###### Abstract

Future Higgs Factories will allow the precise study of $e^{-}e^{+}\rightarrow
q\bar{q}$ with $q=s,c,b,t$ interactions at different energies, from the Z-pole
up to high energies never reached before. In this contribution, we will
discuss the experimental prospects for the measurement of differential
observables in $e^{-}e^{+}\rightarrow b\bar{b}$ and $e^{-}e^{+}\rightarrow
c\bar{c}$ processes at high energies, 250 and 500 GeV, using full simulation
samples and the full reconstruction chain from the ILD concept group. These
processes call for superb primary and secondary vertex measurements, a high
tracking efficiency to correctly measure the vertex charge and excellent
hadron identification capabilities using $dE/\/dx$. This latter aspect will be
discussed in detail together with its implementation within the standard
flavour tagging tools developed for ILD (LCFI+). In addition, prospects
associated with potential improvements using cluster counting techniques
instead of traditional $dE/\/dx$ will be discussed.

###### Contents

  1. 1 Introduction
  2. 2 The ILD Concept and MC Samples
  3. 3 $R_{q}$ and $A_{FB}$ reconstruction at ILD
  4. 4 Optimisation of the $A_{FB}$ reconstruction at ILC
    1. 4.1 Optimisation of the default flavour tagging of ILD: Adding hadron PID as input.
    2. 4.2 Prospects of using a pixel TPC device: From $dE/\/dx$ to $dN/\/dx$
  5. 5 Experimental prospects for $A_{FB}$ and $R_{q}$ at ILC250 and ILC500
  6. 6 Summary and prospects
  7. 7 Appendix
    1. 7.1 Flavour tagging variables
    2. 7.2 Reconstruction performance plots for the ILC500 case

##  1 Introduction

The Standard Model (SM) was successfully confirmed when the Higgs boson was
discovered by the Large Hadron Collider (LHC)[1, 2] experiments. Since then,
the measurement of its properties with the highest accuracy has become a
primary goal. For that reason, in parallel with the LHC data exploitation, the
high-energy accelerator-based particle physics community works towards
constructing the next large accelerator after the LHC. Such a machine will be
an $e^{+}e^{-}$ collider at relatively high energy to achieve the highest
precision on the Higgs physics measurements. Different projects have been
proposed and are under discussion. These are the so-called Higgs factories.

The International Linear Collider (ILC)[3, 4, 5, 6, 7] is one of the Higgs
factories proposed for the future. The ILC operation foresees $e^{+}e^{-}$
collisions at different centres of mass energies ranging from the Z-pole
(Giga-Z) to up to 1 TeV after energy upgrades. The nominal program defines an
initial stage at 250GeV (ILC250), with a luminosity upgrade, and a following
one at 500GeV (ILC500), after an energy upgrade. The ILC also features
polarised beams: $80\%$ for electrons and $30\%$ for positrons. The ILC250
physics program foresees a total integrated luminosity of 2000 fb-1
distributed between four different beam polarisation schemes: $45\%$ in
$P_{\mathrm{e^{-}e^{+}}}=(-0.8,+0.3)$, $45\%$ in
$P_{\mathrm{e^{-}e^{+}}}=(+0.8,+0.3)$, $5\%$ in
$P_{\mathrm{e^{-}e^{+}}}=(-0.8,-0.3)$ and $5\%$ in
$P_{\mathrm{e^{-}e^{+}}}=(+0.8,-0.3)$. The ILC500 program foresees a total
integrated luminosity of 4000 fb-1 distributed among the different
polarisation configurations in the same proportions as the ILC250 case.

The ILD [7, 8] is one of the two proposed detector concepts for the ILC. It is
a highly-hermetic multi-purpose detector designed for the maximal exploitation
of particle flow techniques in event reconstruction.

This document updates the studies presented in [9]. In that work, the
experimental prospects for $e^{-}e^{+}\rightarrow q\bar{q}$ measurements at
ILC250 are studied. The study of such topologies is critical to obtain a
complete picture of the electroweak sector interactions between SM bosons and
fermions, particularly quarks. For instance, models of new physics featuring
extra-dimensions [10, 11, 12] have been proposed to explain the striking mass
hierarchy in the fermion sector. These models predict deviations in the
electroweak sector with modification of the SM couplings and new heavy
resonances $Z^{\prime}$ that interact with the fermions. Furthermore, the
LEP/SLC anomaly in $e^{-}e^{+}\rightarrow b\bar{b}$ is still unexplained [13].
The effects of new physics may differ for different fermion chiralities and
additional terms associated with various mediators (SM $Z$ and $\gamma$ or
beyond SM $Z^{\prime}$ or mixing of these). This motivates the study of quark
pair production in high energy $e^{-}e^{+}$ collisions at past lepton
colliders [14] and at future ones [9].

In [9], two experimental observables have been studied in detail for the
ILC250 using the ILD detector in full simulation. The observables are the
Hadronic Fraction ($R_{q}$) and Forward-Backward Asymmetry ($A_{FB}$). The
observable $R_{q}$ is defined as:

$R_{q}=\frac{\sigma_{e{-}e^{+}\rightarrow q\bar{q}}}{\sigma_{had.}}$ (1)

where $\sigma_{had.}$ is the integration of $\sigma_{q\bar{q}}$ for all quark
flavours except the top quark. In the case of $A_{FB}$ the definition reads:

$A^{q\bar{q}}_{FB}=\frac{\sigma^{F}_{e{-}e^{+}\rightarrow
q\bar{q}}-\sigma^{B}_{e{-}e^{+}\rightarrow
q\bar{q}}}{\sigma^{F}_{e{-}e^{+}\rightarrow
q\bar{q}}+\sigma^{B}_{e{-}e^{+}\rightarrow q\bar{q}}}$ (2)

where $\sigma^{F/B}_{e{-}e^{+}\rightarrow q\bar{q}}$ is the cross-section in
the forward/backward hemisphere as defined by the polar angle $\theta_{q}$.

In this contribution, we report on incorporating an improved flavour tagger
that uses the hadron identification (protons, kaons, pions) as input. In
addition, the study has been extended to include the prospects of the same
study at ILC500. Finally, we include a prospective study of the precisions
that could be reached if a pixel TPC is used for the ILD.

##  2 The ILD Concept and MC Samples

The ILD subdetector layout consists of a high-precision vertex detector (VTX),
silicon tracking systems, a time projection chamber (TPC), a highly granular
calorimeter system (ECAL and HCAL) and a muon catcher. All the aforementioned
subdetectors are placed inside a solenoid providing a magnetic field of $3.5$
T, surrounded by an iron yoke instrumented for muon detection. The ILD TPC
[15, 16] is a large volume time projection chamber that allows continuous 3D
tracking and particle identification. Its baseline design consists of a
barrel-shaped structure with an inner radius of 329 mm and an outer radius
larger than 1808 mm. It provides a single-point resolution of 100 µm over
about 200 readout points and a $dE/\/dx$ resolution of $\sim 4.5\%$. It is
based on $1\times 6$ mm2 pads readout with GEM or MicroMegas technologies.
This document will also explore the potential of an alternative approach: a
pixel TPC. Several technological solutions are under study, and all envision a
four times greater readout density with a pixel size of 300 µm. Simulations
extrapolating beam test results show that an improved relative resolution of
$\sim 3$-$4\%$ will be feasible using cluster counting techniques ($dN/\/dx$)
instead of the traditional approach [16].

This study has been conducted running full simulation via ILCSOFT
v02-02-03111Link: https://github.com/iLCSoft, which merge different software
packages and algorithms that operate in a modular way from MC events to final
reconstruction. All simulations use the ILD-L[8] model, whose geometry,
material and readout systems are implemented in the DD4HEP framework[17],
interfaced with Geant4 toolkit. Both signal and background events are
generated with the WHIZARD[18] event generator at LO. The beam energy
spectrum, beam-beam interaction and QED ISR is generated via Guinea-Pig[19].
Non-perturbative effects, such as the FSR and hadronization, are provided by
Pythia event generator[20].

To reproduce different stages of the ILC operation, two different samples have
been provided by the ILD concept group; one for 250 GeV and one for 500 GeV.
Each set of samples features full polarised beams in different configurations.
The main samples for this study are the $\mathrm{e^{-}_{L}}\mathrm{e^{+}_{R}}$
and $\mathrm{e^{-}_{R}}\mathrm{e^{+}_{L}}$. Background processes from
electroweak bosons, Higgs and top-quark production (at 500 GeV) are studied
using additional MC samples. Backgrounds from the production of lepton pairs
have been ignored since they are expected to be easily identified. All
distributions and results have been reweighted to match the baseline
polarization and full luminosity scenarios for ILC250 and ILC500, so-called
the H20 scenario [21].

The generation of the $e^{-}e^{+}\rightarrow q\bar{q}$ signal events and the
$e^{-}e^{+}\rightarrow Z\gamma\rightarrow q\bar{q}\gamma$ with the Z-boson
generated on-shell are done simultaneously. Following the same recipe as in
[9], we define the signal as the events with acolinearity of the
$q$$\overline{q}$ system (at parton level) smaller than 0.3 and a large
invariant mass of the $q$$\overline{q}$ pair larger than 140 GeV - for the
ILC250 case - or larger than 200 GeV for the ILC500 case (i.e. large enough to
be away of the Z-mass peak).

The signal and background cross sections are listed in Tables 1 and 2 from [9]
for ILC250 and in Tables 1 and 2 for ILC500. Only backgrounds leading to fully
hadronic final states are considered. Backgrounds involving leptons in the
final states are ignored since those are expected to be easily identified.

| $\sigma_{e^{-}e^{+}\rightarrow q\bar{q}}$[fb] | Radiative Return BKG [fb]
---|---|---
Polarisation |  $b$$\overline{b}$ |  $c$$\overline{c}$ |  $q$$\overline{q}$ ($q=uds$) |  $b$$\overline{b}$ |  $c$$\overline{c}$ |  $q$$\overline{q}$ ($q=uds$)
$e_{L}^{-}e_{R}^{+}$ | 611.4 | 1545.9 | 2770.2 | 5506.4 | 5118.0 | 16244.2
$e_{R}^{-}e_{L}^{+}$ | 416.9 | 893.8 | 1728.3 | 3002.5 | 2766.9 | 8773.5
Table 1: Production cross section of signal and background originated by di-boson production.  | $\sigma_{e^{-}e^{+}\rightarrow\,X}$ [fb]
---|---
Polarisation | $X=WW\rightarrow q_{1}\bar{q_{2}}q_{3}\bar{q_{4}}$ | $X=ZZ\rightarrow q_{1}\bar{q_{1}}q_{2}\bar{q_{2}}$ | $X=HZ\rightarrow q\bar{q}H$ | $X=t\bar{t}\rightarrow b\bar{b}q_{1}\bar{q}_{2}q_{3}\bar{q}_{4}$
$e_{L}^{-}e_{R}^{+}$ | 7680.0 | 680.2 | 114.7 | 660
$e_{R}^{-}e_{L}^{+}$ | 33.5 | 271.9 | 73.4 | 254.8
Table 2:  Cross sections at 250 GeV for processes producing at least one pair
of $q$-quarks using fully polarised beams.

##  3 $R_{q}$ and $A_{FB}$ reconstruction at ILD

The reconstruction of the events begins with the track reconstruction
performed by the MarlinTrk framework, part of ILCSoft. Later, Pandora[22] runs
the particle flow algorithm (PFA) that matches the tracking information with
the high-granular calorimetry information following pattern recognition
techniques. The resulting reconstructed objects are denominated particle flow
objects (PFO) and are treated as single particles.

Once the PFOs are reconstructed, the vertex reconstruction, jet reconstruction
and flavour tagging are performed with LCFI+ software tool.

Once all high-level reconstruction objects are obtained, the event
preselection is performed. It is based on a series of kinematical cuts applied
to enrich the data sample with signal events while removing the backgrounds.
The preselection for 250 GeV is the same as described in [9], where the
variables are described, with minor adaptations made for the study of the 500
GeV samples (noted in parenthesis):

  1. 1.

photon veto cuts, rejecting events if:

    1. (a)

at least one of the jets contains only one PFO;

    2. (b)

at least one of the jets contains a reconstructed $\gamma_{cluster}$ with E >
115 GeV (220 GeV);

  2. 2.

events with $\sin\Psi_{acol}$ > 0.3 are rejected;

  3. 3.

events with $m_{jj}$ < 140 GeV (200 GeV) are rejected;

  4. 4.

events with $y_{23}$ > 0.02 (0.007) are rejected.

The most noticeable change for the ILC500 case is performed in the last cut,
which is tightened to reduce the much larger $WW$ background contamination in
the case of left-handed polarisation. This last cut decreases the final
preselection efficiency to $\sim 50\%$ instead of $\sim 75\%$ for the ILC250
case.

The next step is applying the Double Tag (DT) method stated in [9] by which a
cut is performed in the $b_{tag}$ (or $c_{tag}$) likelihood in both jets of
the event. The working points are defined as in [9] such that the mistagging
of the other quarks when performing the tagging of $b$-quark (or $c$-quark) is
smaller than the $\sim 1.5\%$ $(\sim 3\%)$, assuming the same size for the
$b/c/uds$ samples. The DT yields a high-purity selection of the events for
each flavour and is the last necessary step to reconstruct $R_{q}$
measurements.

To study $A_{FB}$, the charge of the jets needs to be reconstructed to fully
determine the direction of the $q\bar{q}$ system. As described in [9], two
independent methods are used for the charge measurement: $Vtx$-method or
$K$-method. The $Vtx$-method counts the charge of all tracks in secondary
vertices in the jet, while the $K$-method only uses the information of tracks
identified as Kaons from the TPC dE/dx PID. The Double Charge (DC) method
performs the final charge measurement, which combines the possible
measurements of both methods and accepts only jets with opposite charges. The
differential, in $\cos\theta$, cross section is reconstructed, and the
$A_{FB}$ extracted from a fit restricted to ($\|\cos\theta\|>0.9$) to avoid
the region where the reconstruction efficiency drops due to insufficient
acceptance of the tracking detectors.

All the methods (Preselection, DT, DC, Fit) have been developed for the
analysis at 250 GeV and are explained in detail in [9] together with a
comprehensive study of the most dominant systematic uncertainties. The same
methods are used for the study at 500 GeV shown in this document. In Appendix
7.2, we collect the most relevant performance plots for the ILC500 case to be
compared with the ones in [9].

##  4 Optimisation of the $A_{FB}$ reconstruction at ILC

###  4.1 Optimisation of the default flavour tagging of ILD: Adding hadron PID
as input.

The flavour tagging is performed by the LCFI+ package [23]. The vertex
reconstruction method distinguishes between vertices and pseudo-vertices,
which are single-track vertices candidates to be merged in a fully
reconstructed vertex in a second iteration of the algorithm. At the end of the
process, if these pseudo-vertices are not merged in a full vertex, they are
kept in the list of objects and treated as single-track vertices.

The total number of vertices and pseudo-vertices is always two or fewer. Four
different categories are defined (A, B, C, D): A are jets without vertices and
up to two pseudo-vertices, B are jets with one vertex and no pseudo-vertices,
C are jets with one vertex, and one pseudo-vertex and D are jets with two
vertices. The flavour content in each category is different as expected due to
the different hadronisation products of heavy quarks: Above 95% of light-quark
jets are in category A, more than 90% of $c$-quark jets are spread between
categories A and B while more than 80% of the $b$-quark jets are split between
categories B, C and D.

To prepare weights for flavour tagging via LCFI+, two main algorithms have to
be run: MakeNTuple and TrainMVA. MakeNTuple prepares the ROOT files with
NTuples with all the different variables that could be used for the flavour
tagging. TrainMVA uses 3 NTuples as input (one for $b$-jets, one for $c$-jets
and one for $uds$-jets) and then runs a classificator based on Boosted
Decision Trees (BDT implemented in ROOT’s TMVA). The input is the signal data
separated into the different categories described above. A Marlin
processor222Link: https://github.com/marherje/LCIO_Extraction.git was designed
ad hoc to do this selection.

New variables were built using the TPC’s PID capabilities via $dE/\/dx$ to
improve the flavour tagging. The new variables were a count of kaons, protons
and pion candidates. The charged kaon, proton and pion IDs are defined by
constructing likenesses variables using a statistical distance measurement,
comparing theoretical and experimental values. for instance, the definition of
kaonness, $\Delta_{dE/\/dx-K}$, is defined as:

$\Delta_{dE/\/dx-K}=\left(\frac{dE/dx_{exp}-dE/dx_{K,BB}}{\Delta
dE/dx_{exp}}\right),$ (3)

where $dE/dx_{K,BB}$ is the theoretical value given by the Bethe-Bloch
formula, $dE/dx_{exp}$ is the experimental value from the TPC’s measurements
and $\Delta dE/dx_{K,exp}$ is the error associated to that measurement as
implemented in the official ILD reconstruction software. This definition is
extended to other hadron types (pionness, protonness). This distance can be
modelled as normally distributed. It can be constructed concerning each
particle’s theoretical values, hence having the Gaussian centred in 0 for each
case (kaonness, pionness, protonness). To have a proper distribution, all
tracks with momenta below 3 GeV and $\|\cos\theta\|>0.95$ were discarded
before selecting particle candidates. Low-momenta tracks are removed because
the Bethe-Bloch curves at low energies for pions and kaons overlap. Removal of
very forward/backward tracks is for a loss of geometrical acceptance of the
TPC. We count as candidate particles of each type those whose absolute
likeness value is below 1.5, which minimises overlapping with other particles.
All these new variables and selections were implemented in a beta version of
LCFI+ that is already available and waiting for an official pull to the main
GitHub repository of LCFI+. This process counts only tracks associated with a
secondary vertex (or pseudo-vertex).The final three new variables added to
MakeNTuple are named: dEdxNKaonSec, dEdxNProtonSec and dEdxNPionSec, as they
represent the number of each particle in secondary vertices.

There is a direct correlation between quark flavour and kaon production via
B-mesons ($b$-quark) and D-mesons ($b$-quark and $c$-quark) decays which
motivate using kaon PID in flavour tagging. Identification of kaons is also
useful for the charge reconstruction as stated in Sec. 3, which is especially
important in the case of $c$-quark. Using $dE/\/dx$ is strongly motivated
given the expected separation power of high momenta tracks (above 3 GeV)(see
[8], Figure 8.6).

The standard training process runs once through the four vertexing categories
using the same configuration of the BDTs. Still, in the case of this
optimisation, independent training has been performed for each category. To
ensure that the BDT configuration is optimal for each different scenario
(energy, category and variable selections), a Particle Swarm Optimisation
(PSO) was run. The PSO[24] is a parameter-free, stochastic, bio-inspired
algorithm that searches for the optimal configuration of a problem by doing an
iterative scanning of configurations for the problem. The particles are
positions in the space of configurations for the given problem, which move to
a new position after each iteration. It requires the definition of a Function
Of Merit (FOM) as scoring to determine the movement of the particles after
each iteration. In this case, a PSO software[25] has been adapted and extended
to a 3-class classifier with filters suitable to work with LCFI+333Link:
https://github.com/marherje/PSOforLCFIPlus.git. The space of configurations
for flavour tagging are the hyper-parameters of the BDTs, and the particles of
the PSO are different selections of such hyper-parameters. The different
hyper-parameters for the BDTs are the number of trees, the maximum number of
leaves for each tree, the shrinkage, the bagging fraction and the number of
bins (for the physical input variables’ histograms). The function of merit of
choice for this study was the average value of the integral of the Receiving
Operating Characteristic (ROC) curve for the three different flavour
categories ($b$, $c$, $uds$) from the test sample. To avoid overfitting to the
test sample, two different statistical tests (Kolmogorov-Smirnov[26, 27, 28]
and Anderson-Darling[29, 30, 31]) were run each time a particle finished its
training, comparing the BDTs output between the test and training samples. In
both statistical tests, a threshold value was set upon observation, always
keeping the p-value over the standard $p>0.05$ criteria but also checking that
the fluctuations in the FOM values were below 1.5 %. The complete list of
variables used in each category is in Table 3 of the Appendix 7.1, where the
bold text are the new variables and the rest are original LCFI+ variables as
introduced in [23].

###  4.2 Prospects of using a pixel TPC device: From $dE/\/dx$ to $dN/\/dx$

A better PID resolution is expected when using $dN/\/dx$ instead of $dE/\/dx$.
This will be possible with a pixel-based TPC. According to [16], an
improvement of $\sim 30-40\%$ in the kaon/pion separation power is achievable
for tracks with momentum between 3-50 GeV, with the separation power as
defined in [8] and in the following equation:

$\eta_{A,B}(p)=\frac{\lvert\mu_{A}(p)-\mu_{B}(p)\rvert}{\sqrt{\frac{1}{2}(\sigma_{A}^{2}(p)+\sigma_{B}^{2}(p)}},$
(4)

where $\mu_{(}p)$ and $\sigma$ are the mean and standard deviations of the
gaussian fits to the experimental value of $dE/\/dx$, defined as a function of
the momenta $p$ of each bin. However, the $dN/\/dx$ reconstruction is not yet
implemented in the ILD software, so it is not possible to introduce a
$dN/\/dx$ PID variable for the flavour tagging and charge measurement
directly, as for $dE/\/dx$. Instead, we apply a correction to the likenesses
distributions by the expected value. This is done by a dedicated Marlin
processor that emulates a $25\%$ reduction in standard deviation in the
likenesses distributions. This $25\%$ reduction in the standard deviation in
the likenesses distributions is equivalent to a to a $\sim 33\%$ improvement
in the separation power, In addition, in the flavour tagging case, an Ideal
PID resolution has been considered by reducing the standard deviation of the
simulated likeness distributions by a $99\%$, compared with the full
simulation.

Figure 1: Kaonness distribution of secondary tracks in c-quark jets at ILC250
for protons, kaons and pions. The continuous line represents the real
$dE/\/dx$ distribution, while the dotted line represents the $dN/\/dx$
estimation.

Once the expected improvement in resolution for the $dN/\/dx$ case is
accounted for, the full analysis is repeated. Given that now the likeness
distributions offer a better separation between types of hadrons (see Fig.1),
the criteria to accept a track as kaon candidate requires an absolute value of
the kaonness smaller than 1.0 instead of 1.5. Besides the separation criteria,
the optimisation process is the same, including the PSO for each possible
scenario. In Fig.2, the effects on flavour tagging when adding the three new
variables from kaon, protons and pions are shown for the case of
$dE/\/dx$,$dN/\/dx$ and ideal PID with 100% efficiency for both 250 and 500
GeV. In Fig.3, the impact of moving from $dE/\/dx$ to $dN/\/dx$, envisioning a
pixel PID, is shown for 250 and 500 GeV.

|
---|---
|
Figure 2: Prospects on flavour tagging (purity vs efficiency). On top row: effects of b-tagging in the left plot and c-tagging in the right plot, both at 250 GeV. The second row shows the result for 500 GeV. The panel below represents the ratio between the new taggings and the default selection on all plots.  |
---|---
|
Figure 3: Prospects on charged kaon selection (purity vs efficiency) comparing
the use of $dN/\/dx$ instead of $dE/\/dx$. On top row: effects in
$b$$\overline{b}$ (left) and $c$$\overline{c}$ (right) events, both at 250
GeV. The second row shows the result for 500 GeV.

##  5 Experimental prospects for $A_{FB}$ and $R_{q}$ at ILC250 and ILC500

Three different scenarios have been studied: reconstruction without TPC kaon
ID, a reconstruction using TPC Kaon ID (via $dE/\/dx$) for charge measurement
as well as adding $dE/\/dx$ in the flavour tagging and a reconstruction using
TPC Kaon ID (via $dN/\/dx$) for charge measurement as well as adding $dN/\/dx$
in the flavour tagging, being the later an estimation as described in section
4.2. These three scenarios are covered for both 250 and 500 GeV, for the cases
of $P_{\mathrm{e^{-}e^{+}}}=(-0.8,+0.3)$ and
$P_{\mathrm{e^{-}e^{+}}}=(+0.8,-0.3)$. A comparison for each case using only
statistical uncertainties has also been plotted. The results are summarised in
Fig. 4.

|
---|---
|
Figure 4: Prospects on $A_{FB}$ and $R_{q}$ experimental measurements at
ILC250 (left column) and ILC500 (right column). In the first row, the
estimations assume only statistical uncertainties. In the second row, the
systematic uncertainties are also included.

##  6 Summary and prospects

In this document, an update of the previous work at ILC250 [9] has been done,
extending it to ILC500. Furthermore, a detailed study of the impact of the TPC
PID capabilities at ILD in both scenarios has been covered, including
prospects of using cluster counting ($dN/\/dx$) with a pixel TPC. Adding PID
from TPC measurements into the flavour tagging software for $b$ and $c$-quark
only provides a residual improvement since the main flavour-tagging variables
are related to the vertex reconstruction. However, the TPC PID plays a key
role in the reconstruction of $A_{FB}$. Prospects of a pixel TPC using
$dN/\/dx$ for PID envision a noticeable impact in the charge reconstruction of
the $q$$\overline{q}$ systems, especially for the $c$-quark case, where a
factor $\sim 2$ improvement in the experimental precision is obtained.

Ongoing work by the authors of this note is the evaluation of experimental
sensitivity of BSM models predicting new heavy resonances $Z^{\prime}$[10, 11,
12], which could affect the predictions of $R_{q}$ and $A_{FB}$ at ILC250/500.

The ILC program also envisions a run at the Z-pole, the GigaZ mode[32]. With
that run. one would expect at least one order of magnitude of improvements for
the Z-couplings measurements of $b$-quark and $c$-quark compared to those of
SLC and LEP and give a complete picture of the quark EW couplings when
combining measurements at Giga-Z, ILC250 and ILC500. However, since a study at
the Giga-Z would involve the development of new simulations and methodologies,
it is considered out of this work’s scope.

## Acknowledgements

We would like to thank the LCC generator working group and the ILD software
working group for providing the simulation and reconstruction tools and
producing the Monte Carlo samples used in this study. This work has benefited
from computing services provided by the ILC Virtual Organization, supported by
the national resource providers of the EGI Federation and the Open Science
GRID.

The authors are funded by the Generalitat Valenciana (Spain) under the grant
number CIDEGENT/2020/21. AI also acknowledges the financial support from the
MCIN with funding from the European Union NextGenerationEU and Generalitat
Valenciana in the call Programa de Planes Complementarios de I+D+i (PRTR 2022)
Project Si4HiggsFactories, reference ASFAE$/2022/015$.

##  7 Appendix

###  7.1 Flavour tagging variables

Cat. |  Variables
---|---
A |  trk1d0sig trk2d0sig trk1z0sig trk2z0sig trk1pt_jete trk2pt_jete jprobr2 jprobr25sigma jprobz2 jprobz25sigma d0bprob2 d0cprob2 d0qprob2 z0bprob2 z0cprob2 z0qprob2 dEdxNKaonSec dEdxNPionSec dEdxNProtonSec
B |  trk1d0sig trk2d0sig trk1z0sig trk2z0sig trk1pt_jete trk2pt_jete jprobr2 jprobz2 vtxlen1_jete vtxsig1_jete vtxdirang1_jete vtxmom1_jete vtxmass1 vtxmult1 vtxmasspc vtxprob d0bprob2 d0cprob2 d0qprob2 z0bprob2 z0cprob2 z0qprob2 trkmass nelectron nmuon dEdxNKaonSec dEdxNPionSec dEdxNProtonSec
C |  trk1d0sig trk2d0sig trk1z0sig trk2z0sig trk1pt_jete trk2pt_jete jprobr2 jprobz2 vtxlen1_jete vtxsig1_jete vtxdirang1_jete vtxmom1_jete vtxmass1 vtxmult1 vtxmasspc vtxprob 1vtxprob vtxlen12all_jete vtxmassall dEdxNKaonSec dEdxNPionSec dEdxNProtonSec
D |  trk1d0sig trk2d0sig trk1z0sig trk2z0sig trk1pt_jete trk2pt_jete jprobr2 jprobz2 vtxlen1_jete vtxsig1_jete vtxdirang1_jete vtxmom1_jete vtxmass1 vtxmult1 vtxmasspc vtxprob vtxlen2_jete vtxsig2_jete vtxdirang2_jete vtxmom2_jete vtxmass2 vtxmult2 vtxlen12_jete vtxsig12_jete vtxdirang12_jete vtxmom_jete vtxmass vtxmult 1vtxprob dEdxNKaonSec dEdxNPionSec dEdxNProtonSec
Table 3: Variables used in the TMVA’s BDT in each of the categories in LCFI+.
This selection of variables is the same for the case with $dE/\/dx$, $dN/\/dx$
or ideal PID scenarios. The bold text represent the new variables while the
rest are original LCFI+ variables introduced in [23].

###  7.2 Reconstruction performance plots for the ILC500 case

|
---|---
|
Figure 5: Extracted $\epsilon_{q}$ and $R_{q}$ using the DT method for $e^{-}e^{+}\rightarrow c\bar{c}$ (left) and $e^{-}e^{+}\rightarrow b\bar{b}$ (right). For the estimation of $R_{q}$ the comparison with the case in which the Monte Carlo is used to estimate the $\epsilon_{q}$ is included, showing no difference with the DT expectations.  |
---|---
Figure 6: Distributions for $P_{chg.}$ for the $Vtx$-method (blue) and the $K$-method (red) for $c$$\overline{c}$ (left) and $b$$\overline{b}$ (right). The results are shown for different beam polarisation scenarios using different types of lines. This figure is the correspondence to Figure 15 from [9] but for ILC500 and using the $dE/\/dx$ information also for the flavour tagging.  |
---|---
Figure 7: Distribution of the different selection efficiencies for $c$$\overline{c}$ (left) and $b$$\overline{b}$ (right) for the $A_{FB}$ measurement, as described in Eq. 20-22 [9] For the $c$$\overline{c}$ ($b$$\overline{b}$) case, the $Cat.1$ corresponds to only the $K$-method ($Vtx$-method) applied and $Cat.2$ to only the $Vtx$-method ($K$-method) applied. This figure is the correspondence to Figure 18 from [9] but for ILC500 and using the $dE/\/dx$ information also for the flavour tagging.  |
---|---
|
Figure 8: Fit (red graph) of the final distributions (red points) to the
function described in Equation 23 from [9] The fit is performed between
$-0.9<\cos\theta<0.9$ to avoid the regions with a large correction due to
efficiency and acceptance losses. The result of the fit is extended to
$[-1,1]$ for the $A_{FB}$ estimation. The LO calculations are represented in
the green graphs. The forward-backward asymmetry after the full reconstruction
and the LO prediction are compared and are well in agreement with statistical
uncertainties. This figure is the correspondence to Figure 19 from [9] but for
ILC500 and using the $dE/\/dx$ information also for the flavour tagging.

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