Measurement of Hadron Production
in $\uppi^{-}\text{-\,C}$ Interactions at 158 and 350
$\mbox{Ge\kern-1.00006ptV}\kern-1.00006pt/\kern-0.50003ptc$
with at the CERN SPS CERN-EP-2022-194 Phys. Rev. C We present a measurement of
the momentum spectra of $\uppi^{\pm}$, K±, p±, $\Lambda$, $\bar{\Lambda}$ and
K${}^{0}_{\text{S}}$ produced in interactions of negatively charged pions with
carbon nuclei at beam momenta of 158 and 350
$\mbox{Ge\kern-1.00006ptV}\kern-1.00006pt/\kern-0.50003ptc$. The total
production cross sections are measured as well. The data were collected with
the large-acceptance spectrometer of the fixed target experiment at the CERN
SPS. The obtained double-differential $p$-$p_{\text{T}}$ spectra provide a
unique reference data set with unprecedented precision and large phase-space
coverage to tune models used for the simulation of particle production in
extensive air showers in which pions are the most numerous projectiles.

##  1 Introduction

When cosmic rays collide with the nuclei of the atmosphere, they initiate a
cascade of secondary particles called air shower. The interpretation of
cosmic-ray data from air-shower arrays such as KASCADE-Grande [1], IceTop [2],
Telescope Array [3] or the Pierre Auger Observatory [4] relies to a large
extent on the understanding of these particle cascades in the atmosphere,
specifically on the correct modeling of hadron-air interactions that occur
during shower development. However, it is a well-established fact that air
shower simulations using current state-of-the-art models of high-energy
hadronic interactions produce significantly less muons than observed in data
[5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17].

The majority of muons in air showers are created in decays of charged pions
when the energy of the pion is low enough such that its decay length is
smaller than its interaction length in air. The projectiles creating these
pions are typically produced at equivalent beam energies below a TeV [18, 19,
20] which is well within the reach of current accelerators. However, only a
very limited amount of data exists on the interactions of the most numerous
projectile in air showers, the $\uppi$-meson [21].

In this paper, we present new data from the experiment at the CERN SPS on the
particle production in interactions of pion beams at 158 and 350
$\mbox{Ge\kern-1.00006ptV}\kern-1.00006pt/\kern-0.50003ptc$ with a thin carbon
target (used as a proxy for nitrogen, the most abundant nucleus in air). After
a brief introduction to the experiment in Section 2, we will describe the
various data analysis steps that lead to the results presented in this paper.
These steps are sketched in Fig. 1 for a better orientation of the flow of the
analysis and the corresponding sections in which they are described in this
article. The processing and selection of data and simulations are introduced
in Section 3. The three main analyses of the cross section, $\text{V}^{0}$
decays, and identified charged particles are explained in Sections 4, LABEL:,
5, LABEL:, and 6, respectively. The calculation of the particle spectra and
the estimation of their uncertainties is outlined in Section 7, where the
measured production spectra of $\uppi^{\pm}$, K±, p±, $\Lambda$,
$\bar{\Lambda}$ and K${}^{0}_{\text{S}}$ are presented. In Section 8 we
conclude by comparing our measurements to the predictions of hadronic
interaction models used for the modeling of air showers.

simulation data  $\uppi^{-}\text{-\,C}$ Geant (Section 3.2)
$\text{h}^{-}\text{-\,C}$ (Section 2) $\uppi^{-}$ beam selection (Section 3.3)
$\sigma(\uppi^{-}\text{-\,C})$ (Section 4) $\Lambda$, $\bar{\Lambda}$,
K${}^{0}_{\text{S}}$ (Section 7.5) p, $\bar{\text{p}}$, $\uppi^{\pm}$, K±
(Section 7.5) reconstruction (Section 3.1) event and track selection (Sections
3.3 and 3.4) cross section analysis (Section 4) $\text{V}^{0}$ analysis
(Section 5) $\mathrm{d}E/\mathrm{d}x$ analysis (Section 6)
correctioncorrection (Section 7.1)tune (Section 7.2)correction (Section 7.1)
Figure 1: Schematic view of the analysis presented in this paper.

##  2 Experimental Setup and Data Taking

The data reported in this paper were taken in 2009 with the instrument, a
wide-acceptance hadron spectrometer at the CERN SPS on the H2 beam line of the
CERN North Area [22]. The experimental setup used to record
$\uppi^{-}\text{-\,C}$ interactions is shown in Fig. 2.

The main part of the detector consists of five Time Projection Chambers (TPCs)
which were inherited from NA61’s predecessor, the NA49 experiment [23]. Two
Vertex TPCs (VTPC-1 and VTPC-2) are located inside the magnetic field produced
by two superconducting dipole magnets. For the measurements presented in this
paper the two magnets were operated at full electric current providing a field
of 1.5 and 1.1 T, respectively. Two Main-TPCs are located downstream of the
VTPCs to measure particles bent in the left and right hemispheres (MTPC-L and
MTPC-R). An additional small TPC is placed between VTPC-1 and VTPC-2, covering
the very-forward region, and is referred to as the gap-TPC (GTPC). The
combined bending power of the magnets is 9 T m and the coordinates on a track
are measured with a precision of a few 100 $\upmu$m. The resolution of the
measurement of particle momenta in the TPCs depends on the track topology,
i.e. on the overall track length and the number of position measurements [24].
Typical values for the momentum resolution are
$\sigma(p)/p^{2}=7{\times}10^{-4}\,(\text{GeV}/c)^{-1}$ for low-momentum
tracks measured only in the VTPC-1 ($p\lesssim 8$
$\mbox{Ge\kern-1.00006ptV}\kern-1.00006pt/\kern-0.50003ptc$) and
${\sim}3{\times}10^{-3}\,(\text{GeV}/c)^{-1}$ for tracks traversing the full
detector up to the MTPCs ($p\gtrsim 8$
$\mbox{Ge\kern-1.00006ptV}\kern-1.00006pt/\kern-0.50003ptc$).

Figure 2: Experimental Setup of the NA61/SHINE experiment [22] (configuration
for the $\uppi^{-}\text{-\,C}$ data taking). The coordinate system used in
this paper is indicated on the lower left. The incoming beam direction is
along the $z$ axis. The magnetic field bends charged particle trajectories in
the $x$-$z$ (horizontal) plane. The drift direction in the TPCs is along the
$y$ (vertical) axis. A zoomed view of the beam and trigger instrumentation is
shown as an elliptical inset at the bottom.

For the study reported in this paper, the SPS delivered a secondary hadron
beam originating from interactions of 400
$\mbox{Ge\kern-1.00006ptV}\kern-1.00006pt/\kern-0.50003ptc$ primary protons
impinging on a 10-cm-long beryllium target. The negatively charged hadrons
($\text{h}^{-}$) produced in these interactions were transported through the
H2 beam line to the experiment. A beam momentum of 158
$\mbox{Ge\kern-1.00006ptV}\kern-1.00006pt/\kern-0.50003ptc$ was requested for
the first part of data taking and 350
$\mbox{Ge\kern-1.00006ptV}\kern-1.00006pt/\kern-0.50003ptc$ for the second
part. At these momenta the negatively charged beam particles are mostly
$\uppi^{-}$ mesons. They are identified by a differential ring-imaging
Cherenkov detector (CEDAR) [25] and the fraction of pions was measured to be
${\sim}95\%$ for 158
$\mbox{Ge\kern-1.00006ptV}\kern-1.00006pt/\kern-0.50003ptc$ and ${\sim}100\%$
for 350 $\mbox{Ge\kern-1.00006ptV}\kern-1.00006pt/\kern-0.50003ptc$ (see Fig.
2 in [26]). The CEDAR signal is recorded during data taking and then used as
an offline selection cut.

Downstream of the CEDAR three proportional chambers are used as Beam Position
Detectors (BPDs) to measure the trajectories of the incoming particles. Two
scintillation counters (S1 and S2) and three veto counters (V0, V1 and V1′)
define the beam trigger with the coincidence

$\text{T}_{\text{beam}}=\text{S1}\land\text{S2}\land\overline{\text{V0}}\land\overline{\text{V1}}\land\overline{\text{V1}^{\prime}},$
(1)

see inset in Fig. 2. The beam trigger is a _zero bias trigger_. Furthermore,
an interaction trigger,

$\text{T}_{\text{int}}=\text{T}_{\text{beam}}\land\overline{\text{S4}},$ (2)

is defined as the anti-coincidence of the incoming beam particle and S4, a
scintillation counter with a diameter of 2 cm placed between the VTPC-1 and
VTPC-2 along the beam trajectory at about 3.7 m from the target. If an
inelastic interaction occurs then the produced particles typically have
momenta considerably lower than the beam momentum and are thus bent away from
the beam trajectory such that no particle reaches S4. The anti-coincidence
with S4, therefore, serves as a _minimum-bias interaction trigger_.

During data taking a prescaled fraction of the $\text{T}_{\text{beam}}$ and
${\text{T}_{\text{int}}}$ signals can trigger the data acquisition system to
read out the TPCs and write a raw event to disk. For most of the data taking
period, the pre-scaling was 0.4% for the zero-bias beam trigger and 100% for
the minimum-bias interaction trigger.

The target used for this study was an isotropic graphite plate with a
thickness along the beam axis of 2 cm and a density of $\rho=1.840$ g/cm3,
equivalent to about 4% of a nuclear interaction length. 90% of data was
recorded with the target inserted and 10% with the target removed. The latter
data was used to subtract interactions that took place in the material
upstream and downstream of the target. In total, 5.5 million events were
recorded with the target inserted at a beam momentum of 158
$\mbox{Ge\kern-1.00006ptV}\kern-1.00006pt/\kern-0.50003ptc$ and 4.5 million
events at 350 $\mbox{Ge\kern-1.00006ptV}\kern-1.00006pt/\kern-0.50003ptc$.

##  3 Data Processing and Selection

###  3.1 Reconstruction

The raw data recorded during data taking are processed with the standard
reconstruction chain (based on tried-and-tested algorithms developed by the
NA49 collaboration) to obtain high-level physics information such as the
charge and momentum of the produced particles. Firstly, charge clusters are
identified in the raw data of the TPCs and their three-dimensional positions
are reconstructed from the centroids in drift time and in position on the TPC
readout pads. A pattern recognition combines these clusters to form local
track segments in each TPC separately. The local track segments are matched to
_global tracks_ for which the track parameters (track position at reference
plane, charge and three-momentum) are fitted.

The beam trajectory before the target is determined using the position
measurements from the BPDs. The position of the nominal _main interaction
vertex_ is estimated as the intersection of the reconstructed beam trajectory
with a plane located at the center $z$-position of the 2 cm-carbon target.
Global tracks compatible with this position are then re-fitted with this
vertex hypothesis leading to _main-vertex track candidates_ with track
parameters at the nominal interaction vertex. Furthermore, the $z$-position of
the main interaction vertex is reconstructed by fitting a common origin of
these global tracks along the beam trajectory. Depending on the track
multiplicity, the resolution along the $z$-axis for the vertex position
obtained in this way exceeds the target thickness, therefore better main-
vertex-constrained track-momenta can be obtained by using the nominal
interaction vertex in the middle of the target instead. Yet, the reconstructed
vertex position is useful for the rejection of obvious out-of-target
interactions during the event selection (see below).

In addition to the main-vertex hypothesis, each combination of positively and
negatively charged global tracks within one event is investigated for a common
origin downstream of the target from the decay of a long-lived neutral
particle (so-called $\text{V}^{0}$ events). All pairs with a distance of
closest approach $\leq 1$ cm anywhere along their trajectory downstream of the
target are re-fitted with the constraint to originate from a common vertex
resulting in _$\text{V}^{0}$ candidates_ that will be analyzed in Section 5.

The results of the reconstruction are stored in a dedicated Root-based [27]
output format [28] for further processing during data analysis.

###  3.2 Simulation

Through this analysis, we will use simulations of the measurement to correct
the raw-data spectra for various distortions originating from the detector
acceptance, re-interactions with the detector material and within the target,
feed-down from weak decays etc. For this purpose we simulated
$\uppi^{-}\text{-\,C}$ interactions at both beam momenta with the hadronic
event generators Epos 1.99 [29] and QGSJet II-04 [30] with the crmc program
[31]. The 1.99 version of the Epos model was used rather than its newer “LHC”
variant as the former is better tuned to interactions at SPS energies [32].
The particles produced by these generators are then passed to a simulation of
the passage of particles through the material and magnetic field of the setup
using the Geant 3.21 package [33]. The hits generated in the active detector
volumes are digitized to produce the same raw information as for real data and
an interaction trigger is simulated by checking whether any of the charged
particles hit the S4 counter. The simulated information is then processed with
the same reconstruction algorithms discussed in the previous section and the
results are stored in the same output format as the reconstructed data.

###  3.3 Event Selection

We apply the following event-selection criteria111The event selection criteria
used here are identical to the ones described in Ref. [26] for the same data
set discussed in this paper. We therefore refer to Sections 2 and 3 of that
paper for a detailed description of the event selection. to obtain a set of
high-quality interaction triggers.

Pion projectiles are selected with the CEDAR (see previous section) and a
pile-up of interactions is avoided by rejecting events in which the S1
scintillator detected another beam particle within $\pm 2\,\upmu$s of the time
of the interaction trigger. Furthermore, it is required that the direction of
the beam was well measured with the three beam position detectors. These three
cuts select high-quality $\uppi^{-}$ projectiles based on measurements from
the beam detectors upstream of the target.

Further event selection criteria define the subset of $\uppi^{-}$-projectiles
with an interaction in the target. Here we analyze the particles produced in
events recorded with the minimum-bias trigger. Furthermore, we reject events
with an interaction vertex reconstructed far from the center of the target
($|\Delta z|>17$ cm), since such events mostly originate from interactions
outside of the target.

With these criteria we select $2.8{\times}10^{6}$ and $2.6{\times}10^{6}$
minimum-bias triggers recorded with an inserted C-target at 158 and 350
$\mbox{Ge\kern-1.00006ptV}\kern-1.00006pt/\kern-0.50003ptc$, respectively. By
construction, due to the restriction to events with a reconstructed vertex
close to the target position, only very few events recorded with a removed
target survive the selection (${\lesssim}7{\times}10^{3}$ events in each data
set). The sum of simulated events with an inserted target is
$7.2{\times}10^{6}$ and $6.0{\times}10^{6}$ for beam momenta of 158 and 350
$\mbox{Ge\kern-1.00006ptV}\kern-1.00006pt/\kern-0.50003ptc$ respectively.

###  3.4 Track Selection

The set of selection criteria given below is applied to the tracks measured in
the TPCs. They are constructed to assure good quality of the momentum
measurements and to select regions of the detector with a solid understanding
of the detection efficiency.

##### Track Quality

The total number of clusters on the track must be $\geqslant 25$ and the sum
of clusters in both VTPCs must be $\geqslant 12$, or the number of clusters in
the GTPC must be $\geqslant 6$. These cuts assure a good momentum
determination in either the VTPCs within the magnetic field, or a large lever
arm for forward tracks measured with a combination of GTPC and MTPC [34].

##### Fiducial Acceptance

For each particle charge, we restrict the analysis to bins in azimuthal angle,
total momentum and transverse momentum, ($\phi$, $p$, $p_{\text{T}}$), in
which the track selection efficiency is larger than 90% for simulated
tracks.222Since the acceptance is only a property of the track momentum and
direction (not of the beam momentum or track multiplicity), it can be
determined with high statistical accuracy averaging all available simulation
sets. A total of $2.1{\times}10^{7}$ simulated events were used to construct
the acceptance map. This number is higher than the sum of simulated events
mentioned in Section 3.3, since for the acceptance study we also included
events generated with the DPMJet 3.06 model, which were not used otherwise for
this study. This cut mainly removes tracks at the edges of the detector for
which the efficiency drops rapidly, as illustrated in the example shown in the
top-left panel of Fig. 3. Since the TPCs have a larger width than height (cf.
lower left panel in Fig. 3), the acceptance is not uniform in $\phi$ and
especially poor along $\pm\pi/2$. But since neither the beam nor the target
are polarized, the particle yields are independent of $\phi$ and we can thus
restrict the analysis to $\phi$ regions where the detection efficiency is near
100%. The _fiducial volume_ selected by the acceptance cut therefore leads to
a near-geometric acceptance that is given by the number of accepted
$\phi$-bins [34]. The acceptance for positively charged tracks is visualized
in Fig. 3.

##### Origin at Primary Vertex

Furthermore, we require that the distance between the position of the
intercept of the extrapolation of the track (reconstructed without vertex-
constraint) to the interaction plane and the position of the main interaction
vertex must be smaller than 4 cm in both horizontal and vertical direction.
This cut removes out-of-target interactions and tracks from particle decays
(“feed-down”).

Furthermore, we require that the distance between the extrapolation of the
track (reconstructed without vertex-constraint) to the interaction plane and
the interaction point must be smaller than 4 cm in both horizontal and
vertical direction. This cut removes out-of-target interactions and tracks
from particle decays (“feed-down”).

Figure 3: _Top left:_ Example of the charged particle multiplicity at
$(p,p_{\text{T}})=(5.3,0.4)$
$\mbox{Ge\kern-1.00006ptV}\kern-1.00006pt/\kern-0.50003ptc$ as a function of
track angle $\phi$. Red squares are simulated data and black circles are
measurements. _Bottom left:_ Event display illustrating the aspect-ratio of
the TPC chambers (blue boxes) and showing reconstructed tracks (red lines),
clusters (yellow points) and the primary interaction vertex (orange point).
_Right:_ 3D view of the acceptance definition for positively charged tracks.
Each ($\phi$, $p$, $p_{\text{T}}$) bin which does not satisfy the acceptance
criteria is empty.

After the track selection, the measured tracks are split into two subsets
called right-side tracks (RSTs) and wrong-side tracks (WSTs). The former group
is defined as the tracks that bend away from the beam axis, while the latter
as the tracks that bend towards the beam axis333The terminology of wrong- and
right-side tracks dates back to NA49 and reflects the fact that the tilt angle
of the pads in the VTPCs were optimized to measure the right-side topologies.,
i.e. $p_{x}q>0$ (RST) and $p_{x}q<0$ (WST), where $p_{x}=p\sin\theta\cos\phi$
and $p$ and $q$ are the momentum and charge of the track. These conditions
simplify to $q\cos\phi>0$ for RSTs and $q\cos\phi<0$ for WSTs for the usual
case of forward production in the laboratory frame, $\theta>0$. The
distinction between the two track topologies is motivated by the fact that a
right- and a wrong-side track with the same $p$ and $p_{\text{T}}$, cross
different regions of the detector, which has important implications for the
particle reconstruction and identification step based on the
$\mathrm{d}E/\mathrm{d}x$ measurements (see Section 6). Furthermore, this
subdivision defines two independent data sets that can be compared to estimate
the systematic uncertainties of the measured particle multiplicities, see
Section 7.4.2.

##  4 Cross Section Analysis

The measurement of the production cross section444The production cross section
is defined by
$\sigma_{\text{prod}}=\sigma_{\text{tot}}-\sigma_{\text{el}}-\sigma_{\text{qe}}$,
where $\sigma_{\text{tot}}$ denotes the total cross section,
$\sigma_{\text{el}}$ the coherent elastic cross section and
$\sigma_{\text{qe}}$ the quasi-elastic cross section. Quasi-elastic
interactions are processes in which no new particles are produced, but the
target nucleus is fragmented. in $\uppi^{-}\text{-\,C}$ interactions closely
follows the analysis procedure detailed in Ref. [35]. The experimental
interaction cross section is measured by counting the number of interaction
triggers, $N(\text{T}_{\text{beam}}\wedge{\text{T}_{\text{int}}})$, within the
recorded zero-bias beam triggers, $N(T_{\text{beam}})$, to obtain the
interaction trigger probability

$P_{\text{${\text{T}_{\text{int}}}$}}=\frac{N(\text{T}_{\text{beam}}\wedge{\text{T}_{\text{int}}})}{N(\text{T}_{\text{beam}})}.$
(3)

The interaction probability in the carbon target is then obtained by
correcting for out-of-target interactions via

$P_{\text{int}}=\frac{P_{{\text{T}_{\text{int}}}}^{\text{I}}-P_{{\text{T}_{\text{int}}}}^{\text{R}}}{1-P_{{\text{T}_{\text{int}}}}^{\text{R}}},$
(4)

where the trigger probabilities measured with the target removed are denoted
by a superscript R and the ones with the target inserted with a superscript I.

The interaction trigger cross section is given by

$\sigma_{\text{trig}}=\frac{m_{A}}{L\,\rho\,N_{\text{A}}}\ln\left(\frac{1}{1-P_{\text{int}}}\right),$
(5)

where $N_{\text{A}}$ is Avogadro’s number and $\rho$, $A$ and $L$ are the
density, molar mass and length of the target, respectively. The logarithmic
term,
$\ln\left(\frac{1}{1-P_{\text{int}}}\right)=P_{\text{int}}+\frac{1}{2}P_{\text{int}}^{2}+\frac{1}{3}P_{\text{int}}^{3}+\cdots$,
accounts for the exponential attenuation of the beam inside the target.

The experimentally accessible interaction trigger cross section can be related
to the production cross section by correcting for the residual contributions
to $\sigma_{\text{trig}}$ originating from elastic and quasi-elastic
scattering ($\sigma_{\text{el}}$ and $\sigma_{\text{qe}}$). Furthermore, a
correction for inelastic interactions, to which the interaction trigger is not
sensitive to, is needed

$\sigma_{\text{prod}}=\left(\sigma_{\text{trig}}-\sigma_{\text{el}}\,f_{\text{el}}-\sigma_{\text{qe}}\,f_{\text{qe}}\right)\frac{1}{f_{\text{prod}}},$
(6)

where $f_{\text{el}}$, $f_{\text{qe}}$, and $f_{\text{prod}}$ are the
fractions of elastic, quasi-elastic, and production events that lead to an
${\text{T}_{\text{int}}}$ trigger. $f_{\text{el}}$ and $f_{\text{qe}}$ thus
give the fraction of false-positive interaction triggers from (quasi-)elastic
interactions and $1-f_{\text{prod}}$ is the fraction of false-negative
production interactions.

Table 1: List of correction factors used in Eq. 6 to convert the trigger cross section to the production cross section. The quoted uncertainties are systematic uncertainties compared to which the statistical uncertainties are negligible. The fractions $f$ are given for a trigger radius of 0.9 and 0.6 cm (see text) for the beam energies of 158 and 350 $\mbox{Ge\kern-1.00006ptV}\kern-1.00006pt/\kern-0.50003ptc$, respectively. beam momentum | 158 $\mbox{Ge\kern-1.00006ptV}\kern-1.00006pt/\kern-0.50003ptc$ | 350 $\mbox{Ge\kern-1.00006ptV}\kern-1.00006pt/\kern-0.50003ptc$
---|---|---
$\sigma_{\text{el}}$/mb |  $35.1$ p m 0.8 |  $36.4$ p m 0.9
$\sigma_{\text{qe}}$/mb |  $12.5$ p m 0.3 |  $12.0$ p m 0.3
$f_{\text{el}}$ | 0.0012 | 0.00
$f_{\text{qe}}$ |  $0.27$ p m 0.03 |  $0.06$ p m 0.02
$f_{\text{prod}}$ |  $0.91$ p m 0.02 |  $0.87$ p m 0.02

We derived model predictions for $\sigma_{\text{el}}$ and $\sigma_{\text{qe}}$
by performing a Glauber calculation [36] of $\uppi^{-}\text{-\,C}$
interactions using a fit to previously measured cross sections of
$\uppi^{-}\text{-\,C}$ [37] as an input. The resulting cross sections are
listed in Table 1 where the quoted systematic uncertainty originates from
different assumption on the inelastic screening in $\uppi^{-}\text{-\,C}$
interactions [38].

The fractions $f$ depend on the chosen interaction trigger condition. We use
the FTFP_BERT physics list of Geant 4.9.4.p01 [39] to estimate $f_{\text{el}}$
and $f_{\text{qe}}$. For systematic checks, we assumed that the angular
distribution of quasi-elastic scattering in $\uppi^{-}\text{-\,C}$ is very
similar to free pion-nucleon scattering and can thus be modeled using the
elastic slope $B_{\text{ela}}$ in $\uppi^{-}\text{-\,C}$ scattering.
$f_{\text{prod}}$ was estimated by generating interactions with the hadronic
models Fluka 2011.2.9 [40], Epos 1.99 [29], QGSJet01 [41], QGSJet II-03 [30],
Venus 4.12 [42], Sibyll 2.1 [43] Urqmd 1.3.1 [44, 45] using the INTTEST mode
of Corsika [46]. The arithmetic mean of the different predictions of
$f_{\text{prod}}$ is used to correct the data, and the maximum and minimum
values as an estimate of the systematic uncertainty.

In previous cross-section analyses within we used the absence of a signal in
the S4 counter to define an interaction. But, as can be seen in Fig. 20 in the
appendix, especially for the 350
$\mbox{Ge\kern-1.00006ptV}\kern-1.00006pt/\kern-0.50003ptc$ data studied in
this paper, the radius of 1 cm of the S4 would lead to a large model-dependent
correction for $\sigma_{\text{prod}}$ with Eq. 6. Therefore, we instead use
the GTPC to define an offline interaction trigger by requiring the absence of
a track within a radius $r_{\text{trig}}$ from the beam extrapolation at 3.7 m
downstream of the target (i.e. at the $z$-position of the S4 plane, located
before the GTPC). We choose the trigger radius which minimizes the quadratic
sum of statistical and systematic uncertainty of the measured cross section
leading to an $r_{\text{trig}}$ of 0.9 and 0.6 cm for beam energies of 158 and
350 $\mbox{Ge\kern-1.00006ptV}\kern-1.00006pt/\kern-0.50003ptc$ respectively,
see Ref. [47]. The analysis is performed on the zero-bias beam trigger data
with $4.8\,(2.0)\,{\times}10^{5}$ and $6.5\,(2.2)\,{\times}10^{5}$ events at
158 and 350 $\mbox{Ge\kern-1.00006ptV}\kern-1.00006pt/\kern-0.50003ptc$,
respectively, where the number in brackets refers to the data taken with a
removed target.

With the above definition of the offline interaction trigger, we find an
interaction probability of

$P_{\text{int}}=0.0293\pm 0.0003\,(\text{stat.}),\quad
p_{\text{beam}}=158\,\mbox{Ge\kern-1.00006ptV}\kern-1.00006pt/\kern-0.50003ptc$
(7)

and

$P_{\text{int}}=0.0284\pm 0.0003\,(\text{stat.}),\quad
p_{\text{beam}}=350\,\mbox{Ge\kern-1.00006ptV}\kern-1.00006pt/\kern-0.50003ptc$
(8)

and the cross section fractions as listed in Table 1.

Figure 4: Production cross section in $\uppi^{-}\text{-\,C}$ interactions
(red) and $\uppi^{+}\text{-\,C}$ interactions (blue). Measurements by are
shown as filled circles, previous measurements from [48, 49, 50, 51] are
indicated by open symbols. Predictions from the Glauber model with different
values of the inelastic screening are shown as dashed lines ($\lambda=0.1$,
0.3, 0.5, 0.7 and 0.9 from top to bottom).

Evaluating Eq. 6 with the trigger cross section derived via Eq. 5 from these
measurements as well as with the correction factors listed in Table 1, leads
to our estimates of the production cross section in $\uppi^{-}\text{-\,C}$
interactions of

$\sigma_{\text{prod}}=\left(172\pm 2\,(\text{stat.})\,\pm
4\,(\text{sys.})\right)\text{mb},\quad
p_{\text{beam}}=158\,\mbox{Ge\kern-1.00006ptV}\kern-1.00006pt/\kern-0.50003ptc$
(9)

and

$\sigma_{\text{prod}}=\left(178\pm 2\,(\text{stat.})\,\pm
5\,(\text{sys.})\right)\text{mb},\quad
p_{\text{beam}}=350\,\mbox{Ge\kern-1.00006ptV}\kern-1.00006pt/\kern-0.50003ptc.$
(10)

The systematic uncertainty includes contributions from the inefficiency of the
GTPC, uncertainties of the detector simulation, and the uncertainties of the
correction factors quoted in Table 1. The largest contribution of about 3.5 mb
originates from the model uncertainty of $f_{\text{prod}}$. In Appendix A we
provide the efficiency map corresponding to the two beam energies such that in
the future it will be possible to recalculate $f_{\text{prod}}$ with a
different set of models, and possibly smaller systematic uncertainty.

The obtained cross sections are presented in Fig. 4 along with the theoretical
prediction using the Glauber theory [36] for several assumptions on the
inelastic screening parameter $\lambda$ [38] (from top to bottom
$\lambda=0.1$, 0.3, 0.5, 0.7 and 0.9) and previous measurements [48, 49, 50,
51, 52].555Here we use Glauber predictions of $\sigma_{\text{prod}}$,
$\sigma_{\text{tot}}$ and $\sigma_{\text{inel}}$ to scale all measurements to
$\sigma_{\text{prod}}$. The measurements of $\sigma_{\text{tot}}$ from Refs.
[50] and [51] are multiplied by
$(\sigma_{\text{prod}}/\sigma_{\text{tot}})_{\text{Glauber}}\sim 0.77$ and the
measurement of $\sigma_{\text{inel}}$ from Ref. [49] is multiplied by
$(\sigma_{\text{prod}}/\sigma_{\text{inel}})_{\text{Glauber}}\sim 0.91$. As
can be seen, the only previous data set on the production cross section in
$\uppi^{-}\text{-\,C}$ interactions from Ref. [48] agrees well with our
measurement and the total uncertainty of the cross sections presented here
matches the statistical precision of this old measurement (no systematic error
was quoted in this reference). Furthermore, our measurement agrees well with
the Glauber predictions for a broad range of inelastic screening assumptions,
but small values are disfavored and the preferred range is within
$0.5<\lambda<0.9$.

##  5 $\text{V}^{0}$ analysis

As a first step of the analysis of particle spectra, we investigate the
production of the neutral weakly-decaying particles $\Lambda$, $\bar{\Lambda}$
and K${}^{0}_{\text{S}}$. The production spectra of these strange particles
provide important constraints for the tuning of hadronic interaction models,
see e.g. Ref. [53]. Moreover, a good knowledge of these spectra is also
important to distinguish particles created directly in $\uppi^{-}\text{-\,C}$
interactions from particles originating from weak decays, see Section 7.2.

Neutral weakly decaying particles with an average decay length ($c\tau$) of
the order of a few to tens of centimeters can be detected by experiment
through their charged decay products. This type of particle is traditionally
called $\text{V}^{0}$, because of its neutral charge and the V-shaped decay
topology. Although the $\text{V}^{0}$ itself does not create a track in the
TPCs, the products of its decay may do, allowing us to reconstruct the
position of the decay vertex and, by using the momenta of the daughter tracks,
reconstruct the properties of the parent particle.

In Table 2 we list the three $\text{V}^{0}$ particles studied in this work
together with the branching ratio of the decay channels investigated here.

Table 2: List of the $\text{V}^{0}$ particles measured in this work, together with their mass and decay length. The last two columns give the decay channel used for the reconstruction as well as its branching ratio [37].  $\text{V}^{0}$ | mass [GeV/$c^{2}$] |  $c\tau$ [cm] | decay | branching ratio
---|---|---|---|---
$\Lambda$ | 1.1157 | 7.89 | p+ | $\uppi^{-}$ | 63.9$\%$
$\bar{\Lambda}$ | 1.1157 | 7.89 |  $\uppi^{+}$+ | $\bar{\text{p}}$ | 63.9$\%$
K${}^{0}_{\text{S}}$ | 0.4976 | 2.68 |  $\uppi^{+}$+ | $\uppi^{-}$ | 69.2$\%$

$\text{V}^{0}$ candidates are selected by calculating the distance of closest
approach (dca) for each combination of one positively and one negatively
charged track. To assure a good momentum resolution, both tracks should in
total have more than 30 clusters each and more than 15 clusters in the VTPCs.
Each combination with a $\text{dca}<2$ cm downstream of the main vertex is re-
fitted under a common vertex hypothesis. To increase the signal-to-background
ratio, the reconstructed $\text{V}^{0}$ momentum is then extrapolated to the
vertex plane and the radial impact parameter,
$b_{r}=\sqrt{\smash[b]{(b_{x}/2)^{2}+b_{y}^{2}}}$, is required to be $\leq 2$
cm. Here $b_{x}$ and $b_{y}$ denote the coordinates of the impact point in the
target plane with respect to the main vertex and the factor
$\nicefrac{{1}}{{2}}$ accounts for the fact that the resolution in the $xz$
bending plane is approximately twice as large as in the $yz$-plane. The
efficiency of this cut is $\geq 90\%$ in all of the phase space bins studied
here.

A further reduction of the background is possible by selecting events with a
reconstructed decay vertex far from the target, as the background arises
mostly from combinations involving non-$\text{V}^{0}$ tracks from the main
vertex. A large distance cut, $d_{\text{min}}$, improves the purity of the
sample, but also diminishes the selection efficiency,
$\varepsilon=\exp(-\gamma c\tau/d_{\text{min}})$ for an ideal detector.
Therefore an optimal selection distance that minimizes the uncertainty of the
extracted signal is chosen, depending on the $\text{V}^{0}$ particle type and
momentum as described in Ref. [54].

For the measurement of the multiplicity of $\text{V}^{0}$ particles, we study
the distribution of the invariant mass $m_{\text{inv}}$ of combinations of
candidate tracks,

$m_{\text{inv}}^{2}=(\mathbf{p_{+}}+\mathbf{p_{-}})^{2}=(m_{+}^{2}+m_{-}^{2})\,c^{4}+2\,(E_{+}E_{-}-\vec{p}_{+}\vec{p}_{-}\,c^{2}),$
(11)

in bins of $p$ and $p_{\text{T}}$ (cf. Appendix B). Here the subscripts $+$
and $-$ refer to the positively and negatively charged daughter particles,
$\mathbf{p_{\pm}}$ denote their four-momenta and $\vec{p}_{\pm}$ the
reconstructed three-momenta of candidate tracks. Eq. 11 is evaluated for three
mass combinations $m_{\pm}$ corresponding to the main $\Lambda$,
$\bar{\Lambda}$ and K${}^{0}_{\text{S}}$ decays listed in Table 2 and the
energies $E_{\pm}$ are calculated accordingly via
$E_{\pm}=\sqrt{\smash[b]{m_{\pm}^{2}c^{4}+|\vec{p}_{\pm}|^{2}c^{2}}}$.
Examples of invariant mass distributions are shown in Fig. 5. As can be seen,
the “real” combinations peak around the mass of the $\text{V}^{0}$ whereas the
background of unrelated track combinations exhibits a broad and nearly flat
distribution.

The number of $\text{V}^{0}$s is extracted in a Poissonian likelihood fit [55]
describing the invariant mass distribution as the sum of the signal template
and the background function. For this purpose, the signal distribution is
modeled using the shape of the reconstructed $\text{V}^{0}$ as predicted by
the detector simulation and for the background distribution a polynomial
function is used. We found that a second-degree polynomial provides a
satisfactory description of the data in the chosen $m_{\text{inv}}$ interval.
A third-degree polynomial was used to estimate the systematic uncertainties of
the background shape description (see Section 7.4.2). These signal and
background templates are illustrated as blue and red histograms in Fig. 5. The
number of $\text{V}^{0}$s produced in the fitted $p$-$p_{\text{T}}$ bin is
then inferred from the normalization of the fitted distributions.

\begin{overpic}[clip,rviewport=0.02 0 0.98
1,height=161.8885pt]{Plots/mass_Data158_t0_ph1_h0_x1_y3_nolabel}
\put(16.0,86.0){$\Lambda$ }
\put(-2.0,50.0){\rotatebox{90.0}{\scalebox{0.8}{entries}}}
\end{overpic}\begin{overpic}[clip,rviewport=0.07 0 0.98
1,height=161.8885pt]{Plots/mass_Data158_t0_ph1_h1_x1_y3_nolabel}
\put(13.0,86.0){$\bar{\Lambda}$ }
\end{overpic}\begin{overpic}[clip,rviewport=0.07 0 0.98
1,height=161.8885pt]{Plots/mass_Data158_t0_ph1_h2_x2_y4_nolabel}
\put(13.0,86.0){K${}^{0}_{\text{S}}$ } \end{overpic}

Figure 5: Examples of the fitted $m_{\text{inv}}$ distributions for the 158
$\mbox{Ge\kern-1.00006ptV}\kern-1.00006pt/\kern-0.50003ptc$ data set with
target inserted for $\Lambda$ (left), $\bar{\Lambda}$ (middle) and
K${}^{0}_{\text{S}}$ (right). The black markers show the measured
$m_{\text{inv}}$ distributions. The curves show the results of the fit with
the signal in blue, the background in red and the total in gray. On the bottom
of each plot we show the residual distributions, i.e. the difference $\Delta$
between the observed and fitted number of entries in units of the
uncertainties $\sigma$ of the observed number. The $\langle p\rangle$ and
$\langle p_{\text{T}}\rangle$ of the phase space bin are indicated on the top
of each panel.

##  6 $\mathrm{d}E/\mathrm{d}x$ analysis

The spectra of identified particles produced at the main vertex are obtained
by extracting the average particle yields in each kinematic bin from the
measured distributions of the energy loss of tracks in the TPCs. For this
purpose, we calculate the truncated mean [56],
$\langle\mathrm{d}E/\mathrm{d}x\rangle_{j}=\nicefrac{{2}}{{N_{j}}}\sum_{i=1}^{N_{j}/2}\mathrm{d}E/\mathrm{d}x_{ij}$,
of the charges $\mathrm{d}E/\mathrm{d}x_{ij}$ of the 50% of clusters with the
lowest charge among the $N_{j}$ clusters detected along each particle track
$j$. As in the Bethe formula [57], $\langle\mathrm{d}E/\mathrm{d}x\rangle$
depends on $p/m=\beta\gamma$ and can thus be used to identify particles of
different masses $m$ at a particular momentum $p$. The two-dimensional
distribution of energy deposit and momenta of selected main-vertex tracks at a
beam momentum of 158
$\mbox{Ge\kern-1.00006ptV}\kern-1.00006pt/\kern-0.50003ptc$ are displayed in
Fig. 6 along with a Bethe-inspired parametrization of the mean energy deposit
for the particles considered here, i.e. electrons, pions, kaons, protons, and
deuterons and their anti-particles.

Figure 6: Energy deposit vs momentum of negatively and positively charged
tracks for the 158 $\mbox{Ge\kern-1.00006ptV}\kern-1.00006pt/\kern-0.50003ptc$
data set. The dashed lines indicate the average energy deposit for each
particle type (see Fig. 25 in the appendix for
$p_{\text{beam}}=350\leavevmode\nobreak\
\mbox{Ge\kern-1.00006ptV}\kern-1.00006pt/\kern-0.50003ptc$).

An example of the distribution of the $\langle\mathrm{d}E/\mathrm{d}x\rangle$
of tracks in a particular $p$ and $p_{\text{T}}$ bin is shown in Fig. 7 for
negatively and positively charged particles. The production yields of
different particle types are determined by fitting the distribution with the
sum of $\langle\mathrm{d}E/\mathrm{d}x\rangle$-templates for electrons, pions,
kaons, protons, and deuterons. The shapes of these templates are based on
previous studies from and [58, 59, 60] which describe the distribution of
energy deposit of each particle type by the sum of asymmetric Gaussians taking
into account the distribution of $N_{j}$ (number of clusters per track) and
the distribution of momenta within the bin.

The fitting is performed using a binned maximum-likelihood method. In the
general case, 10 particle fractions (5 particle types and 2 charges) and 10
model parameters are taken as free parameters of the fit. Most of these model
parameters are nuisance parameters that allow the mean and width of the
distributions to stray away from the global $\mathrm{d}E/\mathrm{d}x$
parametrization. In that way, residual systematic offsets of the calibration
of cluster charges in different parts of the detector can be corrected.

As illustrated in Fig. 7, in general the $\mathrm{d}E/\mathrm{d}x$-fits lead
to a very satisfactory description of the data. The example shown here is
close to a momentum where the mean $\mathrm{d}E/\mathrm{d}x$ values of protons
and kaons as well as the one of electrons and deuterons overlap (cf. Fig. 6).
In case of a near-complete degeneracy between the fitting templates of
different masses in certain momentum bins (“Bethe crossings”), the fitted
yields of these ambiguous particles are excluded from the final results.

\begin{overpic}[clip,rviewport=0 0 1
0.95,width=212.47617pt]{Plots/dist_158_v0_c0_x13_y3_b} \put(16.0,91.0){$q=-1$}
\put(2.0,50.0){\rotatebox{90.0}{\scalebox{1.0}{entries}}}
\end{overpic}\begin{overpic}[clip,rviewport=0 0 1
0.95,width=212.47617pt]{Plots/dist_158_v0_c1_x13_y3_b} \put(16.0,91.0){$q=+1$}
\put(2.0,50.0){\rotatebox{90.0}{\scalebox{1.0}{entries}}} \end{overpic}

Figure 7: Example of the $\mathrm{d}E/\mathrm{d}x$ distributions for one phase
space bin ($\langle
p\rangle=2.19\,\mbox{Ge\kern-1.00006ptV}\kern-1.00006pt/\kern-0.50003ptc$ and
$\langle
p_{\text{T}}\rangle=0.35\,\mbox{Ge\kern-1.00006ptV}\kern-1.00006pt/\kern-0.50003ptc$)
of the 158 $\mbox{Ge\kern-1.00006ptV}\kern-1.00006pt/\kern-0.50003ptc$ data
set. The black markers show the measured distributions and the colored areas
show the result of the $\mathrm{d}E/\mathrm{d}x$ fit for different particles.
Negatively and positively charged particles are shown on the left and right,
respectively. Fit residuals are shown in the lower panels in units of the
statistical uncertainty of the data distribution.

##  7 Derivation of the Particle Spectra

The $\text{V}^{0}$ and $\mathrm{d}E/\mathrm{d}x$ analyses described in the
previous two sections result in the number $\hat{n}$ of identified tracks
($\uppi^{\pm}$, K±, p, $\bar{\text{p}}$, $\Lambda$, $\bar{\Lambda}$, and
K${}^{0}_{\text{S}}$) in bins of $p$ and $p_{\text{T}}$ for each of the two
data taking modes, i.e. with the target inserted and removed. For the choice
of binning, see Appendix B.

Given these measurements, the double-differential particle production spectra
can be computed for each particle as

$\frac{1}{N_{\text{prod}}}\frac{\mathrm{d}^{2}n}{\mathrm{d}p\,\mathrm{d}p_{\text{T}}}=\frac{\mbox{$C_{\text{sim}}$}}{\Delta
p\,\Delta
p_{\text{T}}}\,\frac{\hat{n}_{\text{I}}-b\,\hat{n}_{\text{R}}}{\hat{N}_{\text{I}}-b\,\hat{N}_{\text{R}}}.$
(12)

This quantity is the track multiplicity per production event in a certain
$p$-$p_{\text{T}}$ bin, sometimes also referred to as _average multiplicity_.
Here $\Delta p$ and $\Delta p_{\text{T}}$ denote the widths of the phase-space
bin and the transverse component $p_{\text{T}}$ of the momentum vector
$\mathbf{p}$ is defined with respect to the beam direction
${\mathbf{u}}_{\text{b}}$,
$p_{\text{T}}=\sqrt{{\mathbf{p}}^{2}-({\mathbf{u}}_{\text{b}}\,{\mathbf{p}})^{2}}$.
$C_{\text{sim}}$ is a correction factor derived from simulations that will be
discussed in the next section. The indexes I and R refer to target inserted
and removed, respectively, and $\hat{N}$ is the number of selected minimum-
bias interaction-trigger events. The factor $b$ is the so-called target-
removed factor and it normalizes both target inserted and removed data set to
the same number of beam particles. Given the number of events,
$N_{\text{T}_{\text{int}}}$, and the probability of a beam particle to produce
an interaction trigger, $P_{\text{T}_{\text{int}}}$, the number of beam
particles for a given data set can be estimated as
$N_{\text{beam}}=N_{\text{T}_{\text{int}}}/P_{\text{T}_{\text{int}}}$.
$P_{\text{T}_{\text{int}}}$ is measured from zero-bias beam triggers, see Eq.
3. The target-removed factor is given by
$b=N_{\text{beam}}^{\text{I}}/N_{\text{beam}}^{\text{R}}$ and its value is
about five, corresponding to the time spent in target-removed configuration
during data taking.

###  7.1 Correction Factors

The selected number of events $\hat{N}$ and the estimated number of particles
$\hat{n}$ in a given phase space bin are biased estimators of the true number
of production events $N_{\text{prod}}$ and the true number of produced
particles $n$. These biases are corrected for with the help of simulations for
which it is easy to determine the ratio of generated and measured spectra,

$\mbox{$C_{\text{sim}}$}=\left(\frac{n_{\text{gen}}}{N_{\text{gen}}}\right)\big{/}\left(\frac{\hat{n}}{\hat{N}}\right),$
(13)

where $n$ and $N_{\text{prod}}$ are the values from the event generator,
$\hat{n}$ and $N_{\text{T}_{\text{int}}}$ are obtained after the detector
simulation, event reconstruction, and event and track selection. Two simulated
data sets generated with different hadronic interaction models (see Section 3)
are used for systematic studies.

To gain further insights into the different contributions to the correction
factor it is useful to split $C_{\text{sim}}$ into event- and particle-
contribution factors, referred to as $\alpha$ and $\beta$ in the following,

$\mbox{$C_{\text{sim}}$}=\left(\frac{\hat{N}}{N_{\text{gen}}}\right)\big{/}\left(\frac{\hat{n}}{n_{\text{gen}}}\right)=\alpha/\beta.$
(14)

The $\alpha$ factor is a property of the data set as a whole and thus depends
only on the beam energy, whereas the $\beta$ factor depends on the phase space
bin and on the particle type. We get $\alpha=0.872\pm 0.004$ and $0.732\pm
0.012$ for beam momenta of 158 and 350
$\mbox{Ge\kern-1.00006ptV}\kern-1.00006pt/\kern-0.50003ptc$, respectively.
These values were determined from the arithmetic average of the two simulated
data sets, with the upper uncertainty range corresponding to the result
predicted by Epos 1.99 and the lower range to the one from QGSJet II-04. These
values give the product of the efficiency of the vertex-$z$ cut (cf. Section
3.3) and the efficiency of the minimum-bias interaction trigger (cf. Section
2). The former amounts to approximately 0.975 with a good agreement between
real and simulated data. Therefore, $\alpha$ is dominated by the trigger
efficiency and is thus mainly a model-dependent correction, since the
efficiency depends on the fraction of events with a high-momentum charged
particle triggering the S4 scintillator. Overall, the two hadronic generators
used here show a good overall agreement in their predictions of $\alpha$. In
the following, we will use the average value for the correction and the
difference for systematic uncertainty. A slightly larger uncertainty would
result if the standard deviation of efficiencies of all the models shown in
Fig. 20 of the appendix would be used as an estimate of the systematic
uncertainty. Note that for the majority of phase-space bins, the total
correction for the trigger efficiency is small, as it affects both $\hat{N}$
and $N_{\text{gen}}$. The correction is mostly relevant at high momenta when
the trigger efficiency affects $N_{\text{gen}}$ but not $\hat{N}$ since the
conservation of the beam momentum does not allow for the simultaneous presence
of a high-momentum particle in the TPCs and another high-momentum particle
hitting the S4 scintillator.

The $\beta$ correction factors as a function of $p$ and $p_{\text{T}}$ are
given by the ratio of the generated and measured number of tracks. They are
shown in Fig. 8 for $\uppi^{\pm}$, K±, p±, $\Lambda$, $\bar{\Lambda}$, and
K${}^{0}_{\text{S}}$. Note that the $\beta$ factors for the $\text{V}^{0}$s
also include the effect of the $\text{V}^{0}$ cuts, presented in Section 5.

\begin{overpic}[clip,rviewport=0 0.143 1
1,width=195.12767pt]{Plots/fac_158_All_beta_c0_p1} \put(20.0,53.0){$\pi^{+}$}
\end{overpic}\begin{overpic}[clip,rviewport=0 0.143 1
1,width=195.12767pt]{Plots/fac_158_All_beta_c1_p1} \put(20.0,53.0){$\pi^{-}$}
\end{overpic}\begin{overpic}[clip,rviewport=0 0.143 1
1,width=195.12767pt]{Plots/fac_158_All_beta_c0_p2} \put(20.0,53.0){K${}^{+}$}
\end{overpic}\begin{overpic}[clip,rviewport=0 0.143 1
1,width=195.12767pt]{Plots/fac_158_All_beta_c1_p2} \put(20.0,53.0){K${}^{-}$}
\end{overpic}\begin{overpic}[clip,rviewport=0 0 1
1,width=195.12767pt]{Plots/fac_158_All_beta_c0_p3} \put(20.0,63.0){p${}^{+}$}
\end{overpic}\begin{overpic}[clip,rviewport=0 0 1
1,width=195.12767pt]{Plots/fac_158_All_beta_c1_p3} \put(20.0,63.0){p${}^{-}$}
\end{overpic}\begin{overpic}[clip,rviewport=0 0 1
1.1,width=429.28616pt]{Plots/beta158} \put(0.0,22.0){$\Lambda$ }
\put(33.0,22.0){$\bar{\Lambda}$ } \put(67.0,22.0){K${}^{0}_{\text{S}}$ }
\end{overpic}

Figure 8: $\beta$ correction factors for the 158
$\mbox{Ge\kern-1.00006ptV}\kern-1.00006pt/\kern-0.50003ptc$ data set (see Fig.
23 in the appendix for $p_{\text{beam}}=350\leavevmode\nobreak\
\mbox{Ge\kern-1.00006ptV}\kern-1.00006pt/\kern-0.50003ptc$).

The geometrical acceptance of the detector is the dominant contribution to the
$\beta$ factor at large $p_{\text{T}}$. Most of the overall structure visible
in the $\beta$ plots is due to the acceptance and reflects the aspect ratio of
the TPCs (rectangular in the $xy$ plane) and the bending of particles in the
magnetic field. At particle momenta of ${\sim}5$
$\mbox{Ge\kern-1.00006ptV}\kern-1.00006pt/\kern-0.50003ptc$, the TPCs provide
full coverage for $p_{\text{T}}\leq 0.2$
$\mbox{Ge\kern-1.00006ptV}\kern-1.00006pt/\kern-0.50003ptc$ increasing up to
$p_{\text{T}}\leq 1$
$\mbox{Ge\kern-1.00006ptV}\kern-1.00006pt/\kern-0.50003ptc$ at $\sim$30
$\mbox{Ge\kern-1.00006ptV}\kern-1.00006pt/\kern-0.50003ptc$. Due to the
requirement on the number of clusters in the different TPCs (cf. Section 3.4),
the coverage decreases at higher momenta due to the forward gap between the
TPCs and at lower momenta due to the strong bending of particles in the
magnetic field. The fiducial acceptance cuts assure that these effects are
taken into account by a purely geometrical correction and the corresponding
efficiency is simply given by the number of fiducial $\phi$ bins in the map of
the ($\phi$, $p$, $p_{\text{T}}$) acceptance.

Other corrections subsumed in $\beta$ are related to the efficiencies of the
event selection which, as mentioned above, partially cancels out with the
$\alpha$ correction. By construction, the reconstruction and selection
efficiency within the fiducial acceptance is $>90\%$, but typically $>95\%$.
$\beta$ also includes the effects of bin migration due to the finite momentum
resolution ($n_{\text{gen}}$ is counted in a bin of generated $p$ and
$p_{\text{T}}$ whereas $\hat{n}$ in a bin of reconstructed momenta). In most
of the bins, this correction is at the sub-percent level, as the bin width is
much larger than the momentum resolution. Only at $p\gtrsim 40$
$\mbox{Ge\kern-1.00006ptV}\kern-1.00006pt/\kern-0.50003ptc$ this correction
becomes significant but stays $\leq 5\%$ over the whole phase space studied
here.

Finally, $\beta$ also includes corrections for “feed-down”, i.e. the fact that
a fraction of the measured particles are not produced in the main target
interaction, but instead, in processes like interactions of secondary
particles inside the target or in the detector material, or the decay of
unstable particles which are produced in the main interaction. Therefore, in
contrast to the corrections discussed before, which are efficiency corrections
($\hat{n}/n_{\text{gen}}<1$), the feed-down correction gives rise to
$\hat{n}/n_{\text{gen}}>1$. The particles most affected by feed-down are
protons and anti-protons for which a substantial fraction originates from weak
decays, most prominently the decays of $\Lambda$ and $\bar{\Lambda}$, leading
to $\beta$ factors $>1$ in $p$-$p_{\text{T}}$ regions where the other
efficiency-type corrections are near unity (see the p± panels in Fig. 8). Due
to the potentially large model dependence of this correction, it is treated
with special care as detailed in the next section.

###  7.2 Feed-down from weak decays

According to the two hadronic generators used here, the contribution of weak
decays is negligible for K±, but typically several percent for $\uppi^{\pm}$
and up to 20% for p±, depending on the phase-space bin. K${}^{0}_{\text{S}}$
decays are responsible for $\gtrsim 70\%$ of the feed-down to charged pions
and $\Lambda$ and $\bar{\Lambda}$ decays dominate the feed-down to p and
$\bar{\text{p}}$ respectively. The $\Lambda(\bar{\Lambda})$ and
K${}^{0}_{\text{S}}$ production varies substantially among different hadronic
event generators and therefore the model dependence on the $\beta$ correction
for $\uppi^{\pm}$ and p± could be very large. To avoid the corresponding large
systematic uncertainties, we use here the measured spectra of $\Lambda$,
$\bar{\Lambda}$, and K${}^{0}_{\text{S}}$ to correct the feed-down
contribution from these particles.

The procedure adopted for this purpose is based on a re-weighting of the
simulated particles which are produced from the decay of $\Lambda$,
$\bar{\Lambda}$, and K${}^{0}_{\text{S}}$, see also Ref. [35]. This weight is
determined by the ratio, $R$, between the reconstructed spectra of these
spectra in data and simulation. In Fig. 9 we show examples of the ratios for
the three $\text{V}^{0}$ particles for the model Epos 1.99 and beam momentum
of 158 $\mbox{Ge\kern-1.00006ptV}\kern-1.00006pt/\kern-0.50003ptc$. A fit of
$R$ as a function of $p$ and $p_{\text{T}}$ is performed by using a log-normal
function in $p$ which parameters are interpolated as a function of
$p_{\text{T}}$ by a second-degree polynomial function. This parametrization is
shown as colored lines in Fig. 9. The weight given for the simulated particles
is then computed by using the parametrization of the ratio between measured
and generated spectra. Note that since $R$ is calculated from the ratio of
reconstructed $\text{V}^{0}$ spectra without feed-down correction from heavier
baryons such as $\Omega$ and $\Xi$, it also adjusts the feed-down contribution
from $\text{V}^{0}$s produced in decays of these baryons.

\begin{overpic}[clip,rviewport=0 0 1 1,width=134.42113pt]{Plots/par_158_h0_m0}
\put(18.0,84.0){$\Lambda$ } \end{overpic}\begin{overpic}[clip,rviewport=0 0 1
1,width=134.42113pt]{Plots/par_158_h1_m0} \put(18.0,84.0){$\bar{\Lambda}$ }
\end{overpic}\begin{overpic}[clip,rviewport=0 0 1
1,width=134.42113pt]{Plots/par_158_h2_m0} \put(18.0,84.0){K${}^{0}_{\text{S}}$
} \end{overpic}

Figure 9: Ratio between the measured and generated spectra, $R$, as a function
of $p$ (Epos 1.99) for $\Lambda$, $\bar{\Lambda}$, and K${}^{0}_{\text{S}}$
particles. The markers show different $p_{\text{T}}$ bins and the colored
lines show the result of the parametrization. On the bottom of each panel the
differences $\Delta$ between the observed value of $R$ and the parametrization
divided by its uncertainty $\sigma$ are shown.

Concerning the weak feed-down for the particle spectra of $\text{V}^{0}$s, the
correction is negligible for K${}^{0}_{\text{S}}$, but can reach up to 25% for
$\Lambda$ and $\bar{\Lambda}$, in which case the decaying particles are mostly
charged and neutral $\Xi$ baryons as well as $\Omega^{\pm}$. Since we did not
measure the production spectra of these particles, this correction is fully
model-dependent with correspondingly larger uncertainties than achieved for
$\uppi^{\pm}$ and p± from the primary interaction.

###  7.3 Integration over $p_{\text{T}}$

For some purposes it is useful to calculate the single-differential spectra,
$1/N\,\text{d}n/\text{d}p$, by integrating the double-differential spectra
given by Eq. 12 over $p_{\text{T}}$. Since the measured spectra do not cover
the full $p_{\text{T}}$ range, an extrapolation is needed to perform this
integration. We therefore fit the double-differential spectra as a function of
$p_{\text{T}}$ for each $p$ bin and then use the integral of the fitted
function to extrapolate the measured spectra to full phase space in
$p_{\text{T}}$.

We found that a Gaussian function convoluted with an exponential one gives a
very satisfactory description of the spectra. For the purpose of evaluating
the systematic uncertainty of the extrapolation we also used an exponential in
transverse mass,
$m_{\text{T}}=\sqrt{\smash[b]{p_{\text{T}}^{2}\,c^{2}+m^{2}\,c^{4}}}$, which
describes the data equally well. The $p_{\text{T}}$-integrated spectra are
computed by summing the measured spectra over all the available $p_{\text{T}}$
bins and adding the integral of the fitted function over the remaining
$p_{\text{T}}$ range. Single-differential spectra are calculated only for $p$
bins where the fraction of the extrapolation is smaller than 5% of the total
for the charged hadrons and 20% for the $\text{V}^{0}$ particles.

###  7.4 Uncertainties

\begin{overpic}[clip,rviewport=0 0 1
1,width=195.12767pt]{Plots/syst_158_c0_p1_std} \put(2.0,68.0){$\pi^{+}$}
\end{overpic}\begin{overpic}[clip,rviewport=0 0 1
1,width=195.12767pt]{Plots/syst_158_c1_p1_std} \put(2.0,68.0){$\pi^{-}$}
\end{overpic}

\begin{overpic}[clip,rviewport=0 0 1
1,width=195.12767pt]{Plots/syst_158_c0_p2_std} \put(2.0,68.0){K${}^{+}$}
\end{overpic}\begin{overpic}[clip,rviewport=0 0 1
1,width=195.12767pt]{Plots/syst_158_c1_p2_std} \put(2.0,68.0){K${}^{-}$}
\end{overpic}

\begin{overpic}[clip,rviewport=0 0 1
1,width=195.12767pt]{Plots/syst_158_c0_p3_std} \put(2.0,68.0){p${}^{+}$}
\end{overpic}\begin{overpic}[clip,rviewport=0 0 1
1,width=195.12767pt]{Plots/syst_158_c1_p3_std}
\put(2.0,68.0){$\bar{\text{p}}$} \end{overpic}

\begin{overpic}[clip,rviewport=0.03 0 0.97
1,height=114.90903pt]{Plots/syst_158_h0} \put(20.0,16.0){$\Lambda$ }
\end{overpic}\begin{overpic}[clip,rviewport=0.1 0 0.97
1,height=114.90903pt]{Plots/syst_158_h1} \put(15.0,16.0){$\bar{\Lambda}$ }
\end{overpic}

\begin{overpic}[clip,rviewport=0.12 0 0.97
1,height=114.90903pt]{Plots/syst_158_h2} \put(13.0,16.0){K${}^{0}_{\text{S}}$
} \end{overpic}

Figure 10: Systematic uncertainties of the single-differential spectra
$\text{d}n/\text{d}p$ for the charged hadrons (top three rows) and
$\text{V}^{0}$s (bottom row) as a function of momentum for the data set
recorded with
$p_{\text{beam}}=158\mbox{Ge\kern-1.00006ptV}\kern-1.00006pt/\kern-0.50003ptc$
(see Fig. 24 in the appendix for $p_{\text{beam}}=350\leavevmode\nobreak\
\mbox{Ge\kern-1.00006ptV}\kern-1.00006pt/\kern-0.50003ptc$).

####  7.4.1 Statistical uncertainties

The statistical uncertainties of the measured spectra, Eq. 12, are dominated
by the statistical uncertainties of $\hat{n}_{\text{I}}$ originating from the
$\mathrm{d}E/\mathrm{d}x$-fit in case of charged hadrons and from the
invariant-mass-fit for $\text{V}^{0}$ particles. These are in general larger
than the simple Poisson uncertainties,
$\sigma(\hat{n}_{\text{I}})\gtrsim\sqrt{\hat{n}_{\text{I}}}$. Since the number
of target-removed tracks is substantially smaller than the target-inserted
ones, the statistical uncertainties on $b\,n^{\text{R}}$ can be neglected.
Furthermore, the statistical uncertainty of $C_{\text{sim}}$ due to the
limited number of simulated tracks is taken into account, but it constitutes
only a minor contribution to the overall uncertainty.

####  7.4.2 Systematic uncertainties

The contributions of different sources to the overall systematic uncertainty
of the charged hadron and $\text{V}^{0}$ spectra as a function of $p$ are
displayed in Fig. 10 and are listed in the following with the corresponding
label in the figure given in brackets.

##### Modeling of the energy loss distributions ($\mathrm{d}E/\mathrm{d}x$)

The uncertainties related to the $\mathrm{d}E/\mathrm{d}x$ model used for the
fit to establish the fractions of $\uppi^{\pm}$, K±, and p± are estimated by
repeating the fit with different configurations of the model of the
$\mathrm{d}E/\mathrm{d}x$ distribution including different assumptions of the
center position of the Gaussian constraints on the mean value of the
$\mathrm{d}E/\mathrm{d}x$ of the six particles (moved separately by two
standard deviations in both directions) and different assumptions of the shape
and signal dependence of the distribution (see Ref. [54] for more details).
The resulting spectra obtained with these model variations are compared to the
standard spectra and the size of the uncertainties is taken as the differences
between the extreme cases and the standard one.

##### Minimum Bias Interaction Trigger Efficiency (T2)

Distortions of the spectra due to the minimum bias interaction trigger are
corrected for with the $C_{\text{sim}}$ correction factor, see Eq. 14. Since
both, the number of events and the number of tracks are affected by the
trigger, the model differences were estimated by combining both $\alpha$ and
$\beta$ factors. To compute the systematic uncertainties, we first isolate the
contributions of the trigger efficiency to the correction factor,
$\alpha_{T2}$ and $\beta_{T2}$, and then compute the factor
$\alpha_{T2}/\beta_{T2}$ for the two event generators separately. The relative
differences between the extreme values of $\alpha_{T2}/\beta_{T2}$ and the
average one are used to define the relative systematic uncertainties.

##### Main-Vertex Cut (vtx Z)

The model dependence of $C_{\text{sim}}$ due to the cut on the $z$ position of
the main vertex in the event selection is evaluated analogously to the
systematics of the interaction trigger.

##### Feed Down (FD)

The systematic uncertainties of the feed-down correction are estimated by
comparing the differences between the corrections predicted by the two
different event generators. The corresponding systematic uncertainties for the
feed down to $\text{V}^{0}$s are substantial, but since the predictions of the
feed down to $\uppi^{\pm}$, K±, and p± from $\Lambda$, $\bar{\Lambda}$, and
K${}^{0}_{\text{S}}$ are constrained to the $\text{V}^{0}$ data, the feed-
down-related systematics for the charged hadron spectra are small.

##### Event Topology (RST/WST)

The data set is subdivided into two statistically independent subsets based on
the sign of the product $q\cos\phi$, where $q$ denotes the charge and $\phi$
is the azimuthal angle (see Section 3.4). Since neither the beam nor the
target is polarized, we expect identical spectra for the two data sets.
Differences between the particle spectra derived for the two data sets are
small ($\lesssim 3\%$ in most of the phase space bins), but larger than the
statistical uncertainties. We interpret these differences as an evaluation of
the residual disagreement between the measured data and the idealized detector
simulation originating from e.g. calibration uncertainties not present in the
simulated events and add them in quadrature to the other contributions.

##### Selection of $\text{V}^{0}$ candidates ($\text{V}^{0}$ sel.)

The minimum number of clusters required for the daughter tracks of the
$\text{V}^{0}$ selection is changed from 30 to 20 and both, the signal
extraction and the calculation of $C_{\text{sim}}$, are repeated. This cut
variation results in slightly different $\text{V}^{0}$ spectra and the
differences are conservatively added to the overall systematic uncertainty of
the $\text{V}^{0}$ spectra.

#####  $m_{\text{inv}}$ Background Model (BG)

The shape of the background used for the $m_{\text{inv}}$ fit of
$\text{V}^{0}$ candidates is changed from a second-degree to a third-degree
polynomial. The systematic uncertainty related to the background subtraction
is then estimated as the relative difference between the particle
multiplicity, obtained by the new background function, and the standard one.

Figure 11: Production spectra of $\uppi^{\pm}$, K± and p($\bar{\text{p}}$) in
$\uppi^{-}\text{-\,C}$ interactions at $p_{\text{beam}}=158$
$\mbox{Ge\kern-1.00006ptV}\kern-1.00006pt/\kern-0.50003ptc$. For each bin in
momentum $p$, the spectrum was multiplied by $10^{m}$ with the value of $m$
shown on the right. Error bars show the statistical uncertainties and are most
of the times smaller than the marker size.

###  7.5 Results

The measured double-differential spectra of $\uppi^{\pm}$, K±, p±, $\Lambda$,
$\bar{\Lambda}$, and K${}^{0}_{\text{S}}$ spectra in $\uppi^{-}\text{-\,C}$
interactions at 158 and 350
$\mbox{Ge\kern-1.00006ptV}\kern-1.00006pt/\kern-0.50003ptc$ are shown in Figs.
11, 12, and 13. The single-differential, $p_{\text{T}}$-integrated spectra are
displayed in Figs. 15 and 16 and will be discussed in more detail in the next
Section. Tables of the measured spectra can be downloaded at [61].

Figure 12: Production spectra of $\uppi^{\pm}$, K± and p($\bar{\text{p}}$) in
$\uppi^{-}\text{-\,C}$ interactions at $p_{\text{beam}}=350$
$\mbox{Ge\kern-1.00006ptV}\kern-1.00006pt/\kern-0.50003ptc$. For each bin in
momentum $p$, the spectrum was multiplied by $10^{m}$ with the value of $m$
shown on the right. Error bars show the statistical uncertainties and are most
of the times smaller than the marker size.

Figure 13: Production spectra of $\Lambda$, $\bar{\Lambda}$ and
K${}^{0}_{\text{S}}$ in $\uppi^{-}\text{-\,C}$ interactions at
$p_{\text{beam}}=158$ and $350$
$\mbox{Ge\kern-1.00006ptV}\kern-1.00006pt/\kern-0.50003ptc$. For each bin in
momentum $p$, the spectrum was multiplied by $10^{m}$ with the value of $m$
shown on the right. Error bars show the statistical uncertainties.

##  8 Discussion

Figure 14: Ratio of particle spectra for data (points with statistical error
bars) and models (lines). Ratios at beam energies 158 and 350
$\mbox{Ge\kern-1.00006ptV}\kern-1.00006pt/\kern-0.50003ptc$ are displayed on
the left and right column respectively.

For a first assessment of the validity of hadronic interaction models used in
air-shower simulations, we compare the ratio of measured particle spectra to
the model predictions in Fig. 14. Solid lines denote recent model-tunes [62,
63, 64, 65, 66] based on LHC and fixed target data, whereas dashed lines show
previous versions of these models [43, 29]. Most of the models agree
reasonably well with the measured pion charge ratio shown in the first row of
Fig. 14, but both versions of the Sibyll model overpredict the ratio at high
particle momenta for the 158
$\mbox{Ge\kern-1.00006ptV}\kern-1.00006pt/\kern-0.50003ptc$ data set. The kaon
charge ratio in the second row is best described by Epos LHC. All models
underpredict the antiproton-to-proton ratio at low momenta shown in the third
row and the shape of the momentum-dependence of this ratio with two inflection
points is best reproduced by Epos 1.99. We compare the production of charged
to neutral kaons in the fourth row by computing the ratio of
$\frac{1}{2}(\text{K}^{+}+\text{K}^{-})$ to K${}^{0}_{\text{S}}$, where K is
the shorthand for the production spectrum of the particles. This ratio is
expected to be unity from simple arguments based on the counting of valence
quarks of the beam and target. Indeed all models but the old version of Sibyll
predict a value close to 1 whereas our data suggest values in the range of 1.2
to 1.3. The difference to the expectation of 1 is about three times the
systematic uncertainty assigned to the integrated kaon spectra, see Figs. 10
and 24. Finally, the ratio of $\bar{\Lambda}$ to $\Lambda$ baryons is best
described by the Epos 1.99 model, whereas all recent model re-tunes slightly
overpredict the production of $\bar{\Lambda}$ at intermediate momenta of
$p/p_{\text{beam}}\sim 0.1$.

Figure 15: Comparison of the $p_{\text{T}}$-integrated particle production
spectra of $\uppi^{\pm}$, K± and p($\bar{\text{p}}$) at 158
$\mbox{Ge\kern-1.00006ptV}\kern-1.00006pt/\kern-0.50003ptc$ with predictions
of hadronic interaction models. The data is shown as black circles with
statistical error bars. Systematic uncertainties are displayed by gray
rectangles.

Figure 16: Comparison of the $p_{\text{T}}$-integrated particle production
spectra of $\uppi^{\pm}$, K± and p($\bar{\text{p}}$) at 350
$\mbox{Ge\kern-1.00006ptV}\kern-1.00006pt/\kern-0.50003ptc$ with predictions
of hadronic interaction models. The data is shown as black circles with
statistical error bars. Systematic uncertainties are displayed by gray
rectangles.

Figure 17: Comparison of the $p_{\text{T}}$-integrated particle production
spectra of $\Lambda$, $\bar{\Lambda}$ and K${}^{0}_{\text{S}}$ at 158 and 350
$\mbox{Ge\kern-1.00006ptV}\kern-1.00006pt/\kern-0.50003ptc$ with predictions
of hadronic interaction models. The data is shown as black circles with
statistical error bars. Systematic uncertainties are displayed by gray
rectangles.

Furthermore, we compare the $p_{\text{T}}$-integrated data to predictions from
hadronic interaction models in Figs. 15, 16, and 17. As can be seen, none of
the models provides a satisfactory description of our data and each of the
recent re-tunes has its own deficiencies. Of all the models, Sibyll 2.3c gives
the worst description of the charged pion spectra and it under-predicts
$\bar{\text{p}}$ and $\bar{\Lambda}$ production. QGSJet II-04 fails
spectacularly to reproduce kaon-production in $\uppi^{-}\text{-\,C}$
interactions and also produces too few $\bar{\text{p}}$ and $\bar{\Lambda}$
particles. In many aspects, the previous version of the Epos model (Epos 1.99)
gives a better prediction of our data than the current Epos LHC version. In
particular, Epos 1.99 provides the best description of the charged pion
spectra and a near-spot-on prediction of $\bar{\text{p}}$ production, whereas
the newer version of the model gives the best match to our $\bar{\Lambda}$
measurements. It will be interesting to see air shower predictions of future
versions of these models that describe all aspects of our data.

For a further qualitative analysis of the relevance of this measurement to the
“muon puzzle” in cosmic-ray-induced air showers, it is useful to recall that
the key to model muon production in air showers is to correctly predict the
fraction $f$ of the energy that remains in the hadronic cascade in each
interaction and is not lost to the electromagnetic component via $\uppi^{0}$
production. In a simplified model with the production of only charged and
neutral pions, this fraction is $f=2/3$ and after $n$ interactions $(2/3)^{n}$
of the initial energy is left in the hadronic component. Muons are produced
when the pions reach low energies and decay, which happens after about $n=8$
generations of interactions for an air shower induced by a primary of
$10^{20}$ eV [67]. In a more realistic scenario the energy transfer to the
hadronic component is $f=(2/3+\Delta)$, where $\Delta$ accounts for hadronic
particles without dominant electromagnetic decay channels such as $\uprho^{0}$
mesons [68, 63] or baryons [29]. Then a fraction of
$(2/3+\Delta)^{n}\approx(2/3)^{n}\,(1+3/2\;n\,\Delta)$ of the initial cosmic-
ray energy can produce muons after $n$ interactions and only if the value of
$\Delta$ is accurately known throughout the whole chain of interactions, there
is hope for a precise prediction of the muon number in air showers.

The production of $\uprho^{0}$ mesons in $\uppi^{-}\text{-\,C}$ interactions
has already been addressed by in Ref. [69] and the integrated production
spectrum gives $\Delta_{\uprho^{0}}=\big{(}7.7\pm 0.1(\text{stat.})\pm
0.2(\text{syst.})\big{)}\%$ at 158
$\mbox{Ge\kern-1.00006ptV}\kern-1.00006pt/\kern-0.50003ptc$. Here we can
comment on the baryon production for which the best proxy is the production of
anti-protons since protons can also originate from target fragmentation. The
average energy fraction transferred to anti-protons is displayed in Fig. 18,
and is obtained by integrating the measured $p\,\mathrm{d}n/\mathrm{d}p$
spectra including an extrapolation to the full beam momentum [54]. This gives
$\Delta_{\bar{\text{p}}}=\big{(}1.59\pm 0.01(\text{stat.})\;\pm
0.07(\text{syst.})\pm 0.01(\text{mod.})\big{)}\%$ and $\big{(}1.76\pm
0.01(\text{stat.})\;\pm 0.08(\text{syst.})\pm 0.35(\text{mod.})\big{)}\%$ at
158 and 350 $\mbox{Ge\kern-1.00006ptV}\kern-1.00006pt/\kern-0.50003ptc$, where
the last of the three quoted uncertainties is due to the model-dependence of
the extrapolation to full beam momentum. Note that the anti-proton fraction
constrains the production of p, $\bar{\text{p}}$, n, and $\bar{\text{n}}$,
i.e. naively $\Delta_{\text{baryon}}\sim 4\Delta_{\bar{\text{p}}}$.
Numerically we find $\Delta_{\text{baryon}}\sim\Delta_{\uprho^{0}}$, i.e. both
processes are about equally important for the evolution of air showers. Our
measurement can be used to normalize the model differences at low energies
(cf. Fig. 18), leaving then only the energy-evolution of
$\Delta_{\text{baryon}}$ as the remaining uncertainty of baryon production in
air showers.

Figure 18: Energy fraction transferred to anti-protons as derived from the
measurement presented in this article (data points) and as predicted by
hadronic interaction models over the whole range of beam energies relevant for
air showers.

##  9 Summary

In this article, we presented a new measurement of particle production in
interactions of negatively charged pions with carbon nuclei at beam momenta of
158 and 350 $\mbox{Ge\kern-1.00006ptV}\kern-1.00006pt/\kern-0.50003ptc$. We
estimated the production cross section and determined the double-differential
$p$-$p_{\text{T}}$ spectra of produced $\uppi^{\pm}$, K±, p±, $\Lambda$,
$\bar{\Lambda}$, and K${}^{0}_{\text{S}}$. This measurement provides a unique
reference data set with unprecedented precision and large phase-space coverage
to enable future tuning of models used for the simulation of particle
production in extensive air showers in which pions are the most numerous
projectiles. None of the current state-of-the art hadronic interaction models
describes the measured particle spectra well. A tuning of these models to
match the measurements from at SPS energies will significantly reduce the
uncertainties in predictions of muons in air showers.

## Acknowledgments

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
138136/138152), 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,
2019/33/B/ST9/03059 and 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 Russian Science
Foundation (grant 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).

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

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##  Appendix A Cross Section

The interaction trigger is set by an offline requirement on the absence of
tracks within a radius $r_{\text{trig}}$ with respect to the beam particle 3.7
m downstream of the target, i.e. after passing through the magnetic field of
the first superconducting dipole magnet. To ease a possible re-analysis of the
measured trigger cross section with different choice of model corrections, we
show a visual representation of the efficiency of this requirement as a
function of curvature $q/p$ and transverse momentum $p_{\text{T}}$ in Fig. 19.
These efficiency maps are available electronically at [61]. The cumulative
efficiencies for our choice of models is shown in Fig. 20 as a function of
$r_{\text{trig}}$. These efficiencies can be thought of as the result of
folding the $(q/p,p_{\text{T}})$-distribution of a specific process with the
two-dimensional efficiency map.

Figure 19: Efficiency of the interaction trigger for particles with a certain
curvature $q/p$ and transverse momentum $p_{\text{T}}$ for the two beam
energies 158 (left, trigger radius 0.9 cm) and 350
$\mbox{Ge\kern-1.00006ptV}\kern-1.00006pt/\kern-0.50003ptc$ (right, trigger
radius 0.6 cm).

Figure 20: Fraction of events leading to a forward particle within a given
radial trigger distance from the beam. Columns show from left to right
simulations of elastic, quasi-elastic and production interactions. The top row
shows the fractions for beam energies of 158
$\mbox{Ge\kern-1.00006ptV}\kern-1.00006pt/\kern-0.50003ptc$ and the bottom row
for 350 $\mbox{Ge\kern-1.00006ptV}\kern-1.00006pt/\kern-0.50003ptc$.

##  Appendix B Binning

The data analysis is performed by splitting the data into 2-dimensional phase-
space bins of the $p$ and $p_{\text{T}}$ variables. For the charged hadron
analysis a unique phase-space binning was defined. The $p$ intervals are
nearly uniform in $\lg p$. Only small adjustments were done to move the
crossing points of the energy deposit function of different particles closer
to the center of the bins. Since some of these bins in the crossing regions
will be removed from the analysis, this strategy has been effective to reduce
the number of removed bins. The average width of the $\lg p$ intervals is
$\Delta\lg\left(p/(\mbox{Ge\kern-1.00006ptV}\kern-1.00006pt/\kern-0.50003ptc)\right)=0.1$.
Concerning the $p_{\text{T}}$ intervals, the bin width increases with
$p_{\text{T}}$ from $\Delta p_{\text{T}}=0.1$ to $0.5$
$\mbox{Ge\kern-1.00006ptV}\kern-1.00006pt/\kern-0.50003ptc$. In Fig. 21 we
show the phase-space binning used for the charged hadron analysis.

Because the $\text{V}^{0}$ analysis is done independently for the $\Lambda$,
$\bar{\Lambda}$, and K${}^{0}_{\text{S}}$, the phase-space binning is not
required to be unique. However, because the statistics is similar for
$\Lambda$ and $\bar{\Lambda}$, the same binning was defined for these two
particles. For either $\Lambda(\bar{\Lambda})$ or K${}^{0}_{\text{S}}$ the $p$
intervals vary from
$\Delta\lg\left(p/(\mbox{Ge\kern-1.00006ptV}\kern-1.00006pt/\kern-0.50003ptc)\right)=0.2$
to $0.3$. Concerning the $p_{\text{T}}$ intervals, the widths vary from
$\Delta p_{\text{T}}=0.2$ to $0.8$. In Fig. 22 we show the binning used for
the $\text{V}^{0}$ analysis.

Figure 21: Illustration of the phase-space binning used for the charged hadron
analysis. Figure 22: Illustration of the phase-space binning used for the
$\text{V}^{0}$ analysis. The plot on the left show the binning used for
$\Lambda$ and $\bar{\Lambda}$, and the plot on the right shows the binning
used for K${}^{0}_{\text{S}}$.

##  Appendix C Additional Plots for the 350
$\mbox{Ge\kern-1.00006ptV}\kern-1.00006pt/\kern-0.50003ptc$ Data Set

For a more concise flow of the main part of this article, some of the plots
are only shown for the 158
$\mbox{Ge\kern-1.00006ptV}\kern-1.00006pt/\kern-0.50003ptc$ data set. In this
appendix, the counterparts for the 350
$\mbox{Ge\kern-1.00006ptV}\kern-1.00006pt/\kern-0.50003ptc$ data are
presented.

\begin{overpic}[clip,rviewport=0 0.143 1
1,width=195.12767pt]{Plots/fac_350_All_beta_c0_p1} \put(20.0,53.0){$\pi^{+}$}
\end{overpic}\begin{overpic}[clip,rviewport=0 0.143 1
1,width=195.12767pt]{Plots/fac_350_All_beta_c1_p1} \put(20.0,53.0){$\pi^{-}$}
\end{overpic}\begin{overpic}[clip,rviewport=0 0.143 1
1,width=195.12767pt]{Plots/fac_350_All_beta_c0_p2} \put(20.0,53.0){K${}^{+}$}
\end{overpic}\begin{overpic}[clip,rviewport=0 0.143 1
1,width=195.12767pt]{Plots/fac_350_All_beta_c1_p2} \put(20.0,53.0){K${}^{-}$}
\end{overpic}\begin{overpic}[clip,rviewport=0 0 1
1,width=195.12767pt]{Plots/fac_350_All_beta_c0_p3} \put(20.0,63.0){p${}^{+}$}
\end{overpic}\begin{overpic}[clip,rviewport=0 0 1
1,width=195.12767pt]{Plots/fac_350_All_beta_c1_p3} \put(20.0,63.0){p${}^{-}$}
\end{overpic}\begin{overpic}[clip,rviewport=0 0 1
1.1,width=429.28616pt]{Plots/beta350} \put(0.0,22.0){$\Lambda$ }
\put(33.0,22.0){$\bar{\Lambda}$ } \put(67.0,22.0){K${}^{0}_{\text{S}}$ }
\end{overpic}

Figure 23: $\beta$ correction factors for the 350
$\mbox{Ge\kern-1.00006ptV}\kern-1.00006pt/\kern-0.50003ptc$ data set .

\begin{overpic}[clip,rviewport=0 0 1
1,width=195.12767pt]{Plots/syst_350_c0_p1_std} \put(2.0,68.0){$\pi^{+}$}
\end{overpic}\begin{overpic}[clip,rviewport=0 0 1
1,width=195.12767pt]{Plots/syst_350_c1_p1_std} \put(2.0,68.0){$\pi^{-}$}
\end{overpic}

\begin{overpic}[clip,rviewport=0 0 1
1,width=195.12767pt]{Plots/syst_350_c0_p2_std} \put(2.0,68.0){K${}^{+}$}
\end{overpic}\begin{overpic}[clip,rviewport=0 0 1
1,width=195.12767pt]{Plots/syst_350_c1_p2_std} \put(2.0,68.0){K${}^{-}$}
\end{overpic}

\begin{overpic}[clip,rviewport=0 0 1
1,width=195.12767pt]{Plots/syst_350_c0_p3_std} \put(2.0,68.0){p${}^{+}$}
\end{overpic}\begin{overpic}[clip,rviewport=0 0 1
1,width=195.12767pt]{Plots/syst_350_c1_p3_std}
\put(2.0,68.0){$\bar{\text{p}}$} \end{overpic}

\begin{overpic}[clip,rviewport=0.03 0 0.97
1,height=114.90903pt]{Plots/syst_350_h0} \put(20.0,16.0){$\Lambda$ }
\end{overpic}\begin{overpic}[clip,rviewport=0.1 0 0.97
1,height=114.90903pt]{Plots/syst_350_h1} \put(15.0,16.0){$\bar{\Lambda}$ }
\end{overpic}

\begin{overpic}[clip,rviewport=0.12 0 0.97
1,height=114.90903pt]{Plots/syst_350_h2} \put(13.0,16.0){K${}^{0}_{\text{S}}$
} \end{overpic}

Figure 24: Systematic uncertainties of the single-differential spectra
$\text{d}n/\text{d}p$ for the charged hadrons (top three rows) and
$\text{V}^{0}$s (bottom row) as a function of momentum for the data set
recorded with
$p_{\text{beam}}=350\mbox{Ge\kern-1.00006ptV}\kern-1.00006pt/\kern-0.50003ptc$
.

Figure 25: Energy deposit vs momentum of negatively and positively charged
tracks for the 350 $\mbox{Ge\kern-1.00006ptV}\kern-1.00006pt/\kern-0.50003ptc$
data set. The dashed lines indicate the average energy deposit for each
particle type .

The Collaboration

H. Adhikary 13, K.K. Allison 30, N. Amin 5, E.V. Andronov 25, T. Antićić 3,
I.-C. Arsene 12, Y. Balkova 18, M. Baszczyk 17, D. Battaglia 29, S. Bhosale
14, A. Blondel 4, M. Bogomilov 2, Y. Bondar 13, N. Bostan 29, A. Brandin 24,
A. Bravar 27, W. Bryliński 21, J. Brzychczyk 16, M. Buryakov 23, M. Ćirković
26, M. Csanad  7,8, J. Cybowska 21, T. Czopowicz 13,21, A. Damyanova 27, N.
Davis 14, H. Dembinski 5, A. Dmitriev  23, W. Dominik 19, P. Dorosz 17, J.
Dumarchez 4, R. Engel 5, G.A. Feofilov 25, L. Fields 29, Z. Fodor 7,20, M.
Friend 9, A. Garibov 1, M. Gaździcki 6,13, O. Golosov 24, V. Golovatyuk  23,
M. Golubeva 22, K. Grebieszkow 21, F. Guber 22, A. Haesler 27, M. Haug 5, S.N.
Igolkin 25, S. Ilieva 2, A. Ivashkin 22, A. Izvestnyy 22, S.R. Johnson 30, K.
Kadija 3, N. Kargin 24, N. Karpushkin 22, E. Kashirin 24, M. Kiełbowicz 14,
V.A. Kireyeu 23, H. Kitagawa 10, R. Kolesnikov 23, D. Kolev 2, A. Korzenev 27,
Y. Koshio 10, V.N. Kovalenko 25, S. Kowalski 18, B. Kozłowski 21, A.
Krasnoperov 23, W. Kucewicz 17, M. Kuchowicz 20, M. Kuich 19, A. Kurepin 22,
A. László 7, M. Lewicki 20, G. Lykasov 23, V.V. Lyubushkin 23, M. Maćkowiak-
Pawłowska 21, I.C. Mariş 5, Z. Majka 16, A. Makhnev 22, B. Maksiak 15, A.I.
Malakhov 23, A. Marcinek 14, A.D. Marino 30, K. Marton 7, H.-J. Mathes 5, T.
Matulewicz 19, V. Matveev 23, G.L. Melkumov 23, A. Merzlaya 12, A.O. Merzlaya
16, B. Messerly 31, Ł. Mik 17, A. Morawiec 16, S. Morozov 22, Y. Nagai  8, T.
Nakadaira 9, M. Naskręt 20, S. Nishimori 9, V. Ozvenchuk 14, O. Panova 13, V.
Paolone 31, O. Petukhov 22, I. Pidhurskyi 6, R. Płaneta 16, P. Podlaski 19,
B.A. Popov 23,4, B. Porfy 7,8, M. Posiadała-Zezula 19, R.R. Prado 5, D.S.
Prokhorova 25, D. Pszczel 15, S. Puławski 18, J. Puzović 26, M. Ravonel 27, R.
Renfordt 18, D. Röhrich 11, E. Rondio 15, M. Roth 5, Ł. Rozpłochowski 14, M.
Rumyantsev 23, M. Ruprecht 5, A. Rustamov 1,6, M. Rybczynski 13, A. Rybicki
14, K. Sakashita 9, K. Schmidt 18, A.Yu. Seryakov 25, P. Seyboth 13, Y.
Shiraishi 10, M. Słodkowski 21, P. Staszel 16, G. Stefanek 13, J. Stepaniak
15, M. Strikhanov 24, H. Ströbele 6, T. Šuša 3, M. Szuba 5, R. Szukiewicz 20,
A. Taranenko 24, A. Tefelska 21, D. Tefelski 21, V. Tereshchenko 23, A. Toia
6, R. Tsenov 2, L. Turko 20, T.S. Tveter 12, R. Ulrich 5, M. Unger 5, M.
Urbaniak 18, F.F. Valiev 25, D. Veberič 5, V.V. Vechernin 25, V. Volkov 22, A.
Wickremasinghe 31,28, K. Wójcik 18, O. Wyszyński 13, A. Zaitsev 23, E.D.
Zimmerman 30, A. Zviagina 25, and R. Zwaska 28

${}^{\phantom{2}1}$ National Nuclear Research Center, Baku, Azerbaijan
${}^{\phantom{2}2}$ Faculty of Physics, University of Sofia, Sofia, Bulgaria
${}^{\phantom{2}3}$ Ruđer Bošković Institute, Zagreb, Croatia
${}^{\phantom{2}4}$ LPNHE, University of Paris VI and VII, Paris, France
${}^{\phantom{2}5}$ Karlsruhe Institute of Technology, Karlsruhe,
25 Germany
${}^{\phantom{2}6}$ University of Frankfurt, Frankfurt, Germany
${}^{\phantom{2}7}$ Wigner Research Centre for Physics of the
27 Hungarian Academy of Sciences, Budapest, Hungary
${}^{\phantom{2}8}$ Eötvös Loránd University, Budapest, Hungary
${}^{\phantom{2}9}$ Institute for Particle and Nuclear Studies, Tsukuba,
9 Japan
10 Okayama University, Japan
11 University of Bergen, Bergen, Norway
12 University of Oslo, Oslo, Norway
13 Jan Kochanowski University in Kielce, Poland
14 Institute of Nuclear Physics, Polish Academy of
14 Sciences, Cracow, Poland
15 National Centre for Nuclear Research, Warsaw,
15 Poland
16 Jagiellonian University, Cracow, Poland
17 AGH - University of Science and Technology,
17 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 Institute for Nuclear Research, Moscow, Russia
23 Joint Institute for Nuclear Research, Dubna, Russia
24 National Research Nuclear University (Moscow
24 Engineering Physics Institute), Moscow, Russia
25 St. Petersburg State University, St. Petersburg, Russia
26 University of Belgrade, Belgrade, Serbia
27 University of Geneva, Geneva, Switzerland
28 Fermilab, Batavia, USA
29 University of Notre Dame, Notre Dame , USA
30 University of Colorado, Boulder, USA
31 University of Pittsburgh, Pittsburgh, USA