Kaon-proton strong interaction at low relative momentum via femtoscopy in Pb-Pb collisions at the LHC

In quantum scattering processes between two particles, aspects characterizing the strong and Coulomb forces can be observed in kinematic distributions of the particle pairs. The sensitivity to the interaction potential reaches a maximum at low relative momentum and vanishing distance between the two particles. Ultrarelativistic heavy-ion collisions at the LHC provide an abundant source of many hadron species and can be employed as a measurement method of scattering parameters that is complementary to scattering experiments. This study confirms that momentum correlations of particles produced in Pb-Pb collisions at the LHC provide an accurate measurement of kaon-proton scattering parameters at low relative momentum, allowing precise access to the $ {K}^{-} p\rightarrow {K}^{-} p$ process. This work also validates the femtoscopic measurement in ultrarelativistic heavy-ion collisions as an alternative to scattering experiments and a complementary tool to the study of exotic atoms with comparable precision. In this work, the first femtoscopic measurement of momentum correlations of ${K}^{-} p\ ({K}^{+}\overline{p})$ and ${K}^{+}p ({K}^{-}\overline{p})$ pairs in Pb-Pb collisions at centre-of-mass energy per nucleon pair of $\sqrt{s_{\rm NN}} = 5.02$ TeV registered by the ALICE experiment is reported. The components of the ${K}^{-} p$ complex scattering length are extracted and found to be $\Re f_0=-0.91\pm~{0.03}$(stat)$^{+0.17}_{-0.03}$(syst) and $\Im f_0 = 0.92\pm~{0.05}$(stat)$^{+0.12}_{-0.33}$(syst). The results are compared with chiral effective field theory predictions as well as with existing data from dedicated scattering and exotic kaonic atom experiments.


Introduction
The study of kaon-nucleon (KN) and antikaon-nucleon (KN) interactions is essential to understand lowenergy quantum chromodynamics (QCD). The Goldstone boson nature of the kaon and its role as an active degree of freedom within effective field theories are used to describe processes within the chiral SU(3) dynamics at low energies. Since perturbation theory is not applicable in this energy regime, experimental data are essential to constrain the currently available effective theories [1]. In general, three experimental techniques can be used to access hadron-hadron interaction, such as KN, which will be discussed below. Their schematic illustration is shown in Fig. 1. A direct measurement of the KN interaction was first possible in scattering experiments with secondary K − beams impinging on hydrogen targets. Such experiments allow the precise control of both the initial (incoming) and final (outgoing) states, see Fig. 1 a. Furthermore, polarised beams and targets can be used [2]. However, scattering measurements can only be performed for pairs of charged and long-lived particles, because of the technical challenges of creating beams of other particle species, especially involving strangeness or antimatter [3]. Finally, attaining zero relative momentum for strange hadrons, such as kaons, is also challenging. Current K − p measurements can only reach a kaon momentum of approximately 100 MeV/c with respect to the proton at rest [4][5][6][7][8][9][10]. Therefore, the present accuracy of the K − p cross sections and branching ratios at threshold does not sufficiently constrain the scattering amplitudes.
Another way of studying the low-energy KN interaction is by observing the shifts and widths of energy levels in exotic kaonic atoms [11], where a negative kaon replaces an electron, see Fig. 1 b. In particular, kaonic hydrogen and kaonic deuterium can be formed in dedicated experimental setups and measurements of their X-ray deexcitation spectra provide a link to the K − p strong interaction at zero relative momentum [12]. When the negative hadron, such as K − , transits between the lowest energy states, its wave function overlaps with the nucleus and the presence of the strong force modifies the energy levels and their widths due to coupling to intermediate states before absorption [13,14]. The measurement of the energy shift and width of the 1S state of the kaonic hydrogen line can be connected to the K − p scattering length ( f 0 ) through the Trueman-Deser formula [15,16]. However, it requires model-dependent cor-rections to account for the presence of the isospin-breaking couplings and the Coulomb interaction [16]. The scattering length is a complex number related to both the elastic and inelastic cross sections of the interaction. The current benchmark result is from chiral effective field theory (χ EFT calculations) [17,18] anchored to the kaonic hydrogen data [19,20] collected by the SIDDHARTA experiment [21] at the DAΦNE electron-positron collider [22], and estimates the real part ℜ f 0 = −0.65 ± 0.10 fm 1 and the imaginary part ℑ f 0 = 0.81 ± 0.15 fm of the scattering length. Historically, the scattering parameters extracted from the kaonic hydrogen experiments pointed to an attractive interaction, while a repulsive K − p scattering length in free space was measured via scattering experiments. Currently, both results are explained simultaneously by the presence of the Λ(1405) resonance located just below the KN threshold [23]. Similar experiments have been performed with other hadrons, such as pions [14,24] and antiprotons [25], providing precise measurements of the interaction at the kinematic threshold.
On the other hand, the repulsive nature of the K + p interaction is well established. It is described by an s-wave [26] scattering length with a negative real part (ℜ f 0 = −0.308 ± 0.003) and the Coulomb interaction [27,28].
Scattering experiments are special in that both the initial and final states are fixed, which implies that only the K − p → K − p process can be accessed. This is not the case for ultrarelativistic particle collisions since only the K − p final state is fixed and different initial states are allowed, such as K 0 n and πΣ, which are referred to as coupled channels. Collisions of nuclei at the Large Hadron Collider (LHC) provide an abundant source of various particle species (including strange hadrons [29]), in both small (pp, p-Pb) and large (Pb-Pb) systems. In a single central Pb-Pb collision, the number of produced particles reaches thousands at midrapidity (|y| < 0.5) [30][31][32]. Moreover, since the baryon chemical potential for ultrarelativistic heavy-ion collisions at LHC energies approaches zero, a nearly equal amount of matter and antimatter are produced [33]. Therefore, they are an ideal environment for conducting interaction studies involving both strangeness and antimatter [34][35][36], as shown in Fig. 1 c. A recent breakthrough in the field was achieved by analysing the kaon-proton momentum correlations [37] using the technique of femtoscopy in high-energy pp collisions at the ALICE experiment [38], showing clear evidence for the presence of coupled channels in the measured correlation function. In particular, a clear signature for the opening at the threshold of the K 0 n isospin breaking channel is observed due to the mass difference between the pairs.
In order to reproduce the experimental data, theoretical models need to describe the Coulomb interaction, the coupled-channels effects, and the threshold energy difference among the isospin multiplets used in the calculation [39]. This can be done in a realistic KN-πΣ-πΛ coupled-channel potential which can be constructed starting from chiral SU(3) dynamics [18]. Alternatively, the KN interaction can also be characterized by the Lednický-Lyuboshitz analytical model where the s-wave scattering parameters are used to describe the measurements [40,41].
It has been recently predicted that the strength of the coupled channels is significantly reduced for source sizes larger than 3 fm, because channels other than K − p have a considerable magnitude only around a distance of 1 fm [39]. It is therefore expected that for larger sources, such as those created in heavy-ion collisions, their effect on the correlation function should be significantly reduced or become asymptotically small. Therefore, the analysis of K − p pairs in Pb-Pb collisions, as shown in this work for the first time, can help to disentangle the K − p → K − p process from contributions stemming from on-shell coupled channels at the reaction threshold.

Experimental setup and particle selection
The ALICE apparatus [38] is designed to measure the particles produced in proton and lead (Pb) collisions at the LHC [42]. The central barrel consists of a large solenoid magnet used to generate homogeneous magnetic fields up to 0.5 T along the beam directions and a set of detectors located in the interior surrounding the beam axis. Additional detectors are located outside the central barrel in the forward beam direction. A detailed description of the performance of the ALICE detector can be found in Ref. [43].
In this work, the minimum-bias Pb-Pb data at a collision energy per nucleon-nucleon pair of √ s NN = 5.02 TeV collected by ALICE in the LHC Run 2 (2015) are used. The recording of the collisions products was triggered using the V0 detector consisting of two arrays of 32 scintillator counters covering pseudorapidity (η) ranges of 2.8 < η < 5.1 (V0A) and −3.7 < η < −1.7 (V0C). A coincident signal in both detectors consistent with the collision occurring at the centre of the ALICE detector was required. The centrality of the collision, expressed in percentages of the total hadronic cross section, was determined using the amplitudes of the signals in the V0 detectors following the procedure described in Ref. [44] and classified into six intervals of the Pb-Pb cross section: 0-5%, 5-10%, 10-20%, 20-30%, 30-40%, and 40-50%. Those intervals together with the corresponding charged-particle multiplicity densities at midrapidity dN ch /dη are listed in Table 1. Only collisions within ±10 cm of the nominal interaction point along the beam axis are considered in this analysis in order to achieve uniform tracking and particle identification performance. For track reconstruction and hadron identification the analysis uses information provided by the Inner Tracking System (ITS) [38], the Time Projection Chamber (TPC) [38,46], and the Time-of-Flight detector (TOF) [47], located inside the solenoid magnet.
The ITS is composed of six cylindrical layers of silicon detectors and it is used to determine the locations of the primary vertex by extrapolation of primary tracks.
The TPC is a large (volume of 88 m 3 ) cylindrical detector filled with gas and is used as the main tracking and particle identification detector of ALICE. The central electrode divides the TPC into two halves. At the end of each half there is a readout plane composed of 18 sectors (with full azimuthal angle ϕ coverage). Each sector contains 159 padrows arranged radially. A track signal in the TPC consists of space points (referred to as clusters), each reconstructed in one of the padrows. A Kalman fit is performed on a set of clusters in order to obtain the parameters of a given track. In this analysis, the determination of momentum is performed using the track curvature from the TPC detector only.
TOF is a cylindrical detector composed of Multigap Resistive Plate Chambers (MRPC) located at r ∼ = 380 cm from the beam axis. They are also arranged into 18 azimuthal sectors. The detector measures the arrival of particles with a precision of the order of 50 ps.
Acceptance is restricted to |η| < 0.8. Selected kaons have transverse momenta of 0.19 < p T < 0.45 GeV/c, while for protons the interval is 0.5 < p T < 4.0 GeV/c. The distances of closest approach (DCA) of a Track selection proton track to the primary vertex in the transverse (DCA XY ) and longitudinal (DCA Z ) directions are required to be less than 0.0105 + 0.0350/(p T /(GeV/c) −1.1 ) cm and 3.2 cm, respectively. These selections are imposed to reduce the contamination from secondary tracks originating from weak decays and from interaction with the detector material. All selections are summarized in Table 2.
Particle identification is based on energy loss dE/dx in the TPC and time of flight in the TOF. The difference between the measured and expected signals can be expressed in units of detector resolution called N σ . For protons, the combined signal from TOF and TPC is allowed to differ from the expected one by N σ = 3 √ 2. For kaons, five selection criteria (depending on the momentum p) were introduced, as detailed in Table 2. The selection criteria are optimised to obtain a high-purity sample without compromising the efficiency. The identification purity of the data sample is estimated from detailed Monte Carlo (MC) simulations using the HIJING [48] event generator coupled to the GEANT3 [49] transport package and found to be above 98.5% for both the kaon and proton samples. The fraction of protons originating from weak decays is determined to be 5.6% and negligible for kaons (< 0.05%). An additional method based on template fits to the distributions of the DCA Z is used to determine the contribution from weak decays in the proton sample. The fraction of 7.9% obtained for secondary protons is averaged with the fraction from MC, resulting in a contamination of 6.5% of weak decays in the sample.
The identified tracks from each event are combined into pairs to form the distribution A(k * ). Two-particle detector acceptance effects, including track splitting, track merging, as well as effects coming from γ → e + e − conversion, contribute to the measured distributions. The following selections are applied to reduce these effects. For pairs of tracks within the relative pseudorapidity of the two tracks |∆η| < 0.01, an exclusion on the fraction of merged points is introduced. The merged fraction is defined as the ratio of the number of steps of 1 cm considered in the TPC radius range for which the distance between the tracks is less than 3 cm, to the total number of steps. Pairs with a merged fraction above 3% are removed. The e + e − pairs originating from photon conversions can be misidentified as real kaon-proton pairs and it is necessary to remove spurious correlations arising from such pairs. These pairs are removed if their invariant mass, assuming the electron mass for both particles, is less than 0.002 GeV/c 2 , and the relative polar angle, ∆θ , between the two tracks is less than 0.008 rad.

Data analysis
The femtoscopic measurements are based on the sensitivity of two-particle correlations at low relative momentum to the space-time separation of the particle emitters due to the effects of quantum statistics and/or final state interactions (FSI), i.e. Coulomb and strong forces, depending on the pair under consideration [50,51]. The femtoscopic correlation function, as shown in Fig. 1 c, in the pair rest frame (PRF) can be written as [52,53] where | k * | = k * is half the kaon-proton pair relative momentum, which is equal to the momentum of the first particle in the PRF, where the pair centre of mass is at rest ( p 1 = − p 2 ). The two-particle emitting source function is denoted as S( r * ), Ψ( k * , r * ) is the pair wave function, and r * is the relative separation vector. The pair wave function depends on the interactions between the two hadrons [40,41]. In the case of equal emission time in the PRF, the Bethe-Salpeter amplitude in the Lednický-Lyuboshitz formalism [40,41,54], described in detail in Appendix A.1, coincides with a stationary solution of the scattering problem with reversed time direction in the emission process. The interaction then depends only on the scattering length, f 0 , and the effective range of the interaction, d 0 . In this work, the spin-averaged scattering parameters are obtained, i.e. the real and imaginary parts, ℜ f 0 and ℑ f 0 , respectively, of the scattering length. Moreover, the zero-effective-range approximation (d 0 = 0) is used in the analysis.
The experimental correlation function, C(k * ) = N A(k * )/B(k * ), is constructed as a ratio between the measured distribution of pair relative momenta A(k * ) and the uncorrelated distribution B(k * ) obtained using the mixed-event technique [55,56], where N is a free normalization factor which is constrained by C(k * ) = 1 at 300 < k * < 800 MeV/c. Effects of the scattering length as well as of the size of the femtoscopic source are studied in the region of k * < 150 MeV/c. Deviations from unity of C(k * ) in this region indicate the presence of an interaction between the studied particles. In general, the attractive or repulsive forces manifest themselves in values of the correlation function greater or smaller than one, respectively [57].
The femtoscopic correlation functions for K − p and K + p pairs and their charge conjugates K + p and K − p are measured in six centrality intervals [44]. The centrality intervals and the corresponding chargedparticle multiplicity densities at midrapidity dN ch /dη are listed in Table 1. Since no deviations, apart from statistical ones, between pairs and corresponding charge conjugate pairs are observed, they are combined and denoted as K + p ⊕ K − p for same-and K − p ⊕ K + p for opposite-charge combinations, respectively.
In Pb-Pb collisions, particles are also correlated due to the collective expansion of the system. This generates a slope in the correlation function [58] which appears to be well described by both hydrodynamic simulations [59] coupled to the statistical hadronisation code THERMINATOR 2 [60] or using the AMPT model [61]. These background correlations are therefore subtracted before fitting the correlation functions by employing a first-order polynomial fit to the AMPT simulated data in the first two centrality intervals (0-10%) and a fit to the THERMINATOR 2 simulated data for the remaining centrality classes (10-50%). The fixed value of the impact parameter used for 0-10% centrality interval in THERMINA-TOR 2 was not reproducing the distribution of measured average charged-particle multiplicity.
The extraction of the scattering length parameters of K − p ⊕ K + p pairs from the measured correlation functions uses a dedicated fitting procedure based on the Lednický-Lyuboshitz formalism. The assumption for the source function S( r * ) is that, for each centrality, it is a three-dimensional spheroid with a Gaussian density profile in the PRF [56,62,63] with a common width parameter for sameand opposite-charge pairs R Kp , referred to as the source size or femtoscopic radius. This assumption is based on hydrodynamic models [64,65] of the matter produced in ultrarelativistic heavy-ion collisions which provide quantitative agreement [66] with collective phenomena [67], particle transversemomentum spectra [68], and femtoscopy of both identical [56,69] and non-identical particles [63]. The femtoscopic radii follow the previously observed linear scaling with dN ch /dη 1/3 and the transversemass scaling behaviour [69][70][71][72][73]. This allows one to fix the source functional form in Eq. (1) and use the Lednický-Lyuboshitz formalism for measurements of the interaction between any pair of particles that can be detected in the final state. Moreover, the fit is further constrained by using the measured scattering length values for K + p ⊕ K − p pairs, established from a partial wave analysis to be ℜ f 0 = −0.308 ± 0.003 fm [28]. Finally, there are six different femtoscopic radii corresponding to each centrality class and two, real and imaginary, components of the K − p ⊕ K + p scattering length. In a simultaneous fit to all measured correlation functions, the fitting procedure determines the six radii from the K + p correlations and the real and imaginary parameters of the strong interaction scattering length from the K − p correlations.
The fitting procedure also accounts for the purity of the sample, defined as the percentage of properly identified primary particle pairs originating from the three-dimensional Gaussian profile of the source, following the method described in Ref. [74]. This method was successfully employed in the pion-kaon femtoscopic analysis by ALICE [63]. The value for the purity parameter was estimated to be 87% and depends on the misidentification, secondary contamination from weak decays, and percentage of kaons and protons that come from strongly decaying resonances constituting the long tails in the source distribution, outside the Gaussian core.
The measured correlation function is also affected by the finite momentum resolution of the ALICE detector. The detector response matrix is obtained using a detailed Monte Carlo simulation of the detector based on HIJING [48] coupled to GEANT3 [49] transport code and is used to smear the theoretical calculations.

Systematic uncertainties
The final systematic uncertainties are the maximum and minimum deviations from the reference values obtained for different variations of the analysis. The relative values of the important sources of uncorrelated systematic uncertainty are listed in Table 3. One of the most significant sources of uncertainty comes from background estimation. The analysis considers two variations of the default option. The first one (Background 1) defines the background as a first-order polynomial fit to the raw data in the k * region 0.35-0.6 GeV/c instead of predictions from the AMPT model in the first two centrality intervals. The second one (Background 2) estimates the background from the average of the first-order polynomial fit to the raw data and the AMPT prediction in the first two centrality intervals and the average of functions provided by the THERMINATOR 2 code and the AMPT model in the other four centrality intervals. The pair purity value used for scaling data points is also varied. The default value equals 87%, the effect of possible additional sources of impurities is evaluated and found to be smaller than 7%.

Results
The correlation functions for opposite-charge particle pairs (K − p ⊕ K + p), after division by the AMPT or THERMINATOR 2 baselines, are shown in Fig. 2. The correlation functions for same-charge particle pairs (K + p ⊕ K − p) are presented as inserts. The cyan band represents the result of the simultaneous fit to all obtained correlation functions with the Lednický-Lyuboshitz model [40,41]. The orange band corresponds to predictions from χ EFT calculations. These are computed using the Kyoto model [39] and evaluated with the CATS framework [80]. The Kyoto model is based on a chiral effective SU(3) calculation in which all coupled channels, such as K 0 n and πΣ are included. At present, this model is able to describe both the ALICE results from proton-proton collisions [37] and the SIDDHARTA measurement [18]. The bottom panels of Fig. 2 show the difference between the data and the fit (model) divided by the statistical uncertainty of the data, σ stat .
The following effects can be observed: the K − p ⊕ K + p pairs show an attractive Coulomb interaction for small k * . The effect is opposite for K + p ⊕ K − p pairs. The influence of the repulsive strong interaction manifests as correlation functions reaching values below unity in the region of k * ≈ 20 − 50 MeV/c and becomes more pronounced towards more peripheral events, i.e., smaller source sizes. As predicted in Ref. [39], features of the correlation function related to the coupled channels, observed in the analysis of pp collisions [37], are negligible here. Neither the cusp structure at 58 MeV/c due to the presence of the isospin-breaking channel K 0 n → K − p nor the enhancement due to the coupled channels below threshold enhancing the correlation above unity in the intermediate k * range are visible in the correlation function in Pb-Pb.
The common femtoscopic radii R Kp for same-and opposite-charge pairs obtained from the Lednický- Lyuboshitz fit are provided in Fig. 2 as well. They increase from around 5 fm for peripheral events to almost 9 fm for central events, and all are larger than 3 fm where the predicted effect of coupled channels is reduced or negligible [39]. The radii scale linearly with the cube root of the mean charged-particle multiplicity density dN ch /dη 1/3 , as observed for pion-pion [73], kaon-kaon [56], and pion-kaon [63] pairs. The scattering length parameters obtained from the fit are ℜ f 0 = −0.91 ± 0.03(stat) +0. 17 −0.03 (syst) fm and ℑ f 0 = 0.92 ± 0.05(stat) +0. 12 −0.33 (syst) fm. The obtained parameters of the scattering length are compared with the available experimental values as well as model calculations [18,[75][76][77][78][79] in the left panel of Fig. 3. Numerical values of those parameters are also provided in Tab. 4. The ALICE results are compatible with them within uncertainties 2 . Up until this point, the world's best experimental data on Kp scattering are mainly from exotic kaonic atoms, where the interaction at the threshold is measured, and from scattering experiments. Theory predictions and calculations are based on χ EFT models.
Moreover, the Lednický-Lyuboshitz formalism is also used to compute femtoscopic correlation functions 2 Note that systematic uncertainties are not provided for some of the older results.  Figure 3: Left: scattering parameters obtained from the Lednický-Lyuboshitz fit compared with available world data and theoretical calculations. Statistical uncertainties are represented as bars and systematic uncertainties, if provided, as boxes. Right: experimental femtoscopic correlation function for K − p ⊕ K + p pairs in the 30-40% centrality interval, together with various Lednický-Lyuboshitz calculations obtained using the scattering length parameters from Refs. [17,18,[75][76][77][78][79] and the source radius from this analysis. The statistical and systematic uncertainties of the measured data points are added in quadrature and shown as vertical bars.
using scattering length parameters from previous measurements and theory predictions. They are then compared with the experimental data and the deviations in units of χ 2 /ndf are obtained. The result of such a procedure is shown in Fig. 3 (right), while the χ 2 /ndf values are presented in Table 4. The Kyoto model, which captures well the structures related to coupled channels in pp collisions, reproduces the data trends in all measured Pb-Pb centrality intervals, confirming that the coupled channels are fundamental in the description of small sources but have a negligible influence on correlation functions at large source sizes [39]. However, the model still requires further development as the resulting χ 2 /ndf = 2.8 is slightly worse than the best calculations using the Lednický-Lyuboshitz analytical approach.

Summary
In this work, we present the first measurement of non-identical particle femtoscopy of K − p ⊕ K + p and K + p ⊕ K − p pairs performed in Pb-Pb collisions at √ s NN = 5.02 TeV in six centrality intervals. Up to now, the existing experimental data on K − p scattering allowed us to study the interaction at a fixed range, either in the asymptotic regime, where the effect of the coupled channels is not present, or at very small ranges (R Kp ∼ 1 fm) where they are dominant and difficult to disentangle from the K − p → K − p channel. The analysis of the Pb-Pb data with the ALICE detector addresses these issues by testing the K − p interaction as a function of the source size in the range R Kp ∈ (4.9, 8.9) fm. In femtoscopy, a precise knowledge of the source from which hadrons are emitted is needed to measure the interaction using quantum scattering theory. In the case of pp collisions, where this is of the order of 1 fm, very short interaction distances are probed. In Pb-Pb collisions, the larger average distance of several fm between the hadrons probes the asymptotic form of the two-particle wave function, as demonstrated in this work. The obtained values of R Kp follow a linear scaling with the cube root of averaged chargedparticle multiplicity density and the transverse-mass scaling. This is in agreement with other femtoscopic measurements in heavy-ion collisions, as expected from the hydrodynamic models. The resulting correlation functions are fitted simultaneously to extract the complex scattering length whose components are

A.1 Lednický-Lyuboshitz formalism
The pair wave function in Eq. (1), Ψ( k * , r * ), depends on the two-particle interaction. Kaons interact with protons via the strong and, for charged kaons, also the Coulomb force. In such a scenario, the interaction