aainstitutetext: Tsung-Dao Lee Institute and School of Physics and Astronomy,
Shanghai Jiao Tong University, 800 Dongchuan Road, Shanghai, 200240
Chinabbinstitutetext: Shanghai Key Laboratory for Particle Physics and
Cosmology, Key Laboratory for Particle Astrophysics and Cosmology (MOE),
Shanghai Jiao Tong University, Shanghai 200240, Chinaccinstitutetext: Amherst
Center for Fundamental Interactions, Department of Physics, University of
Massachusetts, Amherst, MA 01003ddinstitutetext: Kellogg Radiation Laboratory,
California Institute of Technology, Pasadena, CA 91125 USAeeinstitutetext: CAS
Key Laboratory of Theoretical Physics, Institute of Theoretical Physics,
Chinese Academy of Sciences, Beijing 100190, P. R. Chinaffinstitutetext:
School of Physical Sciences, University of Chinese Academy of Sciences,
Beijing 100049, P.R. Chinagginstitutetext: Center for High Energy Physics,
Peking University, Beijing 100871, Chinahhinstitutetext: School of Fundamental
Physics and Mathematical Sciences, Hangzhou Institute for Advanced Study,
UCAS, Hangzhou 310024, Chinaiiinstitutetext: International Centre for
Theoretical Physics Asia-Pacific, Beijing/Hangzhou, China

# Probing Extended Scalar Sectors with Precision $e^{+}e^{-}\to Zh$ and Higgs
Diphoton Studies

Michael J. Ramsey-Musolf e, f, g, h, i Jiang-Hao Yu c Jia Zhou
<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

We compute the one-loop corrections to $\sigma(e^{+}e^{-}\to Zh)$ arising from
representative extended Standard Model scalar sector scenarios. According to
the new scalar $\mathrm{SU\left(2\right)_{L}}$ representations, we consider
the inert doublet, real and complex triplet, quintuplet, and septuplet models.
With the sub-percent level precision expected for prospective future $e+e-$
collider measurements of $\sigma(e^{+}e^{-}\to Zh)$, studies of the
Higgsstrahlung process will probe extended scalar sector particle spectrum and
interactions in a manner complementary to direct searches at the Large Hadron
Collider and possible future $pp$ colliders. We also compare with the
sensitivity of future Higgs diphoton decay rate measurements. We find that the
$\sigma(e^{+}e^{-}\to Zh)$ and $\Gamma(h\to\gamma\gamma)$ complementarity is
particularly pronounced for the complex triplet model.

††preprint: ACFI-T21-01

##  1 Introduction

The discovery of the Higgs boson at the Large Hadron Collider (LHC) in 2012
was an important milestone in high energy physics, with completing the
particle spectra and validating the Higgs mechanism in the Standard Model
(SM). It is known that there must be new physics beyond the SM (BSM), for
example, to account for dark matter and matter-antimatter asymmetry in the
universe. Given that the Higgs boson plays essential roles in various BSM
scenarios, it is natural to expect BSM will influence properties of the Higgs
boson, and thus lead to deviations of the Higgs properties from the SM
predictions.

After the second run of the LHC, several of the Higgs couplings have been
measured and found to agree with the SM predictions within about 15% accuracy
ATLAS:2019slw ; Sirunyan:2018koj . At the high luminosity LHC, measurements of
the Higgs couplings are expected to be improved to around 5 - 10% precision
level Cepeda:2019klc ; deBlas:2019rxi ; Abada:2019lih , which can yield BSM
sensitivity comparable to that of direct searches for BSM at the LHC. Thus to
probe BSM significantly beyond the LHC direct searches one would require the
precision measurements of the Higgs boson couplings at percent-level accuracy.
Achieving such precision would require new facilities like the lepton
collider, since the lepton collider has the advantage of clean signatures and
high statistics samples of the Higgs boson, i.e., a Higgs factory. Several
proposals of such Higgs factories have been made, including the Circular
Electron Positron Collider (CEPC) in China, the International Linear Collider
(ILC) in Japan, and the Future Circular Collider with $e^{+}e^{-}$ (FCC-ee)
and the Compact Linear Collider (CLIC) at CERN.

The CEPC, ILC, and FCC-ee plan to operate their (first) runs at a center of
mass energy of around 240 - 250 GeV. With this energy, the dominant Higgs
production channel is the Higgs-strahlung process $e^{+}e^{-}\to Zh$. These
Higgs factories (with $\sqrt{s}=$240 GeV for the CEPC and FCC-ee
$\sqrt{s}=$250 GeV for the ILC) expect to obtain similar integrated
luminosities and should provide similar physics sensitivities, assuming
adoption of optimized detectors and analyses CEPCStudyGroup:2018ghi ;
Abada:2019zxq ; Barklow:2015tja ; Fujii:2017vwa ; Fujii:2019zll . At the CEPC,
with the expected integrated luminosity of 5.6 ab-1, over one million Higgs
boson are expected to be produced. With such number of events, the CEPC will
be able to measure the Higgs boson couplings to the $Z$ boson pair with an
accuracy of $0.25$% TheATLAScollaboration:2014ewu ; CMS:2013xfa ;
CEPCStudyGroup:2018ghi , more than 10 times better than the expected accuracy
at the HL-LHC. Such a precision measurement offers unprecedented discovery
potential for BSM connected with the Higgs boson.

Extended scalar sectors constitute a well-motivated class of such BSM
scenarios. According to the $\mathrm{SU\left(2\right)_{L}}$ quantum number of
a given representation, the extended scalar could be classified as real or
complex singlet, doublet, real or complex triplet, quadruplet, and so on.
These new scalars usually affect the electroweak symmetry breaking pattern and
in principle may catalyze a first order electroweak phase transition (EWPT)
Profumo:2007wc ; Espinosa:2011ax ; Niemi:2018asa ; Niemi:2020hto ;
Barger:2007im ; Barger:2008jx ; Papaefstathiou:2020iag . Furthermore, in some
cases these scalars could provide the dark matter candidates Cirelli:2005uq ;
McDonald:1993ex ; Burgess:2000yq ; Diaz:2015pyv ; Ma:2006km ; Barbieri:2006dq
; Banerjee:2016vrp ; FileviezPerez:2008bj ; Araki:2011hm ; Hambye:2009pw ;
AbdusSalam:2013eya ; Chao:2018xwz  or mechanism for neutrino mass generation
Konetschny:1977bn ; Magg:1980ut ; Schechter:1980gr ; Cheng:1980qt . In this
work, we consider the following specific cases: inert doublet, real triplet,
complex triplet, quintuplet and septuplet. These new scalars are expected to
be discovered at the future HL-LHC runs, which has been extensively studied in
various literature Diaz:2015pyv ; Banerjee:2016vrp ; FileviezPerez:2008bj ;
Blank:1997qa ; Chen:2008jg ; Konetschny:1977bn ; Magg:1980ut ;
Schechter:1980gr ; Cheng:1980qt ; Du:2018eaw ; Chao:2018xwz .

In these extended scalar models, the Higgs portal term of the generic form
$|\Phi|^{2}|H|^{2}$ usually exists. Its presence could allow for a first order
EWPT. In this case the scalar potential in the extended scalar sector is
directly relevant to the EWPT. We expect the $e^{+}e^{-}\to Zh$ process could
help determine important aspects of the extended scalar potential, which is in
general quite difficult to measure at the HL-LHC. These new scalars modify the
Higgs couplings to the Z pair through radiative correction with new scalars
running in the loop. These radiative corrections depend on the new scalar mass
spectra, the gauge interactions of the new scalars, and the scalar Higgs
couplings in the scalar potential. Given the fact that gauge interactions are
determined by gauge invariance, one could extract information on the scalar
Higgs couplings by measuring the total rate and distributions of the
$e^{+}e^{-}\to Zh$ process, once the new scalar bosons are discovered and the
scalar mass spectra are determined.

On the other hand, it is possible the HL-LHC may yield no direct evidence for
new scalars. Nevertheless, it is possible that deviations of the Higgs boson
properties from the SM predictions could well provide the first evidence for
BSM. Even in this case, the $e^{+}e^{-}\to Zh$ process could help us
indirectly probe the scalar sector, and thus tell us the information on the
scalar mass spectrum.

In this context, it is important to emphasize that the Higgs diphoton process
$h\gamma\gamma$ provides a theoretically clean, complementary, indirect probe
of an extended scalar sector at colliders. The target of the future lepton
collider - CEPC & FCC-ee - precision at this channel is a few percent (refer
to Table 1) CEPCStudyGroup:2018ghi ; Abada:2019zxq . Unlike the $e^{+}e^{-}\to
Zh$ process, only charged scalars from the extended scalar sector contribute
to $h\gamma\gamma$ via triangle loops. In this context one expects
complementarity on the parameter space of the Higgs portal couplings between
the $e^{+}e^{-}\to Zh$ and Higgs diphoton processes. We explore this
complementarity for each model scenario under consideration in this paper. We
find that this complementarity is most pronounced for the complex triplet
model, whereas it is less evident for the other scenarios considered here.

In what follows, we systematically classify the scalar sector according to
their $\mathrm{SU\left(2\right)_{L}}$ quantum number, and obtain the form of
the Higgs potential. Then we perform one-loop calculation of the
$e^{+}e^{-}\to Zh$ process and separate the BSM contribution from the SM one
in the $\overline{\mathrm{MS}}$ and the On-shell schemes. Calculations of the
complete SM one-loop corrections are reported in Refs. Fleischer:1982af ;
Kniehl:1991hk ; Denner:1992bc  111The complete SM one-loop corrections consist
of the weak and QED corrections, which are individually gauge invariant. In
Refs. Fleischer:1982af ; Kniehl:1991hk ; Denner:1992bc , the real photon
emission - photon initial-state radiation (ISR) - in the QED correction was
approximated with soft photon bremsstrahlung formalism. The emitted photon is
deemed as soft if its energy $E_{\gamma}$ in the rest frame satisfies
$0<E_{\gamma}<\delta_{s}\frac{\sqrt{s}}{2}$, where $\delta_{s}$ is the soft
cutoff parameter. $\delta_{s}\sim 0.1$ was used in Fleischer:1982af ;
Denner:1992bc . Ref. Kniehl:1991hk  also approximated the hard photon
contribution as a convolution of the reduced cross section with a photon
radiator in the hard/collinear region, so that the sum of the soft and hard
photon contribution was free of the choice of $\delta_{s}$. In principle, the
hard/non-collinear region should also be included even though its contribution
may not be numerically significant.. At LEP200 energies the weak correction in
the $\alpha$-parameterization 222The weak relative correction in the
$G_{\mu}$-parameterization $\delta_{\text{weak}}^{G_{\mu}}$ differs from the
one in the $\alpha$-parameterization $\delta_{\text{weak}}^{\alpha}$ by a
constant shift $\Delta r$ that
$\delta_{\text{weak}}^{G_{\mu}}=\delta_{\text{weak}}^{\alpha}-\Delta r$, where
$\Delta r$ summarizes the one-loop electroweak corrections to muon decay
Sirlin:1980nh . has a positive sign and a magnitude of a few percent. The sign
of the correction becomes negative at higher energies Denner:1992bc . The full
one-loop radiative corrections to $e^{+}e^{-}\to Zh/H^{0}A^{0}$ processes in
the inert doublet model has been recently calculated Abouabid:2020eik . The
Higgs diphoton decay rate including the BSM contribution can be readily
obtained by means of the well known loop functions. We investigate the
projected sensitivities to the scalar mass splitting and scalar Higgs
couplings given the expected precision levels at the lepton colliders.

Our study is organized as follows. In Section 2 we discuss the generic form of
the multiplet ${\Phi}_{n}$ in representation $j$ and the corresponding scalar
potentials (explicit expressions are given in Appendix B). The NLO corrections
from the scalar multiplets to the process $e^{+}e^{-}\rightarrow Zh$ are
calculated in $\overline{\mathrm{MS}}$ scheme in Section 3. Numerical results
are presented in Section 4, where the constraints on the scalar potential
parameters are obtained from the comparison of the $Zh$ cross section
measurement to that of the Higgs diphoton decay rate assuming the expected
CEPC integrated luminosity. Finally we summarize the significance of the study
in Section 5. We also add Feynman rules for the models studied in Appendix A,
the expressions for self energy and vertex corrections in Appendix C, and
calculation in on-shell renormalization scheme used to compare with the that
in $\overline{\mathrm{MS}}$ in Appendix D.

##  2 Extended Scalar Sector

In an extended scalar sector, an additional scalar multiplet, either charged
or singlet under
$\mathrm{SU\left(2\right)_{L}}\times\mathrm{U\left(1\right)_{Y}}$, is
introduced. We could systematically classify the scalar multiplet based on its
transformation under the $\mathrm{SU\left(2\right)_{L}}$ group. Symbolically
one can write the scalar multiplet as ${\Phi}$ as opposed to SM Higgs doublet
written as ${H}$. For a $\mathrm{SU\left(2\right)_{L}}$ $n=2j+1$ multiplet
$\Phi$ following the representation in Ref. Chao:2018xwz ; Pilkington:2016erq
, using the notation $(j,j_{3})$ as the subscript of the field it is written
as

$\Phi=\left[\begin{array}[]{c}\phi_{j,j}\\\ \phi_{j,j-1}\\\ .\\\ .\\\
\phi_{j,-j}\\\ \end{array}\right]\,,\qquad{\textrm{with electric charge}}\quad
Q_{i}=j_{3}+\frac{Y}{2},$ (1)

where it has $2j+1$ component fields, each having electric charge $Q_{i}$. It
is convenient to consider the associated conjugate $\overline{\Phi}$ instead
of $\Phi^{\ast}$

${\overline{\Phi}}_{j,m}=(-1)^{j-m}\Phi^{\ast}_{j,-m}\ \ \ .$ (2)

Since $\Phi$ and ${\overline{\Phi}}$ transform in the same way under
$\mathrm{SU\left(2\right)_{L}}$, it is convenient to build the
$\mathrm{SU\left(2\right)_{L}}$ invariant using $\Phi$ and
${\overline{\Phi}}$. For integer isospin $j$, the scalar multiplet $\Phi$ can
be real or complex, while for half integer isospin $\Phi$ is always complex
for any value of hypercharge $Y$. If $\Phi$ is a real multiplet (integer
isospin), there is a redundancy $\Phi=\overline{\Phi}$ such that the realness
condition $\phi_{j,m}=(-1)^{j-m}\phi^{*}_{j,-m}$ is satisfied. If $\Phi$ is a
complex multiplet (all kinds of isospin), each component represents one degree
of freedom field, and it can be decomposed into two real multiplets as follows

$\Phi=\frac{1}{\sqrt{2}}\left(S+iA\right),\qquad\overline{\Phi}=\frac{1}{\sqrt{2}}\left(S-iA\right)\,,$
(3)

where both $S$ and $A$ satisfy the realness condition $S=\overline{S}$ and
$A=\overline{A}$. Thus the scalar extension with a complex multiplet $\Phi$ is
equivalent to a model of two real multiplets $S$ and $A$.

For a general $2j+1$ multiplet $\Phi$ with the hypercharge $Y$, the kinetic
term reads

$\mathcal{L}_{\mathrm{kin}}=\left(D_{\mu}{H}\right)^{\dagger}\left(D_{\mu}{H}\right)+\left(D_{\mu}{\Phi}\right)^{\dagger}\left(D^{\mu}{\Phi}\right).$
(4)

The covariant derivative for the multiplet can be written in a general form

$D_{\mu}=\partial_{\mu}+ig_{2}W_{\mu}^{a}T^{a}+ig_{1}\frac{Y}{2}B_{\mu},\qquad\left(a=1,2,3\right)$
(5)

where $T^{a}$ are the $\mathrm{SU\left(2\right)_{L}}$ generators. It is
convenient to define the raising and lowering generators and the charged $W$
bosons accordingly

$T^{\pm}=T^{1}\pm iT^{2},\qquad W_{\mu}^{\pm}=W_{\mu}^{1}\mp iW_{\mu}^{2}.$

The other two physical neutral bosons are obtained via the diagonalization in
such a way that

$\left[\begin{tabular}[]{c}$Z_{\mu}$\\\
$A_{\mu}$\end{tabular}\right]=\left[\begin{tabular}[]{cc}$c_{W}$&$-s_{W}$\\\
$s_{W}$&$c_{W}$\end{tabular}\right]\left[\begin{tabular}[]{c}$W_{\mu}^{3}$\\\
$B_{\mu}$\end{tabular}\right],$

with $s_{W},~{}c_{W}$ the sine and cosine of the SM weak mixing angle
$\theta_{W}$. Since in the representation $j$ the generator
$T^{a}~{}\left(a=\pm,3\right)$ is a $\left(2j+1\right)\times\left(2j+1\right)$
matrix333See the generic form of the generators $T^{a}$ in Appendix. A, which
adopts the representation in Ref. Chao:2018xwz ., the
$\left(n,n^{\prime}\right)$ element of the last two terms in Eq. (5) reads

$\displaystyle\left(g_{2}W_{\mu}^{a}T^{a}+g_{1}\frac{Y}{2}B_{\mu}\right)_{n,n^{\prime}}$
$\displaystyle=$
$\displaystyle\left(g_{2}W_{\mu}^{+}T^{+}+g_{2}W_{\mu}^{-}T^{-}+g_{2}W_{\mu}^{3}T^{3}+g_{1}\frac{Y}{2}B_{\mu}\right)_{n,n^{\prime}}$
(6) $\displaystyle=$
$\displaystyle\left(g_{1}\frac{Y}{2}B_{\mu}-g_{2}W_{\mu}^{3}m\right)\delta_{n^{\prime},n}+g_{2}W_{\mu}^{+}\sqrt{\left(j-m\right)\left(j+m+1\right)}\delta_{n^{\prime},n+1}$
$\displaystyle+g_{2}W_{\mu}^{-}\sqrt{\left(j+m\right)\left(j-m+1\right)}\delta_{n^{\prime},n-1},$

where $n=j+m+1$ and $m=-j,\ldots,j$. In particular, the hypercharge $Y$ takes
$1$ for doublet, $2$ for complex triplet, and $0$ for real triplet, quintuplet
and septuplet, respectively. Therefore, the gauge Feynman rule are quite
generic for scalar multiplets, see Appendix A.

Although the kinetic Lagrangian is quite generic for a scalar multiplet, the
scalar potential is representation dependent. To construct the general scalar
potential, one could proceed Chao:2018xwz ; Pilkington:2016erq  to build
$\mathrm{SU\left(2\right)_{L}}$ invariants by first pairing the field
ingredients into irreducible representations and finally into
$\mathrm{SU\left(2\right)_{L}}$ invariants. The pairing of the fields has the
possibility of

$\displaystyle{\textrm{Pairing: }}\
\overline{H}H,\quad\Phi\Phi,\quad\overline{\Phi}\Phi,\quad\overline{\Phi}~{}\overline{\Phi}.$
(7)

For half integer isospin multiplet, additional pairing
$HH,H\Phi,\overline{H}~{}\overline{\Phi}$ appears with $Y=1,-3$, and similarly
$HH,\overline{H}\Phi,H\overline{\Phi}$ for $Y=-1,+3$. From these building
blocks, the $\mathrm{SU\left(2\right)_{L}}$ invariants can be constructed. For
any multiplet, it must contain the following universal terms

$\displaystyle\left[\left(\Phi{\Phi}\right)_{J}\left({\overline{\Phi}}\,{\overline{\Phi}}\right)_{J}\right]_{0},\quad\left({\overline{H}}H\right)_{0}\left({\overline{\Phi}}\Phi\right)_{0},\quad\left[\left({\overline{H}}H\right)_{1}\left({\overline{\Phi}}\Phi\right)_{1}\right]_{0}.$
(8)

In the case of $Y=0$, there are additional terms

$\displaystyle\left[\left({\overline{H}}H\right)_{1}\left({\Phi}\Phi\right)_{1}\right]_{0},\
\ {\textrm{for $j$ = half
integer}};\quad\left({\overline{H}}H\right)_{0}\left({\Phi}\Phi\right)_{0},\ \
{\textrm{for $j$ = integer}};$ (9)

and the scalar self interactions

$\displaystyle\left[\left(\Phi{\Phi}\right)_{J}\left({\Phi}\,{\Phi}\right)_{J}\right]_{0},\quad\left[\left({\overline{\Phi}}{\Phi}\right)_{J}\left({\Phi}\,{\Phi}\right)_{J}\right]_{0}.$
(10)

For other special multiplet, there could have additional invariant terms. The
potentials in the specific models are given in Appendix B.

##  3 NLO Calculation in $\overline{\mathrm{MS}}$ Scheme

The LO process $e^{+}e^{-}\rightarrow Zh$ is depicted in Fig. 1.

Figure 1: LO process: $e^{-}(p_{1})~{}+~{}e^{+}(p_{2})\rightarrow
Z(k_{1})~{}+~{}h(k_{2})$

The expression for the matrix element squared has the following form:

$\left|{\bf M}_{e^{+}e^{-}\rightarrow
Zh}^{\mathrm{LO}}\right|^{2}=\frac{2e^{4}\left(g_{v}^{2}+g_{a}^{2}\right)}{s_{W}^{2}c_{W}^{2}}\cdot\frac{tu+2sM_{Z}^{2}-M_{h}^{2}M_{Z}^{2}}{\left(s-M_{Z}^{2}\right)^{2}+M_{Z}^{2}\Gamma_{Z}^{2}},$
(11)

where $s_{W}=\sin\theta_{W},~{}c_{W}=\cos\theta_{W}$ with $\theta_{W}$ the
weak mixing angle, and

$g_{v}=\frac{-1/2+2s_{W}^{2}}{2s_{W}c_{W}},~{}g_{a}=-\frac{1}{4s_{W}c_{W}}$

are the vector and axial-vector couplings to to the $e^{+}e^{-}Z$ vertex,
respectively. The Mandelstam variables are defined as

$\left(p_{1}+p_{2}\right)^{2}=\left(k_{1}+k_{2}\right)^{2}=s,\,\left(p_{1}-k_{1}\right)^{2}=\left(p_{2}-k_{2}\right)^{2}=t,\,\left(p_{1}-k_{2}\right)^{2}=\left(p_{2}-k_{1}\right)^{2}=u,$

and the relation $s+t+u=M_{Z}^{2}+M_{h}^{2}$. The total cross section reads

$\sigma=\frac{1}{4}\frac{1}{48\pi
s^{2}}\frac{e^{4}\left(g_{v}^{2}+g_{a}^{2}\right)}{s_{W}^{2}c_{W}^{2}}\cdot\frac{\left[\kappa\left(s,M_{Z}^{2},M_{h}^{2}\right)+12sM_{Z}^{2}\right]\kappa\left(s,M_{Z}^{2},M_{h}^{2}\right)}{\left(s-M_{Z}^{2}\right)^{2}+M_{Z}^{2}\Gamma_{Z}^{2}},$
(12)

where the front $\frac{1}{4}$ is the spin average over the initial states, and
the function $\kappa$ takes the form:

$\kappa\left(x,y,z\right)=\sqrt{x^{2}+y^{2}+z^{2}-2xy-2xz-2yz}.$

In general one loop corrections to the $e^{+}e^{-}\rightarrow Zh$ process
shown in Fig. 2 consist of vertex corrections to the initial and finial
vertex, self energy corrections to the mediated propagator, and box
corrections with virtual particles in loop in contact with both initial and
final states, respectively. The interaction between the extended scalars and
initial fermions is negligible due to its suppressed Yukawa couplings of
$\mathcal{O}\left(\frac{m_{f}}{M_{W}}\right)$. Therefore, in what follows, the
NLO contribution originated from extended scalar loop corrections to the $Zh$
production consists of self energy and vertex corrections to the vector and
scalar bosons, as indicated in Fig. 2 for the diagrams with hatching blobs.

Figure 2: General one loop contributions to $e^{+}e^{-}\rightarrow Zh$
amplitude. The blobs denote one loop irreducible diagrams, and these hatched
indicate where the extended scalar loop corrections can take place.

Loop corrections alone are often UV divergent and need be renormalized in
certain renormalization scheme. We use dimensional regularization in
$d=4-2\epsilon$ spacetime dimensions and the modified minimal subtraction
($\overline{\mathrm{MS}}$) scheme to perform the calculation. Here, we use the
terms with caret $~{}\hat{}~{}$ denote the renormalized quantities, which are
the bare quantities subtracting the part proportional to
$\frac{1}{\bar{\epsilon}}=\frac{1}{\epsilon}-\gamma_{E}+\log 4\pi$ by
definition. In this scheme, the one loop amplitude can be written as below

$\displaystyle i{\bf M}_{e^{+}e^{-}\rightarrow Zh}^{\mathrm{NLO}}$
$\displaystyle=i{\bf M}_{e^{+}e^{-}\rightarrow Zh}^{\mathrm{tree}}+i{\bf
M}_{e^{+}e^{-}\rightarrow Zh}^{\mathrm{self}}+i{\bf M}_{e^{+}e^{-}\rightarrow
Zh}^{\mathrm{vert}},$ (13)
$\displaystyle=-i\frac{\hat{e}^{2}\hat{M}_{Z}}{\hat{s}\hat{c}}\hat{\rho}_{NC}\left(s\right)\bar{v}\left(p_{2}\right)\gamma^{\mu}\left(g_{v}^{eff}-g_{a}^{eff}\gamma_{5}\right)u\left(p_{1}\right)\epsilon_{\mu}\left(k_{1}\right)+i{\bf
M}_{Z^{*}\rightarrow Zh}^{\mathrm{vert}}+i{\bf M}_{\gamma^{*}\rightarrow
Zh}^{\mathrm{vert}},$

where the last two terms on the right hand side denote the contribution of
$ZZh$ and $\gamma Zh$ vertex corrections, and $\hat{s},\hat{c}$ are sine and
cosine of the weak mixing angle at 1-loop level in $\overline{\mathrm{MS}}$
scheme and can be expressed in terms of the $\overline{\mathrm{MS}}$ masses
such that $\hat{c}^{2}=1-\hat{s}^{2}=\hat{M}_{W}^{2}/\hat{M}_{Z}^{2}$. The
effective vector and axial-vector coupling $g_{v}^{eff},g_{a}^{eff}$ have the
following form

$g_{v}^{eff}=\frac{I_{W,e}^{3}-2\hat{\kappa}\left(s\right)\hat{s}^{2}Q_{e}}{2\hat{s}\hat{c}},\quad
g_{a}^{eff}=\frac{I_{W,e}^{3}}{2\hat{s}\hat{c}},$ (14)

where the factor $\hat{\kappa}\left(s\right)$ modifying $\hat{s}^{2}$
originates from the $\gamma-Z$ mixing tensor and

$\hat{\kappa}\left(s\right)=1-\frac{\hat{c}}{\hat{s}}\frac{\hat{\Sigma}_{T}^{\gamma
Z}\left(s\right)}{s}.$

The factor $\hat{\rho}_{NC}\left(s\right)$ takes the form

$\displaystyle\hat{\rho}_{NC}\left(s\right)=$
$\displaystyle\frac{1}{s-\hat{M}_{Z}^{2}+\hat{\Sigma}_{T}^{ZZ}\left(s\right)}\left(1+\frac{1}{2}\delta\hat{Z}_{ZZ}+\frac{1}{2}\delta\hat{Z}_{h}\right),$
(15) $\displaystyle=$
$\displaystyle\frac{1}{s-M_{Z}^{2}+i\Gamma_{Z}M_{Z}}\left(1+\frac{\mathrm{Re}\hat{\Sigma}_{T}^{ZZ}\left(M_{Z}^{2}\right)-\mathrm{Re}\hat{\Sigma}_{T}^{ZZ}\left(s\right)}{s-M_{Z}^{2}}+\frac{1}{2}\delta\hat{Z}_{ZZ}+\frac{1}{2}\delta\hat{Z}_{h}\right)$
$\displaystyle+\mathcal{O}\left(\alpha^{2},\frac{\Gamma_{Z}^{2}}{M_{Z}^{2}}\right)$

where we expand the expression in the second(third) line and omit terms of
$\mathcal{O}\left(\alpha^{2}\right)$ or
$\mathcal{O}\left(\frac{\Gamma_{Z}^{2}}{M_{Z}^{2}}\right)$ using

$\hat{M}_{Z}^{2}=M_{Z}^{2}+\mathrm{Re}\hat{\Sigma}_{T}^{ZZ}\left(M_{Z}^{2}\right),\quad\Gamma_{Z}M_{Z}=\mathrm{Im}\hat{\Sigma}_{T}^{ZZ}\left(s\right),$

and

$\delta\hat{Z}_{ZZ}=\hat{Z}_{ZZ}-1=-\mathrm{Re}\left.\frac{\partial\hat{\Sigma}_{T}^{ZZ}\left(k^{2}\right)}{\partial
k^{2}}\right|_{k^{2}=M_{Z}^{2}},\quad\delta\hat{Z}_{h}=\hat{Z}_{h}-1=-\mathrm{Re}\left.\frac{\partial\hat{\Sigma}^{h}\left(k^{2}\right)}{\partial
k^{2}}\right|_{k^{2}=M_{h}^{2}},$

with $\hat{Z}_{ZZ},~{}\hat{Z}_{h}$ the finite residues of $Z$ and $h$
propagators. The self energy functions in all scalar multiplet models that are
relevant in our calculation are listed in Appendix C.2.

We separate the vertex corrections in Eq. (13) because they cannot fully
factorize to Born (extra Lorentz structures other than that of the LO
amplitude). It should be noted that the vertex corrections alone are free of
UV divergences, and thus can be written individually apart from the Born-like
structure. Therefore, the total NLO corrections at matrix element square level
can be obtained as below

$\left|{\bf M}\right|_{\mathrm{tot,corr}}^{2}=\left|{\bf
M}_{e^{+}e^{-}\rightarrow Zh}^{\mathrm{NLO}}\right|^{2}-\left|{\bf
M}_{e^{+}e^{-}\rightarrow Zh}^{\mathrm{LO}}\right|^{2},$ (16)

where ${\bf M}_{e^{+}e^{-}\rightarrow Zh}^{\mathrm{LO}}$ is the amplitude at
the leading order. And the amplitude square of ${\bf M}_{e^{+}e^{-}\rightarrow
Zh}^{\mathrm{NLO}}$ is

$\left|{\bf M}_{e^{+}e^{-}\rightarrow
Zh}^{\mathrm{NLO}}\right|^{2}=\frac{\hat{e}^{4}\hat{M}_{Z}^{2}\left(g_{v,eff}^{2}+g_{a,eff}^{2}\right)}{M_{Z}^{2}\hat{s}^{2}\hat{c}^{2}}\left|\hat{\rho}_{NC}\right|^{2}\left(tu+2sM_{Z}^{2}-M_{h}^{2}M_{Z}^{2}\right)+\ldots,$
(17)

where the ellipsis denotes other contributions that are at NLO and NNLO level.
For the NLO contribution it is contained in the interference between the
vertex term and the first term in the last line of Eq. (13) and in particular
the exact NLO level contribution reads

$\displaystyle 2~{}\mathrm{Re}\left[\left({\bf M}_{Z^{*}\rightarrow
Zh}^{\mathrm{vert}}+{\bf M}_{\gamma^{*}\rightarrow
Zh}^{\mathrm{vert}}\right)\cdot\left({\bf M}_{e^{+}e^{-}\rightarrow
Zh}^{\mathrm{tree}}\right)^{*}\right]$ (18) $\displaystyle=$
$\displaystyle-\frac{1}{16\pi^{2}}\frac{4~{}e^{3}}{M_{Z}s_{W}c_{W}}\sum_{s,\left(M_{1},M_{2}\right)}\left(\frac{g_{ssh}g_{ssZ}^{2}\left(g_{v}^{2}+g_{a}^{2}\right)}{\left(s-M_{Z}^{2}\right)^{2}+M_{Z}^{2}\Gamma_{Z}^{2}}+\frac{g_{ssh}g_{ss\gamma}g_{ssZ}~{}g_{v}\left(s-M_{Z}^{2}\right)}{s\left[\left(s-M_{Z}^{2}\right)^{2}+M_{Z}^{2}\Gamma_{Z}^{2}\right]}\right)\times$
$\displaystyle
2\mathrm{Re}\Bigg{\\{}\left(tu+2sM_{Z}^{2}-M_{h}^{2}M_{Z}^{2}\right)\left(2C_{24s}\left(M_{1},M_{2}\right)-\frac{1}{2}B_{0}\left(k_{2},M_{2},M_{2}\right)\right)$
$\displaystyle+\left(s+M_{Z}^{2}-M_{h}^{2}\right)\left(M_{h}^{2}M_{Z}^{2}-tu\right)\left(C_{22s}\left(M_{1},M_{2}\right)-C_{23s}\left(M_{1},M_{2}\right)\right)\Bigg{\\}},$

where the expressions of $C_{22s}\left(M_{1},M_{2}\right)$,
$C_{23s}\left(M_{1},M_{2}\right)$ and $C_{24s}\left(M_{1},M_{2}\right)$ can be
found in Appendix C.3, and $\sum_{s,\left(M_{1},M_{2}\right)}$ denotes the sum
of all possible scalar couplings and corresponding pair of scalar masses
$\left(M_{1},M_{2}\right)$. $g_{ssh},~{}g_{ssZ},~{}g_{ss\gamma}$ are the
couplings of scalar-scalar-Higgs, scalar-scalar-$Z$ and scalar-scalar-photon,
respectively. The Feynman rules of the multiplets in the gauge coupling sector
can be found in Appendix A. The masses in the pair $\left(M_{1},M_{2}\right)$
are degenerate ($M_{1}=M_{2}$) in all scalar models for each induced scalar
triangle loop except for that in the inert doublet model which has the two
masses assigned differently.

##  4 Numerical Results in $\overline{\mathrm{MS}}$ Scheme

In this section, we present the contour comparison between relative
corrections to $Zh$ production and the Higgs diphoton ($h\gamma\gamma$) decay
rate induced by scalar multiplet loop contributions, with an effort to
delineate the prospective BSM parameter space constraints implied by each
process. The relative correction by the scalar-induced loop contribution to
the $e^{+}e^{-}\rightarrow Zh$ process is defined as follows,

$\delta\sigma_{Zh}=\frac{\sigma_{\mathrm{BSM}}^{\mathrm{1-loop}}}{\sigma_{\mathrm{SM}}^{\mathrm{LO}}},$
(19)

where $\sigma_{\mathrm{BSM}}^{\mathrm{1-loop}}$ is the total cross section
based on the matrix element square term in Eq. (16), and
$\sigma_{\mathrm{SM}}^{\mathrm{LO}}$ is the LO total cross section given in
Eq. (12). There are three commonly-used input-parameter schemes for the
calculation of EW radiative corrections: the $\alpha\left(0\right)$,
$\alpha\left(M_{Z}\right)$ and $G_{\mu}$ schemes, respectively. All are suited
for both the $\overline{\mathrm{MS}}$ and OS renormalization schemes. Each
scheme has its advantages and disadvantages. The most suitable choice depends
on the nature of process under consideration. In general, the
$\alpha\left(M_{Z}\right)$ and $G_{\mu}$ schemes render the SM EW corrections
free of explicit large logarithms involving light fermion masses and are,
therefore, preferable over the $\alpha\left(0\right)$ scheme if no external
photons are involved in the process. A detailed discussion of these input-
parameter schemes as well as the relation between one another can be found in
Ref. Denner:2019vbn . In the present context, the one-loop contribution from
the extended scalar sector particles in our study can be performed
independently of the overall SM EW contribution, and the result is manifestly
free from light fermion mass logarithms. Moreover, the impact on
$\sigma(e^{+}e^{-}\to Zh)$ of the renormalized extended scalar sector loop
contribution is less sensitive to the choice of input parameter scheme than is
the SM one-loop contribution. Hence, for simplicity we use the
$\alpha\left(0\right)$ input-parameter scheme and set the corresponding SM
input parameters as follows:

$\displaystyle\alpha^{-1}=\left(\frac{e^{2}}{4\pi}\right)^{-1}=137.036,$ (20)
$\displaystyle
M_{W}=80.385~{}\mathrm{GeV},~{}M_{Z}=91.1876~{}\mathrm{GeV},~{}\Gamma_{Z}=2.4952~{}\mathrm{GeV},~{}M_{h}=125.1~{}\mathrm{GeV}.$

At one-loop level in $\overline{\mathrm{MS}}$ scheme the vector boson masses
($M_{V},~{}\left(V=W,Z\right)$), mixing angle ($\hat{s},\hat{c}$) and the
electric charge ($\hat{e}$) are

$\displaystyle\hat{M}_{V}^{2}=M_{V}^{2}+\mathrm{Re}\hat{\Sigma}_{T}^{VV}\left(M_{V}^{2}\right),$
(21)
$\displaystyle\hat{c}^{2}=1-\hat{s}^{2}=\frac{\hat{M}_{W}^{2}}{\hat{M}_{Z}^{2}},$
$\displaystyle\hat{e}=e\left(1-\frac{1}{2}\delta\hat{Z}_{\gamma\gamma}-\frac{1}{2}\frac{\hat{s}}{\hat{c}}\delta\hat{Z}_{Z\gamma}\right),$

where the expressions of
$\delta\hat{Z}_{\gamma\gamma},~{}\delta\hat{Z}_{Z\gamma}$ can be found in Eq.
(84) and the self energy functions are given in Appendix C.2.

Turning now to the $h\to\gamma\gamma$ process, the relative correction by
scalar-induced loop contributions to the Higgs diphoton decay rate is defined
below

$\delta
R_{h\gamma\gamma}=\frac{\Gamma^{\mathrm{BSM}+\mathrm{SM}}_{h\rightarrow\gamma\gamma}-\Gamma^{\mathrm{SM}}_{h\rightarrow\gamma\gamma}}{\Gamma^{\mathrm{SM}}_{h\rightarrow\gamma\gamma}},$
(22)

where the expression of Higgs diphoton decay width is given by,

$\displaystyle\Gamma^{\mathrm{BSM}+\mathrm{SM}}_{h\rightarrow\gamma\gamma}=$
$\displaystyle\frac{G_{F}\alpha^{2}M_{h}^{3}}{128\sqrt{2}\pi^{3}}\left|\sum_{f}N_{c}Q_{f}^{2}g_{hff}A_{1/2}^{h}\left(\tau_{f}\right)+g_{hWW}A_{f}^{h}\left(\tau_{W}\right)-\sum_{s}\frac{M_{W}}{g_{2}M_{s}^{2}}g_{ss\gamma}^{2}g_{ssh}A_{0}^{h}\left(\tau_{s}\right)\right|^{2},$
(23)

with the loop functions $A_{1/2}^{h},~{}A_{1}^{h},~{}A_{0}^{h}$ defined as
Djouadi:2005gj

$\displaystyle A_{1/2}^{h}\left(\tau_{i}\right)$
$\displaystyle=-2\tau_{i}\left[1+\left(1-\tau_{i}\right)\mathcal{F}\left(\tau_{i}\right)\right]$
(24) $\displaystyle A_{1}^{h}\left(\tau_{i}\right)$
$\displaystyle=2+3\tau_{i}+3\tau_{i}\left(2-\tau_{i}\right)\mathcal{F}\left(\tau_{i}\right)$
$\displaystyle A_{0}^{h}\left(\tau_{i}\right)$
$\displaystyle=-\tau_{i}\left[1-\tau_{i}\mathcal{F}\left(\tau_{i}\right)\right],$

and the function $\mathcal{F}\left(\tau_{i}\right)$ reads

$\mathcal{F}\left(\tau_{i}\right)=\left\\{\begin{array}[]{ll}\left[\sin^{-1}\left(\sqrt{\frac{1}{\tau_{i}}}\right)\right]^{2},&\tau_{i}\geq
1;\\\
-\frac{1}{4}\left[\ln\left(\frac{1+\sqrt{1-\tau_{i}}}{1-\sqrt{1-\tau_{i}}}\right)-i\pi\right]^{2},&\tau_{i}<1;\end{array}\right.$

and $\tau_{i}=4M_{i}^{2}/M_{h}^{2}~{}\left(i=f,W,s\right)$.
$\Gamma^{\mathrm{SM}}_{h\rightarrow\gamma\gamma}$ shares the same expression
with $\Gamma^{\mathrm{BSM}+\mathrm{SM}}_{h\rightarrow\gamma\gamma}$ but the
last term in the module square in Eq. (23). One can see that only charged
scalars contribute to the last term in Eq. (23) since $g_{ss\gamma}$ vanishes
for neutral scalars.

Table 1 shows the estimated precision for $\sigma(Zh)$ production and
$h\gamma\gamma$ decay signal rate for the CEPC and FCC-ee at $\sqrt{s}=240$
GeV as well as the ILC at $\sqrt{s}=250$ GeV. Note that all three exhibit
roughly the same level of precision for $\sigma(Zh)$ measurements. Besides,
the uncertainty on the measurement of the Higgs production via gluon and
$b\bar{b}$ fusion in $h\to\gamma\gamma$ decay channel at the HL-LHC is also
listed in the right end column in the table, whose precision is a few percent
and somewhat better than those estimated in the future Higgs factories even
though the $\sigma\left(Zh\right)$ measurement is much less precise. In what
follows, provide numerical results assuming the CEPC projections
CEPCStudyGroup:2018ghi  for purposes of concrete illustration 444The precision
for the Higgs production via the gluon and $b\bar{b}$ fusion in
$h\to\gamma\gamma$ channel is applied in addition in the complex triplet model
to illustrate a straightforward improvement for the constraints in the
parameter space. . In the following, we categorize our result into two cases
where the constraints on the BSM parameter space by the $Zh$ production are
complementary and degenerate to that by the Higgs diphoton decay rate,
respectively. We assume the constraints on the quartic Higgs (portal)
couplings from the perturbativity in each model are subject to the relation

$\lambda_{i}\left(\mu\right)\lesssim\frac{\lambda_{\text{FP}}}{3},$ (25)

where $\lambda_{\text{FP}}=12.1\ldots$ is the value at a fixed point where
$\beta_{\lambda}=0$ . The renormalization scale
$\mu\in\left[M_{Z},\Lambda\right]$, and $\Lambda$ is the cutoff scale of the
theory. This relation is an extension to the approximate perturbativity
constraint on the couplings in the complex singlet model in Ref.
Gonderinger:2012rd  which is based on the work in Ref. Riesselmann:1996is .

Table 1:  Estimated precision of CEPC, FCC-ee and ILC for $\sigma(Zh)$ and $h\to\gamma\gamma$ signal rate. The projected precision for the CEPC and FCC-ee are taken from Refs. CEPCStudyGroup:2018ghi ; Abada:2019lih  and those for the ILC are from Table 3 in Ref. Durieux:2017rsg , which are rescaled from the luminosity samples of 250 fb-1 in the ILC operating Scenarios Barklow:2015tja ; Bambade:2019fyw ; Fujii:2019zll . All quoted values are are for statistical uncertainties at the 68% confidence level. The total uncertainty (stat + syst) on the measurement for $h\to\gamma\gamma$ with Higgs production via gluon and $b\bar{b}$ fusion at the HL-LHC Cepeda:2019klc  is listed in the last column on the right in comparison of that at the CEPC, FCC-ee and ILC.  Measurement | CEPC | FCC-ee | ILC | HL-LHC
---|---|---|---|---
|  (240 GeV, 5.6 ab-1) |  (240 GeV, 5 ab-1) |  (250 GeV, 2 ab-1) |  (14 TeV, 3 ab-1)
$\sigma\left(Zh\right)$ | $0.50\%$ | $0.50\%$ | $0.71\%$ | –
$\sigma\times\mathrm{BR}\left(h\rightarrow\gamma\gamma\right)$ | $6.8\%$ | $9.0\%$ | $12\%$ |  $4\%$ (ggF+bbH)

It is worth mentioning that in order to verify that our result is independent
of scheme choice, we also perform the one-loop calculation in an alternative
renormalization scheme - on-shell renormalization scheme and obtain the
consistent result with that in $\overline{\mathrm{MS}}$ scheme. One can refer
to Appendix D for a detailed procedure of on-shell renormalization scheme.

###  4.1 Complementary with Higgs Diphoton Decays

#### Complex Triplet

The complex scalar triplet model with the hypercharge of the extended scalar
triplet $Y=2$ is a component of the type-II seesaw model Konetschny:1977bn ;
Magg:1980ut ; Schechter:1980gr ; Cheng:1980qt . The neutrino masses are
obtained through the Yukawa interaction term in the type-II seesaw model below

$\mathcal{L}_{\text{Yuk}}=h_{ij}\overline{L^{Ci}}i\tau_{2}\Delta
L^{j}+\text{h.c.},$ (26)

where $h_{ij}$ is the neutrino Yukawa coupling that is complex and symmetric,
$L_{i}=\left(\nu_{i},e_{i}\right)_{L}^{T}$ $\left(i=e,\mu,\tau\right)$ is a
left-handed lepton doublet, and $\Delta$ is a $2\times 2$ representation of
the complex triplet fields (see expression below (66)). After electroweak
symmetry breaking (EWSB), the neutrinos acquire masses due to non-vanishing
triplet vacuum expectation value (VEV) $v_{\Delta}\neq 0$, and the neutrino
mass matrix reads

$m_{\nu,ij}=\sqrt{2}h_{ij}v_{\Delta}.$ (27)

Meanwhile the minimization of the potential (66) gives rise to the triplet VEV

$v_{\Delta}\approx\frac{\mu
v_{\phi}^{2}}{\sqrt{2}\left[M^{2}+\frac{\lambda_{4}+\lambda_{5}}{2}v_{\phi}^{2}\right]},$
(28)

when $v_{\Delta}/v_{\phi}\ll 1$. One can see that for fixed
$\lambda_{4},\lambda_{5}$, $v_{\Delta}\rightarrow 0$ with $M/v_{\phi}\gg 1$.
Being constrained by the $\rho$ parameter the upper bound of the triplet VEV
$v_{\Delta}\lesssim 3~{}\mathrm{GeV}$ Du:2018eaw , so that the sine of the
mixing angles 555There are three mixing angles
$\left(\alpha,~{}\beta_{0},~{}\beta_{\pm}\right)$ respectively for the neutral
and singly charged components of the triplet; the doubly charged component
$H^{\pm\pm}$, it is already in a mass eigenstate. See details in Ref.
Du:2018eaw . are highly suppressed since $\propto v_{\Delta}/v_{\phi}\ll 1$.
Therefore, in what follows, we consider the scalar triplet unmixed with the SM
Higgs doublet after EWSB, as being treated in the other models with zero VEV
in the subsequent subsection.

We observe that the complex triplet makes the most interesting case in our
study, as it exhibits the complementary sensitivity between the $Zh$ and
$h\gamma\gamma$ decay rate precision measurements. In this model, there are
three free parameters that enter the one-loop calculation, which are
$\lambda_{4},~{}\lambda_{5}$ and $M_{\Delta}$, respectively. The scalar
triplet masses are given by

$\displaystyle M_{H^{\pm\pm}}^{2}$
$\displaystyle=M_{\Delta}^{2}-\frac{\lambda_{5}v_{\phi}^{2}}{2},$ (29)
$\displaystyle M_{H^{\pm}}^{2}$
$\displaystyle=M_{\Delta}^{2}-\frac{\lambda_{5}v_{\phi}^{2}}{4},$
$\displaystyle M_{H}^{2}$ $\displaystyle=M_{A}^{2}=M_{\Delta}^{2},$

with $M_{H^{\pm\pm}}$, $M_{H^{\pm}}$, $M_{H}$ and $M_{A}$ the masses of the
doubly and singly charged, CP-even and CP-odd neutral scalar, respectively. In
Fig. 3 the contours of $\left|\delta\sigma_{Zh}\right|\leq 0.5\%$ ($1\sigma$
cyan band with dashed central and outer lines) and $\left|\delta
R_{h\gamma\gamma}\right|\leq 6.8\%$ ($1\sigma$ orange band with solid central
and outer lines) are plotted in $\left(\lambda_{4},\lambda_{5}\right)$ plane
with $M_{\Delta}=400,~{}800~{}\mathrm{GeV}$. To illustrate a straightforward
improvement for probing region of parameter space in Higgs diphoton decay
channel when increasing the precision for the measurement in this channel, a
contour of $\left|\delta R_{h\gamma\gamma}\right|\leq 4\%$ ($1\sigma$ grey
band with dashdotted central and outer lines) corresponding to the precision
at the HL-LHC is also plotted. A shrinking of the band width constraining
parameter $\lambda_{4}$ is apparent with smaller uncertainties on the Higgs
diphoton decay measurement.

The ellipse encompassing the crossed range between the two bands on each plot
outlines the constraints of the parameter $\lambda_{4}$ and $\lambda_{5}$ at
$95\%$ confidence level when combining $\left|\delta\sigma_{Zh}\right|\leq
0.5\%$ and $\left|\delta R_{h\gamma\gamma}\right|\leq 6.8\%$ and using
$\chi^{2}$ test for 2 degree-of-freedom. One can see a cross-like shape for
the two contours, indicating complementary results between the two processes
and thereby putting the most stringent constraints one another. We find that
as the mass of the neutral component ($M_{\Delta}$) increases the cyan band
(with dashed lines) as the contour from $\left|\delta\sigma_{Zh}\right|\leq
0.5\%$ gets flatter and becomes less dependent of the parameter $\lambda_{4}$
666One should compare the flatness of the cyan band (with dashed lines) in the
same area of $\left(\lambda_{4},\lambda_{5}\right)$ plane. Compared to the
left-hand subfigure in Fig. 3, the cyan band in the right-hand subfigure in a
zoom-in range of $\lambda_{4}\in\left(-3,3\right)$ appears fairly flat. . This
can be explained as follows. As one can see in Eq. (29) the mass difference of
differently charged components is proportional to $\lambda_{5}$. Performing
the large mass expansion in the range of $s/M_{\Delta}^{2}\ll 1$ to the self
energy and vertex loop functions, we find that the dominant contribution is
from the $WW$ self energy contribution that enters $\hat{s}$ and $\hat{c}$
(see first and second lines in Eq. (21)) due to the mass splitting between
scalars with different number of charges and the mass difference is solely
proportional to $\lambda_{5}$ (see expression of
$\Sigma_{T}^{WW}\left(k^{2}\right)$ in (79) and apply large mass expansion to
$B_{0}$ functions). On the other hand, as for the orange band, when neglecting
the mass difference in the loop function $A_{0}^{h}\left(\tau_{i}\right)$
(which is valid in the large mass limit as the case we discuss above using
large mass expansion), the minimization of the $\delta R_{h\gamma\gamma}$
(i.e., the third term in the modular square in Eq. (23) vanishes) gives rise
to a simple linear relation between $\lambda_{4}$ and $\lambda_{5}$ that
$10\lambda_{4}+\lambda_{5}=0$, which roughly reflects the central position of
the orange band in Fig. 3. In other words, the complementary parameter space
probes by the precision measurement for $\sigma\left(Zh\right)$ and
$h\to\gamma\gamma$ decay rate are realized in two aspects:

  1. 1.

The BSM contribution to $\sigma\left(Zh\right)$ is dominated by the $WW$ self
energy via differently charged scalar in loops which is susceptible to the
variation of the mass splitting parameter ($\lambda_{5}$), compared to other
types of corrections.

  2. 2.

The BSM contribution to $h\to\gamma\gamma$ decay rate involves triple Higgs
couplings with two charged Higgs that have a stronger dependence on the
parameter $\lambda_{4}$ than on the other couplings (see Feynman rules in
Table 4), making it more susceptible to variation in $\lambda_{4}$.

Therefore, the complementarity is observed when exploring the parameter space
in the plane $\left(\lambda_{4},\lambda_{5}\right)$. Furthermore, from the
ellipses constraining the parameter $\lambda_{4}$ and $\lambda_{5}$ at $95\%$
confidence level one also can see that the constraint on $\lambda_{5}$ is
stronger than that on $\lambda_{4}$ with increase of $M_{\Delta}$.

Figure 3: Contour comparison of $\left|\delta\sigma_{Zh}\right|\leq 0.5\%$
($1\sigma$ cyan band with dashed central and outer lines) and $\left|\delta
R_{h\gamma\gamma}\right|\leq 6.8\%$ ($1\sigma$ orange band with solid central
and outer lines) in complex triplet model with $M_{\Delta}=400~{}\mathrm{GeV}$
(left) and $M_{\Delta}=800~{}\mathrm{GeV}$ (right), respectively. The ellipse
on each plot denotes the $95\%$ confidence level for the parameter
$\lambda_{4}$ and $\lambda_{5}$ with corresponding fixed $M_{\Delta}$. In
addition, the contour of $\left|\delta R_{h\gamma\gamma}\right|\leq 4\%$
($1\sigma$ grey band with dashdotted central and outer lines) corresponding to
the accuracry at the HL-LHC via gluon and $b\bar{b}$ fusion is added for the
comparison, where one can see that with higher precision on $h\to\gamma\gamma$
decay rate a better constraint on the parameter $\lambda_{4}$ can be
archieved.

Instead of fixing the neutral scalar mass, one may fix one of the two coupling
parameters and investigate the contours of $\delta\sigma_{Zh}$ and $\delta
R_{h\gamma\gamma}$ at $1\sigma$ level in the neutral scalar mass and coupling
parameter plane. First we can make slice of the contours in the
$\left(M_{\Delta},\lambda_{5}\right)$ plane with fixed $\lambda_{4}$. Judging
from Fig. 3, the boundary of $\lambda_{4}$ with
$M_{\Delta}=400~{}\mathrm{GeV}$ is $\left|\lambda_{4}\right|\sim 1$ and that
with $M_{\Delta}=800~{}\mathrm{GeV}$ is $\left|\lambda_{4}\right|\sim 3$ which
are still within the allowed perturbativity range (25); while the constraint
on parameter $\lambda_{5}$ keeps it in a small range
$\left|\Delta\lambda_{5}\right|\lesssim 0.4$
($\left|\lambda_{5}\right|\lesssim 0.2$ for small $\lambda_{4}$).

Fig. 4 shows the two contours of $\left|\delta\sigma_{Zh}\right|\leq 0.5\%$
and $\left|\delta R_{h\gamma\gamma}\right|\leq 6.8\%$ with the fixed
$\lambda_{4}=1,~{}3$, respectively. The contour of
$\left|\delta\sigma_{Zh}\right|\leq 0.5\%$ constantly restricts $\lambda_{5}$
within a thin strip region, while the contour of $\left|\delta
R_{h\gamma\gamma}\right|\leq 6.8\%$ puts a lower bound $M_{\Delta}$ region, so
that in the overlapping area of the two contours the lower bound of the mass
$M_{\Delta}$ is approximately $400~{}\mathrm{GeV}$ for $\lambda_{4}=1$ and
$800~{}\mathrm{GeV}$ for $\lambda_{4}=3$. Similar lower bound of $M_{\Delta}$
can be obtained for negative $\lambda_{4}$, i.e., $\lambda_{4}=-1,-3$,
respectively. This is consistent with the result in Fig. 3, where the left
figure ($M_{\Delta}=400~{}\mathrm{GeV}$) has the boundary of
$\left|\lambda_{4}\right|\lesssim 1$ and the right figure
($M_{\Delta}=800~{}\mathrm{GeV}$) has the boundary of
$\left|\lambda_{4}\right|\lesssim 3$.

Figure 4: Contour comparison of $\left|\delta\sigma_{Zh}\right|\leq 0.5\%$
(cyan band outlined with dashed lines) and $\left|\delta
R_{h\gamma\gamma}\right|\leq 6.8\%$ (orange area outlined with solid line) in
complex triplet model in plane $\left(M_{\Delta},\lambda_{5}\right)$ with
$\lambda_{4}=1$ (left) and $\lambda_{4}=3$ (right).

In addition to the constraints from perturbativity (25), vacuum stability and
perturbative unitarity also impose boundaries in the parameter space. In
general any of the above three sources of constraints impacts all the
couplings in the potential. For instance, in Ref. Du:2018eaw  it was shown
that vacuum stability and perturbative unitarity at tree-level with fixed
values of $\lambda_{2}=0.2$ and $\lambda_{3}=0$ require the parameter space in
the $\left(\lambda_{4},\lambda_{5}\right)$ plane to be mostly in the 4th
quadrant, and initial condition ($\lambda_{i}$ fixed at the starting scale)
should be set first for the renormalization group equations (RGEs) to test the
perturbativity with running scales, which are both illustrated in Fig. 1
therein. On the other hand, the constraints by the precision measurement on
$Zh$ and $h\gamma\gamma$ decay rate only rely on the Higgs portal couplings
$\left(\lambda_{4},\lambda_{5}\right)$ in addition to the triplet mass
spectrum, which therefore, complement that by perturbativity, vacuum stability
and perturbative unitarity.

As discussed above that the upper bound of the triplet VEV $v_{\Delta}\lesssim
3$ GeV is strictly constrained by the $\rho$ parameter and the sine of the
mixing angles are highly suppressed so that one can consider the scalar
triplet and the SM doublet fields do not mix together and treat the virtual
scalar triplet loop corrections to the $Zh$ production separately. On the
other hand, as can be seen in (27) that the neutrino masses derived from the
type-II seesaw model depend on the neutrino Yukawa couplings $h_{ij}$ and the
triplet VEV $v_{\Delta}$. The sum of neutrino masses has an upper bound, and
in the context of a minimal 7-parameter model ($\mathrm{\Lambda CDM}+\sum
m_{\nu}$) one obtains $\sum m_{\nu}<0.12$ eV (95%CL) from the combination of
Planck 2018 and BAO Aghanim:2018eyx ; Zyla:2020zbs  data sets. The parameters
$h_{ij}$ and $v_{\Delta}$, however, are not individually rigorously
constrained (e.g., $v_{\Delta}$ could be in the KeV or GeV regime), because
their product yields the light neutrino mass matrix. For a fixed $v_{\Delta}$,
a larger (smaller) $h_{ij}$ corresponds to a larger (smaller) value for
neutrino masses, and vice versa.

As discussed in Ref. Du:2018eaw , this interplay of $h_{ij}$ and $v_{\Delta}$
affects the sensitivity of collider probes of the complex triplet model. At
the LHC and a prospective 100 TeV $pp$ collider, the dominant discovery
channels could be $H^{++}H^{--}$ and $H^{\pm\pm}H^{\mp}$ depending on the
value of $v_{\Delta}$ Du:2018eaw . The doubly charged Higgs $H^{\pm\pm}$ could
decay into same-sign di-leptons (di-bosons) and the branching ratio
Br$\left(H^{\pm\pm}\to l^{\pm}l^{\pm}\right)$ (Br$\left(H^{\pm\pm}\to
W^{\pm}W^{\pm}\right)$) depends on the parameters $M_{\Delta}$, $\lambda_{5}$,
and $h_{ij}$ ($M_{\Delta}$, $\lambda_{5}$, and $v_{\Delta}$). The singly
charged Higgs $H^{\pm}$ has the decay vertex $H^{\pm}hW^{\mp}$ and
Br$\left(H^{\pm}\to hW^{\pm}\right)$ depends on the parameters $M_{\Delta}$,
$\lambda_{4}$, $\lambda_{5}$ and $v_{\Delta}$. In Ref. Du:2018eaw  the regions
of $>5\sigma$ significance 777The significance is defined as
$\frac{\text{Sig}}{\sqrt{\text{Sig+Bkg}}}$ with Sig and Bkg the total signal
and background event numbers. are obtained using the discovery channels
$H^{++}H^{--}$ and $H^{\pm\pm}H^{\mp}$ with $H^{\pm\pm}$ decaying into same-
sign di-leptons and/or di-bosons in the $\left(M_{\Delta},v_{\Delta}\right)$
plane with fixed neutrino masses ($m_{\nu_{l}}=0.01$ eV for $l=e,\mu,\tau$)
and Higgs portal couplings ($\lambda_{4}=0,\lambda_{5}=-0.1$) (see Fig. 7
therein).

In our study, we find that the parameter $\lambda_{5}$ could be tightly
constrained ($\left|\Delta\lambda_{5}\right|\lesssim 0.4$) by the precision
measurement for the $Zh$ cross section since $\sigma\left(Zh\right)$ is
sensitive to the mass splitting between the scalar components. Moreover,
measurements of $\sigma\left(Zh\right)$ and $\Gamma(H\to\gamma\gamma)$ provide
complementary information, as illustrated in Fig. 4. The Higgs di-photon decay
rate is relatively insensitive to $\lambda_{5}$ but strongly-dependent on
$\lambda_{4}$. Therefore, with additional information about the Higgs portal
parameters one might further delineate the discovery regions in the
$\left(M_{\Delta},v_{\Delta}(h_{ij})\right)$ plane for given values of the
neutrino masses.

To illustrate, we take two benchmark points from Ref. Du:2018eaw  to show the
consistency on the Higgs portal parameter space settings between the findings
in our study and in probing the discovery channels with neutrino masses
involved, which read

$M_{\Delta}$ | $M_{Z}$ | $M_{h}$ | $m_{\nu}$ | $v_{\Delta}$ | $\lambda_{2}$ | $\lambda_{3}$ | $\lambda_{4}$ | $\lambda_{5}$
---|---|---|---|---|---|---|---|---
400 & 800 GeV | 91.1876 GeV | 125 GeV | 0.01 eV |  $10^{-4}$ GeV | 0.2 | 0 | 0 | -0.1

We stress that the value of $M_{h}$ that is off 0.1 GeV compared to our
settings in (20) ($125.1$ GeV for loop correction vs. $125$ GeV for the
benchmark points) does not noticeably affect other settings. The mass of the
neutrino is equal in the three generations. These two benchmark points
represent two mass scales (low and high $M_{\Delta}$ region) and are covered
by various overlapping discovery regions in the
$\left(M_{\Delta},v_{\Delta}\right)$ plane (see Fig. 7 in Ref. Du:2018eaw ).
In the analyses of Fig. 3 and Fig. 4 we already observe that with the choice
of $M_{\Delta}=400~{}(800)~{}\mathrm{GeV}$, the regions of $\lambda_{4}$ and
$\lambda_{5}$ constrained by the precision measurement of $Zh$ and
$h\gamma\gamma$ decay rate approximate $\left|\lambda_{4}\right|\lesssim
1~{}(3)$ and $\left|\lambda_{5}\right|\lesssim 0.2$, which encompass
$\left(\lambda_{4},\lambda_{5}\right)=\left(0,-0.1\right)$ for the two
benchmark points in (4.1).

###  4.2 Degenerate with Higgs Diphoton Decay Sensitivity

In this subsection the models of interest including the inert doublet, real
triplet, quintuplet and septuplet are studied, and likewise the constraints on
the parameter space in each model are outlined in terms of the respective
precision measurements on the $Zh$ production and Higgs diphoton decay rate.
We find that in these models it gives less complementary sensitivity between
the $Zh$ and Higgs diphoton decay rate precision measurement, showing
(partial) degeneracy in constraining the parameter spaces between the $Zh$ and
$h\gamma\gamma$ processes. In particular, there is a large degree of
degeneracy on constraints of the parameter space between the two processes in
the real triplet model; while in the inert doublet, quintuplet and septuplet
models, since there are more free parameters it depends on which sliced
parameter plane that one looks into to determine the degree of degeneracy.

#### Inert Doublet

The inert doublet model is the simplest two Higgs doublet model (THDM) that
has an imposition of $\mathcal{Z}_{2}$ symmetry and therefore the lightest
neutral component in the extended scalar sector could be a potential weakly
interacting massive particle (WIMP) dark matter candidate 888There are two
neutral components: CP-even $H^{0}$ and CP-odd $A^{0}$. One can choose either
one as the dark matter candidate as long as it’s the lightest particle. The
convention in the studies is take $H^{0}$ the lightest particle by requiring
the parameter $\lambda_{5}$ non-positive (see Eq. (30) for the mass
definition).. The extended scalar sector also decouple to the SM fermions due
to its vanishing VEV. The interplay between the dark matter phenomenology and
the EWPT in the context of the inert double model has been studied in a
variety of spectra Borah:2012pu ; Gil:2012ya ; Blinov:2015vma ;
Chowdhury:2011ga ; Cline:2013bln . It shows that a strongly first order
electroweak phase transition (SFOEWPT) requires a large mass splitting between
the dark matter candidate particle and the other extended scalars, and the
Higgs funnel regime ($M_{H^{0}}\sim M_{h}/2$) is the only region of parameter
space that can successfully saturate the dark matter abundance and provide a
SFOEWPT Borah:2012pu ; Gil:2012ya ; Blinov:2015vma .

In the inert double model there are four free parameters that enter the one-
loop calculation, which are
$\\{\mu_{2}^{2},~{}\lambda_{3},~{}\lambda_{4},~{}\lambda_{5}\\}.$ The inert
scalar masses are given by

$\displaystyle M_{H^{\pm}}^{2}$
$\displaystyle=\mu_{2}^{2}+\frac{1}{2}\lambda_{3}v_{\phi}^{2},$ (30)
$\displaystyle M_{H^{0}}^{2}$
$\displaystyle=\mu_{2}^{2}+\frac{1}{2}\lambda_{L}v_{\phi}^{2},$ $\displaystyle
M_{A^{0}}^{2}$ $\displaystyle=\mu_{2}^{2}+\frac{1}{2}\lambda_{A}v_{\phi}^{2},$

with $\lambda_{L,A}=\left(\lambda_{3}+\lambda_{4}\pm\lambda_{5}\right)$. Since
only the charged component ($H^{\pm}$) contributes to the correction to the
Higgs diphoton decay (refer to the last term on the right hand side of Eq.
(23) where $g_{ss\gamma}\equiv 0$ for neutral scalars), it leaves us two
parameters to plot the two contours of $\left|\delta\sigma_{Zh}\right|\leq
0.5\%$ and $\left|\delta R_{h\gamma\gamma}\right|\leq 6.8\%$ in the same
plane, which are $\\{\mu_{2},\lambda_{3}\\}$. By fixing $\lambda_{4}$ and
$\lambda_{5}$, Fig. 5 shows the two contours by $Zh$ and Higgs diphoton
precision measurements in the $\left(\lambda_{3},M_{H^{\pm}}\right)$ plane,
where $M_{H^{\pm}}$ is rescaled from $\mu_{2}$ according the first relation in
Eq. 30. One can see that the contour by Higgs diphoton precision measurement
is unchanged because it depends on no other parameters but $\lambda_{3}$ and
$M_{H^{\pm}}\left(\mu_{2}\right)$. The contour by $Zh$ precision measurement
is however dependent on the choice of parameters $\lambda_{4}$ and
$\lambda_{5}$, as illustrated in Fig. 5. We find that compared to both
vanishing $\lambda_{4}$ and $\lambda_{5}$ which makes a somewhat central-sit
contour, the positive $\lambda_{4}$ pushes the contour downward and the
negative upward, and non-vanishing $\lambda_{5}$ gives rise to higher lower
boundary of $M_{H^{\pm}}$. Note that the sign of $\lambda_{5}$ alone does not
change the shape of contour due to the fact that the two neutral components
interchange the mass definition in Eq. (30) while the Feynman rules involving
these two particles are identical (refer to the Feynman rule in Appendix A).

Figure 5: Contours in $\left(\lambda_{3},M_{H^{\pm}}\right)$ plane for
$\left|\delta\sigma_{Zh}\right|\leq 0.5\%$ (cyan area outlined with dashed
lines) and $\left|\delta R_{h\gamma\gamma}\right|\leq 6.8\%$ (orange area
outlined with solid lines) in inert double model. From left to right panel,
$\lambda_{4}$ and $\lambda_{5}$ are set at different values. The positive and
negative $\lambda_{4}$ shifts the contour by $Zh$ precision measurement
downward and upward accordingly. The non-zero $\lambda_{5}$ makes greater
lower bound of $M_{H^{\pm}}$ compared to vanishing $\lambda_{5}$.

On the other hand, one observes that the inert scalars decouple from the Higgs
diphoton process when setting $\lambda_{3}=0$ because
$g_{H^{+}H^{-}h}\propto\lambda_{3}$. In the meantime, $\mu_{2}$ represents the
mass of the physical charged Higgs $H^{\pm}$. In this case we consider the
constraints of $\lambda_{4}$ and $\lambda_{5}$ by $Zh$ future precision are
utterly complementary with that by Higgs diphoton future precision. Hence, one
can plot the $Zh$ contours in $\\{\lambda_{4},\lambda_{5}\\}$ parameter space
by fixing $\lambda_{3}=0$ and varying $M_{H^{\pm}}\left(\mu_{2}\right))$. Fig.
6 shows contours of $\left|\delta\sigma_{Zh}\right|\leq 0.5\%$ with
$M_{H^{\pm}}=300,~{}500,~{}700~{},900~{}\mathrm{GeV}$ in the inert double
model, where the area within the red dashed square denotes the perturbativity
area of $\left(\lambda_{4},~{}\lambda_{5}\right)$ Hambye:1996wb . The contours
of $\delta R_{h\gamma\gamma}$ is missing in this case due to $\delta
R_{h\gamma\gamma}\equiv 0$ when $\lambda_{3}=0$.

Figure 6: Contours in $\left(\lambda_{4},\lambda_{5}\right)$ plane for
$\left|\delta\sigma_{Zh}\right|\leq 0.5\%$ in inert double model with
$\mu_{2}=300,~{}500,~{}700,~{}900~{}\mathrm{GeV}$, which correspond to
concentric areas in yellow, green, cyan and orange, and are respectively
outlined with dashed, dotted, dash-dotted and solid lines. With the increase
of $\mu_{2}$ the contour area gets larger and eventually excesses the region
of perturbativity of the $\lambda$ parameters in the scalar potential. The
allowed region of perturbativity is dictated within the red dashed box
according to (25).

In order to see if the constraints on the parameters in our study give region
of interest where the SFOEWPT occurs, we exploit the benchmark models (BMs)
specified in Ref. Blinov:2015vma  and mark the three benchmark points "BM1,
BM2, BM3" corresponding to the three benchmark scenarios in the
$\left(\lambda_{3},M_{H^{\pm}}\right)$ plane along with the contours by
$h\gamma\gamma$ decay and $Zh$ precision measurement in Fig. 7. The legends
"L1, L2, L3" on the right hand side of the figure denote the contours by $Zh$
precision measurement (i.e., $\left|\delta\sigma_{Zh}\right|\leq 0.5\%$) with
the fixed $\lambda_{4}$ and $\lambda_{5}$ in accordance with that in "BM1,
BM2, BM3", respectively. One can find the input parameters for the three
benchmark scenarios in Table 2. As can be seen from Fig. 7, all three
benchmark points are excluded from the contours by both the $h\gamma\gamma$
decay and $Zh$ precision measurement at 1-$\sigma$ confidence level. This
tells us that with the expected CEPC precision the constrained parameter space
may further exclude some region that has SFOEWPT and is permitted
phenomenologically elsewhere like the benchmark scenarios used above.

Table 2: Input parameters for the three benchmark scenarios given in Ref.Blinov:2015vma . Masses are in unit $\mathrm{GeV}$. Couplings ($\lambda_{3},~{}\lambda_{4},~{}\lambda_{5}$) are valued at scale $Q=246~{}\mathrm{GeV}$ with finite temperature correction. | $M_{H^{0}}$ | $M_{A^{0}}$ | $M_{H^{\pm}}$ | $\lambda_{3}$ | $\lambda_{4}$ | $\lambda_{5}$
---|---|---|---|---|---|---
BM1 | 66 | 300 | 300 | 3.3 | -1.7 | -1.5
BM2 | 200 | 400 | 400 | 4.6 | -2.3 | -2.0
BM3 | 5 | 265 | 265 | 2.7 | -1.4 | -1.2
Figure 7: Contours in $\left(\lambda_{3},M_{H^{\pm}}\right)$ plane for
$\left|\delta\sigma_{Zh}\right|\leq 0.5\%$ (red, cyan and yellow areas
outlined with dotdashed, dotted and dashed lines and labelled with L1, L2 and
L3) and $\left|\delta R_{h\gamma\gamma}\right|\leq 6.8\%$ (orange area
outlined with solid lines). L1, L2 and L3 respectively denote the choice of
$\lambda_{4}$ and $\lambda_{5}$ in accordance with that in BM1, BM2 and BM3,
which are listed in Table 2. The three benchmark points BM1, BM2 and BM3 are
marked with red crosses.

#### Real Triplet

The real triplet model has the extended scalar sector a real triplet with the
neutral component a potential WIMP dark matter candidate if the triplet VEV is
vanishing FileviezPerez:2008bj . Due to small mass splitting between the
charged and neutral triplet scalars ($\Delta M=166~{}\mathrm{MeV}$)
Cirelli:2005uq ; Cirelli:2007xd  in the large scalar mass regime
$M_{\Sigma}\gg M_{W}$, the direct search for the dark matter involves the
process of charged scalar pair production accompanying the decay to neutral
scalar and a soft $\pi^{\pm}$, the so-called disappearing charge tracks
(DCTs). Experimentally, the DCT signature is profiled in search for the
charginos at the LHC Aad:2013yna ; CMS:2014gxa ; Aaboud:2017mpt ;
Sirunyan:2018ldc . The reach of the real triplet with DCT signature at the 13
TeV LHC and a possible future 100 TeV $pp$ collider is studied in
Chiang:2020rcv . On the other hand, the Higgs portal parameter $\lambda_{3}$
may play a role in EWPT. The EWPT in the real triplet has been thoroughly
studied in Niemi:2018asa ; Niemi:2020hto  using dimensional reduction - a
three dimensional effective field theory (DR3EFT) that allows non-perturbative
lattice simulation.

In the real triplet model the free parameters from the potential that enter
our calculation are the mass of the scalar triplet $M_{\Sigma}$ and
$\lambda_{3}$, where we have omitted the mass splitting difference between the
neutral and charged components at loop level since the splitting is small
($\Delta M=166~{}\mathrm{MeV}$ Cirelli:2005uq ; Cirelli:2007xd ) compared to
the range of the mass investigated ($M_{\Sigma}\gtrsim 200~{}\mathrm{GeV}$).
In this case, one has the contour plots in the parameter space $M_{\Sigma}$
versus $\lambda_{3}$ for both $\left|\delta\sigma_{Zh}\right|\leq 0.5\%$ and
$\left|\delta R_{h\gamma\gamma}\right|\leq 6.8\%$ shown in Fig. 8, and can see
that the contour by the $Zh$ measurement largely overlaps with that by the
Higgs diphoton decay rate measurement. When $M_{\Sigma}\lesssim
400~{}\mathrm{GeV}$ the Higgs diphoton measurement reduces the positive bound
of $\lambda_{3}$ while for the rest the $Zh$ measurement gives slight better
constraints upon these two parameters. It is interesting to mention that the
triplet dark matter direct search gives rise to much narrower window than the
Higgs diphoton decay for $\lambda_{3}$ when $M_{\Sigma}$ is in $\mathrm{TeV}$
regime (see Figure 7 in Ref. Chiang:2020rcv ). Given the first order EWPT
region provided in Niemi:2018asa ; Niemi:2020hto , one can compare this region
with the constraints on the triplet mass $M_{\Sigma}$ and the Higgs portal
coupling $\lambda_{3}$ by the precision measurement for $Zh$ production and
$h\gamma\gamma$ decay rate at the CEPC to see in which area of the parameter
space does it occur the first order EWPT, which is illustrated in Fig. 9,
where the red strip represents the up-to-date first order EWPT region 999The
first order EWPT region obtained in Ref. Niemi:2018asa  is a subset of that
calculated in Ref. Niemi:2020hto . It has the same lower bound of the region
aligned with the larger region given in the latter paper. Both papers
performed dimensional reduction to an effective three-dimensional theory
(DR3EFT) to identify the phase transition regions. The former paper integrates
out the triplet field, assuming it is (super)heavy so that there is an area in
which the dimensional reduction (DR) no longer holds; while this area can be
accessible with the treatment in the latter paper and the result is validated
with lattice simulation. ($M_{\Sigma}\in[100,400]~{}\mathrm{GeV}$) calculated
in Niemi:2020hto  101010The red region represents the one-step first order
phase transition $O\rightarrow\phi$. In Niemi:2020hto  there are three thin
strips adjacent and above the red region leading to two-step transitions
$O\rightarrow\Sigma\rightarrow\phi$ with $\Sigma\rightarrow\phi$ a first order
and $O\rightarrow\Sigma$ a crossover, a first order or a second order EWPT.
See Figure 1 in Niemi:2020hto  for the details.. As can be seen that with the
CEPC required accuracy the first order EWPT could be excluded by precision
measurement of the $h\gamma\gamma$ decay rate with $M_{\Sigma}\lesssim
350~{}\mathrm{GeV}$ while it is permitted in reference to the $Zh$ measurement
solely.

Figure 8: Contours in $\left(\lambda_{3},M_{\Sigma}\right)$ plane for
$\left|\delta\sigma_{Zh}\right|\leq 0.5\%$ (cyan area outlined with dashed
lines) and $\left|\delta R_{h\gamma\gamma}\right|\leq 6.8\%$ (orange area
outlined with solid lines) in real triplet model. Figure 9: Contours same as
that in Fig. 8 with superposition of the red region denoting the first order
phase transition presented in Ref. Niemi:2020hto .

#### Quintuplet and Septuplet

The quintuplet and septuplet are the high dimensional EW scalar multiplets
($n=5,7$) carrying zero hypercharge ($Y=0$) so as to have the neutral
components which could be potential WIMP dark matter candidates. The EW scalar
multiplets of dimension $n$ are widely studied and discussed in Hambye:2009pw
; AbdusSalam:2013eya ; Chao:2018xwz ; Pilkington:2016erq . Ref. Chao:2018xwz
updated the most general renormalizable potentials reported in previous
studies Hambye:2009pw ; AbdusSalam:2013eya , and focused on $n=5,7$ cases to
illustrate the dark matter phenomenology. In what follows, we follow the
representation of the $\mathrm{SU\left(2\right)_{L}}$ EW scalar multiplet and
the convention of the formulism adopted in Ref. Chao:2018xwz .

The quintuplet and septuplet have the same parameters that appear in the one-
loop calculation and the difference is the fold of the scalar components
respectively being $n=5$ and $n=7$. The scalar masses are given by

$\displaystyle
M_{S_{A}}^{2}=M_{A}^{2}+\frac{1}{2}\lambda_{1}v^{2}+\frac{2}{\sqrt{n}}M_{B}^{2}+\frac{1}{\sqrt{n}}\lambda_{3}v^{2},$
(31) $\displaystyle
M_{S_{B}}^{2}=M_{A}^{2}+\frac{1}{2}\lambda_{1}v^{2}-\frac{2}{\sqrt{n}}M_{B}^{2}-\frac{1}{\sqrt{n}}\lambda_{3}v^{2},$

where the subscripts $A,~{}B$ denote the type of the scalars, and the mass of
each type is equal regardless of the number of charges carried at tree level.
Analogous to that in the real triplet model the radiative correction gives
rise to small mass splitting between the charged and neutral component when
$M_{S_{A,B}}\gg M_{W}$ that $M_{S_{A,B}^{m}}-M_{S_{A,B}^{0}}=m^{2}\Delta M$
($m=-\frac{n-1}{2},\ldots,\frac{n-1}{2}$, $\Delta M=166~{}\mathrm{MeV}$
Cirelli:2005uq ; Cirelli:2007xd ). There are four free parameters from the
potential that enter our calculation. They are the couplings
$\lambda_{1},~{}\lambda_{3}$ and the masses either the unphysical
$M_{A},~{}M_{B}$ or the physical $M_{S_{A}},~{}M_{S_{B}}$.

(a) Quintuplet

(b) Septuplet

Figure 10: Contour comparison for $\left|\delta\sigma_{Zh}\right|\leq 0.5\%$
(cyan band outlined with dashed lines) and $\left|\delta
R_{h\gamma\gamma}\right|\leq 6.8\%$ (orange band outlined with solid lines)
for quintuplet and septuplet with fixed values of the physical masses
$\left(M_{S_{A}},~{}M_{S_{B}}\right)=\left(300,300\right),~{}\left(500,300\right),~{}\left(1000,300\right)~{}\mathrm{GeV}$,
respectively.

We choose the physical masses $\left(M_{S_{A}},M_{S_{B}}\right)$ to be fixed
and perform the contour comparison for the two processes in the $n=5,~{}7$
scalar model. Fig. 10 illustrates the comparisons with three sets of fixed
$\left(M_{S_{A}},M_{S_{B}}\right)$. As we can see that the overlapping area
varies with the setting of $\left(M_{S_{A}},M_{S_{B}}\right)$, and the two
contours become more degenerate as increases of the mass difference between
the type A and type B scalars. When the two physical masses are set to be
equal, one observes that the $h\gamma\gamma$ decay rate constraint appears as
a thin straight vertical band in the $\left(\lambda_{1},\lambda_{3}\right)$
plane, meaning the result is independent of the parameter $\lambda_{3}$. This
is due to the natural of the tri-couplings $g_{s_{i}^{m}s_{i}^{-m}\gamma}$ and
$g_{s_{i}^{m}s_{i}^{-m}h}$ ($i=A,~{}B$, the tri-couplings correspond to
notation of vertices in Table 3 and 4 in Appendix A), which have the form

$g_{S_{i}^{m}S_{i}^{-m}\gamma}=-em,\quad
g_{S_{A}^{m}S_{A}^{-m}h}=-\left(\lambda_{1}+\frac{2}{\sqrt{n}}\lambda_{3}\right),\quad
g_{S_{B}^{m}S_{B}^{-m}h}=-\left(\lambda_{1}-\frac{2}{\sqrt{n}}\lambda_{3}\right),$
(32)

with $m=0,\ldots,\frac{n-1}{2}$. When $M_{S_{A}}=M_{S_{B}}$, the loop function
$A_{0}^{h}\left(\tau_{s}\right)$ in Eq. (23) and (24) returns the same value
for both $S_{A}$ and $S_{B}$, then the sum of the same charged components
makes the exact cancellation of $\lambda_{3}$ between type $A$ and $B$
scalars.

In Ref. Chao:2018xwz  it was found that the effective coupling is rather small
$\left|\lambda_{\text{eff}}\right|\ll 1$ when saturating the observed relic
density and evading the direct detection limits by LUX Akerib:2016vxi ,
PandaX-II Cui:2017nnn  and XENONT1 Aprile:2018dbl . The effective coupling
$\lambda_{\text{eff}}$ is a linear combination of parameter $\lambda_{1}$ and
$\lambda_{3}$. One could take the septuplet for illustration to determine the
constraints on the parameter space in connection with the dark matter
phenomenology. The effective coupling in accord with our definition for the
septuplet reads

$\lambda_{\text{eff}}=\lambda_{1}-\frac{2}{\sqrt{7}}\lambda_{3}.$ (33)

The neutral component of $S_{A}$ is in general taken as the dark matter
candidate. According to the findings in Chao:2018xwz , if we assume the
septuplet is the only candidate of dark matter, then its mass should be $\sim
9$ TeV in the one-species scenario for vanishing $\lambda_{\text{eff}}$. In
Fig. 11 it shows contours for $\left|\delta\sigma_{Zh}\right|\leq 0.5\%$ and
$\left|\delta R_{h\gamma\gamma}\right|\leq 6.8\%$ compared to the line for
vanishing $\lambda_{\text{eff}}$ with the mass of the dark matter
$M_{S_{A}}=9$ TeV and the choice of $M_{S_{B}}$ being $300~{}\mathrm{GeV}$
111111$M_{S_{B}}$ is a free parameter with no constraint in dark matter
phenomenology.. As can be seen that the line denoting a vanishing
$\lambda_{\text{eff}}$ goes through the overlapping area of the two contours
and almost aligns with the central line of the contour area by $\left|\delta
R_{h\gamma\gamma}\right|\leq 6.8\%$ (the contour with $\delta
R_{h\gamma\gamma}\equiv 0$). This is not surprising when the dark matter mass
is in multi-TeV regime and $M_{S_{B}}/M_{S_{A}}\ll 1$. Referring to Eq. (23)
one of the solutions to minimize $\delta R_{h\gamma\gamma}$ in (22) is a
vanishing last term in (23) which is proportional to
$g_{S_{B}^{m}S_{B}^{-m}h}$ since $\frac{M_{W}}{g_{2}M_{S_{A}}^{2}}\ll 1$ when
$M_{S_{A}}$ is in multi-TeV regime. And one can see that
$g_{S_{B}^{m}S_{B}^{-m}h}=-\lambda_{\text{eff}}$ for septuplet. Therefore, one
can conclude that the constraint on parameter space in the dark matter
phenomenology is consistent with that by the precision measurement of $Zh$
production and $h\gamma\gamma$ decay rate.

Figure 11: Contours by $\left|\delta\sigma_{Zh}\right|\leq 0.5\%$ (cyan band
outlined with dashed lines) and $\left|\delta R_{h\gamma\gamma}\right|\leq
6.8\%$ (orange band outlined with solid lines) compared to the line (red
solid) for $\lambda_{\text{eff}}=0$ for septuplet with fixed values of
$M_{S_{A}}=9$ TeV and $M_{S_{B}}=300~{}\mathrm{GeV}$.

##  5 Conclusion

In this work we calculated the one-loop corrections on the Higgs-strahlung
process $e^{+}e^{-}\to Zh$ in the presence of an extended scalar sector. In
the case of zero or tiny VEV for the neutral scalar components, the BSM
contribution may be computed separately from the SM EW corrections, making the
calculation simpler than one might otherwise expect. We first classify the
extended scalar sector according to their $\mathrm{SU\left(2\right)_{L}}$
representation. We then exhibit universal analytic expressions for the one-
loop corrections in the $\overline{\mathrm{MS}}$ and on-shell schemes, and
subsequently apply the results to representative scalar sector extensions.

Since the Higgs-strahlung process $e^{+}e^{-}\to Zh$ and the Higgs diphoton
decay process $h\to\gamma\gamma$ are relatively sensitive to the BSM, we
investigated the BSM contributions running in loops to these two independent
processes to obtain the constraints on the scalar potentials in the extended
scalar models. The projected precision at the CEPC for the inclusive
$\sigma\left(Zh\right)$ and $h\to\gamma\gamma$ decay rate were assumed for
purposes of concrete illustration. We note that the precision for the $Zh$
measurement with the Higgs decaying to bottom quark pair further reduces
compared to the inclusive Higgs production. For instance, the estimated
precision of $\sigma\left(Zh\right)\times\mathrm{BR}\left(h\to
b\bar{b}\right)$ is 0.27% at the CEPC, which is approximately a half of the
precision for the inclusive one. One might use narrow width approximation
(NWA) to estimate the BSM contributions to the Higgs-strahlung process
accompanying $h\to b\bar{b}$. Provided the extended scalar sector does not
interact with the bottom quark, the BSM contributions to the full process in
NWA would be simply broken down to the BSM loop effects to
$\sigma\left(Zh\right)$ and BR$\left(h\to b\bar{b}\right)$ at LO. In this
scenario the contour area of $\left|\delta\sigma_{Zh}\right|$ in the Fig. 3 \-
11 constraining the parameter space in the extended scalar sector shrinks
accordingly with higher precision whereas the complementarity or degeneracy
between $Zh$ production and $h\to\gamma\gamma$ decay rate remains
qualitatively. Nevertheless, due to the NWA accuracy Uhlemann:2008pm  and the
Yukawa coupling between the extended scalars and the bottom quark subject to
$\mathcal{O}\left(\frac{m_{b}}{M_{W}}\right)$, the resulting uncertainty might
offset the improvement of the precision by introducing the subsequent $h\to
b\bar{b}$ channel.

Based on the numerical results, we found the following generic features in
extended scalar models. First, similar to the oblique $T$ parameter, the $Zh$
cross section rate is sensitive to the mass splitting between different
components of the multiplet. In the real triplet, quintuplet, and septuplet
models there is no mass splitting at tree level 121212In principle, there can
be mass splitting at tree level in the quintuplet and septuplet models, see
Eq. (16) and (17) in Ref. Chao:2018xwz  for generic definition of mass
eigenvalues, where the term proportional to $\lambda_{2}$ leads to mass
splitting between differently charged scalars. In our study, the $\lambda_{2}$
contribution is omitted, which is necessary for the neutral scalar to be the
DM candidate in the models.. In these cases, $\sigma(Zh)$ and the
$h\to\gamma\gamma$ signal rate are sensitive to similar regions of model
parameter space. The situation differs for the complex triplet model, wherein
the mass splitting arises at tree level and the two precision Higgs
observables provide complementary parameter space probes 131313 Refer to the
explanation in the case of the complex triplet model in Section 4.1 for cases
with mass splittings. When there is no mass splitting, the scalar mass can be
a direct input without mass splitting parameter(s), so that the BSM
contribution to the self energy in $\sigma\left(Zh\right)$ depends on the
scalar mass only. As both self energy and vertex corrections decrease with the
scalar mass, the vertex corrections to $\sigma\left(Zh\right)$ and
$h\to\gamma\gamma$ decay rate are sensitive to the triple Higgs couplings in a
similar manner with relatively large scalar masses.. Second, if the mass
spectrum in the multiplet has been identified at a hadron collider, one could
further extract information on the new scalar-Higgs couplings from the
precision Higgs observables analyzed here, because they depend on the new
scalar masses, gauge couplings (fixed by gauge invariance), and extended
scalar potential couplings. In principle, should the neutral component of the
new scalar electroweak multiplet contribute to the dark matter relic density,
these precision Higgs observables may further constrain the mass and
interactions of some portion of the dark matter.

More specifically,

  * •

In one class of models, such as the real scalar models, both the
$e^{+}e^{-}\to Zh$ and diphoton processes probe similar parameter regions in
the scalar potential. Thus, for example, should one observable yield a
significant difference from SM expectations while the other fall in agreement
with the SM, a particular model in this class may be disfavored.

  * •

In another class of models, such as the complex triplet model, both the
$e^{+}e^{-}\to Zh$ and $h\to\gamma\gamma$ processes probe complementary
regions of parameter space. To again follow our example hypothetical
experimental outcome, agreement with the SM in one observable, coupled with
disagreement in the other measurement could be accommodated in this case.

One may also perform the foregoing analysis of $\sigma(Zh)$ and the
$h\to\gamma\gamma$ signal rate for other versions of the two Higgs doublet
model and singlet scalar models. In these extended scenarios, the new scalar
may obtain a non-zero VEV, and thus the new scalar contribution cannot in
general be extracted from the SM one. For $\sigma(Zh)$ in particular, one must
calculate the full one-loop corrections, including both weak and QED
corrections. Such a computation for $\sigma(Zh)$ in the Minimal Supersymmetric
Standard Model Heinemeyer:2015qbu  and Two Higgs Doublet Model Xie:2018yiv
have recently appeared. The radiative corrections for $\sigma\left(Zh\right)$
in the Minimal Dilaton Model (MDM) 141414The Minimal Dilaton Model is an
extension of the SM by introducing one singlet scalar called dilaton. was
considered in Cao:2014ita , where partial contributions were neglected due to
their size in the MDM151515The negligible contributions include: (a) the
correction mediated by $t^{\prime}$ quark; (b) loops involving the $sZZ$
interaction but no possible large self-couplings among the scalars.. Looking
to the future, it would be interesting to compare the $Zh$ cross section in
the other extended scalar models with and without vanishing VEVs.

###### Acknowledgements.

J.H.Y. was supported by the National Natural Science Foundation of China
(NSFC) under Grants No. 12022514 and No. 11875003. J.H.Y. was also supported
by National Key Research and Development Program of China Grant No.
2020YFC2201501 and the NSFC under Grants No. 12047503. MJRM and JZ were
supported in part under U.S. Department of Energy contract No. DE-SC0011095.
MJRM was also supported in part under National Natural Science Foundation of
China grant No. 19Z103010239.

##  Appendix A Feynman Rules

The Feynman rules with regard to the scalar multiplets coupling to gauge
bosons are introduced via the kinetic term in the Lagrange
$\mathcal{L}_{\mathrm{kin}}^{{\Phi}}\left(D_{\mu}{\Phi}\right)^{\dagger}\left(D^{\mu}{\Phi}\right)$
with the covariant derivative defined in Eq. (5), and the
$\mathrm{SU\left(2\right)_{L}}$ generators $T^{a}$ in representation $j$ have
the following generic matrix form,

$\displaystyle
T^{+}=\left(\begin{array}[]{ccccccc}0&\sqrt{2j}&0&0&\ldots&0&0\\\
0&0&\sqrt{2(2j-1)}&0&\ldots&0&0\\\ 0&0&0&\sqrt{6(j-1)}&\ldots&0&0\\\
\vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\vdots\\\
0&0&0&0&\ldots&\sqrt{2(2j-1)}&0\\\ 0&0&0&0&\ldots&0&\sqrt{2j}\\\
0&0&0&0&\ldots&0&0\end{array}\right),$ (41) $\displaystyle
T^{-}=\left(\begin{array}[]{ccccccc}0&0&0&\ldots&0&0&0\\\
\sqrt{2j}&0&0&\ldots&0&0&0\\\ 0&\sqrt{2(2j-1)}&0&\ldots&0&0&0\\\
0&0&\sqrt{6(j-1)}&\ldots&0&0&0\\\
\vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\vdots\\\
0&0&0&\ldots&\sqrt{2(2j-1)}&0&0\\\
0&0&0&\ldots&0&\sqrt{2j}&0\end{array}\right),$ (49) $\displaystyle
T^{3}=\left(\begin{array}[]{cccccc}j{}&~{}0{}&~{}0{}&~{}\ldots{}&~{}0{}&~{}0\\\
0{}&~{}j-1{}&~{}0{}&~{}\ldots{}&~{}0{}&~{}0\\\
0{}&~{}0{}&~{}j-2{}&~{}\ldots{}&~{}0{}&~{}0\\\
\vdots{}&~{}\vdots{}&~{}\vdots{}&~{}\vdots{}&~{}\vdots{}&~{}\vdots\\\
0{}&~{}0{}&~{}0{}&~{}\ldots{}&~{}-j+1{}&~{}0\\\
0{}&~{}0{}&~{}0{}&~{}\ldots{}&~{}0{}&~{}-j\end{array}\right).$ (56)

Expanding the kinetic Lagrange $\mathcal{L}_{\mathrm{kin}}^{{\Phi}}$ with the
specific form of generator $T^{a}$ in representation $j$, it is easy to read
off the trilinear and quartic couplings. The generic gauge Feynman rules are
listed in Table. 3 below for the scalar multiplets,

Table 3: Gauge Feynman rules for the EW scalar multiplets Vertices | Couplings
---|---
$S^{\mp Q}S^{\pm Q\mp 1}W_{\mu}^{\pm}$ |  $ig_{2}N_{S^{\mp}Q}N_{S^{\pm Q\mp 1}}\sqrt{2}\sqrt{\left(j+m\right)\left(j-m+1\right)}\left(k_{S^{\mp Q}}-k_{S^{\pm Q\mp 1}}\right)_{\mu}$, $(1\leq Q\leq j+\frac{Y}{2})$
$S^{Q}S^{-Q}W_{\mu}^{+}W_{\nu}^{-}$ | $\pm 2g_{2}^{2}\left[j\left(j+1\right)-m^{2}\right]g_{\mu\nu}$
$S_{a}^{Q}S_{b}^{-Q}Z_{\mu}$ | $\left(i\right)^{\delta_{ab}}\left(g_{2}c_{W}m+\left(-1\right)^{\delta_{ab}}g_{1}s_{W}\frac{Y}{2}\right)\left(k_{S^{Q}}-k_{S^{-Q}}\right)_{\mu}$
$S_{a}^{Q}S_{b}^{-Q}\gamma_{\mu}$ | $\left(i\right)^{\delta_{ab}}\left(g_{2}s_{W}m-\left(-1\right)^{\delta_{ab}}g_{1}c_{W}\frac{Y}{2}\right)\left(k_{S^{Q}}-k_{S^{-Q}}\right)_{\mu}$
$S^{Q}S^{-Q}Z_{\mu}Z_{\nu}$ | $2\left(g_{2}c_{W}m-g_{1}s_{W}\frac{Y}{2}\right)^{2}g_{\mu\nu}$
$S^{Q}S^{-Q}\gamma_{\mu}\gamma_{\nu}$ | $2\left(g_{2}s_{W}m+g_{1}c_{W}\frac{Y}{2}\right)^{2}g_{\mu\nu}$
$S^{Q}S^{-Q}Z_{\mu}\gamma_{\nu}$ | $2\left(g_{2}c_{W}m-g_{1}s_{W}\frac{Y}{2}\right)\left(g_{2}s_{W}m+g_{1}c_{W}\frac{Y}{2}\right)g_{\mu\nu}$

where the normalized factor $N_{S^{\mp}Q}N_{S^{\pm Q\mp
1}}=\left\langle{\Phi^{j\pm m}}\right|\left.{\Phi^{j\pm m\mp
1}}\right\rangle$, $Q=m+\frac{Y}{2}$ is the charge of the scalar component and
positive definite in the first vertex that is indicated in parenthesis. The
Feynman rules in the scalar potential are more complicated as they are
representation dependent. In what follows, these Feynman rules are given in
Table 4 for each model under consideration.

Table 4: Feynman rules in the scalar potential for Inert Doublet, Real/Complex Triplet, Quintuplet and Septuplet Models Models | vertices | couplings
---|---|---
Inert Doublet: ${\Phi}=\begin{bmatrix}H^{+}\\\ \frac{1}{\sqrt{2}}\left(H^{0}+iA^{0}\right)\end{bmatrix}$ | $H^{+}H^{-}h$ | $-\lambda_{3}v_{\phi}$
$H^{0}H^{0}h$ | $-\left(\lambda_{3}+\lambda_{4}+\lambda_{5}\right)v_{\phi}$
$A^{0}A^{0}h$ | $-\left(\lambda_{3}+\lambda_{4}-\lambda_{5}\right)v_{\phi}$
$H^{+}H^{-}hh$ | $-\lambda_{3}$
$H^{0}H^{0}hh$ | $-\left(\lambda_{3}+\lambda_{4}+\lambda_{5}\right)$
$A^{0}A^{0}hh$ | $-\left(\lambda_{3}+\lambda_{4}-\lambda_{5}\right)$
Real Triplet: ${\Phi}=\begin{bmatrix}H^{+},~{}H^{0},~{}-H^{-}\end{bmatrix}^{T}$ | $H^{+}H^{-}h,~{}H^{0}H^{0}h$ | $-\lambda_{3}v_{\phi}$
$H^{+}H^{-}hh,~{}H^{0}H^{0}hh$ | $-\lambda_{3}$
Complex Triplet: ${\Phi}=\begin{bmatrix}H^{++}\\\ H^{+}\\\ \frac{1}{\sqrt{2}}\left(H+iA\right)\end{bmatrix}$ | $H^{++}H^{--}h$ | $-\lambda_{4}v_{\phi}$
$H^{+}H^{-}h$ | $-\left(\lambda_{4}+\frac{\lambda_{5}}{2}\right)v_{\phi}$
$HHh,~{}AAh$ | $-\left(\lambda_{4}+\lambda_{5}\right)v_{\phi}$
$H^{++}H^{--}hh$ | $-\lambda_{4}$
$H^{+}H^{-}hh$ | $-\left(\lambda_{4}+\frac{\lambda_{5}}{2}\right)$
$HHhh,~{}AAhh$ | $-\left(\lambda_{4}+\lambda_{5}\right)$
Quintuplet ($n=5$) & Septuplet ($n=7$): ${\Phi}=\begin{bmatrix}S_{A}^{j}+iS_{B}^{j}\\\ iS_{A}^{j-1}-S_{B}^{j-1}\\\ \vdots\\\ S_{A}^{-j}+iS_{B}^{-j}\end{bmatrix}$ | $S_{A}^{m}S_{A}^{-m}h,~{}\left(m=0,\ldots,j\right)$ | $-\left(\lambda_{1}+\frac{2}{\sqrt{n}}\lambda_{3}\right)v_{\phi}$
$S_{B}^{m}S_{B}^{-m}h,~{}\left(m=0,\ldots,j\right)$ | $-\left(\lambda_{1}-\frac{2}{\sqrt{n}}\lambda_{3}\right)v_{\phi}$
$S_{A}^{m}S_{A}^{-m}hh,~{}\left(m=0,\ldots,j\right)$ | $-\left(\lambda_{1}+\frac{2}{\sqrt{n}}\lambda_{3}\right)$
$S_{B}^{m}S_{B}^{-m}hh,~{}\left(m=0,\ldots,j\right)$ | $-\left(\lambda_{1}-\frac{2}{\sqrt{n}}\lambda_{3}\right)$

##  Appendix B Potentials in Specific Models

Let us discuss the models with the extended scalar sector of dimension $n$.
For the models under consideration in the context ($n=2,3,5,7$), there is an
imposition of $\mathcal{Z}_{2}$ symmetry to have stable neutral component as
DM candidate161616$\mathcal{Z}_{2}$ symmetry is protected in complex triplet
model when the triplet has vanishing VEV..

1\. Real and Complex Singlet model Barger:2007im ; Barger:2008jx : In this
case, usually $Y=0$ is taken, and thus additional term exists:

$\displaystyle\left({\overline{H}}H\right)_{0}\Phi,\quad\left({\overline{\Phi}}\Phi\right)\Phi,\quad\Phi^{3}.$
(57)

The scalar potential is

  * •

real scalar singlet

$\displaystyle V\left(H,{\Phi}\right)=$
$\displaystyle\frac{m^{2}}{2}{H}^{\dagger}{H}+\frac{\lambda}{4}\left({H}^{\dagger}{H}\right)^{2}+\frac{\delta_{1}}{2}{H}^{\dagger}{H}{\Phi}+\frac{\delta_{2}}{2}{H}^{\dagger}{H}{\Phi}^{2}$
(58)
$\displaystyle+\left(\frac{\delta_{1}m^{2}}{2\lambda}\right){\Phi}+\frac{\kappa_{2}}{2}{\Phi}^{2}+\frac{\kappa_{3}}{3}{\Phi}^{3}+\frac{\kappa_{4}}{4}{\Phi}^{4}.$

  * •

complex scalar singlet

$\displaystyle V\left(H,{\Phi}\right)=$
$\displaystyle\frac{m^{2}}{2}{H}^{\dagger}{H}+\frac{\lambda}{4}\left({H}^{\dagger}{H}\right)^{2}+\left(\frac{\left|\delta_{1}\right|e^{i\phi_{\delta_{1}}}}{4}{H}^{\dagger}{H}{\Phi}+\mathrm{h.c.}\right)+\frac{\delta_{2}}{2}{H}^{\dagger}{H}\left|{\Phi}\right|^{2}$
(59)
$\displaystyle+\left(\frac{\left|\delta_{3}\right|e^{i\phi_{\delta_{3}}}}{4}{H}^{\dagger}{H}{\Phi}^{2}+\mathrm{h.c.}\right)+\left(\left|a_{1}\right|e^{i\phi_{a_{1}}}{\Phi}+\mathrm{h.c.}\right)+\left(\frac{\left|b_{1}\right|e^{i\phi_{b_{1}}}}{4}{\Phi}^{2}+\mathrm{h.c.}\right)$
$\displaystyle+\frac{b_{2}}{2}\left|{\Phi}\right|^{2}+\left(\frac{\left|c_{1}\right|e^{i\phi_{c_{1}}}}{6}{\Phi}^{3}+\mathrm{h.c.}\right)+\left(\frac{\left|c_{2}\right|e^{i\phi_{c_{2}}}}{6}{\Phi}\left|{\Phi}\right|^{2}+\mathrm{h.c.}\right)$
$\displaystyle+\left(\frac{\left|d_{1}\right|e^{i\phi_{d_{1}}}}{8}{\Phi}^{4}+\mathrm{h.c.}\right)+\left(\frac{\left|d_{3}\right|e^{i\phi_{d_{3}}}}{8}{\Phi}^{2}\left|{\Phi}\right|^{2}+\mathrm{h.c.}\right)+\frac{d_{2}}{4}\left|{\Phi}\right|^{4}$

2\. Two Higgs Doublet model Ma:2006km ; Barbieri:2006dq ; Branco:2011iw : For
the complex doublet $n=2,Y=+1$, additional terms appear

$\displaystyle\left[\left(HH\right)_{1}\left({\overline{\Phi}}~{}{\overline{\Phi}}\right)_{1}\right]_{0},\quad\left[\left(HH\right)_{1}\left({\overline{H}}~{}{\overline{\Phi}}\right)_{1}\right]_{0}\quad\left[\left(H\Phi\right)_{1}\left({\overline{\Phi}}~{}{\overline{\Phi}}\right)_{1}\right]_{0}.$
(60)

If $n=2,Y=-1$, we have the above expression with $\Phi$ replaced by
$\overline{\Phi}$. Thus the scalar potential in the two Higgs doublet model is
written as

$\displaystyle V\left({H},{\Phi}\right)=$
$\displaystyle~{}\mu_{1}^{2}~{}{H}^{\dagger}{H}+\mu_{2}^{2}~{}{\Phi}^{\dagger}{\Phi}+\mu_{12}^{2}\left({\Phi}^{\dagger}{H}+\mathrm{h.c.}\right)+\lambda_{1}\left({H}^{\dagger}{H}\right)^{2}+\lambda_{2}\left({\Phi}^{\dagger}{\Phi}\right)^{2}$
(61)
$\displaystyle+\lambda_{3}\left({H}^{\dagger}{H}\right)\left({\Phi}^{\dagger}{\Phi}\right)+\lambda_{4}\left({H}^{\dagger}{\Phi}\right)\left({\Phi}^{\dagger}{H}\right)+\left[\frac{\lambda_{5}}{2}\left({H}^{\dagger}{\Phi}\right)^{2}+\mathrm{h.c.}\right],$

where the soft $\mathcal{Z}_{2}$ breaking term (the one $\propto\mu_{12}^{2}$)
is dropped in the inert doublet model.

3\. Real Triplet model Blank:1997qa ; FileviezPerez:2008bj ; Chen:2008jg :
Here we consider $Y=0$. For real triplet $n=3,Y=0$, there are additional term

$\displaystyle\left[\left({\overline{H}}H\right)_{1}\Phi\right]_{0}.$ (62)

Thus the general potential is

$\displaystyle V\left({H},{\Phi}\right)=$
$\displaystyle~{}\mu_{1}^{2}~{}{H}^{\dagger}{H}+\frac{\mu_{2}^{2}}{2}~{}{\Phi}^{\dagger}{\Phi}+\lambda_{1}\left({H}^{\dagger}{H}\right)^{2}+\frac{\lambda_{2}}{4}\left({\Phi}^{\dagger}{\Phi}\right)^{2}+\frac{\lambda_{3}}{2}\left({H}^{\dagger}{H}\right)\left({\Phi}^{\dagger}{\Phi}\right).$
(63)

4\. Complex Triplet model AbdusSalam:2013eya ; Konetschny:1977bn ; Magg:1980ut
; Schechter:1980gr ; Cheng:1980qt : For the complex triplet, there is another
possibility with $n=3,Y=\pm 2$, additional terms appear

$\displaystyle\left[\left({\overline{H}}H\right)_{1}\overline{\Phi}\right]_{0},\quad\left[\left({\overline{H}}H\right)_{1}\Phi\right]_{0}.$
(64)

The general potential is

$\displaystyle V\left({H},{\Phi}\right)=$
$\displaystyle~{}\mu_{1}^{2}~{}{H}^{\dagger}{H}+\mu_{2}^{2}~{}{\Phi}^{\dagger}{\Phi}+\lambda_{1}\left({H}^{\dagger}{H}\right)^{2}+\lambda_{2}\left({\Phi}^{\dagger}{\Phi}\right)^{2}+\lambda_{3}\left|{\Phi}^{\dagger}T^{a}{\Phi}\right|^{2}$
(65)
$\displaystyle+\lambda_{4}{H}^{\dagger}{H}{\Phi}^{\dagger}{\Phi}+\lambda_{5}{H}^{\dagger}\tau^{a}{H}{\Phi}^{\dagger}T^{a}{\Phi}.$

It is customary to write the complex scalar triplet fields in a $2\times 2$
representation $\Delta$. Following the notation of Du:2018eaw  with
$Y=2$171717$Y=-2$ gives the multiplet that is conjugate to the one with
$Y=2$., the potential is written

$\displaystyle V\left({H},\Delta\right)=$
$\displaystyle-m^{2}{H}^{\dagger}{H}+M^{2}\mathrm{Tr}\left(\Delta^{\dagger}\Delta\right)+\left(\mu{H}^{\mathrm{T}}i\tau_{2}\Delta^{\dagger}{H}+\mathrm{h.c.}\right)+\lambda_{1}\left({H}^{\dagger}{H}\right)^{2}+\lambda_{2}\mathrm{Tr}^{2}\left(\Delta^{\dagger}\Delta\right)$
(66)
$\displaystyle+\lambda_{3}\mathrm{Tr}\left(\Delta^{\dagger}\Delta\Delta^{\dagger}\Delta\right)+\lambda_{4}\left({H}^{\dagger}{H}\right)\mathrm{Tr}\left(\Delta^{\dagger}\Delta\right)+\lambda_{5}{H}^{\dagger}\Delta\Delta^{\dagger}{H},$

with

$\Delta=\left(\begin{array}[]{cc}\frac{\Delta^{+}}{\sqrt{2}}&H^{++}\\\
\frac{1}{\sqrt{2}}\left(\delta+v_{\Delta}+i\eta\right)&-\frac{\Delta^{+}}{\sqrt{2}}\end{array}\right)$

The relation between the $3\times 1$ and $2\times 2$ representation is

$\Delta={\Phi}\cdot\tau^{a}.$

5\. Complex Quadruplet model AbdusSalam:2013eya : For complex triplet
$n=4,Y=\pm 1$, additional terms appear

$\displaystyle\left[\left(HH\right)_{1}\left({\overline{H}}~{}{\overline{\Phi}}\right)_{1}\right]_{0},\quad\left[\left(HH\right)_{1}\left({\overline{H}}{\Phi}\right)_{1}\right]_{0},$
(67)

and for $Y=\pm 3$

$\displaystyle\left[\left(HH\right)_{1}\left(H{\overline{\Phi}}\right)_{1}\right]_{0},\quad\left[\left(HH\right)_{1}\left(H{\Phi}\right)_{1}\right]_{0}.$
(68)

The scalar potential is

$\displaystyle V\left({H},{\Phi}\right)=$
$\displaystyle~{}\mu_{1}^{2}~{}{H}^{\dagger}{H}+\mu_{2}^{2}~{}{\Phi}^{\dagger}{\Phi}+\lambda_{1}\left({H}^{\dagger}{H}\right)^{2}+\lambda_{2}\left({\Phi}^{\dagger}{\Phi}\right)^{2}+\lambda_{3}\left|{\Phi}^{\dagger}T^{a}{\Phi}\right|^{2}$
(69)
$\displaystyle+\alpha{H}^{\dagger}{H}{\Phi}^{\dagger}{\Phi}+\beta{H}^{\dagger}\tau^{a}{H}{\Phi}^{\dagger}T^{a}{\Phi}+\gamma\left[\left({H}^{T}\epsilon\tau^{a}{H}\right)\left({\Phi}^{T}CT^{a}{\Phi}\right)^{\dagger}+\text{h.c.}\right],$

where $C$ is an antisymmetric matrix defined as

$CT^{a}C^{-1}=-T^{aT},$

and its explicit form as well as that of $\epsilon$ can be found in appendix
(6.2) in AbdusSalam:2013eya .

6\. Complex and Real Quintuplet and Septuplet models Chao:2018xwz : For
complex $n=5,7$ and $Y=0$, additional terms appear

$\displaystyle\left[\left(\Phi{\Phi}\right)_{J}\Phi\right]_{0},\quad\left[\left(\Phi{\Phi}\right)_{J}{\overline{\Phi}}\right]_{0}.$
(70)

The scalar potential is

$\displaystyle V\left({H},{\Phi}\right)=$
$\displaystyle~{}-\mu^{2}~{}{H}^{\dagger}{H}+M_{A}^{2}\left({\Phi}^{\dagger}{\Phi}\right)+\left[M_{B}^{2}\left({\Phi}{\Phi}\right)_{0}+\mathrm{h.c.}\right]+\lambda\left({H}^{\dagger}{H}\right)^{2}$
(71)
$\displaystyle+\lambda_{1}\left({H}^{\dagger}{H}\right)\left({\Phi}^{\dagger}{\Phi}\right)+\lambda_{2}\left[\left(\overline{{H}}{H}\right)_{1}\left(\overline{{\Phi}}{\Phi}\right)_{1}\right]+\left[\lambda_{3}\left(\overline{{H}}{H}\right)_{0}\left(\overline{{\Phi}}{\Phi}\right)_{0}+\mathrm{h.c.}\right]$

It was, however, noted that for the quintuplet ($n=5$) the stability of the DM
is ensured by either the imposition of $\mathcal{Z}_{2}$ symmetry or a heavy
mass scale $\Lambda$ well above the Planck scale inversely in a non-
renormalizable dimension five operator; while for the septuplet ($n=7$) DM
stability could be ensured by choosing $\Lambda$ below the Planck scale.

7\. Generic models Chao:2018xwz : For generic multiplets other than special
cases above, the general potential (except the mass terms) is written as

  * •

Integer isospin $j$

$\displaystyle V\left({H},{\Phi}\right)=$
$\displaystyle\sum_{J=0}^{2j}\lambda_{J}\left[\left({\Phi}{{\Phi}}\right)_{J}\left({\overline{\Phi}}\,{\overline{\Phi}}\right)_{J}\right]_{0}+\alpha\left({H}^{\dagger}{H}\right)\left({\Phi}^{\dagger}{\Phi}\right)+\beta\left[\left({\overline{H}}{H}\right)_{1}\left({\overline{\Phi}}{\Phi}\right)_{1}\right]_{0}$
(72)
$\displaystyle+\delta_{0,Y}\Bigg{\\{}\sum_{\kappa=0}^{2j}\left\\{\lambda_{\kappa}^{\prime}\left[\left({\Phi}{\Phi}\right)_{\kappa}\left({\Phi}{\Phi}\right)_{\kappa}\right]_{0}+\lambda_{\kappa}^{\prime\prime}\left[\left(\overline{{\Phi}}{\Phi}\right)_{\kappa}\left({\Phi}{\Phi}\right)_{\kappa}\right]_{0}+\mathrm{h.c.}\right\\}+\left[\gamma\left(\overline{{H}}{H}\right)_{0}\left({\Phi}{\Phi}\right)_{0}+\mathrm{h.c.}\right]$
$\displaystyle+\delta_{0,j\left(\text{mod
2}\right)}\left[\mu_{1}\left({\Phi}\left({\Phi}{\Phi}\right)_{j}\right)_{0}+\mu_{2}\left(\overline{{\Phi}}\left({\Phi}{\Phi}\right)_{j}\right)_{0}+\text{h.c.}\right]\Bigg{\\}}$

  * •

Half integer isospin $j$

$\displaystyle V\left({H},{\Phi}\right)=$
$\displaystyle\sum_{J=1}^{2j}\lambda_{J}\left[\left({\Phi}{{\Phi}}\right)_{J}\left({\overline{\Phi}}\,{\overline{\Phi}}\right)_{J}\right]_{0}+\alpha\left({H}^{\dagger}{H}\right)\left({\Phi}^{\dagger}{\Phi}\right)+\beta\left[\left({\overline{H}}{H}\right)_{1}\left({\overline{\Phi}}{\Phi}\right)_{1}\right]_{0}$
(73)
$\displaystyle+\delta_{0,Y}\Bigg{\\{}\sum_{\kappa=1}^{2j}\left\\{\lambda_{\kappa}^{\prime}\left[\left({\Phi}{\Phi}\right)_{\kappa}\left({\Phi}{\Phi}\right)_{\kappa}\right]_{0}+\lambda_{\kappa}^{\prime\prime}\left[\left(\overline{{\Phi}}{\Phi}\right)_{\kappa}\left({\Phi}{\Phi}\right)_{\kappa}\right]_{0}+\mathrm{h.c.}\right\\}$
$\displaystyle+\left\\{\gamma\left[\left(\overline{{H}}{H}\right)_{1}\left({\Phi}{\Phi}\right)_{1}\right]_{0}+\mathrm{h.c.}\right\\}\Bigg{\\}}+\delta_{1,Y}\Bigg{\\{}\Big{\\{}\gamma_{1}\left[\left({H}{\Phi}\right)_{j+\frac{1}{2}}\left(\overline{{\Phi}}~{}\overline{{\Phi}}\right)_{j+\frac{1}{2}}\right]_{0}\delta_{0,j-\frac{1}{2}~{}\left(\text{mod
2}\right)}$
$\displaystyle+\gamma_{2}\left[\left({H}{\Phi}\right)_{j-\frac{1}{2}}\left(\overline{{\Phi}}~{}\overline{{\Phi}}\right)_{j-\frac{1}{2}}\right]_{0}\delta_{0,j+\frac{1}{2}~{}\left(\text{mod
2}\right)}+\text{h.c.}\Big{\\}}+\Big{\\{}\kappa_{1}\left[\left({H}{H}\right)_{1}\left(\overline{{\Phi}}~{}\overline{{\Phi}}\right)_{1}\right]_{0}+\text{h.c.}\Big{\\}}\Bigg{\\}}$
$\displaystyle+\delta_{-1,Y}\Bigg{\\{}\Big{\\{}\gamma_{1}\left[\left({H}\overline{{\Phi}}\right)_{j+\frac{1}{2}}\left({\Phi}{\Phi}\right)_{j+\frac{1}{2}}\right]_{0}\delta_{0,j-\frac{1}{2}~{}\left(\text{mod
2}\right)}+\gamma_{2}\left[\left({H}\overline{{\Phi}}\right)_{j-\frac{1}{2}}\left({\Phi}{\Phi}\right)_{j-\frac{1}{2}}\right]_{0}\delta_{0,j+\frac{1}{2}~{}\left(\text{mod
2}\right)}$
$\displaystyle+\text{h.c.}\Big{\\}}+\Big{\\{}\kappa_{1}\left[\left({H}{H}\right)_{1}\left({\Phi}{\Phi}\right)_{1}\right]_{0}+\text{h.c.}\Big{\\}}\Bigg{\\}}$

##  Appendix C Self Energy and Vertex Corrections

Since in the models under consideration the neutral components in the scalar
multiplet sector decouple to that in the SM, one can treat the scalar
multiplet loop contributions independently. The self energy and vertex loop
contributions are written in terms of the one-, two- and three-point
integrals. In the following the relevant scalar and vector/tensor integrals in
D-dimension ($D=4-2\epsilon$), self energy and vertex corrections in each
model are given, respectively.

###  C.1 The N-point Integrals with $N\leq 3$

The scalar one-point integral is

$\frac{i}{16\pi^{2}}A_{0}\left(m\right)=\mu^{2\epsilon}\int\frac{d^{D}l}{\left(2\pi\right)^{D}}\frac{1}{l^{2}-m^{2}},$
(74)

and $A_{0}\left(m\right)$ has simple form

$A_{0}\left(m\right)=m^{2}\left(\frac{1}{\epsilon}+1+\log\frac{\mu^{2}}{m^{2}}\right)+\mathcal{O}\left(\epsilon\right).$

The scalar and vector two-point integrals $B_{0,1}$ are defined as

$\frac{i}{16\pi^{2}}\left(B_{0},B_{\mu}\right)\left(k,m_{1},m_{2}\right)=\mu^{2\epsilon}\int\frac{d^{D}l}{\left(2\pi\right)^{D}}\frac{\left(1,l_{\mu}\right)}{\left(l^{2}-m_{1}^{2}\right)\left[\left(l+k\right)^{2}-m_{2}^{2}\right]},$
(75)

with $B_{\mu}=k_{\mu}B_{1}$ and

$\displaystyle B_{0}\left(k,m_{1},m_{2}\right)$
$\displaystyle=\frac{1}{\epsilon}-\int_{0}^{1}dx\log\frac{x^{2}k^{2}-x\left(k^{2}+m_{1}^{2}-m_{2}^{2}\right)+m_{1}^{2}-i\epsilon}{\mu^{2}}+\mathcal{O}\left(\epsilon\right),$
$\displaystyle B_{1}\left(k,m_{1},m_{2}\right)$
$\displaystyle=\frac{1}{2k^{2}}\left[A_{0}\left(m_{1}\right)-A_{0}\left(m_{2}\right)-B_{0}\left(k,m_{1},m_{2}\right)\left(k^{2}+m_{1}^{2}-m_{2}^{2}\right)\right].$

The scalar and (rank 2) tensor three-point integrals are

$\frac{i}{16\pi^{2}}\left(C_{0},C_{\mu\nu}\right)\left(k_{1},k_{2},m_{1},m_{2},m_{3}\right)=\mu^{2\epsilon}\int\frac{d^{D}l}{\left(2\pi\right)^{D}}\frac{\left(1,l_{\mu}l_{\nu}\right)}{\left(l^{2}-m_{1}^{2}\right)\left[\left(l+k_{1}\right)^{2}-m_{2}^{2}\right]\left[\left(l+k_{1}+k_{2}\right)^{2}-m_{3}^{2}\right]},$
(76)

with

$C_{\mu\nu}=k_{1\mu}k_{1\nu}C_{21}+k_{2\mu}k_{2\nu}C_{22}+\left(k_{1\mu}k_{2\nu}+k_{2\mu}k_{1\nu}\right)C_{23}+g_{\mu\nu}C_{24},$

where $C_{22},~{}C_{23},~{}C_{24}$ are the relevant contributions entering our
calculation and their explicit expressions with specific arguments are given
in C.3.

###  C.2 Gauge Boson Self energies

In the following the gauge boson self energies involving the virtual scalar
multiplet contributions are given in the models below.

  1. 1.

Inert Double

$\displaystyle\Sigma_{T}^{ZZ}\left(k^{2}\right)$
$\displaystyle=\frac{1}{16\pi^{2}}\frac{1}{3}\times\Bigg{\\{}+\frac{e^{2}\left(c_{W}^{2}-s_{W}^{2}\right)^{2}}{4c_{W}^{2}s_{W}^{2}}\left[4M_{H^{\pm}}^{2}B_{0}\left(0,M_{H^{\pm}},M_{H^{\pm}}\right)+\frac{2}{3}k^{2}+\left(k^{2}-4M_{H^{\pm}}^{2}\right)B_{0}\left(k,M_{H^{\pm}},M_{H^{\pm}}\right)\right]$
$\displaystyle+\frac{e^{2}}{4c_{W}^{2}s_{W}^{2}}\bigg{[}-\frac{\left(M_{H^{0}}^{2}-M_{A^{0}}^{2}\right)^{2}}{k^{2}}-\frac{M_{H^{0}}^{2}-M_{A^{0}}^{2}}{k^{2}}\left[M_{H^{0}}^{2}B_{0}\left(0,M_{H^{0}},M_{H^{0}}\right)-M_{A^{0}}^{2}B_{0}\left(0,M_{A^{0}},M_{A^{0}}\right)\right]$
$\displaystyle+2\left[M_{H^{0}}^{2}B_{0}\left(0,M_{H^{0}},M_{H^{0}}\right)+M_{A^{0}}^{2}B_{0}\left(0,M_{A^{0}},M_{A^{0}}\right)\right]+\frac{2}{3}k^{2}+\frac{\left(M_{H^{0}}^{2}-M_{A^{0}}^{2}\right)^{2}}{k^{2}}B_{0}\left(k,M_{H^{0}},M_{A^{0}}\right)$
$\displaystyle+\left(k^{2}-2M_{H^{0}}^{2}-2M_{A^{0}}^{2}\right)B_{0}\left(k,M_{H^{0}},M_{A^{0}}\right)\bigg{]}\Bigg{\\}}$
$\displaystyle\Sigma_{T}^{\gamma Z}\left(k^{2}\right)$
$\displaystyle=\frac{1}{16\pi^{2}}\frac{1}{3}\times\Bigg{\\{}+\frac{e^{2}\left(c_{W}^{2}-s_{W}^{2}\right)}{2c_{W}s_{W}}\left[4M_{H^{\pm}}^{2}B_{0}\left(0,M_{H^{\pm}},M_{H^{\pm}}\right)+\frac{2}{3}k^{2}+\left(k^{2}-4M_{H^{\pm}}^{2}\right)B_{0}\left(k,M_{H^{\pm}},M_{H^{\pm}}\right)\right]\Bigg{\\}}$
$\displaystyle\Sigma_{T}^{\gamma\gamma}\left(k^{2}\right)$
$\displaystyle=\frac{1}{16\pi^{2}}\frac{1}{3}\times\Bigg{\\{}+e^{2}\left[4M_{H^{\pm}}^{2}B_{0}\left(0,M_{H^{\pm}},M_{H^{\pm}}\right)+\frac{2}{3}k^{2}+\left(k^{2}-4M_{H^{\pm}}^{2}\right)B_{0}\left(k,M_{H^{\pm}},M_{H^{\pm}}\right)\right]\Bigg{\\}}$
$\displaystyle\Sigma_{T}^{WW}\left(k^{2}\right)$
$\displaystyle=\frac{1}{16\pi^{2}}\frac{1}{3}\times\frac{e^{2}}{4s_{W}^{2}}\Bigg{\\{}-\frac{\left(M_{H^{\pm}}^{2}-M_{H^{0}}^{2}\right)^{2}}{k^{2}}B_{0}\left(0,M_{H^{\pm}},M_{H^{0}}\right)-\frac{\left(M_{H^{\pm}}^{2}-M_{A^{0}}^{2}\right)^{2}}{k^{2}}B_{0}\left(0,M_{H^{\pm}},M_{A^{0}}\right)$
$\displaystyle+4M_{H^{\pm}}^{2}B_{0}\left(0,M_{H^{\pm}},M_{H^{\pm}}\right)+2M_{H^{0}}^{2}B_{0}\left(0,M_{H^{0}},M_{H^{0}}\right)+2M_{A^{0}}^{2}B_{0}\left(0,M_{A^{0}},M_{A^{0}}\right)+\frac{4}{3}k^{2}$
$\displaystyle+\frac{1}{k^{2}}\left[\left(M_{H^{\pm}}-M_{H^{0}}\right)^{2}-k^{2}\right]\left[\left(M_{H^{\pm}}+M_{H^{0}}\right)^{2}-k^{2}\right]B_{0}\left(k,M_{H^{\pm}},M_{H^{0}}\right)$
$\displaystyle+\frac{1}{k^{2}}\left[\left(M_{H^{\pm}}-M_{A^{0}}\right)^{2}-k^{2}\right]\left[\left(M_{H^{\pm}}+M_{A^{0}}\right)^{2}-k^{2}\right]B_{0}\left(k,M_{H^{\pm}},M_{A^{0}}\right)\Bigg{\\}}$
$\displaystyle\Sigma^{H}\left(k^{2}\right)$
$\displaystyle=\frac{1}{16\pi^{2}}\times\Bigg{\\{}+\lambda_{3}A_{0}\left(M_{H^{\pm}}\right)+\lambda_{3}^{2}v_{\phi}^{2}B_{0}\left(k,M_{H^{\pm}},M_{H^{\pm}}\right)$
$\displaystyle+\frac{1}{2}\left[\lambda_{L}A_{0}\left(M_{H^{0}}\right)+\lambda_{L}^{2}v_{\phi}^{2}B_{0}\left(k,M_{H^{0}},M_{H^{0}}\right)+\lambda_{A}A_{0}\left(M_{A^{0}}\right)+\lambda_{A}^{2}v_{\phi}^{2}B_{0}\left(k,M_{A^{0}},M_{A^{0}}\right)\right]\Bigg{\\}}$
(77)

  2. 2.

Real Triplet ($V_{1}V_{2}=ZZ,\gamma Z,WW,\gamma\gamma$)

$\displaystyle\Sigma_{T}^{V_{1}V_{2}}\left(k^{2}\right)$
$\displaystyle=\frac{1}{16\pi^{2}}~{}\frac{1}{3}~{}g_{ssV_{1}}g_{ssV_{2}}\left[4M_{S}^{2}B_{0}\left(0,M_{S},M_{S}\right)+\frac{2}{3}k^{2}+\left(k^{2}-4M_{S}^{2}\right)B_{0}\left(k,M_{S},M_{S}\right)\right],$
(78) $\displaystyle\Sigma^{H}\left(k^{2}\right)$
$\displaystyle=\frac{1}{16\pi^{2}}\frac{3}{2}\left[\lambda_{3}A_{0}\left(M_{S}\right)+\lambda_{3}^{2}v^{2}B_{0}\left(k,M_{S},M_{S}\right)\right].$

  3. 3.

Complex Triplet

$\displaystyle\begin{aligned}
\Sigma_{T}^{ZZ}\left(k^{2}\right)&=\frac{1}{16\pi^{2}}\frac{1}{3}\times\Bigg{\\{}+\frac{e^{2}s_{W}^{2}}{c_{W}^{2}}\left[4M_{H^{\pm}}^{2}B_{0}\left(0,M_{H^{\pm}},M_{H^{\pm}}\right)+\frac{2}{3}k^{2}+\left(k^{2}-4M_{H^{\pm}}^{2}\right)B_{0}\left(k,M_{H^{\pm}},M_{H^{\pm}}\right)\right]\\\
&+\frac{e^{2}\left(c_{W}^{2}-s_{W}^{2}\right)^{2}}{c_{W}^{2}s_{W}^{2}}\left[4M_{H^{\pm\pm}}^{2}B_{0}\left(0,M_{H^{\pm\pm}},M_{H^{\pm\pm}}\right)+\frac{2}{3}k^{2}+\left(k^{2}-4M_{H^{\pm\pm}}^{2}\right)B_{0}\left(k,M_{H^{\pm\pm}},M_{H^{\pm\pm}}\right)\right]\\\
&+\frac{e^{2}}{c_{W}^{2}s_{W}^{2}}\left[4M_{\Delta}^{2}B_{0}\left(0,M_{\Delta},M_{\Delta}\right)+\frac{2}{3}k^{2}+\left(k^{2}-4M_{\Delta}^{2}\right)B_{0}\left(k,M_{\Delta},M_{\Delta}\right)\right]\Bigg{\\}}\end{aligned}$
(79) $\displaystyle\begin{aligned} \Sigma_{T}^{\gamma
Z}\left(k^{2}\right)&=\frac{1}{16\pi^{2}}\frac{1}{3}\times\Bigg{\\{}-\frac{e^{2}s_{W}}{c_{W}}\left[4M_{H^{\pm}}^{2}B_{0}\left(0,M_{H^{\pm}},M_{H^{\pm}}\right)+\frac{2}{3}k^{2}+\left(k^{2}-4M_{H^{\pm}}^{2}\right)B_{0}\left(k,M_{H^{\pm}},M_{H^{\pm}}\right)\right]\\\
&+\frac{2e^{2}\left(c_{W}^{2}-s_{W}^{2}\right)}{c_{W}s_{W}}\left[4M_{H^{\pm\pm}}^{2}B_{0}\left(0,M_{H^{\pm\pm}},M_{H^{\pm\pm}}\right)+\frac{2}{3}k^{2}+\left(k^{2}-4M_{H^{\pm\pm}}^{2}\right)B_{0}\left(k,M_{H^{\pm\pm}},M_{H^{\pm\pm}}\right)\right]\Bigg{\\}}\end{aligned}$
$\displaystyle\begin{aligned}
\Sigma_{T}^{\gamma\gamma}\left(k^{2}\right)&=\frac{1}{16\pi^{2}}\frac{1}{3}\times\Bigg{\\{}+e^{2}\left[4M_{H^{\pm}}^{2}B_{0}\left(0,M_{H^{\pm}},M_{H^{\pm}}\right)+\frac{2}{3}k^{2}+\left(k^{2}-4M_{H^{\pm}}^{2}\right)B_{0}\left(k,M_{H^{\pm}},M_{H^{\pm}}\right)\right]\\\
&+4e^{2}\left[4M_{H^{\pm\pm}}^{2}B_{0}\left(0,M_{H^{\pm\pm}},M_{H^{\pm\pm}}\right)+\frac{2}{3}k^{2}+\left(k^{2}-4M_{H^{\pm\pm}}^{2}\right)B_{0}\left(k,M_{H^{\pm\pm}},M_{H^{\pm\pm}}\right)\right]\Bigg{\\}}\end{aligned}$
$\displaystyle\begin{aligned}
\Sigma_{T}^{WW}\left(k^{2}\right)&=\frac{1}{16\pi^{2}}\frac{1}{3}\times\frac{e^{2}}{s_{W}^{2}}\Bigg{\\{}-\frac{\left(M_{H^{\pm}}^{2}-M_{\Delta}^{2}\right)^{2}}{k^{2}}B_{0}\left(0,M_{H^{\pm}},M_{\Delta}\right)-\frac{\left(M_{H^{\pm\pm}}^{2}-M_{H^{\pm}}^{2}\right)^{2}}{k^{2}}B_{0}\left(0,M_{H^{\pm\pm}},M_{H^{\pm}}\right)\\\
&+2M_{\Delta}^{2}B_{0}\left(0,M_{\Delta},M_{\Delta}\right)+4M_{H^{\pm}}^{2}B_{0}\left(0,M_{H^{\pm}},M_{H^{\pm}}\right)+2M_{H^{\pm\pm}}^{2}B_{0}\left(0,M_{H^{\pm\pm}},M_{H^{\pm\pm}}\right)+\frac{4}{3}k^{2}\\\
&+\frac{1}{k^{2}}\left[\left(M_{H^{\pm}}-M_{\Delta}\right)^{2}-k^{2}\right]\left[\left(M_{H^{\pm}}+M_{\Delta}\right)^{2}-k^{2}\right]B_{0}\left(k,M_{H^{\pm}},M_{\Delta}\right)\\\
&+\frac{1}{k^{2}}\left[\left(M_{H^{\pm\pm}}-M_{H^{\pm}}\right)^{2}-k^{2}\right]\left[\left(M_{H^{\pm\pm}}+M_{H^{\pm}}\right)^{2}-k^{2}\right]B_{0}\left(k,M_{H^{\pm\pm}},M_{H^{\pm}}\right)\Bigg{\\}}\end{aligned}$
$\displaystyle\begin{aligned}
\Sigma^{H}\left(k^{2}\right)&=\frac{1}{16\pi^{2}}\times\Bigg{\\{}+\left(\lambda_{4}+\frac{\lambda_{5}}{2}\right)A_{0}\left(M_{H^{\pm}}\right)+\left(\lambda_{4}+\frac{\lambda_{5}}{2}\right)^{2}v_{\phi}^{2}B_{0}\left(k,M_{H^{\pm}},M_{H^{\pm}}\right)\\\
&+\lambda_{4}A_{0}\left(M_{H^{\pm\pm}}\right)+\lambda_{4}^{2}v_{\phi}^{2}B_{0}\left(k,M_{H^{\pm\pm}},M_{H^{\pm\pm}}\right)\\\
&+\left(\lambda_{4}+\lambda_{5}\right)A_{0}\left(M_{\Delta}\right)+\left(\lambda_{4}+\lambda_{5}\right)^{2}v_{\phi}^{2}B_{0}\left(k,M_{\Delta},M_{\Delta}\right)\Bigg{\\}}\end{aligned}$

  4. 4.

Quintuplet ($n=5$) and Septuplet ($n=7$)

$\displaystyle\Sigma_{T}^{V_{1}V_{2}}\left(k^{2}\right)=$
$\displaystyle\frac{1}{16\pi^{2}}\frac{1}{3}\times\Bigg{\\{}\sum_{i=A}^{B}\left(\sum_{m=0}^{m=\frac{n-1}{2}}g_{S_{i}^{m}S_{i}^{-m+q}V_{1}}\cdot
g_{S_{i}^{m}S_{i}^{-m+q}V_{2}}\right)\Big{[}4M_{S_{i}}^{2}B_{0}\left(0,M_{S_{i}},M_{S_{i}}\right)+\frac{2}{3}k^{2}$
$\displaystyle+\left(k^{2}-4M_{S_{i}}^{2}\right)B_{0}\left(k,M_{S_{i}},M_{S_{i}}\right)\Big{]}\Bigg{\\}},$
$\displaystyle\left(q=1~{}\mathrm{for}~{}V_{1}V_{2}=WW,~{}q=0~{}\mathrm{for}~{}V_{1}V_{2}=ZZ,\gamma
Z,\gamma\gamma\right)$ $\displaystyle\Sigma^{h}\left(k^{2}\right)=$
$\displaystyle\frac{1}{16\pi^{2}}\frac{n}{2}\sum_{i=A}^{B}\Bigg{[}-g_{S_{i}S_{i}hh}A_{0}\left(M_{S_{i}}\right)+g_{S_{i}S_{i}h}^{2}B_{0}\left(k,M_{S_{i}},M_{S_{i}}\right)\Bigg{]},$
$\displaystyle\left(g_{S_{i}S_{i}h}=g_{S_{i}S_{i}hh}\cdot
v,~{}g_{S_{A}S_{A}hh}=-\lambda_{1}-\frac{2}{\sqrt{n}}\lambda_{3},~{}g_{S_{B}S_{B}hh}=-\lambda_{1}+\frac{2}{\sqrt{n}}\lambda_{3}\right)$
(80)

###  C.3 Three-point Tensor Integrals

The vertex contribution is given in Eq. (18). In addition to $B_{0}$ given in
C.1, the explicit expressions of $C_{22s},~{}C_{23s},~{}C_{24s}$ are given
below. The subscript ${}^{\prime}s^{\prime}$ and arguments
$\left(M_{1},M_{2}\right)$ are short for the full arguments for the three-
point functions that
$C_{..s}\left(M_{1},M_{2}\right)=C_{..}\left(k_{1},k_{2},M_{1},M_{2},M_{2}\right)$,
and
$k_{1}^{2}=M_{Z}^{2},~{}k_{2}^{2}=M_{h}^{2},~{}\left(k_{1}+k_{2}\right)^{2}=s$.
The masses $\left(M_{1},M_{2}\right)$ represent the masses of the scalar
components in each model, where $M_{1}=M_{2}$ for each single vertex
contribution except in the Inert Doublet when two neutral components as
virtual particles in the loop and one should take
$\left(M_{1},M_{2}\right)=\left(M_{H^{0}},M_{A^{0}}\right),~{}\left(M_{A^{0}},M_{H^{0}}\right)$
respectively for two types of vertex correction.

$\displaystyle C_{22s}\left(M_{1},M_{2}\right)$
$\displaystyle=\frac{1}{2\left[M_{h}^{4}+\left(M_{Z}^{2}-s\right)^{2}-2M_{h}^{2}\left(M_{Z}^{2}+s\right)\right]^{2}}\Bigg{\\{}2M_{Z}^{2}\left[M_{h}^{4}+\left(M_{Z}^{2}-s\right)^{2}-2M_{h}^{2}\left(M_{Z}^{2}+s\right)\right]$
$\displaystyle-\frac{1}{s}\left(M_{h}^{2}-M_{Z}^{2}-s\right)\left[M_{h}^{4}+\left(M_{Z}^{2}-s\right)^{2}-2M_{h}^{2}\left(M_{Z}^{2}+s\right)\right]\left[A_{0}\left(M_{1}\right)-A_{0}\left(M_{2}\right)\right]$
$\displaystyle-6M_{Z}^{2}\left[M_{Z}^{2}\left(M_{h}^{2}-M_{Z}^{2}+s\right)+\left(M_{1}^{2}-M_{2}^{2}\right)\left(M_{h}^{2}+M_{Z}^{2}-s\right)\right]B_{0}\left(k_{1},M_{1},M_{2}\right)$
$\displaystyle-\bigg{[}M_{h}^{6}+\left(M_{Z}^{2}-s\right)^{2}\left(3M_{Z}^{2}-s\right)-M_{h}^{4}\left(5M_{Z}^{2}+3s\right)+M_{h}^{2}\left(M_{Z}^{4}+3s^{2}\right)$
$\displaystyle-2\left(M_{1}^{2}-M_{2}^{2}\right)\left(M_{h}^{4}+M_{h}^{2}\left(4M_{Z}^{2}-s\right)+\left(M_{Z}^{2}-s\right)^{2}\right)\bigg{]}B_{0}\left(k_{2},M_{2},M_{2}\right)$
$\displaystyle+\bigg{[}\left(M_{h}^{2}-3M_{Z}^{2}\right)\left(M_{h}^{2}-M_{Z}^{2}\right)^{2}-\left(3M_{h}^{4}+M_{Z}^{4}\right)s+\left(3M_{h}^{2}+5M_{Z}^{2}\right)s^{2}-s^{3}$
$\displaystyle+\frac{1}{s}\left(M_{1}^{2}-M_{2}^{2}\right)\left(\left(M_{h}^{2}-M_{Z}^{2}\right)^{3}-5\left(M_{h}^{4}-M_{Z}^{4}\right)s+\left(7M_{h}^{2}-M_{Z}^{2}\right)s^{2}-3s^{3}\right)\bigg{]}B_{0}\left(k_{1}+k_{2},M_{1},M_{2}\right)$
$\displaystyle+2\bigg{[}M_{Z}^{4}\left(M_{h}^{4}-2M_{h}^{2}\left(M_{Z}^{2}-2s\right)+\left(M_{Z}^{2}-s\right)\right)+\left(M_{1}^{4}+M_{2}^{4}\right)\left(M_{h}^{4}+2M_{h}^{2}\left(2M_{Z}^{2}-s\right)+\left(M_{Z}^{2}-s\right)^{2}\right)$
$\displaystyle-2M_{Z}^{2}\left(M_{h}^{4}\left(-2M_{1}^{2}+M_{2}^{2}\right)+\left(M_{Z}^{2}-s\right)^{2}\left(M_{1}^{2}-2M_{2}^{2}\right)+M_{h}^{2}\left(M_{Z}^{2}+s\right)\left(M_{1}^{2}+M_{2}^{2}\right)\right)$
$\displaystyle-2M_{1}^{2}M_{2}^{2}\left[M_{h}^{4}+2M_{h}^{2}\left(2M_{Z}^{2}-s\right)+\left(M_{Z}^{2}-s\right)^{2}\right]\bigg{]}C_{0}\left(k_{1},k_{2},M_{1},M_{2},M_{2}\right)\Bigg{\\}}$
$\displaystyle C_{23s}\left(M_{1},M_{2}\right)$
$\displaystyle=\frac{1}{2\left[M_{h}^{4}+\left(M_{Z}^{2}-s\right)^{2}-2M_{h}^{2}\left(M_{Z}^{2}+s\right)\right]^{2}}\Bigg{\\{}\left(M_{h}^{2}+M_{Z}^{2}-s\right)\left[M_{h}^{4}+\left(M_{Z}^{2}-s\right)^{2}-2M_{h}^{2}\left(M_{Z}^{2}+s\right)\right]$
$\displaystyle-\frac{1}{s}\left(M_{h}^{2}-M_{Z}^{2}+s\right)\left(M_{h}^{4}+\left(M_{Z}^{2}-s\right)^{2}-2M_{h}^{2}\left(M_{Z}^{2}+s\right)\right)\left[A_{0}\left(M_{1}\right)-A_{0}\left(M_{2}\right)\right]$
$\displaystyle+\bigg{[}M_{Z}^{2}\left(-M_{h}^{4}+5\left(M_{Z}^{2}-s\right)^{2}-4M_{h}^{2}\left(M_{Z}^{2}+s\right)\right)$
$\displaystyle-\left(M_{1}^{2}-M_{2}^{2}\right)\left(M_{h}^{4}+\left(M_{Z}^{2}-s\right)^{2}+2M_{h}^{2}\left(5M_{Z}^{2}-s\right)\right)\bigg{]}B_{0}\left(k_{1},M_{1},M_{2}\right)$
$\displaystyle-\bigg{[}M_{h}^{6}+2\left(M_{Z}^{2}-s\right)^{3}-2M_{h}^{4}\left(3M_{Z}^{2}+2s\right)+M_{h}^{2}\left(3M_{Z}^{4}-8M_{Z}^{2}s+5s^{2}\right)$
$\displaystyle-6\left(M_{1}^{2}-M_{2}^{2}\right)M_{h}^{2}\left(M_{h}^{2}+M_{Z}^{2}-s\right)\bigg{]}B_{0}\left(k_{2},M_{2},M_{2}\right)$
$\displaystyle+\bigg{[}M_{h}^{6}-\left(M_{Z}^{2}-s\right)^{2}\left(3M_{Z}^{2}+2s\right)-M_{h}^{4}\left(5M_{Z}^{2}+4s\right)+M_{h}^{2}\left(7M_{Z}^{4}-4M_{Z}^{2}s+5s^{2}\right)$
$\displaystyle+\left(M_{1}^{2}-M_{2}^{2}\right)\left(\left(M_{h}^{2}-M_{Z}^{2}\right)^{2}+2\left(-3M_{h}^{4}+M_{h}^{2}M_{Z}^{2}+2M_{Z}^{4}\right)s+\left(3M_{h}^{2}-5M_{Z}^{2}\right)s^{2}+2s^{3}\right)\bigg{]}B_{0}\left(k_{1}+k_{2},M_{1},M_{2}\right)$
$\displaystyle+2\bigg{[}M_{Z}^{2}\left(\left(M_{Z}^{2}-s\right)^{3}+M_{h}^{4}\left(M_{Z}^{2}+2s\right)-M_{h}^{2}\left(2M_{Z}^{4}-3M_{Z}^{2}s+2s^{2}\right)\right)$
$\displaystyle+3\left(M_{1}^{4}+M_{2}^{4}\right)M_{h}^{2}\left(M_{h}^{2}+M_{Z}^{2}-s\right)+M_{1}^{2}\left(M_{h}^{2}-M_{Z}^{2}+s\right)\left(M_{h}^{4}+2M_{h}^{2}\left(2M_{Z}^{2}-s\right)+\left(M_{Z}^{2}-s\right)^{2}\right)$
$\displaystyle+2M_{2}^{2}\left(\left(M_{Z}^{2}-s\right)^{3}-M_{h}^{4}\left(2M_{Z}^{2}+s\right)+M_{h}^{2}\left(M_{Z}^{4}-3M_{Z}^{2}s+2s^{2}\right)\right)$
$\displaystyle-6M_{1}^{2}M_{2}^{2}M_{h}^{2}\left(M_{h}^{2}+M_{Z}^{2}-s\right)\bigg{]}C_{0}\left(k_{1},k_{2},M_{1},M_{2},M_{2}\right)\Bigg{\\}}$
$\displaystyle C_{24s}\left(M_{1},M_{2}\right)$
$\displaystyle=\frac{1}{4\left[M_{h}^{4}+\left(M_{Z}^{2}-s\right)^{2}-2M_{h}^{2}\left(M_{Z}^{2}+s\right)\right]}\Bigg{\\{}M_{h}^{4}+\left(M_{Z}^{2}-s\right)^{2}-2M_{h}^{2}\left(M_{Z}^{2}+s\right)$
$\displaystyle-\left[M_{Z}^{2}\left(M_{h}^{2}-M_{Z}^{2}+s\right)+\left(M_{1}^{2}-M_{2}^{2}\right)\left(M_{h}^{2}+M_{Z}^{2}-s\right)\right]B_{0}\left(k_{1},M_{1},M_{2}\right)$
$\displaystyle+M_{h}^{2}\left(M_{h}^{2}-M_{Z}^{2}-s+2M_{1}^{2}-2M_{2}^{2}\right)B_{0}\left(k_{2},M_{2},M_{2}\right)$
$\displaystyle+\left[s\left(-Mh^{2}-Mz^{2}+s\right)-\left(M_{1}^{2}-M_{2}^{2}\right)\left(M_{h}^{2}-M_{Z}^{2}+s\right)\right]B_{0}\left(k_{1}+k_{2},M_{1},M_{2}\right)$
$\displaystyle+2\bigg{[}M_{h}^{2}M_{Z}^{2}s+\left(M_{1}^{4}+M_{2}^{4}\right)M_{h}^{2}+M_{1}^{2}M_{h}^{2}\left(M_{h}^{2}-M_{Z}^{2}-s\right)+M_{2}^{2}\left(\left(M_{Z}^{2}-s\right)^{2}-M_{h}^{2}\left(M_{Z}^{2}+s\right)\right)$
$\displaystyle-2M_{1}^{2}M_{2}^{2}M_{h}^{2}\bigg{]}C_{0}\left(k_{1},k_{1},M_{1},M_{2},M_{2}\right)\Bigg{\\}}$
(81)

##  Appendix D On-shell Renormalization Scheme

The on-shell (OS) renormalization conditions consist of two parts: (a) the
pole of the propagator defines the real part of the renormalized mass; (b) the
real part of the residue of the propagator is equal to one. The renormalized
quantities and renormalization constants are in accordance with the scheme of
Denner Denner:1991kt , and are defined as follows

$\displaystyle M_{W,0}^{2}$ $\displaystyle=M_{W}^{2}+\delta M_{W}^{2},$ (82)
$\displaystyle M_{Z,0}^{2}$ $\displaystyle=M_{Z}^{2}+\delta M_{Z}^{2},$
$\displaystyle M_{h,0}^{2}$ $\displaystyle=M_{h}^{2}+\delta M_{h}^{2},$
$\displaystyle W_{0}^{\pm}$
$\displaystyle=Z_{W}^{1/2}W^{\pm}=\left(1+\frac{1}{2}\delta
Z_{W}\right)W^{\pm},$ $\displaystyle\begin{pmatrix}Z_{0}\\\
A_{0}\end{pmatrix}$
$\displaystyle=\begin{pmatrix}Z_{ZZ}^{1/2}&Z_{Z\gamma}^{1/2}\\\ Z_{\gamma
Z}^{1/2}&Z_{\gamma\gamma}\end{pmatrix}\begin{pmatrix}Z\\\
A\end{pmatrix}=\begin{pmatrix}1+\frac{1}{2}\delta Z_{ZZ}&\frac{1}{2}\delta
Z_{Z\gamma}\\\ \frac{1}{2}\delta Z_{\gamma Z}&1+\frac{1}{2}\delta
Z_{\gamma\gamma}\end{pmatrix}\begin{pmatrix}Z\\\ A\end{pmatrix}$
$\displaystyle h_{0}$ $\displaystyle=Z_{h}^{1/2}h=\left(1+\frac{1}{2}\delta
Z_{h}\right)h,$

where quantities with subscript $0$ denote the bare parameters.

In terms of the renormalized self-energy functions which are denoted with
caret (one should differentiate this with the caret notation in
$\overline{\mathrm{MS}}$ scheme), the OS conditions are given

$\displaystyle\mathrm{Re}~{}\hat{\Sigma}_{T}^{WW}\left(M_{W}^{2}\right)=\delta
M_{W}^{2},\quad\mathrm{Re}\left.\frac{\partial\hat{\Sigma}_{T}^{WW}\left(k^{2}\right)}{\partial
k^{2}}\right|_{k^{2}=M_{W}^{2}}=0,$ (83)
$\displaystyle\mathrm{Re}~{}\hat{\Sigma}_{T}^{ZZ}\left(M_{Z}^{2}\right)=\delta
M_{Z}^{2},\quad\mathrm{Re}\left.\frac{\partial\hat{\Sigma}_{T}^{ZZ}\left(k^{2}\right)}{\partial
k^{2}}\right|_{k^{2}=M_{Z}^{2}}=0,$
$\displaystyle\hat{\Sigma}_{T}^{\gamma\gamma}\left(0\right)=0,\quad\mathrm{Re}\left.\frac{\partial\hat{\Sigma}_{T}^{\gamma\gamma}\left(k^{2}\right)}{\partial
k^{2}}\right|_{k^{2}=0}=0,$
$\displaystyle\mathrm{Re}~{}\hat{\Sigma}_{T}^{\gamma
Z}\left(M_{Z}^{2}\right)=\hat{\Sigma}_{T}^{\gamma Z}\left(0\right)=0,$
$\displaystyle\mathrm{Re}~{}\hat{\Sigma}^{h}\left(M_{h}^{2}\right)=\delta
M_{h}^{2},\quad\mathrm{Re}\left.\frac{\partial\hat{\Sigma}^{h}\left(k^{2}\right)}{\partial
k^{2}}\right|_{k^{2}=M_{h}^{2}}=0.$

The mass and wave function renormalization counterterms are then determined by
the following relations

$\displaystyle\delta
M_{W}^{2}=\mathrm{Re}~{}\Sigma_{T}^{WW}\left(M_{W}^{2}\right),\quad\delta
Z_{W}=-\mathrm{Re}\left.\frac{\partial\Sigma_{T}^{WW}\left(k^{2}\right)}{\partial
k^{2}}\right|_{k^{2}=M_{W}^{2}},$ (84) $\displaystyle\delta
M_{Z}^{2}=\mathrm{Re}~{}\Sigma_{T}^{ZZ}\left(M_{Z}^{2}\right),\quad\delta
Z_{ZZ}=-\mathrm{Re}\left.\frac{\partial\Sigma_{T}^{ZZ}\left(k^{2}\right)}{\partial
k^{2}}\right|_{k^{2}=M_{Z}^{2}},$ $\displaystyle\delta
Z_{\gamma\gamma}=-\mathrm{Re}\left.\frac{\partial\Sigma_{T}^{\gamma\gamma}\left(k^{2}\right)}{\partial
k^{2}}\right|_{k^{2}=0},$ $\displaystyle\delta Z_{\gamma
Z}=-2~{}\mathrm{Re}~{}\frac{\Sigma_{T}^{\gamma
Z}\left(M_{Z}^{2}\right)}{M_{Z}^{2}},\quad\delta
Z_{Z\gamma}=2\frac{\Sigma_{T}^{\gamma Z}\left(0\right)}{M_{Z}^{2}},$
$\displaystyle\delta
M_{h}^{2}=\mathrm{Re}~{}\Sigma^{h}\left(M_{h}^{2}\right),\quad\delta
Z_{h}=-\mathrm{Re}\left.\frac{\partial\Sigma^{h}\left(k^{2}\right)}{\partial
k^{2}}\right|_{k^{2}=M_{h}^{2}},$

where the expressions of the unrenormalized self energies for each scalar
multiplet model can be found in Appendix C.2.

The counter terms of self-energy that are relevant to our calculation read

$\includegraphics[scale={0.6},valign={c}]{figure/self_ct}=\left\\{\begin{tabular}[]{l}$-i~{}\delta\Sigma_{T}^{WW}=-i\left[\left(k^{2}-M_{W}^{2}\right)\delta
Z_{W}-\delta M_{W}^{2}\right]$,\\\
$-i~{}\delta\Sigma_{T}^{ZZ}=-i\left[\left(k^{2}-M_{Z}^{2}\right)\delta
Z_{ZZ}-\delta M_{Z}^{2}\right]$,\\\ $-i~{}\delta\Sigma_{T}^{\gamma
Z}=-\frac{i}{2}\left[\left(k^{2}-M_{Z}^{2}\right)\delta
Z_{Z\gamma}+k^{2}\delta Z_{\gamma Z}\right]$,\end{tabular}\right.$ (85)

so that one can obtain the renormalized self-energies according to

$\hat{\Sigma}_{T}^{V_{1}V_{2}}\left(k^{2}\right)=\Sigma_{T}^{V_{1}V_{2}}\left(k^{2}\right)+\delta\Sigma_{T}^{V_{1}V_{2}}.$

The self-energy contributions in terms of matrix element square factorize to
Born, which read

$\displaystyle 2~{}\mathrm{Re}\left[\left(\delta{\bf
M}_{\mathrm{self}}^{ZZ}+\delta{\bf M}_{\mathrm{self}}^{\gamma
Z}\right)\cdot{\bf M}_{LO}^{*}\right]$ (86) $\displaystyle=$ $\displaystyle
2\frac{2e^{4}}{s_{W}^{2}c_{W}^{2}}\left[-\frac{\left(g_{v}^{2}+g_{a}^{2}\right)\left(s-M_{Z}^{2}\right)}{\left(s-M_{Z}^{2}\right)^{2}+M_{Z}^{2}\Gamma_{Z}^{2}}\cdot\hat{\Sigma}_{T}^{ZZ}\left(s\right)-\frac{g_{v}}{s}\cdot\hat{\Sigma}_{T}^{\gamma
Z}\left(s\right)\right]\frac{tu+2sM_{Z}^{2}-M_{h}^{2}M_{Z}^{2}}{\left(s-M_{Z}^{2}\right)^{2}+M_{Z}^{2}\Gamma_{Z}^{2}}$

It should be noted that the vertex contribution per se is UV finite. Using OS
renormalization scheme, one has to cross check if the corresponding counter
term contribution is also UV finite. The vertex counter terms have the
following contributions,

$\includegraphics[scale={0.6},valign={c}]{figure/vert_ct}=\left\\{\begin{tabular}[]{ll}$ieg_{\mu\nu}\cdot\delta_{\mathrm{ct}}^{ZZh}$:&$\delta
Z_{e}+\frac{2s_{W}^{2}-c_{W}^{2}}{c_{W}^{2}}\frac{\delta
s_{W}}{s_{W}}+\frac{1}{2}\frac{\delta M_{W}^{2}}{M_{W}^{2}}+\frac{1}{2}\delta
Z_{h}+\delta Z_{ZZ}$\\\ $ieg_{\mu\nu}\cdot\delta_{\mathrm{ct}}^{\gamma
Zh}$:&$\frac{1}{2}\delta Z_{Z\gamma}$\end{tabular}\right.$ (87)

where $\delta Z_{e}$ and $\delta s_{W}^{2}/s_{W}^{2}$ can be written in terms
of the mass and field renormalization constants (FRCs) that

$\delta Z_{e}=-\frac{1}{2}\left(\delta
Z_{\gamma\gamma}+\frac{s_{W}}{c_{W}}\delta
Z_{Z\gamma}\right),\quad\frac{\delta
s_{W}}{s_{W}}=-\frac{1}{2}\frac{c_{W}^{2}}{s_{W}^{2}}\left(\frac{\delta
M_{W}^{2}}{M_{W}^{2}}-\frac{\delta M_{Z}^{2}}{M_{Z}^{2}}\right).$ (88)

It is clear that both vertex counter terms in Eq.(87) are UV finite.

In addition to the final vertex ($(Z/\gamma)Zh$) counter term, one has to take
the initial vertex ($e^{-}e^{+}Z$) counter term into account, even though
there is no scalar loop corrections to the initial vertex. The counter term of
the initial vertex $e^{-}e^{+}Z$ has the following form.

$\includegraphics[scale={0.6},valign={c}]{figure/init_ct}=ie\gamma_{\mu}\left(\delta
c_{v}-\delta c_{a}\gamma^{5}\right)$ (89)

$\displaystyle\delta c_{v}=\delta g_{v}+\frac{g_{v}}{2}\delta
Z_{ZZ}-\frac{Q_{e}}{2}\delta Z_{\gamma Z},$ $\displaystyle\delta c_{a}=\delta
g_{a}+\frac{g_{a}}{2}\delta Z_{ZZ},$

where the variations of $\left(g_{v},~{}g_{a}\right)$ read

$\displaystyle\delta g_{v}=\frac{I_{W,e}^{3}}{2s_{W}c_{W}}\left(\delta
Z_{e}+\frac{s_{W}^{2}-c_{W}^{2}}{c_{W}^{2}}\frac{\delta
s_{W}}{s_{W}}\right)-\frac{s_{W}}{c_{W}}Q_{e}\left(\delta
Z_{e}+\frac{1}{c_{W}^{2}}\frac{\delta s_{W}}{s_{W}}\right),$
$\displaystyle\delta g_{a}=\frac{I_{W,e}^{3}}{2s_{W}c_{W}}\left(\delta
Z_{e}+\frac{s_{W}^{2}-c_{W}^{2}}{c_{W}^{2}}\frac{\delta s_{W}}{s_{W}}\right),$

with the quantum numbers for electron taking $I_{W,e}^{3}=-1/2,~{}Q_{e}=-1$.
It is worth noting that the initial vertex counter term contribution is as
well UV finite.

The vertex counter term stand alone contributions in terms of matrix element
square take the form below,

$\displaystyle 2~{}\mathrm{Re}\left[\left(\delta{\bf
M}_{\mathrm{vert,ct}}^{ZZh}+\delta{\bf M}_{\mathrm{vert,ct}}^{\gamma
Zh}+\delta{\bf M}_{\mathrm{vert,ct}}^{e^{-}e^{+}Z}\right){\bf
M}_{LO}^{*}\right]$ (90) $\displaystyle=$ $\displaystyle
2\frac{2e^{4}}{s_{W}^{2}c_{W}^{2}}\left(tu+2sM_{Z}^{2}-M_{h}^{2}M_{Z}^{2}\right)\times$
$\displaystyle\left\\{\frac{\left(g_{v}^{2}+g_{a}^{2}\right)\delta_{\mathrm{ct}}^{ZZh}}{\left(s-M_{Z}^{2}\right)^{2}+M_{Z}^{2}\Gamma_{Z}^{2}}+\frac{g_{v}\left(s-M_{Z}^{2}\right)\delta_{\mathrm{ct}}^{\gamma
Zh}}{s\left[\left(s-M_{Z}^{2}\right)^{2}+M_{Z}^{2}\Gamma_{Z}^{2}\right]}+\frac{\delta
c_{v}g_{v}+\delta
c_{a}g_{a}}{\left(s-M_{Z}^{2}\right)^{2}+M_{Z}^{2}\Gamma_{Z}^{2}}\right\\}.$

## References

  * (1) ATLAS Collaboration, Combined measurements of Higgs boson production and decay using up to $80$ fb-1 of proton–proton collision data at $\sqrt{s}=$ 13 TeV collected with the ATLAS experiment, .
  * (2) CMS Collaboration, A. M. Sirunyan et. al., Combined measurements of Higgs boson couplings in proton–proton collisions at $\sqrt{s}=13\,\text{Te}\text{V}$, Eur. Phys. J. C 79 (2019), no. 5 421 [1809.10733].
  * (3) M. Cepeda et. al., Report from Working Group 2: Higgs Physics at the HL-LHC and HE-LHC, CERN Yellow Rep. Monogr. 7 (2019) 221–584 [1902.00134].
  * (4) J. de Blas et. al., Higgs Boson Studies at Future Particle Colliders, JHEP 01 (2020) 139 [1905.03764].
  * (5) FCC Collaboration, A. Abada et. al., FCC Physics Opportunities: Future Circular Collider Conceptual Design Report Volume 1, Eur. Phys. J. C 79 (2019), no. 6 474.
  * (6) CEPC Study Group Collaboration, M. Dong et. al., CEPC Conceptual Design Report: Volume 2 - Physics & Detector, 1811.10545.
  * (7) FCC Collaboration, A. Abada et. al., FCC-ee: The Lepton Collider: Future Circular Collider Conceptual Design Report Volume 2, Eur. Phys. J. ST 228 (2019), no. 2 261–623.
  * (8) T. Barklow, J. Brau, K. Fujii, J. Gao, J. List, N. Walker and K. Yokoya, ILC Operating Scenarios, 1506.07830.
  * (9) K. Fujii et. al., Physics Case for the 250 GeV Stage of the International Linear Collider, 1710.07621.
  * (10) LCC Physics Working Group Collaboration, K. Fujii et. al., Tests of the Standard Model at the International Linear Collider, 1908.11299.
  * (11) Projections for measurements of Higgs boson signal strengths and coupling parameters with the ATLAS detector at a HL-LHC, .
  * (12) CMS Collaboration, Projected Performance of an Upgraded CMS Detector at the LHC and HL-LHC: Contribution to the Snowmass Process, in Community Summer Study 2013: Snowmass on the Mississippi, 7, 2013.  1307.7135.
  * (13) S. Profumo, M. J. Ramsey-Musolf and G. Shaughnessy, Singlet Higgs phenomenology and the electroweak phase transition, JHEP 08 (2007) 010 [0705.2425].
  * (14) J. R. Espinosa, T. Konstandin and F. Riva, Strong Electroweak Phase Transitions in the Standard Model with a Singlet, Nucl. Phys. B 854 (2012) 592–630 [1107.5441].
  * (15) L. Niemi, H. H. Patel, M. J. Ramsey-Musolf, T. V. Tenkanen and D. J. Weir, Electroweak phase transition in the real triplet extension of the SM: Dimensional reduction, Phys. Rev. D 100 (2019), no. 3 035002 [1802.10500].
  * (16) L. Niemi, M. J. Ramsey-Musolf, T. V. I. Tenkanen and D. J. Weir, Thermodynamics of a Two-Step Electroweak Phase Transition, Phys. Rev. Lett. 126 (2021), no. 17 171802 [2005.11332].
  * (17) V. Barger, P. Langacker, M. McCaskey, M. J. Ramsey-Musolf and G. Shaughnessy, LHC Phenomenology of an Extended Standard Model with a Real Scalar Singlet, Phys. Rev. D 77 (2008) 035005 [0706.4311].
  * (18) V. Barger, P. Langacker, M. McCaskey, M. Ramsey-Musolf and G. Shaughnessy, Complex Singlet Extension of the Standard Model, Phys. Rev. D 79 (2009) 015018 [0811.0393].
  * (19) A. Papaefstathiou and G. White, The Electro-Weak Phase Transition at Colliders: Confronting Theoretical Uncertainties and Complementary Channels, 2010.00597.
  * (20) M. Cirelli, N. Fornengo and A. Strumia, Minimal dark matter, Nucl. Phys. B 753 (2006) 178–194 [hep-ph/0512090].
  * (21) J. McDonald, Gauge singlet scalars as cold dark matter, Phys. Rev. D 50 (1994) 3637–3649 [hep-ph/0702143].
  * (22) C. Burgess, M. Pospelov and T. ter Veldhuis, The Minimal model of nonbaryonic dark matter: A Singlet scalar, Nucl. Phys. B 619 (2001) 709–728 [hep-ph/0011335].
  * (23) M. A. Díaz, B. Koch and S. Urrutia-Quiroga, Constraints to Dark Matter from Inert Higgs Doublet Model, Adv. High Energy Phys. 2016 (2016) 8278375 [1511.04429].
  * (24) E. Ma, Verifiable radiative seesaw mechanism of neutrino mass and dark matter, Phys. Rev. D 73 (2006) 077301 [hep-ph/0601225].
  * (25) R. Barbieri, L. J. Hall and V. S. Rychkov, Improved naturalness with a heavy Higgs: An Alternative road to LHC physics, Phys. Rev. D 74 (2006) 015007 [hep-ph/0603188].
  * (26) S. Banerjee and N. Chakrabarty, A revisit to scalar dark matter with radiative corrections, JHEP 05 (2019) 150 [1612.01973].
  * (27) P. Fileviez Perez, H. H. Patel, M. Ramsey-Musolf and K. Wang, Triplet Scalars and Dark Matter at the LHC, Phys. Rev. D 79 (2009) 055024 [0811.3957].
  * (28) T. Araki, C. Geng and K. I. Nagao, Dark Matter in Inert Triplet Models, Phys. Rev. D 83 (2011) 075014 [1102.4906].
  * (29) T. Hambye, F.-S. Ling, L. Lopez Honorez and J. Rocher, Scalar Multiplet Dark Matter, JHEP 07 (2009) 090 [0903.4010]. [Erratum: JHEP 05, 066 (2010)].
  * (30) S. S. AbdusSalam and T. A. Chowdhury, Scalar Representations in the Light of Electroweak Phase Transition and Cold Dark Matter Phenomenology, JCAP 05 (2014) 026 [1310.8152].
  * (31) W. Chao, G.-J. Ding, X.-G. He and M. Ramsey-Musolf, Scalar Electroweak Multiplet Dark Matter, JHEP 08 (2019) 058 [1812.07829].
  * (32) W. Konetschny and W. Kummer, Nonconservation of Total Lepton Number with Scalar Bosons, Phys. Lett. B 70 (1977) 433–435.
  * (33) M. Magg and C. Wetterich, Neutrino Mass Problem and Gauge Hierarchy, Phys. Lett. B 94 (1980) 61–64.
  * (34) J. Schechter and J. Valle, Neutrino Masses in SU(2) x U(1) Theories, Phys. Rev. D 22 (1980) 2227.
  * (35) T. Cheng and L.-F. Li, Neutrino Masses, Mixings and Oscillations in SU(2) x U(1) Models of Electroweak Interactions, Phys. Rev. D 22 (1980) 2860.
  * (36) T. Blank and W. Hollik, Precision observables in SU(2) x U(1) models with an additional Higgs triplet, Nucl. Phys. B 514 (1998) 113–134 [hep-ph/9703392].
  * (37) M.-C. Chen, S. Dawson and C. Jackson, Higgs Triplets, Decoupling, and Precision Measurements, Phys. Rev. D 78 (2008) 093001 [0809.4185].
  * (38) Y. Du, A. Dunbrack, M. J. Ramsey-Musolf and J.-H. Yu, Type-II Seesaw Scalar Triplet Model at a 100 TeV $pp$ Collider: Discovery and Higgs Portal Coupling Determination, JHEP 01 (2019) 101 [1810.09450].
  * (39) J. Fleischer and F. Jegerlehner, Radiative Corrections to Higgs Production by $e^{+}e^{-}\to ZH$ in the Weinberg-Salam Model, Nucl. Phys. B 216 (1983) 469–492.