Electroweak Phase Transition in an Inert Complex Triplet Model

We study the dynamics of electroweak phase transition in a simple extension of the Standard Model where the Higgs sector is extended by adding an $SU(2)_L$-triplet with hypercharge Y=2. By making random scans over the parameters of the model, we show that there are regions consistent with constraints from collider experiments and the requirement for a strong first-order electroweak phase transition which is needed for electroweak baryogenesis. Further, we also study the power spectrum of the gravitational waves which can be generated due to the first-order phase transitions. Moreover, the detectability of these gravitational waves, via future space-based detectors, is discussed.


I. INTRODUCTION
Cosmological electroweak phase transition (EWPT) is interesting for numerous reasons. It can be a source for primordial magnetic fields [1], generate detectable background gravitational waves [2], affect the abundance of thermal relic densities for candidate dark matter particles [3], and perhaps most importantly, lead to suitable preconditions for baryogenesis [4].
This is consistent with astronomical measurements of light-element abundances, assuming the standard Big Bang Nucleosynthesis [4,6]. This measured value represents one of the big unresolved particle physics puzzles. It quantifies the matter-antimatter asymmetry of the universe [4,7]. The mechanism(s) behind the asymmetry needs to be determined and understood. About half a century ago, A. D. Sakharov proposes three early universe conditions needed to be satisfied for a successful baryon asymmetry generation [8]: i) baryon number violation, ii) C and CP violation, and iii) departure from thermal equilibrium. These are in principle possible within the framework of electroweak phase transition at the early universe called electroweak baryogenesis (EWBG). The realisation of EWBG within the standard model (SM) of particle physics turned out to be problematic according to lattice simulations [9]. It was found that only cross-over, instead of strongly first-order, phase transitions are possible at the early universe for the observed value of the Higgs boson mass. This indicates that some physics beyond the SM is essential. One of the simplest class of models beyond the SM which may lead to strong first order phase transitions and successful EWBG, can be made by adding an electroweak scalar SU (2) L -multiplet to the SM Higgs sector. * mj kazemi@sbu.ac.ir † abdussalam@sbu.ac.ir A global Z 2 -symmetry is imposed in constructing this so-called inert multiplet models. With the Z 2 -symmetry, the lightest neutral component of the new scalar multiplet can be considered as a candidate for dark matter [10,18]. The EWPT and its gravitational wave signatures have been well-studied within the framework of the Singlet [11], Doublet [12,13], and real Triplet (Y=1) [13,14] cases of scalar-multiplet class of models. Another well motivated representation of SU (2) L group is the complex Triplet (Y=2), which could also be used for explaining the smallness of neutrino mass in Type-II seesaw mechanism [15]. In this article we address the EWPT of the Inert Complex Triplet model by scanning its parameter space with experimental constraints from Higgs signal strengths imposed. We analyse the parameter space regions that could lead to a first-order EWPT. We also study the power spectra of gravitational waves which could be generated following the first-order transitions. Observation of such gravitational waves, such as by future space-based gravitational wave detectors [16], could yield information that is complementary to collider and dark matter experiments. This paper is organised as follows. In Sec. II we briefly review the Inert Complex Triplet model, describe the relevant parameters for the EWPT, and the theoretical constraints taken into account. In Sec. III we describe the results from collider experiments regarding Higgs decay to diphoton and used these to constrain the parameter space of the Inert Complex Triplet model. Finally, in Sec. IV and Sec. V we present the numerical analyses of the model in light of EWPT and gravitational waves generation respectively.

II. THE MODEL
We extend the SM Higgs sector by adding one complex scalar SU L (2) triplet, ∆, with hypercharge Y = 2, 27 Feb 2021 and impose a Z 2 discrete symmetry, under which ∆ → −∆ and all other fields unchanged. The most general scalar potential, symmetric under Z 2 , involving this triplet and the standard SU L (2) Higgs doublet, can be written in the following form [17,18], Here, τ a and T a are the SU (2) generators in fundamental and 3's representation respectively. These are normalised such that Tr[τ a , τ b ] = 1 2 δ ab and Tr(T a T b ) = 1 2 δ ab . Explicitly, τ a = 1 2 σ a where σ a s are Pauli matrices and T a s are We require that ∆ be odd under Z 2 symmetry so that the neutral component will not acquire any vacuum expectation value. There is an electroweak symmetry-breaking minimum at zero temperature, with H T = (0, v/ √ 2) and ∆ T = (0, 0, 0). In this case, the tree-level fielddependent masses of standard model particles are same as in the SM, and the masses of the component of the additional Triplet scalar are given by This model has five real parameters in addition to those of the SM. However, only three of them, i.e. the triplet mass parameter and the doublet-triplet couplings, appear in the tree-level triplet masses. Thus only µ ∆ , λ H∆ and λ (2) H∆ parameters are relevant for electroweak phase transition dynamics, at one-loop approximation (see Sec. III).
In the next section, we first study some constrains which come from collider phenomenology on these parameters (or equivalently on the mass spectrum of the Triplet scalar at φ = v = 246 GeV). After this, we will study the parameter space to find regions that can lead to (i) strong first order phase transitions, and (ii) detectable gravitational waves. While sampling the parameter space, we apply theoretical constraints, checking that unitarity and vacuum stability conditions on the triplet self-couplings are satisfied before applying those from the Higgs signal strength measurements. For a stable vacuum, the scalar potential should be bounded from below along all possible field directions. At the tree level, the vacuum stability requirement leads to [18], H∆ .
In what follows, we begin by considering the implications of h → γγ and h → Zγ limits on the parameter regions of the Inert Complex Triplet model which from now on we address as the inert triplet model (ITM).

III. EXCLUDING PARAMETER SPACE VIA HIGGS DECAY RATES
The branching ratios of the Higgs decays in the ITM differ from the SM ones. As such the Higgs decay measurement or limits can be used as a probe for ITM. Specifically, the Higgs-to-diphoton channel, h → γγ, because of its relatively clean signature at the Large Hadron Collider (LHC), could play an important role for this purpose. Here we analyse the ITM parameter space by using recent ATLAS and CMS results for the Higgs-to-diphoton signal strength. We find that a significantly large region of the parameter space are excluded via these recent data.
To study the ITM contributions to h → γγ decay rate, we address the ratio, Here the fact that the gluon-gluon fusion is the dominant channel for Higgs production were used. Moreover, since σ(gg → h) is the same in both the ITM and SM, the R γγ reduces to [19] In similar way, for Zγ decay channel, an analogous quantity, R Zγ , can be defined as Within the SM, many channels contribute to the total decay width of the Higgs boson. The most important ones for m h = 125 GeV are bb, cc, τ + τ − , ZZ * , W W * , γγ, Zγ and gg. Hence the total Higgs decay widths is approximately given by: In the ITM, the total decay width of the Higgs can be modified with respect to the SM, since the charged scalars exchanged in loops give extra contributions to the h → γγ and h → Zγ amplitudes [20]. In addition, the total decay width changes due to the existence of additional The decay rates of these additional decay channels, when they are kinematically open, 2m ϕ < m h , are given by where The partial widths of the tree-level Higgs decays into SM particles, and the loop-mediated decay into gg in the ITM are equal to the corresponding ones in the SM; for completeness they are summarised in Appendix A. The h → Zγ and h → γγ SM processes get modified within the ITM. In SM these decays are dominated by contributions from the W gauge boson and top quark loops while in the ITM, the couplings of the Higgs doublet to the triplet scalars modify these decays via the following one-loop diagrams.
FIG. 1: Feynman diagrams for charged scalar particles contributing to h → γγ or h → Zγ.
Following the general results for spin-0, spin-1/2 and spin-1 contributions to these decay rates [20,23] which can be obtained using the Feynman rules listed in Ref. [22], the modified decays in ITM are given by [24] Here, A γγ , s w = sin θ w , and c w = cos θ w . θ w is the Weinberg mixing angle and the coupling constants are given by The loop functions A γγ (0, 1/2, 1) and A Zγ (0, 1/2, 1) are given in Appendix B. Rγγ = 0.99 +0.14 −0.14 (ATLAS) FIG. 3: The regions of triplet mass spectrum which are consistent with the measured values of R γγ in ATLAS and CMS, for µ ∆ equals to 100, 150, 200 and 250 (GeV). The solid black contour lines represent R Zγ . The blues and green bands capture the 1 σ uncertainties for the ATLAS and CMS results. The light-blue band is for the CMS' observed value of R γγ whose central value is represented by the solid blue line. The light-green band is for the ATLAS' observed value of R γγ whose central value is represented by the broken green line.
In the Fig.2 we plot the branching rations for some regions of parameter space. It turns out that when the triplet decay channels, h → AA, h → SS, h → ∆ + ∆ − and h → ∆ ++ ∆ −− are kinematically allowed, their partial widths dominate over the partial widths of decays into SM particles. Therefore, in this case, the value of R γγ deviates significantly from R γγ = 1. This is not consistent with experimental results. Conversely, when these decay channels are kinematically closed, (m A , m S , m ∆ + , m ∆ ++ > m h /2), the total width of h is slightly modified with respect to the SM case, since the branching ratios of h → γγ and h → Zγ, which are the only processes that receive contributions from triplet scalars, are of the order of 10 −2 .
For the numerical analysis, we scan the parameter space of the ITM in the range 10 GeV ≤ m S , m ∆ ++ ≤ 500 GeV, for some specific values of µ ∆ ; 100, 150 , 200, 250 (GeV). We then compare the values of R γγ obtained with the most recent measurements by the ATLAS [26] and CMS [27] collaborations: In fact we have found that the R γγ enhancement is only possible when m S , m ∆ ++ > m h /2. In Fig.3 we illustrate the regions of the parameter space allowed by these experimental constraints. The coloured bands represented the regions within the reported experimental uncertainties. We also superimpose the contour lines which represent the values of R Zγ . The decay h → Zγ has not been discovered [28], but R Zγ is rather bounded from above to be less than 3.6 at 95% C.L. [29].
In the next section we discuss how these experimental results can constrain the properties of electroweak phase transitions and the gravitational wave spectra that could follow.

IV. DYNAMICS OF THE EWPT
In order to study the electroweak phase transition, we need to follow the evolution of the Higgs vacuum expectation value, i.e. the minimum of the Higgs effective potential over the thermal history of the universe. For this, we use the standard techniques of finite temperature field theory [30]. The one-loop level Higgs effective potential at finite temperatures can be written as where the tree-level potential is given by The zero-and finite-temperature corrections at one-loop, i.e. the Coleman-Weinberg potential V CW (φ) [31,32] and V T (φ, T ) are respectively given by [33,34] Here i = {W, Z, t, b, h, A, S, ∆ + , ∆ ++ }, F i represents the fermionic number, n i is the number of degrees of freedom of the different species of particles, n i = {6, 3, 12, 12, 1, 1, 1, 2, 2}, A first-order phase transition happens when the effective potential has two minima of the same value at some critical temperature, T c . In such case the system can transit between the vacua via thermal fluctuations or quantum tunnelling. This transition physically means creation of spherically symmetric regions of true vacuum, bubbles of the broken phase, expanding in the background of the false vacuum.
In the standard EWBG scenario, the SM fermions interact with the bubble walls in a CP-violating manner. This leads to a chiral asymmetry production in front of the bubble wall which can subsequently turn to baryon generation via sphaleron processes which convert the chiral asymmetry to baryon asymmetry [4,35]. The generated baryons could then fall into the growing bubble. This EWBG mechanism could work for explaining matter-antimatter asymmetry of the universe if the generated baryon asymmetry is not washed out by sphalerons inside the bubble [4,35]. This condition requires that [4,36] φ c T c > 1 where φ c is the Higgs vacuum expectation value at the critical temperature, T c . We numerically check this condition within the ITM by randomly scanning the ITM parameters allowed within the ranges

V. GRAVITATIONAL WAVE SPECTRUM
In this section we briefly review the processes for gravitational waves (GW) production following a first-order phase transition and then represent our result for the ITM extension of the SM.
There are three main sources for gravitational waves due to an electroweak phase transition: 1) The collisions of the bubble walls and subsequent shocks in the plasma [40][41][42][43][44][45][46], 2) The sound waves in the plasma generated after the bubbles have collided but before expansion has dissipated the kinetic energy in the plasma [52][53][54][55], and 3) The magneto-hydrodynamical (MHD) turbulence in the plasma formed after the bubbles have collided [56][57][58][59][60]. Generically, the three processes happen or coexist and thus the corresponding contributions to the gravitational waves' power spectrum, Ωh 2 (f ), must linearly combine at least approximately so that where Ω col h 2 , Ω sw h 2 and Ω tur h 2 represents the corresponding contribution from bubble collisions, sound waves and turbulence respectively. Now, let us briefly review each of these contributions in detail and estimate the predictions for ITM. Analytical studies and also numerical simulations show that in order to estimate the gravitational wave power spectrum due to a specific extension of the SM, one needs to supply at least three parameters [38,39]: i) the ratio of released latent heat from the transition to the energy density of the plasma background, α; ii) the time scale of the phase transition, H * /β; and, iii) the bubble wall velocity, v b . Using the effective potential and its derivatives at nucleation temperature, T n , the parameter α reads [38] where V f is the value of the potential in the unstable vacuum, V EW is the value of the potential in the final vacuum, and ρ R is the energy density of radiation bath, ρ R = g * π 2 T 4 n /30. The time scale of the phase transition can be calculated as [38] T =Tn (12) where S 3 (T ) is the 3D Euclidean action of the critical bubble. The last ingredient from the phase transition is the velocity of the bubble wall v b . The exact calculation of bubble wall velocity is more complicated since one need to consider bubble interaction with the background plasma. However, the bubble wall velocity can be estimated in terms of α as [51] v In fact, the above expression for the bubble wall velocity provides only a lower bound on the true wall velocity [45]. For some SM extensions, it has been checked that replacing the above approximation even with v b = 1, which is more appropriate for a very strong transition, does not significantly modify the results on gravitational wave signals [47]. For our analyses, we use the above approximation so that the parameters α and β are sufficient in order to calculate the GW signals. Now based on the numerical simulations, the peak frequency of GW generated by bubble collision [45], sound waves [54,55] and Kolmogorov-type turbulence [59] are respectively given by Tn 100 g * 100 Here g * is the number of relativistic degrees of freedom in the plasma at T n . The energy densities of the GW spectrum, corresponding to each of the three contributions, are given by Here h * is the Hubble rate at nucleation temperature, h * = 16.5 µHz Tn 100 GeV (g * /100) 1 6 and κ cool , κ sw and κ turb are efficiency factors.
The relative importance of each contribution to GW generation is encoded in the efficiency factors. These depend strongly on the dynamical details of the phase transition. In this regard, the velocity of the bubble wall plays a key role. Depending on the velocity of bubble wall, there are two regimes; when the wall velocity is relativistic or not. Moreover, in the relativistic regime, there are two different scenarios. First, whether the bubble wall reaches a terminal velocity (non-runaway scenario) or, second, the bubble wall accelerates without bound (runaway scenario). To calculate the GW spectrum, it is important to know which of the aforementioned scenarios apply. For this, the critical value α ∞ can be used to distinguish between these two scenarios [38,48] Here c a = n a /2 (c a = n a ) and n a is the number of degrees of freedom for boson (fermion) species and ∆m 2 a is the squared mass difference of particles between two phases at the nucleation temperature.
For non-runaway scenarios, α < α ∞ , the bubble wall velocity, v b , remains subliminal and the available energy is transformed into fluid motion. So the dominant contributions to GW come from sound waves and MHD turbulence, h 2 Ω GW h 2 Ω sw + h 2 Ω turb , with the efficiency factors given by [38] Here ≈ 0.05 and κ, in the small and large v b limits, is approximately given by The full expressions for κ are given in Ref. [48].
For runaway scenario, α > α ∞ , the excess vacuum energy density leads to bubble acceleration and v b is bounded only by the speed of light, v b = 1. In this case, all the three GW sources contribute with efficiency factors where in this case κ is given by [47,48] For more accuracy in the calculation of GW spectra, we consider two corrections which were discovered in recent studies [61,62]. Firstly, we consider the correction of efficiency factors for strong transitions. The values of κ given above, are from a semi-analytical hydrodynamic analysis. These are good estimations of κ only for relatively weak transitions with α 1. For strong transitions and small v b , a recent simulation found that κ as specified in eq.(21) gives an overestimation [61]. Using the numerical results in [61], we refine the estimation of the efficiency factor [63]. Secondly, we consider an additional suppression factor Υ in the Ωh 2 sw , which originates from the finite lifetime, τ sw , of the sound waves [62] For the classical approach, τ sw → ∞ is usually assumed and that corresponds to the asymptotic Υ → 1. The lifetime τ sw can be considered as the time scale when the turbulence develops, approximately given by [64,65] where R * is the mean bubble separation and is related to β through the relation R * = (8π) 1/3 v b /β for an exponential bubble nucleation [66]. Further, the analysis performed in Ref. [66] was based on Minkowski spacetime. For an analysis based on an expanding universe, see [62]. The denominatorŪ f is the root-mean-squared fluid velocity which can be obtained from hydrodynamic analyses asŪ f = (3κ ν α/4) [37,66]. In Fig.5, we explicitly show the effect of these corrections for our analyses for a typical point the ITM parameters space. Considering the above corrections, we show in Fig.6 the GW power spectra for selected points with various φ c /T c values. We find that the peak frequencies and strengths of the gravitational wave signals are strongly correlated with the strength of the phase transition.  Fig.4, i.e. the points with strong first-order EWPT. The sensitivity region for prospective GW detectors such as eLISA, BBO and DECIGO are also shown. It can be seen that the intensity of GW signal increases with the strength of the phase transition, i.e. φ c /T c . For comparison we also show the sensitivity regions for SKA and EPTA detectors which cannot probe any part of the parameters space of inert complex triplet model.
Next, in order to assess the detectability of the GW signal by a given detector, one needs to consider the signal-to-noise ratio (SNR) over the running time of the detector, t obs , which is given by [16], where h 2 Ω Sens (f ) represents the sensitivity of the detector. The interval of integration, [f min , f max ], is the frequency bandwidth of the detector. The factor δ, which indicate the number of independent channels for the GWs detector, is equal to 2 for BBO and U-DECIGO, and is equal to 1 for the rest. We consider t obs = 5 year, for all the detectors. Whenever SNR turns out to be larger than some threshold value, SNR > SNR thr , then one can assert that the experiment under consideration will be able to detect the GW signal. The method of quantifying SNR thr is briefly described in [16]. For example, the SNR threshold for discovery at eLISA is 10 or 50, depending on the operating configuration [16]. Here we compute the signal-to-noise ratio for the eLISA, LISA, BBO, DECIGO and U-DECIGO detectors. The results are shown in Fig 7. Based on the results peresented in Fig.7, the computed SNRs of eLISA are less than its threshold, 10, which means these gravitational waves are not detectable by eLISA. The biggest SNRs are associated with U-DECIGO. In fact, for points with φ c /T c ∼ 3, calculatons concerning the U-DECIGO leads to a SNR ∼ 100. However, since the SNR thr of U-DECIGO is not known (to the best of our knowledge), the question remains whether it can detect such GW arising from EWPT with φ c /T c ∼ 3 or not.

VI. SUMMARY
We have investigated the cosmological electroweak phase transition in an inert triplet scalar extension of the SM model. We found that there are regions of parameter space which can yield a strong first-order electroweak phase transition and at the same time consistent with recent LHC results on Higgs to diphoton decay rate.
In principle, a first order cosmological phase transition can lead to a background stochastic GW. This, besides collider phenomenology, can be used to probe the parameter space of particle physics model beyond the SM. In this regard, considering the recent treatment of GW spectrum estimation [61,62], we study the GW signals generated after the first order electroweak phase transitions within the framework of the inert triplet model.
Based on the signal-to-noise ratio analyses, we have found that very sensitive GW detectors will be needed for detecting the inert triplet model signal. Probing the GW signals of this model will be difficult or maybe impossible for prospective space-based GW detectors with less sensitive configurations compared to U-DECIGO.
We also compute the H → Zγ decay rate in this model, which can be used as probe, at future collider experiments, such as High Luminosity LHC and other colliders with higher center-of-mass energies [28].

APPENDIX A
For completeness, below we summarise the decay widths of the Higgs boson, which are same in both SM and ITM. Decay to leptons-. In the Born approximation, the partial decay width of h to any fermion channel is [20,67] where N c , the color factor, is 1 for leptons and 3 for quarks.