Investigation of K+K− interactions via femtoscopy in Pb-Pb collisions at √sNN = 2.76 TeV at the CERN Large Hadron Collider

This is a self-archived version of an original article. This version may differ from the original in pagination and typographic details. Author(s): Title: Year: Version: Copyright: Rights: Rights url: Please cite the original version: CC BY 4.0 https://creativecommons.org/licenses/by/4.0/ Investigation of K+K− interactions via femtoscopy in Pb-Pb collisions at √sNN = 2.76 TeV at the CERN Large Hadron Collider ©2023 CERN, for the ALICE Collaboration Published version ALICE Collaboration ALICE Collaboration. (2023). Investigation of K+K− interactions via femtoscopy in Pb-Pb collisions at √sNN = 2.76 TeV at the CERN Large Hadron Collider. Physical Review C, 107, Article 054904. https://doi.org/10.1103/PhysRevC.107.054904 2023 PHYSICAL REVIEW C 107, 054904 (2023) Investigation of K+K−interactions via femtoscopy in Pb-Pb collisions at √sNN =2.76 TeV at the CERN Large Hadron Collider S. Acharya et al.∗ (ALICE Collaboration) (Received 9 December 2022; accepted 7 February 2023; published 3 May 2023) Femtoscopic correlations of nonidentical charged kaons (K+K−) are studied in Pb-Pb collisions at a center-of- mass energy per nucleon-nucleon collision √sNN =2.76 TeV by ALICE at the CERN Large Hadron Collider. One-dimensional K+K−correlation functions are analyzed in three centrality classes and eight intervals of particle-pair transverse momentum. The Lednický and Luboshitz interaction model used in the K+K−analysis includes the final-state Coulomb interactions between kaons and the final-state interaction through a0(980) and f0(980) resonances. The mass of f0(980) and coupling were extracted from the fit to K+K−correlation functions using the femtoscopic technique. The measured mass and width of the f0(980) resonance are consistent with other published measurements. The height of the φ(1020) meson peak present in the K+K−correlation function rapidly decreases with increasing source radius, qualitatively in agreement with an inverse volume dependence. A phenomenological fit to this trend suggests that the φ(1020) meson yield is dominated by particles produced directly from the hadronization of the system. The small fraction subsequently produced by final-state interactions could not be precisely quantified with data presented in this paper and will be assessed in future work. DOI: 10.1103/PhysRevC.107.054904 I. INTRODUCTION Femtoscopy is a tool for measuring the space-time geometry of the particle emission region in high-energy collisions of protons and ions [1,2]. It is based on the measurement of two-particle momentum correlation functions (CF) which are determined by final-state interactions (FSI) between the emitted particles and effects of quantum statistics in the case of identical species [1,3,4]. The technique was traditionally used to determine the size of the emission region and its dependence on the particle-pair transverse momentum and transverse mass, and on the event multiplicity [5,6]. Recently, there has been a great interest in studying the interaction of particles using femtoscopy methods along with a parametrization of the size of the particle emitting source [7–16]to analyze the measured correlation functions. There has also been interest in the femtoscopic correlations of pairs of nonidentical kaons involving a neutral kaon K0 SK±[17–19]. An important complement to these studies is the measurement of the K+K−correlations. The existing results in this area are rather scarce [5,20,21]. This is due to the complexity of the measurements and the subsequent complicated analysis. In comparison to identical kaons, the interaction between K+ and K−in the final state is much more complex. It includes ∗Full author list given at the end of the article. Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI. the Coulomb interaction and the strong interaction through the near-threshold f0(980) (I=0 isospin state) and a0(980) (I=1 isospin state) resonances, and the strong p-wave FSI through the φ(1020) meson. The properties of the scalar mesons a0(980) and f0(980), discovered in the mid-1960s and in the beginning of the 1970s, respectively [22–26], are still subject of research. The idea of the nature of these mesons as a quark-antiquark pair [27] is supplemented by a state in the form of a KKmolecule [28] and even a tetraquark state [29]. Recently, in Ref. [19], it has been shown that the study of the magnitude of the correlation strength (λ)inK0 SK0 Sand K0 SK±pairs allows one to conclude that the observed difference in measured λis compatible with the a0(980) resonance being a tetraquark state. However, it should be noted that, according to the results of the latest ALICE work on f0(980) in pp collisions at √s= 5.02 TeV [30], the model descriptions assuming a tetraquark (uuss), KK molecule, and ss disagree with the experimental measurement. The first measurement of K+K−correlations was carried out for Pb-Pb collisions at the CERN Super Proton Synchrotron (SPS) [5]. It was shown that the theoretical K+K−correlation functions calculated by using a finitesize Coulomb wave function with the radius extracted from identical kaon correlations were noticeably greater than the measured ones. However, by taking into account the contribution due to strong interactions, a reasonably good description of the data was obtained at small relative momenta. Preliminary results from the analysis of unlike-sign kaon femtoscopic correlations in Au-Au collisions at √sNN = 200 GeV were reported by the STAR Collaboration [20,21]. 2469-9985/2023/107(5)/054904(16) 054904-1 ©2023 CERN, for the ALICE Collaboration S. ACHARYA et al. PHYSICAL REVIEW C 107, 054904 (2023) The experimental one-dimensional K+K−correlation function in terms of the invariant momentum difference was compared with the theoretical prediction based on the Lednický–Luboshitz approach [1]. The measured K+K−CF could be described by the theoretical calculations using a Gaussian function to model the source with the size parameters extracted from the fit of the correlation function of identical charged kaons. To account for additional physical effects not included in the theoretical function, the calculated correlation function CFtheor was scaled according to CF = (CFtheor −1)λ−1, where the correlation strength parameter λwas obtained from the fit to the like-sign kaon correlation function. The STAR study showed that the model could qualitatively reproduce the general structure of the measured K+K−correlation function both at low relative momenta q<200 MeV/c, where correlations are determined by the interplay of the Coulomb and s-wave strong interactions, and in the φ(1020) resonance region. In this work, K+K−femtoscopic correlations are studied in Pb-Pb collisions at a center-of-mass energy per nucleonnucleon collision √sNN =2.76 TeV at the CERN Large Hadron Collider (LHC) by the ALICE Collaboration [31]. The physics goals of the present study are as follows: (1) extraction of the f0(980) mass and coupling parameters based on the fit to the K+K−correlation function using the Lednický- Luboshitz model [1] and on the assumption that the source radii of K+K−pairs are the same as those of K±K±[6]; (2)testofthea0(980) mass and coupling parameters used in the K0 SK±femtoscopy study [17,18]; (3) investigation of how the height of the φ(1020) peak in the CF changes with the source radius in order to shed light on the nature of the production of the φ(1020) meson in heavy-ion collisions. The organization of this article is as follows. In Sec. II,the event and track selection criteria are described. In Sec. III,the theoretical and experimental details of the correlation functions and the fitting procedure are discussed. The results of the analysis are shown in Sec. IV, and a summary is provided in Sec. V. II. EVENT AND CHARGED-PARTICLE RECONSTRUCTION AND SELECTION The analysis presented in this paper used a sample of about 4×107minimum bias Pb-Pb collisions at √sNN =2.76 TeV collected with the ALICE detector in the LHC run 1 period (2009–2013). Monte Carlo (MC) simulations were used for correcting the obtained CFs for track momentum resolution. In the simulations, particles from Pb-Pb collision events were generated with the HIJING [32] general-purpose event generator and were propagated through the ALICE detector using the GEANT3[33] transport code. The total number of MC events used in this analysis was about 4 ×106. Most of the event and track selection criteria in the current analysis are thesameasin[6]. Events were classified according to their centrality determined using the measured signal amplitudes in the V0 detectors [34–36], which consist of two arrays of scintillator counters installed on each side of the interaction point and covering the pseudorapidity intervals 2.8<η<5.1(V0A) and −3.7<η<−1.7(V0C)[36]. Charged particles were reconstructed and identified with the detectors located within a solenoidal magnet that provides a uniform field of 0.5 T along the beam direction. Charged particle tracking was performed using the inner tracking system (ITS) [37] and the time projection chamber (TPC) [38]. The ITS consists of six cylindrical layers of silicon detectors, located at radii between 4 and 43 cm. The ITS and TPC cover the pseudorapidity range |η|<0.9 for all vertices located within the interaction diamond [31]. The ITS provides high spatial resolution in determining the primary (collision) vertex and the distance of closest approach (DCA) of a track to the primary vertex. The primary-vertex position along the beam direction (zcoordinate in the ALICE reference frame) was required to be within ±10 cm from the center of the ALICE detector to ensure uniform tracking performance. The TPC is the main part of the ALICE apparatus and was designed to track and identify charged particles in the high particle-density environment of heavy-ion collisions at the LHC. The TPC is a 5 m long cylindrical gas detector with a volume close to 90 m3and with full acceptance in the pseudorapidity range |η|<0.9. The particle momenta were determined using tracks reconstructed with the TPC and constrained to originate from the primary vertex. In order to reduce the number of secondaries, primary tracks were selected based on DCA to the primary vertex. Additional track selections based on the quality of the track momentum fit and the number of detected space points in the TPC were used. Each track was required to have at least 80 (out of a maximum of 159) associated space-points in the TPC. Track pairs sharing more than 5% of TPC clusters were rejected [6]. Particle identification (PID) was carried out using both the TPC and the time of flight (TOF) [39] detectors in the pseudorapidity range |η|<0.8. The TOF is a cylindrical detector with a radius of about 3.7 m. The total area of the active part of the TOF is about 141 m2. The main unit of the ALICE TOF detector is the multigap resistive plate chamber (MRPC) strip detector. The usable area of each MRPC is about 120 ×7.4cm 2. The ALICE TOF array was assembled from 1593 MRPC strips, subdivided into 18 azimuth sectors. For TPC PID, a parametrized Bethe-Bloch formula for a particle with a given charge, mass, and momentum was used to calculate the expected specific energy loss (dE/dx)inthe detector. The deviation between the measured and expected dE/dxvalues was required to be within a certain number of standard deviations (Nσ,TPC) relative to the dE/dxresolution of the TPC [40]. A similar Nσ,TOF selection was applied for particle identification with the TOF. In this case, the expected time of flight for a particle with a given mass was calculated from the track length and momentum measured with the tracking detectors. A detailed description of the particle identification is given in [41]. The selection criteria which were used for kaon selection in the TPC and TOF are shown in Table I. The purity of kaons is larger than 99% for tracks with momentum greater than 0.45 GeV/c[6]. To estimate the charged kaon purity for p<0.45 GeV/c, the measured dE/dxdistribution was used [42]. First, the measured dE/dx distributions in track momentum intervals were considered, and the contributions of electrons, pions, kaons, and protons 054904-2 INVESTIGATION OF K+K−INTERACTIONS VIA … PHYSICAL REVIEW C 107, 054904 (2023) TABLE I. Charged kaon selection criteria. pT0.14 <pT<1.5 GeV/c |η|<0.8 DCAtransverse to primary vertex <2.4cm DCAlongitudinal to primary vertex <3.0cm Nσ,TPC (for p<0.4 GeV/c)<2 Nσ,TPC (for 0.4<p<0.45 GeV/c)<1 Nσ,TPC (for p>0.45 GeV/c)<3 Nσ,TOF (for 0.45 <p<0.8 GeV/c)<2 Nσ,TOF (for 0.8<p<1.0 GeV/c)<1.5 Nσ,TOF (for 1.0<p<1.5 GeV/c)<1.0 Number of track points in TPC ⩾80 χ2/Nclusters of the track fit ⩽4 were parametrized via Gaussian fits. Next, an estimate of the purity of kaons for momentum p<0.45 GeV/cwas made using this parametrization. The estimated single kaon purity as a function of momentum pis shown in Fig. 1(left panel) for different centrality intervals. The obtained values of purity decrease from semiperipheral (30–50%) to central (0–10%) collisions. The resulting kaon pair purity as a function of pair transverse momentum kT=| pT,1+ pT,2|/2 for different centralities is shown in Fig. 1(right panel). The pair purity distribution is wider and its values are larger on average than for the single-kaon purity. The value of the pair purity is higher than 99% for K+K−pairs in the considered kTinterval. The main contamination for K+K−pairs comes from γ→e+e−conversions. It should be noted that to reduce this effect, the identification of kaons with the TOF starts when the charged kaon momentum is larger than 0.45 GeV/cinstead of 0.5 GeV/cas it was in the identical kaon femtoscopy analysis published in [6]. In addition, a more stringent selection on Nσwas applied in the momentum interval where the contamination from e+e−pairs is expected to be large (see Table I). III. ANALYSIS TECHNIQUE Two-particle momentum correlations are defined as C( q)=A( q)/B( q), where A( q) is the measured distribution of same-event pair momentum difference  q= p1− p2, p1 and  p2are the momentum of the first and second particle in the pair, and B( q) is the reference distribution of pairs from mixed events. The mixed-event pair distribution was obtained by mixing particles from events with similar centrality and vertex positions along the beam direction. The correlation function is measured as a function of q=√| q|2−q2 0, where q0=E1−E2is determined by the energies E1,E2of the correlating particles. The correlation function is normalized to unity such that C(q)→1 in the absence of a correlation signal. A. Theoretical description of K+K−correlation function This analysis studies femtoscopic correlations of particles produced in Pb-Pb collisions using the two-particle correlation function. The measured K+K−CF was fitted with a theoretical correlation function calculated within the Lednický-Lyuboshitz approach [1,43–45]. The K+and K− particles were assumed to be correlated in the final state due to the Coulomb interaction, to strong interactions through the near-threshold resonances a0(980) and f0(980), and to the pwave strong interaction through the φ(1020) meson resonance [45]. In the calculations, the correlations are conveniently expressed as a function of the single particle momentum in the pair rest frame (PRF), k∗=| k∗|. Note that, in the case of pairs of particles with equal masses, k∗is related to the momentum difference qas k∗=q/2. For particle production occurring at a small enough phase-space density, the correlations of two particles emitted with small k∗are dominated by the effects of their mutual final-state interaction and, if particles under consideration are identical, by quantum statistics. These correlations depend on the PRF temporal (t∗) and spatial 0.3 0.35 0.4 0.45 0.5 0.55 )c (GeV/p 0.96 0.98 1 purity ± K 10%− 0 30%−10 50%−30 = 2.76 TeV NN sPb −ALICE Pb 0.4 0.6 0.8 1 1.2 )c (GeV/ T k 0.96 0.98 1 pair purity - K + K = 2.76 TeV NN sPb −ALICE Pb 10%− 0 30%−10 50%−30 FIG. 1. Left: Single-kaon purity as a function of particle momentum pfor 0–10%, 10–30%, and 30–50% centrality intervals. Points are shifted along the xaxis for clarity. Right: K+K−pair purity as a function of pair transverse momentum kT. Systematic uncertainties are shown by bars. Statistical uncertainties are smaller than the size of the markers. 054904-3 S. ACHARYA et al. PHYSICAL REVIEW C 107, 054904 (2023) ( r∗) separation of the particle emission points. Usually, one can neglect the temporal separation and in such equal-time approximation [1,46] these effects are described by properly symmetrized wave function . Assuming sufficiently smooth behavior of single-particle spectra in a narrow correlation region of small k∗(smoothness assumption) [47], one can write the K+K−correlation function at a given  k∗and the total pair three-momentum P=| P|as CFSI( k∗, P)=d r∗ α Sα P( r∗, k∗)αα − k∗( r∗)2,(1) where the sum is done over the two intermediate channels α=K+K−and β=K0K0, denoted by the index α.Itis implied that particles are produced in a complex process with equilibrated spin and isospin projections. The separation distribution (source function) Sα P( r∗, k∗) is then independent of these projections so that its channel index αcan be omitted. Assuming possible position-momentum correlations at particle freeze-out, the source function can be parametrized as [45] SP( r∗, k∗)=exp −r∗2 4R2+b r∗ k∗/[8π3/2R3exp(b2k∗2R2)], (2) where Ris the Gaussian source radius, and bis a  r∗- k∗ correlation parameter. Typically, b≈0.25 [45], the  r∗- k∗correlation in the low k∗region (k∗<1/R) can be neglected, and Eq. (2) reduces to the usual spherically symmetric Gaussian parametrization. Outside the range of the strong interaction potential and at a sufficiently small k∗, one may account only for the s-wave strong interaction and write [44,46] αα − k∗( r∗)=N(η)e−i k∗ r∗F(−iη, 1,iξ)+fαα c(k∗) G(ρ,η) r∗, (3) βα − k∗( r∗)=N(η)fβα c(k∗)μβ μα eik∗ βr∗ r∗,(4) where N(η)=eiδc(η)√Ac(η), η=(k∗a)−1,ais the twoparticle Bohr radius including the sign of the interaction (for K+K−a=−109.6fm),ρ=k∗r∗,μα=mK+/2, k∗= k∗ α, and μβ=mK0/2, k∗ βare the respective reduced masses, and K+and K0momenta in PRF, δc=arg (1 +iη)isthe Coulomb s-wave phase shift, Ac(η)=2πη[e2πη −1]−1is the Coulomb penetration (Gamow) factor, Fis the confluent hypergeometric function, and  Gis a combination of the regular and singular s-wave Coulomb functions [48]. The s-wave scattering amplitudes fαα cdue to the short-range interaction renormalized by the long-range Coulomb forces are the elements of a 2 ×2matrix[44,46,49] ˆ fc=(ˆ K−1−iˆ kc)−1.(5) Here, ˆ Kis a symmetric matrix and ˆ kcis a diagonal matrix in the channel representation kαα c=Ac(η)k∗−2ih(η)/a,kββ c= k∗ 2with k∗ 2being the kaon momentum in PRF of the inelastic channel, where the function h(η) is expressed through the digamma function ψas h(η)=[ψ(iη)−ψ(−iη)+ln η2]/2. TABLE II. a0(980) and f0(980) square masses (in GeV2/c4)and coupling parameters (in GeV). Model m2 f0m2 a0γf0→K+K−γf0→ππ γa0→K+K−γa0→πη Martin [51] 0.9565 0.9487 0.792 0.199 0.333 0.222 Antonelli [52] 0.9467 0.9698 2.763 0.5283 0.4038 0.3711 Achasov1 [53] 0.9920 0.9841 1.305 0.2684 0.5555 0.4401 Achasov2 [54] 0.9920 1.0060 1.305 0.2684 0.8365 0.4580 The elements of the ˆ K−1matrix in the channel flavor representation α=K+K−,β=K0K0are expressed through elements K−1 Iof the diagonal matrix ˆ K−1in the representation of the channel isospin I=0,1as[44] (ˆ K−1)αα =(ˆ K−1)ββ =1 2K−1 0+K−1 1,(6) (ˆ K−1)βα =(ˆ K−1)αβ =−1 2K−1 0−K−1 1.(7) The latter are assumed to be dominated by the isoscalar f0(980) and isovector a0(980) resonances, so K0(k∗)=γf0→KK m2 f0−s−iγf0→ππkππ ,(8) K1(k∗)=γa0→KK m2 a0−s−iγa0→πηkπη ,(9) where s=4(m2 K+k∗2), ma0and mf0are the masses of the a0(980) and f0(980) resonances, respectively (see Table II), γf0→KK ,γf0→ππ and γa0→KK ,γa0→πη are the respective couplings, and kπη,kππ are the decay pion momenta in the respective channels. To take into account the deviation of the spherical waves from the true scattered waves in the inner region of the shortrange potential, a correction CKK should be applied (see Eq. (153) in [46]) CKK =−2πSP(0,k∗)Ac(η)fαα c2dαα 0 +fβα c2dββ 0+2Rfαα cfβα∗ cdβα 0,(10) where dαα 0=2Rd( ˆ K−1)αα/dk∗2. An additional contribution to the K+K−correlation function due to the p-wave strong interaction through the φ(1020) meson resonance was also taken into consideration. The usual femtoscopic correlation formalism of Eq. (1)usingthe smoothness assumption at small k∗should be modified at the φ→K+K−decay momentum k0=127 MeV/cto account for substantial  r− kcorrelations quantified by the parameter b≈0.25 in Eq. (2). As a result, the φ(1020) contribution to the correlation function is exponentially suppressed by the normalization factor in Eq. (2) and, neglecting a small Coulomb correction, becomes [45] CFSI φ=SP(0,k∗)6π k∗Rdfαα φ dk∗+ α 2Ik∗ αfαα∗ φ dfαα φ dk∗, (11) 054904-4 INVESTIGATION OF K+K−INTERACTIONS VIA … PHYSICAL REVIEW C 107, 054904 (2023) 0 .95 1 1.05 1.1 1.15 c<0.4 GeV/ T k0.3< 1 1.1 1.2 ALICE =2.76 TeV NN sPb − Pb − K + K 1 1.2 1.4 c<0.6 GeV/ T k0.5< Data Fit c<1.0 GeV/ T k0.8< 10%− 0 30%−10 50%−30 0 0.1 0.2 0.3 0.4 0 0.1 0.2 0.3 0.4 0 0.1 0.2 0.3 0.4 )c (GeV/q )q(C FIG. 2. The K+K−experimental correlation functions corrected for nonflat baselines according to Eq. (14) as a function of pair relative momentum q. The CFs are presented in three centrality classes (rows): 0–10%, 10–30%, and 30–50% and three pair transverse momentum kTbins (columns): (0.3–0.4), (0.5–0.6), and (0.8–1.0) GeV/c. Statistical (bars) and systematic (boxes) uncertainties are shown. The red line shows the fit of the CF with the Lednický-Lyuboshitz parametrization [Eq. (13)] using free parameters (mass and couplings) for f0(980) and Achasov [54] parameters for a0(980) in the 0 <q<0.5 GeV/crange. The dashed-dotted lines correspond to the baseline from Eq. (14). where fαα φ=±[(α/k∗ α)(α/k∗)]1/2mφ/(m2 φ−s−imφ), =αα+is the total φ(1020) width, =0.168is the partial width of the φ(1020) decays to the channels other than the KKones, and α∼k∗3 α. The sign ±corresponds to α=αand β, respectively. The expression for fαα φfollows from Eqs. (5)–(7) with the substitution ˆ K−1→ˆ kLˆ K−1ˆ kLin Eqs. (6) and (7) due to nonzero orbital angular momentum L=1. Here, ˆ kis a diagonal matrix in the channel flavor representation, kαα =k∗,kββ =k∗ 2; in the channel isospin representation, k00 =k11 =(kαα +kββ)/2, k01 =k10 =(kαα − kββ)/2. The matrix ˆ kˆ K−1ˆ kis diagonal in the channel isospin representation (ˆ k−1ˆ Kˆ k−1)00 =γφ→KK/[m2 φ−s−imφ] and (ˆ k−1ˆ Kˆ k−1)11 →0. Equation (11) has the same structure as the s-wave correction in the inner region [43] and can thus be substituted by Eq. (10) multiplied by the p-wave factor 2L+1=3. By neglecting the difference between the K+K− and K0K0channel momenta, the φ(1020) contribution can be rewritten as CFSI φ . =12πSP(0,k∗)|fαα|2 (μαα/k∗) . =6πSP(0,k∗)(−) μαk0k2 0−k∗2/μα−i2. (12) Here, the usage of the nonrelativistic Breit-Wigner expression in the last equality is motivated by the narrow φ(1020) width allowing one to neglect the momentum dependence of α. The total correlation function is determined as a sum of the s-wave term described in Eqs. (1)–(10) and the p-wave term FSI described by Eq. (11) and the direct φ(1020) meson production C(p1,p2)=1+λCFSI a0,f0(p1,p2)+Cφ(p1,p2), Cφ(p1,p2)=adirectCdirect φ(p1,p2)+aFSICFSI φ(p1,p2),(13) where Cdirect φcan be described by a nonrelativistic Breit- Wigner function [50], CFSI φis calculated from Eq. (11), and adirect and aFSI are coefficients that determine the ratio of φ(1020) mesons produced directly and due to FSI, respectively. Within the experimental accuracy, CFSI φcan be described by a nonrelativistic Breit-Wigner function. Therefore, this function was used to fit Cφ(where Cφis the nonrelativistic Breit-Wigner function) instead of the two separate terms adirectCdirect φand aFSICFSI φ. The ratio of the contributions of φ(1020) meson production from the direct mechanism adirect and from the coalescence mechanism aFSI is discussed in Sec. IV B. B. The K+K−correlation function and fitting procedure The correlation functions were measured in eight pair transverse momentum kTintervals: (0.2–0.3), (0.3–0.4), (0.4– 0.5), (0.5–0.6), (0.6–0.7), (0.7–0.8), (0.8–1.0), and (1.0–1.3) GeV/cand three centrality classes: 0–10%, 10–30%, 30– 50%. As an example, the K+K−correlation functions for three of these eight kTbins in different centrality classes are shown in Fig. 2. The measured correlation functions were normalized to unity in the region of 0.35 <q<0.5GeV/cand corrected for momentum resolution as described in Sec. III C. In this figure, one can see the main features of the K+K− 054904-5 S. ACHARYA et al. PHYSICAL REVIEW C 107, 054904 (2023) femtoscopic correlation function: Coulomb attraction at very small q(q<0.05 GeV/c) resulting in C(q)>1, suppression (C(q)<1) due to strong final-state interactions via the formation of the near-threshold resonances a0(980) and f0(980) in 0.05 <q<0.2GeV/c, and the narrow φ(1020) resonance peak at qaround 0.25 GeV/c. The measured correlation function was corrected for the nonflat baseline Dbefore the fit. The baseline is fitted in awideqrange (0.35 <q<1.0GeV/c) using a first-order polynomial function D(q)=κ(1 +aq),(14) where κis a normalization factor, and ais a free parameter of the fit. The observed nonflat baseline effect is almost negligible for the most central collisions and at low kT, while it becomes significant at low multiplicities and high transverse momenta. The nonflat baseline could be associated with the manifestation of minijets in peripheral collisions. A similar effect was observed for K±K±in Pb-Pb collisions at √sNN = 2.76 TeV [6]. The changing trend of the slope in different kTintervals could be reproduced qualitatively by HIJING simulations but the magnitude of the slope in the simulation was different than the one in the data in all considered kT intervals. Therefore, the MC was not used to estimate the nonflat baseline effect in this analysis. The fit of the calculated correlation function to the measured distributions allows one to constrain the masses and coupling parameters of the a0(980) and f0(980) resonances. The a0(980) parameters were fixed using the Achasov model [54]intheK0 SK±femtoscopic correlation analysis of Pb- Pb collisions at √sNN =2.76 TeV and of pp collisions at √s=7TeV[17,18]. Therefore, only the parameters of the f0(980) resonance were studied in this work. Three possible sets of values of f0(980) parameters proposed by theoretical models (Marin [51], Antonelli [52], and Achasov [53,54], see Table II) were considered. The source radii for K+K−pairs obtained by using all these theoretical models are inconsistent with the source radii parameters obtained in the identical sign K±K±analysis [6], while there are no physical reasons for this difference. Thus, in this study the parameters of the f0(980) resonance have been estimated by treating them as free parameters in the fits and constraining the source radii of K+K−pairs to be consistent with those obtained from the analysis of identical sign K±K±correlations. The parameter λfor K+K−does not necessarily have to be equal to the parameter λfor K±K±. In the case of identical charged kaon correlations, the λparameter decreases with increasing kT, which can be attributed to a non-Gaussian shape of the source [6]. Instead, the K+K−correlation function is determined by the contribution of the Coulomb and strong FSI, which are not very sensitive to a possible non-Gaussian shape of the source. The fits of the K+K−experimental correlation function for three different kTand centrality intervals with the Lednický-Lyuboshitz parametrization using free parameters (mass and couplings) for f0(980) and Achasov [54] parameters for a0(980) are shown in Fig. 2. Systematic uncertainties of CF were estimated using correlation functions obtained with different magnetic field orientations in the detector. As seen from Fig. 2,the f0(980) FSI parametrization obtained TABLE III. Summary of relative systematic uncertainties on R and λparameters. The symbol ‘-’ means that the contribution from the given source is negligible. The ranges reported for each specific source and for the total uncertainty reflect the fact that the uncertainty values depend on centrality and kTintervals. Only systematic uncertainties whose statistical significance level exceeds 68% according to the Barlow criterion were considered. Sources of systematic uncertainty R(%) λ(%) Single particle selection 0–10 0–12 Purity – 0–0.5 Baseline fit range 0–4 0–3 Momentum resolution 3–9 4–25 Total (quad. sum) 3–14 4–28 in this analysis gives an excellent description of the data in the interval 0 <q<0.35 GeV/c, where its contribution is relevant. The corresponding χ2/ndf are in the range from 1 to 2. C. Finite momentum resolution Finite track momentum resolution causes the reconstructed relative momentum qrec of a pair to differ from the true value qtrue. This is accounted for through the use of the response matrix M(qtrue,qrec) generated with HIJING simulations of Pb-Pb collisions at √sNN =2.76 TeV. To account for this effect, the theoretical CF can be smeared through the response matrix according to [55] C(qrec)=qtrue C(qtrue)M(qtrue,qrec) qtrue M(qtrue,qrec).(15) This smearing was applied directly in the fit of the measured CF to the theoretical one. The momentum resolution effect was included in the theoretical CF used to fit the data. In general, the momentum resolution reduces the height of the correlation function peak and makes it wider. D. Systematic uncertainties Possible sources and estimated values of systematic uncertainties of the source radius Rand correlation strength λ parameters are presented in this section. Systematic uncertainties of the mass and coupling parameter of the f0(980) meson are also discussed. The total systematic uncertainty sys = ii sys2(16) was taken as the square-root of the quadratic sum of all systematic contributions i sys =|y0−yi var|from the fit and the selection criteria (Table III). Here, y0refers to the value obtained with the default criteria (central value), yi var is a value with some variation iof particle selections and fit criteria. The Barlow factor [56] was also considered to estimate a statistical 054904-6 INVESTIGATION OF K+K−INTERACTIONS VIA … PHYSICAL REVIEW C 107, 054904 (2023) 0.2 0.4 0.6 0.8 1 1.2 )c (GeV/ T k 4 6 8 (fm) R ALICE = 2.76 TeV NN sPb −Pb − K + 10% K− 0 30% −10 50% −30 ± K ± 10% K− 0 30% −10 50% −30 0.2 0.4 0.6 0.8 1 1.2 (GeV/c) T k 0.5 1 1.5 λ ALICE = 2.76 TeV NN sPb −Pb − K + 10% K− 0 30% −10 50% −30 ± K ± 10% K− 0 30% −10 50% −30 FIG. 3. R(left panel) and λ(right panel) parameters as a function of pair transverse momentum kTextracted in K+K−analysis with free parameters (mass and couplings) for f0(980) and Achasov [54] parameters for a0(980). The parameters are compared to those obtained for identical charged kaons [6]. Statistical (bars) and systematic (boxes) uncertainties are shown. significance level of each deviation as B=y0−yi var σ2 0+σ2 var −2ρσ0σvar ,(17) where σ0indicates the statistical uncertainty of the central value, σvar is the statistical uncertainty of yi var,ρcharacterizes correlation between y0and yi var. The variation iis included in the systematic uncertainty evaluation if Bis larger than unity. The effect of the track selections was investigated by varying the criteria shown in Table I. The DCA and PID selection values were varied by ±10%. These variations resulted up to a 10–12% contribution to the systematic uncertainties for the R and λparameters (see Table III). The residual contamination from other particle species in the K+K−pair signal was found to have a minimal effect on the extracted parameters because of the high purity (see Fig. 1) of selected kaons, which was better than 99% for a pair of kaons. The systematic effect due to the choice of the baseline fit range was estimated by varying the qinterval in which the fit is performed. The standard fit range of the baseline was 0.35 <q<1.0GeV/c, and the upper limit of qwas changed to 0.7 and 1.3 GeV/c. The estimated uncertainty depends on the centrality and the kTinterval, and its maximum and minimum values are reported in Table III. Changing the range of the fit slightly changes the femtoscopic parameters extracted from the correlation function. However, after correcting the correlation function for the nonflat baseline, the influence of the fit range variation was found to be negligible. The effect of finite momentum resolution on femtoscopic radii and λparameters and the related systematic uncertainty were studied by modifying the width of momentum resolution distribution in the response matrix described in Sec. III C. The width of the qtrue vs qrec distribution determined by M(qtrue,qrec) was varied maximally (±10%) to account for its influence on the CF without distorting its shape. The resulting systematic uncertainties are reported in Table III. The systematic uncertainties of the γf0→KK and γf0→ππ couplings are determined by their possible maximal deviation from the default values providing the best CF fit under condition of having close radii for K+K−to those obtained for pairs of identical kaons, and a constraint on the γf0→KK /γf0→ππ ratio to be in accordance with the predictions of the models [51–54]. IV. RESULTS AND DISCUSSION A. Source size and correlation strength In the CF fit, the Rparameters for K+K−correlations were constrained to be compatible with the corresponding parameters for the same sign K±K±pairs within statistical uncertainties since there are no physical reasons for them to be different as was discussed above. Figure 3shows the obtained radii R(left panel) and correlation strengths λ(right panel) as a function of kTfor the 0–10%, 10–30%, and 30–50% centrality intervals. The parameters extracted from K+K− correlations are compared to those obtained for K±K±pairs. The Rparameters from K+K−correlations shown in Fig. 3 (left panel) are by construction consistent with those from identical kaon pairs, due to the constraint applied in the fit. This constraint was implemented by minimizing χ2/Nthat was calculated for each Rvalue according to χ2 R/N= N  i=1 [Ri(K±K±)−Ri(K+K−)]2 σ2 i ,(18) where iin Riruns over eight kTvalues for each centrality bin (N=8) and σi=√σ2 K±K±+σ2 K+K−is the statistical uncertainty of the difference between the extracted radius parameters for K+K−and K±K±correlations. The obtained χ2 R/Nvalues are 1.5 for 0–10%, 0.5 for 10–30%, and 1.1 for 30–50% centralities. 054904-7 S. ACHARYA et al. PHYSICAL REVIEW C 107, 054904 (2023) The correlation strength parameters for K+K−pairs tend to be slightly larger than those for K±K±for kT>0.7GeV/c. In particular, for K+K−pairs the λparameter does not show the decreasing trend with increasing kTthat was found for identical charge K±K±correlations. This difference could be due to the fact that the decreasing trend in the identical sign result is due to a non-Gaussian shape of the source and to the fact that the K+K−pairs are less sensitive to it. The values of λare about 0.7, i.e., lower than the ideal value of unity. This can be due to the contribution of kaons from K∗decays and from other long-lived resonances distorting the spatial kaon source distribution with respect to an ideal Gaussian, which is assumed in the fit function [6]. Mass and coupling parameters for the f0(980) meson were extracted in this analysis using Eq. (8) with the constraint on the K+K−radii to be close to the corresponding K±K± radii as was explained above. The resulting mass and coupling parameters with statistical and systematic uncertainties are mf0=967 ±3±7MeV/c2,γf0→KK =0.34 ±0.07 ± 0.10 GeV, γf0→ππ =0.089 ±0.018 ±0.026 GeV. The obtained f0(980) mass is consistent within uncertainties with its PDG value (mf0=990 ±20 MeV/c2)[57]. The ratio of kaon to pion couplings is equal to γf0→KK /γf0→ππ =3.82 ±1.07 and is consistent with those shown in Table II, which are in the range from 4 to 5 for all models. The full width of the f0(980) meson estimated in this work is f0=43.81 ± 8.76 ±6.90 MeV/c2and is also consistent with the PDG value (f0=10−100 MeV/c2). B. The φ(1020) meson peak height versus radius Figure 2shows that the height of the peak related to φ(1020) meson decays depends on both kTand centrality. Since it is also known that the source radius changes with kT and centrality, it is interesting to investigate how the height of the peak changes with the radius. The height of the φ(1020) meson peak can be measured from the correlation function minus unity at a relative momentum qthat corresponds to the φ(1020) meson mass as Cφ=Cq=m2 φ−4m2 K=2k0−1.(19) The Cφvalue as a function of the K+K−radius is presented in Fig. 4. It should be noted that it was corrected for both the λ parameter magnitude and the momentum resolution. It can be assumed that the mechanism of the φ(1020) meson production consists of at least two processes. The first is the direct production of φ(1020) mesons at the time of hadronization of the system. The second is the regeneration via the K+K− FSI in the subsequent hadronic phase leading to resonance formation [45]. To estimate the contribution from directly produced φ(1020) mesons [adirect in Eq. (13)], the following reasoning can be applied. The correlation function is defined as a ratio of the signal to the background. According to statistical models of hadron production [58], the φ(1020) meson yield (signal) is proportional to the volume of the φ(1020) meson production region. The combinatorial background is proportional to the square of the multiplicity, which in its turn is proportional 3 3.5 4 4.5 5 5.5 6 (fm)R 1− 10 1 )R( φ C φ Direct φ Direct and FSI − K + K 10% − 0 30% −10 50% −30 = 2.76 TeV NN sPb −ALICE Pb FIG. 4. Height of φ(1020) meson peak (Cφ) as a function of source radius R, for three centrality classes. Statistical uncertainties are shown by bars. Systematic uncertainties are smaller than the size of the markers. Blue solid line corresponds to the fit of CF with Cdirect φ=const/R3. Red dashed line corresponds to the fit with the second line in Eq. (13). to the source volume. Therefore, the height of the φ(1020) meson peak in the K+K−correlation function is expected to rapidly decrease with increasing Rin qualitative accordance with an inverse volume dependence, i.e., as 1/R3. Based on these arguments, it is possible to fit the height of the φ(1020) meson peak in the K+K−correlation function with Cdirect φ= const/R3asshowninFig.4. This fit function describes the data very well, indicating that the FSI contribution [aFSI in Eq. (13)] to the production of φ(1020) mesons is quite small. The influence of FSI can be estimated from simple model considerations. The interaction of kaons in the final state depends on the separation of their production points, which is affected by collective flow and resonance decays leading to the  r- kcorrelation characterized by the parameter b[45]. The K+K−FSI contribution could be approximately described by Eq. (11) and Eq. (4) in [45] CFSI φ=const ×exp−b2k2 0R2 R3.(20) Assuming that the direct production of φ(1020) mesons is described by the inverse volume dependence 1/R3while the FSI production is described by Eq. (20), the K+K−correlation function can be fit with the second line in Eq. (13). The parameter bis fixed to the value about 0.25 obtained from the blast-wave model estimation [45]. The results of the calculation are presented in Fig. 4. The height of the φ(1020) meson peak in the K+K−correlation function is fit with Cdirect φ=const/R3(blue curve in Fig. 4). The constant is equal to 15.08 ±0.44. The value of χ2/ndf =13.45/23 = 0.58. The results of the fit of the data to the second line in Eq. (13) are also shown in Fig. 4. The red curve corresponds to the fit with parameters of the fit adirect =0.75 ±0.16, aFSI = 0.25 ±0.16, and χ2/ndf =11.07/21 =0.53. The resulting fraction of directly produced φ(1020) mesons adirectCdirect φ/Cφ varies from 0.7 to 0.8 with increasing Rand is within the range expected from the integrated hydrokinetic model [59]. 054904-8 INVESTIGATION OF K+K−INTERACTIONS VIA … PHYSICAL REVIEW C 107, 054904 (2023) 41Gauhati University, Department of Physics, Guwahati, India 42Helmholtz-Institut für Strahlen- und Kernphysik, Rheinische Friedrich-Wilhelms-Universität Bonn, Bonn, Germany 43Helsinki Institute of Physics (HIP), Helsinki, Finland 44High Energy Physics Group, Universidad Autónoma de Puebla, Puebla, Mexico 45Horia Hulubei National Institute of Physics and Nuclear Engineering, Bucharest, Romania 46Indian Institute of Technology Bombay (IIT), Mumbai, India 47Indian Institute of Technology Indore, Indore, India 48INFN, Laboratori Nazionali di Frascati, Frascati, Italy 49INFN, Sezione di Bari, Bari, Italy 50INFN, Sezione di Bologna, Bologna, Italy 51INFN, Sezione di Cagliari, Cagliari, Italy 52INFN, Sezione di Catania, Catania, Italy 53INFN, Sezione di Padova, Padova, Italy 54INFN, Sezione di Pavia, Pavia, Italy 55INFN, Sezione di Torino, Turin, Italy 56INFN, Sezione di Trieste, Trieste, Italy 57Inha University, Incheon, Republic of Korea 58Institute for Gravitational and Subatomic Physics (GRASP), Utrecht University/Nikhef, Utrecht, Netherlands 59Institute of Experimental Physics, Slovak Academy of Sciences, Košice, Slovak Republic 60Institute of Physics, Homi Bhabha National Institute, Bhubaneswar, India 61Institute of Physics of the Czech Academy of Sciences, Prague, Czech Republic 62Institute of Space Science (ISS), Bucharest, Romania 63Institut für Kernphysik, Johann Wolfgang Goethe-Universität Frankfurt, Frankfurt, Germany 64Instituto de Ciencias Nucleares, Universidad Nacional Autónoma de México, Mexico City, Mexico 65Instituto de Física, Universidade Federal do Rio Grande do Sul (UFRGS), Porto Alegre, Brazil 66Instituto de Física, Universidad Nacional Autónoma de México, Mexico City, Mexico 67iThemba LABS, National Research Foundation, Somerset West, South Africa 68Jeonbuk National University, Jeonju, Republic of Korea 69Johann-Wolfgang-Goethe Universität Frankfurt Institut für Informatik, Fachbereich Informatik und Mathematik, Frankfurt, Germany 70Korea Institute of Science and Technology Information, Daejeon, Republic of Korea 71KTO Karatay University, Konya, Turkey 72Laboratoire de Physique des 2 Infinis, Irène Joliot-Curie, Orsay, France 73Laboratoire de Physique Subatomique et de Cosmologie, Université Grenoble-Alpes, CNRS-IN2P3, Grenoble, France 74Lawrence Berkeley National Laboratory, Berkeley, California, USA 75Lund University Department of Physics, Division of Particle Physics, Lund, Sweden 76Nagasaki Institute of Applied Science, Nagasaki, Japan 77Nara Women’s University (NWU), Nara, Japan 78National and Kapodistrian University of Athens, School of Science, Department of Physics, Athens, Greece 79National Centre for Nuclear Research, Warsaw, Poland 80National Institute of Science Education and Research, Homi Bhabha National Institute, Jatni, India 81National Nuclear Research Center, Baku, Azerbaijan 82National Research and Innovation Agency - BRIN, Jakarta, Indonesia 83Niels Bohr Institute, University of Copenhagen, Copenhagen, Denmark 84Nikhef, National institute for subatomic physics, Amsterdam, Netherlands 85Nuclear Physics Group, STFC Daresbury Laboratory, Daresbury, United Kingdom 86Nuclear Physics Institute of the Czech Academy of Sciences, Husinec- ˇ Rež, Czech Republic 87Oak Ridge National Laboratory, Oak Ridge, Tennessee, USA 88Ohio State University, Columbus, Ohio, USA 89Physics department, Faculty of science, University of Zagreb, Zagreb, Croatia 90Physics Department, Panjab University, Chandigarh, India 91Physics Department, University of Jammu, Jammu, India 92Physics Program and International Institute for Sustainability with Knotted Chiral Meta Matter (SKCM2), Hiroshima University, Hiroshima, Japan 93Physikalisches Institut, Eberhard-Karls-Universität Tübingen, Tübingen, Germany 94Physikalisches Institut, Ruprecht-Karls-Universität Heidelberg, Heidelberg, Germany 95Physik Department, Technische Universität München, Munich, Germany 96Politecnico di Bari and Sezione INFN, Bari, Italy 97Research Division and ExtreMe Matter Institute EMMI, GSI Helmholtzzentrum für Schwerionenforschung GmbH, Darmstadt, Germany 98Saga University, Saga, Japan 054904-15 S. ACHARYA et al. PHYSICAL REVIEW C 107, 054904 (2023) 99Saha Institute of Nuclear Physics, Homi Bhabha National Institute, Kolkata, India 100School of Physics and Astronomy, University of Birmingham, Birmingham, United Kingdom 101Sección Física, Departamento de Ciencias, Pontificia Universidad Católica del Perú, Lima, Peru 102Stefan Meyer Institut für Subatomare Physik (SMI), Vienna, Austria 103SUBATECH, IMT Atlantique, Nantes Université, CNRS-IN2P3, Nantes, France 104Sungkyunkwan University, Suwon City, Republic of Korea 105Suranaree University of Technology, Nakhon Ratchasima, Thailand 106Technical University of Košice, Košice, Slovak Republic 107The Henryk Niewodniczanski Institute of Nuclear Physics, Polish Academy of Sciences, Cracow, Poland 108The University of Texas at Austin, Austin, Texas, USA 109Universidad Autónoma de Sinaloa, Culiacán, Mexico 110Universidade de São Paulo (USP), São Paulo, Brazil 111Universidade Estadual de Campinas (UNICAMP), Campinas, Brazil 112Universidade Federal do ABC, Santo Andre, Brazil 113University of Cape Town, Cape Town, South Africa 114University of Houston, Houston, Texas, USA 115University of Jyväskylä, Jyväskylä, Finland 116University of Kansas, Lawrence, Kansas, USA 117University of Liverpool, Liverpool, United Kingdom 118University of Science and Technology of China, Hefei, China 119University of South-Eastern Norway, Kongsberg, Norway 120University of Tennessee, Knoxville, Tennessee, USA 121University of the Witwatersrand, Johannesburg, South Africa 122University of Tokyo, Tokyo, Japan 123University of Tsukuba, Tsukuba, Japan 124University Politehnica of Bucharest, Bucharest, Romania 125Université Clermont Auvergne, CNRS/IN2P3, LPC, Clermont-Ferrand, France 126Université de Lyon, CNRS/IN2P3, Institut de Physique des 2 Infinis de Lyon, Lyon, France 127Université de Strasbourg, CNRS, IPHC UMR 7178, F-67000 Strasbourg, France, Strasbourg, France 128Université Paris-Saclay Centre d’Etudes de Saclay (CEA), IRFU, Départment de Physique Nucléaire (DPhN), Saclay, France 129Università degli Studi di Foggia, Foggia, Italy 130Università del Piemonte Orientale, Vercelli, Italy 131Università di Brescia, Brescia, Italy 132Variable Energy Cyclotron Centre, Homi Bhabha National Institute, Kolkata, India 133Warsaw University of Technology, Warsaw, Poland 134Wayne State University, Detroit, Michigan, USA 135Westfälische Wilhelms-Universität Münster, Institut für Kernphysik, Münster, Germany 136Wigner Research Centre for Physics, Budapest, Hungary 137Yale University, New Haven, Connecticut, USA 138Yonsei University, Seoul, Republic of Korea 139Zentrum für Technologie und Transfer (ZTT), Worms, Germany 140Affiliated with an institute covered by a cooperation agreement with CERN 141Affiliated with an international laboratory covered by a cooperation agreement with CERN †Also at: Max-Planck-Institut für Physik, Munich, Germany. ‡Also at: Italian National Agency for New Technologies, Energy and Sustainable Economic Development (ENEA), Bologna, Italy. §Also at: Dipartimento DET del Politecnico di Torino, Turin, Italy. Deceased. ¶Also at: An institution covered by a cooperation agreement with CERN. #Also at: Department of Applied Physics, Aligarh Muslim University, Aligarh, India. **Also at: Institute of Theoretical Physics, University of Wroclaw, Poland. 054904-16