What is the iε for the S-matrix? PDF
Preview What is the iε for the S-matrix?
SpringerBriefs in Physics Holmfridur Sigridar Hannesdottir · Sebastian Mizera What is the iε for the S-matrix? SpringerBriefs in Physics SeriesEditors BalasubramanianAnanthanarayan,CentreforHighEnergyPhysics,Indian InstituteofScience,Bangalore,Karnataka,India EgorBabaev,DepartmentofPhysics,RoyalInstituteofTechnology,Stockholm, Sweden MalcolmBremer,H.H.WillsPhysicsLaboratory,UniversityofBristol,Bristol,UK XavierCalmet,DepartmentofPhysicsandAstronomy,UniversityofSussex, Brighton,UK FrancescaDiLodovico,DepartmentofPhysics,QueenMaryUniversityof London,London,UK PabloD.Esquinazi,InstituteforExperimentalPhysicsII,UniversityofLeipzig, Leipzig,Germany MaartenHoogerland,UniversityofAuckland,Auckland,NewZealand EricLeRu,SchoolofChemicalandPhysicalSciences,VictoriaUniversityof Wellington,Kelburn,Wellington,NewZealand DarioNarducci,UniversityofMilano-Bicocca,Milan,Italy JamesOverduin,TowsonUniversity,Towson,MD,USA VesselinPetkov,Montreal,QC,Canada StefanTheisen,Max-Planck-InstitutfürGravitationsphysik,Golm,Germany CharlesH.T.Wang,DepartmentofPhysics,UniversityofAberdeen, Aberdeen,UK JamesD.Wells,PhysicsDepartment,UniversityofMichigan,AnnArbor,MI,USA AndrewWhitaker,DepartmentofPhysicsandAstronomy,Queen’sUniversity Belfast,Belfast,UK SpringerBriefsinPhysicsareaseriesofslimhigh-qualitypublicationsencompassing the entire spectrum of physics. 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HolmfridurSigridarHannesdottir SebastianMizera SchoolofNaturalSciences SchoolofNaturalSciences InstituteforAdvancedStudy InstituteforAdvancedStudy Princeton,NJ,USA Princeton,NJ,USA ISSN 2191-5423 ISSN 2191-5431 (electronic) SpringerBriefsinPhysics ISBN 978-3-031-18257-0 ISBN 978-3-031-18258-7 (eBook) https://doi.org/10.1007/978-3-031-18258-7 ©TheAuthor(s),underexclusivelicensetoSpringerNatureSwitzerlandAG2022 Thisworkissubjecttocopyright.AllrightsaresolelyandexclusivelylicensedbythePublisher,whether thewholeorpartofthematerialisconcerned,specificallytherightsoftranslation,reprinting,reuse ofillustrations,recitation,broadcasting,reproductiononmicrofilmsorinanyotherphysicalway,and transmissionorinformationstorageandretrieval,electronicadaptation,computersoftware,orbysimilar ordissimilarmethodologynowknownorhereafterdeveloped. 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ThisSpringerimprintispublishedbytheregisteredcompanySpringerNatureSwitzerlandAG Theregisteredcompanyaddressis:Gewerbestrasse11,6330Cham,Switzerland Contents 1 Introduction ................................................... 1 References ..................................................... 12 2 UnitarityImpliesAnomalousThresholds ......................... 17 2.1 HolomorphicUnitarityEquation .............................. 17 2.2 NormalandAnomalousThresholds ........................... 19 2.3 MassShiftsandDecayWidths ............................... 22 2.4 HolomorphicCuttingRules .................................. 25 References ..................................................... 27 3 PrimerontheAnalyticS-matrix ................................. 31 3.1 FromLoopMomentatoSchwingerParameters ................. 31 3.2 WhereAretheBranchCuts? ................................. 36 3.3 WhereAretheSingularities? ................................. 38 3.4 PhysicalInterpretations ..................................... 40 3.5 LefschetzThimbles ......................................... 43 3.6 ContourDeformations ...................................... 46 3.7 Discontinuity,ImaginaryPart,andUnitarityCuts ............... 50 References ..................................................... 54 4 SingularitiesasClassicalSaddlePoints ........................... 57 4.1 ParametricRepresentation ................................... 57 4.2 ThresholdsandLandauEquations ............................ 61 4.3 ComplexifyingWorldlines ................................... 68 4.4 WhenIstheImaginaryPartaDiscontinuity? ................... 70 References ..................................................... 74 5 BranchCutDeformations ....................................... 79 5.1 AnalyticityfromBranchCutDeformations ..................... 79 5.2 ExampleI:NecessityofDeformingBranchCuts ................ 83 5.2.1 BoxDiagram ........................................ 83 5.2.2 AnalyticExpression .................................. 86 5.2.3 DiscontinuitiesandImaginaryParts .................... 89 v vi Contents 5.2.4 UnitarityCutsinthes-Channel ........................ 92 5.2.5 Discussion .......................................... 94 5.3 ExampleII:DisconnectingtheUpper-andLower-HalfPlanes .... 96 5.3.1 External-MassSingularities ........................... 96 5.3.2 TriangleDiagram .................................... 97 5.3.3 AnalyticExpression .................................. 100 5.3.4 DiscontinuitiesandImaginaryParts .................... 104 5.3.5 UnitarityCutsintheu-Channel ........................ 106 5.3.6 UnitarityCutsinthes-Channel ........................ 109 5.3.7 Discussion .......................................... 112 5.4 ExampleIII:SummingoverMultipleDiagrams ................. 114 References ..................................................... 116 6 GlimpseatGeneralizedDispersionRelations ..................... 117 6.1 StandardFormulation ....................................... 117 6.2 Schwinger-ParametricDerivation ............................. 119 6.2.1 DiscontinuityVersion ................................. 120 6.2.2 Imaginary-PartVersion ............................... 122 References ..................................................... 124 7 FluctuationsAroundClassicalSaddlePoints ...................... 127 7.1 ThresholdExpansion ........................................ 127 7.1.1 BulkSaddles ........................................ 128 7.1.2 BoundarySaddles .................................... 132 7.2 BoundontheTypeofSingularitiesfromAnalyticity ............. 136 7.3 AnomalousThresholdsThatMimicParticleResonances ......... 138 7.4 AbsenceofCodimension-2Singularities ....................... 139 7.5 Examples ................................................. 141 7.5.1 NormalThresholds ................................... 142 7.5.2 One-LoopAnomalousThresholds ...................... 143 References ..................................................... 148 8 Conclusion .................................................... 151 Reference ...................................................... 153 AppendixA:ReviewofSchwingerParametrization ................... 155 Chapter 1 Introduction Imprints of causality on the S-matrix remain largely mysterious. In fact, there is not even an agreed-upon definition of what causality is supposed to entail in the firstplace,withdifferentnotionsincludingmicrocausality(vanishingofcommuta- torsatspace-likeseparations),macrocausality(onlystableparticlescarryingenergy- momentumacrosslongdistances),Bogoliubovcausality(localvariationsofcoupling constantsnotaffectingcausally-disconnectedregions),ortheabsenceofShapirotime advances.Atthemechanicallevel,thereispresentlynocheckthatcanbemadeon S-matrixelements thatwouldguarantee thatitcame fromacausalscatteringpro- cessinspace-time.Motivatedbytheintuitionfrom(0+1)-dimensionaltoymodels, where causality implies certain analyticity properties of complexified observables [1,2],itisgenerallybelievedthatitsextrapolationtorelativistic(3+1)-dimensional S-matrices will involve similar criteria [3–7]. Converting this insight into precise resultshasprovenenormouslydifficult,leavinguswitharealneedformakingana- lyticitystatementssharper,especiallysinceitisexpectedthattheyimposestringent conditionsonthespaceofallowedS-matrices.Progressinsuchdirectionsincludes [8–32], often under optimistic assumptions on analyticity. This work takes a step towardsansweringanevenmorebasicquestion:howdoweconsistentlyupliftthe S-matrixtoacomplex-analyticfunctioninthefirstplace? Complexification.LetusdecomposetheS-matrixoperatorintheconventionalway, S =1+iT,intoitsnon-interactingandinteractingpartsandcallthecorresponding matrixelementsT.WewanttoaskhowtoextendTtoafunctionofcomplexMandel- staminvariantsTC.Forexample,for2→2scatteringTC(s,t)wouldbeafunction inthetwo-dimensionalcomplexspaceC2parametrizedbythecenter-of-massenergy squareds andthemomentumtransfersquaredt. Amongmanymotivationswecanmentionexploitingcomplexanalysistoderive physicalconstraintsviathetheoryofdispersionrelations,complexangularmomenta, oron-shellrecursionrelations;see,e.g.,[2–7].Anotherincentivestemsfromthecon- jecturalpropertyoftheS-matrixcalledcrossingsymmetry,whichcanbesummarized bythefollowingpracticalproblem.Letussaythatweperformedadifficultcomputa- ©TheAuthor(s),underexclusivelicensetoSpringerNatureSwitzerlandAG2022 1 H.S.HannesdottirandS.Mizera,WhatistheiεfortheS-matrix?, SpringerBriefsinPhysics, https://doi.org/10.1007/978-3-031-18258-7_1 2 1 Introduction Fig.1.1 AnalyticstructureofthematrixelementTC(s,t∗)for2→2scatteringofthelightest state of mass M in theories with a mass gap in the complex s-plane at sufficiently small fixed t =t∗<0.Therearetwosetsofbranchcuts(thicklines)correspondingtonormalthresholdsinthe s-channel(s>4M2)andu-channel(u>4M2ors<−t∗).TheamplitudeisrealintheEuclidean regionbetweenthem,whichcanalsofeaturesingle-particlepoles.Thecausalwayofapproaching thephysicalchannelsisindicatedwitharrows.Thepurposeofthisworkistoinvestigatehowthis picturegeneralizestomorerealistictheories tionforthepositron-electronannihilationprocesse+e− →γγatagivennumberof loops.Thequestioniswhetherwecanrecyclethisresulttoobtaintheanswerforthe crossedprocess,Comptonscatteringγe− →γe−,“forfree”,i.e.,byanalyticcon- tinuation. Unfortunately, the two S-matrix elements are defined indisjointregions ofthekinematicspace:fors >0ands <0respectively,soinordertoevenponder suchaconnection,oneisforcedtouplifts toacomplexvariable. Alas, complexifying the S-matrix opens a whole can of worms because it now becomesamulti-valuedfunctionwithanenormously-complicatedbranchcutstruc- ture.Notwithstandingthisobstruction,alotofprogressinunderstandingtheanalytic structurehasbeenmadefor2→2scatteringofthelighteststateintheorieswitha massgapatlowmomentumtransfer,see,e.g.,[4,33].Anoften-invokedapplication isthepionscatteringprocessππ →ππ[34–43].Thissetupgivesrisetotheclassic pictureofthecomplexs-planeforsufficientlysmallphysicalt =t∗ <0illustrated inFig.1.1.Inthistoymodel,therearebranchcutsextendingalongtherealaxiswith s >4M2 responsiblefors-channelresonancesandsimilarlyfors <−t∗ fortheu- channelones(bymomentumconservations+t +u =4M2,sou >4M2,whereM isthemassofthelightestparticle),withpossiblepolesresponsibleforsingle-particle exchanges.Inprinciple,thisstructurecanbearguedfornon-perturbatively,see,e.g., [44]. It turns out that, in this case, the causal matrix element T in the s-channel is obtainedbyapproachingTCfromtheupper-halfplane: T(s,t∗)= lim TC(s+iε,t∗) (1.1) ε→0+ fors >4M2.Similarly,theu-channelneedstobeapproachedfromthes−iεdirec- tion. Because of the branch cut, it is important to access the physical region from 1 Introduction 3 the correct side: the opposite choice would result in the T-matrix with anti-causal propagation.Establishingsuchanalyticitypropertieshingesontheexistenceofthe “Euclidean region”, which is the interval −t∗ <s <4M2 where the amplitude is realandmeromorphic;see,e.g.,[33]. Acloselyrelatedquestioniswhethertheimaginarypartoftheamplitude, (cid:2) (cid:3) ImT(s,t∗)= 1 T(s,t∗)−T(s,t∗) (1.2) 2i forphysicals isalwaysequaltoitsdiscontinuityacrosstherealaxis (cid:2) (cid:3) DiscsTC(s,t∗)=εl→im0+ 21i TC(s+iε,t∗)−TC(s−iε,t∗) . (1.3) Recall that the former is the absorptive part of the amplitude related to unitarity, whilethelatterentersdispersionrelations.Sofar,theonlywayforarguingwhy(1.2) equals (1.3) relies on the application of the Schwarz reflection principle when the Euclideanregionispresent,butwhetherthisequalitypersistsinmoregeneralcases isfarfromobvious. It might be tempting to draw a parallel between (1.1) and the Feynman iε pre- scription,thoughatthisstageitisnotentirelyclearwhythetwoshouldberelated: one gives a smallimaginary part to the external energy, while the other one tothe propagators. So what is the connection between (1.1) and causality? One of the objectives of this work is studying this relationship and delineating when (1.1) is validandwhenitisnot. More broadly, the goal of this paper is to investigate the extension of Fig. 1.1 to more realistic scattering processes, say those in the Standard Model (possibly including gravity or other extensions), that might involve massless states, UV/IR divergences,unstableparticles,etc.Littleisknownaboutgeneralanalyticityprop- erties of such S-matrix elements. The most naive problem one might expect that the branch cuts in Fig. 1.1 start sliding onto each other and overlapping, at which momenttheEuclideanregionnolongerexistsandmanyofthepreviousarguments breakdown.Butatthisstage,whywouldwenotexpectothersingularitiesthatused to live outside of the s-plane to start contributing too? What then happens to the iεprescriptionin (1.1)?Clearly,beforestartingtoanswersuchquestionsweneed to understand the meaning of singularities of the S-matrix in the first place. This questionistightlyconnectedtounitarity. Unitarityandanalyticity.UnitarityoftheS-matrix,SS† =1,encodesthephysical principleofprobabilityconservation.ExpandedintermsofT andT†,itimpliesthe constraint 1(T −T†)= 1TT†. (1.4) 2i 2 Thisstatementisusefulbecauseitallowsustorelatetheright-handsidetothetotal cross-section,inaresultknownastheopticaltheorem;see,e.g.,[5].However,inorder tobeabletoprobecomplex-analyticpropertiesofT andmanifestallitssingularities, itismuchmoreconvenienttoexpresstheright-handsideasaholomorphicfunction.
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