Prospects for charged Higgs searches at the LHC

The goal of this report is to summarize the current situation and discuss possible search strategies for charged scalars, in non-supersymmetric extensions of the Standard Model at the LHC. Such scalars appear in Multi-Higgs-Doublet models (MHDM), in particular in the popular Two-Higgs-Doublet model (2HDM), allowing for charged and additional neutral Higgs bosons. These models have the attractive property that electroweak precision observables are automatically in agreement with the Standard Model at the tree level. For the most popular version of this framework, Model~II, a discovery of a charged Higgs boson remains challenging, since the parameter space is becoming very constrained, and the QCD background is very high. We also briefly comment on models with dark matter which constrain the corresponding charged scalars that occur in these models. The stakes of a possible discovery of an extended scalar sector are very high, and these searches should be pursued in all conceivable channels, at the LHC and at future colliders.


Introduction
In the summer of 2012 an SM-like Higgs particle (h) was found at the LHC [1,2]. As of today its properties agree with the SM predictions at the 20% level [3,4]. Its mass derived from the γ γ and Z Z channels is 125.09 ± 0.24 GeV [5]. However, the SM-like limit exists in various models with extra neutral Higgs scalars. A charged Higgs boson (H + ) would be the most striking signal of an extended Higgs sector, for example with more than one Higgs doublet. Such a discovery at the LHC is a distinct possibility, with or without supersymmetry. However, a charged Higgs particle might be rather hard to find, even if it is abundantly produced.
We here survey existing results on charged-scalar phenomenology, and discuss possible strategies for further searches at the LHC. Such scalars appear in Multi-Higgs-Doublet models (MHDM), in particular in the popular Two-Higgs-Doublet model (2HDM) [6,7], allowing for charged and more neutral Higgs bosons. We focus on these models, since they have the attractive property that electroweak precision observables are automatically in agreement with the Standard Model at the tree level, in particular, ρ = 1 [8][9][10].
The production rate and the decay pattern would depend on details of the theoretical model [6], especially the Yukawa interaction. It is useful to distinguish two cases, depending on whether the mass of the charged scalar (M H ± ) is below or above the top mass. Since an extended Higgs sector naturally leads to Flavor-Changing Neutral Currents (FCNC), these would have to be suppressed [11,12]. This is normally achieved by imposing discrete symmetries in modeling the Yukawa interactions. For example, in the 2HDM with Model II Yukawa interactions a Z 2 symmetry under the transformation 1 → 1 , 2 → − 2 is assumed. In this case, the B → X s γ data constrain the mass of H + to be above approximately 480 GeV [13]. A recent study concludes that this limit is even higher, in the range 570-800 GeV [14]. Our results can easily be re-interpreted for this new limit. Alternatively, if all fermion masses are generated by only one doublet ( 2 , Model I) there is no enhancement in the Yukawa coupling of H + with down-type quarks and the allowed mass range is less constrained. The same is true for the Model X (also called Model IV or lepton-specific 2HDM) [15,16], where the second doublet is responsible for the mass of all quarks, while the first doublet deals with leptons. Charged Higgs mass below O(M Z ) has been excluded at LEP [17]. Low and high values of tan β are excluded by various theoretical and experimental model-dependent constraints.
An extension of the scalar sector also offers an opportunity to introduce additional CP violation [18], which may facilitate baryogenesis [19].
Charged scalars may also appear in models explaining dark matter (DM). These are charged scalars not involved in the spontaneous symmetry breaking, and we will denote them as S + . Such charged particles will typically be members of an "inert" or "dark" sector, the lightest neutral member of which is the DM particle (S). In these scenarios a Z 2 symmetry will make the scalar DM stable and forbid any charged-scalar Yukawa coupling. Consequently, the phenomenology of the S + , the charged component of a Z 2 -odd doublet, is rather different from the one in usual 2HDM models. In particular, S + may become long-lived and induce observable displaced vertices in its leptonic decays. This is a background-free experimental signature and would allow one to discover the S + at the LHC.
The SM-like scenario (also referred to as the "alignment limit") observed at the LHC corresponds to the case when the relative couplings of the 125 GeV Higgs particle to the electroweak gauge bosons W/Z with respect to the ones in the SM are close to unity. We will assume that this applies to the lightest neutral, mainly CP-even Higgs particle, denoted h. Still there are two distinct options possible -with and without decoupling of other scalars in the model. In the case of decoupling, very high masses of other Higgs particles (both neutral and charged) arise from the soft Z 2 breaking term in the potential without any conflict with unitarity.
The focus of this paper will be the Z 2 -softly broken 2HDM, but we will also briefly discuss models with more doublets. In such models, one pair of charged Higgs-like scalars (H + H − ) would occur for each additional doublet. We also briefly describe scalar dark matter models.
This work arose as a continuation of activities around the workshops "Prospects for Charged Higgs Discovery at Colliders", taking place every 2 years in Uppsala. The paper is organized as follows. In Sects. 2-4 we review the basic theoretical framework. In Sect. 5 we review charged Higgs decays, and in Sect. 6 we review charged-Higgs production at the LHC. Section 7 is devoted to an overview of different experimental constraints. Proposed search channels for the 2HDM are presented in Sect. 8, whereas in Sects. 9 and 10 we discuss models with several doublets, and models with dark matter, respectively. Section 11 contains a brief summary. Technical details are collected in appendices.

Potential and states
The general 2HDM potential allows for various vacua, including CP-violating, charge breaking and inert ones, leading to distinct phenomenologies. Here we consider the case when both doublets have non-zero vacuum expectation values. CP violation, explicit or spontaneous, is possible in this case.

The potential
We limit ourselves to studying the softly Z 2 -violating 2HDM potential, which reads Apart from the term m 2 12 , this potential exhibits a Z 2 symmetry, The most general potential contains in addition two more quartic terms, with coefficients λ 6 and λ 7 , and violates Z 2 symmetry in a hard way [6]. The parameters λ 1 -λ 4 , m 2 11 and m 2 22 are real. There are various bases in which this potential can be written, often they are defined by fixing properties of the vacuum state. The potential (2.1) can lead to CP violation, provided m 2 12 = 0.

13)
The strict SM-like limit corresponds to sin(β − α) = 1, however, the experimental data from the LHC [3,4] allow for a departure from this limit 1 down to approximately 0.7, which we are going to allow in our study.
In the following analysis, the gauge couplings to neutral Higgs bosons are also involved. They differ from the SM coupling by the factor (V = W ± , Z ) (2.14) In particular, for H 1 , this factor becomes cos(β − α 1 ) cos α 2 .
In the CP-conserving case, we have Note that the couplings (2.11) and (2.14) are given by unitary matrices, and hence satisfy sum rules. Furthermore, for any j, the relative couplings of (2.11) (the expression in the square brackets) and (2.14) satisfy the following relation [23]: These relations are valid for both the CP-conserving and the CP-violating cases.

Theoretical constraints
The 2HDM is subject to various theoretical constraints. First, it has to have a stable vacuum, 2 what leads to so-called positivity constraints for the potential [24, 29,30], V ( 1 , 2 ) > 0 as | 1 |, | 2 | → ∞. Second, we should be sure to deal with a particular vacuum (a global minimum) as in some cases various minima can coexist [31][32][33].
Other types of constraints arise from requiring perturbativity of the calculations, tree-level unitarity [34][35][36][37][38] and perturbativity of the Yukawa couplings. In general, imposing tree-level unitarity has a significant effect at high values of tan β and M H ± , by excluding such values. These constraints limit the absolute values of the λ parameters as well as tan β, the latter both at very low and very high values. This limit is particularly strong for a Z 2 symmetric model [33,39,40]. The dominant one-loop corrections to the perturbative unitarity constraints for the model with softly broken Z 2 symmetry are also available [41].
The electroweak precision data, parametrized in terms of S, T and U [42][43][44][45][46][47][48], also provide important constraints on these models. Table 1 The most popular models of the Yukawa interactions in the 2HDM (also referred to as "Types"). The symbols u, d, refer to upand down-type quarks, and charged leptons of any generation. Here, 1 Explicitly, for the charged Higgs bosons in Model II, we have for the coupling to the third generation of quarks [6] H + bt : where V tb is the appropriate element of the CKM matrix. For other Yukawa models the factors tan β and cos β will be substituted according to Table 6 in Appendix B. As mentioned above, the range in α (or α 1 ) is π , which can be taken as [−π, 0], [−π/2, π/2] or [0, π]. This is different from the MSSM, where only a range of π/2 is required [50], −π/2 ≤ α ≤ 0. The spontaneous breaking of the symmetry and the convention of having a positive value for v means that the sign (phase) of the field is relevant. This doubling of the range in the 2HDM as compared with the MSSM is the origin of "wrong-sign" Yukawa couplings.

Charged Higgs-boson decays
This section presents an overview of the different H + decay modes, illustrated with branching ratio plots for parameter sets that are chosen to exhibit the most interesting features. Branching ratios required for modes considered in Sects. 8-10 are calculated independently.
As discussed in [6,[51][52][53][54][55], a charged Higgs boson can decay to a fermion-antifermion pair, or to a neutral Higgs boson and a gauge boson, and their charge conjugates. Below, we consider branching ratios mainly for the CPconserving case. For the lightest neutral scalar we take the mass M h = 125 GeV. Neither experimental nor theoretical constraints are here imposed (they have significant impacts, as will be discussed in subsequent sections). For the calculation of branching ratios, we use the software 2HDMC [55] and HDECAY [53,56]. As discussed in [56], branching ratios are calculated at leading order in the 2HDM parameters, but we include QCD corrections according to [57][58][59], and three-body modes via off-shell extensions of H + → tb, H + → hW + , H + → H W + and H + → AW + . The treatment of three-body decays is according to Ref. [52].
For light charged Higgs bosons, M H ± < m t , Model II is excluded by the B → X s γ constraint discussed in Sect. 7. For Model I (which in this region is not excluded by B → X s γ ), the open channels have fermionic couplings proportional to cot β. The gauge couplings (involving decays to a W + and a neutral Higgs) are proportional to sin(β − α) or cos(β − α), whereas the corresponding Yukawa couplings depend on the masses involved, together with tan β.
The CP-violating case for the special channel H + → H 1 W + is presented in Sect. 5.4.

Branching ratios vs. tan β
Below, we consider branching ratios, assuming for simplicity M H ± = M A , in the low-and high-mass regions.

Light H + (M H ± < m t )
For a light charged Higgs boson, such as might be produced in top decay, the tb and W h channels would be closed, and the τ ν and cs channels would dominate. The relevant Yukawa couplings are given by tan β and the fermion masses involved. With scalar masses taken as follows: we show in Fig. 1 branching ratios for the different Yukawa models.
Since the τ ν and cs couplings for Model I are the same, the branching ratios are independent of tan β, as seen in the left panel. For Models X and II the couplings to c and τ have different dependences on tan β, and consequently the branching ratios will depend on tan β. In the case of Model Y, the cs channel is for tan β > √ m c /m s controlled by the term m s tan β, which dominates over the τ ν channel at high tan β.  Fig. 2), the dominant decay rates are to the heaviest fermion-antifermion pair and to W together with h or H (for the considered parameters, both h and H are kinematically available). Model X differs in having an enhanced coupling to tau leptons at high tan β; see Table 6 in Appendix B. If the decay to W h is kinematically not accessible, the τ ν mode may be accessible at high tan β.

Heavy H
For Model II (right part of Fig. 2), the dominant decay rates are to the heaviest fermion-antifermion pair at low and high values of tan β, with hW or H W dominating at medium tan β (if kinematically available). At high tan β it is the downtype quark that has the dominant coupling. Hence, modulo phase space effects, the τ ν rate is only suppressed by the mass ratio (m τ /m b ) 2 . Model Y differs from Model II in not having enhanced coupling to the tau at high values of tan β.  (Fig. 3), 3 and 30 (Fig. 4), together with the neutral-sector masses (note that here we take M H ± = M A ), and we consider the two values sin(β −α) = 1 and 0.7, corresponding to different strengths of the gauge couplings (2.13). The picture from Figs. 1 and 2 is confirmed: at low masses, the τ ν channel dominates, whereas at higher masses, the tb channel will compete against hW and H W , if these channels are kinematically open and not suppressed by some particular values of the mixing angles.
Of course, for tan β = 1 (Fig. 3), all four Yukawa models give the same result. Qualitatively, the result is simple. At low masses, the τ ν and cs channels dominate, whereas above the t threshold, the tb channel dominates. There is, however, some competition with the hW and H W channels. Similar results hold for sin(β −α) = 1, the only difference being that the H W branching ratio rises faster with mass, and the hW mode disappears completely in this limit. Even below the hW threshold, branching ratios for three-body decays via an off-shell W can be significant [52]. The strength of the hW channel is proportional to cos 2 (β − α), and it is therefore absent for sin(β − α) = 1 (not shown).
At higher values of tan β (Fig. 4 Here, we present the case of sin(β − α) = 0.7. The case of sin(β −α) = 1 is similar, the main difference is a higher H W branching ratio, while the hW channel disappears. It should be noted that three-body channels that proceed via hW and H W can be important also below threshold, if the tb channel is closed.

Top decay to H + b
A light charged Higgs boson may emerge in the decay of the top quark . The partial width, relative to its maximum value, is given by the quantity cos 2 α 2 sin 2 (β − α 1 ) + sin 2 α 2 , (5.8) which is shown in Fig. 6. We note that there is no dependence on the mixing angle α 3 . If α 3 = 0 or ±π/2, then CP is conserved along the axis α 2 = 0 with H 1 = h.
In the alignment limit,  Hence, the H + → H 1 W + decay crucially depends on some deviation from this limit. We note that the V V H 1 cou-pling is proportional to cos α 2 cos(β − α 1 ). Thus, the deviation of the square of this coupling from unity (which represents the SM-limit) is given by Eq. (5.8). Note that the experimental constraint (on the deviation of the coupling squared from unity) is 15-20% at the 95% CL [3,4].
For comparison, a recent study of decay modes that explicitly exhibit CP violation in Model II [60], compatible with all experimental constraints, considers tan β values in the range 1.3 to 3.3, with parameter points displaced from the alignment limit by ( α 1 /π ) 2 + ( α 2 /π ) 2 ranging from 1.5 to 83.2% (the one furthest away has a negative value of α 1 ).
This decay channel is also interesting for Model I [61].

H + production mechanisms at the LHC
This section describes H + production and detection channels at the LHC. Since a charged Higgs boson couples to mass, it will predominantly be produced in connection with heavy fermions, τ , c, b and t, or bosons, W ± or Z , and likewise We shall here split the discussion of possible H + production mechanisms into two mass regimes, according to whether the charged Higgs boson can be produced (in the on-shell approximation) in a top decay or whether it could decay to a top and a bottom quark. These two mass regimes will be referred to as "low" and "high" M H ± mass, respectively.
While discussing such processes in hadron-hadron collisions one should be aware that there are two approaches to the treatment of heavy quarks in the initial state. One may take the heavy flavors as being generated from the gluons, then the relevant number of active quarks is N f = 4 (or sometimes 3). Alternatively, the b-quark can be included as a constituent of the hadron, then an N f = 5 parton density should be used in the calculation of the corresponding cross section. These two approaches are referred to as the 4-flavor and 5-flavor schemes, abbreviated 4FS and 5FS. This should be kept in mind when referring to the lists of possible subprocesses initiated by heavy quarks and the corresponding figures in the following discussion. Below, we will use the notation q , Q and Q to denote quarks which are not b-quarks. We only indicate b-quarks when they couple to Higgs bosons, thus enhancing the rate.
For some discussions it is useful to distinguish "bosonic" and "fermionic" production mechanisms, since the former, corresponding to final states involving only H + and W − , may proceed via an intermediate neutral Higgs, and thus depend strongly on its mass; see, e.g., Ref. [62].

Production processes
Below, we list all important H + production processes represented in Figs. 11, 12, 13, and 14 in the 5FS. 4 4 Charge-conjugated processes are not shown separately. Higgs radiation from initial-state quarks is not shown explicitly.
A first comparison of the H + → tb signal with the tt background [65] (in the context of the MSSM) concluded that the signal could not be extracted from the background. More optimistic conclusions were reached for the H + → τ + ν channel [70,71], again in the context of the MSSM.
The first study [72] of the fermionic process (6.2) pointed out that there is a double counting issue (see Sect. 6.1.2). Subsequently, it was realized [73,87] that the gb → H +t process could be described as gg → H +t b, where a gluon splits into bb and one of these is not observed. As mentioned above, this approach is in the recent literature referred to as the four-flavor scheme (4FS) whereas in the fiveflavor scheme (5FS) one considers b-quarks as proton constituents.
NLO QCD corrections to the gb → H +t cross section have been calculated [77,78,86], and the resulting scale dependence studied [78,79], both in the 5FS and the 4FS. In a series of papers by Kidonakis [80,82,85], soft-gluon corrections have been included at the "approximate NNLO" order and found to be significant near threshold, i.e., for heavy H + . A recent study [86] is devoted to total cross sections in the intermediate-mass region, M H + ∼ m t , providing a reliable interpolation between low and high masses.
These fixed-order cross section calculations have been merged with parton showers [81,83,84,88], both at LO and NLO, in the 4FS and in the 5FS. The 5FS results are found to exhibit less scale dependence [84].
In addition to the importance of the tt channel at low mass, the following processes containing two accompanying b jets (see Fig. 8) are important at high charged-Higgs mass: There are also processes with a single H + and two jets (see Fig. 9): In this particular case, with many possible gauge boson couplings, one of the final-state jets could be a b.
In addition, single H + production can be initiated by a b-quark, as illustrated in Fig. 10.
Feynman diagrams for the production processes (6.4). If the line has no arrow, it represents either a quark or an antiquark Feynman diagrams for the production processes (6. 6) In the 5FS, single H + production can also take place from c and s quarks, typically accompanied by a gluon jet [89][90][91][92] ( Fig. 11): Similarly, one can consider cb initial states. At infinite order the 4FS and the 5FS should only differ by terms of O(m b ), but the perturbation series of the two schemes are organized differently. Some authors (see, e.g., Ref. [83]) advocate combining the two schemes according to the "Santander matching" [93]: with the relative weight factor since the difference between the two schemes is logarithmic, and in the limit of M H ± m b the 5FS should be exact. 12 Feynman diagrams for the production processes (6.9)

The double counting and NWA issues
A b-quark in the initial state may be seen as a constituent of the proton (5FS), or as resulting from the gluon splitting into bb (4FS). Adding gg → bbg → bH +t (with one b possibly not detected) and gb → H +t in the 5FS one may therefore commit double counting [94,95]. The resolution lies in subtracting a suitably defined infrared-divergent part of the gluon-initiated amplitude [88]. 6 The problem can largely be circumvented by choosing either the 5FS or the 4FS. For a more pragmatic approach, see Refs. [97,98]. A related issue is the one of low-mass H + production via t-quark decay, gg, qq → tt followed by t → H + b (witht a spectator), usually treated in the Narrow Width Approximation (NWA). The NWA, however, fails the closer the top and charged Higgs masses are, in which case the finite top width needs to be accounted for, which in turn implies that the full gauge invariant set of diagrams yielding gg, qq → H + bt has to be computed. A considerable effort has been made to understand this implementation; see also Refs. [99][100][101].

Production cross sections
In this section, predictions for single Higgs production at 14 TeV for the CP-conserving 2HDM, Models I and II (valid also for X and Y) are discussed.
In Fig. 15, pp → H + X cross sections for the main production channels are shown at leading order, sorted by the parton-level mechanism [62]. 7 The relevant partonic channels can be categorized as: Fig. 7 Fig. 8 a, Fig. 7 The charge-conjugated channels are understood to be added unless specified otherwise. No constraints are imposed here, neither from theory (like positivity, unitarity), nor from experiments.
The CTEQ6L (5FS) parton distribution functions [119] are adopted here, with the scale μ = M H . Three values of tan β are considered, and M H and M A are held fixed at (M H , M A ) = (500, 600) GeV. Furthermore, we consider the CP-conserving alignment limit, with sin(β − α) = 1. The bosonic cross section is accompanied by a next-to-leading order QCD K -factor enhancement [120].
Several points are worth mentioning: • To any contribution at fixed order in the perturbative expansion of the gauge coupling, the three cross sections are to be merged with regards to the interpretation in different flavor schemes, as discussed above. In the following, we focus on the first fermionic channel in the 5FS at tree level.  proportional to m t and those proportional to m b ; see Table 6. • Models X and Y will have the same production cross sections as Models I and II, respectively, but the sensitivity in the τ ν-channel would be different. While recent studies (see Sect. 6.1.1) provide a more accurate calculation of the gb → H +t cross section than what is given here, they typically leave out the 2HDM model-specific s-channel (possibly resonant) contribution to the cross section.
In Fig. 16, the bosonic charged-Higgs production cross section vs. M 3 ≡ M A for a set of CP-conserving parameter points that satisfy the theoretical and experimental constraints [62] (see also [121,122] Low values of tan β are enhanced for the bosonic mode due to the contribution of the t-quark in the loop, whereas the modulation is due to resonant A production. In the CPviolating case, this modulation is more pronounced [62]. As summarized by the LHC Top Physics Working Group the pp → tt cross section has been calculated at nextto-next-to leading order (NNLO) in QCD including resummation of next-to-next-to-leading logarithmic (NNLL) softgluon terms with the software Top++2.0 [123][124][125][126][127][128][129]. The decay width (t → bW + ) is available at NNLO [130][131][132][133][134][135][136], while the decay width (t → bH + ) is available at NLO [137].

Experimental constraints
Here we review various experimental constraints for charged Higgs bosons derived from different low (mainly B-physics) and high (mainly LEP, Tevatron and LHC) energy processes. Also some relevant information on the neutral Higgs sector is presented. Some observables depend solely on H + exchange, and are thus independent of CP violation in the potential, whereas other constraints depend on the exchange of neutral Higgs bosons, and are sensitive to the CP violation introduced via the mixing discussed in Sect. 2.2. Due to the possibility of H + , in addition to W + exchange, we are getting constraints from a variety of processes, some at tree and some at the loop level. In addition, we present general constraints coming from electroweak precision measurements, S, T , the muon magnetic moment and the electric dipole moment of the electron. The experimental constraints listed below are valid only for Model II, if not stated otherwise. 8 Also, some of the constraints are updated, with respect to those used in the studies presented in later sections.
The charged-Higgs contribution may substantially modify the branching ratios for τ ν τ -production in B-decays [140]. An attempt to describe various τ and B anomalies (also W → τ ν) in the 2HDM, Model III, with a novel ansatz relating upand down-type Yukawa couplings, can be found in [141]. This analysis points towards an H + mass around 100 GeV, with masses of other neutral Higgs bosons in the range 100-125 GeV. A similar approach to describe various low-energy anomalies by introducing additional scalars can be found in [142]. Here, a lepton-specific 2HDM (i.e., of type X) with non-standard Yukawa couplings has been analyzed with the second neutral CP-even Higgs boson light (below 100 GeV) and a relatively light H + , with a mass of the order of 200 GeV.

Low-energy constraints
As mentioned above, several decays involving heavy-flavor quarks could be affected by H + in addition to W + -exchange. Data on such processes provide constraints on the coupling (represented by tan β) and the mass, M H ± . Below, we discuss the most important ones.

Constraints from H + tree-level exchange B → τ ν τ (X):
The measurement of the branching ratio of the inclusive process B → τ ν τ X [143] leads to the following constraint, at the 95% CL: This is in fact a very weak constraint (a similar result can be obtained from the leptonic tau decays at tree level [144]). A more recent measurement for the exclusive case gives BR(B → τ ν τ ) = (1.14 ± 0.27) × 10 −4 [145]. 9 With a Standard Model prediction of (0.733 ± 0.141) × 10 −4 [147], 10 we obtain Interpreted in the framework of the 2HDM at tree level, one finds [148][149][150] Two sectors of the ratio tan β/M H ± are excluded. Note that this exclusion is relevant for high values of tan β.
are sensitive to H + -exchange, and they lead to constraints similar to the one following from B → τ ν τ X [152]. In fact, there has been some tension between BaBar results [151,153,154] and both the 2HDM (II) and the SM. These ratios have also been measured by Belle [155,156] and LHCb [157]. Recent averages [141,158] are summarized in Table 2, together with the SM predictions [159][160][161]. They are compatible at the 2σ -3σ level. A comparison with the 2HDM (II) concludes [155] that the results are compatible for tan β/M H ± = 0.5/GeV. However, in view of the high values for M H ± required by the B → X s γ constraint, uncomfortably high values of tan β would be required. The studies 9 The error of the B → τ ν measurement, given by HFAG [146] and released after the PDG 2014 [145], is slightly lower: (1.14 ± 0.22) × 10 −4 . 10 We have added in quadrature symmetrized statistical and systematic errors.  [158] 0 .388 ± 0.047 0.300 ± 0.010 R(D) [141] 0 .408 ± 0.050 0.297 ± 0.017 given for Model II in Sect. 8.3 do not take this constraint into account. D s → τ ν τ : Severe constraints can be obtained, which are competitive with those from B → τ ν τ [162].

Constraints from H + loop-level exchange
B → X s γ : The B → X s γ transition may also proceed via charged Higgs-boson exchange, which is sensitive to the values of tan β and M H ± . The allowed region depends on higherorder QCD effects. A huge effort has been made devoted to the calculation of these corrections, the bulk of which are the same as in the SM [164][165][166][167][168][169][170][171][172][173][174][175][176][177][178][179][180][181][182][183]. They are now complete up to NNLO order. On top of these, there are 2HDM-specific contributions [13,[184][185][186][187][188] that depend on M H ± and tan β. The result is that mass roughly up to M H ± = 480 GeV is excluded for high values of tan β [13], with even stronger constraints for very low values of tan β. Recently, a new analysis [189] of Belle results [190] concludes that the lower limit is 540 GeV. Also note the new result of Misiak and Steinhauser [14] with lower limit in the range 570-800 GeV; see Fig. 17 (right) for high tan β and high H + masses. We have here adopted the more conservative value of 480 GeV, however, our results can easily be re-interpreted for this new limit. Constraints from B → X s γ decay for lower H + masses are presented in Fig. 19 together with other constraints.
For low values of tan β, the constraint is even more severe. This comes about from the charged-Higgs coupling to b and t quarks (s and t) containing terms proportional to m t / tan β and m b tan β (m s tan β). The product of these two couplings determine the loop contribution, where there is an intermediate t H − state, and leads to terms proportional to m 2 t / tan 2 β (responsible for the constraint at low tan β) and m t m b (responsible for the constraint that is independent of tan β). For Models I and X, on the other hand, both these couplings are proportional to cot β. Thus, the B → X s γ constraint is in these models only effective at low values of tan β. 11 [192][193][194][195][196][197]. Recent values for the oscillation parameters m d and m s are given in Ref. [198], only at very low values of tan β do they add to the constraints coming from B → X s γ .

Other precision constraints
T and S: The precisely measured electroweak (oblique) parameters T and S correspond to radiative corrections, and are (especially T ) sensitive to the mass splitting of the additional scalars of the theory. In papers [47,48] general expressions for these quantities are derived for the MHDMs and by confronting them with experimental results, in particular T , strong constraints are obtained on the masses of scalars. In general, T imposes a constraint on the splitting in the scalar sector, a mass splitting among the neutral scalars gives a negative contribution to T , whereas a splitting between the charged and neutral scalars gives a positive contribution. A recent study [199] also demonstrates how RGE running may induce contributions to T and S. Current data on T and S are given in [145].
The muon anomalous magnetic moment: We are here considering heavy Higgs bosons (M 1 , M H ± 100 GeV), with a focus on the Model II, therefore, according to [39,200,201], the 2HDM contribution to the muon anomalous magnetic moment is negligible even for tan β as high as ∼40 (see, however, [202]).

The electron electric dipole moment:
The bounds on electric dipole moments constrain the allowed amount of CP violation of the model. For the study of the CP-non-conserving Model II presented in Sect. 8.3, the bound [203] (see also [204]): |d e | 1 × 10 −27 [e cm], (7.5) was adopted at the 1σ level (More recently, an order-ofmagnitude stronger bound has been established [205]). The contribution due to neutral Higgs exchange, via the two-loop Barr-Zee effect [206], is given by Eq. (3.2) of [204].

Summary of low-energy constraints
A summary of constraints of the 2HDM Model II coming from low-energy physics performed by the "Gfitter" group [207] is presented in Fig. 19. The more recent inclusion of higher-order effects pushes the B → X s γ constraint up to around 480 GeV [13] or even higher, as discussed above. See also Refs. [198,208,209].

High-energy constraints
Most bounds on charged Higgs bosons are obtained in the low-mass region, where a charged Higgs might be produced in the decay of a top quark, t → H + b, with the LEP 95% CL exclusion 95% CL excluded regions (Reprinted with kind permission from EPJC and the authors of "Gfitter" [207] Fig. 4, it vanishes in the alignment limit.

Charged-Higgs constraints from LEP
The branching ratio R b ≡ Z →bb / Z →had would be affected by Higgs exchange. Experimentally R b = 0.21629 ± 0.00066 [145]. The contributions from neutral Higgs bosons to R b are negligible [22], however, charged Higgs-boson contributions, as given by [210], Eq. (4.2), exclude low values of tan β and low M H ± . See also Fig. 19.
LEP and the Tevatron have given limits on the mass and couplings, for charged Higgs bosons in the 2HDM. At LEP a lower mass limit of 80 GeV that refers to the Model II scenario for BR(H + → τ + ν) + BR(H + → cs) = 100% was derived. The mass limit for BR(H + → τ + ν) = 100% is 94 GeV (95% CL), and for BR(H + → cs) = 100% the region below 80.5 as well as the region 83-88 GeV are excluded (95% CL). Search for the decay mode H + → AW + with A → bb, which is not negligible in Model I, leads to the corresponding M H ± limit of 72.5 GeV (95% CL) if M A > 12 GeV [17].

Search for charged Higgs at the Tevatron
A D0 analysis [211] with an integrated luminosity 1 fb −1 has been performed for t → H + b, with H + → cs and H + → τ + ν. In the SM one has BR(t → W + b) = 100% with W → lν/q q. The presence of a sizable BR(t → H + b) would change these ratios. For the optimum case of BR(H + → q q) = 100%, upper bounds on BR(t → H + b) between 19 and 22% were obtained for 80 GeV < M H ± < 155 GeV. In [211] the decay H + → q q was assumed to be entirely H + → cs. But these limits on BR(t → H + b) also apply to the case of both H + → cs and H + → cb having sizable BRs, as discussed in [212]. This is because the search strategy merely requires that H + decays to quark jets.
An alternative strategy was adopted in the CDF analysis [213] with an integrated luminosity 2.2 fb −1 . A direct search for the decay H + → q q was performed by looking for a peak centered at M H ± in the di-jet invariant mass distribution, which would be distinct from the peak at M W arising from the SM decay t → W + b with W → q q. For the optimum case of BR(H + → q q) = 100%, upper bounds on BR(t → H + b) between 32 and 8% were obtained for 90 GeV < M H ± < 150 GeV. No limits on BR(t → H + b) were given for the region 70 GeV < M H ± < 90 GeV due to the large background from W → q q decays. For the region 60 GeV < M H ± < 70 GeV, limits on BR(t → H + b) between 9 and 12% were derived.
A search for charged-Higgs production has also been carried out by D0 [214] at higher masses, where H + → tb. Bounds on cross section times branching ratio have been obtained for Models I and III, in the range 180 GeV ≤ M H ± ≤ 300 GeV, for tan β = 1 and tan β > 10.

LHC searches for charged Higgs
A search for t → H + b followed by the decay H + → cs at the LHC (7 TeV) has been performed by the ATLAS collaboration with 4.7 fb −1 [217]. Assuming BR(H + → cs) = 100%, the derived upper limits on BR(t → H + b) are 5.1, 2.5 and 1.4% for M H ± = 90 GeV, 110 GeV and 130 GeV, respectively. These limits are superior to those from the Tevatron search [213], and exclude a sizable region of the Yukawacoupling plane, 12 not excluded by B → X s γ . The recent data from CMS [215] on the production in the tt channel of light charged Higgs bosons decaying to cs at the collision energy of 8 TeV and with an integrated luminosity 19.7 fb −1 show no deviation from the SM. Assuming BR(H + → cs) = 100%, the derived upper limits on BR(t → H + b) are 1.2-6.5% for M H ± in the range (90-160 GeV); see Fig. 20. The data points are found to be consistent with the signal-plus-background hypothesis for a charged Higgs-boson mass of 150 GeV for a best-fit branching fraction value of (1.2 ± 0.2)% including both statistical and systematic errors. The local observed significance is 2.4σ (1.5σ including the look-elsewhere effect).  Similarly, constraints are obtained by ATLAS [218] from the 8 TeV measurements at the LHC, with luminosity 19.5 fb −1 . Results for low-and high-mass H + are shown in Fig. 22, for BR(t → H + b) × BR(H + → τ + ν) (left) and for σ ( pp → H + t + X ) × BR(H + → τ + ν) (right), respectively.
In Fig. 23 (left) CMS results [216] for the case BR(H + → tb) = 100% are presented. Results of a recent ATLAS analysis, performed using a multi-jet final state for the process gb → t H − are presented in Fig. 23 (right). An excess of events above the background-only hypothesis is observed across a wide mass range, amounting to up to 2.4σ .
In addition, ATLAS provides limits on the s-channel production cross section, via the decay mode H + → tb for heavy charged Higgs bosons (masses from 0.4 TeV to 3 TeV), for two categories of final states; see Fig. 24.
It should be noted that in all these figures, "expectations" are a measure of the instrumental capabilities, and the amount of data. In fact, theoretical (model-dependent) expectations can be significantly lower. In particular, in Model I and Model II, the branching ratio for H + → τ + ν is at high masses very low, see Fig. 4. Thus, these models are not yet constrained by the high-mass results shown in Figs. 21 and 22 [221]. However, for Model X the τ + ν branching ratio is sufficiently high for these searches to be already relevant.

Summary of search for charged scalars at high energies
The LEP lower limits on the mass for light H + are 80.5-94 GeV, depending on the assumption on the H + decaying 100% into cs, bs or cs + bs channels. For low-mass H + , ∼80 (90)-160 GeV, limits for the top decay to H + b were derived at the Tevatron and the LHC (ATLAS and CMS) at the level of a few per cent (5.1-1.2%) for the assmption of 100% decay to cs. CMS results on For heavy H + the region between 200 and 600 GeV was studied at LHC for σ ( pp → t (b)H + ) × BR(H + → τ + ν). A special search for an s-channel resonance with mass of H + up to 3 TeV with the decay mode to tb was performed by ATLAS.
Some excesses at 2.4σ for H + mass equal to 150 GeV, as well as for masses between 220 and 320 GeV, are reported by CMS [215,216] and for a very wide H + mass range 200-600 GeV by ATLAS [219].

LHC constraints from the neutral Higgs sector
After the discovery in 2012 of the SM-like Higgs particle with a mass of 125 GeV, measurements of its properties lead to serious constraints on the parameters space of the 2HDM, among others on the mass of the H + .
Constraints on the gauge coupling of the lightest neutral 2HDM Higgs boson were recently obtained by ATLAS [222] for j = 2, 3, and require it to be below the stronger 95% CL obtained by ATLAS or CMS in the scans described in Sect. 8.3.

Further search for H + at the LHC
Here, a discussion of possible search strategies for charged scalars at the LHC is presented. The stakes of a possible discovery from an extended scalar sector are very high, these searches should be pursued in all conceivable channels. Some propositions are described below, separately for low and high masses of the H + boson. As discussed in previous sections, a light charged Higgs boson is only viable in Models I and X. In the more familiar Model II (and also Y), the B → X s γ constraint enforces M H ± 480 GeV or even higher for all values of tan β [13].

Channels for M H ± m t
For low M H ± mass, the proposed searches can be divided into two categories, based on single H + production or H + H − pair production. For all channels presented here, τ ν decays of charged Higgs bosons are the recommended ones.

Single H + production
In Ref. [226] processes with a single H + were studied for Models I and X. Here, the production mechanism depends on the H + bt Yukawa coupling, proportional to 1/ tan β, thus falling off sharply at high tan β. Concentrating on processes without neutral-Higgs-boson intermediate states 13 [Eqs. (6.4)-(6.6)], it was found that for 30 fb −1 of integrated luminosity the reach at the 95% CL allows exploring low values of tan β, up to about 10. At higher values of tan β, the Model I branching ratio for t → H + b becomes too small (see Fig. 5) for the search to be efficient. In Table 3 we present promising parameters for two proposed channels from this analysis.

H + H − pair production
Charged Higgs-boson pair production, see Eq. (6.10) and Figs. 13 and 14, can be sensitive also to higher values of tan β [226]. This will require resonant production via H j decaying to H + H − , and assuming an enhancement of the coupling between charged and neutral Higgs bosons. In Table 4 we present channels which would be viable in the case of resonant intermediate H j states, as represented by the mechanism of Fig. 13a (i

The channel H + → W + H j → W + bb
A study [230] of the process pp → H +t in Model II, where the charged Higgs boson decays to a W and the observed Higgs boson at 125 GeV, which in turn decays to bb, for charged-Higgs mass up to about 500 GeV, concludes that an integrated luminosity of the order of 3000 fb −1 is required for a viable signal. This search channel has recently been re-examined for Model II, for high charged-Higgs mass and neutral-Higgs masses all low [231]. The discovered 125 GeV Higgs boson is taken to be H , the heavier CP-even one. Thus, the charged one could decay to W H, W A and W h. The dominant production mode at high charged-Higgs mass is from the channel (6.2), bg → H +t → H +b W − . There will thus be at least three b quarks in the final state, two of which will typically come from one of the neutral Higgs bosons: Interesting parameter regions are identified for tan β = O(1), and sin(β−α) close to 0 (the discovered Higgs boson is the H ), where a signal can be extracted over the background. The fact that the H + is heavy, means that the W + from its decay will be highly boosted. This fact is exploited to isolate the signal. Since the interesting region has tan β = O(1), this channel remains relevant also for other Yukawa types.
Relaxing CP conservation, this option has been explored in [121]. Imposing the theoretical and experimental constraints discussed in Sects. 3 and 7, one finds a surviving parameter space that basically falls into two regions: (i) low tan β, with non-negligible CP violation and a considerable Table 5 Suggested points selected from the allowed parameter space [121]. Note that M 3 is not an independent parameter. Furthermore, μ = 200 GeV has been studied (see also Ref. [62]). A priori, there is a considerable tt background. However, imposing a series of kinematical cuts, it is found that this background can be reduced to a manageable level, yielding sensitivities of the order of 2-5 for a number of events of the order of 10-20, with an integrated luminosity of 100 fb −1 at 14 TeV. A more sophisticated experimental analysis could presumably improve on this. The more promising parameter points are presented in Table 5. No point was found at higher values of tan β ( 2), within the now allowed range of M H ± . While the above analysis focused on the bosonic production mode, where resonant production via H 2 or H 3 is possible, a study of the fermionic mode, pp(gg) → H +t bX → ttbbX, (8.3) has been performed for the maximally symmetric 2HDM, which is based on the SO(5) group, and has natural SM alignment [232]. In this analysis, the "stransverse" mass, M T 2 [233], is exploited, and it is found that by reconstructing at least one top quark, a signal can be isolated above the SM background.

The channel H + → τ + ν
The H + → τ + ν channel is traditionally believed to have little background. However, a recent study of Model II finds [62] that this channel can only be efficiently searched for at some future facility at a higher energy. This is due to a combination of many effects. At high mass (M H ± 480 GeV) the production rate goes down, whereas a variety of multijet processes also give events with an isolated τ and missing momentum.

The channel H + → tb
As discussed in Sect. 5, except for particular parameter regions allowing the H + → W + H j modes, at high values of M H ± the tb channel is the dominant one. This channel has long been ignored because of the enormous QCD background, but methods are being developed to suppress this, as exemplified in Ref. [219].

Exploiting top polarization
At high masses the bg → H − t production mechanism is dominant. If the H + decays fully hadronically and its mass is known, then semileptonic decays of the top quark can be analyzed in terms of its polarization. Such studies can yield information on tan β, since this parameter determines the chirality of the H + tb coupling [234][235][236][237][238].

Other scenarios
Various scenarios for additional Higgs bosons have been discussed in the literature. These typically assume CP conservation. Several scenarios [239,240] and channels have recently been presented, mostly focussing on the neutral sector, in particular the phenomenology of the heavier CP-even state, H . In the "Scenario D (Short cascade)" of Ref. [239], it is pointed out that if H is sufficiently heavy, it may decay as H → H + W − , or even as H → H + H − . A version of the former is discussed above, in Sect. 8.3, for Model II. In "Scenario E (Long cascade)", it is pointed out that for heavy H + , one may have the chain H + → AW + → H Z W + or H + → H W + , whereas a heavy A may allow A → H + W − → H W + W − . The modes H + → H W + and H + → AW + have also recently been discussed in Refs. [241,242].
The class of bW production mechanisms qb → q H + b depicted in Fig. 10 has been explored in Ref. [243], where it is pointed out that in the alignment limit, with neutral Higgs masses close, M A M H , there is a strong cancellation among different diagrams. Thus, if M A should be light, this mechanism would be numerically important. It is also suggested that the p T -distribution of the b-jet may be used for diagnostics of the production mechanism.

Multi-Higgs-doublet models
Multi-Higgs Doublet Models (MHDM) are models with n scalar SU(2) doublets, where n ≥ 3 [191]. The n = 1 case corresponds to the Standard Model, the n = 2 case corresponds to the 2HDM, the main topic of this paper. New phenomena will appear for n ≥ 3, for which we below often use the abbreviation MHDM. The MHDM has the virtue of predicting ρ = 1 at tree level, as does the 2HDM. In the MHDM there are n − 1 charged-scalar pairs, H + i . We shall The Yukawa interaction of an H + i , i = 1, . . . , n − 1 is described by the Lagrangian: It applies to the 2HDM (n = 2); then the Fs given in Table 6 of Appendix A coincide with the F 1 in the above equation. In general, the F D i , F U i and F L i are complex numbers, which are defined in terms of an n × n matrix U , diagonalizing the mass matrix of the charged scalars. 14 It is evident that the branching ratios of the charged Higgs bosons, H + i depend on the parameters F D i , F U i and F L i . In the case of the 2HDM this shrinks to a single parameter, tan β, which determines these three couplings. This implies that certain combinations are constrained, for example, in Model II we have, for each i, |F D i F U i | = 1. As in the 2HDM (Models II and Y), an important constraint on the mass and couplings of H + in the MHDM is provided by the decay B → X s γ . However, here, even a light H + (i.e., M H ± m t ) is still a possibility, because of a cancellation between the loop contributions from the different scalars. Recently, 2σ intervals in the F D 1 −F U 1 parameter space for M H ± = 100 GeV were derived from B → X s γ [49,244,245], assuming |F U 1 | < 1, in order to comply with constraints from Z → bb.
The fully active 3HDMs with two softly broken discrete Z 2 symmetries have two pairs of charged Higgs bosons, H ± 1 and H ± 2 , studied in [246]. Depending on the Z 2 parity assignment, there are different Yukawa interactions. In each of these, the phenomenology of the charged Higgs bosons is in the CP-conserving case described by five parameters: the masses of the charged Higgs bosons, two ratios of the Higgs vev's tan β and tan γ and a mixing angle θ C between H + 1 and H + 2 . The BR(B → X s γ ) is determined by W + , H + 1 and H + 2 loop contributions. The scenario with masses of O(100 GeV) for the charged Higgs bosons is allowed. Therefore, the search for a light charged Higgs boson, which in some Yukawa models dominantly decays into cb, may allow one to distinguish 3HDMs from 2HDMs. Some results are presented in Fig. 26  < m t is allowed (also by the Tevatron and LEP2). The region of 80 GeV< M H ± 1 < 90 GeV is not constrained by current LHC searches for t → H + b followed by the dominant decay H + → cs/cb, and this parameter space is only weakly constrained from LEP2 and Tevatron searches; see Fig. 26 (left). Any future signal in this region could readily be accommodated by H ± 1 from a 3HDM. A Monte Carlo simulation of the H + 1,2 signals and W + background via the processes gg, qq → tbH − 1,2 and gg, qq → tbW − , respectively, followed by the corresponding di-jet decays is shown in Fig. 26 (right). The charged Higgs-boson signals should be accessible at the LHC, pro- vided that b-tagging is enforced so as to single out the cb component above the cs one (see the following subsection). Therefore, these (multiple) charged Higgs-boson signatures can be used not only to distinguish between 2HDMs and 3HDMs but also to identify the particular Yukawa model realizing the latter. Some benchmark points are provided in [246].

Enhanced H + → cb branching ratio
In the 2HDM, the magnitude of BR(H + → cb) is always less than a few percent, with the exception of Model Y (see Fig. 1), since the decay rate is suppressed by the small CKM element V cb ( V cs ).
In this limit, the ratio of BR(H + → cb) and BR(H + → cs) can be expressed as follows: In Ref. [248] the magnitude of BR(H + → cb) as a function of the couplings F U 1 , F D 1 and F L 1 was studied, updating the numerical study of [15]. As an example, in Fig. 27 (left), being real and negative (positive). Increased sensitivity can be achieved by requiring also a b-tag on the jets from the decay of H + . In Fig. 27 (right) for M H ± = 120 GeV we show contours of BR(t → H + b) × BR(H + → cb), starting from 0.2%, accessible at the LHC. In this case, a large part of the region of |F D 1 | < 5 could be probed, even for |F U 1 | < 0.2. In summary, a distinctive signal of H + from a 3HDM for M H + m t could be a sizable branching ratio for H + → cb. A dedicated search for t → H + b and H + → cb, in which the additional b-jet originating from H + is tagged, would be a well-motivated and (possibly) straightforward extension of the ongoing searches with the decay H + → cs.

Models with charged scalars and DM candidates
It is possible that the issues of dark matter (DM) and mass generation are actually related. 16 Such models must of course contain a Standard-Model-like neutral Higgs particle, with mass at M h = 125 GeV. Additionally, there appear charged Higgs particles and other charged (as well as neutral) scalars. In these models, the DM relic density provides a constraint on the charged scalars.
In order to have a stable DM candidate some Z 2 symmetry is typically introduced, under which an SU(2) singlet or doublet involving the DM particle, is odd. The Z 2 -odd scalars are often called dark scalars. Among them, the lightest neutral one is a DM candidate. Below, we will denote the charged ones S + , and the neutral ones A and S, with S being the lightest. In some models, there may be several scalars, then referred to as S + i and S i . Typically, the charged scalars of these models have some features in common with the charged Higgs of Model I (and X). They do not couple to the b and s quarks (in fact, they do not couple to any fermion), and thus are not affected by the B → X s γ constraint. Hence, they can be rather light. LEP searches for charginos can be used to establish a lower mass bound of about 70 GeV [250] for such charged scalars.
The first model which allows this relationship between the Higgs and DM sectors was introduced many years ago [29], and will here be referred to as the "Inert Doublet Model", or IDM [251][252][253]. 17 Here, one SU(2) doublet ( 1 ) plays the same role as the SM scalar doublet, the other one with zero vacuum expectation value does not couple to fermions. An extension of this model with an extra doublet [254][255][256][257] or a singlet [258] allows also for CP violation. This improves the prospects for describing baryogenesis [19].
Alternatively, a Z 2 -odd scalar SU(2) singlet S [259][260][261][262] mixed with a Z 2 -odd scalar SU(2) doublet 2 may provide a framework for dark matter. 18 It was shown in Refs. [263,264] that the high-energy theory leading to electroweak-scale scalar DM models can be a non-SUSY SO(10) Grand Unified Theory (GUT) [265]. Indeed, the discrete Z 2 symmetry, which makes DM stable, could be an unbroken discrete remnant of some underlying U(1) gauge subgroup [266][267][268]. Unlike in the IDM, in the GUT-induced scalar DM scenario the lightest dark scalar is predicted by RGEs to be dominantly singlet.
A corresponding charged dark scalar S + can be searched for in the decays mediated by a virtual or real W ± , where S i is a neutral dark scalar, and f and f denote SM fermions. The production at the LHC of S + S − pairs, pp → S + S − , and of S + together with neutral dark scalars pp → S + S i was investigated [107,[269][270][271]. Because of relic density and electroweak precision measurement constraints, S + and S tend to be close in mass, M S ± − M S m W . In such regions of parameter space, and in the limit of massless fermions, the S + width is [272] so the S + will be long-lived and travel a macroscopic distance.

The inert doublet model, IDM
The Inert Doublet Model can be defined in terms of the potential 17 The initial motivation was to provide a mechanism for neutrino mass generation. 18 This is in contrast to the mechanism discussed in [258], where the singlet has a non-zero vev and is not related to the dark matter.
This is the same potential as in Eq. (2.1), but without the Z 2breaking term proportional to m 2 12 , and hence with λ 5 real; see Eq. (2.10).
The charged-scalar mass coming from Eq. (10.3) is given by The parameter λ 3 also governs the coupling of the S + to the Higgs particle h. Perturbative unitarity constraints on other lambdas, together with precision data on the electroweak parameters S and T , limit masses of the dark scalars to less than 600 GeV for S and less than 700 GeV for A and S + , for |m 2 22 | below If the model is required to saturate the relic DM abundance, then M S ± has to be below approximately 300 GeV, or else above ∼500 GeV. In the latter case, its mass is very close to that of the DM particle. On the other hand, if the model is not required to saturate the DM relic abundance, only not to produce too much DM, then the charged-scalar mass is less constrained [273].
Analyses based on an extensive set of theoretical and experimental constraints on this model have recently been performed, both at tree level [273,274] and at loop level [26]. Collider as well as astroparticle data limits were included, the latter in the form of dark matter relic density as well as direct detection data. A minimal scale of 45 GeV for the dark scalar mass, and a stringent mass hierarchy M S ± > M A are found [273]. Parameter points and planes for dark scalar pair production S + S − for the current LHC run are proposed, with S + masses in the range 120-450 GeV [273]. It is found that the decay S + → W + S dominates, and M S ± − M S > 100 MeV. A heavier S + benchmark (M S ± > 900 GeV) is also proposed [26].

The CP-violating inert doublet model, IDM2
If the above model is extended with an extra doublet, one can allow for CP violation, like in the 2HDM [254,255]. There are then a total of three doublets, one of which is inert in the sense that it has no vacuum expectation value, and hence no coupling to fermions. This kind of model will have two charged scalars, one (H + ) with fermionic couplings and phenomenology similar to that of the 2HDM (but the constraints on the parameter space will be different, due to the extra degrees of freedom), and one (S + ) with a phenomenology similar to that of the IDM.
The charged scalar S + could be light, down to about 70 GeV. Its phenomenology has been addressed in [272]. The allowed ranges for the DM particle are similar to those found for the IDM, a low-to-intermediate-mass region up to about 120 GeV, and a high-mass region above about 500 GeV.
The decay modes of the additional charged scalar are the same as in the IDM, it decays either to a W and a DM particle, or to a Z and the neutral partner of the DM particle [272]. In the low-to-intermediate S + mass region, the W and Z could be virtual.

One or two inert doublets within 3HDM
In the context of 3HDMs, models with one or two inert doublets were considered, involving new charged Higgs and/or charged inert scalars. (The IDM2 model discussed above, is one such case, allowing for CP violation.) The richer inert particle spectrum for the case with two inert doublets enables a variety of co-annihilation channels of the DM candidate, including those with two different pairs of charged inert bosons [256,257]. This allows one to relieve the tension in current experimental constraints from Planck, LUX and the LHC. As a consequence, new DM mass regions open up, both at the light (M S 50 GeV) and heavy (360 GeV M S 500 GeV) end of the spectrum, which are precluded to the IDM and are in turn testable at the LHC. Concerning LHC phenomenology of visible channels, a smoking gun signature of the model with two inert doublets is a new decay channel of the next-to-lightest inert scalar into the scalar DM candidate involving (off-shell) photon(s) plus missing energy [275], which is enabled by S + i W − loops. The hallmark signal for the model with one inert doublet would be significantly increased H + → W + Z and W + γ decay rates (in which a key role is played by loops involving the S + state) with respect to the IDM [276]. This new phenomenology is compliant with the most up-to-date constraints on the respective parameter spaces, both experimental and theoretical [277].

SO(10) and the GUT-induced scalar DM scenario
The GUT-induced scalar DM scenario with minimal particle content includes a Higgs boson in a 10 and the DM in a 16 representation of SO (10). One identifies here the Z 2 symmetry as the matter parity [263,264], defined as Since the matter parity P M is directly related to the breaking of B − L , the dark sector actually consists of scalar partners of the SM fermions which carry the same gauge quantum numbers as the MSSM squarks and sleptons. In this scenario the origin and stability of DM, the non-vanishing neutrino masses via the seesaw mechanism [278][279][280][281][282] and the baryon asymmetry of the Universe via leptogenesis [283] all spring from the same source -the breaking of the SO(10) gauge symmetry.
The IDM represents just one particular corner of parameter space of the general P M -odd scalar DM scenario in which the DM is predominantly doublet. A theoretically better motivated particle spectrum can be obtained by renormalization group (RG) evolution of the model parameters from the GUT scale to m Z [264], in a direct analogy with the way the particle spectrum is obtained in the constrained minimal supersymmetric standard model (CMSSM). 19 In the GUT-induced scalar DM model, the EWSB may occur due to the existence of dark scalar couplings to the Higgs boson. Moreover, such couplings can lower the stability bound and accommodate also a Higgs mass around 125 GeV [287].
Below the GUT scale M G and above the EWSB scale the model is described by the scalar potential for the doublets and the singlet: V = V IDM + terms bilinear and quartic in S, (10.6) invariant under and with V IDM defined by Eq. (10.3). The charged-scalar mass of this model is determined by V IDM and given by Eq. (10.4). The mass degeneracy of S and the next-to-lightest neutral scalar, denoted S NL , is a generic property of the scenario and follows from the underlying SO(10) gauge symmetry. It implies a long lifetime for S NL , which provides a clear experimental signature of a displaced vertex in the decays S NL → S + − at the LHC [288]. 10.5 Dark charged-scalar phenomenology at the LHC Compared to the H + of 2HDM models, dark S + production lacks some primary parton-level processes, since it has to be q q W + S + S Fig. 28 Feynman diagram for associated dark S + S production at the LHC produced in association with a neutral dark-sector particle, as illustrated in Fig. 28, or else pair produced via γ , Z or H j in the s-channel (see Fig. 13).
The early literature [107,252,269,270] on the IDM was mostly aimed at guiding the search for evidence on the model, via the production of the charged member, S + , together with a neutral one. It focused on a DM mass of the order of 60-80 GeV, and a charged state S + with a mass of the order of 100-150 GeV. For these masses, the production cross sections are at 14 TeV of the order of 100-500 fb, and the two- [270] and three-lepton [107] channels were advocated. For an update on the allowed parameter space and proposed benchmark points; see Ref. [273].
While the S + S channel has the highest cross section, because of the larger phase space, its discovery is challenging. The S + would decay to an (invisible) S, plus a virtual W + , giving a two-jet or a lepton-neutrino final state. The overall signal would thus be jets or an isolated charged lepton (from the W ) plus missing transverse energy. If instead the heavier neutral state A is produced together with the S + , some of its decays (via a virtual Z ) would lead to two-lepton and three-lepton final states [107,270]. Various cuts would permit the extraction of a signal against the tt and W Z background. In the case of the very similar phenomenology of the IDM2, a study of M S = 75 GeV with 100 fb −1 of data concludes that the best S + search channel is in the hadronic decay of the W , leading to two merging jets plus missing transverse energy [272], pp → j + MET. (10.8) Recently, also the four-lepton modes for S + masses in the range 98-160 GeV have been studied [289], and one addressed constraints on the model from existing data on SUSY searches [290], considering two S + masses, 85 and 150 GeV. Another recent study of the S + S and S + A channels [291] concludes that the di-jet channel may offer the best prospects for discovery, but that a luminosity of 500 fb −1 would be needed for an S + mass up to 150 GeV, whereas 1 and 2 ab −1 for masses of 200 and 300 GeV.
In the above SO(10) scenario the mass difference M S ± − M S turns out to be less than m W and at the leading order the allowed decays are only those given by Eq. (10.1) with a virtual W .
In some cases S + is so long-lived (decay length 1 mm), due to an accidental mass degeneracy between S + and S that it may decay outside the detector. Those experimental signatures are in principle background free and allow S + to be discovered at the LHC up to masses M S ± 300 GeV.
To study charged-scalar pair production in the SO(10) scenario at the LHC, parameter points with distinctive phenomenologies are proposed [263,264,271,[286][287][288]. For some parameter points, the S + decays inside the tracker of an LHC experiment. The experimental signature of those points is that the charged track of S + breaks into a charged lepton track and missing energy.

Summary and outlook
Since the summer of 2012 we are in the final stage of confirmation of the foundation of the SM. However, so far there is no clear clue for a further direction. Various SM-like models with extra Higgs scalars exist. A charged Higgs boson (H + ) would be the most striking signal of a Higgs sector with more than one Higgs doublet. Such a discovery at the LHC is a distinct possibility, with or without supersymmetry. However, a charged Higgs particle might be hard to find, even if it is abundantly produced.
For masses of the charged scalar below 500 GeV, a variety of 2HDM models remain viable, with H + decaying either to heavy flavors or to τ + ν. Some of these have Model I-type Yukawa couplings, others arise in models that accommodate dark matter. Above 500 GeV, also Model II would be a possible interpretation. Here, the most "natural" decay modes would be to tb and H j W + , where H j could be any of the three neutral Higgs bosons.
If a signal were to be found, one of the first questions would be whether it is the charged Higgs of the MSSM or not. We note that the MSSM mass spectrum is very constrained; the heavier states should be close in mass. Secondly, the Yukawa couplings would at tree level be those of Model II. This means that the low-mass region would be severely constrained by B → X s γ , unless there is some cancellation of the H + contribution. A natural candidate would be a squark-chargino loop. But lower bounds on squark and chargino masses make this hard to arrange.
The charged Higgs boson can also be part of a higher representation. Additionally, in higher representations one could have doubly charged H ++ , and also more "exotic" decay modes. For example, with a Higgs triplet, one could have [292] H + → W + Z at tree level. Note however, a recent ATLAS analysis [293] excludes charged Higgs between 240 and 700 GeV if H + → W + Z is the dominant decay mode. This could be the case for the Georgi-Machacek model [294]. This process is also possible in the 3HDM discussed in Sect. 9.1, but then only being generated at the one-loop level.
For the above-mentioned H + → H j W + decay modes, there are two competing effects. (i) Decay to the lightest state H 1 (or h) benefits from a non-negligible phase space, but vanishes in the alignment limit; see Eq. (2.13) and Fig. 6. (ii) Decays to the heavier ones, H 2 and H 3 (or H and A), where couplings are not suppressed, suffer from a small phase space, since various constraint (T , in particular) force these masses to be close to M H ± . In view of the convergence of measurements pointing to a CP-conserving Higgs sector, and alignment, the parameter space for the H + → H j W + decay modes is shrinking, and at high masses the tb mode may be the most promising one. However, the QCD background is very challenging, so improved analysis techniques could turn out to be very beneficial.
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Appendix A: Field decompositions
Breaking the electroweak symmetry spontaneously, we assume that the electrically neutral components of the Higgs doublets have non-zero expectation values, cf. Eq. (2.3). By assuming that they are real and positive, we define a basis in which Note that the introduced parameter tan β has no a priori connection to the Yukawa interaction. The decompositions for 1 and˜ 1 = −i † 1 σ 2 T = iσ 2 * 1 are given by and similarly for 2 and˜ 2 .

Appendix B: Yukawa couplings for the 2HDM
For completeness, we summarize in this appendix the definition of Yukawa couplings in the general 2HDM employed for the analysis in this paper. Below, we also give a comparison with other notations.

B.1 Our notation
Assuming the SM fermion content (without right-handed neutrinos), couplings of the fermions to two scalar doublets (a = 1, 2) may be written in a completely general setting as −L Yukawa = Q L a F D a D R + Q L a F U a U R + L L a F L a L R + h.c., (B.1)  Table 7 Dictionary of notations. "HHG": Higgs Hunter's Guide [6]. "BHP": Barger, Hewett, Phillips [296]. "G": Grossman [191], "AS": Akeroyd, Stirling [15]. The (*) denotes interchange 1 ↔ 2 . "ARS": Atwood, Reina, Soni [297]. "AKTY": Aoki, Kanemura, Tsumura, Yagyu [298]. "BFLRSS": Branco, Ferreira, Lavoura, Rebelo, Sher, Silva [7] 1 2   This work  HHG  BHP  G, AS  ARS  AKTY  BFLRSS   u, d,  I  I  I  I ( * )  -I  I   d,  u  II  II  II  II  -II  II   u, d,  u, d,  III  ---III  - There are various ways fermions can couple to the Higgs doublets, leading to different Yukawa couplings. Since an extended Higgs sector naturally leads to FCNC, these would have to be suppressed. This is normally achieved by imposing discrete symmetries in modeling the Yukawa interactions, as for example Z 2 symmetry under the transformation 1 → 1 , 2 → − 2 . There are four such possible models with Natural Flavor Conservation (NFC): all fermions couple only to one doublet (conventionally taken to be 2 ), or one fermion (U , D, L) couples to one doublet, the other two to the other doublet. 20 Still other Yukawa models are being considered, where all fermions couple to both doublets (Model III), leading to tree-level FCNC processes. This issue is discussed in Appendix B.3 below.
In the three-generation case with discrete symmetry imposed on the Yukawa Lagrangian, such that each righthanded fermionic state interacts with only one scalar doublet, we have for the fermion mass eigenstates where we used a notation like in Eq. (B.1), with N referring to the neutrinos. Here, P L and P R are chirality projection operators. The couplings F D , F U defining models of Yukawa interactions are given in Table 6 for the notation that is used in this paper. Note the appearance of the V C K M matrix. 20 Avoiding FCNC at tree level may not be sufficient, however. One should also investigate stability of these conditions under radiative corrections [295].
We can write the charged-Higgs Lagrangian for one generation in the simplified form (neglecting elements of the CKM matrix) In the limit that in the above equation the second term dominates (for example, for the third generation, with m t m b ) these couplings are the same as for Model II, for moderate values of tan β.

B.2 Various notations
The 1981 paper by Hall and Wise [299] may have been the first to introduce "Model I" and "Model II". They were introduced in analogy with the later convention of "The Higgs Hunter's Guide" (see below), but with the role of 1 and 2 interchanged. An early paper distinguishing quarks and leptons in this respect, was that of Barnett, Senjanovic, Wolfenstein, and Wyler [300]. They define models IA, IB, IIA, IIB.
The definitions of "Model I" and "Model II" presented above coincide with those of the "Higgs Hunter's Guide" [6]. Barger, Hewett and Phillips [296] defined additional models, where quarks and leptons couple differently. Also Grossman [191], Akeroyd and Stirling [15] discussed such models, under different names. Aoki, Kanemura, Tsumura, and Yagyu [298] introduced "Model X" and "Model Y" to avoid the ambiguity previously associated with "Model III". We have adopted the latter notation in this paper.
In Table 7 we present a "dictionary" of notations for the five models.

B.3 Minimal flavor violation
In the most general version of the 2HDM, the fermionic couplings of the neutral scalars are non-diagonal in flavor, leading to FCNC at the tree level.
In Refs. [301,302], the authors propose the so-called aligned 2HDM by fixing the matrices F F a in Eq. (B.1), for a = 1 and a = 2, to be pairwise proportional, Thus, there is no FCNC at tree level. The aligned 2HDM is just the most general minimally flavor-violating (MFV) renormalizable 2HDM, with the lowest order in the couplings Y F . Following Ref. [303], the most general MFV ansatz is given by the expansion This simple form of F D 1 and F U 2 can be assumed without loss of generality. But even if the higher-order terms in F D 2 and F U 1 are not included at tree level, they are generated by radiative corrections. This is ensured by the RG invariance of the MFV hypothesis which is implemented by the flavor SU (3) 3 symmetry. Thus, the functional form of Eq. (B.7) is preserved, only the coefficients i and i change and become related via the RG equations. In view of this, it is also clear that setting all coefficients to zero leads to heavy fine-tuning. Thus, in general there is no Yukawa alignment within the MFV framework.
In Ref. [295], the stability of the various tree-level implementations is discussed. In the MFV case, the FCNC induced by higher-order terms are under control, since even when the coefficients in Eq. (B.7) are of O(1) the expansion is rapidly convergent due to small CKM matrix elements and small quark masses [303].
The higher-dimensional operators which are Z 2 invariant may still induce new FCNC and further flavor protection is needed [295], e.g. via the MFV hypothesis. This problem already occurs in the case of one Higgs doublet [304][305][306].