Type-I 2HDM under the Higgs and Electroweak Precision Measurements

We explore the extent to which future precision measurements of the Standard Model (SM) observables at the proposed $Z$-factories and Higgs factories may have impacts on new physics beyond the Standard Model, as illustrated by studying the Type-I Two-Higgs-doublet model (Type-I 2HDM). We include the contributions from the heavy Higgs bosons at the tree-level and at the one-loop level in a full model-parameter space. While only small $\tan\beta$ region is strongly constrained at tree level, the large $\tan\beta$ region gets constrained at loop level due to $\tan\beta$ enhanced tri-Higgs couplings. We perform a multiple variable global fit with non-alignment and non-degenerate masses. We find that the allowed parameter ranges could be tightly constrained by the future Higgs precision measurements, especially for small and large values of $\tan\beta$. Indirect limits on the masses of heavy Higgs bosons can be obtained, which can be complementary to the direct searches of the heavy Higgs bosons at hadron colliders. We also find that the expected accuracies at the $Z$-pole and at a Higgs factory are quite complementary in constraining mass splittings of heavy Higgs bosons. The typical results are $|\cos(\beta-\alpha)|<0.05, |\Delta m_\Phi |<200\ {\rm GeV}$, and $\tan\beta \gtrsim 0.3$. The reaches from CEPC, FCCee and ILC are also compared, for both Higgs and $Z$-pole precision measurements. Comparing to the Type-II 2HDM, the 95\% C.L. allowed range of $\cos(\beta-\alpha)$ is larger, especially for large values of $\tan\beta$.


Contents
1 Introduction The discovery of the Higgs boson at the CERN Large Hadron Collider [1,2] has profound implications in our understanding of physics at short distances. It not only verifies the mechanism for the spontaneous electroweak symmetry breaking (EWSB), but also establishes a self-consistent theory, the "Standard Model (SM)", that could be valid to an exponentially high scale, perhaps to the Planck Scale. Indeed, all the current measurements at the electroweak (EW) scale of a few hundred GeV seem to indicate the observed Higgs boson to be a SM-like elementary scalar. When high energy physics advances to the next level, it is thus a natural and pressing question to ask whether in Nature there are other Higgs bosons, associated with a new physics scale as predicted in many extended theories beyond the Standard Model (BSM). As such, searching for new Higgs bosons at the current and future facilities should be of high priority. One of the well-motivated extensions is the Two-Higgs-doublet model (2HDM) [3]. After the EWSB with the EW gauge bosons absorbing three Goldstone bosons, there are five massive spin-zero states left in the spectrum (h, H, A, H ± ), among which h is assumed to be the SM-like Higgs boson. 1 Extensive searches for the additional Higgs bosons have been actively carried out, especially at the LHC [4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19][20][21][22][23]. In the absence of signal observation at the LHC experiments, this would imply either the new Higgs bosons are much heavier and essentially decoupled from the SM, or their interactions with the SM particles are highly suppressed and the couplings of the SM-like Higgs accidentally aligned with the SM predictions [24,25]. In either situation, it would be challenging to directly produce those states in the current and near-future experiments.
Alternatively, precision measurements of SM observables and the Higgs properties could lead to relevant insights into new physics. The recent proposals of construction of a Higgs factory, including the Circular Electron Positron Collider (CEPC) in China [26,27], the electron-positron stage of the Future Circular Collider (FCCee) at CERN (previously known as TLEP [28][29][30]), and the International Linear Collider (ILC) in Japan [31], could shed light on new physics in the pursuit of precision Higgs measurements. With about 10 6 Higgs bosons expected at the Higgs factory, one would be able to reach sub-percentage precision determination of the Higgs properties, and thus to be sensitive to new physics associated with the Higgs boson. As an integrated part of the circular collider program, one would like to return to the Z-pole. With about 10 10 − 10 12 Z bosons, the achievable precisions on the SM observables could be improved by a factor of 20 − 200 over the Large Electron Positron (LEP) Collider results [32]. Such an unprecedented precision would hopefully lead to hints of new physics associated with the EW sector.
There is a plethora of articles in the literature to study the effects of the heavy Higgs states on the SM observables [3]. In particular, a few current authors performed a study focusing on the Type-II 2HDM [33]. We found that the expected accuracies at the Z-pole and at a Higgs factory are quite constraining to mass splittings of heavy Higgs bosons. The reach in the heavy Higgs masses and couplings can be complementary to the direct searches of the heavy Higgs bosons at hadron colliders [34][35][36][37][38]. In this paper, we extend the previous study by examining the Type-I 2HDM. There are interesting and qualitative differences in those two models. One of the most distinctive features comes from the coupling pattern of the Higgs bosons to the SM fermions. Relevant to our studies are the Yukawa couplings of the SM-like Higgs boson h. The deviations from the SM predictions scale with a factor cot β in Type-I, while the scaling factor for down-type fermions goes like tan β in Type-II. In our analyses, we include the tree-level corrections to the SM-like Higgs couplings and one-loop level contributions from heavy Higgs bosons. We perform a global fit in the full modelparameter space. In particular, we study the extent to which the parametric deviations from the alignment and degenerate masses can be probed by the precision measurements. We will comment on the results whenever there is a difference between Type-I and Type-II.
The rest of the paper is organized as follows. An overview of the Higgs and electroweak precision observables at future e + e − colliders is given in Section 2, which will serve as inputs for our analyses of the Type-I 2HDM. We present the Type-I 2HDM and the one-loop corrections in Section 3. In Section 4, we impose the set of theoretical constraints to the Higgs boson masses and self-couplings. In Section 5, the constraints from the direct LHC searches for heavy Higgs bosons are presented for the Type-I 2HDM, under the current LHC runs and the future projected HL-LHC sensitivities. Section 6 shows our main results from the global fit, for the cases of mass degeneracy and non-degeneracy of heavy Higgs bosons. We summarize our results and draw conclusions in Section 7. Some useful analytic formulae and approximate treatments are given in Appendix A.

Higgs Observables at Future Lepton Colliders
The SM has been tested to a high precision from the measurements at the Z-pole from LEP-I [32], at the Tevatron [39] and the LHC [40]. It has been demonstrated that the EW and Higgs precision measurements can impose strong constraints on new physics models [41,42]. In this section, we closely follow the approach adopted in Ref. [33], and list the projected precision achievable by several proposed future e + e − machines on Z-pole and Higgs measurements.
These measurements are expected to be significantly improved by a new run at the Zpole at future lepton colliders with a much larger data sample [26,[28][29][30]43]. The expected precision on the measurements of α s , ∆α Table 1 in Ref. [33]. The corresponding accuracy for S, T and U at 1σ level in CEPC is estimated to be ∆S = ±0.0246, ∆T = ±0.0255, ∆U = ±0.0208. (2.1) In our analyses as detailed in a later section, the S, T and U contours at a 95% Confidence Level (C.L.) are adopted to constrain the 2HDM parameter spaces, using the χ 2 -fit with error-correlation matrices . A Higgs factory with e + e − collisions at a center-of-mass energy of 240-250 GeV exploits the Higgsstrahlung process e + e − → hZ. (2.2) Owing to the clean experimental condition and well-constrained kinematics at the lepton colliders, both the inclusive cross section σ(hZ) independent of the Higgs decays, and the exclusive channels of individual Higgs decays in terms of σ(hZ) × BR, can be measured to remarkable precisions. The invisible decay width of the Higgs boson can also be sensitively probed. In addition, the cross sections of vector boson fusion processes for the Higgs production (W W, ZZ → h) grow with the center of mass energy logarithmically. While their rates are still rather small at 240-250 GeV, at higher energies in particular for a linear collider, such fusion processes become significantly more important and can provide crucial complementary information. For √ s > 500 GeV, tth production can also be utilized as well. We list the running scenarios of various machines in terms of their center of mass energies and the corresponding integrated luminosities, as well as the estimated precisions of relevant Higgs measurements that are used in our global analyses in Tab. 1. These expected results in the table serve as the input values for the later studies in this paper in constraining the theoretical parameters in the BSM Higgs sector. Comparing to the values used in earlier study of Ref. [33], the main update is the h → γγ precision at FCCee, which is 9% instead of 4% because of different simulation methods [44]. We only include the rate information for the Higgsstrahlung Zh and the W W fusion process in our global fit. Some other measurements, such as the angular distributions, the diboson process e + e − → W W , can provide additional information in addition to the rate measurements alone [45][46][47].

Type-I Two-Higgs-Doublet Model
A generic 2HDM consists of two SU(2) L scalar doublets Φ i (i = 1, 2) with a hyper-charge assignment Y = +1/2 After the EWSB, each doublet obtains a vacuum expectation value (vev) v i (i = 1, 2) with v 2 1 + v 2 2 = v 2 = (246 GeV) 2 , and v 2 /v 1 = tan β. The 2HDM Lagrangian for the Higgs sector is given by with the CP-conserving potential  [27], FCCee program with 5 ab −1 integrated luminosity [48,49], and ILC with various center-of-mass energies [50]. and a soft Z 2 symmetry breaking term m 2 12 . One of the four neutral components and two of the four charged components are eaten by the SM gauge bosons Z, W ± after the EWSB, providing their masses. The remaining physical mass eigenstates are two CP-even neutral Higgs bosons h and H, with m h < m H , one CP-odd neutral Higgs boson A, plus a pair of charged Higgs bosons H ± . Instead of the eight parameters appearing in the Higgs potential m 2 11 , m 2 22 , m 2 12 , λ 1,2,3,4,5 , a more convenient set of the parameters is v, tan β, α, m h , m H , m A , m H ± , m 2 12 , where α is the rotation angle diagonalizing the CP-even Higgs mass matrix. We choose m h = 125 GeV to be the SM-like Higgs boson.
The Type-I 2HDM is characterized by the choice of the Yukawa couplings to the SM fermions and they are of the form After the EWSB, the effective Lagrangian for the light CP-even Higgs couplings to the SM particles can be parameterized as with i indicating the individual Higgs coupling. Their values at the tree level are We adopt the sign convention β ∈ (0, π/2), β − α ∈ [0, π], so that sin(β − α) ≥ 0. Note that comparing to the Type-II 2HDM, in which up-type Yukawa couplings are proportional to cot β and bottom-type and lepton Yukawa couplings are proportional to tan β, all the tree-level Yukawa couplings in the Type-I 2HDM are proportional to cot β. Therefore, κ tree f are enhanced comparing to the SM values only at low tan β < 1 region. At the leading order, the CP-even Higgs couplings to the SM gauge bosons are g hV V = sin(β − α), and g HV V = cos(β − α). The current measurements of the Higgs boson properties from the LHC are consistent with the SM Higgs boson interpretation. There are two wellknown limits in 2HDM that would lead to a SM-like Higgs sector. The first situation is the alignment limit [24,51] of cos(β − α) = 0, in which the light CP-even Higgs boson couplings are identical to the SM ones, regardless of the other scalar masses, potentially leading to rich BSM physics. For sin(β − α) = 0, the opposite situation occurs with the heavy H being identified as the SM Higgs boson. While it is still a viable option for the heavy Higgs boson being the observed 125 GeV SM-like Higgs boson [52,53], the allowed parameter space is being squeezed with the stringent direct and indirect experimental constraints. Therefore, in our analyses below, we identify the light CP-even Higgs h as the SM-like Higgs with m h fixed to be 125 GeV. The other well-known case is the "decoupling limit", in which the heavy mass scales are all large: m A,H,H ± 2m Z [54], so that they decouple from the low energy spectrum. For masses of heavy Higgs bosons much larger than λ i v 2 , cos(β − α) ∼ O(m 2 Z /m 2 A ) under perturbativity and unitarity requirement. Therefore, the light CP-even Higgs boson h is again SM-like. Although it is easier and natural to achieve the decoupling limit by taking all the other mass scales to be heavy, there would be little BSM observable effects given the nearly inaccessible heavy mass scales. We will thus mainly focus on the alignment limit.
While the couplings hgg, hγγ and hZγ are absent at the tree-level in both the SM and the 2HDM, they are generated at the loop-level. In the SM, hgg, hγγ and hZγ all receive contributions from fermions (mostly top quark) running in the loop, while hγγ and hZγ receive contributions from W -loop in addition [55]. In 2HDM, the corresponding hf f and hW W couplings that enter the loop corrections need to be modified to the corresponding 2HDM values. Expressions for the dependence of κ g , κ γ and κ Zγ on κ V and κ f can be found in Ref. [56]. There are, in addition, loop corrections to κ g , κ γ and κ Zγ from extra Higgs bosons in 2HDM.
The triple couplings among Higgs bosons themselves are relevant for the loop corrections. When omitting the O(cos 2 (β − α)), they read with M 2 ≡ m 2 12 /(sin β cos β), C Φ = 2(1) for Φ = H ± (A). One notable difference between the Type-I study here and our former Type-II study [33] is that those terms proportional to cos(β − α) could play a more important role in the Type-I 2HDM. In the Type-II 2HDM, Yukawa couplings have both cot β enhancement and tan β enhancement at the tree level, which tightly constraints the range of cos(β − α) when Higgs precision measurements are considered. In the Type-I 2HDM, all Yukawa couplings are proportional to cot β at the leading order, with no large tan β enhancement. The viable range of cos(β − α) could be larger when tree-level effects are included. However, given that loop corrections induced by the triple Higgs couplings λ hΦΦ , λ hHH would have interesting tan β enhancement that competes with the tree-level corrections, which are usually subdominant in Type-II 2HDM, especially in the large tan β region when the tree level effects dominate. Once loop effects are included, the range of cos(β − α) at the tree-level loosely constrained large tan β region in the Type-I 2HDM shrinks significantly.
With the degenerate masses of m Φ ≡ m H = m A = m H ± and the alignment limit of cos(β − α) = 0, we can introduce a new parameter of λv 2 defined as which is the parameter that enters the Higgs self-couplings and relevant for the loop corrections to the SM-like Higgs boson couplings. This parameter could be used interchangeably with m 2 12 as we will do for convenience. For the rest of our analysis, we take the input parameters v = 246 GeV and m h = 125 GeV. The remaining free parameters are tan β , cos(β − α) , m H , m A , m H ± and λv 2 . (3.12) Although these six parameters are independent of each other, their allowed ranges under perturbativity, unitarity, and stability consideration are correlated. For simplicity, one often begins with the degenerate case where all heavy Higgs boson masses are set the same. We will explore both the degenerate case and deviation from that, the non-degenerate case, specified as As such, there will be four independent parameters for the degenerate case, and five for the non-degenerate case if assuming ∆m A = ∆m C . With the current LHC Higgs boson measurements [57][58][59][60][61][62], deviations of the Higgs boson couplings from the decoupling and alignment limits are still allowed at about 10% level. All tree-level deviations from the SM Higgs boson couplings are parametrized by only two parameters: tan β and cos(β − α). Once additional loop corrections are included, dependence on the heavy Higgs boson masses as well as λv 2 also enters. In our following analyses, we study the combined contributions to the couplings of the SM-like Higgs boson with both tree-level and loop corrections. The calculations of κ's are performed with full electroweak one-loop corrections, 2 as discussed in details in Ref. [33].

Theoretical Constraints
Heavy Higgs loop corrections will involve the Higgs boson masses and self-couplings, as λ 1−5 in Eq. (3.3). These parameters are constrained by theoretical considerations such as vacuum stability [63], perturbativity, and partial wave unitarity [64].
• Vacuum Stability • Unitarity The details of a 0 i,± were shown in Refs. [34,64]. In what follows, we will discuss the constraints in several different cases.

Case 1: alignment limit with degenerate heavy Higgs masses
Theoretical constraints do not depend on the Yukawa structure at the leading order. Detailed discussions were included in the previous work of [33,34]. In general, λv 2 is constrained to be which gives −0.258 < λ = −λ 4 = −λ 5 < 5.949 and 0 < λ 3 < 6.207. tan β dependence enters as Eq. (4.5a) mainly comes from the requirement of vacuum stability, while Eq. (4.5b) is due to the partial wave unitarity. In Fig. 1, we present contours to illustrate the theoretical constraints on tan β versus the other model parameters as discussed in the beginning of Sec. 4. As shown in the left panel, for λv 2 = 0, m H/H ± = 800 GeV, tan β is unconstrained. The allowed range of tan β quickly shrinks as λv 2 increases. The degenerate case under consideration for this section is shown by the outer contour ∆m A = m A − m H/H ± = 0.

Case 2: alignment limit with non-degenerate heavy Higgs masses
For non-degenerate heavy Higgs masses with the mass splittings ∆m A(C) = m A(H ± ) − m H , two special cases are of particular interest m A = m H ± and m H ± = m H . The theoretical constraints for m A = m H ± have been discussed in [33]. Here we focus on the other case m H ± = m H in which the Z-pole constraints are automatically satisfied.
The strongest constraints on parameters are imposed by the partial wave unitarity, in particular, a 0 1,+ ≤ 1 2 , which primarily sets limits on the value of tan β, and a 0 4,− ≤ 1 2 , which constrains the mass splitting ∆m A = m A − m H/H ± . The allowed range of tan β for various ∆m A are plotted in the left panel of Fig. 1, which is not very sensitive to the mass splitting. It can be well approximated by Eq. (4.5b). For large mass splitting, namely m 2 A − m 2 H πv 2 , a 0 4,− ≤ 1 2 sets a strong upper limit on λv 2 , which is given by This explains the straight right boundary in the left panel of Fig. 1 as ∆m A 100 GeV. For ∆m A > 0, the range of λv 2 shrinks more for larger ∆m A : from about (600 GeV) 2 with ∆m A = 0 to about (300 GeV) 2 for ∆m A = 150 GeV.

Case 3: non-alignment limit with degenerate heavy Higgs masses
The theoretical constraints also limit the range of cos(β − α), as shown in the right panel of Fig. 1 for the degenerate case with different values of λv 2 for m Φ = 800 GeV. A larger value of λv 2 leads to a more relaxed range of cos(β − α), but a stronger constraint on tan β. A larger value of m Φ leads to a smaller region in cos(β − α). One interesting feature is a symmetry: for cos(β − α) → − cos(β − α), tan β → cot β, which is evident in the right panel of Fig. 1 as well.

Current and Expected LHC Search Bounds
The heavy Higgs bosons in the 2HDM have been searched for at the LHC Run-I and Run-II via various channels. The direct searches include the decay channels of τ τ [4,5] 12]. Since the limits from heavy Higgs decays to µµ and bb are always similar but weaker comparing to the τ τ channel, we will only show constraints from τ τ channel here. In addition, the leading decay modes of heavy neutral Higgs bosons is A/H → tt, which was known to have strong signal-background interference effects [65]. Recent studies of the LHC 8 TeV search sensitivities via this channel can be found in Refs. [66], utilize the lineshape of tt invariant mass distribution [67][68][69], and the experimental searches were made in Ref.
[18]. Knowing the search limit at √ s = 8 TeV or √ s = 13 TeV, the associated limit at √ s = 14 TeV could be estimated through s scaling relation [67] To use the published cross-section times branching ratio limits to directly constrain the 2HDM parameter space, we work with the SusHi package [70] for the production cross-section at the NNLO level, and our own improved 2HDMC code, which adds loop-level effects to the public 2HDMC code [71], for the branching ratios.
In the left panel of Fig. 2, we present the 95% C.L. limits of the neutral Higgs boson searches in the m A/H − tan β plane for the degenerate case under the current LHC Run-II searches (shaded region enclosed by the dashed lines) as well as the projected HL-LHC search limits (region enclosed by the solid lines). We have chosen the non-alignment case of cos(β − α) = 0.025, in which H → hh, A → hZ, and W W/ZZ channel contribute. Unlike the Type-II 2HDM case in which there are very strong experimental search limits from τ τ channel at large tan β, for the Type-I 2HDM, large tan β region is basically unconstrained since all the Yukawa couplings are proportional to 1/ tan β. For the low tan β ∼ 0.1 region, τ τ mode has the best reach: the current LHC Run II excludes heavy Higgs mass up to about 550 GeV, In Fig. 3, we present the 95% C.L. excluded region in the cos(β − α) − tan β plane for the LHC 13 TeV searches [10] and for the projected HL-LHC 14 TeV sensitivity via the A → hZ channel. The results were shown for two fixed heavy CP-odd Higgs boson masses of m A = 800 GeV and m A = 2000 GeV, respectively. For the case of m A = 800 GeV, a narrow band for cos(β − α) −0.1 or | cos(β − α)| 0.02 is allowed by the HL-LHC data. In addition, the large-tan β regions of tan β 5 and tan β 20 are also allowed at the current LHC and the future HL-LHC searches, which is due to the suppressed production cross sections of σ(gg → A) and σ(bb → A) in the Type-I 2HDM case. For the m A = 2000 GeV case, the current and the future LHC searches for the A → hZ can only exclude the regions with small input values of tan β, due to the suppressed production cross section for heavy Higgs bosons. Note that the strong constraints usually present in the Type-II case at large tan β is again absent here in the Type-I case, due to the 1/ tan β dependence of the Higgs Yukawa couplings. 6 Fitting Results

Global fit framework
With the Higgs precision measurements summarized in Tab. 1, we performed a global fit to determine the allowed parameter space of Type-I 2HDM. With the same method described in Ref. [33], we construct the χ 2 with the profile likelihood method, is the signal strength for various Higgs search channels, σ µ i is the estimated error for each process. Usually, the correlations among different σ × Br are not provided and are thus assumed to be zero. For future colliders, µ obs i are set to be unity in the current analyses, assuming no deviations from the SM observables. 3 In this work, we will focus specifically on CEPC, and also compare the reaches of three future lepton colliders, including ILC and FCCee.
The overall χ 2 is calculated by substituting the κ's defined in Eq. (3.5) into corresponding µ BSM i . In the rest of the analyses, we determine the allowed parameter region at the 95% Confidence Level (C.L.). For the one-, two-or three-parameter fit, the corresponding ∆χ 2 = χ 2 − χ 2 min at 95% C.L. is 3.84, 5.99 or 7.82, respectively.

Case with Degenerate Heavy Higgs Masses
We will first show our results in the case with degenerate heavy Higgs masses, m Φ = m H = m A = m H ± , which satisfies the Z-pole physics constraints automatically.

Constraints in the cos(β − α) − tan β plane
For the case with degenerate masses, we first show the result in Fig. 4 of the two-parameter global fit in the cos(β − α) − tan β plane at 1-loop level for m Φ = 800 GeV and √ λv 2 = 300 GeV. The red region represents the overall allowed region with the CEPC precision measurements at 95% C.L. at one-loop level, while the black dashed line represents the allowed region at tree level. Individual constraints from hbb, hcc, hτ τ , hZZ and hgg are also shown by colored solid lines. The constraints from hW W and hγγ are much weaker due to worse experimental precisions, hence they are not shown in Fig. 4. Compared with the tree-level dashed line, the allowed red region at the loop level has quite different behaviors at both large and small tan β regions. For small tan β, the overall allowed region is mainly constrained by κ f due to the large cot β enhancement of fermion Yukawa couplings at the tree level, while the constraints from κ Z is weak despite its high precision. This makes the outline of red region close to that of the tree level. However, the individual fermion lines show peculiar distortion away from the tree level result. Such modification is related to √ λv 2 term in the triple Higgs couplings λ hφφ , and Yukawa couplings. The effect is proportional to cot 2 β, therefore more pronounced when tan β is small. In particular, the hbb coupling plays an important role in the distortion from the tree-level results due to the large top Yukawa coupling in the top loop contributions.
In the meantime, all couplings at one-loop level significantly deviate from those of tree level ones at large tan β. For the Type-I 2HDM there is no tan β enhancement at tree level, and the main constraint at large tan β region comes from the precise hZZ coupling measurement. At one-loop level, the strongest constraint is still from hZZ coupling, because all the SM-Higgs couplings receive a universal tan β enhancement from the Higgs field renormalization   We perform a three-parameter fit for cos(α−β), tan β, and √ λv 2 . Owing to the one more free parameter in the fit and the correlation among the parameters, the allowed region could be different from that of the two-parameter fit. Figure 5 shows the fitting results for a fixed value m Φ = 800 GeV with various values √ λv 2 = 0, 100, 200, 300 GeV (left panel) and m Φ = 2000 GeV with √ λv 2 = 100, 400, 500, 600 GeV (right panel), indicated by different colored lines. With three-parameter fit, √ λv 2 ≥ 400 GeV for m Φ = 800 GeV is excluded. In general, including loop corrections shrinks the allowed parameter space, especially for large and small tan β. For larger √ λv 2 , there will be a larger asymmetry with respect to cos(β − α) = 0. The asymmetry at small tan β is the result from the loop-level hbb couplings because of the large top Yukawa contribution. At large tan β, on the other hand, it is from hZZ coupling, because the triple Higgs couplings as in Eq. (3.8) have terms proportional to cos(β −α). There is also no decoupling effect for regions with non-zero cos(β − α): the allowed region is smaller with larger m Φ . This is because a large part of regions is outside of the theoretically allowed region, shown by the colored shaded region in Fig. 5.

Constraints on Heavy Scalar Masses
To see how precision measurement constrains heavy scalar masses, we also explore the global fit in three parameters by fixing λv 2 (or m 2 12 ). Figure 6 shows the fitting results at 95% C.L. in the m Φ − tan β plane under CEPC precision for √ λv 2 = 0 GeV (left panel) and 300 GeV (right panel) for the degenerate mass case. Green, blue and red curves (stars) represent the constraints (best fit point) for cos(β − α) = −0.01, 0, 0.01 respectively. The theoretical allowed regions are also shown in the shaded areas with the same color. For √ λv 2 = 0, the constraint on the heavy Higgs mass is rather loose. Once tan β 1, all mass values greater than m h are allowed. For nonzero cos(β − α), large tan β are excluded.
It is also interesting to see how future precision measurements constrain the soft Z 2 breaking parameter m 2 12 . Figure 7 shows the fitting results similar to Fig. 6 but for fixing value of m 12 instead of fixing λv 2 . For m 12 = 0, m Φ = √ λv 2 is constrained to be less than around 250 GeV. For larger values of m 12 , the rather narrow region in the plane as seen in the right panel indicates a strong correlation between m Φ and tan β, approximately scaled as tan β ∼ (m Φ /m 12 ) 2 , which minimizes the corresponding λv 2 value and thus its loop effects.  Comparing with the expected direct search limits of heavy Higgs bosons at the HL-LHC, as shown in Section 5, we see that the indirect sensitivities of the SM-like Higgs precision measurements to the heavy Higgs masses and values of tan β obtained here complement the direct search very well. While the direct searches usually have better reach in the mass of the heavy Higgs bosons, its sensitivity reduced greatly for large tan β given the suppressed Yukawa couplings. The indirect reach, on the other hand, tightly constraints the large tan β region when away from the alignment limit, given the enhanced tri-Higgs couplings which enter the corrections to the SM-like Higgs couplings at the loop level.

Case with Non-degenerate Heavy Higgs Masses
In this section, we go beyond the mass degenerate case and investigate how the Higgs coupling precision measurements could constrain the mass splittings among m H , m A and m H ± , and how it complements the Z-pole precision measurements. For Z-pole precision measurements, we fit for the oblique parameters S, T and U , including the correlation between those oblique parameters, as described in detail in [33].
In Fig. 8, we explore the constrained region in ∆m Φ = m A/H ± − m H (upper panels) and ∆m Φ = m A − m H/H ± (lower panels) under alignment limit for various values of m H . m H ± = m A (m H ± = m H ) is assumed in the former (latter) case to satisfy the Z-pole constraints. Left and right panels correspond to √ λv 2 = 0, and 300 GeV, respectively. Shaded colored regions are used to indicate theoretical preferred regions.  √ λv 2 is set to 0 (left panels) and 300 GeV (right panels). Shaded colored regions are used to indicate theoretical preferred regions.
In all panels of Fig. 8, the allowed regions show a sudden cutoff at certain value of ∆m Φ , in particular for tan β 1. This feature mainly comes from the loop-level corrections to κ Z .  Figure 9. Allowed region (red) from two-parameter fitting results at 95% C.L. in the ∆m A − ∆m C plane from the individual Higgs coupling measurement, for tan β = 0.2 (left), 1 (middle), 7 (right) under the alignment limit, with m H = 800 GeV and √ λv 2 = 0. For individual coupling constraint, the dashed line represents −σ limit, while the solid line represents the +σ limit. Regions between the solid and dashed curves are the allowed region. For κ γ , region above the line is allowed. Also shown is the theoretically allowed region shaded in grey color. responsible for the sudden cutoff. For √ λv 2 = 0 GeV, λ hHH = −m 2 h /2v is small and ∆m 2 Φ /v 2 would dominate, resulting in a symmetric bound of ∆m Φ . For a larger value of √ λv 2 , ∆m Φ λ hHH /v 2 becomes more important, resulting in an asymmetric bound ∆m Φ . Such behaviour appears for all loop-level Higgs couplings, with κ Z becomes the most constraining one at tan β 1. The case of m H ± = m H = m A − ∆m Φ is similar.
In Fig. 9, we show the constraints in ∆m C ≡ m H ± − m H versus ∆m A ≡ m A − m H parameter space, with constraints from individual coupling indicated by the colored curves, and the 95% C.L. global fit region indicated by the red shaded region with the best fit point indicated by the black star. Other parameters are chosen as m H = 800 GeV, √ λv 2 = 0 and tan β = 0.2(left), 1(middle), 7(right) under the alignment limit. For each individual coupling constraint, the dashed line is for the −σ limit, while the solid line is for the +σ limit. The range between the two lines is the survival region. Under the alignment limit, κ Z is independent of tan β as apparent in the figure. The light grey shadow region is the region preferred by the theoretical considerations.
For the Type-I 2HDM under alignment limit, all the Higgs fermion couplings are cot β enhanced and Higgs-vector boson couplings are tan β-independent. At small tan β as shown in the left panel of Fig. 9, ∆m A and ∆m C are strongly constrained to be close to 0, due to the constraints from κ b , κ c and κ τ , dominantly. The blue κ b constraint has a different shape comparing to that of κ c , κ τ , mainly due to the top quark vertex correction. For larger tan β, the fermion couplings constraints are reduced. As a result, κ Z provide the dominant constraint, as shown in the middle and right panels of Fig. 9, which is less constraining. ∆m A,C is constrained to be less than about 200 GeV for tan β = 1 and less than about 400 GeV for tan β = 7, which is quite different from the Type-II 2HDM, which gets tightly constrained in large tan β as well. For tan β > 7, the survived red region does not change significantly. The √ λv 2 = 0, while the right one for √ λv 2 = 300 GeV. tan β is chosen to be 0.5 (orange), 1 (green), 2 (blue), 7 (red). Also shown are the allowed regions by theoretical constraints, which are indicated with shadows of the same color codes. typical survived regions for ∆m A , ∆m C are (−300, 400) GeV, (−200, 300) GeV respectively at large tan β, and generally can be extended to (−500,600) GeV and (−300, 400) GeV. Compared to the Type-II 2HDM [33], the Type-I 2HDM has a more restricted region at small tan β and much relaxed region at large tan β.
In Fig. 10, we show global fit results at 95% C.L. in the ∆m A − ∆m C plane with varying tan β under the alignment limit of cos(β − α) = 0 for m H = 800 GeV. The left panel is for √ λv 2 = 0, and the right panel is for √ λv 2 = 300 GeV. Also shown in color shaded region are the theoretically preferred regions. In general, the range for ∆m A and ∆m C gets bigger for larger tan β and smaller λv 2 . In particular, for tan β = 7, the allowed ranges of ∆m A and ∆m C shrink to a narrow range around −100 GeV.
In Fig. 11, we take Z-pole precision into account as well. The left panels show Higgs precision (solid curves) and Z-pole precision (dashed curves) for different cos(β − α) values: −0.01 (green curve), 0 (blue curve) and 0.01 (red curve). The right panels show the combined fitting results. Shaded region is used to indicate the theoretical constraints at cos(β − α) = 0. While the Z-pole precision measurements are more constraining in the mass difference of ∆m A and ∆m C in general, they can always be satisfied for ∆m The Higgs precision measurements, on the other hand, could provide an upper limit on |∆m A,C |. When combined together, a more restrictive range of ∆m A,C can be achieved. In Fig. 13, we show the 95% C.L. constraints on the ∆m A − ∆m C plane with both Higgs and Z-pole precision measurements under alignment limit. Left panel is for the individual constraints while the right panel show the combined fit. While ILC has better Higgs precision reach, FCCee is slightly better for Z-pole reach. Combined together, reaches of three machine options are similar.

Summary and Conclusions
With the discovery of the SM-like Higgs boson at the LHC, searching for additional Higgs bosons beyond the SM is strongly motivated from both theoretical and experimental considerations. In the absence of signals from the direct searches in the LHC experiments, it would be prudent to seek for complementary means, in particular, from the precision measurements of the SM observables which are sensitive to BSM new physics. In this paper, extending the previous work on the Type-II 2HDM [33], we performed a comprehensive study for the Type-I 2HDM from the impacts of the precision measurements of the SM observables at the proposed Z-factories and Higgs factories on the extended Higgs sector.
First, we listed the latest expected accuracies on determining the EW observables at the Z-pole and the Higgs factories (Tab. 1 in Section 2), as a general guidance and inputs for the following studies. We gave a brief summary for Type-I 2HDM in Section 3 to specify the model and set the scope for the rest of the paper by introducing the degenerate and nondegenerate cases. We then presented the existing constraints on the model parameters from theoretical considerations (Fig. 1 in Section 4) and the LHC bounds from the current searches and the future expectations ( Fig. 2 and Fig. 3  just the tree-level deviations, or loop corrections under the alignment limit, and with the assumption of degenerate masses of the heavy Higgs bosons. A recent study [33] extended the previous work to have included the general one-loop effects in the Type-II 2HDM. In our analyses, we extended the existing results by including the tree-level and one-loop level effects of non-degenerate Higgs masses in the Type-I 2HDM. The main results of the paper were presented in Section 6, where we performed a global fit to the expected precision measurements in the full model-parameter space. We first illustrated the simple case with degenerate heavy Higgs masses as in Fig. 4 with the expected CEPC precision. We found that in the parameter space of cos(β − α) and tan β, the largest 95% C.L. range of | cos(β −α)| 0.05 could be achieved for tan β between 5−10, with smaller and larger values of tan β tightly constrained by κ g,c,b,τ and κ Z , respectively. For the Type-I 2HDM, large tan β regions were always less restricted since all Yukawa couplings are cot β-enhances. When including loop-level corrections, the large tan β regions got additional tan β-enhanced constraints from triple Higgs couplings terms proportional to cos(β − α) tan β as in Eq. (3.10). Varying heavy Higgs masses and λv 2 , as shown in Fig. 5, the significant loop-level effect shown again and shifted the 95% C.L. region. The low tan β results in Figs. 4 and 5 are similar to those in Type-II, but constraints are stronger since all couplings are cot β-enhanced around cos(β − α) = 0.
The limits on the heavy Higgs masses also depend on tan β, λv 2 and cos(β − α), as shown in Fig. 6 and alternatively in Fig. 7 varying m 2 12 . While the most relaxed limits can be obtained under cos(β − α) = 0 with small λv 2 , deviation away from cos(β − α) = 0 leads to tighter constraints, especially for the allowed range of tan β. The reach seen in the m Φ − tan β plane is complementary to direct non-SM Higgs search limits at the LHC and future pp colliders as in Fig. 2, especially in the large tan β region when the direct search limits are relaxed.
It is important to explore the extent to which the parametric deviations from the degenerate mass case can be probed by the precision measurements. Figure 8 showed the allowed deviation for ∆m Φ with the expected CEPC precision and Fig. 9 demonstrated the constraints from the individual decay channels of the SM Higgs boson. As shown in Fig. 10, the Higgs precision measurements alone constrain ∆m A,C to be less than about a few hundred GeV, with tighter constraints achieved for small m H , large λv 2 and small values of tan β. Z-pole measurements, on the other hand, constrain the deviation from m H ± ∼ m A,H . We found that the expected accuracies at the Z-pole and at a Higgs factory are quite complementary in constraining mass splittings. While Z-pole precision is more sensitive to the mass splittings between the charged Higgs and the neutral ones (either m H or m A ), Higgs precision measurements in addition could impose an upper bound on the mass splitting between the neutral ones. Combining both Higgs and Z-pole precision measurements, the mass splittings are constrained even further, as shown in Fig. 11, especially when deviating from the alignment limit. In summary, Higgs precision measurements are more sensitive to parameters like cos(β − α), tan β, √ λv 2 and the masses of heavy Higgs bosons. We found that except for cancelations in some correlated parameter regions, the allowed ranges are typically (7.1) We mostly presented our results adopting the expecations of the CEPC precision on Higgs and Z-pole measurements. The comparison among different proposed Higgs factories of CEPC, FCCee and ILC are illustrative and are shown in Figs. 12 and 13. While the ILC with different center-of-mass energies has slightly better reach in the Higgs precision fit, the FCCee has slightly better reach in the Z-pole precision fit.
While the precision for the Higgs coupling measurements with the LHC program is expected to be at the order of few percent, the precision measurements of the SM observables at the proposed Z and Higgs factories would significantly advance our understanding of the electroweak physics and shed lights on possible new physics beyond the SM, and could therefore be complementary to the direct searches at the LHC and future hadron colliders.