Probing electroweak baryogenesis induced by extra bottom Yukawa coupling via EDMs and collider signatures

We study the prospect of probing electroweak baryogenesis driven by an extra bottom Yukawa coupling ρbb in a general two Higgs doublet model via electric dipole moment (EDM) measurements and at the collider experiments. The parameter space receives meaningful constraints from 125 GeV Higgs h boson signal strength measurements as well as several heavy Higgs boson searches at the Large Hadron Collider (LHC). In addition, we show that the asymmetry of the CP asymmetry of inclusive B → Xsγ decay would provide complementary probe. A discovery is possible at the LHC via bg → bA → bZh process if |ρbb| ∼ 0.15 and 250 GeV≲ mA ≲ 350 GeV, where A is CP odd scalar. For mA> 2mt threshold, where mt is the top quark mass, one may also discover bg → bA →btt¯\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ bt\overline{t} $$\end{document} at the high luminosity LHC run if an extra top Yukawa coupling |ρtt| ∼ 0.5, though it may suffer from systematic uncertainties. For completeness we study gg →tt¯A\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ t\overline{t}A $$\end{document}→tt¯bb¯\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ t\overline{t}b\overline{b} $$\end{document} but find it not promising.


Introduction
The existence of the matter-antimatter asymmetry is unswervingly established over the years by various cosmological observations such as the cosmic microwave background anisotropies and big-bang nucleosynthesis [1]. It has been understood that the Universe started with equal number of baryons and antibaryons, but later evolved into baryon dominated Universe dynamically via a mechanism called baryogenesis. A successful baryogenesis requires three necessary conditions namely, baryon number violation, charge conjugation (C) and charge conjugation-parity (CP) violation and, departure from thermal equilibrium, laid out by Sakharov in 1967 [2]. A plethora of baryogenesis scenarios have been proposed so far to account for the observed baryon asymmetry of the Universe (BAU), however, its origin is still unclear. After the discovery of the 125 GeV Higgs boson at the Large Hadron Collider (LHC) [3], a significant attention has been directed in particular to electroweak baryogenesis (EWBG) [4, 1 mechanism for its direct connections to Higgs physics and, its testability at the ongoing experiments. The Standard Model (SM) belongs to this class, however the CP violation is too small and, the electroweak symmetry breaking is not strongly first order phase transition (EWPT) to drive departure from thermal equilibrium.
While we do not have any strong experimental evidence of new physics yet, multi-Higgs sector is the natural consequence of most ultraviolet (UV) theories due to enlarged JHEP11(2020)025 symmetries. Whatever the fundamental theory might be, their effective descriptions at O(100) GeV scale should resemble the SM in light of the latest experimental results. As for the Higgs sector, two cases are conceivable: one is that all the new scalars are much heavier than O(100) GeV scale, thereby the Higgs sector is effectively reduced to the SM, while the other is that new scalars have O(100) GeV masses but their couplings to the gauge bosons and fermions are SM like, mimicking the SM. From the viewpoint of new physics discovery potential, it is timely to consider the latter case and investigate whether the aforementioned cosmological issue can be solved or not. Since we have already confirmed the existence of the Higgs doublet in nature, it is tempting to us to think of additional Higgs doublets in analogy with the fact that all the fermions come in three copies.
The general two Higgs doublet model (g2HDM) is one of the simplest renormalizable low-energy models where the scalar sector of the SM is extended by an extra scalar doublet. 2 Without the presence of discrete symmetry, in g2HDM, both the scalar doublets couple with up-and down-type fermions at tree level. In the mass eigenbasis of the fermions (F ), one has two independent Yukawa couplings λ F ij and ρ F ij , where the former is real and diagonal that are responsible for the fermion mass generation, while the latter is complex and non-diagonal. Such complex couplings can provide additional CP violating sources beyond the usual Cabibbo-Kobayashi-Maskawa (CKM) framework [92] of the SM.
EWBG in g2HDM is widely investigated in refs. [79][80][81][82][83][84][85][86]. This model can simultaneously accommodate the strong first-order EWPT and sufficient amount of CP violation which the SM fails to provide. The most natural EWBG scenario in g2HDM would be the case in which BAU is driven by the extra top Yukawa coupling (ρ tt ) of O(0.01 − 1) in magnitude with moderate size of the CP phase (ρ tt -EWBG) [84]. The devoted collider study of this scenario is conducted in ref. [93].
As a complementary study to ρ tt -EWBG, the present authors consider a scenario in which the CP phase of ρ tt is approximately zero and the extra bottom Yukawa coupling (ρ bb ) plays a dominant role in generating BAU (ρ bb -EWBG) [85]. It is demonstrated that BAU can reach the observed level if |Imρ bb | 0.058 with generous assumptions on a Higgs bubble wall profile. There exist several direct and indirect search constraints on the parameter space for ρ bb -EWBG such as h boson signal strength measurements, heavy Higgs searches at the LHC. The ρ bb -EWBG can be discovered at the LHC via bg → bA → bZH (or bg → bH → bZA) process if |Im(ρ bb )| ∼ O(0.1) [94]. However, the process requires that m A > m H + m Z and ρ tt to be negligibly small to avoid constraints from flavor physics [94]. In addition, for m A > 2m t , the process gets dilution from A → tt decay if ρ tt is nonvanishing. Also, it would be extremely difficult to probe the phase of ρ bb at the LHC since its information is lost in pp collision.
In this paper we show that the electron EDM measurement and asymmetry of CP asymmetry of the B → X s γ decay offer exquisite probes for Im(ρ bb ). We also analyze the prospect of discovery at the LHC. In particular we study the discovery potential of ρ bb -EWBG via bg → bA → bZh and bg → bA → btt processes at 14 TeV LHC. Purpose of this paper is to find possible direct and indirect signatures and correlation between them in probing the parameter space for the ρ bb -EWBG.

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Induced by ρ bb , the bg → bA → bZh process can be searched at the LHC via pp → bA + X → bZh + X (X is inclusive activities) followed by Z → + − ( = e, µ) and h → bb decays, constituting same flavor opposite sign dilepton pair and three b-tagged jets. While the bg → bA → bZh process can be induced ρ bb , the bg → bA → btt process requires both ρ bb and ρ tt to be nonvanishing. The latter process can be searched via pp → bA + X → btt + X with at least one top decays semileptonically, constituting three b-tagged jets, at least one charged lepton (e and µ) and missing transverse energy signature (denoted as 3b1 process). These processes provide the sensitive probes for the parameter space of ρ bb -EWBG, which is complementary to ref. [94].
For the sake of completeness we also investigate the discovery prospect of the gg → ttA → ttbb process, which is induced by nonzero ρ bb and ρ tt . At the LHC the process can be searched via pp → ttA + X → ttbb + X, with at least one top decaying semileptonically. As ρ tt gets involved in both bg → bA → btt and gg → ttA → ttbb, the processes would provide complementary probes also for ρ tt -EWBG.
In the following, we outline the formalism in section 2, followed by a detailed discussion on the available parameter space and potential indirect probes in section 3. We discuss the discovery potential of ρ bb -EWBG at the LHC in section 4 and summarize our results with some outlook in section 5.

Framework
The particle content of g2HDM is the SM plus additional Higgs doublet. In general, this model induces flavor-changing neutral current (FCNC) processes mediated by the neutral Higgs bosons at tree level. It is common to impose a Z 2 symmetry to suppress the FCNC processes to be consistent with various flavor physics data. Though this setup works well, having the Z 2 symmetry implies that the model has some specific UV theories such as supersymmetric models. Since we do not try to connect the model to any specific UV completions, we do not impose the Z 2 symmetry or something similar, which enables us to discuss physics at O(100) GeV scale in wider perspective. In this bottom-up approach, the tree-level FCNC processes are possible as long as the experimental data allow, and sources of CP violation are much richer than 2HDMs with some discrete symmetries.
The most general two Higgs doublet potential can be written in the Higgs basis as [95,96] Each Higgs doublet fields is expressed as

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where v( 246 GeV) is the vacuum expectation value, h is the SM-like Higgs boson, G 0,± are the Nambu-Goldstone bosons, H and A are the CP-even and -odd Higgs bosons, respectively, and H ± are the charged Higgs bosons. From the minimization condition with respect to Φ, it follows that µ 2 11 = −η 1 v 2 /2. For simplicity, we assume CP-conserving Higgs sector at tree level. 3 The second minimization condition with respect to Φ gives µ 2 12 = η 6 v 2 /2. The mixing angle γ between the CP-even bosons h and H satisfies the relations [96] An alignment limit is defined as c γ = 0 and s γ = −1, where c γ and s γ are shorthands for cos γ and sin γ respectively. One can express the masses of h, H, A and H ± in terms of the parameters in eq. (2.1): Note that in the alignment limit, one has m 2 h = η 1 v 2 and m 2 2 22 . In contrast to m h , the masses of the extra Higgs bosons are controlled by η i v 2 and µ 2 22 , where η i denotes some linear combinations of the η couplings. As is well known, magnitudes of the heavy Higgs loop contributions can become sizable if η i v 2 µ 2 22 , which is necessary for achieving the strong first-order EWPT.
Purpose of this paper is to probe the parameter space for EWBG driven by the extra bottom Yukawa ρ bb . It is found that a successful EWBG requires |Im(ρ bb )| 0.058 [85]. In the following we shall show that the parameter space receives meaningful constraints from several direct and indirect searches. The most stringent constraint on Im(ρ bb ) arises from electron EDM and ∆A CP of B(B → X s γ). In addition, coupling strength measurements of h and heavy Higgs searches at the LHC would also provide the complementary probes. In JHEP11(2020)025 addition to these constraints, we also study potential signatures at the LHC. We primarily focus on three searches at the 14 TeV LHC bg → bA → bZh , 4 gg → ttA → ttbb and bg → bA → btt (for the discussion on latter two processes see also refs. [105] 5 ). Induced by ρ bb the bg → bA → bZh process can be searched at the LHC if c γ is nonzero and m A > m h + m Z . On the other hand, gg → ttA → ttbb process requires ρ tt and ρ bb both nonvanishing with mild dependence on c γ . The final process bg → bA → btt also depends both on ρ bb and ρ tt but needs m A > 2m t . Together with electron EDM and ∆A CP of B(B → X s γ), these processes can probe significant part of the parameter space for ρ bb -EWBG.

Parameter space
Let us find the allowed parameter space for m A , m H and m H ± . The parameters in eq. (2.1) are required to satisfy perturbativity, tree-level unitarity and vacuum stability conditions, for which we utilized the public tool 2HDMC [112]. We express the quartic couplings η 1 , 22 , γ, and v as [95] The quartic couplings η 2 and η 7 do not enter scalar masses, nor the mixing angle γ. Therefore in our analysis we take v, m h , and γ, m A , m H , m H ± , µ 22 , η 2 , η 7 as the phenomenological parameters. Further, to save computation time, we randomly generated these parameters in the following ranges: JHEP11(2020)025 m h = 125 GeV with c γ = 0.1. 8 The randomly generated parameters are then fed into 2HDMC for scanning. 2HDMC utilizes [112] m H ± and Λ 1−7 as the input parameters in the Higgs basis whereas v 246 GeV. In order to match the 2HDMC convention, we identify η 1−7 as Λ 1−7 and, take −π/2 ≤ γ ≤ π/2. For the positivity conditions of the Higgs potential of eq. (2.1), the parameter η 2 > 0 along with other more involved conditions implemented in 2HDMC. We further conservatively demand |η i | ≤ 6.
Next we impose the stringent oblique T parameter [113] constraint, which restricts hierarchical structures among the scalar masses m H , m A and m H ± [114,115], and hence η i s. Utilizing the expression given in ref. [115] the points that passed unitarity, perturbativity and positivity conditions from 2HDMC, are further required to satisfy the T parameter constraint within the 2σ error [116]. 9 These points are denoted as "scanned points". We plot the scanned points in the m A -m H and m A -m H ± planes in the left and right panels of figure 1, which illustrates that significant amounts of the allowed points exists. A more detailed discussions on the scanning procedure can be found in refs. [118,119]. At this point, we have not yet required that EWPT should be strongly first order, and not all points are compatible with the ρ bb EWBG.
To find the constraints on ρ bb and ρ tt and, subsequently analyze the potential of future probes we choose three benchmark points (BPs) from the scanned points in figure 1, which are summarized in table 1. Here, we also demand that the chosen parameter sets give rise to the strong first-order EWPT. The BPa and BPb are chosen such that m A < 2m t . Since there is no suppression from B(A → tt), such a choice would enhance the discovery potential of bg → bA → bZh and gg → ttA → ttbb . For the BPc, where m A > 2m t , the bg → bA → btt process 10 can provide additional probes for the parameter space. Further, for all three BPs, A is assumed to be lighter than H and H ± to forbid A → ZH and JHEP11(2020)025  Table 1. Parameter values of three benchmark points chosen from the scanned points in figure 1, which are consistent with the strong first-order EWPT. A → H ± W ∓ decays and boost the discovery potential of these processes to some extent. Heavier A are indeed possible, but the cross sections are reduced due to rapid fall in the parton luminosity. We also remark that one requires [4][5][6][7][8][9][10][11][12][13][14] sub-TeV m A , m H ± and m H for the strong first-order EWPT, which is required for conventional sub-TeV EWBG  (for high-scale EWBG, see, e.g., refs. [87][88][89]). In the following we will scrutinize the relevant constraints on ρ bb and ρ tt . For simplicity, we assume that ρ ij except for ρ bb , ρ tt and ρ ee are negligibly small so as not to affect our main discussion. The impacts of nonzero ρ ij would be discussed later part of the paper.

Flavor constraints
There exist several constraints from flavor physics that restricts the parameter space. In particular, the following three observables are relevant: (i) the branching ratio measurement of B → X s γ (B(B → X s γ)), (ii) the asymmetry of the CP asymmetry between the charged and neutral B → X s γ decays (∆A CP ) and (iii) the B q -B q (q = d, s) mixings.
Let us first focus on B(B → X s γ). Non-zero ρ bb and ρ tt modify B(B → X s γ) via top quark and charged Higgs boson loop. The modification is parametrized by the (LO) 7,8 (y H + ) can be found in ref. [120]. The current world average of B(B → X s γ) exp extrapolated to the photon-energy cut E 0 = 1.6 GeV is found by the HFLAV Collaboration to be (3.32 ± 0.15) × 10 −4 [123]. The next-to-next-to LO (NNLO) B(B → X s γ) prediction in the SM for the same photonenergy cut is (3.36 ± 0.23) × 10 −4 [124]. In order to find the constraint, we adopt the prescription outlined in ref. [125] and define Based on our LO Wilson coefficients, we further express and take m W and m b (m b ) respectively as the matching scale and the low-energy scales. Finally, we demand R theory to remain within the 2σ error of R exp . In figure 2 the excluded regions are shown as the purple shaded regions in the Re(ρ bb )-Im(ρ bb ) plane for three BPs.
Here, we assume ρ tt = 0.5. Flavor constraints on ρ tt is moderately strong, with B d,s -B d,s mixings providing the most stringent constraint on ρ tt for 500 m H + 650 GeV, which is the ballpark mass ranges of m H ± for all the three BPs. The B q -B q mixing amplitude M q 12 receives modification from the charged Higgs and W bosons loop with t quark. Utilizing the expression for B q -B q mixing in type-II 2HDM [126], it is found in ref. [122] that with m t and m W being the masses of the top quark and W bosons. The expressions for I W W , I W H and I HH are respectively given by [122] Under the assumption on the ρ F ij couplings made in our analysis, we have M q 12 /M q SM = C Bq . Allowing 2σ errors on C B d and C Bs we find that B s,d -B d,s mixings exclude |ρ tt | 0.9 for BPa and BPb and, |ρ tt | 1 for BPc.
One of the most powerful probes of Im(ρ bb ) is the direct CP asymmetry A CP [129] of B → X s γ. It is advocated in ref. [130], however, that ∆A CP is even more sensitive to the CP-violating couplings, which is defined as [130] 78 . The largerΛ 78 would make the constraint stronger. We also remark that we utilize the LO Wilson coefficients in eq. (3.6) as a first approximation for simplicity. Note that the excluded regions by ∆A CP measurement in figure 2 is asymmetric and constrains positive Im(ρ bb ) more stringently. This is solely due to our choice of ρ tt = 0.5. If we take ρ tt = −0.5, the blue shaded regions would flip and exclude the negative regions of Im(ρ bb ).
We note in passing that if ρ tt is also complex, ∆A CP can be zero if the complex phases of ρ tt and ρ bb are aligned, i.e., Im(ρ tt ρ bb )=0, equivalently, Reρ bb /Reρ tt = −Imρ bb /Imρ tt . Such a phase alignment is discussed in ref. [86].

EDMs
The complex phase of ρ bb is severely constrained by EDMs of the electron, neutron, and atoms, etc. Currently, the most stringent experimental bound comes from EDM of thorium monoxide (ThO), which is approximately given by where d e is the electron EDM and C S is the coefficient of the nuclear spin-independent interaction (NSID), which are respectively defined as In our scenario, d e is predominantly induced by two-loop Barr-Zee diagrams as depicted in figure 3, which are decomposed into the three parts: where φ = h, H, A for the first two terms and φ = H ± for the last term. Let us denote the contribution of i-species to d φγ e as (d φγ e ) i . If ρ bb is the only source of CP violation, d e (d φγ e ) b . With Imρ bb required by ρ bb -EWBG mechanism, d e is so large that one cannot avoid the ACME18 bound as noted in ref. [85]. This fact suggests two options: (i) the alignment limit (c γ → 0) and (ii) cancellation mechanism. As discussed in ref. [86], however, the first option may not be consistent with EWBG in g2HDM since the BAU would be suppressed with decreasing c γ . We thus consider the second option. Even though we identify the parameter space for the cancellation in ref. [85], we do not show its detail there, and moreover, d φW e , which can come into play in the cancellation region, is missing. We therefore update our previous analysis taking all the relevant contributions into consideration.
If there exist more than two CP -violating phases, we could tune the parameters in such a way that d e becomes small. While it is nothing more than the parameter turning, we still classify the cancellation parameter space into two kind. We call a cancellation structured cancellation if it happens when the hierarchical structures of the ρ ij matrices JHEP11(2020)025 closely resemble those of the SM Yukawa matrices, and anything else is unstructured cancellation. It is revealed in ref. [86] that the parameter space of ρ tt -EWBG accommodates the structured cancellation. We here scrutinize the type of the cancellation in ρ bb -EWBG.
Following a method adopted in ref. [86], we split (d φγ e ) b into two parts as 12 with α em and Q f representing the fine structure constant and electric charges of f , respectively, and and g(τ ) are the loop functions and their explicit forms are shown in appendix A. In our notation, the sign of e is positive. In the wave parenthesis in eq. (3.23), the upper sign is for up-type fermions and the lower is for down-type fermions, respectively. For c γ 1 and In the ρ bb -EWBG scenario, Im(ρ ee ρ tt ) = ρ tt Im(ρ ee ). To make our discussion on the cancellation mechanism simpler, we consider a case in which Im(ρ ee ρ * bb ) 0 so that When ρ ee is nonzero, the primary contribution could be (d φγ e ) W as inferred from the fact that the φ-γ-γ vertex in d φγ e is more or less common to the h → 2γ decay. Noting that the W -loop has only the "mix" contribution since the Higgs couplings to the W bosons are the gauge couplings, one may find [133] (d Hγ where (3.27) It is found that c = 1.0 × 10 −3 for c γ = 0.1, ρ tt = 0.5, m h = 125 GeV and m H = 405 GeV. Therefore, the cancellation is possible but unstructured since c deviates much from the unity JHEP11(2020)025 as opposed to the ρ tt -EWBG scenario [86]. Once this accidental cancellation happens, other contributions could become relevant. On the grounds of dimensional analysis, one can find that d φZ e is suppressed by the Z boson coupling to the electron, g Zee = 1/4 − sin θ W 0.02 with θ W representing the weak mixing angle, while d φW e is not and becomes leading contribution. The dominant contribution in d φW e comes from the diagrams involving the top and bottom loops, which amounts to [86,134,135] (d φW where m b = 0 and V tb is the (33) element of the CKM matrix, which is close to one [1]. J 1 is the loop function listed in appendix A. In general, this contribution has the ρ bb dependence but vanishes in the case of m b = 0. Note that (d φW e ) t/b is absent in the softly-broken Z 2 2HDMs. For one of the ρ bb -EWBG parameter points, e.g., Im(ρ bb ) = −0.15, one would get (d φW e ) t/b 1.2 × 10 −29 e cm in the cancellation region specified by eq. (3.27), together with m H ± = 391 GeV and ρ tt = 0.5, which slightly exceeds the ACME18 bound. Therefore, the allowed region is not exactly determined by the cancellation condition but it occurs in its vicinity, as we show in our numerical analysis conducted below. It should be noted that even though ρ tt is real in ρ bb -EWBG, its magnitude can be constrained by the electron EDM due to the proportionality of ρ tt Im(ρ ee ). Now we move on to discuss the C S contribution. We estimate C S using the CPviolating 4-fermion interactions between the quarks and electron defined as where C qe = φ=h,H,A g S φqq g P φēe /m 2 φ (explicit forms of g S φqq and g P φēe are shown in appendix A) With those, C S is estimated as [136] Note that for c γ 1 and m H m A , C qe for up-and down-type quarks are, respectively, cast into the form [86] Therefore, the dependences of the CP-violating phases are the same as those of (d φγ e ) extr u and (d φγ e ) extr d , respectively. In our numerical analysis, we parametrize ρ f f , except for ρ tt , as Re(ρ ee ) = −r(λ e /λ b )Re(ρ bb ) and Im(ρ ee ) = −r(λ e /λ b )Im(ρ bb ). Though the CP-violating phases in the first and second generations of ρ F matrices have nothing to do with ρ bb -EWBG, we fix them through the above relations. However, the effects of the extra CP violation are too small to affect our cancellation mechanism in ρ bb -EWBG.
In figure 4, |d ThO | and its details are shown as functions of r. We take BPa for the Higgs spectrum and set c γ = 0.1, Re(ρ bb ) = 0, Im(ρ bb ) = −0.15 and ρ tt = 0.5 as an example JHEP11(2020)025  of the ρ bb -EWBG scenario. As seen, the magnitude of α ThO C S is much smaller than that of d e , we thus can use the ACME18 bound of |d e | < 1.1 × 10 −29 e cm, which is represented by the horizontal dotted line in black, to constrain the parameter space. As discussed in eq. (3.27), the cancellation happens in d φγ e at around r 1×10 −3 , which is the consequences of (d φγ e ) t + (d φγ e ) b + (d φγ e ) W 0. At this point, d φW e becomes dominant and |d e | exceeds the ACME 18 bound. Nevertheless, the cancellation is still at work at around r 0.7 × 10 −3 . Similar to this case, we can always find cancellation regions in the cases of BPb and BPc as well, and thus conclude that ρ bb -EWBG scenario is still consistent with the ACME18 bound. Note that here we set c γ = 0.1 while finding the constraints from ACME18 to illustrate Im(ρ bb ) ∼ 0.15 is still allowed for ρ tt ∼ 0.5 for all the three BPs. In the rest of the paper, however, we ignore the c γ dependence since it is insensitive to our collider study.
Here, we briefly discuss the EDMs of neutron and Mercury. Their current experimental values are respectively given by [137,138]  On the theoretical side, the neutron EDM based on QCD sum rules is estimated as [139],

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where g 3 is the SU(3) C gauge coupling and d C q are the quark chromo EDMs defined by the operator L CEDM = −(i/2)d C q G µνq σ µν γ 5 q with G µν representing SU(3) C field strength tensor. We note that even though the cancellation mechanism can work in d n as well, it does not occur at the cancellation point of d e . Using the same input parameters as in figure 4 with r = 0.7 × 10 −3 , we obtain |d n | = 2.4 × 10 −29 e cm, which is nearly 3 orders of magnitude below the current bound. For the mercury EDM, on the other hand, we estimate it using formulas in refs. [140][141][142] assuming d I Hg defined in ref. [141] and find that |d Hg | = 8.0 × 10 −31 e cm, which is smaller than the current bound by about 1 order of magnitude.
We note in passing that a future measurement of the proton EDM could be another good prober of ρ bb -EWBG. The experimental sensitivity of the proton EDM at IBS-CAPP [143] and BNL [144] is |d p | ∼ 10 −29 e cm. As is the case of neutron EDM discussed above, the proton EDM can be estimated by use of the QCD sum rules as [139] With this, it is found that |d p | = 6.1 × 10 −29 e cm for the parameters used in figure 4 with r = 0.7 × 10 −3 . Therefore, the future measurement of d p could access the ρ bb -EWBG parameter space regardless of the d e cancellation.

Direct search limits
There exist several direct search limits from ATLAS and CMS that may restrict the parameter space of ρ bb , even for c γ = 0 and ρ tt = 0. The coupling ρ bb receives several constraints from heavy Higgs boson searches at the LHC. In particular, refs. [145][146][147][148][149] are relevant to our study. We find that the most stringent constraint arises from CMS search involving heavy Higgs boson production in association with at least one b-jet and decaying into bb pair based on 13 TeV 35.7 fb −1 data [145]. The CMS search provides a model independent 95% CL upper limits on the σ(pp → bA/H + X) · B(A/H → bb) in the mass range beginning from 300 GeV to 1300 GeV. We first extract 13 corresponding 95% CL upper limit σ(pp → bA/H + X) · B(A/H → bb) for our three BPs. Taking a reference |ρ bb | value, we then estimate the production cross sections of pp → bA/H + X at the leading order (LO) utilizing Monte Carlo event generator MadGraph5_aMC@NLO [150] (denoted as Mad-Graph5_aMC) with the default parton distribution function (PDF) NN23LO1 set [151] for the BPs. As the analysis does not veto additional activity in the event [145], we therefore include contributions from gg → bbA/H along with bg → bA/H while estimating the cross sections. These cross sections are finally rescaled by |ρ bb | 2 × B(A/H → bb), assuming B(A/H → bb) = 100%, to obtain the corresponding 95% CL upper limits on |ρ bb |. It is found that |ρ bb | 0.6 is excluded for BPa at 95% CL and likewise, the regions where |ρ bb | 0.7 are ruled out for both BPb and BPc. These upper limits are rather weak and would be further weakened by B(A → Zh) and B(A/H → tt). The limits are even weaker from a similar search performed by ATLAS [146]. We note that while estimating the upper limit JHEP11(2020)025 on ρ bb we set all ρ ij = 0 for simplicity. In general, we remark that nonzero ρ ij would further alleviate these upper limits. Further, ρ bb coupling can induce pp → t(b)H ± process which is proportional to V tb (see eq. (2.7)). These processes are extensively searched by ATLAS [147] and CMS [148,149] with H + /H − → tb/tb decays. We find that the constraints are weaker for all the three BPs, however, as we see below these searches would provide sensitive probe to ρ tt . The effective model is implemented in the FeynRules 2.0 [152] framework.
We now turn to constraints on ρ tt . As ρ tt can also induce V tb , the searches pp →t(b)H + followed by H + → tb [147][148][149] would also be relevant. The ATLAS search [147] is based on 36 fb −1 √ s = 13 TeV dataset, which provides model independent 95% CL upper limit on σ(pp →tbH + ) × B(H + → tb) from m H ± = 200 GeV and 2 TeV. Similar searches are also performed by CMS based on √ s = 13 TeV 35.9 fb −1 dataset [148,149]. These searches provide 95% CL upper limit on σ(pp →tH + ) × B(H + → tb) for m H ± = 200 GeV and 3 TeV in leptonic [148] and, combining leptonic and all-hadronic final states [149]. Like before, the nonvanishing ρ tt enhanced by V tb can induce such process, leading to stringent constraints. To find the constraints, as done before, we calculate the cross sections σ(pp → tbH + ) × (H + → tb) at LO for a reference |ρ tt | for the three BPs via MadGraph5_aMC. These cross sections are then rescaled by |ρ tt | 2 × B(H + → tb) to get the corresponding 95% CL upper limits on |ρ tt |. The extracted (footnote 13) 95% CL upper limits from ATLAS search [147] on ρ tt for the three BPs are |ρ tt | 0.7, 0.8 and 1, respectively, while the limits from CMS [149] are much stronger, which read as |ρ tt | 0.6, 0.61 and 0.61, respectively. We remark that the constraints from CMS search with leptonic final state [148] is mildly weaker than the search with combined leptonic and all-hadronic final states [149]. We also note that all the ρ ij except for ρ tt are assumed to be zero when extracting the upper limits for the sake of simplicity. Therefore if other ρ ij are turned on, the limits on ρ tt in general becomes weaker due to dilution from other branching ratios of H ± .
The ATLAS [106] and CMS [107] search for heavy Higgs via gg → H/A → tt would also constrain ρ tt . The ATLAS [106] result is based on 20.3 fb −1 data at 8 TeV, which provides exclusion limits on tan β vs m A (or, m H ) in type-II 2HDM framework starting from m A and m H = 500 GeV. The CMS search is based on 35.9 fb −1 data at √ s = 13 TeV, which provides upper limit on coupling modifier (see ref. [107]  Moreover, ρ tt would also receive constraint from CMS search for SM four-top production [153]. The search is performed with 13 TeV 137 fb −1 dataset and provides 95% CL upper limits on σ(pp → ttA/ttH) × B(A/H → tt) for 350 GeV ≤ m A/H ≤ 650 GeV. The search also includes contributions from σ(pp → tW A/H, tqA/H) followed by A/H → tt, which can also be induced by ρ tt . To understand how strong the constraints could be, we generate these cross sections at LO by MadGraph5_aMC for a reference value of |ρ tt | setting all other ρ ij = 0, and then rescale simply by |ρ tt | 2 × B(A/H → tt). Again, given the masses of BPs, the σ(pp → ttA) × B(A → tt) search can constrain only BPc, for which |ρ tt | 0.8 excluded at 95% CL. However, σ(pp → ttH) × B(H → tt) can constrain all three BPs, and we find that the regions of |ρ tt | 0.9, 0.8 and 1 are excluded at 95% CL, respectively, where B(A/H → tt) = 100% is assumed. However, it should be noted that the presence of B(A/H → bb) and B(A → Zh) would alleviate the limits. As in the case of gg → H/A → tt, the search here also assumes that m A and m H are decoupled from each other. Therefore, one expects the limits to be mildly stronger for all the three BPs.
We finally conclude that for Re(ρ bb ) ∼ 0, c γ = 0.1, |Im(ρ bb )| ∼ 0.15 and ρ tt ∼ 0.5 are well allowed by the current measurements for the mass spectrum under consideration. We take these values as representative values for our analysis with Re(ρ bb ) = 0. 14 Under the assumptions, i.e., setting all ρ ij = 0 except ρ bb and ρ tt , the total decay width of A can be nicely approximated as the sum of the partial widths of A → bb and A → Zh for BPa and BPb and, A → bb, A → Zh and A → tt for BPc. The total decay widths of A are 0.4 GeV, 0.5 GeV and 4.97 GeV respectively for the three BPs with |Im(ρ bb )| ∼ 0.15 and ρ tt ∼ 0.5. The corresponding branching ratios are presented in table 2. Note that when calculating decay widths and branching ratios of A, we neglect tiny loop induced decays such as A → γγ, A → Zγ etc.

The bg → bA → bZh process
In this subsection we study the discovery potential of bg → bA → bZh process at 14 TeV LHC. The process can be searched via pp → bA+X → bZh+X followed by Z → + − ( = e, µ) and h → bb, comprising three b-jets, same flavor opposite sign lepton pair (denoted 14 As discussed before, both ρ bb and ρtt can induce pp → t(b)H ± process, hence, the constraints from ref. [149] would become stronger if both the couplings are nonvanishing. However, we have checked that ρtt ∼ 0.5 is allowed for |Im(ρ bb )| ∼ 0.15 for all the three BPs. as the bZh process). There are several SM backgrounds for this final state topology. The dominant backgrounds are tt+jets, Drell-Yan+jets (DY+jets), W t+jets, ttZ+jets, tth, tZ+jets, with subdominant contributions arise from four-top (4t), ttW , tW h, tW Z and W Z+jets. Backgrounds from W W +jets and ZZ+jets are negligibly small and hence not included. Note that one can also search for h → γγ or h → τ τ ; however, we do not find them as promising.
The signal and background event samples are generated in pp collision with √ s = 14 TeV CM energy at LO by MadGraph5_aMC with NN23LO1 PDF set as done before and then interfaced with Pythia 6.4 [154] for hadronization and showering and finally fed into Delphes 3.4.2 [155] for fast detector simulation adopting default ATLAS-based detector card. We adopt MLM scheme [156,157] for matrix element and parton shower merging. Note that we have not included backgrounds from the fake and non-prompt sources in our analysis. Such backgrounds are not properly modeled in Monte Carlo simulations and requires data to estimate such contributions.
The LO tt+jets and W t+jets cross sections are normalized to NNLO (next-to-next-to LO) with NNLL (next-to-next-to leading logarithmic) corrections by factors 1.84 [158] and 1.35 [159] respectively. We normalize the DY+jets background cross section to the NNLO QCD+NLO EW one by factor 1.27, which is obtained by utilizing FEWZ 3.1 [160,161]. The LO ttZ,tZ+ jets, tth, 4t and ttW − (ttW + ) cross sections are adjusted to NLO ones by K-factors 1.56 [162] [164] respectively, but tW Z and tW h both are kept at LO. Finally, the background W − Z+jets is normalized to NNLO by factor 2.07 [165]. For simplicity we assume the same QCD correction factors for the conjugate processes tZj and W + Z+jets, while the signal cross sections are kept at LO.
To reduce backgrounds, we adopt following event selection criteria: each event should contain a same flavor opposite sign lepton pair and at least three b-tagged jets. The transverse momenta (p T ) of the leading and subleading leptons should be > 28 GeV and > 25 GeV respectively, whereas p T for all three b-jets should be > 20 GeV. The pseudorapidity (|η|) of the leptons and all three b-jets are required to be < 2.5. reconstructed by anti-k T algorithm with radius parameter R = 0.4. The separation ∆R between any two b-jets, a b-jet and a lepton and between two leptons should be > 0. 4. In order to reduce the tt+jets background, we veto events having missing transverse energy (E miss T ) > 35 GeV. The invariant mass of the two same flavor opposite charge leptons (m ) is needed to remain between 76 < m < 100 GeV, i.e., the Z boson mass window. We then apply invariant mass for two b-jets m bb in a event. As there are at least three b-jets in a event, more than one m bb combinations are possible; the one closest to m h is selected and JHEP11(2020)025 required to remain within |m h − m bb | < 25 GeV. Further, we require the invariant mass m bb constructed from the two same flavor opposite charge leptons and b-jets combination that passes the m bb selection to be within |m A − m bb | < 50 GeV. The normalized m and m bb distributions before application of any selection cuts are presented in figure 5 while the same for m bb is shown in figure 6. We adopt the b-tagging efficiency and c-and light-jets misidentification efficiencies of Delphes ATLAS based detector card. The background cross sections after selection cuts of the three benchmark points are summarized in table. 3, while the signal cross sections along with their corresponding significances with the integrated luminosity L = 300 and 1000 fb −1 are presented in table 4. The statistical significances are estimated using Z = 2[(S + B) ln(1 + S/B) − S] [167], where S and B are the numbers of the signal and background events.
Let us take a closer look at table 4. We find very promising the discovery potential with sufficiently large S/B ratio, especially for m A < 2m t . The achievable significance for the BPa and BPb are ∼ 2.9σ (∼ 5.3σ) and ∼ 5.4σ (∼ 9.8σ) respectively with 300 (1000) fb −1 integrated luminosity. The BPc requires larger dataset due to fall in parton luminosity and suppression from B(A → tt) decay and ∼ 2.2σ is possible with 1000 fb −1 but could reach up to ∼ 3.8σ with the full high luminosity LHC (HL-LHC) dataset (3000 fb −1 integrated luminosity).

The bg → bA → btt process
We now discuss the discovery potential of pp → bA+X → btt+X, followed by semileptonic decay of at least one top quark, constituting three b-jets, at least one charged lepton (e and µ) and missing transverse energy (E miss T ) signature, which we denote as 3b1 signature. Note that bg → bA → btt is only possible for BPc as m A < 2m t but for BPa and BPb one can have 3b1 signature via pp → bH + X → btt + X. However, such signatures will be mild for the former two BPs due to suppression from B(H → AZ). Further, bg →tH + →ttb process may also contribute to the same final state topologies, if at least one of the top decays semileptonically. Such contribution could be moderate for all three BPs. In our analysis, however, we neglect them for simplicity.
There exist several SM backgrounds. The dominant backgrounds are tt+jets, t-and schannel single-top (tj), W t, with subdominant backgrounds from tth and, ttZ productions. Further, small contributions come from Drell-Yan+jets, W +jets, four-top (4t), ttW , tW h, which are collectively denoted as "Others". We do not include backgrounds originating from non-prompt and fake sources. These backgrounds are not properly modeled in Monte Carlo event generators and one requires data to estimate such contributions.
Here we follow the same event generation procedure for signal and backgrounds as in previous subsections, i.e., via MadGraph5_aMC followed by hadronization and showering in Pythia and with Delphes ATLAS based detector card for fast detector simulation. The LO tt+jets background cross section is normalized up to the NNLO by a factor of 1.84 while t-and s-channel single-top cross sections are normalized by factors of 1.2 and 1.47, respectively [166]. The LO W t+jets background is normalized to the NLO cross section by a factor of 1.  as that between any two b-jets are required to be greater than 0.4. The E miss T in each event should be larger than 35 GeV. Note that in our exploratory study we do not optimize the selection cuts such as p T , η, E miss T and H T for simplicity. The signal and total background cross sections along with different components after the selection cuts are shown in table 5.
The corresponding significance Z for the integrated luminosities 300 (3000) fb −1 for the BPc is ∼ 3.5σ (∼ 7.9σ). For the BPa and BPb from pp → bH + X → btt + X are ∼ 2.4σ and ∼ 2.8σ respectively for 3000 fb −1 . Note that here we have not considered the systematic uncertainties associated with the backgrounds, which could be considerable in particular for the largest tt+ jets backgrounds. As S B for the 3b1 process, in the presence of systematic uncertainties the Z formula simply becomes Z ≈ S/ B + σ 2 B . The σ B denotes the systematic uncertainty that depends on the factor between control sample and the background in the signal region. The value of σ B is very much analysis 15 dependent. Here we assume that the systematic uncertainty arise only from tt+ jets backgrounds for simplicity and neglect the same for other subdominant backgrounds and take two different for σ B for illustration. E.g., if σ B = √ B the significance of BPc reduces to ∼ 5.5σ whereas for σ B = 0.1 B the significances goes below 1σ with HL-LHC dataset. Similarly, for σ B = √ B the significances of BPa and BPb from pp → bH +X → btt+X reduces to ∼ 1.7σ and ∼ 2σ respectively but much below 1σ if σ B = 0.1 B. Therefore we remark that the 3b1 process is promising, but one needs precise understanding of the background systematics.

The gg → ttA → ttbb process
We now briefly discuss the discovery potential of gg → ttA → ttbb process. The process can in principle probe the parameter space for ρ bb -EWBG mechanism. We search this process via pp → ttA + X → ttbb + X, followed by at least one top quark decaying semileptonically i.e., with four b-jets, at least one charged lepton and E miss T signature. The final state topology receives mild contributions from inclusive pp → W tA + X and pp → tjA + X processes. As we show below, the signature is not promising as opposed to bg → bA → btt process.
We generate events at LO as in 3b1 process, i.e., via MadGraph5_aMC followed by hadronization and showering in Pythia and finally incorporate the detector effects of Delphes ATLAS based detector card. The dominant backgrounds arise from the tt+ jets, Single-top and W t+ jets, whereas tth, ttZ and 4t constitute subdominant backgrounds. We assume the same QCD corrections factor as in 3b1 process for simplicity.
To reduce the background, we use the following event selection cuts. The events are selected so that they contain at least one lepton (e and µ), at least four jets with at least four are b-tagged and some missing E miss T (denoted as 4b1 process). The lepton is required to have p T > 25 GeV and |η| < 2.5. For any jet in the event p T > 20 GeV and |η| < 2.5. E miss T in each event is required to be greater than 35 GeV. The separation ∆R between any two jets as well as that between a jet and a lepton should be larger than 0.4. Finally, we construct all possible combinations of the invariant mass m jj from the four leading jets and demand that the one closest to m A should lie between |m A − m jj | < 50 GeV. The impact of these cuts on the signal and background processes are summarized in table 6.
We find that the achievable significance for all three BPs of the gg → ttA → ttbb process are below ∼ 1σ with 3000 fb −1 integrated luminosity, which is rather low. This means that no meaningful constraints can be extracted unless both ATLAS and CMS data are added. It should be remarked that since we use the same QCD correction factors for the backgrounds as in bg → bA → btt process, there are greater uncertainties in these cross sections.
Before closing this section, we discuss the impact of the other ρ ij couplings. So far, we have set all ρ ij = 0 except ρ bb and ρ tt . Presence of the other ρ ij couplings open up other decay modes of A, which in principle may reduce the achievable significances summarized in previous subsections. For instance, if ρ τ τ ∼ λ τ , it would induce A → τ + τ − decay. However, JHEP11(2020)025 the significances remain practically same for all the three BPs. Moreover, ρ tc ∼ 0.3-0.4 is still allowed by current data and would induce cg → tA/tH → ttc (same-sign top) [93,168] (see also refs. [169,170] 16 ) and cg → tA/tH → ttc (triple-top) signature, which might emerge in the Run-3 of LHC.

Discussion and summary
We have analyzed the available constraints and prospect of probing EWBG driven by the extra bottom Yukawa coupling ρ bb at the ongoing and future experiments. The parameter space receives meaningful constraints from h boson coupling measurements, B(B → X s γ), ∆A CP of B → X s γ , electron EDM measurement and heavy Higgs searches at the LHC. We primarily focused on sub-TeV m A , m H and m H ± with mixing angle c γ ∼ 0.1, which is required by EWBG 85]. The constraints would be improved, e.g., on c γ , ρ bb and ρ tt at the HL-LHC [85,174] if no discovery is made. This would allow us to probe even larger part of the parameter space of ρ bb -EWBG.
Taking three benchmark points for illustration, two below 2m t threshold and one above, we have shown that a discovery is possible at the LHC via ρ bb induced bg → bA → bZh process for m A < 2m t . We find that the process may emerge in the Run 3 of LHC if 250 GeV m A 350 GeV. With a simple rescaling of the significances in table 4, we find that |Im(ρ bb )| 0.05 and 0.04 can be excluded for BPa and BPb with full HL-LHC dataset. Those are below the nominal value |Im(ρ bb )| 0.058 required EWBG. For m A > 2m t , a discovery may happen via bg → bA → btt . Note that one may also have the bg → bH → bhh process, which could be sensitive if H is lighter than A and H ± . The process is being studied elsewhere. A discovery of the bg → bA → btt process is possible via 3b1 signature if m A ∼ 450 GeV and ρ tt ∼ 0.5 but requires controlling of the systematics of tt-jets background. Additionally, we have also investigated the potential for gg → ttA → ttbb process but find it below the sensitivity even at the HL-LHC.
In principle the pp → bA+ X process can also be induced by ρ bd , ρ db , ρ bs and ρ sb at the LHC. However, due to severe constraints arising from B d and B s mixings [176] their impacts are typically inconsequential. In addition if the charm quark gets misidentified as b-jet, a sizable ρ cc can also mimic similar signature in pp collision via cg → cA → ctt process. We remark that such possibilities can be disentangled by the simultaneous application of band c-tagging on the final state topologies [177].
Although the discovery is possible at the LHC, to attribute it to ρ bb -EWBG mechanism is beyond the scope of LHC as information of the CP-violating phase of ρ bb is lost in pp collision. In this regard, ∆A CP of B(B → X s γ) would provide very sensitive probe for the Im(ρ bb ) even though the observable has uncertainties associated with the hadronic parameterΛ 78 . While finding the constraints in figure 2, we assumedΛ 78 = 89 MeV, which is the average of 17 MeV ≤Λ 78 ≤ 190 MeV [130]. However, ifΛ 78 is taken as its upper range, the constraint becomes much severe for Im(ρ bb ). Furthermore, on the experimental side, projected Belle II accuracy of ∆A CP measurement is ∼ 5% [175]. Therefore, more precise estimation ofΛ 78 together with Belle II measurement can stringently probe the parameter JHEP11(2020)025 space of Im(ρ bb ) unless the CP-violating phases of ρ tt and ρ bb are aligned [86] in which ∆A CP = 0. In such a case, measurements of EDMs play a pivotal role in probing Im(ρ bb ).
The unprecedented electron EDM constraint set by ACME Collaboration in 2018 reduces most EWBG scenarios to despair. We updated our previous analysis done in ref. [85] including all the relevant Barr-Zee diagrams. Because of the significant contributions arising from the diagrams involving ρ ee , the cancellation mechanism can be effective. It was found that the electron EDM cancellation in ρ bb -EWBG belongs to the unstructured cancellation category in which the diagonal hierarchical structures of ρ F ij are much different from those of the SM Yukawa couplings, which is in stark contrast to the case in ρ tt -EWBG that can accommodate the structured cancellation [86]. Nonetheless, the viable parameter space of ρ bb -EWBG still exists. Besides the extreme fine tuning of the parameters, ρ bb -EWBG would be confirmed or ruled out if the electron EDM is improved down to ∼ 10 −30 e cm level. Moreover, as discussed in section 3.2, the future measurement of the proton EDM could play a complementary role in probing ρ bb -EWBG.
So far we have not discussed the uncertainties. 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. In general, the LO bg → bA/bH processes have ∼ 25−30% scale uncertainties for m A/H ∼ (300−400) GeV if bottom quark with p T > (15− 30) GeV and, |η| < 2.5 [178] (see also [179][180][181]). It has been found that [182] the LO cross sections calculated with LO PDF set CTEQ6L1 [183] 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 same level of uncertainties. It has also been found that [182] for µ F ≈ m A (or m H ) the corrections to the LO cross sections could be large negative (∼ −70%), whereas for the choice of µ F ≈ m A /4 (or m H /4) the corrections are mild; which indicates that the µ F ≈ m A /4 is the relevant factorization scale. Furthermore, the cross section uncertainties from factorization scale and renormalization choices are found to be particularly small at µ R = m A and if 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 [182]. In addition, our analysis does not include PDF uncertainties, which could be in general significant for any bottom-quark initiated process as discussed, e.g., in ref. [184]. Detailed discussions on different PDFs and associated uncertainties for the LHC are summarized in ref. [185]. These typically would induce some uncertainties in our results. We leave out the detailed estimation of these uncertainties for future work.
In summary, we have explored the possibility of electroweak baryogenesis induced by the extra bottom Yukawa coupling ρ bb via direct and indirect signatures at the collider experiments. We find that the discovery is possible at the high luminosity LHC if 250 GeV m A 350 GeV. We also find that heavier mass ranges can also be probed via bg → bA → btt but the process is associated with larger uncertainties. While LHC can indeed discover the process, however, the information of the CP-violating phase of ρ bb can only be probed via ∆A CP of B(B → X s γ) or the EDM measurements of the electron, neutron and mercury though the latter two have the less probing power to date. For completeness we also studied gg → ttA → ttbb process and found that it is not promising. In conclusion, together with the electron EDM measurement and/or ∆A CP of B(B → X s γ) JHEP11(2020)025 decay, the discovery of bg → bA → btt process may help us to understand the mechanism behind the observed matter-antimatter asymmetry of the Universe.