Relaxed constraints on the heavy scalar masses in 2HDM

In the wake of new scalar searches at LHC in various channels, it is interesting to investigate the sacrosanctity of the constraints on the masses and couplings of the heavier scalars in a two-Higgs-doublet model (2HDM). We consider the effects of new physics beyond a 2HDM encoded in terms of bosonic dim-6 operators. Although these constraints are mostly immune to such new physics, we demonstrate for a specific class of bosonic operators, the constraints on the masses of the exotic scalars from cascade decays can get substantially relaxed. We present such effects for both degenerate and hierarchical mass spectra of the heavier scalars in 2HDM. Some decay channels of the new scalars vanish at the alignment limit in the tree-level 2HDM. But the inclusion of dim-6 terms can lead to significant cross-sections for such processes. It is also pointed out that observation of such processes can no longer rule out the alignment limit if such dim-6 operators are present.


I. INTRODUCTION
Even after the discovery of a Higgs boson [1,2] whose characteristics resemble that of the standard model (SM) Higgs, the dynamics of electroweak symmetry breaking and the structure of the scalar sector remains an open question. The non-cancellation of quadratic divergence of Higgs mass under the framework of SM has motivated plethora of beyondstandard model (BSM) theories for decades. The two-Higgs-doublet model (2HDM) is an archetype of an extended scalar sector, theoretically well-motivated from the viewpoint of supersymmetry, composite Higgs models, etc. For example, in supersymmetric models [3] the motivation behind a second Higgs doublet is twofold: firstly to cancel chiral anomalies created by the superpartners of such scalars, and secondly from the requirement of the superpotential to be holomorphic. 2HDMs arising in the framework of composite Higgs [4], Little Higgs [5], Twin Higgs [6] have also been studied in the literature. Even keeping the hierarchy problem aside, it is often deployed to explain issues of electroweak baryogenesis [7], flavour anomalies [8], neutrino mass [9], dark matter [10], etc.
In light of measurements of the signal strengths of the observed Higgs, any model with a scalar sector beyond the SM must contain a CP-even neutral scalar whose couplings are aligned to that of the SM Higgs boson. Such an alignment can be realised when the new scalars which mix with the SM-like Higgs, are decoupled from the mass spectrum of SM a la Applequist-Carrazone [11,12]. The 'alignment without decoupling' scenario becomes viable only for models with additional scalar doublets [12][13][14][15][16][17][18]. In such cases, the scalars can have masses below a TeV, i.e., well within the reach of LHC. Thus, along with the signal strengths of the SM-like Higgs, the direct bounds on the masses of exotic scalars also play a pivotal role in constraining the parameter space of 2HDM. Such bounds also depend on the specifications of the Yukawa sector of the models. Non-observation of such new scalars rule out a significant region of parameter space in the 'alignment without decoupling' scenarios.
Also, some decay channels involving exotic scalars remain absent at the alignment limit in a CP-conserving 2HDM [19]. If the LHC discovers any new scalar state in one of these channels, the interpretation involving a CP-conserving 2HDM would readily imply a deviation from the alignment limit.
If new physics beyond 2HDM exists as a decoupled sector from the mass scale of 2HDM, the effects of such new physics can be encoded in the higher-dimensional operators in an effective theory where the fields of 2HDM constitute the low-energy spectrum [20][21][22]. Such an effective theory is dubbed as two-Higgs-doublet model effective field theory (2HDMEFT) in the literature. Several aspects of such effective theories for various extended scalar sectors have been addressed in the literature [23][24][25][26][27]. A complete basis of the 6-dim operators in 2HDMEFT has been introduced only recently [22]. It has also been shown that such 6-dim operators are capable of masking the true alignment limit in a 2HDM, by modifying various decay channels of the SM-like Higgs boson [28]. In the present paper, we have investigated the role of such 6-dim terms while extracting the LHC constraints on the masses of the new scalars in a 2HDM. We consider different mass spectra of these new scalars allowed from the theoretical constraints and measurements of the oblique parameters. The constraints ensuing from different searches for the heavy scalars at the LHC and possible deviations in the presence of 6-dim terms have been illustrated.
In Sec. II we briefly review the theoretical framework of a general 2HDM. In Sec. III we discuss the theoretical as well as phenomenological constraints on the parameter space of a 2HDM relevant to this work. In Sec. IV we introduce the 6-dim terms that have been considered in this work, along with the modified couplings of the scalars. In Sec. V we present the benchmark scenarios to illustrate the effect of such 6-dim terms on the parameter space of the 2HDM and eventually conclude in Sec. VI.

II. 2HDM: A REVIEW
The two scalar doublets are defined as: with I = 1, 2. Here φ ± I , ρ I , η I , and v I denote the charged, neutral CP-even and neutral CPodd degrees of freedom (d.o.f.) and the vacuum expectation value (vev) of the I-th doublet respectively.
Before spontaneous symmetry breaking (SSB), the tree-level 2HDM Lagrangian, augmented with 6-dim operators, assumes the form where, Here, c i is the Wilson coefficient of the 6-dim operator O i and f is the scale of new physics beyond the tree-level 2HDM. The terms proportional to λ 6,7 are called 'hard-Z 2 violating' because they lead to a quadratically divergent amplitude for ϕ 1 ↔ ϕ 2 transition [29] and they also lead to CP-violation in the scalar sector [30]. But it is possible to realise the CP-conserving limit with non-zero values of λ 6,7 as well [19]. In this paper we contain our discussion to the CP-conserving 2HDM, and we take λ 6,7 = 0. The electroweak symmetry is broken by the vacuum expectation values (vev), namely v 1 and v 2 corresponding to the the two doublets ϕ 1,2 respectively. This leads to the mixing of similar types of degrees of freedom pertaining to ϕ 1,2 . In the CP-conserving case, the mass matrices of the neutral CP-even and odd scalars and the charged scalars are diagonalised by the following field rotations: Here, h, H are the neutral CP-even physical d.o.f., whereas A and H ± are the neutral CP-odd and charged d.o.f respectively. As it can be seen from eq. (2.4), β is the mixing angle of the charged and CP-odd sectors and it is given by β = tan −1 (v 2 /v 1 ). α is the mixing angle of the CP-even neutral scalars and can be expressed as with M 2 ρ being the mass-squared matrix in the neutral CP-even sector. In this paper, we assume h to be the SM-like Higgs with a mass of m h ∼ 125.09 GeV and m H > m h . The case of m H < m h has been explored in the literature as well [16]. It was shown that the tree-level Higgs-mediated flavour-changing neutral currents (FCNC) appear in models where more than one scalar doublet give mass to the same kind of SM fermions [31,32]. Such a situation can be avoided under the framework of various discrete symmetries, for example, a Z 2 symmetry [31,32]. There are four possible ways in which such a Z 2 charge assignment of the SM fermions can be done, namely the type- Type-II scenario is also dubbed as the MSSM-like case due to similarity in the Yukawa sectors. Type-III and -IV are sometimes also referred to as flipped and lepton-specific scenarios respectively.
Due to the rotation in the scalar sector following eq. (2.4), the couplings of the SM gauge bosons and fermions to the SM-like Higgs boson are rescaled compared to the corresponding SM values. After SSB the Yukawa sector of the 2HDM can be written as, with, κ f = √ 2M f /v for f = U, D, L and, U, D, L and ν represent the up-type, down-type quarks, charged leptons and neutrinos in their mass bases respectively. The generation indices of the fermionic fields have been suppressed in eq. (2.7). As mentioned earlier, the measurement of the signal strengths of the as well as the production of h in both gg and bb-fusion substantially increase. It is mainly due to the measurement of the processes like gg → γγ, bb, V V * and V h → bb, that the parameter space of a type-II 2HDM in the cos(β − α) − tan β plane is quite strongly constrained. The impact of the measurement of the Higgs signal strengths in each individual search channels on the cos(β − α) − tan β plane has been discussed in ref. [38]. It should be mentioned that the coupling multipliers of the SM-like Higgs also becomes close to unity when sin(β +α) = 1 i.e., at the so-called 'wrong-sign Yukawa' limit [39] for type-II, -III, and -IV 2HDM. Though, with better measurement of the processes like V h → bb, h → Υγ [40,41] the fate of the wrong-sign Yukawa region will be decided in near future.
It is also clear from eq. (2.7) that, though the coupling multipliers of the SM-like Higgs become unity at the alignment limit, the couplings of the exotic Higgses with SM fermions can be non-zero. The HV V coupling becomes identically zero at the alignment limit, protecting an explanation for the matter-antimatter asymmetry, with A → ZH being its smoking gun signature at LHC [43,44]. In general, the importance of Higgs cascade decays as the possible probes of an extended scalar sector have been discussed in the literature [45][46][47][48][49][50] and A → ZH decay is dubbed as a 'golden channel' in this context [51].

III. CONSTRAINTS ON 2HDM PARAMETER SPACE
We work with the 2HDM parameters in the physical basis which consists of {m h , m H , Along with m h = 125.09 GeV, v = 246 GeV and λ 6,7 = 0 we are left with six free parameters. The theoretical constraints are discussed below.
• Vacuum stability The stability of the EW vacuum in a 2HDM at the tree-level is ensured if [12], It can be shown that at the alignment limit, the first two conditions are satisfied if m 2 12 = m 2 H s β c β . Along with that, the last two criteria are satisfied if respectively. This means for degenerate masses of the new scalars the last two criteria are automatically satisfied if the first two are satisfied. For hierarchical mass spectrum, the mass of the exotic scalars cannot be arbitrarily different.
• Perturbativity The perturbativity of the quartic couplings is satisfied if λ i 4π. At the alignment limit this implies for t β 1, • Unitarity Tree-level unitarity of the S-matrix requires the eigenvalues of the 2 → 2 scattering matrix to be less than 8π. At the alignment limit for m 2 12 ∼ m 2 H s β c β , this implies that the differences between the masses of the new scalars have to be v.
• Oblique parameters The new scalars in 2HDM contributes to the oblique parameters through their couplings to the massive gauge bosons [52][53][54]. At the alignment limit such contributions to the T -parameter assume the form, As eq. (3.2) suggests, this anomalous contribution to the T -parameter vanishes at the limit for type-I (II) 2HDM [55]. Also the searches for Z → AH → ττ ττ lead to the constraint m H + m A 208 GeV [56]. The charged scalar mediates flavour-violating processes such as The measurement of the width of B d → X s γ leads to the most stringent constraint on the charged scalar mass for type-II 2HDM, m H + 480 GeV [57,58]. For type-I 2HDM, the constraint from meson decays is comparatively less stringent, m H + 160 GeV [57]. Based on the similarity in couplings of the scalars to the quarks, the constraints on charged scalar mass for type-I and -II 2HDM can also be used for type-IV and -III cases respectively. Though we do not consider this as a hard bound for our purpose as it can be ameliorated in several extensions of 2HDM [59].

IV. COUPLINGS OF THE HEAVIER SCALARS IN 2HDMEFT
We contain our discussion only to the bosonic operators of 2HDMEFT for simplicity. Phenomenology of the fermionic dim-6 terms will be reported elsewhere. As discussed in ref. [28], In presence of such operators, the non-diagonal kinetic terms arise after SSB [22]. In order to get rid of such terms, one needs to rescale the neutral CP-even d.o.f. i.e., ρ 1 and ρ 2 . This implies that the physical neutral CP-even scalars in presence of these operators are rescaled compared to the tree-level 2HDM.
x 1 , x 2 and y can be written in terms of the Wilson coefficients of the operators appearing in eq. (4.1) and the scale of new physics beyond 2HDM. The analytical forms of x 1 , x 2 and y can be found in Appendix A. Eq. (4.2) dictates that any coupling involving at least one h or H field are modified compared to 2HDM at the tree-level. For example, Many of the sum rules involving various gauge couplings, which hold in 2HDM at the treelevel, are no longer valid in the presence of 6-dim operators [4]. These sum rules can play an important role in deciphering new physics beyond 2HDM. For instance, in 2HDM at the treelevel, the sum rule κ 2 hV V +κ 2 HV V = 1 holds true, but in presence of the dim-6 terms mentioned in eq. (4.1), . If another CP-even neutral scalar, H is discovered after h(125), the measurement of its decay width and Br(H → W W ) will facilitate the verification of such a sum rule. A deviation from κ 2 hV V + κ 2 HV V = 1 will point to a departure from the CP-conserving 2HDM. If κ 2 hV V + κ 2 HV V < 1, then it may indicate towards CP-violating 2HDM or CP-conserving NHDM (N > 2). Even the dim-6 operators in 2HDMEFT can lead to κ 2 hV V + κ 2 HV V < 1. On contrary, neither CP-violating 2HDM nor NHDM can lead to κ 2 hV V +κ 2 HV V > 1. Though, such a scenario can be interpreted in terms of the dim-6 terms of 2HDMEFT. At the CP-conserving limit with λ 6,7 = 0 the similar argument for tree-level 2HDM is valid in this context, whereas in general λ 6,7 = 0 will follow the argument for CP-violating 2HDM.

V. BENCHMARK SCENARIOS
Following the discussions in Section III in the context of oblique parameters, for the hierarchical mass spectrum, we consider either m A = m H ± or m H = m H ± . The limit m A = m H is highly constrained from the measurement of S, T parameters and the decays of H and A into each other are kinematically forbidden.
So the mass spectra under scrutiny for the hierarchical case are [49]: suggests, Br(A → ττ ) attains the smallest value for type-III case among all the Yukawa types. In both type-I and -III the Aττ coupling is proportional to cot β as opposed to tan β in type-II and -IV. Br(A → ττ ) is even smaller in type-III 2HDM compared to type-I 2HDM because Br(A → bb) becomes larger in the latter case. On inclusion of the 6-dim operators, the region excluded from gg → A → ττ is not significantly altered for type-I, -II and -IV cases.
In the context of Yukawa types, a similar pattern in the excluded region can be seen for mass spectra C2 as depicted in fig. 2. It can also be seen that the excluded region from A → ZH is small in the case C2 compared to C1. For mass spectrum C2, the decay channel A → W ± H ∓ becomes kinematically viable and has branching ratio almost similar to that of  The dotted blue region is disfavoured from the theoretical constraints, viz. stability, perturbativity and unitarity.
It is to be noted that, according to eq. (2.7), for a particular value of | cos(β − α)|  fig. 9.  So far we have considered the phenomenology of only the neutral scalars. Now we comment on a few effects of the 6-dim terms in 2HDMEFT on the decay modes of the charged scalars. We calculate the production cross-section of the charged scalar following ref. [66] as it was recommended in ref. [67]. For m H ± > m t , the key production channel of H ± is through the process pp → H ± t. The H ± tb coupling multiplier depends on the value of tan β and the top and bottom quark masses. Following the tan β-dependence of σ(H ± t) we rescale σ(H ± t) at tan β = 30 with the appropriate numerical factor to obtain the cross-sections at significantly [48]. The cascade decay channels with other Higgses as intermediate states can be interesting as well [48,69]. In fig. 7 we have shown the contours of σ(H ± t)Br(H ± → hW ± ) in BP1 of 2HDMEFT for the 2HDM mass spectrum corresponding to cases C1 and C4 as In fig. 10  The couplings of h(125) will be even more precisely measured in the future experiments.
For instance, the coupling multipliers κ hγγ and κ hW W are to be measured with an accuracy of ∼ 5 − 7% and ∼ 4 − 6% respectively at HL-LHC with luminosity ∼ 3 ab −1 [70]. It can push a 2HDM, especially the ones with type-II, -III and -IV Yukawa couplings, further close to cos(β −α) = 0. However, the contribution of dim-6 terms to the signal strengths of h(125) do not decrease with the same scale. As it was also discussed in ref. [28], even at the exact limit cos(β − α) = 0, the effects of the dim-6 terms in masking the true alignment limit can be rather significant.

VI. SUMMARY AND DISCUSSIONS
In the context of the searches for new scalars at LHC, it is an interesting possibility that the exotic scalars in a 2HDM exist below the TeV scale, pertaining to the so-called A becomes larger in 2HDMEFT compared to 2HDM at the tree-level. It is also seen, as it was discussed above, a certain mass range for m H (= m A ) is ruled out even for cos(β − α) = 0 from processes like H → W W , A → Zh, which usually vanish in 2HDM at tree-level. We have also shown in fig. 7 the change in σ×Br for various decay channels of the charged scalar in 2HDMEFT compared to 2HDM at the tree-level at LHC with √ s = 14 TeV.
The key reason for the change in the constraints on 2HDM parameter space upon including dim-6 operators of type ϕ 4 D 2 lies in the redefinition of the CP-even Higgs fields, h, and H.
This way the coupling multipliers involving the CP-even scalars are rescaled compared to 2HDM at the tree-level and lead to a change in the branching ratios of all the processes which involve h and H. It leads to the departure of the 'true' alignment limit from its tree-level 2HDM counterpart, i.e., cos(β − α) = 0. As the projected accuracy of the h(125) coupling measurement at a future version of LHC, such as HL-LHC is at the level 5 − 6%, 2HDMs might get further pushed to the alignment limit. Thus, in the presence of dim-6 operators, even if the couplings of h(125) turn out to be completely aligned with the SM Higgs, the heavier scalars in 2HDM with masses TeV still do not decouple from the SM sector, i.e., their discovery might still be viable. As mentioned earlier, some cascade-type decay channels of the heavier scalars vanish at the alignment limit in the tree-level 2HDM.
It implies that the discovery of a new scalar in such a channel would perhaps rule out the alignment limit in a CP-conserving 2HDM. But if dim-6 operators are present, even if a new scalar is discovered in such channels, it will no longer rule out the alignment limit.
In case of the discovery of the new Higgs(es), the verification of the sum rules involving their coupling multipliers can provide useful information about the nature of the extended Higgs sector. In 2HDMEFT, the redefinition of the CP-even Higgs fields due to ϕ 4 D 2 operators also imply that the sum rules involving these scalars are modified in a certain way. We have discussed how the measurement of sum rules can help distinguishing between various options beyond a CP-even 2HDM.
If new scalars are discovered at the LHC in near future, the correlation of their signal strengths in different channels will be important to determine the exact nature of the underlying scalar sector. In this context, 2HDMEFT can be an efficient framework in quantifying the departure from tree-level 2HDM in various channels, providing an opportunity to narrow down the possible UV-complete scenarios.