Higgs photon associated production in a Two Higgs Doublet Type-II Seesaw Model
at future electron-positron colliders.

B. Ait<EMAIL_ADDRESS>M.
Chabab1†††mchabab@uca.ac.ma(corresponding author) K.
<EMAIL_ADDRESS>
1 LPHEA, Faculty of Science Semlalia, Cadi Ayyad University, P.O.B. 2390
Marrakech, Morocco.

###### Abstract

We studied the one-loop prediction for the single production of a Higgs-like
boson in association with a photon in electron-positron collisions in the
context of two Higgs doublet type-II seesaw model (2HDMcT). We explored to
what extent the new scalars in the (2HDMcT) spectrum affect its production
cross-section, the ratio $R_{\gamma h_{1}}$ as well as the signal strengths
$R_{\gamma Z}$ and $R_{\gamma\gamma}$ when $h_{1}$ is identified with the
observed $SM$ Higgs boson within the $2HDMcT$ delimited parameter space. More
specifically, we focused on $e^{+}e^{-}\to h_{1}\gamma$ process at one-loop,
and analyzed how it evolves under a full set of theoretical constraints and
the available experimental data, including $B\to X_{s}\gamma$ constraint at
95$\%$ C.L. Our analysis showed that these observables are particularly
sensitive to the parameters $\alpha_{1}$, $\lambda_{7}$, $\lambda_{9}$, the
trilinear Higgs couplings and to the charged Higgs masses. We found that
$\sigma(e^{+}e^{-}\to h_{1}\gamma)$ can significantly be enhanced up to
$8.1\,\,10^{-2}$ fb, so exceeding the Standard Model prediction. Additionally,
as a byproduct, we also found that $R_{\gamma h_{1}}$ is entirely correlated
with both the $h_{1}\to\gamma\gamma$ and $h_{1}\to\gamma Z$ signal strengths.

##  1 Introduction

####

The discovery of the Higgs boson, the last missing piece in the completion of
the Standard Model ($SM$), by the ATLAS [1] and CMS [2] collaborations
provided an experimental evidence for the Brout-Englert-Higgs mechanism. Since
then, major ongoing studies have focused on exploring in detail the properties
of the Higgs boson with the aim to enhance our understanding of the
fundamental laws of nature. Although most of the $SM$’s predictions have been
tested successfully to a high level of accuracy[3, 4, 5], it still suffers
from drawbacks since it failed to explain several established physical
phenomena. As examples, the problem of dark matter[6, 7], the hierarchy
problem [8], and the neutrino mass generation [9, 10] do not fit in the SM.

####

To address these issues, a plethora of new physics models have been proposed
in literature. Among these beyond the Standard Model ($BSM$), scenarios
relying on an extended Higgs sectors with new scalar fields, as the popular
two Higgs Doublet Model (2HDM) [11, 12, 13, 14, 15, 16], the Higgs Triplet
Models (HTM) [16, 14] and recently 2HDM augmented by a complex triplet scalar,
dubbed the two Higgs Doublet Type-II Seesaw model (2HDMcT) [17, 18]. Since the
spectra of these models generally predict additional Higgs bosons with novel
features, searching for these new scalars has been actively conducted as a key
focus and one of the major motivations for the current and future experiments,
especially at LHC [19, 20, 21, 22, 23, 24, 25]. Besides, as no direct evidence
for new physics has been seen yet, precision measurements of the Higgs boson
properties and couplings [26, 27, 28, 29, 30] to other new scalars can offer a
promising opportunity for a potential discovery of new physics. Indeed, more
accurate understanding of the Higgs boson and the measurements of its
couplings with high levels of precision is the main goal of future
$e^{+}e^{-}$ colliders [31] such as the International Linear Collider (ILC)
[32, 33], Compact Linear Collider (CLIC) [34, 31], Circular Electron-Positron
Collider (CEPC) [35, 36, 37], and Future Circular Collider (FCC) [38, 39].
Compared to hadron colliders, these colliders, featuring a cleaner
$e^{+}e^{-}$ background, can yield substantial improvements over LHC
measurements [40].

####

The associated production of Standard Model (SM) Higgs boson with a photon,
$e^{+}e^{-}\to h_{1}\gamma$, is well suited to study the Higgs-gauge bosons
couplings such as the $h_{1}\gamma\gamma$ and $h_{1}\gamma Z$ couplings. Since
the production rate can be sizably amplified in the presence of new physics
contributions as compared to the Standard Model, this process may serve as a
potential discovery channel. This process was investigated in the (SM) [41,
42, 43] and recently in some $BSM$ scenarios as the inert Higgs doublet model
[44], (HTM) [45], the minimal supersymmetric standard model (MSSM) [46, 47]
and the effective field theory [48].

####

In this work, we investigate the single production of the neutral Higgs boson
in association with a photon in the electron positron collisions within the
Two Higgs Doublet Type II seesaw Model 2HDMcT. To do that, we implement a full
set of theoretical constraints originated from perturbative unitarity,
electroweak vacuum stability, as well as the experimental Higgs exclusion
limits from LEP, LHC and Tevatron. Since the mass generation from seesaw
mechanism is similar to the Brout-Englert-Higgs mechanism, 2HDMcT model is
appealing, displaying many phenomenological features especially different from
those emerging in 2HDM scalar sector. Apart its broader spectrum than 2HDM’s
one, the doubly charged Higgs $H^{++}$, as a smoking gun of 2HDMcT, $H^{++}$
is currently intensively searched for at ATLAS and CMS, by means of promising
decay channels, such as $H^{++}H^{--}$ and $H_{i}^{+}H_{i}^{-}$ $(i=1,2)$,
decaying to the same sign di-lepton [17]. Additionally, beyond Higgs
phenomenology, it has been demonstrated that interactions between doublet and
triplet fields may induce a strong first order electroweak phase transition,
which provide conditions for the generation of the baryon asymmetry through
electroweak baryongenesis [49].

####

This paper is organized as follows: In Sect.2, we briefly review 2HDMcT model
and mention some theoretical and experimental constraints imposed on the
model. In Sect.3 we study the two processes $e^{+}e^{-}\to h_{1}\gamma$ and
$e^{-}\gamma\to h_{1}e^{-}$ in 2HDMcT and discuss the correlation of the
signal strengths for the one loop induced processes $h_{1}\to\gamma\gamma$ and
$h_{1}\to Z\gamma$ in 2HDMcT. Sect.4 is devoted to our conclusion.

##  2 Two Higgs doublet type-II seesaw model: Brief overview

The two Higgs doublet type II seesaw model has recently gained interest as a
well-motivated extension of the 2HDM. Besides the two Higgs doublets with
hypercharge $Y=+1$, 2HDM is augmented with one colorless triplet field
$\Delta$ transforming under the $SU(2)_{L}$ gauge group as a complex scalar
with $Y_{\Delta}=2$.

$\Phi_{1}=\left(\begin{array}[]{c}\phi_{1}^{+}\\\ \phi_{1}^{0}\\\
\end{array}\right)\quad{\rm,}\quad\Phi_{2}=\left(\begin{array}[]{c}\phi_{2}^{+}\\\
\phi_{2}^{0}\\\ \end{array}\right)\quad{\rm
and}\quad\begin{matrix}\Delta&=&\left(\begin{array}[]{cc}\delta^{+}/\sqrt{2}&\delta^{++}\\\
(v_{t}+\delta^{0}+i\eta_{0})/\sqrt{2}&-\delta^{+}/\sqrt{2}\\\
\end{array}\right)\end{matrix}$ (1)

with $\phi_{1}^{0}=(v_{1}+\psi_{1}+i\eta_{1})/\sqrt{2}$,
$\phi_{2}^{0}=(v_{2}+\psi_{2}+i\eta_{2})/\sqrt{2}$. $v_{1}$, $v_{2}$ and
$v_{t}$ denote the vacuum expectation values of the Higgs doublets and triplet
fields respectively, acquired when the electroweak symmetry is spontaneously
broken, with $\sqrt{v_{1}^{2}+v_{2}^{2}+2v_{t}^{2}}=246$ GeV. Consequently,
eleven physical Higgs states occur in the model spectrum: three CP-even
neutral Higgs bosons $(h_{1},h_{2},h_{3})$, four simply charged Higgs bosons
$(H_{1}^{\pm},H_{2}^{\pm})$, two CP odd Higgs $(A_{1},A_{2})$, and finally two
doubly charged Higgs bosons $H^{\pm\pm}$. For details see Ref.[17].

The most general $SU(2)_{L}\times U(1)_{Y}$ invariant scalar potential in this
model reads : [13, 17]:

$\displaystyle V(\Phi_{1},\Phi_{2},\Delta)$ $\displaystyle=$ $\displaystyle
m_{11}^{2}\Phi_{1}^{\dagger}\Phi_{1}+m_{22}^{2}\Phi_{2}^{\dagger}\Phi_{2}-[m_{12}^{2}\Phi_{1}^{\dagger}\Phi_{2}+{\rm
h.c.}]+\frac{\lambda_{1}}{2}(\Phi_{1}^{\dagger}\Phi_{1})^{2}+\frac{\lambda_{2}}{2}(\Phi_{2}^{\dagger}\Phi_{2})^{2}$
(2) $\displaystyle+$
$\displaystyle\lambda_{4}(\Phi_{1}^{\dagger}\Phi_{2})(\Phi_{2}^{\dagger}\Phi_{1})+\left\\{\frac{\lambda_{5}}{2}(\Phi_{1}^{\dagger}\Phi_{2})^{2}+\big{[}\beta_{1}(\Phi_{1}^{\dagger}\Phi_{1})+\beta_{2}(\Phi_{2}^{\dagger}\Phi_{2})\big{]}\Phi_{1}^{\dagger}\Phi_{2}+{\rm
h.c.}\right\\}$ $\displaystyle+$
$\displaystyle\lambda_{3}(\Phi_{1}^{\dagger}\Phi_{1})(\Phi_{2}^{\dagger}\Phi_{2})+\lambda_{6}\,\Phi_{1}^{\dagger}\Phi_{1}Tr\Delta^{\dagger}{\Delta}+\lambda_{7}\,\Phi_{2}^{\dagger}\Phi_{2}Tr\Delta^{\dagger}{\Delta}$
$\displaystyle+$
$\displaystyle\left\\{\mu_{1}\Phi_{1}^{T}{i}\sigma^{2}\Delta^{\dagger}\Phi_{1}+\mu_{2}\Phi_{2}^{T}{i}\sigma^{2}\Delta^{\dagger}\Phi_{2}+\mu_{3}\Phi_{1}^{T}{i}\sigma^{2}\Delta^{\dagger}\Phi_{2}+{\rm
h.c.}\right\\}+\lambda_{8}\,\Phi_{1}^{\dagger}{\Delta}\Delta^{\dagger}\Phi_{1}$
$\displaystyle+$
$\displaystyle\lambda_{9}\,\Phi_{2}^{\dagger}{\Delta}\Delta^{\dagger}\Phi_{2}+m^{2}_{\Delta}\,Tr(\Delta^{\dagger}{\Delta})+\bar{\lambda}_{8}(Tr\Delta^{\dagger}{\Delta})^{2}+\bar{\lambda}_{9}Tr(\Delta^{\dagger}{\Delta})^{2}$

In this work we assume that $m_{11}^{2}$, $m_{22}^{2}$, $m_{\Delta}^{2}$,
$m_{12}^{2}$, $\lambda_{1,2,3,4,5,6,7,8,9}$, $\bar{\lambda}_{8,9}$,
$\mu_{1,2,3}$, $\beta_{1,2}$ are real parameters. To avoid Higgs mediated
$FCNC_{s}$ at tree level, we consider $Z_{2}$ symmetry where
$\beta_{1}=\beta_{2}=0$. Also the $Z_{2}$ symmetry is softly broken by the bi-
linear terms proportional to $m_{12}^{2}$, $\mu_{1}$, $\mu_{2}$ and $\mu_{3}$
parameters. Thanks to the combination
$v^{2}=v_{1}^{2}+v_{2}^{2}+2v_{t}^{2}=(2\sqrt{2}G_{F})^{-1}$ and the three
minimization conditions, the scalar potential Eq. (2) has seventeen
independent parameters, one choice is :

$\alpha_{1}$, $\alpha_{2}$, $\alpha_{3}$, $m_{h_{1}}$, $m_{h_{2}}$,
$m_{h_{3}}$, $m_{H^{\pm\pm}}$, $\lambda_{1}$, $\lambda_{3}$, $\lambda_{4}$,
$\lambda_{6}$, $\lambda_{8}$, $\bar{\lambda}_{8}$, $\bar{\lambda}_{9}$,
$\mu_{1}$, $v_{t}$ and $\tan\beta$

where $\alpha_{i=1,2,3}$ are the CP-even mixing angles and
$\tan\beta=v_{2}/v_{1}$.

####

In our subsequent analysis $h_{1}$ is identified to the SM-like Higgs boson
with $m_{h}=125$ GeV [1, 2], so the scalar potential is described by sixteen
free parameters. $\mathcal{L}_{\rm Yukawa}$ contains all the Yukawa sector of
the Two Higgs Doublet Model plus one extra Yukawa term emerging from the
triplet field and generating, after spontaneous symmetry breaking, a small
Majorana mass terms for the neutrinos.

$-\mathcal{L}_{\rm Yukawa}\supset-Y_{\nu}L^{T}C\otimes i\sigma^{2}\Delta
L+{\rm h.c.}$ (3)

For the type II (2HDMcT-II), up quarks interact with $\Phi_{2}$, while leptons
and down quarks with $\Phi_{1}$ as :

$\displaystyle-{\cal L}^{II}_{Yukawa}$ $\displaystyle=$ $\displaystyle-
y_{u}\bar{Q}_{L}\tilde{\Phi}_{2}u_{R}-y_{d}\bar{Q}_{L}\Phi_{1}d_{R}-y_{\ell}\bar{L}_{L}\Phi_{1}\ell_{R}+h.c\,,$
(4)

Here $Q_{L}$ and $L_{L}$ represent the left-handed quark and lepton doublets,
$u_{R}$, $d_{R}$, and $\ell_{R}$ are the right-handed up-type quark, down-type
quark, and lepton singlets. $\Phi_{1}$ and $\Phi_{2}$ are the two Higgs
doublets , with $\tilde{\Phi}_{2}=i\sigma_{2}\Phi_{2}^{*}$. $y_{u}$, $y_{d}$,
and $y_{\ell}$ are the Yukawa couplings for the up, down-type quarks and
leptons, respectively.

$\phi$ | $\xi^{u}_{\phi}$ | $\xi^{d}_{\phi}$ | $\xi^{\ell}_{\phi}$
---|---|---|---
$h_{1}$ |  $\frac{\mathcal{E}_{12}}{s_{\beta}}$ |  $-\frac{\mathcal{E}_{11}}{c_{\beta}}$ |  $-\frac{\mathcal{E}_{11}}{c_{\beta}}$
$h_{2}$ |  $\frac{\mathcal{E}_{22}}{s_{\beta}}$ |  $-\frac{\mathcal{E}_{21}}{c_{\beta}}$ |  $-\frac{\mathcal{E}_{21}}{c_{\beta}}$
$h_{3}$ |  $\frac{\mathcal{E}_{32}}{s_{\beta}}$ |  $-\frac{\mathcal{E}_{31}}{c_{\beta}}$ |  $-\frac{\mathcal{E}_{31}}{c_{\beta}}$
Table 1: Yukawa couplings of the $h_{1}$, $h_{2}$, and $h_{3}$ bosons to the
quarks and leptons in 2HDMcT-II

###  2.1 Theoretical and Experimental Constraints

####

The phenomenological analysis in 2HDMcT is performed via implementation of a
full set of theoretical constraints [17, 18] as well as the Higgs exclusion
limits from various experimental measurements at colliders, namely :

  * •

Unitarity : The scattering processes must obey the perturbative unitarity.

  * •

Perturbativity: The quartic couplings of the scalar potential are constrained
by the following conditions:$|\lambda_{i}|<8\pi$ for each $i=1,..,5$.

  * •

Vacuum stability : Boundedness from below $BFB$ arising from the positivity in
any direction of the fields $\Phi_{i}$, $\Delta$.

  * •

Electroweak precision observables: The oblique parameters $S,T$ and $U$ [50,
51] have been calculated in $2HDMcT$ [18]. The analysis of the precision
electroweak data in light of the new PDG mass of the $W$ boson yields [52]:

$\displaystyle S=0.05\pm 0.08,\ \ T=0.09\pm 0.07,\ \ \rho_{ST}=0.92,$

  * •

To further delimit the allowed parameter space, the HiggsTools package [53] is
employed. This ensures that the allowed parameter regions align with the
observed properties of the $125$ GeV Higgs boson ( HiggsSignals [54, 55, 56,
53]) and with the limits from searches for additional Higgs bosons at the LHC
and at LEP ( HiggsBounds [57, 58, 59, 60, 53]).

  * •

Flavour constraints: Flavour constraints are also implemented in our analysis.
We used $B$-physics derived results derived in [18] as well as the
experimental data at 2$\sigma$ [61] displayed in Table 2.

Observable | Experimental result | 95% C.L.
---|---|---
BR($\bar{B}\to X_{s}\gamma$)[18] |  $(3.49\pm 0.19)\times 10^{-4}$[61] | $[3.11\times 10^{-4},3.87\times 10^{-4}]$
Table 2: Experimental result of flavor observable: $\bar{B}\to X_{s}\gamma$ at
95$\%$ C.L.

##  3 $e^{+}e^{-}\to\gamma h_{1}$ and $e^{-}\gamma\to e^{-}h_{1}$ in 2HDMcT

###  3.1 Process topology

The $e^{+}e^{-}\to\gamma h_{1}$ process has been studied in many of beyond the
Standard Model frameworks. In the 2HDMcT, at tree level, the associated
production processes $e^{+}e^{-}\to\gamma h_{1}$ and $e^{-}\gamma\to
e^{-}h_{1}$ are intermediated by the t-channel and s-channel electron exchange
diagrams respectively. However, the former diagrams are suppressed by the
electron mass. At one-loop these diagrams are mediated by the self-energy,
box, and triangle diagrams and hence they are sensitive to all virtual
particles inside the loop including the charged Higgs states, $H^{\pm}_{1}$,
$H^{\pm}_{2}$ and $H^{\pm\pm}$, predicted by the model.

Figure 1: Feynman diagrams involving the charged contributions to
$e^{-}\,e^{+}\,\,\to\,\,\gamma\,h_{1}$ process in 2HDMcT. In $d_{1}$ and
$d_{4}$ diagrams, the loops receive contributions from all SM particles as
well as the charged Higgs bosons $H^{\pm}_{1}$, $H^{\pm}_{2}$ and
$H^{\pm\pm}$. Generic diagrams $d_{2},d_{3}$, depict $h_{1}\gamma V$,
$V=\gamma,\;Z$. Figure 2: Feynman diagrams involving the charged contributions
to $e^{-}\gamma\to e^{-}h_{1}$ process in the 2HDMcT. In diagrams $v_{1}$, we
depict all possible charged particles. Diagrams $s_{1}$ receive contributions
from all SM particles and from $H^{\pm}_{1}$, $H^{\pm}_{2}$ and $H^{\pm\pm}$
bosons.

Figs. 1, 2 and 3 illustrate the generic Feynman diagrams that effectively
contribute to the $e^{+}e^{-}\to\gamma h_{1}$, $e^{-}\gamma\to e^{-}h_{1}$ and
$h_{1}\to\gamma V$ processes, with $V=\gamma,\;Z$. Here $F$ stands for any
fermionic particle, while $S$ denotes the charged states $H^{\pm}_{1}$,
$H^{\pm}_{2}$ and $H^{\pm\pm}$.

Figure 3:  Feynman diagrams for the decay process $h_{1}\to\gamma V$
($V=\gamma,\;Z$) in the 2HDMcT. $a_{4},a_{5}$ illustrate the contributions of
$H^{\pm}_{1}$, $H^{\pm}_{2}$ and $H^{\pm\pm}$.

Our calculation is performed using dimensional regularization along with
FeynArts and FormCalc packages [62, 63]. while the numerical evaluation of the
scalar integrals is done with LoopTools [64, 65]. Note that the gauge
invariance of our final results is assured once contributions from all box and
triangle diagrams are summed. In the subsequent analysis, we introduce the
ratio,

$\displaystyle R_{\gamma h_{1}}\equiv\frac{\sigma(e^{+}e^{-}\to\gamma
h_{1})}{\sigma_{\rm SM}(e^{+}e^{-}\to\gamma H)}$ (5)

which is the total cross-sections in the $2HDMcT$ normalized to the SM one.
Also we define the diphoton and $\gamma Z$ gauge bosons signal strengths,
$R_{\gamma\gamma}$ and $R_{Z\gamma}$,

$R_{\gamma\gamma(Z\gamma)}\equiv\frac{\sigma(gg\to h_{1})\times
Br(h_{1}\to\gamma\gamma(Z\gamma))}{\sigma(gg\to H^{\rm SM})\times Br(H^{\rm
SM}\to\gamma\gamma(Z\gamma))}$ (6)

Compared with the Standard Model, the amplitudes for the loop processes
$e^{-}\gamma\to e^{-}h_{1}$, $e^{+}e^{-}\to\gamma h_{1}$ and $h_{1}\to\gamma
V$ ($V=\gamma,\;Z$) receive additional contributions from the $H^{\pm}_{1}$,
$H^{\pm}_{2}$ and $H^{\pm\pm}$, the charged Higgs bosons predicted by the
model spectrum. These amplitudes are essentially proportionals to the
couplings with $h_{1}$ Higgs boson,

$\displaystyle\bar{\lambda}_{h_{1}H^{\pm}_{1}H^{\pm}_{1}}$ $\displaystyle=$
$\displaystyle\frac{s_{w}}{2em_{w}}\left(2\mathcal{C}_{21}^{2}\left(\lambda_{6}\mathcal{E}_{13}v_{\Delta}+\lambda_{1}v_{1}\mathcal{E}_{11}+\lambda_{3}v_{2}\mathcal{E}_{12}\right)\right.$
(7)
$\displaystyle+2\mathcal{C}_{22}^{2}\left(\lambda_{7}\mathcal{E}_{13}v_{\Delta}+\lambda_{2}v_{2}\mathcal{E}_{12}+\lambda_{3}v_{1}\mathcal{E}_{11}\right)$
$\displaystyle+\mathcal{C}_{23}^{2}\left(2\mathcal{E}_{13}(2\overline{\lambda_{8}}+\overline{\lambda_{9}})v_{\Delta}+\left(2\lambda_{6}+\lambda_{8}\right)v_{1}\mathcal{E}_{11}+\left(2\lambda_{7}+\lambda_{9}\right)v_{2}\mathcal{E}_{12}\right)$
$\displaystyle+\mathcal{C}_{22}\mathcal{C}_{23}\left(\sqrt{2}\lambda_{9}\mathcal{E}_{12}v_{\Delta}+\sqrt{2}\lambda_{9}v_{2}\mathcal{E}_{13}-4\mu_{2}\mathcal{E}_{12}-2\mu_{3}\mathcal{E}_{11}\right)$
$\displaystyle+\mathcal{C}_{21}\left(\right.\mathcal{C}_{23}\left(\sqrt{2}\lambda_{8}\mathcal{E}_{11}v_{\Delta}+\sqrt{2}\lambda_{8}v_{1}\mathcal{E}_{13}-4\mu_{1}\mathcal{E}_{11}-2\mu_{3}\mathcal{E}_{12}\right)$
$\displaystyle\left.+2\mathcal{C}_{22}\left(\lambda_{4}+\lambda_{5}\right)\left(v_{2}\mathcal{E}_{11}+v_{1}\mathcal{E}_{12}\right)\left.\right)\right)$
$\displaystyle\bar{\lambda}_{h_{1}H^{\pm}_{2}H^{\pm}_{2}}$ $\displaystyle=$
$\displaystyle\frac{s_{w}}{2em_{w}}\left(2\mathcal{C}_{31}^{2}\left(\lambda_{6}\mathcal{E}_{13}v_{\Delta}+\lambda_{1}v_{1}\mathcal{E}_{11}+\lambda_{3}v_{2}\mathcal{E}_{12}\right)\right.$
(8)
$\displaystyle+2\mathcal{C}_{32}^{2}\left(\lambda_{7}\mathcal{E}_{13}v_{\Delta}+\lambda_{2}v_{2}\mathcal{E}_{12}+\lambda_{3}v_{1}\mathcal{E}_{11}\right)$
$\displaystyle+\mathcal{C}_{33}^{2}\left(2\mathcal{E}_{13}(2\overline{\lambda_{8}}+\overline{\lambda_{9}})v_{\Delta}+\left(2\lambda_{6}+\lambda_{8}\right)v_{1}\mathcal{E}_{11}+\left(2\lambda_{7}+\lambda_{9}\right)v_{2}\mathcal{E}_{12}\right)$
$\displaystyle+\mathcal{C}_{32}\mathcal{C}_{33}\left(\sqrt{2}\lambda_{9}\mathcal{E}_{12}v_{\Delta}+\sqrt{2}\lambda_{9}v_{2}\mathcal{E}_{13}-4\mu_{2}\mathcal{E}_{12}-2\mu_{3}\mathcal{E}_{11}\right)$
$\displaystyle+\mathcal{C}_{31}\left(\right.\mathcal{C}_{33}\left(\sqrt{2}\lambda_{8}\mathcal{E}_{11}v_{\Delta}+\sqrt{2}\lambda_{8}v_{1}\mathcal{E}_{13}-4\mu_{1}\mathcal{E}_{11}-2\mu_{3}\mathcal{E}_{12}\right)$
$\displaystyle\left.+2\mathcal{C}_{32}\left(\lambda_{4}+\lambda_{5}\right)\left(v_{2}\mathcal{E}_{11}+v_{1}\mathcal{E}_{12}\right)\left.\right)\right)$
$\displaystyle\bar{\lambda}_{h_{1}H^{\pm\pm}H^{\pm\pm}}$ $\displaystyle=$
$\displaystyle\frac{s_{w}}{em_{w}}\left(2\overline{\lambda_{8}}\mathcal{E}_{13}v_{\Delta}+\lambda_{6}v_{1}\mathcal{E}_{11}+\lambda_{7}v_{2}\mathcal{E}_{12}\right)$
(9)

At this stage it is worth noticing that:

  * •

These couplings are mostly fixed by $\alpha_{1}$, $\lambda_{3}$,
$\lambda_{4}$, $\lambda_{7}$ and $\lambda_{9}$ parameters when $\tan\beta$ get
relatively large values.

  * •

Depending on their signs the charged Higgs contributions can either enhance or
suppress the $e^{+}e^{-}\to\gamma h_{1}$ and $h_{1}\to\gamma V$
($V=\gamma,\;Z$) rates.

###  3.2 Results and analysis

####

Given that $h_{1}$ is identified as the Higgs-like boson, observed at the LHC
with $m_{h_{1}}=125$ GeV, we perform scans within the allowed parameter space
by implementation of the theoretical and experimental constraints mentioned
above.
The following set of input parameters is used in the subsequent numerical
analysis :

$\displaystyle\mathcal{P}_{I}=\left\\{\alpha_{1},\alpha_{2},\alpha_{3},m_{h_{1}},m_{h_{2}},\lambda_{1},\lambda_{3},\lambda_{4},\lambda_{6},\lambda_{7},\lambda_{8},\lambda_{9},\bar{\lambda}_{8},\bar{\lambda}_{9},\mu_{1},v_{t},\tan\beta\right\\}$
(10)
$\displaystyle\begin{matrix}m_{h_{1}}=125\,\,\text{GeV},\,\,\,m_{h_{1}}\leq
m_{h_{2}}\leq m_{h_{3}}\leq 1\,\text{TeV},\,\,\,\par 80\,\,\text{GeV}\leq
m_{H^{\pm}_{1}},m_{H^{\pm}_{2}},m_{H^{\pm\pm}}\leq 1\,\text{TeV},\,\,\\\
-2\pi\leq\lambda_{1}\leq 2\pi,\,\,-2\pi\leq\lambda_{3,4}\leq
2\pi,\,\,\,\,\,-2\pi\leq\lambda_{6,7}\leq
2\pi,\,\,\,-2\pi\leq\lambda_{8,9}\leq
2\pi,\,\,-2\pi\leq\bar{\lambda}_{8,9}\leq 2\pi\\\
-\pi/2\leq\alpha_{1}\leq\pi/2,\,\,\alpha_{2}\approx 0,\,\,\,\alpha_{3}\approx
0,\,\,\mu_{1}=v_{t}=1\,\,\text{GeV},\,\,\tan\beta=8\end{matrix}$ (11)

From Eqs. (7), (8), and (9) one can readily see that the effect of
$H_{1}^{\pm}$, $H_{2}^{\pm}$ virtual states on $h_{1}\gamma\gamma$ and
$h_{1}\gamma Z$ couplings introduces a strong sensitivity to $\alpha_{1}$,
$\lambda_{3}$, $\lambda_{4}$, $\lambda_{7}$ and $\lambda_{9}$ , while
$H^{\pm\pm}$ leads to a noticeable dependence on $\alpha_{1}$, $\lambda_{7}$.
Comparatively, dependence on the other model parameters is either mild or
negligible.

####

In Fig. 4, we show the allowed ranges for $(\lambda_{7},\,\,\lambda_{9})$ as
well as with the size of the trilinear couplings
$\bar{\lambda}_{h_{1}H_{1}^{\pm}H_{1}^{\pm}}$,
$\bar{\lambda}_{h_{1}H_{2}^{\pm}H_{2}^{\pm}}$ and
$\bar{\lambda}_{h_{1}H^{\pm\pm}H^{\pm\pm}}$. All generated points passed the
full set of constraints at $2\sigma$. The upper panel displays a significant
dependence between the $(\lambda_{7},\,\,\lambda_{9})$ parameters and the
charged Higgs boson masses; $m_{H^{\pm}_{1}}$ (left), $m_{H^{\pm}_{2}}$
(middle) and $m_{H^{\pm\pm}}$ (right). We can clearly see that the input
ranges for $-2\pi<\lambda_{7}<2\pi$ and $-2\pi<\lambda_{9}<2\pi$ are strongly
reduced to $0<\lambda_{7}<2\pi$ and $-5.25<\lambda_{9}<1.01$ essentially due
to the combination the BFB conditions with the Higgs boson exclusion limits
implemented in HiggsTools. Also, one can easily notice that the parameters
$\lambda_{7}$ and $\lambda_{9}$ have opposite effects on the trilinear
couplings, as illustrated by the middle and lower panels in Fig. 4:
$\lambda_{7}$ increases
$\bar{\lambda}_{h_{1}H^{\pm\pm}H^{\mp\mp}}$,$\bar{\lambda}_{h_{1}H_{1}^{\pm}H_{1}^{\pm}}$,$\bar{\lambda}_{h_{1}H_{2}^{\pm}H_{2}^{\pm}}$,
while $\lambda_{9}$ weaken these trilinear couplings. Similarly compatibility
with all the constraints oblige the parameters $\mu_{2,3}$ to undergo drastic
reductions being confined within the mitigated intervals $0<\mu_{2}<8$ and
$70<\mu_{3}<108$ respectively.

Figure 4: Upper panels: The charged Higgs masses as a function of
$\lambda_{7}$ and $\lambda_{9}$ with $m_{H^{\pm}_{1}}$ (left),
$m_{H^{\pm}_{2}}$ (middle) and $m_{H^{\pm\pm}}$ (right). Middle panels:
correlation between $\bar{\lambda}_{h_{1}H^{\pm}_{1}H^{\mp}_{1}}$ and
$\bar{\lambda}_{h_{1}H^{\pm}_{2}H^{\mp}_{2}}$ (left),
$\bar{\lambda}_{h_{1}H^{\pm}_{1}H^{\mp}_{1}}$ and
$\bar{\lambda}_{h_{1}H^{\pm\pm}H^{\mp\mp}}$ (middle),
$\bar{\lambda}_{h_{1}H^{\pm}_{2}H^{\mp}_{2}}$ and
$\bar{\lambda}_{h_{1}H^{\pm\pm}H^{\mp\mp}}$ (right) as a function of
$\lambda_{7}$. Lower panels: correlation between
$\bar{\lambda}_{h_{1}H^{\pm}_{1}H^{\mp}_{1}}$ and
$\bar{\lambda}_{h_{1}H^{\pm}_{2}H^{\mp}_{2}}$ (left),
$\bar{\lambda}_{h_{1}H^{\pm}_{1}H^{\mp}_{1}}$ and
$\bar{\lambda}_{h_{1}H^{\pm\pm}H^{\mp\mp}}$ (middle),
$\bar{\lambda}_{h_{1}H^{\pm}_{2}H^{\mp}_{2}}$ and
$\bar{\lambda}_{h_{1}H^{\pm\pm}H^{\mp\mp}}$ (right) as a function of
$\lambda_{9}$. Here we assume that $\alpha_{1}=1.42$ and $m_{h_{2}}=673.30$
GeV. The other inputs are the same as in (11). The analysis is performed with
$\chi^{2}_{ST}$ test where only points that are within $95\%$ confidence level
are considered.

Figure 5: Total unpolarized cross-section in fb for $e^{+}e^{-}\to\gamma
h_{1}$ (left) and $e^{-}\gamma\to e^{-}h_{1}$ (right) as a function of center-
of-mass energy, in two benchmark scenarios. Here we assume that
$\alpha_{1}=1.42$ and $m_{h_{2}}=673.30$ GeV, for the other inputs (see Table
3).

####

In Fig. 5, we display the unpolarized cross sections of $e^{-}e^{+}\to
h_{1}\gamma$ and $e^{-}\gamma\to e^{-}h_{1}$, with respect to the Standard
Model one, as a function of $\sqrt{s}$ in the range $200$ to $1000$ GeV. These
production cross sections at the linear collider are analyzed for two
benchmark scenarios, as illustrated in Table 3. All generated points passed
upon the full set of constraints including $B\to X_{s}\gamma$ constraint at
95$\%$ C.L. The corresponding cross sections in the Standard Model are also
plotted (dots in black). Our analysis shows that the cross sections of these
two processes are very sensitive to the parameters of the model, especially
$\alpha_{1}$, $\lambda_{3}$, $\lambda_{7}$ and $\lambda_{9}$, so as expected
they are enhanced in the vicinity of $250$ GeV where they get their largest
values. Besides we also see that they begin to decline when the detrimental
effects of the destructive interference between the Standard Model and 2HDMcT
contributions become more pronounced. At this point, we must stress that the
constructive or destructive interference with the Standard Model contribution
is highly sensitive to the values and signs of the various couplings,
$\bar{\lambda}_{h_{1}H^{\pm\pm}H^{\mp\mp}}$,$\bar{\lambda}_{h_{1}H_{1}^{\pm}H_{1}^{\pm}}$,$\bar{\lambda}_{h_{1}H_{2}^{\pm}H_{2}^{\pm}}$
and on $\alpha_{1}$,$\lambda_{3}$,$\lambda_{7}$ and $\lambda_{9}$ parameters.
Any change in these couplings (values or signs) could significantly affects
the resulting cross sections.

Table 3: Benchmark scenarios used in Fig. 5 where the generated points pass upon the full set of constraints. Bench. | $\lambda_{1}$ | $\lambda_{3}$ | $\lambda_{4}$ | $\lambda_{6}$ | $\lambda_{7}$ | $\lambda_{8}$ | $\lambda_{9}$ | $\bar{\lambda}_{8}$ | $\bar{\lambda}_{9}$ |
---|---|---|---|---|---|---|---|---|---|---
BP1 | $1.31$ | $-0.12$ | $1.70$ | $0.64$ | $0.01$ | $0.28$ | $0.38$ | $2.33$ | $0.54$ |
BP2 | $1.31$ | $0.80$ | $3.06$ | $6.04$ | $2.15$ | $-2.72$ | $0.47$ | $-1.10$ | $3.13$ |

Figure 6: The cross-section $\sigma(e^{+}e^{-}\to\gamma h_{1})$ is depicted as
a function of the parameter $\lambda_{7}$ for two values of the parameter
$\alpha_{1}$: $\alpha_{1}=1.42$ (upper panels), and $\alpha_{1}=1.44$ (lower
panels). The plotted points pass upon all constraints. We used :
$\lambda_{1}=1.31$, $\lambda_{3}=6.15$, $\lambda_{4}=-3.85$ and
$m_{h_{2}}=673.30$ GeV. The other inputs are the same as in (11). The color
coding illustrate the variation of the trilinear couplings
$\bar{\lambda}_{h_{1}H^{\pm}_{1}H^{\mp}_{1}}$(left),
$\bar{\lambda}_{h_{1}H^{\pm}_{2}H^{\mp}_{2}}$(middle) and
$\bar{\lambda}_{h_{1}H^{\pm\pm}H^{\mp\mp}}$ (right panel)

.

Next, we first aim to see whether the diphoton signal strength
$R_{\gamma\gamma}(h_{1})$ agrees with the experimental data within the model’s
parameter space, knowing that, unlike $h_{1}\to\gamma Z$ process, the decay
channel $h_{1}\to\gamma\gamma$, has been measured with high precision [66, 67,
68, 69, 70, 4, 3]. Also, we want to analyze to what extent the cross-section
$\sigma(e^{+}e^{-}\to\gamma h_{1})$ behavior is affected either by the
$2HDMcT$ charged Higgs bosons or by the potential parameters. Fig. 6 and Fig.7
illustrate the variation of these observables as a function of parameter
$\lambda_{7}$, and the trilinear couplings
$\bar{\lambda}_{h_{1}H^{+}_{1}H^{-}_{1}}$,
$\bar{\lambda}_{h_{1}H^{+}_{2}H^{-}_{2}}$ and
$\bar{\lambda}_{h_{1}H^{++}H^{--}}$. It is generally observed that an increase
in $\lambda_{7}$ results in detrimental effect on both $R_{\gamma\gamma}$ and
$\sigma(e^{+}e^{-}\to\gamma h_{1})$. Similar trend is seen if these
observables are plotted versus $\lambda_{3}$.

In Fig.6 we plot the cross-section of $e^{+}e^{-}\to\gamma h_{1}$ process as a
function of the parameter $\lambda_{7}$, and the trilinear Higgs couplings for
two nearly degenerate values of the parameter $\alpha_{1}$. From the upper
panels, with $\alpha_{1}=1.42$, we clearly see that
$\sigma(e^{+}e^{-}\to\gamma h_{1})$ undergoes a notable increase and even
exceeds the Standard Model prediction if $\lambda_{7}<0$, a region excluded by
constraints originating from $BFB$. In this scenario, because of $BFB$
conditions, the contributions of the parameters
$\lambda_{6,7,8,9,\bar{8},\bar{9}}$ are rather predominantly detrimental,
which induce a decrease in $\sigma(e^{+}e^{-}\to\gamma h_{1})$. More
specifically this depreciation stems particularly from the relations,
$\lambda_{7}>0$ and $\lambda_{7}+\lambda_{9}>0$, with $\lambda_{7}$ and
$\lambda_{9}$ exhibiting contrasting effects. However, if we consider a
slightly larger value of the parameter $\alpha_{1}=1.44$, as illustrated by
the lower panels, the cross sections experience a significant enhancement and
can well surpass that of the Standard Model up to $8.06\,\,10^{-2}$ fb, while
still satisfying BFB conditions. On top of that, our results also show that
the charged Higgs masses $m_{H^{\pm}_{1}}$, $m_{H^{\pm}_{2}}$ and
$m_{H^{\pm\pm}}$ are compelled to lie within the very stringent intervals
$[785,790]$ GeV, $[807,990]$ GeV and $[810,995]$ GeV respectively, conforming
the predictions reported in [18].
Next we analyze the signal strengths $R_{\gamma\gamma}$, $R_{\gamma Z}$ and
the ratio $R_{\gamma h_{1}}$ within the model parameter space. First, we plot
in Fig. 7 the signal strength of the Higgs to diphoton decay as a function of
$\lambda_{7}$ parameter, so we observe that $R_{\gamma\gamma}$ is clearly
consistent with the measured signal strength[66, 67, 68, 69, 70, 4, 3] at
$1\sigma$.

Figure 7: The signal strength $R_{\gamma\gamma}(h_{1})$ as a function of
$\lambda_{7}$. The plotted points passed upon all constraints. The color
coding shows the variation of the trilinear coupling
$\bar{\lambda}_{h_{1}H^{\pm}_{1}H^{\mp}_{1}}$(left),
$\bar{\lambda}_{h_{1}H^{\pm}_{2}H^{\mp}_{2}}$(middle) and
$\bar{\lambda}_{h_{1}H^{\pm\pm}H^{\mp\mp}}$ (right panel). The displayed
horizontal lines denote the central and $\pm 1\sigma$ diphoton strength signal
values reported by ATLAS [66, 4] at $13$ TeV. We used the following inputs:
$\alpha_{1}=1.42$, $\lambda_{1}=1.31$, $\lambda_{3}=6.15$, $\lambda_{4}=-3.85$
and $m_{h_{2}}=673.30$ GeV. The other inputs are the same as in (11)

, while the error for $\chi^{2}$ fit is 95.5% C.L.

Then, we investigate how $R_{\gamma h_{1}}$ behaves with respect to
$R_{\gamma\gamma}$ and $R_{\gamma Z}$ and wether these observables are
correlated. One can readily see from Fig. 8 a clear correlation with these two
signal strengths. Additionally, we also analyzed the incidence of the doubly
charged Higgs on these loop induced processes, via its trilinear Higgs
couplings $\bar{\lambda}_{h_{1}H^{\pm\pm}H^{\mp\mp}}$. The results
interestingly indicate that the lighter it is, the more significant the
enhancement in the total cross-section $\sigma(e^{+}e^{-}\to\gamma h_{1})$, as
in $R_{\gamma h_{1}}$ and $R_{\gamma Z}$.

Figure 8: $R_{\gamma h_{1}}$ correlation with $R_{\gamma\gamma}(h_{1})$ (left
panels) and $R_{\gamma Z}(h_{1})$ (right panels) in 2HDMcT. Color coding shows
the variation of $\bar{\lambda}_{h_{1}H^{\pm\pm}H^{\mp\mp}}$. The inputs are
the same as in Fig. 7.

##  4 CONCLUSION

The future $e^{-}e^{+}$ colliders are anticipated to play a vital role in
understanding the nature of the Higgs boson and its coupling to the Standard
Model particles with unprecedented precision. In the present paper, we have
studied the one-loop processes $e^{+}e^{-}\to\gamma h_{1}$ and $e^{-}\gamma\to
e^{-}h_{1}$ in the framework of the two Higgs doublet type-II seesaw model
(2HDMcT) at the $e^{-}e^{+}$ colliders, where the $h_{1}$ Higgs boson is
chosen to replicate the observed Higgs. Within the $2HDMcT$ parameter space
delimited by a full set of constraints, we have shown that the $2HDMcT$
charged scalars can substantially alter the cross sections
$\sigma(e^{+}e^{-}\to\gamma h_{1})$ and $\sigma(e^{-}\gamma\to e^{-}h_{1})$.
Our analysis also observed that these observables are especially sensitive to
the parameters $\alpha_{1}$, $\lambda_{7}$, $\lambda_{9}$, the trilinear Higgs
couplings as to the charged scalars masses. These parameters conspire to
induce significant contributions to $\sigma(e^{+}e^{-}\to h_{1}\gamma)$, up to
$8.1\times 10^{-2}$ fb so exceeding the Standard Model prediction.

Additionally, the charged scalars $H_{1}^{\pm}$, $H_{2}^{\pm}$ and
$H^{\pm\pm}$ also affected the ratio $R_{h_{1}\gamma}$ and the signal strength
$R_{Z\gamma}$, while $R_{\gamma\gamma}$ being consistent with the measured
signal strength at $1\sigma$. Besides, another outcome of our study imply that
these three observables are entirely correlated with each other.

## References

  * [1] ATLAS Collaboration, G. Aad et al., Observation of a new particle in the search for the Standard Model Higgs boson with the ATLAS detector at the LHC, Phys. Lett. B 716 (2012) 1–29, [arXiv:1207.7214].
  * [2] CMS Collaboration, S. Chatrchyan et al., Observation of a New Boson at a Mass of 125 GeV with the CMS Experiment at the LHC, Phys. Lett. B 716 (2012) 30–61, [arXiv:1207.7235].
  * [3] CMS Collaboration, A. Tumasyan et al., A portrait of the Higgs boson by the CMS experiment ten years after the discovery., Nature 607 (2022), no. 7917 60–68, [arXiv:2207.00043].
  * [4] ATLAS Collaboration, G. Aad et al., A detailed map of Higgs boson interactions by the ATLAS experiment ten years after the discovery, Nature 607 (2022), no. 7917 52–59, [arXiv:2207.00092]. [Erratum: Nature 612, E24 (2022)].
  * [5] W. Elmetenawee, Summary of CMS Higgs Physics, arXiv:2401.07650.
  * [6] F. Zwicky, Die Rotverschiebung von extragalaktischen Nebeln, Helv. Phys. Acta 6 (1933) 110–127.
  * [7] V. C. Rubin and W. K. Ford, Jr., Rotation of the Andromeda Nebula from a Spectroscopic Survey of Emission Regions, Astrophys. J. 159 (1970) 379–403.
  * [8] M. J. G. Veltman, The Infrared - Ultraviolet Connection, Acta Phys. Polon. B 12 (1981) 437.
  * [9] T. Kajita, Nobel lecture: Discovery of atmospheric neutrino oscillations, Rev. Mod. Phys. 88 (Jul, 2016) 030501.
  * [10] A. B. McDonald, Nobel lecture: The sudbury neutrino observatory: Observation of flavor change for solar neutrinos, Rev. Mod. Phys. 88 (Jul, 2016) 030502.
  * [11] N. G. Deshpande and E. Ma, Pattern of Symmetry Breaking with Two Higgs Doublets, Phys. Rev. D 18 (1978) 2574.
  * [12] S. L. Glashow and S. Weinberg, Natural conservation laws for neutral currents, Phys. Rev. D 15 (Apr, 1977) 1958–1965.
  * [13] G. C. Branco, P. M. Ferreira, L. Lavoura, M. N. Rebelo, M. Sher, and J. P. Silva, Theory and phenomenology of two-Higgs-doublet models, Phys. Rept. 516 (2012) 1–102, [arXiv:1106.0034].
  * [14] S. Dawson, C. Englert, and T. Plehn, Higgs Physics: It ain’t over till it’s over, Phys. Rept. 816 (2019) 1–85, [arXiv:1808.01324].
  * [15] I. P. Ivanov, Building and testing models with extended Higgs sectors, Prog. Part. Nucl. Phys. 95 (2017) 160–208, [arXiv:1702.03776].
  * [16] S. Weinberg, Baryon- and lepton-nonconserving processes, Phys. Rev. Lett. 43 (Nov, 1979) 1566–1570.
  * [17] B. Ait Ouazghour, A. Arhrib, R. Benbrik, M. Chabab, and L. Rahili, Theory and phenomenology of a two-Higgs-doublet type-II seesaw model at the LHC run 2, Phys. Rev. D 100 (2019), no. 3 035031, [arXiv:1812.07719].
  * [18] B. Ait Ouazghour and M. Chabab, The two Higgs doublet type-II seesaw model: Naturalness and B¯→Xs$\gamma$ versus heavy Higgs masses, Phys. Lett. B 846 (2023) 138241, [arXiv:2305.08030].
  * [19] ATLAS Collaboration, M. Aaboud et al., Search for additional heavy neutral Higgs and gauge bosons in the ditau final state produced in 36 fb-1 of pp collisions at $\sqrt{s}=13$ TeV with the ATLAS detector, JHEP 01 (2018) 055, [arXiv:1709.07242].
  * [20] ATLAS Collaboration, M. Aaboud et al., Search for new phenomena in high-mass diphoton final states using 37 fb-1 of proton–proton collisions collected at $\sqrt{s}=13$ TeV with the ATLAS detector, Phys. Lett. B 775 (2017) 105–125, [arXiv:1707.04147].
  * [21] CMS Collaboration, A. M. Sirunyan et al., Search for a new scalar resonance decaying to a pair of Z bosons in proton-proton collisions at $\sqrt{s}=13$ TeV, JHEP 06 (2018) 127, [arXiv:1804.01939]. [Erratum: JHEP 03, 128 (2019)].
  * [22] ATLAS Collaboration, M. Aaboud et al., Search for a heavy Higgs boson decaying into a $Z$ boson and another heavy Higgs boson in the $\ell\ell bb$ final state in $pp$ collisions at $\sqrt{s}=13$ TeV with the ATLAS detector, Phys. Lett. B 783 (2018) 392–414, [arXiv:1804.01126].
  * [23] CMS Collaboration, A. M. Sirunyan et al., Search for heavy Higgs bosons decaying to a top quark pair in proton-proton collisions at $\sqrt{s}=$ 13 TeV, JHEP 04 (2020) 171, [arXiv:1908.01115]. [Erratum: JHEP 03, 187 (2022)].
  * [24] CMS Collaboration, A. M. Sirunyan et al., Search for MSSM Higgs bosons decaying to $\mu$ \+ $\mu$ $-$ in proton-proton collisions at s=13TeV, Phys. Lett. B 798 (2019) 134992, [arXiv:1907.03152].
  * [25] CMS Collaboration, Search for a charged Higgs boson decaying into a heavy neutral Higgs boson and a W boson in proton-proton collisions at $\sqrt{s}$ = 13 TeV, arXiv e-prints (July, 2022) arXiv:2207.01046, [arXiv:2207.01046].
  * [26] R. S. Gupta, H. Rzehak, and J. D. Wells, How well do we need to measure the Higgs boson mass and self-coupling?, Phys. Rev. D 88 (2013) 055024, [arXiv:1305.6397].
  * [27] J. Baglio and C. Weiland, The triple Higgs coupling: A new probe of low-scale seesaw models, JHEP 04 (2017) 038, [arXiv:1612.06403].
  * [28] ILC physics, detector study Collaboration, J. Tian and K. Fujii, Measurement of Higgs boson couplings at the International Linear Collider, Nucl. Part. Phys. Proc. 273-275 (2016) 826–833.
  * [29] C. F. Dürig, Measuring the Higgs Self-coupling at the International Linear Collider.  PhD thesis, Hamburg U., Hamburg, 2016.
  * [30] T. Liu, K.-F. Lyu, J. Ren, and H. X. Zhu, Probing the quartic Higgs boson self-interaction, Phys. Rev. D 98 (2018), no. 9 093004, [arXiv:1803.04359].
  * [31] A. Arbey et al., Physics at the e+ e- Linear Collider, Eur. Phys. J. C 75 (2015), no. 8 371, [arXiv:1504.01726].
  * [32] ILC International Development Team Collaboration, A. Aryshev et al., The International Linear Collider: Report to Snowmass 2021, arXiv:2203.07622.
  * [33] P. Bambade et al., The International Linear Collider: A Global Project, arXiv:1903.01629.
  * [34] CLIC Physics Working Group Collaboration, E. Accomando et al., Physics at the CLIC multi-TeV linear collider, in 11th International Conference on Hadron Spectroscopy, CERN Yellow Reports: Monographs, 6, 2004.  hep-ph/0412251.
  * [35] CEPC Study Group Collaboration, CEPC Conceptual Design Report: Volume 1 - Accelerator, arXiv:1809.00285.
  * [36] CEPC Study Group Collaboration, M. Dong et al., CEPC Conceptual Design Report: Volume 2 - Physics & Detector, arXiv:1811.10545.
  * [37] CEPC Study Group Collaboration, W. Abdallah et al., CEPC Technical Design Report – Accelerator, arXiv:2312.14363.
  * [38] TLEP Design Study Working Group Collaboration, M. Bicer et al., First Look at the Physics Case of TLEP, JHEP 01 (2014) 164, [arXiv:1308.6176].
  * [39] FCC Collaboration, A. Abada et al., FCC Physics Opportunities: Future Circular Collider Conceptual Design Report Volume 1, Eur. Phys. J. C 79 (2019), no. 6 474.
  * [40] LCC Physics Working Group Collaboration, K. Fujii et al., Tests of the Standard Model at the International Linear Collider, arXiv:1908.11299.
  * [41] A. Abbasabadi, D. Bowser-Chao, D. A. Dicus, and W. W. Repko, Higgs-boson–photon associated production at eē colliders, Phys. Rev. D 52 (Oct, 1995) 3919–3928.
  * [42] A. Djouadi, V. Driesen, W. Hollik, and J. Rosiek, Associated production of Higgs bosons and a photon in high-energy e+e- collisions, Nuclear Physics B 491 (Feb., 1997) 68–102, [hep-ph/9609420].
  * [43] A. Barroso, J. Pulido, and J. C. Romao, HIGGS PRODUCTION AT e+ e- COLLIDERS, Nucl. Phys. B 267 (1986) 509–530.
  * [44] A. Arhrib, R. Benbrik, and T.-C. Yuan, Associated Production of Higgs at Linear Collider in the Inert Higgs Doublet Model, Eur. Phys. J. C 74 (2014) 2892, [arXiv:1401.6698].
  * [45] L. Rahili, A. Arhrib, and R. Benbrik, Associated production of SM Higgs with a photon in type-II seesaw models at the ILC, Eur. Phys. J. C 79 (2019), no. 11 940, [arXiv:1909.07793].
  * [46] S. Heinemeyer and C. Schappacher, Neutral Higgs boson production at e+̂e-̂ colliders in the complex MSSM: a full one-loop analysis, European Physical Journal C 76 (Apr., 2016) 220, [arXiv:1511.06002].
  * [47] M. Demirci, Associated production of Higgs boson with a photon at electron-positron colliders, Phys. Rev. D 100 (2019), no. 7 075006, [arXiv:1905.09363].
  * [48] ILD concept group Collaboration, Y. Aoki, K. Fujii, and J. Tian, Study of $e^{+}e^{-}\to\gamma h$ at the ILC, arXiv:2203.07202.
  * [49] M. J. Ramsey-Musolf, The electroweak phase transition: a collider target, JHEP 09 (2020) 179, [arXiv:1912.07189].
  * [50] M. E. Peskin and T. Takeuchi, Estimation of oblique electroweak corrections, Phys. Rev. D46 (1992) 381–409.
  * [51] W. Grimus, L. Lavoura, O. M. Ogreid, and P. Osland, The Oblique parameters in multi-Higgs-doublet models, Nucl. Phys. B 801 (2008) 81–96, [arXiv:0802.4353].
  * [52] Particle Data Group Collaboration, P. A. Zyla et al., Review of Particle Physics, PTEP 2020 (2020), no. 8 083C01.
  * [53] H. Bahl, T. Biekötter, S. Heinemeyer, C. Li, S. Paasch, G. Weiglein, and J. Wittbrodt, HiggsTools: BSM scalar phenomenology with new versions of HiggsBounds and HiggsSignals, Comput. Phys. Commun. 291 (2023) 108803, [arXiv:2210.09332].
  * [54] P. Bechtle, S. Heinemeyer, O. Stål, T. Stefaniak, and G. Weiglein, $HiggsSignals$: Confronting arbitrary Higgs sectors with measurements at the Tevatron and the LHC, Eur. Phys. J. C 74 (2014), no. 2 2711, [arXiv:1305.1933].
  * [55] P. Bechtle, S. Heinemeyer, O. Stål, T. Stefaniak, and G. Weiglein, Probing the Standard Model with Higgs signal rates from the Tevatron, the LHC and a future ILC, JHEP 11 (2014) 039, [arXiv:1403.1582].
  * [56] P. Bechtle, S. Heinemeyer, T. Klingl, T. Stefaniak, G. Weiglein, and J. Wittbrodt, HiggsSignals-2: Probing new physics with precision Higgs measurements in the LHC 13 TeV era, Eur. Phys. J. C 81 (2021), no. 2 145, [arXiv:2012.09197].
  * [57] P. Bechtle, O. Brein, S. Heinemeyer, G. Weiglein, and K. E. Williams, HiggsBounds: Confronting Arbitrary Higgs Sectors with Exclusion Bounds from LEP and the Tevatron, Comput. Phys. Commun. 181 (2010) 138–167, [arXiv:0811.4169].
  * [58] P. Bechtle, O. Brein, S. Heinemeyer, G. Weiglein, and K. E. Williams, HiggsBounds 2.0.0: Confronting Neutral and Charged Higgs Sector Predictions with Exclusion Bounds from LEP and the Tevatron, Comput. Phys. Commun. 182 (2011) 2605–2631, [arXiv:1102.1898].
  * [59] P. Bechtle, O. Brein, S. Heinemeyer, O. Stål, T. Stefaniak, G. Weiglein, and K. E. Williams, $\mathsf{HiggsBounds}-4$: Improved Tests of Extended Higgs Sectors against Exclusion Bounds from LEP, the Tevatron and the LHC, Eur. Phys. J. C 74 (2014), no. 3 2693, [arXiv:1311.0055].
  * [60] P. Bechtle, D. Dercks, S. Heinemeyer, T. Klingl, T. Stefaniak, G. Weiglein, and J. Wittbrodt, HiggsBounds-5: Testing Higgs Sectors in the LHC 13 TeV Era, Eur. Phys. J. C 80 (2020), no. 12 1211, [arXiv:2006.06007].
  * [61] HFLAV Collaboration, Y. S. Amhis et al., Averages of $b$-hadron, $c$-hadron, and $\tau$-lepton properties as of 2021, Phys. Rev. D 107 (2023) 052008, [arXiv:2206.07501].
  * [62] T. Hahn, Generating Feynman diagrams and amplitudes with FeynArts 3, Comput. Phys. Commun. 140 (2001) 418–431, [hep-ph/0012260].
  * [63] T. Hahn and M. Pérez-Victoria, Automated one-loop calculations in four and d dimensions, Computer Physics Communications 118 (1999), no. 2 153–165.
  * [64] G. J. van Oldenborgh, FF: A Package to evaluate one loop Feynman diagrams, Comput. Phys. Commun. 66 (1991) 1–15.
  * [65] T. Hahn, Feynman Diagram Calculations with FeynArts, FormCalc, and LoopTools, PoS ACAT2010 (2010) 078, [arXiv:1006.2231].
  * [66] ATLAS Collaboration, G. Aad et al., Measurement of the properties of Higgs boson production at $\sqrt{s}=13$ TeV in the $H\to\gamma\gamma$ channel using $139$ fb-1 of $pp$ collision data with the ATLAS experiment, JHEP 07 (2023) 088, [arXiv:2207.00348].
  * [67] ATLAS Collaboration, G. Aad et al., A search for the $Z\gamma$ decay mode of the Higgs boson in $pp$ collisions at $\sqrt{s}$ = 13 TeV with the ATLAS detector, Phys. Lett. B 809 (2020) 135754, [arXiv:2005.05382].
  * [68] CMS Collaboration, A. Tumasyan et al., Search for Higgs boson decays to a Z boson and a photon in proton-proton collisions at $\sqrt{s}$ = 13 TeV, JHEP 05 (2023) 233, [arXiv:2204.12945].
  * [69] CMS, ATLAS Collaboration, G. Aad et al., Evidence for the Higgs boson decay to a $Z$ boson and a photon at the LHC, arXiv:2309.03501.
  * [70] ATLAS Collaboration, G. Aad et al., Search for the $Z\gamma$ decay mode of new high-mass resonances in $pp$ collisions at $\sqrt{s}=13$ TeV with the ATLAS detector, arXiv:2309.04364.


# estimation of optimal encoding ladders for tiled 360$\degree$ VR video in
adaptive streaming systems

###### Abstract

Given the significant industrial growth of demand for virtual reality (VR),
360∘ video streaming is one of the most important VR applications that require
cost-optimal solutions to achieve widespread proliferation of VR technology.
Because of its inherent variability of data-intensive content types and its
tiled-based encoding and streaming, 360∘ video requires new encoding ladders
in adaptive streaming systems to achieve cost-optimal and immersive streaming
experiences. In this context, this paper targets both the provider’s and
client’s perspectives and introduces a new content-aware encoding ladder
estimation method for tiled 360∘ VR video in adaptive streaming systems. The
proposed method first categories a given 360∘ video using its features of
encoding complexity and estimates the visual distortion and resource cost of
each bitrate level based on the proposed distortion and resource cost models.
An optimal encoding ladder is then formed using the proposed integer linear
programming (ILP) algorithm by considering practical constraints. Experimental
results of the proposed method are compared with the recommended encoding
ladders of professional streaming service providers. Evaluations show that the
proposed encoding ladders deliver better results compared to the recommended
encoding ladders in terms of objective quality for 360∘ video, providing
optimal encoding ladders using a set of service provider’s constraint
parameters.

Index Terms—  360∘ video, virtual reality, adaptive streaming, encoding
ladder, optimization

##  1 Introduction

Recent years have witnessed a significant industrial investment in virtual
reality (VR) technology that has motivated technical developments of graphic
cards and head-mounted displays (HMDs) [vrmarket]. Currently, the video
technology field is evolving toward providing immersive VR experiences using
360∘ video streaming. 360∘ video is captured with omnidirectional camera
arrays and the individual camera views are projected onto a sphere. For
backward-compatibility purposes with the existing video coding standards and
streaming pipelines, the spherical videos are mapped onto a planar surface
using projection techniques, such as equi-rectangular projection (ERP). ERP
videos contain full panoramic 360∘ horizontal and 180∘ vertical views of the
scene.

360∘ video streaming is significantly challenging owing to its resource-
intensive encoding and storage requirements to cope with the very high
resolution of its representation. As the VR end-user can only view the field
of view (FoV) of the display device (e.g., HMD, smartphone, tablet or laptop),
called viewport, very high resolution of 360∘ video (e.g., 8K$\times$4K ERP)
is required for transmission in order to achieve high-quality and seamless
video streaming experiences. To reduce both the bitrate consumption of the
end-user and the visual distortion of the viewport, 360∘ video frames can be
divided into self-decodable regions [heymann2005representation,
Grunheit2002du], namely, tiles.

To deliver the tiled 360∘ videos to the end-user devices, adaptive streaming
systems, such as MPEG-dynamic adaptive streaming over HTTP (DASH)
[Isoiec230091undatedlt], provide smooth 360∘ video streaming experiences, but
still require high encoding and storage costs for the tiled 360∘ video. The
_spatial relationship description_ (SRD) [Niamut2016hd] can be used with DASH
systems where the 360∘ video stream is divided into tiles. In the SRD, each
360∘ video is divided into a set of tiles that includes different _bitrate
levels_ of the tiled video. Different bitrate levels share the same video
content but are encoded using various settings, such as the resolution and the
target bitrate for encoding. Each different version is called a
_representation_ , and a set of representations for the video content forms
the _encoding ladder_ which is requested by the DASH client to play the tiled
360∘ video. However, encoding and accumulating a large combination of
representations for each video content might cover a broad range of network
bandwidths such that the end-users can request video streams of appropriate
bitrates, and thus it requires high encoding and storage costs [janEncoding].
Fig. 1 illustrates the different stages from the spherical projection to the
encoding ladder with the different representations of the ERP video.

Fig. 1: Overview of the different formats and representations.

To tackle this problem, cost-optimal encoding ladders are needed for service
providers to deliver tiled 360∘ video content and satisfy network bandwidths.
In fact, tiled 360$\degree$ video provides different rate-distortion (RD)
performance compared to the traditional video content due to different
characteristics of both. In particular, tiling affects the coding efficiency,
because redundancy cannot be exploited over tiles. Furthermore, given its 2D
projection for encoding (e.g., ERP), each tile of the 360$\degree$ video has a
different level of contribution for the overall 360$\degree$ video viewing
quality due to stretching effects caused by the projection [qomexAna,
Jvet2016tw]. To this end, new encoding ladder configurations are required for
the tiled 360$\degree$ videos to provide cost-optimal video streaming service
for VR end-user devices.

Adaptive streaming systems must deal with issues of the delivery of the tiled
360∘ video from two different perspectives, the service provider and the
client. Most recent work focused on the client’s perspective [myIcip2017,
Hosseini2016kv, Corbillon2016js, LeFeuvre2016pg, Graf2017, Frounhoundatedwi]
without considering the service providers’ perspective. More clearly, they
neither provide 360∘ video content-specific encoding ladders nor consider the
resource costs of the content delivery network (CDN), which is a cloud-based
video streaming system that delivers videos to the edge servers so as to
effectively connect to the end-users. Given the different characteristic of
the tiled 360∘ video content (e.g., ERP and tile encoding), recommended
encoding ladders for traditional videos [appleRec, netflixblogPeerTitle], that
are currently used for adaptive streaming systems, might not achieve an
acceptable quality of experience (QoE) [6982310, janEncoding] for the tiled
360∘ video. Using such encoding ladders might also waste CDN resources and the
end-users’ bandwidth.

Our work aims to improve the performance of adaptive 360∘ video streaming
systems, providing guidelines for the design of optimal 360∘ VR video
streaming systems using tiles. To this end, we focus on the configuration of
cost-optimal encoding ladders in adaptive streaming systems by considering
both the provider’s and client’s perspective and develop an encoding ladder
estimation method for tiled 360∘ video streaming, which is the main
contribution of this work. To the best of our knowledge, such encoding ladder
estimation method has not been studied yet. The proposed method deals with
minimizing the distortion of the observed tiled 8K$\times$4K ERP video content
on the client side while reducing the resource costs on the service provider
side, such as storage capacity utilization and computational costs for
encoding. In this context, we categorize the given 360∘ videos using their
extracted features of encoding complexity, estimate their visual distortion
based on the developed distortion model, and calculate the resource costs
using the proposed cost models. The cost-optimal encoding ladder configuration
problem is then solved using the formulated integer linear programming (ILP)
algorithm by considering practical constraints. Our evaluations show that the
proposed cost-optimal encoding ladders using a set of service provider’s
constraint parameters achieve better results compared to the recommended
encoding ladders in terms of objective quality for 360∘.

The remainder of this paper is organized as follows. Related work is detailed
in Section 2. Then, the proposed system model is presented in Section 3.
Experiments to demonstrate the performance of our proposed method are
presented in Section 4. Finally, Section 5 concludes this paper with a summary
and future work.

##  2 Related Works

To define the most suitable encoding ladder for traditional video, an unique
encoding ladder for each given video content is generated for instance by the
engineers at Netflix using the brute-force search algorithm
[netflixblogPeerTitle]. In their research work, each quality-resolution pair
was plotted for a given content at each bitrate level. An upper convex hull of
its RD curve was then selected to define the encoding ladder. Their approach
is very effective concerning QoE for traditional video content. However, it is
neither cost-optimal in the sense of resource consumption of a CDN nor
content-specific and optimized for tiled 360∘ videos.

Similarly, academic researchers demonstrated that the previously defined fixed
encoding ladders such as Apple’s and Netflix’s one-size-fits-all schemes
[appleRec, netflixblogPeerTitle], have critical weaknesses for traditional
video content as described in [Toni_2014_2557652]. Here, the authors defined
an optimal encoding ladder for each video category to improve the performance
of adaptive streaming for traditional videos. The problem was formulated as an
optimization algorithm to find the best bitrate ladder for the given videos by
considering the characteristics of a set of end-users in a given database
without considering encoding and storage costs. The results have shown,
however, that the fixed encoding ladders cannot provide the best objective
quality for given traditional videos and clients’ bandwidth.

Most recent work focused on 360∘ video streaming solutions using tiles in
order to optimize the quality on the client side [Frounhoundatedwi,
myIcip2017, Hosseini2016kv, Corbillon2016js, LeFeuvre2016pg, Graf2017]. The
authors in [myIcip2017] proposed a new adaptive streaming system based on
tiling, integration of the DASH standard and a viewport-aware bitrate level
selection method. In [Hosseini2016kv], an adaptive bandwidth-efficient 360 VR
video streaming system using a divide and conquer approach was presented. The
work is based on a dynamic viewport-aware adaptation technique using tiles,
derived from a hexaface sphere, and the DASH standard. Similar to the previous
work, the authors of [Corbillon2016js] also propose a viewport-adaptive video
delivery system using tiles (cube maps) and different video representations
that differ by their bitrate and different scene regions. Additionally, in
[LeFeuvre2016pg], high-resolution video content is transmitted in tiled
fashion using fixed rectangular tiles. The authors in [Graf2017] presented a
bandwidth efficient adaptive 360∘ video streaming system. The work in
[Frounhoundatedwi] described the bandwidth problem of 360$\degree$ video, and
suggested to use tile-based streaming. Furthermore, their work described the
principles of adaptive streaming of 360$\degree$ video using tiles and
evaluated their system with respect to bitrate overhead, bandwidth, and
quality requirements. However, none of these works are dealing with cost-
optimal encoding ladders on the service provider’s side to reduce storage
capacity utilization and computational costs.

##  3 Proposed System Model

Fig. 2: Schematic diagram of a cloud-based video streaming pipeline for VR
which includes source, media platform, and delivery of the tiled 360∘ video
content.

We consider a cloud-based video-on-demand 360∘ video streaming pipeline for VR
as depicted in Fig. 2. Each captured 360∘ spherical video is mapped to the ERP
representation in 8K$\times$4K resolution for encoding purposes at the source
node. The media platform divides each ERP video into $N$ tiles and estimates
an unique cost-optimal encoding ladder. Each tile is then encoded at various
bitrate levels using multiple encoders with estimated cost-optimal encoding
ladder parameters. Then, the generated bitstreams are divided into a set of
chunks with equal playback duration, encapsulated by the packaging node and
eventually stored on the origin server. Each stored content is then deployed
to the CDN, where the bitstreams are efficiently distributed to the VR end-
users through the edge servers.

Each end-user device contains the tiled DASH-VR player [myIcip2017] to
communicate with the edge servers and to request individual tiles with
appropriate bitrate levels and resolutions from the encoding ladder depending
on the bandwidth availability of the network. For adaptive streaming purposes,
a set of tiles is encoded at the media platform using different encoding
settings. More precisely, let $v$ be an 8K$\times$4K ERP 360$\degree$ video in
the set of videos $\mathcal{V}$. Each $v$ is split into $N$ tiles, each tile
$j$, $j\in\mathcal{T}$, is then encoded at a different bitrate $b_{j}$ and
resolution $r_{j}=w_{j}\times h_{j}$. Hence, the quadruple ($v$,$j$,$b$,$r$)
corresponds to a representation of the video $v\in\mathcal{V}$ for the tile
$j\in\mathcal{T}$, encoded at a target bitrate $b\in\mathcal{B}$ and spatial
resolution $r\in\mathcal{R}$. Note that $v$, $j$, $b$, and $r$ are integer
values and represent the indices of their corresponding sets.

In this context, encoding and accumulating all combinations of the quadruple
($v$,$j$,$b$,$r$) might be very expensive for service providers. Therefore, a
cost-effective optimization is required in order to minimize the service
provider’s resource costs while providing cost-optimal and high quality
360$\degree$ video streaming experience.

For this aim, the proposed estimation method contains four main parts:
classification of the content type, distortion modeling, cost modeling, and
problem formulation. First, we extract spatial and temporal features
($f_{spa}$ and $f_{tmp}$) of the $v$-th video to classify its content type as
described in subsection 3.1. Then, we perform an automatic estimation
procedure for the encoding ladder using distortion and cost models for the
tiled $v$-th video as detailed in subsections 3.2 and 3.3, respectively.
Again, in this encoding ladder estimation process we consider both the client
side (quality distortion) and service provider side (resource costs). Finally,
we formulate the cost-optimal estimation problem for the encoding ladder by
applying certain practical constraints, which is eventually solved using the
proposed ILP algorithm as described in subsection 3.4.

###  3.1 Classification of the content type

To classify the content type from a given set of content types $\mathcal{O}$,
spatial $f_{spa}$ and temporal $f_{tmp}$ complexity features are extracted
from the videos. As each video $v$ has different RD performances at various
resolutions, we can identify two sources of video distortion: spatial down-
sampling and quantization. As a down-sampled version of $v$ suffers from
spatial information loss, the level of information loss depends on the
_spatial complexity_ of each video, which is one of the encoding complexity
features. Moreover, the high-resolution version of a given $v$ requires a
larger amount of bits to reduce its visual distortion. Compared to its low-
resolution version, the high-resolution version has a higher sensitivity for
unpredictable motions, which requires further residuals to avoid visual
distortions. Since predicted residuals are compressed through quantization
which results in quality distortions, _temporal complexity_ is the second
encoding complexity feature. The content type $o$ of each video is then
determined from a given $\mathcal{O}$ by classification using the extracted
two complexity features.

To extract the feature set $\mathcal{F}=\\{f_{spa},f_{tmp}\\}$, we use the
constant rate factor (CRF) encoding. The CRF encoding, unlike the constant
quantization parameter (QP)-based encoding, has the QPs slightly varied across
the time based on the scene complexity, action, and motion. For instance, when
a scene contains a lot of action and motion, a higher compression can be
applied by raising the QP in order to save bitrates. Therefore, the feature
set $\mathcal{F}$ can be extracted from the CRF encoded stream to identify the
encoding complexity of each $v$. For this purpose, the average size of I- and
P- frames can be used as main indicators to determine the complexity features.
As also demonstrated in [chen2016subjective], the size of I-frames expresses
the spatial complexity of each $v$. Thus, we use the normalized version of the
I frame sizes to estimate $f_{spa}$ for a given video. As the average size of
P frames characterizes the amount of residual bits, we use the ratio of the
size of P frames over the size of I-frames as the indicator for $f_{tmp}$.

###  3.2 Distortion modeling

To model the distortion of a given $v$, we model two sources of artifacts, the
compression and spatial scaling artifacts, of the tiled 360$\degree$ video
using its content type and encoding resolution. Both artifacts, which are the
most important distortions that deteriorate QoE, are driven by the encoding
target rate and the adaptation of the video resolution to the target
resolution. With the aim of reducing search complexity, we generate a
continuous distortion model for each content type, as the given parameter
space is too large for a the brute-force search algorithm (e.g., Netflix’s
work in [netflixblogPeerTitle]). To this end, we derive a distortion function
by fitting the two-term power series model using the following fit function:

$FT_{ogB}=k_{og}Z_{B}^{\Omega_{og}}+\Phi_{og},$ (1)

where $k$, $\Omega$, and $\Phi$ are fitting parameters used in the curve
fitting operation for the $o$-th content type, $o\in\mathcal{O}$ and
$\mathcal{O}=\\{o_{1},o_{2},\ldots,o_{|\mathcal{O}|}\\}$, of the $g$-the
resolution, $g\in\mathcal{G}$ and
$\mathcal{G}=\\{g_{1},g_{2},\ldots,g_{|\mathcal{G}|}\\}$, at the tiled ERP
video bitrate $B$. Note that $Z$ is the value of the total bitrate of the
tiled 360$\degree$ video in terms of Mbps (i.e., total bitrate of the ERP
video recomposed of the tiles with bitrate $B$). These parameters for the
proposed distortion model, shown in Table 1, were found using the curve
fitting operator. Note that index number of $o$ and $g$ are listed in
ascending order of their size. The target resolution size is 8K$\times$4K. For
the sake of simplicity and also a lack of variety of 8K 360$\degree$ video
content types, we only distinguish between three content types and
resolutions. Each row and column number of the fitting parameters in the table
represents a different content type and resolution, respectively.

Resolution $\mathcal{G}$ |  | $g_{1}$ | $g_{2}$ | $g_{3}$
---|---|---|---|---
Model |  | Distortion | Data size | Distortion | Data size | Distortion | Data size
|  | $k$ | $\Omega$ | $\Phi$ | $k$ | $\Omega$ | $\Phi$ | $k$ | $\Omega$ | $\Phi$ | $k$ | $\Omega$ | $\Phi$ | $k$ | $\Omega$ | $\Phi$ | $k$ | $\Omega$ | $\Phi$
Content type $\mathcal{O}$ | $o_{1}$ | 1809 | -0.6959 | 5.649 | 0.7613 | 0.9901 | 52.54 | 4002 | -0.7558 | 2.723 | 0.8005 | 0.9859 | 52.25 | 1829 | -0.5587 | -3.266 | 0.8264 | 0.9846 | 214.9
$o_{2}$ | 220.1 | -0.3583 | 6.447 | 0.6467 | 1.003 | 29.36 | 191.9 | -0.2763 | -5.728 | 0.6078 | 1.009 | 71.15 | 480.6 | -0.3643 | -5.728 | 0.5654 | 1.015 | 269
$o_{3}$ | 820.4 | -0.4702 | 6.2 | 0.6631 | 1.001 | 10.69 | 643 | -0.3825 | -2.625 | 0.6691 | 1 | 17.46 | 616.9 | -0.2837 | -23.78 | 0.5943 | 1.012 | 203.8

Table 1: Curve fitting parameters for the proposed distortion and data size
estimation models.

To better reflect the distortion of the 360$\degree$ video at the clients’
side, we estimate the distortion, caused by the mapping of the spherical
content onto the planar surface of the devices (_spherical distortion_), of
the tiled 360$\degree$ video as a target value in the curve fitting using the
_weighted-to-spherically-uniform mean square error_ (WS-MSE) [Jvet2016tw]. WS-
MSE measures the spherical surface using a non-linear weighting in the MSE
calculation. Such weights are calculated using the stretching ratio of the
area that is projected from the planar surface to the spherical surface. The
noise power for the $i$-th representation of the $j$-th tile, $d_{ij}$, can be
formulated as follows:

$d_{ij}=\frac{\underset{\begin{subarray}{c}x\in
W\end{subarray}}{\sum}\underset{\begin{subarray}{c}y\in
H\end{subarray}}{\sum}\left((t_{j}(x,y)-\tilde{t}_{ij}(x,y))^{2}q_{j}(x,y)\right)}{\underset{\begin{subarray}{c}x\in
W\end{subarray}}{\sum}\underset{\begin{subarray}{c}y\in
H\end{subarray}}{\sum}q_{j}(x,y)},$ (2)

where $W\times H$ is the resolution of the reconstructed version of the ERP
360$\degree$ video. Note that $x$ and $y$ denote the pixel coordinates of the
ERP video, $t$ and $\tilde{t}$ stand for the original (i.e., uncompressed) and
reconstructed versions of the $j$-th tile and $q_{j}(x,y)$ represents the
weighting intensity in ($x$, $y$) of the weight distribution of the ERP for
$t_{j}$ which can be calculated according to [Jvet2016tw] with:

$q_{j}(x,y)=cos\frac{(y+0.5-H/2)\pi}{H}.$ (3)

###  3.3 Cost modeling

In this subsection, we develop cost models for the cloud-based video streaming
system in order to minimize the resource costs for encoding workload and
storage capacity utilization at the service providers’ side.

####  3.3.1 Encoding cost

The encoding cost is one of the most expensive computing costs which usually
occurs on the cloud servers and which heavily depends on the video resolution.
To calculate encoding costs, we consider the broken-line model where the same
cost is defined for similar resolutions. To this end, we extend the cost
calculation model used by the Amazon cloud service [amazonPrice] in order to
consider broad range of resolution sizes. The encoding cost $c^{e}$ can be
described for the $j$-th tile of the $i$-th representation as follows:

$c^{e}_{ij}=\begin{cases}\mu_{e},&r_{ij}\leq 720p\\\ 2\mu_{e},&720p<r_{ij}\leq
1080p\\\ 4\mu_{e},&1080p<r_{ij}\leq 4K\\\ 8\mu_{e},&4K<r_{ij}\leq 8K\\\
\end{cases}$ (4)

where $\mu_{e}$ is a constant term for the encoding cost defined by the
service provider and $r_{ij}$ is the resolution of the $j$-th tile in the
$i$-th representation.

####  3.3.2 Storage cost

Additionally, large storage capacity is required to store all encoded tiles
with different representations for adaptive streaming on the server. The
storage cost depends on the data size of the tiled 360$\degree$ video which is
located on the server. Considering a linear cost model where the cost is
proportional to the data size of each tiled 360$\degree$ video stream, the
storage cost $c^{s}$ for the $j$-th tile of the $i$-th representation can be
described as follows:

$c^{s}_{ij}=\mu_{s}bs_{ij},$ (5)

where $\mu_{s}$ is a constant term for storage cost defined by the service
provider and $bs_{ij}$ is the estimated data size of the $j$-th tile in the
$i$-th representation. The data size for each $j$ tile is estimated using the
curve fitting technique similar to the one used for Eq. (1). Parameters for
the equation, shown in Table 1 (Data Size), were found using the curve fitting
operator.

###  3.4 Problem formulation

In order to obtain the cost-optimal encoding ladder $\mathcal{L}^{*}$ for a
given video, a set of representations for $\mathcal{L}^{*}$ is chosen from the
set of the estimated representation $\mathcal{L}$ that minimize both the total
spherical quality distortion of tiles and the total resource cost of the
cloud-based streaming system. For this purpose, we formulate the problem as an
optimization problem using the following practical constraints:

  1. (I)

Bandwidth: In the proposed system, we consider that the encoding ladder needs
to cover a set of given network bandwidth profiles
$\mathcal{P}=\\{p_{1},p_{2},\ldots,p_{|\mathcal{P}|}\\}$ with their minimum
$B^{min}$ and maximum $B^{max}$ bandwidth ranges.

  2. (II)

Computational and storage costs: We set limits for the encoding and storage
costs which are the maximum allowed computational cost $C^{max}$ and storage
cost $S^{max}$ of the streaming system.

  3. (III)

Encoding rate: The bitrate levels of the representations should be spaced
between each other by the minimum step size $\tau$.

Our objective is to provide a low-quality distortion encoding ladder for a
given tiled $v$ at minimum resource costs by considering the above described
constrains. Thus, we formulate the optimization problem as follows:

$\mathcal{L}^{*}:\arg\\!\min_{\mathcal{L}}\sum_{i\in\mathcal{L}}\sum_{p\in\mathcal{P}}\left(\gamma
c_{i}+(1-\gamma)d_{i}\right)a_{ip}$ (6)

with

$c_{i}=\sum_{j\in\mathcal{T}}(c^{e}_{ij}+c^{s}_{ij})\quad c_{i}\in\mathcal{P}$
(7)

and

$d_{i}=\sum_{j\in\mathcal{T}}d_{ij},$ (8)

where $c_{i}$ and $d_{i}$ are the total resource cost and quality distortion
for the $i$-th representation, respectively. In order to have a trade-off
between $c_{i}$ and $d_{i}$, we introduce a pre-defined constant
$\gamma\in[0,1]$ to be assigned by the service-provider. To cover a wide range
of network bandwidths, we introduce a set of network bandwidth profiles in the
problem definition. The decision variable $a_{ip}=\\{0,1\\}$ indicates if the
$i$-th bitrate level for the $p$-th profile of a set of network bandwidth
profiles $\mathcal{P}$ is included _or_ excluded in the encoding ladder for a
given $v$.

Equation (6) minimizes both the overall distortion of the tiled 360$\degree$
video and resource costs of the cloud-based streaming system and is subject to
the following constraints:

$\vspace{-0.8em}\begin{split}B^{min}_{p}\leq b_{i}a_{ip}\leq
B^{max}_{p}\quad\forall i\in\mathcal{L}\ \text{and}\ \forall
p\in\mathcal{P},\end{split}$ (9)
$\sum_{i\in\mathcal{L}}a_{ip}=\lfloor\frac{M\Lambda_{p}}{\sum_{p\in\mathcal{P}}\Lambda_{p}}\rfloor\quad\quad\forall
p\in\mathcal{P},$ (10) $\sum_{p\in\mathcal{P}}a_{ip}\leq 1\quad\quad\forall
i\in\mathcal{L},$ (11)
$\sum_{i\in\mathcal{L}}\sum_{p\in\mathcal{P}}s_{i}a_{ip}\leq S^{max},$ (12)
$\sum_{i\in\mathcal{L}}\sum_{p\in\mathcal{P}}c_{i}a_{ip}\leq C^{max},$ (13)
$\frac{b_{i}a_{ip}}{b^{*}_{n}}\geq\tau,\quad\forall i\in\mathcal{L},\ \forall
n\in\mathcal{L^{*}}\ \text{and}\ \forall p\in P.$ (14)

Equation (9) addresses Constraint (I) for each $p$. Equation (10) sets the
maximum number of representations in the encoding ladder for the $p$-th
profile based on its weighting factor $\Lambda$ and the total number of
representations $M$ in the encoding ladder. The weighting factor $\Lambda$ for
each network profile is shown in Table 2. The constraint of Equation (11)
avoids the selection of the same representation for each profile.
Additionally, Equations (12) and (13) satisfy Constraint (II) by ensuring that
encoded videos for estimated encoding ladders cannot exceed $S^{max}$ and
$C^{max}$. Equation (14) satisfies Constraint (III) by ensuring that the
target bitrate of each selected representation $n$ in the $\mathcal{L}^{*}$ is
spaced by a minimum step size $\tau$.

##  4 Experimental Results

In this section, we investigate the performance of the proposed encoding
ladder estimation method by comparison with the one-size-fits-all schemes
[appleRec, axinomRec, netflixblogPeerTitle] for the tiled 360$\degree$ video,
and evaluate the proposed method under several service provider’s constraints.

###  4.1 Setup

We use as the following six 8K$\times$4K resolution 360$\degree$ ERP video
test sequences: $\mathcal{V}$ = $\\{$Train, Stitched_left_Dancing360_8K,
Basketball, KiteFlite, ChairLift, SkateboardInLot$\\}$ [AdeelAbbasGoPro,
inteldigital, 2016rr]. Each $v\in\mathcal{V}$ was split into $N=10$ tiles
which was obtained as an optimal number in our previous research work in
[myIcip2017]. The encoded bitrate for each tile is equally distributed by
dividing the target bitrate to the $N$ tiles. Their encoding complexity
features and assigned content types are shown in Table 3, which was estimated
using the described method in the Section 3.1. Three content types in the set,
$\mathcal{O}=\\{o_{1},o_{2},o_{3}\\}$, were used to classify the videos using
the estimated complexity features. The Train, Basketball, and ChairLift
sequences were used to model the curve fitting function in Equation (1) and we
evaluate our method using the Stitched_left_Dancing360_8K, KiteFlite, and
SkateboardInLot video sequences. Further, three different resolutions
$\mathcal{G}=\\{3072\times 1536,4096\times 2048,8192\times 4096\\}$ in the
encoding ladders and four different bandwidth profiles $p$ were used as
defined in Table 2 with minimum $B^{min}$ and maximum $B^{max}$ bandwidth
ranges, and $\Lambda$ for each bandwidth profile.

Profiles: | $p_{1}$ | $p_{2}$ | $p_{3}$ | $p_{4}$
---|---|---|---|---
$B^{min}$ (Mbps) | 1 | 3 | 15 | 25
$B^{max}$ (Mbps) | 4 | 20 | 30 | 40
$\Lambda$ | 0.25 | 0.25 | 0.25 | 0.25
Table 2: Network bandwidth profiles.

We focus on the browser-based video streaming use-case which is one of the
core experiments in the ongoing standardization activity [Jtc1sc29wg2016wm].
Since AVC is the only implemented decoder in current available browsers which
can support HMDs, we apply the H.264/AVC standard in our experiments. In this
context, we encoded videos using the FFmpeg software (ver. N-85291) [x264]
with two-pass and 200 percent constrained variable bitrate encoding
configurations. At this stage, it is important to mention that our proposed
method is video codec agnostic; it can be easily utilized with different video
coding standards.

Sequence | $f_{spa}$ | $f_{tmp}$ | $\mathcal{O}$
---|---|---|---
Train | 0.977 | 0.065 | $o_{1}$
Stitched_left_Dancing360_8K | 0.884 | 0.110
Basketball | 0.843 | 0.090 | $o_{2}$
KiteFlite | 0.861 | 0.090
ChairLift | 0.789 | 0.212 | $o_{3}$
SkateboardInLot | 0.827 | 0.521
Table 3: Encoding complexity features and assigned content types for the used
test sequences.

To evaluate our proposed method, the objective quality metrics WS-MSE and WS-
PSNR [Jvet2016tw] were utilized to calculate the quality performance of the
360$\degree$ video. Further, three different one-size-fits-all encoding
ladders (i.e., Apple [appleRec], Axinom [axinomRec], and Netflix
[netflixblogPeerTitle]), which are recommended for traditional videos, were
used as references to investigate the quality performance of our proposed
method. Table 4 shows three reference one-size-fits-all encoding ladders for
their three ERP resolutions and four total target encoding rates. In the
table, resolutions and target encoding rate were calculated by summation of
each tile’s resolution and target encoding rate, respectively.

Apple [appleRec] |  Axinom [axinomRec] |  Netflix [netflixblogPeerTitle]
---|---|---
$Z$ (Mbps) | $W\times H$ |  $Z$ (Mbps) | $W\times H$ |  $Z$ (Mbps) | $W\times H$
45 |  8192 $\times$ 4096 | 45 |  8192 $\times$ 4096 | 43 |  8192 $\times$ 4096
30 |  8192 $\times$ 4096 | 30 |  8192 $\times$ 4096 | 30 |  4096 $\times$ 2048
20 |  4096 $\times$ 2048 | 21 |  4096 $\times$ 2048 | 23.5 |  4096 $\times$ 2048
11 |  3072 $\times$ 1536 | 12 |  3072 $\times$ 1536 | 17.5 |  3072 $\times$ 1536

Table 4: Recommended one-size-fits-all encoding ladders for traditional videos
by service providers.

###  4.2 Performance evaluation

Encoding ladders for our proposed method have been estimated by solving the
formulated ILP algorithm in Section 3.4 using Pyomo (ver. 5.0)
[hart2012pyomo]. We set $\mu_{e}$ and $\mu_{s}$ to $0.017$ and $0.023$,
respectively. These cost values are same as the real cost values in
[amazonPrice].

To derive the distortion function in Equation (1), we calculated the WS-MSE
versus bitrate (in Mbps) performance graphs in Fig. 3 for each resolution of
the videos Train, Basketball, and ChairLift. The results demonstrate the
various performances due to the high diversity in video content
characteristics. As can be seen in the figure, each content type has various
content dependencies for each encoding resolution and bitrate. For instance,
the Train sequence (content type $o_{1}$), which contains the lowest complex
encoding features, achieves a low distortion score compared to content types
$o_{2}$ and $o_{3}$. Because of such diversity, one-size-fits-all schemes,
which are used by almost all research works, cannot provide cost-optimal and
high-quality streaming performances for the tiled 360$\degree$ videos.

051015202530354045Bitrate [Mbps]5101520253035Average WS-
MSETrain8192x40964096x20483072x1536

051015202530354045Bitrate [Mbps]5101520253035Average WS-
MSEBasketball8192x40964096x20483072x1536

051015202530354045Bitrate [Mbps]5101520253035Average WS-
MSEChairLift8192x40964096x20483072x1536

Fig. 3: Average WS-MSE - bitrate curves for sample 8K$\times$4K ERP
360$\degree$ videos with different content type.

Evaluation I: To evaluate the RD performance gain of our encoding ladder
estimation solution, we compare our proposed method with three different
recommended one-size-fits-all schemes of the streaming service providers. As
these ladders were estimated without considering constraints, we set
$\gamma=0$ (in order to focus on distortion only) and exclude other
constraints in equations between (9) and (13) for a fair comparison in this
test.

1015202530354045Bitrate [Mbps]41.542.042.543.043.544.044.545.045.5Average WS-
PSNR [dB]Stitched_left_Dancing360_8KAppleAxinomNetflixProposed

1015202530354045Bitrate [Mbps]3536373839404142Average WS-PSNR
[dB]KiteFliteAppleAxinomNetflixProposed

1015202530354045Bitrate [Mbps]36.537.037.538.038.539.0Average WS-PSNR
[dB]SkateboardInLotAppleAxinomNetflixProposed

Fig. 4: Performance comparison using the RD curves computed with the average
WS-PSNR.

Figure 4 shows the RD curves computed with average WS-PSNR for the
Stitched_left_Dancing360_8K, KiteFlite, and SkateboardInLot sequences. The
results show that our proposed method considerably increases the objective
video quality (i.e., WS-PSNR) compared to the one-size-fits-all schemes at all
times. In particular, the proposed method demonstrates high bitrate savings
between 10-30 Mbps bandwidth ranges for the content types $o_{1}$ and $o_{2}$.
To this end, we notice that one-size-fits-all schemes provide high scores for
the content type $o_{3}$ compared to their scores for content types $o_{1}$
and $o_{2}$.

Evaluation II: We further analyze the performance gain of our method using the
Bjøntegaard metric [Bjontegaard] in Table 5. This metric describes the
distance between two RD curves. In this manner, the bitrate difference, i.e.
BD-rate, was calculated in percentage averaged over the entire range. A
negative BD-rate indicates a decrease of bitrate at the same quality. From the
table, we can notice that the proposed method provides considerable bitrate
savings compared to the recommended encoding ladders at the same bitrates.

Sequence $v$ | Streaming vendor
---|---
Apple | Axinom | Netflix
Stitched_left_Dancing360_8K | -5.557 | -5.885 | -69.253
KiteFlite | -13.876 | -14.436 | -69.178
SkateboardInLot | -1.673 | -1.701 | -1.155

Table 5: BD-rate saving (%) of the proposed method.

Evaluation III: Finally, in the last set of evaluations, we consider a
scenario where the constraints of $S^{max}$ and $C^{max}$ are 8000, $\tau$ =
1.2, and $M$ = 12. In this setup, we use the normalized difference of the
total cost $\Delta CS$ and the distortion $\Delta DS$ (in terms of WS-MSE) in
percentages for evaluation purpose. Table 6 shows the results of the proposed
encoding ladder estimation using resolution-bitrate pairs for $\gamma$ = 0,
$\gamma$ = 0.1, and $\gamma$ = 0.5.

width=1

Sequence $v$ | $\gamma$ | Representation $i$
---|---|---
1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12
Stitched_left_Dancing360_8K | $0.0$ | ($g_{1}$,1.47) | ($g_{1}$,1.78) | ($g_{1}$,2.15) | ($g_{1}$,3.8) | ($g_{1}$,4.6) | ($g_{1}$,5.6) | ($g_{2}$,10.84) | ($g_{2}$,13.11) | ($g_{2}$,15.87) | ($g_{2}$,28.11) | ($g_{3}$,34.01) | ($g_{3}$,41.15)
$0.1$ | ($g_{2}$,1.34) | ($g_{2}$,1.61) | ($g_{2}$,1.95) | ($g_{2}$,2.60) | ($g_{3}$,3.14) | ($g_{3}$,3.80) | ($g_{3}$,6.12) | ($g_{3}$,7.40) | ($g_{3}$,8.96) | ($g_{3}$,17.45) | ($g_{3}$,21.12) | ($g_{3}$,25.55)
$0.5$ | ($g_{2}$,1.00) | ($g_{2}$,1.21) | ($g_{2}$,1.47) | ($g_{2}$,2.36) | ($g_{3}$,2.86) | ($g_{3}$,3.46) | ($g_{3}$,6.12) | ($g_{3}$,7.40) | ($g_{3}$,8.96) | ($g_{3}$,17.45) | ($g_{3}$,21.12) | ($g_{3}$,25.55)
KiteFlite | $0.0$ | ($g_{1}$,1.47) | ($g_{1}$,1.78) | ($g_{2}$,2.15) | ($g_{2}$,3.80) | ($g_{2}$,4.60) | ($g_{3}$,5.56) | ($g_{3}$,10.84) | ($g_{3}$,13.11) | ($g_{3}$,15.87) | ($g_{3}$,28.11) | ($g_{3}$,34.01) | ($g_{3}$,41.15)
$0.1$ | ($g_{1}$,1.47) | ($g_{1}$,1.78) | ($g_{2}$,2.15) | ($g_{2}$,3.80) | ($g_{2}$,4.60) | ($g_{3}$,5.56) | ($g_{3}$,6.73) | ($g_{3}$,8.14) | ($g_{3}$,9.85) | ($g_{3}$,17.45) | ($g_{3}$,21.12) | ($g_{3}$,25.55)
$0.5$ | ($g_{1}$,1.00) | ($g_{1}$,1.21) | ($g_{1}$,1.47) | ($g_{2}$,2.36) | ($g_{2}$,2.86) | ($g_{2}$,3.46) | ($g_{3}$,6.12) | ($g_{3}$,7.40) | ($g_{3}$,8.96) | ($g_{3}$,17.45) | ($g_{3}$,21.12) | ($g_{3}$,25.55)
SkateboardInLot | $0.0$ | ($g_{1}$,1.47) | ($g_{1}$,1.78) | ($g_{1}$,2.15) | ($g_{1}$,3.80) | ($g_{1}$,4.60) | ($g_{1}$,5.56) | ($g_{2}$,10.84) | ($g_{2}$,13.11) | ($g_{2}$,15.87) | ($g_{2}$,28.11) | ($g_{3}$,34.01) | ($g_{3}$,41.15)
$0.1$ | ($g_{1}$,1.47) | ($g_{1}$,1.78) | ($g_{1}$,2.15) | ($g_{1}$,2.86) | ($g_{1}$,3.46) | ($g_{1}$,4.18) | ($g_{1}$,6.12) | ($g_{1}$,7.40) | ($g_{1}$,8.96) | ($g_{1}$,17.45) | ($g_{2}$,21.12) | ($g_{2}$,25.55)
$0.5$ | ($g_{1}$,1.21) | ($g_{1}$,1.47) | ($g_{1}$,1.78) | ($g_{1}$,2.36) | ($g_{1}$,2.86) | ($g_{1}$,3.46) | ($g_{1}$,6.12) | ($g_{1}$,7.40) | ($g_{1}$,8.96) | ($g_{2}$,17.45) | ($g_{2}$,21.12) | ($g_{2}$,25.55)

Table 6: Results of the proposed encoding ladder estimation for $\gamma$ = 0,
$\gamma$ = 0.1, and $\gamma$ = 0.5.

From the results, we observe that the lowest complex content, i.e., content
type $o_{1}$, increases its encoding resolution and decreases its target
encoding rate at the range between $i=2$ and $i=10$ to reduce the total cost
by considering cost and distortion tradeoffs using $\gamma$ = 0.1 and $\gamma$
= 0.5. On the other hand, we observe that the most complex content, i.e.
content type $o_{3}$, decreases both its encoding resolution and target
encoding rate in order to reduce the total cost by considering cost and
distortion tradeoffs using the $\gamma$ = 0.1 and $\gamma$ = 0.5. Table 7
reports the total cost saving and distortion gain with respect to different
$\gamma$. Finally, we would like to mention that, the GNU linear programming
kit (GLPK) for Pyomo was able to solve the formulated ILP algorithm in Section
3.4 using the calculated data in less than one minute on Intel(R) Core(TM)
i7-6700 CPU @ 3.40GHz with 32 GB of RAM.

Sequence $v$ |  $\Delta$cost (%) |  $\Delta$distortion (%)
---|---|---
$\gamma=0.1$ | $\gamma=0.5$ | $\gamma=0.1$ | $\gamma=0.5$
Stitched_left_Dancing360_8K | 37.463 | 39.683 | -13.628 | -42.914
KiteFlite | 33.165 | 39.206 | -9.564 | -25.326
SkateboardInLot | 37.214 | 38.884 | -8.977 | -15.26

Table 7: Total cost saving and distortion gain with respect to $\gamma$=0.0.

##  5 Conclusions

This paper introduced a novel encoding ladder estimation method for tiled 360∘
video streaming systems, considering both the provider’s and client’s
perspectives. To this end, the objective of our proposed method was to provide
cost-optimal and enhanced video streaming experiences for VR end-users. The
developed system included classification of the content type, distortion
modeling, cost modeling, and problem formulation. The performance of our
proposed method was verified in experimental evaluations. The results showed
that our method achieved significant bitrate savings (especially for the
content types $o_{1}$ and $o_{2}$) compared to the one-size-fits-all encoding
ladders which are recommended by streaming service providers. Furthermore, the
developed method can automatically find cost-optimal encoding ladders using
several practical constraints, and provides efficient streaming service for
tiled 360∘ video. As future work, we plan to extend our optimization framework
by considering the number of tiles for a given content type and investigating
the effect of total costs by evaluating the effects of the various constraint
parameters using a larger set of video sequences.

## References