Exploring the low $\tan\beta$ region of two Higgs doublet models at the LHC

Current interpretations of the LHC results on two Higgs doublet models (2HDM) underestimate the sensitivity due to neglecting higher order effects. In this work, we revisit the impact of these effects using the current cross-section times branching ratio limits of the $A\to hZ, H \to VV$ and $H\to hh$ channels. With a degenerate heavy Higgs mass $m_\Phi$, we find that the LHC searches gain sensitivity to the small $\tan\beta$ region after including loop corrections, even close to $\cos(\beta-\alpha)=0$ which is not reachable at tree level for all types of 2HDM. For a benchmark point with $m_\Phi=300$ GeV, $\tan\beta<1.8(1.2)$ can be probed for the Type-I(II) 2HDM model for $\cos(\beta-\alpha)=0$. When the deviation from $\cos(\beta-\alpha)=0$ is larger, the region for which current searches have exclusion potential becomes larger.

the A → Zh, H → V V , and H → hh channels, the tree-level couplings are proportional to cos(β − α). They therefore vanish in the "alignment limit" of cos(β − α) = 0 and, as a result, give no constraint around the cos(β − α) = 0 region [7-9, 11, [24][25][26][27][28]. Loop corrections change this picture substantially, however, and we will find below that with a combination of these channels the region around cos(β − α) = 0 is no longer unreachable. Our study shows that the searches in these channels are sensitive to small tan β values even for cos(β −α) = 0 which is unconstrained at tree level, and we present the updated limits on the parameters of two Higgs doublet models for degenerate heavy Higgs masses. The study shows that the small tan β region can be strongly constrained for all four types of 2HDM, where the experimental limits are applicable.
The paper is structured as follows. In Section 2, we give a brief introduction to 2HDMs, summarizing the experimental and theoretical results for the four decay channels described above, with the detailed formulae at one-loop level given in Appendix A. We present our individual channel analyses as well as the combined results in Section 3. Finally we give our main conclusions in Section 4.

2HDM Higgs sector
For the general CP-conserving 2HDM, there are two SU(2) L scalar doublets Φ i (i = 1, 2) with hyper-charge Y = +1/2, where v i are the vacuum expectation values (vev) of the two doublets after EWSB with v 2 1 + v 2 2 = v 2 = (246 GeV) 2 and tan β ≡ v 2 /v 1 . The Higgs sector Lagrangian of the 2HDM can be written as with a Higgs potential of where we have assumed CP conservation, and a soft Z 2 symmetry breaking term m 2 12 . After EWSB, three Goldstone bosons are eaten by the SM gauge bosons Z, W ± , providing their masses. The remaining physical mass eigenstates are h, H, A and H ± . The usual eight parameters appearing in the Higgs potential are m 2 11 , m 2 22 , m 2 12 , λ 1,2,3,4,5 , and can be transformed to a more convenient choice of the mass parameters: 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. L Yuk in the Lagrangian represents the Yukawa interactions of the two doublets. To avoid tree-level flavor-changing neutral currents (FCNC), all fermions with the same quantum numbers are made to couple to the same doublet [29,30]. By assigning different doublets to different fermions, in general, there are four possible types of Yukawa coupling: Type-I, Type-II, Type-LS (Lepton specific), and Type-FL (Flipped). In this work, the most relevant Yukawa couplings are y b and y t , which matter for the A → Zh with h → bb process and constitute the dominant loop corrections. Thus we will focus on the Type-I and Type-II 2HDMs. The situation in the Type-LS and Type-FL 2HDMs would be similar: Here we take c x = cos x and s x = sin x.

Heavy Higgs Decay Channels
The heavy Higgs decay channels we are interested in and the corresponding tree-level couplings are At the tree-level, all of these couplings vanish in the alignment limit where c β−α = 0. As a consequence, all are currently thought to be irrelevant for constraining 2HDMs in the treelevel alignment limit. However, at one-loop level, the definition of the alignment limit will shift channel-bychannel from the previous definition c β−α = 0. For c β−α = 0 and m H = m A = m H ± = (m 2 12 /s β c β ) ≡ m φ , the one-loop coupling expressions for the relevant vertices can be simplified and given by where ξ t = cot β for both the Type-I and Type-II models, while ξ b = cot β for the Type-I model and ξ b = − tan β for the Type-II model.
are the couplings between fermions and the Z-boson. (PV i ) represents some combination of Passarino-Veltman functions [31] which only depends on the masses. The full expressions in this limit can be found in Appendix A.
In the tree-level alignment limit, the dominant contributions to the above couplings come from the fermion (top and/or bottom) loop. Thus, all of these couplings are related to the Yukawa coupling modifier ξ f . In both the Type-I and Type-II models, ξ t = cot β which will be significantly enhanced in the low tan β region. Together with the large value of m t , the unexplored region around c β−α = 0 at low tan β can thus be probed by these channels.
We calculated the complete expressions for the amplitudes for all decay channels (f f , V V , SV and SS) of all scalars (h, H, A, H ± ) using FeynArts [32] and FormCalc [33], using the 2HDM model files including full 1-loop counter terms and renormalization conditions. All renormalization constants are determined using the on-shell renormalization scheme, except m 2 12 which is renormalized by MS [19]. Note that, for the SV V type couplings, we also include the Lorentz structure p µ 1 p ν 2 /m 2 V in the calculation which does not appear in the treelevel calculation. The µνρσ structure which represents a CP-odd interaction between the scalar and vector bosons can be safely ignored in our calculation, since for the CP-even scalar H, the presence of such structure indicates CP violation which can only come from the CP phase in the CKM matrix in the CP conserving 2HDM. All of these NLO amplitudes are then implemented in 2HDMC [34] to fully determine the branching fractions of different channels and the total width of each particle.

Heavy Higgs search results at LHC Run-II
A variety of searches for heavy Higgs bosons have been conducted by the ATLAS and CMS collaboration. Here we use the published cross-section times branching ratio limits to directly constrain the 2HDM parameter space, with the SusHi package [35] for the production crosssection at NNLO level, and our own improved 2HDMC code, which adds loop-level effects to the public 2HDMC code [34], for the branching ratios. For each Higgs production and decay process of interest, there exist both ATLAS and CMS public results, and these are non-trivial to apply in practice due to the assumption of a particular width in the presentation of the final results. Since we are not modelling a continuous likelihood, we do not combine these results but, instead, take the most constraining, or the limit with the widest region of applicability from the perspective of the Higgs boson widths. For points where the heavy Higgs decay widths are larger than those assumed for the derivation of the published limits, we use the largest available Γ H /m H limit, but overlay plots with a region of applicability to indicate the need for caution in our reinterpretations (a device borrowed from experimental reports such as Fig.7 of [24]). The analyses that we consider include the following:

A → Zh: Both the ATLAS [11] and CMS [24] collaborations have presented results
with h → bb. Here we choose the ATLAS report for reinterpretation, due to the clear statement of decay width Γ A /m A ≤ 10%, and it includes both b-associated and gluon fusion production modes.
As in the H → ZZ case, the heavy Higgs production here is assumed to include all modes. In the report, the cross section times branching ratio limits are estimated based on the assumption that all SM-like Higgs couplings, except for the triple Higgs coupling itself g hhh , are the same as those expected in the SM. We use the acceptance and efficiency information given and then rescale to get the cross section times branching ratio limits for the 2HDM where the SM-like Higgs couplings can depart from their SM values. The CMS results can be found in [28].

Results
To build intuition for the full impact of the one-loop corrections to heavy Higgs decays, we first provide a detailed comparison of the tree-level and one-loop-level results for each channel separately in the cos(β − α) − tan β plane.

The
In Fig. 1, the constrained parameter space is shown in the cos(β − α) − tan β plane, with the benchmark point m A = m H = m H ± = 300 GeV, m 2 H = m 2 12 /(s β c β ) . The left panel is for the Type-I 2HDM and the right one is for the Type-II. The results are shown separately for the b-associated and gluon fusion production modes, as the green and red shadow regions respectively. In the figures the tree-level results are shown with dashed lines. For the Type-I 2HDM, the effects of the limits within the region of applicability can exclude the tan β < 8 region, except for two bands. The central band around cos(β − α) = 0 has a small tree-level AhZ coupling as in Eq. (2.5) and the lower left curve band has a small hbb coupling. The green and red regions represent the one-loop level results excluded by the b-associated production and gluon-fusion production channel respectively. We display the regions of Γ A /m A > 0.1 or Γ H /m H > 0.1 with light blue backgrounds. Generally tan β < 8 is also strongly constrained, but the loop-level effects shift the the allowed region around cos(β − α) = 0 in the small tan β region. The 0.2 < tan β < 2 allowed region is shifted to the right whilst the tan β < 0.2 region is shifted to left. For the Type-II 2HDM, the shifted allowed region at small tan β is similar to that seen in the Type-I scenario. The allowed band for cos(β − α) > 0.3 arises because the hbb coupling becomes small in that region. Meanwhile the constraints in large tan β region are quite different because of the effect of the hbb Yukawa couplings, which affect the b-associated production cross-section.

The H → V V channel
As shown in Eq. (2.6), and Eq. (2.7), the tree-level HV V couplings are only proportional to cos(β − α), and they are therefore independent of the 2HDM model type. At one-loop level, the couplings will become type-dependent through fermion corrections. However, the main difference comes from the production which is similar to the A → hZ case. Hence, we will only show the results for one type and briefly comment on the difference after that.
In Fig. 2, we show the constrained parameter space in the cos(β − α) − tan β plane for the Type-I 2HDM H → ZZ channel (left panel), and for the Type-II H → W W channel (right panel), with the benchmark point m A = m H = m H ± = 300 GeV, m 2 12 /(s β c β ) = m 2 H . For the tree-level results shown with dashed red lines, in the region the limits are applicable, tan β < 2(1) is strongly constrained for the Type-I and Type-II 2HDMs except for the central band around the cos(β − α) = 0 region. At the one-loop level, the excluded regions represented by red shadow are largely separated into two parts. For the H → ZZ channel, one excluded part is at 0.3 < tan β < 5, cos(β − α) < 0 and the other one has tan β < 2 around cos(β − α) = 0. For the Type-II H → W W scenario, one is around tan β = 1 with cos(β − α) < 0 and the other one has tan β < 0.2 around cos(β − α) = 0. The large deviations from the tree-level results in the low tan β region are mainly from the influence of the large triple scalar couplings which give rise to large corrections through Higgs field renormalization as well as the large branching ratio of H → hh. Similar to Fig. 1, the regions of Γ A /m A > 0.1 or Γ H /m H > 0.1 are displayed with light blue backgrounds.
The large tan β region is less constrained because the production cross-sections of the heavy Higgs bosons are much smaller. For the H → ZZ channel, the report from CMS [9] includes both gluon fusion and b-associated production. As a result, the different types of exclusion limit for the HZZ channel are different at large tan β to the case of the A → Zh channel. On the other hand, results are only reported for gluon-fusion production in the case of the HW W channel. Hence the results are quite similar for the different model types. We also note the small red region at large tan β, which comes from the noncontinuous cross section times branching ratio limits in that region.

The H → hh channel
As shown in Eq. (2.8), the Hhh couplings at tree-level are type-independent. At one-loop level, though, the correction is type-dependent. The main differences between types come from the production mode as in previous cases. At large tan β, b-associated production makes a big difference for different types.
The results for the H → hh channel are shown in Fig. 3, where the left panel is for Type-I and the right panel is for Type-II, with the dashed red lines for tree-level results and red regions for loop-level results. Here the benchmark point is still m A = m H = m H ± = 300 GeV, m 2 12 /(s β c β ) = m 2 H . For both of the Yukawa types, there are two allowed regions. One is the region around cos(β − α) = 0, and the other one is the band starting from tan β = 1.5, cos(β − α) = −1 to tan β = 0.01, cos(β − α) = 0. The main feature here is that, in the parameter space where reports limits are applicable (non-blue region), there is nearly no allowed region, especially for tan < 0.3, at one-loop level, which is still allowed at tree-level.

Loop effects summary
Our individual analyses of the channels A → hZ, H → V V /hh have revealed that loop effects can contribute greatly in some regions, especially the small tan β region. Now we display the combined results with the same benchmark point m A = m H = m H ± = 300 GeV, m 2 12 /(s β c β ) = m 2 H . In Fig. 4, the combined results are shown in the cos(β − α)-tan β plane. In the Type-I 2HDM scenario (left panel), the allowed region is generally around cos(β − α) = 0. At treelevel, considering the parameter space where the reported limits are applicable, the allowed regions are tan β > 8, | cos(β − α)| < 0.6, tan β < 8, | cos(β − α)| < 0.02 and smaller tan β with smaller | cos(β − α)|. At one-loop level, for the non-blue region, the allowed region at tan β > 1.8 is similar, while the small tan β region is totally excluded, even if cos(β − α) = 0. The Type-II 2HDM results are displayed in the right panel. The main differences occur at cos(β − α) > 0.3. At the one-loop level, the region tan β < 1.2 around cos(β − α) = 0 is totally excluded except for the blue region. We keep in mind that the blue regions are not currently detectable because of the large heavy Higgs boson decay width in that region.
Further, the combined results are also shown in the m Φ − tan β plane in Fig. 5. Here we take m 2 H = m 2 12 /(s β c β ) and cos(β − α) = 0, ±0.01 as the benchmark parameters. We find that the Type-I and Type-LS models have quite similar results because of their similar hbb coupling, shown in the left panel. For the red line with cos(β − α) = 0, with current published limits the region m Φ < 2m t GeV for tan β < 0.5 can be constrained, and the sensitivity can be extended up to tan β ∼ 3 with lower masses except for the grey region where the decay width is too large and limits are not applicable any more. When cos(β − α) deviates from exactly 0, such as ±0.01 (shown by the blue and green lines), we can see that the constraints in the small tan β region become stronger than those for cos(β − α) = 0. In the right panel, we show the Type-II and Type-FL cases which have similar results. The exclusion limits are also similar to the Type-I and Type LS models, except for the moderately reduced constraints on tan β. With these combined exclusion regions shown in the cos(β − α) − tan β and m Φ − tan β planes, we find that loop effects in the considered channels are important in the small tan β region, especially for the cos(β − α) = 0 region which is excluded by a loop-level analysis except for the space where the limits are not applicable, while the tree-level analysis has no sensitivity. We also find that, even though the loop corrections are usually type-dependent, the difference between loop-and tree-level results becomes relatively type-independent.

Conclusions
Studies of extensions of the Higgs sector of the SM are a promising way to try and address various theoretical and experimental questions following the discovery of the SM-like Higgs boson. In the framework of 2HDMs, we have interpreted current LHC experimental limits on the cross section times branching ratio of the A → Zh, H → V V and H → hh channels at the one-loop level. In previous studies, the limits were reported at tree-level, with no limit for the region around cos(β − α) = 0 because the couplings are proportional to the parameter cos(β − α). At one-loop level, however, we have shown that these results are modified considerably.
Our results for individual channels were displayed in Fig. 1-Fig. 3 in the cos(β −α)−tan β plane, which showed that loop effects can contribute significantly in some regions of parameter space, especially in the small tan β region with cos(β −α) ∼ 0. Through the combined analysis shown in Fig. 4, we find that the region around cos(β − α) = 0 with degenerate heavy Higgs masses m Φ is detectable using these channels. Except for the regions of parameter space where current limits are not applicable due to large decay width, tan β < 1.8(1.2) can be excluded for the Type-I(II) models, for a benchmark point with m Φ = 300 GeV. The combined results in the m Φ − tan β plane were also shown in Fig. 5. Generally the sensitive region is tan β < 4. For cos(β − α) = 0, ±0.01, the sensitive region has m Φ values up to 350 GeV. For large m Φ , the tt decay channel opens, resulting in large heavy Higgs decay widths, and the current reported limits are no longer applicable. Our study also shows that the improvement of the sensitivity through loop corrections is approximately type-independent.

A Coupling Formula
Here are the more detailed equations of Eq. (2.9)-Eq. (2.12) for m H = m A = m H ± = m 2 12 /(s β c β ) ≡ m φ and cos(β − α) = 0, where ξ t = cot β for both the Type-I and Type-II models, while ξ b = cot β for the Type-I model and ξ b = − tan β for the Type-II model.f is the SU(2) partner of f .