Remarks on the analysis of the reaction e(+)e(-) -> Sigma(0)(Sigma)over-bar(0)

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Remarks on the analysis of the reaction e

⁺

e

⁻

→ Σ

⁰

¯Σ

⁰

Göran Fäldt^*

Department of Physics and Astronomy, Uppsala University, Box 516, S-751 20 Uppsala, Sweden (Received 16 August 2020; accepted 4 January 2021; published 5 February 2021) We investigate roads for evaluating model-independent cross-section-distribution functions for the sequential-hyperon decayΣ⁰→ Λγ; Λ → pπ⁻ and its corresponding antihyperon decay. TheΣ⁰and ¯Σ⁰ hyperons are produced in the reaction e^þe⁻→ J=ψ → ¯Σ⁰Σ⁰. Cross-section-distribution functions are calculated using the folding technique, but a comparison with results using the helicity technique is also made.

DOI:10.1103/PhysRevD.103.033001

I. INTRODUCTION

The BESIII experiment[1]is exploring new venues into hyperon physics, based on e^þe⁻annihilation into hyperon- antihyperon pairs. In a recent paper [2], we investigated in some detail the reaction e^þe⁻→ J=ψ → Σ⁰¯Σ⁰ and its associated decay chainsΣ⁰→ Λγ; Λ → pπ⁻and ¯Σ⁰→ ¯Λγ;

¯Λ → ¯pπ^þ. By measuring this process in the vicinity of the J=ψ-vector-charmonium state, one gains information on the strong baryon-antibaryon-decay process of the J=ψ- vector-charmonium state and also, it offers a model- independent way of measuring weak-decay-asymmetry parameters, that in turn could probe CP symmetry [3].

The diagram for the basic reaction e^þe⁻→ J=ψ → Σ⁰¯Σ⁰ is graphed in Fig. 1. Its structure is governed by two vertices. The strength of the lepton-vertex function is determined by a single parameter, the electromagnetic- fine-structure constant αe, but two complex form factors G^ψ_MðsÞ and G^ψ_EðsÞ are needed for the baryonic-vertex function. However, we shall not work with the form factors themselves but with certain combinations thereof: the strength of form factors D_ψðsÞ; the ratio of form-factor magnitudes ηψðsÞ; and the relative phase of form factors ΔΦ_ψðsÞ. These form-factor combinations are defined in AppendixA.

The theoretical description of the annihilation reaction of Fig. 1 can be found in Ref. [4]. Accurate experimental results for the form-factor parametersη_ψ andΔΦ_ψ and the weak-interaction parametersαΛðα_¯ΛÞ for the J=ψ annihilation process are all reported in Ref. [3]. In addition, the

graph can be generalized to include hyperons that decay sequentially.

Our analysis of the cross-section-distribution function for the annihilation reaction e^þe⁻ → J=ψ → Σ⁰¯Σ⁰, fol- lowed by its subsequent hyperon decays, starts from the master formula of Ref.[2], and which is reproduced in the following section. The purpose of our investigation is to find out which coordinate choice would be most convenient when evaluating the master formula, and at the same time being able to compare our result to those of others.

II. MASTER FORMULA

In several previous publications we studied e^þe⁻ annihilation into hyperon pairs Y ¯Y and the subsequent decays of those pairs. Photon as well as charmonium induced annihilaton was considered. In the present investigation we limit ourselves to the hyperon-decay chain Σ⁰→ Λγ;

Λ → pπ⁻, and its corresponding antihyperon-decay chain

¯Σ⁰→ ¯Λγ; ¯Λ → ¯pπ^þ, again when simultaneously occurring in the reaction e^þe⁻→ J=ψ → Σ⁰¯Σ⁰.

In Ref. [2] it was shown that the cross-section- distribution function for a J=ψ induced joint production and subsequent decay of aΣ⁰¯Σ⁰pair can be summarized in the master formula

dσ ¼ dσ

dΩ_Σ⁰ðe^þe⁻→ J=ψ → Σ⁰¯Σ⁰Þ

×

WðξÞ R

dΦðΣ⁰; Λ; p; ¯Σ⁰; ¯Λ; ¯pÞ: ð2:1Þ

As can be seen the master formula involves three factors, describing the annihilation of a lepton pair into a hyperon pair, the folded product of spin densitiesWðξÞ representing hyperon production and decay, and the phase space element of sequential hyperon decays. Each event is specified by a

*goran.faldt@physics.uu.se

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Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI. Funded by SCOAP³.

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nine-dimensional vector ξ ¼ ðθ; ΩΛ; Ωp; Ω_¯Λ; Ω¯pÞ, with θ the scattering angle in the e^þe⁻ → Σ⁰¯Σ⁰ subprocess.

Following Refs. [4,2] we write the cross-section- distribution function for the J=ψ induced annihilation reaction e^þe⁻ → J=ψ → Σ⁰¯Σ⁰ as

dσ

dΩ_Σ⁰ðe^þe⁻ → J=ψ → Σ⁰¯Σ⁰Þ

¼p k

α_ψαg

ðs − m²ψÞ²þ m²ψΓðmψÞD_ψðsÞR; ð2:2Þ

where the strength function D_ψðsÞ is defined in AppendixA, and the structure function R in Appendix B. The electromagnetic-coupling constant α_ψ is determined by the electromagnetic-decay widthΓðJ=ψ → e^þe⁻Þ, and the hadronic-coupling constant αg similarly by the hadronic-decay widthΓðJ=ψ → Σ⁰¯Σ⁰Þ, as illustrated in Fig.2.

The differential-spin-distribution function WðξÞ of Eq. (2.1) is obtained by folding a product of five spin densities,

WðξÞ ¼ hSðn_Σ⁰; n_¯Σ⁰ÞGðn_Σ⁰; n_ΛÞGðn_Λ; npÞ

× Gðn_¯Σ⁰; n_¯ΛÞGðn_¯Λ; n¯pÞi_n; ð2:3Þ in accordance with the prescription of Ref. [5] and of Eq.(5.1). The folding operationh…i_napplies to each of the six hadron spin vectors,n_Σ⁰; …; n¯p.

The function Sðn_Σ⁰; n_¯Σ⁰Þ represents the spin-density distribution for theΣ⁰¯Σ⁰ hyperon pair. This function also depends on the unit vectors l_Σ⁰ and l_¯Σ⁰, which are unit vectors in the directions of motion of the Σ⁰ and ¯Σ⁰ hyperons in the center-of-momentum (c.m.) frame of the event. The four remaining spin-density-distribution functions GðnY₁;nY₂Þ represent spin-density distributions for the hyperon decays Σ⁰→ Λγ; or Λ → pπ⁻, or their antihyperon counterparts.

The spin-decay-distribution functions GðnY₁; nY₂Þ are normalized to unity, which means their spin independent terms are unity. However, for convenience the spin- density-distribution function Sðn_Σ⁰; n_¯Σ⁰Þ is normalized toR.

The phase-space factor, dΦðΣ⁰; Λ; p; ¯Σ⁰; ¯Λ; ¯pÞ of the master equation, describes the normalized phase-space element for the sequential decays of the two baryonsΣ⁰ and ¯Σ⁰,

dΦðΣ⁰; Λ; p; ¯Σ⁰; ¯Λ; ¯pÞ

¼ dΩ_Σ⁰·ΓðΣ⁰→ ΛγÞ ΓðΣ⁰→ allÞ

dΩΛ

4π ·ΓðΛ → pπ⁻Þ ΓðΛ → allÞ

dΩp

4π

·Γð ¯Σ⁰→ ¯ΛγÞ Γð ¯Σ⁰→ allÞ

dΩ_¯Λ

4π ·Γð ¯Λ → ¯pπ⁻Þ Γð ¯Λ → allÞ

dΩ_¯p

4π : ð2:4Þ

The widths are defined in the usual way. ForΓðΣ⁰→ ΛγÞ this means forming an average over theΣ⁰spin directions, and summing over theΛ and γ spin directions. However, since the Σ⁰→ Λγ decay rate is 100% we also have ΓðΣ⁰→ ΛγÞ ¼ ΓðΣ⁰→ allÞ.

The angles Ω_Λ define the direction of motion of the Λ hyperon in the Σ⁰ rest system, the angles Ωp the direction of motion of the p baryon in theΛ rest system, and so on.

III. e⁺e⁻ ANNIHILATION INTO Σ⁰¯Σ⁰ PAIRS The cross-section-distribution function for e^þe⁻ annihilation into aΣ⁰¯Σ⁰pair appears in two places in the master formula of Eq. (2.1). The unpolarized-cross-section- distribution function is a prefactor in the master formula, and the hyperon-spin-density-distribution function enters as a factor in the spin-density-distribution function of Eq.(2.3).

The cross-section distribution for polarized-final-state hyperons was derived in Refs.[2,4] as

FIG. 1. Graph describing the psionic annihilation reaction e^þe⁻→ J=ψ → ¯Σ⁰Σ⁰. The same reaction can also proceed hadronicly via other vector-charmonium states such as ψ⁰ or ψð2SÞ, or electromagnetically via photons.

FIG. 2. Graph describing the reaction e^þe⁻→ ¯Σ⁰Σ⁰, and the subsequent decays,Σ⁰→ Λγ; Λ → pπ⁻and ¯Σ⁰→ ¯Λγ; ¯Λ → ¯pπ^þ. The reaction graphed can, in addition to photons, be mediated by vector charmonia, such as J=ψ, ψ⁰andψð2SÞ. Solid lines refer to baryons, dashed to mesons, and wavy to photons.

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dσ

¼ p 4k

α_ψαgD_ψðsÞ

ðs − m²_ψÞ²þ m²_ψΓðm_ψÞSðn_Σ⁰; n_¯Σ⁰Þ; ð3:1Þ where D_ψðsÞ is the strength function of Eq.(A1),n_Σ⁰ and n_¯Σ⁰ the spin vectors of the Σ⁰ and ¯Σ⁰ hyperons, and Sðn_Σ⁰; n_¯Σ⁰Þ the spin-density-distribution function for the final-state hyperons. This spin-density-distribution function is normalized so that its spin-independent part equals R, with

R ¼ 1 þ η_ψcos²θ; ð3:2Þ according to Eq. (B1). Consequently, summing over the final-state-hyperon polarizations gives the unpolarized cross-section-distribution function

dσ

¼p k

αψαg

ðs − m²_ψÞ²þ m²_ψΓðm_ψÞD_ψðsÞR: ð3:3Þ The branching rate for the decay channel J=ψ → Σ⁰¯Σ⁰ is ð1.07 0.08Þ × 10⁻³, and for the channel J=ψ → Λ ¯Λ it is ð1.89 0.09Þ × 10⁻³ [6].

For a spin-one-half baryon of four-momentum p, the four-vector spin sðpÞ is related to the three-vector-spin direction n, the spin in the rest system, by

sðp; nÞ ¼n_k

M ðjpj; EˆpÞ þ ð0; n⊥Þ: ð3:4Þ Longitudinal and transverse directions of vectors are relative to the ˆp direction.

In the global c.m. system kinematics simplifies. There, three-momenta p and k are defined such that

p_Σ⁰ ¼ −p_¯Σ⁰ ¼ p; ð3:5Þ ke^þ ¼ −ke⁻ ¼ k; ð3:6Þ and the scattering angleθ such that cos θ ¼ ˆp · ˆk. For the Σ⁰and ¯Σ⁰unit vectorsl_Σ⁰andl_¯Σ⁰, we havel_Σ⁰ ¼ −l_¯Σ⁰ ¼ ˆp.

The spin-density-distribution function Sðn_Σ⁰; n_¯Σ⁰Þ is a sum of seven mutually orthogonal contributions[7], Sðn_Σ⁰; n_¯Σ⁰Þ ¼ R þ SN · n_Σ⁰þ SN · n_¯Σ⁰þ T₁n_Σ⁰·ˆpn_¯Σ⁰·ˆp

þ T2n_Σ⁰_⊥·n_¯Σ⁰_⊥þ T3n_Σ⁰_⊥· ˆkn_¯Σ⁰_⊥· ˆk=sin²θ þ T₄ðn_Σ⁰·ˆpn_¯Σ⁰_⊥· ˆk þ n_¯Σ⁰·ˆpn_Σ⁰_⊥· ˆkÞ=sinθ;

ð3:7Þ where N is normal to the scattering plane,

N ¼ 1

sinθˆp × ˆk: ð3:8Þ The six structure functionsR, S, and T of Eq.(3.7)depend on the scattering angleθ, the ratio function ηψðsÞ, and the phase function ΔΦ_ψðsÞ. For their definitions we refer to AppendixB, but be careful, our original definitions were slightly different[7].

IV. ASSORTED SPIN DENSITIES

To be able to calculate the differential-distribution function of Eq. (2.3) we need in addition to the spin- density-distribution function for theΣ⁰¯Σ⁰ final-state pair, the spin-density-distribution functions for the decaysΣ⁰→ Λγ and Λ → pπ⁻, and their antiparticle conjugate decays.

Weak decays of spin-one-half baryons, such as Λ → pπ⁻, involve both S- and P-wave amplitudes, and the spin-density-decay distribution is commonly parame- trized by three parameters, denotedαβγ, and which fulfill a relation

α²þ β²þ γ²¼ 1: ð4:1Þ Details of this description can be found in Refs.[8]or[2].

The spin-density-distribution function Gðn_Λ; npÞ, describing the decay Λ → pπ⁻, is a scalar, which we choose to evaluate in the rest system of the Λ hyperon, to get

GðnΛ; npÞ ¼ 1 þ αΛnΛ·lpþ αΛnp·lpþ nΛ·LΛðnp; lpÞ;

ð4:2Þ with the vector-valued functionL_Λðnp; lpÞ defined as L_Λðnp; lpÞ ¼ γ_Λnpþ ½ð1 − γ_ΛÞnp·lplpþ β_Λnp×lp:

ð4:3Þ Here, n_Λ and np are the spin vectors of the Λ hyperon and the p baryon, andlp a unit vector in the direction of motion of the proton in the rest system of theΛ hyperon.

TheΛ indices remind us the parameters refer to a Λ decay.

An important aspect of the spin-density-distribution function is its normalization. The spin-independent term is unity.

The spin-density-distribution function Gðn_¯Λ; n_¯pÞ for the antiparticle-conjugate decay ¯Λ → ¯pπ^þ has exactly the same functional structure as Gðn_Λ; npÞ, but the decay parameters take other numerical values. For CP conserving interactions the asymmetry parameters of the Λ-hyperon decay are related to those of the ¯Λ-hyperon decay by[9,10]

αΛ¼ −α_¯Λ; βΛ¼ −β_¯Λ; γΛ¼ γ_¯Λ: ð4:4Þ

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Numerical values for the weak-interaction parameters are given in Ref. [6]: for the decay Λ → pπ⁻ we have

½α ¼ 0.732 0.014, and ½ϕ ¼ arctanðβ=γÞ ¼ −6.5 3.5;

and for the decay ¯Λ → ¯pπ^þ, we have ½α ¼ −0.758 0.010 0.007.

Next, we turn to the electromagnetic M1 transition Σ⁰→ Λγ. It is caused by a transition-magnetic moment, of strength

μ_ΣΛ¼ eF₂ð0Þ=ðm_Σþ m_ΛÞ: ð4:5Þ The normalized-spin-density-distribution function for a Σ⁰→ Λγ transition to a final state of fixed photon helicity λ_γ is, according to Ref.[2],

G_γðn_Σ⁰; n_Λ; λ_γÞ ¼ 1 − n_Σ⁰·l_γl_γ·n_Λþ λ_γðn_Σ⁰·l_γ− n_Λ·l_γÞ;

ð4:6Þ wherel_γ is a unit vector in the direction of motion of the photon, and lΛ¼ −lγ a unit vector in the direction of motion of theΛ hyperon, both in the rest system of the Σ⁰ baryon. The photon helicities λγ take on the values 1.

We notice that when both hadron spins are parallel or antiparallel to the photon momentum, then the decay probability vanishes, a property of angular-momentum conservation.

Summing, in Eq.(4.6), the contributions from the two photon-helicity states gives the normalized-spin-density- distribution function

Gðn_Σ⁰; n_ΛÞ ¼ 1 − n_Σ⁰·l_γl_γ·n_Λ: ð4:7Þ The normalized-spin-density-distribution function for the conjugate transition, ¯Σ⁰→ ¯Λγ, is obtained by replacing, in expression(4.7), the particle spin vectorsn_Σ⁰ andn_Λby the antiparticle-spin vectorsn_¯Σ⁰ andn_¯Λ.

V. SEQUENTIAL DECAY OF HYPERONS A factor of our master formula for hyperon production and decay, Eq. (2.1), is the differential-spin-distribution functionWðξÞ of Eq.(2.3), which is obtained by folding a product of five spin densities. The folding prescription is especially adapted to spin one-half baryons. A folding operation implies forming an average over intermediate- spin directionsn according to the prescription of Ref.[5], h1i_n¼ 1; hni_n¼ 0; hn · kn · li_n¼ k · l: ð5:1Þ The spin-density distribution Wðn_Σ⁰; npÞ for the decay chain Σ⁰→ Λγ; Λ → pπ⁻ is obtained by folding the product of the spin density distributions in the decay chain.

We obtain

Wðn_Σ⁰; npÞ ¼ hGðn_Σ⁰; n_ΛÞGðn_Λ; npÞi_n_Λ; ð5:2Þ where the two spin-density-distribution functions on the right-hand side are defined in Eqs. (4.7) and (4.2).

Performing the folding operation gives

Wðn_Σ⁰; npÞ ¼ U_Σ⁰þ n_Σ⁰·V_Σ⁰; ð5:3Þ U_Σ0 ¼ 1 þ αΛnp·lp; ð5:4Þ V_Σ⁰ ¼ −l_γ½α_Λl_γ·lpþ np·L_Λðl_γ; −lpÞ; ð5:5Þ and the same for Wðn_¯Σ⁰; n¯pÞ.

VI. PRODUCTION AND DECAY OF Σ⁰¯Σ⁰ PAIRS Now, we come to our final task: production and decay of Σ⁰¯Σ⁰pairs. The starting point is the reaction e^þe⁻ → Σ⁰¯Σ⁰, the spin-density-distribution function of which was calculated in Sec.III, and named Sðn_Σ⁰; n_¯Σ⁰Þ. The spin-density- distribution function Wðn_Σ⁰; npÞ which represents the decay chainΣ⁰→ Λγ; Λ → pπ⁻ was calculated in Sec.V, and so for the antichain-decay function Wðn_¯Σ⁰; n¯pÞ.

The final-state-angular distributions are obtained by folding the spin distributions for production and decay, according to prescription(5.1). Invoking Eq.(3.7)for the production step and Eq. (5.3) and its antidistribution for the decay steps, we get the differential-spin-density- distribution function

WðξÞ ¼ hSðn_Σ⁰; n_¯Σ⁰ÞWðn_Σ⁰; npÞWðn_¯Σ⁰; n_¯pÞi_n

Σ0;n_¯Σ0

¼ RU_Σ⁰U_¯Σ0þ SU_¯Σ⁰N · V_Σ⁰þ SU_Σ⁰N · V_¯Σ⁰ þ T1V_Σ⁰·ˆpV_¯Σ⁰· ˆp þ T2V_Σ⁰_⊥·V_¯Σ⁰_⊥ þ T₃V_Σ⁰_⊥· ˆkV_¯Σ⁰_⊥· ˆk=sin²θ

þ T4ðV_Σ⁰·ˆpV_¯Σ⁰_⊥· ˆk þ V_¯Σ⁰·ˆpV_Σ⁰_⊥· ˆkÞ=sin θ:

ð6:1Þ The functions U_Σ0 andV_Σ⁰ are defined in Sec. B, and

U_Σ0 ¼ 1 þ α_Λnp·lp; ð6:2Þ V_Σ⁰ ¼ −l_γ½α_Λl_γ·lpþ np·L_Λðl_γ; −lpÞ: ð6:3Þ We observe that U_Σ0 depends on the weak interaction parameter αΛ, whereas V_Σ⁰ in addition depends on the parameters βΛ and γΛ through the vector function LΛ, of Eq.(4.3).

The angular distributions of Eq.(6.1), which are the most general ones, still depend on the spin vectorsnpandn_¯p. In case we are satisfied with considering their averages, then the variables U andV simplify,

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U_¯Σ0¼ 1; V_Σ⁰ ¼ −α_Λl_Λ·lpl_Λ;

U_¯Σ0¼ 1; V_¯Σ⁰ ¼ −α_¯Λl_¯Λ·l¯pl_¯Λ: ð6:4Þ

When U_Σ0 ¼ U_¯Σ⁰ ¼ 1 the effect of the folding is to make the replacements n_Σ⁰ → V_Σ⁰ and n_¯Σ⁰ → V_¯Σ⁰ in the spin- density function Sðn_Σ⁰; n_¯Σ⁰Þ of Eq.(3.7). We notice that the U and V variables now are independent of the weak- asymmetry parametersβ_Λ andγ_Λ.

Inserting the expressions of Eq. (6.4) into the spin- density function of Eq.(6.1), we get

WðξÞ ¼ R − α_ΛSN · l_Λl_Λ·lp− α_¯ΛSN · l_¯Λl_¯Λ·l_¯p þ αΛα_¯ΛlΛ·lpl_¯Λ·l¯p½T1lΛ· ˆpl_¯Λ·ˆp þ T₂l_Λ⊥·l_¯Λ⊥þ T₃l_Λ⊥· ˆkl_¯Λ⊥· ˆk=sin²θ

þ T₄ðl_Λ· ˆpl_¯Λ⊥· ˆk þ l_¯Λ·ˆplΛ⊥· ˆkÞ=sin θ: ð6:5Þ

Thus, this is the angular distribution obtained when folding the product of spin densities for production and decay.

These results were previously reported in Ref. [2].

VII. DIFFERENTIAL-SPIN DISTRIBUTIONS A closer inspection of the differential-spin-density- distribution function of Eq. (6.5) shows that the weak- interaction parameters α_Λ and α_¯Λ always come in the combinations α_Λl_Λ·lp or α_¯Λl_¯Λ·l_¯p. Therefore, it is convenient to define the following functions:

λ_Λðθ_ΛpÞ ¼ α_Λl_Λ·lp¼ α_Λcosðθ_ΛpÞ; ð7:1Þ

λ_¯Λðθ_{¯Λ ¯p}Þ ¼ α_¯Λl_¯Λ·l_¯p ¼ α_¯Λcosðθ_{¯Λ ¯p}Þ: ð7:2Þ

Then, the differential-spin-density-distribution function of Eq. (6.5)can be rewritten as

WðξÞ ¼ R − ½λΛQ_Λþ λ_¯ΛQ_¯ΛS

þ λ_Λλ_¯Λ½Q₁T₁þ Q₂T₂þ Q₃T₃þ Q₄T₄; ð7:3Þ

with the argumentξ a nine-dimensional vector ξ ¼ ðθ; Ω_Λ; Ωp; Ω_¯Λ; Ω¯pÞ representing the scattering angle and four directional-unit vectors of particle motion.

The six structure functionsR, S, and T are functions of the scattering angleθ and the ratio of form factors η_ψ. The six kinematic Q functions are functions ofl_Λandl_¯Λ. Their dependencies on the unit vectorslpandl_¯preside solely in the functionsλΛ andλ_¯Λ of Eqs.(7.1) and(7.2).

The analytic expressions for the six functions Qðl_Λ; l_¯ΛÞ are obtained by comparing Eqs. (6.5)and(7.3),

Q_Λ¼ N · l_Λ; Q_¯Λ¼ N · l_¯Λ;

Q₁¼ l_Λ· ˆpl_¯Λ·ˆp;

Q₂¼ lΛ⊥·l_¯Λ⊥;

Q₃¼ l_Λ⊥· ˆkl_¯Λ⊥· ˆk=sin²θ;

Q₄¼ ½lΛ·ˆpl_¯Λ⊥· ˆk þ l_¯Λ· ˆplΛ⊥· ˆk=sin θ: ð7:4Þ Here, longitudinal and transverse components of vectors are defined relative to ˆp, the direction of motion of the Σ⁰ hyperon.

The differential-spin-density distribution of Eq. (7.3), and the angular functions above, depend on a number of unit vectors; ˆp and −ˆp are unit vectors along the directions of motion of theΣ⁰and the ¯Σ⁰in the c.m. system; ˆk and − ˆk are unit vectors along the directions of motion of the incident electron and positron in the c.m. system;l_Λandl_¯Λ are unit vectors along the directions of motion of theΛ and

¯Λ in the rest systems of the Σ⁰and the ¯Σ⁰; andlpandl¯pare unit vectors along the directions of motion of the p and the

¯p in the rest frames of the Λ and the ¯Λ.

VIII. SPIN POLARIZATIONS

As an application we shall now investigate what can be learned by concentrating our attention to the proton leg.

The spin-density-distribution functionWðnp; n_¯pÞ for final- state-spin vectorsnpandn¯p is described by Eq.(6.1). We start by averaging over the final-state antiproton-spin directionsn¯p, to get

WðnpÞ ¼ hWðnp; n¯pÞi_n_¯p

¼ ðXaþ XbÞ þ aXaþ bXb; ð8:1Þ where the functions X_a and X_b are defined as

X_a¼ R − λ_¯ΛQ_¯ΛS; ð8:2Þ X_b¼ −λΛQ_ΛS þ λΛλ_¯Λ½Q1T1þ Q2T2þ Q3T3þ Q4T4;

ð8:3Þ and the functions a and b as

a ¼ αΛnp·lp; ð8:4Þ b ¼ −np·LΛð−lΛ; −lpÞ: ð8:5Þ Since the vector-valued functionL_Λð−l_Λ; −lpÞ is defined in Eq.(4.3), and X¼ Xaþ XbequalsWðξÞ of Eq.(7.3), it follows that in this particular case the final-state-proton polarizationPp becomes

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Pp¼ ðα_ΛX_alp− XbL_Λð−l_Λ; −lpÞÞ=X: ð8:6Þ Hence, apart from theΛ weak-interaction parameters, the final-state-proton-polarization vector is built on the vectors lp andL_Λ.

It is instructive to compare this result with the spin- density-distribution function GðnΛ; npÞ of Eq. (4.2) describing the decayΛ → pπ⁻. For aΛ hyperon of initial- state poarization nΛ¼ PΛ, the spin-density-distribution function reads

GðP_Λ; npÞ ¼ 1 þ α_ΛP_Λ·lpþ α_Λnp·lpþ P_Λ·L_Λðnp; lpÞ;

ð8:7Þ

and implies a proton polarization

Pp¼ ðαΛlp− LΛð−PΛ; −lpÞÞ=ð1 þ αΛPΛ·lpÞ: ð8:8Þ We immediately notice the similarity between the final- state-proton polarizations of Eqs. (8.6)and(8.8).

However, it should be remembered that the spin polarization Pp of Eq.(8.6)is only one of many possible.

IX. GLOBAL ANGULAR FUNCTIONS The differential-spin-density distribution(6.5)is a function of several unit vectors. In order to handle them we need a common coordinate system, which we call global and define as follows. The scattering plane of the reaction e^þe⁻→ Σ⁰¯Σ⁰is spanned by the unit vectors ˆp ¼ lΣ⁰ and ˆk ¼ le, as measured in the c.m. system. The scattering plane makes up the xz plane, with the y axis along the normal to this plane. We choose a right-handed coordinate system with basis vectors

ez¼ ˆp;

ey ¼ 1

sinθð ˆp × ˆkÞ;

ex ¼ 1

sinθð ˆp × ˆkÞ × ˆp; ð9:1Þ and where the initial-state-lepton momentum is decomposed as

ˆk ¼ sin θexþ cos θez: ð9:2Þ The reason we call this coordinate system global is that we use it whenever studying a subprocess of the e^þe⁻ annihilation.

In spherical xyz coordinates the unit vectors l_Λ andl_¯Λ associated with the directions of motion of the Λ and ¯Λ hyperons are

l_Λ¼ ðcos ϕ_Λsinθ_Λ; sin ϕ_Λsinθ_Λ; cos θ_ΛÞ;

l_¯Λ¼ ðcos ϕ_¯Λsinθ_¯Λ; sin ϕ_¯Λsinθ_¯Λ; cos θ_¯ΛÞ: ð9:3Þ

However, in order to make our formulas more transparent we introduce the notations lΛ¼ E ¼ ðEx; E_y; E_zÞ and l_¯Λ¼ F ¼ ðFx; F_y; F_zÞ. In this Cartesian notation, the expressions for kinematic functions QðlΛ; l_¯ΛÞ of Eq.(7.4) are

Q_Λ¼ Ey; Q_¯Λ¼ Fy;

Q₁¼ EzF_z; Q₂¼ ExF_xþ EyF_y;

Q₃¼ ExF_x; Q₄¼ ExF_zþ EzF_x: ð9:4Þ

Inserting them into Eq.(7.3), the differential-spin-density- distribution function becomes

WðξðΩÞÞ

¼ 1þη_ψcos²θ − ffiffiffiffiffiffiffiffiffiffiffiffi 1−η²_ψ q

sinðΔΦ_ψÞsinθcosθ½λ_ΛE_yþλ_¯ΛF_y þλ_Λλ_¯Λ½ð1þη_ψÞEzF_zþsin²θðExF_x−EzF_z−η_ψE_yF_yÞ þ ffiffiffiffiffiffiffiffiffiffiffiffiffiffi

1−η_ψ² q

cosðΔΦ_ψÞsinθcosθðExF_zþEzF_xÞ: ð9:5Þ

Now, the phase-space-angular variables are hidden inside theEðθ_Λ; ϕ_ΛÞ and Fðθ_¯Λ; ϕ_¯ΛÞ functions.

The differential-spin-density-distribution functionWðξÞ of Eq.(9.5)involves two parameters related to the e^þe⁻ → Σ⁰¯Σ⁰ reaction that can be determined by data: the ratio of form factorsη_ψ, and the relative phase of form factors ΔΦψ. In addition, the distribution functionWðξÞ depends on the weak-asymmetry parametersα_Λ andα_¯Λof the two Lambda-hyperon decays. The dependencies on the weak- asymmetry parametersβ and γ drop out, when final-state- proton and antiproton spins are unidentified.

An important conclusion to be drawn from the differential distribution of Eq.(9.5)is that when the phaseΔΦ_ψ is small, the parametersα_Λandα_¯Λare strongly correlated and therefore difficult to separate. In order to contribute to the experimental precision value of α_Λ and α_¯Λ a nonzero value ofΔΦψ is required.

X. GLOBAL2 ANGULAR FUNCTIONS In the global2-coordinate system, the scattering plane of the reaction e^þe⁻ → Σ⁰¯Σ⁰ is still spanned by the unit vectorsˆp ¼ l_Σ⁰and ˆk ¼ le, as measured in the c.m. system, and with scattering angle cosθ ¼ ˆk · ˆp. Again, the scattering plane makes up the x⁰z⁰plane, and the y⁰axis is normal to this plane. In the x⁰y⁰z⁰ coordinate system we choose a right-handed set of basis vectors

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e⁰z¼ ˆk;

e⁰y¼ 1

sinθð ˆk × ˆpÞ;

e⁰x¼ 1

sinθð ˆk × ˆpÞ × ˆk: ð10:1Þ We observe that by this definition the xyz and x⁰y⁰z⁰ coordinate bases are related by an interchange of the ˆp and ˆk momenta. Moreover, in global2 coordinates the final- state-hyperon momentum can be decomposed as

ˆp ¼ sin θe⁰xþ cos θe⁰z; ð10:2Þ withN ¼ −e⁰ynormal to the scattering plane, forN defined in Eq.(3.8).

In spherical x⁰y⁰z⁰ coordinates the unit vectorsl_Λandl_¯Λ associated with the directions of motion of the Λ and ¯Λ hyperons are

l_Λ¼ ðcos ϕ⁰_Λsinθ⁰_Λ; sin ϕ⁰_Λsinθ⁰_Λ; cos θ⁰_ΛÞ;

l_¯Λ¼ ðcos ϕ⁰_¯Λsinθ⁰_¯Λ; sin ϕ⁰_¯Λsinθ⁰_¯Λ; cos θ⁰_¯ΛÞ; ð10:3Þ and similarly for the unit vectorslpandl_¯p. More generally, we use the prime notation for vectors in global2 coordinates,lΛ¼ E⁰¼ ðE⁰x; E⁰_y; E⁰_zÞ and l_¯Λ¼ F⁰ ¼ ðF⁰x; F⁰_y; F⁰_zÞ.

In order to be able to determine the spin-density- distribution function in terms of the angles of Eqs. (10.3), first we need to determine the angular dependencies of the six kinematic functions Qðl_Λ; l_¯ΛÞ of Eq.(7.4). In principle, this is straightforward but it turns out to be more involved than for the global case, since some of the Qðl_Λ; l_¯ΛÞ functions will depend on the scattering angleθ.

The basis vectors of Eqs.(10.1)and(9.1)are related by ex¼ − cos θe⁰xþ sin θe⁰z;

ey¼ −e⁰y;

ez¼ sin θe⁰xþ cos θe⁰z: ð10:4Þ From this relation one obtains a corresponding relation for the xyz components F_k, and the x⁰y⁰z⁰components F⁰_k, of the directional unit vectorl_¯Λ¼ F associated with the ¯Λ hyperon,

F_x ¼ − cos θF⁰xþ sin θF⁰z; F_y ¼ −F⁰y;

F_z¼ sin θF⁰xþ cos θF⁰z; ð10:5Þ and the same for theΛ hyperon case.

The new set of the six QðlΛ; l_¯ΛÞ functions of Eq.(7.4) is obtained by replacing global-vector components by global2-vector components, which give

Q_Λ¼ −E⁰y; Q_¯Λ¼ −F⁰y;

Q₁¼ ðsin θE⁰xþ cos θE⁰zÞðsin θF⁰xþ cos θF⁰zÞ;

Q₂¼ Q3þ E⁰yF⁰_y;

Q₃¼ ð− cos θE⁰xþ sin θE⁰zÞð− cos θF⁰xþ sin θF⁰zÞ;

Q₄¼ ð− cos θE⁰xþ sin θE⁰zÞðsin θF⁰xþ cos θF⁰zÞ

þ ðsin θE⁰xþ cos θE⁰zÞð− cos θF⁰xþ sin θF⁰zÞ: ð10:6Þ This global2 set of functions has a decidedly more complex dependence on the scattering angle θ than the global set of Eq. (9.4), which is independent of the scattering angle.

The differential-distribution function as defined in Eq.(7.3) now takes the form

WðξðΩ⁰ÞÞ

¼ 1þη_ψcos²θ þ ffiffiffiffiffiffiffiffiffiffiffiffi 1−η²_ψ

q sinðΔΦ_ψÞsinθcosθ½λ_ΛE⁰_yþλ_¯ΛF⁰_y

þλ_Λλ_¯Λ½ð1þη_ψÞQ₁þsin²θððQ₃−Q₁Þþη_ψðQ₃−Q₂ÞÞ þ ffiffiffiffiffiffiffiffiffiffiffiffiffiffi

1−ηψ2

q

cosðΔΦψÞsinθcosθQ4; ð10:7Þ with the functions QðΩ⁰Þ of Eq. (10.6).

This ends our involvement with x⁰y⁰z⁰ global2 coordinates, since the xyz coordinates seem considerably easier to work with.

XI. CROSS-SECTION DISTRIBUTIONS The differential-cross-section-distribution function WðξðΩÞÞ of Eq.(7.3) is a Cartesian scalar. Its argument ξ is a nine-dimensional vector ξ ¼ ðθ; Ω_Λ; Ωp; Ω_¯Λ; Ω¯pÞ, which represents the scattering angle and four directional- unit vectors of particle motion. In view of the findings of the previous sections, we propose evaluating this cross- section-distribution function in the global xyz coordinate system of Eq.(9.1), and so for each event.

The desired expression for the cross-section-distribution function WðξðΩÞÞ is in our global coordinates already known, and displayed in Eq.(9.5), where the symbol Ω refers to spherical angles,Ω ¼ ðθ; ϕÞ, in the xyz coordinate system.

It might be remembered we introduced the notation E ¼ lΛandF ¼ l_¯Λ, with Cartesian components as defined in Eq.(9.3). A unit vector such aslΛ, which is a unit vector in the direction of motion of the Λ hyperon in the rest system of the Σ⁰ hyperon, can be expressed in either Cartesian xyz or spherical-angular variables,

l_Λ¼ ðl_Λx; l_Λy; l_ΛzÞ ¼ ðcos ϕ_Λsinθ_Λ; sin ϕ_Λsinθ_Λ; cos θ_ΛÞ:

ð11:1Þ

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The decomposition into spherical coordinates needs to be known since in our treatment the phase-space element dΩΛ

is expressed in terms of spherical-angular variables.

It was already noticed in Sec. VII that the angular variables Ωp and Ω_¯p only appear in the multiplicative parameters λ_Λðθ_ΛpÞ and λ_¯Λðθ_{¯Λ ¯p}Þ of Eqs. (7.1)and (7.2).

Averaging these parameters overΩporΩ_¯pgive a vanishing result, as e.g.,

Z dΩp

4π α_Λcosðθ_ΛpÞ ¼ 0: ð11:2Þ The phase-space element dΦðξÞ of Eq.(2.4), associated with the spin-density-distribution function WðξÞ, is nine- dimensional as the dimensionality of the ξ vector. The corresponding nine-dimensional-cross-section distribution is that of the master formula Eq.(2.1). Certainly, it should be possible to determine the weak-interaction parameters α_Λðα_¯ΛÞ, and the amplitude parameters η_ψðsÞ and ΔΦ_ψðsÞ, from this distribution.

Lower dimensional cross-section distributions may con- tain as much information as the nine-dimensional one. To investigate this claim let us integrate over the antihyperon anglesΩ_¯pandΩ_¯Σ⁰. The result is a five-dimensional cross- section-distribution function

Z dΩ_¯p 4π

dΩ_¯Σ⁰

4π WðξÞ ¼ Wðξ⁰Þ; ð11:3Þ withξ⁰¼ ðθ; Ω_Λ; ΩpÞ, and

Wðξ⁰Þ ¼ R − λ_ΛQ_ΛS;

¼ Rðθ; ηψÞ − αΛN · lΛlΛ·lpSðθ; ΔΦψÞ: ð11:4Þ Thus, we realize the five-dimensional cross-section- distribution function contains as much information as the nine-dimensional one.

A further reduction of phase-space into a three-dimensional space can be obtained by integrating over the hyperon anglesΩ_Λ, giving

Z dΩ_Λ

4π Wðξ⁰Þ ¼ Wðξ⁰⁰Þ; ð11:5Þ withξ⁰⁰¼ ðθ; ΩpÞ, and

Wðξ⁰⁰Þ ¼ R − λΛQ_ΛS;

¼ Rðθ; η_ψÞ −1

3α_ΛN · lpSðθ; ΔΦ_ψÞ: ð11:6Þ Since N ¼ ey it follows that N · lp¼ lpy¼ sin ϕpsinθp. Again we are forced to conclude that the three-dimensional phase-space harbors as much information as the nine- dimensional one.

We end our investigation with a remark on polarization.

If we integrate the cross-section-distribution function over the antiparticle leg, which will then be the polarizations of theΣ⁰andΛ baryons? After the integration, we get the Σ⁰ polarization from Eq.(3.7)

P_Σ⁰ ¼ S=RN; ð11:7Þ

and, similarly, theΛ polarization can be picked out from Eq.(6.5)

P_Λ¼ −S=RN · l_Λl_Λ: ð11:8Þ Thus, the polarization of theΣ⁰is directed along the normal to the scattering plane, and the polarization of the Λ directed along its own momentum.

XII. SUMMARY

This is a study of joint production and simultaneous sequential decay ofΣ⁰¯Σ⁰ pairs produced in e^þe⁻ annihilation. It starts from a master formula which is a product of three factors, describing: the annihilation of a lepton pair into a hyperon pair, the spin-density distribution WðξÞ representing the spin dependence in hyperon production and decay, and the phase-space element in sequential hyperon decay. Each measured event is specified by a nine-dimensional vector ξ ¼ ðθ; Ω_Λ; Ωp; Ω_¯Λ; Ω_¯pÞ, with θ the scattering angle in the e^þe⁻→ Σ⁰¯Σ⁰subprocess.

The dynamics of the process is described by four unit- three vectors lp, lΛ, l¯p, l_¯Λ, directed along the directions of motion of the final state baryonsðΩp; Ω_Λ; Ω_¯p; Ω_¯ΛÞ. We have arranged so that the spin-density-distribution function can be written as

WðξÞ ¼ R − ½λΛQ_Λþ λ_¯ΛQ_¯ΛS

þ λ_Λλ_¯Λ½Q₁T₁þ Q₂T₂þ Q₃T₃þ Q₄T₄:

ð12:1Þ Here, the six functions R, S, and T are functions of the scattering angleθ and the ratio of form factors ηψ, whereas the six functions Q are functions of l_Λ and l_¯Λ, and of ˆp ¼ lΣ⁰and ˆk ¼ le. The unit vectorslpandl_¯ponly enter the weak-asymmetry functionsλΛandλ_¯Λof Eqs.(7.1)and(7.2).

It remains to connect the four kinematic unit vectors to measured quantities. To this end we imbed Cartesian- coordinate systems in our events. Then, with the Lambda hyperon as an example,

l_Λ¼ ðl_Λx; l_Λy; l_ΛzÞ ¼ ðcos ϕ_Λsinθ_Λ; sin ϕ_Λsinθ_Λ; cos θ_ΛÞ:

ð12:2Þ Our preferred coordinate system is named global and has the xz plane as scattering plane, andˆp along the z direction.

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In global coordinates the building blocks of the spin-density- distribution function WðξÞ in Eq. (12.1) have the simple structure mentioned above. In particular, the six Q functions are independent of the scattering angle θ.

An alternative to global coordinates is helicity-like coordinates, when the x⁰z⁰ plane is the scattering plane, andk directed along the z⁰ axis. Several of the Q functions now depend on the scattering angle θ in a complex way, even though the two coordinate systems are related by a rotation.

ACKNOWLEDGMENTS

I would like to thank Karin Schönning and Andrzej Kupsc for their kind help and interest.

APPENDIX A: BARYON FORM FACTORS The diagram in Fig.1describes the annihilation reaction e⁻ðk₁Þe^þðk₂Þ → Yðp₁Þ ¯Yðp₂Þ and involves two vertex functions: one of them leptonic, the other one baryonic.

The strength of the lepton-vertex function is determined by the fine-structure constantαe, but two complex form factors G^ψ_MðsÞ and G^ψ_EðsÞ are needed for a proper parametrization of the baryonic vertex function, as of Ref.[4]. The values of these form factors vary with energy, s¼ ðp₁þ p₂Þ².

The strength of the baryon form factors is measured by the function D_ψðsÞ,

D_ψðsÞ ¼ sjG^ψMj²þ 4M²jG^ψEj²; ðA1Þ with the M-variable representing the hyperon mass. The ratio of form factors is measured byη_ψðsÞ,

ηψðsÞ ¼sjG^ψ_Mj²− 4M²jG^ψ_Ej²

sjG^ψMj²þ 4M²jG^ψEj²; ðA2Þ withη_ψðsÞ satisfying −1 ≤ η_ψðsÞ ≤ 1. The relative phase of form factors is measured byΔΦ_ψðsÞ,

G^ψ_E

G^ψ_M¼ e^iΔΦ^ψ^ðsÞ

G^ψ_E G^ψ_M

: ðA3Þ

A model involving both strong and electromagnetic amplitudes, and simultaneously describing the J=ψ decays into baryon-antibaryon pairs, J=ψ → Y ¯Y, is investigated in Ref. [11]. The model parameters are determined by fitting to available experimental data. For the parameters we need, those of the decay J=ψ → Σ⁰¯Σ⁰, experimental data exist [12]. In particular, ½ηψ ¼ −0.467 0.014 and

½ΔΦ_ψ ¼ 0.092 0.030.

APPENDIX B: STRUCTURE FUNCTIONS The six structure functions R, S, and T of Eq. (3.7) depend on the scattering angleθ, the ratio function ηψðsÞ, and the phase functionΔΦ_ψðsÞ. To be specific[4,7],

R ¼ 1 þ η_ψcos²θ; ðB1Þ

S ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffi 1 − η²_ψ q

sinθ cos θ sinðΔΦ_ψÞ; ðB2Þ

T₁¼ η_ψþ cos²θ; ðB3Þ

T₂¼ −η_ψsin²θ; ðB4Þ

T3¼ ð1 þ ηψÞsin²θ; ðB5Þ T₄¼ ffiffiffiffiffiffiffiffiffiffiffiffiffi

1 − η²_ψ q

sinθ cos θ cosðΔΦ_ψÞ: ðB6Þ The parameters η_ψ and ΔΦ_ψ are defined in Eqs. (A2) and (A3). The function T3 of Eq. (B5) differs from the corresponding functionT₃of Ref.[2]by the sin²θ factor.

Similarly, the function T₄ of Eq. (B6) differs from the corresponding functionT₄of Ref.[2] by the sinθ factor.

APPENDIX C: INTRODUCING ANGULAR VARIABLES

The angular functions QðlΛ; l_¯ΛÞ of Eq.(7.4) and the λ parameters of Eqs. (7.1)and(7.2) are expressed in terms of unit vectors such aslp and lΛ, which are not directly measurable but which must be calculated. We suggest the following approach.

For each event we imbed the particle momenta in its c.m.

system and with coordinate axes as defined in Eq.(9.1).

For theΣ⁰hyperon the components of the momentum are, by definition,

ˆp_Σ⁰ ¼ ð0; 0; 1Þ: ðC1Þ Then, let us consider the proton and the hyperon of the final state, with momentappandpΛin the c.m. system. In the rest system of the Lambda hyperon Lp denotes the proton momentum, which is given by the expression

Lp¼ ppþ BΛppΛ; ðC2Þ

B_Λp¼ 1 m_Λ

1

E_Λþ mΛp_Λ·pp− E_Λ

: ðC3Þ

Now, the length of the vectorLpis well known, being the momentum in the hyperon decayΛ → πN, and therefore

jLpj ¼ 1

2m_Λ½ðm²_Λþ m²_π− m²_NÞ²− 4m²_Λm²_π¹⁼²: ðC4Þ Hence, the unit vector lp appearing in our equations should be

lp¼ Lp=jLpj; ðC5Þ

¼ ðcos ϕpsinθp; sin ϕpsinθp; cos θpÞ: ðC6Þ