#  Status of the two-Higgs-doublet model
in light of the CDF $m_{W}$ measurement

Soojin Lee<EMAIL_ADDRESS>Department of Physics, Konkuk University,
Seoul 05029, Republic of Korea  Kingman Cheung<EMAIL_ADDRESS>Department of Physics, Konkuk University, Seoul 05029, Republic of Korea
Department of Physics, National Tsing Hwa University, Hsinchu 300, Taiwan
Center for Theory and Computation, National Tsing Hua University, Hsinchu 300,
Taiwan  Jinheung Kim<EMAIL_ADDRESS>Department of Physics, Konkuk
University, Seoul 05029, Republic of Korea  Chih-Ting Lu<EMAIL_ADDRESS>Department of Physics and Institute of Theoretical Physics, Nanjing Normal
University, Nanjing, 210023, China  Jeonghyeon Song<EMAIL_ADDRESS>Department of Physics, Konkuk University, Seoul 05029, Republic of Korea

###### Abstract

The most recent the $W$-boson mass measurement by the CDF Collaboration with a
substantially reduced uncertainty indicated a large deviation from the
standard model prediction, as large as $7\sigma$ if taken literally. In terms
of the Peskin-Takeuchi parameters of $S$ and $T$, the central values would
shift to larger values. It has profound consequences in searching for physics
beyond the SM. We show in the framework of two-Higgs-doublet Models the effect
of the new $m^{\rm CDF}_{W}$ measurement on the parameter space. Combined this
$m^{\rm CDF}_{W}$ measurement with other constraints including theoretical
requirements, flavor-changing neutral currents in $B$ physics, maintaining the
perturbativity of all couplings until the cutoff scale $\Lambda_{c}=1$ TeV,
Higgs precision data, and direct collider search limits from the LEP,
Tevatron, and LHC experiments, we find upper bounds on the masses of the heavy
Higgs bosons $M_{H,A,H^{\pm}}\lesssim 1000-1100$ GeV in type-I, II, X, and Y
for the normal scenario and $M_{H^{\pm},A}\lesssim 410-440$ GeV in type I and
X for the inverted scenario. Another important finding is that type-II and
type-Y in the inverted scenario are completely excluded. Such unprecedented
findings imply that the upcoming LHC run can readily close out the still-
available parameter space.

Higgs Physics, Beyond the Standard Model, electroweak precision data

###### Contents

  1. I Introduction
  2. II Review of 2HDM
  3. III Scanning strategies
  4. IV Characteristic features of the normal scenario
    1. IV.1 type-I and type-X
    2. IV.2 type-II and type-Y
  5. V Characteristic features of the inverted scenario
  6. VI Conclusions

##  I Introduction

The standard model (SM) of electroweak theory with $SU(2)\times U(1)$ gauge
symmetry is highly successful in explaining almost all the observations in
particle physics experiments, especially after discovering the final piece of
SM, the Higgs boson. Therefore, we believe that the SM is at least a good
effective theory for the energy scale below TeV. On the other hand, theorists
have not stopped extending the SM to answer the outstanding questions in
particle physics, such as the neutrino mass and mixing, matter-antimatter
asymmetry in the Universe and dark matter. Hence, if any experiment at the
energy scale below about 1 TeV observes an anomaly, it would shake the
foundation of the SM.

Very recently, the CDF collaboration at Fermilab reported the most precise $W$
boson mass measurement: $m^{\text{CDF}}_{W}=80.4335\pm 0.0094$ GeV CDF:2022hxs
. The total uncertainty is less than $10$ MeV and the central value is about
$76.5$ MeV larger than the SM prediction: $m^{\text{SM}}_{W}=80.357\pm 0.006$
GeV ParticleDataGroup:2020ssz ). We have about $7\sigma$ standard deviation
from the SM prediction. Before the CDF run-II result, the world average of
$m_{W}$ measurements was just $1.8\sigma$ standard deviation from the SM
expectation ParticleDataGroup:2020ssz . Even though more careful cross-checks
of the systematic uncertainties among CDF run-II analysis and other $W$ boson
mass measurements such as LEP ALEPH:2013dgf , LHCb LHCb:2021bjt , ATLAS
ATLAS:2017rzl , and D0 D0:2012kms  are still needed, the validity of the SM is
questioned and possible explanations from new physics models are pursued after
this new measured $m_{W}$ value.

The global fit to the electroweak precision data (EWPD) is one of the most
powerful tools to quantify the consistency of the SM and point in the
direction of new physics Haller:2018nnx ; deBlas:2021wap . Therefore, the new
global electroweak fit should be the first step to approaching this $W$ boson
mass anomaly. According to Ref. Lu:2022bgw ; Strumia:2022qkt ; deBlas:2022hdk
; Fan:2022yly ; Paul:2022dds ; Gu:2022htv ; Asadi:2022xiy ; Endo:2022kiw ;
Balkin:2022glu ; Carpenter:2022oyg ; Du:2022fqv , it is hard to explain
$m^{\text{CDF}}_{W}$ without going beyond the SM. An efficient way to
parametrize the impact of new physics on the global EW fits is a set of
Peskin-Takeuchi oblique parameters $S$, $T$, and $U$ Peskin:1990zt ;
Marciano:1990dp ; Kennedy:1990ib . If all of these three parameters can vary,
the new fits show that $S$ and $T$ can keep as before, but the $U$ increases
substantially such that $S=0.06\pm 0.10$, $T=0.11\pm 0.12$, $U=0.13\pm 0.09$
Lu:2022bgw . However, the contributions to $U$ can only appear in the
dimension-eight operator and most new physics models have tiny contributions
to $U$. Therefore, setting $U=0$ while varying $S$ and $T$ is reassumed,
resulting in both $S$ and $T$ moving to large and positive values: $S=0.14\pm
0.08$ and $T=0.26\pm 0.06$ Lu:2022bgw . Based on these changes of $S$ and $T$,
some new physics models including two-Higgs-doublet model (2HDM) and its
extensions 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 , the Higgs triplet model
Cheng:2022jyi ; Du:2022brr ; Kanemura:2022ahw ; Mondal:2022xdy ; Borah:2022obi
, supersymmetry Yang:2022gvz ; Du:2022pbp ; Tang:2022pxh ; Athron:2022isz ;
Zheng:2022irz ; Ghoshal:2022vzo , leptoquarks Athron:2022qpo ; Cheung:2022zsb
; Bhaskar:2022vgk , seesaw mechanisms of neutrino mass and their extensions
Blennow:2022yfm ; Arias-Aragon:2022ats ; Liu:2022jdq ; Chowdhury:2022moc ;
Popov:2022ldh , vector-like leptons or vector-like quarks Lee:2022nqz ;
Kawamura:2022uft ; Crivellin:2022fdf ; Nagao:2022oin , the standard model
effective field theory (SMEFT) deBlas:2022hdk ; Fan:2022yly ;
Bagnaschi:2022whn ; Paul:2022dds ; Gu:2022htv ; DiLuzio:2022xns ; Endo:2022kiw
; Balkin:2022glu ; Cirigliano:2022qdm , and others Yuan:2022cpw ;
Strumia:2022qkt ; Cacciapaglia:2022xih ; Sakurai:2022hwh ; Heckman:2022the ;
Krasnikov:2022xsi ; Peli:2022ybi ; Perez:2022uil ; Wilson:2022gma ;
Zhang:2022nnh ; Carpenter:2022oyg ; Du:2022fqv  are proposed to explain this
$W$ boson mass anomaly. Especially, some of them also try to explain the long-
standing tension in muon $g-2$ measurement Athron:2022qpo ; Du:2022pbp ;
Tang:2022pxh ; Lee:2022nqz ; Han:2022juu ; Kawamura:2022uft ; Cheung:2022zsb ;
Nagao:2022oin ; Chowdhury:2022moc ; Arcadi:2022dmt ; Bhaskar:2022vgk  which
has been confirmed from Fermilab Muong-2:2021ojo .

In this work, we pursue a comprehensive study of the 2HDM in light of the new
CDF $m_{W}$ measurement. We study not only four famous tree-level flavor-
conserved types (type-I, type-II, type-X, and type-Y) but also two scenarios
for the observed Higgs boson, the normal scenario (NS) and inverted scenario
(IS). We impose all the theoretical and experimental constraints, including
the stability of the scalar potential, the electroweak precision data (EWPD)
with the new CDF $m_{W}$ measurement, the Higgs precision data, and the direct
search bounds for new scalar bosons at the LEP, Tevatron, and LHC. In
addition, we study the RGE evolutions of the model parameters, and demand the
stability of the 2HDM potential up to 1 TeV, i.e., the cutoff scale
$\Lambda_{\rm c}$ above 1 TeV. Combining $\Lambda_{\rm c}>1{\;{\rm TeV}}$ with
the other constraints puts the upper bounds on the masses of new Higgs bosons
at about 1 TeV. The upper bound on the charged Higgs boson mass by
$\Lambda_{\rm c}>1{\;{\rm TeV}}$ causes a conflict with the lower bound on
$M_{H^{\pm}}$ from $b\to s\gamma$ Arbey:2017gmh ; Misiak:2017bgg  for type-II
and type-Y, especially in the IS. Consequently, type-II and type-Y in the IS
are excluded in light of the CDF $m_{W}$ measurement. This is one of the most
salient results in our paper. A total of eight cases, four types in the NS and
IS, are thoroughly investigated by scanning the whole parameter space without
any conditions on the masses and couplings. And the results give apparent
signals for the future collider phenomenologies: (i) type-I has the most
survival parameter points for both NS and IS; (ii) the light charged Higgs
boson with mass below the top quark mass is viable in type-I of the NS; (iii)
light neutral Higgs bosons, CP-even and CP-odd, are still allowed for type-I
and type-X.

The rest of this paper is arranged as follows. We first briefly review the
2HDM in Sec. II. The parameter scanning strategies are introduced in Sec. III,
and the allowed mass ranges of new Higgs bosons are also shown. We then
discuss the characteristic features of the normal and inverted scenarios of
2HDMs in Sec. IV and Sec. V, respectively. Finally, we conclude in Sec. VI.

##  II Review of 2HDM

In the 2HDM, there exist two complex $SU(2)_{L}$ Higgs scalar doublet fields,
$\Phi_{1}$ and $\Phi_{2}$ Branco:2011iw :

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

where $v_{1}$ and $v_{2}$ are the nonzero vacuum expectation values of
$\Phi_{1}$ and $\Phi_{2}$, respectively. The electroweak symmetry is broken by
$v=\sqrt{v_{1}^{2}+v_{2}^{2}}=246{\;{\rm GeV}}$. We define the ratio of two
vacuum expectation values to be $\tan\beta=v_{2}/v_{1}$. For simplicity, we
use the simplified notation of $s_{x}=\sin x$, $c_{x}=\cos x$, and $t_{x}=\tan
x$ ($x=\beta,\beta-\alpha,\cdots$) in what follows.

We additionally impose a discrete $Z_{2}$ symmetry, under which
$\Phi_{1}\to\Phi_{1}$ and $\Phi_{2}\to-\Phi_{2}$, to avoid the flavor-
changing-neutral-current (FCNC) at tree leve Glashow:1976nt ; Paschos:1976ay .
The scalar potential with softly broken $Z_{2}$ and CP invariance is

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

where the $m^{2}_{12}$ term softly breaks the $Z_{2}$ parity. The model has
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 $H^{\pm}$. The weak eigenstates in Eq. (3) are linear combinations of
physical Higgs bosons through two mixing angles $\alpha$ and $\beta$, which
are referred to Ref. Song:2019aav . A significant relation is the SM Higgs
boson ${h_{\rm SM}}$ with $h$ and $H$:

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

In the 2HDM, the observed Higgs boson at a mass of 125 GeV can be either $h$
or $H$, which is called the normal scenario (NS) and the inverted scenario
(IS), respectively:

NS: $\displaystyle\leavevmode\nobreak\ m_{h}=m_{125};$ (6) IS:
$\displaystyle\leavevmode\nobreak\ M_{H}=m_{125},$

where $m_{125}$ is the observed Higgs boson mass. For the consistency with the
Higgs precision data at the LHC, the Higgs alignment limit has drawn a lot of
attention, where $h={h_{\rm SM}}$ in the NS and $H={h_{\rm SM}}$ in the IS.
Even though the limit simplifies the phenomenology of the new Higgs bosons
such that $H\to WW/ZZ$, $A\to Zh$, and $H^{\pm}\to W^{\pm(*)}h$ are prohibited
at tree level, it may interfere with the observation of the 2HDM. Therefore,
we pursue the allowed parameter space without any conditions on the masses and
couplings. Only the theoretical and experimental constraints will be taken
into account to restrict the parameter space.

We take the physical parameter basis of

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

where six are free parameters from Eq. (6). The range of $\beta-\alpha$ is set
to be $[-\pi/2,\pi/2]$, following the notation of the public codes such as
2HDMC Eriksson:2009ws , HiggsSignals Bechtle:2020uwn , and HiggsBounds
Bechtle:2020pkv , which results in $c_{\beta-\alpha}>0$. Since one of our main
results is from the RGE evolution of the quartic couplings, we present the
quartic couplings Das:2015mwa

$\displaystyle\lambda_{1}$ $\displaystyle=$
$\displaystyle\frac{1}{v^{2}c_{\beta}^{2}}\left[c_{\alpha}^{2}M_{H}^{2}+s_{\alpha}^{2}m_{h}^{2}-t_{\beta}m_{12}^{2}\right],$
(8) $\displaystyle\lambda_{2}$ $\displaystyle=$
$\displaystyle\frac{1}{v^{2}s_{\beta}^{2}}\left[s_{\alpha}^{2}M_{H}^{2}+c_{\alpha}^{2}m_{h}^{2}-\frac{1}{t_{\beta}}m_{12}^{2}\right],$
$\displaystyle\lambda_{3}$ $\displaystyle=$
$\displaystyle\frac{1}{v^{2}}\left[2M_{H^{\pm}}^{2}+\frac{s_{2\alpha}}{s_{2\beta}}(M_{H}^{2}-m_{h}^{2})-\frac{m_{12}^{2}}{s_{\beta}c_{\beta}}\right],$
$\displaystyle\lambda_{4}$ $\displaystyle=$
$\displaystyle\frac{1}{v^{2}}\left[M_{A}^{2}-2M_{H^{\pm}}^{2}+\frac{m_{12}^{2}}{s_{\beta}c_{\beta}}\right],$
$\displaystyle\lambda_{5}$ $\displaystyle=$
$\displaystyle\frac{1}{v^{2}}\left[\frac{m_{12}^{2}}{s_{\beta}c_{\beta}}-M_{A}^{2}\right],$

where $M^{2}=m_{12}^{2}/(s_{\beta}c_{\beta})$.

According to the $Z_{2}$ parity of the fermion singlets, there are four
different types in the 2HDM, type-I, type-II, type-X, and type-Y. According to
the type, the Yukawa couplings to the SM fermions are different, but the gauge
couplings remain the same. We parameterize the Yukawa Lagrangian as

$\displaystyle\mathscr{L}_{\rm Yuk}$ $\displaystyle=$
$\displaystyle-\sum_{f}\left(\frac{m_{f}}{v}\xi^{h}_{f}\bar{f}fh+\frac{m_{f}}{v}\xi^{H}_{f}\bar{f}fH-i\frac{m_{f}}{v}\xi^{A}_{f}\bar{f}\gamma_{5}fA\right)$
$\displaystyle-\left\\{\dfrac{\sqrt{2}V_{ud}}{v}H^{+}\overline{u}\left(m_{u}\xi^{A}_{u}\text{P}_{L}+m_{d}\xi^{A}_{d}\text{P}_{R}\right)d+\dfrac{\sqrt{2}m_{\ell}}{v}H^{+}\xi^{A}_{\ell}\overline{\nu}_{L}\ell_{R}+{\rm
H.c.}\right\\},$

where $\xi^{h,H,A}_{f}$, the Yukawa coupling modifiers, are presented in Table
1. Since we shall consider a total of eight cases, four types in the NS and
IS, we shall use a simple notation of NS-I (IS-I) for type-I in the NS (IS),
for instance.

|  $\xi^{h}_{u}$ |  $\xi^{h}_{d}$ |  $\xi^{h}_{\ell}$ |  $\xi^{H}_{u}$ |  $\xi^{H}_{d}$ |  $\xi^{H}_{\ell}$ |  $\xi^{A}_{u}$ |  $\xi^{A}_{d}$ |  $\xi^{A}_{\ell}$
---|---|---|---|---|---|---|---|---|---
Type-I | $\frac{c_{\alpha}}{s_{\beta}}$ | $\frac{c_{\alpha}}{s_{\beta}}$ | $\frac{c_{\alpha}}{s_{\beta}}$ | $\frac{s_{\alpha}}{s_{\beta}}$ | $\frac{s_{\alpha}}{s_{\beta}}$ | $\frac{s_{\alpha}}{s_{\beta}}$ | $\frac{1}{t_{\beta}}$ | $-\frac{1}{t_{\beta}}$ | $-\frac{1}{t_{\beta}}$
Type-II | $\frac{c_{\alpha}}{s_{\beta}}$ | $-\frac{s_{\alpha}}{c_{\beta}}$ | $-\frac{s_{\alpha}}{c_{\beta}}$ | $\frac{s_{\alpha}}{s_{\beta}}$ | $\frac{c_{\alpha}}{c_{\beta}}$ | $\frac{c_{\alpha}}{c_{\beta}}$ | $\frac{1}{t_{\beta}}$ | $t_{\beta}$ | $t_{\beta}$
Type-X | $\frac{c_{\alpha}}{s_{\beta}}$ | $\frac{c_{\alpha}}{s_{\beta}}$ | $-\frac{s_{\alpha}}{c_{\beta}}$ | $\frac{s_{\alpha}}{s_{\beta}}$ | $\frac{s_{\alpha}}{s_{\beta}}$ | $\frac{c_{\alpha}}{c_{\beta}}$ | $\frac{1}{t_{\beta}}$ | $-\frac{1}{t_{\beta}}$ | $t_{\beta}$
Type-Y | $\frac{c_{\alpha}}{s_{\beta}}$ | $-\frac{s_{\alpha}}{c_{\beta}}$ | $\frac{c_{\alpha}}{s_{\beta}}$ | $\frac{s_{\alpha}}{s_{\beta}}$ | $\frac{c_{\alpha}}{c_{\beta}}$ | $\frac{s_{\alpha}}{s_{\beta}}$ | $\frac{1}{t_{\beta}}$ | $t_{\beta}$ | $-\frac{1}{t_{\beta}}$
Table 1: The Yukawa coupling modifiers in four types of the 2HDM.

Before scanning the whole parameter space, we study the implication of the
oblique parameter $T$ on the masses of new Higgs bosons in a simple setup, the
normal scenario in the Higgs alignment limit. We consider two different
Peskin-Takeuchi oblique parameter sets with $U=0$, before and after the new
CDF $m_{W}$ measurement, denoted by “PDG” and “CDF” respectively:

PDG: $\displaystyle\leavevmode\nobreak\ S=0.05\pm 0.08,\quad T=0.09\pm
0.07,\quad\rho_{ST}=0.92,$ (10) CDF: $\displaystyle\leavevmode\nobreak\
S=0.15\pm 0.08,\quad T=0.27\pm 0.06,\quad\rho_{ST}=0.93,$

where $\rho_{ST}$ is the correlation between $S$ and $T$.

Figure 1:  Allowed regions of $(\Delta M_{H},\Delta M_{A})$ by the oblique
parameters of $S$ and $T$ (red at $1\sigma$ and yellow in $2\sigma$),
$|\lambda_{3}|<4\pi$ (light blue), and $|\lambda_{3}|<1$ (green), where
$\Delta m\equiv m-M_{H^{\pm}}$. The left (right) panel shows the PDG (CDF)
result. We set $M_{H^{\pm}}=1.5{\;{\rm TeV}}$ and $M^{2}=M_{A}^{2}$ in the
normal scenario with $s_{\beta-\alpha}=1$.

As shall be shown, one of the most important results of the CDF-updated $S/T$
is that there exist the upper bounds on the masses of new Higgs bosons even in
the NS, which is not expected. This is due to the combined constraints from
the theoretical stability, the new $S/T$, and $\Lambda_{\rm c}>1{\;{\rm
TeV}}$. To understand why very heavy new Higgs bosons are excluded, we present
the allowed regions of $(\Delta M_{H},\Delta M_{A})$ for
$M_{H^{\pm}}=1.5{\;{\rm TeV}}$ by the oblique parameters $S$ and $T$ (red),
$|\lambda_{3}|<4\pi$ (light blue), and $|\lambda_{3}|<1$ (green) in Fig. 1,
where $\Delta m\equiv m-M_{H^{\pm}}$. The PDG result is in the left panel, and
the CDF result is in the right panel. For illustration, we take a simple
setup, the NS in the Higgs alignment limit. The condition of $M^{2}=M_{A}^{2}$
is motivated to avoid excessively fast running of the quartic couplings in the
RGE studies of the 2HDM Das:2015mwa .

While the PDG $T$ parameter allows the total mass degeneracy of
$M_{H^{\pm}}=M_{A}=M_{H}$ at $2\sigma$ level, the CDF-updated $T$ prohibits
it. Instead, the CDF result leaves two separate regions of the hyperbola-like
shape. Despite the absence of the total mass degeneracy, the oblique $T$
parameter from the CDF result still permits heavy masses of new Higgs bosons:
when $M_{H}\approx M_{H^{\pm}}$ ($M_{A}\approx M_{H^{\pm}}$), very heavy
$M_{A}$ ($M_{H}$) is feasible. When applying the perturbativity of
$\lambda_{3}$ (light blue), however, large mass gaps of $\Delta M_{H}$ and
$\Delta M_{A}$ are forbidden. When narrowing the range into $|\lambda_{3}|<1$,
there is no overlap with the allowed space by the CDF-updated $T$ parameter.
The $|\lambda_{3}|$ above one at the electroweak scale invokes its fast
running under the RGE evolution, thus bringing the breakdown of the theory at
the low energy scale by the RGE. Therefore, heavy Higgs bosons with masses
above about 1 TeV cannot simultaneously satisfy the theoretical stability, the
CDF-updated $T$ parameter, and $\Lambda_{\rm c}>1{\;{\rm TeV}}$.

##  III Scanning strategies

We study the implication of the theoretical and experimental constraints
through random scanning of the model parameters. The scanning ranges in the NS
and IS are

NS: $\displaystyle M_{H}\in\left[130,2000\right]{\;{\rm
GeV}},\;M_{A}\in\left[15,2000\right]{\;{\rm GeV}},$ (11) $\displaystyle
s_{\beta-\alpha}\in\left[0.8,1.0\right],\,\,m_{12}^{2}\in\left[0,1000^{2}\right]{\;{\rm
GeV}}^{2},$ IS: $\displaystyle m_{h}\in\left[15,120\right]{\;{\rm
GeV}},\;M_{A}\in\left[15,2000\right]{\;{\rm GeV}},$ $\displaystyle
c_{\beta-\alpha}\in\left[0.8,1.0\right],\,\,m_{12}^{2}\in\left[0,1000^{2}\right]{\;{\rm
GeV}}^{2}.$

The range of $s_{\beta-\alpha}$ is set comfortably, although the most updated
results on the Higgs coupling modifiers are $\kappa_{Z}>0.86$ and
$\kappa_{W}>0.94$ with $\kappa_{W,Z}\leq 1$ at 95% C.L. Aad:2019mbh . The
parameters in Eq. (11) are independent of the type, but $M_{H^{\pm}}$ and
$t_{\beta}$ are different because of the FCNC observables Arbey:2017gmh ;
Misiak:2017bgg : $M_{H^{\pm}}$ in type-II and type-Y is tightly constrained to
be $M_{H^{\pm}}\gtrsim 580{\;{\rm GeV}}$; $M_{H^{\pm}}$ in type-I and type-X
can be as light as about $100{\;{\rm GeV}}$. Therefore, we take the following
ranges for $M_{H^{\pm}}$ and $t_{\beta}$:

type-I & type-X: $\displaystyle\leavevmode\nobreak\
M_{H^{\pm}}\in\left[80,2000\right]{\;{\rm GeV}},\quad
t_{\beta}\in\left[1,50\right],$ (12) type-II & type-Y:
$\displaystyle\leavevmode\nobreak\ M_{H^{\pm}}\in\left[580,2000\right]{\;{\rm
GeV}},\quad t_{\beta}\in\left[0.5,50\right].$

With the prepared random parameter points, we cumulatively impose the
following steps for the theoretical and experimental constraints:

Step-(i) Theory+FCNC:

We require a parameter point to satisfy the theoretical stabilities and the
FCNC results, by using the public code 2HDMC-v1.8.0 Eriksson:2009ws .

  1. 1.

Higgs potential being bounded from below Ivanov:2006yq ;

  2. 2.

Perturbative unitarity of the amplitudes of scalar-scalar, scalar-vector, and
vector-vector scatterings at high energies Kanemura:1993hm ; Akeroyd:2000wc ;

  3. 3.

Perturbativity of the quartic couplings Branco:2011iw ; Chang:2015goa ;

  4. 4.

Vacuum stability Barroso:2013awa .

  5. 5.

FCNC observables: In type-I and type-X, the viable space of
$(M_{H^{\pm}},t_{\beta})$ is determined practically by $b\to s\gamma$
Arbey:2017gmh ; Misiak:2017bgg . In type-II and type-Y, on the other hand,
$\Delta M_{B_{s}}$ HFLAV:2019otj  is more important in constraining the region
with $M_{H^{\pm}}\gtrsim 650{\;{\rm GeV}}$.

Step-(ii) EWPD:

We calculate the Peskin-Takeuchi oblique parameters in the 2HDM He:2001tp ;
Grimus:2008nb ; Zyla:2020zbs , and compare them with the oblique parameters
fitted to the EWPD before and after the CDF $m_{W}$ measurement. Since we
perform two parameter ($S$ and $T$) fitting under the assumption of $U=0$, we
require $\chi^{2}<5.9$.

Step-(iii) RGEs for $\Lambda_{\rm c}>1{\;{\rm TeV}}$:

We require that the model should be valid at least up to 1 TeV. Using the
RGE’s Cheng:1973nv ; Komatsu:1981xh ; Branco:2011iw ; Das:2015mwa ;
Basler:2017nzu , we run the dimensionless parameters in the 2HDM, including
the gauge couplings, the quartic couplings in the scalar potential, and the
Yukawa couplings of the top quark, bottom quark, and tau lepton. The initial
conditions of the gauge couplings and the Yukawa couplings are set at the top
quark mass scale $m_{t}=173.34{\;{\rm GeV}}$ Oredsson:2018vio . As increasing
the energy scale, we check the unitarity and stability conditions. If any
condition is broken at the energy scale below 1 TeV, we exclude the parameter
point. In other words, the cutoff scale $\Lambda_{\rm c}$ of the model should
be above 1 TeV. We use the public code 2HDME-v1.2 Oredsson:2018vio  at one-
loop level.

Step-(iv) Collider:

The collider constraints consist of two categories, the Higgs precision data
and the direct search bounds at the LEP, Tevatron, and LHC. To check the
consistency with the Higgs precision, we use HiggsSignals-v2.6.2
Bechtle:2020uwn , which yields the $\chi^{2}$ output for 111 Higgs observables
Aaboud:2018gay ; Aaboud:2018jqu ; Aaboud:2018pen ; Aad:2020mkp ;
Sirunyan:2018mvw ; Sirunyan:2018hbu ; CMS:2019chr ; CMS:2019kqw . Since there
are six model parameters, the number of degrees of freedom is 105. We demand
that the $p$-value be larger than 0.05. For the consistency with the direct
searches at high energy collider, we use the public code, HiggsBounds-v5.10.2
Bechtle:2020pkv . For each process at the LEP, Tevatron, and LHC, we calculate
the cross section in the model, and compare it with the upper bound on the
cross section at the 95% C.L. When the model prediction is above the observed
upper bound, we exclude the parameter point.

Figure 2:  Allowed regions of $(M_{H},M_{A})$ in the NS (upper panels) and
$(m_{h},M_{A})$ in the IS (lower panels) by the theoretical stabilities and
the $b\to s\gamma$ constraint. The results in type-I and type-X are in the
left panels, while those in type-II and type-Y are in the right panels. The
color codes indicate $M_{H^{\pm}}$.

We perform the random scan over the entire six-dimensional parameter space.
For each type in the NS and IS (eight cases), we obtained $10^{6}$ parameter
points that satisfy Step-(i), which is to be the reference when computing the
survival probabilities. Before proceeding to the subsequent steps, we
investigate the implications of Step-(i) on the masses of new Higgs bosons. In
Fig. 2, we present the parameter points in $(M_{H},M_{A})$ that pass Step-(i),
where the color codes denote $M_{H^{\pm}}$. The results in the NS (IS) are in
the upper (lower) panels and those at type-I/X (type-II/Y) are in the left
(right) panels.

The theoretical stabilities and the FCNC observables significantly restrict
the masses of new Higgs bosons. In the NS, the low mass regions with
$M_{A},M_{H},M_{H^{\pm}}\lesssim 750{\;{\rm GeV}}$ are uniformly permitted,
without a correlation among the masses. For the high mass above 750 GeV,
however, Step-(i) demands a considerable correlation among the masses,
preferring small mass gaps and thus the mass degeneracy. The higher the mass
scales are, the smaller the mass gaps will be. In the IS, Step-(i) already
puts the upper bounds on $M_{A}$ and $M_{H^{\pm}}$, below about $750{\;{\rm
GeV}}$.

| EWPD | $\Lambda_{\rm c}>1{\;{\rm TeV}}$ | Collider | EWPD | $\Lambda_{\rm c}>1{\;{\rm TeV}}$ | Collider
---|---|---|---|---|---|---
| Normal scenario | Inverted scenario
type-I | PDG | 12.98% | 5.13% | 0.60% | 7.20% | 5.08% | 0.85%
CDF | 4.42% | 1.31% | 0.14% | 1.30% | 0.72% | 0.19%
type-II | PDG | 10.76% | 0.43% | 0.20% | 2.14% | 0 | 0
CDF | 3.36% | 0.03% | 0.01% | 0.69% | 0 | 0
type-X | PDG | 12.98% | 5.13% | 0.18% | 7.20% | 5.08% | 0.03%
CDF | 4.42% | 1.31% | 0.03% | 1.30% | 0.72% | 0.01%
type-Y | PDG | 10.76% | 0.43% | 0.20% | 2.14% | 0 | 0
CDF | 3.36% | 0.03% | 0.01% | 0.69% | 0 | 0
Table 2: Survival probabilities at each step in the NS and IS for all the four
types, with respect to the parameter points that pass Step (i). For the EWPD,
we adopt two different schemes of the oblique parameters, before and after the
CDF-updated $m_{W}$ measurement, denoted by “PDG” and “CDF”.

Now we cumulatively impose the constraints of Step-(ii), Step-(iii), and
Step-(iv). In Table 2, we present the survival probabilities of the random
scanning in Eqs. (11) and (12)111The survival probabilities depend on the
scanning ranges. For smaller ranges, the probabilities increase. at each step
in the NS and IS for all the four types, with respect to the parameter points
that pass Step-(i). One of the most important features is that the combination
of all the theoretical and experimental constraints restricts the model more
severely for the CDF result than for the PDG: see two figures in the right
panels.

Let us compare the overall differences between the NS and IS. In the NS, all
the types survive the final Step-(iv), even though the survival probabilities
are very small, of the order of 0.1% (0.01%) for the PDG (CDF) result. We find
that type-I has the largest survival probabilities while type-II and type-Y
have the smallest probabilities. In the IS, type-II and type-Y are totally
excluded.222To confirm this finding, we increased the number of points in
random scanning. But no parameter point survived for type-II and type-Y in the
IS. The decisive factor is the combination of $M_{H^{\pm}}>580{\;{\rm GeV}}$
from $b\to s\gamma$ and the requirement of $\Lambda_{\rm c}>1{\;{\rm TeV}}$.
The constraint from $b\to s\gamma$ demands heavy Higgs masses, which brings
out too fast running of the quartic couplings. Type-I and type-X are still
allowed in the IS.

Now we study the differences between the PDG and CDF results. As shown in
Table 2, the survival probabilities are very different, which are much small
for the CDF result in all the eight cases. The probability gap between the PDG
and CDF results widens sharply at Step-(ii). It is coming from the tension
that the EWPD prefer sizable $\Delta M_{A}$ and $\Delta M_{h,H}$ but
$\Lambda_{\rm c}>1{\;{\rm TeV}}$ favors the mass degeneracy Das:2015mwa .
Step-(iv) including the Higgs precision data and the direct search bounds also
reduces the parameter space substantially, in both cases with the PDG and CDF
results.

Figure 3:  In type-I of the normal scenario, the allowed parameter space of
$(M_{H},M_{A})$ after Step-(ii) (EWPD), Step-(iii) (Step(ii)+RGE), and
Step-(iv) (Step(iii)+Higgs precision+Direct searches), with the color code
indicating $M_{H^{\pm}}$. The upper panels are for PDG while the lower panels
for CDF.

To demonstrate the origin of the differences between the PDG and CDF in more
detail, we present the parameter points of $(M_{H},M_{A})$ that survive each
step. In Fig. 3, we consider type-I in the NS. At Step-(ii), the PDG and CDF
cases yield similar shapes also in the heavy mass regions. But the size is
different: the CDF result generates a slimmer area with substantial mass gaps,
while the PDG case allows a broader area, including $M_{H}=M_{A}=M_{H^{\pm}}$.
Existence or absence of the total mass degeneracy decides whether to save the
heavy mass region when imposing $\Lambda_{\rm c}>1{\;{\rm TeV}}$. Therefore,
the combination of the theoretical constraint with $\Lambda_{\rm c}>1{\;{\rm
TeV}}$ puts the upper bounds on the masses of new Higgs bosons in the CDF
case. Applying the constraints from the Higgs precision data and the direct
collider searches at Step-(iv) reduces the viable parameter points, but the
upper bounds on new Higgs bosons remain intact.

Figure 4:  In type-II of the normal scenario, the allowed parameter space of
$(M_{H},M_{A})$ after Step-(ii), Step-(iii), and Step-(iv), with the color
code indicating $M_{H^{\pm}}$. The upper panels are for PDG while the lower
panels for CDF.

Figure 4 presents the same results for type-II, where $M_{H^{\pm}}>580{\;{\rm
GeV}}$ due to the constraints from FCNC. As in type-I, the EWPD before and
after the CDF $m_{W}$ measurement yield similar shapes in $M_{H}$ vs $M_{A}$
with color code of $M_{H^{\pm}}$. Sizable mass gaps are also required in the
CDF case. A dramatic difference occurs when we further impose $\Lambda_{\rm
c}>1{\;{\rm TeV}}$, which puts the upper bounds on the masses of new Higgs
bosons and the lower bounds on $M_{H}$ and $M_{A}$. At the final Step-(iv), a
large portion of the parameter points is further excluded, but the mass bounds
of new Higgs bosons remain almost intact.

In summary, we present the allowed mass ranges in the NS:

NS-I: $\displaystyle M_{H^{\pm}}\in[87,1060]{\;{\rm GeV}},\quad
M_{H}\in[130,971]{\;{\rm GeV}},\quad M_{A}\in[28,977]{\;{\rm GeV}};$ (13) NS-
II: $\displaystyle M_{H^{\pm}}\in[600,1139]{\;{\rm GeV}},\quad
M_{H}\in[427,1127]{\;{\rm GeV}},\quad M_{A}\in[470,1125]{\;{\rm GeV}};$ NS-X:
$\displaystyle M_{H^{\pm}}\in[110,1060]{\;{\rm GeV}},\quad
M_{H}\in[130,971]{\;{\rm GeV}},\quad M_{A}\in[48,977]{\;{\rm GeV}};$ NS-Y:
$\displaystyle M_{H^{\pm}}\in[600,1139]{\;{\rm GeV}},\quad
M_{H}\in[427,1127]{\;{\rm GeV}},\quad M_{A}\in[470,1125]{\;{\rm GeV}};$

and in the IS:

IS-I: $\displaystyle M_{H^{\pm}}\in[166,436]{\;{\rm GeV}},\quad
m_{h}\in[47,120]{\;{\rm GeV}},\quad M_{A}\in[55,411]{\;{\rm GeV}};$ (14) IS-X:
$\displaystyle M_{H^{\pm}}\in[170,435]{\;{\rm GeV}},\quad
m_{h}\in[64,120]{\;{\rm GeV}},\quad M_{A}\in[28,409]{\;{\rm GeV}}.$

It is noteworthy that a light charged Higgs boson with mass around 100 GeV is
feasible in the NS-I Cheung:2022ndq .

##  IV Characteristic features of the normal scenario

In this section, we study the characteristics of the finally allowed
parameters after the CDF $m_{W}$ measurement, focusing on the NS. Since the
results of type-I (type-II) are similar to those of type-X (type-Y), we show
the results of type-I and type-X (type-II and type-Y) together. We shall
sequentially present the parameter points according to the steps when needed.
For a more detailed understanding, we separate Step-(iv) into two, one for the
Higgs precision data and the other for the direct search bounds at the LHC. In
summary, the order of the cumulative steps is as follows: (1) Theory+FCNC; (2)
EWPD for the oblique parameters of $S$ and $T$; (3) RGE study for
$\Lambda_{\rm c}>1{\;{\rm TeV}}$; (4) Higgs precision data through
HiggsSignals; (5) Direct search bounds through HiggsBounds.

###  IV.1 type-I and type-X

Figure 5:  $M_{H}$ vs $M_{A}$ with color code of $M_{H^{\pm}}$ in type-I (left
panel) and type-X (right panel) in the CDF case. We adopt the normal scenario.

First, we take a closer look at the allowed masses at the final Step-(iv). In
Fig 5, we present $M_{H}$ vs $M_{A}$ with color code of $M_{H^{\pm}}$ in
type-I (left panel) and type-X (right panel). In the heavy mass region with
$M_{H},M_{A}\gtrsim 700{\;{\rm GeV}}$, a strong correlation exists among the
masses of new Higgs bosons. We found that both $\Delta M_{H}$ and $\Delta
M_{A}$ are clustered around $\Delta M_{H,A}\simeq-100{\;{\rm GeV}}$ and
$\Delta M_{H,A}\simeq 100{\;{\rm GeV}}$: more points are in the negative
$\Delta M_{H,A}$ region. For $M_{H},M_{A}\lesssim 700{\;{\rm GeV}}$, there is
little correlation among masses. Compared with the results at the Step-(i) as
shown in Fig. 2, however, similar mass scales are weakly favored in the end.

Figure 6:  Allowed regions of $(|s_{\beta-\alpha}|,t_{\beta})$ in type-I
(upper panels) and type-X (lower panels), with color code indicating
$M_{H^{\pm}}$ for the normal scenario. We compare the results before (left
panels) and after (right panels) the CDF $m_{W}$ measurement.

Now we move on to $|s_{\beta-\alpha}|$ vs $t_{\beta}$. Figure 6 presents their
correlation in type-I (upper panels) and type-X (lower panels), with color
code indicating $M_{H^{\pm}}$. We compare the result in the PDG case (left
panels) with that in the CDF case (right panels). For $|s_{\beta-\alpha}|$,
the Higgs precision data play a vital role in limiting it as
$|s_{\beta-\alpha}|\gtrsim 0.88$ in type-I and $|s_{\beta-\alpha}|\gtrsim
0.90$ in type-X, which are almost the same both before and after the CDF
$m_{W}$ anomaly. Interestingly, the upper bound on $t_{\beta}$ is considerably
small in the CDF case. In type-I, most of the surviving points have
$t_{\beta}\lesssim 45$ in the CDF case, which was $t_{\beta}\lesssim 50$ in
the PDG case. The larger $t_{\beta}$ is, the more fermiphobic the new Higgs
bosons are in type-I because all the fermion Yukawa couplings to $H$, $A$, and
$H^{\pm}$ are inversely proportional to $t_{\beta}$. So, the reduced upper
bounds on $t_{\beta}$ increase the LHC discovery potential of new Higgs bosons
through fermionic production and decay channels. In type-X, the points with
$t_{\beta}\gtrsim 30$ are all excluded in the CDF case: in the PDG case,
however, $t_{\beta}\gtrsim 35$ is prohibited. We also observe that the heavy
charged Higgs boson with mass above $\sim 700{\;{\rm GeV}}$ demands the almost
exact Higgs alignment limit.

Figure 7:  Allowed parameter points of $(s_{\beta-\alpha},t_{\beta})$ in
type-I (left panel) and type-X (right panel) of the normal scenario that pass
the RGE (gray) and the Higgs precision data (yellow).

An intriguing difference between type-I and type-X appears in the positive and
negative $s_{\beta-\alpha}$ regions,333In the scheme of $s_{\beta-\alpha}>0$,
as usually adopted in many phenomenology studies, positive and negative
$s_{\beta-\alpha}$ correspond to the positive and negative $c_{\beta-\alpha}$,
respectively. when imposing the Higgs precision data. Figure 7 shows
$s_{\beta-\alpha}$ vs $t_{\beta}$ in type-I (left panel) and type-X (right
panel). We separately display the parameter points satisfying $\Lambda_{\rm
c}>1{\;{\rm TeV}}$ (gray) and the points consistent with the Higgs precision
data (yellow): yellow points are on top of gray points. At Step (iii), both
the positive and negative $s_{\beta-\alpha}$ have almost symmetric regions.
Type-I almost keeps the symmetric shape when imposing the Higgs precision
data, but type-X does not. The part with negative $s_{\beta-\alpha}$ is
excluded chiefly, except for the Higgs alignment limit.

Such a difference between type-I and type-X comes from the tau Yukawa coupling
to the Higgs boson $h$ in Table 1:

type-I: $\displaystyle\leavevmode\nobreak\
\xi^{h}_{\tau}=\frac{c_{\alpha}}{s_{\beta}}=s_{\beta-\alpha}+\frac{c_{\beta-\alpha}}{t_{\beta}},$
(15) type-X: $\displaystyle\leavevmode\nobreak\
\xi^{h}_{\tau}=-\frac{s_{\alpha}}{c_{\beta}}=s_{\beta-\alpha}-t_{\beta}c_{\beta-\alpha}.$
(16)

In type-I, the $c_{\beta-\alpha}$ term is suppressed by large $t_{\beta}$ so
the negative $s_{\beta-\alpha}$ does not change $|\xi^{h}_{\tau}|$ much. In
type-X, however, the $c_{\beta-\alpha}$ term is proportional to $t_{\beta}$
and thus non-negligible. Negative $s_{\beta-\alpha}$ excessively increases
$|\xi^{h}_{\tau}|$ so that the branching ratio of $h\to{\tau^{+}\tau^{-}}$
exceeds the experimental bound. Most of the negative $s_{\beta-\alpha}$ region
is excluded, except for almost the exact alignment limit. The appearance of
the “arm” region in the positive $s_{\beta-\alpha}$ case of type-X can also be
explained by Eq. (16). When $t_{\beta}\simeq 2/c_{\beta-\alpha}$,
$\xi^{h}_{\tau}(\simeq s_{\beta-\alpha}-2)$ approaches $-1$ when
$s_{\beta-\alpha}\simeq 1$. We have the wrong-sign limit for the tau lepton.
It is to be compared with the wrong-sign limit in type-II where both the
bottom quark and tau lepton have the Yukawa couplings with the wrong sign.

Figure 8:  In type-X of the normal scenario, the allowed parameter points of
$(M_{A},t_{\beta})$ at the final Step-(iv). The color code denotes
$M_{H^{\pm}}$.

Finally, we find that $M_{A}$ in type-X is strongly constrained by the direct
search bounds at the LHC. In Fig. 8, we present $M_{A}$ vs $t_{\beta}$, with
the color code indicating $M_{H^{\pm}}$. Particularly for $t_{\beta}\lesssim
10$, the intermediate $M_{A}$ around $200{\;{\rm GeV}}$ is excluded. A major
reason is that the masses of both $H$ and $A$ are similar to $M_{H^{\pm}}$:
see Fig. 3. As $H$ and $A$ are within the LHC reach, several direct search
processes exclude the $M_{A}\sim 200{\;{\rm GeV}}$ region. The first three
important processes, in order of importance, are $t\to H^{\pm}b\to\tau\nu b$
ATLAS:2018gfm , $pp\to H\to{\tau^{+}\tau^{-}}$ CMS:2019qcx ; ATLAS:2020pgp ,
and $gg\to A\to Zh\to\ell\ell{b\bar{b}}$ ATLAS:2020zms .

###  IV.2 type-II and type-Y

Figure 9:  In type-II and type-Y of the normal scenario, the allowed parameter
points of $(\Delta M_{H},\Delta M_{A})$ at the Step-(ii) in the left panel, at
the Step-(iii) in the middle panel, and at the Step-(iv) in the right panel,
where $\Delta m\equiv m-M_{H^{\pm}}$. The color code denotes $M_{H^{\pm}}$.

At every step, the viable parameter points in type-II are very similar (almost
indistinguishable from the eyes) to those in type-Y. So, the results in this
subsection apply to both type-II and type-Y.

In type-II and type-Y, the biggest impact comes from the condition of
$\Lambda_{\rm c}>1{\;{\rm TeV}}$. Figure 9 shows the allowed parameter points
of $(\Delta M_{H},\Delta M_{A})$ at the Step-(ii) in the left panel, at the
Step-(iii) in the middle panel and at the Step-(iv) in the right panel. The
EWPD data after the CDF $m_{W}$ measurement allow the hyperbola-shape with a
sufficiently large mass gaps of $\Delta M_{H}$ and $\Delta M_{A}$: see Fig.
1.444Negative $\Delta M_{H,A}$ region is different from Fig. 1. The difference
is attributed to the assumption of the Higgs alignment limit in Fig. 1. Our
main results do not depend on any prerequisites for the model parameters.
Imposing $\Lambda_{\rm c}>1{\;{\rm TeV}}$ drastically curtails the parameter
space of $(\Delta M_{H},\Delta M_{A})$, excluding the mass gaps above around
200 GeV. In addition, the positive $\Delta M_{H,A}$ case is more severely
restricted by $\Lambda_{\rm c}>1{\;{\rm TeV}}$, leaving a much smaller allowed
region. It is because the heavy Higgs bosons invoke fast running of the
quartic couplings, failing the unitarity and vacuum stability at the energy
scale below 1 TeV. The constraints from the collider data do not change
$\Delta M_{H}$ vs $\Delta M_{A}$ much. these.

Figure 10:  For type-II and type-Y in the normal scenario, the allowed
parameter points of $(\Delta M_{H},\Delta M_{A})$ in the left panel and
$(|s_{\beta-\alpha}|,t_{\beta})$ in the right panel at the final Step-(iv)
with $M_{H^{\pm}}>800{\;{\rm GeV}}$ from the $b\to s\gamma$ constraint. The
color code denotes $M_{H^{\pm}}$.

Finally, a brief discussion on the constraint from $b\to s\gamma$ is in order
here. We took a conservative bound on the charged Higgs boson mass as
$M_{H^{\pm}}\gtrsim 580{\;{\rm GeV}}$ for consistency with the FCNC
observables. But the bound considerably increases to about 800 GeV if we adopt
the calculation of the NNLO QCD corrections to $\mathcal{B}(\bar{B}\to
X_{s}\gamma)$ in the SM without the interpolation in the charm quark mass
Misiak:2020vlo . There still exist certain parameter points like $800{\;{\rm
GeV}}\lesssim M_{H^{\pm}}\lesssim 1100{\;{\rm GeV}}$. But the survival rate is
very small: only 34 parameter points out of $10^{6}$ points, which satisfy
Step-(i), pass the final Step-(iv) for $M_{H^{\pm}}>800{\;{\rm GeV}}$.
Obviously, the limited $M_{H^{\pm}}$ restricts other model parameters. Figure
10 shows the finally viable parameter points in $(M_{H},M_{A})$ (left panel)
and in $(|s_{\beta-\alpha}|,t_{\beta})$ (right panel), with color code of
$M_{H^{\pm}}$. To thoroughly investigate the status of type-II and type-Y with
$M_{H^{\pm}}>800{\;{\rm GeV}}$, we additionally generated parameter points
focusing on this particular region. Both $M_{H}$ and $M_{A}$ (left panel) are
accordingly constrained to be $[670,1000]{\;{\rm GeV}}$, slightly relaxed than
$M_{H^{\pm}}$. The restriction to $|s_{\beta-\alpha}|$ and $t_{\beta}$ (right
panel) are stronger. The Higgs alignment limit is almost exact, within about
0.2%, and $t_{\beta}$ is in $[1,5]$.

##  V Characteristic features of the inverted scenario

As shown in Table 2, type-II and type-Y in the IS are excluded by the
condition of $\Lambda_{\rm c}>1{\;{\rm TeV}}$. Therefore, we focus on type-I
and type-X in this section.

Figure 11:  Allowed parameter points of $(m_{h},M_{A})$ after imposing
$\Lambda_{\rm c}>1{\;{\rm TeV}}$ (left panels), the Higgs precision data
(middle panels), and the direct search bounds (right panels) for type-I (upper
panel) and type-X (lower panel) in the inverted scenario. The color code
denotes $M_{H^{\pm}}$.

The first remarkable feature of the IS is the considerably different survival
probabilities between type-I and type-X. In Fig. 11, we show the parameter
points of $(m_{h},M_{A})$ at the steps of $\Lambda_{\rm c}>1{\;{\rm TeV}}$
(left panels), the Higgs precision data (middle panels), and the direct search
bounds (right panels) for type-I (upper panels) and type-X (lower panels) in
the IS. At the RGE step, type-I and type-X yield almost the same results. A
significant difference arises when imposing the Higgs precision data: the
allowed points in type-X are more sparsely dense than those in type-I. The
direct search bounds further widen the difference. In type-X, only about 0.03%
of the parameter space survives, which is to be compared with about 0.77% in
type-I. The leptophilic nature of type-X, being more severe for large
$t_{\beta}$, plays an essential role in this considerable curtailment. A
significant result at the final step is that in type-I, there exist some
parameter points with $m_{h}$ below $m_{125}/2$, while in type-X, all the
parameter points with $m_{h}<62.5{\;{\rm GeV}}$ are excluded.

Figure 12:  Allowed parameter points of $(\Delta m_{h},\Delta M_{A})$ at the
final step for type-I in the inverted scenario, with color code of
$M_{H^{\pm}}$.

The second noteworthy feature is the strong correlation of $M_{A}$ and $m_{h}$
to $M_{H^{\pm}}$. Figure 12 shows the allowed parameter points of $(\Delta
m_{h},\Delta M_{A})$ at the final step for type-I in the inverted scenario:
the results in type-X are very similar, with sparse density. First of all,
only the negative $\Delta m_{h}$ and negative $\Delta M_{A}$ are allowed. As
we have a relatively light mass spectrum for all the new Higgs bosons, the LHC
is expected to have discovery potential. Another unanticipated aspect is the
approximate mass degeneracy of $M_{H^{\pm}}\simeq M_{A}$ for heavy
$M_{H^{\pm}}$. For $M_{H^{\pm}}\gtrsim 350{\;{\rm GeV}}$, $|\Delta
M_{A}|<40{\;{\rm GeV}}$. We expect that this region is quite elusive at the
LHC, especially for large $t_{\beta}$: the production rates of $A$ and
$H^{\pm}$ through gluon fusion are suppressed by large $t_{\beta}$; the decay
of $H^{\pm}\to AW^{\pm*}\to Af\bar{f}^{\prime}$ accompanying soft jets or
leptons is also challenging to probe.

Figure 13:  Allowed parameter points of $(|s_{\beta-\alpha}|,t_{\beta})$ at
the final step for type-I (left panel) and type-X (right panel) in the
inverted scenario, with color code of $M_{H^{\pm}}$.

Finally, we present $|s_{\beta-\alpha}|$ vs $t_{\beta}$ in Fig. 13 for type-I
(left panel) and type-X (right panel) of the IS. The difference between type-I
and type-X is considerable. In type-I, the deviation from the Higgs alignment
limit is sizable. The value of $|s_{\beta-\alpha}|$ can be as large as about
0.48. In addition, large $t_{\beta}$ around 50 is also permitted. In type-X,
however, both $|s_{\beta-\alpha}|$ and $t_{\beta}$ are more severely limited,
like $|s_{\beta-\alpha}|\lesssim 0.2$ and $t_{\beta}\lesssim 30$. Only a small
deviation from the Higgs alignment limit is permitted. Moreover, the very
large $t_{\beta}$ region in this leptophilic model is strongly constrained by
the direct searches at the LHC, as in the NS.

##  VI Conclusions

The recent $m_{W}^{\rm CDF}$ measurement by the CDF Collaboration has a
significant impact on searching for physics beyond the standard model. In
terms of the Peskin-Takeuchi parameters of $S$ and $T$, the central values
shift to larger values: $S=0.15\pm 0.08$, $T=0.27\pm 0.06$ with $U=0$, which
has profound consequences in searching for physics beyond the SM. We have
illustrated in the framework of two-Higgs-doublet Models, including type-I,
type-II, type-X, and type-Y, the effects on the still-valid parameter space
due to the new $m_{W}^{\rm CDF}$ together with other constraints, which
include theoretical requirements (potential bounded from below, unitarity,
perturbativity, vacuum stability), flavor-changing neutral currents in $B$
physics, maintaining the perturbativity of the model until the cutoff scale
$\Lambda_{c}=1$ TeV, Higgs precision data, and direct collider search limits
from the LEP, Tevatron, and LHC.

The most unprecedented consequence is the emergence of upper bounds on the
masses of the heavy Higgs boson $M_{H,A,H^{\pm}}\lesssim 1000-1100$ GeV in the
normal scenario and $M_{H^{\pm},A}\lesssim 410-440$ GeV in the inverted
scenario. Such interesting findings imply that the upcoming LHC run can
readily close out the still-available parameter space.

Before closing, a few more findings from our study are offered as follows:

  1. 1.

The updated fit on the oblique parameters of $S$ and $T$ indicates that the
new $m_{W}$ measurement requires larger mass splittings among the isospin
components in multiplet models.

  2. 2.

In subsequent steps of imposing constraints on the parameters, we found that
the survival probabilities for the case of using the new $m_{W}^{\rm CDF}$ are
much smaller than the case of using the PDG value, thus implying more
restriction on physics beyond the SM (not only for 2HDM). This behavior has
been demonstrated in all four types and two scenarios of 2HDM.

  3. 3.

The $\tan\beta$ is bounded from above more severely when the new $m_{W}^{\rm
CDF}$ value is used in the normal scenario. Roughly, $\tan\beta\lesssim
45,\,6,\,30,\,10$ for type-I, II, X, and Y, respectively. On the other hand,
$\tan\beta$ is not bounded in type I and $\tan\beta\lesssim 30$ for type X in
the inverted scenario. There is no parameter space surviving all the
requirements in type-II and type-Y of the inverted scenario.

  4. 4.

If we raise the cutoff scale $\Lambda_{c}$ beyond 1 TeV, the restriction on
the parameter space would become more severe.

###### Acknowledgements.

We would like to thank Dr. Jing-Hwan Cho of National Institute for
Mathematical Sciences in Korea for helping the numerical computation. K.C. was
supported by MoST with grant numbers MOST-110-2112-M-017-MY3. The work of JK,
SL, and JS is supported by the National Research Foundation of Korea, Grant
No. NRF-2022R1A2C1007583.

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# Resource-Efficient HAPS-RIS Enabled Beyond-Cell Communications

Safwan Alfattani, Animesh Yadav, Halim Yanikomeroglu, Abbas Yongaçoglu This
research has been supported in part by King AbdulAziz University, Saudi
Arabia, and in part by Huawei Canada Company Ltd. S. Alfattani is with King
AbdulAziz University, Saudi Arabia (e-mail: smalfattani@kau.edu.sa). A. Yadav
is with the Department of ECET, Minnesota State University, Mankato, MN,
56001, USA (e-mail: animesh.yadav@mnsu.edu). H. Yanikomeroglu is with the
Department of Systems and Computer Engineering, Carleton University, Ottawa,
ON, K1S 5B6, Canada (e-mail: halim@sce.carleton.ca). A. Yongacoglu is with the
University of Ottawa, ON, Canada (e-mail: yongac@uottawa.ca).

###### Abstract

In the future, urban regions will encounter a massive number of capacity-
hungry devices. Relying solely on terrestrial networks for serving all UEs
will be a cost-ineffective approach. Consequently, with the anticipated supply
and demand mismatch, several UEs will be unsupported. To offer service to the
left-out UEs, we employ an energy-efficient and cost-effective _beyond-cell
communications_ approach, which uses reconfigurable intelligent surfaces (RIS)
on a high-altitude platform station (HAPS). Particularly, unsupported UEs will
be connected to a dedicated control station (CS) through RIS-mounted HAPS. A
novel resource-efficient optimization problem is formulated that maximizes the
number of connected UEs, while minimizing the total power consumed by the CS
and RIS. Since the resulting problem is a mixed-integer nonlinear program
(MINLP), a low-complexity two-stage algorithm is developed. Numerical results
demonstrate that the proposed algorithm outperforms the benchmark approach in
terms of the percentage of connected UEs and the resource-efficiency (RE).
Also, the results show that the number of connected UEs is more sensitive to
transmit power at the CS than the HAPS size.

###### Index Terms:

high-altitude platform station (HAPS), reconfigurable intelligent surfaces
(RIS), non-terrestrial network (NTN), quality-of-service (QoS)

##  I Introduction

Future wireless networks are expected to face unprecedented demands for
continuous and ubiquitous connectivity due to the increasing number of mobile
users, the evolving deployment of Internet-of-Things (IoT) networks, and the
growing number of novel use cases. Increasing base station (BS) density and
using relays may be a straightforward solution to cope with this unforeseen
situation, but it comes at the cost of high capital and operational
expenditures. Alternatively, a green and an energy-efficient solution that
utilizes reconfigurable intelligent surfaces (RIS) in terrestrial networks
(TNs) has been introduced [1]. RIS constitutes a large number of low-cost and
nearly passive elements, which can be configured to reflect the incident radio
frequency (RF) signal to a desired direction [2, 1].

However, the deployment of RIS in terrestrial environments involves several
challenges, such as placement inflexibility and channel impairments including
excessive path loss and shadowing effects. Instead, in [3], we proposed the
integration of RIS on aerial or stratospheric platforms and discussed its
prospects for wireless systems and services. The main motivations behind this
proposition include better channel links, wider coverage and dynamic
placement. Moreover, since energy consumption is a critical issue in aerial
platforms, equipping them with a massive number of active antennas would
exaggerate the issue. Alternatively, mounting aerial platforms with RIS can
provide an energy-efficient solution [4, 5, 6, 7, 8]. In addition, due to the
favorable wireless channel conditions in non-terrestrial networks (NTN), RIS
can support panoramic full-angle reflection serving wider areas with strong
line-of-sight (LoS) links [9, 4]. Further, in [10], we showed that RIS-mounted
HAPS has the potential to outperform RIS-aided terrestrial networks and other
RIS-mounted aerial platforms (e.g., UAV). Indeed, although the path-loss from
a terrestrial node to a UAV is significantly lower than that to a HAPS, the
reflection gain offered by HAPS-RIS is much higher, due to the typical large
size of HAPS.

To reap the benefits of RIS-mounted HAPS (HAPS-RIS), in [11], we proposed a
novel beyond-cell communications approach. This approach offers service to
stranded terrestrial user equipment devices, whose either channel conditions
are below the required quality-of-service (QoS), or are located in a cell with
fully loaded BS. In particular,  stranded UEs get service from a dedicated
ground control station (CS) through the HAPS-RIS. We showed that HAPS-RIS can
work in tandem with legacy TNs to support unserved UEs. We also discussed the
optimal power and RIS units allocation design schemes to maximize the system
throughput and the worst UE rate.

Previous works on RIS-assisted communications consider serving all UEs by
optimizing only the RIS phase shifts and the transmit power [1, 11]. However,
due to practical limitations on system resources, including the transmit power
of the CS and HAPS size (equivalently, the number of RIS units), it might be
infeasible to serve all unsupported UEs by CS through HAPS-RIS, especially,
for a large set of UEs. Accordingly, we formulated a novel optimization
problem to maximize the resource-efficiency (RE) of the system.

  * •

A novel resource-efficient optimization problem that maximizes the percentage
of connected UEs while minimizing the usage of RIS units and transmit power is
formulated.

  * •

Since the resulting problem is a mixed-integer nonlinear program (MINLP) and
hard to solve, a low complexity two-stage algorithm is proposed to solve it.

  * •

Through numerical results, we study the impact of HAPS size and QoS
requirements of UEs on the percentage of connected UEs and demonstrate
significant improvements in RE of the system.

The rest of the letter is organized as follows. In Section II, the system
model is described. Section III presents the problem formulation. The proposed
solution and the algorithm for solving the optimization problem are discussed
in Section IV. Numerical results and discussion are presented in Section V.
Finally, Section VI concludes the letter.

Figure 1: System model for HAPS-RIS assisting beyond cell communications.

##  II System Model

We consider a typical urban region consists of $K$ UEs, $L$ terrestrial
BSs111BSs have down-tilted antennas dedicated to terrestrial UEs, and can’t
generally communicate to HAPS., a single HAPS-RIS, and one CS222If the number
of unsupported users increases, possible solutions may include increasing the
CS transmit power or antenna gain or the number of CS in the HAPS coverage
area., as depicted in Fig. 1. The UEs are assumed to suffer from severe
shadowing and blockages, and NLoS paths, which are typical characteristics of
propagation media in urban regions. Based on channel conditions between the
BSs and the UEs, and the maximum serving capacity of the BSs, a set of
$\mathcal{K}_{1}=\\{1,\ldots,K_{1}\\}$ UEs will be supported by direct links
from the BSs (referred to as within-cell communications [11]). The set of
remaining UEs $\mathcal{K}_{2}=\\{1,\ldots,K_{2}\\}$, which cannot form direct
connection with the terrestrial BS will be served by the CS via HAPS-RIS
(referred to as beyond-cell communications [11]). The CS is located somewhere
in the HAPS coverage area333According to 3GPP standards [12], a CS (gateway)
should be deployed to ensure minimum elevation angle of 5∘, and the maximum
Euclidean distance of 229 km between HAPS and CS . Note that
$\mathcal{K}=\mathcal{K}_{1}\cup\mathcal{K}_{2}$. We assume that the CS serves
the stranded UEs in set $\mathcal{K}_{2}$ using orthogonal subcarriers, and
hence, there will be no inter-UE interference. Further, both within-cell and
beyond-cell communications occur in two orthogonal frequency bands, while
keeping the subcarrier bandwidth $B_{\rm UE}$ same for both types of
communications. As a result, the signals from within-cell UEs will not
interfere with the signals from beyond-cell UEs and vice versa.

We assume that within-cell UEs are connected optimally with terrestrial BSs,
and hence, our focus in this work is on beyond-cell communications.
Accordingly, the received signal at UE $k\in\mathcal{K}_{2}$ on a given
subcarrier can be expressed as

$y_{k}=\sqrt{P^{\rm CS}_{k}}h_{k}\mathrm{\Phi}_{k}\;x_{{k}}+w_{k},$ (1)

where $x_{{k}}$ and $P^{\rm CS}_{k}$ denote the transmitted signal and power
of UE $k$, respectively. $w_{k}$ denotes the additive white Gaussian noise
(AWGN) $\sim\mathcal{CN}(0,N_{0}B_{\rm UE})$, where $N_{0}$ is the noise power
spectral density. $h_{k}$ denotes the effective channel gain from the CS to
the HAPS-RIS and from the HAPS-RIS to UE $k$, and is given by

$h_{k}=\sqrt{G^{\rm CS}G_{r}^{k}(\textsf{PL}^{\rm{CS-
HAPS}-\textit{k}})^{-1}},$ (2)

where $G^{\rm CS}$ denotes the antenna gain of the control station, and
$G_{r}^{k}$ is the receiver antenna gain of UE $k$. $\textsf{PL}^{\text{CS-
HAPS}-k}=\textsf{PL}^{\rm{CS}-\rm{HAPS}}\textsf{PL}^{\text{HAPS}-k}$ denotes
the effective path loss between the CS and UE $k$ via HAPS-RIS. Since the
amplitude and phase responses of RIS reflecting units are frequency-dependent
[13] and each UE uses different subcarrier, each UE is assigned with a
distinct and dedicated set of RIS units tuned to its respective subcarrier.
Accordingly,  $\mathrm{\Phi}_{k}$ represents the reflection gain of the RIS
corresponding to UE $k$, and is expressed as

$\mathrm{\Phi}_{k}=\sum_{i=1}^{N_{k}}\rho_{i}e^{-j\left(\phi_{i,k}-\theta_{i}-\theta_{i,k}\right)},$
(3)

where $\rho_{i}$ denotes the reflection loss corresponding to RIS unit $i$.
$\theta_{i}$ and $\theta_{i,k}$ represent the corresponding phases between RIS
unit $i$ and both the control station and UE $k$, respectively. $\phi_{i,k}$
represents the adjusted phase shift of RIS unit $i$ corresponding to user $k$,
and $N_{k}$ represents the total number of RIS units allocated to UE $k$.

Using (1), the signal-to-noise ratio (SNR) at UE $k$ is written as

$\gamma_{k}=\frac{P_{k}^{\rm
CS}\left|h_{k}\mathrm{\Phi}_{k}\right|^{2}}{N_{0}B_{\rm UE}},$ (4)

and the corresponding achievable rate can be expressed as

$R_{k}=B_{\rm UE}\log_{2}(1+\gamma_{k}).$ (5)

##  III Problem Formulation

Since the required power for the flight and control system of a HAPS is
dependent on its type and size, for generalization, we focus on the power
consumption due to communication payload. Using power efficiently at the HAPS-
RIS for activating the RIS units, and at the CS for transmitting signals offer
sustainable network and longer refueling intervals for HAPS. Also, in case of
a large UE density or strict QoS requirement, it might be infeasible to serve
all unserved UEs by the CS via HAPS-RIS. Therefore, we need to select as many
UEs that can be supported by the CS, while using as minimum system resources
(transmit power and RIS units) as possible. Accordingly, we define a novel
performance metric known as resource efficiency as follows:

###### Definition 1

Resource efficiency (RE) of the beyond-cell communication system, $\eta$, is
defined as the ratio between the percentage of served UEs and the average
power consumption in dBm by each supported UE, which includes the consumption
towards signal transmission and RIS units configuration:

$\eta=\dfrac{\frac{1}{K}\Big{(}K_{1}+\sum_{k=1}^{K_{2}}u_{k}\Big{)}}{\frac{1}{|\mathcal{U}|}\sum_{k=1}^{K_{2}}\Big{(}P_{k}^{\rm
CS}u_{k}+P_{\rm RIS}N_{k}u_{k}\Big{)}},$ (6)

where $u_{k}$ is the indicator variable, if $u_{k}=1$, user $k$ will get
service by HAPS-RIS, otherwise not if $u_{k}=0$. In the denominator, the terms
$\sum_{k=1}^{K_{2}}P_{k}^{\rm CS}u_{k}$ and $\sum_{k=1}^{K_{2}}P_{\rm
RIS}N_{k}u_{k}$ represent the total transmit power consumption at the CS and
the total power consumed by the RIS units for all supported UEs, respectively.
$P_{\rm RIS}$ denotes the consumed power by each RIS unit for phase shifting,
which is dependent on the RIS configuration technology and its resolution.
Finally, $|\mathcal{U}|$ denotes the cardinality of the set, which constitutes
the UEs supported by the CS via HAPS-RIS.

In the following, we formulate the resource-efficient UEs maximization
problem. It can be expressed as

$\displaystyle\max_{u_{k},\mathrm{\Phi}_{k},N_{k},P_{k}^{\rm CS}}\eta$ (7a)
$\displaystyle\text{ s.t. }L\leq L_{\rm max},$ (7b) $\displaystyle\quad
R_{k}\geq u_{k}R_{\rm th},\enspace\forall k=1,2,\ldots,K_{2},$ (7c)
$\displaystyle\quad\sum_{k=1}^{K_{2}}u_{k}N_{k}\leq N_{\rm max},$ (7d)
$\displaystyle\quad{\color[rgb]{0,0,0}\phi_{i,k}\in\\{0,\Delta\phi,2\Delta\phi,\ldots,(2^{b}-1)\Delta\phi\\},\enspace\forall
i=1,2,\ldots,N_{k},}$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad\qquad{\color[rgb]{0,0,0}\forall
k=1,2,\ldots,K_{2},}$ (7e)
$\displaystyle\quad\sum_{k=1}^{K_{2}}u_{k}P_{k}^{\rm CS}\leq P_{\rm max}^{\rm
CS},$ (7f)
$\displaystyle\quad{\color[rgb]{0,0,0}N_{k}\in\\{0,1,\ldots,N_{k,\rm
max}\\},\text{ \enspace$\forall k=1,2,\ldots,K_{2},$}}$ (7g)
$\displaystyle\quad 0\leq P_{k}^{\rm CS}\leq u_{k}P_{k,\rm max}^{\rm
CS},\enspace\forall k=1,2,\ldots,K_{2},$ (7h) $\displaystyle\quad
u_{k}\in\\{0,1\\},$ (7i)

where $P^{\rm CS}_{\rm max}$ denotes the maximum available transmit power at
the CS. $N_{k,\rm max}$ and $P_{k,\rm max}^{\rm CS}$ denote the maximum number
of RIS units and the amount of power allocated to UE $k$, respectively. (7b)
limits the number of BSs in the area. Constraint (7c) guarantees that each
selected UE satisfies the minimum rate requirement of $R_{\rm th}$. Constraint
(7d) ensures the total number of RIS units allocated to the supported UEs does
not exceed the maximum number of available RIS units, $N_{\rm max}$, which is
limited by the HAPS size. Constraint (7) determines the discrete range of the
adjustable phase shifts values for each RIS unit, where
$\Delta\phi_{i,k}=2\pi/2^{b}$ with $b$ as the number of bits used to uniformly
quantize the phase shifts of each RIS element. Constraint (7f) guarantees that
the total allocated power by the CS to the selected UEs is less than the
maximum available power. Finally, constraints (7g) and (7h) ensure fair
allocation of both RIS units and CS power, respectively, to each UE.

##  IV Proposed Solution

Since problem (7) is an MINLP, it is hard to solve jointly with the involved
variables in an optimal manner with lower complexity. Therefore, we decouple
the problem into two subproblems and develop a low-complexity algorithm to
solve it suboptimally. The main step of the proposed algorithm involves
solving (7) in two stages.

In the first subproblem, we maximize the numerator of (7a) by maximizing the
number of UEs, which can establish direct connection with TNs (i.e., by
setting $L=L_{\rm max}$444This value is dependent on the communication
frequency, and the statistics of the users density and their rate demands. It
is also determined by operator’s expenditure analysis.), and then finding the
set of maximum feasible UEs $\in\mathcal{K}_{2}$, which can be supported by
the CS via HAPS-RIS. To this end, we first sort the channel gains of all UEs.
Secondly, under the assumptions of equal power allocation to each UE and
perfect reflection of each RIS unit, we allocate the minimum required number
of RIS units to each UE as follows:

$N_{k}=\Biggl{\lceil}\sqrt{\frac{N_{0}B_{\rm UE}(2^{\frac{R_{\rm min}}{B_{\rm
UE}}}-1)}{P_{k}^{\rm CS}\left|h_{k}\right|^{2}}}\Biggr{\rceil}.$ (8)

The initial RIS units allocation starts with UEs with the best channel
conditions until all RIS units are utilized. As a result, a set $\mathcal{U}$
with the largest number of feasible UEs is determined.

In the second subproblem, the denominator of (7a) is minimized by optimally
allocating the CS power and RIS units to each UE belongs to set $\mathcal{U}$.
This is accomplished by solving the following optimization problem:

$\displaystyle\min_{\mathrm{\Phi}_{k},N_{k},P_{k}^{\rm
CS}}\sum_{k=1}^{|\mathcal{U}|}P_{k}^{\rm CS}+P_{\rm RIS}N_{k}$ (9a)
$\displaystyle\text{ s.t. }{\color[rgb]{0,0,0}R_{k}\geq R_{\rm
th}},\enspace\forall k=1,2,\ldots,|\mathcal{U}|,$ (9b)
$\displaystyle\quad\sum_{k=1}^{K_{2}}N_{k}\leq N_{\rm max},$ (9c)
$\displaystyle\quad{\color[rgb]{0,0,0}\phi_{i,k}\in\\{0,\Delta\phi,2\Delta\phi,\ldots,(2^{b}-1)\Delta\phi\\},\enspace\forall
i=1,2,\ldots,N_{k},}$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad\qquad{\color[rgb]{0,0,0}\forall
k=1,2,\ldots,K_{2},}$ (9d) $\displaystyle\quad\sum_{k=1}^{K_{2}}P_{k}^{\rm
CS}\leq P_{\rm max}^{\rm CS},$ (9e)
$\displaystyle\quad{\color[rgb]{0,0,0}N_{k}\in\\{0,1,\ldots,N_{k,\rm
max}\\},\enspace\forall k=1,2,\ldots,|\mathcal{U}|,}$ (9f) $\displaystyle\quad
0\leq P_{k}^{\rm CS}\leq P_{k,\rm max}^{\rm CS},\enspace\forall
k=1,2,\ldots,|\mathcal{U}|.$ (9g)

Without loss of generality, problem (9) can be re-written as

$\displaystyle\min_{\mathrm{\Phi}_{k},N_{k},P_{k}^{\rm
CS}}\sum_{k=1}^{|\mathcal{U}|}P_{k}^{\rm CS}+P_{\rm RIS}N_{k}$ (10a)
$\displaystyle\text{ s.t. }\frac{1}{\gamma_{k}}\leq\frac{1}{\gamma_{\rm
min}},\enspace\forall k=1,2,\ldots,|\mathcal{U}|,$ (10b)
$\displaystyle\quad\quad\eqref{p1_throu:c3}-\eqref{p1_throu:c7}.$ (10c)

Due to the involvement of integer and discrete variables $N_{k}$ and
$\phi_{i,k}$, respectively, problem (10) is a challenging problem and is
difficult to solve optimally in polynomial time. To solve it efficiently, we
relax $N_{k}$ and $\phi_{i,k}$ to be continuous variables. After this
relaxation, the objective and the constraints of (10) become posynomials555A
posynomial function is a special function refers to a sum of positive
monomials[14]., which can be solved optimally using geometric programming (GP)
technique [14]. At the solution of the relaxed problem, the final solution to
(10) is obtained by approximation as $N_{k}\approx\lceil N^{*}_{k}\rceil$. The
pseudo-code of the proposed two-stage algorithm is described in Algorithm 1.
Note that in practice $\phi_{i,k}$ should be selected from a set of discrete
phase shifts. The cardinality of the set is based on the resolution of the
phase shift. The selection optimization of the discrete phase shifts is still
an open problem. We investigated this problem in [15] and showed that with our
proposed algorithm, the performance gap between continuous phase shifts and a
set of two discrete phase shifts is about 30%. The impact of relaxing
$\phi_{i,k}$ and the performance gap can be significantly reduced with a set
of large number of phase shifts.

#### Complexity Analysis

The iteration-complexity of Algorithm 1 is the same as of the standard
barrier-based interior-point method with accuracy error $\epsilon$, and is in
the order of
$\mathcal{O}(2.5\sqrt{|\mathcal{U}|+1}\ln\frac{6(|\mathcal{U}|+1)\Delta}{\epsilon})$,
where the term $\Delta$ is associated to a perturbation in the feasible set
[16]. However, the complexity of solving the original formulated problem
optimally without relaxations and without decoupling them into two subproblems
is computationally prohibited as it requires searching all possible
combinations of RIS unit allocation and phase shifts configurations, and can
be given as $\mathcal{O}\big{(}(2^{b})^{N_{\rm
max}}\sum_{k=1}^{K_{2}}\prod_{i=1}^{N_{k,\rm max}}\binom{N_{\rm
max}}{i}\big{)}$.

Algorithm 1 Efficient maximization of connected UEs

1:Set $L=L_{\rm max}$ and obtained set $\mathcal{K}_{2}$.

2:Input: $h_{k},k\in\\{1,2,\ldots,K_{2}\\}$.

3:Sort $K_{2}$ UEs in descending order based on their channel gains
$K_{S2}\leftarrow K_{2}$

4:for $k=1\dots K_{S2}$ do

5: while $\sum_{k=1}^{K_{S2}}N_{k}\leq N_{\rm max}$ do

6: $P_{k}^{\rm CS}=P_{\rm max}^{\rm CS}/{\color[rgb]{0,0,0}K_{S2}}$

7: Obtain initial $N_{k}$ from (8).

8: end while

9:end for

10:Stage-1 Output: Selected UEs $\mathcal{U}$.

11:Solve optimally (10) for the set $\mathcal{U}$.

12:Stage-2 Output: $P_{k}^{\text{CS}*}$ and $N^{*}_{k}$ ($\forall
k=1,\ldots,|\mathcal{U}|$).

##  V Numerical Results and Discussion

In this section, we present and discuss the performance of the proposed
algorithm. For the purpose of comparison, we employ a benchmark approach,
which first allocates equal power to all UEs and then selects the UEs with the
highest channel gains to be served first with the minimum number of RIS units
until the QoS requirement is satisfied.

In the simulation setup, we consider an urban square area of dimension 10 km
by 10 km, with $L_{\rm max}=4$ terrestrial BSs serving $K=100$ uniformly
randomly distributed UEs with a minimum separation distance of 100 m among
them. We assume that the terrestrial BSs are optimally placed in the
considered region. The channel gains between all the UEs and the terrestrial
BSs are obtained by adopting the 3GPP standards [17]. The carrier frequency is
set to $f_{c}=2$ GHz with shadowing standard deviation $\sigma=8$ dB. Unless
stated otherwise, the minimum rate for a direct connection between a UE and a
BS is $R_{\rm th}=2$ Mb/s. The maximum available bandwidth to each BS is
$B_{\rm BS}=50$ MHz, and subcarrier bandwidth for each UE is set to $B_{\rm
UE}=2$ MHz. Accordingly, a UE will be connected to a terrestrial BS that
provides the highest data rate, and the collection of such UEs forms the set
$\mathcal{K}_{1}$. Consequently, the collection of stranded UEs forms the
other set $\mathcal{K}_{2}$.

On the other hand, the effective channel gains from the CS to all the UEs in
set $\mathcal{K}_{2}$ through HAPS-RIS are obtained using the standardized
3GPP channel model between a HAPS and terrestrial nodes in urban environments
[12, 10]. This model considers dry air atmospheric attenuation, and
corresponding atmosphere parameters are selected based on the mean annual
global reference atmosphere [18]. Further, we assume each UE has 0 dB antenna
gain, and noise power spectral density $N_{0}=-174$ dBm/Hz. We further set
$P_{\rm max}^{\rm CS}=$ 33 dBm, and $G^{\rm CS}=$ 43.2 dBm [12] in all of the
simulations, unless stated otherwise. The values of other parameters are as
follow: $P_{\rm RIS}=7.8$ mW[1], $P_{k,\rm max}^{\rm CS}=30$ dBm, and
$N_{k,\rm max}=50,000$ units.

###  V-A Resource-efficiency maximization

Fig. 2 plots the normalized RE obtained using Algorithm 1 (on the left-hand
side y-axis) and the percentage of connected UEs ( on the right-hand side
y-axis) versus different values of minimum rate requirement $R_{\rm min}$. It
also compares the performance of Algorithm 1 with the benchmark approach. The
maximum number of RIS units mounted on HAPS is set to $N_{\rm max}=220,000$
units. It can be observed that as the QoS (represented by $R_{\rm min}$)
increases, the percentage of connected UEs and the RE drops. However, this
performance degradation is more significant in terms of the percentage of
connected UEs than the RE. Furthermore, we observed that the RE obtained using
Algorithm 1 significantly outperforms the one obtained using the benchmark
approach. This is due to the fact that Algorithm 1 optimizes allocation of
both power and RIS units to UEs.

Figure 2: Resource-efficiency performance of connected UEs for different
$R_{\rm min}$.

###  V-B Percentage of connected UEs

Figure 3: Maximizing connected UEs performance for different $N_{\rm max}$.
Figure 4: Maximizing connected UEs performance for different $P_{\rm max}^{\rm
CS}$.

Figs. 3 and 4 plot and compare the percentage of connected UEs obtained
through Algorithm 1, the benchmark approach, and within-cell communication
approach for different values of the number of RIS units $N_{\rm max}$
available at the HAPS, and maximum power $P_{\rm max}^{\rm CS}$ available at
the CS, respectively.

In Fig. 3, to study the impact of $N_{\rm max}$, we consider only the second
term (i.e., $P_{\rm RIS}N_{k}$) of the objective function in (10a). The
selected range of $N_{\rm max}$ is set between 10,000 and 220,000 units. This
range corresponds to a total RIS area between 9 $\rm m^{2}$ and 198 $\rm
m^{2}$ at carrier frequency of 2 GHz 666This represents a limited area on a
typical HAPS surface, as the length of an airship is between 100-200 m,
whereas aerodynamic HAPS have wingspans between 35 m and 80 m. The size of
each RIS unit is $(0.2\lambda)^{2}$ [19].. In Fig. 4, only the first term of
(10a) (i.e., $P_{k}^{\rm CS}$) is considered to study the effect of $P_{\rm
max}^{\rm CS}$ on the percentage of connected UEs. The maximum transmit power
of CS is set to vary between 30 dBm and 35 dBm, and $N_{\rm max}$ is set to
150,000 units. It can be observed from the figures that the percentage of
connected UEs increases with the maximum power of the CS $P_{\rm max}^{\rm
CS}$, and the maximum number of RIS units $N_{\rm max}$ (or the size of HAPS).
These behaviours are intuitive as making  more system resources ($P_{\rm
max}^{\rm CS}$ and $N_{\rm max}$) available allows more number of stranded
users to be served by the CS via HAPS-RIS.

Figs. 3 and 4 also show that the performance of the proposed approach is 1-3 %
higher than the benchmark approach. Moreover, $P_{\rm max}^{\rm CS}$ has more
significant impact than $N_{\rm max}$ on the percentage of connected UEs. By
increasing $P_{\rm max}^{\rm CS}$ by 2 dB, the percentage of connected UEs
increases about 4%, whereas doubling of the RIS size is needed to achieve the
same increase in the percentage of connected UEs.

Furthermore, it can be observed from the figures that 76% UEs are served
through within-cell communication approach, and the CS supports the remaining
UEs via HAPS-RIS. Hence, beyond-cell communications via HAPS-RIS is able to
complement TNs.

##  VI Conclusion

In this letter, we investigated the resources optimization of beyond-cell
communications that use HAPS-RIS technology to complement TNs by supporting
unserved UEs. In particular, given the limitations of the CS power and HAPS-
RIS size, it might not be feasible to support all unserved UEs. Therefore, we
formulated a novel resource-efficient optimization problem that simultaneously
maximizes the percentage of connected UEs using minimal CS power and RIS
units. The results show the capability of the beyond-cell communications
approach to support a larger number of UEs. Further, the results show the
superiority of the proposed solutions over the benchmark approach and
demonstrate the impact of the HAPS size and the QoS requirement on the
percentage of connected UEs and the efficiency of the system.

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  *[HAPS]: high-altitude platform station
  *[RIS]: reconfigurable intelligent surfaces
  *[NTN]: non-terrestrial network
  *[QoS]: quality-of-service
  *[IoT]: Internet-of-Things
  *[BS]: base station
  *[RF]: radio frequency
  *[CS]: control station
  *[RE]: resource-efficiency
  *[UEs]: user equipment device
  *[MINLP]: mixed-integer nonlinear program
  *[BSs]: base station
  *[SNR]: signal-to-noise ratio