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Possible effect of mixed phase and deconfinement upon spin correlations in the pairs generated in relativistic heavy-ion collisions

. Spin correlations for the ΛΛ and Λ¯Λ pairs, generated in relativistic heavy-ion collisions, and related angular correlations at the joint registration of space-parity nonconserving hadronic decays of two hyperons are theoretically analyzed. The correlation tensor components can be derived from the double angular distribution of products of two decays by the method of “moments”. The properties of the “trace” of the correlation tensor (a sum of three diagonal components), determining the angular correlations as well as the relative fractions of the triplet and singlet states of respective pairs, are discussed. In the present report, spin correlations for two identical (ΛΛ) and two non-identical (Λ¯Λ) particles are generally considered within the conventional model of one-particle sources, implying that correlations vanish at suﬃciently large relative momenta. However, under these conditions (especially at ultrarelativistic energies), for two non-identical particles (Λ¯Λ) the two-particle annihilation sources – quark-antiquark and two-gluon ones – start playing a noticeable role and lead to the diﬀerence of the correlation tensor from zero. In particular, such a situation may arise, when the system passes through the “mixed phase” and – due to the multiple production of free quarks and gluons in the process of deconﬁnement of hadronic matter – the number of two-particle sources strongly increases.


Introduction
Spin correlations for the ΛΛ and ΛΛ pairs, generated in heavy ion collisions, and respective angular correlations at the joint registration of hadronic decays of two hyperons with space parity nonconservation give important information about the character and mechanism of multiple processes. The advantage of the ΛΛ and ΛΛ systems over other ones is due to the fact that the P -odd decays Λ → p + π − andΛ →p + π + serve as effective analyzers of spin state of the Λ andΛ particles. In connection with this, spin correlations in the ΛΛ and ΛΛ systems can be rather easily distinguished and studied experimentally by the method of "moments" over the background of a large amount of produced secondary particles. This fact is especially meaningful for the investigations of multiple generation at modern and future ion colliders like RHIC, LHC and NICA, since the polarization parameters -especially for the ΛΛ pair -are very sensitive to the scenario of process after the act of collision of relativistic heavy ions. IOP Conf. Series: Journal of Physics: Conf. Series 1137 (2019) 012034 IOP Publishing doi:10.1088/1742-6596/1137/1/012034 2 2. General structure of the spin density matrix of the pairs ΛΛ and ΛΛ The spin density matrix of the ΛΛ and ΛΛ pairs, just as the spin density matrix of two spin-1/2 particles in general, can be presented in the following form [1,2,3]: in doing so, tr (1,2)ρ (1,2) = 1.
HereÎ is the two-row unit matrix,σ = (σ x ,σ y ,σ z ) is the vector Pauli operator (x, y, z → 1, 2, 3), P 1 and P 2 are the polarization vectors of first and second particle (P 1 = σ (1) , are the correlation tensor components. In the general case T ik = P 1i P 2k . The tensor with components C ik = T ik −P 1i P 2k describes the spin correlations of two particles .
3. Spin correlations and angular correlations at joint registration of decays of two Λ particles into the channel Λ → p + π − It is essential that any decay of an unstable particle may serve as an analyzer of its spin state .
The normalized angular distribution at the decay Λ → p + π − takes the form: Here P Λ is the polarization vector of the Λ particle, n is the unit vector along the direction of proton momentum in the rest frame of the Λ particle, α Λ is the coefficient of P -odd angular asymmetry ( α Λ = 0.642 ). The decay Λ → p + π − selects the projections of spin of the Λ particle onto the direction of proton momentum; the analyzing power equals ξ = α Λ n . Now let us consider the double angular distribution of flight directions for protons formed in the decays of two Λ particles into the channel Λ → p + π − , normalized by unity ( the analyzing powers are ξ 1 = α Λ n 1 , ξ 2 = α Λ n 2 ). It is described by the following formula [2,3]: where P 1 and P 2 are polarization vectors of the first and second Λ particle, T ik are the correlation tensor components, n 1 and n 2 are unit vectors in the respective rest frames of the first and second Λ particle, defined in the common ( unified ) coordinate axes of the c.m. frame of the pair The polarization parameters can be determined from the angular distribution of decay products by the method of moments -as a result of averaging the combinations of trigonometric functions of proton ( antiproton ) flight angles over the double angular distribution of flight directions for products of two decays [2,3] .
The angular correlation, integrated over all angles except the angle between the vectors n 1 and n 2 and described by the formula [2,3,4,5] d w(cos θ) is determined only by the "trace" of the correlation tensor T = W t − 3W s ( W s and W t are relative fractions of the singlet state and triplet states, respectively ), and it does not depend on the polarization vectors ( single-particle states may be unpolarized ).  4. Correlations at the joint registration of the decays Λ → p + π − andΛ →p + π + Due to CP invariance, the coefficients of P -odd angular asymmetry for the decays Λ → p + π − andΛ →p + π + have equal absolute values and opposite signs: αΛ = −α Λ = −0.642 . The double angular distribution for this case is as follows [2,3]: (here −α Λ = +αΛ and −α 2 Λ = +α Λ αΛ ) . Thus, the angular correlation between the proton and antiproton momenta in the rest frames of the Λ andΛ particles is described by the expression: where θ is the angle between the proton and antiproton momenta .

Spin correlations at the generation of ΛΛ pairs in multiple processes
Further we will use the model of one-particle sources [6], which is the most adequate one in the case of collisions of relativistic ions .
As far as two Λ particles ( identical particles ) are concerned, the spin correlations and angular correlations at their decays were considered previously, within the model of one-particle sources, in the works [2,7] -taking into account Fermi statistics and final-state interaction ( see also the detailed analysis in the paper [8] ).
In the present report, we will be interested in spin correlations at the decays of ΛΛ pairs . In the framework of the model of independent one-particle sources, spin correlations in the ΛΛ system arise only on account of the difference between the interaction in the final triplet state ( S = 1 ) and the interaction in the final singlet state . At small relative momenta, the s-wave interaction plays the dominant role as before, but, contrary to the case of identical particles (ΛΛ), in the case of non-identical particles (ΛΛ) the total spin may take both the values S = 1 and S = 0 at the orbital momentum L = 0 . In doing so, the interference effect, connected with quantum statistics, is absent .
If the sources emit unpolarized particles, then, in the case under consideration, the correlation function describing momentum-energy correlations has the following structure ( in the c.m. frame of the ΛΛ pair ): The spin density matrix of the ΛΛ pair is given by the formula : and the components of the correlation tensor are as follows: here the contributions of final-state triplet and singlet ΛΛ interaction are determined by the expression obtained in the works [2,7] .
In this case the angular correlations in the decays Λ → p + π − ,Λ →p + π + , connected with the final-state interaction, are absent : 6. Angular correlations in the decays Λ → p + π − andΛ →p + π + and the "mixed phase" Thus, at sufficiently large relative momenta (for example, k m π ) one should expect that the angular correlations in the decays Λ → p + π − andΛ →p + π + , connected with the interaction of the Λ andΛ hyperons in the final state ( i.e. with one-particle sources ) are absent . But, if at the considered energy the dynamical trajectory of the system passes through the so-called "mixed phase", then the two-particle sources, consisting of the free quark and antiquark , start playing a noticeable role . For example, the process ss → ΛΛ may be discussed .
In this process, the charge parity of the pairs ss and ΛΛ is equal to C = (−1) L+S , where L is the orbital momentum and S is the total spin of the fermion and antifermion. Meantime, the CP parity of the fermion-antifermion pair is CP = (−1) S+1 .
In the case of one-gluon exchange, CP = 1, and then S = 1, i.e. the ΛΛ pair is generated in the triplet state; in doing so, the "trace" of the correlation tensor T = 1 .
Even if the frames of one-gluon exchange are overstepped, the quarks s ands, being ultrarelativistic, interact in the triplet state ( S = 1 ) . In so doing, the primary CP parity CP = 1, and, due to the CP parity conservation, the ΛΛ pair is also produced in the triplet state. Let us denote the contribution of two-quark sources by x . Then at large relative momenta Apart from the two-quark sources, there are also two-gluon sources being able to play a comparable role. Analogously with the annihilation process γγ → e + e − [9], in this case the "trace" of the correlation tensor is described by the formula ( here the process g g → ΛΛ is implied ): where β is the velocity of Λ ( andΛ) in the c.m. frame of the ΛΛ pair, θ is the angle between the momenta of one of the gluons and Λ in the c.m. frame (see [9]). At small β ( β ≪ 1 ) the ΛΛ pair is produced in the singlet state ( total spin S = 0, T = −3 ), whereas at β ≈ 1 -in the triplet state ( S = 1, T = 1 ). Let us remark that at ultrarelativistic velocities β ( i.e. at extremely large relative momenta of Λ andΛ ) both the two-quark and two-gluon mechanisms lead to the triplet state of the ΛΛ pair ( T = 1 ).

Summary
So, it is surely advisable to investigate the spin correlations of ΛΛ and ΛΛ pairs produced in relativistic heavy ion collisions ( see also, e.g., [8,10] ) .
The spin correlations, as well as the momentum-energy ones, make it possible to determine the space-time characteristics of the multiple particle generation region . In doing so, the best way of studying the spin correlations is the method of angular correlations -method of moments.
Finally, it should be emphasized that, in the general case, the appearance of angular correlations in the decays Λ → p + π − andΛ →p + π + with the nonzero values of the "trace" of the correlation tensor T at large relative momenta of the Λ andΛ particles may testify to the passage of the system through the "mixed phase" [8,10] .