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[b,c]Subhojit Roy

# Prospects of a Strong first-order Electroweak Phase Transition in the
$Z_{3}$-NMSSM

Arindam Chatterjee  AseshK Datta

###### Abstract

We have studied the viability and possible patterns of a strong first-order
electroweak phase transition (SFOEWPT) within the $Z_{3}$-symmetric Next-to-
Minimal Supersymmetric Standard Model (NMSSM), in view of the latest
experimental results from the dark matter (DM) sector, Higgs sector and the
searches of the lighter chargino and neutralinos at the Large Hadron Collider
(LHC). We show that the region of parameter space with relatively small
$\mu_{\mathrm{eff}}$ that favors an SFOEWPT has started to get excluded from
the searches at the LHC and various DM experiments. However, there still
remain phenomenologically much involved and compatible regions for an SFOEWPT
that are yet not sensitive to the latest LHC and DM searches. We further
estimate the production of stochastic gravitational waves (GW) from an SFOEWPT
within and without the bag model and the prospects of detecting such signals
at various future/proposed GW experiments.

##  1 Introduction

Electroweak Baryogenesis (EWBG) is the process giving rise to the observed
baryon asymmetry of the Universe (BAU) at the electroweak scale. This process
requires a strong first-order phase transition (FOPT) to occur during the
electroweak phase transition (EWPT) to met one of the well-known Sakharov’s
conditions, that is, the system has to go out of equilibrium. However, such an
FOPT (and hence EWBG) is not possible in the Standard Model (SM) which can be
traced to the fact that the experimentally observed SM-like Higgs boson has
been found to be too heavy ($m_{h_{\mathrm{SM}}}\approx 125$ GeV) for the
purpose. It is well known that popular supersymmetry (SUSY) extensions of the
SM, viz., the Minimal SUSY SM (MSSM), its next-to-minimal version (NMSSM),
etc., could provide a favorable setup for EWBG to take place. An effective
Higgs potential that favours an FOPT in the early Universe can be found in the
presence of an extended Higgs sector and other scalar degrees of freedom (in
particular, the top squarks) of these scenarios. However, null results from
the searches for suitable light top squarks at the LHC have made the situation
extremely tight for an strong first-order electroweak phase transition
(SFOEWPT) to take place in the MSSM. On the other hand, the presence of a
gauge singlet scalar field makes the NMSSM more favorable for SFOEWPT.

In this paper, we examine the possibilities of an SFOEWPT in the
$Z_{3}$-symmetric NMSSM ($Z_{3}$-NMSSM) after meeting the latest constraints
from the LHC and the DM sector. The effective higgsino mass parameter
($\mu_{\mathrm{eff}}$) is the most important parameter of the present model
that connects the phenomenology of the DM, the LHC and the EWPT sectors. This
is since apart from $\mu_{\mathrm{eff}}$ there are a few other model
parameters that appear both in the Higgs and the electroweakino sectors. We
will find that such a connection raises the tantalising possibility that
relatively light higgsinos with masses below a few hundred GeV, in the
presence of light singlet-like scalars and a lighter singlino and/or a bino,
can somewhat comfortably escape the latest LHC constraints from their searches
[1, 2]. In particular, we would like to explore how small a
$\mu_{\mathrm{eff}}$ might still be feasible in the light of the latest
experimental constraints given that an SFOEWPT prefers the same and which is
also motivated by ‘naturalness’. In addition to this, stochastic gravitational
waves (GW) can be generated from the dynamics of the nucleated bubbles during
an SFOEWPT. In this work, we estimate the GW spectra for a chosen set of
benchmark scenarios and study the prospects of detecting GWs in future
proposed experiments.

The present work is organized as follows. We provide a short description of
the $Z_{3}$-NMSSM scenario in section 2. We discuss our results in section 3
where we present a few benchmark scenarios that exhibit SFOEWPT but are
excluded from the LHC searches a for the electroweakinos. Later on we
demonstrate few scenarios that pass latest constraints from various
experiments. We discuss how SFOEWPT is realized even for a scenarios with a
relatively small $\mu_{\mathrm{eff}}$. Finally, we present the prospects of
detecting the GW in future proposed GW experiments in these viable benchmark
scenarios. In section 4 we provide an overview of our present work and a
forecast for the future. The details of this work can be found in reference
[3].

##  2 The theoretical framework of the $Z_{3}$-NMSSM

The superpotential of the $R$-parity conserving $Z_{3}$-NMSSM is given by

${\cal W}={\cal
W}_{\mathrm{MSSM}}|_{\mu=0}+\lambda\widehat{S}\widehat{H}_{u}\cdot\widehat{H}_{d}+{\kappa\over
3}\widehat{S}^{3}\,,$ (1)

where ${\cal W}_{\mathrm{MSSM}}|_{\mu=0}$ is the MSSM superpotential without
the higgsino mass term (the $\mu$-term). $\widehat{H}_{u},\widehat{H}_{d}$ and
$\widehat{S}$ denote the $SU(2)$ Higgs doublet superfields and the gauge
singlet superfield, respectively. The (real) scalar field of the singlet
superfield $\widehat{S}$ develops a non-zero vacuum expectation value (vev),
$v_{\\!{}_{S}}$, during EWPT which generates an effective $\mu$-term as
$\mu_{\rm eff}=\lambda v_{\\!{}_{S}}/\sqrt{2}$. The soft SUSY-breaking
Lagrangian is given by

$-\mathcal{L}^{\rm soft}=-\mathcal{L_{\rm MSSM}^{\rm
soft}}|_{B\mu=0}+m_{S}^{2}|S|^{2}+(\lambda A_{\lambda}SH_{u}\cdot
H_{d}+\frac{\kappa}{3}A_{\kappa}S^{3}+{\rm h.c.})\,,$ (2)

where $m_{S}$ denotes the soft SUSY-breaking mass of the singlet scalar field,
‘$S$’, $A_{\lambda}$ and $A_{\kappa}$ are trilinear soft parameters with mass
dimensions one. These complex scalar fields can be expressed as

$\displaystyle H_{u}=\begin{pmatrix}H_{u}^{+}\\\
\tfrac{1}{\sqrt{2}}\left(h_{u}+ia_{u}\right)\end{pmatrix},\quad\quad
H_{d}=\begin{pmatrix}\tfrac{1}{\sqrt{2}}\left(h_{d}+ia_{d}\right)\\\
H_{d}^{-}\end{pmatrix},\quad\quad S=\frac{1}{\sqrt{2}}\left(s+i\sigma\right),$
(3)

where $\langle h_{u}\rangle=v_{u}$, $\langle h_{d}\rangle=v_{d}$ and $\langle
s\rangle=v_{s}$ are the vevs of the real components ($CP$-even) of the neutral
scalar fields at zero temperature. Note that
$\sqrt{v_{u}^{2}+v_{d}^{2}}=v\simeq 246$ GeV with $\tan\beta=v_{u}/v_{d}$. The
present model consists of three $CP$-even, two $CP$-odd and two charged
physical Higgs states. The neutralino sector of the $Z_{3}$-symmetric NMSSM
consists of five neutralinos which are mixtures of bino ($\widetilde{B}$),
wino ($\widetilde{W}^{0}_{3}$), two higgsinos ($\widetilde{H}^{0}_{d}$,
$\widetilde{H}^{0}_{u}$) and a singlino ($\widetilde{S}$), the last one being
the fermionic component of the singlet superfield $\widehat{S}$ appearing in
the superpotential of equation 1. The chargino sector of the $Z_{3}$-symmetric
NMSSM and MSSM is exactly similar.

##  3 Results

The latest theoretical and experimental constraints eliminate a certain region
of parameter space that exhibit SFOEWPT. The theoretical constraints ensure
the allowed region of parameter space free from the unphysical global minimum
of the scalar potential, tachyonic states and the evolutions of various model
couplings with energy do not encounter Landau poles, etc. The experimental
constraints include those coming from the hunt for new physics at the
colliders, as well as from the Higgs, DM and flavour sectors. Further, we
ensure that after the SFOEWPT, which facilitates EWBG, the system finally ends
up in the physical electroweak minimum at zero temperature. In this work, we
use publicly available packages like NMSSMTools (v5.5.3), CheckMATE (v2.0.34),
SModelS (v2.1.1), HiggsBounds (v5.8.0), HiggsSignals (v2.5.0), and
CosmoTransitions (v2.0.6) for our numerical analysis and take into account all
the above mentioned constraints.

We present a few benchmark scenarios in which a SFOEWPT is realized and which
satisfy all other theoretical and experimental constraints but are excluded
from the searches of the lighter electroweakinos at the LHC. A small value of
$\mu_{\mathrm{eff}}$ that prefers SFOEWPT draws a significant constraints from
the latest DMDD and LHC searches [1]. Our goal has been to identify the
smallest possible values of $\mu_{\mathrm{eff}}$ (except for the scenarios
when the higgsino-like neutralino/chargino is the lightest of all the
electroweakinos) that are allowed under different circumstances. We also
present a few allowed scenarios with optimally light higgsinos that satisfies
all the constraints and favor an SFOEWPT.

Inputs/Observables | BP-D1 | BP-D2 | BP-D3
---|---|---|---
$\lambda,\,\kappa,\,\tan\beta$ | $0.683,\,0.060,\,4.77$ | $0.547,\,0.044,\,2.87$ | $0.565,\,0.071,\,2.87$
$A_{\lambda},\,A_{\kappa},\,\mu_{\mathrm{eff}},\,M_{1}$ (GeV) | $-1352.3,\,134.5,\,-274.4,\,478.8$ | $978.4,\,-110.0,\,308.0,\,460.3$ | $963.5,\,-112.5,\,308.0,\,-57.2$
$m_{\widetilde{Q}_{3}},\,m_{\widetilde{U}_{3},A_{t}}$ (GeV) |  2956.7, 3378.3, $-1019.7$ | 3710.8, 3562.8, 2204.0 | 3710.8, 3562.8, 2204.0
$m_{\chi_{1,2,3,4}^{0},\chi_{1}^{\pm}}$ (GeV) | 60.9, -304.3, 307.9, 479.4, -284.1 |  60.6, 312.7, $-338.3$, 468.1, 316.3 |  $-59.6$, 91.1, 327.2, $-338.4$, 316.3
$m_{h_{1}},\,m_{h_{2}},\,m_{a_{1}},\,m_{H^{\pm}}$ (GeV) | 79.2, 124.4, 126.6, 1359.0 | 78.1, 122.2, 109.5, 963.8 | 86.9, 123.0, 142.6, 963.6
$\Omega h^{2}$ | $4.9\times 10^{-4}$ | $4.4\times 10^{-4}$ | $4.8\times 10^{-3}$
$\sigma^{\rm SI}_{\chi^{0}_{1}-p(n)}\times\xi$ (cm2) | $4.5(4.6)\times 10^{-47}$ | $2.4(2.5)\times 10^{-47}$ | $2.5(2.6)\times 10^{-47}$
$\sigma^{\rm SD}_{\chi^{0}_{1}-p(n)}\times\xi$ (cm2) | $3.5(3.2)\times 10^{-42}$ | $7.6(5.8)\times 10^{-43}$ | $1.9(1.5)\times 10^{-43}$
First $T_{c}$ (GeV) / Transition type | 129.4 / 1st-order | 151.5 / 1st-order | 165.7 / 1st-order
$\\{h_{d},h_{u},s\\}_{{}_{\text{False\\_vac.}}}$ (GeV) | $\\{0,~{}0,~{}0\\}$ | $\\{0,~{}0,~{}0\\}$ | $\\{0,~{}0,~{}0\\}$
$\\{h_{d},h_{u},s\\}_{{}_{\text{True\\_vac.}}}$ (GeV) | $\\{25.5,145.6,-474.4\\}$ | $\\{0,0,539.9\\}$ | $\\{0,0,557.5.9\\}$
Second $T_{c}$ (GeV) / Transition type | $-$ | 112.7 / 2nd-order | 105.6 / 1st-order
$\\{h_{d},h_{u},s\\}_{{}_{\text{False\\_vac.}}}$ (GeV) | $-$ | $\\{0,~{}0,661.7\\}$ | $\\{0,~{}0,662.3\\}$
$\\{h_{d},h_{u},s\\}_{{}_{\text{True\\_vac.}}}$ (GeV) | $-$ | $\\{9.5,31.5,668.2\\}$ | $\\{12.8,41.6,669.0\\}$
$T_{n}$ (GeV) / (Nucleation) Transition type | $-$ | 96.2 / 1st-order | 55.9 / 1st-order
$\\{h_{d},h_{u},s\\}_{{}_{\text{False\\_vac.}}}$ (GeV) | $-$ | $\\{0,~{}0,~{}0\\}$ | $\\{0,~{}0,~{}0\\}$
$\\{h_{d},h_{u},s\\}_{{}_{\text{True\\_vac.}}}$ (GeV) | $-$ | $\\{67.0,197.8,774.8\\}$ | $\\{68.1,199.2,759.2\\}$
$\gamma_{{}_{\rm{EW}}}=\Delta_{SU(2)}/T_{n}$ | $-$ | $2.2$ | $3.8$
CheckMATE result | Excluded | Excluded | Excluded
$r$-value | 1.12 | 1.01 | 2.13
Analysis ID |  CMS$\\_$SUS$\\_$16$\\_$039 |  CMS$\\_$SUS$\\_$16$\\_$039 |  CMS$\\_$SUS$\\_$16$\\_$039
Signal region ID |  SR$\\_$A30 |  SR$\\_$A30 |  SR$\\_$G05
Table 1: Benchmark scenarios allowed by all relevant theoretical and
experimental constraints except for those from the LHC searches for the
electroweakinos.

In table 1, we present a few benchmark points that exhibit SFOEWPT but are
ruled out by the electroweakino searches at the LHC. A dedicated CheckMATE
analysis excludes BP-D1 (with $r$=$1.12$) and BP-D2 (with $r$=$1.01$) via a
CMS analysis [4] of 35.6 fb-1 data in the $3\ell+\not{E}_{T}$ final state
where an opposite-sign, same-flavor lepton ($e$ or $\mu$)-pair originates from
the decay of an on-shell $Z$-boson coming from the decay of a heavier
higgsino-like neutralino. Latest LHC analyses for the same final state which
exploit 139 fb-1 of data would anyway exclude more convincingly (i.e., with a
larger $r$-value) such scenarios with $\mu_{\mathrm{eff}}$ as small as 275
GeV. In the final states with more than 3 leptons signal region of the same
CMS analysis [4] excludes BP-D3 rather emphatically ($r=2.13$). Note that
BP-D2 and BP-D3 benchmark scenarios are cosmologically good points, however
BP-D1 is not. In BP-D1, the system does not nucleate successfully to the true
minimum phase and hence it would remain trapped at the metastable false
minimum ($\\{0,0,0\\}$).

Input/Observables | BP-A1 | BP-A2 | BP-A3 | BP-A4
---|---|---|---|---
$\lambda,\,\kappa,\,\tan\beta$ | 0.609, 0.326, 1.98 | 0.609, 0.326, 1.98 | 0.633, 0.216, 1.79 | 0.523, 0.041, 3.65
$A_{\lambda},A_{\kappa},\mu_{\mathrm{eff}},M_{1}$ (GeV) | 477.0, 38.7, 421.8, 480.1 | 477.0, 37.8, 421.8, -365.1 | -558.7, 46.3, -398.7, 286.3 | -1253.9, 138.1, -334.5, -143.8
$m_{\widetilde{Q}_{3}},\,m_{\widetilde{U}_{3}},\,A_{t}$ (GeV) | 4262.7, 3450.4, -639.2 | 4262.7, 3450.4, -639.2 | 3950.3, 3544.4, 1372 | 2292, 3435.8, 3862.4
$m_{\chi_{1,2,3,4}^{0},\chi_{1}^{\pm}}$ (GeV) | 396, -446, 477, 510, 432 | -361, 415, -448, 493, 432 | 285, -290, -422, -427, -412 | -61, -139, -359, 360, -345
$m_{h_{1}},\,m_{h_{2}},\,m_{a_{1}},\,m_{H^{\pm}}$ (GeV) | 122.6, 449.2, 75.01 816.5 | 122.7, 449.0, 79.0, 821.4 | 126.9, 288.5, 84.8, 800.9 | 74.0, 124.7, 121.0, 1293.3
$\Omega h^{2}$ | $3.78\times 10^{-4}$ | 0.107 | 0.119 | $1.96\times 10^{-3}$
$\sigma^{\rm SI}_{\chi^{0}_{1}-p(n)}\times\xi$ (cm2) | $1.2(1.3)\times 10^{-46}$ | $7.2(7.6)\times 10^{-48}$ | $1.2(1.2)\times 10^{-46}$ | $4.1(4.3)\times 10^{-47}$
$\sigma^{\rm SD}_{\chi^{0}_{1}-p(n)}\times\xi$ (cm2) | $4.6(4.5)\times 10^{-44}$ | $9.4(7.3)\times 10^{-42}$ | $3.5(2.8)\times 10^{-42}$ | $1.1(0.8)\times 10^{-41}$
First $T_{n}$ (GeV) | 946.6 / 1st-order | 945.6 / 1st-order | 644.3 / 1st-order | 116.9 / 1st-order
$\\{h_{d},h_{u},s\\}_{{}_{\text{False\\_vac.}}}$ (GeV) | $\\{0,~{}0,~{}0\\}$ | $\\{0,~{}0,~{}0\\}$ | $\\{0,~{}0,~{}0\\}$ | $\\{0,~{}0,~{}0\\}$
$\\{h_{d},h_{u},s\\}_{{}_{\text{True\\_vac.}}}$ (GeV) | $\\{0,~{}0,64.9\\}$ | {0, 0, 66.2 } | $\\{0,0,-104.8\\}$ | $\\{30.3,113.8,-877.4\\}$
Second $T_{n}$ (GeV) | 90.2 / 1st-order | 86.2 / 1st-order | 94.5 / 1st-order | $-$
$\\{h_{d},h_{u},s\\}_{{}_{\text{False\\_vac.}}}$ (GeV) | $\\{0,0,1000.9\\}$ | $\\{0,0,1000.8\\}$ | $\\{0,0,-914.9\\}$ | $-$
$\\{h_{d},h_{u},s\\}_{{}_{\text{True\\_vac.}}}$ (GeV) | $\\{44.2,86.9,1000.6\\}$ | $\\{57.1,112.5,1000.3\\}$ | $\\{48.5,85.6,-914.8\\}$ | $-$
$\gamma_{{}_{\rm{EW}}}=\Delta_{SU(2)}/T_{n}$ | 1.08 | 1.46 | 1.04 | 1.01
CheckMATE result | Allowed | Allowed | Allowed | Allowed
$r$-value | 0.03 | 0.08 | 0.14 | 0.55
Analysis ID |  CMS$\\_$SUS$\\_$16$\\_$039 |  CMS$\\_$SUS$\\_$16$\\_$039 |  CMS$\\_$SUS$\\_$16$\\_$039 |  CMS$\\_$SUS$\\_$16$\\_$039
Signal region ID |  SR$\\_$A01 |  SR$\\_$A08 |  SR$\\_$A28 |  SR$\\_$A31
Table 2: Benchmark scenarios (with successful nucleation) allowed by all
relevant theoretical and experimental constraints including the recent ones
from the LHC searches for the electroweakinos.

Thus, table 1 indicates how different types of spectra for the light higgsino-
like electroweakinos (i.e., smaller $\mu_{\mathrm{eff}}$), which aid SFOEWPT
and are allowed by all relevant constraints including those coming from the DM
sector, get excluded by the LHC analyses with only $\sim 36$ fb-1 of data even
when the latter’s sensitivities to the targeted final states deteriorate
significantly for the kind of spectra in use. We choose our benchmark
scenarios in such a way that those end up with $r\gtrsim 1$. Such ‘r’-value
reflects how relatively light electroweakinos are still allowed before they
start attracting bounds from the LHC analyses. Of course, the latest LHC
analyses with 139 fb-1 of data are likely to increase these mass-bounds
($\mu_{\mathrm{eff}}$) but have not yet been included in the recast package.

In table 2, we present a few benchmark scenarios that exhibit an SFOEWPT but
now also pass the latest constraints coming from electroweakino searches at
the LHC as well. This pushes $\mu_{\mathrm{eff}}$ up which impedes a
successful SFOEWPT from nucleating. The SFOEWPT now prefers to follow a two-
step pattern. This is common when the trivial and global minima in the field
space are widely separated which is the case here given that a larger value
for $\mu_{\mathrm{eff}}$ corresponds to a larger $v_{\\!{}_{S}}$ at $T=0$ for
a given value of $\lambda$, which is a characteristic that regulates the field
separation at $T_{c}$. In the dark sector, it is also found that to comply
with the latest DMDD constraints, the bino/singlino-like LSP must satisfy
various DMDD blind-spot conditions [1].

Figure 1: GW energy density spectrum for the benchmark scenarios BP-A1 (top,
left), BP-A2 (top, right), BP-A3 (bottom, left) and BP-A4 (bottom, right)
illustrated against the detection sensitivities of some GW experiments like
LISA, TianQin, aLigo+, Taiji, ALIA, BBO and U-DECIGO.

The GW (frequency) spectra for these benchmark points BP-A1 to BP-A4 are
presented in figure 1. In all these cases, the individual contributions from
sound waves (within the bag model) and turbulence are shown with broken lines,
in green and red colors, respectively. The total GW spectra within (beyond)
the bag model, are denoted by solid (broken) black lines. The peak of GW
spectrum lies within the sensitivity of various future proposed GW detectors.
For all these benchmark scenarios, the quantity signal-to-noise ratio (SNR)
that measures the detectability of the GW signal is way below than the
detectability criterion of the LISA experiment.

##  4 Summary and outlook

The work sheds light on what the recent searches of the electroweakinos at the
LHC have to say about the viability of SFOEWPT in the $Z_{3}$-NMSSM while
complying with all constraints from various pertinent theoretical and
experimental sectors, including the DM sector experimental finding. While an
SFOEWPT is favored in the range of a few hundreds of a GeV of
$\mu_{\mathrm{eff}}$, the latest constraints are tending to push
$\mu_{\mathrm{eff}}$ steadily above such a range. The general goal of such a
study could then be to check if there is a meeting ground somewhere in the
middle where both constraints are simultaneously complied with. A middle
ground can thus only be found if the reported constraints from the LHC could
be evaded under circumstances that have not been considered explicitly by the
LHC experiments. The two-step phase transition is a more likely phenomenon for
$\mu_{\mathrm{eff}}$ on the larger side with the first transition occurs along
the singlet field direction followed by the second one in the SU(2)-field
directions. Stochastic GW signals originating from FOEWPT in these scenarios
are likely to remain too weak to get detected in the LISA experiment.

New studies on the searches of electroweakinos at relatively higher masses and
in difficult scenarios like the compressed ones using the data coming from the
recently terminated run 2 phase of LHC and from the HL-LHC would continue to
shed light on the motivated region of parameter space for EWBG within the
$Z_{3}$-NMSSM. A further study is required to estimate the BAU in the allowed
scenarios that have been discussed in this work. Furthermore, there are
several scopes to improve the theoretical calculations of bubble wall profile,
$CP$-violation, solving transport equations, GW spectrum, etc. which are
important for the accurate estimation of the EWBG preferred region of NMSSM
parameter space. A complementarity/interplay search among GW, DM and LHC
physics is likely to be rather intriguing and we reserve it for future works.

## References

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