2022 107 25 May

Constraining the $\rm\overline{K}N$ coupled channels

ALICE Collaboration ALICE Collaboration

The interaction of $\rm{K}^{-}$with protons is characterised by the presence
of several coupled channels, systems like $\rm\overline{K}^{0}$n and
$\uppi\Sigma$ with a similar mass and the same quantum numbers as the
$\rm{K}^{-}$p state. The strengths of these couplings to the $\rm{K}^{-}$p
system are of crucial importance for the understanding of the nature of the
$\Lambda(1405)$ resonance and of the attractive $\rm{K}^{-}$p strong
interaction. In this article, we present measurements of the $\rm{K}^{-}$p
correlation functions in relative momentum space obtained in pp collisions at
$\sqrt{s}~{}=~{}13$ TeV, in p–Pb collisions at
$\sqrt{s_{\mathrm{NN}}}~{}=~{}5.02$ TeV, and (semi)peripheral Pb–Pb collisions
at $\sqrt{s_{\mathrm{NN}}}~{}=~{}5.02$ TeV. The emitting source size, composed
of a core radius anchored to the $\rm{K}^{+}$p correlation and of a resonance
halo specific to each particle pair, varies between 1 and 2 fm in these
collision systems. The strength and the effects of the $\rm\overline{K}^{0}$n
and $\uppi\Sigma$ inelastic channels on the measured $\rm{K}^{-}$p correlation
function are investigated in the different colliding systems by comparing the
data with state-of-the-art models of chiral potentials. A novel approach to
determine the conversion weights $\omega$, necessary to quantify the amount of
produced inelastic channels in the correlation function, is presented. In this
method, particle yields are estimated from thermal model predictions, and
their kinematic distribution from blast-wave fits to measured data. The
comparison of chiral potentials to the measured $\rm{K}^{-}$p interaction
indicates that, while the $\uppi\Sigma$–$\rm{K}^{-}$p dynamics is well
reproduced by the model, the coupling to the $\rm\overline{K}^{0}$n channel in
the model is currently underestimated.

##  1 Introduction

The interaction between antikaons ($\rm\overline{K}$) and nucleons (N) is one
of the main building blocks of low-energy effective field theories aiming at
describing the non-perturbative regime of the strong interaction with
strangeness degrees of freedom. The $\rm\overline{K}N$ interaction is
characterised by the presence of several coupled channels, such as
$\rm\uppi^{0}\Lambda$ and $\uppi\Sigma$ [1, 2], with the same quantum numbers
as $\rm\overline{K}N$ and an invariant mass below that of the
$\rm\overline{K}N$ system [3, 4, 5, 6] (i.e. sub-threshold). This coupled
channel dynamics is responsible for the inelastic component of the
$\rm\overline{K}N$ interaction, accounting for transitions like
$\rm\overline{K}N\xleftrightarrow{}\rm\uppi^{0}\Lambda,\uppi\Sigma$ [7, 8].

While the coupling of $\rm\overline{K}N$ to $\rm\uppi^{0}\Lambda$ is
negligible [9, 10], the one to $\uppi\Sigma$ is dominant in the sub-threshold
region. At low energy, the dynamics between the $\rm\overline{K}N$ and the
$\uppi\Sigma$ channel leads to the formation of the $\Lambda(1405)$ resonance,
approximately 27 MeV below the threshold. The internal structure of the
$\Lambda(1405)$ has been widely investigated and currently it is the only
accepted molecular state arising from the interplay of the
$\rm\overline{K}N$–$\uppi\Sigma$ poles [11, 12, 13]. Measurements of the
$\Lambda(1405)$ spectral shape in the strong decays to the $\uppi\Sigma$ final
state can provide information on the $\rm\overline{K}N$–$\uppi\Sigma$
coupling, but direct access is mainly hampered by the limited amount of data,
by the presence of interference terms in the total amplitude due to baryonic
resonances in the final state [14, 15, 16, 17, 18, 19], and ultimately by
differences in the theoretical description of the $\rm\overline{K}N$
interaction below threshold. In fact, the behaviours of the $\rm\overline{K}N$
interaction and of the coupling to the $\uppi\Sigma$ channel in the sub-
threshold region are anchored to the knowledge of such interaction above and
at the threshold. The measurement of kaonic hydrogen [20] provides the most
precise constraint on the $\rm\overline{K}N$ interaction at the threshold, but
the coupled channel dynamics is included in the extracted $\rm\overline{K}N$
scattering parameters only indirectly [3].

Scattering experiments available above the $\rm\overline{K}N$ threshold, in
which the initial state (typically $\rm{K}^{-}$p) is fixed [21, 22, 23, 24,
25], represent the largest source of constraints for the available theoretical
approaches [26, 27, 28, 29, 30, 31, 32, 33, 34, 35]. Access to the
$\rm\overline{K}N$ interaction and to the different inelastic couplings is
achieved by measurements of elastic and inelastic cross sections of the
different final states. In the specific case of the $\rm{K}^{-}$p system,
another inelastic channel, $\rm\overline{K}^{0}$n, appears approximately 5 MeV
above the $\rm{K}^{-}$p threshold, due to the mass difference between neutral
kaons and neutrons and their charged isospin partners ($\rm{K}^{-}$, p). A
cusp structure in the total $\rm{K}^{-}$p cross section is predicted to occur
at the opening of this coupled channel [36], but no evidence in scattering
data has been observed yet due to the insufficient statistical precision [23,
25, 37]. As a consequence, the theoretical description of the dynamics between
the $\rm{K}^{-}$p and $\rm\overline{K}^{0}$n channels is currently not fully
constrained by experimental data.

Recently, this scenario changed due to the measurements of correlations of
($\rm{K}^{-}$p $\oplus$ $\rm{K}^{+}$$\overline{\mathrm{p}}$) (for brevity
$\rm{K}^{-}$p) pairs in pp collisions at centre-of-mass energies of
$\sqrt{s}~{}=~{}5.02$ TeV, 7 TeV and 13 TeV by the ALICE Collaboration, which
provided the first experimental evidence for the opening of the
$\rm\overline{K}^{0}$n channel and the most precise data on the $\rm{K}^{-}$p
interaction down to zero relative momenta [38]. The femtoscopy technique [39,
40] applied in small colliding systems, in which particles are emitted by a
source with typical size of about 1 fm, can easily access the short-range part
of the strong interaction in which the coupled channel dynamics plays an
important role. The measured correlation functions show great sensitivity to
the underlying strong potential [41, 42, 43, 44, 45] and to the different
inelastic channels, affecting both the shape and magnitude of the femtoscopic
signal [46, 8, 7]. The recent results by the ALICE Collaboration on
$\rm{K}^{-}$p pairs in central Pb–Pb collisions showed quantitatively that the
contributions of ($\rm\overline{K}^{0}$n $\oplus$
K${}^{0}\overline{\mathrm{n}}$) (hereafter $\rm\overline{K}^{0}$n) and
$\uppi\Sigma$ channels are negligible for inter-particle distances of 5 fm and
above [47]. The results obtained in pp and Pb–Pb collisions clearly suggest
the possibility to constrain the $\rm{K}^{-}$p interaction and the different
inelastic coupling by measuring $\rm{K}^{-}$p pairs in different colliding
systems.

In this article, the femtoscopic measurements of ($\rm{K}^{+}$p $\oplus$
$\rm{K}^{-}$$\overline{\mathrm{p}}$) (hereafter $\rm{K}^{+}$p) and
$\rm{K}^{-}$p pairs obtained in three different centrality intervals in p–Pb
collisions at collision energy per nucleon pair
$\sqrt{s_{\mathrm{NN}}}~{}=~{}5.02$ TeV and in three different centrality
intervals in peripheral Pb–Pb collisions at
$\sqrt{s_{\mathrm{NN}}}~{}=~{}5.02$ TeV are presented. Different centrality
intervals were chosen to probe different radii of the particle emitting
source. Results from the reanalysis of the femtoscopic data obtained in
minimum bias pp collisions at $\sqrt{s}~{}=~{}13$ TeV shown in Ref. [38] are
also included. In this study, the data from Ref. [38] are corrected for the
finite experimental momentum resolution and analysed again, employing the same
procedure used for the other colliding systems. The same-charge pairs,
$\rm{K}^{+}$p, for which the interaction is well known and no inelastic
channels are present [48, 49], are used as a benchmark to extract information
on the emitting source size needed to evaluate the correlation function. The
measured correlations of $\rm{K}^{-}$p are compared with state-of-the-art
chiral potentials [8, 50] derived in a coupled channel approach. A detailed
investigation on the strong coupling to $\uppi\Sigma$ and
$\rm\overline{K}^{0}$n channels is performed in the three different colliding
systems.

This paper is organised as follows: a short description of the ALICE detector
can be found in Section 2 while in Section 3 the event and track selection
criteria are discussed together with the particle identification technique and
the systematic uncertainties evaluation on the correlation function. In
Section 4, the analysis and the modeling of the correlation function are
introduced and the determination of the emitting source size is described; the
obtained results are presented and discussed in Section 5. Finally, in Section
6, conclusions and future outlooks are provided.

##  2 The ALICE detector

A detailed description of the ALICE experimental setup can be found in Refs.
[51, 52] and references therein. The main sub-detectors used in this analysis
are: the V0 detectors [53], the Inner Tracking System (ITS) [54], the Time
Projection Chamber (TPC) [55] and the Time-Of-Flight (TOF) detector [56]. The
ITS, TPC and TOF are located inside a solenoidal magnet that provides a
uniform field of 0.5 T directed along the beam direction. All the detectors in
the central barrel region used for this analysis (ITS, TPC and TOF) cover the
full azimuth and have a pseudorapidity coverage of $|\eta|<0.9$.

The V0 detector consists of two arrays of scintillation counters placed on
either side of the interaction point: one covering the pseudorapidity interval
$2.8<\eta<5.1$ (V0A) and the other one covering $-3.7<\eta<-1.7$ (V0C). The
scintillator arrays have an intrinsic time resolution better than 0.5 ns.
Their timing information is used in coincidence for offline rejection of
events produced by the interaction of the beams with residual gas in the
vacuum pipe. The V0 scintillators are used to determine the collision
centrality from the measured charged-particle multiplicity [57, 58, 59].

The ITS, designed to provide high-resolution track points close to the beam
line, is composed of three subsystems of silicon detectors placed around the
interaction region with a cylindrical symmetry. The Silicon Pixel Detector
(SPD) is the closest subsystem to the beam pipe and is made of two layers of
pixel detectors. The third and the fourth layers are formed by silicon drift
detectors, while the outermost two layers are equipped with double-sided
silicon strip detectors.

The TPC, which is the main tracking detector, consists of a hollow cylinder
whose axis coincides with the nominal beam axis and which surrounds the ITS.
The charged-particle tracks are then built by combining the hits in the ITS
and up to 159 reconstructed space points in the TPC. The momentum component
transverse to the beam pipe ($p_{\rm T}$) of charged particles is
reconstructed from the curvature of the tracks within the TPC, which is
permeated by a magnetic field. The TPC is also used for particle
identification (PID) via the measurement of the specific energy loss
(d$E$/d$x$) in its gas volume.

The TOF detector is based on the multi-gap resistive plate chambers technology
and is located, with a cylindrical symmetry, at an average radial distance of
380 cm from the beam axis. The particle velocity $\beta$ of each particle can
be determined by using the time-of-flight measurement, the track length, and
the associated momentum, allowing for PID at intermediate momenta.

##  3 Event and track selection

The data samples used for the measurements presented in this paper were
recorded by ALICE in 2015, 2016, 2017, and 2018 during the LHC pp runs at
$\sqrt{s}~{}=~{}13$ TeV, the p–Pb runs at $\sqrt{s_{\mathrm{NN}}}~{}=~{}5.02$
TeV and the Pb–Pb runs at $\sqrt{s_{\mathrm{NN}}}~{}=~{}5.02$ TeV. A minimum
bias trigger was used during all the data taking, which required coincident
signals in both V0 scintillators to be synchronous with the beam crossing time
defined by the LHC clock. Events with multiple primary vertices identified
with the SPD are tagged as pile-up and excluded from the analysis to achieve
the best PID performance.

The primary vertex position was determined from tracks reconstructed in the
ITS and TPC as described in Ref. [52] and only events with a reconstructed
primary vertex position along the beam axis ($V_{z}$) within 10 cm from the
nominal interaction point along the beam direction are selected. This
procedure ensures a uniform detector coverage within $|\eta|<0.8$. After the
application of the event selection criteria, about 1$\times 10^{9}$ minimum
bias pp events, about 8$\times 10^{8}$ minimum bias p–Pb collisions in the
0–100% multiplicity interval and about 6.5$\times 10^{7}$ minimum bias Pb–Pb
collisions in the 60–90% centrality interval, which corresponds to most
peripheral events, have been analysed.

Small collision systems are affected by the presence of mini-jets background,
which might affect the femtoscopic signal [38, 60]. To reduce the contribution
from the mini-jet background, the events were classified according to their
transverse sphericity ($S_{\mathrm{T}}$), an observable which is known to be
correlated with the number of hard parton–parton interactions in each event
[61]. An event with hard parton–parton interactions will generally produce a
jet-like event topology that yields low sphericity, while an event dominated
by soft parton–parton interactions can yield higher sphericity. To reduce the
strong mini-jet background at low momenta, only events with $S_{\mathrm{T}}$,
defined as in Ref. [62], larger than 0.7 were considered in the pp analysis.
Generally, the mini-jet background is most pronounced in pp while in p–Pb and
peripheral Pb–Pb the events are typically isotropic. However, for consistency,
the same selection on $S_{\mathrm{T}}$ was applied to the p–Pb and semi-
peripheral Pb–Pb data.

Charged kaon and proton candidate tracks were selected from charged-particle
tracks reconstructed in the TPC, with the additional constraint that the track
originates from the primary vertex [52]. Candidate tracks were selected in the
range $|\eta|<~{}$0.8. To assure a good $p_{\rm T}$ resolution and to remove
wrongly reconstructed tracks from the sample, only tracks with at least 70
space points were selected. To suppress contributions of secondary kaons and
protons in the sample, the reconstructed tracks were required to have a
distance of closest approach to the primary vertex (DCA) along both the beam
($z$) and transverse ($x,y$) directions smaller than 1 cm. In order to reduce
the hadron misidentification, kaon candidates with 0.15 $<$ $p_{\rm T}$ $<$
1.4 GeV$\kern-1.49994pt/\kern-1.19995ptc$ and proton candidates with 0.4 $<$
$p_{\rm T}$ $<$ 3 GeV$\kern-1.49994pt/\kern-1.19995ptc$ were selected.

For particle identification, both the TPC and the TOF detectors were employed.
Kaons (protons) with transverse momenta up to 0.4 (0.8)
GeV$\kern-1.49994pt/\kern-1.19995ptc$ were identified using only the TPC
information by requiring that the average $\textrm{d}E/\textrm{d}x$
(calculated using the Bethe–Bloch parameterisation) is within three standard
deviations ($\sigma$) from the expected average at a given momentum for the
particle mass hypothesis. For kaons with $p_{\rm T}$ $>$ 0.4
GeV$\kern-1.49994pt/\kern-1.19995ptc$ and protons with $p_{\rm T}$ $>$ 0.8
GeV$\kern-1.49994pt/\kern-1.19995ptc$, a similar method was applied for the
particle identification using the TOF, where, on top of TPC selection, a
3$\sigma$ selection on the expected time of flight for a given particle at a
given momentum was applied. Tracks for which the PID was ambiguous were
discarded. To remove the large fraction of e+e- pairs that can affect the
extraction of the correlation function of the opposite-charge pairs, a
selection on the $p_{\rm T}$ of kaons and protons of each charge was applied:
kaon candidates were excluded if 0.3 $<$$p_{\rm T}$ $<$ 0.4
GeV$\kern-1.49994pt/\kern-1.19995ptc$, while proton candidates were excluded
in the interval between 0.6 $<$ $p_{\rm T}$ $<$ 0.8
GeV$\kern-1.49994pt/\kern-1.19995ptc$. It should be noted that cutting in this
$p_{\rm T}$ region does not influence the correlation signal as the
statistical uncertainty of the data is sufficiently small in each investigated
$k^{*}$ interval. The quoted $p_{\rm T}$ intervals correspond to the regions
in which the electron $\textrm{d}E/\textrm{d}x$ band merges with the band of
the kaons and protons, respectively. The purity of the selected particle
samples was determined by Monte Carlo simulations based on PYTHIA 8 [63] for
the pp analysis, on EPOS [64] for p–Pb, and on HIJING [65] for Pb–Pb. The
purity is larger than 99% in the considered $p_{\rm T}$ intervals for all the
analysed data.

##  4 Analysis of the correlation function

###  4.1 Experimental correlation function

The main observable in this work is the two-particle correlation function.
Experimentally, the correlation function $C(k^{*})^{\rm measured}$ is
constructed as [40]

$\displaystyle C(k^{*})^{\rm measured}=\mathcal{N}\dfrac{A(k^{*})}{B(k^{*})},$
(1)

where $k^{*}$ is the magnitude of the momentum of each of the particles in the
pair reference frame. The numerator $A$($k^{*}$) is the measured distribution
of pairs from the same event, while $B$($k^{*}$) is the reference distribution
of pairs from mixed events. The pairs in the denominator are formed by mixing
particles from one event with particles from a pool of up to seven other
events with a comparable number of charged particles at midrapidity [66], a
distance between primary vertex coordinate $V_{z}$ along the beam axis $\Delta
V_{z}\leq$ 2 cm and with similar $S_{\mathrm{T}}$ as for the numerator.

The $\mathcal{N}$ parameter is chosen such that the mean value of the
correlation function equals unity for 600 $<$ $k^{*}$ $<$ 1000
MeV$\kern-1.49994pt/\kern-1.19995ptc$ where the correlation function is flat.
Reconstruction biases, such as the merging of tracks that are very close to
each other, were evaluated by means of Monte Carlo simulations and found to be
negligible in all the collision systems and centralities considered in this
analysis. The measured $k^{*}$ is not identical to the true relative momentum
of the pair due to effects of momentum resolution [67]. Hence, to compare the
experimental results with theoretical predictions, an unfolding of the data is
required. This was done by applying a Bayesian unfolding method [68] both to
the to the $A$($k^{*}$) and $B$($k^{*}$) distributions. The finite
experimental momentum resolution modifies the measured correlation functions
at most by 6% in the first $k^{*}$ interval and less than 1% for $k^{*}$
around 300 MeV$\kern-1.49994pt/\kern-1.19995ptc$. The resulting $C(k^{*})^{\rm
corrected}$ correlation function can be described by

$C(k^{*})^{\rm
corrected}=(a+b\,k^{*})\left(1+\lambda_{\mathrm{genuine}}(C(k^{*})^{\mathrm{genuine}}-1)+\sum_{ij}\lambda_{ij}\left(C_{ij}(k^{*})-1\right)\right).$
(2)

The correlation function is dominated by the contribution of the genuine
$\rm{K}^{+}$p or $\rm{K}^{-}$p interaction, $C(k^{*})^{\mathrm{genuine}}$. It
is weighted by the corresponding $\lambda_{\mathrm{genuine}}$ parameter, which
describes the purity and fractions of primary kaons and protons in the sample
[41]. Other contributions $i,j$ originating from incorrectly identified
particles, from particles stemming from weak decays (such as protons from
$\Lambda\rightarrow\mathrm{p}\uppi^{-}$), and from particles coming from long-
lived resonances (such as kaons from $\phi$
(1020)$\rightarrow$$\rm{K}^{+}$$\rm{K}^{-}$) give rise to the $C_{ij}(k^{*})$
correlation functions. Since weak decays occur typically some centimetres away
from the collision vertex, a negligible final-state interaction between their
decay products and the primary particles under study can be assumed. A similar
argument can be applied also to particles coming from long-lived resonances
($c\tau>5\,\mathrm{fm}$). Hence, the resulting correlation functions
$C_{ij}(k^{*})$ originating from the different combinations of primary,
secondary, and misidentified particles are considered independent of $k^{*}$.
The correlations due to misidentifications are evaluated experimentally and
their contribution depends on the fraction of primary pairs of the sample
(around 70% for kaon–proton pairs), on the fraction of $\phi$ (1020) that
decay into kaon pairs and are identified as primary kaons (around 6%), and on
the purity of primary pairs in the analysed sample (around 98% for kaon–proton
pairs). These fractions are determined by fitting Monte Carlo templates to the
measured DCAxy distributions of kaons and protons, similarly to what is
described in Ref. [38]. All the contributions related to misidentified kaons
and protons are encapsulated into the $\lambda_{ij}$ parameter of Eq. (2).
Finally, an additional contribution related to energy-momentum and charge
conservation is present in $C(k^{*})^{\rm corrected}$. This contribution is
described by a linear baseline $(a+b\,k^{*})$ whose coefficients are fixed by
fitting the measured correlation function in a $k^{*}$ region where the short-
range effects are negligible. For $\rm{K}^{+}$p the parameters are fixed in
the 300 $<~{}k^{*}~{}<$ 400 MeV$\kern-1.49994pt/\kern-1.19995ptc$ interval. In
the $\rm{K}^{-}$p case, the baseline parameters are fixed using the data in
the 180 $<~{}k^{*}~{}<$ 270 MeV$\kern-1.49994pt/\kern-1.19995ptc$ region.

Once all the above contributions have been taken into account, the genuine
correlation $C(k^{*})^{\mathrm{genuine}}$ ($C(k^{*})$ from now on for
simplicity) is obtained. The systematic uncertainties on $C(k^{*})$ were
evaluated for each $k^{*}$ interval by varying event and track selection
criteria. The event sample was varied by changing the selection on the $V_{z}$
position from $\pm$10 cm to $\pm$7 cm and by varying the sphericity of the
accepted events from $S_{\mathrm{T}}$ $>$ 0.7 to $S_{\mathrm{T}}$ $>$ 0.6 and
$S_{\mathrm{T}}$ $>$ 0.8. Systematic uncertainties related to the track
selection criteria were studied by varying the selection on the DCAxy within
the experimental resolution. Systematic uncertainties related to other
variations related to track selections were found to be negligible. To study
systematic effects related to particle identification, the number of standard
deviations around the expected energy loss for kaons and protons in the TPC
and, similarly, for the time of flight in the TOF detector was modified from
3$\sigma$ to 2$\sigma$. For each source, the systematic uncertainty was
estimated as the root-mean-square (RMS) of all the deviations in each $k^{*}$
interval. The total systematic uncertainty was calculated as the square root
of the quadratic sum of each source’s contribution and amounts to about 3% up
to $k^{*}$ $<$ 500 MeV$\kern-1.49994pt/\kern-1.19995ptc$. The uncertainties
related to the correction procedure described in the previous paragraph are
propagated into the measured data points by means of the bootstrap technique
[69].

The $C(k^{*})$ of $\rm{K}^{+}$p are shown in Fig. 1 for pp collisions at
$\sqrt{s}~{}=~{}13$ TeV and in Fig. 2 and Fig. 3 for p–Pb and Pb–Pb at
$\sqrt{s_{\mathrm{NN}}}~{}=~{}5.02$ TeV, respectively. In Fig. 2 and Fig. 3,
each panel corresponds to a different centrality, as indicated in the legend
of the figures. The data shown in Fig. 1, already presented in [38], are
corrected as described above. Similarly, the $C(k^{*})$ for $\rm{K}^{-}$p are
shown in Fig. 4 for pp collisions, and in Fig. 5 and Fig. 6 for p–Pb at
$\sqrt{s_{\mathrm{NN}}}~{}=~{}5.02$ TeV and Pb–Pb at
$\sqrt{s_{\mathrm{NN}}}~{}=~{}5.02$ TeV, respectively. Also in this case the
data obtained in pp collisions have already been shown in Ref. [38] and are
reported here after the corrections described above.

Figure 1: $\rm{K}^{+}$p ($\rm{K}^{+}$p $\oplus$
$\rm{K}^{-}$$\overline{\mathrm{p}}$) correlation function obtained in pp
collisions at $\sqrt{s}~{}=~{}13$ TeV. The measured data points are taken from
Ref. [38] and are corrected for finite experimental momentum resolution and
for residual correlations as described in Section 4.1. Measured data are shown
by the black markers, the vertical error bars and the boxes represent the
statistical and systematic uncertainties, respectively. The red band in the
upper panel represents the model calculation and its systematic uncertainty as
described in the text. The $r_{\mathrm{core}}$ and $r_{\rm eff}$ values of the
source are reported with their statistical and systematical uncertainties,
respectively. Bottom panels represent the data-to-model comparison.  Figure 2:
$\rm{K}^{+}$p ($\rm{K}^{+}$p $\oplus$ $\rm{K}^{-}$$\overline{\mathrm{p}}$)
correlation functions obtained in p–Pb collisions at
$\sqrt{s_{\mathrm{NN}}}~{}=~{}5.02$ TeV in the 0–20% (left), 20–40% (middle)
and 40–100% (right) centrality intervals. The measurement is shown by the
black markers. The vertical error bars and the boxes represent the statistical
and systematic uncertainties, respectively. The red band in the upper panels
represents the model calculation and its systematic uncertainty as described
in the text. The $r_{\mathrm{core}}$ and $r_{\rm eff}$ values of the source
are reported with their statistical and systematical uncertainties,
respectively. Bottom panels represent the data-to-model comparison. Figure 3:
$\rm{K}^{+}$p ($\rm{K}^{+}$p $\oplus$ $\rm{K}^{-}$$\overline{\mathrm{p}}$)
correlation functions obtained in Pb–Pb collisions at
$\sqrt{s_{\mathrm{NN}}}~{}=~{}5.02$ TeV in the 60–70% (left), 70–80% (middle)
and 80–90% (right) centrality intervals. The measurement is shown by the black
markers. The vertical error bars and the boxes represent the statistical and
systematic uncertainties, respectively. The red band in the upper panels
represents the model calculation and its systematic uncertainty as described
in the text. The $r_{\mathrm{core}}$ and $r_{\rm eff}$ values of the source
are reported with their statistical and systematical uncertainties,
respectively. Bottom panels represent the data-to-model comparison.

In Fig. 4, Fig. 5, and Fig. 6, the structure that can be seen in the
$\rm{K}^{-}$p correlation function at $k^{*}$ around 240
MeV$\kern-1.49994pt/\kern-1.19995ptc$ is consistent with the $\Lambda$(1520)
($\overline{\Lambda}$(1520)) which decays into a $\rm{K}^{-}$p
($\rm{K}^{+}$$\overline{\mathrm{p}}$) pair of relative momentum $k^{*}=243$
MeV$\kern-1.49994pt/\kern-1.19995ptc$ [70]. As already observed in pp
collisions [38], the correlation function for $\rm{K}^{-}$p exhibits a clear
structure between 50 and 60 MeV$\kern-1.49994pt/\kern-1.19995ptc$ visible for
all the colliding systems, centralities and energies. The position of the
structure at $k^{*}=59$ MeV$\kern-1.49994pt/\kern-1.19995ptc$ is consistent
with the opening of the $\mathrm{\overline{K}^{0}n}$ channel corresponding to
the kaon measured momentum in the laboratory frame $p_{\mathrm{lab}}=89$
MeV$\kern-1.49994pt/\kern-1.19995ptc$ [23, 25, 37].

Figure 4: ($\rm{K}^{-}$p $\oplus$ $\rm{K}^{+}$$\overline{\mathrm{p}}$)
correlation functions obtained in pp collisions at $\sqrt{s}~{}=~{}13$ TeV.
The measured data points are taken from [38] and have been corrected for
finite experimental momentum resolution and for residual correlations as
described in Section 4.1. Measured data are shown by the black markers, the
vertical error bars and the boxes represent the statistical and systematic
uncertainties, respectively. The red and blue bands in the upper panels
represents the model calculations and its systematic uncertainty as described
in the text. The $r_{\mathrm{core}}$ and $r_{\rm eff}$ values of the source
are reported with their statistical and systematical uncertainties. Bottom
panels represent the data-to-model comparison as described in the text.
Figure 5: ($\rm{K}^{-}$p $\oplus$ $\rm{K}^{+}$$\overline{\mathrm{p}}$)
correlation functions obtained in p–Pb collisions at
$\sqrt{s_{\mathrm{NN}}}~{}=~{}5.02$ TeV in the 0–20% (left), 20–40% (middle)
and 40–100% (right) centrality intervals. The measurement is shown by the
black markers, the vertical error bars and the boxes represent the statistical
and systematic uncertainties, respectively. The red and blue bands in the
upper panels represent the model calculations and their systematic uncertainty
as described in the text. The $r_{\mathrm{core}}$ and $r_{\rm eff}$ values of
the source are reported with their statistical and systematical uncertainties,
respectively. Bottom panels represent the data-to-model comparison as
described in the text.  Figure 6: ($\rm{K}^{-}$p $\oplus$
$\rm{K}^{+}$$\overline{\mathrm{p}}$) correlation functions obtained in Pb–Pb
collisions at $\sqrt{s_{\mathrm{NN}}}~{}=~{}5.02$ TeV in the 60–70% (left),
70–80% (middle) and 80–90% (right) centrality intervals. The measurement is
shown by the black markers, the vertical error bars and the boxes represent
the statistical and systematic uncertainties respectively. The red and blue
bands in the upper panels represent the model calculations and their
systematic uncertainty as described in the text. The $r_{\mathrm{core}}$ and
$r_{\rm eff}$ values of the source are reported with their statistical and
systematical uncertainties, respectively. Bottom panels represent the data-to-
model comparison as described in the text.

###  4.2 Modeling the correlation function with inelastic channels

For the modeling of the correlation function in the $\rm{K}^{-}$p system, the
Koonin–Pratt formula [40] is modified to take into account the presence of the
coupled channels $j=$ $\rm\uppi^{0}\Lambda$, $\uppi\Sigma$
($\rm\uppi^{-}\Sigma^{+}$, $\rm\uppi^{+}\Sigma^{-}$,
$\rm\uppi^{0}\Sigma^{0}$), and $\rm\overline{K}^{0}$n in the following way
[71, 7, 8]:

$\displaystyle C_{\rm{K}^{-}p}(k^{*})$ $\displaystyle=\int
d^{3}r^{*}S_{\rm{K}^{-}p}(r^{*})|\psi_{\rm{K}^{-}p}(k^{*},r^{*})|^{2}+\sum_{j}\omega_{j}\int
d^{3}r^{*}S_{j}(r^{*})|\psi_{j}(k^{*},r^{*})|^{2}$
$\displaystyle=C_{\rm{K}^{-}p}^{\rm el.}(k^{*})+C_{\rm\overline{K}^{0}n}^{\rm
inel.}(k^{*})+C_{\rm\uppi^{-}\Sigma^{+}}^{\rm
inel.}(k^{*})+C_{\rm\uppi^{+}\Sigma^{-}}^{\rm
inel.}(k^{*})+C_{\rm\uppi^{0}\Sigma^{0}}^{\rm
inel.}(k^{*})+C_{\rm\uppi^{0}\Lambda}^{\rm inel.}(k^{*}).$ (3)

The first integral on the right-hand side of the equation describes the
elastic processes $\rm{K}^{-}p\xleftrightarrow{}\rm{K}^{-}p$ in which the
initial state produced in the collision and the final measured pair are the
same. The second integral in Eq. (4.2) represents the explicit contributions
stemming from inelastic processes
$(j=\rm\uppi^{0}\Lambda,~{}\uppi\Sigma,~{}\rm\overline{K}^{0}n)\rightarrow\rm{K}^{-}p$,
in which initial and final states are different but share the same quantum
numbers. The functions $S_{\rm{K}^{-}p}(r^{*})$ and $S_{j}(r^{*})$ represent
the emitting source profiles, as a function of the relative distance $r^{*}$
in the pair rest frame, for $\rm{K}^{-}$p pairs and for the $j$ inelastic
channels, respectively. Details on the modeling of the source are available in
Section 4.3.

One of the main ingredients needed to calculate $C_{\rm{K}^{-}p}(k^{*})$ are
the elastic $\psi_{\rm{K}^{-}p}(k^{*},r^{*})$ and the inelastic
$\psi_{j}(k^{*},r^{*})$ wave functions to be evaluated in a coupled channel
approach by solving the multi-channel Schrödinger equation.

Since the coupled channel dynamics mostly acts at inter-particle distances of
the order of 1 fm, the inelastic terms shown in Eq. (4.2) should be relevant
for femtoscopic measurements performed in small colliding systems like pp,
p–Pb, peripheral and semi-peripheral Pb–Pb. It has been shown that the probed
source sizes in such small systems are around 1 fm [72] and the explicit
inclusion of the inelastic correlations is needed to reproduce the data [38,
8, 67, 60]. These inelastic terms can modify the shape and the strength of the
total correlation function as shown in Refs. [46, 8, 7]. Channels opening
below the threshold, such as $\rm\uppi^{0}\Lambda$ and $\uppi\Sigma$,
introduce an enhancement of the femtoscopic signal at low momenta and channels
opening above threshold, such as $\rm\overline{K}^{0}$n, lead to the
appearance of a cusp-like structure at the corresponding $k^{*}$ value [8, 7].
These modifications and the corresponding coupled channel contributions become
negligible when the correlation function is measured in large colliding
systems, such as central and semi-central Pb–Pb collisions, for which the
source can reach radii of about 5 fm and above. Under these circumstances, the
correlation function is mostly driven by the elastic contribution, given by
the first term in Eq. (4.2).

To properly account for the coupled channel correlations, additional
quantities are needed: the conversion weights $\omega_{j}$. These conversion
weights are related to the number of pairs in each inelastic channel $j$
originating from primary particle produced in the collision and can hence be
defined as $\omega_{j}=\omega^{\rm prod}_{j}$. The yields of pairs in the $j$
channel used to compute $\omega^{\rm prod}_{j}$ are obtained in this analysis
by using the statistical thermal model implemented in the Thermal-FIST (TF)
package [73]. Since the sizes of the source in pp, p–Pb, and semi-peripheral
Pb–Pb are small, the canonical statistical model (CSM) with incomplete
equilibration of strangeness as implemented in TF [74] is used in this study
($\gamma_{\mathrm{s}}$-CSM). The three parameters of the model (i.e. the
chemical freeze-out temperature $T_{\rm ch}$, the strangeness saturation
parameter $\gamma_{\mathrm{s}}$, and the volume at midrapidity d$V$/d$y$) are
extracted from the parameterisation provided in Ref. [74] at the average
multiplicity of charged particles
($\langle\mathrm{d}N_{\mathrm{ch}}/\mathrm{d}\eta\rangle$) measured by ALICE
for the different collision systems, energies, and centrality intervals
considered in this analysis [75, 76, 77]. The values of the parameters and
their estimated uncertainties are summarised in Table 1. The uncertainties
assigned to the different parameters are evaluated starting from the
systematic uncertainties assigned to the measured
$\langle\mathrm{d}N_{\mathrm{ch}}/\mathrm{d}\eta\rangle$ and propagated
through the parameterisation. For fixed freeze-out parameters, the TF
framework provides the yields of the primary particles composing the pairs in
each channel $j$. The final amount of pairs in the $j$ channel, $N_{j}$, is
given by the product between the primary yields of the particles in the
considered pair.

Table 1: Parameters used in the $\gamma_{\mathrm{s}}$-CSM model of Thermal-FIST [74, 73] for different colliding systems and centrality intervals. The average $\mathrm{d}N_{\mathrm{ch}}/\mathrm{d}\eta$ ($\EuScript{M}=\langle\mathrm{d}N_{\mathrm{ch}}/\mathrm{d}\eta\rangle$) and the associated systematic uncertainties corresponding to the various colliding systems are taken from Refs. [75, 76, 77]. The total uncertainties assigned to $T_{\rm ch}$, $\gamma_{\mathrm{s}}$ and d$V$/d$y$ are obtained as described in the text.  System | $\EuScript{M}$ |  $T_{\rm ch}$ (MeV) | $\gamma_{\mathrm{s}}$ | d$V$/d$y$ (fm3)
---|---|---|---|---
pp, $\sqrt{s}$ =13 TeV | 6.94${}^{+0.10}_{-0.08}$ | 171$\pm$1 | 0.78$\pm$0.06 | 16.66$\pm$1.39
p–Pb, 0–20% | 35.42$\pm$1.44 | 167$\pm$1 | 0.86$\pm$0.33 | 85.01$\pm$7.08
p–Pb, 20–40% | 23.12$\pm$0.52 | 168$\pm$1 | 0.83$\pm$0.20 | 55.49$\pm$4.62
p–Pb, 40–100% | 9.88$\pm$0.42 | 170$\pm$1 | 0.79$\pm$0.09 | 23.71$\pm$1.98
Pb–Pb, 60–70% | 96.3$\pm$5.8 | 164$\pm$1 | 0.95$\pm$0.59 | 231.12$\pm$19.26
Pb–Pb, 70–80% | 44.9$\pm$3.4 | 166$\pm$1 | 0.88$\pm$0.43 | 107.76$\pm$8.98
Pb–Pb, 80–90% | 17.52$\pm$1.89 | 169$\pm$1 | 0.81$\pm$0.15 | 42.05$\pm$3.50

The kinematic distributions of the produced pairs are obtained with toy Monte
Carlo simulations using a Blast-Wave (BW) parameterisation [78] of the $p_{\rm
T}$ of each particle composing the pair of interest. The parameters of the BW
are taken from Ref. [79] for p–Pb data and from Ref. [80] for Pb–Pb data, and
are assumed to describe also the spectra of particles not measured by ALICE
(e.g. neutrons and $\Sigma$s). The parameters used for the most peripheral
p–Pb centrality intervals are also used to describe the spectra of particles
produced in pp collisions at $\sqrt{s}~{}=~{}13$ TeV. Once the momentum
distributions of the two particles are generated, only pairs in the $j$
channel with relative momentum $k^{*}$ below 200 MeV/$c$ were selected and
taken into consideration. The obtained distribution is then integrated over
transverse-momentum space and the BW yields are extracted. These yields, for
the $\rm{K}^{-}$p and different $j$-pairs, are finally rescaled by the primary
TF $N_{j}$ results in order to account for the proper produced abundances in
each colliding system. The corresponding production weights $\omega^{\rm
prod}_{j}$ are obtained by dividing these final yields with the total
production of $\rm{K}^{-}$p pairs, considered as the reference for this study.
The relative production weights $\omega^{\rm prod}_{j}$ are reported in Table
2 for the different centralities and collision systems. The systematic
uncertainties on the $\omega^{\rm prod}_{j}$ reported in the table were
evaluated by varying the parameters in the $\gamma_{\mathrm{s}}$-CSM model
implemented in TF ($\sigma^{\rm TF}$) by $\pm 1\sigma^{\rm TF}$ and by varying
the upper limit of the relative momenta $k^{*}$ of the produced pairs by $\pm
100$ MeV/$c$. The latter variation represents the dominant contribution to the
uncertainty for pairs produced below the $\rm{K}^{-}$p threshold. The
systematic uncertainty assigned to the $\omega^{\rm prod}_{j}$ is around 20%
for $\rm{K}^{-}$p and $\rm\overline{K}^{0}$n pairs and around 50% for the
other coupled channel pairs. Since the productions of the three different
species in the isospin triplet of $\uppi$ and $\Sigma$ particles are similar
in TF, in the following analysis a single $\omega^{\rm prod}_{j}$ for
$\uppi\Sigma$, evaluated as the average of the three channels, will be used.

Table 2: Production weights $\omega^{\rm prod}_{j}$ for
$j=\rm\uppi^{0}\Lambda,~{}\uppi\Sigma,~{}\rm\overline{K}^{0}n$ pairs for
minimum bias pp collisions at $\sqrt{s}~{}=~{}13$ TeV and for three different
centrality intervals in p–Pb and Pb–Pb collisions at
$\sqrt{s_{\mathrm{NN}}}~{}=~{}5.02$ TeV. The average
$\mathrm{d}N_{\mathrm{ch}}/\mathrm{d}\eta$
($\EuScript{M}=\langle\mathrm{d}N_{\mathrm{ch}}/\mathrm{d}\eta\rangle$) and
the associated systematic uncertainties used to compute the production yield
of each species, as described in Section 4.2, are also reported in Refs. [75,
76, 77]. For the calculation, an upper limit of $k^{*}$ of 200 MeV/$c$ for the
pairs was imposed. The systematic uncertainties associated to each
$\omega^{\rm prod}_{j}$ are evaluated as described in Section 4.2.

Pairs pp p–Pb Pb–Pb $\sqrt{s}~{}=~{}13$ TeV, MB 0–20% 20–40% 40–100% 60–70%
70–80% 80–90% $\EuScript{M}$ = $6.94^{+0.10}_{-0.08}$ $\EuScript{M}$ =
$35.42\pm 1.44$ $\EuScript{M}$ = 23.12$\pm 0.52$ $\EuScript{M}$ = 9.88$\pm
0.42$ $\EuScript{M}$ = 96.3$\pm 5.8$ $\EuScript{M}$ = 44.9$\pm 3.4$
$\EuScript{M}$ = 17.52$\pm 1.89$ $\rm{K}^{-}$p 1.00 1.00 1.00 1.00 1.00 1.00
1.00 $\rm\overline{K}^{0}$n 0.97 $\pm$ 0.20 0.98 $\pm$ 0.20 0.97 $\pm$ 0.20
0.99 $\pm$ 0.20 0.99 $\pm$ 0.20 0.99 $\pm$ 0.20 0.99 $\pm$ 0.20
$\rm\uppi^{-}\Sigma^{+}$ 1.41 $\pm$ 0.70 1.41 $\pm$ 0.70 1.35 $\pm$ 0.67 1.27
$\pm$ 0.63 1.46 $\pm$ 0.73 1.38 $\pm$ 0.69 1.30 $\pm$ 0.65
$\rm\uppi^{+}\Sigma^{-}$ 1.42 $\pm$ 0.71 1.42 $\pm$ 0.71 1.35 $\pm$ 0.67 1.29
$\pm$ 0.64 1.47 $\pm$ 0.73 1.39 $\pm$ 0.69 1.31 $\pm$ 0.65
$\rm\uppi^{0}\Sigma^{0}$ 1.37 $\pm$ 0.68 1.41 $\pm$ 0.70 1.38 $\pm$ 0.70 1.22
$\pm$ 0.61 1.46 $\pm$ 0.73 1.38 $\pm$ 0.69 1.31 $\pm$ 0.65
$\rm\uppi^{0}\Lambda$ 1.96 $\pm$ 0.93 2.07 $\pm$ 1.03 1.96 $\pm$ 0.93 1.86
$\pm$ 0.93 1.48 $\pm$ 0.74 1.40 $\pm$ 0.70 1.32 $\pm$ 0.66

###  4.3 Experimental characterization of the emitting source $S(r^{*})$

A fundamental ingredient entering in the evaluation of the theoretical
correlation function in Eq. (4.2) is the emitting source for the elastic part
$S(r^{*})$ and for the inelastic terms $S_{j}(r^{*})$. Recently, a data-driven
analysis based on p–p correlations measured in pp collisions provided a model
for the emitting source of baryon–baryon pairs in small colliding systems
[72]. The source is composed of two components: a Gaussian core with a common
radius $r_{\rm core}$, which scales with the transverse mass $m_{\mathrm{T}}$
of the pair due to possible collective effects (e.g. radial flow) and an
exponential tail stemming from short-lived resonances ($c\tau\approx$1–2 fm)
strongly decaying into the particles composing the pair of interest. In the
baryon–baryon sector, the determination of $r_{\rm core}$ is anchored to the
p–p correlation since its underlying strong interaction is the best known. The
emitting source obtained with this model was used in several baryon–baryon and
baryon–antibaryon femtoscopic measurements [45, 43, 60] to study the
underlying strong interaction.

In the meson–baryon sector the role played by the p–p interaction in
constraining the source for baryon–baryon pairs is overtaken in this study by
the $\rm{K}^{+}$p system since the interaction is well known [49] and coupled
channels are not present [38]. The influence of short-lived resonances
($c\tau<5$ fm) on the source is quantified by evaluating the yields of each
resonance with TF [73] and by extracting the decay kinematics using transport
model dynamics implemented in EPOS [64]. According to these calculations,
which entail the full decay chain from heavy to light particles, around 52%
(36%) of the total $\rm{K}^{+}$(p) yield is primordial. The relevant
contributions of short-lived resonances decaying into $\rm{K}^{+}$are
summarised in Table 3. For these calculations, every hadron consisting of
light and strange quarks was taken into account. As already introduced in
Section 4, due to its large lifetime of 46 fm, the $\phi$ (1020) is considered
in this analysis as a primary particle, and its contribution is taken into
account as done for secondary particles. The resonance yields for the proton
are the same as reported in Ref. [72].

Table 3: List of resonances contributing at least 1% to the yield of $\rm{K^{+}}$. These fractions are computed with Thermal-FIST for pp minimum bias collisions at $\sqrt{s}~{}=~{}13$ TeV, and are used also for the other collision systems. Resonances | Fraction ($\%$)
---|---
$\rm{K^{*}(892)^{0}}$ | 21
$\rm{K^{*}(892)^{+}}$ | 11
$\rm{a_{0}(980)^{+}}$ | 1
$\rm{K^{*}_{2}(1430)^{0}}$ | 1
$\rm{K^{*}_{1}(1270)^{0}}$ | 1
$\phi$ (1020) | 6

The weighted average of the lifetimes of the resonances feeding into
$\rm{K}^{+}$(p) is 3.66 fm/$c$ (1.65 fm/$c$), while the weighted average of
the masses is 1.05 GeV/$c^{2}$ (1.36 GeV/$c^{2}$). The decay kinematics are
extracted from the EPOS transport model by generating high-multiplicity pp
events at $\sqrt{s}=13$ TeV and selecting primordial $\rm{K}^{+}$(p) and the
resonances feeding into the particle of interest. The particle resonance
cocktail is required to reproduce the weighted average of the masses
(lifetimes) of the resonances feeding into $\rm{K}^{+}$or p. Using as an input
the fraction of primordial $\rm{K}^{+}$and p, the weighted average of the
lifetimes/masses and the corresponding decay kinematics, the source is modeled
employing a dedicated Monte Carlo procedure, details of which are described in
Ref. [72].

In order to extract the core source size, the genuine $\rm{K}^{+}$p
correlation function $C(k^{*})$ can be modeled using the following equation

$C(k^{*})=\EuScript{N}\times(C_{\rm{K}^{+}p}(k^{*})),$ (4)

where the normalization constant, $\EuScript{N}$, is a free parameter of the
fit and is introduced to take into account any remaining contributions which
survive the correction procedure described in Section 4.1.

The correlation function $C(k^{*})$ is fitted with Eq. (4) in the range
$0<k^{*}<250$ MeV/$c$. A variation of $\pm 30$ MeV/$c$ for the upper $k^{*}$
value is applied and accounted for in the systematic uncertainties. The
theoretical correlation function, $C_{\rm{K}^{+}p}(k^{*})$, is evaluated using
the CATS framework [81]. The strong potential for the $\rm{K}^{+}$p is
constructed based on the scattering amplitude in chiral SU(3) dynamics [49],
using the procedure developed in Ref. [50]. The Coulomb interaction in the
same-charge pair is taken into account in the final potential directly using
CATS. The source is modeled including the effects of short-lived resonances
via a dedicated Monte Carlo procedure implemented in CATS, details are given
in Ref. [72]. In order to account for the uncertainty related to the
correction procedure, the $\lambda$ parameter estimation, and the baseline,
the data were fitted several times. Each time either the $\lambda$ parameter
or the parameters of the baseline defined in Eq. (2) were varied by $\pm
1\sigma$. The radii obtained from the fits which minimise the $\chi^{2}$ per
number of degrees of freedom ($\chi^{2}$/NDF) represent the $r_{\rm core}$ for
$\rm{K}^{+}$p pairs. In order to evaluate the systematic uncertainty on
$r_{\rm core}$, half of the maximum variations among the different radii
extracted from the multiple fits are considered. Possible biases in the input
hadronic spectrum were cross checked by comparing the resonance yields
obtained from TF with model calculations based on Ref. [82]. The yields
obtained with both methods agree within 5%, for the short-lived as well as the
long-lived resonances. Therefore, in order to account properly for
discrepancies due to the model choice, the fraction of resonances that decay
into K and p was varied by $\pm$10%. Comparing the two different models, the
extracted values for the weighted average of the lifetimes (masses) were found
to be consistent within 10%. Finally, the $k^{*}$ cutoff in the EPOS transport
code, which is used to extract the decay kinematics, is varied from 200
MeV/$c$ to 150 and 250 MeV/$c$. The limit in $k^{*}$ is used to accept pairs
lying within the kinematic range for which femtoscopic effects are expected.
All the variations described here are bootstrapped [69] to obtain a systematic
uncertainty related to resonance contributions.

In Fig. 1, the results obtained for the measured $\rm{K}^{+}$p correlation in
pp collisions at $\sqrt{s}~{}=~{}13$ TeV are shown. The data are fitted with
Eq. (4) and a similar fitting procedure has been performed in p–Pb collisions
(see Fig. 2) and Pb–Pb collisions (see Fig. 3). The data in all three
colliding systems are well reproduced by the assumed $\rm{K}^{+}$p
interaction. The red band in the upper panels represents the model calculation
and its systematic uncertainty. The lower panel of each figure shows the
difference between data and model evaluated in the middle of each $k^{*}$
interval, and divided by the statistical uncertainty of data
(n${}_{\sigma_{\rm stat}}$). The width of the bands represents the
n${}_{\sigma_{\rm stat}}$ range associated to the model systematic variations.
The reduced $\chi^{2}$ are also shown. The values of $r_{\mathrm{core}}$ are
reported on the figures along with the corresponding effective Gaussian source
size $r_{\rm eff}$, which describes the resulting core source with the
resonance contributions. The associated statistical and systematic
uncertainties are also reported. This study shows an agreement between the
results obtained from p–p correlations in Ref. [72] and the extracted $r_{\rm
core}$ for $\rm{K}^{+}$p pairs and the $\rm{K}^{+}$p system is used to fix the
core radius for both the elastic channel $S_{\rm{K}^{-}p}(r^{*})$ and the
inelastic channels $S_{j}(r^{*})$ of the $\rm{K}^{-}$p interaction as
described in Eq. (4.2).

The sources of other pairs involving kaons and nucleons, such as
$S_{\rm{K^{-}p}}(r^{*})$ and $S_{\rm{\rm\overline{K}^{0}n}}(r^{*})$, are
modeled assuming the same feed-down from resonances and decay kinematics as
described for the $\rm{K}^{+}$p pairs. The emitting source for the inelastic
correlations containing pions,
$S_{\rm\uppi^{-}\Sigma^{+}}(r^{*}),~{}S_{\rm\uppi^{+}\Sigma^{-}}(r^{*}),~{}S_{\rm\uppi^{0}\Sigma^{0}}(r^{*})$,
and $S_{\rm\uppi^{0}\Lambda}(r^{*})$, are treated separately from the source
of the elastic $\rm{K}^{-}$p and charge-exchange $\rm\overline{K}^{0}$n terms
due to the presence of pions [38]. The resonance contributions to the $\uppi$
part of
$S_{\rm\uppi^{-}\Sigma^{+}}(r^{*}),~{}S_{\rm\uppi^{+}\Sigma^{-}}(r^{*}),~{}S_{\rm\uppi^{0}\Sigma^{0}}(r^{*})$,
and $S_{\rm\uppi^{0}\Lambda}(r^{*})$ are assumed to be equivalent. The feed-
down from resonances to $\Sigma$ and $\Lambda$ baryons is similar, hence the
sources for the $\uppi\Sigma$ and $\rm\uppi^{0}\Lambda$ contributions are
modeled using resonances and decay kinematics related to $\Lambda$ particles
as in Ref. [72], together with the ones necessary for the pions. Around 28%
(36%) of the total $\uppi$ ($\Sigma$/$\Lambda$) yields are primordial. The
contributions to the $\uppi$ yield stemming from the different resonances are
summarised in Table 4. The remaining resonances, not explicitly quoted in
Table 4, contribute at the sub percent level and largely decay into a nucleon
and a $\uppi$. Around 12% (including the yield from $\omega$ (782)) of the
total $\uppi$ yield stems from strongly decaying resonances with a lifetime
larger than 5 fm/$c$. Due to the long lifetime, these contributions are
included in the weak feed-down embedded in the $\lambda$ parameters,
consistent with the handling of the $\rm{K}^{+}$stemming from the $\phi$
decay.

Table 4: List of resonances which contribute at least 1% to the yield of $\uppi^{+}$. The fractions are computed with Thermal-FIST for pp minimum bias collisions at $\sqrt{s}~{}=~{}13$ TeV, and used also for the other collision systems. Resonances | Fraction (%)
---|---
$\rm{\rho(770)^{0}}$ | 9.0
$\rm{\rho(770)^{+}}$ | 8.7
$\rm{\omega}$(782) | 7.7
$\rm{K^{*}({892})^{+}}$ | 2.3
$\rm{\overline{K}^{*}({892})^{0}}$ | 2.6
$\rm{b_{1}({1235})^{0}}$ | 1.9
$\rm{a_{2}({1320})^{+}}$ | 1.5
$\rm{\eta}$ | 1.5
$\rm{a_{1}({1260})^{+}}$ | 1.4
$\rm{f_{2}({1270})}$ | 1.4
$\rm{a_{0}({980})^{+}}$ | 1.4
$\rm{h_{1}({1170})}$ | 1.2

The weighted average of the lifetimes of the resonances feeding into $\uppi$
($\Sigma$/$\Lambda$) is 1.5 fm/$c$ (4.7 fm/$c$), while the weighted average of
the masses is 1.13 GeV/$c^{2}$ (1.46 GeV/$c^{2}$). The decay kinematics are
extracted by using the transport model dynamics implemented in EPOS, as done
for $\rm{K}^{+}$.

Following this procedure, the source terms for the coupled channel
contributions as well as for the genuine $\rm{K}^{-}$p correlation in Eq.
(4.2) are evaluated by using the size of the Gaussian core source from the
$\rm{K}^{+}$p results as a starting value in the fit procedure and by taking
into account the specific feed-down from resonances. Similarly to the
$\rm{K}^{+}$p modeling, Eq. (4) is used to model the measured $\rm{K}^{-}$p
correlation in the three colliding systems.

The theoretical correlation function, including the elastic and inelastic
contributions, is evaluated within the CATS framework using the wave functions
obtained from state-of-the-art chiral coupled channel
$\rm\overline{K}N$–$\uppi\Sigma$–$\rm\uppi^{0}\Lambda$ potentials [50] based
on the scattering amplitudes in Refs. [36, 83] (hereafter chiral model). The
boundary conditions on the outgoing $\rm{K}^{-}$p wave function properly take
into account the contributions from the inelastic channels
$\rm\overline{K}^{0}$n, $\uppi\Sigma$ and $\rm\uppi^{0}\Lambda$ [8]. The
parameters of the chiral model are tuned to reproduce the available scattering
data and the kaonic hydrogen constraints from the SIDDHARTA experiment [20].

##  5 Results and discussion

The results for $\rm{K}^{-}$p pairs are shown in the upper panel of Fig. 4,
Fig. 5 and Fig. 6 for pp, p–Pb and Pb–Pb collisions, respectively. The effect
of the coupled channels on the measured correlation function can be seen
explicitly by comparing the data in the three colliding systems. For the
centrality interval 60–70% in Pb–Pb collisions (left panel in Fig. 6), in
which the largest source size, considered in this study, is achieved, no clear
evidence of the expected $\rm\overline{K}^{0}$n opening cusp is observed in
the data. The disappearance of this cusp in large colliding systems is
expected as described in [8, 46] based on the coupled channel formulation. The
lack of a cusp in the correlation function is also observed in $\rm{K}^{-}$p
femtoscopic measurements performed in central Pb–Pb collisions [47] in which
the emitting source size can exceed radii of 5 fm. As the source size
decreases, moving to more peripheral Pb–Pb collisions (middle and right panels
in Fig. 6), the cusp structure at $k^{*}\approx 59$ MeV/$c$ becomes more
evident. The opening of the $\rm\overline{K}^{0}$n channel is noticed as well
in the measured correlation function in pp (Fig. 4) and in the three
centrality classes in p–Pb collisions (Fig. 5), in which the core source size
is around 1 fm.

The results are compared with the theoretical predictions from the chiral
potentials described in Section 4. The blue bands visible in Fig. 4, in Fig.
5, and Fig. 6 are obtained by fixing the production weights $\omega^{\rm
prod}_{j}$ to the expected values in Table 2. An agreement with the data is
achieved in particular for the colliding systems with the larger source sizes
as in the first panel of Fig. 6. As the source size decreases, moving to more
peripheral Pb–Pb collisions, to p–Pb and pp systems, the theoretical band
clearly underestimates the data in the region of the $\rm\overline{K}^{0}$n
cusp. To quantify this deviation, a scaling factor $\alpha_{j}$ is introduced
in the definition of the conversion weights
$\omega_{j}=\alpha_{j}\times\omega^{\rm prod}_{j}$. The red bands in Fig. 4,
in Fig. 5, and Fig. 6 are obtained by letting $\alpha_{j}$ free for the
$\rm\overline{K}^{0}$n and $\uppi\Sigma$ contributions, while keeping the
production weights fixed. The scaling factor corresponding to the
$\rm\uppi^{0}\Lambda$ channel is kept at unity due to the negligible coupling
of this channel to the $\rm{K}^{-}$p system [9, 10, 7, 8]. The bottom panels
of Fig. 4, Fig. 5, and Fig. 6 show the data-to-model difference expressed in
terms of number of statistical standard deviations ($\sigma_{\mathrm{stat}}$)
of the data. The width of the bands represents the n${}_{\sigma_{\rm stat}}$
range associated to the model systematic variations. Blue squares show the
n${}_{\sigma_{\mathrm{stat}}}$ distribution when the $\alpha_{j}$ are fixed to
unity, while red squares show the results when $\alpha_{j}$ for
$j=\rm\overline{K}^{0}\rm n$, $\uppi\Sigma$ are free parameters of the fit.
The $\chi^{2}$ over the number of degree of freedom ($\chi^{2}$/NDF) of the
fit obtained by using the two approaches are also shown. Clearly, the current
tune of the chiral model is unable to properly describe the opening of the
inelastic $\rm\overline{K}^{0}$n channel.

The results of the extracted $\alpha_{j}$ for $j=\rm\overline{K}^{0}\rm n$,
$\uppi\Sigma$ are shown in Fig. 7 as a function of the $r_{\rm core}$ radius,
for all colliding systems. The black and red bands represent the uncertainties
coming from the yield estimates in TF and the variations applied in the BW
kinematics for $\rm\overline{K}^{0}$n and $\uppi\Sigma$ production weights,
respectively. The vertical error bars in the figure represent the statistical
uncertainty of the extracted parameter, while the boxes represent the
systematic uncertainty on the conversion weights obtained by repeating the fit
several times and by varying within statistical and systematic uncertainties
$r_{\rm core}$ and the amount of resonances that contribute to the source
distribution. The $\uppi\Sigma$ scaling factors (red squares) obtained at
different core radii are compatible with unity, indicating that this coupling
to $\rm{K}^{-}$p is well modeled by the underlying chiral interaction. The
extracted scaling factor for the $\rm\overline{K}^{0}$n coupling (black
circles) significantly deviates from unity for values of $r_{\rm core}$ below
1.5 fm.

As can be seen in Fig. 7, for Pb–Pb collisions at 60–70% centrality
($r_{\mathrm{{core}}}$ = $1.53\pm 0.05\pm 0.24$ fm), the scaling factor
$\alpha_{\rm\overline{K}^{0}\rm n}$ is consistent with unity. Both fit
approaches shown in the left panel of Fig. 6 can be used to describe the data,
as can be seen in the $\rm n_{\sigma_{\rm stat}}$ distribution. These results
are expected since the 60–70% centrality corresponds to the largest source
size ($r_{\rm core}=1.53$ fm) considered in this work, in which the coupled
channel effects on the correlation are negligible. As the source decreases and
as the $\rm\overline{K}^{0}$n cusp appears, the fit assuming free scaling
factors provides a better description of the data and the extracted
$\alpha_{\rm\overline{K}^{0}n}$ are larger than unity. This deviation is
likely due to the fact that a direct experimental constraint of the
$\rm{K}^{-}$p to $\rm\overline{K}^{0}$n coupling was not available before the
measurements presented here and for the similar analysis performed in minimum
bias events in pp collisions at $\sqrt{s}=5.02$, 7, and 13 TeV [38]. The
deviation from unity of $\alpha_{\rm\overline{K}^{0}\rm n}$ indicates that the
transition between the $\rm{K}^{-}$p and the $\rm\overline{K}^{0}$n channel
currently implemented in the chiral model model is too weak. Since other
observables are also affected by the coupling of $\rm{K}^{-}$p and
$\rm\overline{K}^{0}$n, it is necessary in future studies to update the $S=-1$
meson-baryon scattering amplitude of
$\rm\overline{K}N$–$\uppi\Sigma$–$\uppi\Lambda$ system by including the
present correlation function measurements in addition to the available kaonic
hydrogen and scattering data. Moreover, the observed strong negative
correlation between the coupling weights $\alpha_{\rm\overline{K}^{0}n}$ and
$\alpha_{\uppi\Sigma}$ indicates that a revision of the full coupled-channel
$\rm{K}^{-}$p potential is required to describe the measured correlation
function.

The data presented in this work provide unique constraints to pin down the
coupling strength to the $\rm\overline{K}^{0}$n channel and indicate the first
difference between the chiral model and experimental data on the $\rm{K}^{-}$p
interaction. A fine tuning of the chiral model is beyond the scope of this
article since it requires a more detailed investigation on how much variation
of the two isospin components of the $\rm\overline{K}N$ interaction is allowed
in order to still fulfil the scattering data and kaonic hydrogen constraints,
above and below threshold, respectively.

Figure 7: Scaling factor ($\alpha_{j}$) for $\rm\overline{K}^{0}$n (black
circles) and $\uppi\Sigma$ (red squares) extracted from the different fits of
the $\rm{K}^{-}$p correlation function as a function of the core radius
$r_{\mathrm{core}}$ extracted for pp, p–Pb and Pb–Pb collisions. The vertical
error bars and boxes represent the statistical and systematic uncertainties on
the extracted parameters, respectively. The widths of the boxes represent the
systematic uncertainty associated to each extracted $r_{\mathrm{core}}$. The
black and red bands represent the uncertainty coming from the yield estimates
in TF and the variations applied in the BW kinematics summed in quadrature as
described in the text for $\rm\overline{K}^{0}$n and $\uppi\Sigma$,
respectively.

##  6 Summary

This study contains a detailed and comprehensive analysis of the $\rm{K}^{+}$p
and $\rm{K}^{-}$p correlation function in different colliding systems. In the
$\rm{K}^{-}$p correlation function, it is possible to observe the opening of
the $\rm\overline{K}^{0}$n channel in pp, p–Pb, and peripheral Pb–Pb
collisions ($r_{\mathrm{core}}\lesssim$ 1.5 fm), while it is less pronounced
in semi-central Pb–Pb collisions, hence for larger distances. By using the
strong potential for $\rm{K}^{+}$p pairs based on scattering amplitudes in
chiral SU(3) dynamics and the Coulomb potential, it was possible to extract
the common core radius ($r_{\rm core}$) of the Gaussian source, which,
together with an exponential tail, is used to model the particle emitting
source of $\rm{K}^{+}$p. The same $r_{\rm core}$ is used to model the source
for $\rm{K}^{-}$p. In this case, the source is composed of the elastic
$\rm{K}^{-}$p and the charge-exchange $\rm\overline{K}^{0}$n terms ($S_{\rm
KN}$), and the inelastic correlations containing a pion term
($S_{\uppi\Sigma/\rm\uppi^{0}\Lambda}$). The accurate source description
allows us to study the coupling strength associated with each of the inelastic
channels as a function of the source size. In order to assess the $\omega_{j}$
conversion parameters of the different coupled channels present in the
$\rm{K}^{-}$p system, a novel approach was used to estimate the contributions
of the inelastic channels to the measured correlation function. The
$\omega_{j}$ parameters can be expressed as the product of two terms:
$\omega_{j}^{\rm prod}$, which takes into consideration the production yield
of each particle pairs, and the scaling factor $\alpha_{j}$. The latter should
be equal to unity if the coupling strength is correctly estimated within the
Kyoto model. From the fits to the measured correlation functions with the
state-of-the-art Kyoto model, calculated within the coupled channel approach,
it is possible to observe that the dynamics of the coupled channels is under
control in the case of $\uppi\Sigma$, while the deviation from unity of
$\alpha_{\rm\overline{K}^{0}\rm n}$ indicates that the transition between the
$\rm{K}^{-}$p and the $\rm\overline{K}^{0}$n channel, as currently implemented
in the Kyoto model, is too weak. Hence, the data presented in this work
provide a unique constraint to pin down the coupling strength to the
$\rm\overline{K}^{0}$n channel.

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The results on the amount of primordial protons and $\Lambda$ and the
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