Complementarity in direct searches for additional Higgs bosons at the LHC and the International Linear Collider

We discuss complementarity of discovery reaches of heavier neutral Higgs bosons and charged Higgs bosons at the LHC and the International Linear Collider (ILC) in two Higgs doublet models (2HDMs). We perform a comprehensive analysis on their production and decay processes for all types of Yukawa interaction under the softly-broken discrete symmetry which is introduced to avoid flavour changing neutral currents, and we investigate parameter spaces of discovering additional Higgs bosons at the ILC beyond the LHC reach. We find that the 500 GeV run of the ILC with the integrated luminosity of 500 fb^{-1} shows an advantage for discovering the additional Higgs bosons in the region where the LHC cannot discover them with the integrated luminosity of 300 fb^{-1}. For the 1 TeV run of the ILC with the integrated luminosity of 1 ab^{-1}, production processes of an additional Higgs boson associated with the top quark can be useful as discovery channels in some parameter spaces where the LHC with the integrated luminosity of 3000 fb^{-1} cannot reach. It is emphasized that the complementary study at the LHC and the ILC is useful not only to survey additional Higgs bosons at the TeV scale, but also to discriminate types of Yukawa interaction in the 2HDM.


I. INTRODUCTION
In July 2012, both the ATLAS and CMS Collaborations announced the observation of a long-sought new particle with a mass approximately at 126 GeV [1,2]. Further measurements of the properties of this new particle manifest consistency with the Higgs boson in the standard model (SM) within the errors which are not small up to now [3][4][5][6]. It makes the SM much closer to its triumph in explaining electroweak symmetry breaking. However, this does not necessarily mean that the SM is fundamentally correct. There is no theoretical principle to justify the minimal Higgs sector with only one Higgs doublet in the SM, and many new physics models beyond the SM predict non-minimal Higgs sectors. Therefore, it is very important to determine the Higgs sector in order to understand the structure of the new physics model by future experiments at the LHC and the International Linear Collider (ILC) [7,8].
The two Higgs doublet model (2HDM) is one of the simplest extensions of the SM Higgs sector, which is useful in both exploring the phenomenology of extended Higgs sectors and interpreting experimental results from searches for additional Higgs bosons. Some of the new physics models contain two Higgs doublets, such as the minimal supersymmetric extension of the SM (MSSM) [9][10][11], models for extra CP phases, models for electroweak baryogenesis [12][13][14], and models for radiative neutrino mass generation mechanism [15][16][17]. In general, the extension with additional doublet fields causes flavour changing neutral currents (FCNCs), which are strongly bounded by experimental data. In order to avoid such dangerous FCNCs, different quantum number should be assigned to each doublet field [18]. This can be attained by introducing a softly-broken discrete symmetry under which Φ 1 → +Φ 1 and Φ 2 → −Φ 2 , where Φ 1 and Φ 2 are the two doublet fields 1 . In this case, there can be four types of Yukawa interaction, depending on the assignment of charges of the discrete symmetry [22,23]. In the 2HDMs, there are two CP-even neutral scalars h and H, one CP-odd neutral scalar A, and a pair of charged scalars H ± . We assume that the lighter CP-even neutral scalar h is the discovered SM-like Higgs boson with the mass of about 126 GeV. Additional neutral and charged Higgs bosons have rich phenomenology and serve as a cornerstone for physics beyond the SM.
In the literature, there have been many discussions on various types of 2HDMs and their signatures at the LHC [24][25][26][27]. For a recent systematic study on the theory and phenomenology of 2HDMs, we refer to Ref. [28] and references therein. In light of the recent data collected at the LHC 7-8 TeV run, many possibilities for explanation of the current data of several decay channels for the observed Higgs boson are explored in the framework of the 2HDMs [29][30][31][32][33][34][35][36][37][38][39][40][41][42][43][44][45][46]. Furthermore, the parameter regions in the 2HDMs have been constrained by direct searches for additional Higgs bosons at the LHC [47,48]. For the future run of the LHC with the collision energy of 14 TeV, additional Higgs bosons are expected to be detected as long as their masses are smaller than 350 GeV to 800 GeV, depending on the scenario of the 2HDMs for the integrated luminosity of 300 fb −1 [49].
The ILC is a future electron-positron linear collider with the collision energies to be from 250 GeV to 1 TeV [7,8]. The ILC can be used for precision measurements of the masses and couplings of the SM particles. We can expect that the first run of the ILC with the collision energy at 250 GeV is capable of measuring the properties of the discovered SMlike Higgs boson with a considerable level. By the combination of the results with higher collision energies up to 1 TeV, all the coupling constants with the discovered Higgs boson can be measured with excellent accuracies. For instance, the Higgs couplings with weak gauge bosons can be measured by better than 1%, the Yukawa coupling constants can be measured by percent levels, and the triple Higgs boson coupling can be measured by a ten percent level [49,50]. Such precision measurements of coupling constants of the discovered Higgs boson can make it possible to perform fingerprinting of extended Higgs sectors when deviations from the SM predictions are detected, because each extended Higgs sector predicts a different pattern in deviations of coupling constants [49][50][51][52][53]. However, the deviations in the coupling constants of the SM-like Higgs boson from the SM predictions can be smaller than those detectable at the ILC, even when additional Higgs bosons are not too heavy.
At the ILC, the direct searches can also be well performed for new particles in the models beyond the SM as long as kinematically accessible. Additional Higgs bosons can be produced mainly in pair if the sum of the masses is less than the collision energy, via e + e − → hA [54], e + e − → HA [55] and e + e − → H + H − [55]. For the collision energy below the threshold of the pair production, single production processes of new additional Higgs bosons can be used too, although the production cross sections are not large. The single charged Higgs boson production has been studied in the framework of the MSSM [56,57]. Preliminary detection possibilities were studied at linear colliders, and their analysis shows that in the parameter space beyond the kinematic limit for pair production, single production of H ± associated with the top quark turns out to be a useful channel in studying the charged Higgs boson phenomenology [57]. QCD corrections to the process e + e − →tbH + and its charge conjugate counterpart have been studied in the MSSM in Ref. [58]. The single production processes of additional neutral Higgs bosons have been studied in Ref. [59], and QCD corrections to the e + e − → QQH and e + e − → QQA processes are calculated in Refs. [60,61] where Q = t and b. The discovery potential for additional Higgs bosons through single and pair production processes at linear collider are evaluated in the MSSM [62], which is useful in distinguishing the MSSM from the other models.
In this paper, we perform a comprehensive analysis on the production and decay processes of additional Higgs bosons for all types of Yukawa interaction under the discrete symmetry.
The parameter space of discovering additional Higgs bosons at the LHC is shown for all types of Yukawa interaction in the 2HDM according to the analysis given in Ref. [49]. We then examine detailed signatures of additional Higgs bosons for all types of Yukawa interaction at the ILC. We find that the complementary study at the LHC and the ILC is useful not only to survey additional Higgs bosons at the TeV scale, but also to discriminate types of Yukawa interaction in the 2HDM.
The paper is organized as follows. In Sec. II, we introduce the 2HDMs and the different types of Yukawa interaction. In Sec. III, we present a brief summary of theoretical and experimental (flavour and collider) constraints on the additional neutral and charged Higgs bosons. Our study on the future prospects of the LHC searches are also presented in this section. Sec. IV is devoted to our systematic analysis on the ILC search for the additional Higgs bosons. Based on several benchmark scenarios, further discussions on the prospects of the direct searches of additional Higgs bosons at future collider experiments are given in Sec. V. Finally, we draw a conclusion in Sec. VI.
where m 2 1 , m 2 2 , λ 1−4 are real parameters while m 2 3 , λ 5−7 are complex in general. For the most general 2HDM, the presence of Yukawa interactions leads to the FCNCs via tree-level Higgs-mediated diagrams which is not phenomenologically acceptable. To avoid such FCNCs, we consider 2HDMs with discrete Z 2 symmetry, under which the two doublets are transformed as Φ 1 → +Φ 1 and Φ 2 → −Φ 2 [18,[63][64][65]. For the SM fermions, four sets of parity assignment under the Z 2 transformation are possible [22], which is summarized in Table I. Because of these types of Yukawa interaction, the 2HDM with Z 2 parity contains a variety of phenomenology with quarks and leptons.
To preserve the discrete symmetry, we hereafter restrict ourselves with the Higgs potential in Eq. (1) with vanishing λ 6 and λ 7 which induce the explicit breaking of the symmetry.
On the other hand, the presence of the m 2 3 term induces the soft breaking of the symmetry characterized by the soft-breaking scale M 2 = m 2 3 /(sin β cos β) [66]. Therefore, we allow the m 2 3 term and the soft breaking of the Z 2 symmetry. Furthermore, we consider the CP-conserving scenario for simplicity by taking m 2 3 and λ 5 to be real. After the electroweak symmetry breaking and after the three Nambu-Goldstone bosons are absorbed by the Higgs mechanism, five physical states are left; two CP-even neutral Higgs bosons, h and H; one CP-odd neutral Higgs boson, A; and charged Higgs bosons, H ± . Masses of these scalars are obtained by solving the stability conditions of the potential in Eq. (1) [10]. In addition to the four kinds of masses m h , m H , m A and m H ± as well as the soft-breaking parameter M 2 , the remaining two parameters are chosen as follows. One is tan β = v 2 /v 1 , the ratio of the vacuum expectation values (VEVs) of the two doublet fields, . The other is α, a mixing angle for diagonalizing the mass matrix for the neutral CP-even component.
The limit of sin(β − α) = 1 is called the SM-like limit where the light CP-even scalar h behaves as the SM Higgs boson [67]. We take h as the observed SM-like Higgs boson with The input parameters of the model are v, m h , m H , m A , m H ± , M, α and β. In terms of these parameters, the quartic coupling constants are expressed as [66] The interactions of the Higgs bosons to weak gauge bosons are common among the types of Yukawa interaction. Feynman rules for these interactions are read out from the Lagrangian [10,11]; and hAZ µ : where p µ and p ′ µ are outgoing four-momenta of the first and the second scalars, respectively, and g Z = g W / cos θ W .

B. Type of Yukawa interaction
The Yukawa interactions of the 2HDM Higgs bosons to the SM fermions are written as where R and L represent the right-handed and left-handed chirality of fermions, respectively.
Φ f =u,d,ℓ is chosen from Φ 1 or Φ 2 to make the interaction term Z 2 invariant, according to the Table I. The Type-I 2HDM is the case that all the quarks and charged leptons obtain the masses from v 2 , and the Type-II 2HDM is that up-type quark masses are generated by v 2 but the masses of down-type quarks and charged leptons are generated by v 1 . In the Type-X 2HDM, both up-and down-type quarks couple to Φ 2 while charged leptons couple to Φ 1 .
The last case is the Type-Y 2HDM where up-type quarks and charged leptons couple to Φ 2 while up-type quarks couple to Φ 1 . We note that the Type-II 2HDM is predicted in the context of the MSSM [9,10] and that the Type-X 2HDM is used in some of radiative seesaw models [16,17].
In terms of the mass eigenstates, Eq. (5) is rewritten as where P R,L are the chiral projection operators. The coefficients ξ f φ are summarized in Table II.  [27]. c θ = cos θ, and s θ = sin θ for θ = α, β.
In the SM-like limit, all the φV V vertices in Eq. (3) and φhV in Eq. (4) in which one additional Higgs boson is involved disappear, where φ represents H, A or H ± . On the other hand, the Yukawa interactions of additional Higgs boson remain even in this limit.
Therefore, Yukawa interactions of the additional Higgs bosons are very important for the decay and production processes of additional Higgs bosons in this limit.

C. Decay widths and decay branching ratios
For each type of Yukawa interaction, the decay widths and branching ratios of additional Higgs bosons can be calculated for given values of tan β, sin(β − α) and the masses. The total decay widths of additional Higgs bosons are necessary for the consistent treatment of the production and decays of additional Higgs bosons. We refer to Ref. [27] where the total decay widths are discussed in details for sin(β − α) ≃ 1. Explicit formulae for all the partial decay widths can be found, e.g., in Ref. [27]. Here, we review the characteristic behaviors of the decays of additional Higgs bosons in each type of Yukawa interaction by presenting numerical results of the branching ratios. For simplicity, we set sin(β − α) = 1, the SM-like limit. In this limit, the decay modes of H → W + W − , ZZ, hh as well as A → Zh are absent. Decay branching ratios of the SM-like Higgs boson become completely the same as those in the SM at the leading order, so that we cannot distinguish models by the precision measurement of the couplings of the SM-like Higgs boson 2 . As we discuss later, the branching ratios can drastically change if sin(β − α) is slightly deviated from unity.     The decay branching ratios of H and A are almost unchanged from the results for 125 GeV, but those of H ± are changed due to the new decay mode tb. This decay mode dominates for all the tan β regions for the Type-I, Type-II and Type-Y, and for tan β 10 for Type-X.
The τ ν mode can be dominant and sub-dominant (∼ 0.3) for tan β 10 for Type-X and Type-II, respectively.
In Fig. 3, the same branching ratios are evaluated for m H = m A = m H ± = M = 500 GeV.
In this case, the tt mode opens in the decays of H and A. The tt decay dominates in all the tan β regions for Type-I, tan β 5 for Type-II, Type-X and Type-Y, while the other modes are suppressed accordingly. The decays of H ± are similar to those in the 250 GeV cases.

III. CONSTRAINTS ON 2HDM PARAMETERS
In this section, we briefly review the theoretical and experimental constraints on the parameters in the 2HDMs.
A. Constraints on the Higgs potential from perturbative unitarity and vacuum stability First, we introduce the constraints on the parameters by theoretical arguments, namely perturbative unitarity and vacuum stability. The tree-level unitarity requires the scattering amplitudes to be perturbative [72], i.e. |a i | < 1/2 [10], where a i are the eigenvalues of the S-wave amplitudes of two-to-two elastic scatterings of the longitudinal component of weak gauge bosons and the Higgs boson. In the 2HDM with the softly-broken Z 2 symmetry, this condition gives constraints on the quartic couplings in the Higgs potential [73][74][75]. The eigenvalues for 14 × 14 scattering matrix for neutral states are given as [73], and for singly charged states, one additional eigenvalue is added [74], Second, the requirement of vacuum stability that the Higgs potential must be bounded from below gives [76][77][78] The parameter space of the model is constrained by these conditions on the coupling constants in the Higgs potential.

B. Constraints on the Higgs potential from electroweak precision observables
Further constraints on the Higgs potential of the 2HDM are from the electroweak precision measurements. The S, T and U parameters are defined to disentangle new physics effects in the radiative corrections to the gauge bosons two-point functions [79]. Those are sensitive to the effects of Higgs bosons through the loop corrections [80,81]. The T parameter corresponds to the ρ parameter, which is severely constrained by experimental observations as ρ = 1.0005 +0.0007 −0.0006 where U = 0 is assumed [71]. Because of this constraint, the mass splitting among the additional Higgs bosons are constrained in the 2HDM with the light SM-like Higgs boson [82,83].

C. Flavour constraints on m H ± and tan β
Flavour experiments constrain the 2HDM through the H ± contribution to the flavour mixing observables by tree-level or loop diagrams [27,84,85]. Since the amplitudes of these processes contain the Yukawa interaction, constraints from the flavour physics strongly depends on the type of Yukawa interaction. In Ref. [86], the limits on the general couplings by flavour physics are translated to the limits in the (m H ± , tan β) plane in each type of Yukawa interaction in the 2HDM. See also recent studies in Refs. [87][88][89].
The strong exclusion limit is provided from the measurements of the branching ratio of B → X s γ processes [90]. For Type-II and Type-Y, a tan β-independent lower limit of m H ± 380 GeV is obtained [91] by combining with the NNLO calculation [92]. On the other hand, for Type-I and Type-X, tan β 1 is excluded for m H ± 800 GeV, but no lower bound on m H ± can be obtained.
For all types of Yukawa interaction, lower tan β regions (tan β ≤ 1) are also excluded for m H ± 500 GeV by the measurement of B 0 d -B 0 d mixing [90], because of the universal couplings of H ± to the up-type quarks.
Constraints for larger tan β regions are obtained only in the Type-II 2HDM by using the leptonic meson decay processes [90], B → τ ν [93] and D s → τ ν [94]. This is because for Type-X and Type-Y (Type-I). For Type-II, upper bounds of tan β are given at around 30 for m H ± ≃ 350 GeV and around 60 for m H ± ≃ 700 GeV [86]. 100 GeV to 300 GeV, respectively. For the H ± search at the Tevatron, the decay modes of H ± → τ ν and H ± → cs have been investigated using the production from the top quark decay of t → bH ± [103][104][105]. Upper bounds on the decay branching ratio B(t → bH ± ) have been obtained, which can be translated into the bound on tan β in various scenarios. In the Type-I 2HDM, for H ± heavier than the top quark, upper bounds on tan β have been obtained to be from around 20 to 70 for m H ± from 180 GeV to 190 GeV, respectively [104].
At the LHC, direct searches for the additional Higgs bosons have been performed by using the recorded events at a center-of-mass energy of 7 TeV with 4.9 fb −1 and 8 TeV with 19.7 fb −1 in 2011 and 2012, respectively. The CMS experiment has searched H and A decaying to the τ + τ − final state, and upper limits on tan β have been obtained for the MSSM scenario or the Type-II 2HDM from 4 to 60 for m A from 140 GeV to 900 GeV, respectively [106]. Similar searches have been also performed by ATLAS [107]. In Type-II and Type-Y 2HDMs, the CMS experiment has also searched the bottom-quark associated production of H or A which decays into the bb final state [108], and has obtained the upper bounds on tan β; i.e., tan β 16 (28)  According to Refs. [49,51], we evaluate the expected discovery potential of additional Higgs bosons at the LHC with the integrated luminosity of L = 300 fb −1 and 3000 fb −1 by using the signal and background analysis for various channels [112], which are combined with the production cross sections and the decay branching ratios for each type of Yukawa interaction. Processes available for the searches are • H/A(+bb) inclusive and associated production followed by the H/A → τ + τ − decay [113].
For the production cross sections, we utilize the Born-level cross sections convoluted with the CTEQ6L parton distribution functions [120]. The scales of the strong coupling constant and parton distribution functions are chosen to the values used in Ref. [11,121]. For the last process, we follow the analysis in Ref. [118] by re-evaluating the signal events for the different mass, and combine the statistical significance of all channels for the decay patterns of 4τ .
The similar analysis on the HH ± and AH ± production processes resulting the signature of 3τ plus large missing transverse momentum gives comparable exclusion curves to the 4τ analysis [118].
In For Type-I, H/A production followed by their τ + τ − decay can be probed for the parameter regions of tan β 3 and m H,A ≤ 350 GeV, where the inclusive production cross section is enhanced by the relatively large top Yukawa coupling and also the τ + τ − branching ratio is sizable. The tH ± production followed by the H ± → tb decay can be used to search H ± in relatively smaller tan β regions. The mass reach for the discovery of H ± can be up to 800 GeV for tan β 1 (2) for the integrated luminosity of 300 fb −1 (3000 fb −1 ).
For Type-II, the inclusive and the bottom-quark-associated production processes of H/A followed by the τ + τ − decay or the bb decay can be used to search H and A in relatively large tan β regions. They can also be used in relatively small tan β regions with m H,A 350 GeV.
Because of the difficulty of separating the signal from the SM background, the lighter mass regions (200 ∼ 300 GeV) may not be excluded with the 300 fb −1 data as loopholes are seen in the figure. H ± can be probed by the tH ± production followed by the H ± → tb analysis given in Ref. [112] for each process, where the signal events are rescaled to the prediction in each case [49,51], except the 4τ process for which we follow the analysis in Ref. [118]. Thick solid lines are the expected exclusion contours by L = 300 fb −1 data, and thin dashed lines are for L = 3000 fb −1 data. For Type-II, the regions indicated by circles may not be excluded by This process is also relevant for Type-II, but the constraint is weaker than H/A → τ + τ − mode. The search of H ± is similar to that for Type-II.
If all the curves are combined by assuming that all the masses of additional Higgs bosons are the same, the mass below 400 GeV (350 GeV) can be excluded by the 300 fb −1 data, and the mass below 550 GeV (400 GeV) can be excluded by the 3000 fb −1 data for any value of tan β for Type-II and Type-Y (Type-X). Only for Type-I, a universal mass bound cannot be given, namely the regions with tan β 5 (10) cannot be excluded by the 300 fb −1 (3000 fb −1 ) data. However, in the general 2HDM, the mass spectrum of additional Higgs boson is less constrained, and has more degrees of freedom. Therefore, we can still find allowed parameter regions where we keep m H to be relatively light but taking m A (≃ m H ± ) rather heavy for the rho parameter constraint [83]. Thus, the overlaying of these exclusion curves for different additional Higgs bosons may be applied to only the case with m H = m A = m H ± .
At the LHC, the discovery reach of H ± is extensive in all types of Yukawa interaction, because of the large cross section of the gb → tH ± process followed by the H ± → tb decay. If H ± is discovered at the LHC, the determination of its mass would follow immediately [112,122]. Hence, the next progress would be the determination of the type of Yukawa interaction. At the LHC, although some methods have been proposed by using the observables related to the top-quark spin [122,123], we could not completely distinguish the types of Yukawa interaction, because the Type-I and Type-X, or Type-II and Type-Y posses the same coupling structure for the tbH ± interaction. Therefore, we have to look at the other process like the neutral Higgs boson production processes. However, as we have seen in We also note that the above results are obtained in the SM-like limit, sin(β − α) = 1.
However, in the general 2HDM, sin(β − α) is also a free parameter. It is known that a deviation from the SM-like limit induces decay modes of H → W + W − , ZZ, hh as well as A → Zh [10,[124][125][126][127]. Especially, for Type-I with a large value of tan β, branching ratios of these decay modes can be dominant even with a small deviation from the SMlike limit [27,125]. These processes open when the collision energy is above the sum of the masses of the two scalars. For energies below the threshold, the single production processes, e + e − → H(A)ff and e + e − → H ± ff ′ are the leading contributions [56]. The single production processes are enhanced when the relevant Yukawa coupling constants of φff ( ′ ) are large. The cross sec-tions of these processes have been studied extensively [8,56,57,62], mainly for the MSSM or for the Type-II 2HDM.
Here, we give numerical results in the general 2HDMs but with softly-broken discrete symmetry with all types of Yukawa interaction. We consider the processes of e + e − → bbH, e + e − → ttH, e + e − →tbH + . In Fig. 7, cross sections of e + e − → τ − νH + are shown as a function of m H ± for various situations in the same manner as Fig. 5. In the first row, cross sections of e + e − → τ − νH + are plotted for tan β = 1, 3, 10, 30 and 100 at the ILC √ s = 250 GeV. For energies below the threshold, √ s < 2m H ± , the single production process can be sizable for Type-II and Type-X, due to the enhanced τ νH ± couplings by tan β. In the second row, for √ s = 500 GeV, there is a sharp edge at around m H ± = 180 GeV for Type-I, Type-Y and also for Type-II and Type-X with small tan β, because the decay of H ± → tb opens. In the third row, for m H ≥ 500 GeV, HA pair production is kinematically forbidden, and the single production becomes the leading mechanism. In all types, the Yukawa couplings of H and A to the top quark are suppressed for large tan β.
In Fig. 9, cross sections of e + e − →tbH + are plotted as a function of m H ± . In the first row, the results for √ s = 500 GeV are shown. For m t + m b ≤ m H ± ≤ 250 GeV, the pair production e + e − → H + H − followed by the decay of H − →tb gives the largest contribution.
The cross section of e + e − → H + H − does not depend on tan β, but only the branching ratio of the decay H ± → tb does. For m H ± ≤ m t − m b and √ s ≥ 2m t , there is a production mechanism oftbH + from e + e − → tt followed by the decay of t → bH + . The partial decay width of t → bH ± can be found e.g. in Ref. [27]. For m H ± ≥ 250 GeV, only the single production mechanism contributes for Type-II and Type-Y, which is enhanced by cot β via the top quark Yukawa coupling or by tan β via the bottom quark Yukawa coupling. In the second row, the same results but for √ s = 1 TeV are shown.

B. Contour Plot
Now we discuss the collider signatures of additional Higgs boson production at the ILC.
Both the pair and single production processes of additional Higgs bosons tend to result in four-particle final-states (including neutrinos) when the decays of the additional Higgs bosons are taken into account. To evaluate the net production rates of them, the production cross sections and the decay branching ratios of additional Higgs bosons have to be taken into account consistently. We calculate the cross sections of various four-particle final-states The figures in the third row are for Type-X. The 4τ signature can be expected for large tan β regions even below the pair production mass threshold. The detailed studies for the 4τ signature can be found in Ref. [129]. For relatively small tan β regions, 4b or 4t signature is expected depending on the masses of H and A. In between, 2b2τ or 2t2τ signature can have sizable rates.  including τ + τ − , gg and cc can be expected because all these decay branching ratios are comparably sizable. To avoid too much overlapping, we ignore the curves for the signatures including cc, which are however comparable with those of the 4g, 2g2τ and 4τ signatures.
For m H/A 350 GeV, the 4t and 2t2b signatures are expected to appear for tan β 10.
In Fig. 11, contour plots of the four-particle production cross sections through H ± are shown in the (m H ± , tan β) plane in the same manner as Fig. 10.
The figures in the first row are for Type-I. For m H ± 180 GeV below the H ± → tb threshold, H ± → τ ν and cs are the dominant decay modes, as illustrated in Fig. 1.
Therefore, the τ ντ ν, τ νcs and cscs signatures are expected to appear as long as For m H ± 180 GeV and √ s ≥ 350 GeV, H ± can be produced through the decay of top quarks in the top quark pair production process. In the middle column at √ s = 500 GeV, the signature of tbτ ν comes from this contribution followed by the decay of H ± → τ ν. For m H ± 180 GeV, the dominant decay mode quickly switches into tb. Therefore the tbtb signature becomes the largest.
The figures in the second row are for Type-II. For the mass below the tb threshold, H + H − pair production tends to be the τ ντ ν signature in the large tan β regions, and the τ νcs, cscs signatures in the medium to small tan β regions. In addition, because of the large Yukawa coupling of top quarks, single tbH ± production followed by H ± → τ ν and cs decays gives sizable tbτ ν and tbcs signatures, respectively. On the other hand, for the mass above the tb threshold, the tbtb signature is the dominant signature for any values of tan β because of the enhanced tbH ± Yukawa interaction. The tbτ ν and τ ντ ν signatures are still visible in large tan β regions, because of the large H ± → τ ν branching ratio.
The figures in the third row are for Type-X. As is the case for Type-II, for the mass below the tb threshold, the τ ντ ν signature in the large tan β regions, and the τ νcs, cscs signatures in the medium to small tan β regions are expected. Through the tbH ± production which is sizable only in the small and medium tan β regions, the tbτ ν and tbcs signatures are expected to be seen. Above the tb threshold, the signatures are tbtb for small and medium tan β and τ ντ ν for large tan β. In between, tbτ ν can also be large.
The figures in the fourth row are for Type-Y. In this case, for the mass below the tb threshold the dominant decay mode of H ± is cb for large tan β. Therefore, cbcb signature is expected for large tan β regions. In small tan β regions, τ ν and cs would be the dominant. Therefore, τ ντ ν, τ νcs and cscs signatures are expected to be significant. To avoid overlapped plotting, we ignore to plot the contours which include the cs mode. Above the tb threshold, since the tb decay mode is dominant for any values of tan β, the tbtb signature would be the only visible mode.

C. SM background processes
Here, we discuss the SM background processes and their cross sections. In Table III, total cross sections without kinematical cuts are calculated by Madgraph [128]. The crosssection for the signatures including gluons is neglected, because the partonic calculation is meaningless unless an infrared safe observable is defined, such as the cross-section for jets production. In general, for the four-particle production processes, the SM background cross The cross section of the 4t production is very small in the SM, see Table III. Therefore, a clean signature can be expected to be detected in this mode. However, because of the decays of top quarks, more complicated background processes can be involved, and the event reconstruction is not straightforward. Detailed studies on the signal and background processes for tbtb production can be found in Ref. [57], and the signal-to-background analysis for the 4τ production can be found in Ref. [129] with the reconstruction method of the masses of additional Higgs bosons.  We take six sets of (m φ , tan β) as benchmark scenarios, where m φ represents the common mass of H, A and H ± , namely m φ = 220 GeV and 400 GeV, and tan β = 2, 7 and 20, for all types of Yukawa interaction. We fix the value of sin(β − α) to be unity. In Table IV, we summarize the expected signatures of H/A and H ± to be observed at the LHC with 300 fb −1 , 3000 fb −1 and at the ILC with √ s = 500 GeV, according to our estimation in the last sections for the benchmark scenarios with m φ = 220 GeV. In Table V  signatures of H/A and H ± are summarized at the LHC with 300 fb −1 , 3000 fb −1 and at the ILC with √ s = 1 TeV for the benchmark scenarios with m φ = 400 GeV. We note again that at the ILC signatures are assumed to be detected by a criterion whether the cross section is greater than 0.1 fb. We present the results for each type of Yukawa interaction, Type-I to Type-Y from the left column to right column, respectively.
In Table IV, the expected signals are summarized for each benchmark scenario with a relatively light mass, m φ = 220 GeV. Let us look at the scenario of (m φ , tan β) = (220 GeV, 20  tbtb   TABLE V: The similar table as Table IV, but for m φ = 400 GeV. ILC1TeV represents the ILC run of 1 TeV. Next, we discuss the scenario of (m φ , tan β) = (400 GeV, 7). At the LHC with 300 fb −1 , no signature is discovered for all types of Yukawa interaction at all. At the LHC 3000 fb −1 , the signals of Type-II, Type-X and Type-Y can be discovered with different signatures, while Type-I cannot be seen. At the ILC, all types are observed with different signatures.
Therefore, the complete discrimination or exclusion needs the ILC in this scenario too.
Finally, we discuss the scenario of (m φ , tan β) = (400 GeV, 2). At the LHC with 300 fb −1 , only the H ± → tb signature is predicted for all types of Yukawa interaction. The situation does not change even with 3000 fb −1 . Therefore, the signals for all types of Yukawa interaction can be discovered, but the type cannot be discriminated at the LHC. At the ILC, tbtb signature is observed for the pair and single production of H ± for all types of Yukawa interaction. For the neutral Higgs bosons, for Type-I and Type-X only the 4t signature is observed, while 4t and 2t2b signatures are observed for Type-II and Type-Y. Therefore, at the ILC, we are able to discriminate the type of Yukawa interaction as either Type-I or Type-X, or either Type-II or Type-Y. However, precision measurements of the number of signal events at the ILC could be used for further discrimination.
To summarize, the additional Higgs bosons can be discovered for all the benchmark scenarios by the combination of searches at the LHC and ILC. Furthermore, the type of Yukawa interaction can be separated by looking at the pattern of the observed signatures.
For the scenarios with (m φ , tan β) = (220 GeV, 20), (400 GeV, 20) and (400 GeV, 7), the ILC is necessary for the complete separation of the type of Yukawa interaction. For the scenario with (m φ , tan β) = (400 GeV, 2), the LHC cannot discriminate the type of Yukawa interaction, while at the ILC two groups of the type, Type-I or Type-X and Type-II or Type-Y can be separated by looking at the difference of signatures, and further discrimination may be possible by precision measurements of the number of signal events. Therefore, the LHC and the ILC are complementary for additional Higgs boson searches and also for discrimination the type of Yukawa interaction in the 2HDM. Furthermore, the determination of tan β can be performed through the observation of the branching ratio or the total decay widths of additional Higgs bosons [130][131][132][133].
We briefly give a comment for the cases with m φ < 200 GeV and m φ > 500 GeV. For In our discussion above, the SM-like limit, sin(β − α) = 1, has been commonly assumed in the benchmark scenarios in Tables IV and V. We here discuss the case in which the SM-like limit is slightly relaxed, i.e., sin 2 (β − α) = 0.9 to 0.99. The pattern of branching ratios of additional Higgs bosons drastically changes in this case: see for example Fig. 2

in
Ref. [27] for sin 2 (β − α) = 1 and Fig. 3 in Ref. [27] for sin 2 (β − α) = 0.96. In particular, for sin 2 (β − α) = 0.96, H can decay into weak gauge bosons, whose decay branching ratios can easily be substantially large. Consequently, our discussion above can be changed. We may expect that the discovery signal of H can be clearer in this case because of the decay into weak gauge boson pairs. The analysis for such a case will be separately performed in the future. We also note that if sin 2 (β − α) is slightly less than unity, the coupling constants of the SM-like Higgs boson with the SM particles differ from the SM predictions. The pattern of the deviations depends on the type of Yukawa interactions. Therefore, by detecting the pattern by precision measurements of the coupling constants of the SM-like Higgs boson at the ILC, we can fingerprint the specific type of Yukawa interaction in the 2HDM [49,51].
Notice that fingerprinting of the model by using the measurement of SM-like Higgs boson coupling constants is powerful as long as sin 2 (β − α) is less than unity by more than 1%. If the deviation is much smaller, we cannot fingerprint the 2HDM by looking at the SM-like Higgs boson coupling constants. In such a case, namely the SM-like limit, only the direct searches for the additional Higgs bosons at the LHC and the ILC are useful.
Finally, we mention the case where our assumption of the common mass for additional Higgs bosons is relaxed. In general, masses of additional Higgs bosons are given by whereλ i represent specific combinations of λ coupling constants. Our assumption is basically reasonable when additional Higgs bosons are heavy enough, because their masses are basically given by the unique scale M, the scale of soft breaking of the discrete symmetry.
When their masses are around the electroweak scale, they can be varied by the contribution of the termλ i v 2 without contradicting the constraints from the rho parameter and also from perturbative unitarity etc. In this case, the signals from neutral Higgs boson processes and those from charged Higgs boson processes are independent. However, even in such a case, we can repeat the discussion of discrimination of the type of Yukawa interaction by using Tables IV and V, although the situation becomes more complicated.

VI. CONCLUSIONS
In this paper, we have studied the direct searches of additional Higgs bosons in the general 2HDM with the Z 2 symmetry imposed to avoid FCNCs. We have considered the possible four types of Yukawa interaction which are determined by generic charge assignment of the Z 2 parity to the SM fermions.
We have discussed the prospect of direct searches for the additional Higgs bosons at the LHC, and stressed that the exclusion potential is extensive but not conclusive. It means that by taking into account the wide parameter space of the general 2HDM, there are possibilities that the LHC can discover only part of the additional Higgs bosons or that even the LHC cannot discover any additional Higgs boson but the ILC can discover.
We have studied the collider signatures of additional Higgs boson production by evaluating the production cross sections as well as the decay branching ratios of additional Higgs bosons at the ILC for the all types of Yukawa interaction. We find that various signatures can be expected depending on the type of Yukawa interaction, the masses of additional Higgs bosons and tan β. Thus, as long as the additional Higgs bosons are kinematically accessible, their production can be detected at the ILC, and further details around the additional Higgs bosons, i.e. the type of Yukawa interaction and the model parameters can be studied.
Therefore, the searches at the ILC would be a useful complementary survey even after the LHC results.