∎

[$\star$]t1This work was carried out in the framework of the ILD concept
group<EMAIL_ADDRESS>e3On leave from Tohoku University, Sendai,
Japan

11institutetext: IFIC, Universitat de València and CSIC, C./ Catedrático José
Beltrán 2, E-46980 Paterna, Spain 22institutetext: Université Paris-Saclay,
CNRS/IN2P3, IJCLab, 91405 Orsay, France 33institutetext: Department of
Physics, National Taiwan University, Taipei, Taiwan 10617, R.O.C.

# Probing Gauge-Higgs Unification models at the ILC with di-quark forward-
backward asymmetry at center-of-mass energies above the $Z$ mass. t1

A. Irlesaddr1,e1  J.P. Márquezaddr1  R. Pöschladdr2  F. Richardaddr2  A.
Saibeladdr1
H. Yamamotoe3,addr1  N. Yamatsuaddr3

(Received: date / Accepted: date)

###### Abstract

The International Linear Collider (ILC) will allow the precise study of
$e^{-}e^{+}\rightarrow q\bar{q}$ interactions at different center-of-mass
energies from the $Z$-pole to 1 TeV. In this paper, we discuss the
experimental prospects for measuring differential observables in
$e^{-}e^{+}\rightarrow b\bar{b}$ and $e^{-}e^{+}\rightarrow c\bar{c}$ at the
ILC baseline energies, 250 and 500 GeV. The study is based on full simulation
and reconstruction of the International Large Detector (ILD) concept. Two
gauge-Higgs unification models predicting new high-mass resonances beyond the
Standard Model are discussed. These models predict sizable deviations of the
forward-backward observables at the ILC running above the $Z$ mass and with
longitudinally polarized electron and positron beams. The ability of the ILC
to probe these models via high-precision measurements of the forward-backward
asymmetry is discussed. Alternative scenarios at other energies and beam
polarization schemes are also discussed, extrapolating the estimated
uncertainties from the two baseline scenarios.

###### Keywords:

ILC Beyond Standard Model Higgs-Boson

††journal: Eur. Phys. J. C

##  1 Introduction

The Standard Model (SM) is a successful theory, well-established
experimentally and theoretically. With the discovery of the Higgs boson
CMS:2012qbp ; ATLAS:2012yve , the structure of the SM seems to be confirmed.
However, the SM cannot explain many of its seemingly arbitrary features. An
example is the striking mass hierarchy in the fermion sector. Moreover, while
the dynamics of the SM gauge bosons, the photon, $W$ and $Z$ bosons, and
gluons are governed by the gauge principle, the dynamics of the Higgs boson
are different and unique in the SM. The SM does not predict the strength of
the Higgs couplings of quarks and leptons, nor the Higgs self-couplings. Large
quantum corrections must be canceled by fine-tuning the parameters to match
the measured Higgs boson mass. One possible solution to this issue, achieving
stabilization of the Higgs mass against quantum corrections, appears when the
Higgs boson is associated with the zeroth mode of a dimension-five component
of extensions of the SM gauge group. These models are referred to as gauge-
Higgs unification (GHU) models.

The two most precise determinations of $\sin^{2}\theta_{\text{eff}}$ by the
LEP and SLC differ by $3.7$ standard deviations, and neither agrees with the
SM prediction ALEPH:2005ab ; Djouadi:2006rk . In particular, the LEP value was
extracted from the forward-backward asymmetry measurement for $b$-quarks in
LEP1 data, and is nearly three standard deviations away from the value
predicted by the SM. Clarifying this anomaly and exploring the possibility of
BSM physics motivates the study of quark pair production in high energy
$e^{-}e^{+}$ collisions at future colliders both at the $Z$ boson mass and
higher energies. In the SM, these interactions are mediated by the photon, $Z$
boson, and their interference. Some BSM theories predict deviations of these
bosons’ couplings or even sizable new contributions to these processes from
new mediators (such as heavy $Z^{\prime}$ resonances). These deviations would
be accessible experimentally by performing high precision measurements of
$e^{-}e^{+}\rightarrow q\overline{q}$ observables at different center-of-mass
energies ($\sqrt{s}$). The work presented here is based on the study of such
processes at the ILC.

In parallel to the exploitation of data from the Large Hadron Collider (LHC),
the high-energy accelerator-based particle physics community is working
towards the next large collider after the LHC. It will be a high-energy
$e^{-}e^{+}$ collider. Various projects have been proposed and are under
discussion. These “Higgs Factories” are designed for the precise scrutiny of
the Higgs sector and search for new physics through precision measurements. We
here discuss in detail the International Linear Collider (ILC) which we
consider the most mature from a technological point of view, having produced a
technical design report (TDR) in 2013 Behnke:2013xla ; Baer:2013cma ;
Adolphsen:2013jya ; Adolphsen:2013kya ; Behnke:2013lya . The ILC plans for a
comprehensive high-precision physics program based on collisions of polarized
electron and positron beams at a center-of-mass energy of 250 “ILC250” and 500
GeV “ILC500”. Operation around the $t$$\overline{t}$ production threshold, the
$Z$-mass, and at 1 TeV are also proposed. The studies discussed in this
document are based on full simulations of the International Large Detector
(ILD) concept Behnke:2013lya ; ILD:2020qve . The ILD is one of the detectors
proposed for collecting and exploiting the ILC data. It has been optimized to
perform high-precision measurements at ILC250 and ILC500.

Other proposals for Higgs factories, not discussed in this document, are the
Compact Linear Collider (CLIC CLIC:2016zwp ; CLIC:2018fvx ; CLICdp:2018cto ),
the Future Circular $e^{-}e^{+}$ Collider (FCCee FCC:2018byv ; FCC:2018evy ),
the Circular electron-positron Collider (CEPC CEPCStudyGroup:2018rmc ;
CEPCStudyGroup:2018ghi ) and the Cool Copper Collider (C3 Vernieri:2022fae ).
The CLIC and C3 are linear colliders with baseline design collision energies
of 380, 1000, and 3000 GeV (for CLIC) or 250 and 550 GeV (for C3). Both
foresee using longitudinal polarization only for the electron beam. The FCCee
and CEPC are electron-positron circular colliders featuring only
longitudinally unpolarized beams and with baseline collision energies at the
$Z$ mass and 240 GeV with an eventual upgrade to 365 GeV, at least for the
FCCee case. Intermediate energy stages around the $WW$ production threshold
are also envisioned.

The content of this article is structured as follows. Section 2 briefly
describes the two GHU models used as benchmarks and discusses the sensitivity
of the forward-backward asymmetry, $A_{FB}$, in $e^{-}e^{+}\rightarrow
b\bar{b}$ and $e^{-}e^{+}\rightarrow c\bar{c}$ processes at different energies
and beam polarization scenarios. Section 3 describes the ILC, the ILD, and the
experimental framework for the study presented here. Section 4 presents a
detailed discussion of the methodology and prospects for precisely measuring
$A^{c}_{FB}$ and $A^{b}_{FB}$ at ILC250 and ILC500, based on previous full
simulation studies Irles:2023ojs ; Irles:2023nee . Section 5 discusses the
potential of indirect constrains on the two aforementioned GHU theories
through the measurement of $A_{FB}$. Conclusions and prospects are discussed
in Section 6.

##  2 Theory

###  2.1 Gauge-Higgs Unification Models

The $SU(3)_{c}\times SO(5)_{W}\times U(1)_{X}$ GHU models on a Randall-Sundrum
(RS) warped space have been studied as candidates to explain physics beyond
the SM, where $SO(5)_{W}\times U(1)_{X}$ contains the electroweak gauge
symmetry $SU(2)_{L}\times U(1)_{Y}$ Agashe:2004rs ; Medina:2007hz ;
Hosotani:2008tx ; Funatsu:2014fda ; Funatsu:2019xwr ; Funatsu:2020haj ;
Funatsu:2021yrh ; Funatsu:2023jng ; Yamatsu:2023bde . In this scenario, the
Higgs field naturally appears as a fluctuation mode of the Aharonov-Bohm phase
$\theta_{H}$ in the extra dimension. Gauge symmetry stabilizes the Higgs boson
mass against quantum corrections. With suitable parameter choices, low energy
predictions of this gauge theory can match the SM. In this paper, two specific
GHU models are considered: the $A$ Funatsu:2014fda  and $B$ Funatsu:2019xwr
models.

Both models predict massive neutral vector bosons $Z^{\prime}$, which are the
Kaluza-Klein (KK) modes of $\gamma$, $Z$, and $Z_{R}$ (the $SU(2)_{R}$ gauge
boson). Here, we simply describe the models and parameter sets that are used
in this paper. For the full description see Refs. Agashe:2004rs ;
Medina:2007hz ; Hosotani:2008tx ; Funatsu:2014fda ; Funatsu:2019xwr ;
Funatsu:2020haj ; Funatsu:2021yrh ; Funatsu:2023jng ; Yamatsu:2023bde .

In the $A$ model Funatsu:2014fda , quark-lepton multiplets are introduced in
the vector representation of $SO(5)$. The non-observation of $Z^{\prime}$ at
the LHC implies limits $\theta_{H}\lesssim 0.09$, $m_{KK}\gtrsim 9$ TeV
Funatsu:2021yrh . Whether the coupling of the $Z^{\prime}$ bosons is stronger
to right- or left-handed fermions depends strongly on the sign of the bulk
mass of fermions. The couplings of the right-handed fermions to the
$Z^{\prime}$ are large since positive bulk masses of fermions are chosen to
avoid large deviations of the $Z$ boson couplings to fermions. In this paper,
two parameter sets of the $A$ model ($A_{1}$ and $A_{2}$) are adopted as
benchmark points Funatsu:2017nfm : 111The parameter sets $A_{1}$ and $A_{2}$
correspond to $A_{2}$ and $A_{3}$ in Funatsu:2017nfm , respectively.
$A_{1}:\theta_{H}=0.0917,m_{KK}=8.81$ TeV$\rightarrow m_{Z^{1}}=7.19$ TeV;
$A_{2}:\theta_{H}=0.0737,m_{KK}=10.3$ TeV$\rightarrow m_{Z^{1}}=8.52$ TeV,
where $Z^{1}$ is the first KK $Z$ boson. The masses of the first KK $\gamma$
and $Z_{R}$ bosons are similar to those of the first KK $Z$ boson. The
parameters are chosen such that the couplings of the SM-like $Z$ boson to
fermions (other than the top quark) agree with current measurements within one
part in $10^{4}$. The values of $m_{KK}$ and $\theta_{H}$ for which the top
quark and the Higgs boson masses and electroweak symmetry breaking can be
realized are strongly restricted. The $m_{KK}$ value in the parameter set
$A_{2}$ is at the maximum reachable in the $A$ model.

|  |
---|---|---
|  |
Figure 1: Predicted differences for the $A_{FB}$ observable in $e^{-}e^{+}$
collisions at several $\sqrt{s}$ between different GHU models and the SM. The
expectations for different final states for $c$-quark or $b$-quark pair
production and different longitudinal beam polarizations are compared for each
energy.

In the $B$ model Funatsu:2019xwr , which is inspired by grand unification, the
quark-lepton multiplets are introduced in spinor, vector, and singlet
representations of $SO(5)$. More specifically, this model is constructed as a
low-energy effective description of the $SO(11)$ gauge-Higgs grand unification
model on the RS warped space Hosotani:2015hoa ; Furui:2016owe , and the
representations of $SU(3)_{c}$, $SO(5)_{W}$, and $U(1)_{X}$ that can be
introduced are strongly restricted by $SO(11)$ gauge symmetry. The non-
observation of $Z^{\prime}$ and $W^{\prime}$ signals at LHC implies limits
$\theta_{H}\lesssim 0.10$, $m_{KK}\gtrsim 13$ TeV Funatsu:2021yrh . In this
paper, six parameter sets of the $B$ model, $B_{j}^{\pm}$ ($j=1,2,3$), are
adopted as benchmark points Funatsu:2023jng ; Yamatsu:2023bde  222The
parameter sets $B_{1}^{\pm}$, $B_{2}^{\pm}$, and $B_{3}^{\pm}$ correspond to
$A^{\pm}$, $B^{\pm}$, and $C^{\pm}$ in Funatsu:2023jng ; Yamatsu:2023bde ,
respectively. :
$B_{1}^{\pm}$: $\theta_{H}=0.10,m_{KK}=13$ TeV$\rightarrow m_{Z^{1}}=10.2$
TeV;
$B_{2}^{\pm}$: $\theta_{H}=0.07,m_{KK}=19$ TeV$\rightarrow m_{Z^{1}}=14.9$
TeV;
$B_{3}^{\pm}$: $\theta_{H}=0.05,m_{KK}=25$ TeV$\rightarrow m_{Z^{1}}=19.6$
TeV;
where the superscripts $\pm$ indicate the sign of the lepton bulk masses. The
bulk masses of the quarks are negative in all parameter sets since very light
vector-like quarks appear when the bulk masses of the quarks are taken to be
positive. Model parameters are chosen such that $Z$ couplings to fermions
(other than the top quark) agree with the SM within one part in $10^{3}$.
These models make use of the so-called 4D bare Weinberg angle (a projection of
the electroweak mixing angle), which is defined such that all the parameter
sets in the $B$ model predict values of $A_{FB}$
$(e^{-}e^{+}\rightarrow\mu^{-}\mu^{+})$ compatible with current best
measurements Funatsu:2020haj . For $B_{j}^{+}$, the coupling of right-handed
electron to the $Z^{\prime}$ boson is larger than that of the left-handed
electron; for $B_{j}^{-}$, the coupling of left-handed electron to
$Z^{\prime}$ boson is larger than that of the right-handed electron.

###  2.2 Forward-backward asymmetry predictions in GHU

The forward-backward asymmetry, $A_{FB}$, is defined as

$A^{q}_{FB}=\frac{\sigma^{F}-\sigma^{B}}{\sigma^{F}+\sigma^{B}},$ (1)

where $\sigma^{F/B}$ is the $e^{-}e^{+}\rightarrow q\bar{q}$ cross-section in
the forward (F) and backward (B) hemisphere as defined by the polar angle of
the quark $\theta_{q}$ in the nominal center-of-mass reference frame and with
respect to the electron beam direction. For each of the models described
above, $A_{FB}$ has been calculated at leading order for $b$ and $c$-quark
production in several $e^{-}e^{+}$ scenarios at various center-of-mass
energies and beam polarizations. We use the $(P_{e^{-}},P_{e^{+}})$ notation
for beam polarization, in which the first term is for the electron and the
second for the positron beam, a negative sign signifies a left-handed
polarization, and 0 corresponds to un-polarized and $\pm$1 to fully polarized
beams.

The deviations from the SM value of $A_{FB}$ induced by the different models
are shown in Fig. 1 at 250, 500, and 1000 GeV, with and without ILC-like beam
polarization. The expected differences increase with energy and show large
variations depending on the model and the beam polarization. At 250 GeV, the
largest deviations occur in $b$-quark pair production with $(+0.8,-0.3)$ beam
polarization, with the highest values for the $A$ models. At 500 GeV,
$c$-quark pair production also shows large deviations in $(+0.8,-0.3)$ for the
$A$ models and in $(-0.8,+0.3)$ for the $B$ models. At 1 TeV, most models show
sizeable deviations for at least one of the discussed channels.

##  3 Experimental framework

###  3.1 The International Linear Collider and the International Large
Detector

The International Linear Collider (ILC) is a linear electron-positron collider
that will produce collisions at several energies, and feature a high degree of
longitudinal polarization for both beams. This article focuses on collisions
at center-of-mass energies of 250 GeV and 500 GeV (ILC250 and ILC500) in the
baseline running scenario, so-called H20-staged Bambade:2019fyw . The
H20-staged scenario assumes different integrated luminosities split between
left-handed and/or right-handed electron and positron beams. In addition, we
also briefly discuss other scenarios such as operation at the $Z$-pole
(ILCGigaZ) and 1 TeV (ILC1000). This information is summarized in the Tab. 1.

Table 1: Considered integrated luminosities,$\int\mathcal{L}$, and beam polarization degree scenarios considered in this work. The second row gives the degree of beam polarization for electrons and positrons. The third row shows the split of the total integrated luminosities when operating with opposite sign polarization (OSP) or same sign polarization (SSP) beams. | ILCGigaZ | ILC250 | ILC500 | ILC1000
---|---|---|---|---
$\int\mathcal{L}$ $[fb^{-1}]$ | 100 | 2000 | 4000 | 8000
$(|P_{e^{-}}|,|P_{e^{+}}|)$ | (0.8,0.3) | (0.8,0.3) | (0.8,0.3) | (0.8,0.2)
OSP|SSP [$\%$] | 40|10 | 45|5 | 40|10 | 40|10

The International Large Detector (ILD) is one of the detectors proposed for
collecting and exploiting the ILC data. The ILD design is optimized for the
reconstruction of final state particles using Particle Flow techniques
Brient:2002gh ; Thomson:2009rp . ILD consists of inner vertexing and tracking
systems and high granularity calorimeters within a 3.5 T solenoid, followed by
an instrumented flux return used to identify muons. A detailed description of
the different subsystems and the proposed technological solutions can be found
in Refs. Behnke:2013lya ; ILD:2020qve . The tracking systems in the current
ILD design are briefly discussed due to their crucial role in the studies
presented in this paper.

The vertexing and tracking systems are based on silicon sensors and a time
projection chamber (TPC). The vertex detector (VTX) is the closest to the beam
pipe, spanning radii from 16 to 60 mm. Its design is optimized to provide a
single hit resolution of 3 µm. The ILC bunch train structure allows for power-
pulsed operation, reducing power consumption and cooling requirements by one
to two orders of magnitude. Low-mass passive cooling technologies can
therefore be used, resulting in a material budget of around 0.15% of a
radiation length per layer, thereby minimizing multiple scattering.

Silicon tracking systems follow the VTX detector: the silicon internal tracker
(SIT) covers the central region, and the forward tracking detector (FTD)
extends the coverage to lower angles closer to the beam axis. The SIT also
features a barrel geometry and covers the region between 16 and 164 degrees
with respect to the beam axis. The FTD comprises disks perpendicular to the
beam axis and is designed to cover the low-angle region down to 4.8 degrees,
complementing the SIT coverage between 16 and 32 degrees. The TPC is a large
volume time projection chamber allowing continuous 3D tracking and charged
particle identification based on the specific energy loss $dE/\/dx$. It has a
length of 4 m and spans radii from 329 mm to 1808 mm, providing up to 220
track measurements with a position resolution in the $r-\phi$ plane of around
100 µm and a $dE/\/dx$ resolution of approximately 4.5$\%$ with pad-based
readout. An alternative approach read out by 55 µm pixels has the potential
for improved performance. 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 $dE/\/dx$
approach LCTPC:2022pvp .

###  3.2 Event simulation

The results in this paper are based on full simulation samples of ILC250 and
ILC500 provided by the ILD concept group. All simulations use the ILD-L
ILD:2020qve  model, whose geometry, material, and readout are implemented in
DD4HEP Frank:2014zya , interfaced with Geant4 Agostinelli:2002hh ;
Allison:2006ve ; Allison:2016lfl . Inactive materials describing cables,
support structures, and services are accounted for in the simulation and
reconstruction. SM signal and background events are generated with the WHIZARD
v2.8.5 Kilian:2007gr  event generator at LO, including QED ISR in perturbative
leading-logarithmic approximation. Parton shower and hadronization effects are
simulated by the Pythia 6.4 event generator Sj_strand_2006 . The beam energy
spectrum and beam-beam interaction producing incoherent $e^{-}e^{+}$
background pairs are generated with Guinea-Pig Schulte:1998au . Other
background sources, such as $\gamma\gamma$ to low $p_{T}$ hadrons, are
generated separately and overlaid on the simulated events Chen_1994 . A
description of the whole procedure to generate all SM processes is given in
Berggren:2021sju . Each set of samples features fully longitudinally polarized
beams in various configurations. Samples with realistic polarization scenarios
are obtained by merging these samples with appropriate weights.

This work is based on the analysis of $e^{-}e^{+}\rightarrow q\overline{q}$
events at high center-of-mass energies Irles:2023ojs ; Irles:2023nee . In the
SM, at $\sqrt{s}$ larger than the $Z$ boson mass, these processes are
sensitive to $\gamma$ and $Z$-couplings. However, due to QED ISR, the center-
of-mass energy of the $e^{-}e^{+}$ system may be reduced with respect to the
$\sqrt{s}$ of the collider. Furthermore, if the energy radiated in the QED ISR
process is large enough, the $e^{-}e^{+}$ system may undergo a radiative
return to the $Z$-pole, producing an on-shell $Z\rightarrow q\overline{q}$
process.

The ISR changes the kinematic event properties, so such events can be
distinguished from the high mass signal events. It is therefore treated as a
background, which we name “$\gamma q\overline{q}$” or “radiative-return”. For
a formal separation between the signal and radiative-return processes, we
define as signal those events with a quark-pair invariant mass larger than 140
GeV for ILC250 (200 GeV for ILC500) and acolinearity smaller than 0.3, with
acolinearity defined as in Eq. 7 from Irles:2023ojs , using a simplified
definition of the acolinearity

$\displaystyle\sin{\Psi_{acol}}=\frac{|\vec{p_{q}}\times\vec{p}_{\bar{q}}|}{|\vec{p_{q}}|\cdot|\vec{p}_{\bar{q}}|},$
(2)

with $\cos(\Psi_{acol})<0$.

Other sources of backgrounds are due to hadronically decaying di-boson or, for
ILC500, top-quark pair production. The cross sections for all these processes,
involving $q=u,d,s,c,b$ in the final state, are summarized in Tab. 2.

Table 2: Expected cross sections for signal ($q\overline{q}$) and main
background processes.

Process | ILC250 | ILC500
---|---|---
$\sigma$ [pb] | $(-0.8,+0.3)$ | $(+0.8,-0.3)$ | $(-0.8,+0.3)$ | $(+0.8,-0.3)$
$q\overline{q}$ | 17.2 | 6.4 | 2.9 | 2.0
$\gamma q\overline{q}$ | 60.2 | 39.3 | 16.3 | 9.5
$q\overline{q}q\overline{q}$ (no Higgs) | 16.2 | 1.5 | 8.2 | 0.6
$HZ\rightarrow q\bar{q}H$ | 0.2 | 0.1 | 0.07 | 0.05
$b\bar{b}q_{1}\bar{q}_{2}q_{3}\bar{q}_{4}$ | – | – | 0.4 | 0.2

###  3.3 Event reconstruction

The ILD track reconstruction Gaede_2014 ; ILD:2020qve  is based on pattern
recognition algorithms carried out independently in the different parts of the
tracker systems. This is followed by the combination of all the track
candidates and segments for a final refit performed with a Kalman filter. The
entire process is implemented in the MarlinTrk framework, which is part of the
ILCSoft toolkit. The resulting tracks are combined with calorimeter hits in
the Pandora particle flow algorithm (PFA) Marshall:2015rfa  to produce a set
of particle flow objects (PFO), each of which should correspond to a final-
state particle.

####  3.3.1 Vertex and jet reconstruction

Once the PFOs are reconstructed, vertex reconstruction, jet reconstruction,
and jet flavor tagging are performed using LCFIPlus Suehara:2015ura . The
primary vertex of the event is found in a tear-down procedure, starting with
all tracks and gradually removing tracks less compatible with being associated
with the primary vertex hypothesis. In a second step, LCFIPlus performs an
iterative reconstruction of secondary vertices. Jets are reconstructed using
the $VLC$ algorithm Boronat:2016tgd  for $e^{-}e^{+}$ colliders in exclusive
two-jet mode. This algorithm includes additional beam jets which are
disregarded.

####  3.3.2 Flavor tagging

The $b$ and $c$-quark tagging are performed by LCFIPlus using boosted decision
trees (BDTs) based on sensitive variables from tracks, vertices, and charged-
hadron identification using TPC information. BDTs are trained using
reconstructed jets in $e^{-}e^{+}\rightarrow q\bar{q}$ events at
$\sqrt{s}=250$ or 500 GeV accordingly Irles:2023nee . Particle Swarm
Optimization 488968  with combined Kolmogorov-Smirnoff kolmogorov1933sulla ;
smirnov1948table ; hodges1958significance  and Anderson-Darling
cf37c5fc-d933-3771-9586-9d6b4b285d8b ; doi:10.1080/01621459.1954.10501232 ;
engmann2011comparing  tests are applied to optimize the BDTs performance while
avoiding overtraining. The resulting BDT working points are chosen to limit
the mistagging of the other quarks to $\sim 1.5\%$ $(\sim 3\%)$ in the case of
$b$-quark ($c$-quark) tagging.

####  3.3.3 Jet-charge measurement

The measurement of the differential cross-sections and $A_{FB}$ requires
measuring the jet charges, i.e. the separation of jets originating from quarks
and anti-quarks. Two methods are used to measure the jet-charge Irles:2023ojs
: the Vertex Charge and Kaon Charge.

#### Vertex Charge, $Vtx$-method

The Vertex Charge is defined as the sum of the charges of all tracks
associated with reconstructed displaced vertices within a jet. The perfect
application of the method requires that all charged tracks from $b$ or
$c$-hadron decays are correctly measured and associated with the jet’s
secondary vertex.

The tracking efficiency in ILD for isolated tracks with momentum above 1 GeV,
transverse momentum above 100 MeV and $|\cos\theta|<0.85$ is estimated to be
close to 100% and better than 99% in the very forward direction ILD:2020qve .
However, when reconstructing secondary vertices in dense jets, loss of tracks
or incorrect association with secondary vertices may not be negligible. It has
been estimated that up to $\sim 5\%$ of the tracks may be missed during vertex
reconstruction, especially in the very forward and backward regions
Irles:2023ojs . This is particularly relevant for $b$-quark jets due to the
large number of secondary tracks per jet: most jets have between two and five
(and in rare cases up to fifteen) displaced tracks. In contrast, most
$c$-quark jets contain two displaced tracks. The difference in performance for
the two flavors can be qualified by $P_{chg}$, the probability that the jet
charge reproduces the sign of the charge of the initial quark. In the central
regions $P^{b}_{chg}\sim 0.83$, decreasing to 0.75 for polar angles
$lvert\cos\theta\rvert\gtrsim 0.8$, while $P^{c}_{chg}\sim 0.9$ for all
angles. These numbers are similar for ILC250 and ILC500.

#### Kaon Charge, $K$-method

The charge of the jet can also be estimated by the charge sum of all kaons
associated to displaced vertices inside the jet. These kaons are produced in
decays of $D$-mesons and $B$-hadrons. The performance of this method is
limited mostly by charged-kaon identification capabilities and physics
processes such as $B^{0}-\bar{B}^{0}$ oscillations. The baseline ILD design
foresees a $K^{\pm}/\pi^{\pm}$ separation power of $\sim$3333The separation
power is calculated as defined in Eq. 8.3 of Ref. ILD:2020qve  but an
improvement is expected when using alternative TPC designs with larger
pixelation allowing cluster-counting reconstruction, $dN/\/dx$, in place of
the traditional $dE/\/dx$ LCTPC:2022pvp . We implemented this feature of
improved reconstruction in our analysis, such that the TPC track
reconstruction offers $K^{\pm}/\pi^{\pm}$ separation power up to 4 using
$dN/\/dx$ measurements for tracks with momentum between $\sim(3-30)$ GeV. A
detailed description of the discriminating variables and procedure can be
found in Irles:2023nee  where it is shown that using $dE/\/dx$ for single
charged-kaon identification in displaced tracks inside jets, we can get up to
an $\sim 80\%$ efficiency with $\sim 80\%$ purity for the $b$$\overline{b}$
case at ILC250 and ILC500, an $\sim 80\%$ efficiency with $\sim 90\%$ purity
for the $c$$\overline{c}$ case at ILC250 and an $\sim 80\%$ efficiency with
$\sim 80\%$ purity for the $c$$\overline{c}$ case at ILC500 (because of the
higher momentum of displaced tracks compared with ILC250). However, for the
$dN/\/dx$ at the ILC250 scenario, we can work at the points with $90\%$
efficiency and $\leq 95\%$ for both final states. At ILC500, the purity is
slightly worse due to the higher momentum of the displaced tracks but still
above the $90\%$.

For the $dN/\/dx$ method, used as baseline in this study, $P_{chg}$ was
estimated to be:
$P^{b}_{chg}\sim 0.75$ for $b$-quark, ILC250;
$P^{c}_{chg}\sim 0.95$ for $c$-quark, ILC250;
$P^{b}_{chg}\sim 0.7$ for $b$-quark, ILC500;
$P^{c}_{chg}\sim 0.8$ for $c$-quark, ILC500;
for polar angles within the TPC angular acceptance ($|\cos\theta|\lesssim
0.8$), and showing a rapid drop in performance in more forward regions.

####  3.3.4 Event preselection

Once events are reconstructed, a preselection is based on a series of
kinematic cuts applied to enrich the data sample of $q$$\overline{q}$
($q=u,d,s,c,b$) signal events, expected to show back-to-back two-jet
kinematics, while removing the backgrounds. The largest background
contamination is the radiative-return background associated with QED ISR. Most
QED ISR will be collinear to the beam and, if energetic enough, will result in
a reconstructed invariant mass of the two-jet system significantly smaller
than $\sqrt{s}$, making it straightforward to filter such events. However, a
non-negligible fraction of the radiative-return events, $\sim 10\%$ according
to the existing simulations, will radiate photons inside the detector volume,
requiring other kinematic variables, defined in the following, to be used to
filter these events.

We define two new objects, $\gamma_{clus,j}$, by clustering all uncharged PFOs
within each jet ($j$). We denote the most energetic of these two as
$\gamma_{clus.}$. The reconstructed energy and angles of $\gamma_{clus.}$ will
be used for the event selection. Another variable used is $y_{23}$, the
distance (as defined by the $VLC$ algorithm) at which the event transitions
from a two to a three-jet topology. We also use the invariant mass of the two-
jet system ($m_{jj}$). The selection requirements for ILC250 are summarized in
the following, with the modifications for ILC500 shown inside parentheses:

  1. 1.

photon veto cuts, rejecting events if:

    1. (a)

at least one of the jets contains only one PFO;

    2. (b)

if the energy of $\gamma_{clus.}$ is larger than 115 GeV (220 GeV) or it is
reconstructed in the forward region $|\cos\theta|>0.97$;

  2. 2.

events with $(\sin\Psi_{acol})_{jj}$ > 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.

Cuts 1 to 3 are designed to reduce the contamination from the radiative-return
backgrounds, and cut 4 is most effective against di-boson backgrounds. The
last cut is significantly tightened at ILC500 to reduce the much larger $WW$
background contamination in the case of left-handed electron beam
polarization. This cut reduces the efficiency from $\sim 75\%$ to $\sim 55\%$
while keeping the background over signal $B/S$ ratio below $3-5\%$. At ILC500,
cut 2 (on acolinearity) reduces the $t$$\overline{t}$ contribution to the 1%
B/S level, and cut 4 further reduces the $t$$\overline{t}$ background to
negligible levels. The event selection procedure has been studied in detail
and documented in Ref. Irles:2023ojs .

##  4 Experimental reconstruction of the forward-backward observable using
full simulation tools at ILC250 and ILC500

This section describes the most critical aspects of the experimental
reconstruction of the observable forward-backward asymmetry $A^{q}_{FB}$ by
measuring $\frac{d\sigma}{d\cos\theta}$. This work is based on a previous ILD
study Irles:2023ojs ; Irles:2023nee . The method starts with a preselection
that results in a highly pure $q$$\overline{q}$ sample, followed by the double
flavor tagging (Sec 4.1) which selects $b$-quark (or $c$-quark) events, and
ends with the double charge (Sec. 4.2) measurement to distinguish between
quark and anti-quark jets. The differential cross-section is extracted from
the measurement of the total number of events reconstructed as a function of
the quark-jet scattering angle $\theta$:

$\displaystyle\frac{dN}{d\cos\theta}$ $\displaystyle=$
$\displaystyle\mathcal{L}\left[\varepsilon_{pre}\varepsilon_{DTC}\frac{d\sigma}{d\cos\theta}+\varepsilon_{bkg}\frac{d\sigma_{bkg}}{d\cos\theta}\right]$
(3)

where $\mathcal{L}$ is the integrated luminosity and the $\varepsilon$
variables are the different selection efficiencies, defined as a function of
$|\cos\theta|$. The signal preselection efficiency ($\varepsilon_{pre}$) and
the background selection efficiency ($\varepsilon_{bkg}$) are estimated using
only Monte Carlo data.

###  4.1 Double Tagging method

The Double Tag method ($DT$) ALEPH:2005ab  is based on the comparison of
single and double flavor-tagged samples for the simultaneous extraction of the
tagging efficiency, $\epsilon_{q}$ and the hadronic cross-section fraction
$R_{q}$ 444The hadronic cross-section fraction is defined as
$R_{q}=\frac{\sigma_{e^{-}e^{+}\rightarrow q\bar{q}}}{\sigma_{had.}}$ (4)
where $\sigma_{had.}$ is defined as $\sigma_{e^{-}e^{+}\rightarrow q\bar{q}}$
integrated over all quark flavors except the top-quark.. The method is applied
once we have preselected an enriched $q$$\overline{q}$ sample reconstructed as
two jets. We next apply flavor tagging to all jets, and extract two numbers:
the fraction of all jets tagged as being of flavor $q$, and the fraction of
events in which both jets are tagged as flavor $q$. The exact formulation is
described in Ref. Irles:2023ojs  for a fully differential analysis, in
contrast to the integral analyses performed in the past. By comparing these
two ratios for $b$-quark and $c$-quark, we can simultaneously measure the
efficiency of the flavor tagging algorithm ($\epsilon_{q}$) and $R_{q}$ for
both flavors.

Although the method is based on data comparisons, some initial hypotheses,
based on simulations, are required. For instance, it is assumed to be an
almost background-free analysis (or to have a perfectly modeled background).
This was easier to achieve at LEP and SLC running at the $Z$-pole. For data
taken in the continuum above the $Z$-pole, a tighter preselection is required
to minimize the background contribution of the radiative return or di-boson
hadronic decays. The method also assumes knowledge of the mis-tagging
efficiencies, the probability of tagging a true $q^{\prime}$ as $q$. The size
of these mis-tagging efficiencies and their uncertainties will directly impact
the uncertainty of the measurements. This factor was one of the dominant
sources of uncertainty for the measurements at LEP and SLC, with less
relevance in the latter case due to its improved flavor-tagging capabilities
(especially for the $c$-quark). Finally, the method assumes that the quark-
tagging efficiencies are symmetric between the two sides of the detector
(positive and negative $\cos\theta$). However, effects such as the resolution
on the primary vertex reconstruction, inhomogeneities in the detector layout
or performance, or kinematic variations of the back-to-back topology due to
hard gluon radiation may introduce correlations. These correlations can be
parametrized by (1+$\rho_{q}$), known in the literature as the hemisphere or
jet angular correlation. With this notation, the double-tag efficiency is
given by $\epsilon_{q}^{2}\cdot(1+\rho_{q})$. This parameter is the only one
that is calculable only using Monte Carlo simulations, and it was considered a
significant source of uncertainty in past experiments. It was less of an issue
at SLC than at LEP thanks to the more precise primary vertex determination.
The jet angular correlation is expected to be negligible in the case of the
ILD at ILC Irles:2023ojs  except in the forward detector region, where the
tagging efficiency drops due to limited acceptance. In a fully differential
analysis, these regions and others affected by detector issues can be removed
from the study when performing a fit and be substituted by extrapolations.

The preselection and $DT$ selection efficiencies for the two flavors are shown
in Tab. 3 for the ILC250 and ILC500.

###  4.2 Double Charge method

The Double Charge method ($DC$) requires a pure $q$$\overline{q}$ sample (with
$q$ being only one flavor, $c$ or $b$) for its application. It consists of the
measurement of the charge of each jet using one of the methods described in
Sect. 3.3.3. Only events with two oppositely charged jets are accepted. The
experimental determination of the probability $P_{chg}$ that the jet charge
reproduces the sign of the charge of the quark of the hard scattering is
straightforward Irles:2023ojs . For the $c/b$-quark cases and different ILC
running scenarios, these probabilities are reported in Sect. 3.3.3. Since the
$Vtx$-method and the $K$-method show similar values of $P_{chg}$, it is also
possible to use mixed cases in which opposite jets use different methods. For
the $b$-quark, the most efficient method is the $Vtx$-method: $\sim 95\%$ of
the events have at least one of the jets with its charge measured by the
$Vtx$-method; and in $\sim 63\%$ of the cases, both charges have been measured
with this method. For the $c$-quark case, the $K$-method is the most
efficient, and the corresponding efficiencies are $\sim 95\%$ and $\sim 50\%$.
In Ref. Irles:2023ojs , three different categories are defined and studied for
each flavor:

  1. 1.

The charge of both jets has been measured with the most efficient method.

  2. 2.

Only one jet has no measurement of its charge with the most efficient method
but does have it with the other method.

  3. 3.

Both jets have their charge measured with the less efficient method only.

The $DC$ measurements minimize the migrations between hemispheres due to
wrongly measured jet charge. However, these migrations are sizable even with
the $DC$, especially for the $b$-quark with high left-handed electron-beam
polarization. The reason is two-fold: the relatively small $P_{chg}$ for the
$b$-quark case and the large value of $A_{FB}$ that produces much more
migration from the forward to the backward region than from backward to
forward. This has a sizable impact on the reconstructed differential
distribution, with localized deviations of the differential cross-section of
up to a factor three. These migrations can be easily corrected using the
measured values of $P_{chg}$, which give the probability that the charge has
been wrongly measured in both jets, hence flipping the sign of $\cos\theta$
Irles:2023ojs .

The efficiency of the full selection, including preselection, $DT$, and $DC$
selection for the two flavors, are shown in Tab. 3 for the ILC250 and ILC500
cases.

Table 3: Efficiencies and $B/S$ estimated using ILD full simulation and
reconstruction.

| Eff(B/S)$[\%]$
---|---
| ILC250$(-0.8,+0.3)$ | ILC250$(+0.8,-0.3)$
Preselection: | $udscb:$ 75.8(5.6) | $udscb:$ 75.9(4.8)
+$DT$ |  $c:$ 9.5(0.6) |  $b:$ 38.1(0.9) |  $c:$ 9.4(0.5) |  $b:$ 38.1(1.3)
+$DC$ |  $c:$ 3.7(0.5) |  $b:$ 14.7(0.8) |  $c:$ 3.7(0.5) |  $b:$ 14.7(1.2)
| ILC500$(-0.8,+0.3)$ | ILC500$(+0.8,-0.3)$
Preselection: | $udscb:$ 54.4(7.4) | $udscb:$ 54.5(5.0)
+$DT$ |  $c:$ 10.9(0.3) |  $b:$ 29.7(0.3) |  $c:$ 10.8(0.3) |  $b:$ 29.5(0.3)
+$DC$ |  $c:$ 3.2(0.3) |  $b:$ 12.1(0.3) |  $c:$ 3.1(0.5) |  $b:$ 12.1(1.3)

|
---|---
|
Figure 2: Comparison of parton level and fully reconstructed
$d\sigma/d\cos\theta$ distributions for different ILC scenarios and the $b$
and $c$-quark.

###  4.3 Estimation of $A_{FB}$

Once the $DT$ and $DC$ methods have been applied, all the inputs needed to
extract the efficiency of double tagging and charge measurement,
$\varepsilon_{DTC}$, and to infer $d\sigma/d\cos\theta$ are available. The
efficiency is estimated for every flavor and each $DC$ category, and the
statistical uncertainties are propagated accordingly Irles:2023ojs . The
correction for the efficiencies is performed differentially as a function of
$|\cos\theta|$.

The result of the entire correction procedure is shown in Fig. 2 for the
different flavors and running scenarios using SM samples.

For the estimation of $A_{FB}$, the function

$\displaystyle\frac{d\sigma}{d\cos\theta}=S\left(1+\text{cos}^{2}\theta\right)+A\cos\theta$
(5)

is fit to the reconstructed distributions. Note that this function neglects
the SM tensorial contribution $T\times\text{sin}^{2}\theta$, which is very
small given the large boost of $b$ and $c$-quarks. The fit is performed in the
range of $|\cos\theta|<0.9$, avoiding the very forward regions in which the
acceptance decreases. The $A_{FB}$ is extracted by extrapolating the fitted
function to the full range of $\cos\theta$, and the statistical uncertainty is
estimated for the expected integrated luminosity, including the uncertainties
of the correction methods. The expected statistical uncertainties for the
various running scenarios are shown in Fig. 3.

Figure 3: Estimated statistical uncertainties on $A^{c}_{FB}$ and $A^{b}_{FB}$
using ILD full simulation and reconstruction at ILC250 and ILC500.

###  4.4 Systematic uncertainties

At ILC250 and ILC500, the expected statistical uncertainties on $A_{FB}$ are
at the level of a few per mil. A comprehensive study of the leading
experimental systematic uncertainties is reported in Ref. Irles:2023ojs . The
most significant systematic uncertainties on $A_{FB}$ are due to the
preselection efficiency, the hadronization/fragmentation modeling, the angular
correlations due to QCD effects, and the knowledge of the beam polarization.
These are reported in the following. However, we emphasize that the expected
size of these systematic uncertainties is negligible compared to the
statistical uncertainties expected at ILC when running above the $Z$-pole.

####  4.4.1 Pre-selection efficiency

Although this efficiency cancels in the numerator and denominator of the
integral calculation of $A_{FB}$, the preselection efficiency cannot be
neglected in a full differential analysis since it affects the shape of the
differential measurement. The impact of this uncertainty has been evaluated by
producing pseudo-data distributions applying uncorrelated 10$\%$ relative
variations in bin-by-bin preselection efficiencies. This uncertainty is
propagated to $A_{FB}$, giving rise to a relative uncertainty on $A_{FB}$ of
$\leq 0.1\%$. Uncertainties from background mis-modeling become mostly
negligible, $\simeq 0.01\%$, due to the very efficient background rejection.

####  4.4.2 Hadronization/fragmentation modeling

Uncertainties on $A_{FB}$ related to fragmentation are expected to be
negligible thanks to the $DT$ and $DC$ methods, for which different
fragmentation models affect only the mis-tagging efficiency. Furthermore, the
mis-tagging rates will be much lower than at past experiments thanks to the
high performance expected for modern flavor-tagging algorithms using modern
statistical and machine-learning techniques and the progress in detector
technologies.

####  4.4.3 Angular correlations

Full simulation studies suggest that the value of $\rho_{q}$ at ILC250 is
smaller than 0.2% throughout most of the detector volume Irles:2023ojs . The
tracking system in the ILD simulation is symmetric, and no coherent noise is
simulated, indicating that a non-zero $\rho_{q}$ value can only be the result
of occasional mis-measurements of the primary vertex or hard QCD radiation
diluting the back-to-back configuration of the di-jet system. The small value
of $\rho_{q}$ suggests that both effects can be effectively controlled. This
results from the small beam size (jet angular correlations due to a misplaced
common vertex are suppressed) and an excellent tracking system. Moreover, a
high tagging efficiency can significantly reduce jet angular correlations. To
consider the impact of angular correlations due to QCD radiation, we assume
they contribute an uncertainty of $\lesssim 0.1\%\cdot A_{FB}$, following Ref.
AlcarazMaestre:2020fmp , after having introduced acolinearity cuts in our
definition of the signal and selection procedure. A full assessment of this
effect would require simulations based on NLO QCD.

####  4.4.4 Beam polarization

Beam polarization uncertainties Karl:424633  influence the accuracy of
precision measurements. For the measurement of $A_{FB}$, this uncertainty
affects the $b$-quark and $c$-quark flavors differently, and has a non-
negligible impact only on the right polarization scenario for the b-quark; in
this scenario, we expect an uncertainty contribution of $0.15\%\cdot A_{FB}$
at ILC250, and somewhat smaller at ILC500 Irles:2023ojs .

##  5 Statistical discrimination power for GHU models at ILC

The detailed studies described in the previous section result in a realistic
estimation of the uncertainties on $A_{FB}$ for the $b$ and $c$-quark at
ILC250 and ILC500, with existing detector models and reconstruction tools. We
have shown that the statistical uncertainties on $A_{FB}$ expected at ILC
running above the $Z$-pole are much larger than any systematic effects, which
are therefore ignored in this section. Furthermore, we assume Gaussian
uncertainties and uncorrelated measurements. This second approximation is
motivated by the nature of the analysis, in which the $DT$ and $DC$ methods
lead to the selection of fully independent samples for the different flavors
and beam polarizations. Statistically independent MC simulations have been
used to analyze the various polarization scenarios.

To probe the potential of ILC to indirectly search for new physics, the
significance when comparing two models, $i$ and $j$, is defined as

$d_{ij}=\frac{|A_{FB,i}-A_{FB,j}|}{\Delta A_{FB,j}}$ (6)

with $A_{FB,i/j}$ being the $A_{FB}$ predicted at leading order by model $i$
or $j$, as explained in Sect. 2. The $\Delta A_{FB,j}$ corresponds to the
expected statistical uncertainty of the forward-backward measurement at ILC,
obtained as explained in Sect. 4. In addition, the systematic uncertainties
arising from the limited knowledge of the exact value of the couplings in the
SM prediction are added in quadrature to $\Delta A_{FB,j}$. For these
estimates, we used different assumptions on the uncertainties on the Z-boson
couplings to fermions and propagated them to the leading order calculation of
$A_{FB}$. The different assumptions on these uncertainties extracted from
Refs. ParticleDataGroup:2020ssz ; ILCInternationalDevelopmentTeam:2022izu
are:
C: current uncertainties, from LEP and SLC measurements;
R: expected uncertainties from measurements of radiative return events at
ILC250;
Z: expected uncertainties from measurements at a dedicated ILCGigaZ run.

The probability for each $d\geq d_{ij}$ case and the discovery power
discrimination are calculated in terms of the number of standard deviations
from the null hypothesis for the model $i$. This is performed for each
measurement and combined following the multivariate Gaussian formalism.

Fig. 4 shows the expected discrimination power between the different GHU
models and the SM for the different ILC running stages above the $Z$-pole
foreseen by the H20-staged plan, ILC250 and ILC500. We have included two extra
cases:

  * •

ILC250${}^{\blacklozenge}(no~{}pol.)$: a hypothetical case of an ILC250
operating with un-polarized beams and assuming a total integrated luminosity
equal to the baseline ILC250 scenario (2000 fb-1). The (◆) symbol is used to
distinguish this case from the other studies, which use the nominal ILC beam
polarization. This scenario can be compared with current circular collider
proposals. Still, one should bear in mind that the simulated beam conditions
are those of ILC and that the simulated ILD model has been optimized for the
full range of ILC energies. Dedicated studies with full simulation of the beam
conditions and detector models designed for circular colliders would be
required to extract more definitive conclusions.

  * •

ILC1000*: we have extrapolated statistical uncertainties from ILC500 to 8000
fb-1 at ILC1000, split according to the different polarization conditions as
described in Tab. 1. The (*) symbol is used to distinguish this case from the
three other cases, which use full simulation samples and reconstruction. At
ILC1000, compared with ILC250 and ILC500, the experimental challenges are
expected to be slightly different, with much more collimated jets and vertices
and the possible presence of new backgrounds. However, no significant
differences are expected in overall detector performance since the ILD has
already been optimized to operate at ILC1000 (see Behnke:2013lya ;
ILCInternationalDevelopmentTeam:2022izu  and references therein).

Figure 4: Statistical discrimination power between the GHU models described in
the text and in Funatsu:2017nfm ; Funatsu:2020haj ; Funatsu:2023jng  and the
SM. Different running scenarios of ILC are compared:
ILC250${}^{\blacklozenge}(no~{}pol.)$ (hypothetical case with no beam
polarization and 2000 fb-1 of integrated luminosity), ILC250 (2000 fb-1),
ILC500 (4000 fb-1), and ILC1000* (8000 fb-1, not using full simulation studies
but extrapolations of uncertainties from ILC500). Three different assumptions
for the $Z$-fermion couplings uncertainties are considered
ILCInternationalDevelopmentTeam:2022izu : C for current knowledge; R for
expected knowledge after the full ILC250 program and the study of $Z$-fermion
couplings from radiative return events, and Z for expected knowledge after a
full ILCGigaZ program.

###  5.1 Requirements on the precision of the $Z$-fermion couplings
measurements

Even if large deviations in the $A_{FB}$ observables are measured at ILC250 or
at higher energy in $e^{-}e^{+}\rightarrow q\bar{q}$, one could still be
unable to distinguish the contribution of new resonances from deviations from
the SM $Z$-boson couplings. For this reason, high-precision measurements of
the fermionic $Z$ couplings are required. An ILCGigaZ run would allow the
precise determination of all $Z$-boson couplings to fermions (except the top
quark). Full simulation experimental studies are yet to be performed at the
ILC and other Higgs Factories. First studies Irles:2019xny ;
ILCInternationalDevelopmentTeam:2022izu  show that an improvement of up to two
orders of magnitude (including systematic uncertainties) could be obtained at
the ILC for these couplings compared with the LEP and SLC. These studies
include discussions of the most significant systematic uncertainties, which
are expected to be 2-10 times larger than the statistical ones. Moreover,
studies presented in Ref. ILCInternationalDevelopmentTeam:2022izu  suggest
that even at ILC250, such couplings could be measured with about one order of
magnitude higher precision than at LEP/SLC by studying the radiative return to
the $Z$-pole.

The impact of assuming improved precision on these couplings, according to the
scenarios explained above, is shown in Fig. 4 in the different columns $C$,
$R$, and $Z$. It shows that at ILC250, the precision on the $Z$ couplings from
a study of radiative return events allows one to approach the five standard
deviations ($\sigma$) level for the discrimination of $A$ models vs SM and get
almost $3\sigma$ for the $B_{1}^{\pm}$. With ILCGigaZ precision, the
discrimination power would be enhanced, allowing full discrimination of the
$A$ models. When including the higher energy stages of ILC, the difference
between the two possible scenarios for improved $Z$ coupling precisions
becomes less critical due to the larger size of the new resonance
contributions.

###  5.2 Importance of beam polarization and high energy reach

The ILC offers high-energy beams with a high degree of longitudinal
polarization. This provides direct access to the different helicity amplitudes
at different energies. This is particularly important for models predicting
deviations in the right-handed electroweak couplings, which are less
constrained by existing measurements. The comparison between the first columns
of Fig. 4, shows that the high degree of electron and positron beam-
polarization at ILC250 offers a gain comparable to a factor two in integrated
luminosity with respect to ILC250${}^{\blacklozenge}(no~{}pol.)$. Fig. 1 shows
that most of the sensitivity at ILC250 comes from the configuration with
right-handed electron beam and left-handed positron polarization. An increase
of statistics of this run from 900 fb-1 to $\sim 2.5\times 900$ fb-1 would
provide $5\sigma$ discrimination power for the $A_{1}$, $A_{2}$ and the
$B_{1}^{\pm}$ GHU models (predicting $m_{KK}=8.81,~{}10.3$ and $13$ TeV
respectively).

Figure 5: Similar to Fig. 4 but fixing the R case for the $Z$-fermion
couplings. Three scenarios for the positron beam polarization are shown. For
the ILC250${}^{\blacklozenge}(no~{}pol.)$ we continue assuming no polarization
for any beam.

The production of an intense electron beam with a high degree of polarization
is expected to be technically possible, while the positron source poses some
technological challenges. To face these challenges, two options are being
considered PositronWG ; ILCInternationalDevelopmentTeam:2022izu . The baseline
undulator-based positron source allows the production of polarized positron
beams, while the other electron-driven concept would provide un-polarized
positron beams. The baseline option provides $30\%$ positron beam polarization
at ILC250 and ILC500 and $20\%$ at ILC1000. An upgrade could increase the
polarization to 60$\%$. We perform the exercise of comparing several scenarios
for the positron beam polarization. The result is shown in Fig. 5 where we
compare the discrimination power assuming three positron beam polarization
scenarios: 0$\%$, 30$\%$ and 60$\%$555For simplicity we use $30\%$ positron
beam polarization also for the middle column for the ILC 1 TeV case. However,
it is not the baseline configuration, which foresees $20\%$ positron beam
polarization., all of them using extrapolated uncertainties from the R
scenario for $Z$-fermion couplings. This figure shows that having electron-
beam polarization alone already makes a sizable improvement in sensitivity
compared with no polarization at all (comparison of the
ILC250${}^{\blacklozenge}(no~{}pol.)$ case and the first column of ILC250 in
the plot). Adding positron-beam polarization enhances the sensitivity at low
energies. However, at high energies, the positron-beam polarization becomes
less critical. As observed in Fig. 1, the sensitivity depends on the
polarization and increases with the $\sqrt{s}$. Including the ILC500 program
in the ILC250 expectations allows the inspection of all studied GHU models up
to the $B_{2}^{\pm}$ models, which predict $m_{kk}=19$ TeV. Adding also the
estimate for ILC1000, the discrimination of $B_{2}^{\pm}$ ($m_{kk}=25$ TeV)
will also be within reach.

---
Figure 6: Similar to Fig. 4 but fixing the R case for the $Z$-fermion
couplings. Instead, three different scenarios for the charged hadron particle
identification capabilities (PID) are considered: O for no PID used, E for PID
based on the ILD baseline $dE/\/dx$ reconstruction, and N for an optimized TPC
design with higher granularity and cluster counting reconstruction $dN/\/dx$.

###  5.3 The role of charged-hadron identification with the ILD TPC

In this study, we have assumed a novel TPC design that will allow cluster
counting ($dN/\/dx$) for charged-hadron identification capabilities, with
better resolution than using the mean energy loss per distance ($dE/\/dx$),
described in detail in Sect. 3.3. We also consider two other scenarios: a
first without any charged-hadron identification (although the TPC tracking
capabilities are still used for track reconstruction) and a second one using
the traditional $dE/\/dx$ for particle identification. In Ref. Irles:2023nee ,
it has been estimated that not using charged-hadron identification for the
kaon selection could increase the statistical uncertainty by up to a factor of
two, especially for the $c$-quark case, which largely relies on the
$K$-method, reducing the power of discovery of GHU in these signatures in the
baseline program of ILC. The expectations for these three cases are shown in
Fig. 6, assuming the $R$ scenario for the $Z$ couplings. The benefits of using
charged-hadron identification capabilities with the ILD TPC become clear,
especially at the lower energy stage of ILC, since it allows the use of two
methods for charge determination, $Vtx$-method and $K$-method. The
discrimination power differences between using $dE/\/dx$ and $dN/\/dx$ are
moderate and depend on the different models: the ones predicting larger
deviations for the $c$-quark case are more sensitive to moderate improvements
in the $K$-method.

|
---|---
|
Figure 7: Statistical discrimination power between GHU models after different
ILC stages. For completeness, a hypothetical ILC250 stage assuming no
longitudinal beam polarization is included. The
ILC250${}^{\blacklozenge}(no~{}pol.)$, ILC250, and ILC500 estimations are
performed using full simulation studies. The ILC1000* is obtained from
extrapolations of the ILC500 studies.

###  5.4 Discrimination power between GHU models

In Fig. 7, the statistical discrimination between the different GHU models is
shown for different ILC scenarios. These four plots show the benefits of high
longitudinal polarization for both beams and, especially, the benefits of the
energy reach foreseen for the ILC. With operation at 500 GeV and above, almost
full differentiation between the different models will be possible, allowing
for detailed scrutiny of potential contributions from heavy resonances.

##  6 Summary and outlook

The search for new physics at the LHC and at future electron-positron
colliders requires a global approach. Searching for new resonances is and will
be addressed by a combination of direct searches for such resonances and the
precise measurement of observables whose deviations with respect to SM
predictions are sensitive to new physics. The ILC program will provide a broad
range of experimental measurements that can be used to probe for BSM physics.
The GHU models discussed here are especially sensitive to deviations in the
electroweak observables at high $\sqrt{s}$ with polarized beams.

This study uses forward-backward asymmetries of $b$ and $c$-quarks in high-
energy electron-positron collisions to demonstrate the sensitivity to GHU-
inspired models. The experimental input is based on detailed simulations of
the ILD at center-of-mass energies of 250 and 500 GeV and extrapolated to 1000
GeV. More specifically, we present the fully differential cross-section
$d\sigma/d\cos\theta$ from which the $A^{b}_{FB}$ and $A^{c}_{FB}$ are
inferred. Studies at ILC250 and ILC500 have been completed, showing that a
per-mil level of statistical precision is achievable. Experimental systematic
uncertainties have been briefly discussed. These are found to be sub-dominant,
thanks to: a) the excellent vertexing and flavor-tagging capabilities expected
at the ILC, b) the use of fully differential measurements; and c) the use of
double-tagging and double-charge measurements that reduce the use of Monte
Carlo tools to the minimum when addressing modeling uncertainties, such as the
hadronization uncertainties.

Moreover, the ILD also offers the critical capability of providing charged-
kaon identification over a broad momentum spectrum. This enhances the
statistical efficiency of these measurements, and, in particular, it provides
a factor $\sim 2$ improvement for $A^{c}_{FB}$. This analysis uses a simple
approach based on selection cuts in kinematic distributions for the background
rejection. Machine learning techniques like multi-classifying neural networks
can improve the event selection and construct control regions for the
backgrounds CMS:2018hnq . This will further improve the sensitivity of the
analysis.

The analysis used currently available tools for tracking, vertexing, particle
flow, jet reconstruction, and flavor tagging. In particular, the flavor-
tagging algorithm is based on multivariate analysis using boosted decision
trees as classifiers Suehara:2015ura . Advanced machine-learning methods such
as graph neural networks are expected to significantly improve jet flavor
identification performance ATLAS:2022rkn .

The GHU models described in Refs. Funatsu:2017nfm ; Funatsu:2020haj ;
Funatsu:2023jng  show high expected sensitivity for the $A^{b}_{FB}$ and
$A^{c}_{FB}$ observables. The expected sensitivity increases with the energy
and depends on the electron and positron beam polarization. These GHU models
predict new massive $Z^{\prime}$ resonances and deviations of all SM
$Z$-fermion couplings. They are constructed not to induce significant changes
in the EW precision observables measured at past lepton colliders and agree
with the non-observation of $Z^{\prime}$ at LHC.

We show that the ILC operating polarized beams colliding at 250 GeV and 500
GeV could provide full discrimination power (at the $5\sigma$ level) between
these models and the SM, through $A^{b}_{FB}$ and $A^{c}_{FB}$ measurements.
The ILC250 case has also been compared with an ILC250 without beam
polarization. For the latter case, at least a factor of two of integrated
luminosity is required to get similar prospects.

The ILC program also considers a run at high energies of 1000 GeV. Detailed
studies of these scenarios, with optimized designs of the ILD, would be
required to provide realistic estimations of the uncertainties expected in
these cases. Instead, the exercise of extrapolating from the ILC500 full
simulation studies has been performed. Including the predictions for ILC1000,
full discovery potential for different GHU models predicting Kaluza-Klein
resonance with masses up to 25 TeV will be possible.

It is important to remark on the intrinsic importance of precisely
constraining the couplings of the $Z$-boson to all fermions without deviations
caused by heavy resonances predicted by BSM. This can be done at dedicated
runs at the $Z$-pole (ILCGigaZ) or with a dedicated analysis at ILC250 using
radiative return events ILCInternationalDevelopmentTeam:2022izu . The
importance of getting updated estimations for such couplings with at least one
order of magnitude better precision than current measurements is also shown in
this study.

This study is based on leading-order accurate simulations, and calculations
should be updated, including next-to-leading order contributions in QCD and EW
at the theory and Monte Carlo simulation level. This is out of the scope of
this work due to the non-existence of such simulations or GHU predictions at
the moment.

Measurements for different fermions, various types of observables, under
different $\sqrt{s}$ and beam-polarization conditions will allow a deep
investigation of such theories and the disentanglement of potential new
effects: deviations of the $Z$-$f$ couplings, contributions from new heavier
resonances, mixing effects, etc. In particular, future colliders will provide
multiple differential cross-section measurements for different processes that
can be used to set limits on the SM via effective-field interpretation models.
In the Standard Model Effective Field Theory (SMEFT) Brivio:2017vri , the
study of the implications of new physics is performed indirectly through the
introduction of modifications to the SM at very high energies. The strength of
the modification of SM couplings is determined by the energy scale at which
the new physics is expected and the coupling strengths expressed by the Wilson
coefficients, which can be fitted, or the exclusion limits on the SMEFT
parameters and, therefore, BSM physics can be set. The SMEFT approach has the
advantage of allowing the simulation of the impact of the modification of
couplings for several physics processes simultaneously. Therefore, a global
analysis of the data with different processes is possible, accounting for
their correlations.

###### Acknowledgements.

We are grateful to the ILD Publication and Speakers Bureau, particularly K.
Kawagoe, M. Berggren, D. Jeans and K. Fuji for their work and support as the
editorial board team. We thank Y. Hosotani for his indispensable input. We
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. A.I. and
J.P.M acknowledge the support by the PlanGenT program from the Generalitat
Valenciana (Spain) with the grant number CIDEGENT$/2020/021$. A.I.
acknowledges 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 Si4HTopF with reference ASFAE$/2022/015$). A.S.
acknowledges support from the Alexander von Humboldt Foundation through the
Feodor Lynen Research Fellowship. A.I, J.P.M and A.S acknowledge the funding
from PID2021-122134NB-C21. N.Y. acknowledges the support by the Ministry of
Science and Technology of Taiwan under Grant No. MOST-111-2811-M-002-047-MY2.

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