Single Higgs production in association with a photon at electron-positron colliders in extended Higgs models

We study associated Higgs production with a photon at electron-positron colliders, $e^+e^-\to h\gamma$, in various extended Higgs models, such as the inert doublet model (IDM), the inert triplet model (ITM) and the two Higgs doublet model (THDM). The cross section in the standard model (SM) is maximal around $\sqrt{s}=$250 GeV, and we present how and how much the new physics can enhance or reduce the production rate. We also discuss the correlation with the $h\to\gamma\gamma$ and $h\to Z\gamma$ decay rates. We find that, with a sizable coupling to a SM-like Higgs boson, charged scalars can give considerable contributions to both the production and the decay if their masses are around 100 GeV. Under the theoretical constraints from vacuum stability and perturbative unitarity as well as the current constraints from the Higgs measurements at the LHC, the production rate can be enhanced from the SM prediction at most by a factor of two in the IDM. In the ITM, in addition, we find a particular parameter region where the $h\gamma$ production significantly increases by a factor of about six to eight, but the $h\to\gamma\gamma$ decay still remains as in the SM. In the THDM, possible deviations from the SM prediction are minor in the viable parameter space.


I. INTRODUCTION
High-energy e + e − colliders as a Higgs factory have been discussed for a long time in order to identify the Higgs sector; i.e., a mechanism of electroweak symmetry breaking and its relation to the physics beyond the standard model (SM). Especially, after the discovery of the 125 GeV Higgs boson in pp collisions at the Large Hadron Collider (LHC), physics potential for precise measurements of the Higgs couplings at √ s = 240 − 250 GeV have been extensively studied for realization of the International Linear Collider (ILC) [1][2][3] as well as the lepton collision option of the Future Circular Collider (FCC-ee) [4] and the Circular Electron Positron Collider (CEPC) [5]. Such collision energies are optimal to study associated Higgs production with a Z boson, e + e − → hZ, whose total cross section is maximal around √ s = 250 GeV in the SM.
On the other hand, it is less known that the cross section for associated Higgs production with a photon, e + e − → hγ, also has a peak around √ s = 250 GeV in the SM. Since the process is protected by the electromagnetic gauge symmetry and the tree-level contribution is highly suppressed by the electron mass, it is essentially loop-induced, and hence the cross section is rather small, O(0.1) fb at √ s = 250 GeV. However, because the signal is very clean; i.e., a monochromatic photon in the final state, the above future colliders with the planned luminosity may be able to observe the signal. Moreover, new physics can substantially enhance the production rate relative to the SM case; i.e., the process have a potential to explore physics beyond the SM. In this work, we study how and how much new physics can enhance (or reduce) the hγ signal at future e + e − colliders.
The process was studied in the SM long time ago [6][7][8]. The next-to-leading order QCD corrections were recently estimated in Ref. [9], showing negligible impact at lower center-of-mass energies √ s < 300 GeV. On the other hand, new physics effects to the process have not been fully explored yet. There are several studies in literatures in the context of anomalous Higgs boson couplings or an effective field theory [10][11][12][13][14][15] as well as concrete new physics models such as supersymmetric models [8,[16][17][18] and the inert doublet model (IDM) [19].
In this article, we study the e + e − → hγ process in various extended Higgs models, including the IDM studied in Ref. [19], the inert triplet model (ITM) and two Higgs doublet models (THDMs) as distinctive benchmark models for the process. We calculate the loop-induced amplitudes at the leading order (LO) by employing and extending the H-COUP program [20], which evaluates the renormalized gauge invariant vertex functions for the SM-like Higgs boson in various extended Higgs models. Through a systematic study of the different models, we would like to obtain more general information on new physics effects in the signal. We begin with a brief review of the e + e − → hγ cross section in the SM in Sec. II. After introducing each extended Higgs model in Sec. III, we present the cross sections in each model, and also discuss the correlations with the h → γγ and h → Zγ decay rates in Sec. IV. The results are shown not only at √ s = 250 GeV but also at √ s = 500 GeV in anticipation of the energy extension of the ILC or the Compact Linear Collider (CLIC) [21]. The summary is given in Sec. V.
Explicit formulae for the hγγ and hZγ vertex functions are listed in Appendix.

II. THE PROCESS OF e + e − → hγ IN THE SM
We briefly review the total cross section for e + e − → hγ in the SM [6][7][8][9]. As mentioned in Sec. I, the process is loop-induced, and we evaluate it at LO. The details of the calculation will be discussed in Sec. IV. Figure 1 shows the total cross section σ(e + e − → hγ) in the SM as a function of the collision energy √ s. 1 One can observe an interesting peak-dip structure due to negative interference among different contributions [8,9]. To understand the structure, we decompose all the contributions into two categories in the 't Hooft-Feynman gauge; one is contributions from triangle hV γ (V = γ or Z) loops, the other is from box loops, shown in blue and green lines, respectively, in Fig. 1 loop. Combining all the contributions at the amplitude level, the total cross section eventually has the first peak at √ s ∼ 250 GeV and the second one at √ s ∼ 500 GeV. As we see, the production rate is very sensitive to the magnitudes of each amplitude as well as the relative phases among them. Therefore, slight modifications of the SM interactions as well as loop contributions from new particles can substantially enhance (or reduce) the production rate. We note that the relative magnitudes between the triangle and box contributions are rather different at different energies, and hence new physics can affect the total rates differently at different collision energies, e.g.
between at √ s = 250 GeV and 500 GeV, as seen below.
Before turning to the next section, we discuss how new physics can affect the process. In extended Higgs models, the SM-like Higgs interactions can be modified due to the mixing with other neutral scalars. The effect is described as the tree-level scaling factors the ratio of the Higgs coupling in a new physics model to the one in the SM, denoted as red blobs in diagrams (a), (b) and (c) in Fig. 2. In addition, the process can be modified by additional triangle loops of charged scalars; see the diagram (d). In the following, we study such effects in each extended Higgs model.

III. EXTENDED HIGGS MODELS
As discussed above, in extended Higgs models, the e + e − → hγ process can be altered through 1) the modified couplings of the SM-like Higgs boson and/or 2) charged scalar loops. In order to study such new physics effects on the process, we consider three distinctive models: the inert doublet model (IDM), the inert triplet model (ITM) with the hypercharge Y = 1, and the two Higgs doublet model (THDM). At the tree level, while in the first two models the SM-like Higgs couplings to SM fermions and weak bosons are the same as in the SM, in the THDM those are modified by the mixing of the scalar fields. While all the three models include singly charged scalars, the ITM includes doubly charged scalars as well. We briefly describe the three models in order. We give the Higgs potential and define the relevant parameters for each model. We also describe theoretical and experimental constraints on the parameters. The most important experimental constraint comes from the h → γγ signal at the LHC, which will be discussed exceptionally in the next section.

A. Inert doublet model
In the IDM [22,23], an isospin doublet field Φ 2 with hypercharge Y = 1/2 is added to the SM Higgs doublet field Φ 1 . The model imposes an exact Z 2 symmetry under which Φ 2 is odd while all the other fields are even. The general Higgs potential under the Z 2 symmetry is given by where all the parameters can be taken to be real. Due to the Z 2 symmetry, only Φ 1 can have a non-zero vacuum expectation value (VEV) v. The two doublet fields Φ 1,2 are parameterized as where G ±,0 are Nambu-Goldstone bosons to be absorbed by weak bosons. h is the SM-like Higgs boson with the mass of 125 GeV, while H, A and H ± are Z 2 -odd inert scalar bosons. After imposing the stationary condition, masses of the scalar bosons are written in terms of the parameters in the potential as with λ ± 345 ≡ λ 3 + λ 4 ± λ 5 . We conventionally choose the following five parameters in the IDM: Because of the exact Z 2 symmetry, none of the inert scalars have direct couplings to SM fermions.
The couplings of the SM-like Higgs boson to SM fermions and weak bosons are exactly the same as in the SM; i.e., in Eq. (1). 2 While the couplings of the charged scalars to gauge bosons are the SM gauge couplings, the coupling to the SM-like Higgs boson is given by We note that, if λ 3 = 0 (or m 2 H + = µ 2 2 ), the predictions for e + e − → hγ in the IDM and in the SM are identical at the LO.
The following theoretical and experimental constraints on the above parameters are taken into account [26,27]. Vacuum stability [22] and existence of the inert vacuum [28] require and respectively. We evaluate constraints from perturbative unitarity adopting the formulae given in Refs. [29,30].
LEP-I precision measurements exclude the possibility that weak bosons decay into a pair of inert scalars; i.e., m H + + m H,A > m W , 2m H + > m Z , and m H + m A > m Z . Moreover, the electroweak precision data, especially the T parameter implies that a mass difference between charged scalars (H ± ) and a neutral inert scalar (H or A) must not be too large [23,31]. We note that direct searches for additional scalars in collider experiments have been often done in the context of THDMs or supersymmetric models, in which the scalar particles couple to SM fermions.
Therefore, careful re-interpretations are necessary to constrain the inert models. LEP-II chargino search results are naively interpreted as m H + > 70 − 90 GeV [32], while neutralino search results where m H < m A is assumed [33,34]. The constraints from the 8-TeV LHC data have been also studied recently [31,35].
In our study, the value of the coupling g hH + H − in Eq. (7); i.e., λ 3 , is one of the relevant parameters. Figure 3(left) shows allowed regions of λ 3 as a function of λ 2 in the IDM after imposing the above theoretical constraints, where we take λ 4 = λ 5 = 0 for simplicity. λ 3 is bounded below by the vacuum stability (green region) and above by the existence of the inert vacuum (red). The constraint from the inert vacuum depends on the charged scalar mass. Those constrains depend on λ 2 , which is bounded above by perturbative unitarity (blue).
We note that the inert doublet model has been often studied in the context of dark matter [23,36,37] since the lightest neutral inert scalar can be stable due to the discrete Z 2 symmetry, but we do not consider such property in this work. See, e.g. Ref. [31], for dark matter constraints.

B. Inert triplet model
Instead of a doublet field as in the IDM, we introduce an isospin triplet field ∆ with hypercharge Y = 1, and impose an exact Z 2 symmetry under which ∆ is odd while all the other SM fields are even [38,39]. The Higgs potential is given by 3 (10) where all the parameters are real, and the fields are parameterized by In addition to singly charged scalars, the model contains doubly charged scalars. The mass spectrum for the inert scalars is given by while the mass of the SM-like Higgs boson is m 2 h = −2µ 2 1 = v 2 λ 1 . We choose the following five parameters in the ITM: Similar to the IDM, the inert scalars do not couple to SM fermions, and the SM-like Higgs boson interacts with SM fermions and weak bosons as in the SM; i.e., The couplings of the charged scalars to the SM-like Higgs boson is given by We note that, if λ 5 = 0, the doubly and singly charged scalars have the same mass and the same coupling to the SM-like Higgs boson. If λ 3 = λ 5 = 0; i.e., m 2 H ++ = m 2 H + = µ 2 2 , the predictions for e + e − → hγ in the ITM and in the SM are identical at the LO.
The above parameters should satisfy the following theoretical and experimental constraints.
With the squared masses in Eq. (12) positive, vacuum stability [40] and existence of the inert vacuum require and respectively. We evaluate constraints from perturbative unitarity adopting the formulae given in Refs. [40][41][42]. While the constraints from LEP precision measurements are similar to the ones on the IDM described above [43], direct searches for the ITM at the LHC was studied in Ref. [44] but have not been fully explored yet.
Similar to the IDM, one of the relevant parameters is λ 3 . In Fig. 3(right) we show allowed regions of λ 3 as a function of λ 2 in the ITM after imposing the theoretical constraints from vacuum stability (green region), existence of the inert vacuum (red), and perturbative unitarity (blue), where we take λ 2 = λ 4 and λ 5 = 0 for simplicity. 4 The allowed region of λ 3 in the ITM is slightly smaller than that in the IDM.
We note that the ITM with hypercharge Y = 0 does not contain doubly charged scalars, and hence the situation is very similar to that in the IDM in our study. Similar to the IDM, we do not consider the dark matter property; see Refs. [38,39,44].

C. Two Higgs doublet models
The THDM is similar to the IDM, but instead of imposing an exact Z 2 symmetry, the model allows to softly break a Z 2 symmetry, which still avoids flavor changing neutral currents at tree level [46]. The Higgs potential in the THDM is given by where m 2 3 and λ 5 can be real by assuming the CP conservation. The two doublet fields Φ 1,2 are parameterized as where v 1 and v 2 are the VEVs of the Higgs doublet fields with v = (v 2 1 + v 2 2 ) 1/2 . The mass eigenstates of the Higgs fields are defined as follows: where R(θ) is a rotation matrix with a mixing parameter θ and tan β = v 2 /v 1 . After imposing two stationary conditions for h 1,2 , masses of the physical Higgs bosons and the mixing angle α are expressed by where s θ and c θ represent sin θ and cos θ, and M 2 ≡ m 2 3 /s β c β describes the soft breaking scale of the Z 2 symmetry, and M 2 ij are the mass matrix elements for the CP -even scalar states in the basis of (h 1 , h 2 )R(β): We choose the following free parameters in the THDM: Unlike the inert models, additional Higgs bosons in the THDM couple with SM fermions through the Yukawa interaction. We can define four types of the interactions under the softly-broken Z 2 symmetry; Type-I, II, X and Y, depending on the Z 2 charge assignment for the right-handed fermions [47][48][49][50][51][52]. In all the four types of the THDMs, the up-type Yukawa interaction is modified in the same manner. In the Type-II and Type-Y THDMs, on the other hand, the down-type Yukawa can be enhanced by tan β.
where ζ f = cot β for the up-type Yukawa coupling. The h-H + -H − coupling is given by For later comparison, λ 3 is defined in the same manner as in Eq. (7) in the IDM.

IV. NUMERICAL RESULTS
In this section, we describe the formalism of our calculations, and show the numerical results at √ s = 250 GeV as well as at √ s = 500 GeV.

A. Loop calculations by using the H-COUP program
We consider the process where the four-momenta (k,k, p, p h ) and helicities (σ,σ, λ) are defined in the center-of-mass (CM) frame of the collision. The amplitudes are given by where the vertex function Γ ν can be decomposed in terms of the form factors C σ i as [8] Γ ν = γ µ C σ 1 g µν p · k − p µ k ν + C σ 2 g µν p ·k − p µkν + C σ 3 g µν .
C σ i sum all the one-loop contributions as where C hee,σ i denotes the hee vertex contributions, C hV γ,σ i (V = γ, Z) are the triangle vertex contributions (diagrams (a,b,d) in Fig. 2), and C box,σ i is the box contributions (diagram (c)). Note that, due to the gauge invariance, C σ 3 is zero after summing all the contributions. The differential cross section is where θ is the scattering angle between the electron and the photon in the CM frame, and t, u = −(s − m 2 h )(1 ∓ cos θ)/2. In the following, we briefly describe how we evaluate the form factors C σ i for each loop diagram by using the H-COUP program [20]. In the m e = 0 limit, the one-loop hee vertex in extended Higgs models deviates from the SM prediction only by the mixing effects as the κ V scaling factor due to weak-boson loops. However, the hee vertex diagrams are non-zero only for C σ 3 ; i.e., do not contribute to the physical observable. In order to check the gauge invariance, we evaluate C hee,σ with q = k +k being the momentum of the s-channel off-shell photon or Z boson. As shown in Fig. 2, Γ µν hV γ includes top and W loops with the scaling factors κ f and κ V as well as charged scalar loops in extended Higgs models. C hV γ,σ 1,2,3 can be evaluated in terms of Γ 1,2 hV γ as where g σ V ee are the SM γee and Zee couplings. One can find the explicit expressions for Γ 1,2 hγγ and Γ 1,2 hZγ in Appendix. For validity of our program, we numerically check the gauge invariance; i.e., Γ 1 hV γ = p · q Γ 2 hV γ for top and charged scalar loops independently, and C σ 3 = 0 for weak-boson loops after summing all the contributions. When q 2 = 0 or m 2 Z , namely the case for the h → γγ or h → Zγ decay, we can also check Γ 1 hV γ = p · q Γ 2 hV γ even for W -boson loops.
B. Results at √ s = 250 GeV We show numerical results for the cross sections at √ s = 250 GeV in the IDM, the ITM, and the THDM, in order. We also discuss correlations between the e + e − → hγ production and the h → γγ decay as well as the h → Zγ decay. In order to see deviations from the predictions in the SM, we evaluate the ratios of the total cross sections and the partial decay rates where V = γ or Z, and σ NP and Γ NP are evaluated in each extended Higgs model. (instead of µ 2 2 ) as independent parameters for our illustration. We fix the other parameters, m H , m A and λ 2 , so as to avoid the constraints discussed in Sec. III A. We note that our results agree with the previous study in Ref. [19].
In Fig. 4(left), we show total cross sections for e + e − → hγ in the IDM as a function of λ 3 . In When λ 3 is negative (positive), the production is enhanced (reduced) from the SM prediction. The same triangle H ± loops appear in the h → γγ process, which can modify the observed Higgs decay rate. One can find the explicit partial decay rate in the IDM in Refs. [27,71,72], which is implemented in H-COUP [25]. In Fig. 4(right), we show correlations between ∆R γγ and ∆R hγ , defined in Eqs. (35) and (34), respectively. There are clear positive correlations. As the e + e − → hγ process is more intricate than h → γγ, namely off-shellness of one of the photons and the additional hZγ and box contributions, the slope of the correlations differs by the charged Higgs boson mass; |∆R hγ | > |∆R γγ | for m H + < 150 GeV, while ∆R hγ ∼ ∆R γγ for m H + 150 GeV.
In Fig. 4(right), we also present the h → γγ signal strength from the LHC Run-I data with the 1σ and 2σ uncertainties, µ γγ = 1.14 +0.19 −0.18 [73], by vertical blue lines. This provides the stronger constraints for light charged Higgs bosons. For instance, for the case of m H + = 100 GeV, −1.2 λ 3 0.6 is allowed at 95% confidence level (CL). If we observe ∆R γγ ∼ 0 with smaller uncertainties Similar to Fig. 4(left) in the IDM, we show total cross sections in the ITM as a function of Fig. 5(left). The production rate is highly enhanced due to the considerable contributions from the doubly charged scalars. This is simply because photons couples to particles with the electromagnetic charge. The vertex functions Γ 1,2 hγγ in Eqs. (A6) and (A7) are proportional to S Q 2 S , and hence, for the λ 5 = 0 case, the contribution from the charged scalar loops of the hγγ vertex in the ITM is five times larger than that in the IDM at the amplitude level.
A remarkable difference from the IDM is that, even for positive λ 3 , the production rate can be significantly enhanced if the charged Higgs bosons are light. This is because the charged scalar contributions overwhelm all the other contributions. A caveat is that there is an upper bound on λ 3 from the requirement of the existence of the inert vacuum, which depends on the masses of the inert scalars, as discussed in Sec. III B. The excluded regions are indicated by dotted lines. We also note that, even in the IDM, we naively expect that the cross section could be enhanced if we take much larger positive λ 3 . However, again, such parameter region is not allowed by the theoretical constraints.
Once c β−α deviates from zero, the scaling factors κ f and κ V deviate from one. In order to see this mixing effect, we take tan β = 2 and vary c β−α from 0 to 0.3, which is allowed in the current constraints from the Higgs coupling measurements [80,81]. We note that a negative c β−α is also allowed, and the results are very similar to the positive case. Unlike the inert models, not only the hV γ contributions but also the box contributions deviate from the SM due to the scaling factors of the SM-like Higgs couplings. Therefore, even for λ 3 = 0, the cross section in the THDM differs from that in the SM if c β−α = 0. Except such small mixing effects, the qualitative behaviors are very similar to the case in the IDM, namely negative (positive) λ 3 can enhance (reduce) the production rate.
As shown in filled circles in Fig. 6(left), however, the theoretical constraints in the THDM are [Right] Correlations between ∆R(h → γγ) and ∆R(e + e − → hγ). Vertical blue lines indicate the h → γγ signal strength from the LHC Run-I data with the 1σ and 2σ uncertainties.
much severer than those in the IDM; the vacuum stability requires λ 3 > 0.1 for tan β = 2. As mentioned in Sec. III C, although such light charged Higgs bosons with tan β = 2 are allowed in the Type-I and Type-X models for the B-physics experiments [63], the current LHC data require higher tan β values to avoid the lower limit on the mass, m H + > 160 GeV [62]. On the other hand, for larger tan β, the allowed region with the theoretical constraints in Fig. 6(left) becomes smaller due to the unitarity constraint. Therefore, in the THDMs, there is a tension between the allowed parameter region and the preferred region to observe large deviations in the e + e − → hγ signal.
In Fig. 6(right), we present correlations between ∆R γγ and ∆R hγ , where we still keep tan β = 2 as an illustration. Although we vary c β−α as 0 < c β−α < 0.3, this mixing effect is small, and hence the correlations look similar to ones in the IDM in Fig. 4(right), except no allowed parameter region for positive ∆R γγ and ∆R hγ . In short, the production rate in the THDM tends to reduce from the SM prediction after imposing the theoretical constraints, and the magnitude of the deviation is minor after imposing all the current experimental constraints.
Before turning to the case for √ s = 500 GeV, we mention the h → Zγ decay, which is closely related to the h → γγ decay. As seen in Fig. 7, ∆R γγ and ∆R Zγ are strongly correlated in the IDM and the ITM. The correlation in the THDM is very similar to that in the IDM, and hence we omit it. The sensitivity to the charged scalar loops for the h → Zγ decay is much weaker than for the h → γγ decay. Moreover, from the experimental point of view, the observation of the h → Zγ decay is much more difficult than the h → γγ decay; the current LHC data only give the upper limit on the production cross section times the branching ratio for pp → h → Zγ as about seven times the SM prediction [82,83]. An interesting thing in the ITM is that, for the case of m H + = m H ++ = 90 GeV with λ 3 = 1.5, leading to the large enhancement for the hγ production, we can retain the h → γγ decay as in the SM, but the h → Zγ decay should be suppressed by about 70% from the SM prediction. In other words, if we observe the h → Zγ decay as in the SM in future experiments, we could not expect such hγ enhancement.
C. Results at √ s = 500 GeV As we have seen in Fig. 1, the e + e − → hγ process in the SM has the second peak around √ s = 500 GeV, whose cross section is about a half of that at √ s = 250 GeV. 5 Here, we repeat the exact same analyses in the previous subsection, but consider the collision energy at 500 GeV, where one can find rather different parameter dependences of the cross sections and the ∆R γγ - 5 The K factor of the next-to-leading order QCD corrections is about −0.4% and 14% at √ s = 250 and 500 GeV, respectively [9].

V. SUMMARY
We studied Higgs production associated with a photon at e + e − colliders in the IDM, the ITM and the THDM as distinctive new physics benchmark scenarios. The cross section in the SM is maximal around √ s = 250 GeV, and we presented how and how much the new physics can enhance or reduce the production rate. We also discussed the correlations with the h → γγ and h → Zγ decay rates. We found that, with a sizable coupling to a SM-like Higgs boson, charged scalars can give considerable contributions to both the production and the decay if their masses are around 100 GeV. Under the theoretical constraints from vacuum stability and perturbative unitarity as well as the current constraints from the Higgs measurements at the LHC, the production rate can be enhanced from the SM prediction at most by a factor of two in the IDM. In the ITM, in addition, we found a particular parameter region, m H ++ = m H + ∼ 90 GeV with λ 3 ∼ 1.5, where the hγ production significantly increases by a factor of about six to eight, but still retaining the h → γγ decay as in the SM. For such parameter region, on the other hand, the h → Zγ decay is suppressed. In the THDM, possible deviations from the SM prediction are minor in the viable parameter space. We also showed that the dependence of the cross section on the model parameters is rather different at √ s = 500 GeV, leading to a possibility to access more information on the Higgs sector.
As a final remark, we would like to point out that new physics can affect not only the total cross section but also the differential distribution, which would be important for realistic simulation of the signal events to compare with the background.
The 1PI diagram contributions from fermion, W -boson and charged scalar loops are expressed, respectively, as follows: FIG. 10: Real and imaginary parts of the loop function (C 12 + C 23 )(p 2 1 , p 2 2 , p 2 h ; m S , m S , m S ) as a function of the mass of the scalar particle m S . External momenta are fixed at p 2 1 = 0, p 2 2 = (0, 250, 500 GeV) 2 , and where v f = I f /2 − Q f s 2 W and g Z = g 2 + g 2 . Q S and I S 3 denote electric charge and isospin for charged scalars, respectively. The counter term contributions in an improved on-shell scheme [69], which is applied in H-COUP, are written by We note that extra Higgs loop contributions do not appear in the counter terms. Similar to the hγγ vertex functions, κ f , κ V and g hSS * are given in each model.
For better understanding of our results in the main text, we look at the m S dependence of Γ 2 hγγ,S for the e + e − → hγ production and the h → γγ decay. As discussed in Sec. IV A, Γ 1 hγγ,S is related to Γ 2 hγγ,S due to the gauge invariance. For the kinematical configuration p 2 1 = 0, p 2 2 = s, and p 2 h = m 2 h , C 2 can be expressed by only the C 0 and B 0 functions as In Fig. 10, we show behaviors of the loop function C 2 (≡ C 12 + C 23 ) as a function of the mass of scalar particle m S , where √ s is fixed at 0, 250 and 500 GeV, denoted by black, red and blue lines, respectively. For m S > √ s/2, the loop function becomes pure real, and goes to zero in the large mass limit.