Decays of Higgs Bosons in the Standard Model and Beyond
DDecays of Higgs Bosons in the Standard Model and Beyond
Seong Youl Choi, ∗ Jae Sik Lee, , , † Jubin Park , ‡ Department of Physics and RIPC, Jeonbuk National University, Jeonju 54896, Korea Department of Physics, Chonnam National University, Gwangju 61186, Korea IUEP, Chonnam National University, Gwangju 61186, Korea APCTP, Pohang, Gyeongbuk 37673, KoreaFebruary 1, 2021
Abstract
We make a substantially updated review and a systematic and comprehensive analysis of thedecays of Higgs bosons in the Standard Model (SM) and its three well-defined prototype extensionssuch as the complex singlet extension of the SM (cxSM), the four types of two Higgs-doubletmodels (2HDMs) without tree-level Higgs-mediated flavour-changing neutral current (FCNC) andthe minimal supersymmetric extension of the SM (MSSM). We summarize the state-of-art oftheoretical predictions for the decay widths of the SM Higgs boson and those appearing in itsextensions taking account of all possible decay modes. We incorporate them to study and analyzedecay patterns of CP-even, CP-odd, and CP-mixed neutral Higgs bosons and charged ones. Weput special focus on the properties of a neutral Higgs boson with mass about 125 GeV discoveredat the LHC and present constraints obtained from precision analysis of it. We also work out theproperties of the extended Higgs sectors which can be probed by present and future high-energyand high-precision experiments. This review is intended to be as self-contained as possible aimingat providing practical information for studying decays of Higgs bosons in the SM and beyond. ∗ [email protected] † [email protected] ‡ [email protected] a r X i v : . [ h e p - ph ] J a n ontents Potential and mixing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.2.2
Higgs-boson interactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.3 Two Higgs Doublet Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102.3.1
Potential and mixing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102.3.2
Interactions of Higgs bosons with massive vector bosons . . . . . . . . . . . . . . 122.3.3
Interactions of Higgs bosons with the SM fermions . . . . . . . . . . . . . . . . . 122.3.4
Higgs-boson self-interactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132.4 Minimal Supersymmetric Extension of the SM . . . . . . . . . . . . . . . . . . . . . . 152.4.1
Potential and mixing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152.4.2
Interactions of Higgs bosons with the SM particles and self-interactions . . . . . 172.4.3
Interactions of Higgs bosons with the SUSY particles . . . . . . . . . . . . . . . 17 H → f ¯ f . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223.2 Decays into two massive vector bosons: H → V V with V = Z, W . . . . . . . . . . . . 273.3 Decays into a lighter scalar boson and a vector boson and into two lighter scalar bosons: H → ϕV , ϕϕ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 303.4 Decays into two gluons: H → gg . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 303.5 Decays into two photons: H → γγ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 363.6 Decays into a vector boson Z and a photon: H → Zγ . . . . . . . . . . . . . . . . . . . 413.7 Numerical results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 443.7.1 Anatomy of Higgs boson decays with M H = 125 . GeV . . . . . . . . . . . . . . 453.7.2
Decays of heavy Higgs bosons in CP-violating 2HDMs . . . . . . . . . . . . . . . 47 . A.1 Input parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68A.2 Running of the strong coupling constant and quark masses . . . . . . . . . . . . . . . . 69
B Supersymmetric Contributions to the
Hgg , Hγγ , and
HZγ
Form Factors 71C QCD Corrections to Γ( H → γγ ) : C sf ( τ ) and C pf ( τ )
72D Input parameters for the most general 2HDM potential 74E Cubic Higgs-boson self-couplings in 2HDMs 75 Introduction
Since the discovery of a resonance with a mass of approximately 125 GeV at the Large Hadron Collider(LHC) in 2012 [1, 2], the substantial subsequent studies of its properties with the data set collectedduring the LHC Run 1 period from 2009 to 2012 and the LHC Run 2 period from 2015 to 2018 havefirmly confirmed the compatibility of the resonance with the spin-zero and parity-even SM Higgs bosonwhich appears in the spontaneously broken gauge theory where the electroweak interactions are governedby the SU(2) L × U(1) Y symmetry group [3, 4, 5]. At the present time we are on a watershed peak forexploring a new territory of particle physics through the Higgs landscape.The rates and differential measurements of all the possible production and decay channels of theresonance state so far are consistent with those predicted in the SM within experimental and theoreticaluncertainties [7, 8]. The mass of the Higgs boson has been measured at the per-mille precision level,mainly through the high-resolution decay modes with four-lepton and di-photon final states [9, 10].Furthermore, the couplings of the Higgs boson not only to the gauge bosons and but also to the chargedfermions of the third generation [11, 12, 13, 14, 15] and, recently, to the muons [16, 17] were estab-lished independently and unambiguously. Based on the observational facts, we will call the discoveredresonance particle as the SM-like Higgs boson wherever appropriate in the following.Nevertheless, the couplings of the SM-like Higgs boson to the electrons and lighter quarks of thefirst and second generations and its cubic and quartic self-couplings defining the profile of the Higgspotential are yet to be established and measured independently. Furthermore, a more complex Higgssector associated with additional states has not been ruled out. Therefore, it is not yet firmly establishedwhether the SM-like Higgs boson is indeed the only elementary scalar state as in the SM, whether thereexist more additional elementary scalar particles, or even whether it is a composite particle with internalstructure or not.Conceptually, the SM itself with the Higgs boson could be weakly interacting well above the weakscale of v = 246 GeV without violating unitarity and so with no need for new physics. However the Higgsboson mass is influenced subtly by the presence of heavy particles and it receives quantum correctionsdestabilizing the weak scale and requiring a delicate fine-tuning of apparently unrelated parameters.This so-called naturalness or hierarchy problem [18, 19, 20] has been the key argument for expectingnew physics to be revealed at the TeV scale. To mention just a few, new theoretical frameworks basedon a fermion-boson symmetry called supersymmetry [21, 22, 23], a collective symmetry between theSM particles and heavier partners as in Little Higgs theories [24, 25, 26] or an effective reduction ofthe Planck scale to the TeV scale as in extra-dimension models [27, 28, 29, 30] have been proposed andintensively investigated.In addition to alleviating the hierarchy problem, new scenarios involving extensions of the Higgssector generically have been proposed and investigated to account for the dark matter (DM) abun-dance [31, 32], the mass-antimatter asymmetry of the Universe with new sources of the charge-parity(CP) symmetry breakdown [33, 34, 35], the tiny but non-vanishing neutrino masses [36], inflation [37],etc. Such models with additional scalars can provide us with solid platforms for exploring new Higgsboson signals concretely and comprehensively since, in each scenario, Higgs bosons exhibit their owndistinctive features in their couplings to gauge bosons, fermions and those among themselves.After the successful completion of Run 1 and Run 2, the LHC is presently in the second long shutdown period while undergoing important upgrades for its high luminosity phase. Much larger datasets are to be collected during the Run 3 period and, ultimately, during the operation period of thehigh-luminosity LHC (HL-LHC) and they will enable us to explore new physics beyond the SM (BSM) Note that the gauge and fermion sectors of the SM have been already well probed with great precision both theoret-ically and experimentally as can be checked with the reference book Ref. [6].
3y performing more challenging as well as more precise measurements. In light of such promisingexperimental prospects at the LHC and at other future high-energy and high-precision experiments, itis quite timely and worthwhile to perform a systematic and comprehensive review and analysis of thedecays of Higgs bosons including the state-of-art of theoretical calculations known up to now, not onlyin the SM but also in various BSM scenarios with unique features in their extended Higgs sectors.Certainly it is formidable to review all theoretical and experimental aspects of Higgs sectors in allBSM models proposed so far in a single report with limited space. Unavoidably, we restrict ourselvesin this review to the SM and the three well-defined prototype BSM models with extended Higgs sectorspossessing their own characteristic features and broad implications. Specifically, in addition to theSM, we consider the following representative examples: ( i ) the complex singlet extension of the SM(cxSM) [38, 39, 40, 41, 42, 43, 44, 45, 46], ( ii ) the four types of two Higgs doublet models (2HDMs) [47]with natural flavour conservation at the tree level and the so-called ρ parameter close to unity abidingby the stringent experimental constraint on it, and ( iii ) the minimal supersymmetric extension of theSM (MSSM) [48, 49, 50, 51, 52].Several related previous reviews on Higgs physics in the SM and the MSSM can be found in Refs. [53,54, 55, 56, 57]. A few tailer-made sophisticated computational packages have been developed for themass spectra and decay widths of neutral and charged Higgs bosons in the SM and the MSSM withreal parameters [58, 59, 60] and with explicit CP violation [61, 62, 63, 64, 65]. This review updates theprevious works substantially by including two more popular BSM models in addition to the MSSM andallowing for complex parameters leading to CP-violating phenomena. We perform a systematic andcomprehensive analysis for the decays of neutral and charged Higgs bosons in those three prototypeBSM models as well as in the SM, expecting more complete reviews on the Higgs sectors of many otherBSM scenarios to come out timely in step with more advanced experimental developments.In this review, we try to contain all the relevant information needed to implement the most up-to-date theoretical calculations of Higgs decays. We aim to make it be of practical use especially forincorporating the state-of-art of the corrections beyond the leading order. We also emphasize that ourapproach is largely model independent and it can be easily applicable to Higgs bosons appearing inBSM models not explicitly mentioned in this review.This review is organized as follows. Section 2 is devoted to reviewing the Higgs sectors of the SM andthree extended scenarios - cxSM, 2HDMs and MSSM - without imposing any constraints on the modelparameters. We work out the analytic structure of the Higgs potential and mixing. Also worked out arethe Higgs interactions with gauge bosons, the SM fermions, and new scalars and fermions as well as theHiggs-boson self interactions. We review and update the decays of neutral Higgs bosons in Section 3 andthose of charged Higgs bosons in Section 4. We provide explicit analytical expressions of the individualpartial decay widths as precisely as possible by including the state-of-art of theoretical calculations. InSection 5, we present the constraints on the couplings of the SM-like Higgs boson weighing about 125GeV obtained from global fits to the LHC precision Higgs data. Conclusions are made in Section 6.To make this review self-contained, we include several appendices to provide supplemental materials.Appendix A includes a summary of the SM parameters used for the numerical estimates of the Higgsdecay widths and a description of the running of the strong coupling constant and heavy quark masses.In Appendix B, the supersymmetric contributions to the loop-induced couplings of the Higgs bosonto two gluons, two photons and Zγ are presented and Appendix C is devoted to the presentationof the QCD corrections to the partial width of the Higgs-boson decay to two photons. Finally, inAppendix D, we present expressions for the most general 2HDM potential parameters in terms of themasses of charged and neutral Higgs bosons and the elements of the orthogonal matrix describing themixing among neutral Higgs bosons and, in Appendix E, we apply them for deriving cubic Higgs-bosonself-couplings. 4 Standard Model and Beyond
In this section, we derive and describe the basic form of Higgs boson masses and mixing as well astheir interactions in the SM, cxSM, 2HDMs and MSSM. The derived analytical results are utilizedcomprehensively in the sequential sections for the detailed review of the decays of neutral and chargedHiggs bosons and also for the model-independent precision study of the SM-like neutral Higgs bosonwhich has been extensively probed at the LHC since its discovery in 2012.
The self-interactions of the SM Higgs boson and its interactions with the massive vector bosons arederived from the Higgs Lagrangian: L Higgs = ( D µ Φ) † ( D µ Φ) − V SM (Φ) , (1)where Φ denotes a complex SU(2) L doublet Higgs field with hypercharge Y = 1 / D µ Φ = (cid:18) ∂ µ − ig τ a W aµ − ig (cid:48) B µ (cid:19) Φ= (cid:18) ∂ µ − i ( gW µ + g (cid:48) B µ ) − ig ( W µ − iW µ ) − ig ( W µ + iW µ ) ∂ µ + i ( gW µ − g (cid:48) B µ ) (cid:19) Φ , (2)in terms of the SU(2) L and U(1) Y gauge couplings g and g (cid:48) , respectively, the three SU(2) L gauge bosons W , , µ , and the single U(1) Y gauge boson B µ with the usual three 2 × τ = (cid:18) (cid:19) , τ = (cid:18) − ii (cid:19) , τ = (cid:18) − (cid:19) . (3)And the renormalizable SM Higgs potential V SM (Φ) is given by V SM (Φ) = µ (Φ † Φ) + λ (Φ † Φ) , (4)with µ < , v + H ) T / √ v = (cid:112) − µ /λ and the realscalar field H after rotating away three Goldstone modes and using W ± µ = ( W µ ∓ iW µ ) / √ Z µ = ( gW µ − g (cid:48) B µ ) / (cid:112) g + g (cid:48) , we can render the kinetic term of the Higgs Lagrangian in Eq. (1) intothe form expanded as( D µ Φ) † ( D µ Φ) = 12 ( ∂ µ H )( ∂ µ H ) + M W W + µ W µ − + 12 M Z Z µ Z µ (5)+ gM W (cid:18) W + µ W µ − + 12 c W Z µ Z µ (cid:19) H + 1 v (cid:18) M W W + µ W µ − + M Z Z µ Z µ (cid:19) H , in the unitary gauge. We use the abbreviation s W ≡ sin θ W for the sine of the weak mixing angle θ W and c W ≡ cos θ W , t W ≡ sin θ W / cos θ W , etc. The masses of the massive gauge bosons W and Z aregiven by M W = gv/ M Z = M W /c W with v = (cid:0) √ G F (cid:1) − / ≈
246 GeV fixed by the Fermi constant G F , which is determined with a precision of 0 . L and U(1) Y gauge couplings are g = e/s W and g (cid:48) = g t W = e/c W , respectively, where the5agnitude of the electron electric charge e = 2 √ πα with α being the fine structure constant. On theother hand, the SM Higgs potential takes the form of V SM ( H ) = − v M H + 12 M H H + 13! (cid:18) M H v (cid:19) v H + 14! (cid:18) M H v (cid:19) H , (6)which is completely fixed in terms of v and the Higgs mass M H with the replacements of µ = − λv and λ = M H / v .The Higgs interactions with the SM fermions are derived by considering the following Yukawainteractions −L Y = U R h u Q T ( iτ ) Φ − D R h d Q T ( iτ ) (cid:101) Φ − E R h e L T ( iτ ) (cid:101) Φ + h . c . , (7)where (cid:101) Φ = iτ Φ ∗ and Q T = ( U L , D L ) and L T = ( ν L , E L ) with U and D standing for the threeup- and down-type quarks, respectively, and ν and E for the three neutrinos and charged leptons,respectively, in the weak eigenstate basis. And the 3 × h u,d,e . TakingΦ = (0 , v + H ) T / √ −L H ¯ ff = (cid:88) f = u,d,c,s,t,b,e,µ,τ m f v H f f , (8)with the masses m f = h f v/ √ In this subsection, as the first BSM example, we consider a model in which the SM is extended byadding a complex SU(2) L × U(1) Y singlet (cxSM). Potential and mixing
When a complex scalar singlet field S is added to the SM Higgs sector [38, 39, 40, 41, 42, 43, 44, 45, 46],the most general renormalizable scalar potential takes the form [38] V (Φ , S ) = µ (Φ † Φ) + λ (Φ † Φ) + (cid:0) δ Φ † Φ S + c.c. (cid:1) + δ Φ † Φ | S | + (cid:0) δ Φ † Φ S + h.c. (cid:1) + ( a S + c.c. ) + (cid:0) b S + c.c. (cid:1) + b | S | + (cid:0) c S + c.c. (cid:1) + (cid:0) c S | S | + c.c. (cid:1) + (cid:0) d S + c.c. (cid:1) + d | S | + (cid:0) d S | S | + c.c. (cid:1) , (9)where all terms containing odd powers of S are eliminated by imposing a discrete Z symmetry under S → − S . Furthermore, imposing a global U(1) symmetry eliminates all terms containing complexcoefficients. One may allow a soft U(1)-breaking b term to avoid massless CP-odd Goldstone bosonwhich is not phenomenologically viable. And then, in order to avoid the cosmological domain wallproblem caused by the presence of the b term, one may additionally include the U(1)- and Z -breakinglinear term proportional to a . The resulting cxSM scalar potential takes the form V (Φ , S ) = µ (Φ † Φ) + λ (Φ † Φ) + δ Φ † Φ | S | + b | S | + d | S | + (cid:0) a S + b S + c.c. (cid:1) , (10)in terms of the original couplings in Eq. (9), or, alternatively [66], V (Φ , S ) = µ (Φ † Φ) + µ | S | + λ (Φ † Φ) + λ | S | + λ Φ † Φ | S | + (cid:0) a S + b S + c.c. (cid:1) , (11)6n terms of a more systematic parameter set of 5 real parameters of µ , and λ , , and 2 complex massiveparameters of a = | a | e iφ a and b = | b | e iφ b .By parameterizing the SU(2) L doublet Φ and singlet S asΦ = (cid:18) G +1 √ ( v + φ + iG ) (cid:19) ; S = e iξ √ v s + s + i ( v a + a )] , (12)we obtain the following three tadpole conditions for minimizing the potential: (cid:20) µ + λ v + 12 λ v sa (cid:21) v = 0 , (cid:20) µ + λ v sa + 12 λ v + 2 (cid:60) e( b e iξ ) (cid:21) v s − (cid:61) m( b e iξ ) v a + √ (cid:60) e( a e iξ ) = 0 , (cid:20) µ + λ v sa + 12 λ v − (cid:60) e( b e iξ ) (cid:21) v a − (cid:61) m( b e iξ ) v s − √ (cid:61) m( a e iξ ) = 0 , (13)with the abbreviation v sa = v s + v a . The mass terms of the scalar states are given by V cxSM , mass = 12 ( φ s (cid:48) a (cid:48) ) M φs (cid:48) a (cid:48) , (14)in terms of a real and symmetric 3 × M decomposed into the two parts: M = X + 2 λ v λ vv sa λ vv sa X + 2 λ v sa
00 0 X + 2 (cid:60) e( b e iξ ) c γ − (cid:61) m( b e iξ ) s γ −(cid:60) e( b e iξ ) s γ − (cid:61) m( b e iξ ) c γ −(cid:60) e( b e iξ ) s γ − (cid:61) m( b e iξ ) c γ −(cid:60) e( b e iξ ) c γ + (cid:61) m( b e iξ ) s γ . (15)The two parameters of X , appearing in the diagonal components of the first term are defined by X ≡ µ + λ v + 12 λ v sa , X ≡ µ + λ v sa + 12 λ v . (16)In Eq. (14), the primed scalar fields s (cid:48) and a (cid:48) are related to the original scalar fields s and a throughthe rotation (cid:18) s (cid:48) a (cid:48) (cid:19) = (cid:18) c γ s γ − s γ c γ (cid:19) (cid:18) sa (cid:19) , (17)with c γ = cos γ = v s /v sa and s γ = sin γ = v a /v sa . Note that X = 0 always to have the non-zerovev of v , as can be checked with the first tadpole condition in Eq. (13). On the other hand, only inthe U(1)-conserving case with both a = 0 and b = 0, X = 0 to have the non-vanishing vevs of v a and v s giving rise to the massless Goldstone boson a (cid:48) . In this case, the singlet vacuum takes the U(1)symmetric vev of v sa = (cid:112) v s + v a while each of the vevs remains undetermined.In some cases, instead of the discrete Z symmetry, a different discrete Z (cid:48) symmetry under theinterchange S ↔ S ∗ is imposed. In this case, the U(1)-breaking part of the scalar potential reads (cid:0) a S + b S + c.c (cid:1)(cid:12)(cid:12) Z (cid:48) = 2 (cid:2) (cid:60) e( a ) S + (cid:60) e( b ) ( S − A ) (cid:3) , (18)7ith S = S + iA = [( v s + s ) + i ( v a + a )] / √
2. Note that, if the potential has the Z (cid:48) symmetry, just tworeal parameters are sufficient for parameterizing the U(1)-breaking part of the potential. Assuming a and b to be real with no loss of generality, the tadpole conditions of the scalar potential become X v = 0 , ( X + 2 b ) v s + √ a = 0 , ( X − b ) v a = 0 . (19)Assuming v (cid:54) = 0, which forces X = 0, the mass-squared matrix simplifies into the form M = λ v λ vv sa λ vv sa X + 2 λ v sa + 2 b c γ − b s γ − b s γ X − b c γ . (20)When v s (cid:54) = 0 and v a (cid:54) = 0 guaranteeing s γ (cid:54) = 0, the CP symmetry is spontaneously broken and allthe three states mix. In this case, one may parameterize the potential with a set of 7 parameters of { λ , λ , λ , v, v sa , tan γ, b } with the relations µ = − λ v − λ v sa , X = 2 b , a = − √ b v s , (21)where the second relation is solved to give µ = 2 b − λ v sa − λ v .When v s (cid:54) = 0 and v a = 0, the angle γ = 0, i.e. s (cid:48) = s and a (cid:48) = a and the scalar mixing occurs onlybetween the two states of φ and s with the pseudoscalar mass-squared M a = − √ a v s − b . (22)In this 2-state mixing case, the scalar potential can be parameterized with a set of 7 parameters of { λ , λ , λ , v, v s , a , b } with the relations µ = − λ v − λ v sa , X = − √ a v s − b , (23)where the second relation is solved to give µ = −√ a /v s − b − λ v sa − λ v . Incidentally, if v s = 0in addition to v a = 0, the parameter a should vanish due to the second tadpole condition in Eq. (19)and X = µ + λ v / M φ = 2 λ v , M s = µ + λ v / b , M a = µ + λ v / − b , (24)and the 6 parameters { λ , λ , λ , v, µ , b } can be employed for describing the scalar potential. Varioustypes of vacua in cxSM are summarized in Table 1.Finally, without loss of generality, the orthogonal 3 × O diagonalizing the mass-squared matrix in Eq. (20) is defined through( φ, s (cid:48) , a (cid:48) ) Tα = O αi ( H , H , H ) Ti , (25)such that O T M O = diag( M H , M H , M H ) with the increasing ordering of M H ≤ M H ≤ M H .8able 1: Vacua in cxSM imposing Z (cid:48) : the parameters a and b are real and v (cid:54) = 0 is taken. FromRef. [66].vacua X a possible set of inputs miscellaneous relations v a (cid:54) = 0 & v s (cid:54) = 0 2 b { λ , λ , λ , v, v sa , tan γ, b } a = − √ b v s v a (cid:54) = 0 & v s = 0 2 b { λ , λ , λ , v, v a , b } a = 0 ; v sa = v a , c γ = 0 , s γ = 1 v a = 0 & v s (cid:54) = 0 −√ a /v s − b { λ , λ , λ , v, v s , a , b } v sa = v s , c γ = 1 , s γ = 0 v a = 0 & v s = 0 µ + λ v / { λ , λ , λ , v, µ , b } a = 0 ; v sa = 0 , c γ → , s γ → Higgs-boson interactions
The interactions of the three Higgs bosons with the SM fermions and the massive vector bosons aregiven by −L H i ¯ ff = (cid:88) f = u,d,c,s,t,b,e,µ,τ m f v g SH i ¯ ff H i f f ; L HV V = g M W (cid:18) W + µ W − µ + 12 c W Z µ Z µ (cid:19) (cid:88) i =1 g HiV V H i , (26)with the normalized dimensionless couplings simply given by g SH i ¯ ff = g HiV V = O φi . (27)And the cubic and quartic couplings are given by the self-interaction term of the scalar potential: −L self = λ vφ + 2 λ v sa φ s (cid:48) + 2 λ vφ ( s (cid:48) + a (cid:48) ) + 4 λ v sa s (cid:48) ( s (cid:48) + a (cid:48) )+ 14 λ φ + λ φ ( s (cid:48) + a (cid:48) ) + λ ( s (cid:48) + 2 s (cid:48) a (cid:48) + a (cid:48) ) ≡ v (cid:88) i ≥ j ≥ k =1 g HiHjHk H i H j H k + (cid:88) i ≥ j ≥ k ≥ l =1 g HiHjHkHl H i H j H k H l , (28)where the normalized cubic and quartic couplings of the three Higgs mass eigenstates are g HiHjHk = (cid:88) α ≤ β ≤ γ =1 { O αi O βj O γk } g αβγ ,g HiHjHkHl = (cid:88) α ≤ β ≤ γ ≤ δ =1 { O αi O βj O γk O δl } g αβγδ , (29)with i, j, k, l = 1 , ,
3, the cubic weak-eigenstate couplings g φφφ = λ , g φφs (cid:48) = 2 λ v sa v ,g φs (cid:48) s (cid:48) = g φa (cid:48) a (cid:48) = 2 λ , g s (cid:48) s (cid:48) s (cid:48) = g s (cid:48) a (cid:48) a (cid:48) = 4 λ v sa v , (30) Here, the indices α , β , γ , and δ count the Higgs weak eigenstates of φ , s (cid:48) , and a (cid:48) and inequalities between them implythat the cubic and quartic terms in the Higgs potential are ordered in the weak eigenstates. g φφφφ = λ / , g φφs (cid:48) s (cid:48) = g φφa (cid:48) a (cid:48) = λ ,g s (cid:48) s (cid:48) s (cid:48) s (cid:48) = g s (cid:48) s (cid:48) a (cid:48) a (cid:48) / g a (cid:48) a (cid:48) a (cid:48) a (cid:48) = λ . (31)In Eq. (29), the expressions within the curly brackets {· · ·} need to be symmetrized with respect to theindices i, j, k, l and divided by the corresponding symmetry factors in cases where two or more indicesare the same. For example, { O αi O βj O γk } can explicitly be evaluated as follows: { O αi O βj O γk } ≡ N S (cid:16) O αi O βj O γk + O αi O βk O γj + O αj O βi O γk + O αj O βk O γi + O αk O βi O γj + O αk O βj O γi (cid:17) , (32)with N S = 3! = 6 when i = j = k , N S = 1 when ( i, j, k ) = (3 , , N S = 2! = 2 in all the othercases. In this subsection, we give a detailed description of the models in which the SM is extended by addingone more SU(2) L doublet (2HDMs) while taking the same gauge group SU(3) C × SU(2) L × U(1) Y as inthe SM. Potential and mixing
The general 2HDM scalar potential containing two complex SU(2) L doublets of Φ and Φ with thesame hypercharge Y = 1 / V = µ (Φ † Φ ) + µ (Φ † Φ ) + m (Φ † Φ ) + m ∗ (Φ † Φ )+ λ (Φ † Φ ) + λ (Φ † Φ ) + λ (Φ † Φ )(Φ † Φ ) + λ (Φ † Φ )(Φ † Φ )+ λ (Φ † Φ ) + λ ∗ (Φ † Φ ) + λ (Φ † Φ )(Φ † Φ ) + λ ∗ (Φ † Φ )(Φ † Φ )+ λ (Φ † Φ )(Φ † Φ ) + λ ∗ (Φ † Φ )(Φ † Φ ) , (33)in terms of 2 real and 1 complex dimensionful quadratic couplings and 4 real and 3 complex dimensionlessquartic couplings. With the parameterization of two scalar doublets Φ , asΦ = (cid:18) φ +11 √ ( v + φ + ia ) (cid:19) ; Φ = e iξ (cid:18) φ +21 √ ( v + φ + ia ) (cid:19) , (34)and denoting v = v cos β = vc β and v = v sin β = vs β with v = (cid:112) v + v , one may remove µ , µ ,and (cid:61) m( m e iξ ) from the 2HDM potential using three tadpole conditions: µ = − v (cid:20) λ c β + 12 λ s β + c β s β (cid:60) e( λ e iξ ) (cid:21) + s β M H ± ,µ = − v (cid:20) λ s β + 12 λ c β + c β s β (cid:60) e( λ e iξ ) (cid:21) + c β M H ± , (cid:61) m( m e iξ ) = − v (cid:2) c β s β (cid:61) m( λ e iξ ) + c β (cid:61) m( λ e iξ ) + s β (cid:61) m( λ e iξ ) (cid:3) , (35)with the square of the charged Higgs-boson mass M H ± = − (cid:60) e( m e iξ ) c β s β − v c β s β (cid:2) λ c β s β + 2 c β s β (cid:60) e( λ e iξ ) + c β (cid:60) e( λ e iξ ) + s β (cid:60) e( λ e iξ ) (cid:3) . (36)10hen, including the vacuum expectation value v , in general we need the following 13 parameters plus1 sign: v , t β , | m | ; λ , λ , λ , λ , | λ | , | λ | , | λ | ; φ + 2 ξ , φ + ξ , φ + ξ , sign[cos( φ + ξ )] , (37)to fully specify the general 2HDM scalar potential. Here m = | m | e iφ and λ , , = | λ , , | e iφ , , and we note that sin( φ + ξ ) is fixed by the CP-odd tadpole condition if the CP phases φ + 2 ξ , φ + ξ and φ + ξ are given together with | m | , | λ , , | , v , and t β and, accordingly, cos( φ + ξ ) is determinedup to a two-fold ambiguity. One may take the convention with ξ = 0 corresponding to re-defining the1 quadratic and 3 quartic complex parameters without loss of generalityThe 2HDM Higgs potential includes the mass terms which can be cast into the form consisting oftwo parts V , mass = M H ± H + H − + 12 ( φ φ a ) M φ φ a , (38)in terms of the charged Higgs boson H − , two neutral scalars φ , , and one neutral pseudoscalar a afterabsorbing the charged and neutral Goldstone bosons G − and G in the 2-state mixings of the twocharged scalars and two neutral pseudoscalars as (cid:18) φ − φ − (cid:19) = (cid:18) c β − s β s β c β (cid:19) (cid:18) G − H − (cid:19) ; (cid:18) a a (cid:19) = (cid:18) c β − s β s β c β (cid:19) (cid:18) G a (cid:19) . (39)And the 3 × M is given by M = M A s β − s β c β − s β c β c β
00 0 1 + M λ , (40)with (reinstating the relative phase ξ for the sake of generality) M A = M H ± + (cid:20) λ − (cid:60) e( λ e iξ ) (cid:21) v , (41)and the second part expressed in terms of the quartic couplings as M λ v = λ c β + 2 (cid:60) e( λ e iξ ) s β λ c β s β + (cid:60) e( λ e iξ ) c β −(cid:61) m( λ e iξ ) s β +2 (cid:60) e( λ e iξ ) s β c β + (cid:60) e( λ e iξ ) s β −(cid:61) m( λ e iξ ) c β λ c β s β + (cid:60) e( λ e iξ ) c β λ s β + 2 (cid:60) e( λ e iξ ) c β −(cid:61) m( λ e iξ ) c β + (cid:60) e( λ e iξ ) s β +2 (cid:60) e( λ e iξ ) s β c β −(cid:61) m( λ e iξ ) s β −(cid:61) m( λ e iξ ) s β −(cid:61) m( λ e iξ ) c β −(cid:61) m( λ e iξ ) c β −(cid:61) m( λ e iξ ) s β , (42)where the abbreviation λ = ( λ + λ ) / v = 2 M W /g , a = − s β a + c β a and H + = − s β φ +1 + c β φ +2 . We need to specify, therefore, the 13 parameters plus 1 sign listed in Eq. (37) tofix the mass-squared matrix.Once the real and symmetric mass-squared matrix is given, the orthogonal 3 × O is defined through ( φ , φ , a ) Tα = O αi ( H , H , H ) Ti , (43)such that O T M O = diag( M H , M H , M H ) with the increasing ordering of M H ≤ M H ≤ M H .11 .3.2 Interactions of Higgs bosons with massive vector bosons
The cubic interactions of the neutral and charged Higgs bosons with the massive gauge bosons Z and W ± are described by the three interaction Lagrangians: L HV V = g M W (cid:18) W + µ W − µ + 12 c W Z µ Z µ (cid:19) (cid:88) i g HiV V H i , (44) L HHZ = g c W (cid:88) i>j g HiHjZ Z µ ( H i ↔ ∂ µ H j ) , (45) L HH ± W ∓ = − g (cid:88) i g HiH + W − W − µ ( H i i ↔ ∂ µ H + ) + h . c . , (46)respectively, where X ↔ ∂ µ Y = X∂ µ Y − ( ∂ µ X ) Y , i, j = 1 , , g HiV V , g HiHjZ and g HiH + W − are given in terms of the neutral Higgs-boson 3 × O by (note thatdet( O ) = ± O ): g HiV V = c β O φ i + s β O φ i ,g HiHjZ = sign[det( O )] ε ijk g HkV V ,g HiH + W − = c β O φ i − s β O φ i − iO ai , (47)leading to the following sum rules: (cid:88) i =1 g HiV V = 1 and g HiV V + | g HiH + W − | = 1 for each i = 1 , , . (48) Interactions of Higgs bosons with the SM fermions
Without loss of generality, the Yukawa couplings in 2HDMs could be cast into the form [68]: −L Y = h u u R Q T ( iτ ) Φ − h d d R Q T ( iτ ) (cid:16) η d (cid:101) Φ + η d (cid:101) Φ (cid:17) − h l l R L T ( iτ ) (cid:16) η l (cid:101) Φ + η l (cid:101) Φ (cid:17) + h . c . , (49)where (cid:101) Φ i = iτ Φ ∗ i and Q T = ( u L , d L ) and L T = ( ν L , l L ) with u and d standing for three up- anddown-type quarks, respectively, and l for three charged leptons. We note that there is a freedom toredefine the two linear combinations of Φ and Φ to eliminate the coupling of the up-type quarks toΦ [69]. The 2HDMs are classified according to the values of η l , and η d , as in Table 2.By identifying the couplings in terms of the vev v and the mixing angle β as h u = √ m u v s β ; h d = √ m d v η d c β + η d s β ; h l = √ m l v η l c β + η l s β , (50)we obtain the following Lagrangians −L H i ¯ ff = m u v (cid:20) ¯ u (cid:18) O φ i s β − i c β s β O ai γ (cid:19) u (cid:21) H i Here we take the convention with ξ = 0 and the couplings h u,d,l are supposed to be real without loss of generality. η d η d η l η l m d v (cid:20) ¯ d (cid:18) η d O φ i + η d O φ i η d c β + η d s β − i η d s β − η d c β η d c β + η d s β O ai γ (cid:19) d (cid:21) H i + m l v (cid:20) ¯ l (cid:18) η l O φ i + η l O φ i η l c β + η l s β − i η l s β − η l c β η l c β + η l s β O ai γ (cid:19) l (cid:21) H i , (51)for the interactions of neutral Higgs bosons with fermion pairs and −L H ± ¯ ud = − √ m u v (cid:18) c β s β (cid:19) ¯ u P L d H + − √ m d v (cid:18) η d s β − η d c β η d c β + η d s β (cid:19) ¯ u P R d H + − √ m l v (cid:18) η l s β − η l c β η l c β + η l s β (cid:19) ¯ ν P R l H + + h . c . , (52)for the interactions of the charged Higgs boson with fermion pairs. Higgs-boson self-interactions
Given the orthogonal mixing matrix O diagonalizing the mass-squared matrix of the neutral Higgsbosons, the cubic and quartic Higgs-boson self-couplings are given in terms of the Higgs mass eigenstatesby [71, 72, 73, 74]: −L H = v (cid:88) i ≥ j ≥ k =1 g HiHjHk H i H j H k + v (cid:88) i =1 g HiH + H − H i H + H − , (53) −L H = (cid:88) i ≥ j ≥ k ≥ l =1 g HiHjHkHl H i H j H k H l + (cid:88) i ≥ j =1 g HiHjH + H − H i H j H + H − + g H + H − H + H − ( H + H − ) , (54)where the normalized cubic and quartic weak-eigenstate couplings are g HiHjHk = (cid:88) α ≤ β ≤ γ =1 { O αi O βj O γk } g αβγ , g HiH + H − = (cid:88) α =1 O αi g αH + H − , (55) g HiHjHkHl = (cid:88) α ≤ β ≤ γ ≤ δ =1 { O αi O βj O γk O δl } g αβγδ , Here, the indices α , β , and γ count the Higgs weak eigenstates of φ , φ , and a and inequalities between them implythat the cubic and quartic terms in the Higgs potential are ordered in the weak eigenstate basis. HiHjH + H − = (cid:88) α ≤ β =1 { O αi O βj } g αβ H + H − . (56)We note again that, in the above equations (55) and (56), the expressions within the curly brackets {· · ·} need to be symmetrized fully with respect to the indices i, j, k, l and divided by the correspondingsymmetry factors in cases where two or more indices are the same as in, for example, Eq. (32).For the sake of completeness, we present all the effective cubic and quartic Higgs–boson self–couplingsof the Higgs weak eigenstates. The cubic self-couplings of the neutral Higgs bosons are given by g φ φ φ = c β λ + 12 s β (cid:60) e λ ,g φ φ φ = s β λ + s β (cid:60) e λ + 32 c β (cid:60) e λ ,g φ φ φ = c β λ + c β (cid:60) e λ + 32 s β (cid:60) e λ ,g φ φ φ = s β λ + 12 c β (cid:60) e λ ,g φ φ a = − s β c β (cid:61) m λ −
12 (1 + 2 c β ) (cid:61) m λ ,g φ φ a = − (cid:61) m λ − s β c β (cid:61) m ( λ + λ ) ,g φ φ a = − s β c β (cid:61) m λ −
12 (1 + 2 s β ) (cid:61) m λ ,g φ aa = s β c β λ + c β λ − c β (1 + s β ) (cid:60) e λ + 12 s β ( s β − c β ) (cid:60) e λ + 12 s β c β (cid:60) e λ ,g φ aa = s β c β λ + s β λ − s β (1 + c β ) (cid:60) e λ + 12 s β c β (cid:60) e λ + 12 c β ( c β − s β ) (cid:60) e λ ,g aaa = s β c β (cid:61) m λ − s β (cid:61) m λ − c β (cid:61) m λ , (57)with the abbreviation λ = ( λ + λ ). The effective cubic couplings g αH + H − read: g φ H + H − = 2 s β c β λ + c β λ − s β c β λ − s β c β (cid:60) e λ + s β ( s β − c β ) (cid:60) e λ + s β c β (cid:60) e λ ,g φ H + H − = 2 s β c β λ + s β λ − s β c β λ − s β c β (cid:60) e λ + s β c β (cid:60) e λ + c β ( c β − s β ) (cid:60) e λ ,g aH + H − = 2 s β c β (cid:61) m λ − s β (cid:61) m λ − c β (cid:61) m λ . (58)On the other hand, the quartic couplings for the neutral Higgs bosons are g φ φ φ φ = 14 λ , g φ φ φ φ = 12 (cid:60) e λ , g φ φ φ φ = 12 λ + 12 (cid:60) e λ ,g φ φ φ φ = 12 (cid:60) e λ , g φ φ φ φ = 14 λ ,g φ φ φ a = − c β (cid:61) m λ , g φ φ φ a = − c β (cid:61) m λ − s β (cid:61) m λ ,g φ φ φ a = − s β (cid:61) m λ − c β (cid:61) m λ , g φ φ φ a = − s β (cid:61) m λ , The relative phase ξ could be reinstated by replacing λ with λ e iξ and λ , with λ , e iξ , if necessary. φ φ aa = 12 s β λ + 12 c β λ − c β (cid:60) e λ − s β c β (cid:60) e λ ,g φ φ aa = − s β c β (cid:60) e λ + 12 s β (cid:60) e λ + 12 c β (cid:60) e λ ,g φ φ aa = 12 c β λ + 12 s β λ − s β (cid:60) e λ − s β c β (cid:60) e λ ,g φ aaa = s β c β (cid:61) m λ − s β c β (cid:61) m λ − c β (cid:61) m λ ,g φ aaa = s β c β (cid:61) m λ − s β (cid:61) m λ − s β c β (cid:61) m λ ,g aaaa = 14 g H + H − H + H − , (59)together with the quartic coupling of the charged Higgs bosons given by g H + H − H + H − = s β λ + c β λ + s β c β ( λ + λ ) + 2 s β c β (cid:60) e λ − s β c β (cid:60) e λ − s β c β (cid:60) e λ . (60)Finally, the remaining quartic couplings involving the charged Higgs boson pairs, g αβ H + H − , are given by g φ φ H + H − = s β λ + 12 c β λ − s β c β (cid:60) e λ ,g φ φ H + H − = − s β c β λ − s β c β (cid:60) e λ + s β (cid:60) e λ + c β (cid:60) e λ ,g φ φ H + H − = c β λ + 12 s β λ − s β c β (cid:60) e λ ,g φ aH + H − = 2 s β c β (cid:61) m λ − s β c β (cid:61) m λ − c β (cid:61) m λ ,g φ aH + H − = 2 s β c β (cid:61) m λ − s β (cid:61) m λ − s β c β (cid:61) m λ ,g aaH + H − = g H + H − H + H − . (61) In this subsection, we give a brief review of the minimal supersymmetric extension of the SM (MSSM)with our particular focus on the MSSM Higgs sector. For general reviews on the MSSM, see, forexample, Refs. [48, 49, 50, 51, 52].
Potential and mixing
The superpotential of the MSSM is written as W MSSM = (cid:98) U C h u (cid:98) Q · (cid:98) H u + (cid:98) D C h d (cid:98) H d · (cid:98) Q + (cid:98) E C h e (cid:98) H d · (cid:98) L + µ (cid:98) H u · (cid:98) H d , (62)where (cid:98) H u,d are the two Higgs chiral superfields, and (cid:98) Q , (cid:98) L , (cid:98) U C , (cid:98) D C and (cid:98) E C are the left-handed doubletand right-handed singlet superfields related to up- and down-type quarks and charged leptons. TheYukawa couplings h u,d,e are in general 3 × µ parameter that mixes the two Higgs supermultiplets, which has to be of the electroweak orderfor a natural realization of the electroweak symmetry breaking mechanism without any significant finetuning. See Table 3 for the full particle contents appearing in the MSSM. Here and in the following, we introduce a product between SU(2) L doublets as defined by A · B = A T iτ B = (cid:15) ab A a B b with (cid:15) = − (cid:15) = +1. Q = T + Y and (cid:102) W ± = ( (cid:102) W ∓ i (cid:102) W ) / √ (cid:102) W = (cid:102) W .spin-0 spin- spin-1 colour T T Y B L (cid:101) Q L = (cid:32) (cid:101) U L (cid:101) D L (cid:33) Q L = (cid:18) U L D L (cid:19) (cid:18) +1 / − / (cid:19) + + (cid:101) U ∗ R (cid:0) U C (cid:1) L ∗ − − (cid:101) D ∗ R (cid:0) D C (cid:1) L ∗ − (cid:101) L L = (cid:18) (cid:101) ν L (cid:101) E L (cid:19) L L = (cid:18) ν L E L (cid:19) (cid:18) +1 / − / (cid:19) − (cid:101) E ∗ R (cid:0) E C (cid:1) L − (cid:101) g g (cid:102) W + (cid:102) W (cid:102) W − W + W W − +10 − (cid:101) B B H d ) = (cid:18) H (cid:101) H − (cid:19) (cid:101) H L = (cid:32) (cid:101) H (cid:101) H − (cid:33) L (cid:18) +1 / − / (cid:19) − H u ) = (cid:18) H +2 (cid:101) H (cid:19) (cid:101) H L = (cid:32) (cid:101) H +2 (cid:101) H (cid:33) L (cid:18) +1 / − / (cid:19) + −L soft = 12 (cid:16) M (cid:101) B (cid:101) B + M (cid:102) W i (cid:102) W i + M ˜ g a ˜ g a + h . c . (cid:17) + (cid:101) Q † (cid:102) M Q (cid:101) Q + (cid:101) L † (cid:102) M L (cid:101) L + (cid:101) U † (cid:102) M U (cid:101) U + (cid:101) D † (cid:102) M D (cid:101) D + (cid:101) E † (cid:102) M E (cid:101) E + M H u H † u H u + M H d H † d H d + (cid:16) Bµ H u · H d + h . c . (cid:17) + (cid:16) (cid:101) U † a u (cid:101) Q · H u + (cid:101) D † a d H d · (cid:101) Q + (cid:101) E † a e H d · (cid:101) L + h . c . (cid:17) . (63)Here M , , are the soft SUSY-breaking masses associated with one U(1) Y gaugino (cid:101) B , three SU(2) L gauginos (cid:102) W i , and eight SU(3) C gauginos (cid:101) g a , respectively. In addition, M H u,d and Bµ are the softmasses related to the Higgs doublets H u,d and their bilinear mixing. Finally, (cid:102) M Q,L,D,U,E are the 3 × a u,d,e are the corresponding 3 × Hence, in addition to the µ term, the unconstrained CP-violating MSSM contains 109 Alternatively, the soft Yukawa mass matrices a u,d,e may be defined by the relation: ( a u,d,e ) ij = ( h u,d,e ) ij ( A u,d,e ) ij ,where the parameters ( A u,d,e ) ij are generically of order M SUSY in gravity-mediated SUSY breaking models. M , , : 3 × , (cid:102) M Q,L,D,U,E : 5 × , a u,d,e : 3 ×
18 (9) = 54 (27) ,M H u,d : 2 × ,B : 2 (1) = 2 (1) , (64)where, in each line, the number of CP phases is separately counted in parentheses for the correspondingsoft-SUSY breaking parameters.By identifying H u = Φ and H d = (cid:101) Φ = iτ Φ ∗ , one may obtain the same form of the Higgspotential as in the 2HDMs: V MSSM = µ (Φ † Φ ) + µ (Φ † Φ ) + m (Φ † Φ ) + m ∗ (Φ † Φ ) + λ (Φ † Φ ) + λ (Φ † Φ ) + λ (Φ † Φ )(Φ † Φ ) + λ (Φ † Φ )(Φ † Φ ) + λ (Φ † Φ ) + λ ∗ (Φ † Φ ) (65)+ λ (Φ † Φ )(Φ † Φ ) + λ ∗ (Φ † Φ )(Φ † Φ ) + λ (Φ † Φ )(Φ † Φ ) + λ ∗ (Φ † Φ )(Φ † Φ ) , with the potential parameters given in terms of the µ and the soft SUSY-breaking parameters as wellas the SU(2) L and U(1) Y gauge couplings by µ = M H d + | µ | , µ = M H u + | µ | , m = Bµ ,λ = λ = 18 ( g + g (cid:48) ) , λ = 14 ( g − g (cid:48) ) , λ = − g ,λ = λ = λ = 0 . (66)Note that the quartic couplings λ , , , are solely determined by the gauge couplings and λ , , arevanishing at the tree level. However, the quartic couplings λ , , receive significant radiative correctionsfrom scalar-top and scalar-bottom loops and, especially in the presence of CP-violating phases in thesoft SUSY-breaking terms, the CP-violating mixing among the three neutral Higgs states are induced[75, 76, 77, 78, 79, 80, 81]. In this case, as in the 2HDMs, the orthogonal 3 × O has tobe introduced for diagonalizing the 3 × φ , φ , a ) Tα = O αi ( H , H , H ) Ti (67)with the increasing ordering of M H ≤ M H ≤ M H . Interactions of Higgs bosons with the SM particles and self-interactions
The interactions of Higgs bosons with massive vector bosons and those among themselves in the MSSMare formally the same as in the 2HDMs. And the Higgs-boson interactions with the SM fermions arethe same as those in the type-II 2HDM.
Interactions of Higgs bosons with the SUSY particles
For the sake of completeness and reference, although no serious analyses on them are presented in thepresent review, we fix the convention for the interactions of Higgs bosons with the supersymmetric(SUSY) particles such as charginos, neutralinos and sfermions. We recall Φ = (cid:16) φ +1 , √ ( v + φ + ia ) (cid:17) T and Φ = e iξ (cid:16) φ +2 , √ ( v + φ + ia ) (cid:17) T together with a = − s β a + c β a . L H (cid:101) χ + (cid:101) χ − = − g √ (cid:88) i,j =1 3 (cid:88) k =1 H k (cid:101) χ − i (cid:16) g SH k ˜ χ + i ˜ χ − j + iγ g PH k ˜ χ + i ˜ χ − j (cid:17) (cid:101) χ − j , (68)with the normalized Higgs-chargino-chargino couplings g SH k ˜ χ + i ˜ χ − j = 12 (cid:110) [( C R ) i ( C L ) ∗ j G φ k + ( C R ) i ( C L ) ∗ j G φ k ] + [ i ↔ j ] ∗ (cid:111) ,g PH k ˜ χ + i ˜ χ − j = i (cid:110) [( C R ) i ( C L ) ∗ j G φ k + ( C R ) i ( C L ) ∗ j G φ k ] − [ i ↔ j ] ∗ (cid:111) , (69)where G φ k = ( O φ k − is β O ak ) and G φ k = ( O φ k − ic β O ak ). The two different unitary 2 × C L ) iα and ( C R ) iα are required to diagonalize the chargino mass matrix M C = (cid:32) M √ M W c β √ M W s β µ (cid:33) , (70)in the ( ˜ W − , ˜ H − ) L and ( ˜ W + , ˜ H + ) L bases with the convention ˜ H − L = ˜ H − and ˜ H + L = ˜ H +2 in such a waythat C R M C C † L = diag { m ˜ χ ± , m ˜ χ ± } , (71)with the increasing ordering of m ˜ χ ± ≤ m ˜ χ ± . Explicitly, the mixing matrices relate the electroweakeigenstates to the mass eigenstates, via˜ W − L = (cid:88) i =1 , ( C L ) ∗ i ˜ χ − iL , ˜ H − L = (cid:88) i =1 , ( C L ) ∗ i ˜ χ − iL , ˜ W − R = (cid:88) i =1 , ( C R ) ∗ i ˜ χ − iR , ˜ H − R = (cid:88) i =1 , ( C R ) ∗ i ˜ χ − iR . (72)Note that the convention ˜ H − L ( R ) = ˜ H − is adopted with the subscripts 1 and 2 being associated withthe Higgs supermultiplets leading to the tree-level mass generation of the down- and up-type quarks,respectively, see Table 3. We recall that we take the following abbreviations throughout this paper: s β ≡ sin β , c β ≡ cos β , t β ≡ tan β , s β ≡ sin 2 β , c β ≡ cos 2 β , s W ≡ sin θ W , c W ≡ cos θ W , etc.The interactions of three neutral Higgs bosons with neutralinos, which are mixtures of 2 neutralgauginos and 2 neutral higgsinos, are described by the following Lagrangian: L H (cid:101) χ (cid:101) χ = − g (cid:88) i,j =1 3 (cid:88) k =1 H k (cid:101) χ i (cid:16) g SH k ˜ χ i ˜ χ j + iγ g PH k ˜ χ i ˜ χ j (cid:17) (cid:101) χ j , (73)with the normalized Higgs-neutralino-neutralino couplings g SH k ˜ χ i ˜ χ j = 12 (cid:60) e[( N ∗ j − t W N ∗ j )( N ∗ i G φ k − N ∗ i G φ k ) + ( i ↔ j )] ,g PH k ˜ χ i ˜ χ j = − (cid:61) m[( N ∗ j − t W N ∗ j )( N ∗ i G φ k − N ∗ i G φ k ) + ( i ↔ j )] , (74)where i, j = 1-4 for the four neutralino states and k = 1-3 for the three neutral Higgs bosons. Oneunitary 4 × × M N = M − M Z c β s W M Z s β s W M M Z c β c W − M Z s β c W − M Z c β s W M Z c β c W − µM Z s β s W − M Z s β c W − µ , (75)in the ( (cid:101) B, (cid:102) W , (cid:101) H , (cid:101) H ) L basis into a diagonal matrix as N ∗ M N N † = diag ( m (cid:101) χ , m (cid:101) χ , m (cid:101) χ , m (cid:101) χ ) , (76)with the increasing mass ordering of m (cid:101) χ ≤ m (cid:101) χ ≤ m (cid:101) χ ≤ m (cid:101) χ . The single neutralino mixing matrix N iα relates the left-handed and right-handed electroweak eigenstates to the left-handed and right-handedmass eigenstates via ( (cid:101) B, (cid:102) W , (cid:101) H , (cid:101) H ) TαL = N ∗ iα ( (cid:101) χ , (cid:101) χ , (cid:101) χ , (cid:101) χ ) TiL and( (cid:101) B, (cid:102) W , (cid:101) H , (cid:101) H ) TαR = N iα ( (cid:101) χ , (cid:101) χ , (cid:101) χ , (cid:101) χ ) TiR , (77)respectively.The interactions of charged Higgs bosons with charginos and neutralinos are described by the fol-lowing Lagrangian: L H ± (cid:101) χ i (cid:101) χ ∓ j = − g √ (cid:88) i =1 2 (cid:88) j =1 H + (cid:101) χ i (cid:16) g SH + (cid:101) χ i (cid:101) χ − j + iγ g PH + (cid:101) χ i (cid:101) χ − j (cid:17) (cid:101) χ − j + h . c . , (78)with the normalized couplings of the charged Higgs boson with a chargino and a neutralino g SH + (cid:101) χ i (cid:101) χ − j = 12 (cid:110) s β (cid:104) √ N ∗ i ( C L ) ∗ j − ( N ∗ i + t W N ∗ i )( C L ) ∗ j (cid:105) + c β (cid:104) √ N i ( C R ) ∗ j + ( N i + t W N i )( C R ) ∗ j (cid:105)(cid:111) ,g PH + (cid:101) χ i (cid:101) χ − j = i (cid:110) s β (cid:104) √ N ∗ i ( C L ) ∗ j − ( N ∗ i + t W N ∗ i )( C L ) ∗ j (cid:105) − c β (cid:104) √ N i ( C R ) ∗ j + ( N i + t W N i )( C R ) ∗ j (cid:105)(cid:111) , (79)expressed in terms of the chargino and neutralino mixing matrices.The neutral Higgs–sfermion–sfermion interactions can be written in terms of the sfermion masseigenstates as L H (cid:101) f (cid:101) f = v (cid:88) f = u,d,l (cid:88) i =1 (cid:88) j,k =1 , g H i (cid:101) f ∗ j (cid:101) f k ( H i (cid:101) f ∗ j (cid:101) f k ) , (80)where the couplings of the Higgs bosons with sfermions in the mass eigenstate basis v g H i (cid:101) f ∗ j (cid:101) f k = (cid:88) α = φ ,φ ,a (cid:88) β,γ = L,R (cid:16) Γ α (cid:101) f ∗ (cid:101) f (cid:17) βγ O αi U (cid:101) f ∗ βj U (cid:101) fγk , (81)expressed in terms of the 2 × α (cid:101) f ∗ (cid:101) f in the weak-eigenstatebasis, of which the explicit form is given later in Eq. (88), and the 3 × × O and U (cid:101) f , with the convention of α = ( φ , φ , a ) = (1 , , β, γ = L, R , i = ( H , H , H ) =191 , ,
3) and j, k = 1 , L H ± (cid:101) f (cid:101) f (cid:48) = v (cid:88) ( f,f (cid:48) )=( u,d ) , ( ν,l ) (cid:88) j,k =1 , g H + (cid:101) f ∗ j (cid:101) f (cid:48) k ( H + (cid:101) f ∗ j (cid:101) f (cid:48) k ) + h . c ., (82)where the couplings of the charged Higgs boson with sfermions in the mass eigenstate basis v g H + (cid:101) f ∗ j (cid:101) f (cid:48) k = (cid:88) β,γ = L,R (cid:16) Γ H + (cid:101) f ∗ (cid:101) f (cid:48) (cid:17) βγ U (cid:101) f ∗ βj U (cid:101) f (cid:48) γk , (83)expressed in terms of the 2 × H + (cid:101) f ∗ (cid:101) f in the weak-eigenstate basis, of which the explicit form is given later in Eqs. (89) and (90), and the 2 × U (cid:101) f with the convention of β, γ = L, R and j, k = 1 ,
2. The unitary 2 × U (cid:101) f is obtained by diagonalizing the 2 × (cid:102) M f for f = t, b and τ in sucha way that U (cid:101) f † (cid:102) M f U (cid:101) f = diag ( m (cid:101) f , m (cid:101) f ) , (84)with the increasing mass ordering of m (cid:101) f ≤ m (cid:101) f . The mixing matrix U (cid:101) f relates the sfermion electroweakeigenstates (cid:101) f L,R to the sfermion mass eigenstates (cid:101) f , via( (cid:101) f L , (cid:101) f R ) Tα = U (cid:101) fαi ( (cid:101) f , (cid:101) f ) Ti . (85)Explicitly, the stop and sbottom mass-squared matrices is written in the ( (cid:101) q L , (cid:101) q R ) electroweak basis as (cid:102) M q = t,b = (cid:32) M (cid:101) Q + m q + c β M Z ( T qz − Q q s W ) h ∗ q v q ( A ∗ q − µR q ) / √ h q v q ( A q − µ ∗ R q ) / √ M (cid:101) R + m q + c β M Z Q q s W (cid:33) , (86)with the third-generation left- and right-sfermion soft SUSY-breaking mass-squared M (cid:101) Q and M (cid:101) R = (cid:101) U , (cid:101) D ,a cubic soft-breaking term A q , T tz = − T bz = 1 / Q t = 2 / Q b = − / v b = v , v t = v , R b = tan β = v /v , R t = cot β , and the Yukawa coupling h q of the quark q . Similarly, the stau mass-squared matrixis written in the ( (cid:101) τ L , (cid:101) τ R ) electroweak basis as (cid:102) M τ = (cid:32) M (cid:101) L + m τ + c β M Z ( s W − / h ∗ τ v ( A ∗ τ − µ tan β ) / √ h τ v ( A τ − µ ∗ tan β ) / √ M (cid:101) E + m τ − c β M Z s W (cid:33) , (87)derived directly from the sbottom mass-squared mass matrix by replacing b by τ and (cid:101) Q , (cid:101) R by (cid:101) L , (cid:101) E , and taking Q τ = −
1. Incidentally, the mass of the tau sneutrino (cid:101) ν τ is simply given by m (cid:101) ν τ = (cid:113) M (cid:101) L + c β M Z , as it has no right–handed counterpart in the MSSM unlike the squark andcharged slepton cases.For the sake of completeness and explicit analytic and numerical calculations, we present the ex-plicit form of the Higgs–sfermion–sfermion couplings in the electroweak-interaction basis for the third-generation sfermions. The 2 × α (cid:101) f ∗ (cid:101) f are given in the ( (cid:101) f L , (cid:101) f R ) basis with f = t, b, τ, ν τ and α = a, φ , φ byΓ a (cid:101) b ∗ (cid:101) b = 1 √ (cid:18) i h ∗ b ( s β A ∗ b + c β µ ) − i h b ( s β A b + c β µ ∗ ) 0 (cid:19) , Γ φ (cid:101) b ∗ (cid:101) b = (cid:32) −| h b | vc β + (cid:0) g + g (cid:48) (cid:1) vc β − √ h ∗ b A ∗ b − √ h b A b −| h b | vc β + g (cid:48) vc β (cid:33) , φ (cid:101) b ∗ (cid:101) b = (cid:32) − (cid:0) g + g (cid:48) (cid:1) vs β √ h ∗ b µ √ h b µ ∗ − g (cid:48) vs β (cid:33) , Γ a (cid:101) t ∗ (cid:101) t = 1 √ (cid:18) i h ∗ t ( c β A ∗ t + s β µ ) − i h t ( c β A t + s β µ ∗ ) 0 (cid:19) , Γ φ (cid:101) t ∗ (cid:101) t = (cid:32) − (cid:0) g − g (cid:48) (cid:1) vc β √ h ∗ t µ √ h t µ ∗ − g (cid:48) vc β (cid:33) , Γ φ (cid:101) t ∗ (cid:101) t = (cid:32) −| h t | vs β + (cid:0) g − g (cid:48) (cid:1) vs β − √ h ∗ t A ∗ t − √ h t A t −| h t | vs β + g (cid:48) vs β (cid:33) , Γ a (cid:101) τ ∗ (cid:101) τ = 1 √ (cid:18) i h ∗ τ ( s β A ∗ τ + c β µ ) − i h τ ( s β A τ + c β µ ∗ ) 0 (cid:19) , Γ φ (cid:101) τ ∗ (cid:101) τ = (cid:32) −| h τ | vc β + ( g − g (cid:48) ) vc β − √ h ∗ τ A ∗ τ − √ h τ A τ −| h τ | vc β + g (cid:48) vc β (cid:33) , Γ φ (cid:101) τ ∗ (cid:101) τ = (cid:32) − ( g − g (cid:48) ) vs β √ h ∗ τ µ √ h τ µ ∗ − g (cid:48) vs β (cid:33) , Γ a (cid:101) ν ∗ τ (cid:101) ν τ = 0 , Γ φ (cid:101) ν ∗ τ (cid:101) ν τ = − (cid:0) g + g (cid:48) (cid:1) vc β , Γ φ (cid:101) ν ∗ τ (cid:101) ν τ = 14 (cid:0) g + g (cid:48) (cid:1) vs β . (88)The 2 × H + (cid:101) u ∗ (cid:101) d is given in the ( (cid:101) u L , (cid:101) d R ) basis byΓ H + (cid:101) u ∗ (cid:101) d = (cid:32) √ ( | h u | + | h d | − g ) vs β c β h ∗ d ( s β A ∗ d + c β µ ) h u ( c β A u + s β µ ∗ ) √ h u h ∗ d v (cid:33) , (89)and the couplings of the charged Higgs boson with a tau sneutrino and a stau given byΓ H + (cid:101) ν ∗ τ (cid:101) τ L = 1 √ | h τ | − g ) vs β c β , Γ H + (cid:101) ν ∗ τ (cid:101) τ R = h ∗ τ ( s β A ∗ τ + c β µ ) . (90) Without loss of generality, the Lagrangian describing the interactions of a generic neutral Higgs boson H with two fermions, which is applicable for all the models described in the previous section, can bewritten as L Hff = − m f v H ¯ f (cid:16) g SH ¯ ff + ig PH ¯ ff γ (cid:17) f , (91)in terms of the normalized scalar and pseudoscalar couplings of g SH ¯ ff and g PH ¯ ff with m f denoting thefermion mass and v ≈
246 GeV. The Lagrangian describing the interactions of Higgs bosons withmassive gauge bosons Z and W ± can be written as L HV V = g M W (cid:18) g HWW W + µ W − µ + g HZZ c W Z µ Z µ (cid:19) H , (92)in terms of the normalized couplings of g HWW and g HZZ with g = e/s W the SU(2) L gauge coupling, s W ≡ sin θ W , c W ≡ cos θ W , t W ≡ sin θ W / cos θ W , etc. And, if not mentioned otherwise, we set g HWW = g HZZ = g HV V in the following.In the presence of Higgs/scalar bosons ϕ ’s lighter than H , the neutral Higgs boson H can decay intoa lighter Higgs boson and a massive vector boson and also into two lighter Higgs/scalar bosons. The21nteraction Lagrangian describing these types of decays could be cast into the expressions: L HϕZ = g c W g HϕZ Z µ ( H ↔ ∂ µ ϕ ) , L Hϕ ± W ∓ = − g g Hϕ + W − W − µ ( H i ↔ ∂ µ ϕ + ) + h . c . , L self ⊃ − v (cid:88) i ≥ j g Hϕiϕj Hϕ i ϕ j = − v (cid:2) g Hϕ ϕ Hϕ + (cid:0) g Hϕ ϕ Hϕ ϕ + g Hϕ ϕ Hϕ (cid:1) + · · · (cid:3) . (93)Note that the scalar states ϕ i and ϕ j are ordered in the last expression so as to avoid the couplingssuch as g Hϕ ϕ with a wrong ordering of the scalar states.It is noteworthy that we are assuming the neutral Higgs boson H to be a general CP-mixed state,i.e. a scalar-pseudoscalar mixture. H → f ¯ f Including the radiative corrections known up to now, the Higgs decay width into fermions can beorganized as [57]:Γ( H → f ¯ f ) = N fC m f v β f M H π (cid:104) β f | g SH ¯ ff | (cid:16) δ QCD + δ f : St + δ f elw + δ f mixed (cid:17) + | g PH ¯ ff | (cid:16) δ QCD + δ f : Pt (cid:17)(cid:105) , (94)where β f ≡ (cid:112) − κ f with κ f = M f /M H and the colour factor N fC = 3 for quarks and 1 for leptons. The lepton pole mass is taken for m f while, for the Higgs decays into quarks, the MS quark mass m q ( M H )is used. In passing, we recall that 1 /v = √ G F . The electroweak corrections to the pseudoscalar parthave not been included expecting them to be evaluated with reliable precision in the near future.The pure QCD corrections for the Higgs decays into a quark pair q ¯ q consist of a universal partof δ QCD and two types of flavour- and parity-dependent parts of δ q : St and δ q : Pt which are given by[82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95] δ QCD = 5 . α s ( M H ) π + (35 . − . N F ) (cid:18) α s ( M H ) π (cid:19) +(164 . − . N F + 0 . N F ) (cid:18) α s ( M H ) π (cid:19) +(39 . − . N F + 9 . N F − . N F ) (cid:18) α s ( M H ) π (cid:19) ,δ q : St = (cid:18) α s ( M H ) π (cid:19) (cid:20) . −
23 log M H M t + 19 log m q ( M H ) M H (cid:21) ,δ q : Pt = (cid:18) α s ( M H ) π (cid:19) (cid:20) . − log M H M t + 16 log m q ( M H ) M H (cid:21) , (95)where N F counts the flavour number of quarks lighter than H . The QCD coupling strength α s andthe running MS quark mass m q ( M H ) are defined at the scale of the Higgs mass to absorb large masslogarithms. In this review, we denote the pole mass of the fermion f by M f and its running mass by m f ( µ ). δ f elw = 32 απ Q f (cid:32) − log M H M f (cid:33) + G F √ π (cid:26) k f M t + M W (cid:20) − s W log c W (cid:21) − M Z (6 v Z ¯ ff − a Z ¯ ff ) (cid:27) , (96)where v Z ¯ ff = I f / − Q f s W and a Z ¯ ff = I f / I f denoting the third component of the electroweakisospin and Q f the electric charge of the fermion f . We refer to Eq. (B.9) for the Z couplings withfermions. The large logarithm log M H /M f can be absorbed in the running fermion mass as in the QCDcorrections. For decays into leptons and light quarks, the coefficient k f = 7 while it is 1 for b and t quarks. The electroweak corrections are below the percent level for f = b, c while they are of O (1-5)%for f = τ, µ .The mixed corrections evaluated by means of low-energy theorems could be cast into the expressions[101, 102, 103]: δ q mixed = − G F M t √ π (cid:18)
32 + ζ (cid:19) α s ( M t ) π for light quarks ,δ b,t mixed = − G F M t √ π ζ ) α s ( M t ) π for b and t , (97)in the next-to-next-to-leading-order (NNLO) with ζ = π /
6. The full mixed QCD-electroweak correc-tions δ q mixed turn out to be smaller than O (0.1)% [104].In the left column of Fig. 1, taking g SH ¯ ff = 1 and g PH ¯ ff = 0, we show the decay widths of a Higgsboson H into a pair of b quarks, c quarks, tau leptons, and muons for varying M H . For the decays H → b ¯ b and H → c ¯ c , the lower dashed lines are for the decay widths in the leading order (LO) whilethe upper (black) solid lines are for those taking full account of the QCD and electroweak (ELW)corrections. The decay widths including only the electroweak corrections are denoted by the lower (red)solid lines. For the decays H → τ + τ − and H → µ + µ − , the dashed lines are for the decay widths in theLO and the sold lines are for the decay widths including the electroweak corrections. The behaviourdoes not alter much for other choices of ( g SH ¯ ff , g PH ¯ ff ) as far as | g SH ¯ ff | + | g PH ¯ ff | = 1 since the QCDcorrection δ QCD , which is common in the scalar and pseudoscalar contributions to the Higgs decay widthinto quarks, dominates.In each frame of the right column of Fig. 1, the corresponding full decay widths are shown in thelow mass region of 120 GeV < M H <
130 GeV for the three choices of ( g SH ¯ ff , g PH ¯ ff ) = (1 , , / √ , / √
2) denoted by the solid, dashed and dotted lines, respectively. The pure scalar casewith ( g SH ¯ ff , g PH ¯ ff ) = (1 ,
0) has a slightly smaller width compared to the pure pseudoscalar case with( g SH ¯ ff , g PH ¯ ff ) = (0 ,
1) due to the kinematical suppression factor of β f .Incidentally, in the leading order (LO), taking the consideration of double off-shell effects, the decaywidth of a Higgs boson into a top-quark pair t ¯ t , each of which subsequently decays into bW + and ¯ bW − ,is given by [105] Γ LO ( H → t ( p t ) ¯ t (¯ p t ) → b ¯ bW + W − ) = N tC m t ( M H ) v g M H π F , (98)where the dimensionless quantity F is given by an integrated function of the pole masses of b and t quarks, the W -boson mass, and the top-quark total width Γ t as F = (cid:90) λ / H λ / t λ / t (cid:104) | λ S | (cid:12)(cid:12) g SH ¯ tt (cid:12)(cid:12) + | λ P | (cid:12)(cid:12) g PH ¯ tt (cid:12)(cid:12) (cid:105) (cid:18) p t + ¯ p t M W + p t ¯ p t M W (cid:19) × p t + M b − M W ( p t − M t ) + M t Γ t ¯ p t + M b − M W (¯ p t − M t ) + M t Γ t dp t d ¯ p t , (99)23 S =
1, g p = + ELWLO + ELW + QCD
200 400 600 800 100002468101214 M H [ GeV ] Γ ( H → b b ) [ M e V ] g S =
1, g p = S =
0, g p = S = g p = /
120 122 124 126 128 1302.252.302.352.402.452.50 M H [ GeV ] Γ ( H → b b ) [ M e V ] g S =
1, g p = + ELWLO + ELW + QCD
200 400 600 800 10000.00.10.20.30.40.50.60.7 M H [ GeV ] Γ ( H → c c ) [ M e V ] g S =
1, g p = S =
0, g p = S = g p = /
120 122 124 126 128 1300.1140.1160.1180.120 M H [ GeV ] Γ ( H → c c ) [ M e V ] g S =
1, g p = + ELW
200 400 600 800 10000.00.51.01.52.0 M H [ GeV ] Γ ( H → ττ ) [ M e V ] g S =
1, g p = S =
0, g p = S = g p = /
120 122 124 126 128 1300.2400.2450.2500.2550.2600.2650.2700.275 M H [ GeV ] Γ ( H → ττ ) [ M e V ] g S =
1, g p = + ELW
200 400 600 800 100002468 M H [ GeV ] Γ ( H → μμ ) [ k e V ] g S =
1, g p = S =
0, g p = S = g p = /
120 122 124 126 128 1300.840.860.880.900.920.940.960.981.00 M H [ GeV ] Γ ( H → μμ ) [ k e V ] Figure 1: (Left) Decay widths of a neutral Higgs boson with mass M H into b ¯ b , c ¯ c , τ + τ − , and µ + µ − fromtop to bottom taking g SH ¯ ff = 1 and g PH ¯ ff = 0. See the main text for details. (Right) In each row, thecorresponding full decay widths are shown for the three choices of ( g SH ¯ ff , g PH ¯ ff ) = (1 ,
0) (solid), (0 , / √ , / √
2) (dotted) in the low mass region around M H = 125 GeV. The vertical lineslocate the positions of M H = 125 . λ H = 1 + ( p t ) M H + (¯ p t ) M H − p t M H − p t M H − p t ¯ p t M H ,λ t = (cid:18) − M b p t − M W p t (cid:19) − M b p t M W p t ,λ ¯ t = (cid:18) − M b ¯ p t − M W ¯ p t (cid:19) − M b ¯ p t M W ¯ p t ,λ S = (cid:26)(cid:104) (1 + λ / H ) − ( p t − ¯ p t ) /M H (cid:105) / − (cid:104) (1 − λ / H ) − ( p t − ¯ p t ) /M H (cid:105) / (cid:27) / ,λ P = (cid:26)(cid:104) (1 + λ / H ) − ( p t − ¯ p t ) /M H (cid:105) / + (cid:104) (1 − λ / H ) − ( p t − ¯ p t ) /M H (cid:105) / (cid:27) / . (100)After integrating over p t and ¯ p t , we can recast the LO decay width intoΓ LO ( H → t ¯ t → b ¯ bW + W − ) ≡ (cid:12)(cid:12) g SH ¯ tt (cid:12)(cid:12) T LO S + (cid:12)(cid:12) g PH ¯ tt (cid:12)(cid:12) T LO P , (101)and, taking account of the radiative corrections as well as double off-shell effects, the Higgs decay widthinto a top quark pair Γ( H → t ∗ ¯ t ∗ ) has been estimated as follows:Γ( H → t ∗ ¯ t ∗ ) = (cid:12)(cid:12) g SH ¯ tt (cid:12)(cid:12) T LO S (cid:0) δ QCD + δ t : St + δ t elw + δ t mixed (cid:1) + (cid:12)(cid:12) g PH ¯ tt (cid:12)(cid:12) T LO P (cid:0) δ QCD + δ t : Pt (cid:1) . (102)When M H > M t , taking p t = ¯ p t = M t leads to λ S = λ / H = β t = (1 − M t /M H ) / and λ P = 1 and,neglecting the kinematical b -quark mass in the t → bW process, we reach the following factorized formfor the LO decay widthΓ LO ( H → t ¯ t → b ¯ bW + W − ) (cid:12)(cid:12) M H > M t = Γ LO ( H → t ¯ t ) (cid:18) Γ LO ( t → bW )Γ t (cid:19) , (103)using the narrow-width approximation (NWA) denoted by δ ( p − m ) = lim Γ → m Γ π p − m ) + m Γ , (104)and the LO decay widths for H → t ¯ t and t → bW given byΓ LO ( H → t ¯ t ) = N tC m t ( M H ) v β t M H π (cid:104) β t (cid:12)(cid:12) g SH ¯ tt (cid:12)(cid:12) + (cid:12)(cid:12) g PH ¯ tt (cid:12)(cid:12) (cid:105) , (105)Γ LO ( t → bW ) = g M t π (cid:18) − M W M t (cid:19) (cid:18) M t M W (cid:19) , (106)respectively.In Fig. 2, we show the decay width Γ( H → t ∗ ¯ t ∗ ) as a function of M H , taking ( g SH ¯ tt , g PH ¯ tt ) = (1 , g SH ¯ tt , g PH ¯ tt ) = (0 ,
1) (lower), respectively. We take Γ t = Γ LO ( t → bW ) assuming that atop quark decays 100% into a b quark and a W boson and Γ LO ( H → t ¯ t → b ¯ bW + W − ) converges toΓ LO ( H → t ¯ t ) in the high M H limit. Practically, far above the top-quark-pair threshold with M H > H → t ∗ ¯ t ∗ ) from the kinetic edge region of25 OLO + ELWLO + ELW + QCD g HttS =
1, g
HttP =
260 280 300 320 340 36010 - - M H [ GeV ] Γ ( H → tt ) [ G e V ] LOLO + ELWLO + ELW + QCD g HttS =
1, g
HttP =
300 400 500 600 700 800 900 100001020304050 M H [ GeV ] Γ ( H → tt ) [ G e V ]
350 360 370 380 390 4000.51.01.52.02.5 g HttS =
0, g
HttP = LOLO + QCD
260 280 300 320 340 36010 - - M A [ GeV ] Γ ( H → tt ) [ G e V ] LOLO + QCD g HttS =
0, g
HttP =
300 400 500 600 700 800 900 100001020304050 M H [ GeV ] Γ ( H → tt ) [ G e V ]
340 350 360 370 380 390 4000246810
Figure 2: Decay widths of a neutral Higgs boson with mass M H into t ∗ ¯ t ∗ or Γ( H → t ∗ ¯ t ∗ ) taking( g SH ¯ tt , g PH ¯ tt ) = (1 ,
0) (upper) and ( g SH ¯ tt , g PH ¯ tt ) = (0 ,
1) (lower). In the left panels, the vertical lines locatethe top-quark-pair thresholds and the (magenta) dash-dotted lines are for the corresponding full 2-bodydecay widths of Γ( H → t ¯ t ). Note that, in the right panels, we switch from Γ( H → t ∗ ¯ t ∗ ) to Γ( H → t ¯ t )from M H = 500 GeV and above. (cid:112) p t + (cid:112) ¯ p t ∼ M H , assuming that the intermediate top quarks are reconstructed by requiring on-shellconditions of p t (cid:39) M t and ¯ p t (cid:39) M t . Finally, for decays into two different fermions such as charginos or neutralinos or for flavour changingdecays such as H → b ¯ s , one may write the effective interaction as L Hff (cid:48) = − g ¯ ff (cid:48) H ¯ f ( g S H ¯ ff (cid:48) + ig P H ¯ ff (cid:48) γ ) f (cid:48) + h . c . , (107)without loss of generality. Then, in the LO, the decay width may take a form ofΓ LO ( H → f ¯ f (cid:48) ) = N ff (cid:48) C g ff (cid:48) M H λ / ff (cid:48) π (cid:104) (1 − κ − κ (cid:48) )( | g S H ¯ ff (cid:48) | + | g P H ¯ ff (cid:48) | ) − √ κκ (cid:48) ( | g S H ¯ ff (cid:48) | − | g P H ¯ ff (cid:48) | ) (cid:105) , (108)where κ ≡ m f /M H , κ (cid:48) ≡ m f (cid:48) /M H and λ ff (cid:48) = (1 − κ − κ (cid:48) ) − κκ (cid:48) . The colour factor N ff (cid:48) C = 3 for quarksand 1 for leptons, charginos, and neutralinos. For the decays into neutralinos (cid:101) χ j and (cid:101) χ k , we need tomultiply the factor of 4 / (1 + δ jk ) with δ jk = 1 for an identical Majorana neutralino pair. See Fig. 3 forthe M H dependence of the normalized LO decay width of Γ LO ( H → f ¯ f (cid:48) ) / ( g ff (cid:48) N ff (cid:48) C ) for various choicesof the couplings | g S H ¯ ff (cid:48) | and | g P H ¯ ff (cid:48) | and the fermion masses with m f = m f (cid:48) . When m f (cid:54) = m f (cid:48) , we observe For example, when M H = 1000 GeV, we find that Γ( H → t ∗ ¯ t ∗ ) / Γ( H → t ¯ t ) takes the values of 1 .
07 and 1 .
10 for( g SH ¯ tt , g PH ¯ tt ) = (1 ,
0) and ( g SH ¯ tt , g PH ¯ tt ) = (0 , g Hff'S |= | g Hff'P |= | g Hff'S |=| g Hff'P |= / | g Hff'S |= | g Hff'P |=
200 400 600 800 1000010203040 M H [ GeV ] Γ L O / ( g ff ' · N C ) [ G e V ] Figure 3: Γ LO ( H → f ¯ f (cid:48) ) / ( g ff (cid:48) N ff (cid:48) C ) as functions of M H for the three choices of ( | g S H ¯ ff (cid:48) | , | g P H ¯ ff (cid:48) | ) = (1 , | g S H ¯ ff (cid:48) | , | g P H ¯ ff (cid:48) | ) = (1 / √ , / √
2) (blue dashed), and ( | g S H ¯ ff (cid:48) | , | g P H ¯ ff (cid:48) | ) = (0 ,
1) (red dotted).We have taken m f = m f (cid:48) = 0 , ,
300 GeV from left to right. Note that there is no dependence onthe choice of the couplings when m f = m f (cid:48) = 0 as far as | g S H ¯ ff (cid:48) | + | g P H ¯ ff (cid:48) | remains the same.that the LO decay width locates between the pseudoscalar (red dotted) and scalar (black solid) casesif the position of the mass threshold m f + m f (cid:48) is the same. H → V V with V = Z, W
Taking the full consideration of double off-shell effects, the Higgs decay width into two massive vectorbosons is given by [106, 107]Γ LO ( H → V ∗ V ∗ ) = 1 π (cid:90) ω V (cid:15) V d y ( y − + (cid:15) V (cid:90) ( √ ω V −√ y ) (cid:15) V d x ( x − + (cid:15) V × δ V g HV V G F M H √ π λ / (cid:18) , xω V , yω V (cid:19) (cid:20) λ (cid:18) , xω V , yω V (cid:19) + 12 xyω V (cid:21) , (109)where δ W = 2, δ Z = 1, ω V = 1 /κ V = M H /M V , (cid:15) V = Γ V /M V , and λ ( a, b, c ) = ( a − b − c ) − bc . When M V < M H < M V as in the case of the 125 GeV Higgs boson, the off-shell effects of one of the twovector bosons are negligible and the decay width readsΓ LO ( H → V V ∗ ) = δ V V ∗ π (cid:90) ( √ ω V − (cid:15) V d x ( x − + (cid:15) V × δ V g HV V G F M H √ π ω V λ / ( ω V , x,
1) [ λ ( ω V , x,
1) + 12 x ] , (110)with δ V V ∗ = 2. See Fig. 4 for comparisons of Γ LO ( H → V ∗ V ∗ ) and Γ LO ( H → V V ∗ ). Incidentally, onemay neglect all the off-shell effects for a heavy Higgs boson with M H > M V and the decay width takes27
00 110 120 130 140 1500.050.100.5015 M H [ GeV ] Γ L O ( H S M ⟶ WW ) [ M e V ]
100 110 120 130 140 1500.0010.0100.1001 M H [ GeV ] Γ L O ( H S M ⟶ ZZ ) [ M e V ] Figure 4: Comparisons of Γ LO ( H → V ∗ V ∗ ) (solid) and Γ LO ( H → V V ∗ ) (dashed). The vertical linescorrespond to M H = 125 . g HV V = 1 is taken. V = ZV = W
200 300 400 500 600 70002468101214 M H [ GeV ] δ e l w V [ % ] Figure 5: The electroweak correction factors δ V = Z,W elw obtained from the complete electroweak correc-tions of O ( α ) to the decay processes of H → ZZ → e − e + µ − µ + (solid) and H → W W → ν e e + µ − ¯ ν µ (dashed). Obtained by combining the results presented in Figs. 7 and 8 of Ref. [60].a simple form: Γ LO ( H → V V ) = δ V g HV V G F M H √ π β V (cid:2) − κ V + 12 κ V (cid:3) , (111)where β V = √ − κ V with κ V = M V /M H . 28 OLO + ELW
100 120 140 160 180 20010 - - - M H [ GeV ] Γ ( H → WW ) [ G e V ] LOLO + ELW
200 400 600 800 1000050100150200250300 M H [ GeV ] Γ ( H → WW ) [ G e V ]
200 250 300 350 400 450 500010203040
LOLO + ELW
100 120 140 160 180 20010 - - - M H [ GeV ] Γ ( H → ZZ ) [ G e V ] LOLO + ELW
200 400 600 800 1000050100150 M H [ GeV ] Γ ( H → ZZ ) [ G e V ]
200 250 300 350 400 450 50005101520
Figure 6: Decay widths of a neutral Higgs boson with mass M H into W ∗ W ∗ (upper) and into Z ∗ Z ∗ (lower) or Γ( H → V V → f ) taking g HV V = 1. In the left panels, the vertical lines locate the vector-boson-pair thresholds and the (magenta) dash-dotted lines are for the corresponding 2-body decaywidths of Γ( H → V V ) = Γ LO ( H → V V )(1 + δ V elw ).Beyond the leading order including radiative corrections, we estimate the radiatively–corrected decaywidth into two vector bosons by introducing a correction factor asΓ( H → V V → f ) = Γ LO ( H → V ∗ V ∗ ) (cid:0) δ V elw (cid:1) . (112)For the electroweak correction factors δ V = Z,W elw , we adopt the complete electroweak corrections of O ( α )to the Higgs decays into four fermions through intermediate W and Z bosons, supplemented by thecorrections originating from heavy-Higgs effects and final-state radiation [60]. We note that theseelectroweak corrections are applicable even near and below the gauge-boson-pair thresholds where thenarrow-width approximation (NWA) is not applicable. The corrections amount to about 3% or less for M H = 125 GeV, as can be checked in Fig. 5.It has been estimated that missing corrections beyond O ( α ) make the theoretical calculations forthe inclusive decay rates into four fermions uncertain by the amount of 0.5% [108, 109].In Fig. 6, we show the decay widths of a neutral Higgs boson with mass M H into W ∗ W ∗ (upper) andinto Z ∗ Z ∗ (lower). Note that we include the electroweak corrections directly read off from Fig. 5 in therange of M H between 120 GeV and 700 GeV. Since the electroweak corrections grow as M H increasesand become unphysical beyond M H ∼
600 GeV, we show only the LO widths for M H beyond 700 GeV.29 .3 Decays into a lighter scalar boson and a vector boson and into twolighter scalar bosons: H → ϕV , ϕϕ In the presence of multiple Higgs bosons, the decay of a heavier Higgs boson H into a lighter neutralHiggs boson ϕ and a massive gauge boson Z may occur and, considering the case of a virtual Z ∗ , itsdecay width is given by an integral form asΓ LO ( H → ϕZ ∗ ) = G F M H g HϕZ √ π (cid:90) ( √ ω −√ ω ϕ ) d x (cid:15) Z λ / ( ω, ω ϕ , x ) ω π [( x − + (cid:15) Z ] , (113)with ω = M H /M Z and ω ϕ = M ϕ /M Z . When M H is larger than M ϕ + M Z , using the Z -boson narrow-width approximation, it reduces toΓ LO ( H → ϕZ ) = G F M H √ π g HϕZ λ / (1 , κ ϕ , κ Z ) , (114)where κ ϕ = M ϕ /M H and κ Z = M Z /M H . And, a heavier Higgs boson H might also decay into a lightercharged Higgs boson ϕ ± and a massive gauge boson W ∓ with its decay width given by an integral formas Γ LO ( H → ϕ ± W ∓∗ ) = G F M H | g Hϕ + W − | √ π (cid:90) ( √ ω −√ ω ± ) d x (cid:15) W λ / ( ω, ω ± , x ) ω π [( x − + (cid:15) W ] , (115)where Γ LO ( H → ϕ ± W ∓∗ ) = Γ LO ( H → ϕ + W −∗ ) = Γ LO ( H → ϕ − W + ∗ ) with ω = M H /M W and ω ± = M ϕ ± /M W . When M H is larger than M ϕ ± + M W , in the W -boson narrow-width approximation,it reduces to Γ LO ( H → ϕ ± W ∓ ) = G F M H √ π | g Hϕ + W − | λ / (1 , κ ϕ , κ W ) , (116)where κ ϕ = M ϕ ± /M H and κ W = M W /M H .Finally, when a heavier Higgs boson decays into a pair of lighter neutral Higgs bosons of ϕ i and ϕ j or into a pair of sfermions, in the LO, we have for the decay widthΓ LO (cid:16) H → ϕ i ϕ j , (cid:101) f ∗ i (cid:101) f j (cid:17) = N ϕ v |G| πM H λ / (1 , κ i , κ j ) , (117)where ( N ϕ , G ) = (1 + δ ij , g Hϕiϕj ) or ( N fC , g H (cid:101) f ∗ i (cid:101) f j ) and κ i = M ϕ i , (cid:101) f i /M H . The decay width into a pair oflighter charged Higgs bosons is given by taking ( N ϕ , G ) = (1 , g HH + H − ).In Fig. 7, we show the LO decay width of Γ LO ( H → ϕ − W + ∗ ) (upper) which is the same as Γ LO ( H → ϕ + W −∗ ), Γ LO ( H → ϕZ ∗ ) (middle), and Γ LO ( H → ϕ i ϕ j ) (lower). For the mass of a lighter scalar boson ϕ , we take three values of 50 GeV, 200 GeV, and 300 GeV for H → ϕV ∗ for illustration. For the decays H → ϕ i ϕ j , we take M ϕ i = M ϕ j = 50 GeV, 100 GeV, 200 GeV, and 300 GeV. All the relevant couplingsare taken to be 1 in the numerical analyses. H → gg By introducing two form factors, without loss of generality, the amplitude for the decay process H → gg can be written as M abggH = − α s ( M H ) M H δ ab π v (cid:110) S g ( M H ) ( (cid:15) ∗ ⊥ · (cid:15) ∗ ⊥ ) − P g ( M H ) 2 M H (cid:104) (cid:15) ∗ (cid:15) ∗ k k (cid:105) (cid:111) , (118)where a and b ( a, b = 1 to 8) are indices of the eight generators in the SU(3) adjoint representation, k , the four momenta of the two gluons and (cid:15) , the wave vectors of the corresponding gluons, (cid:15) µ ⊥ =30
100 200 300 400 50010 - M H [ GeV ] Γ L O ( H ⟶ φ ± W ∓ ) [ G e V ]
200 400 600 800 1000050100150200250300 M H [ GeV ] Γ L O ( H ⟶ φ ± W ∓ ) [ G e V ]
50 200 300
100 200 300 400 50010 - - M H [ GeV ] Γ L O ( H ⟶ φ Z ) [ G e V ]
200 400 600 800 1000050100150200250300 M H [ GeV ] Γ L O ( H ⟶ φ Z ) [ G e V ]
200 400 600 800 1000051015 M H [ GeV ] Γ L O ( H ⟶ φ i φ j ) [ G e V ] Figure 7: The LO decay widths of a neutral Higgs boson with mass M H into ϕ − W + ∗ (upper), ϕZ ∗ (middle), and ϕ i ϕ j (lower) taking | g Hϕ + W − | = 1, g HϕZ = 1, and N ϕ |G| = 1, respectively. In theupper and middle panels, the (magenta) dash-dotted lines are for the corresponding 2-body decaywidths and we are taking M ϕ / GeV = 50 ,
200 , and 300 from left to right. While in the lower panel, M ϕ / GeV = 50 , ,
200 , and 300 are taken. (cid:15) µ − k µ ( k · (cid:15) ) /M H , (cid:15) µ ⊥ = (cid:15) µ − k µ ( k · (cid:15) ) /M H and (cid:104) (cid:15) (cid:15) k k (cid:105) ≡ (cid:15) µνρσ (cid:15) µ (cid:15) ν k ρ k σ . Retaining only thedominant contributions from third–generation quarks and introducing ∆ S g and ∆ P g to parameterizecontributions from the triangle loops in which non-SM coloured particles are running, the scalar andpseudoscalar form factors are given by S g ( M H ) = (cid:88) f = b,t g SH ¯ ff F sf ( τ f ) + ∆ S g ; P g ( M H ) = (cid:88) f = b,t g PH ¯ ff F pf ( τ f ) + ∆ P g , (119) See Appendix B for ∆ S g and ∆ P g in the MSSM. sf ( t ) t F pf ( t ) t F ( t ) t F ( t ) t -1-0.500.511.522.53 0 2 4 6 8 10 -1-0.500.511.522.53 0 2 4 6 8 1002468101214 0 2 4 6 8 10 -1-0.500.511.522.53 0 2 4 6 8 10 Figure 8: Behaviour of the real (solid) and imaginary (dashed) parts of the four form factors of F sf , F pf , F and F versus τ = s/ m with √ s = M H and m being the mass of the particle running in thetriangle loops. The form factors are defined explicitly in the main text where they first appear, seeEqs. (120), (129), and (B.3). The vertical lines denote the mass threshold above which M H > m .where τ f ≡ M H / M f is defined by using the pole masses of the bottom and top quarks. The formfactors F sf and F pf can be expressed by F sf ( τ ) = τ − [1 + (1 − τ − ) f ( τ )] , F pf ( τ ) = τ − f ( τ ) , (120)in terms of a so-called scaling function f ( τ ) which stands for the integral function f ( τ ) = − (cid:90) d yy ln [1 − τ y (1 − y )] = (cid:40) arcsin ( √ τ ) : τ ≤ , − (cid:104) ln (cid:16) √ τ + √ τ − √ τ −√ τ − (cid:17) − iπ (cid:105) : τ ≥ . (121)It is clear that the imaginary parts of the form factors appear for Higgs-boson masses greater than twicethe mass of the coloured particle running in the loop, i.e., τ ≥
1. In the limit τ → F sf (0) = 2 / F pf (0) = 1, see Fig. 8.The decay width of the process H → gg may be cast into the formΓ( H → gg ) = M H α S π v (cid:104) | S g ( M H ) | (cid:16) δ g : S QCD + δ g : S elw (cid:17) + | P g ( M H ) | (cid:16) δ g : P QCD + δ g : P elw (cid:17)(cid:105) , (122)including the QCD and electroweak corrections. The QCD correction of δ g : S QCD is known up to theNLO including the full quark mass dependence [110] and up to the N LO in the limit of heavy topquarks [111, 112, 113]: δ g : S QCD = (cid:18) − N F + ∆ g : Sm (cid:19) α s ( M H ) π W ZZ tt
100 200 300 400 500 - - M H [ GeV ] δ e l w g : S [ % ] Figure 9: Behaviour of the two-loop electroweak corrections of δ g : S elw for Γ( H → gg ) versus M H . Thevertical lines locate M H = 125 . W W , ZZ , and t ¯ t thresholds. From Ref. [122].+ (cid:20) . − . N F + 0 . N F + (2 .
375 + 0 . N F ) log M H M t (cid:21) (cid:18) α s ( M H ) π (cid:19) + (cid:20) . − . N F + 52 . N F − . N F +(66 .
66 + 14 . N F − . N F ) log M H M t +(6 .
53 + 1 . N F − . N F ) log M H M t (cid:21) (cid:18) α s ( M H ) π (cid:19) , (123)with N F counting the flavour number of quarks lighter than H and ∆ g : Sm ≈ .
024 for the NLO quark-mass effects from the top, bottom and charm quarks [110]. Taking M H = 125 GeV, we find δ g : S QCD =0 .
64 + 0 .
20 + 0 .
02 for the NLO, NNLO, and N LO corrections. . On the other hands, the correction δ g : P QCD is known up to the NLO including the full quark mass dependence [110] and up to the NNLO inthe limit of heavy top quarks [115]: δ g : P QCD = (cid:18) − N F + ∆ g : Pm (cid:19) α s ( M H ) π + (cid:18) . − . N F + 0 . N F + N F log M H M t (cid:19) (cid:18) α s ( M H ) π (cid:19) , (124)with ∆ g : Pm ≈ .
04. Taking M H = 125 GeV, we find δ g : P QCD (cid:39) .
66 + 0 .
22 for the NLO and NNLO terms.The electroweak corrections δ g : S elw of O ( G F M t ) are given by [103, 116, 117] δ g : S elw = G F M t √ π . (125) Our numerical estimation of 0 .
64 for the NLO QCD correction is smaller than that presented in Ref. [57] by theamount of about 0 .
03 which might be responsible for our smaller estimations for the decay widths into two gluons andquarks by the amount of about 1% than those given in the literature such as Ref. [114], see Table 5 Hbb S / g Htt S = LOLO + ELWLO + ELW + QCD
200 400 600 800 1000020406080 M H [ GeV ] Γ ( H ⟶ gg ) / ( g H tt S ) [ M e V ] g Hbb S / g Htt S = LOLO + ELWLO + ELW + QCD
100 120 140 160 180 200 220 2400.00.51.01.52.02.53.0 M H [ GeV ] Γ ( H ⟶ gg ) / ( g H tt S ) [ M e V ] g HbbS / g HttS = HbbS / g HttS = HbbS / g HttS =
200 400 600 800 1000020406080 M H [ GeV ] Γ ( H ⟶ gg ) / ( g H tt S ) [ M e V ] g HbbS / g HttS = HbbS / g HttS = HbbS / g HttS =
100 120 140 160 180 200 220 2400.00.51.01.52.02.53.0 M H [ GeV ] Γ ( H ⟶ gg ) / ( g H tt S ) [ M e V ] Figure 10: (Upper) Normalized decay widths of a neutral Higgs boson with mass M H into gg taking g SH ¯ bb /g SH ¯ tt = 1, ∆ S g = 0, and g PH ¯ qq = ∆ P g = 0. In the right panel, the low M H region is magnified. TheLO (blue dashed), LO+ELW (lower red solid), and LO+ELW+QCD (upper black solid) results areseparately shown. (Lower) The same as in the upper panels but for three values of g SH ¯ bb /g SH ¯ tt = 1 ,
10 ,and 20 taking full account of the electroweak and QCD corrections. The vertical lines in the right panelslocate the position M H = M t .Note that the above O ( G F M t ) electroweak corrections increase the gluonic decay width only by theamount of about 0 . δ g : S elw ∼
5% below the
W W threshold, see Fig. 9. On the otherhand, the other electroweak corrections δ g : P elw may be given by [123, 124] δ g : P elw (cid:39) − (cid:16) η g : P elw (cid:17) G F M t √ π , (126)where the first factor of 7 counts the contribution of the SU(2) L doublet which, in the decoupling limit,plays the role of the SM SU(2) L doublet including the 125 GeV Higgs boson. And the second factorof η g : P elw denotes the model-dependent BSM contribution. In the type-II 2HDM, for example, it is givenby η g : P elw = 10 / tan β in the infinite top quark mass approximation [124]. These electroweak correctionsreduce the gluonic decay width at the percent level.We have addressed all the known QCD and electroweak corrections to the decay width of a neutralHiggs boson into two gluons. The theoretical uncertainties due to the unknown higher-order QCD andNLO electroweak corrections are estimated as 3% and 1%, respectively [108, 114].In the upper panels of Fig. 10, we show the normalized decay widths of Γ( H → gg ) / ( g SH ¯ tt ) at the LO(blue dashed), including only the electroweak corrections (red solid), and including both the electroweak34 Hbb P / g Htt P = LOLO + ELWLO + ELW + QCD
200 400 600 800 1000020406080100 M H [ GeV ] Γ ( H ⟶ gg ) / ( g H tt P ) [ M e V ] g Hbb P / g Htt P = LOLO + ELWLO + ELW + QCD
100 120 140 160 180 200 220 24001234567 M H [ GeV ] Γ ( H ⟶ gg ) / ( g H tt P ) [ M e V ] g HbbP / g HttP = HbbP / g HttP = HbbP / g HttP =
200 400 600 800 1000020406080100 M H [ GeV ] Γ ( H ⟶ gg ) / ( g H tt P ) [ M e V ] g HbbP / g HttP = HbbP / g HttP = HbbP / g HttP =
100 120 140 160 180 200 220 24001234567 M H [ GeV ] Γ ( H ⟶ gg ) / ( g H tt P ) [ M e V ] Figure 11: The same as in Fig. 10 while taking g SH ¯ qq = 0 and g PH ¯ qq (cid:54) = 0 with ∆ S g = ∆ P g = 0.and QCD corrections (black solid). We assume all the pseudo-scalar couplings of g PH ¯ qq are vanishing and∆ S g = 0. The electroweak corrections δ g : S elw are directly read off from Fig. 9 in the range of M H between100 GeV and 500 GeV. For M H >
500 GeV, we simply neglect the electroweak corrections, expectingthe corrections for heavier masses to be evaluated with reliable precision in the near future if necessary.In the right panel, we magnify the low M H region and locate the position M H = M t with a thin verticalline where, as M H grows, N F changes from 5 to 6, causing a discontinuity in the QCD corrections. Inthe lower panels, we show the normalized decay widths including the available electroweak and QCDcorrections and taking three values of g SH ¯ bb /g SH ¯ tt = 1 (black solid), 10 (blue dashed), and 20 (red dotted),considering the situation in which the bottom Yukawa coupling is enhanced as in the 2HDM II for largetan β . Note that the black solid lines in the lower panels are the same as those in the upper ones.In Fig. 11, the alternative choice is made to show Γ( H → gg ) / ( g PH ¯ tt ) assuming all the scalarcouplings of g SH ¯ qq are vanishing and ∆ P g = 0. For the electroweak corrections δ g : P elw , we take η g : P elw = 0.Compared to the scalar case in which the form factor f sf ( τ ) is involved, the rise near the top-quark-pairthreshold is sharper and bigger due to the behaviour of the real and imaginary parts of the form factor f pf ( τ ) around τ = 1, see Fig. 8. In the low M H region, we find that the decay widths are larger by aboutthe factor of [ f pf (0) /f sf (0)] ∼ b -quark contributions are neglected. As the coupling ratios g S,PH ¯ bb /g S,PH ¯ tt increase, the contributions from b -quark loops become comparable to and larger than thosefrom t -quark loops and, in this case, the decay widths are nearly the same especially around M H = 100GeV as can be checked with the dashed and dotted lines in the lower-right panels of Figs. 10 and 11.35 .5 Decays into two photons: H → γγ The amplitude for the radiative decay process H → γγ , playing a crucial role in the discovery of theHiggs boson at the LHC, can be written as M γγH = − α (0) M H π v (cid:110) S γ ( M H ) ( (cid:15) ∗ ⊥ · (cid:15) ∗ ⊥ ) − P γ ( M H ) 2 M H (cid:104) (cid:15) ∗ (cid:15) ∗ k k (cid:105) (cid:111) , (127)in terms of the two form factors of S γ and P γ . Here k , and (cid:15) , are the four–momenta and wavevectors of the two photons, respectively, as in the decay H → gg . Note that the electromagnetic finestructure constant in the coupling should be taken at the scale q = 0 since the final–state photons arereal. Retaining only the dominant contributions from third–generation fermions and the charged gaugebosons W ± and introducing two residual form factors ∆ S γ and ∆ P γ to parameterize contributionsfrom the triangle loops in which non-SM charged particles are running, the scalar and pseudoscalarform factors are given by S γ ( M H ) = 2 (cid:88) f = b,t,τ N fC Q f g SH ¯ ff F sf ( τ f ) − g HWW F ( τ W ) + ∆ S γ ; P γ ( M H ) = 2 (cid:88) f = b,t,τ N fC Q f g PH ¯ ff F pf ( τ f ) + ∆ P γ , (128)where N fC = 3 for quarks and N fC = 1 for charged leptons, respectively. For the τ lepton and the W boson, τ τ = M H / M τ and τ W = M H / M W , respectively. On the other hand, for quarks, τ q isdefined in terms of the running quark mass at the scale of M H /
2, i.e. τ q = M H / m q ( M H / where m q is normalized as m q ( M q ) = M q . Note that the choice of the scale µ q = M H / m q ( M H /
2) = M q at the threshold where M H = 2 M q and makes the full two-loop QCD correctionsremain small in the entire range of the variable τ q by effectively absorbing all relevant large logarithmsinto the running mass. The form factor F is given by F ( τ ) = 2 + 3 τ − + 3 τ − (2 − τ − ) f ( τ ) , (129)which takes the value of 7 in the limit τ →
0, see Fig. 8 for the τ dependence of the form factor. In theLO, the decay width of the radiative process is given byΓ LO ( H → γγ ) = M H α π v (cid:2) | S γ ( M H ) | + | P γ ( M H ) | (cid:3) , (130)with the fine structure constant α = α (0) (cid:39) / b - and t -quarkcontributions as: F sf ( τ q ) −→ F sf ( τ q ) (cid:20) C sf ( τ q ) α s ( M H ) π (cid:21) ; F pf ( τ q ) −→ F pf ( τ q ) (cid:20) C pf ( τ q ) α s ( M H ) π (cid:21) . (131)The scaling factors C sf and C pf approach − τ →
0, see Fig. 12. See Appendix B for ∆ S γ and ∆ P γ in the MSSM. For m q ( µ ), see Appendix A. For a detailed description of the scaling factors of C sf ( τ ) and C pf ( τ ), see Appendix C. e ( C sf ) Im ( C sf ) - τ C s f Re ( C pf ) Im ( C pf ) - τ C p f Figure 12: Behaviour of the real (solid) and imaginary (dashed) parts of the scaling factors C sf ( τ )(left) and C pf ( τ ) (right) for the QCD corrections to the decay of a Higgs boson into two photons.Figure 13: The two-loop electroweak corrections δ γ : S elw (solid line) to the part of the decay width of aHiggs boson into two photons via the scalar form factor. Also shown are the QCD corrections (dottedline) and the sum of the two types of corrections (dash-dotted line). From Ref. [140].The two-loop electroweak corrections of δ γ : S elw to the scalar part of the decay width via the scalarform factor have been calculated in Refs. [122, 138, 139, 140], see Fig. 13. The electroweak correctionsto the pseudoscalar part is also available: δ γ : P elw = − G F M t √ π (cid:16) η γ : P elw (cid:17) , (132)where, similarly as in the decay H → gg , the factor 4 counts the contribution of the SM SU(2) L doubletin the decoupling limit and η γ : P elw the model-dependent BSM contribution. In the type-II 2HDM, forexample, η γ : P elw = 7 / tan β in the infinite top quark mass approximation [123, 124], suppressed for largetan β . 37 QCD γ : P δ QCD γ : S
200 400 600 800 1000 - H [ GeV ] δ Q CD γ [ % ] Figure 14: Behaviour of the two-loop QCD corrections to the decay width of a Higgs boson into twophotons: δ γ : S QCD (solid) and δ γ : P QCD (dashed). We take g HWW = g SH ¯ tt = g SH ¯ bb = g SHττ = 1 for δ γ : S QCD while g PH ¯ bb = g PHττ = 0 for δ γ : P QCD . Both of the residual form factors ∆ S γ and ∆ P γ are taken to be vanishing.Incorporating all the QCD and electroweak corrections, one may writeΓ( H → γγ ) = M H α π v (cid:104) | S γ ( M H ) | (cid:16) δ γ : S QCD + δ γ : S elw (cid:17) + | P γ ( M H ) | (cid:16) δ γ : P QCD + δ γ : P elw (cid:17)(cid:105) . (133)Note that the electroweak corrections are directly from Fig. 13 and Eq. (132) while the QCD correctionsenter through the scaling factors C sf ( τ q ) and C pf ( τ q ). For M H = 125 . δ γ : S QCD increase the decay width into two photons by about 2%. On the other hand, the electroweakcorrections δ γ : S elw decrease the decay width by about 2% , almost canceling the NLO QCD correctionsto the corresponding part, see Fig. 13. In Fig. 14, we show the QCD corrections δ γ : S QCD (solid) and δ γ : P QCD (dashed) with varying M H . For the scalar QCD correction δ γ : S QCD , we take the SM values of g HWW = g SH ¯ tt = g SH ¯ bb = g SHττ = 1 with ∆ S γ = 0. While, for the pseudoscalar QCD correction δ γ : P QCD , weassume a scenario in which the pseudoscalar form factor P γ is dominated by the top-quark contributiontaking g PH ¯ bb = g PHττ = ∆ P γ = 0. At M H = 2 M t , the pseudoscalar QCD correction δ γ : P QCD diverges dueto the singular property of C pf at τ = 1. Around M H = 600 GeV where the large cancellation occursbetween the W -boson and top-quark contributions, the scalar QCD correction δ γ : S QCD is relatively largeand it could vary between about − . . g H ¯ tt coupling squared taking g HWW /g SH ¯ tt = g SH ¯ bb ,Hττ /g SH ¯ tt = 1 and g PH ¯ ff = ∆ S γ = ∆ P γ = 0. This reduces tothe SM decay width of a Higgs particle with mass M H when g H ¯ tt = 1. For M H ≤
170 GeV, we apply theelectroweak corrections δ γ : S elw directly read off from Fig. 13 and we simply neglect them above M H = 170GeV, expecting the electroweak corrections to be evaluated with high precision for heavy Higgs bosonsif necessary. Below the W -boson-pair threshold, the W -loop contributions become dominant, leadingto the sharp rise as M H approaches 2 M W . Passing M H = 2 M W from below, the real part of the In the general case with arbitrary g HWW and g S,PH ¯ ff couplings, the scaling factors C sf ( τ q ) and C pf ( τ q ) should be takeninto account at the amplitude level to incorporate the corresponding QCD corrections properly. Hbb ( H ττ ) S / g HttS = LOLO + ELWLO + ELW + QCD g HWW / g HttS =
200 400 600 800 10000.000.050.100.150.200.250.30 M H [ GeV ] Γ ( H ⟶ γγ ) / ( g H tt S ) [ M e V ] g Hbb ( H ττ ) S / g HttS = LOLO + ELWLO + ELW + QCD g HWW / g HttS =
100 120 140 160 180 2000.000.020.040.060.080.10 M H [ GeV ] Γ ( H ⟶ γγ ) / ( g H tt S ) [ M e V ] g Hbb ( H ττ ) S / g HttS = Hbb ( H ττ ) S / g HttS = Hbb ( H ττ ) S / g HttS = Hbb ( H ττ ) S / g HttS =
50 g
HWW / g HttS =
200 400 600 800 10000.000.050.100.150.200.250.30 M H [ GeV ] Γ ( H ⟶ γγ ) / ( g H tt S ) [ M e V ] g Hbb ( H ττ ) S / g HttS = Hbb ( H ττ ) S / g HttS = Hbb ( H ττ ) S / g HttS = Hbb ( H ττ ) S / g HttS =
50 g
HWW / g HttS =
100 120 140 160 180 2000.000.020.040.060.080.10 M H [ GeV ] Γ ( H ⟶ γγ ) / ( g H tt S ) [ M e V ] g Hbb ( H ττ ) S / g HttS = Hbb ( H ττ ) S / g HttS = Hbb ( H ττ ) S / g HttS = Hbb ( H ττ ) S / g HttS =
50 g
HWW =
200 400 600 800 10000.000.050.100.150.200.250.30 M H [ GeV ] Γ ( H ⟶ γγ ) / ( g H tt S ) [ M e V ] g Hbb ( H ττ ) S / g HttS = Hbb ( H ττ ) S / g HttS = Hbb ( H ττ ) S / g HttS = Hbb ( H ττ ) S / g HttS =
50 g
HWW =
100 120 140 160 180 2000.00000.00050.00100.00150.00200.00250.00300.0035 M H [ GeV ] Γ ( H ⟶ γγ ) / ( g H tt S ) [ M e V ] Figure 15: (Upper) Normalized decay widths of a neutral Higgs boson with a mass M H into γγ taking g HWW /g SH ¯ tt = g SH ¯ bb ,Hττ /g SH ¯ tt = 1 and g PH ¯ ff = 0. In the right panel, the low M H region is magnified.The LO (blue dashed), LO+ELW (red solid), and LO+ELW+QCD (black solid) results are separatelyshown. (Middle) The same as in the upper panels but for the four values of g SH ¯ bb /g SH ¯ tt = 1 , ,
20 ,and 50 taking full account of the electroweak and QCD corrections. g HWW /g SH ¯ tt = 1 is taken. (Lower)The same as in the middle panels but with g HWW = 0. The vertical lines in the right panels locate theposition M H = 170 GeV beyond which the electroweak corrections are ignored. In all panels, we take∆ S γ = ∆ P γ = 0. W -loop contributions decreases but its imaginary part starts to be developed. As a result, the Higgsdecay width continues to increase with increasing M H until the Higgs mass meets the top-quark-pairthreshold, M H = 2 M t . Passing the top-quark-pair threshold, the newly-developed imaginary part ofthe t -quark loop contributions starts to cancel that of the W -loop contributions. We note that, beyond M H = 2 M t , the cancellation between the W -loop and t -quark loop contributions broadly occurs and it39 Hbb ( H ττ ) P / g HttP = LOLO + ELWLO + ELW + QCD
200 400 600 800 10000.000.050.100.150.200.250.300.35 M H [ GeV ] Γ ( H ⟶ γγ ) / ( g H tt P ) [ M e V ] g Hbb ( H ττ ) P / g HttP = LOLO + ELWLO + ELW + QCD
100 120 140 160 180 2000.0000.0020.0040.0060.0080.010 M H [ GeV ] Γ ( H ⟶ γγ ) / ( g H tt P ) [ M e V ] g Hbb ( H ττ ) P / g HttP = Hbb ( H ττ ) P / g HttP = Hbb ( H ττ ) P / g HttP = Hbb ( H ττ ) P / g HttP =
200 400 600 800 10000.000.050.100.150.200.250.300.35 M H [ GeV ] Γ ( H ⟶ γγ ) / ( g H tt P ) [ M e V ] g Hbb ( H ττ ) P / g HttP = Hbb ( H ττ ) P / g HttP = Hbb ( H ττ ) P / g HttP = Hbb ( H ττ ) P / g HttP =
100 120 140 160 180 2000.0000.0020.0040.0060.0080.010 M H [ GeV ] Γ ( H ⟶ γγ ) / ( g H tt P ) [ M e V ] Figure 16: Normalized decay widths of a neutral Higgs boson with a mass M H into γγ via pseudoscalarform factor P γ with ∆ P γ = 0. In this case, only the fermion couplings of g PH ¯ ff with f = t, b, τ arerelevant.leads to a dip around M H = 600 GeV. Specifically we find S γ ( M H = 600 GeV) (cid:39) ( − . − . i ) g HWW + (1 .
404 + 2 . i ) g SH ¯ tt , (134)with negligible contributions from the b -quark and τ -lepton loops. In the middle panels of Fig. 15, weshow the variation depending on g SH ¯ bb /g SH ¯ tt = g SHττ /g SH ¯ tt still taking g HWW /g SH ¯ tt = 1. The (black) solidlines represent the same case as in the upper panels taking full account of the QCD and electroweakcorrections. And, in the lower panels, we show the results taking g HWW = 0. The last case may applyto the heavy neutral Higgs bosons appearing in the 2HDMs and/or MSSM when their couplings to themassive vector bosons are naturally suppressed and almost vanishing [141, 142].In Fig. 16, the alternative choice is made to show the normalized decay width Γ( H → γγ ) / ( g PH ¯ tt ) assuming all the scalar couplings of g SH ¯ ff are vanishing and, again, taking ∆ S γ = ∆ P γ = 0. Notethat, in this case, only the fermion loops are contributing. For the electroweak corrections δ γ : P elw , wetake η γ : P elw = 0. At M H = 2 M t , the decay width diverges because of the singular property of the QCDcorrections. In the lower panels, we show the dependence on g PH ¯ bb /g PH ¯ tt = g PHττ /g PH ¯ tt for the four valuesof 1 , ,
20, and 50 taking full account of the electroweak and QCD corrections. In the right panels, asthe same as in Fig. 15, we magnify the low M H regions.40 t = M Z2 t2 τ t G s f ( τ t , λ t ) λ t = M Z2 t2 τ t G p f ( τ τ , λ τ ) λ W = M Z2 W2 τ W G ( τ W , λ W ) Figure 17: Behaviour of the real (solid) and imaginary (dashed) parts of the form factors G sf ( τ t , λ t )and G pf ( τ t , λ t ) (upper) and that of G ( τ W , λ W ) (lower). We recall the relations τ x = M H / M x and λ x = M Z / M x . The vertical lines at τ x = 1 denote the mass threshold above which M H > M x . Z and a photon: H → Z γ
The amplitude for the decay process H → Zγ can be written as [63] M ZγH = − α πv (cid:26) S Zγ ( M H , M Z ) [ k · k (cid:15) ∗ · (cid:15) ∗ − k · (cid:15) ∗ k · (cid:15) ∗ ] − P Zγ ( M H , M Z ) (cid:104) (cid:15) ∗ (cid:15) ∗ k k (cid:105) (cid:27) , (135)with the two form factors of S Zγ and P Zγ . Here k , are the momenta of the Z boson and the photon(we note that 2 k · k = M H − M Z ), (cid:15) , are their polarization vectors, and (cid:104) (cid:15) (cid:15) k k (cid:105) ≡ (cid:15) µναβ (cid:15) µ (cid:15) ν k α k β .Retaining only the dominant contributions from third–generation fermions and W ± and introducingtwo residual form factors ∆ S Zγ and ∆ P Zγ to parameterize contributions from the triangle loops inwhich non-SM particles are running, the scalar and pseudoscalar form factors are given by S Zγ ( M H , M Z ) = 2 (cid:88) f = t,b,τ N fC Q f I f − s W Q f s W c W g SH ¯ ff G sf ( τ f , λ f ) − s W g HWW G ( τ W , λ W ) + ∆ S Zγ ,P Zγ ( M H , M Z ) = 2 (cid:88) f = t,b,τ N fC Q f I f − s W Q f s W c W g PH ¯ ff G pf ( τ f , λ f ) + ∆ P Zγ , (136) See Appendix B for the explicit forms of ∆ S Zγ and ∆ P Zγ in the MSSM. I u,ν = +1 / I d,e = − / τ x = M H / m x and λ x = M Z / m x , respectively. The loop functionsare given by: G sf ( τ f , λ f ) = I ( τ f , λ f ) − I ( τ f , λ f ) ; G pf ( τ f , λ f ) = I ( τ f , λ f ) , (137) G ( τ W , λ W ) = c W (cid:26) (cid:2) τ W ( t W −
1) + ( t W − (cid:3) I ( τ W , λ W ) + 4(3 − t W ) I ( τ W , λ W ) (cid:27) , where I , are functions of the two variables of τ and λ and they are expressed as I ( τ, λ ) = 12( λ − τ ) + 12( λ − τ ) [ f ( τ ) − f ( λ )] + λ ( λ − τ ) [ g ( τ ) − g ( λ )] ,I ( τ, λ ) = − λ − τ ) [ f ( τ ) − f ( λ )] , (138)in terms of the f ( τ ) function, defined in Eq. (121), and the function g ( τ ) which is defined as g ( τ ) = (cid:113) τ − √ τ ) : τ ≤ , (cid:113) τ − τ (cid:104) ln (cid:16) √ τ + √ τ − √ τ −√ τ − (cid:17) − iπ (cid:105) : τ ≥ . (139)The explicit τ x dependence of the form factors G sf and G pf for x = t and G for x = W is shown inFig. 17, clearly exhibiting the development of their imaginary parts beyond M H > M x with λ t,W < LO ( H → Zγ ) = α (0) G F M W s W π M H (cid:18) − M Z M H (cid:19) (cid:16)(cid:12)(cid:12) S Zγ ( M H , M Z ) (cid:12)(cid:12) + (cid:12)(cid:12) P Zγ ( M H , M Z ) (cid:12)(cid:12) (cid:17) . (140)A detailed description of the scalar and pseudoscalar form factors in the framework of MSSM taken asa specific BSM model is given in Appendix B.The QCD corrections turn out to be less than 0 .
3% [146, 147, 148]. While the theoretical uncer-tainties of the electroweak corrections have been estimated as ∼
5% [114] which constitutes the largesttheoretical uncertainty involved in the decay widths of the SM Higgs boson.In the upper panels of Fig. 18, we show the LO decay width into a vector boson Z and a photonnormalized to the g H ¯ tt coupling squared taking g HWW /g SH ¯ tt = 1 and g PH ¯ ff = ∆ S Zγ = ∆ P Zγ = 0 for thefour values of g SH ¯ bb ,Hττ /g SH ¯ tt = 1 , ,
20, and 50. This reduces to the SM decay width of a Higgs particleweighing M H when g HWW /g SH ¯ tt = g SH ¯ bb ,Hττ /g SH ¯ tt = 1 together with g H ¯ tt = 1. Below the W -boson-pairthreshold, the W -loop contributions become dominant, leading to the sharp rise as M H approaches2 M W . Passing the W -pair threshold M H = 2 M W from below, the real part of the W -loop contributionscontinues to grow, see the lower panel of Fig. 17, leading to another sharp rise. Passing M H = 2 M t , the t -quark loop contributions starts to cancel that of the W -loop contributions as shown in the upper-leftpanel of Fig. 18. In the lower panels, we show the results taking g HWW = 0 for the four values of g SH ¯ bb ,Hττ /g SH ¯ tt = 1 , , ,
50. This case may apply to the heavy neutral Higgs bosons appearing in the2HDMs and/or MSSM when their couplings to the massive vector bosons are suppressed and they arealmost vanishing [141, 142].In Fig. 19, the alternative pseudoscalar choice is made to show Γ( H → Zγ ) / ( g PH ¯ tt ) taking the fourvalues of g PH ¯ bb /g PH ¯ tt = g PHττ /g PH ¯ tt = 1 , , ,
50 with ∆ S Zγ = ∆ P Zγ = 0. Here we assume all the scalar We take the pole masses of top and bottom quarks for τ t,b and λ t,b . Hbb ( H ττ ) S / g HttS = Hbb ( H ττ ) S / g HttS = Hbb ( H ττ ) S / g HttS = Hbb ( H ττ ) S / g HttS = HWW / g HttS =
200 400 600 800 10000.00.20.40.60.81.0 M H [ GeV ] Γ L O ( H ⟶ Z γ ) / ( g H tt S ) [ M e V ] g Hbb ( H ττ ) S / g HttS = Hbb ( H ττ ) S / g HttS = Hbb ( H ττ ) S / g HttS = Hbb ( H ττ ) S / g HttS =
50 g
HWW / g HttS =
100 120 140 1600.00.10.20.30.4 M H [ GeV ] Γ L O ( H ⟶ Z γ ) / ( g H tt S ) [ M e V ] g Hbb ( H ττ ) S / g HttS = Hbb ( H ττ ) S / g HttS = Hbb ( H ττ ) S / g HttS = Hbb ( H ττ ) S / g HttS =
50 g
HWW =
200 400 600 800 10000.000.020.040.060.080.10 M H [ GeV ] Γ L O ( H ⟶ Z γ ) / ( g H tt S ) [ M e V ] g Hbb ( H ττ ) S / g HttS = Hbb ( H ττ ) S / g HttS = Hbb ( H ττ ) S / g HttS = Hbb ( H ττ ) S / g HttS =
50 g
HWW =
100 150 200 250 300 3500.0000.0050.0100.015 M H [ GeV ] Γ L O ( H ⟶ Z γ ) / ( g H tt S ) [ M e V ] Figure 18: (Upper) Normalized decay widths of a neutral Higgs boson with a mass M H into Zγ at theLO taking g HWW /g SH ¯ tt = 1 and g PH ¯ ff = ∆ S Zγ = ∆ P Zγ = 0 for four values of g SH ¯ bb ,Hττ /g SH ¯ tt = 1 , , M H region is magnified. (Lower) The same as in the upper panelsbut taking g HWW = 0. g Hbb ( H ττ ) P / g HttP = Hbb ( H ττ ) P / g HttP = Hbb ( H ττ ) P / g HttP = Hbb ( H ττ ) P / g HttP =
200 400 600 800 10000.000.020.040.060.080.10 M H [ GeV ] Γ L O ( H ⟶ Z γ ) / ( g H tt P ) [ M e V ] g Hbb ( H ττ ) P / g HttP = Hbb ( H ττ ) P / g HttP = Hbb ( H ττ ) P / g HttP = Hbb ( H ττ ) P / g HttP =
100 150 200 250 300 3500.000.010.020.030.040.05 M H [ GeV ] Γ L O ( H ⟶ Z γ ) / ( g H tt P ) [ M e V ] Figure 19: Normalized decay widths of a neutral Higgs boson with a mass M H into Zγ via pseudoscalarform factor P Zγ at the LO taking ∆ P Zγ = 0. In this case, only the fermion couplings of g PH ¯ ff with f = t, b, τ are relevant.couplings of g SH ¯ ff are vanishing. Note that, in this case, only the fermion loops are contributing as thepseudoscalar state does not couple to gauge bosons at the tree level. In the right panel, as the same asin Fig. 18, we magnify the low M H regions. Compared to the scalar case shown in Fig. 18, it is clearly43able 4: The leading M H dependence and the ballpark values of normalized decay widths of the neutralHiggs boson H far above the mass threshold of its decay products. Specifically, we have taken M H = 1TeV. For the radiative loop-induced decays, the scalar (upper) and pseudoscalar (lower) contributionsare shown separately. For the scalar contributions to H → γγ and H → Zγ which are dominatedby t -quark and W -boson loops, we assume g SH ¯ tt = g HWW = 1. See Table 11 for the MS quark masses m q ( M H ) while we refer to Eq. (A.5) for m q ( µ = M H / M H dependence Γ nomalized M H =1TeV [GeV] Reference Figure H → f ¯ f M H (cid:104) m q ( M H ) m t ( M H ) (cid:105) Figs. 1 and 2 (cid:104) M τ ,µ m t ( M H ) (cid:105) H → W W ( ZZ ) M H
300 (150) Fig. 6 H → ϕV M H
300 Fig. 7 H → ϕϕ /M H H → gg M H (cid:12)(cid:12)(cid:12) F sf (cid:16) M H M t (cid:17)(cid:12)(cid:12)(cid:12) × − Fig. 10 M H (cid:12)(cid:12)(cid:12) F pf (cid:16) M H M t (cid:17)(cid:12)(cid:12)(cid:12) × − Fig. 11 H → γγ M H (cid:12)(cid:12)(cid:12) F sf (cid:16) M H [4 m t ( M H / (cid:17) − F (cid:16) M H M W (cid:17)(cid:12)(cid:12)(cid:12) × − Fig. 15 M H (cid:12)(cid:12)(cid:12) F pf (cid:16) M H [4 m t ( M H / (cid:17)(cid:12)(cid:12)(cid:12) × − Fig. 16 H → Zγ M H (cid:12)(cid:12)(cid:12) . G sf (cid:16) M H M t , M Z M t (cid:17) − G (cid:16) M H M W , M Z M W (cid:17)(cid:12)(cid:12)(cid:12) × − Fig. 18 M H (cid:12)(cid:12)(cid:12) G pf (cid:16) M H M t , M Z M t (cid:17)(cid:12)(cid:12)(cid:12) × − Fig. 19shown in Fig. 19 that the scalar and pseudoscalar decays exhibit quite distinct patterns, in particular,around the t -pair threshold.Before moving to the last subsection for numerical results obtained by analysing the decays of severalneutral Higgs bosons, we provide Table 4 in which the leading M H dependence and the ballpark valuesof normalized decay widths at M H = 1 TeV are shown for all the decay modes elaborated up to thissubsection. Closing this section dedicated to a detailed study of neutral Higgs boson decays, we present the resultsof two numerical analyses of ( i ) the decays of a neutral Higgs boson with its mass fixed to 125 . ii ) the decays of heavy neutral Higgs bosons which are mixturesof CP-even and CP-odd states.In the first numerical analysis, we extend the SM in somewhat model-independent way by allowing forthe pseudoscalar as well as scalar couplings of the 125 . Anatomy of Higgs boson decays with M H = 125 . GeV
In this subsubsection, we present all the details of calculating the decay widths of a neutral Higgs bosonby taking M H = 125 . g SH ¯ ff = g HV V = 1 and g PH ¯ ff = ∆ S g ,γ ,Zγ = ∆ P g ,γ ,Zγ = 0.For the decays of a neutral Higgs boson with M H = 125 . b and c quarks and muons andtau leptons, the relevant radiative corrections are given numerically by δ QCD = +0 .
203 + 0 .
037 + 0 . − .
001 = +0 .
241 ; δ b : St = +0 . , δ c : St = +0 .
019 ; δ b elw = − . , δ c elw = +0 .
004 ; δ τ elw = − . , δ µ elw = − . δ b mixed = − . , δ c mixed = − .
000 ; δ b tot = +0 . , δ c tot = +0 .
264 ; δ τ tot = − . , δ µ tot = − . . (141)Incidentally, we have δ b : Pt = 0 .
018 and δ c : Pt = 0 .
030 which contribute to the pseudoscalar part.For the decays H → W W, ZZ , from Fig. 5, we have two electroweak corrections as δ W elw = 0 . , δ Z elw = 0 . . (142)For the decay H → gg , the scalar and pseudoscalar form factors and the relevant radiative correctionsare given numerically apart from the scalar and pseudoscalar Higgs–fermion–fermion couplings by S g = 0 . g SH ¯ tt + ( − .
043 + 0 . i ) g SH ¯ bb + ∆ S g ; P g = 1 . g PH ¯ tt + ( − .
049 + 0 . i ) g PH ¯ bb + ∆ P g ; δ g : S QCD = 0 . , δ g : S elw = 0 . δ g : P QCD = 0 . , δ g : P elw = − . , (143)where δ g : S elw is from Fig. 9 and we set η g : P elw = 0 for δ g : P elw . In the SM, we have S g = 0 .
646 + 0 . i and P g = 0. Similarly for the decay H → γγ , in terms of the HW W and Higgs–fermion–fermion couplings,we obtain the following scalar and pseudoscalar form factors S γ = − . g HWW + 1 . g SH ¯ tt + ( − .
021 + 0 . i ) g SH ¯ bb + ( − .
023 + 0 . i ) g SHττ + ∆ S γ ; P γ = 2 . g PH ¯ tt + ( − .
023 + 0 . i ) g PH ¯ bb + ( − .
025 + 0 . i ) g PHττ + ∆ P γ ; δ γ : S QCD = 0 . , δ γ : S elw = − .
016 ; δ γ : P QCD = 0 . , δ γ : P elw = − . , (144)45able 5: Partial and total decay widths of the SM Higgs in MeV taking M H = 125 . δ Γ defined by δ Γ ≡ (cid:0) Γ This Review − Γ Ref . [ ] (cid:1) / Γ Ref . [ ] . Also presented are theoretical uncertainties (THUs) of thepartial and total decay widths from missing higher orders estimated around M H = 125 GeV, see Tables178 and 182 in Ref. [114].Γ( H → b ¯ b ) Γ( H → W W ) Γ( H → gg ) Γ( H → τ τ ) Γ( H → c ¯ c )This Review 2 .
370 9 . × − . × − . × − . × − Ref. [114] 2 .
387 9 . × − . × − . × − . × − δ Γ [%] − . − . − . − . − . ± . ± . ± . ± . ± . H → ZZ ) Γ( H → γγ ) Γ( H → Zγ ) Γ( H → µµ ) Γ tot This Review 1 . × − . × − . × − . × − . . × − . × − . × − . × − . δ Γ [%] − . − . − . − . − . ± . ± . ± . ± . ± . C sf,pf ( τ t,b ) into the formfactors F sf,pf ( τ t,b ) and the electroweak correction δ γ : S elw is from Fig. 13. For δ γ : P elw , we set η γ : P elw = 0. For thenumerical estimate of δ γ : S QCD , more precisely, we take the SM values of g HWW = g SH ¯ tt = g SH ¯ bb = g SHττ = 1.While, for the numerical estimate of δ γ : P QCD , we assume a scenario in which the pseudoscalar form factor P γ is dominated by the top-quark contribution taking g PH ¯ bb = g PHττ = 0, i.e. neglecting the b – and τ –loop contributions. In the SM, we have S γ = − .
558 + 0 . i and P γ = 0.Finally, for the decay H → Zγ , the scalar and pseudoscalar form factors are given by S Zγ = − . g HWW + 0 . g SH ¯ tt + ( − .
105 + 0 . i ) g SH ¯ bb + ( − . . i ) g SHττ + ∆ S Zγ ; P Zγ = 0 . g PH ¯ tt + ( − .
021 + 0 . i ) g PH ¯ bb + ( − . . i ) g PHττ + ∆ P Zγ , (145)in terms of the HW W and Higgs–fermion–fermion couplings. In the SM, we have S Zγ = − . . i and P Zγ = 0. The H → Zγ decay width is estimated in the LO.In Table 5, we show the partial and total decay widths of the SM Higgs boson with M H =125 . g SH ¯ ff = g HV V = 1 and g PH ¯ ff = ∆ S g ,γ ,Zγ = ∆ P g ,γ ,Zγ = 0. For a quantita-tive comparison with those presented in Ref. [114], we introduce δ Γ ’s, which are defined by δ Γ ≡ (cid:0) Γ This Review − Γ Ref . [ ] (cid:1) / Γ Ref . [ ] for each decay mode, for being contrasted with theoretical un-certainties (THUs) given in Ref. [114]. In terms of δ Γ / THU, we find excellent agreement for H → gg , τ τ , Zγ , γγ , µµ and marginal consistency for H → b ¯ b , ZZ , between our analysis and thatin Ref. [114]. On the other hand, we find δ Γ / | THU | (cid:39) − H → W W and H → c ¯ c . To our bestknowledge, the H → W W discrepancy mainly comes from our treatment of the electroweak correctionof δ W elw which is smaller than that given in the improved Born approximation (IBA) around M H = 120GeV, see the lower panels of Figs. 7 and 8 in Ref. [60]. This also explains our smaller estimate ofΓ( H → ZZ ). The largest contribution to the discrepancy of Γ tot comes from H → b ¯ b with the second(third) largest one from H → W W ( gg ). We note that our estimations of the decay widths into quarksand gluons are smaller than those in Ref. [114]. This might come from an overestimation of the leadingterm of the QCD corrections of δ QCD and δ g : S QCD in the literature, see Eq. (123) and explanations below. Note that we use the same values for all the input parameters as in Ref. [114]. M H = 125 . B ( H → b ¯ b ) B ( H → W W ) B ( H → gg ) B ( H → τ τ ) B ( H → c ¯ c )This Review 5 . × − . × − . × − . × − . × − Ref. [114] 5 . × − . × − . × − . × − . × − THU+PU [%] 1 . . . . . B ( H → ZZ ) B ( H → γγ ) B ( H → Zγ ) B ( H → µµ ) Γ tot [MeV]This Review 2 . × − . × − . × − . × − . . × − . × − . × − . × − . . . . . . M H = 125 . H → b ¯ b, W W, τ τ, ZZ, γγ and µµ . While it is about 7% for H → gg, c ¯ c , and Zγ . The total decay width is determined with about 2% error. In Fig. 20, thebranching ratios (BRs) are shown in the Higgs-boson mass range between 120 GeV and 130 GeV. Foreach BR line, the band width represents the corresponding total uncertainty. Decays of heavy Higgs bosons in CP-violating 2HDMs
In this subsubsection, we study the decays of heavy neutral Higgs bosons appearing in BSM models.To be specific, we choose the type-I 2HDM identifying the lightest neutral Higgs boson as the SM-like125 . λ , , in the Higgs potential. In this scenario, with no much need ofdecoupling the heavier Higgs bosons, all the branching ratios and the total decay width of the lightestHiggs boson remain consistent with those of the SM Higgs within the ranges allowed by the currentLHC Higgs precision data [151].To fix all the relevant couplings of three neutral Higgs bosons, one may start from the orthogonal3 × O describing the mixing among them. For CPV scenarios in 2HDMs, the three neutralHiggs bosons do not carry definite CP parities and they become mixtures of CP-even and CP-oddstates. In this case, without loss of generality, the mixing matrix can be parameterized as O = − s α c α c α s α
00 0 1 c η s η − s η c η c ω s ω − s ω c ω = − s α c η c α c ω + s α s η s ω c α s ω − s α s η c ω c α c η s α c ω − c α s η s ω s α s ω + c α s η c ω − s η − c η s ω c η c ω , (146) Here we take the abbreviations such as cos α = c α , sin α = s α , etc. [GeV] H M
120 121 122 123 124 125 126 127 128 129 130 B r an c h i ng R a t i o -4 -3 -2 -1
10 1 L HC H I GG S XS W G bb tt mm ccgg gg ZZWW g Z Figure 20: Higgs boson branching ratios and their uncertainties for the mass range around 125 GeV.The plot is taken from Ref. [114].introducing a CP-conserving (CPC) mixing angle α and two CPV angles ω and η . We recall that themixing matrix O relates the electroweak eigenstates ( φ , φ , a ) to the mass eigenstates ( H , H , H )via ( φ , φ , a ) Tα = O αi ( H , H , H ) Ti , with the ordering of M H ≤ M H ≤ M H . Assuming the lightest Higgs boson is purely CP even ortaking s η = 0 and c η = 1, the mixing matrix takes the simpler form of O | s η =0 ,c η =1 = − s α c α c ω c α s ω c α s α c ω s α s ω − s ω c ω . (147)Note that, in the CP-conserving case, one of the heavy Higgs boson is purely CP odd and its coupling toa pair of massive gauges bosons is identically vanishing. We observe H is purely CP odd when | c ω | = 1while H is CP odd when | s ω | = 1. Plugging the above expression of O into Eq. (47), the couplings ofthree neutral Higgs bosons to a pair of massive vector bosons are given by g H V V = s β − α ≡ √ − (cid:15) , g H V V = c β − α c ω ≡ δ , g H V V = c β − α s ω ≡ δ , (148)with δ + δ = (cid:15) . We note that the two mixing angles are determined as follows s α = −√ − (cid:15) c β + δ c ω s β , c α = √ − (cid:15) s β + δ c ω c β ,c ω = δ δ + δ = δ (cid:15) , s ω = δ δ + δ = δ (cid:15) . (149)48n terms of the couplings δ = g H V V and δ = g H V V together with t β . And then, the Yukawa couplingsof the three neutral Higgs bosons are determined by g SH i ¯ uu = g SH i ¯ dd = g SH i ¯ (cid:96)(cid:96) = O φ i /s β ; − g PH i ¯ uu = g PH i ¯ dd = g PH i ¯ (cid:96)(cid:96) = O ai /t β , (150)where u and d stand for the up- and down-type quarks, respectively, and (cid:96) for three charged leptons.To summarize, in the scenario under consideration, all the Yukawa couplings of the two heavy Higgsbosons could be fixed by giving their couplings to the massive vector bosons. On the other hand,depending on sign[ δ /c ω ], all the Yukawa couplings of the lightest Higgs boson are determined by O φ /s β = c α /s β = √ − (cid:15) ± √ (cid:15)/t β which, especially for large t β , approaches the SM value of 1 asquickly as the g H V V = √ − (cid:15) coupling when (cid:15) goes to zero. This is the very reason why we choosethe type-I 2HDM for our numerical study avoiding conflicts with the current LHC Higgs precisiondata [151].For our numerical study, we vary t β but, for δ , , we are taking g H V V = δ = √ (cid:15) (cid:18) M H M H (cid:19) , g H V V = δ = √ (cid:15) (cid:18) M H M H (cid:19) , (151)reflecting the behaviour of (cid:15) which is suppressed by the quartic powers of the heavy Higgs-boson massesin the leading order [151]. With the above parameterizations of δ , and taking c ω > δ /c ω = c β − α = √ (cid:15) > s ω = M H / ( M H + M H ) leading to a maximal CPV mixing between thetwo heavy Higgs bosons when they are degenerate. For (cid:15) or the largest possible value of δ , , we choosea value which is a little bit lager than the lower 1 σ error of C v in the CPC4 fit: (cid:15) = 0 . , (152)having in mind the relation g H V V = √ − (cid:15) ≥ √ − (cid:15) (cid:39) − (cid:15) . For the masses of Higgs bosons, wetake M H = 125 . , M H = M H −
50 GeV , M H < M H ± ∼ , (153)with M H varied. This choice may result in the simpler decay pattern by forbidding or suppressing thedecay channels of H , → H ± W ∓∗ , H , → H ± H ∓ , H → H H , H → H H , etc. By assuming veryheavy charged Higgs boson, also neglected are the contributions from the charged-Higgs-boson loops tothe decay processes of the heavy neutral Higgs bosons into γγ and Zγ . Finally, for H , → H H ,we take | g H H H | = | g H H H | = 0 .
1. For the rigorous treatment of the cubic H H H and H H H self-couplings expressed in terms of the masses of charged and neutral Higgs bosons and the elementsof the mixing matrix O , see Appendix E.In Fig. 21, we show the decay widths and branching ratios of the two heavy Higgs bosons in thetype-I 2HDM taking t β = 5. For CPC, H is taken to be CP odd with g H V V = 0 and, accordingly, thedecays of H into W W , ZZ , and H H are forbidden. Incidentally, we note that H → H Z decay isalso forbidden since g H H Z = g H V V . On the other hand, for CPV, there are no forbidden decay modesas long as they are kinematically allowed. In the CPC case, the total decay widths of CP-even H andCP-odd H are largely enhanced at the H H and t ¯ t thresholds, respectively. While, in the CPV case,both of the thresholds contribute to the total decay widths as shown in the upper panels of Fig. 21.For CPC, we further observe that the couplings of the CP-even state of H to fermions are identicallyvanishing when g SH ¯ ff = 0. It does happen at s α = 0 or (cid:15) = 1 / (1 + t β ), see Eq. (149). This explains why See Section 5.2 and Table 10 therein. Note C v ≡ C w = C z , see Eq. (165). For the details of the contributions from the charged-Higgs-boson loops to the neutral Higgs boson decays into twophotons in the 2HDM and the MSSM, see Appendix E. M H [ GeV ] Γ t o t [ G e V ] CPCType I 2HDM : tan β = 5 M H [ GeV ] Γ t o t [ G e V ] CPV tt bb ττ WWZZ H H M H2 [ GeV ] B ( H ) µµ H Z M H2 [ GeV ] B ( H ) µµ H ZH Z M H3 [ GeV ] B ( H ) H ZH Z M H3 [ GeV ] B ( H ) -3 -2 -1 -3 -2 -1 -6 -5 -4 -3 -2 -1 -6 -5 -4 -3 -2 -1 -6 -5 -4 -3 -2 -1 -6 -5 -4 -3 -2 -1 Figure 21: The total decay widths and branching ratios of the two heavy Higgs bosons in the type-I2HDM taking t β = 5 and | g H H H | = | g H H H | = 0 .
1. In the left CPC panels, we take δ = g H V V =2 (cid:15) ( M H /M H ) = (cid:15) and δ = g H V V = 0. While, in the right CPV panels, we take δ = g H V V = (cid:15) ( M H /M H ) and δ = g H V V = (cid:15) ( M H /M H ). In the both CPC and CPV cases, we take (cid:15) = 0 . H state at M H = [2 (cid:15) (1 + t β )] / M H (cid:39) √ t β GeV as found in the middle-left panels of Fig. 21 and Fig. 22 around M H = 160 GeV and385 GeV, respectively. We note that the branching ratios of fermionic decay modes H , → t ¯ t, b ¯ b, τ τ, µµ are smaller for the larger value of t β since the corresponding decay widths are suppressed by the factorof ∼ /t β . For large values of M H , , the numerical results are consistent with the observation thatthe decay width of fermionic decay modes is proportional to M H , while that of bosonic decay ones is inversely proportional to M H , especially with the parameterization of Eq. (151) for H , → V V , H Z . In CPC, note that we take δ = 2 (cid:15) ( M H /M H ) = (cid:15) with δ = 0. M H [ GeV ] Γ t o t [ G e V ] CPCType I 2HDM : tan β = 30 M H [ GeV ] Γ t o t [ G e V ] CPV ttbb ττµµ
WWZZ H H M H2 [ GeV ] B ( H ) H Z M H2 [ GeV ] B ( H ) H ZH Z M H3 [ GeV ] B ( H ) H ZH Z M H3 [ GeV ] B ( H ) -3 -2 -1 -3 -2 -1 -6 -5 -4 -3 -2 -1 -6 -5 -4 -3 -2 -1 -6 -5 -4 -3 -2 -1 -6 -5 -4 -3 -2 -1 Figure 22: The same as in Fig. 21 but for t β = 30.On the other hand, the decay width Γ( H → H Z ) is not suppressed by the heavy mass M H becauseof the relation g H H Z = g H V V . In our numerical study, it is suppressed since we have taken the smallmass difference between H and H of M H − M H = 50 GeV < M Z . Otherwise, it may increase inproportion to M H . The effective couplings of the charged Higgs boson H ± to quarks and leptons are described by theinteraction Lagrangian: L H ± f ↑ f ↓ = √ H + f ↑ (cid:16) m f ↑ v g f ↑ P L + m f ↓ v g f ↓ P R (cid:17) f ↓ + h . c . , (154)51here P L,R = (1 ∓ γ ) / f ↑ , f ↓ ) = ( t, b ) , ( c, s ) , ( ν τ , τ ) , ( ν µ , µ ), etc. The masses of the up- anddown-type fermions are denoted by m f ↑ and m f ↓ , respectively. On the other hand, the interaction ofthe charged Higgs boson with a massive gauge boson W and a neutral Higgs boson H is given by L HH ± W ∓ = − g g HH + W − W − µ ( H i ↔ ∂ µ H + ) + h . c . , (155)with the convention X ↔ ∂ µ Y ≡ X ( ∂ µ Y ) − ( ∂ µ X ) Y .When a charged Higgs boson decays into quarks, the decay width is given byΓ( H + → f ↑ f ↓ ) = N f ↑ f ↓ C M H ± πv λ / (cid:0) , κ f ↑ , κ f ↓ (cid:1) (cid:26) (cid:0) − κ f ↑ − κ f ↓ (cid:1) (cid:16) m f ↑ g f ↑ + m f ↓ g f ↓ (cid:17) − √ κ f ↑ κ f ↓ m f ↑ m f ↓ g f ↑ g f ↓ (cid:27) (1 + δ QCD ) , (156)including the QCD correction factor δ QCD with κ f ↑ ,f ↓ = M f ↑ ,f ↓ /M H ± N f ↑ f ↓ C = 3. Note that the MSquark masses such as m t ( M H ± ) and m b ( M H ± ) are taken for m f ↑ and m f ↓ , respectively. While, for thedecays into leptons, the charged lepton pole masses are used for m f ↓ together with m f ↑ = 0, N f ↑ f ↓ C = 1,and δ QCD = 0. A charged Higgs boson may decay into a lighter neutral Higgs boson H and a massivegauge boson W and the LO decay width is given byΓ LO ( H + → HW + ∗ ) = G F M H ± | g HH + W − | √ π (cid:90) ( √ ω ± −√ ω ) d x (cid:15) W λ / ( ω ± , ω, x ) ω ± π [( x − + (cid:15) W ] , (157)with ω = M H /M W and ω ± = M H ± /M W . When M H ± is larger than the sum M H + M W , it reduces toΓ LO ( H + → HW + ) = G F M H ± √ π | g HH + W − | λ / (1 , κ H , κ W ) , (158)where κ H = M H /M H ± and κ W = M W /M H ± . The electroweak corrections within the 2HDM frameworkhave been calculated, specifically for the process of H ± → W ± h, W ± H, W ± A [155, 156, 157]. They areof moderate size and numerically stable if a process- and gauge-independent renormalization scheme ischosen [157].Finally, in the LO, the decay widths of a charged Higgs boson H + into a chargino (cid:101) χ + j and a neutralino (cid:101) χ i are given byΓ LO ( H + → (cid:101) χ + j (cid:101) χ i ) = g M H ± λ / (1 , κ i , κ j )16 π (159) × (cid:104) (1 − κ i − κ j )( | g SH + (cid:101) χ i (cid:101) χ − j | + | g PH + (cid:101) χ i (cid:101) χ − j | ) − √ κ i κ j ( | g SH + (cid:101) χ i (cid:101) χ − j | − | g PH + (cid:101) χ i (cid:101) χ − j | ) (cid:105) , with κ i = m (cid:101) χ i /M H ± and κ j = m (cid:101) χ ± j /M H ± and the LO decay widths into a pair of sfermions byΓ LO ( H + → (cid:101) f i (cid:101) f (cid:48)∗ j ) = N ff (cid:48) C v | g H + (cid:101) f ∗ i (cid:101) f (cid:48) j | πM H ± λ / (1 , κ i , κ j ) , (160)where κ i = M (cid:101) f i /M H ± , κ j = M (cid:101) f (cid:48) j /M H ± , and N ff (cid:48) C = 3 and 1 for squarks and sleptons, respectively.52or a numerical example, we take the 2HDMs in which the relevant couplings are given by H + → t ¯ b , c ¯ s : g f ↑ = 1 t β , g f ↓ = − t β (I , III) , t β (II , IV) ; H + → ντ + , νµ + : g f ↑ = 0 , g f ↓ = − t β (I , IV) , t β (II , III) ; H + → HW + : | g HH + W − | = 1 − g HV V (I , II , III , IV) , (161)depending on the 2HDM type as denoted by I, II, III or IV. For the decay H + → HW + , we set M H = 125 . | g HH + W − | = 0 . For the strange quark mass, we take M s = 93 MeV and m s ( µ ) = m c ( µ ) / .
72 [6] for its pole and running MS masses, respectively.For the charged Higgs–boson decay H + → t ¯ b , we take the contribution of the off-shell top quarkinto account:Γ LO ( H ± → t ( p t ) ¯ b → b ¯ bW + ) = N tbC v g M H ± π (cid:90) ( M H ± − M b ) M W λ / H ± p t (cid:18) − M W p t (cid:19) (cid:18) p t M W (cid:19) × (1 − α t − α ¯ b ) [ m t ( M H ± ) g t + m b ( M H ± ) g b ] − √ α t α ¯ b m t ( M H ± ) m b ( M H ± ) g t g b ( p t − M t ) + M t Γ t dp t , (162)where the kinematical b -quark mass is neglected in the t → bW + decay process and the triangle function λ H ± is given by λ H ± = 1 + α t + α b − α t − α ¯ b − α t α ¯ b , (163)with α t = p t /M H ± and α ¯ b = M b /M H ± . When M H ± > M t + M b , using Eqs. (104) and (106), we haveΓ LO ( H ± → t ¯ b → b ¯ bW + ) = Γ LO ( H ± → t ¯ b ) Γ LO ( t → bW )Γ t , (164)in a factorized form.In the upper panels of Fig. 23, we show the decay widths of a charged Higgs boson into two quarks, t ( ∗ ) ¯ b (left) and c ¯ s (right) in the type-I and type-III 2HDMs taking account of the QCD corrections.In these models, g t,c = − g b,s = 1 /t β and the top-quark and charm-quark contributions dominate andthe decay widths scale as 1 /t β . In the lower panels of Fig. 23, we consider the type-II and -IV 2HDMsin which g t,c = 1 /t β and g b,s = t β . For low values of t β , the top-quark and charm-quark contributionsdominate and the decay widths scale as 1 /t β . On the other hand, for high values of t β , the bottom-quarkand strange-quark contributions are enhanced by the factor of t β while the top-quark and charm-quarkcontributions are suppressed by the factor of 1 /t β . The bottom-quark and strange-quark contributionsstart to dominate when t β is larger than (cid:112) m t /m b (cid:39) . (cid:112) m c /m s (cid:39) .
4, respectively. Whenthe bottom-quark and strange-quark contributions dominate, the decay widths scale as ( t β / and( t β / , respectively, compared to the case with t β = 1 and these factors are responsible for thesignificant change of the decay widths for high t β values, as can be checked by comparing the lines with t β = 1 and those with t β = 10 and 30.In Fig. 24, we show the LO decay widths of a charged Higgs boson with mass M H ± into τ + ν inthe type-I/IV 2HDMs (left) and in the type-II/III ones (right) taking t β = 0 . See Eq. (52) and Table 2. For this, we take g HV V (cid:39) .
95 adopting a little bit lager value than the lower 1 σ error of C v in the CPC4 fit, seeSection 5.2 and Table 10 there in. At M H ± = 500 GeV, we switch from the three-body decay width Γ ( H + → b ¯ bW + ) to the two-body decay widthΓ( H + → t ¯ b ) because of the same reasons as in H → t ∗ ¯ t ∗ described previously. an β = β = β = β = / III
200 400 600 800 100010 - - - - M H + [ GeV ] Γ ( H + ⟶ t b ) [ G e V ] tan β = β = β = β = / III
200 400 600 800 100010 - - - - M H + [ GeV ] Γ ( H + ⟶ c s ) [ G e V ] tan β = β = β = β = / IV
200 400 600 800 100010 - - - - M H + [ GeV ] Γ ( H + ⟶ t b ) [ G e V ] tan β = β = β = β = / IV
200 400 600 800 100010 - - - - M H + [ GeV ] Γ ( H + ⟶ c s ) [ G e V ] Figure 23: (Upper) Decay widths of a charged Higgs boson with a mass M H ± into t ¯ b (left) and c ¯ s (right) in the type-I and type-III 2HDMs. QCD corrections are taken into account. For tan β , we takefour values of 0 . tan β = β = β = β = / IV
200 400 600 800 100010 - - M H + [ GeV ] Γ ( H + ⟶ ν τ τ + ) [ G e V ] tan β = β = β = β = / III
200 400 600 800 100010 - - M H + [ GeV ] Γ ( H + ⟶ ν τ τ + ) [ G e V ] Figure 24: (Upper) Decay widths of a charged Higgs boson with a mass M H ± into τ + ν in the type-I/IV2HDMs (left) and in the type-II/III ones (right). For tan β , we take four values of 0 . g τ = 1 /t β and t β , respectively. Compared to the case of H + → τ + ν , the decaywidths Γ( H + → µ + ν ) are simply suppressed by the large factor of m τ /m µ ∼ M H ± into HW + taking54
00 400 600 800 1000051015202530 M H + [ GeV ] Γ L O ( H + ⟶ H W + ) [ G e V ]
100 150 200 250 300 35010 - - M H + [ GeV ] Γ L O ( H + ⟶ H W + ) [ G e V ] Figure 25: Decay widths of a charged Higgs boson with a mass M H ± into HW + taking M H = 125 . | g HH + W − | = 0 .
1. In the right panel, we magnify the low M H region and compare with the two-bodydecay width (dash-dotted magenta line). The vertical line locates the position M H ± = M H + M W . M H = 125 . | g HH + W − | = 0 .
1. In the right panel, we magnify the low M H region covering thecase with a virtual W + ∗ and we compare the decay width with the prediction of the two-body decaywidth (dash-dotted magenta line). We note that they are nearly identical for M H ± > M H + M W , asexpected.In Fig. 26, we show the total decay widths of a charged Higgs boson for four values of t β : tan β = 0 . H + → t ¯ b , H + → c ¯ s , H + → τ ν , H + → µν , and H + → HW + with M H = 125 . | g HH + W − | = 0 . H + → t ¯ b ) scales as 1 /t β ,independently of the 2HDM types. As the value of t β grows, the total decay width monotonicallydecreases in the type-I and type-III 2HDMs. In contrast, in the type-II and type-IV 2HDMs, thedecay width reaches the minimum around t β = 8, where the top-quark and bottom-quark contributionsbecome comparable, and then it starts to increases as t β grows further. Based on these observations, wecan work out the behaviour of the total decay width of a heavy charged Higgs boson for large values of t β : the dominant contribution comes from the H + → HW + decay mode with the subleading/competingcontributions from H + → t ¯ b . We identify that there exists an additional subleading contribution from H + → τ ν for t β = 10 and 30 in the type-II and type-III 2HDMs. On the other hand, for low valuesof t β , the H + → t ¯ b decay mode dominates the total decay width with the subleading contributionfrom H + → HW + which amounts to about 30 GeV at M H ± = 1 TeV taking M H = 125 . | g HH + W − | = 0 .
1, see Fig. 25.Secondly, we consider the case of a charged Higgs boson lighter than t quark, see the right panels ofFig. 26. For large values of t β , the H + → τ ν decay mode dominates in the type-II and type-III 2HDMs.In the type-I 2HDM, the H + → τ ν decay mode still dominates with the subleading contributions from H + → c ¯ s though both of them are suppressed by 1 /t β . In the type-IV 2HDM, the H + → c ¯ s decaymode dominates where the strange-quark contribution is enhanced by t β . For small values of t β , the H + → τ ν decay mode mostly dominates, leading to the larger decay widths for t β = 0 . | g τ | = 1 /t β .In Fig. 27, we show the branching ratios of a charged Higgs boson varying its mass between 100GeV and 1 TeV in the type-I and type-III 2HDMs taking tan β = 0 .
5, 1, 10, and 30 from top tobottom. We consider 5 decay modes of H + → t ¯ b (red solid), H + → c ¯ s (magenta solid), H + → τ ν an β = β = β = β =
200 400 600 800 1000050100150200 M H ± [ GeV ] Γ ( H ± ) [ G e V ] tan β = β = β = β =
100 200 300 400 50010 - - - - M H ± [ GeV ] Γ ( H ± ) [ G e V ] tan β = β = β = β =
200 400 600 800 1000050100150200 M H ± [ GeV ] Γ ( H ± ) [ G e V ] tan β = β = β = β =
100 200 300 400 50010 - - - - M H ± [ GeV ] Γ ( H ± ) [ G e V ] tan β = β = β = β =
200 400 600 800 1000050100150200 M H ± [ GeV ] Γ ( H ± ) [ G e V ] tan β = β = β = β =
100 200 300 400 50010 - - - - M H ± [ GeV ] Γ ( H ± ) [ G e V ] tan β = β = β = β =
200 400 600 800 1000050100150200 M H ± [ GeV ] Γ ( H ± ) [ G e V ] tan β = β = β = β =
100 200 300 400 50010 - - - - M H ± [ GeV ] Γ ( H ± ) [ G e V ] Figure 26: Total decay widths of a charged Higgs boson with a mass M H ± in the type-I, type-II,type-III, and type-IV 2HDMs from top to bottom. We are taking tan β = 0 . M H region.(blue solid), H + → µν (green solid), and H + → HW + (black dashed) with M H = 125 . g t,c = | g b,s | = 1 /t β while those to leptons are given by g µ,τ = − /t β (type I) and g µ,τ = t β (type III).We observe that, especially for large values of t β , the heavy charged Higgs boson dominantly decaysinto HW + (black dashed) since Γ( H + → t ¯ b ) is suppressed as t β grows. And, when the H + → HW + R ( H + → tb ) BR ( H + → cs ) BR ( H + → τν τ ) BR ( H + → μν μ ) BR ( H + → H W + ) Type I ( tan β = )
200 400 600 800 1000110 - - - - - - - M H + [ GeV ] B R ( H + ) BR ( H + → tb ) BR ( H + → cs ) BR ( H + → τν τ ) BR ( H + → μν μ ) BR ( H + → H W + ) Type III ( tan β = )
200 400 600 800 1000110 - - - - - - - M H + [ GeV ] B R ( H + ) Type I ( tan β = )
200 400 600 800 1000110 - - - - - - - M H + [ GeV ] B R ( H + ) Type III ( tan β = )
200 400 600 800 1000110 - - - - - - - M H + [ GeV ] B R ( H + ) Type I ( tan β = )
200 400 600 800 1000110 - - - - - - - M H + [ GeV ] B R ( H + ) Type III ( tan β = )
200 400 600 800 1000110 - - - - - - - M H + [ GeV ] B R ( H + ) Type I ( tan β = )
200 400 600 800 1000110 - - - - - - - M H + [ GeV ] B R ( H + ) Type III ( tan β = )
200 400 600 800 1000110 - - - - - - - M H + [ GeV ] B R ( H + ) Figure 27: Branching ratios of a charged Higgs boson with a mass M H ± in the type-I (left) and type-III(right) 2HDMs taking tan β = 0 .
5, 1, 10, and 30 from top to bottom. We consider 5 decay modes of H + → t ¯ b (red solid), H + → c ¯ s (magenta solid), H + → τ ν (blue solid), H + → µν (green solid), and H + → HW + (black dashed) with M H = 125 . t β in the type-III 2HDM. In thetype-III 2HDM, we note that B ( H + → τ ν ) becomes comparable to B ( H + → t ¯ b ) around t β ∼
10 and itbecomes larger as t β grows.In Fig. 28, we show the branching ratios of a charged Higgs boson for its mass between 100 GeVand 1 TeV in the type-II and type-IV 2HDMs taking tan β = 0 .
5, 1, 10, and 30 from top to bottom.57 R ( H + → tb ) BR ( H + → cs ) BR ( H + → τν τ ) BR ( H + → μν μ ) BR ( H + → H W + ) Type II ( tan β = )
200 400 600 800 1000110 - - - - - - - M H + [ GeV ] B R ( H + ) BR ( H + → tb ) BR ( H + → cs ) BR ( H + → τν τ ) BR ( H + → μν μ ) BR ( H + → H W + ) Type IV ( tan β = )
200 400 600 800 1000110 - - - - - - - M H + [ GeV ] B R ( H + ) Type II ( tan β = )
200 400 600 800 1000110 - - - - - - - M H + [ GeV ] B R ( H + ) Type IV ( tan β = )
200 400 600 800 1000110 - - - - - - - M H + [ GeV ] B R ( H + ) Type II ( tan β = )
200 400 600 800 1000110 - - - - - - - M H + [ GeV ] B R ( H + ) Type IV ( tan β = )
200 400 600 800 1000110 - - - - - - - M H + [ GeV ] B R ( H + ) Type II ( tan β = )
200 400 600 800 1000110 - - - - - - - M H + [ GeV ] B R ( H + ) Type IV ( tan β = )
200 400 600 800 1000110 - - - - - - - M H + [ GeV ] B R ( H + ) Figure 28: The same as in Fig. 27 but for the type-II (left) and type-IV (right) 2HDMs.We consider the same 5 decay modes as in Fig. 27. In the type-II and type-IV models, the chargedHiggs couplings to the (right-handed) up-type quarks are again given by g t,c = 1 /t β but those to thedown-type fermions are by g b,s = g µ,τ = t β (type II) and g b,s = 1 / | g µ,τ | = t β (type IV). We observe that,especially for large values of t β , the heavy charged Higgs boson dominantly decays into HW + (blackdashed) eventually. But, compared to the type-I and type-III 2HDMs where the top-quark contributionsto the H + → t ¯ b decay always dominate and the decay width decreases as t β grows, the dominance ofthe H + → HW + decay mode develops rather slowly because the bottom-quark contributions take overthe dominance around t β = 8 and the partial width Γ( H + → t ¯ b ) increases as t β grows in the type-II58
200 400 600 800 1000050100150200 M H + [ GeV ] Γ L O ( H + ⟶ H W + ) [ G e V ] Figure 29: Decay widths of a charged Higgs boson with a mass M H ± into H (cid:48) W + for three values of M H (cid:48) = 300 GeV (blue dashed), 500 GeV (red dotted), and 700 GeV (magenta dash-dotted) taking | g H (cid:48) H + W − | = 1. For comparisons, Γ LO ( H + → HW + ) with M H = 125 . | g HH + W − | = 0 . t β in the type-II 2HDM and wefind that, specifically for t β = 30, the three decay modes of H + → t ¯ b , H + → HW + , and H + → τ ν arecompeting in the region of 350 < ∼ M H ± / GeV < ∼ τ ν and/or c ¯ s before the H + → t ∗ ¯ b decaychannel opens and starts to dominate. Exceptions occur in the type-I and type-III 2HDMs when t β islarge, see the lowest panels of Fig. 27. Specifically, for t β = 30 in the type-I model, the charged Higgsboson dominantly decays into HW + ∗ in the narrow region of M H ± between 150 GeV and 180 GeV.We confirm that the behaviour of the branching ratio of each decay mode versus M H ± does notdepend on the 2HDM type when t β = 1 as it should be: see the tan β = 1 panels in Figs. 27 and 28.Before closing this section, we address the case in which there exists another neutral Higgs boson H (cid:48) with its mass smaller than M H ± and the decay H + → H (cid:48) W + is kinematically allowed. In the 2HDMframework, identifying the lightest neutral Higgs boson as the SM-like H with M H = 125 . | g HH + W − | = 1 − g HV V = 0 . H (cid:48) and a charged vector boson W takes almost the maximum value of 1 due to the sumrules given in Eq. (48). Furthermore, the decay width grows by the cubic powers of the charged Higgs-boson mass. These combined properties result in a large width for the decay mode H ± → H (cid:48) W ± when H ± is heavy enough. When M H ± > ∼
950 GeV, we observe that the decay width Γ LO ( H + → H (cid:48) W + ) islarger than 200 GeV for a H (cid:48) boson with M H (cid:48) = 300 GeV, see the blue dashed line in Fig. 29. Evenfor H (cid:48) as heavy as 700 GeV, we note that the decay width could be comparable to Γ LO ( H + → HW + )when M H ± ∼ Ever since the discovery of a SM-like Higgs boson in the year 2012 [1, 2], the era of Higgs-bosonprecision studies, which was termed as Higgcision in Refs. [158, 159, 160], has begun. In this section59e present highlights from global fits of the Higgs boson couplings to all the 7 TeV, 8 TeV, and 13TeV data available up to the Summer 2018, based on the works [158, 159, 160]. For several differenttypes of approaches with their own merits on the global fits of the Higgs couplings, see, for example,Refs. [114, 161, 162, 163, 164, 165, 166, 167, 168, 169, 170, 171, 172, 173, 174]. For some other earlyworks based on model-independent approach, we refer to Refs. [175, 176, 177, 178, 179, 180, 181, 182,183, 184, 185, 186, 187, 188, 189, 190, 191].Assuming generation independence for the normalized Yukawa couplings of g S,PH ¯ ff , we use the following C notations for the couplings in the global fits: C Su = g SH ¯ uu , C Sd = g SH ¯ dd , C S(cid:96) = g SH ¯ ll ; C w = g HWW , C z = g HZZ ; C Pu = g PH ¯ uu , C Pd = g PH ¯ dd , C P(cid:96) = g PH ¯ ll . (165)We further keep the custodial symmetry between the W and Z bosons which leads to the relation C v ≡ C w = C z . Each theoretical signal strength can be written in a product form as (cid:98) µ ( P , D ) (cid:39) (cid:98) µ ( P ) (cid:98) µ ( D ) , (166)where P = ggF , VBF , VH , ttH denote the production mechanisms and D = γγ, ZZ ( ∗ ) , W W ( ∗ ) , b ¯ b, τ + τ − the decay channels, which are experimentally clean and/or dominant for M H (cid:39)
125 GeV. More ex-plicitly, in the LO, the production signal strengths are given in terms of the relevant form factors andcouplings by (cid:98) µ (ggF) = | S g ( M H ) | + | P g ( M H ) | | S g SM ( M H ) | , (cid:98) µ (VBF) = (cid:98) µ (VH) = g HWW,HZZ , (cid:98) µ (ttH) = (cid:0) g SH ¯ tt (cid:1) + (cid:0) g PH ¯ tt (cid:1) , (167)and the decay signal strengths by (cid:98) µ ( D ) = B ( H → D ) B ( H SM → D ) , (168)with the branching fraction of each decay mode defined by B ( H → D ) = Γ( H → D )Γ tot ( H ) + ∆Γ tot . (169)Note that an arbitrary non-SM contribution ∆Γ tot to the total decay width is introduced. We observeΓ tot ( H ) becomes the SM total decay width when g SH ¯ ff = 1, g PH ¯ ff = 0, g HWW,HZZ = 1, and ∆ S γ,g,Zγ =∆ P γ,g,Zγ = 0.On the experimental side, we use the direct Higgs signal strength data collected at the Tevatron andthe LHC. Specifically, we use 3 signal strengths measured at the Tevatron, see Table 7. At the LHCwith the center-of-mass energies of 7 and 8 (7 ⊕
8) TeV, the signal strengths obtained from a combinedATLAS and CMS analysis [194] are used, see Table 8. On the other hand, the 13 TeV data are still givenseparately by ATLAS and CMS and in different production and decay channels. Under this situation,to derive the combined signal strengths of various channels, we use a simple χ method assuming thateach distribution is Gaussian. The results are shown in Table 9. For the details of the 13 TeV data sets used, see Appendix B of Ref. [160] and references therein. (Tevatron: 1.96 TeV)
The signal strengths data from Tevatron (10.0 fb − at 1.96 TeV). Channel Signal strength µ M H (GeV) Production mode χ (each)c.v ± error ggF VBF VH ttHTevatron (Nov. 2012)Combined H → γγ [192] 6 . +3 . − .
125 78% 5% 17% - 2.60Combined H → W W ( ∗ ) [192] 0 . +0 . − .
125 78% 5% 17% - 0.03VH tag H → bb [193] 1 . +0 . − .
125 - - 100% - 0.67 χ (subtot): 3.30 Table 8: (LHC: 7 ⊕ Combined ATLAS and CMS data on signal strengths from Table 8 ofRef. [194]. Decay modeProduction mode H → γγ H → ZZ ( ∗ ) H → W W ( ∗ ) H → bb H → τ + τ − ggF 1 . +0 . − . . +0 . − . . +0 . − . - 1 . +0 . − . VBF 1 . +0 . − . . +1 . − . . +0 . − . - 1 . +0 . − . WH 0 . +1 . − . - 1 . +1 . − . . +0 . − . − . +1 . − . ZH 0 . +3 . − . - 5 . +2 . − . . +0 . − . . +2 . − . ttH 2 . +1 . − . - 5 . +1 . − . . +1 . − . − . +3 . − . χ (subtot): 19.93Table 9: (LHC: 13 TeV) Combined ATLAS and CMS (13 TeV) data on signal strengths. The µ deccombined ( µ prodcombined ) represents the combined signal strength for a specific decay (production) channelby summing all the production (decay) modes, and χ are the corresponding minimal chi-squarevalues. In the VH/WH row, the production mode for H → γγ and H → ZZ ( ∗ ) is VH while it is WHfor H → W W ( ∗ ) and H → τ + τ − ; for the remaining decay mode H → b ¯ b , we combine the two signalstrengths from WH and VH, see Table XII in Ref. [160]. Decay modeProduction mode H → γγ H → ZZ ( ∗ ) H → W W ( ∗ ) H → bb H → τ + τ − µ prodcombined χ ( χ )ggF 1 . +0 . − . . +0 . − . . +0 . − . . +2 . − . . +0 . − . . +0 . − . . +0 . − . . +0 . − . . +0 . − . - 1 . +0 . − . . +0 . − . . +0 . − . . +0 . − . . +1 . − . . +0 . − . . +1 . − . . +0 . − . . +1 . − . . +0 . − . . +1 . − . . +0 . − . . +0 . − . . +0 . − . - 0 . +0 . − . - 0 . +0 . − . . +0 . − . - 1 . +0 . − . . +0 . − . . +0 . − . . +0 . − . µ deccombined . +0 . − . . +0 . − . . +0 . − . . +0 . − . . +0 . − . . +0 . − . χ ( χ ) 6.83(5.72) 9.13(8.88) 9.48(7.32) 1.56(1.51) 3.58(3.20) 30.58(27.56) . GeV
In this subsection, we present a few representative results obtained from performing LO analysis ofthe direct Higgs data collected at the Tevatron and the LHC by considering CP-conserving (CPC)scenarios only. Note that, in the most general
CPC scenario, one may vary all the 7 parameters of C Su , C Sd , C S(cid:96) , C v , ∆ S g , ∆ S γ , and ∆Γ tot while taking vanishing pseudoscalar couplings and form factors as61able 10: (CPC) The best-fitted values in various CP conserving fits and the corresponding chi-square per degree of freedom and goodness of fit. The p -value for each fit hypothesis against the SMnull hypothesis is also shown. For the SM, we obtain χ = 53 . χ /dof = 53 . /
64, and so thegoodness of fit = 0 . Cases
CPC1 CPC2 CPC4 CPCN4
Varying ∆Γ tot ∆ S γ C Su , C Sd , C Su , C v Parameters ∆ S g C S(cid:96) , C v ∆ S γ , ∆ S g C Su . +0 . − . . +0 . − . . +0 . − . − . +0 . − . − . +0 . − . C Sd . +0 . − . C S(cid:96) . +0 . − . C v . +0 . − . . +0 . − . . +0 . − . . +0 . − . . +0 . − . ∆ S γ − . +0 . − . − . +0 . − . − . +0 . − . . +0 . − . . +0 . − . ∆ S g . +0 . − . − . +0 . − . − . +0 . − . . +0 . − . . +0 . − . ∆Γ tot (MeV) − . +0 . − . χ /dof p -value 0.124 0.379 0.554 0.583 Figure 30:
CPC1 : ∆ χ from the minimum versus ∆Γ tot with only ∆Γ tot varying in the fit. FromRef. [160]. C Pu = C Pd = C P(cid:96) = ∆ P γ = ∆ P g = 0.Our goal is to provide constraints on the couplings of the neutral Higgs boson, which was discoveredat the LHC, without much loss of generality when it is interpreted in various frameworks beyond theSM. Accordingly, we consider the four CPC fits listed in Table 10 in which the second low explicitlyshows the varying parameters of each fit.Referring to Ref. [160] for more detailed explanations, we offer highlights of each fit as follows: • CPC1 : The best-fit value for the residual total decay width ∆Γ tot is ∆Γ tot = − . +0 . − . MeVwhich is 1 . σ below zero. At 95% confidence level (CL), on the other hand,∆Γ tot = − . +0 . − . MeV , (170)62igure 31: CPC4 : (Upper) The confidence-level (CL) regions of the fit by varying C v , C Su , C Sd , and C S(cid:96) . The contour regions shown are for ∆ χ ≤ . .
99 (red+green), and 11 .
83 (red+green+blue)above the minimum, which correspond to confidence levels of 68.3%, 95%, and 99.7%, respectively. Thebest-fit points are denoted by triangles. (Lower) ∆ χ from the minimum versus Yukawa couplings.From Ref. [160].as shown in Fig. 30. Using the upper error as the upper limit, we obtain the constraint ∆Γ tot ≤ . B ( H → nonstandard) ≤ . • CPC2 : The best-fit point (∆ S γ , ∆ S g ) = ( − . , . Hγγ and
Hgg form factors of | S γ | and | S g | ,respectively. We note that the error of ∆ S g is ± . − .
043 to the real part of S g , see Eq. (143), alerting that we havealready reached the sensitivity to probe the sign of the bottom-quark Yukawa coupling in gluonfusion. • CPC4 : We observe that the possibility for the top-quark Yukawa coupling to be negative hasbeen entirely ruled out as shown clearly in the left upper and lower panels of Fig. 31. And, asalready anticipated in the
CPC2 fit, the bottom-quark Yukawa coupling C Sd prefers the positivesign to the negative one, see the middle panels of Fig. 31. It is more clear from the middle lowerpanel that the point C Sd = − χ > C Sd = +1. The current dataprecision is yet insufficient for showing any preference for the sign of tau-Yukawa coupling, asshown in the right panels of Fig. 31. • CPCN4 : In this fit, there are 4 degenerate minima with ∆ S g ∼ , ∓ . C Su ∼ ± | S g | ∼ | . C uS + ∆ S g | ∼ .
7, see Table 10. Note that ∆ S γ can compensate the sign change in C Su allowing it to be about − S γ ∼ +3 .
5. This couldbe understood by noting the relation | S γ | ∼ | − . C v + 1 . C uS + ∆ S γ | ∼ .
5. We further observethat the negative top-quark Yukawa coupling is allowed only when there exist additional particles For this reason, we have considered the minimum around C S(cid:96) = +1 only in Table 10. Γ ≤
ΔΓ ≤ / ΔΓ ≤ / M φ [ GeV ] g H φφ ΔΓ ≤
ΔΓ ≤ / ΔΓ ≤ / M φ [ GeV ] | g H φ Z | Figure 32: (Left) The allowed parameter space on the ( M ϕ , | g Hϕϕ | ) plane from ∆Γ tot ≤ .
38 MeV at95% CL. Also shown are future prospects assuming two and four times stronger constraints on ∆Γ tot .(Right) The same as in the left panel but for ( M ϕ , | g HϕZ | ).running in the H - γ - γ loop with the size of contributions equal to two times the SM top-quarkcontribution within about 10 %. This tuning on the couplings and form factors will become moreand more severe as more data are accumulated at the LHC. In the cxSM where the SM is extended by adding a complex SU(2) L singlet, there could be a light Higgsboson ϕ mainly from the singlet sector and the 125.5 GeV Higgs boson H may couple to a pair of themthrough the singlet-doublet mixing term in the potential. When kinematically allowed or 2 M ϕ < . H decays into a pair of light scalars and, in this case, ∆Γ tot ≤ .
38 MeV may provide constraintson the mass M ϕ of the scalar particle and the absolute value of the coupling g Hϕϕ at 95% CL. We find | g Hϕϕ | < ∼ .
005 for M ϕ < ∼
40 GeV, see the left panel of Fig. 32. In 2HDMs, assuming that the scalar ϕ is CP odd and the Higgs boson H with M H = 125 . H → ϕZ ∗ to constrain the coupling g HϕZ depending on M ϕ . We find | g HϕZ | < ∼ . M ϕ < ∼
50 GeV,see the right panel of Fig. 32.In the MSSM, there still exists a room for the lightest neutralino (cid:101) χ to be light with its mass twotimes smaller than M H = 125 . tot ≤ .
38 MeV may provide 95% CLconstraints on the mass m (cid:101) χ and the absolute value of the coupling g SH (cid:101) χ (cid:101) χ assuming H is purely CPeven. We find | g SH (cid:101) χ (cid:101) χ | < ∼ .
01 for m (cid:101) χ < ∼
45 GeV, see Fig. 33.From Table 10, we find | ∆ S γ | < ∼ . | ∆ S g | < ∼ .
06 in
CPCN4 . Exploiting these constraints,we derive constraints on the H couplings to a pair of charged Higgs bosons, lighter charginos, and thelightest stops/sbottoms. For the SUSY contributions to ∆ S γ and ∆ S g , we refer to Appendix B.In the left panel of Fig. 34, we show the contour lines for | ∆ S γ | = 0 . , . , . m (cid:101) χ ± > ∼
390 GeV by the current LHC Higgs datawhen the relevant coupling is assumed to be 1. The lower-bound constraint on m (cid:101) χ ± linearly increases64 Γ ≤
ΔΓ ≤ / ΔΓ ≤ / m χ ∼ [ GeV ] g H χ ∼ χ ∼ S Figure 33: The allowed parameter space on the ( m (cid:101) χ , | g SH (cid:101) χ (cid:101) χ | ) plane from ∆Γ tot ≤ .
38 MeV at 95%CL. Also shown are future prospects assuming two and four times stronger constraints on ∆Γ tot .
200 400 600 800 1000 1200 1400 16000.00.20.40.60.81.01.21.4 m χ ± ∼ [ GeV ] | g H χ + ∼ χ - ∼ |
200 400 600 800 10000246810 M H ± [ GeV ] | g HH + H - | Figure 34: (Left) Contour lines for | ∆ S γ | = 0 . , . , . m (cid:101) χ ± , | g H (cid:101) χ +1 (cid:101) χ − | )plane assuming the lighter-chargino-loop contributions dominate ∆ S γ . (Right) The same as in the leftpanel but on the ( M H ± , | g HH + H − | ) plane now assuming ∆ S γ is dominated by the contributions fromcharged-Higgs loops. For the reference value of | ∆ S γ | = 0 .
4, we are taking the 1 σ error of the CPCN4 fit.as the bound on | ∆ S γ | becomes stronger. In the right panel of Fig. 34, we show the same contour linesnow assuming ∆ S γ is dominated by the charged-Higgs loops. In the 2HDM and the MSSM, keeping65 .06 0.03 0.015
500 1000 1500 20000246810 m f ∼ [ GeV ] | g H f ∼ f ∼ | Figure 35: Contour lines for | ∆ S g | = 0 . , . , .
015 from left to right on the ( m (cid:101) t , | g H (cid:101) t ∗ (cid:101) t | ) planeassuming ∆ S g is dominated by the contributions from the lighter-stop loops. For the reference value of | ∆ S g | = 0 .
06, we are taking the 1 σ error of the CPCN4 fit.the most significant three contributions when M H , > ∼ v , we find the g H H + H − coupling is given by g H H + H − ∼ (cid:18) M H ± (cid:19) + 5 (cid:34) (cid:18) t β (cid:19) − (cid:18)
400 GeV M (cid:19) (cid:32) t β − t β (cid:33) (cid:18) t β (cid:19) (cid:35) (cid:32) t β t β (cid:33) (cid:20) (cid:60) e( m e iξ )(100 GeV) (cid:21) − (cid:18)
400 GeV M (cid:19) (cid:32) t β t β (cid:33) (cid:18) t β (cid:19) (cid:60) e( λ e iξ ) , (171)under the assumption that H is purely CP even state. Note that the second and the third termscontain contributions enhanced by the factors of t β and t β and, for the t β -enhanced contributions,we are taking M H ∼ M H ≡ M and c β − α = √ (cid:15) = (cid:112) δ + δ = √ (cid:15) M H /M with (cid:15) = 0 . S γ ( H ± ) → / M H ± , M H , (cid:29) M H , v , (cid:60) e( m e iξ ), see Eq. (E.8) and the two equations following it. Note that theconstraint on M H ± becomes two times stronger when the bound on | ∆ S γ | becomes four times stronger.In Fig. 35, we show the contour lines for | ∆ S g | = 0 . , . , .
015 assuming that it is dominated by thecontributions from the lighter-stop loops. The current LHC Higgs data constrain the mass of the lighterstop as m (cid:101) t > ∼ In this review, we have calculated and discussed in detail all the decay widths and branching fractions ofHiggs bosons in the frameworks of the SM and its BSM extensions such as cxSM, 2HDMs with naturalflavour conservation, and MSSM. We allow for CP-violating complex phases as generally as possible See the first relation in Eq. (E.8) in Appendix E. See also https://pdglive.lbl.gov/DataBlock.action?node=S046STP. . Acknowledgment
We thank Abdesslam Arhrib, Eri Asakawa, Gabriela Barenboim, Francesca Borzumati, Cristian Bosch,Marcela Carena, Jung Chang, Kingman Cheung, Kiwoon Choi, Debajyoti Choudhury, Byung-chulChung, Brian Cox, Manuel Drees, Birgit Eberle, John Ellis, Christoph Englert, Jeffrey Forshaw, AyresFreitas, Benedikt Gaissmaier, Kaoru Hagiwara, Tie-Jiun Hou, Ran Huo, Jan Kalinowski, PyungwonKo, Yi Liao, M.Luisa L´opez-Iba˜nez, Chih-Ting Lu, David Miller, James Monk, Margarete Muhlleit-ner, Junya Nakamura, Chan Beom Park, Yvonne Peters, Apostolos Pilaftsis, Christian Schwanen-berger, Stefano Scopel, Eibun Senaha, Jeonghyeon Song, Wan Young Song, Michael Spira, Yue-LinSming Tsai, Po-Yan Tseng, Oscar Vives, Carlos Wagner, and Peter Zerwas for fruitful collaborations.This work was supported by the National Research Foundation (NRF) of Korea Grant No. NRF-2016R1E1A1A01943297 (J.S.L. and J.P.) and No. NRF-2018R1D1A1B07051126 (J.P.). The work ofS.Y.C was supported in part by Basic Science Research Program through the NRF of Korea Grant No.NRF-2016R1D1A3B01010529 and in part by the CERN-Korea theory collaboration.
Appendices
This section consists of five appendices. Appendix A is for a summary of the SM parameters usedfor the numerical estimates of the Higgs decay widths and a description of the running of the strong67oupling constant and quark masses. Appendix B is for the supersymmetric contributions to the loop-induced couplings of the Higgs boson to two gluons, two photons and Zγ . Appendix C is for the QCDcorrections to the partial width of the Higgs-boson decay to two photons. We work out the relationsamong the parameters of the most general 2HDM in Appendix D and we apply them for deriving cubicHiggs-boson self-couplings in Appendix E. A Standard Model Parameters
In this appendix, we summarize the SM parameters used for the estimation of decay widths of Higgsbosons. And we also show the running of the strong coupling constant and quark masses.
A.1 Input parameters
The SM parameters used for the estimation of decay widths of Higgs bosons are [6, 114]: • Gauge coupling strengths α s ( M Z ) = 0 . ± . ,α (0) = 1 / . ,α ( M W ) = 1 / . (A.1) • Electroweak parameters M W = (80 . ± . , Γ W = (2 . ± . M Z = (91 . ± . , Γ Z = (2 . ± . G F = 1 . × − GeV − . (A.2)The vev v of the SM Higgs field is given by v = (cid:0) √ G F (cid:1) − / (cid:39) .
22 GeV. And we treat thesquare of the sine of the weak mixing angle s W = 0 . g ( M W ) = e/s W (cid:39) . g (cid:48) ( M W ) = e/c W (cid:39) . e = e ( M W ) = 2 (cid:112) πα ( M W ) =0 . c W = 0 . • Lepton masses M µ = (105 . ± . , M τ = (1776 . ± .
16) MeV . (A.3) • Quark masses M t = (172 . ±
1) GeV ,m b ( m b ) = (4 . ± .
03) GeV ,m c (3 GeV) = (0 . ± . . (A.4)Note that the pole mass is used for t quark while, for b and c quarks, MS masses are used. The electron and u, d -quark masses are not included as their masses are too tiny to influence the numerical analysesmade in this work. For the s -quark mass, we take M s = 93 MeV and m s ( µ ) = m c ( µ ) / .
72 [6]. α s ( µ ), m c ( µ ), m b ( µ ), and m t ( µ ). M + q and M q − are introduced for decouplingeffects from matching the (effective) theory with N F − N F at the scale M q . µ [GeV] α s ( µ ) m c ( µ ) [GeV] m b ( µ ) [GeV] m t ( µ ) [GeV]1 . + ( M + c ) 3 . × − . × . × . × . × − . × − . × . × .
18 ( m b ) 2 . × − . × − . × . × . − ( M b − ) 2 . × − . × − . × . × . + ( M + b ) 2 . × − . × − . × . ×
10 1 . × − . × − . × . ×
20 1 . × − . × − . × . ×
30 1 . × − . × − . × . ×
40 1 . × − . × − . × . ×
50 1 . × − . × − . × . ×
60 1 . × − . × − . × . ×
70 1 . × − . × − . × . ×
80 1 . × − . × − . × . ×
90 1 . × − . × − . × . ×
100 1 . × − . × − . × . ×
110 1 . × − . × − . × . ×
120 1 . × − . × − . × . × . . × − . × − . × . ×
130 1 . × − . × − . × . ×
140 1 . × − . × − . × . ×
150 1 . × − . × − . × . ×
160 1 . × − . × − . × . ×
170 1 . × − . × − . × . × . − ( M t − ) 1 . × − . × − . × . × . + ( M + t ) 1 . × − . × − . × . ×
180 1 . × − . × − . × . ×
190 1 . × − . × − . × . ×
200 1 . × − . × − . × . ×
300 1 . × − . × − . × . ×
400 9 . × − . × − . × . ×
500 9 . × − . × − . × . × . × − . × − . × . × A.2 Running of the strong coupling constant and quark masses
We neglect the running of the SU(2) L and U(1) Y electroweak couplings and the leptons masses, measuredexperimentally with great precision. For the running of the strong coupling strength α s ( µ ) and the MSquark masses m q ( µ ), we use the most recent version of RunDec [196, 197] in which five-loop correctionsof the QCD beta function and four-loop decoupling effects are included. The results are shown in Fig. 36and Table 11. We note that α s (125 . . m t ( M t ) = 161 . b quark, the three-loop conversion relation is taken to give M b = 4 .
93 GeV [6]. On the other hand,for the pole mass of c quark, we take the relation between the on-shell charm-quark and bottom-quarkmasses, giving M c = M b − .
41 GeV = 1 .
52 GeV [6, 114, 198].69 μ [ GeV ] α S ( μ ) m tLO ( μ ) m t ( μ )
10 50 100 500 1000140160180200220 μ [ Gev ] T opqua r k M a ss [ G e V ] m b ( μ ) m c ( μ ) m b ( m b )= c ( )= m bLO ( μ ) μ [ GeV ] M S - M a ss [ G e V ] Figure 36: Running of the strong coupling constant α s ( µ ) (upper) and the MS quark masses m t ( µ )(middle) and m b,c ( µ ) (lower). In the upper frame, the vertical line locate the position of M H . In themiddle and lower frames, we also show m LO t,b ( µ ) used in the calculations of Γ( H → γγ ) and the verticallines locate the positions of the pole masses of M t and M b,c . And, in the lower frame, the input valuesfor b and c quark masses are denoted by bullets, see Eq. (A.4). The open circles in the middle and lowerframes denote the positions m LO t ( µ = M t ) = M t and m LO b ( µ = M b ) = M b , respectively, with M t = 172 . M b = 4 .
93 GeV. 70pecifically for the loop-induced decay H → γγ , we use the running masses m t,b ( µ ) which, in theLO, are given by m LO q ( µ ) = M q (cid:20) α s ( µ ) α s ( M q ) (cid:21) / (33 − N F ) . (A.5)Note that m q ( M q ) = M q as denoted by open circles in the middle and lower frames in Fig. 36. B Supersymmetric Contributions to the
H gg , H γγ , and
H Z γ
Form Factors
In this appendix, we present the contributions to the loop-induced
Hgg , Hγγ , and
HZγ form factorsfrom the triangle diagrams in which charginos, charged and/or coloured sfermions, and/or chargedHiggs bosons are running.In the minimal supersymmetric extension of the SM (MSSM), the form factors of ∆ S g,γ and ∆ P g,γ denoting new MSSM contributions to the Hgg, Hγγ vertices are given by:∆ S gi = − (cid:88) (cid:101) f j = (cid:101) t , (cid:101) t , (cid:101) b , (cid:101) b g H i (cid:101) f ∗ j (cid:101) f j v m (cid:101) f j F ( τ i (cid:101) f j ) , ∆ P gi = 0 ; (B.1)∆ S γi = √ g (cid:88) f = (cid:101) χ ± , (cid:101) χ ± g SH i ¯ ff vm f F sf ( τ if ) − (cid:88) (cid:101) f j = (cid:101) t , (cid:101) t , (cid:101) b , (cid:101) b , (cid:101) τ , (cid:101) τ N fC Q f g H i (cid:101) f ∗ j (cid:101) f j v m (cid:101) f j F ( τ i (cid:101) f j )+ g HiH + H − v M H ± F ( τ iH ± ) , ∆ P γi = √ g (cid:88) f = (cid:101) χ ± , (cid:101) χ ± g PH i ¯ ff vm f F pf ( τ if ) , (B.2)where τ ix = M H i / m x with i = 1 , , N fC = 3 for (s)quarksand N fC = 1 for status, respectively. The form factor F ( τ ) is given by F ( τ ) = τ − [ − τ − f ( τ )] , (B.3)which takes the value of 1 / τ = 0.On the other hand, the form factors of ∆ S Zγ and ∆ P Zγ denoting new MSSM contributions to the HZγ vertices may take forms of [63]∆ S Zγi = −√ gc W s W (cid:88) j,k v m (cid:101) χ ± j f (cid:16) m (cid:101) χ ± j , m (cid:101) χ ± k , m (cid:101) χ ± k (cid:17) v Z (cid:101) χ + j (cid:101) χ − k g SH i (cid:101) χ + k (cid:101) χ − j − (cid:88) f = t,b,τ N fC Q f c W s W (cid:34)(cid:88) j,k g H i (cid:101) f ∗ j (cid:101) f k g Z (cid:101) f ∗ k (cid:101) f j v C ( m (cid:101) f j , m (cid:101) f k , m (cid:101) f k ) (cid:35) +2 g H i H + H − c W s W v M H ± I ( τ iH ± , λ H ± ) , ∆ P Zγi = −√ i gc W s W (cid:88) j,k v m (cid:101) χ ± j g (cid:16) m (cid:101) χ ± j , m (cid:101) χ ± k , m (cid:101) χ ± k (cid:17) v Z (cid:101) χ + j (cid:101) χ − k g PH i (cid:101) χ + k (cid:101) χ − j , (B.4)71here τ iH ± = M H i / M H ± with i = 1 , , λ H ± = M Z / M H ± , and N fC = 3 for squarks and N fC = 1 forstaus, respectively. For the explicit form of the three loop functions of f ( m , m , m ), g ( m , m , m ),and C ( m , m , m ), we refer to [199]. Note that they implicitly depend on M H i and M Z . For the Higgscouplings to SUSY particles, see subsubsection 2.4.3 and the relevant Z -boson interactions are given bythe following Lagrangian terms: • Z -sfermion-sfermion L Z (cid:101) f (cid:101) f = − ig Z g Z (cid:101) f ∗ j (cid:101) f i (cid:16) (cid:101) f ∗ j ↔ ∂ µ (cid:101) f i (cid:17) Z µ , (B.5)where g Z = e/ ( s W c W ) and g Z (cid:101) f ∗ j (cid:101) f i = I f U (cid:101) f ∗ Lj U (cid:101) fLi − Q f s W δ ij , (B.6)with I u,ν = +1 / I d,e = − / • Z -chargino-chargino [200] L Z (cid:101) χ + (cid:101) χ − = − g Z (cid:101) χ − i γ µ (cid:16) v Zχ + i (cid:101) χ − j − a Zχ + i (cid:101) χ − j γ (cid:17) (cid:101) χ − j Z µ , (B.7)where the vector and axial–vector couplings are given by v Zχ + i (cid:101) χ − j = 14 (cid:104) ( C L ) i ( C L ) ∗ j + ( C R ) i ( C R ) ∗ j (cid:105) − c W δ ij ,a Zχ + i (cid:101) χ − j = 14 (cid:104) ( C L ) i ( C L ) ∗ j − ( C R ) i ( C R ) ∗ j (cid:105) . (B.8)For completeness, we recall that the Z -boson couplings to the quarks and leptons are given by L Z ¯ ff = − g Z ¯ f γ µ (cid:0) v Z ¯ ff − a Z ¯ ff γ (cid:1) f Z µ , (B.9)with v Z ¯ ff = I f / − Q f s W and a Z ¯ ff = I f / I f and the electric charge Q f of each fermion f . C QCD Corrections to Γ( H → γγ ) : C sf ( τ ) and C pf ( τ ) The scaling factors C sf ( τ ) and C pf ( τ ) for the QCD corrections to the decay width of a Higgs boson H into two photons might be given by [110] C sf ( τ ; ρ ) = C H ( τ ) + C H ( τ ) (cid:20) log τ + log 4 ρ (cid:21) ; C pf ( τ ; ρ ) = C A ( τ ) + C A ( τ ) (cid:20) log τ + log 4 ρ (cid:21) , (C.1)where τ = M H / m q ( µ q ) with the renormalization scale µ q = M H /ρ . As demonstrated in Section 3.5,we take ρ = 2. We note again the running mass m q is normalized as m q ( M q ) = M q .The C H ( τ ) function is given via the following relation [134] F sf ( τ ) C H ( τ ) = − θ (1 + θ + θ + θ )3 (1 − θ ) (cid:104)
108 Li ( θ ) + 144 Li ( − θ ) −
64 Li ( θ ) ln θ Eqs. (10) and (12) in Ref. [133] contain typos, see footnote 3 of Ref. [134]. We also correct the typos in the expressionsgiven in APPENDIX A of Ref. [110].
64 Li ( − θ ) ln θ + 14 Li ( θ ) ln θ + 8 Li ( − θ ) ln θ + 112 ln θ +4 ζ ln θ + 16 ζ ln θ + 18 ζ (cid:105) + 2 θ (1 + θ ) − θ ) (cid:104) −
32 Li ( − θ ) + 16 Li ( − θ ) ln θ − ζ ln θ (cid:105) − θ (7 − θ + 7 θ )3 (1 − θ ) Li ( θ ) + 16 θ (3 − θ + 3 θ )3 (1 − θ ) Li ( θ ) ln θ + 4 θ (5 − θ + 5 θ )3 (1 − θ ) ln(1 − θ ) ln θ + 2 θ (3 + 25 θ − θ + 3 θ )9 (1 − θ ) ln θ + 8 θ (1 − θ + θ )3 (1 − θ ) ζ + 8 θ (1 − θ ) ln θ − θ (1 + θ )(1 − θ ) ln θ − θ − θ ) , (C.2)where θ is a τ -dependent function defined by θ ≡ θ ( τ ) = √ − τ − − √ − τ − + 1 . (C.3)The three values, ζ , ζ and ζ , of the Riemann’s zeta function are given by ζ = π , ζ = 1 . , ζ = π , (C.4)and the polylogarithm function is defined by a power series in a complex variable z as follows Li n ( z ) = ∞ (cid:88) k =1 z k k n , n = 1 , , , · · · . (C.5)For analytic continuation to the complex τ plane, the replacement τ → τ + 0 i is understood. The C H ( τ ) function is given via [134] F sf ( τ ) C H ( τ ) = 2 τ − (cid:20) τ + ( τ − f ( τ ) − ( τ − τ d f ( τ )d τ (cid:21) , (C.6)with the function f ( τ ) defined in Eq. (121).On the other hand, the C A ( τ ) and C A ( τ ) functions are given via the following relations [134] F pf ( τ ) C A ( τ ) = − θ (1 + θ )(1 − θ ) (1 + θ ) (cid:26)
72 Li ( θ ) + 96 Li ( − θ ) − (cid:104) Li ( θ ) + Li ( − θ ) (cid:105) ln θ + 283 Li ( θ ) ln θ + 163 Li ( − θ ) ln θ + 118 ln θ + 83 ζ ln θ + 323 ζ ln θ + 12 ζ (cid:27) + θ (1 − θ ) (cid:104) −
563 Li ( θ ) −
643 Li ( − θ ) + 16 Li ( θ ) ln θ + 323 Li ( − θ ) ln θ + 203 ln(1 − θ ) ln θ − ζ ln θ + 83 ζ (cid:105) + 2 θ (1 + θ )3 (1 − θ ) ln θ ; (C.7) F pf ( τ ) C A ( τ ) = 2 τ − (cid:20) f ( τ ) − τ d f ( τ )d τ (cid:21) , (C.8) It is not yet clear whether ζ is given in a compact form or not, unlike ζ , . Li n ( z ) has a branch cut discontinuity in the complex z plane running from 1 to ∞ . D Input parameters for the most general 2HDM potential
In this appendix, we work out the relations among the parameters needed to fully specify the mostgeneral 2HDM potential, the masses of neutral and charged Higgs bosons, and the mixing matrix O .In subsection 2.3, we demonstrate that one needs all the elements of the following set of parameters I = { v , t β , | m | ; λ , λ , λ , λ , | λ | , | λ | , | λ | , φ + 2 ξ , φ + ξ , φ + ξ ; sign[cos( φ + ξ )] } , (D.1)to fully specify the most general 2HDM scalar potential, see Eq. (37). The set I contains 13 parametersplus 1 sign with sin( φ + ξ ) being determined by the third CP-odd tadpole condition in Eq. (35).Alternatively to the set I , one may use the following equivalent set I (cid:48) = { v , t β , (cid:60) e( m e iξ ) ; (D.2) λ , λ , λ , λ , (cid:60) e( λ e iξ ) , (cid:60) e( λ e iξ ) , (cid:60) e( λ e iξ ) , (cid:61) m( λ e iξ ) , (cid:61) m( λ e iξ ) , (cid:61) m( λ e iξ ) } . The above set I (cid:48) contains 10 parameters for the real and complex quartic couplings λ − and any 7of them, in principle, can be traded with the 4 masses of charged and neutral Higgs bosons and the 3independent angles of the 3 × O by judiciously exploiting Eq. (36) and thematrix relation O T M O = diag( M H , M H , M H ). By choosing (cid:61) m( λ e iξ ), (cid:60) e( λ e iξ ), and (cid:60) e( λ e iξ )as independent input parameters, one may use the following set of more physical parameters: P = { v , t β , (cid:60) e( m e iξ ) ; M H , M H , M H , M H ± , { O } ; (cid:61) m( λ e iξ ) , (cid:60) e( λ e iξ ) , (cid:60) e( λ e iξ ) } . (D.3)Explicitly, we find that the 7 quartic couplings of λ , λ , λ , λ , (cid:60) e( λ e iξ ), (cid:61) m( λ e iξ ), and (cid:61) m( λ e iξ )in the set I (cid:48) can be expressed in terms of M H , , , M H ± , and the elements of the mixing matrix O inthe set P as follows: λ = s β v c β (cid:60) e( m e iξ ) + O φ v c β M H + O φ v c β M H + O φ v c β M H − t β (cid:60) e( λ e iξ ) + t β (cid:60) e( λ e iξ ) ,λ = c β v s β (cid:60) e( m e iξ ) + O φ v s β M H + O φ v s β M H + O φ v s β M H + 14 t β (cid:60) e( λ e iξ ) − t β (cid:60) e( λ e iξ ) ,λ = 1 v s β c β (cid:60) e( m e iξ ) + 2 v M H ± + O φ O φ v s β c β M H + O φ O φ v s β c β M H + O φ O φ v s β c β M H − t β (cid:60) e( λ e iξ ) − t β (cid:60) e( λ e iξ ) ,λ = − v s β c β (cid:60) e( m e iξ ) − v M H ± + O a v M H + O a v M H + O a v M H − t β (cid:60) e( λ e iξ ) − t β (cid:60) e( λ e iξ ) , (cid:60) e( λ e iξ ) = − v s β (cid:60) e( m e iξ ) − O a v M H − O a v M H − O a v M H − t β (cid:60) e( λ e iξ ) − t β (cid:60) e( λ e iξ ) , (cid:61) m( λ e iξ ) = − O φ O a v c β M H − O φ O a v c β M H − O φ O a v c β M H − t β (cid:61) m( λ e iξ ) , (cid:61) m( λ e iξ ) = − O φ O a v s β M H − O φ O a v s β M H − O φ O a v s β M H − t β (cid:61) m( λ e iξ ) . (D.4) For M , see Eq. (40) and the two subsequent relations following it.
74e find our results are consistent with those presented in Refs. [201, 202]. We note that λ can beobtained from λ or vice versa by exchanging c β ↔ s β , φ ↔ φ , and λ ↔ λ . The same observationcould be applied for (cid:61) m( λ e iξ ) and (cid:61) m( λ e iξ ) which are vanishing when each Higgs boson is purelyCP-even or CP-odd state and (cid:61) m( λ e iξ ) = 0. About λ and λ , we note that λ + λ is independentof (cid:60) e( m e iξ ) and M H ± , the neutral Higgs mass contributions to λ ( λ ) are involved with only theCP-even (CP-odd) components of each Higgs boson, and the contributions from (cid:60) e( λ e iξ ) and (cid:60) e( λ e iξ )are in common. In passing, we check that Eq. (41) for the difference between λ / (cid:60) e( λ e iξ ) issatisfied by noting the relation O a M H + O a M H + O a M H = M A . E Cubic Higgs-boson self-couplings in 2HDMs
In this appendix, we apply the relations among the 2HDM input parameters obtained in Appendix Dto derive cubic Higgs-boson self-couplings when the lightest Higgs boson is purely CP even as assumedin subsubsection 3.7.2 by taking the following O matrix: O = − s α c α c ω c α s ω c α s α c ω s α s ω − s ω c ω . (E.1)In this case, using Eq. (55), Eq. (57) for the cubic self–couplings of the Higgs weak eigenstates, andEq. (D.4) for the conversion relations, we find the couplings of the heavy neural Higgs bosons H , toa pair of the lightest Higgs bosons are given by g H H H = − c ω s β (cid:104) s β c β − α + 6 c β s β − α c β − α − s β c β − α (cid:105) (cid:18) (cid:60) e( m e iξ ) v (cid:19) − c ω s β (cid:104) s β c β − α + 2 c β s β − α c β − α − s β c β − α (cid:105) (cid:20)(cid:18) M H v (cid:19) + 12 (cid:18) M H v (cid:19)(cid:21) − s ω c β − α s β (cid:61) m( λ e iξ ) − c ω s β − α c β − α s β (cid:60) e( λ e iξ ) + 3 c ω s β − α c β − α c β (cid:60) e( λ e iξ ) ,g H H H = − s ω s β (cid:104) s β c β − α + 6 c β s β − α c β − α − s β c β − α (cid:105) (cid:18) (cid:60) e( m e iξ ) v (cid:19) − s ω s β (cid:104) s β c β − α + 2 c β s β − α c β − α − s β c β − α (cid:105) (cid:20)(cid:18) M H v (cid:19) + 12 (cid:18) M H v (cid:19)(cid:21) + c ω c β − α s β (cid:61) m( λ e iξ ) − s ω s β − α c β − α s β (cid:60) e( λ e iξ ) + 3 s ω s β − α c β − α c β (cid:60) e( λ e iξ ) . (E.2)We note that g H H H /c ω = g H H H /s ω when (cid:61) m( λ e iξ ) = 0 and M H = M H are taken and thecontributions proportional to the input quartic couplings are suppressed by the factor of c β − α . Wefurther note that the couplings g H H H and g H H H are vanishing in the limit where g H V V = g H V V = 0or c β − α = 0. Otherwise, they are non-vanishing. To be more specific, as in subsubsection 3.7.2, we take δ = √ (cid:15) (cid:18) M H M H (cid:19) , δ = √ (cid:15) (cid:18) M H M H (cid:19) , (E.3)together with Eq. (E.1). These specific choices of δ , fully fix all the elements of the 3 × O in terms of the masses of the neutral Higgs bosons and the parameter (cid:15) through s α = −√ − (cid:15) c β + √ (cid:15) s β , c α = √ − (cid:15) s β + √ (cid:15) c β , Note we are taking c ω > s ω > ω = M H (cid:113) M H + M H , s ω = M H (cid:113) M H + M H , (E.4)where (cid:15) = (cid:15) (cid:18) M H M H + M H M H (cid:19) . (E.5)Incidentally, we note s β − α = √ − (cid:15) and c β − α = √ (cid:15) . Taking (cid:15) = 0 . M H = 125 . M H ∼ M H , we find the couplings of the heavy neural Higgs bosons H , to a pair of the lightest Higgsbosons are given by g H H H (cid:39) g H H H (cid:39) − . − . s β (cid:18) (cid:60) e( m e iξ ) M H (cid:19) , (E.6)keeping the two most significant contributions: the first term comes from − s ω c β − α M H v and the secondone from − s ω s β c β − α (cid:60) e( m e iξ ) v . We observe that it is easy to achieve | g H H H | = | g H H H | = 0 . (cid:12)(cid:12) (cid:60) e( m e iξ ) (cid:12)(cid:12) /s β ∼ M H . Incidentally, we find g H H H = 2 s β ( s β s β − α c β − α + c β c β − α ) (cid:18) (cid:60) e( m e iξ ) v (cid:19) + 12 s β (cid:104) s β s β − α + 2 s β s β − α c β − α + 2 c β c β − α (cid:105) (cid:18) M H v (cid:19) + c β − α s β (cid:60) e( λ e iξ ) − c β − α c β (cid:60) e( λ e iξ ) , (E.7)which becomes the SM coupling of M H / v when c β − α = 0, see Eq. (6).Finally, we address the contributions from the charged-Higgs-boson loops to the decay processes ofneutral Higgs bosons into two photons in the 2HDM and the MSSM which are mentioned in subsubsec-tion 3.7.2 and subsection 5.3. Using Eq. (55), Eq. (58), and Eq. (D.4), we obtain the following couplingsof neutral Higgs bosons to a pair of charged Higgs bosons: g H H + H − = 2 s β − α (cid:18) M H ± v (cid:19) + 4 s β ( s β s β − α + c β c β − α ) (cid:18) (cid:60) e( m e iξ ) v (cid:19) + 1 s β ( s β s β − α + 2 c β c β − α ) (cid:18) M H v (cid:19) + c β − α s β (cid:60) e( λ e iξ ) − c β − α c β (cid:60) e( λ e iξ ) ,g H H + H − = 2 c ω c β − α (cid:18) M H ± v (cid:19) − c ω s β ( c β s β − α − s β c β − α ) (cid:18) (cid:60) e( m e iξ ) v (cid:19) − c ω s β (2 c β s β − α − s β c β − α ) (cid:18) M H v (cid:19) − s ω s β (cid:61) m( λ e iξ ) − c ω s β − α s β (cid:60) e( λ e iξ ) + c ω s β − α c β (cid:60) e( λ e iξ ) ,g H H + H − = 2 s ω c β − α (cid:18) M H ± v (cid:19) − s ω s β ( c β s β − α − s β c β − α ) (cid:18) (cid:60) e( m e iξ ) v (cid:19) − s ω s β (2 c β s β − α − s β c β − α ) (cid:18) M H v (cid:19) + 2 c ω s β (cid:61) m( λ e iξ ) − s ω s β − α s β (cid:60) e( λ e iξ ) + s ω s β − α c β (cid:60) e( λ e iξ ) . (E.8)76n the 2HDM as well as in the MSSM, the contributions from the charged-Higgs-boson loops to theneutral Higgs boson decays into two photons enter through the form factor ∆ S γi ( H ± ) = g HiH + H − v M H ± F (cid:18) M H i M H ± (cid:19) . (E.9)In the infinite charged-Higgs-boson mass limit, using F (0) = 1 /
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