diff --git "a/1dE4T4oBgHgl3EQfaAyc/content/tmp_files/2301.05061v1.pdf.txt" "b/1dE4T4oBgHgl3EQfaAyc/content/tmp_files/2301.05061v1.pdf.txt"
new file mode 100644--- /dev/null
+++ "b/1dE4T4oBgHgl3EQfaAyc/content/tmp_files/2301.05061v1.pdf.txt"
@@ -0,0 +1,5781 @@
+PREPRINT
+Electroweak Phase Transition in a Right-Handed
+Neutrino Superfield Extended NMSSM
+Pankaj Borah,a Pradipta Ghosh,a Sourov Royb and Abhijit Kumar Sahab
+aDepartment of Physics, Indian Institute of Technology Delhi, Hauz Khas 110 016, India
+bSchool of Physical Sciences, Indian Association for the Cultivation of Science, 2A & 2B Raja
+S.C. Mullick Road, Kolkata 700 032, India
+E-mail: Pankaj.Borah@physics.iitd.ac.in, tphyspg@physics.iitd.ac.in,
+tpsr@iacs.res.in, psaks2484@iacs.res.in
+Abstract: Supersymmetric models with singlet extensions can accommodate single- or
+multi-step first-order phase transitions (FOPT) along the various constituent field direc-
+tions. Such a framework can also produce Gravitational Waves, detectable at the upcom-
+ing space-based interferometers, e.g., U-DECIGO. We explore the dynamics of electroweak
+phase transition and the production of Gravitational Waves in an extended set-up of the
+Next-to-Minimal Supersymmetric Standard Model (NMSSM) with a Standard Model sin-
+glet right-handed neutrino superfield. We examine the role of the new parameters compared
+to NMSSM on the phase transition dynamics and observe that the occurrence of a FOPT,
+an essential requirement for Electroweak Baryogenesis, typically favours a right-handed
+sneutrino state below 125 GeV. Our investigation shows how the analysis can offer com-
+plementary probes for physics beyond the Standard Model besides the collider searches.
+arXiv:2301.05061v1  [hep-ph]  12 Jan 2023
+
+Contents
+1
+Introduction
+2
+2
+The Model
+5
+2.1
+A convenient basis choice
+8
+2.2
+Higher order contributions
+10
+2.3
+Contributions from non-zero temperature
+12
+3
+Choice of parameters
+13
+3.1
+Experimental Constraints
+15
+4
+The EWPT and its Properties
+17
+4.1
+PT in the NMSSM + one RHN model
+19
+4.2
+Numerical Results
+20
+4.3
+GW spectrum from SFOPT in the NMSSM + one RHN model
+28
+5
+Summary and Conclusion
+36
+A Field dependent mass matrices
+38
+A.1 CP-even neutral scalars squared mass matrix
+40
+A.2 CP-odd neutral scalars squared mass matrix
+41
+A.3 Uncoloured charged scalars squared mass matrix
+43
+A.4 Neutralino mass matrix
+45
+A.5 Chargino mass matrix
+45
+B Neutral scalar mass matrices after the EWSB
+46
+B.1
+CP-even mass squared elements
+46
+B.2
+CP-odd mass squared elements
+47
+C Counter terms
+47
+D Daisy coefficients
+48
+E Minimization conditions
+49
+– 1 –
+
+1
+Introduction
+Baryon asymmetry of the Universe is a precisely measured quantity by Planck experiment
+[1]. Different kinds of proposals pertaining to baryon asymmetry production mechanism in
+the early Universe are prevalent in literature (for a brief summary see Ref. [2]). In recent
+times, baryon asymmetry production during the Electroweak Phase Transition (EWPT),
+known as the Electroweak Baryogenesis (EWBG) [3] has gained particular attention. The
+EWBG occurs around the TeV scale and has the potential to be probed in collider ex-
+periments [4–6]. Irrespective of different baryon asymmetry generation mechanisms, the
+Sakharov conditions [7], namely, (i) baryon number violation, (ii) charge (C) and charge-
+parity (CP) violation and (iii) deviation from thermal equilibrium must be satisfied.
+It is well known that the Standard Model (SM) of particle physics fails to provide a
+sufficient departure from thermal equilibrium [8, 9]. Moreover, C and CP violations in the
+SM are not adequate enough to yield the observed baryon asymmetry of the Universe [8, 9].
+In principle, a strong first-order EWPT (SFOEWPT) in the early Universe can pave the
+way for the EWBG by allowing sufficient out-of-equilibrium processes [10]. The SM of par-
+ticle physics with the observed Higgs mass ∼ 125 GeV [11, 12], shows a smooth cross-over
+pattern along the Higgs field direction without any PT [13–15] and thus, fails to accom-
+modate the EWBG. This issue can be circumvented by introducing new scalar degrees of
+freedom having sizeable coupling with the SM Higgs boson. In general, the strength of the
+EW phase transition is determined by both the high and low-temperature behaviour of the
+scalar potential. Computation of critical temperature reveals displacement of the global
+minimum for a scalar potential when expressed as a function of the temperature (T) of
+the Universe. However, a correct description of the EWPT requires the study of bubble
+nucleation dynamics since PT proceeds via the nucleation of bubbles [16]. The dynamics of
+bubble nucleation, during the first-order EWPT, can yield stochastic Gravitational Waves
+(GWs) in the early Universe [17–22] that may appear detectable at different GW experi-
+ments. In fact, the search for GWs for probing different kinds of beyond the SM (BSM)
+frameworks has long been practised (see Refs. [23–26] for some of the recent works).
+Supersymmetric models, having a rich scalar sector compared to the SM, carry the
+necessary ingredients for exhibiting an SFOEWPT. The PT properties in the Minimal
+Supersymmetric Standard Model (MSSM) (see Ref.
+[27] for a review) are exercised in
+Refs. [28–37]. It is shown in Ref. [37] that a strong EWPT with a 125 GeV Higgs boson
+favours a hierarchical stop sector in the MSSM, i.e., one of two stops appears to be much
+heavier than the EW scale while the lighter one remains around O(100 GeV) [36, 37]. The
+presence of such a light stop enhances the Higgs production rate through gluon-gluon fusion
+[37, 38] and confronts constraints from LHC data [11, 12]. This tension, nevertheless, can
+be alleviated by considering a light neutralino with a mass lower than about 60 GeV [37].
+However, once again it is challenged by the LHC data of Higgs invisible decay width [39–42]
+and neutralino searches from the stop decay [43–45]. Besides, the MSSM also suffers from
+a new kind of naturalness problem known as the µ-problem [46] and, just like the SM, is
+incapable of accommodating non-zero neutrino masses and mixing [47, 48] in its original
+– 2 –
+
+form1.
+The Next-to-Minimal Supersymmetric Standard Model (NMSSM) [54] provides a dy-
+namical solution to the µ-problem, a challenge that has plagued the MSSM. In the NMSSM,
+the scalar sector of the MSSM is further enriched by the presence of a gauge singlet scalar
+S. Studies related to EWPT in the NMSSM can be found in Refs. [55–60]. It has been
+observed [55–60] that in the NMSSM soft supersymmetry (SUSY) breaking term involving
+S and Higgs doublets assists to form the potential barrier even at T = 0 in contrast to the
+MSSM where T ̸= 0 effects are essential for barrier formation. Thus, the PT dynamics is
+more involved in the NMSSM where one needs to consider a three-dimensional field space
+spanned by three2 CP-even scalar fields.
+The EWPT could occur either in single-step or multi-step.
+In the NMSSM, both
+single-step and multi-step phase transitions are possible as discussed in Ref.
+[59, 60].
+These studies [59, 60] rely on an effective field theory set-up after integrating out heavy
+stops which yield potentially large contributions to the one-loop effective potential. Such
+an effective-theory-based approach reduces the number of degrees of freedom participating
+in the EWPT dynamics. Refs. [59, 60] also showed that the NMSSM can accommodate
+EWBG in some region corners of the NMSSM parameter space.
+Shifting our attention to non-zero neutrino masses and mixing [47, 48, 53], another
+experimentally established BSM signature, both MSSM and NMSSM, are futile just like
+the SM. Extensions of these models with additional ingredients, e.g., right-handed (RH)
+neutrinos, however, offer a simple elegant way to accommodate massive neutrinos using
+the popular type-I see-saw mechanism [62–65]. Supersymmetric type-I seesaw mechanism,
+where the MSSM superfield content is extended with RH-neutrino superfield(s) is well stud-
+ied, see for example, Refs. [66–68]. Incorporating RH-neutrino superfield(s) in the NMSSM
+provides a minimal model [69] where, apart from accommodating none-zero neutrino masses
+and mixing, one also gets a solution for the µ-problem3. In such a framework, non-zero
+neutrino masses appear through three sources: (i) type-I seesaw mechanism involving RH-
+neutrino(s), generally known as the “canonical seesaw”, (ii) type-I and type-III seesaw
+involving gauginos, popularly known as the “gaugino seesaw” and, (iii) seesaw involving
+higgsinos, better known as “higgsino seesaw” [69]. The last two pieces arise when left-
+handed (LH) and RH sneutrinos acquire vacuum expectation values (VEVs), i.e., R-parity
+gets spontaneously broken [73, 74] and effective bilinear R-parity-violating [49] terms are
+generated. For this study, for simplicity, we considered the NMSSM framework extended
+with one RH-neutrino superfield. One, however, needs at least two RH-neutrino superfields
+to accommodate the neutrino data, leaving the lightest one massless [75, 76]. The chosen
+simple framework, nevertheless, offers a nice platform to investigate the PT dynamics and
+subsequently the predictions for GW emission, besides providing the correct scale for the
+1MSSM extended with new superfields or new symmetries or R-parity violation [49] (see Refs. [50–52]
+for further reading) can accommodate neutrino data [47, 48, 53]. R-parity is defined as RP = (−1)3B+L+2s
+where L(B) denotes the lepton (baryon) number and s represents the spin.
+2The PT dynamics in guided by a two-dimensional field space in the MSSM [35, 61].
+3An alternative minimal framework, known as µνSSM [70–72], also solves the µ-problem and satisfies
+the neutrino oscillation data simultaneously, even at the tree-level [72].
+– 3 –
+
+neutrino mass and the atmospheric mass-square difference [47, 48, 53]. We plan to inves-
+tigate the possible correlations between the neutrino sector and the PT dynamics in the
+context of NMSSM with more than one family of RH-neutrino superfields in a forthcoming
+publication.
+Restoring the discussion of PT dynamics, the electrically neutral uncoloured scalar
+sector of the NMSSM extended with one RH-neutrino superfield set-up possesses fourteen
+degrees of freedom, including the neutral Goldstone mode. However, as we will see later
+in section 2, the effective degrees of freedom appear to be eight owing to weak couplings
+of the LH-sneutrino states with the remaining states. Out of these eight, only four are
+CP-even in nature and actively participate in the PT dynamics. Hence, the concerned field
+space is four-dimensional for the chosen framework. This enhanced field space compared
+to the MSSM (two-dimensional due to two Higgses) and the NMSSM (three-dimensional
+owing to two Higgs doublets and one singlet), facilitates the study of EWPT, via single
+steps and multi-steps.
+In the numerical frontier, we adopt a benchmark-based analysis and finally select a
+few benchmark points (BPs) that appear promising from the viewpoint of EWBG and
+also exhibit distinct (single-step or two-steps) PT properties in the early Universe along
+the various constituent field directions.
+In the later part, we exploit some of the BPs
+in order to further investigate the role of new parameters that appear in the setup due
+to the presence of RH-neutrino superfield in the PT dynamics. We also consider various
+relevant experimental constraints, e.g., collider, charged-lepton flavour violation, etc., while
+choosing our BPs. In fact, null experimental evidence of sparticles to date has put stringent
+lower bounds on the concerned states, especially the coloured ones [77–80]. Thus, for the
+analysis of EWPT, we integrate such heavy states out and work in the context of a simplified
+effective model rather than considering the full NMSSM + one RH-neutrino framework.
+We have adopted both the critical and nucleation temperature analyses to describe the PT
+properties in our model. This is crucial since earlier studies, e.g., Ref. [60], have reported
+that the analysis of PT, solely based on critical temperature calculation does not provide
+a complete picture. In fact, the critical temperature analysis does not confirm whether
+a PT has indeed taken place or not.
+A first-order phase transition (FOPT) proceeds
+via bubble nucleation and hence computation of nucleation probability and subsequently,
+nucleation temperature are vital to correctly describe the pattern of a FOPT. Finally, we
+discuss the detection prospects of all our BPs in the forthcoming GW interferometers and
+find that the future space-based experiments: namely, U-DECIGO and U-DECIGO-corr
+[81, 82], have the required sensitivities to test a few of our BPs. This possibility gives
+a complementary detection scope for the NMSSM + one RH-neutrino set-up beyond the
+conventional experimental searches, e.g., collider, neutrino, flavour, etc.
+The paper is organised as follows. In section 2 we discuss the model setup. Next in
+section 3, we talk about the relevant model parameters that are important for studying
+the PT properties and the possible experimental constraints. Subsequently in section 4, we
+present the dynamics of EWPT in detail along with our numerical findings. This section
+also addresses the production of the GW and the testability of our framework in upcoming
+space-based interferometers, e.g., U-DECIGO. Finally, we summarize our analysis and
+– 4 –
+
+conclude in section 5. Some useful formulae and relations are relegated to the appendices.
+2
+The Model
+The superpotential for the chosen framework is given by
+W = W ′
+MSSM + λ �S �Hu · �Hd + κ
+3
+�S3 + Y i
+N �
+N �Li · �Hu + λN
+2
+�S �
+N �
+N,
+(2.1)
+where i = 1, 2, 3 denotes the generation indices. Eq.(2.1) is nothing but the Z3 symmet-
+ric NMSSM superpotential, extended with one Right-Handed Neutrino (RHN) superfield
+( ˆN), keeping the initial Z3 symmetry unbroken. Here W ′
+MSSM denotes the MSSM super-
+potential (see reviews [27, 83–85]) without the bilinear µ-term, ˆHu = ( ˆH+
+u , ˆH0
+u)T , ˆHd =
+( ˆH0
+d, ˆH−
+d )T , ˆLi = (ˆνi, ˆli)T are the SU(2)L doublet up-type Higgs, down-type Higgs, and
+lepton superfields, respectively and the “·” notation is used to express SU(2) product, e.g.,
+ˆLi · ˆHu = ˆνi ˆH0
+u − ˆli ˆH+
+u . The superpotential in Eq. (2.1) cannot be made invariant under
+a global U(1) symmetry, e.g., U(1) of the Lepton number. This in turn ensures the disap-
+pearance of a Nambu-Goldstone boson which results from the spontaneous breaking of a
+global symmetry. The ˆN is considered to be odd under RP while the ˆS transforms as even.
+RP is violated spontaneously in this model when, along with the other neutral scalars, the
+RH-sneutrino ( �
+N) also acquires a non-zero VEV. These VEVs yield the effective µ-term
+(µ = λ⟨S⟩), the effective bilinear RP -violating couplings (ϵi = Y i
+N⟨ �
+N⟩), and the Majorana
+mass term for the RHN (λN⟨S⟩). One should note the presence of four extra couplings
+(three neutrino Yukawa couplings Y 1,2,3
+N
+and another trilinear coupling λN) in Eq. (2.1),
+apart from the known Z3 invariant NMSSM couplings, λ and κ (see for example Refs.
+[54, 86]).
+We would like to re-emphasize here that with only one ˆN, of course, one cannot
+reproduce the observed neutrino mass squared differences and mixing [47, 48, 53], even after
+including loop corrections [76]. However, even this simple choice can predict the absolute
+mass scale and atmospheric mass squared difference for the active neutrinos, besides giving
+interesting information about the EWPT and GW, the primary goals of this article. We
+plan to explore the possible correlations between neutrino observable with the EWPT and
+GW sectors in the context of a two or three ˆN scenario [69] in future work.
+Following Eq. (2.1), in a similar way, we can write down Lsoft, the piece of Lagrangian
+density that contains soft-SUSY breaking terms:
+−Lsoft = −L′soft + m2
+S S∗S + M2
+N �
+N∗ �
+N +
+�
+λAλS Hu · Hd + h.c.
+�
++
+�κAκ
+3 S3 + (ANYN)i �Li · Hu �
+N + AλN λN
+2
+S �
+N �
+N + h.c.
+�
+,
+(2.2)
+where L′
+soft contains the MSSM soft-supersymmetry breaking terms, excluding the Bµ term
+[27, 83–85, 87, 88]. The remaining terms are typical to that of the Z3 symmetric NMSSM,
+except the terms involving �
+N. Soft terms, as depicted in Eq. (2.2), are written in the
+framework of supergravity mediated SUSY breaking [89]. All the trilinear A-terms and
+the soft squared masses are assumed to lie in the TeV regime and consequently, all VEVs
+– 5 –
+
+are expected to appear also in the same regime. In other words, the scale of RHN mass,
+which is determined solely by the scale of soft-SUSY breaking terms will also lie in the
+TeV regime assuming λN ∼ O(1). This assures neutrino mass generation via the TeV
+scale seesaw mechanism which is also testable at colliders [90–95]. Further, the TeV scale
+seesaw immediately suggests Y i
+N ∼ O (10−6 − 10−7) and left-handed sneutrino VEVs,
+⟨�νi⟩ ∼ O (10−4 − 10−5) GeV. These values of Y i
+N, ⟨�νi⟩ indicate (i) tiny RP violation (∼ O
+(10−3 − 10−4) GeV, typical for the bilinear RP violation [96]) and, (ii) weak mixing of the
+left-handed leptons and sleptons (neutral and charged) with the concerned sectors, e.g.,
+charged and neutral gauginos, higgsinos, Higgses, right-handed neutrino and sneutrino,
+etc.
+One can use the advantage of such weak mixing to perform a simplified analysis
+without the loss of generality, e.g., using a set of four fields (Hu, Hd, S, ˜N) instead of seven
+(Hu, Hd, S, �
+N, �Li) while investigating the PT phenomena.
+The tree-level neutral scalar potential is the sum of F-term (VF ), D-term (VD) and
+the soft-SUSY breaking terms and is given by
+Vtree = VF + VD + Vsoft,
+(2.3)
+where Vsoft ≡ −Lsoft is given by Eq. (2.2). VF , following the usual prescription from Eq.
+(2.1), is written as
+VF
+=
+��� − λH0
+uH0
+d + κS2 + λN
+2
+�
+N2���
+2
++ |Y i
+N|2 |H0
+u|2| �
+N|2 + |λ|2|S|2|H0
+u|2
++
+���
+3
+�
+i=1
+Y i
+N �νiH0
+u + λNS �
+N
+���
+2
++
+���
+3
+�
+i=1
+Y i
+N �νi �
+N − λSH0
+d
+���
+2
+,
+(2.4)
+and VD, again using the standard procedure is read as
+VD = g2
+1 + g2
+2
+8
+�
+|H0
+d|2 +
+3
+�
+i=1
+|�νi|2 − |H0
+u|2
+�2
+,
+(2.5)
+with g1, g2 as the U(1)Y , SU(2)L gauge couplings, respectively.
+The neutral CP-even scalar components4, after the EW-symmetry breaking (EWSB),
+develop the following zero-temperature VEVs:
+⟨H0
+u⟩ = vu, ⟨H0
+d⟩ = vd, ⟨S⟩ = vS, ⟨�νi⟩ = vi, ⟨ �
+N⟩ = vN,
+i = 1, 2, 3
+or
+e, µ, τ.
+(2.6)
+The first three VEVs are typical to the NMSSM while the last two VEVs appear for the
+chosen framework as a consequence of the spontaneous RP violation. One can use these
+VEVs to trade off the concerned soft squared masses as depicted in Eq. (2.2). The VEVs
+vS, vN, being governed by the TeV scale soft-terms, also lie in the same regime whereas vi
+appears to be much smaller ∼ O(100 MeV) for vN, vS ∼ O(1 TeV) [74]. Generation of the
+neutrino mass via a TeV scale seesaw mechanism, as already advocated, however, offers
+4Here we adhere to CP-conservation. Further, we do not consider the possibility of charge and colour-
+breaking minima for this study (see e.g., Ref. [97] in the context of the NMSSM) and hence, assign vanishing
+VEVs to charged and coloured scalars.
+– 6 –
+
+a more stringent constraint on vi (∼ O (10−4 − 10−5) GeV), similar to models studied
+in Refs. [73, 98–101]. One can write down minimization conditions for vN, vi, using Eq.
+(2.3), as:
+∂Vtree
+∂ �
+N
+���
+VEVs as Eq. (2.6) = λNvN
+�
+λvuvd + κv2
+S + λN
+2 v2
+N
+�
++ |Y i
+N|2v2
+uvN
++λNvS
+�
+3
+�
+i=1
+Y i
+Nvivu + λNvSvN
+�
++
+3
+�
+i=1
+Y i
+Nvi
+�
+3
+�
+j=1
+Y j
+NvjvN − λvSvd
+�
++M2
+NvN +
+3
+�
+i=1
+(ANYN)ivivu + ANλNvSvN,
+∂Vtree
+∂ �νi
+���
+VEVs as Eq. (2.6) = Y i
+Nvu
+�
+3
+�
+j=1
+Y j
+Nvjvu + λNvSvN
+�
++ Y i
+NvN
+�
+3
+�
+j=1
+Y j
+NvjvN − λvSvd
+�
++
+3
+�
+j=1
+m2
+�Lijvj + (ANYN)ivuvN + g2
+1 + g2
+2
+4
+�
+�v2
+d +
+3
+�
+j=1
+v2
+j − v2
+u
+�
+� vi,
+(2.7)
+where m2
+�Lij denotes soft-squared masses for sleptons [27, 83–85] and all the concerned
+parameters are assumed to be real. It is apparent from Eq.(2.7) that if one neglects terms
+like Y i
+NY j
+N, Y i
+Nvi for smallness, then vS → 0 suggests vN → 0 and consequently vi → 0.
+Thus, a non-zero vS is indirectly connected to a non-zero vi. The smallness of vi, compared
+to vu, vd, also assures that one can still safely use the MSSM relations v2 = v2
+u + v2
+d and
+tan β = vu/vd.
+The presence of tiny but non-zero Y i
+N, vi, as already stated, generates mixing between
+left-handed neutrinos and neutral gauginos. These new mixing terms in the EW sector
+enhance the size of neutral scalar, neutral pseudoscalar, charged scalar, neutral fermion and
+charged fermion mass matrices. Being explicit, RP -violating mixing of H0
+u, H0
+d, S states
+with �
+N and three families of �νi, enlarges the NMSSM CP-even and CP-odd neutral scalar
+mass matrices from 3 × 3 to 7 × 7. Similar augmentation appears (i) in the charged scalar
+sector (2 × 2 in the NMSSM to 8 × 8 due to RP -violating mixing of H±
+u , H∓
+d states with
+the three families of left- and right-handed charged sleptons), (ii) in the neutral fermion
+sector (5 × 5 in the NMSSM to 9 × 9 due to RP -violating mixing among neutral gauginos,
+�
+H0u, �
+H0
+d, �S states with the right-handed neutrino and the three families of left-handed
+neutrinos), and (iii) in the charged fermion sector (2×2 in the NMSSM to 5×5 due to RP -
+violating mixing among the charged higgsino, gaugino states with the three families of the
+left- and right-handed handed leptons). However, because of tiny values of Y i
+N, vi, one can
+easily decompose the aforesaid mass matrices in blocks for approximate analytical studies.
+For example, for all practical purposes, the neutral scalar mass matrix can be decomposed
+into two diagonal blocks: (i) a 4 × 4 one consisting of CP-even H0
+u, H0
+d, S, �
+N states, (ii)
+another 3 × 3 one consisting of CP-even left-handed sneutrino states, and off-diagonal
+blocks containing tiny mixing terms between the two aforementioned states. A similar
+observation holds true for the neutral pseudoscalar, charged scalar, neutralino and chargino
+mass matrices, which can be effectively considered as having dimensions 3 × 3, 2 × 2, 6 × 6
+– 7 –
+
+and 2×2, respectively5, without any loss of generality, leaving the almost pure left-handed
+CP-odd sneutrino, charged slepton, left-handed neutrino and charged leptons states aside.
+For the purpose of analyzing the chosen model numerically, it is convenient to express the
+aforesaid mass matrices in the extended Higgs basis [102–109] which will be introduced
+subsequently. Entries of these mass matrices are detailed in appendix A, along with the
+full uncoloured scalar potential.
+2.1
+A convenient basis choice
+We have already introduced the tree-level neutral scalar potential in Eq.(2.3), using Eqs.(2.2),
+(2.4) and (2.5). However, to study the phenomena of PT we need to move beyond the tree-
+level contribution. For this purpose, as we already mentioned, it is useful to work in the
+extended Higgs basis [102–109], given as:
+Hd =
+� 1
+√
+2(cβHSM − sβHNSM) +
+i
+√
+2(−cβG0 + sβANSM)
+−cβG− + sβH−
+�
+,
+Hu =
+�
+sβG+ + cβH+
+1
+√
+2(sβHSM + cβHNSM) +
+i
+√
+2(sβG0 + cβANSM)
+�
+,
+S =
+1
+√
+2(HS + iAS),
+�
+N
+=
+1
+√
+2(NR + i NI),
+(2.8)
+where cβ(sβ) = cos β(sin β) with tan β = vu/vd.
+Note that one trades off the scalar,
+the pseudoscalar and the charged components of the relevant four fields {Hu, Hd, S, �
+N}
+with the four neutral CP-even interaction states (HSM, HNSM, HS, NR), three CP-odd
+interaction states (ANSM, AS, NI), one charged Higgs pairs (H±), along with the neutral
+and charged Goldstone modes (G0, G±) in the extended Higgs basis.
+This particular
+basis choice assures the SM-like couplings between HSM with the up-type SM fermions,
+the down-type SM fermions and the SM vector bosons. In addition, the aforementioned
+basis choice also predicts vanishing couplings between HS, NR with the same aforesaid SM
+states. Furthermore, from Eq. (2.8), in the light of Eq.(2.6) and v2 = v2
+u + v2
+d, one can
+see that ⟨HSM⟩ =
+√
+2v, ⟨HNSM⟩ = 0, ⟨HS⟩ =
+√
+2vS and ⟨NR⟩ =
+√
+2vN, i.e., non-vanishing
+VEVs appear only in certain field directions leaving the SM-direction undisturbed. These
+interaction states later mix to produce the mass eigenstates. However, one of the CP-even
+states with a mass in the ballpark of 125 GeV (see Ref.
+[110] and references therein)
+contains the predominant HSM component. This alignment between the 125 GeV SM-like
+Higgs in the mass basis and HSM of the extended Higgs basis implies negligible admixing
+among various states in the extended Higgs basis. Mathematically, after the EWSB, in the
+HSM, HNSM, HS, NR basis:
+|M2
+S,1i| ≪ |M2
+S,ii − M2
+S,11|,
+(2.9)
+5One can easily identify the remaining three neutralinos and three chargions, lying at the bottom of the
+mass spectrum, as three LH-neutrino dominated states and the charged leptons, e, µ, τ.
+– 8 –
+
+where i = 2, 3, 4 and M2
+S,1i, the entries of the CP-even scalar squared mass matrix, are
+given in appendix B. It is now apparent that in order to satisfy Eq.(2.9) one either needs
+small M2
+S,1i or large |M2
+S,ii − M2
+S,11|, i.e., decoupling of HSM from the three remaining
+states. The latter, in terms of the mass eigenstates, predicts three significantly heavier
+states dominated by HNSM, HS, NR compositions, and one ∼ O(125 GeV) state controlled
+by HSM composition. In reality, for the SFOEWPT, singlet-like states lighter than 125
+GeV are favoured. Besides, heavier singlet-dominated states create a kind of “push-down”
+effect [71, 111] which makes it difficult to achieve an SM-like Higgs state around 125 GeV.
+Thus, for our numerical studies, we consider regions of the parameter space that can
+accommodate one or more singlet-like states lighter than 125 GeV. These light singlet-
+dominated states are helpful in accommodating a 125 GeV SM-like Higgs through the
+“push-up effect” [71, 111].
+One can use Eq.(2.9) subsequently to derive a few approximate relations, useful for
+parameter space scanning. For example, using appendix B and assuming M2
+S,11 = m2
+h125,
+the condition M2
+S,12 → 0, i.e., vanishing mixing between the HSM and HNSM states, implies
+λ2 ≃ m2
+h125 − m2
+Z cos 2β
+2v2 sin2 β
+.
+(2.10)
+As mh125, mZ (mass of the SM Z0-boson), v are known, λ approximately appears to be a
+function of tan β only. A similar relation like Eq.(2.10) holds also for the NMSSM [60].
+Applying the same procedure to minimize the mixing between HSM and HS states, i.e.,
+M2
+S,13 → 0, one gets
+M2
+A ≃
+4µ2
+sin2 2β
+�
+1 − κ
+2λsin 2β + λλNv2
+N
+4µ2
+sin 2β
+�
+,
+(2.11)
+choosing M2
+A ≃
+2µ
+sin 2β
+�
+Aλ + κµ
+λ + λλNv2
+N
+2µ
+�
+6. The last term in the Eq. (2.11) appears due
+to mixing with the RH-sneutrino. In the limit of κ ≪ λ, using Eq. (2.11), it turns out that
+M2
+A ≃ M2
+H ≃ M2
+H± ≃ 4µ2csc2 2β
+�
+1 + λλNv2
+N
+4µ2
+sin 2β
+�
+where MH represents mass of a state
+with dominant HNSM contribution. The presence of vN shows that these mass eigenstates
+possess contributions from the RH-sneutrino. These kinds of mixing may appear sizable
+depending on λN and vS values.
+Adopting a similar analysis for M2
+S,14 → 0, i.e., effacing the mixing between HSM and
+NR states, it is hardly possible to get a simple relation. A light state below 80 GeV with
+dominant RH-sneutrino contribution hints for a sizable mixing between the HSM and NR
+states. This effect, via one-loop, makes it easy to assure a 125 GeV SM-like Higgs, even
+with stop mass below O(1 TeV) [112]. By choosing the parameters carefully, one can of
+course consider a heavier stop mass to secure a 125 GeV SM-like Higgs having negligible
+admixing with a lighter RH-sneutrino-dominated state. This is precisely what we have
+done while scanning the parameter space since a lighter sneutrino, as also stated earlier,
+is advantageous for SFOEWPT. We will discuss this aspect in detail later. We note in
+6At the limit λN → 0, Eq. (2.11) reproduces the known NMSSM result [60]. If one further considers
+κ → 0, Eq. (2.11) matches the well-known MSSM relation [27].
+– 9 –
+
+passing that so far we have discussed only the tree-level aspects of the scalar potential.
+In reality, the scalar potential receives considerable contributions from radiative effects
+involving various SM particles and their SUSY partners [54, 113–115].
+Some of these
+higher-order contributions have observable consequences, e.g., effects of the top and stop
+loops to procure a 125 GeV SM-like Higgs.
+2.2
+Higher order contributions
+It is relevant to investigate various sources critically before implementing higher-order ef-
+fects arising from the different SM and BSM states on the tree-level scalar potential. The
+effect of higher-order contributions, especially via SUSY partners, is crucial for yielding the
+observed SM mass spectrum, e.g., the Higgs mass. These effects, however, are diluted for
+the analysis of EWPT. Hence, we concentrate only on the leading one-loop effects which
+can arise from various SM and BSM sources.
+Regarding the latter, one needs to con-
+sider the following facts: (i) BSM Higgs masses, i.e., states with dominant HNSM, HS, NR,
+ANSM, AS and NI components, must not remain very far from the EW scale for a suc-
+cessful SFOEWPT and, (ii) hitherto unseen experimental evidence of SUSY searches have
+set lower limits on sparticle masses. These limits are stringent for the coloured sector,
+e.g., gluinos and squarks, >∼ O (1 TeV) (see, for example, the latest CMS [77–79, 116]
+and ATLAS [80, 117–119] limits). On the other hand, for the uncoloured sparticles, e.g.,
+sleptons, LH-sneutrinos, etc, experimental lower bounds are rather flexible [120–122]. For
+convenience, however, we consider heavy sleptons and LH-sneutrinos, >∼ O (1 TeV), for
+this study7. A careful range of relevant parameters was considered so that even with these
+heavy sleptons one can satisfy the latest result on the anomalous magnetic moment of
+muon [123] which typically favours the aforesaid states to be lighter than a TeV.
+With the above mentioned facts and assumptions, one ends up with a situation where
+one encounters >∼ O (1 TeV) sleptons, LH-sneutrinos, squarks & gluinos together with
+other BSM states, e.g., scalar and pseudoscalar Higgses, neutralinos, and charginos, in the
+ballpark of the EW scale. Clearly, now one can integrate out these >∼ O (1 TeV) states
+to yield an effective theory with BSM scalar, pseudoscalar, charged Higgses, neutralinos,
+charginos and, of course, the SM particles. Here we would like to point out again that
+the neutralino and the chargino sector for the concerned model are enhanced compared
+to the NMSSM, owing to the presence of Y i
+N in the superpotential (see Eq. (2.1)) and
+non-zero LH-sneutrino VEVs (see Eq. (2.6)). However, these parameters are compelled to
+remain tiny (∼ O (10−6 − 10−7) and ∼ O (10−4 − 10−5) GeV), thanks to the constraints
+arising from the neutrino experiments and the assumption of a TeV scale seesaw. A similar
+observation, as already stated, also holds true for the BSM Higgs sector. In summary, the
+effective number of contributing states are four CP-even Higgses (S0
+i ), three CP-odd Higgses
+(P 0
+i ), two charged Higgses (H±), six neutralinos (�χ0
+i ), two charginos (�χ±
+i ), charged and the
+neutral Goldstone bosons (G±, G0), and, the relevant SM particles (t, W ±, Z0)8. This
+7Unlike the coloured sector, >∼ O (1 TeV) sleptons and sneutrinos do not introduce large higher-order
+corrections to the scalar sector owing to small values of the concerned lepton Yukawa couplings.
+8Contributions from the remaining SM fermions are sub-leading due to the sizes of concerned Yukawa
+couplings.
+– 10 –
+
+set of nineteen particles including the two Goldstone bosons, together with the t, W ±, Z0,
+will be considered as the dynamical degrees of freedom needed for the current study. One
+can derive parameters of the aforesaid effective theory through the renormalization group
+equation and subsequently, by matching onto the complete model at some intermediate
+scale Λ which we fixed at mt, the top mass. The leading contribution to the tree-level
+potential Vtree obtained using this procedure is
+∆V = ∆λ2
+2 |Hu|4,
+(2.12)
+where ∆λ2 at one-loop level is given by [124–127],
+∆λ2 =
+3
+8π2 y4
+t
+�
+log
+�
+M2
+�t
+m2
+t
+�
++ A2
+t
+M2
+�t
+�
+1 −
+A2
+t
+12M2
+�t
+��
+.
+(2.13)
+Here yt is the top Yukawa coupling evaluated using the running top quark mass, M�t =
+√m�t1m�t2 depicts the geometric mean of two stop masses and At is the soft trilinear coupling
+between Higgs and stops (appears within L′
+soft of Eq. (2.2) [27]). One can of course write
+down contributions like the one shown in Eq. (2.12) for other scalar states, e.g., Hd. Such
+a term, however, appears due to mixing between Hu and Hd through the effective µ-term
+and is usually sub-leading compared to the one shown in Eq. (2.12), as long as µ ≪ M�t
+9
+and tan β value appears not too large. The quantity ∆λ2 is crucial to accommodate a 125
+GeV SM-like Higgs and can be estimated using the same.
+The leftover degrees of freedom also contribute to the potential (see Eq. (2.3)) through
+radiative corrections. Their collective contributions are given by Coleman-Weinberg po-
+tential [129]
+V 1−loop
+CW
+=
+1
+64π2
+�
+i=B,F
+(−1)Finim4
+i (φα)
+�
+log
+�m2
+i (φα)
+Λ2
+�
+− Ci
+�
+,
+(2.14)
+where i = B (F), i.e., bosons (fermions), ni represents the relevant degrees of freedom,
+FB = 0 (FF = 1), Ci is a constant with a value of 3/2 (1/2) for scalars, fermions, longitu-
+dinally polarized vector bosons (transversely polarized vector bosons), Λ is the aforesaid
+intermediate energy scale, fixed at mt and, m2
+i (φα) = m2
+i (HSM, HNSM, HS, NR) denotes
+field-dependent masses. The latter is estimated from Vtree + ∆V (see Eq. (2.3) and Eq.
+(2.12)). Contributions from Vtree are detailed in appendix A. The set of involved Bs are
+given by S0
+1,..,4, P 0
+1,2,3, H±, G0, G±, Z0, W ± with nB = 4×1, 3×1, 2, 1, 2, 3, 2×3, depend-
+ing on the nature of the concerned state, i.e., scalar or complex scalar or massless bosons
+or massive vector bosons. A similar approach for the fermions give F = �χ0
+1,..,9, �χ±
+4,5, t with
+nB = 9 × 2, 2 × 2, 3 × 4 considering their electric and colour charges. One should note
+that the presence of G0, G± in the Coleman-Weinberg potential yields divergent contribu-
+tions. However, these can be effaced by using an infrared regulator. Finally, putting all
+9Such a choice helps one parameterize radiative contributions from stops effectively, even beyond the
+one-loop order [125, 127, 128].
+– 11 –
+
+these pieces, i.e., Vtree (see Eq. (2.3)), ∆V (see Eq. (2.12)) and V 1−loop
+CW
+(see Eq. (2.14))
+together, one obtains the effective scalar potential as
+Veff = Vtree + V 1−loop
+CW
++ ∆V.
+(2.15)
+Inclusion of Coleman-Weinberg contributions (see Eq. (2.15)) to the tree-level scalar poten-
+tial, however, changes the position of physical minima and masses. To restore the original
+position for the physical minima, keeping M2
+S,13, M2
+S,14 → 0 and maintaining the mass of
+the CP-even scalar state with leading HSM composition at 125 GeV, one needs to intro-
+duce appropriate counterterms, encapsulated within another contributor Vct. The latter is
+normally related to a redefinition of the entries of −Lsoft (see Eq. (2.2)) [130–132] which
+are depicted in appendix C. The counterterms are, thus, not arbitrary but fixed by the
+aforesaid criteria. Mathematically,
+∂
+∂φi
+�
+Veff + Vct
+����
+φi=⟨φi⟩ = 0 and
+∂2
+∂φi∂φj
+�
+Veff + Vct
+����
+φi=⟨φj⟩ = 0,
+(2.16)
+with φi = {HSM, HNSM, HS, NR}. One can figure out ⟨φi⟩ using Eq. (2.6) and Eq. (2.8).
+We note in passing that till now we have discussed modifications of the tree-level scalar
+potential from higher order effects at vanishing temperature, i.e., T = 0. In reality, however,
+one also needs to include contributions arising from T ̸= 0 which we will address now.
+2.3
+Contributions from non-zero temperature
+The one-loop temperature-dependent potential is given by [133]
+V 1−loop
+T̸=0
+= T 4
+2π2
+�
+i=B,F
+(−1)FiniJB/F
+�m2
+i (φα, T)
+T 2
+�
+,
+(2.17)
+where T represents the temperature, symbols FF,B, nF,B are the same as discussed in the
+context of Eq. (2.14), m2
+i (φα, T) depicts thermal field-dependent masses of the ith degrees
+of freedom as:
+m2
+i (φα, T) = m2
+i (φα) + ciT 2,
+(2.18)
+with ci representing the concerned Daisy coefficients [133–137]. These coefficients appear
+non-vanishing for bosons and are given in appendix D. Finally, JB/F , i.e., the thermal
+function, is defined as
+JB/F
+�
+x2 ≡ m2
+i (φα, T)
+T 2
+�
+= ±
+� ∞
+0
+dy y2 log
+�
+1 ∓ e−√
+x2+y2�
+,
+(2.19)
+where + (−) sign is for bosons (fermions). One should note that at the m2 ≫ T 2 limit,
+where “m” depicts a generic mass term, JB/F suffers an exponential suppression from
+Boltzmann factor. These repressions ensure that massive degrees of freedom, e.g., squarks,
+gluinos, etc., that are already integrated out (see subsection 2.2), do not affect T ̸= 0
+corrections.
+– 12 –
+
+Clubbing all the pieces together, i.e., tree-level scalar potential, one-loop contributions
+via Coleman-Weinberg potential, and contributions from the finite temperature part, one
+gets the finite temperature effective scalar potential at the one-loop order as
+VT = Vtree + ∆V + V ′1−loop
+CW
++ Vct + V 1−loop
+T̸=0
+≡ VT (φ, T),
+(2.20)
+where V ′1−loop
+CW
+has a form similar to Eq. (2.14) but replacing m2
+i (φα) with thermal masses
+m2
+i (φα, T), as depicted in Eq. (2.18). We will use Eq. (2.20) to inquire about the PT
+properties. We note in passing that the components of VT have explicit gauge dependence
+[138–140]. Besides, V 1−loop
+CW
+(see Eq. (2.14)), and hence V ′1−loop
+CW
+, also has renormalization
+scale (Λ) dependence which could dominate over the gauge dependence [141]. To note,
+we have worked in the Landau gauge while computing the one-loop corrected potential at
+both zero and non-zero temperatures.
+So far we have discussed different pieces of the scalar potential needed to study the
+PT dynamics. Now we will address how and to which extent various model parameters
+can affect the same.
+3
+Choice of parameters
+The set of new parameters, compared to the NMSSM, are
+Y i
+N, λN, vN, (ANYN)i, AλN λN,
+(3.1)
+using Eq. (2.1), Eq. (2.3), Eq. (2.6), and replacing soft-SUSY breaking square mass term
+M2
+N with the corresponding VEV. Now, as already discussed, Y i
+Ns are associated with the
+neutrino mass generation through a TeV scale seesaw and thus, are constrained to be small.
+These Y i
+N values, for TeV-scale trilinear terms, predicts (ANYN)i ∼ O (10−3 − 10−4) GeV.
+The latter is also related to the smallness of vi, i.e, the LH-sneutrino VEVs (see Eq. (2.6)),
+as guided by a TeV scale seesaw mechanism and neutrino data. Hence, for the PT analysis,
+we can neglect these tiny parameters, i.e., vi, Y i
+N, (ANYN)i, without any loss of generality
+as they have negligible effects on the PT dynamics. Now from the discussion of section
+2, it is evident that relevant “bare” parameters for the uncoloured scalar potential after
+trading (see appendix E for details) soft-squared masses with the corresponding VEVs (see
+Eq. (2.6)) are,
+λ, λN, κ, vu, vd, vS, vN, Aλ, Aκ, AλN .
+(3.2)
+One can redefine this list further by trading vu, vd with v =
+�
+v2u + v2
+d, tan β = vu/vd and
+vS with µ = λvS. As v = 174 GeV is known, Eq.(3.2) can be re-casted as
+λ, λN, κ, tan β, µ, vN, Aλ, Aκ, AλN .
+(3.3)
+One can also trade parameter vN with the RH-neutrino mass term MN ∝ λNvN. Similar
+trading is also possible for Aλ with MA, using a relation given in subsection 2.1. We,
+however, do not use MA, MN for the parameter space scanning. Parameter λ can also
+be exchanged using Eq.(2.10).
+The same parameter can also be constrained using an
+– 13 –
+
+upper-bound on the tree-level SM-like Higgs mass [54, 142, 143], given as m2
+Z(cos2 2β +
+g−2
+2 λ2 sin2 2β). This helps us to consider small tan β ≲ 5 and λ ∼ O(0.1) or higher such
+that one gets a significant contribution to the tree-level SM-like Higgs mass10.
+The ranges of other parameters are also guided by certain aspects, e.g., in order to
+avoid the presence of Landau pole [144, 145] below the GUT scale, i.e., 1016 GeV, one
+needs to consider λ, κ values carefully at the EW scale such that
+√
+λ2 + κ2 <∼ 0.7 [54].
+Besides, smaller values of κ ∼ O(10−2) are favoured as a stronger PT along a particular
+field direction prefers smaller values of the quartic coupling (e.g., κ for PT along the
+HS direction) and larger values of the cubic coupling (e.g., Aκ for a PT along the HS
+direction), leading to an enhanced barrier height along that specific direction. A small
+value of κ, together with a small Aκ value11, as already discussed in subsection 2.1, assure
+the presence of light CP-even and CP-odd states below 125 GeV. These light states help
+to procure a 125 GeV SM-like Higgs via the “push-up” [71, 111] effect. It is evident that
+one needs to consider Aκ values carefully as for this parameter larger values are favourable
+for the PT dynamics while smaller ones are useful in fixing the SM-like Higgs mass around
+125 GeV. Tree-level mass of the singlet-dominated CP-even state, using Eq.(3.3) and Eq.
+(B.1), is
+M2
+S,33 ≡ m2
+HS = −λλNAλN v2
+N
+2µ
++ κAκµ
+λ
++ 4κ2µ2
+λ2
++ λ2v2Aλ sin 2β
+µ
+.
+(3.4)
+This reduces to the known NMSSM result [143] at the limit λN → 0 with a O(λ2) correc-
+tion12. It is apparent from Eq. (3.4) that how different parameters appear instrumental
+in determining the mass of a CP-even singlet-dominated state in this framework. We con-
+sider κ > 0, Ak < 0 in this study to ensure the formation of a barrier along the HS field
+direction. The parameter µ plays a vital role in the PT dynamics and, as given in Eq.
+(3.4), is also crucial for the mass and composition of a singlet-like state. Ref. [59] suggests
+that a strong EWPT favours µ ≲ 300 GeV for the Z3 invariant NMSSM. We consider
+similar ranges for µ in our analysis which also obey the “naturalness” criteria and the LEP
+chargino bound [148–151], i.e., |µ| >∼ 103.5 GeV. This range of µ values, together with the
+choice of λ ∼ O(0.1), suggests a value for vS not too far from the EW scale as required to
+yield a sizable impact on the EWPT from the singlet sector. A similar observation holds
+true for the RH-sneutrino VEV vN. The parameter vS also determines the mass term for
+RH-neutrino, i.e., ∝ λNvS which is constrained to be around a TeV as non-zero neutrino
+masses in the chosen framework arise through a TeV scale seesaw. The adaptation of a
+TeV scale seesaw also put some bounds on the parameter λN that is expected to be at
+most O(1) to avoid the existence of Landau pole below the GUT scale. The requirement
+of having stronger PT along the NR field direction, however, suggests smaller values of
+λN. This behaviour, is similar to κ, as addressed before. The role played by λN in the
+10Lower λ values suggest reduced tree-level mass and hence, needs larger corrections from the stop sector.
+In the NMSSM, considering the perturbative nature of λ up to the scale of the Grand Unified Theory (GUT)
+one gets λ ≲ 0.7, in the limit of κ ≪ λ [54].
+11These ranges of κ, Aκ are guided by the well-known U(1)PQ, U(1)R limits [143, 146, 147] for the
+NMSSM.
+12This term appear to be sub-leading for small λ, tan β values together with vS ≪ v.
+– 14 –
+
+PT dynamics is somewhat non-trivial and will be addressed later in detail. The remaining
+parameters, Aλ, AλN are connected to the scale of vS, vN and thus, are expected to be in
+the ballpark of a TeV. These parameters, i.e., Aλ, AλN also affect tree-level masses of the
+CP-even and CP-odd scalar states as detailed in appendix B. In this analysis we consider
+Aλ > 0 and AλN < 0. The latter choice helps to efface the possible existence of a tachyonic
+state in the CP-odd scalar sector (see Eq. (B.2)). We note in passing that so far we have
+presented a qualitative discussion in the context of the chosen independent parameters,
+as depicted in Eq. (3.3). For finding BPs through numerical analysis, one, however, also
+needs to consider all the relevant present and anticipated experimental bounds which we
+will address in the next subsection.
+3.1
+Experimental Constraints
+A viable phenomenological analysis must satisfy all the concerned experimental limits, the
+existing and the projected ones. The inclusion of these bounds reduces the size of the
+available parameter space. In this analysis, apart from considering sensitivity reaches of
+the existing [152–154] and upcoming [81, 155, 156] GW detection setups, we also consid-
+ered constraints arising from (i) analysis of the SM-like Higgs boson properties and BSM
+Higgs searches at colliders, (ii) other BSM searches at the colliders, (iii) flavour-violating
+processes, (iv) neutrino experiments, (v) muon anomalous magnetic moment, etc. In order
+to employ these constraints in our numerical analysis, we first implemented the concerned
+model in SARAH 4.14.5 [157–164]. Subsequently, we use SPheno-4.0.5 [158, 162, 164–171]
+to get the mass spectrum and decay widths. The output of SPheno-4.0.5 also provides
+branching fractions for various flavour-violating processes, BSM contributions to the muon
+anomalous magnetic moment [166], several LHC observables like reduced Higgs couplings,
+etc. We will now discuss the aforesaid constraints one by one in further detail.
+(i) Analysis of the SM-like Higgs boson properties and BSM Higgs searches at colliders:
+Here one needs to consider two aspects: (a) SM-like Higgs analyses, and (b) the BSM Higgs
+searches. Concerning the first, important constraints appear from the measured mass, i.e.,
+≈ 125 GeV [42, 172], and couplings [39–42, 173–177]. We have used these results to assure
+the existence of an SM-like 125 GeV Higgs in our analysis. Besides, to assure the SM-
+like nature we also put a lower limit (80%) on the Hu composition of the 125 GeV mass
+eigenstate.
+Regarding the BSM Higgs searches, i.e., for states with leading HNSM, HS
+components, and the charged Higgs, we consider the concerned experimental bounds, see
+for example Ref. [178] and references therein. We used HiggsBounds [179] 5.10.2 [180] to
+implement experimental constraints from the SM and BSM Higgs searches in our numerical
+study.
+(ii) Other BSM searches at the colliders: We already discussed in subsection 2.2 that
+we are working in an effective framework after integrating out heavy degrees of freedom like
+gluinos, squarks and even charged sleptons and LH-sneutrinos. We consider these states
+to remain heavier than 1 TeV. Such assumptions, especially for gluinos and squarks are
+supported by the experimental findings. In this study, we consider gluino mass >∼ 1.8 TeV
+and squark masses >∼ 1.2 TeV. These choices are guided by the present CMS [77–79, 116]
+and ATLAS [80, 117–119] observations. Experimental lower bounds on the charged slepton
+– 15 –
+
+and LH-sneutrino masses are somewhat less [120–122]. However, we also considered them
+to be heavier than a TeV and integrate them out. In our numerical study, the lightest
+neutralino mass varies from 3 GeV to 120 GeV. However, this does not contradict any
+experimental bounds, e.g., SM-like Higgs decaying to a pair of neutralinos, (see for example
+Refs. [80, 121, 181–184]) as its predominant composition (≳ 90%) is from the singlino and
+the RH-neutrino. For charginos, we used a lower bound of 103.5 GeV [148–151] in our
+analysis. It is important to note that experimental lower bounds are often interpreted in
+the context of simplified models and hence, they may not directly restrict the concerned
+model parameter space.
+(iii) Flavour-violating processes: The presence of BSM states can significantly enhance
+branching fractions (BR) of certain flavour-violating processes, e.g., B → Xsγ, B0
+s → µ+µ−
+(see Refs. [185–191] and references therein), etc., compared to the SM predictions. One can
+minimize these new contributions by taking tan β <∼ 5 and fixing squarks, gluinos, sleptons,
+etc., masses to be heavier than a TeV. However, finite BSM contributions to these pro-
+cesses still appear through the EW scale uncoloured neutral scalars, neutral pseudoscalars,
+charged scalars, charginos and neutralinos, as required for the EWPT. Thus, we consider
+the following 2σ bounds
+BR(B → Xsγ) = (3.49 ± 0.38) × 10−4,
+BR(B0
+s → µ+µ−) = (3.45 ± 0.58) × 10−9.
+(3.5)
+We note in passing that BR(B → Xsγ), BR(B0
+s → µ+µ−) also receive extra contributions
+due to R-parity breaking [192, 193]. However, given the framework of a TeV scale seesaw,
+the size of R-parity violating couplings, i.e., Y i
+NvN, appears to be ∼ O(10−3 − 10−4) GeV
+and hence, hardly yield any significant contributions.
+We consider charged Yukawa couplings to be diagonal for this work which helps to
+bypass constraints from the flavour-violating Higgs decays [194, 195]. One can also con-
+sider slepton soft squared masses to be diagonal to minimize mixing among sleptons (both
+charged and neutral). With these choices, the effective bilinear R-parity violating cou-
+plings, i.e., Y i
+NvN, and the LH-sneutrino VEVs appear to be main sources for the various
+charged lepton flavour violating (cLFV) processes like µ → eγ, µ → eee, etc. However, the
+scale of these couplings, i.e., ∼ O(10−3 − 10−4) GeV, as required for a TeV scale seesaw,
+can easily evade these bounds. This behaviour is very similar to the SUSY models with
+bilinear R-parity violation [196–198]. We note in passing that in our numerical studies
+we emphasized on the cLFV processes for the µ over the similar ones from τ as the con-
+cerned existing and upcoming experimental sensitivities are much more stringent for µ.
+Nevertheless, we also include constraints for cLFV processes involving a τ in our analysis,
+e.g., BR(τ → µγ) < 4.4 × 10−8 [199]. The µ-based cLFV bounds included in the current
+analysis are given by
+BR(µ → eγ) < 4.2 × 10−13 [200],
+BR(µ → eee) < 1.0 × 10−12 [201],
+CR(µN → eN∗) < 7 × 10−13 [202],
+– 16 –
+
+where CR(µN → eN∗) represents muon to electron conversion ratio in atomic nuclei with
+N (N∗) representing the nucleus in the normal (excited) state. The given number, i.e.,
+7 × 10−13 is for the gold nuclei.
+(iv) Neutrino experiments: With one generation of RH-neutrino, as already stated in
+section 2, it is not possible to accommodate the experimentally observed three-flavour neu-
+trino masses and mixing [47, 48, 53], even with the inclusion of loop effects [76]. Thus, one
+will get one massive and two nearly massless neutrinos in this model. Nevertheless, even in
+such a scenario, we used constraints from the atmospheric mass squared difference ∆m2
+atm,
+i.e., 2.430(−2.574) × 10−3 − 2.593(−2.410) × 10−3 eV2 for normal (inverted) hierarchy, and
+the sum of three neutrino masses ≲ 1.2 eV [1, 203].
+(v) Muon anomalous magnetic moment: Just like the flavour violating processes, the
+anomalous magnetic moment of muon also receives extra contributions over the SM from
+new parameters and the BSM states (see Refs. [204, 205] and references therein).
+The
+recent comprehensive SM prediction of the muon anomaly is 116591810 (43) × 10−11 (0.37
+ppm) [206] while the experimental average13 is 116592061(41) × 10−11 (0.35 ppm). These
+numbers, adding errors in quadrature, gives ∆aµ = (251±59)×10−11 which is arising from
+the BSM sources. This, in 4σ span, gives (1.5 − 48.7) × 10−10. The BSM contributions,
+especially involving charged sleptons states below a TeV [208–210], can affect this process
+significantly and can easily accommodate the latest experimental observation [123]. In our
+analysis, as already discussed in subsection 2.2, we kept charged slepton masses around a
+TeV. Nevertheless, by playing with the other concerned parameters we checked that the
+aforesaid ∆aµ range, i.e., (1.5 − 48.7) × 10−10 is not violated in our BPs. In fact, the
+choice of slepton, squark masses around a TeV or more yields suppressed cLFV processes
+and smaller BSM contributions to the anomalous magnetic moment of the muon. All the
+chosen BPs respect all the five aforesaid classes of constraints. We now discuss this study’s
+key objectives in detail, i.e., PT properties and GW production.
+4
+The EWPT and its Properties
+As we already discussed, understanding the EWPT properties in the early Universe in
+a Particle Physics model has twofold advantages. Firstly, it can be confirmed whether
+the model carries the prospect to explain the origin of EWBG at some corner of the
+parameter space. Secondly, it provides scope to test the model at GW detectors beyond
+the conventional BSM searches. One of the prerequisites of EWBG is the FOPT with
+sufficient strength along the SU(2)L field directions so that it can suppress the processes
+which wash out the baryon asymmetry after it is produced, namely SU(2)L sphalerons [2].
+The same FOPT may yield a detectable amount of GWs that could be accessible by future
+GW interferometers.
+The structure of the thermal effective potential for a PT reveals that at very high
+temperatures the Universe would be in a symmetric phase with the relevant field (say φi)
+being located at zero. As the Universe cools down, the symmetric vacuum may disappear
+13Here we have used combined experimental average obtained from the FNAL [123] and the BNL E821
+[207] results.
+– 17 –
+
+and the corresponding field values could be finite. Additionally, a second minimum can
+be formed at some higher field value which becomes degenerate with the previous one at
+T = Tc, known as critical temperature. At temperature below Tc, the transition from the
+high-T VEVs (say v′
+X) to the low-T VEVs (say vX) can take place. Here X = u, d, S, i, N
+as depicted in Eq. (2.6). We should note here that a high-T (low-T) phase means an
+unstable (stable) vacuum below Tc or above nucleation temperature. Therefore, to have
+an in-depth understanding of PT dynamics, an estimate of critical temperature Tc and the
+strength of PT are enormously important.
+Theoretically, the critical temperature can be obtained from the following equality:
+VT (v′
+X, Tc) = VT (vX, Tc),
+(4.1)
+where v′
+X and vX represent high-T and low-T VEVs, respectively, along a particular field
+direction. We also need to ensure the existence of high- and low-T vacua which can be
+confirmed by the following equalities,
+∂φαVT (v′
+X, Tc) = 0, ∂φαVT (vX, Tc) = 0,
+(4.2)
+where φα = {HSM, HNSM, HS, NR}. In many cases, including ours, analytical solutions of
+Eq. (4.1) and Eq. (4.2) are almost impossible to derive in order to obtain the estimates of
+the relevant parameters to study the PT properties. We have used the publicly available
+package cosmoTransitions [211] to carry out the numerical calculation for our model in
+consideration.
+A FOPT proceeds via bubble nucleation and the nucleation rate (Γ) per unit volume
+(V ) at finite temperature is given by Γ
+V ∝ T 4e−SE/T , where SE is the three-dimensional
+effective Euclidean action known as bounce action. The criterion which set the condition
+for the onset of bubble nucleation is given by [16, 212],
+SE(Tn)
+Tn
+≃ 140,
+(4.3)
+where Tn is the nucleation temperature. If it happens that the quantity SE(Tn)
+Tn
+> 140, then
+the transition does not occur due to low tunnelling probability.
+As mentioned earlier, we use cosmoTransitions [211] to compute SE and Tn, which
+also allows for estimating the probability of a transition taking place. Since we have four-
+dimensional field space, relevant to EWPT, a detailed scan of the model parameter space
+is challenging and numerically expensive as well. Therefore in the present work, we first
+provide a representative BP-based study which will be detailed subsequently. We will see
+that such BPs are sufficient to understand the parameter space of NMSSM + one RHN
+framework that can potentially give rise to an SFOPT and can also be interesting from
+the viewpoint of EWBG. Subsequently, we discuss the impact of new parameters in the
+present setup compared to the NMSSM on PT strength along different field directions by
+providing a scan of the relevant parameter spaces.
+Before we proceed further, let us now define different criteria to consider a PT to be a
+strong one. Conventionally, in the critical temperature analysis, the order parameter that
+– 18 –
+
+decides the fate of PT is given by,
+γc ≡ vc(Tc)
+Tc
+=
+�
+⟨HSM⟩2 + ⟨HNSM⟩2
+Tc
+≳ 1.0,
+(4.4)
+where vc(Tc) denotes VEVs of the SU(2)L Higgs fields, i.e., HSM, HNSM, at Tc. For the nu-
+cleation temperature calculation, we define an SFOPT along the respective field directions
+as follows:
+• Along SU(2)L doublet Higgs direction:
+∆φSU(2)
+Tn
+=
+�
+(
+�
+HlT
+SM
+�
+−
+�
+HhT
+SM
+�
+)2 + (
+�
+HlT
+NSM
+�
+−
+�
+HhT
+NSM
+�
+)2
+Tn
+≳ 1.0
+(4.5)
+• Along the SU(2)L singlet Higgs and the RH-sneutrino direction:
+∆φS
+Tn
+=
+�
+(
+�
+HlT
+S
+�
+−
+�
+HhT
+S
+�
+)2
+Tn
+≳ 1.0 ;
+∆φ �
+N
+Tn
+=
+�
+(
+�
+NlT
+R
+�
+−
+�
+NhT
+R
+�
+)2
+Tn
+≳ 1.0,
+(4.6)
+where
+∆φSU(2)
+Tn
+, ∆φS
+Tn
+and
+∆φ �
+N
+Tn
+represent PT strength along the SU(2)L-doublet, SU(2)L-
+singlet and the RH-sneutrino field direction, respectively. The notation,
+�
+ΦlT�
+denotes the
+low temperature minimum while
+�
+ΦhT�
+is the high temperature minimum of a scalar field
+(Φ) before nucleation. A favourable condition to yield the observed baryon asymmetry
+of the Universe via the EWBG is (
+�
+HhT
+SM
+�
+,
+�
+HhT
+NSM
+�
+) = (0, 0) with
+∆φSU(2)
+Tn
+≳ 1. In con-
+trast, when (
+�
+HhT
+SM
+�
+,
+�
+HhT
+NSM
+�
+) ̸= (0, 0), the sphaleron processes outside the bubble gets
+substantially suppressed which lead to inefficient production of the baryon asymmetry of
+the Universe from the EWBG.
+4.1
+PT in the NMSSM + one RHN model
+As we already specified, the field space relevant to the PT analysis is four-dimensional in
+the present framework. This opens up the possibility of obtaining a richer PT pattern
+compared to the case of the NMSSM. We define the high-temperature symmetric vacuum
+of the scalar potential as Ω0. In principle, one can have many distinct PT patterns in the
+whole parameter region of the NMSSM + one RHN framework. Here we summarise a few
+such possibilities that advocate some unique PT patterns along the various field directions:
+• Type-I: As already stated, at T ≫ Tc, the Universe remains in the symmetric phase
+where each of the four fields has zero VEV. The simplest possibility for a PT is that
+at critical temperature the symmetry-breaking minimum of the total scalar potential
+appears only along the HSM direction. Then the PT happens from symmetric to the
+broken phase directly in that direction. We denote this by Ω0
+PT
+−−→ ΩHSM where ΩHSM
+represents the vacuum along SM Higgs direction.
+• Type-IIa: This pattern involves displacement of the HS field VEV (at T > Tc) from
+the initial zero value as the Universe cools down. We label it as Type II. Below Tc,
+the PT occurs along both the HSM and HS field directions. We denote this particular
+pattern (IIa) as Ω0 → Ω′
+HS
+PT
+−−→ ΩHSM + ΩHS.
+– 19 –
+
+• Type-IIb: This is similar to the earlier case where for T > Tc, a shift of the HS field
+value from zero vacuum appears. Below the critical temperature, the transition also
+takes place along the HS direction only and is represented by Ω0 → Ω′
+HS
+PT
+−−→ ΩHS.
+• Type-IIc: This case also falls under the Type II category. However, below critical
+temperature, the PT happens along both HS and NR field directions as indicated by
+Ω0 → Ω′
+HS
+PT
+−−→ ΩHS + ΩNR.
+• Type-IIIa: In this category, for T > Tc, the shifts of HSM and HS VEVs from the
+initial zero values take place. When T < Tc, PT also occurs along the same field
+directions. This pattern is represented by Ω0 → Ω′
+HSM + Ω′
+HS
+PT
+−−→ ΩHSM+ΩHS.
+• Type-IIIb: In this category, at T > Tc, the behaviour of the scalar potential is
+similar to the last one. However, at T < Tc, the PT occurs along HSM, HS and NR
+directions as indicated by Ω0 → Ω′
+HSM + Ω′
+HS
+PT
+−−→ ΩHSM +ΩHS+ΩNR.
+• Type-IV: This category is defined to indicate a particular PT pattern where at a
+T > Tc, the symmetric vacuum of the total scalar potential gets displaced along
+the S and NR field directions. The PT occurs below Tc along any of the four field
+directions.
+As described earlier, any BP showing either of the type-I or type-IIa PT pattern is
+preferred in view of efficient EWBG, provided the corresponding PT strength satisfies the
+condition
+∆φSU(2)
+Tn
+≳ 1. Whereas, the rest of the types as listed above may not lead to
+EWBG due to non-satisfaction of either of the conditions,
+��
+HhT
+SM
+�
+,
+�
+HhT
+NSM
+��
+̸= (0, 0) or
+∆φSU(2)
+Tn
+≳ 1. The PT types that do not favour EWBG, can be still interesting if it triggers
+an SFOPT along the SU(2)L doublet or singlet field directions and subsequently radiates
+GW at a detectable amount.
+4.2
+Numerical Results
+As earlier mentioned, we would like to begin with a benchmark-based study of EWPT in
+the present work. In the later part, we will be discussing explicitly the dependence of new
+parameters in the current setup compared to the NMSSM. We first tabulate six BPs in
+Table 1 that are consistent with all relevant theoretical and experimental constraints, as
+discussed in subsection 3.1. We select the BPs in such a way that they show distinct PT
+characteristics with some of them favouring EWBG and carrying good to moderate detec-
+tion prospects at GW detectors. Note that we have four soft-SUSY breaking parameters
+(i.e., Aλ, Aκ, AλN , AN) in our model. We discuss the possible role of all the A− parameters
+in section 3. Recall that one of the soft parameters AN does not contribute much to the
+PT dynamics since it is always associated with the tiny neutrino Yukawa coupling Y i
+N as
+earlier clarified. We keep AN above the TeV scale for all BPs, which ensures slepton masses
+≳ O(1 TeV). In Table 1, we provide the eigenvalues of the four CP-even mass eigenstates,
+i.e., mh125, mH, mHS, m �
+N, corresponding to each BPs. The leading composition in these
+states are coming from the HSM, HNSM, HS and NR fields, respectively. We have explicitly
+– 20 –
+
+checked that all the BPs evade the relevant experimental bounds as detailed in subsection
+3.1. Nevertheless, we have explicitly shown values of the various flavour-violating processes
+∆aµ and ∆m2
+atm for the sake of completeness. In Table 2 and Table 3, we have summarised
+the PT outputs of the BPs as obtained from the cosmoTransitions [211] package. Below
+we discuss the PT characteristics for each of the BPs in detail.
+BP-I
+BP-II
+BP-III
+BP-IV
+BP-V
+BP-VI
+tan β
+2.90
+2.74
+2.90
+5.77
+4.79
+5.86
+λ
+0.416
+0.412
+0.416
+0.384
+0.118
+0.111
+κ
+0.022
+0.019
+0.022
+0.012
+0.013
+0.051
+λN
+0.146
+0.142
+0.146
+0.130
+0.260
+0.238
+Y 1
+N × 107
+0.9
+0.65
+1.1
+1.0
+3.6
+4.3
+Y 2
+N × 107
+0.9
+0.65
+1.1
+1.0
+3.6
+4.3
+Y 3
+N × 107
+0.9
+0.65
+1.1
+1.0
+3.6
+4.3
+Aλ [GeV]
+775.48
+705.32
+775.48
+1184.87
+988.08
+920.08
+Aκ [GeV]
+-62.75
+-25.37
+-95.61
+-107.08
+-11.70
+-41.61
+AλN [GeV]
+-349.68
+-337.77
+-326.60
+-363.16
+-1358.30
+-1528.57
+AN [GeV]
+-16000.0
+-12000.0
+-8500.0
+-12000.0
+-6500.0
+-5000.0
+µ [GeV]
+224.56
+220.86
+224.56
+203.12
+153.59
+162.64
+vN [GeV]
+308.80
+325.21
+284.50
+386.45
+136.57
+355.66
+v1 × 104 [GeV]
+1.0
+0.55
+1.0
+1.2
+1.0
+1.0
+v2 × 104 [GeV]
+1.0
+0.55
+1.0
+1.2
+1.0
+1.0
+v3 × 104 [GeV]
+1.0
+0.55
+1.0
+1.2
+1.0
+1.0
+mh125 [GeV]
+126.02
+124.80
+125.64
+125.63
+126.28
+124.05
+mH [GeV]
+772.36
+718.07
+772.73
+1213.76
+897.40
+1012.14
+mHS [GeV]
+83.60
+88.98
+69.48
+109.54
+97.31
+195.41
+m �
+N [GeV]
+48.60
+51.65
+51.89
+27.65
+65.18
+115.63
+BR(B → Xsγ) × 104
+3.61
+3.70
+3.62
+3.47
+3.59
+3.55
+BR(B0
+s → µ+µ−) × 109
+3.24
+3.26
+3.24
+3.19
+3.20
+3.19
+BR(µ → eγ) × 1030
+394
+0.61
+4.98
+51.4
+404
+173
+BR(µ → eee) × 1029
+113.0
+363.7
+44.6
+53.9
+2.04
+2.04
+CR(µN → eN∗) × 1028
+1.81
+0.11
+2.49
+4.43
+4.85
+7.31
+∆m2
+atm × 103 eV2
+2.51
+2.57
+2.58
+2.54
+2.58
+2.46
+∆aµ × 1010
+3.88
+0.75
+3.42
+1.94
+1.54
+3.24
+Table 1.
+The representative BPs that we will use to study the PT patterns in the present
+framework.
+Apart from the parameters mentioned above, we fix the gaugino mass parameters
+M1 = 300 GeV, M2 = 2M1, M3 = 6M1, trilinear soft coupling At around 2 TeV. We also consider
+RH-slepton soft masses above 1 TeV and squarks soft masses M �
+Qi, M�uc
+i , M �
+dc
+i all above 1.2 TeV.
+With the chosen values of parameters Y i
+N, vi and AN, the LH-sneutrino and LH-slepton masses also
+appear in the ballpark of a TeV. As already stated in subsection 3.1, suppressed cLFV processes
+and smaller BSM contributions to the anomalous magnetic moment of muon are evident now due to
+slepton, squark masses around a TeV or more. In fact, for BP-II, ∆aµ remains below the aforesaid
+4σ range. CR(µ N → e N ∗) value is estimated for the gold nuclei.
+• BP-I and BP-II : Out of these two representative BPs, BP-I shows an SFOPT along
+both the SU(2)L-doublet and singlet field directions.
+On the other hand, we obtain a
+weaker FOPT for BP-II in the SU(2)L doublet directions whereas a stronger one along the
+SU(2)L singlet direction. In Figure 1, we have shown the evolution of the phase structures
+along the HSM (left) and the HS (right) field directions as a function of temperature for
+BP-I. The critical temperature for BP-I is 117.8 GeV as noted in Table 2. Above the critical
+– 21 –
+
+BP-I
+BP-II
+BP-III
+Transition Type
+Type-IIa
+Type-IIa
+Type-IIIa
+vc/Tc
+1.30 (In); 0 (Out)
+0.73 (I); 0 (O)
+1.83 (I); 0.61 (O)
+∆φSU(2)/Tn
+1.58
+0.81
+1.28
+∆φS/Tn
+4.70
+1.16
+7.61
+∆φ �
+N/Tn
+0
+0
+0
+Tc (GeV)
+117.8
+127.2
+101.6
+Tn (GeV)
+109.9
+126.7
+82.9
+high-Tn VEVs
+(0, 0, 113.8, 0)
+(0, 0, 341.6, 0)
+(105.8, 32.5, 88.8, 0)
+low-Tn VEVs
+(173.1, 9.5, 631.3, 0)
+(102.3, 11.3, 488.7, 0)
+(208.1, 4.8, 719.7, 0)
+high-Tc VEVs
+(0, 0, 72.6, 0)
+(0, 0, 333.1, 0)
+(62.4, 20.9, 35.6, 0)
+low-Tc VEVs
+(152.9, 11.8, 572.5, 0)
+(92.5, 10.1, 467.7, 0)
+(186.4, 10.6, 625.4, 0)
+Table 2. The PT properties for first three BPs as tabulated in Table 1.
+temperature HSM is located at zero (as pointed by the legend phase 3, red coloured, in
+Figure 1). At T = Tc, we find another degenerate minimum along the same field direction,
+which is ⟨HSM⟩ = 152.9 GeV (as marked by phase 2, green coloured). The black coloured
+line with the arrow connects the high-T and low-T VEVs indicating a possible FOPT. The
+bubble nucleation occurs afterwards and it ends at 109.9 GeV which we have highlighted
+in orange colour (also labelled as phase 1). A similar pattern can be observed along HS
+direction too as shown in the right panel of Figure 1. The interesting point to mention
+here is that the ⟨HS⟩ starts to get displaced from zero value even at a temperature above
+Tc. This is in contrast to the evolution of phase structure along HSM direction for this
+particular BP. The BP-II shows similar characteristics although the strong PT occurs only
+along the HS direction. The high-temperature behaviour of the total scalar potential leads
+us to identify the PT properties for both BP-I and BP-II as Type-IIa. For BP-I, we observe
+from Table 2, that the PT strength at T = Tc is greater than one inside the bubble and
+zero outside the bubble. Therefore a baryon number may be generated in the broken phase
+and the wash-out effects are likely to be suppressed. In view of this, BP-I is favoured in
+order to address EWBG. However, BP-II shows a weaker FOPT in the SU(2)L doublet
+directions and hence is not suitable to address the question of EWBG. In subsection 4.3 we
+will discuss the strength of emitted GW spectrum during bubble nucleation for both BP-I
+and BP-II in view of the proposed sensitivities of a few forthcoming GW experiments.
+• BP-III: The BP-III falls into Type-IIIa category.
+It implies that at a temperature
+above Tc, both HSM and HS attain non-zero VEVs. The critical temperature for this BP
+comes out to be 101.6 GeV. At this temperature, the presence of two degenerate vacua is
+noticed having nonzero field values for both SU(2)L doublet and singlet fields, which set
+the possibility of a PT. We obtain SFOPT along both the SU(2)L-doublet and singlet field
+directions where the PT strength turns out to be larger than one. However, the quantity φc
+Tc
+becomes non-zero both inside and outside the bubble. This gives rise to a stronger wash-
+out effect which is likely to suppress the yield of baryon asymmetry and hence seemingly
+disfavored in view of EWBG. Nevertheless, it carries good detection prospects in the GW
+detectors due to relatively larger PT strength ∆φS
+Tn
+compared to BP-I.
+– 22 –
+
+0
+50
+100
+150
+200
+250
+⟨HSM⟩ [GeV]
+0
+50
+100
+150
+200
+250
+300
+T [GeV]
+phase3
+phase2
+phase1
+phase0
+0
+200
+400
+600
+800
+⟨HS⟩ [GeV]
+0
+50
+100
+150
+200
+250
+300
+T [GeV]
+phase3
+phase2
+phase1
+phase0
+Figure 1.
+Phase structures as a function of temperature along the HSM and HS field directions
+for BP-I. Different colours represent the locations of a particular field as a function of temperature.
+The black coloured line with the arrow connects two degenerate phases at T = Tc and the direction
+of the arrow indicates a possible FOPT.
+0
+200
+400
+600
+⟨HS⟩ [GeV]
+0
+200
+400
+600
+800
+1000
+T [GeV]
+phase3
+phase2
+phase1
+phase0
+−500
+−400
+−300
+−200
+−100
+0
+⟨NR⟩ [GeV]
+0
+200
+400
+600
+800
+1000
+T [GeV]
+phase3
+phase2
+phase1
+phase0
+Figure 2.
+Phase structures as function of temperature along HS and NR field directions for
+BP-IV. Different colours show the evolution of minimum along a particular field direction with
+temperature. The line with the arrow connects two degenerate phases at T = Tc and the direction
+of the arrow indicates a possible FOPT.
+• BP-IV: The BP-IV in Table 1 shows type-IIc PT pattern. The numerical estimates of
+the relevant parameters that govern the PT dynamics for BP-IV are listed in Table 3. We
+find SOFPT along both the HS and NR directions. Clearly, this BP is not preferred to
+address EWBG. In Figure 2, we show the phase structure along HS and NR directions for
+BP-IV as a function of temperature. At temperature above Tc = 184.5 GeV, HS takes a
+non-zero field value which is the typical type-II feature. The black coloured line with arrow
+in Figure 2 connects two degenerate phases at the critical temperature and paves the way
+for the PTs in the respective singlet field directions.
+• BP-V: This BP is unique in the sense that we obtain FOPT below the critical tempera-
+ture along the directions of SU(2)L fields, HS and NR at the same time. This BP falls into
+– 23 –
+
+the type-III category since at temperature above Tc, we find high-T VEV to be non-zero for
+both HSM and HS fields. Although this particular BP shows FOPT along HSM direction,
+the strength is relatively weaker as can be seen from Table 3. Therefore, the possibility
+of EWBG remains unlikely for this BP. Nevertheless, we obtain SFOPT along HS and ˜N
+directions in contrast to weaker FOPT in the HSM direction.
+BP-IV
+BP-V
+BP-VI
+Transition Type
+Type-IIc
+Type-IIIb
+Type-IV
+vc/Tc
+0.0 (In) ; 0.0 (Out)
+0.0 (I); 0.0 (O)
+2nd: 0.54 (In); 0.0 (Out)
+1st: 0.0 (In) ; 0.0 (Out)
+∆φSU(2)/Tn
+0
+0.04
+1st: 0 ; 2nd: 0.57
+∆φS/Tn
+1.01
+1.56
+1st: 0; 2nd: 0
+∆φ �
+N/Tn
+2.81
+1.71
+1st: 0.2; 2nd: 0.13
+Tc (GeV)
+184.5
+177.9
+2nd: 206.3
+1st: 232.8
+Tn (GeV)
+165.8
+144.3
+2nd: 204.6
+1st: 232.6
+high-Tn VEVs
+(0, 0, 529.9, 0)
+(137.9, 3.5, 1606.9, 0)
+2nd: (0, 0, 2087.9, −720.9)
+1st: (0, 0, 2087.7, −845.4)
+low-Tn VEVs
+(0, 0, 696.6, −465.28)
+(143.2, 0, 1832.7, 247.2)
+2nd: (117.2, 0, 2088.1, −747.6)
+1st: (0, 0, 2087.7, -807.2)
+high-Tc VEVs
+(0, 0, 459.9, 0)
+(0, 0, 1484.6, 0)
+2nd: (0, 0, 2087.9, −724.6)
+1st: (0, 0, 2087.7, −846.2)
+low-Tc VEVs
+(0, 0, 671.2, −429.5)
+(0, 0, 1827.6, 275.9)
+2nd: (112.3, 0, 2088.1, −749.3)
+1st: (0, 0, 2087.4, −808.5)
+Table 3. The PT properties for the last three BPs as tabulated in Table 1.
+• BP-VI: So far, for all the BPs we have obtained single-step FOPT. In contrast, BP-VI
+shows a two-step FOPT. The outputs are tabulated in Table 3. In both steps, the high-
+temperature behaviour of the scalar potential closely follows the Type-IV pattern. On the
+other hand, in the first step FOPT occurs along the NR direction only, while in the second
+step, we find FOPT in both the NR and HSM directions. Note that, this BP shows a weaker
+FOPT and hence, is not suitable for the EWBG.
+Recall from section 3 that the new physics parameters, relevant for the study of PT in
+the current framework are {λN, AλN , vN} compared to the Z3 symmetric NMSSM. In the
+subsequent analysis, we like to inquire about the impact of these new parameters on the
+PT strength along different field directions. Also, note that a FOPT apparently favours a
+lighter RH-sneutrino-like state below 125 GeV as we observe from the BP-based study of
+PT and their outcomes. This characteristic is likely to be further confirmed while we vary
+the new parameters and obtain the sensitivity of PT strength on these parameters.
+First, in Figure 3 we show the impact of vN (left) and λN (right) on the PT strength
+vc
+Tc . In each of the sub-figures, we have fixed the other relevant parameters as in BP-I of
+Table 1. We find the PT strength decreases with the rise of both vN and λN. We repeat
+the analysis for the same BP as shown in the top panel of Figure 4 considering nucleation
+temperature calculation. In particular, we estimate the PT strength in the SU(2)L field
+– 24 –
+
+150
+200
+250
+300
+350
+400
+vN (GeV)
+0.9
+1.0
+1.1
+1.2
+1.3
+1.4
+1.5
+vc/Tc
+0.14
+0.16
+0.18
+0.20
+0.22
+0.24
+0.26
+λN
+0.5
+0.6
+0.7
+0.8
+0.9
+1.0
+1.1
+1.2
+1.3
+vc/Tc
+Figure 3. These plots show the dependence of PT strength on vN (left) and λN (right) in the Tc
+calculation. Parameters Y i
+N, vi and AN have no significant effect in PT dynamics and thus, we keep
+their values ∼ O(10−7), ∼ O(10−4 GeV), ∼ O(1 TeV), respectively. Other relevant parameters are
+fixed as in BP-I of Table 1, except vN and λN.
+directions, i.e., ∆φSU(2)/Tn as function of vN and λN and notice similar trends as in
+Figure 3. Now a smaller λN or vN implies lighter sneutrino following the CP-even mass
+matrices mentioned in Appendix B. Hence, Figures 3 and 4 further reinforce the fact that
+a comparatively lighter RH-snuetrino is indeed preferred to trigger a possible FOPT along
+the SU(2)L doublet field directions in the present framework. Now the remaining new
+parameter AλN is expected to show a minor impact on the ∆φSU(2)/Tn. This is because it
+is not directly connected to the relevant terms at the tree level in the Lagrangian involving
+the SU(2) doublet Higgs fields. Indeed, in our analysis, we have found that the ∆φSU(2)/Tn
+remains more or less unaltered upon varying AλN as shown in the bottom panel of Fig. 4.
+Next, we like to examine the impact of the new physics parameters as earlier specified
+on the PT strength along SU(2)L-singlet field direction ∆φS/Tn while the other parameters
+are set according to BP-III of Table 1. In top panel of Figure 5 we depict the variation of
+∆φS/Tn as function of vN (left) and λN (right). We observe that the quantity ∆φS/Tn
+increases upon lowering λN when vN is fixed. In the other case when we fix λN and vary
+vN, the ∆φS/Tn gets enhanced for a smaller vN. Once again, these observations further
+strengthen our earlier finding that a lighter RH-sneutrino below 125 GeV is favoured for
+the occurrence of an SFOPT in the SU(2)L-singlet, i.e., HS direction as well. On the
+other hand, we also notice that the ∆φS/Tn increases with the rise of AλN as shown in
+the bottom panel of Figure 5. Note that AλN is appearing as the coefficient of the cubic
+interaction S �
+N �
+N (see Eq. (2.2)). Hence a larger AλN is expected to increase the barrier
+height which results in a stronger ∆φS/Tn.
+Previously, we have found that BP-IV provides us with a SOFPT along the NR field
+direction ∆φ �
+N/Tn. We would like to utilize this particular BP to enquire about the de-
+pendence of new parameters on ∆φ �
+N/Tn. In top left of Figure 6, we show the dependence
+of ∆φ �
+N/Tn on vN. We find that for vN ≲ 500 GeV, the ∆φ �
+N/Tn remains more or less
+– 25 –
+
+320
+340
+360
+380
+400
+vN (GeV)
+1.0
+1.1
+1.2
+1.3
+1.4
+1.5
+∆φSU(2)/Tn
+0.14
+0.16
+0.18
+0.20
+0.22
+0.24
+0.26
+λN
+0.6
+0.8
+1.0
+1.2
+1.4
+1.6
+∆φSU(2)/Tn
+−1400 −1200 −1000 −800
+−600
+−400
+−200
+AλN
+0.08
+0.09
+0.10
+0.11
+0.12
+0.13
+0.14
+0.15
+0.16
+∆φSU(2)/Tn
+Figure 4. These plots show the dependence of PT strength ∆φSU(2)/Tn on vN (top left), λN (top
+right) and AλN (bottom) along the SU(2)L field direction, in the Tn calculation. Here, orders of
+parameters Y i
+N, vi and AN are chosen as in Figure 3 and the other relevant parameters are fixed as
+in BP-I of Table 1, except vN, λN and AλN .
+constant, however, decreases while we increase vN further. Additionally, from top right
+of Figure 6 the ∆φ �
+N/Tn gets reduced as well upon increasing λN. The reason for this is
+twofold. As we mentioned earlier, a smaller λN leads to lighter RH-sneutrino states below
+125 GeV which in turn enhances the ∆φ �
+N/Tn. Moreover, a smaller λN also assists in
+increasing the barrier height and hence results in enhanced ∆φ �
+N/Tn. In bottom panel of
+Figure 6, we have shown the ∆φ �
+N/Tn strength gets enhanced upon increasing AλN . This
+is once again caused by the enhanced barrier height for a larger AλN similar to the earlier
+case.
+After examining the individual dependence of new parameters on PT strength, we
+now give a random scan on new physics parameters highlighting the region allowed by
+the experimental constraints and favouring an SFOPT along SU(2)L field directions. We
+vary (λN, vN) and fix the other relevant parameters in Eq. (3.3) following BP-I. However,
+orders of parameters Y i
+N, vi and AN are chosen as ∼ O(10−7), ∼ O(10−4 GeV), ∼ O(1
+– 26 –
+
+150
+200
+250
+300
+350
+400
+vN (GeV)
+2
+3
+4
+5
+6
+∆φS/Tn
+0.14
+0.16
+0.18
+0.20
+0.22
+0.24
+0.26
+λN
+1
+2
+3
+4
+5
+∆φS/Tn
+−800
+−700
+−600
+−500
+−400
+−300
+−200
+AλN
+0.8
+0.9
+1.0
+1.1
+1.2
+1.3
+1.4
+∆φS/Tn
+Figure 5. These plots show the dependence of PT strength ∆φS/Tn on vN (top left), λN (top
+right) and AλN (bottom) along the SU(2)L-singlet field direction, in the Tn calculation. Here,
+orders of parameters Y i
+N, vi and AN are chosen as in Figure 3 and the other relevant parameters
+are fixed as in BP-III of Table 1, except vN, λN and AλN .
+TeV), respectively, as they hardly affect the PT dynamics. We have randomly generated
+pairs of (λN, vN) and pass through all the experimental bounds mentioned in subsection
+3.1. We first sort out the points that pass all the experimental constraints as shown in
+green colour in Figure 7. Next, we apply the condition of SFOPT along the SU(2)L field
+direction and pin down the points that favour SFOPT only and SFOPT with possible
+EWBG having minimal wash-out effects. We have marked them in Figure 7 by coloured
+‘▲’ and ‘■’, respectively. These points depict the variation of ∆φSU(2)/Tn in the vN - λN
+plane.
+Next in Figure 8, we made a scenario similar to that of Figure 7, however, in the HS
+field direction in the context of BP-IV, as shown in Table 3. Here, points which undergo
+SFOPT are marked by ‘⋆’. We also compute the ∆φS/Tn strength and find that the
+∆φS/Tn strength is maximum when both λN and vN are small, which is in agreement with
+our earlier observations.
+– 27 –
+
+400
+500
+600
+700
+800
+900
+1000
+vN (GeV)
+2.0
+2.2
+2.4
+2.6
+2.8
+3.0
+∆φ�
+N/Tn
+0.15
+0.20
+0.25
+0.30
+0.35
+λN
+0.0
+0.5
+1.0
+1.5
+2.0
+2.5
+3.0
+∆φ�
+N/Tn
+−1000
+−800
+−600
+−400
+−200
+AλN
+2.6
+2.8
+3.0
+3.2
+3.4
+3.6
+3.8
+∆φ�
+N/Tn
+Figure 6. These plots show the dependence of PT strength ∆φ �
+N/Tn on vN (top left), on λN
+(top right) and on AλN (bottom) along the NR direction, in the Tn calculation. Here, orders of
+parameters Y i
+N, vi and AN are chosen as in Figure 3 and the other relevant parameters are fixed as
+in BP-IV of Table 1, except vN, λN and AλN .
+Finally, in Figure 9 we perform an analogous exercise to show the variation of ∆φ �
+N/Tn
+in the vN - λN plane. In this case, we have utilized the BP-IV of Table 3 once again to fix
+the other relevant parameters, except vN and λN. The green-coloured points are allowed
+by the various experimental constraints as stated in subsection 3.1. We mark the points
+that favour SFOPT in the NR direction by coloured ‘⋆’. Once again, we find that the
+∆φ �
+N/Tn is maximum for simultaneous lower values of vN and λN, consistent with our
+earlier findings.
+4.3
+GW spectrum from SFOPT in the NMSSM + one RHN model
+A cosmological FOPT can produce GWs in the early Universe that contains information
+about the strength of different model parameters. In the preceding section, we have dis-
+cussed different PT characteristics in the proposed framework and computed the relevant
+quantities that determine the strength of a PT. In the current section, we will be talking
+– 28 –
+
+220
+240
+260
+280
+300
+320
+340
+vN (GeV)
+0.135
+0.140
+0.145
+0.150
+0.155
+0.160
+0.165
+0.170
+λN
+tan β = 2.90, λ = 0.416, κ = 0.022, µ = 224.56 GeV,
+Aλ = 775.48 GeV, Aκ = −62.75 GeV, AλN = −349.68 GeV
+Allowed by experimental bounds
+SFOPT and no EWBG
+SFOPT and possible EWBG
+1.50
+1.75
+2.00
+2.25
+2.50
+2.75
+3.00
+3.25
+3.50
+∆φSU(2)/Tn
+Figure 7. This figure shows variations of ∆φSU(2)/Tn in the vN - λN plane. The green-coloured
+points pass all the experimental constraints as discussed in subsection 3.1. The points favoured for
+SFOPT along the SU(2)L field direction without and with EWBG are marked by coloured ‘▲’ and
+‘■’ symbols, respectively. Orders of parameters Y i
+N, vi and AN are chosen as in Figure 3 and the
+other relevant parameters are fixed as in BP-I of Table 1.
+about the production of GW and its detection prospects within our model setup.
+As we have mentioned earlier, a FOPT is characterized by critical temperature Tc,
+and nucleation temperature Tn. The critical temperature indicates the moment when the
+location of the global minimum changes from one vacuum phase to another. However,
+the critical temperature analysis does not assure that the associated PT is indeed taking
+place. On the other hand, FOPT proceeds via bubble nucleation, and hence calculation of
+nucleation temperature is very crucial in order to obtain the phenomenological parameters
+that are important from the standpoint of estimating GW spectra. When the nucleation
+happens, at a temperature below Tc, the probability of tunnelling Γ(T) from the false
+vacuum to the true one is given by [213],
+Γ(T) ≈ T 4
+� SE
+2πT
+�3/2
+e− SE
+T ,
+(4.7)
+where SE is the bounce action corresponding to the critical bubble and can be written as
+[212],
+SE =
+� ∞
+0
+4πr2dr
+�
+VT (φ, T) + 1
+2
+�dφ(r)
+dr
+�2�
+,
+(4.8)
+with r being the radial coordinate and φ corresponding to the scalar dynamical fields
+present in a model framework. The scalar field solution φ can be derived by solving the
+– 29 –
+
+340
+360
+380
+400
+420
+440
+460
+vN
+0.12
+0.13
+0.14
+0.15
+0.16
+0.17
+0.18
+0.19
+0.20
+λN
+tan β = 5.77, λ = 0.384, κ = 0.012, µ = 203.12 GeV,
+Aλ = 1184.87 GeV, Aκ = −107.08 GeV, AλN = −363.16 GeV
+Allowed by experimental bounds
+SFOPT along HS direction
+1.0
+1.2
+1.4
+1.6
+1.8
+2.0
+2.2
+∆φS/Tn
+Figure 8. This figure shows variations of ∆φS/Tn in the vN - λN plane. The green-coloured points
+pass all the experimental constraints as discussed in subsection 3.1. The points favoured for SFOPT
+along the HS field direction are marked by coloured ‘⋆’. Orders of parameters Y i
+N, vi and AN are
+chosen as in Figure 3 and the other relevant parameters are fixed following BP-IV of Table 1.
+classical field equation [212, 214, 215]
+d2φ
+dr2 + 2
+r
+dφ
+dr = dVT (φ, T)
+dr
+,
+(4.9)
+and subsequently applying proper boundary conditions:
+dφ
+dr = 0 when r → 0 and φ(r) →
+φfalse when r → ∞, where φfalse represents the four-dimensional field values at the false
+vacua. We reiterate here that in order to solve the differential equation and the bounce
+action numerically, we have implemented our model in the cosmoTransitions [211] pack-
+age.
+The essential parameters that are required for the estimation of GW spectra from
+FOPT are relative change in energy density during the PT (α), and the inverse of the
+duration of the PT (β). Both the parameters, α, and β, are defined at the nucleation
+temperature Tn. The first parameter, α, is computed from [216],
+α = ∆ρ
+ρrad
+,
+(4.10)
+where ∆ρ is the released latent heat and it is expressed as [217],
+∆ρ =
+�
+VT (φ0, T) − T dVT (φ0, T)
+dT
+�
+T=Tn
+−
+�
+VT (φn, T) − T dVT (φn, T)
+dT
+�
+T=Tn
+,
+(4.11)
+– 30 –
+
+340
+360
+380
+400
+420
+440
+460
+vN (GeV)
+0.12
+0.13
+0.14
+0.15
+0.16
+0.17
+0.18
+0.19
+0.20
+λN
+tan β = 5.77, λ = 0.384, κ = 0.012, µ = 203.12 GeV,
+Aλ = 1184.87 GeV, Aκ = −107.08 GeV, AλN = −363.16 GeV
+Allowed by experimental bounds
+SFOPT along �
+N direction
+1.6
+1.8
+2.0
+2.2
+2.4
+2.6
+2.8
+∆φ�
+N/Tn
+Figure 9. This figure shows variations of ∆φ �
+N/Tn in the vN - λN plane. The green-coloured
+points pass all the experimental constraints as discussed in subsection 3.1. The points favoured
+for SFOPT along the NR field direction are marked by coloured ‘⋆’. Orders of parameters Y i
+N, vi
+and AN are chosen as in Figure 3 and the other relevant parameters are fixed following BP-IV of
+Table 1.
+with φ0 and φn represent, in our case, the four-dimensional field values at the false and true
+vacua, respectively, and VT (φ, T) is the finite-temperature effective potential as mentioned
+in Eq. (2.20). We should note here that the quantity ∆ρ measures the strength of a PT, the
+larger value of the same corresponds to a stronger FOPT. In Eq. (4.10), ρrad corresponds
+to the radiation energy in the plasma and it is expressed as, ρrad = π2g∗
+30 T 4
+n, with g∗ being a
+temperature-dependent quantity that counts the total number of relativistic energy degrees
+of freedom.
+The parameter β is defined as [218],
+β
+H∗
+= T d
+dT
+�SE
+T
+� �����
+T=T∗
+≡ T d
+dT
+�SE
+T
+� �����
+T=Tn
+,
+(4.12)
+where H∗ is the expansion rate of the Universe during the PT and T∗ stands for the PT
+temperature. We have considered T∗ ≃ Tn in the present work. We have tabulated the
+obtained values of α and β in Table 4 for different BPs shown in Table 1. As stated earlier,
+the quantity α is proportional to the energy released during the PT and hence a larger PT
+strength should lead to a larger α value. In fact, this is exactly the case where we find the
+largest α for the BP-III (see Table 4) having ∆φS/Tn = 7.61 (see Table 2.) We obtain the
+lowest α for the first-step PT of BP-VI since the corresponding ∆φ �
+N/Tn is weakest among
+all as can be seen from Tables 2 and 3.
+– 31 –
+
+BPs
+α
+β/H∗
+BP-I
+0.0456
+37535.2
+BP-II
+0.0121
+143931.0
+BP-III
+0.0870
+11729.8
+BP-IV
+0.0101
+7596.0
+BP-V
+0.0027
+4611.3
+BP-VI-I
+0.0002
+516911.0
+BP-VI-II
+0.0017
+63837.8
+Table 4. Estimates of the parameters α and β as defined in Eq. (4.10) and Eq. (4.12), respectively
+for the six BPs listed in Table 1. Note that the BP-VI-I shows two-step PT patterns and we have
+made the estimates of α and β in both steps.
+There are mainly three different processes that trigger the emission of GWs in a FOPT:
+(i) bubble wall collisions, (ii) sound waves, and (iii) magneto-hydrodynamic (MHD) tur-
+bulence in the plasma.
+Therefore, the total energy spectrum of the emitted GW can
+approximately be given as a sum of these three contributions [155, 219],
+ΩGWh2 ≈ Ωcolh2 + Ωswh2 + Ωturh2, respectively,
+(4.13)
+where, h = H0/(100 km · sec−1 · Mpc−1) [220] with H0 corresponding to Hubble’s constant
+at the present epoch. The contribution to the total GW energy density from the bubble
+wall collision can be computed using the envelope approximation and it can be estimated
+as a function of frequency “f” as [221],
+Ωcolh2 = 1.67 × 10−5
+� β
+H∗
+�−2 � κcα
+1 + α
+�2 �100
+g∗
+�1/3 � 0.11v3
+w
+0.42 + v2w
+�
+3.8 (f/fcol)2.8
+1 + 2.8 (f/fcol)3.8 ,
+(4.14)
+where vw is the bubble wall velocity and κc is the efficiency factor of bubble collision, given
+as,
+κc =
+0.715α + 4
+27
+�
+3α
+2
+1 + 0.715α
+.
+(4.15)
+The red-shifted peak frequency fcol [221] is expressed as (with the approximation T∗ ≈ Tn),
+fcol = 16.5 × 10−6
+�f∗
+β
+� � β
+H∗
+� �
+Tn
+100 GeV
+� � g∗
+100
+�1/6
+Hz,
+(4.16)
+where the fitting function, f∗/β, at the time of the PT is given by,
+f∗
+β =
+0.62
+1.8 − 0.1vw + v2w
+.
+(4.17)
+In order to obtain a GW spectrum with higher strength, it is generally assumed that the
+expanding bubbles attain a relativistic terminal velocity in the plasma and we consider
+– 32 –
+
+vw ≃ 1 in our calculations 14. However, there is a note of caution that runway bubble walls
+are generally undesirable in view of the successful yield of a sizeable amount of EWBG 15.
+The contribution to the total GW density from sound waves can be parameterized as
+[230–233],
+Ωswh2 = 2.65×10−6 Υ(τsw)
+� β
+H∗
+�−1
+vw
+� κswα
+1 + α
+�2 � g∗
+100
+�1/3 � f
+fsw
+�3 �
+7
+4 + 3 (f/fsw)2
+�7/2
+,
+(4.18)
+where κsw is the efficiency factor for the sound wave contribution representing the fraction
+of the energy (latent heat) that gets converted into the bulk motion of the plasma and
+subsequently emits gravitational waves as given by (in the limit vw → 1)
+κsw ≃
+�
+α
+0.73 + 0.083√α + α
+�
+.
+(4.19)
+The quantity fsw corresponds to the present peak frequency for the sound wave contribution
+to the total GW energy density, expressed as
+fsw = 1.9 × 10−5
+� 1
+vw
+� � β
+H∗
+� �
+Tn
+100 GeV
+� � g∗
+100
+�1/6
+Hz.
+(4.20)
+The parameter Υ(τsw) appears due to the finite lifetime of the sound waves which suppresses
+their contributions to the GW energy density as written as
+Υ(τsw) = 1 −
+1
+√1 + 2τswH∗
+,
+(4.21)
+with τsw being the lifetime of the sound waves. The onset of the turbulence takes place at
+this timescale and disrupts the sound wave source. Following Ref. [232], we write τsw ≈
+R∗/U f, where R∗ = (8π)1/3 vw/β and U f =
+�
+3κswα/4 are the mean bubble separation and
+the root-mean-squared fluid velocity which can be obtained from a hydrodynamic analysis,
+respectively.
+At the time of PT, the plasma is fully ionized and due to the resulting MHD turbulence,
+it leads to another source of GWs. The MHD turbulence contribution to the total GW
+energy density is modelled as [235]
+Ωturh2 = 3.35 × 10−4
+� β
+H∗
+�−1
+vw
+�κturα
+1 + α
+�3/2 �100
+g∗
+�1/3
+�
+�
+(f/ftur)3
+[1 + (f/ftur)]11/3 �
+1 + 8πf
+h∗
+�
+�
+� ,
+(4.22)
+14A precise determination of bubble wall velocity is non-trivial [222–226] and out of scope of the present
+analysis. Instead, we consider here vw as an input parameter.
+15Recently, an improved analysis on bubble wall dynamics has reported that EWBG may be possible
+even for supersonic vw [227–229] which is in contrast with our traditional notion.
+– 33 –
+
+U-DECIGO
+U-DECIGO-corr
+DECIGO-corr
+BP-I
+BP-II
+BP-III
+10-7
+10-5
+0.001
+0.100
+10
+10-28
+10-23
+10-18
+10-13
+10-8
+Figure 10.
+Prediction of GW energy density as a function of the frequency for the first three BPs
+as shown in Table 1. We have also highlighted the regions that indicate the proposed sensitivities
+of the GW experiments namely U-DECIGO and U-DECIGO corr [81, 82]. The sensitivity curves
+for DECIGO and U-DECIGO with correlation analyses are taken from Ref. [234].
+.
+where h∗ = 16.5 ×
+�
+Tn
+100 GeV
+� �
+g∗
+100
+�1/6
+Hz, the inverse Hubble time during GW production,
+red-shifted to today. The peak frequency ftur is given by,
+ftur = 2.7 × 10−5 1
+vw
+� β
+H∗
+� �
+Tn
+100 GeV
+� � g∗
+100
+�1/6
+Hz.
+(4.23)
+We set κtur = ϵκsw where ϵ stands for the fraction of the bulk motion which is turbulent.
+Simulations suggest κtur = 0.1κsw which we have considered in our numerical calculations.
+U-DECIGO
+U-DECIGO-corr
+DECIGO-corr
+BP-IV
+BP-V
+BP-VI-I
+BP-VI-II
+10-7
+10-5
+0.001
+0.100
+10
+10-28
+10-23
+10-18
+10-13
+10-8
+Figure 11. Prediction of GW energy density as a function of the frequency for the last three BPs
+from Table 1. We have also highlighted the regions that indicate the proposed sensitivities of the
+GW experiments namely DECIGO-corr, U-DECIGO and U-DECIGO corr [81, 82].
+With these details, in Figure 10 we present the estimates of GW energy density spec-
+trum as a function of frequency for the first three BPs as shown in Table 1. The predictions
+– 34 –
+
+of ΩGWh2 for the last three BPs of Table 1 are shown in Figure 11. We notice from Eq.
+(4.14), Eq. (4.18) and Eq. (4.22), that each individual contribution to the total GW energy
+density, ΩGWh2 (as defined in Eq.(4.13)) is an increasing function of α 16. This feature in
+turn makes ΩGWh2 rise as well for a relatively larger α. In contrast, a larger
+β
+H∗ reduces
+the amount of ΩGWh2. Earlier, in Table 4, we observed that BP-III yields the largest value
+of α among the six BPs of Table 1 with relatively smaller
+β
+H∗ ratio. Consequently, we find
+the corresponding peak amplitude of ΩGWh2 to be ∼ O(10−17) for BP-III, which turns
+out to be the largest as well. This feature is depicted in Figure 10. The lowest peak am-
+plitude of ΩGWh2 that we obtain is for the first-step PT of BP-VI which is ∼ O(10−25)
+as shown in Figure 11. The massive suppression to ΩGWh2 for BP-VI-I is caused by the
+simultaneous presence of a large
+β
+H∗ value together with a small α value as shown in Table
+4. The second-step PT of BP-VI produces a peak having amplitude ∼ O(10−22) which is
+relatively less suppressed due to a smaller value of
+β
+H∗ compared to BP-VI-I as shown in
+Table 4.
+In view of such estimates, the proposed future GW interferometers namely U-DECIGO
+and U-DECIGO correlation have the required sensitivities to probe all the BPs, except
+BP-VI-I, considered in our analysis including BP-I which is preferred in order to address
+EWBG. We also find it pertinent to mention that the peak frequency of each contribution
+to GW energy density is linearly proportional to the ratio
+β
+H∗ as evident from Eqs. (4.16),
+(4.20) and (4.23).
+It is numerically found that the frequency fmax where ΩGWh2 (see
+Eq.(4.13)) attains maximum, also emerges to be an increasing function of
+β
+H∗ ratio. As
+already noted in Table 4, that BP-VI-I produces the largest
+β
+H∗ ratio among all the BPs.
+This makes the peak frequency fmax of the corresponding GW spectrum for BP-VI-I the
+largest among all BPs.
+−1.5
+−1.0
+−0.5
+0.0
+0.5
+log10α
+2
+3
+4
+5
+6
+7
+log10(β/H∗)
+SFOPT and possible EWBG
+SFOPT and no EWBG
+1.5
+2.0
+2.5
+3.0
+3.5
+∆φSU(2)/Tn
+−1.5
+−1.0
+−0.5
+0.0
+0.5
+log10α
+2
+3
+4
+5
+6
+7
+log10(β/H∗)
+SFOPT and possible EWBG
+SFOPT and no EWBG
+20
+40
+60
+80
+100
+120
+140
+160
+Tn
+Figure 12. Values of α and
+β
+H∗ as a function of ∆φSU(2)/Tn (right) and nucleation temperature Tn
+(left) for the points in Figure 7 that satisfy the criteria of SFOPT with possible EWBG (depicted
+by coloured ‘■’) and SFOPT without EWBG (depicted by coloured ‘▲’).
+16For α ≫ 1, Ωcolh2, Ωswh2 and Ωturh2 are expected to turn insensitive to the change of α.
+– 35 –
+
+Earlier in Figure 7 we have identified points in the vN − λN plane that exhibits strong
+PT along the SU(2)L doublet direction, i.e., ∆φSU(2)/Tn > 1, with and without favouring
+EWBG as highlighted by coloured ‘■’ and ‘▲’ symbols, respectively. Recollect that, in
+order to prepare Figure 7, we have utilised the fixed values of the other relevant independent
+parameters as in BP-I, except vN and λN. In Figure 12, we show the estimates of α and
+β/H∗, corresponding to the same parameter corner, that is relevant to estimate ΩGWh2 as a
+function of ∆φSU(2)/Tn (left) and the nucleation temperature Tn (right), respectively. Note
+that we are giving particular emphasis on analysing Figure 7 further to compute the GW
+energy density since it offers the scope of realising EWBG while exhibiting ∆φSU(2)/Tn > 1
+(traceable at GW interferometers) at the same time. The Figure 12 illustrates the fact that
+the points, favoured for EWBG require relatively higher β/H∗ and lower α values compared
+to the points that do not favour EWBG. This essentially suppresses the peak amplitude
+of ΩGWh2 for the points favouring EWBG and simultaneously increase the peak frequency
+fmax. The right panel of Figure 12 indicates that a lower Tn tends to increase α which
+in turn enhance the ∆φSU(2)/Tn leading to larger Ωpeak
+GW h2. Such features are imprinted
+in Figure 13 where we have shown the estimates of ΩGWh2 as a function of f for both
+the coloured ‘■’ and ‘▲’ shaped points, present in Figure 7. We clearly observe that the
+points which are not favoured for possible EWBG, produce a larger amount of ΩGWh2 at a
+particular f and may even fall within the sensitivity curves of LISA [236] and BBO [156].
+However, the discovery scopes of those points purely depend on the signal-to-noise ratio of
+the corresponding experiments [237].
+10−5
+10−4
+10−3
+10−2
+10−1
+100
+101
+102
+103
+f [Hz]
+10−28
+10−25
+10−22
+10−19
+10−16
+10−13
+10−10
+10−7
+ΩGWh2(f)
+LISA
+BBO
+DECIGO-corr
+U-DECIGO
+U-DECIGO-corr
+BP-I
+SFOPT and EWBG
+0.020
+0.025
+0.030
+0.035
+0.040
+0.045
+α
+10−5
+10−4
+10−3
+10−2
+10−1
+100
+101
+102
+103
+f [Hz]
+10−28
+10−25
+10−22
+10−19
+10−16
+10−13
+10−10
+10−7
+ΩGWh2(f)
+LISA
+BBO
+DECIGO-corr
+U-DECIGO
+U-DECIGO-corr
+SFOPT and no EWBG
+0.2
+0.4
+0.6
+0.8
+1.0
+1.2
+1.4
+1.6
+α
+Figure 13. GW spectra for the points that show SFOPT in the SU(2)L doublet field directions
+with (left) and without (right) possible EWBG. Note that these points are marked by ‘ ■’ and ‘ ▲’
+in Figure 7. For both figures, we keep α as a variable.
+5
+Summary and Conclusion
+In the present work, we have addressed the properties of EWPT in the RHN superfield
+extended setup of Z3 invariant NMSSM. The RHN extended Z3 invariant NMSSM is cap-
+tivating due to its ability to provide solutions to the µ−problem of the MSSM and non-
+– 36 –
+
+vanishing neutrino masses and mixing simultaneously. In particular, we consider the case
+where both the LH- and RH-sneutrino receive non-zero VEVs, leading to a spontaneous R-
+parity-violating scenario. We have worked in an effective field theory set-up by integrating
+out the heavier squarks, gluinos, as well as sleptons. Additionally, a simple parametriza-
+tion of the TeV scale seesaw dictates the LH-sneutrino fields to weakly couple to the other
+relevant fields and thus, is expected to contribute negligibly to the PT dynamics. These
+facts effectively lead to a four-dimensional field space spanned by the four CP-even Higgses
+which is of interest in order to explore the PT characteristics in the present framework.
+Without going into the numerical details, one can naively anticipate that in the current
+setup having a four-dimensional field space, the PT dynamics is likely to be more involved
+than in the NMSSM where the relevant field space is three-dimensional.
+The EWPT
+properties and estimate of GW spectrum in the NMSSM have been extensively studied in
+literature where the roles of NMSSM parameters on the PT strength are also detailed. In
+this work, we scrutinize the role served by the new parameters that appear in theory due
+to the presence of the RHN superfield on the PT dynamics. In particular, we find that
+three new parameters λN, AλN and vN leave a non-trivial impact on determining the PT
+strength.
+In the beginning, we describe the model details and successively develop the tools re-
+quired to study the behaviour of the scalar potential as a function of temperature. We
+then demonstrate the possible experimental constraints that are of utmost importance to
+obtain a viable parameter space. Specifically, we undertake constraints arising from the
+validation of SM Higgs boson properties, BSM Higgs and SUSY searches at colliders, var-
+ious flavour-violating processes, neutrino experiments and the muon anomalous magnetic
+moment. Since extensive scanning of full parameter space considering a four-dimensional
+field space, relevant for PT is numerically challenging, we first adopt a benchmark-based
+analysis. We provide six BPs that pass through all the experimental constraints and exhibit
+distinct kinds of FOPT patterns along the different field directions. We have discussed the
+PT dynamics corresponding to each BP in detail.
+An SFOPT is a prerequisite for EWBG with distinct high-temperature behaviour of
+the total scalar potential along the SU(2)L field directions. We have shown that BP-I is
+the preferred BP that exhibits the essential features required for a possible EWBG. On
+the other hand, BP-II - BP-V showing SFOPT along the different SU(2)L doublet and
+singlet field directions in single-step, however, are not suitable for successful EWBG. We
+find multi-step FOPT for BP-VI. All the BPs listed have one particular feature in common
+which is the preference for a lighter RH-sneutrino-dominated state below 125 GeV for the
+occurrence of a FOPT. Next, we utilize a few of the BPs to inquire about the role of new
+parameters on PT strength. Two of the new parameters vN and λN show similar impacts
+on the PT strength along either of the SU(2)L doublet or singlet field directions. It turns
+out that the PT strength increases with the decrease of either vN or λN. The remaining
+parameter AλN has a minor role in the PT along SU(2)L doublet field directions whereas
+the PT strengths in the SU(2)L singlet field directions get enhanced with the increase of
+|AλN |. The possible reasons for such unique properties are associated with the impact of
+the new parameters on the barrier height in the constituent field directions and also the
+– 37 –
+
+lightness of the RH-sneutrino state.
+Finally, we examine the testability of the BPs by computing the GW energy density
+corresponding to each BP. We have considered all possible sources that trigger GW emis-
+sion in a FOPT namely, bubble wall collisions, sound waves and magneto-hydrodynamic
+turbulence. The highest peak amplitude of the GW energy density that we obtain is for
+BP-III which lies within the proposed sensitivity of DECIGO correlation data. The peak
+amplitude of ΩGWh2 for other BPs is relatively weaker, however, within the reach of U-
+DECIGO and U-DECIGO-corr sensitivities. It is to be noted that a TeV scale canonical
+seesaw model with RHN weakly coupled to SM particles is extremely difficult to probe
+at collider experiments. Our analysis infers an alternative albeit promising pathway to
+validate a TeV scale seesaw model at future GW interferometers beyond colliders.
+In the present work, we have not performed an exact prediction of the baryon asym-
+metry of the Universe.
+Instead, we find the corner of the parameter space that shows
+SFOPT along the SU(2)L doublet field directions and facilitates EWBG. Improvement of
+our analysis is possible by precise computation of bubble wall profile, bubble wall velocity,
+and CP-violation that decide the final amount of baryon asymmetry of the Universe, which
+is also correlated with NMSSM + RHN model parameters. In an R-parity violating theory
+like the present one, gravitino can be a potential decaying dark matter candidate. Future
+works may also include investigating the correspondence between gravitino dark matter
+phenomenology and NMSSM + RHN parameter space, favouring an SFOPT.
+Acknowledgements
+P. B. acknowledges the financial support received from the Indian Institute of Technology,
+Delhi (IITD) as a Senior Research Fellow. P. G. acknowledges the IITD SEED grant sup-
+port IITD/Plg/budget/2018-2019/21924, continued as IITD/Plg/budget/2019-
+2020/173965, IITD Equipment Matching Grant IITD/IRD/MI02120/208794, and
+Start-up Research Grant (SRG) support SRG/2019/000064 from the Science and Engi-
+neering Research Board (SERB), Department of Science and Technology, Government of
+India. A.K.S. is supported by NPDF grant PDF/2020/000797 from the SERB, Govern-
+ment of India. P.B. and A.K.S. also acknowledge Mikael Chala, Bo-Qiang Lu, Jiang Zhu
+and Kaius Loos for useful communications regarding cosmoTransitions code.
+A
+Field dependent mass matrices
+Our numerical studies are based on the field-dependent masses (see subsection 2.1). The
+corresponding scalar squared mass terms are evaluated at T = 0 using the tree-level un-
+coloured scalar potential Vscalar (see below), including only the dominant higher-order con-
+tributions ∆V (see Eq. (2.12)). Mathematically, for the uncoloured scalar squared mass
+matrices
+M2
+X,ij = M2
+φαφβ(HSM, HNSM, HS, NR) ≡ ∂2Vscalar
+∂φα∂φβ
+����
+φα̸=0
+,
+(A.1)
+where X = S (for the CP-even neutral scalar) or A (for the CP-odd neutral scalar) and
+i, j = 1, ....., 7. Further, φα(β) = HSM, HNSM, HS, NR, ℜ(�ν1,2,3) for the CP-even neutral
+– 38 –
+
+scalar and φα(β) = ANSM, AS, G0, NI, ℑ(�ν1,2,3) for the CP-even neutral scalar, respectively.
+For the uncoloured electrically charged scalar, X = C with i, j = 1, ....., 8 and φα(β) ≡ C+ =
+H+, G+, �e+
+L, �µ+
+L, �τ +
+L , �e+
+R, �µ+
+R, �τ +
+R . Here, we have used
+�νi = ℜ�νi + iℑ�νi
+√
+2
+≡ νRi + i�νIi
+√
+2
+with
+i = 1, 2, 3 ≡ e, µ, τ.
+(A.2)
+The full uncoloured scalar potential is given by
+Vscalar =
+�����
+3
+�
+i=1
+Y i
+N �νi �
+N − λSH0
+d
+�����
+2
++
+������
+3
+�
+i,j=1
+Y ij
+e �li�ec
+j − λSH0
+u
+������
+2
++
+������
+Y i
+NH0
+u �
+N −
+3
+�
+j=1
+Y ij
+e H−
+d �ec
+j
+������
+2
++
+����λHu · Hd + κS2 + λN
+2
+�
+N 2
+����
+2
++
+�����
+3
+�
+i=1
+Y i
+N �Li · Hu + λNS �
+N
+�����
+2
++
+�����
+3
+�
+i=1
+Y ij
+e Hd · �Li
+�����
+2
++
+������
+λSH+
+u −
+3
+�
+i,j=1
+Y ij
+e �νi�ec
+j
+������
+2
++
+�����λSH−
+d −
+3
+�
+i=1
+Y i
+N�li �
+N
+�����
+2
++
+������
+3
+�
+j=1
+Y ij
+e H0
+d�ec
+j − Y i
+NH+
+u �
+N
+������
+2
++ g2
+1
+8 (|Hd|2 − |Hu|2 + |�Li|2 − 2|�ec
+i|2)2 + g2
+2
+2
+3
+�
+a=1
+�
+H†
+d
+τ a
+2 Hd + H†
+u
+τ a
+2 Hu + �L†
+i
+τ a
+2
+�Li
+�2
++ m2
+Hd|Hd|2 + m2
+Hu|Hu|2 + m2
+S|S|2 + M 2
+N| �
+N|2 +
+3
+�
+i,j=1
+m2
+�Lij �Lm∗
+i
+�Lm
+j +
+3
+�
+i,j=1
+m2
+�ec
+ij�ecm∗
+i
+�ecm
+j
++
+3
+�
+i=1
+(AeYe)ijHd · �Li�ec
+j + λAλSHu · Hd + (ANYN)i�Li · Hu �
+N + κAκ
+3 S3 + λNAλN
+2
+S �
+N 2
++ h.c.
+(A.3)
+Here Y ij
+e
+belongs to W ′
+MSSM (see Eq.
+(2.1)) and m2
+Hd, m2
+Hu, m2
+�Lij, m2
+�ec
+ij, (AeYe)ij are
+encapsulated within −L′
+soft (see Eq. (2.2)). Further, i, j are generation indices, τ as are
+Pauli spin matrices and m = 1, 2, as per the standard notation (see Refs. [27, 83–85, 87, 88]
+for details).
+In a similar way, one can derive field-dependent mass matrices for the uncoloured elec-
+trically neutral and electrically charged fermions, i.e., neutralinos and charginos, directly
+from the superpotential W (see Eq. (2.1)). Mathematically, the generic mass term for the
+neutralino sector and the chargino sector are given by
+− 1
+2
+�
+ψ0T
+i M0ijψ0
+j + h.c.
+�
+,
+−1
+2(ψ+, ψ−)T Mχ±(ψ+, ψ−) + h.c.,
+(A.4)
+respectively. Here basis for the neutralino sector is given by ψ0T = { �B0, �
+W 0
+3 , �H0
+d, �H0
+u, �S, N,
+ν1, ν2, ν3} involving neutral U(1)Y , SU(2)L gauginos ( �B0, �
+W 0
+3 ), neutral higgsinos ( �H0
+d, �H0
+u),
+singlino (�S), RH-neutrino (N) and LH-neutrinos (ν1,2,3). For charginos, including charged
+SU(2)L gauginos (�
+W ±), charged higgsinos ( �H+
+u , �H−
+d ) and charged leptons (e±
+L, R, µ±
+L, R,
+τ ±
+L, R), one gets ψ+T = {�
+W +, �H+
+u , e+
+R, µ+
+R, τ +
+R } and ψ−T = {�
+W −, �H−
+d , e−
+L, µ−
+L, τ −
+L }, respec-
+– 39 –
+
+tively. We will start with the scalar mass squared matrices and will discuss the fermionic
+sector subsequently.17
+A.1
+CP-even neutral scalars squared mass matrix
+In the basis HSM, HNSM, HS, NR, ℜ(�ν1,2,3), non-zero entries of the symmetric M2
+S,ij are
+M2
+S,11 ≃
+1
+16vuvd
+�
+8λvSv2 (Aλ + κvS) + 2λ2vuvd
+�
+−4
+�
+v2 + 2v2
+S
+�
++ H2
+NSM + 4H2
+S + 3H2
+SM
+�
++vuvd
+�
+3∆λ2 + G2� �
+H2
+NSM + 3H2
+SM
+�
+−4 cos 2β
+�
+v2 cos 2β
+�
+2λvS (Aλ + κvS) + vuvd
+�
+G − 2λ2�
+vu
+�
++ 3∆λ2vuvdH2
+SM
+�
++vuvd
+�
+(4 sin 2β
+�
+3∆λ2HNSMHSM − 2λHS
+�√
+2Aλ + κHS
+��
+−3
+�
+∆λ2 + G − 2λ2� �
+2 sin 4β HNSMHSM + cos 4β
+�
+H2
+NSM − H2
+SM
+�� ��
+− 1
+2v
+�λNv
+2
+sin 2β
+�
+N2
+R − 2v2
+N
+��
+,
+(A.5)
+M2
+S,12 ≃
+1
+16vuvd
+�2v2 sin 4β
+�
+2λvS (Aλ + κvS) + vuvd
+�
+G − 2λ2��
+vuvd
+−8λ cos 2β HS
+�√
+2Aλ + κHS
+�
++ 3 sin 4β
+�
+∆λ2 + G − 2λ2� �
+H2
+NSM − H2
+SM
+�
+−6 cos 4β HNSMHSM
+�
+∆λ2 + G − 2λ2�
++ 2HNSMHSM
+�
+3∆λ2 + G + 2λ2�
++6∆λ2 sin 2β
+�
+H2
+NSM + H2
+SM
+� �
+− 1
+4
+�
+λ cos 2β λN
+�
+N2
+R − 2v2
+N
+� �
+,
+(A.6)
+M2
+S,13 ≃ λ2HSHSM − 1
+2λ
+�√
+2Aλ + 2κHS
+�
+(cos 2β HNSM + sin 2β HSM) ,
+(A.7)
+M2
+S,14 ≃ −1
+2λλNNR (cos 2β HNSM + sin 2β HSM),
+(A.8)
+M2
+S,1 (4+i) ≃ 1
+2NRY i
+N
+�√
+2AN sin β + HS (λ cos β + λN sin β)
+�
+,
+(A.9)
+M2
+S,22 ≃
+1
+16vuvd
+�
+8λvSv2 (Aλ + κvS) + vuvd (3∆λ2 + G)
+�
+3H2
+NSM + H2
+SM
+�
++2λ2vuvd
+�
+−4
+�
+v2 + 2v2
+S
+�
++ 3H2
+NSM + 4H2
+S + H2
+SM
+�
++4 cos 2β
+�
+v2 cos 2β
+�
+2λvS (Aλ + κvS) + vdvu
+�
+G − 2λ2��
++ 3∆λ2vdH2
+NSMvu
+�
++vdvu
+�
+4 sin 2β
+�
+2λHS
+�√
+2Aλ + κHS
+�
++ 3∆λ2HNSMHSM
+�
++3
+�
+∆λ2 + G − 2λ2� �
+2 sin 4β HNSMHSM + cos 4β
+�
+H2
+NSM − H2
+SM
+�� ��
++ 1
+8v
+�
+cos β cot βλN
+�
+λv(cos 4β + 3) sec3 βv2
+N + 4λv sin β tan β N2
+R
+��
+,
+(A.10)
+17While writing field-dependent masses, we ignore terms that are quadratic in vi, Y i
+N and terms like
+3�
+i=1
+viY i
+N, keeping in mind their smallness. Besides, as already stated, these terms do not play any crucial
+role in the EWPT. Nevertheless, we have kept all these terms in our numerical analysis.
+– 40 –
+
+M2
+S,23 ≃ 1
+2λ
+�√
+2Aλ + 2κHS
+�
+(sin 2β HNSM − cos 2β HSM) + λ2HNSMHS,(A.11)
+M2
+S,24 ≃ 1
+2λλNNR (sin 2β HNSM − cos 2β HSM),
+(A.12)
+M2
+S,2 (4+i) ≃ 1
+2NRY i
+N
+�√
+2AN cos β + HS (cos β λN − λ sin β )
+�
+,
+(A.13)
+M2
+S,33 ≃ λvuvd (Aλ + 2κvS)
+vS
++ κ
+�
+Aκ
+�√
+2HS − vS
+�
++ κ
+�
+3H2
+S − 2v2
+S
+��
+− λ2v2
++λ2
+2
+�
+H2
+NSM + H2
+SM
+�
+− λκ cos 2β HNSMHSM + λκ
+2 sin 2β
+�
+H2
+NSM − H2
+SM
+�
++ 1
+2vS
+�
+λN
+�
+(κ + λN) vS
+�
+N2
+R − 2v2
+N
+�
+− v2
+NAλN
+��
+,
+(A.14)
+M2
+S,34 ≃ 1
+2λNNR
+�√
+2AλN + 2 (κ + λN) HS
+�
+,
+(A.15)
+M2
+S,3 (4+i) ≃ 1
+2Y i
+NNR (HNSM (λN cos βλ sin β ) + HSM (λ cos β + λN sin β )),
+(A.16)
+M2
+S,44 ≃
+1
+4vN
+�
+− 2λλN cos 2β HNSMHSMvN+λ sin 2β vNλN
+�
+H2
+NSM − H2
+SM + 2v2�
++λNvN
+�
+2AλN
+�√
+2HS − 2vS
+�
++ 2 (κ + λN) H2
+S
+�
++λNvN
+�
+λN
+�
+3N2
+R − 2v2
+N
+�
+− 4 (κ + λN) v2
+S
+� �
+,
+(A.17)
+M2
+S,4 (4+i) ≃ 1
+2Y i
+N
+�√
+2AN (cos β HNSM + sin β HSM) + HS
+�
+HNSM (λN cos β − λ sin β)
++HSM (λ cos β + λN sin β)
+��
+,
+(A.18)
+M2
+S,(4+i) (4+j) ≃ δij
+8
+�
+− 2G sin 2β HNSMHSM − G cos 2β
+�
+H2
+NSM − H2
+SM
+�
+−8vNY i
+N (vu (AN + λNvS) + λvdvS)
+vi
+− 2Gv2 cos 2β
+�
+−1
+4g2
+2 (sin β HNSM − cos β HSM) 2,
+(A.19)
+where we have used G = g2
+1 + g2
+2, v2
+u + v2
+d = v2 and i = 1, 2, 3 are generational indices.
+A.2
+CP-odd neutral scalars squared mass matrix
+In the basis ANSM, AS, G0, NI, ℑ(�ν1,2,3), non-zero entries of the symmetric M2
+A,ij are
+M2
+A,11 ≃
+1
+16vdvu
+�
+8λvSv2 (Aλ + κvS) + Gvdvu
+�
+H2
+NSM − H2
+SM
+�
++2λ2vdvu
+�
+−4
+�
+v2 + 2v2
+S
+�
++ H2
+NSM + 4H2
+S + 3H2
+SM
+�
++ ∆λ2vdvu
+�
+3H2
+NSM + H2
+SM
+�
++4 cos 2β
+�
+v2 cos 2β
+�
+2λvS (Aλ + κvS) + vuvd
+�
+G − 2λ2��
++ ∆λ2vuvdH2
+NSM
+�
++vuvd
+�
+4 sin 2β
+�
+2λHS
+�√
+2Aλ + κHS
+�
++ ∆λ2HNSMHSM
+�
++
+�
+∆λ2 + G − 2λ2� �
+2 sin 4β HNSMHSM + cos 4β
+�
+H2
+NSM − H2
+SM
+�� ��
++ 1
+8v
+�
+cos β cot β λN
+�
+λv(cos 4β + 3) sec3 β v2
+N + 4λv sin β tan β N2
+R
+��
+,
+(A.20)
+– 41 –
+
+M2
+A,12 ≃ 1
+2λHSM
+�√
+2Aλ − 2κHS
+�
+,
+(A.21)
+M2
+A,13 ≃ 1
+16
+�2v2 sin 4β
+�
+2λvS (Aλ + κvS) +
+�
+G − 2λ2�
+vuvd
+�
+vuvd
+−8λ cos 2β HS
+�√
+2Aλ + κHS
+�
++ 2∆λ2 sin 2β
+�
+H2
+NSM + H2
+SM
+�
++2HNSMHSM
+�
+∆λ2 + G − 2λ2�
+− 2 cos 4β HNSMHSM
+�
+∆λ2 + G − 2λ2�
++ sin 4β
+�
+∆λ2 + G − 2λ2�
+(HNSM − HSM) (HNSM + HSM)
+�
+− 1
+4v
+�
+λλNv cos 2β
+�
+N2
+R − 2v2
+N
+� �
+,
+(A.22)
+M2
+A,14 ≃ −1
+2λλNHSMNR,
+(A.23)
+M2
+A,1 (4+i) ≃ −1
+2Y i
+NNR
+�√
+2AN cos β + HS (cos β λN − λ sin β )
+�
+,
+(A.24)
+M2
+A,22 ≃ λvuvd (Aλ + 2κvS)
+vS
+− κAκ
+�√
+2HS + vS
+�
+− 1
+2λ2 �
+2v2 + H2
+NSM + H2
+SM
+�
++λκ cos 2β HNSMHSM + λκ sin β cos β
+�
+H2
+SM − H2
+NSM
+�
++ κ2 �
+H2
+S − 2v2
+S
+�
+− 1
+2vS
+�
+λN
+�
+(v2
+NAλN + vS
+�
+2v2
+N (κ + λN) + N2
+R (κ − λN)
+� ��
+,
+(A.25)
+M2
+A,23 ≃ −1
+2λHNSM
+�√
+2Aλ − 2κHS
+�
+,
+(A.26)
+M2
+A,24 = −1
+2λNNR
+�√
+2AλN − 2κHS
+�
+,
+(A.27)
+M2
+A,2 (4+i) ≃ Y i
+N
+2 NR (HNSM (λN cos β − λ sin β) + HSM (λ cos β + λN sin β)),
+(A.28)
+M2
+A,33 ≃
+1
+16vuvd
+�
+8λvSv2 (Aλ + κvS) − Gvuvd
+�
+H2
+NSM − H2
+SM
+�
++2λ2vuvd
+�
+−4
+�
+v2 + 2v2
+S
+�
++ 3H2
+NSM + 4H2
+S + H2
+SM
+�
++ ∆λ2vuvd
+�
+H2
+NSM + 3H2
+SM
+�
+−4 cos 2β
+�
+v2 cos 2β
+�
+2λvS (Aλ + κvS) + vuvd
+�
+G − 2λ2��
++ ∆λ2vdH2
+SMvu
+�
++vuvd
+�
+4 sin 2β
+�
+∆λ2HNSMHSM − 2λHS
+�√
+2Aλ + κHS
+��
+−
+�
+∆λ2 + G − 2λ2� �
+2 sin 4β HNSMHSM + cos 4β
+�
+H2
+NSM − H2
+SM
+�� ��
+− 1
+2v
+�
+λλNv sin β cos β
+�
+N2
+R − 2v2
+N
+��
+,
+(A.29)
+M2
+A,34 ≃ 1
+2λλNHNSMNR,
+(A.30)
+M2
+A,3 (4+i) ≃ −Y i
+N
+2 NR
+�√
+2AN sin β + HS (λ cos β + sin β λN)
+�
+,
+(A.31)
+– 42 –
+
+M2
+A,44 ≃
+1
+4vN
+�
+2λλN cos 2βHNSMHSMvN
++λλN sin 2βvN
+�
+−H2
+NSM + H2
+SM + 2v2�
++λNvN
+�
+− 2AλN
+�√
+2HS + 2vS
+�
++ 2 (λN − κ) H2
+S
+�
++λNvN
+�
+λN
+�
+N2
+R − 2v2
+N
+�
+− 4v2
+S (κ + λN)
+��
+,
+(A.32)
+M2
+A,4 (4+i) ≃ Y i
+N
+2
+�
+HS
+�
+HNSM (λ sin β + λN cos β) + HSM (λN sin β − λ cos β)
+�
+−
+√
+2AN (cos βHNSM + sin β HSM)
+�
+,
+(A.33)
+M2
+A, (4+i)(4+j) ≃ − δij
+8vj
+�
+G cos 2βvj
+�
+H2
+NSM − H2
+SM + 2v2�
++ 2Gvj sin 2β HNSMHSM
++8v sin β vNY j
+N (AN + λNvS) + 8λv cos β vNY j
+NvS
+�
+−1
+4g2
+2 (sin β HNSM − cos β HSM)2,
+(A.34)
+M2
+A, 56 ≃ Y 1
+NY 2
+N
+2
+(cos βHNSM + sin βHSM)2 ,
+(A.35)
+M2
+A, 57 ≃ Y 1
+NY 3
+N
+2
+(cos βHNSM + sin βHSM)2
+(A.36)
+M2
+A,67 ≃ Y 2
+NY 3
+N
+2
+(cos β HNSM + sin β HSM)2 .
+(A.37)
+where we have used G = g2
+1+g2
+2, v2
+u+v2
+d = v2 and i = 1, 2, 3 are generational indices. At the
+physical vacuum, i.e.,
+�
+⟨HSM⟩, ⟨HNSM⟩, ⟨HS⟩, ⟨NR⟩
+�
+=
+�√
+2v, 0,
+√
+2vS,
+√
+2vN
+�
+, neglecting
+terms like v2
+i , Y i2
+N ,
+3�
+i=1
+viY i
+N, the Goldstone mode appears massless and decouples from the
+other CP-odd states.
+A.3
+Uncoloured charged scalars squared mass matrix
+Non-zero entries for the uncoloured symmetric charged scalar mass squared matrix, i.e.,
+C+MCC−, in the basis C+ = H+, G+, �e+
+L, �µ+
+L, �τ +
+L , �e+
+R, �µ+
+R, �τ +
+R are
+M2
+C,11 ≃ 1
+16
+�
+2 cos 2βg2
+1
+�
+2 sin 2βHNSMHSM + cos 2β(2v2 + H2
+NSM − H2
+SM)
+�
++g2
+2
+�
+(1 + cos 4β)H2
+NSM + 2 sin 4βHNSMHSM − (−3 + cos 4β)H2
+SM + 2v2(1 + cos 4β)
++4 cos 2β
+�
+− 4v2λ2(3 + cos 4β) + 2λ2(4H2
+S + (−1 + cos 4β)H2
+SM) − 16λ2v2
+s
++2∆λ2 sin2 2βH2
+SM
++4(λ2 sin2 2β + ∆λ22 cos4 β)H2
+SM + 4(λ2 sin 4β − 4∆λ2 cos3 β sin β)HNSMHSM
++4λvSAλ(3 + cos 4β) csc β sec β + 4λ sin β
+�
+2HS(
+√
+2Aλ + κHS + λNN2
+R)
+�
++4λ(3 + cos 4β) csc 2β(2κv2
+S + v2
+NλN)
+�
+,
+(A.38)
+– 43 –
+
+M2
+C,12 ≃ 1
+16
+� �
+2 cos 4β(2λ2 − G − ∆λ2) + 2(2λ2 + g2
+1 − g2
+2 + ∆λ2)
+�
+HNSMHSM
++2 sin 2β∆λ2(H2
+NSM + (1 + 2 sin2 β)H2
+SM)
++ sin 4β
+�
+(G − 2λ2)(2v2 + H2
+NSM − H2
+SM) + (H2
+NSM − H2
+SM)∆λ2
+�
++8λ cos 2β
+�
+κH2
+S + Aλ(
+√
+2HS − 2vS) − 2κv2
+S + λN
+2 (N2
+R − 2v2
+N)
+� �
+,(A.39)
+M2
+C1, (2+i) ≃ δij
+4
+�
+vj
+� �
+g2
+2 cos 2β + 2(Y ij
+e )2 sin2 β
+�
+HNSM + sin 2β
+�
+g2
+2 − (Y ij
+e )2�
+HSM
+�
+−2Y j
+NNR
+�√
+2AN cos β + cos β
+�
+λNHS − Y ij
+e (sin βHNSM
+− cos βHSM)
+�
++ λ sin βHS
+��
+,
+(A.40)
+M2
+C,1 (5+i) ≃ − 1
+√
+2AeY ij
+e vj sin β − 1
+2Y ij
+e Y j
+N sin β(NR) (cos βHNSM + sin βHSM) ,
+(A.41)
+M2
+C,22 ≃ 1
+16
+�
+2G cos2 β(H2
+SM − 2v2) + 4λ2 sin2 2β(H2
+SM − 2v2) + 8λ2(H2
+S − 2v2
+S)
++
+�
+− 2g2
+1 cos2 2β + g2
+2(cos 4β − 3) + (cos 4β − 1)(2λ2 − ∆λ2)
+�
+H2
+NSM
++2
+�
+sin 4β(2λ2 − G)8 cos β sin3 β∆λ2
+�
+HSMHNSM + 4λ sin 2β
+�
+− 2κH2
+S
++4κv2
+S + Aλ
+�
+−2
+√
+2HS + 4vS − λN(N2
+R − 2v2
+N)
+� ��
+,
+(A.42)
+M2
+C2, (2+i) ≃ δij
+4
+�
+vj
+� �
+−g2
+2 cos 2β − 2(Y ij
+e )2 sin2 β
+�
+HSM + sin 2β
+�
+g2
+2 − (Y ij
+e )2�
+HNSM
+�
+−2Y j
+NNR
+�√
+2AN sin β + sin β
+�
+λNHS − Y ij
+e (sin βHNSM
+− cos βHSM)
+�
+− λ cos βHS
+��
+,
+(A.43)
+M2
+C,2 (5+i) ≃ −(AeYe)ij
+√
+2
+vj cos β − 1
+2Y ij
+e Y j
+NNR
+�
+cos2 βHNSM + sin 2β
+2
+HSM
+�
+, (A.44)
+M2
+C,(2+i)(2+j) ≃ m2
+�Lij + δij
+8
+�
+(g2
+1 − g2
+2)(cos 2β(H2
+SM − H2
+NSM) − 2 sin 2βHSMHNSM)
++4(Y ij
+e )2(cos βHSM − sin βHNSM)2
+�
+,
+(A.45)
+M2
+C,(2+i)(5+j) ≃ δij(AeYe)ij
+√
+2
+(cos βHSM − sin βHNSM)
+−δijλY ij
+e
+2
+(cos βHNSM + sin βHSM)HS,
+(A.46)
+M2
+C,(5+i)(5+j) ≃ m2
+�ec
+ij − δij
+4
+�
+g2
+1(cos 2β(H2
+SM − H2
+NSM) − 2 sin 2βHSMHNSM)
+−2(Y ij
+e )2(cos βHSM − sin βHNSM)2
+�
+,
+(A.47)
+– 44 –
+
+where we have used G = g2
+1+g2
+2, v2
+u+v2
+d = v2 and i = 1, 2, 3 are generational indices. At the
+physical vacuum, i.e.,
+�
+⟨HSM⟩, ⟨HNSM⟩, ⟨HS⟩, ⟨NR⟩
+�
+=
+�√
+2v, 0,
+√
+2vS,
+√
+2vN
+�
+, neglecting
+terms like v2
+i , Y i2
+N ,
+3�
+i=1
+viY i
+N, the Goldstone mode appears massless and decouples from the
+other charged states.
+A.4
+Neutralino mass matrix
+In the basis of ψ0T = { �B0, �
+W 0
+3 , �H0
+d, �H0
+u, �S, N, ν1, ν2, ν3}, the matrix M0 (see Eq. (A.4)) is
+given as
+M0 =
+�
+�
+�
+M6×6 m6×3
+mT
+3×6 03×3
+�
+�
+� ,
+(A.48)
+where we have used ⟨�νi⟩ = vi (see Eq. (2.6)) as the LH-sneutrinos are not dynamical in
+nature (see subsection 2.1). Further, matrices mT
+3×6 and M6×6, using Eq. (2.8), are given
+as
+mT
+3×6 =
+�
+�
+�
+�
+�
+�
+�
+�
+�
+−g1ve
+√
+2
+g2ve
+√
+2 0 Y 1
+NNR
+√
+2
+0 Y 1
+N
+√
+2Y
+−g1vµ
+√
+2
+g2vµ
+√
+2 0 Y 2
+NNR
+√
+2
+0 Y 2
+N
+√
+2Y
+−g1vτ
+√
+2
+g2vτ
+√
+2 0 Y 3
+NNR
+√
+2
+0 Y 3
+N
+√
+2Y
+�
+�
+�
+�
+�
+�
+�
+�
+�
+,
+(A.49)
+with Y =
+�
+sβHSM + cβHNSM
+�
+and the symmetric matrix M6×6 is given as,
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+M1
+0
+− g1
+2 X
+g1
+2 Y
+0
+0
+M2
+g2
+2 X
+− g2
+2 Y
+0
+0
+0
+− λ
+√
+2HS − λ
+√
+2Y
+0
+0
+− λ
+√
+2X
+0
+√
+2κHS
+λN
+2
+√
+2NR
+λN
+√
+2HS
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+,
+(A.50)
+where we have omitted symmetric entries, i.e., M0ij = M0ji for ̸= j and X =
+�
+cβHSM −
+sβHNSM
+�
+.
+A.5
+Chargino mass matrix
+Using a similar approach, in the basis ψ+T = {�
+W +, �H+
+u , e+
+R, µ+
+R, τ +
+R } and ψ−T = {�
+W −,
+�H−
+d , e−
+L, µ−
+L, τ −
+L }, the matrix M± is given as
+M± =
+�
+0 XT
+X
+0
+�
+,
+(A.51)
+– 45 –
+
+where the 5 × 5 matrix X is given by
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+M2
+g2
+√
+2Y
+0
+0
+0
+g2
+√
+2X
+λ
+√
+2HS
+−Y 11
+e ve −Y 22
+e ve −Y 33
+e vτ
+g2ve −
+Y 1
+N NR
+√
+2
+Y 11
+e
+√
+2 X
+0
+0
+g2vµ −
+Y 2
+N NR
+√
+2
+0
+Y 22
+e
+√
+2 X
+0
+g2vτ −
+Y 3
+N NR
+√
+2
+0
+0
+Y 33
+e
+√
+2 X
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+.
+(A.52)
+Here we have used Y ij
+e
+= Y ii
+e δij.
+B
+Neutral scalar mass matrices after the EWSB
+Weak couplings among the LH-handed sneutrino states and the remaining states, as already
+discussed in section 2, suggest that one can safely decouple the LH-sneutrino-dominated
+states from the CP-even and CP-odd scalar squared mass matrices without any loss of
+generality.
+After the aforesaid detachment, both CP-even and CP-odd scalar squared
+mass matrices appear to be 4 × 4 in size.
+The full 7 × 7 squared mass matrices are
+given in subsections A.1 & A.2, including LH-sneutrino states. In this section, squared
+mass matrices of the CP-even and the CP-odd Higgses are given after the EW symmetry
+breaking, i.e., using relations given in subsections A.1 & A.2 and considering ⟨HSM⟩ =
+√
+2v,
+⟨HNSM⟩ = 0, ⟨HS⟩ =
+√
+2vS, ⟨NR⟩ =
+√
+2vN, ⟨ANSM⟩ = 0, ⟨AS⟩ = 0, ⟨NI⟩ = 0 (see
+subsection 2.1). For the CP-even states, we consider the {HSM, HNSM, HS, NR} basis while
+for the CP-odd ones we use {ANSM, AS, G0, AN} basis.
+B.1
+CP-even mass squared elements
+M2
+S,11 = λ2v2 sin2 2β + (g2
+1 + g2
+2)v2
+2
+cos2 2β, M2
+S,12 = λ2v2
+2
+sin 4β − (g2
+1 + g2
+2)v2
+4
+sin 4β,
+M2
+S,13 = 2λ2vvS − λv(Aλ + 2κvS) sin 2β,
+M2
+S,14 = −λλNvN sin 2β,
+M2
+S,22 = 2λvS(Aλ + κvS) csc 2β + λλNv2
+N csc 2β − λ2v2 sin2 2β + (g2
+1 + g2
+2)v2
+2
+sin2 2β,
+M2
+S,23 = −λv(Aλ + 2κvS) cos 2β,
+M2
+S,24 = −λλNvvN cos 2β,
+M2
+S,33 = κvS(Aκ + 4κvS) + λv2Aλ
+2vS
+sin 2β − λNv2
+NAλN
+2vS
+,
+M2
+S,34 = λNvNAλN + 2λNκvSvN + 2λ2
+NvSvN,
+M2
+S,44 = λ2
+Nv2
+N,
+(B.1)
+where we have used the symmetric nature of these entries, i.e., M2
+S,ij = M2
+S,ji for i ̸= j.
+– 46 –
+
+B.2
+CP-odd mass squared elements
+M2
+A,11 = λλNv2
+N csc 2β + 2λvS(Aλ + κvS) csc 2β,
+M2
+A,12 = λvAλ − 2λκvvS,
+M2
+A,13 = 0,
+M2
+A,14 = −λλNvvN,
+M2
+A,22 = κ(2λv2 sin 2β − 3vSAκ) + λv2Aλ
+2vS
+sin 2β − λNv2
+N
+2vS
+(AλN + 4κvS),
+M2
+A,23 = 0,
+M2
+A,24 = 2λNκvSvN − λNvNAλN ,
+M2
+A,33 = 0,
+M2
+A,34 = 0,
+M2
+A,44 = λλNv2 sin 2β − 2λNvS(AλN + κvS),
+(B.2)
+where we have used the symmetric nature of these entries, i.e., M2
+A,ij = M2
+A,ji for i ̸= j.
+C
+Counter terms
+As already addressed in subsection 2.2, after including Coleman-Weinberg contributions
+(see Eq. (2.15)), counter terms are necessary to restore the original physical minima and
+masses. These terms are encapsulated within Vct which is written as
+Vct = δm2
+Hd |Hd|2 + δm2
+Hu |Hu|2 + δm2
+S |S|2 + δM2
+N | �
+N|2 + δλAλ (SHu · Hd + h.c.)
++δλNAλN (S �
+N �
+N + h.c.) + δλ2
+2
+|Hu|4,
+(C.1)
+where δm2
+Hd, δm2
+Hu, δm2
+S, δM2
+N , δλAλ, δλNAλN , δλ2 are counter terms corresponding to en-
+tries given by Eq. (2.2) and Eq. (2.12). Entries corresponding to δm2
+Hd, δm2
+Hu are encapsu-
+lated within −L′
+soft of Eq. (2.2). In order to maintain the location of the physical minima
+solutions for the counter-terms must satisfy the following relations:
+δm2
+Hd
+=
+1
+√
+2v
+�
+tan β
+∂Veff
+∂HNSM
+− ∂Veff
+∂HSM
+�
++ µ sec2 β
+2λv
+∂2Veff
+∂HS∂HSM
+,
+δm2
+Hu
+= csc2 β
+4vλ
+∂
+∂HSM
+�√
+2λ(cos 2β − 2) Veff + 2µ∂Veff
+∂HS
++ 2λv ∂Veff
+∂HSM
+�
+− 1
+√
+2v cot β
+∂Veff
+∂HNSM
+,
+δm2
+S
+= λ
+2µ
+∂
+∂HS
+�
+v ∂Veff
+∂HSM
++ vN
+∂Veff
+∂NR
+−
+√
+2 Veff
+�
+,
+δM2
+N
+= − 1
+2vN
+∂
+∂NR
+�√
+2 Veff − 2µ
+λ
+∂Veff
+∂HS
+�
+,
+δλAλ
+= csc 2β
+v
+∂2Veff
+∂HS∂HSM
+,
+δλNAλN
+= − 1
+2vN
+∂2Veff
+∂HS∂NR
+,
+δλ2 = csc4 β
+4v3
+∂
+∂HSM
+�√
+2 Veff − 2v ∂Veff
+∂HSM
+�
+.
+(C.2)
+– 47 –
+
+Identifying δλ2 as a counter term for ∆λ2, a quartic coupling among Hu as given in Eq.
+(2.12), seems inconsistent. However, in reality, ∆λ2 is connected to the soft SUSY-breaking
+terms as the estimation of ∆λ2 includes soft SUSY-breaking terms of the stop sector (see
+Eq. (2.13)).
+D
+Daisy coefficients
+The Daisy coefficients [133–137], ci, using Eq. (2.18) is given by
+ci = m2
+i (φα, T) − m2
+i (φα)
+T
+,
+(D.1)
+and can be estimated using the high-temperature limit, i.e., T 2 ≫ m2 (m depicts a generic
+mass term involved in the calculation) [133], of the thermal corrections from V T̸=0
+1−loop (see
+Eq. (2.17)) as
+1
+T 2
+∂2V 1−loop
+T̸=0
+∂φi∂φj
+.
+(D.2)
+Daisy coefficients are calculated at the T 2 ≫ m2 limit which helps to efface gauge depen-
+dence for these coefficients although V 1−loop
+T̸=0
+, as already discussed in subsection 2.3, has
+explicit gauge dependence. The form of Eq. (D.2), except the 1/T 2 factor, looks similar to
+relations that are conventionally used for the computation of i, j-th entry of the different
+scalar mass matrices from the concerned potential. For the calculation of Daisy coefficients
+we use V 1−loop
+T̸=0
+as a function of m2
+i (φα) and not as a function of m2
+i (φα, T). However,
+while computing V 1−loop
+T̸=0
+and V ′1−loop
+CW
+(see Eq. (2.20)) we use thermal masses m2
+i (φα, T).
+Expanding thermal function JB/F (see Eq. (2.19)), in the limit T 2 ≫ m2, one gets in the
+leading order [137, 238]
+V T̸=0
+1−loop
+∼
+T 2
+48
+�
+2
+�
+i=B
+nim2
+i +
+�
+i=F
+nim2
+i
+�
+,
+(D.3)
+where B(F) represents boson (fermion) and ni depicts the associated degrees of freedom,
+as already detailed in subsection 2.2. It is also apparent from Eq. (D.3) that contribu-
+tions from the bosonic sources are the leading ones. Also, as detailed in Ref.[137], cubic
+contributions in the V T̸=0
+1−loop appears only via bosons. Further, quartic contributions from
+fermions are suppressed compared to the same from bosons and do not affect the shift in
+the VEVs [137]. Thus, we neglect contributions from the relevant fermionic sources (see
+Ref. [239] for a similar discussion in the context of the NMSSM.). In light of Eq. (D.2)
+and Eq. (D.3), non-zero Daisy coefficients are given below where field-dependent masses
+– 48 –
+
+are considered as a function of all bosonic degrees of freedom.
+cHSMHSM = cG0G0 = λ2
+4 + (3m2
+Z + 4m2
+W )
+8v2
++ m2
+Z
+4v2 sin2 θw cos2 β + m2
+t
+4v2 + ∆λ2
+4v2 ,
+cHSMHNSM = cHNSMG0 = m2
+t
+4v2
+1
+tan2 β + ∆λ2 sin 2β
+8
+− m2
+Z
+8v2 sin2 θw sin 2β,
+cHNSMHNSM = cANSMANSM = λ2
+4 + (m2
+Z + 4m2
+W )
+8v2
++
+m2
+t
+4v2 tan2 β + m2
+Z
+4v2 sin2 θw sin2 β
++ ∆λ2
+4 cos2 β,
+cHSHS = λ2 + κ2
+2
++ λ2
+N
+8 , cASAS = λ2 + κ2
+3
++ λ2
+N
+12 , cNRNR = λ2
+N
+4 , cNINI = λ2
+N
+6 ,
+cH+H− = λ2
+6 + (m2
+Z + 8m2
+W )
+24v2
+− m2
+Z
+4v2 sin2 θw sin2 β +
+m2
+t
+4v2 tan2 β
+1
+tan2 β + ∆λ2
+4 cos2 β,
+cH+G− =
+m2
+t
+4v2 tan2 β
+1
+tan2β + ∆λ2 sin 2β
+8
+− m2
+Z
+8v2 sin2 θw sin 2β,
+cG+G− = λ2
+6 + (7m2
+Z + 8m2
+W )
+24v2
+− m2
+Z
+4v2 sin2 θw sin2 β + m2
+t
+4v2 + ∆λ2
+4 sin2 β,
+(D.4)
+where mW , mZ represent masses for the W ±, Z0 bosons, respectively and θw is Weinberg
+angle [110].
+Longitudinal modes of the massive gauge bosons also yield non-zero Daisy coefficients
+[240, 241]
+cW +
+L W −
+L = cW 3
+LW 3
+L = 5
+2g2
+2,
+cBLBL = 13
+6 g2
+1,
+(D.5)
+where W ±
+L , W 3
+L, BL correspond to longitudinal modes of the SM SU(2)L, U(1)Y gauge
+bosons. These results are the same as the Z3-invariant NMSSM as gauge sector of the
+chosen NMSSM + one RH-neutrino framework remains exactly the same as the Z3-invariant
+NMSSM. Finally, at T ̸= 0 the photon (γ) also gets a temperature-dependent mass, i.e., a
+non-vanishing longitudinal component, which should also be included in the field-dependent
+mass matrix used to evaluate eigenvalues of the electrically neutral EW gauge bosons, γ, Z0
+at T ̸= 0.
+m2
+ZLγL(HSM, HNSM, HS, NR, T) =
+�
+�
+�
+g2
+2
+H2
+SM+H2
+NSM
+4
++ 5
+2g2
+2T 2
+−g1g2
+H2
+SM+H2
+NSM
+4
+−g1g2
+H2
+SM+H2
+NSM
+4
+g2
+1
+H2
+SM+H2
+NSM
+4
++ 13
+6 g2
+1T 2
+�
+�
+� .
+(D.6)
+E
+Minimization conditions
+As already stated in section 3 that one can trade different soft-masses, i.e., m2
+Hu, m2
+Hd,
+m2
+�Lij, m2
+S, M2
+N (see Eq. (2.2)) with the corresponding VEVs (see Eq. (2.6)) using min-
+imization conditions of the Vtree (see Eq. (2.3)). One can also use the neutral part of
+Vscalar as depicted in Eq. (A.3). Mathematically, the minimization condition gives a set of
+– 49 –
+
+equations like
+�∂Vtree
+∂Xi
+�����
+X=⟨X⟩
+= 0,
+(E.1)
+where Xi = H0
+u, H0
+d, �νi, S, �
+N, and ⟨X⟩ represents all the concerned VEVs as given in Eq.
+(2.6). In detail, assuming all superpotential couplings (see Eq. (2.1)) to be real, one gets
+�∂Vtree
+∂H0u
+�����
+VEVs
+= λvd
+�
+λvuvd − κv2
+S − λN
+2 v2
+N
+�
++ Y i2
+N v2
+Nvu + λ2v2
+Svu + m2
+Huvu
++
+3
+�
+j=1
+Y j
+Nvj
+� 3
+�
+i=1
+Y i
+Nvivu + λNvSvN
+�
++ g2
+1 + g2
+2
+4
+�
+v2
+d +
+3
+�
+i=1
+v2
+i − v2
+u
+�
+vu
++ λAλvSvd +
+3
+�
+i=1
+(ANYN)ivivN,
+(E.2)
+�∂Vtree
+∂H0
+d
+�����
+VEVs
+= λvu
+�
+λvuvd − κv2
+S − λN
+2 v2
+N
+�
++ λvS
+�
+λvSvd −
+3
+�
+i=1
+Y i
+NvivN
+�
++ g2
+1 + g2
+2
+4
+�
+v2
+d +
+3
+�
+i=1
+v2
+i − v2
+u
+�
+vd + m2
+Hdvd + λAλvSvu,
+(E.3)
+�∂Vtree
+∂ �νi
+�����
+VEVs
+= Y i
+Nvu
+�
+�
+3
+�
+j=1
+Y j
+Nvjvu + λNvSvN
+�
+� + Y i
+NvN
+�
+�
+3
+�
+j=1
+Y j
+NvjvN − λvdvS
+�
+�
++ g2
+1 + g2
+2
+4
+�
+v2
+d +
+3
+�
+i=1
+v2
+i − v2
+u
+�
+vi + (ANYN)ivuvN +
+3
+�
+j=1
+m2
+�Lijvj,
+(E.4)
+�∂Vtree
+∂S
+�����
+VEVs
+= 2κvS
+�
+−λvuvd + κv2
+S + λN
+2 v2
+N
+�
++ λvd
+�
+λvSvd −
+3
+�
+i
+Y i
+NvivN
+�
++ λNvN
+� 3
+�
+i=1
+Y i
+Nvivu + λNvSvN
+�
++ λ2v2
+uvS + m2
+SvS + λAλvuvd
++ κAκv2
+S + λNAλN
+2
+v2
+N,
+(E.5)
+�∂Vtree
+∂ �
+N
+�����
+VEVs
+= λNvN
+�
+−λvuvd + κv2
+S + λN
+2 v2
+N
+�
++ λNvS
+� 3
+�
+i=1
+Y i
+Nvivu + λNvSvN
+�
++
+3
+�
+j=1
+Y j
+Nvj
+� 3
+�
+i=1
+Y i
+NvivN − λvSvd
+�
++ Y i2
+N v2
+uvN + M2
+NvN
++
+3
+�
+i=1
+(ANYN)ivivu + λAλN vSvN.
+(E.6)
+– 50 –
+
+References
+[1] Planck collaboration, Planck 2018 results. VI. Cosmological parameters, Astron.
+Astrophys. 641 (2020) A6 [1807.06209].
+[2] J. M. Cline, Baryogenesis, in Les Houches Summer School - Session 86: Particle Physics
+and Cosmology: The Fabric of Spacetime, 9, 2006, hep-ph/0609145.
+[3] G. W. Anderson and L. J. Hall, The Electroweak phase transition and baryogenesis, Phys.
+Rev. D 45 (1992) 2685.
+[4] D. Curtin, P. Meade and C.-T. Yu, Testing Electroweak Baryogenesis with Future Colliders,
+JHEP 11 (2014) 127 [1409.0005].
+[5] M. Artymowski, M. Lewicki and J. D. Wells, Gravitational wave and collider implications of
+electroweak baryogenesis aided by non-standard cosmology, JHEP 03 (2017) 066
+[1609.07143].
+[6] T. Modak and E. Senaha, Probing Electroweak Baryogenesis induced by extra bottom
+Yukawa coupling via EDMs and collider signatures, JHEP 11 (2020) 025 [2005.09928].
+[7] A. D. Sakharov, Violation of CP Invariance, C asymmetry, and baryon asymmetry of the
+universe, Pisma Zh. Eksp. Teor. Fiz. 5 (1967) 32.
+[8] M. B. Gavela, P. Hernandez, J. Orloff, O. Pene and C. Quimbay, Standard model CP
+violation and baryon asymmetry. Part 2: Finite temperature, Nucl. Phys. B 430 (1994) 382
+[hep-ph/9406289].
+[9] W. Bernreuther, CP violation and baryogenesis, Lect. Notes Phys. 591 (2002) 237
+[hep-ph/0205279].
+[10] D. E. Morrissey and M. J. Ramsey-Musolf, Electroweak baryogenesis, New J. Phys. 14
+(2012) 125003 [1206.2942].
+[11] ATLAS collaboration, Observation of a new particle in the search for the Standard Model
+Higgs boson with the ATLAS detector at the LHC, Phys. Lett. B 716 (2012) 1 [1207.7214].
+[12] CMS collaboration, Observation of a New Boson at a Mass of 125 GeV with the CMS
+Experiment at the LHC, Phys. Lett. B 716 (2012) 30 [1207.7235].
+[13] K. Kajantie, M. Laine, K. Rummukainen and M. E. Shaposhnikov, Is there a hot
+electroweak phase transition at mH ≳ mW ?, Phys. Rev. Lett. 77 (1996) 2887
+[hep-ph/9605288].
+[14] K. Rummukainen, M. Tsypin, K. Kajantie, M. Laine and M. E. Shaposhnikov, The
+Universality class of the electroweak theory, Nucl. Phys. B 532 (1998) 283
+[hep-lat/9805013].
+[15] F. Csikor, Z. Fodor and J. Heitger, Endpoint of the hot electroweak phase transition, Phys.
+Rev. Lett. 82 (1999) 21 [hep-ph/9809291].
+[16] A. Mazumdar and G. White, Review of cosmic phase transitions: their significance and
+experimental signatures, Rept. Prog. Phys. 82 (2019) 076901 [1811.01948].
+[17] R. Apreda, M. Maggiore, A. Nicolis and A. Riotto, Gravitational waves from electroweak
+phase transitions, Nucl. Phys. B 631 (2002) 342 [gr-qc/0107033].
+[18] C. Grojean, G. Servant and J. D. Wells, First-order electroweak phase transition in the
+standard model with a low cutoff, Phys. Rev. D 71 (2005) 036001 [hep-ph/0407019].
+– 51 –
+
+[19] D. J. Weir, Gravitational waves from a first order electroweak phase transition: a brief
+review, Phil. Trans. Roy. Soc. Lond. A 376 (2018) 20170126 [1705.01783].
+[20] J. Ellis, M. Lewicki and J. M. No, On the Maximal Strength of a First-Order Electroweak
+Phase Transition and its Gravitational Wave Signal, JCAP 04 (2019) 003 [1809.08242].
+[21] T. Alanne, T. Hugle, M. Platscher and K. Schmitz, A fresh look at the gravitational-wave
+signal from cosmological phase transitions, JHEP 03 (2020) 004 [1909.11356].
+[22] C. Caprini et al., Detecting gravitational waves from cosmological phase transitions with
+LISA: an update, JCAP 03 (2020) 024 [1910.13125].
+[23] F. Costa, S. Khan and J. Kim, A two-component dark matter model and its associated
+gravitational waves, JHEP 06 (2022) 026 [2202.13126].
+[24] A. Ahriche, K. Hashino, S. Kanemura and S. Nasri, Gravitational Waves from Phase
+Transitions in Models with Charged Singlets, Phys. Lett. B 789 (2019) 119 [1809.09883].
+[25] V. R. Shajiee and A. Tofighi, Electroweak Phase Transition, Gravitational Waves and Dark
+Matter in Two Scalar Singlet Extension of The Standard Model, Eur. Phys. J. C 79 (2019)
+360 [1811.09807].
+[26] A. Alves, T. Ghosh, H.-K. Guo, K. Sinha and D. Vagie, Collider and Gravitational Wave
+Complementarity in Exploring the Singlet Extension of the Standard Model, JHEP 04
+(2019) 052 [1812.09333].
+[27] H. E. Haber and G. L. Kane, The Search for Supersymmetry: Probing Physics Beyond the
+Standard Model, Phys. Rept. 117 (1985) 75.
+[28] J. R. Espinosa, M. Quiros and F. Zwirner, On the electroweak phase transition in the
+minimal supersymmetric Standard Model, Phys. Lett. B 307 (1993) 106 [hep-ph/9303317].
+[29] M. Carena, M. Quiros and C. E. M. Wagner, Opening the window for electroweak
+baryogenesis, Phys. Lett. B 380 (1996) 81 [hep-ph/9603420].
+[30] D. Delepine, J. M. Gerard, R. Gonzalez Felipe and J. Weyers, A Light stop and electroweak
+baryogenesis, Phys. Lett. B 386 (1996) 183 [hep-ph/9604440].
+[31] M. Laine and K. Rummukainen, The MSSM electroweak phase transition on the lattice,
+Nucl. Phys. B 535 (1998) 423 [hep-lat/9804019].
+[32] J. M. Cline and G. D. Moore, Supersymmetric electroweak phase transition: Baryogenesis
+versus experimental constraints, Phys. Rev. Lett. 81 (1998) 3315 [hep-ph/9806354].
+[33] C. Balazs, M. Carena, A. Menon, D. E. Morrissey and C. E. M. Wagner, The
+Supersymmetric origin of matter, Phys. Rev. D 71 (2005) 075002 [hep-ph/0412264].
+[34] C. Lee, V. Cirigliano and M. J. Ramsey-Musolf, Resonant relaxation in electroweak
+baryogenesis, Phys. Rev. D 71 (2005) 075010 [hep-ph/0412354].
+[35] M. Carena, G. Nardini, M. Quiros and C. E. M. Wagner, The Baryogenesis Window in the
+MSSM, Nucl. Phys. B 812 (2009) 243 [0809.3760].
+[36] A. Menon and D. E. Morrissey, Higgs Boson Signatures of MSSM Electroweak
+Baryogenesis, Phys. Rev. D 79 (2009) 115020 [0903.3038].
+[37] M. Carena, G. Nardini, M. Quiros and C. E. M. Wagner, MSSM Electroweak Baryogenesis
+and LHC Data, JHEP 02 (2013) 001 [1207.6330].
+– 52 –
+
+[38] D. Curtin, P. Jaiswal and P. Meade, Excluding Electroweak Baryogenesis in the MSSM,
+JHEP 08 (2012) 005 [1203.2932].
+[39] ATLAS collaboration, Search for associated production of a Z boson with an invisibly
+decaying Higgs boson or dark matter candidates at √s =13 TeV with the ATLAS detector,
+Phys. Lett. B 829 (2022) 137066 [2111.08372].
+[40] CMS collaboration, Search for invisible decays of the Higgs boson produced via vector boson
+fusion in proton-proton collisions at √s=13 TeV, Phys. Rev. D 105 (2022) 092007
+[2201.11585].
+[41] ATLAS collaboration, Search for invisible Higgs-boson decays in events with vector-boson
+fusion signatures using 139 fb−1 of proton-proton data recorded by the ATLAS experiment,
+JHEP 08 (2022) 104 [2202.07953].
+[42] CMS collaboration, A portrait of the Higgs boson by the CMS experiment ten years after
+the discovery, Nature 607 (2022) 60 [2207.00043].
+[43] K. Krizka, A. Kumar and D. E. Morrissey, Very Light Scalar Top Quarks at the LHC, Phys.
+Rev. D 87 (2013) 095016 [1212.4856].
+[44] A. Delgado, G. F. Giudice, G. Isidori, M. Pierini and A. Strumia, The light stop window,
+Eur. Phys. J. C 73 (2013) 2370 [1212.6847].
+[45] R. Gr¨ober, M. M. M¨uhlleitner, E. Popenda and A. Wlotzka, Light Stop Decays:
+Implications for LHC Searches, Eur. Phys. J. C 75 (2015) 420 [1408.4662].
+[46] J. E. Kim and H. P. Nilles, The mu Problem and the Strong CP Problem, Phys. Lett. B 138
+(1984) 150.
+[47] I. Esteban, M. C. Gonzalez-Garcia, M. Maltoni, T. Schwetz and A. Zhou, The fate of hints:
+updated global analysis of three-flavor neutrino oscillations, JHEP 09 (2020) 178
+[2007.14792].
+[48] M. C. Gonzalez-Garcia, M. Maltoni and T. Schwetz, NuFIT: Three-Flavour Global Analyses
+of Neutrino Oscillation Experiments, Universe 7 (2021) 459 [2111.03086].
+[49] L. J. Hall and M. Suzuki, Explicit R-Parity Breaking in Supersymmetric Models, Nucl.
+Phys. B 231 (1984) 419.
+[50] M. Hirsch and J. W. F. Valle, Supersymmetric origin of neutrino mass, New J. Phys. 6
+(2004) 76 [hep-ph/0405015].
+[51] R. Barbier et al., R-parity violating supersymmetry, Phys. Rept. 420 (2005) 1
+[hep-ph/0406039].
+[52] R. N. Mohapatra et al., Theory of neutrinos: A White paper, Rept. Prog. Phys. 70 (2007)
+1757 [hep-ph/0510213].
+[53] P. F. de Salas, D. V. Forero, S. Gariazzo, P. Mart´ınez-Mirav´e, O. Mena, C. A. Ternes et al.,
+2020 global reassessment of the neutrino oscillation picture, JHEP 02 (2021) 071
+[2006.11237].
+[54] U. Ellwanger, C. Hugonie and A. M. Teixeira, The Next-to-Minimal Supersymmetric
+Standard Model, Phys. Rept. 496 (2010) 1 [0910.1785].
+[55] M. Pietroni, The Electroweak phase transition in a nonminimal supersymmetric model,
+Nucl. Phys. B 402 (1993) 27 [hep-ph/9207227].
+– 53 –
+
+[56] A. T. Davies, C. D. Froggatt and R. G. Moorhouse, Electroweak baryogenesis in the
+next-to-minimal supersymmetric model, Phys. Lett. B 372 (1996) 88 [hep-ph/9603388].
+[57] S. J. Huber, T. Konstandin, T. Prokopec and M. G. Schmidt, Electroweak Phase Transition
+and Baryogenesis in the nMSSM, Nucl. Phys. B 757 (2006) 172 [hep-ph/0606298].
+[58] W. Huang, Z. Kang, J. Shu, P. Wu and J. M. Yang, New insights in the electroweak phase
+transition in the NMSSM, Phys. Rev. D 91 (2015) 025006 [1405.1152].
+[59] J. Kozaczuk, S. Profumo, L. S. Haskins and C. L. Wainwright, Cosmological Phase
+Transitions and their Properties in the NMSSM, JHEP 01 (2015) 144 [1407.4134].
+[60] S. Baum, M. Carena, N. R. Shah, C. E. M. Wagner and Y. Wang, Nucleation is more than
+critical: A case study of the electroweak phase transition in the NMSSM, JHEP 03 (2021)
+055 [2009.10743].
+[61] M. Carena, G. Nardini, M. Quiros and C. E. M. Wagner, The Effective Theory of the Light
+Stop Scenario, JHEP 10 (2008) 062 [0806.4297].
+[62] P. Minkowski, µ → eγ at a Rate of One Out of 109 Muon Decays?, Phys. Lett. B 67 (1977)
+421.
+[63] T. Yanagida, Horizontal gauge symmetry and masses of neutrinos, Conf. Proc. C 7902131
+(1979) 95.
+[64] M. Gell-Mann, P. Ramond and R. Slansky, Complex Spinors and Unified Theories, Conf.
+Proc. C 790927 (1979) 315 [1306.4669].
+[65] R. N. Mohapatra and G. Senjanovic, Neutrino Mass and Spontaneous Parity
+Nonconservation, Phys. Rev. Lett. 44 (1980) 912.
+[66] J. Hisano, T. Moroi, K. Tobe and M. Yamaguchi, Lepton flavor violation via right-handed
+neutrino Yukawa couplings in supersymmetric standard model, Phys. Rev. D 53 (1996)
+2442 [hep-ph/9510309].
+[67] J. Hisano, T. Moroi, K. Tobe, M. Yamaguchi and T. Yanagida, Lepton flavor violation in
+the supersymmetric standard model with seesaw induced neutrino masses, Phys. Lett. B 357
+(1995) 579 [hep-ph/9501407].
+[68] J. R. Ellis, J. Hisano, M. Raidal and Y. Shimizu, A New parametrization of the seesaw
+mechanism and applications in supersymmetric models, Phys. Rev. D 66 (2002) 115013
+[hep-ph/0206110].
+[69] R. Kitano and K.-y. Oda, Neutrino masses in the supersymmetric standard model with
+right-handed neutrinos and spontaneous R-parity violation, Phys. Rev. D 61 (2000) 113001
+[hep-ph/9911327].
+[70] D. E. Lopez-Fogliani and C. Munoz, Proposal for a Supersymmetric Standard Model, Phys.
+Rev. Lett. 97 (2006) 041801 [hep-ph/0508297].
+[71] N. Escudero, D. E. Lopez-Fogliani, C. Munoz and R. Ruiz de Austri, Analysis of the
+parameter space and spectrum of the mu nu SSM, JHEP 12 (2008) 099 [0810.1507].
+[72] P. Ghosh and S. Roy, Neutrino masses and mixing, lightest neutralino decays and a solution
+to the mu problem in supersymmetry, JHEP 04 (2009) 069 [0812.0084].
+[73] C. S. Aulakh and R. N. Mohapatra, Neutrino as the Supersymmetric Partner of the
+Majoron, Phys. Lett. B 119 (1982) 136.
+– 54 –
+
+[74] A. Masiero and J. W. F. Valle, A Model for Spontaneous R Parity Breaking, Phys. Lett. B
+251 (1990) 273.
+[75] S. F. King, Large mixing angle MSW and atmospheric neutrinos from single right-handed
+neutrino dominance and U(1) family symmetry, Nucl. Phys. B 576 (2000) 85
+[hep-ph/9912492].
+[76] A. Ibarra and G. G. Ross, Neutrino phenomenology: The Case of two right-handed
+neutrinos, Phys. Lett. B 591 (2004) 285 [hep-ph/0312138].
+[77] CMS collaboration, Search for top squarks decaying via the four-body mode in single-lepton
+final states from Run 2 of the LHC, CMS-PAS-SUS-21-003 (2022) .
+[78] CMS collaboration, Search for top squark pair production in a final state with one or two
+tau leptons in proton-proton collisions at √s = 13 TeV, CMS-PAS-SUS-21-004 (2022) .
+[79] CMS collaboration, Search for supersymmetry in final states with a single electron or muon
+using angular correlations and heavy object tagging in proton-proton collisions at
+√s = 13 TeV, CMS-PAS-SUS-21-007 (2022) .
+[80] ATLAS collaboration, SUSY Summary Plots March 2022, ATL-PHYS-PUB-2022-013
+(2022) .
+[81] H. Kudoh, A. Taruya, T. Hiramatsu and Y. Himemoto, Detecting a gravitational-wave
+background with next-generation space interferometers, Phys. Rev. D 73 (2006) 064006
+[gr-qc/0511145].
+[82] K. Yagi and N. Seto, Detector configuration of DECIGO/BBO and identification of
+cosmological neutron-star binaries, Phys. Rev. D 83 (2011) 044011 [1101.3940].
+[83] H. P. Nilles, Supersymmetry, Supergravity and Particle Physics, Phys. Rept. 110 (1984) 1.
+[84] M. F. Sohnius, Introducing Supersymmetry, Phys. Rept. 128 (1985) 39.
+[85] S. P. Martin, A Supersymmetry primer, Adv. Ser. Direct. High Energy Phys. 18 (1998) 1
+[hep-ph/9709356].
+[86] M. Maniatis, The Next-to-Minimal Supersymmetric extension of the Standard Model
+reviewed, Int. J. Mod. Phys. A 25 (2010) 3505 [0906.0777].
+[87] I. Simonsen, A Review of minimal supersymmetric electroweak theory, hep-ph/9506369.
+[88] M. Drees, R. Godbole and P. Roy, Theory and phenomenology of sparticles: An account of
+four-dimensional N=1 supersymmetry in high energy physics. World Scientific Publishing
+Co Pvt Ltd, 2004.
+[89] L. J. Hall, J. D. Lykken and S. Weinberg, Supergravity as the Messenger of Supersymmetry
+Breaking, Phys. Rev. D 27 (1983) 2359.
+[90] S. Roy and B. Mukhopadhyaya, Some implications of a supersymmetric model with R-parity
+breaking bilinear interactions, Phys. Rev. D 55 (1997) 7020 [hep-ph/9612447].
+[91] W. Porod, M. Hirsch, J. Romao and J. W. F. Valle, Testing neutrino mixing at future
+collider experiments, Phys. Rev. D 63 (2001) 115004 [hep-ph/0011248].
+[92] M. Hirsch, A. Vicente and W. Porod, Spontaneous R-parity violation: Lightest neutralino
+decays and neutrino mixing angles at future colliders, Phys. Rev. D 77 (2008) 075005
+[0802.2896].
+– 55 –
+
+[93] A. Bartl, M. Hirsch, A. Vicente, S. Liebler and W. Porod, LHC phenomenology of the mu
+nu SSM, JHEP 05 (2009) 120 [0903.3596].
+[94] J. Fidalgo, D. E. Lopez-Fogliani, C. Munoz and R. Ruiz de Austri, The Higgs sector of the
+µνSSM and collider physics, JHEP 10 (2011) 020 [1107.4614].
+[95] P. Ghosh, D. E. Lopez-Fogliani, V. A. Mitsou, C. Munoz and R. Ruiz de Austri, Probing
+the µ-from-ν supersymmetric standard model with displaced multileptons from the decay of a
+Higgs boson at the LHC, Phys. Rev. D 88 (2013) 015009 [1211.3177].
+[96] M. Hirsch, M. A. Diaz, W. Porod, J. C. Romao and J. W. F. Valle, Neutrino masses and
+mixings from supersymmetry with bilinear R parity violation: A Theory for solar and
+atmospheric neutrino oscillations, Phys. Rev. D 62 (2000) 113008 [hep-ph/0004115].
+[97] M. E. Krauss, T. Opferkuch and F. Staub, Spontaneous Charge Breaking in the NMSSM -
+Dangerous or not?, Eur. Phys. J. C 77 (2017) 331 [1703.05329].
+[98] H. M. Georgi, S. L. Glashow and S. Nussinov, Unconventional Model of Neutrino Masses,
+Nucl. Phys. B 193 (1981) 297.
+[99] A. Santamaria and J. W. F. Valle, Spontaneous R-Parity Violation in Supersymmetry: A
+Model for Solar Neutrino Oscillations, Phys. Lett. B 195 (1987) 423.
+[100] A. Santamaria and J. W. F. Valle, Supersymmetric Majoron Signatures and Solar Neutrino
+Oscillations, Phys. Rev. Lett. 60 (1988) 397.
+[101] A. Santamaria and J. W. F. Valle, Solar Neutrino Oscillation Parameters and the Broken R
+Parity Majoron, Phys. Rev. D 39 (1989) 1780.
+[102] M. Carena, H. E. Haber, I. Low, N. R. Shah and C. E. M. Wagner, Alignment limit of the
+NMSSM Higgs sector, Phys. Rev. D 93 (2016) 035013 [1510.09137].
+[103] H. Georgi and D. V. Nanopoulos, Suppression of Flavor Changing Effects From Neutral
+Spinless Meson Exchange in Gauge Theories, Phys. Lett. B 82 (1979) 95.
+[104] J. F. Donoghue and L. F. Li, Properties of Charged Higgs Bosons, Phys. Rev. D 19 (1979)
+945.
+[105] L. Lavoura and J. P. Silva, Fundamental CP violating quantities in a SU(2) x U(1) model
+with many Higgs doublets, Phys. Rev. D 50 (1994) 4619 [hep-ph/9404276].
+[106] F. J. Botella and J. P. Silva, Jarlskog - like invariants for theories with scalars and
+fermions, Phys. Rev. D 51 (1995) 3870 [hep-ph/9411288].
+[107] J. F. Gunion and H. E. Haber, The CP conserving two Higgs doublet model: The Approach
+to the decoupling limit, Phys. Rev. D 67 (2003) 075019 [hep-ph/0207010].
+[108] J. F. Gunion, H. E. Haber, G. L. Kane and S. Dawson, The Higgs Hunter’s Guide, vol. 80.
+CRC Pressv, 2000.
+[109] J. F. Gunion, H. E. Haber, G. L. Kane and S. Dawson, Errata for the Higgs hunter’s guide,
+hep-ph/9302272.
+[110] Particle Data Group collaboration, Review of Particle Physics, PTEP 2022 (2022)
+083C01.
+[111] P. Ghosh, D. E. Lopez-Fogliani, V. A. Mitsou, C. Munoz and R. Ruiz de Austri, Probing
+the µνSSM with light scalars, pseudoscalars and neutralinos from the decay of a SM-like
+Higgs boson at the LHC, JHEP 11 (2014) 102 [1410.2070].
+– 56 –
+
+[112] K. Huitu and H. Waltari, Higgs sector in NMSSM with right-handed neutrinos and
+spontaneous R-parity violation, JHEP 11 (2014) 053 [1405.5330].
+[113] J. P. Derendinger and C. A. Savoy, Quantum Effects and SU(2) x U(1) Breaking in
+Supergravity Gauge Theories, Nucl. Phys. B 237 (1984) 307.
+[114] S. F. King and P. L. White, Resolving the constrained minimal and next-to-minimal
+supersymmetric standard models, Phys. Rev. D 52 (1995) 4183 [hep-ph/9505326].
+[115] M. Masip, R. Munoz-Tapia and A. Pomarol, Limits on the mass of the lightest Higgs in
+supersymmetric models, Phys. Rev. D 57 (1998) R5340 [hep-ph/9801437].
+[116] CMS collaboration, Search for top squark production in fully-hadronic final states in
+proton-proton collisions at ���s = 13 TeV, Phys. Rev. D 104 (2021) 052001 [2103.01290].
+[117] ATLAS collaboration, Search for squarks and gluinos in final states with one isolated
+lepton, jets, and missing transverse momentum at √s = 13 with the ATLAS detector, Eur.
+Phys. J. C 81 (2021) 600 [2101.01629].
+[118] ATLAS collaboration, Search for R-parity violating supersymmetry in a final state
+containing leptons and many jets with the ATLAS experiment using √s = 13 TeV
+proton-proton collision data, ATLAS-CONF-2021-007 (2021) .
+[119] ATLAS collaboration, Search for supersymmetry in final states with missing transverse
+momentum and three or more b-jets in 139 fb−1 of proton−proton collisions at √s = 13
+TeV with the ATLAS detector, 2211.08028.
+[120] ATLAS collaboration, Search for direct stau production in events with two hadronic
+τ-leptons in √s = 13 TeV pp collisions with the ATLAS detector, Phys. Rev. D 101 (2020)
+032009 [1911.06660].
+[121] CMS collaboration, Search for electroweak production of supersymmetric particles in final
+states containing hadronic decays of WW, WZ, or WH and missing transverse momentum,
+CMS-PAS-SUS-21-002 (2021) .
+[122] ATLAS collaboration, Search for direct pair production of sleptons and charginos decaying
+to two leptons and neutralinos with mass splittings near the W-boson mass in √s = 13 TeV
+pp collisions with the ATLAS detector, 2209.13935.
+[123] Muon g-2 collaboration, Measurement of the Positive Muon Anomalous Magnetic Moment
+to 0.46 ppm, Phys. Rev. Lett. 126 (2021) 141801 [2104.03281].
+[124] J. R. Ellis, G. Ridolfi and F. Zwirner, On radiative corrections to supersymmetric Higgs
+boson masses and their implications for LEP searches, Phys. Lett. B 262 (1991) 477.
+[125] H. E. Haber and R. Hempfling, The Renormalization group improved Higgs sector of the
+minimal supersymmetric model, Phys. Rev. D 48 (1993) 4280 [hep-ph/9307201].
+[126] J. A. Casas, J. R. Espinosa, M. Quiros and A. Riotto, The Lightest Higgs boson mass in the
+minimal supersymmetric standard model, Nucl. Phys. B 436 (1995) 3 [hep-ph/9407389].
+[127] M. Carena, J. R. Espinosa, M. Quiros and C. E. M. Wagner, Analytical expressions for
+radiatively corrected Higgs masses and couplings in the MSSM, Phys. Lett. B 355 (1995)
+209 [hep-ph/9504316].
+[128] G. Lee and C. E. M. Wagner, Higgs bosons in heavy supersymmetry with an intermediate
+mA, Phys. Rev. D 92 (2015) 075032 [1508.00576].
+– 57 –
+
+[129] S. R. Coleman and E. J. Weinberg, Radiative Corrections as the Origin of Spontaneous
+Symmetry Breaking, Phys. Rev. D 7 (1973) 1888.
+[130] X.-J. Bi, L. Bian, W. Huang, J. Shu and P.-F. Yin, Interpretation of the Galactic Center
+excess and electroweak phase transition in the NMSSM, Phys. Rev. D 92 (2015) 023507
+[1503.03749].
+[131] S. J. Huber, T. Konstandin, G. Nardini and I. Rues, Detectable Gravitational Waves from
+Very Strong Phase Transitions in the General NMSSM, JCAP 03 (2016) 036 [1512.06357].
+[132] L. Bian, H.-K. Guo and J. Shu, Gravitational Waves, baryon asymmetry of the universe
+and electric dipole moment in the CP-violating NMSSM, Chin. Phys. C 42 (2018) 093106
+[1704.02488].
+[133] L. Dolan and R. Jackiw, Symmetry Behavior at Finite Temperature, Phys. Rev. D 9 (1974)
+3320.
+[134] D. A. Kirzhnits and A. D. Linde, Symmetry Behavior in Gauge Theories, Annals Phys. 101
+(1976) 195.
+[135] R. R. Parwani, Resummation in a hot scalar field theory, Phys. Rev. D 45 (1992) 4695
+[hep-ph/9204216].
+[136] J. R. Espinosa, M. Quiros and F. Zwirner, On the phase transition in the scalar theory,
+Phys. Lett. B 291 (1992) 115 [hep-ph/9206227].
+[137] P. B. Arnold and O. Espinosa, The Effective potential and first-order phase transitions:
+Beyond leading-order, Phys. Rev. D 47 (1993) 3546 [hep-ph/9212235].
+[138] N. K. Nielsen, On the Gauge Dependence of Spontaneous Symmetry Breaking in Gauge
+Theories, Nucl. Phys. B 101 (1975) 173.
+[139] H. H. Patel and M. J. Ramsey-Musolf, Baryon Washout, Electroweak Phase Transition, and
+Perturbation Theory, JHEP 07 (2011) 029 [1101.4665].
+[140] L. Di Luzio and L. Mihaila, On the gauge dependence of the Standard Model vacuum
+instability scale, JHEP 06 (2014) 079 [1404.7450].
+[141] M. Laine, M. Meyer and G. Nardini, Thermal phase transition with full 2-loop effective
+potential, Nucl. Phys. B 920 (2017) 565 [1702.07479].
+[142] M. Drees, Supersymmetric Models with Extended Higgs Sector, Int. J. Mod. Phys. A 4
+(1989) 3635.
+[143] J. R. Ellis, J. F. Gunion, H. E. Haber, L. Roszkowski and F. Zwirner, Higgs Bosons in a
+Nonminimal Supersymmetric Model, Phys. Rev. D 39 (1989) 844.
+[144] L. D. Landau, A. A. Abrikosov and I. M. Khalatnikov, An asymptotic expression for the
+photon Green function in quantum electrodynamics, Dokl. Akad. Nauk SSSR 95 (1954)
+1177.
+[145] L. D. Landau, On the quantum field theory, Niels Bohr and the Development of Physics, ed.
+W. Pauli. Pergamon Press, London, 1955.
+[146] B. A. Dobrescu, G. L. Landsberg and K. T. Matchev, Higgs boson decays to CP odd scalars
+at the Tevatron and beyond, Phys. Rev. D 63 (2001) 075003 [hep-ph/0005308].
+[147] B. A. Dobrescu and K. T. Matchev, Light axion within the next-to-minimal supersymmetric
+standard model, JHEP 09 (2000) 031 [hep-ph/0008192].
+– 58 –
+
+[148] L3 collaboration, Search for charginos and neutralinos in e+e− collisions at √s = 189
+GeV, Phys. Lett. B 472 (2000) 420 [hep-ex/9910007].
+[149] ALEPH collaboration, Absolute mass lower limit for the lightest neutralino of the MSSM
+from e+e− data at √s up to 209 GeV, Phys. Lett. B 583 (2004) 247.
+[150] DELPHI collaboration, Searches for supersymmetric particles in e+e− collisions up to 208
+GeV and interpretation of the results within the MSSM, Eur. Phys. J. C 31 (2003) 421
+[hep-ex/0311019].
+[151] OPAL collaboration, Search for chargino and neutralino production at √s = 192 - 209
+GeV at LEP, Eur. Phys. J. C 35 (2004) 1 [hep-ex/0401026].
+[152] L. Lentati et al., European Pulsar Timing Array Limits On An Isotropic Stochastic
+Gravitational-Wave Background, Mon. Not. Roy. Astron. Soc. 453 (2015) 2576
+[1504.03692].
+[153] NANOGRAV collaboration, The NANOGrav 11-year Data Set: Pulsar-timing Constraints
+On The Stochastic Gravitational-wave Background, Astrophys. J. 859 (2018) 47
+[1801.02617].
+[154] LIGO Scientific, VIRGO collaboration, Characterization of the LIGO detectors during
+their sixth science run, Class. Quant. Grav. 32 (2015) 115012 [1410.7764].
+[155] C. Caprini et al., Science with the space-based interferometer eLISA. II: Gravitational
+waves from cosmological phase transitions, JCAP 04 (2016) 001 [1512.06239].
+[156] V. Corbin and N. J. Cornish, Detecting the cosmic gravitational wave background with the
+big bang observer, Class. Quant. Grav. 23 (2006) 2435 [gr-qc/0512039].
+[157] F. Staub, SARAH, 0806.0538.
+[158] F. Staub, T. Ohl, W. Porod and C. Speckner, A Tool Box for Implementing
+Supersymmetric Models, Comput. Phys. Commun. 183 (2012) 2165 [1109.5147].
+[159] F. Staub, SARAH 3.2: Dirac Gauginos, UFO output, and more, Comput. Phys. Commun.
+184 (2013) 1792 [1207.0906].
+[160] H. Dreiner, K. Nickel, W. Porod and F. Staub, Full 1-loop calculation of BR(B0
+s,d → ℓ¯ℓ) in
+models beyond the MSSM with SARAH and SPheno, Comput. Phys. Commun. 184 (2013)
+2604 [1212.5074].
+[161] F. Staub, SARAH 4 : A tool for (not only SUSY) model builders, Comput. Phys. Commun.
+185 (2014) 1773 [1309.7223].
+[162] M. D. Goodsell, K. Nickel and F. Staub, Two-Loop Higgs mass calculations in
+supersymmetric models beyond the MSSM with SARAH and SPheno, Eur. Phys. J. C 75
+(2015) 32 [1411.0675].
+[163] F. Staub, Exploring new models in all detail with SARAH, Adv. High Energy Phys. 2015
+(2015) 840780 [1503.04200].
+[164] M. D. Goodsell and F. Staub, Unitarity constraints on general scalar couplings with
+SARAH, Eur. Phys. J. C 78 (2018) 649 [1805.07306].
+[165] W. Porod, SPheno, a program for calculating supersymmetric spectra, SUSY particle decays
+and SUSY particle production at e+e− colliders, Comput. Phys. Commun. 153 (2003) 275
+[hep-ph/0301101].
+– 59 –
+
+[166] W. Porod and F. Staub, SPheno 3.1: Extensions including flavour, CP-phases and models
+beyond the MSSM, Comput. Phys. Commun. 183 (2012) 2458 [1104.1573].
+[167] W. Porod, F. Staub and A. Vicente, A Flavor Kit for BSM models, Eur. Phys. J. C 74
+(2014) 2992 [1405.1434].
+[168] M. Goodsell, K. Nickel and F. Staub, Generic two-loop Higgs mass calculation from a
+diagrammatic approach, Eur. Phys. J. C 75 (2015) 290 [1503.03098].
+[169] M. D. Goodsell and F. Staub, The Higgs mass in the CP violating MSSM, NMSSM, and
+beyond, Eur. Phys. J. C 77 (2017) 46 [1604.05335].
+[170] J. Braathen, M. D. Goodsell and F. Staub, Supersymmetric and non-supersymmetric
+models without catastrophic Goldstone bosons, Eur. Phys. J. C 77 (2017) 757 [1706.05372].
+[171] M. D. Goodsell and R. Moutafis, How heavy can dark matter be? Constraining colourful
+unitarity with SARAH, Eur. Phys. J. C 81 (2021) 808 [2012.09022].
+[172] ATLAS collaboration, Measurement of the Higgs boson mass in the H → ZZ∗ → 4ℓ decay
+channel using 139 fb−1 of √s = 13 TeV pp collisions recorded by the ATLAS detector at the
+LHC, 2207.00320.
+[173] ATLAS collaboration, Constraints on Higgs boson production with large transverse
+momentum using H → b¯b decays in the ATLAS detector, Phys. Rev. D 105 (2022) 092003
+[2111.08340].
+[174] ATLAS collaboration, Measurements of Higgs boson production cross-sections in
+the H → τ +τ − decay channel in pp collisions at √s = 13 TeV with the ATLAS detector,
+JHEP 08 (2022) 175 [2201.08269].
+[175] ATLAS collaboration, Direct constraint on the Higgs-charm coupling from a search for
+Higgs boson decays into charm quarks with the ATLAS detector, Eur. Phys. J. C 82 (2022)
+717 [2201.11428].
+[176] ATLAS collaboration, Measurement of the properties of Higgs boson production at √s = 13
+TeV in the H → γγ channel using 139 fb−1 of pp collision data with the ATLAS
+experiment, 2207.00348.
+[177] ATLAS collaboration, Combined measurement of the total and differential cross sections in
+the H → γγ and the H → ZZ∗ → 4ℓ decay channels at √s = 13 TeV with the ATLAS
+detector, ATLAS-CONF-2022-002 (2022) .
+[178] ATLAS and CMS collaboration, Rare and BSM Higgs Searches, CMS-CR-2022-179
+(2022) .
+[179] P. Bechtle, O. Brein, S. Heinemeyer, G. Weiglein and K. E. Williams, HiggsBounds:
+Confronting Arbitrary Higgs Sectors with Exclusion Bounds from LEP and the Tevatron,
+Comput. Phys. Commun. 181 (2010) 138 [0811.4169].
+[180] P. Bechtle, D. Dercks, S. Heinemeyer, T. Klingl, T. Stefaniak, G. Weiglein et al.,
+HiggsBounds-5: Testing Higgs Sectors in the LHC 13 TeV Era, Eur. Phys. J. C 80 (2020)
+1211 [2006.06007].
+[181] CMS collaboration, Search for electroweak production of charginos and neutralinos in
+proton-proton collisions at √s = 13 TeV, JHEP 04 (2022) 147 [2106.14246].
+[182] CMS collaboration, Search for charging-neutralino production in events with Higgs and W
+– 60 –
+
+bosons using 137 fb−1 of proton-proton collisions at √s = 13 TeV, JHEP 10 (2021) 045
+[2107.12553].
+[183] ATLAS collaboration, Search for charginos and neutralinos in final states with two boosted
+hadronically decaying bosons and missing transverse momentum in pp collisions at √s = 13
+TeV with the ATLAS detector, Phys. Rev. D 104 (2021) 112010 [2108.07586].
+[184] ATLAS collaboration, SUSY March 2021 Summary Plot Update,
+ATL-PHYS-PUB-2021-007 (2021) .
+[185] S. Bertolini, F. Borzumati and A. Masiero, New Constraints on Squark and Gluino Masses
+from Radiative b Decays, Phys. Lett. B 192 (1987) 437.
+[186] S. Bertolini, F. Borzumati, A. Masiero and G. Ridolfi, Effects of supergravity induced
+electroweak breaking on rare B decays and mixings, Nucl. Phys. B 353 (1991) 591.
+[187] M. Ciuchini, G. Degrassi, P. Gambino and G. F. Giudice, Next-to-leading QCD corrections
+to B → Xsγ in supersymmetry, Nucl. Phys. B 534 (1998) 3 [hep-ph/9806308].
+[188] F. Borzumati, C. Greub and Y. Yamada, Beyond leading order corrections to B → Xsγ at
+large tan β: The Charged Higgs contribution, Phys. Rev. D 69 (2004) 055005
+[hep-ph/0311151].
+[189] G. Degrassi, P. Gambino and P. Slavich, QCD corrections to radiative B decays in the
+MSSM with minimal flavor violation, Phys. Lett. B 635 (2006) 335 [hep-ph/0601135].
+[190] F. Domingo and U. Ellwanger, Updated Constraints from B Physics on the MSSM and the
+NMSSM, JHEP 12 (2007) 090 [0710.3714].
+[191] A. Arbey, M. Battaglia, F. Mahmoudi and D. Mart´ınez Santos, Supersymmetry confronts
+Bs → µ+µ− : Present and future status, Phys. Rev. D 87 (2013) 035026 [1212.4887].
+[192] O. C. W. Kong and R. Vaidya, tan beta enhanced contributions to b → s + γ in SUSY
+without R-parity, hep-ph/0411165.
+[193] Y.-G. Xu, R.-M. Wang and Y.-D. Yang, Probe R-parity violating supersymmetry effects in
+B → K∗ℓ+ℓ− and Bs → ℓ+ℓ−decays, Phys. Rev. D 74 (2006) 114019 [hep-ph/0610338].
+[194] ATLAS collaboration, Searches for lepton-flavour-violating decays of the Higgs boson in
+√s = 13 TeV pp collisions with the ATLAS detector, Phys. Lett. B 800 (2020) 135069
+[1907.06131].
+[195] CMS collaboration, Search for lepton-flavor violating decays of the Higgs boson in the µτ
+and eτ final states in proton-proton collisions at √s = 13 TeV, Phys. Rev. D 104 (2021)
+032013 [2105.03007].
+[196] M. Frank, Bilinear R-parity violation parameters from µ → eγ, Phys. Rev. D 62 (2000)
+015006.
+[197] K.-m. Cheung and O. C. W. Kong, µ → eγ from supersymmetry without R parity, Phys.
+Rev. D 64 (2001) 095007 [hep-ph/0101347].
+[198] D. F. Carvalho, M. E. Gomez and J. C. Romao, Charged lepton flavor violation in
+supersymmetry with bilinear R-parity violation, Phys. Rev. D 65 (2002) 093013
+[hep-ph/0202054].
+[199] BaBar collaboration, Searches for Lepton Flavor Violation in the Decays τ ± → e±γ and
+τ ± → µ±γ, Phys. Rev. Lett. 104 (2010) 021802 [0908.2381].
+– 61 –
+
+[200] MEG collaboration, Search for the lepton flavour violating decay µ+ → e+γ with the full
+dataset of the MEG experiment, Eur. Phys. J. C 76 (2016) 434 [1605.05081].
+[201] SINDRUM collaboration, Search for the Decay µ+ → e+e+e−, Nucl. Phys. B 299 (1988) 1.
+[202] SINDRUM II collaboration, A Search for muon to electron conversion in muonic gold,
+Eur. Phys. J. C 47 (2006) 337.
+[203] M. M. Ivanov, M. Simonovi´c and M. Zaldarriaga, Cosmological Parameters and Neutrino
+Masses from the Final Planck and Full-Shape BOSS Data, Phys. Rev. D 101 (2020) 083504
+[1912.08208].
+[204] A. Kobakhidze, M. Talia and L. Wu, Probing the MSSM explanation of the muon g-2
+anomaly in dark matter experiments and at a 100 TeV pp collider, Phys. Rev. D 95 (2017)
+055023 [1608.03641].
+[205] R. S. Hundi, Constraints from neutrino masses and muon (g-2) in the bilinear R-parity
+violating supersymmetric model, Phys. Rev. D 83 (2011) 115019 [1101.2810].
+[206] T. Aoyama et al., The anomalous magnetic moment of the muon in the Standard Model,
+Phys. Rept. 887 (2020) 1 [2006.04822].
+[207] Muon g-2 collaboration, Final Report of the Muon E821 Anomalous Magnetic Moment
+Measurement at BNL, Phys. Rev. D 73 (2006) 072003 [hep-ex/0602035].
+[208] M. Chakraborti, S. Heinemeyer and I. Saha, The new “MUON G-2” result and
+supersymmetry, Eur. Phys. J. C 81 (2021) 1114 [2104.03287].
+[209] M. Badziak and K. Sakurai, Explanation of electron and muon g − 2 anomalies in the
+MSSM, JHEP 10 (2019) 024 [1908.03607].
+[210] F. Domingo and U. Ellwanger, Constraints from the Muon g-2 on the Parameter Space of
+the NMSSM, JHEP 07 (2008) 079 [0806.0733].
+[211] C. L. Wainwright, CosmoTransitions: Computing Cosmological Phase Transition
+Temperatures and Bubble Profiles with Multiple Fields, Comput. Phys. Commun. 183
+(2012) 2006 [1109.4189].
+[212] A. D. Linde, Decay of the False Vacuum at Finite Temperature, Nucl. Phys. B 216 (1983)
+421.
+[213] C. Grojean and G. Servant, Gravitational Waves from Phase Transitions at the Electroweak
+Scale and Beyond, Phys. Rev. D 75 (2007) 043507 [hep-ph/0607107].
+[214] I. Affleck, Quantum Statistical Metastability, Phys. Rev. Lett. 46 (1981) 388.
+[215] A. D. Linde, On the Vacuum Instability and the Higgs Meson Mass, Phys. Lett. B 70
+(1977) 306.
+[216] M. Kamionkowski, A. Kosowsky and M. S. Turner, Gravitational radiation from first order
+phase transitions, Phys. Rev. D 49 (1994) 2837 [astro-ph/9310044].
+[217] J. Kehayias and S. Profumo, Semi-Analytic Calculation of the Gravitational Wave Signal
+From the Electroweak Phase Transition for General Quartic Scalar Effective Potentials,
+JCAP 03 (2010) 003 [0911.0687].
+[218] A. Nicolis, Relic gravitational waves from colliding bubbles and cosmic turbulence, Class.
+Quant. Grav. 21 (2004) L27 [gr-qc/0303084].
+– 62 –
+
+[219] J. Ellis, M. Lewicki and J. M. No, Gravitational waves from first-order cosmological phase
+transitions: lifetime of the sound wave source, JCAP 07 (2020) 050 [2003.07360].
+[220] DES collaboration, Dark Energy Survey Year 1 Results: A Precise H0 Estimate from DES
+Y1, BAO, and D/H Data, Mon. Not. Roy. Astron. Soc. 480 (2018) 3879 [1711.00403].
+[221] R. Jinno and M. Takimoto, Gravitational waves from bubble collisions: An analytic
+derivation, Phys. Rev. D 95 (2017) 024009 [1605.01403].
+[222] M. Dine, R. G. Leigh, P. Y. Huet, A. D. Linde and D. A. Linde, Towards the theory of the
+electroweak phase transition, Phys. Rev. D 46 (1992) 550 [hep-ph/9203203].
+[223] J. Ignatius, K. Kajantie, H. Kurki-Suonio and M. Laine, The growth of bubbles in
+cosmological phase transitions, Phys. Rev. D 49 (1994) 3854 [astro-ph/9309059].
+[224] G. D. Moore and T. Prokopec, How fast can the wall move? A Study of the electroweak
+phase transition dynamics, Phys. Rev. D 52 (1995) 7182 [hep-ph/9506475].
+[225] G. D. Moore and T. Prokopec, Bubble wall velocity in a first order electroweak phase
+transition, Phys. Rev. Lett. 75 (1995) 777 [hep-ph/9503296].
+[226] G. D. Moore, Electroweak bubble wall friction: Analytic results, JHEP 03 (2000) 006
+[hep-ph/0001274].
+[227] J. M. Cline and K. Kainulainen, Electroweak baryogenesis at high bubble wall velocities,
+Phys. Rev. D 101 (2020) 063525 [2001.00568].
+[228] G. C. Dorsch, S. J. Huber and T. Konstandin, A sonic boom in bubble wall friction, JCAP
+04 (2022) 010 [2112.12548].
+[229] G. C. Dorsch, S. J. Huber and T. Konstandin, On the wall velocity dependence of
+electroweak baryogenesis, JCAP 08 (2021) 020 [2106.06547].
+[230] M. Hindmarsh, S. J. Huber, K. Rummukainen and D. J. Weir, Gravitational waves from the
+sound of a first order phase transition, Phys. Rev. Lett. 112 (2014) 041301 [1304.2433].
+[231] M. Hindmarsh, Sound shell model for acoustic gravitational wave production at a first-order
+phase transition in the early Universe, Phys. Rev. Lett. 120 (2018) 071301 [1608.04735].
+[232] M. Hindmarsh, S. J. Huber, K. Rummukainen and D. J. Weir, Shape of the acoustic
+gravitational wave power spectrum from a first order phase transition, Phys. Rev. D 96
+(2017) 103520 [1704.05871].
+[233] H.-K. Guo, K. Sinha, D. Vagie and G. White, Phase Transitions in an Expanding Universe:
+Stochastic Gravitational Waves in Standard and Non-Standard Histories, JCAP 01 (2021)
+001 [2007.08537].
+[234] K. Nakayama and J. Yokoyama, Gravitational Wave Background and Non-Gaussianity as a
+Probe of the Curvaton Scenario, JCAP 01 (2010) 010 [0910.0715].
+[235] C. Caprini, R. Durrer and G. Servant, The stochastic gravitational wave background from
+turbulence and magnetic fields generated by a first-order phase transition, JCAP 12 (2009)
+024 [0909.0622].
+[236] LIGO Scientific collaboration, Advanced LIGO: The next generation of gravitational
+wave detectors, Class. Quant. Grav. 27 (2010) 084006.
+[237] A. Kosowsky, M. S. Turner and R. Watkins, Gravitational radiation from colliding vacuum
+bubbles, Phys. Rev. D 45 (1992) 4514.
+– 63 –
+
+[238] D. Curtin, P. Meade and H. Ramani, Thermal Resummation and Phase Transitions, Eur.
+Phys. J. C 78 (2018) 787 [1612.00466].
+[239] P. Athron, C. Balazs, A. Fowlie, G. Pozzo, G. White and Y. Zhang, Strong first-order phase
+transitions in the NMSSM - a comprehensive survey, JHEP 11 (2019) 151 [1908.11847].
+[240] M. E. Carrington, The Effective potential at finite temperature in the Standard Model, Phys.
+Rev. D 45 (1992) 2933.
+[241] D. Comelli and J. R. Espinosa, Bosonic thermal masses in supersymmetry, Phys. Rev. D
+55 (1997) 6253 [hep-ph/9606438].
+– 64 –
+