Hadronic structure functions in the e(+) e(-) -> (Lambda)over-bar Lambda reaction

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Physics Letters B

www.elsevier.com/locate/physletb

Hadronic structure functions in the e ⁺ e ⁻ → ¯ ^reaction

Göran Fäldt

^∗

, Andrzej Kupsc

DivisionofNuclearPhysics,DepartmentofPhysicsandAstronomy,UppsalaUniversity,Box 516,75120Uppsala,Sweden

a r t i c l e i n f o a b s t ra c t

Articlehistory:

Received28February2017

Receivedinrevisedform16April2017 Accepted4June2017

Availableonline8June2017 Editor:J.-P.Blaizot

Keywords:

Hadronproductionine⁻e⁺interactions Hadronicdecays

Cross-section distributionsare calculated forthereactione⁺e⁻→ ^J/ψ→ ¯(→ ¯^p

π

⁺)(→^p

π

⁻),and relatedannihilationreactionsmediatedbyvectormesons.Thehyperon-decaydistributionsdependona number ofstructurefunctions thatarebilinear inthe,possiblycomplex,psionic formfactorsG^ψ_M and G^ψ_E oftheLambdahyperon.Therelativesizeandrelativephaseoftheseformfactorscan beuniquely determined from theunpolarized joint-decay distributions ofthe Lambda and anti-Lambda hyperons.

Alsothedecay-asymmetryparametersofLambdaandanti-Lambdahyperonscanbedetermined.

©^{2017TheAuthors.PublishedbyElsevierB.V.ThisisanopenaccessarticleundertheCCBYlicense} (http://creativecommons.org/licenses/by/4.0/).FundedbySCOAP³.

1. Introduction

Twohadronicformfactors,commonlycalledGM

(

s

)

andGE

(

s

)

, areneededforthedescriptionoftheannihilationprocesse⁻e⁺→

 ¯ 

, Fig. 1a, andby varying the c.m. energy√

s, their numerical valuescan inprinciple be determined forall s valuesabove

 ¯ 

threshold.Forthegeneralcaseofannihilationviaanintermediate photon,thejoint

(

→^p

π

⁻

) ¯ (

→ ¯^p

π

⁺

)

decaydistributionswere calculated and analyzed in Ref. [1], using methods developed in [2,3].Recently,afirstattempttocalculatethehyperonformfactors GM

(

s

)

andGE

(

s

)

inthetime-likeregionwasreportedinRef.[4].

Previously, theinteresting specialcaseof annihilationthrough an intermediate J

/ψ

or

ψ(

2S

)

, Fig. 1b, hasbeen investigated in severaltheoretical [5,6]andexperimental papers [7–9].This pro- cess has also been used for determination of the anti-Lambda decay-asymmetry parameter and for CP symmetry tests in the hyperon system. A precise knowledge of the Lambda decay- asymmetry parameter is needed for studies of spin polarization in



⁻,



⁻,and



⁺_c decays.

Presently, a collected data sample of 1

.

31×^{109 J}

/ψ

events [10] by the BESIII detector[11] permitshigh-precision studies of spincorrelations.

Intheexperimental workreferredtoabove,thejoint-hyperon- decaydistributionsconsideredarenotthemostgeneralonespos- sible, butseem to be curtailed.Incomplete distribution functions donotpermitareliabledeterminationoftheformfactorsandwe

*

Correspondingauthor.

E-mailaddresses:goran.faldt@physics.uu.se(G. Fäldt), andrzej.kupsc@physics.uu.se(A. Kupsc).

Fig. 1. Graphdescribingthereactione⁺e⁻→ ¯^{;a)generalcase,andb)mediated} bythe J/ψresonance.

thereforesuggesttofittheexperimentaldatatothegeneraldistri- butiondescribedin[1],andfurtherelaboratedbelow.

Since thephoton andthe J

/ψ

are both vector particles,their corresponding annihilation processeswillbe similar. Infact, by a simplesubstitution,thecross-sectiondistributionsinRef.[1],valid inthephotoncase,aretransformedintodistributionsvalidinthe J

/ψ

case,butexpressedinthecorrespondingpsionicformfactors G^ψ_M andG^ψ_E.

http://dx.doi.org/10.1016/j.physletb.2017.06.011

0370-2693/©^{2017TheAuthors.PublishedbyElsevierB.V.ThisisanopenaccessarticleundertheCCBYlicense}(http://creativecommons.org/licenses/by/4.0/).Fundedby SCOAP³.

In order to specify events and compare measured data with theoretical predictions, we need distribution functions expressed insome specific coordinate system. For this purpose we employ thecoordinatesystemintroducedin[1].Manyinvestigations em- ploydifferentcoordinatesystemsfortheLambdaandanti-Lambda decays,acustomwhichinouropinioncanleadtoconfusion.

Ourcalculationisperformedintwosteps.Aftersomeprelimi- nariesweturntotheinclusiveprocess ofleptonannihilationinto polarized hyperons. The results obtained are the starting point for the calculation of exclusive annihilation, i.e. the distribution forthe hyperon-decay products. Our method of calculation con- sistsin multiplying thehyperon-production distribution withthe hyperon-decaydistributions,averagingoverintermediatehyperon- spindirections.Themethodisreferredtoasfolding.

2. Basicnecessities

Resolvingthehyperonvertexin Fig. 1auncoversa numberof contributions. The one of interest to usis described by the dia- gramofFig. 1b,wherebythephotoninteractionwiththehyperons is mediated by the J

/ψ

vector meson, andthe coupling of the initial-stateleptonstothe J

/ψ

relatedtothedecay J

/ψ

→^e+e−. Fora J

/ψ

decaythrough anintermediate photon,tensorcou- plingscan be ignored.Thus, theeffectivecouplingofthe J

/ψ

to theleptonsisthesameasthatforthephoton,providedwereplace theelectricchargee_em byacouplingstrengthe_ψ,



^e_μ

(

k₁

,

k₂

) = −

^ieψ

γ

_μ

,

(2.1)

withe_ψ determinedbythe J

/ψ

→^{e+e−decay(see}Appendix A).

Atthe J

/ψ

-hyperonvertextwo form factorsare possible and theyarebothconsidered.WefollowRef.[1]inwritingthehyperon vertexas



^_μ

(

p₁

,

p₂

) = −

^ieg



G^ψ_M

γ

μ

−

^2M

Q²

(

G^ψ_M

−

^Gψ_E

)

Q_μ



,

(2.2)

withP=^p1+^p2,andQ =^p1−^p2,andM theLambdamass.The argumentoftheformfactorsequalss=^{P2.Thecouplingstrength} eginEq.(2.2)isdeterminedbythehadronic-decayratefor J

/ψ

→

 ¯ 

(seeAppendix A).

InRef.[1]polarizationsandcross-sectiondistributionswereex- pressed in terms of structure functions, themselves functions of theform factors G^ψ_M andG^ψ_E.Here, we shall introduce combina- tionsofformfactorscalledD,

α

,and



,whichareemployedby theexperimentalgroups[7–9]aswell.

Thestrengthofformfactorsismeasuredby D

(

s

)

,

D

(

s

) =

^s

 

^GψM

 

²

+

^4M2

 

^GψE

 

²

,

(2.3) afactorthatmultipliesallcross-sectiondistributions.Theratioof formfactorsismeasuredby

α

,

α ₌

s

 

^Gψ_M

 

²

−

^4M2

 

^Gψ_E

 

² s

 

^Gψ_M

 

²

+

^4M2

 

^Gψ_E

 

²

,

(2.4)

with

α

satisfying−¹≤

α

_≤1.Therelativephaseofformfactorsis measuredby



,

G^ψ_E

G^ψ_M

=

^ei



 

G^ψ_E G^ψ_M

 

 .

(2.5)

The diagram ofFig. 1 representsa J

/ψ

exchange of momen- tum P . J

/ψ

beingavectormeson,itspropagatortakestheform

g_μν

−

^PμP_ν

/

m²_ψ

s

−

^m2_ψ

+

^imψ

(ψ ) ,

(2.6)

wherem_ψ isthe J

/ψ

mass,and

(ψ)

thefull widthofthe J

/ψ

. However,sincethe J

/ψ

couplestoconservedleptonandhyperon currents,thecontributionfromthePμ Pν termvanishes.Inconclu- sion,the matrixelement fore⁺e⁻ annihilationthrough aphoton willbestructurallyidenticaltothatforannihilationthrougha J

/ψ

providedwemakethereplacement

e_ψe_g

s

−

^m2_ψ

+

^im2_ψ

(ψ ) →

^e2em

s

,

(2.7)

whereeemistheelectriccharge.

3. Crosssectionfore⁺e⁻→ (^s1) ¯(s2)

Our first taskis to calculate the cross-section distribution for e⁺e⁻ annihilationintopolarizedhyperons.From thesquaredma- trixelement|M|^{2 forthisprocessweremoveafactor}

K

ψ,toget d

σ =

¹

2s

K

ψ

| M

red

|

²dLips

(

k1

+

k2

;

p1

,

p2

),

(3.8) withdLips thephase-spacefactor,withs=^P2,andwith

K

ψ

=

^e

2 ψe²_g

(

s

−

^m2_ψ

)

²

+

^m2_ψ



²

(

m_ψ

) .

(3.9) Thesquareofthereducedmatrixelementcanbefactorizedas

 _M

_red

(

e⁺e⁻

→ (

s1

)(

s2

)) 

²

=

L

·

K

(

s1

,

s2

),

(3.10) withL

(

k₁

,

k₂

)

andK

(

p₁

,

p₂;^s1

,

s₂

)

leptonandhadrontensors,and s1 ands2 hyperonspinfour-vectors.

Leptontensorincludingaveragesoverleptonspins;

L_νμ

(

k₁

,

k₂

) =

¹₄^Tr



γ

ν

/

k₁

γ

μk

/

₂



=

^k1νk_2μ

+

^k2νk_1μ

−

¹₂^sgνμ

.

(3.11) Hadrontensorforpolarizedhyperons;

K_νμ

(

s₁

,

s₂

) =

^Tr





^_ν

( /

p₁

+

^M

)

¹₂

(

1

+ γ

₅

/

s₁

)

× 

^_μ

( /

p2

−

^M

)

¹₂

(

1

+ γ

5

/

s2

)



/

e²_g

,

(3.12)

withp1ands1 momentumandspinfortheLambda hyperonand p2 ands2 correspondinglyfortheanti-Lambdahyperon.Thetrace itselfissymmetricinthetwohyperonvariables.

The spin four-vector s

(

p

,

n

)

of a hyperon of mass M, three- momentump,andspindirectionn initsrestsystem,is

s

(

p

,

n

) =

^n

M

( |

^p

|;

^Ep

ˆ ) + (

⁰

;

ⁿ⊥

).

(3.13) Here,longitudinalandtransversedesignationsrefertothep direc-ˆ tion; n_=ⁿ· ˆ^{p andn}⊥=ⁿ− ˆ^p

(

n· ˆ^p

)

are parallel andtransverse components of the spin vector n. Also, observe that the four- vectorsp ands areorthogonal,i.e. p·^s

(

p

)

=^0.

Fortheevaluationofthematrixelementweturntotheglobal c.m. system wherekinematics simplifies. Here, three-momenta p andk aredefinedsuchthat

p1

= −

^p2

=

^p

,

(3.14)

k1

= −

k2

=

k

,

(3.15)

andscatteringangleby,

cos

θ = ˆ

^p

· ˆ

^k

.

(3.16)

Thephase-spacefactorbecomes dLips

(

k1

+

^k2

;

^p1

,

p2

) =

^p

32

π

²kd

,

(3.17)

withp= |^p|^andk= |^k|^.

The matrix element in Eq.(3.10) can be written as a sum of four terms that depend on the hyperon spin directions in their respectiverestsystems,n₁andn₂,

 _M

_red

(

e⁺e⁻

→ (

^s1

)(

s2

))

²

=

^sD

(

s

)



H⁰⁰

(

0

,

0

) +

^H05

(

n₁

,

0

) +

^H50

(

0

,

n₂

) +

^H55

(

n₁

,

n₂

)

 .

(3.18) Thepolarization distributions H^ab are eachexpressedin termsof structurefunctionsthatdependonthescatteringangle

θ

,theratio function

α (

s

)

, andthe phasefunction

(

s

)

. There are sixsuch structurefunctions,

R =

¹

+ α

cos²

θ,

(3.19)

S = 

1

− α

²sin

θ

cos

θ

sin

(),

(3.20)

T

1

= α ₊

cos²

θ,

(3.21)

T

2

= − α

sin²

θ,

(3.22)

T

3

=

¹

+ α ,

(3.23)

T

4

= 

1

− α

²cos

θ

cos

().

(3.24)

The definitions andnotations are slightlydifferentfrom those of Ref.[1].Inparticular,afactorsD

(

s

)

hasbeenpulledoutfromthe structurefunctions,andplacedinfrontofthesumofthepolariza- tiondistributionsofEq.(3.18).

Thepolarizationdistributions H^abare,

H⁰⁰

= R

(3.25)

H⁰⁵

= S



1

sin

θ (

p

ˆ × ˆ

^k

) ·

ⁿ1

(3.26) H⁵⁰

= S



1

sin

θ (

^p

ˆ × ˆ

^k

) ·

ⁿ2

(3.27) H⁵⁵

=

T

1n1

· ˆ

^pn2

· ˆ

^p

+ T

2n1⊥

·

ⁿ2⊥

+ T

3n_1⊥

· ˆ

^kn2⊥

· ˆ

^k

+ T

4



n1

· ˆ

pn2⊥

· ˆ

k

+

n2

· ˆ

pn1⊥

· ˆ

k



(3.28)

Transversecomponentsn_1⊥andn_2⊥areorthogonaltotheLambda hyperonmomentum p intheglobalc.m. system. Also,transverse n_⊥ andlongitudinaln_= ˆ^p·n polarization componentsenterdif- ferently, since they transform differently under Lorentz transfor- mations.

Allpolarization observables,single anddouble,canbedirectly readoffEqs.(3.25)–(3.28),andtherearenootherpossibilities.The setofscalarproductsinvolvingn1andn2iscomplete.Asanexam- ple,the Lambda-hyperon polarization isobtainedfrom Eq.(3.26) whichshowsthatthepolarizationisdirectedalongthenormalto thescatteringplane,^pˆ× ˆ^{k,andthatthevalue ofthe}polarization is

P_

(θ ) = _R ^S =

√

1

− α

²cos

θ

sin

θ

1

+ α

cos²

θ

^sin

()

(3.29)

From Eq. (3.27) we conclude that the polarization of the anti- Lambda isexactlythe same,butthen oneshould rememberthat p is the momentum of the Lambda hyperon but −^{p that ofthe} anti-Lambda.

Fig. 2. Graph describing the reaction e⁺e⁻→ (→^pπ⁻) ¯(→ ¯^pπ⁺).

4. Foldingofdistributions

Our next taskis tocalculate the cross-section distributionfor e⁺e⁻ annihilation into hyperon pairs, followed by the hyperon decays intonucleon–pion pairs.This reactionis described by the connecteddiagramofFig. 2.

Again,we extracta prefactor,

K = K

ψ

K

1

K

2,fromthesquared matrixelement,writing

| M |

²

= K | M

red

|

²

.

(4.30)

The prefactor originates, as before withthe propagator denomi- nators. Due to the smallness of the hyperon widths each of the hyperonpropagatorscan,aftersquaring,beapproximatedas,

K

i

=

¹

(

p²_i

−

^M2

)

²

+

^M2



²

(

M

) =

²

π

2M

(

M

) δ(

p²_i

−

^M2

).

(4.31) Effectively, this approximation puts the hyperons on their mass shells.

Hyperon-decay distributions are obtained by a folding calcu- lation, whereby hyperon-production and -decay distributions are multiplied together andaveragedover theintermediate hyperon- spindirections.ItwasprovedinRef.[2]thatthefoldingprescrip- tion gives the same result as the evaluation of the connected- Feynman-diagram expression. Hence, summing over final hadron spins,

| M |

²

= 

±s1,±s2

  _M (

e⁺e⁻

→ (

^s1

) ¯ (

s2

))

²

×  _M ((

s1

) →

^p

π

⁻

)

²

 _M ( ¯ (

s2

) → ¯

^p

π

⁺

)

²



n1n2

.

(4.32) Productionanddecaydistributionsare,

 _M (

e⁺e⁻

→ (

^s1

) ¯ (

s2

))

²

=

^L

·

^K

(

s1

,

s2

),

(4.33)

 _M ((

s₁

) →

^p

π

⁻

^)

²

=

^R[1

− α

1l₁

·

^s1

/

l_]

,

(4.34)

 _M ( ¯ (

s₂

) → ¯

^p

π

⁺

) 

²

=

^R[1

− α

2l₂

·

^s2

/

l_]

,

(4.35) withl_ thedecay momentumin the Lambda restsystem. R_ is determinedbytheLambdadecayrate.

The notation in Eq. (4.34) is the following; s₁ denotes the Lambda four-spinvector,l1 thefour-momentumofthedecaypro- ton,and

α

1thedecay-asymmetryparameter.Similarlyfortheanti- LambdahyperonparametersofEq.(4.35).

We evaluate the hyperon-decay distributions in the hyperon- restsystems,where

 _M ((

s1

) →

^p

π

⁻

)

²

=

^R



1

+ α

1

ˆ

_l1

·

ⁿ1



,

(4.36)

 _M ( ¯ (

s₂

) → ¯

^p

π

⁺

) 

²

=

^R



1

+ α

2

ˆ

_l2

·

ⁿ2



,

(4.37)

where ˆl1=^l1

/

l_ is the unit vector in the direction of the pro- ton momentum in the Lambda-rest system, andcorrespondingly fortheanti-Lambdacase.

Angular averages in Eq.(4.32) are calculated accordingto the prescription

 (

n

·

l

)

n



n

=

l

.

(4.38)

The foldingof theproduction distributions, Eqs. (3.25)–(3.28), withthedecaydistributions,Eqs.(4.36,4.37),yields

| M

red

|

²

=

^sD

(

s

)

R²_



G⁰⁰

+

^G05

+

^G50

+

^G55

,

(4.39)

withtheG^ab functionsdefinedas

G⁰⁰

= R ,

(4.40)

G⁰⁵

= α

1

S



1

sin

θ (

p

ˆ × ˆ

^k

) · ˆ

^l1

,

(4.41)

G⁵⁰

= α

2

S



1

sin

θ (

p

ˆ × ˆ

^k

) · ˆ

^l2

,

(4.42)

G⁵⁵

= α

1

α

2

T

1

ˆ

l₁

· ˆ

^p

ˆ

l₂

· ˆ

^p

+ T

2

ˆ

l_1⊥

· ˆ

^l2⊥

+ T

3

ˆ

_l1⊥

· ˆ

^k

ˆ

_l2⊥

· ˆ

^k

+ T

4

 ˆ

_l1

· ˆ

^p

ˆ

_l2⊥

· ˆ

^k

+ ˆ

^l2

· ˆ

^p

ˆ

_l1⊥

· ˆ

^k



.

(4.43)

Thus, we concludethe connection between joint-hadronproduc- tionandjoint-hadrondecaydistributionssimplytobe,

G^ab

(ˆ

l1

;ˆ

^l2

) =

^Hab

(

n1

→ α

1

ˆ

_l1

;

ⁿ2

→ α

2

ˆ

_l2

).

(4.44) Werepeatthenotation;p andk aremomentaofLambdaand electronintheglobalc.m. system;l1 andl2 aremomentaofpro- tonandanti-proton inLambda andanti-Lambdarestsystems;or- thogonalmeansorthogonaltop;andstructurefunctions

R

^,

S

^,and

T

^{arefunctionsof}

θ

,

α

,and



.Theangularfunctionsmultiply- ingthestructurefunctionsformasetofsevenmutuallyorthogonal functions,whenintegratedovertheprotonandanti-protondecay angles.

5. Crosssectionfore⁺e⁻→ (→^p

π

⁻) ¯(→ ¯^p

π

⁺)

Ourlast taskis to find the properly normalized cross-section distribution.Westartfromthegeneralexpression,

d

σ ₌

¹

2s

K | M

red

|

^2dLips

(

k1

+

^k2

;

^q1

,

l1

,

q2

,

l2

),

(5.45) with dLips the phase-space density for four final-state particles.

Theprefactor

K

containsonthemassshelldeltafunctionsforthe twohyperons.Thiseffectivelyseparatesthephasespaceintopro- ductionand decayparts. Repeating themanipulations ofRef. [2]

weget d

σ =

¹

64

π

²

p k

α

g

α

ψ

(

s

−

^m2_ψ

)

²

+

^m2_ψ



²

(ψ )







_¯



²

(

M

) ·

·

⎛

⎝

D

(

s

) 

a,b

G^ab

⎞

⎠

d



_d



1d



2

,

(5.46)

withk andp theinitial- andfinal-statemomenta;



thehyperon scatteringangleintheglobalc.m.system;



1and



2decayangles measured in the restsystems of



and ¯;



_ and



_¯ channel widths;and

(

M

)

and

(ψ)

totalwidths.

Integration over the angles



1 makes the contributions from thefunctionsG⁰⁵ andG⁵⁵disappear[2],andcorrespondingly for the angles



2.Integration over both angular variables results in thecross-sectiondistributionforthereactione⁺e⁻→^J

/ψ

→  ¯^.

Suppose weintegrate over theangles



2.Then, thepredicted hyperon-decaydistributionbecomesproportionaltothesum G⁰⁰

+

^G05

= R 

1

+ α

1P_

· ˆ

^l1



,

(5.47)

P_

= ^S

R ,

(5.48)

withthepolarization P_asinEq.(3.29),andthepolarization vec- torP directedalongthenormaltothescatteringplane

6. Differentialdistributions

Wefirstdefineourcoordinatesystem.Thescatteringplanewith thevectorsp andk makeup thexz-plane, withthe y-axisalong thenormaltothescatteringplane.Wechoosearight-handedco- ordinatesystemwithbasisvectors

ez

= ˆ

^p

,

(6.49)

ey

=

¹

sin

θ (

p

ˆ × ˆ

k

),

(6.50)

e_x

=

¹

sin

θ (

p

ˆ × ˆ

^k

) × ˆ

^p

.

(6.51)

Expressedintermsofthemtheinitial-statemomentum

k

ˆ =

^sin

θ

e_x

+

^cos

θ

e_z

.

(6.52)

Thiscoordinatesystemisusedforfixingthedirectionalangles ofthedecayprotonintheLambdarestsystem,andthedecayanti- protonintheanti-Lambdarestsystem.Thesphericalanglesforthe protonare

θ

1 and

φ

1,andthecomponentsoftheunitvectorindi- rectionofthedecay-protonmomentumare,

ˆ

l1

= (

cos

φ

1sin

θ

1

,

sin

φ

1sin

θ

1

,

cos

θ

1

),

(6.53) sothat

ˆ

l_1⊥

= (

^cos

φ

1sin

θ

1

,

sin

φ

1sin

θ

1

,

0

).

(6.54) Themomentumofthedecayprotonisbydefinitionl1=^lˆ_l1.This same coordinate system is used for defining the directional an- glesofthedecayanti-protonintheanti-Lambdarestsystem,with sphericalangles

θ

2 and

φ

2.

Now,wehaveallingredientsneededforthecalculationofthe G functionsofEqs.(4.40)–(4.43),thefunctionsthatintheendde- terminethecross-sectiondistributions.

An event of the reaction e⁺e⁻→ (→^p

π

⁻

) ¯ (

→ ¯^p

π

⁺

)

is specified by the fivedimensional vector ξ= (θ,



1

, 

2

)

,and the differential-cross-sectiondistributionassummarizedbyEq.(4.39) reads,

d

σ ∝ W (ξ )

dcos

θ

d



1d



2

.

At the moment, we are not interested in the absolute normal- izationofthedifferentialdistribution.Thedifferential-distribution function

W(ξ)

^{isobtainedfromEqs.}(4.40)–(4.43)andcanbeex- pressedas,

W (ξ ) = F

0

(ξ ) + α F

5

(ξ ) + α

1

α

2

 F

1

(ξ ) + 

1

− α

²cos

() F

2

(ξ ) + α F

6

(ξ )



+ 

1

− α

²sin

() ( α

1

F

3

(ξ ) + α

2

F

4

(ξ )) ,

(6.55) usingasetofsevenangularfunctions

F

k

(ξ )

definedas:

F

0

(ξ ) =

1

F

1

(ξ ) =

^sin2

θ

sin

θ

1sin

θ

2cos

φ

1cos

φ

2

+

^cos2

θ

cos

θ

1cos

θ

2

F

2

(ξ ) =

^sin

θ

cos

θ (

sin

θ

1cos

θ

2cos

φ

1

+

^cos

θ

1sin

θ

2cos

φ

2

) F

3

(ξ ) =

^sin

θ

cos

θ

sin

θ

1sin

φ

1

F

4

(ξ ) =

^sin

θ

cos

θ

sin

θ

2sin

φ

2

F

5

(ξ ) =

^cos2

θ

F

6

(ξ ) =

^cos

θ

₁cos

θ

₂

−

^sin2

θ

sin

θ

₁sin

θ

₂sin

φ

₁sin

φ

₂

.

(6.56) ThedifferentialdistributionofEq.(6.55) involvestwoparame- tersrelatedtothee⁺e⁻→  ¯^{processthatcanbedeterminedby} data: theratio ofform factors

α

,and therelative phase of form factors



. In addition, the distribution function

W(ξ)

^{can be} usedtoextractseparately



and ¯decay-asymmetryparameters:

α

1 and

α

2,andhenceallowingadirecttestofCPconservationin thehyperondecays.

Thetermproportional tosin

()

inEq.(6.55)originateswith Eqs.(4.41)and(4.42),andcanberewrittenas,

S (θ ) ( α

1sin

θ

1sin

φ

1

+ α

2sin

θ

2sin

φ

2

) ,

withthe structure function

S

^{definedby Eq. (3.20). The relation} betweenthe structure functionsand the polarization P

(θ )

was discussed in Sect. 3, where it was shown that the polarization, P_

(θ )

ofEq.(3.29),andthepolarization vector,ey,arethe same for Lambda and anti-Lambda hyperons. This information tells us that



is polarized along the normal to the production plane, andthat the polarization vanishes at

θ

=^{0◦, 90◦ and 180◦. The} maximumvalue ofthepolarizationisforcos

θ

= ±¹

/(

2+

α )

,and

|^P

(θ )| <

²₃sin

()

.

It should be stressed that thesimplified distributions usedin previousanalyses,suchasRef.[9],assumethehyperonstobeun- polarized and therefore terms containing P

(θ )

are missing. In fact, such decay distributions, only permit the determination of two parameters: the ratioof form factors

α

, andthe product of hyperon-asymmetryparameters

α

1

α

2.

Inouropinion,theformulaspresentedinthislettershouldbe employed for the exclusive analysisof the newBESIII data [10].

Duetohugeandcleandatasamples:

(

440675±⁶⁷⁰

)

J

/ψ

→  ¯

and

(

31119±¹⁸⁷

) ψ(

3686

)

→  ¯^{,precisionvaluesforthedecay-} hadronic-formfactorscould beextractedaswellasprecisionval- uesfor



and ¯ decay-asymmetryparameters.The formulaspre-

sentedcouldeasilybegeneralizedtoneutrondecaysofthe



and to production of other J=¹

/

2 hyperons with analogous decay modes.

Acknowledgements

WearegratefultoTordJohanssonwhoprovidedtheinspiration forthiswork.

Appendix A

The coupling of the initial-state leptons to the J

/ψ

vector meson is determined by the decay J

/ψ

→^{e+e−. Assuming the} decay to go via an intermediate photon, Fig. 1b, we can safely ignore any tensor coupling. The vector coupling of the J

/ψ

to leptons is thereforethe same as forthe photon, if replacing the electric charge e_em by a coupling strength e_ψ. From the decay

J

/ψ

→^{e+e−onederives}

α

_ψ

₌

e²_ψ

/

4

π ₌

3

(

J

/ψ →

^e+e−

)/

m_ψ

.

(A.57) In a similar fashion we relate the strength e_g of J

/ψ

cou- pling to the hyperonsto the decay J

/ψ

→  ¯^{. In analogywith} Eq.(A.57)weget

α

g

=

^e2g

/

4

π =

³



(

1

+

^2M2

/

m²_ψ

)



1

−

^4M2

/

m²_ψ



₋1

× (

J

/ψ →  ¯)/

m_ψ

.

(A.58)

When the



mass M is replaced by thelepton mass m_l=^{0 we} recoverEq.(A.57).

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