Entanglement in joint $\Lambda \bar{\Lambda}$ decay; cont

We have previously investigated joint $\Lambda \bar{\Lambda}$ decay in the reaction $e^+ e^- \rightarrow \gamma \Lambda(\rightarrow p\pi^-) \bar{\Lambda}(\rightarrow \bar{p}\pi^+)$. The cross-section-distribution functions encountered were relativistically covariant and expressed in terms of scalar products of the four-momentum vectors of the particles involved. In the present, sequel investigation, we show that by working with three-momentum scalars instead results could possibly become more transparant.


I. INTRODUCTION
The BABAR Collaboration [1] has measured initial-state-radiation in the annihilation reaction e + e − → γΛ(→ pπ − ) Λ(→ pπ + ).Such measurements are interesting since they offer opportunities to determine electromagnetic form factors of the Lambda hyperons in the time-like region.
A theoretical analysis of this annihilation reaction is presented in refs.[2] and [3].A complete determination of the cross-section-distribution functions, including those describing joint Lambda anti-Lambda decays, is also given in ref. [4].The arguments of those functions, and there are several, are the scalar products of the four-momentum vectors of the particles involved.To determine all those functions is a formidable task, and we have therefore decided on an alternative approach, to work with three-momentum vectors instead.
Replacing four-dimensional arguments by three-dimensional ones also requires considerable work, but this work is worth-while, as we shall see.The result is transparent.

II. CROSS-SECTION DISTRIBUTION
Our notation follows Pilkuhn [5].The cross-section distribution for the reaction e + e − → γΛ(→ pπ − ) Λ(→ pπ + ) is written as where the average over the squared matrix element indicates summation over final proton and anti-proton spins and average over initial electron and positron spins.The definitions of the particle momenta are explained in fig. 1.
We remove some trivial factors from the squared matrix element, collected in a factor denoted K,

III. PREVIOUS ANALYSIS
We start where our previous analysis ended, ref. [4], but before we can do so it is necessary to repeat some of the important definitions and results.
The form factors of the hyperon-electromagnetic couplings are denoted G 1 and G 2 , a standard choice.The designations of particle four-momenta can be seen in the Feynman diagrams of fig. 1.
The cross-section-distribution function, or rather the covariant square of the annihilation matrix element |M red | 2 , is the contraction of hadronic H µν and leptonic L µν tensors, Its right-hand side is naturally decomposed as with suffixes RS and R S referring to Λ and Λ decay constants, with R the spin-independent and S the spin-dependent one.
From the structure of the lepton tensor, eq.(24) of ref. [4], one concludes that each of the M XY functions has two parts, where the A XY factor is obtained by contracting the hadron tensor with the symmetric tensor k 1µ k 1ν + k 2µ k 2ν , and the B XY factor by contraction with the tensor g µν .For details see ref. [4].The weight factors a y and b y are defined in appendix A.
The functions A XY and B XY are bilinear forms of G 1 and G 2 , and we expand them accordingly, for A XY ,

IV. PREVIOUS RESULTS
The leading term of eq.( 4) is M RR as it is independent of variables that relate to spin dependences in the hyperon decay distributions.We have with Q = p 1 − p 2 .Furthermore, Thus, the distribution function M RR does not depend on the decay momenta l and q of the Lambda hyperons.
Next in order are terms linear in the spin variables, with det(abcd) = ǫ αβγδ a α b β c γ d δ and The expressions for the spin-spin contributions are more complicated.We have for the A contribution and for the B contribution The functions A SS and B SS describe the joint-decay distributions of the Lambda and anti-Lambda hyperons.The distributions are correlated, i.e., they cannot be written as a product of Lambda and anti-Lambda distribution functions.Our distribution functions are explicitly covariant, as they are expressed in terms of the four-momentum vectors of the participating particles.It is not necessary to work in several coordinate systems, as in refs.
[1] and [3].Another important point is that our calculation correctly counts the number of intermediate hyperon states.

V. REFERENCE FRAMES
The cross-section distribution function of sect.IV involves expressions that are functions of scalar products of particle four-momenta.To determine the scalar product of two fourvectors requires knowledge of those vectors in one and the same reference frame.Our task in this section is to demonstrate how this is achieved.
The gamma three-momentum q, and electron three-momentum k, are momenta defined in the e + e − centre-of-momentum (c.m.) reference frame, in which q• k = cos θ.We refer to this frame as S 0 .In S 0 electron and positron four-momenta are with ǫ the common lepton energy.With ω the gamma energy, the gamma four-momentum is denoted q = ω(1, q).Furthermore, the four-momenta of Lambda and anti-Lambda are p 1 = (E 1 , p 1 ) and p 2 = (E 2 , p 2 ).Now, we shall not perform our calculations in S 0 but in S 1 which is the c.m. frame of the Λ Λ pair.We indicate variables in this frame by a prime, so that with E Λ = p 2 Λ + M 2 Λ and f a unit vector.The Λ Λ c.m. energy W = 2E Λ may be obtained from the identity The next question concerns the relation between frames S 1 and S 0 .Since p 1 +p 2 = −q in S 0 , we argue that S 1 can be reached from S 0 through a boost along the direction of motion of the gamma, and of magnitude, and with Lorentz-transformation (LT) coefficient Also, note that v is the relative velocity between two reference frames, it is not a particle velocity.
A Lorentz boost from S 0 to S 1 leads to new four-momentum vectors for the initial state leptons, namely and with n and N by definition Relations ( 21) and ( 22) are identical to those introduced by the BaBar collaboration [1].
The photon radiated in our annihilation process carries energy ω and three-momentum q = ωn , when observed in S 0 .A boost from S 0 to S 1 , sends this vector into q ′ = ω ′ n, with However, we should not forget the decay products of the hyperons, the antiproton and the proton.In the rest system S 2 of the Lambda the proton is represented by the four-vector with g a unit vector, and with decay parameters p g and E g .
Similarly, in the rest system S 3 of the anti-Lambda the anti-proton is represented by the four-vector with h a unit vector, and decay parameters p h = p g and E h = E g .A passage from S 3 to S 1 is achieved by a Lorentz boost with velocity v h and direction f, whereas a passage from S 2 to S 1 is achieved by a Lorentz boost with velocity v g and direction −f.
The boost equations for the massive hyperons are well known.Vectors orthogonal to the boost velocity, v = vn, are unchanged, those parallel are changed according to the Lorentz-transformation prescription The inverse-transformation equations, going from S 1 to S 0 , are obtained by changing the sign of the velocity, from n to −n.
After this elementary discussion we are ready for the proton and antiproton fourmomentum vectors in S 1 ; for the proton the transverse-vector component p g⊥ being with p g⊥ • f = 0; and for the antiproton Our calculations make use of the shorthand notations,

VI. CALCULATING CO-FACTORS
Co-factors can be identified in the A XY and the B XY functional distributions of sect.
IV.The results are co-factors expressed in terms of scalar products of four-vector momenta.
However, our goal was to find simpler expressons, and this by evaluating all scalars in one and the same reference frame, the c.m. reference frame of the Λ Λ pair.
We have evaluated cross-section distributions for two sets of form-factor parameters.
The two sets have attached form-factor sets, that we indicate by different letters, such that (G 1 , G 2 ) ⇒ {K} and (G M , G E ) ⇒ {L} .We start with the K set and return to the L set in sect.VII.
The much needed Ω functions are defined in appendix B. The spin-correlation functions are equally important.Their definitions are, prefactors of the above-mentioned equations are quite easily obtained, and equals In the S 1 frame the determinant boils down to Hence, components of the vectors p ′ g or k ′ 2 along f will not contribute to the value of the determinant, so that by eq.( 28) we may replace p ′ g by p g .For k ′ 2 we make recourse to eq.( 20).All this yields, The co-factors that relate to B XY of eq.( 5) vanish, With the co-factors of eqs.( 48) and (49) in hand we can determine A RS and A SR from eq.( 6).The related functions B RS and B SR vanish identically.In agreement with refs.[2], C. Doubly spin-dependent co-factors Now, the co-factors are suffixed ASS and BSS.Those suffixed ASS are obtained by analysing A SS of eq.(13); and similarly for B XY .The relation between the two sets of functions, {K XY Z i } and {L XY Z i } becomes and with the parameter Z defined in eq.( 37).The most notible fact of the new set is that several co-factors vanish; whereas L ASS 3 = 0.
Co-factors suffixed ARR; Co-factors suffixed BRR; Co-factors suffixed ARS and ASR; The co-factors that relate to B XY of eq.( 5) vanish, With these co-factors in hand we can determine A RS and A SR from Eq.( 6) whereas the related functions B RS and B SR vanish.