Exploring the N$\Lambda$-N$\Sigma$ coupled system with high precision correlation techniques at the LHC

The interaction of $\Lambda$ and $\Sigma$ hyperons (Y) with nucleons (N) is strongly influenced by the coupled-channel dynamics. Due to the small mass difference of the $\rm N \Lambda$ and $\rm N \Sigma$ systems, the sizeable coupling strength of the $\rm N \Sigma\leftrightarrow N \Lambda$ processes constitutes a crucial element in the determination of the N$\Lambda$ interaction. In this letter we present the most precise measurements on the interaction of p$\Lambda$ pairs, from zero relative momentum up to the opening of the $\rm N \Sigma$ channel. The correlation function in the relative momentum space for $\mathrm{p}\Lambda\oplus\overline{\mathrm{p}}\overline{\Lambda}$ pairs measured in high-multiplicity triggered pp collisions at $\sqrt{s}~=~13$ TeV at the LHC is reported. The opening of the inelastic N$\Sigma$ channels is visible in the extracted correlation function as a cusp-like structure occurring at relative momentum $k^{*}$ = 289 MeV/$c$. This represents the first direct experimental observation of the $\rm N \Sigma\rightarrow N \Lambda$ coupled channel in the p$\Lambda$ system. The correlation function is compared with recent chiral effective field theory calculations, based on different strengths of the $\rm N \Sigma\leftrightarrow N \Lambda$ transition potential. A weaker coupling, as possibly supported by the present measurement, would require a more repulsive three-body NN$\Lambda$ interaction for a proper description of the $\Lambda$ in-medium properties, which has implications on the nuclear equation of state and for the presence of hyperons inside neutron stars.


Introduction
The proton-Lambda (pΛ) system is one of the best-known examples in hadron physics where the role of coupled-channel dynamics is crucial for the understanding of the two-body and three-body interaction, both in vacuum and at finite nuclear densities [1][2][3][4]. The coupling between the nucleon-Sigma (NΣ) and NΛ systems arises from these pairs having the same strangeness content and a small mass difference, and it is responsible for the dominant attractive pΛ interaction in the spin-triplet state of coupled-channel potentials [3,5,6].
The attractive nature of the interaction between a proton and a Λ was established from measurements of binding energies of light Λ-hypernuclei [7,8] and scattering experiments at low energies [9][10][11]. However, the available scattering cross sections are characterised by large uncertainties. Moreover, they are limited to hyperon momenta above p lab ∼100 MeV/c. Thus, a reliable determination of standard quantities like scattering lengths, which provide a simple quantitative measure for the strength of an interaction, is practically impossible. Furthermore, in the region p lab ≈ 640 MeV/c, where the nΣ + and pΣ 0 channels open, the momentum resolution of the existing data is poor [12,13]. Calculations based on NΛ-NΣ coupled-channel potentials [2,3,6] predict a narrow but sizeable enhancement of the pΛ cross section in that region which reflects the strength of the channel coupling and also that of the NΣ interaction. However, because of the poor resolution of the mentioned scattering data, the presence of such a structure could not be confirmed. New pΛ data that became available recently [14] cover only energies well above the NΣ threshold. Experimental observations of a cusp-like structure at the NΣ threshold stem only from studies of the pΛ invariant mass (IM) spectrum in strangeness exchange processes such as K − d → π − pΛ [15,16] and more recently from measurements of the reaction pp → K + pΛ [17,18].
It is known that the strength of the NΣ ↔ NΛ conversion is relevant for the behaviour of Λ hyperons in infinite nuclear matter [19][20][21]. This has been emphasised in a recent study of the YN interaction based on chiral effective field theory (χEFT) [3]. Specifically, this work discussed the interplay between the NΣ ↔ NΛ conversion, the in-medium properties of the Λ and the role played by three-body forces. The abundant data on hypernuclei allowed the determination of the average attraction (−30 MeV) experienced by a Λ hyperon within symmetric nuclear matter at the nuclear saturation density [22]. However, the interaction of hyperons with the surrounding nucleons at larger baryonic densities is not known empirically. The outcome of pertinent calculations depends on the employed NΛ and NNΛ interactions in vacuum. These contributions are directly correlated to the NΣ ↔ NΛ conversion, as the parameters driving the coupling strength in the theory can be tuned differently while still reproducing the existing experiments [3]. For example, compared to the original version of the next-to-leading order (NLO) χEFT (NLO13) [2], the revisited version (NLO19) [3] involves a weaker NΣ ↔ NΛ transition potential. However, it leads to practically identical results for NΛ two-body scattering, but to an enhanced attractive behaviour in the medium. This points to a stronger repulsive three-body force needed within the latter realisation. The interplay between the NΛ and NNΛ interaction is relevant to the debated presence of Λ hyperons inside the core of neutron stars (NS) [22][23][24]. The hyperon puzzle originates from the contraposition between the energetically favoured production of hyperons in the interior of NS [25] and the subsequent softening of the corresponding equation of state (EoS). The latter does not support the existence of the heaviest observed NS of up to 2.2 solar masses [26][27][28]. Applications of the NLO19 χEFT potentials in calculations of the EoS [4] demonstrated that a repulsive genuine NNΛ interaction suppresses the appearance of Λ hyperons inside NS, giving a more quantitative reference for the solution of the hyperon puzzle. Thus new experimental data of high precision providing constraints on the NΣ ↔ NΛ dynamics are needed.
Recent studies of two-particle correlations in pp, p-Pb and Pb-Pb collisions have been successful in studying the final-state interaction (FSI) and in delivering high precision data on particle pairs of limited accessibility using traditional experimental techniques [29][30][31][32][33][34][35][36][37][38][39]. Performing such measurements in small collision systems results in a stronger sensitivity of the experimental correlation to the coupled-channel dynamics, as recently proven by means of pK − correlations [35,40,41]. In this letter we present the combined measurement of pΛ and pΛ pairs in pp collisions with a high-multiplicity (HM) trigger at √ s =13 TeV [42,43].

Data analysis
The relevant observable in this analysis is the two-particle correlation function C(k * ). This is related to an effective particle emission source S(r * ) and to the wave function Ψ( k * , r * ) of the particle pair, by means of the relation C(k * ) = S(r * )|Ψ( k * , r * )| 2 d 3 r * [44], where the relative distance r * and relative momentum q * = 2k * are evaluated in the pair rest frame. The experimental correlation is defined as where N(k * ) is the distribution of pairs where both reconstructed particles are measured in the same event, M(k * ) is the reference distribution of uncorrelated pairs sampled from different (mixed) events and N is a normalisation factor. The uncorrelated sample in the denominator, M(k * ), is obtained by combining particles from one event with particles from a set of other events. The two events are required to have comparable number of charged particles at midrapidity and a similar primary vertex coordinate V z along the beam axis (z).
The ALICE experiment excels in correlation studies thanks to its good tracking and particle identification (PID) [42,43]. These capabilities are related to the three subdetectors, the inner tracking system (ITS) [45], the time projection chamber (TPC) [46] and the time-of-flight detector (TOF) [47]. The event trigger is based on the measured amplitude in the V0 detector system, consisting of two arrays of plastic scintillators located at forward (2.8 < η < 5.1) and backward (−3.7 < η < −1.7) pseudorapidities [48]. The selected HM events correspond to 0.17% of all events with at least one measured charged particle within |η| < 1 (INEL> 0). This condition results in an average of 30 charged particles in the range |η| < 0.5 [34]. Compared to a minimum-bias trigger, HM events provide not only a larger number of particles per event, but an overall higher production rate of particles containing strangeness, such a Λ hyperons [49]. Consequently, the HM sample offers a tenfold increase in the amount of pΛ pairs reconstructed below k * of 200 MeV/c, leading to a total of 1.3 million pairs within the same event sample. The reconstructed primary vertex (PV) of the event is required to have a maximal displacement with respect to the nominal interaction point of 10 cm along the beam axis, in order to ensure a uniform acceptance. Pile-up events with multiple primary vertices are removed following the procedure described in [29,30,33,34]. The final number of selected HM events reaches approximately 10 9 . Charged particles, such as protons and pions, are directly measured, while the Λ candidates are reconstructed based on the IM of the decay products. The correlation functions obtained for particles (pΛ) and anti-particles (pΛ) are identical within uncertainties, thus the final result is presented as their weighted sum pΛ ⊕ pΛ.
Both the protons and the Λ candidates are reconstructed using the procedure described in [30], while the related systematic uncertainties are evaluated by varying the kinematic and topological observables used in the reconstruction. For the purpose of correlation studies it is essential to differentiate between primary particles, which participate in the FSI, and secondary (feed-down) particles, which stem from weak or electromagnetic decays. Experimentally, the former can be selected by demanding the particle candidates to be close to the PV of the event, while the latter have to be associated with a secondary vertex within the event. In the following text, the systematic variations are enclosed in parentheses. The primary proton candidates are selected in the momentum interval 0.5 (0.4, 0.6) < p T < 4.05 GeV/c and |η| < 0.8 (0.77, 0.85). To improve the quality of the tracks a minimum of 80 (70, 90) out of the 159 possible spatial points (hits) inside the TPC are required. The candidates are selected by comparing the measurements in the TPC and TOF detectors to the expected distributions for a proton candidate. The agreement is expressed in terms of the detector resolution σ (n PID σ ). For protons with p T < 0.75 GeV/c the n PID σ is evaluated only based on the energy loss and track measurements in the TPC, while for p T > 0.75 GeV/c a combined TPC and TOF PID selection is applied (n PID σ = n 2 σ ,TPC + n 2 σ ,TOF ). The n PID σ of the accepted candidates is required to be within 3 (2.5, 3.5). To reject non-primary particles the distance of closest approach (DCA) to the PV of the tracks is required to be less than 0.1 cm in the transverse plane and less than 0.2 cm along the beam axis. Nevertheless, due to the limited resolution of the reconstruction, the selected primary proton candidates will contain certain amount of secondaries, stemming from weak decays, and misidentifications. These contributions are extracted using Monte Carlo (MC) template fits to the measured distributions of the DCA to the PV [29]. The resulting proton purity is 99.4% with a 82.3% fraction of primaries.
The Λ candidates are reconstructed via the weak decay Λ → pπ − . The secondary daughter tracks are subject to similar selection criteria as for the primary protons. In addition, the daughter tracks are required to have a DCA to the PV of at least 0.05 (0.06) cm. The DCA of the corresponding Λ candidates to the PV has to be below 1.5 (1.2) cm. The cosine of the pointing angle (CPA) between the vector connecting the PV to the decay vertex and the three-momentum of the Λ candidate is required to be larger than 0.99 (0.995). To reject unphysical secondary vertices, reconstructed with tracks stemming from collisions corresponding to different crossings of the beam, the decay tracks are required to possess a hit in one of the SPD or SSD detectors or a matched TOF signal [31]. The final Λ candidates are selected in a 4 MeV/c 2 mass window around the nominal mass [50], where the width of the IM peak is c.a. 1.6 MeV/c 2 . The number of primary and secondary contributions for Λ are extracted similarly as for protons, using the CPA as an observable for the template fits. The average fraction of primary Λ hyperons is 57.6 (52.1, 60.6)% and 19.2 (15.4, 21.9)% originate from the electromagnetic decays of Σ 0 . The number of Σ 0 particles is related to their ratio to the Λ hyperons, which is fixed to 0.33 (0.27, 0.40). These values are based on predictions from the isospin symmetry, thermal model calculations using the Thermal-FIST package [51] and measurements of the corresponding production ratios [52][53][54]. Further, each of the weak decays of Ξ − and Ξ 0 contributes with 11.6 (13.5)% to the yield of Λ hyperons. The purity of Λ and Λ was extracted by fitting, as a function of k * , the IM spectra of candidates selected in the mixed-event sample. The fits were performed in the IM range of 1088 to 1144 MeV/c 2 using a double Gaussian for the signal and a third-order spline for the background. The result was averaged for k * < 480 MeV/c, leading to a purity P Λ = 95.3%. The systematic variations include a modelling of the signal using the sum of three Gaussians, leading to a purity of 96.3%. The effect of misidentified Λ candidates (Λ) can be accounted for by the relations where the signal is decomposed into its ingredients, weighted by the corresponding λ parameters and corrected for the non-FSI baseline B(k * ).
Such a decomposition is required [29], as the experimental signal contains correlations complementing the genuine pΛ signal C pΛ (k * ). In the present analysis the contribution C pΛ related to misidentified Λ candidates (Λ) is explicitly measured and subtracted from the total correlation C exp (k * ). This is achieved by performing a sideband analysis [32], which relies on purposefully selecting Λ candidates incompatible with the true Λ mass by more than 5σ .
The corrected correlation C corrected (k * ) has an effective Λ purity of 100%, and the remaining contributions (Eq. 3) are the genuine signal of interest C pΛ , the residual (feed-down) correlation C p(Σ 0 ) of Λ particles originating from the decay of a Σ 0 , the residual signal C p(Ξ) related to Ξ (Ξ − ⊕ Ξ 0 ) decaying into Λ, other sub-dominant (flat) sources of feed-down correlations C ff ≈ 1, and contamination Cp Λ stemming from misidentified protons. Each of these contributions is weighted by a statistical factor λ , evaluated as the product of the purities and fractions (primary or secondary) of the set particles [29]. These weight factors  Table 1. The contribution Cp Λ cannot be modelled, however the associated λp Λ is only 0.6%, justifying the assumption λp Λ Cp Λ ≈ λp Λ within the uncertainties of C corrected (k * ). By contrast, the residual correlations C p(Σ 0 ) and C p(Ξ) are significant, but in these cases their interactions with protons can be described by theory. Recent correlation studies of the pΣ 0 system showed that this interaction is rather weak [32]. This channel is modelled assuming either a flat function or employing the same χEFT calculations used for the genuine pΛ interaction [3]. The contribution from the pΞ (pΞ − ⊕ pΞ 0 ) channel is modelled employing the lattice potentials from the HAL QCD collaboration [55]. They were experimentally validated by comparison with precision measurements of pΞ − correlations [33,34]. The residual contributions C p(Σ 0 ) (k * ) and C p(Ξ) (k * ) are obtained by transforming the corresponding genuine correlation functions to the basis of the pΛ interaction, using the formalism described in [29] and [56] applied to the phase space of the measured pairs.
The non-FSI background (baseline) is parameterised by a third-order polynomial B(k * ) constrained to be flat at k * → 0 and fitted to the data (Eq. 3). By default, the fit is performed for k * ∈ [0, 456] MeV/c, with systematic variations of the upper limit to 432 and 480 MeV/c. Further, due to the expectation of a flat baseline at low k * , a systematic cross-check has been performed by assuming the hypothesis of a constant B(k * ) and fitting the correlation function for k * below 336 MeV/c.
The correlation function (Eq. 3) is given as a function of the measured k * , which is not identical to the true relative momentum of the pair due to the effects of momentum resolution. Thus, to compare the experimental results with theoretical predictions an unfolding of the data is required. Both the same-and mixed-event samples (N(k * ), M(k * )) are biased by the resolution of the detector. They relate to their true underlying distributions by and where T (k * , k * true ) is the detector response matrix. The latter is a two-dimensional matrix corresponding to the probability of having a true value k * true given a measured k * . By using a full scale simulation of the detector, involving Pythia 8 [57] as an event generator and Geant3 [58] to model the detector response, the matrix T (k * , k * true ) has been determined. The resulting spread in the distribution of k * for a fixed k * true is, on average, 4.2 MeV/c. Using N true (k * true ) = M true (k * true )C(k * true ) and defining W (k * , k * true ) = T (k * , k * true )M true (k * true )/M(k * ), Eq. 1 becomes equivalent to In the present analysis the unfolding is performed as a two-step process, first obtaining M true (k * ) from Eq. 5, second using Eq. 6 to obtain C true (k * ). Each step is performed by using a cubic spline to parameterise the true functions, which are fitted to their measured counterparts. The splines are defined for k * < 1000 MeV/c, using a total of 32 knots. The quality of the procedure is validated by transforming the unfolded functions backwards using Eq. 5 and Eq. 6, which ideally should restore the input distributions (χ 2 = 0). In case the resulting χ 2 per data point is larger than 0.2, the value of each C true (k * ) bin is perturbed using a bootstrap procedure [59], until a better χ 2 is achieved. This is iteratively repeated until obtaining the desired precision, and until no single bin deviates by more than half of their uncertainty.

Results and discussion
The corrected and unfolded experimental correlation function for pΛ ⊕ pΛ is shown in Figs. 1 and 2.
The correlation function is measured with high-precision in the low momentum region down to k * = 6 MeV/c, in contrast to existing pΛ scattering data which cover the region k * >60 MeV/c. The precision achieved for k * <110 MeV/c is better than 1%, which corresponds to an improvement of factor up to 25 compared to previous scattering data [9][10][11]. The theoretical correlation functions in Eq. 3 were evaluated using the CATS framework [60]. The size of the emitting source employed in the calculation was fixed from independent studies of proton pairs [30], which demonstrate a common primordial (core) Gaussian source for pp and pΛ pairs when the contribution of strongly decaying resonances is explicitly accounted for [30]. This source exhibits a pronounced m T dependence and considering the average transverse mass m T = 1.55 GeV of the measured pΛ pairs a corresponding core source radius of r core ( m T ) = 1.02 ± 0.04 fm is obtained. The total source function can be approximated by an effective Gaussian emission source of size 1.23 fm. The genuine pΛ correlation function is modelled by χEFT hyperonnucleon potentials, considering the leading-order (LO) interaction [1] and two NLO versions (NLO13 [2] and NLO19 [3]). For the NLO interactions the variation with the underlying cut-off parameter Λ (cf. Ref. [2]) is explored, while Λ =600 MeV is chosen as a default value. Both NLO versions provide an excellent description of the available scattering data, having a χ 2 ≈ 16 for the considered 36 data points [3].  Table 2, where the left two columns show the n σ only in the low momentum region, and the right two columns represent the deviation evaluated for k * ∈ [0, 300] MeV/c.
The presented results are the first direct experimental evidence of the NΣ ↔ NΛ coupling in a two-body final state. The signal of the cusp is determined by the properties of the interaction, and further modified by the relative amount of NΣ and pΛ initial state pairs leading to the final state (measured) pΛ pairs. The amount of initial state pairs was fixed by the above-mentioned Σ:Λ ratio, enabling a direct test of the strong interaction. The LO chiral potential [1] predicts a too small NΣ cusp with respect to the measurement, the green line in Fig. 1, while both NLO interactions provide a satisfactory description of the cusp structure. On the other hand, in the momentum region below 110 MeV/c there is a tension between the data and the theory predictions for all considered interactions. In particular, the results for the two NLO potentials are not that well in line with the measured correlation function, despite of the fact that these interactions reproduce the low-energy pΛ scattering data perfectly [3]. The best result is provided by the NLO19 potential with Λ =600-650 MeV, though the deviation of n σ = 3.2 from the experiment is substantial. For NLO13 this deviation is even larger and amounts to n σ = 4.2. Further, it is observed that for NLO13 and NLO19 the best agreement with the data is achieved within the same range of cut-off values (550-650 MeV) which also provide the best description of the available scattering and hypertriton data [2,3]. The deviation between data and predictions, expressed in terms of n σ . The fit is performed using NLO13 (red) χEFT potentials with cut-off Λ =600 MeV [2,3] and using a cubic baseline (dark grey). The residual pΞ − ⊕ pΞ 0 (pink) and pΣ 0 (royal blue) correlations are modelled using, respectively, a lattice potential from the HAL QCD collaboration [33,55] and a χEFT potential [2]. Both contributions are plotted relative to the baseline, while in panel b) the strong interaction of pΣ 0 is neglected. The reduced χ 2 , for k * < 300 MeV/c, amounts to 2.2 in case a) and to 1.9 in case b).
The discrepancy between the data and χEFT at low momenta could be an indication for a weaker genuine pΛ interaction, but it could also signal that the pΣ 0 correlation is very small. As visible in the right panels of Figs. 1, 2 and Table 2, adopting the hypothesis of a negligible pΣ 0 correlation leads to a better agreement with the present pΛ data (n σ = 2.2). At the moment it is impossible to differentiate between these two cases because the existing direct measurement of the pΣ 0 channel is not precise enough for drawing pertinent conclusions [32]. The pΣ 0 measurement is compatible with both the NLO predictions (of a weakly attractive pΣ 0 interaction) and with a flat correlation (negligible pΣ 0 interaction). A precision measurement of the genuine pΣ 0 channel, expected to be achieved in the upcoming LHC Run 3 [61], should provide clarification. Then the actual strength of the NΛ interaction can be pinned down in a model independent way by a dedicated theoretical analysis of the pΛ data.
All the conclusions of the present analysis remain the same under the alternative hypothesis of a constant baseline, or in case the deviation is evaluated for k * < 300 MeV/c. Within that momentum region, the NLO19 provides a satisfactory description of the data, with a deviation of n σ = 2.8, while the NLO13 still results in a larger discrepancy (n σ = 3.6).  Fig. 1, where the pΛ interaction is modelled using NLO19 (cyan) χEFT potentials with cut-off Λ =600 MeV [2,3]. This leads to an improved description of the low momentum region. The reduced χ 2 , for k * < 300 MeV/c, equals 2.0 in case the pΣ 0 is modelled by χEFT (panel a) and 1.8 in case the pΣ 0 final state interaction is ignored (panel b).

Summary
In conclusion, two-particle correlation techniques were used to study the final state interaction in the NΣ ↔ NΛ coupled system. This was achieved by studying the pΛ correlation function at low relative momenta with an unprecedented precision. The significance of the coupling of pΛ to NΣ is manifested as a cusp-like enhancement present at the corresponding threshold energy, which is the first direct experimental observation of this structure. Further, using different modellings for the pΣ 0 feed-down leads to a statistically significant modification of the measured pΛ correlation, implying an indirect sensitivity to the genuine pΣ 0 correlation. In the momentum range k * ∈ [110, 300] MeV/c all of the tested NLO χEFT interactions are compatible with the data, however a significant deviation is present at lower values. The detailed analysis, presented in Table 2, reveals a deviation of at least n σ = 3.2, for k * < 110 MeV/c, for the considered χEFT interactions. The result for NLO19 exhibits an overall better compatibility, compared to the NLO13 prediction. The former involves a weaker NΣ ↔ NΛ transition potential and a more attractive two-body interaction of the Λ hyperon in the medium. This requires a stronger repulsive NNΛ three-body force, which leads to a stiffening of the EoS at large densities [4] and a disfavoured production of these strange hadrons in neutron stars. The presented data provide an opportunity to improve the theoretical calculations for the NΣ ↔ NΛ coupled system, including the low-energy properties of NΛ. The successful use of correlation techniques in the two-body sector can be extended to measure directly the three-body correlations [62]. The increased amount of statistics during the third running period of the LHC [61] will allow for such measurements. The deviation, expressed in terms of n σ , between data and prediction for the different interaction hypotheses of pΛ and pΣ 0 , evaluated for k * ∈ [0, 110] MeV/c (first two columns) and k * ∈ [0, 300] MeV/c (last two columns). The default values correspond to the fit with a cubic baseline and the values in parentheses represent the results using a constant baseline. The default interaction (in bold) is the χEFT NLO19 potential with cut-off Λ =600 MeV [3]. Each row corresponds to a different variant of the χEFT interaction used for evaluating the pΛ correlation. The first and third column correspond to the case of modelling the pΣ 0 using χEFT, while the second and fourth column represent the case of negligible pΣ 0 final state interaction.