UMN-TH-4220/23

# Top Yukawa Coupling Determination at High Energy Muon Collider

Zhen Liu  School of Physics and Astronomy, University of Minnesota,
Minneapolis, MN, 55455, USA Kun-Feng Lyu  School of Physics and Astronomy,
University of Minnesota, Minneapolis, MN, 55455, USA Ishmam Mahbub  School of
Physics and Astronomy, University of Minnesota, Minneapolis, MN, 55455, USA
Lian-Tao Wang  Department of Physics, University of Chicago, Chicago, IL,
60637, USA Enrico Fermi Institute, University of Chicago, Chicago, IL 60637,
USA Kavli Institute for Cosmological Physics, University of Chicago, Chicago,
IL 60637, USA

###### Abstract

The Top Yukawa coupling profoundly influences several core mysteries linked to
the electroweak scale and the Higgs boson. We study the feasibility of
measuring the Top Yukawa coupling at high-energy muon colliders by examining
the high-energy dynamics of the weak boson fusion to top quark pair processes.
A deviation of the Top Yukawa coupling from the Standard Model would lead to a
modified $VV\rightarrow t\bar{t}$ process, violating unitarity at high energy.
Our analysis reveals that utilizing a muon collider with a center-of-mass
energy of 10 TeV and an integrated luminosity of 10 ab-1 allows us to
investigate the Top Yukawa coupling with a precision surpassing 1.5%, more
than one order of magnitude better than the precision from $t\bar{t}h$ channel
at muon colliders. This precision represents a notable enhancement compared to
the anticipated sensitivities of the High-Luminosity LHC (3.4%) and those at
muon colliders derived from the $t\bar{t}H$ process.

###### Contents

  1. I Introduction
  2. II Theoretical Framework
    1. II.1 Perturbative unitarity in the $W_{L}^{+}W_{L}^{-}\rightarrow t\bar{t}$ Process
    2. II.2 Dimension-Six Operator
    3. II.3 Vector-Like Quarks
    4. II.4 Electroweak Gauge Boson PDFs
  3. III Relevant Channels at Muon Collider
    1. III.1 Polarized Partonic Cross Section
    2. III.2 Convolution with PDFs
  4. IV Projected Sensivities
    1. IV.1 Top Yukawa Precision
    2. IV.2 VLQ Model
  5. V Conclusion
  6. A Helicity Amplitude
  7. B Signal vs Background
  8. C Parameter Constraints on VLQ

##  I Introduction

Following the discovery of the Higgs boson in 2012 [1, 2], the elucidation of
its properties, particularly its interactions with other Standard Model (SM)
fields, has become one of the top priorities of particle physics. Substantial
efforts at the LHC have focused on quantifying Higgs interactions with
fermions, gauge bosons, and self-couplings. Nonetheless, precision
measurements at hadron colliders face constraints due to substantial QCD
backgrounds. Muon colliders (MuCs) have emerged as an exciting venue for high-
precision Higgs exploration. Recent research accentuates their potential,
combining the precision of lepton colliders with high center-of-mass energies,
thereby allowing for exploration at scales of 10 TeV or higher in an
environment with low background [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14].
Admittedly, one inherent challenge for muon colliders is the short lifetime of
muons. The high-energy physics community is working on tackling the challenges
and the research to enable a future muon collider, in particular, through an
ionization cooling scheme.

A crucial physics objective for future colliders is the precise measurement of
Higgs couplings. The Top Yukawa coupling, one of the least constrained
parameters in the SM, holds significance for Higgs research, with its deep
connection to the profound puzzle of naturalness and Higgs vacuum stability.
The recent LHC measurement for Top Yukawa is $y_{t}=1.16^{+0.24}_{-0.35}$
[15].

This study emphasizes the measurement of the Top Yukawa coupling at muon
colliders. The dominance of electroweak gauge boson fusion at muon colliders
results from the logarithmic growth of the electroweak parton distribution
function (PDF) with energy [16, 17, 18, 5, 19, 20, 21, 22]. We adopt a
factorization approach to compute cross-sections, integrating the enhanced
collinear splitting with the vector boson hard scattering. Factorization and
resummation ensure a cross-section devoid of collinear divergences in high-
energy limits. Our initial assumption only considers a Higgs coupling shift
$y_{t}\rightarrow y_{t}\left(1+\delta_{yt}\right)$ through a single SMEFT
operator $\mathcal{O}_{y}^{t}=H^{\dagger}H\bar{Q}\tilde{H}t_{R}$.
Additionally, we examine a scenario involving a heavy singlet vector-like
quark.

The paper is structured as follows: Section II introduces our theoretical
framework, Section III details various partonic channels, Section IV presents
the results, and Section V contains our conclusions.

##  II Theoretical Framework

The Top Yukawa coupling can be directly measured from processes with final
states containing Higgs boson, such as $ht\bar{t}$. One can also access the
Top Yukawa coupling with processes with Higgs boson in the intermediate state
but not appearing in the external legs. The scattering amplitude of a process
with longitudinal gauge bosons as external states, such as $VV\rightarrow
t\bar{t}$, could grow with energy if a coupling deviates from the SM. As a
result, such an amplitude eventually violates unitarity at a high-energy
scale, implying the breakdown of the low-energy description and the appearance
of new physics. Similar study on the gauge boson scattering has been carried
on in [23, 24, 25, 26, 27] to probe the dynamics of electroweak symmetry
breaking and various new physics sectors.

In this section, we will first examine the effects of the anomalous Top
Yuakawa coupling in the $W_{L}^{+}W_{L}^{-}\rightarrow t\bar{t}$ channel and
focus on the energy dependence of the scattering amplitude. We then discuss
some possible UV models that can generate such Top Yukawa coupling, and
further discuss the electroweak PDF which would be convoluted with the
$VV\rightarrow t\bar{t}$ cross-section to obtain the total signal rate at muon
colliders.

###  II.1 Perturbative unitarity in the $W_{L}^{+}W_{L}^{-}\rightarrow
t\bar{t}$ Process

Figure 1: The Feynman diagrams for the process $W^{+}W^{-}\rightarrow
t\bar{t}$.

In this section, we consider the channel $W^{+}_{L}W^{-}_{L}\rightarrow
t\bar{t}$ to illustrate the role of perturbative unitarity in the process. Our
analysis of $t\bar{t}$ production in muon colliders involves initial state
vector boson WW, ZZ, Z$\gamma$ and $\gamma\gamma$, but we choose
$W^{+}_{L}W^{-}_{L}\rightarrow t\bar{t}$ as our example here since it is the
dominant channel in our analysis. As shown in Fig. 1, the Feynman diagrams for
the process can be divided into three categories. The first is the $t$-channel
diagram via the exchange of the $b$ quark. The second is the $s$-channel
diagram mediated by the neutral vector boson, and the last diagram is mediated
by the Higgs boson in the $s$-channel. The leading order contribution for the
amplitude of the first and second diagrams can be estimated to be
$O(E^{2}/m_{W}^{2})$ by dimensional analysis for $t,\bar{t}$ helicity of
$(\pm,\mp)$ and $O(E/m_{W})$ for $t,\bar{t}$ helicity of $(\pm,\pm)$. For the
case of opposite outgoing helicity of the quarks, once we add the
contributions from individual diagrams, the leading $O(E^{2}/m_{W}^{2})$
behavior of each diagram is canceled through gauge symmetry, leaving a
constant term as would be needed for unitarity. While for the case of the same
quark helicity, the leading order energy scaling is not eliminated without
adding the contribution from the Higgs-mediated diagram. After combining all
four diagrams, the energy growth behavior is precisely canceled and
perturbative unitarity in this process is restored. Such a cancellation is
expected from the Goldstone boson equivalence theorem. In the high energy
limit, we have:

$\mathcal{M}_{W_{L}^{+}W_{L}^{-}\rightarrow
t\bar{t}}=\mathcal{M}_{\phi^{+}\phi^{-}\rightarrow
t\bar{t}}\left[1+O\left(\dfrac{m_{W}^{2}}{E^{2}}\right)\right]$ (1)

The amplitude for $\mathcal{M}_{\phi^{+}\phi^{-}\rightarrow t\bar{t}}$ in SM
is at most a constant. Hence, we do not expect unitarity-violating behavior in
the $W^{+}_{L}W^{-}_{L}\rightarrow t\bar{t}$ process.

If the Top Yukawa is shifted,

$y_{t}\rightarrow y_{t}(1+\delta_{yt}),$ (2)

only the amplitude of the Higgs-mediated diagram will be modified from the SM,
and one can show that in the high-energy limit, the total scattering amplitude
is

$\mathcal{M}(W^{+}W^{-}\rightarrow
t\bar{t})=\,\frac{m_{t}}{\nu^{2}}\delta_{yt}\sqrt{s}\quad\text{for}\quad\sqrt{s}\gg
m_{t}.$ (3)

In order to preserve unitarity, or more precisely, to maintain the validity of
a weakly coupled EFT, there must be a cutoff. To estimate this cut-off, we can
follow the standard partial wave analysis. The J = 0 partial wave amplitude is

$a_{0}=\frac{1}{32\pi}\int_{-1}^{1}d\cos\theta|\mathcal{M}|$ (4)

Unitarity requires the amplitude $|a_{0}|\leq 1$. Hence, we obtain the cutoff
as (ignoring the constant term)

$\Lambda_{\rm BSM}<\frac{16\pi v^{2}}{m_{t}\,\delta_{yt}}.$ (5)

The current precision on the Top Yukawa coupling $y_{t}$ is around
$\mathcal{O}(10)\%$. Therefore, the cutoff scale is well above the center of
mass energy of a 10 TeV muon collider. We note that the parameterization in
Eq. 2 is (overly) simplified. We can also characterize the deviation in terms
of the contribution from new physics. In the following, we describe two
approaches to parameterize such contributions.

###  II.2 Dimension-Six Operator

The most straightforward way to express the deviation of the Top Yukawa
coupling from the SM is in terms of the higher dimensional operators in SMEFT
[28, 29, 30], generated after integrating out heavy new physics [31, 32]. One
can assume the leading EFT operator is
$\mathcal{O}_{y}^{t}=\left(H^{\dagger}H\right)\bar{Q}_{L}\tilde{H}t_{R}$ with
the corresponding Wilson coefficient $c_{y}^{t}$.111For simplicity, we
consider the $c_{y}^{t}$ being real here. One can study consistently defined
CP-violation effects in detail in differential decays, which will not affect
the inclusive rate that we focus on here. After Electroweak Symmetry Breaking
(EWSB) and combining it with the Top Yukawa coupling term, we get

$\mathcal{L}\supset-\left[m_{t}+\left(\dfrac{m_{t}}{v}-\dfrac{c_{y}^{t}v^{2}}{\sqrt{2}}\right)h-\dfrac{3c_{y}^{t}v}{2\sqrt{2}}h^{2}--\dfrac{3c_{y}^{t}}{2\sqrt{2}}h^{3}\right]\left(\bar{t}_{L}t_{R}+\bar{t}_{R}t_{L}\right)$
(6)

in the unitary gauge. After imposing the top pole-mass physical boundary
condition, the Top Yukawa coupling is shifted by

$\delta_{yt}=-\dfrac{c^{t}_{y}v^{3}}{\sqrt{2}m_{t}}.$ (7)

The additional terms in Eq. 6 are higher dimensional operators describing
interactions between the top quark pair and two or three Higgs bosons. Since
we are considering the $VV\rightarrow t\bar{t}$ process, only the single Higgs
coupling would enter the calculations. Beyond the scenario only the operator
$O_{y}^{t}$ is turned on, other operators could also enter the process. Such
an assumption, while minimal, may be oversimplifying. In addition to
$O_{y}^{t}$, the UV models often generate multiple operators at the same time.
Next, we present one simple UV scenario to generate the effective anomalous
Yukawa coupling in the IR.

###  II.3 Vector-Like Quarks

Models with Vector-like quarks (VLQs), as a simple extension of SM, are
motivated in a broad class of new physics theories (see, e.g., Refs. [33, 34,
35] and references therein). As seen from its name, it is similar to quarks,
transforming as a triplet under the SU(3)C group. Its left and right-handed
components carry the same color and electroweak charge. It is possible to
write a mass term independent of the Higgs VEV. Therefore, it has a decoupling
limit and it is consistent with the electroweak precision and Higgs precision
data. We can exploit its mixing with the third generation quarks to obtain the
effective non-SM Yukawa coupling at low energies.

We start from the simple case that the VLQ is an SM electroweak SU(2) singlet
[36] and has the same quantum numbers as right-handed top quark, $t_{R}$.
Denoting the one flavor VLQ as $T$, and its left and right-handed components
are $T_{L}$ and $T_{R}$. The relevant Lagrangian is

$\mathcal{L}\supset
i\bar{T}\not{D}T-\left(\lambda_{1}\bar{Q}_{L}\tilde{H}t_{R}+\lambda_{2}\bar{Q}_{L}\tilde{H}T_{R}+M_{1}\bar{T}_{L}t_{R}+M_{2}\bar{T}_{L}T_{R}\right)\
.$ (8)

However, we can always combine $t_{R}$ and $T_{R}$ to get a new right-handed
quark, which we will still denote as $T_{R}$. This field redefinition can
remove the mass mixing term between $T_{L}$ and $t_{R}$. The resulting
Lagrangian is

$\mathcal{L}\supset
i\bar{T}\not{D}T-\left(\lambda_{t}\bar{Q}_{L}\tilde{H}t_{R}+\lambda_{0}\bar{Q}_{L}\tilde{H}T_{R}+M_{0}\bar{T}_{L}T_{R}\right)\
.$ (9)

After EWSB we can write down the mass matrix as follows

$-\mathcal{L}\supset\begin{pmatrix}\bar{t}_{L}&\bar{T}_{L}\end{pmatrix}\begin{pmatrix}\lambda_{t}v/\sqrt{2}&\hskip
14.22636pt\lambda_{0}v/\sqrt{2}\\\ 0&M_{0}\end{pmatrix}\begin{pmatrix}t_{R}\\\
T_{R}\end{pmatrix}+\text{h.c.}=\bar{F}_{L}MF_{R}+\text{h.c.}$ (10)

in which we abbreviate the left-handed and right-handed fermions into the
column vectors, $F_{L}$ and $F_{R}$, respectively. The mass matrix is denoted
as $M$. Note that for fermions, we need separate transformations for the left
and right-handed components, $U_{L}$ and $U_{R}$, respectively, to diagonalize
the mass matrix. We have

$U_{L}MU_{R}^{\dagger}=\Lambda=\begin{pmatrix}m_{t}&\\\
&m_{T}\end{pmatrix}\quad\text{with}\quad
U_{L,R}=\begin{pmatrix}c_{L,R}&s_{L,R}\\\ -s_{L,R}&c_{L,R}\end{pmatrix}$ (11)

where $m_{t}$ and the $M_{T}$ are the physical masses of the top quark and the
VLQ. In the mass basis, the Yukawa matrix can be written as

$U_{L}\begin{pmatrix}\lambda_{t}&\quad\lambda_{0}\\\ 0&\quad
0\end{pmatrix}U_{R}^{\dagger}=\dfrac{\sqrt{2}}{v}\,U_{L}\begin{pmatrix}1&\quad
0\\\ 0&\quad
0\end{pmatrix}U_{L}^{\dagger}\Lambda=\dfrac{\sqrt{2}}{v}\begin{pmatrix}c_{L}^{2}m_{t}&-c_{L}s_{L}m_{T}\\\
-c_{L}s_{L}m_{T}&s_{L}^{2}m_{T}\end{pmatrix}\ .$ (12)

On the other hand, using the identity
$U_{L}MM^{\dagger}U_{L}^{\dagger}=\Lambda^{2}$, we can obtain the relative
shift for the Top Yukawa coupling

$\delta_{yt}=-s_{L}^{2}=-\dfrac{\left(\lambda_{t}^{2}+\lambda_{0}^{2}\right)v^{2}/2-m_{t}^{2}}{m_{T}^{2}-m_{t}^{2}}.$
(13)

In the limit $m_{T}\gg m_{t}$, we have $\delta_{yt}\sim v^{2}/m_{T}^{2}$, as
expected from the correction induced by a dimension-6 operator. Moreover, the
fermion field rotation would modify the coupling between the top quark and the
gauge field. Making the replacement that $t_{L}\rightarrow
c_{L}t_{L}-s_{L}T_{L}$, we get

$\begin{split}\dfrac{g}{\sqrt{2}}W^{+}_{\mu}\,\bar{t}_{L}\gamma^{\mu}b_{L}\rightarrow&\dfrac{g}{\sqrt{2}}W^{+}_{\mu}\,\left(c_{L}\,\bar{t}_{L}\gamma^{\mu}b_{L}-s_{L}\,\bar{T}_{L}\gamma^{\mu}b_{L}\right)\
,\\\
\dfrac{g}{\cos\theta_{w}}\left(\dfrac{1}{2}-\dfrac{2}{3}\sin^{2}\theta_{w}\right)Z_{\mu}\bar{t}_{L}\gamma^{\mu}t_{L}\rightarrow&\dfrac{g}{\cos\theta_{w}}\left(\dfrac{1}{2}-\dfrac{2}{3}\sin^{2}\theta_{w}\right)Z_{\mu}\\\
&\left[c_{L}^{2}\bar{t}_{L}\gamma^{\mu}t_{L}-c_{L}s_{L}\left(\bar{t}_{L}\gamma^{\mu}T_{L}+\bar{T}_{L}\gamma^{\mu}t_{L}\right)+s_{L}^{2}\bar{T}_{L}\gamma^{\mu}T_{L}\right]\\\
\dfrac{g}{\cos\theta_{w}}\left(-\dfrac{2}{3}\sin^{2}\theta_{w}\right)Z_{\mu}\bar{T}_{L}\gamma^{\mu}T_{L}\rightarrow&\dfrac{g}{\cos\theta_{w}}\left(-\dfrac{2}{3}\sin^{2}\theta_{w}\right)Z_{\mu}\\\
&\left[s_{L}^{2}\bar{t}_{L}\gamma^{\mu}t_{L}+c_{L}s_{L}\left(\bar{t}_{L}\gamma^{\mu}T_{L}+\bar{T}_{L}\gamma^{\mu}t_{L}\right)+c_{L}^{2}\bar{T}_{L}\gamma^{\mu}T_{L}\right].\end{split}$
(14)

Apart from the modified $Zt\bar{t}$ coupling, there are also new off-diagonal
couplings of the form $V-T-t$, with $V$ being the charged and neutral gauge
bosons. These couplings would modify the electroweak oblique parameters $T$
and $S$, thus being highly constrained. However, one can always add more
vector-like quarks carrying the same charge as the bottom quark to compensate.
Alternatively, we can integrate out the heavy quark and obtain the
corresponding matched EFT Lagrangian. At tree level, we get

$\dfrac{\lambda^{2}_{0}}{M_{0}^{2}}\bar{Q}_{L}\tilde{H}i\not{D}\left(\tilde{H}^{\dagger}Q_{L}\right)$
(15)

After EWSB, this operator leads to the corrections to the top couplings
consistent with the mixing effect discussed above at the leading order. The
single operator in Eq. 15 can be converted into the operators in Warsaw basis
of SMEFT via the equation of motion. We get

$\dfrac{\lambda^{2}_{0}}{M_{0}^{2}}\left[y_{t}H^{\dagger}H\,\bar{Q}_{L}\tilde{H}t_{R}+\dfrac{1}{4}H^{\dagger}i\overleftrightarrow{D_{\mu}}H\,\bar{Q}_{L}\gamma^{\mu}Q_{L}-\dfrac{1}{4}H^{\dagger}\sigma^{a}i\overleftrightarrow{D_{\mu}}H\,\bar{Q}_{L}\sigma^{a}\gamma^{\mu}Q_{L}\right]$
(16)

Compared to the coupling modifications discussed in previous sections, apart
from the $\mathcal{O}_{y}^{t}$ operator, two more operators with correlated
coefficients appear and modify the coupling of the charged and neutral current
of the third-generation quarks. These correlations have implications when we
interpret our results in later sections.

###  II.4 Electroweak Gauge Boson PDFs

Vector Boson Fusion (VBF) production is one of the most important production
channels at high-energy colliders. For example, the Higgs boson can be
produced by VBF, in addition to the $s$-channel Higgsstrahlung channels at
future lepton colliders. As the center of mass energy $\sqrt{s}$ increases,
the VBF channel becomes more and more important than the $s$ channel. For muon
colliders with benchmark $\sqrt{s}=3$ or 10 TeV, the VBF processes dominate
the others for Higgs production. In many cases, a high-energy muon collider
can be considered a gauge boson collider; the virtual gauge bosons split off
the incoming muon and anti-muon beams, then participate in hard scatterings.

There are a couple of challenges in computing the VBF process accurately at
high-energy colliders. Splitting a muon into a virtual gauge boson plus
remnant lepton is highly forward at high energies, and complexities arise for
massive gauge bosons. One useful approximation is to treat the nearly on-shell
gauge boson as the parton within the incoming muon. Analogous to the proton
case, which can be seen as the collection of separate partons in the QCD
perturbative energy scale, the total collision cross section is the sum of the
convoluted partonic cross section with the parton distribution function. A
factorization procedure similar to hadron collisions can be applied here. The
full VBF process can be decomposed into the muon splitting part and the gauge
boson scattering sub-process. One can derive the PDF for the virtual gauge
bosons through the splitting function. Theoretically, it requires solving the
DGLAP equation and matching between the massless splitting and massive gauge
bosons. The matching can only be meaningfully done with higher-order
calculation information, which is beyond the scope of this study. The gauge
boson scattering sub-processes can be convoluted with the PDF to obtain the
cross-section. The total cross-section can be expressed as

$\sigma(\mu^{+}\mu^{-}\rightarrow F+X)\ =\ \int_{\tau_{\rm min}}^{\tau_{\rm
max}}d\tau\sum_{ij}\frac{\mathcal{L}_{ij}}{d\tau}\hat{\sigma}(ij\rightarrow
F),$ (17)

where $X$ refers to the collinear remnant partons, $F$ stands for the
collection of the final state particles from the hard scattering, and $i$, $j$
label the intermediate virtual gauge bosons. $\tau=\hat{s}/{s}$ is the
fraction of the partonic center of mass energy. The parton luminosity function
is defined as

$\frac{d\mathcal{L}_{ij}}{d\tau}\ =\
\frac{1}{1+\delta_{ij}}\int_{\tau}^{1}\frac{d\xi}{\xi}\left[f_{i}\left(\xi,\mu_{f}\right)f_{j}\left(\frac{\tau}{\xi},\mu_{f}\right)+i\leftrightarrow
j\right]$ (18)

in which $f_{i}(\xi,\mu_{f})$ is the PDF for the parton $i$ carrying a
fraction of the longitudinal momentum of the incoming muon at the
factorization scale $\mu_{f}$. In the following computation, we set both the
factorization scale $\mu_{f}$ and the renormalization scale equal to
$\sqrt{\hat{s}}/2$ [18].

The photon PDF is the largest among all gauge bosons due to its huge flux
starting from a low energy scale. However, photon-initiated channels make
little contribution to the signal part of our analysis which comes from the
Higgs exchange diagram. Our main signal channel is from the longitudinal gauge
boson initial states, especially $W_{L}^{+}W_{L}^{-}\rightarrow t\bar{t}$. The
longitudinal gauge boson ($W_{L},Z_{L}$) PDF has no scale dependence at the
leading order [37, 38, 39], which can be seen as the remnant from the
electroweak symmetry breaking. The suppression from the longitudinal gauge
boson PDF can be compensated by the large partonic cross section. These
features will be apparent in Section III and key to our sensitivity analysis
of the deviation of Top Yukawa coupling.

##  III Relevant Channels at Muon Collider

At a high-energy muon collider, the primary production channel of top quark
pairs is from the VBF process. In this section, we will compute the SM
partonic cross-sections for the processes $VV\rightarrow t\bar{t}$ and then
present the results after convolution with the muon PDF. FeynCalc[40] was
first used to calculate the helicity amplitude and the cross-sections. We then
modify the vertices in the SM UFO model files within MadGraph5_aMC$@$NLO [41,
42] as cross-check. The gauge boson PDF is computed at the truncated to
leading logarithmic order as done in [5].

###  III.1 Polarized Partonic Cross Section

Figure 2: The SM partonic cross-section for $VV\rightarrow t\bar{t}$ as a
function of the center of mass energy $\sqrt{\hat{s}}$ with different initial
helicities states.

We first present the SM predictions of the partonic cross sections in Fig. 2,
where all the possible initial polarization states of the vector bosons are
separately shown for the range of center-of-mass energies from the $t\bar{t}$
threshold to 10 TeV. First, let us look at the $WW$ initial states, the most
dominant $t\bar{t}$ production channel. The figure shows that the primary
channel is the one in which both incoming $W^{\pm}$ are longitudinally
polarized. The subdominant contribution to the cross-section occurs for right-
handed $W^{-}$ with longitudinal $W^{+}$ as well as the CP conjugated
combination. In the high energy regime, we observe from the figure that the
cross-sections of the channels with initial polarization of $(W^{+},W^{-})$ to
be $(0,0)$, $(0,+)$, $(-,0)$, $(-,+)(+,-)$ scale as $\hat{s}^{-1}$ and the
cross-sections of the four other channels scale as $\hat{s}^{-2}$. Therefore,
the amplitude squared term does not scale for $(0,0)$, $(0,+)$, $(-,0)$,
$(-,+)(+,-)$ and for $(+,0)$, $(0,-)$, $(-,-)$, $(+,+)$ scales as
$\hat{s}^{-1}$.

$W^{+}$ | $W^{-}$ | ($t$, $\bar{t}$)
---|---|---
(+,+) | $(-,+)$ | $(+,-)$ | $(-,-)$
0 | 0 | $\hat{s}^{-1}$ | $\hat{s}^{0}$ | $\hat{s}^{0}$ | $\hat{s}^{-1}$
+ | 0 | $\hat{s}^{-2}$ | $\hat{s}^{-1}$ | $\hat{s}^{-1}$ | $\hat{s}^{-2}$
$-$ | 0 | $\hat{s}^{-2}$ | $\hat{s}^{-1}$ | $\hat{s}^{-1}$ | $\hat{s}^{0}$
0 | + | $\hat{s}^{0}$ | $\hat{s}^{-1}$ | $\hat{s}^{-1}$ | $\hat{s}^{-2}$
+ | + | $\hat{s}^{-1}$ | $\hat{s}^{-2}$ | $\hat{s}^{-2}$ | $\hat{s}^{-3}$
$-$ | + | $\hat{s}^{-1}$ | $\hat{s}^{0}$ | $\hat{s}^{-2}$ | $\hat{s}^{-1}$
0 | $-$ | $\hat{s}^{-2}$ | $\hat{s}^{-1}$ | $\hat{s}^{-1}$ | $\hat{s}^{-2}$
+ | $-$ | $\hat{s}^{-1}$ | $\hat{s}^{0}$ | $\hat{s}^{-2}$ | $\hat{s}^{-1}$
$-$ | $-$ | $\hat{s}^{-3}$ | $\hat{s}^{-2}$ | $\hat{s}^{-2}$ | $\hat{s}^{-1}$
Table 1: The high energy $\hat{s}$ scaling of the squared amplitude for
$W^{+}W^{-}\rightarrow t\bar{t}$ where helicities of the outgoing top quarks
and incoming $W$ bosons are explicitly shown. Here we have ignored the
potential additional logarithmic dependence. For the fermion helicities, we
use $\pm$ to denote their helicity.

To illustrate the scaling mentioned, we show the explicit $\hat{s}$ dependence
of the amplitude squared term of all possible helicity configurations for the
$W^{+}(h_{i})+W^{-}(h_{j})\rightarrow t(h_{k})+\bar{t}(h_{l})$ processes in
Table 1, where the helicity of the particle is shown in the brackets. Here,
$h_{i},h_{j}$ can take value of $1$, 0 or $-1$ while $h_{k}$ and $h_{l}$ can
be $\pm 1/2$ (labelled as $\pm$ in the table for brevity). The tree-level
helicity amplitudes are invariant under the CP transformation

$W^{+}(h_{i})\rightarrow W^{-}(-h_{i}),\quad W^{-}(h_{j})\rightarrow
W^{+}(-h_{j}),\quad
t(h_{k})\rightarrow\bar{t}(-h_{k}),\quad\bar{t}(h_{l})\rightarrow t(-h_{l})$
(19)

which is manifest in the table. For example, if one compares the third and
fourth row, $(W^{+}(-),W^{-}(0))$ is transformed to $(W^{+}(0),W^{-}(+))$ and
the $\hat{s}$ power is indeed swapped under the CP transformation of the top
quark pairs.

We now focus on the scaling of helicity amplitude for
$W_{L}^{+}W_{L}^{-}\rightarrow t\bar{t}$ due to its significance as the
standard model background as well as the source of the signal. 222$t\bar{t}$
can also be produced through the $s$-channel process, but it is by far
subdominant at high energy muon colliders, and easily distinguishable from the
signal process considered here due to its energetic top quark jet, high
invariant mass, and low rapidity of the $t\bar{t}$ system. Let us first
consider $t,\bar{t}$ helicity $(\pm,\pm)$. For the Top Yukawa coupling we are
probing, this channel is the largest source of signal in our analysis due to
the Higgs exchange diagram. Each individual diagram for this helicity scales
as $\sqrt{\hat{s}}$ as seen from Eqs. 44, 45, 46 and 47 and Eqs. 56, 57, 58
and 59 in the high energy limit. Once we add all four diagram the leading
terms cancels and in SM the amplitude scales as 1/$\sqrt{\hat{s}}$ which is
the next leading order term in all these amplitudes. Note that the signal
comes only from the Higgs diagram scaling as $\sqrt{\hat{s}}$ and so its
interference with SM therefore does not scale. We elaborate interference and
relevant scaling in Section IV.1.

The leading contribution of $W^{+}_{L}W^{-}_{L}\rightarrow t\bar{t}$ comes
from the final quark helicity $t(+)\bar{t}(-)$ and so we now focus on
$t,\bar{t}$ helicity $(\pm,\mp)$ to understand their behavior. As shown by the
helicity amplitude in Appendix A, the amplitudes of the processes with final
quarks helicities $(-,+)$ or $(+,-)$ approach a constant in the high energy
limit given by

$\displaystyle\mathcal{M}\left(W^{+}_{L}W^{-}_{L}\rightarrow
t(+)\bar{t}(-)\right)$
$\displaystyle=-\frac{2m_{t}^{2}}{v^{2}}\cot\dfrac{\theta}{2}+\dfrac{g^{\prime
2}}{3}\sin\theta\ +\mathcal{O}(1/\hat{s}),$ (20)
$\displaystyle\mathcal{M}\left(W^{+}_{L}W^{-}_{L}\rightarrow
t(-)\bar{t}(+)\right)$ $\displaystyle=\dfrac{3g^{2}+g^{\prime
2}}{12}\sin\theta+\mathcal{O}(1/\hat{s})\ .$ (21)

where $\theta$ is the angle between $W^{+}$ and $t$ in the partonic center of
mass frame. The leading order behavior of these amplitudes can be understood
from the perspective of the Goldstone boson equivalence theorem and from the
process $\phi^{+}\phi^{-}\rightarrow t\bar{t}$. Let us first consider
$\phi^{+}\phi^{-}\rightarrow t(+)\bar{t}(-)$. At leading order, the
$s$-channel process has contribution only from the U(1) field $B_{\mu}$
exchange and no contribution from the $SU(2)_{L}$ field, which will be of
order $O(m_{t}/\sqrt{\hat{s}})$ for this helicity configuration. The
$s$-channel $B_{\mu}$ diagram is proportional to $g^{\prime 2}$ and the Wigner
function $d^{1}_{0,1}=\sin\theta/\sqrt{2}$. The process also has a $t$-channel
contribution from $b$-quark exchange which will come from terms in Lagrangian
of the form $y_{t}\bar{b}_{L}t_{R}\phi^{-}$ and its hermitian conjugate. This
explains why in Eq. 20, we observe $t$-channel behavior proportional to
$y_{t}^{2}$. Now, let us consider the channel $\phi^{+}\phi^{-}\rightarrow
t(-)\bar{t}(+)$. In the leading order, the $s$-channel diagram is mediated by
$SU(2)_{L}$ charge neutral vector boson $W_{\mu}^{3}$ with coupling $g$ and
U(1) field $B_{\mu}$ with coupling $g^{\prime}$. Since we assume $y_{b}=0$ in
our calculation, the leading order $t$-channel contribution is of order
$\mathcal{O}(m_{t}/\sqrt{\hat{s}})$ and explains the absence of $t$-channel
behavior in Eq. 21. Due to a $t$-channel enhancement which is absent in other
channels in the leading order, $W_{L}^{+}W_{L}^{-}\rightarrow t(+)\bar{t}(-)$
dominates top quark pair production from W-bosons and serve as the largest SM
backgrounds in our analysis.

Furthermore, we numerically compute the amplitude of this channel by
decomposing the top current into left-handed current and right-handed current
parts separately. Explicitly, each $s$-channel diagram can be decomposed into
the two subdiagrams with $\bar{t}_{L}\gamma^{\mu}t_{L}V_{\mu}$ and
$\bar{t}_{R}\gamma^{\mu}t_{R}V_{\mu}$ with $V$ being either photon or the
neutral $Z$ boson. The $t$-channel diagram only contains the left-handed
current $\bar{t}_{L}\gamma^{\mu}b_{L}$ plus its conjugation
$\bar{b}_{L}\gamma^{\mu}t_{L}$. It turns out that the summation of the left-
handed current consisting of the $\bar{t}_{L}\gamma^{\mu}t_{L}$ part from $s$
channel diagram and the $t$ channel diagram dominates over the summation of
the right-handed current part $\bar{t}_{R}\gamma^{\mu}t_{R}$ from the $s$
channel diagrams. This can be understood as follows. First, in the
$t(+)\bar{t}(-)$ final helicity state, the Higgs diagram does not contribute,
and the $t-$channel diagram only contains the left-handed current. In the high
energy limit, the $s$-channel diagram exchanging $Z/\gamma$ bosons can be
treated as restoring into $W_{\mu}^{3}$ field because the $W$ boson pairs can
only interact with $W_{\mu}^{3}$ via the non-abelian self-interaction. While
$W_{\mu}^{3}$ field can only couple to left-handed current. The right-handed
current can be only from the finite $Z$ boson mass effect finally manifested
by the $g^{\prime 2}$ terms. Such behavior has a significant impact on the VLQ
model, which will be discussed later on.

For $ZZ$-fusion channels, the partonic cross section is invariant under the CP
transformation and the exchange of initial particle states. Consequently, only
four distinct behaviors exist for all possible initial Z boson helicity
combinations. Longitudinally polarized $Z$ boson states also dominate the
partonic cross-section. The subdominant channels consist of one longitudinal
polarized $Z$ boson plus the other transverse polarized $Z$ boson. Both
$t$-channel and $u$-channel diagrams are possible, resulting in both forward
and backward enhancement of the differential distribution. This could also be
understood because the top quark exchange is symmetric with respect to
$\theta=\pi/2$. This contrasts with the $WW$-fusion case where the $t$-channel
diagram exchanging the $b$ quark prefers the forward direction, and there is
no $u$-channel diagram. For production processes starting from transverse
helicity modes, the cross-section from the opposite helicity initial state is
larger than that of the same helicity.

For the diphoton-initiated processes, there are only two distinct
combinations, with either the same or opposite helicities of the photons. One
can see that the opposite helicity case (orange curve) is larger than the
same-helicity case (red curve) in the high energy limit. Same-helicity process
scales as $\hat{s}^{-1}$ asymptotically while the opposite-helicity process
scales as $\hat{s}^{-1}\log\hat{s}$. A similar pattern also shows up in the
$Z\gamma$ partonic cross section. It should be emphasized that the photon PDF
from muon is around one order smaller than that from the electron at the same
factorization scale, due to the two orders difference in the muon and electron
mass. This results in decreased signal rate of diphoton initiated process at
muon collider.

###  III.2 Convolution with PDFs

Figure 3: The differential distribution for $\mu^{+}\mu^{-}\rightarrow
t\bar{t}+X$ versus the invariant mass $m_{\bar{t}t}$ at a 3 TeV and a 10 TeV
MuC.

Having computed the partonic cross-section for all possible polarizations of
the gauge boson, the next step is to convolute them with the electroweak gauge
boson PDFs. We work with two benchmarks of the center of mass energies for the
future muon collider: one is $\sqrt{s}=3$ TeV with an integrated luminosity of
1 $\text{ab}^{-1}$, and the other one with $\sqrt{s}=$ 10 TeV with an
integrated luminosity of 10 $\text{ab}^{-1}$. In this section, we first
present the differential distribution with the invariant mass of the top quark
pair in Fig. 3. The orders of $t\bar{t}$ cross-section contributions are $WW$,
$\gamma\gamma$, $Z\gamma$ and $ZZ$ initial states. As we have seen in Fig. 2,
the partonic cross section of $WW$ is over two orders larger than that of
$\gamma\gamma$ although the photon PDF is larger than $W$ boson. Due to the
smallness of the $Z$ boson PDF, $ZZ$-fusion channel is suppressed while
$Z\gamma$ sits in between. Beyond the simple PDF treatment, we also evaluate
the importance of the interference effects between different electroweak gauge
bosons. After all, the PDF treatment is a quasi-real approximation of the
intermediate states, which are fully interfering. From the view of the full
$2\rightarrow 4$ process $\mu^{+}\mu^{-}\rightarrow\mu^{+}\mu^{-}t\bar{t}$,
the intermediate off-shell gauge boson can be either $Z$ boson or the photon.
One needs to first sum up the amplitudes of both processes before squaring
them, namely

$\mathcal{M}=\sum_{i,j}\mathcal{M}_{V_{+,i}V_{-,j}}=\mathcal{M}_{ZZ}+\mathcal{M}_{\gamma\gamma}+\mathcal{M}_{Z\gamma}+\mathcal{M}_{\gamma
Z}$ (22)

where we denote $V_{+}$ as the neutral gauge boson radiated off $\mu^{+}$ and
$V_{-}$ from $\mu^{-}$. After squaring, multiple interference terms should be
handled carefully. Remembering that in computing the gauge boson PDF, there is
the term $f_{Z\gamma}$ which accounts for the mixing from the splitting
amplitude $\mathcal{M}^{\rm split}_{\mu\rightarrow\mu\gamma}$ and
$\mathcal{M}^{\rm split}_{\mu\rightarrow\mu Z}$. Hence, the total cross
section for $\mu^{+}\mu^{-}\rightarrow\mu^{+}\mu^{-}t\bar{t}$ should be

$\sigma\sim\sum\int
dx_{1}dx_{2}f_{i_{-}j_{-}}(x_{1})f_{i_{+}j_{+}}(x_{2})\mathcal{M}_{i_{-}i_{+}}\mathcal{M}^{*}_{j_{-}j_{+}}$
(23)

For example, if $i_{-}=j_{-}=\gamma$, $f_{i_{-}j_{-}}(x)$ equals to
$f_{\gamma}(x)$. The case of $i_{-}\neq j_{-}$ corresponds to $f_{Z\gamma}$
function. We use the brackets to enclose $Z\gamma$ in Fig. 3 to show their
contributions. It turns out that the interference terms are small compared to
the diagonal terms where $i_{\pm}=j_{\pm}$.

The angular distributions are shown in Fig. 4. The left panel is for the 3 TeV
MuC, and the right one is for 10 TeV MuC. The dominant channel $W^{+}W^{-}$
favors the forward direction due to the $t$-channel diagram exchanging
intermediate $b$ quark. For the neutral gauge bosons channels, they are
symmetric with respect to $\theta=\pi/2$.

Figure 4: The differential angular distribution for $\mu^{+}\mu^{-}\rightarrow
t\bar{t}+X$ at muon colliders with $\sqrt{s}$ = 3 TeV and $\sqrt{s}$ = 10 TeV.

##  IV Projected Sensivities

After exploring both the rate and the kinematics, we perform signal analysis
and derive the sensitivity of the Top Yukawa coupling deviations for 3 TeV and
10 TeV muon colliders. This section presents the projected sensitivities for
the anomalous Top Yukawa coupling and VLQ model.

###  IV.1 Top Yukawa Precision

We show the relative change of the partonic cross section compared to the SM
case due to the anomalous Top Yukawa coupling presented on the left panel of
Fig. 5. Since the dominant signal comes from the WW channel after convoluting
with PDF, so we will concentrate on this channel first to understand the
leading contribution to the signal sensitivity. We can decompose the squared
amplitude as

$|\mathcal{M}_{\rm tot}|^{2}=|\mathcal{M}_{\rm SM}+\delta_{yt}\mathcal{M}_{\rm
sig}|^{2}=|\mathcal{M}_{\rm SM}|^{2}+2\delta_{yt}\,{\rm Re}\mathcal{M}_{\rm
SM}^{*}\mathcal{M}_{\rm sig}+\delta_{yt}^{2}|\mathcal{M}_{\rm sig}|^{2},$ (24)

here $\delta_{yt}$ is the shift in Top Yukawa, and $\mathcal{M}_{\rm sig}$ is
the signal which will come from Higgs mediated diagram.

Figure 5:  The deviation of $WW\to t\bar{t}$ cross-section from the Standard
Model prediction value for benchmark choices of shift in Top Yukawa
$\delta_{yt}$ as a function of (left panel:) partonic center of mass energy
and (right panel:) scattering angle in the CM frame for a fixed partonic
center of mass energy of 2 TeV. The solid and dashed lines are with and
without the quadratic contributions, respectively.

The dashed lines in Fig. 5 show the signal-to-background ratio with linear in
$\delta_{yt}$ contributions (as a typical choice for EFT analysis), and the
solid lines include the terms proportional to $\delta_{yt}^{2}$. Here
$\hat{\sigma}_{\rm SM}$ is the sum of all SM cross sections with all
helicities. A constructive interference between the SM background and the
anomalous signal for $\delta_{yt}<0$ exists, and similarly, a destructive
interference for $\delta_{yt}>0$ exists at linear order. As seen from the
figure, if we only keep the linear term, while the modification to the
$s$-channel Higgs amplitude scales as $\sqrt{\hat{s}}$ as shown in Eq. 3,
$\Delta\hat{\sigma}/\hat{\sigma}_{\rm SM}$ approaches a constant
asymptotically. The ratio grows linearly with $\hat{s}$ in the high energy
limit if we also include the quadratic term. As we will show later, the
sensitivity is dominated by the low invariant mass, linear regime. This
implies the requirement for having systematic control at the sub-percent
level. As an electroweak machine, it should be achievable, and we also want to
emphasize here the importance of precision calculation.

To understand the leading order behavior of these ratios, one should first
remember that only the cross section with final top quark helicities
$(\pm,\pm)$ receives contributions from $\delta_{yt}$ in the process
$W_{L}^{+}W_{L}^{-}\rightarrow t\bar{t}$. For the other two helicities, the
Higgs exchange diagram is absent. More explicitly, we have

$\dfrac{\Delta\hat{\sigma}}{\hat{\sigma}_{\rm
SM}}\sim\dfrac{\sum_{\lambda=\pm}2\delta_{yt}\text{Re}\mathcal{M}_{\lambda\lambda}^{h}\mathcal{M}^{*}_{\lambda\lambda}+\delta_{yt}^{2}|\mathcal{M}_{\lambda\lambda}^{h}|^{2}}{\sum|\mathcal{M}|^{2}}\
.$ (25)

In the numerator, $\mathcal{M}_{\lambda\lambda}$ refers to all the SM
amplitude with the top quark having the same helicities.
$\mathcal{M}_{\lambda\lambda}^{h}$ denotes the amplitude for the Higgs-
mediated diagram alone, which scales as $\sqrt{\hat{s}}$ in the high-energy
limit. From Table 1, we can infer that the amplitude of
$W_{L}^{+}W_{L}^{-}\rightarrow t(\pm)\bar{t}(\pm)$ scales like
$1/\sqrt{\hat{s}}$ in the high energy regime while the term proportional to
$\delta_{yt}$ coming from the Higgs diagram scale as $\sqrt{\hat{s}}$. Thus
the interference between these two terms will have no scaling behavior. On the
other hand, the sum of the squared amplitudes in SM approaches constant at the
leading order. Therefore, the ratio of the linear term in $\delta_{yt}$ of
$\Delta\hat{\sigma}$ and $\hat{\sigma}_{\rm SM}$ becomes asymptotically flat
since both have a leading order constant term. The term quadratic in
$\delta_{yt}$ scales as $\hat{s}$ and so the ratio including both terms
increases linearly with $\hat{s}$ asymptotically.

Figure 6: The significance of various bins in ranges of $m_{t\bar{t}}$ and
$\cos\theta$ at 3 TeV and 10 TeV MuC. For illustration, we take $\delta_{yt}$
to be 0.015 for 10 TeV muon collider and 0.1 for 3 TeV muon collider, which is
close to their one-sigma sensitivity. It can be seen from the figure that most
of the sensitivity is coming from the interference term, near the threshold
where the SM cross-section peaks and the quadratic term contribution is
negligible.

The corresponding angular distribution of
$\Delta\hat{\sigma}/\hat{\sigma}_{\rm SM}$ is shown on the right panel of Fig.
5 for the partonic $\sqrt{\hat{s}}=2$ TeV. Note that $\mathcal{M}_{\rm sig}$
has no angular dependence due to the spin 0 Higgs boson exchange. Therefore
the angular dependence of the interference term is purely from the SM terms,
and the term with quadratic $\delta_{yt}$ is flat. After dividing by the SM
background contribution, the peak of the ratio occurs in the backward
direction since the signal $\Delta\sigma$ is flatly distributed while the
background favors the forward region.

To derive the sensitivity of the Top Yukawa coupling for the two muon collider
benchmarks, we convolute the partonic signal cross sections with the PDF. For
our analysis, we choose to bin the phase space in terms of the top quark pair
invariant mass $m_{t\bar{t}}$ and the outgoing polar angle in the CM frame
with respect to the $\mu^{+}$ direction. Bin-by-bin signal significance after
convolution is shown in Fig. 6, where the range of $\cos\theta$ is divided
into eight equally sized bins, and $m_{t\bar{t}}$ is divided into 50 GeV bins
to take into account finite resolution effects. From the figure, it is clear
that our sensitivity will come from the linear term, which peaks near the
threshold.

Figure 7: The $\Delta\chi^{2}$ as a function of $\delta_{yt}$ after summation
over bins for $\sqrt{s}$ = 3 TeV (left) and $\sqrt{s}$ = 10 TeV (right).
Contributions from linear terms (interference piece) and quadratic terms are
included in the $\Delta\chi^{2}$ analysis for the solid curves, and dashed
curves only include the linear terms. For the solid curves, the
$\Delta\chi^{2}$ asymmetry results from including the quadratic term.  Figure
8: The reach plot for $\delta_{yt}$ at 1 $\sigma$ sensitivity on various
colliders with their best projections through various processes. The bound on
the Top Yukawa shift factor $\delta_{yt}$ from (top to the bottom) the current
LHC [15, 43], future High Luminosity LHC [44], proposed ILC [45, 46],CLIC
[47], FCC-ee and FCC-hh [48]. In the sub-panel, we show the zoom-in version to
compare different bars in the smaller range.

We perform a chi-square test to set bounds on anomalous Top Yukawa coupling.
The results for the $\chi^{2}$ analysis are presented in Fig. 7 for which we
require the number of events in each bin to be at least 10. We compute the
sensitivity $\Delta\chi_{i}^{2}=S_{i}^{2}/B_{i}$ using the corresponding
number of events for each bin. It is important to note that the signal
($S_{i}$) contains both the interference term and the quadratic term although
the contribution of the quadratic term is negligible at the energy scale being
considered. The total $\Delta\chi^{2}$ is obtained by summing over all the
bins. This procedure is equivalent to a template double differential line-
shape fit.

For a top quark pair, the branching ratios to various final states are $44\%$
(hadronic), $44\%$ (semileptonic), $11\%$ (fully leptonic)[49, 50].333Here we
split the $\tau$ final states into hadronic ones and leptonic ones with
corresponding $\tau$ branch fractions. In our final state considerations, we
conservatively take only the fully hadronic and semi-leptonic $t\bar{t}$, as
they are fully reconstructable without any sizable ambiguities. We also impose
a cut on the outgoing $t$ angle $\theta$ with respect to $W^{+}$ in the
partonic center of mass frame:

$10^{\circ}\ <\ \theta\ <\ 170^{\circ},$ (26)

to ensure high reconstruction efficiency and avoid the forward regions where
shielding is needed due to beam-induced backgrounds. These cuts do not notably
change our sensitivity since most of the signal significance comes from the
central bins in angular distribution, as seen in Fig. 13.

The $\chi^{2}$ analysis presented in Fig. 7 shows that the reach of
$\delta_{yt}$ at one sigma C.L. is around $(-6\%,9\%)$ for 3 TeV MuC and
$(-1.3\%,1.4\%)$ for 10 TeV MuC. The corresponding 2 sigma C.L. exclusion are
around $(-11\%,19\%)$ and $(-2.4\%,3.1\%)$ for 3 and 10 TeV MuC, respectively
444If one use only the semi-leptonic $t\bar{t}$, the $95\%$ C.L. limit on the
anomalous top coupling is projected to be $5.6\%$. This is highly consistent
with the simulation-based study [51] at a 10 TeV MuC. . It should be noted
that in Fig. 7, we include the contributions from both the linear and
quadratic terms of the signal. The resulting weight of these terms to the
total $\Delta\chi^{2}$ is:

$\displaystyle\Delta\chi^{2}\ $ $\displaystyle=\ 2.0\times
10^{2}\delta_{yt}^{2}-1.2\times 10^{3}\delta_{yt}^{3}+3.9\times
10^{3}\delta_{yt}^{4}\ \ \ \text{(3TeV)}$ (27) $\displaystyle\Delta\chi^{2}\ $
$\displaystyle=\ 5.6\times 10^{3}\delta_{yt}^{2}-4.8\times
10^{4}\delta_{yt}^{3}+2.6\times 10^{5}\delta_{yt}^{4}\ \ \ \text{(10TeV)}$
(28)

As can be seen, the inclusion of quadratic term results in producing
asymmetric $\Delta\chi^{2}$ distribution. Furthermore, at one sigma C.L for
both these colliders, if we neglect the quadratic term, $\delta_{yt}$ reach
would be 7% for 3 TeV and 1.33% for 10 TeV case. Fig. 7 also shows the results
for only including the interference effects in dashed lines, which is
conventionally done for EFT analysis. We can see the span of the sensitivity
is very similar to the results, including the quadratic terms. One can obtain
the corresponding $\Delta\chi^{2}$ expressions by dropping the last two terms
in the above equations.

For more general Higgs precision fits, the relative contributions from the
$WW$-fusion and $ZZ$-fusion channels need to be specified. Here, we also write
down the $\Delta\chi^{2}$ in the widely used $\kappa$-framework [52, 53, 54,
55, 12]. Defining $\kappa_{W}=g_{WWh}/g^{\rm SM}_{WWh}$,
$\kappa_{Z}=g_{ZZh}/g^{\rm SM}_{ZZh}$, $\kappa_{t}=y_{t}/y_{t}^{\rm SM}$ in
the kappa-framework, we further perform the chi-square analysis for 3 TeV
collider:

$\begin{split}\Delta\chi^{2}\ &=\ 2.2\times
10^{2}(\kappa_{W}\kappa_{t}-1)^{2}+2.9(\kappa_{Z}\kappa_{t}-1)^{2}-23(\kappa_{W}\kappa_{t}-1)(\kappa_{Z}\kappa_{t}-1)\\\
&-1.2\times
10^{3}(\kappa_{W}\kappa_{t}-1)^{3}+10(\kappa_{Z}\kappa_{t}-1)^{3}+92(\kappa_{W}\kappa_{t}-1)^{2}(\kappa_{Z}\kappa_{t}-1)\\\
&-1.3\times
10^{2}(\kappa_{W}\kappa_{t}-1)(\kappa_{Z}\kappa_{t}-1)^{2}+3.1\times
10^{3}(\kappa_{W}\kappa_{t}-1)^{4}+38(\kappa_{Z}\kappa_{t}-1)^{4}\\\
&+6.9\times
10^{2}(\kappa_{W}\kappa_{t}-1)^{2}(\kappa_{Z}\kappa_{t}-1)^{2}\end{split}$
(29)

and for 10 TeV:

$\begin{split}\Delta\chi^{2}\ &=6.2\times
10^{3}(\kappa_{W}\kappa_{t}-1)^{2}+1\times
10^{2}(\kappa_{Z}\kappa_{t}-1)^{2}-6.6\times
10^{2}(\kappa_{W}\kappa_{t}-1)(\kappa_{Z}\kappa_{t}-1)\\\ &-4.7\times
10^{4}(\kappa_{W}\kappa_{t}-1)^{3}+4.4\times
10^{2}(\kappa_{Z}\kappa_{t}-1)^{3}+4.0\times
10^{3}(\kappa_{W}\kappa_{t}-1)^{2}(\kappa_{Z}\kappa_{t}-1)\\\ &-5.1\times
10^{3}(\kappa_{W}\kappa_{t}-1)(\kappa_{Z}\kappa_{t}-1)^{2}+2.1\times
10^{5}(\kappa_{W}\kappa_{t}-1)^{4}+2.6\times
10^{3}(\kappa_{Z}\kappa_{t}-1)^{4}\\\ &+4.7\times
10^{4}(\kappa_{W}\kappa_{t}-1)^{2}(\kappa_{Z}\kappa_{t}-1)^{2}\end{split}$
(30)

Similarly, one can obtain the interference alone contribution by dropping all
terms other than the first lines of the above equations. These equations also
reveal that the sensitivity reach is dominated by the $WW$ channel. Note that
here, we assume no forward muon tagging, which means the $ZZ$-fusion and
$WW$-fusion are indistinguishable at the analysis level. In other words, they
share the same backgrounds. If one can tag the forward muons and hence reduce
the $ZZ$-fusion backgrounds, the relative importance and contribution will
change.

The $\delta_{yt}$ reach at one sigma C.L. is be (-6%,8%) for 3TeV MuC and
(-1.2%,1.4%) for 10TeV MuC. In Fig. 8, we show the one sigma sensitivity on
$\delta_{yt}$ at the current LHC, HL-LHC, ILC-500, ILC-100, CLIC, FCC-ee, and
FCC-hh, and compare them with our results at muon colliders of 3 and 10 TeV.
At present, the LHC experimental constraint is at the $O(10\%)$ level. The
future 100 TeV FCC-hh can probe the Top Yukawa coupling to around 1%. We show
here the muon collider can achieve comparable sensitivity.

###  IV.2 VLQ Model

Figure 9:  The relative shift of the partonic cross section for the WW channel
in VLQ model as a function of $\sqrt{\hat{s}}$ and the polar angle
$\cos\theta$.

Now we study this channel in the VLQ model, where Top Yukawa modification
correlate with other coupling modifications. Analogous to our previous
discussion, we first focus on the deviation in the partonic cross-section of
the $W^{+}W^{-}\rightarrow t\bar{t}$ channel which is shown in Fig. 9 for
$s_{L}^{2}=0.01$. Since this is a UV-complete model, we keep all the higher
order terms of $\delta_{yt}$ or $s_{L}^{2}$. We observe that the relative
change decreases when the center-of-mass energy increases from the thresh-hold
energy and becomes almost flat as $\sqrt{\hat{s}}$ exceeds around 1 TeV. To
understand this result, first, we note that the dominant (helicity) amplitude
for the SM background is asymptotically flat. Since we are working in the
weakly-coupled UV-completed model, the amplitude should also be unitarized.
This explains why the two distributions are relatively flat. It is interesting
to note that the angular distribution of the BSM-to-SM ratio
$\Delta\hat{\sigma}/\hat{\sigma}_{\rm SM}$ is also nearly flat, which suggests
that the signal follows the shape of the background. In other words, the VLQ
signal shifts the SM cross-section.

The shifting can be explained by noting that the leading order contribution
for $W^{+}_{L}W^{-}_{L}\rightarrow t\bar{t}$ is dominated by the left-handed
current as we have emphasized in Section III.1. For our VLQ analysis, the
$t_{L}$ field is rescaled by $c_{L}$. Hence the amplitudes of the two channels
are shifted by a common factor of $-s_{L}^{2}$. This explains why the relative
shifting factor for the cross section is approaching $-2s_{L}^{2}$ in the high
energy regime and is almost independent of the theta angle.

On the other hand, we can also compute using the Goldstone boson equivalence
theorem. As discussed in Section II, if there is only a single operator EFT
$\mathcal{O}_{y}^{t}$, a new contact vertex $\phi^{+}\phi^{-}t\bar{t}$ is
generated, leading to the energy growing behavior. While in the VLQ case, two
other operators can generate the new coupling
$i\left(\partial_{\mu}\phi^{+}\bar{t}_{L}\gamma^{\mu}b_{L}-\partial_{\mu}\phi^{-}\bar{b}_{L}\gamma^{\mu}t_{L}\right)$
that modify the $t$-channel diagram amplitude which also has the energy
growing behavior. The correlated Wilson coefficients guarantee the
cancellation of individual unitarity violations.

Figure 10:  $S/\sqrt{B}$ for $s_{L}^{2}$ = 0.01 for both a 3 TeV (left) and a
10 TeV (right) MuC. We cut the whole space in terms of the invariant mass
$m_{t\bar{t}}$ and the angular variable $\cos\theta$. Different colors denote
the bin range of $\cos\theta$.

Figure 11: The $\Delta\chi^{2}$ as a function of $s_{L}^{2}=-\delta_{yt}$
after summation over bins for $\sqrt{s}$ = 3 TeV (left) and $\sqrt{s}$ = 10
TeV (right) in VLQ model.

In Fig. 10, we show the $S/\sqrt{B}$ distribution over the top quark pair
invariant mass $m_{t\bar{t}}$ after dividing the phase space into various
bins. Curves with different colors refer to the angular bins. The peak values
for the significance occur at low $m_{t\bar{t}}$ regime as well as the forward
region. As $S/B$ is almost a constant, we get stronger sensitivity in the
forward direction and low $m_{t\bar{t}}$ region where the statistics are
large.

Figure 12: The 95% C.L. exclusion sensitivity for the parameters in VLQ model.
The blue dashed line is from the top quark pair production and the Higgs
decaying to di-gluon measurement [13]. The orange and pink dashed lines refer
to the exclusion sensitivity at 95% exclusion limit. The light-yellow region
denotes the strong coupling regime where the UV Yukawa coupling $\lambda_{0}$
is larger than $O(10)$. We expect the region for $m_{T}<5$ TeV can be covered
by the direct search, as the $T\bar{T}$ pair production is kinematically
allowed.

We follow the same procedure described in the previous section to estimate the
projected sensitivity and show the results in Fig. 11. The anomalous Top
Yukawa coupling shift can be tested to around 0.5% at 3 TeV MuC and 0.1% at 10
TeV MuC at one sigma level. In Fig. 12, we show various 95% exclusion regions
on the $m_{T}-\delta_{yt}$ plane. The blue, orange, and pink dashed lines
refer to the bound from the Higgs decaying to di-gluon precision measurement
at 10 TeV MuC, the $t\bar{t}$ channel at 3 TeV MuC and 10 TeV MuC
respectively. The coupling is strong in the light-yellow region, violating
perturbative unitarity in the Higgs-mediated four-fermi scattering process,
with the UV Yuakawa coupling $\lambda_{0}>O(10)$ in Eq. 13. The shaded region
for $m_{T}<5$ TeV is denoted as “Direct Search”, where the $T\bar{T}$
production is kinematically allowed. We also made an estimate of the reach of
the single heavy VLQ search at the 10 TeV MuC, and this channel can exclude
$\delta_{yt}$ to be around one percent level. However, given the sensitivity
dependence on the detailed realization of VLQ, e.g., the existence of heavy
bottoms, we do not show the estimation here. For a similar reason, we also do
not show electroweak precision from future Z-pole programs in this plot.

##  V Conclusion

In this work, we have examined the prospects for measuring the Top Yukawa
coupling at upcoming high-energy muon colliders, focusing on two benchmark
center-of-mass energies: 3 TeV and 10 TeV. We thoroughly investigated the
partonic cross sections of the processes $VV\rightarrow t\bar{t}$ for various
helicities and delved into the differential kinematical distributions. Upon
convolution with the electroweak gauge boson PDFs, it becomes evident that the
dominant partonic channel is the scattering of longitudinal $W$ bosons into a
right-handed top quark and a left-handed anti-top-quark.

We utilized two scenarios to characterize deviations from the Standard Model
(SM) Top Yukawa coupling. We first considered the case in which a single
dimension-six EFT operator can encapsulate the influences of new physics,
denoted as $\mathcal{O}_{y}^{t}$. Here, the primary effect is an anomalous
deviation in the Top Yukawa coupling. Consequently, the amplitude of
$W^{+}W^{-}\rightarrow t\bar{t}$ rises with the partonic center of mass
energy, leading to significant discrepancies from SM predictions at larger
$m_{t\bar{t}}$. In our second approach, taking inspiration from a UV-completed
model, we introduced a pair of singlet vector-like quarks. Post electroweak
symmetry breaking, a mixture occurs between the top quark and the heavy VLQ,
altering the Top Yukawa coupling. In this scenario, the amplitude or cross-
section’s relative shift remains mostly flat across a broad range of
$m_{t\bar{t}}$.

Constraints on the Top Yukawa deviation can be established, assuming future
measurements agree with SM. Segmenting the entire phase space according to the
production polar angle and the invariant mass of the top quark pair allows for
chi-square analysis. For the EFT model, the Top Yukawa measurement precision
approaches the percent level, providing valuable measurements of Top Yukawa
beyond HL-LHC. The results are summarized in Fig. 8. Furthermore, we provide
chi-square data for global fit to the Higgs signal, in Eqs. 27, 28, 29 and 30.
Importantly, compared to the conventional channel of $t\bar{t}h$ [13, 14], our
method is more than one order of magnitude better in projected precision and
also sensitive to the sign of the Top Yukawa deviation. Conversely, in the VLQ
model, this precision is further improved to nearing per mil level. These
high-precision measurement also requires precision calculation from the theory
side to ensure their robustness. We estimated the leading order perturbative
correction would be at %-level.

Acknowledgements: We want to thank K. Hagiwara, T. Han, Y. Ma, P. Meade and A.
Wulzer for their helpful discussions, and J. Fan and M. Reece for
participation in the earlier part of this study. K.F.L., Z.L., and I.M. are
supported in part by the DOE grant DE-SC0022345 and DOE grant DE-SC0011842.
Z.L., K.F.L., and LTW acknowledge the Aspen Center for Physics where the final
part of this work is completed, which is supported by National Science
Foundation grant PHY-2210452. The work of L.T.W. is supported by the DOE grant
DE-SC-0013642. The data associated with the figures in this paper can be
accessed via GitHub.

##  Appendix A Helicity Amplitude

In this section, we focus on the amplitude of $W^{+}W^{-}\rightarrow t\bar{t}$
and its analytic expressions. The total amplitude for a given helicity
configuration can be written as the sum of contributions from four channels:

$\mathcal{M}_{h_{W^{+}}h_{W^{-}},h_{t}h_{\bar{t}}}\ =\
\mathcal{M}^{\gamma}+\mathcal{M}^{Z}+\mathcal{M}^{b}+\mathcal{M}^{h}$ (31)

where the $h$ in the subscript refers to the corresponding helicities.
$\mathcal{M}^{\gamma/Z/h}$ refer to the $s$-channel contribution and
$\mathcal{M}^{b}$ refer to the $t$-channel diagram. We will compute the
amplitude in the massless $b$-quark limit. Also, we choose the incoming
$W^{+}$ to be aligned along the z-axis and define the scattering angle
$\theta$ as the polar angle between $W^{+}$ and the $t$-quark in the partonic
center-of-mass frame. Thus, with this definition, we have in the massless $b$
quark limit:

$s=4E_{t}^{2}=4E_{W}^{2}\ \ ,\ \ t\
=-\frac{s}{4}(\beta_{t}^{2}+\beta_{W}^{2}-2\beta_{t}\beta_{W}\cos\theta)$ (32)

Here, $E_{t,W}$ is the center of mass energy of each particle, and
$\beta_{t,W}^{2}=1-4m_{t,W}^{2}/s$. In this section, all the variables are
partonic variables, and we will drop the hat for abbreviation as there is no
ambiguity. For our calculations, we use the explicit form of four-spinors and
polarization vectors in the center of mass frame and calculate the helicity
amplitudes. Wigner d-functions are explicitly given by:

$\displaystyle d_{0,0}^{2}\
=\frac{1}{2}\left(3\cos^{2}\theta-1\right),~{}~{}~{}d_{2,0}^{2}\
=\sqrt{\frac{3}{8}}\sin^{2}\theta$ (33) $\displaystyle d_{1,0}^{2}\
=-d_{-1,0}^{2}\ =\ -\sqrt{\frac{3}{8}}\sin 2\theta$ (34) $\displaystyle
d_{2,1}^{2}\ =-d_{-2,-1}^{2}\ =\
-\frac{1}{2}\left(1+\cos\theta\right)\sin\theta$ (35) $\displaystyle
d_{2,-1}^{2}\ =-d_{-2,1}^{2}\ =\
-\frac{1}{2}\left(1-\cos\theta\right)\sin\theta$ (36) $\displaystyle
d_{1,1}^{2}\ =d_{-1,-1}^{2}\ =\
\frac{1}{2}\left(2\cos^{2}\theta+\cos\theta-1\right)$ (37) $\displaystyle
d_{1,-1}^{2}\ =d_{-1,1}^{2}\ =\
\frac{1}{2}\left(-2\cos^{2}\theta+\cos\theta+1\right)$ (38) $\displaystyle
d_{1,1}^{1}\ =d_{-1,-1}^{1}\ =\ \frac{1}{2}\left(1+\cos\theta\right)$ (39)
$\displaystyle d_{1,-1}^{1}\ =d_{-1,1}^{1}\ =\
\frac{1}{2}\left(1-\cos\theta\right)$ (40) $\displaystyle d_{1,0}^{1}\
=-d_{-1,0}^{1}\ =\ -\sqrt{\frac{1}{2}}\sin\theta$ (41) $\displaystyle
d_{0,0}^{1}\ =\cos\theta,~{}~{}~{}d_{0,0}^{0}\ =1$ (42) $\displaystyle
d_{m,m^{\prime}}^{j}\ =(-1)^{m-m^{\prime}}d^{j}_{m^{\prime},m}\ =\
d_{-m^{\prime},-m}^{j}\ $ (43)

  * •

$(00,++)$

$\displaystyle\mathcal{M}_{00,++}^{\gamma}\ $ $\displaystyle=\ \frac{2}{3}\
\left[2\sqrt{2}G_{F}m_{t}s_{w}^{2}s^{1/2}\beta_{W}\left(3-\beta_{W}^{2}\right)\right]d_{0,0}^{1}$
(44) $\displaystyle\mathcal{M}_{00,++}^{Z}\ $ $\displaystyle=\ \frac{2}{3}\
\left[\frac{1}{\sqrt{2}}G_{F}m_{t}\left(\frac{3}{2}-4s_{w}^{2}\right)\frac{s^{3/2}}{s-M_{Z}^{2}}\beta_{W}\left(3-\beta_{W}^{2}\right)\right]d_{0,0}^{1}$
(45) $\displaystyle\mathcal{M}_{00,++}^{b}\ $ $\displaystyle=\
\frac{1}{2\sqrt{2}}G_{F}m_{t}\frac{s^{3/2}}{t}\left[\beta_{t}\left(\frac{1}{3}+\beta_{W}^{2}\right)+\beta_{W}\left(1-\beta_{W}^{2}\right)d^{1}_{0,0}-\frac{4}{3}\beta_{t}d^{2}_{0,0}\right]$
(46) $\displaystyle\mathcal{M}_{00,++}^{h}\ $ $\displaystyle=\
\frac{1}{{2}}G_{F}y_{t}v\frac{s^{3/2}}{s-m_{H}^{2}}\beta_{t}\left(1+\beta_{W}^{2}\right)$
(47)

  * •

$(00,+-)$

$\displaystyle\mathcal{M}_{00,+-}^{\gamma}\ $ $\displaystyle=\ -\frac{2}{3}\
\left[2G_{F}s_{w}^{2}s\beta_{W}\left(3-\beta_{W}^{2}\right)\right]d_{0,1}^{1}$
(48) $\displaystyle\mathcal{M}_{00,+-}^{Z}\ $ $\displaystyle=\ \ -\frac{2}{3}\
\left[\frac{1}{2}G_{F}\left(\frac{3}{2}\left(1-\beta_{t}\right)-4s_{w}^{2}\right)\frac{s^{2}}{s-M_{Z}^{2}}\beta_{W}\left(3-\beta_{W}^{2}\right)\right]d_{0,1}^{1}$
(49) $\displaystyle\mathcal{M}_{00,+-}^{b}\ $ $\displaystyle=\
\frac{1}{2}G_{F}\frac{s^{2}}{t}(1-\beta_{t})\left[\left(\beta_{W}\beta_{t}-\frac{1}{2}\beta_{W}\left(1-\beta_{W}^{2}\right)\right)d^{1}_{0,1}+\sqrt{\frac{2}{3}}\beta_{t}d^{2}_{0,1}\right]$
(50) $\displaystyle\mathcal{M}_{00,+-}^{h}\ $ $\displaystyle=\ 0$ (51)

  * •

$(00,-+)$

$\displaystyle\mathcal{M}_{00,-+}^{\gamma}\ $ $\displaystyle=\ \frac{2}{3}\
\left[2G_{F}s_{w}^{2}s\beta_{W}\left(3-\beta_{W}^{2}\right)\right]d_{0,-1}^{1}$
(52) $\displaystyle\mathcal{M}_{00,-+}^{Z}\ $ $\displaystyle=\ \frac{2}{3}\
\left[\frac{1}{2}G_{F}\left(\frac{3}{2}\left(1+\beta_{t}\right)-4s_{w}^{2}\right)\frac{s^{2}}{s-M_{Z}^{2}}\beta_{W}\left(3-\beta_{W}^{2}\right)\right]d_{0,-1}^{1}$
(53) $\displaystyle\mathcal{M}_{00,-+}^{b}\ $ $\displaystyle=-\
\frac{1}{2}G_{F}\frac{s^{2}}{t}\left(1+\beta_{t}\right)\left[\left(-\beta_{W}\beta_{t}-\frac{1}{2}\beta_{W}\left(1-\beta_{W}^{2}\right)\right)d^{1}_{0,-1}+\sqrt{\frac{2}{3}}\beta_{t}d^{2}_{0,-1}\right]$
(54) $\displaystyle\mathcal{M}_{00,-+}^{h}\ $ $\displaystyle=\ 0$ (55)

  * •

$(00,--)$

$\displaystyle\mathcal{M}_{00,--}^{\gamma}\ $ $\displaystyle=\ -\frac{2}{3}\
\left[2\sqrt{2}G_{F}m_{t}s_{w}^{2}s^{1/2}\beta_{W}\left(3-\beta_{W}^{2}\right)\right]d_{0,0}^{1}$
(56) $\displaystyle\mathcal{M}_{00,--}^{Z}\ $ $\displaystyle=\ -\frac{2}{3}\
\left[\frac{1}{\sqrt{2}}G_{F}m_{t}\left(\frac{3}{2}-4s_{w}^{2}\right)\frac{s^{3/2}}{s-M_{Z}^{2}}\beta_{W}\left(3-\beta_{W}^{2}\right)\right]d_{0,0}^{1}$
(57) $\displaystyle\mathcal{M}_{00,--}^{b}\ $ $\displaystyle=\
-\frac{1}{2\sqrt{2}}G_{F}m_{t}\frac{s^{3/2}}{t}\left[\beta_{t}\left(\frac{1}{3}+\beta_{W}^{2}\right)+\beta_{W}\left(1-\beta_{W}^{2}\right)d^{1}_{0,0}-\frac{4}{3}\beta_{t}d^{2}_{0,0}\right]$
(58) $\displaystyle\mathcal{M}_{00,--}^{h}\ $ $\displaystyle=\
-\frac{1}{{2}}G_{F}y_{t}v\frac{s^{3/2}}{s-m_{H}^{2}}\beta_{t}\left(1+\beta_{W}^{2}\right)$
(59)

  * •

$(-0,++)$

$\displaystyle\mathcal{M}_{-0,++}^{\gamma}\ $ $\displaystyle=\ \frac{2}{3}\
\left[8\sqrt{2}G_{F}m_{t}m_{W}s_{w}^{2}\beta_{W}\right]d_{-1,0}^{1}$ (60)
$\displaystyle\mathcal{M}_{-0,++}^{Z}\ $ $\displaystyle=\ \frac{2}{3}\
\left[2\sqrt{2}G_{F}m_{t}m_{W}\left(\frac{3}{2}-4s_{w}^{2}\right)\frac{s}{s-M_{Z}^{2}}\beta_{W}\right]d_{-1,0}^{1}$
(61) $\displaystyle\mathcal{M}_{-0,++}^{b}\ $ $\displaystyle=\
-\frac{1}{\sqrt{2}}G_{F}m_{t}m_{W}\frac{s}{t}\left[\left(-\beta_{t}\beta_{W}-\beta_{t}-\beta_{W}\left(1-\beta_{W}\right)\right)d_{-1,0}^{1}+\frac{2}{\sqrt{3}}\beta_{t}d_{-1,0}^{2}\right]$
(62) $\displaystyle\mathcal{M}_{-0,++}^{h}\ $ $\displaystyle=\ 0$ (63)

  * •

$(-0,+-)$

$\displaystyle\mathcal{M}_{-0,+-}^{\gamma}\ $ $\displaystyle=\ -\frac{2}{3}\
\left[8G_{F}m_{W}s_{w}^{2}s^{1/2}\beta_{W}\right]d_{-1,1}^{1}$ (64)
$\displaystyle\mathcal{M}_{-0,+-}^{Z}\ $ $\displaystyle=\ -\frac{2}{3}\
\left[2G_{F}m_{W}\left(\frac{3}{2}\left(1-\beta_{t}\right)-4s_{w}^{2}\right)\frac{s^{3/2}}{s-M_{Z}^{2}}\beta_{W}\right]d_{-1,1}^{1}$
(65) $\displaystyle\mathcal{M}_{-0,+-}^{b}\ $ $\displaystyle=\
\frac{1}{2}G_{F}m_{W}\frac{s^{3/2}}{t}(1-\beta_{t})\left[\left(-\beta_{W}\left(1-\beta_{W}\right)+\beta_{t}\beta_{W}\right)d^{1}_{-1,1}+\beta_{t}d^{2}_{-1,1}\right]$
(66) $\displaystyle\mathcal{M}_{-0,+-}^{h}\ $ $\displaystyle=\ 0$ (67)

  * •

$(-0,-+)$

$\displaystyle\mathcal{M}_{-0,-+}^{\gamma}\ $ $\displaystyle=\ \frac{2}{3}\
\left[8G_{F}s_{w}^{2}m_{W}s^{1/2}\beta_{W}\right]d_{-1,-1}^{1}$ (68)
$\displaystyle\mathcal{M}_{-0,-+}^{Z}\ $ $\displaystyle=\ \frac{2}{3}\
\left[2G_{F}m_{W}\left(\frac{3}{2}\left(1+\beta_{t}\right)-4s_{w}^{2}\right)\frac{s^{3/2}}{s-M_{Z}^{2}}\beta_{W}\right]d_{-1,-1}^{1}$
(69) $\displaystyle\mathcal{M}_{-0,-+}^{b}\ $ $\displaystyle=\
-\frac{1}{2}G_{F}m_{W}\frac{s^{3/2}}{t}(1+\beta_{t})\left[\left(-\beta_{W}\left(1-\beta_{W}\right)-\beta_{t}\beta_{W}\right)d^{1}_{-1,-1}+\beta_{t}d^{2}_{-1,-1}\right]$
(70) $\displaystyle\mathcal{M}_{-0,-+}^{h}\ $ $\displaystyle=\ 0$ (71)

  * •

$(-0,--)$

$\displaystyle\mathcal{M}_{-0,--}^{\gamma}\ $ $\displaystyle=\ -\frac{2}{3}\
\left[8\sqrt{2}G_{F}m_{t}m_{W}s_{w}^{2}\beta_{W}\right]d_{-1,0}^{1}$ (72)
$\displaystyle\mathcal{M}_{-0,--}^{Z}\ $ $\displaystyle=\ -\frac{2}{3}\
\left[2\sqrt{2}G_{F}m_{t}m_{W}\left(\frac{3}{2}-4s_{w}^{2}\right)\frac{s}{s-M_{Z}^{2}}\beta_{W}\right]d_{-1,0}^{1}$
(73) $\displaystyle\mathcal{M}_{-0,--}^{b}\ $ $\displaystyle=\
\frac{1}{\sqrt{2}}G_{F}m_{t}m_{W}\frac{s}{t}\left[\left(\beta_{t}\beta_{W}+\beta_{t}-\beta_{W}(1-\beta_{W})\right)d_{-1,0}^{1}+\frac{2}{\sqrt{3}}\beta_{t}d_{-1,0}^{2}\right]$
(74) $\displaystyle\mathcal{M}_{-0,--}^{h}\ $ $\displaystyle=\ 0$ (75)

The corresponding amplitude for $h_{W^{+}}=0$, $h_{W^{-}}=+$ can be found be
CP transforming the above expressions.

  * •

$(+0,++)$

$\displaystyle\mathcal{M}_{+0,++}^{\gamma}\ $ $\displaystyle=\
\frac{2}{3}\left(8\sqrt{2}G_{F}m_{t}m_{W}s_{w}^{2}\beta_{W}\right)d_{1,0}^{1}$
(76) $\displaystyle\mathcal{M}_{+0,++}^{Z}\ $ $\displaystyle=\ \frac{2}{3}\
\left[2\sqrt{2}G_{F}m_{t}m_{W}\left(\frac{3}{2}-4s_{w}^{2}\right)\frac{s}{s-M_{Z}^{2}}\beta_{W}\right]d_{1,0}^{1}$
(77) $\displaystyle\mathcal{M}_{+0,++}^{b}\ $ $\displaystyle=\
\frac{1}{\sqrt{2}}G_{F}m_{t}m_{W}\frac{s}{t}\left[\left(\beta_{t}\beta_{W}-\beta_{t}+\beta_{W}(1+\beta_{W})\right)d^{1}_{1,0}-\frac{2}{\sqrt{3}}\beta_{t}d^{2}_{1,0}\right]$
(78) $\displaystyle\mathcal{M}_{+0,++}^{h}\ $ $\displaystyle=\ 0$ (79)

  * •

$(+0,+-)$

$\displaystyle\mathcal{M}_{+0,+-}^{\gamma}\ $ $\displaystyle=\ -\frac{2}{3}\
\left[8G_{F}s_{w}^{2}m_{W}s^{1/2}\beta_{W}\right]d_{1,1}^{1}$ (80)
$\displaystyle\mathcal{M}_{+0,+-}^{Z}\ $ $\displaystyle=\ -\frac{2}{3}\
\left[2G_{F}m_{W}\left(\frac{3}{2}\left(1-\beta_{t}\right)-4s_{w}^{2}\right)\frac{s^{3/2}}{s-M_{Z}^{2}}\beta_{W}\right]d_{1,1}^{1}$
(81) $\displaystyle\mathcal{M}_{+0,+-}^{b}\ $ $\displaystyle=\
-\frac{1}{2}G_{F}m_{W}\frac{s^{3/2}}{t}(1-\beta_{t})\left[\left(\beta_{W}(1+\beta_{W})-\beta_{t}\beta_{W}\right)d_{1,1}^{1}-\beta_{t}d^{2}_{1,1}\right]$
(82) $\displaystyle\mathcal{M}_{+0,+-}^{h}\ $ $\displaystyle=\ 0$ (83)

  * •

$(+0,-+)$

$\displaystyle\mathcal{M}_{+0,-+}^{\gamma}\ $ $\displaystyle=\ \frac{2}{3}\
\left[8G_{F}s_{w}^{2}m_{W}s^{1/2}\beta_{W}\right]d_{1,-1}^{1}$ (84)
$\displaystyle\mathcal{M}_{+0,-+}^{Z}\ $ $\displaystyle=\ \frac{2}{3}\
\left[2G_{F}m_{W}\left(\frac{3}{2}\left(1+\beta_{t}\right)-4s_{w}^{2}\right)\frac{s^{3/2}}{s-M_{Z}^{2}}\beta_{W}\right]d_{1,-1}^{1}$
(85) $\displaystyle\mathcal{M}_{+0,-+}^{b}\ $ $\displaystyle=\
\frac{1}{2}G_{F}m_{W}\frac{s^{3/2}}{t}(1+\beta_{t})\left[\left(\beta_{W}(1+\beta_{W})+\beta_{t}\beta_{W}\right)d_{1,-1}^{1}-\beta_{t}d^{2}_{1,-1}\right]$
(86) $\displaystyle\mathcal{M}_{+0,-+}^{h}\ $ $\displaystyle=\ 0$ (87)

  * •

$(+0,--)$

$\displaystyle\mathcal{M}_{+0,--}^{\gamma}\ $ $\displaystyle=\ -\frac{2}{3}\
[8\sqrt{2}G_{F}m_{t}m_{W}s_{w}^{2}\beta_{W}]d_{1,0}^{1}$ (88)
$\displaystyle\mathcal{M}_{+0,--}^{Z}\ $ $\displaystyle=\ -\frac{2}{3}\
\left[2\sqrt{2}G_{F}m_{t}m_{W}\frac{s}{s-M_{Z}^{2}}\left(\frac{3}{2}-4s_{w}^{2}\right)\beta_{W}\right]d_{1,0}^{1}$
(89) $\displaystyle\mathcal{M}_{+0,--}^{b}\ $ $\displaystyle=\
-\frac{1}{\sqrt{2}}G_{F}m_{t}m_{W}\frac{s}{t}\left[\left(-\beta_{t}\beta_{W}+\beta_{t}+\beta_{W}(1+\beta_{W})\right)d^{1}_{1,0}-\frac{2}{\sqrt{3}}\beta_{t}d^{2}_{1,0}\right]$
(90) $\displaystyle\mathcal{M}_{+0,--}^{h}\ $ $\displaystyle=\ 0$ (91)

Amplitude for $h_{W^{+}}=0$, $h_{W^{-}}=-$ are related by CP transformation.

  * •

$(++,++)$

$\displaystyle\mathcal{M}_{++,++}^{\gamma}\ $ $\displaystyle=\ -\frac{2}{3}\
\left[8\sqrt{2}G_{F}m_{t}m_{W}^{2}s_{w}^{2}s^{-1/2}\beta_{W}\right]d_{0,0}^{1}$
(92) $\displaystyle\mathcal{M}_{++,++}^{Z}\ $ $\displaystyle=\ -\frac{2}{3}\
\left[2\sqrt{2}G_{F}m_{t}m_{W}^{2}\left(\frac{3}{2}-4s_{w}^{2}\right)\frac{s^{1/2}}{s-M_{Z}^{2}}\beta_{W}\right]d_{0,0}^{1}$
(93) $\displaystyle\mathcal{M}_{++,++}^{b}\ $ $\displaystyle=\
2\sqrt{2}G_{F}m_{t}m_{W}^{2}\frac{s^{1/2}}{t}\left[\frac{1}{6}\left(\beta_{t}-3\beta_{W}\right)+\frac{d_{0,0}^{1}}{2}(\beta_{t}-\beta_{W})+\frac{d_{0,0}^{2}}{3}\beta_{t}\right]$
(94) $\displaystyle\mathcal{M}_{++,++}^{h}\ $ $\displaystyle=\
2G_{F}\beta_{t}y_{t}vm_{W}^{2}\frac{\sqrt{s}}{s-m_{H}^{2}}$ (95)

  * •

$(++,+-)$

$\displaystyle\mathcal{M}_{++,+-}^{\gamma}\ $ $\displaystyle=\ \frac{2}{3}\
\left[8G_{F}m_{W}^{2}s_{w}^{2}\beta_{W}\right]d_{0,1}^{1}$ (96)
$\displaystyle\mathcal{M}_{++,+-}^{Z}\ $ $\displaystyle=\ \frac{2}{3}\
\left[2G_{F}m_{W}^{2}\left(\frac{3}{2}\left(1-\beta_{t}\right)-4s_{w}^{2}\right)\frac{s}{s-M_{Z}^{2}}\beta_{W}\right]d_{0,1}^{1}$
(97) $\displaystyle\mathcal{M}_{++,+-}^{b}\ $ $\displaystyle=\
G_{F}m_{W}^{2}\frac{s}{t}(1-\beta_{t})\left[\beta_{W}d_{0,1}^{1}-\sqrt{\frac{1}{3}}\beta_{t}d_{0,1}^{2}\right]$
(98) $\displaystyle\mathcal{M}_{++,+-}^{h}\ $ $\displaystyle=\ 0$ (99)

  * •

$(++,-+)$

$\displaystyle\mathcal{M}_{++,-+}^{\gamma}\ $ $\displaystyle=\ -\frac{2}{3}\
\left[8G_{F}m_{W}^{2}s_{w}^{2}\beta_{W}\right]d_{0,-1}^{1}$ (100)
$\displaystyle\mathcal{M}_{++,-+}^{Z}\ $ $\displaystyle=\ -\frac{2}{3}\
\left[2G_{F}m_{W}^{2}\left(\frac{3}{2}\left(1+\beta_{t}\right)-4s_{w}^{2}\right)\frac{s}{s-M_{Z}^{2}}\beta_{W}\right]d_{0,-1}^{1}$
(101) $\displaystyle\mathcal{M}_{++,-+}^{b}\ $ $\displaystyle=\
-G_{F}m_{W}^{2}\frac{s}{t}\left(1+\beta_{t}\right)\left[\beta_{W}d_{0,1}^{1}-\sqrt{\frac{1}{3}}\beta_{t}d_{0,1}^{2}\right]$
(102) $\displaystyle\mathcal{M}_{++,-+}^{h}\ $ $\displaystyle=\ 0$ (103)

  * •

$(++,--)$

$\displaystyle\mathcal{M}_{++,--}^{\gamma}\ $ $\displaystyle=\ \frac{2}{3}\
\left[8\sqrt{2}G_{F}m_{t}m_{W}^{2}s_{w}^{2}s^{-1/2}\beta_{W}\right]d_{0,0}^{1}$
(104) $\displaystyle\mathcal{M}_{++,--}^{Z}\ $ $\displaystyle=\ \frac{2}{3}\
\left[2\sqrt{2}G_{F}m_{t}m_{W}^{2}\left(\frac{3}{2}-4s_{w}^{2}\right)\frac{s^{1/2}}{s-M_{Z}^{2}}\beta_{W}\right]d_{0,0}^{1}$
(105) $\displaystyle\mathcal{M}_{++,--}^{b}\ $ $\displaystyle=\
-2\sqrt{2}G_{F}m_{t}m_{W}^{2}\frac{s^{1/2}}{t}\left[\frac{1}{6}\left(\beta_{t}+3\beta_{W}\right)-\frac{d_{0,0}^{1}}{2}(\beta_{t}+\beta_{W})+\frac{d_{0,0}^{2}}{3}\beta_{t}\right]$
(106) $\displaystyle\mathcal{M}_{++,--}^{h}\ $ $\displaystyle=\
-{2}G_{F}\beta_{t}y_{t}vm_{W}^{2}\frac{\sqrt{s}}{s-m_{H}^{2}}$ (107)

Amplitude for $h_{W^{+}}=-$, $h_{W^{-}}=-$ are related by CP transformation.

  * •

$(+-,++)$

$\displaystyle\mathcal{M}_{+-,++}^{\gamma}\ $ $\displaystyle=\ 0$ (108)
$\displaystyle\mathcal{M}_{+-,++}^{Z}\ $ $\displaystyle=\ 0$ (109)
$\displaystyle\mathcal{M}_{+-,++}^{b}\ $ $\displaystyle=\
\frac{4}{\sqrt{3}}G_{F}m_{t}m_{W}^{2}\frac{s^{1/2}}{t}\beta_{t}d_{2,0}^{2}$
(110) $\displaystyle\mathcal{M}_{+-,++}^{h}\ $ $\displaystyle=\ 0$ (111)

  * •

$(+-,+-)$

$\displaystyle\mathcal{M}_{+-,+-}^{\gamma}\ $ $\displaystyle=\ 0$ (112)
$\displaystyle\mathcal{M}_{+-,+-}^{Z}\ $ $\displaystyle=\ 0$ (113)
$\displaystyle\mathcal{M}_{+-,+-}^{b}\ $ $\displaystyle=\
-\sqrt{2}G_{F}m_{W}^{2}\frac{s}{t}\beta_{t}(1-\beta_{t})d_{2,1}^{2}$ (114)
$\displaystyle\mathcal{M}_{+-,+-}^{h}\ $ $\displaystyle=\ 0$ (115)

  * •

$(+-,-+)$

$\displaystyle\mathcal{M}_{+-,-+}^{\gamma}\ $ $\displaystyle=\ 0$ (116)
$\displaystyle\mathcal{M}_{+-,-+}^{Z}\ $ $\displaystyle=\ 0$ (117)
$\displaystyle\mathcal{M}_{+-,-+}^{b}\ $ $\displaystyle=\
\sqrt{2}G_{F}m_{W}^{2}\frac{s}{t}\beta_{t}(1+\beta_{t})d_{2,-1}^{2}$ (118)
$\displaystyle\mathcal{M}_{+-,-+}^{h}\ $ $\displaystyle=\ 0$ (119)

  * •

$(+-,--)$

$\displaystyle\mathcal{M}_{+-,--}^{\gamma}\ $ $\displaystyle=\ 0$ (120)
$\displaystyle\mathcal{M}_{+-,--}^{Z}\ $ $\displaystyle=\ 0$ (121)
$\displaystyle\mathcal{M}_{+-,--}^{b}\ $ $\displaystyle=\
-\frac{4}{\sqrt{3}}G_{F}m_{t}m_{W}^{2}\frac{s^{1/2}}{t}\beta_{t}d_{2,0}^{2}$
(122) $\displaystyle\mathcal{M}_{+-,--}^{h}\ $ $\displaystyle=\ 0$ (123)

Amplitude for $h_{W^{+}}=-$, $h_{W^{-}}=+$ are related by CP transformation.

##  Appendix B Signal vs Background

In this section, we show the dependence of the significance on the angular
variable $\cos\theta$ and the top quark invariant mass $m_{t\bar{t}}$ in the
center-of-mass frame for both the 3 TeV and 10 TeV colliders respectively.
Polar angle $\theta$ is defined in the partonic frame and is the angle between
$t$-quark and $W^{+}$ boson.

Figure 13:  The significance for 3 TeV (left) and 10 TeV (right) MuC at
various ranges of $m_{t\bar{t}}$ as a function of polar angle. We chose
$\delta_{yt}=0.1$ for 3 TeV collider and $\delta_{yt}=0.015$ for 10 TeV
collider which is near the one sigma C.L. limit for each collider.

We show angular dependence of signal significance for a $\sqrt{s}$ = 10 TeV
and $\mathcal{L}_{int}$ = 10 $ab^{-1}$ collider and for a $\sqrt{s}$ = 3 TeV
and $\mathcal{L}_{int}$ = 1 $ab^{-1}$ collider in Fig. 13 for various
$m_{t\bar{t}}$ ranges using 50 GeV bins 555As different lines correspond to
different sizes of the $m_{t\bar{t}}$ energy, we choose to represent the
result in such a normalization.. In this figure, background $B$ refers to the
prediction from SM, and signal $S$ refers to the deviation from SM value due
to $\delta_{yt}$. The fact that the significance primarily comes from the
interference term is explicit here by noticing higher significance near the
threshold. In both figures, one can notice that the sign of significance near
the forward region is opposite to the rest for lower energy ranges. This
pattern can also be observed in Fig. 6. The origin of this change of sign in
the angular distribution can be attributed to the change of sign for the
helicity amplitude $\mathcal{M}_{00,++}$ and $\mathcal{M}_{00,--}$ at
$\cos\theta\approx 0.7$ and $\sqrt{\hat{s}}=400$ GeV. Since the signal is
mostly coming from the interference between the above two channels with the
contribution from the Higgs exchange (which has a flat angular dependence),
the signal follows the shape of $\mathcal{M}_{00,++}$, $\mathcal{M}_{00,--}$
and this can explain why we observe the pattern in Fig. 13. At higher
energies, we will have the contribution from quadratic $\delta_{yt}^{2}$ term,
which is always positive, and so the pattern is not explicit in this region.
Another point to note here is that the significance mainly comes from the
central bins in angular distribution, so angle cuts in the forward region do
not significantly affect our $\Delta\chi^{2}$ analysis.

##  Appendix C Parameter Constraints on VLQ

The VLQ model modifies the couplings between the top quark and the gauge
bosons. The modified Higgs-top coupling would make the precision Higgs
observable deviate from the SM predictions, such as the Higgs decay modes to
diphoton and di-gluon. In SM, the Higgs boson decay into diphoton occurs via
the $W$ boson loop and charged fermion loop [56, 57, 58], and Higgs decay to
di-gluon receives contributions from quark loops [56, 59]. Here we focus on
the $h\rightarrow gg$ channel since it is more sensitive to the shift
$s_{L}^{2}$ and only considers the $t$ and $T$ loop, since the heavy quark
dominates this loop-induced coupling. In the VLQ model, we need to consider
the $T$-loop contribution and the shift in Yukawa couplings for the quark
loops. Hence in total, the amplitude of Higgs to di-gluon mode before QCD
correction is

$\mathcal{M}_{q}^{\lambda\lambda^{\prime}ab}=\left[(1-s_{L}^{2})F_{f}(\beta_{t})+s_{L}^{2}F_{f}(\beta_{T})\right]\dfrac{\alpha_{s}g\left(k_{1}\cdot
k_{2}g^{\mu\nu}-k_{1}^{\mu}k_{2}^{\nu}\right)\frac{1}{2}\delta^{ab}\epsilon_{\mu}(k_{1})^{\lambda}\epsilon_{\nu}(k_{2})^{\lambda^{\prime}}}{2\pi
m_{W}}$ (124)

where,

$\beta_{t}=\dfrac{4m_{t}^{2}}{m_{h}^{2}},\quad\beta_{T}=\dfrac{4m_{T}^{2}}{m_{h}^{2}},$
(125)

$\begin{split}F_{f}(\beta)&=-2\beta\left[1+(1-\beta)f(\beta)\right],\end{split}$
(126)

and

$f(\beta)=\arcsin^{2}\left(\beta^{-\frac{1}{2}}\right)\quad\textrm{for}\quad\beta\geq
1.$ (127)

Since the new physics does not change the kinematics, we can compute the
excess squared amplitude and constrain the new physics parameters. Using
$\delta\Gamma/\Gamma=\delta|\mathcal{M}|^{2}/|\mathcal{M}_{\rm SM}|^{2}$ and
inserting the projected precision for the Higgs decay width to di-gluon at
future colliders, we can derive the corresponding allowed region in the
parameter plane of the VLQ model under consideration.

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