Measurement of the production cross section of 31
$\mbox{Ge\kern-1.00006ptV}\\!/\\!c$ protons on carbon via beam attenuation in
a 90-cm-long target CERN-EP-2020-198 Phys. Rev. D The production cross section
of 30.92 $\mbox{Ge\kern-1.00006ptV}\\!/\\!c$ protons on carbon is measured by
the spectrometer at the CERN SPS by means of beam attenuation in a copy
(replica) of the 90-cm-long target of the T2K neutrino oscillation experiment.
The employed method for direct production cross-section estimation minimizes
model corrections for elastic and quasi-elastic interactions. The obtained
production cross section is $\sigma_{\mathrm{prod}}\leavevmode\nobreak\
=\leavevmode\nobreak\ 227.6\leavevmode\nobreak\ \pm\leavevmode\nobreak\
0.8\mathrm{(stat)}\leavevmode\nobreak\ _{-\leavevmode\nobreak\
3.2}^{+\leavevmode\nobreak\ 1.9}\mathrm{(sys)}\leavevmode\nobreak\
{-\leavevmode\nobreak\ 0.8}\mathrm{(mod)}$ mb. It is in agreement with
previous results obtained with a thin carbon target, while providing improved
precision with a total fractional uncertainty of less than 2$\%$. This direct
measurement will reduce the uncertainty of the T2K neutrino flux prediction.

##  I Introduction

(SPS Heavy Ion and Neutrino Experiment) [1] is a fixed-target experiment at
the CERN Super Proton Synchrotron (SPS). It has a rich physics program
covering three different fields of research: strong-interaction physics,
cosmic ray physics and neutrino physics. ’s neutrino program includes hadron
production measurements for the Tokai-to-Kamioka (T2K) long-baseline neutrino
oscillation experiment in Japan [2] and Fermilab neutrino experiments on the
NuMI and LBNF beam lines: MINER$\nu$A [3], NO$\nu$A [4] and DUNE [5]. These
measurements are used to reduce the systematic uncertainties associated with
the prediction of the corresponding (anti)neutrino fluxes. A new calculation
of the production cross section for protons on carbon (p + C) via a
measurement of the number of beam particles remaining after interactions in a
replica of the 90-cm-long T2K graphite target is presented here.
The collaboration uses the following classification of nuclear interactions
based on the type of particles produced in the process111The same definitions
of nuclear interaction types are used by Bellettini et al. in [6], where
measurements of proton-nuclei cross-sections are described and interpreted..
An elastic process, that is coherent nuclear elastic scattering, is one in
which no new particles are produced. Except for coherent nuclear scattering,
all other processes due to the strong interaction are labeled as inelastic.
Furthermore, the inelastic interactions are subdivided into two groups: quasi-
elastic and production processes. Quasi-elastic processes result in the
fragmentation of the target nucleus. In production interactions, new hadrons
are produced. The probability for each of these three processes is governed by
the corresponding cross section and the total cross section is:

$\displaystyle\sigma_{\mathrm{tot}}=\sigma_{\mathrm{el}}+\sigma_{\mathrm{inel}},$
(1)

where

$\displaystyle\sigma_{\mathrm{inel}}=\sigma_{\mathrm{qe}}+\sigma_{\mathrm{prod}}.$
(2)

The experiment collected hadron production data for T2K during the runs in
years 2007, 2009 and 2010. A 30.92 $\mbox{Ge\kern-1.00006ptV}\\!/\\!c$ proton
beam and both thin and thick (90-cm-long T2K replica) targets were used.
Primary hadron interactions in the target were examined using a 2-cm-thin
carbon target, with a thickness equal to 4$\%$ of a nuclear interaction
length, $\lambda_{I}$. Both single and multiple re-interactions inside the
target were simultaneously studied using a long graphite target which is
equivalent to about 2$\lambda_{I}$. A schematic view of this target is shown
in Fig. 1. It is a copy of the T2K target - a 90-cm-long cylinder with a
1.3-cm radius. Previously obtained results from the hadron production
measurements can be found in [7, 8, 9, 10, 11, 12, 13]. They include inelastic
and production cross-section measurements from the thin target runs and hadron
yields from both the thin and replica target data sets. Their full
incorporation in the T2K neutrino flux simulation is ongoing. However, it has
already been reported that thin target results reduced the systematic
uncertainty of the (anti)neutrino fluxes in T2K to about 10$\%$ [14], and
further reduction to around 5$\%$ was achieved after incorporating the 2009
replica target charged pion results [15]. Even stronger constraints (about
4$\%$) are expected after including the measured charged-pion, kaon and proton
yields from the 2010 replica target data.

The focus of this paper is a measurement with the T2K replica target from the
2010 run, namely the extraction of the production cross section based on the
number of beam protons that scattered elastically or quasi-elastically in the
long target. This result is to be included in the future in the hadron
interaction rate re-weighting procedure that is part of the T2K neutrino flux
prediction. The goal of the current measurement is to improve the precision of
the previously obtained production cross-section results from the 2007 and
2009 thin target data [8, 11], thus reducing the associated systematic
uncertainty on the T2K flux prediction.

Figure 1: The T2K replica target with the upstream aluminum flange (1) and the
downstream plastic ring (2) used to support the target.

The structure of this paper is as follows: An overview of the spectrometer is
given in Sec. II. Section III describes the analysis procedure. The estimation
of systematic uncertainties is presented in Sec. IV, followed by a discussion
of the Monte Carlo models in Sec. V. Section VI reports the resulting
production cross section and compares it to previous measurements. A brief
conclusion closes the paper.

##  II The experiment

The experiment is located on a secondary beam line, the H2 beam line, of the
CERN SPS. The 400 $\mbox{Ge\kern-1.00006ptV}\\!/\\!c$ primary SPS proton beam
first strikes a target 535 m upstream of the experiment to produce a secondary
beam. Then, along the H2 line, a magnet system is employed to select the
desired particle momentum according to particle rigidity. Further beam
identification is performed by the detectors of the trigger system.
A schematic view of the setup and its coordinate system is given in Fig. 2.
The beam direction and size are checked by a set of scintillator counters -
S1, S2 and S3 used in coincidence and $V_{0}$ and $V_{1}^{P}$ used in anti-
coincidence. The S1 counter provides a timing reference (start signal) for the
experiment. Sitting just 0.5 cm upstream of the target, the S3 counter selects
particles that hit the target. Both veto counters, $V_{0}$ and $V_{1}^{P}$,
have 1-cm holes centered around the beam line and allow for selection of a
narrow beam. Beam particle identification is done by two Cherenkov detectors -
Cherenkov Differential Counter with Achromatic Ring Focus [16] (CEDAR) and
Threshold Cherenkov Counter (THC). For particles of the same momentum, only a
certain particle type will produce Cherenkov light, depending on the gas
pressure in the Cherenkov detectors. This allows for particle-type selection
and can provide an estimate of the beam composition at a given energy. The
trajectory of every incoming beam particle is reconstructed by a set of three
Beam Position Detectors: BPD-1, 2 and 3. Each BPD consists of two orthogonal
multi-wire proportional chambers that are comprised of a sense wire plane and
a cathode strip plane. The BPDs measure the position of the beam particle in
the transverse plane with a precision of 200 $\mu$m [1].
After the beam hits the target, the resulting particles go through the
spectrometer. During the 2010 run it comprised five Time Projection Chambers
(TPC) and three Time-of-Flight (ToF) scintillator walls. Two of the TPCs,
Vertex TPC1 (VTPC-1) and Vertex TPC2 (VTPC-2), are placed inside two
superconducting dipole magnets. Their combined maximum bending power is 9 T m.
Both VTPC-1 and VTPC-2 have one section on the left and one on the right side
of the beam line forming a 24-cm-wide gap centered around the beam. A third
TPC, the Gap TPC (GTPC), covers the forward region standing between the two
Vertex TPCs and is in the residual magnetic field of the two dipole magnets.
Further downstream of VTPC-2 are Main TPC Left (MTPC-L) and Main TPC Right
(MTPC-R). ’s TPC system provides precise momentum, charge and specific energy
loss (${\mathrm{d}}E\\!/\\!{\mathrm{d}}x$) measurements. Additional particle
identification, below 8 $\mbox{Ge\kern-1.00006ptV}\\!/\\!c$, is done with the
Time-of-Flight detectors: two side ToF-Left (ToF-L) and ToF-Right (ToF-R)
walls and a third ToF-Forward (ToF-F) wall. The ToF-F detector is crucial for
the T2K hadron production measurements. It consists of 80 plastic scintillator
bars oriented vertically and arranged in ten separate modules. The dimensions
of each bar are Width $\times$ Height $\times$ Length = 10 $\times$ 120
$\times$ 2.5 cm3. Two photomultiplier tubes (PMTs) on the top and the bottom
of each bar read out the produced signal.

(a)

(b)

Figure 2: Top (2(a)) and side (2(b)) view of the experimental setup during the
2010 T2K replica target data-taking. The orientation of the coordinate system
is shown in the bottom-left corner. The beam is aligned with the $z$-axis and
comes from the left.

##  III Analysis

###  A Data collected

During the year 2010 data-taking period, carried out two types of replica
target runs. The target was placed at the entrance of the TPC system,
longitudinally aligned along the $z$-axis. The majority of events, 9 $\times
10^{6}$, were collected using a 1.2 T m magnetic field configuration - also
referred to as the "low" magnetic field. These events were used to measure the
hadron yields from the surface of the T2K replica target, and the
corresponding results were published in [13]. In between the low-magnetic
field runs, the dipole magnets were operated at their maximum 9 T m bending
power and another 1.2 $\times 10^{6}$ events were recorded. This allowed for
better detection of beam protons that scattered elastically or quasi-
elastically and passed through the target. Analysis of this maximum magnetic
field data set is the subject of this paper.

###  B Target position and alignment

A dedicated study of the target position, target tilt, its alignment with
respect to the BPDs, and the BPD-TPC alignment was carried out for the
analysis of the low magnetic field replica target data and it is described in
detail in [13]. The resulting values and their uncertainties are given in Tab.
1 and were adopted here for the maximum magnetic field data analysis.

|  | x (cm) | y (cm) | z (cm) |  $t_{x}$ (mrad) |  $t_{y}$ (mrad)
---|---|---|---|---|---|---
| Value | 0.15 | 0.12 | -657.5 | 0.0 | 0.0
| Uncertainty | 0.03 | 0.02 | 0.1 | 0.3 | 0.3
Table 1: Target upstream position and target tilt in the NA61/SHINE coordinate
system (see Fig. 2) and their uncertainties [13]. The target tilt is a measure
of the longitudinal alignment of the target to the nominal beam direction, the
$z$-axis, and is given separately for the $xz$-plane ($t_{x}$) and the
$yz$-plane ($t_{y}$).

###  C Event selection

To maximize the precision of the beam track reconstruction, each selected beam
particle is required to be detected in all three BPDs. To ensure that the
proton beam strikes the target and to reduce systematic biases caused by beam
divergence, all events must also satisfy the so-called "T3 trigger" condition.
This corresponds to the following signals in the scintillator counters and the
Cherenkov detectors as described in Sec. II:

T3 = S1 $\>\wedge\>$ S2
$\>\wedge\>\overline{V_{0}}\>\wedge\>\overline{V}^{p}_{1}\>\wedge\>$ CEDAR
$\>\wedge\>\overline{\mathrm{THC}}$

Due to the strong magnetic field and because the 90-cm-long replica is about
67 cm upstream of VTPC-1 and the upstream magnet, the fringe magnetic field in
the target area is non-negligible. Therefore, operation of the scintillator
counter closest to the target, the S3 counter, was affected by the magnetic
field and the signal from this scintillator was removed from the trigger.
Event selection is finalized by a requirement that the reconstructed path of
the beam particle must have passed through the whole target length. The mean
proton beam divergence during the data taking was about 0.3 mrad. This final
cut removes events where the beam particle hits the target close to its edge
and at a relatively large angle. With full event selection applied, the
($x-y$) profile of selected beam tracks extrapolated to the $z$ position of
the target upstream face is shown in Fig. 3, where the black circle indicates
the boundary of the target front-face area. Since in T2K the beam profile
changes on a run-by-run basis, perfect agreement between the beam properties
for a given hadron production measurement and the T2K beam is not feasible.
However, a comparison of the radial distributions of the two beams in given
runs can be found in Fig. 3 in Ref. [13], where the T2K beam profile is wider
than the used T3 beam profile.

Figure 3: Beam profile at the upstream target face with respect to the BPD
measurements. The black circle represents the target upstream contour and has
a 1.3-cm radius. The mesh structure originates from the finite width and pitch
between the strips on the cathode planes of the BPDs.

###  D Track selection

For all of the particles that go through the TPCs, track selection is designed
to extract high-energy beam particles that have passed through the whole
target without producing any new hadron.
Track reconstruction in the software framework requires that each track must
have more than 5 clusters in at least one TPC. Furthermore, tracks are
selected only if they have a reconstructed momentum. Then, they are
extrapolated backwards from the TPCs to the target surface. Extrapolation
continues until the track hits the downstream target face or the minimum
distance between the target body surface is reached. If this distance is less
than three times the radial uncertainty on the extrapolated position, the
track is selected. This requirement removes most particles that do not come
from the target, but are products of decays or interactions outside the
target. Also, only positively charged particles that exit the target from the
downstream target face are analyzed.
The removal of off-time TPC tracks is ensured by a requirement that tracks
must be detected by the ToF-F wall. Such functionality of the ToF-F detector
is possible since its acquisition time window is 100 ns, while during the
data-taking period the mean time difference between beam particles was around
120 $\mu$s and the readout time of the TPCs is 50 $\mu$s. An example of a
pile-up event is given in Fig. 4, where tracks of two particles are
reconstructed by the TPCs in the same readout period. Both particles passed
through the target without producing new hadrons. Their reconstructed momentum
is approximately the same and is also close to the proton beam momentum of
30.92 $\mbox{Ge\kern-1.00006ptV}\\!/\\!c$. Momentum conservation requires that
one of these two particles is a pile-up event. The particle that first entered
the spectrometer passed the event selection and had a recorded ToF-F hit. The
second beam particle, the off-time particle, came afterwards and was not
detected by the ToF-F wall as it hit the same ToF-F scintillator bar while the
signal of the first particle was being processed. Hence, the ToF-F hit
requirement allows for the rejection off-time TPC tracks. The case of an off-
time beam particle that produces new hadrons in the target is discussed under
the systematic uncertainty study in Sec. IV F.

(a)

(b)

Figure 4: Top (4(a)) and side (4(b)) view of the tracks of two high-energy
particles going through the TPCs. The beam comes from the left. The track
points are shown in green and the reconstructed TPC tracks are given in red.
The two tracks are indistinguishable in the $xz$-plane shown on plot 4(a).
Plot 4(b), illustrating the $yz$-plane, shows that only one of the particles
is detected by the ToF-F wall on the right. This is marked by a big red dot.
The other track is of an off-time particle that scattered elastically or
quasi-elastically in the target and did not produce a signal in the ToF
detector (busy processing the previous hit). The $y$ position of the ToF-F hit
is reconstructed based on the timing difference between the signals of the two
PMTs at the ends of the hit scintillator bar.

The ToF-F detector is not used further in the track selection. It could not be
employed in particle identification as the ToF-F resolution is insufficient
for particle separation in the energy range of interest. However, the specific
energy loss, ${\mathrm{d}}E\\!/\\!{\mathrm{d}}x$, measurements alone provide
good discrimination between protons and pions, as shown in Fig. 5. A linear
graphical cut is used to separate the two particle species.

Figure 5: Distribution of energy loss in the TPCs as a function of momentum.
Good separation between $\pi^{+}$ and protons is observed. The red line shows
the graphical cut applied in the track selection.

The most restrictive requirement in the selection is that track momentum must
be larger than 29.73 $\mbox{Ge\kern-1.00006ptV}\\!/\\!c$ (the value of this
cut is explained below). This cut value takes into account the continuous
energy loss due to ionization in the target and the recoil energy in quasi-
elastic interactions. The ionization losses inside the target follow a Landau
distribution. Simulations were made to examine this distribution for 30.92
$\mbox{Ge\kern-1.00006ptV}\\!/\\!c$ protons going through a 90-cm-long
graphite target with a density of 1.83 g/$\mathrm{cm}^{3}$. Its median was
found to be $E_{\mathrm{ion}}=315$ MeV.
Assuming elastic and quasi-elastic interactions with either the carbon nuclei
or with the nucleons inside the nuclei, the kinetic energy of the recoiling
particle is given by

$\displaystyle T_{R}=\frac{|t|}{2M}=\frac{p^{2}\theta^{2}}{2M},$ (3)

where $t$ is the four-momentum transfer, $M$ is the rest mass of the recoiling
particle, $p$ is the momentum of the beam, and $\theta$ is the scattering
angle between the incoming and outgoing particle. In elastic and quasi-elastic
interactions, the scattered particle stays intact with most of its energy, and
an approximation of the four-momentum transfer is $t\sim p^{2}\theta^{2}$.
From Eq. 3, the kinetic energy of the recoiling particle is largest when the
recoil energy is transferred to a single nucleon, that is a knock-out
reaction. A value of 30 mrad, at the tail of the scattering angle distribution
for tracks with energy above 30 GeV, is assigned to $\theta$. Thus, the
calculated maximal kinetic energy of the recoiling particle in quasi-elastic
events is 459 MeV. The length of the T2K replica target amounts to two nuclear
interaction lengths and one can expect, on average, two nuclear scatterings
for a particle passing through the whole target length. Taking this into
account, the expected energy range of an outgoing elastically or quasi-
elastically scattered beam particle, $E_{\mathrm{out}}$, is

$\displaystyle E_{\mathrm{out}}\geq
E_{\mathrm{beam}}-E_{\mathrm{ion}}-2T_{R},$ (4)

where $E_{\mathrm{beam}}$ is the total energy of the beam particles. In terms
of proton momenta, Eq. 4 transforms to $p\geq 29.73$
$\mbox{Ge\kern-1.00006ptV}\\!/\\!c$.
In elastic and quasi-elastic events, alongside the high-energy selection
candidate, low-energy nuclear fragments, electrons, protons, neutrons and de-
excitation photons can be produced. Those that exit the target and get
reconstructed in the TPCs222Neutral particles cannot be detected with a TPC
are bent away from the beam line and the spectrometer itself by the strong
magnetic field and do not reach the ToF detectors. Particle identification for
such tracks is not feasible since their energy loss measurements lie in a
crossover region between electrons and $\pi^{-}$ or protons and $\pi^{+}$.
This is depicted in the low-momenta range in Fig. 6 which shows the energy
loss vs momentum distribution for tracks that are reconstructed together with
a high-energy selection candidate. Energy loss of the selected candidate is
not plotted, thus at larger momenta Fig. 6 shows off-time beam particles,
primarily $\pi^{+}$ and protons, that have not undergone production
interaction inside the target and have passed through the TPCs. An example of
such a track is shown in Fig. 4. In order to collect the maximum number of
elastically and quasi-elastically scattered particles without damaging the
purity of the selection, it is required that every reconstructed track in the
same event as the selection candidate, if any, must have a momentum larger
than the calculated 29.73 $\mbox{Ge\kern-1.00006ptV}\\!/\\!c$ cut value.
Therefore, interactions are discarded in cases where low-energy particles are
detected by the TPCs, even if there is also a high-energy proton track in the
same event.

Figure 6: Distribution of energy loss in the TPCs as a function of momentum
for tracks produced alongside the high-energy selection candidate. On the left
is the distribution for negatively charged particles, q<0, while on the right
is the distribution for positively charged ones, q>0\. In the low-momentum
regions on both plots separation between particle species is troublesome. For
positive particles at momenta around the 30.92
$\mbox{Ge\kern-0.90005ptV}\\!/\\!c$ proton beam momentum, there are two
distinct peaks. These are off-time beam particles that have survived the
nuclear interactions inside the target - off-time protons (lower) and
$\pi^{+}$ (higher) in ${\mathrm{d}}E\\!/\\!{\mathrm{d}}x$.

The numbers of remaining events and tracks in the course of the selection
process are given in Tab. 2.

Event selection | ($10^{6}$)
---|---
Total | 1.235
T3 trigger | 0.965
BPD measurement | 0.790
$\mathrm{R}_{\mathrm{upstream}}$ | 0.766
$\mathrm{R}_{\mathrm{downstream}}$ | 0.766

Track selection | ($10^{6}$)
---|---
Distance from target | 2.839
Charge | 1.848
ToF-F hit | 0.313
Target exit point | 0.195
${\mathrm{d}}E\\!/\\!{\mathrm{d}}x$ measurement | 0.176
$p\geq 29.73$ $\mbox{Ge\kern-1.00006ptV}\\!/\\!c$ | 0.108

Table 2: Number of events and tracks after consecutive application of the
selection criteria. The notations $R_{\textrm{upstream}}$ and
$R_{\textrm{downstream}}$ indicate the extrapolated beam track passed through
the upstream and would have passed through the downstream target face in the
case there were no interactions inside the target.

Phase space in $\\{p,\theta\\}$ of selected high-energy tracks is plotted in
Fig. 7. The majority of tracks have polar angles below a few milliradians and
momenta around 30.6 $\mbox{Ge\kern-1.00006ptV}\\!/\\!c$, which corresponds to
elastic scattering. The fractional momentum resolution for high-energy tracks
having momenta larger than 29.73 $\mbox{Ge\kern-1.00006ptV}\\!/\\!c$ is
$5\times 10^{-3}$. Since the precision of track momentum reconstruction
depends on the number of points on track, it is worth mentioning that no
restrictions on this number are applied. In this analysis, trajectories of
selected particles can be divided into two topologies based on the segments
they have in different TPCs. The majority of tracks leave clusters in GTPC,
VTPC-2 and MTPC-L. Less than 10$\%$ of the selected tracks only go through the
GTPC and MTPC-L. For both topologies, track selection indirectly implies that
the number of clusters in GTPC is the maximum possible, i.e 7, and in the
MTPC-L their number is more than 20, out of 90 possible. Tracks that pass
through the VTPC-2 turn out to have more than 20 clusters in that chamber, out
of a maximum of 72. A typical selected event, i.e., a single track passing
through the spectrometer, is shown in Fig. 8. This track has a total of 125
measured points, hits the ToF-F wall, and has momentum of 30.34
$\mbox{Ge\kern-1.00006ptV}\\!/\\!c$.

Figure 7: Distribution of selected high-energy tracks in $\\{p,\theta\\}$. The
abscissa range starts at the 29.73 $\mbox{Ge\kern-0.90005ptV}\\!/\\!c$
momentum cut value.  Figure 8: Top view of a selected event - a single track
going through GTPC, VTPC-2 and MTPC-L. The green markers, 125 in total,
represent the measured points along the track. The red line is the
reconstructed particle trajectory starting from the target on the left to the
ToF-F wall on the right. The track’s reconstructed momentum is
$p\leavevmode\nobreak\ =\leavevmode\nobreak\ 30.34$
$\mbox{Ge\kern-0.90005ptV}\\!/\\!c$.

###  E Survival probability

The probability, $P_{\mathrm{surv}}$, that a beam particle avoids a production
interaction inside the 90-cm-long target and the production cross section
$\sigma_{\mathrm{prod}}$ are related via:

$\displaystyle P_{\mathrm{surv}}=\frac{\text{Number of selected
tracks}}{\text{Number of beam particles}}=e^{-L\cdot
n\cdot\sigma_{\mathrm{prod}}},$ (5)

where $L$ is the target length and $n$ is the number density of the target
nuclei. Following the event and track selection procedures described in Secs.
III C and III D the number of selected beam particles is 766,164 and the
number of selected tracks is 108,378. Before extracting the production cross
section, the probability $P_{\mathrm{surv}}$ is corrected for various effects.

###  F Correction factors

Two separate corrections are applied to the probability $P_{\mathrm{surv}}$: a
Monte Carlo-based acceptance correction and a data-based ToF-F efficiency
correction. Their product gives the total correction factor.
Approximately 6 $\times 10^{6}$ events were simulated using Geant4 version
10.4.p03 [17, 18, 19]. The beam profile from data was used to generate the
Monte Carlo beam profile. The Monte Carlo correction is defined as

$\displaystyle C^{MC}=\frac{\text{Number of non-production simulated
events}}{\text{Number of selected reconstructed events}},$ (6)

where the numerator is the number of simulated beam protons that pass thought
the whole target length and do not produce new hadrons inside it. The
denominator is the number of reconstructed Monte Carlo events that are
accepted as elastic or quasi-elastic by the same selection procedure that is
applied to the data. Hence, by construction, any variation in the event and
track selection has an impact on the magnitude of the MC correction factor. On
the other hand, in the simulation one must accurately reproduce the data-
taking environment, including the detector geometry. Thereby, the
reconstruction algorithm in the Monte Carlo and the data case is run under
similar conditions and a proper estimate of the MC correction can be obtained.
Furthermore, the MC correction is sensitive to changes related to the
generation of particle interactions inside the target, which is model
dependent. All of the above-mentioned biases are addressed in Secs. IV and V.
The calculated MC correction for the reference Monte Carlo sample (QBBC
physics list from Geant4) is $C^{MC}=1.035$.
The ToF-F efficiency correction factor relies on the calculation of the
efficiency of the ToF-F scintillator bars from the data. In the Monte Carlo,
the response of the ToF detectors is not simulated. Simply if a simulated
track passes through the ToF-F wall, a ToF-F hit is assigned to it. The
procedure to calculate the ToF-F efficiency is described in [13]. The
efficiency of each scintillator bar in the ToF-F wall is defined as the ratio
of the number of tracks that hit that bar to the number of tracks that reach
the end of MTPCs and are extrapolated to the ToF-F $z$-position of that
scintillator bar. Then, the ToF-F efficiency in the phase space of the
selected tracks is calculated as

$\displaystyle\epsilon^{\mathrm{tof}}=\frac{n_{\mathrm{sel}}}{\sum_{s}n_{\mathrm{sel}}^{s}/\epsilon^{s}},$
(7)

where $n_{\mathrm{sel}}$ is the total number of selected data tracks,
$n_{\mathrm{sel}}^{s}$ is the number of selected data tracks that hit bar s
and $\epsilon^{s}$ is the efficiency of this bar s. The summation goes over
every active bar. The sample of all selected tracks produced signals in around
10 scintillator bars. The estimated ToF-F efficiency is around 96$\%$333The
efficiency value is reduced by a quality cut on the timing difference between
the signals of the top and bottom PMTs of each scintillator bar. The cut is
implemented in order to improve separation between particles for the mass
squared particle identification. Even though this particle identification
method is not applicable in the current analysis, the quality cut remains in
the assessment of the ToF-F efficiency.. Its uncertainty serves as an estimate
of the ToF-F systematic uncertainty.

###  G Production cross section

Using Eq. 5 for the survival probability and applying the correction factors
from Eqs. 6 and 7, the production cross section $\sigma_{\mathrm{prod}}$
becomes

$\displaystyle\sigma_{\mathrm{prod}}=-ln\left(P_{\mathrm{surv}}\times
C^{MC}\times\frac{1}{\epsilon^{\mathrm{tof}}}\right)/(L\cdot
N_{A}\cdot\rho/\mu),$ (8)

where the number density, $n=N_{A}\cdot\rho/\mu$, is given in terms of
Avogadro’s number $N_{A}$, the target material density and molar mass, $\rho$
and $\mu$.

##  IV Systematic uncertainties

As a result of many cross-checks and a broad overview of previous analyses
within , several sources of systematic uncertainty were identified. In the
current section, experimental systematic uncertainties are discussed, while
the physics model uncertainty is presented in Sec. V. The magnitude of each
systematic effect is calculated as the deviation from unity of the ratio of
the recalculated production cross section to the standard, nominal one.

###  A Target density

The target density enters into the production cross-section calculation via
the number density as shown in Eq. 8. The reported target density is $1.83\pm
0.01$ g/$\mathrm{cm}^{3}$. The uncertainty of the target density is propagated
to the production cross-section result. The corresponding effect is $\pm
0.6\%$.

###  B Backward track extrapolation

The target position and tilt, discussed in Sec. III B, have a finite precision
that can induce biases when extrapolating TPC tracks backwards to the target
surface. To study this bias, in the analysis codes the target position and
tilt were changed within the calibration uncertainties. The data were
reprocessed resulting in supplemental estimates of the survival probability
and thus production cross-section values. The effects of the target position
and tilt change in each axis and plane are added in quadrature to form the
total uncertainty, which was found to be around $\pm 0.1\%$.

###  C Beam spot size on upstream target face

Since many re-interactions can take place inside the 90-cm-long replica
target, tracks that hit the outer edge of the target are more likely to be
scattered outside of the target before reaching its downstream end than those
that hit the target center. To study this effect, for both data and Monte
Carlo, the radius of the beam profile on the target front face is reduced from
1.28 cm to 0.65 cm, cutting down 10$\%$ of the selected events. Both the
survival probability $P_{\textrm{surv}}$ and the MC correction $C^{MC}$ are
recalculated and lead to additional production cross-section estimates. The
assigned systematic uncertainty is $-0.2\%$.

###  D Particle identification of low-energy products

In elastic and quasi-elastic events, besides the high-energy proton, low-
energy nuclear fragments, electrons, protons, neutrons and de-excitation
photons can be produced. In reality, the species of the above-mentioned
charged particles cannot be resolved (see Fig. 6) and therefore the
interactions in which they are produced are not selected. However, simulations
provide particle identification (PID) that can be consulted when calculating
the Monte Carlo correction factor described in Sec. III F. To facilitate this
study, matching of simulated and reconstructed tracks is performed and one
knows the true particle type corresponding to each reconstructed track in the
MC sample. Then, reconstructed MC events, in which a high-energy proton is
produced alongside low-energy particles, such as protons, electrons, and
nuclear fragments, can be flagged as elastic or quasi-elastic. Given the high-
energy proton in such flagged events meets all other selection criteria, those
events, as well as selected events with a single reconstructed high-energy
proton track, will be used in the estimation of the MC correction factor.
Their combined number will become the denominator in Eq. 6. Overall, the
described procedure mimics the case, where PID is possible for all charged
particles produced in an event. Using Monte Carlo PID, the MC correction
factor is recalculated and so is the production cross section. The systematic
uncertainty attributed to particle identification of low-energy products is
$-0.4\%$.

###  E Proton loss

Beam protons that have scattered elastically or quasi-elastically in the
target may miss the ToF-F wall or re-interact in the detector. Removal of the
requirement for a ToF-F hit adds these tracks to the selected sample. To
account for off-time beam particles surviving the interactions in the target,
for any event in which more than one high-energy track is reconstructed, a
single track is accepted. Additionally, to guarantee high-quality track
reconstruction and momentum fit, selected tracks in this study must have a
segment in VTPC-2. With no requirement for a ToF-F detection, but with a
VTPC-2 measurement, the survival probability $P_{\textrm{surv}}$ and the MC
correction $C^{MC}$ are recomputed. The corresponding production cross-section
value is compared to the standard result. A $-0.1\%$ effect is assigned to
hadron loss.

###  F Off-time events

When the triggered beam particle undergoes an elastic or quasi-elastic
interaction, but there is also an off-time event that produces new hadrons in
the target (an off-time production event), the surviving beam particle will
not be selected. The reason is that the products of the two interaction types
cannot be separated. However, the probability of such a combination of events
can be estimated based on the number of cases where both the triggered and the
off-time beam particles scatter elastically or quasi-elastically. This is
possible since the probabilities of any type of interaction in the target is
independent of whether the beam particle is counted by the trigger system or
not. The survival probability estimate from this analysis is used. The number
of beam particles avoiding production interactions in the replica target but
accompanied by products of an off-time beam particle production interaction is
estimated and added to the total number of selected tracks. This alters the
survival probability in Eq. 8. The production cross section is recalculated,
compared to the nominal result and the systematic effect is found to be
$-0.8\%$.

###  G ToF efficiency uncertainty

Particles may also lack a recorded ToF-F hit due to the inefficiency of the
ToF-F detector. Also, ToF-F efficiency is used as a correction factor (see
Sec. III F), and so influences the production cross-section result. For these
reasons, the uncertainty of the calculated ToF-F efficiency is propagated to
the production cross-section uncertainty. The corresponding systematic effect
is $\pm 0.3\%$.

###  H Reconstruction

Potential differences between the spectrometer description in the simulation
and the real detector geometry would affect track reconstruction and could
bias the MC correction factor. To address such an effect, the VTPC-2 position
used for the reconstruction of simulated tracks is shifted by $\pm$ 0.2 mm in
$x$ and $\pm$ 0.3 mm in the $y$-direction. Hence, this uncertainty source
reflects the impact that small shifts in the detector positions have on the
reconstruction of tracks in the MC chain. The size of the shifts are chosen to
be much larger than any observed alignment and residual effects in the
calibration of the data. The other detectors are not moved since the selected
high-energy tracks leave no clusters in VTPC-1, the GTPC covers the forward
region, and the MTPCs are well outside of the magnetic field and are not used
for momentum determination. The Monte Carlo correction factor and the
production cross section are recalculated for each of the produced four MC
samples. Then their deviations from the nominal production cross-section value
are added in quadrature, taking into account the corresponding sign, to form
the asymmetric systematic uncertainty due to reconstruction effects. The
resulting fractional uncertainty is ${}_{-0.8}^{+0.5}\%$.

###  I Track momentum cut

As discussed in Sec. III D, the most constraining selection criterion is the
momentum requirement $p\geq 29.73$ $\mbox{Ge\kern-1.00006ptV}\\!/\\!c$. On one
side, the track momentum reconstruction depends on the precision of TPC track
reconstruction and backward extrapolation. These two factors are separately
studied. On the other hand, the cut value itself depends on the beam momentum
resolution. It is reported that the beam momentum spread is less than 1$\%$
[1]. This spread is in turn propagated to the cut value uncertainty. Then the
cut value is varied within its estimated uncertainty and the track selection
is repeated. The change in track selection reflects on both the survival
probability $P_{\textrm{surv}}$ and the MC correction $C^{MC}$. The
corresponding systematic effect is $\pm 0.2\%$. In this evaluation, no
uncertainty is attributed to the value of the energy loss due to ionization
inside the target.

##  V Physics model uncertainty

Variations in Monte Carlo modeling of physics processes can lead to different
MC correction factors. To estimate the corresponding uncertainty, three Geant4
physics lists were used: QBBC, FTFP_BERT and QGSP_BIC. The treatment of
elastic processes for FTFP_BERT and QGSP_BIC is identical, but it differs for
QBBC. In general, modeling of inelastic, and so of quasi-elastic, interactions
changes within each physics list. However, in the range of interest, which is
above $p\geq 29.73$ $\mbox{Ge\kern-1.00006ptV}\\!/\\!c$, QBBC and FTFP_BERT
both use the Fritiof parton model. The reference Monte Carlo model used in
this analysis is the one employed in QBBC. The other two physics lists are
used to evaluate the physics model systematic uncertainty. A separate Monte
Carlo set using the Fluka 2011.2c.5 package [20, 21, 22] was also prepared.
Normalized momentum, $p$, and polar angle, $\theta$, distributions for
reconstructed data and MC events are shown in Fig. 9. For Fig. 9(a), all
tracks that have more than 30 points in the TPCs are processed, while Fig.
9(b) is for tracks that pass all of the selection criteria. Table 3 presents
production cross sections calculated using the survival probability extracted
from the data, applying Monte Carlo corrections calculated with each of the
three Geant4 lists as well as with Fluka 2011.2c.5. Previous production cross-
section analyses used a Geant4-based MC correction, while the latest results
for hadron production in the T2K replica target rely on Fluka 2011.2c.5
simulations for the MC correction of particle yields. Furthermore, the T2K
neutrino flux simulations feature Fluka as a generator of the hadronic
interactions inside the target. For these reasons, and as a cross-check, the
additional Fluka 2011.2c.5 Monte Carlo was produced and corresponding MC
correction was applied to the data. The Fluka-corrected production cross
section is in best agreement with the FTFP_BERT one. Overall, production
cross-section estimates with MC corrections from the three Geant4 models and
the Fluka2011 model show good agreement. The maximum magnitude of the
uncertainty due to modeling of physics interactions is found for the QGSP_BIC
list, and it is $-0.4\%$.

(a)

(b)

Figure 9: Normalized momentum and polar angle distributions for all
reconstructed tracks having more than 30 points in the detector (9(a)), and
for selected high-energy tracks (9(b)). Presented are distributions for the
three Geant4 physics lists (QBBC, FTFP_BERT, QGSP_BIC) and Fluka 2011.2c.5.
The color scheme is given on each plot. The sharp cut in the momentum
distribution on the left-hand plot (9(b)) corresponds to the $p\geq 29.73$
$\mbox{Ge\kern-0.90005ptV}\\!/\\!c$ track selection requirement.

Physics model | MC correction | Measured | 1 - $\sigma_{\mathrm{prod}}^{\mathrm{ref}}$/$\sigma_{\mathrm{prod}}$
---|---|---|---
used for correction | $C^{MC}$ |  $\sigma_{\mathrm{prod}}$ (mb) | ($\%$)
QBBC | 1.035 | 227.6 | -
FTFP_BERT | 1.036 | 227.5 | \- 0.04
QGSP_BIC | 1.042 | 226.8 | \- 0.4
Fluka 2011.2c.5 | 1.037 | 227.4 | \- 0.09
Table 3: Monte Carlo correction, the measured production cross section
obtained using the corresponding MC correction and assigned fractional
uncertainty using Geant4 version 10.4.p03 and Fluka 2011.2c.5 interaction
generators. QBBC, FTFP_BERT, QGSP_BIC are the employed three physics lists
from Geant4. The reference MC model is QBBC.

##  VI Results and comparisons

A production cross section for 30.92 $\mbox{Ge\kern-1.00006ptV}\\!/\\!c$
protons on carbon was estimated based on the number of beam particles that
pass through the 90-cm-long T2K replica target, interact elastically or quasi-
elastically inside it, and leave a track in the spectrometer. The result,
including statistical, systematic, and model uncertainties, is

$\displaystyle\mathrm{\sigma_{prod}=227.6\pm
0.8(stat)\>_{-3.2}^{+1.9}(sys)\>{-0.8}(mod)\>mb}.$ (9)

A list of all identified uncertainty sources and their fractional magnitudes
is given in Tab. 4. The reported production cross-section measurement is in
agreement with previous results for p + C at 31
$\mbox{Ge\kern-1.00006ptV}\\!/\\!c$ [8, 11], while providing better precision,
as shown in Fig. 10 and Tab. 5. For comparison, the production cross-section
measurements for p + C at 60 $\mbox{Ge\kern-1.00006ptV}\\!/\\!c$ by 444 Part
of ’s hadron production measurements for the Fermilab neutrino experiments
[23] and by Carroll et al. [24] are also given.

Uncertainty source | Fractional size ($\%$)
---|---
Target density |  $\pm$0.6
Backward track extrapolation |  $\pm$0.1
Beam spot size on upstream target face | -0.2
PID of low-energy products | -0.4
Proton loss | -0.1
Off-time events | -0.8
ToF efficiency |  $\pm$0.3
Reconstruction | ${}_{-0.8}^{+0.5}$
Track momentum cut |  $\pm$0.2
Physics model | -0.4
Statistical |  $\pm$0.4
Table 4:  Uncertainty sources considered in the current production cross-section measurement and their fractional size. The discussion of each item is given in Secs. IV and V. Figure 10: The result of this analysis compared to the production cross-section measurements for p + C interactions at different beam momenta. Alongside the thin-target results, production cross section by Carroll et al. at 60 $\mbox{Ge\kern-0.90005ptV}\\!/\\!c$ [24] is shown. Experiment |  $p_{\mathrm{beam}}$ ($\mbox{Ge\kern-1.00006ptV}\\!/\\!c$) |  $\sigma_{\mathrm{prod}}$ (mb) |  $\Delta_{\mathrm{total}}$ (mb)
---|---|---|---
NA61/SHINE 2010 | 31 | 227.6 | ${}_{-3.4}^{+2.1}$
NA61/SHINE 2007 | 31 | 229.3 |  $\pm$ 9.2
NA61/SHINE 2009 | 31 | 230.7 | ${}_{-4.6}^{+7.0}$
NA61/SHINE 2016 | 60 | 226.9 | ${}_{-6.4}^{+4.1}$
Carroll et al. | 60 | 222 |  $\pm$ 7
Table 5: Result of this analysis compared to the production cross-section
thin-target measurements by [8, 11, 23] and Carroll et al. [24] for p + C
interactions at different beam momenta. The total uncertainty
($\Delta_{\mathrm{total}}$) is the statistical, systematic and model
uncertainties added in quadrature.

##  VII Summary

This paper presents a direct method for a production cross-section measurement
using the attenuation of beam particles in a thick medium. The production
cross section of 30.92 $\mbox{Ge\kern-1.00006ptV}\\!/\\!c$ protons on carbon
has been obtained using data collected by the collaboration with the 90-cm-
long T2K replica target. About 2$\%$ total uncertainty is estimated by careful
studies of statistical, systematic and model uncertainties. The resulting
production cross section is in agreement with previous measurements and has
the smallest total uncertainty. It will allow for a more accurate hadron
interaction rate re-weighting of the T2K neutrino flux prediction, as its
estimation is based on the simultaneous selection of both elastic and quasi-
elastic processes inside the T2K replica target.

## Acknowledgments

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

This work was supported by the Hungarian Scientific Research Fund (grant NKFIH
123842/123959), the Polish Ministry of Science and Higher Education (grants
667/N-CERN/2010/0, NN 202 48 4339, NN 202 23 1837 and DIR/WK/2016/2017/10-1),
the National Science Centre Poland (grants 2014/14/E/ST2/00018, 2014/15/B/ST2
/ 02537 and 2015/18/M/ST2/00125, 2015/19/N/ST2 /01689, 2016/23/B/ST2/00692,
2017/ 25/N/ ST2/ 02575, 2018/30/A/ST2/00226, 2018/31/G/ST2/03910), the Russian
Science Foundation, grant 16-12-10176 and 17-72-20045, the Russian Academy of
Science and the Russian Foundation for Basic Research (grants 08-02-00018,
09-02-00664 and 12-02-91503-CERN), the Russian Foundation for Basic Research
(RFBR) funding within the research project no. 18-02-40086, the National
Research Nuclear University MEPhI in the framework of the Russian Academic
Excellence Project (contract No. 02.a03.21.0005, 27.08.2013), 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
(grant GA 1480/8-1), the Bulgarian Nuclear Regulatory Agency and the Joint
Institute for Nuclear Research, Dubna (bilateral contract No. 4799-1-18/20),
Bulgarian National Science Fund (grant DN08/11), Ministry of Education and
Science of the Republic of Serbia (grant OI171002), Swiss Nationalfonds
Foundation (grant 200020117913/1), ETH Research Grant TH-01 07-3 and the Fermi
National Accelerator Laboratory (Fermilab), a U.S. Department of Energy,
Office of Science, HEP User Facility managed by Fermi Research Alliance, LLC
(FRA), acting under Contract No. DE-AC02-07CH11359 and the IN2P3-CNRS
(France).

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The Collaboration

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

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