DIV=last advisor,advisors,byadvisor mvbookbookvolumesurl disposition title
subtitle

# Double/hit separation and $\mathrm{d}E/\mathrm{d}x$ resolution of a time
projection chamber with GEM readout

This document is a collaborative effort by the LCTPC Collaboration.

Y. Aoki  High Energy Accelerator Research Organization (KEK), 1-1 Oho,
Tsukuba, Ibaraki 305-0801, Japan  D. Attié  CEA Saclay, Institute of research
into the fundamental laws of the Universe (IRFU), 91191 Gif-sur-Yvette, France
T. Behnke  Deutsches Elektronen-Synchrotron DESY, Notkestr. 85, 22607 Hamburg,
Germany  A. Bellerive  Carleton University, Department of Physics, 1125
Colonel By Drive, Ottawa, ON K1S 5B6, Canada  O. Bezshyyko  Taras Shevchenko
National University of Kyiv, 64/13, Volodymyrska Street, City of Kyiv 01601,
Ukraine  D.S. Bhattacharya  CEA Saclay, Institute of research into the
fundamental laws of the Universe (IRFU), 91191 Gif-sur-Yvette, France  _now
at_ University of Würzburg, Institute for Theoretical Physics and
Astrophysics, Am Hubland, 97074 Würzburg, Germany  P. Bhattacharya  Saha
Institute of Nuclear Physics, Applied Nuclear Physics Division, 1/AF, Sector
1, Bidhan Nagar, Kolkata 700064, India  _now at_ Adamas University, Department
of Physics, Adamas Knowledge City, Barasat-Barrackpore Road, 24 Parganas
North, Kolkata 700126, India  S. Bhattacharya**footnotetext: retired Saha
Institute of Nuclear Physics, Applied Nuclear Physics Division, 1/AF, Sector
1, Bidhan Nagar, Kolkata 700064, India  Y. Chang  Institute of High Energy
Physics, Chinese Academy of Sciences, 19B Yuquan Road, Shijingshan District,
Beijing 100049, P.R. China  P. Colas  CEA Saclay, Institute of research into
the fundamental laws of the Universe (IRFU), 91191 Gif-sur-Yvette, France  G.
De Lentdecker  Inter University Institute for High Energies (ULB-VUB),
Pleinlaan 2, 1050 Brussels, Belgium  K. Dehmelt  Deutsches Elektronen-
Synchrotron DESY, Notkestr. 85, 22607 Hamburg, Germany  _now at_ State
University of New York at Stony Brook, Department of Physics and Astronomy,
100 Nicolls Road, Stony Brook, NY 11794-3800, USA  K. Desch  University of
Bonn, Physikalisches Institut, Nußallee 12, 53115 Bonn, Germany  R. Diener
Deutsches Elektronen-Synchrotron DESY, Notkestr. 85, 22607 Hamburg, Germany
M. Dixit  Carleton University, Department of Physics, 1125 Colonel By Drive,
Ottawa, ON K1S 5B6, Canada  TRIUMF, 4004 Wesbrook Mall, Vancouver, BC V6T 2A3,
Canada  U. Einhaus  Deutsches Elektronen-Synchrotron DESY, Notkestr. 85, 22607
Hamburg, Germany  O. Fedorchuk  Deutsches Elektronen-Synchrotron DESY,
Notkestr. 85, 22607 Hamburg, Germany  I. Fleck  University of Siegen,
Department of Physics, Emmy Noether Campus, Walter-Flex Str. 3, 57068 Siegen,
Germany  K. Fujii  High Energy Accelerator Research Organization (KEK), 1-1
Oho, Tsukuba, Ibaraki 305-0801, Japan  T. Fusayasu  Saga University,
Department of Physics, 1 Honjo-machi, Saga 840-8502, Japan  S. Ganjour  CEA
Saclay, Institute of research into the fundamental laws of the Universe
(IRFU), 91191 Gif-sur-Yvette, France  P. Gros  Saga University, Department of
Physics, 1 Honjo-machi, Saga 840-8502, Japan  _now at_ Queen’s University,
Department of Physics, Engineering Physics & Astronomy, Stirling Hall, 64
Bader Lane, Kingston, Ontario K7L 3N6, Canada  P. Hayman  Carleton University,
Department of Physics, 1125 Colonel By Drive, Ottawa, ON K1S 5B6, Canada  K.
Ikematsu  Saga University, Department of Physics, 1 Honjo-machi, Saga
840-8502, Japan  _now at_ High Energy Accelerator Research Organization (KEK),
1-1 Oho, Tsukuba, Ibaraki 305-0801, Japan  L. Jönsson  Lund University,
Department of Physics, Div. of Particle Physics, Box 117, 221 00 Lund, Sweden
J. Kaminski  University of Bonn, Physikalisches Institut, Nußallee 12, 53115
Bonn, Germany  Y. Kato  Kindai University, Department of Physics, 3-4-1
Kowakae, Higashi-Osaka, Osaka 577-8502, Japan  C. Kleinwort  Deutsches
Elektronen-Synchrotron DESY, Notkestr. 85, 22607 Hamburg, Germany  P.M. Kluit
National Institute for Subatomic Physics (Nikhef), P.O. Box 41882, 1009 DB
Amsterdam, The Netherlands  M. Kobayashi  High Energy Accelerator Research
Organization (KEK), 1-1 Oho, Tsukuba, Ibaraki 305-0801, Japan  U. Krämer
Deutsches Elektronen-Synchrotron DESY, Notkestr. 85, 22607 Hamburg, Germany
B. Lundberg  Lund University, Department of Physics, Div. of Particle Physics,
Box 117, 221 00 Lund, Sweden  M. Lupberger  University of Bonn, Physikalisches
Institut, Nußallee 12, 53115 Bonn, Germany  N. Majumdar  Saha Institute of
Nuclear Physics, Applied Nuclear Physics Division, 1/AF, Sector 1, Bidhan
Nagar, Kolkata 700064, India  P. Malek  Deutsches Elektronen-Synchrotron DESY,
Notkestr. 85, 22607 Hamburg, Germany  T. Matsuda  High Energy Accelerator
Research Organization (KEK), 1-1 Oho, Tsukuba, Ibaraki 305-0801, Japan  U.
Mjörnmark  Lund University, Department of Physics, Div. of Particle Physics,
Box 117, 221 00 Lund, Sweden  S. Mukhopadhyay  Saha Institute of Nuclear
Physics, Applied Nuclear Physics Division, 1/AF, Sector 1, Bidhan Nagar,
Kolkata 700064, India  F. Müller  Deutsches Elektronen-Synchrotron DESY,
Notkestr. 85, 22607 Hamburg, Germany  A. Münnich  Deutsches Elektronen-
Synchrotron DESY, Notkestr. 85, 22607 Hamburg, Germany  _now at_ European XFEL
GmbH, Holzkoppel 4, 22869 Schenefeld, Germany  J. Nakajima  High Energy
Accelerator Research Organization (KEK), 1-1 Oho, Tsukuba, Ibaraki 305-0801,
Japan  S. Narita  Iwate University, 3-18-8 Ueda, Morioka, Iwate 020-8550,
Japan  K. Negishi  Iwate University, 3-18-8 Ueda, Morioka, Iwate 020-8550,
Japan  T. Ogawa  High Energy Accelerator Research Organization (KEK), 1-1 Oho,
Tsukuba, Ibaraki 305-0801, Japan  A. Oskarsson  Lund University, Department of
Physics, Div. of Particle Physics, Box 117, 221 00 Lund, Sweden  L. Österman
Lund University, Department of Physics, Div. of Particle Physics, Box 117, 221
00 Lund, Sweden  D. Peterson  Cornell University, Laboratory for Elementary-
Particle Physics (LEPP), Ithaca, NY 14853, USA  H. Qi  Institute of High
Energy Physics, Chinese Academy of Sciences, 19B Yuquan Road, Shijingshan
District, Beijing 100049, P.R. China  M. Riallot  CEA Saclay, Institute of
research into the fundamental laws of the Universe (IRFU), 91191 Gif-sur-
Yvette, France  C. Rosemann  Deutsches Elektronen-Synchrotron DESY, Notkestr.
85, 22607 Hamburg, Germany  S. Roth  RWTH Aachen University, Department of
Physics, Institute III B, Otto-Blumenthal-Str., 52074 Aachen, Germany  P.
Schade  Deutsches Elektronen-Synchrotron DESY, Notkestr. 85, 22607 Hamburg,
Germany  O. Schäfer  Deutsches Elektronen-Synchrotron DESY, Notkestr. 85,
22607 Hamburg, Germany  R.D. Settles  Max Planck Institute for Physics (Werner
Heisenberg Institute), Föhringer Ring 6, 80805 Munich, Germany  A.N. Shirazi
University of Siegen, Department of Physics, Emmy Noether Campus, Walter-Flex
Str. 3, 57068 Siegen, Germany  A. Shoji  Iwate University, 3-18-8 Ueda,
Morioka, Iwate 020-8550, Japan  O. Smirnova  Lund University, Department of
Physics, Div. of Particle Physics, Box 117, 221 00 Lund, Sweden  A. Sugiyama
Saga University, Department of Physics, 1 Honjo-machi, Saga 840-8502, Japan
T. Takahashi  Hiroshima University, Graduate School of Advanced Sciences of
Matter, 1-3-1 Kagamiyama, Higashi-Hiroshima, Hiroshima 739-8530, Japan  J.
Timmermans  National Institute for Subatomic Physics (Nikhef), P.O. Box 41882,
1009 DB Amsterdam, The Netherlands  M. Titov  CEA Saclay, Institute of
research into the fundamental laws of the Universe (IRFU), 91191 Gif-sur-
Yvette, France  D. Tsionou  Deutsches Elektronen-Synchrotron DESY, Notkestr.
85, 22607 Hamburg, Germany  A. Vauth  Deutsches Elektronen-Synchrotron DESY,
Notkestr. 85, 22607 Hamburg, Germany  _now at_ University of Hamburg,
Department of Physics, Institute of Experimental Physics, Luruper Chaussee
149, 22761 Hamburg, Germany  T. Watanabe  Kogakuin University, Laboratory for
High Energy Physics, 2665-1 Nakano-machi, Hachioji, Tokyo 192-0015, Japan  U.
Werthenbach  University of Siegen, Department of Physics, Emmy Noether Campus,
Walter-Flex Str. 3, 57068 Siegen, Germany  M. Wu  Deutsches Elektronen-
Synchrotron DESY, Notkestr. 85, 22607 Hamburg, Germany  _now at_ Radboud
University, Institute for Mathematics, Astrophysics and Particle Physics, P.O.
Box 9010, 6500 GL Nijmegen, The Netherlands  Y. Yang  Inter University
Institute for High Energies (ULB-VUB), Pleinlaan 2, 1050 Brussels, Belgium  R.
Yonamine  Inter University Institute for High Energies (ULB-VUB), Pleinlaan 2,
1050 Brussels, Belgium  _now at_ High Energy Accelerator Research Organization
(KEK), 1-1 Oho, Tsukuba, Ibaraki 305-0801, Japan  Z. Yuan  Institute of High
Energy Physics, Chinese Academy of Sciences, 19B Yuquan Road, Shijingshan
District, Beijing 100049, P.R. China  K. Yumino  High Energy Accelerator
Research Organization (KEK), 1-1 Oho, Tsukuba, Ibaraki 305-0801, Japan  F.
Zhang  Hubei University of Technology, 28 Nanli Road, Hong-shan District,
Wuchang, Wuhan, Hubei Province 430068, P.R. China

###### Abstract

A time projection chamber (TPC) with micropattern gaseous detector (MPGD)
readout is investigated as main tracking device of the International Large
Detector (ILD) concept at the planned International Linear Collider (ILC). A
prototype TPC equipped with a triple gas electron multiplier (GEM) readout has
been built and operated in an electron test beam. The TPC was placed in a
$1\text{\,}\mathrm{T}\text{/}$ solenoidal field at the DESY II Test Beam
Facility, which provides an electron beam up to
$6\text{\,}\mathrm{GeV}\text{/}\mathrm{c}$. The performance of the readout
modules, in particular the spatial point resolution, is determined and
compared to earlier tests. New studies are presented with first results on the
separation of close-by tracks and the capability of the system to measure the
specific energy loss $\mathrm{d}E/\mathrm{d}x$. This is complemented by a
simulation study on the optimization of the readout granularity to improve
particle identification by $\mathrm{d}E/\mathrm{d}x$.

Keywords: Time Projection Chambers (TPC), Micropattern Gaseous Detectors
(MPGD), Gas Electron Multipliers (GEM), International Linear Collider (ILC),
International Large Detector (ILD)

##  1 Introduction

The LCTPC collaboration has built and operated a versatile prototype for the
ILD TPC [1].111See https://www.lctpc.org for more information about LCTPC.
This prototype TPC, including its performance and the principles of its
construction, has been described in detail in [2]. In the present publication,
results from a second test/beam campaign with an upgraded set of readout
modules are presented. The findings in the previous paper have been validated.
New studies are shown on measurements of the double/hit separation power and
on the capabilities to do particle identification via a measurement of the
specific energy loss $\mathrm{d}E/\mathrm{d}x$.

In section 2, after a brief overview of the experimental setup, the updates in
the experiment compared to the previous measurement campaign are discussed.
Section 3 gives an overview of the methods used in the reconstruction of the
test/beam data. Results from an analysis to re/establish the basic performance
are shown in section 4. In section 5 two new algorithms to optimize the
double/hit separation are introduced. In section 6 a first determination of
the resolution of the specific energy loss measurement is presented. These
results are extrapolated to the proposed large TPC at the ILD and compared to
theoretical expectations. Furthermore, in a simulation study two approaches to
measure the specific energy loss are compared, with a focus on their
performance in dependence of the readout granularity.

##  2 Experimental setup

The TPC prototype consists of a light/weight field cage, providing an active
gaseous volume of $57\text{\,}\mathrm{cm}\text{/}$ length and
$72\text{\,}\mathrm{cm}\text{/}$ diameter, and an end plate which can house up
to $7$ identical readout modules. The prototype has been constructed such that
it can be used to test the different readout technologies that are under
discussion for use at the ILC. Details about the construction of the prototype
can be found in [3]. The construction has been supported by the EUDET and AIDA
projects.222For more information on EUDET and AIDA see https://www.eudet.org/
and https://aida2020.web.cern.ch/aida2020/node/283.html, respectively.

The TPC prototype is operated at the DESY II Test Beam Facility [4], which
provides an electron beam with an adjustable momentum between
$1\text{\,}\mathrm{GeV}\text{/}\mathrm{c}$ and
$6\text{\,}\mathrm{GeV}\text{/}\mathrm{c}$. The facility hosts a
superconducting solenoid magnet (PCMAG) [5] that can provide a magnetic field
of up to $1\text{\,}\mathrm{T}\text{/}$. The dimensions of the prototype TPC
have been chosen to exactly fit into the solenoid bore. Before traversing the
TPC, the electron beam has to pass through the magnet, which presents around
$20\text{\,}\mathrm{\char 37\relax}\text{/}$ of a radiation length $X_{0}$. A
movable stage, specifically constructed for the demands of this type of
research, can move the solenoid including the TPC horizontally and vertically
perpendicular to the electron beam, and rotate the system relative to the beam
around the vertical axis.

The measurements presented here were taken with the TPC prototype equipped
with a triple GEM readout system. The gas used was the so/called T2K gas
mixture of $95\text{\,}\mathrm{\char 37\relax}\text{/}$ argon,
$3\text{\,}\mathrm{\char 37\relax}\text{/}$ tetrafluoromethane
($\mathrm{CF}_{4}$), and $2\text{\,}\mathrm{\char 37\relax}\text{/}$ isobutane
($i\mathrm{C}_{4}\mathrm{H}_{10}$) [6]. The chamber was operated at
atmospheric pressure. Unless stated otherwise, the presented measurements were
taken with $5\text{\,}\mathrm{GeV}\text{/}\mathrm{c}$ electron momentum, a
magnetic field of $1\text{\,}\mathrm{T}\text{/}$ and a drift field of
$240\text{\,}\mathrm{V}\text{/}\mathrm{cm}$, which is close to the maximum of
the electron drift velocity for this gas mixture.

###  2.1 The GridGEM module

Figure 1: Explosion view of the module, showing the GEM foils, the ceramic
mounting grids and the readout and back plane.

The defining characteristic of the system is the amplification by a triple GEM
stack positioned on top of a finely segmented readout plane, as shown in
figure 1. The GEM foils [7] are spaced and kept under tension by a system of
thin ceramic frames. The readout plane is segmented into pads with a pitch of
$1.26\text{\times}5.85\text{\,}\mathrm{mm}\text{/}$, arranged in $28$ rows.
This results in a total of $4828$ pads per module. The readout electronics [8]
is based on the ALTRO ASIC [9], originally developed for the ALICE experiment
at CERN and modified by the Lund University to be used for GEM readout. The
results presented in [2] were based on a previous generation of GEM based
readout modules. Details on that module generation can be found in [10].

When operating the first generation system, the following issues were
identified and are subsequently addressed in the second generation system,
presented here. First, although the earlier system was operated stably in a
test beam over several weeks, there was concern about potential weaknesses in
its high voltage stability. These were addressed by a number of measures. The
procedure to glue the GEMs on the frames was improved and the quality of the
glue joint was closely monitored. This ensures a continuous glue line with an
average width of $\sim 0.5\text{\,}\mathrm{mm}\text{/}$ without interruptions.
The design of the interface between GEM foil and ceramic frame was changed
such that there is always a minimum distance of
$0.15\text{\,}\mathrm{mm}\text{/}$ between a GEM hole and the ceramic frame.
This avoids glue spillage into GEM holes, which is suspected to lead to
instabilities. The electrode layout of the GEM foils and their connection
scheme to the external high voltage supply were revised to minimize high field
areas at the edges of the GEMs.

Figure 2: Picture of the stretching tool. The left part positions the ceramic
frame, holding it with a vacuum system. The right part tensions the GEM foil
and is positioned on top of the left part after the glue has been deposited on
the ceramic frame by a robotized dispenser.

A second issue observed in the previous modules was related to the flatness of
the GEM foils. Measurements showed deflections of the GEM foils of up to
$300\text{\,}\mathrm{\SIUnitSymbolMicro m}\text{/}$ peak/to/peak, which
translate into significant gain inhomogeneities across a GEM module. To
provide a more controllable and reproducible procedure to tension the GEM
foils while gluing, a new stretching tool, shown in figure 2, was designed and
built. The tension is controlled by a system of springs that ensures a
consistent and uniform force. In addition, the tool allows for a precise and
reproducible alignment between the GEM foil and the ceramic frame. The tension
is held during the gluing and curing process and is only released at the very
end. Using this approach, the GEM deflections could be reduced significantly,
which directly improves the gain homogeneity.

(a) Measured GEM height deviations.

(b) Calculated module gain deviations.

Figure 3: Effects of the new GEM mounting method. 3(a) Distribution of the
measured height deviation for the old (2013) and new (2016) GEM modules. 3(b)
The corresponding calculated effective gain of the system, normalized to the
nominal gain for perfectly flat GEMs.

In figure 3(a) the distribution of the measured GEM height deviations is shown
for the old and the new modules. The scale of the deflections, given by the
RMS of the distribution, has been reduced by about a factor of $2$ from
$96\text{\,}\mathrm{\SIUnitSymbolMicro m}\text{/}$ to
$41\text{\,}\mathrm{\SIUnitSymbolMicro m}\text{/}$. A simulation, following
the arguments and using the methods described in [11], was used to translate
the measured GEM deflections into an impact on the effective gain of the
system. The resulting distribution of the normalized gain is shown in figure
3(b). The gain deviations, again given by the RMS, could also be reduced by
more than a factor of $2$ from $8.4\text{\,}\text{/}$ to
$3.6\text{\,}\text{/}$.

###  2.2 Test-beam configuration

Three new modules were built and installed into the end plate of the TPC
prototype in a way that allows to measure tracks over a length of about
$50\text{\,}\mathrm{cm}\text{/}$ on a total of $84$ pad rows. The TPC with all
three modules installed showed stable performance for the full three week
test/beam period. In total, $9.5\text{\cdot}{10}^{6}$ events were recorded,
mostly at an electron momentum of $5\text{\,}\mathrm{GeV}\text{/}\mathrm{c}$.
The distance between the beam and the readout plane was varied between
$0\text{\,}\mathrm{cm}\text{/}$ and $57\text{\,}\mathrm{cm}\text{/}$. For the
data presented in this paper the beam was kept parallel to the readout plane,
i.e. $\theta\approx$$$, and perpendicular to the pad rows of the readout. The
angular spread of the beam after passing through the magnet wall was about
$None$.

The objectives of this test beam were to evaluate the stability and
reproducibility of the performance of the detector, measure the
$\mathrm{d}E/\mathrm{d}x$ resolution and to study the double/hit separation of
the TPC prototype. Therefore, two different data sets were taken: One to study
the performance of the detector under nominal conditions and one including a
thin target in front of the TPC, to produce multiple particles out of a single
beam electron to study the double/hit separation in multi/track events. In
addition to measurements using the default high/gain and -stability voltage
settings, the former data set includes a subset using a setting intended to
minimize ion/backflow. According to [12], the gain of the default voltage
setting is estimated to be $\sim 2000$. The charge measurements with both GEM
settings at the test beam show that the gain of the minimum/ion/backflow
setting is about $20\text{\,}\mathrm{\char 37\relax}\text{/}$ lower. As in
previous measurements, the readout electronics was run at a sampling rate of
$20\text{\,}\mathrm{MHz}\text{/}$ with a gain of
$12\text{\,}\mathrm{mV}\text{/}\mathrm{fC}$ and a peaking time of
$120\text{\,}\mathrm{ns}\text{/}$.

(a) Sketch of the target location.

(b) Yield of suitable track pairs.

Figure 4: Setup for the measurement of multi/track events. 4(a) Sketch of the
setup overlaid with an event from a Geant4 simulation. 4(b) Simulated yield of
track pairs suitable for double/hit studies for the options to install the
target outside the magnet or inside directly in front of the TPC prototype, in
dependence on the target thickness as a fraction of a radiation length
$X_{0}$. The fraction of suitable events determined from measured data is
shown in comparison.

For the second data set, a stainless steel target was installed inside the
magnet in front of the TPC as shown in the sketch in figure 4(a). Figure 4(b)
shows the fraction of suitable events as a function of the thickness of the
target, for $5\text{\,}\mathrm{GeV}\text{/}\mathrm{c}$ beam electrons. For
events to be included they need to contain at least two tracks reconstructed
in the TPC. The distance between two of the tracks has to be less than
$3\text{\,}\mathrm{mm}\text{/}$ along the first two or more pad rows at the
entry point of the TPC, which is usually fulfilled for tracks originating from
an interaction of a beam electron with the steel target. This ensures at least
two merged hits from two different tracks. In addition, at least
$40\text{\,}\mathrm{\char 37\relax}\text{/}$ of the remaining hits of these
tracks have to be clearly separated. This corresponds to a distance between
the tracks of $7\text{\,}\mathrm{mm}\text{/}$ or more. The figure shows
results from Geant4 [13, 14, 15] simulations for two different locations of
the target: one inside the solenoid magnet right in front of the TPC prototype
and one outside of the magnet directly in front of it.

According to the simulation, the highest yield of events suitable for the
double/hit separation studies is expected for a target with a thickness of
about $50\text{\,}\mathrm{\char 37\relax}\text{/}$ $X_{0}$, placed inside the
magnet right in front of the TPC prototype. Therefore, a target made of
$8\text{\,}\mathrm{mm}\text{/}$ thick stainless steel, which corresponds to
$47\text{\,}\mathrm{\char 37\relax}\text{/}$ $X_{0}$, was installed in the
setup. Figure 4(b) also shows the fraction of events fulfilling the selection
criteria determined from the data measured with this setup. Very good
agreement between measurement and simulation is found.

##  3 Reconstruction methods

The MarlinTPC software package [16] was used to reconstruct and analyze the
data. This package is based on the linear collider software suite [17, 18,
19]. The individual steps of the reconstruction are introduced briefly in the
following.

The amplified electron avalanche after the GEM stack is collected on the pads
of the segmented readout anode. The signal on a single pad is called a pulse.
The readout electronics measures the collected charge in time bins. The charge
of a pulse is the sum of the ADC counts in these bins from the rise of the
signal over a start threshold to its fall under a stop threshold. In addition
to the thresholds, a minimum number of samples over threshold is required. The
arrival time of each pulse is derived from the inflection point of its rising
edge.

The combination of several pulses on a pad row by a row/based clustering
algorithm is called a hit. The coordinate along the pad rows is determined
using a charge/weighted center/of/gravity method. The charge of the hit is the
sum of the charge of the contributing pulses. For the time of the hit, the
time of the largest contributing pulse is used, since the time information of
the adjacent, often much smaller, pulses is affected by a number of systematic
effects.

To combine the reconstructed hits to tracks, an iterative Hough transformation
[20] is used in the following, unless stated otherwise. The track parameters
are determined by a fit using the General Broken Lines method [21]. As track
model either a helix, for data taken with magnetic field, or a straight line,
for data taken without magnetic field, is used.

Two coordinate systems are used in the reconstruction and analysis, one global
for the whole prototype and one local to each readout module. The global
coordinate system is Cartesian. The $z$/̄axis is perpendicular to the readout
plane pointing from the anode towards the cathode. The $xy$/̄plane is parallel
to the readout plane with the $y$/̄axis pointing upwards and the $x$/̄axis in
the direction of the electron beam. Since the shape of each module is an
annular segment, the local coordinate system is cylindrical with its origin at
the center of the annulus. The $z$/̄direction is identical to that of the
global coordinate system. The position parallel to the readout plane is
defined by the radial coordinate $r$ and the azimuth angle $\varphi$. The
coordinate $r\varphi$ describes the corresponding distance along the
circumference of a circle with radius $r$.

##  4 Overall performance of the TPC

The performance of the new readout module generation has been studied and
compared to the previous generation, based on the data taken in the new
test/beam campaign and the data presented in [2], respectively. For both data
sets, the same settings were used in reconstruction and analysis, with the
exception of an adjustment of the minimum number of pulse samples over
threshold to account for different operating conditions in the new
measurement. The data taken with the previous module generation have been re-
analyzed with the current software version. The results differ slightly from
the original analysis, since in the re-analysis the lower boundary of the fit
range of the function describing the spatial resolution was changed from
$70\text{\,}\mathrm{mm}\text{/}$ to $90\text{\,}\mathrm{mm}\text{/}$.

###  4.1 Drift velocity

A key parameter for a TPC is the electron drift velocity in the drift volume.
It has been determined from the reconstructed mean arrival time for several
measurement runs at various positions of the beam along the drift direction.
The position of the beam is determined from the position of the movable stage,
since no absolute reference is available.

Figure 5: Drift velocity measurement for the new module generation (2016) and
the previous one (2013). Shown is the beam position versus reconstructed
arrival time of the hit signal for $B=$1\text{\,}\mathrm{T}\text{/}$$ at the
default drift field of $240\text{\,}\mathrm{V}\text{/}\mathrm{cm}$ including
linear fits to determine the drift velocity.

Figure 5 shows the beam position inside the drift volume plotted against the
signal arrival time. The slope of a linear fit to the data points corresponds
to the drift velocity. The fit results are given in table 1, including a
comparison with values from a Magboltz [22] simulation. The errors on the
measured values originate mainly from the uncertainties of the stage position
measurement and less from the uncertainty of the time measurement. Other
potential uncertainties are not considered in the fit. The simulation errors
are negligible in comparison to the errors of the measured values. The
measured values agree reasonably well with the corresponding results of the
simulation and are similarly consistent between the two data sets. The
remaining discrepancy can be explained by uncertainties in the gas conditions,
i.e. pressure, temperature and contaminants. Using the resulting
drift/velocity values, the time measurement of the reconstructed hits is
converted to the spatial coordinate in the drift direction.

data set |  $v_{\mathrm{d}}$ [$\mathrm{mm}\text{/}\mathrm{\SIUnitSymbolMicro s}$]
---|---
| measurement | simulation
2013 | $77.28\pm 0.04$ | $76.87$
2016 | $75.97\pm 0.03$ | $76.45$
Table 1: The resulting slope of the linear fits in figure 5 and the
corresponding simulated values of the drift velocity. The statistical
uncertainties of the simulation are negligible compared to the measurement
errors and have therefore been omitted.

###  4.2 Pad response function

Another key parameter for the performance is the pad response. The pad
response function (PRF) describes the average shape of the charge distribution
of a hit on the pad plane. Its width is defined primarily by the diffusion of
the primary electrons in the drift volume and the defocusing of the charge
avalanche in the GEM stack. Therefore, the shape of the distribution should
follow a convolution of a Gaussian and a box function with the width of the
pad pitch. The latter accounts for the uniform probability distribution of the
initial position of the electron relative to the pad.

(a) Shape of pad response function.

(b) Transverse diffusion measurement.

Figure 6: Pad response function and diffusion measurement. 6(a) The PRF in
simulation and measurement at $\sim 20\text{\,}\mathrm{mm}\text{/}$ drift
distance in a $240\text{\,}\mathrm{V}\text{/}\mathrm{cm}$ drift field. Also
shown is a fit to the measured data. 6(b) The width of the PRF versus drift
distance including a fit of equation 1 to extract the transverse diffusion
constant. Results are shown for the 2016 test beam using the new module
generation and the 2013 test beam using the previous one.

In reality, the tails of this distribution are wider than expected from the
above model, as can be seen in the measured PRF shown in figure 6(a) for the
example of a drift distance of $\sim 20\text{\,}\mathrm{mm}\text{/}$. This can
be explained by induction from drifting electrons ending up on adjacent pads.
This induced signal has an equal positive and negative part but the ALTRO
readout only records negative charge signals and cuts off the positive part.
Thus, a non-zero signal is recorded. The size of this induction signal on the
adjacent pads has been estimated to be around $10\text{\,}\mathrm{\char
37\relax}\text{/}$. As can be seen in the figure, a simulation including the
induction effect describes the tails in the measured values only partially.
Therefore, to extract the width of the PRF, the fit of the simple model
described above is performed only within a range of $\pm 3$ standard
deviations around the maximum, as shown in the figure. As visible, this
results in a realistic value for the width of the distribution.

data set |  $\sigma_{\mathrm{PRF,0}}$ [$\mathrm{\SIUnitSymbolMicro m}\text{/}$] |  $D_{\mathrm{t}}$ [$\mathrm{\SIUnitSymbolMicro}\mathrm{m}\mathrm{/}\sqrt{\mathrm{c}\mathrm{m}}$]
---|---|---
|  | measurement | simulation
2013 | $578.2\pm 1.9$ | $102.87\pm 0.38$ | $97.49\pm 0.70$
2016 | $530.8\pm 2.0$ | $99.06\pm 0.40$ | $95.48\pm 0.63$
Table 2: The resulting parameters of equation 1 in the combined fits to the
PRF width and the $r\varphi$/̄resolution, as described in the text. Also shown
are the corresponding simulated values of $D_{\mathrm{t}}$.

In figure 6(b) the resulting PRF width $\sigma_{\mathrm{PRF}}$ is plotted
against the drift distance. The behavior can be described by the function [23]

$\sigma_{\mathrm{PRF}}(z)=\sqrt{\bigl{(}\sigma_{\mathrm{PRF,0}}\bigr{)}^{2}+\bigl{(}D_{\mathrm{t}}\bigr{)}^{2}\cdot
z}\enskip,$ (1)

which allows to determine the transverse diffusion constant $D_{\mathrm{t}}$.
Since this is also required as an input to the fit of the transverse point
resolution $\sigma_{r\varphi}$, as is explained in section 4.3, a combined
$\chi^{2}$/̄fit of $\sigma_{\mathrm{PRF}}$ with equation 1 and
$\sigma_{r\varphi}$ with equation 2 was performed. The fit results regarding
the PRF width are summarized in table 2, including simulated values of the
corresponding diffusion constants, taking into account differences in measured
gas pressure, temperature and contamination with air and water vapor. As
expected from the Magboltz simulation, the diffusion measured in the 2016 data
set is smaller than in the 2013 data set, mostly due to a difference in the
gas pressure. However, the measured diffusion constants from both data sets
are significantly larger than the corresponding simulated results. As
discussed in [10], the simulation only describes the ideal case of perfectly
homogeneous and parallel electric and magnetic fields. The inhomogeneity of
the magnetic field in the measurement setup introduces field transverse
components, which increase the spread of the drifting electrons and thus can
explain the systematic differences between measured and simulated results.

The intrinsic width $\sigma_{\mathrm{PRF,0}}$ at zero drift distance, which is
caused by the defocusing of the electron avalanche in the GEM stack and signal
induction on the pad plane, shows a significant difference between the two
measurements, with a smaller width in the 2016 data. Qualitatively this result
is consistent with the reduced deflections of the GEMs used in the new
modules. Deflections of the GEMs lead to inhomogeneous electric fields in the
transfer and induction regions between the GEMs and the pad plane, creating
transverse field components. Equivalent to the magnetic field inhomogeneity in
the drift region, these field distortions increase the charge spread in the
amplification region. This results in an increased width of the PRF
independent of the drift distance.

###  4.3 Point resolution

The point resolution for the single/hit position is determined from the width
of the residual distribution as explained in detail in [2]. The residual is
the distance of a hit to the corresponding track. Here, the resolution is
evaluated in the directions $r\varphi$ and $z$, which correspond to the
distance of a hit to its track along a pad row and along the drift direction,
respectively.

The dependence of the point resolution $\sigma_{r\varphi/z}(z)$ on the drift
distance $z$ is described by the equation [24]

$\sigma_{r\varphi/z}(z)=\sqrt{\bigl{(}\sigma_{r\varphi/z,0}\bigr{)}^{2}+\frac{\bigl{(}D_{\mathrm{t/l}}\bigr{)}^{2}}{N_{\mathrm{eff}}\cdot
e^{-Az}}\cdot z}\enskip,$ (2)

where $\sigma_{r\varphi/z,0}$ denotes the intrinsic resolution of the setup,
due to diffusion in the GEM stack and the finite pad width, and
$N_{\mathrm{eff}}$ the effective number of primary electrons. The term
$e^{-Az}$, with the electron attachment coefficient $A$, describes the loss of
signal electrons during drift by attachment to gas molecules, primarily due to
oxygen contamination of the gas. This equation only holds for tracks parallel
to the readout plane and perpendicular to the pad rows.

Since in the equation the diffusion constants $D_{\mathrm{t/l}}$ and
$N_{\mathrm{eff}}$ appear as the quotient
$\bigl{(}D_{\mathrm{t/l}}\bigr{)}^{2}/N_{\mathrm{eff}}$, they are fully
correlated in a fit and only the quotient can be determined. Thus, the
longitudinal diffusion constant $D_{\mathrm{l}}$ was provided by a Magboltz
simulation and used to calculate $N_{\mathrm{eff}}$ from the fit result. As
described in the previous section, the transverse diffusion constant
$D_{\mathrm{t}}$ can also be determined by a fit of equation 1 to the PRF
width. Due to the observed systematic difference of this measurement to the
simulated result, the measured value is assumed to be correct. Therefore, to
resolve the correlation of $D_{\mathrm{t}}$ and $N_{\mathrm{eff}}$, a combined
fit of equations 1 and 2 to the PRF width and the $r\varphi$/̄resolution,
respectively, is performed. Since equation 2 is not a good description of the
$r\varphi$/̄resolution at short drift distances, it is fitted only to the data
points with a drift distance larger than $90\text{\,}\mathrm{mm}\text{/}$.

The parameters $N_{\mathrm{eff}}$ and $A$ are also strongly correlated.
Therefore, it would be preferable to determine the attachment coefficient
directly from the signal charge measurement. However, in the 2013 data the
charge measurement is affected by environmental effects and no charge
calibration is available for this data. Thus, this approach is not reliable
here and both parameters are left free in the fits, to ensure equal treatment
of the two data sets.

(a) $r\varphi$-resolution.

(b) $z$-resolution.

Figure 7: Comparison of the measured point resolution for current (2016) and
previous (2013) module generation in 7(a) $r\varphi$\- and 7(b)
$z$/̄direction, both as a function of the drift distance. The lines represent
fits of equation 2.

In figure 7, comparisons of the point resolution in the $r\varphi$ and $z$
directions are shown between the 2013 test/beam campaign with the
first/generation modules and the 2016 test/beam campaign with the new modules.
The plots include the fits of equation 2 to the measured resolution, as
described above. The fit results are summarized in tables 3 and 4. A very good
agreement between the two campaigns is found in the measurements and the
fitted results, indicating a good stability and reproducibility of the
performance of the system.

data set |  $\sigma_{r\varphi,0}$ [$\mathrm{\SIUnitSymbolMicro m}\text{/}$] | $N_{\mathrm{eff}}$ |  $A$ [$\text{/}\mathrm{cm}$] |  $D_{\mathrm{t}}$ [$\mathrm{\SIUnitSymbolMicro}\mathrm{m}\mathrm{/}\sqrt{\mathrm{c}\mathrm{m}}$]
---|---|---|---|---
2013 | $68.1\pm 1.6$ | $43.4\pm 2.8$ | $0.0054\pm 0.0011$ | $102.87\pm 0.38$
2016 | $66.2\pm 2.1$ | $38.7\pm 3.0$ | $0.0040\pm 0.0013$ | $99.06\pm 0.40$
Table 3: The resulting parameters of equation 2 in the combined fits to the PRF width and the $r\varphi$/̄resolution, as described in the text. The results for $D_{\mathrm{t}}$ are the same as already shown in table 2.  data set |  $\sigma_{z,0}$ [$\mathrm{\SIUnitSymbolMicro m}\text{/}$] | $N_{\mathrm{eff}}$ |  $A$ [$\text{/}\mathrm{cm}$] |  $D_{\mathrm{l}}^{\mathrm{sim}}$ [$\mathrm{\SIUnitSymbolMicro}\mathrm{m}\mathrm{/}\sqrt{\mathrm{c}\mathrm{m}}$]
---|---|---|---|---
2013 | $288.4\pm 0.6$ | $39.9\pm 1.4$ | $0.0050\pm 0.0006$ | $221.6\pm 1.6$
2016 | $286.0\pm 0.7$ | $38.6\pm 1.3$ | $0.0033\pm 0.0006$ | $227.1\pm 1.4$
Table 4: The results of fits of equation 2 to the $z$/̄resolution.
$N_{\mathrm{eff}}$ is calculated from the fit parameters using the simulated
values of $D_{\mathrm{l}}$ given in the last column.

#### Minimal ion backflow

Ions from the GEM amplification flowing back into the sensitive TPC volume can
cause field distortions and electron attachment. A gating GEM system [25] is
foreseen for the ILD TPC that will capture the ions produced in the
amplification while having a very high transparency to drifting electrons. In
addition to using the gate, the amount of ions flowing from the amplification
into the drift volume will be minimized by the GEM/voltage settings described
in the following.

During the 2016 test/beam campaign a data set was taken with altered GEM
voltage settings based on the studies in [12], as shown in figure 8(a), which
should minimize the ion backflow (MIBF). The main idea is that the GEM facing
the sensitive volume is operated at a lower voltage to minimize the ion
production at this stage. To compensate and ensure a sufficient total gain,
the remaining two GEMs are operated at higher voltages. The ions produced here
will be mostly captured by the GEM facing the sensitive volume. The average
hit charge measured with the MIBF settings was about $20\text{\,}\mathrm{\char
37\relax}\text{/}$ lower than observed with the nominal voltages.

(a) Comparison of module voltages.

(b) Comparison of $r\varphi$/̄resolution.

Figure 8: Minimum ion backflow studies: 8(a) Comparison of the nominal GEM
voltage settings used to collect most of the test/beam data and the settings
corresponding to the MIBF conditions. 8(b) Comparison of the point resolution
in $r\varphi$ for nominal and MIBF GEM/voltage settings.

In figure 8(b) the point resolution in the $r\varphi$/̄direction is plotted
against the drift distance. Good agreement is observed between the
measurements under MIBF conditions and the default voltage settings, except
for a slightly worse resolution at small drift distances with the MIBF
settings. This is interpreted as a more pronounced hodoscope effect [26],
likely caused by a combination of the lower amplification and the slightly
lower diffusion of the electron avalanche within the GEM stack due to the
changed fields. Still, this shows that the MIBF settings only have a small
influence on the achievable point resolution and therefore represent a viable
parameter set for the operation of the ILD TPC.

#### Extrapolation to the ILD TPC

To judge the performance of a large TPC in the final detector, equation 2 is
used to extrapolate the measured point resolution to the characteristics of
the ILD TPC, i.e. to a magnetic field of $3.5\text{\,}\mathrm{T}\text{/}$ and
a maximum drift length of $2.35\text{\,}\mathrm{m}\text{/}$. As input to the
extrapolation, $\sigma_{r\varphi,0}$ and $N_{\mathrm{eff}}$ are taken from the
fit of equation 2 to the measured resolution of the 2016 test/beam data given
in table 3. A Magboltz simulation has been performed to determine the
transverse diffusion constant at the ILD magnetic field, giving
$D_{\mathrm{t}}=$31.8\pm
0.4\text{\,}\mathrm{\SIUnitSymbolMicro}\mathrm{m}\mathrm{/}\sqrt{\mathrm{c}\mathrm{m}}$$.

Figure 9: Extrapolation of the point resolution to the ILD TPC dimensions with
a magnetic field of $3.5\text{\,}\mathrm{T}\text{/}$ and a maximum drift
length of $2.35\text{\,}\mathrm{m}\text{/}$. Each curve includes a $\pm
2\text{\,}\sigma$ confidence band. The extrapolation based on the measured
electron loss due to attachment is denoted _fit with $e^{-}$ loss_. For the
other graph (_fit with $e^{-}$ loss removed_) the attachment coefficient was
set to zero while keeping the other parameters unchanged.

The results of the extrapolation are shown in figure 9 for two cases. The
first case assumes the electron attachment rate as determined from the
measurement, where a nearly constant oxygen contamination of $\sim
65\text{\,}\mathrm{\mathrm{ppm}}\text{/}$ was observed. For the other case,
the electron attachment rate is set to zero to show the ultimate resolution
possible under perfect operating conditions. For both cases $\pm
2\text{\,}\sigma$ confidence bands are shown, which are derived from the
errors of the fitted parameters and their correlations. For the case in which
the attachment is set to zero, its uncertainty is — in contrast to the
previous analysis in [2] — also set to zero, resulting in a lower uncertainty
overall.

The extrapolation shows that — if contamination of the gas is minimized — a
point resolution of $100\text{\,}\mathrm{\SIUnitSymbolMicro m}\text{/}$ can be
reached over almost the full drift length in the ILD TPC, fulfilling the
requirement set for the ILD TPC. Experience of running experiments using a
TPC, e.g. T2K [6] and ALICE [27], indicates that an oxygen contamination of
less than $1\text{\,}\mathrm{\mathrm{ppm}}\text{/}$ is a realistic assumption.

##  5 Double-hit separation in $r\varphi$

In this section, the performance of the double/hit separation between two
tracks is studied using different reconstruction techniques on the multi/track
data set described in section 2.2.

The _default hit reconstruction_ algorithm uses the charge deposited on each
pad and computes its position in $r\varphi$ using a charge weighted center of
gravity method. It includes a simple local minimum algorithm to identify
double/hit candidates before the track finding, but no hit separation is
performed. A hit is tagged as a double/hit candidate, if the charge
distribution on the pads includes a dip in charge amplitude of at least one
ADC count in comparison to the adjacent pads. In this double/hit study, the
_triplet track finder_ [28] — a local track finding algorithm using triplet
chains — is used as the default algorithm on these hits.

In order to improve the double/hit and track separation in the pattern
recognition, two new methods have been developed and implemented in MarlinTPC.
The first method, called _pad pulse road search_ , is a combined track and hit
finding algorithm based on local road search with pulses. The second method,
called _PRF hit separation_ , is a hit splitting algorithm, where the pad
response function is fitted to the charge distribution of a (double-) hit
candidate.

#### Pad pulse road search

The pad pulse road search is a combined track and hit finding algorithm taking
the global event topology into account. It was adapted specifically for the
data taken with the prototype TPC in order to improve the double/hit
separation and works directly on the individual pulse signals on the pads of a
module.

In a first step, seed pulses are identified that have the maximum absolute
charge in their neighborhood, which is a region in a pad row of a size
reflecting the average transverse diffusion. Starting from these seed pulses,
linear roads with a width of a few times the pad pitch are defined between
pairs of yet unused seed pulses in different rows, provided they exceed a
minimum distance. Pulses inside a road are added and the road is iteratively
refined using a linear or a helical fit, resulting in a so/called pad pulse
segment. Ambiguous pulses, which belong to more than one pad pulse segment,
are removed and the segments refitted. Then the ambiguous pulses are
reassigned to the segment with the closest distance in the $r\varphi$ plane.

The hit finding is performed locally per row. Pulses are added to the maximum
pulse not yet assigned to a hit until another pad pulse segment starts or
there is a gap. The hit position is determined as in the default hit finding
procedure described in section 3. The _integrated hit separation_ of the pad
pulse finder assigns ambiguous pulses to hits based on proximity. In the case
of close-by seed pulses, the corresponding hits can only grow outwards,
leading to biases in the position determination.

Track segments are defined and fitted based on all hits of a pad pulse
segment. In a last step, matching track segments on different readout modules
are combined to track candidates, which are passed to the track fitting. A
detailed description of this combined track and hit finding method is given in
[29].

#### PRF hit separation

Figure 10: Schematic illustration of a fit of the sum of two pad response
functions to the signal of a merged hit.

The PRF hit separation method is shown schematically in the example in figure
10. After a merged hit candidate has been identified, a fit is performed using
the sum of two single/hit PRFs, which are convolutions of a Gaussian and a box
function. In this fit, the sigma of the two Gaussian functions is fixed
according to the specific drift distance. It is determined from a fit of
equation 1 to the measured PRF width, as shown in figure 6(b). The width of
the box functions is equal to the pad pitch. The mean positions are free
parameters, as are the amplitudes to adjust for different hit charges.

A series of requirements, listed in table 5, are used in order to identify
double/hit versus single/hit candidates and ensure a proper reconstruction. To
exclude signals from $\delta$/̄electrons, the fit is not performed if the hit
charge is significantly larger than average. If the charge of a hit after the
separation is too small, it will be rejected. If the number of pulses
contributing is too small, the fit becomes unstable. If it is too large, the
signal most probably does not come from a single or double hit but from other
effects like $\delta$/̄electrons or more than two overlapping tracks. Finally,
the $\chi^{2}$ of the fit of the single- and the double/hit PRF are compared
and there are limits on their accepted values.

parameter | accepted range | comment
---|---|---
charge of single hit $Q$ [a.u.] | $$600$<Q<$20,000$$ | average $Q\approx$2000$$
number of pulses $N$ | $$4$<N<$15$$ | for single hit: $N\leq$7$$
$\chi^{2}$ of a fit of the single/hit PRF | $\chi^{2}_{\mathrm{single}}<$10$$ |
$\chi^{2}$ of a fit of the double/hit PRF | $\chi^{2}_{\mathrm{double}}<$52$$ |
Table 5: The cut values used in the PRF hit separation.

The PRF hit separation can be used either on the tagged double/hit candidates
of the local hit search of the standard reconstruction or to improve the
handling of ambiguous pulses in the hit finding of the pad pulse road search.

#### Performance comparison

The default reconstruction is compared to the results from the pad pulse road
search. In an additional step, the PRF hit separation is run on the double/hit
candidates identified by the pad pulse road search to further improve the
double/hit separation.

(a) Measurement.

(b) Simulation.

Figure 11: Comparison between different hit and track reconstruction
algorithms and their effect on the double/hit separation 11(a) for measured
test/beam data and 11(b) for simulated tracks, both times for a drift distance
of $200\text{\,}\mathrm{mm}\text{/}$. Shown are the triplet track finder with
default hit reconstruction and the pad pulse road search with its integrated
and with the PRF hit separation. The results of the fitted curves are listed
in table 6. The curves show the fits of equation 3 to the data points.

Figure 11 shows the efficiency of the double/hit separation, using the methods
described above, as a function of the distance between the two tracks at the
location of the hit, for 11(a) test/beam data and 11(b) tracks from a Monte
Carlo simulation, both at a drift distance of $\sim
200\text{\,}\mathrm{mm}\text{/}$. The efficiency is defined as the ratio of
the number of separated hits to that of all double/hit candidates considered
at the respective distance between tracks. The distance between two tracks
corresponding to an efficiency of $50\text{\,}\mathrm{\char 37\relax}\text{/}$
defines the double/hit separation capability of the respective method. These
values were determined from a $\chi^{2}$ fit of the following function to the
measured separation efficiency:

$f\left(\Delta_{r\varphi}\right)=\frac{A}{2}\left(1+\operatorname{erf}\left(\frac{\Delta_{r\varphi}-M}{\sqrt{2}\cdot\sigma}\right)\right)\enskip.$
(3)

Here, $\Delta_{r\varphi}$ is the distance between tracks in the $r\varphi$
direction, $\operatorname{erf}(x)$ is the error function and $A$, $M$ and
$\sigma$ are fit parameters denoting the maximum efficiency at large track
distance, the inflection point and the standard deviation, respectively.

The fitted values of the parameters for the examples in figure 11 are given in
table 6. The results from the pad pulse road search algorithm show a
significant improvement in comparison to the results from the default
reconstruction. The combination of the pad pulse road search with the PRF hit
separation further improves the double/hit separation. Overall, an improvement
of the double/hit separation from around $5\text{\,}\mathrm{mm}\text{/}$ to
less than $2\text{\,}\mathrm{mm}\text{/}$ is observed in data and simulation.
This is consistent with the studies using parallel laser tracks in [30], where
efficient two/track separation was achieved for distances down to $\sim 1.5$
times the pad pitch.

| measured data | simulation
---|---|---
algorithm | $A$ | $M$ | $\sigma$ | $A$ | $M$ | $\sigma$
triplet track finder + default hit reconstruction | $0.91$ | $5.02$ | $0.74$ | $0.95$ | $4.87$ | $0.52$
pad pulse road search + integrated hit separation | $0.95$ | $2.29$ | $0.68$ | $0.98$ | $2.17$ | $0.75$
pad pulse road search + PRF hit separation | $0.96$ | $1.90$ | $0.48$ | $0.99$ | $1.65$ | $0.60$
Table 6: Parameter values of the fits of equation 3 in figure 11 for the
different hit and track reconstruction algorithms.

(a) Measurement.

(b) Simulation.

Figure 12: Comparison of the double/hit separation for three different
combinations of hit and track reconstruction algorithms 12(a) for measured
test/beam data and 12(b) for simulated tracks. Shown are the triplet track
finder with default hit reconstruction and the pad pulse road search with its
integrated as well as with the PRF hit separation. The bars depict the $\pm
1\text{\,}\sigma$ width of equation 3 describing the efficiency in dependence
of the track distance from zero to full separation.

To study the impact of the charge diffusion in the TPC volume on the hit
separation, the performance has been determined for different drift distances.
A comparison of the different methods is shown in figure 12 for 12(a) data and
12(b) the simulation. At short drift distances, the pad pulse road search
performs better in combination with the PRF hit separation than with its
integrated hit separation. At larger distances, where the diffusion effect
dominates, the performance of both methods is similar. Overall, an improvement
by more than a factor of two relative to the default hit reconstruction can be
seen. A more detailed description of this study is given in [31].

##  6 Resolution of the specific energy loss measurement

The specific energy loss per unit length ($\mathrm{d}E/\mathrm{d}x$) of a
charged particle at a certain momentum depends on its rest mass. Thus, by
measuring the $\mathrm{d}E/\mathrm{d}x$ and the momentum, the Bethe/Bloch
curve can be used to infer the mass and identify the particle species [32].
The specific energy loss is determined by measuring the charge deposited on
each readout row, which is proportional to the energy loss. The mean energy
loss, as given by the Bethe/Bloch formula, is determined from the resulting
Laudau/like distribution of all charge measurements along the trajectory.

In the analysis presented here, certain quality criteria are applied to ensure
that only hits with an unbiased charge measurement contribute to the
$\mathrm{d}E/\mathrm{d}x$ estimation. Before any other cuts, $15$ rows
containing too many noisy or dead channels, mostly due to faulty connections
of the readout cables, are excluded from the analysis, reducing the number of
usable rows from $84$ to $69$. Then, hits containing one or more dead channels
are rejected. Additionally, hits at the module edges, i.e. hits containing a
pulse on a pad at either end of a row and all hits on the first and last row
on each module, are ignored, as are hits on the two rows adjacent to the
central bar of the ceramic frame. If a hit is tagged as a multi/hit candidate
it is not taken into account as well. The pulse finder identifies these
multi/hit candidates based on the presence of a double/pulse structure in
time, which is equivalent to the drift direction. Also the hit finding
algorithm checks for double hits, both in the drift direction and the
transverse plane. In the drift direction, hits are tagged if any pad contains
more than one pulse within the allowed window of arrival time. In the
transverse plane, double/hit candidates are identified by looking for a local
minimum in the charge distribution between adjacent pads. After all cuts, an
average of $56.5$ hits per track is used for the $\mathrm{d}E/\mathrm{d}x$
calculation. The standard deviation of this value is $2.3$ hits.

For each of these hits, the charge $Q_{\mathrm{hit}}$ is normalized to the
path length $\Delta s$ of the track segment over the corresponding pad row as
an estimate of the energy loss:

$\frac{\Delta E}{\Delta x}\propto\frac{Q_{\mathrm{hit}}}{\Delta s}\enskip.$
(4)

To gain a stable estimator for the mean energy loss of a track, several
methods have been studied, including truncated and trimmed averages, all
yielding very similar results. In the end, the best resolution was achieved
with the truncated mean of $75\text{\,}\mathrm{\char 37\relax}\text{/}$ of the
smallest samples, cutting off the tail towards high charge values.

(a) Distribution of $\Delta E/\Delta x$.

(b) Distribution of $\mathrm{d}E/\mathrm{d}x$.

Figure 13: Distributions of the energy loss for one measurement run for 13(a)
individual samples and 13(b) the average of each track.

Figure 13(a) shows the distribution of the energy loss per sample $\Delta
E/\Delta x$ for all valid hits of one measurement run. For each track, the
average energy loss is calculated as the truncated mean of all the hits
belonging to this track, i.e. $\mathrm{d}E/\mathrm{d}x={\langle\Delta E/\Delta
x\rangle}_{$75$}$. The resulting distribution of $\mathrm{d}E/\mathrm{d}x$ for
all tracks of one run is plotted in figure 13(b). As a result of the Landau
tail, a slight enhancement of the falling edge of the distribution compared to
a Gaussian distribution can be observed. For this reason, the relative
$\mathrm{d}E/\mathrm{d}x$ resolution is calculated from the RMS and mean of
the distribution. Since the relative resolution is found to be independent of
the drift distance, the weighted average of $15$ consecutive measurement runs
at different drift distances is used to obtain a better estimate. This results
in a $\mathrm{d}E/\mathrm{d}x$ resolution of
$\sigma_{\mathrm{d}E/\mathrm{d}x}=$(8.95\pm 0.02\pm
0.14)\text{\,}\mathrm{\char 37\relax}\text{/}$$, where the first uncertainty
is the standard error of the weighted mean of the individual measurements. The
second error is the standard deviation of these measurements, which can be
interpreted as a systematic uncertainty.

To extrapolate to the $\mathrm{d}E/\mathrm{d}x$ resolution achievable at the
final ILD TPC, a method relying exclusively on the test/beam data has been
applied. For the ILD TPC with an outer radius of
$1770\text{\,}\mathrm{mm}\text{/}$, the foreseen number of pad rows is $220$.
In order to acquire tracks with a higher number than the $56.5$ hits per track
available from the test/beam data, hits from tracks in multiple events are
combined randomly. This allows to emulate tracks with an arbitrary number of
hits. Each hit is used only once during this process to avoid introducing
correlations. Twenty-five ensembles of pseudo tracks with lengths from $4$ up
to $220$ hits are created in this way. For each pseudo/track length the
$\mathrm{d}E/\mathrm{d}x$ resolution is extracted in the way described above,
including the combination of consecutive measurement runs.

Figure 14: Extrapolation of the $\mathrm{d}E/\mathrm{d}x$ resolution to the
ILD TPC dimensions, based on beam/test data.

The result is shown in figure 14. Naively, one could expect a square/root
dependence of the resolution on the number of hits. Instead, former
experiments have found that power laws
$\sigma_{\mathrm{d}E/\mathrm{d}x}=\sigma_{0}\cdot N^{-k}$ with exponents
$k<$0.5$$ describe their data best [33, 34, 35, 36]. Therefore, a fit of a
power law is performed on the extrapolated resolution, including only the
statistical errors, with the exponent and the constant $\sigma_{0}$ as free
parameters, resulting in $k=$0.480\pm 0.003$$. From this fit, the
$\mathrm{d}E/\mathrm{d}x$ resolution in the ILD TPC with up to $220$ hits is
expected to be $(4.41\pm 0.02\pm 0.10)\text{\,}\mathrm{\char
37\relax}\text{/}$ for tracks with a polar angle of $\theta=$$$ with respect
to the $z$/̄axis. The first error is the uncertainty calculated from the fit
results. As above, the second error is the standard deviation of the
extrapolation results gained from the individual measurements, which is
interpreted as a systematic uncertainty. For tracks with $\theta\neq$$$, the
effective track length in the TPC increases as $L\propto 1/\sin\theta$ while
the number of samples stays fixed, resulting in an improved resolution
$\sigma_{\mathrm{d}E/\mathrm{d}x}\propto L^{-k^{\prime}}$ with
$k^{\prime}=$0.32$$ [33], as long as the full number of hits is available.

A comparison of the interpolated curve in figure 14 with the direct
measurement of real tracks in the prototype reveals a discrepancy of about
$3\text{\,}\sigma$. This may indicate a bias in the method used for
extrapolation. A potential source of this bias are correlations between hits
on the same row in multiple consecutive events, that are assigned to the same
pseudo/track. Under this assumption, pseudo/tracks with few hits should be
least affected by this bias. Therefore, to estimate the effect of the bias it
is assumed that the fitted value of $\sigma_{0}$ is correct. The exponent $k$
is adjusted such that the function coincides with the direct measurement. This
results in $k=$0.465\pm 0.004$$ and a resolution for the ILD TPC that is
worsened by $\sim 0.3\text{\,}\mathrm{\char 37\relax}\text{/}$ absolute to
$(4.71\pm 0.02\pm 0.14)\text{\,}\mathrm{\char 37\relax}\text{/}$, still
allowing to fulfill the goal of a $\mathrm{d}E/\mathrm{d}x$ resolution of
better than $5\text{\,}\mathrm{\char 37\relax}\text{/}$ defined in [1].

###  6.1 Optimization of the readout granularity for $\mathrm{d}E/\mathrm{d}x$

To explore possible ways to improve the particle identification via the
specific energy loss, a simulation study was carried out to investigate the
performance of two methods to measure $\mathrm{d}E/\mathrm{d}x$, in dependence
of the granularity of the readout. The usual approach to measure
$\mathrm{d}E/\mathrm{d}x$ by determining the number of generated electrons,
i.e. the total charge, is compared to an alternative approach of counting the
number of primary ionizing interactions instead. In the first approach — here
called _charge summation_ — the number of electrons per path length follows a
Landau/like distribution. The long tail of this distribution worsens the
correlation of the measured average energy loss and the particle species. The
second approach — called _cluster counting_ — relies on the number of ionizing
interactions, i.e. ionization clusters, of the incident particle, which
follows a Poisson distribution with a significantly smaller width. This
results in a better correlation and particle identification power, as shown in
a previous simulation study presented in [37].

The number of ionizing interactions of the incident particle can be determined
by reconstructing and counting charge clusters on a highly granular readout.
If the pad or pixel size and the diffusion are smaller than the typical
distance between charge clusters, charge deposits on the readout plane are
correlated and can be combined to reconstructed clusters. Unfortunately, it is
not possible to meet these conditions fully, in particular a low enough
diffusion, to reach a counting efficiency close to $100\text{\,}\mathrm{\char
37\relax}\text{/}$. However, the study in [37] has shown, that even with a
cluster counting efficiency of only $20\text{\,}\mathrm{\char
37\relax}\text{/}$ the resulting separation power is better than by measuring
the energy loss by charge summation. This novel approach has not yet been
applied in an experiment, but developments in low/noise ASICs and compact
packaging may facilitate this in a future detector.

In the present study the performance of the two approaches is compared using
the particle separation power. The separation power $S_{AB}$ for two particle
types $A$ and $B$ is defined as

$S_{AB}=\frac{\left|\mu_{A}-\mu_{B}\right|}{\sqrt{\frac{1}{2}\left(\sigma^{2}_{A}+\sigma^{2}_{B}\right)}}\enskip.$
(5)

Here, $\mu_{A/B}$ are the estimators for the energy loss — usually the mean
$\mathrm{d}E/\mathrm{d}x$ — and $\sigma_{A/B}$ are the corresponding
measurement accuracies for particle types $A$ and $B$, respectively. In the
denominator the combined width from $\sigma_{A}$ and $\sigma_{B}$ is used
since this study is performed without considering any specific experiment
background. Therefore, there is no signal or background particle type and both
are compared using an equal number of particles of type $A$ and $B$.

Figure 15: The ionization density of pions and kaons in the simulated
argon/based gas mixture. The third graph shows the relative ionization density
of the two particle species on the axis to the right.

Following equation 5, the performance of particle separation is proportional
to the difference of the average ionization. As shown in figure 15, the
relative ionization of different particle species depends on the momentum.
Throughout the following comparisons, the separation power is evaluated at the
maximum of the ionization difference between pions and kaons of about
$16\text{\,}\mathrm{\char 37\relax}\text{/}$ at a momentum of
$3\text{\,}\mathrm{GeV}\text{/}\mathrm{c}$. To compare the simulation with the
results from test/beam measurements with relativistic electrons, a conversion
between the measured $\mathrm{d}E/\mathrm{d}x$ resolution and the pion/kaon
separation power was performed. This conversion takes into account the
dependence on the number of samples to adjust for the difference in track
length, as well as the differences between the ionization densities of
electrons, pions and kaons at the respective momenta. For the test/beam
measurement presented in this work, this results in a separation power of
$S_{\pi K}\approx$1.5$$ for the simulated track length of
$300\text{\,}\mathrm{mm}\text{/}$. Equivalent conversions are performed for
results with other pad/based readout systems [38, 39] as well as a pixel/based
readout system with a pitch of $55\text{\,}\mathrm{\SIUnitSymbolMicro
m}\text{/}$ [40]. The latter is considered twice, once with an anode coverage
of $95\text{\,}\mathrm{\char 37\relax}\text{/}$, comparable to the pad/based
systems, and once with about $60\text{\,}\mathrm{\char 37\relax}\text{/}$
coverage, comparable to the current pixel/based prototypes.

Figure 16: Simulation of the charge deposition in T2K gas at
$1\text{\,}\mathrm{T}\text{/}$ magnetic field on a readout with
$220\text{\,}\mathrm{\SIUnitSymbolMicro m}\text{/}$ wide pads. Measured charge
values are shown from low to high in dark to bright red. The yellow line shows
the path of the simulated incident minimum ionizing particle at
$1\text{\,}\mathrm{m}\text{/}$ drift distance. The green squares depict the
position of the drifted electrons arriving at the GEM stack, with their
respective size being proportional to their randomized amplification. The blue
crosses represent the cluster centers reconstructed by the source/extractor
software.

The MarlinTPC framework, introduced in section 3, has been used for the
simulation and the reconstruction. The detailed simulation operates at the
level of individual electrons from primary ionization and entails all steps of
the electron propagation in the detecor, i.e. ionization, drift, amplification
in a triple GEM stack, projection onto the readout plane and finally the
digitization in the readout electronics. For the reconstruction of the
ionization clusters, the source/extractor software package [41] has been
integrated into the MarlinTPC reconstruction. The simulation is performed for
square pads, varying the pad pitch between
$55\text{\,}\mathrm{\SIUnitSymbolMicro m}\text{/}$ and
$6\text{\,}\mathrm{mm}\text{/}$. Figure 16 shows an example of the resulting
charge deposition on pads with a pitch of
$220\text{\,}\mathrm{\SIUnitSymbolMicro m}\text{/}$. Also shown are the
incident particle track, the corresponding signal electrons and the
reconstructed cluster centers. In the simulation the gain of the GEM stack was
adjusted by varying the GEM voltages between $230\text{\,}\mathrm{V}\text{/}$
and $280\text{\,}\mathrm{V}\text{/}$ per GEM, resulting in overall gains
between $\sim 500$ and $62,000$. GEM voltages above
$280\text{\,}\mathrm{V}\text{/}$ were not used since this is considered the
limit of stable operation without frequent discharges. For each pad size and
method, the gain resulting in the best separation power was used.

(a) Pion-kaon separation power.

(b) $\mathrm{d}E/\mathrm{d}x$ resolution.

Figure 17: Results of the simulation study showing the performance of the
charge measurement and the cluster counting methods at two drift lengths as a
function of the size of the square pads. 17(a) The separation power between
pion and kaon tracks of $300\text{\,}\mathrm{mm}\text{/}$ length. 17(b) The
$\mathrm{d}E/\mathrm{d}x$ resolution for relativistic electrons with a track
length of $1300\text{\,}\mathrm{mm}\text{/}$. Also included are several values
from measurements with different systems: AsianGEM [38], GridGEM (this work),
Micromegas [39], GridPix [40].

Figure 17 shows the results of the simulations for both
$\mathrm{d}E/\mathrm{d}x$ measurement methods in terms of 17(a) the achievable
separation power between pion and kaon tracks of
$300\text{\,}\mathrm{mm}\text{/}$ length and 17(b) the resolution for
electrons with a momentum of $5\text{\,}\mathrm{GeV}\text{/}\mathrm{c}$ and a
track length of $1300\text{\,}\mathrm{mm}\text{/}$, equivalent to the ILD TPC.
Both are shown as a function of the size of the readout pads, once for a drift
distance of $200\text{\,}\mathrm{mm}\text{/}$ and once for a longer drift
distance of $1000\text{\,}\mathrm{mm}\text{/}$, always at a magnetic field of
$1\text{\,}\mathrm{T}\text{/}$. These parameters are comparable to the
situation at the prototype TPC with a maximum drift length of
$570\text{\,}\mathrm{mm}\text{/}$ and a magnetic field of
$1\text{\,}\mathrm{T}\text{/}$. The overall diffusion is also similar to the
conditions at a potential ILD TPC, which would have a larger drift length of
over $2000\text{\,}\mathrm{mm}\text{/}$ but operates at a higher magnetic
field of $3.5\text{\,}\mathrm{T}\text{/}$.

For very small readout pads the cluster counting method yields a very good
separation power, reaching up to $S=$2.3$$ for a drift distance of
$200\text{\,}\mathrm{mm}\text{/}$ and a pad size of
$110\text{\,}\mathrm{\SIUnitSymbolMicro m}\text{/}$. Above
$300\text{\,}\mathrm{\SIUnitSymbolMicro m}\text{/}$ the separation power
quickly drops with increasing pad size and becomes nearly independent of the
drift distance.

For the charge summation method the separation power rises with decreasing pad
size, from about $S=$1.5$$ with $6\text{\,}\mathrm{mm}\text{/}$ pads to a
value of $1.75$ or $2.1$, depending on the drift distance. At very small pad
sizes the separation power decreases again as the charge is distributed over
many pads and falls below the noise threshold. This is compensated to some
degree by a small drift distance and thus lower diffusion, resulting in a
separation power of $2.1$ at an optimum pad size of
$440\text{\,}\mathrm{\SIUnitSymbolMicro m}\text{/}$ for a drift distance of
$200\text{\,}\mathrm{mm}\text{/}$. As is presented in [23], former experiments
suggest the separation power measured by charge summation to behave as
$S\propto(\text{pad\\-/size})^{\alpha}$, with $\alpha$ between $-0.12$ and
$-0.15$. This dependence is included as the purple line in figure 17, and is
compatible with the simulation for large pads, until the aforementioned
threshold effect starts to play a role.

The simulation agrees very well with the performance of the test/beam
measurements with pad/based systems with pad sizes around
$6\text{\,}\mathrm{mm}\text{/}$. It describes the increase in separation power
with decreasing pad size, including the transition region going from
macroscopic pads and charge summation to very small pixels and cluster
counting. At a granularity of $55\text{\,}\mathrm{\SIUnitSymbolMicro
m}\text{/}$ the simulation with cluster counting reaches a separation power
similar to the result from the current pixel/based readout systems.

This study demonstrates that to perform particle identification using
$\mathrm{d}E/\mathrm{d}x$, a pad size of about a factor of $10$ smaller than
currently used could improve the separation power by about
$15\text{\,}\mathrm{\char 37\relax}\text{/}30\text{\,}\mathrm{\char
37\relax}\text{/}$, depending on the drift distance. To do particle
identification by cluster counting, the granularity has to be increased by
another factor of two to improve the particle separation power, with the
benefit of being more independent of the drift distance and the corresponding
diffusion. These results show the direction a further study could take to
optimize the particle identification. In the final pad/size optimization,
additional factors such as the pattern/recognition performance, the spatial
resolution and the overall number of channels will have to be taken into
account.

##  7 Conclusion

A new generation of readout modules based on a triple GEM stack were operated
in a large prototype TPC at the DESY test beam. The construction procedures
for this generation of modules have been improved resulting in a more
controlled and reproducible production, while at the same time ensuring a
higher quality. The performance of the new readout modules was studied. The
comparison to measurements with the previous readout generation show a good
reproducibility of the performance. The extrapolation of the spatial single-
point resolution confirmed that the requirements set for the ILD TPC can be
fulfilled, if the gas contamination is minimized.

Furthermore, two new approaches have been implemented to identify and split
hits of close/by tracks. This includes a pad pulse road search as combined hit
and track finder as well as a hit splitting algorithm based on fitting pad
response functions to the double/hit candidates. The methods were studied on
simulated and measured data and compared to the existing method of hit
reconstruction. The combination of both new methods leads to an improvement of
the double/hit separation power by more than a factor of two, down to a track
distance well below $2$ times the pitch of the readout pads. For the pad pitch
of $1.26\text{\,}\mathrm{mm}\text{/}$ in the $r\varphi$ direction, this
fulfills the requirement of $2\text{\,}\mathrm{mm}\text{/}$ for the ILD TPC.

Finally, the resolution of the specific energy loss measurement was determined
for the test/beam data, resulting in a resolution of $(8.95\pm 0.02\pm
0.14)\text{\,}\mathrm{\char 37\relax}\text{/}$. The extrapolation of these
results to the final ILD TPC yields a resolution of $(4.71\pm 0.02\pm
0.14)\text{\,}\mathrm{\char 37\relax}\text{/}$, which meets the goal of
$5\text{\,}\mathrm{\char 37\relax}\text{/}$ for the ILD TPC. A simulation
study examining ways to improve the particle identification by specific energy
loss was performed. It indicates that with smaller pads of about
$500\text{\,}\mathrm{\SIUnitSymbolMicro m}\text{/}$ pitch a separation power
up to $30\text{\,}\mathrm{\char 37\relax}\text{/}$ better could be achieved.
With even smaller pads below $300\text{\,}\mathrm{\SIUnitSymbolMicro
m}\text{/}$ pitch, a cluster counting method could be employed, further
improving the separation power.

## Acknowledgments

This material is based upon work supported by the National Science Foundation
of the U.S.A. under Grant No. 0935316, was supported by the Japan Society for
the Promotion of Science (JSPS) KAKENHI Grant No. 23000002 and was funded by
the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) –
491245950. The research leading to these results has received funding from the
European Commission under the 6th Framework Programme “Structuring the
European Research Area”, contract no. RII3-026126, and under the FP7 Research
Infrastructures project AIDA, grant agreement no. 262025.

Special thanks go to Y. Makida, M. Kawai, K. Kasami and O. Araoka of the KEK
IPNS cryogenic group, and A. Yamamoto of the KEK cryogenic center for their
support in the configuration and installation of the superconducting PCMAG
solenoid.

The measurements leading to these results have been performed at the Test Beam
Facility at DESY Hamburg (Germany), a member of the Helmholtz Association. The
authors would like to thank the technical team at the DESY II accelerator and
test beam facility for the smooth operation of the test beam and the support
during the test/beam campaign.

The contributions to the experiment by the University of Lund, KEK, Nikhef and
CEA are gratefully acknowledged.

## Disclaimer

Any opinions, findings, and conclusions or recommendations expressed in this
material are those of the authors and do not necessarily reflect the views of
any of the funding agencies involved. No warranty expressed or implied is made
with regard to any information or its use in this paper.

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to split $I_{2}$ into a sequence $I^{\prime}_{2}<\dots<I^{\prime}_{k}$ with
holes, and get (A.2) in the general case. To show this, let
$H_{1}=\\{a,\dots,b-1\\}$ and $I_{2}=\\{b,\dots,c\\}$. Then relation (A.19)
reads

$C_{n\dots c+1,b-1\dots a,1}=-[C_{n\dots b,1},C_{a\dots b\dots
c}]_{q}+C_{n\dots a,1}C_{b\dots c}+C_{a\dots b-1}C_{n\dots c+1,1}.$

Applying $\overline{r}_{c}$, according to Proposition 3.6, we get

$C_{n\dots c+2,c,b-1\dots a,1}=-[C_{n\dots b,1},C_{a\dots b\dots
c-1,c+1}]_{q}+C_{n\dots a,1}C_{b\dots c-1,c+1}+C_{a\dots b-1}C_{n\dots
c+2,c,1}.$

Thus we have successfully created $I^{\prime}_{2}=\\{b,\dots,c-1\\}$,
$I^{\prime}_{3}=\\{c+1\\}$ and a hole $H_{2}=\\{c\\}$ such that
$I_{2}=I^{\prime}_{2}I^{\prime}_{3}$. We can reproduce this as much as we need
to produce as many holes as we want in $I_{2}$.

###  A.3 Proof of the morphism property for $\bm{r_{0}}$

First, we show that $r_{0}$ preserves the commutation relations (2.5). We
start with the situation where $r_{0}$ acts non-trivially only on one element
of the commutator, that is, with ${[C_{1\dots k},C_{i\dots j}]=0}$ with $1\neq
i\leq j$. The image by $r_{0}$ is ${[C_{n\dots k+1\,1},C_{i\dots j}]=0}$. This
is satisfied by Lemma 2.13 since if $\\{i,\dots,j\\}$ is included in (resp.
disjoint from) $\\{1,\dots,k\\}$, then is it disjoint from (resp. included in)
$\\{n,\dots,k+1\\}\cup\\{1\\}$.

Next, we consider the relation $[C_{1\dots k},\\!C_{1\dots
k^{\prime}}]\\!=\\!0$, which is sent by $r_{0}$ to ${[C_{n\dots
k+1\,1},\\!C_{n\dots k^{\prime}+1\,1}]\\!=\\!0}$. This latter relation is
satisfied due to (2.29b).

Then we consider (2.6) and (2.7). As for the proof for the other $r_{i}$’s,
the situation where $r_{0}$ leaves invariant the two elements appearing in the
$q$-commutator is easily handled, since $r_{0}$ leaves invariant every
elements appearing in the relation. When $r_{0}$ leaves invariant only one
element in the $q$-commutator, we can apply the same reasoning as around
(A.1), using the formula in Proposition 3.4 and $q$-Jacobi. We omit the
details since they follow the same reasoning as for the other $r_{i}$’s.

We are left with the situation where $r_{0}$ acts non-trivially on the two
elements appearing in the $q$-commutator. This can happen only for relation
(2.6), when $1\in I_{3}$ (so that $I_{3}<I_{2}<I_{1}$). Here we use
Proposition 3.6 to apply $r_{0}$ and get

$C_{I_{1}I_{2}}=-[C_{I_{0}I_{2}\,1},C_{I_{0}I_{1}\,1}]_{q}+C_{I_{1}}C_{I_{2}}+C_{I_{0}I_{1}I_{2}\,1}C_{I_{0}\,1},$

where $I_{0}$ is the subset adjacent to $I_{1}$ and going up to $n$. If
$\\{1\\}$ is adjacent to $I_{2}$, then this is relation (2.19c). The general
case is obtained with the same reasoning as for (A.19).

### Computer programs

We used formal calculation softwares (FORM and Maple) for some proofs done in
this article: the corresponding programs are available upon request to the
authors.

### Acknowledgements

The authors thank A. Lacabanne for fruitful discussions. N. Crampé and L.
Poulain d’Andecy thank LAPTh for its hospitality and are supported by the
international research project AAPT of the CNRS and the ANR Project AHA
ANR-18-CE40-000. This work is partially supported by Université Savoie Mont
Blanc and Conseil Savoie Mont Blanc grant APOINT. We also wish to thank the
anonymous referees for their valuable comments during the course of revision.

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