Sequential hyperon decays in the reaction e + e − → Σ 0 ¯ Σ 0

We report on a study of the sequential hyperon decay Σ 0 → Λ γ ; Λ → p π − and its corresponding antihyperon decay. We derive a multidimensional and model-independent formalism for the case when the hyperons are produced in the reaction e þ e − → Σ 0 ¯ Σ 0 . Cross-section distributions are calculated using the folding technique. We also study sequential decays of single-tagged hyperons.


I. INTRODUCTION
The BESIII experiment [1] has created new opportunities for research into hyperon physics, based on e þ e − annihilation into hyperon-antihyperon pairs. Such possibilities are interesting, and for several reasons: (i) They offer the currently only feasible way for investigating the electromagnetic structure of hyperons [2]. (ii) By measuring in the vicinity of vector-charmonium states, one gains information on the strong baryonantibaryon decay processes of charmonia. (iii) They offer a model-independent method for measuring weak-decay-asymmetry parameters, which can probe CP symmetry [3]. The basic reaction, e þ e − → YȲ, is graphed in Fig. 1. In the continuum region, i.e., in energy regions that do not overlap with energies of vector charmonia like J=ψ, ψ 0 and ψð2SÞ, the production process is dominated by one-photon exchange, e þ e − → γ Ã → YȲ. The reaction amplitude is then governed by the electromagnetic form factors G E and G M . In the vicinity of vector resonances, the electromagnetic form factors are replaced by hadronic form factors G ψ E and G ψ M . However, the shapes of the differential-crosssection distributions are the same in the two cases: all physics of the production mechanism is contained within the form factors, or equivalently, the strength of form factors, D ψ ðsÞ; the ratio of form-factor magnitudes, η ψ ðsÞ; and the relative phase of form factors, ΔΦ ψ ðsÞ.
Analyses of joint-decay distributions of hyperons, such as Λð→ pπ − ÞΛð→pπ þ Þ, enables us to determine the weak-interaction-decay parameters, αβγ. For a complete determination we need to know the bayon-final-state polarizations.
The theoretical description of the annihilation reaction of Fig. 1 is described in Ref. [4], and the corresponding annihilation reaction mediated by J=ψ in Ref. [5]. Accurate experimental results for the form-factor parameters η ψ and ΔΦ ψ and the weak-interaction parameters α Λ ðαΛÞ for the latter annihilation process are all reported in Ref. [3]. A precise knowledge of the asymmetry parameters α Λ ðαΛÞ is needed for studies of spin polarization in Ω − , Ξ − , and Λ þ c decays, and for tests of the Standard Model.
The graph of Fig. 1 can be generalized in the sense that it can include hyperons that decay sequentially. It can also include cases where the produced hyperon is of a different kind than the produced antihyperon, i.e., e þ e − → Y 1Ȳ2 .
In this paper we shall consider annihilation into Σ 0Σ0 pairs, in a way similar to that of Ref. [6]. The Σ 0 decays electromagnetically, Σ 0 → Λγ, and subsequently the Lambda hyperon decays weakly, Λ → pπ − . The interest of such a study is many-fold: (i) The form factors provide information about the production process. So far, literature has focused on electromagnetic form factors whose interpretation is FIG. 1. Graph describing the electromagnetic annihilation reaction e þ e − →ΛΛ. The same reaction can also proceed hadronicly via vector charmonium states such as J=ψ, ψ 0 , or ψð2SÞ, replacing the photon. straightforward [2,7]. However, recent experimental advances call for an interpretation also of the hadronic form factors. In particular, it would be interesting to compare the decay of J=ψ into various hyperon-antihyperon pairs with the corresponding decays of other vector charmonia. (ii) The BESIII Collaboration plans to perform a first measurement of the branching fraction of the Σ 0 Dalitz decay Σ 0 → Λγ Ã , γ Ã → e þ e − using the large data sample available for the e þ e − → J=ψ →Σ 0 Σ 0 process. Then, the most important background will come from e þ e − → J=ψ →Σ 0 Σ 0 ; (Σ 0 → Λγ; Λ → pπ − þ c:c), where one of the photons undergoes external conversion into an e þ e − pair. This is because the branching ratio of the Σ 0 → Λγ, according to QED, is 3 orders of magnitude larger than that of the Dalitz decay. In order to properly account for the background, precise knowledge of the joint angular distribution is required. (iii) It can provide an independent measurement of the Lambda asymmetry parameters α Λ and αΛ. (iv) It can provide a first test of strong CP symmetry in the Σ 0 → Λγ decay [8]. Our calculation is performed in steps. First, we review some important facts; the spin structure of the e þ e − → Σ 0Σ0 annihilation reaction [4]; the classical αβγ description of hyperon decays [9]; the description of the electromagnetic Σ 0 → Λγ decay, both for real and virtual photons [6,10]. The virtual photons decay into Dalitz lepton pairs. An important element of our calculation is the factorization of the squared amplitudes into a spin-independent fractional decay rate and a spin-density distribution.
Following these reviews we demonstrate how the folding method of Ref. [11] is adapted to sequential decays. Both simple and double decay chains are treated. Finally, we join production and decay steps to give the cross-section distributions.
The information we are hoping to gain resides in the angular distributions, and we are therefore not overly concerned with absolute normalizations, although they may be obtained without too much effort.

II. BARYON FORM FACTORS
The diagram in Fig. 1 describes the annihilation reaction e − ðk 1 Þe þ ðk 2 Þ → Yðp 1 ÞȲðp 2 Þ and involves two vertex functions: one of them leptonic, the other one baryonic. The strength of the lepton-vertex function is determined by the electric charge e e , but two form factors G M ðsÞ and G E ðsÞ are needed for describing the baryonic vertex function. Here, s ¼ ðp 1 þ p 2 Þ 2 with p 1 and p 2 as defined in Fig. 1.
The strength of the baryon form factors is measured by the function DðsÞ, ð2:1Þ with the M-variable representing the hyperon mass. The ratio of form factors is measured by ηðsÞ, with ηðsÞ satisfying −1 ≤ ηðsÞ ≤ 1. The relative phase of form factors is measured by ΔΦðsÞ, In Ref. [5] annihilation in the region of the J=ψ and ψð2SÞ masses is considered. The photon propagator of Fig. 1 is then replaced by the appropriate vector-meson propagator.
Our first task is to review the calculation of the crosssection distribution for e þ e − annihilation into baryonantibaryon pairs, with baryon-four-vector polarizations s 1 and s 2 [4,5]. From the squared matrix element of this process, jMj 2 , we remove a factor e 4 e =s 2 , which is the square of the propagator, and get dσ ¼ 1 2s e 4 e s 2 jM red ðs 1 ; s 2 Þj 2 dLipsðk 1 þ k 2 ; p 1 ; p 2 Þ; ð3:1Þ with s ¼ ðp 1 þ p 2 Þ 2 , and dLips denotes the phase-space element of Ref. [12], as described in Appendix A. For a baryon of momentum p the four-vector spin s is related to the three-vector spin n, the spin in the rest system, by sðp; nÞ ¼ n k M ðjpj; EpÞ þ ð0; n ⊥ Þ: ð3:2Þ Longitudinal and transverse directions of vectors are relative to thep direction.
In the global c.m. system kinematics simplifies. There, three-momenta p and k are defined such that with p ¼ jpj and k ¼ jkj.
The matrix element in Eq. (3.1) can be written as a sum of terms that depends on the baryon and antibaryon spin directions in their respective rest systems, n 1 and n 2 , jM red ðe þ e − → Yðs 1 ÞȲðs 2 ÞÞj 2 ¼ sDðsÞSðn 1 ; n 2 Þ; ð3:7Þ with the strength function DðsÞ defined in Eq. (2.1). We call a function such as Sðn 1 ; n 2 Þ a spin density. In the present case, the spin density is a sum of seven mutually orthogonal contributions [4], where N is the normal to the scattering plane, The six structure functions R, S, and T of Eq. (3.8) depend on the scattering angle θ, the ratio function ηðsÞ, and the phase function ΔΦðsÞ. Their detailed expressions are given in Appendix C.
The This result tells us that the baryon is polarized and that its polarization is directed along the normal to the scattering plane,p ×k, and that the value of the polarization is From Eq. (3.8) we conclude that there is a corresponding result for the antibaryon, but it should then be remembered that p is the momentum of the baryon, but −p that of the antibaryon. Baryon and antibaryon polarizations in e þ e − annihilation were first discussed by Dubničkova et al. [13], but with results slightly different from ours, and later by Czyż et al. [14]. For details see Ref. [4].

IV. WEAK BARYON DECAYS
Weak decays of spin one-half baryons, such as Λ → pπ − , involve two amplitudes, one S-wave and one P-wave amplitude, and the decay distribution is commonly parametrized by three parameters, denoted αβγ, and which fulfill a relation Details of this description can be found in Refs. [15] or [4,9]. Since we shall encounter several weak baryon decays of the same structure as the Λ → pπ − decay, we shall use a generic notation, c → dπ, for those decays.
The matrix element describing the decay of a polarized c baryon into a polarized d baryon is Mðc → dπÞ ¼ūðp d ; s d ÞðA þ Bγ 5 Þuðp c ; s c Þ; ð4:2Þ with p and s with appropriate indices denoting momenta and spin four-vectors of the baryons. The square of this matrix element we factorize, writing where n c and n d are the spin vectors of baryons c and d in their rest frames, Eq. (3.2). The R-factor is a spin independent factor, defined by where Φðc → dπÞ ¼ Φðm c ; m d ; m π Þ is the phase-space volume of Appendix B. We refer to Rðc → dπÞ as the fractional decay rate, since it is a decay rate per unit phase space. Further inspection of Eq. (4.3) tells us that Γðc → dπÞ is defined as an average over the spins of both initial-and final-state baryon. The spin-density-distribution function, Gðn c ; n d Þ of Eq. (4.3), is a Lorentz scalar, which we choose to evaluate in the rest system of the mother baryon, c, The vector l d is a unit vector in the direction of motion of the daughter baryon, d, in the rest system of mother baryon c. The indices on the αβγ parameters remind us they characterize baryon c. A spin density is normalized if the spin-independent term is unity. We observe an important symmetry, Since the spin of baryon d is usually not measured, the interesting spin-density is obtained by taking the average over the spin directions n d , For an initial state polarization P c we put n c ¼ P c , and get an angular distribution known from the weak hyperon decay Λ → pπ − [4,9]. The matrix element describing the decay of a polarizedc (anti)baryon into a polarizedd (anti)baryon is similar to that of Eq. The relation between the parameters A, B and A 0 , B 0 is clarified in Refs. [16,17].
The square of the antibaryon matrix element of Eq. (4.10) is factorized exactly as the baryon-matrix element of Eq. where nc and nd are the spin vectors of baryonsc andd in their rest systems. The functions Rðc →dπÞ and Gðnc; ndÞ are tied to hyperonsc andd in exactly the same way as those tied to hyperons c and d, Eqs. (4.4) and (4.5), or to be specific, Gðc;dÞ ¼ 1 þ αcnc · ld þ αcnd · ld þ nc · Lcðnd; ldÞ:

ð4:12Þ
For CP conserving interactions the asymmetry parameters of the hyperon pair c, d are related to those of antihyperon pairc,d by [16,17] Electromagnetic transitions such as Σ 0 → Λγ and Ξ 0 → Σ 0 γ are readily investigated in e þ e − annihilation. The electromagnetic Σ 0 → Λ transition is caused by the fourvector current [12] This transition current is gauge invariant, inasmuch as k · J ¼ 0. In fact, the F 1 ðk 2 Þ and F 2 ðk 2 Þ contributions are each, by themselves, gauge invariant. For real photons k 2 ¼ 0 and the F 1 contribution vanishes, since F 1 itself vanishes, F 1 ð0Þ ¼ 0. Thus, for this case it is sufficient to consider the F 2 term. We denote by μ cd , the strength of the M1 magnetic transition. As a consequence, the expression for the matrix element for any electromagnetic Σ 0 → Λγ like decay, becomes where s c and s d are the spin four-vectors of the two baryons.
It is convenient to write the square of this matrix element on the form with H μν γ ðkÞ referred to as the hadron tensor. We have also made use of the simplifying identity valid for real photons. Summation over the two photon-spin directions entails replacing e μ ðkÞe ⋆ ν ðkÞ by −g μν . This leads to X e γ jM γ ðc → dγÞj 2 ¼ Rðc → dγÞG γ ðn c ; n d Þ; ð5:6Þ and again n c and n d are the spin vectors of baryons c and d in their rest systems. Photon polarizations are summed over. There are also electromagnetic transitions between charged baryons, but in this section we limit ourselves to electromagnetic transitions between neutral baryons. The factorization of Eq. (5.6) is chosen so that the fractional decay rate Rðc → dγÞ is the unpolarized part of Eq. (5.6) and its G γ ðn c ; n d Þ factor the normalized spindensity-distribution function. Here, unpolarized means averaged over the spin directions of both initial and final baryons.
The fractional decay rate, Rðc → dγÞ of Eq. (5.7), has the same structure as the corresponding one for weak baryon decays, Eq. (4.4), The electromagnetic decay width is where ω is the photon energy. Remember, that this width is obtained after averaging over both initial and final baryon spin states. The spin-density-distribution function of Eq. (5.6) involves an implicit summation over photon polarizations. For such a case where l γ is a unit vector in the direction of motion of the photon, and l d ¼ −l γ a unit vector in the direction of motion of baryon d, both in the rest system of baryon c. 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. We also notice that expression (5.9) cannot be written in the αβγ representation of Eqs. (4.5) and (4.6).
When the spin of the final-state baryon d is not measured, the relevant spin density is obtained by forming the average over the spin directions n d , Thus, the decay-distribution function is independent of the initial-state baryon spin vector n c .
The normalized spin density corresponding to the antiparticle-matrix element of Eq. (5.12) is the same as that corresponding to the particle matrix element of Eq. (5.3), as given in Eq. (5.9), provided we replace the particle spin vectors n c and n d by the antiparticle spin vectors nc and nd.
The possibility to search for P-violating admixtures in the electromagnetic decay Σ 0 → Λγ was advocated by Nair et al. [8]. Such contributions are created by making the substitution in the decay amplitude of Eq. (5.3). This substitution is gauge invariant and changes the normalized spin density with asymmetry parameter Similarly, the decay width of Eq. (5.8) is changed into Parity violating admixtures in the antiparticle decaȳ Σ 0 →Λγ can also be simulated by the substitution of Eq. (5.13). Replacing the parameter b byb, the spin density for the antiparticle decay becomes The P-violating interference term now enters with the opposite sign. If CP is conserved thenb ¼ −b. For a full discussion of P and CP conservation in this context we refer to Ref. [8].

VI. ELECTROMAGNETIC HYPERON DECAYS: VIRTUAL PHOTONS
The leptonic decay Σ 0 → Λe þ e − is a small fraction of the electromagnetic decay Σ 0 → Λγ [18,19]. The lepton pair of the leptonic decay is interpreted as the decay product of a virtual, massive photon. This pair is often referred to as a Dalitz lepton pair.
The form factors F 1 ðk 2 Þ and F 2 ðk 2 Þ have been calculated in chiral perturbation theory [20,21]. The form factor F 1 ðk 2 Þ remains small for virtual photons and it is therefore reasonable to neglect its contribution.
The steps to follow in order to find the cross-section distribution for virtual photons are well known. The square of the reduced matrix element is written as where H μν e is the hadron tensor and L μν the lepton tensor. The hadron tensor can be extracted from Eq. (5.4), We need the square of M e for fixed baryon spins but summed over lepton spins. The summation over lepton spins leads to a lepton tensor, ð6:3Þ where k 1 and k 2 are the lepton momenta, and k ¼ k 1 þ k 2 the four momentum of the virtual photon. Next, we integrate over the lepton momenta. For this purpose we rewrite the phase-space element as with k 2 ¼ m 2 γ and dLipsðk; k 1 ; k 2 Þ the phase-space element for the lepton pair, as in Appendix B.
The integration over the lepton phase space affects only the lepton tensor. Thus, we note that ð6:5Þ and similarly for hk 2μ k 2ν i, with brackets denoting integration over lepton phase space, dLipsðk; k 1 ; k 2 Þ, and h1i denoting the phase-space volume itself. The term proportional to k μ k ν in Eq. (6.5) vanishes due to gauge invariance. As a consequence, we get as average of the lepton tensor, ð6:6Þ The lepton tensor L μν of Eq. (6.6) comes with a factor ð−g μν Þ. Contracting it with the hadron tensor H μν ðc → dgÞ, with g representing the virtual photon, is equivalent to summing over photon polarizations. We write The factorization is chosen so that Rðc → dgÞ is spin independent, and so that the spin-independent term of Gðn c ; n d Þ is unity. The functions R and G are easily calculated. Neglecting terms unimportant for the Σ 0 → Λγ transition, we get for the fractional decay rate of Eq. (6.8), where Φðc → dgÞ ¼ Φðm c ; m d ; m γ Þ is the phase-space volume. For m γ ¼ 0 we recover Rðc → dγÞ for real photons, Eq. (5.7). Again neglecting terms unimportant for the Σ 0 → Λγ transition, the properly normalized spin density reads Gðn c ; n d Þ ¼ 1 − n c · l γ l γ · n d : ð6:10Þ Thus, it is in this approximation also equal to the normalized spin density for real photons, Eq. (5.9). The exact expressions for R and G are given in Appendix D. Next, we combine the matrix elements for the transitions c → dg and g → e þ e − , g representing a virtual photon of mass m γ .
Since the lepton tensor of Eq. (6.7) lacks spin dependence, so that Gðg → e þ e − Þ ¼ 1, we have the spin-density relation ð6:11Þ and a corresponding R-factor relation

12Þ
The function Rðg → e þ e − Þ collects the remains, the lepton tensor of Eq. (6.7) multiplied by the propagator 1=k 4 of Eq. (6.1), ð6:13Þ This expression comes with the phase-space element where k is the four-momentum of the virtual photon and Deviations from the Dalitz-distribution function of Eq. (6.13) signals the importance of electromagnetic form factors in the virtual photon exchange.

VII. FOLDING
Our general aim is to calculate the cross-section distributions for e þ e − annihilation into Σ 0Σ0 pairs that subsequently decay, as Σ 0 → Λ → p orΣ 0 →Λ →p, and as illustrated in Fig. 2. The first step in this endeavour is to perform the folding of a product of spin densities, a technique especially adapted to spin one-half baryons.
A folding procedure implies forming an average over intermediate-spin directions n according to the prescription h1i n ¼ 1; hni n ¼ 0; hn · kn · li n ¼ k · l: ð7:1Þ For more details see Ref. [11].
In the present case there are five spin densities; the annihilation spin density Sðn Σ ; nΣÞ of Eq. (3.8); the spin densities of the electromagnetic and weak decays, Eqs. (5.9) and (4.5), with L Λ ðn p ; l p Þ defined in Eq. (4.6); and the antihyperon versions of the last two spin densities. Remember that the symbol l represents a unit vector. The spin density for the Σ 0 → p transition is obtained by folding a product of spin densities. Averaging over the Lambda and final-state proton spins, according to the folding prescription Eq. (7.1), gives us ð7:4Þ We notice that this spin density does not depend on the asymmetry parameters β Λ and γ Λ , a consequence of the average over the final-state-proton-spin directions.
To the baryon decay chain Σ 0 → Λ → p there is a corresponding antibaryon decay chainΣ 0 →Λ →p, and a corresponding transition-spin density.
To go from the baryon to the antibaryon case, we simply replace the baryon variables by their antibaryon counterparts, n Σ → nΣ, α Λ → αΛ, etc.
The inclusion of parity violation in the Σ 0 → Λγ decay is straightforward. We simply replace GðΣ 0 → ΛγÞ of Eq. (7.2) by of Eq. (5.14), and get From this expression the angular distribution in the decay of a Σ 0 of polarization P Σ is obtained by the substitution n Σ → P Σ . The angular distribution in the decay of an unpolarized Σ 0 hyperon becomes ð1 − ρ Σ α Λ l γ · l p Þ. Hence, as a consequence of parity violation the crosssection distribution acquires a small angular dependent term.

VIII. SINGLE CHAIN DECAYS
Single-chain decays of Σ 0 hyperons can be studied in the e þ e − annihilation into Σ 0Σ0 pairs, provided theΣ 0 is somehow identified, e.g., as a missing hyperon [6]. The spin-density state of the Σ 0 will then be obtained from Eq. (3.8) as ð8:1Þ A Σ 0 hyperon in a state of polarization P Σ , subject to the condition jP Σ j ≤ 1, is characterized by a normalized spindensity function, ð8:2Þ Therefore, by Eq. (8.1), it follows that If a Σ 0 hyperon of polarization P Σ undergoes an electromagnetic decay, Σ 0 → Λγ, we can determine the spin-density distribution of the Λ hyperon by folding the initial state Σ 0 spin density of Eq. (8.2) with the Σ 0 decay distribution of Eq. (5.9), to get ð8:4Þ with l Λ ¼ −l γ and a Λ polarization Consequently, the Λ polarization is directed along the Λ momentum l Λ , a fact which is independent of the initial Σ 0 hyperon spin. Let us now consider also the weak decay of the Λ-hyperon, Λ → pπ − , which is described by the spin density G Λ ðn Λ ; n p Þ of Eq. (4.5). Since the spin of the final-state proton is usually not measured, we form the average over the proton-spin directions. Then, the spindensity-distribution function of Eq. (8.4) is expanded to ð8:6Þ The decay chain Σ 0 → Λγ → pπ − makes part of our annihilation process and it is therefore of interest to investigate what additional information may be obtained by measuring the spin of the final-state proton. Thus, instead of the spin density of Eq. (7.6) we investigate the spin density GðΣ 0 → pÞ ¼ hGðΣ 0 → ΛγÞGðΛ → pπ − Þi n Λ : ð8:7Þ Invoking the vector-function identity of Eq. (4.6) we get GðΣ 0 → pÞ ¼ 1 þ α Λ n p · l p − n Σ · l γ ½α Λ l γ · l p þ n p · L Λ ðl γ ; −l p Þ: ð8:8Þ Finally, the spin-density-distribution function for the final state proton is obtained as ð8:9Þ This result describes a proton polarization which is V p =U p . It is explicitly dependent on α Λ , but there is a hidden dependence on β Λ and γ Λ in the vector function L Λ .

IX. PRODUCTION AND DECAY OF Σ 0Σ0 PAIRS
Now, we come to the main task of our investigation: production and decay of Σ 0Σ0 pairs. The starting point is the reaction e þ e − → Σ 0Σ0 , the spin-density distribution of which was calculated in Sec. III. We name it Sðn Σ ; nΣÞ. The explicit expression is given by Eq. (3.8), with n 1 , n 2 replaced by n Σ ; nΣ.
The final-state-angular distributions are obtained by folding the spin distributions for production and decay, according to prescription (7.1). Invoking Eq. (3.8) for the production step and Eqs. (9.1) and its antidistribution for the decay steps, we get the angular distribution W ΣΣ ðl a Þ ¼ hSðn Σ ; nΣÞW Σ ðn Σ ; n p ÞWΣðnΣ; npÞi n Σ ;nΣ where l a represents the ensemble of l values in the decays.
The angular distributions of Eq. (9.4) still depend on the spin vectors n p and np which are difficult to measure. If we are willing to consider proton-and antiproton-spin averages, then variables U and V simplify, ð9:5Þ Since U Σ ¼ UΣ ¼ 1 the effect of the folding is to make, in the spin-density function Sðn Σ ; nΣÞ of Eq. (3.8), the replacements n Σ → V Σ and nΣ → VΣ. We notice that the U and V variables are independent of the weak-asymmetry parameters β Λ and γ Λ . Their dependence is hidden in the vector function L Λ ðl γ ; −l p Þ of Eq. (9.3), and which is absent in Eq. (9.4).
Inserting the expressions of Eq. (9.5) into the spindensity function of Eq. (9.4) we get Thus, this is the angular distribution obtained when folding the product of spin densities for production and decay.

X. DIFFERENTIAL DISTRIBUTIONS
Explicit expressions for the structure functions R, S, and T are given in Appendix C. With their help we can rewrite the differential distribution function of Eq. (9.6) as cosðΔΦÞ sin θ cos θF 9 ; ð10:1Þ where the argument ξ of the angular functions is a ninedimensional vector ξ ¼ ðθ; Ω Λ ; Ω p ; ΩΛ; ΩpÞ.
The ten angular functions F k ðξÞ are defined as F 0 ðξÞ ¼ 1; The cross-section distribution (9.6), and also the ten The differential distribution function WðξÞ of Eq. (10.1) involves two parameters related to the e þ e − → Σ 0Σ0 reaction that can be determined by data: the ratio of form factors η, and the relative phase of form factors ΔΦ. In addition, the distribution function WðξÞ depends on the weak-asymmetry parameters α Λ and αΛ of the two Lambda-hyperon decays. The dependence on the weakasymmetry parameters β and γ drops out, since final-stateproton and antiproton spins are not measured.
An important conclusion to be drawn from the differential distribution of Eq. (10.1) is that when the phase ΔΦ is small, the parameters α Λ and αΛ are strongly correlated and therefore difficult to separate. In order to contribute to the experimental precision of α Λ and αΛ a nonzero value of ΔΦ is required.
The sequential differential decay distribution of a singletagged Σ 0 produced in e þ e − annihilation can be obtained form Eq. (10.1) by suitably integrating over the angular variables ΩΛ and Ωp. As a result we get the differential distribution for Σ 0 production and decay, sinðΔΦÞ sin θ cos θ × cos θ Λp sin θ Λ sin ϕ Λ dΩdΩ Λ dΩ p : ð10:3Þ Here, θ is the Σ 0 production angle, θ Λp the relative angle between the vectors l Λ and l p , and θ Λ and ϕ Λ the directional angles of l Λ in the global coordinate system of Appendix E. From the angular distribution of Eq. (10.3) we can determine the product α Λ sinðΔΦÞ, and from the corresponding Σ 0 distribution the product αΛ sinðΔΦÞ. In this application the final-state-proton spin can be included in a formula of finite length. From Eq. (9.4) we get with n p the final-state-proton-spin vector, and the function L Λ ð−l Λ ; −l p Þ defined in Eq. (4.6), and dependent on the weak interaction parameters β Λ and γ Λ . Important information can be retrieved from Eq. (10.3). Denoting its right-hand side W Σ , and forming the average over the final-state-phase space, we get The correlation between the scattering angle θ and the angle θ Np , with cos θ Np ¼ N · l p , can also be determined, and Thus, knowledge of the weak interaction parameter α Λ , and the ratio of form factors η, allows us to determine the relative phase Φ between form factors, by considering the ratio of expressions (10.6) and (10.5). Since the absolute value of cross sections are usually unknown it is essential to consider cross-section ratios for information.

XI. CROSS-SECTION DISTRIBUTIONS
We shall now consider the phase-space imbedding of the differential-distribution function of Eq. (10.1). We start with the cross-section-distribution function for creation of a pair of baryons, e þ e − → Σ 0Σ0 . Combining Eqs. (3.1), (3.6), and (3.7), we get where Ω are the baryon scattering angles in the c.m. system. Next we consider the propagator factors associated with the sequential decays of the baryons Σ 0 andΣ 0 produced in the e þ e − annihilation process. These sequential decays are illustrated in Fig. 2. There are three factors associated with the square of each propagator. Let us consider the decay c → dg, where g can represent a pion or a photon. Other decay modes are also possible to incorporate. Then, we have Here, the first factor comes from squaring the propagator in the Feynman diagram; the second factor from dividing the phase-space element into a product of two-body phasespace elements; and the third factor is the reduced matrix element squared for the decay c → dg, and the product of the normalized spin density G c and the fractional decay rate R c . The fractional decay rate R c is defined in Eq. (5.7) as where Φ is the two-body phase-space volume, and Γðc → dgÞ the channel width for the decay c → dg. It was defined to be spin averaged for both initial and final baryon states. However, in a sequential decay both final spin-state contributions must be included. This is achieved by multiplying Rðc → dγÞ by a factor of 2. This factor can be incorporated in the channel width Γðc → dgÞ, reinterpreting it to include the sum over final baryon spin states. Finally, we observe that giving as a consequence a P factor with Ω c the angular variable in the rest system of baryon c. In our application index c represents one of the four mother hyperons Σ 0 ; Λ andΣ 0 ;Λ. Similarly, index d represents one of the four daughter hyperons Λ, p andΛ;p. The differential-distribution function WðξÞ of Eq. (10.1) is obtained by folding a product of five spin densities Folding involves averages over spin directions, but as remarked, cross-section distributions require summing over the spin directions. Thus, an average over the spin density Sðn Σ 0 ; nΣ0Þ is accompanied by an extra factor of 4, and it is not normalized to unity either but to R. The folding formula Eq. (11.6) combined with Eqs. (11.1) and (3.11) gives the master equation This readily understood structure agrees with that reported in Ref. [11].
Since spin densities are normalized, except for the annihilation density Sðn Σ 0 ; nΣ0Þ, the overall normalization condition reads This normalization is checked explicitly in single-chainsequential decay in Ref. [6]. When the Σ 0 → Λγ reaction is involved, and the photon is real, then the channel width in Eq. (11.7) is for all practical purposes equal to the total width, ΓðΣ 0 → ΛγÞ ¼ ΓðΣ 0 → allÞ.
We also point out that for virtual photons the crosssection distribution of Eq. (11.7) receives an additional lepton factor, where m γ is the virtual photon mass, m 2 γ ¼ k 2 , and R the Dalitz function

ð11:10Þ
The e þ e − annihilation reactions described above are all concerned with annihilation through ordinary photons, as illustrated in Fig. 1. However, the same reactions can be initiated by other vector mesons as well. Of special interest is the J=ψ case, which is treated in Ref. [5], and which is accessible to the BESIII experiment. By making the replacement in the photon-induced reaction, Eq. (11.1), we get the crosssection-distribution formula for annihilation through the J=ψ meson. The meaning of the parameters α ψ and α g is explained in Ref. [5]. This replacement is equivalent to replacing in the master formula of Eq. (11.7) the corresponding photon-induced e þ e − → Σ 0Σ0 annihilation cross section by the J=ψ induced cross section, ð11:12Þ
In the present investigation we analyze the sequential hyperon decay Σ 0 → Λγ; Λ → pπ − , and its corresponding sequential antihyperon decay, again when simultaneously taking place in the reaction e þ e − → Σ 0Σ0 . The structure of the cross-section distribution for annihilation, whether via one-photon states or vector-charmonium states, is the same.
The aim of the present investigation was, among other things, to discuss how to relate measured observables to observables used in theoretical analyses. In studies of reactions like e þ e − → Σ 0Σ0 ðΛΛÞ it is customary to use different coordinate systems for observables referring to the Σ 0 ðΛÞ andΣ 0 ðΛÞ hyperons, whereas we prefer the use of a single-common-global-coordinate system.
The cross-section distribution for production and subsequent decay of a Σ 0Σ0 pair is described by an easy to understand master formula, ð12:1Þ already encountered in Ref. [4]. The master formula is a product of three factors, describing the annihilation of lepton pairs into hyperon pairs, the folded product of spin densities representing hyperon production and decay, and the phase space of sequential hyperon decays. Each event is specified by a nine-dimensional vector ξ ¼ ðθ; Ω Λ ; Ω p ; ΩΛ; ΩpÞ, with θ the scattering angle in the e þ e − → Σ 0Σ0 subprocess. According to Eq. (3.11), the cross-section distribution for the reaction e þ e − → Σ 0Σ0 can in the one-photon approximation be written as where α e is the fine-structure constant, and R a function defined by the equation Two complex form factors, G M ðsÞ and G E ðsÞ, are needed for a unique characterization of the Σ 0Σ0 γ-electromagneticvertex function, but it is more convenient to work with the three real combinations thereof, DðsÞ, ηðsÞ, and ΔΦðsÞ defined in Sec. II.
The reaction e þ e − → Σ 0Σ0 can also be initiated by a vector-charmonium state, such as the J=ψ. Since the photon and the J=ψ are both vector mesons the structures of the corresponding cross-section distributions will be similar. In fact we obtain the J=ψ cross-section distribution by making the replacement in the photon-induced cross-section distribution. The constant α ψ is determined by the electromagnetic-decay width ΓðJ=ψ → e þ e − Þ, and the constant α g similarly by the hadronic-decay width ΓðJ=ψ → Σ 0Σ0 Þ. The redefined real form-factor functions are now denoted D ψ ðsÞ, η ψ ðsÞ, and ΔΦ ψ ðsÞ.
The differential-spin-distribution function WðξÞ of Eq. (12.1) is obtained by folding a product of five spin densities, ð12:5Þ in accordance with the prescription of Eq. (7.1). The folding operation h…i n applies to each of the six hadron spin vectors, n Σ 0 ; …; np. The function Sðn Σ 0 ; l Σ 0 ; nΣ0; lΣ0Þ represents the spindensity distribution for the hyperon pair produced in the e þ e − → Σ 0Σ0 reaction, and l Σ 0 and lΣ0 are unit vectors in their directions of motion in the global c.m. system. The four remaining spin-density distributions Gðn Λ ; n p Þ etc., represent spin-density distributions for the hyperon decays Σ 0 → Λγ, Λ → pπ − , or their antihyperon counterparts. The simplest spin-density-decay distribution of the four is that for the decay Σ 0 → Λγ, ð12:6Þ The decay-distribution functions Gðn Y 1 ; n Y 2 Þ are normalized to unity, i.e., the spin independent terms are unity, but the density-distribution function Sðn Σ 0 ; nΣ0Þ is normalized to R.
The phase-space factor, dΦðΣ 0 ; Λ; p;Σ 0 ;Λ;pÞ of the master equation, describes the normalized phase-space element for the sequential decays of the two baryons Σ 0 andΣ 0 , dΦðΣ 0 ; Λ; p;Σ 0 ;Λ;pÞ The widths are defined in the usual way. For ΓðΣ 0 → ΛγÞ this means forming an average over the Σ 0 spin direction, and summing over the Lambda and gamma spin directions. The angles Ω Λ define the direction of motion of the Λ hyperon in the Σ 0 rest system. And so on. In addition to the reactions mentioned above we have also calculated the cross-section distributions obtained when one of the final-state photons is materialized as a Dalitz e þ e − pair. We have also investigated how parity violating contributions affect the Σ 0 → Λγ amplitude.
where index CM refers to the two-body reaction e þ e − → YȲ, and index Y to each of the four intermediatestate hyperon decays, in their respective hyperon rest systems.

APPENDIX B: PHASE-SPACE VOLUME
The Lorentz invariant two-body phase-space element is by definition Integration exploiting the delta functions leads to where ffiffi ffi s p ¼ M, k c the momentum, and Ω c the angular variable, both in the c.m. system. In terms of the mass variables The phase-space volume Φ is obtained from Eq. (B2) by integration over dΩ c , For equal masses m 1 ¼ m 2 ¼ m the value of the phasespace volume becomes ΦðM; m; mÞ ≡ h1i ¼ 1 8π APPENDIX C: STRUCTURE FUNCTIONS The six structure functions R, S, and T of Eq. (3.8) depend on the scattering angle θ, in the c.m. system, the ratio function ηðsÞ, and the phase function ΔΦðsÞ. To be specific [4,5], q sin θ cos θ sinðΔΦÞ; ðC2Þ The parameters η and ΔΦ are defined in Eqs. (2.2) and (2.3).

APPENDIX D: DECAY INTO VIRTUAL GAMMA
The squared matrix element jMðc → dgÞj 2 for the decay of a baryon c into a baryon d and a virtual gamma g of mass m γ is given in Eq. (6.8). It can be factorized into factors Rðc → dgÞ and G g ðn c ; n d Þ. The exact expression for the fractional width is with 2m e ≤ m γ ≤ ðm c − m d Þ. In the limit m γ ¼ 0 we recover Rðc → dγÞ for real photons, Eq. (5.7). The exact expression for the normalized spin density is Here, we can without qualm put B ¼ −1 and C ¼ 0. In this limit we recover the normalized spin density for real photons, Eq. (5.9).

APPENDIX E: ANGULAR FUNCTIONS
The cross-section distribution (9.6) is a function of two hyperon unit vectors: l Λ , the direction of motion of the Lambda hyperon in the rest system of the Sigma hyperon, and l p the direction of motion of the proton in the rest system of the Lambda hyperon. Plus the corresponding vectors for the antihyperon chain. In order to handle these vectors we introduce a common global coordinate system, which we define as follows.
The scattering plane of the reaction e þ e − → Σ 0Σ0 is spanned by the unit vectorsp ¼ l Σ andk ¼ l e , as measured in the c.m. system. The scattering plane makes up the xz-plane, with the y-axis along the normal to the scattering plane. We choose a right-handed coordinate system with basis vectors e z ¼p; e y ¼ 1 sin θ ðp ×kÞ; Expressed in terms of them the initial-state lepton momentum becomes k ¼ sin θe x þ cos θe z : ðE2Þ This coordinate system is used for defining the directional angles of the Lambda and the proton. The directional angles of the Lambda hyperon in the Sigma hyperon rest system are l Λ ¼ ðcos ϕ Λ sin θ Λ ; sin ϕ Λ sin θ Λ ; cos θ Λ Þ; ðE3Þ whereas the directional angles of the proton in the Lambda hyperon rest system are l p ¼ ðcos ϕ p sin θ p ; sin ϕ p sin θ p ; cos θ p Þ: ðE4Þ And so for the antihyperons. An event of the reaction e þ e − → Σ 0Σ0 , Σ 0 → Λ → p; Σ 0 →Λ →p is specified by a nine-dimensional vector ξ ¼ ðθ; Ω Λ ; Ω p ; ΩΛ; ΩpÞ. The differential-cross-section distribution is proportional to a function WðξÞ, which according to Eq. (10.1) can be decomposed as The set of ten angular functions, F 0 ðξÞ − F 9 ðξÞ, are defined in Eq. (10.2). The scalar products needed for their determination are as follows: We understand that the remaining scalar products are obtained from those above by the substitution ðΛ; pÞ → ðΛ;pÞ. With the scalar products of Eq. (E6) in hand one quickly determines the ten angular functions F k ðξÞ of Eq. (10.2), F 0 ðξÞ ¼ 1; F 1 ðξÞ ¼ cos 2 θ; F 2 ðξÞ ¼ sin θ Λ sin θ p cosðϕ Λ − ϕ p Þ þ cos θ Λ cos θ p ; F 3 ðξÞ ¼ sin θΛ sin θp cosðϕΛ − ϕpÞ þ cos θΛ cos θp; F 4 ðξÞ ¼ cos θ Λ cos θΛ; F 5 ðξÞ ¼ sin θ Λ sin ϕ Λ ; F 6 ðξÞ ¼ sin θΛ sin ϕΛ; F 7 ðξÞ ¼ sin θ Λ sin θΛ cosðϕ Λ − ϕΛÞ; F 8 ðξÞ ¼ sin θ Λ cos ϕ Λ sin θΛ cos ϕΛ; F 9 ðξÞ ¼ cos θ Λ sin θΛ cos ϕΛ þ sin θ Λ cos ϕ Λ cos θΛ: The differential distribution of Eq. (E5) involves two parameters related to the e þ e − → Σ 0Σ0 reaction that can be determined by data: the ratio of form factors η, and the relative phase of form factors ΔΦ. In addition, the distribution function WðξÞ depends on the weak-decay parameters α Λ and αΛ of the two Λ hyperon decays. The dependence on the weak decay parameters β and γ drops out, when final-state proton and antiproton spins are not measured.

APPENDIX F: FINDING ANGULAR VARIABLES
The angular functions and differential distributions of the previous appendix are expressed in terms of unit vectors such as l p and l Λ , which are not directly measurable but which must be calculated. We suggest the following approach.
For each event we embed the particle momenta in its c.m. system and with coordinate axes as defined in Eq. (E1). For the Σ 0 hyperon the components of the momentum are, by definition,p Σ 0 ¼ ð0; 0; 1Þ: Then, let us consider the final-state proton with momentum p p in the c.m. system. In the rest system of the Lambda hyperon the momentum of the same proton is denoted L p , and given by the expression Now, the length of the vector L p is well known, being the momentum in the decay Λ → πN, and therefore