Confronting the fourth generation two Higgs doublet model with the phenomenology of heavy Higgs bosons

A sequential fourth generation is known to be excluded because the non-decoupling contribution to $\kappa_g$, the Higgs coupling modifier with a gluon pair, is unacceptably large. Recently a new way to save the model was suggested in the Type-II two Higgs doublet model: if the Yukawa couplings of down-type fermions have wrong-sign, the contributions from $t'$ and $b'$ to $\kappa_g$ are cancelled. We study the theoretical and experimental constraints on this model, focusing on the heavy Higgs bosons. Two constraining features are pointed out. First the exact wrong-sign limit does not allow the alignment, which makes the perturbative unitarity for the scalar-scalar scattering put the upper bounds on the heavy Higgs boson masses like $M_H, M_A \lesssim 920$ GeV and $M_{H^\pm} \lesssim 620$ GeV. Secondly, the Yukawa couplings of the fourth generation fermions to the heavy Higgs bosons are generically large as being proportional to the heavy fermion mass and, for the down-type fermions, to $\tan\beta$ as well. The gluon fusion productions of $H$ and $A$ through the fourth generation quark loops become significant. We found that the current LHC data on $pp \to Z Z$ for $H$ along with the theoretical and indirect constraints exclude the model at leading order.


Introduction
Our dearest wish for new physics (NP) beyond the standard model (SM) has not been fulfilled yet since the dedicated searches for new particles at the LHC found no new signals and various precision data are consistent with the SM predictions.The usual strategy for a NP theory is to hide it in the decoupling limit: very heavy new particles cannot be observed at the current 13 TeV LHC nor significantly contribute to the precision data including the Higgs signals.One important exception is a sequential fourth generation model where the new chiral fermions (t , b , ν and τ ) acquire their masses via the same Higgs mechanism.This model addresses one of the most fundamental questions−whether there exist only three fermion generations in the universe [1][2][3].The discovery of the SMlike Higgs boson with a mass of 125 GeV [4,5] excludes this model since the contributions of the heavy fourth generation quarks to the gluon fusion production of the Higgs boson do not decouple but saturate to a constant value which is unacceptably large.
Based on the observation that the NP contribution to the Higgs coupling modifier with a gluon pair, κ g , is proportional to the sum of the Higgs coupling modifiers of t and b , i.e., (κ t + κ b ) when M t , M b m h , the authors in Ref. [6] found that the sequential fourth generation model can survive the Higgs precision constraint if the down-type quark Yukawa coupling has opposite sign to that of the up-type quark, i.e., κ b = −κ t .More interesting is that in the exact wrong-sign limit where all of the down-type fermions have opposite Higgs coupling to the up-type fermion, the new contributions to κ γγ and κ Zγ also vanish.The wrong-sign Yukawa couplings for the down-type fermions cannot be realized in the SM with only one Higgs doublet [7]: the Higgs sector should be extended as in the two Higgs doublet model (2HDM) [8][9][10][11].Authors in Ref. [6] showed that the Type-II 2HDM with a sequential fourth generation satisfies not only the Higgs signal strength measurements but also the oblique parameters ∆S and ∆T .Follow-up studies focused on top quark dipole moments [12], dark matter [13], and lepton flavor changing in the Higgs boson decays [14].
Albeit the appeal of the model in satisfying the Higgs precision constraint through such a simple remedy, this model has a potentially dangerous spot, the phenomenology of the heavy Higgs bosons.The 2HDM has five physical Higgs bosons, the light CP-even scalar h, the heavy CP-even scalar H, the CP-odd pseudoscalar A, and two charged Higgs bosons H ± .With a sequential fourth generation, the Yukawa couplings of the fourth generation fermion F with all the Higgs bosons are proportional to the fermion mass M F , which are naturally very large.In addition, the down-type fermion Yukawa couplings with H and A, Y H/A b , are proportional to tan β while the up-type fermion couplings are inversely proportional to tan β, where tan β is the ratio of two vaccum expectation values (VEVs) of two Higgs doublet fields.The cancellation of t and b contributions to the hgg coupling does not occur to the Hgg and Agg couplings.At the LHC, H and A can be copiously produced.
Another concern is from the exact wrong-sign limit, the key that allows the sequential fourth generation.It relates two mixing angles α and β through α+β = π/2, where α is the mixing angle between h and H. Unless tan β becomes very large, the exact wrong-sign limit cannot approach the alignment limit of sin(β − α) = 1.Since too large tan β breaks the perturbativity of Y H/A b , the exact wrong-sign limit brings about significant deviation from the alignment limit.One important consequence is the different dependence of the scalar quartic couplings on the heavy Higgs bosons masses from the alignment limit.As shall be shown, this difference causes a serious result on the perturbative unitarity of scalarscalar scattering, the upper bounds on the heavy Higgs boson masses.The decoupling limit cannot be achieved along with the exact wrong-sign limit.The model does not have a safety zone.We need a comprehensive study on the phenomenology of the model including the heavy Higgs bosons, which is our main purpose.
This paper is organized as follows.In Sec. 2, we review the Type-II 2HDM with a sequential fourth generation in the exact wrong-sign limit.The Higgs coupling modifiers in this model are compared with those in the alignment limit, with special focus on κ V .Section 3 deals with the theoretical constraints on the scalar sector: the bounded-frombelow potential, unitarity, perturbativity, and vacuum stability.Here we shall explicitly show that the behavior of λ 3 about M H is very different from that in the alignment limit.In Sec. 4, we study the indirect constraints from the electroweak oblique parameters and the Higgs precision data.The observed κ V restricts tan β significantly.Based on the narrowed parameter space from the theoretical and indirect constraints, we study the decay and production of H and A in Sec. 5.Here we shall point out that above the b b threshold, the total decay widths of both H and A become so wide like Γ H/A tot ∼ M H/A .The ordinary analysis based on σ × B does not work here.We suggest a method to probe this very fat resonance, not relying on the total event counting.The production cross sections of the gluon fusion of H and A are also studied.In Sec. 6, we consider direct search results for new particles at the LEP and LHC, e + e − → 4b, 4τ, 2b2τ and pp → τ τ , ZZ, Zh.Two smoking-gun signals, Zh for A and ZZ for H, are to be elaborated, drawing a rather strong conclusion that this model at leading order is excluded by the combination of theoretical and experimental constraints.In Sec. 7, we summarize and conclude.

Review of the 2HDM-SM4
We consider a 2HDM with a sequential fourth generation in the exact wrong-sign limit.The Higgs sector and the fermion sector are extended by introducing two complex SU (2) L Higgs doublet scalar fields, Φ 1 and Φ 2 [15], and a sequential fourth generation respectively: where i = 1, 2, and v 1,2 is the nonzero VEV of Φ 1,2 .Note that the anomaly cancellation condition [16,17] requires the existence of the fourth generation leptons.When parametrizing GeV, which generates the electroweak symmetry breaking.Its orthogonal combination H 2 = −s β Φ 1 + c β Φ 2 acquires zero VEV.For simplicity of notation, we take s x = sin x, c x = cos x, and t x = tan x.
In order to avoid flavor changing neutral current (FCNC) at tree level, a discrete Z 2 symmetry is imposed, under which Φ 1 → Φ 1 and Φ 2 → −Φ 2 [18,19].Then the most general scalar potential with CP invariance and softly broken Z 2 symmetry is The CP invariance requires all of the parameters to be real.Note that the soft Z 2 symmetry breaking parameter m 2 12 can be negative.There are five physical Higgs bosons, the light CP-even scalar h, the heavy CP-even scalar H, the CP-odd pseudoscalar A, and two charged Higgs bosons H ± .They are related with the weak eigenstates in Eq. (2.1) via where z 0 and w ± are the Goldstone bosons to be eaten by the Z and W bosons, respectively.The rotation matrix R(θ) is We consider the normal scenario where the observed Higgs boson is the lighter CP-even h, although the other scenario with M H = 125 GeV is still allowed by the Higgs precision data [20,21].The Yukawa couplings are different according to the Z 2 parity of the fermions.We fix that Q L → Q L and L L → L L under the Z 2 symmetry, where Q L and L L are the left-handed quark and lepton doublets, respectively.Then each right-handed fermion field only couples to one scalar doublet field.There are four different ways to assign the Z 2 symmetry on the right-handed fermion fields, leading to four different types in the 2HDM, Type-I, Type-II, Type-X, and Type-Y.We parameterize the Yukawa interactions with the neutral Higgs bosons as Note that κ f is the Higgs coupling modifier, parameterizing the NP effects on the Higgs couplings: While κ f 's are different according to the 2HDM type, κ V and ξ V (V = W ± , Z) have the common leading order expressions of Because the observed Higgs boson at a mass of 125 GeV is very SM-like, the so-called alignment limit is usually adopted in the 2HDM1 , defined by α = β − π 2 (alignment limit) (2.8) where u = t, t , ν and d = b, b , τ, τ .With a sequential fourth generation, however, this alignment limit does not guarantee a SM-like Higgs boson because of the large contribution from the fourth generation fermions to the loop induced couplings of the Higgs boson, especially to the κ g : where τ f = m 2 h /4m 2 f , F = t , b , and the expression for the loop function A h 1/2 (τ ) is referred to Ref. [23].It is known that A h 1/2 (τ f ) approaches the value of 4/3 when m f m h .In the alignment limit (κ t = κ b = 1) with M F m h , therefore, the value of κ g approaches 3. We cannot but conclude that a sequential fourth generation in the SM or the aligned 2HDM is excluded by the Higgs precision data.
Based on the observation that δκ g is proportional to (κ t + κ b ) for M F m h and the current LHC data cannot determine the sign of κ b yet, the exact wrong-sign limit [6,11,24] is suggested, given by α = π 2 − β (exact wrong-sign limit) (2.10) The wrong-sign Yukawa couplings for the down-type fermions cannot be realized in the SM where there exists only one scalar doublet field: all of the Yukawa couplings can be set positive by chiral rotation.We need an additional Higgs doublet field, which can be minimally realized in the 2HDM.Among four types of the 2HDM, only Type-II can accommodate the exact wrong-sign limit where both b and τ have opposite Yukawa couplings to t and ν .In what follows, 2HDM-SM4 denotes the Type-II 2HDM with a sequential fourth generation in the exact wrong-sign limit.
More surprising feature of the exact wrong-sign limit is that new contributions from the sequential fourth generation fermions to κ γγ and κ Zγ are also suppressed in the heavy M F limit [6]: ) where Q f is the electric charge of the fermion f , N f C is the color factor, and (T f 3 ) L is the isospin projection of the left-handed f L .
In the 2HDM, however, there exist other Higgs bosons, H, A, and H ± .The exact wrong-sign condition simplifies ξ H,A u,d , defined in Eq. (2.5), into (2.12) An immediate concern is that the Yukawa couplings of b and τ with H and A, proportional to the heavy fermion masses, can be dangerously large especially in the large t β limit.Theoretical principles and collider experiments associated with H and A shall constrain the model significantly.
We now specify the model parameters in the 2HDM-SM4.In the scalar potential sector, there are 7 free parameters of m 2  12 , t β , and λ 1,••• ,5 , after applying the tadpole conditions for m 11 and m 22 .Equivalently we can take the physical parameters of m h , M H , M A , M H ± , m 2 , α and β, where m 2 = m 2  12 /(s β c β ) is chosen because of its efficiency to show the invariance under the reparameterization in the space of Lagrangian [25].Since m h = 125 GeV is known and the exact wrong-sign limit relates α and β as α + β = π/2, there are five free parameters in the scalar sector.In the fourth generation fermion sector, only their masses are unknown because their gauge and Yukawa couplings are the same as the SM fermions.In summary, the 2HDM-SM4 has the following model parameters: (2.13) Brief comments on the fourth generation fermion mass M F and M H ± are in order here.For M F , there are two kinds of constraints working in the opposite way, one from the unitarity and the other from the direct searches.First, the perturbative unitarity for the fermion-fermion scattering puts the upper mass bound as m q 550 GeV and m 1.2 TeV [26,27].On the other hand, the direct searches for t and b at the LHC put the lower bounds of M t 680 GeV under the assumption that the produced t and b decay into a SM quark accompanied by a W or Z boson [28].If the mixing between the SM quarks and the fourth generation quarks is extremely small like V i4 10 −7 , however, no limits can be set [28].The CDF collaboration took the assumption of specific flavor-mixing rates and put the lower bound on M t , M b 335 − 385 GeV [29,30].With general flavor mixing, the lower mass bounds were recalculated to be as low as 290 GeV [31,32].The fourth generation leptons have weaker bounds as m τ > 100.8 GeV and m ν > 41 GeV [33].The charged Higgs boson mass in a Type-II 2HDM is most strongly constrained by the FCNC process B → X s γ.The updated next-to-next-to-leading order SM prediction of B SM ( B → X s γ) [34,35] and the recent Bell result [36] got closer, yielding M H ± > 570 (440) GeV for t β 2 at 95% (99%) C.L.For t β 2, the lower bound on M H ± increases significantly.Note that the fourth generation quarks do not affect the process B → X s γ under the assumption of V 4i 10 −7 .

Theoretical constraints on the scalar potential
The quartic coupling constants in the scalar potential V Φ can be rewritten in terms of the physical mass parameters, which are in the exact wrong-sign limit They are constrained by the following theoretical conditions: 1.The scalar potential V Φ should be bounded from below in any direction, requiring [37,38] 2. The tree-level perturbative unitarity demands [39][40][41] where a i,± (i = 1, • • • , 6) are the eigenvalues of the T matrix for the S-wave amplitudes of the scalar-scalar scattering, given by (λ 3 ± λ 5 ) .

The perturbativity of scalar quartic couplings requires
4. The vacuum of V Φ should be global, which happens if and only if [42] where In the exact wrong-sign limit, the theoretical constraints are more difficult to satisfy than in the alignment limit.Crucial is λ 3 , which shows different dependence on M H : , in the exact wrong-sign limit; In both limits, λ 3 has the same dependence on M2 H ± but has opposite signs for the M 2

H
terms.Since M H ± in a Type-II 2HDM should be very heavy to explain the b → sγ result, heavy M H in the exact wrong-sign limit shall easily increase λ 3 above 4π.In the alignment limit, the negative M 2 H contribution cancels the positive M 2 H ± contribution to some extent, being able to control the λ 3 value.
Figure 1 presents the theoretically allowed parameter space.In the left panel, we show the allowed parameter space 2 of (M H , m) for M H ± = 440 GeV and t β = 4 (blue region), M H ± = 580 GeV and t β = 3, 4 (red region), and M H ± = 620 GeV and t β = 4 (black region).We also set M A = 0.9M H in order to explain the oblique ∆T parameter.Two cases of t β = 3 and t β = 4 for the same M H ± = 580 GeV show that small t β allows more freedom for m, but does not significantly change the allowed range for M H .   very large, only a fine linear line along M H = m is allowed, irrespective of M H ± .The most important result is that theoretical constraints put an upper bound on M H , of which the M H ± dependence is small and the t β dependence is negligible.The charged Higgs boson mass is also bounded from above.In the right panel of Fig. 1, we show the theoretically allowed parameter space of (M H , M H ± ) for t β = 4.We consider three cases of M A = M H , M A = 1.1 M H , and M A = 0.9 M H to explain the oblique parameter ∆T .It can be seen that M H cannot exceed about 920 GeV and M H ± should be smaller than 620 GeV.Since the t β dependence on the upper bounds is negligible as shown in the left panel of Fig. 1, the presence of the upper bounds on M H , m, and M H ± is a generic feature of the 2HDM-SM4.This result is unexpected.In the so-called normal mass hierarchy scenario where the observed 125 GeV scalar is the lighter h, it is usually expected that the theory can hide in the decoupling region.However the exact wrong-sign limit does not allow decoupling of the theory.This is a unique feature of this model.

Indirect Constraints from the oblique parameters and the Higgs precision data
In this section we narrow down the parameter space by applying the indirect constraints from the electroweak oblique parameters and the Higgs precision data.

Electroweak oblique parameters ∆S and ∆T
The oblique parameters ∆S and ∆T get affected by the sequential fourth generation fermions [26,43] as well as by new scalar bosons [44,45].The current experimental data are consistent with the SM values, given by [33] ∆S = 0.05 ± 0.10, ∆T = 0.08 ± 0.12, ( where the parameter ∆T is more sensitive to new particles.It is well known that new contributions to ∆T are suppressed when the new particles running in the self-energy diagrams of gauge bosons have the same masses.We require that M t M b , M τ M ν , and two masses among M H , M A , and M H ± are degenerate.We also note that the contributions from H and A to the oblique parameter ∆T are negative [15] while those from the fourth generation fermions are positive [26,43,45].Large mass splittings in the scalar and fourth generation fermion sectors are allowed if exquisite cancellation occurs.In this work, however, we do not consider the conspiracy between the new scalar sector and the new fermion sector in explaining the oblique parameters.

Higgs precision data
We take the combined analysis of ATLAS and CMS on the Higgs coupling modifier κ i based on the LHC Run 1 data, corresponding to integrated luminosities per experiment of 5 fb −1 at √ s = 7 TeV and 20 fb −1 at √ s = 8 TeV [46] 3 .The analysis is based on a few assumptions, each of which constrains κ i 's differently.The 2HDM-SM4 model belongs to the category where new loop couplings beyond the SM (BSM) are allowed.The analysis result is that all of κ i 's are consistent with the SM value.For the allowed values of κ i 's in this category, we refer the reader to Fig. 15 in Ref. [46].The exact wrong-sign limit naturally explains the SM-like κ t , |κ b,τ |, κ g and κ γ .One minor concern is |κ b |, of which the maximum value at 2σ level is about 5% smaller than one.Since a small deviation from the exact wrong-sign limit can easily accommodate this result, we stick to the exact wrong-sign limit as our reference point.On the other hand, κ V can significantly deviate from one.In Fig. 2, we show κ V and ξ V as a function of t β .In Fig. 3, we summarize the constraints from the oblique parameter ∆T at 2σ level, the Higgs coupling modifier κ V at 2σ level, and the perturbativity of ) is fixed to be 300 GeV.The observed κ V allows t β 3, while the perturbativity of Y H/A b ,τ requires t β 8.The oblique parameter ∆T demands that the mass of an up-type fourth generation fermion be similar to that of the corresponding down-type fermion like ∆M/M 20%.
In addition, the experimental result on exotic Higgs decay B h BSM ≤ 0.34 [46] has an important implication on the mass of the pseudoscalar A. In the 2HDM, M A is a free parameter so that A can be light enough to kinematically allow h → AA.The observed oblique parameter ∆T can be explained by the mass degeneracy M H M H ± .When writing L ⊃ λ hAA AAh/2, the partial decay rate is In the general 2HDM, λ hAA is unknown for the given m h and M A because of an additional free parameter m 2 .In the exact wrong-sign limit (α + β = π/2), however, the m 2 term is proportional to c β+α and thus vanishes.The value of λ hAA is determined by t β and M A as The observed B h BSM ≤ 0.34 is translated into Γ(h → AA) Γ h,SM tot /2, which can be satisfied when 0.9 GeV).This small t β region is excluded by the observed κ V : see Fig. 2. In summary, a light pseudoscalar boson with M A m h /2 in the 2HDM-SM4 is excluded by the observed B h BSM .
5 Decay and Production of H and A

Decays
In this section, we discuss the decay and production of neutral heavy Higgs bosons, H and A. Considering the theoretical and direct search bounds on M F , the perturbative unitarity of the scalar-scalar scattering, and the b → sγ constraint altogether, we take the following benchmark scenario: Kinematically, the decays of H → AA and H → H + H − are prohibited.Note that the decays of H → W ± H ∓ and A → Zh are possible through the interaction Lagrangian of where g Z = g/c θ W , θ W is the weak mixing angle, and φ In Fig. 4, we show the branching ratios of H as a function of M H for t β = 4 and m = M H . Before the W W threshold, the dominant decay mode of H is into b b, followed by τ τ and gg modes because of the t β enhancement in Y H b,τ .After the W W/ZZ threshold, non-zero ξ V = c β−α (c β−α ≈ 0.47 for t β = 4) yields dominant decay of H into W W and ZZ, because of the longitudinal polarization enhancement in the heavy scalar decay into a massive gauge boson pair [49].
For M H > 2m h , the decay of H → hh becomes also important.The triple Higgs coupling λ Hhh in the exact wrong-sign limit is With the sizable c β−α and the condition of m M H from the theoretical constraints shown in Fig. 1, the branching ratio of H → hh is substantial.Above the t t threshold, H → t t mode turns on, but not dominantly because Ht t coupling is inversely proportional to t β .For the same reason, H → t t mode is also minor even after the t t threshold.After the 200 300 400 500 600 700 800 900 0.1 much larger branching ratio than the τ τ mode, unlike the case of H.It is because of the larger gg loop function for the CP-odd scalar than that for the CP-even scalar.When m Z + m h < M A < 2M b , the decay of A → Zh is dominant, contrary to the alignment limit.In this mass range, the decay into gg is still important.We expect a sizable cross section of gluon fusion production of A. The branching ratios of the t t and t t modes are not as large as that of the Zh mode because both At t and At t couplings are inversely proportional to t β .After the b b and τ τ threshold, A decays into b b dominantly, followed by the τ τ mode.The next dominant mode is into Zh with B(A → Zh) ∼ O(1)%.
In Fig. 6, we show the total decay widths of H and A as a function of M H and M A for t β = 3 and t β = 4 in the benchmark scenario.For both H and A, Γ tot increases rapidly with its mass, especially above the b b threshold.The dependence of Γ H/A tot on t β is interesting.Before the b b threshold, the larger t β is, the smaller the total decay width is.This behavior is due to the fact that the major decay modes before the b b threshold, H → W W/ZZ and A → Zh, have the partial decay rates all proportional to ξ 2 V which decreases with increasing t β : see Fig. 2.After the b b threshold, on the other hand, the dominant decay rate Γ(H/A → b b ) is proportional to t 2 β .Another crucial question related to Γ tot is whether H and A can be probed as a narrow resonance so that the number of the new events is proportional to the production cross section times the branching ratio.For reference, the LHC criteria as a narrow resonances is Γ/M ≤ 1% in the γγ channel [50,51], Γ/M ≤ 0.5% in the ZZ channel [47,52], and Γ/M ≤ 15% in the dijet channel [53].When t β = 4, for example, heavy Higgs bosons like M H 350 GeV and M A 300 GeV do not belong to the narrow width category, which requires going beyond σ × B. Particularly above the b b threshold, both Γ H tot and Γ A tot are so large like Γ tot ∼ M H .It is almost impossible to observe a mass peak in this mass region.
We need a new strategy.
In order to deal with the very large width case like Γ H/A tot ∼ M H/A , two points should be considered.First, we need full calculation of σ(pp → ij) including the SM continuum background in order not to miss the significant interference.The second point is that new events spread out over multiple m ij bins, not only to the bin of m ij = M .Without the possibility of observing a mass peak, we may rely on counting total events, which requires a very good understanding of the background.Or we can utilize the results of usual analysis of the resonance searches for the excess in the invariant mass bins, which takes the following steps: (i) events are collected in a specific m ij bin; (ii) the number of the events in the bin is compared with that of the expected background; (iii) no excess leads to the upper bounds on σ × B(X → ij) for a possible new particle with mass m ij .Since multiple m ij bins get affected by a single new particle when Γ ∼ M , we need to calculate the excess of each m ij bin nearby M and compare with the upper bounds on σ × B(A → Zh) for the corresponding bin.The size of each m ij bin, ∆M , depends on the experimental analysis.Since our dσ NP already includes the SM contributions, the excess over the SM backgrounds corresponds to the difference between the full dσ NP and dσ SM .In order to compared with the excess in the m ij ∈ [M, M + ∆M ] bin, therefore, we calculate the partially integrated cross section ∆σ NP given by We shall use this method when constraining very heavy A and H in the Zh and ZZ final states, respectively.

Productions
We study the production of H and A. The production cross section of H/A with a small width is [55] σ(pp where p is a parton in a proton and C pp is the pp dimensionless partonic integral.Because C b b/C gg ∼ 0.01 at the 13 TeV LHC and moderate t β ∈ [3,8] implies not too large Hb b and Ab b couplings, we consider only the gluon fusion production.Figure 7 presents the gluon fusion production cross sections of H and A at the 13 TeV LHC as a function of M H and M A respectively.We consider t β = 3 and t β = 4, and include the NNLO K factor by using HIGLU package [54].In the whole mass ranges, the production cross section is large: for M H , M A = 600 GeV, for example, σ(pp → H/A) ∼ O(10) pb.The pseudoscalar A has larger cross section than H, because of larger loop function for gg.As t β increases, σ(gg → H/A) increases because the contribution from the b quark in the loop is enhanced.
For the CP-odd A, the threshold effects at M A = 2m t and M A = 2M b are more prominent due to the cusp structure of the real part of the gg form factor A A 1/2 (τ ) [56].We caution the reader that the production cross section in Eq. (5.5) for M H/A 2M b gives just a rough estimation.When Γ tot ∼ M H/A , the production of H/A itself is not meaningful.We need  The gluon fusion production cross sections of H and A as functions of M H and M A at the 13 TeV LHC for t β = 3 and t β = 4.We apply NNLO K factor to the heavy Higgs resonance production by using HIGLU package [54].
to set the final states ij and to perform the full calculation of σ(pp → ij) including the SM continuum background because the interference effects crucially depend on the final state.

Constraints from the direct searches
In this section, we study the constraints from the direct searches for neutral scalar bosons at high energy colliders.The allowed region for the charged Higgs boson mass, 570 M H ± 620 GeV, is very difficult to probe because of extremely small production rate of H ± [68].Our main target channels are summarized in Table 1.Brief comments on γγ, µ + µ − , and b b modes are in order here.Although the very clean γγ mode is searched for from m γγ = 65 (70) GeV by the ATLAS (CMS) experiments [69,70], it does not constrain the model since the branching ratio is very small: B(A → γγ) ∼ O(10 −5 ) for M A = 65 GeV.The data in the µ + µ − channel [71] are also insufficient yet because of the extremely small B(H/A → µ + µ − ) O 10

Constraints from the direct searches on A
Meaningful constraint is from the neutral Higgs boson searches at the LEP through e + e − → H i H j (H i,j = h, H, A) in the framework of the Minimal Supersymmetric Standard Model [57].The analysis was based on four different decay channels of 4b, 2b2τ , and 4τ .Since the searches span a center of mass energies from 91 GeV to 209 GeV, the heavy CP-even H cannot be produced kinematically.8: σ(gg → A)×B(A → τ τ ) as a function of M A at the 13 TeV LHC for t β = 3 and t β = 4, We apply NNLO K factor to the heavy Higgs resonance production by using HIGLU package [54].For comparison, we also show the 95% C.L. upper limits on σ × B(φ → τ τ ) data [60,61].
We now consider the resonance searches in the τ τ channel at the LHC.Both ATLAS and CMS experiments presented their results based on the Run-1 [58,59] and Run-2 [60,61] data.Since the LHC Run-1 data at √ s = 7+8 TeV put weaker constraints, we focus on the Run-2 results.In Fig. 8, we show σ ×B(A → τ τ ) in the 2HDM-SM4 at the 13 TeV LHC.As shall be shown in the next sub-section, the pseudoscalar A gets much stronger constraint from the τ τ channel than the CP-even H.It is partly because the gluon fusion production of A is more efficient due to larger loop function than that of H: see Fig. (a) Figure 9: Feynman diagrams for the gg → Zh process in the 2HDM-SM4.Here q denotes all of the four generation quarks, including t and b .
The smoking-gun signature for A at the LHC is the Zh channel, followed by Z → and h → b b/τ τ [67,75,76].In the SM, pp → Zh proceeds mainly through q q → Zh, mediated by Z * .The gluon fusion production also occurs in the SM through the quark triangle diagram and the quark box diagram, (a) and (c) in Fig. 9.In the SM, σ(gg → Zh) is small, about 10% of q q annihilation process, mainly because of the destructive interference between the triangle and box diagrams [77].In the 2HDM-SM4, there are four kinds of new contributions: (i) σ(q q → Zh) is reduced by the factor of κ In order to see the interference effects in detail, we split the scattering amplitude into the Z-triangle part (M Z ), the A-triangle part (M A ), and the box part (M ).The parton-level cross section becomes where θ * is the scattering angle in the center-of-mass frame and denotes the proper summation and average over helicities and colors.Figure 10 shows the individual contributions to the m Zh distribution of the full gg → Zh for t β = 4 and M A = 800 GeV with Γ A tot = 2.0 TeV at the 13 TeV LHC.We use K q q = 1.31 and K gg = 2.1 to match up with the updated Higgs calculation results [73,74], and we assume the same K factor for the NP calculation.The cross section only from the Z triangle diagrams, dσ Z , shows threshold < l a t e x i t s h a 1 _ b a s e 6 4 = " g m / g C t 8 5 6 a < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " / u j f z U j q 2 a N g 8 R q 7 n L j S 0 X w Q B X e v Y n 3 5 n 2 s u q p 5 6 9 L O 4 I + 8 z x 8 4 x I o 4 < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " J 8 / A E b s p 7 L < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " A g 8 r x X 1 4 / A E b s p 7 L < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " A g 8 r x X 1 4 / A E b s p 7 L < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " A g 8 r x X 1 4  A-triangle, box diagrams, and their interference terms.We use K q q = 1.31 and K gg = 2.1 [73,74].
behaviors around m Zh 2m t , 2M t , 2M b .The contribution from t and b is rather small because the corresponding transition amplitude is proportional to the axial vector coupling of the Z boson, g Zf f A = −(T f 3 ) L /2 [74].For almost degenerate masses of t and b , two contributions are cancelled.On the other hand, dσ has large signal rate in the heavy m Zh range because the opposite sign between g Zb b A and g Zt t A is compensated by the opposite sign between Y h b and Y h t .The dσ A /dm Zh shows a very wide resonance shape, resulting in small signal rate.Both dσ Z and dσ ZA yield destructive interference in the whole m Zh region, large enough to almost cancel dσ Z .The interference between the A-triangle and box diagrams is constructive for m Zh ≤ M A while destructive for m Zh ≥ M A , a typical peak-dip structure [78].In summary, the contributions from the interference are as large as non-interference ones.
In Fig. 11, we present the total invariant mass distribution in the SM and two 2HDM-SM4 cases: one with t β = 4 and M A = 800 GeV (the pink line) and another with t β = 3 and M A = 500 GeV (the blue line).We use K q q = 1.31 and K gg = 2.1 [73,74].For both NP cases, we calculate the full Feynman diagrams in Fig. 9.The total m Zh distribution for t β = 4 and M A = 800 GeV, of which the individual contributions are shown in Fig. 10, shows a peculiar shape with two thresholds of t t and b b followed by a very slow down hill.This bizarre distribution is the consequence of large interference effects.The case with t β = 3 and M H = 500 GeV with Γ A tot = 13.8GeV yields a prominent peak over the SM main background of q q → Zh.The LHC experiments cannot miss the peak.Indeed the total cross section of pp → Zh for t β = 3 and M A = 500 GeV is about 24.
Figure 11: Invariant mass m Zh distribution for pp → Zh process at √ s = 13 TeV.We separately present q q contribution and gg contributions in the SM and 2HDM-SM4.We consider two NP cases: (i) t β = 3 and M A = 500 GeV; (ii) t β = 4 and M A = 800 GeV in the benchmark scenario in Eq. (5.1).We use K q q = 1.31, and K gg = 2.1 [73,74].
does not help to allow the model when M A 2M b : (i) the resonance peak becomes more prominent because of smaller width for larger t β in this mass range as shown in Fig. 6; (ii) t β 4 for M A < 2M b is already excluded by the τ τ resonance searches as in Fig. 8.When M A 2M b , the width becomes too wide to show a resonance peak.We calculate the partially integrated cross section for the excess in each m Zh bin, defined in Eq.(5.4), and compare with the ATLAS result of the upper bounds on σ × B(A → Zh) [75].We find that t β = 4 and M A = 800 GeV is still allowed.This conclusion is valid for larger t β and M A > 2M b .

Constraints from the direct searches for H
For CP-even H, we first study the constraint from the resonance searches in the τ τ channel based on the ATALS and CMS Run-2 data [60,61].Figure 12 shows σ × B(H → τ τ ) in the 2HDM-SM4 at the 13 TeV LHC.In the benchmark scenario, we consider two cases of t β = 3 and t β = 4.For the K factor, we take the NNLO result from HIGLU package [54].Compared with the rapid drop of B(H → τ τ ) as a function of M H in Fig. 4, σ ×B(H → τ τ ) decreases slowly.This is because of sizable cross section of the gluon fusion production of H for heavy M H : see Fig. 7.The current LHC data in the τ τ final states meaningfully exclude M H , though less than the pseudoscalar A. Adopting the weaker constraint between two experiment results, M H < 180 GeV (M H < 350 GeV) is excluded for t β = 3 (t β = 4) at the 95% C.L.
The smoking-gun signature for the CP-even H is from the ZZ final state [62,63].In the SM, the production of a Z boson pair is mainly through the q q annihilation.The gluon 200 300 400 500 600 700 800 900 10  We apply NNLO K factor to the heavy Higgs resonance production by using HIGLU package [54].We also show the ATLAS and CMS 95% C.L. upper bounds on σ × B(φ → τ τ ). (a) Figure 13: Feynman diagrams for the gg → ZZ process in the 2HDM-SM4.Here q denotes all of the four generation quarks, including t and b .
fusion production via the quark loops is sub-leading, of which the cross section is about 10% of the Drell-Yan process.In the 2HDM-SM4, the Drell-Yan process q q → ZZ is not affected since the CKM mixing V 4i is extremely suppressed.The gluon fusion process has three kinds of Feynman diagrams as shown in Figure 14 shows the non-interference contributions to dσ/dm ZZ (gg → ZZ) from the h-triangle, H-triangle, and box diagrams as a function of m ZZ at the 13 TeV LHC.We set t β = 4 and M H = 900 GeV and include K gg = 1.8 and K q q = 1.5.The definition of dσ i is given in Eq. (6.1).The box diagrams (dσ ) yield the continuum background with monotonically decreasing slope against m ZZ , but the fourth generation quarks in the loop            < l a t e x i t s h a 1 _ b a s e 6 4 = " d P X y q 8 2 k q 2 m D A s Q 8 l 6 s d 6 t j 0 V r x S p n j s E f W J 8 / N M q W r g = = < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " d P X y q 8 2 k q 2 m D A s Q 8 l 6 s d 6 t j 0 V r x S p n j s E f W J 8 / N M q W r g = = < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " d P X y q 8 2 k q 2 m D A s Q 8 l 6 s d 6 t j 0 V r x S p n j s E f W J 8 / N M q W r g = = < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " d P X y q 8 2 k q 2 m D A s Q total < l a t e x i t s h a 1 _ b a s e 6 4 = " B X g M I e H C D 5 y A I 7 P b S V u l X Z y l h P g = " 3 s e y d c M r Z s 7 g D 7 z P H w S 5 j s g = < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " B X g M I e H C D 5 y A I 7 P b S V u l X Z y l h P g = " 3 s e y d c M r Z s 7 g D 7 z P H w S 5 j s g = < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " B X g M I e H C D 5 y A I 7 P b S V u l X Z y l h P g = " 3 s e y d c M r Z s 7 g D 7 z P H w S 5 j s g = < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " B X g M I e H C D 5 y A I 7 P b S V u l X Z y l h P g = " The differential cross section for gg → ZZ against m ZZ in the 2HDM-SM4 at the 13 TeV LHC.We set t β = 4 and M H = 900 We show individual contributions from the h-triangle, H-triangle, and box diagrams.We include K gg = 1.8 [79] and K q q = 1.5 [80].slow the slope around the b b threshold.In terms of the total signal rate, the contribution from the box diagrams is dominant.The H-triangle diagrams yield a very wide resonance peak around M A = 900 GeV.In the range of m ZZ 1 TeV, the contribution from the H-triangle diagrams is as large as that from the box diagrams.On the other hand, the h-triangle diagrams make a negligible contribution in the whole range of m ZZ .The total distribution denoted by the black line shows some deviation from the simple sum of dσ + dσ H + dσ h , especially around m ZZ ∼ 700 GeV.Obviously the interference plays a role.
In Fig. 15, we show the three kinds of interference effects, dσ ij defined in Eq. (6.1), that are too small to show together with non-interference dσ i in a single plot.First the interference between the h-triangle and box diagrams, σ h , is destructive and largest.We observe the threshold effects around 2m t and 2M b .The interference dσ H shows a very asymmetric and wide peak-dip structure [81], practically like a wide peak of constructive interference.The opposite sign between dσ h and dσ H is attributed to the opposite sign between the hb b and Hb b couplings.Finally dσ hH is destructive but negligible.
Figure 16 shows the total invariant mass m ZZ distributions of q q → ZZ in the SM, gg → ZZ in the SM and the 2HDM-SM4.We consider three NP cases: (i) t β = 3 and M H = 500 GeV; (ii) t β = 4 and M H = 900 GeV; (i) t β = 7 and M H = 900 GeV.It is clear to see that the case of t β = 3 and M H = 500 GeV, which was allowed by the τ τ constraint in Fig. 12, yields a very outstanding peak because of relatively narrow width and large production rate of gg → H. Apparently, the current upper bound on σ × B(φ → ZZ) excludes this case.Note that the whole parameter space with M H < 2M b (where H remains as a prominent peak) is excluded since larger t β yields larger production cross  adopt the method suggested at the end of Sec.5.1, utilizing the 95% C.L. upper limits on σ × B(H → ZZ) for each invariant mass bin.Since the CMS collaboration presented the results for Γ X = 100 GeV, we take the CMS results [62].
In Fig. 17, we show the partially integrated cross section from NP effects only, ∆σ NP , for various m ZZ bins.The size of bins are based on the CMS results [62].We set M H = 900 GeV and consider two cases of t β = 4 and t β = 7.The red region is excluded by the 95% C.L. upper bounds on σ × B(H → ZZ).Both t β = 4 and t β = 7 cases are excluded by the bin of m ZZ ∈ [1, 1.5] TeV.Larger t β , apart from the breakdown of the perturbativity of Y H b , does not help to decrease the signal rate of NP since the gluon fusion production cross section increases with larger t β .Note that we cannot increase M H substantially further since M H = 900 GeV is almost maximally allowed value by the theoretical constraint: see Fig. 1.In summary, the current LHC data on the ZZ channel along with theoretical constraints exclude the 2HDM-SM4 at leading order.

Conclusions
We have studied the theoretical and phenomenological constraints on the Type-II two Higgs doublet model with a sequential fourth fermion generation in the exact wrong-sign limit, the 2HDM-SM4.The SM fermion sector is extended to accommodate an additional chiral fermion generation of which the masses are generated by the same Higgs mechanism.Upon the absence of new fermion signals at high energy colliders, the fourth generation fermion F should be heavy, which greatly enhances the hF F couplings.The loop induced Higgs coupling modifiers such as κ g , κ γ , and κ Zγ become SM-like if the up-type and down-type fermions have opposite Higgs coupling modifiers: κ up = 1 and κ down = −1.We call this the exact wrong-sign limit, which can be realized in the Type-II 2HDM.
We have studied the following constraints: (i) the theoretical constraints on the scalar potential from the bounded-from-below potential, unitarity, perturbativity, and the vacuum stability; (ii) the LHC and Tevatron search bounds on the fourth generation fermion masses; (iii) the B → X s γ bound on M H ± ; (iv) the electroweak oblique parameters ∆S and ∆T ; (v) the observed Higgs coupling modifiers including the exotic Higgs decay rate; (vi) the LEP searches for e + e − → Ah; (vii) the LHC scalar resonance searches in τ τ , Zh and ZZ modes.Based on this comprehensive and thorough study, we come to the conclusion that the 2HDM-SM4 is excluded at leading order.Two features of the 2HDM-SM4 play the crucial role here, the exact wrong-sign limit and the very large Yukawa couplings of the down-type fourth generation fermions with H and A, Y H/A b ,τ = tan βM F /v.The exact wrong-sign limit requiring β + α = π/2 constrains the model very differently from the alignment limit of β − α = π/2.First the theoretical constraints do not allow the decoupling limit, which puts the upper bounds like M H , M A 920 GeV and M H ± 620 GeV.Secondly, the constraint on tan β from the Higgs precision data works in the opposite way to that from the unitarity of Y H/A b ,τ = tan βM F /v: the observed κ V (= s β−α ) at 95% C.L. requires tan β 3 while the unitarity of Y H/A b ,τ requires tan β 8.The allowed value of tan β ∈ [3,8] is not large enough to achieve the alignment limit: κ V (= s β−α ) deviates significantly from one and thus c β−α has sizable value.Therefore, the usual 2HDM safety zone in the alignment and decoupling limit cannot be reached in the 2HDM-SM4.
These indirect constraints set the basic characteristics of the decays of H and A. Two features are to be noted.First the sizable c β−α leads to significant decays of H → W W/ZZ/hh and A → Zh, which would have been absent in the alignment limit.The smoking-gun signatures of A and H are the LHC direct searches in the Zh and ZZ channels, respectively.Since their total decay widths are not small enough to adopt the narrow width approximation, we performed the full calculation including the SM continuum background without resorting to σ × B. The interference effects turned out to be very important.In the gg → Zh process, they are as much as the non-interference ones.We found that both A and H with the mass below the b b threshold are excluded by the current LHC data at the 95% C.L. We would have seen a prominent mass peak that the LHC experiment cannot miss.Above the b b threshold, Γ H/A tot becomes very large, being compatible with the mass M H/A .The invariant mass distributions spread very widely.Beyond relying on the total event rates, we suggested a method to utilize the upper bounds on σ × B on each invariant mass bin, through the partially integrated cross section.Although the pseudoscalar A with M A > 2M b is allowed by the current LHC data in the Zh channel, the CP-even heavy scalar H is excluded by the current LHC data in the ZZ channel.
We conclude that the 2HDM-SM4 is excluded by the combination of the theoretical and experimental constraints at leading order.Before finishing, however, we would like to point out that all of the analysis are performed at leading order and thus the conclusion may be relaxed if we consider the next-to-leading order corrections.The sizable cos(β − α), the multiplying factor for the HV V and AZh couplings at tree level, plays the decisive role in excluding the model.With the fourth generation fermions running in the loop, both HV V and AZh couplings can be meaningfully different from the tree level results.The constraints from the LHC resonance searches in the ZZ and Zh final states may be alleviated.

Figure 2 :
Figure 2: κ V and ξ V as a function of t β .

Figure 3 :
Figure 3: The excluded regions of (t β , ∆M DU ) by the observed Higgs coupling modifier κ V at 2σ level, the oblique parameter ∆T at 2σ level, and the perturbativity of Y H/A b .Here ∆M DU = M b − M t = M τ − M ν for the fixed M U = 300 GeV.For the oblique ∆T parameter, we set M H = M A = M H ± = 600 GeV.

Figure 4 :
Figure 4: The branching ratios of H as a function of M H for t β = 4.We set M t = 300 GeV, M b = 340 GeV, M ν = 430 GeV, M τ = 380 GeV, m = M H , M A = M H , and M H ± = 580 GeV.

Figure 5 :
Figure 5: The branching ratios of A as a function of M A for t β = 4.We set M t = 300 GeV, M b = 340 GeV, M ν = 430 GeV, M τ = 380 GeV, m = M H , M A = M H , and M H ± = 580 GeV.

Figure 6 :
Figure 6: Total decay width of H and A as a function of their masses for two t β cases, t β = 3 and t β = 4.We set m = M H , M t = 300 GeV, M b = 340 GeV, M ν = 430 GeV, and M τ = 380 GeV, and do not include the decays of H → H + H − and H → AA.

Figure 7 :
Figure7: The gluon fusion production cross sections of H and A as functions of M H and M A at the 13 TeV LHC for t β = 3 and t β = 4.We apply NNLO K factor to the heavy Higgs resonance production by using HIGLU package[54].

− 4 .
The b b mode constrains a NP model only in the heavy mass range of m b b ≥ 800 GeV [72] due to the huge QCD background, where both B(H → b b) and B(A → b b) are extremely small because of the dominant decays into b b . σ(gg→A)⨯BR(A→ττ)[pb]

2 V
; (ii) B(h → b b/τ τ ) in the exact wrong-sign limit is increased because of smaller Γ(h → W W/ZZ) than in the SM but the same Γ(h → b b/τ τ ); (iii) the fourth generation quarks contribute to all of the loop diagrams for gg → Zh; (iv) new triangle diagrams mediated by A appear.In what follows, we call Fig. 9(a) the Z-triangle diagram, Fig. 9(b) the A-triangle diagram, and Fig. 9(c) the box diagram.

3 5 n 2 s u q p 5 6 9 L
O 4 I + 8 z x 8 4 x I o 4 < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " J 8 T U K O 6 i m 6 x 3 C 9 9 M D 7 Y i j m b H c 7 w D p d w A 1 3 o A Y E n e I E 3 e D e e j V f j Y 1 F X z a h 6 O 4 Q / M D 5 / A B w T n 2 A = < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " 6 / e e f b i g B b y B d + P Z e D U + j M / Z a M U o d w 7 B H I y v X 6 s t n o Y = < / l a t e x i t > d A < l a t e x i t s h a 1 _ b a s e 6 4 = " A g 8 r x X 1 4 5 p b H h T d 5 M U 0 A 5 i a Y 2 0 r T o P 2 Q 4 o i v I j W P g r L A w g x M r A x r / B T T N A y 0 m W T n f f 5 7 P P S x g V 0 j S / t d r K 6 t r 6 R n 1 T 3 9 r e 2 d 0 z 9 g / 6 I k 4 5 J j 0 c s 5 g P P S Q I o x H p S S o Z G S a c o N B j Z O B N r 2 f + 4 J 5 w Q e P o T m Y J c U I 0 j m h A M Z J K c o 0 z u 7 w j z w h j 8 U O h N 3 1 Y K R 5 D e F r Y g o 5 D 5 O Z r T o P 2 Q 4 o i v I j W P g r L A w g x M r A x r / B T T N A y 0 m W T n f f 5 7 P P S x g V 0 j S / t d r K 6 t r 6 R n 1 T 3 9 r e 2 d 0 z 9 g / 6 I k 4 5 J j 0 c s 5 g P P S Q I o x H p S S o Z G S a c o N B j Z O B N r 2 f + 4 J 5 w Q e P o T m Y J c U I 0 j m h A M Z J K c o 0 z u 7 w j z w h j 8 U O h N 3 1 Y K R 5 D e F r Y g o 5 D 5 O Z r T o P 2 Q 4 o i v I j W P g r L A w g x M r A x r / B T T N A y 0 m W T n f f 5 7 P P S x g V 0 j S / t d r K 6 t r 6 R n 1 T 3 9 r e 2 d 0 z 9 g / 6 I k 4 5 J j 0 c s 5 g P P S Q I o x H p S S o Z G S a c o N B j Z O B N r 2 f + 4 J 5 w Q e P o T m Y J c U I 0 j m h A M Z J K c o 0 z u 7 w j z w h j 8 U O h N 3 1 Y K R 5 D e F r Y g o 5 D 5 O Z r T o P 2 Q 4 o i v I j W P g r L A w g x M r A x r / B T T N A y 0 m W T n f f 5 7 P P S x g V 0 j S / t d r K 6 t r 6 R n 1 T 3 9 r e 2 d 0 z 9 g / 6 I k 4 5 J j 0 c s 5 g P P S Q I o x H p S S o Z G S a c o N B j Z O B N r 2 f + 4 J 5 w Q e P o T m Y J c U I 0 j m h A M Z J K c o 0 z u 7 w j z w h j 8 U O h N 3 1 Y K R 5 D e F r Y g o 5 D 5 O Z

Figure 10 :
Figure 10: Invariant mass m Zh distribution for gg → Zh production for t β = 4 and M A = 800 GeV at √ s = 13 TeV.We plot individual contributions from the Z-triangle,

m
8 pb, far above the upper bound on σ × B(A → Zh) 0.854 pb at M A = 500 GeV.Larger t β Zh [GeV] < l a t e x i t s h a 1 _ b a s e 6 4 = " g m D g C h E W x x A y 4 V J E 9 5 y W 7 H L M D 7 0 )⨯BR(H→ττ)[pb]

Fig. 13 :
(a) the triangle diagrams mediated by h; (b) the triangle diagrams mediated by H; (c) the box diagrams.New contributions are from the fourth generation quarks running in the loops and from the H-triangle diagram.
4 m n I U E e l l x Q U 5 P N P K E I a x 0I 8 r W K i / O z I U S T m N A l 0 Z I T W W i 9 5 M / M 9 z U x V e e h n l S a o I x / N F Y c q g i u E s E j i k g m D F p p o g L K i + F e I x E g g r H Z y p Q 3 A W v 7 x M O o 2 6 Y 9 e d u 0 a 1 e V X G U Q H H 4 B S c A w d c g C a 4 B S 3 Q B hg 8 g m f w C t 6 M J + P F e D c + 5 q U r R t l z B P 7 A + P w B Z b + X z w = = < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " 4 m n I U E e l l x Q U 5 P N P K E I a x 0I 8 r W K i / O z I U S T m N A l 0 Z I T W W i 9 5 M / M 9 z U x V e e h n l S a o I x / N F Y c q g i u E s E j i k g m D F p p o g L K i + F e I x E g g r H Z y p Q 3 A W v 7 x M O o 2 6 Y 9 e d u 0 a 1 e V X G U Q H H 4 B S c A w d c g C a 4 B S 3 Q B hg 8 g m f w C t 6 M J + P F e D c + 5 q U r R t l z B P 7 A + P w B Z b + X z w = = < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " 4 m n I U E e l l x Q U 5 P N P K E I a x 0I 8 r W K i / O z I U S T m N A l 0 Z I T W W i 9 5 M / M 9 z U x V e e h n l S a o I x / N F Y c q g i u E s E j i k g m D F p p o g L K i + F e I x E g g r H Z y p Q 3 A W v 7 x M O o 2 6 Y 9 e d u 0 a 1 e V X G U Q H H 4 B S c A w d c g C a 4 B S 3 Q B hg 8 g m f w C t 6 M J + P F e D c + 5 q U r R t l z B P 7 A + P w B Z b + X z w = = < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " 4 m n I U E e l l x Q U 5 P N P K E I a x 0I 8 r W K i / O z I U S T m N A l 0 Z I T W W i 9 5 M / M 9 z U x V e e h n l S a o I x / N F Y c q g i u E s E j i k g m D F p p o g L K i + F e I x E g g r H Z y p Q 3 A W v 7 x M O o 2 6 Y 9 e d u 0 a 1 e V X G U Q H H 4 B S c A w d c g C a 4 B S 3 Q B h g 8 g m f w C t 6 M J + P F e D c + 5 q U r R t l z B P 7 A + P w B Z b + X z w = = < / l a t e x i t > d /dm ZZ [pb/GeV]< l a t e x i t s h a 1 _ b a s e 6 4 = " e i P w R A A b Y P A I n s E r e D O e j B f j 3 f i Y l 6 4 Y R c 8 R + A P j 8 w c k y 5 0 A < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " e i P w R A A b Y P A I n s E r e D O e j B f j 3 f i Y l 6 4 Y R c 8 R + A P j 8 w c k y 5 0 A < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " e i P w R A A b Y P A I n s E r e D O e j B f j 3 f i Y l 6 4 Y R c 8 R + A P j 8 w c k y 5 0 A < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " e i P w R A A b Y P A I n s E r e D O e j B f j 3 f i Y l 6 4 Y R c 8 R + A P j 8 w c k y 5 0 A < / l a t e x i t > t = 4, M H = 900 GeV < l a t e x i t s h a 1 _ b a s e 6 4 = " B t s e h 8 a m 9 3 o j k f 8 J b g h C y 6 n Q 1 0 z t H e N 0 r P 7 s TJ P o j 1 S f y Y y G m k 9 i E I z G V G 8 1 d N e L v 7 n d V P s H / s Z l 0 m K I N l 4 U T 8 V N s Z 2 X o 3 d 4 w o Y i o E h l C l u / m q z W 6 o o Q 1 Ng 0 Z T g T p / 8 l 7 Q P q q 5 T d S 8 P y v X T S R 0 F s k 1 2 y B 5 x y R G p k w Z p k h Z h 5 I E 8 k R f y a j 1 a z 9 a b 9 T 4 e n b E m m S 3 y C 9 b H N / 8 c m i 8 = < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " B t s e h 8 a m 9 3 o j k f 8 J b g h C y 6 n Q 1 0 z t H e N 0 r P 7 s TJ P o j 1 S f y Y y G m k 9 i E I z G V G 8 1 d N e L v 7 n d V P s H / s Z l 0 m K I N l 4 U T 8 V N s Z 2 X o 3 d 4 w o Y i o E h l C l u / m q z W 6 o o Q 1 Ng 0 Z T g T p / 8 l 7 Q P q q 5 T d S 8 P y v X T S R 0 F s k 1 2 y B 5 x y R G p k w Z p k h Z h 5 I E 8 k R f y a j 1 a z 9 a b 9 T 4 e n b E m m S 3 y C 9 b H N / 8 c m i 8 = < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " B t s e h 8 a m 9 3 o j k f 8 J b g h C y 6 n Q

a j 1 a z 9 a b 9 T
4 e n b E m m S 3 y C 9 b H N / 8 c m i 8 = < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " B t s e h 8 a m 9 3 o j k f 8 J b g h C y 6 n Q

a j 1 a z 9 a b 9 T
4 e n b E m m S 3 y C 9 b H N / 8 c m i 8 = < / l a t e x i t > d h

Figure 14 :
Figure 14:The differential cross section for gg → ZZ against m ZZ in the 2HDM-SM4 at the 13 TeV LHC.We set t β = 4 and M H = 900 We show individual contributions from the h-triangle, H-triangle, and box diagrams.We include K gg = 1.8[79] and K q q = 1.5[80].

Figure 17 :
Figure 17: Partially integrated cross section of NP effects only, ∆σ NP defined in Eq. (5.4), for various m ZZ bins.We set M H = 900 GeV and consider t β = 4 and t β = 7.The red region is excluded by the 95% C.L. upper bounds on σ × B(H → ZZ).
Above the W W/ZZ (Zh) and below b b threshold, H (A) decays dominantly into W W/ZZ/hh (Zh).After the b b threshold, the enhanced Yukawa coupling Y H/A b makes the decay of H/A → b b dominant.Because of the sizable Γ(H/A → gg) and non-negligible B(H/A → τ τ ), the LHC direct searches in the τ τ channel impose the strong constraint: for t β = 4, M A < 2M b and M H < 350 GeV are excluded; for t β = 3, M A 350 GeV and M H < 180 GeV are excluded at the 95% C.L.
When t β is

Table 1 :
The CP-odd scalar boson A is produced in association with h, mediated by the Z boson.With the observed Higgs boson mass of 125 GeV, the Summary 7. Another reason is larger branching ratio of A → τ τ because of the absence of A → W W/ZZ/hh.Both ATLAS and CMS experiments exclude the parameter space with t β ≥ 4 and M A ≤ 2M b .When t β = 3 (the minimum value of t β allowed by the observed κ V ), two experiments yield different lower bounds on M A .Upon the absence of a combined ATLAS and CMS analysis yet, we take a conservative stance on constraining the model, i.e., adopting the weaker constraint between two experiment results.For t β = 3, M A 350 GeV is excluded at the 95% C.L.