Implication of the 750 GeV diphoton resonance on two-Higgs-doublet model and its extensions with Higgs field
aa r X i v : . [ h e p - ph ] M a r Implication of the 750 GeV diphoton resonance ontwo-Higgs-doublet model and its extensions with Higgs field
Xiao-Fang Han , Lei Wang , Department of Physics, Yantai University, Yantai 264005, P. R. China IFIC, Universitat de Val ` e ncia-CSIC,Apt. Correus 22085, E-46071 Val ` e ncia, Spain Abstract
We examine the implication of the 750 GeV diphoton resonance on the two-Higgs-doublet modelimposing various theoretical and experimental constraints. The production rate of two-Higgs-doublet model is smaller than the cross section observed at the LHC by two order magnitude. Inorder to accommodate the 750 GeV diphoton resonance, we extend the two-Higgs-doublet model byintroducing additional Higgs fields, and focus on two different extensions, an inert complex Higgstriplet and a real scalar septuplet. With the 125 GeV Higgs being agreement with the observeddata, the production rate for the 750 GeV diphoton resonance can be enhanced to 0.6 fb for theformer and 4.5 fb for the latter. The results of the latter are well consistent with the 750 GeVdiphoton excess at the LHC.
PACS numbers: 12.60.Fr, 14.80.Ec, 14.80.Bn . INTRODUCTION Very recently, the ATLAS and CMS collaborations have reported an excess of events in thediphoton channel with an invariant mass of about 750 GeV [1]. The local significance of thissignal is at the 3 σ level for ATLAS and slightly less for CMS. The approximate productioncross section times branching ratio is 4.47 ± ± t ¯ t [4], diboson or dilepton channels, which gives a challenge to the the possible new physicsmodel accommodating the 750 GeV diphoton resonance. Some plausible explanations ofthis excess have already appeared [2, 5–7]. The two-Higgs-doublet model (2HDM) can notproduce the enough large cross section to accommodate the 750 GeV diphoton resonance.Ref. [7] introduces some additional vectors-like quarks and leptons to 2HDM in order toenhance the production rate of 750 GeV diphoton resonance.In this paper, we first examine the implication of the 750 GeV diphoton resonance onthe two-Higgs-doublet model imposing various theoretical and experimental constraints. Wegive the allowed mass ranges of the pseudoscalar and charged Higgs for m H = 750 GeV,and find that the production rate of 2HDM is smaller than the cross section observed at theLHC by two order magnitude. Finally, in order to explain the 750 GeV diphoton excess,we extend the two-Higgs-doublet model by introducing additional Higgs fields, and focus ontwo different extensions, an inert complex Higgs triplet and a real scalar septuplet. We findthat the production rate for the 750 GeV diphoton resonance can reach 0.6 fb for the formerand 4.5 fb for the latter with the 125 GeV Higgs being consistent with the observed data.Our work is organized as follows. In Sec. II we recapitulate the two-Higgs-doublet model.In Sec. III we introduce the numerical calculations, and examine the implications of the750 GeV diphoton resonance on the 2HDM after imposing the theoretical and experimentalconstraints. In Sec. IV, we respectively add an inert complex Higgs triplet and a real scalarseptuplet to 2HDM, and discuss the production rate for the 750 GeV diphoton resonance.Finally, we give our conclusion in Sec. V. 2 I. TWO-HIGGS-DOUBLET MODEL
The general Higgs potential is written as [8]V = m (Φ † Φ ) + m (Φ † Φ ) − h m (Φ † Φ + h . c . ) i + k † Φ ) + k † Φ ) + k (Φ † Φ )(Φ † Φ ) + k (Φ † Φ )(Φ † Φ )+ (cid:20) k † Φ ) + h . c . (cid:21) + h k (Φ † Φ )(Φ † Φ ) + h . c . i + h k (Φ † Φ )(Φ † Φ ) + h . c . i . (1)Here we focus on the CP-conserving case where all k i and m are real, and take k = k = 0.This can be realized by introducing a discrete Z symmetry, and m is a soft-breaking term.The two complex scalar doublets have the hypercharge Y = 1,Φ = φ +11 √ ( v + φ + ia ) , Φ = φ +21 √ ( v + φ + ia ) . (2)Where the electroweak vacuum expectation values (VEVs) v = v + v = (246 GeV) ,and the ratio of the two VEVs is defined as usual to be tan β = v /v . After spontaneouselectroweak symmetry breaking, there are five physical Higgses: two neutral CP-even h and H , one neutral pseudoscalar A , and two charged scalar H ± .The general Yukawa interaction can be given as L Y = − √ f (cid:2) z f sin( β − α ) + ρ f cos( β − α ) (cid:3) h f + − √ f (cid:2) z f cos( β − α ) − ρ f sin( β − α ) (cid:3) H P R f + i √ Q f ) ¯ f ρ f A P R f − ¯ u (cid:2) V ρ d P R − ρ u † V P L (cid:3) dH + − ¯ ν (cid:2) ρ ℓ P R (cid:3) ℓH + + H . c .. (3)Where f = u , d, ℓ , and z f = √ m f /v , while ρ matrices are free and have both diagonaland off-diagonal elements. For the aligned 2HDM [9], ρ = √ m f κ f /v , which leads that thecouplings of neutral Higgs bosons normalized to the SM Higgs boson are give by y hV = sin( β − α ) , y hf = sin( β − α ) + cos( β − α ) κ f ,y HV = cos( β − α ) , y Hf = cos( β − α ) − sin( β − α ) κ f ,y AV = 0 , y Au = − iγ κ u , y Ad,ℓ = iγ κ d,ℓ . (4)Where V denotes Z and W . We fix κ u = 1 / tan β , which denotes that a special basis istaken where there is no the up-type quark coupling to Φ [10, 11].3 II. NUMERICAL CALCULATIONS AND RESULTS OF 2HDMA. numerical calculations
We use [12] to implement the theoretical constraints from the vacuum stability,unitarity and coupling-constant perturbativity [13], and calculate the oblique parameters ( S , T , U ) and δρ . HiggsBounds-4.1.4 [14, 15] is employed to implement the exclusion constraintsfrom the neutral and charged Higgses searches at LEP, Tevatron and LHC at 95% confidencelevel. The in-house code is used to calculate the B → X s γ , ∆ m B s , and R b . The experimentalvalues of electroweak precision data, B → X s γ , ∆ m B s are taken from [16] and R b from [17].Since we focus on the implications of 750 GeV diphoton resonance on the 2HDM, we fix m h = 125 GeV, m H = 750 GeV, | sin( β − α ) | = 1, κ d = κ ℓ =0. The last three choices cannaturally accommodate the non-observation of excesses for diboson, dijet and dilepton. Wescan randomly the parameters in the following ranges:375 GeV ≤ m H ± ≤ , . ≤ tan β ≤ , − (2000 GeV) ≤ m ≤ (2000 GeV) . (5)Since the heavy CP-even Higgs coupling to the top quark is proportional to 1 / tan β forcos( β − α ) =0, we take the small tan β to avoid the sizable suppression of this coupling. Wetake m A ≃ m H which leads that the 750 GeV diphoton resonance is from both H and A ,and the more large cross section may be obtained. We define R γγ ≡ σ ( gg → H ) × Br ( H → γγ ) + σ ( gg → A ) × Br ( A → γγ ) . (6)In this paper we will introduce additional Higgs fields to 2HDM, and some multi-chargedscalars give very important contributions to the CP-even Higgs decay into γγ . Therefore,we give the general formulas for the CP-even Higgs decay into γγ [18],Γ( H → γγ ) = α m H π v (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)X i y i N ci Q i F i (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) , (7)with N ci , Q i are the color factor and the electric charge respectively for particles running inthe loop. The dimensionless loop factors for particles of spin given in the subscript are F ( τ ) = 2+3 τ +3 τ (2 − τ ) f ( τ ) , F / ( τ ) = − τ [1+(1 − τ ) f ( τ )] , F ( τ ) = τ [1 − τ f ( τ )] , (8)4 m A (GeV) m H ( G e V ) – FIG. 1: The scatter plots of surviving samples projected on the planes of m A versus m H ± . Allthe samples are allowed by the theoretical constraints. The crosses (red) and bullets (black) arerespectively allowed and excluded by the oblique parameters and ∆ ρ . where τ = 4 m i /m H and y i is from L = − m t v y t ¯ ttH + 2 m W v y W W + W − H − m φ v y φ φφH. (9) f ( τ ) = [sin − (1 / √ τ )] , τ ≥ − [ln( η + /η − ) − iπ ] , τ < B. results and discussions
First we examine the allowed mass range of pseudoscalar and charged Higgs for m H = 750GeV after imposing the theoretical constraints, oblique parameters and ∆ ρ , where we scan m A in the range of 375 GeV ≤ m A ≤ m A versus m H ± . m A and m H ± are favored in the range of 700 GeV and 800GeV. In addition, the pseudoscalar and charged Higgs masses are allowed to have sizabledeviations from 750 GeV for the small mass splitting between them. Also the pseudoscalarmass is allowed to have sizable deviation from 750 GeV for m H ± is around 750 GeV.Now we calculate R γγ taking m A ≃ m H = 750 GeV. Fig. 1 shows that m H ± is requiredto be larger than 650 GeV for m A ≃ m H = 750 GeV. For the decay H → γγ , the form factor5f scalar-loop is generally much smaller than that of fermion-loop. Further, 4 m H ± /m H hassizable deviation from 1 where the peak of form factor appears. Therefore, the contributionof the charged Higgs to the decay H → γγ is much smaller than that of top quark unless thetop quark Yukawa coupling is sizably suppressed by a large tan β . The W boson does notgive the contribution to the decay H → γγ since the H couplings to gauge bosons are zerofor cos( β − α ) = 0. The decays H → H + H − , AA, AZ are kinematically forbidden, and thewidths of H → W W, ZZ, b ¯ b, τ ¯ τ are zero due to cos( β − α ) = 0 and κ d = κ ℓ = 0. Also the H coupling to hh is zero for cos( β − α ) = 0, and we will give the detailed explanation in theAppendix A. The width of H → H ± W ∓ can be comparable to H → t ¯ t for the small chargedHiggs mass. However, m H ± is required to be larger than 650 GeV, which leads the width tobe sizably suppressed by the large phase space. Therefore, the decay H → t ¯ t dominates thetotal width of the heavy Higgs.In this paper we focus on the CP-conserving case where all the couplings constants ofHiggs potential are taken to be real. Therefore, the pseudoscalar A coupling to hh is zero.The pseudoscalar has no decays A → W W, ZZ, hh , and the widths of A → hZ, b ¯ b, τ ¯ τ are zero due to cos( β − α ) = 0 and κ d = κ ℓ = 0. Therefore, A → t ¯ t is the dominantdecay channel. Since the charged Higgs and gauge boson do not give the contributions to H → γγ , the top quark plays the dominant contributions to A → γγ . In our calculations,we use to calculate the total widths of H and A including various possible decaychannels.In Fig. 2, we project the surviving samples on the planes of R γγ versus tan β and m H ± versus tan β . The left panel shows the R γγ increases with decreasing of tan β since the σ ( gg → H ) and σ ( gg → A ) are proportional to 1 / tan β , and the dependence Br ( H → γγ )and Br ( A → γγ ) on tan β can be canceled to some extent by the widths of t ¯ t and γγ channels. However, due to the constraints of R b , B → X s γ and ∆ m B s , tan β is favored tobe larger than 1, which leads that the maximal value of R γγ is 0.015 fb. The correlationsamong R γγ , tan β and m H ± are shown in the right panel. IV. EXTENDING 2HDM WITH ADDITIONAL HIGGS FIELD
In the minimal version of 2HDM, the production rate of 750 GeV diphoton resonanceonly can reach 0.015 fb, which is smaller than cross section observed by CMS collaboration6 .0020.0040.0060.0080.010.0120.0140.016 1 1.5 2 2.5 3 3.5 4 tan b R gg (f b ) gg < 0.01 fb0.01 fb < R gg < 0.015 fb0.001 fb < R gg < 0.005 fb tan b m H ( G e V ) – FIG. 2: The scatter plots of surviving samples projected on the planes of R γγ versus tan β and m H ± versus tan β . by two order of magnitude. Ref. [7] introduces additional vector-like quark and lepton to2HDM in order to enhance the production rate. Here we will extend 2HDM with additionalHiggs fields and discuss two different extensions, an inert complex Higgs triplet and a realscalar septuplet. A. 2HDM with an inert complex Higgs triplet (2HDM-IHT)
The extension SM with a complex Higgs triplet is proposed in [19], called type-II seesawmodel, and ref. [20] also extends SM with an inert complex Higgs triplet. Here we addan inert complex SU(2) L triplet scalar field ∆ with Y = 2 to the 2HDM imposing an Z symmetry in which the triplet is assigned to be odd and the others even. The VEV oftriplet scalar field is zero to keep the Z symmetry unbroken. The potential of triplet fieldis written as V = M T r (∆ † ∆) + λ T r (∆ † ∆) + λ ( T r ∆ † ∆) + λ Φ † ∆∆ † Φ + λ (Φ † Φ ) T r (∆ † ∆)+ λ ′ Φ † ∆∆ † Φ + λ ′ (Φ † Φ ) T r (∆ † ∆) , (11)where ∆ = δ + / √ δ ++ ( δ r + iδ i ) / √ − δ + / √ . (12)7fter the two Higgs doublet Φ and Φ acquire the VEVs, the last four terms in Eq. (11)will give the additional contributions to the masses of components in the triplet, respectively.At the tree-level, the Higgs triplet masses are given as, m δ ±± = M + 12 v ( λ c β + λ ′ s β ) m δ ± = M + 12 v ( λ c β + λ ′ s β ) + 14 v ( λ c β + λ ′ s β ) m δ r = m δ i = M + 12 v ( λ c β + λ ′ s β ) + 12 v ( λ c β + λ ′ s β ) . (13)The charged Higgs triplet scalars couplings to the h and H are given as, hδ + δ − : − v ( λ ′ c α s β − λ s α c β + 2 λ ′ c α s β − λ s α c β ) Hδ + δ − : − v ( λ ′ s α s β + λ c α c β + 2 λ ′ s α s β + 2 λ c α c β ) hδ ++ δ −− : − v ( λ ′ c α s β − λ s α c β ) Hδ ++ δ −− : − v ( λ ′ s α s β + λ c α c β ) . (14) B. 2HDM with a real scalar septuplet (2HDM-RSS)
The extension of SM with the complex and real scalar septuplet has been studied in [21]and [22]. Here we introduce a real scalar septuplet to the 2HDM assuming that the septupletdoes not develop the VEV. The potential of the septuplet field Σ is written as V = M Σ † Σ + λ (Σ † Σ) + λ
48 (Σ † T a T b Σ) + λ (Φ † Φ )(Σ † Σ) + λ ′ (Φ † Φ )(Σ † Σ) , (15)where Σ = 1 √ (cid:0) T +++ , T ++ , T + , T , T − , T −− , T −−− (cid:1) T . (16)After the two Higgs doublet Φ and Φ develop the VEVs, the last two terms in Eq. (15)will give the additional contributions to the masses of all components of Σ. The septupletscalars are degenerate at the tree-level and their mass are m = M + 12 v ( λ c β + λ ′ s β ) . (17)The charged components of septuplet couplings to the h and H are, hT + T − = hT ++ T −− = hT +++ T −−− : − v ( λ ′ c α s β − λ s α c β ) HT + T − = HT ++ T −− = HT +++ T −−− : − v ( λ ′ s α s β + λ c α c β ) . (18)8 . calculations and discussions In the original 2HDM, the production rate for the 750 GeV diphoton resonance increaseswith decreasing of tan β . However, the R b and B flavor observables disfavor tan β to besmaller than 1. Therefore, in the following calculations we will taketan β = 1 , sin( β − α ) = 1 . (19)Further, in order to forbid the new charged Higgses altering the 125 GeV Higgs decay into γγ via one-loop effects, we require the light CP-even Higgs couplings to these charged Higgsesto be zero, which leads for Eq. (19), λ = − λ ′ and λ = − λ ′ for 2HDM − IHT , (20) λ = − λ ′ for 2HDM − RSS . (21)For Eq. (19) and Eq. (20), the triplet scalar masses in the 2HDM-IHT become degenerate, m δ ±± = m δ ± = m δ r = m δ i = M . (22)The H couplings to the charged components of triplet are, Hδ + δ − : − (cid:18) λ + 12 λ (cid:19) v, Hδ ++ δ −− : − λ v. (23)Similarly, the septuplet scalar masses of the 2HDM-RSS are m Σ = M. (24)The H couplings to the charged components of septuplet are, HT + T − = HT ++ T −− = HT +++ T −−− : − λ v. (25)For the Higgs triplet and septuplet, the mass splitting among the components can beinduced by loop corrections, and the charged components are very slightly heavier thanthe neutral components [20, 22]. These mass splittings are negligibly small, and the twoextensions can be well consistent with the oblique parameters. Both the Higgs triplet andseptuplet have no interactions with fermions, and the interactions with gauge bosons haveto contain two components simultaneously, which makes them to be hardly constrained bythe low energy observables and collider experimental searches. For the inert scalars, the9ulti-lepton + E/ T is regarded as one of the most promising channels at the LHC. Takingthe inert Higgs triplet model as an example, the charged scalars can be produced at theLHC through Drell-Yan process, q ¯ q → δ + δ − , δ ++ δ −− , δ ± δ ∓∓ , (26)and the charged scalars can have the following cascade decay assuming m δ ±± > m δ ± >m δ i > m δ r ( δ r is the stable particle), δ ±± → δ ± W ± ( ∗ ) ( W ± ( ∗ ) → l ± ν ) ,δ ± → W ± ( ∗ ) δ r → l ± ν δ r ,δ ± → W ± ( ∗ ) δ i → l ± ν δ r Z ( ∗ ) ( Z ( ∗ ) > l ± l ∓ ) . (27)The ATLAS and CMS collaborations have searched the 2 l + E/ T [23, 24], 3 l + E/ T [24, 25]and 4 l + E/ T [24], and set the limits on the next-to-lightest neutralino, the lightest-neutralinoand chargino in the supersymmetric model. The lower bound of their masses can be up tohundreds of GeV for the large mass splittings. The ATLAS and CMS searches for the multi-lepton + E/ T signals rely on triggers that require p T >
20 GeV for the transverse momentumof one lepton at least. The produced leptons tend to become soft with the decreasing ofthe mass splittings, and the soft leptons are difficult to be detected due to the lepton p T requirements of the search. For example, using the 3 l + E/ T signal at the 14 TeV LHC,the 300 fb − of data is required to discover these supersymmetric spectra with dark massbetween 40 GeV and 140 GeV for the mass splittings drop down to 9 GeV [26]. In thispaper, the charged and neutral components of the inert scalar multiplets are degenerateat the tree-level, and the loop corrections only make the charged components to be veryslightly heavier than the neutral components [20, 22]. For such small mass splitting, theleptons of the multi-lepton + E/ T event are very soft. Therefore, the inert scalars are freefrom the constraints of the ATLAS and CMS searches for the multi-lepton + E/ T at the 8 TeVLHC, and even difficult to be detected at the 14 TeV LHC with the more high integratedluminosity. Note that in order to enhance the 750 GeV Higgs decay into diphoton, in thispaper we take the masses of the inert scalars to be in the range of 375 GeV and 500 GeV.Such range of mass will lead the relic density to be much smaller than the observed valuealthough the lightest neutral component is stable [20]. Therefore, to produce the observed10 .10.20.30.40.50.6 380 400 420 440 460 480 500 l = 4 p M (GeV) R gg (f b ) M = 375 GeV R gg (f b ) l l M (GeV)
FIG. 3: R γγ versus M , R γγ versus λ and λ versus M in the 2HDM with an inert complex Higgstriplet. In the right panel R γγ < < R γγ < < R γγ < relic density [27], some other dark matter sources need be introduced, which is beyond thescope of this paper.For tan β = 1 and sin( β − α ) = 1, the H couplings to the charged Higgs H ± of the original2HDM is zero. Therefore, the H ± does not give the contributions to the decay H → γγ .For 2HDM-IHT, the doubly charged and singly charged Higgses δ ±± and δ ± give additionalcontributions to the decay H → γγ , which are sensitive to the mass M and the couplingconstants λ , λ . The degenerate masses of the triplet scalars are taken to be larger than375 GeV, which makes the 750 GeV Higgs decays into triplet scalars to be kinematicallyforbidden. The perturbativity requires the absolute values of λ and λ in the quartic termsto be smaller than 4 π . The stability of the potential favors λ and λ to be larger than0, and gives the lower bound of λ and λ for they are smaller than zero [28]. Thus wetake 0 < λ < π and fix λ = 3 [29]. Because δ ±± has an electric charge of ±
2, the δ ±± contributions are enhanced by a relative factor 4 in the amplitude of H → γγ , see Eq.(7), which can help δ ±± contributions dominate over the other particle contributions. Sincethere are the same sign between Hδ ++ δ −− and Hδ + δ − , the δ ±± and δ ± contributions areconstructive each other.In the Fig. 3, we project the samples of 2HDM-IHT on the planes of R γγ versus M , R γγ versus λ and λ versus M . The left panel shows that R γγ has a peak around M = 375GeV, and decreases rapidly with increasing of M . The characteristic is determined by theform factor F ( τ ) in the H → γγ . R γγ can reach 0.6 fb for M = 375 GeV and λ = 4 π , but11 l = 4 p M (GeV) R gg (f b ) M = 375 GeV R gg (f b ) l l M (GeV)
FIG. 4: R γγ versus M , R γγ versus λ and λ versus M in the 2HDM with a real scalar septuplet.In the right panel R γγ < < R γγ < < R γγ < is still much smaller than the cross section for the 750 GeV diphoton resonance observed byCMS and ATLAS collaborations.For the 2HDM-RSS, T ±±± , T ±± and T ± give additional contributions to the decay H → γγ , which are sensitive to the mass M and the coupling λ . The perturbativity requiresthe absolute value of λ to be smaller than 4 π , and λ > T ±±± of 2HDM-RSS has anelectric charge of ±
3, and the T ±±± contributions are enhanced by a relative factor 9 in theamplitude of H → γγ , which makes T ±±± contributions to dominate over the other particlecontributions. Further, since there are the same sign between HT +++ T −−− , HT ++ T −− and HT + T − , their contributions are constructive each other. Therefore, the width of H → γγ of 2HDM-RSS can be much larger than that of 2HDM-ITH, approximate 8 times for thesame Higgs coupling and mass. In the Fig. 4, we project the samples of 2HDM-RSS on theplanes of R γγ versus M , R γγ versus λ and λ versus M . R γγ can reach 4.6 fb for M = 375GeV and λ = 4 π , which is approximate 8 times of the maximal value of of 2HDM-IHT.The right panel shows that R γγ > M <
400 GeV and λ >
6, and R γγ > M <
380 GeV and λ >
10. Therefore, 2HDM-RSS can accommodate the 750 GeVdiphoton resonance observed by the CMS and ATLAS at the LHC.12 . CONCLUSION
In this paper, we first consider various theoretical and experimental constraints, andexamine the implications of the 750 GeV diphoton resonance on the two-Higgs-doubletmodel. We find the pseudoscalar and charged Higgs masses are favored in the range of 700GeV and 800 GeV, and their masses are allowed to have sizable deviations from 750 GeVfor the small mass splitting between them. Also the pseudoscalar mass is allowed to havesizable deviation from 750 GeV for the charged Higgs mass around 750 GeV. In the two-Higgs-doublet model, the production rate for 750 GeV diphoton resonance is smaller thanthe cross section observed at LHC by two order magnitude. In order to accommodate the750 GeV diphoton resonance, we respectively introduce an inert complex Higgs triplet anda real scalar septuplet to the two-Higgs-doublet model. The multi-charged scalars in thesemodels can enhance the branching ratio of H → γγ sizably. The production rate for the 750GeV diphoton resonance can be enhanced to 0.6 fb for 2HDM with an inert Higgs tripletand 4.5 fb for 2HDM with a real scalar septuplet. The latter can give a valid explanationfor the 750 GeV diphoton resonance at the LHC. Acknowledgment
This work has been supported in part by the National Natural Science Foundation ofChina under grant No. 11575152, and by the Spanish Government and ERDF funds fromthe EU Commission [Grants No. FPA2014-53631-C2-1-P, SEV-2014-0398, FPA2011-23778].
Appendix A: The coupling of
Hhh
The scalar potential shown in the Eq. (1) is expressed in the physical basis where bothΦ and Φ have the non-zero VEVs. It is more convenient to understand the coupling Hhh in the Higgs basis where the two scalar doublets are given as H = G +1 √ ( v + ρ + iG ) ≡ Φ c β +Φ s β , H = H +1 √ ( ρ + iA ) ≡ − Φ s β +Φ c β . (A1)In the Higgs basis, the H field has a VEV v =246 GeV, and the VEV of H field is zero.13he scalar potential in the physical basis (as shown in the Eq. (1) ) can be expressed inthe Higgs basis [30], V = Y H † H + Y H † H + Y [ H † H + h . c . ] + Z ( H † H ) + Z ( H † H ) + Z ( H † H )( H † H )+ Z ( H † H )( H † H ) + n Z ( H † H ) + (cid:2) Z ( H † H ) + Z ( H † H ) (cid:3) H † H + h . c . o , (A2)where the Y i are real linear combinations of the m ij and the Z i are real linear combinationsof the λ i . For λ = λ = 0, we simply have [30] Z ≡ λ c β + λ s β + λ s β , (A3) Z ≡ λ s β + λ c β + λ s β , (A4) Z i ≡ s β (cid:2) λ + λ − λ (cid:3) + λ i , (for i = 3 , , (A5) Z ≡ − s β (cid:2) λ c β − λ s β − λ c β (cid:3) , (A6) Z ≡ − s β (cid:2) λ s β − λ c β + λ c β (cid:3) , (A7)where c β = cos 2 β , s β = sin 2 β and λ = λ + λ + λ .The H + and A are the mass eigenstates of the charged Higgs boson and CP-odd Higgsboson, and their masses are given by m H + = Y + Z v ,m A = Y + ( Z + Z − Z ) v . (A8)The physical CP-even Higgs bosons h and H are the linear combination of ρ and ρ , H = ρ cos( β − α ) − ρ sin( β − α ) ,h = ρ sin( β − α ) + ρ cos( β − α ) . (A9)For cos( β − α ) = 0, there is no mixing of h and H , which requires Z = 0 and leads to H = − ρ , h = ρ . (A10)For the Eq. (A10), the other terms except for the Z term in the Eq. (A2) do not producethe coupling of Hhh . Therefore, the H coupling to hh is zero for cos( β − α ) = 0.Note that cos( β − α ) denotes the coupling of H and gauge bosons normalized to SMHiggs. Both cos( β − α ) and the coupling of Hhh are the physical observables and basis-independent. Therefore, for cos( β − α ) = 0, the coupling of Hhh equals to zero in the14hysical basis and Higgs basis. In fact, in the physical basis, the Higgs potential shown inEq. (1) gives the coupling of
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