Probing the charged Higgs boson at the LHC in the CP-violating type-II 2HDM

We present a phenomenological study of a CP-violating two-Higgs-doublet Model with type-II Yukawa couplings at the Large Hadron Collider (LHC). In the light of recent LHC data, we focus on the parameter space that survives the current and past experimental constraints as well as theoretical bounds on the model. Once the phenomenological scenario is set, we analyse the scope of the LHC in exploring this model through the discovery of a charged Higgs boson produced in association with a W boson, with the former decaying into the lightest neutral Higgs and a second W state, altogether yielding a b\bar b W^+W^- signature, of which we exploit the W^+W^- semileptonic decays.


Introduction
One of the main tasks of the Large Hadron Collider (LHC) experiments is to study and understand the mechanism of Electroweak Symmetry Breaking (EWSB). Recently, the ATLAS and CMS collaborations released the results of the search for the Higgs bosons with more than 10 fb −1 data collected at 7 TeV in 2011 and at 8 TeV in 2012 [1,2]. Both experiments have recorded an excess of events above the expected background in different decay channels (mainly γγ, ZZ and W + W − ) at a mass near 125 GeV. The excess is compatible with the Standard Model (SM) Higgs boson. Complementary evidence is also provided by the updated combination of the Higgs searches performed by the CDF and D0 collaborations at the Tevatron [3].
Investigating the Higgs mechanism in the framework of the SM constitutes a major effort [4]. However, the minimal choice of Higgs sector is (so far) arbitrary. Even if the existence of a scalar resonance, compatible with an SM-like Higgs, has already been uncovered by the current data, one must take advantage of the unique opportunity to test the phenomenology of more complicated Higgs models.
Much effort has been dedicated over the years to the study of extended Higgs sectors. In this paper we consider the two-Higgs-doublet Model (2HDM) with type-II Yukawa sector. This model is one of the most popular extensions of the Higgs sector due to its strong connection with a tree-level Minimal Supersymmetric Standard Model (MSSM) [5,6], which is one of the most accredited proposals for solving some theoretical inconsistencies of the SM. As is well known, the Higgs sector of the MSSM is quite well constrained in terms of the number of free parameters on which the masses depend [7][8][9][10][11][12][13][14][15][16][17][18][19][20][21][22]. It is possible that the Higgs sector lies in a lower mass range than the superpartners of the SM particles. In this regard, the 2HDM should be explored as an effective low-energy MSSM-like Higgs sector.
While a tree-level MSSM Higgs sector is strictly CP-conserving (no mixing is allowed between the scalar and pseudo-scalar Higgs components), it has been shown that a CPviolating effective Higgs sector could be produced by loop corrections under specific circumstances [23]. Accordingly, in this paper we adopt a bottom-up approach by considering a CP-violating 2HDM with type-II Yukawa couplings.
Due to its complicated Yukawa structure, a CP-violating parameter space must be carefully constrained by theoretical arguments and experimental data. Therefore, our first aim is to provide a detailed analysis of the allowed parameter space in the light of recent LHC results. We will show that very little CP-conserving parameter space survives these data. This exploration of the allowed parameter space has been addressed recently by several authors, from different points of view [24][25][26][27][28]. Where there is overlap, we compare our results with those obtained by these authors.
Regarding phenomenology, the only way to unambiguously probe the existence of a Higgs sector with two doublets arises through the discovery of a charged Higgs boson, since this particle is the hallmark of such a structure of the Higgs sector. Hence, our second aim is to profile a charged Higgs boson in the surviving parameter space via a detailed study of its production cross-section and decay Branching Ratios (BRs).
Then, our third aim is to study the scope of the LHC in discovering a charged Higgs state. In this respect, it is well known that the production of a single charged Higgs state at a hadron collider proceeds in association with either top/bottom quarks or scalar/vector bosons [29,30]. By taking into account the recent experimental excess observed by ATLAS and CMS [1,2] and the Tevatron [3] (i.e., corresponding to a light Higgs with a mass of ≈ 125 GeV), we propose a search strategy for a charged Higgs boson produced in association with a W boson and decaying into a bbW final state. In particular, we show that an appropriate choice of the selection cuts would allow the discovery of such a particle despite the considerable tt dominated background. This paper is organised as follows: in section 2 we give an overview of the considered model, in section 3 we analyse the allowed parameter space in the light of both theoretical and experimental constraints, in section 4 we present the main phenomenological results, in section 5 we briefly comment on possible future developments, and in section 6 we present our conclusions. In the appendix we discuss the decoupling limit for the CP-violating type-II 2HDM.

The model
We describe here our parametrisation of the 2HDM with type-II Yukawa couplings. The Higgs sector is defined by the presence of two Higgs doublets, with one (Φ 2 ) field coupled to the u-type quarks, and the other (Φ 1 ) to the d-type quarks and charged leptons [5].
We take the 2HDM potential to be The Z 2 symmetry will be respected by the quartic terms (there are no λ 6 or λ 7 terms), and Flavour-Changing Neutral Currents (FCNCs) are constrained [31].
We parametrise the Higgs fields as . (2. 2) The real and non-negative Vacuum Expectation Values (VEVs) for the Higgs doublets are v 1 = vc β and v 2 = vs β , with c β = cos β and s β = sin β, and the ratio defines CP violation is allowed, and it is realised by means of the fact that λ 5 and m 2 12 are complex numbers. All three neutral states will then mix, with the physical Higgs particles H i (i = 1, 2, 3) related to the weak fields η j (j = 1, 2, 3) of Eq. (2.2) by In terms of the non-diagonal mass-squared matrix M 2 , determined from second derivatives of the above potential, we have The 3 × 3 mixing matrix R governing the neutral sector will be parametrised in terms of the angles α 1 , α 2 and α 3 as in [32,33]: where c 1 = cos α 1 , s 1 = sin α 1 , etc., and For these angular ranges, we have c i ≥ 0, s 3 ≥ 0, whereas s 1 and s 2 may be either positive or negative. We will use the terminology "general 2HDM" as a reminder that CP violation is allowed. With all three masses different, there are three limits of no CP-violation, i.e., with two Higgs bosons that are CP-even and one that is odd. In the above notation, the three limits are [34]: In the general CP-violating case, the neutral sector is conveniently described by these three mixing angles, together with two masses (M 1 , M 2 ), tan β (the ratio between the two Higgs VEVs) and the parameter µ 2 = Re m 2 12 /(2 cos β sin β). From Eq. (2.5), it follows that When CP is violated, both (M 2 ) 13 and (M 2 ) 23 will be non-zero. In fact, they are related by (M 2 ) 13 = tan β(M 2 ) 23 . (2.10) From these two equations, (2.9) and (2.10), we can determine M 3 from M 1 , M 2 , the angles (α 1 , α 2 , α 3 ) and tan β [32]: Providing also M H ± and µ 2 , all the λ's are consequently determined. Since the lefthand side of (2.9) can be expressed in terms of the parameters of the potential (see, for example, [35]), we can solve these equations and obtain the λ's in terms of the rotation matrix, the neutral mass eigenvalues, µ 2 and M H ± . The explicit expressions are given in Ref. [34].
The interest in allowing for CP violation lies in the fact that it may be helpful for baryogenesis [36]. Also, from a more pragmatic point of view, it opens up a bigger parameter space, and allows certain couplings to be larger.

Yukawa and gauge couplings
For the type-II 2HDM, and for the third generation, the neutral-sector Yukawa couplings are (assuming all fields incoming): Likewise, the charged-Higgs couplings are [5] H + bt : The H j ZZ (H j W + W − ) coupling is, relative to that of the SM, given by [cos βR j1 + sin βR j2 ], for j = 1. (2.14) Note that when H 1 is CP-odd (H 1 = A), then c 2 = 0 and this vector coupling vanishes [26]. Finally, the H j H + W − coupling is given by [35] H j H ± W ∓ :

Constraining the parameter space
The multi-dimensional type-II 2HDM parameter space is severely restricted by a variety of theoretical and experimental constraints, which are discussed in the following.

Theoretical constraints
We impose three classes of theoretical constraints: • Positivity: In order to have a stable potential, we impose positivity, V (Φ 1 , Φ 2 ) > 0 as |Φ 1 |, |Φ 2 | → ∞ [37][38][39]. Additionally, we must insist on a non-trivial solution to Eq. (2.11): M 2 3 > 0 and M 2 ≤ M 3 . While positivity may be satisfied for the given parameters of the potential, the considered minimum need not be the global one. However, it has been shown that, if a local charge-conserving minimum exists, then there can be no charge-breaking minimum [40][41][42]. Nevertheless, the potential of the 2HDM can have more than one charge-conserving minimum. We therefore check that the minimum obtained is the global one, following the approach of Ref. [43].
• Tree-level unitarity: We also impose tree-level unitarity on Higgs-Higgs scattering [44][45][46][47][48]. These conditions have a rather dramatic effect at "large" values of tan β and M H ± , though some tuning of µ can extend the allowed range to larger values of tan β [49].
• Perturbativity: We impose the following upper bound on all λ's: with ξ = 0.8, meaning |λ i | < ∼ 10. The effect of this is to restrict large values of the masses, unless the soft parameter µ is comparable to M 2 and M H ± .
For illustrations of how these theory constraints cut into the parameter space, see Refs. [34,35].

Experimental constraints
Below, we list the different experimental constraints that are important. The SM predictions of the flavour observables quoted in this subsection are obtained using SuperIso v3.2 [50,51].
• B → X s γ: This rare FCNC inclusive decay receives contributions from the charged Higgs boson that can be comparable to the W ± contribution in the SM. Since the charged Higgs state always contributes positively to the corresponding BR, it is an effective tool to probe the type-II 2HDM. The most up-to-date SM prediction for this decay, at the Next-to-Next-to-Leading Order (NNLO), gives [50][51][52][53][54][55][56][57][58][59][60][61][62][63][64][65]: while the combined experimental value from HFAG points to a larger value [66]: For type-II Yukawa interactions, which we consider here, light charged Higgs bosons are excluded by this observable. The actual limit is sensitive to higher-order QCD effects and is of the order of 300 GeV, being more severe at low values of tan β [59,65,67,68]. Recently, a higher-order calculation [69] concludes that the 95% C.L. is at 380 GeV. While adopting the more conservative limit of 300 GeV, we shall also discuss this more restrictive one.
• B u → τ ν τ : In contrast to the b → sγ transitions, where the charged Higgs state participates in loop diagrams, the process B u → τ ν τ can be mediated by H ± already at tree level. Since this decay is helicity suppressed in the SM, whereas there is no such suppression for spinless H ± exchange in the limit of large tan β, these two contributions can be of similar magnitude [70]. The 2HDM contribution factorises in the ratio R Bτ ν as compared to the SM value. This decay suffers from uncertainties from f B and V ub , and using f B = 192.8 ± 9.9 MeV [71], and the combined value |V ub | = (3.92 ± 0.46) × 10 −3 [72], the SM BR evaluates numerically to [50,51]: The SM prediction is compared to the current HFAG value 1 [66] BR(B u → τ ν τ ) exp = (1.64 ± 0.34) × 10 −4 (3.5) by forming the ratio In the framework of the 2HDM this leads to the exclusion of two sectors of the ratio tan β/M H ± [65,70,[74][75][76][77][78].
• B → Dτ ν τ : Compared to B u → τ ν τ , the semi-leptonic decays B → D ν have the advantage of depending on |V cb |, which is known to greater precision than |V ub |. In addition, the BR(B → Dτ ν τ ) is about 50 times larger than the BR(B u → τ ν τ ) in the SM. The experimental determination remains however very complicated due to the presence of at least two neutrinos in the final state. The ratio where the 2HDM contributes only to the numerator, allows one to reduce some of the theoretical uncertainties. The SM prediction for this ratio is [50,51] ξ SM D ν = (29.7 ± 3) × 10 −2 , (3.8) and the most recent experimental result by the BaBar collaboration is [79] This ratio is also sensitive to a light charged Higgs boson, and leads to complementary constraints to the ones following from B u → τ ν τ [65,78,[80][81][82].
• D s → τ ν τ : Constraints on a light charged Higgs can be obtained, competitive with those obtained from B u → τ ν τ [83]. The main uncertainty here is due to the decay constant f Ds . The SM prediction for this decay is [50,51]: using f Ds = 248 ± 2.5 MeV [84], and the current world average of the experimental measurements gives [66]: • B d,s → µ + µ − : These decays are helicity suppressed in the SM and can receive sizeable enhancement or depletion from Higgs-mediated contributions. At large tan β, the non-observation of these decay modes imposes a lower bound on the charged Higgs boson mass [85,86]. The most stringent limits for their BRs were reported very recently by the LHCb collaboration [87]: at 95% C.L. Combining these values with the limits from ATLAS and CMS [88] results in even stronger bounds: The SM predictions for these branching ratios are [50,51,89]: with BR(B s → µ + µ − ) being the more constraining. The largest uncertainty is from f Bs , we used f Bs = 234 ± 10 MeV for our evaluation. In the type-II 2HDM, the experimental limits can be reached for very large values of the Yukawa couplings and small charged Higgs boson masses. The constraining power of B d,s → µ + µ − in this study is hence rather limited as compared to the other flavour observables. • R b : The branching ratio R b ≡ Γ Z→bb /Γ Z→had would also be affected by Higgs boson exchange. The contributions from neutral Higgs bosons to R b are negligible [35], however, charged Higgs boson contributions, via the H ± bt Yukawa coupling, as given by [96], Eq. (4.2), exclude low values of tan β and low M H ± .
• pp → H j X: Two aspects of the recent neutral Higgs searches at the LHC are considered [97,98]: -The production and subsequent decay of a neutral Higgs to γγ, around M = 125 GeV is taken to be within a factor of 2 from the SM rate. Assuming the dominant production to be via gluon fusion, this can be approximated as 0.5 ≤ R γγ ≤ 2, where we define .
We take into account (1) the modified scalar coupling to the fermion or W in the loop, (2) the pseudoscalar contribution, and (3) the charged Higgs contribution on the γγ side. This condition (3.18) mainly constrains the Yukawa couplings of H 1 . In particular, the (dominant) H 1 tt contribution to the loop integrals should be comparable to that of the SM, meaning where P (τ ) represents the ratio of the pseudoscalar and the scalar contributions to the loop integral [6], with τ t = M 2 1 /(4m 2 t ). At low tan β there is some freedom, either s 2 1 c 2 2 or s 2 2 should be of order unity, whereas at high tan β this constraint requires α 1 ±π/2 and α 2 0.
-The production and subsequent decay, dominantly via ZZ and W W , is constrained in the mass range 130 GeV < ∼ M < ∼ 600 GeV. We consider the quantity , (3.20) for j = 2, 3 and require it to be below the stronger 95% CL obtained by ATLAS or CMS. This constraint thus affects the product of the Yukawa and gauge couplings of H 2 and H 3 (see Eqs. (2.12) and (2.14)). In the limit of unchanged total width, this implies where η is the 95% CL on σ/σ SM . For tan β of the order of unity, the first factor is "small" when R 2 j1 is of order unity, whereas the second factor is "small" when |R j3 | is of order unity.
For larger values of tan β, we may substitute α 1 ±π/2 and α 2 0 from the above consideration, whereupon the constraint (3.21) takes the form: which is not very strong. In particular, it is automatically satisfied at large tan β.
For the total widths of H 2 and H 3 entering in the numerator of (3.20), we include also H j → H 1 H 1 and H j → H 1 Z, in addition to the familiar decay modes of H 1 . The relevant couplings can be found in [99] and [35].
• T and S: For the electroweak "precision observables" T and S, we impose the bounds |∆T | < 0.10, |∆S| < 0.10 [100], at the 1-σ level, within the framework of Refs. [101,102]. While S is not very restrictive, T gets a positive contribution from a splitting between the masses of charged and neutral Higgs bosons, whereas a pair of neutral ones give a negative contribution.
• Electron Electric Dipole Moment (EDM): The bound on the electron EDM constrains the allowed amount of CP violation of the model. We adopt the bound [103] (see also [104]): at the 1-σ level. The contribution due to neutral Higgs exchange, via the two-loop Barr-Zee effect [105], is given by Eq. (3.2) of [104].
In contrast to the MSSM, in the 2HDM, an additional contribution to the muon anomalous magnetic moment arises only at the two-loop level. Since we are considering heavy Higgs bosons (M 1 , M H ± > ∼ 100 GeV) therefore, according to [68,106,107], the 2HDM contribution to the muon anomalous magnetic moment is negligible even for tan β as large as ∼ 40.
The above constraints are not independent. Therefore, we do not attempt to add their contributions to an overall χ 2 , but rather require that none of them should be violated by more than 2σ. The LHC constraints are imposed at the quoted 95% C.L.
Among these constraints, basically b → sγ requires M H ± > ∼ 300 (380) GeV, while the different B-meson constraints impose additional constraints at low and high values of tan β (the latter are basically excluded anyway, by the unitarity constraints). The T constraint prevents the masses of the neutral and charged Higgs bosons from being very different, and thus effectively provides a cut-off at high masses.

Two scenarios
In order to develop some intuition for the viable parts of the parameter space, we shall here consider two scenarios. In both of them, we take the lightest neutral Higgs boson mass to be M 1 = 125 GeV. Furthermore: For scenario 1, we will consider a range of charged Higgs boson masses, from 300 to 600 GeV (for 700 GeV, only a few viable parameter points are found), and a range of tan β values. We will typically consider small values of tan β, of the order of 1. High values lead (for fixed µ) to conflict with the unitarity constraints.
Scenario 2 is rather fine-tuned. The masses M 2 and M H ± have to be very close to µ, in order to avoid conflict with the unitarity constraints. It is discussed in some detail in the appendix. There, it is shown that in addition to the SM-like region of α 1 ±π/2 with α 2 0 (H 2 or H 3 being odd under CP), there is also another region, with α 1 0 and α 2 ±π/2 (with H 1 being odd, see [108,109]).

Scenario 1
We shall first consider separately the LHC constraints on H 1 → γγ (an allowed range) and H 2,3 → W + W − (an upper bound). For fixed additional input parameters, M H ± = 300, 500 GeV, tan β = 1, µ = 200 GeV, (3.26) we show in fig. 1 the result of a scan over 10 6 sets of (α 1 , α 2 , α 3 ). The blue regions satisfy all the constraints discussed above, as well as one of these LHC constraints, whereas the red regions do not satisfy the LHC constraint considered, but all other constraints discussed in sections 3.1 and 3.2. The allowed regions are for these parameters rather independent of M H ± in the range 300-400 GeV, but start shrinking around 500 GeV and vanish around M H ± ∼ 600 GeV. (If we allow ξ = 1, see Eq. (3.1), the allowed values of M H ± reach out to about 700 GeV.) The underlying checkered pattern is due to the positivity constraint, together with M 2 3 ≥ M 2 2 . When we impose the LHC constraints discussed above [97,98] (scanning now over 5 · 10 6 points), we obtain the blue regions for the effects of the H 1 → γγ (left) and the H 2,3 → W + W − (right) constraint. For the considered case of tan β = 1, M 2 = M H ± = 300 GeV we see that these only very marginally overlap.
Imposing then both of these LHC constraints, we show in figure 2 the resulting surviving parameter space in green on top of the red regions, now for M 2 = 400 GeV, and two values of charged-Higgs mass, 300 and 500 GeV. Note that although the LHC experiments exclude an SM Higgs with a mass from about 130 GeV to about 600 GeV, there are still viable regions of parameter space for the second (and third) Higgs state to be in this region, since it may couple more weakly than the SM Higgs boson. A striking first observation is that the allowed regions are very much reduced, only values close to α 1 = ±π/4 are now allowed for the low value of M H ± , and a somewhat higher value for the higher value of M H ± . This is a little different from the results reported recently for the CP-conserving model, where it was found that only a region around α = 0 is allowed [24]. We recall that in the particular CP-conserving limit of α 2 = α 3 = 0 (corresponding to the heaviest one, H 3 , being odd under CP), α = 0 corresponds to α 1 = ±π/2. That region is here found to violate unitarity. For a complementary view of the allowed region, we show in fig. 3 the corresponding projections onto the α 2 -α 3 plane. Much of the region with α 2 < 0 is populated, whereas most of the region with α 2 > 0 is excluded. All values of α 3 are represented. When we impose the LHC constraints, the main characteristic is that α 2 becomes more restricted, as was also seen in fig. 2, and by Barroso et al [27]. Two regions of α 3 -values are represented: values close to 0 or π/2 (in both cases for α 2 close to zero).
A comparison with Arhrib et al [28] indicates that we find a more constrained region, presumably because of our tighter constraint on R γγ . In addition, we find that some points are excluded because of conflict with the electron EDM.
For higher values of tan β, several things change. At some point, also negative values of α 1 become allowed (not shown), and the allowed ranges of α 1 move towards ±π/2 as tan β increases. At tan β = 2 and 3, the allowed region has shrunk by a factor of 2 to 3, compared to tan β = 1.
For lower values of M 2 , the impact of the LHC constraints is more severe, but allowed points are found, for example also for M 2 = 150 GeV. For higher values of M 2 , the allowed region is restricted to some neighborhood of α 1 = π/4 and α 2 = 0, with α 3 close to 0 or π/2.
In view of the results shown in fig. 2, let us comment on the special limit Then, the rotation matrix can be simplified as 28) and the physical states are related to the "weak" states η j as

29b)
We recall that, with type-II Yukawa couplings, η 1 couples to down-type quarks whereas η 2 couples to up-type quarks. Thus, in the limit (3.27), the lightest neutral Higgs boson couples coherently to the b-and the t-quarks (with strengths proportional to 1/ cos β and 1/ sin β, respectively, and thus, for tan β = 1, with the same strength). The heavier Higgs bosons, however, have a CP-even content given by η 1 − η 2 (note the minus sign). For tan β = 1 only the pseudoscalar component, η 3 , of these heavier Higgs bosons will couple to fermions. We note that two CP-conserving limits are contained in this scenario: H 3 = A for α 3 = 0 and H 2 = A for α 3 = π/2.

Scenario 2
Again, we start out with an overview of the allowed regions of parameter space in the absence of LHC constraints. This is presented in red, in fig. 4, for tan β = 10 and two values of M 2 , namely 400 (left) and 600 (right) GeV. The populated regions are at (α 1 , α 2 ) (±π/2, 0) and (0, ±π/2), the decoupling regions which are discussed in the appendix. They are seen to shrink considerably as the masses are increased from 400 to 600 GeV.

CP-conserving limits
Solutions also exist in CP-conserving limits. With When H 2 is CP-odd (H 2 = A) and tan β = 5, we find allowed solutions at M = 400 GeV and 600 GeV. But compared to the case H 3 = A, the mass range for M 3 is more constrained.
We do not find any solution when H 1 is CP-odd (H 1 = A). The crucial LHC constraint is the quantity R γγ . Others [25,26] have argued for an interpretation of the 125 GeV excess in terms of a pseudoscalar, but we are not able to confirm this. We find points which satisfy all other constraints, but not the LHC constraints. Another recent study agrees with this [110].  The unitarity constraint plays an important role in delimiting high values of both tan β and M H ± . This constraint requires the λ's to be small, which to some extent is achieved by taking the "soft" mass parameter µ large. In fact, in the co-called decoupling limit, discussed for the CP-conserving case in [111], and for the present case in the appendix, one can respect the unitarity constraints for large masses, provided µ is tuned to these masses: (3.30) For moderate values of tan β (3 − 5), that limit also requires β ∼ α 1 , α 2 ∼ 0, α 3 arbitrary. (3.31) For large values of tan β ( > ∼ 5), this evolves into the region (α 1 , α 2 ) ∼ (π/2, 0). Furthermore, an additional region opens up for large masses and large tan β, leading to Decoupling 1: (α 1 , α 2 ) ∼ (±π/2, 0), (3.32a) with α 3 arbitrary. Two comments are here in order: (i) because of the periodicity of the trigonometric functions, regions at α i −π/2 and α i +π/2 are connected; (ii) the SM limit requires α 1 ∼ β, and is thus contained in the region "Decoupling 1". In view of the above discussion, in order to determine the maximally allowed ranges of tan β and M H ± , we scan over some range in µ, starting at the geometric mean (3.33)

The experimental constraints
In fig. 6 we show allowed regions in the tan β-M H ± plane. Again, the larger red region is allowed in the absence of recent LHC results, whereas the green region shows what remains compatible with these data. We note some reduction in the range of charged Higgs masses. Also, at high tan β, the masses M 2 and M H ± tend to be close, as discussed above. The "fractal" appearance of these plots is in part due to the finite number of points in the scans. Some could also reflect genuine "islands" in parameter space.
In fig. 7 we show typical values of M 3 . Note that for each point in the allowed part of this plane, some ranges of α's are allowed (see the previous subsection). Each set of α's corresponds to a particular value of M 3 . The values plotted here are those first encountered in a random scan over α's. We see that as M 2 and M H ± increase, also typical values of M 3 increase.
By allowing a larger value of the perturbativity cut-off ξ of Eq. (3.1), higher masses of M H ± would be allowed. For example, ξ = 1 permits masses above 600 GeV. Also the unitarity and the electroweak parameter T constrain this high-mass region. Which of these gives the strongest limits depends on the other parameters.

Benchmark analysis
In this section we study the profile of the charged Higgs boson at the LHC in view of the allowed parameter space analysis. For this, we start by studying a set of candidate benchmark points P i that allow us to synthesise the main features of the surviving models.
As we have shown in the previous sections, this model depends on eight parameters. However, since the µ parameter does not directly participate in the phenomenology of interest, if not specified otherwise, we will consider µ = 200 GeV hereafter. The exception will be the high-tan β case, where µ has to be carefully tuned in order to respect the unitarity constraint. Then, we consider points that pass the constraints, at least in the charged Higgs mass range 300 − 600 GeV. We are then left with five parameters: tan β, α 1 , α 2 , α 3 and M 2 .
In table 1, we list a set of 10 candidate benchmark points: we consider the most illustrative four of them for determining cross sections and relevant BRs, the rest will be discussed only qualitatively.
This set of points has distinctive characteristics in the phenomenology, as will be shortly made clear. Also, we note that P 9 and P 10 are nearly CP-conserving, with α 2 α 3 0. All the others correspond to CP-violating scenarios.  The model has been implemented through the LanHEP module [112] (see [113] for details) and the following analysis has been performed by means of the CalcHEP package [114,115]. Furthermore, we have used the CTEQ6.6M [116] set of five-flavour parton distribution functions (PDFs). Due to their relevance at hadron colliders, the effective ggH i , γγH i and γZH i vertices have been implemented by means of a link between CalcHEP and LoopTools [117], and the numerical results have been cross-checked against the analytical results in [118].

Charged and lightest neutral Higgs bosons: BRs
This subsection is devoted to an analysis of the BRs of the charged Higgs state in the allowed parts of the parameter space. In fact, since we are mainly interested in signatures of a charged Higgs boson produced in association with a W boson, which involve model dependent couplings, it is of fundamental importance to establish some characteristic features of the BRs for some specific points of parameter space. In this regard, we consider four points from table 1 and we determine the most important decay modes. We consider only BRs > 10 −4 , rates below this value are not of phenomenological relevance. Then, we have six decay modes: W H 1 , W H 2 , W H 3 , tb, ts, τ ν τ , displayed in fig. 8 for selected benchmark points. In addition, we remark that the results are presented for an illustrative range of M H = 300 − 600 GeV, while recalling that the allowed range is always bounded in the range M H ∼ 380 − 450 GeV for an intermediate choice of tan β.
First, we consider two points associated to the choice of tan β = 1 in fig. 8, P 2 and P 6 . The dominant decay mode is always tb, and this feature is even reinforced when the masses M 1 and M 2 are closer (P 2 ), with respect to the case in which they are well separated (P 6 ). However, it is important to remark that the W H i branching fractions, when allowed by phase space, are always ∼ O(0.1) and never smaller than ∼ O(0.01). In particular, if M H ± > 400 GeV, then the BR for W H 1 is ∼ O(0.1), this assures that the suppression brought about by this decay mode is never stronger than about an order of magnitude for a rather large value of M H ± . The result does not hold for the W ± H 1 case when tan β = 2 (see fig. 8, P 7 and P 8 ). In fact, it strongly depends on the choice of point in parameter space: for P 8 this decay mode is suppressed down to ∼ O(0.01), while for P 7 its branching franction is restored to ∼ O(0.1) because increasing the mixing between η 1 and η 3 via |α 2 /π| = −0.03 → −0.07 increases its CP-odd component, and this effect leads to an enhancement.
Another feature of the tan β = 2 choice is that the W ± H 2 decay mode is always dominant as compared to the tb one when M H ± is large (though not always allowed),  Since we are interested in the phenomenology of the charged Higgs boson produced in association to vector bosons, it is important to understand the properties of the lightest neutral Higgs. For this, we conclude this subsection by presenting a relevant set of H 1 BRs. As is clear from table 2, the three most important decay mode are always the bb, gg and W W * ones. Since the first is the phenomenologically simplest among the three, we will only consider this decay channel for studying combined H 1 signatures.

Charged Higgs: single production mechanisms
In this subsection we study single charged Higgs production at the LHC for two choices of energy, √ s = 8 TeV and √ s = 14 TeV. Considering the partonic amplitudes, we have three main mechanisms to produce a single charged Higgs boson from hadrons, i.e., associated with bosons (B) or fermions (F): . 9a); (4.1) In fig. 9 we show the main partonic contributions to the three production channels. Note that the process in fig. 9b will generally be disfavoured for two reasons. First, the quark density inside the proton is lower than the gluon density, so this channel is suppressed in this respect, owing to the typical H ± masses considered (recall that Figure 9: Single charged Higgs production channels at parton level. √ s = 8 or 14 TeV). Second, the intermediate W boson will be largely off-shell, also inducing significant depletion of the production rates. In contrast, the channel in fig. 9a receives contributions from both quark and gluon initiated processes and can further be resonant in the s-channel (for some of our benchmarks). In principle, the box contribution from heavy fermions should be included in the set, for it has been proven in several scenarios [119][120][121] that it can lead to O(10 − 100%) corrections when tan β ∼ O (10). Still, in the phenomenological scenario that we propose to be tested at LHC (tan β ∼ O(1)) this inclusion is totally irrelevant for our conclusions. The mode in fig. 9c also benefits from counting on two subchannels, though it is never resonant (as M H ± > m t −m b for the model considered here).
We consider now the four benchmark points P 2 , P 6 , P 7 and P 8 from table 1 and in fig. 10 we plot the cross sections against the charged Higgs mass for the aforementioned production channels. In the first place, we confirm that the associated production with a neutral scalar is disfavoured. Secondly, the remaining production mechanisms are always within a range of an order of magnitude at most. Again, we remark that the results are presented for an illustrative range of M H = 300 − 600 GeV, while recalling that the allowed range is always restricted to the range M H ∼ 380 − 450 GeV for an intermediate choice of tan β.
As regards the fermion-associated production mechanism of fig. 9c, we remark that it only depends on the values of M H ± and tan β (see Eq. (2.13)), and there is a considerable reduction when moving from tan β = 1 to tan β = 2 (roughly a factor 2) due to the fact that the dominant contribution in the coupling is ∼ m t / tan β, hence the ratio of VEVs acts as a reduction factor. The cross section of the fermion-associated contribution at tan β = 1 is ∼ 10 − 10 2 (10 2 − 10 3 ) fb when √ s = 8 (14) TeV and it is mostly inversely proportional to tan β. The scope of the fermion-associated production mechanism in extracting a H ± → W bb signature (see below) has been analysed already in the literature, albeit in the MSSM, see [122], and we will revisit it in a CP-violating type-II 2HDM in a future publication.
Instead, here, we concentrate on vector-boson-associated production. The corresponding cross sections show a complicated behaviour with respect to different choices of parameters. We start our analysis by considering the channel with a final H ± W ∓ state. From fig. 10 (P 2 and P 6 ) we see that a choice of tan β = 1 plus a low-Higgs-masses scenario (P 2 : M 2 ∼ M 3 ∼ 300 − 400 GeV) has a cross section ∼ 10 − 10 2 (10 2 − 10 3 ) fb when √ s = 8 (14) TeV, and that it is dominant (competitive) with respect to the fermion-associated production. On the other hand, we see that the choice of a high M 2 (P 6 ) and M 3 (≥ 500 GeV) favour the contribution from the parton-level channel gg → H i=2,3 → H ± W ∓ proceeding through the on-shell H i=2,3 , and this results in a cross section that is always dominant and even enhanced when m W + M H ± < M i=2,3 , i.e. ∼ 10 2 − 10 3 (10 3 − 10 4 ) fb when √ s = 8 (14) TeV.
These qualitative conclusions also hold when tan β = 2. Despite an overall suppression by one order of magnitude due to the increased value of tan β, from fig. 10 (P 7 ) we see that on-shell production is not taking place (the H 2 and H 3 masses are too light). Hence, starting from a cross section of ∼ 10 2 (10 3 ) fb when √ s = 8 (14) TeV at the lower M H ± scale we find a rate of ∼ 10 (10 2 ) fb when √ s = 8 (14) TeV for high values of M H ± . However, the on-shell production is important for P 8 , where we can see an interplay between the on-shell H 2 and H 3 (M 3 ∼ 460 GeV) production at M H ± ∼ 300 GeV being realised by a "double-shoulder" shaped line. In fact, when M H ± ∼ 320 (380) GeV the H 2 (H 3 ) on-shell production is switched off. In this framework, the cross section is ∼ 10 (10 2 ) fb when √ s = 8 (14) TeV, but it is increased by an order of magnitude when M H ± < 380 GeV. The high-tan β benchmark points give rather low production cross sections. For P 9 and P 10 we find cross sections roughly half and a tenth, respectively, of that for P 8 . Since the latter, after the cuts discussed below, does not yield any useful signal, we have not explored P 9 and P 10 any further.
Finally, we briefly comment on the sub-dominant channel with a W -mediated H ± H i=1,3 final state. This channel is unlikely to have interesting phenomenological implication at the LHC, at least at the early running stage: when √ s = 8 TeV and with integrated luminosity L int = 10 fb −1 the cross section is typically just above the threshold for producing a few events. Since it is not competitive with the other production channels, we will not study this mechanism.
In the next subsection we consider the neutral-Higgs-mediated production mechanism for analysing the scope of the LHC in discovering such a state in the allowed parameter space.

pp → H ± W ∓ : significance analysis
In this subsection we analyse the significance of single charged Higgs boson production in association with gauge bosons for the set of benchmark points in table 1 (except P 9 and P 10 ). All figures herein refer to an integrated luminosity of 100 fb −1 .
Among the different charged Higgs decay modes, we have chosen to study H ± → W ± H i . The decay chain with the H i → bb intermediate decay is numerically favoured, so that we adopt it here, hence the complete H ± decay chain is Therefore, we are interested in the significance (Σ ∼ S/ √ B) analysis of a 2b + 2W final state produced via a single charged Higgs. The most important background at the LHC for this final state is top quark pair production. However, we will show that a systematic reduction of this background is possible.
Notice that one may worry here about the contribution of the fermionic charged Higgs decay chain H + → tb → W + bb, (4.5) as it yields an irreducible final state that is identical to the chosen one that could be defined as part of either the signal or the background. Under any circumstances, though, we believe that our top-mass veto (see below) will render this contribution negligible, so we omit it here 3 . A b-tagging efficiency of ∼ 70% has been assumed for each b-(anti)quark in the final state, and a full reconstruction efficiency has been assumed with respect to the W bosons. Among the possible decay patterns of the two W bosons, the semileptonic one was chosen, i.e., one hadronic and one leptonic decay, allowing for the full reconstruction of the events (unlike the fully leptonic decay mode) and a neater environment than the fully hadronic decay mode.
Hence, the overall selected process for the signal is the following: For each benchmark point, 2 · 10 4 unweighted events were produced. Regarding the top background, 4.5 · 10 6 unweighted events (with generation cuts) have been simulated in CalcHEP. For both signal and background the standard set of CTEQ6.6M [116] PDFs with scale Q = √ s were employed. For emulating a real LHC-prototype detector, a Gaussian smearing was included to take into account the electromagnetic energy resolution of 0.15/ √ E and the hadronic energy resolution of 0.5/ √ E. We describe now the overall strategy for the background reduction procedure. A first set of cuts includes typical detector kinematic acceptances and standard intermediate object reconstruction, such as W → jj and H 1 → bb (cuts 1-3). Further, a t-(anti)quark reconstruction is used as "top veto" (cut 4). Led by the consideration that a b quark pair stemming from the Higgs boson is boosted (unlike the almost back-to-back pair from tt), we define the last cut of the following set (cut 5): 1) Kinematics: standard detector cuts with η the pseudorapidity and ∆R = (∆η) 2 + (∆φ) 2 .
2) light Higgs reconstruction: 5) same-hemisphere b quarks: In  After these rather generic cuts are imposed, more signal-based selections can improve the significance. The main consideration of the following analysis is that the charged Higgs mass can equivalently be reconstructed by either the invariant mass of the four jets (2b+2j), M (bbjj), or the transverse mass of the b jets, the lepton and the MET, M T (bb ν). Let's focus on the M (bbjj)-M T (bb ν) plane: for the signal, either of the two variables (if not both) will always reconstruct the correct charged Higgs boson mass, thus producing a cross-like shape in the plane defined by the two masses. In contrast, the background events To determine which is the better of the two proposed strategies and what is the optimal value for M lim , we studied the effects of C squ and C sng for several values of M lim . Results are shown in tables 4 and 5 for the points P 2 and P 4 , respectively.
Clearly, a higher value for M lim results in an increase of the significance, the top background is reduced more than the signal. It is important to note that for low charged Higgs masses, C squ seems to perform better than the single cut. However, this is strickly true for M H ± 310 GeV only: if a further selection is imposed, restricting the evaluation   Table 5: Comparison between C sng and C squ vs M lim for P 4 : surviving events and significance with respect to the background.
of the significance to the peak-region only peak cut: the significance obtained by imposing C sng , when calculated for all the other charged Higgs boson mass values, is always higher than the one obtained by imposing C squ . Here, M = min M (bbjj), M T (bb ν) when eq. (4.12) is employed, while M = M (bbjj) when eq. (4.13) is employed. For the following analysis, the value M lim = 600 GeV has been chosen as well as the selection C sng , this choice provides the best significance and a narrower peak while keeping a sufficient number of signal events (> 10). Should the surviving signal events be less than 10, it would then be advisable to choose instead the squared cut C squ for the higher survival probability of the signal events (despite the lower significance and the broader peak).
The invariant mass distributions for the points P 2 , P 3 , P 4 , P 5 , and P 7 are plotted in Figs. 12-16, each for two values of the charged Higgs mass.  Table 6: Surviving events and their significance after the single cut of eq. (4.13) and after the peak selection of eq. (4.14), for all points of table 1, except P 9 and P 10 .
As regards P 1 , P 6 and P 8 , no choice of allowed M H ± produces any appreciable signal after the whole set of cuts, hence we will not discuss them any further.
From fig. 12 we learn that a choice of tan β = 1 (though a rather low value of M 2 disfavours the production cross section) is enough to produce a visible signal, even for M H ± = 310 GeV with the signal peak lying over the background. However, it is clear that the signal suffers from the selection cuts, and this critical situation is eased up only when M H ± = 390 GeV. In fact, for higher allowed M H ± masses, the signal contains a conspicuous number of events (∼ 10 in the peak bin), and it is always above the background.
If we consider the points P 3 , P 4 and P 5 the same circumstances occur: the higher M H ± , the more visible and clear the signal with respect to the background. Moreover, from P 3 (M 2 = 350 GeV) to P 5 (M 2 = 450 GeV) the production cross section increases, as shown in figs. 13-15. If we consider the point P 7 (which has tan β = 2) in fig. 16, we do not note any change from the previous considerations: even when tan β grows we still have a considerable production cross section in the allowed parameter space, and the signal would be observable with respect to the background, at least for M H ± = 390 GeV.

Possible future scenarios
We shall here discuss possible future experimental developments, and consider their implications for the model, in particular for the proposed benchmarks. The basic question is of course: which experimental efforts are required to exclude the 2HDM altogether? We shall below adress a couple of LHC-related aspects of this question.

5.1
Higher and more constrained rates for gg → H 1 → γγ Several authors have recently argued that the LHC experiments point to an overall rate for pp → H → γγ that is somewhat high compared to the SM prediction. In our parameter scans in section 3 we generously allowed the ratio R γγ of Eq. (3.18) to satisfy 0.5 ≤ R γγ ≤ 2.0. We shall here briefly comment on how the parameter space is further constrained when we only allow the upper range: 1.5 ≤ R γγ ≤ 2.0 (5.1) For the case shown in Fig. 2, namely tan β = 1 and M 2 = 400 GeV (and two values of  M H ± ), we show in Fig. 17 how the LHC-allowed region gets constrained. Such a development can have dramatic consequences for the model: benchmark points P 1 , P 6 , P 7 and P 8 would be excluded. In figure 18 we show the remaining allowed regions in the tan β-M H ± plane, which exhibits a significant reduction, as compared with figure 6. While the scans have limited statistics, there is an indication that the remaining allowed parameter space starts fragmenting into disconnected regions.  Presumably, the search for an SM-like Higgs will continue in the mass range from around 130 GeV and up. To a first (rough) approximation, a Higgs in this mass region is produced via gluon fusion, and decays via W W (or ZZ) bosons. Assuming these upper bounds are tightened, it is interesting to see how the allowed parameter space behaves. In Fig. 19 we show how the allowed parameter space shrinks if we assume that the upper bound on a Higgs-like particle, represented by the quantity R ZZ of Eq. (3.20), is lowered by a factor 0.5. Again, we see a rather dramatic impact of such a development.

Conclusions
For the channel pp → H ± W ∓ → W + W − bb, we have established a set of 7 benchmarks for the CP-violating 2HDM with type-II Yukawa interactions. These points all have M 1 = 125 GeV, low tan β, they all violate CP, and allow for a range of charged-Higgs masses. A set of cuts is proposed, that will reduce the tt background to a tolerable level, and allow for the detection of a signal in the W W → jj ν channel. Some of the proposed benchmark points lead to enhanced H ± W ∓ production cross sections due to resonant production via H 2 or H 3 in the s-channel. Most of the proposed points are in the interior of some allowed domain in the α space, and thus robust with respect to minor modifications of the experimental constraints. Some of the benchmark points are vulnerable to a higher value of R γγ . However, the points P 2 , P 3 , P 4 and P 5 are not endangered.
It should also be noted that the proposed channel only benefits from favourable production cross sections and branching ratio at low values of tan β. In this region, the charged Higgs mass is constrained to the range ∼ 380-470 GeV.

A.2 Large tan β
The case of large values of tan β requires special attention. Because of over-all factors which were left out in Eq. (A.3), the first on them, Eq. (A.3a) must be satisfied to a much higher degree than the others (a factor 1/c 2 β is involved). This means that the expression