m (1) withF2=dA+A2,i.e.nopowersofgandNappearineitherF2orin A .Intermsoftheseneweldswehavethateachinteractionvertexnowcomeswithanoverall 20

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factorofN,whilethegluonandquarkpropagatorsappearwithafactorof1=Neach.Specializingforthevacuumbubblediagramssuchastheonesshowningure 1-8 ,onecanassociatetoallvacuumdiagramsatwo-dimensionalsurfacedescribedbya Figure1-8. Vacuumbubbles(a)Threefaces,sixedges,zeroholes,andzerohandles(b)Oneface,sixedges,onehandle,andzeroholes(c)twofaces,fouredges,onehole,andzerohandles. polygonwithacertainnumberoffaces,edgesandboundaries1.Theinteriorofeachcolorloop(i.e.theclosedlinesthatproduceafactorofkk=N)correspondstoafaceofthepolygon.Eachgluonpropagatorcorrespondstoanedgeandeachquarkpropagatorcorrespondstotheboundaryofaholeoredgeofapolygon.Fromthelagrangianin( 1 )weseethatbothgluonandquarkpropagatorsprovideafactorofg2=NwhileeachinteractionvertexgivesN=g2.Denotingthenumberoffaces,edges,andverticesbyF,E,andVrespectively,eachgraphproducesafactorof NF)]TJ /F4 7.97 Tf 6.58 0 Td[(E+VN (1) whereisthetopologicalnumberknownastheEulercharacteristic.IntermsofthenumberofhandlesH,andthenumberofboundariesB(quarkpropagatorsinourcase),isalsogivenby =2)]TJ /F2 11.955 Tf 11.95 0 Td[(2H)]TJ /F2 11.955 Tf 11.95 0 Td[(B (1) 1Noticethatthediagramsingure 1-8 areobtainedbyamputatingtheexternallegsfromthediagramsingure 1-7 andwehaveshrunkthewidthoftheedgetomakethepolygondescriptionmoreevident. 21

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whichisatopologicalinvariant.Therefore,inthelargeNlimit,thegraphsthatwilldominatearetheoneswithH=0,andB=0,i.e.planardiagrams(nohandles)withoutquarkloops.Inotherwords,largeNQCDisdescribedbyapureYang-Millstheorywithplanardiagramsonly. 1.2PlanarOpenStringAmplitudesandLargeNQCDWealreadysaw,throughsomeexamplesintheprevioussection,thatplanardiagramsalwayscomewithafactorof(g2N)nwherenisanintegerpositivenumber.Also,sinceonlytheplanardiagramssurvivethelargeNlimit,thesumofthedominantdiagramsforlargeNQCDhastheform 1Xn=0cn(g2YMN)n=1Xn=0cnn (1) whereg2YMNistheso-called'tHooftcouplingwhichischosentoremainxedaswetaketheN!1limitinordertogiveasensibleseriesexpansion.Wewillnowreviewtheperturbativeexpansioninopenstringtheoryandshowhowtodescribethesumofeldtheoryplanardiagrams( 1 )asthelowenergylimitofthesumofplanarmultiloopopenstringdiagrams.Webeginwithanopenstringwhoseend-pointscanlieonanytwodifferentbranesofastackofNcoincident2Dp-branes,say,branesiandj.Wethereforewanttocomputethescatteringamplitudeforthisopenstringtobeginandendinthe[ij]conguration.Sincethisisquantummechanics,weneedtosumallpossiblecongurationscompatiblewiththeinitialandnalopenstringstretchingbetweenD-branesiandj.Wenowrecallthatwhenanopenstringsplits,thestrengthofthatinteractioniscontrolledbytheopenstringcouplingconstantgo.Thus,onepossibilityisthatthestringsimplypropagateswithoutsplittingonce.Thisprocessisschematically 2WewantcoincidentDp-branesinordertohavemasslessstatesinthelowenergylimitwhichwewilllateridentifywiththegluonsoftheeldtheory.Sincestringshavetension,anyniteseparationamonganytwoD-braneswillproducemassivestatesevenaftertakingtheeldtheorylimit0!0 22

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showningure 1-9 andithasthetopologyofadisk.Thiscontributestotheamplitudewithamereconstanttermd0,i.e.itdoesnotdependongooronN.Considernowthe Figure1-9. Afreelypropagatingopenstringdiagram.Thetopologyofthisdiagramisthatofadisk.Theindicesiandjrepresentthetwoend-pointsoftheopenstringattachedtoD-branesiandjrespectively. casewherethestringsplitsintotwoandthetwonewend-pointsgetbothattachedtoathirdD-brane,saythek-thD-brane.Thisisnowshowningure 1-10 andithasthetopologyofanannulus.SincethatintermediateD-branecouldbeanyoftheNavailableD-branesandweneedtoincludeinoursumallpossibleintermediatecongurations,thetotalcontributionfromthisprocessisd1g2oN.Theg2ofactorcomesfromthesplittingandre-joiningoftheopenstringandd1isjustaconstant.Ifweaddonemoresplittingwiththesubsequentrejoiningasshowningure 1-11 ,wewillhaveacontributionofd2(g2oN)2.Wethereforeseethat,ifweaddmoresplittingsandrejoiningsbycreatingmorenewboundariesintheinterioroftheoriginalstrip,wecanwritethefullamplitudeas 1Xn=0dn(g2oN)n=1Xn=0dnn=g0() (1) fromwhichweseethattheconvergenceofthisexpressioniscontrolledbythe'tHooftcouplingg2oNifweidentifytheopenstringcouplingwiththeYang-Millsone,i.e.go=gYM.ThisistrueifthelowenergylimitoftheopenstringtheoryisanSU(N)gaugetheory3.Tondthecompleteamplitude,weneedtoincludeallpossiblenewcongurationswhichdonotcreatenewboundaries,astheoneshowningure 3Fortheopenstringtheorywestudyinthisthesis,calledtheNS+openstring,thisiscertainlythecase. 23

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1-12 .Inthiscase,wehaveanextrafactorofg2oduetothenewinteractions,butthenumberofboundarieshasdecreasedbyone.Thismeansthatthisdiagramnowgivesg6oN=(g2oN)3N)]TJ /F4 7.97 Tf 6.58 0 Td[(2,whichisthensubleadingbyafactorof1=N2inthelargeNexpansionwithxed.Thistypeofdiagramsareknownasnon-planardiagramssincewecannotdrawthemontheplanewithoutmakingthestripscross.Byaddingmorestripsinthisfashion,oneisintroducingafactorofg2o=N==N2eachtime,thereforethefullexpressionfortheamplitudeincludingallplanarandnon-planarcontributionsis A=g0()+g1()1 N2+g2()1 N4+ (1) Therefore,intheN!1limitwithheldxed,onlythersttermabovecontributestotheamplitude,i.e.onlytheplanarmultiloopopenstringdiagramssurviveinthislimit. Figure1-10. Openstringplanarone-loopdiagram.Sincethereisoneinternalboundaryandtwointeractionverticesthisdiagramisg2oN. Figure1-11. Openstringplanartwo-loopdiagram.Therearetwointernalboundariesandfourinteractionvertices,thusthisdiagramisg4oN2=(g2oN)2. Finally,weseethattheexpansionforlargeNQCD( 1 )andtheoneforplanaropenstringmultiloopdiagrams( 1 )havethesameformprovidedthatthestringcouplingweusebegg2YMN.Thus,gmustbeheldxedinthelargeNlimit. 24

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Figure1-12. Openstringnon-planarloopdiagram.Duetonon-planarity,thisdiagramhasonlyoneinternalboundary.Countingthenumberofinteractionverticesandinternalboundariesthisdiagramgivesg6oN=(g2oN)3N)]TJ /F4 7.97 Tf 6.58 0 Td[(2. Moreover,ifwecomputethescatteringamplitudeofmasslessvectorstatesinanopenstringtheorywhoselowenergyspectrumisthespectrumofpureYang-Mills(i.e.SU(N)vectorbosons),the0!0limitof( 1 )mustbecometheseriesin( 1 ).Undoubtedly,wewillalsoneedthatourstringmasslessstatesdescribefourdimensionalgaugebosonsas0!0sincestringtheoriesareusuallydenedindimensionshigherthanfour.Thisisachieved,forexample,whenthespacetimedimensionsoftheD-branesisfour,i.e.,whentheopenstringendpointsareattachedtoD3-branes4. 4Anotherpossibilityistobasethesumofplanardiagramsonthesubcriticalopenstringproposedin[ 7 ]wherethestringamplitudesaredirectlydenedinfourdimensions.Yetanotherapproachistousethecriticalstringin10Dandcompactifyingthesixextradimensions.However,theD-braneapproachseemssuperiorbecauseitintroduceslessarbitrariness. 25

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CHAPTER2ANALYTICCONTINUATIONANDRENORMALIZATIONOFTHEONE-LOOPM-GLUONAMPLITUDEThesumofmultiloopplanaropenstringdiagramsproposedin[ 8 9 ]anddiscussedintheintroductionchapter,needswelldenedconvergentintegralrepresentationsforthescatteringamplitudes.Thisbecomescrucial,forexample,whenattemptingtousenumericalmethodstoperformthesum.Stringamplitudesforanarbitrarynumberoftheexternalstatesusuallycomeintheformofintegralrepresentationsthataredivergentinvariouscornersoftheintegrationregionwhichcorrespondtorealandspuriousdivergences.Thespuriousdivergenceshavebeenextensivelystudiedfortheamplitudeofexternaltachyonicstates[ 1 2 10 ]andnormallyananalyticcontinuationschemeisnecessarytoprovideaniteanswer.Inthischapterweextendthesestudiestothecaseofmasslessvectorstates. 2.1TheGNSRegulatorandtheRemovalofSpuriousDivergencesWewillseeindetailinsections 2.1.1 and 2.1.2 thattheintegralexpressionfortheM-gluonplanarone-loopamplitudeisplaguedwithlinearandlogarithmicdivergencesincertaincornersoftheintegrationregion.Theseinnitiessimplyarisefromtheuseofanintegralrepresentationoutsideitsdomainofconvergence[ 1 ].Thepointwewouldliketostresshereisthatthesedivergencesareadirectconsequenceofmomentumconservation,andifweallowforPMi=1pi=p6=0,wecanregulateandtracktheeffectsofthesedivergences.Finally,weanalyticallycontinuetheintegralstop=0attheveryendofourcalculations.Wewillseethatthistechniqueleadstophysicallymeaningfulconsequencessuchasgaugeinvariancebecauseitallowstoprovethatmasslessvectorbosonsremainmasslessatoneloop.Itwillalsoprovetobeimportantwhenwestudythehighenergyregimesinchapters 3 and 4 .Thistechniquewasproposedandused,inthecontextoftheoldDualResonancesModels(DRM),byPeterGoddard[ 2 ]andAndreNeveuandJoelScherk[ 1 ].ThevariablepthatrepresentsthesuspensionofmomentumconservationisreferredtoastheGoddard-Neveu-ScherkorGNSregulatorforshort[ 7 ]. 26

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Wewillrstbeginbywritingthefullopenstringplanarone-loopamplitudeforthescatteringofMgluonsfortheevenG-parityNeveu-Schwarz(NS+)model[ 11 12 ].Bygluonswereallymeanmasslessvectorbosonstringstatessincethesestateswillbecomethegluonsinthelimitinggaugetheory.Weshouldalsoclarifythatanoverallgrouptheoryfactoroftr(TaTbTcTd),comingfromtheSU(N)Chan-Patonfactors,isimplicitinallofouramplitudes.Havingsaidthis,theproperlynormalizedM-gluonamplitudeis(gp 20)Mtimes MM=1 2(M+M)-222(M)]TJ /F4 7.97 Tf 0 -8.19 Td[(M) (2) where MM=Zdw wMYi=2dyi yiw)]TJ /F4 7.97 Tf 6.58 0 Td[(1=2)]TJ /F2 11.955 Tf 9.29 0 Td[(1 40lnwD=2exp(0Xi
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D-braneprojectionproposedin[ 13 ].Adifferentwaywouldbebymeansofanorbifoldprojectionwhichwediscussbrieylateroninsection 3.5 .Wewillalsoseethattheorbifoldprojectionproducesthesameanswerintheeldtheory(i.e.lowenergy)limitasthenonabelianone,buttheydifferathighenergyasexpected.ThenonabelianD-braneprojection[ 13 ]producesanextrafactorof(1w1=2)10)]TJ /F4 7.97 Tf 6.59 0 Td[(D)]TJ /F4 7.97 Tf 6.59 0 Td[(Sintheintegrandabovewhichwealsoneedtoinclude,whereSisthenumberofscalarsremainingaftertheprojection.Certainly,forthecaseweareinterestedatwhichislargeNQCD,therearenomasslessscalarsinthespectrum,sowewouldneedS=0.However,wewillleaveSarbitraryinordertomakeourexpressionsmoregeneral.ThefactorsinvolvingtheJacobi1functionhavetheinniteproductrepresentation Yi
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for)]TJ /F1 11.955 Tf 12.62 0 Td[(theycontributewithoppositesigns.IntheF2picture,thedifferenceofthetwotracesprojectsouttheoddG-paritystates.Finallywepresenttheone-loopamplitudeinthenaturalcylindervariables,i=lnyi=lnwandlnq=22=lnw, M+M=2M1 820D=2ZMYk=2dkZ10dq q)]TJ /F5 11.955 Tf 9.29 0 Td[( lnq(10)]TJ /F4 7.97 Tf 6.59 0 Td[(D)=2P+(q)Yl
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evaluatedwithcontractions: hPli=p 20Xiki"1 2cotil+1Xn=12q2n 1)]TJ /F2 11.955 Tf 11.95 0 Td[(q2nsin2nil# (2) hPiPli)-222(hPiihPli=1 4csc2il)]TJ /F6 7.97 Tf 16.35 14.94 Td[(1Xn=1n2q2n 1)]TJ /F2 11.955 Tf 11.96 0 Td[(q2ncos2nil (2) hHiHji++(ji)=1 2sinji)]TJ /F2 11.955 Tf 11.96 0 Td[(2Xrq2rsin2rji 1+q2r=1 22(0)4(0)3(ji) 1(ji) (2) hHiHji)]TJ /F3 11.955 Tf 17.05 -4.94 Td[()]TJ /F2 11.955 Tf 7.09 1.79 Td[((ji)=cosji 2sinji)]TJ /F2 11.955 Tf 11.96 0 Td[(2Xnq2nsin2nji 1+q2n=1 23(0)4(0)2(ji) 1(ji). (2) Wehaveabbreviatedji=j)]TJ /F5 11.955 Tf 12.46 0 Td[(iandwehaveagainsuppressedspace-timeindices.Finallytherangeofintegrationis 0=1<2<
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Momentumconservationimplies2k1k2=(k1+k2)2=0,thus MBose2=Z10[dq]Z0d"1 4csc2)]TJ /F6 7.97 Tf 16.36 14.95 Td[(1Xn=1n2q2n 1)]TJ /F2 11.955 Tf 11.95 0 Td[(q2ncos2n# (2) fromwhichweseethatthersttermisclearlydivergentinthe0,regions.However,byusingtheGNSregulatorwewillshowthatthisisaspuriousdivergence.Inordertoobtainabetterintegralrepresentationwecouldintegratebypartsandusethatasouranalyticcontinuationprocedure,howeverthatisnotthepathwefollowhere.Instead,wesuspendmomentumconservationintheintermediatestepsbyusingtheGNSregulatorp(whichwetreatasacomplexvariable)sothatnowwehave2k1k2=p2insteadof2k1k2=p2=0whichmakesintegralrepresentationperfectlyconvergentfor0p2>1.Wethenanalyticallycontinuetheintegraltop!0attheend.Noticethatthereisonlyoneangularintegrationinthetwogluonfunctionwhichwillallowustoperformthetheanalyticcontinuationtop=0ratherstraightforwardlyasweshallnowsee.Writingtheamplitude( 2 )again,butthistimewiththepregulatoron,reads MBose2=Z10[dq]Z0d[sin]0p2"1Yn=11)]TJ /F2 11.955 Tf 11.96 0 Td[(2q2ncos2+q4n (1)]TJ /F2 11.955 Tf 11.95 0 Td[(q2n)2#0p2"1 4csc2)]TJ /F6 7.97 Tf 16.36 14.94 Td[(1Xn=1n2q2n 1)]TJ /F2 11.955 Tf 11.95 0 Td[(q2ncos2n# (2) Expandingtheinniteproductuptorstorderinp2isenoughforourpurposes.Doingthisandperformingaresummationyields "1Yn=11)]TJ /F2 11.955 Tf 11.95 0 Td[(2q2ncos2+q4n (1)]TJ /F2 11.955 Tf 11.96 0 Td[(q2n)2#0p2=1+0p21Xm=11 m2q2m 1)]TJ /F2 11.955 Tf 11.95 0 Td[(q2m(1)]TJ /F2 11.955 Tf 11.95 0 Td[(cos2m)+O(p2) (2) 31

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therefore MBose2=Z10[dq]1 4Z0d[sin]0p2)]TJ /F4 7.97 Tf 6.59 0 Td[(2+ (2) )]TJ /F6 7.97 Tf 15.69 14.94 Td[(1Xn=1n2q2n 1)]TJ /F2 11.955 Tf 11.95 0 Td[(q2nZ0d[sin]0p2cos2n+ (2) +0p21Xm=11 m2q2m 1)]TJ /F2 11.955 Tf 11.95 0 Td[(q2m1 4Z0d[sin]0p2)]TJ /F4 7.97 Tf 6.59 0 Td[(2(1)]TJ /F2 11.955 Tf 11.96 0 Td[(cos2m)+ (2) )]TJ /F5 11.955 Tf 9.3 0 Td[(0p21Xm,n=11 m2q2m 1)]TJ /F2 11.955 Tf 11.96 0 Td[(q2mn2q2n 1)]TJ /F2 11.955 Tf 11.95 0 Td[(q2nZ0d[sin]0p2cos2n(1)]TJ /F2 11.955 Tf 11.95 0 Td[(cos2m)# (2) Withouttheregulator,theonlyproblematictermhereistherstone,sinceputtingp2=0intheintegrandshowsalineardivergenceintheintegration.HoweverifweassumethatRe(p2)>1wehave 1 4Z0d[sin]0p2)]TJ /F4 7.97 Tf 6.59 0 Td[(2=1 4\(1=2)\(0p2=2)]TJ /F2 11.955 Tf 11.96 0 Td[(1=2) \(0p2=2)=)]TJ /F5 11.955 Tf 10.5 8.09 Td[(0p2 4+O(p4) (2) Thus,takingtherighthandsidetobetheanalyticcontinuationoftheleft-handsideasp!0,wehaveaconvergentexpression.Therestoftheintegralsarecompletelyconvergentevenifwesetp2=0intheirintegrands.Thuswenowhaveanewexpressionwhichwetakeittobetheanalyticcontinuationof( 2 )top!0,thatreads MBose2=Z10[dq]")]TJ /F5 11.955 Tf 10.5 8.09 Td[(0p2 4+0p21Xn=12q2n 1)]TJ /F2 11.955 Tf 11.96 0 Td[(q2nn 2n+0p21 41Xm=11 m2q2m 1)]TJ /F2 11.955 Tf 11.95 0 Td[(q2m2m++0p21Xm=11 m2q2m 1)]TJ /F2 11.955 Tf 11.95 0 Td[(q2m1Xn=1n2q2n 1)]TJ /F2 11.955 Tf 11.95 0 Td[(q2n 2n,m#=0p2Z10[dq]")]TJ /F2 11.955 Tf 10.49 8.09 Td[(1 4+1Xn=1q2n 1)]TJ /F2 11.955 Tf 11.96 0 Td[(q2n+1Xm=1q2m 1)]TJ /F2 11.955 Tf 11.96 0 Td[(q2m+1Xn=12q4n (1)]TJ /F2 11.955 Tf 11.95 0 Td[(q2n)2#=0p2Z10[dq]")]TJ /F2 11.955 Tf 10.49 8.09 Td[(1 4+1Xn=12q2n (1)]TJ /F2 11.955 Tf 11.96 0 Td[(q2n)2# (2) 32

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whichshowsthat,notonlythelimitp!0isnite,butthatitisactuallyzero.Thisisverywelcomeheresincethevanishingofthetwo-gluonfunctionguaranteesthatthegluonremainsmasslessinperturbationtheory,whichisaconsequenceofgaugeinvariance.Thecompletetwo-gluonamplitudefortheNS+stringwasworkedoutbyThorn[ 7 ]withtheresult: MNS,+20p2Z[dq]+24)]TJ /F2 11.955 Tf 10.49 8.08 Td[(1 2+41Xn=1q2n (1)]TJ /F2 11.955 Tf 11.96 0 Td[(q2n)2+41Xr=1=2q2r (1+q2r)235 (2) MNS,)]TJ /F4 7.97 Tf -13.4 -8.6 Td[(20p2Z[dq])]TJ /F9 11.955 Tf 9.08 15.5 Td[("41Xn=1q2n (1)]TJ /F2 11.955 Tf 11.95 0 Td[(q2n)2+41Xn=1q2n (1+q2n)2# (2) whichshowsthattheanalyticallycontinuedresulttop!0isagainzerointhetwo-gluonfunctionfortheNS+string.TomotivatethegeneralresultfortheM-gluonstringamplitudeatoneloop,letusconsiderthefourgluonfunction.Inordertobeabletocomparevariouscalculationsthatwewillperforminthisthesis,wewillfocusonaparticularpolarizationstructureofthe4-gluonamplitude,namelythecoefcientof1423.ThemainreasontodothisisthatthisstructuredominatesintheReggelimit(s!withtxed)attreelevel.Thereforewehavetofocusonitinordertoextracttheone-loopcorrectiontotheReggetrajectorywhichwedoinchapter 3 .Attreelevel,the4-gluonamplitudefortheNSstring(forthepolarizationaboveandomittingnumericalcoefcients)is Mtree4=g2\(1)]TJ /F5 11.955 Tf 11.96 0 Td[(0s)\()]TJ /F5 11.955 Tf 9.3 0 Td[(0t) \()]TJ /F5 11.955 Tf 9.3 0 Td[(0s)]TJ /F5 11.955 Tf 11.95 0 Td[(0t) (2) Atone-loopthegeneralformofthe4-gluonamplitudeis M4=1 2)]TJ /F3 11.955 Tf 5.48 -9.68 Td[(M+4)-222(M)]TJ /F4 7.97 Tf 0 -7.89 Td[(4 (2) 33

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with M+4=241 80D=2Z10dq q)]TJ /F5 11.955 Tf 9.3 0 Td[( lnq(10)]TJ /F4 7.97 Tf 6.59 0 Td[(D)=2q)]TJ /F4 7.97 Tf 6.59 0 Td[(1(1)]TJ /F2 11.955 Tf 11.95 0 Td[(w1=2)10)]TJ /F4 7.97 Tf 6.59 0 Td[(D)]TJ /F4 7.97 Tf 6.59 0 Td[(SQ1r(1+q2r)8 Q1n(1)]TJ /F2 11.955 Tf 11.96 0 Td[(q2n)8Z4Yk=2dkYi
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Aspointedoutbefore,theexpressions( 2 )and( 2 )divergeinvariouscornersofintegrationregionoverk.Wealreadyencounteredadivergenceofthelineartypeinthe2-gluonamplitudeduetothebehaviorof(csc)2neartheendpoints0,.WeshowedthatthisdivergencewasspuriousanditwastreatedbysuspendingmomentumconservationinPMi=1pi=p6=0withtheuseoftheGNSregulatorp.Afterthat,wewereabletoidentifytheintegralin( 2 )astheEulerBetafunctionwhichallowedustoanalyticallycontinuethelefthandsidetothecompletecomplexp-plane.Undoubtedly,forthethreeandhighergluonamplitudesaclosedformispracticallyimpossibletoobtain.However,ourapproachtotheproblemwillnotbetotrytodothis,buttoextractthedivergentcontributionsfromthesingularregionsandtracktheconsequencesoftheseseeminglydivergentterms.Whatwewillndisthattheanalyticcontinuationtop=0ofthelineardivergencespreciselycombineandgivethetreeamplitudefollowingthestepsof[ 1 ].Althoughthecoefcientofthistermisaninnitenumber(whichcanalsobeviewedduetothepresenceoftheclosedstringtachyonwhichintroducesasingularityintheq0region),thefactthatitisproportionaltothetreeamplitudeallowsusabsorbthedivergenceintotheopenstringcouplingconstant2.Wewillthenndthatthelogarithmicallydivergentcorners,whencontinuedtop=0alsoproducetermsproportionaltothetreeamplitude,althoughinthiscasethecoefcientinfrontofitisanitenumberandthesecornerswillsimplycorrectthetreeamplitudebyaniteamount.Wewillnowmakethesestatementsexplicitwiththefollowingcalculations. 2.1.1LinearDivergencesWewillnowextracttheleadingdivergencesinthekintegrationsatxedqandshowthattheyarelineardivergencesintherelevantangularvariables.Weconstructthenecessarycountertermstocanceltheseinnitiesandshowthatafteranalytic 2Asasidenote,wecouldsaythattheinfraredphysicsofclosedstringtachyongetscoarsegrainedandabsorbedintotheopenstringcouplingconstant. 35

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continuation,thelimitp!0oftheangularintegralsisnite3.WewillfollowcloselytheanalysisdonebyNeveuandScherk[ 1 ]adaptedforourcase,theNS+stringwithmasslessvectorbosonsintheexternalstates,andshowthatnotonlythelimitisnitebutalsothatitscontinuationtop!0givespreciselythetreeamplitude.Thisallowsustoabsorbthecorrespondingcountertermsintotheopenstringcoupling.FortheM-pointplanarone-loopamplitude,theintegrationregioninRQkdkisgivenby0<2<3<
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Wecanstudytheverticesofthe(M)]TJ /F2 11.955 Tf 13.28 0 Td[(1)-simplexbyrememberingthattheycorrespondtothecongurationinparameterspacewhereallthevertexoperatorscoincide.Forinstance,wecanexaminetheonewhereMM)]TJ /F4 7.97 Tf 6.59 0 Td[(120bystudyingtheM0limitandperformingthechanges j)]TJ /F4 7.97 Tf 6.58 0 Td[(1j^j)]TJ /F4 7.97 Tf 6.59 0 Td[(1j=3,M (2) Forthe4-gluonamplitudeandkeepingonlythemostdivergenttermsinthekintegrationswehave Yi
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propagationofaclosedstringtachyon.Incontrasttosuperstringtheorieswherethiskindofdivergencesareabsentduetosupersymmetry(orbytheGSOprojectionintheRNSformulation)theNS+modelisrenormalizableandthecancellationofthesedivergencesisachievedwiththeintroductionofcounter-termsjustasintheearlydaysofthedualresonancemodels.Wenowproceedtocancelthisandalloftheotherlineardivergenceswhichcomefromalltheverticesofthesimplex4withonesinglecounter-term.Wesubtractandaddbackthefollowingcounter-term: C+4241 820D=2Z10[dq]+Z4Yk=2dkYi
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Thus Yi
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Asp!0,theonlynon-zerocontributionstotheintegralcomefromonlytwocorners[ 1 ]:4andrisneareitherr=1orr=x.Eachcornergivesthesameanswerwhichis,therefore: I!2asp!0 (2) Hence Z4Yk=2dkYi
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divergence,andthefactthatitisproportionaltothetreeamplitudeallowsustoabsorbthisdivergenceintoacouplingconstantrenormalization.Theremarkablefeatureofthiscounter-termisthatitallowstocancelboth,theq=0singularity,andthespuriouslineardivergencesofthekintegrationsatthesametime.Thisisaconsequenceofthefunctionalformofthecorrelatorh^P1^PMisincethe)]TJ /F4 7.97 Tf 6.58 0 Td[(2divergenttermsthatarisefromthecsc2functionsonlycomefromtheq=0partofWickexpansionofh^P1^PMi.Wethusnowhaveanewexpressionfreeofboththespuriouslineardivergences6inthekvariablesandtheonefromq=0: M+4!M+4)]TJ /F2 11.955 Tf 11.95 0 Td[(C+4 (2) NowweneedtoaddresstheM)]TJ /F4 7.97 Tf 0 -7.88 Td[(4partoftheamplitude.Noticein( 2 )thatthepresenceofD-branes,whichbringsthelogarithmicfactor)]TJ /F7 7.97 Tf 6.58 0 Td[( lnq(10)]TJ /F4 7.97 Tf 6.59 0 Td[(D)=2,makestheq-integrationcompletelynitenearq=0aslongasD<8andhencethereisnoneedforacountertermfortheM)]TJ /F4 7.97 Tf 0 -7.88 Td[(4partoftheamplitude7.However,wedoneedtodealwiththesamelinearandlogarithmicdivergencesinthekintegrationasintheM+4case.Fortheleadingdivergences,thenaturalchoiceforacancelingtermwouldbethesameoneweusedfortheM+4casewithh^P1^PMi+replacedbyh^P1^PMi)]TJ /F1 11.955 Tf 7.09 -4.34 Td[(,butwehavenotbeenabletoobtaintheanalyticcontinuationtop=0forsuchanexpression.Themaindifcultycomesfromthefactthattheh^P1^PMi)]TJ /F1 11.955 Tf 10.41 -4.34 Td[(correlatorsinvolvecotjifunctionswhichchangesignintheintegrationregion0<2<
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However,sinceweonlyneedtocancelthelineardivergences,wesimplychoosethesamecorrelatorasbefore.i.e.h^P1^PMi+andonlyadaptthecounter-termtothe(-)amplitudebyintegratingwiththe[dq])]TJ /F1 11.955 Tf 10.41 -4.34 Td[(measure.Thismeansthatwechoose: C)]TJ /F4 7.97 Tf 0 -7.89 Td[(4241 820D=2Z10[dq])]TJ /F9 11.955 Tf 9.08 11.34 Td[(Z4Yk=2dkYi
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operatorsbutonecometogetherinparameterspace.Itisawellknownfactthatthesedivergencescorrespondtoloopcorrectionstothemassoftheexternalstates.Sincewearedealingwithmasslessstringstates,weexpectthesedivergencestobecompletelyabsentaftercontinuationtop=0becausethegluonmustremainmasslessinperturbationtheoryduetogaugeinvariance.Wewillindeedndthisresultforthe4-gluonamplitude.Themechanicsoftheprocedureisverywellillustratedbythe4-gluonamplitudeandwillallowustoseehowtoextenditforanarbitrarynumberofgluons.RecallthatfortheM-gluonamplitudetheintegrationregionoverkisan(M)]TJ /F2 11.955 Tf -425.18 -23.91 Td[(1)-simplexwhichhasM!=(2!(M)]TJ /F2 11.955 Tf 12.17 0 Td[(2)!)edges(seegure 2-1 forthe4-gluonamplitudeinwhichcasethereare6edges).Eachoftheseedgescorrespondtoprocesseswhereanopenstringloopisinsertedbetweentwostringstates.IfoneofthesestatescorrespondtooneoftheMexternalstates,wethushavethesituationwhereaninternalpropagatorgetsevaluatedon-shellproducinganinnity.Beforeproceedingwiththeanalysisoftheseinnities,letusdosomecounting.Weseethatthereare M! 2!(M)]TJ /F2 11.955 Tf 11.96 0 Td[(2)!)]TJ /F2 11.955 Tf 11.95 0 Td[(M=M 2(M)]TJ /F2 11.955 Tf 11.95 0 Td[(3) (2) edgesleftwhichdonotcorrespondtoradiativecorrectionstotheexternallegs.Therefore,thenumberofedgesthatcorrespondtoaloopinsertionintheinternalchannelsoftheM-gluonamplitudehastobegivenbyequation( 2 ).Ontheotherhand,weknowthatthenumberofplanarchannelsinanM-pointamplitudeisM=2(M)]TJ /F2 11.955 Tf 9.48 0 Td[(3)whichpreciselymatchesthenumberabove.Letusnowfocusonthe4-gluonamplitude.Thishasfour(outofsix)edgesthatshouldcorrespondtoanopenstringloopinsertedforeachexternalleg8.Wenowstudyoneofthem,namelytheedge320whichishighlightedingure 2-2 This 8Theothertwoedgesevidentlycorrespondtoloopinsertionsineachofthetwoplanarchannels:sandt.Thet-channelwillbeimportantinchapter 3 sinceitisthedominantoneintheReggelimitandallowsustoobtaintheone-loopcorrectiontothegluonReggetrajectory. 43

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Figure2-2. Theedgecorrespondingto320isshownasthehighlightedlineinthegure.Thisregioncorrespondstoaloopinsertioninoneoftheexternalstateswhichforcesthepropagatorforstatenumber4tobeevaluatedon-shellproducingadivergence correspondstotheregionwherethevertexoperatorsassociatedwithexternalstates1,2and3getclosetogetherinparameterspaceanditreectsaradiativecorrectiontothemassoftheexternalleg4.Toanalyzeit,itisconvenienttomakethechange23^2andstudythesmall3behavior,namely Yi
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hCorri')]TJ /F3 11.955 Tf 5.48 -9.68 Td[(P14+0thH1H4i2(1+0t)1 4232+hH1H4i21 232(0s)2 22)]TJ /F5 11.955 Tf 13.15 8.09 Td[(02(s+t)2 23=1 423")]TJ /F3 11.955 Tf 5.48 -9.69 Td[(P14+0thH1H4i2(1+0t) (1)]TJ /F2 11.955 Tf 12.89 3.16 Td[(^2)2+hH1H4i2 (0s)2 ^2(1)]TJ /F2 11.955 Tf 12.89 3.16 Td[(^2))]TJ /F5 11.955 Tf 13.15 8.09 Td[(02(s+t)2 (1)]TJ /F2 11.955 Tf 12.9 3.16 Td[(^2)!#=1 423"P14(1+0t) (1)]TJ /F2 11.955 Tf 12.9 3.16 Td[(^2)2+hH1H4i2 0t(1+0t) (1)]TJ /F2 11.955 Tf 12.89 3.16 Td[(^2)2+(0s)2 ^2(1)]TJ /F2 11.955 Tf 12.9 3.16 Td[(^2))]TJ /F5 11.955 Tf 13.15 8.09 Td[(02(s+t)2 (1)]TJ /F2 11.955 Tf 12.9 3.16 Td[(^2)!# (2) FromQ4k=2dk=d33d^2andequations( 2 )and( 2 )weseethattheleadingbehavioroftheintegraloverthethreeanglesseparatesintothreeindependentintegrals.Theintegrationoverthekvariablesin( 2 )thenbecome Z4Yk=2dk"Yi
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Therestofthetermsalreadyhaveanexplicitfactorofpinfront,sowecansimplyputp=0intheirintegrandsobtaining: IP')]TJ /F6 7.97 Tf 30.96 14.95 Td[(1Xn=12q2n 1)]TJ /F2 11.955 Tf 11.95 0 Td[(q2nn" 2n20k4p+20k4pZ0dcos2n"ln)]TJ /F6 7.97 Tf 16.66 14.95 Td[(1Xm=11 m2q2m 1)]TJ /F2 11.955 Tf 11.95 0 Td[(q2m(1)]TJ /F2 11.955 Tf 11.95 0 Td[(cos2m)##')]TJ /F6 7.97 Tf 30.96 14.95 Td[(1Xn=12q2n 1)]TJ /F2 11.955 Tf 11.95 0 Td[(q2nn20k4p" 2n)]TJ /F2 11.955 Tf 13.15 8.09 Td[(Si(2n) 2n+1Xm=11 m2q2m 1)]TJ /F2 11.955 Tf 11.96 0 Td[(q2m 2n,m#'20k4p1Xn=1q2n 1)]TJ /F2 11.955 Tf 11.95 0 Td[(q2n)]TJ /F5 11.955 Tf 9.29 0 Td[(+Si(2n))]TJ /F2 11.955 Tf 20.46 8.09 Td[(2q2n 1)]TJ /F2 11.955 Tf 11.96 0 Td[(q2nThenewterminthesum,Si(2n),makesthesumconvergeratherfastatxedqsothereisnothingpotentiallydangerouscomingfromthisterm.Hence,thesmallpbehavioroftheP(4))-222(P(4)Ccontributionis =1 41 20k4p(1+0t)20k4p1Xn=1q2n 1)]TJ /F2 11.955 Tf 11.95 0 Td[(q2n)]TJ /F5 11.955 Tf 9.3 0 Td[(+Si(2n))]TJ /F2 11.955 Tf 20.46 8.09 Td[(2q2n 1)]TJ /F2 11.955 Tf 11.95 0 Td[(q2nZ10d^2^)]TJ /F7 7.97 Tf 6.58 0 Td[(0s2(1)]TJ /F2 11.955 Tf 12.89 3.16 Td[(^2))]TJ /F7 7.97 Tf 6.59 0 Td[(0t)]TJ /F4 7.97 Tf 6.59 0 Td[(2=1 4(1+0t)1Xn=1q2n 1)]TJ /F2 11.955 Tf 11.96 0 Td[(q2n)]TJ /F5 11.955 Tf 9.3 0 Td[(+Si(2n))]TJ /F2 11.955 Tf 20.45 8.09 Td[(2q2n 1)]TJ /F2 11.955 Tf 11.95 0 Td[(q2n\(1)]TJ /F5 11.955 Tf 11.95 0 Td[(0s)\()]TJ /F2 11.955 Tf 9.29 0 Td[(1)]TJ /F5 11.955 Tf 11.96 0 Td[(0t) \()]TJ /F5 11.955 Tf 9.29 0 Td[(0s)]TJ /F5 11.955 Tf 11.95 0 Td[(0t)= 41Xn=1q2n 1)]TJ /F2 11.955 Tf 11.96 0 Td[(q2n1)]TJ /F2 11.955 Tf 13.15 8.09 Td[(Si(2n) +2q2n 1)]TJ /F2 11.955 Tf 11.95 0 Td[(q2n\(1)]TJ /F5 11.955 Tf 11.95 0 Td[(0s)\()]TJ /F5 11.955 Tf 9.3 0 Td[(0t) \()]TJ /F5 11.955 Tf 9.3 0 Td[(0s)]TJ /F5 11.955 Tf 11.96 0 Td[(0t) (2) fromwhichweseethatthiscountertermisalsoproportionaltothetreeamplitude. 48

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Beforegoingonandcomputetheintegralover4forthe(4)2)]TJ /F5 11.955 Tf 12.67 0 Td[((4)2Ctermin( 2 ),letusrstcalculatetheintegralover^2thatmultipliesit.Thisis Z10d^2^)]TJ /F7 7.97 Tf 6.59 0 Td[(0s2(1)]TJ /F2 11.955 Tf 12.89 3.16 Td[(^2))]TJ /F7 7.97 Tf 6.58 0 Td[(0t 0t(1+0t) (1)]TJ /F2 11.955 Tf 12.9 3.16 Td[(^2)2+(0s)2 ^2(1)]TJ /F2 11.955 Tf 12.9 3.16 Td[(^2))]TJ /F5 11.955 Tf 13.15 8.09 Td[(02(s+t)2 (1)]TJ /F2 11.955 Tf 12.9 3.16 Td[(^2)!=0t(1+0t)Z10d^2^)]TJ /F7 7.97 Tf 6.59 0 Td[(0s2(1)]TJ /F2 11.955 Tf 12.9 3.16 Td[(^2))]TJ /F7 7.97 Tf 6.58 0 Td[(0t)]TJ /F4 7.97 Tf 6.59 0 Td[(2+(0s)2Z10d^2^)]TJ /F7 7.97 Tf 6.59 0 Td[(0s)]TJ /F4 7.97 Tf 6.59 0 Td[(12(1)]TJ /F2 11.955 Tf 12.89 3.16 Td[(^2))]TJ /F7 7.97 Tf 6.59 0 Td[(0t)]TJ /F4 7.97 Tf 6.59 0 Td[(1)]TJ /F5 11.955 Tf 9.3 0 Td[(02(s+t)2Z10d^2^)]TJ /F7 7.97 Tf 6.58 0 Td[(0s2(1)]TJ /F2 11.955 Tf 12.89 3.15 Td[(^2))]TJ /F7 7.97 Tf 6.59 0 Td[(0t)]TJ /F4 7.97 Tf 6.58 0 Td[(1=0t(1+0t)\(1)]TJ /F5 11.955 Tf 11.95 0 Td[(0s)\()]TJ /F2 11.955 Tf 9.29 0 Td[(1)]TJ /F5 11.955 Tf 11.96 0 Td[(0t) \()]TJ /F5 11.955 Tf 9.29 0 Td[(0s)]TJ /F5 11.955 Tf 11.95 0 Td[(0t)+(0s)2\()]TJ /F5 11.955 Tf 9.3 0 Td[(0s)\()]TJ /F5 11.955 Tf 9.3 0 Td[(0t) \()]TJ /F5 11.955 Tf 9.3 0 Td[(0s)]TJ /F5 11.955 Tf 11.95 0 Td[(0t))]TJ /F5 11.955 Tf 9.3 0 Td[(02(s+t)2\(1)]TJ /F5 11.955 Tf 11.95 0 Td[(0s)\()]TJ /F5 11.955 Tf 9.3 0 Td[(0t) \(1)]TJ /F5 11.955 Tf 11.95 0 Td[(0s)]TJ /F5 11.955 Tf 11.96 0 Td[(0t)=)]TJ /F5 11.955 Tf 9.3 0 Td[(0t\(1)]TJ /F5 11.955 Tf 11.96 0 Td[(0s)\()]TJ /F5 11.955 Tf 9.3 0 Td[(0t) \()]TJ /F5 11.955 Tf 9.3 0 Td[(0s)]TJ /F5 11.955 Tf 11.95 0 Td[(0t))]TJ /F5 11.955 Tf 11.96 0 Td[(0s\(1)]TJ /F5 11.955 Tf 11.96 0 Td[(0s)\()]TJ /F5 11.955 Tf 9.3 0 Td[(0t) \()]TJ /F5 11.955 Tf 9.3 0 Td[(0s)]TJ /F5 11.955 Tf 11.95 0 Td[(0t)+0(s+t)\(1)]TJ /F5 11.955 Tf 11.95 0 Td[(0s)\()]TJ /F5 11.955 Tf 9.3 0 Td[(0t) \()]TJ /F5 11.955 Tf 9.3 0 Td[(0s)]TJ /F5 11.955 Tf 11.96 0 Td[(0t)=0\(1)]TJ /F5 11.955 Tf 11.96 0 Td[(0s)\()]TJ /F5 11.955 Tf 9.3 0 Td[(0t) \()]TJ /F5 11.955 Tf 9.3 0 Td[(0s)]TJ /F5 11.955 Tf 11.95 0 Td[(0t)[)]TJ /F2 11.955 Tf 9.3 0 Td[(t)]TJ /F2 11.955 Tf 11.95 0 Td[(s+s+t] (2) =0 (2) Thustheintegralover4ofthehH1H4itermin( 2 )doesnotneedtobecomputedsinceitismultipliedbyzeroanyhow!Theimmediatequestioniswhetherwewouldhaveobtainedthisresultalsoifwehadkeptp6=0whenweperformedtheWickcontractionstogethCorri.Theanswerisyes,althoughitisnottotallyobvioussinceifthisfactorvanishesasO(p),thenwedohaveanon-vanishingcontributionfromthistermduetotheO(p)]TJ /F4 7.97 Tf 6.58 0 Td[(1)factorcomingfromthe3integration(seeequation( 2 )).However,weperformthiscalculationintheappendixandshowthatitactuallyvanishesasO(p2),thereforetheargumentwegivehereissafeandcorrect.Afteralltheseintermediatecalculations,wecannallywritethecontinuationof( 2 )top=0,whichis = 41Xn=1q2n 1)]TJ /F2 11.955 Tf 11.96 0 Td[(q2n1)]TJ /F2 11.955 Tf 13.15 8.08 Td[(Si(2n) +2q2n 1)]TJ /F2 11.955 Tf 11.96 0 Td[(q2n\(1)]TJ /F5 11.955 Tf 11.96 0 Td[(0s)\()]TJ /F5 11.955 Tf 9.3 0 Td[(0t) \()]TJ /F5 11.955 Tf 9.3 0 Td[(0s)]TJ /F5 11.955 Tf 11.95 0 Td[(0t)i.e.,/ 41Xn=1q2n 1)]TJ /F2 11.955 Tf 11.95 0 Td[(q2n1)]TJ /F2 11.955 Tf 13.15 8.08 Td[(Si(2n) +2q2n 1)]TJ /F2 11.955 Tf 11.96 0 Td[(q2nfTreeg (2) 49

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whichshowsthatitscompletekinematicdependenceisexactlythesameasthetreeamplitude.Thus,thiscountertermcanalsobeabsorbedintoa(nite)couplingrenormalization.Wearethusnowreadytowritethecompleteniteexpressionfortheone-loopamplitude,wheremomentumconservationisexact,forfourgluons.Thisreads Z04Yi=2(i+1)]TJ /F5 11.955 Tf 11.96 0 Td[(i)di"Yi
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where IZ4Yk=2dkYi
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34suggeststhattheydonotcontributetotheregimewheretislargeandsisheldxedeither.Inspectingequations( 2 )through( 2 ),itisnaturaltoconjecturethatthisstructurewillremainvalidforanarbitrarynumberofexternalgluons.Theanalyticcontinuationtop=0oftheCcountertermwasproventobecomethetreeamplitudeandtosuccessfullycanceltheleadingdivergencesforthescatteringofanarbitrarynumberofexternaltachyonsintheoriginaldualresonancemodels[ 1 ].Itisthusplausibletobelievethat,sinceitworkedforthe2,3and4gluonamplitudes,itwillcontinuetodosoforanarbitrarynumberofexternalgluons10.Itwouldbeinterestingtoshowthisexplicitlyforthe5-gluoncase.Also,thefactthatthereisamatchbetweenthenumberofedgesandthenumberofloopinsertionsininternalchannelsplusthenumberofexternallegs(seeequation( 2 )),suggeststhattheBcountertermscanbeconstructedinthesamesystematicalwayweusedhereforthe4-gluoncase. 2.3D8andD9BranesandReggeSlopeRenormalizationThepresenceoftheDp-branesinallofthepreviouscalculationsensuredthattherewerenosubleadingdivergencesasq!0aslongasp<8duetoanextralogarithmicfactorthatcomesfromtheloopmomentumintegration,i.e.theonlydivergencesinthislimitwereofthetypeR0dq=q2whichareduetothepresenceoftheclosedstringtachyons11.However,wecanrenormalizeawaythesedivergencesbyacouplingconstantrenormalizationashasbeenshowninsection 2.1.1 10Forthe3-gluoncase,seereference[ 7 ].11RecallthattheevenG-parityprojectionwasperformedontothestatesofopenstringsector.Theclosedstringsectorhasbeenkeptintactandthetachyonisstillpresent.AssuggestedbyThorn[ 7 ],weactuallywantthistachyontoremainpresentsinceitcouldhelptoresolvenon-perturbativeissuesofQCDsinceitmightbesignalingthedirectionofthetruevacuumofQCDaftertheresummationoftheplanardiagrams. 52

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InthissectionwewouldliketobrieyconsiderthecasesforD8andD9branesforwhichthelogarithmicfactorisnotenoughtoavoidthesubleadingdivergence.However,itispossibletoseethatwecanagainabsorbthisnewdivergencebyarenormalizationoftheonlyparameterleftinthetheory:theReggeslopeparameter0.RecallthatinordertohaveReggesloperenormalization,wemustbeabletoshowthatthecountertermneededtosubtractthesubleadingdivergenceCslsatises Csl/@ @0Atree4 (2) Considernowthebosonicpartoftheactionthatcontributestotheworld-sheetpathintegralwith ZDXIe)]TJ /F4 7.97 Tf 6.59 0 Td[(SZDXIexp)]TJ /F2 11.955 Tf 10.5 8.09 Td[(T 2Zd2@XI@XI (2) Fromhereweseethat @ @0ZDXIe)]TJ /F4 7.97 Tf 6.59 0 Td[(S=)]TJ /F2 11.955 Tf 14.84 8.09 Td[(1 02@ @TZDXIexp)]TJ /F2 11.955 Tf 10.49 8.09 Td[(T 2Zd2@XI@XI=1 202ZDXIZd2@XI@XIexp)]TJ /F2 11.955 Tf 10.49 8.08 Td[(T 2Zd2@XI@XI (2) inwhichwerecognize12 VD(k=0)=@XI@XI (2) asthevertexoperatorfortheemissionofazero-momentumdilaton.Therefore,takingthederivativewithrespecttotheReggeslope0correspondstotheamplitudefortheemission(absorption)ofaclosedstringdilatoninto(from)thevacuum.Now,thesubleadingdivergencepresentforD9-branecase(space-llingD-brane)isoftheformRdq qanditcorrespondspreciselytotheemissionofaclosedstringdilatonatzero 12HerethestringtensionTandtheReggeslope0arerelatedbyT=0)]TJ /F20 6.974 Tf 8.53 0 Td[(1. 53

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momentum[ 15 ].Thus,thefateofthissubleadingdivergenceisalsotoberenormalizedaway,althoughthistime,throughtherenormalizationof0.Althoughwedonotproveithere,weexpectsomethingsimilartooccurfortheD8-branecase. 54

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CHAPTER3REGGETRAJECTORYANDITSFIELDTHEORYLIMITInthischapterweintendtodeepenourunderstandingofperturbativeopenstringamplitudesandtheirconnectionswithgaugetheoryinthelowenergylimit.Westudythisstring/eldconnectionbyfocusingontheone-loopcorrectiontotheleadingopenstringReggetrajectory.Writingthetrajectoryfunctionas(t)=1+0t+(t)with=O(g2)weseethatthe0!0limitofshouldreectthereggeizationofthegaugeparticle(i.e.thegluon)intheeldtheorywhichwasindeedtheresultobtainedin[ 16 ].Morespecically,whentheopenstringsareattachedtoastackofNcoincidentDp-branes,weobtainedthatthelowenergylimitof(t)givesexactlythesameanswerobtainedindimensionallyregularizedeldtheorywithareggeizedgluon.Wealsostudy(t)intheregimewhentisawayfromzerowhichisrelevanttoopenstringphysics,anditwillalsohelpustomakeaconnectionwiththehardscatteringregimeinchapter 4 3.1TheOne-LoopOpenStringReggeTrajectoryTheReggelimitisbydenitionthelimitwheretheMandelstamvariablesislargecomparedtothetypicalmassscalesofasystemwhiletisheldxed1.Asanexample,considerthetreelevelamplitudeforthescatteringofbosonicopenstringtachyons A(s,t)=g2Z10dxx)]TJ /F7 7.97 Tf 6.58 0 Td[(0s(1)]TJ /F2 11.955 Tf 11.95 0 Td[(x))]TJ /F7 7.97 Tf 6.59 0 Td[(0t=\(1)]TJ /F5 11.955 Tf 11.96 0 Td[(0s)\(1)]TJ /F5 11.955 Tf 11.95 0 Td[(0t) \(2)]TJ /F5 11.955 Tf 11.95 0 Td[(0s)]TJ /F5 11.955 Tf 11.96 0 Td[(0t) (3) 1SincethemassspectrumoftheNS+openstringsectorisgivenby0m2=0,1,2,...,thetypicalmassscaleissetby0)]TJ /F20 6.974 Tf 8.53 0 Td[(1.Thus,theReggelimitimpliesthatweconsider0jsj1. 55

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WecanobtaintheReggelimit)]TJ /F5 11.955 Tf 9.3 0 Td[(0s1of( 3 )asfollows.UsingStirling'sapproximation\(x)p 2x1=2+xe)]TJ /F4 7.97 Tf 6.59 0 Td[(xforx!1,wehave \(1)]TJ /F5 11.955 Tf 11.96 0 Td[(0s)\(1)]TJ /F5 11.955 Tf 11.95 0 Td[(0t) \(2)]TJ /F5 11.955 Tf 11.95 0 Td[(0s)]TJ /F5 11.955 Tf 11.96 0 Td[(0t)=)]TJ /F2 11.955 Tf 39.91 8.09 Td[(1 0s+0t)]TJ /F2 11.955 Tf 11.95 0 Td[(1\(1)]TJ /F5 11.955 Tf 11.96 0 Td[(0s)\(1)]TJ /F5 11.955 Tf 11.96 0 Td[(0t) \(1)]TJ /F5 11.955 Tf 11.95 0 Td[(0s)]TJ /F5 11.955 Tf 11.96 0 Td[(0t))]TJ /F2 11.955 Tf 36.23 8.09 Td[(\(1)]TJ /F5 11.955 Tf 11.96 0 Td[(0t) 0s+0t)]TJ /F2 11.955 Tf 11.95 0 Td[(1()]TJ /F5 11.955 Tf 9.3 0 Td[(0s))]TJ /F7 7.97 Tf 6.58 0 Td[(0s+1=2e0s ()]TJ /F5 11.955 Tf 9.3 0 Td[(0s)]TJ /F5 11.955 Tf 11.95 0 Td[(0t))]TJ /F7 7.97 Tf 6.59 0 Td[(0s)]TJ /F7 7.97 Tf 6.59 0 Td[(0t+1=2e0s+0t=)]TJ /F2 11.955 Tf 9.29 0 Td[(\(1)]TJ /F5 11.955 Tf 11.96 0 Td[(0t)()]TJ /F5 11.955 Tf 9.3 0 Td[(0s)0t)]TJ /F4 7.97 Tf 6.59 0 Td[(1e0t(1+t=s)0t)]TJ /F4 7.97 Tf 6.59 0 Td[(1=2e)]TJ /F7 7.97 Tf 6.59 0 Td[(0t()]TJ /F2 11.955 Tf 9.3 0 Td[(1)]TJ /F2 11.955 Tf 11.96 0 Td[(t=s+1=0s))]TJ /F4 7.97 Tf 6.59 0 Td[(1=\(1)]TJ /F5 11.955 Tf 11.96 0 Td[(0t)()]TJ /F5 11.955 Tf 9.3 0 Td[(0s)0t)]TJ /F4 7.97 Tf 6.59 0 Td[(1[1+O(t=s)] (3) Thus,weseethetreelevelamplitudeintheReggelimitisbasicallyoftheform A(s,t)(t)s(t) (3) where(t)isalinearfunctionoft.Thisiswhatiscalled(linear)Reggebehavioranditisacommonfeatureofstringy-likeobjects2.Thequantity(t)iscalledtheleadingReggetrajectory.Whenincludingloopcorrectionstotheamplitude,wealsoexpectReggebehaviorbutthetrajectoryfunction(t)andthecoefcient(t)shouldbemodied.Sincethestrengthofthemodicationtothefullamplitudeiscontrolledbythecouplingg2whichissmallinperturbationtheory,weexpectalsoasmallmodicationtothefunctions(t)and(t).Thesecanbereadofffromthelargesbehavioratxedtoftheoneloopamplitude[ 17 19 ],thus ((t)+)s(t)+s+slns+s, (3) isjustthecoefcientofslnsintheoneloopamplitude.ItispossibletoshowthatthelargesbehaviorofMMin( 2 )and( 2 )forthefourgluoncaseiscontrolledbytheregion23,410or23and4.Thepolarization 2Thisfeaturewasoneofthemainmotivationsthatleadtothebirthofstringtheory,rstconceivedasatheoryofhadrons.HadronsdohavethisstringybehavioralthoughnowweunderstanditduetotheformationofQCDuxtubesamongquarks. 56

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factorsoftheReggecontributiontothetreeamplitudeare2314,sowepickoutthosetermsinthecorrelator h^P1^P2^P3^P4i!2314hP2P3ihP1P4i)-222(hP2P3ihH1H4i220k1k4hP1P4ihH2H3i220k2k3+402hH2H3ihH1H4ihk1H1k2H2k3H3k4H4i!2314hP2P3ihP1P4i)-222(hP2P3ihH1H4i220k1k4hP1P4ihH2H3i220k2k3+402hH2H3ihH1H4i(k1k2k3k4hH1H2ihH3H4i)]TJ /F2 11.955 Tf 9.3 0 Td[(k1k3k2k4hH1H3ihH2H4i+k1k4k2k3hH2H3ihH1H4i!2314(hP2P3i+0thH2H3i2)(hP1P4i+0thH1H4i2)hH2H3ihH1H4i(02s2hH1H2ihH3H4i)]TJ /F5 11.955 Tf 19.26 0 Td[(02(s+t)2hH1H3ihH2H4i (3) Inthelimit32!0,41!wehave: hP1P4i1 4()]TJ /F5 11.955 Tf 11.96 0 Td[(4)2,hP2P3i1 4(3)]TJ /F5 11.955 Tf 11.95 0 Td[(2)2hH1H4i1 2()]TJ /F5 11.955 Tf 11.96 0 Td[(4),hH2H3i1 2(3)]TJ /F5 11.955 Tf 11.95 0 Td[(2),hH1H3i=(3)hH1H2i=(3)]TJ /F5 11.955 Tf 11.95 0 Td[(32)(3))]TJ /F5 11.955 Tf 11.96 0 Td[(320(3)+232 200(3)hH3H4i=()]TJ /F5 11.955 Tf 11.96 0 Td[(3)]TJ /F2 11.955 Tf 11.95 0 Td[(()]TJ /F5 11.955 Tf 11.96 0 Td[(4))=(3+()]TJ /F5 11.955 Tf 11.95 0 Td[(4))(3)+()]TJ /F5 11.955 Tf 11.96 0 Td[(4)0(3)+()]TJ /F5 11.955 Tf 11.95 0 Td[(4)2 200(3)hH2H4i=()]TJ /F5 11.955 Tf 11.96 0 Td[(3+32)]TJ /F2 11.955 Tf 11.95 0 Td[(()]TJ /F5 11.955 Tf 11.96 0 Td[(4))=(3+()]TJ /F5 11.955 Tf 11.95 0 Td[(4))]TJ /F5 11.955 Tf 11.96 0 Td[(32)(3)+()]TJ /F5 11.955 Tf 11.96 0 Td[(4)]TJ /F5 11.955 Tf 11.96 0 Td[(32)(3)+()]TJ /F5 11.955 Tf 11.95 0 Td[(4)]TJ /F5 11.955 Tf 11.95 0 Td[(32)2 200(3)hH1H2ihH3H4i)-222(hH1H3ihH2H4i32()]TJ /F5 11.955 Tf 11.95 0 Td[(4)((3)00(3))]TJ /F5 11.955 Tf 11.96 0 Td[(02(3)) (3) Puttingtheseformsintothecorrelatorwehave h^P1^P2^P3^P4i2314(1+0t)2 16232()]TJ /F5 11.955 Tf 11.95 0 Td[(4)2+1 4(0s)2((3)00(3))]TJ /F5 11.955 Tf 11.96 0 Td[(02(3)))]TJ /F5 11.955 Tf 9.3 0 Td[(022(3)2st+t2 432()]TJ /F5 11.955 Tf 11.96 0 Td[(4) (3) 57

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Thesdependentfactorsinthefourstring1-loopdiagraminvolvethecombination: (43) (2) (42) (3)= (3+()]TJ /F5 11.955 Tf 11.96 0 Td[(4)) (3)]TJ /F5 11.955 Tf 11.96 0 Td[(32) (3)]TJ /F5 11.955 Tf 11.96 0 Td[(32+()]TJ /F5 11.955 Tf 11.95 0 Td[(4)) (31)( +()]TJ /F5 11.955 Tf 11.95 0 Td[(4) 0+()]TJ /F5 11.955 Tf 11.96 0 Td[(4)2 00=2)( )]TJ /F5 11.955 Tf 11.95 0 Td[(32 0+232 00=2) ( +()]TJ /F5 11.955 Tf 11.95 0 Td[(4)]TJ /F5 11.955 Tf 11.95 0 Td[(32) 0+()]TJ /F5 11.955 Tf 11.95 0 Td[(4)]TJ /F5 11.955 Tf 11.95 0 Td[(32)2 00=2) exp)]TJ /F5 11.955 Tf 9.3 0 Td[(32()]TJ /F5 11.955 Tf 11.95 0 Td[(4) 02 2)]TJ /F5 11.955 Tf 13.15 8.09 Td[( 00 =expf32()]TJ /F5 11.955 Tf 11.96 0 Td[(4)(ln )00g (43) (2) (42) (3))]TJ /F7 7.97 Tf 6.59 0 Td[(0sexpf)]TJ /F5 11.955 Tf 15.27 0 Td[(0s32()]TJ /F5 11.955 Tf 11.96 0 Td[(4)(ln )00g (3) Meanwhilethetdependenceisgivenbythefactor (4) (32) (42) (3))]TJ /F7 7.97 Tf 6.59 0 Td[(0t32()]TJ /F5 11.955 Tf 11.96 0 Td[(4) 2(3))]TJ /F7 7.97 Tf 6.58 0 Td[(0t (3) Takings!,the32,4)]TJ /F2 11.955 Tf 12.93 3.16 Td[(^4integralsaredominatedbysmall230,^40,andhencecanbedonebyusing Z0dZ0d()ae)]TJ /F7 7.97 Tf 6.58 0 Td[(=Z0d Z0dae)]TJ /F7 7.97 Tf 6.59 0 Td[(=)]TJ /F4 7.97 Tf 6.59 0 Td[(a)]TJ /F4 7.97 Tf 6.58 0 Td[(1Z20daln2 e)]TJ /F7 7.97 Tf 6.59 0 Td[(\(a+1))]TJ /F4 7.97 Tf 6.58 0 Td[(a)]TJ /F4 7.97 Tf 6.58 0 Td[(1ln,!1 (3) 58

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Weusethisformulafor=()]TJ /F5 11.955 Tf 9.3 0 Td[(0s)()]TJ /F2 11.955 Tf 9.3 0 Td[([ln ]00)anda=)]TJ /F5 11.955 Tf 9.3 0 Td[(0t)]TJ /F2 11.955 Tf 11.95 0 Td[(2,)]TJ /F5 11.955 Tf 9.3 0 Td[(0t)]TJ /F2 11.955 Tf 11.96 0 Td[(1,)]TJ /F5 11.955 Tf 9.3 0 Td[(0t: M+41623141 820D=2Z10dq q2)]TJ /F5 11.955 Tf 9.29 0 Td[( lnq(10)]TJ /F4 7.97 Tf 6.59 0 Td[(D)=2(1)]TJ /F2 11.955 Tf 11.95 0 Td[(w1=2)10)]TJ /F4 7.97 Tf 6.59 0 Td[(D)]TJ /F4 7.97 Tf 6.59 0 Td[(SQr(1+q2r)8 Qn(1)]TJ /F2 11.955 Tf 11.96 0 Td[(q2n)8Zd3Z0d32d^4 32^4 2(3)!)]TJ /F7 7.97 Tf 6.58 0 Td[(0texpf)]TJ /F5 11.955 Tf 15.27 0 Td[(0s32^4(ln )00g(1+0t)2 16232^24+1 4(0s)2(+(3)00+(3))]TJ /F5 11.955 Tf 11.95 0 Td[(02+(3)))]TJ /F5 11.955 Tf 11.96 0 Td[(022+(3)2st+t2 432^4423141 820D=2Z10dq q2)]TJ /F5 11.955 Tf 9.3 0 Td[( lnq(10)]TJ /F4 7.97 Tf 6.58 0 Td[(D)=2(1)]TJ /F2 11.955 Tf 11.96 0 Td[(w1=2)10)]TJ /F4 7.97 Tf 6.58 0 Td[(D)]TJ /F4 7.97 Tf 6.58 0 Td[(SQr(1+q2r)8 Qn(1)]TJ /F2 11.955 Tf 11.95 0 Td[(q2n)8\()]TJ /F5 11.955 Tf 9.3 0 Td[(0t)()]TJ /F5 11.955 Tf 9.3 0 Td[(0s)1+0tln()]TJ /F5 11.955 Tf 9.3 0 Td[(0s)Zd3()]TJ /F5 11.955 Tf 9.3 0 Td[( 2(3)[ln ]00)0t)]TJ /F2 11.955 Tf 13.15 8.09 Td[(1 4(1+0t)[)]TJ /F2 11.955 Tf 11.29 0 Td[(ln ]00)]TJ /F5 11.955 Tf 11.95 0 Td[(0t+(3)00+(3))]TJ /F5 11.955 Tf 11.96 0 Td[(02+(3) [)]TJ /F2 11.955 Tf 11.29 0 Td[(ln ]00+20t2+(3)M)]TJ /F4 7.97 Tf 0 -7.89 Td[(4423141 820D=2Z10dq q24)]TJ /F5 11.955 Tf 9.3 0 Td[( lnq(10)]TJ /F4 7.97 Tf 6.59 0 Td[(D)=2(1+w1=2)10)]TJ /F4 7.97 Tf 6.58 0 Td[(D)]TJ /F4 7.97 Tf 6.58 0 Td[(SQn(1+q2n)8 Qn(1)]TJ /F2 11.955 Tf 11.95 0 Td[(q2n)8\()]TJ /F5 11.955 Tf 9.3 0 Td[(0t)()]TJ /F5 11.955 Tf 9.3 0 Td[(0s)1+0tln()]TJ /F5 11.955 Tf 9.3 0 Td[(0s)Zd3()]TJ /F5 11.955 Tf 9.3 0 Td[( 2(3)[ln ]00)0t)]TJ /F2 11.955 Tf 10.49 8.09 Td[(1 4(1+0t)[)]TJ /F2 11.955 Tf 11.29 0 Td[(ln ]00)]TJ /F5 11.955 Tf 11.96 0 Td[(0t)]TJ /F2 11.955 Tf 7.08 1.79 Td[((3)00)]TJ /F2 11.955 Tf 7.08 2.95 Td[((3))]TJ /F5 11.955 Tf 11.96 0 Td[(02)]TJ /F2 11.955 Tf 7.08 2.95 Td[((3) [)]TJ /F2 11.955 Tf 11.29 0 Td[(ln ]00+20t2)]TJ /F2 11.955 Tf 7.08 2.95 Td[((3) (3) Itisworthrecallingatthispointthatthetotalplanar1-loopamplitudeisgivenby M1loop4=(gp 20)4M+4)-222(M)]TJ /F4 7.97 Tf 0 -7.88 Td[(4 2. (3) Toextracttheone-loopcorrectiontotheReggetrajectorycorrectly,werstneedtoknowtheReggebehaviorofthetree: MTree)]TJ /F2 11.955 Tf 28.56 0 Td[(2g22314\()]TJ /F5 11.955 Tf 9.3 0 Td[(0t)()]TJ /F5 11.955 Tf 9.3 0 Td[(0s)1+0t (3) Combiningthetreeamplitudewiththeone-loopamplitudeinthislimit,wehave MTree4+M1loop4)]TJ /F2 11.955 Tf 28.56 0 Td[(2g22314\()]TJ /F5 11.955 Tf 9.3 0 Td[(0t)()]TJ /F5 11.955 Tf 9.3 0 Td[(0s)1+0t(1+(t)ln()]TJ /F5 11.955 Tf 9.3 0 Td[(0s)), (3) 59

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where=(+)]TJ /F2 11.955 Tf 11.95 0 Td[()]TJ /F2 11.955 Tf 7.08 -4.34 Td[()=2with +=)]TJ /F2 11.955 Tf 10.5 8.09 Td[(8g202)]TJ /F4 7.97 Tf 6.59 0 Td[(D=2 (82)D=2Z10dq q)]TJ /F5 11.955 Tf 9.3 0 Td[( lnq(10)]TJ /F4 7.97 Tf 6.58 0 Td[(D)=2P+(q)Z0d()]TJ /F5 11.955 Tf 9.3 0 Td[( 2()[ln ]00)0t0t [ln ]00X+)]TJ /F2 11.955 Tf 13.15 8.09 Td[(1 4()]TJ /F5 11.955 Tf 9.3 0 Td[( 2()[ln ]00)0t[)]TJ /F2 11.955 Tf 11.29 0 Td[(ln ]00+1 4sin2 (3) )]TJ /F2 11.955 Tf 11.07 -4.94 Td[(=)]TJ /F2 11.955 Tf 10.5 8.09 Td[(8g202)]TJ /F4 7.97 Tf 6.59 0 Td[(D=2 (82)D=2Z10dq q)]TJ /F5 11.955 Tf 9.3 0 Td[( lnq(10)]TJ /F4 7.97 Tf 6.58 0 Td[(D)=2P)]TJ /F2 11.955 Tf 7.09 1.79 Td[((q)Z0d()]TJ /F5 11.955 Tf 9.3 0 Td[( 2()[ln ]00)0t0t [ln ]00X)]TJ /F3 11.955 Tf 9.74 -4.93 Td[()]TJ /F2 11.955 Tf 13.15 8.08 Td[(1 4()]TJ /F5 11.955 Tf 9.3 0 Td[( 2()[ln ]00)0t[)]TJ /F2 11.955 Tf 11.29 0 Td[(ln ]00+1 4sin2. (3) wherewehavedened X1 4[)]TJ /F2 11.955 Tf 11.29 0 Td[(ln ]002+()00())]TJ /F5 11.955 Tf 11.95 0 Td[(02())]TJ /F2 11.955 Tf 11.96 0 Td[(22()[)]TJ /F2 11.955 Tf 11.29 0 Td[(ln ]00 (3) ItisremarkablethatthefunctionsXareinfactindependentof.Thiscanbeshownmoreeasilybywritingthemintermsofthesn(),cn()anddn()Jacobifunctionsandusingthemanyidentitiesthesefunctionsandtheirderivativessatisfy[ 20 ].Thesin)]TJ /F4 7.97 Tf 6.58 0 Td[(2termsintheintegrandsabovecamefromthecounterterms( 2 )and( 2 )neededtocanceltheleadingdivergencesintheRQ4k=2dkintegralswhicharelineardivergences.Howaboutthelogarithmicdivergencesandtheircorrespondingcountertermsintroducedinsection 2.1.2 ?IntheReggelimitthesecountertermsareinfactnotneeded.ThereasonisthattheregionthatdominatesintheReggelimit,i.e.:4,23,isnotentirelydivergent.Infact,thisregiononlyshowslineardivergencesinthekvariables,whilelogarithmicdivergencesareabsent.Figure 3-1 makesthispicturemoreclear.Thelogarithmicdivergencesoccurinthefouredgeshighlightedinorange,whilethedominantregionintheReggelimitistheedgeshowninyellow.FromthereweclearlyseethatthedominantregionintheReggelimit(yellowline)touchesthedivergentregionsonlyatthevertices,whichdivergelinearly.Thismeansthatweonlyneedtheleadingcountertermstomaketheintegralrepresentationof(t)convergent. 60

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Figure3-1. Integrationregionoverthekvariables.TheedgehighlightedinyellowistheregionthatdominatesintheReggelimit,whiletheonesinorangecorrespondtotheplaceswheretheamplitudeisdivergent.ThefactthedominantregionintheReggelimit(yellowline)overlapswiththelogarithmicallydivergentregions(orangelines)onlyattheverticesimpliesthatweonlyneedtheleadingcountertems(whicharisefromthevertices)tocomputetheone-loopcorrectiontotheleadingReggetrajectory. Sincetheleadingcounterterms( 2 )and( 2 )wesubtractedhereamounttoevaluatethefunctions ()and()atq=0,wecanalsothinkofthesin)]TJ /F4 7.97 Tf 6.58 0 Td[(2counterterminexpressions( 3 )and( 3 )tohavearisenfromthefollowingreplacements: C=sin,+!C=1 2sin)]TJ /F5 11.955 Tf 9.3 0 Td[( 2C[ln C]00=11 4(1+0t)[)]TJ /F2 11.955 Tf 11.29 0 Td[(ln C]00+0tC00C)]TJ /F5 11.955 Tf 11.96 0 Td[(00C [)]TJ /F2 11.955 Tf 11.29 0 Td[(ln C]00)]TJ /F2 11.955 Tf 11.95 0 Td[(20t2C=1 4sin2 (3) 61

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Moreover,usingtheGNSregulatorP,thesesubtractiontermsareactuallyzeroafteranalyticcontinuationtoP=0 Z20d(sin=2)P2)]TJ /F4 7.97 Tf 6.59 0 Td[(2=\(1=2)\((P2)]TJ /F2 11.955 Tf 11.96 0 Td[(1)=2) \(P2=2)!0,forP!0. (3) FurthermoreaspointedoutbyNeveuandScherk[ 1 ],whent=0theintegralofthelasttermsintheexpressionsforgiveszero: Z0d[ln ]00+1 sin2=([ln ]0)]TJ /F2 11.955 Tf 11.95 0 Td[(cot)j0= 1Xn=14q2nsin2 1)]TJ /F2 11.955 Tf 11.95 0 Td[(2q2ncos2+q4n!0=0 (3) Itwillbeconvenienttoreplaceeachsin)]TJ /F4 7.97 Tf 6.58 0 Td[(2termby[)]TJ /F2 11.955 Tf 11.29 0 Td[(ln ]00.Inadditionwecallthequantitiesinsquarebracketsin( 3 )and( 3 )X,sowecanrewritethesetwoequationsas +=)]TJ /F2 11.955 Tf 10.5 8.09 Td[(8g202)]TJ /F4 7.97 Tf 6.59 0 Td[(D=2 (82)D=2Z10dq q2)]TJ /F5 11.955 Tf 9.3 0 Td[( lnq(10)]TJ /F4 7.97 Tf 6.59 0 Td[(D)=2(1)]TJ /F2 11.955 Tf 11.95 0 Td[(w1=2)10)]TJ /F4 7.97 Tf 6.59 0 Td[(D)]TJ /F4 7.97 Tf 6.59 0 Td[(SQr(1+q2r)8 Qn(1)]TJ /F2 11.955 Tf 11.96 0 Td[(q2n)8Z0d()]TJ /F5 11.955 Tf 9.3 0 Td[( 2()[ln ]00)0t0t [ln ]00X+)]TJ /F2 11.955 Tf 13.15 8.09 Td[(1 4[()]TJ /F5 11.955 Tf 9.3 0 Td[( 2()[ln ]00)0t)]TJ /F2 11.955 Tf 11.96 0 Td[(1][)]TJ /F2 11.955 Tf 11.29 0 Td[(ln ]00 (3) )]TJ /F2 11.955 Tf 17.05 -4.94 Td[(=)]TJ /F2 11.955 Tf 10.5 8.09 Td[(8g202)]TJ /F4 7.97 Tf 6.59 0 Td[(D=2 (82)D=2Z10dq q24)]TJ /F5 11.955 Tf 9.3 0 Td[( lnq(10)]TJ /F4 7.97 Tf 6.58 0 Td[(D)=2(1+w1=2)10)]TJ /F4 7.97 Tf 6.59 0 Td[(D)]TJ /F4 7.97 Tf 6.59 0 Td[(SQn(1+q2n)8 Qn(1)]TJ /F2 11.955 Tf 11.96 0 Td[(q2n)8Z0d()]TJ /F5 11.955 Tf 9.3 0 Td[( 2()[ln ]00)0t0t [ln ]00X)]TJ /F3 11.955 Tf 9.74 -4.93 Td[()]TJ /F2 11.955 Tf 13.15 8.08 Td[(1 4[()]TJ /F5 11.955 Tf 9.3 0 Td[( 2()[ln ]00)0t)]TJ /F2 11.955 Tf 11.96 0 Td[(1][)]TJ /F2 11.955 Tf 11.29 0 Td[(ln ]00 (3) Writtenthiswayformallyvanishast!0,whichisexpectedsincethemasslessopenstringstateisagaugeparticleandshouldremainmassless.ThecorrectedReggetrajectoryis (t)=1+0t+1 2(+)]TJ /F2 11.955 Tf 11.96 0 Td[()]TJ /F2 11.955 Tf 7.09 -4.94 Td[()1+0t+(t) (3) 62

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WeclosethissectionbynotingadramaticsimplicationofX: X+=1 44(0)43(0)4)]TJ /F2 11.955 Tf 13.15 8.09 Td[(E 4(0)43(0)2+E2 23(0)4=4q+O(q2) (3) X)]TJ /F2 11.955 Tf 17.05 -4.93 Td[(=)]TJ /F2 11.955 Tf 10.49 8.09 Td[(1 44(0)43(0)4+E2 23(0)4=O(q2) (3) E= 63(0)23(0)4+4(0)4)]TJ /F5 11.955 Tf 13.15 8.08 Td[(0001(0) 01(0)= 2)]TJ /F2 11.955 Tf 11.95 0 Td[(2q+O(q2) (3) wherewehaveshownontheextremerightofeachequationitssmallqbehavior.Remarkably,Xturnouttobeindependentof!Themostefcientwaytoshowthisistoexpressand)]TJ /F2 11.955 Tf 9.29 0 Td[([ln ]00intermsoftheJacobianellipticfunctionssn,cn,anddn.Thenoneexploitsthemanyidentitiesthesefunctionsandtheirderivativessatisfy[ 20 ].Forlaterusewelisttheexpansions 3(0)=Yn(1)]TJ /F2 11.955 Tf 11.96 0 Td[(q2n)Yr(1+q2r)2 (3) 4(0)=Yn(1)]TJ /F2 11.955 Tf 11.96 0 Td[(q2n)Yr(1)]TJ /F2 11.955 Tf 11.96 0 Td[(q2r)2 (3) 0001(0) 01(0)=)]TJ /F2 11.955 Tf 9.3 0 Td[(1+24Xnq2n (1)]TJ /F2 11.955 Tf 11.95 0 Td[(q2n)2 (3) wheresumsovernareoverpositiveintegersandthoseoverrareoverhalfoddintegers. 3.2SmalltBehaviorof(t),FieldTheoryLimitandGluonReggeizationTheeldtheorylimitofstringtheoryiscontrolledbythezeroslopelimit0!0.Inopenstringtheory,thisentailsanalyzingthebehaviorofphysicalquantitiesatlowmomentum0klkm1.Inthissection,wewishtostudythesmalltbehavioroftheopenstringReggetrajectory.Theoneloopcorrectiontothistrajectoryshouldreecttheone-loopReggeizationofthegluoningaugetheory.Thepartof(t)analyticint(i.e.integerpowers)receivescontributionsfromthewholerangeofq,butnon-analyticbehaviorinatt=0isproducedbyintegrationofqnear1.Thusitisconvenientinthissectiontotransformtothewvariableslnw=22=lnq=)]TJ /F2 11.955 Tf 9.3 0 Td[(2i=,viatheJacobiimaginarytransformation,andourfocuswillbeonthesmallwcontributionto.Themeasurechangeisq)]TJ /F4 7.97 Tf 6.58 0 Td[(1dq!22(wln2w))]TJ /F4 7.97 Tf 6.58 0 Td[(1dw,andthe 63

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partitionfunctionschangeaccordingto P+q)]TJ /F4 7.97 Tf 6.59 0 Td[(1(1)]TJ /F2 11.955 Tf 11.96 0 Td[(w1=2)10)]TJ /F4 7.97 Tf 6.58 0 Td[(D)]TJ /F4 7.97 Tf 6.58 0 Td[(SQr(1+q2r)8 Qn(1)]TJ /F2 11.955 Tf 11.95 0 Td[(q2n)8=2 lnw4w)]TJ /F4 7.97 Tf 6.59 0 Td[(1=2(1)]TJ /F2 11.955 Tf 11.96 0 Td[(w1=2)10)]TJ /F4 7.97 Tf 6.59 0 Td[(D)]TJ /F4 7.97 Tf 6.59 0 Td[(SQr(1+wr)8 Qn(1)]TJ /F2 11.955 Tf 11.95 0 Td[(wn)8 (3) P)]TJ /F3 11.955 Tf 17.05 1.8 Td[(24(1+w1=2)10)]TJ /F4 7.97 Tf 6.59 0 Td[(D)]TJ /F4 7.97 Tf 6.59 0 Td[(SQn(1+q2n)8 Qn(1)]TJ /F2 11.955 Tf 11.96 0 Td[(q2n)8=2 lnw4w)]TJ /F4 7.97 Tf 6.59 0 Td[(1=2(1+w1=2)10)]TJ /F4 7.97 Tf 6.59 0 Td[(D)]TJ /F4 7.97 Tf 6.59 0 Td[(SQr(1)]TJ /F2 11.955 Tf 11.96 0 Td[(wr)8 Qn(1)]TJ /F2 11.955 Tf 11.96 0 Td[(wn)8 (3) fromwhichwendthesmallwbehavior )]TJ /F5 11.955 Tf 9.3 0 Td[( lnq5)]TJ /F4 7.97 Tf 6.58 0 Td[(D=2dq qP++P)]TJ ET q .478 w 165.04 -219.83 m 209.56 -219.83 l S Q BT /F2 11.955 Tf 184.38 -231.02 Td[(2dw 2w)]TJ /F2 11.955 Tf 9.3 0 Td[(2 lnw1+D=2w)]TJ /F4 7.97 Tf 6.58 0 Td[(1=2 (3) )]TJ /F5 11.955 Tf 9.29 0 Td[( lnq5)]TJ /F4 7.97 Tf 6.59 0 Td[(D=2dq qP+)]TJ /F2 11.955 Tf 11.95 0 Td[(P)]TJ ET q .478 w 164.85 -255.33 m 209.56 -255.33 l S Q BT /F2 11.955 Tf 184.28 -266.52 Td[(2(D+S)]TJ /F2 11.955 Tf 11.95 0 Td[(2)dw 2w)]TJ /F2 11.955 Tf 9.3 0 Td[(2 lnw1+D=2 (3) NextweturntotheJacobitransformsofthedependentfactors. 1ilnw 2,p w=)]TJ /F2 11.955 Tf 9.3 0 Td[(i)]TJ /F2 11.955 Tf 9.29 0 Td[(2 lnw1=2exp)]TJ /F5 11.955 Tf 9.3 0 Td[(2lnw 221(,q) (3) 01(0,p w)=)]TJ /F2 11.955 Tf 9.3 0 Td[(2 lnw3=201(0,q) (3) (,q)=1(,q) 0(0)=i)]TJ /F2 11.955 Tf 9.3 0 Td[(2 lnwexp2lnw 221(ilnw=2,p w) 01(0,p w)= )]TJ /F2 11.955 Tf 11.29 0 Td[(lnwexp)]TJ /F5 11.955 Tf 10.49 8.09 Td[(()]TJ /F5 11.955 Tf 11.96 0 Td[()lnw 22(1)]TJ /F2 11.955 Tf 11.96 0 Td[(w=)(1)]TJ /F2 11.955 Tf 11.95 0 Td[(w1)]TJ /F7 7.97 Tf 6.59 0 Td[(=)1Yn=1(1)]TJ /F2 11.955 Tf 11.96 0 Td[(wn+=)(1)]TJ /F2 11.955 Tf 11.95 0 Td[(wn+1)]TJ /F7 7.97 Tf 6.59 0 Td[(=) (1)]TJ /F2 11.955 Tf 11.95 0 Td[(wn)2 (3) )]TJ /F5 11.955 Tf 13.38 8.09 Td[(@2 @2ln =)]TJ /F2 11.955 Tf 11.29 0 Td[(lnw 2+ln2w 21Xn=0wn+= (1)]TJ /F2 11.955 Tf 11.96 0 Td[(wn+=)2+wn+1)]TJ /F7 7.97 Tf 6.58 0 Td[(= (1)]TJ /F2 11.955 Tf 11.95 0 Td[(wn+1)]TJ /F7 7.97 Tf 6.59 0 Td[(=)2 (3) 64

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Wehavewritten anditsdoublelogarithmicderivativeinawaythatismanifestlysymmetricunder!)]TJ /F5 11.955 Tf 11.96 0 Td[(.ContinuingwithourlistofJacobitransforms, E=)]TJ /F2 11.955 Tf 11.29 0 Td[(lnw 1223(0,p w)43(0,p w)+42(0,p w)+0001(0.p w) 01(0,p w))]TJ /F2 11.955 Tf 17.48 8.09 Td[(12 lnw (3) X+=)]TJ /F2 11.955 Tf 10.5 8.09 Td[(lnw 2443(0,p w)42(0,p w) 4)]TJ /F2 11.955 Tf 13.15 8.09 Td[(E )]TJ /F2 11.955 Tf 10.5 8.09 Td[(lnw 2323(0,p w)42(0,p w) 4+E2 2)]TJ /F2 11.955 Tf 10.49 8.08 Td[(lnw 2243(0,p w) (3) X)]TJ /F2 11.955 Tf 17.05 -4.93 Td[(=)]TJ /F9 11.955 Tf 11.29 16.85 Td[()]TJ /F2 11.955 Tf 10.5 8.08 Td[(lnw 2443(0,p w)42(0,p w) 4+E2 2)]TJ /F2 11.955 Tf 10.5 8.08 Td[(lnw 2243(0,p w) (3) X+)]TJ /F2 11.955 Tf 11.95 0 Td[(X)]TJ /F2 11.955 Tf 9.08 -4.93 Td[(=2)]TJ /F2 11.955 Tf 10.5 8.08 Td[(lnw 2443(0,p w)42(0,p w) 4)]TJ /F2 11.955 Tf 9.16 8.08 Td[(E )]TJ /F2 11.955 Tf 10.5 8.08 Td[(lnw 2323(0,p w)42(0,p w) 4 (3) X++X)]TJ /F2 11.955 Tf 13.07 -4.93 Td[(=)]TJ /F2 11.955 Tf 10.49 8.09 Td[(E )]TJ /F2 11.955 Tf 10.5 8.09 Td[(lnw 2323(0,p w)42(0,p w) 4+2E2 2)]TJ /F2 11.955 Tf 10.5 8.09 Td[(lnw 2243(0,p w) (3) Finallyweneedthesmallwbehaviorofthesecombinationsoffunctions: 3(0,p w)=Yn(1)]TJ /F2 11.955 Tf 11.96 0 Td[(wn)Yr(1+wr)21+2w1=2+O(w) (3) 2(0,p w)=2w1=8Yn(1)]TJ /F2 11.955 Tf 11.95 0 Td[(wn)Yr(1+wr)22w1=8(1+O(w)) (3) 0001(0,p w) 01(0,p w)=)]TJ /F2 11.955 Tf 9.3 0 Td[(1+24Xnwn (1)]TJ /F2 11.955 Tf 11.96 0 Td[(wn)2)]TJ /F2 11.955 Tf 21.91 0 Td[(1+O(w),E1+O(w1=2lnw) (3) X+)]TJ /F2 11.955 Tf 11.95 0 Td[(X)]TJ /F3 11.955 Tf 17.04 -4.94 Td[(8w1=2lnw 241+O1 lnw (3) X++X)]TJ /F3 11.955 Tf 17.04 -4.94 Td[(2 2lnw 22(1+O(w1=2)) (3) X+andX)]TJ /F1 11.955 Tf 10.41 -4.34 Td[(entertheintegrandof+)]TJ /F2 11.955 Tf 11.95 0 Td[()]TJ /F1 11.955 Tf 10.41 1.8 Td[(inthecombination P+X+)]TJ /F2 11.955 Tf 11.96 0 Td[(P)]TJ /F2 11.955 Tf 7.08 1.79 Td[(X)]TJ /F2 11.955 Tf 17.14 -4.94 Td[(=(X+)]TJ /F2 11.955 Tf 11.95 0 Td[(X)]TJ /F2 11.955 Tf 7.08 -4.94 Td[()P++P)]TJ ET q .478 w 219.24 -526.46 m 263.76 -526.46 l S Q BT /F2 11.955 Tf 238.57 -537.65 Td[(2+(X++X)]TJ /F2 11.955 Tf 7.08 -4.94 Td[()P+)]TJ /F2 11.955 Tf 11.96 0 Td[(P)]TJ ET q .478 w 335.81 -526.46 m 380.52 -526.46 l S Q BT /F2 11.955 Tf 355.24 -537.65 Td[(2 (3) 8w1=2lnw 24P++P)]TJ ET q .478 w 240.55 -557.96 m 285.06 -557.96 l S Q BT /F2 11.955 Tf 259.88 -569.15 Td[(2+2 2lnw 22P+)]TJ /F2 11.955 Tf 11.95 0 Td[(P)]TJ ET q .478 w 365.14 -557.96 m 409.85 -557.96 l S Q BT /F2 11.955 Tf 384.57 -569.15 Td[(2 (3) 65

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and )]TJ /F5 11.955 Tf 9.29 0 Td[( lnq5)]TJ /F4 7.97 Tf 6.59 0 Td[(D=2dq q(P+X+)]TJ /F2 11.955 Tf 11.96 0 Td[(P)]TJ /F2 11.955 Tf 7.09 1.79 Td[(X)]TJ /F2 11.955 Tf 7.09 -4.94 Td[()dw w)]TJ /F2 11.955 Tf 9.3 0 Td[(2 lnw)]TJ /F4 7.97 Tf 6.59 0 Td[(3+D=2"4+D+S)]TJ /F2 11.955 Tf 11.96 0 Td[(2 22 lnw2# (3) ThusthecontributionofthesmallwregiontothecorrectiontotheReggetrajectory,whichcanproducenonanalyticbehaviorast!0,is )]TJ /F2 11.955 Tf 29.76 8.09 Td[(8g202)]TJ /F4 7.97 Tf 6.59 0 Td[(D=2 (82)D=2Z0dw w)]TJ /F2 11.955 Tf 9.3 0 Td[(2 lnw)]TJ /F4 7.97 Tf 6.58 0 Td[(3+D=2Z0d()]TJ /F5 11.955 Tf 9.3 0 Td[( 2()[ln ]00)0t0t [ln ]00"2+D+S)]TJ /F2 11.955 Tf 11.96 0 Td[(2 222 lnw2#)]TJ /F2 11.955 Tf 13.15 8.08 Td[(D+S)]TJ /F2 11.955 Tf 11.95 0 Td[(2 82 lnw4[()]TJ /F5 11.955 Tf 9.3 0 Td[( 2()[ln ]00)0t)]TJ /F2 11.955 Tf 11.95 0 Td[(1][)]TJ /F2 11.955 Tf 11.29 0 Td[(ln ]00. (3) Todiscussthesmalltbehaviorof(t),weexaminethetdependentfactorofthesmallwintegrand.Wesetw=e)]TJ /F4 7.97 Tf 6.59 0 Td[(Tand=x,andnotethatatlargeT>e1= )]TJ /F5 11.955 Tf 11.95 0 Td[( 2()[ln ]00ex(1)]TJ /F4 7.97 Tf 6.58 0 Td[(x)T(1)]TJ /F2 11.955 Tf 11.96 0 Td[(e)]TJ /F4 7.97 Tf 6.58 0 Td[(xT)2(1)]TJ /F2 11.955 Tf 11.96 0 Td[(e)]TJ /F4 7.97 Tf 6.59 0 Td[((1)]TJ /F4 7.97 Tf 6.58 0 Td[(x)T)21 T+e)]TJ /F4 7.97 Tf 6.59 0 Td[(xT (1)]TJ /F2 11.955 Tf 11.95 0 Td[(e)]TJ /F4 7.97 Tf 6.58 0 Td[(xT)2+e)]TJ /F4 7.97 Tf 6.59 0 Td[((1)]TJ /F4 7.97 Tf 6.58 0 Td[(x)T (1)]TJ /F2 11.955 Tf 11.96 0 Td[(e)]TJ /F4 7.97 Tf 6.59 0 Td[((1)]TJ /F4 7.97 Tf 6.59 0 Td[(x)T)2 (3) Thisquantityisraisedtothepower0t=)]TJ /F5 11.955 Tf 9.3 0 Td[(0jtjsincewelimitourselvestot<0,wheretheintegralrepresentationofisvalid.Thefactore)]TJ /F4 7.97 Tf 6.59 0 Td[(x(1)]TJ /F4 7.97 Tf 6.58 0 Td[(x)T0jtjlimitsthequantityx(1)]TJ /F2 11.955 Tf 12.25 0 Td[(x)T<1=0jtj.Nonanalyticbehaviorintast!0canbeproducedbyintegrationovertheregion1
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Thenthesmallw(orlargeT)contributiontosimpliesto )]TJ /F2 11.955 Tf 29.76 8.09 Td[(8g202)]TJ /F4 7.97 Tf 6.59 0 Td[(D=2 (82)D=2Z1=x(1)]TJ /F4 7.97 Tf 6.59 0 Td[(x)dT2 T)]TJ /F4 7.97 Tf 6.59 0 Td[(2+D=2Z10dx)]TJ /F5 11.955 Tf 9.3 0 Td[(0tT0jtje)]TJ /F4 7.97 Tf 6.59 0 Td[(x(1)]TJ /F4 7.97 Tf 6.58 0 Td[(x)T0jtj)]TJ /F2 11.955 Tf 10.49 8.09 Td[(D+S)]TJ /F2 11.955 Tf 11.95 0 Td[(2 42 T2hT0jtje)]TJ /F4 7.97 Tf 6.59 0 Td[(x(1)]TJ /F4 7.97 Tf 6.58 0 Td[(x)T0jtj)]TJ /F2 11.955 Tf 11.95 0 Td[(1i)]TJ /F2 11.955 Tf 29.76 8.09 Td[(8g202)]TJ /F4 7.97 Tf 6.59 0 Td[(D=2 (4)D=2)]TJ /F5 11.955 Tf 9.3 0 Td[(0t 4\()]TJ /F2 11.955 Tf 9.3 0 Td[(2+0t+D=2)2 \(D)]TJ /F2 11.955 Tf 11.95 0 Td[(4+20t)Z1duu2)]TJ /F7 7.97 Tf 6.59 0 Td[(0t)]TJ /F4 7.97 Tf 6.59 0 Td[(D=2e)]TJ /F4 7.97 Tf 6.58 0 Td[(u0jtj)]TJ /F2 11.955 Tf 10.49 8.09 Td[(D+S)]TJ /F2 11.955 Tf 11.95 0 Td[(2 4Z1duu)]TJ /F4 7.97 Tf 6.58 0 Td[(D=2\(0t+D=2)2 \(D+20t)u)]TJ /F7 7.97 Tf 6.58 0 Td[(0te)]TJ /F4 7.97 Tf 6.59 0 Td[(u0jtj)]TJ /F2 11.955 Tf 13.15 8.09 Td[(\(D=2)2 \(D) (3) Therstintegralisoftheform(putting=0jtj=)]TJ /F5 11.955 Tf 9.3 0 Td[(0t) Z1duuae)]TJ /F4 7.97 Tf 6.59 0 Td[(u=)]TJ /F4 7.97 Tf 6.59 0 Td[(a)]TJ /F4 7.97 Tf 6.59 0 Td[(1Z1duuae)]TJ /F4 7.97 Tf 6.59 0 Td[(u)]TJ /F4 7.97 Tf 6.59 0 Td[(a)]TJ /F4 7.97 Tf 6.59 0 Td[(1\(a+1),a+1>0 (3) Ifa+10,afewintegrationbypartsshowsthat Z1duuae)]TJ /F4 7.97 Tf 6.59 0 Td[(u=)]TJ /F4 7.97 Tf 6.59 0 Td[(a)]TJ /F4 7.97 Tf 6.59 0 Td[(1Z1duuae)]TJ /F4 7.97 Tf 6.59 0 Td[(u)]TJ /F4 7.97 Tf 6.59 0 Td[(a)]TJ /F4 7.97 Tf 6.58 0 Td[(1\(a+1)+P(,)e)]TJ /F7 7.97 Tf 6.59 0 Td[( (3) wherePisapolynomialin(andhenceapolynomialint)withdependentcoefcients.IntheexpressionfortheterminvolvingPismultipliedbyafunctionanalyticatt=0andsocontributesintegerpowersoftto.Thusthenonanalyticfractionalpower,thoughalwayspresent,dominatesthebehavioronlyifa+1>0.FortherstintegralthismeansD<6.ThesubtractionterminthesecondintegralisniteprovidedD>2,whichisnotaseriousrestriction.Theanalysisofthatintegralagainleadstoanexpressionoftheform( 3 ),withthesubtractiontermsimplycancelingtheconstantterminthepowerseriesarisingfromP(,).Collectingtheresultsforbothintegralswendthesmalltbehavior )]TJ /F2 11.955 Tf 25.77 8.09 Td[(2g202)]TJ /F4 7.97 Tf 6.59 0 Td[(D=2 (4)D=2\()]TJ /F2 11.955 Tf 9.3 0 Td[(2+0t+D=2)2 \(D)]TJ /F2 11.955 Tf 11.95 0 Td[(4+20t)()]TJ /F5 11.955 Tf 9.3 0 Td[(0t)0t)]TJ /F4 7.97 Tf 6.59 0 Td[(2+D=2\(3)]TJ /F5 11.955 Tf 11.96 0 Td[(0t)]TJ /F2 11.955 Tf 11.96 0 Td[(D=2))]TJ /F2 11.955 Tf 9.3 0 Td[((D+S)]TJ /F2 11.955 Tf 11.96 0 Td[(2)\(0t+D=2)2 \(D+20t)()]TJ /F5 11.955 Tf 9.3 0 Td[(0t)0t)]TJ /F4 7.97 Tf 6.58 0 Td[(1+D=2\(1)]TJ /F5 11.955 Tf 11.96 0 Td[(0t)]TJ /F2 11.955 Tf 11.96 0 Td[(D=2)+O(0t) (3) )]TJ /F2 11.955 Tf 25.77 8.09 Td[(2g202)]TJ /F4 7.97 Tf 6.59 0 Td[(D=2 (4)D=2\()]TJ /F2 11.955 Tf 9.3 0 Td[(2+D=2)2 \(D)]TJ /F2 11.955 Tf 11.95 0 Td[(4)()]TJ /F5 11.955 Tf 9.29 0 Td[(0t))]TJ /F4 7.97 Tf 6.58 0 Td[(2+D=2\(3)]TJ /F2 11.955 Tf 11.96 0 Td[(D=2)+O(0tln()]TJ /F5 11.955 Tf 9.3 0 Td[(0t)). (3) 67

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Intherstline,weshowthesecondtermproportionalto(D+S)]TJ /F2 11.955 Tf 12.26 0 Td[(2),whichisdownbyafactorof0tcomparedtothersttermtostressthatthenumberofmasslessscalarsSdoesnotgureintheleadingsmalltbehavior.WiththeO(0tln()]TJ /F5 11.955 Tf 9.3 0 Td[(0t))termwestressthatthedisplayedrsttermisdominantonlyforD<6,andwhenD>6itisonlysignicantwhenD=2isnotaninteger.Thecontributionstofromwawayfromzerobyaniteamountareanalyticint,i.e.aseriesofintegerpowersoftstartingwiththerstpower,since(0)=0.Toproperlyregulateinfrareddivergences(whichwould,amongotherthings,invalidatetheintegralrepresentationfor),wewouldalsoneedtostipulatethatD>4,soinpracticethisformulagivesthedominantsmalltbehavioronlyintherange4
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Forlaterconvenience,wewritethepartitionfunctionsas P+=16(1)]TJ /F2 11.955 Tf 11.96 0 Td[(w1=2)10)]TJ /F4 7.97 Tf 6.58 0 Td[(D)]TJ /F4 7.97 Tf 6.58 0 Td[(S3(0)4 01(0)4P)]TJ /F2 11.955 Tf 17.05 1.79 Td[(=16(1+w1=2)10)]TJ /F4 7.97 Tf 6.59 0 Td[(D)]TJ /F4 7.97 Tf 6.59 0 Td[(S2(0)4 01(0)4 (3) wherewehaveusedthefactthat Q1r(1+q2r)8 Q1n(1)]TJ /F2 11.955 Tf 11.96 0 Td[(q2n)8=16q3(0)4 01(0)4,Q1n(1+q2n)8 Q1n(1)]TJ /F2 11.955 Tf 11.95 0 Td[(q2n)8=16q2(0)4 01(0)4 (3) Expandingaboutq=0wehave: P+)]TJ /F2 11.955 Tf 11.95 0 Td[(P)]TJ /F3 11.955 Tf 10.4 1.79 Td[('P+'q)]TJ /F4 7.97 Tf 6.59 0 Td[(1)]TJ /F5 11.955 Tf 9.3 0 Td[(2 lnq10)]TJ /F4 7.97 Tf 6.59 0 Td[(D)]TJ /F4 7.97 Tf 6.59 0 Td[(S (3) P+X+)]TJ /F2 11.955 Tf 11.95 0 Td[(P)]TJ /F2 11.955 Tf 7.08 1.8 Td[(X)]TJ /F3 11.955 Tf 13.06 -4.93 Td[('4)]TJ /F5 11.955 Tf 9.29 0 Td[(2 lnq10)]TJ /F4 7.97 Tf 6.59 0 Td[(D)]TJ /F4 7.97 Tf 6.59 0 Td[(S (3) [)]TJ /F2 11.955 Tf 11.29 0 Td[(ln ]00=csc2+O(q2) (3) log()]TJ /F5 11.955 Tf 9.3 0 Td[( 2(log )00)=16q2sin4+O(q4) (3) Thesmallqcontributiontothetrajectoryfunction( 3 )becomes (t)'16g202)]TJ /F4 7.97 Tf 6.59 0 Td[(D=2 (82)D=20tZ0dq q)]TJ /F5 11.955 Tf 9.3 0 Td[( lnq25)]TJ /F4 7.97 Tf 6.58 0 Td[(5D=2)]TJ /F4 7.97 Tf 6.59 0 Td[(2SZ0de)]TJ /F7 7.97 Tf 6.59 0 Td[(0jtj16q2sin4sin2+g202)]TJ /F4 7.97 Tf 6.58 0 Td[(D=2 (82)D=2Z0dq q2)]TJ /F5 11.955 Tf 9.3 0 Td[( lnq25)]TJ /F4 7.97 Tf 6.59 0 Td[(5D=2)]TJ /F4 7.97 Tf 6.58 0 Td[(2SZ0dhe)]TJ /F7 7.97 Tf 6.58 0 Td[(0jtj16q2sin4)]TJ /F2 11.955 Tf 11.95 0 Td[(1icsc2 (3) Althoughthepresenceoftheexplicitfactoroftinfrontoftherstintegralabovemakesthistermthedominantone,acarefulanalysisshowsthatthistermislargerthanthesecondonebyafactorofp t,notbyafactoroft.Thereforewestudytheexpression IZ0dq q)]TJ /F2 11.955 Tf 9.29 0 Td[(1 lnqaZ0dsin2e)]TJ /F4 7.97 Tf 6.59 0 Td[(16q2j0tjsin4 (3) inthelimitj0tj!1wherea25)]TJ /F2 11.955 Tf 12.61 0 Td[(5D=2)]TJ /F2 11.955 Tf 12.61 0 Td[(2S.Performingthechangeofvariableu=)]TJ /F2 11.955 Tf 11.29 0 Td[(lnqwehave IZ1)]TJ /F4 7.97 Tf 8 0 Td[(lndu1 uaZ0dsin2exp)]TJ /F2 11.955 Tf 11.29 0 Td[(expln(16j0tjsin4))]TJ /F2 11.955 Tf 11.95 0 Td[(2u (3) 69

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followedbythechangeu=ln(160jtjsin4),yields IZ0dsin2f()1)]TJ /F4 7.97 Tf 6.58 0 Td[(aZ1ln=f()d)]TJ /F4 7.97 Tf 6.59 0 Td[(aexp[)]TJ /F2 11.955 Tf 11.29 0 Td[(exp[(1)]TJ /F2 11.955 Tf 11.96 0 Td[(2)f()]] (3) wheref()ln(160jtjsin4).Atxed,theexponentialfactorrestrictstheintegrationovertotherange1=2<<1asjtj!1,andsincetherearenosingularitiescomingfromtheendpointsoftheintegrationwehave I'1 lnj0tja)]TJ /F4 7.97 Tf 6.59 0 Td[(1Z0dsin2Z11=2d)]TJ /F4 7.97 Tf 6.59 0 Td[(a (3) 21 lnj0tja)]TJ /F4 7.97 Tf 6.59 0 Td[(12a)]TJ /F4 7.97 Tf 6.59 0 Td[(1 a)]TJ /F2 11.955 Tf 11.96 0 Td[(1 (3) Puttingthisresultinto( 3 )wenallyhave (t)'g20(820)1)]TJ /F4 7.97 Tf 6.59 0 Td[(D=2 24)]TJ /F2 11.955 Tf 11.95 0 Td[(5D=2)]TJ /F2 11.955 Tf 11.95 0 Td[(2S0t2 lnj0tj24)]TJ /F4 7.97 Tf 6.59 0 Td[(5D=2)]TJ /F4 7.97 Tf 6.58 0 Td[(2S (3) asjtj!1. 3.4NumericalAnalysisandGraphicsWepresenthereanumericalanalysisofthetrajectoryfunction(t)inbothlimitsstudied,largetandsmalltaswellasinthefullrangebetweenthesetwoasymptoticregions.Wewillseethatthenumericalcomputationoftheintegralrepresentationofthetrajectoryfunction(t)matcheswellwiththeasymptoticanalyticalpredictionsdescribedinsections 3.2 and 3.3 .Atlargetweexpect(t)tobehavelike( 3 ).Asanumericalexample,letusconsiderthecasewhenthereisthemaximumnumberofscalarscirculatingintheloopS=10)]TJ /F2 11.955 Tf 11.96 0 Td[(D,inD=5i.e.openstringsendingonaD4-brane.Weexpectatlarget: (t)lead'0t42 32 lnj0tj3=2 (3) where4g202)]TJ /F4 7.97 Tf 6.58 0 Td[(D=2=(82)D=2.Forplottingconvenience,weshowinFigure 3-2 boththenumericalevaluationof(t)=0tdirectlyfromequation( 3 )representedbythedots, 70

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andweshowtheleadingestimate (t)lead 0t42 32 lnj0tj3=2 (3) bythesolidline.Fromthegureweseethatthenumericalevaluationofthefulltrajectoryfunction( 3 )approachesthepredictedbehavioratlargejtj,althoughtheapproachisveryslowduetothelogarithmicdependence.Duetocomputational Figure3-2. Thedotscorrespondtothedirectnumericalintegrationof(t)=(t)whilethesolidlineisthepredictedbehavioratlargejtjbothasafunctionofln()]TJ /F5 11.955 Tf 9.3 0 Td[(0t).Atlargevaluesof0jtjthenumericalintegrationapproachesthepredictedbehaviorfrombelow. limitationsoftheintegrationroutine,themaximumvalueof)]TJ /F5 11.955 Tf 9.3 0 Td[(0tforwhichthetrajectoryfunctioncouldbenumericallyevaluatedwasoftheorderof)]TJ /F5 11.955 Tf 9.3 0 Td[(0t105whichcorrespondstotheupperlimitln(105)11.5inthehorizontalaxisshowninthegures.Performingatoftheexactnumericalevaluationusingtheleadingandthenextthreesubleadingcorrectionsweobtain 0t12.892 lnj0tj3=2)]TJ /F2 11.955 Tf 11.96 0 Td[(7.742 lnj0tj5=2+4.842 lnj0tj7=2)]TJ /F2 11.955 Tf 11.96 0 Td[(0.852 lnj0tj9=2 (3) 71

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whoseleadingtermistobecomparedwiththepredictedleadingbehavior (t)lead 0t42 32 lnj0tj3=213.162 lnj0tj3=2 (3) whichisingoodagreementwiththeanalyticprediction.Figure 3-3 showsthettingfunction( 3 )togetherwiththedatapoints. Figure3-3. Thetoftheleadingandthreesubleadingcorrectionstothedatapointsispresentedasthesolidline.Hereweseethattheagreementbetweenthedatapointsandthetincreasesforlargervaluesof0tasexpected,anditisalreadygoodstartingatln(0t)6sincewehaveincludedsubleadingcorrections. Nowweturntothesmalltbehavior.Ashasbeennotedinsection3,thisisbeststudiedinthewvariablessinceitisthew0regionthatdominatesthenonanalyticbehavioratsmallt.Weneedtheexactformof(t)which,usingtheintegralrepresentationasafunctionofwis: (t)=Z10dw 2w)]TJ /F2 11.955 Tf 9.3 0 Td[(2 lnw)]TJ /F4 7.97 Tf 6.59 0 Td[(3+D=2Z0d()]TJ /F5 11.955 Tf 9.3 0 Td[( 2()[ln ]00)0t0t [ln ]00(P+X+)]TJ /F2 11.955 Tf 11.96 0 Td[(P)]TJ /F2 11.955 Tf 7.09 1.79 Td[(X)]TJ /F2 11.955 Tf 7.09 -4.94 Td[())]TJ /F2 11.955 Tf 10.49 8.08 Td[(1 4(P+)]TJ /F2 11.955 Tf 11.96 0 Td[(P)]TJ /F2 11.955 Tf 7.09 1.8 Td[()h()]TJ /F5 11.955 Tf 9.3 0 Td[( 2()[ln ]00)0t)]TJ /F2 11.955 Tf 11.95 0 Td[(1i[)]TJ /F2 11.955 Tf 11.29 0 Td[(ln ]00 (3) 72

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Toevaluateitnumericallyweneedtheexactw-dependentformsoftheexpressionslistedinsection 3.2 ,thepartitionfunctionsin( 3 ),andalso: [)]TJ /F2 11.955 Tf 11.29 0 Td[(ln ]00=lnw 2223(0,p w)22(0,p w)24(ilnw=2,p w) 21(ilnw=2,p w))]TJ /F2 11.955 Tf 13.16 8.09 Td[(2E )]TJ /F2 11.955 Tf 11.29 0 Td[(lnw 223(0,p w) (3) ThenumericalintegrationroutineproducestheplotinFigure 3-4 fortherange0t2[)]TJ /F2 11.955 Tf 9.3 0 Td[(1,)]TJ /F2 11.955 Tf 9.3 0 Td[(0.01]whichsuggeststhat(t)approacheszerowithinniteslopeasitshouldreadingofffrom( 3 ).Nonetheless,wewouldliketoseewhetherthenumericsisreallyproducingthisbehavioratsmallt.Inordertodothis,weneedtozoominnearzeroinFigure 3-4 inwhichcasewedothefollowing:weseparateoutthesmallwregionas Z10dq[]=Zq00dq[]+Zw00dw[] (3) Ifforinstance,wetakeq0=0.8,bymeansofw=e22=lnqweseethattheupperlimit Figure3-4. ThesmalltbehaviorisshownforD=5.Weexpect(t)togotozerowithinniteslopewhichcanbeappreciatedfromthisgure. inthewintegrationisw0410)]TJ /F4 7.97 Tf 6.59 0 Td[(39whichisverysmallandallowsonetousethe 73

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Figure3-5. Zoominsmallt.Theexpected()]TJ /F5 11.955 Tf 9.3 0 Td[(0t)1=2behaviorneart=0canbeappreciatedmoreclearlyinthisgure.Thesolidlineisthepredictedasymptoticbehavior(t)=)]TJ /F2 11.955 Tf 21.91 0 Td[(23=24()]TJ /F5 11.955 Tf 9.3 0 Td[(0t)1=2ast!0whichmatcheswellwiththedatapointsinthislimit. asymptoticexpression( 3 )with and[)]TJ /F2 11.955 Tf 11.29 0 Td[(ln ]00beingalsoapproximatedby: Tex=2(1)]TJ /F4 7.97 Tf 6.59 0 Td[(x)T(1)]TJ /F2 11.955 Tf 11.95 0 Td[(e)]TJ /F4 7.97 Tf 6.59 0 Td[(xT)(1)]TJ /F2 11.955 Tf 11.95 0 Td[(e)]TJ /F4 7.97 Tf 6.58 0 Td[((1)]TJ /F4 7.97 Tf 6.59 0 Td[(x)T) (3) [)]TJ /F2 11.955 Tf 11.29 0 Td[(ln ]00'1 2T+T2e)]TJ /F4 7.97 Tf 6.59 0 Td[(xT (1)]TJ /F2 11.955 Tf 11.96 0 Td[(e)]TJ /F4 7.97 Tf 6.59 0 Td[(xT)2+e)]TJ /F4 7.97 Tf 6.58 0 Td[((1)]TJ /F4 7.97 Tf 6.59 0 Td[(x)T (1)]TJ /F2 11.955 Tf 11.96 0 Td[(e)]TJ /F4 7.97 Tf 6.59 0 Td[((1)]TJ /F4 7.97 Tf 6.58 0 Td[(x)T)2 (3) asshowninequation( 3 ).Thenumericalevaluationof( 3 )inthiscaseisshowninFigure 3-5 .Atofthedatapointsincludingtheleadingandsubleadingpowersturnsouttobe )]TJ /F2 11.955 Tf 21.92 0 Td[(262.82()]TJ /F5 11.955 Tf 9.3 0 Td[(0t)1=2)]TJ /F2 11.955 Tf 11.95 0 Td[(323.260t (3) Theanalyticexpressionfortheleadingbehaviorisgivenby( 3 )whichinthiscasebecomes )]TJ /F2 11.955 Tf 23.11 8.09 Td[((2)D=2 2\()]TJ /F2 11.955 Tf 9.3 0 Td[(2+D=2)2 \(D)]TJ /F2 11.955 Tf 11.95 0 Td[(4)()]TJ /F5 11.955 Tf 9.3 0 Td[(0t)D=2)]TJ /F4 7.97 Tf 6.58 0 Td[(2\(3)]TJ /F2 11.955 Tf 11.96 0 Td[(D=2))]TJ /F2 11.955 Tf 21.91 0 Td[(23=24()]TJ /F5 11.955 Tf 9.3 0 Td[(0t)1=2 (3) forD=5.Fromhereweseethatwehavegoodagreementwith( 3 )since23=24=275.52.Wenishthissectionbyshowingalargerrangeintinwhichthetwoasymptotic 74

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Figure3-6. Alargerrangethatincludesbothlargeandsmalltbehaviorisshown.Inthisplotitispossibletoseethetwoasymptoticregionswithsomeaccuracy.Thelargetregiongrowsast=(lnt)3=2asdescribedinSection5.Althoughitisnotcompletelyevidentfromthisgure,(t)isgoingtozerowithinniteslopeas()]TJ /F5 11.955 Tf 9.3 0 Td[(0t)1=2intheregionneart=0(seeFigure 3-5 ) regionst0andt!canbevisualized.Althoughitisnotevidentfromtheplot,Figure 3-6 showsthetwoasymptoticregionswehavedescribedabovei.e.,()]TJ /F5 11.955 Tf 9.3 0 Td[(0t)1=2ast!0and0t=(lnj0tj)3=2ast!. 3.5OrbifoldProjectionWediscussverybrieyanalternativeproceduretoeliminatethemasslessscalarscirculatingtheloop,namelybyprojectingthemoutoftheusingtheso-calledorbifoldprojection.ItconsistsbasicallyindemandingthatthewekeeponlythestatesthatareevenunderaIn,bIr!)]TJ /F2 11.955 Tf 25.75 0 Td[(aIn,)]TJ /F2 11.955 Tf 9.3 0 Td[(bIrforthecomponentsI=D+S,D+S+1,,10.Thus,ifwewanttohavepureYang-Millstheoryinthelowenergy(0!0)limit,i.e.S=0(noscalars),wedemandthisconditionforallthetransversecomponentstotheD-brane.Thisimpliesthatinthepartitionfunctionsinequations( 2 )and( 2 )nowbecome P+q)]TJ /F4 7.97 Tf 6.59 0 Td[(1(1)]TJ /F2 11.955 Tf 11.96 0 Td[(w1=2)10)]TJ /F4 7.97 Tf 6.59 0 Td[(D)]TJ /F4 7.97 Tf 6.58 0 Td[(SQr(1+q2r)8 Qn(1)]TJ /F2 11.955 Tf 11.95 0 Td[(q2n)8=2 lnw4w)]TJ /F4 7.97 Tf 6.59 0 Td[(1=2(1)]TJ /F2 11.955 Tf 11.96 0 Td[(w1=2)10)]TJ /F4 7.97 Tf 6.59 0 Td[(D)]TJ /F4 7.97 Tf 6.59 0 Td[(SQr(1+wr)8 Qn(1)]TJ /F2 11.955 Tf 11.95 0 Td[(wn)8P)]TJ /F3 11.955 Tf 17.05 1.8 Td[(24(1+w1=2)10)]TJ /F4 7.97 Tf 6.59 0 Td[(D)]TJ /F4 7.97 Tf 6.59 0 Td[(SQn(1+q2n)8 Qn(1)]TJ /F2 11.955 Tf 11.96 0 Td[(q2n)8=2 lnw4w)]TJ /F4 7.97 Tf 6.58 0 Td[(1=2(1+w1=2)10)]TJ /F4 7.97 Tf 6.59 0 Td[(D)]TJ /F4 7.97 Tf 6.58 0 Td[(SQr(1)]TJ /F2 11.955 Tf 11.95 0 Td[(wr)8 Qn(1)]TJ /F2 11.955 Tf 11.95 0 Td[(wn)8 75

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getmodiedasfollows: P+!q)]TJ /F4 7.97 Tf 6.58 0 Td[(11 2"Qr(1+q2r)8 Qn(1)]TJ /F2 11.955 Tf 11.95 0 Td[(q2n)8+q(10)]TJ /F4 7.97 Tf 6.58 0 Td[(D)]TJ /F4 7.97 Tf 6.58 0 Td[(S)=4)]TJ /F5 11.955 Tf 9.3 0 Td[( 4lnq(D+S)]TJ /F4 7.97 Tf 6.59 0 Td[(10)=2Qr(1+q2r)D+S)]TJ /F4 7.97 Tf 6.59 0 Td[(2Qn(1+q2n)10)]TJ /F4 7.97 Tf 6.58 0 Td[(D)]TJ /F4 7.97 Tf 6.58 0 Td[(S Qn(1)]TJ /F2 11.955 Tf 11.96 0 Td[(q2n)D+S)]TJ /F4 7.97 Tf 6.59 0 Td[(2Qr(1)]TJ /F2 11.955 Tf 11.96 0 Td[(q2r)10)]TJ /F4 7.97 Tf 6.59 0 Td[(D)]TJ /F4 7.97 Tf 6.59 0 Td[(S#P)]TJ /F3 11.955 Tf 17.05 1.79 Td[(!241 2"Qn(1+q2n)8 Qn(1)]TJ /F2 11.955 Tf 11.95 0 Td[(q2n)8+)]TJ /F5 11.955 Tf 9.3 0 Td[( lnq(D+S)]TJ /F4 7.97 Tf 6.59 0 Td[(10)=2Qn(1+q2n)D+S)]TJ /F4 7.97 Tf 6.58 0 Td[(2Qr(1+q2r)10)]TJ /F4 7.97 Tf 6.59 0 Td[(D)]TJ /F4 7.97 Tf 6.59 0 Td[(S Qn(1)]TJ /F2 11.955 Tf 11.96 0 Td[(q2n)D+S)]TJ /F4 7.97 Tf 6.59 0 Td[(2Qr(1)]TJ /F2 11.955 Tf 11.96 0 Td[(q2r)10)]TJ /F4 7.97 Tf 6.58 0 Td[(D)]TJ /F4 7.97 Tf 6.59 0 Td[(S# (3) Itisworthnoticingthatinthecaseofthemaximalnumberofscalarscirculatingtheloop,i.e.D+S=10,themodiedpartitionfunctionsbecome P+!q)]TJ /F4 7.97 Tf 6.59 0 Td[(1Qr(1+q2r)8 Qn(1)]TJ /F2 11.955 Tf 11.96 0 Td[(q2n)8 (3) P)]TJ /F3 11.955 Tf 17.05 1.79 Td[(!24Qn(1+q2n)8 Qn(1)]TJ /F2 11.955 Tf 11.96 0 Td[(q2n)8 (3) whichareidenticaltothepartitionfunctionsinthecasewithoutorbifoldprojections3.ThenewReggetrajectoryisthen (t)=)]TJ /F2 11.955 Tf 10.49 8.08 Td[(4g202)]TJ /F4 7.97 Tf 6.59 0 Td[(D=2 (82)D=2Z10dq q)]TJ /F5 11.955 Tf 9.3 0 Td[( lnq(10)]TJ /F4 7.97 Tf 6.59 0 Td[(D)=2Z0d()]TJ /F5 11.955 Tf 9.3 0 Td[( 2()[ln ]00)0t0t [ln ]00(P+X+)]TJ /F2 11.955 Tf 11.95 0 Td[(P)]TJ /F2 11.955 Tf 7.08 1.8 Td[(X)]TJ /F2 11.955 Tf 7.08 -4.93 Td[())]TJ /F2 11.955 Tf 10.49 8.09 Td[(1 4(P+)]TJ /F2 11.955 Tf 11.96 0 Td[(P)]TJ /F2 11.955 Tf 7.08 1.79 Td[()h()]TJ /F5 11.955 Tf 9.3 0 Td[( 2()[ln ]00)0t)]TJ /F2 11.955 Tf 11.95 0 Td[(1i[)]TJ /F2 11.955 Tf 11.29 0 Td[(ln ]00 (3) withthePfunctionsdenedaboveandtherestisthesameasbefore.Bywritingthenewpartitionfunctionsintheoriginalwvariables Porb+=1 2w1=2)]TJ /F2 11.955 Tf 9.3 0 Td[(2 lnw4Qr(1+wr)8 Qn(1)]TJ /F2 11.955 Tf 11.96 0 Td[(wn)8+Qr(1+wr)D+S)]TJ /F4 7.97 Tf 6.59 0 Td[(2Qr(1)]TJ /F2 11.955 Tf 11.95 0 Td[(wr)10)]TJ /F4 7.97 Tf 6.58 0 Td[(D)]TJ /F4 7.97 Tf 6.58 0 Td[(S Qn(1)]TJ /F2 11.955 Tf 11.96 0 Td[(wn)D+S)]TJ /F4 7.97 Tf 6.58 0 Td[(2Qn(1+wn)10)]TJ /F4 7.97 Tf 6.58 0 Td[(D)]TJ /F4 7.97 Tf 6.58 0 Td[(S (3) Porb)]TJ /F2 11.955 Tf 18.72 2.96 Td[(=1 2w1=2)]TJ /F2 11.955 Tf 9.3 0 Td[(2 lnw4Qr(1)]TJ /F2 11.955 Tf 11.96 0 Td[(wr)8 Qn(1)]TJ /F2 11.955 Tf 11.96 0 Td[(wn)8+Qr(1)]TJ /F2 11.955 Tf 11.95 0 Td[(wr)D+S)]TJ /F4 7.97 Tf 6.58 0 Td[(2Qr(1+wr)10)]TJ /F4 7.97 Tf 6.58 0 Td[(D)]TJ /F4 7.97 Tf 6.58 0 Td[(S Qn(1)]TJ /F2 11.955 Tf 11.96 0 Td[(wn)D+S)]TJ /F4 7.97 Tf 6.58 0 Td[(2Qn(1+wn)10)]TJ /F4 7.97 Tf 6.58 0 Td[(D)]TJ /F4 7.97 Tf 6.58 0 Td[(S (3) weseethatthelowenergylimit0!0isnotmodiedsincethisregimeisgovernedbythew0behaviorwhichdoesnotchangeaswecanseebyexpandingthenew 3Whichinturncoincideswiththenon-abelianD-braneprojectionsintheD+S=10caseaswell 76

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partitionfunctionsinthislimit,where Porb1 w1=2)]TJ /F2 11.955 Tf 9.3 0 Td[(2 lnw41(D+S)]TJ /F2 11.955 Tf 11.95 0 Td[(2)w1=2+O(w)whichisthesameasymptoticbehaviorthatthenonabelianD-braneconstructionprovides.Largetlimit Thenewpartitionfunctionsslightlymodifythehightbehaviorinthiscasecomparedtotheonewherethenon-abelianD-braneprojectionwasused.Thisisbecausethelargetbehaviorof(t)isdominatedbytheregionq0inwhichPbehaveslightlydifferent: P+1 2qP)]TJ /F3 11.955 Tf 10.4 1.79 Td[(8)]TJ /F5 11.955 Tf 9.3 0 Td[( lnq(D+S)]TJ /F4 7.97 Tf 6.58 0 Td[(10)=2 (3) UsingthatX+4q+O(q2)andX)]TJ /F3 11.955 Tf 10.41 -4.34 Td[(O(q2)wehave P+)]TJ /F2 11.955 Tf 11.95 0 Td[(P)]TJ /F3 11.955 Tf 10.4 1.8 Td[(P+1 2q+O(lnpq) (3) P+X+)]TJ /F2 11.955 Tf 11.95 0 Td[(P)]TJ /F2 11.955 Tf 7.08 1.79 Td[(X)]TJ /F3 11.955 Tf 17.04 1.79 Td[(2+O(lnpq) (3) wherep(10)]TJ /F2 11.955 Tf 11.95 0 Td[(D)]TJ /F2 11.955 Tf 11.95 0 Td[(S)=20.Usingtheseexpansionswehave (t)'32g202)]TJ /F4 7.97 Tf 6.59 0 Td[(D=2 (82)D=20tZ0dq q)]TJ /F5 11.955 Tf 9.3 0 Td[( lnq(10)]TJ /F4 7.97 Tf 6.59 0 Td[(D)=2Z0de)]TJ /F7 7.97 Tf 6.59 0 Td[(0jtj16q2sin4sin2+2g202)]TJ /F4 7.97 Tf 6.59 0 Td[(D=2 (82)D=2Z0dq q2)]TJ /F5 11.955 Tf 9.3 0 Td[( lnq(10)]TJ /F4 7.97 Tf 6.59 0 Td[(D)=2Z0dhe)]TJ /F7 7.97 Tf 6.59 0 Td[(0jtj16q2sin4)]TJ /F2 11.955 Tf 11.96 0 Td[(1icsc2 (3) Inthelargejtjlimit,wehave I1Z0dq q)]TJ /F2 11.955 Tf 9.3 0 Td[(1 lnqZ0de)]TJ /F7 7.97 Tf 6.59 0 Td[(0jtj16q2sin4sin2 21 lnj0tj)]TJ /F4 7.97 Tf 6.58 0 Td[(12)]TJ /F4 7.97 Tf 6.59 0 Td[(1 )]TJ /F2 11.955 Tf 11.96 0 Td[(1 (3) I2Z0dq q2)]TJ /F2 11.955 Tf 9.3 0 Td[(1 lnqZ0dhe)]TJ /F7 7.97 Tf 6.58 0 Td[(0jtj16q2sin4)]TJ /F2 11.955 Tf 11.96 0 Td[(1icsc2)]TJ /F2 11.955 Tf 21.92 0 Td[(43=2()]TJ /F5 11.955 Tf 9.3 0 Td[(0t)1=22 lnj0tj (3) 77

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where(10)]TJ /F2 11.955 Tf 12.93 0 Td[(D)=2.Weseethat,eventhoughI2islargerthanI1byafactorof()]TJ /F5 11.955 Tf 9.3 0 Td[(0t)1=2inthislimit,theexplicitfactorof0tinfrontofI1makesthiscontributiontodominate.Therefore (t)'32g202)]TJ /F4 7.97 Tf 6.59 0 Td[(D=2 (82)D=22(8)]TJ /F4 7.97 Tf 6.59 0 Td[(D)=2 8)]TJ /F2 11.955 Tf 11.96 0 Td[(D0t1 lnj0tj(8)]TJ /F4 7.97 Tf 6.58 0 Td[(D)=2 (3) Fromthisexpressionweseethatnumberofscalarscirculatingtheloopdonothaveaneffectinthehigh-energybehaviorof(t).ThiscontrastswiththenonabelianD-braneprojectionwhichdoeshaveaneffectathigh-energyaswecanseefromequation( 3 ).Theseareinterestingdifferencesworthinvestigatingmoreindepthinthefuture.ItisamusingtotakethecriticalcaseinwhichthedimensionalityoftheD-braneisD!8: (t)'32g202)]TJ /F4 7.97 Tf 6.59 0 Td[(D=2 (82)D=22(8)]TJ /F4 7.97 Tf 6.59 0 Td[(D)=2 8)]TJ /F2 11.955 Tf 11.95 0 Td[(D0t1 lnj0tj(8)]TJ /F4 7.97 Tf 6.58 0 Td[(D)=2g2 1287020t1 8)]TJ /F2 11.955 Tf 11.96 0 Td[(D+ln(ln()]TJ /F5 11.955 Tf 9.3 0 Td[(0t))+ (3) Thisisthecasewhentheintegralsthatdene(t)divergeduetolongdistanceeffectsofclosedstringdilatons,andrequiresloperenormalizationtocancelthem.Itisworthnoticingthatweobtainthesameanswer(althoughwithadifferentnumericalcoefcient)ifweusethenonabelianD-branesprojectioninthemaximalcaseSmax=10)]TJ /F2 11.955 Tf 10.79 0 Td[(D.Inotherwords,theorbifoldprojectiondoesnotseemtoaffectthehigh-energybehaviorofthestringamplitudebutthenonabelianonedoesaffectit. 78

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CHAPTER4HARDSCATTERINGThehardscatteringregime,alsoknownasxed-anglehigh-energylimit,fortachyonscatteringinthebosonicopenstringtheorywasstudiedlongagoforanarbitrarynumberloops[ 25 ].Theirmainresultwasthat,similarlytothestudyofclosedstringsin[ 26 ],theamplitudeforfourexternaltachyonshadasaddlepointatallgenusandthusitcouldbeevaluatedatthatpointgivingananswerforanygenus.Alinearrelationamongtheamplitudesinthehighenergyregimeforallexternalstateswasconjecturedin[ 27 ].However,ithasbeenpointedoutbyMoellerandWest[ 28 ],usingtheso-calledgrouptheoreticapproach,thatthelinearrelationssuggestedin[ 27 ]maynotbevalidforexternalstringstatesotherthantachyons.Adetailedone-loopanalysisforthebosonicstringisalsoprovidedinreference[ 28 ].GrossandManes[ 25 ],extendingtotheopenstringthesemi-classicalanalysisof[ 26 ],foundthattheopenstringplanaramplitudedoesnotpossesssaddlepointsintheinteriorofmodulispace,incontrarytothenon-planarcase.Thereforetheonlyregionsthatcouldpotentiallygivethedominantbehaviorathighenergiesandxedanglefortheplanaramplitudearetheboundariesofmodulispace.EventhoughtheywereworkinginthebosonicstringinD=26thisresultextendsimmediatelytotheNS+openstringinD=10becausethedominantdependenceintheexternalmomentaiscontainedinthefactorQi
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Anadditionalmotivationtostudythisregimeisthat,togetherwiththeReggeregimestudiedinchapter 3 ,itcomplementsourunderstandingofthehighenergydynamicsofperturbativeopenstringphysicsfortheNS+model. 4.1TheOneLoopAmplitudeForthepurposeofcomparingtheresultsofthissectionwiththeReggelimitobtainedinchapter 3 ,wefocusonthepolarizationstructurethatdominatesinReggeregimewhichis1423.Thecoefcientofthispolarizationisgivenby M4=1 2)]TJ /F3 11.955 Tf 5.48 -9.68 Td[(M+4)-222(M)]TJ /F4 7.97 Tf 0 -7.9 Td[(4 (4) whereM+4andM)]TJ /F4 7.97 Tf 0 -7.87 Td[(4areshowninequations( 2 )and( 2 )whichwerepeathereforconvenience: M+4=21 80D=2Z10dq q)]TJ /F5 11.955 Tf 9.3 0 Td[( lnq(10)]TJ /F4 7.97 Tf 6.58 0 Td[(D)=2q)]TJ /F4 7.97 Tf 6.59 0 Td[(1(1)]TJ /F2 11.955 Tf 11.95 0 Td[(w1=2)10)]TJ /F4 7.97 Tf 6.59 0 Td[(D)]TJ /F4 7.97 Tf 6.59 0 Td[(SQ1r(1+q2r)8 Q1n(1)]TJ /F2 11.955 Tf 11.95 0 Td[(q2n)8Z4Yk=2dkYi
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Forthepolarizationstructureweareinterestedinhere,thefullcombinationofcontractionsthatmultipliesthispolarizationis h^P1^P2^P3^P4i!)]TJ /F3 11.955 Tf 5.47 -9.69 Td[(hP2P3i+0thH2H3i2)]TJ /F3 11.955 Tf 12.95 -9.69 Td[(hP1P4i+0thH1H4i2+hH2H3ihH1H4i)]TJ /F5 11.955 Tf 5.48 -9.68 Td[(02s2hH1H2ihH3H4i)]TJ /F5 11.955 Tf 19.26 0 Td[(02(s+t)2hH1H3ihH2H4i (4) andfromnowon,everytimewewriteh^P1^P2^P3^P4iinthischapter,wewillalwaysmeantherighthandsideof( 4 ).Fromchapter 2 ,weknowthat( 4 )divergesatq=0.Thiscomesfromthefactthatboth ()andh^P1^PMiareoforderoneinpowersofq.Wealsoknowfromthatchapterthat( 4 )and( 4 )divergeintheRQkdkintegrationandneedthecountertermsintroducedthereinordertocanceltheleading(linear)andsubleading(logarithmic)divergences.Wewillseeinthischapterthat,incontrastwiththeReggeasymptotiawhereweonlyneededtheleadingcounterterm,weneedallthecountertermsintroducedinchapter 2 inordertostudythehardscatteringregimecorrectly.Therefore,ourstartingpointwillbethecompletelyrenormalized4-gluonamplitude( 2 )whichwewriteagainhereforconvenience: Mren4Z10dq q)]TJ /F5 11.955 Tf 9.3 0 Td[( lnq5)]TJ /F4 7.97 Tf 6.59 0 Td[(D=2[I(q))]TJ /F2 11.955 Tf 11.95 0 Td[(C(q))]TJ /F2 11.955 Tf 11.96 0 Td[(B(q)] (4) where IZ4Yk=2dkYi
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with P+q)]TJ /F4 7.97 Tf 6.58 0 Td[(1(1)]TJ /F2 11.955 Tf 11.95 0 Td[(w1=2)10)]TJ /F4 7.97 Tf 6.59 0 Td[(D)]TJ /F4 7.97 Tf 6.59 0 Td[(SQr(1+q2r)8 Qn(1)]TJ /F2 11.955 Tf 11.96 0 Td[(q2n)8 (4) P)]TJ /F3 11.955 Tf 17.05 1.79 Td[(24(1+w1=2)10)]TJ /F4 7.97 Tf 6.58 0 Td[(D)]TJ /F4 7.97 Tf 6.58 0 Td[(SQn(1+q2n)8 Qn(1)]TJ /F2 11.955 Tf 11.95 0 Td[(q2n)8 (4) andwerecallthattheexpressionsfortheBkfunctionsarelistedinequations( A )through( A )inappendix A .Inequations( 4 ),( 4 ),( 4 ),and( 4 )whichcontainthedependenceontheexternalgluonmomenta,theGNSregulatorp=PMi=1piisassumedtobesettozero,i.e.momentumconservationisexactsincewehavecanceledallpossibledivergencesinthekintegrals.Forinstance,theproductofthe (ji)functionsin( 4 )forthe4-gluonamplitude,whichinterestsusinthischapterreads Yi
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Therefore,thedominantregionsas0s!atxed(hardscattering)aretheoneswhereV0.Allinall,andincludingthecounterterms,wenowwritethefullrenormalized4-gluonamplitudeforM+4as1 M+4,ren=21 80D=2Z10dq q)]TJ /F5 11.955 Tf 9.3 0 Td[( lnq(10)]TJ /F4 7.97 Tf 6.58 0 Td[(D)=2P+(q)Z4Yk=2dkhe)]TJ /F7 7.97 Tf 6.59 0 Td[(0sVh^P1^P2^P3^P4i+)]TJ /F2 11.955 Tf 11.96 0 Td[(e)]TJ /F7 7.97 Tf 6.59 0 Td[(0sV0h^C1^C2^C3^C4i)]TJ /F2 11.955 Tf 19.26 0 Td[(B+i (4) whereV0isbydenitionVinequation( 4 )evaluatedatq=0,whichis V0=lnsin43sin2 sin42sin3)]TJ /F5 11.955 Tf 11.95 0 Td[(lnsin4sin32 sin42sin3 (4) andthecountertermB+isthesumofthe+termsin( A ). 4.2TheHardScatteringLimit 4.2.1HardScatteringLimitThroughOneLoopThe0s!limitwithxedcannowbeextractedbyndingtheregionswhereVVs)]TJ /F5 11.955 Tf 12.06 0 Td[(Vtisamaximum,andexpandingandintegratingVaroundthesedominantregions.AsitwasrstobservedbyGrossandManesfortheopensuperstring[ 25 ]inatspace,allthedominantcriticalpointsfortheone-loopplanaramplitudelieontheboundaryoftheintegrationregion.SincetheexponentialdependenceontheexternalmomentaintheNS+string( 4 )isthesameasthatforthesuperstring,thesameholdstruehere.Therefore,inthisarticlewewillstudyallpossibleboundaryregionsthatproduceacontributionwhichisnotexponentiallysuppressedas0s!atxed.Then,wewillcomparealltherelevantcontributionsinthislimitandextracttheleading 1WewillcomebacktotheM)]TJ /F12 9.963 Tf 9.49 -3.62 Td[(partoftheamplitudelaterinthischapter. 83

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dominantone.Inordertostudytheseregionsbetter,afteraresummationwecanwrite ln ()=lnsin+21Xn=11 nq2n 1)]TJ /F2 11.955 Tf 11.96 0 Td[(q2n(1)]TJ /F2 11.955 Tf 11.96 0 Td[(cos2n) (4) anddening xsin43sin2 sin42sin3 (4) wecanwriteVas V=lnx)]TJ /F5 11.955 Tf 11.96 0 Td[(ln(1)]TJ /F2 11.955 Tf 11.96 0 Td[(x)+21Xn=11 nq2n 1)]TJ /F2 11.955 Tf 11.96 0 Td[(q2n(Sn)]TJ /F5 11.955 Tf 11.96 0 Td[(Tn) (4) where Sn2cosn(2)]TJ /F5 11.955 Tf 11.96 0 Td[(43)[cosn(42+3))]TJ /F2 11.955 Tf 11.96 0 Td[(cosn(2+43)]Tn2cosn(42+3)[cosn(2)]TJ /F5 11.955 Tf 11.95 0 Td[(43))]TJ /F2 11.955 Tf 11.96 0 Td[(cosn(2+43)] (4) From( 4 )weimmediatelyrecognizethatatq=0,theremainingtermispreciselytheonethatwendforthetreelevelamplitudesincewehaveasanoverallfactortheintegral: Z10dxe)]TJ /F7 7.97 Tf 6.58 0 Td[(0sV=Z10dxe)]TJ /F7 7.97 Tf 6.58 0 Td[(0s(lnx)]TJ /F7 7.97 Tf 6.59 0 Td[(ln(1)]TJ /F4 7.97 Tf 6.59 0 Td[(x))=Z10dxx)]TJ /F7 7.97 Tf 6.59 0 Td[(0s(1)]TJ /F2 11.955 Tf 11.96 0 Td[(x))]TJ /F7 7.97 Tf 6.58 0 Td[(0t (4) Becauseofthisfact,andmotivatedbytheanalysisin[ 1 ],theintegralsaremoreeasilyanalyzedbygoingtothefollowingvariables:2 r(3)=sin43 sin3,x(2)=sin43sin2 sin42sin3 (4) 2ThischangeofvariableswasrstusedbyNeveuandScherk[ 1 ]whentheywerestudyingtheplanarone-loopamplitudeformesonsinthecontextoftheoriginaldualresonancemodels.Thisallowedthemtoprovethattheleadingdivergenceatone-loopwasproportionaltotheBornterm(treeamplitude),thusprovidingevidenceofrenormalizabilityinthosemodels.Sincethecountertermusedin[ 1 ]arisesfromthedivergenceatq0,andprovedtobeproportionaltothetreeamplitude,itsoundedverylikelythatthesesetofvariableswasalsogoingtobeusefulinourcalculationsfortheNS+stringinthehardscatteringregime. 84

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ThevariablexallowsustoseethatQi
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Figure4-1. (a)Anopenstringone-loopamplitudeisobtainedbyintegratingovertheradiusoftheannulusdiagram.(b)Theregionofintegrationwheretheradiusqgetsclosetozerobecomesindistinguishablefromtheopenstringtreeamplitudewhichhasthetopologyofadisk. Thersttwotermsin( 4 )areindependentoftheintegrationvariableskandq,sowecantakethemoutoftheintegralsas e)]TJ /F7 7.97 Tf 6.59 0 Td[(0sVe0s[ln()]TJ /F7 7.97 Tf 6.58 0 Td[()+(1)]TJ /F7 7.97 Tf 6.59 0 Td[()ln(1)]TJ /F7 7.97 Tf 6.59 0 Td[()]e)]TJ /F7 7.97 Tf 6.58 0 Td[(0s[(1)]TJ /F21 5.978 Tf 5.75 0 Td[()3 2(x)]TJ /F4 7.97 Tf 6.59 0 Td[(xc)2+2q2(S1)]TJ /F7 7.97 Tf 6.59 0 Td[(T1)] (4) Since<0,theterminsidethesquarebracketsoftherstexponentialispositivedenitegivingtheoverallexponentialsuppressionexpf)]TJ /F5 11.955 Tf 15.28 0 Td[(0jsjf()gwheref()=ln()]TJ /F5 11.955 Tf 9.3 0 Td[()+(1)]TJ /F5 11.955 Tf 12.73 0 Td[()ln(1)]TJ /F5 11.955 Tf 12.73 0 Td[()fortheamplitudeas0s!.Thisisthewellknownexponentialfalloffcharacteristicofstringyamplitudesinthehardscatteringlimit.Moreover,itisidenticaltothetreelevelbehavior.Thereasonisthat,asq!0,theholeoftheannulusshrinkstoapointthusmakingitindistinguishablefromthediskamplitude.Figure 4-1 showsthisdiagrammatically.Wenowre-write( 4 )as e)]TJ /F7 7.97 Tf 6.58 0 Td[(0sVe)]TJ /F7 7.97 Tf 6.59 0 Td[(0jsjf()e)]TJ /F7 7.97 Tf 6.59 0 Td[(0s[(1)]TJ /F21 5.978 Tf 5.76 0 Td[()3 2(x)]TJ /F4 7.97 Tf 6.58 0 Td[(xc)2+2q2(S1)]TJ /F7 7.97 Tf 6.59 0 Td[(T1)] (4) 86

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where f()ln()]TJ /F5 11.955 Tf 9.3 0 Td[()+(1)]TJ /F5 11.955 Tf 11.96 0 Td[()ln(1)]TJ /F5 11.955 Tf 11.95 0 Td[() (4) S1)]TJ /F5 11.955 Tf 11.96 0 Td[(T1=2(sin22+sin243))]TJ /F2 11.955 Tf 11.96 0 Td[(2)]TJ /F2 11.955 Tf 5.48 -9.68 Td[(sin24+sin232)]TJ /F2 11.955 Tf 9.3 0 Td[(2(1)]TJ /F5 11.955 Tf 11.96 0 Td[())]TJ /F2 11.955 Tf 5.48 -9.68 Td[(sin242+sin23 (4) andwhereisalsoimportanttostressthat,atleadingorder,thecombinationS1)]TJ /F5 11.955 Tf 12.5 0 Td[(T1mustbeevaluatedatthevaluewherethecrossratioxextremizesVi.e.:atx=xc=s s+t=(1)]TJ /F5 11.955 Tf 12.17 0 Td[())]TJ /F4 7.97 Tf 6.59 0 Td[(1asdictatedbyequation( 4 ).Therefore,wecansimplify( 4 )using( 4 )withthereplacement !)]TJ /F2 11.955 Tf 25.77 8.09 Td[(sin4sin32 sin43sin2 (4) whichyields (S1)]TJ /F5 11.955 Tf 11.96 0 Td[(T1)x=xc=)]TJ /F2 11.955 Tf 9.3 0 Td[(8sin32sin3sin42sin4 (4) Fromthefactthatfortheplanaramplitudetheivariablesareordered,i.e.0234,weseethat(S1)]TJ /F5 11.955 Tf 12.58 0 Td[(T1)x=xcisanegativenumber.WehavementionedearlierthatweonlyhavenumericalevidencethatVnegative-deniteintheintegrationregion.However,from( 4 )and( 4 )weseeanalyticallythatthisistrueatleastalongthesurfacex=sin2sin43 sin42sin3=xcwhichwilldominateattheend.Afterwritingtheintegralsinthenewsetofvariablesgivenin( 4 )wecanmakethismoreexplicitaswewillshownext.Nowwegoaheadandestimatetheleadingbehaviorthatcomesfromthexxc,q0region.Startingfrom( 4 ),whichwere-writehereforconvenience, M+4,ren=21 80D=2Z10dq q)]TJ /F5 11.955 Tf 9.3 0 Td[( lnq(10)]TJ /F4 7.97 Tf 6.59 0 Td[(D)=2P+(q)Z4Yk=2dkhe)]TJ /F7 7.97 Tf 6.58 0 Td[(0sVh^P1^P2^P3^P4i+)]TJ /F2 11.955 Tf 11.95 0 Td[(e)]TJ /F7 7.97 Tf 6.58 0 Td[(0sV0h^C1^C2^C3^C4i)]TJ /F2 11.955 Tf 19.26 0 Td[(B+i (4) 87

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Theapproximationsfortheexponentialsinsidethesquarebracketsnearthecriticalsurfacearegivenby( 4 ),fromwherewealsoseethattheintegraloverxiswellapproximatedbyaGaussianintegrationinthe0s!limit.Theintegrationoverqisdominatedbytheend-pointq=0whichdemandsthatweexpandtherestoftheintegrandasapowerseriesinq.Aswewillseebelow,weneedtoexpandtheintegrandbeyondleadingorderinqinordertoextractthecorrectleadingbehavior.Wethenneedtheexpansions: P+=q)]TJ /F4 7.97 Tf 6.59 0 Td[(1(1)]TJ /F2 11.955 Tf 11.96 0 Td[(w1=2)10)]TJ /F4 7.97 Tf 6.59 0 Td[(D)]TJ /F4 7.97 Tf 6.59 0 Td[(S(1+8q+O(q2)) (4) h^P1^P2^P3^P4i+=a0+a1q+O(q2) (4) B+(q)=b1q+O(q2) (4) wherea0=h^C1^C2^C3^C4i,which,intermsoftheoriginalkvariablesis 16a0=csc232csc24(1+0t)2+csc4csc32[(0s)2csc2csc43)]TJ /F2 11.955 Tf 9.29 0 Td[((0u)2csc3csc42] (4) andwithasimilar(butmorecumbersome)expressionfora1.Withtheseexpansions,andintegratingoverthenewvariables(,r,x)wehave M+4,ren'21 80D=220)]TJ /F4 7.97 Tf 6.59 0 Td[(2D)]TJ /F4 7.97 Tf 6.58 0 Td[(2Se)]TJ /F7 7.97 Tf 6.59 0 Td[(0jsjf()ZRddrdxjJje)]TJ /F7 7.97 Tf 6.59 0 Td[(0s(1)]TJ /F21 5.978 Tf 5.75 0 Td[()3 2(x)]TJ /F4 7.97 Tf 6.59 0 Td[(xc)2Z0dq q2)]TJ /F2 11.955 Tf 9.3 0 Td[(1 lnqpha0(e)]TJ /F4 7.97 Tf 6.58 0 Td[(20sq2(S1)]TJ /F7 7.97 Tf 6.59 0 Td[(T1))]TJ /F2 11.955 Tf 11.96 0 Td[(1)+q(a1+8a0)e)]TJ /F4 7.97 Tf 6.58 0 Td[(20sq2(S1)]TJ /F7 7.97 Tf 6.59 0 Td[(T1)+b1q+i (4) wherep=15)]TJ /F2 11.955 Tf 12.13 0 Td[(3D=2)]TJ /F2 11.955 Tf 12.13 0 Td[(SandjJjistheJacobianforthetransformationd3d2=jJjdrdxwhichreads jJj=x)]TJ /F4 7.97 Tf 6.59 0 Td[(2r[sin4]2(r2+2rcos4+1))]TJ /F4 7.97 Tf 6.59 0 Td[(1(r2 x2+2r xcos4+1))]TJ /F4 7.97 Tf 6.59 0 Td[(1 (4) 88

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TheintegrationregionRin( 4 )is0<<,0
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whichistheleadingtermofI1asanexpansioninpowersof(ln))]TJ /F4 7.97 Tf 6.58 0 Td[(1.SimilarlyforI2,weperformthechangeu=)]TJ /F2 11.955 Tf 11.29 0 Td[(lnqyielding I2=Z1)]TJ /F4 7.97 Tf 8 0 Td[(lnduu)]TJ /F4 7.97 Tf 6.59 0 Td[(pexp[)]TJ /F5 11.955 Tf 9.3 0 Td[(exp[)]TJ /F2 11.955 Tf 9.3 0 Td[(2u]]=(ln)1)]TJ /F4 7.97 Tf 6.58 0 Td[(pZ1)]TJ /F4 7.97 Tf 8 0 Td[(ln=lnd)]TJ /F4 7.97 Tf 6.59 0 Td[(pexp[)]TJ /F2 11.955 Tf 11.29 0 Td[(exp[(1)]TJ /F2 11.955 Tf 11.95 0 Td[(2)ln]] (4) As!1weseethattheexponentialfactorexp[)]TJ /F2 11.955 Tf 11.3 0 Td[(exp[(1)]TJ /F2 11.955 Tf 12.08 0 Td[(2)ln]]effectivelycutstheintegrationrangeto1=2<<1therefore,forsmallbutxedwehave I2'(ln)1)]TJ /F4 7.97 Tf 6.59 0 Td[(pZ11=2d)]TJ /F4 7.97 Tf 6.59 0 Td[(p=1 p)]TJ /F2 11.955 Tf 11.95 0 Td[(12 lnp)]TJ /F4 7.97 Tf 6.58 0 Td[(1 (4) Withtheseapproximationsfortheqintegration,theamplitudein( 4 )nowbecomes M+4,ren'21 80D=220)]TJ /F4 7.97 Tf 6.58 0 Td[(2D)]TJ /F4 7.97 Tf 6.59 0 Td[(2Se)]TJ /F7 7.97 Tf 6.58 0 Td[(0jsjf()ZRddrdxe)]TJ /F7 7.97 Tf 6.59 0 Td[(0s(1)]TJ /F21 5.978 Tf 5.75 0 Td[()3 2(x)]TJ /F4 7.97 Tf 6.59 0 Td[(xc)2jJj")]TJ 9.3 10.56 Td[(p 20s2 ln0jsjp(S1)]TJ /F5 11.955 Tf 11.96 0 Td[(T1)1=2a0+1 p)]TJ /F2 11.955 Tf 11.95 0 Td[(12 ln0jsjp)]TJ /F4 7.97 Tf 6.59 0 Td[(1(a1+8a0)# (4) Asmentionedabove,thexintegralisverywellapproximatedbyaGaussianinthe0s!limit.Thus,atleadingorder,wehave Z10dxe)]TJ /F7 7.97 Tf 6.58 0 Td[(0s(1)]TJ /F21 5.978 Tf 5.76 0 Td[()3 2(x)]TJ /F4 7.97 Tf 6.58 0 Td[(xc)2h(x)'h(xc)Z1dxe)]TJ /F7 7.97 Tf 6.59 0 Td[(0s(1)]TJ /F21 5.978 Tf 5.75 0 Td[()3 2(x)]TJ /F4 7.97 Tf 6.59 0 Td[(xc)2'h(xc)s 2 0s(1)]TJ /F5 11.955 Tf 11.96 0 Td[()3 (4) whereh(x)simplytracksthecompletedependenceontheoriginalkvariablesoftherestoftheintegrandin( 4 ).Therefore,wenowhave M+4,ren'21 80D=220)]TJ /F4 7.97 Tf 6.58 0 Td[(2D)]TJ /F4 7.97 Tf 6.59 0 Td[(2Se)]TJ /F7 7.97 Tf 6.58 0 Td[(0jsjf()s 2 0s(1)]TJ /F5 11.955 Tf 11.96 0 Td[()3)]TJ 9.3 10.56 Td[(p 20s2 ln0jsjpZRddrjJj(S1)]TJ /F5 11.955 Tf 11.95 0 Td[(T1)1=2a0+1 p)]TJ /F2 11.955 Tf 11.96 0 Td[(12 ln0jsjp)]TJ /F4 7.97 Tf 6.58 0 Td[(1Z0dZ10drjJj(a1+8a0)# (4) 90

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Theintegralsoverrandcannotbeevaluatedinclosedform,butwecansimplifytheexpressionaboveabitfurtherbyinspectingtheleadingtermsinthelarge0slimitwith=)]TJ /F2 11.955 Tf 9.3 0 Td[(t=sheldxed.Werstnoticethatbothfunctionsa0anda1contain(0s)2terms,thereforeitwouldseemthattherstoftheintegralsin( 4 )woulddominateinthelarge0slimit.Thisis,however,nottrue.From( 4 )weseethattheintegrandsin( 4 )needtobeevaluatedatx=sin43sin2 sin42sin3=xc=(1)]TJ /F5 11.955 Tf 11.65 0 Td[())]TJ /F4 7.97 Tf 6.59 0 Td[(1.ThefullexpressionforthefactorjJja0intermsofthenewvariables,r,xthatentersinbothintegrandsisgivenby 16jJja0=r)]TJ /F4 7.97 Tf 6.59 0 Td[(1[sin])]TJ /F4 7.97 Tf 6.59 0 Td[(2x)]TJ /F4 7.97 Tf 6.59 0 Td[(2(1+0t)2x2 (1)]TJ /F2 11.955 Tf 11.96 0 Td[(x)2+(0s)2x 1)]TJ /F2 11.955 Tf 11.96 0 Td[(x)]TJ /F5 11.955 Tf 11.95 0 Td[(02(s+t)2x2 1)]TJ /F2 11.955 Tf 11.96 0 Td[(x (4) Fromhere,wecanreadilyseethatthecoefcientof(0s)2insidethesquarebracketsaboveis x (1)]TJ /F2 11.955 Tf 11.96 0 Td[(x)2(1)]TJ /F2 11.955 Tf 11.96 0 Td[(x(1)]TJ /F5 11.955 Tf 11.95 0 Td[())2 (4) whichvanishespreciselyatthevaluex=xc=(1)]TJ /F5 11.955 Tf 12.02 0 Td[())]TJ /F4 7.97 Tf 6.59 0 Td[(1.Therefore,a0reallycontributeslinearlyinsinthehardscatteringlimit,notquadratically.Ontheotherhand,thefactorjJja1whichentersinthesecondintegralin( 4 ),whenevaluatedatx=xc,becomes jJja1=x)]TJ /F4 7.97 Tf 6.58 0 Td[(2(r2+2rcos+1))]TJ /F4 7.97 Tf 6.59 0 Td[(1(r2 x2+2r xcos+1))]TJ /F4 7.97 Tf 6.58 0 Td[(1(0s)2(1)]TJ /F5 11.955 Tf 11.96 0 Td[()rsin2+0s 2rg(r,) (4) where g(r,)1+4r2+r4)]TJ /F2 11.955 Tf 11.96 0 Td[(4r2)]TJ /F2 11.955 Tf 11.96 0 Td[(2r4+2r4+2r22++2r(2)]TJ /F5 11.955 Tf 11.96 0 Td[()(1+r2)]TJ /F5 11.955 Tf 11.96 0 Td[(r2)cos+2r2(1)]TJ /F5 11.955 Tf 11.95 0 Td[()cos2 (4) thus,thecontributionfroma1doesgoesasa1()]TJ /F5 11.955 Tf 9.29 0 Td[(0s)2inthehardscatteringlimitanddominatesovertheonefroma0.Therefore,theleadingbehavioroftherenormalized 91

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M+4,renamplitudeis M+4,ren'21 80D=220)]TJ /F4 7.97 Tf 6.59 0 Td[(2D)]TJ /F4 7.97 Tf 6.58 0 Td[(2Se)]TJ /F7 7.97 Tf 6.59 0 Td[(0jsjf()s 2 0s(1)]TJ /F5 11.955 Tf 11.95 0 Td[()31 p)]TJ /F2 11.955 Tf 11.95 0 Td[(12 ln0jsjp)]TJ /F4 7.97 Tf 6.59 0 Td[(1Z0dZ10drjJja1'21 80D=220)]TJ /F4 7.97 Tf 6.59 0 Td[(2D)]TJ /F4 7.97 Tf 6.58 0 Td[(2Se)]TJ /F7 7.97 Tf 6.59 0 Td[(0jsjf()p )]TJ /F2 11.955 Tf 9.3 0 Td[(21 p)]TJ /F2 11.955 Tf 11.95 0 Td[(12 ln0jsjp)]TJ /F4 7.97 Tf 6.59 0 Td[(1(1)]TJ /F5 11.955 Tf 11.95 0 Td[()3=2()]TJ /F5 11.955 Tf 9.29 0 Td[(0s)3=2F()where F()Z0dZ10drrsin2 (r2+2rcos+1)(r2(1)]TJ /F5 11.955 Tf 11.95 0 Td[()2+2r(1)]TJ /F5 11.955 Tf 11.96 0 Td[()cos+1) (4) WecannowwriteamoresuccinctexpressionforthenalbehavioroftherenormalizedM+partofamplitudeinthehardscatteringlimitas M+4,ren'G()e)]TJ /F7 7.97 Tf 6.59 0 Td[(0jsjf()1 ln0jsjp)]TJ /F4 7.97 Tf 6.58 0 Td[(1()]TJ /F5 11.955 Tf 9.3 0 Td[(0s)3=2 (4) with G()21 80D=22p)]TJ /F4 7.97 Tf 6.59 0 Td[(120)]TJ /F4 7.97 Tf 6.58 0 Td[(2D)]TJ /F4 7.97 Tf 6.59 0 Td[(2S p)]TJ /F2 11.955 Tf 11.95 0 Td[(1()]TJ /F2 11.955 Tf 9.3 0 Td[(2)1=2(1)]TJ /F5 11.955 Tf 11.95 0 Td[()3=2F() (4) Notethat,sinceinthehardscatteringlimitbothsandtarelargecomparedto0)]TJ /F4 7.97 Tf 9.38 0 Td[(1,wehaveln()]TJ /F5 11.955 Tf 9.3 0 Td[(0s)=ln()]TJ /F5 11.955 Tf 9.3 0 Td[(0t)1+O(1 ln()]TJ /F7 7.97 Tf 6.59 0 Td[(0t)),thusatleadingorderwecanwrite( 4 )alsoas M+4,ren'G()e)]TJ /F7 7.97 Tf 6.59 0 Td[(0jsjf()1 ln0jtjp)]TJ /F4 7.97 Tf 6.58 0 Td[(1()]TJ /F5 11.955 Tf 9.3 0 Td[(0s)3=2 (4) ThisformwillbeusefulwhenwecomparetheseresultswiththeReggebehavioroftheamplitude.Thisisdoneinsection 4.2.3 .Wenowturntotheanalysisoftheq0regionfortheM)]TJ /F1 11.955 Tf 10.41 -4.34 Td[(partoftheamplitude.Thispartoftheamplitudereads M)]TJ /F4 7.97 Tf 0 -7.89 Td[(4,ren=21 80D=2Z10dq q)]TJ /F5 11.955 Tf 9.3 0 Td[( lnq(10)]TJ /F4 7.97 Tf 6.58 0 Td[(D)=2P)]TJ /F2 11.955 Tf 7.09 1.8 Td[((q)Z4Yk=2dkhe)]TJ /F7 7.97 Tf 6.59 0 Td[(0sVh^P1^P2^P3^P4i)]TJ /F3 11.955 Tf 9.74 -4.93 Td[()]TJ /F2 11.955 Tf 11.96 0 Td[(e)]TJ /F7 7.97 Tf 6.59 0 Td[(0sV0h^C1^C2^C3^C4i)]TJ /F2 11.955 Tf 19.26 0 Td[(B)]TJ /F9 11.955 Tf 7.08 8.34 Td[(i (4) 92

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FromhereweseethattheonlydifferenceswithrespecttotheM+caselieonthepartitionfunctionP)]TJ /F2 11.955 Tf 7.08 1.8 Td[((q)andthecorrelatorh^P1^P2^P3^P4i)]TJ /F1 11.955 Tf 7.08 -4.34 Td[(.Theexponentialfactorsarethesameasbefore.Fromequations( 4 )and( 4 )wehave P)]TJ /F2 11.955 Tf 7.09 1.8 Td[((q)=24+O(q2) (4) h^P1^P2^P3^P4i)]TJ /F2 11.955 Tf 17.04 -4.94 Td[(=h^P1^P2^P3^P4i)]TJ /F4 7.97 Tf 0 -7.9 Td[(q=0+O(q2) (4) Expandingaboutthecriticalsurface(x,q)=(xc,0)again,theamplitude( 4 )becomes M)]TJ /F4 7.97 Tf 0 -7.9 Td[(4,ren'21 80D=2e)]TJ /F7 7.97 Tf 6.59 0 Td[(0jsjf()24Z4Yk=2dke)]TJ /F7 7.97 Tf 6.59 0 Td[(0s(1)]TJ /F21 5.978 Tf 5.75 0 Td[()3 2(x)]TJ /F4 7.97 Tf 6.59 0 Td[(xc)2Z0dq q)]TJ /F5 11.955 Tf 9.3 0 Td[( lnq(10)]TJ /F4 7.97 Tf 6.58 0 Td[(D)=2he)]TJ /F4 7.97 Tf 6.59 0 Td[(20sq2(S1)]TJ /F7 7.97 Tf 6.59 0 Td[(T1)h^P1^P2^P3^P4i)]TJ /F4 7.97 Tf 0 -7.89 Td[(q=0)-222(h^C1^C2^C3^C4i+e)]TJ /F4 7.97 Tf 6.58 0 Td[(20sq2(S1)]TJ /F7 7.97 Tf 6.59 0 Td[(T1)O(q2)i (4) Weagainrecallthattheintegraloverthekvariablesisdominatedbythetwodimensionalsurfacex=sin43sin2 sin42sin3=(1)]TJ /F5 11.955 Tf 12.04 0 Td[())]TJ /F4 7.97 Tf 6.59 0 Td[(1=xc.Theintegraloverthecrossratioxthenbecomesagaussianwhich,atleadingorder,demandsthatweevaluatetheexpressioninsidethesquarebracketsaboveatsin43sin2 sin42sin3=(1)]TJ /F5 11.955 Tf 12.29 0 Td[())]TJ /F4 7.97 Tf 6.58 0 Td[(1.Itwillbeagainconvenienttoseparatethes2partofh^P1^P2^P3^P4i)]TJ /F4 7.97 Tf 0 -7.88 Td[(q=0as h^P1^P2^P3^P4i)]TJ /F4 7.97 Tf 0 -7.89 Td[(q=0=As2+Bs+C (4) and,fromequations( 4 )and( 4 ),weobtain A1 16cot4cot322cot4cot32)]TJ /F2 11.955 Tf 11.96 0 Td[(cot2cot43)]TJ /F2 11.955 Tf 11.95 0 Td[((1)]TJ /F5 11.955 Tf 11.96 0 Td[()2cot3cot42 (4) Evaluatingthisexpressiononthecriticalsurfaceimpliesthatwemakethereplacement=)]TJ /F4 7.97 Tf 10.5 4.82 Td[(sin4sin32 sin43sin2.Remarkably,onecanseethatAvanishesinthiscase,yielding h^P1^P2^P3^P4i)]TJ /F4 7.97 Tf 0 -7.89 Td[(q=0!Bs+C (4) onthecriticalsurface.Thecoefcientofs2ofthecountertermh^C1^C2^C3^C4ialsovanishesonthissurfaceasderivedinequations( 4 )and( 4 ).Giventhesefacts,wecannowestimatethecontributionsfromtherestofthetermsin( 4 )asfollows.The 93

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integrationovertherstterminsidethesquarebracketsin( 4 )hasthesameformas( 4 ),thustogetherwiththesfactorcomingfromthecorrelator( 4 )itbehavesass(log0jsj)D=2)]TJ /F4 7.97 Tf 6.59 0 Td[(4.Duetothelackoftheexponentialfactorinfrontit,thecontributionfromthecountertermh^C1^C2^C3^C4iisexponentiallysuppressed.ThiswasexpectedheresincethiscountertermisnotnecessarytomakethebehavioroftheM)]TJ /F1 11.955 Tf 10.41 -4.34 Td[(amplitudeconvergentneartheq=0region4.Wecanalsoestimatethecontributionfromalltherestoftermsintheexpansioninpowersofqbyrecallingthattheexponentialfactorexpf)]TJ /F2 11.955 Tf 15.27 0 Td[(20sq2(S1)]TJ /F5 11.955 Tf 12.73 0 Td[(T1)gin( 4 )cutsofftheeffectiverangeoftheqintegraltos)]TJ /F4 7.97 Tf 6.58 0 Td[(1=2.Thus,sincewehaveanexpansioninevenpowersofq,theintegralwillproduceacontributions)]TJ /F4 7.97 Tf 6.58 0 Td[(1=2(s)]TJ /F4 7.97 Tf 6.58 0 Td[(1=2)2n)]TJ /F4 7.97 Tf 6.58 0 Td[(1(log0jsj)D=2)]TJ /F4 7.97 Tf 6.59 0 Td[(4=s)]TJ /F4 7.97 Tf 6.59 0 Td[(n(log0jsj)D=2)]TJ /F4 7.97 Tf 6.59 0 Td[(4withn1.Themaximumpowerofsthatcouldcomefromthecorrelatorh^P1^P2^P3^P4i)]TJ /F1 11.955 Tf 10.41 -4.33 Td[(iss2.Thus,eveniftherearenocancellationsofthesetermsonthecriticalsurface,theleadingbehaviorcomingfromO(q2)termsin( 4 )iss(log0jsj)D=2)]TJ /F4 7.97 Tf 6.58 0 Td[(4.Finally,from( 4 ),wealreadyknowthattheintegraloverthecrossratioxproducesanoverallfactorofs)]TJ /F4 7.97 Tf 6.59 0 Td[(1=2.Thus,puttingeverythingtogether,wehavethattheleadingbehaviorofM)]TJ /F4 7.97 Tf 0 -7.88 Td[(4)-252(C4inthehardscatteringlimitis M)]TJ /F4 7.97 Tf 0 -7.89 Td[(4,rene)]TJ /F7 7.97 Tf 6.58 0 Td[(0jsjf()1 log0jsjD=2)]TJ /F4 7.97 Tf 6.59 0 Td[(4()]TJ /F5 11.955 Tf 9.3 0 Td[(0s)1=2 (4) whichisdenitelysubleadingwithrespecttoM+4,renin( 4 ).Nowweturntothestudyofthecontributionsfromotherregionsthatwehavenotanalyzedyet.Asmentionedbefore,sincetheasymptoticbehaviorisgovernedbycriticalpointsofthesecondkind,i.e.boundaryregionsofthemodulispace,wealsoneedtoexaminetheregionwhereq!1.TothisenditisconvenienttoperformtheJacobiimaginarytransformationq=expf22=lnwg,whichmapstheq1regiontow0. 4Thiscountertermishowevernecessarytocancelspuriousdivergencesfromcertainregionsinthekintegrals.Thecontributionsfromtheseregionswillbeanalyzedseparatelyattheendofthissection. 94

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Usingthecorrespondingtransformationsonthe1(j)function,wehave 1ilnw 2,p w=)]TJ /F2 11.955 Tf 9.29 0 Td[(i)]TJ /F2 11.955 Tf 9.3 0 Td[(2 lnw1=2exp)]TJ /F5 11.955 Tf 9.3 0 Td[(2lnw 221(,q) (4) 01(0,p w)=)]TJ /F2 11.955 Tf 9.3 0 Td[(2 lnw3=201(0,q) (4) gives (,q)=1(,q) 0(0)=i)]TJ /F2 11.955 Tf 9.3 0 Td[(2 lnwexp2lnw 221(ilnw=2,p w) 01(0,p w)= )]TJ /F2 11.955 Tf 11.29 0 Td[(lnwexp)]TJ /F5 11.955 Tf 10.5 8.09 Td[(()]TJ /F5 11.955 Tf 11.96 0 Td[()lnw 22(1)]TJ /F2 11.955 Tf 11.95 0 Td[(w=)1Yn=1(1)]TJ /F2 11.955 Tf 11.96 0 Td[(wn+=)(1)]TJ /F2 11.955 Tf 11.95 0 Td[(wn)]TJ /F7 7.97 Tf 6.58 0 Td[(=) (1)]TJ /F2 11.955 Tf 11.95 0 Td[(wn)2 (4) Weareinterestedinthesmallwbehaviorofln ,therefore ln =ln )]TJ /F2 11.955 Tf 11.29 0 Td[(lnw)]TJ /F5 11.955 Tf 13.15 8.08 Td[(()]TJ /F5 11.955 Tf 11.95 0 Td[()lnw 22+ln(1)]TJ /F2 11.955 Tf 11.96 0 Td[(w=)+1Xn=1ln(1)]TJ /F2 11.955 Tf 11.95 0 Td[(wn+=)(1)]TJ /F2 11.955 Tf 11.96 0 Td[(wn)]TJ /F7 7.97 Tf 6.59 0 Td[(=) (1)]TJ /F2 11.955 Tf 11.96 0 Td[(wn)2=ln )]TJ /F2 11.955 Tf 11.29 0 Td[(lnw)]TJ /F5 11.955 Tf 13.15 8.09 Td[(()]TJ /F5 11.955 Tf 11.95 0 Td[()lnw 22+O(w) (4) Keepingthersttwotermsisagoodapproximationaslongasisnottooclosetozeroor.Usingtheapproximation( 4 )wehave jVjlnw s2Xi
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behaviorcomingfromtheseplaces.Thisisaverycomplicatedproblemsincetheanalyticapproximationsturnouttobedifculttoanalyze.However,wecanestimatetheircontributionsandshowthattheyaresubleadingwithrespecttotheonefromq0asfollows.Acrucialpointisthataccordingto[ 25 ]allthestationarypointsfortheplanaramplitudelieontheboundaryoftheintegrationregion.Sincewearenowawayfromeitherq=0andq=1andfocusingonallpossiblestationaryregionsthatcouldcomefromtheRdkintegral,allweneedtoanalyzearetheboundaryregionsinthekvariables.Recallthatinthehardscatteringlimittheimportantfactoristheonegiveninequation( 4 )whichwewriteagainhere Yi
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vertexoperatorcoincidecorrespondtothefourverticesingure( 2-1 ).Theseare: 2=3=4=0 (4) 2=3=0,4= (4) 2=0,3=4=,and (4) 2=3=4= (4) Letusanalyzeoneoftheregionswhereallfourvertexoperatorscoincide,say,thevertex( 4 ).Itisconvenientheretomakethechanges4=,3=3,2=2withsmallandexpandeverythinginpowersof.In[ 25 ]theauthorsalsoanalyzetheseregionsandpointoutthattheasymptoticbehavioroftheamplitudedoesnotdependononlyforthesuperstring.Thereasonforthisisthattheregionsinmodulispacewhere0producedivergencesthatareduetothepresenceoftachyonswhichareabsentinthesuperstring.DuetotheformofSm)]TJ /F5 11.955 Tf 12.61 0 Td[(Tmtherewillonlybeevenpowersin.Inthiscasewehave Sm)]TJ /F5 11.955 Tf 11.96 0 Td[(Tm=8m2 s((s+t)2)]TJ /F2 11.955 Tf 11.95 0 Td[((s+t2)3)2+O(4) (4) Theimportantpointisthat,sincewehavetoevaluatethisexpressionatthecriticalsurfacex=xc,wehave x=sin43sin2 sin42sin3=(1)]TJ /F5 11.955 Tf 11.96 0 Td[(3)2 3(1)]TJ /F5 11.955 Tf 11.96 0 Td[(2)+O(2)=1 1+t=s!2=s3 s+t)]TJ /F2 11.955 Tf 11.95 0 Td[(t3+O(2) (4) Pluggingthisinto( 4 )makestheentirecoefcientmultiplying)]TJ /F4 7.97 Tf 6.59 0 Td[(2in( 4 )tovanish!Therefore,onthecriticalsurfacewehaveSm)]TJ /F5 11.955 Tf 12.57 0 Td[(TmO(4).Theexponentialfactor( 4 )thenhasitslargestcontributiontotheintegralfors41,whichimpliesthattheeffectiverangeforeachvariableiiss)]TJ /F4 7.97 Tf 6.59 0 Td[(1=4.Becausewehaveatripleintegralovertheseangles,thetotalcontributionfromthemeasureiss)]TJ /F4 7.97 Tf 6.59 0 Td[(3=4.SincexisO(1)in,thesmallcornerstudiedherestillcontainsthetwodimensionalplanex=xc=1 1+t=s,which 97

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wealreadyknowthatitcontributeswithafactorofs)]TJ /F4 7.97 Tf 6.59 0 Td[(1=2e)]TJ /F7 7.97 Tf 6.59 0 Td[(0jsjf().Recallthatthes)]TJ /F4 7.97 Tf 6.59 0 Td[(1=2factorcomesfromthegaussianapproximationofthecross-ratioalongthisplane.Therestoftheintegrandonlyinvolvesthecontractionsh^P1^P2^P3^P4i.Notethatourexpressionforthegluonamplitudeincludescountertermsthateliminateallpossibledivergencesintheintegrals.Inparticular,thecorneroftheintegrationregionweareconsideringhereispreciselyoneoftheplacesthatoriginallyproduceddivergences.Theseweretakencarebythecounter-term5thatinvolvestakingtheh^P1^P2^P3^P4icorrelatoratq=0.Thus,startingfromequation( 4 ),weseethatweneedtoestimatethecontributionof e)]TJ /F4 7.97 Tf 6.59 0 Td[(20sP1n=11 nq2n 1)]TJ /F19 5.978 Tf 5.76 0 Td[(q2n(Sn)]TJ /F7 7.97 Tf 6.58 0 Td[(Tn)h^P1^P2^P3^P4i)-222(h^C1^C2^C3^C4i (4) fromtheregioninconsideration.Inthisregionwehavethats(Sn)]TJ /F5 11.955 Tf 9.76 0 Td[(Tn)O(1),thereforetheprefactorofaboveisanumberoforderone.Now,expandingtheexpressionh^P1^P2^P3^P4i,givenby( 4 ),inthisregionwehave h^P1^P2^P3^P4i=1)]TJ /F4 7.97 Tf 6.59 0 Td[(4+2)]TJ /F4 7.97 Tf 6.58 0 Td[(2+O(1) (4) wherethecoefcient1aboveturnsouttobe 1=s2 162()]TJ /F2 11.955 Tf 9.3 0 Td[(1+3)(2)]TJ /F5 11.955 Tf 11.96 0 Td[(3))]TJ /F2 11.955 Tf 41.42 8.09 Td[(s2()]TJ /F2 11.955 Tf 9.3 0 Td[(1+)2 16()]TJ /F2 11.955 Tf 9.3 0 Td[(1+2)(2)]TJ /F5 11.955 Tf 11.95 0 Td[(3)3+()]TJ /F2 11.955 Tf 9.3 0 Td[(1+s)2 16(2)]TJ /F5 11.955 Tf 11.96 0 Td[(3)2 (4) Noticethatthiscoefcientlacksqdependence,whichmeansthattheh^C1^C2^C3^C4itermwillhavetheexactsamecoefcientinitsexpansioninpowersof.Now,sincewemustdemandtheexpressionabovetosatisfythecondition( 4 )inordertolayonthecriticalsurface,itissomewhatremarkablethatthes2terminbothcoefcients1and2vanishes.Aswewillnowshow,thismakesthiscontributiontobesmallerthantheonecomputedfromtheq0region,thereforeitissubleadingwithrespectthatone. 5Thiscounter-termhasatwo-foldpurposesinceitalsocancelsthedivergenceRdq q2forsmallqwhichisre-interpretedasarenormalizationofthecoupling.ThisistheonlydivergenceintheqintegrationaslongastheDp-branehasp<7 98

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Ifthishadnotbeenthecase,thisregionwouldhavedominatedinthehardscatteringregimeandentireleadingbehaviorwouldhavebeenmuchhardertoobtain.Moreover,itistheq0regiontheonethatprovidesthecorrectasymptoticbehaviorforstwhichmatcheswiththeReggelimitathightaswewillshowinthenextsection.Allinall,thetotalcontributionfromthiscornerhasthefollowingstructure:(i)thecross-ratioxcontributeswithafactorofs)]TJ /F4 7.97 Tf 6.59 0 Td[(1=2e)]TJ /F7 7.97 Tf 6.59 0 Td[(0jsjf()asseenabove;(ii)sincetherelevantrangeforeachivariableiss)]TJ /F4 7.97 Tf 6.59 0 Td[(1=4,thetripleintegralprovidesafactorofs)]TJ /F4 7.97 Tf 6.58 0 Td[(3=4;(iii)therestoftheintegrand,namelythecorrelatorsh^P1^P2^P3^P4iandh^C1^C2^C3^C4i,behaveass)]TJ /F4 7.97 Tf 6.59 0 Td[(4s2ontheplanex=xc.Therefore,thetotalestimateiss)]TJ /F4 7.97 Tf 6.58 0 Td[(1=2e)]TJ /F7 7.97 Tf 6.58 0 Td[(0jsjf()s)]TJ /F4 7.97 Tf 6.59 0 Td[(3=4s2=s3=4e)]TJ /F7 7.97 Tf 6.59 0 Td[(0jsjf()whichdenitelysmallerthantheoneobtainedin( 4 )inthelargeslimitwhichcamefromtheq0region.Italsostraightforwardtoshow,afterasuitablechangeofvariables,thattheotherthreevertices( 4 ),( 4 ),and( 4 )giveanidenticalcontribution.Analestimationweneedtoobtainistheonefromtheregionsthatcorrespondtoa3-particlecoincidence,thatis,theregionswhereallbutoneofthevertexoperatorscoincideinthemodulispace.Therearefouroftheseregionsandtheycorrespondtofouroftheedgesintheintegrationdomaindepictedingure 2-1 .Alloftheedgescorrespondingtoaloopinsertiononaneternallegwillproduceanimportantcontributioninthehardscatteringlimit.Theseare:2=3=0,2=3=4,2=)]TJ /F5 11.955 Tf 12.31 0 Td[(4=0and3=4=.Letusfocusontherstone.Inthiscaseitisagainconvenienttodene2=2and3=andexpandforsmallvaluesof.Inthiscasewehave x=2+O() (4) andtheanalogousconditionto( 4 )hereis2=(1)]TJ /F5 11.955 Tf 11.98 0 Td[())]TJ /F4 7.97 Tf 6.58 0 Td[(1atleadingorder.ExpandingSn)]TJ /F5 11.955 Tf 11.96 0 Td[(Tninpowersofyields Sn)]TJ /F5 11.955 Tf 11.96 0 Td[(Tn=2n(2(1)]TJ /F5 11.955 Tf 11.95 0 Td[())]TJ /F2 11.955 Tf 11.96 0 Td[(1)sin(2n4)+O(2) (4) 99

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fromwhereseethattheO()termvanishesonthecriticalsurface.TheO(2)doesnotvanishthere,consequentlySm)]TJ /F5 11.955 Tf 12.14 0 Td[(Tm=O(2).Thisimpliesthatnowtheeffectiverangeofeachivariableinthiscorneroftheintegrationregionisis)]TJ /F4 7.97 Tf 6.58 0 Td[(1=2.Expandingagainh^P1^P2^P3^P4ias h^P1^P2^P3^P4i=1)]TJ /F4 7.97 Tf 6.58 0 Td[(2+O(1) (4) Itwillbeagainconvenienttowritetheleadingcoefcient1asapolynomialinsas 1=b0+b1s+b2s2 (4) Asbefore,wesingleoutthecoefcientofthes2termb2forwhichweobtain b2=(1+2()]TJ /F2 11.955 Tf 11.96 0 Td[(1))22(0,q)43(0,q)23(4,q)24(0,q)4 16(2)]TJ /F2 11.955 Tf 11.95 0 Td[(1)221(4,q)201(0,q)2 (4) Fromhereweseeimmediatelythatthiscoefcientvanishesif2=(1)]TJ /F5 11.955 Tf 12.35 0 Td[())]TJ /F4 7.97 Tf 6.58 0 Td[(1,whichispreciselythevaluethat2acquiresonthecriticalsurfacex=xc=(1)]TJ /F5 11.955 Tf 12.09 0 Td[())]TJ /F4 7.97 Tf 6.59 0 Td[(1.Therefore,whenweevaluateh^P1^P2^P3^P4ionthatsurfacewehave 1=b0+b1s (4) andlike-wiseforh^C1^C2^C3^C4isinceinthatcasetheonlydifferenceisthattheJacobithetafunctionsneedtobeevaluatedatq=0givingtheresult csc24(1+2()]TJ /F2 11.955 Tf 11.96 0 Td[(1))2 16(2)]TJ /F2 11.955 Tf 11.96 0 Td[(1)22 (4) whichofalsovanishesforthesamevalue2=(1)]TJ /F5 11.955 Tf 12 0 Td[())]TJ /F4 7.97 Tf 6.59 0 Td[(1.Withthesetworesults,weseethatthebiggestcontributionfromthecorrelatorsgoeslike)]TJ /F4 7.97 Tf 6.59 0 Td[(2ss2.Sincex=O(1),thecriticalplaneisagaincontainedinthecornerweareanalyzing,producingagainafactorofs)]TJ /F4 7.97 Tf 6.58 0 Td[(1=2e)]TJ /F7 7.97 Tf 6.58 0 Td[(0jsjf().Inviewofthat2<3,inthiscornerwealsohavethatd2d2isO(2)thusgivingafactorofs)]TJ /F4 7.97 Tf 6.58 0 Td[(1.Puttingeverythingtogether,wehavethatthecorner320producesatotalcontributions2s)]TJ /F4 7.97 Tf 6.58 0 Td[(1=2e)]TJ /F7 7.97 Tf 6.58 0 Td[(0jsjf()s)]TJ /F4 7.97 Tf 6.59 0 Td[(1=s1=2e)]TJ /F7 7.97 Tf 6.59 0 Td[(0jsjf().Itis 100

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alsostraightforwardtocheckthattheotherthreecornersleftproducethesameanswer.Therefore,weseeagainthattheseregionsproducesubleadingbehaviorwithrespectto( 4 ).Summarizing,wehaveanalyzedallboundaryregionsthatproducecontributionswhicharenotexponentiallysuppressed,i.e.theregionscomprisedofallpossiblestationarypointsofV(seeequations( 4 )and( 4 )).Thisanalysisyieldedthattheleadingbehaviorcomesfromtheboundaryatq=0andalongthetwodimensionalsurfacegivenbyx=(1)]TJ /F5 11.955 Tf 11.99 0 Td[())]TJ /F4 7.97 Tf 6.59 0 Td[(1wherexisdenedin( 4 ).Therefore,quotingtheresultfrom( 4 ),thebehavioroftherenormalized4-gluonamplitudefortheNS+openstringinthehardscatteringregimeis MHard4'G()e)]TJ /F7 7.97 Tf 6.59 0 Td[(0jsjf()(log0jtj)1)]TJ /F4 7.97 Tf 6.59 0 Td[(p(0jsj)3=2 (4) whereG()containsthefulldependenceinthescatteringangleandisgivenby G()2g21 80D=22p)]TJ /F4 7.97 Tf 6.59 0 Td[(120)]TJ /F4 7.97 Tf 6.59 0 Td[(2D)]TJ /F4 7.97 Tf 6.58 0 Td[(2S p)]TJ /F2 11.955 Tf 11.96 0 Td[(1()]TJ /F2 11.955 Tf 9.3 0 Td[(2)1=2(1)]TJ /F5 11.955 Tf 11.96 0 Td[()3=2F() (4) with F()Z0dZ10drrsin2 (r2+2rcos+1)(r2(1)]TJ /F5 11.955 Tf 11.95 0 Td[()2+2r(1)]TJ /F5 11.955 Tf 11.96 0 Td[()cos+1) (4) TheexpressionforF()givenin( 4 )isconvergentintheentirerangeofparameters<<0anditonlydivergeswhenapproacheszeroinwhichcaseitdivergeslogarithmicallyin.WewillseethatthisispreciselywhatisneededinordertorecovertheReggebehaviorfromthehardscatteringlimit.Inconclusion,theamplitudeisexponentiallysuppressedathighenergiesasexpectedforstringyamplitudesinthisregime,butwealsohaveanextralogarithmicfalloffproductofthepresenceoftheD-branes.ThefulldependenceincontainedinthefunctionG()willbecrucialinordertomakecontactwiththeresultsin[ 16 ]becausetakingthe)]TJ /F2 11.955 Tf 9.3 0 Td[(t=s=!0limitoftheamplitudeinthehardscatteringregimegivenin 101

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( 4 )shouldreproducethehightlimitoftheReggebehavior.Wewillshowinsection 4.2.3 thatthislimitisindeedrecovered. 4.2.2ComparisonwithTreeAmplitudeThetreeamplitudeforthispolarizationis Mtree4)]TJ /F2 11.955 Tf 21.92 0 Td[(g21423\(1)]TJ /F5 11.955 Tf 11.96 0 Td[(0s)\()]TJ /F5 11.955 Tf 9.3 0 Td[(0t) \()]TJ /F5 11.955 Tf 9.29 0 Td[(0s)]TJ /F5 11.955 Tf 11.95 0 Td[(0t) (4) UsingStirling'sapproximation\(1+x)'xxe)]TJ /F4 7.97 Tf 6.59 0 Td[(x(2x)1=2,thehigh-energyxed-anglelimitis(omittingthepolarizationvectorsandthefactorofg2) MTree4p 2()]TJ /F5 11.955 Tf 9.3 0 Td[(0s)1+0t()]TJ /F5 11.955 Tf 9.3 0 Td[(0t))]TJ /F4 7.97 Tf 6.59 0 Td[(1=2)]TJ /F7 7.97 Tf 6.59 0 Td[(0t(1+t=s)1=2+0s+0tp 2()]TJ /F5 11.955 Tf 9.3 0 Td[(0s)1=2()]TJ /F5 11.955 Tf 9.3 0 Td[())]TJ /F4 7.97 Tf 6.58 0 Td[(1=2(1)]TJ /F5 11.955 Tf 11.95 0 Td[()1=2e)]TJ /F7 7.97 Tf 6.58 0 Td[(0jsjf() (4) wheref()ln()]TJ /F5 11.955 Tf 9.3 0 Td[()+(1)]TJ /F5 11.955 Tf 12.06 0 Td[()ln(1)]TJ /F5 11.955 Tf 12.06 0 Td[()0.Theratiooftheone-loopamplitudetothetreeoneinthisregimeis Mloop Mtree/()]TJ /F5 11.955 Tf 9.3 0 Td[(0s)3=2e)]TJ /F7 7.97 Tf 6.59 0 Td[(0jsjf()2 ln0jtj15)]TJ /F4 7.97 Tf 6.59 0 Td[(3D=2)]TJ /F4 7.97 Tf 6.58 0 Td[(S ()]TJ /F5 11.955 Tf 9.3 0 Td[(0s)1=2e)]TJ /F7 7.97 Tf 6.59 0 Td[(0jsjf()/)]TJ /F5 11.955 Tf 21.92 0 Td[(0s2 lnj0tj15)]TJ /F4 7.97 Tf 6.59 0 Td[(3D=2)]TJ /F4 7.97 Tf 6.58 0 Td[(S (4) Therefore,havingcomputedtheexactleadingpowerofsmultiplyingtheexponentialfalloffallowsustoassertthattheplanarone-loopamplitudedominatesoverthetreeamplitudeasshownin( 4 ). 4.2.3RecoveryoftheReggeBehavioratHightInordertoseewhetherthereisamatchingbetweenthehardscatteringandReggeregimesweproceedasfollows.RecallthattheReggelimitisobtainedbytakingstobelargecomparedto0)]TJ /F4 7.97 Tf 8.88 0 Td[(1whilekeepingttobexed,whereasthehardscatteringregimeisobtainedbytakingbothsandtlargecomparedto0)]TJ /F4 7.97 Tf 8.88 0 Td[(1whilemaintainingtheratio=)]TJ /F2 11.955 Tf 9.3 0 Td[(t=sxed.Therefore,weexpectthatwhensislargecomparedtotinthehardscatteringlimit( 4 ),thismatcheswiththeReggelimitwhent0)]TJ /F4 7.97 Tf 8.88 0 Td[(1.TheReggelimit 102

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intheNS+modelwasobtainedin[ 16 ]withtheresult MRegge4)]TJ /F2 11.955 Tf 21.92 0 Td[(g2()]TJ /F5 11.955 Tf 9.3 0 Td[(0s)1+0t\()]TJ /F5 11.955 Tf 9.3 0 Td[(0t)log()]TJ /F5 11.955 Tf 9.3 0 Td[(0s)(t) (4) where(t)isgivenby (t)=Cg2Z10dq q)]TJ /F5 11.955 Tf 9.3 0 Td[( lnq(10)]TJ /F4 7.97 Tf 6.58 0 Td[(D)=2Z0d()]TJ /F5 11.955 Tf 9.3 0 Td[( 2[ln ]00)0t0t [ln ]00(P+X+)]TJ /F2 11.955 Tf 11.95 0 Td[(P)]TJ /F2 11.955 Tf 7.08 1.8 Td[(X)]TJ /F2 11.955 Tf 7.08 -4.93 Td[())]TJ /F2 11.955 Tf 10.49 8.09 Td[(1 4(P+)]TJ /F2 11.955 Tf 11.95 0 Td[(P)]TJ /F2 11.955 Tf 7.08 1.8 Td[()h()]TJ /F5 11.955 Tf 9.3 0 Td[( 2()[ln ]00)0t)]TJ /F2 11.955 Tf 11.95 0 Td[(1i[)]TJ /F2 11.955 Tf 11.29 0 Td[(ln ]00 (4) Asmentionedabove,weneedtotakethelimit0t1in( 4 ).UsingStirling'sapproximation\()]TJ /F5 11.955 Tf 9.3 0 Td[(0t)p 2()]TJ /F5 11.955 Tf 9.3 0 Td[(0t))]TJ /F4 7.97 Tf 6.58 0 Td[(1=2)]TJ /F7 7.97 Tf 6.58 0 Td[(0te0tandthefactthatforlarge0ttheone-looptrajectoryfunctionbecomes[ 16 ] (t)0t[log()]TJ /F5 11.955 Tf 9.3 0 Td[(0t)]1)]TJ /F4 7.97 Tf 6.59 0 Td[(p (4) wehavethat MRegge40t1g2()]TJ /F5 11.955 Tf 9.3 0 Td[(0s)1+0t()]TJ /F5 11.955 Tf 9.3 0 Td[(0t)1=2)]TJ /F7 7.97 Tf 6.58 0 Td[(0te0tlog()]TJ /F5 11.955 Tf 9.3 0 Td[(0s)[log()]TJ /F5 11.955 Tf 9.3 0 Td[(0t)]1)]TJ /F4 7.97 Tf 6.58 0 Td[(p (4) Wenowexpecttorecoverthisresultbytakingthestlimitin( 4 ).Thisamountstotakethe!0limitofF()denedin( 4 )andthenputtingthisbackinto( 4 ).Forconvenience,wewritethisintegralhereagain F()=Z10drZ0drsin2 (r2+2rcos+1)(r2(1)]TJ /F5 11.955 Tf 11.95 0 Td[()2+2r(1)]TJ /F5 11.955 Tf 11.96 0 Td[()cos+1) (4) Thisintegralconvergesinthewholerange<<0butitgetslargerandlargerasapproacheszero.Recallthat=)]TJ /F2 11.955 Tf 9.3 0 Td[(t=ssothisispreciselythelimitwewanttostudy.Byputting=0intheintegrandof( 4 ),weseethattheonlysingularregionistheonegivenbyandr1.ThereisanalternativewaytonotethatthisistherelevantregionintheReggelimit.Recallthatthesaddlepointwhichdominatesinthehighenergylimitisgivenbyx=(1)]TJ /F5 11.955 Tf 12.2 0 Td[())]TJ /F4 7.97 Tf 6.58 0 Td[(1.Fromthedenitions( 4 )wenotethatthe 103

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regionandr1corresponds6tox=rsin2 sin421whichispreciselythelocationofthedominantsaddleas!0.ThisperfectlymatcheswiththefactthattheReggebehavioroftheamplitudeisobtainedfromtheregion23,4forwhichwehavex1)]TJ /F5 11.955 Tf 12.11 0 Td[(32()]TJ /F5 11.955 Tf 12.11 0 Td[(4)csc23.Thus,intheintegrandabove,letusreplace(1)]TJ /F5 11.955 Tf 12.12 0 Td[()byx)]TJ /F4 7.97 Tf 6.58 0 Td[(1fornotationalconvenience.Therefore,sincetherelevantregionforintegralaboveisgivenbyr!x!1,wefocusonthecornerrx,,thus F()Zx+x)]TJ /F7 7.97 Tf 6.59 0 Td[(drZ)]TJ /F7 7.97 Tf 6.59 0 Td[(d()]TJ /F5 11.955 Tf 11.95 0 Td[()2 ((x)]TJ /F2 11.955 Tf 11.95 0 Td[(1)2+x()]TJ /F5 11.955 Tf 11.96 0 Td[()2)((r=x)]TJ /F2 11.955 Tf 11.96 0 Td[(1)2+()]TJ /F5 11.955 Tf 11.96 0 Td[()2)2Z0 (x)]TJ /F2 11.955 Tf 11.95 0 Td[(1)2+x2=)]TJ /F2 11.955 Tf 9.3 0 Td[(2lnj1)]TJ /F2 11.955 Tf 11.96 0 Td[(xj+ln((1)]TJ /F2 11.955 Tf 11.96 0 Td[(x)2+2) (4) Therefore,as!0forxed,wehave F())]TJ /F2 11.955 Tf 28.56 0 Td[(2(ln()]TJ /F5 11.955 Tf 9.3 0 Td[())]TJ /F2 11.955 Tf 11.95 0 Td[(ln(1)]TJ /F5 11.955 Tf 11.96 0 Td[())2ln()]TJ /F5 11.955 Tf 9.3 0 Td[(0s) (4) Puttingthisresultbackinto( 4 )gives G()'4g21 80D=22p)]TJ /F4 7.97 Tf 6.59 0 Td[(120)]TJ /F4 7.97 Tf 6.58 0 Td[(2D)]TJ /F4 7.97 Tf 6.59 0 Td[(2S p)]TJ /F2 11.955 Tf 11.96 0 Td[(1()]TJ /F2 11.955 Tf 9.3 0 Td[(2)1=2ln()]TJ /F5 11.955 Tf 9.3 0 Td[(0s) (4) Theexponentialfactore)]TJ /F7 7.97 Tf 6.59 0 Td[(0jsjf()in( 4 )canalsobewrittenas e)]TJ /F7 7.97 Tf 6.58 0 Td[(0jsjf()=()]TJ /F5 11.955 Tf 9.3 0 Td[()0s(1)]TJ /F5 11.955 Tf 11.95 0 Td[()0s(1)]TJ /F7 7.97 Tf 6.59 0 Td[()=()]TJ /F5 11.955 Tf 9.3 0 Td[())]TJ /F7 7.97 Tf 6.59 0 Td[(0t(1+t=s)0s(1)]TJ /F5 11.955 Tf 11.95 0 Td[()0t (4) whichinthest(!0)limitthenbecomes e)]TJ /F7 7.97 Tf 6.58 0 Td[(0jsjf()!()]TJ /F5 11.955 Tf 9.3 0 Td[())]TJ /F7 7.97 Tf 6.58 0 Td[(0te0t (4) 6Recallthattheoriginal4variablewasrenamedwhichiswhatweusehere 104

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Pluggingalltheseapproximationsbackintothefullhardscatteringamplitudein( 4 )yields MHard4stg21 80D=22p+120)]TJ /F4 7.97 Tf 6.58 0 Td[(2D)]TJ /F4 7.97 Tf 6.59 0 Td[(2S p)]TJ /F2 11.955 Tf 11.95 0 Td[(1()]TJ /F2 11.955 Tf 9.3 0 Td[(2)1=2ln()]TJ /F5 11.955 Tf 9.29 0 Td[(0s)()]TJ /F5 11.955 Tf 9.3 0 Td[())]TJ /F7 7.97 Tf 6.59 0 Td[(0te0t(log0jtj)1)]TJ /F4 7.97 Tf 6.59 0 Td[(p(0jsj)3=2 (4) Finally,replacing=)]TJ /F2 11.955 Tf 9.3 0 Td[(t=shereoneobtains MHard4stg2ln()]TJ /F5 11.955 Tf 9.3 0 Td[(0s)()]TJ /F5 11.955 Tf 9.3 0 Td[(0s)1+0t()]TJ /F5 11.955 Tf 9.3 0 Td[(0t)1=2)]TJ /F7 7.97 Tf 6.59 0 Td[(0te0t(log0jtj)1)]TJ /F4 7.97 Tf 6.58 0 Td[(p (4) whichmatchesexactlywiththeexpectedresult( 4 ).InthecaseD=4andS=0wherewehavefourdimensionalpureYang-Millsinthe!0limitthelogarithmicfactoris(log0jtj))]TJ /F4 7.97 Tf 6.59 0 Td[(8 4.3HighEnergyScatteringintheTypeISuperstringAsabyproductofthehardscatteringstudiesintheprevioussection,weperformhereasimilarstudyforthesuperstring[ 29 ].WestudytheReggeandhardscatteringlimitoftheone-loopamplitudeformasslessopenstringstatesinthetypeItheory.Forhardscatteringwendtheexactcoefcientmultiplyingtheknownexponentialfalloffintermsofthescatteringangle,withoutrelyingonasaddlepointapproximationfortheintegrationoverthecrossratio.Thisbypassestheissuesofestimatingthecontributionsfromatdirections,aswellasthosethatarisefromuctuationsofthegaussianintegrationaboutasaddlepoint.Thisresultallowsforastraightforwardcomputationofthesmall-anglebehaviorofthehardscatteringregimeandwendcompleteagreementwiththeReggelimitathighmomentumtransfer,asexpected.Thehardscatteringlimitofopenstringsatone-loopwasstudiedlongagobyAlessandrini,AmatiandMorel[ 17 ]intheearlydaysofstringtheoryandsubsequentlyby[ 25 30 ].Theyanalyzedthe(2+2)non-planaramplitude(i.e.theannuluswithtwoexternalparticlesattachedtoeachboundary)andfoundthecharacteristicexponentialfalloffnowknownforstringyamplitudes.Thecoefcientmultiplyingthisexponential 105

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behavior,whichcontainsdependenceonthescatteringangle,involvesthewell-knownproblemofinversioninthetheoryofellipticmodularfunctions.Asaconsequenceofthis,theangulardependenceintheaforementionedcoefcientcouldonlybeexpressedintermsofaninniteseries.Forthecaseoftheplanarandnon-orientableamplitudes,GrossandManes[ 25 ]studiedtheirbehavioralsointhehardscatteringlimitandfoundthat,contrarytothe(2+2)non-planarcase,theydonotpossessadominantsaddlepointintheinterioroftheintegrationregion.Moreover,theywereabletoshowthatthedominantcontributionscomefromtheboundariesofthisregion,theonewheretheannulusshrinkstoapointbeingthedominantinthiscase.Thestudyofthexed-anglelimitoftheone-loopamplitudeindifferentsituationshasbeencarriedoutbymanyauthors[ 25 31 33 ],butasfarasweawareof,webelievethattheexactdependenceonthescatteringanglefortheamplitudewestudyherehasnotbeenworkedoutintheliteratureinclosedform.Weorganizethisshortarticleasfollows:insection2wereviewthecalculationoftheReggelimitofthesumoftheplanarandnon-orientablediagramsofthetypeItheory.Forthiswealsocomputeitslargemomentumtransferlimit(jtj!1)inordertomakeacomparisonwiththesmallanglebehaviorofthehardscatteringlimitwhichwealsoreviewinthissection.Insection3,bymakinguseofanidentityfoundin[ 34 ],wecomputetheexactformofthecoefcientthatmultipliestheexponentialfalloffofthehardscatteringamplitude.Thispermitsastraightforwardevaluationofthethehardscatteringamplitudeinthelimitwherets,whichindeedmatcheswiththeReggebehaviorcomputedinsection2. 4.3.1ReggeBehavioratOne-LoopWebeginbycomputingtheReggelimit,i.e.wetakes!withtheldxedoftheone-loopamplitudefortypeIopensuperstrings.ThedetailsofthecalculationarebasicallythesameastheonesfortheNS+stringcomputedin[ 16 ]withtheonly 106

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differencebeingthenatureofthecancellationofdivergencesduetothepropagationofclosedstringtachyonsanddilatons.IntheNS+model,theremnantsofclosedstringtachyondivergenceswerecanceledbytheinclusionofacountertermwhich,afteranalyticcontinuationusingamomentumregulator(theGNSregulator),turnedouttobezerointheReggelimit.Thewould-besubleadingdivergencesduetoclosedstringdilatonsweresimplyabsentwiththeinclusionofDp-branesaslongasp<7.TheamplitudeforfourmasslessvectorstatesismuchsimplerinthesuperstringcomparedtotheNS+model,becauseintheformerthefullpolarizationstructurecanbefactoredoutoftheloopintegration,whereasinthelattereachcombinationofpolarizationvectorsmustbeworkedoutseparately.FortheSO(32)gaugegrouptheplanarandnon-orientableone-loopdiagramscombinetogiveaniteexpression[ 35 ]andwefocusourattentiononthiscaseinthisarticle.Theamplitudeforeachdiagram(planarandnon-orientable)wascomputedlongtimeago(seeforinstance[ 15 ])andfortheSO(32)gaugegrouptheycanbecombinedas AP+AN=163g4GPKZ10dq qF(q2))]TJ /F2 11.955 Tf 11.96 0 Td[(F()]TJ /F2 11.955 Tf 9.3 0 Td[(q2) (4) with F(q2)=ZR3Yi=1diYi
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region23and4.Writing Yi
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correctiontotheopenstringReggetrajectory.Since ((t)+)s(t)+s(t)+s(t)logs+s(t), (4) thenewtrajectoryis(t)new=1+0t+g2(t).GiventhatwearealsointerestedinrecoveringtheReggebehaviorfromthehard-scatteringlimit,weneedtoextractthelargetlimitof(t).Inordertodosore-writetheintegralas (t)0tZ10dq qZ0e0tln()]TJ /F7 7.97 Tf 6.59 0 Td[( 2[ln ]00)[)]TJ /F2 11.955 Tf 11.29 0 Td[(ln ]00)]TJ /F4 7.97 Tf 11.18 0 Td[(1)]TJ /F2 11.955 Tf 11.95 0 Td[(e0tln()]TJ /F7 7.97 Tf 6.59 0 Td[( 2N[ln N]00)[)]TJ /F2 11.955 Tf 11.29 0 Td[(ln N]00)]TJ /F2 11.955 Tf 11.95 0 Td[(1 (4) thus,theintegralisdominatedbythecriticalpointsofln()]TJ /F5 11.955 Tf 9.29 0 Td[( 2[ln ]00)andln()]TJ /F5 11.955 Tf 9.29 0 Td[( 2N[ln N]00).Noticethatnowweonlyhaveatwo-dimensionalintegrationregion,forwhichthecriticalpointsshouldbeeasiertoanalyzeinprinciple.Theleadingcontributioncomesfromtheq0region,thuswealsoneedtheapproximations: [)]TJ /F2 11.955 Tf 11.29 0 Td[(ln ]00[)]TJ /F2 11.955 Tf 11.3 0 Td[(ln N]00csc2ln()]TJ /F5 11.955 Tf 9.3 0 Td[( 2[ln ]00))]TJ /F2 11.955 Tf 23.91 0 Td[(ln()]TJ /F5 11.955 Tf 9.3 0 Td[( 2N[ln N]00)16q2sin4 (4) Noticethattheregions0,alsoproduceimportantcontributionstotheintegralforlargetandneedtobeanalyzedseparately.Forthispurposewewouldneedthecorrespondingasymptoticexpressionsforthefunctions and Nandtointegrateoverthefullrange0
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Notethatwehavealsodenedtheintegralabovebyanalyticalcontinuation(t!it)asin[ 25 ].Therefore,wewishtoobtainthelargejtjbehavioroftheexpression (t)i0tZ)]TJ /F7 7.97 Tf 6.59 0 Td[(dsin2Z0dq qeitaq2)]TJ /F2 11.955 Tf 11.95 0 Td[(e)]TJ /F4 7.97 Tf 6.58 0 Td[(itaq2 (4) forxedwitha=16sin4.Wehavealsointroducedthecutofftostressthefactthatweneedtoexaminethecontributionsfromtheregionswhere0,separately.Performingthechangeatq2uwehave (t)i0tZ)]TJ /F7 7.97 Tf 6.59 0 Td[(sin2diZ2ta0du usinu (4) Sinceissmallbutxedwecantaketheupperlimitoftheuintegraltobe1inthejtj!1limit.Alsointhislimitthedependenceintheuintegraldisappearswhichallowsustosendtozero,thus (t))]TJ /F5 11.955 Tf 28.56 0 Td[(0tZ0sin2dZ10du usinu=)]TJ /F5 11.955 Tf 9.3 0 Td[(0t2 4 (4) Therefore,continuingbacktot!)]TJ /F2 11.955 Tf 24.57 0 Td[(itwehave (t)i0tast! (4) Finally,ast!,combiningequations( 4 )and( 4 )yields AP+AMi()]TJ /F5 11.955 Tf 9.3 0 Td[(0s)1+0t\()]TJ /F5 11.955 Tf 9.3 0 Td[(0t)ln()]TJ /F5 11.955 Tf 9.3 0 Td[(0s)0t=i()]TJ /F5 11.955 Tf 9.3 0 Td[(0s)1+0t\(1)]TJ /F5 11.955 Tf 11.95 0 Td[(0t)ln()]TJ /F5 11.955 Tf 9.29 0 Td[(0s) (4) WecoulduseStirling'sapproximation\(1)]TJ /F5 11.955 Tf 13.13 0 Td[(0t)p 2()]TJ /F5 11.955 Tf 9.3 0 Td[(0t)1=2)]TJ /F7 7.97 Tf 6.58 0 Td[(0te0tvalidfor)]TJ /F5 11.955 Tf 9.3 0 Td[(0t1,whichyields AP+AMi()]TJ /F5 11.955 Tf 9.3 0 Td[(0s)1+0t()]TJ /F5 11.955 Tf 9.3 0 Td[(0t)1=2)]TJ /F7 7.97 Tf 6.59 0 Td[(0te0tln()]TJ /F5 11.955 Tf 9.3 0 Td[(0s) (4) 110

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Toconclude,wetakeamomenttoanalyzetheregionswhere0,whicharealsoimportantasjtjbecomeslarge.Usingthefollowingexpressionforthelogarithmof ln ()=lnsin+21Xn=11 mq2m 1)]TJ /F2 11.955 Tf 11.95 0 Td[(q2m(1)]TJ /F2 11.955 Tf 11.95 0 Td[(cos2m) (4) onecanseethat ln()]TJ /F5 11.955 Tf 9.3 0 Td[( 2[ln ]00))]TJ /F2 11.955 Tf 23.91 0 Td[(ln()]TJ /F5 11.955 Tf 9.3 0 Td[( 2N[ln N]00)O(4) (4) Thus,themaincontributionatlargetcomesfromtheregionwhereisoftheorderof()]TJ /F5 11.955 Tf 9.3 0 Td[(0t))]TJ /F4 7.97 Tf 6.59 0 Td[(1=4.Aroughestimationfromtheseregionsgives(t)()]TJ /F5 11.955 Tf 9.3 0 Td[(0t))]TJ /F4 7.97 Tf 6.58 0 Td[(3=4whichissubleadingwithrespecttotheq0contributiongivenin( 4 ). 4.3.2HardScatteringatOneLoopThehigh-energylimitatxedscatteringanglefortheone-loopamplitudewasrstcomputedby[ 17 ]intheearlydaysofstringtheoryinthecontextoftheolddualresonancemodels.There,thecomputationwasdoneforthenon-planaramplitudewhichhadadominantsaddlepointintheinterioroftheintegrationregion.In[ 25 ],GrossandManesshowedthatonlythe(2+2)non-planaramplitude(i.e.theamplitudewithtwoparticlesoneachboundaryoftheannulus)hasasaddlepointintheinterioroftheregionofintegration.Theplanar,non-orientableandthe(3+1)non-planaramplitudesdonotpossessadominantsaddlepointintheinterior,butpointsintheboundaryoftheregiondogivesub-dominantcontributions(withrespecttothe(2+2)non-planar)fromtheboundariesoftheregionofintegration.Theyalsoshowedthattheleadingcontributionforthesumoftheplanarandnon-orientablediagramscomesfromtheregionwhereq0andthecrossratioxsin2sin43 sin42sin3isapproximately(1+t=s))]TJ /F4 7.97 Tf 6.59 0 Td[(1inequation( 4 ).Webeginthissectionbyre-calculatingtheleadingbehaviorknownintheliteratureusingthesaddlepointapproximationforthecross-ratioalthoughusingadifferentsetofintegrationvariables[ 1 ]where2!x,3!rsin43=sin3.Starting 111

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fromequations( 4 )and( 4 )therelevantfactorintheintegrandinthislimitis Yi
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resultwhichwealreadyencounteredin( 4 ).Allinall,forthecoefcientof1423,weobtain: AP+AMsueE0s )]TJ /F2 11.955 Tf 9.3 0 Td[(2 (1)]TJ /F5 11.955 Tf 11.96 0 Td[()3()]TJ /F5 11.955 Tf 9.3 0 Td[(0s))]TJ /F4 7.97 Tf 6.59 0 Td[(1=2F()s2(1+t=s)eE0()]TJ /F5 11.955 Tf 9.3 0 Td[(0t)1=2()]TJ /F5 11.955 Tf 9.3 0 Td[(0s))]TJ /F4 7.97 Tf 6.58 0 Td[(1=2(1+t=s))]TJ /F4 7.97 Tf 6.59 0 Td[(3=2()]TJ /F5 11.955 Tf 9.3 0 Td[(0s))]TJ /F4 7.97 Tf 6.58 0 Td[(1=2F()()]TJ /F5 11.955 Tf 9.3 0 Td[(0s)3=2eE0()]TJ /F5 11.955 Tf 9.3 0 Td[()1=2(1)]TJ /F5 11.955 Tf 11.95 0 Td[())]TJ /F4 7.97 Tf 6.58 0 Td[(1=2F() (4) whichshowstheusualexponentialsuppressioneE0(E0>0)factorandwherethefunctionF()isgivenby F()=Z10drZ0drsin2 (r2+2rcos+1)(r2(1)]TJ /F5 11.955 Tf 11.95 0 Td[()2+2r(1)]TJ /F5 11.955 Tf 11.96 0 Td[()cos+1) (4) Anfewremarksareimportanttonoteaboutthisintegral.Since<<0itisconvergentinthisentirerangebutitdivergesfor=0.Asgetsclosertozero,theintegralbecomeslargerandlargerandweneedtoestimatehowitdivergesinordertoextractthecorrectsmallbehavior.Aswewillshowinthenextsection,wehavethat F())]TJ /F2 11.955 Tf 21.91 0 Td[(2ln()]TJ /F5 11.955 Tf 9.29 0 Td[()+2ln(1)]TJ /F5 11.955 Tf 11.95 0 Td[()2ln()]TJ /F5 11.955 Tf 9.3 0 Td[(0s)forst (4) whichprovidesthelogarithmthatappearsintheReggelimitoftheamplitudein( 4 ).Writingtheexponentialfactoras eE0=()]TJ /F5 11.955 Tf 9.3 0 Td[(0s)0t()]TJ /F5 11.955 Tf 9.3 0 Td[(0t))]TJ /F7 7.97 Tf 6.59 0 Td[(0t(1+t=s)0s+0t (4) wehave AP+AMi()]TJ /F5 11.955 Tf 9.3 0 Td[(0s)1+0t()]TJ /F5 11.955 Tf 9.29 0 Td[(0t)1=2)]TJ /F7 7.97 Tf 6.58 0 Td[(0t(1+t=s)0s+0t)]TJ /F4 7.97 Tf 6.59 0 Td[(1=2F() (4) whichcompletesthehardscatteringlimitoftheone-loopamplitude. 4.3.3RecoveryoftheReggeLimitThehigh-energybehavioratxedanglegiveninEq.( 4 )usesagaussianapproximationaroundthedominantsaddlepointgivenbyxc=(1)]TJ /F5 11.955 Tf 12.98 0 Td[())]TJ /F4 7.97 Tf 6.58 0 Td[(1.Wewill 113

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nowcalculatethislimitusingadifferentmethodwhichdoesnotrequirethegaussianapproximationbutinsteadwewillcomputetheintegraloverthexvariableinanexactclosedform.However,westillneedtoapproximatetheexponentforsmallqbutthisisnottooserioussincethisistheonlyplaceintheqintegrationwherethereisdominantcriticalpoint[ 25 ].Onecouldregardthecalculationweperforminthissectionasacomputationofthegaussianapproximationincludingallthepossibleuctuationsaroundthesaddle.Thisallowsustobypasstheissueofcomputingthecontributionscominganyotherregioninthekintegrationssincewewillbecomputingthistripleintegralinexactform.Startingfrom( 4 ),weobtain Yi
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gaugegroup.Wesimplyquotetheanswerhere I=ZYkdkx)]TJ /F7 7.97 Tf 6.59 0 Td[(0s(1)]TJ /F2 11.955 Tf 11.96 0 Td[(x))]TJ /F7 7.97 Tf 6.59 0 Td[(0t/1 0@ @00\()]TJ /F5 11.955 Tf 9.3 0 Td[(0s)\()]TJ /F5 11.955 Tf 9.3 0 Td[(0t) \(1)]TJ /F5 11.955 Tf 11.95 0 Td[(0s)]TJ /F5 11.955 Tf 11.96 0 Td[(0t) (4) Usingthisandtheresultfortheintegraloverqgivenineq.( 4 )wehave AP+ANi02su1 0@ @002\()]TJ /F5 11.955 Tf 9.3 0 Td[(0s)\()]TJ /F5 11.955 Tf 9.29 0 Td[(0t) \(1)]TJ /F5 11.955 Tf 11.96 0 Td[(0s)]TJ /F5 11.955 Tf 11.96 0 Td[(0t) (4) Wecannowtakethelimits,t!holdingt=sxeddirectlyinsidethebracketstoobtain AP+ANi02su1 0@ @0h02()]TJ /F5 11.955 Tf 9.3 0 Td[(0s))]TJ /F4 7.97 Tf 6.58 0 Td[(1+0t()]TJ /F5 11.955 Tf 9.3 0 Td[(0t))]TJ /F4 7.97 Tf 6.59 0 Td[(1=2)]TJ /F7 7.97 Tf 6.58 0 Td[(0t(1+t=s))]TJ /F4 7.97 Tf 6.58 0 Td[(1=2+0s+0tii()]TJ /F5 11.955 Tf 9.3 0 Td[(0s)1=2()]TJ /F5 11.955 Tf 9.3 0 Td[())]TJ /F4 7.97 Tf 6.59 0 Td[(1=2)]TJ /F7 7.97 Tf 6.59 0 Td[(0t(1)]TJ /F5 11.955 Tf 11.96 0 Td[()1=2+0s+0t[1+20s(ln()]TJ /F5 11.955 Tf 9.3 0 Td[()+(1)]TJ /F5 11.955 Tf 11.95 0 Td[()ln(1)]TJ /F5 11.955 Tf 11.96 0 Td[())] (4) Takingagain)]TJ /F5 11.955 Tf 9.3 0 Td[(0s1,weendupwith AP+ANi()]TJ /F5 11.955 Tf 9.3 0 Td[(0s)3=2()]TJ /F5 11.955 Tf 9.3 0 Td[())]TJ /F4 7.97 Tf 6.58 0 Td[(1=2(1)]TJ /F5 11.955 Tf 11.95 0 Td[()1=2e0s[ln()]TJ /F7 7.97 Tf 6.59 0 Td[()+(1)]TJ /F7 7.97 Tf 6.58 0 Td[()ln(1)]TJ /F7 7.97 Tf 6.59 0 Td[()][ln()]TJ /F5 11.955 Tf 9.3 0 Td[()+(1)]TJ /F5 11.955 Tf 11.96 0 Td[()ln(1)]TJ /F5 11.955 Tf 11.95 0 Td[()]TorecovertheReggebehaviorwetakestabove.Theexponentialbecomes e0s[ln()]TJ /F7 7.97 Tf 6.58 0 Td[()+(1)]TJ /F7 7.97 Tf 6.59 0 Td[()ln(1)]TJ /F7 7.97 Tf 6.59 0 Td[()]=()]TJ /F5 11.955 Tf 9.3 0 Td[())]TJ /F7 7.97 Tf 6.59 0 Td[(0t(1)]TJ /F5 11.955 Tf 11.95 0 Td[()0s+0t()]TJ /F5 11.955 Tf 9.3 0 Td[(0s)0t()]TJ /F5 11.955 Tf 9.3 0 Td[(0t))]TJ /F7 7.97 Tf 6.59 0 Td[(0te0t (4) andthelastfactorbecomes [ln()]TJ /F5 11.955 Tf 9.3 0 Td[()+(1)]TJ /F5 11.955 Tf 11.95 0 Td[()ln(1)]TJ /F5 11.955 Tf 11.96 0 Td[()]ln()]TJ /F5 11.955 Tf 9.3 0 Td[()=)]TJ /F2 11.955 Tf 9.3 0 Td[(t=s[ln()]TJ /F5 11.955 Tf 9.3 0 Td[(0t))]TJ /F2 11.955 Tf 11.96 0 Td[(ln()]TJ /F5 11.955 Tf 9.29 0 Td[(0s)]()]TJ /F5 11.955 Tf 9.29 0 Td[(0t)()]TJ /F5 11.955 Tf 9.3 0 Td[(0s))]TJ /F4 7.97 Tf 6.59 0 Td[(1ln()]TJ /F5 11.955 Tf 9.3 0 Td[(0s) (4) Therefore,theReggelimitathightis AP+ANi()]TJ /F5 11.955 Tf 9.3 0 Td[(0s)3=2()]TJ /F5 11.955 Tf 9.3 0 Td[())]TJ /F4 7.97 Tf 6.58 0 Td[(1=2()]TJ /F5 11.955 Tf 9.29 0 Td[(0s)0t()]TJ /F5 11.955 Tf 9.3 0 Td[(0t))]TJ /F7 7.97 Tf 6.58 0 Td[(0te0t()]TJ /F5 11.955 Tf 9.3 0 Td[(0t)()]TJ /F5 11.955 Tf 9.3 0 Td[(0s))]TJ /F4 7.97 Tf 6.58 0 Td[(1ln()]TJ /F5 11.955 Tf 9.29 0 Td[(0s)i()]TJ /F5 11.955 Tf 9.3 0 Td[(0s)1+0t()]TJ /F5 11.955 Tf 9.3 0 Td[(0t)1=2)]TJ /F7 7.97 Tf 6.58 0 Td[(0te0tln()]TJ /F5 11.955 Tf 9.29 0 Td[(0s) (4) whichisexactlytheresultwefoundin( 4 ). 115

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Wenishthissectionbyshowingthattheresultin( 4 )canalsobeobtainedfromtheapproximateexpressionin( 4 )byanalyzingthesmallbehaviorofF()asanticipatedin( 4 ).WebelieveitisinstructivetodothisbecausewearealsointerestedinthesmallbehaviorofthehardscatteringlimitoftheNS+stringmodelinthecontextof[ 8 9 16 ]wherewecannotaffordtheluxuryofhavinganexactexpressionforthecoefcientoftheexponentialfalloff.Forconveniencewewritethisintegralhereagain F()=Z10drZ0drsin2 (r2+2rcos+1)(r2(1)]TJ /F5 11.955 Tf 11.95 0 Td[()2+2r(1)]TJ /F5 11.955 Tf 11.96 0 Td[()cos+1) (4) Asmentionedabove,F()divergesas!0.Theonlysingularregioninthislimitisandr1.Itisstraightforwardtoseethisbyrecallingthat,intermsofthecrossratiox,thethedominantsaddlepointisgivenbyxc=(1)]TJ /F5 11.955 Tf 12.55 0 Td[())]TJ /F4 7.97 Tf 6.59 0 Td[(1.ThisperfectlymatcheswiththefactthattheReggebehavioroftheamplitudeisobtainedfromtheregion23,4sincex32()]TJ /F5 11.955 Tf 12.36 0 Td[(4)whichgivestheleadingbehavior[ 16 19 ].Thus,theReggelimitoccurswhenx!1.Thereforeinthesmallscatteringanglelimit,theintegralaboveissingularwhere,rx,thus F()Zx+x)]TJ /F7 7.97 Tf 6.59 0 Td[(drZ)]TJ /F7 7.97 Tf 6.59 0 Td[(d()]TJ /F5 11.955 Tf 11.95 0 Td[()2 ((x)]TJ /F2 11.955 Tf 11.95 0 Td[(1)2+x()]TJ /F5 11.955 Tf 11.96 0 Td[()2)((r=x)]TJ /F2 11.955 Tf 11.96 0 Td[(1)2+()]TJ /F5 11.955 Tf 11.96 0 Td[()2)2Z0 (x)]TJ /F2 11.955 Tf 11.95 0 Td[(1)2+x2=)]TJ /F2 11.955 Tf 9.3 0 Td[(2lnj1)]TJ /F2 11.955 Tf 11.96 0 Td[(xj+ln((1)]TJ /F2 11.955 Tf 11.96 0 Td[(x)2+2) (4) Therefore,as!0forxed,wehave F())]TJ /F2 11.955 Tf 28.56 0 Td[(2(ln()]TJ /F5 11.955 Tf 9.3 0 Td[())]TJ /F2 11.955 Tf 11.95 0 Td[(ln(1)]TJ /F5 11.955 Tf 11.96 0 Td[())2ln()]TJ /F5 11.955 Tf 9.3 0 Td[(0s) (4) asanticipatedin( 4 ). 116

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CHAPTER5CONCLUSIONSUsingtheapproachpioneeredin[ 1 2 ],whereanalyticcontinuationfortachyonscatteringamplitudesisachievedbysuspendingtotalmomentumconservationbyaniteamountp(wecallthistheGNSregulator),inchapter 2 weextendedtheirworktoplanarone-loopfourmasslessvectorstatesintheNS+stringmodelandwealsosuggestthesystematicsofextensiontoanarbitrarynumberofexternalstates.Thecountertermsnecessarytocancelthedivergencesinthefourgluonamplitudeareallproportionaltothetreeamplitudeafteranalyticcontinuationtop!0,thereforeoneisleftwithaniteexpressionaftercouplingconstantrenormalization.WebelievethattheextensiontoM-gluonisstraightforwardandfollowsthesamesystematics.Inchapter 3 haveunderstoodonemoreaspectabouttherelationbetweenthedynamicsofopenstringsandtheircorrespondinggaugeeldtheorylimit,namelywehaveextractedtheleadingReggetrajectory(t)=1+0t+(t)fortheplanaramplitudeofmasslessvectoropenstringsstateswhoseendpointslieonastackofNDp-branesincludingone-loopcorrections.Whenstudyingthet!0limit,besidesconrmingthatthecorrectiongoestozeroasareectionofthemasslessnessoftheopenstringgluon,wehavealsoshownthatitbehavesas)]TJ /F2 11.955 Tf 23.66 0 Td[(Cg2(0t)(D)]TJ /F4 7.97 Tf 6.58 0 Td[(4)=2=(D)]TJ /F2 11.955 Tf 12.66 0 Td[(4)whichispreciselytheresultobtainedinDdimensionalgaugetheory.IntheD!4case,whichcorrespondstoaD3-branewherethedynamicsoftheopenstringsdescribethe4-dimensionalSU(N)gauge-theoryin'tHooft'splanarlimit,thet!0limitof(t)is D!4)]TJ /F2 11.955 Tf 26.57 8.09 Td[(g2 422 D)]TJ /F2 11.955 Tf 11.95 0 Td[(4+ln()]TJ /F5 11.955 Tf 9.29 0 Td[(0t) (5) whichpreciselymatchestheresultsknownfromone-loopgaugetheorycalculations.Wehavealsostudiedthelimitt!of(t)whereweobtained: (t)'g20(820)1)]TJ /F4 7.97 Tf 6.59 0 Td[(D=2 24)]TJ /F2 11.955 Tf 11.95 0 Td[(5D=2)]TJ /F2 11.955 Tf 11.95 0 Td[(2S0t2 lnj0tj24)]TJ /F4 7.97 Tf 6.59 0 Td[(5D=2)]TJ /F4 7.97 Tf 6.58 0 Td[(2S (5) 117

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SincethemaximumnumberofscalarscirculatingintheloopisSmax=10)]TJ /F2 11.955 Tf 12.91 0 Td[(Dthe1-loopcorrectionstayssmallerthanthetreetrajectoryforarbitrarilylarge)]TJ /F5 11.955 Tf 9.3 0 Td[(0taslongasD<8.WehaveseenonceagaintheusefulnessoftheGNSregulator1whendealingwiththespuriousdivergencesencounteredintheintegrationover.Thesubtraction( 3 )isverysimple,anditisactuallyzeroafteranalyticcontinuationtop=0.Whenweorganizedtheintegrandintheformofequations( 3 )and( 3 ),therecognitionthatthequantitiesXdenedin( 3 )and( 3 )wereindependentoftheintegrationvariablesignicantlyimprovedthenumericalcalculationsperformedinSection7.Thexed-anglehighenergylimitofopenstringtheorywasstudiedlongagobyAlessandrini,AmatiandMorel[ 17 ]inthecontextoftheolddualresonancemodels.GrossandManes[ 25 ]deepenedthestudybygivingtheleadingasymptoticbehaviorforanarbitrarynumberofloops.Theirmainresultwasthat,similarlytothestudyofclosedbosonicstringsin[ 26 ],theamplitudeforfourexternaltachyonshadasaddlepointatallgenusandthusitcouldbeevaluatedatthatpointgivingananswerforanygenus.Alinearrelationamongtheamplitudesinthehighenergyregimeforallexternalstateswasconjecturedin[ 27 ].However,ithasbeenpointedoutbyMoellerandWest[ 28 ],usingtheso-calledgrouptheoreticapproach,thatthelinearrelationssuggestedin[ 27 ]maynotbevalidforexternalstringstatesotherthantachyons.Adetailedone-loopanalysisfortheopenbosonicstringisalsoprovidedinreference[ 28 ].AnextensionoftheseresultstotheNeveu-Schwarzsectorofthesuperstringisstilllackingintheliterature,althoughtheauthorsof[ 28 ]alsopointoutthatthisextensionshouldberathersimilartotheresultsfoundbythemforthebosonicstring.Theextensiontothesuperstringcaseiscertainlyoneoftheprojectswewouldliketopursueinthefuture. 1SeeAppendixAin[ 7 ]foranapplicationofthisregulatorinconventionaleldtheory. 118

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Becauseofthesefactsandhavinginmindthatwewanttoperformthemulti-loopsumoftheNS+(planar)openstringdiagrams,itbehoovesustocheckthehigh-energylimitatxedscatteringangleforexternalmasslessstatesinthismodelwhichwedoinchapter 4 .There,wecarriedoutthecomputationatone-loopwithfourexternalmasslessvectorstates(thesebecomethegluonsas0!0)ndingagreementwiththeexpectedexponentialfalloffof[ 25 ].Inaddition,wealsoprovidethecompletekinematicfactorthatmultipliestheexponentialsuppressionintermsofthescatteringangle.Becauseofthedirectrelationbetweenthescatteringangleandtheratio=)]TJ /F2 11.955 Tf 9.3 0 Td[(t=s,thiscomputedfactorallowsustocomparethehardscatteringregimewiththeReggeregimewhent1=0whichoccursfor0.Throughacarefulanalysisofthe0limitfortheaforementionedprefactorwewereabletorecovertheReggeregimeathighmomentumtransfer(t1=0).Finally,bystudyingthehardscatteringlimitofthesumoftheone-loopplanarandnon-orientablediagramsforthetype1superstring,wefoundtheexactdependenceinthescatteringanglethatmultipliestheknownexponentialsuppressionathighenergies[ 29 ],i.e.fors,t1=0.Thisavoidstheissueofhavingtoestimatethecontributionsfromatdirectionsintheangularintegralsandtheuctuationsaroundthesaddlepoint,sincewehaveatourdisposalanexactresultfromtheangularintegralinaclosedform.Thisallowedustocompareboth,thehardscatteringandReggeregimesoftheamplitude,sincetheyshouldcoincideinthelimitofhigh-momentumtransferofthelatterregime.Weindeedconrmedthatthismatchingoccursbymakinguseoftheclosedformoftheangularintegralsgivenin( 4 ).Wewerealsoabletoobtainthisresultfromtheapproximateexpression( 4 )byanalyzingthebehavioroftheintegralF()in( 4 )as!0.AnimmediateextensionofthisworkwouldbetoallowtheopenstringstobeattachedtosmallerdimensionalDp-brane(hereweconsideredthecaseofaspace-llingD-brane)wherethesmallqbehaviorcanbeanalyzedseparatelyforthe 119

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planarandnon-orientablediagramssincetheamplitudesareniteaslongasp<8.Itwouldalsobeinterestingtocheckifourresultsherecouldbealsoappliedtothesituationstudiedin[ 33 ]wheretheyanalyzedthecasewherethetwocollidingopenstringslivedondifferentD-branesseparatedbyaxeddistance. 120

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APPENDIXALOGARITHMICCOUNTERTERMSAsmentionedinchapter 3 ,theexpressionfortheBcountertermismorecumbersomethanthelinearcountertermbecauseitneedstobeadaptedtoeveryedgethatcorrespondtologarithmicspuriousdivergences.Herewelisteachoneofthem: B1=1 40(s+t)42)]TJ /F7 7.97 Tf 6.59 0 Td[(0s43)]TJ /F7 7.97 Tf 6.59 0 Td[(0t)]TJ /F4 7.97 Tf 6.58 0 Td[(132(P(4))-222(P(4)C)(1+0t))]TJ /F4 7.97 Tf 6.59 0 Td[(132++)]TJ /F5 11.955 Tf 5.48 -9.69 Td[(2+(4))]TJ /F5 11.955 Tf 11.95 0 Td[(2+(4)C)]TJ /F5 11.955 Tf 12.95 -9.69 Td[(0t(1+0t))]TJ /F4 7.97 Tf 6.59 0 Td[(132+02s2)]TJ /F4 7.97 Tf 6.59 0 Td[(143)]TJ /F5 11.955 Tf 11.96 0 Td[(02(s+t)2)]TJ /F4 7.97 Tf 6.58 0 Td[(142 (A) B2=1 4()]TJ /F5 11.955 Tf 11.96 0 Td[(3)0(s+t))]TJ /F7 7.97 Tf 6.58 0 Td[(0s43()]TJ /F5 11.955 Tf 11.96 0 Td[(4))]TJ /F7 7.97 Tf 6.59 0 Td[(0t)]TJ /F4 7.97 Tf 6.59 0 Td[(1(P(2))-222(P(2)C)(1+0t)()]TJ /F5 11.955 Tf 11.95 0 Td[(4))]TJ /F4 7.97 Tf 6.59 0 Td[(1++)]TJ /F5 11.955 Tf 5.48 -9.69 Td[(2+(2))]TJ /F5 11.955 Tf 11.95 0 Td[(2+(2)C)]TJ /F5 11.955 Tf 12.95 -9.69 Td[(0t(1+0t)()]TJ /F5 11.955 Tf 11.95 0 Td[(4))]TJ /F4 7.97 Tf 6.59 0 Td[(1+02s2)]TJ /F4 7.97 Tf 6.59 0 Td[(143)]TJ /F5 11.955 Tf 11.95 0 Td[(02(s+t)2()]TJ /F5 11.955 Tf 11.95 0 Td[(3))]TJ /F4 7.97 Tf 6.59 0 Td[(1 (A) B3=1 4()]TJ /F5 11.955 Tf 11.96 0 Td[(42)0(s+t))]TJ /F7 7.97 Tf 6.59 0 Td[(0s2()]TJ /F5 11.955 Tf 11.95 0 Td[(4))]TJ /F7 7.97 Tf 6.59 0 Td[(0t)]TJ /F4 7.97 Tf 6.58 0 Td[(1(P(3))-222(P(3)C)(1+0t)()]TJ /F5 11.955 Tf 11.95 0 Td[(4))]TJ /F4 7.97 Tf 6.59 0 Td[(1++)]TJ /F5 11.955 Tf 5.48 -9.68 Td[(2+(3))]TJ /F5 11.955 Tf 11.95 0 Td[(2+(3)C)]TJ /F5 11.955 Tf 12.95 -9.68 Td[(0t(1+0t)()]TJ /F5 11.955 Tf 11.95 0 Td[(4))]TJ /F4 7.97 Tf 6.59 0 Td[(1+02s2)]TJ /F4 7.97 Tf 6.59 0 Td[(12)]TJ /F5 11.955 Tf 11.95 0 Td[(02(s+t)2()]TJ /F5 11.955 Tf 11.95 0 Td[(42))]TJ /F4 7.97 Tf 6.58 0 Td[(1 (A) B4=1 40(s+t)3)]TJ /F7 7.97 Tf 6.59 0 Td[(0s2)]TJ /F7 7.97 Tf 6.59 0 Td[(0t)]TJ /F4 7.97 Tf 6.58 0 Td[(132(P(4))-222(P(4)C)(1+0t))]TJ /F4 7.97 Tf 6.59 0 Td[(132++)]TJ /F5 11.955 Tf 5.48 -9.68 Td[(2+(4))]TJ /F5 11.955 Tf 11.95 0 Td[(2+(4)C)]TJ /F5 11.955 Tf 12.95 -9.68 Td[(0t(1+0t))]TJ /F4 7.97 Tf 6.59 0 Td[(132+0s2)]TJ /F4 7.97 Tf 6.59 0 Td[(12)]TJ /F5 11.955 Tf 11.96 0 Td[(02(s+t)2)]TJ /F4 7.97 Tf 6.59 0 Td[(13 (A) wherethedenitionsforthe()functionsisfoundinequations( 2 )and( 2 )and P()1 4csc2)]TJ /F6 7.97 Tf 16.35 14.95 Td[(1Xn=1n2q2n 1)]TJ /F2 11.955 Tf 11.96 0 Td[(q2ncos2n (A) ThesubscriptCinP()CissimplyashorthandforP()evaluatedatq=0,i.e.P()C1 4csc2. 121

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Notethat,becauseoftheformoftheintegrandsforthesecounterterms,noneofthemissingularinthe4,23regionwhichisthedominantregioninthelargeslimitwiththeldxed,thereforetheywillnotcontributetotheone-loopcorrectiontotheReggetrajectory.Thisiswhyitwasnotnecessarytoincludethemin[ 16 ].Thefactthattheyarealsonon-singularintheremainingedge,namely20,34suggeststhattheydonotcontributetotheregimewheretislargeandsisheldxedeither. 122

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APPENDIXBANIDENTITYUSINGTHEGNSREGULATORInthissectionweshowthatthecancellationthatoccursin( 2 )isoforderO(p2).Thisfactensuresthatthefermionicpartoflogarithmiccountertermsreallyvanishesafteranalyticcontinuationtop=0.Hadtheexpressionin( 2 )beenO(p)instead,the1=pfactorin( 2 )wouldhaverenderedanonzerocontribution,whichwouldhaveprobablyspoiledtheuseoftheGNSregulatortoshowrenormalizability.WestartbywritingallthekinematicalinvariantsintermsoftheMandelstamvariablessandtwhentotalmomentumconservationisnotsatised,butinsteadwehavePiki=p,thisis 2k3k4=)]TJ /F2 11.955 Tf 9.29 0 Td[(s+2p(k1+k2)+p22k2k4=s+t)]TJ /F2 11.955 Tf 11.95 0 Td[(2k2p (B) withsimilarexpressionsfor2k1k4and2k1k3.Using( B )wehavethat( 2 ),aftersomealgebrabecomes: 0s\()]TJ /F5 11.955 Tf 9.3 0 Td[(0s)\()]TJ /F5 11.955 Tf 9.3 0 Td[(0t) \()]TJ /F5 11.955 Tf 9.29 0 Td[(0s)]TJ /F5 11.955 Tf 11.95 0 Td[(0t)20k4p20k2p 0s+0t)]TJ /F5 11.955 Tf 11.95 0 Td[(0p220k2p 0s+0t (B) whichisoforderO(p2)asrequired. 123

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REFERENCES [1] A.NeveuandJ.Scherk,Nucl.Phys.B36(1972)317. [2] P.Goddard,NuovoCim.A4(1971)349. [3] G.'tHooft,Nucl.Phys.B72(1974)461. [4] E.Witten,Nucl.Phys.B160,57(1979). [5] S.Coleman,/N,.ProceedingsofInt.SchoolofSubnuclearPhysics,PointlikeStructuresInsideandOutsideHadrons,Erice,Italy,Jul31-Aug10,1979. [6] J.Preskill, QCDLectureNotes, http://www.theory.caltech.edu/preskill/notes.html [7] C.B.Thorn,Phys.Rev.D78(2008)085022.[arXiv:0808.0458[hep-th]]. [8] C.B.Thorn,Phys.Rev.D80,086010(2009).[arXiv:0906.3742[hep-th]]. [9] C.B.Thorn,Phys.Rev.D82,065009(2010).[arXiv:1005.2924[hep-th]]. [10] A.Neveu,J.Scherk,Phys.Rev.D1,2355-2359(1970). [11] P.GoddardandR.E.Waltz,Nucl.Phys.B34(1971)99. [12] R.R.MetsaevandA.A.Tseytlin,Nucl.Phys.B298(1988)109. [13] C.B.Thorn,Phys.Rev.D78(2008)106008.[arXiv:0809.1085[hep-th]]. [14] D.J.Gross,A.Neveu,J.ScherkandJ.H.Schwarz,Phys.Rev.D2(1970)697;C.S.Hsue,B.SakitaandM.A.Virasoro,Phya.Rev.D2(1970)2857 [15] M.B.Green,J.H.Schwarz,E.Witten,SuperstringTheory.Vol.2:LoopAmplitudes,AnomaliesAndPhenomenology,Cambridge,Uk:Univ.Pr.(1987)596P.(CambridgeMonographsOnMathematicalPhysics). [16] F.Rojas,C.B.Thorn,Phys.Rev.D84,026006(2011).[arXiv:1105.3967[hep-th]]. [17] V.Alessandrini,D.Amati,B.Morel,NuovoCim.A7,797-823(1972). [18] K.Kikkawa,B.Sakita,M.A.Virasoro,Phys.Rev.184(1969)1701-1713. [19] H.J.Otto,V.N.Pervushin,D.Ebert,Theor.Math.Phys.35(1978)308. [20] A.Erdelyi,W.Magnus,F.Oberhettunger,F.G.Tricomi,HigherTranscendentalFunctions,BatemanManuscriptProject,Volume2,ChapterXIII,McGraw-Hill(1953). [21] Z.Kunszt,A.SignerandZ.Trocsanyi,Nucl.Phys.B411(1994)397[arXiv:hep-ph/9305239]; 124

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[22] D.Chakrabarti,J.QiuandC.B.Thorn,Phys.Rev.D72(2005)065022,arXiv:hep-th/0507280. [23] D.Chakrabarti,J.QiuandC.B.Thorn,Phys.Rev.D74(2006)045018[Erratum-ibid.D76(2007)089901][arXiv:hep-th/0602026]. [24] C.B.Thorn,Phys.Rev.D82(2010)125021.[arXiv:1010.5998[hep-th]]. [25] D.J.Gross,J.L.Manes,Nucl.Phys.B326,73(1989). [26] D.J.Gross,P.F.Mende,Phys.Lett.B197,129(1987) [27] D.J.Gross,Phys.Rev.Lett.60,1229(1988). [28] N.Moeller,P.C.West,Nucl.Phys.B729,1-48(2005).[hep-th/0507152]. [29] F.Rojas,arXiv:1111.7319[hep-th]. [30] L.Clavelli,P.Coulter,S.T.JonesandZ.H.Lin,Phys.Rev.D35,2462(1987). [31] J.L.F.Barbon,Phys.Lett.B382,60(1996)[hep-th/9601098]. [32] G.D'Appollonio,P.DiVecchia,R.RussoandG.Veneziano,JHEP1011,100(2010)[arXiv:1008.4773[hep-th]]. [33] C.Bachas,B.Pioline,JHEP9912,004(1999).[hep-th/9909171]. [34] M.B.GreenandJ.H.Schwarz,Nucl.Phys.B198,441(1982). [35] M.B.GreenandJ.H.Schwarz,Phys.Lett.B151,21(1985).J.Scherk,Nucl.Phys.B31(1971)222-234.A.NeveuandJ.Scherk,Nucl.Phys.B36(1972)155.J.Maldacena,Adv.Theor.Math.Phys.2,231(1998),hep-th/9711200.K.BardakciandC.B.Thorn,Nucl.Phys.B626(2002)287,hep-th/0110301. [36] C.B.Thorn,Nucl.Phys.B637(2002)272[arXiv:hep-th/0203167];Phys.Lett.B35(1971)529. 125

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BIOGRAPHICALSKETCH FranciscoRojasobtainedhisbachelordegreeinelectricalandindustrialengineeringatPonticiaUniversidadCatolicadeChile.Hethencontinuedtostudyphysicsinthesameinstitutionin2004byenrollinginthemaster'sdegreeprogramofferedbytheDepartmentofPhysicstointerestedengineeringstudents.InAugust2006,hearrivedintheUnitedStatestopursuedoctoralstudiesintheoreticalparticlephysicsattheUniversityofFloridaandobtainedhisdegreeinApril2012. 126