Extracting $p\Lambda$ scattering lengths from heavy ion collisions

The source radii, previously extracted by STAR Collaboration from the $p-\Lambda \oplus \bar{p}-\bar{\Lambda}$ and $\bar{p}-\Lambda \oplus p-\bar{\Lambda}$ correlation functions measured in 10% most central Au+Au collisions at top RHIC energy $\sqrt{s_{NN}}=200$ GeV, differ by a factor of 2. The probable reason for this is the neglect of residual correlation effect in the STAR analysis. In the present paper we analyze baryon correlation functions within Lednicky and Lyuboshitz analytical model, extended to effectively account for the residual correlation contribution. Different analytical approximations for such a contribution are considered. We also use the averaged source radii extracted from the hydrokinetic model (HKM) simulations to fit the experimental data. In contrast to the STAR experimental study, the calculations in HKM show both $p\Lambda$ and $p\bar{\Lambda}$ radii to be quite close, as expected from theoretical considerations. Using the effective Gaussian parametrization of residual correlations we obtain a satisfactory fit to the measured baryon-antibaryon correlation function with the HKM source radius value 3.28 fm. The baryon-antibaryon spin-averaged strong interaction scattering length is also extracted from the fit to the experimental correlation function.


I. INTRODUCTION
The heavy ion collision experiments provide a good possibility for a study of the baryonbaryon strong interactions using the Final State Interaction (FSI) correlation technique [1][2][3].
The latter is based on the analysis of the momentum correlations caused by the final state interaction between corresponding baryons produced in the collision. This activity is especially interesting in view of the ongoing nuclear collision experiments at the LHC, which produce great amounts of various particles, including exotic multi-strange, charmed and beauty ones. It allows one to study the fundamental interactions between specific hadron species, which can hardly be achieved by other means. The extraction of this information makes it possible to check the correctness of hadron-hadron strong interaction models, constrain corresponding interaction potentials, and also improve existing cascade models (like UrQMD) by including into them the information about still unknown baryon-antibaryon annihilation cross-sections.
In paper [4] the experimental pΛ and pΛ correlation functions were fitted with Lednický and Lyuboshitz analytical model [1] allowing, in principle, to extract scattering lengths characterizing the two-particle strong interaction. However, apart from the interaction characteristics, the correlation function depends also on the source spatial structure, described in terms of emission source function, being the time-integrated relative distance distribution in the pair rest frame. This fact complicates a study of the particle interaction, as it increases the number of free parameters which enter the fit formula.
To simplify this study, one could calculate the corresponding source functions in realistic models of the collision process, which are known to describe well the experimental observables. The hydrokinetic model [5][6][7] provides successful simultaneous description of a wide class of bulk observables in the heavy ion collision experiments at RHIC and LHC [8].
Moreover, it reproduces well [9] the pion and kaon source functions for semi-central Au+Au collisions at the top RHIC energy [10], including the specific non-Gaussian tails observed in the pair momentum and beam direction projections of the experimental source function. In this article we present the results of fitting the experimental data from [4] within the analyti-cal model [1] where the Gaussian parametrization for the emission source function is utilized, and the corresponding Gaussian radii are extracted from the HKM model simulations.

II. MODELS DESCRIPTION
The STAR collaboration studied [4] baryon-baryon p − Λ ⊕p −Λ and baryon-antibaryon were selected for the analysis.
The experimental correlation function is constructed as the ratio of the distribution of particle momentum in the pair rest frame, k * , in the same events to the analogous distribution in mixed events. Then the measured correlation function C meas is corrected for the pair purity, defined as the fraction of correctly identified primary particle pairs among all the selected ones, to give the corrected function C corr where λ(k * ) is the pair purity. The estimated mean pair purity in the experiment is λ = 17.5 ± 2.5%.
To fit the experimental correlation function the Lednický and Lyuboshitz analytical model [1] is used, which connects the two-particle correlation function C(k * ) with the particle emission source size r 0 and the s-wave strong interaction scattering amplitudes f S (k * ) at a given total pair spin S. In the equal-time approximation, valid on condition |t * 1 − t * 2 | ≪ m 2,1 r * 2 for sign(t * 1 − t * 2 ) = ±1 respectively, the correlation function can be calculated as a square of the wave function Ψ S −k * , representing the stationary solution of the scattering problem with the opposite sign of the vector k * , averaged over the total spin S and the distribution of the relative distances S(r * ): In typical nuclear collisions the source radius can be considered much larger than the range of the strong interaction potential, so Ψ S −k * at small k * can be approximated by the s-wave solution in the outer region: The effective range approximation for the s-wave scattering amplitude is utilized where f S 0 is the scattering length and d S 0 is the effective radius for a given total spin S = 1 or S = 0.
The particles are assumed to be emitted unpolarized (i.e. with the polarization P = 0), so that the fraction of pairs in the singlet state ρ 0 = 1/4(1 − P 2 ) = 1/4, and in the triplet state ρ 1 = 1/4(3 + P 2 ) = 3/4. The pair separation distribution (source function) where r 0 is considered as the effective radius of the source.
Under such assumptions the correlation function can be calculated analytically [1]: where πr 0 in this expression corresponds to the correction accounting for a deviation of Ψ S −k * from the true wave function inside the range of the strong interaction potential. So, the model has quite a large number of parameters, being the scattering lengths f S 0 , which may be complex in general case, the effective radii d S 0 and the source radius r 0 . Although in principle all of them can be determined from the measured data, in each concrete situation the number of free parameters can be reduced by making certain reasonable assumptions about the values of some of them.
In our study the source radius r 0 is extracted from the Gaussian fit to the source functions calculated in hybrid HKM model. The simulation of the full process of evolution of the system formed in nuclear or particle collision in hHKM consists of two stages. The first one is hydrodynamical expansion of thermally and chemically equilibrated matter described within ideal hydrodynamics approximation with the lattice-QCD inspired equation of state [11] (corrected for small but nonzero chemical potentials), which is matched with the hadronresonance gas in chemical equilibrium via cross-over type transition. The second stage consists in gradual system decoupling after loosing chemical and thermal equilibrium. It can be described either within hydrokinetic approach with switching to UrQMD cascade at some space-like hypersurface situated behind the hadronization one, or with sudden switch to UrQMD cascade at the hadronization hypersurface. In current study we choose the second variant of switching to cascade, basing on [8], where the comparison of one-and two-particle spectra, calculated at both types of matching hydro and cascade stages for RHIC and LHC energies, showed a fairly small difference between them.
The model provides particle distribution functions d 6 N d 3 xd 3 p at the chosen switching hypersurface. Using the Monte-Carlo procedure, one generates particle momenta and coordinates according to these distributions, which serve as the input for the UrQMD hadronic cascade.
To perform a specific calculation one should specify the initial conditions for the hydrodynamics stage attributed to the starting proper time τ 0 . These conditions are the initial energy density (or entropy) profile ǫ(r) and the initial rapidity profile (initial flow) y(r). Here we suppose longitudinal boost-invariance and use ǫ(r T ) corresponding to the MC-Glauber model calculated with GLISSANDO code [12]. The maximal energy density ǫ 0 is chosen to reproduce the experimental mean charged particle multiplicity, and the initial flow is supposed to be y T = α r T R 2 (φ) , with α = 0.45 fm for top RHIC energy. Thus, model has only two free parameters ǫ 0 and α. We start hydrodynamics at τ 0 = 0.1 fm/c and work in mid-rapidity region. Sudden switch from hydrodynamics to UrQMD is performed at the isotherm T = 165 MeV. The hadron distribution functions (for each hadron sort i) at the switching hypersurface σ sw are calculated according to the Cooper-Frye formula The source functions S(r * ) are calculated as Here r * i and r * j are the particles space positions, and r * is the particle separation in the pair rest frame, δ ∆ (x) = 1 if |x| < ∆r/2 and 0 otherwise, ∆r is the size of the histogram bin.

III. RESULTS AND DISCUSSION
The pΛ source function projections calculated in HKM together with the corresponding Gaussian fits are presented in Fig. 1. Here the out-side-long coordinate system is used, where the out axis is directed along the pair total momentum in longitudinally co-moving system, the long direction coincides with the beam axis, and the side axis is perpendicular to the latter two ones. One can notice that non-Gaussian tails, observed in pion source function [10], are also present in pΛ case. They are partially related with the averaging over a wide  [4] and two free parameters ℜf 0 , ℑf 0 in our fit. The STAR has obtained pΛ source radius value r 0 = 1.50 ± 0.05 +0.10 −0.12 ± 0.3 fm (red curve), which is ∼ 2 times smaller than the pΛ one, although there is no apparent physical reason for such a difference. Both radii can be expected to have similar values, and the HKM source radius for the baryon-antibaryon case r HKM 0 = 3.621 ± 0.001 fm is expectedly close to the corresponding baryon-baryon one. But at this source radius value the fitting curve (blue) is too narrow to describe the data points.
Still, we think that the real baryon-antibaryon source size is close to the baryon-baryon one, and the apparent difference between them is caused by some additional effect.
The possible reason for the small fitted baryon-antibaryon source radius could be the influence of residual correlations and imperfection of purity correction [4,14]. Constructing the experimental correlation function one usually supposes that only the pairs composed of two primary particles are correlated, and the rest of the pairs, which include secondary or misidentified particles, are supposed to be uncorrelated. However, the correlation can exist between two parents of secondary particles or between the parent of secondary particle and the primary one. In case, when the secondary particle carries most of the momentum of its parent, such a particle can be "residually" correlated with another particle (or its daughter), which was correlated with its parent. The interactions in most of such pairs are unknown, so at the moment there is no possibility to reliably refine the constructed experimental correlation function from the effect of residual correlations. However, one can try to account for the residual correlations at least phenomenologically in analytical model used for fitting of the correlation function.
In case when the measured correlation function is not corrected for purity, the fitted uncorrected correlation function is expressed through the true one in (6) similar to (1): The pair purity λ(k * ) in our calculations is taken in the form λ(k * ) = aλ exp (k * ), where λ exp (k * ) is extracted from the plots provided in [15] as the ratio λ exp (k * ) = (C uncorr (k * ) − 1)/(C(k * ) − 1). To account for possible imperfection of purity determination in the experiment, we also introduce here a normalizing constant a, which is in fact a fitting parameter in our analysis.
The first term in formula (9) corresponds to the pairs of correlated (primary only) particles, and the second one represents the contribution of the uncorrelated pairs, where one or both particles are secondary (or misidentified) ones. Assuming that among the latter there can be residually correlated pairs and taking into account that the effect of baryonantibaryon residual correlations is dominated by the effect of annihilation dip in parent correlations and that this dip is essentially widen in residual correlations, it can be effectively described by a factor (1 − βe −4k * 2 R 2 ) multiplying the second term in (9) for baryonantibaryonic system: thus introducing additional two parameters β > 0 and R ≪ r 0 .
In Fig. 4 one can see the result of fitting the experimentalp−Λ⊕p−Λ correlation function not corrected for purity (the data are taken from [15]) using the analytical expressions  ℑf 0 = 2.02±3.91 fm, extracted from our fit neglecting purity k * -dependence (with λ = const as a free parameter) is consistent with the result, obtained in recent paper [14], where the purity k * -dependence is also neglected. The radius r 0 in [14] is treated as a free parameter, and the account for residual correlations is performed by summarizing the contributions from different parent pairs to the full correlation function, making, however, a series of simplifying assumptions about the parameters r 0i , f 0i and d 0i , which describe each residual correlation. high statistics data from RHIC and LHC will provide measurements of various particle pairs, including baryon-antibaryon ones, allowing to investigate the particle interactions in these pairs. A consistent approach for a wide class of observables will help to understand complex and unknown features of the evolution of heavy ion collisions.

V. ACKNOWLEDGMENTS
The authors are grateful to Iurii Karpenko for his assistance with computer code. The research was carried out within the scope of the EUREA: European Ultra Relativistic Energies Agreement (European Research Group: "Heavy ions at ultrarelativistic energies") and