Enhanced Di-Higgs Production in the Two Higgs Doublet Model

We show that the rate for di-Higgs production at the LHC can be enhanced by a factor as large as 25 compared to the Standard Model value in the two Higgs doublet model while being consistent with the known properties of the observed Higgs boson $h$. There are correlated modifications in $t\overline{t}h$ and resonant $Zh$ production rates, which can serve as tests of this model. Our framework treats both Higgs doublets on equal footing, each with comparable Yukawa couplings to fermions. The Cheng-Sher ansatz for multi-Higgs doublet model is shown to be strongly disfavored by current experiments. We propose a new ansatz for the Yukawa couplings of the Higgs doublets $\Phi_a$ is proposed, where $Y_{ij}^{(a)} = C_{ij}^{(a)}\cdot {\rm min}\{m_i, \, m_j\}/v$, with $C_{ij}^{(a)}$ being order one coefficients, $m_i$ the mass of fermion $i$ and $v$ the electroweak vacuum expectation value. Such a pattern of couplings can explain the observed features of fermion masses and mixings and satisfies all flavor violation constraints arising from the exchange of neutral Higgs bosons. The rate for $\mu \rightarrow e \gamma$ decay and new contributions to CP violation in $B_s-\overline{B}_s$ mixing are predicted to be close to the experimental limits.


Introduction
The observation of a new particle with properties matching that of the Higgs boson predicted by the Standard Model (SM) by the ATLAS and CMS experiments [1,2] has been an important step towards understanding the mechanism of electroweak (EW) symmetry breaking. With this discovery, attention has now shifted to testing whether the Higgs boson exhibits any property that deviates from the Standard Model expectation, and to searching for additional Higgs bosons that may take part in EW symmetry breaking. Experimental results to date, including Run−II data from the LHC, have shown no deviations from the SM. Furthermore, no signals of new resonances which could take part in symmetry breaking have been observed. However, as detailed in this paper, there is plenty of room for new physics in the symmetry breaking sector. Di-Higgs production has not been measured to date, with an upper limit set to about 25 times the SM prediction. tth production measurement allows its rate to be as large as 1.9 times the SM value (or as small as 0.5 of the SM value), and Zh production rate is allowed to be as large as 2 times its SM value. The purpose of this paper is to study these processes and their correlations in the context of the two Higgs doublet model (2HDM).
There are a variety of motivations for extending the SM with the addition of a second Higgs doublet. Supersymmetric models require a second Higgs doublet to generate fermion masses; electroweak baryogenesis can be consistently realized with a second Higgs doublet [3]; TeV scale dark matter can be realized in such extensions [4]; vacuum stability can be maintained all the way to the Planck scale with a second doublet [5] unlike in the SM [6], and small neutrinos masses may by generated as radiative corrections with a second doublet (along with a singlet scalar so that lepton number is broken) [7], to name a few. A second Higgs doublet appears naturally in models with extended symmetries such as leftright symmetric models [8], axion models [9], and grand unified theories. While the second doublet may have a mass of order the scale of higher symmetry breaking, it may also survive down to the TeV scale, in which case its signatures can be observed experimentally. It is to be noted that a second Higgs doublet which participates in the EW symmetry breaking can easily be made consistent with EW precision measurements, as the ρ parameter maintains its tree-level value of 1. The 2HDM also provides a foil to test the properties of the SM Higgs boson.
The phenomenology of 2HDM has been extensively studied over the years [10]. However, most studies restrict the form of the Lagrangian by assuming additional discrete symmetries. The type-II 2HDM, for example, allows only one doublet to couple to up-type quarks, with the second doublet coupling to the down-type quarks and charged leptons. While this is natural in the supersymmetric extension, a discrete Z 2 symmetry has to be assumed to achieve this restriction in other cases. One motivation for such a discrete symmetry is to suppress Higgs-mediated flavor changing neutral currents (FCNC) [11]. However, it has been recognized that there is no need to completely suppress such FCNC [12], an appropriate hierarchy in the Yukawa couplings can achieve the necessary suppression. We present a modified ansatz for the Yukawa couplings of Higgs doublet Φ a , where ij min{m i , m j }/v which is valid for the couplings of each of the Higgs doublet. Here C (a) ij are order one coefficients, m i stands for the mass of fermion i and v 246 GeV is the electroweak vacuum expectation value (VEV). Our modified ansatz can be realized in the context of unification [13]. We refer to the 2HDM with no additional symmetry as simply the two Higgs doublet model (2HDM), with no qualifier, as opposed to type-I or type-II models, which require additional assumptions.
The modified Yukawa coupling ansatz that we propose, viz., Y ij min{m i , m j }/v, improves on the Cheng-Sher (CS) ansatz [12] which assumes Y 174 GeV used in Ref. [12].) The pattern of quark mixings is compatible with both these ansatze, as we shall illustrate. The order one coefficients in the CS ansatz will have to be somewhat smaller than unity for explaining the quark mixings, while in the modified ansatz we present some C (a) ij are slightly larger than unity. We find that the overall goodness to fit is similar in the two cases as regards the CKM mixings. We compare the two ansatze for their consistency with Higgs mediated flavor violation and show that in processes such as K 0 − K 0 mixing, B d,s − B d,s mixing, and especially µ → eγ, the modified ansatz gives a better description of current data. We also find that the rate for µ → eγ decay and new contributions to CP violation in B s − B s mixing in the modified ansatz are close to the experimental limits, which may therefore provide tests of the model.
In the 2HDM framework that we adopt there is no fundamental distinction between the two Higgs doublets. The top quark, bottom quark and tau lepton Yukawa couplingsmost relevant for LHC phenomenology -with both doublets are then comparable, unlike in the case of type-I or type-II 2HDM. We shall see that these additional couplings, especially of the top and bottom quarks, would lead to distinct signatures in the hh, tth and Zh production rates in a correlated manner. Here h is the standard model-like Higgs boson of mass 125 GeV. While there would be some deviations in the properties of h from the SM predictions, in our analysis we ensure that such deviations are within experimental limits. Modifications in hh, tth and Zh production rates will also be correlated with such deviations in the properties of h. Discovery of these correlated modifications would be tests of the model. tth production: Theoretically, tth production process is very interesting, as its rate is proportional to Y 2 t , the square of the top Yukawa coupling with the Higgs boson. Within the SM, Y t is known to a good accuracy, as it is related to the top quark mass. But this proportionality relation is disrupted in the 2HDM we present. In Run−I of the LHC, a not-so-small signal of tth production was observed by the ATLAS and CMS collaborations in several channels. Assuming SM-like branching fractions of the Higgs boson, the tth signal strength normalized to the Standard Model prediction was found to be µ tth = 2.3 +0.7 −0.6 by the combined ATLAS and CMS collaborations with an observed significance of 4.4σ [14]. With data collected in the LHC Run−II with 13 TeV center of mass energy, both ATLAS and CMS have presented results for tth production, with ATLAS quoting a µ value of 1.32 +0.28 −0.26 with an observed significance of 5.8σ [15], and CMS quoting a µ value of 1.26 +0. 31 −0.26 [16] with an observed significance of 5.2σ. (In the SM, tth production cross-section is ∼ 0.509 pb.) As can be seen with the errors associated with these measurements, µ tth can be as large as 1.9 and also as low as 0.5. Such large deviations can be achieved in the 2HDM, as shall be shown below. If any significant deviation in tth production rate is observed at the LHC, the 2HDM framework we present can serve as an excellent platform for explaining it.
Di-Higgs production: including both resonant and non-resonant di-Higgs boson production, which gives ample room for its potential observation at the LHC. While other extensions of the SM, such as the singlet scalar extension, can enhance the di-Higgs production, the enhancement is much smaller compared to the 2HDM, owing primarily to severe constraints on the mixing of the SM Higgs and the singlet scalar from the measured properties of h [22]. Large enhancement for di-Higgs is possible in the 2HDM, since both Higgs bosons couple to top and bottom quarks, which allows for the properties of h to be within observed limits. Zh production: A third di-boson channel of experimental interest is the production of h in association with a Z. The rate for Zh production will also be modified in our 2HDM framework. Recently the ATLAS collaboration has reported a small excess in the pp → A → Zh cross section [23], corresponding to a potential pseudoscalar mass of about 440 GeV. The statistical significance is larger if the pseudoscalar A is produced in association with bottom quarks rather than through gluon fusion, but both production processes show deviations.
The statistical significance of this excess is too low to conclude anything meaningful, but it does raise the question: would it be possible to account for such a possible excess arising from a pseudoscalar resonance within a self-consistent framework? We show that this can indeed be achieved in our 2HDM framework.
We base our numerical analysis of the 2HDM on data set which takes into account the light Higgs boson properties as well as searches for heavy Higgs bosons. We also consider theoretical constraints arising from boundedness, stability and perturbativity of the scalar potential, and ensure that the data respects bounds from B-physics and electroweak precision measurements.
The paper is organized as follows: In Sec. 2, we briefly review the Higgs sector and the Yukawa couplings of the 2HDM. In Sec. 3, we present our modified ansatz for flavor, and in Sec. 4 we show the consistency of the ansatz with FCNC constraints. In Sec. 5, we perform numerical simulations for collider signatures of the 2HDM. Here we discuss enhanced di-Higgs production, as well as modifications in tth and hZ productions. We show correlations among these as well as other modified properties of the 125 GeV Higgs boson h. In Sec. 6 we discuss EW precision constraints, boundedness of the potential and unitarity constraints. Finally in Sec. 7 we conclude.

Brief Review of the Two Higgs Doublet Model
We denote the two SU(2) L doublet scalar fields with hypercharge Y = 1 2 as Φ 1 and Φ 2 . The most general gauge invariant scalar potential of this 2HDM is given in Appendix A, The VEVs of the neutral components of Φ 1,2 are denoted as and the mixing angle is given by One then obtains where ϕ 0 1 and ϕ 0 2 are defined in Eq. (4) and the angle (α − β) is defined as: The field h is identified as the observed Higgs boson of mass 125 GeV. We shall use these results in the discussion of flavor phenomenology as well as collider physics.

Yukawa sector of the 2HDM
As noted in the Introduction, we treat the two Higgs doublets on equal footing. Thus, both doublets will couple to fermions with comparable strengths. If the Yukawa coupling matrices have a certain hierarchy, consistent with the mass and mixing angles of fermions, then Higgs-mediated flavor changing neutral currents can be sufficiently suppressed [12].
This statement will be further elaborated in the next section. The original Higgs doublets Φ 1 and Φ 2 will then have the following Yukawa couplings to fermions: Here Q L = (u, d) T L and ψ = (ν, e) T L are the left-handed quark and lepton doublets, whilẽ Φ a = iτ 2 Φ * a . In the rotated Higgs basis (see Eq. (1)) the Yukawa couplings can be written as Here (Y d ,Ỹ d ), (Y u ,Ỹ u ) and (Y ,Ỹ ) are related to the original Yukawa coupling matrices as Since the VEV of H 2 is zero, the up-quark, down-quark, and charged lepton mass matrices are given by We can diagonalize these mass matrices, which would simultaneously diagonalize the Yukawa coupling matrices of the neutral Higgs bosons of the doublet H 1 . Note, however, that the ϕ 0 1 component of H 1 is not a mass eigenstates. All Higgs-induced FCNC will arise from the Yukawa coupling matrices of the neutral members of H 2 . Thus, in the down quark sector, such FCNC will be proportional toỸ d of Eq. (15), etc. We shall define the matircesỸ d ,Ỹ u andỸ of Eq. (15) in a basis where M u,d, have been made diagonal.
In the type-II 2HDM one assumes Y , invoking a Z 2 symmetry. This would lead to Y d andỸ d being proportional in Eq. (15) (and similarly Y u ∝Ỹ u , Y ∝Ỹ ). Thus, in a basis where Y d is diagonal,Ỹ d will also be diagonal.
As a result, type-II 2HDM would have no Higgs mediated flavor violation at the tree level [11]. In a similar fashion, type-I 2HDM, where one assumes Y In our framework, the coupling matricesỸ u ,Ỹ d andỸ are a priori arbitrary matrices. To be consistent with flavor violation mediated by neutral Higgs bosons, we assume a hierarchy inỸ u similar to the hierarchy in Y u , and so forth. This ansatz will be elaborated in the next section. For collider studies, onlyỸ t ,Ỹ b andỸ τ couplings, defined as the (3,3) elements ofỸ u ,Ỹ d andỸ l in a basis where M u,d, are diagonal, will play a role.

A Modified Ansatz for the Yukawa Couplings
The 2HDM can potentially lead to flavor changing neutral currents (FCNC) mediated by the neutral Higgs bosons at an unacceptable level. As already noted, one could completely suppress tree level FCNC by assuming a discrete Z 2 symmetry [11]. One can also assume alignment of the two sets of Yukawa couplings to alleviate this problem [10,24]. However, compete suppression of Higgs mediated FCNC is not necessary, as emphasized in Ref. [12].
A hierarchical pattern of Yukawa couplings that can generate realistic fermion masses and mixings may suppress excessive FCNC.
In Ref. [12], a particular pattern of the Higgs Yukawa couplings was suggested, referred to as the Cheng-Sher (CS) ansatz. The Yukawa couplings of the two Higgs doublets Φ a are taken to be of the form Here m i is the mass of the ith fermion, v = 246 GeV is the electroweak VEV, and C ij are order one coefficients. We have inserted a factor √ 2 since v is normalized to 246 GeV in our analysis (rather than 174 GeV). This is a well motivated ansatz, as this form can explain qualitatively several features of the CKM mixing angles. With this form of the Yukawa coupling matrices, the mass matrices will also take a similar form. The CKM mixing angles will be given by Here K ij are order one coefficients, expressible in terms of C ij of Eq.
These coefficients are roughly of order one, which provides justification to the ansatz of Eq.
(18). Note, however, that K db ∼ 1/8 in particular, is significantly smaller than order one. We shall present an updated analysis of flavor violation constraints for the CS ansatz in the next section. There we show that with the current data, various order one coefficients C ij appearing in Eq. (cs) will have to be smaller than one, provided that the masses of the additional Higgs bosons of the model are below a TeV. In particular, the constraint from µ → eγ decay sets |C eµ | ≤ 0.12. Constraints from K 0 − K 0 mixing, B d,s − B d,s mixing and D 0 − D 0 mixings on the relevant C ij are also of similar order. While these constraints do not exclude the scenario, they do make the ansatz somewhat less motivated.
In view of the strains faced by the CS ansatz, we propose a modified ansatz, which fares equally well in explaining the pattern of CKM mixing angles, but causes acceptable FCNC mediated by the neutral Higgs bosons. We take the Yukawa couplings of the two Higgs doublets Φ a to fermions to be of the form where C ij are order one coefficients, m i is the mass of the ith fermion and v 246 GeV. The main difference of this ansatz, compared to the CS ansatz of Eq. (18) is that the Yukawa couplings scale linearly with the lighter fermion masses, rather than as the geometric means.
We first show that this form of the couplings can generate reasonable CKM mixings. With the form of Eq. (21) for the Yukawa couplings, the CKM mixing angles would be given by Since the mass hierarchies are stronger in the up-quark sector, this would imply that the CKM angles scale as the mass ratios in the down-quark sector. Using the values of the masses of (d, s, b) quarks evaluated at the top quark mass scale, and using central values of the CKM mixing angles, this leads to the following K ij values: These values are roughly of order one, and the overall goodness to fit to the CKM angles is comparable to that of the CS ansatz (compare Eq. (20) with Eq. (22)). While in the CS ansatz the K ij are somewhat smaller than one, in the modified ansatz they are slightly larger than one.
We turn to the flavor phenomenology of the modified ansatz in the next section. There we derive limits on the C ij parameters in the modified ansatz and compare them with the limits on C ij of the CS ansatz. We find that the modified ansatz allows for all C ij , at least in the CP conserving sector, to be of order one.

Higgs Mediated Flavor Phenomenology
In this section we derive constraints on the C ij parameters for the modified ansatz as well as the CS ansatz arising from FCNC mediated by the neutral Higgs bosons of 2HDM. We write down the Yukawa couplings of the two Higgs doublets in a rotated Higgs basis H 1 and H 2 such that H 2 = 0. The Yukawa Lagrangian is given in Eq. (15). We define the matricesỸ u,d, in a basis where the matrices Y u,d, are diagonal. This is the physical mass eigenbasis. Since both Higgs doublets are treated on equal footing, the rotation that is needed to go to the (H 1 , H 2 ) basis should not change the form or the hierarchy of the original Yukawa couplings of Eq. (14). This is true for the CS ansatz as well as the modified ansatz. Thus the form of theỸ ij should be the same as that of Y (a) ij . The Yukawa couplings of H 2 will then take, in the modified ansatz, the following form: We shall show that Higgs mediated FCNC allows all the C ij appearing in Eq. (24) to be of order one. The main constraints come from K 0 − K 0 mixing, B 0 d,s − B 0 d,s mixing and D 0 − D 0 mixing, mediated by neutral Higgs bosons of 2HDM at the tree-level. The µ → eγ decay, although it arises only at the loop level, is also found to provide important constraints. We now turn to derivations of these constraints.

Constarints from tree level Higgs induced FCNC processes
There are accurate experimental measurements [26] of neutral meson-antimeson mixings in the  parameters from these processes.
We can write down the neutral Higgs boson mediated contributions to ∆F = 2 Hamiltonian responsible for the neutral meson-antimeson mixings as [27,28]: Here Y k ij denote the Yukawa couplings of q i , q j with Higgs mass eigenstate H k , with k taking values (h, H, A), and q i,j represent the relevant quark fields contained in the meson.
The transition matrix element for meson mixing can be expressed as Here the neutral mesons (K 0 , B 0 d , B 0 s , D 0 ) are denoted as φ. We adopt the modified vacuum saturation and factorization to parametrize the matrix elements, but use lattice evaluations of the matrix elements for our numerical study: We use the values: (B 2 , B 4 ) = (0.66, 1.03) for the K 0 system, (0.82, 1.16) for the B 0 d and B 0 s systems, and (0.82, 1.08) for the D 0 system [28,29]. The QCD correction factors of the Wilson coefficients C 2 and C 4 of the effective ∆F = 2 Hamiltonian in going from the heavy Higgs mass scale M H to the hadronic scale µ are denoted by η 2 (µ) and η 4 (µ) in Eq. (26). These correction factors are computed as follows. We can write the ∆F = 2 effective Hamiltonian in the general form as where For computing η 2,4 we consider the new physics scale M H to be 500 GeV. The evolution of the Wilson coefficients from M H down to the hadron scale µ is obtained from Here η = α s (M s )/α s (m t ). For our numerical study, we use the magic numbers a i , b (r,s) i and c (r,s) i from Ref. [30] for the K meson system, from Ref. [31] for the B d,s meson system and from Ref. [32] for the D meson system. With M s = 500 GeV, m t (m t ) = 163.6 GeV and α s (m Z ) = 0.118, we find η = α s (0.5 TeV)/α s (m t ) = 0.883.
At the mass scale of the heavy Higgs bosons, only operators Q 2 and Q 4 are induced.
After evolution to low energies for the K 0 system we find This leads to η 2 (µ) = 2.552, η 4 (µ) = 4.362 at µ = 2 GeV. Note that although non-zero C 3 and C 5 are induced via operator mixing, their coefficients are relatively small.
Following the same procedure, we compute the evolution of the Wilson coefficients for the B 0 d,s system and obtain leading to η 2 (µ) = 1.884, η 4 (µ) = 2.824 at µ = M B .
Similarly, for the D 0 system we find leading to η 2 (µ) = 2.174, η 4 (µ) = 3.620 at µ = M D . In all cases, we see that the induced operators C 3 and C 5 are negligible.
The neutral Higgs contributions will modify both the mass difference ∆M K and the CP violation parameter K . The mass splitting is obtained from the relation ∆m K = 2Re(M K 12 ), while the CP violation parameter K is given by If we now set sin(α−β) = 0.4 and M H = M A = 500 GeV, we get a limit of |Ỹ ds | < 1.8×10 −5 .
as suggested by the modified Yukawa ansatz of Eq. (21), we find C ds < 1.16. If we use instead the CS ansatz and writeỸ ds = would get |C ds | < 0.26. These constraints are tabulated, along with other constraints, in Table 1.
We have also derived the constraints on C ds from ∆m K and | K | by assuming Here K gives a much more stringent constraint on the phase φ. These results are summarized in Table 2 where we present three benchmark points for the phase parameter. While both the CS ansatz and the modified ansatz require a relative small phase, the modified ansatz fares better than the CS ansatz.
If we takeỸ bs to be real as well, the limit is |C bs | ≤ 3.17.
Under the assumption that the Yukawas couplings are also real, we get |C bd | ≤ 11.98 in our modified ansatz.
WhenỸ uc is also real, we get |C uc | < 4.9.
Upper bound on the coefficients Table 1 for the modified ansatz, and compare them with the constraints from the CS ansatz. Since the coefficient C sb is near one, one could expect new sources of CP violation in B s − B s , which is small in the SM.

Upper bound on C ij
Cheng-Sher Ansatz Our Ansatz Table 1: Upper bounds on the coefficients

Constraints from µ → eγ
Here we derive limits on the flavor violating leptonic Yukawa couplings from the loopinduced process µ → eγ. Both Higgs doublets have couplings to charged leptons in our 2HDM framework. As a result, there are one-loop and two-loop Barr Table 2: Upper bound on the coefficients C ds from K 0 −K 0 mixing and measurement of CP  decay in the 2HDM are shown in Fig. 2. We follow the analytic results given in Ref. [33] to evaluate these diagrams. The leading one-loop contribution to µ → eγ has the τ lepton and a neutral scalar inside the loop, with the photon radiated from the internal τ line.
It was pointed out in Ref. [34] some time ago that certain two-loop diagrams may in fact dominate over the one loop contributions owing to smaller chiral suppression. The loop suppression is overcome by a chiral enhancement of the two-loop diagram, relative to the one-loop diagram. This effect was noted by Barr and Zee [35] in the context of electric dipole moments.
In Fig. 3, we compute the branching ratio Br (µ → eγ) as a function of the mixing angle sin(α − β) in two scenarios; one following CS Yukawa coupling ansatz, and the other following our proposed modified ansatz. For these plots we setỸ t = 1. As can be seen from the figure, for order one coefficient C eµ in the CS ansatz, most of the parameter space Here we setỸ t = 1, and C eµ = C µe = 1.
Thus we see that the modified Yukawa ansatz fares better as regards the Higgs mediated FCNC compared to the CS ansatz. We have already noted that both ansatze give reasonable values of the CKM matrix. Since in the modified ansatz C eµ is close to one, we would expect the decay µ → eγ to be potentially observable.

Collider Implications of the 2HDM
In this section we analyze the implications of the 2HDM at colliders. We pay special attention to the allowed parameter space of the model from observed properties of the 125 GeV h boson, and investigate possible deviations in tth, di-Higgs and Zh production rates.
As we shall see, in spite of the consistency of the h boson with SM predictions, ample room remains for the above-mentioned signals deviating from the SM.

Higgs observables at the LHC
The propertis of the 125 GeV Higgs boson h in various production modes and decays into various final states at LHC so far seem to agree with the SM predictions. But uncertainties still remain in some of these measurements. This encourages to explore potential deviations in certain observables, and their correlations. One possibility that we have explored is to see if the new Yukawa couplings of the top quark in the 2HDM can lead to deviations in the tth production cross section at the LHC. This has recently been observed by the CMS [16] and ATLAS [15] collaborations. We numerically analyze the effects of anomalous top and bottom (and tau) Yukawa couplings on the tth production as well as the signal strengths of Higgs boson decay modes for h → γγ, W W, ZZ, bb, ττ , Zγ. Then we try to identify the parameter space which is consistent with both the recent ATLAS and CMS results from the LHC Run-2 (37 f b −1 ) data. Then remaining within the allowed parameter region, we analyze possible conspicuous signals such as enhanced di-Higgs boson production allowed by the 2HDM.
The parameter space of our model relevant for LHC study is spanned by the three new Yukawa couplings of t, b and τ in the rotated Higgs basis, mass of the heavy neutral Higgs boson H and the mixing angle α − β: 1 There are several search channels for the 125 GeV h [14] at the LHC by ATLAS and CMS collaborations. These results can give strong bounds on the free parameters of the 2HDM affecting Higgs observables. While the properties of the h boson are consistent with SM expectations, there is still enough room to look for new physics. To characterize the Higgs boson yields, the signal strength µ is defined as the ratio of the measured Higgs boson rate to its SM prediction. For a specific Higgs boson production channel and decay rate into specific final states, the signal strength is expressed as: where one can find the SM partial decay widths in Ref. [37].
In order to study the constraints from the current LHC data, the scaling factors which show deviations in the Higgs coupling from the SM in the 2HDM are defined as: where the loop function are given by: In the 2HDM, the charged Higgs boson will also contribute to h → γγ and h → Zγ decay via loops, in addition to the top quark and W boson loops. These contributions to h → γγ and h → Zγ are negligible, as long as the mass of H ± is kept above 300 GeV to be consistent with the current experimental constraints [26,[38][39][40]. The Run-1 data reported by ATLAS and CMS collaborations have been combined 1 and analyzed using the signal strength formalism and the results are presented in Ref. [14].

Recently, ATLAS and CMS collaborations have reported the results [17] on Higgs searches
based on 36 f b −1 data at 13 TeV LHC. The individual analysis by each experiment examines a specific Higgs boson decay mode corresponding to the various production processes which are h → γγ [41][42][43][44], h → ZZ [45][46][47][48], h → W W [49][50][51], h → τ τ [52,53], h → bb [54,55] and h → Zγ [56,57]. In Fig 4, we have used the most updated constraints on signal strengths reported by ATLAS and CMS collaboration for all individual production and decay modes as shown in Table 3 at 95% confidence level. As a consistency check, we have used also the combined signal strength value for a specific decay mode considering all the production modes and which is relatively stable with the results of Fig. 4.

Deviations in h Yukawa couplings and LHC constraints
4 shows the contour plot of µ tth in {Ỹ b , sin(α − β)} plane for a fixed value ofỸ t = 1.25. Fig. 4 clearly indicates that within the 2HDM, tth can be produced upto 1.9 times the SM predicted cross-section at the LHC, satisfying all the current experimental constraints from the 125 GeV Higgs boson searches within our model as we allow a variation ofỸ t between −2 and 2. As we shall see in the next section, µ tth will be further constrained from experimental limits on heavy Higgs boson searches. It is also to be noted that tth production rate can be as low as 0.5 times weaker than the SM predicted values within the 2HDM. We see from Fig. 4 that µ tth value gets suppressed if either sin(α − β) orỸ t is negative as shown in Eq. 49. Due to the different interference patterns between Yukawas (Ỹ t ,Ỹ b ,Ỹ τ ) and the mixing sin(α − β) in different decay modes, these plots are not symmetric about the central axes. It should be noted that, within our model there are additional modes of tth production via SM Higgs h production in association with the pseudoscalar A or heavy Higgs H, followed by the decay of A and/or H to tt. Since the hA production via quark fusion is dictated by the coupling ZhA, which is suppressed by sin(α − β), and also due to the significant loss in quark luminosity compared to gluon luminosity in the production, we have found its contribution in tth production to be less than 1%. On the other hand, hH or hA production via gluon gluon fusion will occur through triangle and box diagrams with top quarks. Due to the destructive interference between these two diagrams, the production rate will be small. In addition, there will be another suppression in the subsequent branching ratios for H → tt, or A → tt. Although its contribution to the total tth production is found to be less than 2%, we take these effects into account. The white shaded region in Fig. 4 simultaneously satisfies all the experimental constraints.
We have also scanned the parameter space for negativeỸ t and found that the most of the parameter space is ruled out by the current experimental constraints provided that tth production is decreased with compared to the SM. We also calculate the signal strength for Zγ channel which is well consistent with the available experimental data [56,57]. The signal strength in Zγ channel can vary from 0.7 to 1.4 satisfying all the constraints. Considering the best possible scenario in the available allowed parameter space for the Higgs boson production associated with a top quark pair, followed by the Higgs boson decays to W W and ZZ , the limit will go upto 1.6 times of the SM, where we allow |Ỹ t | values upto 2, since it will be suppressed by an extra cos 2 (α − β) term due to the different hW W and hZZ coupling and which can simultaneously explain the recently reported ATLAS and CMS results [15,16] on tth production. The two-dimensional best-fit of the signal strength modifiers for the processes tth, h → bb versus tth, h → V V , (V = W, Z) (left) and tth, h → τ + τ − versus tth, h → V V , (V = W, Z) (right) is shown in Fig. 5. Three benchmark points (BP), mentioned later in next sub-section, are also shown in this contour plot.  Table 4: Current summary of the observed signal strength µ measurements and tth production significance from individual analyses and the combination as reported by ATLAS and CMS collaboration [15,16].

Constraints from measurements of flavor violating Higgs boson couplings
Constraint from the exotic decay of top quark t → hc : In the 2HDM, an exotic top quark decay t → hc will be generated for non-zeroỸ tc u as well asỸ ct u . The decay branching ratio for t → hc is: Here we adopt Γ t = 1.41 GeV for the total decay rate of the top quark. The current experimental bound at the 95% C.L. [58] is reported as: Our model is well consistent with this constraint as it predicts very suppressed branching ratio BR(t → hc) = 7.33 × 10 −6 sin 2 (α − β) for order one coefficient C tc .

Constraints from lepton flavor violating Higgs boson decays:
Searches for the lepton flavor violating Higgs boson decays h → eτ, h → µτ constrain the lepton flavor violating Yukawa couplingsỸ eτ e andỸ µτ e . CMS collaboration recently reported their updated results with an integrated luminosity of 35.9 fb −1 at √ s = 13 TeV: at 95% C.L. [59]. Our model predicts the branching ratio BR(h → µτ ) to be where the total decay rate of the Higgs boson is given by Γ h = 4.1 MeV. Similarly the 2HDM predicts Hence lepton flavor violating Higgs boson decays do not put any significant bound on the parameters of our 2HDM framework.

CP-even Higgs phenomenology
Here we turn to the production and decay of the heavy neutral scalar, H, in the context However, we see that di-Higgs mode is the most dominant decay in all scenarios. In Fig.   7, we have shown the branching ratios to different decay modes of H as a function of the mass M H . Here we fix the value of three Yukawas and mixing as:Ỹ t = 1.25,Ỹ b = −0.09, Y τ = 10 −3 , and sin(α − β) = 0.5) to be consistent with the Fig. 4 constraints from properties of h. Throughout our analysis, the charged Higgs boson (H ± ) mass is kept almost degenerate with H and well above 300 GeV to be consistent with the current experimental constraints [26]. We also keep the mass splitting between charged and neutral member of the H 2 to be below 100 GeV [10].

Di-Higgs boson production
The di-Higgs production has drawn a lot of attentions [22,[62][63][64][65][66][67][68][69][70] since it is the golden channel to directly probe the triple Higgs-boson self-interaction within the SM, and therefore, tests the EW symmetry breaking mechanism. In the SM, the 125 GeV Higgs boson is pair produced through a triangle and a box diagram. The di-Higgs boson production rate in the SM is very small mainly due the smallness of the strength of individual diagrams, and also due to the negative interference between the triangle and box diagrams. At the 13 TeV LHC, the hh production cross section is about 33.5 fb, which is almost 1295 times weaker than the single Higgs production and it cannot be measured at current luminosity owing to the small branching fraction of h decaying into ZZ * and W W * and the large SM background. In the SM, di-Higgs production is thoroughly studied in Ref. [71]. Within the 2HDM framework, extra contribution to di-Higgs production arises from the decay of H after being resonantly produced mainly via gluon gluon fusion process at the LHC. Also, change in the tth coupling compared to the SM could give a significant deviation on di-Higgs production cross-section. These effects could significantly enhance the di-Higgs production rate and make it testable at the LHC. Therefore, it is important to analyze how largethe cross section can be, consistent with SM Higgs boson properties.
The most promising signal for di-Higgs search is the bbγγ, since it benefits from the large branching ratio of h → bb decay (∼ 58%) and also due to the clean diphoton signal (due to high m γγ resolution) on top of the smooth continuum diphoton SM background. On the other hand, due to the higher branching ratios of the SM Higgs boson decays to bb and τ + τ − , the bbbb and bbτ + τ − channels have larger signals, but they suffer from large QCD background.
The signal strength relative to the SM expectation µ hh is defined as following: where, In the 2HDM, di-Higgs (hh) production will occur both resonantly and non-resonantly.
Non-resonant di-Higgs (hh) production will be largely affected by the deviation of tth and hhh couplings, whereas the resonant production of hh, absent in SM, will occur significantly due to the large Hhh coupling which exists at the tree level. In Fig. 8, we show the correlation between tth enhancement and hh production enhancement when compared to  Fig. 8 clearly indicates that we can satisfy enhanced µ tth value of upto 1.9 and the µ hh value as big as 25 depending upon the heavier Higgs mass. This is significant enough to observe the hh pair production in the upcoming run or at the high luminosity LHC. These plots signify that if tth signal strength remains higher than the SM, the 2HDM is a great platform to explain it with a smoking gun signal of di-Higgs production at the LHC. In a recent study [73], it is shown that the SM like di-Higgs production with a cross section of 33.45 fb can be observed with 3.6σ significance [73], if LHC luminosity is upgraded to 3 ab −1 . IN our 2HDM scenario, the enhanced di-Higgs production rate is so large that it can potentially be observed with the 100 fb −1 LHC luminosity, which is close to the data set currently analyzed. The newly proposed future hadron-hadron circular collider (FCC-hh) or super proton-proton collider (SppC), designed to operate at 100 TeV centre of mass energy, can easily probe most of the parameter space in 2HDM through the hh pair production [74][75][76].  Table 5: Sample points on parameter space and corresponding µ hh and µ tth .
A few benchmark points within the 2HDM and the corresponding µ hh and µ tth are summarized in Table 5. Let us focus on one of the benchmark points, say BP1 in detail.
Since κ t is enhanced by a factor ∼ 37%, resonant SM Higgs boson production will also be enhanced in gluon gluon fusion. But, due to tiny enhancement in the hbb coupling and a decrease in the effective couplings hγγ, hW W, hZZ and hττ , the branching ratios for the decay modes h → γγ, W W, ZZ, ττ will be suppressed to ∼ 52%, 75%, 75%, 84% respectively. Overall, production times branching ratio will get adjusted within the signal strength constraints at 95% confidence level.
Since the effective hZZ or hW W couplings are suppressed in these scenario via mixing, it may lead to tension in V BF , W h and Zh production and subsequent decay of h to W W or ZZ, as the signal strength will be suppressed naively by a factor of cos 4 (α − β) due to mixing. Although, there is a huge uncertainty in these measurements accrording to the updated results of 13 TeV 36 f b −1 data, the central value prefers suppressed signal strength as low as 0.05 [45]. Such a suppression, if needed, can be achieved in our scenario compared to the SM. The suppression factor is 0.56 in our case for BP1. Hence, our allowed parameter space can satisfy all the experimental constraints [17] at 2σ level and simultaneously can lead to enhanced di-Higgs and tth production rates as well. Figure 9: : The two-dimensional best-fit of the signal strengths for ggF and VBF production modes compared to the SM expectations (black spade) looking at h → γγ [41] and h → ZZ → 4l [45] channels. The dashed red and black line represent the 2σ standard deviation confidence region for h → γγ [41] and h → ZZ → 4l [45] channels respectively. Red and blue shaded regions are 2HDM predicted region. Three benchmark points (BP) are also shown in this contour plot.
The two-dimensional best-fit for the signal strengths for gluon gluon fusion (ggF) and vector boson fusion (VBF) production modes compared to the SM expectations (black spade) looking at h → γγ [41] and h → ZZ → 4l [45] channels are shown in Fig. 9 using 36 fb −1 data of 13 TeV LHC. Uncertainty in these two channels are least compared to other channels and hence impose the most stringent limits. ggF is the most dominant and VBF is the second dominant production mode for single h production. In the SM, the ratio of resonant single Higgs boson (h) production in VBF production to the ggF production is ∼ 0.085. In the 2HDM this ratio can deviate largely compared to the SM prediction which is reflected in the Fig. 9. For three of the benchmark points listed, this ratio becomes 0.034, 0.07 and 0.154 respectively. As the branching ratios also deviate from the SM simultaneously, these points lie within the 2σ confidence region for the two-dimensional best-fit of the signal strengths for ggF and VBF production modes.
In two Higgs doublet model with additional discrete Z 2 symmetry, there will be resonant di-Higgs production which has been extensively studied in [77]. The resonant di-Higgs production rate is much larger in our framework compared to the type-II 2HDM. It is easy to understand this difference. In the type-II 2HDM, resonant production cross section of H is suppressed since H has no direct coupling to top quark. Such a coupling is induced proportional to the Higgs mixing angle sin(α − β), which is strongly constrained from the properties of h. In our case, H has direct coupling to the top quark.
The viability of a scenarios where the sign of the b-quark coupling to h is opposite to that of the Standard Model (SM), while other couplings are close to their SM value, has been studied in Ref. [78] in the context of type-I and type-II 2HDM. Our analysis here includes such effects, as we allow both signs forỸ b . Using scans over the full parameter space, subject to basic theoretical and experimental constraints as described previously, we found that a sign change in the down-quark Yukawa couplings can be accommodated in the context of the current LHC data set at 95% C.L. as shown in the benchmark points of Table 5. We have shown that such a scenario is consistent with all LHC observations.

Pseudoscalar Higgs phenomenology
Now we turn to the production and decay of the heavy pseudoscalar-A in the context of the LHC experiments. The most promising signal is the tree-level decay of A into di-boson pairs (Zh, ZH). Although, the decay mode A → Zh arises due to the mixing (sin(α − β)) between heavy Higgs H and SM Higgs h, this leads to a smoking gun signal of the 2HDM in the channel pp → A → Zh → l + l − bb. The pseudoscalar-A has loop induced coupling to a pair of gluons due to its ttA coupling at tree level. It will be then dominantly produced via gluon gluon fusion at the LHC. On the other hand, for sufficiently large bottom Yukawa coupling,Ỹ b ∼ 0.1, another promising mode is the production of A in association with two bottom quarks. Representative leading order Feynman diagrams for pseudoscalar A production in association with b quarks and subsequent decay to Zh is shown in Fig. 10.
After being produced at the LHC, it will dominantly decay to Zh, ZH, tt, bb. The unique Figure 10: Representative leading order Feynman diagrams for pseudoscalar A production in association with b quarks and simulataneous decay to Z and h boson. The diagrams that can be obtained by crossing the initial state gluons, or radiating the Higgs off an antibottom quark are not shown. signals at the LHC will be resonant production of pseudoscalar A and its subsequent decay to SM Higgs in association with Z boson (pp → A → Zh) and di-Higgs production in association with Z boson pp → A → ZH → Zhh. If the mass splitting between H and A is kept small, for large mixing, A → Zh will be the most promising mode and we will focus on this scenario from here on. In Fig. 11, branching ratios of A to different decay modes are shown.
The Higgs boson production in association with a Z boson has drawn a lot of attentions as it is the channel to probe the ZZh coupling in the SM, and therefore, tests the electroweak symmetry breaking mechanism. In the SM, the hZ production rate at the 13 TeV LHC is 869 fb. In the 2HDM, Zh production cross-section can significantly deviate from the SM value. Within the 2HDM, extra contributions to Zh production arises from the decay of A after being resonantly produced via gluon gluon fusion process. Hence the signal strength µ Zh relative to the SM expectation can deviate and can be as large as the current experimental limit [23], large mixing sin (α − β) is allowed within our framework. Contourplost of µ Zh and µ hh in µ tth -M H plane are shown in Fig. 12. For simplicity, we consider the case of degenerate H and A. The white dashed line indicates the different values of µ Zh .
As we can see, there is strong correlation between µ hh , µ tth and µ Zh . Black, pink and cyan colored meshed zones are excluded parameter space from current di-Higgs limit looking at different final states bbγγ, bbbb and bbτ + τ − respectively; red,blue and brown meshed zones

Other collider implications for heavy Higgs searches
For completeness, we discuss the prospects for charged Higgs boson and other pseudoscalar searches at the LHC within the 2HDM. LHC experiments have already set very strong bounds on the singly charged Higgs mass in the low mass region M H + < m t for the pp → H +t b process, assuming the decay H + → τ + ν [26,[38][39][40]. To be consistent with the current experimental constraints, we consider the singly charged Higgs mass well above 300 GeV. However, the smallness of T parameter does not allow for a large mass splitting between H + , H and A. A variety of production mechanism is involved in singly charged Higgs production. It can be pair produced at the LHC via gluon gluon fusion through the Higgs portal, via quark fusion through s channel Z or γ exchange and also via photon initiated processes [79]. H + can also be produced in association with fermions via gluon gluon fusion. There is another production mode of H + in association with the SM Higgs h via s-channel W boson exchange. The significant decay modes of H + are H + → τ + ν, H + → tb, H + → W + h in the higher mass region. We choose mass splitting such that H + → W + H, H + → W + A are not kinematically allowed. In this scenario, it mostly decays to tb since H + → W + h is suppressed partially by the mixing angle sin(α − β). The detailed phenomenology of charged Higgs is beyond the scope of this study. We will give some naive estimates of cross-sections and required luminosity to discover it at the LHC for the three sample points shown in Table 5. background. The tau channel benefits from being rather clean compared to the other channels. hH + production will give rise to tbh signals. The production rate of hH + gets a suppression due to mixing. For the three sample points, the estimated hH + production cross-sections are 0.628 fb, 0.211 fb and 0.049 fb respectively. We consider also the signal pp → H +t b →tbτ + ν. For this process, the total cross-section is 0.043 fb, 0.013 fb and 0.016 fb respectively for the three BP. After naive estimation of the background, we get that we need atleast ∼ ab −1 luminosity at the upcoming run of the LHC for the discovery of H ± .
Similar to H the pseudoscalar A is also produced resonantly via gluon-gluon fusion through the triangle loop with the top quark since A directly couples to t. After being produced resonantly, it mainly decays to tt and bb. These channels are very challenging from the point of background elimination. Due to mixing, it also decays to Zh. We restrict ourselves such that A is not kinematically allowed to decay to W + H − by keeping the mass splitting between A and H very small (∼ 5GeV ). The detailed LHC phenomenology of A and H is beyond of the scope of this study and will be presented in our future work.
bbh coupling is also modified in the 2HDM framework. bbh production rate will change compared to the SM. On the other hand, as bbh coupling plays the most significant role in SM Higgs branching ratio to different decay modes, large deviation from SM bbh coupling is not possible as it could highly constrain the parameter space. For three of the benchmark points used in Table 5, the signal strength for bbh production is µ bbh = 1.002, 1.11 and 0.819 respectively. Similar to tth and bbh production, there will be also ttH and bbH production in this model. For three of the benchmark points, the ttH production rate at the 13 TeV LHC is 2.26 fb, 2.05 fb and 4.13 fb respectively. bbH production will give a unique signature of six b-quarks via the process pp → bbH → bbhh → bbbbbb. The production cross section for bbH at the 13 TeV LHC turns out to be 0.897 fb, 0.140 fb and 0.0478 fb respectively for three of the sample points. This is too small to probe at the 13 TeV LHC for the current luminosity. It requires very high luminosity (∼ ab −1 ) to get the discovery reach limit via these channels.
After being resonantly produced via gluon fusion at the LHC, the heavy Higgs boson H can decay via lepton flavor violating processes such as H → µτ [80,81], which is a very clean signal at the LHC. But, due to the small Yukawa coupling (∼ m µ /v), the branching fraction for H → µτ process is highly suppressed. For a benchmark point (M H = 500 GeV,Ỹ t = −0.3,Ỹ b = −0.2,Ỹ τ = 10 −3 , sin (α − β) = 0.3), the branching ratio for H → µτ process is computed to be 10 −7 . Since the branching ratio is highly suppressed, even if for O(∼ pb) resonant production rate of heavy Higgs H, it requires very high luminosity (∼ ab −1 ) to observe a single event. This is too small to probe at the 13 TeV LHC for the current luminosity.

Electroweak precision constraints, boundedness and unitarity
The oblique parameters S, T and U provide important constrains from electroweak precisiton data on many models beyond the SM. These parameters have been calculated at the one loop in the two Higgs doublet model [82][83][84]. We focus on the scenario where wH, A and H + With this assumption, we scan for the predicted values of S and T as a function of the mixing angle sin(α − β). Our results are shown in Fig. 15. We see that the constraints from S and T parameters [26] allow large parameter space of interest in the 2HDM which can lead to observable signals at the LHC.
To ensure the existence of a stable vacuum, the 2HDM scalar potential must be bounded from below, i.e. it must take positive values for any direction for which the the value of any field tends to infinity. This places some restrictions on the allowed values of the quartic scalar couplings. We require Λ 6 < 3.5 in order to avoid non-perturvative regimes. Note that Λ 6 term is related to M H and sin(α − β) via Eq. (13). As a result, for larger mixing (sin(α − β) ∼ 0.5), we can go upto a mass of ∼ 700 GeV for the mass of the heavy Higgs H, whereas for smaller mixing (sin(α − β) ∼ 0.1), M H can be as large as ∼ 1.5 TeV.
We have also checked all the boundedness conditions [85]. To ensure that the scalar potential is bounded from below, we evaluate the eigenvalues and eigenvectors of the following matrix: where we choose all the quartic couplings to be real. Here we present one set of values of the quartic couplings: Λ 1 = 1.4, Λ 2 = 0.01, Λ 3 = 1, Λ 4 = 0.1, Λ 5 = 0.001, Λ 6 = 3 and Λ 7 = −1.2. For this set, we found all the eigenvalues of the matrix to be {2.0527, −1.75315, 0.649943, −0.0495} . This satisfies the boundedness conditons. For this specific choice, we obtain the three mass eigenvalues for the neutral scalar masses to be { 125 GeV, 751 GeV, 706 GeV } from Eq. (7) and the mixing angle sin(α−β = 0.458. The unitarity bounds in the most general two Higgs doublet model without any discrete Z 2 symmetry has been studied here Ref. [86]. Our choice of parameters are consistent with these unitarity constraints.

Conclusion
In this paper we have undertaken a detailed analysis of the collider implications of the two Higgs doublet model. In our framework, both doublets are treated on equal footing, which implies that they have Yukawa couplings with fermions with comparable strengths. Such a scenario would have Higgs mediated flavor changing neutral currents, which can potentially be excessive. It is customary to assume a discrete symmetry to forbid such FCNC. Here we have proposed a simple ansatz for the Yukawa couplings of each of the Higgs doublets that is consistent with the observed CKM mixing angles and also compatible with FCNC constraints. Our ansatz, where the Yukawa couplings are taken to have a form ij min{m i , m j }/v, with C ij being order one coefficients and m i being the mass of fermion i, is an improvement over the Cheng-Sher (CS) ansatz [12] which assumes We have studied the flavor phenomenology of the new ansatz and shown that all Higgs mediated FCNC are within acceptable limits. We have also compared the FCNC constraints with those from the CS ansatz, and found that the modified ansatz fares much better.
We have studied the collider implications of the 2HDM framework with both Higgs doublets coupling to fermions. The LHC phenomenology is sensitive mostly to the couplings of the top and bottom quarks and the tau lepton. Taking these couplings to be comparable for the two doublets, we have shown that resonant di-higgs boson production rate can be enhanced by a factor of 25 compared to the SM. Such a large deviation is possible, as the properties of the 125 GeV Higgs boson h can be within experimentally allowed range with large Yukawa couplings of the t and b quarks to both doublets.