21

11institutetext:  Dipartimento di Fisica e Astrofisica, Università degli Studi
di Firenze, Via G. Sansone 1, I-50019, Sesto Fiorentino, Italy
22institutetext: INAF/Osservatorio Astrofisico di Arcetri, Largo E.Fermi 5,
I-50125, Firenze, Italy
22email<EMAIL_ADDRESS>
#  Are all metal-poor stars of second-generation?

I. Vanni 11 2 2 S. Salvadori 1122 Á. Skúladóttir 1122

(Received: Day Month Year; Accepted: Day Month Year)

###### Abstract

Hydrodynamical cosmological simulations predict that the metal-free Population
III (Pop III) stars were likely very massive and, therefore, short-lived.
However, they left their chemical imprint on their descendants, which can also
have masses $<0.8\mathrm{M_{\odot}}$ and still be alive today. The Milky Way
stellar halo is one of the oldest and most metal-poor component of the Local
Group and a peculiar class of stars, the so-called Carbon-Enhanced Metal-Poor
(CEMP-no) stars seem to be directly related to Pop III stars. We aim at
revealing if _all_ metal-poor halo stars are true _second-generation stars_ or
if they have also been enriched by the subsequent generations of normal (Pop
II) stars. For this purpose, we compare the measured carbon and iron
abundances of the metal-poor halo stars with the ones predicted by our simple
parametric model, varying the pollution level from Pop III and normal stars.
We find that only the most C-enhanced and Fe-poor stars enclose in their
photospheres the pure imprint of Pop III stars, while, as the [C/Fe]
decreases, the probability of being also polluted by normal Pop II stars
increases.

###### keywords:

stars: abundances – ISM: abundances – Galaxy: halo – cosmology: first stars

##  1 Introduction

The Milky Way stellar halo is a promising region to search for the descendants
of the first (Pop III) stars. Here the oldest and most metal-poor stars have
been spotted, suggesting a link with the chemical elements produced by Pop III
stars. The mass of Pop III stars formed in state-of-the-art hydro-dynamical
simulations cover a wide mass range, $0.1<m_{PopIII}<1000\mathrm{M}_{\odot}$
(e.g Hirano et al., 2014; Greif, 2015; Hirano & Bromm, 2017). However, there
is general theoretical consensus supporting the idea that the first stars were
typically more massive than those forming today, with a characteristic mass
$\geq$ 10$\mathrm{M}_{\odot}$ (e.g. Bromm, 2013). Furthermore, a zero-
metallicity star has never been observed, confirming the massive nature of Pop
III stars (e.g. Rossi et al., 2021). If Pop III stars were typically massive,
most of them would have ended their lives in less than 100 Myrs, exploding as
supernovae (SNe). Thus, the first SNe polluted the Inter-Stellar Medium (ISM)
with their newly synthesized chemical elements. Consequently, normal
Population II (Pop II) stars were able to form in this Pop III-enriched ISM
(e.g. Schneider et al., 2012). Among these “second-generation” stars, those
with masses $<0.8\mathrm{M}_{\odot}$ can survive until today preserving in
their photospheres the chemical signatures of Pop III stars.

The iron-
abundance111[Fe/H]=$\log{\left(\frac{N_{Fe}}{N_{H}}\right)}-\log{\left(\frac{N_{Fe}}{N_{H}}\right)_{\odot}}$.
Solar abundances are from Asplund et al. (2009). of halo stars spans more than
5 dex. Stars with [Fe/H] $<-1$ are called metal-poor and they are separated in
two categories depending on their [C/Fe] values (see Beers & Christlieb,
2005): Carbon-Enhanced Metal-Poor stars (CEMP) when [C/Fe] $\geq+0.7$, and
C-normal otherwise. The CEMP stars can be further divided into CEMP-s(/r) and
CEMP-no stars depending on whether [Ba/Fe] is super- or sub-solar,
respectively. Neutron-capture elements from the slow process, such as barium,
are mainly produced by Asymptotic Giant Branch (AGB) stars and their
enhancement in the photosphere of a star is a signature of mass accretion from
an AGB companion. The C-excess in CEMP-s stars is thus expected to be acquired
during its lifetime, via mass transfer from a binary AGB companion (e.g. Abate
et al., 2015). On the other hand, the C-excess in CEMP-no stars is expected to
be representative of the environment of formation. This is supported by
observations, showing that CEMP-s stars are almost exclusively in binary
systems, which is not the case for CEMP-no stars (e.g. Hansen et al., 2016;
Arentsen et al., 2019).

The fraction of CEMP-no stars increases towards decreasing [Fe/H], suggesting
a direct link between the birth clouds of CEMP-no stars and the chemical
products of Pop III SNe (e.g. Salvadori et al., 2015; Chiaki et al., 2020).
Conversely, the origin of C-normal very metal-poor stars ($\rm[Fe/H]<-2$) is
still highly debated: their chemical abundances are indeed consistent with
either an enrichment from Pop III SNe only (see Hartwig et al., 2019; Welsh et
al., 2021), or from both Pop III and normal Pop II SNe (e.g. De Bennassuti et
al., 2017; Liu et al., 2021).

In this work, we aim at understanding if all very metal-poor stars are true
descendants of Pop III stars. To this end we will take advantage of the high-
resolution measurements of C-enhanced and C-normal metal-poor stars, and
compare those with our theoretical models. We refer the reader to Vanni et al.
(in prep) for a comparison with all the chemical abundance ratios measured in
metal-poor star.

##  2 Sample of metal-poor halo stars

The _halo sample_ , which will be used to compare to our model, includes 132
CEMP-no and C-normal halo stars with [Fe/H] $\leq-2$, that have been observed
with high resolution spectroscopy (R $\geq 30\,000$). In particular, this
sample includes the stars presented in Cayrel et al. (2004) and Yong et al.
(2013), excluding CEMP-s stars, as well as extremely metal-poor stars, [Fe/H]
$\leq-3$, from: Christlieb et al. (2004); Norris et al. (2007); Caffau et al.
(2011); Hansen et al. (2014); Keller et al. (2014); Frebel & Norris (2015);
Bonifacio et al. (2015); Li et al. (2015); Starkenburg et al. (2018);
Bonifacio et al. (2018); François et al. (2018); Aguado et al. (2019);
Ezzeddine et al. (2019). These chemical abundances are not corrected for the
non-LTE effects.

The stars of our halo sample cover a wide iron range, $\rm-7.1<[Fe/H]<-2$, and
their metallicity distribution function peaks at [Fe/H]$\sim-3$ (see e.g.
Placco et al., 2014).

Figure 1: Chemical abundance pattern of the _halo sample_ in three categories:
CEMP-no stars with [Fe/H] $\leq-4$ (top), CEMP-no with $-4<$ [Fe/H] $\leq-2$
(middle), and C-normal stars with [Fe/H] $\leq-2$ (bottom). Filled points with
error bars represent the mean and the standard deviation computed without
considering upper/lower limits, while the empty points (top panel) with arrows
including upper and/or lower limits. Shaded areas show the range of the
measured chemical abundances, including also upper/lower limits.

In Fig. 1 we show the average abundance ratios, standard deviation, and
maximum range of the scatter for stars in the halo sample, divided among:
CEMP-no stars (red) with [Fe/H] $<-4$ (upper panel) and with $-4<$ [Fe/H]
$<-2$ (middle), and finally C-normal stars with $\rm[Fe/H]<-2$ (gray, lower
panel). All the stars in the sample have carbon (and iron) abundance
measurements, while for the other elements the measurements are sometimes
limited to few stars (e.g. O or Zn, which are not available for the CEMP-no
stars in our sample with $-4<$ [Fe/H] $<-2$). Furthermore, the majority of the
most iron-poor stars only have upper/lower limits of abundance values. For
this reason, in the top panel of Fig. 1 we include the mean chemical abundance
ratios obtained by both including and excluding the upper/lower limits.
Furthermore, the upper/lower limits were used to evaluate the range of the
scatter. For example, for the C-normal stars we have also used the Caffau et
al. (2011) and Starkenburg et al. (2018) stars which have, respectively,
[C/Fe]$<+0.7$ and [C/Fe]$<+1.0$.

In Fig. 1 we first notice that the standard deviation and scatter in the
abundance ratios are highest for the most Fe-poor CEMP-no stars (top panel),
while they gradually decrease towards lower panels, being smallest for
C-normal stars (see also Cayrel et al. 2004). The mean abundance ratios of the
light elements C, N and O, are generally higher for CEMP-no stars than for
C-normal stars, even if we consider stars in the same [Fe/H] range. On the
other hand, the abundance ratios of the elements heavier than oxygen are
consistent within the error bars between the different stellar classes.

##  3 The model

Here we briefly summarize the simple and general parametric study presented by
Salvadori et al. (2019) and further implemented by Vanni et al. (in prep). The
model aims to chemically characterize the descendants of Pop III stars: long-
lived stars formed in environments predominantly polluted by Pop III SNe, i.e.
where the metals from Pop III SNe account for $\geq 50\%$ of metals in the
ISM.

Following the results of hydrodynamical cosmological simulations (e.g. Hirano
et al., 2014), we assume that a single Pop III star can form in the primordial
star-forming haloes. Then, we evaluate the chemical enrichment of the ISM
after: (i) the injection of heavy elements by Pop III SNe with different
explosion energies and progenitor masses; and (ii) the subsequent contribution
of ”normal” Pop II stars exploding as core-collapse SNe. We adopt the yields
by Heger & Woosley (2002) for very massive Pop III stars,
$m=[140-260]\mathrm{M}_{\odot}$, that explode as Pair Instability SNe (PISN,
$E_{SN}=10^{52}-10^{53}$ erg) and by Heger & Woosley (2010) for intermediate
mass Pop III stars, $m=[10-100]\mathrm{M}_{\odot}$, exploding as faint, core-
collapse, high-energy SNe and hypernovae, whose explosion energy is
respectively equal to $E_{SN}=(0.6,1.2,3.0,10.0)\times 10^{51}$ erg. For Pop
II stars we adopt the yields of Woosley & Weaver (1995) and Limongi & Chieffi
(2018) and show both results to account for the uncertainty due to the choice
of the stellar evolutionary model (see e.g. Nomoto et al., 2013).

The main unknowns related to early cosmic star formation and metal enrichment
are encapsulated into the free parameters, which are varied to obtain the most
general model. The model takes into consideration: the fraction of gas
converted into stars, or the star formation efficiency,
$f_{*}=[10^{-4}-10^{-1}]$; metals retained into the ISM, parameterized with
the dilution factor $f_{dil}$; and the mass fraction of metals injected into
the ISM by Pop III stars with respect to the total, $f_{PopIII}=[100-50]\%$.
It turns out that the iron abundance of the ISM depends on $f_{PopIII}$; the
yields of Pop III, $Y^{III}(m,E_{SN})$, and Pop II stars, $Y^{II}(m,Z)$; and
on the ratio between the first two free parameters, $f_{*}/f_{dil}$.
Conversely, [C/Fe] is not affected by $f_{*}$ nor by $f_{dil}$ (see Salvadori
et al. 2019 for details).

##  4 Results

Figure 2: Carbon-to-iron ratios of PopIII descendants. Colours show different
level of Pop III enrichment: 100% (yellow), 90% (orange), 70% (pink), 50%
(purple). Each panel shows different explosion energies of Pop III SNe (see
labels): from faint SNe ($E\sim 6\times 10^{50}$ erg) to hypernovae ($E\sim
10^{52}$ erg) in the mass range $[10-100]\mathrm{M}_{\odot}$; as well as PISN
($E=10^{52-53}$ erg) for $[140-260]\mathrm{M}_{\odot}$. Star symbols show the
measured abundances of CEMP-no (filled) and C-normal stars (open). The dotted
line is the threshold for the carbon enhancement, $\rm[C/Fe]>+0.7$. Note that
the yellow area for the hypernovae progenitor case is hidden behind the other
colours.

In Fig. 2 we compare the predicted [C/Fe] and [Fe/H] values for the
descendants of the first stars (shaded areas) with the chemical abundances
measured in the _halo sample_ (points). The colours denote different fraction
of metals from Pop III SNe: from the yellow area, where the descendants are
completely imprinted by Pop III ($f_{popIII}=100\%$), to the purple area,
where they are equally imprinted by Pop III and Pop II ($f_{popIII}=50\%$)
stars. The different panels show the results for different explosion energies
of Pop III SNe: from the least energetic faint SNe to the most energetic
PISNe.

The maximum [C/Fe] that is predicted by the models (Fig. 2) decreases with:
(i) increasing explosion energy of Pop III SNe; and (ii) increasing
contribution of Pop II SNe to the chemical enrichment. Furthermore, we see
that CEMP-no halo stars with [C/Fe] $\geq+2.5$ and [Fe/H] $\leq-4$ can only be
reproduced by models accounting for a 100% enrichment from Pop III stars,
which implies that these objects are true second-generation stars. Indeed,
even a 10% pollution by Pop II stars is sufficient to lower the maximum to
[C/Fe]$<+2.5$.

Which kind of Pop III SNe imprinted these highly C-enhanced ultra iron-poor
stars? Different Pop III SN models are able to enrich the ISM with high [C/Fe]
and low [Fe/H]: the lightest PISN progenitors, M $\sim
140-150\mathrm{M}_{\odot}$, and massive Pop III stars,
$40\mathrm{M}_{\odot}\leq\mathrm{M}\leq 100\mathrm{M}_{\odot}$, exploding as
faint SNe or core-collapse SNe. Still, our models show that the abundance
ratios of the other chemical elements measured in these CEMP-no ultra iron-
poor stars (Fig. 1) are not consistent with a light PISN enrichment, but
rather support a scenario where these stars have been imprinted by massive Pop
III stars exploding as low-energy SNe.

We can now try to understand if all metal-poor stars are truly second-
generation objects. At higher $\rm[Fe/H]>-4$, the observed properties of both
CEMP-no and C-normal halo stars can be reproduced by several models (Fig. 2).
At $\rm[Fe/H]>-4$, only the hypernovae case is not able to reproduce the
[C/Fe] values measured in CEMP-no stars with or without Pop II contribution.
To disentangle this degeneracy of different Pop III SN models, we would need
to study the abundance ratios of other chemical elements. This will be done in
a complementary work, Vanni et al. (in prep).

The probability for a star to be uniquely ($100\%$) or predominantly ($>50\%$)
imprinted by the chemical products of Pop III SNe decreases as [C/Fe]
decreases. Furthermore, an increasing contribution from Pop II stars reduces
the scatter in [C/Fe] predicted by different models, in agreement to what is
observed (Fig. 1). Indeed, we can see in Fig. 2 that the scatter in [C/Fe]
spans over $\sim 5$ dex for pure Pop III star descendants, and only $\sim 2$
dex for the descendants enriched by Pop III stars at a 50% level. Thus, the
specific chemical features of the ISM enriched by different Pop III SN types
are mostly washed out if Pop III and Pop II stars equally contribute to the
ISM enrichment. This is a strong indication that a group of stars that show
very similar chemical abundances, i.e. a small star-to-star scatter like
C-normal halo stars, have likely been polluted by Pop III and normal Pop II
stars at the same level, or even predominantly by Pop II stars.

##  5 Conclusions

From comparing our model results with data, we propose simple diagnostics that
exploit measured carbon and iron abundances to identify which metal-poor halo
stars are the descendants of Pop III stars ($>50\%$ of the metals in their ISM
of formation) and to estimate the level of imprint from normal Pop II SNe.
These are our key findings:

  * •

CEMP-no stars with [C/Fe] $\geq+2.5$ are second-generation stars solely
imprinted by massive first stars exploding with low to normal energy
($E_{SN}\leq 1.2\times 10^{51}$ erg).

  * •

CEMP-no stars with $\rm[C/Fe]<+2.5$ are likely Pop III star descendants, which
have been mainly polluted by Pop III SNe with a non-negligible $\sim(10-30)\%$
contribution from Pop II stars.

  * •

The majority of C-normal stars have been enriched by normal Pop II stars at
$\gtrsim 50\%$ level, and hence they do not show strong chemical features left
by Pop III SNe.

Conversely, our models show that Pop III SNe can also enrich the ISM to low
carbon values. The [C/Fe] values of the pure descendants of Pop III SNe
exploding with high-energy ($E_{SN}\geq 3\times 10^{51}$ erg) are mostly
located at [C/Fe] $\lesssim 0$. Therefore, we expect to find rare C-normal
stars that are direct descendants of these very energetic Pop III SNe, thus
showing peculiar chemical abundance patterns. This is consistent with the
recent discoveries of primordial hypernova descendants (see Skúladóttir et
al., 2021; Placco et al., 2021).

Ultimately, our simple and general parametric model can be applied to
interpret the chemical abundances measured in both ancient stars and in high-
redshift absorption systems (see Salvadori et al. in this volume). According
to our predictions, a C-excess measured in gaseous environments at high
redshifts ($z>4$) is a solid proof of chemical enrichment driven by Pop III
stars.

The observed scatter of chemical abundance ratios is another key diagnostic.
Our model shows a larger scatter with a higher fraction of Pop III
contribution. This is supported by observations of metal-poor stars in the
Galactic halo (Fig. 1), and we predict that this is also true in the ISM
observed at high redshifts.

By using a powerful combination of models and data, we have exploited C and Fe
abundances to identify first star descendants and quantify their Pop III
enrichment. However, using only these two elements is not sufficient to break
degeneracies between different models of Pop III SNe (mass and energy). In an
upcoming study, we will thus expand the work presented here to include all the
chemical elements available, exploiting to the fullest the very metal-poor
stars to better understand the first stars in the Universe.

###### Acknowledgements.

This project has received funding from the European Research Council (ERC)
under the European Union’s Horizon 2020 research and innovation programme
(grant agreement No 804240). I.V. and S.S. acknowledge support from the PRIN-
MIUR17, prot. n. 2017T4ARJ5.

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$K^{0}_{S}$ meson production in inelastic p+p interactions at 158 beam
momentum measured by at the CERN SPS CERN-EP-2021-112 The production of
$K^{0}_{S}$ mesons in inelastic p+p collisions at beam momentum 158
($\sqrt{s_{NN}}=17.3$ ) was measured with the spectrometer at the CERN Super
Proton Synchrotron. Double-differential distributions were obtained in
transverse momentum and rapidity. The mean multiplicity of $K^{0}_{S}$ was
determined to be $0.162\pm 0.001(stat.)\pm 0.011(sys.)$. The results on
$K^{0}_{S}$ production are compared with model predictions (, SMASH 2.0, PHSD
and UrQMD 3.4 models) as well as with published world data.

##  1 Introduction

Kaons contain strange or anti-strange valence quarks, which are both not
present in the initial state of collisions between nucleons and nuclei. Thus
kaon production implies the creation of a strange and anti-strange quark pair.
Collisions between nuclei proceed via the formation of a rapidly expanding
high energy density fireball [1]. At sufficiently high collision energy, the
evolution of the fireball is expected to proceed via an intermediate partonic
phase, the quark-gluon plasma (QGP). Thus, investigating these reactions will
shed light on the differences between hadronic and partonic matter and the
characteristics of the phase transition between them. The study of kaon
production in p+p collisions is important not only as a reference for possible
modifications of strangeness production in nucleus-nucleus collisions [2] but
also for understanding strangeness production in elementary interactions. It
was predicted that the onset of deconfinement is located in the few energy
range [3]. In order to explore this region systematically studies observables
indicative of the QGP by a two-dimensional scan in collision energy and
nuclear mass number of the colliding nuclei. Since 2009 has collected data on
p+p, p+Pb, Pb+Pb, Be+Be, Ar+Sc and Xe+La interactions in the energy range
13A-158A [4]. Results on identified hadron spectra measurements can be found
in Ref. [5, 7, 6, 8, 9]. In this paper we present the first results of
$K_{S}^{0}$ production in p+p collisions at 158 , which will be used later as
the reference for comparison with $K_{S}^{0}$ production at lower energies and
constitutes the first step of the energy scan of $K_{S}^{0}$ production in p+p
interactions. After the energy scan we will perform the nuclear mass scan
(heavier systems) and results will be compared with p+p collisions. Thanks to
high statistics, large acceptance and good resolution the results presented
here have significantly higher precision than previously published data at the
SPS energies [10, 11, 12, 13, 14, 15, 16, 17].

The paper is organised as follows. In section 2, details of the detector
system are presented. Section 3 is devoted to the description of the analysis
method. The results are shown in section 4. In section 5, they are compared to
published world data and model calculations. Section 6 closes the paper with a
summary and outlook.

The following variables and definitions are used in this paper. The particle
rapidity $y$ is calculated in the proton-proton collision center of mass
system (cms), $y=0.5ln[(E+cp_{L})/(E-cp_{L})]$, where $E$ and $p_{L}$ are the
particle energy and longitudinal momentum, respectively. The transverse
component of the momentum is denoted as $p_{T}$. The momentum in the
laboratory frame is denoted $p_{lab}$ and the collision energy per nucleon
pair in the centre of mass by $\sqrt{s_{NN}}$.

##  2 Experimental setup

The collaboration uses a large acceptance spectrometer located in the CERN
North Area. The schematic layout of the detector during the p+p 158 data-
taking is shown in Fig. 1. A detailed description of the full detector can be
found in Ref. [18], while the details on the performance of the simulation in
describing the detector performance across different kinematic variables as
well as its inefficiencies can be found in Ref. [19].

Figure 1:  (Color online) The schematic layout of the NA61/SHINE experiment at
the CERN SPS during p+p 158 data taking (horizontal cut, not to scale). The
beam and trigger detector configuration used for data taking in 2009 is shown
in the inset (see Refs. [19, 20] for detailed description). The chosen
coordinate system is drawn on the lower left: its origin lies in the middle of
the VTPC-2, on the beam axis.

The main components of the spectrometer used in this analysis are four large
volume Time Projection Chambers (TPC). Two of them, the vertex TPCs (VTPC-1
and VTPC-2), are located in the magnetic fields of two super-conducting dipole
magnets with a maximum combined bending power of 9 Tm which corresponds to
about 1.5 T and 1.1 T fields in the upstream and downstream magnets,
respectively. Two large main TPCs (MTPC-L and MTPC-R) and two walls of pixel
Time-of-Flight (ToF-L/R) detectors are positioned symmetrically to the
beamline downstream of the magnets. A GAP-TPC (GTPC) is placed between VTPC-1
and VTPC-2 directly on the beamline. It closes the gap between the beam axis
and the sensitive volumes of the other TPCs. The TPCs are filled with Ar and
CO2 gas mixtures. Particle identification in the TPCs is based on measurements
of the specific energy loss (${\text{d}}E\\!/\\!{\text{d}}x$) in the chamber
gas.

Secondary beams of positively charged hadrons at 158 are produced from 400
proton beams extracted from the SPS accelerator. Particles of the secondary
hadron beam are identified by two Cherenkov counters, a CEDAR-N [21] and a
threshold counter (THC). The CEDAR counter, using a coincidence of six out of
the eight photomultipliers placed radially along the Cherenkov ring, provides
identification of protons, while the THC, operated at a pressure lower than
the proton threshold, is used in anti-coincidence in the trigger logic. A
selection based on signals from the Cherenkov counters allowed one to identify
beam protons with a purity of about 99%. A set of scintillation (S1 and S2),
veto (V0 and V1) and Cherenkov counters (C1 and C2) and beam position
detectors (BPDs) upstream of the spectrometer provide timing reference,
identification, and position measurements of incoming beam particles. The
trigger scintillation counter S4 placed downstream of the target has a
diameter of 2 cm and is used to select events with collisions in the target
area by the absence of a charged particle hit.

A cylindrical target vessel of 20.29 cm length and 3 cm diameter was situated
upstream of the entrance window of VTPC-1 (centre of the target z = -581 cm in
the NA61/SHINE coordinate system). The vessel was filled with liquid hydrogen
corresponding to an interaction length of 2.8%. The ensemble of the vessel and
liquid hydrogen constitute the "Liquid Hydrogen Target" (LHT). Data were taken
with full and empty LHT.

Interactions in the target are selected with the trigger system by requiring
an incoming beam proton and no signal from the S4 counter. This minimum bias
trigger is based on the disappearance of the beam proton downstream of the
target.

##  3 Analysis

###  3.1 Data set

In 2009, 2010 and 2011 the detector registered about $5.75\times 10^{7}$ p+p
interactions at 158 . For the analysis, the range of the z-position of the
main vertex was selected to cover mostly the LHT (see Sec. 3.3) in order to
maximize the number of good events and minimize the contamination by off-
target interactions. Figure 2 shows the distributions of reconstructed vertex
z positions in the target-inserted and the target-removed sample as blue and
red histograms, respectively. The target-removed sample was normalised in the
range -450 < z < -300 cm to the same number of reconstructed events as in the
target-inserted sample. The normalised ratio of events in the range -590 < z <
-572 cm is small at the level of 2.9%, and therefore no correction for non-
target interactions was applied. In order to estimate the possible systematic
biases related to the contamination by off-target interactions, the event
selection window of the z-position of the main vertex was varied (see Sec.
3.8).

Figure 2: Distributions of the z-coordinate of the reconstructed vertex for
events recorded with target full (blue histogram) and target empty (red
histogram). Two vertical black lines at position -450 and -300 cm show the
range which is used for histogram normalisation.

###  3.2 Analysis method

Details of the track and vertex reconstruction procedures can be found in
Refs. [19, 20, 22]. In the following section, the criteria for the selection
of events, of tracks and of the $K^{0}_{S}$ decay topology are enumerated.
Then the simulation-based procedure will be described, which is used to
quantify the losses due to reconstruction inefficiencies and the limited
geometrical acceptance.

###  3.3 Event selection

The selection criteria for inelastic p+p interactions are the following:

  * (i)

An interaction was accepted by the trigger logic (see Refs. [19, 20]).

  * (ii)

Beam particle trajectory measured in at least three planes out of four of
BPD-1 and BPD-2 and in both planes of BPD-3.

  * (iii)

The primary interaction vertex fit converged.

  * (iv)

Z position of the interaction vertex (fitted using the beam trajectory and TPC
tracks) not farther away than 9 cm from the center of the Liquid Hydrogen
Target.

The final number of events that satisfy all the above selection criteria is
$2.86\times 10^{7}$.

###  3.4 Track and topology selection

Neutral strange particles are detected and measured by means of their weak
decays into a pair of charged particles. The $K^{0}_{S}$ decays into
$\pi^{+}+\pi^{-}$ with a branching ratio of 69.2% [23]. The decay particles
form the so-called $V^{0}$ topology. $K^{0}_{S}$ decay candidates ($V^{0}$s)
are obtained by pairing all positively and negatively charged pions. The
corresponding tracks are required to have a distance of closest approach
between the two trajectories of less than 1 cm. The tracks of the decay pions
and the $V^{0}$ topology are subject to the following additional selection
criteria:

  * (i)

For each track, the minimum number of measured clusters in VTPC-1 and VTPC-2
was required to be 15.

  * (ii)

All pion tracks must have a measured specific energy loss
(${\text{d}}E\\!/\\!{\text{d}}x$) in the TPCs within $\pm 3\sigma$ around the
nominal Bethe-Bloch value for charged pions. Here $\sigma$ represents the
typical standard deviation of a Gaussian fitted to the
${\text{d}}E\\!/\\!{\text{d}}x$ distribution of pions. Since only small
variations of $\sigma$ were observed for different bins and beam momenta, a
constant value $\sigma=0.052$ is used [24]. This selection criteria is applied
only for experimental data, not for MC simulated data (see below).

  * (iii)

The orientation of the $V^{0}$ decay plane with respect to the magnetic field,
quantified by $|cos\Phi|$ (see Fig. 3), is required to lie in the ranges
$|cos\Phi|<0.95$ for $-0.25<y<0.25$, $|cos\Phi|<0.9$ for $0.25<y<0.75$,
$|cos\Phi|<0.8$ for $0.75<y<1.25$ and $|cos\Phi|<0.5$ for larger $y$ ($y$ is
the ($K^{0}_{S}$) rapidity). This criterion removes $V^{0}$s for which the
determination of the momenta of the decay products and the decay vertex
position suffer from large uncertainties.

  * (iv)

The distance |$\Delta$z| between the primary production vertex and the
$K^{0}_{S}$ decay vertex is required to lie in the rapidity dependent range
|$\Delta$z| > $e^{3.1+0.42\cdot y}$.

  * (v)

Spurious $K^{0}_{S}$ candidates are rejected by an elliptic cut on the impact
parameters of the daughter tracks, which are relative to the $K^{0}_{S}$ decay
vertex, in x ($b_{x}$) and y ($b_{y}$) direction,
$\left(\frac{b_{x}}{2}\right)^{2}+{b_{y}}^{2}<0.25$ cm.

Figure 3: Definition of angle $\Phi$ used for $V^{0}$ selection. $\Phi$ is
defined as the angle between the vectors y’, and n, where y’ is the vector
perpendicular to the momentum of the $V^{0}$-particle which lies in the plane
spanned by the y-axis and the $V^{0}$-momentum vector, and n is a vector
normal to the decay plane.

The quality of the aforementioned track and topology selection criteria is
illustrated in Fig. 4. The population of $K^{0}_{S}$ decay candidates is shown
as a function of the two Armenteros-Podolansky variables $p^{Arm}_{T}$ and
$\alpha^{Arm}$ [25] before (left) and after (right) all track and topology
selection criteria. The quantity $p^{Arm}_{T}$ is the transverse momentum of
the decay particles with respect to the direction of motion of the $V^{0}$
candidate and $\alpha^{Arm}=(p^{+}_{L}-p^{-}_{L})/(p^{+}_{L}+p^{-}_{L})$,
where $p^{+}_{L}$ and $p^{-}_{L}$ are the longitudinal momenta of the
positively and negatively charged $V^{0}$ daughter particles, measured with
respect to the $V^{0}$’s direction of motion. After applying all cuts, a
contamination by $\Lambda$’s of roughly 7% persists (Fig. 4 right). However,
the $\Lambda$ background below the $K^{0}_{S}$ mass peak is small and amounts
to 0.5%.

Figure 4: Armenteros-Podolanski plots for $V^{0}$ candidates without (left)
and with all track and topology selection criteria (right).

###  3.5 $K^{0}_{S}$ yields

The double differential yield of $K^{0}_{S}$ was determined by studying the
invariant mass distributions of the accepted pion pairs in bins of rapidity
and transverse momentum (examples are presented in Fig. 5). True decays will
appear as a peak over a smooth background. The $K^{0}_{S}$ yield was
determined in each bin using a fit function that describes both the signal and
the background. A Lorentzian function was used for the signal:

$L(m)=A\frac{1}{\pi}\frac{\frac{1}{2}\Gamma}{(m-m_{0})^{2}+(\frac{1}{2}\Gamma)^{2}}\leavevmode\nobreak\
,$ (1)

where A is the normalization factor, $\Gamma$ is the full width at the half
maximum of the signal peak, and $m_{0}$ is the mass parameter. The background
contribution is described by a polynomial function of $2^{nd}$ order. Figure 5
shows examples of $\pi^{+}\pi^{-}$ mass distributions after all $V^{0}$
selection cuts as the red histograms for real events (left) and for simulated
events (right). Clearly, the background outside the $K^{0}_{S}$ peak is small.
The width and mass of the $K^{0}_{S}$ peak are well reproduced by the
simulation, thus indicating that the correction factors used for calculating
the final bin-by-bin $K^{0}_{S}$ multiplicities are reliable.

The procedure of fitting the histograms proceeds in three steps. In the first
step, the background outside the signal peak ([0.475-0.525]
$\mbox{Ge\kern-1.00006ptV}\\!/\\!c^{2}$) is fitted with a polynomial of
$2^{nd}$ order. This step is necessary to obtain starting values for the
parameters of the background function. In the next step, a fit of the full
invariant mass spectrum is performed with the sum of the Lorentzian and the
background functions. The initial parameter values for the background function
are taken from the previous step, while the mass parameter is fixed to the PDG
value of $m_{0}=0.497614(24)$ $\mbox{Ge\kern-1.00006ptV}\\!/\\!c^{2}$ [23] and
the width was allowed to vary between 0.01 and 0.03
$\mbox{Ge\kern-1.00006ptV}\\!/\\!c^{2}$. Finally, in the last step, all
parameters were free, and the fitting region was [0.35-0.7]
$\mbox{Ge\kern-1.00006ptV}\\!/\\!c^{2}$. The fitted polynomial background
function is shown by the blue curve, and the fitted Lorentzian signal function
by the red curve in Fig. 5. In order to minimize the sensitivity of the
$K^{0}_{S}$ yield to the integration window, the uncorrected number of
$K^{0}_{S}$ was calculated by subtracting bin-by-bin the fitted background and
summing the background-subtracted signal in the mass window $m_{0}\pm 3\Gamma$
(dashed vertical lines), where $m_{0}$ is the fitted mass of the $K^{0}_{S}$.
The latter agrees with the PDG value within statistical uncertainties. Figure
5 shows that the simulation reproduces the central value of the $K^{0}_{S}$
mass distribution and somewhat underestimate its width. To calculate the
signal from the simulation, the $\Gamma$ parameter fitted to the simulation
was used. Thus a possible bias due to differences between the data and
simulation is reduced, see Sec. 3.8.

Figure 5: The invariant mass distribution of $K^{0}_{S}$ candidates for
experimental data (left) and MC (right). The dashed vertical lines indicate
the regions over which the $K^{0}_{S}$ signal was integrated. The signal data
points are shown in red, the fitted background in blue and the total fit
results in red. The uncertainties are smaller than symbol size and not visible
on the plots. Mass resolutions obtained from the fits are: $\sigma=(0.01026\pm
0.00002)\leavevmode\nobreak\ \mbox{Ge\kern-1.00006ptV}\\!/\\!c^{2}$ for the
experimental data and $\sigma=(0.00819\pm 0.00001)\leavevmode\nobreak\
\mbox{Ge\kern-1.00006ptV}\\!/\\!c^{2}$ for the MC.

Uncorrected bin-by-bin $K^{0}_{S}$ multiplicities and their statistical
uncertainties are shown in Fig. 6 (bottom).

###  3.6 Correction factors

A detailed Monte Carlo simulation was performed to compute the correction for
losses due to the trigger bias, geometrical acceptance, reconstruction
efficiency, as well as the selection cuts applied in the analysis. The
correction factors are based on $9.5\times 10^{7}$ inelastic p+p events at 158
produced by the event generator [26, 27]. Particles in the generated events
were tracked through the apparatus using the Geant3 package [28]. The TPC
response was simulated by dedicated software packages which take into account
all known detector effects. The simulated events were reconstructed with the
same software as used for real events and the same selection cuts were applied
and ${\text{d}}E\\!/\\!{\text{d}}x$ identification was replaced by matching of
simulated and reconstructed tracks. The branching ratio of $K^{0}_{S}$ decays
are taken into account in the Geant3 software package. For each $y$ and
$p_{T}$ bin, the correction factor $c_{MC}(y,p_{T})$ was calculated as:

$c_{MC}(y,p_{T})=\left.\frac{n_{MC}^{gen}(y,p_{T})}{N_{MC}^{gen}}\right/\frac{n_{MC}^{acc}(y,p_{T})}{N_{MC}^{acc}}\leavevmode\nobreak\
,$ (2)

where:

  * -

$n_{MC}^{gen}(y,p_{T})$ is the number of $K^{0}_{S}$ generated in a given
($y,p_{T}$) bin,

  * -

$n_{MC}^{acc}(y,p_{T})$ is the number of reconstructed $K^{0}_{S}$ in a given
($y,p_{T}$) bin. To derive this numbers the invariant mass distribution of the
reconstructed $\pi^{+}$ and $\pi^{-}$ track pairs that pass all selection
requirements was formed. The number of reconstructed $K^{0}_{S}$ is then
obtained by following the same extraction procedure as for real data,
described in Sec. 3.5.

  * -

$N_{MC}^{gen}$ is the number of generated inelastic p+p interactions
($9.5\times 10^{7}$),

  * -

$N_{MC}^{acc}$ is the number of accepted p+p events ($5.4\times 10^{7}$).

The loss of the $K^{0}_{S}$ mesons due to the ${\text{d}}E\\!/\\!{\text{d}}x$
cut is corrected with an additional factor:

$c_{dE/dx}=\frac{1}{\epsilon^{2}}=1.005\leavevmode\nobreak\ ,$ (3)

where $\epsilon=0.9973$ is the probability for the pions to be detected within
$\pm 3\sigma$ around the nominal Bethe-Bloch value.

The double-differential yield of $K^{0}_{S}$ per inelastic event in bins of
($y,p_{T}$) is calculated as follows:

$\frac{d^{2}n}{dy\,dp_{T}}(y,p_{T})=\frac{c_{dE/dx}\cdot
c_{MC}(y,p_{T})}{\Delta y\,\Delta
p_{T}}\cdot\frac{n_{K^{0}_{S}}(y,p_{T})}{N_{events}}\leavevmode\nobreak\ ,$
(4)

where:

  * -

$c_{dE/dx}$, $c_{MC}(y,p_{T})$ are the correction factors described above,

  * -

$\Delta y$ and $\Delta p_{T}$ are the bin widths,

  * -

$n_{K^{0}_{S}}(y,p_{T})$ is the uncorrected number of $K^{0}_{S}$, obtained by
the signal extraction procedure described in Sec. 3.5,

  * -

$N_{events}$ is the number of events after cuts.

###  3.7 Statistical uncertainties

The statistical uncertainties of the corrected double-differential yields (see
Eq. 4) receive contributions from the statistical uncertainty of the
correction factors $c_{MC}(y,p_{T})$ and the statistical uncertainty of the
uncorrected number of $K^{0}_{S}$ ($\Delta N_{K^{0}_{S}}(y,p_{T})$). The
statistical uncertainty of the former receives two contributions, the first,
$\alpha$, caused by the loss of inelastic interactions due to the event
selection and the second, $\beta$, connected with the loss of $K^{0}_{S}$
candidates due to the $V^{0}$ selection:

$c_{MC}(y,p_{T})=\left.\frac{n_{MC}^{gen}(y,p_{T})}{N_{MC}^{gen}}\right/\frac{n_{MC}^{acc}(y,p_{T})}{N_{MC}^{acc}}=\left.\frac{N_{MC}^{acc}}{N_{MC}^{gen}}\right/\frac{n_{MC}^{acc}(y,p_{T})}{n_{MC}^{gen}(y,p_{T})}=\frac{\alpha}{\beta(y,p_{T})}\leavevmode\nobreak\
,$ (5)

The error of $\alpha$ is calculated assuming a binomial distribution:

$\Delta\alpha=\sqrt{\frac{\alpha(1-\alpha)}{N_{MC}^{gen}}}\leavevmode\nobreak\
,$ (6)

The error of $\beta$ is calculated according to formula:

$\Delta\beta(y,p_{T})=\sqrt{\left(\frac{\Delta
n_{MC}^{acc}(y,p_{T})}{n_{MC}^{gen}(y,p_{T})}\right)^{2}+\left(\frac{n_{MC}^{acc}(y,p_{T})\cdot\Delta
n_{MC}^{gen}(y,p_{T})}{(n_{MC}^{gen}(y,p_{T}))^{2}}\right)^{2}}\leavevmode\nobreak\
,$ (7)

where $\Delta n_{MC}^{acc}(y,p_{T})$ is the uncertainty of the fit, and
$\Delta n_{MC}^{gen}(y,p_{T})=\sqrt{n_{MC}^{gen}(y,p_{T})}$. The equation for
$\Delta c_{MC}(y,p_{T})$ can be written as:

$\Delta
c_{MC}(y,p_{T})=\sqrt{\left(\frac{\Delta\alpha}{\beta}\right)^{2}+\left(-\frac{\alpha\cdot\Delta\beta}{\beta^{2}}\right)^{2}}\leavevmode\nobreak\
,$ (8)

Figure 6: Top: Monte-Carlo correction factors (see Eq. 2) in each ($y,p_{T}$)
bin (left) and corresponding statistical uncertainties (right). Bottom:
uncorrected bin-by-bin multiplicities of $K^{0}_{S}$ (left) and statistical
uncertainties (right).

The statistical uncertainties $\Delta n_{K^{0}_{S}}(y,p_{T})$ of the corrected
number of $K^{0}_{S}$ are:

$\Delta\frac{d^{2}n}{dydp_{T}}(y,p_{T})=\sqrt{\left(\frac{c_{dE/dx}\cdot
c_{MC}(y,p_{T})}{N_{events}\,\Delta y\,\Delta p_{T}}\right)^{2}\Delta
n_{K^{0}_{S}}^{2}(y,p_{T})+\left(\frac{c_{dE/dx}\cdot
n_{K^{0}_{S}}(y,p_{T})}{N_{events}\,\Delta y\,\Delta p_{T}}\right)^{2}\Delta
c_{MC}^{2}(y,p_{T})}\leavevmode\nobreak\ .$ (9)

###  3.8 Systematic uncertainties

Three possible sources of the systematic uncertainties related to event
selection criteria, the track and $V^{0}$ selection criteria and the signal
extraction procedure, are included.

The following effects were considered in the calculation of the systematic
uncertainties:

  * (i)

The uncertainties related to event selection criteria were estimated by
performing the analysis with the following changes:

    * –

Simulations were done with and without the S4 trigger condition. One half of
the difference between these two results was taken as a contribution to the
systematic uncertainty, which is 3-10%.

    * –

Vertex z position was changed from -590 < z (cm) < -572 to -588 < z (cm) <
-574. The uncertainty due to variation of the selection window was estimated
to be up to 2%.

  * (ii)

The uncertainties related to track and $V^{0}$ selection criteria were
estimated by performing the analysis with the following changes compared to
the original values:

    * –

the minimum required number of clusters in both VTPCs for $K^{0}_{S}$ decay
products was changed from 15 to 10 and 20 yielding a possible bias up to 4%,

    * –

the standard ${\text{d}}E\\!/\\!{\text{d}}x$ cut used for identification of
$K^{0}_{S}$ decay products was changed from $\pm 3\sigma$ to $\pm 2.5\sigma$
and $\pm 3.5\sigma$ from the nominal Bethe-Bloch value yielding a possible
bias up to 5%,

    * –

DCA cut for daughter tracks at the $V^{0}$ decay vertex was changed from 1 cm
to 0.5 cm and 1.5 cm yielding a possible bias up to 4%,

    * –

the impact parameter cut for the daughters tracks was varied by 50%:
$\left(\frac{b_{x}}{2}\right)^{2}+(b_{y})^{2}<0.125$ and
$\left(\frac{b_{x}}{2}\right)^{2}+(b_{y})^{2}<0.375$ yielding a possible bias
up to 2%,

    * –

the $\Delta$z cut was changed from |$\Delta$z| < $e^{3.1+0.42\cdot y}$ to
|$\Delta$z| < $e^{2.96+0.47\cdot y}$ and |$\Delta$z| < $e^{3.24+0.38\cdot y}$
yielding a possible bias up to 3%,

    * –

the $cos\Phi$ cut was varied with respect to the nominal values yielding a
possible bias up to 3%. The range of cut values is listed in Table 1.

  * (iii)

The uncertainty due to the signal extraction procedure was estimated by:

    * –

changing the background fit function from a $2^{nd}$ order to a $3^{rd}$ order
polynomial yielding a possible bias up to 4%,

    * –

changing the invariant mass range over which the uncorrected number of
$K^{0}_{S}$ was integrated from $m_{0}\pm 3\Gamma$ to $\pm 2.5\Gamma$ and $\pm
3.5\Gamma$ yielding a possible bias up to 2%,

    * –

calculating the uncorrected number of $K^{0}_{S}$ as the sum of entries after
background fit subtraction instead of the integral of the Lorentzian signal
function yielding a possible bias up to 4%,

    * –

changing the region of the fit from [0.35-0.7]
$\mbox{Ge\kern-1.00006ptV}\\!/\\!c^{2}$ to [0.4-0.65]
$\mbox{Ge\kern-1.00006ptV}\\!/\\!c^{2}$ yielding a possible bias up to 3%.

Maximal $|cos\Phi|$ allowed
---
$y_{min}$ | $y_{max}$ | original | new lower | new upper
-0.25 | 0.25 | 0.95 | 0.925 | 0.975
0.25 | 0.75 | 0.9 | 0.85 | 0.95
0.75 | 1.25 | 0.8 | 0.75 | 0.85
1.25 | 1.75 | 0.5 | 0.4 | 0.6
Table 1: Numerical values for $cos\Phi$ cut used for systematic uncertainties
calculation.

The maximum deviations are determined for every group of possible sources,
which contribute to the systematic uncertainty. The systematic uncertainty was
calculated as the square root of the sum of squares of the described possible
biases assuming that they are uncorrelated. This procedure was used to
estimate systematic uncertainties of all final quantities presented in this
paper - yield in each ($y,p_{T}$) bin, inverse slope parameter of transverse
momentum spectrum, yield in each rapidity bin and mean multiplicity.

###  3.9 Mean lifetime measurements

The reliability of the $K^{0}_{S}$ reconstruction and of the correction
procedure can be validated by studying the lifetime distribution of the
analysed $K^{0}_{S}$. The lifetime of each $K^{0}_{S}$ candidate was
calculated from the $V^{0}$ path length and its velocity. The lifetime
distributions corrected for experimental biases (Sec. 3.6) in all 7 rapidity
bins were fitted by an exponential distribution to obtain proper lifetimes
(see Fig. 7). The obtained ratio of the measured mean lifetime to the PDG [23]
value $c\tau_{PDG}=2.6844$ cm is shown in Fig. 8 as a function of rapidity.
The measured $K^{0}_{S}$ lifetime agrees within uncertainties with the PDG
value and thus confirms the quality of the analysis.

Figure 7: (Color online) Corrected lifetime distribution for $K^{0}_{S}$
mesons produced in inelastic p+p interaction at 158 . The curves show the
result of the exponential fit function used to obtain the mean lifetime.
Statistical uncertainties are smaller than marker size and not visible on the
plot.  Figure 8: (Color online) Mean lifetime obtained from fits to the
lifetime distributions of Fig. 7 with statistical (vertical bars) and
systematic (shaded boxes) uncertainties versus the rapidity $y$.

##  4 Results

This section presents the new results on inclusive $K^{0}_{S}$ meson spectra
in inelastic p+p interactions at beam momentum 158 . The spectra refer to
weakly decaying $K^{0}_{S}$ mesons produced in strong interaction processes
and are corrected for experimental biases and the branching ratio.

###  4.1 Transverse momentum spectra

Double differential $K^{0}_{S}$ yields listed in Table 2 represent the main
result of this paper. Yields are determined in 8 consecutive rapidity bins in
the interval $-1.75<y<2.25$ and 6 transverse momentum bins in the interval
$0.0<p_{T}\leavevmode\nobreak\ (\GeVc)<1.8$. The transverse momentum
distributions are shown in Fig. 9.

Figure 9: (Color online) Double-differential $K^{0}_{S}$ spectra in inelastic p+p interaction at 158 in bins of ($y,p_{T}$) as obtained from Eq. 4. Measured points are shown as red full circles. The solid red curve is obtained from a fit to the data points using the exponential function Eq. 10. Statistical uncertainties are indicated by vertical bars (for some points smaller than the symbol size). Red shaded boxes show systematic uncertainties. Only statistical uncertainties are used in the fits as they are uncorrelated bin-to-bin. The numerical values of the data points are listed in Table 2.  | $y$
---|---
$p_{T}$ () | (-1.75;-1.25) | (-1.25;-0.75) | (-0.75;-0.25) | (-0.25;0.25)
(0.0;0.3) |  30.1 $\pm$ 3.5 $\pm$ 5.6 |  43 $\pm$ 1 $\pm$ 3 |  50.3 $\pm$ 0.5 $\pm$ 1.9 |  55.7 $\pm$ 0.3 $\pm$ 2.2
(0.3;0.6) |  33 $\pm$ 2 $\pm$ 2 |  55.4 $\pm$ 0.8 $\pm$ 2.1 |  70.3 $\pm$ 0.6 $\pm$ 2.7 |  75.2 $\pm$ 0.4 $\pm$ 3.1
(0.6;0.9) |  13.9 $\pm$ 0.8 $\pm$ 0.8 |  24.6 $\pm$ 0.4 $\pm$ 1.1 |  32.1 $\pm$ 0.3 $\pm$ 1.5 |  35.6 $\pm$ 0.3 $\pm$ 1.5
(0.9;1.2) |  3.9 $\pm$ 0.3 $\pm$ 0.4 |  7.4 $\pm$ 0.2 $\pm$ 0.5 |  10.8 $\pm$ 0.2 $\pm$ 0.5 |  12.2 $\pm$ 0.2 $\pm$ 0.5
(1.2;1.5) |  0.8 $\pm$ 0.1 $\pm$ 0.2 |  2.07 $\pm$ 0.08 $\pm$ 0.15 |  3.32 $\pm$ 0.09 $\pm$ 0.15 |  3.7 $\pm$ 0.1 $\pm$ 0.2
(1.5;1.8) |  0.22 $\pm$ 0.05 $\pm$ 0.07 |  0.52 $\pm$ 0.04 $\pm$ 0.04 |  0.89 $\pm$ 0.04 $\pm$ 0.04 |  1.15 $\pm$ 0.05 $\pm$ 0.05
| $y$
$p_{T}$ () | (0.25;0.75) | (0.75;1.25) | (1.25;1.75) | (1.75;2.25)
(0.0;0.3) |  54.4 $\pm$ 0.3 $\pm$ 2.3 |  45.6 $\pm$ 0.3 $\pm$ 2.1 |  32.0 $\pm$ 0.3 $\pm$ 2.0 |  17.2 $\pm$ 0.3 $\pm$ 1.5
(0.3;0.6) |  71.4 $\pm$ 0.4 $\pm$ 3.0 |  57.9 $\pm$ 0.3 $\pm$ 2.5 |  39.9 $\pm$ 0.3 $\pm$ 1.7 |  20.1 $\pm$ 0.4 $\pm$ 1.0
(0.6;0.9) |  32.6 $\pm$ 0.3 $\pm$ 1.5 |  24.2 $\pm$ 0.2 $\pm$ 1.1 |  15.7 $\pm$ 0.3 $\pm$ 0.8 |  6.3 $\pm$ 0.4 $\pm$ 0.5
(0.9;1.2) |  10.7 $\pm$ 0.2 $\pm$ 0.5 |  7.0 $\pm$ 0.2 $\pm$ 0.4 |  3.9 $\pm$ 0.2 $\pm$ 0.4 |  1.2 $\pm$ 0.3 $\pm$ 0.5
(1.2;1.5) |  3.34 $\pm$ 0.10 $\pm$ 0.19 |  2.14 $\pm$ 0.09 $\pm$ 0.14 |  0.86 $\pm$ 0.13 $\pm$ 0.15 |
(1.5;1.8) |  0.91 $\pm$ 0.05 $\pm$ 0.09 |  0.51 $\pm$ 0.05 $\pm$ 0.07 |  0.34 $\pm$ 0.09 $\pm$ 0.10 |
Table 2: Double differential $K^{0}_{S}$ yields in bins of ($y,p_{T}$). The
first uncertainty is statistical, while the second one is systematic.

The transverse momentum spectra can be described by an exponential function:

$f(p_{T})=A\cdot
p_{T}\cdot\exp\left(\frac{\sqrt{p_{T}^{2}+m_{0}^{2}}}{T}\right)\leavevmode\nobreak\
,$ (10)

where $m_{0}$ is the mass of the $K^{0}_{S}$ and $T$ is the inverse slope
parameter. Fits of Eq. 10 to the data points provide the values of $T$ in each
rapidity bin which are listed in Table 3 and in the legend of the panels in
Fig. 9.

###  4.2 Rapidity distribution and mean multiplicity

Kaon yields in each rapidity bin were obtained from the corresponding measured
transverse momentum distributions. The small fraction of $K^{0}_{S}$ at high
$p_{\text{T}}$ outside of the acceptance was determined using Eq. 10. The
resulting $\frac{dn}{dy}$ spectrum of $K^{0}_{S}$ mesons produced in inelastic
p+p interactions at 158 is plotted in Fig. 10.

Figure 10: (Color online) Rapidity distribution $dn/dy$ obtained by $p_{\text{T}}$-integration. Statistical uncertainties are shown by vertical bars (often smaller than the marker size), while a red shaded boxes indicates systematic uncertainties. The black dotted lines show the connection line between the last two points on both sides, and the grey areas are the contributions of the extrapolation to the mean multiplicity of $K^{0}_{S}$ mesons. The numerical data are listed in Table 3. $y$ | T (MeV) | $\frac{dn}{dy}\times 10^{-3}$ | extrapolated fraction (%)
---|---|---|---
(-1.75;-1.25) |  162.3 $\pm$ 3.1 $\pm$ 7.0 |  24.8 $\pm$ 1.2 $\pm$ 1.5 | 0.21
(-1.25;-0.75) |  174.6 $\pm$ 1.0 $\pm$ 2.3 |  40.0 $\pm$ 0.4 $\pm$ 1.6 | 0.35
(-0.75;-0.25) |  184.7 $\pm$ 0.7 $\pm$ 0.7 |  50.6 $\pm$ 0.3 $\pm$ 1.8 | 0.56
(-0.25;0.25) |  186.5 $\pm$ 0.6 $\pm$ 0.8 |  55.4 $\pm$ 0.2 $\pm$ 2.1 | 0.58
(0.25;0.75) |  181.5 $\pm$ 0.6 $\pm$ 0.8 |  52.3 $\pm$ 0.2 $\pm$ 2.1 | 0.47
(0.75;1.25) |  170.1 $\pm$ 0.6 $\pm$ 1.6 |  41.3 $\pm$ 0.2 $\pm$ 1.8 | 0.31
(1.25;1.75) |  163.1 $\pm$ 1.0 $\pm$ 2.0 |  27.9 $\pm$ 0.2 $\pm$ 1.3 | 0.23
(1.75;2.25) |  149.7 $\pm$ 2.5 $\pm$ 4.2 |  13.9 $\pm$ 0.2 $\pm$ 0.8 | 3.33
Table 3: First column shows the rapidity range. In the second column the
values of the inverse slope parameter are listed with its statistical and
systematic uncertainties. The third column shows the numerical values of the
$p_{\text{T}}$-integrated yields presented in Fig. 10 with statistical and
systematic uncertainties. In the last column contribution to the dn/dy in % of
the extrapolation to the unmeasured transverse momentum region.

The mean multiplicity of $K^{0}_{S}$ mesons was calculated as the sum of
measured points in Fig. 10 and the integral below linear functions through the
last two measured points on both sides representative for the unmeasured
region (for rapidity $y<-1.75$ and $y>2.25$). The statistical uncertainty of
$\langle K^{0}_{S}\rangle$ was calculated as the square root of the sum of the
squares of the statistical uncertainties of the contributing bins. The
systematic uncertainty was calculated as the square root of squares of
systematic uncertainty described in Sec. 3.8 and half of the extrapolated
yield. To estimate the systematic uncertainty of the method used to determine
the mean multiplicity of $K^{0}_{S}$, the rapidity distribution was also
fitted using a single Gaussian or two Gaussians symmetrically displaced from
midrapidity. The deviations of the results of these fit from $\langle
K^{0}_{S}\rangle$ is included as a contribution to the final systematic
uncertainty.

The numerical value of the total yield of $\langle K^{0}_{S}\rangle$ is:

$\langle K^{0}_{S}\rangle=0.162\pm 0.001(stat.)\pm 0.011(sys.)$ (11)

##  5 Comparison with published world data and model predictions

This section compares the new measurement of $K^{0}_{S}$ production in
inelastic p+p interactions at 158 with publicly available world data as well
as with predictions from microscopic and statistical models [26, 27], UrQMD
3.4 [29, 30], SMASH 2.0 [31] and PHSD [32, 33].

The $K^{0}_{S}$ rapidity spectrum from is compared in Fig. 11 to the results
from Brick $et$ $al.$ at 147 [11] as well as with predictions obtained from
$K^{+}$ and $K^{-}$ yields published by for inelastic p+p interactions at 158
[5]. These predictions are based on valence-quark counting arguments [34] and
lead to the formula $\frac{1}{4}(N_{K^{+}}+3\cdot N_{K^{-}})$. Such a model
was applied earlier for p+C [35] and for p+Be [36] interactions. The measured
$K^{0}_{S}$ yields are seen to agree with this prediction and with the
measurement of Ref. [11] within statistical errors.

Figure 11: (Color online) Rapidity distribution $dn/dy$ of $K^{0}_{S}$ mesons
in inelastic p+p interactions at 158 . Solid red-coloured circles correspond
to the results (systematic uncertainties not shown on the plot), open circles
are mirrored values, black squares represent results from Brick $et$ $al.$ at
FNAL [11] and blue full diamonds show results obtained from the formula
$\frac{1}{4}(N_{K^{+}}+3\cdot N_{K^{-}})$ using charged kaon yields recently
measured by at the same beam momentum [5].

Figure 12 presents a comparison of the measurements with predictions of the ,
PHSD, SMASH 2.0 and UrQMD 3.4 models. Only describes the experimental data
fairly well. All other models overpredict the $K^{0}_{S}$ yield by $10-20\%$.
The shape of the rapidity distribution is also reproduced by the PHSD model.

Figure 12: (Color online) Comparison of the results on the $K^{0}_{S}$
rapidity distribution with predictions of theoretical models. Coloured red
circles show the new measurements of . The black curves show predictions of
models: (solid), PHSD (long dashed), SMASH 2.0 (short dashed) and UrQMD 3.4
(dotted).

The mean multiplicity of $K^{0}_{S}$ mesons in p+p collisions measured by at
$\sqrt{s_{NN}}=17.3$ is compared in Fig. 13 with the world data in the range
from 3 - 32 [37, 38, 39, 40, 41, 42, 10, 12, 13, 11, 14, 15, 16, 17, 43, 44].
The measured values are seen to rise linearly with collision energy
$\sqrt{s_{NN}}$.

Figure 13: (Color online) Collision energy dependence of mean multiplicity of
$K^{0}_{S}$ mesons produced in p+p interactions. The solid coloured red circle
shows the measurement by presented with its systematic uncertainty. The
results published by other experiments are shown by open diamonds coloured in
blue (CERN), open stars coloured in green (FNAL), open triangle pointing up
coloured in orange (BNL), open triangle pointing down coloured in magenta
(SERPUKHOV) and open square coloured in black (SLAC). All the data points are
shown with combined statistical and systematic uncertainties. The references
for the plotted data points are given in the text.

##  6 Summary

This paper presents the new measurement of $K^{0}_{S}$ meson production via
its $\pi^{+}\pi^{-}$ decay mode in inelastic p+p collisions at beam momentum
158 ($\sqrt{s_{NN}}=17.3$ ). Spectra of transverse momentum (up to 1.8 ), as
well as a distribution of rapidity (from -1.75 to 2.25), are presented. The
mean multiplicity, obtained from the $p_{\text{T}}$-integrated and
extrapolated rapidity distribution, is $(0.162\pm 0.001\pm 0.011)$, where the
first uncertainty is statistical and the second systematic. The rapidity
distribution is in agreement with results from other experiments at nearby
beam momenta. Mean multiplicity from model calculations deviate by up to
$20\%$ from the measurements. The model provides the best predictions for the
experimental data. The results of $K^{0}_{S}$ production in proton-proton
interactions presented in this paper significantly improve, with their high
statistical precision, the knowledge of strangeness production in elementary
interactions.

## Acknowledgements

We would like to thank the CERN EP, BE, HSE and EN Departments for the strong
support of NA61/SHINE.

This work was supported by the Hungarian Scientific Research Fund (grant NKFIH
123842/123959), the Polish Ministry of Science and Higher Education (grants
667/N-CERN/2010/0, NN 202 48 4339 and NN 202 23 1837), the National Science
Centre Poland (grants 2014/14/E/ST2/00018, 2014/15/B/ST2/ 02537 and
2015/18/M/ST2/00125, 2015/19/N/ST2/01689, 2016/23/B/ST2/00692, DIR/WK/
2016/2017/ 10-1, 2017/ 25/N/ ST2/ 02575, 2018/30/A/ST2/00226,
2018/31/G/ST2/03910, 2019/34/H/ST2/00585, 2016/21/D/ST2/01983), WUT ID-UB, the
Russian Science Foundation, grant 16-12-10176 and 17-72-20045, the Russian
Academy of Science and the Russian Foundation for Basic Research (grants
08-02-00018, 09-02-00664 and 12-02-91503-CERN), the Russian Foundation for
Basic Research (RFBR) funding within the research project no. 18-02-40086, the
Ministry of Science and Higher Education of the Russian Federation, Project
"Fundamental properties of elementary particles and cosmology" No
0723-2020-0041, the European Union’s Horizon 2020 research and innovation
programme under grant agreement No. 871072, the Ministry of Education,
Culture, Sports, Science and Technology, Japan, Grant-in-Aid for Scientific
Research (grants 18071005, 19034011, 19740162, 20740160 and 20039012), the
German Research Foundation DFG (grants GA 1480/8-1 and project 426579465), the
Bulgarian Ministry of Education and Science within the National Roadmap for
Research Infrastructures 2020-2027, contract No. D01-374/18.12.2020, Ministry
of Education and Science of the Republic of Serbia (grant OI171002), Swiss
Nationalfonds Foundation (grant 200020117913/1), ETH Research Grant TH-01 07-3
and the Fermi National Accelerator Laboratory (Fermilab), a U.S. Department of
Energy, Office of Science, HEP User Facility managed by Fermi Research
Alliance, LLC (FRA), acting under Contract No. DE-AC02-07CH11359 and the
IN2P3-CNRS (France).

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The Collaboration

A. Acharya 10, H. Adhikary 10, K.K. Allison 26, N. Amin 5, E.V. Andronov 22,
T. Antićić 3, V. Babkin 20, M. Baszczyk 14, S. Bhosale 11, A. Blondel 4, M.
Bogomilov 2, Y. Bondar 10, A. Brandin 21, A. Bravar 24, W. Bryliński 18, J.
Brzychczyk 13, M. Buryakov 20, O. Busygina 19, A. Bzdak 14, H. Cherif 6, M.
Ćirković 23, M. Csanad  7,8, J. Cybowska 18, T. Czopowicz 10,18, A. Damyanova
24, N. Davis 11, M. Deliyergiyev 10, M. Deveaux 6, A. Dmitriev  20, W. Dominik
16, P. Dorosz 14, J. Dumarchez 4, R. Engel 5, G.A. Feofilov 22, L. Fields 25,
Z. Fodor 7,17, A. Garibov 1, M. Gaździcki 6,10, O. Golosov 21, V. Golovatyuk
20, M. Golubeva 19, K. Grebieszkow 18, F. Guber 19, A. Haesler 24, S.N.
Igolkin 22, S. Ilieva 2, A. Ivashkin 19, S.R. Johnson 26, K. Kadija 3, N.
Kargin 21, E. Kashirin 21, M. Kiełbowicz 11, V.A. Kireyeu 20, V. Klochkov 6,
V.I. Kolesnikov 20, D. Kolev 2, A. Korzenev 24, V.N. Kovalenko 22, S. Kowalski
15, M. Koziel 6, B. Kozłowski 18, A. Krasnoperov 20, W. Kucewicz 14, M. Kuich
16, A. Kurepin 19, D. Larsen 13, A. László 7, T.V. Lazareva 22, M. Lewicki 17,
K. Łojek 13, V.V. Lyubushkin 20, M. Maćkowiak-Pawłowska 18, Z. Majka 13, B.
Maksiak 12, A.I. Malakhov 20, A. Marcinek 11, A.D. Marino 26, K. Marton 7,
H.-J. Mathes 5, T. Matulewicz 16, V. Matveev 20, G.L. Melkumov 20, A.O.
Merzlaya 13, B. Messerly 27, Ł. Mik 14, S. Morozov 19,21, Y. Nagai  8, M.
Naskręt 17, V. Ozvenchuk 11, O. Panova 10, V. Paolone 27, O. Petukhov 19, I.
Pidhurskyi 6, R. Płaneta 13, P. Podlaski 16, B.A. Popov 20,4, B. Porfy 7,8, M.
Posiadała-Zezula 16, D.S. Prokhorova 22, D. Pszczel 12, S. Puławski 15, J.
Puzović 23, M. Ravonel 24, R. Renfordt 6, D. Röhrich 9, E. Rondio 12, B.T.
Rumberger 26, M. Rumyantsev 20, A. Rustamov 1,6, M. Rybczynski 10, A. Rybicki
11, S. Sadhu 10, A. Sadovsky 19, K. Schmidt 15, I. Selyuzhenkov 21, A.Yu.
Seryakov 22, P. Seyboth 10, M. Słodkowski 18, P. Staszel 13, G. Stefanek 10,
J. Stepaniak 12, M. Strikhanov 21, H. Ströbele 6, T. Šuša 3, A. Taranenko 21,
A. Tefelska 18, D. Tefelski 18, V. Tereshchenko 20, A. Toia 6, R. Tsenov 2, L.
Turko 17, M. Unger 5, D. Uzhva 22, F.F. Valiev 22, D. Veberič 5, V.V.
Vechernin 22, A. Wickremasinghe 27,25, K. Wójcik 15, O. Wyszyński 10, A.
Zaitsev 20, E.D. Zimmerman 26, and R. Zwaska 25

1 National Nuclear Research Center, Baku, Azerbaijan
2 Faculty of Physics, University of Sofia, Sofia, Bulgaria
3 Ruđer Bošković Institute, Zagreb, Croatia
4 LPNHE, University of Paris VI and VII, Paris, France
5 Karlsruhe Institute of Technology, Karlsruhe, Germany
6 University of Frankfurt, Frankfurt, Germany
7 Wigner Research Centre for Physics of the Hungarian Academy of Sciences,
Budapest, Hungary
8 ELTE Institute of Physics, Eötvös Loránd University, Budapest, Hungary
9 University of Bergen, Bergen, Norway
10 Jan Kochanowski University in Kielce, Poland
11 Institute of Nuclear Physics, Polish Academy of Sciences, Cracow, Poland
12 National Centre for Nuclear Research, Warsaw, Poland
13 Jagiellonian University, Cracow, Poland
14 AGH - University of Science and Technology, Cracow, Poland
15 University of Silesia, Katowice, Poland
16 University of Warsaw, Warsaw, Poland
17 University of Wrocław, Wrocław, Poland
18 Warsaw University of Technology, Warsaw, Poland
19 Institute for Nuclear Research, Moscow, Russia
20 Joint Institute for Nuclear Research, Dubna, Russia
21 National Research Nuclear University (Moscow Engineering Physics
Institute), Moscow, Russia
22 St. Petersburg State University, St. Petersburg, Russia
23 University of Belgrade, Belgrade, Serbia
24 University of Geneva, Geneva, Switzerland
25 Fermilab, Batavia, USA
26 University of Colorado, Boulder, USA
27 University of Pittsburgh, Pittsburgh, USA