Hadronic structure functions in the e^+ e^- \rightarrow \barΛ Λ reaction
HHadronic structure functions in the e + e − → ¯ ΛΛ reaction G¨oran F¨aldt a , Andrzej Kupsc a a Division of Nuclear Physics, Department of Physics and Astronomy, Uppsala University, Box 516, 75120 Uppsala, Sweden
Abstract
Cross-section distributions are calculated for the reaction e + e − → J /ψ → ¯ Λ ( → ¯ p π + ) Λ ( → p π − ), and related annihilation reactionsmediated by vector mesons. The hyperon-decay distributions depend on a number of structure functions that are bilinear in the,possibly complex, psionic form factors G ψ M and G ψ E of the Lambda hyperon. The relative size and relative phase of these formfactors can be uniquely determined from the unpolarized joint-decay distributions of the Lambda and anti-Lambda hyperons. Alsothe decay-asymmetry parameters of Lambda and anti-Lambda hyperons can be determined. Keywords:
Hadron production in e − e + interactions, Hadronic decays
1. Introduction
Two hadronic form factors, commonly called G M ( s ) and G E ( s ), are needed for the description of the annihilation pro-cess e − e + → Λ ¯ Λ , Fig. 1a, and by varying the c.m. energy √ s ,their numerical values can in principle be determined for all s values above Λ ¯ Λ threshold. For the general case of annihilationvia an intermediate photon, the joint Λ ( → p π − ) ¯ Λ ( → ¯ p π + ) de-cay distributions were calculated and analyzed in Ref.[1], usingmethods developed in [2, 3]. Recently, a first attempt to calcu-late the hyperon form factors G M ( s ) and G E ( s ) in the time-likeregion was reported in Ref. [4].a) γe + ( k ) e − ( k ) ¯Λ( p )Λ( p ) b) J/ψe + ( k ) e − ( k ) ¯Λ( p )Λ( p ) Figure 1: Graph describing the reaction e + e − → ¯ ΛΛ ; a) genaral case, and b)mediated by the J /ψ resonance. Email addresses: [email protected] (G¨oran F¨aldt), [email protected] (Andrzej Kupsc)
Previously, the interesting special case of annihilationthrough an intermediate J /ψ or ψ (2 S ), Fig. 1b, has been in-vestigated in several theoretical [5, 6] and experimental papers[7, 8, 9]. This process has also been used for determination ofthe anti-Lambda decay-asymmetry parameter and for CP sym-metry tests in the hyperon system. A precise knowledge of theLambda decay-asymmetry parameter is needed for studies ofspin polarization in Ω − , Ξ − , and Λ + c decays.Presently, a collected data sample of 1 . × J /ψ events[10] by the BESIII detector [11] permits high-precision studiesof spin correlations.In the experimental work referred to above, the joint-hyperon-decay distributions considered are not the most generalones possible, but seem to be curtailed. Incomplete distributionfunctions do not permit a reliable determination of the formfactors and we therefore suggest to fit the experimental data tothe general distribution described in [1], and further elaboratedbelow.Since the photon and the J /ψ are both vector particles, theircorresponding annihilation processes will be similar. In fact, bya simple substitution, the cross-section distributions in Ref. [1],valid in the photon case, are transformed into distributions validin the J /ψ case, but expressed in the corresponding psionic formfactors G ψ M and G ψ E .In order to specify events and compare measured datawith theoretical predictions, we need distribution functions ex-pressed in some specific coordinate system. For this purpose weemploy the coordinate system introduced in [1]. Many inves-tigations employ di ff erent coordinate systems for the Lambdaand anti-Lambda decays, a custom which in our opinion canlead to confusion.Our calculation is performed in two steps. After some pre-liminaries we turn to the inclusive process of lepton annihila-tion into polarized hyperons. The results obtained are the start-ing point for the calculation of exclusive annihilation, i.e. thedistribution for the hyperon-decay products. Our method ofcalculation consists in multiplying the hyperon-production dis- a r X i v : . [ h e p - ph ] M a y ribution with the hyperon-decay distributions, averaging overintermediate hyperon-spin directions. The method is referred toas folding.
2. Basic necessities
Resolving the hyperon vertex in Fig.1a uncovers a numberof contributions. The one of interest to us is described by thediagram of Fig.1b, whereby the photon interaction with thehyperons is mediated by the J /ψ vector meson, and the cou-pling of the initial-state leptons to the J /ψ related to the decay J /ψ → e + e − .For a J /ψ decay through an intermediate photon, tensor cou-plings can be ignored. Thus, the e ff ective coupling of the J /ψ to the leptons is the same as that for the photon, provided wereplace the electric charge e em by a coupling strength e ψ , Γ e µ ( k , k ) = − ie ψ γ µ , (2.1)with e ψ determined by the J /ψ → e + e − decay (see Appendix).At the J /ψ -hyperon vertex two form factors are possible andthey are both considered. We follow Ref. [1] in writing thehyperon vertex as Γ Λ µ ( p , p ) = − ie g (cid:104) G ψ M γ µ − MQ ( G ψ M − G ψ E ) Q µ (cid:105) , (2.2)with P = p + p , and Q = p − p , and M the Lambda mass.The argument of the form factors equals s = P . The couplingstrength e g in Eq.(2.2) is determined by the hadronic-decay ratefor J /ψ → Λ ¯ Λ (see Appendix).In Ref. [1] polarizations and cross-section distributions wereexpressed in terms of structure functions, themselves functionsof the form factors G ψ M and G ψ E . Here, we shall introduce com-binations of form factors called D , α , and ∆Φ , which are em-ployed by the experimental groups [7, 8, 9] as well.The strength of form factors is measured by D ( s ), D ( s ) = s (cid:12)(cid:12)(cid:12) G ψ M (cid:12)(cid:12)(cid:12) + M (cid:12)(cid:12)(cid:12) G ψ E (cid:12)(cid:12)(cid:12) . (2.3)a factor that multiplies all cross-section distributions. The ratioof form factors is measured by α , α = s (cid:12)(cid:12)(cid:12) G ψ M (cid:12)(cid:12)(cid:12) − M (cid:12)(cid:12)(cid:12) G ψ E (cid:12)(cid:12)(cid:12) s (cid:12)(cid:12)(cid:12) G ψ M (cid:12)(cid:12)(cid:12) + M (cid:12)(cid:12)(cid:12) G ψ E (cid:12)(cid:12)(cid:12) , (2.4)with α satisfying − ≤ α ≤
1. The relative phase of formfactors is measured by ∆Φ , G ψ E G ψ M = e i ∆Φ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) G ψ E G ψ M (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . (2.5)The diagram of Fig. 1 represents a J /ψ exchange of momen-tum P . J /ψ being a vector meson, its propagator takes the form g µν − P µ P ν / m ψ s − m ψ + im ψ Γ ( ψ ) , (2.6) where m ψ is the J /ψ mass, and Γ ( ψ ) the full width of the J /ψ .However, since the J /ψ couples to conserved lepton and hy-peron currents, the contribution from the P µ P ν term vanishes.In conclusion, the matrix element for e + e − annihilation througha photon will be structurally identical to that for annihilationthrough a J /ψ provided we make the replacement e ψ e g s − m ψ + im ψ Γ ( ψ ) → e em s , (2.7)where e em is the electric charge.
3. Cross section for e + e − → Λ ( s ) ¯ Λ ( s ) Our first task is to calculate the cross-section distribution for e + e − annihilation into polarized hyperons. From the squaredmatrix element |M| for this process we remove a factor K ψ , toget d σ = s K ψ |M red | dLips( k + k ; p , p ) , (3.8)with dLips the phase-space factor, with s = P , and with K ψ = e ψ e g ( s − m ψ ) + m ψ Γ ( m ψ ) . (3.9)The square of the reduced matrix element can be factorized as (cid:12)(cid:12)(cid:12) M red ( e + e − → Λ ( s ) Λ ( s )) (cid:12)(cid:12)(cid:12) = L · K ( s , s ) , (3.10)with L ( k , k ) and K ( p , p ; s , s ) lepton and hadron tensors,and s and s hyperon spin four-vectors.Lepton tensor including averages over lepton spins; L νµ ( k , k ) = Tr (cid:2) γ ν / k γ µ / k (cid:3) = k ν k µ + k ν k µ − sg νµ . (3.11)Hadron tensor for polarized hyperons; K νµ ( s , s ) = Tr (cid:104) Γ Λ ν ( / p + M ) (1 + γ / s ) × Γ Λ µ ( / p − M ) (1 + γ / s ) (cid:105) / e g , (3.12)with p and s momentum and spin for the Lambda hyperonand p and s correspondingly for the anti-Lambda hyperon.The trace itself is symmetric in the two hyperon variables.The spin four-vector s ( p , n ) of a hyperon of mass M , three-momentum p , and spin direction n in its rest system, is s ( p , n ) = n (cid:107) M ( | p | ; E ˆ p ) + (0; n ⊥ ) . (3.13)Here, longitudinal and transverse designations refer to the ˆ p di-rection; n (cid:107) = n · ˆ p and n ⊥ = n − ˆ p ( n · ˆ p ) are parallel andtransverse components of the spin vector n . Also, observe thatthe four-vectors p and s are orthogonal, i.e. p · s ( p ) = p and k are defined such that p = − p = p , (3.14)2 = − k = k , (3.15)and scattering angle by, cos θ = ˆ p · ˆ k . (3.16)The phase-space factor becomesdLips( k + k ; p , p ) = p π k d Ω , (3.17)with p = | p | and k = | k | .The matrix element in Eq.(3.10) can be written as a sum offour terms that depend on the hyperon spin directions in theirrespective rest systems, n and n , (cid:12)(cid:12)(cid:12) M red ( e + e − → Λ ( s ) Λ ( s )) (cid:12)(cid:12)(cid:12) = sD ( s ) (cid:104) H (0 , + H ( n , + H (0 , n ) + H ( n , n ) (cid:105) . (3.18)The polarization distributions H ab are each expressed in termsof structure functions that depend on the scattering angle θ , theratio function α ( s ), and the phase function ∆Φ ( s ). There are sixsuch structure functions, R = + α cos θ, (3.19) S = √ − α sin θ cos θ sin( ∆Φ ) , (3.20) T = α + cos θ, (3.21) T = − α sin θ, (3.22) T = + α, (3.23) T = √ − α cos θ cos( ∆Φ ) . (3.24)The definitions and notations are slightly di ff erent from thoseof Ref. [1]. In particular, a factor sD ( s ) has been pulled outfrom the structure functions, and placed in front of the sum ofthe polarization distributions of Eq.(3.18).The polarization distributions H ab are, H = R (3.25) H = S (cid:34) θ ( ˆ p × ˆ k ) · n (cid:35) (3.26) H = S (cid:34) θ ( ˆ p × ˆ k ) · n (cid:35) (3.27) H = (cid:26) T n · ˆ pn · ˆ p + T n ⊥ · n ⊥ + T n ⊥ · ˆ kn ⊥ · ˆ k + T (cid:18) n · ˆ pn ⊥ · ˆ k + n · ˆ pn ⊥ · ˆ k (cid:19)(cid:27) (3.28)Transverse components n ⊥ and n ⊥ are orthogonal to theLambda hyperon momentum p in the global c.m. system. Also,transverse n ⊥ and longitudinal n (cid:107) = ˆ p · n polarization compo-nents enter di ff erently, since they transform di ff erently underLorentz transformations.All polarization observables, single and double, can be di-rectly read o ff Eqs.(3.25-3.28), and there are no other possibili-ties. The set of scalar products involving n and n is complete.As an example, the Lambda-hyperon polarization is obtained from Eq.(3.26) which shows that the polarization is directedalong the normal to the scattering plane, ˆ p × ˆ k , and that thevalue of the polarization is P Λ ( θ ) = SR = √ − α cos θ sin θ + α cos θ sin( ∆Φ ) (3.29)From Eq.(3.27) we conclude that the polarization of the anti-Lambda is exactly the same, but then one should remember that p is the momentum of the Lambda hyperon but − p that of theanti-Lambda.
4. Folding of distributions
Our next task is to calculate the cross-section distribution for e + e − annihilation into hyperon pairs, followed by the hyperondecays into nucleon-pion pairs. This reaction is described bythe connected diagram of Fig. 2.Again, we extract a prefactor, K = K ψ K K , from thesquared matrix element, writing |M| = K|M red | . (4.30)The prefactor originates, as before with the propagator denom-inators. Due to the smallness of the hyperon widths each of thehyperon propagators can, after squaring, be approximated as, K i = p i − M ) + M Γ ( M ) = π M Γ ( M ) δ ( p i − M ) . (4.31)E ff ectively, this approximation puts the hyperons on their massshells. J/ψ Λ( p )¯Λ( p ) e + ( k ) e − ( k ) ¯ p ( l ) π + ( q ) π − ( q ) p ( l ) Figure 2: Graph describing the reaction e + e − → Λ ( → p π − ) ¯ Λ ( → ¯ p π + ). Hyperon-decay distributions are obtained by a folding cal-culation, whereby hyperon-production and -decay distributionsare multiplied together and averaged over the intermediatehyperon-spin directions. It was proved in Ref.[2] that thefolding prescription gives the same result as the evaluation ofthe connected-Feynman-diagram expression. Hence, summingover final hadron spins, |M| = (cid:88) ± s , ± s (cid:28)(cid:12)(cid:12)(cid:12) M ( e + e − → Λ ( s ) ¯ Λ ( s )) (cid:12)(cid:12)(cid:12) × (cid:12)(cid:12)(cid:12) M ( Λ ( s ) → p π − ) (cid:12)(cid:12)(cid:12) (cid:12)(cid:12)(cid:12) M ( ¯ Λ ( s ) → ¯ p π + ) (cid:12)(cid:12)(cid:12) (cid:29) n n . (4.32)Production and decay distributions are, (cid:12)(cid:12)(cid:12) M ( e + e − → Λ ( s ) ¯ Λ ( s )) (cid:12)(cid:12)(cid:12) = L · K ( s , s ) , (4.33)3 (cid:12)(cid:12) M ( Λ ( s ) → p π − ) (cid:12)(cid:12)(cid:12) = R Λ [1 − α l · s / l Λ ] , (4.34) (cid:12)(cid:12)(cid:12) M ( ¯ Λ ( s ) → ¯ p π + ) (cid:12)(cid:12)(cid:12) = R Λ [1 − α l · s / l Λ ] , (4.35)with l Λ the decay momentum in the Lambda rest system. R Λ isdetermined by the Lambda decay rate.The notation in Eq.(4.34) is the following; s denotes theLambda four-spin vector, l the four-momentum of the decayproton, and α the decay-asymmetry parameter. Similarly forthe anti-Lambda hyperon parameters of Eq.(4.35).We evaluate the hyperon-decay distributions in the hyperon-rest systems, where (cid:12)(cid:12)(cid:12) M ( Λ ( s ) → p π − ) (cid:12)(cid:12)(cid:12) = R Λ (cid:104) + α ˆ l · n (cid:105) , (4.36) (cid:12)(cid:12)(cid:12) M ( ¯ Λ ( s ) → ¯ p π + ) (cid:12)(cid:12)(cid:12) = R Λ (cid:104) + α ˆ l · n (cid:105) , (4.37)where ˆ l = l / l Λ is the unit vector in the direction of the protonmomentum in the Lambda-rest system, and correspondingly forthe anti-Lambda case.Angular averages in Eq.(4.32) are calculated according to theprescription (cid:10) ( n · l ) n (cid:11) n = l . (4.38)The folding of the production distributions, Eqs.(3.25-3.28),with the decay distributions, Eqs.(4.36-4.37), yields |M red | = sD ( s ) R Λ (cid:20) G + G + G + G (cid:21) , (4.39)with the G ab functions defined as G = R , (4.40) G = α S (cid:34) θ ( ˆ p × ˆ k ) · ˆ l (cid:35) , (4.41) G = α S (cid:34) θ ( ˆ p × ˆ k ) · ˆ l (cid:35) , (4.42) G = α α (cid:26) T ˆ l · ˆ p ˆ l · ˆ p + T ˆ l ⊥ · ˆ l ⊥ + T ˆ l ⊥ · ˆ k ˆ l ⊥ · ˆ k + T (cid:18) ˆ l · ˆ p ˆ l ⊥ · ˆ k + ˆ l · ˆ p ˆ l ⊥ · ˆ k (cid:19)(cid:27) . (4.43)Thus, we conclude the connection between joint-hadron pro-duction and joint-hadron decay distributions simply to be, G ab (ˆ l ; ˆ l ) = H ab ( n → α ˆ l ; n → α ˆ l ) . (4.44)We repeat the notation; p and k are momenta of Lambda andelectron in the global c.m. system; l and l are momenta of pro-ton and anti-proton in Lambda and anti-Lambda rest systems;orthogonal means orthogonal to p ; and structure functions R , S , and T are functions of θ , α , and ∆Φ . The angular functionsmultiplying the structure functions form a set of seven mutu-ally orthogonal functions, when integrated over the proton andanti-proton decay angles.
5. Cross section for e + e − → Λ ( → p π − ) ¯ Λ ( → ¯ p π + ) Our last task is to find the properly normalized cross-sectiondistribution. We start from the general expression,d σ = s K |M red | dLips( k + k ; q , l , q , l ) , (5.45)with dLips the phase-space density for four final-state particles.The prefactor K contains on the mass shell delta functions forthe two hyperons. This e ff ectively separates the phase spaceinto production and decay parts. Repeating the manipulationsof Ref.[2] we getd σ = π pk α g α ψ ( s − m ψ ) + m ψ Γ ( ψ ) Γ Λ Γ ¯ Λ Γ ( M ) ·· D ( s ) (cid:88) a , b G ab d Ω Λ d Ω d Ω , (5.46)with k and p the initial- and final-state momenta; Ω Λ the hy-peron scattering angle in the global c.m. system; Ω and Ω decay angles measured in the rest systems of Λ and ¯ Λ ; Γ Λ and Γ ¯ Λ channel widths; and Γ ( M ) and Γ ( ψ ) total widths.Integration over the angles Ω makes the contributions fromthe functions G and G disappear [2], and correspondinglyfor the angles Ω . Integration over both angular variables re-sults in the cross-section distribution for the reaction e + e − → J /ψ → Λ ¯ Λ .Suppose we integrate over the angles Ω . Then, the predictedhyperon-decay distribution becomes proportional to the sum G + G = R (cid:16) + α P Λ · ˆ l (cid:17) , (5.47) P Λ = SR , (5.48)with the polarization P Λ as in Eq.(3.29), and the polarisationvector P Λ directed along the normal to the scattering plane
6. Di ff erential distributions We first define our coordinate system. The scattering planewith the vectors p and k make up the xz -plane, with the y -axisalong the normal to the scattering plane. We choose a right-handed coordinate system with basis vectors e z = ˆ p , (6.49) e y = θ ( ˆ p × ˆ k ) , (6.50) e x = θ ( ˆ p × ˆ k ) × ˆ p . (6.51)Expressed in terms of them the initial-state momentumˆ k = sin θ e x + cos θ e z . (6.52)This coordinate system is used for fixing the directional an-gles of the decay proton in the Lambda rest system, and the4ecay anti-proton in the anti-Lambda rest system. The spheri-cal angles for the proton are θ and φ , and the components ofthe unit vector in direction of the decay-proton momentum are,ˆ l = (cos φ sin θ , sin φ sin θ , cos θ ) , (6.53)so that ˆ l ⊥ = (cos φ sin θ , sin φ sin θ , . (6.54)The momentum of the decay proton is by definition l = l Λ ˆ l .This same coordinate system is used for defining the directionalangles of the decay anti-proton in the anti-Lambda rest system,with spherical angles θ and φ .Now, we have all ingredients needed for the calculation ofthe G functions of Eqs.(4.40-4.43), the functions that in the enddetermine the cross-section distributions.An event of the reaction e + e − → Λ ( → p π − ) ¯ Λ ( → ¯ p π + )is specified by the five dimensional vector ξ = ( θ, Ω , Ω ),and the di ff erential-cross-section distribution as summarized byEq.(4.39) reads, d σ ∝ W ( ξ ) dcos θ d Ω d Ω . At the moment, we are not interested in the absolute normaliza-tion of the di ff erential distribution. The di ff erential-distributionfunction W ( ξ ) is obtained from Eqs.(4.40-4.43) and can be ex-pressed as, W ( ξ ) = F ( ξ ) + α F ( ξ ) + α α (cid:16) F ( ξ ) + √ − α cos( ∆Φ ) F ( ξ ) + α F ( ξ ) (cid:17) + √ − α sin( ∆Φ ) ( α F ( ξ ) + α F ( ξ )) , (6.55)using a set of seven angular functions F k ( ξ ) defined as: F ( ξ ) = F ( ξ ) = sin θ sin θ sin θ cos φ cos φ + cos θ cos θ cos θ F ( ξ ) = sin θ cos θ (sin θ cos θ cos φ + cos θ sin θ cos φ ) F ( ξ ) = sin θ cos θ sin θ sin φ F ( ξ ) = sin θ cos θ sin θ sin φ F ( ξ ) = cos θ F ( ξ ) = cos θ cos θ − sin θ sin θ sin θ sin φ sin φ . (6.56)The di ff erential distribution of Eq. (6.55) involves two pa-rameters related to the e + e − → Λ ¯ Λ process that can be deter-mined by data: the ratio of form factors α , and the relative phaseof form factors ∆Φ . In addition, the distribution function W ( ξ )can be used to extract separately Λ and ¯ Λ decay-asymmetryparameters: α and α , and hence allowing a direct test of CPconservation in the hyperon decays.The term proportional to sin( ∆Φ ) in Eq. (6.55) originateswith Eqs.(4.41) and (4.42), and can be rewritten as, S ( θ ) ( α sin θ sin φ + α sin θ sin φ ) , with the structure function S defined by Eq.(3.20). The relationbetween the structure functions and the polarization P Λ ( θ ) was discussed in Sect. 3, where it was shown that the polarization, P Λ ( θ ) of Eq.(3.29), and the polarization vector, e y , are the samefor Lambda and anti-Lambda hyperons. This information tellsus that Λ is polarized along the normal to the production plane,and that the polarization vanishes at θ = ◦ , 90 ◦ and 180 ◦ . Themaximum value of the polarization is for cos θ = ± / (2 + α ),and | P Λ ( θ ) | < sin( ∆Φ ).It should be stressed that the simplified distributions used inprevious analyses, such as Ref.[9], assume the hyperons to beunpolarized and therefore terms containing P Λ ( θ ) are missing.In fact, such decay distributions, only permit the determinationof two parameters: the ratio of form factors α , and the productof hyperon-asymmetry parameters α α .In our opinion, the formulas presented in this letter shouldbe employed for the exclusive analysis of the new BESIII data[10]. Due to huge and clean data samples: (440675 ± J /ψ → Λ ¯ Λ and (31119 ± ψ (3686) → Λ ¯ Λ , precision valuesfor the decay-hadronic-form factors could be extracted as wellas precision values for Λ and ¯ Λ decay-asymmetry parameters.The formulas presented could easily be generalized to neutrondecays of the Λ and to production of other J = / Appendix
The coupling of the initial-state leptons to the J /ψ vector me-son is determined by the decay J /ψ → e + e − . Assuming thedecay to go via an intermediate photon, Fig.1b, we can safelyignore any tensor coupling. The vector coupling of the J /ψ toleptons is therefore the same as for the photon, if replacing theelectric charge e em by a coupling strength e ψ . From the decay J /ψ → e + e − one derives α ψ = e ψ / π = Γ ( J /ψ → e + e − ) / m ψ . (6.57)In a similar fashion we relate the strength e g of J /ψ cou-pling to the hyperons to the decay J /ψ → Λ ¯ Λ . In analogy withEq.(6.57) we get α g = e g / π = (cid:18) (1 + M / m ψ ) (cid:113) − M / m ψ (cid:19) − × Γ ( J /ψ → Λ ¯ Λ ) / m ψ . (6.58)When the Λ mass M is replaced by the lepton mass m l = Acknowledgments
We are grateful to Tord Johansson who provided the inspira-tion for this work.
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