Fingerprinting the extended Higgs sector using one-loop corrected Higgs boson couplings and future precision measurements

We calculate radiative corrections to a full set of coupling constants for the 125 GeV Higgs boson at the one-loop level in two Higgs doublet models with four types of Yukawa interaction under the softly-broken discrete $Z_2$ symmetry. The renormalization calculations are performed in the on-shell scheme, in which the gauge dependence in the mixing parameter which appears in the previous calculation is consistently avoided. We first show the details of our renormalizaton scheme, and present the complete set of the analytic formulae of the renormalized couplings. We then numerically demonstrate how the inner parameters of the model can be extracted by the future precision measurements of these couplings at the high luminosity LHC and the International Linear Collider.


I. INTRODUCTION
The LHC Run-I has confirmed the existence of a Higgs boson (h) [1,2], whose properties are in agreement with those of the standard model (SM) within the uncertainties of the current data [3][4][5][6][7][8]. Thanks to the discovery of the Higgs boson, the SM was established as an effective theory to describe physics at the scale of electroweak symmetry breaking. In spite of the success of the SM, there are many motivations to consider new physics beyond the SM such as to solve the gauge hierarchy problem and to explain phenomena like neutrino oscillation, dark matter and baryon asymmetry of the Universe. There have been various new physics models proposed, some of which predict new particles at the electroweak to TeV scales. However, currently none of such new particles has been discovered yet. Their discovery is one of the main tasks of the LHC Run-II, which will start its operation in 2015.
Even though the Higgs boson shows SM like properties, the Higgs sector can be extended from the minimal form with only an isospin doublet field. Indeed, there is no theoretical reason for the hypothesis of the minimal structure for the Higgs sector. Thus there are possibilities for extended Higgs sectors such as those with additional iso-singlets, doublets, and/or triplets. These extended Higgs sectors can also be consistent with all the current LHC data in some portions of their parameter space.
Extended Higgs sectors are often introduced in various new physics models. For example, the Minimal Supersymmetric SM (MSSM) requires the Higgs sector with two doublet fields [9,10].
Multi Higgs structures are also studied in the context of additional CP violating phases [11] and also realization of the strong first order phase transition [12], both of which are required for successful electroweak baryogenesis [13]. Models with the Type-II seesaw scenario are motivated to generate tiny neutrino masses by introducing a triplet field [14]. An additional singlet is required in the Higgs sector of the models with spontaneous breakdown of the U (1) B−L symmetry [15][16][17], which may be related to the mechanism of neutrino mass generation [18]. Introduction of an additional unbroken symmetry into an extended Higgs sector, such as a discrete Z 2 symmetry [19,20] or a global U (1) symmetry [21], can provide candidates of dark matter. Under the Z 2 or the global U (1) symmetry, if some of the scalar fields are assigned to be odd or to be charged, respectively, they cannot decay into a pair of SM particles so that the lightest one is stable. Such an unbroken symmetry can also be embedded into models with a radiative generation of neutrino masses [18,[22][23][24][25][26][27], where the existence of tiny neutrino masses and dark matter can be explained by the same origin of the symmetry. Therefore, a characteristic Higgs sector appears in each new physics model.
There are several important properties which characterize the structure of the Higgs sector.
First of all, it is important to know the number of scalar multiplets and their representations.
Second, does it respect new symmetries (global or discrete/exact or softly-broken)? Third, the mass of the second Higgs boson generally contains information of the new scale which does not appear in the SM. Fourth, the strength of the coupling constants among extra Higgs bosons provides information of the dynamics of the Higgs potential which is essentially important to understand nature of electroweak symmetry breaking. Finally, the decoupling property [28] of extra Higgs bosons is closely connected to physics beyond the SM. Therefore, by future measurements of these properties, the Higgs sector can be reconstructed, and the direction of new physics beyond the SM can be determined.
The direct search of extra Higgs bosons can provide a clear evidence to a non-minimal Higgs sector. The current data accumulated from previous collider experiments such as LEP [29,30] and Tevatron [31][32][33][34][35][36] have already given lower bounds for masses of the extra Higgs bosons. At the LHC Run-I, in spite of the discovery of a Higgs boson with the mass of 125 GeV, no extra Higgs boson has been found, and the parameter space for additional light Higgs bosons has been constrained to the considerable extent in regions with relatively smaller masses of the extra Higgs bosons [37][38][39][40][41][42][43][44][45][46][47][48][49]. At the LHC Run-II, with the energy of 13-14 TeV and the integrated luminosity of 300 fb −1 , wider regions of masses of the extra Higgs bosons will be surveyed.
In addition to direct searches, new physics models beyond the SM have also been indirectly investigated by utilizing precision measurements of various physics observables such as the oblique parameters at LEP/SLC experiments [50]. Flavour experiments have also been used to constrain the mass of charged Higgs bosons which appears in extended Higgs sectors [51,52]. Now that the measured couplings of the Higgs boson h with the SM particles are consistent with the predictions in the SM within the uncertainties, it is time to consider fingerprinting of extended Higgs sectors [53,54] by calculating radiative corrections to the predictions of those observables which will be measured with more precision at future experiments such as the LHC Run-II, the high luminosity (HL)-LHC [55-57] with the integrated luminosity of 3000 fb −1 and future lepton colliders like the International Linear Collider (ILC) [58,59]. In new physics models with extended Higgs sectors, the coupling constants of h with the SM particles are generally predicted with deviations from the SM predictions due to field mixing and loop contributions of non-SM particles. Although no deviation has been found up to now in the Higgs boson couplings within the uncertainty of the current data, a deviation could be found in future experiments where more precise measurements will be attained. We then are able to indirectly obtain information of the second Higgs boson from these deviations. Furthermore, a pattern of these deviations strongly depends on the structure of the Higgs sector, so that by comparing theoretical predictions of the Higgs couplings in various new physics models with future experimental data the shape of the Higgs sector can be determined indirectly. In order to compare the theory predictions to future precision data at the HL-LHC and also the ILC, where coupling constants are expected to be measured typically by a few percent or better accuracy, evaluations of the Higgs boson couplings including radiative corrections are inevitable.
There are many studies for radiative corrections in extended Higgs sectors in the literature.
In this paper, we study electroweak radiative corrections to the coupling constants of the 125 GeV Higgs boson h in the THDM [79] with the softly-broken Z 2 symmetry [80]. Under the Z 2 symmetry, four types of Yukawa interactions [81][82][83][84] are possible depending on the assignment of the Z 2 charges into quarks and leptons. We investigate radiative corrections to the full set of Higgs boson couplings (hW W , hZZ, htt, hbb, hτ τ , hhh, hγγ, hZγ and hgg) at the one-loop level in all types of the THDMs. We employ an improved on-shell renormalization scheme in our renormalization calculation where the gauge dependence in the calculation of the mixing angle in the previous studies is eliminated 1 . We then evaluate deviations in these coupling constants from the SM predictions under the constraint of current experimental data and theoretical bounds such as vacuum stability and perturbative unitarity.
Furthermore, we investigate how we can extract information of the inner parameters such as the mass of the second Higgs boson and mixing angles when the scale factors κ X are experimentally determined with the expected uncertainties at the HL-LHC and the ILC, where κ X are the ratios of the measured couplings hXX from the SM predictions. Evaluating κ X at the one-loop level in the THDMs, we discuss the possibility to measure properties of the Higgs sector using the future precision data by fingerprinting, and finally we determine the structure of the Higgs sector.
This paper is organized as follows. In Sec. II, we define the Lagrangian of THDMs, and give formulae for the Higgs boson masses and the Higgs boson couplings at the tree level. After that, 1 According to Ref. [85], the gauge dependence exists in a renormalization of a mixing angle. In Sec. III, we explain renormalization in the electroweak sector, the Yukawa sector, and the Higgs sector in the THDMs. We also discuss the modified renormalization scheme. In Sec. IV, we give formulae of renormalized Higgs couplings and loop induced decay rates. We numerically estimate decoupling properties and non-decoupling effects of our one-loop calculations in the section. In Sec. V, we demonstrate how we can extract inner parameters by using future precision data.
Discussions and conclusions are given in Sec. VI.

A. Lagrangian
In this section, we define the Lagrangian in the THDM with the softly-broken Z 2 symmetry, where the Higgs sector is composed of two isospin doublet scalar fields Φ 1 and Φ 2 . The charge assignment for the Z 2 symmetry is shown in Table I. The following Lagrangian is modified from the SM: where L kin , L Y and V are respectively the kinetic Lagrangian, the Yukawa Lagrangian and the scalar potential. Throughout the paper, we assume the CP invariance in the Higgs sector.
First, the kinetic Lagrangian is given by where D µ is the covariant derivative: with W a µ (a =1-3) and B µ being the SU (2) L and U (1) Y gauge bosons, respectively. The two doublet fields can be parameterized as where v 1 and v 2 are the vacuum expectation values (VEVs) for Φ 1 and Φ 2 , which satisfy The ratio of the two VEVs is defined as tan β = v 2 /v 1 . The mass eigenstates for the scalar bosons are obtained by the following orthogonal transformations as where G ± and G 0 are the Nambu-Goldstone bosons absorbed by the longitudinal component of W ± and Z, respectively. The mixing angle α is expressed in terms of the mass matrix elements for the CP-even scalar states as shown in Eqs. (18)- (21). As the physical degrees of freedom, we have a pair of singly-charged Higgs boson H ± , a CP-odd Higgs boson A and two CP-even Higgs bosons h and H. We define h as the observed Higgs boson with the mass of about 125 GeV.
In terms of the mass eigenbasis of the Higgs fields, the interaction terms among the Higgs bosons and the weak gauge bosons are given by where coefficients of the Scalar-Scalar-Gauge vertex g φ 1 φ 2 V and those of the Scalar-Scalar-Gauge- Next, we discuss the Yukawa Lagrangian. The most general form under the Z 2 symmetry is given by where Φ u,d,e are either Φ 1 or Φ 2 . Depending on the Z 2 charge assignment, there are four types of Yukawa interactions [81,82], which we call as Type-I, Type-II, Type-X and Type-Y [84]. The interaction terms are expressed in terms of the mass eigenstates of the Higgs bosons as where ξ f h and ξ f H are defined by and ξ f in each type of Yukawa interactions are given in The Higgs potential under the softly-broken Z 2 symmetry and the CP invariance is given by The tadpole terms for h 1 and h 2 are respectively calculated as whereλ ≡ λ 3 + λ 4 + λ 5 , and M describes the soft breaking scale of the Z 2 symmetry: We note that M 2 can be taken to be both positive and negative values. By requiring the tree level tadpole conditions; i.e., T 1 = T 2 = 0, m 2 1 and m 2 2 can be eliminated in the Higgs potential. The squared masses of H ± and A are calculated as Those for the CP-even Higgs bosons and the mixing angle α are given by where M 2 ij (i, j = 1, 2) are the mass matrix elements for the CP-even scalar states in the basis of (h 1 , h 2 )R(β): Thus, ten parameters in the potential (v 1,2 , m 2 1-3 and λ 1-5 ) can be described by the eight physical parameters m h , m H , m A , m H ± , α, β, v and M 2 , and two tadpoles T 1 and T 2 which are taken to be zero at the tree level. The quartic couplings λ 1 -λ 5 in the potential are then rewritten in terms of the physical parameters as We here define the so-called scaling factors to describe deviations in the Higgs boson couplings from the SM prediction as follows: where g SM hV V , y SM hf f and λ SM hhh are the hV V , hff and hhh coupling constants in the SM, respectively, and those with THDM in the superscript are corresponding predictions in the THDM. The scaling factors for loop induced couplings can also be defined by where Γ(h → XY ) SM and Γ(h → XY ) THDM are respectively the decay rates of the h → XY mode in the SM and in the THDM. At the tree level, the scaling factors are given by We can see that all the scaling factors become unity when sin(β − α) = 1 is taken, so that we call this limit as the SM-like limit [86].
It is convenient to introduce a parameter x defined as where x → 0 corresponds to the SM-like limit. We note that in the MSSM, the sign of x is determined to be negative due to supersymmetric relations [10]. Because the current LHC data suggest that the observed Higgs boson is SM-like, the case with |x| ≪ 1 describes such a situation.
In this case, we obtain As it has already been pointed out in Ref. [53], looking at the correlation between κ f and κ f ′ (f = f ′ ) is quite useful to distinguish the four types of Yukawa interactions.
In Fig. 1, we show the tree level predictions on the ∆κ E -∆κ D plane (left panels) and ∆κ E -∆κ U plane (right panels) in the four types of Yukawa interactions, where ∆κ X = κ X − 1. The subscripts E, D and U respectively represent the flavour independent charged leptons, down-type quarks and up-type quarks. In this plot, we take |x| = 0.2, 0.14 and 0.028, and the sign of x is set to be negative (positive) for upper (lower) panels. As it can be seen, the predictions for the four types of Yukawa interacitons appear in different quadrants of the ∆κ E -∆κ D plane. Therefore, at least from the tree level result, we can discriminate the type of Yukawa interaction in the THDM by looking at the measured values of ∆κ E and ∆κ D .
In Ref. [76], one-loop corrected Yukawa couplings have been calculated in the four types of Yukawa interactions in the THDM. It has been clarified that the predictions in the four types of Yukawa interactions are well separated on the ∆κ E -∆κ D plane at the one-loop level even if we scan the inner parameters under the constraints from perturbative unitarity and vacuum stability.

B. Vacuum stability and perturbative unitarity
A set of quartic coupling constants in the Higgs potential λ 1 -λ 5 is constrained by taking into account vacuum stability and perturbative unitarity as follows.
First, we require that the Higgs potential is bounded from below in any direction with a large scalar field value. The sufficient condition to keep such a stability of the vacuum is given by [19,87,88] Second, the perturbative unitarity bound [89][90][91][92] is given by requiring that all the independent eigenvalues of the T matrix a 0 i,± (i = 1-6) for the S-wave amplitude of the elastic scatterings of 2-body boson states are satisfied as where each of a 0 i,± is given by [90][91][92] a 0 1,± = 1 32π 3 a 0 6,± = 1 16π (λ 3 ± λ 5 ).
In Fig. 2, we show the allowed parameter region on the m Φ -sin(β − α) plane (m Φ ≡ m H ± = m A = m H ) from the constraints of vacuum stability and unitarity. It is seen that a large mass of additional Higgs bosons is allowed in a case with sin(β − α) ≃ 1. As another view of this figure, we can extract the scale of the mass of the second Higgs boson from the precise measurement of κ V using Eq. (27). For example, if 1% deviation in the hV V coupling is found at future collider experiments, then the second Higgs boson should exist below about 800 GeV.

C. The oblique parameters
The S, T and U parameters proposed by Peskin and Takeuchi [93] are modified in the THDM from those predicted in the SM due to the additional Higgs boson loop contributions and modified values of the SM-like Higgs boson coupling constants [60]. We define the differences of S, T and The loop functions are given by where In the case of p 2 = 0, the F 5 function is expressed by which gives zero in the case of m 1 = m 2 . Therefore, it is seen that ∆T becomes zero when x = 0 and m A = m H ± or x = 0 and m H = m H ± is taken.

D. Flavour Constraints
The mass of H ± can be constrained from various B physics processes, because contributions from the SM W -boson mediation are replaced by H ± . In most of the cases, the constraint from the b → sγ process provides the most stringent lower limit on m H ± [51,52]. In Ref. [52], the branching ratio ofB → X s γ has been calculated at the next-to-next-to-leading order in the Type-I and Type-II THDMs. A lower bound has been found to be m H ± 380 GeV at 95% confidence level (CL) in the Type-II THDM with tan β 2. A stronger bound for m H ± is obtained for smaller values of tan β. On the other hand, in the Type-I THDM, the bound from b → sγ is important in the case with low tan β; e.g., m H ± 200 (800) GeV is excluded at 95% CL in the case of tan β = 2 (1). When we consider the case with tan β 2.5, the bound on m H ± is weaker than the lower bound from the direct search at LEP, namely, about 80 GeV [94]. The similar bounds as those given in the Type-II and Type-I THDMs can be obtained in the Type-Y and Type-X THDMs, respectively, because of the same structure of quark Yukawa interactions.
For a large tan β case, bounds from B → τ ν [95,96], τ → µνν [96,97] and the muon anomalous magnetic moment [98,99] can be more important as compared to the bound from b → sγ in the Type-II THDM. For example, the lower limit on m H ± to be about 400 GeV is given at 95% CL in the case of tan β 50 in the Type-II THDM [96].
For a small tan β case, the B 0 -B 0 mixing is getting important to obtain a severe constraint on m H ± in the THDMs. In the case of tan β = 1, m H ± 500 GeV is exluded at 95% CL in all the types of THDMs [100]. This gives the stronger (weaker) bound than that from b → sγ in the Type-II and Type-Y (Type-I and Type-X) THDMs.

E. Direct searches for additional Higgs bosons at the LHC (7-8 TeV)
The neutral Higgs bosons in the MSSM have been searched in the τ + τ − decay mode in the gluon fusion and bottom quark associated productions [37,38] from the two parameters (κ F and κ V ) fit analysis based on Ref. [101]. The scaling factors for the loop induced Higgs boson couplings κ g and κ γ have also been measured under the assumptions of from the two parameters (κ g and κ γ ) fit analysis based on Ref. [101]. We can see that all the SM predictions (κ X = 1) are included within the 2-σ uncertainty of the measured scaling factors, where the current 1-σ uncertainties of the scaling factors are typically of O(10%).
These scaling factors are expected to be measured more precisely at future collider experiments such as the HL-LHC and the ILC. In Ldt (fb −1 ) 300/expt 3000/expt 250+500 1150+1600 250+500+1000 1150+1600+2500 been calculated. The one-loop corrected hZZ and hhh couplings have been evaluated in Ref. [75] in the Type-II THDM, and the hff couplings have been calculated in Ref. [76] in the four types of THDMs.
We perform renormalization calculations based on the on-shell scheme which has been applied in Ref. [75] 2 . However, it has been pointed out that there remains gauge dependence in the determination of the counter term of β in Ref. [85]. We thus construct a new renormalization scheme for β to get rid of the gauge dependence. As pointed out later in the paper, the gauge dependence is not completely removed, but shifted to a sector which does not contribute to the investigated couplings.
First, we prepare a set of independent counter terms by shifting all the relevant bare parameters in the Lagrangian. We then give the renormalized one-and two-point functions which are written in terms of the contributions from 1PI diagrams and counter terms. After that, we set the same number of renormalization conditions as the number of independent counter terms to determine them.

A. Parameter shift and renormalized functions
We first perform the parameter shift of the electroweak sector and Yukawa sector as the following where ϕ = H ± , A, H and h. The wave functions for the SM gauge bosons B µ and W a µ and the SM left (right) handed fermions ψ L (ψ R ) are shifted as We can then write down the renormalized two point functions for each particle. In the following, Π XY (p 2 ) and Π 1PI XY (p 2 ) respectively denote the renormalized two point functions and the 1PI diagram contributions for fields X and Y with the external momentum p µ . The analytic formulae for the 1PI diagram contributions are given in Appendix C. For the gauge boson two point functions W + W − , ZZ, γγ and the Z-γ mixing, we havê where The renormalized fermion two point function is expressed by the following two parts: with In Eq. (62), and Π 1PI f f,S are the vector, axial vector and scalar parts of the 1PI diagram contributions at the one-loop level, respectively.
For the scalar sector, we first define shifts in the weak eigenbasis of the scalar fields: whereZ even ,Z odd andZ ± are arbitrary real 2 × 2 matrices. We then express shifts of the scalar fields in the mass eigenbasis as where we introduce Z even ≡ R(−α)Z even R(α) and Z odd/± ≡ R(−β)Z odd/± R(β). We define the matrix elements of them as follows: We note that in Ref. [75], the above matrices are chosen to be a symmetric form; i.e., δC Hh = δC hH , δC GA = δC AG and δC G + H − = δC H + G − . In this paper, we do not take the symmetric form, and we use the additional degrees of freedom to remove the gauge dependence in the renormalization of δβ as it will be discussed in Sec. III-D. Finally, we can express the shifts of the scalar fields by For the scalar sector, we have the renormalized one-point function for h and H aŝ where The renormalized two-point functions are expressed aŝ and those of the scalar mixings are given bŷ B. Renormalization conditions in the electroweak gauge sector The renormalization of the electroweak parameters can be done in the same way as in the SM, because the number of parameters to describe the electroweak observables are the same in the THDM. This nature is also applied to models based on the SU (2) L × U (1) Y gauge symmetry with ρ = 1 at the tree level 3 .
We apply the electroweak on-shell scheme based on Ref.
[103] to our model. There are five counter terms in the electroweak sector; i.e., δm 2 W , δm 2 Z , δα em , δZ W and δZ B . Therefore, we need the following five renormalization conditions to determine them: d dp 2Π γγ (p 2 ) whereΓ γee µ is the renormalized photon-electron-positron vertex. From the above conditions, we obtain . The other counter terms are also determined by The counter term for the VEV δv is also obtained through the tree level relation: We here note that the fermion-loop contribution to Π 1PI γγ (0) ′ is given by where Q f is the electric charge of a fermion f , N f c is the color factor: N f c = 3 (1) for f being quarks (leptons), and ∆ is the divergent part of the loop integral as defined in Eq. (B23) in Appendix B.
In order to avoid to input the light quark masses, we can use the following relation obtained from Eqs. (58) and (88) where ∆α em is the shift of the structure constant that we can quote the experimental value. In the right hand side of the above equation, the light fermion mass dependence in Π 1PI γγ (m 2 Z )/m 2 Z is of order m 2 f /m 2 Z , so that we can neglect it.

C. Renormalization conditions in the Yukawa sector
In the Yukawa sector, there are three counter terms δm f , δZ f V and δZ f A . To determine them, we impose the following three conditions for the fermion two point functions [76]: we obtain

D. Renormalization conditions in the Higgs potential
There are totally 21 counter terms in the Higgs potential, namely, the counter terms for two tadpoles δT h and δT H , four mass parameters δm 2 ϕ (ϕ = H ± , A, H and h), two mixing angles δα and δβ, four wave function factors δZ ϕ , six wave function mixing factors δC ij , and δM 2 4 . First, we impose two tadpole conditions at the one-loop level, i.e., We then obtain Second, eight on-shell conditions for the two-point functions: which determine the following eight counter terms and Three counter terms δα, δC hH and δC Hh related to the mixing between the CP-even scalar states are determined by imposing the following three conditionŝ They give Three counter terms δβ, δC AG and δC GA related to the mixing between the CP-odd scalar states are determined by three conditions. Similar to the CP-even sector, we first impose the following two conditions asΠ We then obtain In order to determine three counter terms, we need to impose one more renormalization condition in addition to that given in Eq. (110). This third condition can be used to remove the gauge dependence in δβ which was already mentioned in the beginning of this section. To define such a condition, we separateΠ 1PI AG (p 2 ) into the gauge dependent (G.D.) part and the gauge independent (G.I.) part asΠ Then, we imposed the third condition as Using Eq. (111), the remaining two counter terms are also determined: We note that inΠ 1PI AG (0) only the G.D. part is survived; i.e.,Π 1PI AG (0) =Π 1PI AG (0) G.D. . As it can be seen in Eqs. (114) and (115), there still remains the gauge dependence in δC AG and δC GA . However, they do not appear in the following calculations for the renormalization of the Higgs boson couplings. Instead of applying the above renormalization scheme for δβ, we can apply the MS scheme in which the gauge dependence can also be removed at the one-loop level as discueed in Ref. [85]. In the following discussion, we apply the renormalized tan β determined by Eq. (113).
Two counter terms δC H + G − and δC G + H − for the mixing between the singly-charged scalar states are determined by requiring the vanishment of the mixing between G ± and H ± at p 2 = 0 and p 2 = m 2 H ± :Π We obtain Until here, we did not discuss the determination of δM 2 . As adopted in Ref. [75], we apply the minimal subtraction scheme for δM 2 , where it is determined so as to absorb only the divergent part in the hhh vertex at the one-loop level, that is

A. Analytic expressions
In the previous section, all the counter terms are determined by the set of renormalization conditions. Now, we can evaluate the one-loop corrected Higgs boson couplings hW W , hZZ, hff and hhh. In addition to the above couplings, we also give formulae for the loop induced decay rates h → γγ, h → Zγ and h → gg.
The renormalized hV V , hff and hhh vertices are expressed aŝ where Γ tree hXX , δΓ hXX and Γ 1PI hXX are the contributions from the tree level, the counter terms and the 1PI diagrams for the hXX vertices, respectively. In the above expressions, p 1 and p 2 (q = p 1 + p 2 ) are the incoming momenta of particle X (outgoing momentum for h).
For the hV V and hff vertices, the indices i and j label the following form factors: The tree-level contributions are given as The counter-term contributions are δΓ hhh = 6 δλ hhh + 3 2 δZ h + λ Hhh (δα + δC h ) , where The counter terms δξ f h appearing in the Yukawa couplings are expressed in terms of δβ and δα as listed in Table III. We define the renormalized scaling factors in the following way: The momentum q 2 is fixed to be (m V + m h ) 2 , m 2 h and (2m h ) 2 forκ V ,κ f andκ h , respectively, in the following discussion.
We can see that there appears the term

B. Numerical evaluations
In the following, we show numerical results for the Higgs boson couplings at the one-loop level.
We use the following inputs [94]: We first show the case of the SM-like limit x = 0. In this case, the deviations in the Higgs boson couplings purely comes from the additional Higgs boson loop effects. We note that the We can see that all the deviations approach to zero in the large mass region due to the decoupling theorem [28].
In Fig. 4, we show the deviation in the Higgs boson couplings ∆κ V (upper-left), ∆κe f (upperright), ∆κ 2 γ/Zγ (lower-left) and ∆κ h (lower-right) as a function of m Φ . We take M 2 = 0 and tan β = 1 for all panels. In this case, the magnitude of deviations increase when m Φ becomes ∆κ τ +18% +10% +5% +18% +18% ∆κ b +18% +10% +5% +18% +18%  In this section, we investigate how we can fingerprint the THDMs using the one-loop corrected Higgs boson couplings and also future precision measurements of these couplings at the HL-LHC and the ILC. We carefully see how the tree level analysis for the model discrimination discussed in Sec. II or in Ref. [53] can be improved by the analysis with radiative corrections. Furthermore, we demonstrate how the inner parameters such as x, tan β and masses of additional Higgs bosons can be extracted from the measurement of the couplings for the Higgs boson h. In our analysis below, we assume that the deviations in scale factors of the Higgs boson couplings are measured as expected in Table IV. We also assume that the SM values of these coupling constants are well predicted without large uncertainties which mainly come from QCD corrections 5 .
Let us suppose that ∆κ V , ∆κ τ and ∆κ b are measured at the HL-LHC and the ILC500. We consider five benchmark sets for the central values of (∆κ V , ∆κ τ , ∆κ b ) as listed in From the tree level analysis in Fig. 1, these benchmark sets indicate that the Higgs sector is the THDM with the Type-II (Type-I) Yukawa interaction assuming x ≃ cos(β − α) < 0 (x > 0). In order to further discriminate Type-I or Type-II, we need additional information to determine the sign of x such as the measurement of ∆κ c , namely, if ∆κ c is given to be a negative (positive) value, then we can completely determine the Yukawa interaction to be Type-II (Type-I). In the following, we consider the case of ∆κ c < 0, so that we assume the case of the Type-II THDM. In Fig. 5, we show the allowed parameter regions on the x-tan β, x-m Φ , m Φ -ζ and m Φ -tan β planes from the left to right panels, where we define The parameters  Table IV.
For Set A in Fig. 5, let us first explain the behavior of the red points on the x-tan β plane. In this case, −2.4% < ∆κ V < −1.6% is allowed at the ILC500, which can be explained by taking The point here is that the sign of one-loop effect is negative, and it is proportional to the factor ζ 2 . Therefore, the allowed region above x ≃ −0.18 is explained from the one-loop contribution with a non-zero value of ζ. On the other hand, the one-loop correction to κ τ is given by the same form as for κ V as given in Eq. (132), so that the difference ∆κ τ − ∆κ V is approximately given by the same form −x tan β as that given at the tree level. Now from the measurement, since the difference is determined with the uncertainty, −x tan β is also fixed at the one-loop level. We thus can understand the shape of the allowed region of this plot. Although for ∆κ b the top quark, the bottom quark and H ± loop diagrams give an additional contribution as shown in Eq. (134), this is not so significant in the scanned regions. As a consequence for Set A, when the measurement at the ILC500 is assumed, the allowed value of x and tan β can be determined to be about from −0.22 to −0.12 and from 1 to 2, respectively. On the other hand at the HL-LHC, ∆κ V = 0 is included within the 1-σ uncertainty. Thus, x ≃ 0 is still allowed, so that the value of tan β is not determined at all because of the relation tan β ≃ −∆κ τ /b /x. In addition, we can only extract the lower limit of x to be about −0.22.
Next, we discuss the behavior of the second panel for Set A in Fig. 5. As we mentioned in the above, the vertical axism Φ measures the size of one-loop contribution to the deviation in the Higgs boson couplings. At the ILC500, in the region with x ≃ −0.20, the value ofm Φ is determined to be a smaller value, butm Φ ≃ 0 is not included because of the constraint from vacuum stability.
This can be understood that the deviation from the tree level mixing is dominant in this case.
On the other hand, when the value of x approaches to zero, a sizable value ofm Φ is extracted, in which the deviation driven by the one-loop contribution becomes more important to compensate the reduced contribution from the tree level mixing. In addition, the upper limit ofm Φ to be about 450 GeV is determined by the constraint from perturbative unitarity. At the HL-LHC, although the blue plots are spread over the region with x ≃ 0 as we observed in the x-tan β plot, the upper and lower limit ofm Φ is given by the constraint from unitarity and vacuum stability, respectively.
The third panel for Set A in The panels shown in the second and third rows in Fig. 5 display the allowed parameter regions for Set B and Set C, respectively, where the central value of ∆κ τ (= ∆κ b ) is taken to be smaller than that of Set A, while ∆κ V is taken to be the same. By looking at the panels for the x-tan β plane, we can see that a smaller value of |x| is preferred as compared to the case for Set A. Furthermore, a smaller value of tan β is favored in addition to a smaller value of |x| as seen in the result at the ILC500. These tendencies can be understood in such a way that the deviations in Yuakwa couplings are proportional to −x tan β at the tree level. Because of the smaller value of |x|, the deviation in κ V cannot be explained only from the tree level contribution, so that the one-loop effect is necessary to compensate the tree level contribution. That is the reason why the red points in the second and the third panels for Set B and Set C are given in the upper region which does not includem Φ ≃ 0 and ζ ≃ 0. Therefore, the non-decoupling effect can be extracted at the ILC500 for these two benchmark sets. From the results of ILC500, the upper limit on m Φ is extracted to be about 950 GeV and 800 GeV for Set B and Set C, respectively.
The panels shown in the fourth and fifth rows in center and bottom panels. The 1-σ uncertainty of κ γ is assumed to be 2% as expected at the HL-LHC. The cyan and red points satisfy the benchmark sets within the 1-sigma uncertainty at the HL-LHC and ILC500 given in Eq. (140), respectively. For the panels shown in the second and the third columns, the vertical axis inputs; i.e., ∆κ V , ∆κ τ and ∆κ b . In Fig. 6, we show how the extraction can be improved by adding information of κ γ in addition to the above three inputs. The panels shown in the first row are the same as those shown in the first row in Fig. 5, which are displayed in order to compare the results with κ γ . The panels displayed in the second, third and fourth rows respectively show the allowed region for Set A with the central value of κ γ of 0.98, 1.00 and 1.02 within the 1-σ uncertainty of ±2% as expected at the HL-LHC (see Table II). Because the accuracy of the measurement of κ γ at the ILC500 is not better than that of the best value at the HL-LHC, 2%, we also use 2% for the analysis at the ILC500. As we see Eq. (137), the H ± loop contribution to the decay rate of the h → γγ mode gives a different dependence of the non-decouplingness from that in ∆κ V and ∆κ f , which is not proportional tom Φ , but proportional to ζ, so that the non-decouplingness ζ can be expected to be extracted more precisely depending on the measured value of κ γ . In fact, we can observe that ζ is determined more precisely to be 0.5 ζ 1.0, 0.25 ζ 1.1 and 0.2 ζ 0.5 at the ILC500 for the cases with the central value of κ γ = 0.98, κ γ = 1.00 and κ γ = 1.02, respectively, as compared to the case without κ γ (0.2 ζ 1.2). The determination ofm Φ is also improved, becausem Φ is given as a function of ζ. We note that smaller values of ζ andm Φ are favored in the case of the larger central value of κ γ , because the H ± loop effect gives a destructive contribution to the W boson loop contribution.
In Fig. 7, we also show the allowed parameter region with additional information of κ γ for Set We have found that the inner parameters of the THDM can be determined to a considerable extent as long as κ V will be measured with the deviation about 1%. The extraction of the inner parameters using the ILC500 is much better than that using the HL-LHC. That is mainly due to the good accuracy of the hV V coupling measurement at the ILC500 whose uncertainty is expected to be less than 1%. Although we have only demonstrated the results for Set A to Set E assuming the true Higgs sector is of the Type-II THDM, the similar analysis can be performed straightforwardly in the other types of THDM or the other extended Higgs sectors, and the extraction of inner parameters is expected to be attained as well in these models. Our study given in this paper shows that the numerical evaluation of the Higgs boson couplings at the one-loop level in extended Higgs sectors is essentially important to indirectly determine the structure of the Higgs sector by using the future precision data. In addition, it also shows that in addition to the HL-LHC where especially hγγ can be measured precisely future lepton colliders such as the ILC are absolutely necessary for our purpose of determining the structure of the Higgs sector from the measurement of the coupling constants of the discovered Higgs boson h.
Although we have discussed fingerprinting by using κ V , κ τ , κ b and κ γ , the information of κ c , κ t and κ h is also important to determine the Higgs sector more deeply. In particular, the measurement of the top Yukawa coupling is important not only to determine the nature of the top quark, the heaviest matter particle, but also to test the new physics scenarios based on the composite models.
The measurement of the hhh coupling is essentially important not only to determine the nature of the Higgs potential but also to test, for instance, the new physics models with strongly first order phase transition. Although at the HL-LHC the cross section of the double Higgs production process is expected to be measured at a few times 10% it seems to be hopeless to extract the information of the hhh coupling sufficiently accurately. On the other hand, at the ILC with √ s = 1 TeV the hhh coupling can be measured with the 13% accuracy [59,106], which is sufficient precision to test the strong first order phase transition which is required for successful electroweak baryogenesis.
We conclude that the combination of the future data for all kinds of the couplings for the Higgs boson h and their theory predictions with radiative corrections in various extended Higgs sectors is a promissing way to determine the structure of the Higgs sector and further to access new physics beyond the SM, even if a new particle was not directly discovered in the future experiments.
The coefficients g φV 1 V 2 , g φ 1 φ 2 V and g φ 1 φ 2 V 1 V 2 are listed in Table VI, where we use g Z = g/c W in this table and below. Throughout Appendix, we use the shortened notation of the mixing angles, s β−α = sin(β − α) and c β−α = cos(β − α).
From the Higgs potential, we obtain the scalar trilinear and the scalar quartic couplings. When we use the following notation for these couplings  These coefficients are given by The four point couplings are given by

Appendix B: Loop Functions
The Passarino-Veltman functions [107] are quite useful to systematically express the one-loop functions. First, we define A, B and C functions: where D = 4 − 2ǫ, and µ is a dimensionful parameter to keep the mass dimension four in the k-integral. The propagators are defined by The vector and the tensor functions for B and C are expressed in terms of the following scalar functions: By counting the mass demension of the above functions, we can find that the divergent part is contained in A, B 0 , B 1 , B 21 , B 22 and C 24 . All the scalar functions are expressed by the divergent part and finite part as where and the divergent part ∆ is given by with γ E being the Euler constant. It is convenient to define the following functions [108]: contritbutions. We denote the fermionic-and bosonic-loop contributions by the subscript of F and B, respectively. Throughout this section, we use the shortened notation of the Passarino-Veltman functions [107] as

One-point functions
The 1PI tadpole diagrams for h and H are calculated by

Two-point functions
The 1PI diagram contributions to the scalar boson two point functions are calculated as The 1PI diagram contributions to the gauge boson two point functions are calculated as where the fermion-loop contributions are the same as those in the SM.