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The 2-Higgs-Doublet Model (2HDM) belongs to the simplest extensions of the Standard Model (SM) Higgs sector that are in accordance with theoretical and experimental constraints. In order to be able to properly investigate the experimental Higgs data and, in the long term to distinguish between possible models beyond the SM, precise predictions for...

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... application of the renormal- ization scheme worked out in Ref. [29]. In Appendix A we show in detail how this scheme works and in particular we present its extension from the SM case [29] to the 2HDM. The generic diagrams contributing to the self-energies defined in this 'alternative tadpole' scheme, called Σ tad in the following, are shown in Fig. 1. Besides the generic one-particle irreducible (1PI) diagrams depicted by the first two topologies in Fig. 1, they also contain the tadpole diagrams connected to the self-energies through the CP-even Higgs bosons h and H that are represented by the third topology. The application of the tadpole scheme alters the structure of the mass ...

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... this scheme works and in particular we present its extension from the SM case [29] to the 2HDM. The generic diagrams contributing to the self-energies defined in this 'alternative tadpole' scheme, called Σ tad in the following, are shown in Fig. 1. Besides the generic one-particle irreducible (1PI) diagrams depicted by the first two topologies in Fig. 1, they also contain the tadpole diagrams connected to the self-energies through the CP-even Higgs bosons h and H that are represented by the third topology. The application of the tadpole scheme alters the structure of the mass counterterms and of the off-diagonal wave function renormalization constants 4 such that now the ...

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... are neither IR divergences nor real corrections. The generic diagrams for the virtual corrections and the counterterm are depicted in Fig. 9. The 1PI diagrams contributing to the vertex corrections are given by the triangle diagrams with scalars, fermions, massive gauge bosons and ghost particles in the loops, as shown in the first three rows of Fig. 10, and by the diagrams involving four-particle vertices (last four diagrams of Fig. 10). The corrections to the external leg in Fig. 9 (b) vanish due to the OS renormalization of the H. The mixing contributions (c) and (d) vanish because of the Ward identity for the OS Z boson. The counterterm amplitude is given ...

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... corrections and the counterterm are depicted in Fig. 9. The 1PI diagrams contributing to the vertex corrections are given by the triangle diagrams with scalars, fermions, massive gauge bosons and ghost particles in the loops, as shown in the first three rows of Fig. 10, and by the diagrams involving four-particle vertices (last four diagrams of Fig. 10). The corrections to the external leg in Fig. 9 (b) vanish due to the OS renormalization of the H. The mixing contributions (c) and (d) vanish because of the Ward identity for the OS Z boson. The counterterm amplitude is given ...

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... the µ * denote the polarization vectors of the outgoing Z bosons with four-momentum p 3 and p 4 , respectively. If the tadpole scheme is applied, the HZZ vertex is modified by additional Figure 11: Additional vertex diagrams in the tadpole scheme contributing to the decay H → ZZ. ...

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... contributions, which lead to further diagrams, that have to be taken into account in the computation of the decay width. They are shown in Fig. 11. As the formula for the vertex corrections and counterterms in terms of the scalar one-, two-and three-point functions are quite lengthy, we do not display them explicitly ...

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... Fig. 13 we show the relative NLO corrections for H ± → W ± h, ∆Γ H ± W ± h , as a function of the charged Higgs boson mass for various renormalization schemes. We denote them as proc : process-dependent p ...

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... Fig. 13 (left) we show results for the process-dependent renormalization and for some representatives of the process-independent schemes, the pOS o , the p c and for comparison also the KOSY c scheme. As can be inferred from the left plot, the process-dependent renormalization leads to much larger NLO corrections than the other schemes. The ...

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... can increase the LO width by more than a factor of three. For the process-independent renormalization schemes on the other hand, the NLO corrections are much milder and vary between about −11 to 20% depending on the renormalization scheme and the charged Higgs mass value (and discarding the unphysical KOSY scheme). This can be inferred from Fig. 13 (right) which displays the results for the process-independent schemes, where the β renormalization is performed both through the charged and through the CP-odd sector. 12 Provided that the same choice for the β renormalization is made, the OS tadpole-pinched scheme, pOS, leads to results closer to the KOSY scheme than the p tadpole-pinched ...

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... Fig. 14 we show the relative NLO corrections for H ± → W ± h as a function of the LO width for all generated scenarios compatible with the applied theoretical and experimental constraints. The colours indicate the results for the process-dependent scheme, the p tadpole- pinched schemes, the OS tadpole-pinched schemes and the KOSY c scheme. The ...

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... Fig. 15 we show the relative NLO corrections for the process H ± → W ± H with the parameters given by Scen3, Eq. (5.14). In the plotted m A range the LO decay width, which does not depend on m A , is given by Γ LO = 4.0568 GeV. In the left plot we have included the results for the process-dependent renormalization, for pOS o , p c and KOSY c . ...

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... for the NLO process, Eq. (4.11). The contributions from the angular counterterms δα and δβ come with the factor 1/t β−α , which is numerically very small in the SM-like limit h ≡ H SM . Therefore any difference in the renormalization schemes for the angles will barely manifest itself in the total NLO corrections. The zoomed in region in Fig. 15 (right) again shows that the KOSY scheme is closer to pOS than to the other schemes and that the usage of the OS scale in δβ is less sensitive to a change of the renormalization scheme, while the renormalization of β via the charged sector is less sensitive to a scale change than the one through the CP-odd sector. 14 The small mA mass range is ...

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... Fig. 16 In the left plot the process-dependent renormalization is included. Additionally we show representatives for process-independent schemes, the pOS o , the p c and the KOSY c scheme. Again the counterterm definition via tauonic heavy Higgs decays leads to much larger corrections than the other schemes. In the investigated mass range it ...

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... in the investigated mass range, the theoretical uncertainty due to missing higher order corrections can be estimated to be of less than a percent to around 6% based on a scale change, and it ranges from the permille level to about 4% when estimated from the change of the β renormalization scheme, discarding the numerically unstable process-dependent scheme. Figure 17 shows the relative NLO corrections ∆Γ H→ZZ for H → ZZ as a function of the LO width for all generated scenarios compatible with the applied theoretical and experimental constraints. The colours indicate the results for the various renormalization schemes. ...

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... applying the renormalization condition depicted diagrammatically in Fig. 18, the shift can be interpreted as a connected tadpole diagram, containing the Higgs tadpole and its propagator at zero momentum ...

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... not belong to the definition of the mass counterterms and wave function renormalization constants. With the redefinition of the 1PI self-energy as iΣ tad These results can be generalized to the gauge boson and fermion sectors. The application of the tadpole scheme hence requires a redefinition of the self-energies as depicted diagrammatically in Fig. 19. In the gauge and fermion sectors this implies that the tadpole diagrams of the scalar Higgs bosons that couple to the gauge boson and fermion, respectively, have to be included in their self-energy. Furthermore, in the scalar sector the tadpole counterterms drop out of the definition of the wave function renormalization constants and ...

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## Citations

... The general or type-III 2HDM has been shown to be able to explain the flavour changing charged current for ( ), whilst respecting constraints from the lepton flavour violating (LFV) decays → and → 3 [32]. The 2HDM may also provide an explanation for things such as LHC data [33][34][35][36][37], B-anomalies [38][39][40][41][42], and decays of Higgs bosons [43][44][45][46][47][48][49][50][51], to name a few. ...

GM2Calc is a leading tool for calculating precise contributions to $a_\mu$ in the Minimal Supersymmetric Standard Model. In this proceeding we detail GM2Calc version 2 where it is extended so it can calculate two-loop contributions to $a_\mu$ in the Two-Higgs Doublet Model (2HDM), based on the work in Ref. [1]. The 2HDM is a simple model, yet it is one of the few single field extensions of the Standard Model which is able to explain the muon $g-2$ anomaly. We demonstrate the powerful and flexible 2HDM capabilities of GM2Calc2, which include the most precise contributions in the literature and allow the user to work in their favourite type of the 2HDM as well as use complex and lepton flavour violating couplings. With its multiple interfaces and input flexibility, GM2Calc2 is a powerful tool both as a standalone code and as part of a larger code toolchain.

... The analytical results of decay rates of all the additional Higgs bosons will be implemented in a new version of our developing program H-COUP v3 [46]. There are also important previous works done by several other groups for the higher-order corrections to decays of the additional Higgs bosons [39,[47][48][49][50][51][52]. ...

... In the improved on-shell renormalization scheme, the masses of the additional Higgs bosons, the mixing angles and the wave function renormalization constants are renormalized by imposing the on-shell conditions for the Higgs bosons in the mass eigenstates. The gauge dependences, which appear in the renormalization of mixing angles [80], are removed by applying the pinch technique [23,50]. For the calculation of the decays of the CP-odd Higgs boson, we do not need to renormalize M 2 , while it can be renormalized by the minimal subtraction [17]. ...

We calculate radiative corrections to decay rates of CP-odd Higgs boson $A$ for various decay modes in the four types of two Higgs doublet models with the softly broken discrete $Z_{2}$ symmetry. The decay branching ratios are evaluated at the next-to-leading order for electroweak corrections and the next-to-next-to-leading order for QCD corrections. We comprehensively study the impact of the electroweak corrections on the decay rates and the branching ratios. We find that the radiative corrections can sizably modify the branching ratios, especially for the $A\to Zh$ decay mode in the nearly alignment scenario, where coupling constants of the SM-like Higgs boson $h$ are close to those in the standard model. We also show correlations between the branching ratios of $A$ and the scaling factor of the SM-like Higgs boson coupling including higher-order corrections. In addition, we show characteristic predictions on the decay pattern depending on the types of the Yukawa interaction, by which we can discriminate the types of the Yukawa interaction in future collider experiments.

... We can extract information on the structure such as the number of additional Higgs fields and their representations from the pattern of the deviations [5]. Furthermore, the mass scale of extra Higgs bosons can be deduced from the size of the deviations [6][7][8][9][10][11][12][13][14][15]. Because the precise measurements of cross sections, decay branching ratios (BRs) and the width of the Higgs boson will be performed at future collider experiments such as the high luminosity LHC (HL-LHC) [16], International Linear Collider (ILC) [17][18][19][20], Circular Electron Positron Collider (CEPC) [21] and Future Circular Collider (FCC-ee) [22], precise predictions of these observables including radiative corrections are necessary to compare these measurements. ...

... [14,62] for a CP-odd Higgs boson and in Refs. [12,[63][64][65][66] for singly charged Higgs bosons. ...

We investigate the impact of electroweak (EW), scalar and QCD corrections to the full set of decay branching ratios of an additional CP-even Higgs boson ($H$) in the two Higgs doublet model with a softly broken $Z_2$ symmetry. We employ the improved gauge independent on-shell scheme in the renormalized vertices. We particularly focus on the scenario near the alignment limit in which couplings of the discovered 125 GeV Higgs boson ($h$) coincide with those of the standard model while the $Hhh$ coupling vanishes at tree level. The renormalized decay rate for $H\to hh$ can significantly be changed from the prediction at tree level due to non-decoupling loop effects of additional Higgs bosons, even in the near alignment case. We find that the radiative corrections to the branching ratio of $H\to hh$ can be a few ten percent level in the case with the masses of additional Higgs bosons being degenerate under the constraints of perturbative unitarity, vacuum stability and the EW precision data. Further sizable corrections can be obtained for the case with a mass difference among the additional Higgs bosons.

... The scheme was applied to various extensions of the SM (see e.g. [11,12]). For the CxSM a brief description follows. ...

... We start with the rotation angle α. Previous works [11,13] lead us to the conclusion that a scheme that is simultaneously stable (in the sense that the NLO corrections do not become unreasonably large) and gauge independent can be built by combining the one proposed in Ref. [14,15] with the gauge dependence handled by the use of the pinch technique [16,17]. The scheme proposed in [14,15] introduces a shift in α, the angle from the rotation matrix R α , ...

The search for dark matter (DM) at colliders is founded on the idea of looking for something invisible. There are searches based on production and decay processes where DM may reveal itself as missing energy. If nothing is found, our best tool to constrain the parameter space of many extensions of the Standard Model (SM) with a DM candidate is the Higgs boson. As the measurements of the Higgs couplings become increasingly precise, higher-order corrections will start to play a major role. The tree-level contribution to the invisible decay width provides information about the portal coupling. Higher-order corrections also gives us access to other parameters from the dark sector of the Higgs potential that are not present in the tree-level amplitude. In this work we will focus on the complex singlet extension of the SM in the phase with a DM candidate. We calculate the one-loop electroweak corrections to the decay of the Higgs boson into two DM particles. We find that the corrections are stable and of the order of a few percent. The present measurement of the Higgs invisible branching ratio, BR$(H \to$ invisible $) < 0.11$, already constrains the parameter space of the model at leading order. We expect that by the end of the LHC the experimental measurement will require the inclusion of the electroweak corrections to the decay in order to match the experimental accuracy. Furthermore, the only competing process, which is direct detection, is shown to have a cross section below the neutrino floor.

... In order to compare theoretical predictions with future precision measurements, theoretical calculations comparable with expected experimental accuracy are inevitable. Radia-tive corrections to the SM-like Higgs boson vertices have been studied in various Higgs sectors such as the model with a real isospin singlet Higgs field (HSM) [32][33][34][35], two Higgs doublet models (2HDMs) [36][37][38][39][40][41][42][43][44][45][46][47][48][49], the inert doublet model (IDM) [50,51] and so on. In order to see differences in the prediction among these models, it is quite important to calculate the renormalized SM-like Higgs boson vertices in a consistent and systematic way. ...

... In the wide range of extended Higgs models, there are mixings among Higgs bosons, and the gauge dependence appears in the renormalization of these mixing angles. We apply the pinch technique to remove the gauge dependence in the renormalized vertex functions [32,41,43]. ...

We present the cross section for $$e^{+}e^{-}\rightarrow hZ$$ e + e - → h Z with arbitrary sets of electron and Z boson polarizations at the full next-to-leading order in various extended Higgs models, such as the Higgs singlet model (HSM), the inert doublet model (IDM) and the two Higgs doublet model (2HDM). We systematically perform complete one-loop calculations to the helicity amplitudes in the on-shell renormalization scheme, and present the full analytic results as well as numerical evaluations. The deviation $$\Delta R^{hZ}$$ Δ R hZ in the total cross section from its standard model (SM) prediction is comprehensively analyzed, and the differences among these models are discussed in details. We find that new physics effects appearing in the renormalized hZZ vertex almost govern the behavior of $$\Delta R^{hZ}$$ Δ R hZ , and it takes a negative value in most cases. The possible size of $$\Delta R^{hZ}$$ Δ R hZ reaches several percent under the theoretical and experimental bounds. We also analyze the deviation $$\Delta R^{hZ}_{XY}$$ Δ R XY hZ in the total cross section times decay branching ratios of the discovered Higgs boson by utilizing the program. It is found that the four types of 2HDMs can be discriminated by analyzing the correlation between $$\Delta R^{hZ}_{\tau \tau }$$ Δ R τ τ hZ and $$\Delta R^{hZ}_{bb}$$ Δ R bb hZ and those between $$\Delta R^{hZ}_{\tau \tau }$$ Δ R τ τ hZ and $$\Delta R^{hZ}_{cc}$$ Δ R cc hZ . Furthermore, the HSM and the IDM can be discriminated from the 2HDMs by measuring $$\Delta R^{hZ}_{WW}$$ Δ R WW hZ . These signatures can be tested by precision measurements at future Higgs factories such as the International Linear Collider.

... Ref. [42] for a review or the relevant chapters of Ref. [43]). Although the 2HDM has been studied for many decades and the only Higgs boson discovered is closely SM-like [44,45], interest in this model has not waned, and recent progress has been achieved e.g. on LHC interpretations [46][47][48][49][50], B-physics [51][52][53][54][55], theoretical constraints [56][57][58][59][60], electroweak phase transitions in the early universe [61][62][63][64][65], precision calculations of Higgs decays [66][67][68][69][70][71][72][73][74]. Remarkably, the 2HDM is one of the very few single field extensions of the Standard Model that can explain the deviation between a Exp µ and a SM µ [75] while satisfying existing constraints from collider physics searches and other observables. ...

We present an extension of the GM2Calc software to calculate the muon anomalous magnetic moment ($a_\mu^{\text{BSM}}$) in the Two-Higgs Doublet Model. The Two-Higgs Doublet Model is one of the simplest and most popular extensions of the Standard Model. It is one of the few single field extensions that can give large contributions to $a_\mu^{\text{BSM}}$. It is essential to include two-loop corrections to explain the long standing discrepancy between the Standard Model prediction and the experimental measurement in the Two-Higgs Doublet Model. The new version GM2Calc 2 implements the state of the art two-loop calculation for the general, flavour violating Two-Higgs Doublet Model as well as for the flavour aligned Two-Higgs Doublet Model and the type I, II, X and Y flavour conserving variants. Input parameters can be provided in either the gauge basis or the mass basis, and we provide an easy to use SLHA-like command-line interface to specify these. Using this interface users may also select between Two-Higgs Doublet Model types and choose which contributions to apply. In addition, GM2Calc 2 also provides interfaces in C++, C, Python and Mathematica, to make it easy to interface with other codes.

... The renormalization of the fermion sector as well as any treatment of infrared divergence is not necessary for this particular process. We perform the renormalization of the Higgs sector in the DDP of the N2HDM according to the procedure presented in ref. [57] for the 2HDM and in ref. [58] for the broken phase of the N2HDM. Although most of the parameters in the Higgs sector of the DDP are common with those of the broken phase, we describe the renormalization of all parameters in order to make the paper self-contained. ...

... The latter was originally proposed by Fleischer and Jegerlehner, in ref. [62], for the SM. The ATS was also discussed in detail for the CP-conserving 2HDM in ref. [57] and for the broken phase of the N2HDM in ref. [58]. We will just briefly review the two schemes for completeness. ...

... Hence in the ATS, one needs to insert tadpole diagrams in all amplitudes for which the original vertices contain one of the VEVs in addition to the usual one-particle irreducible diagrams. This general consequence is shown by focusing on specific amplitudes in ref. [57] for the 2HDM and in ref. [58] for the N2HDM. ...

A bstract
The Higgs invisible decay width may soon become a powerful tool to probe extensions of the Standard Model with dark matter candidates at the Large Hadron Collider. In this work, we calculate the next-to-leading order (NLO) electroweak corrections to the 125 GeV Higgs decay width into two dark matter particles. The model is the next-to-minimal 2-Higgs-doublet model (N2HDM) in the dark doublet phase, that is, only one doublet and the singlet acquire vacuum expectation values. We show that the present measurement of the Higgs invisible branching ratio, BR( H → invisible < 0 . 11), does not lead to constraints on the parameter space of the model at leading order. This is due to the very precise measurements of the Higgs couplings but could change in the near future. Furthermore, if NLO corrections are required not to be unphysically large, no limits on the parameter space can be extracted from the NLO results.

... In order to compare theoretical predictions with future precision measurements, theoretical calculations compatible with expected experimental accuracy are inevitable. Radiative corrections to the SM-like Higgs boson vertices have been studied in various Higgs sectors such as the model with a real isospin singlet Higgs field (HSM) [32][33][34][35], two Higgs doublet models (2HDMs) [36][37][38][39][40][41][42][43][44][45][46][47][48][49], the inert doublet model (IDM) [50,51] and so on. In order to see differences in the prediction among these models, it is quite important to calculate the renormalized SM-like Higgs boson vertices in a consistent and systematic way. ...

... In the wide range of extended Higgs models, there are mixings among Higgs bosons, and the gauge dependence appears in the renormalization of these mixing angles. We apply the pinch technique to remove the gauge dependence in the renormalized vertex functions [32,41,43]. ...

We present the cross section for $e^{+}e^{-}\to hZ$ with arbitrary sets of electron and $Z$ boson polarizations at the full next-to-leading order in various extended Higgs models, such as the Higgs singlet model (HSM), the inert doublet model (IDM) and the two Higgs doublet model (2HDM). We systematically perform complete one-loop calculations to the helicity amplitudes in the on-shell renormalization scheme, and present the full analytic results as well as numerical evaluations. The deviation $\Delta R^{hZ}$ in the total cross section from its standard model (SM) prediction is comprehensively analyzed, and the differences among these models are discussed in details. We find that new physics effects appearing in the renormalized $hZZ$ vertex almost govern the behavior of $\Delta R^{hZ}$, and it takes a negative value in most cases. The possible size of $\Delta R^{hZ}$ reaches several percent under the theoretical and experimental bounds. We also analyze the deviation $\Delta R^{hZ}_{XY}$ in the total cross section times decay branching ratios of the discovered Higgs boson by utilizing the $\texttt{H-COUP}$ program. It is found that the four types of 2HDMs can be discriminated by analyzing the correlation between $\Delta R^{hZ}_{\tau\tau}$ and $\Delta R^{hZ}_{bb}$ and those between $\Delta R^{hZ}_{\tau\tau}$ and $\Delta R^{hZ}_{cc}$. Furthermore, the HSM and the IDM can be discriminated from the 2HDMs by measuring $\Delta R^{hZ}_{WW}$. These signatures can be tested by precision measurements at future Higgs factories such as the International Linear Collider.

... Refs. [47,[49][50][51][52][53][54]). ...

... Artificially large corrections in the MS/FJTS scheme for mixing angles have, for instance, also been found in scenarios of the Two-Higgs-Doublet Models with large Higgs-boson masses in Refs. [52,53,78,79]. In general, this typically occurs when the full-theory loop expansion of an EFT parameter has non-uniform scaling behaviour in the heavy-mass limit. ...

Building on an older method used to derive non-decoupling effects of a heavy Higgs boson in the Standard Model, we describe a general procedure to integrate out heavy fields in the path integral. The derivation of the corresponding effective Lagrangian including the one-loop contributions of the heavy particle(s) is particularly transparent, flexible, and algorithmic. The background-field formalism allows for a clear separation of tree-level and one-loop effects involving the heavy fields. Using expansion by regions the one-loop effects are further split into contributions from large and small momentum modes. The former are contained in Wilson coefficients of effective operators, the latter are reproduced by one-loop diagrams involving effective tree-level couplings. The method is illustrated by calculating potential non-decoupling effects of a heavy Higgs boson in a singlet Higgs extension of the Standard Model. In particular, we work in a field basis corresponding to mass eigenstates and properly take into account non-vanishing mixing between the two Higgs fields of the model. We also show that a proper choice of renormalization scheme for the non-standard sector of the underlying full theory is crucial for the construction of a consistent effective field theory.

... In this paper, we dedicate ourselves to investigating the impacts of one-loop EW corrections to various charged Higgs boson decays in THDMs with softly-broken Z 2 symmetry. As earlier works for calculations of higher order corrections to charged Higgs boson decays, NLO EW corrections to the charged Higgs boson decay into the W ± boson and the CP-odd Higgs boson H ± → W ± A [103,104], the decay into the W ± boson and the CP-even Higgs bosons H ± → W ± h/H [105,106] and the loop-induced decays at LO, H ± → W ± V (V = Z, γ) [107][108][109][110][111][112][113][114][115][116][117][118] have been studied. For charged Higgs boson decays into a pair of fermions, the NLO QCD corrections have been studied in THDMs [119,120]. ...

... In the Higgs sector, there are six free parameters given in Eq. (12). Together with the wave function gauge dependence appears in renormalization of mixing angles [152], which are resolved by applying the pinch technique [56,106]. The remaining parameter in the Higgs sector, M 2 , is renormalized by the minimal subtraction [50]. ...

... We describe the one-loop corrected decay rates for the charged Higgs boson decay into the W ± bosons and scalar bosons H ± → W ± φ (φ = h, H, A), with NLO EW corrections. They can be expressed by [103][104][105][106] ...

We calculate the next-to-leading order (NLO) electroweak (EW) corrections to decay rates of charged Higgs bosons for various decay modes in the four types of two Higgs doublet models (THDMs) with the softly broken discrete Z_2 symmetry. Decay branching ratios of charged Higgs bosons are evaluated including NLO EW corrections, as well as QCD corrections up to next-to-next-to-leading order (NNLO). We comprehensively study impacts of the NLO EW corrections to the branching ratios in nearly alignment scenarios where the couplings constants of the Higgs boson with the mass of 125 GeV are close to those predicted in the standard model. Furthermore, in the nearly alignment scenario, we discuss whether or not the four types of THDMs can be distinguished via the decays of charged Higgs bosons. We find that characteristic predictions of charged Higgs branching ratios can be obtained for all types of the THDMs, by which each type of the THDMs are separated, and information on the internal parameters of the THDMs can be extracted from the magnitudes of the various decay branching ratios.