2021 233 7 November

K${}^{0}_{\rm S}$K${}^{0}_{\rm S}$ and K${}^{0}_{\rm S}$K± femtoscopy

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

Femtoscopic correlations with the particle pair combinations K${}^{0}_{\rm
S}$K${}^{0}_{\rm S}$ and K${}^{0}_{\rm S}$K± are studied in pp collisions at
$\sqrt{s}=5.02$ and $13$ TeV by the ALICE experiment. At both energies, boson
source parameters are extracted for both pair combinations, by fitting models
based on Gaussian size distributions of the sources, to the measured two-
particle correlation functions. The interaction model used for the
K${}^{0}_{\rm S}$K${}^{0}_{\rm S}$ analysis includes quantum statistics and
strong final-state interactions through the $f_{0}(980)$ and $a_{0}(980)$
resonances. The model used for the K${}^{0}_{\rm S}$K± analysis includes only
the final-state interaction through the $a_{0}$ resonance. Source parameters
extracted in the present work are compared with published values from pp
collisions at $\sqrt{s}=$ 7 TeV and the different pair combinations are found
to be consistent. From the observation that the strength of the K${}^{0}_{\rm
S}$K${}^{0}_{\rm S}$ correlations is significantly greater than the strength
of the K${}^{0}_{\rm S}$K± correlations, the new results are compatible with
the $a_{0}$ resonance being a tetraquark state of the form
$(q_{1},\overline{q_{2}},s,\overline{s})$, where $q_{1}$ and $q_{2}$ are $u$
or $d$ quarks.

##  1 Introduction

Identical boson femtoscopy, especially identical charged $\pi^{\pm}\pi^{\pm}$
femtoscopy, has been used extensively over the years to study experimentally
the space-time geometry of the collision region in high-energy proton and
heavy-ion collisions [1]. More recently, identical-kaon femtoscopy studies
have been reported for a variety of colliding systems, energies and kaon
pairs: K${}^{0}_{\rm S}$K${}^{0}_{\rm S}$ pairs in Au–Au collisions at
$\sqrt{s_{\rm NN}}=0.2$ TeV by the STAR collaboration [2], K${}^{0}_{\rm
S}$K${}^{0}_{\rm S}$ and K±K± pairs in pp collisions at $\sqrt{s}=7$ TeV and
Pb–Pb collisions at $\sqrt{s_{\rm NN}}=2.76$ TeV by the ALICE collaboration
[3, 4, 5]. Identical-kaon femtoscopy gives information about the collision
region that is complementary to that obtained in identical-pion femtoscopy in
that it probes the hotter region where strange quarks are produced and extends
the momentum range over which the femtoscopy analysis can be applied. Also,
the kaon analyses are expected to offer a cleaner signal compared to pions, as
they are less affected by resonance decays.

Non-identical kaon femtoscopy with K${}^{0}_{\rm S}$K± pairs was first
measured by ALICE in pp collisions at $\sqrt{s}=7$ TeV and Pb–Pb collisions at
$\sqrt{s_{\rm NN}}=2.76$ TeV [6, 7]. Although the space-time geometry of the
kaon source can be extracted with these pairs, the main emphasis of non-
identical kaon femtoscopy is to obtain information about the strong final-
state interaction (FSI) between the kaons. For the identical kaon cases the
interactions are, for K±K±: quantum statistics, Coulomb interaction, and for
K${}^{0}_{\rm S}$K${}^{0}_{\rm S}$: quantum statistics, FSI through
$f_{0}(980)$ and $a_{0}(980)$ threshold resonances [2]. For the K${}^{0}_{\rm
S}$K±, the only FSI is through the $a_{0}$ resonance. Note that “threshold
resonances”, like the $a_{0}$ and $f_{0}$, are resonances where the sum of the
masses of the decay particles is very close in value to the mass of the
resonance. A non-resonant FSI in the K${}^{0}_{\rm S}$K${}^{0}_{\rm S}$ pair
is expected to be small compared with the resonant $f_{0}$ and $a_{0}$ FSI and
can be neglected to first order [2, 8]. The only pair-wise interaction
expected for a K${}^{0}_{\rm S}$K± pair at freeze out from the collision
system is a FSI through the $a_{0}$ resonance. This is because there is no
quantum statistics enhancement for non-identical kaons, no Coulomb effect
since one of the kaons is neutral, and no strong FSI through the $f_{0}$
resonance since the kaon pair is in an isospin-1 state, as is the $a_{0}$,
whereas the $f_{0}$ is isospin-0 and thus isospin would not be conserved.

Another feature of the K${}^{0}_{\rm S}$K± FSI through the $a_{0}$ resonance
is that since the $a_{0}$ has zero strangeness, and the K${}^{0}_{\rm S}$ is
composed of a linear combination of the K0 and ${\rm\overline{K^{0}}}$, only
the ${\rm\overline{K^{0}}}$K+ pair from K${}^{0}_{\rm S}$K+ and the K0K- pair
from K${}^{0}_{\rm S}$K- can form $a_{0}$ resonances in order to conserve zero
strangeness. This feature allows the K0 and ${\rm\overline{K^{0}}}$ sources to
be studied separately. However, it was concluded in the previous ALICE
K${}^{0}_{\rm S}$K± publications that there is no significant difference in
the source parameters between K${}^{0}_{\rm S}$K+ and K${}^{0}_{\rm S}$K- [6,
7].

Lastly, the K${}^{0}_{\rm S}$K± FSI allows the properties of the $a_{0}$
resonance itself to be studied. This is interesting in its own right since
many works exist in the literature discussing the possibility that the $a_{0}$
could be a 4-quark state, i.e. a tetraquark. It was first suggested in 1977
that experimentally-observed low-lying mesons, such as the $a_{0}$, are part
of a SU(3) tetraquark nonet using a quark model [9]. A later follow-up
calculation was published reinforcing this work using lattice QCD calculations
[10]. Since then, there have been a number of QCD studies of these mesons that
fall in the categories of QCD-inspired models, for example Refs. [11, 12, 13,
14], and lattice QCD calculations, for example Refs. [15, 16, 17]. An
interesting result that was found from the previous measurements in comparing
the strengths, i.e. $\lambda$-parameters, of the K${}^{0}_{\rm
S}$K${}^{0}_{\rm S}$ and K${}^{0}_{\rm S}$K± correlations with each other was
that the strength of the K${}^{0}_{\rm S}$K${}^{0}_{\rm S}$ correlations is
significantly larger than the strength of the K${}^{0}_{\rm S}$K± correlations
measured in $\sqrt{s}=7$ TeV pp collisions. It was suggested that this could
be an indication that the $a_{0}$ is a tetraquark state [6, 7].

In light of the interesting results from the $\sqrt{s}=7$ TeV pp measurements,
the main motivations to extend the measurements to $\sqrt{s}=$ 5.02 and 13 TeV
pp collisions are the following:

  * •

In these new measurements, we investigate the collision-energy dependence of
the $\lambda$ difference between K${}^{0}_{\rm S}$K${}^{0}_{\rm S}$ and
K${}^{0}_{\rm S}$K±. A lack of a dependence on the center-of-mass energy would
be consistent with the tetraquark interpretation of the $a_{0}(980)$.

  * •

The previous paper compared results for $\lambda$ that were obtained and
published seven years apart, i.e. K${}^{0}_{\rm S}$K${}^{0}_{\rm S}$ in PLB
from 2012 [3] and K${}^{0}_{\rm S}$K± in PLB from 2019 [6], and that were
analyzed in different ways. The K${}^{0}_{\rm S}$K${}^{0}_{\rm S}$ analysis
from the 2012 paper was done in several multiplicity ranges which had to be
averaged in order to compare with the minimum-bias 2019 K${}^{0}_{\rm S}$K±
result. However, in the present paper the new K${}^{0}_{\rm S}$K${}^{0}_{\rm
S}$ and K${}^{0}_{\rm S}$K± analyses were done at the same time and using the
same kinematic ranges. By carrying out simultaneous measurements of
K${}^{0}_{\rm S}$K${}^{0}_{\rm S}$ and K${}^{0}_{\rm S}$K± this results in a
better comparison with each other.

  * •

In this new analysis a detailed calculation of the effect of long-lived
resonances on the $\lambda$ parameter is presented to better establish that
this contamination is not responsible for the $\lambda$ difference.

##  2 Description of experiment and data selection

Data taken by the ALICE experiment [18] in the LHC Run 2 period (2015–2018)
were employed in the present analysis. This analysis used both $\sqrt{s}=$
5.02 TeV and 13 TeV reconstructed minimum bias triggered pp collisions, giving
about $0.5\times 10^{9}$ and $1.5\times 10^{9}$ events, respectively. Monte
Carlo (MC) simulations were used for determining selection values, momentum
resolution and purity studies, and for the baseline underlying the signal for
the case of the K${}^{0}_{\rm S}$K${}^{0}_{\rm S}$ analyses. In the MC
calculations, particles from pp collision events simulated by the general-
purpose generator PYTHIA8 [19] with the Monash 2013 tune [20] were transported
through a GEANT3 [21] model of the ALICE detector. The total numbers of MC
events used in the $\sqrt{s}=$ 5.02 and 13 TeV analyses were about $0.7\times
10^{9}$ and $1.2\times 10^{9}$, respectively.

The V0 detectors, which consist of two arrays of scintillators located along
the beamline and covering the full azimuth [22, 23] were used for triggering
and event selection. Charged particles were reconstructed and identified with
the central barrel detectors located within a solenoid magnet with a field
strength of magnitude $B=0.5$ T. Charged particle tracking was performed using
the Time Projection Chamber (TPC) [24] and the Inner Tracking System (ITS)
[18]. The momentum determination for K± was made using only the TPC. The ITS
allowed for high spatial resolution in determining the primary collision
vertex, which was used to constrain the TPC tracks. An average momentum
resolution of less than 10 MeV/$c$ was typically obtained for the charged
tracks of interest in this analysis [25]. The primary vertex was obtained from
the ITS, the position being constrained along the beam direction to be within
$\pm 10$ cm of the center of the ALICE detector. In addition to the standard
track quality selections [25], the selections based on the quality of track
fitting and the number of detected tracking points in the TPC were used to
ensure that only well-reconstructed tracks were taken into account in the
analysis [26, 24, 25].

Particle Identification (PID) for reconstructed tracks was carried out using
both the TPC and the Time-Of-Flight (TOF) detectors in the pseudorapidity
range $|\eta|<0.8$ [26, 27]. For the PID signal from both detectors, a value
($N_{\sigma}$) was assigned to each track denoting the number of standard
deviations between the measured track information and expected values,
assuming a mass hypothesis, divided by the detector resolution [5, 26, 27,
25]. For TPC PID, a parametrized Bethe-Bloch formula was used to calculate the
specific energy loss $\left<{\rm d}E/{\rm d}x\right>$ in the detector expected
for a particle with a given charge, mass and momentum. For PID with TOF, the
particle mass was used to calculate the expected time-of-flight as a function
of track length and momentum.

Other event selection criteria were also applied. The event must have one
accepted possible K${}^{0}_{\rm S}$K${}^{0}_{\rm S}$ or K${}^{0}_{\rm S}$K±
pair. Pile-up events were rejected using the standard ALICE pile-up rejection
method [26]. Pile-up effects were also investigated by performing the analysis
using only low-luminosity data-taking periods. No significant difference was
found in the extracted $R$ and $\lambda$ parameters compared with the higher
count-rate runs used.

###  2.1 Kaon selection

The methods used to select and identify individual K${}^{0}_{\rm S}$ and K±
particles are the same as those used for the ALICE K${}^{0}_{\rm
S}$K${}^{0}_{\rm S}$ [3] and K±K± [4] analyses in pp collisions at
$\sqrt{s}=7$ TeV, and are described in the following sections.

####  2.1.1 K${}^{0}_{\rm S}$ reconstruction

Using an invariant mass technique, the neutral K${}^{0}_{\rm S}$ vertices and
parameters are reconstructed and calculated from pairs of detected $\pi^{+}$
$\pi^{-}$ tracks. Single-particle selection criteria for the K${}^{0}_{\rm S}$
and the pions, for example particle momentum ($p$), transverse momentum
($p_{T}$), and pseudorapidity ($\eta$), are shown in Table 1.

Table 1: Single-particle selection criteria. Neutral kaon selection | Value
---|---
Daughter $p_{\rm T}$ |  $>0.15$ GeV/$c$
Daughter $|\eta|$ | $<0.8$
Daughter DCA (3D) to primary vertex |  $>0.4$ cm
Daughter TPC $N_{\rm\sigma}$ | $<3$
Daughter TOF $N_{\rm\sigma}$ (for $p>0.8$ GeV/$c$) | $<3$
$\left|\eta\right|$ | $<0.8$
DCA (3D) $\pi^{+}$ to $\pi^{-}$ |  $<0.3$ cm
DCA (3D) of K${}^{0}_{\rm S}$ to primary vertex |  $<0.3$ cm
Decay length (3D, lab frame) |  $<30$ cm
Decay radius (2D, lab frame) |  $>0.2$ cm
Cosine of pointing angle | $>0.99$
Invariant mass |  $0.485<m<0.510$ GeV/$c^{\rm 2}$
Charged kaon selection | Value
$p_{\rm T}$ |  $0.15<p_{\rm T}<1.2$ GeV/$c$
$|\eta|$ | $<0.8$
Transverse DCA to primary vertex |  $<2.4$ cm
Longitudinal DCA to primary vertex |  $<3.0$ cm
$N_{\rm\sigma}^{TOF}$ with valid TOF signal and $p>0.5$ GeV/$c$ | $<2$
$N_{\rm\sigma}^{TPC}$ if no TOF signal for all $p_{\rm T}$ | $<2$
Kalman fit $\chi^{2}/N^{\rm clus}$ | $\leq 4$

Most of the topological selection criteria ($\pi^{+}$ $\pi^{-}$ distance-of-
closest-approach (DCA), $\pi$-vertex DCA, K${}^{0}_{\rm S}$ DCA, and decay
length) were chosen to optimize purity and statistical significance. If two
reconstructed K${}^{0}_{\rm S}$ particles share a daughter track, both are
removed from the analysis. The selection criteria in this analysis are
comparable to or stricter than those in other K${}^{0}_{\rm S}$ analyses;
strict selection criteria are favored to increase the sample purity.

A candidate K${}^{0}_{\rm S}$ vertex with a reconstructed invariant mass
within $0.485<m(\pi^{+}\pi^{-})<0.510$ GeV/$c^{2}$ is identified as a
K${}^{0}_{\rm S}$. In this range, the single-K${}^{0}_{\rm S}$ purity is
measured to be $98\pm 1\%$ for the $k_{\rm T}$ interval of $0.5<k_{T}<0.7$
GeV/$c$, where $k_{\rm T}=|\overrightarrow{p_{\rm T1}}+\overrightarrow{p_{\rm
T2}}|/2$, and where $\overrightarrow{p_{\rm T1}}$ and $\overrightarrow{p_{\rm
T2}}$ are the transverse momenta of the particles in the pair. The purity here
is defined as Signal/(Signal + Background) and is calculated by fitting a
fourth-order polynomial to the background in the combined invariant mass
intervals 0.4–0.45 GeV/$c^{2}$ and 0.55–0.6 GeV/$c^{2}$ and using the bin
contents of the invariant mass histogram as the “Signal + Background”. No
selection on $p_{\rm T}$ is employed in this analysis for K${}^{0}_{\rm S}$.
Having a pair purity less than unity will be reflected in the lowering of the
$\lambda$ parameter, which can later be corrected for purity, however the
K${}^{0}_{\rm S}$ purity is very close to unity for this analysis.

####  2.1.2 K± identification

As mentioned above, charged kaons are selected using the TPC and TOF detectors
with the same methods as employed in Refs. [4, 5]. The quality of the track is
determined by the $\chi^{2}/N^{\rm clus}$ value for the Kalman fit to the
reconstructed position of the TPC clusters ($N^{\rm clus}$ is the number of
clusters attached to the track). The track is rejected if the value is larger
than 4.0. The selection criteria used for the charged kaon selection in the
TPC and TOF are shown in Table 1. In the table, $N_{\rm\sigma}^{TPC}$ and
$N_{\rm\sigma}^{TOF}$ are the numbers of standard deviations the TPC energy-
loss and TOF signal are away from their predicted values divided by detector
resolution, respectively.

The average charged kaon purity is found using PYTHIA8 MC simulations to be
$91\pm 1\%$ in the $k_{\rm T}$ range used in this analysis, i.e. $0.5<k_{\rm
T}<0.7$ GeV/$c$. This is in agreement with the charged kaon purity found by
the ALICE collaboration in Ref. [5].

###  2.2 Two-track selection

Experimental two-track effects, such as the merging of two real tracks into
one reconstructed track and the splitting of one real track into two
reconstructed tracks, is a challenge for femtoscopic studies. These effects
are observed for tracks with small average separation in the TPC. For each
pair of like-sign tracks, which could be pions from two K${}^{0}_{\rm S}$
decays, or the pion from a K${}^{0}_{\rm S}$ decay and the same-charged K±
track, the distance between the tracks was calculated at up to nine positions
throughout the TPC (every 20 cm along the radial direction from 85 cm to 245
cm) and then averaged. When comparing the distribution of the average
separation of track pairs from single events with the distribution from pairs
constructed of tracks from different events (mixed events), a splitting
enhancement is seen in the same-events for average separations approaching
zero. For the distribution of mixed-event tracks, the primary vertex position
for each track was subtracted from each track point to mock them up as coming
from the same event. To minimize this splitting effect, this analysis demanded
that the tracks must have an average TPC separation of at least 13 cm.

##  3 Two-particle correlation function

This analysis studies the momentum correlations of K${}^{0}_{\rm
S}$K${}^{0}_{\rm S}$ and K${}^{0}_{\rm S}$K± pairs using the two-particle
correlation function, defined as $C(k^{*})=A(k^{*})/B(k^{*})$, where
$A(k^{*})$ is the measured distribution of real pairs from the same event and
$B(k^{*})$ is the reference distribution of pairs from mixed events. The
quantity $k^{*}$ is the momentum of one of the particles in the pair rest
frame, and for the general case of two particles with unequal mass, $m_{1}$
and $m_{2}$, is given by

$k^{*}=\sqrt{\frac{w^{2}-m_{1}^{2}m_{2}^{2}}{2w+m_{1}^{2}+m_{2}^{2}}}$ (1)

where,

$w\equiv(q_{inv}^{2}+m_{1}^{2}+m_{2}^{2})/2.$ (2)

The square of the invariant momentum difference $q_{\rm
inv}^{2}=|\vec{p_{1}}-\vec{p_{2}}|^{2}-|E_{1}-E_{2}|^{2}$ is most conveniently
evaluated with the momenta and energies of the two particles measured in the
lab frame. Note that $m_{1}=m_{2}$ gives $k^{*}=q_{\rm inv}/2$. The
denominator $B(k^{*})$ is formed by mixing particles from each event with
particles from ten other events in the same $z$-vertex bin (2 cm width) and of
similar event multiplicity. A $k^{*}$ bin size of 20 MeV/$c$ was used in all
cases.

As mentioned earlier, correlation functions are calculated for minimum bias
events and a $k_{\rm T}$ range from 0.5–0.7 GeV/$c$. This closely reproduces
the conditions for the kaon femtoscopy measurements with K${}^{0}_{\rm
S}$K${}^{0}_{\rm S}$ and K${}^{0}_{\rm S}$K± pairs published by ALICE for pp
collisions at $\sqrt{s}=7$ TeV with which the present results will be compared
[3, 6]. The $k_{\rm T}$ range used encompasses the peak in the $k_{\rm T}$
distributions at each collision energy. Also, the pseudorapidity density of
charged particles at midrapidity, $dN_{ch}/d\eta$, is found to be small in pp
collisions and has a weak dependence on $\sqrt{s}$, measured to be $5.91\pm
0.45$, $6.01^{+0.20}_{-0.12}$, and $7.60\pm 0.50$ for $\sqrt{s}=5.02$, 7 and
13 TeV, respectively, where the uncertainties are the statistical and
systematic uncertainties added in quadrature [28, 29, 30].

Figure 1 shows an example of raw experimental K${}^{0}_{\rm S}$K${}^{0}_{\rm
S}$ correlation functions along with the resulting distributions from PYTHIA8
simulations normalized in the $k^{*}$ region 0.6–0.8 GeV/$c$ for $\sqrt{s}=$
5.02 and 13 TeV. Note that the PYTHIA8 calculations do not contain FSI or
femtoscopic correlations. Figure 2 shows an example of raw experimental
K${}^{0}_{\rm S}$K± correlation functions plotted with baseline fits for
various functions (see below) for $\sqrt{s}=$ 5.02 and 13 TeV. The raw
correlation functions from the data are enhanced for $k^{*}<0.1$ GeV/$c$ due
to quantum statistics and the FSI of the $f_{0}$ and $a_{0}$ and slightly
suppressed in the region $0.1<k^{*}<0.4$ GeV/$c$ due to the FSI for
K${}^{0}_{\rm S}$K${}^{0}_{\rm S}$. For K${}^{0}_{\rm S}$K± the FSI of the
$a_{0}$ produces similar but smaller enhancements and suppressions in the same
general $k^{*}$ regions. For $k^{*}>0.4$ GeV/$c$ a non-flat baseline is
observed in both cases. PYTHIA8 fairly describes the non-flat baseline of the
experimental correlation functions for K${}^{0}_{\rm S}$K${}^{0}_{\rm S}$, and
is thus used to take out the effect of the non-flat baseline by dividing the
raw experimental correlation functions by the PYTHIA8 correlation functions in
those cases. This is a similar method as was used for the $\sqrt{s}=$ 7 TeV pp
K${}^{0}_{\rm S}$K${}^{0}_{\rm S}$ measurements [3].

Figure 1: Example K${}^{0}_{\rm S}$K${}^{0}_{\rm S}$ correlation functions
along with the resulting distributions from PYTHIA8 simulations in pp
collisions at $\sqrt{s}=5.02$ (left) and $13$ TeV (right).

Figure 2: Example K${}^{0}_{\rm S}$K± correlation functions plotted with fits
of Eqs. 3, 4 and 5 in pp collisions at $\sqrt{s}=5.02$ (left) and $13$ TeV
(right).

Unlike the case for K${}^{0}_{\rm S}$K${}^{0}_{\rm S}$, which has a relatively
large signal compared with the baseline, for the K${}^{0}_{\rm S}$K±
correlation functions PYTHIA8 is not sensitive enough to model the baseline
sufficiently well with respect to the significantly smaller enhancement and
suppression produced by the FSI of the $a_{0}$ alone. Examples of this are
shown in Fig. 3, which compares the raw $\sqrt{s}=5.02$ and 13 TeV pp
correlation functions from data with those from PYTHIA8. For these correlation
functions, quadratic, exponential and Gaussian functions are used to model the
baseline, as was done in Ref. [6] for the measurement in pp collisions at
$\sqrt{s}=$ 7 TeV, of the forms

$C_{\rm quadratic}(k^{*})=a(1-bk^{*}+ck^{*2})$ (3)

$C_{\rm exponential}(k^{*})=a(1+b\exp(-ck^{*}))$ (4)

$C_{\rm Gaussian}(k^{*})=a(1+b\exp(-ck^{*2}))$ (5)

where $a$, $b$ and $c$ are parameters that are fitted to the experimental
$C(k^{*})$ simultaneously with the FSI model (see Section 4). As shown in Fig.
2, the quadratic, Gaussian and exponential functions all describe the data
well in the $k^{*}$ range of $\sim$0.3–0.6 GeV/$c$.

Figure 3: Example K${}^{0}_{\rm S}$K± correlation functions along with the
resulting distributions from PYTHIA8 simulations in pp collisions at
$\sqrt{s}=5.02$ (left) and 13 TeV (right).

##  4 Fitting the correlation functions to extract the source parameters

###  4.1 K${}^{0}_{\rm S}$K${}^{0}_{\rm S}$

The K${}^{0}_{\rm S}$K${}^{0}_{\rm S}$ correlation functions were fitted with
the Lednický parameterization [2] which incorporates quantum statistics with
strong FSI. FSI arise in the K${}^{0}_{\rm S}$K${}^{0}_{\rm S}$ channels due
to the near-threshold resonances, $a_{0}(980)$ and $f_{0}(980)$. This
parameterization is based on the model by R. Lednický and V.L. Lyuboshitz [31,
32].

The general form of the fit function is:

$C_{\rm Lednicky}(k^{*})=1+\lambda
e^{-4k^{*2}R^{2}}+\lambda\alpha\left[\left|\frac{f(k^{*})}{R}\right|^{2}+\frac{4\mathcal{R}f(k^{*})}{\sqrt{\pi}R}F_{1}(2k^{*}R)-\frac{2\mathcal{I}f(k^{*})}{R}F_{2}(2k^{*}R)+\Delta
C\right]$ (6)

where

$F_{1}(z)=\int^{z}_{0}dx\frac{e^{x^{2}-z^{2}}}{z};\hskip
8.53581ptF_{2}(z)=\frac{1-e^{-z^{2}}}{z}.$ (7)

The scattering amplitude is

$f(k^{*})=\frac{f_{0}(k^{*})+f_{1}(k^{*})}{2}$ (8)

where

$f_{\rm I}(k^{*})=\frac{\gamma_{\rm I}}{m_{\rm I}^{2}-s-i(\gamma_{\rm
I}k^{*}+\gamma_{\rm I}^{\prime}k_{\rm I}^{\prime})},$ (9)

$f(k^{*})$ is the s-wave K${}^{0}\overline{\rm K^{0}}$ scattering amplitude
whose contributions are the isoscalar $f_{0}$ and isovector $a_{0}$
resonances; $\alpha$ is set to 0.5 assuming symmetry in K0 and $\overline{\rm
K^{0}}$ production; $R$ is the radius parameter; and $\lambda$ is the
correlation strength. In Eq. 9, $I=$0 or 1 for the $f_{0}$ or $a_{0}$, $m_{\rm
I}$ is the mass of the resonance, and $\gamma_{\rm I}$ and $\gamma_{\rm
I}^{\prime}$ are the couplings of the resonances to their decay channels.
Also, $s=4(m_{K}^{2}+k^{*2})$ and $k_{\rm I}^{\prime}$ denotes the momentum in
the second decay channel. The K${}^{0}\overline{\rm K^{0}}$ s-wave scattering
amplitude depends on the $f_{0}$ and $a_{0}$ resonance mass and decay
couplings, which have been measured [33]. The parameter set used in the
present analysis is shown in Table 2. The quantity $\Delta C$ is a correction
for small source sizes found in pp collisions [2], and is given by:

$\Delta
C=\frac{1}{\sqrt{\pi}R^{3}}\left[|f_{0}(k^{*})|^{2}(\frac{3}{\gamma_{0}}+\frac{1}{\gamma_{1}})+|f_{1}(k^{*})|^{2}(\frac{1}{\gamma_{0}}+\frac{3}{\gamma_{1}})\right].$
(10)

Table 2: The $f_{0}$ and $a_{0}$ masses and coupling parameters used in the present analysis, all in GeV. $m_{f_{0}}$ | $\gamma_{f_{0}K\bar{K}}$ | $\gamma_{f_{0}\pi\pi}$ | $m_{a_{0}}$ | $\gamma_{a_{0}K\bar{K}}$ | $\gamma_{a_{0}\pi\eta}$
---|---|---|---|---|---
0.967 | 0.34 | 0.089 | 1.003 | 0.8365 | 0.4580

Figure 4 shows example fits of Eq. 6 to the ratio of the data to PYTHIA8
correlation functions for K${}^{0}_{\rm S}$K${}^{0}_{\rm S}$ measured in pp
collisions at $\sqrt{s}=$ 5.02 TeV and 13 TeV. Statistical uncertainties are
shown as error bars, and systematic uncertainties are shown as boxes. The
statistical uncertainties from PYTHIA8 were propagated to those on the data
points. The fits of Eq. 6 to the correlation function ratios provide a good
description of the data, typically giving $\chi^{2}/$ndf values close to
unity. The $\chi^{2}/$ndf values of the fits to the K${}^{0}_{\rm
S}$K${}^{0}_{\rm S}$ correlation functions are 1.3 and 2.5, respectively, for
the left and the right figures. The large $\chi^{2}/$ndf value for the fit
shown in the right figure mostly reflects a combination of the small
statistical uncertainties in the data and the deviation of the fit in the
region $k^{*}>0.5$ GeV/c.

Figure 4: Example fits of Eq. 6 to the ratio of the data to PYTHIA8
correlation functions for K${}^{0}_{\rm S}$K${}^{0}_{\rm S}$ in pp collisions
at $\sqrt{s}=$ 5.02 TeV (left) and 13 TeV (right). Statistical uncertainties
are shown as error bars and systematic uncertainties are shown as boxes.

###  4.2 K${}^{0}_{\rm S}$K±

The K${}^{0}_{\rm S}$K± correlation functions were fitted with the expression:

$C(k^{*})=C_{\rm Lednicky2}(k^{*})C_{\rm baseline}(k^{*})$ (11)

where $C_{\rm Lednicky2}(k^{*})$ is a modified version of Eq. 6, and $C_{\rm
baseline}(k^{*})$ is Eq. 3, Eq. 4 or Eq. 5.

The modified form of the Lednický FSI fit function used is:

$C_{\rm
Lednicky2}(k^{*})=1+\left(\frac{\lambda\alpha}{2}\right)\left[\left|\frac{f(k^{*})}{R}\right|^{2}+\frac{4\mathcal{R}f(k^{*})}{\sqrt{\pi}R}F_{1}(2k^{*}R)-\frac{2\mathcal{I}f(k^{*})}{R}F_{2}(2k^{*}R)+\Delta
C^{\prime}\right].$ (12)

The scattering amplitude is:

$f(k^{*})=\frac{\gamma_{a_{0}\rightarrow
K\overline{K}}}{m_{a_{0}}^{2}-s-i(\gamma_{a_{0}\rightarrow
K\overline{K}}k^{*}+\gamma_{a_{0}\rightarrow\pi\eta}k_{\pi\eta})}.$ (13)

Note that the form of the FSI term in Eq. 12 differs from the form of the FSI
term for K${}^{0}_{\rm S}$K${}^{0}_{\rm S}$ correlations, Eq. 6, by a factor
of $1/2$ due to the non-identical particles in K${}^{0}_{\rm S}$K±
correlations and thus the absence of the requirement to symmetrize the
wavefunction. The K0K- or $\overline{\rm K^{0}}$K+ s-wave scattering amplitude
depends only on the $a_{0}$ resonance mass and decay couplings. The ones used
in this analysis are shown in Table 2. The correction due to small source
sizes, $\Delta C^{\prime}$, now becomes:

$\Delta
C^{\prime}=\frac{2}{\sqrt{\pi}R^{3}}\frac{|f(k^{*})|^{2}}{\gamma_{a_{0}\rightarrow
K\overline{K}}}.$ (14)

The fitting strategy is to make a 5-parameter fit of Eq. 11 to the
K${}^{0}_{\rm S}$K± experimental correlation functions to extract $R$,
$\lambda$, $a$, $b$ and $c$ for each baseline functional form.

Figure 5 shows examples of correlation functions divided by the Gaussian
baseline function, Eq. 5, with fits of Eq. 11 for K${}^{0}_{\rm S}$K±, i.e.
summed over K${}^{0}_{\rm S}$K+ and K${}^{0}_{\rm S}$K-. The $a_{0}$ FSI
parameterization coupled with the Gaussian baseline assumption is seen to give
a good representation of the signal region of the data, i.e. reproducing the
enhancement in the $k^{*}$ region 0.0–0.1 GeV/$c$ and the small dip in the
region 0.1–0.3 GeV/$c$. The average $\chi^{2}/$ndf for these fits to the
correlation functions are 1.04 for the left figure and 1.13 for the right
figure. Fits to the data with similarly good $\chi^{2}/$ndf values are also
found using the exponential and quadratic baselines.

Figure 5: Example fits using Eq. 11 to the ratio of the data in pp collisions
at $\sqrt{s}=$ 5.02 TeV (left) and 13 TeV (right) to Eq. 5. Statistical
uncertainties are shown as error bars and systematic uncertainties are shown
as boxes.

###  4.3 Systematic uncertainties

Table 3 shows the total systematic uncertainties of the extracted $R$ and
$\lambda$ parameters from the K${}^{0}_{\rm S}$K${}^{0}_{\rm S}$ and
K${}^{0}_{\rm S}$K± analyses. The total systematic uncertainty is generally
higher than the statistical one. The total systematic uncertainty is taken as
the square-root of the quadratic sum of the systematic uncertainty from the
fit and the selection criteria.

The fit systematic uncertainty is the combined systematic uncertainty due to
the various baseline assumptions and varying the $k^{*}$ fit range. For
K${}^{0}_{\rm S}$K${}^{0}_{\rm S}$, it is calculated from the standard
deviation of the extracted source parameters from six $k^{*}$ fit ranges:
0.0–0.3, 0.0–0.4, 0.0–0.5, 0.0–0.6, 0.0–0.7 and 0.0–0.8 GeV/$c$. For
K${}^{0}_{\rm S}$K± it is calculated from the standard deviation of using the
three baseline functions in four $k^{*}$ fit ranges: 0.0–0.3, 0.0–0.4, 0.0–0.5
and 0.0–0.6 GeV/$c$. The fit values shown in Table 3 are the average values
over these $k^{*}$ ranges.

The selection systematic uncertainty is the systematic uncertainty related to
the various selection criteria applied in the data analysis. To determine
this, single particle selection criteria were varied by $\sim\pm 10$%, and the
value chosen for the minimum separation distance of like charge-sign tracks
was varied by $\sim 20$%. The uncertainties in the purity corrections to the
$\lambda$ parameters, mentioned earlier, are also included in the selection
systematic uncertainty. Taking the upper-limit values of the variations to be
conservative, this led to additional uncertainties of $4\%$ for $R$ and $8\%$
for $\lambda$. As seen in Table 3, the fit systematic uncertainty tends to be
comparable to or larger than the selection systematic uncertainty, reflecting
the scale of uncertainties in determining the non-femtoscopic baseline in pp
collisions. The “total quadratic uncertainty” is the square-root of the
quadratic sum of the “statistical uncertainty” column and the “total
systematic uncertainty” column.

###  4.4 Momentum resolution

Finite track momentum resolution can smear the relative momentum correlation
functions used in this analysis. This effect was taken into account using
PYTHIA8+GEANT MC simulations. Two PYTHIA8 correlation functions are built
using the generator-level momentum ($k^{*}_{\rm ideal}$) and the measured
detector-level momentum ($k^{*}_{\rm meas}$). Because PYTHIA8 does not
incorporate final-state interactions, weights are calculated using a
$9^{th}$-order polynomial fit in $k^{*}$ to an experimental correlation
function and used when filling the same-event distributions. These weights are
calculated using $k^{*}_{\rm ideal}$. Then, the ratio of the “ideal”
correlation function to the “measured” one for each $k^{*}$ bin is multiplied
by the data correlation functions before the fit procedure. It is found that,
due to the large $k^{*}$ bin size of 20 MeV/$c$ which is used in the analysis
of pp collisions, the correction has a small effect on the lowest $k^{*}$ bin
with the largest statistical error bars, and a negligible effect on the
remaining bins. Thus, the momentum resolution correction was found to have a
$<2\%$ effect on the extracted fit parameters.

##  5 Results and discussion

The extracted source parameters for K${}^{0}_{\rm S}$K${}^{0}_{\rm S}$ and
K${}^{0}_{\rm S}$K±, where K${}^{0}_{\rm S}$K+ and K${}^{0}_{\rm S}$K- have
been summed over, are shown in Table 3 and in Figure 6. The $\lambda$
parameters are corrected for particle-pair purity. Figure 6 shows comparisons
of the present results for $R$ and $\lambda$ in pp collisions at $\sqrt{s}=$
5.02 and 13 TeV with published two-kaon femtoscopic results measured in pp
collisions at $\sqrt{s}=$ 7 TeV [6].

Table 3: Fit results for average $R$ and $\lambda$ along with statistical and systematic uncertainties. The $\lambda$ parameters are corrected for particle-pair purity. $R$ or $\lambda$ | pp energy | fit | statistical | fit | selection | total | total
---|---|---|---|---|---|---|---
kaon pair | (TeV) | value | uncert. ($\pm$) | systematic | systematic | systematic | quadratic
|  |  |  | uncert. ($\pm$) | uncert. ($\pm$) | uncert. ($\pm$) | uncert. ($\pm$)
$R$ (fm) | 5.02 | 0.926 | 0.045 | 0.031 | 0.037 | 0.048 | 0.066
K${}^{0}_{\rm S}$K${}^{0}_{\rm S}$ |  |  |  |  |  |  |
$\lambda$ | 5.02 | 0.876 | 0.159 | 0.067 | 0.070 | 0.097 | 0.186
K${}^{0}_{\rm S}$K${}^{0}_{\rm S}$ |  |  |  |  |  |  |
$R$ (fm) | 5.02 | 0.865 | 0.025 | 0.088 | 0.037 | 0.095 | 0.098
K${}^{0}_{\rm S}$K± |  |  |  |  |  |  |
$\lambda$ | 5.02 | 0.353 | 0.031 | 0.039 | 0.029 | 0.049 | 0.058
K${}^{0}_{\rm S}$K± |  |  |  |  |  |  |
$R$ (fm) | 13 | 1.039 | 0.032 | 0.072 | 0.042 | 0.083 | 0.089
K${}^{0}_{\rm S}$K${}^{0}_{\rm S}$ |  |  |  |  |  |  |
$\lambda$ | 13 | 0.748 | 0.073 | 0.121 | 0.059 | 0.134 | 0.153
K${}^{0}_{\rm S}$K${}^{0}_{\rm S}$ |  |  |  |  |  |  |
$R$ (fm) | 13 | 0.974 | 0.020 | 0.131 | 0.042 | 0.138 | 0.139
K${}^{0}_{\rm S}$K± |  |  |  |  |  |  |
$\lambda$ | 13 | 0.325 | 0.020 | 0.044 | 0.028 | 0.052 | 0.055
K${}^{0}_{\rm S}$K± |  |  |  |  |  |  |

Figure 6: $R$ (left) and $\lambda$ (right) parameters extracted in the present
analysis from Table 3 compared with published K${}^{0}_{\rm S}$K${}^{0}_{\rm
S}$ and K${}^{0}_{\rm S}$K± results from ALICE 7 TeV pp collisions [6]
averaged over event multiplicity and evaluated at $\langle k_{\rm
T}\rangle=0.6$ GeV/$c$. Statistical uncertainties are shown as error bars and
the systematic uncertainties are shown as boxes.

For the $R$ parameters, two observations can clearly be made: 1) there is no
significant dependence on $\sqrt{s}$, i.e. all extracted values are $\sim 1$
fm, and 2) the values extracted from K${}^{0}_{\rm S}$K${}^{0}_{\rm S}$ and
K${}^{0}_{\rm S}$K± for a given $\sqrt{s}$ agree within uncertainties, as
would be expected. It is expected that $R$ from K${}^{0}_{\rm S}$K${}^{0}_{\rm
S}$ and K${}^{0}_{\rm S}$K± would agree with each other if a) the
K${}^{0}_{\rm S}$ and K± are produced in the same source geometry, and b) Eqs.
6 and 7 properly describe the pair interactions. Point a) is expected to be
true due to isospin invariance of the strong interaction that produces the
kaons in the pp collision, and point b) is supported by the overall good fits
that Eqs. 6 and 7 are seen to give to the experimental correlation functions.
The $R$ parameter is essentially independent of $\sqrt{s}$. While $R$ in
general depends on pseudorapidity density also in pp collisions [28, 29, 30],
the increase expected from the slow logarithmic rise of pseudorapidity density
with $\sqrt{s}$ is well within our experimental uncertainties.

The extracted $\lambda$ parameters in Fig. 6 suggest that: 1) the values do
not depend significantly on $\sqrt{s}$, 2) the values for K${}^{0}_{\rm
S}$K${}^{0}_{\rm S}$ are in the usual range seen in femtoscopy experiments of
$\lambda\sim$ 0.7–0.8, whereas 3) the values for K${}^{0}_{\rm S}$K± are
significantly smaller being $\lambda\sim$ 0.3–0.4, consistent with the 7 TeV
results. Figure 7 shows the difference between purity-corrected $\lambda$
parameters extracted with K${}^{0}_{\rm S}$K${}^{0}_{\rm S}$ and K${}^{0}_{\rm
S}$K± versus $\sqrt{s}$. The propagated total uncertainty is indicated on
these points. Also shown is the weighted average of these points, weighted by
their total uncertainties. It is assumed that the total uncertainties of the
K${}^{0}_{\rm S}$K${}^{0}_{\rm S}$ and K${}^{0}_{\rm S}$K± measurements are
uncorrelated. This is considered a reasonable assumption given the differences
in the kaon pairs and the equations used to extract the source parameters. The
weighted average of the differences is calculated to be $0.419\pm 0.091$,
which is $4.6\sigma$ from zero.

There are three main technical factors that, while having a small effect on
the $R$ parameter, can significantly affect the value of the $\lambda$
parameter: 1) the experimental kaon reconstruction purity, 2) the degree to
which a Gaussian distribution describes the kaon source, and 3) the presence
of kaons originating from the decay of long-lived resonances diluting the
direct-kaon sample [6]. The effect of factor 1) is already corrected for by
having divided the extracted $\lambda$ values by the products of the single-
kaon purities given in Section 2.1. As seen in Figs. 4 and 5, the Lednický
equation, which uses a Gaussian source, fits the experimental correlation
functions well, an observation that is supported by the good $\chi^{2}/$ndf
values given above, minimizing the effect of 2). The effects from factor 3)
are discussed in the following section.

Figure 7: Differences in $\lambda$ parameters extracted in the present
analysis from Table 3 compared with published identical and non-identical kaon
results from ALICE 7 TeV pp collisions averaged over event multiplicity and
evaluated at $\langle k_{T}\rangle=0.6$ GeV/$c$. Total uncertainties are
shown. The weighted average of the differences is shown as a solid red line
and the weighted uncertainty, $\pm\sigma$, shown as red dashed lines.

###  5.1 Effect of presence of long-lived resonances

Table 4 gives a list of mesons from the Review of Particle Physics [34] with
masses $<1500$ MeV/$c^{2}$ that have decay channels into kaons with
significant branching ratios. The two lowest-lying mesons, the K∗(892) and the
$\phi$(1020), are the most abundantly produced and have the narrowest widths,
and so are expected to have the greatest effect on the values of the extracted
kaon source parameters, which reflect both the kaons produced from the decays
of resonances as well as the kaons produced directly from the pp collision.
Since the mean decay lengths of the K∗ and $\phi$ are about $4$ fm and $50$
fm, respectively (see Table 4), these decays result in effective sources of
kaons which are much larger than the expected size of the direct kaon source
in pp collisions of about $1$ fm. Thus, the effect of these should be mostly
to reduce the extracted $\lambda$ parameter. This is due to the correlation
function for the smaller-sized direct source being wider in $k^{*}$ and so
dominating the extracted $R$ [35]. Thus, the kaons from these resonances only
make an overall suppression of the correlation function.

Table 4: List of mesons from the Review of Particle Physics [34] with masses $<1500$ MeV/$c^{2}$ that have decay channels into kaons with significant branching ratios. Name | mass (MeV/$c^{2}$) |  $\Gamma$ (MeV/$c^{2}$) | kaon decays |  $\hslash c/\Gamma$ (fm)
---|---|---|---|---
K∗(892) | 891.67 | 51.4 | K$\pi$ (100%) | 3.839
$\phi$(1020) | 1019.46 | 4.249 | K+K- (49.2%), K${}^{0}_{\rm L}$K${}^{0}_{\rm S}$ (34%) | 46.4
K1(1270) | 1253 | $\sim 100$ | K$\rho$ (42%) | 1.973
K${}^{*}_{0}$(1430) | 1425 | 270 | K$\pi$ ($\sim 100$%) | 0.731
K${}^{*}_{2}$(1430) | 1427.3 | 100 | K$\pi$ (50%), KX (12%) | 1.973

The dilution effect on the $\lambda$ parameter due to the K${}^{*}(892)$ and
$\phi$(1020) decays can be estimated from K∗0/K and $\phi$/K ratio
measurements from ALICE [36, 37, 38, 39]. Table 5 shows the measurements of
these quantities relevant to the present estimate. As shown in the table, the
measured ratios for both K∗0/K and $\phi$/K are independent of the collision
energy and independent of the decay-kaon charge state within the measurement
uncertainties. For the present calculation, the ratios from Refs. [38, 39] are
used since they are taken with an average $p_{\rm T}$ close to the average
$k_{\rm T}$ of 0.6 GeV/$c$ used in this analysis. Whereas the $\phi$ only has
one charge state for each of its decay channels, as seen in Table 4, and is
its own anti-particle, the K∗ has four charge states, and three unique sets of
decay charge-state channels, as shown in Table 6, which is taken into account
in the present calculation. Using the numbers in Tables 4, 5, and 6, the
direct-kaon purity for K+, K- and K${}^{0}_{\rm S}$, defined as $P$(K+),
$P$(K-) and $P$(K${}^{0}_{\rm S}$), respectively, where $P$(K+) = $P$(K-)
$\equiv P$(K±), are calculated to be $P$(K±) $=0.726$, and $P$(K${}^{0}_{\rm
S})=0.757$.

In the calculations, it has been assumed that the ratios K∗0/K± = K∗±/K± and
K∗0/K${}^{0}_{\rm S}$ = K∗±/K${}^{0}_{\rm S}$. The “diluted” $\lambda$
parameters can then be estimated for K${}^{0}_{\rm S}$K${}^{0}_{\rm S}$ as
$P$(K${}^{0}_{\rm S}$)$P$(K${}^{0}_{\rm S}$) = $0.57\pm 0.02$ and for
K${}^{0}_{\rm S}$K± as $P$(K${}^{0}_{\rm S}$)$P$(K±) = $0.55\pm 0.02$. The
effect of these long-lived resonances is seen to be of the same magnitude,
within the uncertainties, for K${}^{0}_{\rm S}$K${}^{0}_{\rm S}$ and
K${}^{0}_{\rm S}$K±. The estimate is $\sim$1–2$\sigma$ lower than the purity-
corrected $\lambda$ values measured with K${}^{0}_{\rm S}$K${}^{0}_{\rm S}$,
however it is $\sim$3–4$\sigma$ larger than the values measured in
K${}^{0}_{\rm S}$K±. Thus, the dilution effects on $\lambda$ by the K∗(892)
and $\phi$(1020) cannot explain the small values for $\lambda$ measured in pp
collisions in K${}^{0}_{\rm S}$K± femtoscopy.

Table 5: ALICE measurements of K∗(892)/K and $\phi$(1020)/K ratios. The uncertainties given are the statistical and systematic uncertainties combined in quadrature. Ref. | collision | K∗(892)/K |  $\phi$(1020)/K | average $p_{\rm T}$ (GeV/$c$) | trigger
---|---|---|---|---|---
[36] | 2.76 TeV pp | K∗0/K- $0.31\pm 0.04$ |  $\phi$/K- $0.11\pm 0.01$ | $\sim 1$ | inelastic
[37] | 7 TeV pp | K∗0/K- $0.35\pm 0.04$ |  $\phi$/K- $0.11\pm 0.02$ | $\sim 1$ | inelastic
[38] | 13 TeV pp | K∗0/K${}^{0}_{\rm S}$ $0.34\pm 0.01$ |  $\phi$/K${}^{0}_{\rm S}$ $0.11\pm 0.01$ | 0.6 | low multiplicity
[39] | 5.02 TeV pp | K∗0/K± $0.29\pm 0.02$ |  $\phi$/K± $0.08\pm 0.02$ | $\sim 0.6$ | inelastic
Table 6: Decay modes of the charge states of the K∗(892). Note that the K0 is made up of 50% K${}^{0}_{\rm S}$ and 50% K${}^{0}_{\rm L}$. K${}^{*}(892)$ charge state | decay channels | comment
---|---|---
K∗+ | K${}^{+}\pi^{0}$, K${}^{0}\pi^{+}$ | each channel 50%
K∗- | K${}^{-}\pi^{0}$, K${}^{0}\pi^{-}$ | each channel 50%
K∗0, $\bar{\rm K}^{*0}$ | K${}^{+}\pi^{-}$, K${}^{-}\pi^{+}$, K${}^{0}\pi^{0}$ | each channel 33.3%

###  5.2 Physics explanations for differences of $\lambda$ parameters

Since the technical factors discussed above affecting the extracted $\lambda$
values should affect the values from K${}^{0}_{\rm S}$K${}^{0}_{\rm S}$ and
K${}^{0}_{\rm S}$K± in the same way, their difference can be ascribed to a
physics effect. It is important to first compare the $\lambda$ parameters
extracted in the present work to those measured in other published KK
femtoscopic studies. In Pb–Pb collisions, $\lambda$ is measured to be
$\sim$0.7 for K${}^{0}_{\rm S}$K${}^{0}_{\rm S}$, K±K± and K${}^{0}_{\rm S}$K±
[7], similar to what is measured for K${}^{0}_{\rm S}$K${}^{0}_{\rm S}$
presented here, and close to the estimate made for the resonance dilution
effect. For K±K± femtoscopy in pp and pPb collisions, $\lambda$ is measured to
be in the range 0.4–0.5, which is smaller than for K${}^{0}_{\rm
S}$K${}^{0}_{\rm S}$ presented here [40]. Note that one expects K±K± to be
somewhat smaller than K${}^{0}_{\rm S}$K${}^{0}_{\rm S}$ on the basis of the
resonance dilution effect, since for K±K± the $\lambda$ is estimated to be
$P$(K±)$P$(K±) = 0.53 as compared with $P$(K${}^{0}_{\rm S}$)$P$(K${}^{0}_{\rm
S}$) = 0.57 estimated for K${}^{0}_{\rm S}$K${}^{0}_{\rm S}$. As to why the
$\lambda$ parameters in K±K± in pp and p–Pb collisions are smaller than in
Pb–Pb collisions, Ref. [40] suggests that this could be due to the kaon source
being more Gaussian in Pb–Pb collisions.

As discussed in Ref. [6], a physics effect that could cause the difference in
$\lambda$ values is related to the possibility that the $a_{0}$ resonance,
that is solely responsible for the FSI in the K${}^{0}_{\rm S}$K± pair, is a
tetraquark state of the form $(q_{1},\overline{q_{2}},s,\overline{s})$ instead
of a diquark state of the form $(q_{1},\overline{q_{2}})$, where $q_{1}$ and
$q_{2}$ are $u$ or $d$ quarks. The strength of the FSI through a tetraquark
$a_{0}$ could be decreased by the small source size of the kaon source, i.e.
$R\sim 1$ fm as measured in this analysis, since $s-\overline{s}$ annihilation
would be enhanced due to the close creation proximity. For a FSI through a
diquark $a_{0}$, with the form $(q_{1},\overline{q_{2}})$, the small source
geometry should not reduce its strength. For the K${}^{0}_{\rm
S}$K${}^{0}_{\rm S}$ case, $\lambda$ would not be affected much by a
tetraquark $a_{0}$ since the enhancement in the correlation function near
$k^{*}\sim 0$ is dominated by the effect of quantum statistics. Note that for
the large kaon source measured in Pb–Pb collisions to have $R\sim 6$ fm, the
situation would be reversed. The large average separation between the kaons
would favor the formation of a tetraquark $a_{0}$ and suppress the formation
of a diquark $a_{0}$, and a larger $\lambda\sim 0.6$ is indeed measured in
that case, as already mentioned above. Thus, we can conclude that, as was the
case with the published $\sqrt{s}=$ 7 TeV result, the present results in pp
collisions at $\sqrt{s}=$ 5.02 and 13 TeV are compatible with the $a_{0}$
being a tetraquark state.

##  6 Summary

In summary, femtoscopic correlations with the particle pair combinations
K${}^{0}_{\rm S}$K${}^{0}_{\rm S}$ and K${}^{0}_{\rm S}$K± are studied in pp
collisions at $\sqrt{s}=5.02$ and $13$ TeV for the first time by the ALICE
experiment at the LHC. By fitting models that assume a Gaussian size
distribution of the kaon source to the experimental two-particle correlation
functions, kaon source parameters are extracted . The model used for the
K${}^{0}_{\rm S}$K${}^{0}_{\rm S}$ case includes quantum statistics and strong
final-state interactions through the $f_{0}$ and $a_{0}$ resonances. The model
used for the K${}^{0}_{\rm S}$K± case involves only the final-state
interaction through the $a_{0}$ resonance. In both cases, the models gave a
good fit to the experimental correlation functions. Source parameters
extracted in the present work are compared with published values from ALICE
measured in pp collisions at $\sqrt{s}=$ 7 TeV and found to be consistent,
i.e. there is no significant dependence of either $R$ or $\lambda$ on the
collision energy. The new results are compatible with the $a_{0}$ resonance
being a tetraquark state due to the $\lambda$ parameter for K${}^{0}_{\rm
S}$K± being significantly smaller than for K${}^{0}_{\rm S}$K${}^{0}_{\rm S}$.

## 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, Austrian Science Fund (FWF): [M 2467-N36] 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), Financiadora de Estudos e Projetos (Finep), Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP) and Universidade Federal do Rio Grande do Sul (UFRGS), Brazil; Ministry of Education of China (MOEC) , Ministry of Science & Technology of China (MSTC) and National Natural Science Foundation of China (NSFC), China; Ministry of Science and Education and Croatian Science Foundation, Croatia; Centro de Aplicaciones Tecnológicas y Desarrollo Nuclear (CEADEN), Cubaenergía, Cuba; Ministry of Education, Youth and Sports of the Czech Republic, Czech Republic; The Danish Council for Independent Research | Natural Sciences, the VILLUM FONDEN 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 und Forschung (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; Istituto Nazionale di Fisica Nucleare (INFN), Italy; Japanese Ministry of Education, Culture, Sports, Science and Technology (MEXT), Japan Society for the Promotion of Science (JSPS) KAKENHI and Japanese Ministry of Education, Culture, Sports, Science and Technology (MEXT)of Applied Science (IIST), 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 Education and Science, National Science Centre and WUT ID-UB, 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, Ministry of Research and Innovation and Institute of Atomic Physics and University Politehnica of Bucharest, Romania; Joint Institute for Nuclear Research (JINR), Ministry of Education and Science of the Russian Federation, National Research Centre Kurchatov Institute, Russian Science Foundation and Russian Foundation for Basic Research, Russia; Ministry of Education, Science, Research and Sport of the Slovak Republic, Slovakia; National Research Foundation of South Africa, South Africa; Swedish Research Council (VR) and Knut & Alice Wallenberg Foundation (KAW), Sweden; European Organization for Nuclear Research, Switzerland; Suranaree University of Technology (SUT), National Science and Technology Development Agency (NSDTA) and Office of the Higher Education Commission under NRU project of Thailand, Thailand; Turkish Energy, Nuclear and Mineral Research Agency (TENMAK), 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 The ALICE Collaboration

S. Acharya143, D. Adamová97, A. Adler75, J. Adolfsson82, G. Aglieri Rinella34,
M. Agnello30, N. Agrawal54, Z. Ahammed143, S. Ahmad16, S.U. Ahn77, I. Ahuja38,
Z. Akbar51, A. Akindinov94, M. Al-Turany109, S.N. Alam16, D. Aleksandrov90, B.
Alessandro60, H.M. Alfanda7, R. Alfaro Molina72, B. Ali16, Y. Ali14, A.
Alici25, N. Alizadehvandchali126, A. Alkin34, J. Alme21, G. Alocco55, T.
Alt69, I. Altsybeev114, M.N. Anaam7, C. Andrei48, D. Andreou92, A.
Andronic146, M. Angeletti34, V. Anguelov106, F. Antinori57, P. Antonioli54, C.
Anuj16, N. Apadula81, L. Aphecetche116, H. Appelshäuser69, S. Arcelli25, R.
Arnaldi60, I.C. Arsene20, M. Arslandok148, A. Augustinus34, R. Averbeck109, S.
Aziz79, M.D. Azmi16, A. Badalà56, Y.W. Baek41, X. Bai130,109, R. Bailhache69,
Y. Bailung50, R. Bala103, A. Balbino30, A. Baldisseri140, B. Balis2, D.
Banerjee4, Z. Banoo103, R. Barbera26, L. Barioglio107, M. Barlou86, G.G.
Barnaföldi147, L.S. Barnby96, V. Barret137, C. Bartels129, K. Barth34, E.
Bartsch69, F. Baruffaldi27, N. Bastid137, S. Basu82, G. Batigne116, B.
Batyunya76, D. Bauri49, J.L. Bazo Alba113, I.G. Bearden91, C. Beattie148, P.
Becht109, I. Belikov139, A.D.C. Bell Hechavarria146, F. Bellini25, R.
Bellwied126, S. Belokurova114, V. Belyaev95, G. Bencedi147,70, S. Beole24, A.
Bercuci48, Y. Berdnikov100, A. Berdnikova106, L. Bergmann106, M.G. Besoiu68,
L. Betev34, P.P. Bhaduri143, A. Bhasin103, I.R. Bhat103, M.A. Bhat4, B.
Bhattacharjee42, P. Bhattacharya22, L. Bianchi24, N. Bianchi52, J. Bielčík37,
J. Bielčíková97, J. Biernat119, A. Bilandzic107, G. Biro147, S. Biswas4, J.T.
Blair120, D. Blau90,83, M.B. Blidaru109, C. Blume69, G. Boca28,58, F. Bock98,
A. Bogdanov95, S. Boi22, J. Bok62, L. Boldizsár147, A. Bolozdynya95, M.
Bombara38, P.M. Bond34, G. Bonomi142,58, H. Borel140, A. Borissov83, H.
Bossi148, E. Botta24, L. Bratrud69, P. Braun-Munzinger109, M. Bregant122, M.
Broz37, G.E. Bruno108,33, M.D. Buckland23,129, D. Budnikov110, H. Buesching69,
S. Bufalino30, O. Bugnon116, P. Buhler115, Z. Buthelezi73,133, J.B. Butt14, A.
Bylinkin128, S.A. Bysiak119, M. Cai27,7, H. Caines148, A. Caliva109, E. Calvo
Villar113, J.M.M. Camacho121, R.S. Camacho45, P. Camerini23, F.D.M. Canedo122,
F. Carnesecchi34,25, R. Caron140, J. Castillo Castellanos140, E.A.R. Casula22,
F. Catalano30, C. Ceballos Sanchez76, P. Chakraborty49, S. Chandra143, S.
Chapeland34, M. Chartier129, S. Chattopadhyay143, S. Chattopadhyay111, T.G.
Chavez45, T. Cheng7, C. Cheshkov138, B. Cheynis138, V. Chibante Barroso34,
D.D. Chinellato123, S. Cho62, P. Chochula34, P. Christakoglou92, C.H.
Christensen91, P. Christiansen82, T. Chujo135, C. Cicalo55, L. Cifarelli25, F.
Cindolo54, M.R. Ciupek109, G. ClaiII,54, J. CleymansI,125, F. Colamaria53,
J.S. Colburn112, D. Colella53,108,33, A. Collu81, M. Colocci34, M.
ConcasIII,60, G. Conesa Balbastre80, Z. Conesa del Valle79, G. Contin23, J.G.
Contreras37, M.L. Coquet140, T.M. Cormier98, P. Cortese31, M.R. Cosentino124,
F. Costa34, S. Costanza28,58, P. Crochet137, R. Cruz-Torres81, E. Cuautle70,
P. Cui7, L. Cunqueiro98, A. Dainese57, M.C. Danisch106, A. Danu68, P. Das88,
P. Das4, S. Das4, S. Dash49, A. De Caro29, G. de Cataldo53, L. De Cilladi24,
J. de Cuveland39, A. De Falco22, D. De Gruttola29, N. De Marco60, C. De
Martin23, S. De Pasquale29, S. Deb50, H.F. Degenhardt122, K.R. Deja144, R. Del
Grande107, L. Dello Stritto29, W. Deng7, P. Dhankher19, D. Di Bari33, A. Di
Mauro34, R.A. Diaz8, T. Dietel125, Y. Ding138,7, R. Divià34, D.U. Dixit19, Ø.
Djuvsland21, U. Dmitrieva64, J. Do62, A. Dobrin68, B. Dönigus69, A.K.
Dubey143, A. Dubla109,92, S. Dudi102, P. Dupieux137, N. Dzalaiova13, T.M.
Eder146, R.J. Ehlers98, V.N. Eikeland21, F. Eisenhut69, D. Elia53, B.
Erazmus116, F. Ercolessi25, F. Erhardt101, A. Erokhin114, M.R. Ersdal21, B.
Espagnon79, G. Eulisse34, D. Evans112, S. Evdokimov93, L. Fabbietti107, M.
Faggin27, J. Faivre80, F. Fan7, A. Fantoni52, M. Fasel98, P. Fecchio30, A.
Feliciello60, G. Feofilov114, A. Fernández Téllez45, A. Ferrero140, A.
Ferretti24, V.J.G. Feuillard106, J. Figiel119, V. Filova37, D. Finogeev64,
F.M. Fionda55, G. Fiorenza34,108, F. Flor126, A.N. Flores120, S. Foertsch73,
S. Fokin90, E. Fragiacomo61, E. Frajna147, A. Francisco137, U. Fuchs34, N.
Funicello29, C. Furget80, A. Furs64, J.J. Gaardhøje91, M. Gagliardi24, A.M.
Gago113, A. Gal139, C.D. Galvan121, P. Ganoti86, C. Garabatos109, J.R.A.
Garcia45, E. Garcia-Solis10, K. Garg116, C. Gargiulo34, A. Garibli89, K.
Garner146, P. Gasik109, E.F. Gauger120, A. Gautam128, M.B. Gay Ducati71, M.
Germain116, P. Ghosh143, S.K. Ghosh4, M. Giacalone25, P. Gianotti52, P.
Giubellino109,60, P. Giubilato27, A.M.C. Glaenzer140, P. Glässel106, E.
Glimos132, D.J.Q. Goh84, V. Gonzalez145, L.H. González-Trueba72, S.
Gorbunov39, M. Gorgon2, L. Görlich119, S. Gotovac35, V. Grabski72, L.K.
Graczykowski144, L. Greiner81, A. Grelli63, C. Grigoras34, V. Grigoriev95, S.
Grigoryan76,1, F. Grosa34,60, J.F. Grosse-Oetringhaus34, R. Grosso109, D.
Grund37, G.G. Guardiano123, R. Guernane80, M. Guilbaud116, K. Gulbrandsen91,
T. Gunji134, W. Guo7, A. Gupta103, R. Gupta103, S.P. Guzman45, L. Gyulai147,
M.K. Habib109, C. Hadjidakis79, H. Hamagaki84, M. Hamid7, R. Hannigan120, M.R.
Haque144, A. Harlenderova109, J.W. Harris148, A. Harton10, J.A.
Hasenbichler34, H. Hassan98, D. Hatzifotiadou54, P. Hauer43, L.B. Havener148,
S.T. Heckel107, E. Hellbär109, H. Helstrup36, T. Herman37, E.G. Hernandez45,
G. Herrera Corral9, F. Herrmann146, K.F. Hetland36, H. Hillemanns34, C.
Hills129, B. Hippolyte139, B. Hofman63, B. Hohlweger92, J. Honermann146, G.H.
Hong149, D. Horak37, S. Hornung109, A. Horzyk2, R. Hosokawa15, Y. Hou7, P.
Hristov34, C. Hughes132, P. Huhn69, L.M. Huhta127, C.V. Hulse79, T.J.
Humanic99, H. Hushnud111, L.A. Husova146, A. Hutson126, J.P. Iddon34,129, R.
Ilkaev110, H. Ilyas14, M. Inaba135, G.M. Innocenti34, M. Ippolitov90, A.
Isakov97, T. Isidori128, M.S. Islam111, M. Ivanov109, V. Ivanov100, V.
Izucheev93, M. Jablonski2, B. Jacak81, N. Jacazio34, P.M. Jacobs81, S.
Jadlovska118, J. Jadlovsky118, S. Jaelani63, C. Jahnke123,122, M.J.
Jakubowska144, A. Jalotra103, M.A. Janik144, T. Janson75, M. Jercic101, O.
Jevons112, A.A.P. Jimenez70, F. Jonas98,146, P.G. Jones112, J.M. Jowett
34,109, J. Jung69, M. Jung69, A. Junique34, A. Jusko112, M.J. Kabus144, J.
Kaewjai117, P. Kalinak65, A.S. Kalteyer109, A. Kalweit34, V. Kaplin95, A.
Karasu Uysal78, D. Karatovic101, O. Karavichev64, T. Karavicheva64, P.
Karczmarczyk144, E. Karpechev64, V. Kashyap88, A. Kazantsev90, U. Kebschull75,
R. Keidel47, D.L.D. Keijdener63, M. Keil34, B. Ketzer43, Z. Khabanova92, A.M.
Khan7, S. Khan16, A. Khanzadeev100, Y. Kharlov93,83, A. Khatun16, A.
Khuntia119, B. Kileng36, B. Kim17,62, C. Kim17, D.J. Kim127, E.J. Kim74, J.
Kim149, J.S. Kim41, J. Kim106, J. Kim74, M. Kim106, S. Kim18, T. Kim149, S.
Kirsch69, I. Kisel39, S. Kiselev94, A. Kisiel144, J.P. Kitowski2, J.L. Klay6,
J. Klein34, S. Klein81, C. Klein-Bösing146, M. Kleiner69, T. Klemenz107, A.
Kluge34, A.G. Knospe126, C. Kobdaj117, T. Kollegger109, A. Kondratyev76, N.
Kondratyeva95, E. Kondratyuk93, J. Konig69, S.A. Konigstorfer107, P.J.
Konopka34, G. Kornakov144, S.D. Koryciak2, A. Kotliarov97, O. Kovalenko87, V.
Kovalenko114, M. Kowalski119, I. Králik65, A. Kravčáková38, L. Kreis109, M.
Krivda112,65, F. Krizek97, K. Krizkova Gajdosova37, M. Kroesen106, M.
Krüger69, D.M. Krupova37, E. Kryshen100, M. Krzewicki39, V. Kučera34, C.
Kuhn139, P.G. Kuijer92, T. Kumaoka135, D. Kumar143, L. Kumar102, N. Kumar102,
S. Kundu34, P. Kurashvili87, A. Kurepin64, A.B. Kurepin64, A. Kuryakin110, S.
Kushpil97, J. Kvapil112, M.J. Kweon62, J.Y. Kwon62, Y. Kwon149, S.L. La
Pointe39, P. La Rocca26, Y.S. Lai81, A. Lakrathok117, M. Lamanna34, R.
Langoy131, K. Lapidus34, P. Larionov34,52, E. Laudi34, L. Lautner34,107, R.
Lavicka115,37, T. Lazareva114, R. Lea142,23,58, J. Lehrbach39, R.C. Lemmon96,
I. León Monzón121, E.D. Lesser19, M. Lettrich34,107, P. Lévai147, X. Li11,
X.L. Li7, J. Lien131, R. Lietava112, B. Lim17, S.H. Lim17, V. Lindenstruth39,
A. Lindner48, C. Lippmann109, A. Liu19, D.H. Liu7, J. Liu129, I.M. Lofnes21,
V. Loginov95, C. Loizides98, P. Loncar35, J.A. Lopez106, X. Lopez137, E. López
Torres8, J.R. Luhder146, M. Lunardon27, G. Luparello61, Y.G. Ma40, A.
Maevskaya64, M. Mager34, T. Mahmoud43, A. Maire139, M. Malaev100, N.M.
Malik103, Q.W. Malik20, S.K. Malik103, L. MalininaIV,76, D. Mal’Kevich94, D.
Mallick88, N. Mallick50, G. Mandaglio32,56, V. Manko90, F. Manso137, V.
Manzari53, Y. Mao7, G.V. Margagliotti23, A. Margotti54, A. Marín109, C.
Markert120, M. Marquard69, N.A. Martin106, P. Martinengo34, J.L. Martinez126,
M.I. Martínez45, G. Martínez García116, S. Masciocchi109, M. Masera24, A.
Masoni55, L. Massacrier79, A. Mastroserio141,53, A.M. Mathis107, O.
Matonoha82, P.F.T. Matuoka122, A. Matyja119, C. Mayer119, A.L. Mazuecos34, F.
Mazzaschi24, M. Mazzilli34, M.A. MazzoniI,59, J.E. Mdhluli133, A.F. Mechler69,
Y. Melikyan64, A. Menchaca-Rocha72, E. Meninno115,29, A.S. Menon126, M.
Meres13, S. Mhlanga125,73, Y. Miake135, L. Micheletti60, L.C. Migliorin138,
D.L. Mihaylov107, K. Mikhaylov76,94, A.N. Mishra147, D. Miśkowiec109, A.
Modak4, A.P. Mohanty63, B. Mohanty88, M. Mohisin KhanV,16, M.A. Molander44, Z.
Moravcova91, C. Mordasini107, D.A. Moreira De Godoy146, I. Morozov64, A.
Morsch34, T. Mrnjavac34, V. Muccifora52, E. Mudnic35, D. Mühlheim146, S.
Muhuri143, J.D. Mulligan81, A. Mulliri22, M.G. Munhoz122, R.H. Munzer69, H.
Murakami134, S. Murray125, L. Musa34, J. Musinsky65, J.W. Myrcha144, B.
Naik133, R. Nair87, B.K. Nandi49, R. Nania54, E. Nappi53, A.F. Nassirpour82,
A. Nath106, C. Nattrass132, A. Neagu20, A. Negru136, L. Nellen70, S.V.
Nesbo36, G. Neskovic39, D. Nesterov114, B.S. Nielsen91, S. Nikolaev90, S.
Nikulin90, V. Nikulin100, F. Noferini54, S. Noh12, P. Nomokonov76, J.
Norman129, N. Novitzky135, P. Nowakowski144, A. Nyanin90, J. Nystrand21, M.
Ogino84, A. Ohlson82, V.A. Okorokov95, J. Oleniacz144, A.C. Oliveira Da
Silva132, M.H. Oliver148, A. Onnerstad127, C. Oppedisano60, A. Ortiz
Velasquez70, T. Osako46, A. Oskarsson82, J. Otwinowski119, M. Oya46, K.
Oyama84, Y. Pachmayer106, S. Padhan49, D. Pagano142,58, G. Paić70, A.
Palasciano53, J. Pan145, S. Panebianco140, J. Park62, J.E. Parkkila127, S.P.
Pathak126, R.N. Patra103,34, B. Paul22, H. Pei7, T. Peitzmann63, X. Peng7,
L.G. Pereira71, H. Pereira Da Costa140, D. Peresunko90,83, G.M. Perez8, S.
Perrin140, Y. Pestov5, V. Petráček37, M. Petrovici48, R.P. Pezzi116,71, S.
Piano61, M. Pikna13, P. Pillot116, O. Pinazza54,34, L. Pinsky126, C. Pinto26,
S. Pisano52, M. Płoskoń81, M. Planinic101, F. Pliquett69, M.G. Poghosyan98, B.
Polichtchouk93, S. Politano30, N. Poljak101, A. Pop48, S. Porteboeuf-
Houssais137, J. Porter81, V. Pozdniakov76, S.K. Prasad4, R. Preghenella54, F.
Prino60, C.A. Pruneau145, I. Pshenichnov64, M. Puccio34, S. Qiu92, L.
Quaglia24, R.E. Quishpe126, S. Ragoni112, A. Rakotozafindrabe140, L.
Ramello31, F. Rami139, S.A.R. Ramirez45, A.G.T. Ramos33, T.A. Rancien80, R.
Raniwala104, S. Raniwala104, S.S. Räsänen44, R. Rath50, I. Ravasenga92, K.F.
Read98,132, A.R. Redelbach39, K. RedlichVI,87, A. Rehman21, P. Reichelt69, F.
Reidt34, H.A. Reme-ness36, Z. Rescakova38, K. Reygers106, A. Riabov100, V.
Riabov100, T. Richert82, M. Richter20, W. Riegler34, F. Riggi26, C. Ristea68,
M. Rodríguez Cahuantzi45, K. Røed20, R. Rogalev93, E. Rogochaya76, T.S.
Rogoschinski69, D. Rohr34, D. Röhrich21, P.F. Rojas45, S. Rojas Torres37, P.S.
Rokita144, F. Ronchetti52, A. Rosano32,56, E.D. Rosas70, A. Rossi57, A. Roy50,
P. Roy111, S. Roy49, N. Rubini25, O.V. Rueda82, D. Ruggiano144, R. Rui23, B.
Rumyantsev76, P.G. Russek2, R. Russo92, A. Rustamov89, E. Ryabinkin90, Y.
Ryabov100, A. Rybicki119, H. Rytkonen127, W. Rzesa144, O.A.M. Saarimaki44, R.
Sadek116, S. Sadovsky93, J. Saetre21, K. Šafařík37, S.K. Saha143, S. Saha88,
B. Sahoo49, P. Sahoo49, R. Sahoo50, S. Sahoo66, D. Sahu50, P.K. Sahu66, J.
Saini143, S. Sakai135, M.P. Salvan109, S. Sambyal103, V. SamsonovI,100,95,
T.B. Saramela122, D. Sarkar145, N. Sarkar143, P. Sarma42, V.M. Sarti107,
M.H.P. Sas148, J. Schambach98, H.S. Scheid69, C. Schiaua48, R. Schicker106, A.
Schmah106, C. Schmidt109, H.R. Schmidt105, M.O. Schmidt34,106, M. Schmidt105,
N.V. Schmidt98,69, A.R. Schmier132, R. Schotter139, J. Schukraft34, K.
Schwarz109, K. Schweda109, G. Scioli25, E. Scomparin60, J.E. Seger15, Y.
Sekiguchi134, D. Sekihata134, I. Selyuzhenkov109,95, S. Senyukov139, J.J.
Seo62, D. Serebryakov64, L. Šerkšnytė107, A. Sevcenco68, T.J. Shaba73, A.
Shabanov64, A. Shabetai116, R. Shahoyan34, W. Shaikh111, A. Shangaraev93, A.
Sharma102, H. Sharma119, M. Sharma103, N. Sharma102, S. Sharma103, U.
Sharma103, A. Shatat79, O. Sheibani126, K. Shigaki46, M. Shimomura85, S.
Shirinkin94, Q. Shou40, Y. Sibiriak90, S. Siddhanta55, T. Siemiarczuk87, T.F.
Silva122, D. Silvermyr82, T. Simantathammakul117, G. Simonetti34, B. Singh107,
R. Singh88, R. Singh103, R. Singh50, V.K. Singh143, V. Singhal143, T.
Sinha111, B. Sitar13, M. Sitta31, T.B. Skaali20, G. Skorodumovs106, M.
Slupecki44, N. Smirnov148, R.J.M. Snellings63, C. Soncco113, J. Song126, A.
Songmoolnak117, F. Soramel27, S. Sorensen132, I. Sputowska119, J. Stachel106,
I. Stan68, P.J. Steffanic132, S.F. Stiefelmaier106, D. Stocco116, I.
Storehaug20, M.M. Storetvedt36, P. Stratmann146, C.P. Stylianidis92, A.A.P.
Suaide122, C. Suire79, M. Sukhanov64, M. Suljic34, R. Sultanov94, V.
Sumberia103, S. Sumowidagdo51, S. Swain66, A. Szabo13, I. Szarka13, U.
Tabassam14, S.F. Taghavi107, G. Taillepied137, J. Takahashi123, G.J.
Tambave21, S. Tang137,7, Z. Tang130, J.D. Tapia TakakiVII,128, M. Tarhini116,
M.G. Tarzila48, A. Tauro34, G. Tejeda Muñoz45, A. Telesca34, L. Terlizzi24, C.
Terrevoli126, G. Tersimonov3, S. Thakur143, D. Thomas120, R. Tieulent138, A.
Tikhonov64, A.R. Timmins126, M. Tkacik118, A. Toia69, N. Topilskaya64, M.
Toppi52, F. Torales-Acosta19, T. Tork79, A. Trifiró32,56, S. Tripathy54,70, T.
Tripathy49, S. Trogolo34,27, V. Trubnikov3, W.H. Trzaska127, T.P.
Trzcinski144, A. Tumkin110, R. Turrisi57, T.S. Tveter20, K. Ullaland21, A.
Uras138, M. Urioni58,142, G.L. Usai22, M. Vala38, N. Valle28, S. Vallero60,
L.V.R. van Doremalen63, M. van Leeuwen92, R.J.G. van Weelden92, P. Vande
Vyvre34, D. Varga147, Z. Varga147, M. Varga-Kofarago147, M. Vasileiou86, A.
Vasiliev90, O. Vázquez Doce52,107, V. Vechernin114, A. Velure21, E.
Vercellin24, S. Vergara Limón45, L. Vermunt63, R. Vértesi147, M. Verweij63, L.
Vickovic35, Z. Vilakazi133, O. Villalobos Baillie112, G. Vino53, A.
Vinogradov90, T. Virgili29, V. Vislavicius91, A. Vodopyanov76, B.
Volkel34,106, M.A. Völkl106, K. Voloshin94, S.A. Voloshin145, G. Volpe33, B.
von Haller34, I. Vorobyev107, N. Vozniuk64, J. Vrláková38, B. Wagner21, C.
Wang40, D. Wang40, M. Weber115, A. Wegrzynek34, S.C. Wenzel34, J.P.
Wessels146, J. Wiechula69, J. Wikne20, G. Wilk87, J. Wilkinson109, G.A.
Willems146, B. Windelband106, M. Winn140, W.E. Witt132, J.R. Wright120, W.
Wu40, Y. Wu130, R. Xu7, A.K. Yadav143, S. Yalcin78, Y. Yamaguchi46, K.
Yamakawa46, S. Yang21, S. Yano46, Z. Yin7, I.-K. Yoo17, J.H. Yoon62, S.
Yuan21, A. Yuncu106, V. Zaccolo23, C. Zampolli34, H.J.C. Zanoli63, N.
Zardoshti34, A. Zarochentsev114, P. Závada67, N. Zaviyalov110, M. Zhalov100,
B. Zhang7, S. Zhang40, X. Zhang7, Y. Zhang130, V. Zherebchevskii114, Y. Zhi11,
N. Zhigareva94, D. Zhou7, Y. Zhou91, J. Zhu109,7, Y. Zhu7, G. ZinovjevI,3, N.
Zurlo142,58

Affiliation Notes

I Deceased

II Also at: Italian National Agency for New Technologies, Energy and
Sustainable Economic Development (ENEA), Bologna, Italy

III Also at: Dipartimento DET del Politecnico di Torino, Turin, Italy

IV Also at: M.V. Lomonosov Moscow State University, D.V. Skobeltsyn Institute
of Nuclear, Physics, Moscow, Russia

V Also at: Department of Applied Physics, Aligarh Muslim University, Aligarh,
India

VI Also at: Institute of Theoretical Physics, University of Wroclaw, Poland

VII Also at: University of Kansas, Lawrence, Kansas, United States

Collaboration Institutes

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

2 AGH University of Science and Technology, Cracow, Poland

3 Bogolyubov Institute for Theoretical Physics, National Academy of Sciences
of Ukraine, Kiev, Ukraine

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

5 Budker Institute for Nuclear Physics, Novosibirsk, Russia

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

7 Central China Normal University, Wuhan, China

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

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

10 Chicago State University, Chicago, Illinois, United States

11 China Institute of Atomic Energy, Beijing, China

12 Chungbuk National University, Cheongju, Republic of Korea

13 Comenius University Bratislava, Faculty of Mathematics, Physics and
Informatics, Bratislava, Slovakia

14 COMSATS University Islamabad, Islamabad, Pakistan

15 Creighton University, Omaha, Nebraska, United States

16 Department of Physics, Aligarh Muslim University, Aligarh, India

17 Department of Physics, Pusan National University, Pusan, Republic of Korea

18 Department of Physics, Sejong University, Seoul, Republic of Korea

19 Department of Physics, University of California, Berkeley, California,
United States

20 Department of Physics, University of Oslo, Oslo, Norway

21 Department of Physics and Technology, University of Bergen, Bergen, Norway

22 Dipartimento di Fisica dell’Università and Sezione INFN, Cagliari, Italy

23 Dipartimento di Fisica dell’Università and Sezione INFN, Trieste, Italy

24 Dipartimento di Fisica dell’Università and Sezione INFN, Turin, Italy

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

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

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

28 Dipartimento di Fisica e Nucleare e Teorica, Università di Pavia, Pavia,
Italy

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

30 Dipartimento DISAT del Politecnico and Sezione INFN, Turin, Italy

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

32 Dipartimento di Scienze MIFT, Università di Messina, Messina, Italy

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

34 European Organization for Nuclear Research (CERN), Geneva, Switzerland

35 Faculty of Electrical Engineering, Mechanical Engineering and Naval
Architecture, University of Split, Split, Croatia

36 Faculty of Engineering and Science, Western Norway University of Applied
Sciences, Bergen, Norway

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

38 Faculty of Science, P.J. Šafárik University, Košice, Slovakia

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

40 Fudan University, Shanghai, China

41 Gangneung-Wonju National University, Gangneung, Republic of Korea

42 Gauhati University, Department of Physics, Guwahati, India

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

44 Helsinki Institute of Physics (HIP), Helsinki, Finland

45 High Energy Physics Group, Universidad Autónoma de Puebla, Puebla, Mexico

46 Hiroshima University, Hiroshima, Japan

47 Hochschule Worms, Zentrum für Technologietransfer und Telekommunikation
(ZTT), Worms, Germany

48 Horia Hulubei National Institute of Physics and Nuclear Engineering,
Bucharest, Romania

49 Indian Institute of Technology Bombay (IIT), Mumbai, India

50 Indian Institute of Technology Indore, Indore, India

51 Indonesian Institute of Sciences, Jakarta, Indonesia

52 INFN, Laboratori Nazionali di Frascati, Frascati, Italy

53 INFN, Sezione di Bari, Bari, Italy

54 INFN, Sezione di Bologna, Bologna, Italy

55 INFN, Sezione di Cagliari, Cagliari, Italy

56 INFN, Sezione di Catania, Catania, Italy

57 INFN, Sezione di Padova, Padova, Italy

58 INFN, Sezione di Pavia, Pavia, Italy

59 INFN, Sezione di Roma, Rome, Italy

60 INFN, Sezione di Torino, Turin, Italy

61 INFN, Sezione di Trieste, Trieste, Italy

62 Inha University, Incheon, Republic of Korea

63 Institute for Gravitational and Subatomic Physics (GRASP), Utrecht
University/Nikhef, Utrecht, Netherlands

64 Institute for Nuclear Research, Academy of Sciences, Moscow, Russia

65 Institute of Experimental Physics, Slovak Academy of Sciences, Košice,
Slovakia

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

67 Institute of Physics of the Czech Academy of Sciences, Prague, Czech
Republic

68 Institute of Space Science (ISS), Bucharest, Romania

69 Institut für Kernphysik, Johann Wolfgang Goethe-Universität Frankfurt,
Frankfurt, Germany

70 Instituto de Ciencias Nucleares, Universidad Nacional Autónoma de México,
Mexico City, Mexico

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

72 Instituto de Física, Universidad Nacional Autónoma de México, Mexico City,
Mexico

73 iThemba LABS, National Research Foundation, Somerset West, South Africa

74 Jeonbuk National University, Jeonju, Republic of Korea

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

76 Joint Institute for Nuclear Research (JINR), Dubna, Russia

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

78 KTO Karatay University, Konya, Turkey

79 Laboratoire de Physique des 2 Infinis, Irène Joliot-Curie, Orsay, France

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

81 Lawrence Berkeley National Laboratory, Berkeley, California, United States

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

83 Moscow Institute for Physics and Technology, Moscow, Russia

84 Nagasaki Institute of Applied Science, Nagasaki, Japan

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

86 National and Kapodistrian University of Athens, School of Science,
Department of Physics , Athens, Greece

87 National Centre for Nuclear Research, Warsaw, Poland

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

89 National Nuclear Research Center, Baku, Azerbaijan

90 National Research Centre Kurchatov Institute, Moscow, Russia

91 Niels Bohr Institute, University of Copenhagen, Copenhagen, Denmark

92 Nikhef, National institute for subatomic physics, Amsterdam, Netherlands

93 NRC Kurchatov Institute IHEP, Protvino, Russia

94 NRC «Kurchatov»Institute - ITEP, Moscow, Russia

95 NRNU Moscow Engineering Physics Institute, Moscow, Russia

96 Nuclear Physics Group, STFC Daresbury Laboratory, Daresbury, United Kingdom

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

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

99 Ohio State University, Columbus, Ohio, United States

100 Petersburg Nuclear Physics Institute, Gatchina, Russia

101 Physics department, Faculty of science, University of Zagreb, Zagreb,
Croatia

102 Physics Department, Panjab University, Chandigarh, India

103 Physics Department, University of Jammu, Jammu, India

104 Physics Department, University of Rajasthan, Jaipur, India

105 Physikalisches Institut, Eberhard-Karls-Universität Tübingen, Tübingen,
Germany

106 Physikalisches Institut, Ruprecht-Karls-Universität Heidelberg,
Heidelberg, Germany

107 Physik Department, Technische Universität München, Munich, Germany

108 Politecnico di Bari and Sezione INFN, Bari, Italy

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

110 Russian Federal Nuclear Center (VNIIEF), Sarov, Russia

111 Saha Institute of Nuclear Physics, Homi Bhabha National Institute,
Kolkata, India

112 School of Physics and Astronomy, University of Birmingham, Birmingham,
United Kingdom

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

114 St. Petersburg State University, St. Petersburg, Russia

115 Stefan Meyer Institut für Subatomare Physik (SMI), Vienna, Austria

116 SUBATECH, IMT Atlantique, Université de Nantes, CNRS-IN2P3, Nantes, France

117 Suranaree University of Technology, Nakhon Ratchasima, Thailand

118 Technical University of Košice, Košice, Slovakia

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

120 The University of Texas at Austin, Austin, Texas, United States

121 Universidad Autónoma de Sinaloa, Culiacán, Mexico

122 Universidade de São Paulo (USP), São Paulo, Brazil

123 Universidade Estadual de Campinas (UNICAMP), Campinas, Brazil

124 Universidade Federal do ABC, Santo Andre, Brazil

125 University of Cape Town, Cape Town, South Africa

126 University of Houston, Houston, Texas, United States

127 University of Jyväskylä, Jyväskylä, Finland

128 University of Kansas, Lawrence, Kansas, United States

129 University of Liverpool, Liverpool, United Kingdom

130 University of Science and Technology of China, Hefei, China

131 University of South-Eastern Norway, Tonsberg, Norway

132 University of Tennessee, Knoxville, Tennessee, United States

133 University of the Witwatersrand, Johannesburg, South Africa

134 University of Tokyo, Tokyo, Japan

135 University of Tsukuba, Tsukuba, Japan

136 University Politehnica of Bucharest, Bucharest, Romania

137 Université Clermont Auvergne, CNRS/IN2P3, LPC, Clermont-Ferrand, France

138 Université de Lyon, CNRS/IN2P3, Institut de Physique des 2 Infinis de
Lyon, Lyon, France

139 Université de Strasbourg, CNRS, IPHC UMR 7178, F-67000 Strasbourg, France,
Strasbourg, France

140 Université Paris-Saclay Centre d’Etudes de Saclay (CEA), IRFU, Départment
de Physique Nucléaire (DPhN), Saclay, France

141 Università degli Studi di Foggia, Foggia, Italy

142 Università di Brescia, Brescia, Italy

143 Variable Energy Cyclotron Centre, Homi Bhabha National Institute, Kolkata,
India

144 Warsaw University of Technology, Warsaw, Poland

145 Wayne State University, Detroit, Michigan, United States

146 Westfälische Wilhelms-Universität Münster, Institut für Kernphysik,
Münster, Germany

147 Wigner Research Centre for Physics, Budapest, Hungary

148 Yale University, New Haven, Connecticut, United States

149 Yonsei University, Seoul, Republic of Korea