Constraining type II 2HDM in light of LHC Higgs searches

We study the implication of the LHC Higgs search results on the Type II Two Higgs-Doublet Model. In particular, we explore the scenarios in which the observed 126 GeV Higgs signal is interpreted as either the light CP-even Higgs h0 or the heavy CP-even Higgs H0. Imposing both theoretical and experimental constraints, we analyze the surviving parameter regions in mH (mh), mA, mH±, tan β and sin(β − α). We further identify the regions that could accommodate a 126 GeV Higgs with cross sections consistent with the observed Higgs signal. We find that in the h0-126 case, we are restricted to narrow regions of sin(β − α) ≈ ±1 with tan β up to 4, or an extended region with 0.55 < sin(β − α) < 0.9 and 1.5 < tan β < 4. The values of mH, mA and mH±, however, are relatively unconstrained. In the H0-126 case, we are restricted to a narrow region of sin(β − α) ~ 0 with tan β up to about 8, or an extended region of sin(β − α) between −0.8 to −0.05, with tan β extended to 30 or higher. mA and mH ± are nearly degenerate due to Δρ constraints. Imposing flavor constraints shrinks the surviving parameter space significantly for the H0-126 case, limiting tan β ≲ 10, but has little effect in the h0-126 case. We also investigate the correlation between γγ, V V and bb/ττ channels. γγ and V V channels are most likely to be highly correlated with γγ: V V ~ 1 for the normalized cross sections.


Introduction
The discovery of a resonance at 126 GeV with properties consistent with the Standard Model (SM) Higgs boson in both the ATLAS [1,2] and CMS experiments [3,4] is undoubtedly the most significant experimental triumph of the Large Hadron Collider (LHC) to date. The nature of this particle, as regards its CP properties and couplings, are currently being established [4][5][6][7]. Though further data would undoubtedly point us in the right direction, at this point it is useful to explore the implication of the current Higgs search results on models beyond the SM. There are quite a few models that admit a scalar particle in their spectrum and many of them can have couplings and decays consistent with the SM Higgs boson. Thus it behooves us to constrain these models as much as possible with the Higgs search results at hand.
One of the simplest extensions of the SM involves enlarged Higgs sectors. This can be done by simply adding more scalar doublets, or considering Higgs sectors with more complicated representations. In the work, we will study the Two Higgs-Doublet Models (2HDM) that involve two scalar doublets both charged under the SM SU(2) L × U(1) Y gauge symmetries [8][9][10][11]. The neutral components of both the Higgs fields develop vacuum -1 -

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expectation values (vev), breaking SU(2) L × U(1) Y down to U(1) em . Assuming no CPviolation in the Higgs sector, the resulting physical spectrum for the scalars is enlarged relative to the SM and includes light and heavy neutral CP-even Higgses (h 0 and H 0 ), charged Higgses (H ± ), and a pseudoscalar A 0 . In addition to the masses, two additional parameters are introduced in the theory: the ratio of the vevs of the two Higgs fields (tan β), and the mixing of the two neutral CP-even Higgses (sin α).
There are many types of 2HDMs, each differing in the way the two Higgs doublets couple to the fermions (for a comprehensive review, see [8]). In this work, we will be concentrating on the Type II case, in which one Higgs doublet couples to the up-type quarks, while the other Higgs doublet couples to the down-type quarks and leptons. This model is of particular interest as it shares many of the features of the Higgs sector of the Minimal Supersymmetric Standard Model (MSSM). This enables us to translate existing LHC MSSM results to this case. Before proceeding, we point out that over the last few months, there have been various studies on the 2HDM based on the recent discovery [12][13][14][15][16][17][18][19][20][21][22][23][24][25]. While most studies concentrated on finding regions of parameter space that admit σ× Br values reported by the LHC experiments in various channels, some also looked at correlations between the various decay channels. The authors of ref. [12] and ref. [13] did the initial study of looking at the tan β − sin α plane where the observed Higgs signal is feasible, interpreting the discovered scalar as either the light or the heavy CP-even Higgs boson. Ref. [14][15][16][17][18][19] fit the observed Higgs signals in various 2HDM scenarios, taken into account theoretical and experimental constraints. Ref. [20] also paid careful attention to various Higgs production modes. Ref. [21] focused on the CP-violating Type II 2HDM. Ref. [22] studied the case of nearly degenerate Higgs bosons. In addition, ref. [23,24] investigated the possibility that the signal could correspond to the pseudoscalar A 0 -in this context, it is worth remarking that ref. [26] considered the pseudoscalar interpretation of the observed 126 GeV resonance and found that while it is strongly disfavored, the possibility is not yet ruled out at the 5σ level. 1 In the present paper, we extended the above analyses by combining all the known experimental constraints (the LEP, Tevatron and the LHC Higgs search bounds, and precision observables) with the theoretical ones (perturbativity, unitarity, and vacuum stability), as well as flavor constraints. A unique aspect of the present work is that our analysis looks at combinations of all parameters of the theory to identify regions that survive all the theoretical and experimental constraints. We further focus on regions that could accommodate the observed Higgs signal as either the light or the heavy CP-even Higgs, and are thus interesting from a collider study perspective. This enables us to draw conclusions about correlations between different masses and mixing angles to help identify aspects of the model that warrant future study.
We start by briefly introducing the structure and parameters of the Type II 2HDM in section 2. In section 3, we discuss the theoretical constraints and experimental bounds, and outline our analysis methodology. In section 4, we present our results for the light CPeven Higgs being the observed 126 GeV SM-like Higgs boson, looking at surviving regions JHEP01(2014)161 Imposing the perturbativity and unitarity bounds, as explained below in section 3.1, typically leads to an upper bound on the masses of H 0 , A 0 and H ± . The couplings of the CP-even Higgses and CP-odd Higgs to the SM gauge bosons and fermions are scaled by a factor ξ relative to the SM value -these are presented in table 1. In order to translate the ATLAS and CMS limits, we need to pay particular attention to the couplings of the light (heavy) CP-even Higgs to the SM gauge bosons (controlling the partial decay width to W W , ZZ as well as γγ channels) and to the top quark (controlling the gluon fusion production cross section), as well as to the bottom quark (controlling the bb partial decay width, which enters the total decay width as well). From table 1, we see that the relevant couplings are proportional to sin(β − α) (cos(β − α)), 1/ sin β and 1/ cos β. Thus, even though it is customary to look at the combination of parameters (sin α, tan β), we present our results in section 4 and 5 using sin(β − α) and tan β as the independent parameters (in addition to the masses of the physical Higgses) to manifest the effects on the Higgs couplings to gauge bosons. Using sin(β − α) instead of sin α has the additional advantage of being basis-independent, as explained in ref. [27][28][29].

Theoretical and experimental constraints
To implement the various experimental and theoretical constraints, we have employed two programs: the 2HDM Calculator (2HDMC) [30] to calculate the Higgs couplings, compute all the decay branching fractions of the Higgses, and implement all the theoretical constraints; and HiggsBounds 3.8 [31] to consistently put in all the experimental constraints on the model. Here, we briefly describe the list of theoretical and experimental bounds that are of interest.
Theoretical constraints: • Vacuum Stability: this implies that the potential should be bounded from below, which is translated to various conditions for the quartic couplings in the Higgs potential [36][37][38]: With eqs. (2.4) and (2.5), the above requirements serve to constrain the Higgs masses and angles.
• Perturbativity: 2HDMC imposes constraints on the physical Higgs quartic couplings, specifically demanding that λ h i h j h k h l < 4π to stay inside the perturbative regime.

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Note that even though these are different from the λs in the Higgs potential in eq. (2.1), we can still use eqs. (2.4) and (2.5) as rough guides to understand the perturbative bounds, as we will do in later sections to explain the features of our results. The top yukawa coupling y t could also become nonperturbative for very small tan β. We require the perturbativity of y t at scales below 1 TeV, which results in tan β 0.35 [39].
• Unitarity: it is well known that in the SM, the scattering cross section for the longitudinal W modes is unitary only if the Higgs exchange diagrams are included. Since the couplings of the Higgs are modified in the 2HDM, we need to ensure unitarity by demanding that the S matrix of all scattering cross sections of Higgs−Higgs and Higgs−V L (where V L is either W L or Z L ) have eigenvalues bounded by 16π [40]. constrain any new physics model that couples to the W and Z. In particular, T imposes a strong constraint on the amount of custodial symmetry breaking in the new physics sector. In the case of 2HDM, the mass difference between the various Higgses are therefore highly constrained [66], which leads to interesting correlations between some of the masses, as will be demonstrated in section 4 and section 5. In our analysis, we require the contribution from extra Higgses to S and T to fall within the 90% C.L. S − T contour, for a SM Higgs reference mass of 126 GeV [67]. In addition, the charged Higgs contributes to Zbb coupling [68], which has been measured precisely at the LEP via the observable R b = Γ(Z → bb)/Γ(Z → hadrons) [69]. Imposing bounds from R b rules out small tan β regions for a light charged Higgs.
We also show the effect on the available parameter spaces once bounds from flavor sector are imposed in addition to the ones described. To do this, we employed the program SuperIso 3.3 [70], which incorporates, among other things, bounds from B → X s γ, [71][72][73][74][75][76][77]. A summary of flavor bounds can be found in ref. [78]. We have used the latest bounds either from PDG [71]  Excluded by: the m H ± versus tan β plane (left panel) and the m H ± versus m h plane (right panel). While B → X s γ excludes m H ± up to about 300 GeV for all tan β, B − → τ −ν τ and ∆M B d provide the strongest constraints at large and small tan β, respectively. The strongest bound on the neutral Higgs mass comes from B s → µ + µ − , which excludes m h at about 50 GeV or lower.
In addition, we included the latest results from BaBar onB → Dτν τ andB → D * τν τ [79], which observed excesses over the SM prediction at about 2 σ level. We treat the observed excesses as upper bounds and take the 95% C.L. range as R(D) < 0.58 and R(D * ) < 0.39. Note that as pointed out in ref. [79], the excesses in both R(D) and R(D * ) can not be simultaneously explained by the Type II 2HDM [80,81]. Other new physics contributions have to enter if the excesses in both R(D) and R(D * ) stay in the future. Flavor constraints on the Higgs sector are, however, typically more modeldependent. Therefore, our focus in this work is mainly on the implication of the Higgs search results on the Type II 2HDM, and we only impose the flavor bounds at the last step to indicate how the surviving regions further shrink.

Analysis method
In our analysis, we considered two scenarios: In certain regions in which very few points are left after all the constraints are imposed, we generated more points with smaller steps. We used the 2HDMC 1.2beta [30] which tested if each parameter point fulfills the theoretical and experimental constraints implemented in HiggsBounds 3.8 [31]. New LHC results that are not included in HiggsBounds 3.8 were implemented in addition. In particular, the CMS results on MSSM Higgs search in τ τ channel [61][62][63][64] were imposed using the cross section limits reverse-engineered from bounds in m A − tan β plane for m max h scenario, as provided in HiggsBounds 4.0 [31]. We also required each parameter point to satisfy the precision constraints, in particular, S and T , as well as R b .
We further required either h 0 or H 0 to satisfy the dominant gluon fusion cross section requirement for γγ, W W and ZZ channels to accommodate the observed Higgs signal at 95% C.L. [4,7]: in which we have taken the tighter limits from the ATLAS and CMS results, as well as the tighter results for the W W and ZZ channel. In the last step, we imposed the flavor bounds on all points that satisfy eq. (3.7) using the SuperIso 3.3 program to study the consequence of the flavor constraints.

Cross sections and correlations
Before presenting the results of the numerical scanning of parameter regions with all the theoretical and experimental constraints imposed, let us first study the tan β and sin(β − α) dependence of the cross sections for the major search channels at the LHC: gg → h 0 → γγ, W W/ZZ. Both production cross sections and decay branching fractions are modified relative to the SM values: for XX = γγ, V V . Note that since the W W and ZZ couplings are modified the same way in the Type II 2HDM, we use V V to denote both W W and ZZ channels. The ratio of the gluon fusion cross section normalized to the SM value can be written as: The expression for the fermion loop functions A 1/2 (τ t,b ) can be found in ref. [66]. The first term in eq. (4.2) is the top-loop contribution, and the second term is the bottom-loop contribution. In the SM, the top-loop contributes dominantly to the gluon fusion diagram, while the bottom-loop contribution is negligibly small. The situation alters in type II 2HDM for large tan β, when the bottom-loop contribution can be substantial due to the enhanced bottom Yukawa [12]. We also rewrite it in sin(β − α), cos(β − α) and tan β in eq. (4.3) to make their dependence explicit.
In the left panel of figure 2, we show contours of σ/σ SM for the gluon fusion: σ/σ SM = 0.5 (green), 1 (red), and 2 (blue). While contours of σ/σ SM ≥ 1 accumulate in sin(β − α) ∼ −1 region, there is a wide spread of the contours for sin(β − α) > 0. For most regions of sin(β − α) < 0, gg → h 0 is suppressed compared to the SM value due to cancellations between the cos(β −α) and sin(β −α) terms in the top Yukawa coupling, as shown in eq. (4.3). Note that we have shown the plots only for tan β ≤ 4 since the model is perturbatively valid only for tan β 4, as will be demonstrated below in the results of the full analysis. The h 0 decay branching fractions h 0 → V V, γγ can be written approximately as where we have explicitly listed the dominant bb and W W/ZZ channels and used "+ . . ." to indicate other sub-dominant SM Higgs decay channels. In the right panel of figure 2, we show contours of Br/Br SM for V V (solid lines) and γγ (dashed lines) channels. Both V V and loop induced (dominantly W -loop) γγ channels exhibit similar parameter dependence on tan β and sin(β − α) since both channels are dominantly controlled by the same h 0 V V coupling. While contours of Br/Br SM 1 appear near sin(β − α) ∼ ±1 for unsuppressed h 0 V V couplings, h 0 → γγ shows some spread for negative sin(β − α) and small tan β due to the correction to top Yukawa in the loop-indued h 0 γγ coupling.
Combining both the production and the decay branching fractions, we present the contours of σ × Br/SM in figure 3 for γγ (left panel) and V V (right panel) for σ × Br/SM = 0.5 (green), 1 (red), and 2 (blue). Once we demand that the cross sections for these processes be consistent with the experimental observation of a 126 GeV Higgs, as given in eq. (3.7), the allowed regions of parameter space split into four distinct regions, as indicated by the shaded gray areas. There are two narrow regions one each at sin(β − α) = ±1 (the gray regions at sin(β − α) = ±1 overlap with the picture frame boundary and are therefore hard to see), one extended region of 0.55 < sin(β−α) < 0.9, and one low tan β region around sin(β − α) ∼ 0.3 for tan β ∼ 0.5. Constraints from R b disfavor this low tan β region and Color map indicates the density of points with red being the most dense region and blue being the least dense region. Also indicated by the small rectangular box is the normalized signal cross section range of γγ between 0.7 and 1.5, and V V channels between 0.6 and 1.3 [4,7]. therefore we will not discuss it further. In what follows, we will display separate plots for positive and negative sin(β−α) to show the different features that appear in these two cases.
In figure 4, we show the correlations for σ × Br/SM for the γγ channel against V V , for negative (positive) values of sin(β − α) in the left (right) panel as a density plot. Color coding is such that the points in red are the most dense (i.e., most likely) and points in blue are the least dense (i.e., less likely). Also indicated by the small rectangular box is the normalized signal cross section range of γγ between 0.7 and 1.5, and V V channels between 0.6 and 1.3, as given in eq. (3.7) [4,7]. Note that the corresponding signal windows in tan β versus sin(β − α) plane are also sketched in figure 3 as the shaded gray regions. For negative sin(β − α), there are two branches: the one along the diagonal line with γγ : V V ∼ 1 and σ γγ 1, which can be mapped on to the sin(β − α) = −1 branch in figure 3. The other branch in the upper-half plane where γγ : V V 2 and σ γγ extends to 2 or larger is strongly disfavored given the current observed Higgs signal region.
For positive values of sin(β − α), the diagonal region is the most probable, with γγ : V V 1 and σ γγ possibly extending over a relatively large range around 1. Branches with σ γγ or σ V V ∼ 0 along the axes are strongly disfavored given the current observation of the Higgs signal.  be accompanied by an excess in the ZZ and W W channels, and this fact serves as an important piece of discrimination for this model as more data is accumulated.
The above analysis illustrates the cross section and decay branching fraction behavior of the light CP-even Higgs when it is interpreted as the observed 126 GeV SM-like Higgs, using the approximate formulae in eqs. (4.2)-(4.4). Note that we have only included the usual SM Higgs decay channels in Γ total in eq. (4.4). While it is a valid approximation in most regions of the parameter space, it might break down when light states in the spectrum open up new decay modes or introduce large loop contributions to either gg → h 0 or h 0 → γγ. In our full analysis presented below with scanning over the parameter spaces, we used the program 2HDMC, which takes into account all the decay channels of the Higgs, as well as other loop corrections to the gluon fusion production or Higgs decays to γγ.

Parameter spaces
Fixing m h = 126 GeV still leaves us with five parameters: three masses, m H , m A , m H ± , and two angles tan β and sin(β −α). Varying those parameters in the ranges given in eqs. consistent with the light CP-even Higgs interpreted as the observed 126 GeV scalar particle, satisfying the cross section requirement of eq. (3.7) for gg → h 0 → γγ, W W/ZZ. The signal regions (two narrow regions at sin(β−α) = ±1, and one extended region with 0.55 < sin(β− α) < 0.9) agree well with the shaded region in figure 3. The small region around sin(β−α) ∼ 0.3, however, disappeared, due to the R b constraint [68]. Regions with tan β 4 are excluded by perturbative bounds since one of λ 1,2 becomes non-perturbative for larger value of tan β (cos β → 0), as shown in eq. (2.4). Consequently, the bottom loop contribution to the gluon fusion production cross section [8] is not a major factor for the h 0 -126 case.
To further explore the flavor constraints, we show in figure 5 the regions enclosed by the black curves being those that survive the flavor bounds. As can clearly be seen, flavor bounds do not significantly impact the surviving signal regions.
The right panel of figure 5 shows the allowed region in the sin(β−α)−m H plane. Imposing all the theoretical constraints, in particular, the perturbativity requirement, translates into an upper bound on m H of around 750 GeV. Higgs search bounds from the LHC removes a large region in negative sin(β−α), mostly from the stringent bounds from W W and ZZ channels for the heavy Higgs. The positive sin(β − α) region is less constrained since gg → H 0 → W W/ZZ are much more suppressed. R b , in addition, excludes part of the positive sin(β − α) region with relatively large m H . Requiring h 0 to fit the observed Higgs signal further narrows down the favored regions, as shown in dark red. For sin(β −α) = ±1, m H could be as large as 650 GeV. For 0.55 sin(β − α) 0.9, m H is constrained to be less than 300 GeV. The correlation between m H and sin(β − α) indicates that if a heavy CP-even Higgs is discovered to be between 300 and 650 GeV, sin(β − α) is constrained to be very close to ±1, indicating the light Higgs has SM-like couplings to the gauge sector. In figure 6, we present the parameter regions for tan β versus m H with sin(β − α) < 0 (left panel) and sin(β − α) > 0 (right panel). Regions with large m H are typically realized for small tan β roughly between 1 and 2. There are also noticeable difference for positive or negative sin(β − α) for regions that survive all the experimental constraints (red regions). Negative sin(β − α) allows larger values of tan β for a given mass of m H . Small values of tan β is disfavored by the perturbativity of top Yukawa coupling [39], R b [68], and the flavor constraints [78]. Figure 7 shows the parameter regions in sin(β − α) versus m H ± (left panel) and m A (right panel). For negative sin(β−α) between −0.5 to −0.1, only regions with m A < 60 GeV survive the LHC Higgs search bounds. This is because H 0 → A 0 A 0 opens up in this region, which leads to the suppression of H 0 → W W/ZZ allowing it to escape the experimental constraints. The corresponding surviving region in 120 GeV < m H ± < 200 GeV is introduced by the correlation between m A and m H ± due to ∆ρ constraints. Imposing the cross section requirement for h 0 to satisfy the Higgs signal region results in three bands in both m A and m H ± , with masses extending all the way to about 800 GeV. Imposing the flavor constraints leaves regions with m H ± 300 GeV viable for sin(β − α) = ±1 or sin(β − α) between 0.55 and 0.9, while even smaller values for m A remain viable at sin(β − α) = ±1.
The allowed regions in the tan β − m H ± and tan β − m A planes share similar features before flavor constraints are taken into account, which are shown in figure 8. The top two panels show the allowed regions in the tan β − m H ± plane for negative and positive sin(β − α), while the lower two panels are for tan β − m A . LEP places a lower bound on the charged Higgs mass around 80 GeV [55,56]. In the signal region for sin(β − α) < 0, both m H ± and m A are less than about 600 GeV, while their masses could be extended to 800 GeV for sin(β−α) > 0 and tan β > 2. The difference between the m A range for different Flavor bounds, as expected, have a marked effect here ruling out any value of m H ± 300 GeV for all values of tan β, mainly due to the b → sγ constraint. For the CP-odd Higgs, only a corner of tan β > 2 and m A < 300 GeV is excluded, due to the combination of flavor and ∆ρ constraints. As shown in figure 6, only relatively light m H 300 GeV is allowed for tan β > 2. The flavor constraints of m H ± 300 GeV is then translated to m A 300 GeV since the difference between m A and m H ± is constrained by ∆ρ considerations when both m h and m H are relatively small. For tan β < 2, m H could be relatively high, which cancels the large contribution to ∆ρ from large m H ± while allowing m A to be light.
In figure 9, we present the parameter regions in the m A − m H ± plane for negative and positive values of sin(β −α). m A and m H ± are uncorrelated for most parts of the parameter space. For sin(β − α) > 0 when m A,H ± could reach values larger than 600 GeV, tan β is at least 2 or larger (see figure 8). m H is restricted to less than 300 GeV in this region, which results in a strong correlation between m A and m H ± due to the ∆ρ constraints. Figure 10 shows the parameter space in the m A −m H plane for negative (left panel) and positive (right panel) sin(β − α). These two masses are largely uncorrelated for either sign of sin(β −α). Note that for sin(β −α) > 0, large m A between 600 − 800 GeV is only possible for small values of m H 250 GeV. This is because the corresponding tan β is larger than 2, which bounds m H from above. The lower-left corners excluded by flavor constraints correspond to the upper-left corners in m A −tan β plots in figure 8, since at least one of m A or m H would need to be relatively heavy to cancel the contribution to ∆ρ from m H ± > 300 GeV.
We conclude this section with the following comments: • If h 0 is the 126 GeV resonance, then the γγ channel is closely correlated with W W/ZZ. Specifically, a moderate excess in γγ should be accompanied by a corresponding excess in W W/ZZ.
• The combination of all theoretical constraints requires tan β 4. Therefore, the bottom-loop enhancement to the gluon fusion [8] is never a major factor. Regions of sin(β − α) and tan β are highly restricted once we require the light CP-even Higgs to be the observed 126 GeV scalar particle: tan β between 0.5 to 4 for sin(β − α) = ±1, tan β between 1.5 to 4 for 0.55 < sin(β − α) < 0.9. The masses of the other Higgses, m H , m A , and m H ± , however, are largely unrestricted and uncorrelated, except for the region where sin(β − α) > 0 and m A,H ± 600 GeV, which exhibits a strong correlation between these two masses.
• The discovery of any one of the extra scalars can largely narrow down the parameter space, in particular, if the masses of those particles are relatively high.
• Flavor bounds do not change the allowed parameter space much except for the charged Higgs mass, which is constrained to lie above 300 GeV.

Cross sections and correlations
It is possible that the 126 GeV resonance discovered at the LHC corresponds to the heavier of the two CP-even Higgses, H 0 . There are a few noticeable changes for the heavy H 0 being the SM-like Higgs boson. First of all, since the coupling of the heavy Higgs to a gauge boson pair is scaled by a factor of cos(β − α) as opposed to sin(β − α), demanding SM-like cross sections for H 0 forces us to consider sin(β − α) ∼ 0, as opposed to sin(β − α) ∼ ±1 in the h 0 -126 case. Secondly, as will be demonstrated below, the bottom contribution to the gluon fusion production could be significantly enhanced since the range of tan β could be much larger compared to the h 0 -126 case. Similar to eqs. (4.2) and (4.3) in section 4, the ratios of the gluon fusion cross sections normalized to the SM can be written approximately as: Contours of σ/σ SM (gg → H 0 ) = 0.5 (green), 1 (red), and 2 (blue) are shown in the left panel of figure 11. H 0 couples exactly like the SM Higgs for sin(β − α) = 0, while deviations from the SM values occur for sin(β − α) away from zero. For sin(β − α) < 0, σ/σ SM (gg → H 0 ) is almost always larger than 1 (except for a small region around sin(β − α) ∼ −1 and tan β 10) while a suppression of the gluon fusion production is possible for positive values of sin(β − α). This is due to cancellations between the sin(β − α) and cos(β − α) terms in the top Yukawa coupling, in particular, for low tan β. The bottom loop contributes significantly when tan β is large, which enhances the gluon fusion production cross section.

, (5.3)
with the contour lines given in the right panel of figure 11. A relative enhancement of the branching fractions over the SM values are observed in extended region of negative sin(β − α), while it is mostly suppressed for positive sin(β − α). Combining the production cross sections and the decay branching fractions, contours of gg → H 0 → XX are given in figure 12 for γγ (left panel) and W W/ZZ channels (right panel). Requiring the cross section to be consistent with the observed Higgs signal: 0.7 − 1.5 for the γγ channel and 0.6 − 1.3 for the W W/ZZ channel, results in two distinct regions: a region close to sin(β − α) ∼ 0, and an extended region of −0.8 sin(β − α) −0.05.  Color coding is the same as in figure 4. Also indicated by the small rectangular box is the normalized signal cross section range of γγ between 0.7 and 1.5, and V V channels between 0.6 and 1.3 [4,7]. Figure 13 shows the correlation between the γγ and V V channels. Most of the points lie along the diagonal: γγ : V V ∼ 1. A second branch of γγ : W W ∼ 2 also appears, which corresponds to the very low tan β < 1 region in figure 12. This region is strongly constrained by R b and flavor bounds, and is therefore not considered further in our study.

Parameter spaces
We now present the results for H 0 -126 case with the full parameter scan, including all the theoretical and experimental constraints. Figure 14 presents the parameter regions in tan β versus sin(β − α). The color coding is the same as in figure 5, except that the signal regions in dark red are those with the heavy CP-even Higgs H 0 interpreted as the observed 126 GeV scalar.
Requiring the heavy CP-even Higgs to satisfy the cross section ranges of the observed Higgs signal results in two signal regions: one region near sin(β − α) ∼ 0 and an extended region of −0.    [53], while the triangle region of 130 m A 400 GeV and tan β 13 is excluded by the LHC searches for the CP-odd Higgs in τ τ mode [58][59][60][61][62][63][64]. For the charged Higgs, small values of m H ± 80 GeV are ruled out by LEP searches on charged Higgs [55,56]. Tevatron and the LHC charged Higgs searches [58][59][60][61][62][63][64]: t → H ± b → τ ν τ b further rule out regions of m H ± 150 GeV and tan β 17. The triangle in m H ± versus tan β plot for 150 GeV m H ± 400 GeV and tan β 13 is translated from the corresponding region in tan β versus m A , due to the correlation between m A and m H ± introduced by ∆ρ, as shown below in figure 17. Imposing the flavor constraints further limits m A 300 GeV, m H ± 300 GeV and tan β 10. correlation, in the H 0 -126 case, both m h and m H are relatively small. m A and m H ± should therefore be highly correlated in order to avoid large custodial symmetry breaking in the Higgs sector. However, there is a small strip of allowed region at m H ± ∼ 100 GeV with m A between 200 − 700 GeV. This region escapes the ∆ρ constraint since for m H ± ∼ m h ∼ m H , the contribution to ∆ρ introduced by the large mass difference between m A and m H ± is cancelled by the (h 0 , A 0 ) loop and (H 0 , A 0 ) loop. Imposing the flavor constraints again limits m H ± to be larger than 300 GeV. m A is constrained to be more than 300 GeV as well due to the correlations.
The right panel of figure 17 shows  figure 15). Such excluded regions for large m A (and large m H ± due to correlation) also appears in the tan β versus m A (m H ± ) plots in figure 16.
We end the section with the following observations: • Contrary to the h 0 -126 case, fixing the heavy CP-even Higgses to be the 126 GeV resonance forces us into a small narrow region of sin(α − β) ∼ 0 with tan β 8 or an extended region of −0.8 sin(α − β) −0.05 with less restrictions on tan β.
• The light CP-even Higgs can have mass of any value up to 126 GeV, with smaller m h only allowed for sin(β − α) ∼ 0. Note that the case of nearly degenerate h 0 and H 0 is allowed, as studied in detail in ref. [22].
• m A and m H ± exhibit a strong correlation: m A m H ± , due to ∆ρ constraints.
• Flavor bounds impose the strong constraints: tan β 10, m h > 50 GeV, and m H ± > 300 GeV. m A is also constrained to be more than 300 GeV due to the correlation between m A and m H ± .

Other Higgs channels
Thus far, we have concentrated on the gluon fusion production mechanism and the dominant γγ, ZZ and W W decay channels for the Higgs. The vector boson fusion channel is another important production channel for the CP-even Higgses. For certain Higgs decay channels, for example, τ τ mode, VBF production is the one that provides the dominant sensitivity due to the excellent discrimination of the backgrounds using the two forward tagging jets and the central jet-veto [82]. Other production channels, V H and ttH associated production, can also be of interest for Higgs decay to bb. In this section, we discuss the cross sections in other search channels for both h 0 and H 0 when they are interpreted as the observed 126 GeV scalar.
In figure 18, we show the normalized cross sections for the W W/ZZ, γγ (left panel) and bb/τ τ (right panel) final states via VBF or V H associated production (both production cross sections are controlled by h 0 V V coupling) in the tan β versus sin(β − α) plane for the h 0 -126 case. For V BF/V H → h 0 → W W/ZZ, both the production and decay are proportional to sin(β − α), resulting in regions highly centered around sin(β − α) ∼ ±1 for any enhancement above the SM value. For the currently preferred gray Higgs signal regions, V BF/V H → h 0 → W W/ZZ is typically in the range of 0.5 − 1 of the SM value.
The current observation of the Higgs signal has been fitted into the signal strength in both the gluon fusion channel and VBF channel for γγ, W W and ZZ final states [4][5][6][7]. Imposing the 95% C.L. contours of the µ ggF +ttH × B/B SM versus µ V BF +V H × B/B SM on top of the one-dimensional gluon fusion signal regions as given in eq. (3.7) does not lead to additional reduction of the signal parameter space, given the VBF channel is relatively loosely constrained.
For V BF/V H → h 0 → bb/τ τ , the cross section is suppressed for most of the regions, except in the neighborhood of sin(β − α) = ±1 where SM rates can be achieved. The current preferred signal regions typically have a suppression of 0.5 or stronger for this bb/τ τ channel. There is also a strong inverse correlation between the W W/ZZ and bb/τ τ channels, since an increase in bb decay branching fraction can only occur at the expense of W W . Given the relatively loose bounds on the signal strength in the bb and τ τ channels from the LHC and the Tevatron experiments [4,[83][84][85][86], imposing the current search results for bb and τ τ channels does not lead to further reduction of the signal parameter space. Figure 19 show the σ × Br/SM plots for V V , γγ, and bb/τ τ channel via VBF/V H production for the H 0 -126 case. The qualitative features of the V V , γγ plot is the same as that of figure 12. The currently favored gray signal regions typically correspond to a normalized cross section of V BF/V H → H 0 → W W/ZZ around 1 as well.
The bb/τ τ channel, however, exhibits a very different behavior. For two regions of −0.6 ≤ sin(β − α) ≤ −0.1 and 0 ≤ sin(β − α) ≤ 0.6 (regions enclosed by the red curves in the right panel of figure 19), a normalized cross section of at least the SM signal strength can be achieved. A strong suppression, sometimes as small as 0.1, can be obtained in the other regions. The currently favored gray signal region near sin(β − α) ∼ 0 corresponds to σ/σ SM of order 1 for V BF/V H → H 0 → bb/τ τ channel, while a suppression as large as 0.5 is possible for the extended regions in negative sin(β − α). The inverse correlation between bb/τ τ and W W channels also appears in the H 0 -126 case. Similar to the h 0 -126 case, imposing the 95% C.L. range for the VBF process for γγ and W W/ZZ channel, as well as the signal strength obtained from the bb and τ τ modes does not lead to further reduction of the signal region.
We also studied gg → h 0 , H 0 → bb/τ τ channel for both the h 0 -126 and H 0 -126 cases, and noticed that for the currently favored Higgs signal regions, a factor of 2 enhancement could be realized. to about 800 GeV. The MSSM relation of m A ∼ m H ± ∼ m H in the decoupling region is also much more relaxed in the Type II 2HDM. No obvious correlation is observed between m A , m H ± , and m H for the h 0 -126 case, except for the region with large m A,H ± 600 GeV. Note also that in the Type II 2HDM with Z 2 symmetry (such that m 12 = 0) that we are considering, with the additional perturbativity and unitarity constraints imposed, there is an upper limit of about 800 GeV for the mass of H 0 , A 0 and H ± . The presence of an upper bound on the heavy Higgs masses reiterates our point that unlike the MSSM, there is no sensible decoupling limit in this case where only one light SM-like Higgs appears in the low energy spectrum with other Higgses heavy and decouple.
Observations of extra Higgses in the future would further pin down the Higgs sector beyond the SM. While the conventional decay channels of Higgses to SM particles continue to be important channels to search for extra Higgses, novel decay channels of a heavy Higgs into light Higgses or light Higgs plus gauge boson could also appear. Future work along the lines of collider phenomenology of multiple Higgs scenarios is definitely warranted.