Further study of the global minimum constraint on the two-Higgs-doublet models: LHC searches for heavy Higgs bosons

The usually considered vacuum of the two-Higgs-doublet model (2HDM) could be unstable if it locates at a local but not global minimum (GM) of the scalar potential. By requiring the vacuum to be a GM, we obtain an additional constraint, namely the GM constraint, on the scalar potential. In this work, we explore the GM constraint on the $CP$-conserving general 2HDM. This constraint is found to put limits on the soft $\mathbb{Z}_2$ breaking mass parameter $m_{12}^2$ and also squeeze the heavy $CP$-even Higgs boson mass into larger values for the $m_{12}^2<0$ case. Combined with the current global signal fits from the LHC measurements of the 125 GeV Higgs boson, we discuss the phenomenological implications for the heavy Higgs boson searches at the LHC.


I. INTRODUCTION
In the studies of new physics beyond the Standard Model (BSM), it is quite often that one has an extended Higgs sector. A simple and well-known example is the two-Higgs-doublet model (2HDM) 1 , which was motivated from several different aspects, such as Supersymmetry [2,3], CP violation [4], and axion models [5]. With an additional Higgs doublet introduced, the Higgs potential in the 2HDM may develop several different minima.
To avoid the vacuum instability, one may impose a global minimum (GM) condition for the desired vacuum.
This leads to new constraints on the Higgs potential, in addition to the conventional bounded-from-below (BFB) constraints and the unitarity bounds. Recently, the GM condition of the 2HDM potential has been analytically formulated in Ref. [28] and tentatively applied to constrain the general 2HDM. It has been demonstrated that the GM constraint can sometimes be robust in constraining the parameter space of the 2HDM. 2 In this work, we further study the GM constraint on the 2HDM, with the focus on the phenomenological implications at the LHC. It turns out that the GM condition is likely to put constraints on the masses of heavy Higgs bosons and the soft Z 2 breaking scale of m 2 12 in the 2HDM. In turns, these constraints are directly connected to the Higgs self-couplings in the 2HDM. From the experimental point of view, the Higgs self-couplings are likely to be probed by the high-luminosity (HL) and/or high-energy (HE) LHC runs, by looking for the Higgs boson pair productions. Since the discovery of the 125 GeV Higgs boson at the LHC, a lot of efforts have been made in probing such processes in different new physics models at the LHC [31][32][33][34][35][36][37][38][39][40][41][42][43][44][45][46][47][48].
The layout of this paper is described as follows. In Sec. II, we revisit the CP -conserving general 2HDM, where we put emphasis on the GM constraint on the 2HDM potential. This constraint, together with the usual tree-level BFB and perturbative unitarity constraints, will be imposed on the 2HDM parameter space. In Sec. III, we consider benchmark models in two different scenarios, namely, the degenerate heavy Higgs boson scenario of M A = M H = M ± , and the heavy Higgs boson spectrum involving exotic decays. It turns out that the GM condition leads to additional restrictions on the parameter space. In Sec. IV, we study the LHC phenomenologies based on the GM constraints on the benchmark models. Since the GM constraint on the m 2 12 parameter will control the Higgs boson self-couplings in the 2HDM, the pair productions of both SM-like and BSM Higgs bosons at the LHC can be relevant to this constraint. The current LHC 13 TeV searches for the Higgs boson pairs, as well as other exotic heavy Higgs boson decay modes are imposed to the benchmark models with the GM constraint taken into account. Finally, we conclude in Sec. V.
[1] for a comprehensive review. 2 Typically in many BSM models with scalar extensions, the GM conditions of the desired vacua can be nontrivial and deserve further studies. The GM conditions of some models have been studied before, such as the Georgi-Machacek model in Ref. [29], the Type II Seesaw model in Ref. [30].

II. THE GENERAL 2HDM AND THE GM CONSTRAINT
A. The general 2HDM The scalar potential of the general 2HDM is written as follows where all the couplings are real for the CP -conserving case. Here, we do not include the Z 2 broken terms of , focusing our study on the potential with softly-broken Z 2 symmetry. The potential in Eq. (1) contains eight parameters, namely m 2 11 , m 2 12 , m 2 22 , and λ 1···5 , which are usually referred to as the parameters in the generic basis. In phenomenological studies, it is usually more convenient to work in the so-called physical basis, including the five physical boson masses of (M h , M H , M A , M ± ), two mixing angles of (α , β), a soft Z 2 breaking mass squared term of m 2 12 and the electroweak VEV v ≈ 246 GeV. In the general 2HDM, there could be tree-level flavor-changing neutral currents (FCNC), which are wellknown constraints on such model. To alleviate the tree-level FCNC process constraints, the SM fermions of a given representation are usually assigned to a single Higgs doublet. We focus on the so-called Type I and Type II Yukawa couplings of with Type I : Type II : Besides, two CP -even Higgs bosons couple to the gauge bosons such that with The current LHC run-I and run-II have measured the signal strengths of the SM-like 125 GeV Higgs boson 3 via different channels . Here, they are combined to obtain the 95 % C.L. regions in the cos(β − α) vs. tan β plane, as shown in Fig. 1. In both Type I and Type II, the alignment limit of β − α = π/2 are favored by the global fit. For the Type I 2HDM with tan β 2, | cos(β − α)| is constrained to be less than about 0.4 with the LHC run-I and run-II data. This is envisioned to be further constrained to be less than 0.2 with the HL-LHC runs in Ref. [77]. For the Type II 2HDM, large/small tan β inputs will enhance the Yukawa couplings ξ d , h /ξ u h . Thus, the region around tan β = 1 accommodates the largest deviation from the alignment. The current LHC run-I and run-II measurements constrain cos(β − α) in the range of (−0.01 , 0.08) approximately with tan β = 1 (except for the wrong-sign Yukawa coupling region [78,79]).

B. The GM constraint on the 2HDM potential
In terms of three SU(2) L invariants of the potential can be re-written as In principle, one can directly minimize the above potential with respect to q 1 , q 2 , and z. However, one should notice that by definition, the three SU(2) L invariants of (q 1 , q 2 , z) satisfy the boundary conditions of Depending on whether the minima are on one of the boundaries in Eq. (8) or not, we can classify the minima into five types, namely, where, e.g., Type A is not on any of the boundaries and Type E is on all the boundaries. All the five types of minima have been solved in Ref. [28] and summarized in Tab. I.
The row in Tab. I containing the explicit forms of φ 1 and φ 2 indicates that Type D is the usually desired vacuum of 2HDM. Type A minima could break U(1) EM , while Type B and Type C minima appear in the so-called inert 2HDM. Type E is a trivial solution which is listed here for completeness.
The solution of Type A is given by where And the corresponding potential minimum is The Type D minimum is determined by ∂V /∂q 1 ,2 = 0: where q ≡ q 1 + q 2 . Given the potential parameters of (m 2 11 , λ 1 , λ 2 ...), Eqs. (13a) and (13b) can be solved with respect to q and β, which can be further converted to q 1 and q 2 according to q 2 /q 1 = tan 2 β and q 1 + q 2 = q. In practical use with physical inputs, Eqs. (13a) and (13b) are commonly used to evaluate m 2 11 and m 2 22 for given tan β and v, together with the quartic couplings determined by: The potential minima for the Type B, Type C and Type D cases can be expressed as follows in the physical basis A premise of imposing the GM constraint is that the potential must be BFB. The BFB conditions 4 for the potential considered in this paper read: In addition, we shall also impose the perturbative unitarity constraints to the 2HDM potential [85,86]. In practice, we use the 2HDMC [87] to scan the 2HDM parameters in the physical basis, and impose all three constraints accordingly.

III. THE GM CONSTRAINTS IN THE PHYSICAL BASIS
Given inputs of (M h , M H , M A , M ± , m 2 12 , α, β, and v) in the physical basis , we can convert them to the potential parameters of (m 2 11 , m 2 12 , m 2 22 , and λ 1···5 ) in the generic basis and, with the method introduced in Sec. II B, infer whether the corresponding potential violates the GM condition or not. In this section, we study the GM constraints on the parameters in the physical basis, focusing on two simple yet illustrative scenarios below: • (i) all the heavy Higgs bosons are mass-degenerate, i.e., M H = M A = M ± ; • (ii) two of the heavy Higgs bosons are mass-degenerate while the remaining is heavier or lighter than the degenerate mass -see Tab. II. Such a mass spectrum allows exotic decays [88].
For scenario (i), we perform a grid scan of M H = M A = M ± from 200 GeV to 1 TeV at a step of 10 GeV, and m 2 12 from 0 to −(500 GeV) 2 at a step of (10 GeV) 2 . The 2HDM mixing angles (α , β) are taken to be consistent with the current LHC constraints on the 125 GeV Higgs boson signal strengths as shown in Fig. 1.
For scenario (ii), we summarize the benchmark models in Tab. II. Taking BP-1 for instance, we perform the grid scan of the heaviest Higgs boson mass M A from 250 GeV to 1 TeV at a step of 10 GeV, and the next heavy Higgs boson mass M H = M ± from 130 GeV up to M A − 100 GeV. The soft Z 2 broken parameter m 2 12 still takes the negative values from 0 to −(500 GeV) 2 at a step of (10 GeV) 2 . In addition, we take cos(β − α) = 0 (known as the alignment limit) in this case for simplicity.
Here, we present the constraints in the (M A , |m 2 12 |) or (M H , |m 2 12 |) plane. The other 2HDM parameters are chosen to be consistent with the global fit to the current LHC measurements of the SM-like Higgs boson signal strengths. In Fig. 2, we present the joint constraints of the BFB, the GM and the perturbative unitarity for the mass-degenerate heavy Higgs boson case of M A = M H = M ± . The corresponding mixing angles of (α , β) are taken to be consistent with the global signal fit of the SM-like Higgs boson signal strengths, for the Type I and Type II models, respectively. As seen from Eqs. (14), the numerators in λ 1 ,2 are always positive with m 2 12 < 0. Accordingly, λ 1 or λ 2 can be enhanced with larger or smaller tan β inputs. As we have checked, the corresponding choices of larger or smaller tan β will be more stringently constrained by the perturbative unitarity bounds. Therefore, we focus on the parameter input of tan β = 1.0 for the later studies of LHC phenomenologies.
In Fig. 3, we present these joint constraints for the heavy Higgs mass spectrum with exotic decays in the (M A , |m 2 12 |) or (M H , |m 2 12 |) plane, with 2HDM mixing angles of cos(β − α) = 0 and tan β = 1.0. The allowed regions by the GM constraints should be the same for both Type I and Type II models, provided that the same 2HDM mixing angles are assumed. The heaviest neutral Higgs boson masses are always labeled as the x-axis. The blue regions represent the largest allowed regions by the grid scan of the next heavy Higgs mass in each benchmark model. Similarly, we also find that large or small parameter inputs of tan β will be more stringently constrained by the perturbative unitarity bounds, as in the previous scenario so we shall also focus on the input of tan β = 1.0 for the exotic heavy Higgs boson decay scenario.

IV. THE PHENOMENOLOGY IMPLICATIONS: HEAVY HIGGS BOSON SEARCHES AT THE LHC
In this section, we will discuss the implications of the GM constraint on the LHC phenomenology of the heavy Higgs boson searches in the general 2HDM. Since we have found that the GM constraint is able to further constrain m 2 12 , in addition to the usual BFB conditions in Eqs. (16), the actually allowed ranges of the Higgs boson self-couplings are further restricted. For the cubic Higgs self-couplings in the physical basis, one may check Ref. [89] for details. Accordingly, one can expect that the GM condition will be relevant to the SM-like Higgs boson pair productions and other heavy Higgs search limits at the LHC.

A. The heavy CP -even Higgs boson decays into SM-like Higgs boson pairs
We study the resonance productions of the SM-like Higgs boson pair productions for the degenerate heavy Higgs boson scenario. The exact results for the one-loop pair production processes at the pp colliders were first studied in Ref. [90]. For the 2HDM case with non-vanishing inputs of cos(β − α), the leading contribution is due to the heavy CP -even Higgs boson resonance H. In the m 2 12 < 0 region, we plot the decay branching fraction of Br[H → hh] for the Type I model (with parameters of cos(β − α) = 0.01 and tan β = 1.0) and the Type II model (with parameters of cos(β − α) = 0.1 and tan β = 1.0) in Fig. 4, for three different inputs of the heavy CP -even Higgs boson masses. The decay branching fractions of Br[H → hh] are apparently suppressed in the Type II model, with a small cos(β − α) input, as compared to the Type I model. With the GM condition, the allowed ranges of |m 2 12 | are further restricted, which were also displayed in Fig. 2 previously. We obtain the heavy CP -even Higgs boson production cross sections at the LHC 13 TeV runs, by using the Sushi package [91]. For the parton distributions, we use NNPDF. Both the heavy resonance searches for H → hh → bbγγ [92] and H → hh → bbbb [93] were taken into account. In Fig. 5, the current LHC 13 TeV search limits to the SM-like Higgs boson pairs via these two channels, as well as the theoretically allowed regions, 300 GeV. The excluded regions are roughly within the theoretically allowed regions with the m 2 12 = 0 input for all four benchmark planes.
When m 2 12 are decreasing to −(120) 2 GeV 2 in the BP-1 and the BP-4 cases, the theoretically allowed regions are merely excluded by the current experimental search limits. In the BP-2 or BP-3 cases, we decrease m 2 12 from 0 down to −(200) 2 GeV 2 , and the parameter regions with smaller M H inputs are already disallowed from the GM constraints in the similar fashion. This can be understood by the potential minima for the Type B, Type C and Type D cases in Eqs. (15), where they are reduced to with the alignment limit of cos(β − α) = 0. To have both conditions of V min ,D < V min ,B and V min ,D < V min ,C hold with a more negative input of m 2 12 , one thus demands a larger input of M H . For the BP-1 and BP-4 cases, that means a more negative input of m 2 12 pushes two heavy Higgs boson masses of M A and M H close to each other. Therefore, one can expect the current experimental searches via the H → ZA mode become more challenging, since the transverse momenta of final-state b-jets and leptons are smaller. Similar situation can be envisioned for the BP-2 and BP-3 cases as well, but for the different decay mode of A → ZH. This suggests that the heavy Higgs boson spectrum involving exotic decay modes may be hidden from the LHC experimental searches, with negative inputs of m 2 12 and the GM constraint taken into account.

V. CONCLUSION
In the scalar potential of the general 2HDM, it is likely that several minima may coexist. The usually considered vacuum can thus become a local minimum, and it may decay into a deeper one. To avoid this vacuum instability at the tree level, we impose the GM condition to the 2HDM potential.
According to our analysis, it turns out that the GM condition can impose a more stringent bound on the m 2 12 parameter when it is in the negative region. Besides, we find that large or small inputs of tan β can impose stringent bounds on the heavy Higgs boson masses for the illustrated cases. Hence, we focus on the parameter input of tan β = 1.0 in our discussion. Two different scenarios in the heavy Higgs boson sector were considered in our analysis. For the mass-degenerate heavy Higgs bosons, we find that the actually expected