Electron (positron) beam polarization by Compton scattering on circularly polarized laser p

G.L.Kotkin1),V.G.Serbo1),andV.I.Telnov2)

1)

NovosibirskStateUniversity,630090,Novosibirsk,Russia2)

BudkerInstituteofNuclearPhysics,630090,Novosibirsk,Russia

May14,2002

Abstract

Inanumberofpapersanattractivemethodoflaserpolarizationofelectrons(positrons)atstorageringsorlinearcollidershasbeenproposed.WeshowthatthesesuggestionsareincorrectandbasedonerrorsinthesimulationofmultipleComptonscatteringandinthecalculationoftheComptonspin-flipcrosssections.Wearguethattheequilibriumpolarizationinthismethodiszero.

1Introduction

ExperimentsatSLChaveshowngreatpotentialofpolarizede±beamsforinvestigationofnewphysicalphenomena.Inallprojectsoffuturee+e−,e−e−,γγandγelinearcolliders[1],electronandpositronbeamswithhighdegreeofpolarizationareforeseen,thoughthisisnotaneasytask.Thatiswhyanynewmethodsforobtainingpolarizede±beamsareverywelcome.

Therearetwowell–knownandrecognizedmethodsforproductionofpolarizedbeamsforlinearcolliders.Inthefirstmethod,electronbeamswithapolarizationof80%(maybeevenhigher)areobtainedusingphotoguns[2].Anothermethodofpolarization,suitablebothforelectronandpositronbeams,isbasedonatwo-stepscheme[3].Atthefirststep,theunpolarizedelectronbeampassesthroughahelicalundulator(orcollideswithcircularlypolarizedlaserlight)andproducesphotonswithmaximumenergyofabout30÷50MeV.Thesephotonshaveahighdegreeofcircularpolarizationinthehighenergypartofthespectrum.Thenthesephotonspassthroughathintungstentargetandproducee+e−pairs.Atthemaximumenergies,theseparticleshaveahighdegreeoflongitudinalpolarization.Theexpectedpolarizationofelectronandpositronbeamsinthismethodis45÷60%[4,5].

Additionaly,inanumberofpapersanewattractivemethodforproductionofpolarizedelectron1beams,basedontheprocessofmultipleComptonscatteringofultra-relativistic

electronsonthecircularlypolarizedlaserphotons,wasdiscussed.Hereoneshoulddistin-guishbetweentwopossibilities:polarizationofthebeamatthecostofalossinintensityandpolarizationwithoutlossofintensity.Thelattercaseisthesubjectofthepresentpaper.

ItiswellknownthatinComptonscatteringtheelectronsofdifferenthelicitiesareknockedoutofthebeamdifferently.Asaconsequence,afterremovalofthescatteredelectrons,theelectronbeamcangetaconsiderablepolarizationattheexpenseofacon-siderablelossofitsintensity.Adetailedconsiderationofthismethodwasgivenin[8].Thoughthebasicideaofthismethodiscorrect,itwasnotusedinpracticebecausethelossesinintensityduringthepolarizationprocessaretoolarge.

InthispaperwecriticallyconsideranotherproposalofbeampolarizationwhichisbasedonmultipleComptonscatteringofelectronsonlaserphotonswithoutlossofinten-sity.ItimpliesthatduringasingleComptonscatteringtheenergylossofanelectronissmallandtheelectronscatteringangleissmallaswell;therefore,thescatteredelec-tronsremaininthebeam.Itmeansthattheelectronenergyisoftheorderof1GeVforlaserlightwithaphotonenergyofabout1eV.Suchproposalsweregiveninpa-pers[9,10,11,12].Theywerecitedinanumberofpapers(seeRefs.[13],forexample)andattractedattentionattheSnowmass2001conference.Itwassuggestedtoimplementthismethodeitheronastoragering(wheretheelectronbeamcollideswithlaserbeamsmanytimesatasinglepoint)oratalinearcollider(wheretheelectronbeamwouldcollidewithlaserbeamsatseveralpoints)withreaccelerationbetweenthem.

ThetheoreticalconsiderationoftheprocessofmultipleComptonscatteringinpa-pers[9]–[12]isbasedontwodifferentapproaches.Thefirstapproachexploitsthefactthatthescatteredelectronsarepolarizedeveniftheinitialelectronbeamisunpolar-ized.Sincethescatteredelectronsdonotleavethebeam,multiplecollisionswithlaserphotonswouldappeartoleadtoagrowthofthemeanelectronbeampolarization.Thequantitativeconsiderationofthisideainpapers[10,11]resultedin

•Conclusion1[10,11].Alongitudinalpolarizationofelectrons(positrons)upto100%canbeachievedinarelativelyshorttime.

InSect.3weexplaintheoriginofthemistakethatleadto“Conclusion1”.Brieflyspeaking,insimulationofmultipleComptonscatteringoneshouldnotonlyconsiderthemultipleComptonscatteringofthesameelectronbutalsotakeintoaccountthefactthatthepolarizationofunscatteredelectronschangesinthelaserwaveaswell.ThecorrectsimulationprocedureformultipleComptonscattering(Sect.2)leadstozeropolarizationofthefinalelectronbeam.

Inthesecondapproach,onlytheequilibriumpolarizationoftheelectronbeamaf-termultiplepassesthroughthelaserbeamwasconsidered.Letw+−andw−+betheprobabilitiesforComptonscatteringwithagivenelectronspin-flip.Itisnotdifficulttoshow(seeSect.2.2)thattheelectronbeamgetsthemaximalequilibriumpolarization(f)ζz=(w−+−w+−)/(w−++w+−).Thecorrespondingprobabilitieshavebeencalculatedinpapers[9,12]withthefollowing

•Conclusion2[9,12].Thelongitudinalpolarizationofelectrons(positrons)aslargeas62.5%canbeachieved.

InSect.4weshowthatConclusion2isduetoanerrorinthecalculationoftheComp-tonspin-flipcrosssections.Theerrorisconnectedtotheincorrecttransitionbetween

2

thecolliderframe(CF)andtherestframeoftheinitialelectron(RFIE).Thecorrectresultcorrespondstow+−=w−+,therefore,thediscussedprocessoflaserpolarizationisimpossible.

Below,inSect.2,wepresentasetofformulaeforComptonscatteringthattakesintoaccounttheparticlepolarizationfromRef.[6],aswellasashortdescriptionofthesimulationprocedureforthemultipleComptonscatteringfromRef.[7],whichareusefulforquantitativeconsiderationofthismethod.Hereweshowthatscatteringoflaserphotonsdonotleadtopolarizationofelectronbeams,infacttheyleadonlytodepolarization.InSect.3and4weexplicitlyshowtheoriginofmistakesinthepreviouspapersonthissubject.

2

2.1

Polarizationoffinalelectrons

PolarizationoffinalelectronsinsingleComptonscattering

e(p)+γ(k)→e(p′)+γ(k′)

(1)

WeconsiderthebasicComptonscattering

intheCF,inwhichanelectronwithenergyE∼1GeVcollidesahead-onwithalaserphotonofenergyω∼1eV.

LetusintroducesomenotationrelatedtotheComptonscattering(1)inCF.Wechoosethequantizationaxis(z-axis)alongtheinitialelectronmomentump(i.e.,anti-paralleltothelaserphotonmomentumk).LetPc=+1bethemeanhelicityofthecircularly

′

)bethepolarizationvectorsofthepolarizedlaserphotons,ζ=(0,0,ζz)andζ′=(0,0,ζz

electronintheinitialandfinalstates.ItisconvenienttodescribetheComptonscatteringbytheinvariants

2pk2p′kx==

pk

,y=

ω′

m2e

x+1

.(4)

ForE∼1GeVandω∼1eV,thevalueofx∼0.015,thereforehereafterweassume

x≪1,

E−E′=ω′≪E.

(5)

InRFIE,theenergyofthelaserphotonxme/2issmallincomparisonwiththeelectron

mass:xme/2≪me;andthereforethetransversemomentaaresmallaswell:

|p′⊥|=|k′⊥|Asaresult,theelectronscatteringangleinCFisverysmall

θe≈

|p′⊥|

2E

≈2ω

(6)

TheComptoncrosssectioninthecolliderframefortheaboveconditionshasbeengivenin[6]:

dσx

[(1+ζzζz′)F1+(ζz+ζz′)PcF2+ζzζz′

F3],(8)2F1

s2

1=

1−y

,F3=−

yr(1−r),c=1−2r,r=

y

N.(10)

++N−

WhentheelectronbeamtravelsthepathdzinthelaserbeamwiththelaserphotondensitynL(z),thechangeofnumbersN±(z)isgivenbythebalanceequationsderivedbythefollowingsimpleconsideration.Areductionofthenumberofelectronswithζz=+1inthebeamisdeterminedbythequantityN+(z)(dw+++dw+−),where

dwζzζ′z=2σ(ζz,ζz

′

)nL(z)dz(11)

istheprobabilitythatanelectronwithacertainζzisscatteredonthepathdzwiththe

transitiontoacertainζz′

(hereweassumealsoacertainPc=+1).Thecoefficient2isduetothefactthattheelectronandthelaserphotontraveltowardseachotherwiththespeedoflight.Ontheotherhand,thesumN+(z)dw+++N−(z)dw−+representsthe

numberofscatteredelectronswithζ′

aresult,thetotalchangeofthez=+1.

Asnumberofelectronswithζz=+1isequalto

dN+(z)=−N+(z)(dw+++dw+−)+N+(z)dw+++N−(z)dw−+

(12)

=−N+(z)dw+−+N−(z)dw−+

and,similarly,

dN−(z)=−N−(z)dw−++N+(z)dw+−.

(13)

Sinceintheconsideredmethodthescatteredelectronsremaininthebeam,thesumN++N−=Nedoesnotchange:

dN+(z)+dN−(z)=0,

(14)whilethemeanlongitudinalpolarization(10),generallyspeaking,changesas

Nedζz=dN+(z)−dN−(z)=−2N+(z)dw+−+2N−(z)dw−+.

(15)

Balanceequations(12)–(15)aresimplifiedintwoparticularcases.First,iftheinitialelectronbeamisunpolarized,N+(z)=N−(z)=Ne/2,itgets(aftertravelingthepathdz)thepolarization

dζz=dw−+−dw+−.(16)Second,letusconsidertheequilibriumpolarizationoftheelectronbeam(whichisachieved

aftermultiplepassingoftheelectronbeamthroughthelaserbeam).Inthiscase,dN+=dN−=0or

N+dw+−=N−dw−+.(17)Fromthisequationoneobtainstheequilibriumpolarizationdegreeoftheelectronbeam

ζw−+−w+−

z(f)=

ζ=−4σ(ζz=+1,ζz′

=−1)nz

L(z)dz

(22)

fromwhichitfollowsthatthepolarization|ζz|isreducedaftertravelingthepathdz.

Notethattheresult(21)isduetothespecificstructureofEq.(8):thecoefficientin

frontofζzPcinthisequationpreciselycoincideswiththecoefficientinfrontofζz′

Pc.

2.3AschemeforsimulationofmultipleComptonscattering

Insomeproblemssuchasconversionofelectronstophotonsatphotoncolliders,lasercooling,etc.,itisnecessarytocalculatebeamparametersaftermultipleactsofComptonscattering.Letanelectronbeamtraversesaregionwherelaserlightisfocused.ItisclearthattheenergiesoftheseelectronsaswellastheirpolarizationsvaryduetoComptonscattering.

However,whentheelectronpassesthroughthelaserbeam,thepolarizationvariesalsoforthoseelectronswhichconservetheirenergiesanddirectionsofmotion(unscatteredelectrons).Thiseffectisduetotheinterferenceoftheincomingelectronwaveandtheelectronwavescatteredatzeroangle.ThechangeintheelectronpolarizationdependsnotonlyontheComptoncrosssectionbutontherealpartoftheforwardComptonamplitudeaswell.SuchaneffectwasconsideredinRef.[7].

BothoftheseeffectsshouldbetakenintoaccountinsimulationofmultipleComptonscattering.Itcanbetakenintoaccountinthefollowingway.TheelectronstateisdefinedbythecurrentvaluesofitsenergyE,thedirectionofitsmomentum(alongthez-axis)anditsmeanpolarizationvectorζ.Theprobabilitytoscatteronthepathdzisequalto

dw=2σ(E,ζz)nL(z)dz,

(23)

whereσ(E,ζz)isthetotalcrosssectionoftheComptonscatteringprocess.Then,asusual,onecansimulatewhetherthescatteringtakesplaceonthispathdzornot.

Ifthescatteringdoestakeplace,then,usingknownformulaefortheComptoncrosssectioninCF(seeRef.[6]),onecancalculateanewvalueoftheelectronpolarizationvectorζ(f)andotherparameters.

Ifthescatteringdoesnotoccur,onestillhastochangetheelectronpolarizationvector.4Thevariationofelectronpolarizationinthelaserwaveforageneralcasewasconsideredin[7].Followingthatpaper,thechangeoftheelectronpolarizationvectoroftheunscatteredelectronis

2

dζx=(Rζy+Iζzζx)Pc2πrenLdz,

2

nLdz,dζy=(−Rζx+Iζzζy)Pc2πre

22

dζz=−I(1−ζz)Pc2πrenLdz,

(24)

(25)(26)

wherethefunctionsI=I(x)andR=R(x)areequalto:

I==

4

22

x

󰀌

1−y

dy=

5

x+1

−

1

(27)

ln(x+1)−

Thenecessityofthisstepcanalsobeseenfromthefollowingconsideration.ThevalueoftheComptoncrosssectiondependsonpolarizationsofelectronandlaserbeams.Iftheelectronbeamwasinitiallyunpolarized,then,aftertheComptonscatteringofoneelectron,therest(unscattered)partofthebeamgetssomepolarization(seeSect.2.2).Thatisjustbecauseelectronswithdifferentpolarizationshavedifferentscatteringprobabilities.Inotherwords,thelaserbeam“selects”preferablyelectronswithacertainpolarization.Inparticular,equation(26)forthelongitudinalpolarizationcanbeobtainfromthebalanceequationsdiscussedabove.

6

2

2

R(x)=

x

󰀌

F(x−1)−󰀉

1++

x

(x2−1)2

3x

󰀆

,

(28)

with

󰀇x

F(x)=

ln|1+t|

0

2

󰀊

σunpol+

πr2eI(x)ζz

′󰀍

,(30)

whereσunpolistheComptoncrosssectionforunpolarizedbeams.Therefore,thescattered

electronbecomespolarizedafterthefirstscattering,anditsmeandegreeofpolarizationis

ζπr2z(f)

=

eI(x)3RemarkonConclusion1

NowwearereadytoshowtheoriginoftheerrorinConclusion1.LetusdescribetheprocedureofthenaivesimulationofthemultipleComptonscattering.WeconsiderthecasewhenintheCFthepolarizationvectorsoftheinitialandfinalelectronshavez-componentsonlyandtheparameterxissmall.ThecorrespondingComptoncrosssectionwithanaccuracyuptothetermsoftheorderofxcanbeeasilyobtainedfrom(8):

σ=

4

4

′

Pc(ζz+ζz).

󰀃

(35)

Iftheinitialelectronisunpolarized(ζz=0)andthelaserphotoniscircularlypolarized

(Pc=+1),then󰀃

4′

σ=,(36)ζz

4′′

i.e.thecrosssectionissomewhatlargerforζz=−1thanforζz=+1.Therefore,thescatteredelectronbecomespolarizedafterthefirstscatteringanditsmeandegreeofpolarizationis

x(f)

ζz=−

(4/x)+N

,(38)

whichcanreach100%forN≫4/x.ThisfactisthebasisforConclusion1.

This“polarization”isnotconnectedwiththeelectronspin-flip,itisduetosomedifferencesinthecrosssections:thepolarizedlaserbeamselectselectronswithacertain(inourcase,negative)polarization.ButsuchanaivesimulationofthemultipleComptonscatteringisincorrectbecauseitdoesnottakeintoaccountthefactthatunscatteredelectronsbecomepolarizedintheoppositedirection.Thecorrectprocedureforthissimulationisdescribedintheprevioussectionandleadstozeropolarization.

4RemarkonConclusion2

InSect.2.1wehaveshownthattheequilibriumpolarizationofelectronsinthediscussedmethodiszero.Below,weshowtheoriginofthemistakethatledtoConclusion2.Weremindthatourresult(21)hasbeenobtainedintheCF.Tothecontrary,theauthorsofConclusion2hadobtainedtheirresult(20)intheRFIE.BelowwedemonstratehowtoobtainourresultinRFIEandshowthattheerrorinConclusion2isconnectedwithinaccuratetransitionfromCFtoRFIE.

Inourconsideration,weusetheelectronpolarizationvectors5ζandζ′,whichinCFhavetheforms

ζ=(0,0,±1),ζ′=(0,0,∓1).(40)

ζp

me

ζ′p′

me

,ζ+p,ζ+p

′′

ItisnotdifficulttoshowthatinRFIEthevectorζhasthesameform,butthevectorζ′hasanotherform:

ζ′⊥

=∓

p′⊥

me

,

′ζz

=∓1−

󰀂

(p′⊥)2

me(E′+me)

≈∓

p′⊥

me(E′+me)

≈ζ′⊥,

(43)

sinceinRFIEwehave|p′|/me≪1.

TheneededComptoncrosssectioninRFIEcanbefoundinthetextbook[15](seeEqs.(87,22)and(87,23)):

dσ

4

󰀂

ω′

dΩ

=

2re

ω

󰀄2󰀅

fz−gz+g⊥

k′⊥

ω+ω′

me

g⊥

k′⊥

m2e

(1−cosϑ)sin2ϑ≈−

󰀉

meω

󰀌2

(1−cosϑ)sin2ϑ,

(47)

=0,(49)

dΩ

whichisinagreementwiththeconclusion(21)inCF.

Thewrongconclusion(20)wasobtainedbecausethesameform(40)wasusedforthevectorζ′bothintheCFandintheRFIE.ItisequivalenttoomittingthelastterminthesquarebracketinEq.(45).

Thus,thecalculations,performedinCFaswellasinRFIE,giveusthesameresult(21).We,therefore,concludethattheclaim(20)isbasedonaninaccuratetransitionfromCFtoRFIE.

9

5Summary

WehaveshownthatthemultipleComptonscatteringofelectronsoncircularlypolarizedlaserphotonsatusualstorageringsorlinearacceleratorsdoesnotleadtopolarizationofelectronbeams.Statementsbysomeauthorsaboutobtainabilityofhighdegreesofpolarizationareexplainedbymistakesintheircalculationprocedures.WehaddiscussionswithE.G.BessonovandA.P.Potylitsyn,andtheyagreedwithourcriticism.

InthispaperwehaveconsideredthelinearComptonscattering(thescatteringofanelectrononasinglelaserphoton).ItistechnicallypossibletorealizeconditionswhichcorrespondtothenonlinearComptonscattering(thescatteringofanelectrononseverallaserphotons).TheeffectivecrosssectionforthenonlinearComptonscatteringfromRef.[16]hasthesamespecificstructureasEq.(8)butwithmuchmorecomplicatedfunctionsF1,2,3.Fromthis,onecaneasilyobtaintheresult(21),whichmeansthattheequilibriumpolarizationofelectronsiszerointhethecaseofthenonlinearComptonscatteringaswell.

Oneadditionalremark.Thereisnopolarizationoftheelectronbeamasawholeintheconsideredscheme,however,itdoesnotclosethepossibilitytouselasersforpolarizationofelectronbeamsinotherschemes.Forexample,ithasbeenshowninRef.[14]thatusingspeciallyarrangedspin-orbitcouplingindampingrings(byaddingasolenoid),apolarizationofabout60%maybereached.ThismethodisbasedonthedifferenceintheComptoncrosssectionsforelectronswithdifferentvaluesoftheirhelicities,onthefactthatscatteredelectronshavelowerenergycomparedtounscatteredelectrons,andondependenceofthespinprecessionangleontheelectronenergy.Thismethodisnotsimple,andistooslowforpreparationofbeamsforlinearcolliders.

Acknowledgement

WeareverygratefultoE.Bessonov,R.Brinkmann,V.Katkov,A.P.Potylitsyn,E.L.Sal-din,A.N.SkrinskyandV.Strakhovenkoforusefuldiscussions.ThisworkissupportedinpartbyINTAS(code00-00679),RFBR(code00-02-17592and00-15-96691)andbySt.Petersburggrant(codeE00-3.3-146).

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