Higher dimensional operators in 2HDM

We present a complete (non-redundant) basis of CP- and flavour-conserving six-dimensional operators in a two Higgs doublet model (2HDM). We include Z_2-violating operators as well. In such a 2HDM effective field theory (2HDMEFT), we estimate how constraining the 2HDM parameter space from experiments can get disturbed due to these operators. Our basis is motivated by the strongly interacting light Higgs (SILH) basis used in the standard model effective field theory (SMEFT). We find out bounds on combinations of Wilson coefficients of such operators from precision observables, signal strengths of Higgs decaying into vector bosons etc. In 2HDMEFT, the 2HDM parameter space can play a significant role while deriving such constraints, by leading to reduced or even enhanced effects compared to SMEFT in certain processes. We also comment on the implications of the SILH suppressions in such considerations.


I. INTRODUCTION
Two Higgs doublet model is the most studied extension of the scalar sector of the Standard model (SM) of particle physics. Inclusion of an additional scalar doublet is necessary in the supersymmetric models. There are other phenomenological motivations for considering nonsupersymmetric versions of this model as well. For example, an additional Higgs doublet can lead to a successful electroweak baryogenesis [1]. Moreover, anomalies in tauonic B-decays can be addressed in a 2HDM [2][3][4].
Non-observation of new fundamental particles at the LHC motivates us to formulate the SM effective Lagrangian below a TeV or so. The recent data implies that the newlydiscovered 125 GeV Higgs boson is SM-like, but still allows for the existence of new scalars at sub-TeV scales. In the alignment limit, 2HDM can accommodate such new scalars so that the extra contributions coming from the renormalisable 2HDM Lagrangian to various processes involving SM particles are reasonably small. This calls for an extensive study of higher dimensional operators in exploring 2HDM phenomenology as their effects can be of the same order of the extra contribution in 2HDM at the tree-level in the alignment limit.
In this paper, we formulate a basis of independent six-dimensional (6-dim) operators assuming 2HDM to be the low-energy theory. We include the Z 2 -violating operators, but, for simplicity, exclude the CP-and flavour-violating ones. The bounds imparted on the Wilson coefficients by the electroweak precision tests (EWPT) are estimated. Such operators also affect the signal strengths of the SM-like Higgs boson decaying into a pair of vector bosons.
We estimate such contributions as in future, better measurements of these signal strengths can further tighten constraints on these Wilson coefficients. Our results also reflect the fact that the inclusion of higher dimensional operators can relax the bounds on the parameter space of renormalisable 2HDM. We also discuss the cases where we have weighed the Wilson coefficients following the SILH prescription as in the SMEFT [5] and mark the resulting changes in the above-mentioned bounds.
The study of a complete set of higher dimensional operators of SM goes long back [6,7], where the authors formulated the basis of 6-dim operators of SM assuming lepton and baryon number conservation. A systematic study of the electroweak precision constraints on the Wilson coefficients of the bosonic 6-dim operators was first done in the framework of the so-called HISZ basis of SMEFT [8]. The problem of writing a complete set of 6-dim operators of SMEFT was revisited in Ref. [9] where the equations of motions (EoM) of all the fields were used to identify 21 redundant operators in the basis of Ref. [6]. The basis for SMEFT introduced by the latter is often referred to as the Warsaw basis. The SILH basis [5] was formulated in the context of scenarios where the hierarchy problem is alleviated due to the existence of a strongly-interacting sector beyond the TeV scale. There are two broad classes of models which can be represented by the SILH Lagrangian. First, the extra-dimensional models where the Higgs boson is a part of the bulk and the rest of the SM fields are part of a brane at low-energy [10]. The other one consists of all the models where the Higgs is a pseudo Nambu-Goldstone boson (pNGB) of the strong sector. The Little Higgs [11] is an archetype of the latter class. The composite Higgs models can be a part of both the classes.
The SILH Lagrangian is described by two scales of new physics, namely f , the compositeness scale and m ρ ∼ g ρ f , the lightest vector boson mass in the strongly-interacting sector, with g ρ being the coupling of new strong sector. An updated review of the study of SMEFT in light of precision electroweak and Higgs signal strength data can be found in Refs. [12,13].
The study of a composite 2HDM in the context of a SO(6)/SO(4) × SO(2) coset was performed in Ref. [14]. Perturbative unitarity bounds on S-matrix elements were extracted and collider phenomenology of this particular model has been explored only recently [15][16][17].
The 2HDM in a Little Higgs scenario was studied in Refs. [18,19]. A few other studies of the composite inert doublet model of dark matter have been carried out as well [20,21]. In Ref. [22], an extension of SMEFT SILH basis incorporating a light singlet scalar along with SM degrees of freedom was introduced. Impact of some of the six-dimensional operators involving two Higgs doublets on exotic decay channels of charged Higgs boson were studied in Ref. [23]. The kinetic terms comprising of four scalars and two derivatives in a N-Higgs doublet scenario were studied recently in Refs. [24][25][26]. An attempt to write down the full set of 6-dim operators in 2HDM was made in Ref. [27] in a Warsaw-like basis. In contrary, our basis is motivated by the SILH basis in SMEFT and is a complete one, as we point out that there is a redundant operator in the basis of Ref. [27]. In addition, as we mentioned earlier, we include the Z 2 -violating operators as well.
We start with introducing the tree level Lagrangian in 2HDM and the corresponding equations of motion and then formulate our basis in Section II. In Section III, we carry out the kinetic and mass diagonalisations of the scalar fields which enable us to write down the effective couplings of those scalars with the vector bosons. Section IV deals with the constraints on the Wilson coefficients coming from the EWPT. In section V, we evaluate the decay widths and signal strengths of the SM Higgs boson decaying into vector boson pairs.
Finally in Section VI we consolidate our results and eventually conclude.

II. CONSTRUCTION OF THE 2HDMEFT
A. The 2HDM Lagrangian and classical EoMs We use the same notation as in Ref. [27] in order to avoid further confusion. The Higgs fields in the doublet notation can be written as: where I = 1, 2. Before the two Higgs fields ϕ 1 , ϕ 2 get vacuum expectation values (vev) the renormalisable 2HDM Lagrangian is given by: where the first three are field strengths of the gauge bosons of SU (3) C , SU (2) L and U (1) Y respectively. The indices a = 1, .., 8 and i = 1, 2, 3 are summed over. The tree-level 2HDM potential is given by: The term with coefficient m 2 12 is called the soft Z 2 -violating term, whereas the ones with λ 6 and λ 7 are called the hard Z 2 -violating terms as they give rise to quadratically divergent contribution to ϕ 1 − ϕ 2 mixing. However, one is allowed to start with non-zero values of λ 6 and λ 7 as long as they can be rotated to λ 6 , λ 7 = 0 using reparametrisation transformations [28,29]. This scenario is referred to as "hidden soft Z 2 -violation". Moreover, in SILH scenarios, one considers the existence of a strongly-interacting sector at ∼ O (1 TeV) which deliver Higgs as a pNGB at low energy. In those cases new resonances at or above ∼ O(1 TeV) take care of the quadratic divergence of Higgs mass, solving the hierarchy problem. The same mechanism will take care of the quadratic divergence in ϕ 1 − ϕ 2 mixing caused by λ 6 and λ 7 for the 2HDMs which are governed by such a strongly-coupled sector at higher energies. That is why in this paper we carry out all calculations keeping λ 6 , λ 7 = 0.
The same explanation holds true for the inclusion of Z 2 -odd higher dimensional operators.
The general Yukawa Lagrangian is given by, For eliminating the redundant operators from the basis of 6-dim operators, one needs to derive the EoMs of the bosonic fields from the tree-level 2HDM Lagrangian. It is necessary to separate out the redundant ones because they do not contribute to the S-matrix elements [30].
While doing that, we neglect the five-dimensional operators [9]. The EoMs are given as:

B. Operator basis
In the universal theories [31], the deviations of the properties of the Higgs boson from SM can be expressed in terms of only the higher-dimensional bosonic operators. Both the Warsaw basis [9] and SILH basis [5] where, L (5) consists of three operators, O (2.7) We have defined our notation as follows: ϕ, ψ and X stand for the two scalar doublets, fermions and gauge field strength tensors respectively. D stands for a derivative. Throughout with the Z 2 -conserving ones, which was not the case for Ref. [27]. So the total number of operators in our basis is more than that of Ref. [27]. We have marked the Z 2 -violating operators in blue colour. The suppressions of these operators in a SILH scenario are given in Appendix A.
• ϕ 4 D 2 The ϕ 4 D 2 operators in our basis are given in Table I. The operators O (1)21 (2) , O (1)12 (2) , (1) , O (2)11 (2) are common to both our basis and the basis introduced in Ref. [27]. O H1H12 , O H2H12 , O T 4 and O T 5 will not appear in the basis if one demands the Z 2 symmetry to be conserved in the 6-dim Lagrangian, which is the case for Ref. [27]. In absence of these two operators, the number of operator in our basis is 11 compared to 12 of Ref. [27]. We will keep these Z 2 -violating operators following the logic of Section II A. The transformations from the basis of Ref. [27] to our basis are: where i = 1, 2 and T denotes total derivative terms containing ϕ and D and E stands for the ϕ 4 , ϕ 6 and ϕ 3 ψ 2 terms which are already included in the basis. In the above, the fourth transformations in eqn. (2.8) points to the fact that there is one redundant operator in the basis of Ref. [27].
• ϕ 2 D 2 X We have included 12 operators of class ϕ 2 D 2 X, which were not there in Ref. [27] and are listed in Table II. We have traded 6 operators of the class ϕ 2 D 2 X for 6 operators of class ϕ 2 X 2 , according to the relations: where, T stands for total derivative terms and, (2.11) and O W ij , as it was done in the SILH basis of SMEFT.
As it is mentioned in Appendix A, in a SILH scenario, O ϕBij and O ϕW ij have suppressions of ∼ 1/(4πf ) 2 , whereas O Bij and O W ij will be suppressed by ∼ 1/m 2 ρ . Both of these operators are of type ϕ 2 D 2 X, but the latter ones are current-current type of operators and can be generated by integrating out suitable resonances which are typically of mass m ρ and couple to both the currents at the tree level.
• ϕ 2 X 2 As it was discussed for ϕ 2 D 2 X, some of the operators of class ϕ 2 X 2 were traded in favour of the previous ones. Rest of the operators in this category are listed in Table III.
• ϕ 6 These are the corrections to the potential of the renormalizable 2HDM and are listed in Appendix C along with the modified minimisation conditions of the potential.
• ϕ 3 ψ 2 These operators lead to the corrections to the Yukawa terms. It is worth noting that we have written all the possible operators without considering any Z 2 charges of either the scalar doublets or the SM fermions. While working in a particular case of either Type I, II, X or Y 2HDM, certain operators of this category have to be put to zero depending on the discrete charges of the scalars and fermions. All the operators of this category are listed in Table IV.
• ϕ 2 ψ 2 D Some operators of this category were traded away in favour of some ϕ 2 D 2 X type of operators using eqn. (2.11). The remaining operators are listed in Table V. These operators contribute to various decay channels of the W and Z bosons.
• ϕψ 2 X These operators represent the dipole moment of the SM fermions under the SM gauge fields and are listed in Table VI, where σ i and t a stand for the Pauli matrices and Gell-Mann matrices respectively. • D 2 X 2 and ψ 4 The counting in these two classes of operators do not change due to the insertion of a second scalar doublet in the theory, hence these operators in our basis are the same as SMEFT.
For the D 2 X 2 type of operators we refer to Appendix B and the list of ψ 4 operators can be found in [9].

III. SCALARS IN 2HDMEFT
A. Kinetic diagonalisation for scalars The kinetic terms for the scalars (except for the charged scalars) will pick up non-diagonal parts when two of the ϕ s of ϕ 4 D 2 type of operators get vevs. where, One has to shift the fields in order to diagonalise the kinetic terms in the following manner:

B. Masses of the scalars
Mass terms of the scalars are modified in the following manner: m 2 ρ,η,φ stand for the mass matrices of the corresponding fields in the tree-level 2HDM, whereas the ∆m 2 s represent the contributions arising from 6-dim operators. The masses of the scalars get two-fold modifications in the EFT, one coming from the ϕ 6 operators, another coming from the shifting of the fields due to the ϕ 4 D 2 operators, where, (12) , In order to arrive at eqns. (3.6) one has to use the minimisation conditions of the modified potential which is given in Appendix C. From eqns. (3.6) it becomes evident that tan β = v 2 /v 1 diagonalises (m 2 η + ∆m 2 ηϕ 6 ) and (m 2 φ + ∆m 2 φϕ 6 ). Moreover in our case, ∆m 2 φϕD = 0, because ϕ 4 D 2 type of operators do not affect the charged scalar kinetic terms. This ensures that the charged scalar Goldstone boson remains massless in O(1/f 2 ). But ∆m 2 ηϕD is nonzero and is given by, This matrix has null determinant, hence one of the eigenvalues of this matrix will be zero.
This ensures that the Goldstone boson for Z remains massless at O(1/f 2 ). But this matrix cannot be diagonalised by tan β = v 2 /v 1 . The rotation which diagonalises the above mass matrix is denoted by some other rotation angle β η , which can be expressed as: In the limit f → ∞, β η = β, i.e., we get back the tree-level 2HDM. The masses of the physical pseudoscalar and the charged scalar in terms of various Wilson coefficients are presented in Appendix D.
In a similar manner, ∆m 2 ρϕ 6 and ∆m 2 ρϕD will both have non-zero values and the rotation needed for diagonalising ∆m 2 ρ will no longer be the same as α in 2HDM at the tree-level. We call the new rotation angle α . The value of α can be determined from the masses of the neutral scalars after fixing the values of relevant Wilson coefficients. If the light and the heavy scalars have masses m h and m H respectively [33], . (3.10) The expressions for M 2 11ρ and M 2 12ρ in our case are given in Appendix D.

IV. CONSTRAINTS FROM ELECTROWEAK PRECISION OBSERVABLES
We consider all the bosonic classes, i.e., ϕ 4 D 2 , ϕ 6 , ϕ 2 D 2 X, ϕ 2 X 2 , and see which ones are do not contribute to the precision observables.
The operators of class D 2 X 2 in our basis are the same as the ones in SMEFT, as they include no Higgs doublets. Operators of type D 2 X 2 contribute to the oblique parameters V , W , Y and Z and can be constrained from measurements at LEP. We shall mention the bounds for completeness.
We have not considered the EWPT constraints on the fermionic operator classes, i.e., ϕ 3 ψ 2 , ϕ 2 ψ 2 D, ϕψ 2 X and ψ 4 . However, we mention that the operators of class ϕ 2 ψ 2 D can be constrained using the EWPT because they lead to various fermionic decay channels of In a SILH scenario, the coupling of the new dynamics is stronger compared to the SM ones, i.e., g SM g ρ 4π [5]. This prescription distinguishes the mass of lightest vector resonance of the strong sector m ρ ∼ g ρ f from its cut-off Λ ∼ 4πf . To show quantitatively what value g ρ can attain in a realistic scenario, in SO(6)/SO(4) × SO(2) composite Higgs model with the third generation quark doublet and t R , both transforming as 4 of SO(4) [14], one finds that g ρ ∼ 3.6 < 4π for m h ∼ 125 GeV. In the remaining part of this paper, we will use the SILH suppressions of the Wilson coefficients for different classes of operators. We have used the shorthand notation of ξ i , where i = 1, .., 5, to describe these suppressions and these are defined in Appendix B. All the bounds we impose on the Wilson coefficients from now on, can be translated to a non-SILH scenario in the limit ξ i → 1/f 2 , f being the scale of new physics.

• Constraints from anomalous TGVs
TGVs [34] involving W bosons are some of the precisely measured quantities in the LEP experiment. Anomalous contribution to TGVs can be parametrised in terms of five parameters which are defined in Appendix E. The operators from class ϕ 2 D 2 X, except O Bij , contribute to these parameters in the following way: We use the bounds from two parameter fit (δg 1 Z and δκ γ ) with λ γ = 0 of anomalous TGVs at 95% confidence level provided by LEP-II collaboration [35]: (4.2) It is worth mentioning here that, λ γ and λ Z get affected by the CP-odd operator W µν W µρ W ν ρ . As we are considering only the CP-even operators in this paper, λ γ = λ Z = 0 in our basis.
Inclusion of the Higgs signal strength data in the fit changes the bounds on anomalous TGVs. The fitted values of δg 1 Z and δκ γ at 95% confidence level [36] become: 3) The 6-dim operators in our basis also contribute to the Higgs signal strengths. So, in order to extract the bounds on the Wilson coefficients using eqn. (4.3), extra care has to be taken.

• Constraints from oblique parameters
We define vacuum polarisation amplitude involving any two vector bosons V I and V J as  [31,37,38]. Among these, only S, T and U can be measured using the Z-pole observables. The kinetic terms of W ± µ and B µ must be normalised before calculating the oblique parameters [31]. Due to the presence of the ϕ 2 X 2 type of operators, the kinetic term of B µ has to be canonically normalised with help of following transformation: However, such transformation need not be done for W ± µ as the corresponding operators do not exist in our basis, as it can be seen in Table III. The contribution of the effective operators to oblique parameter U is zero. As, U ∝ (Π W + W − (0) − Π 33 (0)), non-zero value of U demands a source of isospin-violation in the theory. In our basis, these two polarisation amplitudes get modified by the operators O W ij , but in an identical way, giving U = 0.
However, the other two parameters S and T get non-zero contributions in our basis: where, (2) ). (4.6) As it was mentioned earlier, it can be seen that the operators that contribute to S and T belong to the classes ϕ 2 D 2 X and ϕ 4 D 2 respectively. Bounds at 95% confidence limit on S and T are given by [39]: The oblique parameters V , W , Y and Z can not be measured using the Z-pole observables.
They represent second derivatives of certain polarisation amplitudes. D 2 X 2 type of operators affect the oblique parameters V , W , Y and Z in following manner [5]: whereas the LEP data suggests [31]:  Fig. 1(a), at lower values of tan β, S is proportonal to c 2 β , but as tan β 1, it becomes proportional to (1/ tan 2 β). Fig. 1(b) shows, at lower tan β, S varies as s 2 β and becomes almost constant at higher values of tan β. In Fig. 1(c) and (d), T varies as (1 − 2s 2 β c 2 β ) and 2s 2 β c 2 β respectively. This is reflected as the local minima and maxima in Fig. 1(c) and (d) respectively, around tan β = 1. At large values of tan β, T becomes almost constant for Fig. 1(c), whereas it becomes proportional to 1/ tan 2 β for Fig. 1(d).
In Fig. 2, we graphically illustrate the dependence of S and T on f and m ρ on the S − T plane. We have chosen random sets of values of the Wilson coefficients which contribute to either S or T for a fixed value of tan β. In Fig. 2(a) we have seen that the higher the value of f , the more likely it is to satisfy the bounds of both S and T . In Fig. 2(b), we have shown that the higher the value of g ρ , the easier it is to satisfy the bounds from S parameter.
Both these effects can be read off eqn. (4.5). The blue points in Fig. 2(b) represent the corresponding non-SILH scenario, i.e., g ρ = 1. In Fig. 2(b), the points corresponding to g ρ = 10 have been quenched into a straight line, indicating the fact that if the new sector is more strongly-coupled, it is easier to satisfy the bounds from S parameter compared to a non-SILH scenario for the same set of values of the relevant Wilson coefficients.
To properly disentangle the effect of the 6-dim operators in this context, we have not considered the contributions to S and T coming at one-loop of renormalisable 2HDM in either Fig. 1 or in Fig. 2. After the LHC Run I and II, 2HDMs of type II, X and Y [44] are pushed close to the alignment limit. In this limit, the couplings of one of the neutral scalars with a pair of vector bosons approach their SM values [45][46][47]. However, alignment can be achieved with or without Appelquist-Carazzone decoupling [48] of the new scalars. We are interested in the scenario of 'alignment without decoupling' [33,49,50].
As the decay into γγ, Zγ and gg channels are loop mediated processes in renormalisable 2HDM, there will be interferences of the one loop amplitudes with the ones coming due to higher dimensional operators. The modified matrix elements are given by: one obtains the partial decay width of SM-like Higgs to γγ and Zγ with help of the eqn. (5.1) [51], 3) The amplitudes and relevant Wilson coefficients in our case are given as, It was evident from Section III B that in 2HDMEFT, the tree-level couplings of the scalars to W and Z bosons are described by three angles, i.e., α , β and β η , rather than two, which is the case for tree-level renormalisable 2HDM. The coupling of the physical neutral scalars with the vector bosons are:  [51,54]: along with, where, with δ V = 2, 1 for V = W, Z respectively, and λ(x, y, z) Definitions of c {W W, ZZ, W ∂W, Z∂Z} are given in Appendix E. In our basis they can be written in terms of the Wilson coefficients in the following way: One can see in eqn. (5.10), all the Wilson coefficients are constrained by either S parameter in eqn. (4.5) or anomalous TGVs eqn. (4.1) or the measurement of decay width of Higgs to γγ and Zγ in eqn. (5.5). This happens in SMEFT as well. But these Wilson coefficients appeared with prefactors different from those in eqns. (4.1) and (4.5). This is a remarkable feature of 2HDMEFT. For example, the Wilson coefficient of O W 11 has come with a prefactor of c 2 β and −c β s α in the expressions for S parameter and hW W coupling respectively, and would come with a prefactor of s 2 α in the hhW W coupling. This effect is absent in SMEFT. The following numerical analysis will illustrate this fact.
• BP4 C W ≈ −10 −3 , C B ≈ −2 × 10 −3 , C ϕW ≈ −10 −2 , C ϕB ≈ −10 −3 , tan β = 1 and In Table VII, Here Γ V V,EF T consists of contributions from tree-level SM and 6-dim operators of 2HDMEFT, and can be obtained from eqns. (5.7), (5.8) and (5.9) by putting sin 2 (β − α) = 1. We have also indicated the ratios between y V V and x V V for both W W and ZZ decay channel. Note that in all the cases we have kept c BB ≈ 0, which is constrained at per-mille level by the measurement of Γ(h → γγ). All the BPs satisfy the TGV constraints, S parameter and We have refrained from considering the one-loop effects in h → W W, ZZ to disentangle the effect of EFT in these processes, as we had done in Section IV. The effects of the 6dim operators can be of the same order of the one-loop effects in 2HDM. For example, the percentage change in the decay widths of the processes h → V * V ( * ) → 4f at one-loop order compared to the lowest order is around ∼ 2.7%, for c β−α = 0.1 [55].
In this section we have seen that the operators which are constrained via S parameter, TGVs and the decay width of SM like Higgs to γγ and Zγ, can still be exploited to impart a change on the h → V V decay widths. At the end of LHC Run II, the error in Higgs couplings are expected to decrease upto 4%. The change in decay widths as mentioned in Table VII can be probed in HL-LHC which will be able to probe the hV V couplings to a precision of 2%. The ILC with √ s = 500 GeV will reduce the error in the hV V couplings upto ∼ 1% [56]. If 2HDM is close to the alignment limit, the effects of these 6-dim operators will be significant and will lead to a confusion with the new contributions from 2HDM at tree and loop level effects for the respective processes.

VI. DISCUSSIONS
Non-observation of any beyond standard model particle in the direct searches at LHC motivates us to adhere to the language of effective field theory. In presence of a strongly interacting weak sector just beyond a TeV or so, which we designate in this paper as 'SILH scenario', the hierarchy problem in Higgs mass reduces to a 'Little' hierarchy problem, thereby alleviating the quadratic divergences. The 2HDM can originate from such an underlying strong dynamics. In this case, one need not bother about hard Z 2 -violating terms and hence, we include them all, even in the six-dimensional Lagrangian. In a SILH scenario the Wilson coefficients of the operators come with various suppression factors which we review in Appendix A. In passing, we emphasize that our basis has a wider applicability -it is valid when such a SILH scenario is envisaged or not.
Next, we have let the operators confront the results from the electroweak precision tests and measured values of Higgs production and decay channels concentrating only on bosonic operators of classes ϕ 4 D 2 , ϕ 6 , ϕ 2 D 2 X and ϕ 2 X 2 containing the new scalars. Ensuing bounds on combinations of Wilson coefficients have been extracted. Out of these classes, some of the operators belonging only to the classes ϕ 4 D 2 and ϕ 2 D 2 X were constrained from EWPT. The operators of class ϕ 6 and some of the operators of class ϕ 4 D 2 can be constrained demanding perturbative unitarity. As expected, constraints from EWPT are much tighter than those from considerations of unitarity. We also consider the class D 2 X 2 which contribute to W , Y and Z parameters in EWPT. For completeness, we have mentioned bounds on these as well, but these are the same as in SMEFT. In discussing impacts on Higgs signal strengths, only the operators of the class ϕ 2 D 2 X and ϕ 2 X 2 have been considered for simplicity. As the phenomenology of 2HDMEFT is quite rich, to avoid cluttering of information we have also avoided considering phenomenology of the fermionic operators of classes ϕ 3 ψ 2 , ϕ 2 ψ 2 D, ϕψ 2 X and ψ 4 . These issues will be addressed elsewhere.
An earlier attempt to present a basis of Z 2 -conserving operators for 2HDMEFT was made in Ref. [27] that resembles the Warsaw basis in SMEFT. But we have found one redundant operator in the class ϕ 4 D 2 . In our basis, kinetic mixing of the gauge eigenstates of the scalar were taken care of by field redefinitions and their effect on determining the masses of the physical scalar were also calculated. An important feature of our basis is that the charged scalar mass matrix is still diagonalised by tan β = v 2 /v 1 as in the tree-level 2HDM. But the neutral psuedoscalar sector needs a diagonalisation matrix which is not characterised by the same tan β. So the kinetic diagonalisation changes the pseudoscalar mass matrix, but not the one for the charged scalars. This is reflected in the expressions for masses of the pseudoscalar and the charged scalars that we have presented in Appendix D.
An interesting feature of 2HDMEFT is that, in contrast to SMEFT, here the 2HDM parameter space plays a crucial role while placing bounds on the Wilson coefficients. For example, the same Wilson coefficient can appear with different prefactors in the expressions for precision parameters and for Higgs decay widths. These prefactors depend on the 2HDM parameter space. This happens because unlike SM, in 2HDM the interaction eigenstates of the scalars are not the same as their mass eigenstates. In Section V we have numerically demonstrated this remarkable effect which is absent in SMEFT.
In short, our complete basis of 2HDMEFT will facilitate further studies of 2HDM phenomenology. We have presented constraints on some of the operators and pointed out that such constraints do depend on the 2HDM parameter space. Such dependence can significantly modify some of the predictions of SMEFT. It was also noticed that, in the vicinity of the alignment limit, the effects of the higher dimensional operators in determining the parameter space of 2HDM are not negligible.