Sequential hyperon decays in the reaction e+e− → ΣΣ̄

We report on a study of the sequential hyperon decay $\Sigma^0\rightarrow\Lambda \gamma;\Lambda\rightarrow p\pi^- $ and its corresponding anti-hyperon decay. We derive a multi-dimentional and model-independent formalism for the case when the hyperons are produced in the reaction $e^+e^-\rightarrow\Sigma^0\bar{\Sigma}^0$. 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 + e − annihilation into hyperon-anti-hyperon pairs. Such possibilities are interesting, and for several reasons: • They offer the currently only feasible way for investigating the electromagnetic structure of hyperons [2].
• By measuring in the vicinity of vector-charmonium states, one gains information on the strong baryon-antibaryon decay processes of charmonia.
• They offer a model-independent method for measuring weak-decay-asymmetry parameters, which can probe CP symmetry [3].
The basic reaction, e + 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 and ψ(2S), the production process is dominated by one-photon exchange, e + e − → γ * → 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-cross-section distributions are the same in the two cases: all physics of the production mechanism is contained within the form factors, or equivalently, the ratio of form-factor magnitudes, α ψ , and the relative phase of form factors, ∆Φ ψ .
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 α ψ and ∆Φ ψ and the weak-interaction parameters α Λ (αΛ) for the latter annihilation process are all reported in Ref. [3]. A precise knowledge of the asymmetry parameters α Λ (αΛ) is needed for studies of spin polarization in Ω − , Ξ − , and Λ + 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 type than the produced antihyperon, i.e., e + e − → Y 1Ȳ2 . In this note we shall consider annihilation into Σ 0Σ0 pairs, Ref. [6]. The Σ 0 decays electromagnetically, Σ 0 → Λγ, and subsequently the Lambda hyperon decays weakly, Λ → pπ − .
The interest of this measurement is many-fold: • The form factors provide information about the production process. So far, literature has focused on electromagnetic form factors whose interpretation is straight-forward [2,7]. However, recent experimental advances calls 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.
• The BESIII collaboration plans to perform a first measurement of the branching fraction of the Σ 0 Dalitz decay Σ 0 → Λγ * , γ * → e + e − using the large data sample available for the e + e − → J/ψ →Σ 0 Σ 0 process. Then, the most important background will come from e + e − → J/ψ →Σ 0 Σ 0 ; (Σ 0 → Λγ; Λ → pπ − + c.c), where one of the photons undergoes external conversion into an e + e − pair. This is because the branching ratio of the Σ 0 → Λγ, according to QED, is three 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.
• It can provide an independent measurement of the Lambda asymmetry parameters α Λ and αΛ.
• It can provide a first test of strong CP symmetry in the Σ 0 → Λγ decay Ref. [8].
Our calculation is performed in steps. First, we review some important facts; the spin structure of the e + e − → Σ 0Σ0 annihilation reaction Ref. [4]; the classical αβγ description of hyperon decays Ref. [9]; the description of the electromagnetic Σ 0 → Λγ decay, both for real and virtual photons Refs. [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 − (k 1 )e + (k 2 ) → Y (p 1 )Ȳ (p 2 ) 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 (s) and G E (s) are needed for describing the baryonic vertex function. Here, s = (p 1 +p 2 ) 2 with p 1 and p 2 as defined in Fig.1.
The strength of the baryon form factors is measured by the function D(s), a factor that multiplies all cross-section distributions. The ratio of form factors is measured by η(s), with η(s) satisfying −1 ≤ η(s) ≤ 1. The relative phase of form factors is measured by In Ref. [5] annihilation in the region of the J/ψ and ψ(2S) 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 cross-section distribution for e + e − annihilation into baryon-antibaryon pairs, with baryon-four-vector polarizations s 1 and s 2 [4,5].
From the squared matrix element of this process, |M | 2 , we remove a factor e 4 e /s 2 , which is the square of the propagator, and get dσ = 1 2s with s = (p 1 + p 2 ) 2 , and dLips the phase-space element of Ref. [12]. 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 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 and with scattering angle θ defined by, cos θ =p ·k. (III.8) Furthermore, in the global c.m. system the phase-space factor reads with p = |p| and k = |k|.
The matrix element in Eq.(III.4) can be written as a sum of terms that depend on the baryon and antibaryon spin directions in their respective rest systems, n 1 and n 2 , M red (e + e − → Y (s 1 )Y (s 2 )) 2 = sD(s) S(n 1 , n 2 ), (III. 10) with the strength function D(s) defined in Eq.(II.1). We call a function such as S(n 1 , n 2 ) a spin density. In the present case, the spin density is a sum of seven mutually orthogonal contributions [4], S(n 1 , n 2 ) = R + S N · n 1 + S N · n 2 + T 1 n 1 ·pn 2 ·p + T 2 n 1⊥ · n 2⊥ + T 3 n 1⊥ ·kn 2⊥ ·k where N is the normal to the scattering plane, The seven structure functions R, S, and T of Eq.(III.11) depend on the scattering angle θ, the ratio function η(s), and the phase function ∆Φ(s). Their detailed expressions are given in Appendix B.
The cross-section distribution for polarized final-state hyperons becomes where α e is the fine-structure constant.
If we sum over baryon and antibaryon final-state polarizations we get a well-known result, (III.14) Summing only over the antibaryon polarizations gives 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.(III.11) 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 + e − annihilation were first discussed by Dubnickova et al. [13], but with results slightly different from ours. 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. [14] 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 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.(III.5). The R-factor is a spin independent factor, defined by as the fractional decay rate, since it is a decay rate per unit phase space. Further inspection of Eq.(IV.19) tells us, that Γ(c → dπ) is defined as an average over the spins of both initial and final baryons.
The spin-density-distribution function, G(n c , n d ) of Eq.(IV.19), 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 = 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.(IV.18), The relation between the parameters A, B and A , B is clarified in Refs. [17,18].
and F 2 (k 2 ) contributions are each, by themselves, gauge invariant. We shall ignore the F 1 term, which vanishes for real photons, k 2 = 0, and stay with the F 2 term. We denote by µ cd , the strength of the magnetic-moment 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 (k) 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 (k)e ν (k) by −g µν .
This leads to 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.(V.35) 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 = −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 anti-parallel to the photon momentum, then the decay probability vanishes, a property of angular-momentum conservation. We also notice that expression (V.38) cannot be written in the αβγ representation of Eqs.(IV.21) and (IV.22).
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 anti-particle matrix element corresponding to the particle matrix element of We assume the parameter µ is the same for particle transitions c → d as for anti-particle transitionsc →d.
The normalized spin density corresponding to the antiparticle matrix element of Eq.(V.41) is the same as that corresponding to the particle matrix element of Eq.(V.32), as given in Eq.(V.38), 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 suggested in Ref. [8]. Such contributions are created by making the substitution in the decay amplitude. Moreover, if one can measure hyperon and anti-hyperon sequential decays simultaneously tests for CP violation become possible.
The substitution (V.42) changes the normalized spin density (V.38) into with asymmetry parameter Similarly, the decay width of Eq.(V.37) is changed into Parity violating admixtures in the anti-particle decayΣ 0 →Λγ can also be simulated by the substitution of Eq.(V.42). For generality we replace b byb, and simultaneously remark that hemiticity requiresb = −b . The spin density for the anti-particle decay becomes The P-violating interference term now enters with the opposite sign. If CP is conserved then 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 + e − is a small fraction of the electromagnetic decay Σ 0 → Λγ [15,16]. The lepton pair of the leptonic decay is interpreted as the decay product of a virtual, massive photon. This pair is often called a Dalitz lepton pair.
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.(V.33), We need the square of M e averaged over baryon spins but summed over lepton spins.
The summation over lepton spins leads to a lepton tensor, with k 2 = m 2 γ and dLips(k; k 1 , k 2 ) the phase-space element for the lepton pair, as in Appendix C. However, care should be excercised since in many experiments the efficiency is very sensitive to the lepton momentum.
The integration over the lepton phase space affects only the lepton tensor. Thus, we note that and similarly for k 2µ k 2ν , with brackets denoting integration over lepton phase space, dLips(k; k 1 , k 2 ), and 1 denoting the phase-space volume itself. The term proportional to k µ k ν in Eq.(VI.52) vanishes due to gauge invariance. As a consequence, we get as average of the lepton tensor, (VI.54) The lepton tensor L µν of Eq.(VI.53) comes with a factor (−g µν ). Contracting it with the hadron tensor H µν (c → dg), with g representing the virtual photon, is equivalent to summing over photon polarizations. We write The factorization is chosen so that R(c → dg) is spin independent, and so that the spinindependent term of G(n c , n d ) 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.(VI.55),

VII. FOLDING
Our general aim is to calculate the cross-section distributions for e + 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.
with L Λ (n p , l p ) defined in Eq.(IV.22); and the anti-hyperon 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.(VII.62), gives us 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 anti-baryon decay chain Σ 0 →Λ →p, and a corresponding transition-spin density.
To go from the baryon to the anti-baryon case, we simply replace the baryon variables by their anti-baryon counterparts, n Σ → nΣ, α Λ → αΛ, etc.
The inclusion of parity violation in the Σ 0 → Λγ decay is straightforward. To this end of Eq.(V.43), and get Therefore, as a consequence of parity violation the normalization of the cross-section distribution acquires a small angular dependent term.

VIII. SINGLE CHAIN DECAYS
Single-chain decays of Σ 0 hyperons can be studied in the e + e − annihilation into Σ 0Σ0 pairs, provided theΣ 0 is somehow identified, e.g., as a missing hyperon Ref. [6]. The spindensity state of the Σ 0 will then be obtained from Eq.(III.11) as, A Σ 0 hyperon in a state of polarization P Σ , subject to the condition |P Σ | ≤ 1, is characterized by a normalized spin-density function, Therefore, by Eq.(VIII.68) the hyperon polarization is in the present case equal to P Σ = S N/R.
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.(VIII.69) with the Σ 0 decay distribution of Eq.(V.38), to get with l Λ = −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 (n Λ , n p ) of Eq.(IV.21). Since the spin of the final-state proton is usually not measured, we form the average over the proton spin directions. The spindensity-distribution function of Eq.(VIII.70) is now expanded to 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.(VII.67) we investigate the spin density (VIII.73) Invoking the vector-function identity of Eq.(IV.22) we get Finally, the spin-density-distribution function for the final state proton is obtained as S(n p ) = S(n Σ )S(n Σ , n p ) 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 + e − → Σ 0Σ0 , the spin-density distribution of which was calculated in Sect.3. We name it S(n Σ , nΣ). The explicit expression is given by Eq.(III.11), 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 presciption (VII.62). Invoking Eq.(III.11) for the production step and Eqs.(IX.78) and its anti-distribution for the decay steps, we get the angular distribution W ΣΣ (l a ) = S(n Σ , nΣ)W Σ (n Σ , n p )WΣ(nΣ, np) where l a denotes the ensemble of l values in the decays.
The angular distributions of Eq.(IX.81) still depend on the spin vectors n p and np which are difficult to measure. If we are willing to consider the spin averages, then variables U and V simplify, Since U Σ = UΣ = 1 the effect of the folding is to make, in the spin-density function S(n Σ , nΣ) of Eq.(III.11), 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 (l γ , −l p ) of Eq.(IX.80), and which is absent in Eq.(IX.81).
Inserting the expressions of Eq.(IX.82) into the spin-density function of Eq.(IX.81) we 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 B.
With their help we can rewrite the differential distribution function of Eq.(IX.83) as where the argument ξ of the angular functions is a nine-dimensional vector ξ = (θ, Ω Λ , Ω p , ΩΛ, Ωp).
The ten angular functions F k (ξ) are defined as; An important conclusion to be drawn from the differential distribution of Eq.(X.84) is that when the phase ∆Φ 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 non-zero value of ∆Φ is required.
The sequential differential decay distribution of a single-tagged Σ 0 produced in e + e − annihilation can be obtained form Eq.(X.84) by suitably integrating over the angular vaiables ΩΛ and Ωp. As a result we get the differential distribution for Σ 0 production and decay, 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 E. From the angular distribution of Eq.(X.86) we can determine the product α Λ sin(∆Φ), and from the correspondingΣ 0 distribution the product αΛ sin(∆Φ).

XI. CROSS-SECTION DISTRIBUTIONS
We shall now consider the phase-space imbedding of the differential-distribution function 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 + 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 phase-space 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.(V.36) as where Φ is the two-body phase-space volume, and Γ(c → dg) 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 contribution must be included. This is achieved by multiplying R(c → dγ) by a factor of two. This factor can be incorporated in the channel width Γ(c → dg), 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 the indices is accompanied by an extra factor of four, and it is not normalized to unity either but to R.
The folding formula Eq.(XI.92) combined with Eqs.(XI.87) and (III.14) gives the master equation This readily understood structure agrees with that found in Ref. [11].
Since spin densities are normalized, except for the annihilation density S(n b , nb), the overall normalization condition reads This normalization is checked explicitely in single-chain-sequential decay in Ref. [6].
When the Σ 0 → Λγ reaction is involved, and the photon is real, then the channel width in Eq.(XI.93) is for all practical purposes equal to the total width, Γ(Σ 0 → Λγ) = Γ(Σ 0 → all).
We also point out that for virtual photons the cross-section distribution of Eq.(XI.93) receives an additional lepton factor, where m γ is the virtual photon mass, and R the Dalitz function (XI.96) The e + 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.(XI.87), we get the cross-section-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 is equivalent to replacing in the master formula of Eq.(XI.93), the photon-induced e + e − → Σ 0Σ0 annihilation cross section by the J/ψ induced cross section.
with 2m e ≤ m γ ≤ (m c − m d ). In the limit m γ = 0 we recover R(c → dγ) for real photons, Eq.(V.36). The exact expression for the normalized spin density is, Here, we can without qualm put B = −1 and C = 0. In this limit we recover the normalized spin density for real photons, Eq.(V.38).

Appendix E: Angular functions
The cross-section distribution (IX.83) 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 anti-hyperon 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 + e − → Σ 0Σ0 is spanned by the unit vectorsp = l Σ andk = l e , as measured in the c.m. system. The scattering plane makes up the xzplane, with the y-axis along the normal to the scattering plane. We choose a right-handed whereas the directional angles of the proton in the Lambda hyperon rest system are l p = (cos φ p sin θ p , sin φ p sin θ p , cos θ p ). (E.4) And so for the anti-hyperons.