The mass-degenerate SM-like Higgs and anomaly of $(g-2)_\mu$ in $\mu$-term extended NMSSM

We chose the $\mu$-term extended next-to-minimal supersymmetric standard model ($\mu$NMSSM) for this work, and the phenomenological research is based on the assumption of double Higgs resonance state as the Standard Model (SM)-like Higgs considering the recent $(g-2)_\mu$ result. The study also take into account a variety of experimental results, including direct detection of dark matter (DM) and searching results for sparticles at the Large Hadron Collider (LHC). We study the characteristic of DM confronted with limitations of direct detection experiments. Following that, we concentrate on the properties of the mass-degenerate SM-like Higgs bosons and explaining the anomaly of $(g-2)_\mu$. We conclude that the anomaly of $(g-2)_\mu$ can be explained in the scenario with two mass-degenerate SM-like Higgs, and there are samples that meet all current constraints and outperform SM in fitting Higgs data.


Introduction
The publication of the latest data by the Fermi National Accelerator Laboratory (FNAL) and Brookhaven National Laboratory (BNL) show a 4.2σ discrepancy from the Standard Model (SM) prediction . When the experimental accuracy is further improved, the measurement of the muon anomalous magnetic moment (g − 2) µ is likely to become a breakthrough of new physics [39][40][41][42]. So the interpretation of (g − 2) µ anomaly has also become an important task of new physics. For examples, low-energy supersymmetry (SUSY) can naturally explain the anomaly of (g − 2) µ .
SUSY models have been widely studied due to its multiple theoretical advantages, such as the explanation of the gauge hierarchy problem, the unification of gauge coupling, and natural solutions to the Dark Matter (DM) mystery. As the most economical realization of SUSY, the minimal supersymmetric standard model (MSSM) [132] with R-parity conservation can provide a viable candidate of DM, which is lightest neutralino as the lightest supersymmetric particle (LSP). However, considering the constraints from DM relic density and DM direct detections, MSSM is strongly restricted, and there are also some problems in the MSSM, such as the µ-problem and little hierarchy problem, which can be solved in the next-to-minimal supersymmetric standard model (NMSSM) [133,134]. NMSSM extends the MSSM by one singlet Higgs superfieldŜ, which develops a vacuum expectation value (VEV) to generate an effective µ-term. Furthermore, because of the enlarged Higgs sector, the SM-like Higgs boson mass can be easily interpreted [135]. In the NMSSM with a Z 3 symmetry (Z 3 -NMSSM), to realize a singlino-dominant DM, 2|κ|/λ must be less than 1 [136]. However, the situation is different in the general NMSSM (GNMSSM). We consider a simplified version of GNMSSM, i.e., the µ-term extended Z 3 -NMSSM (µNMSSM) [137]. In the µNMSSM, the mass ratio of singlino and higgsino is no longer equal to 2|κ|/λ, so the values of κ have a wider range compared to the case in the Z 3 -NMSSM [138]. The parameter κ has an important effect on the singlet fields' self-interactions, which can be entirely responsible for the DM density. Therefore, considering the latest DM experimental constraints, the µNMSSM has a wider parameter space than the Z 3 -NMSSM.
In July 2012, both the ATLAS and CMS collaborations announced a scalar with mass near 125 GeV [139,140]. Combined measurements of Higgs boson production cross sections and decay branching fractions show no significant deviations from Standard Model (SM) predictions. However, for the Higgs production process in association with a top quark pair followed by the decay mode to vector boson (H −→ W W ) or photon (H −→ γγ) [141][142][143][144][145], there is a discrepancy between the SM prediction and the experimental data.
Many theories attempt to interpret the observed data at the LHC. In the µNMSSM, either the lightest CP-even Higgs boson H 1 or the next-to-lightest CP-even Higgs boson H 2 can be SM-like with mass near 125 GeV. In fact, it is also possible to have H 1 and H 2 nearly degenerate with mass near 125 GeV [146][147][148][149][150], so that the observed signal at the LHC is actually a superposition of two individual peaks, and these two peaks cannot be independently resolved. In our work we focus on the scenario with two mass-degenerate Higgs boson with mass near 125 GeV in the µNMSSM considering various constraints including the DM relic density and DM direct detection limits. Furthermore, we try to explain the anomaly of (g − 2) µ in this scenario.
The paper is structured as follows. In Section 2, we briefly introduce the basic properties of µNMSSM including the Higgs, neutralino and chargino sectors. Then we study the observables of the 125 GeV Higgs boson at the LHC, DM and (g − 2) µ in the µNMSSM. We also describe our scanning strategy. In Section 3, we investigate the properties of DM confronted with DM direct detection limits. In Section 4, we show the numerical results in the scenario with two mass-degenerate Higgs boson and explain the anomaly of (g − 2) µ .
Finally, we give a summary in Section 5.

Basics of µNMSSM
To address a number of flaws in MSSM, such as the µ problem, we consider the next-to-minimal extension.
Compared with MSSM, NMSSM introduces a singlet Higgs field S. The superpotential of the general NMSSM (GNMSSM) is given by where the Yukawa terms W Yukawa are as follows [134], Among them, λ and κ are dimensionless couplings, µ and µ are supersymmetric mass terms, and ξ F is the supersymmetric tadpole term of mass square dimension.
In the above formula, the µ, µ and ξ F terms break the Z 3 -symmetry, and the κ 3 term conserves the Z 3symmetry while breaking the PQ-symmetry. Some articles use the terms µ and ξ F to explain the tadpole problem [151] and the cosmological domain-wall problem [152][153][154]. To maintain the purpose of our research without sacrificing generality, we set µ = ξ F = 0. This is known as the µ-term extended NMSSM (µNMSSM).
The extra µ-term is related to the non-minimal supergravity coupling χ via the gravitino mass as µ = 3 2 m 3/2 χ, where m 3/2 denotes the gravitino mass [155]. The superpotential and soft supersymmetry-breaking terms of µNMSSM are given below [156,157], The breaking of electroweak symmetry allows Higgs field to obtain non-zero vacuum expected values (vevs) The expression of the Higgs fields can be written as The ratio of the two Higgs doublet vevs defines the parameter tanβ GeV, and the effective µ-term µ ef f is generated by µ ef f = λv s / √ 2.
We set B µ = 0 in the soft breaking term since it plays a minor role in our work, then we get a Higgs potential as follows, By diagonalizing the mass matrix M 2 S and M 2 P using the unitary rotations V and V P , we can get the mass eigenstates H i (i = 1, 2, 3) and A i (i = 1, 2). The mass of the charged Higgs bosons m H ± can be written as follows The neutralino sector in the µNMSSM consists of Bino fieldB 0 , Wino fieldW 0 , Higgsino fieldsH 0 d ,H 0 u and Singlino fieldS 0 . In the basis ψ 0 = −iB 0 , −iW 0 ,H 0 d ,H 0 u ,S 0 , the mass matrix can be given by [156] where θ W denotes the Weinberg angle, M 1 and M 2 denote the gaugino soft breaking masses. By a rotation matrix N , we can get the mass eigenstateχ 0 i (i = 1 − 5), which are labeled in mass-ascending order. It can clearly see that the higgsino mass in the µNMSSM is determined by the parameter µ tot = µ + µ eff , and the singlino mass mainly depends on 2 κ λ µ eff . In the basis ψ ± = W + ,H + u ,W − ,H − d , the chargino mass matrix is written as By rotation matrix U c and V c , we can get the chargino mass eigenstate χ ± i (i = 1, 2). For convenience, we have selected the following independent parameters as input parameters

Observables of the 125 GeV Higgs bosons
The ATLAS and CMS collaborations discovered a 5σ signal for a Higgs-like resonance with mass around 125 GeV [139,140]. However, these signal channels deviate by 1-2σ from the SM prediction. Of course, as the experiment's precision improves, we will be able to confirm the true nature of the discovered Higgs boson.
But firstly, we need to see if the two mass-degenerate Higgs hypothesis can provide a better explanation for the current experiment. The enhancement of final state γγ under the production modes of gluon fusion (ggH) and vector boson fusion (V BF ) is one of the deviations between the current experimental data and the SM prediction.
We usually focus on the product of the production cross section σ i and the decay branching ratio B f , so where i denotes the production modes: gluon fusion (ggH), vector boson fusion (V BF ), associated production with a Z or W boson (VH), and associated production with a top quark pair (ttH), f denotes the decay modes: γγ, W W , τ τ , bb, etc., α denotes the index of the resonance. Eq.(12) is in general case, with a sum over the index of the resonance, and we can specify it to the case of a single resonance. We list major O if in Table 1 and we could arbitrarily expand the table based on the process discovered at the LHC experiment, or we can empty the table that was not discovered at the LHC experiment. The error size of these data varies greatly due to the number of collider events and other factors. Table 1: Rows represent the Higgs boson production modes: gluon fusion (ggH), vector boson fusion (V BF ), associated production with a Z or W boson (V H), and associated production with a top quark pair (ttH).

Columns represent Higgs boson decay modes.
We can select any part of interest or with high experimental precision to study. For example, when we choose the intersection of the first two rows and the first two columns, the determinant of this 2 × 2 matrix is zero if there is only one Higgs resonance, In general case, the equation should be slightly modified as here δ is a factor, representing the degree of deviation. For clarity, this formula can be transformed as follows, This means that if there is only one Higgs resonance, this type of double ratio is strictly equal to 1, and δ = 0.
Whereas, if there are two or more Higgs resonance, the ratio deviates from one, and δ = 0. Note that the observable O if is defined in Eq. (12), and we should sum over the index of the resonance if there are two or more Higgs resonance. In our work, we focus on the scenario with two mass-degenerate Higgs bosons in the µNMSSM considering various constraints.

Dark Matter in the µNMSSM
We have two requirements for DM in the µNMSSM in our work. Firstly, we suppose that there was a large amount of DM in the early universe, and they reached the current Planck observation Ω DM h 2 = 0.120 ± 0.01 as they freezed out [158][159][160]. In this case, the relic density of DMχ 0 1 in the µNMSSM is required to be less than the central value 0.12. Note that the Z 3 -NMSSM employs four parameters: mχ0 1 , λ, κ and tanβ to describe the properties of DM, but the properties of DM in the µNMSSM are described by five parameters: mχ0 1 , λ, κ, tanβ and µ tot [136]. We can change κ to use theχ 0 1χ 0 1 → h s A s process to achieve the correct relic density. Besides, DM based on higgsino in the µNMSSM has obvious advantages since that higgsino-dominated DM is different from the SU(2) singlet dominated DM which does not interact with Z bosons, such as B, W, and S. And the coupling of Z boson with higgsino-dominated DM C Zχ 0 1χ 0 1 in the µNMSSM is [161,162], where N 2 13 − N 2 14 is called 'higgsino asymmetry'. Secondly, spin-independent (SI) and spin-dependent (SD) DM-nucleon cross sections are required to meet experimental limits [163][164][165]. When the squarks are heavy, the SI nucleon scattering is primarily the t-channel process of exchanging Higgs bosons with the cross-section given as [166,167], is the reduced mass of nucleus andχ 0 1 , and C N N Hi is the coupling of Higgs boson H i with nucleon, In the above formula, F . In our case that mass of the heaviest neutral CP-even Higgs is larger than 1.5 TeV, the dominant contributions to the SI cross-section are from the two light Higgs bosons and the relevant couplings Cχ0 1χ 0 1 Hi is given by [138], Then the SD cross section takes the following simple from [168,169], where C N = C p 4.0 pb for the proton and C N = C n 3.1 pb for the neutron and these equations could be helpful for understanding our numerical results in the following sections.

(g − 2) µ in the µNMSSM
In supersymmetry, the correction for the muon anomalous magnetic moment a SUSY µ in the µNMSSM is almost the same as the case in MSSM, the difference is that µ in MSSM is replaced by µ tot in the µNMSSM.
Although at one loop there is also the possibility of a singlino insertion contribution, this contribution is never large when DM constraints are considered, at least for the singlino-dominated DM case [137].
We list the loop contribution items of a SUSY µ [40,170], Referring to Ref. [43], Feynman diagrams for a 1L µ , a are given in Fig. 1 and Fig. 2, respectively.
The subscript tan β means that each n-loop term is proportional to the (tan β) n term, which leads to a 10% correction for higher-order terms if tan β is greater than 50 [51]. If tan β tends to infinity, even if all SUSY particle masses are above TeV scale, there can keep a large correction [74].
It can be seen from Fig. 1 that the correction of one-loop diagram mainly comes from two parts, one part is the exchange of neutralinosχ 0 i and smuonμ, and the other part is the exchange of charginosχ ± i and sneutrinõ ν µ :  where i = 1,...,5, l = 1, 2, k = 1, 2 are the neutralino, smuon and chargino index, respectively. n L il , n R il , c L k , c R k are expressed as follows, where y µ = g 2 m µ /( √ 2m W cos β) is the muon Yukawa coupling. X is the smuon mass rotation matrix. The It can be seen that a 1L µ depends essentially on the bino(wino) masses M 1 (M 2 ), the higgsino mass µ, the left(right) smuon mass m Lµ , m Eµ and tan β, and it hardly depends on A t ,A κ or A λ . There is a simple relationship, a 1L µ is proportional to tan β/M 2 SU SY , where M 2 SU SY refers to the general SUSY mass [53,55,127].

Parameter space and scanning method
We start with a random scan to find the initial point in the parameter space, and then use the Markov chain Monte Carlo (MCMC) scan in EasyScan-1.0.0 [172] to explore the high-dimensional parameter space as follows, We set A λ and other supersymmetric parameters of the first and third generation sleptons, all squarks and gluinos to 2 TeV. We use SARAH-4.14.3 [173][174][175][176] to generate the model files in the µNMSSM and then use SPheno-4.0.4 [177,178] to generate spectrums, which includes the masses of the particles, mixing angles between mass eigenstates and interaction fields and other physical observables. In order to improve the sampling efficiency, we only require that the spectrums should satisfy the following conditions during scanning, whereas we will discuss the effects of DM direct detection limits and the (g − 2) µ anomaly on the parameter space in Section 3 and Section 4, respectively. With these files as input, we can run HiggsBounds and HiggsSignals by issuing the following commands, respectively: ./HiggsBounds LandH effC 5 1 "LandH" indicates that LEP, Tevatron and LHC analyses will be considered by HiggsBounds, "effC" stands for the input format for HiggsBounds, "5" stands for the number of neutral Higgs bosons and "1" stands for the number of charged Higgs boson in the model.
• Results from sparticle searches at the LHC. SModelS-1.2.3 [187][188][189][190][191][192] is used to determine whether a sample is excluded or not by decomposing spectrum and converting it into Simplified Model topologies and then comparing it with these simplified model results interpreting from the LHC. We consider these typical processes pp →χ 0 1,2χ ± 1 ,χ + 1χ − 1 ,μ +μ− and use the package Prospino2 [193][194][195][196] to generate the next-to-leading order cross sections of these processes as inputs for SModels. detections. So, we can see that the SI and SD constraints are complementary in limiting the parameter space of µNMSSM. We notice that there are a few of samples whose SI cross section can be lower than the neutrino floor, and consequently these DM may never be probed in DM direct detections. In the following discussion, we will focus on the parameter space tightly limited by DM direct detection limits in the µNMSSM.   4 shows the samples on the 2κ/λ -µ tot /µ ef f plane, λ -tanβ plane and µ tot -M 2 plane. Samples with red (green) points are allowed (excluded) by DM direct detection limits. In the scenario with two massdegenerate SM-like Higgs bosons, the DM is higgsino-dominated, which is easier to meet the constraints from DM direct detections compared with the scenario with singlino-dominated DM. To avoid the singlino-dominated DM in the µNMSSM, 2κ/λ should be far greater than µ tot /µ ef f , which can be seen clearly from Fig. 4(a). This is significantly different from the case in the Z 3 -NMSSM, which only requires 2κ/λ to be greater than 1. In Fig. 4(b), we find that a large tanβ greater than about 20 is accompanied with a small λ less than about 0.05, which guarantee to realize two mass-degenerate Higgs boson with mass about 125 GeV. In Fig. 4(c), we can see that a larger M 2 is required since that it can avoid the wino-dominated neutralino as the LSP.

Properties of DM confronted with direct detection limits
To consider the DM direct detection limits, we try to find the factor that affect the variation of SI and SD cross section. We find that mχ0 1 is around 50-200 GeV, which is much greater than the neutron mass. So µ r in Eq.(16) change little for different mχ0 1 , σ SI is primarily influenced by the couplings Cχ0 1χ 0 1 Hi and C N N Hi . We propose that the factor A X derived from Eq. (18) has a powerful influence on σ SI : In Fig. 5 we show the effect of A X on σ SI on the left plot, and the effect of higgsino asymmetry N ha = N 2 13 − N 2 14 on σ SD on the right plot. As the size of the cross section decreases, the color of the surviving sample shows a gradient change, which shows that the SI and SD cross sections depend on the variation of A X and N ha . We project samples on tan β -A X plot and M 2 -A X plot in Fig.6. The trend that σ SI decreases with increasing A X holds true for all samples as shown on the right plot. In the horizontal band A X = 0 ∼ 1, a large number of samples are excluded because they have higher DM relic densities leading to higher σ SD and σ SI .
This can be seen from the green samples in the horizontal band A X = 0 ∼ 1 on the left plot. Ωh 2 of samples in this band are much higher than the surrounding ones and so they are excluded by DM direct detection limits. We project samples on N 13 /N 14 -m χ 0 1 /µ tot plot and N 13 -N 14 plot in Fig.7. When N 13 is less than N 14 , m χ 0 1 /µ tot is proportional to N 13 /N 14 . When N 13 is relatively large, however, the relationship between m χ 0 1 /µ tot and N 13 /N 14 is inversely proportional. We can see that the samples with N 13 /N 14 close to one on the left plot of Fig. 7 correspond to the samples with smaller higgsino asymmetry on the right plot of Fig. 5, and they all have safer σ SI . The horizontal band m χ 0 1 /µ tot = 0.7 − 0.9 is severely excluded by DM direct detection limits because it corresponds to a larger higgsino asymmetry. When N 13 /N 14 is smaller or larger, the surviving samples reappear because N 13 and N 14 themselves are much small. This corresponds to the green samples in the left-bottom corner on the right plot in Fig. 7. The right plot supports our conclusion that the samples satisfying DM direct detection limits must have a smaller higgsino asymmetry.

The Mass-degenerate SM-like Higgs bosons and the explanation of
where α = 1, 2, i denotes the production modes: gluon fusion (ggH), vector boson fusion (V BF ), associated production with a Z or W boson (V H), and associated production with a top quark pair (ttH), f denotes the decay modes: γγ, W W , τ τ , bb, etc. We can simply have In the calculations we ignore the interference effect between the two mass degenerate Higgs bosons in the µNMSSM because the mass difference between most of the two Higgs bosons in our collected samples is greater than 50 MeV.
We project the surviving samples on plots of Fig. 8. From Fig. 8, we can see that R h and B bb(W W ) predicted by SM are different from that measured by ATLAS and CMS. B bb(W W ) predicted by SM even deviates beyond the experimental 1σ error. But Samples in the scenario with two mass-degenerate Higgs bosons are much more consistent with measured data than the SM prediction.
Samples with green (red) color indicate that σ h i mainly comes from the contribution of H 1 (H 2 ), and samples with golden color indicate that σ h comes from contributions of both H 1 and H 2 . The pink line stands for one standard deviation observed by ATLAS and CMS. The blue star represents the SM prediction.
To quantify the degree of agreement with the measured data totally, we define where R h,ob if is the measured central value and R h,er if is one standard deviation. When χ 2 is zero, samples are perfectly aligned with measured data. Besides, to compare with the SM prediction conveniently, we define . If C if is smaller than one, the scenario with two mass-degenerate Higgs bosons is more consistent with the measured data. Referring to Ref. [149], we also use the double ratios: Figure 9: All surviving samples in Fig. 5 are projected on plots µ H1 V BF + µ H2 V BF -C V BF,τ τ , D 3 -C ttH,W W and κ -C ggH,γγ . The colorbar is the same as in Fig. 8 We project surviving samples in Fig. 5 on plots of µ H1 V BF + µ H2 V BF -C V BF,τ τ , D 3 -C ttH,W W and κ -C ggH,γγ in Fig. 9. It is clear that there is a strong correlation between i σ H1 i / i σ h i and C if . When the contribution on i σ h i mainly comes from H 1 or H 2 , the values of C if are close to 1, as the samples with red or green shows. That is to say, these samples are consistent with the SM prediction. But there are lots of samples with C if smaller than 1, which means that the assumption with two mass-degenerate Higgs bosons may be true. For example, in Fig. 9(a), when µ H1 V BF + µ H2 V BF is around the experimental measured value µ ex V BF = 1.18, samples have the lowest C V BF,τ τ . Fig. 9(b) shows that C ttH,W W decreases obviously when D 3 is less than 1. In Fig. 9(c), we can see that the SM prediction is better than samples in the scenario with two mass-degenerate Higgs bosons for C ggH,γγ . The reason is that the cross section of V BF in the scenario with two mass-degenerate Higgs bosons is raised compared to SM, but that of ggH is not.
When we take into account the total contribution C tot ≡ i,j C if including the high precision processes (i, j) = (ggH, γγ), (ggH, ZZ), (ggH, W W ), (V BF, γγ), (V BF, W W ), (V BF, τ τ ), (ttH, W W W ) [141,143,144], there exist some samples with C tot < 1, which are more consistent with the measured Higgs data than the SM prediction. We list the relevant parameters of two benchmark points in Table 2. For the DM mechanism that yields a relic density compatible with the observed upper bound, we find that the leading contribution is from the processχ 0 1χ 0  the DM relic density, the Higgs data, sparticle searches at the LHC, and the (g − 2) µ measurement. C tot < 1 indicates that the scenario with two mass-degenerate Higgs bosons is more consistent with the measured data than the SM prediction.
We project surviving samples on plot of m Lµ − µ tot in Fig. 11. We can see that it is difficult to obtain samples satisfying the result of (g − 2) µ experiment when m Lµ is greater than 1000 GeV. This is because that the loop contribution to (g − 2) µ from smuon decreases sharply when smuon is much heavy.
We project surviving samples on plots m H2 − m H1 versus D i (i = 1, 2, 3) in Fig. 12. We can see that the double ratios D i deviate from 1, indicating that there are different contributions from H 1 and H 2 on these ratios. And there are lots of samples in agreement with the recent experimental result of (g − 2) µ . But we note that the deviations are not very outstanding. The reason is that while the production cross sections of two  mass-degenerate Higgs bosons differ, the ratios are similar, i.e., σ H1 ggH /σ H1 V BF ≈ σ H2 ggH /σ H2 V BF . To achieve better results, we can take an aggressive approach and only consider the decay branching ratios instead of the production cross-section:    Fig. 13. We can see that the deviation is more obvious under the new variable D B . There are some samples that typically deviate from 1. From the (g − 2) µ deviation distribution, we find that the recent experimental result from (g − 2) µ can be well explained in the scenario with two mass-degenerate Higgs bosons. Samples with D B around 1 deviate from the center value of (g − 2) µ , but some samples with D B deviation from 1 can satisfy the result of (g − 2) µ much better.

Summary
In the µNMSSM with two mass-degenerate SM-like Higgs bosons, we investigate the properties of DM, the observation of two mass-degenerate Higgs bosons and the explanation of (g − 2) µ anomaly. We first scan the parameter space under current experimental constraints including results of DM direct detection experiments and from LHC searches for sparticles. We find that, • The SI and SD constraints are complementary in limiting the parameter space of µNMSSM. The DM direct detection limits favor the DM is higgsino-dominated.
• The scenario with two mass-degenerate SM-like Higgs bosons is capable of explaining the (g −2) µ anomaly, that is, the scenarios we consider may be an explanation since they are compatible with the current measurements.
• Deviations defined by the double ratios D i (i = 1, 2, 3) are not very outstanding. The reason is that the ratio between the production cross section of H 1 and H 2 are similar. To achieve better results, we define a new variable D B . Samples with D B deviation from 1 can satisfy the result of (g − 2) µ much better.
• As far as the total contribution C tot is concerned, there exist some samples more consistent with the measured Higgs data than the SM prediction.

Acknowledgement
We Besides, this work is powered by the High Performance Computing Center of Henan Normal University.

A Double ratios and components of Higgs boson in the scenario with two mass-degenerate SM-like Higgs bosons
We project surviving samples on plots m H2 − m H1 versus µ V BF , m H1 versus µ V BF , m H2 versus µ V BF in Fig. A1. We can see that the value of µ H1 V BF + µ H2 V BF can be raised to the measured value 1.18 in Fig. A1, but the single contribution µ H1 V BF or µ H2 V BF is all below 1. We also project surviving samples on plots of m H1 -S 1X and m H2 -S 2X in Fig. A2. Comparing Fig. A1(b) with Fig. A2(a) for H 1 and Fig. A1(c) with Fig. A2(b) for H 2 , we can see that when H 1 or H 2 are dominated by H u , its contribution on σ V BF is large. when H 1 or H 2 are dominated by S, its contribution on σ V BF becomes small. As a result, when the contribution on double ratios mainly comes from a single Higgs, the double ratios are around 1 according to Eq. (29). Only if H 1 and H 2 are the mixing of H u and S, the double ratios can deviate from 1.