Semileptonic decays of spin-entangled baryon-antibaryon pairs

A modular representation for the semileptonic decays of baryons originating from spin polarized and correlated baryon-antibaryon pairs is derived. The complete spin information of the decaying baryon is propagated to the daughter baryon via a real-valued matrix. It allows to obtain joint differential distributions in sequential processes involving the semileptonic decay in a straightforward way. The formalism is suitable for extraction of the semileptonic formfactors in experiments where strange-baryon-antibaryon pairs are produced in electron-positron annihilation or in charmonia decays. We give examples such as the complete angular distributions in the $e^+e^-\to \Lambda\bar\Lambda$ process, where $\Lambda\to pe^-\bar{\nu}_e$ and $\bar\Lambda\to\bar{p}\pi^+$. The formalism can also be used to describe the distributions in semileptonic decays of charm and bottom baryons. Using the same principles, the modules to describe electromagnetic and neutral current weak baryon decay processes involving a charged lepton-antilepton pair can be obtained. As an example, we provide the decay matrix for the Dalitz transition between two spin-1/2 baryons.


I. INTRODUCTION
Baryon semileptonic (SL) decays are an important tool to study transitions between ground state baryons. Comparing to the nonleptonic baryon decays where at least three hadronic currents are involved, the SL transition involves only a two-point hadronic vertex and the external W -boson field coupled to the leptonic current. The properties of the hadronic vertices are described by a set of scalar functions, formfactors, that depend on the invariant mass squared of the emitted virtual W -boson. In particular, the semileptonic processes allow to probe the kinematic regions of the formfactors that are dominated by the static properties of the baryons. The recent progress in the lattice quantum chromodynamics gives a hope to determine the properties of the formfactors from the first principles with the accuracy sufficient for a comparison with precise experimental data [1]. Once the hadronic effects are well understood, the SL decays will provide a complementary method to determine Cabbibo-Kobayashi-Maskawa matrix elements [2] and to search for beyond the Standard Model effects such as violation of lepton flavour and charge-conjugation-parity symmetries [3]. In this article, we provide a modular description of the semileptonic de- * varvara.batozskaya@ncbj.gov.pl cays that can be used to extract properties of the formfactors in the experiments using spin entangled baryon-antibaryon pairs.
The helicity amplitude method [4][5][6] that is commonly used in the analyses of semileptonic decays allows to express the angular distributions in an efficient and compact way. The complete process is described as a sequence of two-body decays, where each of them is analysed in the rest frame of the subsequent decaying particle. For a semileptonic decay ℓ , the first decay step B 1 → B 2 W − off-shell is analysed in the B 1 rest frame, whereas W − off-shell → ℓ −ν ℓ is analysed in the W − off-shell rest frame. The resulting expressions for the differential distributions are compact and can be written in a quasi-factorized form.
The formalism also describes joint angular distributions in the semileptonic decays of a spin polarized baryons.
A novel approach to study strange baryon decays is to use hyperon-antihyperon pairs from J/ψ resonances produced in electron-positron annihilations [7]. The complete angular distribution in such processes can be conveniently represented using a product of real-valued matrices that describe the initial spin-entangled baryon-antibaryon state and chains of twobody weak decays. These matrices can be rearranged to describe many decay scenarios in the e + e − → ΛΛ, e + e − → ΞΞ and similar processes [7][8][9][10]. Several high-profile analyses using multidimensional maximum likelihood fits to angular distributions were performed by the electron-positron collider experiment BESIII [11,12] using this modular formalism. These multidimensional analyses have demonstrated increased precision of the decay parameters measurements and enabled to observe effects that were averaged out in previous studies, such as a polarization of the hyperon-antihyperon pair from charmonia decays.
The tagging processes involve only charged particles in the final state, therefore their momenta can be precisely determined. This allows one to reconstruct the momentum of the antineutrino in the semileptonic process and to determine the fourmomentum squared of the lepton pair that is needed to study the dynamics of the process. The polarization of the hyperons is given by the angular distributions in their decays, but usually the polarization of the leptons is not measured. Such double-tag (DT) technique is often used to determine absolute branching fractions in electron-positron collider experiments [14]. With large number of collected events in experiments such as BESIII [15] studies of decay distributions in the semileptonic hyperon decays are possible. A formalism that uses spin correlations and polarization of the produced baryon-antibaryon system is needed to determine the decay parameters with the best precision. The purpose of this report is to extend the approach from Refs. [8,10] to include decay matrices representing the three-body semileptonic processes. Our starting point is the helicity formalism for semileptonic decays from Ref. [6].
We construct a real-valued decay matrix relating the initial and final baryons' spin states, represented by the Pauli matrices. The obtained decay matrix is used to construct the full joint decay distributions of the spin-entangled baryon-antibaryon pair in a modular way.
The paper is organized as follows: in Sec. II and Sec. IV we review the formalism to describe baryon-antibaryon production process and semileptonic decays, respectively. In Sec. V the main result is derived -the spin-density matrix of the daughter baryon in the semileptonic decay. Sec. VI presents modular formulas to describe the angular distributions of the semileptonic hyperon decays. Finally in Sec. VII we collect some numerical results.

II. PRODUCTION PROCESS
In general a state of two spin-1/2 particles e.g. a baryon-antibaryon pair B 1B1 can be written as [8] where a set of four Pauli matrices σ B 1 µ (σB 1 ν ) acting in the rest frame of a baryon B 1 (B 1 ) is used and C µν is a 4×4 real matrix representing polarizations and spin correlations of the baryons. Here we consider mainly baryon-antibaryon systems created in the e + e − → B 1B1 process. However, the formalism can be applied for the pairs from decays of (pseudo)scalar or tensor particles such as ψ(2S), η c , χ c0 , χ c2 → B 1B1 or in a fact to any pair of spin-1/2 particles (for example baryon-baryon, muon-antimuon and others). The spin matrices σ B 1 µ and σB 1 ν are given in the coordinate systems with the axes denotedx 1 ,ŷ 1 ,ẑ 1 andx 3 ,ŷ 3 ,ẑ 3 as shown in Fig. 1. The directions of the two right-handed coordinate systems are related as state. The overall c.m. frame withẑ axis (e.g. for e + e − → B 1B1 it is defined along the positron momentum). The axes in baryon B 1 and antibaryonB 1 rest frames (helicity frames) are denoted (x 1 ,ŷ 1 ,ẑ 1 ) and (x 3 ,ŷ 3 ,ẑ 3 ), respectively. depends in the lowest order on two parameters, α ψ ∈ [−1, 1] and ∆Φ ∈ [−π, π). The elements of the C µν matrix are functions of the baryon B 1 production angle θ 1 in the electronpositron center-of-momentum (c.m.) system. The matrix for the single photon annihilation of unpolarized electrons and positrons is [8]: where the parameters β ψ and γ ψ are expressed via α ψ and ∆Φ as β ψ = 1 − α 2 ψ sin(∆Φ) and γ ψ = 1 − α 2 ψ cos(∆Φ). We will also use a more general formula from Ref. [10] that describes the annihilation processes with polarized electron beams.

III. INVARIANT FORMFACTORS
Let us consider a semileptonic decay of a 1/2 + hyperon B 1 into a 1/2 + baryon B 2 and an off-shell W − -boson decaying to the lepton pair l −ν l with the momenta and masses denoted as 0). The matrix elements due to the vector J V µ and axial-vector J A µ currents in notation from Ref. [6] are: where q µ := (p 1 − p 2 ) µ = (p l + p ν ) µ is the fourmomentum transfer. The fourmomentum squared q 2 ranges from m 2 l to (M 1 − M 2 ) 2 . The formfactors F V,A 1,2,3 (q 2 ) are complex functions of q 2 that describe hadronic effects in the transition. Neglecting possible CP-odd weak phases, the corresponding formfactors are the same for the (l − ,ν l ) and (l + , ν l ) transitions.
To fully determine the hadronic part of a semileptonic decay, the six involved formfactors should be extracted as a function of q 2 . The formfactors are usually parameterized by the axial-vector to vector g av coupling, the weak-magnetism g w coupling and the pseudoscalar g av3 coupling. They are obtained by normalizing to F V 1 (0): For experiments with a limited number of events, the q 2 -dependence of the formfactors is assumed using a model. The standard approach is to include one or more poles of the mesons that have the correct quantum numbers to mix with the W boson and have the masses close to the q 2 range in the decay. Traditionally one pole is explicitly included together with an effective contribution from other poles [16] such as in the Becirevic-Kaidalov (BK) [17] parameterization: where the dominant pole mass M is outside the kinematic region and the parameter α BK represent an effective contribution from the meson poles with higher messes. Here the case α BK = 0 represents the dominant pole contribution. This parameterization gives real-valued formfactors. If more data is available, one or more extra parameters can be added to describe the q 2 distribution. In the hyperon decays the range of q 2 ≤ (M 1 − M 2 ) 2 is limited and in the first order can completely neglect the q 2 dependence using the values of the couplings at the q 2 = 0 point. A better approximation is to include an effective-range parameter r i that represent linear dependence on q 2 : For example, using (5) the effective-range parameter is r i = (1 + α BK )/M 2 . The main takeaway message from the above discussion is that, for practical purposes, the q 2 dependence of an SL formfactor can be represented by one or two parameters. In experiments, these parameters can be determined from the observed distributions. The optimal method for such parametric estimation is the maximum likelihood method using multidimensional unbinned data. We will first construct modular formulas for the angular distributions and then in Sec. VII discuss the attainable statistical uncertainties for the SL formfactors parameters as the function of the number of observed events.
The energy q 0 of the off-shell W − boson and the magnitude of the three-momentum p are the following functions of the q 2 invariant and where The spin direction and subsequent decays of the baryon B 2 and boson W − off-shell are described in two helicity systems denoted R 2 and R W , respectively. The helicity frame R 2 is obtained by performing three active rotations: (a) around theẑ 1 -axis by −ϕ 2 ; (b) a rotation around The axes in the B 1 , B 2 and W − off-shell rest frames (helicity frames: R 1 , R 2 and R W ) are denoted (x 1 ,ŷ 1 ,ẑ 1 ), (x 2 ,ŷ 2 ,ẑ 2 ) and (x W ,ŷ W ,ẑ W ), respectively. the newŷ-axis by −θ 2 ; (c) a rotation around theẑ 2 -axis by +χ 2 , see Fig. 2 [18]. The first two rotations are sufficient to align p 2 with the z-axis and such two-rotations prescription is used e.g. in Ref. [8]. Here we allow for an additional rotation that can be e.g. used to bring the momenta p 2 , p l and p ν to one plane. Initially, we consider the angle χ 2 of this rotation as an arbitrary parameter. The combined (a)-(c) three-dimensional rotation is given by the product of three axial rotations R(χ 2 , −θ 2 , −ϕ 2 ) = R z (χ 2 )R y (−θ 2 )R z (−ϕ 2 ). Subsequently, one then boosts to the B 2 rest frame. The R W frame is defined using the same procedure with the rotation matrix R(χ W , −θ W , −ϕ W ) and the subsequent boost to the W − off-shell rest frame. Since the W − off-shell direction is opposite to B 2 in R 1 , one has ϕ W = π + ϕ 2 and θ W = π − θ 2 . In order to assure that the coordinate systems in R 2 and R W are related as The matching transition amplitude between B 1 and the two daughter particles expressed using the defined above helicity frames is [8,18]: where D J m 1 ,m 2 (Ω 2 ) := D J m 1 ,m 2 (ϕ 2 , θ 2 , −χ 2 ) is the Wigner rotation matrix, where the conven- . The order and the signs of the angles Ω 2 = {ϕ 2 , θ 2 , −χ 2 } in the Wigner functions are opposite to the used in the rotations to define the helicity reference frames. In addition, the normalization factor is different since we allow for three independent rotation angles. The helicity amplitudes H λ 2 ,λ W (q 2 ) are functions of q 2 and depend on the helicities of the daughter particles. The vector and axial-vector helicity amplitudes H λ 2 ,λ W = H V λ 2 ,λ W + H A λ 2 ,λ W are related to the invariant formfactors in the following way: where the remaining helicity amplitudes are obtained by applying the parity operator: The decay W − → l −ν l is described in R W where the emission angles of the l − lepton are θ l and ϕ l . The value of the lepton momentum in this frame is The decay amplitude reads where Ω l = {ϕ l , θ l , 0}. The helicity amplitudes h l λ l λν for the elementary transition to the final lepton pair can be calculated directly by evaluating the Feynman diagrams. The neutrino helicities are λ ν = 1/2 and λ ν = −1/2 for (l − ,ν l ) and (l + , ν l ), respectively. The moduli squared of h l λ l λν are [6]: where here and in the following the upper and lower signs refer to the configurations (l − ,ν l ) and (l + , ν l ), respectively.
The representations in Eqs. (10) and (14) imply that the complete amplitude for the where the λ W sum runs over the four W -boson helicity components {t, −1, 0, +1}. An explicit representation of the amplitude with the angular part separated is where the final expression combines all azimuthal-angle rotations in one term. One can consider two options for selecting χ 2 to define the transversal orientation of the R 2 and R W helicity frames. The first option is to set χ 2 = 0 as in Ref. [8] where the corresponding azimuthal angle of the charged lepton in the R W system is ϕ 0 l . An alternative is to select χ 3-b 2 so thatx 2 is in the decay plane of the semileptonic decay. In this case the momenta of the leptons are in this plane which corresponds to ϕ 3-b l = 0 and the χ 3-b 2 = ϕ 0 l relation holds. The amplitude can be rearranged by inserting a complete spin basis for the baryon B 2 to represent transition between B 1 (κ) and B 2 (λ 2 ): Therefore the angular dependence on Ω 2 can be separated in the amplitude of the complete process. Since usually experiments do not measure polarization of the leptons, it is useful to consider a tensor that describes the W ± -boson decay with the lepton helicities summed over: The interference contribution from λ W = t and λ W = 0 gives an extra minus sign. We write the tensor as: where ε = m 2 l /(2q 2 ). The hermitian matrix for the nonflip transition reads while for the flip transition

V. DECAY MATRIX
Here, we derive a matrix that relates the spin of the baryon B 2 to the spin of the baryon where the state of the lepton pair with the summed spin projections is given by the L λ W ,λ ′ W (q 2 , Ω l ) tensor in Eq. (23). The transition can be represented by a tensor T κκ ′ ,λ 2 λ ′ 2 that describes how the initial spin-density matrix ρ κκ ′ 1 of the baryon B 1 transforms to the density matrix ρ Using Eq. (10) the transition tensor is given as The explicit expression for the phases of the hadronic tensor due to the azimuthal rotations where we use the generic case with Ω 2 = {ϕ 2 , θ 2 , χ 2 } and Ω l = {ϕ l , θ l , 0}. The overall phases of the contraction of the above hadronic tensor and the leptonic tensor in Eq. (24) for the two choices of the orientations of the coordinate systems R 2 and R W are: The two representations are not equivalent but can be written in terms of the tensors evaluated for Ω 0 2 := {ϕ 2 , θ 2 , 0} and Ω 0 l = {0, θ l , 0} as Instead of the helicities, the transition can be written as in Ref. [8] using spin base vectors σ B 1 µ and σ B 2 ν in the mother and daughter reference systems R 1 and R 2 , respectively. The 4 × 4 matrix B µν describes how the decay process transforms the base Pauli matrices: The real coefficients B µν can be obtained by inserting Pauli σ µ matrices for the mother and the daughter baryons in the expression for the tensor T κκ ′ ,λ 2 λ ′ 2 : However, as we show in Appendix B the coefficients can be represented as where R (4) µκ (Ω 2 ) is the 4 × 4 space-like rotation matrix obtained as the direct sum of identity and 3D rotation R(Ω 2 ): where the orientations of the axes of the reference systems are aligned Ω 2 = {0, 0, 0}. They can be obtained by inserting Pauli σ µ matrices for the mother and the daughter baryons in the expression for the tensor The last form involves only real valued tensors T The hadronic part is encoded in the real-valued functions of q 2 : Moreover, the formfactors H − 1 2 1 = H1 2 −1 = 0 reducing number of the functions.
We will represent the b µν matrix as the sum of the non-flip and flip contributions b µν = b nf µν + εb f µν . The cross-section term is written as define the angular distributions for the decay of unpolarized baryon B 1 when the spins of all final particles are summed over. The differential decay rate is obtained by multiplying by the kinematic and spinor normalization factors that depend on q 2 where V P h (q 2 ) = (2π) −5 (4M 1 ) −2 |p l ||p 2 | is the three-body phase space density factor [13].
The momenta |p 2 | and |p l | of the baryon B 2 and the lepton are given in Eqs. (8) and (13) where I µν are complex. We use notation I µν = I nf µν + εI f µν and The first column b i0 of the matrix corresponds to the decay of the spin polarized baryon B 1 . where The decay plane representation which requires three rotation angles for baryon B 2 gives simple formulas for the remaining terms of the decay matrix. The terms of the non-flip contributions for the aligned (with ϕ l = 0) decay matrix b nf µν are: and The terms of the flip contributions for the aligned decay matrix b f µν are: and The terms of the b µν matrix in general form for an arbitrary ϕ l value are given in Appendix C.
They should be used if two rotation angle representation as in Ref. [8] was applied.

VI. JOINT ANGULAR DISTRIBUTIONS
Here we provide examples how to construct modular expressions for the angular distributions of semileptonic decays of baryons. First, using our formalism, we rewrite the results from Ref. [6] for the single baryon B 2 decay. The simplest case is the decay of a spin polarized baryon B 1 → B 2 l −ν l . If the polarization of the final particles is not measured the fully differential angular distribution dΓ ∝ W = V P h (q 2 )(q 2 − m 2 l )Trρ B 2 , where with the baryon B 1 spin state in its rest frame described by the polarization vector C µ0 = (1, P x , P y , P z ). The elements of the decay matrix b B 1 B 2 µ0 (q 2 , Ω l ) := b µ0 (q 2 , Ω l ; ω B 1 B 2 ) are given in Eq. (49). For example, if the initial polarization has only P z component the joint angular distribution for the decay process B 1 → B 2 l −ν l is: where the vector ξ := (θ 2 , ϕ 2 , q 2 , Ω l ) represents a complete set of the kinematic variables describing an event configuration and the parameter vector ω B 1 B 2 represents the polarization P z , the semileptonic couplings in Eq. (4) and the range parameters in Eq. (6). If the baryon The decay matrix a ν0 (θ 4 , ϕ 4 ; α B 2 ) [8] describes the non-leptonic decay B 2 → B 4 π and using the representation from Appendix D is given as:  where θ 4 and ϕ 4 are the helicity angles of B 4 in the R 2 frame and α B 2 is the decay asymmetry parameter. The corresponding angular distribution for charge-conjugated decay mode is obtained by the replacements H B 1 λ 2 λ W → HB 1 λ 2 λ W , g B 1 av/w → gB 1 av/w and swapping between (l − ,ν l ) and (l + , ν l ). Neglecting hadronic CP-violating effects, one has H [19,20]. Now we consider a decay of a spin-entangled baryon-antibaryon system B 1B1 , where the initial state is given by the spin correlation matrix C B 1B1 µν defined in Eq. (1) with The semileptonic decay is tagged by a common decay of the antibaryon B 1 . For hyperon decay studies, a non-leptonic decayB 1 →B 3π is used. One obvious advantage of the studies using baryon-antibaryon pairs is that the charge-conjugated decays, corresponding to theB 1 →B 2 l + ν l and B 1 → B 3 π scenario, can be studied simultaneously.
A common practice is to implicitly combine events corresponding to the charge-conjugated channels in the analyses to determine the decay properties in the CP-symmetry limit. In such analyses, the quantities that are even (odd) with respect to the parity operation have the same (opposite sign) values when combining the two cases. At the same time, the CPsymmetry can be tested by comparing values of the separately determined parameters for the baryon and antibaryon decays. Using as a building block the semileptonic decay matrix one constructs the angular distribution for the case when polarization of baryons B 2 andB 3 is not measured: The matrix B B 1 B 2 µ0 := B µ0 (θ 2 , ϕ 2 , q 2 , Ω l ; ω B 1 B 2 ) describes the semileptonic decay and aB 1B3 ν0 := aν 0 (θ 3 , ϕ 3 ;ᾱ B 1 ) [8] describes the non-leptonic decayB 1 →B 3 π, where θ 3 and ϕ 3 are the helicity angles ofB 3 in theB 1 rest frame andᾱ B 1 is the decay asymmetry parameter. The joint angular distribution for the process is W(ξ; ω) = V P h (q 2 )(q 2 − m 2 l )Trρ B 2B3 , where: with C B 1B1 µν given in Eq. (2) for the annihilation of the unpolarized electron-positron beams. The vectors of the kinematic variables are ξ = (θ 1 , θ 2 , ϕ 2 , q 2 , Ω l , θ 3 , ϕ 3 ) while ξ ′ = (θ 2 , ϕ 2 , q 2 , Ω l ). The full vector of parameters is denoted as ω := (α ψ , ∆Φ, g B 1 av , g B 1 w ,ᾱ B 1 ).

VII. SENSITIVITIES FOR SL FORMFACTORS PARAMETERS
Here we present estimates for the statistical uncertainties of the parameters describing formfactors of selected semileptonic hyperon decays. The derived angular distributions are used to construct the normalized multidimensional probability density function for an event configuration. They are functions of q 2 and the helicity angles, and depend on the formfactor parameters such as g av and g w (4). The parameters can be determined in an experiment using maximum likelihood (ML) method, which guarantees consistency and efficiency properties.
We provide uncertainties of the parameters in the large number of events limit and assuming the detection efficiency does not depend on the kinematic variables as described in Refs. [9,10]. Since the ML estimators are asymptotically normal, the product of their standard  (5) 2.0(9) 125 [13] a Since for Σ + F 1 = 0, the coupling constants g av and g w are defined as deviations, σ, and √ N , where N is the number of the observed events, does not depend on N . The uncertainties are obtained by calculating elements of the Fisher information matrix that is inverted to obtain the covariance matrix for the parameters.
We consider the semileptonic decays of hyperons listed in Table I. We neglect formfactors F V 3 and F A 2 which vainsh in the limit of of the SU(3) flavor symmetry [22]. Equation (11) allows one to estimate the relative contribution of different formfactors to the angular distributions. Based on the g av and g w values from Table I the q 2 dependence of the six helicity amplitudes for the Λ semileptonic decays is shown in Fig. 3(a). To allow a better comparison the amplitudes are multiplied by q 2 . Close to the lower boundary, q 2 = m 2 e , the longitudinal and scalar helicity amplitudes dominate, with H Close to the upper boundary at the zero recoil point, We do not consider the decay Σ − → ne −ν e since the final state includes two neutral particles, neutron and neutrino, making it impossible to fully reconstruct the events. In addition, no measurements exist for the production parameters in the e + e − → Σ −Σ+ process.
The first case is the decay Λ → pe −ν e studied in the exclusive process e + e − → J/ψ → ΛΛ, whereΛ →pπ + is used for tagging. The angular distribution is given by Eq. (62) where the parameters of the production process e + e − → J/ψ → ΛΛ needed to define the spincorrelation-polarization matrix C µν are given in Table II [13,29] and the charge conjugated process that is used to tag the SL decay are given in Table III. We assume the production parameters and the decay parameters of the non-leptonic decays used for the tagging to be well known and fixed. Since the coupling g av3 is multiplied by m e in the transition amplitude [23], we set it to zero because it cannot be determined from experiment with a reasonable uncertainty. In addition the parameters r v,w and r av defined in Eq. (6) are fixed to the values deduced from the ansatz for the s → u transition of Refs. [2,24] and listed in Table IV. The statistical uncertainties σ(g av ) and σ(g w ) for the coupling constants g av and g w , respectively, are given in the first row of Table IV. The main feature is that the uncertainty for the g av coupling is nearly one order of magnitude less than for g w since the latter is suppressed by the q 2 /M 2 1 < (M 1 − M 2 ) 2 /M 2 1 ≈ 0.025 factor (11). The second row corresponds to an independent method to study Λ → pe −ν e using the e + e − → J/ψ → Ξ −Ξ+ process with the Ξ − → [Λ → pe −ν e ]π − sequence andΞ + → [Λ →pπ + ]π − for the tagging. Decay Λ → pe −ν e 1.8 12 1.94 1.28 The modular expression for the angular distribution of such process reads The polarization of the Λ originating from the non-leptonic weak decay Ξ − → Λπ − , is ∼ 40%, to be compared to the root-mean-squared value of the Λ polarization in e + e − → J/ψ → ΛΛ of 11% [10]. However, the uncertainties of the weak couplings are the same for both methods.
To further investigate dependence on the initial polarization of Λ we set ∆Φ = 0 to have the zero polarization, while to obtain maximally polarized Λ we include the longitudinal polarization of the electron beam and use the production matrix C µν from Ref. [10]. The impact of the spin correlations for the uncertainties can be studied by comparing the results using the angular distributions (62) or (64) with full production matrices C µν to the ones where all elements except C µ0 are set to zero. This arrangement assures that the spin correlation terms are excluded. In all these tests the uncertainties of σ(g av ) and σ(g w ) remain unchanged, meaning that the polarization and the spin correlations of the mother hyperon in the decay play almost no role for the measurements of properties of the semileptonic decays to baryons whose polarization is not measured.
The entries from the third row and below in Table IV correspond to the decays where the polarization of the daughter baryon is measured and the angular distributions include the complete B µµ ′ matrices. For example the angular distribution for Ξ − → Λe −ν e measurement in e + e − → J/ψ → Ξ −Ξ+ is Since the uncertainties depend on the values of the weak couplings it is difficult to compare the results for different decays in Table IV. By repeating the studies with variation of ∆Φ and the electron beam polarization some impact is seen for the uncertainties, specially for the g w parameter in Σ + → Λe + ν e . In addition we study the uncertainties for single spin polarized baryon decays with the angular distributions given by Eq. (58). The baryon B 1 polarization vector is set to C µ0 = (1, 0, P y , 0). The results for σ(g av ) and σ(g w ) are shown in Fig. 4. The uncertainty for large P y decreases typically by 20% comparing to the unpolarized case.
Our formalism applies also to the n → pe −ν e decay and it should be equivalent to the approach from Ref. [31] for the single neutron decay. However, we can also describe decay correlations for a spin entangled neutron-neutron pair. As an example we take nn spin singlet state given by the spin correlation matrix C µν = diag (1, −1, 1, 1). The coupling constants g av and g w are 1.2754(13) and 1.853, respectively [13,31]. The q 2 dependence of the formfactors is neglected due to the tiny range, m e < q 2 < M n − M p , of the variable.
The corresponding helicity amplitudes for the neutron beta decay are shown in Fig. 3(b).
The resulting uncertainty of the g av measurement in the double beta decay of the singlet pair is σ(g av ) √ N = 4.3. It should be compared to the uncertainties in the measurements with single neutrons that are shown in Fig. 4(a) as a function of the neutron polarization.
For unpolarized neutron σ(g av ) √ N = 7.4 and it decreases to 4.1 when the polarization is equal one. The flip-contribution to the helicity amplitudes (54) of about 8% was neglected in the estimates. The g w coupling cannot be determined since its contribution to the helicity amplitude H V 1 2 0 is suppressed by a factor q 2 /M 2 n . Moreover, the second amplitude that includes g w , H V 1 2 1 , is suppressed by q 2 and as seen in Fig. 3(b) it is consistent with zero.  ism allows to extract the weak formfactors using complete information available in such experiments. The lepton mass effects as well as polarization effects of the decaying parent hyperon are included in the formalism. The presented modular expressions are applicable to various sequential processes like B 1 → B 2 (→ B 3 +π)+l+ν l that involve a semileptonic decay.
Two conventions for defining transversal directions of the helicity frames were considered.
The daughter baryon spin-density matrix in a semileptonic decay takes the simplest form when expressed using the angles in the decay plane. The two representations are equivalent, provided that one uses the matching set of rotations to define the helicity angles.
We have not included radiative corrections in our estimates but they have to be considered in the experimental analyses. Over the years, the radiative corrections to hadronic β-decays have been extensively studied [32] and the specific applications to the hyperon semileptonic decays are discussed in Ref. [33]. The state-of-the-art in experimental analyses is to use Photos program [34] that is based on leading-logarithmic (collinear) approximation. The procedure is applied to all final particles but the electron(positron) tracks are most affected.
The BESIII experiment has collected 10 10 J/ψ [15] meaning that for semileptonic decays data samples of less than 10 4 events are available. Therefore a rough estimate of the achievable uncertainties with this data set is given by dividing the values in Table IV by  For the studies of semileptonic decays of heavy baryons induced by the quark transitions c → s + l + + ν l or b → c + l − +ν l the previously available formalism [6] is likely sufficient if only beams of polarized baryons are used. This might change in near future with BESIII and Belle II experiments where entangled charmed baryon-antibaryon pairs will be available.
One difference would be a measurement of the polarization for the tagging reactions which probably has to use three-body hadronic weak decays. However, even for the case of single baryon decays our approach provides an easy and flexible way to implement different decay sequences in the event generators that propagate spin information of the decaying baryons.
Such replacement in Eq. (B1) gives: By evaluating the above expression one gets the explicit form for R which is the 4D rotation where the spatial part R jk (Ω) corresponds to the product of the following three axial rotations: The expression for b µν can be deduced by setting R The elements of the real-valued matrix b ρν expressed in terms of amplitudes H are: The matrix elements b µν are interrelated since they are expressed by the four complex amplitudes H λ,λ ′ . Therefore, neglecting the unobservable overall phase there are up to six independent real-valued functions in addition to the unpolarized cross section term b 00 . The b-matrix can be considered as a generalization of Lee-Yang baryon polarization formula [35] which has maximum two independent parameters (see example in Appendix D 2). The terms b i0 /b 00 are discussed in [36] in the context of hadronic decays and are called aligned polarimeter fields α x,y,z . In Appendix D we give the b matrices for few example processes.
Here we consider a decay of spin-1/2 baryon to a spin-1/2 baryon and a pair of pseudoscalar mesons P 1 and P 2 via an intermediate vector meson V e.g. B 1 → B 2 ρ 0 → π + π − .
The decay matrices are obtained as in Appendix D 3 by replacing the dilepton with the pseudoscalars, and the virtual photon with a massive vector meson decaying strongly. Since the initial baryon decays weakly into the intermediate state B 1 → B 2 V * , all vector and axial vector formfactors should be used. The decay V * (q) → P 1 (m 1 , p π )P 2 (m 2 , −p π ) is described in the R V frame where the emission angles of the P 1 pseudoscalar are θ π and ϕ π . The value of the momentum p π is The tensor for the V * → P 1 P 2 decay for the helicities λ V , λ ′ V = {−1, 0, 1} is: Decay plane aligned parameters reduce to the following form The differential decay rate of the process with unpolarized baryon B 1 and the spins of B 2 summed over is where V P h (q 2 ) is the three-body phase space density factor given by the product of the momenta |p 2 | and |p π |.