under which those manifest. We then go beyond to shed light on what the recent
searches of the electroweakinos at the LHC have to say about the viability of
SFOEWPT in the current framework while compatibility with the constraints from
various pertinent theoretical and experimental sectors including the DM sector
is ensured all through. Furthermore, it appears that the GW signals resulting
from the strong FOPTs in these scenarios are likely to remain too weak to be
detected at future dedicated experiments.

As for an outlook, new LHC studies with data from the recently terminated LHC
Run 2 and those that would arrive soon from high luminosity LHC (HL-LHC) are
likely to shed a more unambiguous light on the broad viability of EWBG within
the $Z_{3}$-NMSSM while these continue to explore electroweakinos at larger
masses and in difficult scenarios like the compressed ones. Furthermore,
improvements are possible in the theoretical calculations of several key EWBG
objects, viz., the bubble wall profile, wall velocity and $CP$-violation and
in the dealing of the transport equations which could lend a more accurate
estimate of the relation between the NMSSM parameters and EWBG. With these, a
reassessment of the detectability of such GW signals may be warranted which
might prove the latter’s role as complementary to the LHC searches. A synergy
like this between LHC and GW physics is likely to be rather intriguing and we
reserve such a study for a future work.

##  6 Acknowledgments

AC thanks Harish-Chandra Research Institute (HRI) for hosting him during the
course of this collaborative work. AC also acknowledges partial support from
the Department of Science and Technology, India, through INSPIRE faculty
fellowship (grant no: IFA 15 PH-130, DST/INSPIRE/04/2015/000110). SR is
supported by the funding available from the Department of Atomic Energy (DAE),
Government of India for the Regional Centre for Accelerator-based Particle
Physics (RECAPP) at HRI. SR would like to thank Peter Athron, Sebastian Baum,
Junji Cao, Ulrich Ellwanger, Andrew Fowlie, Tathagata Ghosh, Sabine Kraml,
Indrani Pal, Avik Paul, Krzysztof Rolbiecki, Tim Stefaniak, Di Zhang, Yang
Zhang for helpful discussions and communications. SR also acknowledges the use
of cluster computing available at the High Performance Scientific Computing
facility at HRI and thanks Rajiv Kumar for his technical help at this
facility.

## Appendices

##  Appendix A Matching the NMSSM parameters to those in the THDMS potential

In terms of the $CP$-even Higgs fields $h_{d}$, $h_{u}$ and $s$, the tree-
level $Z_{3}$-NMSSM potential of equation 3, which is relevant for the study
of phase transitions, can be written as Ellwanger:2009dp

$\hskip-8.0ptV_{\rm tree}^{\rm
NMSSM}(h_{d},h_{u},s)=\frac{1}{32}(g_{1}^{2}+g_{2}^{2})\left(h_{d}^{2}-h_{u}^{2}\right)^{2}+\frac{1}{4}\kappa^{2}s^{4}-\frac{1}{2}\lambda\kappa
s^{2}h_{d}h_{u}+\frac{1}{4}\lambda^{2}\left[h_{d}^{2}h_{u}^{2}+s^{2}\left(h_{d}^{2}+h_{u}^{2}\right)\right]\\\
-\frac{1}{\sqrt{2}}\lambda A_{\lambda}sh_{d}h_{u}+\frac{1}{3\sqrt{2}}\kappa
A_{\kappa}s^{3}+\frac{1}{2}m_{H_{d}}^{2}h_{d}^{2}+\frac{1}{2}m_{H_{u}}^{2}h_{u}^{2}+\frac{1}{2}m_{S}^{2}s^{2}.$
(30)

On the other hand, the tree-level $Z_{3}$-symmetric THDMS potential is given
by Elliott:1993ex ; Elliott:1993uc ; Elliott:1993bs ; Athron:2019teq

$\displaystyle V_{\rm tree}^{\rm THDMS}=$
$\displaystyle\frac{1}{2}\lambda_{1}\left|H_{d}\right|^{4}+\frac{1}{2}\lambda_{2}\left|H_{u}\right|^{4}+\left(\lambda_{3}+\lambda_{4}\right)\left|H_{d}\right|^{2}\left|H_{u}\right|^{2}-\lambda_{4}\left|H_{u}^{\dagger}H_{d}\right|^{2}+\lambda_{5}\left|H_{d}\right|^{2}\left|S\right|^{2}$
$\displaystyle+\lambda_{6}\left|H_{u}\right|^{2}\left|S\right|^{2}+\lambda_{7}\left(S^{*2}H_{d}\cdot
H_{u}+{\rm
h.c.}\right)+\lambda_{8}\left|S\right|^{4}+m_{1}^{2}\left|H_{d}\right|^{2}+m_{2}^{2}\left|H_{u}\right|^{2}+m_{3}^{2}\left|S\right|^{2}$
$\displaystyle-m_{4}\left(H_{d}\cdot H_{u}S+{\rm
h.c.}\right)-\frac{1}{3}m_{5}\left(S^{3}+{\rm h.c.}\right).$ (31)

All parameters in equation 30 and 31 are taken to be real as we do not
consider any $CP$-violation in the Higgs sector in this work. We use ${\tt
NMSSMTools}$ to obtain the particle spectrum of the $Z_{3}$-NMSSM at the scale
$M_{\rm SUSY}$. Except for the electroweakinos with masses around a few
hundred GeV, we consider all other SUSY excitations to be much heavier such
that those may be considered effectively decoupled from the physics of phase
transitions. However, to avoid large logarithmic corrections from appearing in
$V_{\rm CW}^{\rm NMSSM}$ due to the top squarks, those are integrated out at
the scale $M_{\rm SUSY}$ in an EFT approach. Below this scale, $V_{\rm
tree}^{\rm NMSSM}$ can be mapped onto $V_{\rm tree}^{\rm THDMS}$. Comparing
equations 30 and 31, after expanding the latter in terms of the component
fields $h_{d},h_{u}$ and ‘$s$’, the matched conditions among the model
parameters of these two scenarios, at the scale $M_{\rm SUSY}$, are given by
Elliott:1993ex ; Elliott:1993uc ; Elliott:1993bs ; Athron:2019teq

$\displaystyle\small{\begin{gathered}\lambda_{1}=\lambda_{2}=\frac{1}{4}\left({g_{1}}^{2}+g_{2}^{2}\right),\quad\lambda_{3}=\frac{1}{4}\left(g_{2}^{2}-g_{1}^{2}\right),\\\
\lambda_{4}=\frac{1}{2}\left(2|\lambda|^{2}-g_{2}^{2}\right),\quad\lambda_{5}=\lambda_{6}=|\lambda|^{2},\quad\lambda_{7}=-\lambda\kappa,\quad\lambda_{8}=|\kappa|^{2},\\\
m_{1}^{2}=m_{H_{d}}^{2},\quad m_{2}^{2}=m_{H_{u}}^{2},\quad
m_{3}^{2}=m_{S}^{2},\quad m_{4}=A_{\lambda}\lambda,\quad
m_{5}=-A_{\kappa}\kappa.\end{gathered}}$ (35)

At one-loop, the only relevant threshold correction that arises (as the top
squarks are integrated out at the scale $M_{\rm SUSY}$) is to $\lambda_{2}$
and is given by Ellis:1991zd ; Haber:1993an ; Casas:1994us ; Carena:1995bx

$\small{\Delta\lambda_{2}=\frac{3y_{t}^{4}A_{t}^{2}}{8\pi^{2}M_{\textup{SUSY}}^{2}}\left(1-\frac{A_{t}^{2}}{12M_{\textup{SUSY}}^{2}}\right),}$
(36)

where $A_{t}$ is the soft-SUSY-breaking top squark-Higgs trilinear coupling in
the scalar potential and $y_{t}$ is the top quark Yukawa coupling, both
defined at the scale $M_{\rm SUSY}$. Note that all the NMSSM parameters are
also provided at the scale $M_{\rm SUSY}$ and in the $\overline{\text{DR}}$
scheme. Thus, after matching, all the THDMS parameters also get defined at the
same scale and in the same renormalization scheme.

##  Appendix B RGEs in the THDMS

We borrow the set of relevant RGEs from reference Kozaczuk:2014kva  which we
use to run the THDMS model parameters from the scale $M_{\rm SUSY}$ to the
scale $m_{t}$. Contributions to the $\beta$-functions from the SM gauge
bosons, the Higgs bosons, the top quark, the higgsinos and the singlino are
included. As for the gauginos, the contribution from the bino is known to be
small (even when $M_{1}$ does not get to be too large, as is the case in our
present analysis) while $M_{2}$ is set at a rather large value. Hence we
ignore their effects. With these, the one-loop $\beta$-functions for the model
parameters $q_{i}$ are given by

$\small{\beta_{q_{i}}=\frac{1}{16\pi^{2}}\frac{\partial}{\partial\ln\Lambda}q_{i}\,,}$
(37)

where ‘$\Lambda$’ is the energy scale. The one-loop RGEs for the quartic
couplings $\lambda_{i},i\in\\{1,2,3,...,8\\}$, the mass parameters ($m_{4,5}$)
and $v_{s}$ appearing in $V_{\rm tree}^{\rm THDMS}$ of equation 31 are as
follows Kozaczuk:2014kva ; Ellwanger:2009dp ; Elliott:1993ex :

$\displaystyle\beta_{\lambda_{1}}=$ $\displaystyle
12\lambda_{1}^{2}+4\lambda_{3}^{2}+4\lambda_{3}\lambda_{4}+2\lambda_{4}^{2}+2\lambda_{5}^{2}-\lambda_{1}(3g_{1}^{2}+9g_{2}^{2})+\frac{3}{4}g_{1}^{4}+\frac{9}{4}g_{2}^{4}+\frac{3}{2}g_{1}^{2}g_{2}^{2}-4\lambda^{4}+4\lambda^{2}\lambda_{1},$
(38) $\displaystyle\beta_{\lambda_{2}}=$ $\displaystyle
12\lambda_{2}^{2}+4\lambda_{3}^{2}+4\lambda_{3}\lambda_{4}+2\lambda_{4}^{2}+2\lambda_{6}^{2}-\lambda_{2}(3g_{1}^{2}+9g_{2}^{2})+\frac{3}{4}g_{1}^{4}+\frac{9}{4}g_{2}^{4}+\frac{3}{2}g_{1}^{2}g_{2}^{2}+12y_{t}^{2}\lambda_{2}-12y_{t}^{4}$
$\displaystyle-4\lambda^{4}+4\lambda^{2}\lambda_{2},$
$\displaystyle\beta_{\lambda_{3}}=$
$\displaystyle(\lambda_{1}+\lambda_{2})(6\lambda_{3}+2\lambda_{4})+4\lambda_{3}^{2}+2\lambda_{4}^{2}+2\lambda_{5}\lambda_{6}-\lambda_{3}(3g_{1}^{2}+9g_{2}^{2})+\frac{3}{4}g_{1}^{4}+\frac{9}{4}g_{2}^{4}-\frac{3}{2}g_{1}^{2}g_{2}^{2}+6y_{t}^{2}\lambda_{3}$
$\displaystyle-4\lambda^{4}+8\lambda^{2}\lambda_{4}+8\lambda^{2}\lambda_{3},$
$\displaystyle\beta_{\lambda_{4}}=$ $\displaystyle
2\lambda_{4}(\lambda_{1}+\lambda_{2}+4\lambda_{3}+2\lambda_{4})+4\lambda_{7}^{2}-\lambda_{4}(3g_{1}^{2}+9g_{2}^{2})+3g_{1}^{2}g_{2}^{2}+6y_{t}^{2}\lambda_{4}+4\lambda^{4}-4\lambda^{2}\lambda_{4},$
$\displaystyle\beta_{\lambda_{5}}=$
$\displaystyle\lambda_{5}(6\lambda_{1}+4\lambda_{5}+8\lambda_{8})+\lambda_{6}(4\lambda_{3}+2\lambda_{4})+8\lambda_{7}^{2}-\frac{1}{2}\lambda_{5}(3g_{1}^{2}+9g_{2}^{2})-12\kappa^{2}\lambda^{2}-4\lambda^{4}+4\kappa^{2}\lambda_{5}+6\lambda^{2}\lambda_{6},$
$\displaystyle\beta_{\lambda_{6}}=$
$\displaystyle\lambda_{5}(4\lambda_{3}+2\lambda_{4})+\lambda_{6}(6\lambda_{2}+4\lambda_{6}+8\lambda_{8})+8\lambda_{7}^{2}-\frac{1}{2}\lambda_{6}(3g_{1}^{2}+9g_{2}^{2})+6y_{t}^{2}\lambda_{6}-16\kappa^{2}\lambda^{2}-4\lambda^{4}$
$\displaystyle+4\kappa^{2}\lambda_{6}+6\lambda^{2}\lambda_{6},$
$\displaystyle\beta_{\lambda_{7}}=$
$\displaystyle\lambda_{7}(2\lambda_{3}+4\lambda_{4}+4\lambda_{5}+4\lambda_{6}+4\lambda_{8})-\frac{1}{2}\lambda_{7}(3g_{1}^{2}+9g_{2}^{2})+3y_{t}^{2}\lambda_{7}+8\kappa\lambda^{3}+4\kappa^{2}\lambda_{7}+6\lambda^{2}\lambda_{7},$
$\displaystyle\beta_{\lambda_{8}}=$ $\displaystyle
2\lambda_{5}^{2}+2\lambda_{6}^{2}+4\lambda_{7}^{2}+20\lambda_{8}^{2}+8(\kappa^{2}+\lambda^{2})\lambda_{8}-16\kappa^{2}-4\lambda^{4},$
$\displaystyle\beta_{m_{4}}=$
$\displaystyle(2\lambda_{3}+4\lambda_{4}+2\lambda_{5}+2\lambda_{6}+4\lambda^{2}+2\kappa^{2}-\frac{9}{2}g_{2}^{2}-\frac{3}{2}g_{1}^{2}+3y_{t}^{2})m_{4}+4\lambda_{7}m_{5},$
$\displaystyle\beta_{m_{5}}=$
$\displaystyle(12\lambda_{8}+6\lambda^{2}+6\kappa^{2})m_{5}+12\lambda_{7}m_{4},$
$\displaystyle\beta_{v_{s}}=$
$\displaystyle-2v_{s}(\kappa^{2}+\lambda^{2}).\quad$

The individual THDMS parameters at the scale $m_{t}$ are then calculated using
the expression888Note that the NMSSM parameters are in the $\overline{\rm DR}$
scheme Ellwanger:2009dp  whereas, for the calculations of phase transitions,
the THDMS parameters are provided in the $\overline{\rm MS}$ scheme
Kozaczuk:2014kva . We ignore the effect of this shift in the scheme as this
would modify the quartic couplings only mildly due to small threshold
corrections. To convince ourselves, we have compared the mass-eigenvalues of
the $CP$-even Higgs mass-squared matrix and the related mixing matrix that are
obtained from CosmoTransitions to the corresponding ones obtained using
NMSSMTools and their agreements are found to be within the level of a few
percent. (see reference Kozaczuk:2014kva  for details)

$\displaystyle q_{i}(m_{t})\simeq q_{i}(M_{\rm
SUSY})-\beta_{q_{i}}\ln\frac{M_{\rm SUSY}}{m_{t}}\,.$ (39)

##  Appendix C Field-dependent masses and the daisy corrections

Here we present the field-dependent mass-squared matrices for the scalar
sector which are derived from equation 31 Elliott:1993ex ; Athron:2019teq .
The $3\times 3$, symmetric matrix (${\cal M}_{H}^{2}$) for the $CP$-even
scalars, in the basis $\\{h_{d},h_{u},s\\}$, is given by

$\tiny{\begin{pmatrix}m_{1}^{2}+\tfrac{3}{2}\lambda_{1}h_{d}^{2}+\tfrac{1}{2}\lambda_{5}s^{2}+\tfrac{1}{2}(\lambda_{3}+\lambda_{4})h_{u}^{2}&-\tfrac{1}{\sqrt{2}}m_{4}{s}+\tfrac{1}{2}\lambda_{7}s^{2}+(\lambda_{3}+\lambda_{4})h_{u}h_{d}&-\tfrac{1}{\sqrt{2}}m_{4}h_{u}+\lambda_{5}h_{d}{s}+\lambda_{7}h_{u}{s}\\\
...&m_{2}^{2}+\tfrac{3}{2}\lambda_{2}h_{u}^{2}+\tfrac{1}{2}\lambda_{6}s^{2}+\tfrac{1}{2}(\lambda_{3}+\lambda_{4})h_{d}^{2}&-\tfrac{1}{\sqrt{2}}m_{4}h_{d}+\lambda_{7}h_{d}{s}+\lambda_{6}h_{u}{s}\\\
...&...&m_{3}^{2}-\sqrt{2}m_{5}{s}+\tfrac{1}{2}\lambda_{5}h_{d}^{2}+\lambda_{7}h_{u}h_{d}+\tfrac{1}{2}\lambda_{6}h_{u}^{2}+3\lambda_{8}s^{2}\end{pmatrix}\;,}$
(40)

whereas, the corresponding one for the $CP$-odd scalars (${\cal M}_{A}^{2}$),
in the same basis as above, can be written as

$\tiny{\begin{pmatrix}m_{1}^{2}+\tfrac{1}{2}\lambda_{1}h_{d}^{2}+\tfrac{1}{2}\lambda_{5}s^{2}+\tfrac{1}{2}(\lambda_{3}+\lambda_{4})h_{u}^{2}&\tfrac{1}{\sqrt{2}}m_{4}{s}-\tfrac{1}{2}\lambda_{7}s^{2}&\tfrac{1}{\sqrt{2}}m_{4}h_{u}+\lambda_{7}h_{u}{s}\\\
...&m_{2}^{2}+\tfrac{1}{2}\lambda_{2}h_{u}^{2}+\tfrac{1}{2}\lambda_{6}s^{2}+\tfrac{1}{2}(\lambda_{3}+\lambda_{4})h_{d}^{2}&\tfrac{1}{\sqrt{2}}m_{4}h_{d}+\lambda_{7}h_{d}{s}\\\
...&...&m_{3}^{2}+\sqrt{2}m_{5}{s}+\tfrac{1}{2}\lambda_{5}h_{d}^{2}-\lambda_{7}h_{u}h_{d}+\tfrac{1}{2}\lambda_{6}h_{u}^{2}+\lambda_{8}s^{2}\end{pmatrix}\;.}$
(41)

On the other hand, the field-dependent $2\times 2$, symmetric mass-squared
matrix for the charged Higgs sector (${\cal M}_{H^{\pm}}^{2}$) in the basis
$\\{h_{d},h_{u}\\}$ is given by

$\small{{\cal
M}_{H^{\pm}}^{2}=\begin{pmatrix}m_{1}^{2}+\tfrac{1}{2}\lambda_{5}s^{2}+\tfrac{1}{2}\lambda_{1}h_{d}^{2}+\tfrac{1}{2}\lambda_{3}h_{u}^{2}&\tfrac{1}{\sqrt{2}}m_{4}s-\tfrac{1}{2}\lambda_{7}s^{2}-\tfrac{1}{2}\lambda_{4}h_{d}h_{u}\\\
...&m_{2}^{2}+\tfrac{1}{2}\lambda_{6}s^{2}+\tfrac{1}{2}\lambda_{3}h_{d}^{2}+\tfrac{1}{2}\lambda_{2}h_{u}^{2}\end{pmatrix}\;.}$
(42)

The mass parameters $m_{1}^{2}$, $m_{2}^{2}$ and $m_{3}^{2}$ are determined
via the minimization conditions (the tadpoles) of $V_{\rm tree}^{\rm THDMS}$
of equation 31 and are given by

$\displaystyle m_{1}^{2}$
$\displaystyle=-\tfrac{1}{2}(\lambda_{3}+\lambda_{4})v_{u}^{2}-\tfrac{1}{2}\lambda_{1}v_{d}^{2}-\tfrac{1}{2}\lambda_{5}v_{\\!{}_{S}}^{2}-\tfrac{1}{2}\lambda_{7}\frac{v_{u}v_{\\!{}_{S}}^{2}}{v_{d}}+\tfrac{1}{\sqrt{2}}m_{4}\frac{v_{u}v_{\\!{}_{S}}}{v_{d}},$
(43) $\displaystyle m_{2}^{2}$
$\displaystyle=-\tfrac{1}{2}\lambda_{2}v_{u}^{2}-\tfrac{1}{2}(\lambda_{3}+\lambda_{4})v_{d}^{2}-\tfrac{1}{2}\lambda_{6}v_{\\!{}_{S}}^{2}-\tfrac{1}{2}\lambda_{7}\frac{v_{d}v_{\\!{}_{S}}^{2}}{v_{u}}+\tfrac{1}{\sqrt{2}}m_{4}\frac{v_{d}v_{\\!{}_{S}}}{v_{u}},$
$\displaystyle m_{3}^{2}$
$\displaystyle=-\tfrac{1}{2}\lambda_{6}v_{u}^{2}-\tfrac{1}{2}\lambda_{5}v_{d}^{2}-\lambda_{8}v_{\\!{}_{S}}^{2}-\lambda_{7}v_{d}v_{u}+\tfrac{1}{\sqrt{2}}m_{4}\frac{v_{u}v_{d}}{v_{\\!{}_{S}}}+\tfrac{1}{\sqrt{2}}m_{5}v_{\\!{}_{S}}.$

Diagonalization of ${\cal M}_{A}^{2}$ in equation 41 leads to two neutral
$CP$-odd scalars and a Goldstone boson. Similarly, diagonalization of ${\cal
M}_{H^{\pm}}^{2}$ in equation 42 results in a charged Higgs boson and a
charged Goldstone boson. The masses of these neutral and charged Goldstone
bosons are zero at the electroweak minima where the $CP$-even fields acquire
values $\\{v_{d},v_{u},v_{\\!{}_{S}}\\}$. However, the gauge-fixing terms in
the Lagrangian alter the tree-level mass matrices. In the Feynman gauge that
we opt for this work, the mass matrices get modified and the Goldstone bosons
no longer remain massless at the electroweak minima. We include these gauge-
dependent contributions to ${\cal M}_{A}^{2}$ and ${\cal M}_{H^{\pm}}^{2}$
which are listed in reference Athron:2019teq .

Note that the one-loop CW potential (of equation 3.2.1) shifts the location of
the electroweak minimum from where it was appearing in the field space for the
tree-level potential. In the $\overline{\rm MS}$ renormalization scheme that
we adopt, one can find suitable counter-terms (as described in reference
Cline:2011mm ) that modify the quadratic terms
($m_{1}^{2}h_{d}^{2}+m_{2}^{2}h_{u}^{2}+m_{3}^{2}s^{2}$) in the potential (of
equation 31) thus ensuring the minimum of the effective potential coincides
with that of the tree-level potential. The accompanying shifts in the mass
parameters of the potential (obtained from equation 43) are given by

$\small{m_{1}^{2}\rightarrow m_{1}^{2}-\frac{1}{v_{d}}\frac{\partial
V_{\text{{CW}}}}{\partial h_{d}}\Bigr{|}_{\begin{subarray}{c}h_{d}=v_{d}\\\
h_{u}=v_{u}\\\ s=v_{\\!{}_{S}}\end{subarray}},\quad m_{2}^{2}\rightarrow
m_{2}^{2}-\frac{1}{v_{u}}\frac{\partial V_{\text{CW}}}{\partial
h_{u}}\Bigr{|}_{\begin{subarray}{c}h_{d}=v_{d}\\\ h_{u}=v_{u}\\\
s=v_{\\!{}_{S}}\end{subarray}},\quad m_{3}^{2}\rightarrow
m_{3}^{2}-\frac{1}{v_{s}}\frac{\partial V_{\text{CW}}}{\partial
s}\Bigr{|}_{\begin{subarray}{c}h_{d}=v_{d}\\\ h_{u}=v_{u}\\\
s=v_{\\!{}_{S}}\end{subarray}}\,,}$ (44)

Note that in the tree-level field-dependent mass-squared matrices (see
equations 40, 41 and 42) the values of $m_{1}^{2}$, $m_{2}^{2}$ and
$m_{3}^{2}$ are without this modification since the latter are solutions of
the corresponding tree-level tadpole equations as presented in equation 43.

In the fermionic sector, we consider the top quark, the bottom quark, the tau
lepton along with the four neutralinos ($\chi_{1,2,3,4}^{0}$) and the one
chargino ($\chi_{{}_{1}}^{\pm}$), since the wino-like states are taken to be
much heavier and hence are decoupled from the physics of phase transitions.
The field-dependent masses of the top quark, the bottom quark and the tau
lepton are given by Ellwanger:2009dp

$\begin{gathered}m_{t}=\tfrac{1}{\sqrt{2}}y_{t}h_{u}\,,\quad
m_{b}=\tfrac{1}{\sqrt{2}}y_{b}h_{d}\,,\quad
m_{\tau}=\tfrac{1}{\sqrt{2}}y_{\tau}h_{d}\,.\end{gathered}$ (45)

The field-dependent $4\times 4$, symmetric neutralino mass matrix in the basis
$\\{\widetilde{B},~{}\widetilde{H}_{d}^{0},~{}\widetilde{H}_{u}^{0},~{}\widetilde{S}\\}$
is given by

$\small{{\cal
M}_{\chi^{0}}=\begin{pmatrix}M_{1}&-\frac{g_{1}h_{d}}{2}&\frac{g_{1}h_{u}}{2}&0\\\
...&0&-\frac{\lambda s}{\sqrt{2}}&-\frac{\lambda h_{u}}{\sqrt{2}}\\\
...&...&0&-\frac{\lambda h_{d}}{\sqrt{2}}\\\ ...&...&...&\sqrt{2}\kappa
s\end{pmatrix}\;.}$ (46)

On the other hand, the field-dependent mass of the higgsino-like chargino is
given approximately by $m_{{{}_{\chi}}_{{}_{1}}^{\pm}}\simeq\frac{\lambda
s}{\sqrt{2}}$. Note that the masses of these electroweakinos are given in
terms of the NMSSM model parameters which are defined at the scale $M_{\rm
SUSY}$. This is acceptable for our purpose since these masses do not appear in
the tree-level potential.999The electroweakino masses, however, contribute to
higher-order (starting at one-loop) corrections to the tree-level potential.
Thus, the consideration of running of these masses amounts to having an even
higher-order correction to the potential. Hence we ignore such a running.

The field-dependent masses of the gauge bosons, $W^{\pm}$ and $Z$, are given
by

$\small{\begin{gathered}m_{W^{\pm}}^{2}=\tfrac{1}{4}g_{2}^{2}\left(h_{u}^{2}+h_{d}^{2}\right),\quad
m_{Z}^{2}=\tfrac{1}{4}\left({g_{1}}^{2}+g_{2}^{2}\right)\left(h_{u}^{2}+h_{d}^{2}\right).\end{gathered}}$
(47)

Note that in the daisy potential of equation 18, $M_{h}^{2}$ and $M_{V}^{2}$
are the eigenvalues of the thermally improved (i.e., Debye-corrected) mass-
squared matrices for the Higgs and the gauge bosons, respectively, i.e.,
generically, $M^{2}=$ eigenvalues$[{\cal M}^{2}+\Delta(T^{2})]$ where
$\Delta(T^{2})=c_{ij}T^{2}$ and $c_{ij}$’s are the so-called daisy
coefficients. From the high temperature expansion of the thermal one-loop
potential $\widetilde{V}_{T}$ (of equation 15) using equation 17, the daisy
coefficients can be found from the following relation:

$\small{c_{ij}=\left.\frac{1}{T^{2}}\frac{\partial^{2}\widetilde{V}_{T}}{\partial\phi_{i}\partial\phi_{j}}\right|_{T^{2}\gg
m^{2}}.}$ (48)

For an FOPT, the daisy correction is especially important since it has an
impact on the all very critical cubic term of the potential at a finite
temperature. Effects of only the scalars and the longitudinal modes of the
vectors are included in this contribution. Thermal contributions to the
transverse modes are suppressed due to gauge symmetry Espinosa:1992kf . With
these in mind, the various daisy coefficients (neglecting the electroweakino
contributions) are as follows Comelli:1996vm ; Basler:2018cwe ;
Carrington:1991hz ; Athron:2019teq :

$\displaystyle c^{H}_{11}=c^{A}_{11}=c^{H^{\pm}}_{11}$
$\displaystyle=\tfrac{1}{24}\left(6\lambda_{2}+4\lambda_{3}+2\lambda_{4}+2\lambda_{6}+6y_{t}^{2}+\frac{3}{2}{g_{1}}^{2}+\frac{9}{2}g_{2}^{2}\right),$
(49a) $\displaystyle c^{H}_{22}=c^{A}_{22}=c^{H^{\pm}}_{22}$
$\displaystyle=\tfrac{1}{24}\left(6\lambda_{1}+4\lambda_{3}+2\lambda_{4}+2\lambda_{5}++6y_{b}^{2}+2y_{\tau}^{2}+\frac{3}{2}{g_{1}}^{2}+\frac{9}{2}g_{2}^{2}\right),$
(49b) $\displaystyle c^{H}_{33}=c^{A}_{33}$
$\displaystyle=\tfrac{1}{24}\left(4\lambda_{5}+4\lambda_{6}+8\lambda_{8}\right),$
(49c)

where the subscripts $\\{1,2,3\\}$ refer to the fields $\\{h_{d},h_{u},s\\}$.
Note that the gauge symmetries plus the discrete $Z_{3}$ symmetry of the model
set the off-diagonal terms of the $\Delta(T^{2})$ matrix to zero (i.e.
$\Delta(T^{2})$ is a diagonal matrix). The longitudinal components of the
gauge bosons receive thermal corrections. For $W^{\pm}$ bosons the correction
is $c^{W^{\pm}}_{L}=2g_{2}^{2}T^{2}$. Thus, the thermally improved mass of the
longitudinally polarized $W^{\pm}$ bosons is given by

$\displaystyle\small{M_{W^{\pm}_{L}}^{2}={1\over
4}g_{2}^{2}(h^{2}_{d}+h^{2}_{u})+2g_{2}^{2}T^{2}.}$ (50)

Similarly, the longitudinal components of the $Z$-boson and the photon ($A$)
fields also receive thermal corrections. Their masses can be determined by
diagonalizing the following matrix:

$\small{\frac{1}{4}(h^{2}_{d}+h^{2}_{u})\begin{pmatrix}g_{2}^{2}&-g_{2}g_{1}\\\
-g_{1}g_{2}&g_{1}^{2}\end{pmatrix}+\begin{pmatrix}2g_{2}^{2}T^{2}&0\\\
0&2g_{1}^{2}T^{2}\end{pmatrix}.}$ (51)

The thermally improved masses of the longitudinally polarized $Z$-boson and
the photon are given by

$\small{M_{Z_{L},\gamma_{L}}^{2}=\frac{1}{8}(g_{2}^{2}+g_{1}^{2})(h_{d}^{2}+h_{u}^{2})+(g_{2}^{2}+g_{1}^{2})T^{2}\pm\delta,}$
(52)

where

$\small{\delta=\sqrt{\frac{1}{64}(g_{2}^{2}+g_{1}^{2})^{2}(h_{d}^{2}+h_{u}^{2}+8T^{2})^{2}-g_{2}^{2}g_{1}^{2}T^{2}(h_{d}^{2}+h_{u}^{2}+4T^{2})}.}$
(53)

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