Maximally Symmetric Two Higgs Doublet Model with Natural Standard Model Alignment

We study the Higgs mass spectrum as predicted by a Maximally Symmetric Two Higgs Doublet Model (MS-2HDM) potential based on the SO(5) group, softly broken by bilinear Higgs mass terms. We show that the lightest Higgs sector resulting from this MS-2HDM becomes naturally aligned with that of the Standard Model (SM), independently of the charged Higgs boson mass and $\tan \beta$. In the context of Type-II 2HDM, SO(5) is the simplest of the three possible symmetry realizations of the scalar potential that can naturally lead to the SM alignment. Nevertheless, renormalization group effects due to the hypercharge gauge coupling $g'$ and third-generation Yukawa couplings may break sizeably this SM alignment, along with the custodial symmetry inherited by the SO(5) group. Using the current Higgs signal strength data from the LHC, which disfavour large deviations from the SM alignment limit, we derive lower mass bounds on the heavy Higgs sector as a function of $\tan\beta$, which can be stronger than the existing limits for a wide range of parameters. In particular, we propose a new collider signal based on the observation of four top quarks to directly probe the heavy Higgs sector of the MS-2HDM during the run-II phase of the LHC.


Introduction
The discovery of a Higgs resonance with mass around 125 GeV at the LHC [1,2] offers an unprecedented opportunity for probing extended Higgs scenarios beyond the Standard Model (SM). Although the measured properties of the discovered Higgs boson show remarkable consistency with those predicted by the SM [3, 4], the current experimental data still leave open the possibility of new physics that results from an extended Higgs sector. In fact, several well-motivated new-physics scenarios require an enlarged Higgs sector, such as supersymmetry [5], in order to address a number of theoretical and cosmological issues, including the gauge hierarchy problem, the origin of the Dark Matter and the baryon asymmetry in our Universe. Here we follow a modest bottom-up approach and consider one of the simplest Higgs-sector extensions of the SM, namely the Two Higgs Doublet Model (2HDM) [6].
The 2HDM contains two complex scalar fields transforming as iso-doublets (2, 1) under the SM electroweak gauge group SU(2) L ⊗ U(1) Y : with i = 1, 2. In this doublet field space Φ 1,2 , the general 2HDM potential reads present our conclusions. Finally, several technical details related to our study have been relegated to Appendices A and B.

Maximally Symmetric Two Higgs Doublet Model Potential
In order to identify all accidental symmetries of the 2HDM potential, it is convenient to introduce the 8-dimensional complex multiplet [7,27,28]: where Φ i = iσ 2 Φ * i (with i = 1, 2) and σ 2 is the second Pauli matrix. We should remark that the complex multiplet Φ satisfies the Majorana property [7]: Φ = CΦ * , where C = σ 2 ⊗ σ 0 ⊗σ 2 is the charge-conjugation matrix, with σ 0 = 1 2×2 being the identity matrix. In terms of the Φ-multiplet, the following null 6-dimensional Lorentz vector can be defined [7,28]: where A = 0, 1, ..., 5 and the six 8 × 8-dimensional matrices Σ A may be expressed in terms of the three Pauli matrices σ 1,2,3 , as follows: We must emphasize here that the bilinear field space spanned by the 6-vector R A realizes an orthochronous SO(1, 5) symmetry group. In terms of the 6-vector R A defined in (2.2), the 2HDM potential V given in (1.2) takes on a simple quadratic form: where M A and L AB are SO (1,5) constant 'tensors' that depend on the mass parameters and quartic couplings of the scalar potential V and their explicit forms may be found in [28][29][30][31]. Requiring that the SU(2) L gauge-kinetic term of the multiplet Φ remains canonical restricts the allowed set of rotations from SO (1,5) to SO (5), 3 where only the spatial components R I (with I = 1, ..., 5) transform, whereas the zeroth component R 0 remains invariant. Consequently, in the absence of the hypercharge gauge coupling g and fermion Yukawa couplings, the maximal symmetry group of the 2HDM is G R 2HDM = SO (5). 3 We note in passing that if the restriction of SU(2)L gauge invariance is lifted, the 2HDM is then equivalent to an ungauged theory with 8 real scalars and so the maximal symmetry group becomes the larger group O(8) [32].
Given the group isomorphy SO(5) ∼ Sp(4)/Z 2 [33], the maximal symmetry group of the 2HDM in the original Φ-field space is [28] 4 G Φ 2HDM = (Sp(4)/Z 2 ) ⊗ SU(2) L , (2.5) in the limit of vanishing g and fermion Yukawa couplings. In fact, as we will see below by explicit construction [cf. (2.8)], it is not difficult to deduce that in the same limit the maximal symmetry group for an n Higgs Doublet Model (nHDM) will be G Φ nHDM = (Sp(2n)/Z 2 ) ⊗ SU(2) L , (2.6) in which case the multiplet Φ becomes a Majorana 4n-dimensional complex vector. It is interesting to note that for the SM with n = 1 Higgs doublet, (2.6) yields the well-known result: G Φ SM = (SU(2) C /Z 2 ) ⊗ SU(2) L , by virtue of the group isomorphy: Sp(2) ∼ SU(2) C , where SU(2) C is the custodial symmetry group originally introduced in [34]. Hence, it is important to stress that (2.6) represents a general result that holds for any nHDM.
We may now identify all maximal symmetries of the 2HDM potential by classifying all proper, improper and semi-simple subgroups of SO(5) in the bilinear R I space. In this way, it was found [7,28] that a 2HDM potential invariant under SU(2) L ⊗ U(1) Y can possess a maximum of 13 accidental symmetries. This symmetry classification extends the previous list of six symmetries reported in [30], where possible custodial symmetries of the theory were not included. Each of the 13 classified symmetries puts some restrictions on the kinematic parameters appearing in the 2HDM potential (1.2). In a specific diagonally reduced bilinear basis [35,36], one has the general restrictions Im(λ 5 ) = 0 and λ 6 = λ 7 , thus reducing the number of independent quartic couplings to seven. In the maximally symmetric SO(5) (∼ Sp(4)/Z 2 ) limit, we have the following relations between the scalar potential parameters [7,28]: Thus, in the SO(5) limit, the 2HDM potential (1.2) is parametrized by a single mass parameter µ 2 and a single quartic coupling λ: It is worth stressing that the MS-2HDM scalar potential in (2.8) is more minimal than the respective potential of the MSSM at the tree level, even in the custodial symmetric limit g → 0. The latter only possesses a smaller symmetry: O(2) ⊗ O(3) ⊂ SO(5), in the 5-dimensional bilinear R I space.

Custodial Symmetries in the MS-2HDM
It is now interesting to discuss the implications of custodial symmetries for the Yukawa sector of the 2HDM. To this end, let us only consider the quark Yukawa sector of the theory, even though it is straightforward to extend our results to the lepton sector as well. The relevant part of the quark-Yukawa Lagrangian in the 2HDM can generally be written down as follows: where Q L ≡ (u L , d L ) T is the SM quark iso-doublet and we have introduced a 12 × 6dimensional non-square Yukawa coupling matrix (2.10) There are three inequivalent realizations of the custodial symmetry. To find these, we introduce the 10 Lie generators κ a (with a = 0, 1, 2, . . . , 9) of the Sp(4) group [28]: with the normalization: Tr(κ a κ b ) = δ ab . In the Φ-space, the Sp(4) generators may be represented as K a = κ a ⊗ σ 0 , in which case K 0 is the generator associated with U(1) Y rotations. Candidate Sp(4) generators of the custodial symmetry are those generators that do not commute with the hypercharge generator K 0 , i.e. K a with a = 4, 5, 6, 7, 8, 9. It is not difficult to see that these six generators, together with K 0 , form three inequivalent realizations of the SU(2) C custodial symmetry [28]: (i) K 0,4,6 , (ii) K 0,5,7 and (iii) K 0, 8,9 . In order to see the implications of the three custodial symmetries (i), (ii) and (iii) for the quark Yukawa sector, we impose a symmetry commutation relation on H after generalizing it for non-square matrices: where H is expressed in the reduced 4 × 2-dimensional space, in which the 3 × 3 flavour space has been suppressed. In addition, we denote with t b = σ b /2 (with b = 1, 2, 3) one of the three 2 × 2 generators of the custodial SU(2) C group. It is then not difficult to check that it holds κ 0 H − H t 3 = 0 4×2 , which implies that the specific block structure of H in (2.10) respects U(1) Y by construction, given the correspondence: κ 0 ↔ t 3 . In detail, imposing (2.12) for the three SU(2) C symmetries, we obtain the following relations among the 3 × 3 up-and down-type quark Yukawa coupling matrices: where θ is an arbitrary angle unspecified by the symmetry constraint (2.12). We should stress again that only for a fully SO(5)-symmetric 2HDM, the three sets of solutions in (3.14) are equivalent. However, this is not in general true for scenarios that happen to realize only subgroups of SO(5), according to the symmetry classification given in [7,28].

Scalar Spectrum in the MS-2HDM
The masses and mixing in the Higgs sector of a general 2HDM are given in Appendix A.
After electroweak symmetry breaking in the MS-2HDM, we have the breaking pattern which gives rise to a Higgs boson H with mass M 2 H = 2λ 2 v 2 , whilst the remaining four scalar fields, denoted hereafter as h, a and h ± , are massless (pseudo)-Goldstone bosons. The latter is a consequence of the Goldstone theorem [37] and can be readily verified by means of (2.7) in (A.5). Thus, we identify H as the SM-like Higgs boson with the mixing angle α = β [cf. (A.7)]. We call this the SM alignment limit, which can be attributed to the SO(5) symmetry of the theory.
In the exact SO(5)-symmetric limit, the scalar spectrum of the MS-2HDM is experimentally unacceptable, as the four massless pseudo-Goldstone particles, viz. h, a and h ± , have sizeable couplings to the SM Z and W ± bosons [cf. (A.9)]. These couplings induce additional decay channels, such as Z → ha and W ± → h ± h, which are experimentally excluded [38]. Nevertheless, as we will see in the next section, the SO(5) symmetry of the original theory may be violated predominantly by RGE effects due to g and thirdgeneration Yukawa couplings, as well as by soft SO(5)-breaking mass parameters, thereby lifting the masses of these pseudo-Goldstone particles.

RGE and Soft Breaking Effects
As discussed in the previous section, the SO(5) symmetry that governs the MS-2HDM will be broken due to g and Yukawa coupling effects, similar to the breaking of custodial symmetry in the SM. Therefore, an interesting question will be to explore whether these effects are sufficient to yield a viable Higgs spectrum at the weak scale. To address this question in a technically natural manner, we assume that the SO(5) symmetry is realized at some high scale µ X . The physical mass spectrum at the electroweak scale is then obtained by the RG evolution of the 2HDM parameters given by (1.2). Using state-of-the-art twoloop RGEs given in Appendix B, we examine the deviation of the Higgs spectrum from the SO(5)-symmetric limit due to g and Yukawa coupling effects. This is illustrated in Figure 1 for a typical choice of parameters in a Type-II realization of the 2HDM, even though the conclusions drawn from this figure have more general applicability. In particular, we obtain the following breaking pattern starting from a SU(2) L -gauged theory: where U(1) em is the electromagnetic group. In other words, RGE-induced g effects only lift the charged Higgs-boson mass M h ± , while the corresponding Yukawa coupling effects also lift slightly the mass of the non-SM CP-even pseudo-Goldstone boson h. However, they still leave the CP-odd scalar a massless (see left panel of Figure 1), which can be identified as a U(1) PQ axion [39]. The deviation of the scalar quartic couplings from the SO(5)-symmetric limit given in (2.7), thanks to g and Yukawa coupling effects, is illustrated in Figure 1 (right panel) for a simple choice of the single quartic coupling λ at the SO(5)-symmetry scale µ X . Figure 1 also shows that g and Yukawa coupling effects are not sufficient to yield a viable Higgs spectrum at the weak scale, starting from a SO(5)-invariant boundary condition at some high scale µ X . To minimally circumvent this problem, we need to include soft SO(5)-breaking effects, by assuming a non-zero value for Re(m 2 12 ) in the 2HDM potential (1.2). In the SO(5)-symmetric limit (2.7) for the scalar quartic couplings, but with Re(m 2 12 ) = 0, we obtain the following mass spectrum [cf. (A.5)]: as well as an equality between the CP-even and CP-odd mixing angles: α = β, thus predicting an exact alignment for the SM-like Higgs boson H. Notice that the alignment limit α = β is independent of the charged Higgs-boson mass M h ± and the value of tan β. This is achieved without decoupling, i.e. without the need to consider the mass hierarchy M h ± v. Hence, in this softly broken SO(5) 2HDM, we get natural SM alignment, without decoupling. 5 It is instructive to analyze this last point in more detail. In the general CP-conserving 2HDM, the CP-even scalar mass matrix can be written down as [40,41] where M 2 a is given in (A.5), λ 34 ≡ λ 3 + λ 4 , and Here we have used the short-hand notation: 3) is the respective 2 × 2 CP-even mass matrix written down in the so-called Higgs eigenbasis [42]. Evidently, the SM alignment limit α → β for the CP-even scalar mixing angle α is obtained, provided the off-diagonal elements of M 2 S in (3.4) vanish, i.e. for C = 0. From (3.7), this yields the quartic equation Since the scalar quartic couplings in the general 2HDM potential (1.2) are independent variables, all the coefficients of (3.8) must vanish identically. Imposing this restriction, we conclude that all natural alignment solutions require that λ 6 = λ 7 = 0. More importantly, the alignment limit is achieved for any value of tan β, since for λ 6 = λ 7 = 0, (3.8) has a solution Strictly speaking, there will be one-loop threshold corrections to the effective MS-2HDM potential, sourced from a non-zero Re(m 2 12 ), which might lead to small misalignments. A simple estimate suggests that these corrections are of order λ 2 /(16π 2 ) and can therefore be safely neglected to a good approximation.
which 'consistently' gives an indefinite answer 0/0. We further note that the alignment condition (3.9) is independent of M a , i.e. our result is similar to that obtained in [25].
In the alignment limit, the two CP-even Higgs masses are given by the diagonal elements of M 2 S in (3.4): On the other hand, in the limit M a v, we can use a seesaw-like approximation in (3.4) to obtain (3. 13) In (3.12) and (3.13), we have also included the possibility of decoupling via a large λ 5 coupling [24]. For large values of tan β, e.g. tan β > ∼ 10, we readily see that (3.12) reduces to M 2 H 2λ 2 v 2 , which again leads to a natural alignment. As noted above, in the SO(5) symmetric limit of the conformal part of the 2HDM as given by (2.7) and (3.2), the SM alignment is achieved for any value of tan β [cf. (3. 9)]. In addition to SO(5), one may now wonder whether there are other classified symmetries of the 2HDM that lead to natural SM alignment, independently of tan β and M a . Indeed, according to the classification given in Table 1 of [28], we observe that there are only two other symmetries which lead to such natural SM alignment: whereas all other scalar potential parameters in (1.2) are zero. After spontaneous electroweak symmetry breaking, symmetry (i) predicts two pseudo-Goldstone bosons (h, a), whilst symmetry (ii) predicts only one pseudo-Goldstone boson, i.e. the CP-even Higgs boson h. However, a non-zero soft SO(5)-breaking mass parameter m 2 12 can be introduced to render the pseudo-Goldstone bosons sufficiently massive, in agreement with present experimental data, similar to the SO(5) case shown in Figure 1 (left panel). Even though the 2HDM scenarios based on the symmetries (i) and (ii) may be analyzed in a similar fashion, our focus here will be on the MS-2HDM based on the SO(5) group. Nevertheless, the results that we will be deriving in the present study are quite generic and could apply to these cases as well.

Misalignment Predictions
As discussed in Section 3, a realistic Higgs spectrum can be obtained by softly breaking the maximal SO(5) symmetry of the 2HDM potential at some high scale µ X by considering Re(m 2 12 ) = 0. As a consequence, there will be some deviation from the alignment limit in the low-energy Higgs spectrum. By requiring that the mass and couplings of the SM-like Higgs boson in our MS-2HDM are consistent with the latest Higgs data from the LHC [3, 4, 43], we can derive predictions for the remaining scalar spectrum and compare them with the existing (in)direct limits on the heavy Higgs sector. Our subsequent numerical results are derived for the Type-II 2HDM scenario, but the analysis could be easily extended to other 2HDM scenarios.
For the SM-like Higgs boson mass, we will use the current 3σ range as allowed by the latest CMS and ATLAS Higgs mass measurements [4,43]: (4.1) For the Higgs couplings to the SM vector bosons and fermions, we use the constraints in the (tan β, β − α) plane derived from a recent global fit for the Type-II 2HDM [21,22]. 6 For a given set of SO(5) boundary conditions µ X , tan β(µ X ), λ(µ X ) , we thus require that the RG-evolved 2HDM parameters at the weak scale must satisfy the above constraints on the lightest CP-even Higgs boson sector. This requirement of alignment with the SM Higgs sector puts stringent constraints on the MS-2HDM parameter space, as shown in Figure 2.
Here the solid, dashed, and dotted blue shaded regions respectively show the 1σ, 2σ and 3σ excluded regions due to misalignment of the mixing angle α from its allowed range derived from the global fit. The shaded red region is theoretically inaccessible, as there is no viable solution to the RGEs in this area. We ensure that the remaining allowed (white) region satisfies the necessary theoretical constraints, i.e. positivity and vacuum stability of the Higgs potential, and perturbativity of the Higgs self-couplings [6]. From Figure 2, we find that there exists an upper limit of µ X 10 8 GeV on the SO(5)-breaking scale of the 2HDM potential, beyond which an ultraviolet completion of the theory must be invoked. Moreover, for 10 5 GeV µ X 10 8 GeV, only a narrow range of tan β values are allowed. For the allowed parameter space of our MS-2HDM as shown in Figure 2, we obtain concrete predictions for the remaining Higgs spectrum. In particular, the alignment condition imposes a lower bound on the soft breaking parameter Re(m 2 12 ), and hence, on the heavy Higgs spectrum. We compare this limit with the existing experimental limits on the heavy Higgs sector of the 2HDM [38], and find that the alignment limits obtained here are more stringent in a wide range of the parameter space. The most severe experimental constraint comes from the charged Higgs sector, which give significant contributions to various flavour observables, e.g. B → X s γ [44,45]. For this, we use the global fit results for the Type-II 2HDM from [21], which includes limits derived from electroweak precision data, as well as flavour constraints from ∆m Bs and B → X s γ relevant for the low tan β region. The comparison of the existing limit on the charged Higgs-boson mass as a function of tan β with our predicted limits from the alignment condition for a typical value of the boundary scale µ X = 3 × 10 4 GeV is shown in Figure 3. It is clear that the alignment limits are stronger than the indirect limits, except in the very small and very large tan β regimes. For tan β 1 region, the indirect limit obtained from the Z → bb precision observable becomes the strictest [21,46]. Similarly, for the large tan β 30 case, the alignment limit can be easily obtained without requiring a large soft-breaking parameter m 2 12 , and therefore, the lower limit on the charged Higgs mass derived from the misalignment condition becomes somewhat weaker in this regime.
From Figure 2, it should be noted that for µ X 10 5 GeV, we cannot go to the decoupling limit M h ± v for tan β 1, while keeping the lightest CP-even Higgs boson within the experimentally allowed range (4.1) and maintaining vacuum stability up to the scale µ X . Therefore, µ X 10 5 GeV also leads to an upper bound on the charged Higgsboson mass M h ± . This is illustrated in Figure 4 for µ X = 10 5 GeV. Here the green shaded regions show the 1σ (dotted), 2σ (dashed) and 3σ (solid) allowed regions, whereas the corresponding red shaded regions are the experimentally exclusion regions at 1σ (dotted), 2σ (dashed) and 3σ (solid). For µ X 10 5 GeV, the decoupling limit can always be achieved for any value of tan β, and hence in this case, there exists only a lower limit on M h ± , as shown by the blue shaded (exclusion) regions in Figure 3.
Similar alignment constraints are obtained for the heavy neutral pseudo-Goldstone bosons h and a, which are predicted to be quasi-degenerate with the charged Higgs boson h ± in the MS-2HDM. Since the current experimental lower limits on the heavy neutral Higgs sector [38] are much weaker than that for the charged Higgs, the alignment constraints are much more stringent in this case and have important consequences for their collider searches, as discussed in the following section.

Collider Signals
In the alignment limit, the couplings of the lightest CP-even Higgs boson are exactly similar to the SM Higgs couplings, while the heavy CP-even Higgs boson only couples to fermions (see Appendix A). Therefore, two of the relevant Higgs production mechanisms at the LHC, namely, the vector boson fusion and Higgstrahlung processes are suppressed for this gaugephobic heavy neutral Higgs boson. As a consequence, the only relevant production channels to probe the neutral Higgs sector of the MS-2HDM are the gluongluon fusion and tth (bbh) associated production mechanisms at low (high) tan β. For the charged Higgs sector of the MS-2HDM, the dominant production mode is the associated production process: gg →tbh + , irrespective of tan β.

Branching Fractions
For our collider analysis, we calculate all the branching ratios of the heavy Higgs sector in the MS-2HDM as a function of their masses using the public C++ code 2HDMC [47]. The results for tan β = 2 and with SO(5)-symmetric boundary conditions at µ X = 3 × 10 4 GeV are shown in Figure 5 for illustration. It is clear that for the heavy neutral Higgs bosons, the tt decay mode is the dominant one over most of the MS-2HDM parameter space. However, this is true only for low tan β 5, since as we go to higher tan β values, the bb decay mode becomes dominant, with a sub-dominant contribution from τ + τ − , whereas the tt mode gets Yukawa suppressed. This is illustrated in Figure 6, where we compare BR(h → tt) and BR(h → bb) for three representative values of tan β = 2 (solid), 5 (dashed) and 10 (dotted). For the charged Higgs boson h +(−) , the tb(tb) mode is the dominant one over the entire parameter space, as shown in Figure 5 for tan β = 2, and this is true even for larger tan β.
Since the heavy CP-even Higgs boson in the MS-2HDM is gaugephobic, most of the existing collider limits [38] derived using the decay modes h → W W, ZZ do not apply. The only existing searches relevant to the MS-2HDM scenario are those based on gg → h → τ + τ − and gg → bbh → bbτ + τ − [48]. However, due to the relatively small branching ratio of h → τ + τ − , the model-independent upper limits derived in [48] are easily satisfied for the heavy Higgs spectrum presented here. To the best of our knowledge, there exist no direct heavy Higgs searches in the most interesting channels for the MS-2HDM scenario, namely, those involving tt and/or bb final states.

tttt Signal
Here we propose a new search channel for the heavy neutral Higgs boson in the MS-2HDM via the tttt final state: gg → tth → tttt . Such four top final states have been proposed before in the context of other exotic searches at the LHC, e.g. composite top [49][50][51], low-scale extra-dimensions [52,53] and SUSY with light stops and gluinos [54]. However, their relevance for heavy Higgs searches have not been explored so far. We note here that the existing 95% CL experimental upper limit on the four top production cross section is 59 fb from ATLAS [55] and 63 fb from CMS [56], whereas the SM prediction for the inclusive cross section of the process pp → tttt + X is 4-5 times smaller [57].
To get a rough estimate of the signal to background ratio for our new four-top signal, we perform a parton-level simulation of the signal and background events at leading order  in QCD using MadGraph5 [58] with CTEQ6L parton distribution functions [59]. For the inclusive SM cross section for the four-top final state, we obtain 11.85 fb, whereas our proposed four-top signal cross sections are found to be comparable or smaller depending on M h and tan β. However, since we expect one of the tt pairs coming from an on-shell h decay to have an invariant mass around M h , we can impose a cut on the invariant mass of one of the tt pairs to significantly reduce the SM background. Even with a very conservative cut of M h − 50 GeV ≤ M tt ≤ M h + 50 GeV, we can already reduce the SM background by a factor of few, which should make it feasible to search the heavy Higgs sector in MS-2HDM using this four-top channel. Our simulation results for three representative values of tan β are shown in Figure 7. All the predicted signal+background cross sections shown here are found to be consistent with the current experimental upper bound [55,56]. The predicted number of events are shown for an integrated luminosity of 300 fb −1 at √ s = 14 TeV LHC. From this preliminary analysis, we find that the heavy Higgs sector in the MS-2HDM can be effectively probed for tan β 5 using this four-top signal. A detailed detector-level analysis of this new signal, including realistic top reconstruction efficiencies, will be presented in a separate dedicated study.
The above analysis is also applicable for the CP-odd Higgs boson a, which has similar production cross section and tt branching fraction as the CP-even Higgs h. However, the tth(a) production cross section as well as the h(a) → tt branching ratio decreases with increasing tan β. This is due to the fact that the htt coupling in the alignment limit is cos α/ sin β ∼ cot β, which is same as the att coupling [cf. (A. 13)]. Thus the high tan β region of the MS-2HDM cannot be searched via the tttt channel proposed above, and one needs to consider the channels involving down-sector Yukawa couplings, e.g. bbbb and bbτ + τ − , which are very challenging in the LHC environment [60]. For instance, the SM bbbb cross section at √ s = 14 TeV LHC is about 140 pb at NLO [61], whereas the pp → bbh → bbbb signal cross section for M h = 300 GeV and tan β = 10 is only about 0.3 pb at NLO, as estimated using the public FORTRAN code SusHi [62]. In practice, one would require a sophisticated jet substructure technique [63,64] to disentangle such a tiny signal from the huge QCD background.
It is worth commenting on few other possible search modes for the heavy Higgs boson. One could consider the process pp → h → tt, but its cross section is found to be much smaller than the SM tt background, even after imposing a M tt cut. Another interesting possibility is the Higgs pair production process pp → h → HH (see e.g. [65]). However, as shown in (A.12), the h → HH decay mode should also vanish in the exact alignment limit α → β, just like the h → V V decay modes. Therefore, this channel is mostly useful only below the tt threshold (cf. Figure 5). However, the lower limits on the heavy Higgs sector, as derived in Section 4 (e.g. Figures 3 and 4) strongly suggest a mass spectrum above the tt threshold, where the h → HH branching fraction drops orders of magnitude below that of h → tt (bb) at low (high) tan β.

ttbb Signal
In the charged Higgs sector, the h +t b coupling in the Type-II 2HDM is given by (A.14), which implies that the branching fraction of h + → tb is almost 100% (cf. Figure 5), inde-pendent of tan β. This leads to mostly ttbb final state at the LHC via gg →tbh + →tbtb .
(5.2) However, the experimental observation of this channel is again very challenging due to large QCD backgrounds and the non-trivial event topology, involving at least four b-jets [60]. Also the higher order QCD corrections to this process are known to be large [66]. Nevertheless, we make a rough comparison of the signal to background for the ttbb mode, which in conjunction with the tttt mode for the neutral Higgs sector, might be able to probe the simplest extended Higgs-doublet sector. The inclusive SM cross section for pp → ttbb is 18 pb at NLO, with roughly 30% uncertainty. In order to reduce this huge QCD background, we use some basic selection cuts on the transverse momentum, rapidity and invariant mass of the b-jets, following the analysis of [66]: Using these nominal cuts, we could reduce the background to about 2.6 pb, without losing much of the signal. Now similar to the four-top case discussed above, we require the invariant mass of one of the tb-pairs to be centered around M h ± . Since the b-tagging efficiency is usually larger than the top-tagging efficiency, we can use a narrower mass window M h ± − 5 GeV ≤ M tb ≤ M h ± + 5 GeV, as compared to the M tt -cut used in Figure 7. After imposing these cuts, our final results for the signal and background are given in Figure 8. From (A.14), we see that the production cross section gg →tbh + will decrease rapidly with increasing tan β due to Yukawa suppression [cf. (A.14)], even though the branching fraction of h + →tb remains close to 100%. Therefore, the ttbb channel is effective for charged Higgs searches in the MS-2HDM only in a narrow range of low tan β, as shown in Figure 8. We note that in the large tan β regime, the dominant signal for the neutral heavy Higgs sector is also ttbb, which has an interference effect with the charged-Higgs signal presented above, since the heavy Higgs spectrum is quasi-degenerate in the MS-2HDM, especially for Re(m 2 12 ) v 2 . However, due to the large QCD background, the prospects of heavy neutral Higgs bosons searches through this channel: gg → tth → ttbb still remain challenging. In particular, the selection of h → bb candidates and the reconstruction of the M bb mass peak is strongly contaminated by combinatorial background at the LHC, because one of the selected b-jets could come from a t-decay. Thus, we conclude that the tttt channel provides the most promising collider signal to probe the heavy Higgs sector in the MS-2HDM, at low values of tan β.

Conclusions
We have analyzed the scalar potential of the Maximally Symmetric Two Higgs Doublet Model based on the SO(5) group and showed that it naturally leads to the so-called SM alignment limit, independently of the heavy Higgs spectrum and the value of tan β. Small deviations from this alignment limit are expected to be induced by RGE effects due to the hypercharge gauge coupling g and third generation Yukawa couplings, which also break the custodial symmetry of the theory. In addition, a non-zero soft SO(5)-breaking mass parameter is required to yield a viable Higgs spectrum consistent with the existing experimental constraints. Employing the current Higgs signal strength data from the LHC, which disfavour large deviations from the alignment limit, we derive important constraints on the 2HDM parameter space. In particular, we predict lower limits on the heavy Higgs spectrum, which prevail the present limits in a wide range of parameter space. Depending on the scale where the maximal symmetry could be realized in nature, we also obtain an upper limit on the heavy Higgs masses in certain cases, which could be completely probed during the run-II phase of the LHC. Finally, we propose a new collider signal with four top quarks in the final state, which can become a valuable observational tool to directly probe the heavy Higgs sector of the 2HDM in the SM alignment limit.

Acknowledgments
A.P. thanks Celso Nishi for a clarifying remark that led us to include the fourth footnote in this article. P.S.B.D. thanks Otto Eberhardt and Martin Wiebusch for helpful discussions regarding their global fit results in [21,22]. The work of P.S.B.D. and A.P. is supported by the Lancaster-Manchester-Sheffield Consortium for Fundamental Physics under STFC grant ST/J000418/1.

A Higgs Spectrum and Couplings in a General 2HDM
Here we will restrict our discussion to 2HDM potentials realizing CP-conserving vacua. In this case, the minimization of a CP-conserving 2HDM potential (1.2) yields the following real non-negative vacuum expectation values (VEVs): where v = v 2 1 + v 2 2 = 246.2 GeV is the SM electroweak VEV and for later convenience, we define tan β ≡ v 2 /v 1 . The two scalar doublets can be expanded in terms of eight real scalar fields as follows: with j = 1, 2. After spontaneous symmetry breaking, there are three Goldstone modes (G ± , G 0 ), which become the longitudinal components of the SM W ± and Z bosons. Thus, there are five remaining physical scalar mass eigenstates: two CP-even (h, H), one CPodd (a) and two charged (h ± ) scalars. The mixing in the CP-odd and charged sectors is governed by the angle β defined above: where c β ≡ cos β, s β ≡ sin β. On the other hand, in the CP-even sector, we have a new mixing angle α: The corresponding physical mass eigenvalues are given by [40,41] where we have defined tan 2α = 2C/(A − B), and with λ 34 = λ 3 + λ 4 . The SM Higgs field is given by From (A.7), the couplings of h and H to the gauge bosons (V = W ± , Z) with respect to the SM Higgs couplings g H SM V V are given by Similarly, unitarity constraints uniquely fix the other Higgs-Higgs-V couplings [5]: (where θ w is the weak mixing angle) in order to satisfy the sum rules [67] For our subsequent discussion, we also write down the h-H-H coupling [23]: Note that the coupling g hHH is proportional to s β−α and so vanishes identically in the alignment limit α → β.
To obtain a phenomenologically acceptable theory, we need to forbid Higgs interactions with tree-level FCNCs. This can be accomplished minimally by imposing appropriate discrete Z 2 symmetries, which will explicitly break in general the custodial symmetries of the theory. By convention, we may take u R to couple to Φ 2 , i.e. h u 1 = 0, and then Φ 1 (Φ 2 ) to couple to d R , with h d 2 = 0 (h d 1 = 0), in a Type-II (Type-I) realization of the 2HDM. As our interest is in the Type-II 2HDM, we only list the Yukawa couplings of the neutral scalars with respect to those of H SM for this class of models [5]: g htt = cos α/ sin β , g hbb = − sin α/ cos β , g Htt = sin α/ sin β , g Hbb = cos α/ cos β , Finally, we also write down the coupling of the charged scalar to the third-generation quarks [5]: (A.14)