successfully nucleates, i.e., at $T=T_{n}$. The arrows in blue color represent
the direction of the FOPT at $T=T_{n}$ in the field space. The small circle in
yellow color represents that an FOPT happens in the scalar potential at that
temperature but its field values do not change.

BM No. | $T_{n}$ (GeV) | $\alpha$ | $\beta/H_{n}$
---|---|---|---
BP3 | 163.1 | $2.3\times 10^{-4}$ | $4.8\times 10^{6}$
116.3 | $0.027$ | 312.7
Table 8: Values of the parameters $T_{n}$, $\alpha$ and $\beta/H_{n}$ (that
control the GW intensity) for the benchmark point BP3, shown in table 6.

Note that, first, a broken phase $\\{0,0,14.8\\}$ appears only along the
$h_{s}-$field direction at $T_{c}=163.2$ GeV with a possibility of an FOPT.
The successful nucleation occurs at $T_{n}=163.1$ GeV. It has been shown in
the right plot of figure 14. The small difference between $T_{c}$ and $T_{n}$
makes the two arrows overlap each other. In the left plot of figure 14, a
yellow circle has been shown at $T=163.1$ GeV as during this FOPT the
$h_{d}-$field does not acquire any non-zero ${\it vev}$, i.e., EW symmetry
remains unbroken. Thus, the first FOPT occurs from
$\\{h_{d}=0,h_{t}=0,h_{s}=0\\}\rightarrow\\{h_{d}=0,h_{t}=0,h_{s}\neq 0\\}$
phase direction. Subsequently, later stage, another minimum form (phase-III),
designated by the notation $\\{h_{d}\neq 0,h_{t}=0,h_{s}=0\\}$. At $T=145.2$
GeV phase-II ($\\{0,0,63.6\\}$) and phase-III ($\\{78.9,0,0\\}$) become
degenerate and for $T<145.2$ GeV phase-III becomes the global minimum of the
scalar potential. The corresponding nucleation occurs at $T=116.3$ GeV. During
this transition both $h_{d}-$ and $h_{s}-$ fields undergo a change, with the
former acquiring a non-zero ${\it vev}$ from zero and the latter developing a
zero ${\it vev}$ from a non-zero ${\it vev}$ value. Note that, at $T=0$,
$h_{d}$ field in the phase-III evolves to the value of 246 GeV and $h_{s}$
remains at zero in this phase. During these phase transitions, the triplet
field always remains very small. Therefore, the system finally evolves to the
correct EW minimum. Note that in this benchmark scenario, the minimum phase-II
is still present at T= 0 where $<h_{s}>\sim 100$ GeV and all other fields
${\it vevs}$ are almost zero. But, the global potential minimum of the
potential is the phase-III, not the phase-II.

The values of the key parameters ($\alpha$, $\beta$), which are related to the
computation of GW spectra generated from the FOPTs for the benchmark scenarios
BP3, are presented in table 8. The GW frequency spectrum of BP3 is presented
in figure 15 using equations 42 to 49. Note that the first FOPT along the
$h_{s}-$field direction happens very quickly (much larger $\beta/H_{n}$ value)
and also the phase transition is not strong enough which corresponds to a much
lower $\alpha$ value. Thus, the contributions of the first FOPT in the
$h_{s}-$field direction to the total GW spectrum are very small compared to
the ones coming from the second FOPT which happens along the $h_{d}$ and
$h_{s}$ field directions simultaneously. Therefore, a spectral peak due to the
first phase transition in this two-step phase transition does not appear in
the plot.

Figure 15: GW energy density spectrum with respect to frequency for the
benchmark point BP3 illustrated against the experimental sensitivity curves of
the future proposed GW detectors such as LISA, SKA, TianQin, Taiji, aLigo+,
BBO, NANOGrav, and U-DECIGO. The black line denotes the total GW energy
density whereas the red (green) broken line represents the contributions from
sound waves (turbulence). Note that in this benchmark scenario the
contribution from the first phase transition along the $h_{s}-$field direction
does not appear in these plots as its contribution is too small.

Similar to figure 11, here also each phase transition process, the
contributions from sound waves and turbulence are shown in different colors
with broken lines. The GW peak amplitude and the peak frequency are mostly
dominated by sound contributions. This GW spectrum is further compared with
the projected sensitivity of some ground- and space-based GW detectors, viz.,
LISA, Taiji, TianQin, aLigo+, SKA, NANOGrav, Big Bang Observer (BBO) and
Ultimate(U)-DECIGO in Fig. 15. The GW intensity obtained for BP3 lies within
the sensitivities of ALIA, BBO and UDECIGO. As expected, the SNR for LISA is
very low for detection. Although the experiments like ALIA, BBO and UDECIGO
can detect such a GW signal.

Therefore, the FOPT along the singlet-direction permissible parameter space we
address in this section is either excluded from the most recent limits from
the various DMDD experiments or will be investigated in the near future. The
parameter space that is still allowed corresponds to a significantly
underabundant DM. It is projected that the HL-LHC will be unable to limit a
substantial portion of the parameter space. Focusing on this model, the
observation of any signal at future DM and GW experiments but nothing at the
HL-LHC would indicate that DM is severely underabundant.

##  6 Summary and conclusion

In spite of many successes of the SM, it cannot explain the smallness of
neutrino masses and the observed baryon asymmetry of the Universe and does not
have a DM candidate. In this article, we examine the $Z_{3}-$symmetric singlet
scalar extended type-II seesaw model as a potential contender for
simultaneously providing solutions to these three deficiencies of the SM. The
type-II seesaw mechanism generates small neutrino masses from the ${\it vev}$
of the scalar triplet. The complex scalar field, $S$, of the model can be a
stable WIMP-like DM particle under the so-called freeze-out mechanism. In
addition, new interactions among the SM Higgs doublet, the complex triplet,
and the complex singlet can induce an FOEWPT in the early Universe, which is
necessary for the solution of the BAU problem via the EWBG mechanism. Such an
FOEWPT can generate stochastic GW that can be detected in future proposed GW
detectors. In this work, we investigate the correlation among the production
of GW, DM phenomenology, and BSM LHC searches within this model.

In this scenario, the scalar DM can annihilate mainly in three ways to freeze
out in the early Universe and those processes are given by: DM
annihilation$\hskip 8.5359pt(a)~{}S~{}S^{*}~{}\to{\rm SM~{}~{}SM}$, (b)
$S~{}S^{*}~{}\to{\rm X~{}~{}Y};~{}~{}~{}\\{\rm
X,Y\\}=\\{H^{0},A^{0},H^{\pm},H^{\pm\pm}\\}$ and DM semi-annihilation (c)
$~{}S~{}S~{}\to S~{}U;~{}~{}\\{U\\}=\\{h^{0},H^{0}\\}$. The coupling of the
scalar DM with the SM Higgs doublet, $\lambda_{SH}(S^{*}S)(H^{\dagger}H)$, and
the triplet, $\lambda_{S\Delta}(S^{*}S){\rm Tr}[\Delta^{\dagger}\Delta]$,
control the DM-annihilation processes (a) and (b), respectively. In contrast,
the DM semi-annihilation processes are initiated via the cubic term
$\mu_{3}(S^{3}+{S^{*}}^{3})$ in the scalar potential. Such semi-annihilation
processes exist because of the $Z_{3}-$discrete symmetry of the model that
settles the singlet as a DM particle. At a fixed value of $m_{s}$, the DMDDSI
cross-section mostly depends on the $\lambda_{SH}$ portal coupling, which has
an upper bound coming from the latest DMDD experiments. When
$m_{s}<m_{h^{0}}$, the same portal coupling $\lambda_{SH}$ predominantly
controls the DM relic density. Such dependencies on $\lambda_{SH}$
significantly restrict the DM parameter space for $m_{s}<m_{h^{0}}$ and only
the resonance region around $m_{h^{0}}$ is still allowed after complying with
the DM relic density and the latest DMDD constraints. The cubic interaction
based on $\mu_{3}$ expands the parameter space for $m_{s}\gtrsim m_{h^{0}}$.
In addition to this, the interaction between the DM and the triplet sector
through the portal coupling $\lambda_{S\Delta}$ opens up a wide range of DM
allowed region of parameter space if $m_{s}\gtrsim
m_{H^{0},A^{0},H^{\pm},H^{\pm\pm}}$. Therefore, the allowed region of DM
parameter space depends on the triplet-like scalar masses, which are
constrained by the most recent direct searches for heavy neutral and charged
scalars at the LHC. Such phenomena show some correlations between DM
phenomenology and the physics at LHC.

It has been pointed out in the literature that the direct searches at the LHC
will be unable to restrict $H^{\pm\pm}$ mass even at 200 GeV with 3 ab-1 data
for moderate $v_{t}$ and reasonably large $\Delta m$ ($>10$ GeV) values, which
corresponds to $\lambda_{4}<0$. So, we focus on this part of the parameter
space in order to investigate the feasibility of FOPT in the scalar potential
at finite temperatures. We find that the strength of an SFOEWPT along the
$h_{d}-$field direction increases with larger $(\lambda_{1}+\lambda_{4})$ and
smaller $m_{H^{\pm\pm}}$ values. For $\lambda_{1}\lesssim 4.0$,
$m_{H^{\pm\pm}}$ greater than 420 GeV is disfavored since the triplet becomes
decoupled from the SM Higgs potential, rendering an SFOEWPT impossible.

Furthermore, we have thoroughly studied the stochastic GW spectra that will
carry the imprints of FOPT from new physics beyond the SM. Relatively light
triplet-like scalars and large effective quartic couplings
$(\lambda_{1}+\lambda_{4})$ can generate GW with large enough intensity to
fall within the projected sensitivity of some ground- and space-based GW
detectors. However, we also show that the decay rate of
$h^{0}\rightarrow\gamma\gamma$ is significantly affected by the SFOEWPT-
favored region of the parameter space, deviating significantly from the SM-
predicted value for large effective quartic couplings and light triplet-like
scalars. The correlation between the produced GW spectra and the signal
strength $\mu_{\gamma\gamma}$ of that decay channel has been pointed out. LHC
precision study of SM Higgs di-photon decay channel has already excluded most
of the parameter space that can generate stochastic GW signal. Any further
narrowing of $\mu_{\gamma\gamma}$ error bars at the HL-LHC will completely
exclude the possibility of FOEWPT induced by large interactions between the SM
Higgs doublet and $SU(2)$ triplet.

The interaction between the complex scalar and the SM Higgs doublet can be
another source of FOPT in this model. The singlet scalar is employed in this
context to serve dual roles as a DM particle and to offer a favorable
condition for the electroweak phase transition to be strongly first-order.
Such modification in the scalar potential can also generate the possibility of
a two-step FOPT, with the first transition happening in the singlet direction
and the second in the $h_{d}-$field direction. The scenario of an FOPT along
the singlet direction prefers relatively large $\lambda_{SH}$. By imposing
such a criteria for the FOPT, we have shown that a significant amount of the
parameter space is already excluded from the latest constraints on the DMDDSI
cross-section from various DM experiments. Our analysis further shows that
this model can generate an FOPT in the $h_{s}-$field direction and escape the
DMDD constraints when the complex scalar DM is significantly underabundant.
Such a low relic density allows the effective $\sigma_{\rm DD}^{\rm SI}$ rates
to stay below the exclusion limit owing to a considerable scaling down. Such a
situation comes into the picture only when for $m_{s}\gtrsim h^{0}$, where the
$\mu_{3}$ dependent DM semi-annihilation process can significantly contribute
to a drop in the relic density. Scaling down is also conceivable for
$m_{s}\gtrsim m_{H^{0},A^{0},H^{\pm},H^{\pm\pm}}$ by increasing
$\lambda_{S\Delta}$. On the other hand, significantly large values of $m_{s}$
are disfavoured for an FOPT. Thus, the $\mu_{3}$ term in the potential plays a
unique role since it still permits a specific region of parameter space that
can comply with DMDDSI rates and also because it can generate a barrier in the
tree-level potential in favour of FOPT along the $h_{s}-$field direction.

However, a more interesting case of a two-step phase transition from the
singlet-doublet interaction is when the phase transition in the $h_{d}-$field
direction is strongly first-order. This is what we need from the EWBG
perspective. We explore this prospect by analysing a representative benchmark
point where $\mu_{\gamma\gamma}$ remains very close to 1 since the various
triplet sectors couplings are fixed at tiny values. We find that such an FOPT
demands relatively large $\lambda_{SH}$ and as a consequence, future DMDD
experiments will investigate the feasibility of generating GW as a result of
an FOPT in this scenario. Thus, focusing on this model, observing any signal
at future DMDD and GW experiments but nothing at the HL-LHC will indicate that
DM is severely underabundant. The absence of new physics signal at the HL-LHC
and various DMDD experiments in future would severely limit the prospects of
detecting GW at future GW experiments.

## Acknowledgments

PG would like to acknowledge Indian Association for the Cultivation of
Science, Kolkata for the financial support. The work of TG and SR is supported
by the funding available from the Department of Atomic Energy (DAE),
Government of India for Harish-Chandra Research Institute (HRI). SR is also
supported by the Infosys award for excellence in research. SR acknowledges the
High-Performance Scientific Computing facility at the Regional Centre for
Accelerator-based Particle Physics (RECAPP) and HRI.

## Appendix

##  Appendix A Tree-level tadpole relations

The mass parameters $\mu_{H}^{2}$, $\mu_{\Delta}^{2}$ and $\mu$ are determined
via the minimization conditions (the tadpoles) of $V_{0}$ of equation 2.3.1
and are given by

$\displaystyle\mu_{H}^{2}$
$\displaystyle=\tfrac{1}{2}(\lambda_{1}+\lambda_{4})v_{t}^{2}+\lambda_{H}v_{d}^{2}-\sqrt{2}\mu
v_{t},$ (60) $\displaystyle\mu_{\Delta}^{2}$
$\displaystyle=-(\lambda_{2}+\lambda_{3})v_{t}^{2}-\tfrac{1}{2}(\lambda_{1}+\lambda_{4})v_{d}^{2}+\frac{\mu
v_{d}^{2}}{\sqrt{2}v_{t}}.$ $\displaystyle\mu$
$\displaystyle=\frac{v_{t}}{\sqrt{2}v_{d}}\frac{-(\lambda_{2}+\lambda_{3})\sin
2\theta_{t}v_{t}^{2}-(\lambda_{1}+\lambda_{4})v_{d}v_{t}\cos
2\theta_{t}+\lambda_{H}v_{d}^{2}\sin
2\theta_{t}}{v_{d}\sin\theta_{t}\cos\theta_{t}-2v_{t}\cos^{2}\theta_{t}+2v_{t}\sin^{2}\theta_{t}}.$

##  Appendix B Set of UV-finite counterterm coefficients

The various coefficients of the counterterm potential, defined in equation 31
are given by,

$\displaystyle\delta_{\lambda_{S\Delta}}=-\frac{1}{4v_{d}v_{t}}DV_{12}\,\,,\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad$
(61a)
$\displaystyle\delta_{\lambda_{H}}=-\frac{1}{8v_{d}^{3}}DV_{1}-\frac{1}{8v_{d}^{2}}DV_{11}\,\,,\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad$
(62a)
$\displaystyle\delta\mu_{H}^{2}=-\frac{3}{4v_{d}}DV_{1}+\frac{1}{4}DV_{11}+\frac{v_{t}}{4v_{d}}DV_{12}\,\,,\quad\quad\quad\quad\quad\quad$
(63a)
$\displaystyle\delta\mu_{\Delta}^{2}=-\frac{3}{4v_{t}}DV_{2}+\frac{1}{4}DV_{22}+\frac{v_{d}}{4v_{t}}DV_{12}\,\,,\quad\quad\quad\quad\quad\quad$
(64a)
$\displaystyle\delta_{\lambda_{2}}=\frac{1}{8v_{t}^{3}}DV_{2}-\frac{1}{8v_{t}^{2}}DV_{22}\,\,,\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad$
(65a)
$\displaystyle\delta_{\lambda_{SH}}=-\frac{1}{2v_{t}^{2}}DV_{33}\,\,.\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad$
(66a)

$DV_{i}=\frac{\partial V_{\text{CW}}}{\partial\phi_{{}_{i}}}$ and
$DV_{ij}=\frac{\partial^{2}V_{\text{CW}}}{\partial\phi_{{}_{i}}\partial\phi_{{}_{j}}}$
where $\phi_{{}_{i}},\phi_{{}_{j}}=\\{h_{d},h_{t},h_{s}\\}$. All the
derivatives are taken at the true EW minima, i.e., $h_{d}=v_{d}$,
$h_{t}=v_{t}$ and $h_{s}=0.$

##  Appendix C Daisy resummation improved thermal field-dependent masses

From the high-temperature expansion of the thermal one-loop potential $V_{T}$
(of equation 33 for bosons), the daisy coefficients can be found from the
following relation:

$\small{c_{ij}=\left.\frac{1}{T^{2}}\frac{\partial^{2}V_{T}}{\partial\phi_{i}\partial\phi_{j}}\right|_{T^{2}\gg
m^{2}}.}$ (67)

The eigenvalues of the thermally improved mass-squared matrices for the Higgs
and the gauge bosons, i.e., $M^{2}(h_{d},h_{t},h_{s},T)=$
eigenvalues$[m^{2}(h_{d},h_{t},h_{s})+\Pi(T^{2})]$ where
$\Pi(T^{2})=c_{ij}T^{2}$ and $c_{ij}$’s are the above described daisy
coefficients. The daisy resummation is particularly significant for an FOPT
because it affects the all-important cubic term of the potential at a finite
temperature. Only the longitudinal modes of the vectors and scalars get
thermal mass corrections. Due to gauge symmetry, thermal effects to the
transverse modes are suppressed Espinosa:1992kf . With these in mind, the
various daisy coefficients are as follows Comelli:1996vm ; Basler:2018cwe ;
Carrington:1991hz :

$\displaystyle c_{H}$
$\displaystyle=\frac{1}{16}({g_{1}}^{2}+3g_{2}^{2})+\frac{1}{4}y_{t}^{2}+\frac{1}{24}(6\lambda_{1}+3\lambda_{4}+12\lambda_{H}+12\lambda_{SH}),$
(68a) $\displaystyle c_{\Delta}$
$\displaystyle=\frac{1}{4}({g_{1}}^{2}+2g_{2}^{2})+\frac{1}{12}(2\lambda_{1}+8\lambda_{2}+6\lambda_{3}+\lambda_{4}+\lambda_{S\Delta}),$
(68b) $\displaystyle c_{S}$
$\displaystyle=\tfrac{1}{12}4\lambda_{S}+2\lambda_{SH}+3\lambda_{S\Delta}.$
(68c)

The field-dependent thermally mass-improved mass-squared matrices for the
$2\times 2$, symmetric matrix (${\cal M}_{H}^{2}$) for the $CP$-even scalars,
in the basis $\\{h_{d},h_{t}\\}$, is given by

$\displaystyle
M_{{H}_{11}}^{2}=\tfrac{1}{2}(\lambda_{1}+\lambda_{4})h_{t}^{2}+3\lambda_{H}h_{d}^{2}+\tfrac{1}{2}\lambda_{SH}h_{s}^{2}-\sqrt{2}\mu
h_{t}-\mu_{SH}^{2}+c_{H}T^{2},$ (69a) $\displaystyle
M_{{H}_{22}}^{2}=\tfrac{1}{2}(\lambda_{1}+\lambda_{4})h_{d}^{2}+3(\lambda_{2}+\lambda_{3})h_{t}^{2}+\tfrac{1}{2}\lambda_{ST}h_{s}^{2}+\mu_{\Delta}^{2}+c_{\Delta}T^{2},\quad\quad\quad$
(69b) $\displaystyle
M_{{H}_{12}}^{2}=M_{{H}_{21}}^{2}=(\lambda_{1}+\lambda_{4})h_{d}h_{t}-\sqrt{2}\mu
h_{d}.\quad\quad\quad\quad\quad\quad\quad\quad$ (69c)

The field-dependent thermally mass-improved mass-squared matrices for the
$2\times 2$, symmetric matrix (${\cal M}_{H}^{2}$) for the $CP$-odd scalars,
in the basis $\\{a_{d},a_{t}\\}$, is given by

$\displaystyle
M_{{A}_{11}}^{2}=\tfrac{1}{2}(\lambda_{1}+\lambda_{4})h_{t}^{2}+\lambda_{H}h_{d}^{2}+\tfrac{1}{2}\lambda_{SH}h_{s}^{2}+c_{H}T^{2}+\sqrt{2}\mu
h_{t}-\mu_{SH}^{2},$ (70a) $\displaystyle
M_{{A}_{22}}^{2}=\tfrac{1}{2}(\lambda_{1}+\lambda_{4})h_{d}^{2}+(\lambda_{2}+\lambda_{3})h_{t}^{2}+\tfrac{1}{2}\lambda_{ST}h_{s}^{2}+\mu_{\Delta}^{2}+c_{\Delta}T^{2},\quad\quad\quad$
(70b) $\displaystyle M_{{A}_{12}}^{2}=M_{{H}_{21}}^{2}=-\sqrt{2}\mu
h_{d}.\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad$
(70c)

The field-dependent thermally mass-improved mass-squared matrices for the
$2\times 2$, diagonal matrix (${\cal M}_{H}^{2}$) for thedark sector, in the
basis $\\{h_{s},a_{s}\\}$, is given by

$\displaystyle
M_{{DM}_{11}}^{2}=3\lambda_{S}h_{s}^{2}+\tfrac{1}{2}\lambda_{SH}h_{d}^{2}+\tfrac{1}{2}\lambda_{S\Delta}h_{t}^{2}+\frac{1}{\sqrt{2}}\mu_{3}h_{s}+\mu_{S}^{2}+c_{S}T^{2},\quad\quad\quad$
(71a) $\displaystyle
M_{{DM}_{22}}^{2}=\lambda_{S}h_{s}^{2}+\tfrac{1}{2}\lambda_{SH}h_{d}^{2}+\tfrac{1}{2}\lambda_{S\Delta}h_{t}^{2}-\frac{1}{\sqrt{2}}\mu_{3}h_{s}+\mu_{S}^{2}+c_{S}T^{2},\quad\quad\quad$
(71b) $\displaystyle
M_{{DM}_{12}}^{2}=M_{{H}_{21}}^{2}=0.\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad$
(71c)

The field-dependent thermally mass-improved mass-squared matrices for the
$2\times 2$, symmetric matrix (${\cal M}_{H{{}^{\pm}}}^{2}$) for the $CP$-even
singly charged Higgs bosons, in the basis $\\{h_{d},h_{t}\\}$, is given by

$\displaystyle
M_{{H{{}^{\pm}}}_{11}}^{2}=\tfrac{1}{2}\lambda_{1}h_{t}^{2}+\lambda_{H}h_{d}^{2}+\tfrac{1}{2}\lambda_{SH}h_{s}^{2}-\mu_{H}^{2}+c_{H}T^{2},\quad\quad\quad\quad\quad\quad\quad$
(72a) $\displaystyle
M_{{H{{}^{\pm}}}_{22}}^{2}=\tfrac{1}{2}\lambda_{1}h_{d}^{2}+(\lambda_{2}+\lambda_{3})h_{t}^{2}+\frac{1}{4}\lambda_{4}h_{d}^{2}+\frac{1}{2}\lambda_{S\Delta}h_{s}^{2}+\mu_{\Delta}^{2}+c_{\Delta}T^{2},\quad$
(72b) $\displaystyle
M_{{H{{}^{\pm}}}_{12}}^{2}=M_{{H}_{21}}^{2}=\frac{1}{2\sqrt{2}}\lambda_{4}h_{d}h_{t}-\mu
h_{d}.\quad\quad\quad\quad\quad\quad\quad\quad\quad$ (72c)

The thermally mass-improved field-dependent mass-squared doubly-charged Higgs
mass is given by

$M_{{\Delta{{}^{\pm\pm}}}}^{2}=\tfrac{1}{2}\lambda_{1}h_{d}^{2}+\lambda_{2}h_{t}^{2}+\tfrac{1}{2}\lambda_{S\Delta}h_{s}^{2}+\mu_{\Delta}^{2}+c_{\Delta}T^{2}\,,\quad\quad\quad\quad$
(73)

In the fermionic sector, we only consider the top quark degrees of freedom and
its field-dependent mass is given by

$\begin{gathered}m_{t}=\tfrac{1}{\sqrt{2}}y_{t}h_{u}\,\quad.\end{gathered}$
(74)

The field-dependent Daisy resummation improved thermal mass squared of the
charged gauge boson (only the longitudinal modes get thermal mass corrections)
field $W^{\pm}$ and the neutral electroweak gauge boson fields $W^{3}$ of
$SU(2)$ and the $B$ of $U(1)$ group are given by,

$\small{\begin{gathered}m_{W^{{}^{\pm}}_{{}_{T}}}^{2}=\tfrac{1}{4}g_{2}^{2}\left(h_{d}^{2}+2h_{t}^{2}\right),\quad\\\
m_{W^{{}^{\pm}}_{{}_{L}}}^{2}=\tfrac{1}{4}g_{2}^{2}\left(h_{d}^{2}+2h_{t}^{2}+\frac{5}{2}g_{2}^{2}T^{2}\right),\\\
m_{W^{{}^{3}}_{{}_{T}}}^{2}=\tfrac{1}{4}g_{2}^{2}\left(h_{d}^{2}+4h_{t}^{2}\right),\\\
m_{B_{{}_{T}}}^{2}=\tfrac{1}{4}g_{1}^{2}\left(h_{d}^{2}+4h_{t}^{2}\right),\\\
m_{{W^{{}^{3}}_{{}_{T}}}{B_{{}_{T}}}}^{2}=-\tfrac{1}{4}g_{1}g_{2}\left(h_{d}^{2}+4h_{t}^{2}\right)\\\
m_{W^{{}^{3}}_{{}_{L}}}^{2}=m_{W^{{}^{3}}_{{}_{T}}}^{2}+\frac{5}{2}g_{2}^{2}T^{2},\\\
m_{B_{{}_{L}}}^{2}=m_{B_{{}_{T}}}^{2}+\frac{17}{6}g_{1}^{2}T^{2},\\\
m_{{W^{{}^{3}}_{{}_{L}}}{B_{{}_{L}}}}^{2}=m_{{W^{{}^{3}}_{{}_{T}}}{B_{{}_{T}}}}^{2}\end{gathered}}$
(75)

The physical masses of the gauge bosons can be found after diagonalizing the
above matrices. Note that the photon field also acquires non-zero mass at
finite temperatures. The zero-temperature field-dependent mass relations can
be found by putting $T=0$ in the above mass relations.

##  Appendix D Feynman diagrams for DM annihilation

$S$$S^{*}$$h^{0}$$h^{0}$$S$$S^{*}$$S$$h^{0}$$h^{0}$$S$$S^{*}$$h^{0},H^{0}$$f/W^{+}/Z/h^{0}$$\overline{f}/W^{-}/Z/h^{0}$
Figure 16: Feynman diagrams for DM annihilate to the SM particles:
$S~{}S^{*}\to$ SM SM .
$S$$S^{*}$$H^{++}/H^{+}/A^{0}/H^{0}$$H^{--}/H^{-}/A^{0}/H^{0}$$S$$S^{*}$$S$$H^{0}$$H^{0}$$S^{*}$$S$$h^{0},H^{0}$$W^{+}/Z/H^{++}/H^{+}/A^{0}/H^{0}$$H^{-}/A^{0}/H^{--}/H^{-}/A^{0}/H^{0}$
Figure 17: Feynman diagrams for DM annihilate to scalar
triplet(s):$S~{}S^{*}\to W^{+}H^{-},ZA^{0},XY$. Here
$\\{X,Y\\}=\\{H^{\pm\pm},H^{\pm},A^{0},H^{0}\\}$.
$S$$S$$S$$S$$h^{0},H^{0}$$S$$S$$S$$S$$h^{0},H^{0}$ Figure 18: Feynman diagrams
for DM semi-annihilation processes: $S~{}S\to S~{}h^{0},S~{}H^{0}$.

##  Appendix E Theoretical constraints

In this section, we highlight some of the key theoretical constraints on our
model parameter space.

###  E.1 Vacuum stability

The scalar potential of the model requires to have a stable vacuum. In other
words, it means that at large field values the potential is bounded from below
so that the scalar fields do not run away. At large field values, the
quadratic and cubic terms in the scalar potential can be neglected in
comparison to the quartic term to determine the stability conditions of the
scalar potential. Using the copositivity conditions of vacuum stability, as
outlined in references Kannike:2012pe ; Yang:2021buu , we obtain the following
relations for our model:

$\displaystyle\lambda_{H}\geq 0,~{}~{}\lambda_{S}\geq 0,$
$\displaystyle\lambda_{2}+\lambda_{3}\geq
0,~{}~{}\lambda_{2}+\frac{\lambda_{3}}{2}\geq 0,$
$\displaystyle\lambda_{1}+2\sqrt{\lambda_{H}(\lambda_{2}+\lambda_{3})}\geq
0,~{}~{}(\lambda_{1}+\lambda_{4})+2\sqrt{\lambda_{H}(\lambda_{2}+\lambda_{3})}\geq
0,$
$\displaystyle(2\lambda_{1}+\lambda_{4})+4\sqrt{\lambda_{H}(\lambda_{2}+\frac{\lambda_{3}}{2})}\geq
0,~{}~{}~{}(\lambda_{2}+\lambda_{3})+\sqrt{(\lambda_{2}+\lambda_{3})(\lambda_{2}+\frac{\lambda_{3}}{2})}\geq
0,$ $\displaystyle\lambda_{SH}+2\sqrt{\lambda_{S}\lambda_{H}}\geq
0,~{}~{}\lambda_{S\Delta}+2\sqrt{\lambda_{S}(\lambda_{2}+\lambda_{3})}\geq 0,$
$\displaystyle\lambda_{S\Delta}+2\sqrt{\lambda_{S}(\lambda_{2}+\frac{\lambda_{3}}{2})}\geq
0~{}.$ (76)

It is also important to note that a stable global minimum sets an upper bound
on the dark scalar cubic coupling, $\mu_{3}$ as $\mu_{3}/m_{S}\lesssim
2\sqrt{\lambda_{S}}$ Belanger:2012zr ; Hektor:2019ote . For $\lambda_{S}=1$,
this relation becomes $\mu_{3}\lesssim 2m_{S}$.

###  E.2 Perturbativity

In order to ensure the validity of standard perturbation theory where the one-
loop level correction to coupling should be smaller than the tree-level
couplings. The quartic and the Yukawa couplings of the interaction Lagrangian
should obey the following upper bounds Lerner:2009xg ; Yang:2021buu  as:

$\displaystyle|\lambda_{H}|\lesssim\frac{2\pi}{3},$
$\displaystyle|\lambda_{1}|\lesssim 4\pi,~{}|\lambda_{2}|\lesssim
2\pi,~{}|\lambda_{3}|\lesssim 2\sqrt{2}\pi,~{}|\lambda_{4}|\lesssim 4\pi,$
$\displaystyle|\lambda_{1}+\lambda_{4}|\lesssim
4\pi,~{}|\lambda_{2}+\lambda_{3}|\lesssim\frac{2\pi}{3},~{}|\lambda_{1}+\frac{\lambda_{4}}{2}|\lesssim
4\pi,~{}|\lambda_{2}+\frac{\lambda_{3}}{2}|\lesssim\pi,$
$\displaystyle|\lambda_{S}|\lesssim\pi,~{}|\lambda_{SH}|\lesssim
4\pi,~{}|\lambda_{S\Delta}|\lesssim 4\pi$ $\displaystyle{\rm
and}~{}~{}~{}|y_{L}|\lesssim\sqrt{4\pi}~{}.$ (77)

###  E.3 Partial wave unitarity

The quartic couplings of the scalar potential also be constrained from tree
level unitarity of the theory, coming from all possible $2\to 2$ scattering
amplitudes which will form the $S$ matrix Lee:1977eg ; Arhrib:2011uy ;
Yang:2021buu . The eigenvalues of the $S$ matrix are bounded from above as:

$\displaystyle|\lambda_{H}|\leq 4\pi,~{}~{}|\lambda_{1}|\leq
8\pi,~{}~{}|\lambda_{2}|\leq 4\pi,$ $\displaystyle|\lambda_{SH}|\leq
8\pi,~{}~{}|\lambda_{S\Delta}|\leq 8\pi,$
$\displaystyle|\lambda_{2}+\lambda_{3}|\leq
4\pi,~{}~{}|\lambda_{1}+\lambda_{4}|\leq 8\pi,$
$\displaystyle|\lambda_{1}-\frac{\lambda_{4}}{2}|\leq
8\pi,~{}~{}|\lambda_{1}+\frac{3\lambda_{4}}{2}|\leq 8\pi,$
$\displaystyle|x_{1,2,3}|\leq 8\pi,$ (78)

where $x_{1,2,3}$ are the roots of the following polynomial equation:

$\displaystyle
24\lambda_{1}^{2}\lambda_{S}+24\lambda_{1}\lambda_{4}\lambda_{S}-12\lambda_{1}\lambda_{S\Delta}\lambda_{SH}-192\lambda_{2}\lambda_{H}\lambda_{S}+16\lambda_{2}\lambda_{SH}^{2}-144\lambda_{3}\lambda_{H}\lambda_{S}+12\lambda_{3}\lambda_{SH}^{2}+6\lambda_{4}^{2}\lambda_{S}$
$\displaystyle-6\lambda_{4}\lambda_{S\Delta}\lambda_{SH}+18\lambda_{H}\lambda_{S\Delta}^{2}+x\Big{(}-12\lambda_{1}^{2}-12\lambda_{1}\lambda_{4}+96\lambda_{2}\lambda_{H}+32\lambda_{2}\lambda_{S}+72\lambda_{3}\lambda_{H}+24\lambda_{3}\lambda_{S}-3\lambda_{4}^{2}$
$\displaystyle+24\lambda_{H}\lambda_{S}-3\lambda_{S\Delta}^{2}-2\lambda_{SH}^{2}\Big{)}+x^{2}(-16\lambda_{2}-12\lambda_{3}-12\lambda_{H}-4\lambda_{S})+2x^{3}=0$
(79)

##  Appendix F Phenomenological constraints

Here, we turn our attention to the phenomenological constraints on our model
parameter space.

###  F.1 Neutrino Oscillation data

Explaining the smallness of neutrino masses is one of the key motivations of
the model under consideration. Hence, we briefly mention the constraints
imposed on the model parameter space coming from neutrino oscillation data.
Neutrino oscillation experiments provide bounds on neutrino sector parameters
such as mass-squared differences and the mixing angles Esteban:2020cvm . The
$3\sigma$ allowed ranges for these observables depend slightly on the ordering
of neutrino masses. Below we list the latest limits from a global fit
Esteban:2020cvm , for the normal ordering (inverted ordering) of neutrino
masses

$\displaystyle\Delta m^{2}_{21}\in[6.82-8.04]\times 10^{-5}{\text{eV}^{2}},$
$\displaystyle\Delta
m^{2}_{3\ell}\in[2.431-2.598]~{}\Big{(}[-2.583,-2.412]\Big{)}\times
10^{-3}{\text{eV}^{2}},$ $\displaystyle\sin^{2}\theta_{12}\in[0.269-0.343],$
$\displaystyle\sin^{2}\theta_{23}\in[0.407-0.618]~{}\Big{(}[0.411-0.621]\Big{)},$
$\displaystyle\sin^{2}\theta_{13}\in[0.02034-0.02430]~{}\Big{(}[0.02053-0.02436]\Big{)},$
$\displaystyle\delta_{\text{CP}}\in[107-403]^{\circ}~{}\Big{(}[192-360]^{\circ}\Big{)}.$
(80)

Only one set of $3\sigma$ allowed values are quoted above for $\Delta
m^{2}_{21}$ and $\sin^{2}\theta_{12}$ since these two observables are
insensitive to the mass ordering of neutrinos. In addition, cosmological
observations put an upper bound on the sum of neutrino masses,
$\sum_{i=1}^{3}m_{i}<0.12$ eV Planck:2018vyg .

###  F.2 Flavor constraints

The Yukawa interactions of the Higgs triplet with the SM lepton doublets,
which generate the non-zero neutrino masses, can also induce charged lepton
flavor violating (cLFV) processes. The cLFV processes can arise both at tree-
level (e.g. $\ell_{\alpha}\to\ell_{\beta}\ell_{\gamma}\ell_{\delta}$) or loop-
level (e.g. $\ell_{\alpha}\to\ell_{\beta}\gamma$), where
$\alpha,\beta,\gamma,\delta$ are flavor indices. The bounds from the cLFV
processes can be quite stringent, and the two channels that impose the
strongest limits are muon decay to $3e$ and $e\gamma$ Dev:2021axj . The
branching fractions (BF) of these two important processes are given by
Kakizaki:2003jk ; Akeroyd:2009nu ; Dinh:2012bp ; Ashanujjaman:2021txz

$\displaystyle{\rm Br}(\mu^{-}\to
e^{+}e^{-}e^{-})=\frac{|(y_{L})^{\dagger}_{ee}(y_{L})_{\mu
e}|^{2}}{4\,G_{F}^{2}m_{H^{\pm\pm}}^{4}}~{},$ $\displaystyle{\rm
Br}(\mu^{-}\to
e^{-}\gamma)=\frac{\alpha|\sum_{i=1}^{3}(y_{L})^{\dagger}_{ei}(y_{L})_{\mu
i}|^{2}}{192\pi
G_{F}^{2}}\left(\frac{1}{m_{H^{\pm}}^{2}}+\frac{8}{m_{H^{\pm\pm}}^{2}}\right)^{2}~{},$

where $(y_{L})_{\alpha\beta}=\frac{\sqrt{2}\,m_{\alpha\beta}^{\nu}}{v_{t}}$.
The experimental upper limits for ${\rm Br}(\mu^{-}\to e^{+}e^{-}e^{-})$ and
${\rm Br}(\mu^{-}\to e^{-}\gamma)$ are $1.0\times 10^{-12}$ Bellgardt:1987du
and $4.2\times 10^{-13}$ TheMEG:2016wtm , respectively. Using them we can
compute the following lower bounds on $v_{t}$ as a function of
$m_{H^{\pm\pm}}$

$v_{t}\gtrsim 0.78\text{--}1.5\,\big{(}0.69\text{--}2.3\big{)}\times
10^{-9}{\rm~{}GeV}\times\bigg{(}\frac{1~{}\rm TeV}{m_{H^{\pm\pm}}}\bigg{)}$

for the normal (inverted) ordering of neutrino masses.

###  F.3 Electroweak precision observables

Since the $CP$-even neutral scalar of the triplet develops an induced ${\it
vev}$ in this model, it can lead to a deviation of the oblique $T$ parameter
from the SM prediction at the tree level itself. In addition, a mass-
splitting, $\Delta m$, between the components of the triplet can further shift
the oblique parameters $S,\,T$ and $U$ at one-loop level from the SM
predictions Chun:2012jw ; Aoki:2012jj . The tree-level masses of $W$ and $Z$
bosons within this model get altered, which leads to the following expression
of the EW $\rho$ parameter

$\displaystyle\rho=\frac{m_{W}^{2}}{m_{Z}^{2}\cos^{2}\theta_{W}}=\frac{v_{d}^{2}+2v_{t}^{2}}{v_{d}^{2}+4v_{t}^{2}}\simeq
1-\frac{2v_{t}^{2}}{v_{d}^{2}},$ (81)

where $v_{d}=\sqrt{v^{2}-2v_{t}^{2}}$ with $v=246$ GeV. $\rho$ is connected to
the $T$ parameter at the tree level by
$T_{\text{tree}}\simeq-2v_{t}^{2}/v_{d}^{2}/\alpha=(1-\rho)/\alpha$. It is
needless to say that the above relations hold only under the assumption
$v_{t}<<v_{d}$. This assumption is a natural choice in type-II seesaw
motivated models since the triplet ${\it vev}$ generates neutrino masses. A
tiny $v_{t}$ will allow us to fit neutrino masses and mixings with small
Yukawa couplings satisfying all flavor constraints. The current measurement of
the EW observables constrain the $\rho$-parameter as : $\rho=1.00038\pm
0.00020$ ParticleDataGroup:2020ssz , which translates into a bound $v_{t}\leq
2.6$ GeV at $3\sigma$ level.

Now, assuming $v_{t}<<v_{d}$, we incorporate the limits from the three EW
oblique parameters $S,\,T$ and $U$ on $\Delta m$. In this paper, we
exclusively consider $v_{t}<<1$ GeV, and in this regime, the correction to the
$T$ parameter is dominated by 1-loop level contributions compared to the small
tree level correction shown above. References Chun:2012jw ; Aoki:2012jj  show
that oblique parameters constrain $\Delta m$ to be below 40 GeV for a large
range of $m_{H^{\pm\pm}}$ values.

## References

  * (1) G. Aad et al. [ATLAS], Phys. Lett. B 716 (2012), 1-29 doi:10.1016/j.physletb.2012.08.020 [arXiv:1207.7214 [hep-ex]].
  * (2) S. Chatrchyan et al. [CMS], Phys. Lett. B 716 (2012), 30-61 doi:10.1016/j.physletb.2012.08.021 [arXiv:1207.7235 [hep-ex]].
  * (3) A. D. Sakharov, Pisma Zh. Eksp. Teor. Fiz. 5 (1967), 32-35 doi:10.1070/PU1991v034n05ABEH002497
  * (4) M. Trodden, Rev. Mod. Phys. 71 (1999), 1463-1500 doi:10.1103/RevModPhys.71.1463 [arXiv:hep-ph/9803479 [hep-ph]].
  * (5) G. W. Anderson and L. J. Hall, Phys. Rev. D 45 (1992), 2685-2698 doi:10.1103/PhysRevD.45.2685
  * (6) P. Huet and A. E. Nelson, Phys. Rev. D 53 (1996), 4578-4597 doi:10.1103/PhysRevD.53.4578 [arXiv:hep-ph/9506477 [hep-ph]].
  * (7) D. E. Morrissey and M. J. Ramsey-Musolf, New J. Phys. 14 (2012), 125003 doi:10.1088/1367-2630/14/12/125003 [arXiv:1206.2942 [hep-ph]].
  * (8) W. Hu and S. Dodelson, Ann. Rev. Astron. Astrophys. 40 (2002), 171-216 doi:10.1146/annurev.astro.40.060401.093926 [arXiv:astro-ph/0110414 [astro-ph]].
  * (9) D. N. Spergel et al. [WMAP], Astrophys. J. Suppl. 170 (2007), 377 doi:10.1086/513700 [arXiv:astro-ph/0603449 [astro-ph]].
  * (10) N. Aghanim et al. [Planck], Astron. Astrophys. 641 (2020), A6 [erratum: Astron. Astrophys. 652 (2021), C4] doi:10.1051/0004-6361/201833910 [arXiv:1807.06209 [astro-ph.CO]].
  * (11) T. Kajita, Rev. Mod. Phys. 88 (2016) no.3, 030501 doi:10.1103/RevModPhys.88.030501
  * (12) A. B. McDonald, Rev. Mod. Phys. 88 (2016) no.3, 030502 doi:10.1103/RevModPhys.88.030502
  * (13) P. Minkowski, Phys. Lett. B 67 (1977), 421-428 doi:10.1016/0370-2693(77)90435-X
  * (14) T. Yanagida, Conf. Proc. C 7902131 (1979), 95-99 KEK-79-18-95.
  * (15) R. N. Mohapatra and G. Senjanovic, Phys. Rev. Lett. 44 (1980), 912 doi:10.1103/PhysRevLett.44.912
  * (16) W. Konetschny and W. Kummer, Phys. Lett. B 70 (1977), 433-435 doi:10.1016/0370-2693(77)90407-5
  * (17) M. Magg and C. Wetterich, Phys. Lett. B 94 (1980), 61-64 doi:10.1016/0370-2693(80)90825-4
  * (18) G. Lazarides, Q. Shafi and C. Wetterich, Nucl. Phys. B 181 (1981), 287-300 doi:10.1016/0550-3213(81)90354-0
  * (19) J. Schechter and J. W. F. Valle, Phys. Rev. D 22 (1980), 2227 doi:10.1103/PhysRevD.22.2227
  * (20) T. P. Cheng and L. F. Li, Phys. Rev. D 22 (1980), 2860 doi:10.1103/PhysRevD.22.2860
  * (21) S. M. Bilenky, J. Hosek and S. T. Petcov, Phys. Lett. B 94 (1980), 495-498 doi:10.1016/0370-2693(80)90927-2
  * (22) J. A. Casas, D. G. Cerdeño, J. M. Moreno and J. Quilis, JHEP 05 (2017), 036 doi:10.1007/JHEP05(2017)036 [arXiv:1701.08134 [hep-ph]].
  * (23) S. Bhattacharya, P. Ghosh, T. N. Maity and T. S. Ray, JHEP 10 (2017), 088 doi:10.1007/JHEP10(2017)088 [arXiv:1706.04699 [hep-ph]].
  * (24) E. Aprile et al. [XENON], Phys. Rev. Lett. 121 (2018) no.11, 111302 doi:10.1103/PhysRevLett.121.111302 [arXiv:1805.12562 [astro-ph.CO]].
  * (25) Y. Meng et al. [PandaX-4T], Phys. Rev. Lett. 127 (2021) no.26, 261802 doi:10.1103/PhysRevLett.127.261802 [arXiv:2107.13438 [hep-ex]].
  * (26) S. Abdollahi et al. [Fermi-LAT], Phys. Rev. D 95 (2017) no.8, 082007 doi:10.1103/PhysRevD.95.082007 [arXiv:1704.07195 [astro-ph.HE]].
  * (27) M. L. Ahnen et al. [MAGIC and Fermi-LAT], JCAP 02 (2016), 039 doi:10.1088/1475-7516/2016/02/039 [arXiv:1601.06590 [astro-ph.HE]].
  * (28) R. Apreda, M. Maggiore, A. Nicolis and A. Riotto, Nucl. Phys. B 631 (2002), 342-368 doi:10.1016/S0550-3213(02)00264-X [arXiv:gr-qc/0107033 [gr-qc]].
  * (29) C. Grojean, G. Servant and J. D. Wells, Phys. Rev. D 71 (2005), 036001 doi:10.1103/PhysRevD.71.036001 [arXiv:hep-ph/0407019 [hep-ph]].
  * (30) D. J. Weir, Phil. Trans. Roy. Soc. Lond. A 376 (2018) no.2114, 20170126 doi:10.1098/rsta.2017.0126 [arXiv:1705.01783 [hep-ph]].
  * (31) A. Alves, T. Ghosh, H. K. Guo and K. Sinha, JHEP 12 (2018), 070 doi:10.1007/JHEP12(2018)070 [arXiv:1808.08974 [hep-ph]].
  * (32) A. Alves, T. Ghosh, H. K. Guo, K. Sinha and D. Vagie, JHEP 04 (2019), 052 doi:10.1007/JHEP04(2019)052 [arXiv:1812.09333 [hep-ph]].
  * (33) A. Alves, D. Gonçalves, T. Ghosh, H. K. Guo and K. Sinha, JHEP 03 (2020), 053 doi:10.1007/JHEP03(2020)053 [arXiv:1909.05268 [hep-ph]].
  * (34) A. Alves, D. Gonçalves, T. Ghosh, H. K. Guo and K. Sinha, Phys. Lett. B 818 (2021), 136377 doi:10.1016/j.physletb.2021.136377 [arXiv:2007.15654 [hep-ph]].
  * (35) A. Chatterjee, A. Datta and S. Roy, JHEP 06 (2022), 108 doi:10.1007/JHEP06(2022)108 [arXiv:2202.12476 [hep-ph]].
  * (36) C. Caprini, M. Chala, G. C. Dorsch, M. Hindmarsh, S. J. Huber, T. Konstandin, J. Kozaczuk, G. Nardini, J. M. No and K. Rummukainen, et al. JCAP 03 (2020), 024 doi:10.1088/1475-7516/2020/03/024 [arXiv:1910.13125 [astro-ph.CO]].
  * (37) E. Witten, Phys. Rev. D 30 (1984), 272-285 doi:10.1103/PhysRevD.30.272
  * (38) C. J. Hogan, Mon. Not. Roy. Astron. Soc. 218 (1986), 629-636
  * (39) J. Ellis, M. Lewicki and J. M. No, JCAP 04 (2019), 003 doi:10.1088/1475-7516/2019/04/003 [arXiv:1809.08242 [hep-ph]].
  * (40) T. Alanne, T. Hugle, M. Platscher and K. Schmitz, JHEP 03 (2020), 004 doi:10.1007/JHEP03(2020)004 [arXiv:1909.11356 [hep-ph]].
  * (41) P. Amaro-Seoane et al. [LISA], [arXiv:1702.00786 [astro-ph.IM]].
  * (42) X. Gong, Y. K. Lau, S. Xu, P. Amaro-Seoane, S. Bai, X. Bian, Z. Cao, G. Chen, X. Chen and Y. Ding, et al. J. Phys. Conf. Ser. 610 (2015) no.1, 012011 doi:10.1088/1742-6596/610/1/012011 [arXiv:1410.7296 [gr-qc]].
  * (43) W. R. Hu and Y. L. Wu, Natl. Sci. Rev. 4 (2017) no.5, 685-686 doi:10.1093/nsr/nwx116
  * (44) J. Luo et al. [TianQin], Class. Quant. Grav. 33 (2016) no.3, 035010 doi:10.1088/0264-9381/33/3/035010 [arXiv:1512.02076 [astro-ph.IM]].
  * (45) G. M. Harry [LIGO Scientific], Class. Quant. Grav. 27 (2010), 084006 doi:10.1088/0264-9381/27/8/084006
  * (46) V. Corbin and N. J. Cornish, Class. Quant. Grav. 23 (2006), 2435-2446 doi:10.1088/0264-9381/23/7/014 [arXiv:gr-qc/0512039 [gr-qc]].
  * (47) H. Kudoh, A. Taruya, T. Hiramatsu and Y. Himemoto, Phys. Rev. D 73 (2006), 064006 doi:10.1103/PhysRevD.73.064006 [arXiv:gr-qc/0511145 [gr-qc]].
  * (48) B. P. Abbott et al. [LIGO Scientific and Virgo], Phys. Rev. Lett. 116 (2016) no.6, 061102 doi:10.1103/PhysRevLett.116.061102 [arXiv:1602.03837 [gr-qc]].
  * (49) B. P. Abbott et al. [LIGO Scientific and Virgo], Phys. Rev. Lett. 119 (2017) no.16, 161101 doi:10.1103/PhysRevLett.119.161101 [arXiv:1710.05832 [gr-qc]].
  * (50) B. P. Abbott et al. [LIGO Scientific and Virgo], Phys. Rev. X 9 (2019) no.3, 031040 doi:10.1103/PhysRevX.9.031040 [arXiv:1811.12907 [astro-ph.HE]].
  * (51) R. Abbott et al. [LIGO Scientific and Virgo], Phys. Rev. X 11 (2021), 021053 doi:10.1103/PhysRevX.11.021053 [arXiv:2010.14527 [gr-qc]].
  * (52) G. Agazie et al. [NANOGrav], Astrophys. J. Lett. 951 (2023) no.1, L8 doi:10.3847/2041-8213/acdac6 [arXiv:2306.16213 [astro-ph.HE]].
  * (53) J. Antoniadis et al. [EPTA], [arXiv:2306.16214 [astro-ph.HE]].
  * (54) X. Qi and H. Sun, Phys. Rev. D 107 (2023) no.9, 095026 doi:10.1103/PhysRevD.107.095026 [arXiv:2104.01045 [hep-ph]].
  * (55) A. Arhrib, R. Benbrik, M. Chabab, G. Moultaka, M. C. Peyranere, L. Rahili and J. Ramadan, Phys. Rev. D 84 (2011), 095005 doi:10.1103/PhysRevD.84.095005 [arXiv:1105.1925 [hep-ph]].
  * (56) P. Fileviez Perez, T. Han, G. y. Huang, T. Li and K. Wang, Phys. Rev. D 78 (2008), 015018 doi:10.1103/PhysRevD.78.015018 [arXiv:0805.3536 [hep-ph]].
  * (57) E. W. Kolb and M. S. Turner, Front. Phys. 69 (1990), 1-547 doi:10.1201/9780429492860
  * (58) A. Hektor, A. Hryczuk and K. Kannike, JHEP 03 (2019), 204 doi:10.1007/JHEP03(2019)204 [arXiv:1901.08074 [hep-ph]].
  * (59) J. M. Alarcon, L. S. Geng, J. Martin Camalich and J. A. Oller, Phys. Lett. B 730 (2014), 342-346 doi:10.1016/j.physletb.2014.01.065 [arXiv:1209.2870 [hep-ph]].
  * (60) A. Alloul, N. D. Christensen, C. Degrande, C. Duhr and B. Fuks, Comput. Phys. Commun. 185 (2014), 2250-2300 doi:10.1016/j.cpc.2014.04.012 [arXiv:1310.1921 [hep-ph]].
  * (61) G. Belanger, F. Boudjema, A. Pukhov and A. Semenov, Comput. Phys. Commun. 176 (2007), 367-382 doi:10.1016/j.cpc.2006.11.008 [arXiv:hep-ph/0607059 [hep-ph]].
  * (62) V. Vaskonen, Phys. Rev. D 95 (2017) no.12, 123515 doi:10.1103/PhysRevD.95.123515 [arXiv:1611.02073 [hep-ph]].
  * (63) J. Ellis, M. Lewicki, M. Merchand, J. M. No and M. Zych, JHEP 01 (2023), 093 doi:10.1007/JHEP01(2023)093 [arXiv:2210.16305 [hep-ph]].
  * (64) S. R. Coleman and E. J. Weinberg, Phys. Rev. D 7 (1973), 1888-1910 doi:10.1103/PhysRevD.7.1888
  * (65) S. P. Martin, Phys. Rev. D 90 (2014) no.1, 016013 doi:10.1103/PhysRevD.90.016013 [arXiv:1406.2355 [hep-ph]].
  * (66) J. Elias-Miro, J. R. Espinosa and T. Konstandin, JHEP 08 (2014), 034 doi:10.1007/JHEP08(2014)034 [arXiv:1406.2652 [hep-ph]].
  * (67) S. Baum, M. Carena, N. R. Shah, C. E. M. Wagner and Y. Wang, JHEP 03 (2021), 055 doi:10.1007/JHEP03(2021)055 [arXiv:2009.10743 [hep-ph]].
  * (68) L. Dolan and R. Jackiw, Phys. Rev. D 9 (1974), 3320-3341 doi:10.1103/PhysRevD.9.3320
  * (69) S. Weinberg, Phys. Rev. D 9 (1974), 3357-3378 doi:10.1103/PhysRevD.9.3357
  * (70) D. A. Kirzhnits and A. D. Linde, Annals Phys. 101 (1976), 195-238 doi:10.1016/0003-4916(76)90279-7
  * (71) J. R. Espinosa, M. Quiros and F. Zwirner, Phys. Lett. B 314 (1993), 206-216 doi:10.1016/0370-2693(93)90450-V [arXiv:hep-ph/9212248 [hep-ph]].
  * (72) R. R. Parwani, Phys. Rev. D 45 (1992), 4695 [erratum: Phys. Rev. D 48 (1993), 5965] doi:10.1103/PhysRevD.45.4695 [arXiv:hep-ph/9204216 [hep-ph]].
  * (73) N. K. Nielsen, Nucl. Phys. B 101 (1975), 173-188 doi:10.1016/0550-3213(75)90301-6
  * (74) R. Fukuda and T. Kugo, Phys. Rev. D 13 (1976), 3469 doi:10.1103/PhysRevD.13.3469
  * (75) M. Laine, Phys. Rev. D 51 (1995), 4525-4532 doi:10.1103/PhysRevD.51.4525 [arXiv:hep-ph/9411252 [hep-ph]].
  * (76) J. Baacke and S. Junker, Phys. Rev. D 49 (1994), 2055-2073 doi:10.1103/PhysRevD.49.2055 [arXiv:hep-ph/9308310 [hep-ph]].
  * (77) J. Baacke and S. Junker, Phys. Rev. D 50 (1994), 4227-4228 doi:10.1103/PhysRevD.50.4227 [arXiv:hep-th/9402078 [hep-th]].
  * (78) M. Garny and T. Konstandin, JHEP 07 (2012), 189 doi:10.1007/JHEP07(2012)189 [arXiv:1205.3392 [hep-ph]].
  * (79) J. R. Espinosa, M. Garny, T. Konstandin and A. Riotto, Phys. Rev. D 95 (2017) no.5, 056004 doi:10.1103/PhysRevD.95.056004 [arXiv:1608.06765 [hep-ph]].
  * (80) H. H. Patel and M. J. Ramsey-Musolf, JHEP 07 (2011), 029 doi:10.1007/JHEP07(2011)029 [arXiv:1101.4665 [hep-ph]].
  * (81) S. Arunasalam and M. J. Ramsey-Musolf, JHEP 08 (2022), 138 doi:10.1007/JHEP08(2022)138 [arXiv:2105.07588 [hep-ph]].
  * (82) J. Löfgren, M. J. Ramsey-Musolf, P. Schicho and T. V. I. Tenkanen, Phys. Rev. Lett. 130 (2023) no.25, 251801 doi:10.1103/PhysRevLett.130.251801 [arXiv:2112.05472 [hep-ph]].
  * (83) M. Laine, M. Meyer and G. Nardini, Nucl. Phys. B 920 (2017), 565-600 doi:10.1016/j.nuclphysb.2017.04.023 [arXiv:1702.07479 [hep-ph]].
  * (84) B. Allen and J. D. Romano, Phys. Rev. D 59 (1999), 102001 doi:10.1103/PhysRevD.59.102001 [arXiv:gr-qc/9710117 [gr-qc]].
  * (85) C. Caprini, M. Hindmarsh, S. Huber, T. Konstandin, J. Kozaczuk, G. Nardini, J. M. No, A. Petiteau, P. Schwaller and G. Servant, et al. JCAP 04 (2016), 001 doi:10.1088/1475-7516/2016/04/001 [arXiv:1512.06239 [astro-ph.CO]].
  * (86) R. G. Cai, Z. Cao, Z. K. Guo, S. J. Wang and T. Yang, Natl. Sci. Rev. 4 (2017) no.5, 687-706 doi:10.1093/nsr/nwx029 [arXiv:1703.00187 [gr-qc]].
  * (87) C. Caprini and D. G. Figueroa, Class. Quant. Grav. 35 (2018) no.16, 163001 doi:10.1088/1361-6382/aac608 [arXiv:1801.04268 [astro-ph.CO]].
  * (88) J. D. Romano and N. J. Cornish, Living Rev. Rel. 20 (2017) no.1, 2 doi:10.1007/s41114-017-0004-1 [arXiv:1608.06889 [gr-qc]].
  * (89) N. Christensen, Rept. Prog. Phys. 82 (2019) no.1, 016903 doi:10.1088/1361-6633/aae6b5 [arXiv:1811.08797 [gr-qc]].
  * (90) J. R. Espinosa, T. Konstandin, J. M. No and G. Servant, JCAP 06 (2010), 028 doi:10.1088/1475-7516/2010/06/028 [arXiv:1004.4187 [hep-ph]].
  * (91) M. S. Turner, E. J. Weinberg and L. M. Widrow, Phys. Rev. D 46 (1992), 2384-2403 doi:10.1103/PhysRevD.46.2384
  * (92) C. L. Wainwright, Comput. Phys. Commun. 183 (2012), 2006-2013 doi:10.1016/j.cpc.2012.04.004 [arXiv:1109.4189 [hep-ph]].
  * (93) C. W. Chiang and B. Q. Lu, JHEP 07 (2020), 082 doi:10.1007/JHEP07(2020)082 [arXiv:1912.12634 [hep-ph]].
  * (94) M. Hindmarsh, S. J. Huber, K. Rummukainen and D. J. Weir, Phys. Rev. D 92 (2015) no.12, 123009 doi:10.1103/PhysRevD.92.123009 [arXiv:1504.03291 [astro-ph.CO]].
  * (95) A. Kosowsky, M. S. Turner and R. Watkins, Phys. Rev. Lett. 69 (1992), 2026-2029 doi:10.1103/PhysRevLett.69.2026
  * (96) A. Kosowsky, M. S. Turner and R. Watkins, Phys. Rev. D 45 (1992), 4514-4535 doi:10.1103/PhysRevD.45.4514
  * (97) A. Kosowsky and M. S. Turner, Phys. Rev. D 47 (1993), 4372-4391 doi:10.1103/PhysRevD.47.4372 [arXiv:astro-ph/9211004 [astro-ph]].
  * (98) D. Bodeker and G. D. Moore, JCAP 05 (2017), 025 doi:10.1088/1475-7516/2017/05/025 [arXiv:1703.08215 [hep-ph]].
  * (99) M. Hindmarsh, S. J. Huber, K. Rummukainen and D. J. Weir, Phys. Rev. Lett. 112 (2014), 041301 doi:10.1103/PhysRevLett.112.041301 [arXiv:1304.2433 [hep-ph]].
  * (100) J. T. Giblin, Jr. and J. B. Mertens, JHEP 12 (2013), 042 doi:10.1007/JHEP12(2013)042 [arXiv:1310.2948 [hep-th]].
  * (101) J. T. Giblin and J. B. Mertens, Phys. Rev. D 90 (2014) no.2, 023532 doi:10.1103/PhysRevD.90.023532 [arXiv:1405.4005 [astro-ph.CO]].
  * (102) K. Schmitz, JHEP 01 (2021), 097 doi:10.1007/JHEP01(2021)097 [arXiv:2002.04615 [hep-ph]].
  * (103) C. Caprini and R. Durrer, Phys. Rev. D 74 (2006), 063521 doi:10.1103/PhysRevD.74.063521 [arXiv:astro-ph/0603476 [astro-ph]].
  * (104) T. Kahniashvili, A. Kosowsky, G. Gogoberidze and Y. Maravin, Phys. Rev. D 78 (2008), 043003 doi:10.1103/PhysRevD.78.043003 [arXiv:0806.0293 [astro-ph]].
  * (105) T. Kahniashvili, L. Campanelli, G. Gogoberidze, Y. Maravin and B. Ratra, Phys. Rev. D 78 (2008), 123006 [erratum: Phys. Rev. D 79 (2009), 109901] doi:10.1103/PhysRevD.78.123006 [arXiv:0809.1899 [astro-ph]].
  * (106) T. Kahniashvili, L. Kisslinger and T. Stevens, Phys. Rev. D 81 (2010), 023004 doi:10.1103/PhysRevD.81.023004 [arXiv:0905.0643 [astro-ph.CO]].
  * (107) C. Caprini, R. Durrer and G. Servant, JCAP 12 (2009), 024 doi:10.1088/1475-7516/2009/12/024 [arXiv:0909.0622 [astro-ph.CO]].
  * (108) L. Kisslinger and T. Kahniashvili, Phys. Rev. D 92 (2015) no.4, 043006 doi:10.1103/PhysRevD.92.043006 [arXiv:1505.03680 [astro-ph.CO]].
  * (109) T. M. C. Abbott et al. [DES], Mon. Not. Roy. Astron. Soc. 480 (2018) no.3, 3879-3888 doi:10.1093/mnras/sty1939 [arXiv:1711.00403 [astro-ph.CO]].
  * (110) M. Hindmarsh and M. Hijazi, JCAP 12 (2019), 062 doi:10.1088/1475-7516/2019/12/062 [arXiv:1909.10040 [astro-ph.CO]].
  * (111) H. K. Guo, K. Sinha, D. Vagie and G. White, JCAP 01 (2021), 001 doi:10.1088/1475-7516/2021/01/001 [arXiv:2007.08537 [hep-ph]].
  * (112) M. B. Hindmarsh, M. Lüben, J. Lumma and M. Pauly, SciPost Phys. Lect. Notes 24 (2021), 1 doi:10.21468/SciPostPhysLectNotes.24 [arXiv:2008.09136 [astro-ph.CO]].
  * (113) U. L. Pen and N. Turok, Phys. Rev. Lett. 117 (2016) no.13, 131301 doi:10.1103/PhysRevLett.117.131301 [arXiv:1510.02985 [astro-ph.CO]].
  * (114) M. Hindmarsh, S. J. Huber, K. Rummukainen and D. J. Weir, Phys. Rev. D 96 (2017) no.10, 103520 [erratum: Phys. Rev. D 101 (2020) no.8, 089902] doi:10.1103/PhysRevD.96.103520 [arXiv:1704.05871 [astro-ph.CO]].
  * (115) J. M. No, Phys. Rev. D 84 (2011), 124025 doi:10.1103/PhysRevD.84.124025 [arXiv:1103.2159 [hep-ph]].
  * (116) K. Huitu, J. Maalampi, A. Pietila and M. Raidal, Nucl. Phys. B 487 (1997), 27-42 doi:10.1016/S0550-3213(97)87466-4 [arXiv:hep-ph/9606311 [hep-ph]].
  * (117) S. Chakrabarti, D. Choudhury, R. M. Godbole and B. Mukhopadhyaya, Phys. Lett. B 434 (1998), 347-353 doi:10.1016/S0370-2693(98)00743-6 [arXiv:hep-ph/9804297 [hep-ph]].
  * (118) E. J. Chun, K. Y. Lee and S. C. Park, Phys. Lett. B 566 (2003), 142-151 doi:10.1016/S0370-2693(03)00770-6 [arXiv:hep-ph/0304069 [hep-ph]].
  * (119) A. G. Akeroyd and M. Aoki, Phys. Rev. D 72 (2005), 035011 doi:10.1103/PhysRevD.72.035011 [arXiv:hep-ph/0506176 [hep-ph]].
  * (120) J. Garayoa and T. Schwetz, JHEP 03 (2008), 009 doi:10.1088/1126-6708/2008/03/009 [arXiv:0712.1453 [hep-ph]].
  * (121) M. Kadastik, M. Raidal and L. Rebane, Phys. Rev. D 77 (2008), 115023 doi:10.1103/PhysRevD.77.115023 [arXiv:0712.3912 [hep-ph]].
  * (122) A. G. Akeroyd, M. Aoki and H. Sugiyama, Phys. Rev. D 77 (2008), 075010 doi:10.1103/PhysRevD.77.075010 [arXiv:0712.4019 [hep-ph]].
  * (123) F. del Aguila and J. A. Aguilar-Saavedra, Nucl. Phys. B 813 (2009), 22-90 doi:10.1016/j.nuclphysb.2008.12.029 [arXiv:0808.2468 [hep-ph]].
  * (124) A. G. Akeroyd and C. W. Chiang, Phys. Rev. D 80 (2009), 113010 doi:10.1103/PhysRevD.80.113010 [arXiv:0909.4419 [hep-ph]].
  * (125) A. Melfo, M. Nemevsek, F. Nesti, G. Senjanovic and Y. Zhang, Phys. Rev. D 85 (2012), 055018 doi:10.1103/PhysRevD.85.055018 [arXiv:1108.4416 [hep-ph]].
  * (126) M. Aoki, S. Kanemura and K. Yagyu, Phys. Rev. D 85 (2012), 055007 doi:10.1103/PhysRevD.85.055007 [arXiv:1110.4625 [hep-ph]].
  * (127) A. G. Akeroyd and H. Sugiyama, Phys. Rev. D 84 (2011), 035010 doi:10.1103/PhysRevD.84.035010 [arXiv:1105.2209 [hep-ph]].
  * (128) C. W. Chiang, T. Nomura and K. Tsumura, Phys. Rev. D 85 (2012), 095023 doi:10.1103/PhysRevD.85.095023 [arXiv:1202.2014 [hep-ph]].
  * (129) E. J. Chun, H. M. Lee and P. Sharma, JHEP 11 (2012), 106 doi:10.1007/JHEP11(2012)106 [arXiv:1209.1303 [hep-ph]].
  * (130) A. G. Akeroyd, S. Moretti and H. Sugiyama, Phys. Rev. D 85 (2012), 055026 doi:10.1103/PhysRevD.85.055026 [arXiv:1201.5047 [hep-ph]].
  * (131) E. J. Chun and P. Sharma, JHEP 08 (2012), 162 doi:10.1007/JHEP08(2012)162 [arXiv:1206.6278 [hep-ph]].
  * (132) P. S. Bhupal Dev, D. K. Ghosh, N. Okada and I. Saha, JHEP 03 (2013), 150 [erratum: JHEP 05 (2013), 049] doi:10.1007/JHEP03(2013)150 [arXiv:1301.3453 [hep-ph]].
  * (133) S. Banerjee, M. Frank and S. K. Rai, Phys. Rev. D 89 (2014) no.7, 075005 doi:10.1103/PhysRevD.89.075005 [arXiv:1312.4249 [hep-ph]].
  * (134) F. del Águila and M. Chala, JHEP 03 (2014), 027 doi:10.1007/JHEP03(2014)027 [arXiv:1311.1510 [hep-ph]].
  * (135) E. J. Chun and P. Sharma, Phys. Lett. B 728 (2014), 256-261 doi:10.1016/j.physletb.2013.11.056 [arXiv:1309.6888 [hep-ph]].
  * (136) S. Kanemura, K. Yagyu and H. Yokoya, Phys. Lett. B 726 (2013), 316-319 doi:10.1016/j.physletb.2013.08.054 [arXiv:1305.2383 [hep-ph]].
  * (137) S. Kanemura, M. Kikuchi, K. Yagyu and H. Yokoya, Phys. Rev. D 90 (2014) no.11, 115018 doi:10.1103/PhysRevD.90.115018 [arXiv:1407.6547 [hep-ph]].
  * (138) S. Kanemura, M. Kikuchi, H. Yokoya and K. Yagyu, PTEP 2015 (2015), 051B02 doi:10.1093/ptep/ptv071 [arXiv:1412.7603 [hep-ph]].
  * (139) Z. Kang, J. Li, T. Li, Y. Liu and G. Z. Ning, Eur. Phys. J. C 75 (2015) no.12, 574 doi:10.1140/epjc/s10052-015-3774-1 [arXiv:1404.5207 [hep-ph]].
  * (140) Z. L. Han, R. Ding and Y. Liao, Phys. Rev. D 91 (2015), 093006 doi:10.1103/PhysRevD.91.093006 [arXiv:1502.05242 [hep-ph]].
  * (141) Z. L. Han, R. Ding and Y. Liao, Phys. Rev. D 92 (2015) no.3, 033014 doi:10.1103/PhysRevD.92.033014 [arXiv:1506.08996 [hep-ph]].
  * (142) D. Das and A. Santamaria, Phys. Rev. D 94 (2016) no.1, 015015 doi:10.1103/PhysRevD.94.015015 [arXiv:1604.08099 [hep-ph]].
  * (143) K. S. Babu and S. Jana, Phys. Rev. D 95 (2017) no.5, 055020 doi:10.1103/PhysRevD.95.055020 [arXiv:1612.09224 [hep-ph]].
  * (144) M. Mitra, S. Niyogi and M. Spannowsky, Phys. Rev. D 95 (2017) no.3, 035042 doi:10.1103/PhysRevD.95.035042 [arXiv:1611.09594 [hep-ph]].
  * (145) Y. Cai, T. Han, T. Li and R. Ruiz, Front. in Phys. 6 (2018), 40 doi:10.3389/fphy.2018.00040 [arXiv:1711.02180 [hep-ph]].
  * (146) D. K. Ghosh, N. Ghosh, I. Saha and A. Shaw, Phys. Rev. D 97 (2018) no.11, 115022 doi:10.1103/PhysRevD.97.115022 [arXiv:1711.06062 [hep-ph]].
  * (147) A. Crivellin, M. Ghezzi, L. Panizzi, G. M. Pruna and A. Signer, Phys. Rev. D 99 (2019) no.3, 035004 doi:10.1103/PhysRevD.99.035004 [arXiv:1807.10224 [hep-ph]].
  * (148) Y. Du, A. Dunbrack, M. J. Ramsey-Musolf and J. H. Yu, JHEP 01 (2019), 101 doi:10.1007/JHEP01(2019)101 [arXiv:1810.09450 [hep-ph]].
  * (149) P. S. Bhupal Dev and Y. Zhang, JHEP 10 (2018), 199 doi:10.1007/JHEP10(2018)199 [arXiv:1808.00943 [hep-ph]].
  * (150) S. Antusch, O. Fischer, A. Hammad and C. Scherb, JHEP 02 (2019), 157 doi:10.1007/JHEP02(2019)157 [arXiv:1811.03476 [hep-ph]].
  * (151) A. Aboubrahim and P. Nath, Phys. Rev. D 98 (2018) no.9, 095024 doi:10.1103/PhysRevD.98.095024 [arXiv:1810.12868 [hep-ph]].
  * (152) T. B. de Melo, F. S. Queiroz and Y. Villamizar, Int. J. Mod. Phys. A 34 (2019) no.27, 1950157 doi:10.1142/S0217751X19501574 [arXiv:1909.07429 [hep-ph]].
  * (153) R. Primulando, J. Julio and P. Uttayarat, JHEP 08 (2019), 024 doi:10.1007/JHEP08(2019)024 [arXiv:1903.02493 [hep-ph]].
  * (154) R. Padhan, D. Das, M. Mitra and A. Kumar Nayak, Phys. Rev. D 101 (2020) no.7, 075050 doi:10.1103/PhysRevD.101.075050 [arXiv:1909.10495 [hep-ph]].
  * (155) E. J. Chun, S. Khan, S. Mandal, M. Mitra and S. Shil, Phys. Rev. D 101 (2020) no.7, 075008 doi:10.1103/PhysRevD.101.075008 [arXiv:1911.00971 [hep-ph]].
  * (156) S. Ashanujjaman and K. Ghosh, JHEP 03 (2022), 195 doi:10.1007/JHEP03(2022)195 [arXiv:2108.10952 [hep-ph]].
  * (157) S. Mandal, O. G. Miranda, G. Sanchez Garcia, J. W. F. Valle and X. J. Xu, Phys. Rev. D 105 (2022) no.9, 095020 doi:10.1103/PhysRevD.105.095020 [arXiv:2203.06362 [hep-ph]].
  * (158) B. Dutta, R. Eusebi, Y. Gao, T. Ghosh and T. Kamon, Phys. Rev. D 90 (2014), 055015 doi:10.1103/PhysRevD.90.055015 [arXiv:1404.0685 [hep-ph]].
  * (159) G. Aad et al. [ATLAS], Eur. Phys. J. C 72 (2012), 2244 doi:10.1140/epjc/s10052-012-2244-2 [arXiv:1210.5070 [hep-ex]].
  * (160) S. Chatrchyan et al. [CMS], Eur. Phys. J. C 72 (2012), 2189 doi:10.1140/epjc/s10052-012-2189-5 [arXiv:1207.2666 [hep-ex]].
  * (161) G. Aad et al. [ATLAS], JHEP 03 (2015), 041 doi:10.1007/JHEP03(2015)041 [arXiv:1412.0237 [hep-ex]].
  * (162) V. Khachatryan et al. [CMS], Phys. Rev. Lett. 114 (2015) no.5, 051801 doi:10.1103/PhysRevLett.114.051801 [arXiv:1410.6315 [hep-ex]].
  * (163) [CMS], CMS-PAS-HIG-14-039.
  * (164) [CMS], CMS-PAS-HIG-16-036.
  * (165) M. Aaboud et al. [ATLAS], Eur. Phys. J. C 78 (2018) no.3, 199 doi:10.1140/epjc/s10052-018-5661-z [arXiv:1710.09748 [hep-ex]].
  * (166) A. M. Sirunyan et al. [CMS], Phys. Rev. Lett. 120 (2018) no.8, 081801 doi:10.1103/PhysRevLett.120.081801 [arXiv:1709.05822 [hep-ex]].
  * (167) M. Aaboud et al. [ATLAS], Eur. Phys. J. C 79 (2019) no.1, 58 doi:10.1140/epjc/s10052-018-6500-y [arXiv:1808.01899 [hep-ex]].
  * (168) G. Aad et al. [ATLAS], JHEP 06 (2021), 146 doi:10.1007/JHEP06(2021)146 [arXiv:2101.11961 [hep-ex]].
  * (169) T. Ghosh, S. Jana and S. Nandi, Phys. Rev. D 97 (2018) no.11, 115037 doi:10.1103/PhysRevD.97.115037 [arXiv:1802.09251 [hep-ph]].
  * (170) H. Baer, A. Mustafayev and X. Tata, Phys. Rev. D 90 (2014) no.11, 115007 doi:10.1103/PhysRevD.90.115007 [arXiv:1409.7058 [hep-ph]].
  * (171) B. Dutta, A. Gurrola, W. Johns, T. Kamon, P. Sheldon and K. Sinha, Phys. Rev. D 87 (2013) no.3, 035029 doi:10.1103/PhysRevD.87.035029 [arXiv:1210.0964 [hep-ph]].
  * (172) B. Dutta, T. Ghosh, A. Gurrola, W. Johns, T. Kamon, P. Sheldon, K. Sinha, K. Wang and S. Wu, Phys. Rev. D 91 (2015) no.5, 055025 doi:10.1103/PhysRevD.91.055025 [arXiv:1411.6043 [hep-ph]].
  * (173) M. A. Ajaib, B. Dutta, T. Ghosh, I. Gogoladze and Q. Shafi, Phys. Rev. D 92 (2015) no.7, 075033 doi:10.1103/PhysRevD.92.075033 [arXiv:1505.05896 [hep-ph]].
  * (174) B. Dutta, K. Fantahun, A. Fernando, T. Ghosh, J. Kumar, P. Sandick, P. Stengel and J. W. Walker, Phys. Rev. D 96 (2017) no.7, 075037 doi:10.1103/PhysRevD.96.075037 [arXiv:1706.05339 [hep-ph]].
  * (175) G. Aad et al. [ATLAS and CMS], JHEP 08 (2016), 045 doi:10.1007/JHEP08(2016)045 [arXiv:1606.02266 [hep-ex]].
  * (176) A. M. Sirunyan et al. [CMS], Eur. Phys. J. C 79 (2019) no.5, 421 doi:10.1140/epjc/s10052-019-6909-y [arXiv:1809.10733 [hep-ex]].
  * (177) A. M. Sirunyan et al. [CMS], Phys. Lett. B 793 (2019), 520-551 doi:10.1016/j.physletb.2019.04.025 [arXiv:1809.05937 [hep-ex]].
  * (178) A. Arhrib, R. Benbrik, M. Chabab, G. Moultaka and L. Rahili, JHEP 04 (2012), 136 doi:10.1007/JHEP04(2012)136 [arXiv:1112.5453 [hep-ph]].
  * (179) J. F. Gunion, H. E. Haber, G. L. Kane and S. Dawson, Front. Phys. 80 (2000), 1-404 SCIPP-89/13.
  * (180) G. Aad et al. [ATLAS], JHEP 07 (2023), 088 doi:10.1007/JHEP07(2023)088 [arXiv:2207.00348 [hep-ex]].
  * (181) A. M. Sirunyan et al. [CMS], JHEP 07 (2021), 027 doi:10.1007/JHEP07(2021)027 [arXiv:2103.06956 [hep-ex]].
  * (182) C. L. Carilli and S. Rawlings, New Astron. Rev. 48 (2004), 979 doi:10.1016/j.newar.2004.09.001 [arXiv:astro-ph/0409274 [astro-ph]].
  * (183) G. Janssen, G. Hobbs, M. McLaughlin, C. Bassa, A. T. Deller, M. Kramer, K. Lee, C. Mingarelli, P. Rosado and S. Sanidas, et al. PoS AASKA14 (2015), 037 doi:10.22323/1.215.0037 [arXiv:1501.00127 [astro-ph.IM]].
  * (184) A. Weltman, P. Bull, S. Camera, K. Kelley, H. Padmanabhan, J. Pritchard, A. Raccanelli, S. Riemer-Sørensen, L. Shao and S. Andrianomena, et al. Publ. Astron. Soc. Austral. 37 (2020), e002 doi:10.1017/pasa.2019.42 [arXiv:1810.02680 [astro-ph.CO]].
  * (185) M. A. McLaughlin, Class. Quant. Grav. 30 (2013), 224008 doi:10.1088/0264-9381/30/22/224008 [arXiv:1310.0758 [astro-ph.IM]].
  * (186) K. Ghorbani and P. H. Ghorbani, J. Phys. G 47 (2020) no.1, 015201 doi:10.1088/1361-6471/ab4823 [arXiv:1804.05798 [hep-ph]].
  * (187) D. Comelli and J. R. Espinosa, Phys. Rev. D 55 (1997), 6253-6263 doi:10.1103/PhysRevD.55.6253 [arXiv:hep-ph/9606438 [hep-ph]].
  * (188) P. Basler and M. Mühlleitner, Comput. Phys. Commun. 237 (2019), 62-85 doi:10.1016/j.cpc.2018.11.006 [arXiv:1803.02846 [hep-ph]].
  * (189) M. E. Carrington, Phys. Rev. D 45 (1992), 2933-2944 doi:10.1103/PhysRevD.45.2933
  * (190) K. Kannike, Eur. Phys. J. C 72 (2012), 2093 doi:10.1140/epjc/s10052-012-2093-z [arXiv:1205.3781 [hep-ph]].
  * (191) G. Belanger, K. Kannike, A. Pukhov and M. Raidal, JCAP 01 (2013), 022 doi:10.1088/1475-7516/2013/01/022 [arXiv:1211.1014 [hep-ph]].
  * (192) R. N. Lerner and J. McDonald, Phys. Rev. D 80 (2009), 123507 doi:10.1103/PhysRevD.80.123507 [arXiv:0909.0520 [hep-ph]].
  * (193) B. W. Lee, C. Quigg and H. B. Thacker, Phys. Rev. D 16 (1977), 1519 doi:10.1103/PhysRevD.16.1519
  * (194) I. Esteban, M. C. Gonzalez-Garcia, M. Maltoni, T. Schwetz and A. Zhou, JHEP 09 (2020), 178 doi:10.1007/JHEP09(2020)178 [arXiv:2007.14792 [hep-ph]].
  * (195) P. S. B. Dev, B. Dutta, T. Ghosh, T. Han, H. Qin and Y. Zhang, JHEP 03 (2022), 068 doi:10.1007/JHEP03(2022)068 [arXiv:2109.04490 [hep-ph]].
  * (196) M. Kakizaki, Y. Ogura and F. Shima, Phys. Lett. B 566 (2003), 210-216 doi:10.1016/S0370-2693(03)00833-5 [arXiv:hep-ph/0304254 [hep-ph]].
  * (197) A. G. Akeroyd, M. Aoki and H. Sugiyama, Phys. Rev. D 79 (2009), 113010 doi:10.1103/PhysRevD.79.113010 [arXiv:0904.3640 [hep-ph]].
  * (198) D. N. Dinh, A. Ibarra, E. Molinaro and S. T. Petcov, JHEP 08 (2012), 125 [erratum: JHEP 09 (2013), 023] doi:10.1007/JHEP08(2012)125 [arXiv:1205.4671 [hep-ph]].
  * (199) U. Bellgardt et al. [SINDRUM], Nucl. Phys. B 299 (1988), 1-6 doi:10.1016/0550-3213(88)90462-2
  * (200) A. M. Baldini et al. [MEG], Eur. Phys. J. C 76 (2016) no.8, 434 doi:10.1140/epjc/s10052-016-4271-x [arXiv:1605.05081 [hep-ex]].
  * (201) M. Aoki, S. Kanemura, M. Kikuchi and K. Yagyu, Phys. Rev. D 87 (2013) no.1, 015012 doi:10.1103/PhysRevD.87.015012 [arXiv:1211.6029 [hep-ph]].
  * (202) P. A. Zyla et al. [Particle Data Group], PTEP 2020 (2020) no.8, 083C01 doi:10.1093/ptep/ptaa104


# Strategies for the Integration of quantum networks for a future quantum
internet

M.I. García-Cid1,2, L. Ortiz1, J. Sáez1, and V. Martín1
1Center for computational Simulation, Universidad Politéctica de Madrid,
Madrid, Spain
2Indra Sistemas, S.A., Av. de Bruselas, 35, 28108, Madrid, Spain

###### Abstract

The great scientific and technological advances that are being carried out in
the field of quantum communications, accompanied by large investment programs
such as EuroQCI, are driving the deployment of quantum network throughout the
world. One of the final long-term objectives is to achieve the development of
a quantum internet that provides greater security in its services and new
functionalities that the current internet does not have. This article analyzes
the possible integration strategies of already deployed networks or in the
process of being deployed in order to reach a future global quantum network.
Two strategies based on the SDN paradigm are proposed, based on a hierarchical
controller scheme and on a distributed model. Each of these approaches shows
pros and cons and could be applicable in different use cases. To define these
strategies, the most relevant deployments of quantum communications networks
carried out to date has been analyzed, as well as the different approaches for
a quantum network architecture and topology, and the various proposed
definitions of what quantum internet is and what are the components that would
make it up in an ideal scenario. Finally, several detected opportunities and
challenges regarding security and technological aspects are presented.

##  1 Introduction

Quantum communications have exponentially evolved since the first testbeds
that were deployed. This evolution has not only been in size but also in the
underlying technologies, with new protocols and some industrialized devices.
The original networks, i.e. DARPA [1], UQCC [2], were ad hoc dark fiber
terrestrial deployments that had between $6$ and $10$ nodes and were capable
of generating around $81.7\leavevmode\nobreak\ kbps$ over
$45\leavevmode\nobreak\ km$.

Currently, terrestrial networks have been expanded, mainly in China, deploying
more than 700 fibre-based quantum links [4], with similar key rates and
maintaining a dark fiber infrastructure model dedicated exclusively to quantum
communications. Due to the proven feasibility of implementing these
technologies covering increasingly greater distances, two major challenges
have been overcome, opening the doors to real quantum communication
infrastructures (QCI) and not just research-oriented deployments. The first
challenge is the deployment of quantum links integrated in classical
communication infrastructures in production, which represents a drastic
improvement in terms of implementation costs. This approach has been
implemented in [3] using the Software Defined Network (SDN) paradigm which
increases control over the network and allows a better noise management,
making easier the coexistence of classical and quantum channels. The second
great challenge has been the launch of Low Earth Orbit (LEO) satellites with a
quantum payload [4] allowing two distant ground stations to be connected. This
began with feasibility studies [5] and it has already been possible to
generate key rates about 48 kbps between two cities located at a distance of
2,600 km, covering a total distance of 4,600 km [4]. With this achievement,
together with the field tests carried out for transmitting qubits between two
distant drones [6], it is possible to begin to consider theoretical
architectures and feasibility studies for satellite constellations acting as
key relay nodes.
It is clear that the technological maturity has increased much in terrestrial
and spacial QCI based on fiber optics and LEO satellites, respectively. In
addition, the availability of standards from standardization bodies such as
ETSI, IEEE or ITU, allows a better integration into telco environments, even
using devices from different providers. All this, makes possible the
deployment of quantum communication networks (QCN) in metropolitan areas to
guarantee the security of communications between, for example, company
headquarters, bank branches, public organization or military buildings,
scattered throughout the city.

In view of the advances that are taking place in the area of quantum
communications and its potential, the quantum internet began to be talked
about. This term began to be used in the late 1990s [7] and has regained
interest in the second decade of the 21st century. Although elements that will
build the future quantum internet and the possible functionalities that it can
provide have been proposed and defined [8], there are still gaps when
referring to the definition of a common architecture for integrating the
current quantum networks that will support the future quantum internet.
That is why, in this article, we define and compare a hierarchical and a
distributed architectural model for the integration of quantum communication
networks. For that, an analysis of the previously mentioned quantum
communication deployments has been performed, as well as of today’s
technological and security challenges that still exist and that must be
solved.

The article is structured as follows, Section 2 describes the components of an
ideal scenario for a quantum network, different topologies for isolated
quantum networks on the Software-Defined Network (SDN) approach. Taking all
the previous concepts into account, Section 3 reviews what is the perspective
of the future quantum internet. Two architectural approaches based on
hierarchical and distributed models for the integration of the different types
of isolated networks are presented in Section 4. Finally, several detected
opportunities and challenges regarding security and technological aspects are
presented in Section 5.

##  2 Isolated Quantum network architectures

When implementing QCI, different strategies have been used, such as the
deployment of ad-hoc dark fiber infrastructures for the exclusive use of
quantum communications or the integration of quantum technologies such as QKD
devices and fiber optics-based quantum links in current communication networks
in production, sharing the infrastructure with the classical links. The first
strategy optimizes the quantum channel for a better performance, but it
implies higher deployment costs and a lower scalability and flexibility of the
network. The second strategy is less expensive, more scalable, flexible and
manageable, but it is affected by the noise of the classic communications and
a network design optimized for classical communications with which it
coexists, decreasing the quantum network performance. However, the later
approach shows a more realistic choice when it comes to large network
deployments, such as those being considered for future networks. Regardless of
the infrastructure deployment strategy (ad-hoc or shared), a QCN is made up of
a series of disruptive elements and technologies that increase security and
functionalities compared to current communication networks. In the same way as
the network components, the types of applicable QCI topologies are also
independent of the deployment strategy in the sense that all the topologies
discussed later in this section are feasible in both cases. However, the
choice of one strategy or another has an impact on the operating parameters of
the network.

###  2.1 Components of a QCI

All network management and control processes, synchronization of quantum
devices and key distillation processes, among others, classic network elements
such as switches, routers, communication channels, servers, etc. are needed.
In addition to the elements of a classical network, in order to have an ideal
QCI, a series of key features are needed: quantum channels (QC), quantum
repeaters (QR), quantum memories (QM) and quantum nodes (QN), as defined
below. This does not mean that it is strictly necessary to have all of these
components in order to have an operative QCI, but rather that having all of
them at the same time, it is possible to have all the capabilities and
functionalities that are expected (today) from these networks and from the
future quantum internet, as we will see in Section 3.

Quantum nodes (QN). Depending on the context in which the quantum nodes are
being discussed, they can be understood from different approaches.

  * •

Approach 1 – Nodes as devices. Quantum nodes as each of the qubit
emitting/receiving devices. In this case, a pair of QKD devices are described
as two quantum nodes connected through a quantum channel.

  * •

Approach 2 – Nodes as locations. Nodes such as those physical locations that
have the capacity to emit and/or receive qubits. In this approach, a quantum
node could have more than one emitter/receiver module that connects it with
other.

  * •

Approach 3 – Nodes as relays. Networks sometimes have intermediate active
and/or passive elements that simply relay the key generated by QKD, switch the
signal path or (de)multiplex it.

In order to later present the architecture for the integration of different
quantum networks, it is also convenient to distinguish a last type of node,
which can be included within any of the three previous approaches, called
border node or network access [9]. A border node is that node within a network
domain that is linked to the border node of a different one, thus enabling the
communication between different network domains. From here on we will use
approach 2 to refer to the nodes of a quantum network.

Quantum channels (QC). Several transmission media for qubits acting as quantum
channels are available: optical fibre, open air, space and water. Each of them
has its own characteristics. When the transmission of photonic qubits is
carried out through optical fiber, the signal loss increases exponentially
with distance, but metropolitan distances with losses less than about $30dB$
are within reach using available technology. In this case, quantum channels
could - under certain circumstances [3] \- share infrastructure with classical
communications in order to minimize deployment costs and increase the
availability of this technology. For its part, free space transmission in
metropolitan areas suffers from turbulence and other atmospheric effects.
Satellite transmission seems the most optimal option for long-range quantum
communications since the qubits would suffer the major disturbance through 10
km of thickness corresponding to the lower atmosphere where the losses are
higher, but when entering the space regime, these losses would drop. However,
this type of channel is not easy to control. Low Earth Orbits are better for
QKD performance, but the visibility period of the satellite is only a few
minutes. Geostationary satellites provide 24/7 visibility, but the great
distance means more dispersion and requires large telescopes for delivering a
better performance. Finally, in recent years feasibility studies have been
carried out to make quantum communications in an aquatic environment [10].

Quantum memories (QM). Normally, in QKD protocols, qubits are received and
measured, storing their final classical value ($0$ or $1$). However, in order
to have applications beyond the conventional QKD, it may be necessary to store
the qubit in its quantum state, that is, before being measured. For that,
quantum memories are necessary. QM are necessary components to store the
entangled qubits in the nodes, preserving their quantum properties and to
carry out operations (quantum gates) on them in order to transmit the quantum
correlations. This is an essential step to support teleportation, a central
application for the creation of the non-classical correlation that are the key
ingredient to support the new capabilities afforded by a QCI.

Quantum repeaters (QR). Qubits cannot be copied as classical signals can,
which is why it is a technological challenge to build a quantum repeater.
Since today there is no QR, an approach based on trusted nodes has been
adopted to reach greater distances. Trusted nodes start from the premise that
intermediate nodes are trustworthy since the information is decrypted and re-
encrypted in those intermediate nodes. In contrast, quantum repeaters do not
require the intermediate or repeater nodes to be trusted and, not only they
increase the distances in quantum communications, but also, they can establish
entanglement between two distant nodes following the teleportation process.

###  2.2 Network topologies

Although an ideal QCI would be composed of all the elements described in the
previous section, the reality is that with the technological maturity level of
QR and QM, current networks are made up of the minimum components deployed and
managed in different ways depending on the network architecture, topology
chosen and the functionality expected from the network. These components are
classical and quantum channels, pairs of QKD modules, optical elements such as
switches, Wave-Division Multiplexers, Time-Division Multiplexers or
Polarization-Division Multiplexers and common passive network components.
Quantum networks can be deployed in equivalent topologies to those found in
classical networks as showed in Figure 1, being each black dot a QN (approach
2). Thus, to date QCI can be found in ring, star, mesh or bus topology or
hybrid configurations of several of these topologies. Being hybrid topologies
those made up of a network backbone connecting different subnetworks, each one
with its own topology.

Figure 1: Quantum network topologies corresponding to (A) ring, (B) star, (C)
mesh and (D) bus.

As an example, without the need for QR, a quantum network in star topology can
be obtained by making use of a passive optical switch as the central element
that redirect qubits to any QN applying a one-to-many approach [11]. And the
absence of QM means that the qubits must be measured upon receiving them,
obtaining and storing the classical output in key management systems (KMS).

###  2.3 Networks based on SDN

QCN based on the Software-Defined Network (SDN) paradigm have been implemented
and tested in the Spanish network MadQCI [3]. In general, SDN facilitates the
management of the network, both of the classical and quantum elements, and
increases its flexibility and scalability, regardless of the chosen network
topology. This approach allows the integration of QKD devices from different
providers. In a trusted node model, each point-to-point link may use a
different QKD protocol (COW, BB84, E91, GG02). The elements seen above would
be located in the lower layer of the model showed in Figure 2, together with
the classical communication devices such as optical switches (OS) and
satellite links to communicate different networks located at long distances.
This layer corresponds to the data plane (DP) and it is where all the hardware
elements and communication infrastructures are located to carry out the
generation of qubits, their detection, processing and storage of the
cryptographic keys in the KMS, in addition to classical traffic. It also
contains the links between QN through quantum channels (QC), but a classical
channel (CC) is also required for synchronization tasks and key distillation
processes.

Figure 2: SDN quantum network. (Left) Data (DP), control (CP) and application
(AP) planes of the SDN model. AP-CP communicate through the North-bound
interface (NBI) and CP-DP through the South-bound interface (SBI). (Right)
Example of SDN quantum network with three QN in star topology and managed by
the Software-defined network controller (SDNC).

All these elements (classical and quantum) are orchestrated in the upper
layer, which is an abstraction layer called the control plane (CP) where the
SDN controller (SDNC) acts, in communication with the data plane through the
South Bound Interface (SBI). The controller is in charge of collecting service
requests from the application plane (management of notifications and
requests), collecting the availability status of the intervening elements
found in the DP (device and link management), elaborate the entire routing
flow to be followed (network topology, routes between applications involved in
the connection and security mechanisms) and orchestrate all the processes that
have to be carried out to provide the requested service (status management).
Although a single controller is capable of managing the entire network, to
ensure the network security and resilience it is convenient to have a
redundant controller.

Finally, through the North Bound Interface (NBI), the control plane receives
the service requests from the application plane (AP) and provides it with the
information necessary to initiate the service between end users. The AP is
also provided with cryptographic keys from the KMS whose forwarding module is
allocated in the DP to encrypt the communication between end users’
applications. This communication is done through the classical channels,
whether they are fiber optics or wireless connections. Each quantum node is
linked to the SDNC by means of an SDN Agent, which is located between the CP
and the AP. Since quantum elements are not involved in the AP, the functional
and non-functional requirements of this layer as well as the network policies
(e.g., access policies) are almost replicable from classical network models,
although depending on the QKD key generation rate of the network or its
availability in the KMS. In addition to all the applications and services that
currently exist in the AP, applications based in QKD-generated keys are added
such as Network Function Virtualization (NFV), IPSEC, OPoT, quantum digital
signatures (QDS), quantum secret sharing (QSS), quantum teleportation,
byzantine agreement or quantum bit commitment, among others. The European
Telecommunications Standards Institute (ETSI) has already published a series
of standards related to QCN based on SDN, where several specifications and
interfaces currently in use are provided. Of particular relevance are the
interfaces defined to communicate the key manager with the applications
allocated in the AP defined in the ETSI GS QKD 004 and 014 standards, the
interface between CP and DP defined in the ETSI GS QKD 015 standard and the
interface between SDNC and a network orchestrator defined in the ETSI GS QKD
018 standard.

##  3 The quantum internet perspective

Although QCI are currently being deployed in isolation in different countries,
there are already initiatives, such as the EuroQCI program leveraged by the
European Commission, which aims to deploy or expand QCI and integrate them to
build the European Quantum Communication Infrastructure [12]. This type of
programs promoting large quantum communication deployments will allow the
future quantum internet (QI) to become a reality. This is why it is of great
importance to analyse the current QCN and to propose a series of architectures
that facilitate the network integration and evolution but, at the same time,
allow the management of increasingly complex and large networks. In this
section, we will briefly review what QI is and what is expected from it in
order to extract the main requirements needed to define the integration models
of isolated quantum networks presented in Section 4.

The understanding of the Internet at the classical level can be formulated in
two ways [13]:

  * •

Statement 1: Set of necessary hardware and software components that compose
it.

  * •

Statement 2: Services of distributed applications that it provides.

Both approaches can be transferred when defining the QI. Thus, QI is
considered as a set of devices with specific functionalities, which provides a
series of services. After the commercialization of QKD devices, some
definitions of QI have been given [8]. Apart from the definitions, some basic
properties that the QI has to fulfill can be summarized as follows:

  * •

Capability to communicate quantum information between network nodes.

  * •

Cover long distances using quantum repeaters, satellites, etc.

  * •

Be able to generate and distribute entangled pairs between network nodes.

  * •

The infrastructure must host classical communications as well.

The property of entanglement plays a central role since it will allow to carry
out quantum teleportation for the generation of non-classical correlations
between users, which is a functionality that has no classical equivalent. To
date it has been achieved the transmission through satellite links of distance
between 1600-2400 km [4]. However, the transmission of entangled pairs at a
reasonable generation rate is still a complex task, which means that it is a
technology that is still in an early research phase.

In Section 2, we specified the basic elements that build the architecture of a
full quantum internet, following Statement 1. Following the approach from
Statement 2, four stages that make up the development of the QI can be
sketched modifying the approach presented in [8]. Each of these phases adds a
new functionality while increasing the difficulty of the developments. As a
summary, Stage 1 are networks based on trusted nodes in which a prepare and
measure QKD protocol is established between trusted contiguous nodes in order
to share a symmetric key between two final end nodes. This is the state of
current QCN. Stage 2 where quantum entanglement is established between nodes
and qubits can be transmitted without the need of intermediate trusted nodes.
This stage must also introduces Quantum Memories to build networks in which
nodes have the ability to store a qubit in a location for certain time. In
Stage 3 networks have a greater temporary storage capacity that favour
functionalities such as distributed computing. And, finally, in Stage 4
networks will be integrated with quantum computers.

##  4 Integration models of isolated quantum networks

Based on the network components, topologies and SDN paradigm reviewed until
now, we are going to define now two possible integration models - Hierarchical
and Distributed - for isolated QCI. In both approaches, we will be referring
to different domains, understanding a domain as a collection of quantum nodes
that are controlled by a SDNC. In this case, a domain could be equivalent to a
local quantum network.

###  4.1 Hierarchical model

When different network domains with their own controller are integrated into a
larger network, a hierarchy of controllers similar to the hierarchy of ISPs
(Internet System Providers) of the current internet is established. In a
specific country, the Level 1 (L1) local controller knows the status of a
specific network domain and is coordinated by a higher-level regional
controller (L2) which knows the status of several L1 domains. Then, the
regional controllers are coordinated by a national controller (L3) which has
the global network state, as showed in Figure 3. More than 3 level can be set
in a hierarchical model.

Figure 3: Hierarchical model of SDN-based networks integration.

Establishing a hierarchy makes it possible to reduce the complexity of the
operations, increase the overall network control and avoids having a single
point of failure that disables the entire network by replicating the higher
levels SDNC [14]. On the other hand, it presents less flexibility in terms of
asset reorganization.

In the diagram showed in Figure 3, four isolated QCN are integrated in a
hybrid topology composed of a backbone that implements a bus topology
including two satellite nodes. Each subnetwork domain connected to the
backbone has a different topology, specifically, two rings, a simple point-to-
point link and a star topology. Each of these subnetworks has its own L1 SDNC,
which are connected to their associated L2 SDNC and these to the general L3
SDNC. In this scenario, L1 SDNC do not have direct communication links between
them but an interface for communicating with upper level controllers, e.g. L1
SDNC with L2 SDNC.

As an example, Figure 4 shows a sequence diagram for a use case in which two
users located in different network domains and separated by a great distance
want to start a videoconference application encrypted with keys generated by
QKD in the context of the described hierarchical architecture.

Figure 4: Sequence diagram of an encrypted videoconference stablished bewteen
two network domains integrated in a hierarchical model.

The example involves the application (App), controller and QKD device within a
quantum network on the origin (domain 1) and destination (domain 2) and a L2
SDNC. The App on domain 1 makes a request for cryptographic material to the
local controller (L1 SDNC), which ask for availability to the QKD device in
the QN. After confirmation, L1 SDNC detects that the destination App is out of
its domain so that it communicates to L2 SDNC. L2 SDNC analyses the status of
both subnetworks and communicates with L1 SDNC of domain 2. Once the
availability of all the intervening elements in DP has been confirmed and the
two local SDNC have been coordinated, L2 SDNC establishes the interdomain
communication route between domain 1 and 2 and each L1 SDNC establishes the
intradomain route inside its own domain. The connection between QN (1) and QN
(2) is established and the QKD protocol is performed establishing a final
secret symmetric key between both QN. The symmetric key is provided to the App
at each location and, once the process is finished, L1 SDNC (1) communicates
to the L2 SDNC the end of the connection and L2 SDNC to L1 SDNC (2). Finally,
the end users are able to establish an encrypted videoconference.

###  4.2 Distributed model

The second SDN-based quantum communications network integration model is the
distributed controller model. In this case, the management and control of the
entire network is decentralized, in such a way that each domain has its own
SDNC in charge of the intradomain services but at the same time each
controller knows the whole network state. The different controllers
communicate with each other, through an East-West Bound Interface (EWBI), to
exchange network information in order to establish a path for quantum
communications between two network end users belonging to different domains.
In this way, each domain can act as an isolated network or as one more segment
of a large QCN. This type of configuration favors the scalability of the
network, since new domains can become part of it, as well as the robustness of
the network, since it prevents single point of failures [15]. If a network
domain suffers a failure, the whole network is capable of reorganizing the
routes to avoid the damaged segment. However, by not having a central
controller, the amount of information that each controller has to handle
increases, thus increasing the complexity of the operations. In addition,
robust synchronization and prioritization mechanisms are required to avoid
simultaneity in key requests that could lead to wrong key distribution.

As in the previous model, four isolated QCN are integrated in the same hybrid
topology composed of a backbone bus topology, as showed in Figure 5. Each of
these subnetworks has its own SDNC, which are connected through the EWBI to
SDNC from other domains, as the one defined in ETSI GS QKD 018 standard.

Figure 5: Distributed model of SDN-based networks integration.

For this scenario, the same use case as for the hierarchical model is depicted
in Figure 6, but in the context of the described distributed architecture.

Figure 6: Sequence diagram of an encrypted videoconference stablished bewteen
two network domains integrated in a distributed model.

The example involves the application, controller and QKD device within a
quantum network on the origin (domain 1) and destination (domain 2). The App
on domain 1 makes a request for cryptographic material to its associated
controller, which ask for availability to the QKD device in the QN. After
confirmation, SDNC from domain 1 communicates to SDNC from domain 2, which in
turns performs its own availability requests with their QN. Once the
availability of all the intervening elements has been confirmed, the two local
SDNC coordinate and agree the communication route along the interdomain
segments (outside domains 1 and 2) and each SDNC establishes the intradomain
route. The connection between QN (1) and QN (2) is established and a QKD
protocol is performed. Once the process is finished, the end of the connection
is agreed between both SDNC. Finally, the key is provided to the App and the
end users establish an encrypted videoconference.

##  5 Security and technological opportunities and challenges

The security risk once quantum computers reach the capabilities necessary to
break current public key cryptosystems such as RSA or Elliptic-curves
families, could be avoided by merging classical internet infrastructures with
QCI with new quantum cryptographic algorithms. This integration will also
bring new functionalities that could not be achieved with classical internet
in isolation. The new functionalities (those that are currently known, since
others might be discovered in the future) and the advantages that the complete
QI will bring over the classical internet can be summarized as follows:

  * •

High, long-term, security without computational assumptions, i.e. Information
Theoretic Security (ITS). Quantum Key Distribution (QKD) makes possible the
creation of shared random symmetric keys with unconditional security.

  * •

Increase in security levels by applying widely studied quantum cryptographic
protocols and/or quantum assisted protocol such as quantum digital signature,
oblivious transfer, quantum secret sharing, as well as hybrid approaches where
classical and quantum cryptographic tools are integrated to provide
cryptoagile systems.

  * •

Increase in the efficiency of quantum cryptographic protocols compared to
their classic counterparts, under the same operating conditions.

  * •

Assisted information transmission through quantum teleportation.

These opportunities, except for teleportation capabilities, are already
feasible and directly transferable into today’s QCN covering metropolitan
areas. However, there are a number of elements or conditions that still
require development. Throughout the previous sections we have already
commented on some of the elements that still have a low technological maturity
such as QR and QM, but there is also a need of implementing quantum
cryptographic solution assuring the end-to-end security of the entire system.
For that, the following needs to be guaranteed:

  * •

the implementation of quantum channels with measures against physical attacks
or side channel attacks;

  * •

the capability to authenticate new cryptographic modules on the network for
the very first time;

  * •

the availability of self-sustained authentication processes during
communications, for which a Wegman-Carter approach has been proposed [16];

  * •

the capability to guarantee the identity of a user who access a service to
which several users have access;

  * •

the design of hybrid key management systems capable of storing and managing
classical and quantum keys;

  * •

the standardization and certification of quantum cryptographic modules;

  * •

the standardization of hybrid cryptosystems combining classical, quantum and
Post-quantum cryptographic mechanisms.

In addition, although standards related to different SDN interfaces on quantum
networks have already been published by ETSI, there is still a need of
standards for a clear interface definition between controllers and between AP-
CP (i.e. EWBI and NBI, respectively) in both hierarchical and distributed
models but also interfaces between KMS to leverage the integration of quantum
communication subnetworks to rise a full QI. All these challenges, point out
the need of international programs such as EuroQCI.

##  6 Conclusions

Terrestrial quantum communications networks have evolved enormously since
their initial approaches. However, two strategies continue to prevail for the
deployment of this type of networks: infrastructures based on dark fiber
dedicated exclusively to quantum communications and infrastructures with the
coexistence of classical and quantum communications. Regardless of the chosen
strategy, networks of different types have been deployed and are capable of
implementing new functionalities compared to classical networks. Although it
is true that many technological and scientific advances are still required in
terms of the development of elements such as quantum repeaters or quantum
memories, to have the ideal quantum internet, the reality is that it is
already technologically possible to integrate quantum cryptographic protocols
such as QKD to guarantee security of information in current cryptographic
systems, at least in metropolitan distances. The vision of a quantum internet
based on the integration of a set of isolated communication networks leveraged
by the EuroQCI program can be facilitated with an SDN perspective, thanks to
the reduction in complexity in the data plane, virtualizing management and
control functions of the physical elements in the CP. Based on the
architecture of each network, they can be integrated through a hierarchical
model of controllers that manage subnets at different levels or a distributed
model where each controller is the owner of a specific domain and is capable
of coordinating with neighboring domains SDNC. Each of these models has pros
and cons. The hierarchical model is able to reduce the complexity of the
operations and provide full network control, however, greater network control
means a loss in flexibility for asset reorganization. In addition, to avoid
having single points of failure, it requires SDNC redundancy at all levels.
The distributed model facilitates the scalability of the network as well as
its robustness, since it prevents single point of failures, but it presents a
major complexity in operations, demanding robust synchronization mechanisms
and prioritization policies. In this area, a series of opportunities have been
detected, such as the need to guarantee end-to-end security of the entire
crypto-system, to develop hybrid key storage systems or the increase in the
level of technological maturity of the elements required for a quantum
internet with full capabilities. In addition, in order to carry out the
integration of isolated QCN in a hierarchical or distributed model, an NBI is
required between the control plane and the application plane, as well as an
EWBI that details the interface between subnetwork controllers.

## Acknowledgments

This project has received funding from Indra Sistemas S.A; EIT Digital co-
funded by European Institute of Innovation and Technology (EIT), a body of the
European Union; and MCIN with funding from European Union NextGenerationEU
(PRTR-C17.I1); funding from the Comunidad de Madrid. Programa de Acciones
Complementarias, Madrid Quantum; from the European Union’s Horizon Europe
research and innovation programme under the project ”Quantum Security Networks
Partnership” (QSNP, grant agreement No 101114043); and the project “EuroQCI
deployment in Spain” (EuroQCI, grant agreement No 101091638).

## References

  * [1] C. Elliott. The DARPA quantum network. In Quantum Communications and cryptography, CRC Press”, pp. 91-110. (2018)
  * [2] M. Sasaki, M. Fujiwara, H. Ishizuka, et al. Field test of quantum key distribution in the tokyo QKD network. Optics express, 19(11), pp. 10387-10409. (2011). DOI: 10.1364/OE.19.010387.
  * [3] A. Aguado, V. Lopez, D. Lopez, M. Peev, A. Poppe, A. Pastor, J. Folgueira and V. Martin. The engineering of software-defined quantum key distribution networks. IEEE Communications Magazine, 57 (7), pp. 20-26. (2019). DOI: 10.1109/MCOM.2019.1800763
  * [4] Y. Chen et al. An integrated space-to-ground quantum communication network over 4,600 kilometres. Nature, 589(7841), pp. 214-219. (2021).
  * [5] P. Villoresi, F. Tamburini, M. Aspelmeyer, et al. Space-to-ground quantum communication using an optical ground station: a feasibility study. In Quantum Communications and Quantum Imaging II, 5551, pp. 113-120. (2004). DOI: 10.1117/12.564296
  * [6] A. Conrad, S. Isaac, R. Cochran, et al. Drone-based quantum key distribution (QKD). In Free-Space Laser Communications XXXIII, 11678, pp. 177-184. (2021). DOI: 10.1117/12.2582376
  * [7] P. Weiss. Quantum internet: The quirks of quantum mechanics may lead to better computer networks. Science News, 155(14), pp. 220-221. (1999).
  * [8] S.D.C. Wehner, D. C. Elkouss, R. Hanson. Quantum internet: A vision for the road ahead. Science (American Association for the Advancement of Science), 362(6412), pp. 303. (2018). DOI:10.1126/science.aam9288
  * [9] A. Ciurana, J. Martínez-Mateo, M. Peev, et al. Quantum metropolitan optical network based on wavelength division multiplexing. Optics express, 22(2), pp. 1576-1593. (2014). DOI: 10.1364/OE.22.001576
  * [10] F. Hufnagel, et al. Characterization of an underwater channel for quantum communications in the Ottawa River. Optics express, 27(19), pp. 26346-26354. (2019). DOI: 10.1364/OE.27.026346
  * [11] Woo, M. K., et al. One to many QKD network system using polarization-wavelength division multiplexing. IEEE Access, 8, pp. 194007-194014. (2020). DOI: 10.1109/ACCESS.2020.3032992
  * [12] European Commission. The European Quantum Communication Infrastructure (EuroQCI) Initiative. Last accessed: 10 Nov. 2023. https://digital-strategy.ec.europa.eu/en/policies/european-quantum-communication-infrastructure-euroqci
  * [13] K.W. Ross and J.F. Kurose. Computer Networking: A Top-Down Approach. Pearson, 7th Edition, ISBN: 9780133594140. (2017).
  * [14] J. Ali and B-H. Roh. An effective hierarchical control plane for software-defined networks leveraging TOPSIS for end-to-end QoS class-mapping. IEEE Access, 8, pp. 88990-89006. (2020). DOI: 10.1109/ACCESS.2020.2993556
  * [15] Y. E. Oktian, S. Lee, H. Lee and J. Lam. Distributed SDN controller system: A survey on design choice. Computer networks, 121, pp. 100-111. (2017). DOI: 10.1016/j.comnet.2017.04.038
  * [16] Abidin, A., and Larsson, J. Å. Direct proof of security of Wegman–Carter authentication with partially known key. Quantum information processing, 13, pp. 2155-2170. (2014). DOI: 10.1007/s11128-013-0641-6