Electroweak baryogenesis via bottom transport: complementarity between LHC and future lepton collider probes

We study the complementarity between the Large Hadron Collider (LHC) and future lepton colliders in probing electroweak baryogenesis induced by an additional bottom Yukawa coupling $\rho_{bb}$. The context is general two Higgs doublet model (g2HDM) where such additional bottom Yukawa coupling can account for the observed baryon asymmetry of the Universe if $\mbox{Im}(\rho_{bb}) \gtrsim 0.058$. We find that LHC would probe the nominal $\mbox{Im}(\rho_{bb})$ required for baryogenesis to some extent via $bg \to bA \to bZh$ process if $300~\mbox{GeV}\lesssim m_A \lesssim 450$ GeV, where $A$ is the CP-odd scalar in g2HDM. We show that future electron positron collider such as International Linear Collider with $500$ GeV and 1 TeV collision energies may offer unique probe for the nominal $\mbox{Im}(\rho_{bb})$ via $e^+ e^- \to Z^*\to A H$ process followed by $A,H \to b \bar b$ decays in four $b$-jets signature. For complementarity we also study the resonant diHiggs productions, which may give an insight into strong first-order electroweak phase transition, via $e^+ e^- \to Z^*\to A H \to A h h$ process in six $b$-jets signature. We find that 1 TeV collision energy with $\mathcal{O}(1)~\text{ab}^{-1}$ integrated luminosity could offer an ideal environment for the discovery.


I. INTRODUCTION
The discovery of the 125 GeV Higgs boson (h) [1] was a truly watershed moment that established the standard model (SM) as a correct effective theory at electroweak scale. While the SM has withstood all experimental tests so far, cosmological problems such as baryon asymmetry of the Universe (BAU) and dark matter still remain open and a more fundamental theory must exist in nature.
For the BAU generation, one has to satisfy so-called Sakharov's conditions [2]: (i) baryon number violation, (ii) C and CP violation, and (iii) departure from thermal equilibrium. One of the compelling ideas to explain BAU is electroweak baryogenesis (EWBG) [3] (for reviews, see Ref. [4]), whose core mechanism is already built-in even in the SM. However, the observed parameters in the SM turn out to be inconsistent with successful EWBG due to the insufficient magnitude of CP violation and absence of a first-order electroweak phase transition (EWPT). Generally, various new-physics models are conceivable to circumvent those two issues. Among them, a general two-Higgs-doublet model (g2HDM) [5] is one of the most attractive models from the viewpoints of renormalizability, generality, and testability. It is shown that extra Yukawa couplings of the second and third generation quarks and leptons, which can be complex and flavor violating, could provide CP violation sufficient for BAU. The most efficient EWBG scenario would be a case that the top quark has the O(1) extra Yukawa coupling, followed by a case in which the sizable top-charm-changing Yukawa coupling is present [6]. Thoroughgoing study of those collider signatures can be found in Refs. [7][8][9][10][11][12][13][14].
Under generous assumptions for bubble wall profiles, the bottom quark could also drive the sufficient BAU if the size of the extra bottom Yukawa coupling is larger than the SM bottom Yukawa coupling to some degree. This bottom-Yukawa-driven EWBG can be significant in a case that the aforementioned Yukawa couplings in the up-type quark sector happen to be real or tiny. In Ref. [15], the present authors studied phenomenological consequences of the bottom-Yukawa-driven EWBG in detail assuming that both extra top and bottom Yukawa couplings are present but the former is real and the latter is roughly twice larger than a necessary bare minimum for BAU. It was found that Large Hadron Collider (LHC) with 1000 fb −1 integrated luminosity could examine the scenario, primarily via the process bg → bA → bZh with final states comprising of 3b-jets and a lepton pair, where A is the CP-odd scalar and h is the 125 GeV Higgs boson in the g2HDM. In Ref. [16] it was also shown that the bg → bA → bZH process would provide a sensitive test for the case of m A > m H + m Z , where H is the CP-even heavy scalar. While these processes provide a unique probe to bottom-Yukawa-driven EWBG, they become insensitive if m A < m Z + m h and/or m A < m Z + m H . Furthermore, if m A > 2m t , an achievable significance diminishes if the extra top Yukawa coupling is O(1).
In this work, we further pursue the bottom-driven EWBG scenario with particular emphasis on complementarity between LHC and the International Linear Collider (ILC). After taking the theoretical and experimental constraints into accounts, we investigate a discovery potential of the EWBG scenario assuming a necessary bare minimum of the extra bottom Yukawa coupling and absence of the extra top Yukawa coupling, which is diagonal parameter space investigated in Ref. [15]. In this scenario, we examine the bg → bA → bZh process at the LHC, and compare with the results in Ref. [15]. We also proceed to study detectability of EWBG signatures at the ILC assuming 500 GeV and 1 TeV center-of-mass (CM) energies. We consider the process e + e − → AH with the A/H → bb decay as well as the H → hh decay, leading to 4b-jets and 6b-jets final states, respectively.
The paper is organized as follows. In Sec. II we outline the model framework and the available parameter space for our study. Sec III is dedicated for finding discovery prospect of the bg → bA → bZh process. In Sec IV we discuss sensitivity of e + e − → AH → 4b. We also study the e + e − → AH → Ahh process and the corresponding vertex correction for the trilinear Hhh coupling. We summarize our results with some discussions in Sec. V.
The Yukawa sector of the g2HDM is given by [17] where P L,R ≡ (1 ∓ γ 5 )/2, V is CKM matrix, i, j = 1, 2, 3 are generation indices, and U = (u, c, t) T , D = (d, s, b) T , L = (e, µ, τ ) T and ν = (ν e , ν µ , ν τ ) T are column vectors in the flavor space. The matrices λ F ij (= √ 2m F i δ ij /v) are real and diagonal, while ρ F ij are in general complex and non-diagonal. It is pointed out in Ref. [19] that electric dipole moment (EDM) of the electron could be suppressed if the diagonal elements of ρ F ij follow the similar hierarchal structures of the SM Yukawa couplings, i.e., |ρ ee /ρ tt | ∼ λ e /λ t , which tempts us to conjecture |ρ ii /ρ jj | ∼ λ i /λ j for all the flavor indices. We however consider somewhat offset parameter space motivated by the successful ρ bb -EWBG mechanism in which Im(ρ bb ) = 0.058( λ b 0.024). Circumvention of the electron EDM constraint in this scenario will be addressed in Sec. II B.
Here we should note that h, H, and A are not CP eigenstates any more when including loop corrections that break CP through Im(ρ ij ). However, the loop corrections are small enough to regard the neutral Higgs bosons as the CP as well as mass eigenstates.
For all practical purposes we turn off all ρ ij except for ρ bb , however their impact will be discussed in Sec. V.
Primary motivation of this article is to probe the nominal value Im(ρ bb ) = 0.058 [15] required for ρ bb -EWBG. In general, LHC would offer exquisite probe via bg → bA → bZh process if Im(ρ bb ) 0.15 [15] but the process requires m A > m Z + m h . The process bg → bA → bZH would also offer sensitive probe if m A > m H + m Z [16]. We note that the dependence of the AZh and AZH couplings on the mixing angle γ can be found from [5] where c W and g 2 are the Weinberg angle and the SU (2) L gauge coupling respectively. As discussed in Ref. [15], the nonzero γ could have non-negligible impacts on ρ bb -EWBG. From the interactions (3) and (4), one can see that the production bg → bA does not depends on γ and the decays A → ZH and A → Zh are scaled by s γ and c γ , respectively. In the vicinity of the alignment limit γ = −π/2, the bg → bA → bZh process would provide more sensitive probe of the mixing angle through c γ . While bg → bA → bZh process can exclude the nominal |Im(ρ bb )| = 0.058 at HL-LHC if m A ∼ 300 GeV, it fails to probe the nominal Im(ρ bb ) above m A > 2m t if ρ tt ∼ 0.5 [15]. Here we shall revisit potential of bg → bA → bZh process to probe nominal Im(ρ bb ) for scenarios where m A > 2m t but for vanishingly small ρ tt . The bg → bA → bZh process would become insensitive for m A < m h + m Z . In such scenarios future lepton colliders such as ILC or FCCee would offer unique probe for ρ bb -EWBG via e + e − → Z * → AH process [20] followed by A/H → bb decays i.e., in four b-jets signature. The signature would also receive contribution from ρ bb induced e + e − → Z * → bbA/H process if A, H decays to bb. We remark that a similar search pp → Z * → AH → bbbb at the LHC would suffer from overwhelming QCD multijets backgrounds, which prevents us from probing our scenario.
Given the fact that the strong first-order EWPT needs O(1) Higgs quartic couplings, triple Higgs couplings φ i φ j φ k could be potentially large. A sensitive probe for Hhh coupling is possible via e + e − → Z * → AH → Ahh process (see Ref. [21] for similar discussion). We study this process in six b-jets signature. The final state signature would receive contribution from e + e − → bbH → bbhh 1 if both the h decays to bb. The Hhh coupling is defined as the coefficient of the h 2 H term in the Higgs potential, from which it follows that [9] with which implies that λ Hhh → 0 as c γ → 0. The approximate expression (6) does not differ from the exact one (5) by more than about 1.5% in our benchmark points (BPs) described below. We also notice that λ Hhh is always negative in our chosen BPs, which could be important when discussing one-loop corrections. We primarily focus on tree-level Hhh coupling however we will discuss higherorder corrections to λ Hhh and its impact on strong firstorder EWPT in Sec. IV B. A probe for Hhh coupling in the context of ρ bb -EWBG would be indeed possible at the LHC via bb → H → hh and bg → bH → bhh. However we have checked that such processes are beyond the scope of the HL-LHC for nominal value |Im(ρ bb )| = 0.058 primarily due to overwhelming SM QCD background such as multi-jets and tt+jets.

B. Constraints and parameter space
Let us find the allowed parameter space for m A , m H and m H ± such that EWBG is possible. As widely known, η i v 2 , where η i are some linear combinations of η's whose magnitude is O(1), should be greater than µ 2 22 in order to induce the strong first-order EWPT, leading to lower bounds of the heavy Higgs bosons. On the other hand, since the quartic couplings are enforced to satisfy perturbativity and tree-level unitarity, their sizes cannot exceed certain values, e.g., 4π, which sets upper bounds of the heavy Higgs bosons. 2 Therefore, typical 1 Similar final signature has been discussed in the context of a softly Z 2 -broken 2HDM in Ref. [22] 2 We also evaluate a scale at which one of η's exceeds 4π, where the theory starts to enter non-perturbative regime. Using one-loop renormalization group equations (RGEs) with m A as an initial value, it is found that Λ non-perturb = (2.0, 1.7, 2.5) TeV in the 3 benchmark points shown in Table I, respectively. Those scales could be roughly doubled if two-loop RGEs are used (see, e.g., Ref. [23]). The parameters in Eq. (1) are required to satisfy perturbativity, tree-level unitarity and vacuum stability conditions, for which we utilized the public tool 2HDMC [24]. We choose three BPs summarized in Table I that satisfy aforementioned three theoretical constraints, electroweak precision measurements, and strong first-order EWPT as needed for EWBG.
Having fixed the BPs, we now turn our attention to constraints on Im(ρ bb ). There exist several indirect and direct searches that can constrain the parameter space for Im(ρ bb ). For nonvanishing ρ tt , Im(ρ bb ) receives meaningful constraints from the branching ratio measurement of B → X s γ (B(B → X s γ)) and the asymmetry of the CP asymmetry between the charged and neutral B → X s γ decays (∆A CP ) [15]. However as we focus on parameter space where ρ tt is small, such constraints practically allow an order of magnitude larger Im(ρ bb ) than that of the nominal value required for ρ bb -EWBG. Therefore we do not discuss such constraint here and redirect readers to Refs. [15] for further details.
The Higgs signal strength measurements by ATLAS and CMS would however provide some constraints primarily due to our choice of c γ = 0. from Eq. (3) as: where . Allowing 2σ error bars on these measurements we show these limits in Fig. 1 in the Re(ρ bb )-Im(ρ bb ) plane by purple (CMS) and cyan (ATLAS) shaded regions. While finding the limits we simply symmetrized the error bars of CMS and ATLAS measurements. For comparison we also overlay the nominal parameter space for ρ bb -EWBG (|Im(ρ bb )| > 0.058) by the red solid lines in Fig. 1. It is clear that κ b measurements are not able to cover the entire ρ bb -EWBG region. This is primary due to the fact that CP-violating term Im(ρ bb ) does not interfere with the SM part, thereby being more suppressed by the mixing angle c γ , as can be seen from Eq. (7). It would be useful to compare the sensitivity of future e + e − collider in probing κ b . In this regard we focus on the ILC, which is expected to measure κ b within 1.1% and 0.58% [27] uncertainties at 1σ in its √ s = 250 GeV (denoted as ILC250) and, combined 250 GeV and 500 GeV data (denoted as ILC500). Allowing 2σ error we illustrate these limits in Fig. 1 by blue dotted and solid lines respectively, where in both cases the white crescent shaped regions within the lines are allowed. For comparison the HL-LHC is expected to measure κ b with ≈ 6% accuracy [28], which we do not show in Fig. 1. It is clear that sufficient parameter space for ρ bb -EWBG would survive even after various precise measurements of hbb coupling.
There also exist some heavy Higgs searches from AT-LAS and CMS that also constrain Im(ρ bb ). E.g., it was found [15,16] that the most relevant constraints arise from heavy neutral Higgs boson production with at least one b-jet followed by bb decay [29] and, heavy charged Higgs searches pp → t(b)H ± with H + /H − → tb/tb decays [30,31] (see also e.g. Refs. [32,33]). As we primarily focus on parameter space where |Im(ρ bb )| ≈ 0.058 and, the fact that such searches excludes Im(ρ bb ) 0.25 [15,16] for the sub-TeV mass range, we refrain a detailed discussion of these here and redirect readers to Refs. [15,16] for further discussion.
Now we discuss EDM constraint on Im(ρ bb ) in light of the latest result of ACME Collaboration [34]. This constraint is so overwhelming that one cannot dodge it without relying on some mechanism in any EWBG scenarios in g2HDM. As briefly mentioned below Eq. (3), the electron EDM could be sufficiently suppressed by the build-in cancellation mechanism. For that end, ρ tt and ρ ee have to be complex and echo the SM-like Yukawa hierarchy. In the ρ bb -EWBG scenario, however, ρ tt is real or small by assumption and the above solution space is the no-go zone. Nonetheless, it is still possible to render the electron EDM small enough to avoid the ACME constraint in concert with ρ bb and ρ ee though the cancellation does not manifest any structure. We do not repeat the analysis here and refer the readers to Ref. [15] for more details.  Without significant improvements in experimental uncertainties all in all we remark that the nominal value |Im(ρ bb )| = 0.058 for ρ bb -EWBG is likely to survive all current and future measurements discussed in this section. For illustration we take |Im(ρ bb )| = 0.058 for all three BPs in our analysis. For all practical purposes we set all ρ ij = 0 except for Im(ρ bb ) however we shall return to the impact of turning other ρ ij couplings in Sec. V. Under the aforementioned assumption and neglecting tiny loop induced decays such as A → γγ, the CP-odd boson A decays practically 100% to bb for BPa, while additional decay mode Zh are open and constitute about 35% and 70% for BPb and BPc, respectively. The respective branching ratios for the three BPs are summarized in Table II. For the CP-even heavy Higgs boson H, it primarily decays to bb, followed by W W and ZZ in BPa. In BPb, the bb and W W modes comprise about 30% branching ratios, followed by hh and ZZ. In BPc, W W is the dominant decay mode, followed by ZZ. The tt channel is also kinematically accessible, which predominates over the bb and hh modes. In addition to the above decay modes, the decays such as H → τ τ , H → cc, etc. would be turned on via nonzero c γ , as can be seen from Eq. (3). Besides tiny loop-induced decays, the respective branching ratios of H for the three BPs are given in Table III. Here in both tables we consider branching ratios with three significant digits.

III. THE bg → bA → bZh PROCESS
We first analyze the prospect of discovering nominal Im(ρ bb ) required for EWBG via bg → bA → bZh process at HL-LHC. The process can be searched at the LHC via pp → bA + X → bZh + X [35] followed by Z → + − ( = e, µ) and h → bb i.e., in signature comprising of a pair of same flavor opposite sign leptons (denoted as the bZh process) and three b-tagged jets. The process requires that m A > m Z + m h . Therefore BPa for which m A < m Z + m h is out of the reach of LHC and we only focus on BPb and BPc. There exist several SM backgrounds such as tt+jets, Drell-Yan+jets (DY+jets), W t+jets, ttZ+jets, tth, tZ+jets, whereas subdominant contributions arise from four-top (4t), ttW , tW h, tW Z and W Z+jets. Backgrounds from W W +jets is negligibly small and hence not included. We remark that a search can also be performed via h → τ τ and h → γγ modes, however, they are not as promising as h → bb.
We generate the signal and SM background event samples at leading order (LO) in pp collision with √ s = 14 TeV CM energy by MadGraph5 aMC@NLO [36] (denoted as MadGraph5 aMC) with default NN23LO1 PDF set [37] then interface with Pythia 6.4 [38] for hadronization and showering and, finally fed into Delphes 3.4.2 [39] for the fast detector simulation incorporating the default ATLAS-based detector card. We follow MLM scheme [40,41] for the matrix element (ME) and parton shower merging. Note that we do not included backgrounds from the fake and non-prompt sources in our analysis which are typically determined from data and are not properly modeled in the Monte Carlo simulations. The effective model is implemented in FeynRules 2.0 [42] framework. The DY+jets background cross section is adjusted to the NNLO QCD+NLO EW one by a factor 1.27, which is estimated by FEWZ 3.1 [43,44], while the tt+jets background is corrected up to NLO by the K factor 1.36 [36]. We also normalize the LO ttZ,tZ+ jets, tth, 4t and ttW − (ttW + ) cross sections to NLO ones by the K-factors 1.56 [45] The LO W − Z+jets background is normalized to NNLO by a factor 2.07 [48]. We assume the same QCD correction factors for the charge conjugate processes tZj and W + Z+jets. The signal cross sections are kept at LO.
In order to find the prospect, we look for event topologies with same flavor opposite sign lepton pair and at least three b-tagged jets. To reduce backgrounds we apply following event selection cuts. The transverse momenta (p T ) of the leading and subleading leptons are required to be > 28 GeV and > 25 GeV respectively, while p T > 20 for all the three b-jets. The pseudo-rapidity (|η|) for all the leptons and b-jets are needed to satisfy |η| < 2.5. Moreover, the separation ∆R between the two leptons, any two b-jets and, a b-jet and a lepton should be ∆R > 0.4. The jets are reconstructed with antik T algorithm via default ATLAS-based detector card of Delphes 3.4.2. We veto events with missing transverse energy (E miss T ) > 35 GeV to reduce the tt+jets background. We further require that the invariant mass of the same flavor opposite charge lepton pair (m ) should remain within 76 < m < 100 GeV, i.e., in the Z boson mass window. The invariant mass of two b-jets m bb in an event to remain within |m h − m bb | < 25 GeV. As each event contains at least three b-jets more than one m bb combinations are possible; the one closest to m h is selected to pass the |m h − m bb | < 25 GeV cut. We finally require the invariant mass m bb constructed from the same flavor opposite charge lepton pair that pass the 76 < m < 100 GeV window and b-jets combination that passes the |m h − m bb | < 25 GeV selection to remain within |m A − m bb | < 80 GeV. Here we adopt the b-tagging and c-and light-jets rejection efficiencies AT-LAS based detector card of Delphes 3.4.2. The signal and background cross sections after the selection cuts for the BPb and BPc are summarized in Table. IV.
We now focus on the achievable significance at HL-LHC using the likelihood for a simple counting experiment [49] Z(n|n pr ) = −2 ln L(n|n pr ) L(n|n) ; L(n|n) = e −nnn n! , (8) where n and n pr are observed and predicted events. For discovery, the signal plus background (s + b) is compared with the background prediction (b) with the requirement Z(s+b|b) > 5, while for the exclusion we demand Z(b|s+ b) > 2 [49]. An evidence would require Z(s + b|b) > 3. Utilizing the signal and background cross sections in Table IV we find that the achievable significance is ∼ 2.9σ for BPb while ∼ 2.5σ for BPc with 3000 fb −1 integrated luminosity. Therefore we conclude that the discovery is beyond the scope of HL-LHC if the Im(ρ bb ) close to its nominal value 0.058 required for ρ bb -EWBG. We note that Im(ρ bb ) ∼ 0.15-0.2 is still allowed by current data for the sub-TeV mass range as we have already discussed in previous section. We also remark that in BPc, for which m A > 2m t , discovery is well within the HL-LHC if one considers Im(ρ bb ) ∼ 0.15. This is different from  TABLE IV. The signal and background cross sections (in fb) of the bZh process after selection cuts for the respective BPs at √ s = 14 TeV LHC. We have assumed |Im(ρ bb )| = 0.058 and set all other ρij = 0 for the signal process. The subdominant backgrounds 4t, ttW , tW h, tW Z and W Z+jets are added together and denoted as "Others". The total background yield (Total Bkg.) is given in the last column. a scenario discussed in Ref. [15] (also referred to as BPc there) in which B(A → Zh) is suppressed due to the dominance of B(A → tt) induced by |ρ tt | = 0.5, hindering the significance from reaching the discovery level. In this section we investigate the potential for eeAH process i.e., e + e − → Z * → AH production with H/A → bb decays in four b-jets signature for two different e + e − collision energy √ s = 500 GeV and 1 TeV. The signature would also receive contribution from e + e − → Z * → bbA/H process for A/H → bb decays, which we have included in our analysis. It is clear from Table I that BPa would be covered by √ s = 500 GeV while BPb and BPc would require √ s = 1 TeV. Although the environment is clean, there indeed exist some SM backgrounds for this process. The dominant backgrounds come from tt, four-jets (4j) which includes Zh productions, with subdominant contribution would arise from ZZ background. The events are generated as in previous section by MadGraph5 aMC followed by showering and hadronization in PYTHIA 6.4, and fed into Delphes 3.4.2 for detector effects. Here we incorporate the default international linear detector card (ILD) of Delphes 3.4.2 for jet reconstruction via anti-k T algorithm with radius parameter R = 0.5 and, for the btagging and misidentification efficiencies of c and lightjets. The events are selected such that it should contain at least four b-jets with all having p T > 20 GeV and |η| < 2.5. The separations between any two b-jets should be ∆R > 0.4. To reduce the backgrounds further, we demand the scalar sum of p T of all four b-jets (H T ) should be > 350 GeV for BPa, while for BPb and c we require  > 600 GeV. For illustration we show the normalized H T distributions in Appendix for BPa and BPb for √ s = 500 GeV and 1 TeV respectively. The signal and backgrounds after selection cuts for √ s = 500 GeV and 1 TeV are respectively summarized in Tables V and VI. We now estimate the significances from the cross sections summarized in Tables V and VI. It is clear that S/B ratios are considerably large for BPa and BPb for the considered CM energies. Utilizing Eq. (8) we find that BPa can be discovered at √ s = 500 GeV CM energy the with ∼ 250 fb −1 integrated luminosity with evidence emerging with as low as ∼ 80 fb −1 data. The BPb would require √ s = 1 TeV run and an evidence may come with 120 fb −1 but discovery needs 350 fb −1 dataset. The BPc is below the sensitivity of even √ s = 1 TeV lepton collider. Here for all three BPs the signal cross sections are estimated with Im(ρ bb ) = 0.058. Therefore we conclude that the nominal value for ρ bb -EWBG can be fully covered up to m A , m H 200 (400) GeV with moderate integrated luminosity in any future lepton collider if it runs with √ s = 500 GeV (1 TeV) CM energy.

B. The six b-jets signature
We now discuss a resonant diHiggs production e + e − → Z * → AH → Ahh in future e + e − colliders. We search this process in which both h decays to bb i.e. in six b-jets signature. Such final state would also receive contribution from process e + e − → AH → bbhh which we have considered as well. For the parameter space described in Table I only BPb and BPc can facilitate e + e − → Z * → AH → Ahh and e + e − → AH → bbhh since m H > 2m h . Note that discovery may already emerge from four b-jets signature discussed in previous subsection while six b-jets signature would provide complementarity for ρ bb -EWBG.
Based on the LO Hhh coupling given in Eq. (5), we first analyze the prospect of e + e − → AH → bbhh process with both h decays bb i.e., in six b-jets signature with all six b-jets having p T > 20 GeV and |η| < 2.4. Here we consider two different CM energy √ s = 500 GeV and 1 TeV for illustration. The CM energies considered would kinematically allow e + e − → AH → bbhh process only for BPb. For event generation we follow the same procedure as in e + e − → Z * → AH process i.e. generate events via MadGraph5 aMC followed by hadronization and showering in Pythia 6.4 and adopting default ILD card of Delphes for fast detector simulation. The corresponding cross sections √ s = 500 GeV (1 TeV) before application of any selection cuts reads as ∼ 0.001 (∼ 0.2) fb for BPb with |Im(ρ bb )| = 0.058. Following the above mentioned selection cuts, we find 0.0078 fb cross section for √ s = 1 TeV, but tiny 0.00003 fb for √ s = 500 GeV. In finding these cross sections we have normalized the B(h → bb) with the modified hbb coupling due to nonvanishing |Im(ρ bb )| = 0.058. While no statistically significant cross section is found for 500 GeV run, however one may have ∼ 8 (∼ 24) events with 1000 (3000) fb −1 integrated luminosity at √ s = 1 TeV. In SM, we find such six b-jets backgrounds to be negligibly small at e + e − collider, providing ideal environment for discovery of such signature. This should be compared with the discovery prospect discussed in Sec. IV for BPb via e + e − → Z * → AH process, which would require √ s = 1 TeV and ∼ 700 fb −1 data. In finding the six b-jets cross section here we have not included uncertainties arising from high b-jet multiplicity. Hence, we remark that our six b-jets cross sections should be treated as exploratory while a more detailed analysis including possible uncertainties arising in e + e − collider would be studied elsewhere.

The vertex correction for Hhh coupling at g2HDM
It is known that one-loop corrections to triple Higgs couplings could be sizable if EWPT is strongly first order [50] (for one-loop calculations to the hhh coupling, see also Ref. [51]). Here we clarify if this argument applies for our Hhh coupling. Dominant one-loop corrections in the c γ → 0 limit are cast into the form Remarkably, the loop correction would not vanish even in the exact alignment limit c γ = 0 due to the presence of the nonzero η 7 , which is in sharp contrast to softlybroken 2HDMs. In our three BPs, moreover, the loop corrections are constructive since η 7 is positive and treelevel λ Hhh is negative. In each case of BPs, we find that One can see that the one-loop corrections are larger than the tree-level values in BPb and BPc. However, this does not necessarily mean that perturbation breaks down since the tree-level Hhh coupling happens to be small by c γ , and moreover, some combinations of quartic couplings at one-loop level could be larger than those at tree level though each of quartic couplings is less than 4π as seen in Table I. As mentioned in Sec. II B, the tree-level unitarity is not violated either. We note that the H → ff (with f being fermions) decays are not expected to receive large enhancement from the one-loop corrections since the two of the three vertices in there are not Higgsself couplings (for h → ff decays, see, e.g., Ref. [52]). Therefore, B(H → hh) would be significantly increased at loop level, leading to much larger possibility for discovery at future lepton colliders.

V. DISCUSSION AND SUMMARY
We have analyzed the prospect of probing EWBG induced by additional bottom Yukawa couplings at the LHC and future e + e − colliders. We primarily focused on the nominal value |Im(ρ bb )| = 0.058 required for ρ bb -EWBG. We show that HL-LHC can offer some probe for such parameter space via bg → bA → bZh process if 300 m A 450 GeV. However, the discovery would be beyond even for HL-LHC. In this regard we show that future e + e − colliders such as ILC or FCCee would offer exquisite discovery prospect via e + e − → Z * → AH process at √ s = 500 GeV and 1 TeV. For parameter space where m A < m h + m Z , the bg → bA → bZh process kinematically insensitive but a 500 GeV run of any e + e − collider can discover the ρ bb -EWBG via e + e − → Z * → AH process with ∼ 250 fb −1 data. The discovery for the same process with heavier m A is also possible when 1 TeV or larger collision energies are available.
For complementarity, we also studied the prospect e + e − → AH → bbhh process in six b-jets signature. Based on our LO order Hhh coupling we found that 1 TeV e + e − collider can indeed discover such a process as long as m H ∼ 300 GeV. It should be noted that the Hhh coupling could get O(100%) one-loop correction owing to the sizable Higgs quartic couplings required by the strong first-order EWPT, increasing the significance for the discovery.
We now briefly discuss the impact of turning on other ρ ij couplings. Current direct and indirect searches still allow |ρ tt | ∼ 0.5 [15] for sub-TeV m A , m H and m H ± . Further ρ tc ∼ 0.3 is also allowed by direct and indirect searches and flavor physics [13]. A nonvanishing ρ tt motivates one to utilize the conventional gg → A/H → tt [53] and gg → ttA/H → tttt [54] gb →tH + →ttb searches [32,33]. For moderate values of ρ tt and ρ bb one may have gg → bA/H → btt signature which could be sensitive at the HL-LHC [15]. In this regard it should be reminded that complex ρ tt and ρ tc each can account for the observed BAU. Dedicated direct and indirect searches for ρ tc -and ρ tt -EWBG mechanism can be found in Refs. [7][8][9][10][11][12][13][14]. In general if such couplings are real they would not play any role in EWBG, however they would aggravate the signatures that we have discussed so far via suppression in the branching ratios of heavy bosons A/H. Nevertheless they would open up several additional direct and indirect probes. Furthermore moderate values of ρ τ τ is still allowed by current data though its impact is not as significant as ρ tt and ρ tc . We leave out a detailed discussion of EWBG driven by multiple ρ ij couplings and subsequent impacts on collider and flavor physics for future work.
As a first estimate, uncertainties arising from factorization scale (µ F ) and renormalization scale (µ R ) dependences are not included in our LO cross section estimations for bg → bA → bZh process. In general, the LO bg → bA processes have ∼ 25 − 30% scale uncertainties for m A ∼ (300 − 400) GeV as discussed in Ref. [55] (see also Refs. [33,[56][57][58]). In addition it has been found that [59] the LO cross sections calculated with LO PDF set CTEQ6L1 [60] have relatively higher factorization scale dependence. Therefore, we remark that the LO cross sections in our analysis, which we estimated with LO NN23LO1 PDF set, might have similar uncertainties. A reasonable choice of the factorization scale and renormalization scale has been proposed in Ref. [59], with µ R = m A and varied from µ R = m A /2 to µ R = 2m A , along with µ F = m A /4 and varied from µ F = m A /8 to µ F = m A /2. There also exist PDF uncertainties for bottom-quark initiated process as discussed in Ref. [61] (see also Ref. [62]). These would typically induce some uncertainties in our results which we leave out for future work.