Positivity Constraints on Spin Observables in Exclusive Pseudoscalar Meson Photoproduction

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by  Xavier Artru

PositivityConstraints on SpinObservables
inExclusivePseudoscalarMesonPhotoproduction
Xavier Artru,1,∗ Jean-Marc Richard,2,† and Jacques Soffer3,‡
1Institut de Physique Nucle´aire de Lyon, CNRS-IN2P3–Universite´ Lyon-I
4, rue Enrico Fermi, 69622 Villeurbanne cedex, France
2Laboratoire de Physique Subatomique et Cosmologie
Grenoble Universite´s–CNRS-IN2P3-Inge´nierie
53, avenue des Martyrs, F–38026 Grenoble Cedex, France
3Centre de Physique The´orique§
CNRS, Luminy Case 907, F-13288 Marseille Cedex 09 - France
(Dated:February9,2008)
Positivityconstraintshaveprovedtobeimportantforspinobservablesofexclusivereactionsinvolvingpolar-
izedinitialandfinalparticles. Attentionisfocusedinthisnoteonthephotoproductionofpseudoscalarmesons
from spin 1/2 baryons, more specifically γN → KΛ,γN → KΣ, for which new experimental data are
becomingavailable.
PACSnumbers:13.88.+e,24.70.+s,25.20.Lj
7
0
0
2
I. INTRODUCTION
n
a
J Positivityconstraintshavebeenwidelystudiedinhadronphysicstodeterminethealloweddomainofphysicalobservables.
0 Theycanbeusedtotesttheconsistencyofvariousavailablemeasurementsandalsothevalidityofsomedynamicalassumptions
2 intheoreticalmodels. Thispowerfultoolhasabroadrangeofapplicationsforthespinobservablesinexclusivereactions,like
πN πN,NN NN,NN ΛΛ and in one-particle inclusive reactions, like pp ΛX,ep eX,νp eX. It
2 alsop→rovidesconstr→aintsforstruct→urefunctions,partondistributions,etc.[1]. Here,wecon→centrateont→heparticula→rreactions
v
γN KY,withY =Λ,Σ,wheretheincomingphotonbeamispolarized,thenucleontargetispolarizedandthepolarization
0 →
oftheoutgoinghyperonismeasured.
5
ManydataareavailableforthereactionγN KY anditsanaloguewithoutstrangenessγN πN. Mostoftheresultsare
0
→ →
6 availablethroughtheDurhamdatabase[2]. AmongtherecentmeasurementsarethoseoftheSAPHIRcollaborationatBonn
0 andtheLEPScollaborationinJapan. Morerecently,thephotoproductionofakaonhasbeenstudiedbytheCLAScollaboration
6 atJlabandtheGRAALcollaborationatGrenoble.Theresultsareabouttobepublished.
0 Inthisarticle, itis remindedthatthe manypossible spinobservablesare notindependent,butconstrainedby identitiesand
/
h inequalities.Theycanbeestablishedinasystematicwaybypowerfulalgebraicmethods.Thephysicscontentoftheseidentities
t andinequalitiescanberevealedbyalternativederivationsbasedonthepositivityofthedensitymatrixorbyconsiderationson
-
l the norm of the polarizationvectors. We also stress that some of the most recent(still preliminary)results seemingly violate
c
theseconstraints.
u
n Severalauthorsstudiedhowasubsetofwell-chosenobservablesenablesonetoreconstructtheamplitudestoanoverallphase.
: Unavoidably,thequestionoftheredundanciesandcompatibilityamongthevariousobservablesisraised,leadingtolistanumber
v
of relationsamongobservables. Our concernis somewhatsimpler: we study to which extenta new observableis compatible
i
X withthepreviousdata,andwhichmarginisleftfortheyet-unknownobservableswhosemeasurementcanbeforeseen.
r In Sec. II, the formalism of the photoproduction reaction γN KY is briefly summarized with some details given in
a Appendix A. In Sec. III, several inequalities relating two or thre→e spin observables are given, and are derived by different
methods.SectionIVisdevotedtoconfronttherecentmeasurementswiththesemodel-independentinequalities.Ourconclusions
aregiveninSec.V.
§UMR6207-UnitemixtedeRechercheduCNRSetdesUniversite´sAix-MarseilleI,Aix-MarseilleIIetdel’Universite´duSudToulon-Var,Laboratoireaffilie´
a`laFRUMAM.
∗Electronicaddress:[email protected]
†Electronicaddress:[email protected]
‡Electronicaddress:[email protected]
II. FORMALISM
The formalism of the photoproductionof pseudoscalar mesons has been studied by several authors, see the pioneer paper
by Chew et al. [3] and [4, 5, 6], with particular attention to the observableswhich are needed for a full reconstructionof the
amplitudes,uptoanoverallphase.
Itisconvenienttoexpressthetransitionmatrix ofreactionγ+N K+Y inthetransversitybasis: π, and n, for
M → | ±i | ±i
theinitialstateand forthefinalstate,where denotesthetransversityofthenucleonorhyperon,i.e.,theprojection 1/2
|±i ± ±
ofitsspinalongthenormaltothescatteringplane,andπ(resp.n)aphotonstatewithlinearpolarizationparallel(resp.normal)
tothescatteringplane. ThesestatesareeigenstatesofΠ,theoperatorofreflectionaboutthescatteringplaneandconservation
ofΠisequivalenttoparityconservation.
Forthisparity-conservingexclusivereaction,therearefourindependenttransversityamplitudes,whichcanbechosenasthe
followingmatrixelementsofthetransitionoperator
M
a1 = + n+ , a2 = n ,
h |M| i h−|M| −i (1)
a3 = + π , a4 = π+ ,
h |M| −i h−|M| i
while + n = n+ = + π+ = π =0.
h |M| −i h−|M| i h |M| i h−|M| −i
Thecompleteknowledgeofthereactionrequires,toanoverallphase,thedeterminationofsevenrealfunctions. Ontheother
hand,onecanextractfromallthepossibleexperimentssixteendifferentquantities,whicharethebilinearproductsofthefour
amplitudes. A well chosenset ofobservablesgiveaccessto the amplitudes(to anoverallphase)withoutdiscrete ambiguities
[5,6].
Ontheexperimentalside,thereareseveralredundantobservables:
theunpolarizeddifferentialcrosssectionI0,
•
thelinearly-polarizedphotonasymmetryΣγ,
•
thepolarized-targetasymmetryAN,
•
thepolarizationPY oftherecoilingbaryon
•
thebaryondepolarizationcoefficientsT andL expressingthecorrelationbetweenthelongitudinalortransverse(inthe
i i
•
scatteringplane)targetpolarizationandthespinoftherecoilbaryon,
thecoefficientsdescribingthetransferofpolarizationfromaphotonbeamtotherecoilbaryon,inparticularO foroblique
i
•
polarizationandC forcircularpolarization,
i
thecoefficientsG,H,E andF ofdoublespincorrelationsbetweenthephotonbeamandthenucleontarget,
•
triplecorrelationscoefficientsifboththebeamandthetargetarepolarizedandthehyperonpolarizationanalyzed.
•
In these definitions, the index i refers to the component in a frame xˆ,yˆ,zˆ attached to each particle: yˆ, the normal to the
{ }
scatteringplane,isthesameforall,andzˆcanbechosenalongthecenter-of-massmomentump,i.e.zˆ=p/p(inphotoproduction
experiments,zˆ= p/pisusuallychosenforthebaryons). Forthefermions,itisconvenienttousearepresentationofthespin
−
operatorsin whichS isdiagonal(bycircularpermutationoftheusualPaulimatrices). Forthephoton,thethreecomponents
y
(S1,S2,S3) are the Stokesparameters, i.e., S3 S (“planarity”)is the polarizationalongxˆ, S1 S (“obliquity”)is the
⊖ ⊘
≡ ≡
polarization along (xˆ +yˆ)/√2 and S2 S is the circular polarization, or helicity. With full (S = 1) polarization, the
⊙
≡ | |
differentialcrosssectioncanbeexpressedas
dσ
dΩ(Sγ,SN,SY)=I0(λµ|ν)SγλSNµ SYν . (2)
Here λ,µ,ν, run from 0 to 3, the summation is understood over repeated indices and the polarizations have been promoted
to four-vectors with S0 = 1. The correspondence with the notation found in literature is given in Appendix A, where the
observablesarelisted,andsomeoftheirsymmetriesdiscussed.
TheangulardistributionI0 andthe productofI0 bya spin observablearebilinearcombinationsofthe fouramplitudes. In
thispaper,thediscussionisfocusedoneightoftheobservableslistedinAppendixA,namely
I0 = a1 2+ a2 2+ a3 2+ a4 2, I0AN = a1 2 a2 2 a3 2+ a4 2
| | | | | | | | | | −| | −| | | |
I0Σγ = a1 2+ a2 2 a3 2 a4 2, I0PY = a1 2 a2 2+ a3 2 a4 2
| | | | −| | −| | | | −| | | | −| | (3)
I0CxY =−2ℑm(a1a∗4−a2a∗3), I0CzY =2ℜe(a1a∗4+a2a∗3),
I0OxY =−2ℜe(a1a∗4−a2a∗3), I0OzY =−2ℑm(a1a∗4+a2a∗3).
whichareaccessiblewithouttargetpolarization,sincetheanalyzingpowerortargetasymmetryAN isequaltothetransferof
normalpolarizationfromthephotontothehyperon.FortheCY andOY observables,theexpressionsgiveninEq.(3)depend
x,z x,z
onthephaseconventionamongthetransversityamplitudes:wefollowhereRef.[5]. AsseeninAppendixA,thereare15other
pairsofobservableswhichareequaloropposite. Thismeansthatisisnotnecessarytomeasurethereactionwithanypossible
beamandtargetpolarization,exceptforcross-checks.
III. RELATIONSAMONGOBSERVABLES
Eachspinobservable isnormalizedtobelongto[ 1,+1]. However,model-independentinequalitiesexistamongobserv-
i
ables,andasaconsequeOnce,thealloweddomainforap−airofobservablesisoftensmallerthantheentiresquare[ 1,+1]2,and
similarlyforatripleofobservables,itisrestrictedtoasub-domainofthecube[ 1,+1]3. Theinequalitiesamon−gobservables
−
areofcourseindependentofthechoiceofamplitudes,e.g.,transversityvs.helicityamplitudes,andindependentoftheparticular
phaseconventionswhichareadoptedforthese amplitudes. Theinequalitiesdonotevendependontheorientationchosenfor
thex-andz-axesinthescatteringplane,sincearotationaboutyˆofthe(xˆ,yˆ,zˆ)frameattachedtoaparticleonlychangesthe
phaseofthetransversityamplitudes.Theinequalitiesalsoremainunchangedifwedefinethespinobservablesinthelaboratory
or Breit frame. In these frames, the helicity and the transverse polarization in the scattering plane do not coincide with the
center-of-massones,butarerelatedbyaWignerrotationaboutyˆ.
InRef.[4],itisindicatedthatseveralpairsofobservables( , )obeyaninequality 2+ 2 1,thatrestrictsthedomain
Oi Oj Oi Oj ≤
totheunitdisk. Examplesare
(Σγ)2+(OY)2 1, (Σγ)2+(OY)2 1, (OY)2+(OY)2 1. (4)
x ≤ z ≤ x z ≤
Forthe7spinobservablesgivenin(3),thereare21pairs,and15ofthemhavethisunit-diskconstraint.
Fortriples,thereareseveralexampleswherethethreeobservablesareconstrainedinsidetheunitball,inparticular
(PY)2+(CY)2+(CY)2 1, (5a) (Σγ)2+(CY)2+(CY)2 1, (5e)
x z ≤ x z ≤
(PY)2+(OY)2+(OY)2 1, (5b) (Σγ)2+(OY)2+(OY)2 1, (5f)
x z ≤ x z ≤
(PY)2+(CY)2+(OY)2 1, (5c) (Σγ)2+(CY)2+(OY)2 1, (5g)
x x ≤ x x ≤
(PY)2+(CY)2+(OY)2 1, (5d) (Σγ)2+(CY)2+(OY)2 1, (5h)
z z ≤ z z ≤
Byprojection,thisunitballgivesaconstraintinsidetheunitdiskforanysubsystemoftwoobservables. Moreinterestingis
thecasewherethedomainforthethreeobservablesismorerestrictedthantheunitcube[ 1,+1]3,butwithoutrestrictionfor
anypair. FortheobservablesofγN KY,onwhichdataexist,itisknown[4,5]thatAN−,PY andΣγ fulfill[4]
→
AN PY 1 Σγ , AN +PY 1+Σγ . (6)
| − |≤ − | |≤
Theselinearrelationsaresimplyobtainedfromthepositivityofthepuretransversitycrosssections a 2 in(3). Theyarealso
i
| |
foundforspinobservablesofinclusivereactions[1,7]. ThedomaincorrespondstoatetrahedronschematicallydrawninFig.1.
Notice that its volumeis only1/3 of the volume of the entire cube, while its projectionis the entire [ 1,+1]2 squareon any
−
face.
z
y
x
FIG.1:(coloronline)Tetrahedrondomainlimitedbytheinequalities(6)fortheobservablesx=AN,y=PY andz=Σγ.
Thereareseveralmethodstoestablishtheaboveinequalities. Themoststraightforwardconsistsofplottingdummyobserv-
ablesfromamplitudeswhoserealandimaginarypartsaregeneratedrandomly.Theplotsclearlyindicatewhetherthefullsquare
[ 1,+1]2 orcube[ 1,+1]3 is entirelyscanned, orthedomainis limitedbya circle, a triangle,a sphere, etc. Thenforthese
− −
pairsortriples,theobservedinequalitiescanbederivedfromtheexplicitexpressionssuchas(3). Withthetransversityampli-
tudes,AN,PY andΣγ havesimpleexpressions,andthetetrahedronconstraintiseasilyseen,whileitislessobviouswithother
choices,e.g.,helicityamplitudes,forwhich,inturn,otherinequalitiesbecomeeasier.
The proofs remain at the level of elementary calculus. For instance, Eq. (5g) can be obtained from the inequality
(V1 V2)2 (V1 + V2 )2 applied to the vectorsV1 = a1 2 a4 2,2 e(a1a∗4),2 m(a1a∗4) and V2 = a3 2
a2 2,−2 e(a2a≤∗3),|2 m| (a|2a∗3)| , with normalization V1 = a1{2| +| a−4 2| an|d Vℜ2 = a2 2 +ℑ a3 2. A}slight variant{,|e.g|. fo−r
|Eq.|(5h)ℜ,consistsofℑcheckingt}hatthefour-vectors(1| A|N;|PY| Σ|γ,|OY C| Y|, OY| |CY)|are|light-like,andhence,since
theirtimecomponents1 AN arepositive,theirsum±istime-like,±inthesazme±waxyastwxo±phoztonscombineintoamassivestate
±
inelementarykinematics.
Another possible starting point is the existence of identities among observables. A consequence already stressed in the
literature[4,5,6]isthatifsixteenmeasurementscanbeexpressedintermsofsevenrealfunctions,theycannotbeindependent
and mustbe constrainedby a numberofrelations. Severalyearsago, G. Goldsteinet al., followingearlierwork byFrøyland
andWorden[4], derivednine quadraticrelationsamongthese parameters. ThemethodofFierz transformation[6] providesa
systematic list of identities. Most of them, however, involve many observables, and hence it is not obvious to convert these
identitiesintoinequalitiesconstrainingthefewobservablesforwhichdataexist.
Anotherpowerfultoolisprovidedbyimposingthepositivityofthedensitymatrixforthedirectorcrossedchannels. Ifthe
γ+N Λ+X reactionisviewedascrossedfromγ+N +Λ X,thedensitymatrixhasdimension8 8,sinceeachof
→ → ×
theincomingparticlehastwospindegreesoffreedom.Thisisreducedto4 4iftheparityofX isidentified.Inourcase,since
×
X =K hasspin0,this4 4densitymatrixhasrank1,andhenceeach2 2subdeterminantvanishes.Thisgives36quadratic
× ×
relationswhicharethoselistedin[6]orlinearcombinationsofthem.Only9ofthemcanbeindependent,sincetheobservables
dependon7independentrealparameters. Howeveritdoesnotmeanthatthe9independentrelationsaloneinducethe27other
ones. Forinstance,thevanishingoftheninesub-determinantsformedbytheintersectionoftwoconsecutivecolumnswithtwo
consecutivelinesinducesthevanishingofalltheotherdeterminantsonlyifthe2nd and3rd columnsarenon-zero. Thisshows
thatthe 2 2sub-determinantsare relatedin a non-linearwayand27ofthemcannotbeexpressedfromthe remaining9 ones
×
withoutdiscreteambiguities.
Moreexplicitly,followingthemethodof[8],thedensitymatrixofγ+N +Λ K canbedefinedas
→
e′,s′ ,s′ R e,s ,s =C e′,s′ † s′ s e,s , (7)
h N Y | | N Yi h N|M | Yih Y|M| Ni
whereC isa normalizationcoefficient. ByconstructionRissemi-positivedefiniteandofrank1. Intermsoftheobservables
forγ+N K+Λ,
→
R=(λµν)σλ σµ [σν]t . (8)
| γ ⊗ N ⊗ Y
Notein(7)thecrossing s s ofthehyperonandthecorrespondingtranspositionofσν in(8). TableIofAppendixA
| Yi ↔ h Y| Y
givesthematrixelementsofR.Forinstance,fromthevanishingofitsco-diagonal2 2sub-determinantsweobtain
×
(1 AN)2 = (Pγ Σ)2+(OY CY)2+(CY OY)2, (9a)
± ± z ± x z ∓ x
(1 PY)2 = (AN Σ)2+(G F)2+(E H)2, (9b)
± ± ∓ ±
(1 Σγ)2 = (PY A)2+(L T )2+(L T )2 . (9c)
z x x z
± ± ± ∓
Complexidentities,i.e.,pairsofrealidentities,whichdonotcontainPY,AN norΣγ canbeobtainedfromthe2 2determi-
×
nantsmadeonlyofnon-diagonalelementsofTable1,forinstance
(E+iG)2+(F +iH)2+(O iC )2+(O iC )2 =0. (10)
z z x x
− −
Manyidentitiescanbelisted[6],buttheyarerelatedbysymmetryrules,inparticular
rotationinthe(x,y)plane,inparticularthesubstitutionx y,y x,
• → →−
permutationoftheλ,µ,ν indices, i.e., ofthe threeparticleswithspin, exceptforthe transpositionofσν. Forinstance,
• Y
Eqs.(9a)and(9c)arerelatedbysuchatransformation,
π/2rotationinthe(1,3)plane:1 3,3 1forthethreeparticlessimultaneously.Thisisnotcompatiblewiththeparity
• → →−
rulessuchas(322) = 0ortheequivalencesofEqs.(A1). However,relaxingparitytemporarily,onecanapplythe(1,3)
|
rotationtoTable1,thenenforceparityconservationtoreducethenewmatrixelements. Forinstance,thistransformation
appliedtoEq.(9c)leadsto
(1+L )2 =(E+C )2+(Σ+T )2+(G+O )2 , (11)
z z x z
substitution0 i1,1 i0foralltheparticles,correspondingtoanimaginaryLorentztransformationoftheσ ’sinthe
µ
• → →
(0,1)plane. Equation(9c),forinstance,istransformedinto
(1 T )2 =(F +C )2+(Σ L )2+(H O )2. (12)
x x z x
− − −
Asanexampleofapplication,Eq.(9a)implies
(1 AN)2 (PY Σγ)2, (13)
± ≥ ∓
whichisequivalentto(6).
Notealsothatsomeinequalitiesjustfollowfromthedefinitionoftheobservables. Equation(5a),forinstance,expressesthe
usualboundonthethreecomponentsofthepolarizationvectoroftherecoilbaryon,whenthephotonis,say,right-handed.Sim-
ilarly,consideringaphotonwithobliquelinearpolarization,oneobtains(5b)ofwhichthelastinequalityin(4)isaconsequence.
Thefirstoftheinequalities(4)isimpliedbythemoreconstraininginequality(5g)whichcanbeunderstoodasfollows:ifthe
reversereactionisperformedwithanhyperonfullypolarizedalongthexaxis,thentheoutgoingphotonreceivesapolarization
whosecomponentsarepreciselyCY,Σγ,OY,leadingto(4). Similarly,forthesecondinequalityof(4),onecanadd(CY)2.
x x z
Fordemonstrating(5f), itisconceivabletorotatetheaxisinthe(x,z)plane,say(x,z) (u,v)suchthatCY ismaximal
andCY vanishes.Thenthereversereactioncanbeenvisaged,withapolarizationalongtheu→axisforthehyperon.uIfthevarious
v
componentsofthepolarizationoftheoutgoingphotonareconsidered,aninequality
(CY)2+(Σγ)2+(OY)2 1, (14)
u u ≤
isdeducedfromwhich(5f)followsbyneglectingthefirsttermandnoticingthat
(OY)2 =(OY)2+(OY)2. (15)
u x z
IV. CONFRONTINGDATA
Inarecentpaper,theLEPScollaborationpublishedresultsforthephotoproductionreactionγn K+Σ−forincidentphoton
energyfrom1.5to2.4GeV[9]. ItisremarkablethatΣγ isclosetoΣγ =1inawiderangeofene→rgiesforcentre-of-massangle
suchthatcos(θ )>0.6. Equation(6)impliesthatAN PY.
cm
TheCLAScollaborationhasstudiedthereactionγp ≃K+Y measuredforcenter-of-massenergiesW between1.6and2.53
GeV andfor 0.85 < cosθc.m < +0.95[10, 11, 12].→Inadditionto thedifferentialcross-section,threespinobservablesare
measured,nam−elyPY andthKedoublecorrelationparametersCY andCY betweenthecircularlypolarizedphotonandtherecoil
x z
baryonspin alongthe directionsxˆ and zˆin the scattering plane. In these preliminarydata, it is observedthat some valuesof
P(CΛΛa)r2e+ve(rCyΛla)r2ge, f0o.r5,ewxahmicphlesePemΛs=inc−o0n.s7i3steanttWwit=hth1e.7v3alGueeVCΛand c1o,srθepKc.omrte=di+n0R.e3f0..[1F2r]o.mOfEcqo.u(r5sae),,wtheisharveseutlot lweaaditsfotor
x z ≤ z ∼
thefinaldatabeforedrawinganydefiniteconclusion,butweurgetheCLASCollaborationtomakesurethattheaboveconstraint
isindeedfulfilled.
Veryrecently,Schumacher[13]stressedthatintheCLASdata
(PY)2+(CY)2+(CY)2 1. (16)
x z ≃
Itfollowsfrom(9a)that
(AN)2 (Σγ)2+(OY)2+(OY)2, (17)
≃ x z
shouldalsobeverified.
The GRAAL collaboration has measured the beam asymmetry Σγ for π0 photoproduction[14]. Large positive values are
often found, for instance Σγ = 0.990 0.069 at incident energy E = 1344MeV and center-of-mass angle θ = 45.1◦.
γ cm
From Eq. (6), this implies the non trivi±al equality AN PY of analyzingpower and recoil nucleon polarization. The same
collaborationisabouttopublishits resultsonthreespin≃observablesforkaonphotoproduction: thehyperonpolarizationPY,
the beam asymmetry Σγ and the correlation coefficients OY and OY between the photonoblique linear polarization and the
x z
polarizationofthehyperon[15]. Theinequalities(4)areseeminglyfarfromsaturation.
V. OUTLOOK
Thenewdataonphotoproductionofpseudoscalarmesons,inparticularγN KY includeseveralspinobservables,which
→
can discriminateamongthe differentmodels. Before undertakinga phenomenologicalanalysis, it is crucialto check that the
variousspin observablesare compatible. The ultimate criterion would consist of obtaining a consistent set of amplitudes. A
moreimmediatetestistocheckwhetherallthepossiblemodel-independentinequalitiesarefulfilledbythedata.
A similar approach can be applied to other exclusive reactions for which several spins are measured, in particular the
strangeness-exchangereaction p¯p ΛΛ using a polarized target. This study has been done at the LEAR facility of CERN
→
andwillprobablyberesumedathigherenergyattheforthcomingFAIRcomplexatDarmstadt. Withminorchanges,thesame
formalism of inequalitiesalso holdsfor the observablesof inclusive reactions, and for the spin-dependentstructure functions
andpartondistributions[1].
Atfirstsight,derivinginequalitiesamongobservablesisamerealgebraicexerciseappliedtothebilinearrelationsexpressing
these observables in terms of the independent amplitudes, once the symmetry constraints have been imposed. In fact, these
inequalities reflect the positivity of the density matrix for the initial and final states of the reaction in the direct and crossed
channels.Foranyreaction,thespinstateintheinitialstate,factorizableorentangled,undergoesaquantumtransformation,and
isthussubmittedtothegeneralrulesonthetransmissionofinformationinelementaryquantumprocesses.
APPENDIXA:OBSERVABLES
The definition of Σγ, AN, ... CY is taken from [16]. They differ in sign with [5] concerning L , G, E, CY and CY.
x x x z
(000) = (333)=1 (A1a) z =(110) = +(223)= G (A1i)
| − | h⊘ i | | −
= yy′ =(300) = (033)= Σγ (A1b) x =(120) = (213)= H (A1j)
h⊖i −h i | − | − h⊘ i | − | −
y = y′ =(030) = (303)=+AN (A1c) z =(210) = (123)=+E (A1k)
h i −h⊖ i | − | h⊙ i | − |
y′ = y =(003) = (330)=+PY (A1d) x =(220) = +(113)=+F (A1l)
h i −h⊖ i | − | h⊙ i | |
zz′ =(011) = (322)=+L (A1e) z′ =(101) = (232)= OY (A1m)
h i | − | z h⊘ i | − | − z
zx′ =(012) = +(321)=+L (A1f) x′ =(102) = +(231)= OY (A1n)
h i | | x h⊘ i | | − x
xz′ =(021) = +(312)=+T (A1g) z′ =(201) = +(132)=+CY (A1o)
h i | | z h⊙ i | | z
xx′ =(022) = (311)=+T (A1h) x′ =(202) = (131)=+CY (A1p)
h i | − | x h⊙ i | − | x
The symbol x′ , for instance, is an intuitivenotation for the correlationbetween the obliquepolarizationof the photon(at
h⊘ i
+π/4)andthepolarizationtowardsxˆofthefinalbaryon.
Inthetransversitybasis,owingto(1),R=R+ R−,whereR−actsinthesubspacespannedby
⊕
π+ , π + , n++ and n , (A2)
| −i | − i | i | −−i
andwhereR+,actingonthecomplementarysubspace,vanishesidentically.
ThematrixR− isgivenbyTableI,and(0+3, 1 i2 1+i2),forinstance,isacompactnotationfor
− |
(011)+(311) i(021) i(321)+i(012)+i(312)+(022)+(322). (A3)
| | − | − | | | | |
TABLEI:Sub-matrixR−ofthedensitymatrixR.
π+− π−+ n++ n−−
π+− (0+3, 0+3|0−3) (0+3, 1−i2|1−i2) (1−i2, 0+3|1−i2) (1−i2, 1−i2|0−3)
π−+ (0+3, 1+i2|1+i2) (0+3, 0−3|0+3) (1−i2, 1+i2|0+3) (1−i2, 0−3|1+i2)
n++ (1+i2, 0+3|1+i2) (1+i2, 1−i2|0+3) (0−3, 0+3|0+3) (0−3, 1−i2|1+i2)
n−− (1+i2, 1+i2|0−3) (1+i2, 0−3|1−i2) (0−3, 1+i2|1−i2) (0−3, 0−3|0−3)
The 16 equivalencesin Eqs.(A1a-p) reflect the invarianceof (λ,µν) underthe substitutions: 0 3, 3 0, 1 i2,
| → − → − →
2 i1forλandµand0 3,3 0,1 i2,2 i1forν. Itcomesfromthevanishingofamplitudesbetweenaneven
→ − →− → − → − →
numberofparticleswithnegativeσ3,expressedby
σ3 σ3 σ3 R=Rσ3 σ3 σ3 = R. (A4)
⊗ ⊗ ⊗ ⊗ −
Usingtheseequivalences,onecansimplifyRbyreplacingm nbym(m=0or1)onceineachboxofTableI. Theresultis
±
equaltoR/2.
ACKNOWLEDGMENTS
Informativediscussionswith ReinhardSchumacher(CLAS), AnnickLleres, Jean-PaulBocquetandDominiqueRebreyend
(GRAAL), TakashiNakano(LEPS),and Oleg Teryaevare gratefullyacknowledged,as well as the commentsby Muhammad
Asghar.
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