Probing electroweak phase transition in the singlet Standard Model via bbγγ and 4l channels

We investigate the prospects for resonant di-Higgs and heavy Higgs production searches at the 14 TeV HL-LHC in the combination of bbγγ and 4l channels, as a probe of a possible first order electroweak phase transition in real singlet scalar extension of the Standard Model. Event selection follows those utilized in the bbγγ and 4l searches by the ATLAS Collaboration, applied to simulation using benchmark parameters that realize a strong first order electroweak phase transition. The output of discriminant analysis is implemented by numerical calculation, optimised by the joint restriction from the two channels. The prospective reach for bbγγ/4l channel could be more competitive in probing the electroweak phase transition at lower/higher resonance masses. With 3 ab−1 integrated luminosity, the combination of the bbγγ and 4l channels can discover/exclude a significant portion of the viable parameter space that realizes a strong first order phase transition when the resonance mass is heavier than 500 GeV.


I. INTRODUCTION
As the last piece of the Standard Model(SM), the Higgs boson was discovered in 2012 at the Large Hardron Collider(LHC) [1,2].The SM has achieved great success in describing a plethora of electroweak and strong interaction phenomena.However, the SM fails to explain a number of observable facts about the universe.Among these is the origin of the cosmic baryon asymmetry Y B = n b /s, whose present value is Y B = (8.59± 0.11) × 10 −11 according to the PLANCK measurement [3].This open puzzle provides one of the strong motivations to chase after a theory of physics beyond the SM.
One of the most well-studied and experimentally testable baryogenesis scenarios is electroweak baryogenesis (EWBG) [4][5][6][7], wherein Y B is generated in conjunction with electroweak symmetry breaking (EWSB).For a review and references, see Ref. [8].As with other baryogenesis scenarios, successful EWBG must satisfy the three "Sakharov criteria" [9,10]: (1) baryon number violation; (2) C and CP-violation; and (3) departure from thermal equilibrium.The first two conditions guarantee that the net baryon number generation is possible.The last one is to ensure the net baryon number not be "washed out" by sphaleron process [11].In EWBG, (3) is satisfied by the occurrence of a strong first order electroweak phase transition (SFOEWPT), which provides the conditions under which new CP-violating interactions generate the asymmetry.Within the SM, given the known mass of Higgs boson, the phase transition is a smooth crossover [10,12].To explain the magnitude of Y B , new physics, especially those including new scalars coupling to the SM Higgs field, is essential to lead to a strong SFOEWPT.Based on general considerations, the presence of a first order EWPT implies that these particles must be lighter than ∼ 800 − 1000 GeV and couple with sufficient strength to the Higgs boson, making them and their interactions a clear target for high energy collider studies [13].
The simplest realization of a SFOEWPT is achieved through adding one real singlet scalar in Higgs sector [14][15][16][17], which is called the real singlet-extended standard model (xSM).The scenario admits two pathways of EWSB: (i) a direct transition to the electroweak broken vacuum ("Higgs phase"), and (ii) a two-step first-order transition with the singlet scalar acquiring a vacuum expectation value (vev) before electroweak symmetry breaking.In either case, the singlet scalar may also obtain a vev in the Higgs phase.In the presence of a non-vanishing singlet vev, the singlet-like mass eigenstate and the SM-like mass eigenstate mix with each other after EWSB.Therefore, the LHC experiments searching for heavy scalar resonance shed light on exploring the viable parameter space for EWPT in the xSM [13].Such experiments can be classified into several categories: (i) Di-Higgs channels with final states including 4b [18,19], bbV V * [20][21][22], bblν lν [21], W W * W W * [23], bbτ τ [24,25] and bbγγ [26,27].(ii) Di-boson channels with semileptonic final states [28,29], hadronic final states [30,31] and leptonic final states [22,32,33].(iii) Di-fermion channels [34].On the other hand, studies have been performed for the parameter space that can realise the SFOEWPT.Heavy resonance mass up to 500 GeV can be observed by bbγγ and 4τ searches at the 14 TeV HL-LHC with lumiosity equal to 3 ab −1 [16].Also, in search for signals with bbW W * final state, the singlet-like scalar mass can reach 350 GeV at the 13 TeV, 3 ab −1 LHC [35].For recent study in xSM, see in Ref. [16,.
Importantly, the singlet-like scalar will decay to conventional SM-like Higgs decay products due to singlet-doublet mixing as well as to di-Higgs final states (if sufficiently heavy).Thus, it is natural to ask how the combination of such "heavy Higgs" searches and resonant di-Higgs searches can further probe the SFOEWPT-viability in the xSM.In this paper, we focus on the HL-LHC collider experimental tests on the xSM, including the pp → h 2 → h 1 h 1 → bbγγ and pp → h 2 → ZZ → 4ℓ final states.Our strategy in doing so is as follows.We perform a parameter scan by requiring parameters to satisfy the theoretical bounds, including perturbativity and stability of the potential, and fix the mass and vev of SM Higgs doublet to the observed values.From the scanned parameter points, we further restrict the doublet-singlet mixing angle θ by considering the current global Higgs searches and constraints from electroweak precision observables (EWPO) 1 .We select benchmark points that produce maximum and minimum bbγγ cross-section with resonance mass ranging from 260 GeV to 800 GeV and can simultaneously produce the SFOEWPT in the early universe.To analyze the perspective sensitivity of HL-LHC on testing these benchmark points, we perform collider simulations of the two aforementioned processes by following ATLAS analysis [26,28] and obtain the signal efficiencies and distributions, while The background distributions are simply rescaled from the result in the experimental paper [26,28].
As a preview, we summarise the main result in this paper: • For resonance mass ≳ 350 GeV, the 4ℓ channel demonstrates much stronger detection capabilities compared with the bbγγ channel.For resonance mass ≳ 600 GeV, this channel covers the whole viable SFOEWPT parameter space in the xSM by 5σ.
• For resonance mass ≲ 350 GeV, the bbγγ channel is relatively more powerful.The 95% C.L. upper limit for the heavy resonance mass reaches 330 GeV and 480 GeV for parameter space with minimum and maximum bbγγ cross section respectively.
• For the combined searches of the 4ℓ and bbγγ final states, if the resonance mass is located between 300 to 750 GeV, it is possible to observe or exclude the xSM SFOEWP-viable parameter space.Moreover, the 95% C.L. and 5σ upper limit lines cover the whole viable parameter space for the resonance mass ≳ 550 GeV and ≳ 600 GeV respectively.
• A strong correlation between the signs of the mixing angle θ and the heavy Higgs to SM-like Higgs coupling g 211 is observed for SFOEWPT-viable parameter space.
Our discussion of the analysis leading to these conclusions is organized as follows: In section II, we introduce the xSM framework and the various restrictions on the parameter space: perturbativity, global Higgs searches and EWPO.In section III, we consider the one-loop effective potential with high-T approximation and perform a general scan over the allowed parameter space to obtain regions compatiable with a SFOEWPT.Section IV performs a parameter scan with constraints from discribed above.Benchmark paramater choices yielding minimum and maximum bbγγ cross sections and SFOEWPT are listed in the end of this section.Section V performs the bbγγ and 4ℓ simulation based on 13 TeV experiments and extrapolates the result to 14 TeV 3000fb −1 .Section VI presents the discrimination between signal and background events.Further it draws conclusions about the sensitivity of exclusion ability.Section VII is dedicated to the conclusions.In the Appendix, subsection IX A details the 1-loop RGE of xSM used for analyzing the perturbativity bound.Subsections IX B and IX C demonstrate the details of the simulation of bbγγ and 4ℓ respectively.Subsection IX D calculates the impact from the current CDF-II W-boson mass measurement on the heavy Higgs mass and the mixing angle.

II. THE XSM MODEL
We consider a minimal extension of SM that includes a gauge singlet real scalar S. The most general renormalizable scalar potential is given by [15] V where H is the SM SU (2) L scalar doublet.The fields H and S obtain vevs after spontaneous EWSB.They may be cast into the form The a 1 and a 2 terms with a non-zero VEV of the singlet introduce mixing and a portal interaction between the SM Higgs and the singlet.The absence of a 1 and b 3 leads to a Z 2 symmetry that protects the singlet from mixing with SM particles if x 0 vanishes, and hence the singlet becomes a DM candidate.However, even if the potential preserves the Z 2 symmetry explicitly, appropriate potential parameters can lead to a spontaneous Z 2 breaking, which yields a non-zero x 0 and mixes the SM Higgs and singlet scalar through the a 2 term, thereby allowing the decay of both h 1 and h 2 to SM particles.For a recent analysis of this possibility, see Ref. [41].A first order EWPT may arise in several ways [15,[58][59][60][61] (a) a one-step transition to the present Higgs phase wherein x 0 = 0; (b) a one-step transition to a EWSB vacuum in which the vevs of both H 0 and S are non-vanishing; (c) a two-step transition, where the vacuum in the first step has H 0 = 0 and S ̸ = 0 while the vacuum at the end of the second step has H 0 ̸ = 0 and S ̸ = 0. Case (a) requires thermal loop contributions from S to generate a sufficiently large barrier between the symmetric and EWSB phases.For scenario (b), the cubic portal term can induce a tree-level barrier for a 1 < 0. The second step of the two-step history (c) will be first order due to the cross-quartic portal term with a 2 > 0. In the following, we study the SFOEWPT and its di-Higgs signal phenomenology.
The minimization (tadpole) conditions in xSM potential can be utilized to trade the two potential parameters µ 2 and b 2 with vevs of the two scalars: The mass matrix of the scalar sector can be derived by taking the second derivatives of the scalar potential and evaluating their value at the scalar vevs: The above scalar mass matrix can be diagonalized by the mixing matrix parameterized by the mixing angle θ: where m h 1 = 125.25 GeV [62] corresponds to the mass of SM-like mass eigenstate.h 2 is the singlet-like mass eigenstate, such that the Higgs fields can be expressed as To guarantee the stability of the potential, the coefficient of the quartic field term should be positive definite, where the quartic field term is expressed as (II.10) The stability condition requires a 2 ≥ −2 √ λb 4 .Furthermore we need to guarantee that the local potential minimum at (v 0 , x 0 ) is the global one, which is checked numerically by evaluating the potential at all possible extreme points.

III. ELECTROWEAK PHASE TRANSITION
The character of EWPT is understood in terms of the finite-T effective potential.It is well-known that in the conventional treatment, V T ̸ =0 eff , which is derived from the gaugedependent 1PI effective action, suffers from gauge-dependence [37][38][39][40].However, in the high temperature expansion, wherein the potential is expanded in powers of the temperature, the leading field-and temperature-dependent term that arises at O(T 2 ) is gauge-independent when evaluating the potential at the tree-level minimum of the fields according to the ℏexpansion prescription [37].For the same reason, the 1-loop zero-temperature Coleman-Weinberg potential, V CW is also gauge independent.Therefore, in this paper, we employ the gauge-independent high-T approximated potential, which is composed of the tree-level potential, zero-temperature Coleman-Weinberg potential, V CW , and the gauge-independent finite-temperature corrections expanded up to O(T 2 ).
We start by introducing the zero-temperature Coleman-Weinberg potential as follows: wherein the summation contains all fields that interact with the scalar fields h and s.The s k is the spin of the particle, and M k is the mass.The Λ is the renormalization scale, which is fixed to be v 0 .The high-T approximation for bosonic fields and fermion fields are expressed according to [15] V scalar High−T = respectively, where the g s/f is the number of degrees of freedom for scalars/fermions.We ignore the field-independent terms and keep the gauge-independent contributions up to the leading order of temperature.The high-T potential approximation then gives where the field-dependent Goldstone, Higgs, and scalar masses are given by Note that after EWSB M 2 g = 0, as can be seen from the tadpole conditions.In the leading order of the high-T approximation, field-dependent thermal corrections appear in terms of quadratic in the scalar fields.By themselves, these thermal mass contributions do not generate a barrier between the origin and the global minima.Such a barrier can arise as follows, corresponding to the scenarios identified in Sec.II: (a) For the onestep direct transition to a pure Higgs phase, thermal loops, generated by the cross-quartic interaction proportional to a 2 , can enhance the SM contributions to the T h 3 -induced barrier; (b) for the one-step transition to the mixed vev-vacuum, the Z 2 breaking interactions proportional to a 1 and b 3 can generate a tree-level barrier, whose effect becomes active as T decreases due to the thermal mass contributions; (c) in the two-step scenario, the crossquartic interaction proportional to a 2 generates a tree-level barrier between the pure singlet and Higgs (or mixed singlet-Higgs) vacuum; the effect of this barrier again depends on T due to the thermal mass contributions to the potential.In the present analysis, we retain only the gauge-invariant thermal mass contributions to the potential, so our parameter choices do not include the direct one-step transition (a).We defer an analysis of this case to future work.
In practice, we compute phase transition-relevant quantities using the CosmoTransitions package [63].We obtain the critical temperature T c at which the broken and unbroken phases are degenerate and the corresponding Higgs vev v c .Both quantities are readily gauge invariant in the leading order high-T approximation.The condition for a strong first-order phase transition is approximately given by (III.9)

IV. CONSTRAINTS ON PARAMETERS AND NUMERICAL RESULTS
In this section, we discuss the details of our parameter scan and all the relevant theoretical and experimental constraints.In the xSM, the potential includes seven parameters: a 1 , a 2 , b 3 , b 4 , λ, µ2 , and b 2 .By utilizing the tadpole condition described in Eqs.(II.4), it is possible to trade two parameters, µ 2 and b 2 , for the vacuum expectation values (vev) of scalars, denoted as v 0 and x 0 .Additionally, we fix the SM Higgs vev v 0 to be 246 GeV and the mass of the SM-like Higgs m h 1 to be 125.25 GeV [62], which reduces the number of free parameters in the theory to five.Since the off-diagonal elements m 2 hs in the mass matrix given by equation (II.6) have a simple linear relation to a 2 , and m 2 h and m 2 s are independent of a 2 , we can express a 2 in terms of the remaining parameters as In practice, we determine whether to choose the plus or minus sign in Eq. (IV.1) based on whether the resulting parameter point achieves the global minimum at (v 0 , x 0 ).If both signs satisfy the criteria, we randomly select one of the solutions.In the meantime, we also require the mass of the heavy Higgs m h 2 > 2m h 1 to enable the di-Higgs decay channel.Consequently, we have five free parameters a 1 , b 3 , b 4 , λ, x 0 to be scanned.We limit the range of the dimensionful parameters a 1 , b 3 , and x 0 to be within 1 TeV, specified as: We choose this range due to the difficulty in finding parameter space that can yield a SFOEWPT with a heavy scalar boson mass m h 2 exceeding 1 TeV 2 .As for x 0 , we only scan the positive branch since the negative branch can be covered by reversing the signs of a 1 and b 3 .
Regarding the dimensionless parameters λ and b 4 , we perform a test scan with their range to be within the naive parturbativity bounds -0 < λ, b 4 < 2π/3 [64].As we have assumed the renormalization scale for these parameters are defined at the electroweak symmetry breaking scale v 0 in obtaining the zero-temperature Coleman-Weinberg potential, we further check whether these dimensionless parameters after running to 10 TeV with one-loop RGEs3 are still within their naive perturbativity bounds: 0 < 6λ, 6b 4 , |a 2 | < 4π.For those around 150,000 parameter points that survive from this perturbativity requirement after running, we find that the upper limits for the scanned dimensionless parameters λ and b 4 to be 1.052 and 0.868 respectively.Therefore to increase the scanning efficiency, we decided to specify ranges for dimensionless parameters to be narrower than given by the naïve perturbativity bounds: We leave the details for full expressions of the renormalization group equations (RGEs) and the relevant values for input parameters in appendix.IX A. During the parameter scan, we assume flat distributions for all the scanned parameters and trace the finite temperature potentials using CosmoTransitions for each obtained parameter point that satisfies the aforementioned theoretical constraints.Subsequently, we focus on selecting points that are capable of generating a SFOEWPT in the early universe.Following this selection process, we find approximately 80,000 parameter points that meet the desired criteria out of 6 million scans.
As for the experimental bounds, the first constraint we consider is from the electroweak precision observables (EWPO).In the xSM, the oblique parameters receive additional contributions that depend on the singlet-like scalar mass m h 2 and its mixing angle sin θ, and a constraint has been derived shows that the upper limit of | sin θ| varies from 0.35 to 0.2 when the resonance mass satisfies 250 GeV < m h 2 < 950 GeV [65].However, the new measurement of m W from CDF experiments indicates a very different parameter region from the previous measurement [66] and receives a lot of debate because of its deviation from the other measurements.Therefore, in what follows, we will utilize the EWPO constraints without including the new value of m W for reasons discussed in detail in Sec.IX D.
Secondly, we incorporate the constraints arising from the measurements of the signal strength of the Standard Model Higgs boson.The most recent results from ATLAS [67] and CMS [68] for LHC Run-2, assuming a global rescaling of the signal strength µ, are as follows: By utilizing these measurements, we deduce that | sin θ| < 0.193 at a 95% confidence level (CL).Thus, it becomes evident that the Higgs signal strength measurement imposes a more stringent constraint than the EWPO.Apart from the SM-like Higgs, we also study the current and future prospective collider phenomenology for the heavy singlet-like resonance production at the 14 TeV HL-LHC.Since m h 2 > 2m h 1 , the decay of h 2 → h 1 h 1 and h 2 → X SM X SM is allowed, where the X SM stands for SM particle whose mass is less than m h 2 /2.Since the haevy Higgs h 2 interact with the SM particles through the mixing with the SM Higgs h, the total decay width of the heavy scalar h 2 can be written as where the Γ SM (m h 2 ) denotes the decay width function of standard model Higgs with a mass of m h 2 .The decay width of h 2 → h 1 h 1 in the equation above is where the tri-Higgs coupling is obtained from the tree-level potential: As a result, the branching ratio for h 2 → h 1 h 1 can be expressed as In this work, we focus on the di-Higgs channel with bbγγ final state and di-boson channel with four lepton final state.The cross-section of pp → h 2 → h 1 h 1 → bbγγ and pp → h 2 → V V * → ℓ lℓ l channels are calculated with narrow width approximation expressed in the following equations, where the production cross section for heavy Higgs can be expressed as , and σ H (m h 2 ) is the SM Higgs production cross section if its mass were to be m h 2 given by the CERN recommendation [69].
To estimate the ability of the HL-LHC in investigating EWPT in the xSM, we collect all the points in the scan that realize SFOEWPT.Among these points, for a given h 2 mass, the cross sections of bbγγ and 4ℓ channels vary from their minimum to the maximum depending on the choice of other parameters.In Fig. 1, we show the maximum and minimum cross sections for consecutive intervals of 50 GeV starting from 250 GeV.Among selected points, benchmarks with given fixed 4ℓ cross sections that maximize/minimize bbγγ cross sections are listed in the table I/table II.The scanned result is illustrated in the Fig. 2, where we plot the cross section for pp → h 2 → h 1 h 1 → b bγγ and pp → h 2 → V V * → 4ℓ to show their correlation.In the following sections, we show the key results from our simulations for the two channels and derive the prospective bounds for future HL-LHC.

V. SIMULATION OF SIGNAL AND BACKGROUND DISTRIBUTIONS
The strategy of signal and background simulation is as follows: two 13 TeV ATLAS analyses channels pp → h 2 → h 1 h 1 [26] and pp → h 2 → V V * [22] where h 1 → b b/γγ and V → ℓ l, are taken as references for event selection and background distribution.A factor of 1.18 is applied to backgrounds to count for collision energy upgrade from 13 TeV to 14 TeV, as commonly used in ATLAS prospective studies.The signal processes at 14 TeV center-of-mass energy are generated for heavy singlet-like mass states, whose mass ranges from 270 GeV to 800 GeV in steps of 50 GeV.The simulation process is presented briefly in the following, with more details of signal cut-flow being given in Appendix IX.
In the bbγγ channel as described in reference [26], the four-body invariant mass m bbγγ is used as the discriminating variable.Events are categorized into four signal-enriched regions according to number of b-tagged jets as well as loose and tight event selection criteria: "1btag loose", "2-btag loose", "1-btag tight" and "2-btag tight".The "1-btag" and "2-btag" means that at least one and two jets should be recognized as b-jets.Acceptances after passing signal-enriched region are in good agreement with the reference analysis, as shown in Table III in Appendix IX.For background, the m bbγγ distributions from reference analysis are extracted.An exponential function is used to fit the m bbγγ curve in each signal-enriched region, and then use the fitted function to generate m bbγγ distributions according to 3 ab −1 luminosity.An ±2% error value is assigned as luminosity calculation uncertainty.The bbγγ distributions of signal and background are shown in Fig. 3.The normalizations of signals on the plots are arbitrary.
In the V V → 4ℓ channel, as described in the reference [22], the four-body invariant mass m 4ℓ is used as the discriminating variable.Events have to pass signal-enriched region selection, which is given with more details in Appendix IX section B).Signal efficiency is in reasonable agreement with the reference analysis.A similar approach as bbγγ analysis is used to simulate background distributions.Fig. 4 shows the m 4ℓ distributions of signal and background processes.The normalization of signals on the plots are arbitrary.
With these distributions and efficiencies in hand, we show in Fig. 5 the derived the expected 95% CL s upper limit for the two individual channels assuming the zero background systematic uncertainties obtained with the python package pyhf [70,71].For the discovery limits, we demand that the Gaussian significance converted by the p 0 value for the background-only test to be larger than 5 given an observation of the signal plus background predicted in the model in future experiments.In the right plot in Fig. 5, we also plot the discovery limit contours for a combined search for different masses of h 2 on the σ h 2 →V V →4ℓ v.s.σ h 2 →h 1 h 1 →bbγγ plane.These contours are obtained pyhf assuming an uncorrelated signal and background for the two channels.

VI. SENSITIVITY AND EXCLUSION LIMITS ANALYSIS
As has been well-studied in earlier work and confirmed in the present analysis, there exists a wide range of xSM parameters compatible with a SFOEWPT.As shown in Fig. 2, the SFOEWPT-compatible choices can lead to non-trivial correlations between σ h 2 →V V →4ℓ and σ h 2 →h 1 h 1 →bbγγ , particularly for the region for smaller cross sections.On the other hand, for FIG. 5.The HL-LHC prospective 95% CL s upper limits and 5σ discovery limit on the production cross-sections times branching ratios for the two channels.We assume systematic uncertainty for all the backgrounds.the larger values of σ h 2 →V V →4ℓ this correlation evaporates.Generally, for a fixed resonance mass , the 4ℓ and bbγγ cross sections are determined by sin θ and g 211 respectively.Therefore, for a fixed resonance mass, σ 4ℓ will vary over a range due to the allowed range for sin θ.The cross sections would distribute from a minimum and a maximum that are labeled by σ min 4ℓ and σ max 4ℓ .Similarly, for a fixed resonance mass and a fixed 4ℓ cross section, the bbγγ cross section is not determined and locates between the maximam σ max 4ℓ;bbγγ and the minimum σ min 4ℓ;bbγγ .In order to carry out a systematic analysis of the LHC reach in this parameter space, we adopt the following procedure.For each resonance mass point, a set of σ h 2 →V V →4ℓ points are selected, which cover the full range between σ min 4ℓ to σ max 4ℓ and distribute uniformly.Then for each fixed σ h 2 →V V →4ℓ point, we choose two parameter sets, corresponding to the maximum value , σ max 4ℓ;bbγγ , and minimum value, σ min 4ℓ;bbγγ , respectively.In total, for each mass point, overall 30∼40 σ h 2 →V V →4ℓ points are selected, which results from 60∼80 number of points on the σ h 2 →V V →4ℓ and σ h 2 →h 1 h 1 →bbγγ plane.
Fig. 6 shows the prospective discovery significance for HL-LHC with 3 ab −1 integrated luminosity, from the V V → 4ℓ channel, and from the h 1 h 1 → bbγγ channel with maximum and minimum cross-section, respectively.The h 1 h 1 → bbγγ channel with a maximum crosssection (red region in the left plot) has, on average larger sensitivity than the minimum cross-section (red region in the right plot).Compared to the h 1 h 1 → bbγγ, the V V → 4ℓ channel (blue region) has better sensitivity, particularly for higher mass resonance signals.The V V → 4ℓ channel can reach to 5-sigma discovery significance over a wide mass range from 300 GeV to 750 GeV.
It is worth emphasizing that the V V → 4ℓ sensitivity is generic for any heavy Higgs that has a V V decay mode and is not specific to the xSM model.Thus, the observation of this mode at a given mass with the significance indicated would be compatible with the SFOEWPT in the xSM but not conclusive.Conversely, the non-observation of this mode would preclude the SFOEWPT-viable xSM with a heavy resonance in a significant portion of parameter space.
On the other hand, the h 1 h 1 → bbγγ channel would provide a diagnostic probe with some sensitivity up to around 500 GeV.In terms of exclusions corresponding to the 2-sigma line on the plots, both h 1 h 1 → bbγγ and V V → 4ℓ channels have some sensitivity.In order to enhance the analysis sensitivity, results from the two channels are therefore combined.The combined discovery significance is shown in Fig. 7.More than half of the considered phase space points can be excluded at HL-LHC with 3 ab −1 integrated luminosity.On the other hand, there are also points below the 1σ line on the plots, which can be further investigated at future colliders.One may also interpret the results discovery or exclusion regions in the plane of g 211 and sin θ, as shown in Fig. 8. From left to right, these plots illustrate the HL-LHC discovery/exclusion potential for -400±10 GeV, 550±10 GeV, 700±10 GeV respectively.All points generate SFOEWPT and satisfy requirements discussed in Sec.IV.Constraints from Higgs measurement at LHC Run II are labeled by grey shadow.Upper limits with 95% C.L. and 5σ significance for the 4ℓ channel are given in all panels.Only the 95% C.L. upper limit for the bbγγ channel is presented in the first two panels due to its relatively weaker probing ability.Points located on the right side of each line would be observed/excluded.The plots manifest that the 4ℓ channel shows a much stronger observation/exclusion reach than the bbγγ channel, especially for heavy resonance masses.
In more detail: the blue curves and black curves represent the 5σ discovery bound and 95% CL s exclusion bound for the ZZ → 4ℓ channel.The parameter space at the right-hand side of the blue curves with bigger | sin θ| and |g 211 | can be discovered with 5σ significance at HL-LHC if the nature is realized by the real singlet model, and the parameter spaces at the right-hand side of the black curves can be excluded with non-observation of the signal.The green and red curves are for 5σ discovery bound and 95% CL s exclusion bound for the h 1 h 1 → bbγγ channel.All these bounds assume an integrated luminosity of 3000 fb −1 .In the plot for m h 2 = 700 GeV, only the blue, black and red curves exist in the parameter space of interest because the bbγγ has no detection capability for this mass region.The shaded regions in each plot are excluded by current LHC Run-II results.
With these results in mind, one can anticipate several possible experimental outcomes: (i) Both modes are observed: If both modes with a resonance mass ≲ 500 GeV (Fig. 6) are observed by 2σ, or the 4ℓ channel reaches 5σ with the bbγγ reaching 2σ, it is expected by, and shows strong evidence for, the SFOEWPT-viable xSM.However, observation of both channels by 5σ would be inconsistent with the SFOEWPT-viable xSM except for m h 2 ∼ 350 GeV.
(ii) Neither channel is observed at the HL-LHC: The non-observation of both the 4ℓ and di-Higgs channels does not rule out the SFOEWPT-viable xSM.However, it implies an intriguing pattern regarding the sign correlation between sin θ and g 211 .Notably, for parameter points remaining unexcluded by the future HL-LHC experiment, a negative sin θ is only consistent with a relatively light h 2 mass.Additionally, in cases of negative sin θ, there is a strong correlation between g 211 and sin θ, as indicated by the narrow strip in the left plot of Fig. 9.We discuss this correlation and Fig. 9 in more detail below.
(iii) The V V → 4ℓ is observed but the h 1 h 1 → bbγγ is not: If the observed resonance satisfies ≳ 350 GeV, the existence of a heavy resonance compatible with the SFOEWPT-xSM would be established but a future more sensitive search for the h 1 h 1 → bbγγ would be needed.On the other hand, if the resonance is ≲ 350 GeV, the observation would be in consistent with xSM prediction and is hence ruled out.
(iv) The h 1 h 1 → bbγγ is observed but the V V → 4ℓ is not: If one observes the di-Higgs channel with bbγγ final states in the small resonance mass region (≲ 350 GeV) and no significant observations for the 4ℓ channel, the result would be consistent with the SFOEWPT in the xSM.On the contrary, if the observation occurs with heavier-mass resonance (≳ 500 GeV), the SFOEWPT-viable xSM prediction would be inconsistent with the observation and, therefore, the xSM FOEWPT would be ruled out.
Note that for each possible outcome, one would need to perform an updated xSM parameter scan that takes into account the various experimental results.FIG. 6. Prospective discovery/exclusion reach for HL-LHC with 3 ab −1 .Horizontal and vertical axes give heavy scalar mass and significance, respectively.Red points in the left (right) panels indicate range of maximum (minimum) σ h 1 h 1 →bbγγ as one varies over the range of σ V V →4l .The significance for the latter is indicated by the blue points.For a fixed M S , σ V V →4l depends mainly on sin θ mainly.All points satisfy the requirement of SFOEWPT.We now comment further on the question of the sign of sin θ and the correlation with other xSM tri-scalar couplings.First, we observe that the independent measurement of the sign of sin θ is not an easy task, and one cannot determine the sign of sin θ with solely the couplings between h 2 and SM fermions or vector bosons, because in this case each propagator of h 2 present in a diagram must contribute to 2 powers of sin θ erasing the sign information.In this case, the sign of the sin θ must be resolved by a global fit that includes physical processes that are sensitive to all the scalar couplings, such as g 221 .As an illustration, in Fig. 9, we show the scatter plot in the g 211 vs g 221 plane for all the points that can give SFOEWPT and simultaneously satisfy all the theoretical and current experimental bounds, including the one from LHC run-II.The color of the points indicates the sign of the mixing angle sin θ, red for positive and blue for negative.One can find that for m h 2 around 400 GeV, the points with the different sign of sin θ tend to populate different parameter space in this 2-D plane, which indicates that the measurement of the two tri-linear couplings could help to disentangle the sign of sin θ in xSM.

VII. CONCLUSION
Exploring the thermal history of electroweak symmetry breaking is an important effort in its own right and may yield clues to solving the baryogenesis problem through EWBG.This scenario requires a SFOEWPT as a pre-condition for generation of the CPV asymmetries that ultimately yield the baryon asymmetry.In the SM, EWSB occurs though a cross-over transition.However, the Higgs portal interactions in the xSM can generate a SFOEWPT for suitable choices of model parameters.The portal interactions are strongly correlated with the strength of the EWPT.Generally, a successful SFOEWPT requires relatively large portal interactions.However, constraints from Higgs measurement, EWPO and perturbativity indicate that the magnitudes of the mixing angle and the portal couplings cannot be arbitrarily large.Therefore, the parameter space is highly constrained.Through our analysis, we find that the remained parameter space has different sensitivies for different detection channels.As is well-known, one of the effective ways to detect the Higgs portal interactions is the heavy scalar resonance search via di-Higgs channel.For the di-Higgs channel, we have considered in particular the bbγγ final state.Another way to probe the Higgs portal interactions is through the di-boson channel.For the di-boson channel, we have considered in particular the 4ℓ final state.A novel aspect of this study is the analysis of the complementarity between these two channels and the discovery/exclusion reach that results from their combination.
Through our analysis We find that the b bγγ channel is able to explore the low mass region with m 2 ≲ 320 GeV effectively and the 4ℓ channel is more suitable for the high mass region with m 2 ≳ 500 GeV.It is also possible that a combination of the two search channels can push the significance above 5σ even though the significance for each individual channel is less than 5σ.In terms of exclusion, one is able to exclude the portion of SFOEWPT parameter space by taking advantage of the combination of b bγγ and 4ℓ channels over roughly m 2 ≲ 750 GeV at the 14 TeV HL-LHC with luminosity 3ab −1 .Moreover, the combination of both channels provides an important diagnostic probe of the SFOEWPTviable xSM.However, signal events associated with minimal b bγγ cross section are difficult to exclude at the HL-LHC, pointing to the need for a future collider with higher energy to obtain a comprehensive probe (see, e.g., Refs.[13,16]).Finally, in the case of combined discovery, there is a strong correlation between g 211 and sin θ.On the other hand, even though a non-observation of both channels would not rule out SFOEWPT-viable xSM, it would put constraints on the sign of the mixing angle and the tri-linear scalar coupling g 211 .In this case, future high-energy collider experiments may needed to further diagnose and probe the different scalar couplings in the model (see, e.g., Refs.[13,16]).We show as an example how the simultaneous measurement of g 211 and g 221 could allow one to disentangle the sign of sin θ in the SFOEWPT-viable xSM.We use the python package PyR@TE 3 [72] to derive the renormalization group equations (RGEs) in the MS scheme at one-loop level and checked that the results in the limiting cases are in agreement with those in literatures [73][74][75][76].The full one-loop RGEs in the form of dX/d log µ = β (1) (X)/(4π) 2 are given as follows: β (1) (g 3 ) = −7g (IX.8) (IX.9) β (1) (IX.10) where g 1 , g 2 , g 3 are U (1) y , SU (2) L and SU (3) c gauge couplings respectively and y t and y b are Yukawa couplings for top and bottom quarks.Their definitions are summarized in the following equations: where q is the U (1) y charge of the field Ψ, T 2 and T 3 are the corresponding generators of the SU (2) and SU (3) groups depending on the representations of the field Ψ.For example, for the quark left-handed doublet Q L , it has q = 1/6, T 2 2 = σ i /2 and T a 3 = λ a /2 with σ i and λ a the Pauli and Gell-Mann matrices respectively.Hi ≡ iσ 2 ij H j , u R and b R are right-handed up-type and down-type quark singlets respectively.
For the initial inputs of SM Yukawa and gauge couplings g 1 , g 2 , g 3 , y t , y b , we use the package mr [77] to convert the input parameters from the On-Shell scheme to the MS scheme and run to the scale µ = 246 GeV.When converting between the On-shell and MS scheme, we use the relations in the order of one-loop QCD and EW in the Standard model and assume that the correction from the heavy Higgs is small provided that the sin θ is within the constraint of EWPO.For the running couplings g 1 , g 2 , g 3 , y t , y b , we implement the oneloop RGEs for consistency, which is not affected by the heavy singlet in xSM.The On-Shell input parameters are obtained from the Particle Data Group [62]: (IX.18)

B. bbγγ channel simulation
We perform a generation of a signal process pp → h 2 → h 1 h 1 → b bγγ by utilizing MadGraph5 [78] in parton level and use Pythia8 [79] to simulate the parton shower process followed by a detector simulating software Delphes3 [80] to simulate the detector response of this process at LHC.In this section, we will follow the ATLAS analysis in Ref. [26] and reproduce the cutflow efficiency compared with the experimental data.
In the detector simulation, photon candidates must satisfy isolation criteria of ∆R ≤ 0.2 for both calorimeter-and track-based isolation.In track-based isolation, the transverse momentum within the cone (p iso T ) counts only the tracks with p T > 1 GeV.Jets are reconstructed using anti-k t clustering algorithm [81] with a radius parameter set to be R = 0.4 and are required to satisfy |η| < 2.5 and p T > 25 GeV.In addition, the efficiency for a b-quark jet to pass the b−tagging requirement is set to be a constant equal to 70% [82] approximately.
The selection is simplified according to Ref. [26]: • Events are required to have at least two photons and two jets with one or two jets recognized as b-jet(s).Any event containing more than two b-jets is rejected.
• The invariant mass of two photon with the highest p T should satisfy 105 GeV < m γγ < 160 GeV.
• For each photon, it is required to have E T /m γγ > 0.35 and 0.25 respectively.
• Four signal regions are defined based on the highest-p T and next-highest p T (They are called "Cen Jets" in Table.III).If p 1 T > 40GeV and p 2 T > 25GeV, the signal is labeled with "loose".If p 1 T > 100GeV and p 2 T > 30GeV, the signal is labeled "tight".Combining with the b-jet number, signal events are classified to four regions, which are "1-btag loose", "1-btag tight", "2-btag loose" and "2-btag tight".
• The dijet system is rescaled by a factor of m H /m jj .
• Loose selection is used for resonances with m X ≤ 400 GeV and the tight selection for resonances with m X ≥ 400 GeV.For tight (loose) selection, we require 335 GeV < m γγjj < 1140 GeV(245 GeV < m γγjj < 610 GeV).
• For events that pass the selection above, we obtain their m bbγγ distribution.[26].The Sim-line shows the simulation result.

C. ZZ → 4ℓ channel simulation and selection
We generate the parton level signal events pp → h 2 → ZZ → ℓ + ℓ − ℓ + ℓ − with Mad-Graph5 [78], then showering with Pythia8 [79] and simulate the detector effect with Delphes3 [80].We follow the ATLAS analysis in Ref. [22], and reproduce similar signal and background efficiencies in that paper.The analysis cut flow is described as followings: • Select the events with two same-flavor opposite-sign lepton pairs.The reconstructed electron (muon) must have p T > 7(5) GeV and |η| < 2.47(2.7).
• For lepton pairs, we denote the invariant mass of the pair closer to Z mass as m 12 and that of the other pair as m 34 .We demand that m 12 and m 34 satisfying the following cuts: 50 GeV < m 12 < 106 GeV m 34 < 116 GeV • We also require that leptons are separated with each other by ∆R > 0.1 if they are the same flavor, and ∆R > 0.2 otherwise.
• For 4µ and 4e events, we veto the events containing opposite-sign lepton pairs with m ℓℓ < 5 GeV.
• We implement the cut on the track-isolation discriminant as the sum of the transverse momenta of tracks within ∆R = 0.3 (0.2) of the muon (electron) candidate excluding the lepton track, divided by the p T of the lepton.Such discriminants are required to be smaller than 0.15.
Compared to Ref. [22], we did not impose the calorimeter-based isolation requirement because they are technically difficult to implement in the Delphes.In practice" we find that the major effect on the signal efficiencies is from the detector efficiencies for leptons, and with the above cuts we are already able to obtain signal efficiencies reasonably close to those in the ATLAS paper [22].In table.IV, we illustrate the effectiveness of our simulation by showing the comparison of signal efficiencies for different lepton channels for a fixed benchmark point m h 2 = 600 GeV, we only compare the final efficiencies because the efficiencies for the intermediate cuts are inaccessible in the experimental paper.xSM can influence EWPO in two ways: rescaling couplings between the SM-like Higgs to SM particles and generating new diagrams involving the single-like Higgs.When parameterizing theoretical predictions of EWPO in the framework of the oblique parameters S, T and U [83,84], the shifted oblique parameters in the xSM can be uniquely determined by the two physical parameters: the mixing angle θ and the mass of the singlet-like Higgs m 2 , which can be expressed in the following formula, ∆O = (cos 2 θ − 1)O SM (m h 1 ) + sin 2 θO SM (m h 2 ) = sin 2 θ O SM (m h 2 ) − O SM (m h 1 ) , (IX.20) where O can be either S, T or U , and O SM (m h 2 ) are the SM expression for O if the Higgs mass were to be m h 2 .Therefore a constraint on the cos θ − m h 2 plane can be derived by constructing the χ 2 with the predicted oblique observables and the experimental data.In our previous work [35], we used the global fitted center values, uncertainties, and correlations for these oblique parameters from the Gfitter group [85] and obtained a lower bound on cos θ as a function of m h 2 .However, the recently updated measurement of m W from the CDF collaboration [86] exhibits a significant deviation from the SM prediction, which indicates a qualitative change in the EWPO constraint for xSM.
To illustrate the potential influence of such a new m W measurement on the EWPO constraint for xSM, we follow the recent global fit result by the HEPfit group [87], where they combine the measurements from LEP2, LHC and Tevatron including the most recent CDFII results to give a new "world average" of W mass as: m W = 80.4133 ± 0.0080 (0.015) GeV, (IX.21) where the number in the parenthesis represents an inflated error for a conservative average.From this updated value of m W , authors in Ref. [87] derive four sets of constraints on the S, T , and U parameters assuming whether one uses the conservative average or not and whether one sets U = 0 or not.As an example, we plot the allowed parameter space at 95% confidence level (CL) in the cos θ − m h 2 plane in the blue region in Fig. 10 assuming U = 0 and m W taking the conservative average, which corresponds to the following bounds and correlations matrix on the S and T parameters: One can find that the allowed parameter space tends to have a very small m h 2 such that the diHiggs decay channel is not available.This is expected, since the updated W boson mass from CDFII indicates a relatively large violation of custodial symmetry, while xSM scalar potential does not explicitly break the custodial symmetry, as a consequence, the new physics contribution to the custodial symmetry breaking starts at one-loop receiving a loop factor suppression, to compensate such a suppression one needs a relatively small singlet-like Higgs mass.However, we need to emphasize here that the above global analysis heavily depends on the method used for combining the W mass measurements from different experiments, given that the newly measured m W has a significant discrepancy compared with all the previous measurements, the ordinary averaging method assuming correlated Gaussian uncertainties may not be appropriate anymore, and the community has not reached a common agreement on the interpretation of the experimental results yet.
Allowed by ST Fit @ 95% CL FIG. 10.The allowed parameter space at 95% CL for xSM using the global fit from Ref. [87] assuming U = 0 and M W with conservative average.The vertical axis start from θ = π/4, above which the mass eigenstate h 1 gets more contribution from the neutral part in the Higgs doublet thus corresponding to the SM-like Higgs.

FIG. 1 .
FIG. 1.The maximum/minimum of the cross sections of bbγγ and 4l channels that realize SFOEWPT.The color bar stands for first-order EWPT strength.The red points in the third panel are the maximum and minimum benchmarks we choose for m h 2 = 402 GeV and m h 2 = 401 GeV in the Tab.I and Tab.II respectively.

Benchmark
FIG.2.Distribution of the cross sections for the bbγγ and 4ℓ channels.The color stands for first-order EWPT strength as defined in Eq. (III.9).The left panel shows all results with m h 2 varying from 260 GeV to 800 GeV.The masses in the middle and right plots are within 400 ± 10 GeV and 700 ± 10 GeV, respectively.

5 FIG. 7 .
FIG.7.Prospective discovery/exclusion reach for HL-LHC with 3 ab −1 from the combination of two channels: (1) the V V → 4ℓ channel, and (2) h 1 h 1 → bbγγ channel with maximum cross section (left panel) and with minimum cross section (right panel).

FIG. 8 .
FIG. 8. Prospective HL-LHC (3 ab −1 ) discovery and exclusion reaches for the coupling g 211 /GeV as a function of sin θ allowed by the SFOEWPT for 400 GeV (the first panel), 550 GeV (the second panel) and 700 GeV (the third panel) resonance masses.Color bars in the panels correspond to the ratio of g 111 /g SM 111 .The area to the right of each line corresponds to the discovery/exclusion reach for a given channel.The solid lines represent HL-LHC exclusion bound as labeled above The grey region is excluded by LHC Run-II.All the points in the plot satisfy theoretical bounds and survive from the EWPT discussed in section.IV.

FIG. 9 .
FIG. 9. Points in the g 211 vs g 221 plane, where we have removed the points that are excluded by current LHC measurements.The red (blue) color indicates that the sign of sin θ for the corresponding point is positive (negative).

TABLE I .
Benchmark parameter choices that maximize the bbγγ channel cross section when the 4ℓ cross section is fixed.

TABLE IV .
[22]al efficiencies for different lepton channels of the process the h 2 → ZZ → 4ℓ, the Exp numbers are obtained from Ref.[22].D. Comments on the constraint from electroweak precision observables (EWPO)