Alignment Limit in 2HDM: Robustness put to test

In a two-Higgs-doublet model (2HDM), at the vicinity of the alignment limit, the extra contributions to the couplings of the SM-like Higgs with other particles can be subdominant to the same coming from the six dimensional operators. In this context, we revisit the alignment limit itself. It is investigated to what extent these operators can mask the actual alignment in a 2HDM. The bosonic operators which rescale the Higgs kinetic terms can lead to substantial change in the parameter space of the model. We find that some other bosonic operators, which are severely constrained from the electroweak precision tests, can also modify the parameter space of 2HDM due to their anomalous momentum structures. A particular kind of Little Higgs model is explored as an example of 2HDM effective field theory in connection with 2HDM alignment. Choosing a suitable benchmark point in a Type-II 2HDM, we highlight the possibility that the exact alignment limit is ruled out at 95% CL in presence of such operators.


I. INTRODUCTION
After discovery of the Higgs boson [1,2], the last missing piece of Standard Model (SM) particle spectrum, the key challenge lies in searching physics beyond the SM. As far as the scalar sector is concerned, the two-Higgs-doublet model is the most studied extension of the SM and of immense importance considering the ongoing searches of new scalar particles in LHC. The measurements of the signal strengths of the SM-like Higgs boson are in quite good agreement with the SM predictions. As a result, the 2HDM is pushed close to the so-called 'alignment limit' [3][4][5][6]. The existence of new physics beyond 2HDM is also possible, making it an interesting question to ask whether such new physics is capable of causing an apparent departure from 'true' alignment in 2HDM.
An exact alignment can be achieved by demanding the Applequist-Carrazone decoupling of the new scalars in 2HDM [3]. Alignment without decoupling [3][4][5][6][7][8][9][10][11][12] is a more interesting scenario because it allows for the existence of exotic scalar particles even within the reach of LHC. Though, in terms of 2HDM parameters this scenario is quite fine-tuned. Keeping this in mind, to encode the effects of new physics beyond 2HDM, we adhere to the language of 2HDM effective field theory (2HDMEFT), which assumes both the Higgs doublets to be the low-energy fields, to investigate possible deviation from the alignment limit.
The complete basis of operators up to dimension six in 2HDMEFT has been presented only recently [13]. An earlier attempt for the same was made in ref. [14]. The basis of ref. [13] is motivated by the SILH basis [15] of SMEFT, whereas ref. [14] follows the Warsaw basis [16].
There are 126 six-dimensional operators in 2HDMEFT under the assumption of CP -, Band L-conservations compared to 53 in Standard Model effective field theory (SMEFT) [16]. 38 of the six-dimensional operators in 2HDMEFT are Z 2 -violating.
One of the key observations of ref. [13] was that the contribution of the 6-dim operators to decay width of the SM-like Higgs boson can supersede the extra contribution due to 2HDM at tree-level compared to SM. Such effects can modify the signal strengths of the SM-like Higgs boson under the framework of 2HDMEFT. It is worth exploring whether such operators are capable of masking the 'true alignment' of the 2HDM. In other words, the allowed parameter space of the model can be significantly altered due to the presence of the 6-dim operators when confronted with the measured signal strengths involving the SM-like Higgs boson. In this paper, we concentrate on the effects of the bosonic operators of 2HDMEFT and the confusion they can lead to in determining the deviations from the alignment limit.
In Section II we introduce the 2HDM Lagrangian and set up the theoretical ground by discussing the way the 6-dim operators affect the couplings of the CP-even neutral scalars.
The existing bounds on the 2HDM parameter space and the choice of parameters relevant for the present work have been discussed in Section III. In Section IV we illustrate the effects of the 6-dim operators on the alignment limit of 2HDM. We summarise and eventually conclude in Section V.

II. RELEVANT BOSONIC OPERATORS
The theoretical motivation of the 2HDM is manyfold. For example, a second Higgs doublet appears in the supersymmetric extensions of SM. Even without a supersymmetric origin, a general 2HDM has been deployed to address issues pertaining to electroweak baryogenesis [17,18], certain flavour anomalies [19,20], certain DM models etc. Moreover, many BSM models predict the existence of other particles along with a second Higgs doublet, for example, supersymmetric models [21][22][23], composite 2HDMs [24], Little Higgs models [25][26][27], composite Inert doublet models [28,29] etc. Such models can be realised as examples of 2HDMEFT [13,14,30] where all the degrees of freedom except the two Higgs doublets are decoupled from the mass spectrum.
We define the two scalar doublets, following the notation of ref. [13] as, (2.1) Before spontaneous symmetry breaking (SSB), the tree-level 2HDM Lagrangian augmented with 6-dim operators assumes the form, where, of φ ± 1,2 and η 1,2 respectively. The Goldstone modes which get absorbed as the longitudinal degrees of freedom of the W ± and Z bosons are orthogonal to the mass eigenstates H ± and A respectively. The rotation required to transform the unphysical fields φ ± 1,2 and η 1,2 into mass eigenstates is denoted by the angle β = tan −1 (v 2 /v 1 ). Similarly, the rotation required to mass-diagonalise the neutral scalars is parametrised by the angle α [4]: where M 2 ijρ is the ij-th element of the mass-squared matrix for the fields ρ 1 and ρ 2 . It is due to these two rotations in the scalar sector, the couplings of the SM-like Higgs boson, h to the massive vector bosons are scaled compared to the SM scenario. In fact all the couplings of SM-like Higgs boson in 2HDM are their SM counterparts times a coupling multiplier. The coupling multipliers for the interaction of the CP -even neutral scalars, i.e., h and H, to a pair of massive gauge bosons are given by: with κ sXX = g sXX /g SM hXX , where g SM hXX is the coupling of Higgs boson to the species X in SM and g sXX is the coupling of CP -even neutral scalars s = h, H, to the species X in 2HDM.
In the present paper we work under the assumption of m H > m h . However, the alternative scenario with m H < m h has also been investigated in literature [6].
In order to suppress the Higgs-mediated flavour-changing neutral current (FCNC), it is important to make sure that no fermion gets mass from both the doublets [31]. This is ensured by the assumption of a Z 2 -symmetry under which: ϕ 1 → ϕ 1 and ϕ 2 → −ϕ 2 .
There can be four ways in which the assignment of the Z 2 -charges to the fermions can be performed, leading to four different kinds of Yukawa interactions, Type-I, II, Lepton-specific and Flipped [32,33]. In this paper, we concentrate on only the first two kinds, which are the most studied ones in literature. For Type-I 2HDM, all the fermions, the u-type and d-type quarks and the charged leptons, get mass from the second doublet. For Type-II, u-type quarks get their masses from ϕ 2 , while ϕ 1 provides masses to d-type quarks and the charged leptons. Similar to hV V couplings, the couplings of the SM-like Higgs to fermion pairs also get modified in 2HDM. The coupling multipliers for the interactions of the neutral scalars with a pair of SM fermions are given as: Type-II : Due to such modifications in the couplings, the production and decay rates of the SM-like Higgs boson in 2HDM differ from those in SM. After the discovery of the 125 GeV Higgs boson, whose properties are in quite good agreement with that of the SM Higgs boson, the experimentally allowed parameter space of 2HDM gets constrained. It can be seen from eqns. (2.5) and (2.6) that all the coupling multipliers are functions of cos(β − α) and tan β. Moreover, in the limit cos(β − α) → 0 all the couplings reduce to corresponding SM values. This is known as the 'alignment limit' of 2HDM and will be discussed in detail in Section III B. If certain additional symmetries are imposed, alignment limit can be achieved as a natural consequence in models with extra Higgs doublets [34,35]. Also, the case of a 2HDM emerging as a low-energy effective theory of a supersymmetric scenario has been explored [36].
These coupling multipliers get further changed in presence of the higher-dimensional operators. So the production rates as well as the decay width of the SM-like Higgs boson get modified. As mentioned earlier, such changes can be larger compared to the extra contribution in 2HDM at tree-level [13]. This in turn affects the extraction of bounds on the parameter space in 2HDM which can significantly modify the alignment limit itself.
In this paper, we intend to study the effects of the higher-dimensional operators on the alignment limit in detail. Here we concentrate on the effects of the bosonic 6-dim operators.
The impact of the fermionic operator will be discussed elsewhere. The complete set of bosonic operators in case of 2HDM can be found in ref. [13]. The power counting based on naive dimensional analysis (NDA) renders some of the operators more suppressed than the rest, thus making some operators more significant when it comes to phenomenological analyses.
The operators under consideration are of types ϕ 4 D 2 , ϕ 2 D 2 X, ϕ 2 X 2 and ϕ 6 . These are the only bosonic operators which involve the scalar fields, hence important for Higgs physics.
As we go on we will see that the first two types of operators are relevant for our discussion.
• ϕ 4 D 2 These operators lead to the rescaling of the kinetic terms of all the Higgs fields, sans the charged scalars. Such effects should be taken care of by appropriate field redefinitions, which lead to the scaling of the couplings of the SM-like Higgs. Some of these operators contribute to the T -parameter. In order to suppress such contributions, it is a common practice, for most of the UV-complete models that lead to 2HDMEFT at low energies, to assume the existence of an unbroken global SO(4) symmetry, under which the two Higgs doublets transform as bidoublets [24]. As these operators contribute to the T -parameter at tree-level, the corresponding Wilson coefficients are constrained at ∼ O(10 −3 ) [13,37,38].
Thus, these operators lead to insignificant changes in the decay widths of the SM-like Higgs in various channels and we neglect them. So, we are left with the following operators [13], Operators O H1H12 and O H2H12 are odd under the Z 2 -symmetry, whereas the rest are even.
In presence of these higher dimensional operators, the angle β still diagonalises the charged and CP -odd scalars as in 2HDM at the tree-level [13]. However, the CP -even neutral scalars ρ 1,2 can no longer be diagonalised by the mixing angle α as these scalar fields have to be redefined in the following way in order to achieve canonically normalised kinetic terms [13]: where ∆ ijρ are defined in Appendix A. This leads to the redefinition of the physical CP-even scalar fields as: x 1 , x 2 and y are functions of the Wilson coefficients of the higher dimensional operators as given in Appendix A. In our notation, c θ ≡ cos θ, s θ ≡ sin θ and t θ ≡ tan θ. Due to this redefinition of fields, couplings of both the CP-even neutral scalars to vector bosons and fermions get modified compared to 2HDM at the tree-level. The coupling multipliers of the SM-like Higgs boson and of the other neutral scalar H are modified as follows: where V = W, Z. These modified coupling multipliers reduce to that in 2HDM at tree-level with f → ∞, as expected. The reason that the coupling multipliers of the hW W and hZZ vertices are the same lies in the fact that we have ignored the T -parameter violating operators in this paper. In presence of these operators the coupling multiplier for hZZ vertex gets additional contributions compared to the hW W vertex. As we neglect these operators, the AV V and Af f vertices do not modify compared to 2HDM at tree-level.
It is interesting to note that these operators lead to additional momentum-dependent terms in the triple scalar vertices. However, in this paper we are mainly interested in the bounds coming from the signal strengths of the SM-like Higgs boson. The detailed phenomenology of the exotic scalars under the framework of 2HDMEFT will be addressed elsewhere.
• ϕ 2 D 2 X and ϕ 2 X 2 There are 12 different operators of type our basis defined as [13]: There are 3 operators of type ϕ 2 X 2 for X = B given as: purpose. Such an operator with W bosons is not present in our basis [13]. We have refrained from discussing the effects of O GGij which can be constrained from σ(gg → h).

III. EXISTING CONSTRAINTS
A. Choice of 2HDM parameters In this paper, we work in the physical basis of the 2HDM in which the complete set , tan β, µ 2 , v} describes the potential of the model. µ 2 is the coefficient of the Z 2 -odd quadratic term in the 2HDM potential appearing in eqn. (2.3) and v ∼ 246 GeV is the electroweak vev. In defining cos(β − α) we use the convention as in ref. [40]. In 2HDM at tree-level, the coupling multipliers are sole functions of cos(β − α) and tan β. It can also be seen from eqns. (2.11) and (4.4) that the changes in the couplings due to the 6-dim operators also depend on these two 2HDM parameters.
Thus, as it was mentioned earlier, the alignment limit in 2HDM and its modifications after the inclusion of the 6-dim operators are best demonstrated on the cos(β − α) − tan β plane.
The hard Z 2 -violating couplings, λ 6 and λ 7 have been set to zero for now, though it is worth mentioning that even non-zero values of λ 6 and λ 7 can be rotated away into λ 6 = λ 7 = 0 exploiting the reparametrisation invariance of the 2HDM Lagrangian as long as certain conditions in terms of other 2HDM parameters are fulfilled [41]. On the cos(β − α) − tan β plane, in case of non-zero values of Br(H → hh), region of lower tan β gets excluded [42].
However, for most of our benchmark points we have chosen m H < 2m h , so that the decay channel H → hh is kinematically forbidden.
The direct bound on the mass of the charged scalars come from the measurements of LEP, which dictates m H ± 72 GeV (80 GeV) for Type-I (II) 2HDM [43]. LEP searches also put constraint on the sum of the masses of neutral exotic scalars, m H + m A 209 GeV [44].
We work under the approximation of m A ∼ m H ± , which ensures that the contribution to T -parameter at one-loop is rather small [45]. The key decay channels of A near m A ∼ 400 GeV are A → Zh, ττ , which rule out regions at lower tan β for both Type-I and Type-II 2HDM [42]. The theoretical bounds of stability, perturbativity and unitarity of the S-matrix [49][50][51] on the renormalisable 2HDM parameter space have been calculated using 2HDMC-1.7.0 [52].
In order to ensure that the stability criteria is respected, we choose for all our benchmark scenarios, We have taken into account all the theoretical as well as experimental constraints on parameters for all the benchmark scenarios while working on the cos(β − α) − tan β plane.
Our choice of parameters makes it easier to illustrate the effects of the 6-dim operators on the 2HDM parameter space.

B. Signal strengths
The signal strength of Higgs boson in a particular channel is defined as,  [8]. Though the situation is comparably relaxed in Type-I 2HDM where the hbb coupling is less sensitive to tan β for tan β 4, allowing quite higher values of cos(β − α) compared to Type-II 2HDM.
Studies of 2HDM parameter space in light of the discovery of the Higgs boson at 125 GeV and future prospect of the searches of exotic scalars have been done in the literature [5,6,40,[72][73][74][75][76][77]. The global fits on the 2HDM parameter space have been performed in ref. [78,79] taking into consideration the Higgs data, exclusion limits on the new scalars, as well as the EWPT and flavour constraints. However, performing such global fits in 2HDMEFT is beyond the scope of our present work.

IV. ALIGNMENT LIMIT WITH 6-DIM OPERATORS
The 6-dim operators affect the production and decay channels of the SM-like Higgs boson in non-trivial ways. The ϕ 4 D 2 operators redefine the Higgs fields, leading to the rescaling of the hV V couplings. In contrary, ϕ 2 D 2 X type of operators can modify the momentum structure of the same. Due to this, even though these operators are highly constrained from the electroweak precession observables, they can lead to significant changes in the Higgs decay widths. We determine allowed regions on the cos(β − α) − tan β plane in presence of these 6-dim operators. To compute the decay width of the SM and exotic scalars into various channels, we have used 2HDMC-1.7.0 [52] incorporating the modified couplings. The production cross sections for various Higgses for both the gluon fusion and bb-associated production modes have been computed up to NNLO in QCD using SusHi-1.6.0 [80,81]. For production cross-section of the SM-like Higgs via vector boson fusion (VBF) and associated production with vector boson (VH) mode we use, σ/σ SM ∼ κ 2 hV V (κ 2 hV V ) for 2HDM (2HDMEFT). We neglect the loop-level effects in the h → V V decay channels which are subleading to the tree-level contributions [82].
In the remaining part of this section we discuss the effects of the 6-dim bosonic operators under consideration on the alignment limit for the Type-I and Type-II 2HDM. We also study the case of a concrete UV-complete model, namely the little Higgs model based on the coset SU (6)/Sp(6).
A. Type-I 2HDM In presence of such operators, the modified hV V coupling multipliers are given in eqn. (2.11). Due to such modifications the signal strength µ V V gg changes. The process h → γγ is mediated by the W ± boson, charged fermions and scalars, thus µ γγ gg also gets modified due to the presence of these operators.
This happens due to the positive sign of the first two terms in the squared bracket. In absence of 6-dim operators, the excluded region on the positive direction of cos(β − α) represents the area where the value of µ ZZ gg is smaller than the experimentally allowed lower limit, i.e., ∼ 0.76, due to small values of s β−α . For BP1, the hV V coupling multiplier becomes larger than the tree-level value of s β−α in both positive and negative c β−α directions.
For instance, at c β−α ∼ 0.5 this leads to ∼ 15% change in the decay width Γ(h → V V ) compared to 2HDM at tree-level. Though, it can be seen from fig. 2(b), the corresponding branching ratio does not differ significantly compared to 2HDM at tree-level. This happens because the change in Γ(h → V V ) is mostly cancelled by the same in the total decay width. However, for cross-sections no such cancellation occurs. For the case under consideration, htt . At t β ∼ 10, in absence of any 6-dim terms, at c β−α ∼ −0.37, κ htt ∼ 0.89, i.e., σ(ggh)/σ(ggh) SM ∼ 0.79, which leads to a decrease of σ(ggh) by ∼ 20%.
For BP2, using relations (4.1) one arrives at,  where the anomalous couplings can be written in terms of the Wilson coefficients of the operators in 2HDMEFT as [13]: with, c x = (−c β s α c x11 + s β c α c x22 + c β−α c x12 )m 2 W /Λ 2 . c xij , where x = W, B, ϕW, ϕB, BB, are the Wilson coefficients of the operators appearing in eqn. (2.12). The decay width of the SM-like Higgs into a pair of gauge bosons gets modified in the following way [83,84]: where, with, with δ V = 2, 1 for V = W, Z respectively, and λ(x, y, z) This kind of operators contributes to the precision observables and various cross sections related to the SM-like Higgs, which in turn put bounds on the coefficients of these operators. Operators this Little Higgs model [85]. In this case the modified hV V coupling multiplier assumes the form: In this paper, we argue that as the experiments only constrain the effective coupling of the SM-like Higgs boson with other SM particles, we are restricting only the 'effective alignment' rather than the 'true alignment' in a 2HDM. These two alignments differ in presence of physics beyond a 2HDM, which can be encoded in the language of 2HDMEFT.
SMEFT is not adequate for such a study as the exotic scalar particles can be light enough so that they are not decoupled from the mass spectrum of the low-energy theory. The difference between the 'effective' and 'true' alignments is sensitive to the Wilson coefficients of these 6-dim operators, the 2HDM parameters and the type of 2HDM considered.
Depending on these, the allowed parameter space on the cos(β − α) − tan β plane can shift, shrink and what is even more interesting, the exact alignment limit can be excluded. We have demonstrated this by the choice of suitable benchmark points [89]. It is noticed that for Type-I 2HDM, the region allowed from measurements of the signal strengths of the SM-like Higgs boson being quite larger compared to other variants of 2HDM, the 6-dim operators are able to inflict substantial changes, often increasing or decreasing the allowed value of cos(β − α) by ∼ O(0.1). In case of Type-II 2HDM, the percentage change in the allowed value of cos(β − α) can be larger than Type-I 2HDM, achieving values ∼ 100% or higher, whereas such changes for Type-I 2DHM reaches values up to ∼ 25%. We have seen that, negative values of the Wilson coefficients lead to a larger allowed range of cos(β − α) in both positive and negative directions in most of the cases. We also studied a particular Little Higgs model with Type-I Yukawa coupling as a UV-complete example of 2HDMEFT.
Generally the impact of ϕ 2 D 2 X type of operators on 2HDM alignment is smaller compared to the ϕ 4 D 2 operators. In Type-II 2HDM, it is also noticed that in the presence of the 6-dim operators, the exact alignment limit, cos(β − α) = 0, can be ruled out for a wide range of values of tan β at 95% CL. All these demonstrate that the 6-dim operators in 2HDMEFT are capable of masking the true alignment.