#  Disentangling the high- and low-cutoff scales
via the trilinear Higgs couplings
in the type I two-Higgs-doublet model

Sin Kyu Kang<EMAIL_ADDRESS>School of Natural Science, Seoul National
University of Sciene and Technology, Seoul 139-743, Korea  Jinheung Kim
<EMAIL_ADDRESS>Department of Physics, Konkuk University, Seoul
05029, Republic of Korea  Soojin Lee<EMAIL_ADDRESS>Department of
Physics, Konkuk University, Seoul 05029, Republic of Korea  Jeonghyeon Song
<EMAIL_ADDRESS>Department of Physics, Konkuk University, Seoul 05029,
Republic of Korea

###### Abstract

The type I two-Higgs-doublet model in the inverted Higgs scenario can retain
the theoretical stability all the way up to the Planck scale. The Planck-scale
cutoff, $\Lambda_{\rm cut}^{\rm Planck}$, directly impacts the mass spectra
such that all the extra Higgs boson masses should be light below about 160
GeV. However, the observation of the light masses of new Higgs bosons does not
indicate the high-cutoff scale because a low-cutoff scale can also accommodate
the light masses. Over the viable parameter points that satisfy the
theoretical requirements and the experimental constraints, we show that the
trilinear Higgs couplings for low $\Lambda_{\rm cut}$ are entirely different
from those for the Planck-scale cutoff. The most sensitive coupling to the
cutoff scale is from the $h$-$h$-$h$ vertex, where $h$ is the lighter CP-even
Higgs boson at a mass below 125 GeV. The gluon fusion processes of $gg\to hh$
and $gg\to AA$ are insensitive to the cutoff scale, yielding a small variation
of the production cross sections, $\mathcal{O}(1)\,{\rm fb}$, according to
$\Lambda_{\rm cut}$. The smoking-gun signature is from the triple Higgs
production of $q\bar{q}^{\prime}\to W^{*}\to H^{\pm}hh$, which solely depends
on the $h$-$h$-$h$ vertex. The cross section for $\Lambda_{\rm cut}=1\,{\rm
TeV}$ is about $10^{3}$ times larger than that for the Planck-scale cutoff.
Since the decay modes of $H^{\pm}\to W^{*}h/W^{*}A$ and $h/A\to bb$ are
dominant, the process yields the $6b+\ell\nu$ final state, which enjoys an
almost background-free environment. Consequently, the precision measurement of
$pp\to H^{\pm}hh$ can probe the cutoff scale of the model.

###### Contents

  1. I Introduction
  2. II Review of type I of 2HDM in the inverted scenario
  3. III Scanning and RGE analysis
  4. IV Trilinear Higgs couplings
  5. V LHC phenomenology
  6. VI Conclusion
  7. A RGEs in the type I

##  I Introduction

Up to today, all of the measurements of the production cross sections of the
standard model (SM) particles at high-energy colliders are in good agreement
with the SM predictions Ellis:2021kzk , including the observed Higgs boson
with a mass of $125{\;{\rm GeV}}$ ATLAS:2020fcp ; ATLAS:2020bhl ; CMS:2020zge
; ATLAS:2021nsx ; CMS:2021gxc ; ATLAS:2020syy ; ATLAS:2021upe ; ATLAS:2020pvn
; CMS:2021ugl ; ATLAS:2020wny ; ATLAS:2020rej ; ATLAS:2020qdt ; ATLAS:2020fzp
; CMS:2020xwi ; ATLAS:2021zwx . Nevertheless, our minds are rarely in
satisfaction with the SM, because of the unsolved questions such as the
naturalness problem, baryogenesis, non-zero neutrino masses, fermion mass
hierarchy, the origin of CP violation in the quark sector, and the identity of
dark matter. We continue our journey in the quest for the ultimate theory.

A crucial question is whether the ultimate theory shall reveal its whole
structure at one energy scale. The answer is much more likely to be _no_ when
looking back on the SM, the only reliable guideline at this moment. We
witnessed the emergence of SM particles in stages. The same phenomena could
happen in the ultimate theory. In other words, the final theory may consist of
multilevel sub-models. The first-stage NP model111Two different structures
exist for the first-stage NP model. It can take over the SM from a high-energy
scale, accommodating new heavy particles with the multi-TeV masses. Or it
coexists with the SM at the electroweak scale so that the new particles have
the intermediate masses.  describes our universe up to a particular energy
scale $\Lambda_{\rm cut}$ and then hands over its role to the second-stage NP
model. $\Lambda_{\rm cut}$ could be as high as the Planck scale or as low as
$10{\;{\rm TeV}}$. Then can an observable distinguish the high and low
$\Lambda_{\rm cut}$? This is the driving question in our paper.

For the first-stage NP model, we consider the two-Higgs-doublet model (2HDM)
since many fundamental questions are closely related to the Higgs sector. The
2HDM provides the answers to some questions. For example, the first-order
electroweak phase transition in the 2HDM can explain the baryogenesis
Turok:1990zg ; Cohen:1991iu ; Zarikas:1995qb ; Cline:1996mga ; Fromme:2006cm .
However, the model cannot address all the fundamental questions, which makes
it a suitable candidate for the first-stage NP model.

Then the next question is how to calculate the energy scale at which the
second-stage NP model appears. A good way is to calculate the cutoff scale of
the 2HDM. Even though the theoretical requirements (unitarity, perturbativity,
and vacuum stability) are satisfied at the electroweak scale, they can be
broken at a higher energy scale $\Lambda_{\rm cut}$ because the parameters
evolve under renormalization group equations (RGE) Machacek:1983tz ;
Machacek:1983fi ; Machacek:1984zw ; Luo:2002ti ; Das:2015mwa ; Haber:1993an ;
Grimus:2004yh . Since it implies the advent of the second-stage NP model, we
call $\Lambda_{\rm cut}$ the cutoff scale of the model.

Figure 1:  Allowed $(M_{A},M_{H^{\pm}})$ with the cutoff scales of
$\Lambda_{\rm cut}=10{\;{\rm TeV}}$ (gray points) and $\Lambda_{\rm
cut}=10^{18}{\;{\rm GeV}}$ (red points) in the inverted type I, over the
viable parameter points that satisfy the theoretical requirements and the
experimental constraints. Two points in cyan yield the same $M_{A}$ and
$M_{H^{\pm}}$.

In the literature, the high-energy scale behavior of the 2HDM has been
extensively studied. Most studies are focused on the impact of high
$\Lambda_{\rm cut}$ on the extra Higgs boson masses Das:2015mwa ;
Chowdhury:2015yja ; Basler:2017nzu ; Krauss:2018thf ; Oredsson:2018yho ;
Aiko:2020atr ; Dey:2021pyn ; Lee:2022gyf ; Kim:2022hvh ; Kim:2022nmm .
However, observing the scalar mass spectrum that high $\Lambda_{\rm cut}$
predicts does not guarantee that $\Lambda_{\rm cut}$ is high. In this paper,
we focus on the type I of the 2HDM in the inverted Higgs scenario where the
heavier CP-even scalar $H$ is the observed Higgs boson Ferreira:2012my ;
Chang:2015goa ; Jueid:2021avn . The model can accommodate the cutoff scale all
the way up to the Planck scale Lee:2022gyf . As illustrated in Fig. 1, the
allowed points of $(M_{A},M_{H^{\pm}})$ for $\Lambda_{\rm cut}=10^{18}{\;{\rm
GeV}}$ (red points) are overlapped with those for $\Lambda_{\rm cut}=10{\;{\rm
TeV}}$ (gray points). Two points in cyan, one for $\Lambda_{\rm cut}=10{\;{\rm
TeV}}$ and the other for $\Lambda_{\rm cut}=10^{18}{\;{\rm GeV}}$, have the
same $M_{A}$ and $M_{H^{\pm}}$. We need an alternative observable to
disentangle the high- and low-cutoff scales. The measurement of $\tan\beta$,
the ratio of two vacuum expectation values of two Higgs doublet fields, is
tricky in type I when $\tan\beta$ is large. In type I, all the Yukawa
couplings of the BSM Higgs bosons are inversely proportional to $\tan\beta$ in
the Higgs alignment limit. So, the main production channels of $gg\to A/h$ and
$t\to H^{\pm}b$ are suppressed by large $\tan\beta$. In addition, the
branching ratios of all the fermionic decay modes are insensitive to
$\tan\beta$ in type I. We need alternative observables to probe the cutoff
scale.

We will show that the trilinear Higgs couplings can play the role. In
particular, the value of the $h$-$h$-$h$ vertex, $\lambda_{hhh}$, is highly
sensitive to the cutoff scale. To probe the trilinear Higgs couplings at the
LHC, we will study the di-Higgs processes of $gg\to hh$ and $gg\to AA$, and
the tri-Higgs processes of $pp\to H^{\pm}hh$. The gluon fusion production of
$gg\to h/H\to hh/AA$ is to be shown insensitive to $\Lambda_{\rm cut}$ because
of the destructive interference between the $h$ and $H$ contributions. We will
show that the tri-Higgs process $pp\to H^{\pm}hh$ is the best to measure the
cutoff scale because the signal rate is solely dependent on $\lambda_{hhh}$,
yielding $\sigma_{\Lambda_{\rm cut}=1{\;{\rm TeV}}}/\sigma_{\Lambda_{\rm
cut}=10^{18}{\;{\rm GeV}}}\sim 10^{3}$. Considering the dominant decays modes
of $H^{\pm}$ and $h$, we will suggest for the first time the $6b+\ell\nu$
final state as an efficient discriminator between the high- and low-cutoff
scales. These are our new contributions.

The paper is organized in the following way. In Sec. II, we briefly review the
type I in the inverted scenario. Section III describes the methods of the
parameter scanning and the calculation of $\Lambda_{\rm cut}$. The
characteristics of the viable parameter points with high $\Lambda_{\rm cut}$
are also presented. In Sec. IV, we calculate the correlation between the
trilinear Higgs couplings and the cutoff scale. Section V deals with the LHC
phenomenology. Based on the study of the branching ratios of the extra Higgs
bosons, we will suggest efficient observables to distinguish the high- and
low-cutoff scales. Finally we conclude in Sec. VI.

##  II Review of type I of 2HDM in the inverted scenario

The 2HDM introduces two $SU(2)_{L}$ complex scalar doublet fields with
hypercharge $Y=1$, $\Phi_{1}$ and $\Phi_{2}$ Branco:2011iw :

$\displaystyle\Phi_{i}=\left(\begin{array}[]{c}w_{i}^{+}\\\\[3.0pt]
\dfrac{v_{i}+\rho_{i}+i\eta_{i}}{\sqrt{2}}\end{array}\right),\quad(i=1,2)$ (3)

where $v_{1}$ and $v_{2}$ are the nonzero vacuum expectation values of
$\Phi_{1}$ and $\Phi_{2}$, respectively. The ratio of $v_{2}/v_{1}$ defines
the mixing angle $\beta$ via $\tan\beta=v_{2}/v_{1}$. For notational
simplicity, we use $s_{x}=\sin x$, $c_{x}=\cos x$, and $t_{x}=\tan x$ in what
follows. The combination of $v_{1}$ and $v_{2}$,
$v=\sqrt{v_{1}^{2}+v_{2}^{2}}=246{\;{\rm GeV}}$, spontaneously breaks the
electroweak symmetry. To prevent flavor-changing neutral currents at tree
level, we impose a discrete $Z_{2}$ symmetry under which $\Phi_{1}\to\Phi_{1}$
and $\Phi_{2}\to-\Phi_{2}$ Glashow:1976nt ; Paschos:1976ay . We allow softly
broken $Z_{2}$ symmetry since it does not affect the RGE of the dimensionless
quartic couplings Haber:2006ue : the hard $Z_{2}$ breaking in the Yukawa
sector causes too fast growth of the scalar quartic couplings in the RG
running Oredsson:2018yho .

For simplicity, we employ a CP-conserving scalar potential that softly breaks
the $Z_{2}$ symmetry, which is given by

$\displaystyle V_{\Phi}=$ $\displaystyle
m^{2}_{11}\Phi^{\dagger}_{1}\Phi_{1}+m^{2}_{22}\Phi^{\dagger}_{2}\Phi_{2}-m^{2}_{12}(\Phi^{\dagger}_{1}\Phi_{2}+{\rm
H.c.})$
$\displaystyle+\frac{1}{2}\lambda_{1}(\Phi^{\dagger}_{1}\Phi_{1})^{2}+\frac{1}{2}\lambda_{2}(\Phi^{\dagger}_{2}\Phi_{2})^{2}+\lambda_{3}(\Phi^{\dagger}_{1}\Phi_{1})(\Phi^{\dagger}_{2}\Phi_{2})+\lambda_{4}(\Phi^{\dagger}_{1}\Phi_{2})(\Phi^{\dagger}_{2}\Phi_{1})$
$\displaystyle+\frac{1}{2}\lambda_{5}\left[(\Phi^{\dagger}_{1}\Phi_{2})^{2}+{\rm
H.c.}\right],$

where the $m_{12}^{2}$ term softly breaks the $Z_{2}$ symmetry. The scalar
potential $V_{\Phi}$ yields five physical Higgs bosons, the lighter CP-even
scalar $h$, the heavier CP-even scalar $H$, the CP-odd pseudoscalar $A$, and a
pair of charged Higgs bosons $H^{\pm}$. Relations of mass eigenstates with
weak eigenstates in terms of two mixing angles of $\alpha$ and $\beta$ are
referred to Ref. Song:2019aav . The SM Higgs boson is a linear combination of
$h$ and $H$, given by

$\displaystyle{h_{\rm SM}}=s_{\beta-\alpha}\,h+c_{\beta-\alpha}\,H.$ (5)

The observed Higgs boson at a mass of $125{\;{\rm GeV}}$ at the LHC
ATLAS:2020fcp ; ATLAS:2020bhl ; CMS:2020zge ; ATLAS:2021nsx ; CMS:2021gxc ;
ATLAS:2020syy ; ATLAS:2021upe ; ATLAS:2020pvn ; CMS:2021ugl ; ATLAS:2020wny ;
ATLAS:2020rej ; ATLAS:2020qdt ; ATLAS:2020fzp ; CMS:2020xwi ; ATLAS:2021zwx
has so far agreed with the predictions for the SM Higgs boson. The SM-like
Higgs boson strongly motivates the Higgs alignment limit in the 2HDM. Two
scenarios exist for the limit, the normal scenario where ${h_{\rm SM}}=h$
(i.e., $s_{\beta-\alpha}=1$) and the inverted scenario where ${h_{\rm SM}}=H$
(i.e., $c_{\beta-\alpha}=1$). In this paper, we concentrate on the inverted
scenario in the Higgs alignment limit:

$\displaystyle M_{H}=125{\;{\rm GeV}},\quad c_{\beta-\alpha}=1.$ (6)

Then we have the following five parameters:

$\displaystyle\\{m_{h},\quad M_{A},\quad M_{H^{\pm}},\quad t_{\beta},\quad
m_{12}^{2}\\},$ (7)

which define one parameter point. Then, the quartic coupling constants in
$V_{\Phi}$ are written as Kanemura:2011sj

$\displaystyle\lambda_{1}$ $\displaystyle=$
$\displaystyle\frac{1}{v^{2}}\left[m_{125}^{2}+t_{\beta}^{2}\left(m_{h}^{2}-M^{2}\right)\right],$
(8) $\displaystyle\lambda_{2}$ $\displaystyle=$
$\displaystyle\frac{1}{v^{2}}\left[m_{125}^{2}+\frac{1}{t_{\beta}^{2}}\left(m_{h}^{2}-M^{2}\right)\right],$
$\displaystyle\lambda_{3}$ $\displaystyle=$
$\displaystyle\frac{1}{v^{2}}\left[m_{125}^{2}-m_{h}^{2}-M^{2}+2M_{H^{\pm}}^{2}\right],$
$\displaystyle\lambda_{4}$ $\displaystyle=$
$\displaystyle\frac{1}{v^{2}}\left[M^{2}+M_{A}^{2}-2M_{H^{\pm}}^{2}\right],$
$\displaystyle\lambda_{5}$ $\displaystyle=$
$\displaystyle\frac{1}{v^{2}}\left[M^{2}-M_{A}^{2}\right],$

where $m_{125}=125{\;{\rm GeV}}$ and $M^{2}=m_{12}^{2}/(s_{\beta}c_{\beta})$.

The Yukawa couplings to the SM fermions are parametrized as

$\displaystyle\mathscr{L}^{\rm Yuk}$ $\displaystyle=$
$\displaystyle-\sum_{f}\left(\frac{m_{f}}{v}\xi^{h}_{f}\bar{f}fh+\frac{m_{f}}{v}\kappa_{f}^{H}\bar{f}fH-i\frac{m_{f}}{v}\xi_{f}^{A}\bar{f}\gamma_{5}fA\right)$
$\displaystyle-\left\\{\dfrac{\sqrt{2}}{v}\overline{t}\left(m_{t}\xi^{A}_{t}\text{P}_{-}+m_{b}\xi^{A}_{b}\text{P}_{+}\right)bH^{+}+\dfrac{\sqrt{2}m_{\tau}}{v}\xi^{A}_{\tau}\,\overline{\nu}_{\tau}P_{+}\tau
H^{+}+{\rm H.c.}\right\\},$

which are different according to the 2HDM type. In this work, we focus on type
I. To facilitate the discussion below, we will call the type I with the
conditions of Eq. (6) the inverted type I. Then the Higgs coupling modifiers
are

$\displaystyle\xi^{H}_{f}=1,\quad\xi^{h}_{t,b,\tau}=\frac{1}{t_{\beta}},\quad\xi^{A}_{t}=-\xi^{A}_{b,\tau}=\frac{1}{t_{\beta}}.$
(10)

The trilinear Higgs couplings as dimensionless parameters are defined by

$\displaystyle\mathscr{L}_{\rm tri}$ $\displaystyle=$
$\displaystyle\sum_{\varphi_{0}=h,H}v\left\\{\frac{1}{3!}\hat{\lambda}_{\varphi_{0}\varphi_{0}\varphi_{0}}\varphi_{0}^{3}+\frac{1}{2}\hat{\lambda}_{\varphi_{0}AA}\,\varphi_{0}A^{2}+\lambda_{\varphi_{0}H^{+}H^{-}}\,\varphi_{0}H^{+}H^{-}\right\\}$
$\displaystyle~{}+\frac{1}{2}\hat{\lambda}_{Hhh}\,vHh^{2}+\frac{1}{2}\hat{\lambda}_{hHH}\,vhH^{2}.$

In the inverted type I, the couplings are Bernon:2015qea ; Bernon:2015wef

$\displaystyle\hat{\lambda}_{HHH}$
$\displaystyle=-\frac{3m_{125}^{2}}{v^{2}},\qquad\hat{\lambda}_{hHH}=0,$ (12)
$\displaystyle\hat{\lambda}_{hhh}$
$\displaystyle=3\hat{\lambda}_{hAA}=3\hat{\lambda}_{hH^{+}H^{-}}=-\frac{3(M^{2}-m_{h}^{2})(t_{\beta}^{2}-1)}{t_{\beta}v^{2}},$
$\displaystyle\hat{\lambda}_{Hhh}$
$\displaystyle=-\frac{m_{125}^{2}+2m_{h}^{2}-2M^{2}}{v^{2}},$
$\displaystyle\hat{\lambda}_{HAA}$
$\displaystyle=-\frac{m_{125}^{2}+2M_{A}^{2}-2M^{2}}{v^{2}},$
$\displaystyle\hat{\lambda}_{HH^{+}H^{-}}$
$\displaystyle=-\frac{m_{125}^{2}+2M_{H^{\pm}}^{2}-2M^{2}}{v^{2}}.$

The Higgs alignment limit makes the trilinear coupling of the observed Higgs
boson $H$ the same as in the SM, $\hat{\lambda}_{HHH}\simeq 0.77$, which is
one of the most important targets to measure at the HL-LHC and future
colliders Plehn:1996wb ; Djouadi:1999rca ; Barger:2013jfa ; Dawson:2015oha ;
Cheung:2020xij ; Jueid:2021qfs . Another remarkable feature is that
$\hat{\lambda}_{hhh}$, $\hat{\lambda}_{hAA}$, and
$\hat{\lambda}_{hH^{+}H^{-}}$ have the common factor of
$t_{\beta}(M^{2}-m_{h}^{2})$ in the large $t_{\beta}$ limit.

##  III Scanning and RGE analysis

Before studying the high-energy behavior of the model via RGE, the preparation
of the allowed parameter points at the electroweak scale is an essential
prerequisite. Therefore, we randomly scan five model parameters in the range
of

$\displaystyle t_{\beta}$ $\displaystyle\in$ $\displaystyle[1,50],\quad
M_{A}\in[10,3000]{\;{\rm GeV}},\quad m_{12}^{2}\in[-3000^{2},3000^{2}]{\;{\rm
GeV}}^{2},$ (13) $\displaystyle M_{H^{\pm}}$ $\displaystyle\in$
$\displaystyle[80,3000]{\;{\rm GeV}},\quad m_{h}\in[10,120]{\;{\rm GeV}},$

and cumulatively impose the following constraints:

  * •

Theoretical requirements
We demand the bounded-from-below Higgs potential Ivanov:2006yq , tree-level
unitarity of scalar-scalar scatterings Ginzburg:2005dt ; Branco:2011iw ;
Kanemura:2015ska ; Arhrib:2000is , perturbativity Chang:2015goa , and the
stability of the CP-conserving vacuum with $v=246{\;{\rm GeV}}$ Ivanov:2008cxa
; Barroso:2012mj ; Barroso:2013awa . We use the public code 2HDMC-v1.8.0
Eriksson:2009ws . For the perturbativity, 2HDMC requires that the magnitudes
of all the quartic couplings among physical Higgs bosons be less than $4\pi$.
However, 2HDMC does not check whether our vacuum is the global minimum of the
potential. We demand the tree-level vacuum stability condition of
Barroso:2013awa

$\displaystyle
m_{12}^{2}\left(m_{11}^{2}-k^{2}m_{22}^{2}\right)\left(t_{\beta}-k\right)>0,$
(14)

where $k=(\lambda_{1}/\lambda_{2})^{1/4}$. The tree-level conditions have been
known to be more than sufficient up to very high scales in the Higgs alignment
limit Basler:2017nzu ; Branchina:2018qlf .

  * •

Peskin-Takeuchi oblique parameters Peskin:1991sw
The oblique parameters of $S$, $T$, and $U$ in the 2HDM He:2001tp ;
Grimus:2008nb ; Zyla:2020zbs  should satisfy the current best-fit results at
95% C.L. ParticleDataGroup:2022pth :

$\displaystyle S$ $\displaystyle=-0.02\pm 0.10,$ (15) $\displaystyle T$
$\displaystyle=0.03\pm 0.12,\quad U=0.01\pm 0.11,$ $\displaystyle\rho_{ST}$
$\displaystyle=0.92,\quad\rho_{SU}=-0.80,\quad\rho_{TU}=-0.93,$

where $\rho_{ij}$ is the correlation matrix.

  * •

Flavor changing neutral currents
We demand that the most recent observables from $B$ physics be satisfied at
95% C.L. Arbey:2017gmh ; Sanyal:2019xcp ; Misiak:2017bgg . We adopt the
results of Ref. Arbey:2017gmh .

  * •

Higgs precision data
We use the public code HiggsSignals-v2.6.2 Bechtle:2020uwn  to check the
consistency with the Higgs precision data. Based on the $\chi^{2}$ value for
111 Higgs observables Aaboud:2018gay ; Aaboud:2018jqu ; Aaboud:2018pen ;
Aad:2020mkp ; Sirunyan:2018mvw ; Sirunyan:2018hbu ; CMS:2019chr ; CMS:2019kqw
with five parameters, we require that the $p$-value be larger than 0.05.

  * •

Direct search bounds
We demand that the model prediction to the cross sections of the direct search
modes for new scalar bosons at the LEP, Tevatron, and LHC should be less than
95% C.L. upper bound on the observed cross sections. The open code
HiggsBounds-v5.10.2 Bechtle:2020pkv  is used.

Brief comments on the recent CDF measurement of the $W$-boson mass CDF:2022hxs
, $m_{W}^{\rm CDF}=80.4335\pm 0.0094{\;{\rm GeV}}$, are in order here. If we
accept $m_{W}^{\rm CDF}$, the oblique parameters change into $S_{\rm
CDF}=0.15\pm 0.08$ and $T_{\rm CDF}=0.27\pm 0.06$ with $U=0$ Lu:2022bgw .
Although $m_{W}^{\rm CDF}$ has important implications on the 2HDM Fan:2022dck
; Zhu:2022tpr ; Lu:2022bgw ; Zhu:2022scj ; Song:2022xts ; Bahl:2022xzi ;
Heo:2022dey ; Babu:2022pdn ; Biekotter:2022abc ; Ahn:2022xeq ; Han:2022juu ;
Arcadi:2022dmt ; Ghorbani:2022vtv ; Broggio:2014mna ; Lee:2022gyf ;
Kim:2022hvh , our main conclusion on the role of trilinear Higgs couplings in
disentangling the high- and low-cutoff scales does not change. Therefore, we
focus on the oblique parameters without the CDF $m_{W}$ measurement.

Over the parameter points that pass the above constraints, we evolve the
following parameters via the RGE, by using the public code 2HDME
Oredsson:2018yho ; Oredsson:2018vio :

$\displaystyle g_{1,2,3},\quad\lambda_{1,\cdots,5},\quad\xi^{h,H,A}_{f},\quad
m_{11},\quad m_{12},\quad m_{22}^{2},\quad v_{1,2}.$ (16)

We include the mixing effects of two scalar doublet fields (with equal quantum
numbers) on $v_{1}$ and $v_{2}$, which bring about the RG running of
$t_{\beta}$. The top quark pole mass scale, $m_{t}^{\rm pole}=173.4{\;{\rm
GeV}}$, is used to match the 2HDM to the SM. The boundary conditions at
$m_{t}^{\rm pole}$ are referred to Ref. Oredsson:2018yho . The $\beta$
functions of the gauge, Yukawa, and quartic couplings at one loop level are
presented in Appendix A. Since the difference between the one-loop and two-
loop RG running is not large,222For randomly selected parameter points, we
compared $\Lambda_{\rm cut}$ at one-loop level with $\Lambda_{\rm cut}$ at
two-loop level. The difference is $\mathcal{O}(10)\%$. we take the one-loop
RGE to efficiently cover all the parameter points.

Now let us describe how we obtained the cutoff scale $\Lambda_{\rm cut}$. For
each parameter point, we perform the RGE evolution up to the next high energy
scale333To cover from the electroweak scale to the Planck scale, we take a
uniform step in $\log(Q)$. and check three conditions, tree-level unitarity,
perturbativity, and vacuum stability.444Note that 2HDME use the perturbativity
condition of $|\lambda_{1,\cdots,5}|<4\pi$ and the tree-level vacuum stability
conditions in Refs. Ivanov:2006yq ; Ivanov:2007de . If all three are
satisfied, we increase the energy scale into the next step. If any condition
is violated, we stop the running and record the energy scale as $\Lambda_{\rm
cut}$. We find that the Landau pole appears at a higher scale than
$\Lambda_{\rm cut}$. We additionally require that the cutoff scale should be
larger than $1{\;{\rm TeV}}$. In what follows, the “viable parameter points”
denote the parameter points that satisfy the aforementioned constraints and
$\Lambda_{\rm cut}>1{\;{\rm TeV}}$.

Figure 2:  $M_{H^{\pm}}$ versus $M_{A}$ in the left panel and $t_{\beta}$
versus $m_{h}$ in the right panel. The color code denotes the cutoff scale
$\Lambda_{\rm cut}$.

Strong correlations exist between the cutoff scale and the model parameters.
In Fig. 2, we present the viable parameter points over the plane of
$(M_{A},M_{H^{\pm}})$ in the left panel and $(m_{h},t_{\beta})$ in the right
panel. The color code denotes the cutoff scale $\Lambda_{\rm cut}$. We sorted
the parameter points according to $\Lambda_{\rm cut}$, and stacked them in
order of $\Lambda_{\rm cut}$, the points with low $\Lambda_{\rm cut}$
underneath and those with high $\Lambda_{\rm cut}$ on top. The overlap is due
to the projection of five-dimensional parameter space in Eq. (7) on a two-
dimensional subspace.

Several remarkable features are shown in Fig. 2. First, the viable parameter
points are pretty limited even with the weak condition of $\Lambda_{\rm
cut}>1{\;{\rm TeV}}$. The upper bounds on $M_{A}$ and $M_{H^{\pm}}$ exist as
$M_{A},M_{H^{\pm}}\lesssim 430{\;{\rm GeV}}$, to which the condition of
$\Lambda_{\rm cut}>1{\;{\rm TeV}}$ plays a critical role. The Peskin-Takeuchi
oblique parameter $T$ is satisfied if $M_{H^{\pm}}\sim M_{A}$ or
$M_{H^{\pm}}\sim m_{h}$, which explains two branches in the left panel of Fig.
2. The lower branch corresponds to $M_{H^{\pm}}\sim m_{h}$, which puts the
upper bound on the charged Higgs boson mass. For the intermediate mass range
of $M_{A},M_{H^{\pm}}\lesssim 200{\;{\rm GeV}}$, there is no particular
correlation between $M_{A}$ and $M_{H^{\pm}}$. However, meaningful
correlations appear outside the box with $M_{A},M_{H^{\pm}}\lesssim 200{\;{\rm
GeV}}$. If $M_{H^{\pm}}\gtrsim 200{\;{\rm GeV}}$ (belonging to the upper
branch), $M_{H^{\pm}}\simeq M_{A}$. If $M_{H^{\pm}}\lesssim 200{\;{\rm GeV}}$
and $M_{A}\gtrsim 200{\;{\rm GeV}}$ (belonging to the lower branch),
$M_{H^{\pm}}\sim 100{\;{\rm GeV}}$. The lighter CP-even Higgs boson mass
$m_{h}$ (right panel of Fig. 2) is heavier than half the observed Higgs boson
mass due to the strong constraint from the exotic Higgs boson decay of $H\to
hh$. For $t_{\beta}$, most values in the scanning are permitted. Although the
density of the allowed $t_{\beta}$ in the scatter plot is lower for larger
$t_{\beta}$, we cannot conclude that large $t_{\beta}$ is disfavored: nature
chooses just one parameter point.

Figure 3:  For $\Lambda_{\rm cut}>10^{18}{\;{\rm GeV}}$, $M_{A}$ versus
$m_{h}$ in the left panel and $t_{\beta}$ versus $M^{2}$ in the right panel.
The color code denotes $M_{H^{\pm}}$ in the left panel and $m_{h}$ in the
right panel.

The second remarkable feature of Fig. 2 is that substantial parameter points
in the inverted type I remain stable all the way up to the Planck scale. Let
us investigate the characteristics of the parameter points with the high-
cutoff scale. In Fig. 3, we show $M_{A}$ versus $m_{h}$ (left panel) and
$t_{\beta}$ versus $M^{2}$ (right panel) after imposing $\Lambda_{\rm
cut}>10^{18}{\;{\rm GeV}}$. The color code in the left (right) panel denotes
$M_{H^{\pm}}$ ($m_{h}$). It is clearly seen that the high-cutoff scale
requires light masses of the extra Higgs bosons like
$M_{A},M_{H^{\pm}}\lesssim 160{\;{\rm GeV}}$. We observe that the charged
Higgs boson is lighter than the top quark for the Planck-scale cutoff.
However, the lower bounds of $M_{A}\gtrsim 10{\;{\rm GeV}}$, $m_{h}\gtrsim
62.5{\;{\rm GeV}}$, and $M_{H^{\pm}}\gtrsim 80{\;{\rm GeV}}$ do not change by
imposing $\Lambda_{\rm cut}>10^{18}{\;{\rm GeV}}$. Similarly, the Planck-scale
cutoff does not affect the values of $t_{\beta}$, as shown in the right panel
of Fig. 3.

The values of $M^{2}$ are also limited even with $\Lambda_{\rm cut}>1{\;{\rm
TeV}}$. Only the positive values of $M^{2}$ are permitted because negative
$M^{2}$ enhances $\lambda_{1}$ through the terms proportional to
$t_{\beta}^{2}$. Large $\lambda_{1}$ at the electroweak scale quickly evolves
into an unacceptably large value, which endangers the global minimum condition
of the vacuum Ivanov:2008cxa ; Barroso:2012mj ; Barroso:2013awa . For
$\Lambda_{\rm cut}>10^{18}{\;{\rm GeV}}$, a unique correlation of $M^{2}\simeq
m_{h}^{2}$ appears as shown in the right panel in Fig. 3. It is also ascribed
to the $t_{\beta}^{2}$ terms in $\lambda_{1}$. The condition of $M^{2}\simeq
m_{h}^{2}$ suppresses the $t_{\beta}^{2}$ terms and thus helps to retain the
stability of the scalar potential.

Although the high-cutoff scale demands light masses of the extra Higgs bosons,
the inverse is not true. All the mass spectra in the left panel of Fig. 3 also
accommodate a low-cutoff scale: see Fig. 1. Even if we observe
$m_{h}=M_{A}=100{\;{\rm GeV}}$ and $M_{H^{\pm}}=140{\;{\rm GeV}}$, for
example, the mass spectrum alone cannot tell whether the cutoff scale is high
or low. The reader may suggest to use $t_{\beta}$ as a discriminator of the
cutoff scale. However, measuring $t_{\beta}$ in type I is challenging at the
LHC, especially when $t_{\beta}\gg 1$. The value of $t_{\beta}$ governs the
fermionic productions (from the top quark decay or gluon fusion via quark
loops) and fermionic decays of the extra Higgs bosons. If $t_{\beta}$ is
large, the fermionic production of the extra Higgs bosons is highly suppressed
because all the Yukawa couplings of the extra Higgs bosons are inversely
proportional to $t_{\beta}$ in type I. The bosonic productions such as
${q\bar{q}}\to Z^{*}\to Ah$ and ${q\bar{q}}\to W^{*}\to H^{\pm}h/A$
Akeroyd:1995hg ; Akeroyd:1998ui ; Barroso:1999bf ; Brucher:1999tx ;
Akeroyd:2003xi ; Arhrib:2008pw ; Berger:2012sy ; Ilisie:2014hea ;
Delgado:2016arn ; Mondal:2021bxa ; Cheung:2022ndq ; Kim:2022nmm  do not give
information about $t_{\beta}$. Moreover, the fermionic decay parts are
insensitive to $t_{\beta}$ because of the same dependence of all the Yukawa
couplings on $t_{\beta}$. If we cannot measure the exact value of large
$t_{\beta}$, it is reasonable to include all the viable parameter points with
$t_{\beta}>10$ when pursuing a way to discriminate the high and low
$\Lambda_{\rm cut}$.

##  IV Trilinear Higgs couplings

In this section, we study the trilinear Higgs couplings to measure
$\Lambda_{\rm cut}$. We consider the following three benchmark points:

BP-1: $\displaystyle\quad m_{h}=70{\;{\rm GeV}},$ $\displaystyle M_{A}$
$\displaystyle=110{\;{\rm GeV}},$ $\displaystyle M_{H^{\pm}}$
$\displaystyle=110{\;{\rm GeV}},$ (17) BP-2: $\displaystyle\quad
m_{h}=100{\;{\rm GeV}},$ $\displaystyle M_{A}$ $\displaystyle=100{\;{\rm
GeV}},$ $\displaystyle M_{H^{\pm}}$ $\displaystyle=140{\;{\rm GeV}},$ BP-3:
$\displaystyle\quad m_{h}=110{\;{\rm GeV}},$ $\displaystyle M_{A}$
$\displaystyle=70{\;{\rm GeV}},$ $\displaystyle M_{H^{\pm}}$
$\displaystyle=140{\;{\rm GeV}},$

all of which accommodate the cutoff scale from $1{\;{\rm TeV}}$ to
$10^{19}{\;{\rm GeV}}$. As discussed in the previous section, we focus on the
large $t_{\beta}$ limit as

Large $t_{\beta}$ case: $\displaystyle t_{\beta}>10.$ (18)

For $m_{12}^{2}$, we incorporate all the values that satisfy the constraints
in Sec. III.

Figure 4:  Trilinear Higgs couplings of $\hat{\lambda}_{hhh}$ (left panel),
$\hat{\lambda}_{Hhh}$ (middle panel), and $\hat{\lambda}_{HH^{+}H^{-}}$ (right
panel) against the cutoff scale $\Lambda_{\rm cut}$. The color code denotes
$t_{\beta}$. We take BP-2 where $m_{h}=M_{A}=100{\;{\rm GeV}}$,
$M_{H^{\pm}}=140{\;{\rm GeV}}$, and $t_{\beta}>10$.

Let us turn into the trilinear Higgs couplings versus $\Lambda_{\rm cut}$. In
Fig. 4, we present $\hat{\lambda}_{hhh}$ (left panel), $\hat{\lambda}_{Hhh}$
(middle panel), and $\hat{\lambda}_{HH^{+}H^{-}}$ (right panel) at the
electroweak scale, as a function of $\Lambda_{\rm cut}$: note that
$\hat{\lambda}_{hAA}=\hat{\lambda}_{hH^{+}H^{-}}=\hat{\lambda}_{hhh}/3$. Here
only the BP-2 results are shown because BP-1 and BP-3 yield similar results
with $\mathcal{O}(10)\%$ differences. The value of $t_{\beta}$ is shown via
the color code. It is impressive that the values of $\hat{\lambda}_{hhh}$,
$\hat{\lambda}_{Hhh}$, and $\hat{\lambda}_{HH^{+}H^{-}}$ for $\Lambda_{\rm
cut}=10^{18}$ are not overlapped with those for $\Lambda_{\rm cut}\lesssim
10^{17}{\;{\rm GeV}}$, although the allowed values for lower $\Lambda_{\rm
cut}$ are considerably spread by the unfixed $t_{\beta}$ and $m_{12}^{2}$. The
most sensitive dependence on $\Lambda_{\rm cut}$ is shown in
$\hat{\lambda}_{hhh}$, which ranges in $[-0.09,\,1.1]$. Since
$\hat{\lambda}_{hhh}=0$ is included, the change of $\hat{\lambda}_{hhh}$
according to $\Lambda_{\rm cut}$ is huge. On the other hand, the variations of
$\hat{\lambda}_{Hhh}$ and $\hat{\lambda}_{HH^{+}H^{-}}$ are small within
$10\%\sim 20\%$.

Figure 5:  Trilinear Higgs couplings of $\hat{\lambda}_{HAA}$ versus the
cutoff scale $\Lambda_{\rm cut}$ for BP-1, BP-2, and BP-3 (left panel) and
$\hat{\lambda}_{HAA}$ around the focus cutoff scale for BP-2 (right panel).
The color code denotes $t_{\beta}$. We include all the viable parameter points
with $t_{\beta}>10$.

In the left panel of Fig. 5, we present $\hat{\lambda}_{HAA}$ versus
$\Lambda_{\rm cut}$ for BP-1, BP-2, and BP-3. Unlike $\hat{\lambda}_{hhh}$,
$\hat{\lambda}_{Hhh}$, and $\hat{\lambda}_{HH^{+}H^{-}}$, the value of
$\hat{\lambda}_{HAA}$ is sensitive to the benchmark point. For high
$\Lambda_{\rm cut}$, $|\hat{\lambda}_{HAA}|$ of BP-1 is about 25 times larger
than that of BP-3. $\hat{\lambda}_{HAA}$ depends on $M_{A}$ and $M^{2}$, not
on $t_{\beta}$. Since $M^{2}\approx m_{h}^{2}$ and $m_{h}\sim m_{125}$ as
shown in Fig. 3, the $M^{2}$ contribution to $\hat{\lambda}_{HAA}$ is nearly
cancelled by the $m_{125}$ contribution. So, the heavier $M_{A}$ is, the
larger $|\hat{\lambda}_{HAA}|$ is.

Figure 6:  $M(\equiv\sqrt{M^{2}})$ versus the cutoff scale $\Lambda_{\rm cut}$
for BP-2 with the color code of $t_{\beta}$. The left panel presents the
results of all the viable parameter points and the right panel shows those
with $|M^{2}-m_{h}^{2}|\leq 10{\;{\rm GeV}}^{2}$.

We observe a special $\Lambda_{\rm cut}$ in Fig. 4 and the right panel of Fig.
5. Around $\Lambda_{\rm cut}\simeq 10^{17}{\;{\rm GeV}}$, all the trilinear
Higgs couplings are almost fixed. Even though we present only the results of
BP-2 in Fig. 4, the same behavior is found in BP-1 and BP-3. The spread
trilinear Higgs couplings are focused on a single cutoff point which we call
$\Lambda_{\rm cut}^{\rm focus}$. Since the allowed value of $M^{2}$ plays a
crucial role in understanding $\Lambda_{\rm cut}^{\rm focus}$, we show
$M(\equiv\sqrt{M^{2}})$ versus the cutoff scale $\Lambda_{\rm cut}$ for BP-2
in Fig. 6. The left panel presents the results of all the viable parameter
points. It is clear to see that the parameter points with $\Lambda_{\rm
cut}=\Lambda_{\rm cut}^{\rm focus}$ satisfy the condition of
$M^{2}=m_{h}^{2}$. We found that the reverse holds true. If we allow small
deviation like $|M^{2}-m_{h}^{2}|\leq 10{\;{\rm GeV}}^{2}$ as in the right
panel of Fig. 6, the cutoff scale converges to $\Lambda_{\rm cut}=\Lambda_{\rm
cut}^{\rm focus}$ for $t_{\beta}=10$ but ranges from $10^{12}{\;{\rm GeV}}$ to
the Planck scale for $t_{\beta}=50$. Then why does the condition of
$M^{2}=m_{h}^{2}$ fix the trilinear Higgs coupling? It is because the
condition removes the $t_{\beta}$ dependence of the trilinear couplings, which
is the main source for their variation. In addition, $M^{2}=m_{h}^{2}$ removes
the dangerous $t_{\beta}^{2}$ terms of $\hat{\lambda}_{1}$, which guarantees
the high-energy scale of $\Lambda_{\rm cut}^{\rm focus}$.

Figure 7:  For the small deviation from the Higgs alignment limit,
$\hat{\lambda}_{hhh}$ versus the cutoff scale $\Lambda_{\rm cut}$ for BP-2
with the color code of $t_{\beta}$. The left (right) panel presents the
results of $|s_{\beta-\alpha}|=0.05$ ($|s_{\beta-\alpha}|=0.1$).

Now we discuss the other 2HDM scenarios. This paper concentrates on the
inverted type I in the Higgs alignment limit. Let us first consider a small
deviation from the alignment. Since the Higgs precision prospect at the future
muon collider associated with the HL-LHC and the Higgs factory at
$\sqrt{s}=250{\;{\rm GeV}}$ is $\delta\kappa_{W}\simeq 0.11\%$
Forslund:2022xjq , we consider two cases of $|s_{\beta-\alpha}|=0.05,\,0.1$.
In Fig. 7, we present $\hat{\lambda}_{hhh}$ versus $\Lambda_{\rm cut}$ for
$|s_{\beta-\alpha}|=0.05$ (left panel) and $|s_{\beta-\alpha}|=0.1$ (right
panel). Here only BP-2 results are shown. For $|s_{\beta-\alpha}|=0.05$, the
behavior of $\hat{\lambda}_{hhh}$ about $\Lambda_{\rm cut}$ remains almost
same as the alignment case. The values of $\hat{\lambda}_{hhh}$ for
$\Lambda_{\rm cut}=10^{18}{\;{\rm GeV}}$ are not overlapped with those for
$\Lambda_{\rm cut}\lesssim 10^{6}{\;{\rm GeV}}$. If we increase the deviation
from the alignment into $|s_{\beta-\alpha}|=0.1$, the band of
$\hat{\lambda}_{hhh}$ widens. The values of $\hat{\lambda}_{hhh}$ for
$\Lambda_{\rm cut}=10^{18}{\;{\rm GeV}}$ are mostly overlapped with the low
cutoff scale, except for $\Lambda_{\rm cut}\lesssim 10{\;{\rm TeV}}$.

Finally, we discuss whether we have similar results for other types or the
normal Higgs scenario where the lighter CP-even Higgs boson is the observed
one. In the inverted Higgs scenario, type II and type Y are excluded by
imposing the condition of $\Lambda_{\rm cut}>1{\;{\rm TeV}}$ Lee:2022gyf
because the constraint from $b\to s\gamma$, $M_{H^{\pm}}>800{\;{\rm GeV}}$
Misiak:2020vlo , contradicts the required light masses of the BSM Higgs
bosons. Type X in the inverted scenario can accommodate $\Lambda_{\rm
cut}=10^{18}{\;{\rm GeV}}$ and show similar behaviors of the trilinear Higgs
couplings about the cutoff scales. In the normal scenario, all four types can
retain the theoretical stability up to $\Lambda_{\rm cut}=10^{18}{\;{\rm
GeV}}$. The high cutoff scale demands the almost exact mass degeneracy among
the BSM Higgs boson masses, i.e., $M=M_{A}=M_{H}=M_{H^{\pm}}$, but does not
put any upper bounds on the masses. So, the question for the normal scenario
is changed: how can we distinguish the high- and low-cutoff scales via
observables if we observe a highly degenerate mass spectrum of the extra Higgs
bosons? It is more challenging than in the inverted scenario due to the heavy
masses and the soft decay products of the BSM Higgs bosons. Nevertheless, the
similar behavior of $\hat{\lambda}_{HHH}$ about $\Lambda_{\rm cut}$ leaves a
motivation for the phenomenological study in future colliders.

##  V LHC phenomenology

Figure 8:  Branching ratios of $h$ (upper panels), $A$ (middle panels), and
$H^{\pm}$ (lower panels) about the cutoff scale $\Lambda_{\rm cut}$. All the
viable parameter points with $t_{\beta}>10$ are included.

For the LHC phenomenology, we first present the branching ratios of $h$ (left
panels), $A$ (middle panels), and $H^{\pm}$ (right panels) as a function of
$\Lambda_{\rm cut}$ in Fig. 8. The results of BP-1, BP-2, and BP-3 are in the
upper, middle, and lower panels, respectively. We include all the viable
parameter points. The first noteworthy feature in Fig. 8 is that the dominant
decay mode of the extra Higgs boson with the given mass spectra is insensitive
to the cutoff scale, even though the two parameters of $t_{\beta}(>10)$ and
$m_{12}^{2}$ are not fixed. The decay of $h$ depends on the hierarchy between
$m_{h}$ and $M_{A}$. When $m_{h}\leq M_{A}$ as in BP-1 and BP-2, the leading
(next-to-leading) decay mode is $h\to bb$ ($h\to{\tau^{+}\tau^{-}}$). However,
if $m_{h}>M_{A}$ as in BP-3, the dominant decay mode is $h\to AZ^{*}$. The
suppressed Yukawa couplings of $h$ by large $t_{\beta}$ enhance the bosonic
decays modes if kinematically open. The decay of $A$ is primarily determined
by the hierarchy between $M_{A}$ and $m_{h}$. For BP-2 and BP-3 with
$M_{A}\leq m_{h}$, the pseudoscalar $A$ dominantly decays into a pair of $b$
quarks with ${\rm Br}(A\to{b\bar{b}})\gtrsim 0.73$. The next-to-leading decay
mode of $A$ is into $gg$. The substantial ${\rm Br}(A\to gg)$ is attributed to
the larger loop amplitudes of a pseudoscalar than those of a scalar for the
spin-1/2 particle contributions Djouadi:2005gj . The third one is
$A\to{\tau^{+}\tau^{-}}$. If $M_{A}>m_{h}$ as in BP-1, however, ${\rm Br}(A\to
hZ^{*})$ becomes the largest, which holds for the entire range of
$\Lambda_{\rm cut}$. The decay into ${b\bar{b}}$ is mostly next-to-
leading.555We caution the reader that the scattered points are overlapped
except for the leading decay mode. For some parameter points, $A\to gg$ can be
next-to-leading.

The charged Higgs boson mainly decays into $hW^{\pm*}$ and $AW^{\pm*}$, for
the three benchmark points. Once kinematically allowed, the bosonic decay
modes are dominant: the $H^{\pm}$-$W^{\mp}$-$h$ vertex is proportional to
$c_{\beta-\alpha}(=1)$; the $H^{\pm}$-$W^{\mp}$-$A$ vertex is originated from
the pure gauge interaction. In BP-3 with $M_{H^{\pm}}>M_{A}$ and
$M_{H^{\pm}}>m_{h}$, the branching ratios of the fermionic modes are below 1%.
If either $m_{h}$ or $M_{A}$ is beyond the kinematic threshold as in BP-1 and
BP-2, the fermionic decay modes become considerable. The leading fermionic
decay mode depends on the charged Higgs bosons mass. For BP-1 where
$M_{H^{\pm}}$ is substantially lighter than the top quark mass,
$H^{\pm}\to\tau\nu$ has the largest branching ratio among the fermionic decay
modes, followed by $H^{\pm}\to cs$. In BP-2 where $M_{H^{\pm}}$ is near to the
top quark mass, $H^{\pm}\to t^{*}b$ becomes the leading fermionic mode,
followed by $H^{\pm}\to\tau\nu$.

Figure 9:  Feynman diagrams of $gg\to hh/AA$ (left panel) and
$q\bar{q}^{\prime}\to W^{*}\to H^{\pm}hh$ (right panel).

To probe the trilinear Higgs couplings at the LHC, we need to consider multi-
Higgs production mediated by Higgs bosons. The first important are the di-
Higgs processes, $gg\to H/h\to hh/AA$. The corresponding Feynman diagram666We
omit the box diagrams from the top quark loop because two factors of
$1/t_{\beta}$ suppress them.  is in the left panel of Fig. 9. The contribution
of $H$ destructively interferes with that of $h$ because the sign of
$\hat{\lambda}_{Hhh}$ and $\hat{\lambda}_{hhh}$ are opposite to each other:
see Fig. 4.

Figure 10:  Cross sections of $gg\to hh$ (left panel) and $gg\to AA$ (right
panel) at the 14 TeV LHC, as a function of $\Lambda_{\rm cut}$. The color code
denotes $t_{\beta}$. The description of the benchmarks is in the main text.

In Fig. 10, we present as a function of $\Lambda_{\rm cut}$ the parton-level
production cross sections for $gg\to hh$ (left panel) and $gg\to AA$ (right
panel) at the 14 TeV LHC over the viable parameter points. All the three
benchmark points in Eq. (17) are considered. The color code denotes
$t_{\beta}$. To calculate the parton-level cross sections, we first obtained
the Universal FeynRules Output (UFO) Degrande:2011ua  by using FeynRules
Alloul:2013bka . Then we interfered the UFO file with MadGraph5-aMC@NLO
Alwall:2011uj  and calculated the cross sections at the 14 TeV LHC with the
NNPDF31_lo_as_0118 parton distribution function set NNPDF:2017mvq .

For the production cross sections of $gg\to hh$, the most crucial factor is
$m_{h}$. The lighter $m_{h}$ is, the larger $\sigma(gg\to hh)$ is. BP-1 yields
the largest cross section. On the contrary, the production cross section of
$gg\to AA$ is larger for heavier $M_{A}$. The BP-1, which has the heaviest
$M_{A}$ among the three benchmark points, has the largest cross section. It
seems contradictory to the kinematic loss by the heavy $M_{A}$. The main
reason is that the dominant contribution to $gg\to AA$ is from $H$ and thus
$\hat{\lambda}_{HAA}$ determines the signal rate. The heavier $M_{A}$ is, the
larger $\hat{\lambda}_{HAA}$ is: see Fig. 5.

The dependence of $\sigma(gg\to hh/AA)$ on $\Lambda_{\rm cut}$ is not large
enough to distinct the high- and low-cutoff scales. Even the optimistic case,
the process of $gg\to hh$ in BP-1 with $t_{\beta}=10$, makes a few fb
difference in the cross sections between $\Lambda_{\rm cut}=1{\;{\rm TeV}}$
and $\Lambda_{\rm cut}=10^{18}{\;{\rm GeV}}$. It is too small to probe at the
HL-LHC with the expected total luminosity of $3{\;{\rm ab}^{-1}}$. For larger
$t_{\beta}$ like 50, the di-Higgs production cross sections become more
insensitive to $\Lambda_{\rm cut}$. The weak dependence of $\sigma(gg\to hh)$
on $\Lambda_{\rm cut}$ is due to the dominant contribution from $H$ and the
destructive interference between the $H$ and $h$ contributions. In the BP-1
with $t_{\beta}=10$ and $\Lambda_{\rm cut}=1{\;{\rm TeV}}$, for example, the
cross section from $H$ alone is $\sigma(gg\to H\to hh)\simeq 36.5{\;{\rm
fb}}$, from $h$ alone is $\sigma(gg\to h\to hh)\simeq 1.1{\;{\rm fb}}$, and
from the interference $\sigma(gg\to hh)_{\rm intf}\simeq-12.1{\;{\rm fb}}$.
The $\hat{\lambda}_{Hhh}$ controls the cross section, but its variation about
$\Lambda_{\rm cut}$ is small.

To single out only one trilinear Higgs coupling, we consider triple Higgs
productions at the LHC. Since the gluon fusion production of the tri-Higgs
process through the top quark loop is suppressed by large $t_{\beta}$, we
concentrate on the tri-Higgs productions mediated by the gauge bosons. Through
the $Z$ boson, we have

$\displaystyle{q\bar{q}}\to Z^{*}\to Ah^{*}\to Ahh,\quad{q\bar{q}}\to Z^{*}\to
A^{*}h\to Ahh,$ (19) $\displaystyle{q\bar{q}}\to Z^{*}\to Ah^{*}\to AAA,$ (20)

and through $W$

$\displaystyle q\bar{q}^{\prime}\to W^{*}\to H^{\pm}h^{*}\to H^{\pm}hh,$ (21)
$\displaystyle q\bar{q}^{\prime}\to W^{*}\to H^{\pm}h^{*}\to H^{\pm}AA,$ (22)
$\displaystyle q\bar{q}^{\prime}\to W^{*}\to H^{\pm}A^{*}\to H^{\pm}Ah.$ (23)

As a representative, we present the Feynman diagram of $q\bar{q}^{\prime}\to
H^{\pm}hh$ in the right panel of Fig. 9. Since all the above processes in Eqs.
(19)–(23) have the same topology of the Feynman diagram, the production cross
sections as a function of $\Lambda_{\rm cut}$ show almost the same behavior.
In BP-2, for example, $\sigma(pp\to Ahh)/\sigma(pp\to H^{\pm}hh)\simeq 0.9$
holds for all $\Lambda_{\rm cut}$.

Figure 11:  Cross sections of $q\bar{q}^{\prime}\to H^{\pm}hh$ at the 14 TeV
LHC as a function of $\Lambda_{\rm cut}$, for BP-1 (left), BP-2 (middle), and
BP-3 (right). The color codes indicate $t_{\beta}$.

In Fig. 11, we present the parton-level production cross sections of
$q\bar{q}^{\prime}\to H^{\pm}hh$ at the 14 TeV LHC for BP-1 (left panel), BP-2
(middle panel), and BP-3 (right panel). The color code denotes $t_{\beta}$.
The difference of the cross section according to $\Lambda_{\rm cut}$ is big
enough to distinguish the high- and low-cutoff scales of the inverted type I.
The ratio of the cross section for $\Lambda_{\rm cut}=1{\;{\rm TeV}}$ to that
for $\Lambda_{\rm cut}=10^{18}{\;{\rm GeV}}$ is more than about $10^{3}$. This
is the most remarkable result of our study. Measuring the signal rate of the
triple Higgs production tells whether the cutoff scale is high or low.

Finally, we discuss the discovery potential at the HL-LHC. Discriminating the
high- and low-cutoff scales through $gg\to hh/AA$ requires the precision
measurement on the cross section within $\sim 1{\;{\rm fb}}$. Let us roughly
estimate the feasibility. The processes of $gg\to hh$ and $gg\to AA$ mainly
yield $4b$ final states because $A/h\to bb$ is leading or next-to-leading.
Resembling the di-Higgs process in the SM, they are challenging to observe for
two reasons. First, the cross section itself is too small. The maximum cross
section, which happens for $\Lambda_{\rm cut}=1{\;{\rm TeV}}$, reaches
$\mathcal{O}(10){\;{\rm fb}}$. It is to be compared with the SM leading order
result of $\sigma(gg\to{h_{\rm SM}}\,{h_{\rm SM}})_{\rm LO}\simeq 17{\;{\rm
fb}}$ at $\sqrt{s}=14{\;{\rm TeV}}$ Baglio:2012np . Since the projected signal
significance of the SM Higgs boson pair production at the HL-LHC with the
total luminosity of $3{\;{\rm ab}}^{-1}$ is $3.0\,\sigma$, when the $bbbb$,
$bb{\tau^{+}\tau^{-}}$, and $bb{\gamma\gamma}$ decay channels are all combined
ATLAS:2018rvj , it is hard to observe $gg\to hh/AA$. The second difficulty
comes from the softer $b$ jets than in the SM di-Higgs process. For $gg\to
hh\to 4b$, the lighter $m_{h}$ than $125{\;{\rm GeV}}$ yields soft $b$ quarks.
For $gg\to AA\to 4b$, most of the viable parameter points in Fig. 3 have
$M_{A}<125{\;{\rm GeV}}$, which generate soft $b$ quarks. As the $b$ tagging
efficiency is reduced, the signal significance decreases as well. In summary,
the di-Higgs process is not efficient to probe the cutoff scale.

The triple Higgs production of $H^{\pm}hh$ has higher discovery potential.
Since the charged Higgs boson mainly decays into $W^{\pm*}A/h$ in the three
benchmark points, followed by $h/A\to bb$, the final state is $6b+\ell\nu$.
This attractive channel has not been studied in the literature.777The final
state of $4j+\ell\nu$ was studied for the vector boson scattering of $WW\to
WW$ Ballestrero:2008gf , and the SM Higgs boson decaying into a fat jet
consisting of $4b/6b/8b$ was studied Jung:2021tym . The main backgrounds are

$\displaystyle t+\bar{t}+\ell\nu$ $\displaystyle\to bj_{b}^{\rm mis}j_{b}^{\rm
mis}+bj_{b}^{\rm mis}j_{b}^{\rm mis}+\ell\nu,$ (24) $\displaystyle
t+\bar{t}+jj$ $\displaystyle\to b\ell\nu+bj_{b}^{\rm mis}j_{b}^{\rm
mis}+j_{b}^{\rm mis}j_{b}^{\rm mis},$

where $j_{b}^{\rm mis}$ is the light jet (from $u$, $d$, $s$, $c$, and $g$)
mistagged as a $b$ quark jet. We calculated the parton-level cross sections of
the backgrounds, with the $b$ tagging and mistagging efficiencies of $P_{b\to
b}=0.7$, $P_{c\to b}=0.05$, and $P_{j\to b}=0.01$. We imposed the selection
cuts of $p_{T}^{b}>20{\;{\rm GeV}}$, $p_{T}^{\ell}>10{\;{\rm GeV}}$,
$|\eta^{\ell,j}|<2.5$, $E_{T}^{\rm miss}>20{\;{\rm GeV}}$, and the separation
$\Delta R_{ii^{\prime}}>0.4$. After the basic selection, the background cross
section from ${t\bar{t}}jj$ is about $8.7{\;{\rm ab}}$ and from
${t\bar{t}}\ell\nu$ is about $3.8\times 10^{-4}{\;{\rm ab}}$. If we impose
additional cuts on the invariant mass of two $b$ jets like
$|m_{bb}-m_{h}|<15{\;{\rm GeV}}$, the backgrounds are negligible. Despite the
almost background-free environment, the high $\Lambda_{\rm cut}$ yields too
small signal rate of $pp\to H^{\pm}hh$ at the 14 TeV LHC.

Exploring the cutoff scale in the inverted type I via the $6b+\ell\nu$ final
state has a better chance in future high-energy colliders such as the Future
hadron-hadron Circular Collider (FCC-hh) at CERN Gomez-Ceballos:2013zzn , the
CEPC Gao:2021bam ; CEPCStudyGroup:2018ghi , and the muon collider
Palmer:2007zzc ; Delahaye:2019omf ; Black:2022cth . Particularly, we have high
expectations for the muon collider with benchmark energies in the range of
$\sqrt{s}=3-30{\;{\rm TeV}}$ and the integrated luminosity of
$\mathcal{L}=10\,(\sqrt{s}/10{\;{\rm TeV}})^{2}{\;{\rm ab}^{-1}}$. The triple
Higgs processes in Eq. (19) and Eq. (20) with ${q\bar{q}}$ replaced by
${\mu^{+}\mu^{-}}$ will be able to disentangle the high- and low-cutoff scales
in the inverted type I.

##  VI Conclusion

Beyond the studies on how high the cutoff scale of a new physics model can go
up, we have pursued an efficient observable to distinguish the high- and low-
cutoff scales. The type I in the 2HDM has been considered for the inverted
scenario where the observed Higgs boson at a mass of $125{\;{\rm GeV}}$ is the
heavier CP-even Higgs boson $H$. We have first obtained the still-available
parameter points that satisfy the theoretical requirements, the experimental
constraints, and the cutoff scale above $1{\;{\rm TeV}}$. The viable parameter
space at the electroweak scale is already limited such that
$M_{A},M_{H^{\pm}}\lesssim 430{\;{\rm GeV}}$ and $m_{h}\gtrsim 62.5{\;{\rm
GeV}}$. Through the calculation of the cutoff scale $\Lambda_{\rm cut}$ of
each viable parameter point by using the RGE, we have shown that the inverted
type I can retain the stability all the way up to the Planck scale.

The condition of $\Lambda_{\rm cut}>10^{18}{\;{\rm GeV}}$ requires the light
masses of the extra Higgs bosons like $M_{A},M_{H^{\pm}}\lesssim 160{\;{\rm
GeV}}$. However, the light masses alone cannot guarantee the high-cutoff scale
because the parameter points with light masses accommodate $\Lambda_{\rm cut}$
from $1{\;{\rm TeV}}$ to $10^{19}{\;{\rm GeV}}$. Targeting at the
phenomenologically challenging case of $t_{\beta}>10$, we have investigated
the trilinear Higgs couplings versus $\Lambda_{\rm cut}$. Although the values
of all the trilinear couplings for $\Lambda_{\rm cut}=1{\;{\rm TeV}}$ are
different from those for $\Lambda_{\rm cut}=10^{18}{\;{\rm GeV}}$,
$\hat{\lambda}_{hhh}$ shows a large variation about $\Lambda_{\rm cut}$,
$\hat{\lambda}_{hhh}\in[-0.09,\,1.1]$. Multi-Higgs boson productions at the
LHC have been studied to probe the cutoff scale. The gluon fusion productions
of the di-Higgs, $gg\to hh/AA$, are not efficient to measure
$\hat{\lambda}_{hhh}$ because the dominant contribution from $H$ dilutes the
$h$ contribution. The most remarkable result is in the tri-Higgs process of
$pp\to H^{\pm}hh$ mediated by the $W$ boson. The signal cross section shows
huge variation according to $\Lambda_{\rm cut}$, like $\sigma_{\Lambda_{\rm
cut}=1{\;{\rm TeV}}}/\sigma_{\Lambda_{\rm cut}=10^{18}{\;{\rm GeV}}}\sim
10^{3}$. The precision measurement of $pp\to H^{\pm}hh$ can indeed distinguish
the high- and low-cutoff scales of the model. Considering the dominant decay
modes of $H^{\pm}\to W^{\pm*}A/h$, $h\to bb$, and $A\to bb$, an efficient
final state of $pp\to H^{\pm}hh$ is $6b+\ell\nu$. Although it enjoys almost
background-free environment at the LHC, the small cross section ($\lesssim
1{\;{\rm ab}}$) for $\Lambda_{\rm cut}=10^{18}{\;{\rm GeV}}$ motivates future
high-energy colliders, especially the muon collider with $\sqrt{s}=3-30{\;{\rm
TeV}}$.

###### Acknowledgements.

The work of J. K., S. L, and J. S is supported by the National Research
Foundation of Korea, Grant No. NRF-2022R1A2C1007583. S. K. K. was supported by
the National Research Foundation of Korea (NRF) grant funded by the Korea
government (MSIT) (No.2019R1A2C1088953).

##  Appendix A RGEs in the type I

Focusing on the type I, we present the one-loop level RGEs Cheng:1973nv ;
Komatsu:1981xh ; Branco:2011iw ; Das:2015mwa ; Basler:2017nzu . The beta
functions of gauge couplings are given by

$\displaystyle 16\pi^{2}\beta_{g_{3}}$ $\displaystyle=-7g_{3}^{3},$ (25)
$\displaystyle 16\pi^{2}\beta_{g_{2}}$
$\displaystyle=\left(-\frac{10}{3}+\frac{n_{d}}{6}\right)g_{2}^{3}=-3g_{2}^{3},$
(26) $\displaystyle 16\pi^{2}\beta_{g_{1}}$
$\displaystyle=\left(\frac{20}{3}+\frac{n_{d}}{6}\right)=7g_{1}^{3},$ (27)

where $n_{d}$ is the number of the scalar doublets of the fermions, so
$n_{d}=2$ in the 2HDM. The running of the quartic couplings of $\lambda_{i}$’s
is different according to the type. First, we write the $\beta$ functions in
terms of the common part $c_{i}$ and the type-dependent part $h_{i}$ as

$\displaystyle 16\pi^{2}\beta_{\lambda_{i}}^{\rm type}$
$\displaystyle=c_{i}+h_{i}^{\rm type},\quad(i=1,\cdots,5).$ (28)

The common parts for $\lambda_{i}$’s are

$\displaystyle c_{1}$
$\displaystyle=12\lambda_{1}^{2}+4\lambda_{3}^{2}+4\lambda_{3}\lambda_{4}+2\lambda_{4}^{2}+2\lambda_{5}^{2}+\frac{3}{4}(3g^{4}+g^{\prime
4}+2g^{2}g^{\prime 2})-3\lambda_{1}(3g^{2}+g^{\prime 2}),$ (29) $\displaystyle
c_{2}$
$\displaystyle=12\lambda_{2}^{2}+4\lambda_{3}^{2}+4\lambda_{3}\lambda_{4}+2\lambda_{4}^{2}+2\lambda_{5}^{2}+\frac{3}{4}(3g^{4}+g^{\prime
4}+2g^{2}g^{\prime 2})-3\lambda_{2}(3g^{2}+g^{\prime 2})$
$\displaystyle\quad+12y_{t}^{2}\lambda_{2}-12y_{t}^{4},$ $\displaystyle c_{3}$
$\displaystyle=(\lambda_{1}+\lambda_{2})(6\lambda_{3}+2\lambda_{4})+4\lambda_{3}^{2}+2\lambda_{4}^{2}+2\lambda_{5}^{2}+\frac{3}{4}(3g^{4}+g^{\prime
4}-2g^{2}g^{\prime 2})-3\lambda_{3}(3g^{2}+g^{\prime 2})$
$\displaystyle\quad+2(3y_{t}^{2}+3y_{b}^{2}+y_{\tau}^{2})\lambda_{3},$
$\displaystyle c_{4}$
$\displaystyle=2(\lambda_{1}+\lambda_{2})\lambda_{4}+8\lambda_{3}\lambda_{4}+4\lambda_{4}^{2}+8\lambda_{5}^{2}+3g^{2}g^{\prime
2}-3(3g^{2}+g^{\prime 2})\lambda_{4}$
$\displaystyle\quad+2(3y_{t}^{2}+3y_{b}^{2}+y_{\tau}^{2})\lambda_{4},$
$\displaystyle c_{5}$
$\displaystyle=2(\lambda_{1}+\lambda_{2}+4\lambda_{3}+6\lambda_{4})\lambda_{5}-3\lambda_{5}(3g^{2}+g^{\prime
2})+2(3y_{t}^{2}+3y_{b}^{2}+y_{\tau}^{2})\lambda_{5}.$

The $h_{i}$’s in type I are

$\displaystyle h_{1}^{\rm I}$ $\displaystyle=h_{3}^{\rm I}=h_{4}^{\rm
I}=h_{5}^{\rm I}=0,$ (30) $\displaystyle h_{2}^{\rm I}$
$\displaystyle=4(3y_{b}^{2}+y_{\tau}^{2})\lambda_{2}-4(3y_{b}^{4}+y_{\tau}^{4}).$

The Yukawa couplings of the top quark, bottom quark, and tau lepton ($y_{t}$,
$y_{b}$, and $y_{\tau}$) are running with the $\beta$ functions of

$\displaystyle 16\pi^{2}\beta_{y_{f}}^{\rm type}$
$\displaystyle=c_{y_{f}}+h_{y_{f}}^{\rm type},\quad(f=t,b,\tau),$ (31)

where the common parts are

$\displaystyle c_{y_{t}}$
$\displaystyle=\left(-8g_{s}^{2}-\frac{9}{4}g^{2}-\frac{17}{12}g^{\prime
2}+\frac{9}{2}y_{t}^{2}\right)y_{t},$ (32) $\displaystyle c_{y_{b}}$
$\displaystyle=\left(-8g_{s}^{2}-\frac{9}{4}g^{2}-\frac{5}{12}g^{\prime
2}+\frac{3}{2}y_{t}^{2}+\frac{9}{2}y_{b}^{2}\right)y_{b},$ (33) $\displaystyle
c_{y_{\tau}}$ $\displaystyle=\left(-\frac{9}{4}g^{2}-\frac{15}{4}g^{\prime
2}+\frac{5}{2}y_{\tau}^{2}\right)y_{\tau},$

and the type-dependent parts are

$h_{y_{t}}^{\rm I}=\left(\frac{3}{2}y_{b}^{2}+y_{\tau}^{2}\right)y_{t},\quad
h_{y_{b}}^{\rm I}=y_{\tau}^{2}y_{b},\quad h_{y_{\tau}}^{\rm
I}=3\left(y_{t}^{2}+y_{b}^{2}\right)y_{\tau}.$ (34)

The initial conditions of the Yukawa coupling are set at the top quark mass
scale $m_{t}^{\rm pole}$ Das:2015mwa  as

$\displaystyle y_{t}(m_{t}^{\rm pole})$
$\displaystyle=\frac{\sqrt{2}m_{t}}{vs_{\beta}}\left\\{1-\frac{4}{3\pi}\alpha_{s}(m_{t})\right\\},$
(35) $\displaystyle y_{b}(m_{t}^{\rm pole})$
$\displaystyle=\frac{\sqrt{2}m_{b}}{vs_{\beta}},$ $\displaystyle
y_{\tau}(m_{t}^{\rm pole})$
$\displaystyle=\frac{\sqrt{2}m_{\tau}}{vs_{\beta}}.$

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