Novel aspects of baryon-antibaryon production in e+e− an- nihilations

Baryon-antibaryon decays of J/ψ and ψ′ produced at an electron-positron collider are presented as a tool to study baryon properties and to test discrete symmetries. We focus on a novel aspect of this process: the transverse polarization of the produced baryons. The polarization was observed for first time by the BESIII Collaboration in e+e− → J/ψ → ΛΛ̄. The observed spin polarization and correlations were used for a direct determination the Λ and Λ̄ decay asymmetries. The result for Λ→ pπ−, the well known Λ polarimeter process, differs by 17(3)% from the established value of 0.642(13). Finally the prospects of the method for studies of weak decays of hyperons and for CP tests in baryon sector are discussed.


Introduction
The well-defined and simple initial state makes baryon-antibaryon pair production at an electron-positron collider an ideal system to test fundamental symmetries in the baryon sector, in particular when the probability of the process is enhanced by a resonance such as the J/ψ [1]. The spin orientations of the baryon and antibaryon are correlated and, for spin onehalf baryons, the pair is produced either with the same or opposite helicities. The transition amplitudes to the respective spin states can acquire a relative phase, ∆Φ due to the strong interaction in the final state, leading to a time-reversal-odd observable: a transverse spin polarization of the baryons [2,3]. This effect had been neglected in analyses of the baryon pairs from the J/ψ and ψ decays [4,5]. We have shown [6] that the same formalism as for the continuum baryon-antibaryon pair production in electron-positron annihilation related to the time-like baryon form factors [7][8][9][10][11] should be used to describe the decays. We have also derived explicit expressions for the joint angular distributions, suitable for the maximum loglikelihood fits, in e + e − → ΛΛ process where the Λ andΛ are reconstructed using their main two body hadronic weak decays. The formalism was applied in the BESIII analysis leading to the first observation of Λ transverse polarization in J/ψ → ΛΛ decay [12].
In the next sections we summarize the formalism, report the results of BESIII proof of concept analysis and discuss some extensions and prospects of the method. spin density matrix: where the set of 2 × 2 base spin matrices σ B µ in the rest frame of a baryon B is defined as: where 1 2 is 2 × 2 identity matrix and σ x , σ y , σ z are Pauli spin matrices. Here, as in Ref. [6], we use the same orientations of the spin quantization axes for B 1 andB 2 with the z direction along B 1 momentum in the overall center-of-mass (c.m.) system of the B 1B2 pair. The y direction is given by the vector product of the incoming electron and the outgoing baryon B 1 momenta. The coefficients C µν depend on the angle θ between the electron and baryon B 1 . At c.m. energies when electron mass could be neglected, a single photon annihilation process could only proceed if the electrons have opposite helicities. The final state baryons can have both ±1/2 helicities. Due to parity conservation out of the four possible helicity transitions only two are independent: A 1/2,1/2 = A −1/2,−1/2 =: h 1 and A 1/2,−1/2 = A −1/2,1/2 =: h 2 [13]. Therefore e + e − → B 1B2 process at fixed c.m. energy is described by two complex form factors. If one is only interested in the not normalized angular distributions only two real parameters are needed which could be defined as: The C µν 4 × 4 matrix is given as [6]: where β ψ = 1 − α 2 ψ sin(∆Φ) and γ ψ = 1 − α 2 ψ cos(∆Φ) implying α 2 ψ + β 2 ψ + γ 2 ψ = 1. The polarization vectors P of B 1 and B 2 has to be in y direction and the value is given by P y = C 02 /C 00 = β ψ sin θ cos θ/(1 + α ψ cos 2 θ).
If a baryon decays weakly, such as the ground state hyperons or charmed baryons, the polarization can be determined using angular distribution of the daughter particles. For example, for the Λ → pπ − decay with the Λ hyperon polarization given by the P vector, the angular distribution of the daughter protons is 1 4π (1 + α − P ·n), wheren is the unit vector along the proton momentum in the Λ rest frame and α − is the asymmetry parameter of the decay [14]. The corresponding parameters α + forΛ →pπ + , α 0 for Λ → nπ 0 , andᾱ 0 for Λ →nπ 0 are defined in the same way [15]. In a general weak hadronic decay of a spin one-half baryon into a spin one-half baryon and pseudoscalar meson: B A → B B + P, both the initial an the final states are represented by a linear combinations of the spin one-half density matrices σ B A µ and σ B B ν Eq. (2), respectively. One can therefore represent the weak decay by a decay matrix, a B A →B B +P µ,ν , which transforms the base matrices [13]: In general there are two parameters to describe the B A → B B + P decay, in addition to the decay asymmetry, If the polarization of the baryon B B is not measured the decay is described only by the a B A →B B +P µ,0 elements of the decay matrix and only the α B A →B B +P parameter is involved. The joint angular distribution of J/ψ → ΛΛ (Λ → f andΛ →f , f = pπ − ) can be therefore written as the trace of the final proton-antiproton density matrix: C µν (cos θ; α ψ , ∆Φ)a Λ→pπ − µ,0 (n 1 ; α − )aΛ →pπ + ν,0 (n 2 ; α + ). (6) where ξ := (cos θ,n 1 ,n 2 ) is the complete set of kinematical variables describing the event configuration in the five dimensional phase space andn 1 (n 2 ) is the unit vector in the direction of the nucleon (antinucleon) in the rest frame of Λ (Λ). It can be written explicitly as [6]: W(ξ; α ψ , ∆Φ, α − , α + ) = C 00 + C 02 · (α − n 1,y + α + n 2,y ) + α − α + C 11 n 1,x n 2,x − C 22 n 1,y n 2,y + C 33 n 1,z n 2,z +C 13 (n 1,x n 2,z + n 1,z n 2,x ) .
The terms multiplied by α − α + represent the contribution from the ΛΛ spin correlations, while the terms multiplied by α − and α + separately represent the contribution from the polarizations. If all three contributions in Eq. (7) are non-zero an unambiguous determination of the parameters α ψ and ∆Φ and the decay asymmetries α − , α + is possible. One should stress that the inclusive measurement of the proton momenta from the Λ → pπ − decay is not sufficient to determine uniquely α − . Integrating Eq. (7) over the unmeasuredp directionn 2 one gets: where the polarization term is written explicitly in terms of ∆Φ. It is clear that only the product α − · sin(∆Φ) can be determined from such inclusive analysis.
Such representation for the angular distribution in Eq. (9) requires M = 72 unique functions g k (π) of the global parameters while Eq. (6) only M = 7. If ∆Φ = 0 the number of such terms reduces to M = 56 in Eq. (9), therefore likely all the decay parameters can be determined even if ∆Φ = 0 (depending whether other parameters are zero). In our Ref. [13] a formalism for spin-3/2 baryons, which can be used to describe e.g. reaction e + e − → ψ → Ω −Ω+ with the subsequent decays, is also given.