The [1,2] Pad'e Amplitudes for $pipi$ Scatterings in Chiral Perturbation Theory

arXiv:hep-ph/0205214v2 12 Jul 2002The[1,2]Pad´eAmplitudesforππScatterings

inChiralPerturbationTheory

Guang-YouQin,W.Z.Deng,Z.G.XiaoandH.Q.Zheng

DepartmentofPhysics,PekingUniversity,Beijing100871,P.R.China

Abstract

Adetailedanalysistothe[1,2]Pad´eapproximationtotheππscat-tering2–loopamplitudesinchiralperturbationtheoryismade.

Keywords:chiralperturbationtheory;ππscattering;Pad´eapproximation-

PACSnumber:11.55.Fv,12.39.Fe,12.38.Cy

Thechiralperturbationtheory(ChPT)([1]–[4])isapowerfultoolin

studyingstronginteractionphysicsatlowenergiesandhasbeenextensively

studiedat1–looplevel[5].The2–loopresultsarealsoavailableinrecent

years([6]–[9]).However,sincethechiralexpansionisanexpansioninterms

oftheexternalmomentum,theperturbationseriestoanyfiniteorderdi-

vergesveryrapidlyathighenergies.Thereforetheviolationofunitarity

getsevenworsefor2–loopamplitudesthanthe1–loopamplitudesathigh

energies.AlsothenumberofparametersintheeffectiveLagrangianwhich

arenotfixedbysymmetryaloneincreasesrapidly.Thereforeincreasingthe

orderoftheperturbationexpansiondoesnotworkatallforthepurposeof

exploringphysicsinthenon-perturbativeregion,orathigherenergiesand

non-perturbativestudiesbecomenecessary.Awidelyusedmethodtorem-

edytheviolationofunitarityisthesocalledPad´eapproximation.1Anice

featureofthePad´eapproximationisthatitrestoresunitaritywithfullre-

spect,atlowenergies,totheavailableinformationfromperturbationtheory.

Therefore,eventhoughitiswellknownthatitviolatescrossingsymmetry,

Pad´eapproximationisconsideredtobeavaluabletoolinexploringphysics

inthenon-perturbativeregion,suchasthepropertiesofphysicalresonances.However,apreviousstudy[12]indicatesthatthe[1,1]Pad´eapproximation

encountersaseriousproblembypredictingspuriousphysicalsheetresonances

(SPSRs).UsuallytheseSPSRslocateatdistantplacesveryfarfromthere-

gionwheretheperturbationresultsarevalid.ThepredictionsofthePad´e

approximantsconstructedfromtheperturbativeamplitudesshould,ofcourse

not,beconsideredasmeaningfulintheregionfarawayfromtheregionwhere

perturbationtheoryremainstobevalid.Onemayfurtherarguethatsince

thoseSPSRsarefarfromtheregionweareconcerningtheuseofthePad´e

approximationisstillacceptableatleastinphenomenologicaldiscussions.

However,theproblemwithPad´eapproximationisnotonlybecauseitpre-

dictsSPSRsinthedistantregiontoofarawaytobeworthwhiletopayany

attention,butalsobecausethoseSPSRsusuallyhavelargecouplingstoππ

whichleadtostronginfluencetotheregionwearereallyinterestedin,and

hencetheirexistencecastsdoubtontheremainingpredictionsofthePad´e

approximantswhichmightotherwisebeassumedasmeaningful,atleastat

quantitativelevel.Theaimofthepresentstudyistofurtherinvestigatethe

Pad´eapproximationfollowingthemethodofRef.[12].Wewillextendthe

workofRef.[12]byalsoanalyzingthe[1,2]Pad´eapproximants,sincethe

2–loopperturbationresultsarealreadyavailable.Oneofthemainmotiva-

tionofthepresentworkistoinvestigatethepossibilitythatthe[1,2]Pad´e

approximantscanrescue,tosomeextent,thebadsituationthe[1,1]Pad´e

approximantsencounter.Theconclusionisrathernegative,aswewillsee

inthefollowingtext.However,webelieveitisstillworthwhiletopresent

ourresults.SincethePad´eapproximationisaverypopularapproximation

methodwidelyusedinphenomenologicaldiscussions,wehopethepresen-

tationofthepresentworkcouldbenefitphysicistswhoareworkinginthe

relatedfieldsofnon-perturbativephysics.

Fortheππ→ππscattering,itiswellknownthattheisospinamplitudes

intheschannelcanbedecomposedas,

TI=0(s,t,u)=3A(s,t,u)+A(t,u,s)+A(u,s,t),

TI=1(s,t,u)=A(t,u,s)−A(u,s,t),

TI=2(s,t,u)=A(t,u,s)+A(u,s,t),(1)

wheres,t,uaretheusualMandelstamvariables,

s=(p1+p2)2/M2π,t=(p3−p1)2/M2π,u=(p4−p1)2/M2π.(2)

InSU(2)×SU(2)chiralperturbationtheorytotwoloops[9],themomentum

2expansionoftheamplitudesamountstoaTaylorseriesin

x2=M2π

64π󰀆1

−1d(cosθ)PJ(cosθ)TI(s,t,u),

cosθ=1+2t

1−TIJ,4(s)TIJ,2(s)+󰀁TIJ,4(s)Perturbationtheorysatisfiestheelasticunitarityrelation,

ImTIJ(s)=ρ(s)|TIJ(s)|2,(9)

ateachorderoftheperturbationexpansioninpowersofthequarkmasses

andexternalmomentum,i.e.,

ImTIJ,2(s)=0,

ImTIJ,4(s)=ρ(s)󰀇TIJ,2(s)󰀉2,

ImTIJ,6(s)=2ρ(s)TIJ,2(s)ReTIJ,4(s),(10)

......

Withtheserelationsitiseasytoprovethatthe[1,2]Pad´eapproximantinEq.(8)satisfieselasticunitarity:ImTIJ[1,2](s)=ρ(s)|TIJ[1,2](s)|2.(11)

InthefollowingwefrequentlyomittheindicesI,JoftheTmatrixfor

simplicityifitcausesnoconfusion.Foranygivenamplitudesatisfyingsingle

channelunitarity,followingthemethodofRefs.[13,14],wedefinetworeal

analyticfunctions˜FandFas

˜F(s)=1

S(s)󰀄

,

F(s)=1

S(s)󰀄

.(12)

Itisobviousthat˜FandρFaretheanalyticcontinuationofcos(2δ)and

sin(2δ),asthescatteringSmatrixisequaltoexp{2iδ}inthephysicalregion.

Accordingto[13,14],wehavethefollowingdispersionrelationsforFand˜F:

sin(2δ)=ρF=ρ(α+󰀅

iβi2iρ(zIIj)S′(zIIj)(s−zIIj)