2018 150 30 May

p–p, p$\mbox{--}\Lambda$ and $\Lambda\mbox{--}\Lambda$ correlations studied
via femtoscopy in pp at $\sqrt{s}=7$ TeV

ALICE Collaboration††thanks: See Appendix B for the list of collaboration
members ALICE Collaboration

We report on the first femtoscopic measurement of baryon pairs, such as p–p ,
p–$\Lambda$ and $\Lambda$–$\Lambda$ , measured by ALICE at the Large Hadron
Collider (LHC) in proton-proton collisions at $\sqrt{s}=7$ TeV. This study
demonstrates the feasibility of such measurements in
$\mathrm{p\kern-0.50003ptp}$ collisions at ultrarelativistic energies. The
femtoscopy method is employed to constrain the hyperon–nucleon and
hyperon–hyperon interactions, which are still rather poorly understood. A new
method to evaluate the influence of residual correlations induced by the
decays of resonances and experimental impurities is hereby presented. The p–p
, p–$\Lambda$ and $\Lambda$–$\Lambda$ correlation functions were fitted
simultaneously with the help of a new tool developed specifically for the
femtoscopy analysis in small colliding systems “Correlation Analysis Tool
using the Schrödinger Equation” (CATS). Within the assumption that in
$\mathrm{p\kern-0.50003ptp}$ collisions the three particle pairs originate
from a common source, its radius is found to be equal to $r_{0}=1.144\pm
0.019\,\mathrm{(stat)}\,^{+0.069}_{-0.012}\,\mathrm{(syst)}$ fm. The
sensitivity of the measured p–$\Lambda$ correlation is tested against
different scattering parameters which are defined by the interaction among the
two particles, but the statistics is not sufficient yet to discriminate among
different models. The measurement of the $\Lambda$–$\Lambda$ correlation
function constrains the phase space spanned by the effective range and
scattering length of the strong interaction. Discrepancies between the
measured scattering parameters and the resulting correlation functions at LHC
and RHIC energies are discussed in the context of various models.

##  1 Introduction

Traditionally femtoscopy is used in heavy-ion collisions at ultrarelativistic
energies to investigate the spatial-temporal evolution of the particle
emitting source created during the collision [1, 2]. Assuming that the
interaction for the employed particles is known, a detailed study of the
geometrical extension of the emission region becomes possible [3, 4, 5, 6, 7,
8, 9, 10].
If one considers smaller colliding systems such as proton-proton
($\mathrm{p\kern-0.50003ptp}$) at TeV energies and assumes that the particle
emitting source does not show a strong time dependence, one can reverse the
paradigm and exploit femtoscopy to study the final state interaction (FSI).
This is especially interesting in the case where the interaction strength is
not well known as for hyperon–nucleon (Y–N) and hyperon–hyperon (Y–Y) pairs
[11, 12, 13, 14, 15, 16, 17, 18].
Hyperon–nucleon and hyperon–hyperon interactions are still rather poorly
experimentally constrained and a detailed knowledge of these interactions is
necessary to understand quantitatively the strangeness sector in the low-
energy regime of Quantum-Chromodynamics (QCD) [19].
Hyperon–nucleon (p–$\Lambda$ and p$\mbox{--}\Sigma$) scattering experiments
have been carried out in the sixties [20, 21, 22] and the measured cross
sections have been used to extract scattering lengths and effective ranges for
the strong nuclear potential by means of effective models such as the
Extended-Soft-Core (ESC08) baryon–baryon model [23] or by means of chiral
effective field theory ($\chi$EFT) approaches at leading order (LO) [24] and
next-to-leading order (NLO) [25]. The results obtained from the above-
mentioned models are rather different, but all confirm the attractiveness of
the $\Lambda$ –nucleon ($\Lambda$–N) interaction for low hyperon momenta. In
contrast to the LO results, the NLO solution claims the presence of a negative
phase shift in the p–$\Lambda$ spin singlet channel for $\Lambda$ momenta
larger than $p_{\Lambda}>600$ MeV/$c$. This translates into a repulsive core
for the strong interaction evident at small relative distances. The same
repulsive interaction is obtained in the p-wave channel within the ESC08 model
[23].
The existence of hypernuclei [26] confirms that the N–$\Lambda$ is attractive
within nuclear matter for densities below nuclear saturation
$\rho_{0}=\,0.16~{}\mathrm{fm}^{-3}$. An average value of
$U(\rho=\rho_{0},k=0)\approx\,-30$ MeV [26], with $k$ the hyperon momentum in
the laboratory reference system, is extracted from hypernuclear data on the
basis of a dispersion relation for hyperons in a baryonic medium at $\rho_{0}$
.
The situation for the $\Sigma$ hyperon is currently rather unclear. There are
some experimental indications for the formation of $\Sigma$–hypernuclei [27,
28] but different theoretical approaches predict both attractive and repulsive
interactions depending on the isospin state and partial wave [29, 23, 25]. The
scarce experimental data for this hypernucleus prevents any validation of the
models.
A $\Xi$–hypernucleus candidate was detected [30] and ongoing measurements
suggest that the N–$\Xi$ inter-action is weakly attractive [31]. A recent work
by the Lattice HAL-QCD Collaboration [32] shows how this attractive
interaction could be visible in the p–$\Xi$ femtoscopy analysis, in particular
by comparing correlation functions for different static source sizes. This
further motivates the extension of the femtoscopic studies from heavy ions to
$\mathrm{p\kern-0.50003ptp}$ collisions since in the latter case the source
size decreases by about a factor of three at the LHC energies leading to an
increase in the strength of the correlation signal [33].
If one considers hyperon–hyperon interactions, the most prominent example is
the $\Lambda$–$\Lambda$ case. The H-dibaryon $\Lambda$–$\Lambda$ bound state
was predicted [34] and later a double $\Lambda$ hypernucleus was observed
[35]. From this single measurement a shallow $\Lambda$–$\Lambda$ binding
energy of few MeV was extracted, but the H-dibaryon state was never observed.
Also recent lattice calculations [36] obtain a rather shallow attraction for
the $\Lambda$–$\Lambda$ state.
The femtoscopy technique was employed by the STAR collaboration to study
$\Lambda$–$\Lambda$ correlations in Au–Au collisions at
$\sqrt{s_{\mathrm{NN}}}=200$ GeV [15]. First a shallow repulsive interaction
was reported for the $\Lambda$–$\Lambda$ system, but in an alternative
analysis, where the residual correlations were treated more accurately [37], a
shallow attractive interaction was confirmed. These analyses demonstrate the
limitations of such measurements in heavy-ion collisions, where the source
parameters are time-dependent and the emission time might not be the same for
all hadron species.
The need for more experimental data to study the hyperon–nucleon,
hyperon–hyperon and even the hyperon–nucleon–nucleon interaction has become
more crucial in recent years due to its connection to the modeling of
astrophysical objects like neutron stars [38, 39, 40, 41]. In the inner core
of these objects the appearance of hyperons is a possible scenario since their
creation at finite density becomes energetically favored in comparison with a
purely neutron matter composition [40]. However, the appearance of these
additional degrees of freedom leads to a softening of the nuclear matter
equation of state (EOS) [42] making the EOS incompatible with the observation
of neutron stars as heavy as two solar masses [43, 44]. This goes under the
name of the ’hyperon puzzle’. Many attempts were made to solve this puzzle,
e.g. by introducing three-body forces for $\Lambda$NN leading to an additional
repulsion that can counterbalance the large gravitational pressure and finally
allow for larger neutron star masses [45, 46, 47, 48]. A repulsive core for
the two body forces would also stiffen the EOS containing hyperons.
In order to constrain the parameter space of such models a detailed knowledge
of the hyperon–nucleon, including $\Sigma$ and $\Xi$ states, and of the
hyperon–nucleon–nucleon interaction is mandatory.
This work presents an alternative to scattering experiments, using the
femtoscopy technique to study the Y–N and Y–Y interactions in
$\mathrm{p\kern-0.50003ptp}$ collisions at $\sqrt{s}=7\,$TeV. We show that
$\mathrm{p\kern-0.50003ptp}$ collisions at the LHC are extremely well suited
to investigate baryon–baryon final state interactions and that the measurement
of the correlation function is not contaminated with the mini-jet background
visible in meson–meson correlations [49, 50]. The extracted p–p , p–$\Lambda$
and $\Lambda$–$\Lambda$ correlations have been compared to the predicted
function obtained by solving the Schrödinger equation exactly by employing the
Argonne $v_{18}$ potential [51] for p–p pairs and different scattering
parameters available in the literature for p–$\Lambda$ and $\Lambda$–$\Lambda$
pairs. The predictions for the correlation function used to fit the data are
obtained with the newly developed CATS framework [52]. A common source with a
constant size is assumed and the value of the radius is extracted.
The work is organized in the following way: in Section II the experiment setup
and the analysis technique are briefly introduced. In Section III the
femtoscopy technique and the theoretical models employed are discussed. In
Section IV the sources of systematic uncertainties are summarized and finally
in Section V the results for the p–p , p–$\Lambda$ and $\Lambda$–$\Lambda$
correlation function are presented.

##  2 Data analysis

In this paper we present results from studies of the p–p , p–$\Lambda$ and
$\Lambda$–$\Lambda$ correlations in $\mathrm{p\kern-0.50003ptp}$ collisions at
$\sqrt{s}=7$ TeV employing the data collected by ALICE in 2010 during the LHC
Run 1. Approximately $3.4\times 10^{8}$ minimum bias events have been used for
the analysis, before event and track selection. A detailed description of the
ALICE detector and its performance in the LHC Run 1 (2009-2013) is given in
[53, 54]. The inner tracking system (ITS) [53] consists of six cylindrical
layers of high resolution silicon detectors placed radially between $3.9$ and
$43$ cm around the beam pipe. The two innermost layers are silicon pixel
detectors (SPD) and cover the pseudorapidity range $|\eta|<2$. The time
projection chamber (TPC) [55] provides full azimuthal coverage and allows
charged particle reconstruction and identification (PID) via the measurement
of the specific ionization energy loss d$E$/d$x$ in the pseudorapidity range
$|\eta|<0.9$. The Time-Of-Flight (TOF) [56] detector consists of Multigap
Resistive Plate Chambers covering the full azimuthal angle in $|\eta|<0.9$.
The PID is obtained by measuring the particle’s velocity $\beta$. The above
mentioned detectors are immersed in a $B=0.5$ T solenoidal magnetic field
directed along the beam axis. The V0 are small-angle plastic scintillator
detectors used for triggering and placed on either side of the collision
vertex along the beam line at $+3.3$ m and $-0.9$ m from the nominal
interaction point, covering the pseudorapidity ranges $2.8<\eta<5.1$ (V0-A)
and $-3.7<\eta<-1.7$ (V0-C).

###  2.1 Event selection

The minimum bias interaction trigger requires at least two out of the
following three conditions: two pixel chips hit in the outer layer of the
silicon pixel detectors, a signal in V0-A, a signal in V0-C [54].
Reconstructed events are required to have at least two associated tracks and
the distance along the beam axis between the reconstructed primary vertex and
the nominal interaction point should be smaller than 10 cm. Events with
multiple reconstructed SPD vertices are considered as pile-up. In addition,
background events are rejected using the correlation between the number of SPD
clusters and the tracklet multiplicity. The tracklets are constrained to the
primary vertex, and hence a typical background event is characterized by a
large amount of SPD clusters but only few tracklets, while a pile-up event
contains a larger number of clusters at the same tracklet multiplicity.

After application of these selection criteria, about $2.5\times 10^{8}$
million events are available for the analysis.

###  2.2 Proton candidate selection

To ensure a high purity sample of protons, strict selection criteria are
imposed on the tracks. Only particle tracks reconstructed with the TPC without
additional matching with hits in the ITS are considered in the analysis in
order to avoid biases introduced by the non-uniform acceptance in the ITS.
However, the track fitting is constrained by the independently reconstructed
primary vertex. Hence, the obtained momentum resolution is comparable to that
of globally reconstructed tracks, as demonstrated in [54].

The selection criteria for the proton candidates are summarized in Tab. 1. The
selection on the number of reconstructed TPC clusters serve to ensure the
quality of the track, to assure a good $p_{\mathrm{T}}$ resolution at large
momenta and to remove fake tracks from the sample. To enhance the number of
protons produced at the primary vertex, a selection is imposed on the
distance-of-closest-approach (DCA) in both beam ($z$) and transverse ($xy$)
directions. In order to minimize the fraction of protons originating from the
interaction of primary particles with the detector material, a low transverse
momentum cutoff is applied [57]. At high $p_{\mathrm{T}}$ a cutoff is
introduced to ensure the purity of the proton sample, as the purity drops
below 80 % for larger $p_{\mathrm{T}}$ due to the decreasing separation power
of the combined TPC and TOF particle identification.

For particle identification both the TPC and the TOF detectors are employed.
For low momenta ($p<0.75$ GeV/$c$) only the PID selection from the TPC is
applied, while for larger momenta the information of both detectors is
combined since the TPC does not provide a sufficient separation power in this
momentum region. The combination of TPC and TOF signals is done by employing a
circular selection criteria
$n_{\sigma,\mathrm{combined}}\equiv\sqrt{(n_{\sigma,\mathrm{TPC}})^{2}+(n_{\sigma,\mathrm{TOF}})^{2}}$,
where $n_{\sigma}$ is the number of standard deviations of the measured from
the expected signal at a given momentum. The expected signal is computed in
the case of the TPC from a parametrized Bethe–Bloch curve, and in the case of
the TOF by the expected $\beta$ of a particle with a mass hypothesis $m$. In
order to further enhance the purity of the proton sample, the $n_{\sigma}$ is
computed assuming different particle hypotheses (kaons, electrons and pions)
and if the corresponding hypothesis is found to be more favorable, i.e. the
$n_{\sigma}$ value found to be smaller, the proton hypothesis and thus the
track is rejected. With these selection criteria a $p_{\mathrm{T}}$-averaged
proton purity of 99 % is achieved. The purity remains above 99 % for
$p_{\mathrm{T}}<2$ GeV/c and then decreases to 80 % at the momentum cutoff of
4.05 GeV/c.

Selection criterion | Value
---|---
Proton selection criteria |
Pseudorapidity | $|\eta|<0.8$
Transverse momentum |  $0.5<p_{\mathrm{T}}<4.05$ GeV/$c$
TPC clusters | $n_{\mathrm{TPC}}>80$
Crossed TPC pad rows |  $n_{\mathrm{crossed}}>70$ (out of 159)
Findable TPC clusters | $n_{\mathrm{crossed}}/n_{\mathrm{findable}}>0.83$
Tracks with shared TPC clusters | rejected
Distance of closest approach $xy$ |  $|\mathrm{DCA}_{xy}|<0.1$ cm
Distance of closest approach $z$ |  $|\mathrm{DCA}_{z}|<0.2$ cm
Particle identification |  $|n_{\sigma,\mathrm{TPC}}|<3$ for $p<0.75$ GeV/$c$
$n_{\sigma,\mathrm{combined}}<3$ for $p>0.75$ GeV/$c$
Lambda selection criteria |
Daughter track selection criteria |
Pseudorapidity | $|\eta|<0.8$
TPC clusters | $n_{\mathrm{TPC}}>70$
Distance of closest approach |  $\mathrm{DCA}>0.05$ cm
Particle identification | $|n_{\sigma,\mathrm{TPC}}|<5$
$V_{0}$ selection criteria |
Transverse momentum |  $p_{\mathrm{T}}>0.3$ GeV/$c$
$\Lambda$ decay vertex |  $|i_{\mathrm{vertex_{\Lambda}}}|<100$ cm, $i$=$x$,$y$,$z$
Transverse radius of the decay vertex $r_{xy}$ | 0.2$<r_{xy}<$100 cm
DCA of the daughter tracks at the decay vertex |  $\mathrm{DCA}(|p,\pi|)<$1.5 cm
Pointing angle $\alpha$ | $\cos\alpha>0.99$
K0 rejection |  $0.48<\mathrm{M}_{\pi^{+}\pi^{-}}<0.515$ GeV/$c^{2}$
$\Lambda$ selection |  $|\mathrm{M}_{\mathrm{p}\pi}-\mathrm{M}_{\Lambda,\mathrm{PDG}}|<4$ MeV/$c^{2}$
Table 1: Proton (top) and $\Lambda$ candidate (bottom) selection criteria.

###  2.3 $\mathbf{Lambda}$ candidate selection

The weak decay $\Lambda\rightarrow\mathrm{p}\pi^{-}$ (BR= 63.9 %, $c\tau=7.3$
cm [58]) is exploited for the reconstruction of the $\Lambda$ candidate, and
accordingly the charge-conjugate decay for the $\overline{\Lambda}$
identification. The reconstruction method forms so-called $V_{0}$ decay
candidates from two charged particle tracks using a procedure described in
[59]. The selection criteria for the $\Lambda$ candidates are summarized in
Tab. 1. The $V_{0}$ daughter tracks are globally reconstructed tracks and, in
order to maximize the efficiency, selected by a broad particle identification
cut employing the TPC information only. Additionally, the daughter tracks are
selected by requiring a minimum impact parameter of the tracks with respect to
the primary vertex. After the selection all positively charged daughter tracks
are combined with a negatively charged partner to form a pair. The resulting
$\Lambda$ vertex $i_{\mathrm{vertex_{\Lambda}}}$, $i$=$x$,$y$,$z$ is then
defined as the point of closest approach between the two daughter tracks. This
distance of closest approach of the two daughter tracks with respect to the
$\Lambda$ decay vertex $\mathrm{DCA}(|p,\pi|)$ is used as an additional
quality criterion of the $\Lambda$ candidate.

The $\Lambda$ momentum is calculated as the sum of the daughter momenta. A
minimum transverse momentum requirement on the $\Lambda$ candidate is applied
to reduce the contribution of fake candidates. Finally, a selection is applied
on the opening angle $\alpha$ between the $\Lambda$ momentum and the vector
pointing from the primary vertex to the secondary $V_{0}$ decay vertex. The
rather broad PID selection of the daughter tracks introduces a residual pion
contamination of the proton daughter sample that in combination with the
charge-conjugate pion of the $V_{0}$ leads to the misidentification of
$K^{0}_{S}$ as $\Lambda$ candidates. These $K^{0}_{S}$ candidates are removed
by a selection on the $\pi^{+}\pi^{-}$ invariant mass.

The reconstructed invariant mass, its resolution and purity are determined by
fitting eight spectra of the same size in $p_{\mathrm{T}}\in[0.3,~{}4.3]$
GeV/$c$ with the sum of two Gaussian functions describing the signal and a
second-order polynomial to emulate the combinatorial background. The obtained
values for the mean and variance of the two Gaussian functions are combined
with an arithmetic average. The determined mass is in agreement with the PDG
value for the $\Lambda$ and $\overline{\Lambda}$ particles [58]. A total
statistics of $5.9\times 10^{6}$ and $5.5\times 10^{6}$ and a signal to
background ratio of 20 and 25 at a $p_{\mathrm{T}}$-averaged purity of 96 %
and 97 % is obtained for $\Lambda$ and $\overline{\Lambda}$ , respectively. It
should be noted that the $\Lambda$ purity is constant within the investigated
$p_{\mathrm{T}}$ range. Finally, a selection on the $p\pi^{-}$
($\overline{\mathrm{p}}\pi^{+}$) invariant mass is applied. To avoid any
contribution from auto-correlations, all $\Lambda$ candidates are checked for
shared daughter tracks. If this condition is found to be true, the $\Lambda$
candidate with the smaller cosine pointing angle is removed from the sample.
If a primary proton is also used as a daughter track of a $\Lambda$ candidate,
the latter is rejected. Figure 1 shows the $p_{\mathrm{T}}$-integrated
invariant mass of the $\Lambda$ and $\overline{\Lambda}$ candidates.

Figure 1: (Color online) Invariant mass distribution of $\mathrm{p}\pi^{-}$
($\overline{\mathrm{p}}\pi^{+}$) to obtain the $\Lambda$ ($\overline{\Lambda}$
) signal. The dashed lines set the selection width used in the analysis.

##  3 The correlation function

The observable of interest in femtoscopy is the two-particle correlation
function, which is defined as the probability to find simultaneously two
particles with momenta $\mathbf{p}_{1}$ and $\mathbf{p}_{2}$ divided by the
product of the corresponding single particle probabilities

$C(\mathbf{p}_{1},\mathbf{p}_{2})\equiv\frac{P(\mathbf{p}_{1},\mathbf{p}_{2})}{P(\mathbf{p}_{1})\cdot
P(\mathbf{p}_{2})}.$ (1)

These probabilities are directly related to the inclusive Lorentz invariant
spectra
$P(\mathbf{p}_{1},\mathbf{p}_{2})=E_{1}E_{2}\frac{\mathrm{d}^{6}N}{\mathrm{d}^{3}p_{1}\mathrm{d}^{3}p_{2}}$
and
$P(\mathbf{p}_{1,2})=E_{1,2}\frac{\mathrm{d}^{3}N}{\mathrm{d}^{3}p_{1,2}}$. In
absence of a correlation signal the value of
$C(\mathbf{p}_{1},\mathbf{p}_{2})$ equals unity.

Approximating the emission process and the momenta of the particles, the size
of the particle emitting source can be studied. Following [2], Eq. (1) can
then be rewritten as

$C(\mathbf{k^{*}})=\int\,d^{3}r^{*}S(r^{*})|\psi(r^{*},\mathbf{k^{*}})|^{2},$
(2)

where $k^{*}$ is the relative momentum of the pair defined as
$k^{*}=\frac{1}{2}\cdot|\mathbf{p}^{*}_{1}-\mathbf{p}^{*}_{2}|$, with
$\mathbf{p}^{*}_{1}$ and $\mathbf{p}^{*}_{2}$ the momenta of the two particles
in the pair rest frame, $S(r^{*})$ contains the distribution of the relative
distance of particle pairs in the pair rest frame (PRF, denoted by the ∗), the
so-called source function, and $\psi(r^{*},\mathbf{k^{*}})$ denotes the
relative wave function of the particle pair. The latter contains the particle
interaction term and determines the shape of the correlation function. In this
work, the p–p correlation function, which is theoretically well understood, is
employed to obtain the required information about the source function and this
information will be used to study the p–$\Lambda$ and $\Lambda$–$\Lambda$
interaction.

In order to relate the correlation function to experimentally accessible
quantities, Eq. (1) is reformulated [2] as

$C(k^{*})=\mathcal{N}\frac{A(k^{*})}{B(k^{*})},$ (3)

The distribution of particle pairs from the same event is denoted with
$A(k^{*})$ and $B(k^{*})$ is a reference sample of uncorrelated pairs. The
latter is obtained using event mixing techniques, in which the particle pairs
of interest are combined from single particles from different events. To avoid
acceptance effects of the detector system, the mixing procedure is conducted
only between particle pairs stemming from events with similar $z$ position of
the primary vertex and similar multiplicity [2]. The normalization parameter
for mixed and same event yields $\mathcal{N}$ is chosen such that the mean
value of the correlation function equals unity for $k^{*}\in[0.2,~{}0.4]$
GeV/$c$.

As correlation functions of all studied baryon–baryon pairs, i.e. p–p ,
p–$\Lambda$ and $\Lambda$–$\Lambda$ , exhibit identical behavior compared to
those of their respective anti-baryon–anti-baryon pairs, the corresponding
samples are combined to enhance the statistical significance. Therefore, in
the following p–p denotes the combination of $\mbox{p--
p}\oplus\overline{\mathrm{p}}\mbox{--}\overline{\mathrm{p}}$, and accordingly
for p–$\Lambda$ and $\Lambda$–$\Lambda$ .

###  3.1 Decomposition of the correlation function

The experimental determination of the correlation function is distorted by two
distinct mechanisms. The sample of genuine particle pairs include
misidentified particles and feed-down particles from strong and weak decays.

In this work a new method to separate all the individual components
contributing to a measured correlation signal is proposed. The correlation
functions arising from resonances or impurities of the sample are weighted
with the so-called $\lambda$ parameters and in this way are taken into account
in the total correlation function of interest

$C(k^{*})=1+\lambda_{\mathrm{genuine}}\cdot(C_{\mathrm{genuine}}(k^{*})-1)+\sum_{ij}\lambda_{ij}(C_{ij}(k^{*})-1),$
(4)

where the $i,j$ denote all possible impurity and feed-down contributions.
These $\lambda$ parameters can be obtained employing exclusively single
particle properties such as the purity and feed-down probability. The
underlying mathematical formalism is outlined in App. A.

For the case of p–p correlation the following contributions must be taken into
account

$\begin{split}\\{\mathrm{pp}\\}=&~{}\mathrm{pp+p_{\Lambda}p+p_{\Lambda}p_{\Lambda}+p_{\Sigma^{+}}p+p_{\Sigma^{+}}p_{\Sigma^{+}}}\\\
&\mathrm{+p_{\Lambda}p_{\Sigma^{+}}+\tilde{p}p+\tilde{p}p_{\Lambda}+\tilde{p}p_{\Sigma^{+}}+\tilde{p}\tilde{p}},\end{split}$
(5)

where $\tilde{X}$ refers to misidentified particles of specie $X$.
$p_{\Lambda}$ and $p_{\Sigma^{+}}$ correspond to protons stemming from the
weak decay of the corresponding hyperons. The
$\Xi\rightarrow\Lambda\pi\rightarrow p\pi\pi$ decays are explicitly considered
in the feed-down contribution of the p–$\Lambda$ correlation and hence are
omitted in Eq. (5) to avoid double counting. As shown in App. A, the fraction
of primary protons and their feed-down fractions are required to calculate the
$\lambda$ parameters of the different contributions to Eq. (5). The
information about the origin of the protons, i.e. whether the particles are of
primary origin, originating from feed-down or from the interactions with the
detector material, is obtained by fitting Monte Carlo (MC) templates to the
experimental distributions of the distance of closest approach of the track to
the primary vertex. The MC templates and the purity are extracted from Pythia
[60] simulations using the Perugia 2011 tune [61], which were filtered through
the ALICE detector and the reconstruction algorithm [53]. The $p_{\mathrm{T}}$
averages are then calculated by weighting the quantities of interest by the
respective particle yields d$N$/d$p_{\mathrm{T}}$. The resulting fraction of
primary protons averaged over $p_{\mathrm{T}}$ is 87 %, with the other 13 % of
the total yield associated to weak decays of resonances and the contribution
from the detector material is found to be negligible.

The feed-down from weakly decaying resonances is evaluated by using cross
sections from Pythia and for the proton sample consists of the $\Lambda$ (70
%) and $\Sigma^{+}$ (30 %) contributions. The individual contributions to the
total correlation function are presented in Tab. 2.

The decomposition of the p–$\Lambda$ correlation function is conducted in a
similar manner as for the p–p pair, however considering the purities and feed-
down fractions of both particles

$\begin{split}\\{\mathrm{p}\Lambda\\}=&~{}\mathrm{p}\Lambda+\mathrm{p}\Lambda_{\Xi^{-}}+\mathrm{p}\Lambda_{\Xi^{0}}+\mathrm{p}\Lambda_{\Sigma^{0}}+\mathrm{p}_{\Lambda}\Lambda+\mathrm{p}_{\Lambda}\Lambda_{\Xi^{-}}+\mathrm{p}_{\Lambda}\Lambda_{\Xi^{0}}+\mathrm{p}_{\Lambda}\Lambda_{\Sigma^{0}}\\\
&+\mathrm{p}_{\Sigma^{+}}\Lambda+\mathrm{p}_{\Sigma^{+}}\Lambda_{\Xi^{-}}+\mathrm{p}_{\Sigma^{+}}\Lambda_{\Xi^{0}}+\mathrm{p}_{\Sigma^{+}}\Lambda_{\Sigma^{0}}+\tilde{\mathrm{p}}\Lambda+\tilde{\mathrm{p}}\Lambda_{\Xi^{-}}+\tilde{\mathrm{p}}\Lambda_{\Xi^{0}}+\tilde{\mathrm{p}}\Lambda_{\Sigma^{0}}\\\
&+\mathrm{p}\tilde{\Lambda}+\mathrm{p}_{\Lambda}\tilde{\Lambda}+\mathrm{p}_{\Sigma^{+}}\tilde{\Lambda}+\tilde{\mathrm{p}}\tilde{\Lambda}.\end{split}$
(6)

The $\Lambda$ purity is obtained from fits to the invariant mass spectra in
eight bins of $p_{\mathrm{T}}$ and defined as $S/(S+B)$, where $S$ denotes the
actual signal and $B$ the background. The feed-down contribution is determined
from MC template fits of the experimental distributions of the cosine pointing
angle, in which a total of four templates are considered corresponding to
direct, feed-down, material and impurity contributions. The production
probability d$N$/d$p_{\mathrm{T}}$ is employed in order to obtain
$p_{\mathrm{T}}$ weighted average values.
Around $73\%$ of the $\Lambda$s are directly produced in the primary
interaction and $23\%$ originate from weakly decaying resonances, which is in
line with the values quoted in [62]. The remaining yield is associated to
combinatorial background and $\Lambda$s produced in the detector material. The
main contribution to the feed-down fraction is expected to originate from the
$\Xi$ states with no preference for the neutral or the charged, respectively.
This hypothesis is supported by Pythia simulations where the secondary
$\Lambda$ particles arise from the weak decay of the $\Xi^{0}$ ($48\%$) and
$\Xi^{\mathrm{\pm}}$ ($49\%$) resonances. The remaining contribution in the
simulation arises from the $\Sigma^{0}$, which however is treated separately.
Since the latter decays electromagnetically almost exclusively into
$\Lambda\gamma$ [58], it has a very short life time and cannot be
experimentally differentiated from the sample of primary $\Lambda$s.
Measurements of the ratio
$R_{\Sigma^{0}/\Lambda}=\sigma_{\Sigma^{0}}/\sigma_{\Lambda}$ have obtained
values around 1/3 [63, 64, 65, 66], however with large uncertainties for
hadronic collisions at high energies. For lack of better estimates the value
of 1/3 is used in the following. The resulting $\lambda$ parameters for the
p–$\Lambda$ pair are shown in Tab. 2.

For the $\Lambda$–$\Lambda$ correlation function the following pair
contributions are taken into account

$\begin{split}\\{\Lambda\Lambda\\}=&~{}\Lambda\Lambda+\Lambda\Lambda_{\Sigma^{0}}+\Lambda_{\Sigma^{0}}\Lambda_{\Sigma^{0}}+\Lambda\Lambda_{\Xi^{0}}+\Lambda_{\Xi^{0}}\Lambda_{\Xi^{0}}+\Lambda\Lambda_{\Xi^{-}}\\\
&+\Lambda_{\Xi^{-}}\Lambda_{\Xi^{-}}+\Lambda_{\Sigma^{0}}\Lambda_{\Xi^{0}}+\Lambda_{\Sigma^{0}}\Lambda_{\Xi^{-}}+\Lambda_{\Xi^{0}}\Lambda_{\Xi^{-}}\\\
&+\tilde{\Lambda}\Lambda+\tilde{\Lambda}\Lambda_{\Sigma^{0}}+\tilde{\Lambda}\Lambda_{\Xi^{-}}+\tilde{\Lambda}\Lambda_{\Xi^{0}}+\tilde{\Lambda}\tilde{\Lambda}.\end{split}$
(7)

The resulting $\lambda$ parameters are shown in Tab. 2. Notable is that the
actual pair of interest contributes only to about one third of the signal,
while pair fractions involving in particular $\Sigma^{0}$ and $\Xi$ give a
significant contribution. The statistical uncertainties of these parameters
are negligible and their influence on the systematic uncertainties will be
evaluated in Sec. 4.

p–p | p–$\Lambda$ | $\Lambda$–$\Lambda$
---|---|---
Pair |  $\lambda$ parameter [%] | Pair |  $\lambda$ parameter [%] | Pair |  $\lambda$ parameter [%]
pp |  74.18 | p$\Lambda$ |  47.13 | $\Lambda\Lambda$ |  29.94
pΛp |  15.52 | p$\Lambda_{\Xi^{-}}$ |  9.92 | $\Lambda\Lambda_{\Sigma^{0}}$ |  19.96
pΛpΛ |  0.81 | p$\Lambda_{\Xi^{0}}$ |  9.92 | $\Lambda_{\Sigma^{0}}\Lambda_{\Sigma^{0}}$ |  3.33
p${}_{\Sigma^{+}}$p |  6.65 | p$\Lambda_{\Sigma^{0}}$ |  15.71 | $\Lambda\Lambda_{\Xi^{0}}$ |  12.61
p${}_{\Sigma^{+}}$p${}_{\Sigma^{+}}$ |  0.15 | p${}_{\Lambda}\Lambda$ |  4.93 | $\Lambda_{\Xi^{0}}\Lambda_{\Xi^{0}}$ |  1.33
pΛp${}_{\Sigma^{+}}$ |  0.70 | p${}_{\Lambda}\Lambda_{\Xi^{-}}$ |  1.04 | $\Lambda\Lambda_{\Xi^{-}}$ |  12.61
$\tilde{\mathrm{p}}$p |  1.72 | p${}_{\Lambda}\Lambda_{\Xi^{0}}$ |  1.04 | $\Lambda_{\Xi^{-}}\Lambda_{\Xi^{-}}$ |  1.33
$\tilde{\mathrm{p}}$pΛ |  0.18 | p${}_{\Lambda}\Lambda_{\Sigma^{0}}$ |  1.64 | $\Lambda_{\Sigma^{0}}\Lambda_{\Xi^{0}}$ |  4.20
$\tilde{\mathrm{p}}$p${}_{\Sigma^{+}}$ |  0.08 | p${}_{\Sigma^{+}}\Lambda$ |  2.11 | $\Lambda_{\Sigma^{0}}\Lambda_{\Xi^{-}}$ |  4.20
$\tilde{\mathrm{p}}\tilde{\mathrm{p}}$ |  0.01 | p${}_{\Sigma^{+}}\Lambda_{\Xi^{-}}$ |  0.44 | $\Lambda_{\Xi^{0}}\Lambda_{\Xi^{-}}$ |  2.65
|  | p${}_{\Sigma^{+}}\Lambda_{\Xi^{0}}$ |  0.44 | $\tilde{\Lambda}\Lambda$ |  4.38
|  | p${}_{\Sigma^{+}}\Lambda_{\Sigma^{0}}$ |  0.70 | $\tilde{\Lambda}\Lambda_{\Sigma^{0}}$ |  1.46
|  | $\tilde{\mathrm{p}}\Lambda$ |  0.55 | $\tilde{\Lambda}\Lambda_{\Xi^{0}}$ |  0.92
|  | $\tilde{\mathrm{p}}\Lambda_{\Xi^{-}}$ |  0.18 | $\tilde{\Lambda}\Lambda_{\Xi^{-}}$ |  0.92
|  | $\tilde{\mathrm{p}}\Lambda_{\Xi^{0}}$ |  0.12 | $\tilde{\Lambda}\tilde{\Lambda}$ |  0.16
|  | $\tilde{\mathrm{p}}\Lambda_{\Sigma^{0}}$ |  0.12 |  |
|  | p$\tilde{\Lambda}$ |  3.45 |  |
|  | p${}_{\Lambda}\tilde{\Lambda}$ |  0.36 |  |
|  | p${}_{\Sigma^{+}}\tilde{\Lambda}$ |  0.15 |  |
|  | $\tilde{\mathrm{p}}\tilde{\Lambda}$ |  0.04 |  |
Table 2: Weight parameters of the individual components of the p–p ,
p–$\Lambda$ and $\Lambda$–$\Lambda$ correlation function.

###  3.2 Detector effects

The shape of the experimentally determined correlation function is affected by
the finite momentum resolution. This is taken into account when the
experimental data are compared to model calculations in the fitting procedure
by transforming the modeled correlation function, see Eq. (15), to the
reconstructed momentum basis.

When tracks of particle pairs involved in the correlation function are almost
collinear, i.e. have a low $k^{*}$, detector effects can affect the
measurement. No hint for track merging or splitting is found and therefore no
explicit selection criteria are introduced.

###  3.3 Non-femtoscopic background

For sufficiently large relative momenta ($k^{*}>200$ MeV/$c$) and increasing
separation distance, the FSI among the particles is suppressed and hence the
correlation function should approach unity. As shown in Fig. 2, however, the
measured correlation function for p–p and p–$\Lambda$ exhibits an increase for
$k^{*}$ larger than about $200$ MeV/$c$ for the two systems. Such non-
femtoscopic effects, probably due to energy-momentum conservation, are in
general more pronounced in small colliding systems where the average particle
multiplicity is low [2]. In the case of meson–meson correlations at ultra-
relativistic energies, the appearance of long-range structures in the
correlation functions for moderately small $k^{*}$ ($k^{*}<200$ MeV/c) is
typically interpreted as originating from mini-jet-like structures [49, 67].

Pythia also shows the same non-femtoscopic correlation for larger $k^{*}$ but
fails to reproduce quantitatively the behavior shown in Fig. 2, as already
observed for the angular correlation of baryon–baryon and anti-baryon–anti-
baryon pairs [57].

Energy-momentum conservation leads to a contribution to the signal which can
be reproduced with a formalism described in [68] and accordingly also
considered in this work. Therefore, a linear function $C(k^{*})_{\mathrm{non-
femto}}=ak^{*}+b$ where $a,b$ are fit parameters, is included to the global
fit as $C(k^{*})=C(k^{*})_{\mathrm{femto}}\times C(k^{*})_{\mathrm{non-
femto}}$ to improve the description of the signal by the femtoscopic model.
The fit parameters of the baseline function are obtained in
$k^{*}\in[0.3,~{}0.5]$ GeV/$c$ for p–p and p–$\Lambda$ pairs. For the case of
the $\Lambda$–$\Lambda$ correlation function, the uncertainties of the data do
not allow to additionally add a baseline, which is therefore omitted in the
femtoscopic fit.

Figure 2: (Color online) The raw correlation function compared to Pythia 6
Perugia 2011 simulations for p–p (left), p–$\Lambda$ (center) and
$\Lambda$–$\Lambda$ (right) pairs.

###  3.4 Modeling the correlation function

####  3.4.1 Genuine correlation function

For the p–p correlation function the Coulomb and the strong interaction as
well as the antisymmetrization of the wave functions are considered [69]. The
strong interaction part of the potential is modeled employing the Argonne
$v_{18}$ [51] potential considering the $s$ and $p$ waves. The source is
assumed to be isotropic with a Gaussian profile of radius $r_{0}$. The
resulting Schrödinger equation is then solved with the CATS [52].

In the case of p–$\Lambda$ and $\Lambda$–$\Lambda$ we employ the Lednický and
Lyuboshitz analytical model [70] to describe these correlation functions. This
model is based on the assumption of an isotropic source with Gaussian profile

$S(r_{0})=\frac{1}{(4\pi
r_{0}^{2})^{3/2}}\text{exp}\left(-\frac{r^{2}}{4r_{0}^{2}}\right),$ (8)

where $r_{0}$ is the size of the source. Additionally the complex scattering
amplitude is evaluated by means of the effective range approximation

$f(k^{*})^{S}=\left(\frac{1}{f_{0}^{S}}+\frac{1}{2}d_{0}^{S}k^{*2}-ik^{*}\right)^{-1},$
(9)

with the scattering length $f_{0}^{S}$, the effective range $d_{0}^{S}$ and
$S$ denoting the total spin of the particle pair. In the following the usual
sign convention of femtoscopy is employed where an attractive interaction
leads to a positive scattering length. With these assumptions the analytical
description of the correlation function for uncharged particles [70] reads

$C(k^{*})_{\rm{Lednicky}}=1+\sum_{S}\rho_{S}\left[\frac{1}{2}\left|\frac{f(k^{*})^{S}}{r_{0}}\right|^{2}\left(1-\frac{d_{0}^{S}}{2\sqrt{\pi}r_{0}}\right)+\frac{2\Re{f(k^{*})^{S}}}{\sqrt{\pi}r_{0}}F_{1}(Q_{\rm{inv}}r_{0})-\frac{\Im{f(k^{*})^{S}}}{r_{0}}F_{2}(Q_{\rm{inv}}r_{0})\right],$
(10)

where $\Re{f(k^{*})^{S}}$ ($\Im{f(k^{*})^{S}}$) denotes the real (imaginary)
part of the complex scattering amplitude, respectively. The
$F_{1}(Q_{\rm{inv}}r_{0})$ and $F_{2}(Q_{\rm{inv}}r_{0})$ are analytical
functions resulting from the approximation of isotropic emission with a
Gaussian source and the factor $\rho_{S}$ contains the pair fraction emitted
into a certain spin state $S$. For the p–$\Lambda$ pair unpolarized emission
is assumed.

The $\Lambda$–$\Lambda$ pair is composed of identical particles and hence
additionally quantum statistics needs to be considered, which leads to the
introduction of an additional term to the Lednický model, as employed e.g. in
[15].

While the CATS framework can provide an exact solution for any source and
local interaction potential, the Lednicky-Lyuboshitz approach uses the known
analytical solution outside the range of the strong interaction potential and
takes into account its modification in the inner region in an approximate way
only. That is why this approach may not be valid for small systems.

####  3.4.2 Residual correlations

Table 2 demonstrates that a significant admixture of residuals is present in
the experimental sample of particle pairs. A first theoretical investigation
of these so-called residual correlations was conducted in [71]. This analysis
relies on the procedure established in [18], where the initial correlation
function of the residual is calculated and then transformed to the new
momentum basis after the decay.

For the p–p channel only the feed-down from the p–$\Lambda$ correlation
function is considered, which is obtained by fitting the p–$\Lambda$
experimental correlation function and then transforming it to the p–p momentum
basis. All contributions are weighted by the corresponding $\lambda$
parameters and the modeled correlation function for this pair
$C_{model,\mathrm{p\mbox{--}p}}(k^{*})$ can be written as

$C_{model,\mathrm{p\mbox{--}p}}(k^{*})=1+\lambda_{\mathrm{pp}}\cdot(C_{\mathrm{pp}}(k^{*})-1)+\lambda_{\mathrm{pp}_{\Lambda}}(C_{\mathrm{pp}_{\Lambda}}(k^{*})-1).$
(11)

All other residual correlations are assumed to be flat.

For the p–$\Lambda$ , residual correlations from the p–$\Sigma^{0}$ , p–$\Xi$
and $\Lambda$–$\Lambda$ pairs are taken into account. As the
$\Lambda$–$\Lambda$ correlation function is rather flat no further
transformation is applied. The p–$\Sigma^{0}$ correlation function is obtained
using predictions from [72].

As the decay products of the reaction $\Xi\rightarrow\Lambda\pi$ are charged
and therefore accessible by ALICE, we measure the p–$\Xi$ correlation
function. The experimental data are parametrized with a phenomenological
function

$C_{\mathrm{p\mbox{--}}\Xi^{-}}(k^{*})=1+\frac{\exp(-k^{*}a_{\Xi})}{k^{*}a_{\Xi}},$
(12)

where the parameter $a_{\Xi}$ is employed to scale the function to the data
and has no physical meaning. Its value is found to be $a_{\Xi}=3.88$ fm.

The modeled correlation function $C_{model,\mathrm{p}\mbox{--}\Lambda}(k^{*})$
for the pair is obtained by

$C_{model,\mathrm{p}\mbox{--}\Lambda}(k^{*})=1+\lambda_{\mathrm{p}\Lambda}(C(_{\mathrm{p}\Lambda}(k^{*})-1)+\lambda_{\mathrm{p}\Lambda_{\Sigma^{0}}}(C_{\mathrm{p}\Lambda_{\Sigma^{0}}}(k^{*})-1)+\lambda_{\mathrm{p}\Lambda_{\Xi^{-}}}(C_{\mathrm{p}\Lambda_{\Xi^{-}}}(k^{*})-1).$
(13)

As the present knowledge on the hyperon–hyperon interaction is scarce, in
particular regarding the interaction of the $\Lambda$ with other hyperons, all
residual correlations feeding into the $\Lambda$–$\Lambda$ correlation
function are considered to be consistent with unity,

$C_{model,\Lambda\mbox{--}\Lambda}(k^{*})=1+\lambda_{\Lambda\Lambda}(C(_{\Lambda\Lambda}(k^{*})-1).$
(14)

It should be noted, that the residual correlation functions, after weighting
with the corresponding $\lambda$ parameter, transformation to the momentum
base of the correlation of interest and taking into account the finite
momentum resolution, only barely contribute to the total fit function.

####  3.4.3 Total correlation function model

The correlation function modeled according to the considerations discussed
above is then multiplied by a linear function to correct for the baseline as
discussed in Sec. 3.3 and weighted with a normalization parameter
$\mathcal{N}$

$C_{tot}(k^{*})=\mathcal{N}\cdot(a+b\cdot k^{*})\cdot C_{model}(k^{*}),$ (15)

where $C_{model}(k^{*})$ incorporates all considered theoretical correlation
functions, weighted with the corresponding $\lambda$ parameters as discussed
in Sec. 3.1 and 3.4.

The inclusion of a baseline is further motivated by the presence of a linear
but non-flat correlation observed in the data outside the femtoscopic region
(see Fig. 2 for $k^{*}\in[0.3,~{}0.5]$ GeV/$c$). When attempting to use a
higher order polynomial to model the background, the resulting curves are
still compatible with a linear function, while their interpolation into the
lower $k^{*}$ region leads to an overall poorer fit quality.

##  4 Systematic uncertainties

###  4.1 Correlation function

The systematic uncertainties of the correlation functions are extracted by
varying the proton and $\Lambda$ candi-date selection criteria according to
Tab. 3. Due to the low number of particle pairs, in particular at low $k^{*}$,
the resulting variations of the correlation functions are in general much
smaller than the statistical uncertainties. In order to still estimate the
systematic uncertainties the data are rebinned by a factor of 10. The
systematic uncertainty on the correlation function is obtained by computing
the ratio of the default correlation function to the one obtained by the
respective cut variation. Whenever this results in two systematic
uncertainties, i.e. by a variation up and downwards, the average is taken into
account. Then all systematic uncertainties from the cut variations are summed
up quadratically. This is then extrapolated to the finer binning of the
correlation function by fitting a polynomial of second order. The obtained
systematic uncertainties are found to be largest in the lowest $k^{*}$ bin.
The individual contributions in that bin are summarized in Tab. 3 and the
resulting total systematic uncertainty accounts to about 4 % for p–p , 1 % for
p–$\Lambda$ and 2.5 % for $\Lambda$–$\Lambda$ . Variations of the proton DCA
selection are not taken into account for the computation of the systematic
uncertainty since it dilutes (enhances) the correlation signal by introducing
more (less) secondaries in the sample. This effect is recaptured by a change
in the $\lambda$ parameter.

Variable | Default | Variation |  p–p [%] |  p–$\Lambda$ [%] |  $\Lambda$–$\Lambda$ [%]
---|---|---|---|---|---
Min. $p_{\mathrm{T}}$ proton ($\mathrm{GeV}\kern-0.50003pt/\kern-0.20004ptc\,$) |  0.5 |  0.4, 0.6 | 1 | 0.2 | -
$|\eta|$ proton |  0.8 |  0.7, 0.9 | 0.4 | 0.2 | -
$n_{\sigma}$ proton |  3 |  2, 5 | 1.8 | 0.2 | -
Proton tracks |  TPC only |  Global | 2.4 | 0 | -
$n_{\mathrm{Cluster}}$ proton |  80 |  90 | 0.3 | 0.1 | -
Min. $p_{\mathrm{T}}$ $V_{0}$ ($\mathrm{GeV}\kern-0.50003pt/\kern-0.20004ptc\,$) |  0.3 |  0.24, 0.36 | - | 0 | 0
cos$(\alpha)$ $V_{0}$ |  0.9 |  0.998 | - | 0 | 1.8
$n_{\sigma}$ $V_{0}$ daughter |  5 |  4 | - | 0.1 | 0.3
$n_{\mathrm{Cluster}}$ $V_{0}$ daughter |  70 |  80 | - | 0.1 | 0.7
$|\eta|$ $V_{0}$ |  0.8 |  0.7, 0.9 | - | 0.6 | 0.8
$\mathrm{DCA}(|p,\pi|)$ (cm) |  1.5 |  1.2 | - | 0.5 | 0
$\mathrm{DCA}>0.05$ (cm) |  0.05 |  0.06 | - | 0.7 | 0.6
Table 3: Selection parameter variation and the resulting relative systematic
uncertainty on the p–p , p–$\Lambda$ and $\Lambda$–$\Lambda$ correlation
function.

###  4.2 Femtoscopic fit

To evaluate the systematic uncertainty of the femtoscopic fit, and hence on
the measurement of the radius $r_{0}$, the fit is performed applying the
following variations. Instead of the common fit, the radius is determined
separately from the p–p and p–$\Lambda$ correlation functions.
$\Lambda$–$\Lambda$ is excluded because it imposes only a shallow constraint
on the radius, in particular since the scattering parameters unconstrained for
the fit. Furthermore, the input to the $\lambda$ parameters are varied by 25
%, while keeping the purity and the fraction of primaries and secondaries
constant since this would correspond to a variation of the particle selection
and thus would require a different experimental sample as discussed above.
Additionally, all fit ranges of both the femtoscopic and the baseline fits are
varied individually by up to 50 % and 10 %, respectively. The lower bound of
the femtoscopic fit is always left at its default value. For the p–$\Lambda$
correlation function the dependence on the fit model is studied by replacing
the Lednický and Lyuboshitz analytical model with the potential introduced by
Bodmer, Usmani, and Carlson [73] for which the Schrödinger equation is
explicitly solved using CATS. Additionally, the fit for the p–p and
p–$\Lambda$ correlation function is performed without the linear baseline. The
radius is determined for 2000 random combinations of the above mentioned
variations. The resulting distribution of radii is not symmetric and the
systematic uncertainty is therefore extracted as the boundaries of the 68 %
confidence interval around the median of the distribution and accounts to
about 4 % of the determined radius.

##  5 Results

Figure 3: (Color online) The p–p (left), p–$\Lambda$ (center) and
$\Lambda$–$\Lambda$ (right) correlation function with a simultaneous fit with
the NLO expansion (red line) for the scattering parameter of p–$\Lambda$ [25].
The dashed line denotes the linear baseline. After the fit is performed the LO
[24] parameter set (green curve) is plugged in for the p–$\Lambda$ system and
the scattering length obtained from [15] for the $\Lambda$–$\Lambda$ system
(cyan curve).

The obtained p–p , p–$\Lambda$ and $\Lambda$–$\Lambda$ correlation functions
are shown in Fig. 3. For each of the correlation functions we do not observe
any mini-jet background in the low $k^{*}$ region, as observed in the case of
neutral [74] and charged [50] kaon pairs and charged pion pairs [49]. This
demonstrates that the femtoscopic signal in baryon–baryon correlations is
dominant in ultrarelativistic $\mathrm{p\kern-0.50003ptp}$ collisions. The
signal amplitude for the p–p and p–$\Lambda$ correlations are much larger than
the one observed in analogous studies from heavy-ion collisions [1, 11, 12,
14], due to the small particle emitting source formed in
$\mathrm{p\kern-0.50003ptp}$ collisions, allowing a higher sensitivity to the
FSI.

In absence of residual contributions and any FSI, the $\Lambda$–$\Lambda$
correlation function is expected to approach 0.5 as $k^{*}$ $\rightarrow$ 0\.
The data of the herewith presented sample is limited, but we can see that the
$\Lambda$–$\Lambda$ correlation exceeds the value expected considering only
quantum statistic effects, which is likely due to the attractive FSI of the
$\Lambda$–$\Lambda$ system [26, 37].

The experimental data are fitted using CATS and hence the exact solution of
the Schrödinger equation for the $\mathrm{p\kern-0.50003ptp}$ correlation and
the Lednický model for the p–$\Lambda$ and $\Lambda$–$\Lambda$ correlation.
The three fits are done simultaneously and this way the source radius is
extracted and different scattering parameters for the p–$\Lambda$ and
$\Lambda$–$\Lambda$ interactions can be tested. While in the case of the p–p
and p–$\Lambda$ correlation function the existence of a baseline is clearly
visible in the data, the low amount of pairs in the $\Lambda$–$\Lambda$
channel do not allow for such a conclusion. Therefore, the baseline is not
included in the model for the $\Lambda$–$\Lambda$ correlation function.

The simultaneous fit is carried out by using a combined $\chi^{2}$ and with
the radius as a free parameter common to all correlation functions. The fit
range is $k^{*}$ $\in[0,~{}0.16]$ GeV/$c$ for p–p and $k^{*}$ $\in[0,~{}0.22]$
GeV/$c$ for p–$\Lambda$ and $\Lambda$–$\Lambda$ . Hereafter we adopt the
convention of positive scattering lengths for attractive interactions and
negative scattering lengths for repulsive interactions. The p–$\Lambda$ strong
interaction is modeled employing scattering parameters obtained using the
next-to-leading order expansion of a chiral effective field theory at a cutoff
scale of $\Lambda=600$ MeV [25]. The simultaneous fit of the p–p , p–$\Lambda$
and $\Lambda$–$\Lambda$ correlation functions yields a common radius of
$r_{0}=1.144\pm 0.019\,\mathrm{(stat)}\,^{+0.069}_{-0.012}\,\mathrm{(syst)}$
fm.

The blue line in the left panel in Fig. 3 shows the result of the femtoscopic
fit to the p–p correlation function using the Argonne $v_{18}$ potential that
describes the experimental data in a satisfactory way. The red curve in the
central panel shows the result of the NLO calculation for p–$\Lambda$ . In the
case of $\Lambda$–$\Lambda$ (right panel), the yellow curve represents the
femtoscopic fit with free scattering parameters. The width of the femtoscopic
fits corresponds to the systematic uncertainty of the correlation function
discussed in Sec. 4.

After the fit with the NLO scattering parameters has converged, the
p–$\Lambda$ correlation function for the same source size is compared to the
data using various theoretically obtained scattering parameters [75, 76, 77,
78, 79, 24, 25, 80, 81] as summarized in Tab. 4. The degree of consistency is
expressed in the number of standard deviations $n_{\sigma}$. The employed
models include several versions of meson exchange models proposed such as the
Nijmegen model D (ND) [75], model F (NF) [76], soft core (NSC89 and NSC97)
[77, 78] and extended soft core (ESC08) [79]. Additionally, models considering
contributions from one- and two-pseudoscalar-meson exchange diagrams and from
four-baryon contact terms in $\chi$EFT at leading [24] and next-to-leading
order [25] are employed, together with the first version of the Jülich
$Y\mbox{--}N$ meson exchange model [80], which in a later version [81] also
features one-boson exchange.

All employed models describe the data equally well and hence the available
data does not allow yet for a discrimination. As an example, we show in the
central panel of Fig. 3 how employing scattering parameters different than the
NLO ones reflects on the p–$\Lambda$ correlation function. The green curve
corresponds to the results obtained employing LO scattering parameters and the
theoretical correlation function is clearly sensitive for $k^{*}$
$\rightarrow$ 0 to the input parameter.

Model |  |  $f_{0}^{S=0}$ (fm) |  $f_{0}^{S=1}$ (fm) |  $d_{0}^{S=0}$ (fm) |  $d_{0}^{S=1}$ (fm) |  $n_{\sigma}$
---|---|---|---|---|---|---
ND [75] |  |  1.77 |  2.06 |  3.78 |  3.18 |  0.7
NF [76] |  |  2.18 |  1.93 |  3.19 |  3.358 |  0.7
NSC89 [77] |  |  2.73 |  1.48 |  2.87 |  3.04 |  0.7
NSC97 [78] |  a |  0.71 |  2.18 |  5.86 |  2.76 |  0.7
b |  0.9 |  2.13 |  4.92 |  2.84 |  0.7
c |  1.2 |  2.08 |  4.11 |  2.92 |  0.7
d |  1.71 |  1.95 |  3.46 |  3.08 |  0.7
e |  2.1 |  1.86 |  3.19 |  3.19 |  0.7
f |  2.51 |  1.75 |  3.03 |  3.32 |  0.7
ESC08 [79] |  |  2.7 |  1.65 |  2.97 |  3.63 |  0.6
$\chi$EFT |  LO [24] |  1.91 |  1.23 |  1.4 |  2.13 |  1.8
NLO [25] |  2.91 |  1.54 |  2.78 |  2.72 |  1.2
Jülich |  A [80] |  1.56 |  1.59 |  1.43 |  3.16 |  0.9
J04 [81] |  2.56 |  1.66 |  2.75 |  2.93 |  1.1
J04c [81] |  2.66 |  1.57 |  2.67 |  3.08 |  0.8
Table 4: Scattering parameters for the p–$\Lambda$ system from various
theoretical calculations [75, 76, 77, 78, 79, 24, 25, 80, 81] and the
corresponding degree of consistency with the experimentally determined
correlation function expressed in numbers of standard deviations $n_{\sigma}$.
The $\chi$EFT scattering parameters are obtained at a cutoff scale
$\Lambda=600$ MeV. The usual sign convention of femtoscopy is employed where
an attractive interaction leads to a positive scattering length.

In order to probe which scattering parameters are compatible with the measured
$\Lambda$–$\Lambda$ correlation function, the effective range and the
scattering length of the potential are varied within $d_{0}\in[0,18]$ fm and
$1/f_{0}\in[-2,5]$ 1/fm, while keeping the renormalization constant
$\mathcal{N}$ as the only free fit parameter. It should be noted that the
resulting variations of $\mathcal{N}$ are on the percent level. The resulting
correlation functions obtained by employing the Lednický and Lyuboshitz
analytical model [70] and considering also the secondaries and impurities
contributions are compared to the data. The degree of consistency is expressed
in the number of standard deviations $n_{\sigma}$, as displayed in Fig. 4
together with an overview of the present knowledge about the $\Lambda$
–$\Lambda$ interaction. For a detailed overview of the currently available
models see e.g. [37], from which we have obtained the collection of scattering
parameters. Additionally to the Nijmegen meson exchange models mentioned
above, the data are compared to various other theoretical calculations. An
exemplary boson-exchange potential is Ehime [82, 83], whose strength is fitted
to the outdated double hypernuclear bound energy, $\Delta
B_{\Lambda\Lambda}=4$ MeV [84] and accordingly known to be too attractive. As
an exemplary quark model including baryon–baryon interactions with meson
exchange effects, the fss2 model [85, 86] is used. Moreover, the potentials by
Filikhin and Gal (FG) [87] and by Hiyama, Kamimura, Motoba, Yamada, and
Yamamoto (HKMYY) [88], which are capable of describing the NAGARA event [89]
are employed.

Figure 4: (Color online) Number of standard deviations $n_{\sigma}$ of the
modeled correlation function for a given set of scattering parameters
(effective range $d_{0}$ and scattering length $f_{0}$) with respect to the
data, together with various model calculations [75, 76, 77, 78, 79, 82, 83,
85, 86, 87, 88] and measurements [15]. The gray shaded area corresponds to the
region where the Lednický model predicts a negative correlation function for
$\mathrm{p\kern-0.50003ptp}$ collisions at $\sqrt{s}=7\,$TeV.

In contrast to the p–$\Lambda$ case, the agreement with the data increases
with every revision of the Nijmegen potential, while the introduction of the
extended soft core slightly increases the deviation. In particular solution
NSC97f yields the overall best agreement with the data. The correlation
function modeled using scattering parameters of the Ehime model which is known
to be too attractive deviates by about 2 standard deviations from the data.
For an attractive interaction (positive $f_{0}$) the correlation function is
pushed from the quantum statistics distribution for two fermions (correlation
function equal to $0.5$ for $k^{*}=\,0$) to unity. As a result within the
current uncertainties the $\Lambda$–$\Lambda$ correlation function is rather
flat and close to 1 and this lack of structure makes it impossible to extract
the two scattering parameters with a reasonable uncertainty. This means that
even by increasing the data by a factor $10$, as expected from the RUN2 data,
it will be very complicated to constrain precisely the region $f_{0}>0$.
As for the region of negative scattering length $f_{0}$ this is connected in
scattering theory either to a repulsive interaction or to the existence of a
bound state close to the threshold and a change in the sign of the scattering
length. Since the $\Lambda$–$\Lambda$ interaction is known to be slightly
attractive above the threshold [35], the measurement of a negative scattering
lengths would strongly support the existence of the H-dibaryon. Notably the
correlation function modeled employing the scattering parameters obtained by
the STAR collaboration in Au–Au collisions at $\sqrt{s_{\mathrm{NN}}}=200$ GeV
[15] and all the secondaries and impurities contributions deviates by 6.8
standard deviations from the data. This is also shown by the cyan curve
displayed in the right panel of Fig. 3 which is obtained using the source
radius and the $\lambda$ parameters from this analysis and the scattering
parameters from [15]. On the other hand these parameters and all those
corresponding to the gray-shaded area in Fig. 4 lead to a negative genuine
$\Lambda$–$\Lambda$ correlation function if the Lednický model is employed.
The total correlation function that is compared to the experimental data is
not negative because the impurities and secondaries contributions lead to a
total correlation function that is always positive. This means that the
combination of large effective ranges and negative scattering lengths
translate into unphysical correlation functions, for small colliding systems
as $\mathrm{p\kern-0.50003ptp}$. This effect is not immediate visible in
larger colliding system such as Au–Au at $\sqrt{s_{\mathrm{NN}}}=200$ GeV
measured by STAR, where the obtained correlation function does not become
negative. This demonstrates that these scattering parameters intervals
combined with the Lednický model are not suited to describe the correlations
functions measured in small systems. One could test the corresponding local
potentials with the help of CATS [52], since the latter does not suffer from
the limitations of the Lednický model due to the employment of the asymptotic
solution. On the other hand we have directly compared the correlation
functions obtained employing CATS and the $\Lambda$–$\Lambda$ local potentials
reported in [37] with the correlation functions obtained using the
corresponding scattering parameters and the Lednický model. For the typical
source radii of $1.3$ fm the deviations are within 10%̇. This disfavours the
region of negative scattering lengths and large effective ranges for the
$\Lambda$–$\Lambda$ correlation.

This study is the first measurement with baryon pairs in
$\mathrm{p\kern-0.50003ptp}$ collisions at $\sqrt{s}=7\,$TeV, while other
femtoscopic analyses were conducted with neutral [74] and charged [50] kaon
pairs and charged pion pairs [49] with the ALICE experiment. The radius
obtained from baryon pairs is found to be slightly larger that that measured
from meson-meson pairs at comparable transverse mass as shown in Fig. 5

Figure 5: (Color online) Comparison of radii obtained for different charged
particle multiplicity intervals in the $\mathrm{p\kern-0.50003ptp}$ collision
system at $\sqrt{s}=7$ TeV [49, 50, 74]. The error bars correspond to
statistical and the shaded regions to the systematic uncertainties. The black
point is the radius obtained in this analysis with p–p , p–$\Lambda$ and
$\Lambda$–$\Lambda$ pairs, while the gray bar corresponds to the range of
covered $m_{\mathrm{T}}$ in this analysis.

##  6 Summary

This paper presents the first femtoscopic measurement of p–p , p–$\Lambda$ and
$\Lambda$–$\Lambda$ pairs in $\mathrm{p\kern-0.50003ptp}$ collisions at
$\sqrt{s}=7$ TeV. No evidence for the presence of mini-jet background is found
and it is demonstrated that this kind of studies with baryon–baryon and anti-
baryon–anti-baryon pairs are feasible. With a newly developed method to
compute the contributions arising from impurities and weakly decaying
resonances to the correlation function from single particles quantities only,
the genuine correlation functions of interest can be extracted from the
signal. These correlation functions contribute with 74 % for p–p , 47 % for
p–$\Lambda$ and 30 % for $\Lambda$–$\Lambda$ to the total signal. A
simultaneous fit of all correlation functions with a femtoscopic model
featuring residual correlations stemming from the above mentioned effects
yields a radius of the particles emitting source of $r_{0}=1.144\pm
0.019\,\mathrm{(stat)}\,^{+0.069}_{-0.012}\,\mathrm{(syst)}$ fm. For the first
time, the Argonne $v_{18}$ $NN$ potential with the $s$ and $p$ waves was used
to successfully describe the p–p correlation and in so obtain a solid
benchmark for our investigation. For the case of the p–$\Lambda$ correlation
function, the NLO parameter set obtained within the framework of chiral
effective field theory is consistent with the data, but other models are also
found to be in agreement with the data. The present pair data in the
$\Lambda$–$\Lambda$ channel allows us to constrain the available scattering
parameter space. Large effective ranges $d_{0}$ in combination with negative
scattering parameters lead to unphysical correlations if the Lednický model is
employed to compute the correlation function. This also holds true for the the
average values published by the STAR collaboration in Au–Au collisions at
$\sqrt{s_{\mathrm{NN}}}=200$ Ge, that are found to be incompatible with the
measurement in $\mathrm{p\kern-0.50003ptp}$ collisions within the Lednický
model.

The larger data sample of the LHC Run 2 and Run 3, where we expect up to a
factor ten and 100 more data respectively, will enable us to extend the method
also to $\Sigma$, $\Xi$ and $\Omega$ hyperons and thus further constrain the
Hyperon–Nucleon interaction.

## Acknowledgements

The ALICE Collaboration would like to thank all its engineers and technicians
for their invaluable contributions to the construction of the experiment and
the CERN accelerator teams for the outstanding performance of the LHC complex.
The ALICE Collaboration gratefully acknowledges the resources and support
provided by all Grid centres and the Worldwide LHC Computing Grid (WLCG)
collaboration. The ALICE Collaboration acknowledges the following funding
agencies for their support in building and running the ALICE detector: A. I.
Alikhanyan National Science Laboratory (Yerevan Physics Institute) Foundation
(ANSL), State Committee of Science and World Federation of Scientists (WFS),
Armenia; Austrian Academy of Sciences and Nationalstiftung für Forschung,
Technologie und Entwicklung, Austria; Ministry of Communications and High
Technologies, National Nuclear Research Center, Azerbaijan; Conselho Nacional
de Desenvolvimento Científico e Tecnológico (CNPq), Universidade Federal do
Rio Grande do Sul (UFRGS), Financiadora de Estudos e Projetos (Finep) and
Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP), Brazil;
Ministry of Science & Technology of China (MSTC), National Natural Science
Foundation of China (NSFC) and Ministry of Education of China (MOEC) , China;
Ministry of Science and Education, Croatia; Ministry of Education, Youth and
Sports of the Czech Republic, Czech Republic; The Danish Council for
Independent Research — Natural Sciences, the Carlsberg Foundation and Danish
National Research Foundation (DNRF), Denmark; Helsinki Institute of Physics
(HIP), Finland; Commissariat à l’Energie Atomique (CEA) and Institut National
de Physique Nucléaire et de Physique des Particules (IN2P3) and Centre
National de la Recherche Scientifique (CNRS), France; Bundesministerium für
Bildung, Wissenschaft, Forschung und Technologie (BMBF) and GSI
Helmholtzzentrum für Schwerionenforschung GmbH, Germany; General Secretariat
for Research and Technology, Ministry of Education, Research and Religions,
Greece; National Research, Development and Innovation Office, Hungary;
Department of Atomic Energy Government of India (DAE), Department of Science
and Technology, Government of India (DST), University Grants Commission,
Government of India (UGC) and Council of Scientific and Industrial Research
(CSIR), India; Indonesian Institute of Science, Indonesia; Centro Fermi -
Museo Storico della Fisica e Centro Studi e Ricerche Enrico Fermi and Istituto
Nazionale di Fisica Nucleare (INFN), Italy; Institute for Innovative Science
and Technology , Nagasaki Institute of Applied Science (IIST), Japan Society
for the Promotion of Science (JSPS) KAKENHI and Japanese Ministry of
Education, Culture, Sports, Science and Technology (MEXT), Japan; Consejo
Nacional de Ciencia (CONACYT) y Tecnología, through Fondo de Cooperación
Internacional en Ciencia y Tecnología (FONCICYT) and Dirección General de
Asuntos del Personal Academico (DGAPA), Mexico; Nederlandse Organisatie voor
Wetenschappelijk Onderzoek (NWO), Netherlands; The Research Council of Norway,
Norway; Commission on Science and Technology for Sustainable Development in
the South (COMSATS), Pakistan; Pontificia Universidad Católica del Perú, Peru;
Ministry of Science and Higher Education and National Science Centre, Poland;
Korea Institute of Science and Technology Information and National Research
Foundation of Korea (NRF), Republic of Korea; Ministry of Education and
Scientific Research, Institute of Atomic Physics and Romanian National Agency
for Science, Technology and Innovation, Romania; Joint Institute for Nuclear
Research (JINR), Ministry of Education and Science of the Russian Federation
and National Research Centre Kurchatov Institute, Russia; Ministry of
Education, Science, Research and Sport of the Slovak Republic, Slovakia;
National Research Foundation of South Africa, South Africa; Centro de
Aplicaciones Tecnológicas y Desarrollo Nuclear (CEADEN), Cubaenergía, Cuba and
Centro de Investigaciones Energéticas, Medioambientales y Tecnológicas
(CIEMAT), Spain; Swedish Research Council (VR) and Knut & Alice Wallenberg
Foundation (KAW), Sweden; European Organization for Nuclear Research,
Switzerland; National Science and Technology Development Agency (NSDTA),
Suranaree University of Technology (SUT) and Office of the Higher Education
Commission under NRU project of Thailand, Thailand; Turkish Atomic Energy
Agency (TAEK), Turkey; National Academy of Sciences of Ukraine, Ukraine;
Science and Technology Facilities Council (STFC), United Kingdom; National
Science Foundation of the United States of America (NSF) and United States
Department of Energy, Office of Nuclear Physics (DOE NP), United States of
America.

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##  Appendix A Derivation of the $\mathbf{\lambda}$ parameters

Let ’X’ be a specific particle type and $X$ is the number of particles of that
species. For each particle different subsets $X_{i}$ are defined, each
representing a unique origin of the particle, where $i=0$ corresponds to the
case of a primary particle, the rest are either particles originating from
feed-down or misidentification. In particular indexes $1\leq i\leq N_{F}$
should be associated with feed-down contributions and $N_{F}+1\leq i\leq
N_{F}+N_{M}$ should be associated with impurities, where $N_{F}$ is the number
of feed-down channels and $N_{M}$ the number of impurity channels. In the
present work we assume that all impurity channels contribute with a flat
distribution to the total correlation, therefore we do not study
differentially the origin of the impurities and combine them in a single
channel, i.e. $N_{M}=1$. Further we define

$X_{F}=\sum_{i-1}^{N_{F}}X_{i},$ (16)

as the total number of particles that stem from feed-down and

$X_{M}=\sum_{N_{F}+1}^{N_{M}}X_{i},$ (17)

as the total number of particles that were misidentified (i.e. impurities).
$X_{0}$ is the number of correctly identified primary particles that are of
interest for the femtoscopy analysis.
The purity $\mathcal{P}$ is the fraction of correctly identified particles,
not necessarily primary, to the total number of particles in the sample (Eq.
18).

$\displaystyle\mathcal{P}(X)=(X_{0}+X_{F})/X.$ (18)

The impurity is

$\displaystyle\mathcal{\bar{P}}(X)=X_{M}/X.$ (19)

For the later discussion it is beneficial to combine the two definitions and
refer to the purity as

$\displaystyle\mathcal{P}(X_{i})=\begin{cases}\mathcal{P}(X)=(X_{0}+X_{F})/X&\text{
for }i\leq N_{F},\\\ \mathcal{\bar{P}}(X)=X_{M}/X&\text{ else.}\end{cases}$
(20)

Another quantity of interest will be the channel fraction $f_{i}$, which is
defined as the fraction of particles originating from the $i$-th channel
relative to the total number of either correctly identified or misidentified
particles:

$\displaystyle f(X_{i})=\begin{cases}X_{i}/(X_{0}+X_{F})&\text{ for }i\leq
N_{F},\\\ X_{i}/X_{M}&\text{ else.}\end{cases}$ (21)

As discussed in the main body of the paper both the purity and the channel
fractions can be obtained either from MC simulations or MC template fits. The
product of the two reads

$P(X_{i})=\mathcal{P}(X_{i})f(X_{i})=\frac{X_{i}}{X}.$ (22)

Next we will relate $\mathcal{P}(X_{i})$ and $f(X_{i})$ to the correlation
function between particle pairs, which is defined as

$C(XY)=\frac{N(XY)}{M(XY)},$ (23)

where $N$ and $M$ are the yields of an ’XY’ particle pair in same and mixed
events respectively. Note that this is a raw correlation function which is not
properly normalized. The normalization is discussed in the main body of the
paper, but is irrelevant in the current discussion and it will be omitted.
Both $N$ and $M$ are yields which can be decomposed into the sum of their
ingredients. Using the previously discussed notion of different channels of
origin

$N(XY)=N\left(\sum_{i,j}X_{i}Y_{j}\right)=\sum_{i,j}N(X_{i}Y_{j}),$ (24)
$M(XY)=M\left(\sum_{i,j}X_{i}Y_{j}\right)=\sum_{i,j}M(X_{i}Y_{j}).$ (25)

Hence the total correlation function becomes:

$\displaystyle C(XY)$
$\displaystyle=\frac{\sum_{i,j}N(X_{i}Y_{j})}{M(XY)}=\sum_{i,j}\frac{N(X_{i}Y_{j})}{M(XY)}\frac{M(X_{i}Y_{j})}{M(X_{i}Y_{j})}=$
(26)
$\displaystyle=\sum_{i,j}\underbrace{\frac{N(X_{i}Y_{j})}{M(X_{i}Y_{j})}}_{C_{i,j}(XY)}\underbrace{\frac{M(X_{i}Y_{j})}{M(XY)}}_{\lambda_{i,j}(XY)}=\sum_{i,j}\lambda_{i,j}(XY)C_{i,j}(XY),$
(27)

where $C_{i,j}(XY)$ is the contribution to the total correlation of the
$i,j$-th channel of origin of the particles ’X,Y’ and $\lambda_{i,j}(XY)$ is
the corresponding weight coefficient. How to obtain the individual functions
$C_{i,j}(XY)$ is discussed in the main body of the paper. The weights
$\lambda_{i,j}$ can be derived from the purities and channel fractions of the
particles ’X’ and ’Y’. This is possible since $\lambda_{i,j}$ depends only on
the mixed event sample for which the underlying assumption is that the
particles are not correlated. In that case the two-particle yield $M(XY)$ can
be factorized and according to Eq. (26) the $\lambda$ coefficients can be
expressed as

$\lambda_{i,j}(XY)=\frac{M(X_{i}Y_{j})}{M(XY)}=\frac{M(X_{i})}{M(X)}\frac{M(Y_{i})}{M(Y)}=P(X_{i})P(Y_{i}).$
(28)

The last step follows directly from Eq. (22) applied to the mixed event
samples of ’X’ and ’Y’. Eq. 22 relates $P$ to the known quantities
$\mathcal{P}$ and $f$, hence the $\lambda$ coefficients can be rewritten as

$\lambda_{i,j}(XY)=\mathcal{P}(X_{i})f(X_{i})\mathcal{P}(Y_{j})f(Y_{j}).$ (29)

We would like to point out that due to the definition of $P(X_{i})$ the sum of
all $\lambda$ parameters is automatically normalized to unity.

##  Appendix B The ALICE Collaboration

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Oravecorg115&A. Ortiz Velasquezorg70&A. Oskarssonorg80&J. Otwinowskiorg117&K.
Oyamaorg81&Y. Pachmayerorg102&V. Pacikorg88&D. Paganoorg138&G. Paićorg70&P.
Palniorg6&J. Panorg141&A.K. Pandeyorg48&S. Panebiancoorg135&V. Papikyanorg1&P.
Pareekorg49&J. Parkorg60&J.E. Parkkilaorg126&S. Parmarorg98&A.
Passfeldorg142&S.P. Pathakorg125&R.N. Patraorg139&B. Paulorg58&H. Peiorg6&T.
Peitzmannorg63&X. Pengorg6&L.G. Pereiraorg71&H. Pereira Da Costaorg135&D.
Peresunkoorg87&E. Perez Lezamaorg69&V. Peskovorg69&Y. Pestovorg4&V.
Petráčekorg37&M. Petroviciorg47&C. Pettaorg28&R.P. Pezziorg71&S. Pianoorg59&M.
Piknaorg14&P. Pillotorg113&L.O.D.L. Pimentelorg88&O. Pinazzaorg53,org34&L.
Pinskyorg125&S. Pisanoorg51&D.B. Piyarathnaorg125&M. Płoskońorg79&M.
Planinicorg97&F. Pliquettorg69&J. Plutaorg140&S. Pochybovaorg143&P.L.M.
Podesta-Lermaorg119&M.G. Poghosyanorg94&B. Polichtchoukorg90&N. Poljakorg97&W.
Poonsawatorg114&A. Poporg47&H. Poppenborgorg142&S. Porteboeuf-
Houssaisorg132&V. Pozdniakovorg75&S.K. Prasadorg3&R. Preghenellaorg53&F.
Prinoorg58&C.A. Pruneauorg141&I. Pshenichnovorg62&M. Puccioorg26&V.
Puninorg106&J. Putschkeorg141&S. Rahaorg3&S. Rajputorg99&J. Rakorg126&A.
Rakotozafindrabeorg135&L. Ramelloorg32&F. Ramiorg134&R. Raniwalaorg100&S.
Raniwalaorg100&S.S. Räsänenorg43&B.T. Rascanuorg69&R. Rathorg49&V.
Ratzaorg42&I. Ravasengaorg31&K.F. Readorg94,org128&K. Redlichorg84orgIV&A.
Rehmanorg22&P. Reicheltorg69&F. Reidtorg34&X. Renorg6&R. Renfordtorg69&A.
Reshetinorg62&J.-P. Revolorg10&K. Reygersorg102&V. Riabovorg96&T.
Richertorg63,org88,org80&M. Richterorg21&P. Riedlerorg34&W. Rieglerorg34&F.
Riggiorg28&C. Risteaorg68&S.P. Rodeorg49&M. Rodríguez Cahuantziorg44&K.
Røedorg21&R. Rogalevorg90&E. Rogochayaorg75&D. Rohrorg34&D. Röhrichorg22&P.S.
Rokitaorg140&F. Ronchettiorg51&E.D. Rosasorg70&K. Roslonorg140&P.
Rosnetorg132&A. Rossiorg29&A. Rotondiorg137&F. Roukoutakisorg83&C.
Royorg134&P. Royorg107&O.V. Ruedaorg70&R. Ruiorg25&B. Rumyantsevorg75&A.
Rustamovorg86&E. Ryabinkinorg87&Y. Ryabovorg96&A. Rybickiorg117&S.
Saarinenorg43&S. Sadhuorg139&S. Sadovskyorg90&K. Šafaříkorg34&S.K.
Sahaorg139&B. Sahooorg48&P. Sahooorg49&R. Sahooorg49&S. Sahooorg66&P.K.
Sahuorg66&J. Sainiorg139&S. Sakaiorg131&M.A. Salehorg141&S. Sambyalorg99&V.
Samsonovorg91,org96&A. Sandovalorg72&A. Sarkarorg73&D. Sarkarorg139&N.
Sarkarorg139&P. Sarmaorg41&M.H.P. Sasorg63&E. Scapparoneorg53&F.
Scarlassaraorg29&B. Schaeferorg94&H.S. Scheidorg69&C. Schiauaorg47&R.
Schickerorg102&C. Schmidtorg104&H.R. Schmidtorg101&M.O. Schmidtorg102&M.
Schmidtorg101&N.V. Schmidtorg69,org94&J. Schukraftorg34&Y.
Schutzorg34,org134&K. Schwarzorg104&K. Schwedaorg104&G. Scioliorg27&E.
Scomparinorg58&M. Šefčíkorg38&J.E. Segerorg16&Y. Sekiguchiorg130&D.
Sekihataorg45&I. Selyuzhenkovorg91,org104&S. Senyukovorg134&E.
Serradillaorg72&P. Settorg48&A. Sevcencoorg68&A. Shabanovorg62&A.
Shabetaiorg113&R. Shahoyanorg34&W. Shaikhorg107&A. Shangaraevorg90&A.
Sharmaorg98&A. Sharmaorg99&M. Sharmaorg99&N. Sharmaorg98&A.I. Sheikhorg139&K.
Shigakiorg45&M. Shimomuraorg82&S. Shirinkinorg64&Q. Shouorg6,org110&K.
Shtejerorg26&Y. Sibiriakorg87&S. Siddhantaorg54&K.M. Sielewiczorg34&T.
Siemiarczukorg84&D. Silvermyrorg80&G. Simatovicorg89&G.
Simonettiorg34,org103&R. Singarajuorg139&R. Singhorg85&R. Singhorg99&V.
Singhalorg139&T. Sinhaorg107&B. Sitarorg14&M. Sittaorg32&T.B. Skaaliorg21&M.
Slupeckiorg126&N. Smirnovorg144&R.J.M. Snellingsorg63&T.W. Snellmanorg126&J.
Sochanorg115&C. Sonccoorg109&J. Songorg18&F. Soramelorg29&S. Sorensenorg128&F.
Sozziorg104&I. Sputowskaorg117&J. Stachelorg102&I. Stanorg68&P.
Stankusorg94&E. Stenlundorg80&D. Stoccoorg113&M.M. Storetvedtorg36&P.
Strmenorg14&A.A.P. Suaideorg120&T. Sugitateorg45&C. Suireorg61&M.
Suleymanovorg15&M. Suljicorg34,org25&R. Sultanovorg64&M. Šumberaorg93&S.
Sumowidagdoorg50&K. Suzukiorg112&S. Swainorg66&A. Szaboorg14&I. Szarkaorg14&U.
Tabassamorg15&J. Takahashiorg121&G.J. Tambaveorg22&N. Tanakaorg131&M.
Tarhiniorg113&M. Tariqorg17&M.G. Tarzilaorg47&A. Tauroorg34&G. Tejeda
Muñozorg44&A. Telescaorg34&C. Terrevoliorg29&B. Teyssierorg133&D.
Thakurorg49&S. Thakurorg139&D. Thomasorg118&F. Thoresenorg88&R.
Tieulentorg133&A. Tikhonovorg62&A.R. Timminsorg125&A. Toiaorg69&N.
Topilskayaorg62&M. Toppiorg51&S.R. Torresorg119&S. Tripathyorg49&S.
Trogoloorg26&G. Trombettaorg33&L. Tropporg38&V. Trubnikovorg2&W.H.
Trzaskaorg126&T.P. Trzcinskiorg140&B.A. Trzeciakorg63&T. Tsujiorg130&A.
Tumkinorg106&R. Turrisiorg56&T.S. Tveterorg21&K. Ullalandorg22&E.N.
Umakaorg125&A. Urasorg133&G.L. Usaiorg24&A. Utrobicicorg97&M. Valaorg115&J.W.
Van Hoorneorg34&M. van Leeuwenorg63&P. Vande Vyvreorg34&D. Vargaorg143&A.
Vargasorg44&M. Vargyasorg126&R. Varmaorg48&M. Vasileiouorg83&A.
Vasilievorg87&A. Vauthierorg78&O. Vázquez Doceorg103,org116&V.
Vecherninorg111&A.M. Veenorg63&E. Vercellinorg26&S. Vergara Limónorg44&L.
Vermuntorg63&R. Vernetorg7&R. Vértesiorg143&L. Vickovicorg35&J.
Viinikainenorg126&Z. Vilakaziorg129&O. Villalobos Baillieorg108&A. Villatoro
Telloorg44&A. Vinogradovorg87&T. Virgiliorg30&V. Vislaviciusorg88,org80&A.
Vodopyanovorg75&M.A. Völklorg101&K. Voloshinorg64&S.A. Voloshinorg141&G.
Volpeorg33&B. von Hallerorg34&I. Vorobyevorg116,org103&D. Voscekorg115&D.
Vranicorg104,org34&J. Vrlákováorg38&B. Wagnerorg22&H. Wangorg63&M. Wangorg6&Y.
Watanabeorg131&M. Weberorg112&S.G. Weberorg104&A. Wegrzynekorg34&D.F.
Weiserorg102&S.C. Wenzelorg34&J.P. Wesselsorg142&U. Westerhofforg142&A.M.
Whiteheadorg124&J. Wiechulaorg69&J. Wikneorg21&G. Wilkorg84&J.
Wilkinsonorg53&G.A. Willemsorg142,org34&M.C.S. Williamsorg53&E.
Willsherorg108&B. Windelbandorg102&W.E. Wittorg128&R. Xuorg6&S. Yalcinorg77&K.
Yamakawaorg45&S. Yanoorg45&Z. Yinorg6&H. Yokoyamaorg131,org78&I.-K.
Yooorg18&J.H. Yoonorg60&V. Yurchenkoorg2&V. Zaccoloorg58&A. Zamanorg15&C.
Zampolliorg34&H.J.C. Zanoliorg120&N. Zardoshtiorg108&A. Zarochentsevorg111&P.
Závadaorg67&N. Zaviyalovorg106&H. Zbroszczykorg140&M. Zhalovorg96&X.
Zhangorg6&Y. Zhangorg6&Z. Zhangorg6,org132&C. Zhaoorg21&V.
Zherebchevskiiorg111&N. Zhigarevaorg64&D. Zhouorg6&Y. Zhouorg88&Z.
Zhouorg22&H. Zhuorg6&J. Zhuorg6&Y. Zhuorg6&A. Zichichiorg27,org10&M.B.
Zimmermannorg34&G. Zinovjevorg2&J. Zmeskalorg112&S. Zouorg6&

## Affiliation notes

org*Deceased

orgIDipartimento DET del Politecnico di Torino, Turin, Italy

orgIIM.V. Lomonosov Moscow State University, D.V. Skobeltsyn Institute of
Nuclear, Physics, Moscow, Russia

orgIIIDepartment of Applied Physics, Aligarh Muslim University, Aligarh, India

orgIVInstitute of Theoretical Physics, University of Wroclaw, Poland

## Collaboration Institutes

org1A.I. Alikhanyan National Science Laboratory (Yerevan Physics Institute)
Foundation, Yerevan, Armenia

org2Bogolyubov Institute for Theoretical Physics, National Academy of Sciences
of Ukraine, Kiev, Ukraine

org3Bose Institute, Department of Physics and Centre for Astroparticle Physics
and Space Science (CAPSS), Kolkata, India

org4Budker Institute for Nuclear Physics, Novosibirsk, Russia

org5California Polytechnic State University, San Luis Obispo, California,
United States

org6Central China Normal University, Wuhan, China

org7Centre de Calcul de l’IN2P3, Villeurbanne, Lyon, France

org8Centro de Aplicaciones Tecnológicas y Desarrollo Nuclear (CEADEN), Havana,
Cuba

org9Centro de Investigación y de Estudios Avanzados (CINVESTAV), Mexico City
and Mérida, Mexico

org10Centro Fermi - Museo Storico della Fisica e Centro Studi e Ricerche
“Enrico Fermi’, Rome, Italy

org11Chicago State University, Chicago, Illinois, United States

org12China Institute of Atomic Energy, Beijing, China

org13Chonbuk National University, Jeonju, Republic of Korea

org14Comenius University Bratislava, Faculty of Mathematics, Physics and
Informatics, Bratislava, Slovakia

org15COMSATS Institute of Information Technology (CIIT), Islamabad, Pakistan

org16Creighton University, Omaha, Nebraska, United States

org17Department of Physics, Aligarh Muslim University, Aligarh, India

org18Department of Physics, Pusan National University, Pusan, Republic of
Korea

org19Department of Physics, Sejong University, Seoul, Republic of Korea

org20Department of Physics, University of California, Berkeley, California,
United States

org21Department of Physics, University of Oslo, Oslo, Norway

org22Department of Physics and Technology, University of Bergen, Bergen,
Norway

org23Dipartimento di Fisica dell’Università ’La Sapienza’ and Sezione INFN,
Rome, Italy

org24Dipartimento di Fisica dell’Università and Sezione INFN, Cagliari, Italy

org25Dipartimento di Fisica dell’Università and Sezione INFN, Trieste, Italy

org26Dipartimento di Fisica dell’Università and Sezione INFN, Turin, Italy

org27Dipartimento di Fisica e Astronomia dell’Università and Sezione INFN,
Bologna, Italy

org28Dipartimento di Fisica e Astronomia dell’Università and Sezione INFN,
Catania, Italy

org29Dipartimento di Fisica e Astronomia dell’Università and Sezione INFN,
Padova, Italy

org30Dipartimento di Fisica ‘E.R. Caianiello’ dell’Università and Gruppo
Collegato INFN, Salerno, Italy

org31Dipartimento DISAT del Politecnico and Sezione INFN, Turin, Italy

org32Dipartimento di Scienze e Innovazione Tecnologica dell’Università del
Piemonte Orientale and INFN Sezione di Torino, Alessandria, Italy

org33Dipartimento Interateneo di Fisica ‘M. Merlin’ and Sezione INFN, Bari,
Italy

org34European Organization for Nuclear Research (CERN), Geneva, Switzerland

org35Faculty of Electrical Engineering, Mechanical Engineering and Naval
Architecture, University of Split, Split, Croatia

org36Faculty of Engineering and Science, Western Norway University of Applied
Sciences, Bergen, Norway

org37Faculty of Nuclear Sciences and Physical Engineering, Czech Technical
University in Prague, Prague, Czech Republic

org38Faculty of Science, P.J. Šafárik University, Košice, Slovakia

org39Frankfurt Institute for Advanced Studies, Johann Wolfgang Goethe-
Universität Frankfurt, Frankfurt, Germany

org40Gangneung-Wonju National University, Gangneung, Republic of Korea

org41Gauhati University, Department of Physics, Guwahati, India

org42Helmholtz-Institut für Strahlen- und Kernphysik, Rheinische Friedrich-
Wilhelms-Universität Bonn, Bonn, Germany

org43Helsinki Institute of Physics (HIP), Helsinki, Finland

org44High Energy Physics Group, Universidad Autónoma de Puebla, Puebla, Mexico

org45Hiroshima University, Hiroshima, Japan

org46Hochschule Worms, Zentrum für Technologietransfer und Telekommunikation
(ZTT), Worms, Germany

org47Horia Hulubei National Institute of Physics and Nuclear Engineering,
Bucharest, Romania

org48Indian Institute of Technology Bombay (IIT), Mumbai, India

org49Indian Institute of Technology Indore, Indore, India

org50Indonesian Institute of Sciences, Jakarta, Indonesia

org51INFN, Laboratori Nazionali di Frascati, Frascati, Italy

org52INFN, Sezione di Bari, Bari, Italy

org53INFN, Sezione di Bologna, Bologna, Italy

org54INFN, Sezione di Cagliari, Cagliari, Italy

org55INFN, Sezione di Catania, Catania, Italy

org56INFN, Sezione di Padova, Padova, Italy

org57INFN, Sezione di Roma, Rome, Italy

org58INFN, Sezione di Torino, Turin, Italy

org59INFN, Sezione di Trieste, Trieste, Italy

org60Inha University, Incheon, Republic of Korea

org61Institut de Physique Nucléaire d’Orsay (IPNO), Institut National de
Physique Nucléaire et de Physique des Particules (IN2P3/CNRS), Université de
Paris-Sud, Université Paris-Saclay, Orsay, France

org62Institute for Nuclear Research, Academy of Sciences, Moscow, Russia

org63Institute for Subatomic Physics, Utrecht University/Nikhef, Utrecht,
Netherlands

org64Institute for Theoretical and Experimental Physics, Moscow, Russia

org65Institute of Experimental Physics, Slovak Academy of Sciences, Košice,
Slovakia

org66Institute of Physics, Homi Bhabha National Institute, Bhubaneswar, India

org67Institute of Physics of the Czech Academy of Sciences, Prague, Czech
Republic

org68Institute of Space Science (ISS), Bucharest, Romania

org69Institut für Kernphysik, Johann Wolfgang Goethe-Universität Frankfurt,
Frankfurt, Germany

org70Instituto de Ciencias Nucleares, Universidad Nacional Autónoma de México,
Mexico City, Mexico

org71Instituto de Física, Universidade Federal do Rio Grande do Sul (UFRGS),
Porto Alegre, Brazil

org72Instituto de Física, Universidad Nacional Autónoma de México, Mexico
City, Mexico

org73iThemba LABS, National Research Foundation, Somerset West, South Africa

org74Johann-Wolfgang-Goethe Universität Frankfurt Institut für Informatik,
Fachbereich Informatik und Mathematik, Frankfurt, Germany

org75Joint Institute for Nuclear Research (JINR), Dubna, Russia

org76Korea Institute of Science and Technology Information, Daejeon, Republic
of Korea

org77KTO Karatay University, Konya, Turkey

org78Laboratoire de Physique Subatomique et de Cosmologie, Université
Grenoble-Alpes, CNRS-IN2P3, Grenoble, France

org79Lawrence Berkeley National Laboratory, Berkeley, California, United
States

org80Lund University Department of Physics, Division of Particle Physics,
Lund, Sweden

org81Nagasaki Institute of Applied Science, Nagasaki, Japan

org82Nara Women’s University (NWU), Nara, Japan

org83National and Kapodistrian University of Athens, School of Science,
Department of Physics , Athens, Greece

org84National Centre for Nuclear Research, Warsaw, Poland

org85National Institute of Science Education and Research, Homi Bhabha
National Institute, Jatni, India

org86National Nuclear Research Center, Baku, Azerbaijan

org87National Research Centre Kurchatov Institute, Moscow, Russia

org88Niels Bohr Institute, University of Copenhagen, Copenhagen, Denmark

org89Nikhef, National institute for subatomic physics, Amsterdam, Netherlands

org90NRC Kurchatov Institute IHEP, Protvino, Russia

org91NRNU Moscow Engineering Physics Institute, Moscow, Russia

org92Nuclear Physics Group, STFC Daresbury Laboratory, Daresbury, United
Kingdom

org93Nuclear Physics Institute of the Czech Academy of Sciences, Řež u Prahy,
Czech Republic

org94Oak Ridge National Laboratory, Oak Ridge, Tennessee, United States

org95Ohio State University, Columbus, Ohio, United States

org96Petersburg Nuclear Physics Institute, Gatchina, Russia

org97Physics department, Faculty of science, University of Zagreb, Zagreb,
Croatia

org98Physics Department, Panjab University, Chandigarh, India

org99Physics Department, University of Jammu, Jammu, India

org100Physics Department, University of Rajasthan, Jaipur, India

org101Physikalisches Institut, Eberhard-Karls-Universität Tübingen, Tübingen,
Germany

org102Physikalisches Institut, Ruprecht-Karls-Universität Heidelberg,
Heidelberg, Germany

org103Physik Department, Technische Universität München, Munich, Germany

org104Research Division and ExtreMe Matter Institute EMMI, GSI
Helmholtzzentrum für Schwerionenforschung GmbH, Darmstadt, Germany

org105Rudjer Bošković Institute, Zagreb, Croatia

org106Russian Federal Nuclear Center (VNIIEF), Sarov, Russia

org107Saha Institute of Nuclear Physics, Homi Bhabha National Institute,
Kolkata, India

org108School of Physics and Astronomy, University of Birmingham, Birmingham,
United Kingdom

org109Sección Física, Departamento de Ciencias, Pontificia Universidad
Católica del Perú, Lima, Peru

org110Shanghai Institute of Applied Physics, Shanghai, China

org111St. Petersburg State University, St. Petersburg, Russia

org112Stefan Meyer Institut für Subatomare Physik (SMI), Vienna, Austria

org113SUBATECH, IMT Atlantique, Université de Nantes, CNRS-IN2P3, Nantes,
France

org114Suranaree University of Technology, Nakhon Ratchasima, Thailand

org115Technical University of Košice, Košice, Slovakia

org116Technische Universität München, Excellence Cluster ’Universe’, Munich,
Germany

org117The Henryk Niewodniczanski Institute of Nuclear Physics, Polish Academy
of Sciences, Cracow, Poland

org118The University of Texas at Austin, Austin, Texas, United States

org119Universidad Autónoma de Sinaloa, Culiacán, Mexico

org120Universidade de São Paulo (USP), São Paulo, Brazil

org121Universidade Estadual de Campinas (UNICAMP), Campinas, Brazil

org122Universidade Federal do ABC, Santo Andre, Brazil

org123University College of Southeast Norway, Tonsberg, Norway

org124University of Cape Town, Cape Town, South Africa

org125University of Houston, Houston, Texas, United States

org126University of Jyväskylä, Jyväskylä, Finland

org127University of Liverpool, Liverpool, United Kingdom

org128University of Tennessee, Knoxville, Tennessee, United States

org129University of the Witwatersrand, Johannesburg, South Africa

org130University of Tokyo, Tokyo, Japan

org131University of Tsukuba, Tsukuba, Japan

org132Université Clermont Auvergne, CNRS/IN2P3, LPC, Clermont-Ferrand, France

org133Université de Lyon, Université Lyon 1, CNRS/IN2P3, IPN-Lyon,
Villeurbanne, Lyon, France

org134Université de Strasbourg, CNRS, IPHC UMR 7178, F-67000 Strasbourg,
France, Strasbourg, France

org135 Université Paris-Saclay Centre d¿Études de Saclay (CEA), IRFU,
Department de Physique Nucléaire (DPhN), Saclay, France

org136Università degli Studi di Foggia, Foggia, Italy

org137Università degli Studi di Pavia, Pavia, Italy

org138Università di Brescia, Brescia, Italy

org139Variable Energy Cyclotron Centre, Homi Bhabha National Institute,
Kolkata, India

org140Warsaw University of Technology, Warsaw, Poland

org141Wayne State University, Detroit, Michigan, United States

org142Westfälische Wilhelms-Universität Münster, Institut für Kernphysik,
Münster, Germany

org143Wigner Research Centre for Physics, Hungarian Academy of Sciences,
Budapest, Hungary

org144Yale University, New Haven, Connecticut, United States

org145Yonsei University, Seoul, Republic of Korea