Benchmarks for Higgs Effective Theory: Extended Higgs Sectors

Precise measurements of SM particles properties at the LHC allows to look for heavy New Physics in the context of an Effective Field Theory (EFT). These searches, however, often rely on kinematic regions where the validity of the EFT may be compromised. In this paper we propose to address this issue by comparing with benchmark models. The connection between models and their manifestations as EFTs at low energies allows us to quantify the breakdown of the EFT, and describe ways to combine different sources of constraints beyond Higgs physics. To illustrate these techniques, in this paper we propose a set of benchmark models based on extensions of the Higgs sector, namely the inclusion of a singlet, a dilaton and generic 2HDMs. We obtain the matching between these models and the EFT involving the Higgs, electroweak bosons and fermions. We then describe current and future indirect and direct constraints, consider the effect of correlations among the coefficients within models, and discuss the validity of the EFT.


Introduction
The discovery of a Higgs boson [1,2] has opened the Era of Higgs Physics. An intense effort is now devoted to measure the properties of this new particle h and determine whether it is the Standard Model (SM) Higgs boson [3][4][5][6]. So far, no evidence for new physics beyond the SM has been observed, which suggests the use of an Effective Field Theory (EFT) approach, that assumes possible new physics beyond the SM modifying the Higgs interactions to be heavy, with a typical scale Λ ∼ TeV. The effects of new physics are then parametrized through higher-dimensional operators constructed from the SM fields, the leading operators appearing at dimension-six [7][8][9][10][11][12]. Some of these operators which affect Higgs properties can be measured in Higgs physics only, while others are also related to electroweak (EW) observables since the Higgs scalar excitation is always associated with the EW symmetry breaking order parameter v. The experimental data from LEP and Tevatron constrain the size of the Wilson coefficients of these operators, and more recently the results from the LHC experiments ATLAS and CMS allow also to constrain the Wilson coefficients associated to the operators that impact Higgs physics [13][14][15][16][17][18][19][20].
The LHC energy reach allows to go beyond the analysis of Higgs signal strengths as a probe of new physics beyond the SM, exploiting the information encoded in kinematical distributions to boost the sensitivity to new physics beyond the SM in the Higgs sector. This has the power to strongly constrain the presence of new physics which lead to a significant enhancement of the signal in certain kinematical regions, typically at high energy. However, while the bounds on the Wilson coefficients of these dimension-six operators which can be extracted from these measurements are indeed very strong even with the limited amount of present data, the validity of the EFT approach in the kinematical regions which dominate these constraints is questionable [21,22] (see also [20]). This would render these constraints meaningless in the context of EFTs. It is therefore crucial to assess in detail the validity of this approach, and a way to do so is by quantitatively studying the breakdown of the EFT by comparing its predictions with those of possible UV completions.
Among possible UV completions, extended scalar sectors provide an attractive arena for the use of a SM effective theory description. Extensions of the SM Higgs sector constitute a simple yet very well-motivated scenario beyond the SM, with important consequences not only for phenomenology but also for EW cosmology, baryogenesis and dark matter. It is plausible that the new scalar states, which we assume to be significantly heavier than the Higgs boson discovered by ATLAS and CMS, cannot be detected directly at LHC either due to them being very heavy or having vanishingly small couplings to SM gauge bosons if the light Higgs is SM-like. However, it may be possible to measure the effect of these new particles via EW precision observables (EWPO) or using the kinematical information in LHC measurements. An EFT of extended Higgs sectors provides a very useful tool to study these effects systematically under the assumption that the new states are significantly heavier than h. Moreover, since extensions of the SM Higgs sector provide a very simple UV completion to such an EFT, this makes it possible to compare the predictions and bounds derived from the EFT with those of its UV completion, therefore probing the range of validity of the EFT. This allows us to assess the reliability of the LHC constraints drawn from the high-energy kinematical regions, as well as those from EWPO, and also to analyze the LHC potential for indirectly probing extended Higgs sectors.
The paper is organized as follows: In Section 2 we introduce the D = 6 effective operators which are relevant for the analysis. In Section 3 we present the matching of these operators with UV models corresponding to extensions of the SM Higgs sector with an extra scalar singlet, with an extra scalar doublet (a Two-Higgs-Doublet-Model) and with a radion/dilaton, and discuss their experimental constraints both from the UV model and from the EFT point of view. In Section 4 we discuss the validity of the EFT in the light of these scenarios. Finally, in Section 5 we summarize our results and discuss their implications. In the Appendices we describe the connection to the various possible D = 6 Lagrangian terms after EW symmetry breaking, parametrized as anomalous Higgs couplings, and give details on the 2HDM relevant to our analysis.

Effective Field Theory for the SM Higgs Field
The Lagrangian for the Standard Model can be supplemented by higher-dimensional operators that parametrize the effects of new physics beyond the SM appearing at energies much larger than an effective scale which may identified with the vev of the Higgs field v. Considering only operators of dimension D ≤ 6 and assuming baryon and lepton number conservation, the most general SU (3) C × SU (2) L × U (1) Y gauge invariant Lagrangian L Eff constructed out of the SM fields has been known for a long time [7][8][9], and may be mapped into various alternative bases of D ≤ 6 independent SM effective operators [10][11][12] (see also the discussion in [23]).
Each experimental measurement will in general constrain only a handful of effective operators of a certain basis. In combining the available constraints from analyses of various physical processes (via e.g. a global fit), some basis choices will be more appropriate than others 1 . As compared to those of [10,12], the basis of operators O i from [11] is wellsuited for analyses of Higgs properties in combination with precision measurements of EW observables. In this basis, the SM effective Lagrangian L Eff may be expressed as (2.1) We adopt here the decomposition in [24,25] and normalize the Wilson coefficientsc i accordingly. The term L SILH of (2.1) corresponds to a certain set of CP -conserving operators involving the Higgs doublet Φ, inspired by scenarios where the Higgs is part of a strongly interacting sector [11] L SILH =c Here λ stands for the Higgs quartic coupling, g , g and g s are respectively the U (1) Y , SU (2) L and SU (3) C coupling constants and T 2k ≡ σ k /2 are the generators of SU(2) in the fundamental representation (σ k being the Pauli matrices). The Hermitian derivative operator ← → D µ is defined as and our conventions for the gauge-covariant derivatives and field-strength tensors (following [25]) are with k ij and f a bc being respectively the structure constants for SU (2) and SU (3). The term L G consists of operators not directly connected to Higgs physics, but that affect the gauge sector through modifications of the gauge boson self-energies and self-interactions, (2.5) The term L CP in (2.1) supplements L SILH and L G with a set of CP-violating operators with the dual field strength tensors defined by Finally, there are further operators contained in L F 1 -operators involving two Higgs fields and a pair of quarks/leptons-, L F 2 -operators involving one Higgs field, a gauge boson and a pair of quarks/leptons-and L F 3 -four-fermion operators-(see e.g. [24,25]). These effective operators are nevertheless not present at leading order in the extensions of the SM we consider in the present work 2 , and so we do not discuss them in the following. Furthermore, we consider CP conserving scenarios for the time being, leaving an analysis of L CP for the future [26]. 2 An exception are certain operators in LF 3 which do get generated in Two-Higgs-Doublet-Model extensions of the SM. These are however proportional to Y 2 a (with Ya a fermion Yukawa coupling), being then negligible for 1 st and 2 nd fermion generations, and so are essentially unconstrained.
After EW symmetry breaking, we can write the SM effective Lagrangian L Eff in the unitarity gauge and in the mass basis, with with s W (c W ) being the sine (cosine) of the weak mixing angle at tree level. In Appendix B, we detail the relation of the D = 6 terms presented here with the possible anomalous Higgs couplings after EW symmetry breaking.
We are now ready to move onto relating the EFT to specific UV completions. In this work we have chosen to focus on the matching of the Wilson coefficients onto models with extended Higgs sectors, where the effects of New Physics are more apparent in deviations of the couplings of the Higgs and electroweak bosons. We describe the results of the matching, as well as the relation between these models and the EFT in the next section.

Higgs EFT from Extended Higgs Sectors
As discussed in the Introduction, an important aspect of the D ≤ 6 SM effective theory from the previous section is its energy range of validity. This is a key issue if one is to reliably confront the predictions of the effective theory with experimental data, in particular those sensitive to energy scales E v. In this sense, extended (non-minimal) scalar sectors provide a very simple renormalizable completion to the D ≤ 6 SM effective theory, and allow for a quantitative assessment of the EFT's energy range of validity.
Moreover, extensions of the SM Higgs sector provide an attractive arena for the use of the SM effective theory: they are a simple scenario beyond the SM, well-motivated from the point of view of EW cosmology and baryogenesis, and may also be regarded as part of a complete theory beyond the Standard Model at the TeV scale, such as Composite Higgs scenarios or low-energy Supersymmetry. Assuming that the new scalar states are significantly heavier than m h , the effective theory of extended Higgs sectors provides a way to study these effects systematically. In particular, it is plausible that the new scalar states from the extended Higgs sector are very hard to probe at LHC (they might be very heavy, or have no decay branching fractions to SM gauge bosons), but their effect might be possible to detect either in precision EW measurements or using differential information in LHC measurements.
Below we construct the D ≤ 6 SM effective theory for various simple extensions of the SM scalar sector: a singlet extension of the SM (the so-called "Higgs portal"), a Two Higgs Doublet Model and an extension of the SM by a dilaton/radion. In each case, we obtain the Wilson coefficients for the D = 6 effective operators by matching to the UV theory, and perform an analysis of their current experimental bounds and future prospects. The results of this construction for the different scenarios is briefly summarized in Table 1.
In the respective matching procedure we demand that the effective action for the UV theory and the D ≤ 6 EFT agree after an expansion in the light degree of freedoms over the new physics mass scale. This results in the matching of the one-light-particle irreducible (1LPI) Green's functions in the full and effective theory -see [29] for a pedagogical introduction for these type of calculations. In the following we will perform an off-shell matching, where we expand in external momenta and the EW mass-scale over the new physics mass scale and keep equation of motion vanishing operators in the calculation until the final projection. Accordingly all propagators of SM fields will be massless after the expansion and we can perform the calculation in the SU (2) L × U (1) Y symmetric phase, quite analogous to the matching calculation performed in [30]. The light degrees of freedoms of the SU (2) L × U (1) Y symmetric phase comprise the gauge singlet B, triplet W a and octet G a fields as well as the scalar doublet Φ and the fermionic doublets and singlets. In practice we compute 1LPI Green's functions with up to 6 Table 1. Leading order at which the various Wilson coefficients for the D = 6 SM effective field theory are generated in each of the scenarios under consideration. In each case, the operator can be generated at Tree-Level (T) or 1-Loop (L). If some operators are generated at Tree-Level, this may lead to the generation of others via operator mixing under 1-loop Renormalization Group evolution (see e.g. [27,28]), which we denote by RG. Operators which are generated at higher order in RG and EFT expansion are denoted with an X.

The Singlet Higgs Portal: Doublet-Singlet Mixing
The addition of a singlet (real or complex) scalar field is arguably the simplest possible extension of the SM. Despite its minimality, this extension of the SM can have important consequences for the stability of the EW vacuum at high energies [31,32], and it could at the same time constitute a "Higgs portal" into a dark/hidden sector [33,34]. It may also have important consequences for Cosmology, potentially accounting for the dark matter relic density [35][36][37] or yielding a first order EW phase transition in the early Universe [38][39][40][41][42][43][44] that could explain the matter-antimatter asymmetry in the Universe through baryogenesis. In addition, it may give rise to interesting collider phenomenology (see e.g. [45][46][47][48]). Altogether, the singlet scalar extension of the SM constitutes a well-motivated scenario, and the interplay of different present and future experimental data to probe it has been widely studied (see [49][50][51] for up-to-date analyses). Let us then consider the SM scalar potential extended by a singlet scalar field s We assume initially that linear and cubic terms in s are absent from V (Φ, s), which may be achieved by means of a discrete/continuous symmetry G in the hidden sector, and focus on the scenario in which the field s develops a vev 3 , s → s + v s . This generates linear and cubic terms in s, which specific relations among these and the rest of parameters in the potential. After EW symmetry breaking, the scalar potential reads to trade the mass parameters µ 2 H , µ 2 S for the vevs. The term m 2 hs hs in (3.2) induces doublet-singlet mixing, leading to two mass eigenstates h 1,2 , the lighter of which (h 1 ) we identify with the discovered 125 GeV Higgs particle. The mixing angle θ and masses are given by The spontaneous breaking of a discrete or global continuous symmetry G would respectively lead to domain wall formation in the early Universe or the existence of massless Goldstone bosons, both features being undesirable in a realistic model. Possible solutions are to consider G to be a spontaneously broken gauge symmetry, or to allow for a small explicit breaking of the symmetry. We will disregard these issues in the following discussion.
Neglecting O(y 4 ) corrections, the couplings of h 1,2 to the W ± and Z bosons read Noting that y ≡ m 2 hs /m 2 s = m 2 hs /m 2 2 + O(y 3 ), we may integrate out the heavy, singletlike state h 2 . At leading order, this generates an O(y 2 ) contribution to h 2 1 V µ V µ , which precisely cancels the O(y 2 ) correction in (3.7). The Higgs-gauge interactions then read We turn now to the fermionic couplings, focusing on the Higgs-top quark interactions with Y t the top-quark Yukawa coupling. This affects the gluon fusion Higgs effective coupling for h 1 , which gets modified w.r.t the SM one The coupling G a µν G µν a h 2 1 (relevant for di-Higgs production) also gets modified, both through an O(y 2 ) correction due to the singlet-doublet mixing and directly via the presence of h 2 , which can mediate the process g g → h 1 h 1 (forŝ/m 2 s 1, beingŝ the partonic center of mass energy for the process, this contribution is however small).
The above discussion may be directly mapped into an SU (2) L × U (1) Y invariant effective field theory for the SM. We consider (3.1) after s develops a vev We then integrate-out the field s, which yields the effective Lagrangian From (2.2), this corresponds toc H = y 2 . The two contributions toc 6 are shown in Figure  1, and cancel by means of m 2 s = 2 λ S v 2 s (yieldingc 6 = 0). Upon EW symmetry breaking, c H modifies the wave function of the Higgs h which universally reduces all couplings of h: e.g. the coupling between h and the weak bosons simply read matching as expected the result from (3.8).
The previous results forc H are left unchanged by the inclusion of explicit linear and cubic terms for s in (3.1). These can however generate a non-zero value forc 6 , since now the cancellation among the diagrams in Figure 1 is not exact. Let us illustrate this by adding a term µ m |Φ| 2 s to (3.11). This term does not alter the minimization conditions, but contributes to the singlet-doublet mixing upon EW symmetry breaking, so that now the mixing is given by leading again toc H = y 2 , and now to λc 6 = 3 y 2 µ m /v s . The parameter µ m /v s ≡ δ measures the relative importance of explicit vs spontaneous symmetry breaking in (3.1).
Let us also comment on the case where the field s does not develop a vev and still the terms linear and cubic in s are absent from (3.1) (the symmetry G from (3.1) remains unbroken). In this case there is no Higgs-singlet mixing, but nevertheless the operator ∂ µ |Φ| 2 ∂ µ |Φ| 2 is generated at 1-loop [52] with a Wilson coefficientc H = ns λ 2 m v 2 96 π 2 m 2 s (being n s the number of singlet scalar degrees of freedom). The universal suppresion of Higgs couplings can then be defined in a similar fashion to the previous case, y 2 =c H .
We now discuss the current experimental constraints on the mass of the singlet-like state and the parameter y, coming from the latest measurements of Higgs boson signal  Table 2. We then perform a combined χ 2 fit with the various µ i ≡ µ i obs /µ i SM and ∆µ i taken from Table 2, and with potential correlations among the different signal strength measurements not included in the fit. This yields an ATLAS and CMS combined limit y < 0.468 at 95% C.L. via ∆χ 2 h (y) = χ 2 h (y) − χ 2 min = 4, as shown in Figure 2 (horizontal solid-black). In addition, we obtain the 95% C.L. exclusion prospects for LHC at 14 TeV with L = 300 fb −1 (horizontal dotted-black) and L = 3000 fb −1 (HL-LHC, horizontal dashed-black). In doing so, we assume that future measurements of Higgs signal strengths will yield µ i = 1, and use the projected CMS sensitivities 4 Table 2 (horizontal solid-black) and from EWPO (blue). The shaded grey region is excluded at 95% C.L. by the combination (EWPO and Higgs signal strengths). The shaded yellow region may also be excluded by constraints from heavy scalar searches at LHC (dashed-dotted black), but these limits can be evaded in the presence of non-SM decays (see text). Also shown is the projected exclusion reach from Higgs signal strengths at the 14 TeV run of LHC with L = 300 fb −1 (horizontal dotted-black) and at HL-LHC with L = 3000 fb −1 (horizontal dashed-black), from measurements of the S and T oblique parameters with ILC-GigaZ and TLEP (blue) and their combination with the HL-LHC exclusion reach (red), and from a precision measurement of the Higgs production cross section in association with a Z boson σ(hZ) (horizontal solid-green lines) at ILC ( √ s = 250 GeV, L = 1150 fb −1 ) and TLEP ( √ s = 240 GeV, L = 500 fb −1 ).
Turning to EWPO, we perform a fit to the oblique parameters S, T, U using the best-fit values and standard deviations from the global analysis of the GFitter Group [64], with a SM reference point with m t = 173 GeV and a 126 GeV Higgs mass. Under the assumption U = 0, this yields being ρ ij the covariance matrix in the S − T plane. The BSM corrections to S and T in the case of singlet-doublet mixing are given by with the functions H S (x) and H T (x) defined in Appendix C of [65]. We then define where ∆O 0 i denote the central values in (3.17) and (σ 2 ) ij ≡ σ i ρ ij σ j , being σ i the S and T standard deviation from (3.17). We show in Figure 2 the 95% C.L. exclusion limit y(m s ) from ∆χ 2 EW (m s , y) (blue), together with the 95% C.L. exclusion limit y(m s ) from the combination of ∆χ 2 h (y) and ∆χ 2 EW (m s , y) (red). We also study the future exclusion reach that can be derived from prospects of measurements of EW precision observables by planned e + e − colliders (see e.g. [66]): Assuming a SM best-fit value, the ILC GigaZ program's expected precision is σ S = 0.017 and σ T = 0.022 [64], while measurement of EWPO at TLEP could yield σ S = 0.007 and σ T = 0.004 [67,68]. Figure 2 includes the 95% C.L. exclusion reach for y(m s ) both for ILC and TLEP (blue), as well as the respective 95% C.L. exclusion reach when combined with HL-LHC (L = 3000 fb −1 ). Figure 2 also shows the potential 95% C.L. exclusion reach from a precise measurement of the Higgs production cross section in association with a Z boson σ(hZ) at ILC with √ s = 250 GeV and L = 1150 fb −1 (with ∆σ(hZ)/σ(hZ) ∼ 0.012) [69] and at TLEP with √ s = 240 GeV and L = 500 fb −1 (with ∆σ(hZ)/σ(hZ) ∼ 0.004) [67]. From Figure 2 it is evident that this precision measurement is potentially the most powerful probe of a non-SM Higgs admixture, significantly surpassing the reach of EWPO measurements.
It is worth stressing that the contributions to the S and T oblique parameters in the Higgs portal scenario could be re-derived in the EFT approach via operator mixing under renormalization group evolution (RGE), as shown e.g. in [70]. Specifically, the running of c H from the matching scale m s down to m Z generates a contribution to S and T , given at leading order by [70,71] which correspond to the leading order contributions in (3.18) for m s m Z .
Finally, Figure 2 includes the latest 95% C.L. limits on y(m s ) (dotted-dashed black line and yellow region) obtained from CMS searches of a heavy neutral scalar decaying to ZZ → 4 , ZZ → 2 2j and ZZ → 2 2ν final states [72] (searches in decays to W W → 2 2ν final states [73] are found not to be as sensitive). These constitute at present the most stringent constraint on y for m s 500 GeV, but we stress that for m s > 2m h = 250 GeV these limits may be weakened/avoided for a significant branching fraction Br(h 2 → h 1 h 1 ).
From the present limits and future prospects discussed above, we can derive current/projected bounds on the size of the Wilson coefficientsc H (m Z ) andc 6 (m Z ) in this scenario. Fromc H (m s ) = y 2 and λc 6 (m s ) = 3 y 2 µ m /v s , we perform a 1-loop RGE [27] (see also [28]) For the case of λc 6 (m Z ), the region leading to a strongly first order EW phase transition as required by EW baryogenesis is shown as a horizontal orange band, while the grey (brown) region is currently excluded at 95% C.L. for δ ≡ µ m /v s = 1 (δ = 5).
Current bounds onc H (m Z ) andc 6 (m Z ) as a function of the new physics scale m s , as well as the projected 95% C.L. exclusion sensitivity from the combination of HL-LHC and GigaZ, and HL-LHC and TLEP EWPO (TeraZ) measurements, are shown in Figure 3.
As recently noted in [70], a sizeablec 6 (m Z ) could in this scenario lead to a strongly first order EW phase transition, potentially allowing for EW baryogenesis in the Early Universe [74][75][76]. Requiring a sufficiently strong first order phase transition imposes a lower bound λc 6 (m Z ) 0.105 5 , while the successful completion of the phase transition sets the upper bound λc 6 (m Z ) 0.211 [75]. As shown in Figure 3 (Right), the present combination of measured Higgs signal strengths and EW precision data rules out a strong EW phase transition, except for very large values δ ≡ µ m /v s 1 combined with relatively low singlet masses m s . These results are somewhat stronger than those of [70], the reason being that the analysis of the EW phase transition in the presence ofc 6 from [74] (with reference [70] follows) considers thermal masses as the only thermal effects in the effective potential, and tends to overestimate the strength of the EW phase transition as compared to a full 1-loop analysis [75]. Finally, while we stress that our results certainly do not rule out a strong EW phase transition in the singlet scalar extension of the SM (see e.g. [43]), they imply that having it originate from the presence ofc 6 is currently challenging, and the entire region of parameter space will be covered by the combination of HL-LHC and GigaZ.
Let us finish this section on the extension of the SM Higgs sector via a singlet scalar by summarizing the results. We have seen that the EFT leads to just two non-zero operators at tree-level,c
We consider here 2HDM scenarios with a Z 2 -symmetry which is at most softly broken, avoiding tree-level Flavor Changing Neutral Currents (FCNCs) in the Yukawa sector [102]. This Z 2 -symmetry leads to four types of 2HDM (see e.g. [83] for details), according to the way in which Φ 1,2 are coupled to the different SM fermion species, as shown in Table 3. The scalar potential for the two scalar doublets Φ 1,2 reads

Up-type Quarks Down-type Quarks Charged Leptons
In the following, we consider CP-conserving scenarios and set λ 5 and µ 2 to be real. The doublets and their vevs at the EW minimum can be written as with v 2 1 + v 2 2 = v = 246 GeV and v 2 /v 1 = tan β. The angle β also parametrizes the rotation to the mass eigenbasis for the charged states G ± , H ± and neutral CP-odd G 0 , A 0 states, with G ± , G 0 being the Goldstone bosons and H ± , A 0 the physical states. We likewise define α to be the mixing angle parametrizing the rotation to the mass eigenbasis for the CP-even neutral states h, H 0 (see Appendix A.1). The µ parameter is responsible for the soft-breaking of the Z 2 -symmetry in (3.25). After EW symmetry breaking, the parameters µ i , λ i in (3.25) may be written in terms of the masses of the physical states m h , m H 0 , m A 0 , m H ± , the mixing angles α, β and the µ parameter, as shown in Appendix A.1.
In order to obtain an EFT for the SM Higgs, we may perform an SU (2) rotation from the field basis Φ 1 , Φ 2 to a basis H 1 , H 2 in which H 1 = v √ 2 and H 2 = 0 (the socalled Higgs basis). This rotation is precisely parametrized by the angle β. After the field rotation, the scalar potential for H 1,2 reads withμ i ,μ andλ i being functions of the original parameters in (3.25). We note that the field rotation may generateλ 6,7 even if initially absent from (3.25) due to the Z 2 -symmetry.
We may then construct an effective Lagrangian for H 1 by matching to the theory with the second doublet H 2 at the scaleμ 2 , assumingμ 2 2 v 2 . We stress however that the doublet H 1 can only be fully identified with the SM Higgs doublet in the alignment limit cos(β − α) ≡ c β−α → 0, where mixing in the CP-even sector is absent (see the discussion in Appendix A.1). Away from alignment (α = β − π/2), these mixing effects lead to tree-level modifications of SM Higgs couplings (see e.g. the discussion in [103]). For a general 2HDM, c β−α = 0 is possible at tree-level 6 [104,105], and indeed the latest analyses of Higgs data by ATLAS and CMS strongly prefer c 2 β−α 1, as shown in section 3.2.3.
In the following, we introduce and discuss two benchmark scenarios for the use of Higgs Effective Theory in 2HDM: An exact alignment scenario c β−α = 0 and a scenario with |c β−α | 1 (MSSM-like scenario). We then analyze the constraints on the 2HDM parameter space obtained from measurement of Higgs signal strengths and EW Oblique Parameters, and the corresponding constraint on the values of the Wilson coefficients of the Higgs EFT.

Benchmark A: Exact
Alignment c β−α = 0 As discussed above, in this scenario mixing effects are absent and H 1 is precisely the SM Higgs doublet Φ. The scalar potential (3.27) simplifies in this limit, sinceλ 6 andμ are both ∝ c β−α and vanish as shown in (A.15)-(A.16). We then match the 2HDM to an SU (2) L ×U (1) Y invariant EFT with the field content of the SM, being H 2 the only massive field which will be decoupled in the matching calculation. The D = 6 effective operators from (2.1) are generated first at 1-loop order, with the corresponding Wilson coefficients given byc The three operators in (3.30) share the common feature that they do not involve derivatives of the Higgs field H 1 . This very property is in fact responsible for the relations among Wilson coefficients in (3.28), since the Feynman diagrams involved in the EFT matching (see Figure 4-Left) do not involve Higgs field derivatives. Upon departure from the alignment limit, mixing in the CP-even sector leads to several effects that are absent for c β−α = 0: First, there is a modification of the couplings of the Higgs boson h to gauge bosons and fermions at tree-level, parametrized in terms of κ-factors for vector bosons κ V ≡ g hVµVν /g SM hVµVν and fermions κ f ≡ g hf f /g SM hf f (for up-type quarks κ u , down-type quarks κ d and leptons κ ). For the various Types of 2HDM from Table 3, these are given in terms of c β−α and t β ≡ tanβ as In addition, away from alignment g H 0 hh (coupling between H 0 and two Higgs bosons h), g H 0 VµVν (coupling between H 0 and two gauge bosons V µ ) and g φVµh (couplings among h, a gauge boson V µ and φ = A 0 , H ± ) are non-zero, giving rise to 1-loop diagrams contributing to the vertices V µ V ν h and V µ V ν V ρ with both heavy (H 0 , A 0 , H ± ) and light (h, V µ ) states running in the loop.
Let us discuss all these effects in an SU (2) L × U (1) Y invariant Higgs EFT approach. We first note that H 1 = v √ 2 and H 2 = 0 in (3.27) imply the relations which may be simply regarded as minimization conditions. The mass matrix for the neutral CP-even states is non-diagonal, and is brought into a diagonal form via the rotation matrix U The deviation from alignment is then parametrized by (λ 6 v 2 )/μ 2 2 , which matches the corresponding expression for c β−α after EW symmetry breaking (see Appendix A.1) recovering the well-known scaling result for c 2 β−α in 2HDMs as the decoupling limit is approached [104].
At tree-level, the EFT matching generates effective operators suppressed at least by 1/μ 4 2 (except forc 6 , which receives a further contribution suppressed only by 1/μ 2 2 ), with -17 -operators of different D contributing to order v 4 /μ 4 2 : The matching procedure yieldsc 6h =λ 2 6 ,c 4h =c H = 0 andc 2h = 1, and use of the relatioñ µ 2 =λ 6 v 2 results in both the first and the third term in (3.40) contributing at order (λ 6 v 2 ) 2 /μ 4 2 . The first term rescales both the SM Higgs kinetic term and its couplings to gauge boson by the same amount, so it does not have a net effect. The third term does however include a rescaling of the SM Higgs kinetic term ∝ c 2 β−α that is not compensated by a similar one in the gauge boson interactions, leading to the well-known tree-level deviation from the SM Higgs couplings to gauge bosons away from alignment, proportional to c 2 β−α . Regardingc 6 , the presence of a non-zeroλ 6 also yields an extra contribution at 1-loop as compared to (3.28). The full result forc 6 away from alignment is Let us also stress that similarly to the tree-level effects just discussed, the extra 1loop diagrams appearing away from alignment (involving both light and heavy degrees of freedom) are proportional to c 2 β−α . Thus the contribution of these diagrams to the various Wilson coefficients from (2.2) and (2.
and f u,d, depending of the 2HDM-Type under consideration, and given by These results reproduce the κ-factors from (3.33-3.36) to leading order in c β−α .

2HDM Constraints: Higgs Signal Strengths & EW Oblique Parameters
The ATLAS and CMS measurements of Higgs boson signal strengths from Table 2 constrain the allowed parameter region of the 2HDM in the plane (c β−α , tan β) (see e.g. the analyses from [84, 86-88, 90, 97]) through a combined χ 2 fit, which in this case treats the different signal strength measurements as independent (not correlated) (3.48) Eachμ i (the expected signal strength for each channel) may be expressed in terms of rescaling κ-factors for the coupling of the Higgs boson to vector bosons κ V , up-type quarks κ u , down-type quarks κ d and leptons κ . The expression of eachμ i corresponding to each process considered in Table 2 in terms of these κ-factors is given in table 4 with κ i defined in (3.33-3.36) and κ γ (κ u , κ V ) = 1.26 κ V − 0.26 κ u , and Γ(κ d , κ V , κ u , κ ) given by Γ(κ d , κ V , κ u , κ ) = 0.2427 κ 2 V + 0.1124 κ 2 u + 0.578 κ 2 d + 0.0637 κ 2 (3.49) We present the 68%, 95% and 99% C.L. limits on the (c β−α , t β ) plane from global fits of the light Higgs boson couplings, given respectively by ∆χ and Flipped (Lower-Right). In each case, the projected 95% C.L. exclusion limit for LHC at 14 TeV with L = 300 fb −1 (dashed-black) and L = 3000 fb −1 (dotted-black), assuming the measured Higgs signal strengths being µ i = 1, is also shown.
2HDM in Figure 5. These are particularly stringent for Type II and Flipped as compared to Lepton-Specific and specially to Type I, due to the different nature of the Higgs coupling to down-type quarks. Figure 5 also includes the projected 95% C.L. exclusion limits for LHC at 14 TeV with L = 300 fb −1 (dashed-black) and L = 3000 fb −1 (HL-LHC, dotted-black) assuming the measured Higgs signal strengths being µ i = 1 and using the projected CMS sensitivities from Section 3.1. We now analyze the constraints from EWPO, performing a fit to the oblique parameters S, T under the assumption U = 0 using the best-fit values and standard deviations from the global analysis of the GFitter Group [64] (m t = 173 GeV and m h = 126 GeV), as shown in (3.17). The 2HDM contributions to ∆S and ∆T are given by [106] with G A,B,C ,Ĝ A,B and F A,B given in Appendix A.2. We note that both ∆S and ∆T are independent of t β . We define and construct a ∆χ 2 EW function as with ∆O 0 i and (σ 2 ) ij defined after (3.19). The results of the fit are shown in Figure 6 for various values of c β−α and m H ± = 400, 600 GeV.
The calculation of the oblique S and T parameters in the 2HDM matches with the D = 6 effective operators present in the EFT. Considering first the alignment limit and expanding (3.50) and (3.51) around x 0 , x A ∼ 1 (as indicated by the present 95% C.L. experimental limits), ∆S and ∆T in the 2HDM read where we note that the functions G A,B,C scale as (1 − x A,0 ) 2 and can be neglected with respect to the log(x 0 /x A ) term in ∆S. By means of (A.13) and (A.14) in Appendix A.1, the above relations can be rewritten as For m 2 H ± v 2 , and upon the replacement m 2 H ± →μ 2 2 , these match the EFT results (3.28) where we have also translated into the Altarelli-Barbieri parametrization [107] for reference. The bounds on these effective operators are [64] c T (m Z ) ∈ [−1.  [27], whose mixing at leading order would yield However, for the 2HDMc H (μ 2 ) = 0 at tree-level, and thus the Wilson coefficient responsible for the contribution to ∆S and ∆T dependant on c 2 β−α isc 6h [26].  [20]. The purple-solid region corresponds to the combined limits.

Alignment limit: complementarity of EWPTs and LHC limits
In the previous section we discussed how LEP electroweak constraints translate into the full theory (3.28), and in turn in the effective theory (3.56) which affected the operators c T and a combinationc W +c B . It is time to move onto the constraints from LHC, the most stringent ones on operators affecting the decay of the Higgs to photons (c γ ) and production through gluon fusion (c g ). Other operators are better determined by looking at other production and decay channels. In particular, three combinations [18] c W −c B , c HW andc HB , (3.60) are constrained with the help of Higgs production in association with a vector boson, and with diboson data [20]. In the 2HDM, the global fit is more constraining than a fit for a general EFT at dimension-six level. In particular, by varying all the free parameters in the 2HDM simultaneously in a fit to Higgs plus diboson data, one obtains the following 95% C.L. regions [20]c HW ∈ (0.0004, 0.02) c g ∈ −(0.00004, 0.000003) c γ ∈ (−0.0006, 0.00003) (3.61) Whereas the constraint on the T parameter favours aλ 5 close toλ 4 , the constraint onc γ from the LHC restricts the values ofλ 3 . The most interesting case is when we compare constraints from LEP and LHC which set limits on the coupling λ 4 and the scale of new physics. Indeed, one can see this interplay between the two sources of data in Figure 7, where we present results in theλ 4 andμ 2 plane. The red-dotted lines enclose the region allowed by constraints on the S parameter, and the black-dashed lines are the corresponding LHC limits from [20]. The purple-solid region corresponds to the combined limits and it is quite constraining as the two regions have little overlap. Note that the preferred region at 95% C.L. is consistent with the presence of light particles. This is due to the slightly non-zero positive value ofc HW in the global LHC fit and the tension with LEP limits, which favour negative values ofc HW . This hint for new physics is gone at the level of 3σ.
Finally, we comment on the 2HDM and the impact on the EW phase transition. As recently shown in [100], a first order EW phase transition strongly favours a large m A 0 −m H 0 splitting, which via (A.16) impliesλ 5 < 0 and sizable. Since this implies that it is not possible to simultaneously have vanishing ∆S and ∆T unlessμ 2 → ∞, it would be possible to observe a deviation from the SM in ∆S or ∆T as the experimental precision increases (see Figure 8).

A Dilaton/Radion Scenario
So far we have discussed scenarios where the operators involving the Higgs and massive vector bosons, such asc W , are generated at loop level. In this section we present a case where they appear at tree-level, through the exchange of a dilaton or a radion scalar particle. We consider two equivalent scenarios: • The extra-dimensional radion: A radion r is the excitation of the graviton perturbation in extra-dimensions along the direction of the extra-dimension. The interactions of the radion with SM particles are then obtained by expanding the metric at linear order in r. We will consider conformal metrics of the form where w(z) is the warp factor, which depends on the extra-dimension z. Minkowski space corresponds to w = 1 and in Anti-deSitter metrics w = 1/(kz), with k the curvature in the extra-dimension. The extra-dimension is compactified, with z stretching between two points which we will call the UV and IR branes, i.e. z ∈ [z U V , z IR ]. The interaction then reads where T µ µ is the trace of the stress-energy tensor. The dimensional reduction of the interaction in (3.63) to a four-dimensional (4D) interaction depends on the localization of the radion and the SM particles, and generally speaking it would look as where now r and T i = T r(T i µν ) are the dimensionally-reduced radion and trace of the stress-energy tensor of species i. Λ corresponds to the compactification scale, and the coefficients c i account for the overlap of different bulk species i with the radion.
In warped extra-dimensions, the radion is a field localized near the IR brane at z IR . Therefore, for fields localized near the IR brane, c i O(1), for fields delocalized in the bulk (with a flat profile) one finds c i 1 z U V z IR w(z) dz and for fields on the UV brane, c i with a a positive number [108]. As an example, in Anti-deSitter (AdS) models, the suppression for bulk fields is w(z)dz = log z U V z IR , which in Randall-Sundrum translates into log(M P /TeV) O(30).
Although massless gauge fields do not contribute to the trace of the stress-energy tensor at tree-level, loop-level couplings are generated by the trace anomaly where i runs over the gauge groups of the SM, SU (3) c × SU (2) L × U (1) Y , α i = g 2 i /4π and g i are the respective coupling constants. The b i are the β function coefficients leading to anomalous scale-invariance violations (dilaton) or, equivalently, contributions from fields localized near the IR brane or bulk running (radion) [109,110]. The values of b i depend on the degree of compositeness of fermions in the SM and possible new CFT contributions. For simplicity we neglect these and use the pure CFT value, b CF T i = 8π 2 /(g 2 i log(z U V /z IR )). In this limit the coefficient of the anomaly is then independent of the value of the gauge coupling

66)
• The dilaton of spontaneous breaking of scale invariance: Assume the existence of a new sector whose couplings are scale invariant, but this scale invariance is spontaneously broken by some dynamics. For example, in QCD scale invariance is spontaneously broken by confinement dynamics, leading to quark condensates. In our example, we are considering a new strongly coupled sector, possibly with similar dynamics as QCD. The Goldstone boson of this spontaneous breaking is a dilaton. Let us denote the dilaton by r as in the radion case. Non-derivative couplings of the dilaton to SM fields are proportional to the explicit breaking of conformal invariance, namely they are of the form with f the scale of the spontaneous breaking of dilatation symmetry and J µ the global conserved current for this symmetry, J µ = T µν x ν . Therefore the coupling of the dilaton to the SM fields take the same form as those of the radion (3.64) [111]. We stress that the quadratic couplings to SM particles do not have the same form for the radion and dilaton [111], but this difference will not affect the matching of dimension-six coefficients.
The radion/dilaton mass is a model-dependent parameter, related to the mechanism of stabilization of the extra-dimension or the explicit breaking of dilatation symmetry. In absence of stabilization the radion is massless but one could stabilize this dilaton/radion in several ways, and no definite prediction of the mass can be drawn unless we specify the mechanism. For example, in the Goldberger-Wise mechanism [112] in warped extradimensions the radion mass is a function of the vev and mass of the stabilizing field [113] and could be very light as well as around the scale of compactification (or spontaneous breaking) f .
We proceed to integrate out the dilaton assuming its mass m r is larger than the scales we are probing with colliders. The effective Lagrangian has the form where T is the trace of the stress-energy tensor. For the Higgs and gauge bosons, where V = −m 2 h |Φ| 2 + λ|Φ| 4 . One can then extract the Wilson coefficients of the effective operatorsc We stress again that for a pure CFT value b CF T i , b 1 α 1 = b 2 α 2 = b 3 α 3 . We could also consider a more general situation, where the coupling of the dilaton/radion is not universal. This amounts to setting different values of the coefficients c i in (3.64) and would lead to a prefactor in the coefficients of the effective operators (3.70), dependent of the degree of overlap of the wavefunctions in the bulk (radion) or participation on the composite dynamics of the species (dilaton).
While the contributions to ∆T vanish in this case at tree-level [114,115], they will be generated via 1-loop RG running. The corresponding expressions forc T (m Z ) andc W (m Z )+ c B (m Z ) are given by (3.58-3.59) with the substitutionμ 2 → m r and usingc H (m Z ) from (3.70). In Figure 8 we show the results on the ∆S − ∆T plane, and compare the correlation in the oblique parameters with that appearing in the ones encountered respectively in the Higgs portal and 2HDM scenarios from Section 3.1 and 3.2.

The validity of the EFT: from operators to total rates and distributions
The EFT approach, where higher-dimensional operators are written as a way to encode effects of New Physics, is a good approximation at low energies and should not be used in an arbitrary range of energies. Indeed, as these operators are suppressed by some New Physics scale, parton level cross-sections would diverge with the parton energy. This growth with energy is a signature that the EFT approach breaks down when one is able to probe the dynamics of the heavy particles one integrates out. In hadron colliders, this corresponds to the moment when the partonic energy √ŝ is comparable with the masses in the loop. Specifically, for a tree-level exchange as in Figure 1, √ŝ M (4.1) The same argument applies to the dilaton exchange, and to fermion couplings in 2HDM as in Figure 4. On the other hand, in loop-induced processes as in Figure 11 the validity of the EFT extends to the threshold to produce a pair of new particles, namely At around these threshold energies, the EFT description should be substituted by the full UV theory including the resonance dynamics. The question we would like to ask is what is the deviation of the EFT predictions from the full UV model, which depends on the process and distribution one is looking at. In this section we present an example of a specially useful distribution [116] to probe New Physics, namely the invariant mass of the Higgs and vector boson system in Higgs associated production, m V H . This invariant mass is the parton energy,ŝ = m 2 V H . In Figure 9 we show how the invariant mass distribution and a correlated variable, the transverse momentum of the leading lepton in Z → for a specific value ofĉ HW . We related this value to a mass scale Λ through We have drawn lines corresponding to the coupling of new physics 1 and 0.5, to illustrate the difference between UV-completions with weak and strong couplings. In Figure 10, we show the distribution at the LHC13 for the SM case, and for a values ofc HW = 0.1, which is excluded by Run1 data [20] but helps to visualize the effect of these operators in the distribution. At current level of exclusion,c HW 0.02, this effect is not visible in the plot. We see that, even when the value of the quartic coupling is large, the validity of the effective theory is very limited and the theory prediction for the distribution should include the UV model dynamics. This illustrates the need of working with benchmark models to interpolate between the EFT analysis and UV models. The distributions are the result of a simulation including PDF effects (NN23LO1) performed in MG5 MC@NLO [117] using the model implementation in [118], based on Feynrules [119] and the UFO [120] format. Note that this distribution does not grow with √ŝ , but falls down due to the PDF effects.
As we have computed the distributions in the context of the EFT, they do not show the quantitative deviation from the EFT to the full UV model, an analysis which would require implementing a form factor in the scattering amplitude, and is beyond the scope of this paper.
Besides this distribution, one can find others sensitive to New Physics effects, e.g. m jj and ∆y jj in Vector Boson Fusion [121], or ∆R hh and p hh T in di-Higgs production [122].

Discussion and Summary
During Run2 of the LHC, interpretation of data using the Effective Field Theory (EFT) approach will become a standard way to communicate results to theorists, as the translation to UV models is more direct than, for example, the use of form factors. The approach based on anomalous couplings and the one considered here are related. New Physics affects the behaviour of SM particles, inducing anomalous couplings, for example involving the Higgs and electroweak bosons. In the context of heavy New Physics, the anomalous couplings can be translated into combinations of higher-dimensional operators as shown in Tables 5 and 6.
The use of the EFT approach, however, is limited to heavy new physics, whereas current LHC limits strive to reach the TeV region, leading to a delicate balance between using as much data as possible and the model-independent EFT approach. The question of what LHC data is suitable to constrain EFTs depends both on the UV completion and the type of signal one is looking at. Benchmark models allow to address these issues, and also to draw correlations with direct searches and other non-LHC sources of indirect constraints such as flavour physics and LEP. In this paper we have taken a first step towards this program by studying the matching between UV completions encompassing extensions of the Higgs sector and the EFT.
This matching allowed us to study the suitability of LHC data to constrain EFTs. For example, one can consider UV completions with large couplings where the validity of the EFT is improved, such as the 2HDM in Section 3.2 with large quartic couplings. But even for weakly-coupled UV completions, the use of the EFT may be justified as long as the LHC data one uses is restricted to small values of √ŝ , below the masses of new particles, as discussed in Section 4.
We have shown the matching of the UV theory with the low-energy coefficients in detail. This matching is straightforward in the case of the tree-level dilaton exchange (Section 3.3) or mixing with a singlet (Section 3.1). For loop-induced dimension-six operators, however, the interplay of with higher order terms (dimension-eight operators) needs to be handled with care as shown in Section 3.2.2.
Another advantage of the comparison with UV completions is to address correlations among the EFT coefficients which are present in models and reduce the number of free parameters in a global analysis. For example, in our benchmarks specific relations among operators can be traced back to the (limited) Lorentz structures one can build up from scalar fields, see discussion around (3.28).
Note that we have not discussed the very interesting possibility of CP-violating effects from, for example, complex parameters in the 2HDM. This deserves further study, as their kinematic distributions and dependence with energy have to be simulated as close as possible to the actual cuts applied by the LHC collaborations.
We stress that since the masses of the physical states m H ± , m A 0 , m H 0 , m h and the mixing angles α, β are obtained upon EW symmetry breaking, the relations (A.3-A.7) only hold in the EW broken theory. expressions for the various oblique parameters in models with an arbitrary number of scalar doublets may be found in [106]), with the functions F A,B , G A,B,C ,Ĝ A,B being Finally, we stress that the potential (A.10) preserves custodial symmetry in the limitλ 4 = λ 5 (see e.g. [123]), which from (A.13) corresponds to m A 0 = m H ± . To see this, instead of expressing the 2HDM scalar potential in terms of H 1,2 , we can introduce the 2 × 2 matrices Φ 1 = (iσ 2 H * 1 , H 1 ), Φ 2 = (iσ 2 H * 2 , H 2 ). The scalar potential for the 2HDM then reads Both Φ 1,2 transform as bi-doublets of a global symmetry SU (2) L × SU (2) R : Φ i → LΦ i R (with L ∈ SU (2) L and R ∈ SU (2) R ). The potential (A.19) is then invariant under a custodial SU (2) L × SU (2) R in the absence of the term proportional toλ 4 −λ 5 .

B Effective Theory Matching in the EW Broken Phase
We now relate the Wilson coefficients from the D = 6 operators for the SM SU (2) L ×U (1) Y invariant effective field theory given in (2.2), (2.5) and (2.6) to the SM effective Lagrangian after EW symmetry breaking. The complete set of relations from the full D ≤ 6 effective Lagrangian (2.1) may be found in [25]. In the following we present the generic relations for the 3-and 4-point interactions involving Higgs/gauge bosons. The CP-conserving 3-point interactions involving at least one light Higgs scalar h are obtained from L SILH and read where we have introduced abelian field-strength tensors W µν , Z µν and F µν for the W -boson, Z-boson and photon respectively, and G a µν is still the non-abelian gluon field-strength tensor. The CP-conserving 4-point interactions involving at least one light Higgs scalar h similarly read We note that the 4-point interactions involving one Higgs h and three gluons (relevant, e.g. for Higgs production in association with a jet via gluon fusion) are already taken into account by the g hgg term in (B.1). In a similar fashion to (B.1) and (B.2), the CP-odd 3and 4-point interactions involving at least one light Higgs scalar h are obtained from L CP , yielding Finally, the 3-and 4-point gauge boson self-interactions receive CP-even contributions from both L SILH and L G (they also receive CP-odd contributions from L CP , which we do not include here). We adopt here the parametrization of triple gauge couplings (TGCs) from [10,124], which yields with g γ 1 = g Z 1 = κ γ = κ Z = 1 and λ γ = λ Z = 0 for the SM. U (1) EM gauge invariance imposes g γ 1 = 1, while the (spontaneously broken) SU (2) L × U (1) Y gauge invariance combined with the restriction to D ≤ 6 effective operators in L Eff lead to the relations [10] These degeneracies imply that there can be at most three independent contributions to anomalous TCGs from L SILH and L G . The operators proportional to g Z 1 , κ Z and κ γ in (B.4) contain one derivative and are obtained from L SILH , which then gives rise at most to two independent contributions to anomalous TGCs. The last two operators in (B.4) contain three derivatives and are obtained from L G , which then gives rise to a sole linearly independent contribution to anomalous TGCs. This is clear for the operatorc 3W O 3W , as it is constructed from three gauge field-strengths and only their abelian parts can contribute to TGCs (after EW symmetry breaking it directly leads to the last two operators in (B.4) with λ γ = λ Z ). The operatorc 2W O DW also leads after EW symmetry breaking to those operators with λ γ = λ Z , but also produces operators of the form These operators however either vanish or reduce to operators from (B.4) for on-shell gauge bosons (as well as for the virtual photon). Therefore, the parametrization of TGCs (B.4) holds. Moreover, we can safely disregard the contribution ofc 2W to λ γ = λ Z in the following discussion, since this operator can be re-casted as (very constrained) fermionic currents and the operators involving the Higgs we have already considered [125]. For the case of quartic gauge couplings (QGCs) we have hzz = g (1) hww + 2g hww = g m W , g hhww = g 2 2 , g hzww = s W c W g   Table 5. Relations between the different couplings appearing in L 3h , L 4h ,L h and the Wilson coefficients for the D ≤ 6 effective operators in L Eff .
where again g W 2 = g γ 2 = g Z 2 = g γZ 2 = 1 and λ W = λ γW = λ γZ = λ W Z = 0 corresponds to the SM case. L 3V Couplings vs SU (2) L × U (1) Y (D ≤ 6) Wilson Coefficients Table 6. The Trilinear and Quartic Gauge Couplings appearing in L 3V and L 4V as a function of the Wilson coefficients for the D ≤ 6 effective operators in L Eff .
The correspondence between the Wilson coefficientsc i for the effective operators from (2.2), (2.5), (2.6) and the various couplings arising after EW symmetry breaking in (B.1-B.4), (B.7) is shown in Tables 5 and 6. In the next section, we compute the values of these couplings by direct matching in the broken EW theory, for the specific setup of Benchmark A in 3.2.1: the 2HDM in the alignment limit c β−α = 0.
B.1 A Specific Example: 2HDM in the Alignment Limit c β−α = 0 Here we will obtain the values of the various couplings from (B.1) for the 2HDM in the alignment limit c β−α = 0, by direct matching in the broken EW theory. We focus on the V µ V ν h interaction vertices, which receive contributions from loops of scalars H ± , A 0 , H 0 (these are shown in Figure 11 for the case V µ = Z µ ). We note that loops involving only SM particles cancel in the matching between full and effective theories and may then be disregarded, while loops involving both SM particles and H ± , A 0 , H 0 scalars vanish in the alignment limit.