The Proton- Ω correlation function in Au+Au collisions at √ s NN GeV

We present the ﬁrst measurement of the proton-Ω correlation function in heavy-ion collisions for central (0-40%) and peripheral (40-80%) Au+Au collisions at √ s NN =200 GeV by the STAR experiment at the Relativistic Heavy-Ion Collider (RHIC). Predictions for the ratio of peripheral collisions to central collisions for the proton-Ω correlation function are sensitive to the presence of a nucleon-Ω bound state. These predictions are based on the proton-Ω interaction extracted from (2+1)-ﬂavor lattice QCD calculations at the physical point. The measured ratio of proton-Ω correlation function from peripheral (small system) to central (large system) collisions is less than unity for relative momentum smaller than 40 MeV/ c . Comparison of our measured correlation ratio with the theoretical calculation slightly favors a proton-Ω bound system with a binding energy of ∼ 27 MeV.

We present the first measurement of the proton-Ω correlation function in heavy-ion collisions for central (0-40%) and peripheral (40-80%) Au+Au collisions at √ s NN =200 GeV by the STAR experiment at the Relativistic Heavy-Ion Collider (RHIC). Predictions for the ratio of peripheral collisions to central collisions for the proton-Ω correlation function are sensitive to the presence of a nucleon-Ω bound state. These predictions are based on the proton-Ω interaction extracted from (2+1)-flavor lattice QCD calculations at the physical point. The measured ratio of proton-Ω correlation function from peripheral (small system) to central (large system) collisions is less than unity for relative momentum smaller than 40 MeV/c. Comparison of our measured correlation ratio with the theoretical calculation slightly favors a proton-Ω bound system with a binding energy of ∼ 27 MeV.

INTRODUCTION
The study of nucleon-nucleon (NN), hyperon-nucleon (YN) and hyperon-hyperon (YY) interactions are of fundamental importance in understanding relativistic heavyion collisions [1,2], modeling of neutron stars [3][4][5][6] and examining the existence of various exotic hadrons [7][8][9]. A significant amount of NN scattering data acquired over the years allows us to construct precise NN potential models [10? ]. The availability of nominal YN scattering data and no scattering data for the multi-strange YY systems makes the task of constructing YN and YY potentials very challenging. With the development of sophisticated computational techniques, it has become possible to carry out first principle calculations based on lattice Quantum Chromodynamics (QCD) to provide constraints on some of the NN, YY and YN interactions [11][12][13][14][15]. Very often the experimental information on the bound states of strange baryons and nucleons (hypernuclei) is used to provide information on YN interactions [16][17][18]. However, this method becomes difficult to use because these measurements are contaminated by many-body effects, which makes it very difficult to extract NΞ, NΩ, YΞ and YΩ interactions.
High-energy heavy-ion collisions produce a sizable number of hyperons in each collision [19], which provides an excellent opportunity to study the NN, YN and YY interactions. Measurement of two-particle correlations at low relative momentum, also known as femtoscopy, have been used to study the space-time dynamics of the source created in heavy-ion collisions. In addition to this, the measurement of two-particle correlations at low relative momentum can also be used to measure final state interactions (FSI) between NN, YN and YY. This approach has been used by the STAR experiment at RHIC to extract the FSI for ΛΛ [20] and antiproton-antiproton [21].
Recent study of (2+1)-flavor lattice QCD simulations for heavy quark masses shows that the nucleon-Ω interaction (NΩ) is attractive at all distances [15]. Using this NΩ interaction, it is shown that the shape of the two particle correlation function at low relative momentum changes substantially with the strength of the NΩ attraction [22]. However, the presence of the Coulomb interaction in the proton-Ω channel makes it difficult to access the strong interaction directly from the measured two-particle correlation function. Therefore, a new measure, namely the ratio of the correlation functions between the peripheral (small) and central (large) collision systems is proposed in Ref [22]. This ratio provides direct access to strong interaction between proton and Ω, independent of the model used for the emission source.
The attractive nature of an NΩ interaction leads to the possible existence of the NΩ dibaryon with strangeness = -3, spin = 2, and isospin = 1/2, which was first proposed in [23]. Such an NΩ dibaryon is the most interesting candidate [23][24][25][26][27] after the H-dibaryon [7]. The Pauli exclusion principle does not apply among quarks in the NΩ dibaryon and it is stable against strong decay [28,29]. Several attempts have been made to estimate the binding energy of the NΩ state in different QCD motivated models [15,30]. The NΩ dibaryon can be produced in high-energy heavy-ion collisions through the coalescence mechanism [31]. For an S-wave bound state of nucleon and Ω, the strong decays to octet-decuplet systems are prohibited by kinematics and those into octetoctet systems (e.g. ΛΞ) are suppressed dynamically due to the D-wave nature [15]. This makes direct searches via the invariant mass method very challenging in heavy-ion collisions. The measurement of the proton-Ω correlation function for peripheral and central Au+Au collisions at √ s N N = 200 GeV, presented in this Letter, will provide insight into the existence of an NΩ dibaryon.

DATA ANALYSIS
STAR is a large acceptance detector at RHIC [32]. The measurements presented in this Letter are from the data taken for Au+Au collisions at √ s NN =200 GeV in 2011 and 2014. 5.30 × 10 8 minimum bias events from 2011 and 8.76 × 10 8 minimum bias events from 2014 were analyzed. The tracking and particle identification for the measurements were provided by the Time Projection Chamber (TPC) [33] and Time-of-Flight (TOF) [34] detectors. These detectors are located in a 0.5 T magnetic field, which allows determination of the momentum and charge of the particles traversing the TPC. Minimum bias triggered events were selected by requiring coincident signals at forward and backward rapidities in the Vertex Position Detectors (VPD) [35] and requiring a signal at mid-rapidity in the TOF. Centrality was determined by the charged particle multiplicity at midrapidity (|η| <0.5) in the TPC. To suppress events from collisions with the beam pipe, the reconstructed primary vertex was required to lie within a 2 cm radial distance from the center of the beam pipe. In addition, the zposition of the vertex was required to lie within ±40 and ±6 cm of the center of the detector for the data from years 2011 and 2014, respectively.

Ω IDENTIFICATION
The TPC was used for tracking, decay topology and identification of particles for Ω (Ω) reconstruction in the pseudorapidity range |η| < 1. To reconstruct the Ω (Ω), the decay channel Ω(Ω) → ΛK − (ΛK + ), with a branching ratio of 67.8%, with subsequent decay Λ(Λ) → pπ − (pπ + ) (branching ratio of 63.9%) was used [36]. The Λ (Λ) candidates were formed from pairs of p (p) and π − (π + ) tracks whose trajectories pointed to a common secondary decay vertex, which was well separated from the Ω (Ω) vertex. These Λ (Λ) candidates were then combined with bachelor K − (K + ) tracks, which points to a common decay vertex well separated from the primary vertex. The decay length (DL) of an Ω (Ω) candidate was required to be larger than 4 cm from the primary vertex. As listed in Table I, additional selection criteria on the distance of closest approach (DCA) between the two Λ (Λ) daughter tracks, between the Λ (Λ) and bachelor track, the Λ (Λ) and the primary vertex position were applied to select Ω (Ω). Furthermore, the pointing angle of Ω (Ω) track with respect to the pri- where r is the position of Ω and primary vertex, respectively and p Ω is the momentum of Ω) were applied to select Ω (Ω). To reduce the combinatorial background, Λ (Λ) candidates were selected in the invariant mass range between 1.112 and 1.120 GeV/c 2 . In addition, the candidates due to misidentification of π − (π + ) tracks as the bachelor K − (K + ) tracks were removed by checking a Ξ hypothesis. The invariant mass distributions of combined Ω andΩ candidates for 0-40% and 40-80% Au+Au collisions at √ s NN =200 GeV for the transverse momentum (p T ) ranges 1.5< p T <2.0 GeV/c and 3.0< p T <3.5 GeV/c are shown in Figure 1(a-d). The signal (S) to sig-nal+background (S + B) ratio, integrated over ±3σ, is 0.2 for the p T range 1.5< p T <2.0 GeV/c and 0.4 for the p T range 3.0< p T <3.5 GeV/c in 0-40% centrality and is 0.3 for the p T range 1.5< p T <2.0 GeV/c and 0.7 for the p T range 3.0< p T <3.5 GeV/c in 40-80% centrality. All candidates with invariant mass between 1.665 and 1.679 GeV/c 2 were used in the analysis.

PROTON IDENTIFICATION
The TOF and TPC detectors were used for proton (antiproton) identification in the pseudorapidity range |η| < 1. The proton tracks were selected if their DCA was less than 0.5 cm to the primary vertex, greater than 20 points were measured out of a maximum of 45, and the number of points used in track reconstruction divided by the number of possible points was greater than 0.52 in order to prevent split tracks. The time of flight of the particles reaching the TOF detector along with the tracking information from the TPC detector was used to calculate the square of the particle mass (m 2 ) to identify protons. Figure 2 shows m 2 from the TOF detector versus momentum from the TPC. All candidates with m 2 between 0.75 and 1.10 (GeV/c 2 ) 2 were used in the analysis.

TWO-PARTICLE CORRELATION FUNCTION
The two-particle correlation function is defined as: where A(k * ) is the distribution of the invariant relative momentum, where k * = | k * | is relative momentum of one of the particles in the pair rest frame, for a proton and Ω pair or anti-proton andΩ pair from the same event. B(k * ) is the reference distribution generated by mixing particles from different events with the same centrality and with approximately the same vertex position along the z-direction. The same single-and pair-particle cuts were applied for real and mixed events. The data analysis was done in nine centrality bins: 0-5%, 5-10%, 10-20%, 20-30%, 30-40%, 40-50%, 50-60%, 60-70% and 70-80% for both same events and mixed events. The final results were combined and presented in two centrality bins: 0-40% and 40-80%. The efficiency and acceptance effects canceled out in the ratio A(k * )/B(k * ). Corrections to the raw correlation functions were applied according to the expression: where the pair purity, P (k * ), was calculated as a product of S/(S + B) for the Ω (Ω) and purity of the proton (antiproton). The selected sample of proton candidates also included secondary protons from Λ, Σ and Ξ decays. The estimated fraction of primary protons (antiprotons) from thermal model [37] studies is 52% (48%) [38]. The purity of the proton sample is obtained as a product of identification probability and fraction of primary protons. The pair purity is 0.2 (0.36) for 0-40% (40-80)% centrality and is constant over the analyzed range of invariant relative momentum.
The effect of momentum resolution on the correlation functions has also been investigated using simulated tracks from Ω decay and tracks for protons, with known momenta, embedded into real events. Correlation functions have been corrected for momentum resolution using the expression: where C(k * ) represents the corrected correlation function, and C in (k * )/C res (k * ) is the correction factor. C in (k * ) was calculated without taking into account the effect of momentum resolution and C res (k * ) included the effect of momentum resolution applied to each Ω and proton candidates. More details related to these corrections can be found in Ref. [39]. The impact of momentum resolution on correlation functions is negligible compared with statistical errors. To study the shape of correlation function for the background, the candidates from the side-bands of invariant mass of Ω are chosen in the range M<1.665 GeV/c 2 and M>1.679 GeV/c 2 . These selected candidates are then combined with the proton tracks from the same event to construct the relative momentum for the same event. The relative momentum for the mixed event is generated by combining the selected candidates from the side-band of invariant mass of Ω with protons from different events with approximately the same vertex position along the z-direction.
The systematic errors for the measured proton-Ω corre-lation function were estimated by varying the following requirements for the selection of Ω candidates: the decay length, DCA of Ω to the primary vertex, pointing angle cuts and mass range, which affect the purity of the Ω sample. The DCA and m 2 requirements were varied to estimate systematic from the proton purity. In addition, systematic errors from normalization and feed-down contributions were also estimated. The systematic errors from different sources were then added in quadrature. The combined systematic errors are shown in Figure 3 as caps for each bin of the correlation function.
Predictions for the proton-Ω correlation function from [22] for proton-Ω interaction potentials V I , V II and V III for a static source with sizes R p = R Ω = 5.0 fm and R p = R Ω = 2.5 fm are also shown in the Figure 3(a) and Figure 3(b). The selected source sizes are not fit to the experimental data. The choice of the potentials in Ref. [22] is based on an attractive NΩ interaction in the 5 S 2 channel from the lattice QCD simulations with heavy u-, d-, s-quarks from Ref. [15]. The potential V II is obtained by fitting the lattice QCD data with a function V (r) = b 1 e −b2r 2 + b 3 (1 − e −b4r 2 )(e −b5r /r) 2 , where b 1 and b 3 are negative and b 2 , b 4 and b 5 are positive, which represents a case with shallow NΩ bound state. Two more potentials V I and V III represent cases without a NΩ bound state and a deep NΩ bound state, respectively. Binding energy (E b ), scattering length (a 0 ) and effective range (r eff ) for the NΩ interaction potentials V I , V II and V III are listed in Table II [22]. The measured correlation functions for P Ω +PΩ are in agreement with the predicted trend for the P Ω correlation functions with inter-action potentials V I , V II and V III for the 0-40% Au+Au collisions as shown in Figure 3(a). However, due to limited statistics at lower k * , strong enhancement due to Coulomb interaction is not visible in the 40-80% Au+Au collisions in Figure 3(b). The measured proton-Ω and antiproton-Ω correlation function includes three effects coming from the elastic scattering in the 5 S 2 channel, the strong absorption in the 3 S 1 channel and the long-range Coulomb interactions. The Coulomb interaction between the positively charged proton and negatively charged Ω introduces a strong enhancement in the correlation function at small k * , as seen in Figure 3. One can remove the Coulomb enhancement using a Gamow correction, however, this simple correction is not good enough to extract the characteristic feature of correlation function from strong interaction. A full correction with source-size dependence is needed to isolate the effect of strong interaction from Coulomb enhancement. Therefore the ratio of correlation function between small and large collision systems, is proposed in [22] as a model-independent way to access the strong interaction with less contamination from the Coulomb interaction.
The ratio of combined proton-Ω and antiproton-Ω correlation function from peripheral (40-80%) to central (0-40%) collisions, defined as R = C 40−80 /C 0−40 , for proton-Ω and antiproton-Ω (P Ω+PΩ) is shown in Figure 4. The systematic uncertainties are propagated from the measured correlation functions for 0-40% and 40-80% centrality and are shown as caps. For the background study, the candidates from the side-bands of Ω invariant mass were combined with protons to construct the correlation function. The same ratio, R, for the background is unity and is shown as open crosses in Figure 4. Previous measurements of source size for π-π, K 0 S -K 0 S , proton-proton and proton-Λ correlations show that the source size decreases as the transverse mass increases [21,[38][39][40][41]. Using this transverse mass dependence [41], the expected source size for proton-Ω is 2-3 fm for peripheral collisions and 3-5 fm for central collisions. The predictions for the ratio of small system to large system from Refs. [22,42] for proton-Ω interaction potentials V I , V II and V III for static source with different source sizes (S,L) = (2,3), (2,4), (2.5, 5) and (3,5) fm, where S and L correspond to small and large collision sytem, are shown in Figure 4(ad). A small variation in the source size does not change the characteristic of the ratio for the choice of three potentials.
Predictions for the ratio of small to large system with the effects of collective expansion are also shown in the Figure 4(e) [22]. The transverse source sizes are taken as R tr p = R tr Ω = 2.5 fm for small system and R tr p = R tr Ω = 5 fm for large system. The temperature at the thermal freeze-out is T p,Ω = 164 MeV for peripheral collisions and T p,Ω = 120 MeV for the central collisions [43,44] and the proper-time at the thermal freeze-out is τ p (τ Ω ) = 3(2) fm/c for the peripheral collisions and τ p (τ Ω ) = 20 (10) fm/c for the central collisions [45].
The predictions with expanding source for the proton-Ω interaction potentials V I and V II are 3σ larger than the data at k * = 20 MeV/c. The prediction for the proton-Ω interaction potential V III with expanding source and static source are within 1σ of the data at k * = 20 MeV/c. As shown in Figure 4, the measured ratios at k * =20 and 60 MeV/c are R= 0.28 ± 0.35 stat ± 0.03 sys (background = 0.96 ± 0.13 stat ) and R= 0.81 ± 0.22 stat ± 0.08 sys (background = 0.97 ± 0.05 stat ), respectively. Comparing these values with the model calculations shown in Figure 5(b) of the Ref. [22], where a bound state with E b ∼27 MeV for the proton-Ω system is assumed in calculation, we conclude that our data favor a positive scattering length for the proton-Ω interactions. The positive scattering length and the measured ratio of proton-Ω correlation function from peripheral to central collisions less than unity for k * < 40 MeV/c favors the proton-Ω interaction potential V III with E b ∼27 MeV for proton and Ω.

CONCLUSIONS
The first measurement of the proton-Ω correlation function in heavy-ion collisions for Au+Au collisions at √ s NN =200 GeV is presented in this Letter. The measured ratio of proton-Ω correlation function from peripheral to central collisions is compared with the predictions based on proton-Ω interaction extracted from (2+1)-flavor lattice QCD simulations. At present, due to limited statistics, it is not possible to extract the interaction parameters. However the measured ratio of proton-Ω correlation function from peripheral to central collisions less than unity for k * < 40 MeV/c within 1σ indicates that the scattering length is positive for the proton-Ω interaction and favors the proton-Ω bound state hypothesis.