Excess of charged over neutral K meson production
in high-energy collisions of atomic nuclei CERN-EP-2023-283 Collisions of
atomic nuclei at relativistic velocities produce new particles, predominantly
mesons containing one valence quark and one valence anti-quark. These
particles are produced in strong interactions, which preserve an approximate
symmetry between up ($u$) and down ($d$) quarks. In the case of $K$ meson
production, if this symmetry were exact, it would result in equal numbers of
charged ($K^{+}$ and $K^{-}$) and neutral ($K^{0}$ and $\overline{K}^{0}$)
mesons in the final state. In this Letter, we report a measurement of the
relative abundance of charged over neutral $K$ meson production in collisions
of argon and scandium nuclei at a center-of-mass energy of 11.9 GeV per
nucleon pair. We find that production of $\mathit{K^{+}}$ and $\mathit{K^{-}}$
mesons at mid-rapidity displays a significant excess of $(23.3\pm 5.7)\%$
relative to that of the neutral $K$ mesons. The origin of this unexpected
excess remains to be elucidated. Keywords: quarks, strong interaction, isospin
symmetry, nucleus-nucleus collisions, strange meson production

One of the main aims of basic research is the understanding of fundamental
constituents and properties of matter, particularly the underlying fundamental
interactions. Collisions of highly energetic atomic nuclei studied at large
circular accelerators serve the specific purpose of understanding the short-
range strong interaction occurring between the quarks, which build up the
participating protons and neutrons.

At collision energies of several GeV (center-of-mass energy per nucleon pair)
or more, it is generally assumed that the colliding nuclei form, on time
scales of the order of $10^{-23}$ seconds, an intermediate system of
deconfined quarks, anti-quarks and gluons characteristic of the early universe
about $10^{-5}$ seconds after the Big Bang. Cooling down, this quark-gluon
plasma ultimately “freezes out” into observable final state particles [1].
Studying these particle production phenomena is one of the main purposes of
high-energy experiments located at CERN (Super Proton Synchrotron, Large
Hadron Collider) and BNL (Relativistic Heavy Ion Collider).

These typically very abundant newly produced particles (hundreds or thousands
per single nucleus-nucleus collision) are predominantly mesons containing one
valence quark ($\mathit{q}$) and one valence anti-quark
($\mathit{\overline{q}}$). The most copiously produced are the up
($\mathit{u}$) and down ($\mathit{d}$) quarks, which also constitute protons
and neutrons, but a large number of strange quarks ($\mathit{s}$), belonging
to the second quark family of the standard model of particle physics is
produced as well. The same is true for the corresponding anti-quarks:
$\mathit{\overline{u}}$, $\mathit{\overline{d}}$, and $\mathit{\overline{s}}$.

The two colliding nuclei are made of protons and neutrons. These contain,
respectively, ($\mathit{uud}$) and ($\mathit{udd}$) valence quarks. While
initial-state valence quarks can also contribute to the production of new
mesons, the dominant contribution to this process comes from creating new
quark-anti-quark pairs. Generally, strong interactions are independent of
quark type (flavor) in the limit of massless quarks. As such, exchanging the
light $u$ and $d$ quarks (or ${\overline{u}}$ and ${\overline{d}}$ anti-
quarks) should not significantly affect the result of the strong interaction.
This feature is known as isospin symmetry. However, this symmetry does not
apply to $s$ quarks. Their production remains suppressed due to their
significantly higher mass (the corresponding quark masses are
$m_{\mathit{u}}=2.16^{+0.49}_{-0.26}$ MeV,
$m_{\mathit{d}}=4.67^{+0.48}_{-0.17}$ MeV, and
$m_{\mathit{s}}=93.4^{+8.6}_{-3.4}$ MeV [2]). The same is valid for the heavy
charm ($c$), beauty ($b$), and top ($t$) quarks, which all remain very
strongly suppressed in nuclear collisions.111Units in this Letter follow the
Particle Data Group (PDG) [2] convention: masses and energies are expressed in
MeV (or GeV), whereas momenta in MeV/$c$ (or GeV/$c$). The relative
differences are given as the ratio of the difference to the mean.

Let us first consider the case when the two nuclei are made of an equal number
of protons and neutrons (and therefore, also of an equal number of valence $u$
and $d$ quarks). Then, isospin symmetry implies that such collisions will
result in closely equal abundances of $\mathit{u}$ and $\mathit{d}$ quarks and
similarly, closely equal abundances of $\mathit{\overline{u}}$ and
$\mathit{\overline{d}}$ anti-quarks.222With rare exceptions, nuclei with an
equal number of protons and neutrons are in the ground state singlets of
isospin [3]. A discussion of phenomenological aspects of the (approximate)
isospin symmetry in particle production can be found in Ref. [4].  This brings
implications for particle production. It is commonly expected [5, 6, 7, 8, 9]
that collisions of such “isospin-symmetric” nuclei will result in closely
similar abundances of mesons of any particular type, for mesons containing
$\mathit{u}$ quarks and those containing $\mathit{d}$ quarks. The same is
expected for mesons containing $\mathit{\overline{u}}$ and
$\mathit{\overline{d}}$ anti-quarks. A good example for this rule should be
$\mathit{K}$ mesons. With their masses differing by less than 1% ($m_{K^{+}}$
= $m_{K^{-}}$ = 493.677 $\pm$ 0.016 MeV, $m_{K^{0}}$ = $m_{\bar{K}^{0}}$=
497.611 $\pm$ 0.013 MeV [2]), similar yields would be anticipated for the
$\mathit{K^{+}}$ ($\mathit{u\overline{s}}$) and $\mathit{K}^{\,0}$
($\mathit{d\overline{s}}$) states. Also, similar abundances would be expected
for $\mathit{K^{-}}$ ($\mathit{s\overline{u}}$) and
$\mathit{\overline{K}}^{\,0}$ ($\mathit{s\overline{d}}$).

In the realistic case of nucleus-nucleus reactions studied experimentally, the
two colliding nuclei are most often made of non-equal numbers of protons and
neutrons. However, even for heavy nuclei with a significant excess of neutrons
(_e.g._ Pb) the number of valence $\mathit{u}$ and $\mathit{d}$ quarks in the
initial state of the reaction differs only by 14%. As a consequence, it is
expected that in general, all types of high-energy nucleus-nucleus collisions
will bring similar abundances of $\mathit{u}$ and $\mathit{d}$ quarks, and
also similar abundances of $\mathit{\overline{u}}$ and $\mathit{\overline{d}}$
anti-quarks in their final state. Correspondingly, one expects similar
abundances for the produced mesons, and in particular, an approximate equality
of charged and neutral $K$ meson yields
($\mathit{K^{+}}\approx\mathit{K}^{\,0}$,
$\mathit{K^{-}}\approx\mathit{\overline{K}}^{\,0}$) is generally assumed.
These expectations are supported by quantitative predictions of statistical
models, which have been, up to now, very successful in the description of
experimental data [10, 11].

In this Letter, we report a measurement of the relative abundance of charged
($\mathit{K^{+}}$, $\mathit{K^{-}}$) over neutral ($\mathit{K}^{\,0}_{S}$)
meson production in the 10% most central collisions of argon ($\mathrm{Ar}$)
and scandium ($\mathrm{Sc}$) nuclei, at center-of-mass energy per nucleon pair
equal to $\sqrt{s_{\mathrm{NN}}}=11.9$ GeV. The Ar+Sc colliding system
comprises 39 protons and 46 neutrons, which means that the corresponding
number of $u$ and $d$ valence quarks differs by 5.5%. Notwithstanding,
production of $\mathit{K^{+}}$ and $\mathit{K^{-}}$ mesons displays an
unexpected excess to that of the $\mathit{K}^{\,0}_{S}$ meson. The
$\mathit{K}^{\,0}_{S}$ meson is a superposition of the two neutral states
$\mathit{K}^{\,0}$ ($\mathit{d\overline{s}}$) and
$\mathit{\overline{K}}^{\,0}$ ($\mathit{s\overline{d}}$), and its yield in the
final state is equal to half of the summed production of these two states from
the strong interaction.333The $\mathit{K}^{\,0}$ and
$\mathit{\overline{K}}^{\,0}$ states are produced in strong interactions, but
they decay through weak interactions. Consequently, in the final state, one
observes linear combinations of the latter known as $\mathit{K}^{\,0}$ “short”
($\mathit{K}^{\,0}_{S}$) and $\mathit{K}^{\,0}$ “long”
($\mathit{K}^{\,0}_{L}$), where “short” and “long” refer to their weak decay
lifetime. For more details, see Ref. [2]. The corresponding excess is
quantified in terms of the ratio of yields of the produced $\mathit{K}$
mesons:

$\mathit{R_{K}}=\frac{\mathit{K}^{+}+\mathit{K}^{-}}{\mathit{K}^{\,0}+\mathit{{\overline{K}}}\leavevmode\nobreak\
^{0}}=\frac{\mathit{K}^{+}+\mathit{K}^{-}}{2\,\mathit{K}^{\leavevmode\nobreak\
\,0}_{S}}\leavevmode\nobreak\ .$ (1)

This ratio remains above unity in the entire kinematic range available for
measurement. Typically, the measured values exceed 1.2. Following our
estimates, the observed surplus of production of $\mathit{K}$ mesons
containing $\mathit{u}$ ($\mathit{\overline{u}}$) over these containing
$\mathit{d}$ ($\mathit{\overline{d}}$) quarks and anti-quarks corresponds to
about four additional charged $\mathit{K}$ particles per central Ar+Sc event.

Figure 1: Comparison of rapidity spectrum of neutral (_K_ ${}^{\mathbf{\,0}}_{S}$) with the average spectrum of charged (_K_ + and _K_ -) mesons. Total uncertainties, calculated as the square root of the sum of squared statistical and systematic uncertainties ($\sqrt{\text{stat}^{2}+\text{sys}^{2}}$) are drawn. For charged $K$ mesons, the total uncertainties were calculated separately for positively and negatively charged, and then propagated. |  |  | statistical | systematic |  total |
---|---|---|---|---|---|---
| $\left(\frac{\mathrm{d}n}{\mathrm{d}y}\right)_{y\hskip 0.85358pt\approx\hskip 0.85358pt0}\mathit{(K^{+})}$ | 3.968 |  $\pm$ 0.041 |  $\pm$ 0.078 |  $\pm$ 0.088 |
| $\left(\frac{\mathrm{d}n}{\mathrm{d}y}\right)_{y\hskip 0.85358pt\approx\hskip 0.85358pt0}\mathit{(K^{-})}$ | 2.034 |  $\pm$ 0.016 |  $\pm$ 0.021 |  $\pm$ 0.026 |
| $\left(\frac{\mathrm{d}n}{\mathrm{d}y}\right)_{y\hskip 0.85358pt\approx\hskip 0.85358pt0}\mathit{(K}^{\,0}_{S}{\mathit{)}}$ | 2.433 |  $\pm$ 0.027 |  $\pm$ 0.102 |  $\pm$ 0.106 |
| charged-to-neutral $\mathit{K}$ meson ratio: |  |  |  |  |
| ${R_{K}}=$ | 1.233 |  $\pm$ 0.016 |  $\pm$ 0.054 |  $\pm$ 0.057 |
Table 1: Densities of charged and neutral _K_ mesons produced at mid-rapidity.
The measurement was performed in the 10% most central Ar+Sc collisions at
$\sqrt{s_{\mathrm{NN}}}=11.9$ GeV, as described in Methods. The excess of
charged over neutral mesons is quantified by the ratio ${R_{K}}$ defined in
Eq. (1).

Production of _K_ mesons in central Ar+Sc collisions at the CERN SPS. The new
experimental results presented in this Letter have been obtained by the
NA61/SHINE fixed-target experiment at the CERN Super Proton Synchrotron [12].
The measurements of $\mathit{K^{+}}$ and $\mathit{K^{-}}$ production in the
10% most central Ar+Sc reactions at $\sqrt{s_{\mathrm{NN}}}=11.9$ GeV have
been published elsewhere [13]. This Letter presents the first measurement of
$\mathit{K}^{\,0}_{S}$ production in nucleus-nucleus collisions from
NA61/SHINE. Earlier data from this experiment, obtained for
$\mathit{K}^{\,0}_{S}$ mesons in $p$+C, $\pi^{-}$+C, $\pi^{+}$+C, $\pi^{+}$+Be
and $p$+$p$ collisions can be found in Refs. [14, 15, 16, 17, 18].

The comparison of the rapidity distribution of $\mathit{K}^{\,0}_{S}$ mesons
to the average of rapidity distributions for $\mathit{K^{+}}$ and
$\mathit{K^{-}}$ mesons is presented in Fig. 1. The rapidity $y$ is a
relativistic generalization of the particle’s longitudinal velocity $v_{L}$,
along the direction of the incoming nuclei: where $c$ is the speed of light.
In this Letter, we define rapidity in the collision center-of-mass system and
positive $y$ corresponds to the direction of the Ar nucleus.

In the entire range of rapidity covered by the measurement, the averaged
charged $\mathit{K}$ mesons prevail significantly over the neutral
$\mathit{K}^{\,0}_{S}$ mesons. To quantify this effect, Table 1 presents the
densities of $\mathit{K^{+}}$, $\mathit{K^{-}}$ and $\mathit{K}^{\,0}_{S}$
production measured at mid-rapidity ($y\approx 0$). Here, the relative excess
of charged mesons is $\mathrm{(23.3\pm 5.7)}$%. Integration of the two
distributions in Fig. 1 over positive rapidity, $y>0$, gives $4.33\pm 0.08$
and $3.22\pm 0.37$ for the production rates per collision of
$\mathit{\frac{K^{+}+K^{-}}{2}}$ and $\mathit{K}^{\,0}_{S}$, respectively
(total uncertainties are given). The resulting difference of $1.11\pm 0.38$
indicates that the surplus of charged ($\mathit{K^{+}}$ and $\mathit{K^{-}}$)
over neutral ($\mathit{K}^{\,0}$ and $\mathit{\overline{K}}^{\,0}$) states is
$2.22\pm 0.76$ at positive rapidity. Under the assumption that the charged-to-
neutral ratio would be similar also at negative rapidity, the total excess
would amount to 4.4$\pm$1.5 additional $K^{+}$ or $K^{-}$ mesons per one
central Ar+Sc collision.

A comparison of distributions of $\mathit{K}^{\,0}_{S}$ with averaged
$\mathit{K^{+}}$ and $\mathit{K^{-}}$ mesons as a function of transverse
momentum $p_{T}$ (the momentum component perpendicular to the direction of the
incoming nuclei) is shown in Fig. 2. Both distributions are integrated over
the rapidity range $0<y<2$. The prevalence of charged over neutral
$\mathit{K}$ mesons is again evident. The insert in the figure shows the
$p_{T}$-dependence of the ratio ${R_{K}}$. The corresponding excess of
$\mathit{K}$ mesons containing $\mathit{u}$, $\mathit{\overline{u}}$ over
these containing $\mathit{d}$, $\mathit{\overline{d}}$ quarks and anti-quarks
remains in the range 6–36% over the considered range of $p_{T}$.

Figure 2: Comparison of transverse momentum spectrum of neutral (_K_
${}^{\mathbf{\,0}}_{S}$) with the average spectrum of charged (_K_ + and _K_
-) mesons. The bottom panel shows the ratio of the two distributions, as
defined in Eq. (1). The meaning of the total uncertainties drawn is the same
as in Fig. 1.

Figure 3: Ratio of charged to neutral _K_ meson yields in nucleus-nucleus
collisions as a function of collision energy. The measurement from NA61/SHINE
is shown as a red dot. The remaining data points come from the compilation
described in the text.

Comparison to previous experiments. Figure 3 shows the comparison between the
present measurement of the ratio ${R_{K}}$ at mid-rapidity, Table 1, and our
compilation based on existing experimental results on $K^{+}$, $K^{-}$,
and $K^{0}_{S}$ production from the CERES [19], STAR BES [20, 21], STAR [22,
23, 24, 25], ALICE [26, 27], NA35 [28, 29, 30], NA49 [31], HADES [32, 33, 34,
35] and FOPI [36, 37] collaborations. We note that the latter compilation
includes not only measurements at mid-rapidity, but also these of total
multiplicities. This may increase the overall spread between the data points.
We also note the sizeable uncertainties of the earlier measurements. These
probably explain the fact that the aforementioned charged-over-neutral anomaly
was never reported before as an experimental observation. Despite these
uncertainties, a consistent picture emerges in the energy range
$5<\sqrt{s_{\mathrm{NN}}}<200$ GeV. With the unique exception of the NA35
experiment, all the data points correspond to an excess of charged over
neutral $\mathit{K}$ mesons. This excess may fade out at much higher energies,
as suggested by ALICE data at $\sqrt{s_{\mathrm{NN}}}=2760$ GeV.

Discussion of the observed excess. It is to be remarked that given the
prevalence of neutrons ($\mathit{udd}$) over protons ($\mathit{uud}$) in the
two colliding nuclei, the contribution of valence quarks to $\mathit{K}$ meson
production would favour a small excess of $\mathit{d}$ over $\mathit{u}$
quarks. Consequently, it would bring the ratio $R_{K}$ below unity, which is
opposite to the observed result. Thus in this context, our observation remains
an unexpected effect. At the same time, it goes against the expectations based
on statistical models of particle production [10, 11].

A known effect of violation of equivalence of charged and neutral $\mathit{K}$
states is the $\mathit{\phi(1020)}$ resonance, which decays more often to
charged ($\mathit{K^{+}K^{-}}$) than neutral
($\mathit{K}^{\,0}_{S}\mathit{K}^{0}_{{L}}$) mesons. However, the contribution
of the $\mathit{\phi(1020)}$ to $K$ meson production at NA61/SHINE energies is
known from measurements performed in $p$+$p$ and Pb+Pb collisions [38, 39, 40,
41]. The interpolation to the Ar+Sc system shows that this contribution is
insufficient to explain the observed excess.

In summary, we have found an unexpected excess of charged over neutral
$\mathit{K}$ meson production in central Ar+Sc collisions at 11.9 GeV center-
of-mass energy per nucleon pair. The experimental data show prevalence in
production of $\mathit{K}$ mesons containing $\mathit{u}$,
$\mathit{\overline{u}}$ over these containing $\mathit{d}$,
$\mathit{\overline{d}}$ quarks and anti-quarks, in the entire kinematic range
available to the measurement. The measured excess corresponds to about two
additional $K^{+}$ or $K^{-}$ mesons produced at positive rapidity. Under
reasonable assumptions, this gives about four additional charged $\mathit{K}$
mesons per central Ar+Sc collision. At mid-rapidity, the relative excess of
charged mesons is $\mathrm{(23.3\pm 5.7)}$%. Although with large
uncertainties, earlier data from other experiments in the collision energy
range $5<\sqrt{s_{\mathrm{NN}}}<200$ GeV are consistent with the present
result.

The origin of this unexpected excess remains to be elucidated.

## Acknowledgments

We would like to thank the CERN EP, BE, HSE and EN Departments for the strong
support of NA61/SHINE. We also gratefully acknowledge discussions with Claudia
Ahdida, Wojciech Broniowski, Marco van Leeuwen, Francesco Giacosa, Krzysztof
Golec-Biernat and Stanisław Mrówczyński at various stages of preparation of
this Letter.

This work was supported by the Hungarian Scientific Research Fund (grant NKFIH
138136/137812/138152 and TKP2021-NKTA-64), the Polish Ministry of Science and
Higher Education (DIR/WK/2016/2017/10-1, WUT ID-UB), the National Science
Centre Poland (grants 2014/14/E/ST2/00018, 2016/21/D/ST2/01983,
2017/25/N/ST2/02575, 2018/29/N/ST2/02595, 2018/30/A/ST2/00226,
2018/31/G/ST2/03910, 2020/39/O/ST2/00277), the Norwegian Financial Mechanism
2014–2021 (grant 2019/34/H/ST2/00585), the Polish Minister of Education and
Science (contract No. 2021/WK/10), 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
20039012,22H04943), 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, Serbian Ministry of Science, Technological Development and
Innovation (grant OI171002), Swiss Nationalfonds Foundation (grant
200020117913/1), ETH Research Grant TH-01 07-3, National Science Foundation
grant PHY-2013228 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).

The data used in this Letter were collected before February 2022.

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Methods

Experimental setup. The SPS Heavy Ion and Neutrino Experiment (SHINE) is a
fixed-target detector operating at the CERN Super Proton Synchrotron (SPS). It
is a multi-purpose spectrometer optimised to study hadron production in
various collisions (hadron-proton, hadron-nucleus and nucleus-nucleus). The
detection setup in its configuration used for the measurements reported here
is described below. Its details and a description of the detector’s
performance can be found in Ref. [12].

The beamline is equipped with an array of beam detectors upstream and
downstream of the target, used to identify and measure the trajectory of the
beam particles and trigger the spectrometer data acquisition. The tracking
devices of the NA61/SHINE spectrometer are Time Projection Chambers (TPCs).
Two Vertex TPCs are placed inside a magnetic field. Two large-volume Main TPCs
measure the charged particle trajectories downstream of the 4.5 Tm magnetic
field. The latter provides the bending power for a precise determination of
particle momenta. The information about the energy losses
($\mathrm{d}E/\mathrm{d}x$) of the charged particles in the TPCs, together
with Time-of-Flight (ToF) measurements, allows for particle identification in
a wide momentum range. The most downstream detector on the beamline is the
Projectile Spectator Detector (PSD). It is used to measure the energy of the
“spectator” remnant of the projectile nucleus, which is closely related to the
collision centrality in nucleus-nucleus reactions.

Physics objects. This Letter compares the production of charged and neutral
$\mathit{K}$ mesons in Ar+Sc collisions at a center-of-mass energy per nucleon
pair of 11.9 GeV. The ${}^{40}_{18}\mathrm{Ar}$ beam had a momentum of
$75A\,\mbox{Ge\kern-1.00006ptV}\\!/\\!c$. The stationary target consisted of
six ${}^{45}_{21}\mathrm{Sc}$ plates, with a total thickness of 6 mm.

A detailed account on the extraction of charged ($\mathit{K^{+}}$,
$\mathit{K^{-}}$) yields can be found in Ref. [13]. Only the neutral
$K^{0}_{S}$ mesons are considered in the present analysis. They can be
detected via their weak decay into two charged pions
$(K^{0}_{S}\rightarrow\pi^{+}\pi^{-})$. The mean lifetime $(c\tau)$ for this
decay is 2.7 cm.

Analysis. Before the analysis of $K^{0}_{S}$ mesons, the recorded Ar+Sc
collision data undergo event and track selection procedures. Event selection
uses information from the beam detectors to ensure the quality of the measured
beam trajectory. It rejects events where more than one beam-target interaction
occurred during the trigger-time window. It also reduces the background from
off-target interactions based on information about the quality of the main
interaction vertex. Finally, it selects the 10% most central collisions using
the information from the PSD. This is realized by selection of the 10% lowest
energy deposits from the spectator remnant of the Ar nucleus. The total number
of recorded collisions (events) was $2.8\cdot 10^{6}$, from which $1.1\cdot
10^{6}$ (39%) remained after all cuts. For more details, especially the
centrality selection, see Ref. [13].

The next step is the reconstruction of the charged particle tracks in the
TPCs. Pattern recognition algorithms combine space points recorded in the TPCs
into tracks. Their curvature and the magnetic field are used to compute the
momenta of the corresponding particles. The minimum number of reconstructed
space points in the VTPCs must be more than 10, and the computed momenta must
be larger than 400 $\mbox{Me\kern-1.00006ptV}/c$ (in the laboratory frame).
The latter selection excludes a large fraction of low-momentum electrons from
the analysis. The known positions of the target and the most probable
intersection point of measured tracks defines the position of the primary
vertex.

_K_ ${}^{\mathbf{0}}_{S}$ reconstruction. Unlike the charged particles, the
neutral $\mathit{K}$ mesons do not leave a measurable track in the detectors.
They are measured by reconstructing their oppositely charged decay products
(daughter particles). The two-body decays of $K^{0}_{S}$ create characteristic
$V$-shaped particle pairs originating at the decay vertex. This topology is
called $V^{0}$. It is searched for with a dedicated $V^{0}$-finder algorithm
that looks for track pairs of particles with opposite charges. These track
pairs are extrapolated backwards until their mutual distance of closest
approach is reached. If this distance is smaller than a given limit value, the
track pair becomes a $V^{0}$ candidate with its origin at the decay vertex.

Two further cuts are placed on the track pairs. The first cut imposes a
minimum value on the angle between the direction of the line joining the
primary and decay vertices and the direction given by the vector sum of the
momenta of the decay daughters. The second condition requires a minimum
distance between the primary and decay vertex (a minimum length of the
decaying particle). The corresponding cut requirements depend on $K^{0}_{S}$
rapidity, and are listed in Extended Data Table 2. Starting with the
approximate decay point, a $V^{0}$-fitter program optimises the decay point
position and the momenta of the decay daughters. Assuming that the daughter
particles are pions, it is straightforward to reconstruct the invariant mass
of the decaying particle (the invariant mass is defined as
$m_{inv}=\sqrt{(\sum{E_{i}})^{2}-(\sum{\overrightarrow{p_{i}}})^{2}}$, where
$E_{i}$ are the energies of the decay products, $\overrightarrow{p_{i}}$ are
their momenta, and $c\equiv 1$ is assumed).

The invariant mass distribution of $V^{0}$ candidates is populated by
$K^{0}_{S}$ and $\Lambda$ decays, photon conversion in some detector material,
and spurious particle crossings. A $K^{0}_{S}$ signal will appear as a peak on
a slowly varying background. For a double-differential $K^{0}_{S}$ analysis,
the momentum space was divided into seven rapidity bins ranging from -1.5 to 2
and nine bins in transverse momentum from 0 to 2.7
$\mbox{Ge\kern-1.00006ptV}\\!/\\!c$. The raw number of $K^{0}_{S}$ in a given
kinematic bin is obtained from fits of appropriate signal and background
functions to the invariant mass distribution of the corresponding $V^{0}$
candidates. The fitted signal function is taken as a Lorentzian and the
background function is a third-order Chebychev polynomial. The integral of the
signal function divided by the bin width is equal to the raw (uncorrected)
number of the reconstructed $K^{0}_{S}$ in a given kinematic bin. Two typical
invariant mass distributions with signal and background fits are shown in
Extended Data Fig. 4.

Corrections. To correct the results for losses due to detection and data
processing inefficiencies, detailed Monte Carlo simulations were performed.
These simulations consisted of Ar+Sc collisions generated by the EPOS model
[42] and particles propagated in the NA61/SHINE detector using the GEANT
framework [43]. The charged particle tracks were reconstructed and analysed
using the same software as used for the experimental data. The branching ratio
of $K^{0}_{S}$ decays was considered in the GEANT framework. The final output
of the simulation consisted of reconstructed $K^{0}_{S}$ multiplicities. The
ratio of the number of $K^{0}_{S}$ simulated and reconstructed was used as a
correction factor in each $y$–$p_{T}$ bin.

Systematic uncertainties of the measured data points were estimated from the
comparison of the results of the entire analysis (including Monte Carlo
simulations and corrections) obtained with varying cut values. The reliability
of the $V^{0}$ reconstruction and $K^{0}_{S}$ fitting procedures can be
scrutinised by studying the $K^{0}_{S}$ lifetime. Extended Data Fig. 5 shows
the computed mean lifetime of $K^{0}_{S}$ in seven rapidity bins. Good
agreement with the average value provided by the PDG [2] is observed.

Transverse momentum distributions. The distributions shown in Extended Data
Fig. 6 represent the final results of the $K^{0}_{S}$ analysis. The
$K^{0}_{S}$ yields are shown as a function of transverse momentum in seven
bins of rapidity. The data points are fitted with the function:

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

in which $A$ is a normalisation factor, $T$ is the inverse slope parameter,
and $m_{0}$ is the $K^{0}_{S}$ mass taken from Ref. [2]. The formula assumes
$c\equiv 1$ for simplicity. The fit functions are plotted as red curves, and
the inverse slope parameters obtained from the fits are reported in the
figure’s legend.

The transverse momentum distributions of charged and neutral $K$ mesons drawn
in Fig. 2 (of the main text) are also fitted with the function defined by Eq.
(2). The bottom panel of the figure presents the ratio of the two fitted
curves, with its uncertainty band obtained by propagation of the total
uncertainties of the data points.

Rapidity distribution. The final $K^{0}_{S}$ yields in each bin of rapidity
were obtained as the integrals of the curves fitted to the respective
transverse momentum spectra, Eq. (2), including extrapolations to unmeasured
regions. Comparison to the alternative method of replacing integrals in the
measured regions by sums of data points brought only a negligible contribution
to the systematic uncertainty. As apparent in Extended Data Fig. 6,
extrapolations were needed only in the first and last rapidity bin. They
amount to 88% and 6.2%, respectively. The large extrapolation in the first bin
of rapidity results in the increase of the total uncertainty of the
corresponding data point shown in Fig. 1 (of the main text). In this figure,
the obtained rapidity distribution of the $K^{0}_{S}$ has been fitted with a
function consisting of two Gaussians with centers displaced by a value of
$\pm\Delta y$ with respect to $y=0$. These Gaussians have the same widths but
may have different amplitudes. The resulting small asymmetry of the fitted
rapidity distribution originates from a combined effect of the mass asymmetry
of the colliding target and projectile nuclei ($A_{\mathrm{target}}=45$ and
$A_{\mathrm{projectile}}=40$) and the selection of central collisions by the
energy measured in the kinematic region of the projectile spectator remnants.
The former favours backward and the latter forward rapidities. The yields of
charged $K$ mesons at mid-rapidity listed in Table 1 (of the main text) were
taken from Ref. [13]. They were determined in the interval $0.0<y<0.2$ as
discussed therein. The yield of neutral $K^{0}_{S}$ mesons at mid-rapidity was
determined at $y=0$ from the aforementioned fit. Its systematic uncertainty
was estimated the same way as for the data points (see above), and its
statistical uncertainty was obtained by propagation of the statistical errors
of the fit. Both statistical and systematic uncertainties of charged and
neutral $K$ yields were propagated into the ratio $R_{K}$. The additional
uncertainty of $R_{K}$ resulting from the difference in the mid-rapidity
definition for charged and neutral mesons was estimated to be 0.5%, which is
about 10% of the total systematic uncertainty.

Extended Data

rapidity bin | $(-1.5,-1)$ | $(-1,-0.5)$ | $(-0.5,0)$ | $(0,0.5)$ | $(0.5,1)$ | $(1,1.5)$ | $(1.5,2)$
---|---|---|---|---|---|---|---
cut value | cosine of angle |  $>$0.999 |  $>$0.9995 |  $>$0.9995 |  $>$0.9995 |  $>$0.9995 |  $>$0.9999 |  $>$0.9999
distance |  $>$5 cm |  $>$5 cm |  $>$7.5 cm |  $>$12.5 cm |  $>$12.5 cm |  $>$15 cm |  $>$12.5 cm
Table 2: Track pair cuts. Values of the cuts on (top row) the cosine of the
angle between the line joining the primary and decay vertex and the direction
of the vector sum of decay daughter momenta, and (bottom row) the distance
between the primary and decay vertex.

Figure 4: Examples of fitted invariant mass distributions. Two studied bins in
rapidity $y$ and transverse momentum $p_{T}$ of the $K^{0}_{S}$ are presented,
left: $y\in(-1.0,-0.5)$, $p_{T}\in(1.2,1.5)$
$\mbox{Ge\kern-0.90005ptV}\\!/\\!c$, right: $y\in(0.5,1.0)$,
$p_{T}\in(1.2,1.5)$ $\mbox{Ge\kern-0.90005ptV}\\!/\\!c$. The bottom panels
show the difference between the experimental data and the fitted
(Signal+Background) distribution, divided by the experimental uncertainty.
Figure 5: Mean lifetime of _K_ ${}^{\mathbf{0}}_{S}$ mesons as a function of
rapidity. The values obtained by NA61/SHINE are divided by the PDG value [2].
Statistical uncertainties are shown by vertical bars and systematic ones by
shaded boxes. Figure 6: _K_ ${}^{\mathbf{0}}_{S}$ transverse momentum spectra
in rapidity bins. Statistical uncertainties are shown by vertical bars and
systematic ones by shaded boxes. Red curves represent fits of the data with
the function defined in Eq. (2).

The NA61/SHINE Collaboration

H. Adhikary  13, P. Adrich  15, K.K. Allison  26, N. Amin  5, E.V. Andronov
22, I.-C. Arsene  12, M. Bajda  16, Y. Balkova  18, D. Battaglia  25, A.
Bazgir  13, S. Bhosale  14, M. Bielewicz  15, A. Blondel  4, M. Bogomilov  2,
Y. Bondar  13, A. Brandin 22, W. Bryliński  21, J. Brzychczyk  16, M. Buryakov
22, A.F. Camino 28, M. Ćirković  23, M. Csanád  8, J. Cybowska  21, T.
Czopowicz  13, C. Dalmazzone  4, N. Davis  14, A. Dmitriev  22, P. von
Doetinchem  27, W. Dominik  19, J. Dumarchez  4, R. Engel  5, G.A. Feofilov
22, L. Fields  25, Z. Fodor  7,20, M. Friend  9, M. Gaździcki  13, O. Golosov
22, V. Golovatyuk  22, M. Golubeva  22, K. Grebieszkow  21, F. Guber  22, S.N.
Igolkin 22, S. Ilieva  2, A. Ivashkin  22, A. Izvestnyy  22, N. Kargin 22, N.
Karpushkin  22, E. Kashirin  22, M. Kiełbowicz  14, V.A. Kireyeu  22, R.
Kolesnikov  22, D. Kolev  2, Y. Koshio 10, V.N. Kovalenko  22, S. Kowalski
18, B. Kozłowski  21, A. Krasnoperov  22, W. Kucewicz  17, M. Kuchowicz  20,
M. Kuich  19, A. Kurepin  22, A. László  7, M. Lewicki  20, G. Lykasov  22,
V.V. Lyubushkin  22, M. Maćkowiak-Pawłowska  21, Z. Majka  16, A. Makhnev  22,
B. Maksiak  15, A.I. Malakhov  22, A. Marcinek  14, A.D. Marino  26, H.-J.
Mathes  5, T. Matulewicz  19, V. Matveev  22, G.L. Melkumov  22, A. Merzlaya
12, Ł. Mik  17, S. Morozov  22, Y. Nagai  8, T. Nakadaira  9, M. Naskret  20,
S. Nishimori  9, A. Olivier  25, V. Ozvenchuk  14, O. Panova  13, V. Paolone
28, O. Petukhov  22, I. Pidhurskyi  13, R. Płaneta  16, P. Podlaski  19, B.A.
Popov  22,4, B. Pórfy  7,8, D.S. Prokhorova  22, D. Pszczel  15, S. Puławski
18, J. Puzović 23†, R. Renfordt  18, L. Ren  26, V.Z. Reyna Ortiz  13, D.
Röhrich 11, E. Rondio  15, M. Roth  5, Ł. Rozpłochowski  14, B.T. Rumberger
26, M. Rumyantsev  22, A. Rustamov  1, M. Rybczynski  13, A. Rybicki  14, D.
Rybka 15, K. Sakashita  9, K. Schmidt  18, A.Yu. Seryakov  22, P. Seyboth  13,
U.A. Shah  13, Y. Shiraishi 10, A. Shukla  27, M. Słodkowski  21, P. Staszel
16, G. Stefanek  13, J. Stepaniak  15, M. Strikhanov 22, H. Ströbele 6, T.
Šuša  3, Ł. Świderski  15, J. Szewiński  15, R. Szukiewicz  20, A. Taranenko
22, A. Tefelska  21, D. Tefelski  21, V. Tereshchenko 22, R. Tsenov  2, L.
Turko  20, T.S. Tveter  12, M. Unger  5, M. Urbaniak  18, F.F. Valiev  22, D.
Veberič  5, V.V. Vechernin  22, O. Vitiuk  20, V. Volkov  22, A.
Wickremasinghe  24, K. Witek  17, K. Wójcik  18, O. Wyszyński  13, A. Zaitsev
22, E. Zherebtsova  20, E.D. Zimmerman  26, A. Zviagina  22, and R. Zwaska  24

† deceased

1 National Nuclear Research Center, Baku, Azerbaijan
2 Faculty of Physics, University of Sofia, Sofia, Bulgaria
3 Ruder Bošković Institute, Zagreb, Croatia
4 LPNHE, Sorbonne University, CNRS/IN2P3, Paris, France
5 Karlsruhe Institute of Technology, Karlsruhe, Germany
6 University of Frankfurt, Frankfurt, Germany
7 HUN-REN Wigner Research Centre for Physics, Budapest, Hungary
8 Eötvös Loránd University, Budapest, Hungary
9 Institute for Particle and Nuclear Studies, Tsukuba, Japan
10 Okayama University, Japan
11 University of Bergen, Bergen, Norway
12 University of Oslo, Oslo, Norway
13 Jan Kochanowski University, Kielce, Poland
14 Institute of Nuclear Physics, Polish Academy of Sciences, Cracow, Poland
15 National Centre for Nuclear Research, Warsaw, Poland
16 Jagiellonian University, Cracow, Poland
17 AGH - University of Science and Technology, Cracow, Poland
18 University of Silesia, Katowice, Poland
19 University of Warsaw, Warsaw, Poland
20 University of Wrocław, Wrocław, Poland
21 Warsaw University of Technology, Warsaw, Poland
22 Affiliated with an institution covered by a cooperation agreement with CERN
23 University of Belgrade, Belgrade, Serbia
24 Fermilab, Batavia, USA
25 University of Notre Dame, Notre Dame, USA
26 University of Colorado, Boulder, USA
27 University of Hawaii at Manoa, Honolulu, USA
28 University of Pittsburgh, Pittsburgh, USA