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 $h^0$ or the heavy CP-even Higgs $H^0$. Imposing both theoretical and experimental constraints, we analyze the surviving parameter regions in $m_H$ ($m_h$), $m_A$, $m_{H^\pm}$, $\tan\beta$ and $\sin(\beta - \alpha)$. 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 $h^0$-126 case, we are restricted to narrow regions of $\sin(\beta-\alpha) \approx \pm 1$ with $\tan\beta$ up to 4, or an extended region with $0.55<\sin(\beta-\alpha)<0.9$ and $1.5<\tan\beta<4$. The values of $m_H$, $m_A$ and $m_{H^\pm}$, however, are relatively unconstrained. In the $H^0$-126 case, we are restricted to a narrow region of $\sin(\beta-\alpha) \sim 0$ with $\tan\beta$ up to about 8, or an extended region of $\sin(\beta-\alpha) $ between $-0.8$ to $-0.05$, with $\tan\beta$ extended to 30 or higher. $m_A$ and $m_{H^\pm}$ are nearly degenerate due to $\Delta\rho$ constraints. Imposing flavor constraints shrinks the surviving parameter space significantly for the $H^0$-126 case, limiting $\tan\beta \lesssim 10$, but has little effect in the $h^0$-126 case. We also investigate the correlation between $\gamma\gamma$, $VV$ and $bb/\tau\tau$ channels. $\gamma\gamma$ and $VV$ channels are most likely to be highly correlated with $\gamma\gamma:VV \sim 1$ for the normalized cross sections.


I. 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]. 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 [7][8][9][10]. The neutral components of both the Higgs fields develop vacuum expectation values (vev), breaking SU(2) L × U(1) Y down to U(1) em . Assuming no CP-violation 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 [7]). 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 [11][12][13][14][15][16][17][18][19][20][21][22][23][24]. 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. [11] and Ref. [12] 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. [13][14][15][16][17][18] fit the observed Higgs signals in various 2HDM scenarios, taken into account theoretical and experimental constraints. Ref. [19] also paid careful attention to various Higgs production modes. Ref. [20] focused on the CP-violating Type II 2HDM. Ref. [21] studied the case of nearly degenerate Higgs bosons. In addition, Ref. [22,23] investigated the possibility that the signal could correspond to the pseudoscalar A 0 -in this context, it is worth remarking that Ref. [25] 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 II. In Sec. III, we discuss the theoretical constraints and experimental bounds, and outline our analysis methodology. In Sec. IV, we present our results for the light CP-even Higgs being the observed 126 GeV SM-like Higgs boson, looking at surviving regions in various combinations of free parameters. In Sec. V, we do the same for the heavy CP-even Higgs as the observed 126 GeV SM-like Higgs boson. In Sec. VI, we explore the implications for the Vector Boson Fusion (VBF) or V H associated production, and decays of Higgs into bb and τ τ channels. We conclude in Section VII.

II. TYPE II 2HDM
In this section, we briefly describe the Type II 2HDM, focusing on the particle content, Higgs couplings, and model parameters. For more details about the model, see Ref. [7] for a recent review of the theory and phenomenology of 2HDM.

A. Potential, Masses and Mixing Angles
Labeling the two SU(2) L doublet scalar fields Φ 1 and Φ 2 , the most general potential for the Higgs sector can be written down in the following form: We impose a discrete Z 2 symmetry on the Lagrangian, the effect of which is to render m 12 , λ 6 , λ 7 = 0 2 . Note that one consequence of requiring m 12 = 0 is that there is no so called decoupling limit in which only one SM-like Higgs appears at low energy while all other Higgses are heavy and decoupled from the low energy spectrum. After electroweak symmetry breaking (EWSB): φ 0 GeV, we are left with six free parameters, which can be chosen as the four Higgs masses (m h , m H , m A , m H ± ), a mixing angle sin α between the two CP-even Higgses, and the ratio of the two vacuum expectation values, tan β = v 2 /v 1 .
Writing the two Higgs fields as: 2 Ref. [14], which also addresses similar issues as in this paper, allowed for a soft breaking of the Z 2 symmetry with m 2 12 = 0. In this paper, we don't consider such soft-breaking terms. the mass eigenstates of the physical scalars can be written as: For our purposes, it is useful to express the quartic couplings λ 1...5 in terms of the physical Higgs masses, tan β and the mixing angle α: Imposing the perturbativity and unitarity bounds, as explained below in Sec. III A, 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 I. In order to translate the ATLAS and CMS limits, we need to pay particular attention to the couplings of the light (heavy) CPeven 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 I, 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 Sec. IV and V 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. [26,27].

A. Theoretical and Experimental Constraints
To implement the various experimental and theoretical constraints, we have employed two programs: the 2HDM Calculator (2HDMC) [28] to calculate the Higgs couplings, compute all the decay branching fractions of the Higgses, and implement all the theoretical constraints; and HiggsBounds 3.8 [29] 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 [32]: With Eqs. (4) and (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. Note that even though these are different from the λs in the Higgs potential in Eq. (1), we can still use Eqs. (4) and (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 [33].
• Unitarity: It is well known that in the SM, the scattering cross section for the longi-  44] 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 [45], which leads to interesting correlations between some of the masses, as will be demonstrated in Sec. IV and Sec. V. 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 [46].
In addition, the charged Higgs contributes to Zbb coupling [47], which has been measured precisely at the LEP via the observable R b = Γ(Z → bb)/Γ(Z → hadrons) [48]. 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 [49], which incorporates, among other things, bounds from B → X s γ, [50][51][52][53][54][55]. A summary of flavor bounds can be found in Ref. [56]. We have used the latest bounds either from PDG [50]  In addition, we included the latest results from BaBar onB → Dτν τ andB → D * τν τ [57], 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. Excluded by: Note that as pointed out in Ref. [57], the excesses in both R(D) and R(D * ) can not be simultaneously explained by the Type II 2HDM [58,59]. 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 model-dependent. 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.

B. Analysis Method
In our analysis, we considered two scenarios: 6 GeV ≤ m h < 126 GeV in steps of 5 GeV.
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 [28] which tested if each parameter point fulfills the theoretical and experimental constraints implemented in HiggsBounds 3.8 [29]. 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 [43] 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 [29]. We also required each parameter point to satisfy the precision constraints, in particular, S and T , as well as 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. [4,6]: 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. (12) using the SuperIso 3.3 program to study the consequence of the flavor constraints.

A. 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. [45]. The first term in Eq. (14) 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 [11]. We also rewrite it in sin(β − α), cos(β − α) and tan β in Eq. (15) to make their dependence explicit.
In the left panel of  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 Combining both the production and the decay branching fractions, we present the contours of σ × Br/SM in Fig. 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. (12), 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 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 Fig. 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. (12) [4,6]. Note that the corresponding signal windows in tan β versus sin(β − α) plane are also sketched in Fig. 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      least 2 or larger (see Fig. 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. GeV.

VV)
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 [7] 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 • 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.

A. 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 Similar to Eqs. (14) and (15) in Sec. IV, the ratios of the gluon fusion cross sections normalized to the SM can be written approximately as: 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.
Br(H 0 → V V, γγ)/Br SM can also be expressed similar to Eq. (16): with the contour lines given in the right panel of Fig. 11.    is the same as in Fig. 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 [4,6].
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 Fig. 12. This region is strongly constrained by R b and flavor bounds, and is therefore not considered further in our study.

B. Parameter Spaces
We now present the results for H 0 -126 case with the full parameter scan, including all the theoretical and experimental constraints.      Fig. 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 Fig. 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. [21].
• 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 ± .

VI. 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 [60]. 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 Fig. 18  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,[61][62][63][64], imposing the current search results for bb and τ τ channels does not lead to further reduction of the signal parameter space. 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.

VII. CONCLUSIONS
In this paper, we presented a detailed analysis of the Type II 2HDM (with an imposed Z 2 symmetry) parameter space, identifying either the light or the heavy CP-even Higgs as the recently discovered resonance at 126 GeV. We scanned the remaining five parameters sin(β − α), tan β, m A , m H ± , and m H or m h while fixing either m h or m H to be 126 GeV.
We took into account all the theoretical constraints, precision measurements, as well as current experimental search limits on the Higgses. We further studied the implications on the parameter space once flavor constraints are imposed. We found unique features in each of these two cases.
In the h 0 -126 case, we are forced into regions of parameter space where sin(β − α) = ±1 with tan β between 0.5 to 4, or an extended region of 0.55 < sin(β − α) < 0.9, with tan β constrained to be in the range of 1.5 to 4. There is, however, a wide range of values that are still allowed for the masses of the heavy CP-even, pseudo scalar and charged Higgses. The Higgs masses are typically not correlated, except when m A,H ± 600 GeV and sin(β −α) > 0 where there is a strong correlation between m A and m H ± because of the ∆ρ constraint.
Imposing flavor constraints further restricts m H ± > 300 GeV. Note that in both cases, the extended region in sin(β − α) is of particular interest, since a deviation of the Higgs coupling to W W and ZZ can be accommodated for the observed Higgs signal at 126 GeV.
We find that in either of these scenarios, one can identify regions of parameter space that pass all theoretical and experimental bounds and still allow a slightly higher than SM rate to diphotons. γγ and W W/ZZ rates are most likely strongly correlated: γγ : V V ∼ 1 for the normalized cross sections.
We further studied the implication for the Higgs production via VBF or V H process, and decays to bb, τ τ channels. We found that in the h 0 -126 case, both V BF/V H → h 0 → bb/τ τ, W W/ZZ could be significantly suppressed in the Higgs signal region. For the