Table Of ContentSpringerBriefs in Physics
Holmfridur Sigridar Hannesdottir ·
Sebastian Mizera
What is the iε
for the S-matrix?
SpringerBriefs in Physics
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·
Holmfridur Sigridar Hannesdottir
Sebastian Mizera
ε
What is the i
for the S-matrix?
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
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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.
