A New Avenue to Charged Higgs Discovery in Multi-Higgs Models

Current searches for the charged Higgs at the LHC focus only on the $\tau\nu$, $cs$, and $tb$ final states. Instead, we consider the process $pp\to \Phi\to W^\pm H^\mp \to W^+ W^- A$ where $\Phi$ is a heavy neutral Higgs boson, $H^\pm$ is a charged Higgs boson, and $A$ is a light Higgs boson, with mass either below or above the $b\bar{b}$ threshold. The cross-section for this process is typically large when kinematically open since $H^\pm \to W^\pm A$ can be the dominant decay mode of the charged Higgs. The final state we consider has two leptons and missing energy from the doubly leptonic decay of the $W^+ W^-$ and possibly additional jets; it is therefore constrained by existing SM Higgs searches in the $W^+ W^-$ channel. We extract these constraints on the cross-section for this process as a function of the masses of the particles involved. We also apply our results specifically to a type-II two Higgs doublet model with an extra Standard-Model-singlet and obtain new and powerful constraints on $m_{H^\pm}$ and $\tan\beta$. We point out that a slightly modified version of this search, with more dedicated cuts, could be used to possibly discover the charged Higgs, either with existing data or in the future.


I. INTRODUCTION
The quest to unveil the mechanism responsible for the breaking of the electroweak symmetry made a huge leap forward with the recent discovery of a scalar particle whose quantum numbers and interactions appear to be compatible, albeit with large uncertainties, with those of the Standard Model (SM) Higgs boson [1]. The presence of a fundamental scalar particle renders electroweak physics sensitive to arbitrarily large scales possibly present in a full theory of electroweak, strong, and gravitational interactions. Solutions to this problem usually entail the introduction of new physics just above the electroweak scale. Amongst others, hints that point to the incomplete nature of the SM are the strong empirical evidence for particle dark matter, the baryon-antibaryon asymmetry of the universe, and the pattern of neutrino masses and mixing. Even before addressing these problems it is important to realize that while the structure of currently observed gauge interactions is completely dictated by the SM gauge groups alone the pattern of electroweak symmetry breaking is not. In particular, within the context of a perturbative (Higgs) mechanism there are absolutely no "symmetry" reasons for introducing a single doublet (besides the empirical observation that such a choice leads directly to the rather successful Cabibbo-Kobayashi-Maskawa pattern of flavor changing and CP violation). Moreover, it is well known that supersymmetry, one of the most popular extensions of the SM that actually addresses some of the above mentioned problems, requires the introduction of a second Higgs doublet. In view of these observations it is clear that understanding how many fundamental scalars are involved in the electroweak spontaneous symmetry breaking mechanism is one of the most pressing questions we currently face. In particular, any model with at least two doublets contain at least two charged Higgs boson (H ± ) and at least two extra neutral Higgses. In this paper we investigate a previously overlooked technique that could uncover a charged Higgs from a multi-Higgs scenario.
Direct charged Higgs production in the top-bottom fusion channel typically has cross-sections O(1 pb) [2] and discovery would be fairly difficult in this channel [3,4]. If the charged Higgs mass is lower than the top mass, it is possible to bypass this problem by looking for charged Higgs bosons in top decays (t → H + b), taking advantage of the very large tt production crosssection. Moreover, most current experimental studies consider only charged Higgs decays to pairs of fermions (H + → τ + ν, H + → cs, and H + → tb). Under these assumptions ATLAS and CMS were able to place bounds on BR(t → H + b) at the 1-5 % level [5][6][7][8] for m H ± < m t 1 . It is well known that the presence of a light neutral Higgs can significantly modify these conclusions. In fact, the H + → W + A decay (A being a neutral CP -even or -odd Higgs boson) can easily dominate the charged Higgs decay width if it is kinematically allowed and the A has non-vanishing mixing with one of the neutral components of a Higgs doublet. Such a light neutral pseudoscalar Higgs (A = a 1 ) has been looked for by BaBar [10,11] in Υ → a 1 γ → (τ τ, µµ)γ decays and by ATLAS [12] and CMS [13] in pp → a 1 → µµ direct production. These bounds are easily evaded by assuming that the lightest neutral Higgs a 1 has a singlet component. Under this condition, in the context of a type-II two Higgs doublet model (2HDM) with an additional singlet, the BR(t → bH + ) can be as large as O(10 %) for tan β < 6 (tan β being the ratio of the vacuum expectation values of the neutral components of the two Higgs doublets) even for a 1 as light as 8 GeV [14]. Trilepton events in tt production can be used to discover at the LHC a charged Higgs produced in top decays and decaying to W ± A with as little as 20 fb −1 integrated luminosity at 8 TeV center of mass energy.
At the LHC the charged Higgs can be alternatively produced in the decay of a heavier neutral Higgs (Φ). Heavy neutral Higgs bosons are dominantly produced in gluon-gluon fusion (ggF) with a significant cross-section, leading to sizable charged Higgs production rates. For our somewhat model independent analysis, we ignore possible mass relations amongst the various Higgs bosons as they depend on the exact Lagrangian of the model. In the presence of a light Higgs A the decay H + → W + A is mostly dominant for m H + < m t and remains comparable to H + → tb otherwise, depending on the values of the various parameters. Note that the H + → W + h 1 decay (we take h 1 be the particle recently discovered at the LHC) vanishes in the limit that h 1 is completely SM-like.
In this study we consider the process pp → Φ → H ± W ∓ → W + W − A as shown in Fig. 1.
The constraints we derive are valid for m A not too far above the bb threshold, where the decay The impact of the experimental cuts depends on the kinematics and is controlled by the masses of the three intermediate Higgs bosons only. We therefore derive constraints on the LHC crosssection for the considered process that depend only on the masses of the relevant particles and not 2 The ATLAS collaboration recently announced the results of a search for a similar process, where the light state A is identified with the 125 GeV CP -even Higgs, dominantly decaying into two separable b-jets [15]. They consider the semileptonic decay of the W W . This was based on the suggestion put forward in Ref. [16]. See also Ref. [17] which includes the non-resonant production of H ± W ∓ . decays to other beyond-the-Standard-Model particles our bounds depend only on m a 2 , m H ± , m a 1 , tan(β), and ϑ A (the mixing angle in the CP -odd sector). A novelty in our analysis is the exclusion of parameter space regions at low tan β. The 8 TeV LHC data analyzed so far allow one, using our approach, to probe only a relatively light charged Higgs (roughly below the tb threshold); in the future, regions in parameter space with a heavy charged Higgs will be accessible as well. We also consider the same scenario but with one of the CP -even states (h 2 ) as the heavy neutral state Φ.
The types of scenario we consider and constrain can easily be consistent with constraints on the custodial symmetry breaking parameter The paper is organized as follows. In Sec. II we discuss the production and decay cross-section for our signal. In particular, after introducing the type-II 2HDM + singlet scenario in Sec. II A we discuss charged (H ± ) and neutral (Φ) Higgs decays in Secs. II B and II C, the gg → Φ production cross-section in Sec. II D, and the total cross-section (production times branching ratios) in Sec. II E.
In Sec. III A we show the upper bound on the total cross-section that we extract from SM Higgs to W W searches. In Sec. III B we specialize the previous results to our reference scenario (type-II 2HDM with an additional singlet, Φ = a 2 and A = a 1 ) and present the new exclusion bounds at low tan β that we extract. Finally, in Sec. IV, we present our conclusions.

II. CHARGED HIGGS PRODUCTION AND DECAY
In the multi-Higgs models containing at least two SU(2) doublets, there can exist a heavy neutral Higgs (Φ) which decays into H ± W ∓ . The process is shown in Fig. 1 with the charged Higgs decaying to a light neutral Higgs A and another W boson. Looking for this process could be the first way the charged Higgs is discovered and its properties measured. This is due to the large when all particles can be on-shell. In this section, we focus on showing how large such a production cross-section times the branching ratios can be, especially in the context of the type-II 2HDM + singlet scenario. In the following subsections, we show that the branching ratios of H ± → W ± A and Φ → H ± W ∓ can be sizable when kinematics allow and the production cross-section of Φ is roughly as large as that of the SM Higgs. Our general cross-section constraints depend only on the masses of the particles involved and will be discussed in the next section. For the specific type-II 2HDM + singlet reference scenario we can constrain physical parameters (the masses; tan β; and ϑ A , the mixing angle in the CP -odd sector) without specifying the Lagrangian in the Higgs sector and we assume no mass relations among the Higgs bosons states.
A. Our example reference scenario: the type-II two Higgs doublet model with an additional SM singlet Considering the type-II 2HDM with one extra complex singlet scalar we define the field-space basis by where S is the SM-singlet and s is its possibly non-zero VEV and tan β = v u /v d . In this convention, h interacts exactly as a SM Higgs in both gauge and Yukawa interactions; H has no coupling to the   gauge boson pairs and interacts with the up-type quarks (down-type quarks and charged leptons) with couplings multiplied by cot β (tan β) relative to the SM Higgs couplings. The orthogonal state to A H and A N is the Z-boson Goldstone mode.
We define an orthogonal matrix U that transforms the CP -even field-space basis states into the CP -even mass eigenstates We define h 1 to be the particle recently discovered at the LHC and do not demand that h i are ordered by mass. We define a mixing angle between the CP -odd mass eigenstates ϑ A by where a 1 is defined to be the lighter state.
The state a 1 is identified with A in our process pp → Φ → W ∓ H ± → W ∓ W ± A. We mainly consider Φ to be the other CP -odd state a 2 but also consider the case where it is one of the CP -even states, defined to be h 2 .
When the mass of a 1 is below the bb threshold the constraints from the decay Υ → a 1 γ → (τ τ, µµ)γ at BaBar and the light scalar search at the LHC (pp → a 1 → µµ) lead to an upper bound on cos ϑ A tan β of about 0.5 [12, 13,24]. We concentrate on two benchmark a 1 masses: 8 and 15 GeV. Our results depend weakly on this mass; therefore, the 8 GeV threshold is representative of masses just below and just above the bb threshold, where the constraint cos ϑ A tan β 0.5 does and does not apply respectively.
In the parameter region where one of the CP -even Higgses h 2,3 is lighter than 150 GeV, the direct search bounds for light neutral Higgses in associated production h i a 1 (i = 2, 3) at LEP-II can be considered [25]. The final states can be, for example, 4b or 2b2τ . However, even for h 2,3 light enough for this associated production to be possible, the cross-section is proportional to the doublet component of a 1 and is usually small in our scenario. The upper bounds in [25] constrain cos 2 (ϑ A ) U 2 iH (i = 2, 3) times branching ratios as a function of the masses, but this can easily be small enough to be consistent with the bounds. We therefore ignore the LEP-II constraint throughout this paper.
The masses of the extra neutral and charged Higgs bosons can affect the custodial symmetry breaking parameter ρ = M 2 W /(M 2 Z cos 2 ϑ W ), where ϑ W is the weak mixing angle. Since we are considering extensions of the Higgs sector involving only SU (2) doublets and singlets, contributions to ∆ρ ≡ ρ − 1 appear only at loop level. In our type-II 2HDM + singlet reference scenario with Φ = a 2 (mostly doublet), A = a 1 (mostly singlet) and the SM-like Higgs boson discovered at the LHC identified with h 1 , ∆ρ depends also on the two remaining CP -even states h 2,3 . For a simple demonstration of the ∆ρ constraint, we assume that one of these two states is completely doublet (the field-space basis state H defined above). Then the main contributions to the vacuum polarization of the W ± by H ± − a 2 and H ± − H loops need to be cancelled by that of the Z by H − a 2 loop. Therefore one can roughly expect the contribution due to the mass difference between H ± and a 2 can be cancelled by that between H and a 2 , while making that of the H ± − H loop to the W boson small. (In the 2HDM, complete contributions to the oblique parameters are well depicted in the Appendix D of the reference [26].) In Fig. 2, we show the H mass range allowed at 95 % C.L. by the present determination of ∆ρ [27] for given masses of a 2 and H ± . The solid (blue) contours give the maximum value of m H required to satisfy the experimental ∆ρ constraint; the dashed (green) contours show the difference between the maximum and minimum m H required and are therefore a measure of the (low) fine tuning between m H and m H ± that we require. We find that in the parameter space where our process is dominant (m Φ − m H ± M W ) the contributions to ∆ρ can easily be compensated by the contributions of other Higgs states, although the fine tuning between m H + and m H increases as m a 2 does. It is quite possible for the H state to remain unconstrained by LHC Higgs searches. Based on this result, we simply ignore the ∆ρ constraint throughout this paper. We also ignore possible mass relations amongst the various Higgs bosons which depend on the exact details of the Higgs sector Lagrangian.

B. Charged Higgs decays
When the charged Higgs is lighter than the top quark (light charged Higgs), investigating only the usual τ ν or cs final states from its decay may not be enough for discovery. This is because the  [27]. This is for the type-II 2HDM + singlet model discussed in the text.
process H + → W + A, whose decay rate is proportional to m 3 H + , can easily dominate over the τ + ν and cs final states. The detailed analysis of the light charged Higgs from the top quark decay in the context of the type-II 2HDM + singlet is shown in Ref. [14], where the lightest CP odd neutral Higgs a 1 is the particle A. The main factors determining the BR(H + → W + a 1 ) are the SU (2) doublet fraction (at the amplitude level) in a 1 (cos ϑ A ) and tan β. According to that analysis, BR(H + → W + a 1 ) rapidly approaches unity for m H + > M W + m a 1 even when the light Higgs a 1 is highly singlet-like, as long as tan β is small. For a charged Higgs heavier than the top quark (heavy charged Higgs), the channel H + → tb opens to compete with the process H + → W + A. In the context of the type-II 2HDM + singlet, we show the dependence of the BR(H + → W + a 1 ) on cos ϑ A and tan β in Fig. 3. For low tan β 5, the value of Γ(H + → tb) is dominantly determined by the (m t /v) 2 cot 2 β term, so the BR(H + → W + a 1 ) increases for larger tan β. (See App. A for the detailed formulae.) Above threshold the ratio of the H + → W + a 1 and H + → tb decay rates is proportional to cos 2 ϑ A tan 2 βm 2 H + . For m a 1 = 8 GeV the constraint cos ϑ A tan β 0.5 applies and hence BR(H + → W + a 1 ) is at most around 30 % for m H + < 400 GeV, increasing for larger charged Higgs masses. On the other hand, The BR(H ± → W ± a 1 ). Off-shell tops and W s are included. Above threshold the ratio of the H + → W + a 1 and H + → tb decay rates is proportional to cos 2 ϑ A tan 2 βm 2 H ± for small tan β ( 5). For m a1 = 8 GeV, the constraint from the decay Υ → a 1 γ at BABAR and the light scalar search at CMS lead to an upper bound on cos ϑ A tan β of about 0.5 [12, 13,24]. In this region the black solid line therefore represents the maximum possible branching ratio. For m a1 = 15 GeV these bounds do not apply. For tan β 7, the tan β dependence of BR( we do not need to consider this bound when a 1 is heavier than about 9 GeV, so in this case the BR(H + → W + a 1 ) can be larger than 0.5, corresponding to larger values of cos ϑ A tan β when m a 1 is set to 15 GeV in Fig. 3. Far above thresholds and at low tan β we have Consequently, BR(H ± → W ± a 1 ) can be still larger than 0.5 even after the on-shell H + → tb decay opens, as long as a 1 is heavier than about 9 GeV. For large values of tan β ( 7) the tan β

C. Heavy neutral Higgs decays
The Φ → W ± H ∓ decay can easily dominate over decays into SM fermions, including top quarks.
In the type II 2HDM + singlet scenario we set A = a 1 and Φ = a 2 (which is the heavy CP -odd Higgs); then Fig. 4 shows how BR(a 2 → H ± W ∓ ) varies with tan β and the various masses. For small tan β, the branching ratio is affected by the partial width a 2 → tt, whose rate depends on cot 2 β. Since we only consider this decay and decays into SM fermions, all taking place via the doublet (A H ) component of a 2 , the sin 2 (ϑ A ) dependence cancels out of all of the branching ratios of a 2 . For our reference type-II 2HDM + singlet scenario we assume the possible decays a 2 → h i Z and a 2 → h i a 1 to be subdominant compared to Higgs. This is in order to reduce the number of parameters relevant for determining cross-section times branching ratios in this reference scenario (to be compared to the general bounds on this cross-section times branching ratios that we derive). The processes a 2 → h i a 1 are model dependent even within the type II 2HDM + singlet scenario. As for the possible decay modes a 2 → h i Z: the more SM-like the 125 GeV particle discovered at the LHC (h 1 ) is (the more h 1 ∼ h, see Sec. II A), the more suppressed the decay to h 1 Z will be. On the other hand the other final states h i>1 Z can reduce the relevant BR(a 2 → H ± W ∓ ) by up to about 1/3, if we consider that the ∆ρ constraint requires very approximate mass degeneracy of H ± and any state significantly overlapping with H (The width to ZH is equal to the width to H + W − if one ignores the phase-space factor). The results we present (e.g. the new bound in the (tan β, m H ± ) plane for m a 2 ∼ 2m t ) are not much affected by the presence of this decay mode and we will neglect it altogether in the following. Far above thresholds we have Finally let us comment on the possibility of taking Φ to be the CP -even state h 2 . As can be seen from the results collected in App. A, the decay rates are very similar to those for a CP -odd Higgs (Φ = a 2 case). For this case we similarly neglect the two-body decays to Za i , a i a j , and h 1 h 1 .
In this case too, the mixing-matrix-element-squared U 2 2H (see Sec. II A) dependence cancels out of all branching ratios and appears only in the production cross-section.

D. Heavy neutral Higgs production
The dominant production mechanism for h SM at the LHC is ggF mediated by quark loops, mainly dominated by the top quark loop due to its large Yukawa coupling. The production crosssection of Φ depends on its modified couplings to up-and down-type quarks. The A H and H interaction states, defined in Sec. II A, have couplings to up-type quarks suppressed by 1/ tan β and couplings to down-type quarks enhanced by tan β. The production of a 2 is also modified at leading order since there are different form factors for the scalar and pseudoscalar couplings; CP -even Higgs bosons couple to fermions via scalar couplings and CP -odd couple via pseudoscalar.
At leading order the ggF production cross-section for a scalar or pseudoscalar φ is proportional to where g φ is the relative coupling to the quark q (relative to that of the SM Higgs) and m q is the quark pole mass. The form factors A φ 1/2 are equal to for scalar and pseudoscalar couplings respectively. The universal scaling function f can be found, for example, in Ref. [28,29]. In the limit τ → 0 the functions A H 1/2 (τ ) and A A 1/2 (τ ) tend to 4/3 and 2 respectively, so the ratio squared tends to 2.25. The K-factors (the ratios of cross-sections to their leading order approximations) are typically around 1.8 and cannot be neglected. In this work, to calculate the CP -odd (A H ) and CP -even (H) doublet production we take the 8 and 14 TeV ggF production cross-sections recommended by the CERN Higgs Working Group [30] (calculated at NNLL QCD and NLO EW) for a SM Higgs of the same mass M and multiply by the ratio where g q = {tan(β), cot(β)} for {down-, up-} type quarks q. (This is also the approach taken in Ref. [31].) We checked the consistency of this approach using the Fortran code HIGLU [32] at At moderate and large tan β (i.e. tan β 5) heavy neutral Higgs production in bottom fusion (bbF, upper right plot in Fig. 1) can be larger than in gluon fusion (ggF, upper left plot in Fig. 1).
In fact, although the probability to find a bottom quark in a proton is small (whereas gluons have the largest parton distribution function at LHC center-of-mass energies), this is compensated by the fact that bbF is an electroweak tree-level process (whereas the ggF is one-loop suppressed).
In the lower plots of Fig. 5 we show the impact of adding the bbF cross-section (calculated using FeynHiggs [33]) to the ggF one for √ s = 14 TeV; clearly the effect is sizable only for large values of tan β 10. Note that at small tan β ggF is large and dominant and that at large tan β bbF controls the cross-section; at intermediate values of tan β ∼ 5 the ggF suppression is not yet compensated by the bbF enhancement and we find relatively small cross-sections.

E. Total cross-sections
Combining the previous results, we can obtain the complete cross-section times branching ratios σ(gg → a 2 → W + W − a 1 ) at 8 TeV in Fig. 6 for various masses and values of tan β and cos ϑ A .
For small tan β, we can easily obtain a total cross-section times branching ratios O(pb), which is comparable to the SM Higgs production times BR(h SM → W + W − ). Hence the LHC Higgs search   result can constrain the maximum total cross-section of our process, as will be discussed in the next section. For very large tan β ( 20) our study is not very sensitive because the tan β dependence of the a 2 production cross-section (responsible for the enhancement of the latter at large tan β) is compensated by the tan β suppression of the branching ratio BR(a 2 → W + W − a 1 ). The complete branching ratios BR(a 2 → W + W − a 1 ) are calculated as outlined in App. B and are shown in Fig. 7.
For comparison, we also show the expected total cross-section at 14 TeV for both Φ = a 2 and Φ = h 2 in Figs. 8 and 9 respectively. Note that in these plots we add the ggF and bbF production cross-sections. The most important effect of adding the latter is the flattening of the total cross-section for tan β 5; therefore, once we achieve sensitivity to tan β ∼ 5 we expect to be sensitive to all values of tan β (depending on cos θ A ). When Φ = h 2 , the complete cross-section σ(gg → h 2 → W + W − a 1 ) divided by the mixing-element-squared U 2 2H is shown. We show that it is possible to have total cross-sections O(10 pb) in some regions of the parameter space. In these figures we include also a heavier charged Higgs masses, above the tb threshold.
In this method of estimating the total cross-section, multiplying the production cross-section by the branching ratios, the non-zero width of the heavy state Φ is neglected. We check that for m Φ above the H ± W ∓ threshold, going beyond the zero-width approximation for Φ is a numerically small effect in the parameter space we consider. Below the H ± W ∓ threshold the finite width effects can be important if the width of Φ is already comparable to the widths of H ± and W ∓ (the dominant contribution can come from Φ going off-shell rather than H ± or W ∓ ). We find that this can only occur at extreme values of tan β ( 20 or ∼ 1 if m Φ > 2m t ). In these cases our method can underestimate the below threshold (off-shell) total cross-section. See App. C for more details of the Φ width. Our zero-width approximation for the heavy state Φ does not affect the limits that lead to extra jets or leptons. We, therefore, expect this search to provide strong constraints on the charged Higgs production mechanism we consider and potentially to offer an avenue to discover a charged Higgs. However, due to the presence of the light Higgs A, the distributions of kinematic variables that we obtain are different from those expected in the SM Higgs search. In order to apply the results in Ref.
[18] we need to calculate how the efficiency of the various cuts adopted in that analysis are affected by the presence of the light Higgs A.
The constraints that we derive are valid for a light Higgs A whose mass is just above the bb  . 9. The complete cross-section times branching ratios σ(gg, bb → h 2 → W + W − a 1 )/U 2 2H at 14 TeV. The contribution from bbF is added to the contribution from ggF. U 2H is the H amplitude in the CP -even state h 2 . This mixing element suppresses the production of h 2 , but (given the assumptions outlined in subsection II C) cancels out of the branching ratios. state with high enough p T to count as a jet will in fact have its final state b quarks cluster into a single jet most of the time. This has been explicitly checked in Ref. [14] (for A → µ + µ − , τ + τ − ) and in Ref. [34] (for A → bb-see Fig. 6 therein). Using MadGraph we checked that for m A up to around 15 GeV, the ∆R angular opening of the two b quarks is small enough to treat the bb system as a single fat jet (obviously for a low enough p T cut and/or a large enough m A , the two final state b quarks can look like two distinct jets).
For m A below the bb threshold A will decay mainly to τ lepton pairs or maybe to charm quark pairs. For example, for A = a 1 , decaying via its A H admixture, decays to τ pairs will dominate until very low tan β ≈ 1.3, where decays to charm pairs begin to overtake [24]. For such decays into charm quarks the opening angle cannot exceed 0.5 and the decay products will mostly be clustered into a single jet. For the decays into τ leptons, the decay products will also mostly be clustered into a single jet and give no additional isolated leptons; the exception is when both τ leptons decay leptonically (about 13 % of the time). In this case there will be no jet and quite possibly extra isolated leptons that would lead to the event not passing the selection criteria in the CMS analysis.
This small effect should not much affect our results.
The CMS collaboration presented exclusion bounds obtained using two different techniques to isolate the signal from the background. The first is a cut-based analysis in which separate sets of kinematic cuts are applied for each different Higgs mass hypothesis. The second is a shape-based analysis applied to the distribution of events in the two-dimensional (m T , m ) plane. In this paper, we apply the cut based analysis of Ref.
[18] to our signal; at this time, we cannot proceed with the shape-based analysis since the CMS note does not provide enough detail.
All of the CMS data are split into four channels depending on whether the two leptons have different or the same flavor (DF, SF) and whether there is zero or one high p T (> 30 GeV) jet (0j, 1j). In each channel the expected background, expected signal, and observed data are given for several SM Higgs mass hypotheses. For each of these hypotheses a different set of cuts is applied.
The cuts used for SM Higgs searches with mass hypotheses 120, 125, 130, 160, 200, and 400 GeV are presented in Tab. 1 of Ref. [18]. Extra cuts are also applied for the SF channels in order to suppress background from Drell-Yan processes.
In this paper, we analyze the 19. 5  In the type-II 2HDM + singlet reference scenario the expected signal in the 0j channel is then where the exact ϑ A dependence has been factored out. Here s H F 0 is the number of expected events for each of the six SM Higgs mass hypotheses H in each channel F0 in Tab. 4 of the CMS note [18].
B H σ H is the production cross-section times branching ratio for that SM Higgs. The production cross-section times branching ratio for gg → a 2 → H ± W ∓ is given by sin 2 ϑ A σ P B P a 2 . In the branching ratio for H ± → a 1 W ± we factor out the cos 2 ϑ A dependence and define A P = Γ(H ± → is the fraction of events that have one more jet (in addition to those from initial or final state QCD radiation) passing the jet selection due to the decay of a 1 . Here these events are therefore removed from the 0j channel and appear in the 1j channel. a HP F ,rel is the relative acceptance for our signal and, for each Higgs mass hypothesis H, is defined as the ratio of the fraction of pp → a 2 → W W a 1 events that survive a given cut H to the fraction of SM Higgs events that survive the same cut. Both of these numbers depend on F since extra cuts are applied in the SF channels. The exact definition of this relative acceptance is a HP F ,rel = (# of events passing the cut / total # of events before the cut) NP(P) (# of events passing the cut / total # of events before the cut) SM(H) .
For the expected signal in the 1j channels, we obtain To obtain the values of a HP F ,rel and x HP F , we used MadGraph 5 [36] where the dominant gg → Φ production is written from FeynRules [37], and obtained consistent results with SHERPA 1.4.0 [38].
Since the kinematic cuts are independent of the interaction couplings (and thus tan β or cos ϑ A ), the a HP F ,rel and x HP F parameters depend only on the Φ, H ± , and A masses. The width of Φ does technically depend on tan β, but the tan β-independent contribution from Φ → H ± W ∓ is dominant whenever it is important (see App. C). The effects of the cuts do not depend strongly depend on the width of H ± . Our simulated events do not include any jets from initial or final state radiation, which is the main source of 1j events in the SM case; using values for x HP F and a HP F ,rel extracted from these simulations, especially in the s C F 1 (1 − x CP F )a CP F ,rel part of the above 1j channel equation, is therefore just a reasonable approximation, since the kinematic effects of initial and final state radiation are neglected. The ratio x HP F measures the fraction of events with n jets (due to initial or final state QCD radiation) that end up in the (n+1)-jets bin due to hadronic decay of the pseudoscalar Higgs into high-p T b-hadrons: in principle we expect the numerical value of this ratio to be different for the cases n = 0 and n = 1. Taking into account that we did not observe a strong sensitivity of the bounds we extract to the precise value of this ratio and that jet isolation requirements would imply a further reduction of the ratio for n = 1 (thus increasing the number of expected signal events and strengthening the exclusion bounds), we believe that Eq. (12) represents a reasonable and conservative approximation. for the set of cuts H and channel F that sets the best limit. The results turn out to be independent of m A to very good accuracy for the range that we consider.
For each 95 % CL s limit H F J , derived as shown in App. D, we obtain the allowed parameter space by imposing We apply whichever of these conditions leads to the best upper limit on the production crosssection times branching ratios for our signal. These limits on cross-section times branching ratios are model independent in the sense that they apply to any model containing Φ, H ± , and A particles and depend only on the masses of these particles. Moreover, they do not depend on the CP nature of the Φ and A Higgs bosons because the Φ is produced on-shell and the structure of the φ → φ V decay (where φ ( ) are spin-0) does not depend on the CP nature of the φ ( ) (see App. A). These cross-section limits are shown in the upper plots in Fig. 10 and they are superimposed on our reference scenario in Fig. 6. When deriving these limits l we assume a fractional systematic error for the expected signal appearing in each channel of 30 %, which we consider to be conservative (see App. D). We find that the limits hardly vary with m A at all for the range that we consider.
The peaks that appear in the left plot are due to us only having data for discrete values of the SM Higgs mass hypothesis. For instance, the most prominent peak corresponds to the Φ mass at which the 400 GeV cuts take over the 200 GeV cuts in providing the best upper limit. Currently only very low values of tan β ( 2) can be constrained in our reference scenario. The strongest constraint is obtained near the tt threshold region, for this reason we choose m a 2 = 360 GeV as a reference point in the detailed parameter space study presented in the next subsection.
If the analysis were to be performed again using a more appropriate set of cuts for each set of masses the suppression due to the relative acceptance (see Eq. (11)) could certainly be reduced. In fact, since the SM Higgs to W W signal and our signal are very similar, it is reasonable to presume that optimized cuts would lead to relative acceptances closer to unity. This would remove the peaks and slightly lower the baseline in the plot in Fig. 10, leading to an order of magnitude improvement on the upper limit in some parts of the parameter space. Existing 8 TeV data could, therefore, be used to probe more moderate values of tan β. Estimating the possible sensitivity of a dedicated search at √ s = 14 TeV is not simple, nonetheless the problem is one of distinguishing a signal over the uncertainty of the background. Assuming that with more data the background determination continues to be statistics limited and assuming that going from 8 to 14 TeV the background crosssection roughly doubles we can very roughly predict that at 14 TeV with 100 fb −1 (500 fb −1 ) of data a dedicated analysis could be sensitive to cross-sections of order 0.6 pb (0.3 pb), to be compared with the kinds of signals predicted in Figs. 8 and 9. A proper analysis would need to be carried out by the experimental groups after collecting more data.
It is also worth pointing out that our x HP F parameter is almost always closer to unity than to zero. In the SM search the limits coming from the 0j and 1j channels are comparable. In our case, however, the best limit almost always comes from the 1j channels, with the 0j channels setting much weaker limits. Almost as many events are moved out of the 1j channels due to the non-zero x HP F than are moved from the 0j into the 1j channels, so the large x HP F does not significantly increase the limits coming from the 1j channels; it just weakens the limits coming from the 0j channels. However, if one were to look at a 2j channel, with the same cuts as in the 0j and 1j channels, but requiring exactly two high p T (> 30 GeV) jets, the situation could be different.
Such a channel would not be useful for the SM Higgs to W W search (the 2j channel discussed in the CMS analysis [18] has completely different cuts and is designed to single out vector boson fusion production) and is therefore not considered in SM searches. However, for our process the probability to have two high p T jets even in the ggF production, one coming from initial or final state radiation and another coming from the A decay, is significant. Such a 2j channel would also likely have a smaller background and could lead to better limits than the 1j channels for which we have data.
If we replace the a 2 with one of the CP -even states, Φ = h 2 , in our type-II 2HDM + singlet scenario the analysis is similar. In this case there is, however, another independent parameter, the H fraction in h 2 , U 2 2H . This affects the production of but not the decays of h 2 under the assumptions outlined in subsection II C.

B. The type-II 2HDM plus singlet case
As explained in the previous section, SM Higgs W W searches allow one to place model independent constraints on a charged Higgs produced in the decay of a heavy neutral Higgs and decaying to W ± A, where A is a generic light neutral Higgs. In this section we apply the results presented in Sec. III A to the special case of a type-II 2HDM with an extra SM singlet. In the context of this model the limits worked out in Sec. III A apply at relatively low tan β ( 2).
In Fig. 11 we show the limits we obtain for m a 2 = 360 GeV. As explained in the previous section  [14] in terms of tan β and m H + when m a2 = 360 GeV. The additionally excluded parameter region by our search for gg → a 2 → W + W − a 1 is shown as purple area surrounded by the thick black line. We choose the mass m a2 = 360 GeV as a reference point since the strongest constraint can be obtained nearby the tt threshold as mentioned in Sec. III A. The red region is excluded by a direct t → bH + → bW + a 1 → bW + τ + τ − search at CDF [39]. The white region is not excluded. Above: the mass of the light neutral Higgs m a1 = 8 GeV, which is constrained by the Υ decay at BaBar and a 1 → µµ at CMS, represented by the light pink region (above the dashed line). The regions excluded by searching for τ ν and cs final states are shown in the blue and green respectively. Below: the mass of the light neutral Higgs m a1 = 15 GeV, which is free from the BaBar and CMS bounds.
searches in the τ ν [5,6,45] and cs [46] final states, respectively. The pink region is excluded by a combination of searches at BaBar [10,11] (Υ 3s → a 1 γ channel) and at the LHC [12, 13] (direct gg → a 1 → µµ production); this pink exclusion only applies for m a 1 just below the bb threshold and not for m a 1 just above. The red area is excluded by a dedicated t → bH + → bW + a 1 → bW + τ + τ − For a 1 below the bb threshold, above the dotted lines is ruled out by negative results from searches for Υ → a 1 γ → (τ τ, µµ)γ [10,11] and gg → a 1 → µµ [12,13], which together roughly constrain tan β cos ϑ A 0.5 [14,24]. Inside the contours (below the lines in the bottom two graphs) is ruled out from the CMS 8 TeV SM Higgs to W W search at 95 % C.L. by our analysis. search at CDF [39]. The purple area surrounded by the thick black solid line is the additional region of parameter space excluded by our study in the gg → a 2 → W + W − a 1 channel.
At lower values of cos 2 ϑ A the exclusion region narrows due to the cos 2 ϑ A dependence of BR(H ± → W ± a 1 ) (see the discussion in Sec. II B). In particular, for (m a 1 , cos 2 ϑ A ) = (8 GeV, 0.1), the light charged Higgs parameter region analyzed in Ref. [14] is completely excluded (if a heavy Higgs with mass m a 2 = 360 GeV is present). On one hand, at low values of tan β 0.03 we lose sensitivity because the a 2 width becomes dominated by a 2 → tt. On the other hand, at large tan β ≥ 10 either the a 2 production cross-section or BR(a 2 → W + W − a 1 ) are suppressed and our search loses sensitivity.
Our study extends also to charged Higgs masses above the tb threshold. Unfortunately, sensitivity in this region is not currently very strong for the following two reasons. First, in this region the H ± → W ± a 1 branching ratio is suppressed at low tan β 2 and very large tan β 10 unless the charged Higgs mass is fairly large (see Fig. 3). Second, as the charged Higgs mass increases, the phase space for the a 2 → H ± W ∓ decay shrinks; this can be compensated by raising the a 2 mass at the price of a reduced production cross-section. In conclusion, we do not currently find appreciable constraints for m H ± 180 GeV. This heavy charged Higgs parameter space could be constrained in the future with more data.
In Fig. 12 we show regions that we exclude in the (m a 2 , cos 2 ϑ A ) plane at fixed values of m H ± ∈ {110, 160} GeV, m a 1 ∈ {8, 15} GeV, and tan β. The region above the dotted line is excluded by direct a 1 searches at BaBar and at the LHC (tan β cos ϑ A 0.5 [14,24]) when m a 1 is just below the bb threshold. The reason for the weakening of the limits for intermediate a 2 masses in Fig. 12 is purely due to the fact that we have data for the cuts corresponding to SM Higgs mass hypotheses of 200 GeV and 400 GeV, but nothing in between. This then causes the peaks of weakening limits in Fig. 10 and the effects can be seen in Fig. 12. (See also Fig. 6.)

IV. CONCLUSIONS
The experimental discovery at the LHC of a particle compatible with the SM Higgs boson is the first step towards a full understanding of the electroweak symmetry breaking mechanism.
Assuming that the particle discovered at the LHC is a fundamental scalar, it becomes imperative In this paper, we study a charged Higgs whose production mechanism relies on a heavy neutral Higgs (Φ) and whose dominant decay is into a light neutral Higgs (A) For m A 2m b , this particle decays dominantly to pairs of b quarks that are detected, at sufficiently high p T , as a single jet. Under these conditions, the final state is simply W + W − plus jets and is, therefore, constrained by SM Higgs searches in the W W channel (this is also mostly true for m A below the bb threshold). For m A 2m b , the A dominantly decays into τ pairs, whose decay products will also mostly be clustered into a singlet jet unless both τ 's decay leptonically. (The latter case provides no extra jets and extra isolated leptons that would lead to the event not passing the selection criteria in the CMS analysis. This may however be another useful signal to search for.) Using existing data on searches for a SM Higgs in the range 128-600 GeV we are able to place constraints on this new physics process. In particular, we find that the upper limit on the production cross-section times branching ratios for the process in (14)  in principle, to arbitrarily large charged Higgs masses. In practice, the parametric dependence of the production cross-section and branching ratios on the charged Higgs mass limits our present sensitivity to m H ± 180 GeV. However, the parameter space with a heavier charged Higgs could be constrained in the future at the 14 TeV LHC. We point out that once the contribution to production from bb fusion is taken into account alongside gg fusion, sensitivity to all values of tan β in our reference scenario should be achieved at the 14 TeV LHC. With 100 fb −1 of data we very roughly estimate that sensitivity to cross-sections of order 0.6 pb would be achieved, to be compared to the kinds of cross-sections predicted in Figs. 8 and 9. A search for the process where the charged Higgs is produced in the same way but goes to tb is also being considered [47].
Finally, let us comment on the possibility that our process might contribute sizably to the total pp → W + W − cross-section. A recent CMS measurement with 3.54 fb −1 of integrated luminosity at 8 TeV, found a slight excess in this channel: 69.9 ± 2.8 ± 5.6 ± 3.1 pb against a SM expectation of 57.3 +2.4 −1.6 pb without the inclusion of the SM Higgs contribution [48]. Even after accounting for this an additional contribution of several pb seems to be required (see for instance Ref. [49] for a possible explanation of this tension in a supersymmetric framework). If this discrepancy survives, the process discussed in this paper could potentially offer a contribution of the correct order of magnitude. Here 1 and 2 label the fermions from the W decay 4 . This formula is valid for A H → H ± W ∓ , H → H ± W ∓ , and H ± → A H W ± . For a 2 → H ± W ∓ and H ± → a 1 W ± , with the conventions defined in Sec. II A, there is a suppression by sin 2 (ϑ A ) and cos 2 (ϑ A ) respectively. Writing the integral in this way, the inner x 1 integration can be performed analytically and the remaining integrand behaves well for numerical integration and the outer integration over x 2 can evaluated numerically very quickly. For completeness, above threshold in the zero-width on-shell approximation we can In this massless fermion approximation we can write where ∆ qq = 5.67 m q is the running mass at the scale M = m A H , and n f is the QCD number of flavours at M .
Further QCD corrections for the scalar and pseudoscalar decays to quarks are derived in Refs. [52] and [53] and summarised in Ref. [29], but these are only valid in the heavy top mass limit, i.e. when the boson is light compared to the top quark.
For charged leptons l 4 This is for one particular charge of W . The equivalent formula for a Z boson is obtained by replacing W → Z everywhere. The formulae in Ref. [29] (2.20) and Ref. [50] (41,58,59) are a factor of 2 too large for the W boson case, whereas the formula for the Z boson case are correct. This is because δZ (as defined in Ref. [29]), rather than being the ratio of the Z and W widths times cos 3 ϑW , contains an extra factor of 1 /2. This is the symmetry factor relevant for the V V decays, but not the φV decays. There is also a typo in the sin 4 ϑW term in δZ in Ref. [29]. 5 This is also for one particular charge of W and the equivalent formula for a Z boson is again obtained by replacing W → Z. This on-shell formula in Ref. [ (A9) Here m b has been neglected in the integrand. The leading QCD correction can be included by using the running mass for the m 2 t factor that appears out front, which comes directly from the Yukawa coupling in the Feynman rule. In the integrand and in the integration limits the running mass is not used (for k t and k b ) so that the threshold appears in the correct place 6 . For three-body decays written in terms of the xs (energies) of two (1 and 2) of the three final states particles (1, 2, and 3) the kinematic limits are, without neglecting any masses, A formula for Γ(A H → t * t * → bW +b W − ) that is valid both above and below threshold can be obtained by doubling Γ(A H → tt * → tbW − ) and using 4γ t in place of γ t . Above threshold in the zero-width on-shell approximation we can write One of the (1 − k l ) factors comes from the matrix-element-squared and the other is the phase-space factor √ λ lν .
The leading QCD correction can be included by using the running masses for the m 2 t , m 2 b , and m b m t factors that appears out front for each of the three terms, which come directly from the Yukawa couplings in the Feynman rule. Elsewhere in the integrand m b has been neglected 8 .
Elsewhere in the integrand and in the integration limits the running masses are not used (for k t and k b ) so that the threshold appears in the correct place. Above threshold in the zero-width on-shell approximation

Off-shell H ±
In the off-shell decay Φ → W ∓ * H ± * → W ∓ * W ± A, the decay widths Γ W and Γ H ± roughly decide which one is preferred to be off-shell. The full decay width of H ± in our type-II 2HDM + singlet reference scenario is shown in comparison with Γ W in Fig. 13. For an H ± with a mass much above the tb threshold the possible three-body decay of Φ through an off-shell H ± needs to be considered.
Here 1 and 2 label the two W s. These formulae are valid for Φ, φ ∈ {A H , H} and are suppressed by mixing angles for other mass eigenstates that are not completely doublet. Appendix D: CL s limits A 1 − α confidence level CL s limit on a signal s is defined by where D is the data and λ is the expected distribution in the signal-plus-background hypothesis (H 0 ) and in the background only hypothesis (H 1 ).
For each channel and set of cuts we have a background B = b ± σ b . The signal-plus-background may be expressed as where s is the expected signal, its statistical error is taken to be √ s, its fractional systematic error is taken to be Σ. In this paper, we set Σ = 30% as a conservative bound.
We approximate everything as Gaussian. We therefore take Where Φ is the cumulative distribution function. For a given b, σ b , D, Σ, and α the 1 − α confidence level limit on s can therefore be found. We call this solution s = l. For our calculation,