Introduction
The standard model (SM) of particle physics successfully explains various experimental data at the electroweak (EW) scale. However, several phenomena cannot be explained within the SM, such as neutrino oscillations, the existence of dark matter, the baryon asymmetry of the Universe, and inflation. These phenomena provide strong motivation for physics beyond the SM (BSM).
The discovery of the 125 GeV Higgs boson established the existence of a Higgs field associated with EW symmetry breaking. Although the SM provides a minimal realization of the Higgs mechanism, there is no fundamental principle that determines the Higgs sector, and its structure remains unknown. In many BSM models, the Higgs sector is extended from the SM by introducing additional scalar fields with various representations.
The real Higgs triplet model (RHTM) [1 ] is a simple extension of the SM with an additional real isospin-triplet scalar field. Extensions of the SM with scalar triplets are qualitatively different from those with singlet or doublet scalars. This is because the vacuum expectation values (VEVs) of triplet fields shift the tree-level value of the EW ρ \rho ρ 1 [4 , 5 , 6 ] . Gravitational-wave signatures associated with first-order phase transitions have also been studied in Refs. [7 , 8 , 9 ] .
The collider phenomenology of the additional Higgs bosons in the RHTM has been studied extensively. Direct searches at the LHC can probe the additional neutral and charged Higgs bosons mainly through pair and associated production, while the diphoton decay of the SM-like Higgs boson provides an indirect constraint on the charged Higgs boson mass [10 , 11 , 12 , 13 , 14 , 15 , 16 , 17 , 18 ] . For a charged Higgs boson lighter than the top quark, searches for exotic top-quark decays provide additional constraints [19 ] . The prospects for direct searches at future electron-positron colliders are discussed in Ref. [20 ] .
The signatures of the RHTM can appear in the SM-like Higgs boson observables. At tree level, the couplings of the SM-like Higgs boson are modified from their SM values through scalar mixing effects. In addition, the additional Higgs bosons affect these couplings through radiative corrections. The precision of the Higgs coupling measurements is expected to reach a few-percent level at the High-Luminosity LHC (HL-LHC) [21 , 22 ] and a per-mille level at future Higgs factories, such as the International Linear Collider [23 , 24 , 25 , 26 ] , the Circular Electron-Positron Collider [27 , 28 ] , and the Future Circular Collider [29 , 30 ] . To fully exploit the precision of these measurements, theoretical predictions with comparable precision are essential, requiring the inclusion of radiative corrections.
New physics effects in the SM-like Higgs boson observables have been widely studied, especially in singlet and doublet extensions. In the two-Higgs doublet model (2HDM), one-loop corrections to the Higgs boson couplings [31 , 32 , 33 , 34 , 35 ] , decays of the SM-like Higgs boson [36 , 37 , 38 ] , and decays of heavy Higgs bosons [39 , 40 , 41 , 42 , 43 , 44 , 45 , 46 ] have been investigated. Similar studies have also been performed in the Higgs singlet model (HSM) [47 , 48 , 49 , 50 , 51 ] . Two-loop corrections have been studied [52 , 53 , 54 , 55 , 56 , 57 , 58 ] . As a public tool for the numerical evaluation of higher-order corrections, the authors of this paper have developed the H-COUP program [59 , 60 , 61 ] . Other public tools, such as 2HDECAY [62 ] , ewN2HDECAY [63 ] , EWsHDECAY [51 ] , Prophecy4f [64 ] , and FlexibleDecay [65 ] , are also available. 2HDECAY, ewN2HDECAY, and EWsHDECAY calculate the full NLO corrections to the two-body decays of all Higgs bosons for a given extended Higgs model. Prophecy4f computes full NLO corrections for specific decay modes, namely the four-body decays of CP-even Higgs bosons into four fermions, in the HSM and 2HDMs. FlexibleDecay incorporates various BSM models and computes higher-order corrections to Higgs boson decays without including BSM loop effects. The main difference between H-COUP and these programs is that H-COUP evaluates the full one-loop electroweak corrections to all major decay modes of both the SM-like Higgs boson and additional Higgs bosons in various extended Higgs models. These calculations are performed using a common on-shell renormalization scheme [31 , 66 ] , in which the scalar mixing angles are renormalized in a gauge-independent way using the pinch technique [67 , 68 , 69 , 70 , 71 , 72 ] .2 Z 2 Z_2 Z 2
In the RHTM, the renormalization procedure in the SM must be extended to accommodate the additional input parameter associated with the triplet VEV. In Ref. [76 ] , a renormalization scheme for the electroweak sector based on the effective weak mixing angle of the electron was proposed for the RHTM. This scheme was used to evaluate the electroweak precision observables (EWPOs) at one-loop level [77 , 78 ] and was applied to the complex Higgs triplet model (CHTM) in Ref. [79 ] . The EWPOs were also studied using a renormalization scheme with the W W W [80 , 81 ] , as well as the star scheme [82 ] . The renormalization of the triplet VEV in the MS scheme was studied in Refs. [83 , 84 , 85 ] , and this approach was used in a two-loop calculation of the Higgs di-photon decay rate [86 ] . A different approach based on scalar mixing angles was developed for the CHTM, and the one-loop corrections to Higgs boson couplings were studied [87 , 88 ] . This scheme was formulated in a gauge-invariant way in the CHTM, and its impact on Higgs boson decays was investigated in Ref. [89 ] .
In this paper, we present a renormalization scheme for the RHTM based on the scalar mixing angles and investigate electroweak one-loop corrections to the decays of the SM-like Higgs boson. The counterterms for the scalar mixing angles are determined by the self-energies, and the gauge dependence is eliminated by the pinch technique [67 , 68 , 69 , 70 , 71 , 72 ] . In particular, the charged scalar sector involves additional diagrams arising from the tree-level H ± W ∓ Z H^\pm W^\mp Z H ± W ∓ Z [45 , 66 , 90 ] . We analytically confirm the cancellation of the gauge dependence in the general R ξ R_\xi R ξ
The decay rates of the SM-like Higgs boson are calculated including NLO EW and higher-order QCD corrections. We find that sizable one-loop effects can arise in the decay rates and the Higgs trilinear coupling, while the tree-level effects are rather small due to the small triplet VEV. The resulting patterns of deviations in decay rates of the SM-like Higgs boson can help distinguish the RHTM from other extended Higgs models.
This paper is organized as follows. In § 2 , we define the RHTM Lagrangian. In § 3 , we discuss the renormalization of the RHTM. The numerical study of the Higgs boson decays is presented in § 4 , and conclusions are given in § 5 . The Feynman rules, analytical formulae for effective Higgs vertices and 1PI functions, and the application of the pinch technique to the RHTM are presented in Appendices.
The real Higgs triplet model
The RHTM contains an S U ( 2 ) L \mathrm{SU}(2)_{\mathrm{L}} SU ( 2 ) L Φ \Phi Φ Y = 1 / 2 Y = 1/2 Y = 1/2 Δ \Delta Δ Y = 0 Y = 0 Y = 0
Φ = ( ϕ + 1 2 ( v ϕ + ϕ + i G 0 ) ) , Δ = 1 2 ( Δ 0 Δ + Δ − − Δ 0 ) with Δ 0 = 1 2 ( v Δ + δ ) , (1) \Phi= \begin{pmatrix} \phi^+ \\ \frac{1}{\sqrt{2}}(v_\phi+ \phi+ iG^0) \end{pmatrix}, \qquad\Delta= \frac{1}{\sqrt{2}}\begin{pmatrix} \Delta^0 & \Delta^+ \\ \Delta^- & -\Delta^0 \end{pmatrix} \quad\text{with} \quad\Delta^0 = \frac{1}{\sqrt{2}}(v_\Delta+ \delta), \tag*{(1)} Φ = ( ϕ + 2 1 ( v ϕ + ϕ + i G 0 ) ) , Δ = 2 1 ( Δ 0 Δ − Δ + − Δ 0 ) with Δ 0 = 2 1 ( v Δ + δ ) , (1) where v = v ϕ 2 + 4 v Δ 2 ≃ 246 G e V v = \sqrt{v_\phi^2 + 4v_\Delta^2} \simeq246\,\mathrm{GeV} v = v ϕ 2 + 4 v Δ 2 ≃ 246 GeV
The Higgs potential
The Higgs potential is given by
V ( Φ , Δ ) = m Φ 2 Φ † Φ + m Δ 2 Tr ( Δ † Δ ) + μ Φ † Δ Φ + λ 1 2 ( Φ † Φ ) 2 + λ 2 2 [ Tr ( Δ † Δ ) ] 2 + λ 3 ( Φ † Φ ) Tr ( Δ † Δ ) , (2) V(\Phi,\Delta) = m_\Phi^2\Phi^\dagger\Phi+ m_\Delta^2\operatorname{Tr}\left(\Delta^\dagger\Delta\right) + \mu\Phi^\dagger\Delta\Phi
+ \frac{\lambda_1}{2}\left(\Phi^\dagger\Phi\right)^2 + \frac{\lambda_2}{2}\left[\operatorname{Tr}\left(\Delta^\dagger\Delta\right)\right]^2 + \lambda_3\left(\Phi^\dagger\Phi\right)\operatorname{Tr}\left(\Delta^\dagger\Delta\right), \tag*{(2)} V ( Φ , Δ ) = m Φ 2 Φ † Φ + m Δ 2 Tr ( Δ † Δ ) + μ Φ † ΔΦ + 2 λ 1 ( Φ † Φ ) 2 + 2 λ 2 [ Tr ( Δ † Δ ) ] 2 + λ 3 ( Φ † Φ ) Tr ( Δ † Δ ) , (2) where all parameters are real, and hence the Higgs potential is CP conserving [16 ] . The tadpole terms for ϕ \phi ϕ δ \delta δ
T ϕ = − ∂ V ∂ ϕ ∣ 0 = − ( m Φ 2 + λ 1 2 v ϕ 2 + λ 3 2 v Δ 2 − μ 2 v Δ ) v ϕ , (3) T_{\phi}=-\left.\frac{\partial V}{\partial\phi}\right|_{0}=-\left(m_{\Phi}^{2}+\frac{\lambda_{1}}{2}v_{\phi}^{2}+\frac{\lambda_{3}}{2}v_{\Delta}^{2}-\frac{\mu}{2}v_{\Delta}\right)v_{\phi}, \tag*{(3)} T ϕ = − ∂ ϕ ∂ V 0 = − ( m Φ 2 + 2 λ 1 v ϕ 2 + 2 λ 3 v Δ 2 − 2 μ v Δ ) v ϕ , (3) T δ = − ∂ V ∂ δ ∣ 0 = − ( m Δ 2 + λ 2 2 v Δ 2 + λ 3 2 v ϕ 2 − μ v ϕ 2 4 v Δ ) v Δ , (4) T_{\delta}=-\left.\frac{\partial V}{\partial\delta}\right|_{0}=-\left(m_{\Delta}^{2}+\frac{\lambda_{2}}{2}v_{\Delta}^{2}+\frac{\lambda_{3}}{2}v_{\phi}^{2}-\frac{\mu v_{\phi}^{2}}{4v_{\Delta}}\right)v_{\Delta}, \tag*{(4)} T δ = − ∂ δ ∂ V 0 = − ( m Δ 2 + 2 λ 2 v Δ 2 + 2 λ 3 v ϕ 2 − 4 v Δ μ v ϕ 2 ) v Δ , (4) where ⋯ ∣ 0 \cdots|_{0} ⋯ ∣ 0 T ϕ = T δ = 0 T_{\phi}=T_{\delta}=0 T ϕ = T δ = 0
m Φ 2 = − λ 1 2 v ϕ 2 − λ 3 2 v Δ 2 + μ 2 v Δ , m Δ 2 = − λ 2 2 v Δ 2 − λ 3 2 v ϕ 2 + μ v ϕ 2 4 v Δ . (5) m_{\Phi}^{2}=-\frac{\lambda_{1}}{2}v_{\phi}^{2}-\frac{\lambda_{3}}{2}v_{\Delta}^{2}+\frac{\mu}{2}v_{\Delta},\qquad m_{\Delta}^{2}=-\frac{\lambda_{2}}{2}v_{\Delta}^{2}-\frac{\lambda_{3}}{2}v_{\phi}^{2}+\frac{\mu v_{\phi}^{2}}{4v_{\Delta}}. \tag*{(5)} m Φ 2 = − 2 λ 1 v ϕ 2 − 2 λ 3 v Δ 2 + 2 μ v Δ , m Δ 2 = − 2 λ 2 v Δ 2 − 2 λ 3 v ϕ 2 + 4 v Δ μ v ϕ 2 . (5) In the following expressions, we retain the tadpole-dependent terms in the mass matrices, which are needed for the one-loop calculation in the standard tadpole scheme (see § 3.1 ).
The mass matrices of the charged and CP-even scalar bosons are given by
M ± 2 = ( μ v Δ μ v ϕ 2 μ v ϕ 2 μ v ϕ 2 4 v Δ ) − ( T ϕ v ϕ 0 0 T δ v Δ ) , M e v e n 2 = ( M 11 2 M 12 2 M 21 2 M 22 2 ) − ( T ϕ v ϕ 0 0 T δ v Δ ) , (6) \mathcal{M}_{\pm}^{2}=
\begin{pmatrix}
\mu v_{\Delta} & \frac{\mu v_{\phi}}{2}\\
\frac{\mu v_{\phi}}{2} & \frac{\mu v_{\phi}^{2}}{4v_{\Delta}}
\end{pmatrix}
-
\begin{pmatrix}
\frac{T_{\phi}}{v_{\phi}} & 0\\
0 & \frac{T_{\delta}}{v_{\Delta}}
\end{pmatrix},
\qquad
\mathcal{M}_{\mathrm{even}}^{2}=
\begin{pmatrix}
\mathcal{M}_{11}^{2} & \mathcal{M}_{12}^{2}\\
\mathcal{M}_{21}^{2} & \mathcal{M}_{22}^{2}
\end{pmatrix}
-
\begin{pmatrix}
\frac{T_{\phi}}{v_{\phi}} & 0\\
0 & \frac{T_{\delta}}{v_{\Delta}}
\end{pmatrix}, \tag*{(6)} M ± 2 = ( μ v Δ 2 μ v ϕ 2 μ v ϕ 4 v Δ μ v ϕ 2 ) − ( v ϕ T ϕ 0 0 v Δ T δ ) , M even 2 = ( M 11 2 M 21 2 M 12 2 M 22 2 ) − ( v ϕ T ϕ 0 0 v Δ T δ ) , (6) where
M 11 2 = λ 1 v ϕ 2 , M 22 2 = λ 2 v Δ 2 + μ v ϕ 2 4 v Δ , M 12 2 = M 21 2 = λ 3 v ϕ v Δ − μ v ϕ 2 . (7) \mathcal{M}_{11}^{2}=\lambda_{1}v_{\phi}^{2},\qquad\mathcal{M}_{22}^{2}=\lambda_{2}v_{\Delta}^{2}+\frac{\mu v_{\phi}^{2}}{4v_{\Delta}},\qquad\mathcal{M}_{12}^{2}=\mathcal{M}_{21}^{2}=\lambda_{3}v_{\phi}v_{\Delta}-\frac{\mu v_{\phi}}{2}. \tag*{(7)} M 11 2 = λ 1 v ϕ 2 , M 22 2 = λ 2 v Δ 2 + 4 v Δ μ v ϕ 2 , M 12 2 = M 21 2 = λ 3 v ϕ v Δ − 2 μ v ϕ . (7) We define the mass eigenstates as
\begin{pmatrix}
\phi^{\pm}\\
\Delta^{\pm}
\end{pmatrix}
=R(\beta)
\begin{pmatrix}
G^{\pm}\\
H^{\pm}
\end{pmatrix},
\qquad
\begin{pmatrix}
\phi\\
\delta
\end{pmatrix}
=R(\alpha)
\begin{pmatrix}
h\\
H
\end{pmatrix},
\qquad
\text{with }R(\theta)=
\begin{pmatrix}
c_{\theta} & -s_{\theta}\\
s_{\theta} & c_{\theta}
\end{pmatrix}, \tag*{(8)}\end{pmatrix}
where c θ = cos θ c_{\theta}=\cos\theta c θ = cos θ s θ = sin θ s_{\theta}=\sin\theta s θ = sin θ
tan β = − 2 v Δ v ϕ , tan 2 α = 2 M 12 2 M 11 2 − M 22 2 = v ϕ v Δ 2 λ 3 v Δ 2 − μ v Δ λ 1 v ϕ 2 − λ 2 v Δ 2 − μ v ϕ 2 / ( 4 v Δ ) . (9) \tan\beta=-\frac{2v_{\Delta}}{v_{\phi}},\qquad
\tan2\alpha=\frac{2\mathcal{M}_{12}^{2}}{\mathcal{M}_{11}^{2}-\mathcal{M}_{22}^{2}}
=\frac{v_{\phi}}{v_{\Delta}}\frac{2\lambda_{3}v_{\Delta}^{2}-\mu v_{\Delta}}{\lambda_{1}v_{\phi}^{2}-\lambda_{2}v_{\Delta}^{2}-\mu v_{\phi}^{2}/(4v_{\Delta})}. \tag*{(9)} tan β = − v ϕ 2 v Δ , tan 2 α = M 11 2 − M 22 2 2 M 12 2 = v Δ v ϕ λ 1 v ϕ 2 − λ 2 v Δ 2 − μ v ϕ 2 / ( 4 v Δ ) 2 λ 3 v Δ 2 − μ v Δ . (9) The tadpoles in the mass eigenbasis are given by
\begin{pmatrix}
T_{h}\\
T_{H}
\end{pmatrix}
=R(-\alpha)
\begin{pmatrix}
T_{\phi}\\
T_{\delta}
\end{pmatrix}
=
\begin{pmatrix}
c_{\alpha}T_{\phi}+s_{\alpha}T_{\delta}\\
-s_{\alpha}T_{\phi}+c_{\alpha}T_{\delta}
\end{pmatrix}. \tag*{(10)}\end{pmatrix}
The mass matrix of the charged scalar bosons becomes
R ( − β ) M ± 2 R ( β ) = ( 0 0 0 m H ± 2 ) − ( T G + G − T G + H − T H + G − T H + H − ) , (11) R(-\beta)\mathcal{M}_{\pm}^{2}R(\beta)=
\begin{pmatrix}
0 & 0\\
0 & m_{H^{\pm}}^{2}
\end{pmatrix}
-
\begin{pmatrix}
T_{G^{+}G^{-}} & T_{G^{+}H^{-}}\\
T_{H^{+}G^{-}} & T_{H^{+}H^{-}}
\end{pmatrix}, \tag*{(11)} R ( − β ) M ± 2 R ( β ) = ( 0 0 0 m H ± 2 ) − ( T G + G − T H + G − T G + H − T H + H − ) , (11) where
m H ± 2 = μ 4 v Δ v 2 (12) m_{H^\pm}^{2}=\frac{\mu}{4v_{\Delta}}v^{2} \tag*{(12)} m H ± 2 = 4 v Δ μ v 2 (12) and
T G + G − = T ϕ v ϕ c β 2 + T δ v Δ s β 2 , T H + H − = T ϕ v ϕ s β 2 + T δ v Δ c β 2 (13) T_{G^+G^-}=\frac{T_{\phi}}{v_{\phi}}c_{\beta}^{2}+\frac{T_{\delta}}{v_{\Delta}}s_{\beta}^{2},\qquad T_{H^+H^-}=\frac{T_{\phi}}{v_{\phi}}s_{\beta}^{2}+\frac{T_{\delta}}{v_{\Delta}}c_{\beta}^{2} \tag*{(13)} T G + G − = v ϕ T ϕ c β 2 + v Δ T δ s β 2 , T H + H − = v ϕ T ϕ s β 2 + v Δ T δ c β 2 (13) T G + H − = T H + G − = − ( T ϕ v ϕ − T δ v Δ ) s β c β . (14) T_{G^+H^-}=T_{H^+G^-}=-\left(\frac{T_{\phi}}{v_{\phi}}-\frac{T_{\delta}}{v_{\Delta}}\right)s_{\beta}c_{\beta}. \tag*{(14)} T G + H − = T H + G − = − ( v ϕ T ϕ − v Δ T δ ) s β c β . (14) The mass matrix of the CP-even scalar bosons becomes
R ( − α ) M e v e n 2 R ( α ) = ( m h 2 0 0 m H 2 ) − ( T h h T h H T H h T H H ) , (15) R(-\alpha)\mathcal{M}_{\mathrm{even}}^{2}R(\alpha)=\begin{pmatrix}m_{h}^{2}&0\\0&m_{H}^{2}\end{pmatrix}-\begin{pmatrix}T_{hh}&T_{hH}\\T_{Hh}&T_{HH}\end{pmatrix}, \tag*{(15)} R ( − α ) M even 2 R ( α ) = ( m h 2 0 0 m H 2 ) − ( T hh T H h T h H T H H ) , (15) where
m h 2 = M 11 2 c α 2 + M 22 2 s α 2 + 2 M 12 2 s α c α , (16) m_{h}^{2}=\mathcal{M}_{11}^{2}c_{\alpha}^{2}+\mathcal{M}_{22}^{2}s_{\alpha}^{2}+2\mathcal{M}_{12}^{2}s_{\alpha}c_{\alpha}, \tag*{(16)} m h 2 = M 11 2 c α 2 + M 22 2 s α 2 + 2 M 12 2 s α c α , (16) m H 2 = M 11 2 s α 2 + M 22 2 c α 2 − 2 M 12 2 s α c α , (17) m_{H}^{2}=\mathcal{M}_{11}^{2}s_{\alpha}^{2}+\mathcal{M}_{22}^{2}c_{\alpha}^{2}-2\mathcal{M}_{12}^{2}s_{\alpha}c_{\alpha}, \tag*{(17)} m H 2 = M 11 2 s α 2 + M 22 2 c α 2 − 2 M 12 2 s α c α , (17) and
T h h = T ϕ v ϕ c α 2 + T δ v Δ s α 2 , T H H = T ϕ v ϕ s α 2 + T δ v Δ c α 2 , T h H = T H h = − ( T ϕ v ϕ − T δ v Δ ) s α c α . (18) T_{hh}=\frac{T_{\phi}}{v_{\phi}}c_{\alpha}^{2}+\frac{T_{\delta}}{v_{\Delta}}s_{\alpha}^{2},\qquad T_{HH}=\frac{T_{\phi}}{v_{\phi}}s_{\alpha}^{2}+\frac{T_{\delta}}{v_{\Delta}}c_{\alpha}^{2},\qquad T_{hH}=T_{Hh}=-\left(\frac{T_{\phi}}{v_{\phi}}-\frac{T_{\delta}}{v_{\Delta}}\right)s_{\alpha}c_{\alpha}. \tag*{(18)} T hh = v ϕ T ϕ c α 2 + v Δ T δ s α 2 , T H H = v ϕ T ϕ s α 2 + v Δ T δ c α 2 , T h H = T H h = − ( v ϕ T ϕ − v Δ T δ ) s α c α . (18) The potential parameters can be written as
μ = 4 m H ± 2 v 2 v Δ , (19) \mu=\frac{4m_{H^\pm}^{2}}{v^{2}}v_{\Delta}, \tag*{(19)} μ = v 2 4 m H ± 2 v Δ , (19) λ 1 = 1 v ϕ 2 ( m h 2 c α 2 + m H 2 s α 2 ) , (20) \lambda_{1}=\frac{1}{v_{\phi}^{2}}\left(m_{h}^{2}c_{\alpha}^{2}+m_{H}^{2}s_{\alpha}^{2}\right), \tag*{(20)} λ 1 = v ϕ 2 1 ( m h 2 c α 2 + m H 2 s α 2 ) , (20) λ 2 = 1 v Δ 2 ( m h 2 s α 2 + m H 2 c α 2 − v ϕ 2 v 2 m H ± 2 ) , (21) \lambda_{2}=\frac{1}{v_{\Delta}^{2}}\left(m_{h}^{2}s_{\alpha}^{2}+m_{H}^{2}c_{\alpha}^{2}-\frac{v_{\phi}^{2}}{v^{2}}m_{H^\pm}^{2}\right), \tag*{(21)} λ 2 = v Δ 2 1 ( m h 2 s α 2 + m H 2 c α 2 − v 2 v ϕ 2 m H ± 2 ) , (21) λ 3 = 2 m H ± 2 v 2 + m h 2 − m H 2 v ϕ v Δ s α c α . (22) \lambda_{3}=\frac{2m_{H^\pm}^{2}}{v^{2}}+\frac{m_{h}^{2}-m_{H}^{2}}{v_{\phi}v_{\Delta}}s_{\alpha}c_{\alpha}. \tag*{(22)} λ 3 = v 2 2 m H ± 2 + v ϕ v Δ m h 2 − m H 2 s α c α . (22) The scalar couplings, such as g ϕ i ϕ j ϕ k g_{\phi_i\phi_j\phi_k} g ϕ i ϕ j ϕ k g ϕ i ϕ j ϕ k ϕ l g_{\phi_i\phi_j\phi_k\phi_l} g ϕ i ϕ j ϕ k ϕ l § 7 .
Kinetic term and Yukawa interaction
The kinetic term is given by
L K i n = ( D μ Φ ) † ( D μ Φ ) + Tr [ ( D μ Δ ) † D μ Δ ] , (23) \mathcal{L}_{\mathrm{Kin}}=(D_{\mu}\Phi)^{\dagger}(D^{\mu}\Phi)+\operatorname{Tr}\left[(D_{\mu}\Delta)^{\dagger}D^{\mu}\Delta\right], \tag*{(23)} L Kin = ( D μ Φ ) † ( D μ Φ ) + Tr [ ( D μ Δ ) † D μ Δ ] , (23) where the covariant derivatives are defined as
D μ Φ = ( ∂ μ − i g τ a W μ a − i g ′ Y B μ ) Φ , D μ Δ = ∂ μ Δ − i g [ τ a W μ a , Δ ] , (24) D_{\mu}\Phi=\left(\partial_{\mu}-ig\tau^{a}W_{\mu}^{a}-ig'YB_{\mu}\right)\Phi,\qquad D_{\mu}\Delta=\partial_{\mu}\Delta-ig\left[\tau^{a}W_{\mu}^{a},\Delta\right], \tag*{(24)} D μ Φ = ( ∂ μ − i g τ a W μ a − i g ′ Y B μ ) Φ , D μ Δ = ∂ μ Δ − i g [ τ a W μ a , Δ ] , (24) where [ A , B ] = A B − B A [A,B]=AB-BA [ A , B ] = A B − B A S U ( 2 ) L \mathrm{SU}(2)_{L} SU ( 2 ) L τ a = σ a / 2 \tau^{a}=\sigma^{a}/2 τ a = σ a /2 σ a \sigma^{a} σ a
m W 2 = g 2 4 v 2 , m Z 2 = g Z 2 4 v ϕ 2 . (25) m_{W}^{2}=\frac{g^{2}}{4}v^{2},\qquad m_{Z}^{2}=\frac{g_{Z}^{2}}{4}v_{\phi}^{2}. \tag*{(25)} m W 2 = 4 g 2 v 2 , m Z 2 = 4 g Z 2 v ϕ 2 . (25) with g Z = g / c W g_{Z}=g/c_{W} g Z = g / c W tan θ W = g ′ / g \tan\theta_{W}=g^{\prime}/g tan θ W = g ′ / g
ρ 0 = m W 2 m Z 2 c W 2 = 1 + tan 2 β . (26) \rho_{0}=\frac{m_{W}^{2}}{m_{Z}^{2}c_{W}^{2}}=1+\tan^{2}\beta. \tag*{(26)} ρ 0 = m Z 2 c W 2 m W 2 = 1 + tan 2 β . (26) The scalar-gauge couplings, such as g ϕ V 1 V 2 g_{\phi V_{1}V_{2}} g ϕ V 1 V 2 g ϕ 1 ϕ 2 V g_{\phi_{1}\phi_{2}V} g ϕ 1 ϕ 2 V g ϕ 1 ϕ 2 V 1 V 2 g_{\phi_{1}\phi_{2}V_{1}V_{2}} g ϕ 1 ϕ 2 V 1 V 2 § 7 .
Since the triplet scalar field Δ \Delta Δ L Y \mathcal{L}_{Y} L Y
L Y ⊃ − ∑ f = u , d , e m f v [ f ˉ f ( ζ h h + ζ H H ) − 2 i I f f ˉ γ 5 f ζ G 0 G 0 ] + [ ( 2 m u v V u d u ˉ P L d − 2 m d v V u d u ˉ P R d ) ( ζ G ± G + + ζ H ± H + ) + h . c . ] − [ 2 m e v ν ˉ P R e ( ζ G ± G + + ζ H ± H + ) + h . c . ] , (27) \begin{aligned}
\mathcal{L}_{Y}\supset{}&-\sum_{f=u,d,e}\frac{m_{f}}{v}\left[\bar{f}f\left(\zeta_{h}h+\zeta_{H}H\right)-2iI_{f}\bar{f}\gamma_{5}f\zeta_{G^{0}}G^{0}\right]\\
&+\left[\left(\frac{\sqrt{2}m_{u}}{v}V_{ud}\bar{u}P_{L}d-\frac{\sqrt{2}m_{d}}{v}V_{ud}\bar{u}P_{R}d\right)\left(\zeta_{G^{\pm}}G^{+}+\zeta_{H^{\pm}}H^{+}\right)+\mathrm{h.c.}\right]\\
&-\left[\frac{\sqrt{2}m_{e}}{v}\bar{\nu}P_{R}e\left(\zeta_{G^{\pm}}G^{+}+\zeta_{H^{\pm}}H^{+}\right)+\mathrm{h.c.}\right], \tag*{(27)}
\end{aligned} L Y ⊃ − f = u , d , e ∑ v m f [ f ˉ f ( ζ h h + ζ H H ) − 2 i I f f ˉ γ 5 f ζ G 0 G 0 ] + [ ( v 2 m u V u d u ˉ P L d − v 2 m d V u d u ˉ P R d ) ( ζ G ± G + + ζ H ± H + ) + h.c. ] − [ v 2 m e ν ˉ P R e ( ζ G ± G + + ζ H ± H + ) + h.c. ] , (27) where I f I_{f} I f f f f V u d V_{ud} V u d ζ X \zeta_{X} ζ X
ζ h = c α / c β , ζ G ± = 1 , ζ G 0 = 1 / c β , (28) \zeta_{h}=c_{\alpha}/c_{\beta},\qquad\zeta_{G^{\pm}}=1,\qquad\zeta_{G^{0}}=1/c_{\beta}, \tag*{(28)} ζ h = c α / c β , ζ G ± = 1 , ζ G 0 = 1/ c β , (28) ζ H = − s α / c β , ζ H ± = − tan β . (29) \zeta_{H}=-s_{\alpha}/c_{\beta},\qquad\zeta_{H^{\pm}}=-\tan\beta. \tag*{(29)} ζ H = − s α / c β , ζ H ± = − tan β . (29) Theoretical constraints
The Higgs potential has to be bounded from below at large field values to ensure the stability of the theory. The necessary and sufficient conditions are given by [11 , 16 ]
λ 1 > 0 , λ 2 > 0 , λ 3 + λ 1 λ 2 > 0. (30) \lambda_{1}>0,\qquad\lambda_{2}>0,\qquad\lambda_{3}+\sqrt{\lambda_{1}\lambda_{2}}>0. \tag*{(30)} λ 1 > 0 , λ 2 > 0 , λ 3 + λ 1 λ 2 > 0. (30) Tree-level unitarity requires that the absolute values of the eigenvalues of the two-to-two scalar scattering matrix do not exceed ζ L Q T = 1 \zeta_{\mathrm{LQT}}=1 ζ LQT = 1 [91 , 92 ] or 1 / 2 1/2 1/2 [93 , 94 ] , depending on the choice of unitarity criterion. In the RHTM, the independent eigenvalues are given by [11 , 16 ]
16 π a 1 = λ 1 , 16 π a 2 = λ 2 , 16 π a 3 = λ 3 , 16 π a 4 ± = 1 4 [ 6 λ 1 + 5 λ 2 ± ( 6 λ 1 − 5 λ 2 ) 2 + 48 λ 3 2 ] . (31) \begin{aligned}
16\pi a_{1}&=\lambda_{1},\qquad16\pi a_{2}=\lambda_{2},\qquad16\pi a_{3}=\lambda_{3},\\
16\pi a_{4}^{\pm}&=\frac{1}{4}\left[6\lambda_{1}+5\lambda_{2}\pm\sqrt{(6\lambda_{1}-5\lambda_{2})^{2}+48\lambda_{3}^{2}}\right]. \tag*{(31)}
\end{aligned} 16 π a 1 16 π a 4 ± = λ 1 , 16 π a 2 = λ 2 , 16 π a 3 = λ 3 , = 4 1 [ 6 λ 1 + 5 λ 2 ± ( 6 λ 1 − 5 λ 2 ) 2 + 48 λ 3 2 ] . (31) The tree-level unitarity bound is given by
∣ a i ∣ < ζ L Q T , ( i = 1 , 2 , 3 , 4 ± ) . (32) |a_{i}|<\zeta_{\mathrm{LQT}},\qquad(i=1,2,3,4^{\pm}). \tag*{(32)} ∣ a i ∣ < ζ LQT , ( i = 1 , 2 , 3 , 4 ± ) . (32) In this paper, we take ζ L Q T = 1 / 2 \zeta_{\mathrm{LQT}}=1/2 ζ LQT = 1/2
Renormalization of the RHTM
In this section, we formulate the renormalization scheme with the scalar mixing in the RHTM at the one-loop level. We first discuss the tadpole renormalization in the standard tadpole (ST) scheme [95 ] and the Fleischer–Jegerlehner (FJ) scheme [96 ] (see also Refs. [45 , 97 ] ). We then describe the renormalization of the Higgs sector, including the scalar masses, field renormalization constants, and mixing angles. Finally, we discuss the renormalization of the gauge sector, which is different from the SM due to ρ 0 ≠ 1 \rho_0 \ne1 ρ 0 = 1
Tadpole
Standard tadpole scheme
In the ST scheme, we shift the bare tadpole as
T S , B = T S + δ T S , (33) T_{S,\mathrm{B}} = T_S + \delta T_S, \tag*{(33)} T S , B = T S + δ T S , (33) where S S S h h h H H H
Γ ^ S = T S + Γ S 1 P I + δ T S , (34) \widehat{\Gamma}_S = T_S + \Gamma_S^{\mathrm{1PI}} + \delta T_S, \tag*{(34)} Γ S = T S + Γ S 1PI + δ T S , (34) where the 1PI one-point functions Γ S 1 P I \Gamma_S^{\mathrm{1PI}} Γ S 1PI § 9 .§ 1 . We impose the following renormalization conditions.
Γ ^ S = 0. (35) \widehat{\Gamma}_S = 0. \tag*{(35)} Γ S = 0. (35) Then, we obtain
T S = 0 , δ T S = − Γ S 1 P I . (36) T_S = 0,\qquad\delta T_S = -\Gamma_S^{\mathrm{1PI}}. \tag*{(36)} T S = 0 , δ T S = − Γ S 1PI . (36) From Eq. (10) , δ T ϕ \delta T_\phi δ T ϕ δ T δ \delta T_\delta δ T δ
δ T ϕ = c α δ T h − s α δ T H , δ T δ = s α δ T h + c α δ T H . (37) \delta T_\phi= c_\alpha\delta T_h - s_\alpha\delta T_H,\qquad\delta T_\delta= s_\alpha\delta T_h + c_\alpha\delta T_H. \tag*{(37)} δ T ϕ = c α δ T h − s α δ T H , δ T δ = s α δ T h + c α δ T H . (37) In the ST scheme, tadpole-inserted diagrams are canceled by the corresponding tadpole counterterm diagrams. Therefore, we do not need to take into account these diagrams in perturbative calculations. On the other hand, the tadpole counterterms appear in the mass and mixing counterterms. We note that the bare tadpoles are not zero in the ST scheme: T S , B = δ T S = − Γ S 1 P I T_{S,\mathrm{B}} = \delta T_S = -\Gamma_S^{\mathrm{1PI}} T S , B = δ T S = − Γ S 1PI Γ S 1 P I \Gamma_S^{\mathrm{1PI}} Γ S 1PI T S , B T_{S,\mathrm{B}} T S , B
Fleischer-Jegerlehner scheme
In the FJ scheme, we shift the bare VEVs as v i , B → v i , B + Δ v i v_{i,\mathrm{B}} \to v_{i,\mathrm{B}} + \Delta v_i v i , B → v i , B + Δ v i i = ϕ , Δ i = \phi, \Delta i = ϕ , Δ
T ϕ , B → T ϕ , B + Δ T ϕ = T ϕ , B + ( T ϕ , B v ϕ , B − M 11 , B 2 ) Δ v ϕ − M 12 , B 2 Δ v Δ , (38) T_{\phi,\mathrm{B}} \to T_{\phi,\mathrm{B}} + \Delta T_\phi= T_{\phi,\mathrm{B}} + \left(\frac{T_{\phi,\mathrm{B}}}{v_{\phi,\mathrm{B}}} - M_{11,\mathrm{B}}^2\right)\Delta v_\phi- M_{12,\mathrm{B}}^2\Delta v_\Delta, \tag*{(38)} T ϕ , B → T ϕ , B + Δ T ϕ = T ϕ , B + ( v ϕ , B T ϕ , B − M 11 , B 2 ) Δ v ϕ − M 12 , B 2 Δ v Δ , (38) T δ , B → T δ , B + Δ T δ = T δ , B − M 21 , B 2 Δ v ϕ + ( T δ , B v Δ , B − M 22 , B 2 ) Δ v Δ . (39) T_{\delta,\mathrm{B}} \to T_{\delta,\mathrm{B}} + \Delta T_\delta= T_{\delta,\mathrm{B}} - M_{21,\mathrm{B}}^2\Delta v_\phi+ \left(\frac{T_{\delta,\mathrm{B}}}{v_{\Delta,\mathrm{B}}} - M_{22,\mathrm{B}}^2\right)\Delta v_\Delta. \tag*{(39)} T δ , B → T δ , B + Δ T δ = T δ , B − M 21 , B 2 Δ v ϕ + ( v Δ , B T δ , B − M 22 , B 2 ) Δ v Δ . (39) where we have retained terms up to O ( Δ v i ) \mathcal{O}(\Delta v_i) O ( Δ v i ) v ϕ , B v_{\phi,\mathrm{B}} v ϕ , B v Δ , B v_{\Delta,\mathrm{B}} v Δ , B T ϕ , B = T δ , B = 0 T_{\phi,\mathrm{B}}=T_{\delta,\mathrm{B}}=0 T ϕ , B = T δ , B = 0
\begin{pmatrix}
\Delta T_{\phi}\\
\Delta T_{\delta}
\end{pmatrix}
=
-\left.\mathcal{M}^{2}_{\mathrm{even}}\right|_{T_{\phi}=T_{\delta}=0}
\begin{pmatrix}
\Delta v_{\phi}\\
\Delta v_{\Delta}
\end{pmatrix}, \tag*{(40)}\end{pmatrix}
where M e v e n 2 \mathcal{M}^{2}_{\mathrm{even}} M even 2 (6) . Since M e v e n 2 ∣ T ϕ = T δ = 0 \left.\mathcal{M}^{2}_{\mathrm{even}}\right|_{T_{\phi}=T_{\delta}=0} M even 2 T ϕ = T δ = 0 R ( α ) R(\alpha) R ( α )
\begin{pmatrix}
\Delta v_h\\
\Delta v_H
\end{pmatrix}
=
\begin{pmatrix}
-\dfrac{\Delta T_h}{m_h^2}\\
-\dfrac{\Delta T_H}{m_H^2}
\end{pmatrix}, \tag*{(41)}\end{pmatrix}
with
\begin{pmatrix}
\Delta v_h\\
\Delta v_H
\end{pmatrix}
=
R(-\alpha)
\begin{pmatrix}
\Delta v_{\phi}\\
\Delta v_{\Delta}
\end{pmatrix},
\qquad
\begin{pmatrix}
\Delta T_h\\
\Delta T_H
\end{pmatrix}
=
R(-\alpha)
\begin{pmatrix}
\Delta T_{\phi}\\
\Delta T_{\delta}
\end{pmatrix}. \tag*{(42)}\end{pmatrix}
The renormalized one-point functions are given by
Γ ^ S = Γ S 1 P I + Δ T S , (43) \hat{\Gamma}_S=\Gamma_S^{1\mathrm{PI}}+\Delta T_S, \tag*{(43)} Γ ^ S = Γ S 1 PI + Δ T S , (43) where S S S h h h H H H
Γ ^ S = 0. (44) \hat{\Gamma}_S=0. \tag*{(44)} Γ ^ S = 0. (44) Then, we obtain Δ T S = − Γ S 1 P I \Delta T_S=-\Gamma_S^{1\mathrm{PI}} Δ T S = − Γ S 1 PI (41) , we obtain
Δ v S = Γ S 1 P I m S 2 . (45) \Delta v_S=\frac{\Gamma_S^{1\mathrm{PI}}}{m_S^2}. \tag*{(45)} Δ v S = m S 2 Γ S 1 PI . (45) From this expression, we can interpret terms with Δ v S \Delta v_S Δ v S
Scalar sector
In the mass eigenbasis, we shift the bare masses as
m S , B 2 = m S 2 + δ m S 2 , (46) m_{S,\mathrm{B}}^2=m_S^2+\delta m_S^2, \tag*{(46)} m S , B 2 = m S 2 + δ m S 2 , (46) where S = h S=h S = h H H H H ± H^\pm H ±
\begin{pmatrix}
h_{\mathrm{B}}\\
H_{\mathrm{B}}
\end{pmatrix}
=
Z_{\mathrm{even}}^{1/2}
\begin{pmatrix}
h\\
H
\end{pmatrix}
=
\begin{pmatrix}
1+\dfrac{\delta Z_{hh}}{2} & \dfrac{\delta Z_{hH}}{2}\\
\dfrac{\delta Z_{Hh}}{2} & 1+\dfrac{\delta Z_{HH}}{2}
\end{pmatrix}
\begin{pmatrix}
h\\
H
\end{pmatrix}, \tag*{(47)}\end{pmatrix}
\begin{pmatrix}
G_{\mathrm{B}}^\pm\\
H_{\mathrm{B}}^\pm
\end{pmatrix}
=
Z_{\pm}^{1/2}
\begin{pmatrix}
G^\pm\\
H^\pm
\end{pmatrix}
=
\begin{pmatrix}
1+\dfrac{\delta Z_{G^\pm G^\pm}}{2} & \dfrac{\delta Z_{G^\pm H^\pm}}{2}\\
\dfrac{\delta Z_{H^\pm G^\pm}}{2} & 1+\dfrac{\delta Z_{H^\pm H^\pm}}{2}
\end{pmatrix}
\begin{pmatrix}
G^\pm\\
H^\pm
\end{pmatrix}. \tag*{(48)}\end{pmatrix}
For the inverse propagators, we impose
Re Γ i i ( m i 2 ) = 0 , Re d Γ i i ( p 2 ) d p 2 ∣ p 2 = m i 2 = 1 , (49) \operatorname{Re}\Gamma_{ii}(m_i^2)=0,\qquad\operatorname{Re}\left.\frac{\mathrm{d}\Gamma_{ii}(p^2)}{\mathrm{d}p^2}\right|_{p^2=m_i^2}=1, \tag*{(49)} Re Γ ii ( m i 2 ) = 0 , Re d p 2 d Γ ii ( p 2 ) p 2 = m i 2 = 1 , (49) Re Γ i j ( m i 2 ) = 0 , Re Γ i j ( m j 2 ) = 0 ( i ≠ j ) , (50) \operatorname{Re}\Gamma_{ij}(m_i^2)=0,\qquad\operatorname{Re}\Gamma_{ij}(m_j^2)=0\quad(i\ne j), \tag*{(50)} Re Γ ij ( m i 2 ) = 0 , Re Γ ij ( m j 2 ) = 0 ( i = j ) , (50) where ( i , j ) (i,j) ( i , j ) ( h , H ) (h,H) ( h , H ) ( G ± , H ± ) (G^\pm,H^\pm) ( G ± , H ± ) p 2 = 0 p^2=0 p 2 = 0 G ± G^\pm G ±
The inverse propagators are given by
Γ i j ( p 2 ) = i [ ( p 2 − m i 2 ) δ i j + Π ^ i j ( p 2 ) ] , (51) \Gamma_{ij}(p^2)=i\left[(p^2-m_i^2)\delta_{ij}+\widehat{\Pi}_{ij}(p^2)\right], \tag*{(51)} Γ ij ( p 2 ) = i [ ( p 2 − m i 2 ) δ ij + Π ij ( p 2 ) ] , (51) where the renormalized self-energies are given by
Π ^ i j ( p 2 ) = Π i j 1 P I ( p 2 ) + Π i j T a d − δ m i 2 δ i j + δ Z i j 2 ( p 2 − m i 2 ) + δ Z j i 2 ( p 2 − m j 2 ) . (52) \widehat{\Pi}_{ij}(p^2)=\Pi_{ij}^{\mathrm{1PI}}(p^2)+\Pi_{ij}^{\mathrm{Tad}}-\delta m_i^2\delta_{ij}+\frac{\delta Z_{ij}}{2}(p^2-m_i^2)+\frac{\delta Z_{ji}}{2}(p^2-m_j^2). \tag*{(52)} Π ij ( p 2 ) = Π ij 1PI ( p 2 ) + Π ij Tad − δ m i 2 δ ij + 2 δ Z ij ( p 2 − m i 2 ) + 2 δ Z j i ( p 2 − m j 2 ) . (52) The tadpole contributions depend on the renormalization scheme. For the CP-even Higgs bosons, we obtain
Π h h T a d ∣ S T = δ T h h = δ T ϕ v ϕ c α 2 + δ T δ v Δ s α 2 , (53) \Pi_{hh}^{\mathrm{Tad}}\big|_{\mathrm{ST}}=\delta T_{hh}=\frac{\delta T_\phi}{v_\phi}c_\alpha^2+\frac{\delta T_\delta}{v_\Delta}s_\alpha^2, \tag*{(53)} Π hh Tad ST = δ T hh = v ϕ δ T ϕ c α 2 + v Δ δ T δ s α 2 , (53) Π H H T a d ∣ S T = δ T H H = δ T ϕ v ϕ s α 2 + δ T δ v Δ c α 2 , (54) \Pi_{HH}^{\mathrm{Tad}}\big|_{\mathrm{ST}}=\delta T_{HH}=\frac{\delta T_\phi}{v_\phi}s_\alpha^2+\frac{\delta T_\delta}{v_\Delta}c_\alpha^2, \tag*{(54)} Π H H Tad ST = δ T H H = v ϕ δ T ϕ s α 2 + v Δ δ T δ c α 2 , (54) Π h H T a d ∣ S T = δ T h H = − ( δ T ϕ v ϕ − δ T δ v Δ ) s α c α , (55) \Pi_{hH}^{\mathrm{Tad}}\big|_{\mathrm{ST}}=\delta T_{hH}=-\left(\frac{\delta T_\phi}{v_\phi}-\frac{\delta T_\delta}{v_\Delta}\right)s_\alpha c_\alpha, \tag*{(55)} Π h H Tad ST = δ T h H = − ( v ϕ δ T ϕ − v Δ δ T δ ) s α c α , (55) Π h h T a d ∣ F J = 6 g h h h Δ v h + 2 g H h h Δ v H , (56) \Pi_{hh}^{\mathrm{Tad}}\big|_{\mathrm{FJ}}=6g_{hhh}\Delta v_h+2g_{Hhh}\Delta v_H, \tag*{(56)} Π hh Tad FJ = 6 g hhh Δ v h + 2 g H hh Δ v H , (56) Π H H T a d ∣ F J = 2 g h H H Δ v h + 6 g H H H Δ v H , (57) \Pi_{HH}^{\mathrm{Tad}}\big|_{\mathrm{FJ}}=2g_{hHH}\Delta v_h+6g_{HHH}\Delta v_H, \tag*{(57)} Π H H Tad FJ = 2 g h H H Δ v h + 6 g H H H Δ v H , (57) Π h H T a d ∣ F J = 2 g h h H Δ v h + 2 g H H h Δ v H . (58) \Pi_{hH}^{\mathrm{Tad}}\big|_{\mathrm{FJ}}=2g_{hhH}\Delta v_h+2g_{HHh}\Delta v_H. \tag*{(58)} Π h H Tad FJ = 2 g hh H Δ v h + 2 g H H h Δ v H . (58) For the charged scalar bosons, we obtain
Π H + H − T a d ∣ S T = δ T H + H − = δ T ϕ v ϕ s β 2 + δ T δ v Δ c β 2 , (59) \Pi_{H^+H^-}^{\mathrm{Tad}}\big|_{\mathrm{ST}}=\delta T_{H^+H^-}=\frac{\delta T_\phi}{v_\phi}s_\beta^2+\frac{\delta T_\delta}{v_\Delta}c_\beta^2, \tag*{(59)} Π H + H − Tad ST = δ T H + H − = v ϕ δ T ϕ s β 2 + v Δ δ T δ c β 2 , (59) Π H + G − T a d ∣ S T = δ T H + G − = − ( δ T ϕ v ϕ − δ T δ v Δ ) s β c β , (60) \Pi_{H^+G^-}^{\mathrm{Tad}}\big|_{\mathrm{ST}}=\delta T_{H^+G^-}=-\left(\frac{\delta T_\phi}{v_\phi}-\frac{\delta T_\delta}{v_\Delta}\right)s_\beta c_\beta, \tag*{(60)} Π H + G − Tad ST = δ T H + G − = − ( v ϕ δ T ϕ − v Δ δ T δ ) s β c β , (60) in the ST scheme. Those in the FJ scheme are given by
Π H + H − T a d ∣ F J = g h H + H − Δ v h + g H H + H − Δ v H , (61) \Pi_{H^+H^-}^{\mathrm{Tad}}\big|_{\mathrm{FJ}}=g_{hH^+H^-}\Delta v_h+g_{HH^+H^-}\Delta v_H, \tag*{(61)} Π H + H − Tad FJ = g h H + H − Δ v h + g H H + H − Δ v H , (61) Π H + G − T a d ∣ F J = g h H + G − Δ v h + g H H + G − Δ v H . (62) \Pi_{H^+G^-}^{\mathrm{Tad}}\big|_{\mathrm{FJ}}=g_{hH^+G^-}\Delta v_h+g_{HH^+G^-}\Delta v_H. \tag*{(62)} Π H + G − Tad FJ = g h H + G − Δ v h + g H H + G − Δ v H . (62) We note that Π G + G − T a d \Pi_{G^+G^-}^{\mathrm{Tad}} Π G + G − Tad
δ m i 2 = Re [ Π i i 1 P I ( m i 2 ) + Π i i T a d ] , δ Z i i = − d d p 2 Re Π i i 1 P I ( p 2 ) ∣ p 2 = m i 2 , (63) \delta m_i^2=\operatorname{Re}\left[\Pi_{ii}^{\mathrm{1PI}}(m_i^2)+\Pi_{ii}^{\mathrm{Tad}}\right],\qquad\delta Z_{ii}=-\left.\frac{\mathrm{d}}{\mathrm{d}p^2}\operatorname{Re}\Pi_{ii}^{\mathrm{1PI}}(p^2)\right|_{p^2=m_i^2}, \tag*{(63)} δ m i 2 = Re [ Π ii 1PI ( m i 2 ) + Π ii Tad ] , δ Z ii = − d p 2 d Re Π ii 1PI ( p 2 ) p 2 = m i 2 , (63) δ Z i j = 2 m i 2 − m j 2 Re [ Π i j 1 P I ( m j 2 ) + Π i j T a d ] . (64) \delta Z_{ij}=\frac{2}{m_i^2-m_j^2}\operatorname{Re}\left[\Pi_{ij}^{\mathrm{1PI}}(m_j^2)+\Pi_{ij}^{\mathrm{Tad}}\right]. \tag*{(64)} δ Z ij = m i 2 − m j 2 2 Re [ Π ij 1PI ( m j 2 ) + Π ij Tad ] . (64) Mixing angles
We shift the bare mixing angles as
α B = α + δ α , β B = β + δ β . (65) \alpha_{\mathrm{B}}=\alpha+\delta\alpha,\qquad\beta_{\mathrm{B}}=\beta+\delta\beta. \tag*{(65)} α B = α + δ α , β B = β + δ β . (65) We determine the mixing-angle counterterms by the off-diagonal components of the wave-function renormalization matrices [31 ] (see also Ref. [73 ] ). For the CP-even sector, we shift the bare fields in the weak basis as
\begin{pmatrix}\phi_{\mathrm{B}}\\ \delta_{\mathrm{B}}\end{pmatrix}
=Z_{\mathrm{even,weak}}^{1/2}
\begin{pmatrix}\phi\\ \delta\end{pmatrix}. \tag*{(66)}\end{pmatrix}
Then, the bare fields in the mass eigenbasis are renormalized as
\begin{pmatrix}h_{\mathrm{B}}\\ H_{\mathrm{B}}\end{pmatrix}
=Z_{\mathrm{even,mass}}^{1/2}
\begin{pmatrix}h\\ H\end{pmatrix}
=R(-\delta\alpha)\widetilde{Z}_{\mathrm{even}}^{1/2}
\begin{pmatrix}h\\ H\end{pmatrix}, \tag*{(67)}\end{pmatrix}
where we have defined Z ~ e v e n 1 / 2 = R ( − α ) Z e v e n , w e a k 1 / 2 R ( α ) \widetilde{Z}_{\mathrm{even}}^{1/2}=R(-\alpha)Z_{\mathrm{even,weak}}^{1/2}R(\alpha) Z even 1/2 = R ( − α ) Z even , weak 1/2 R ( α )
Let us focus on the structure of the UV-divergent parts. Since ϕ \phi ϕ δ \delta δ S U ( 2 ) L \mathrm{SU}(2)_{\mathrm{L}} SU ( 2 ) L Z e v e n , w e a k 1 / 2 Z_{\mathrm{even,weak}}^{1/2} Z even , weak 1/2 Z ~ e v e n 1 / 2 \widetilde{Z}_{\mathrm{even}}^{1/2} Z even 1/2
Z ~ e v e n 1 / 2 = ( 1 + δ Z h h U V 2 δ C h H U V δ C h H U V 1 + δ Z H H U V 2 ) + UV-finite parts . (68) \widetilde{Z}_{\mathrm{even}}^{1/2}
=
\begin{pmatrix}
1+\dfrac{\delta Z_{hh}^{\mathrm{UV}}}{2} & \delta C_{hH}^{\mathrm{UV}}\\
\delta C_{hH}^{\mathrm{UV}} & 1+\dfrac{\delta Z_{HH}^{\mathrm{UV}}}{2}
\end{pmatrix}
+\text{UV-finite parts}. \tag*{(68)} Z even 1/2 = 1 + 2 δ Z hh UV δ C h H UV δ C h H UV 1 + 2 δ Z H H UV + UV-finite parts . (68) Then, from Eq. (67) , we obtain
Z e v e n , m a s s 1 / 2 = ( 1 + δ Z h h U V 2 δ C h H U V + δ α U V δ C h H U V − δ α U V 1 + δ Z H H U V 2 ) + UV-finite parts . (69) Z_{\mathrm{even,mass}}^{1/2}
=
\begin{pmatrix}
1+\dfrac{\delta Z_{hh}^{\mathrm{UV}}}{2} & \delta C_{hH}^{\mathrm{UV}}+\delta\alpha^{\mathrm{UV}}\\
\delta C_{hH}^{\mathrm{UV}}-\delta\alpha^{\mathrm{UV}} & 1+\dfrac{\delta Z_{HH}^{\mathrm{UV}}}{2}
\end{pmatrix}
+\text{UV-finite parts}. \tag*{(69)} Z even , mass 1/2 = 1 + 2 δ Z hh UV δ C h H UV − δ α UV δ C h H UV + δ α UV 1 + 2 δ Z H H UV + UV-finite parts . (69) Comparing with Eq. (47) , we obtain
δ α U V = 1 4 ( δ Z h H U V − δ Z H h U V ) . (70) \delta\alpha^{\mathrm{UV}}=\frac{1}{4}\left(\delta Z_{hH}^{\mathrm{UV}}-\delta Z_{Hh}^{\mathrm{UV}}\right). \tag*{(70)} δ α UV = 4 1 ( δ Z h H UV − δ Z H h UV ) . (70) Similarly, for the charged sector, we obtain
δ β U V = 1 4 ( δ Z G + H + U V − δ Z H + G + U V ) . (71) \delta\beta^{\mathrm{UV}}=\frac{1}{4}\left(\delta Z_{G^{+}H^{+}}^{\mathrm{UV}}-\delta Z_{H^{+}G^{+}}^{\mathrm{UV}}\right). \tag*{(71)} δ β UV = 4 1 ( δ Z G + H + UV − δ Z H + G + UV ) . (71) While the UV-divergent parts of the mixing-angle counterterms are determined, the finite parts are not uniquely fixed and can depend on the choice of renormalization scheme. We impose the relations in Eqs. (70) and (71) not only on the UV-divergent parts but also on the finite parts of the mixing-angle counterterms.
δ α = 1 4 ( δ Z h H − δ Z H h ) , δ β = 1 4 ( δ Z G + H + − δ Z H + G + ) . (72) \delta\alpha=\frac{1}{4}\left(\delta Z_{hH}-\delta Z_{Hh}\right),\qquad
\delta\beta=\frac{1}{4}\left(\delta Z_{G^{+}H^{+}}-\delta Z_{H^{+}G^{+}}\right). \tag*{(72)} δ α = 4 1 ( δ Z h H − δ Z H h ) , δ β = 4 1 ( δ Z G + H + − δ Z H + G + ) . (72) The mixing angle counterterms defined by Eqs. (72) are gauge-dependent. It is known that the gauge dependence of the mixing angles does not cancel in higher-order amplitudes, leading to gauge dependence in physical quantities such as partial decay widths [45 ] . To eliminate the gauge dependence, we define the mixing angle counterterms using the pinch technique [67 , 68 , 69 , 70 , 71 , 72 ] . We introduce gauge-independent pinched self-energies, in which the gauge dependence is canceled by adding pinch terms:
Π i j ( p 2 ) = Π i j 1 P I ( p 2 ) + Π i j T a d + Π i j P T ( p 2 ) . (73) \Pi_{ij}(p^2) = \Pi_{ij}^{\mathrm{1PI}}(p^2) + \Pi_{ij}^{\mathrm{Tad}} + \Pi_{ij}^{\mathrm{PT}}(p^2). \tag*{(73)} Π ij ( p 2 ) = Π ij 1PI ( p 2 ) + Π ij Tad + Π ij PT ( p 2 ) . (73) The pinch terms of the h h h H H H H + H^+ H + G − G^- G −
Π h H P T ( p 2 ) = − g 2 32 π 2 ( 2 p 2 − m h 2 − m H 2 ) [ c G ± h W ∓ c G ∓ H W ± B 0 ( p 2 ; W , W ) + c H ± h W ∓ c H ∓ H W ± B 0 ( p 2 ; W , H ± ) ] \Pi_{hH}^{\mathrm{PT}}(p^2) = -\frac{g^2}{32\pi^2}\left(2p^2-m_h^2-m_H^2\right)\left[c_{G^\pm hW^\mp}c_{G^\mp HW^\pm}B_0(p^2;W,W)
+ c_{H^\pm hW^\mp}c_{H^\mp HW^\pm}B_0(p^2;W,H^\pm)\right] Π h H PT ( p 2 ) = − 32 π 2 g 2 ( 2 p 2 − m h 2 − m H 2 ) [ c G ± h W ∓ c G ∓ H W ± B 0 ( p 2 ; W , W ) + c H ± h W ∓ c H ∓ H W ± B 0 ( p 2 ; W , H ± ) ] − g Z 2 64 π 2 ( 2 p 2 − m h 2 − m H 2 ) c G 0 h Z c G 0 H Z B 0 ( p 2 ; Z , Z ) , (74) -\frac{g_Z^2}{64\pi^2}\left(2p^2-m_h^2-m_H^2\right)c_{G^0hZ}c_{G^0HZ}B_0(p^2;Z,Z), \tag*{(74)} − 64 π 2 g Z 2 ( 2 p 2 − m h 2 − m H 2 ) c G 0 h Z c G 0 H Z B 0 ( p 2 ; Z , Z ) , (74) Π H + G − P T ( p 2 ) = − g 2 64 π 2 ( 2 p 2 − m H ± 2 ) [ c H ± h W ∓ c G ± h W ∓ B 0 ( p 2 ; W , h ) + c H ± H W ∓ c G ± H W ∓ B 0 ( p 2 ; W , H ) + c H ± G 0 W ∓ c G ± G 0 W ∓ B 0 ( p 2 ; W , Z ) ] \Pi_{H^+G^-}^{\mathrm{PT}}(p^2) = -\frac{g^2}{64\pi^2}\left(2p^2-m_{H^\pm}^2\right)\left[c_{H^\pm hW^\mp}c_{G^\pm hW^\mp}B_0(p^2;W,h)
+ c_{H^\pm HW^\mp}c_{G^\pm HW^\mp}B_0(p^2;W,H)
+ c_{H^\pm G^0W^\mp}c_{G^\pm G^0W^\mp}B_0(p^2;W,Z)\right] Π H + G − PT ( p 2 ) = − 64 π 2 g 2 ( 2 p 2 − m H ± 2 ) [ c H ± h W ∓ c G ± h W ∓ B 0 ( p 2 ; W , h ) + c H ± H W ∓ c G ± H W ∓ B 0 ( p 2 ; W , H ) + c H ± G 0 W ∓ c G ± G 0 W ∓ B 0 ( p 2 ; W , Z ) ] − g Z 2 64 π 2 ( 2 p 2 − m H ± 2 ) [ c H + H − Z c H ± G ∓ Z B 0 ( p 2 ; Z , H ± ) + c G + G − Z c H ± G ∓ Z B 0 ( p 2 ; W , Z ) ] , (75) -\frac{g_Z^2}{64\pi^2}\left(2p^2-m_{H^\pm}^2\right)\left[c_{H^+H^-Z}c_{H^\pm G^\mp Z}B_0(p^2;Z,H^\pm)
+ c_{G^+G^-Z}c_{H^\pm G^\mp Z}B_0(p^2;W,Z)\right], \tag*{(75)} − 64 π 2 g Z 2 ( 2 p 2 − m H ± 2 ) [ c H + H − Z c H ± G ∓ Z B 0 ( p 2 ; Z , H ± ) + c G + G − Z c H ± G ∓ Z B 0 ( p 2 ; W , Z ) ] , (75) where
c G ± G 0 W ∓ = c β , c H ± G 0 W ∓ = − s β , c G 0 h Z = c α c β , c G 0 H Z = − s α c β , c_{G^\pm G^0 W^\mp}=c_\beta,\qquad c_{H^\pm G^0 W^\mp}=-s_\beta,\qquad c_{G^0 h Z}=\frac{c_\alpha}{c_\beta},\qquad c_{G^0 H Z}=-\frac{s_\alpha}{c_\beta}, c G ± G 0 W ∓ = c β , c H ± G 0 W ∓ = − s β , c G 0 h Z = c β c α , c G 0 H Z = − c β s α , c G ± h W ∓ = c α c β − 2 s α s β , c H ± h W ∓ = − 2 c β s α − c α s β , c_{G^\pm h W^\mp}=c_\alpha c_\beta-2s_\alpha s_\beta,\qquad c_{H^\pm h W^\mp}=-2c_\beta s_\alpha-c_\alpha s_\beta, c G ± h W ∓ = c α c β − 2 s α s β , c H ± h W ∓ = − 2 c β s α − c α s β , c G ± H W ∓ = − c β s α − 2 c α s β , c H ± H W ∓ = − 2 c α c β + s α s β , c_{G^\pm H W^\mp}=-c_\beta s_\alpha-2c_\alpha s_\beta,\qquad c_{H^\pm H W^\mp}=-2c_\alpha c_\beta+s_\alpha s_\beta, c G ± H W ∓ = − c β s α − 2 c α s β , c H ± H W ∓ = − 2 c α c β + s α s β , c H + H − Z = c 2 W + c β 2 , c H ± G ∓ Z = s β c β , c G + G − Z = c 2 W + s β 2 . (76) c_{H^+H^-Z}=c_{2W}+c_\beta^2,\qquad c_{H^\pm G^\mp Z}=s_\beta c_\beta,\qquad c_{G^+G^-Z}=c_{2W}+s_\beta^2. \tag*{(76)} c H + H − Z = c 2 W + c β 2 , c H ± G ∓ Z = s β c β , c G + G − Z = c 2 W + s β 2 . (76) From Eqs. (72) and (64) , the gauge-independent mixing-angle counterterms are obtained as
δ α = − 1 2 ( m H 2 − m h 2 ) Re [ Π h H ( m h 2 ) + Π h H ( m H 2 ) ] . (77) \delta\alpha=-\frac{1}{2(m_H^2-m_h^2)}\operatorname{Re}\left[\Pi_{hH}(m_h^2)+\Pi_{hH}(m_H^2)\right]. \tag*{(77)} δ α = − 2 ( m H 2 − m h 2 ) 1 Re [ Π h H ( m h 2 ) + Π h H ( m H 2 ) ] . (77) δ β = − 1 2 m H ± 2 Re [ Π H + G − ( 0 ) + Π H + G − ( m H ± 2 ) ] . (78) \delta\beta=-\frac{1}{2m_{H^\pm}^2}\operatorname{Re}\left[\Pi_{H^+G^-}(0)+\Pi_{H^+G^-}(m_{H^\pm}^2)\right]. \tag*{(78)} δ β = − 2 m H ± 2 1 Re [ Π H + G − ( 0 ) + Π H + G − ( m H ± 2 ) ] . (78) Gauge sector
The renormalization of the gauge sector is largely the same as in the SM [98 , 99 ] . However, because ρ 0 ≠ 1 \rho_0 \ne1 ρ 0 = 1 δ β \delta\beta δ β
m V , B 2 = m V 2 + δ m V 2 , e B = e + δ e , v B = v + δ v . (79) m_{V,B}^2=m_V^2+\delta m_V^2,\qquad e_B=e+\delta e,\qquad v_B=v+\delta v. \tag*{(79)} m V , B 2 = m V 2 + δ m V 2 , e B = e + δ e , v B = v + δ v . (79) The bare gauge fields are shifted as
W μ B ± = ( 1 + 1 2 δ Z W W ) W μ ± , (80) W_{\mu\mathrm{B}}^{\pm} = \left(1+\frac{1}{2}\delta Z_{WW}\right)W_{\mu}^{\pm}, \tag*{(80)} W μ B ± = ( 1 + 2 1 δ Z W W ) W μ ± , (80) \begin{pmatrix} Z_{\mu\mathrm{B}} \\ A_{\mu\mathrm{B}} \end{pmatrix}
=
\begin{pmatrix}
1+\frac{1}{2}\delta Z_{ZZ} & \frac{1}{2}\left(\delta C_{Z\gamma}-\frac{\delta s_{W}^{2}}{c_{W}s_{W}}\right) \\
\frac{1}{2}\left(\delta C_{Z\gamma}+\frac{\delta s_{W}^{2}}{c_{W}s_{W}}\right) & 1+\frac{1}{2}\delta Z_{\gamma\gamma}
\end{pmatrix}
\begin{pmatrix} Z_{\mu} \\ A_{\mu} \end{pmatrix}. \tag*{(81)}\end{pmatrix}
These counterterms are not independent. When we take δ Z γ γ \delta Z_{\gamma\gamma} δ Z γ γ δ C Z γ \delta C_{Z\gamma} δ C Z γ δ Z W W \delta Z_{WW} δ Z W W δ Z Z Z \delta Z_{ZZ} δ Z Z Z
δ Z W W = δ Z γ γ + c W s W δ C Z γ , δ Z Z Z = δ Z γ γ + c W 2 − s W 2 c W s W δ C Z γ . (82) \delta Z_{WW}=\delta Z_{\gamma\gamma}+\frac{c_{W}}{s_{W}}\delta C_{Z\gamma},\qquad
\delta Z_{ZZ}=\delta Z_{\gamma\gamma}+\frac{c_{W}^{2}-s_{W}^{2}}{c_{W}s_{W}}\delta C_{Z\gamma}. \tag*{(82)} δ Z W W = δ Z γ γ + s W c W δ C Z γ , δ Z Z Z = δ Z γ γ + c W s W c W 2 − s W 2 δ C Z γ . (82) From the bare relations s W , B 2 = 1 − m W , B 2 / ( m Z , B 2 ρ 0 , B ) s_{W,\mathrm{B}}^{2}=1-m_{W,\mathrm{B}}^{2}/(m_{Z,\mathrm{B}}^{2}\rho_{0,\mathrm{B}}) s W , B 2 = 1 − m W , B 2 / ( m Z , B 2 ρ 0 , B ) v B 2 = m W , B 2 s W , B 2 / ( π α e m , B ) v_{\mathrm{B}}^{2}=m_{W,\mathrm{B}}^{2}s_{W,\mathrm{B}}^{2}/(\pi\alpha_{\mathrm{em},\mathrm{B}}) v B 2 = m W , B 2 s W , B 2 / ( π α em , B )
δ s W 2 = c W 2 ( δ m Z 2 m Z 2 − δ m W 2 m W 2 + δ ρ 0 ρ 0 ) , δ v = v 2 ( δ m W 2 m W 2 + δ s W 2 s W 2 − δ α e m α e m ) , (83) \delta s_{W}^{2}=c_{W}^{2}\left(\frac{\delta m_{Z}^{2}}{m_{Z}^{2}}-\frac{\delta m_{W}^{2}}{m_{W}^{2}}+\frac{\delta\rho_{0}}{\rho_{0}}\right),\qquad
\delta v=\frac{v}{2}\left(\frac{\delta m_{W}^{2}}{m_{W}^{2}}+\frac{\delta s_{W}^{2}}{s_{W}^{2}}-\frac{\delta\alpha_{\mathrm{em}}}{\alpha_{\mathrm{em}}}\right), \tag*{(83)} δ s W 2 = c W 2 ( m Z 2 δ m Z 2 − m W 2 δ m W 2 + ρ 0 δ ρ 0 ) , δ v = 2 v ( m W 2 δ m W 2 + s W 2 δ s W 2 − α em δ α em ) , (83) where δ ρ 0 / ρ 0 = 2 tan β δ β \delta\rho_{0}/\rho_{0}=2\tan\beta\delta\beta δ ρ 0 / ρ 0 = 2 tan β δ β δ m W 2 \delta m_{W}^{2} δ m W 2 δ m Z 2 \delta m_{Z}^{2} δ m Z 2 δ Z γ γ \delta Z_{\gamma\gamma} δ Z γ γ δ C Z γ \delta C_{Z\gamma} δ C Z γ δ e \delta e δ e δ β \delta\beta δ β
The inverse propagators are given by
Γ ^ V V ′ μ ν = − i g μ ν ( p 2 − m V 2 ) δ V V ′ + i Π ^ V V ′ μ ν ( p 2 ) , (84) \widehat{\Gamma}_{VV'}^{\mu\nu}=-ig^{\mu\nu}(p^{2}-m_{V}^{2})\delta_{VV'}+i\widehat{\Pi}_{VV'}^{\mu\nu}(p^{2}), \tag*{(84)} Γ V V ′ μν = − i g μν ( p 2 − m V 2 ) δ V V ′ + i Π V V ′ μν ( p 2 ) , (84) where V , V ′ = W , Z , γ V,V'=W,Z,\gamma V , V ′ = W , Z , γ
Π ^ V V ′ μ ν ( p 2 ) = ( − g μ ν + p μ p ν p 2 ) Π ^ V V ′ T ( p 2 ) + p μ p ν p 2 Π ^ V V ′ L ( p 2 ) . (85) \widehat{\Pi}_{VV'}^{\mu\nu}(p^{2})=\left(-g^{\mu\nu}+\frac{p^{\mu}p^{\nu}}{p^{2}}\right)\widehat{\Pi}_{VV'}^{T}(p^{2})+\frac{p^{\mu}p^{\nu}}{p^{2}}\widehat{\Pi}_{VV'}^{L}(p^{2}). \tag*{(85)} Π V V ′ μν ( p 2 ) = ( − g μν + p 2 p μ p ν ) Π V V ′ T ( p 2 ) + p 2 p μ p ν Π V V ′ L ( p 2 ) . (85) The longitudinal parts are irrelevant to the renormalization of physical particles. Thus, we omit the T T T L L L
Π ^ W W ( p 2 ) = Π W W 1 P I ( p 2 ) + Π W W T a d − δ m W 2 + δ Z W W ( p 2 − m W 2 ) , (86) \widehat{\Pi}_{WW}(p^{2})=\Pi_{WW}^{\mathrm{1PI}}(p^{2})+\Pi_{WW}^{\mathrm{Tad}}-\delta m_{W}^{2}+\delta Z_{WW}(p^{2}-m_{W}^{2}), \tag*{(86)} Π W W ( p 2 ) = Π W W 1PI ( p 2 ) + Π W W Tad − δ m W 2 + δ Z W W ( p 2 − m W 2 ) , (86) Π ^ Z Z ( p 2 ) = Π Z Z 1 P I ( p 2 ) + Π Z Z T a d − δ m Z 2 + δ Z Z Z ( p 2 − m Z 2 ) , (87) \widehat{\Pi}_{ZZ}(p^{2})=\Pi_{ZZ}^{\mathrm{1PI}}(p^{2})+\Pi_{ZZ}^{\mathrm{Tad}}-\delta m_{Z}^{2}+\delta Z_{ZZ}(p^{2}-m_{Z}^{2}), \tag*{(87)} Π Z Z ( p 2 ) = Π Z Z 1PI ( p 2 ) + Π Z Z Tad − δ m Z 2 + δ Z Z Z ( p 2 − m Z 2 ) , (87) Π ^ γ γ ( p 2 ) = Π γ γ 1 P I ( p 2 ) + δ Z γ γ p 2 , (88) \widehat{\Pi}_{\gamma\gamma}(p^{2})=\Pi_{\gamma\gamma}^{\mathrm{1PI}}(p^{2})+\delta Z_{\gamma\gamma}p^{2}, \tag*{(88)} Π γ γ ( p 2 ) = Π γ γ 1PI ( p 2 ) + δ Z γ γ p 2 , (88) Π ^ Z γ ( p 2 ) = Π Z γ 1 P I ( p 2 ) + 1 2 ( δ C Z γ − δ s W 2 s W c W ) ( p 2 − m Z 2 ) + 1 2 ( δ C Z γ + δ s W 2 s W c W ) p 2 . (89) \widehat{\Pi}_{Z\gamma}(p^{2})=\Pi_{Z\gamma}^{\mathrm{1PI}}(p^{2})+\frac{1}{2}\left(\delta C_{Z\gamma}-\frac{\delta s_{W}^{2}}{s_{W}c_{W}}\right)(p^{2}-m_{Z}^{2})+\frac{1}{2}\left(\delta C_{Z\gamma}+\frac{\delta s_{W}^{2}}{s_{W}c_{W}}\right)p^{2}. \tag*{(89)} Π Z γ ( p 2 ) = Π Z γ 1PI ( p 2 ) + 2 1 ( δ C Z γ − s W c W δ s W 2 ) ( p 2 − m Z 2 ) + 2 1 ( δ C Z γ + s W c W δ s W 2 ) p 2 . (89) The tadpole inserted self-energies of the gauge bosons Π V V T a d \Pi_{VV}^{\mathrm{Tad}} Π V V Tad
Π W W T a d ∣ S T = 0 , Π Z Z T a d ∣ S T = 0 , (90) \left.\Pi_{WW}^{\mathrm{Tad}}\right|_{\mathrm{ST}}=0,\qquad
\left.\Pi_{ZZ}^{\mathrm{Tad}}\right|_{\mathrm{ST}}=0, \tag*{(90)} Π W W Tad ST = 0 , Π Z Z Tad ST = 0 , (90) Π W W T a d ∣ F J = − g h W W Δ v h − g H W W Δ v H , Π Z Z T a d ∣ F J = − 2 ( g h Z Z Δ v h + g H Z Z Δ v H ) . (91) \left.\Pi_{WW}^{\mathrm{Tad}}\right|_{\mathrm{FJ}}=-g_{hWW}\Delta v_{h}-g_{HWW}\Delta v_{H},\qquad
\left.\Pi_{ZZ}^{\mathrm{Tad}}\right|_{\mathrm{FJ}}=-2\left(g_{hZZ}\Delta v_{h}+g_{HZZ}\Delta v_{H}\right). \tag*{(91)} Π W W Tad FJ = − g hW W Δ v h − g H W W Δ v H , Π Z Z Tad FJ = − 2 ( g h Z Z Δ v h + g H Z Z Δ v H ) . (91) For the inverse propagators, we impose
Re Γ ^ W W ( m W 2 ) = 0 , Re Γ ^ Z Z ( m Z 2 ) = 0 , (92) \operatorname{Re}\widehat{\Gamma}_{WW}(m_W^2)=0,\qquad\operatorname{Re}\widehat{\Gamma}_{ZZ}(m_Z^2)=0, \tag*{(92)} Re Γ W W ( m W 2 ) = 0 , Re Γ Z Z ( m Z 2 ) = 0 , (92) d d p 2 Re Γ ^ γ γ ( p 2 ) ∣ p 2 = 0 = 1 , Re Γ ^ Z γ ( 0 ) = 0. (93) \left.\frac{\mathrm{d}}{\mathrm{d}p^2}\operatorname{Re}\widehat{\Gamma}_{\gamma\gamma}(p^2)\right|_{p^2=0}=1,\qquad\operatorname{Re}\widehat{\Gamma}_{Z\gamma}(0)=0. \tag*{(93)} d p 2 d Re Γ γ γ ( p 2 ) p 2 = 0 = 1 , Re Γ Z γ ( 0 ) = 0. (93) Then, we obtain
δ m W 2 = Re [ Π W W 1 P I ( m W 2 ) + Π W W T a d ] , δ m Z 2 = Re [ Π Z Z 1 P I ( m Z 2 ) + Π Z Z T a d ] , (94) \delta m_W^2=\operatorname{Re}\left[\Pi_{WW}^{\mathrm{1PI}}(m_W^2)+\Pi_{WW}^{\mathrm{Tad}}\right],\qquad\delta m_Z^2=\operatorname{Re}\left[\Pi_{ZZ}^{\mathrm{1PI}}(m_Z^2)+\Pi_{ZZ}^{\mathrm{Tad}}\right], \tag*{(94)} δ m W 2 = Re [ Π W W 1PI ( m W 2 ) + Π W W Tad ] , δ m Z 2 = Re [ Π Z Z 1PI ( m Z 2 ) + Π Z Z Tad ] , (94) δ Z γ γ = − Re d Π γ γ 1 P I ( p 2 ) d p 2 ∣ p 2 = 0 , δ C Z γ = 2 Re Π Z γ 1 P I ( 0 ) m Z 2 + δ s W 2 s W c W . (95) \delta Z_{\gamma\gamma}=-\left.\operatorname{Re}\frac{\mathrm{d}\Pi_{\gamma\gamma}^{\mathrm{1PI}}(p^2)}{\mathrm{d}p^2}\right|_{p^2=0},\qquad\delta C_{Z\gamma}=\frac{2\operatorname{Re}\Pi_{Z\gamma}^{\mathrm{1PI}}(0)}{m_Z^2}+\frac{\delta s_W^2}{s_Wc_W}. \tag*{(95)} δ Z γ γ = − Re d p 2 d Π γ γ 1PI ( p 2 ) p 2 = 0 , δ C Z γ = m Z 2 2 Re Π Z γ 1PI ( 0 ) + s W c W δ s W 2 . (95) We determine δ α e m \delta\alpha_{\mathrm{em}} δ α em Γ ^ e e γ μ \widehat{\Gamma}^{\mu}_{ee\gamma} Γ ee γ μ
\widehat{\Gamma}^{\mu}_{ee\gamma}(q^2=0,\slashed{p}_1=\slashed{p}_2=m_e)=ie\gamma^\mu. \tag*{(96)}
We obtain,
δ α e m α e m = 2 δ e e = d Π γ γ 1 P I ( p 2 ) d p 2 ∣ p 2 = 0 + 2 s W c W Π Z γ 1 P I ( 0 ) m Z 2 . (97) \frac{\delta\alpha_{\mathrm{em}}}{\alpha_{\mathrm{em}}}=2\frac{\delta e}{e}=\left.\frac{\mathrm{d}\Pi_{\gamma\gamma}^{\mathrm{1PI}}(p^2)}{\mathrm{d}p^2}\right|_{p^2=0}+\frac{2s_W}{c_W}\frac{\Pi_{Z\gamma}^{\mathrm{1PI}}(0)}{m_Z^2}. \tag*{(97)} α em δ α em = 2 e δ e = d p 2 d Π γ γ 1PI ( p 2 ) p 2 = 0 + c W 2 s W m Z 2 Π Z γ 1PI ( 0 ) . (97) In this renormalization scheme, the residue of the gauge boson two-point function is not unity. Therefore, we need to include the derivative of the renormalized self-energy, − Π ^ V V ′ ( m V 2 ) / 2 -\widehat{\Pi}'_{VV}(m_V^2)/2 − Π V V ′ ( m V 2 ) /2 W W W Z Z Z − Π ^ Z γ ( m Z 2 ) / m Z 2 -\widehat{\Pi}_{Z\gamma}(m_Z^2)/m_Z^2 − Π Z γ ( m Z 2 ) / m Z 2 γ \gamma γ Z Z Z Z Z Z
Fermion sector
In the following, we neglect fermion flavor mixing in the one-loop calculations. In the mass-eigenbasis, we shift the bare masses and fields as
m f , B = m f + δ m f , ψ B = ψ + 1 2 ( δ Z V f − δ Z A f γ 5 ) ψ . (98) m_{f,\mathrm{B}}=m_f+\delta m_f,\qquad\psi_{\mathrm{B}}=\psi+\frac{1}{2}\left(\delta Z_V^f-\delta Z_A^f\gamma_5\right)\psi. \tag*{(98)} m f , B = m f + δ m f , ψ B = ψ + 2 1 ( δ Z V f − δ Z A f γ 5 ) ψ . (98) The inverse propagators are given by
\widehat{\Gamma}_{ff}(p)=i(\slashed{p}-m_f)+i\widehat{\Pi}_{ff}(p), \tag*{(99)}
where the fermion self-energies are decomposed into vector, axial, and scalar parts
\widehat{\Pi}_{ff}(p)=\slashed{p}\widehat{\Pi}_{ff,V}-\slashed{p}\gamma_5\widehat{\Pi}_{ff,A}+m_f\widehat{\Pi}_{ff,S}, \tag*{(100)}
and they are given by
Π ^ f f , V ( p 2 ) = Π f f , V 1 P I ( p 2 ) + δ Z V f , (101) \widehat{\Pi}_{ff,V}(p^2)=\Pi_{ff,V}^{\mathrm{1PI}}(p^2)+\delta Z_V^f, \tag*{(101)} Π f f , V ( p 2 ) = Π f f , V 1PI ( p 2 ) + δ Z V f , (101) Π ^ f f , A ( p 2 ) = Π f f , A 1 P I ( p 2 ) + δ Z A f , (102) \widehat{\Pi}_{ff,A}(p^2)=\Pi_{ff,A}^{\mathrm{1PI}}(p^2)+\delta Z_A^f, \tag*{(102)} Π f f , A ( p 2 ) = Π f f , A 1PI ( p 2 ) + δ Z A f , (102) Π ^ f f , S ( p 2 ) = Π f f , S 1 P I ( p 2 ) + Π f f , S T a d − δ Z V f − δ m f m f . (103) \widehat{\Pi}_{ff,S}(p^2)=\Pi_{ff,S}^{\mathrm{1PI}}(p^2)+\Pi_{ff,S}^{\mathrm{Tad}}-\delta Z_V^f-\frac{\delta m_f}{m_f}. \tag*{(103)} Π f f , S ( p 2 ) = Π f f , S 1PI ( p 2 ) + Π f f , S Tad − δ Z V f − m f δ m f . (103) The tadpole inserted self-energy of the fermions Π f f , S T a d \Pi^{\mathrm{Tad}}_{ff,S} Π f f , S Tad
Π f f , S T a d ∣ S T = 0 , Π f f , S T a d ∣ F J = − ζ h Δ v h v − ζ H Δ v H v . (104) \left.\Pi^{\mathrm{Tad}}_{ff,S}\right|_{\mathrm{ST}}=0,\qquad\left.\Pi^{\mathrm{Tad}}_{ff,S}\right|_{\mathrm{FJ}}=-\zeta_h\frac{\Delta v_h}{v}-\zeta_H\frac{\Delta v_H}{v}. \tag*{(104)} Π f f , S Tad ST = 0 , Π f f , S Tad FJ = − ζ h v Δ v h − ζ H v Δ v H . (104) For the inverse propagators, we impose
Re Γ ^ f f ( p ) u ( p ) ∣ p 2 = m f 2 = 0 , p̸ + m f p 2 − m f 2 Re Γ ^ f f ( p ) u ( p ) ∣ p 2 = m f 2 = i u ( p ) . (105) \left.\operatorname{Re}\widehat{\Gamma}_{ff}(p)u(p)\right|_{p^2=m_f^2}=0,\qquad\left.\frac{\not{p}+m_f}{p^2-m_f^2}\operatorname{Re}\widehat{\Gamma}_{ff}(p)u(p)\right|_{p^2=m_f^2}=iu(p). \tag*{(105)} Re Γ f f ( p ) u ( p ) p 2 = m f 2 = 0 , p 2 − m f 2 p + m f Re Γ f f ( p ) u ( p ) p 2 = m f 2 = i u ( p ) . (105) Then, we obtain
δ m f = m f Re [ Π f f , V 1 P I ( m f 2 ) + Π f f , S 1 P I ( m f 2 ) + Π f f , S T a d ] , (106) \delta m_f=m_f\operatorname{Re}\left[\Pi^{\mathrm{1PI}}_{ff,V}(m_f^2)+\Pi^{\mathrm{1PI}}_{ff,S}(m_f^2)+\Pi^{\mathrm{Tad}}_{ff,S}\right], \tag*{(106)} δ m f = m f Re [ Π f f , V 1PI ( m f 2 ) + Π f f , S 1PI ( m f 2 ) + Π f f , S Tad ] , (106) δ Z V f = − Re Π f f , V 1 P I ( m f 2 ) − 2 m f 2 d d p 2 Re [ Π f f , V 1 P I ( p 2 ) + Π f f , S 1 P I ( p 2 ) ] ∣ p 2 = m f 2 , (107) \delta Z_V^f=-\operatorname{Re}\Pi^{\mathrm{1PI}}_{ff,V}(m_f^2)-2m_f^2\left.\frac{\mathrm{d}}{\mathrm{d}p^2}\operatorname{Re}\left[\Pi^{\mathrm{1PI}}_{ff,V}(p^2)+\Pi^{\mathrm{1PI}}_{ff,S}(p^2)\right]\right|_{p^2=m_f^2}, \tag*{(107)} δ Z V f = − Re Π f f , V 1PI ( m f 2 ) − 2 m f 2 d p 2 d Re [ Π f f , V 1PI ( p 2 ) + Π f f , S 1PI ( p 2 ) ] p 2 = m f 2 , (107) δ Z A f = − Re [ Π f f , A 1 P I ( m f 2 ) ] . (108) \delta Z_A^f=-\operatorname{Re}\left[\Pi^{\mathrm{1PI}}_{ff,A}(m_f^2)\right]. \tag*{(108)} δ Z A f = − Re [ Π f f , A 1PI ( m f 2 ) ] . (108) Numerical study
In this section, we perform a numerical study of the decays of the SM-like Higgs boson in the RHTM. The decay rates are expressed in terms of the renormalized vertices given in Appendix § 8 . The expressions derived for the CHTM in [89 ] can be applied to the RHTM with the following replacements:
c h Z Z = c α , c h W W = c β c α − 2 s β s α , (109) c_{hZZ}=c_\alpha,\qquad c_{hWW}=c_\beta c_\alpha-2s_\beta s_\alpha, \tag*{(109)} c h Z Z = c α , c hW W = c β c α − 2 s β s α , (109) where c h V V c_{hVV} c hV V V = Z , W ± V=Z,W^\pm V = Z , W ± g h V V g_{hVV} g hV V h → Z f f ˉ h\to Zf\bar{f} h → Z f f ˉ
Γ L O ( h → Z f f ˉ ) = N c f ( c h Z Z c β 2 ) 2 G F 2 m h m Z 4 24 π 3 ( v f 2 + a f 2 ) F ( ϵ Z ) , (110) \Gamma_{\mathrm{LO}}(h\to Zf\bar{f})=N_c^f\left(\frac{c_{hZZ}}{c_\beta^2}\right)^2\frac{G_F^2m_hm_Z^4}{24\pi^3}(v_f^2+a_f^2)F(\epsilon_Z), \tag*{(110)} Γ LO ( h → Z f f ˉ ) = N c f ( c β 2 c h Z Z ) 2 24 π 3 G F 2 m h m Z 4 ( v f 2 + a f 2 ) F ( ϵ Z ) , (110) where v f = I f / 2 − Q f s W 2 v_f=I_f/2-Q_fs_W^2 v f = I f /2 − Q f s W 2 a f = I f / 2 a_f=I_f/2 a f = I f /2 F ( ϵ Z ) F(\epsilon_Z) F ( ϵ Z ) [89 , 100 ] . The coefficients of the loop functions of the box diagrams, C i B Z C^{BZ}_i C i B Z C i B W C^{BW}_i C i B W h → Z Z ∗ , W W ∗ h\to ZZ^*,WW^* h → Z Z ∗ , W W ∗ § 9 .§ 4 .
The resulting analytic formulae are implemented in the H-COUP program [59 , 60 , 61 ] . The loop functions are numerically evaluated using LoopTools [101 ] . We have numerically checked that the cancellation of the UV divergence of the renormalized quantities. In addition, we have checked that there is no scheme difference between the ST and FJ scheme in the renormalized quantities.3 [36 , 95 , 102 ] . We have numerically checked that the inclusive decay rates are independent of the photon mass. The SM input parameters used in the one-loop calculations are listed in Ref. [61 ] , while their numerical values are taken from the latest results [103 ] .
The remainder of this section is organized as follows. We first discuss the input scheme adopted for the one-loop calculations in the RHTM. We then examine the constraints from electroweak precision observables, collider searches, and Higgs signal measurements. Finally, we investigate deviations in the decay of the SM-like Higgs boson and compare the predictions of the RHTM with those of other extended Higgs models, such as the HSM, IDM, 2HDMs, and CHTM.
Input scheme
We here describe the input scheme and the procedure for generating the model parameters. In the on-shell renormalization scheme, m W m_W m W m W m_W m W G F G_F G F G F G_F G F
G F = 1 2 v 2 ( 1 − Δ r ) with v 2 = m W 2 π α e m ( 1 − m W 2 ρ 0 m Z 2 ) , (111) G_F = \frac{1}{\sqrt{2}v^2(1-\Delta r)} \quad\text{with} \quad v^2 = \frac{m_W^2}{\pi\alpha_{\mathrm{em}}}\left(1-\frac{m_W^2}{\rho_0m_Z^2}\right), \tag*{(111)} G F = 2 v 2 ( 1 − Δ r ) 1 with v 2 = π α em m W 2 ( 1 − ρ 0 m Z 2 m W 2 ) , (111) where Δ r \Delta r Δ r [99 , 104 ]
Δ r = Re Π ^ W W ( 0 ) m W 2 + δ V B , (112) \Delta r = \frac{\operatorname{Re}\widehat{\Pi}_{WW}(0)}{m_W^2}+\delta V_B, \tag*{(112)} Δ r = m W 2 Re Π W W ( 0 ) + δ V B , (112) with [76 ]
δ V B = α e m 4 π s W 2 [ 6 + 10 ( 1 − s W 2 ) − 3 ( R / c W 2 ) ( 1 − 2 s W 2 ) 2 ( 1 − R ) ln R ] with R = m W 2 m Z 2 . (113) \delta V_B = \frac{\alpha_{\mathrm{em}}}{4\pi s_W^2}\left[6+\frac{10(1-s_W^2)-3(R/c_W^2)(1-2s_W^2)}{2(1-R)}\ln R\right] \quad\text{with} \quad R=\frac{m_W^2}{m_Z^2}. \tag*{(113)} δ V B = 4 π s W 2 α em [ 6 + 2 ( 1 − R ) 10 ( 1 − s W 2 ) − 3 ( R / c W 2 ) ( 1 − 2 s W 2 ) ln R ] with R = m Z 2 m W 2 . (113) The weak mixing angle and the W boson mass are written as
s W 2 = 1 2 [ 1 − 1 − 4 A 0 ρ 0 m Z 2 ( 1 − Δ r ) ] , m W 2 = ρ 0 m Z 2 c W 2 with A 0 = π α e m 2 G F . (114) s_W^2 = \frac{1}{2}\left[1-\sqrt{1-\frac{4A_0}{\rho_0m_Z^2(1-\Delta r)}}\right], \quad m_W^2=\rho_0m_Z^2c_W^2 \quad\text{with} \quad A_0=\frac{\pi\alpha_{\mathrm{em}}}{\sqrt{2}G_F}. \tag*{(114)} s W 2 = 2 1 [ 1 − 1 − ρ 0 m Z 2 ( 1 − Δ r ) 4 A 0 ] , m W 2 = ρ 0 m Z 2 c W 2 with A 0 = 2 G F π α em . (114) We note that Eqs. (114) are self-consistent equations because Δ r \Delta r Δ r m W m_W m W Δ r \Delta r Δ r m ‾ W \overline{m}_W m W [36 ]
m ‾ W 2 = ρ 0 m Z 2 c ‾ W 2 with c ‾ W 2 = 1 − s ‾ W 2 , (115) \overline{m}_W^2=\rho_0m_Z^2\overline{c}_W^2 \quad\text{with} \quad\overline{c}_W^2=1-\overline{s}_W^2, \tag*{(115)} m W 2 = ρ 0 m Z 2 c W 2 with c W 2 = 1 − s W 2 , (115) where s ‾ W \overline{s}_W s W
s ‾ W 2 = 1 2 ( 1 − 1 − 4 A 0 ρ 0 m Z 2 ) . (116) \overline{s}_W^2=\frac{1}{2}\left(1-\sqrt{1-\frac{4A_0}{\rho_0m_Z^2}}\right). \tag*{(116)} s W 2 = 2 1 ( 1 − 1 − ρ 0 m Z 2 4 A 0 ) . (116) We evaluate Δ r \Delta r Δ r m ‾ W \overline{m}_W m W m W m_W m W Δ r \Delta r Δ r m ‾ W \overline{m}_W m W s ‾ W \overline{s}_W s W
Let us discuss the RHTM parameter generation. In addition to v v v m h m_h m h
m H , m H ± , α , β . (117) m_H,\qquad m_{H^\pm},\qquad\alpha,\qquad\beta. \tag*{(117)} m H , m H ± , α , β . (117) Since these parameters are subject to nontrivial correlations imposed by the theoretical constraints, it is not numerically efficient to generate parameter points directly in terms of them. Instead, we introduce auxiliary tree-level quantities and generate the parameters according to the corresponding tree-level relations.
First, we introduce v ‾ Δ \overline{v}_\Delta v Δ
v ‾ ϕ 2 + 4 v ‾ Δ 2 = v ‾ 2 with v ‾ 2 = 1 2 G F , (118) \overline{v}_\phi^2+4\overline{v}_\Delta^2=\overline{v}^2
\qquad\text{with}\qquad
\overline{v}^2=\frac{1}{\sqrt{2}G_F}, \tag*{(118)} v ϕ 2 + 4 v Δ 2 = v 2 with v 2 = 2 G F 1 , (118) and obtain β \beta β (9) ,
tan β = − 2 v ‾ Δ v ‾ ϕ . (119) \tan\beta=-\frac{2\overline{v}_\Delta}{\overline{v}_\phi}. \tag*{(119)} tan β = − v ϕ 2 v Δ . (119) Next, we introduce λ ‾ 2 \overline{\lambda}_2 λ 2 λ ‾ 3 \overline{\lambda}_3 λ 3 m H ± m_{H^\pm} m H ±
m H 2 = M ‾ 22 2 + ( M ‾ 12 2 ) 2 M ‾ 22 2 − m h 2 , (120) m_H^2=\overline{\mathcal{M}}_{22}^2+\frac{\left(\overline{\mathcal{M}}_{12}^2\right)^2}{\overline{\mathcal{M}}_{22}^2-m_h^2}, \tag*{(120)} m H 2 = M 22 2 + M 22 2 − m h 2 ( M 12 2 ) 2 , (120) where M ‾ 22 2 \overline{\mathcal{M}}_{22}^2 M 22 2 M ‾ 12 2 \overline{\mathcal{M}}_{12}^2 M 12 2
M ‾ 22 2 = λ ‾ 2 v ‾ 2 + v ‾ ϕ 2 v ‾ 2 m H ± 2 , M ‾ 12 2 = v ‾ ϕ v ‾ Δ ( λ ‾ 3 − 2 m H ± 2 v ‾ 2 ) . (121) \overline{\mathcal{M}}_{22}^2=\overline{\lambda}_2\overline{v}^2+\frac{\overline{v}_\phi^2}{\overline{v}^2}m_{H^\pm}^2,\qquad
\overline{\mathcal{M}}_{12}^2=\overline{v}_\phi\overline{v}_\Delta\left(\overline{\lambda}_3-\frac{2m_{H^\pm}^2}{\overline{v}^2}\right). \tag*{(121)} M 22 2 = λ 2 v 2 + v 2 v ϕ 2 m H ± 2 , M 12 2 = v ϕ v Δ ( λ 3 − v 2 2 m H ± 2 ) . (121) Finally, α \alpha α
sin ( 2 α ) = 2 v ‾ ϕ v ‾ Δ m H 2 − m h 2 ( 2 m H ± 2 v ‾ 2 − λ ‾ 3 ) . (122) \sin(2\alpha)=\frac{2\overline{v}_\phi\overline{v}_\Delta}{m_H^2-m_h^2}\left(\frac{2m_{H^\pm}^2}{\overline{v}^2}-\overline{\lambda}_3\right). \tag*{(122)} sin ( 2 α ) = m H 2 − m h 2 2 v ϕ v Δ ( v 2 2 m H ± 2 − λ 3 ) . (122) Thus, for given { m H ± , λ ‾ 2 , λ ‾ 3 , v ‾ Δ } \{m_{H^\pm},\overline{\lambda}_2,\overline{\lambda}_3,\overline{v}_\Delta\} { m H ± , λ 2 , λ 3 , v Δ } { m H , α , β } \{m_H,\alpha,\beta\} { m H , α , β }
m H 2 > 0 , ( m H 2 − m h 2 ) 2 ≥ 4 ( M ‾ 12 2 ) 2 , (123) m_H^2>0,\qquad\left(m_H^2-m_h^2\right)^2\geq4\left(\overline{\mathcal{M}}_{12}^2\right)^2, \tag*{(123)} m H 2 > 0 , ( m H 2 − m h 2 ) 2 ≥ 4 ( M 12 2 ) 2 , (123) where the second condition ensures ∣ sin ( 2 α ) ∣ ≤ 1 |\sin(2\alpha)|\leq1 ∣ sin ( 2 α ) ∣ ≤ 1 λ ‾ 1 \overline{\lambda}_1 λ 1
λ ‾ 1 = 1 2 v ‾ ϕ 2 [ m H 2 + m h 2 − ( m H 2 − m h 2 ) cos 2 α ] . (124) \overline{\lambda}_1=\frac{1}{2\overline{v}_\phi^2}\left[m_H^2+m_h^2-\left(m_H^2-m_h^2\right)\cos2\alpha\right]. \tag*{(124)} λ 1 = 2 v ϕ 2 1 [ m H 2 + m h 2 − ( m H 2 − m h 2 ) cos 2 α ] . (124) The barred quantities denote tree-level quantities and receive loop corrections when expressed in the G F G_F G F v Δ v_\Delta v Δ
v Δ 2 = s β 2 2 ( 1 + c β 2 ) 1 2 G F ( 1 − Δ r ) . (125) v_\Delta^2=\frac{s_\beta^2}{2(1+c_\beta^2)}\frac{1}{\sqrt{2}G_F(1-\Delta r)}. \tag*{(125)} v Δ 2 = 2 ( 1 + c β 2 ) s β 2 2 G F ( 1 − Δ r ) 1 . (125) Similarly, λ 1 \lambda_1 λ 1 λ 2 \lambda_2 λ 2 λ 3 \lambda_3 λ 3 Δ r \Delta r Δ r λ ‾ i \overline{\lambda}_i λ i λ ‾ i \overline{\lambda}_i λ i λ i \lambda_i λ i
Experimental constraints
First, we discuss the constraint from the measurement of EWPOs. From Eq. (26) , we can see that a nonzero v Δ v_\Delta v Δ ρ 0 \rho_0 ρ 0 4 ρ e x p = 1.0003 ± 0.0009 \rho_{\rm exp}=1.0003\pm0.0009 ρ exp = 1.0003 ± 0.0009 [103 ] , we obtain v Δ ≲ 5.8 v_\Delta\lesssim5.8 v Δ ≲ 5.8 ρ \rho ρ v Δ ≪ v v_\Delta\ll v v Δ ≪ v [105 , 106 , 107 ]
Δ ρ ≈ G F 3 2 π 2 ( m H − m H ± ) 2 , (126) \Delta\rho\approx\frac{G_F}{3\sqrt{2}\pi^2}(m_H-m_{H^\pm})^2, \tag*{(126)} Δ ρ ≈ 3 2 π 2 G F ( m H − m H ± ) 2 , (126) which is positive and therefore further increases the ρ \rho ρ Figure 1 , we show the contour of m H m_H m H λ ˉ 2 \bar{\lambda}_2 λ ˉ 2 λ ˉ 3 \bar{\lambda}_3 λ ˉ 3 m H ± = 200 m_{H^\pm}=200 m H ± = 200 v ˉ Δ = 1 \bar{v}_\Delta=1 v ˉ Δ = 1 v ˉ Δ 2 \bar{v}_\Delta^2 v ˉ Δ 2
Figure 1. (Left) Contours of m H m_H m H λ ˉ 2 \bar{\lambda}_2 λ ˉ 2 λ ˉ 3 \bar{\lambda}_3 λ ˉ 3 m H ± = 200 m_{H^\pm}=200 m H ± = 200 v ˉ Δ = 1 \bar{v}_\Delta=1 v ˉ Δ = 1 W W W m H ± = 200 m_{H^\pm}=200 m H ± = 200 2 σ 2\sigma 2 σ W W W λ ˉ 2 \bar{\lambda}_2 λ ˉ 2 λ ˉ 3 \bar{\lambda}_3 λ ˉ 3
m H 2 − m H ± 2 = v ˉ Δ 2 [ ( λ ˉ 3 − 2 m H ± 2 v 2 ) 2 v 2 m H ± 2 − m h 2 + λ ˉ 2 − 4 m H ± 2 v 2 ] + O ( v ˉ Δ 4 ) . (127) m_H^2-m_{H^\pm}^2=\bar{v}_\Delta^2\left[\left(\bar{\lambda}_3-\frac{2m_{H^\pm}^2}{v^2}\right)^2\frac{v^2}{m_{H^\pm}^2-m_h^2}+\bar{\lambda}_2-\frac{4m_{H^\pm}^2}{v^2}\right]+O(\bar{v}_\Delta^4). \tag*{(127)} m H 2 − m H ± 2 = v ˉ Δ 2 [ ( λ ˉ 3 − v 2 2 m H ± 2 ) 2 m H ± 2 − m h 2 v 2 + λ ˉ 2 − v 2 4 m H ± 2 ] + O ( v ˉ Δ 4 ) . (127) Therefore, the mass difference is small and remains below 1 GeV for v ˉ Δ = 1 \bar{v}_\Delta=1 v ˉ Δ = 1 λ ˉ 3 \bar{\lambda}_3 λ ˉ 3 λ ˉ 2 \bar{\lambda}_2 λ ˉ 2 λ ˉ 2 \bar{\lambda}_2 λ ˉ 2 λ ˉ 3 \bar{\lambda}_3 λ ˉ 3 m H ± m_{H^\pm} m H ± v ˉ Δ \bar{v}_\Delta v ˉ Δ
For the EWPO constraints, we require that m W m_W m W ρ \rho ρ s e 2 s_e^2 s e 2 Γ ( Z → ℓ ℓ ˉ ) \Gamma(Z \to\ell\bar{\ell}) Γ ( Z → ℓ ℓ ˉ ) 1 σ 1\sigma 1 σ [103 ]
∣ Δ m W ∣ ≤ 7.7 M e V , ∣ ρ − 1 ∣ ≤ 9.3 × 10 − 4 , ∣ Δ s e 2 ∣ ≤ 1.3 × 10 − 4 , ∣ Δ Γ ( Z → ℓ ℓ ˉ ) ∣ ≤ 8.5 × 10 − 2 M e V . (128) \left|\Delta m_W\right| \le7.7\ \mathrm{MeV},\quad\left|\rho-1\right| \le9.3 \times10^{-4},\quad\left|\Delta s_e^2\right| \le1.3 \times10^{-4},
\qquad
\left|\Delta\Gamma(Z \to\ell\bar{\ell})\right| \le8.5 \times10^{-2}\ \mathrm{MeV}. \tag*{(128)} ∣ Δ m W ∣ ≤ 7.7 MeV , ∣ ρ − 1 ∣ ≤ 9.3 × 1 0 − 4 , Δ s e 2 ≤ 1.3 × 1 0 − 4 , ΔΓ ( Z → ℓ ℓ ˉ ) ≤ 8.5 × 1 0 − 2 MeV . (128) Here, we have taken the collider average for ∣ Δ s e 2 ∣ \left|\Delta s_e^2\right| Δ s e 2 ∣ Δ Γ ( Z → ℓ ℓ ˉ ) ∣ \left|\Delta\Gamma(Z \to\ell\bar{\ell})\right| ΔΓ ( Z → ℓ ℓ ˉ ) W W W [108 ] .5 2 σ 2\sigma 2 σ Figure 1 , we have evaluated Δ m W = m W R H T M − m W S M \Delta m_W = m_W^{\mathrm{RHTM}} - m_W^{\mathrm{SM}} Δ m W = m W RHTM − m W SM m W = 80.357 + Δ m W m_W = 80.357 + \Delta m_W m W = 80.357 + Δ m W [103 ] .6 W W W ρ 0 \rho_0 ρ 0 v Δ v_\Delta v Δ m H ± m_{H^\pm} m H ± λ 2 \lambda_2 λ 2 λ 3 \lambda_3 λ 3
Next, we review the constraint from the collider searches. When v Δ < O ( 1 ) v_\Delta< \mathcal{O}(1) v Δ < O ( 1 ) [14 , 16 ] . H ± H^\pm H ± H ± → τ ν H^\pm\to\tau\nu H ± → τ ν H H H λ 3 \lambda_3 λ 3 H → Z Z H \to ZZ H → Z Z m H ± < m t m_{H^\pm} < m_t m H ± < m t H ± b ˉ H^\pm\bar{b} H ± b ˉ H ± → W Z H^\pm\to WZ H ± → W Z [19 ] , and v Δ v_\Delta v Δ
Finally, we discuss the constraint from Higgs measurements. We define the coupling modifier for the Higgs decays as
κ X 2 = Γ ( h → X X ) Γ ( h → X X ) S M . (129) \kappa_X^2 = \frac{\Gamma(h \to XX)}{\Gamma(h \to XX)_{\mathrm{SM}}}. \tag*{(129)} κ X 2 = Γ ( h → X X ) SM Γ ( h → X X ) . (129) We use a symmetric bound around the SM value, i.e., κ X = 1 \kappa_X = 1 κ X = 1 [103 , 111 , 112 ] . The 1 σ 1\sigma 1 σ κ X \kappa_X κ X
∣ Δ κ W ∣ ≤ 0.06 , ∣ Δ κ Z ∣ ≤ 0.06 , ∣ Δ κ γ ∣ ≤ 0.07 , ∣ Δ κ g ∣ ≤ 0.07 , ∣ Δ κ b ∣ ≤ 0.13 , ∣ Δ κ τ ∣ ≤ 0.07 , ∣ Δ κ Z γ ∣ ≤ 0.36 , (130) \left|\Delta\kappa_W\right| \le0.06,\quad\left|\Delta\kappa_Z\right| \le0.06,\quad\left|\Delta\kappa_\gamma\right| \le0.07,\quad\left|\Delta\kappa_g\right| \le0.07,
\qquad
\left|\Delta\kappa_b\right| \le0.13,\quad\left|\Delta\kappa_\tau\right| \le0.07,\quad\left|\Delta\kappa_{Z\gamma}\right| \le0.36, \tag*{(130)} ∣ Δ κ W ∣ ≤ 0.06 , ∣ Δ κ Z ∣ ≤ 0.06 , ∣ Δ κ γ ∣ ≤ 0.07 , ∣ Δ κ g ∣ ≤ 0.07 , ∣ Δ κ b ∣ ≤ 0.13 , ∣ Δ κ τ ∣ ≤ 0.07 , ∣ Δ κ Z γ ∣ ≤ 0.36 , (130) where Δ κ X = κ X − 1 \Delta\kappa_X = \kappa_X - 1 Δ κ X = κ X − 1 2 σ 2\sigma 2 σ
The trilinear Higgs coupling is sensitive to loop effects of new particles. The experimental constraint on the coupling modifier is obtained from Higgs-pair production processes by scaling the SM trilinear Higgs coupling as g h h h S M → κ λ e x p g h h h S M g_{hhh}^{\mathrm{SM}} \to\kappa_\lambda^{\mathrm{exp}}g_{hhh}^{\mathrm{SM}} g hhh SM → κ λ exp g hhh SM − 0.71 < κ λ e x p < 6.1 -0.71 < \kappa_\lambda^{\mathrm{exp}} < 6.1 − 0.71 < κ λ exp < 6.1 [113 ] . In this paper, we define the coupling modifier with the zero-momentum effective trilinear Higgs vertex.
κ λ = 1 + Γ ^ h h h ( 0 , 0 , 0 ) − Γ ^ h h h S M ( 0 , 0 , 0 ) Γ h h h t r e e , S M . (131) \kappa_\lambda= 1 + \frac{\widehat{\Gamma}_{hhh}(0,0,0)-\widehat{\Gamma}_{hhh}^{\mathrm{SM}}(0,0,0)}{\Gamma_{hhh}^{\mathrm{tree,SM}}}. \tag*{(131)} κ λ = 1 + Γ hhh tree , SM Γ hhh ( 0 , 0 , 0 ) − Γ hhh SM ( 0 , 0 , 0 ) . (131) This definition isolates the effects of new physics from the SM prediction and approaches unity in the decoupling limit.
Deviation in the SM-like Higgs boson decays
We study the deviations in the SM-like Higgs boson decays at NLO. Following the input scheme described in § 4.1 , we determine { β , m H , α } \{\beta, m_{H}, \alpha\} { β , m H , α } { m H ± , v Δ , λ 2 , λ 3 } \{m_{H^\pm}, v_{\Delta}, \lambda_{2}, \lambda_{3}\} { m H ± , v Δ , λ 2 , λ 3 }
0.1 G e V ≤ v Δ < 3 G e V , 200 G e V ≤ m H ± < 1000 G e V , 0 ≤ λ 2 < 4 π , − 2.5 ≤ λ 3 < 15. (132) \begin{aligned}
0.1\ \mathrm{GeV} &\le v_{\Delta} < 3\ \mathrm{GeV}, \qquad200\ \mathrm{GeV} \le m_{H^\pm} < 1000\ \mathrm{GeV},\\
0 &\le\lambda_{2} < 4\pi, \qquad-2.5 \le\lambda_{3} < 15. \tag*{(132)}
\end{aligned} 0.1 GeV 0 ≤ v Δ < 3 GeV , 200 GeV ≤ m H ± < 1000 GeV , ≤ λ 2 < 4 π , − 2.5 ≤ λ 3 < 15. (132) Results in the RHTM
In the left panel of Figure 2 , we show the prediction for Δ κ γ \Delta\kappa_\gamma Δ κ γ m H ± m_{H^\pm} m H ± λ 3 \lambda_{3} λ 3 m H ± ≲ 400 G e V m_{H^\pm} \lesssim400\ \mathrm{GeV} m H ± ≲ 400 GeV λ 3 \lambda_{3} λ 3 κ γ \kappa_\gamma κ γ g H + H − h ≈ − λ 3 v g_{H^{+}H^{-}h} \approx-\lambda_{3}v g H + H − h ≈ − λ 3 v λ 3 \lambda_{3} λ 3 Δ κ γ \Delta\kappa_\gamma Δ κ γ λ 3 \lambda_{3} λ 3 λ 3 \lambda_{3} λ 3 λ 3 + λ 1 λ 2 > 0 \lambda_{3} + \sqrt{\lambda_{1}\lambda_{2}} > 0 λ 3 + λ 1 λ 2 > 0 Δ κ γ \Delta\kappa_\gamma Δ κ γ + 4 % +4\% + 4%
Figure 2. (Left) Prediction for Δ κ γ \Delta\kappa_\gamma Δ κ γ κ λ \kappa_\lambda κ λ κ γ \kappa_\gamma κ γ κ λ \kappa_\lambda κ λ
In the right panel of Figure 2 , we show the prediction for κ λ \kappa_\lambda κ λ [31 , 33 , 114 ] . In the limit v Δ ≪ v v_\Delta\ll v v Δ ≪ v m h ≪ m H ± , m H m_h \ll m_{H^\pm},m_H m h ≪ m H ± , m H
κ λ ≈ 1 + λ 3 3 v 4 32 π 2 m h 2 m H ± 2 . (133) \kappa_\lambda\approx1+\frac{\lambda_3^3v^4}{32\pi^2m_h^2m_{H^\pm}^2}. \tag*{(133)} κ λ ≈ 1 + 32 π 2 m h 2 m H ± 2 λ 3 3 v 4 . (133) Thus, the loop contribution is positive (negative) for positive (negative) λ 3 \lambda_3 λ 3 λ 3 \lambda_3 λ 3 κ λ \kappa_\lambda κ λ κ λ \kappa_\lambda κ λ (133) is shown by the dot-dashed line in Figure 2 . For m H ± ≲ 600 m_{H^\pm}\lesssim600 m H ± ≲ 600 λ 3 \lambda_3 λ 3 κ λ \kappa_\lambda κ λ λ 3 \lambda_3 λ 3 Δ κ γ \Delta\kappa_\gamma Δ κ γ m H ± ≳ 300 m_{H^\pm}\gtrsim300 m H ± ≳ 300 Δ κ γ \Delta\kappa_\gamma Δ κ γ m H ± ≲ 300 m_{H^\pm}\lesssim300 m H ± ≲ 300
In Figure 3 , we show the prediction for Δ κ Z \Delta\kappa_Z Δ κ Z Δ κ Z \Delta\kappa_Z Δ κ Z v Δ v_\Delta v Δ h → f f ˉ h\to f\bar{f} h → f f ˉ h → V V ∗ h\to VV^* h → V V ∗ δ Z h h \delta Z_{hh} δ Z hh [31 , 33 ] . In the limit v Δ ≪ v v_\Delta\ll v v Δ ≪ v m h ≪ m H ± , m H m_h\ll m_{H^\pm},m_H m h ≪ m H ± , m H h → Z Z ∗ h\to ZZ^* h → Z Z ∗
Figure 3. Prediction for Δ κ Z \Delta\kappa_Z Δ κ Z
κ Z 2 − 1 ≈ − λ 3 2 v 2 64 π 2 m H ± 2 . (134) \kappa_Z^2-1\approx-\frac{\lambda_3^2v^2}{64\pi^2m_{H^\pm}^2}. \tag*{(134)} κ Z 2 − 1 ≈ − 64 π 2 m H ± 2 λ 3 2 v 2 . (134) Thus, Δ κ Z \Delta\kappa_Z Δ κ Z ∣ λ 3 ∣ \lvert\lambda_3\rvert ∣ λ 3 ∣ Δ κ Z \Delta\kappa_Z Δ κ Z (134) is shown by the dot-dashed line in Figure 3 . As shown in Figure 2 , λ 3 \lambda_3 λ 3 Δ κ γ \Delta\kappa_\gamma Δ κ γ κ λ \kappa_\lambda κ λ m H ± ≲ 600 m_{H^\pm}\lesssim600 m H ± ≲ 600
Model discrimination
Let us study the distinguishability of the RHTM from other extended Higgs models, such as the HSM, the IDM, the type-I and type-II 2HDMs (2HDM-I and 2HDM-II), and the CHTM in the heaviest H ± ± H^{\pm\pm} H ±± H ± ± H^{\pm\pm} H ±±
Δ R ( h → X Y ) = Γ ( h → X Y ) Γ ( h → X Y ) S M − 1 , Δ μ ( h → X Y ) = BR ( h → X Y ) BR ( h → X Y ) S M − 1. (135) \Delta R(h \to XY) = \frac{\Gamma(h \to XY)}{\Gamma(h \to XY)_{\mathrm{SM}}} - 1,\qquad\Delta\mu(h \to XY) = \frac{\operatorname{BR}(h \to XY)}{\operatorname{BR}(h \to XY)_{\mathrm{SM}}} - 1. \tag*{(135)} Δ R ( h → X Y ) = Γ ( h → X Y ) SM Γ ( h → X Y ) − 1 , Δ μ ( h → X Y ) = BR ( h → X Y ) SM BR ( h → X Y ) − 1. (135) The decay rates of h h h [59 , 60 , 61 ] (see Ref. [89 ] for the CHTM).
We scan the input parameters defined in Refs. [61 , 89 ] as follows.7
H S M : 400 G e V ≤ m H < 1000 G e V , λ S = 0.1 , μ s = 0 G e V , 0.95 ≤ c α < 1 , 0 ≤ m s < m H + 500 G e V , (136) \begin{aligned}
\mathrm{HSM:}\quad&400\ \mathrm{GeV} \le m_H < 1000\ \mathrm{GeV},\quad\lambda_S = 0.1,\quad\mu_s = 0\ \mathrm{GeV},\\
&0.95 \le c_\alpha< 1,\quad0 \le m_s < m_H + 500\ \mathrm{GeV}, \tag*{(136)}
\end{aligned} HSM : 400 GeV ≤ m H < 1000 GeV , λ S = 0.1 , μ s = 0 GeV , 0.95 ≤ c α < 1 , 0 ≤ m s < m H + 500 GeV , (136) I D M : m H = 60 G e V , m A = m H ± , μ 2 2 = 3581 G e V 2 , 100 G e V ≤ m H ± < 1000 G e V , 0 ≤ λ 2 < 4 π , (137) \begin{aligned}
\mathrm{IDM:}\quad&m_H = 60\ \mathrm{GeV},\quad m_A = m_{H^\pm},\quad\mu_2^2 = 3581\ \mathrm{GeV}^2,\\
&100\ \mathrm{GeV} \le m_{H^\pm} < 1000\ \mathrm{GeV},\quad0 \le\lambda_2 < 4\pi, \tag*{(137)}
\end{aligned} IDM : m H = 60 GeV , m A = m H ± , μ 2 2 = 3581 GeV 2 , 100 GeV ≤ m H ± < 1000 GeV , 0 ≤ λ 2 < 4 π , (137) 2 H D M T y p e - I ( I I ) : 400 ( 670 ) G e V ≤ m H < 1000 G e V , m A = m H ± = m H , 0.99 ( 0.995 ) ≤ s β − α < 1 , 1.5 ( 1 ) ≤ tan β < 10 , 0 ≤ M < m H + 500 G e V , (138) \begin{aligned}
\mathrm{2HDM\ Type\text{-}I\ (II):}\quad&400\ (670)\ \mathrm{GeV} \le m_H < 1000\ \mathrm{GeV},\quad m_A = m_{H^\pm} = m_H,\\
&0.99\ (0.995) \le s_{\beta-\alpha} < 1,\quad1.5\ (1) \le\tan\beta< 10,\\
&0 \le M < m_H + 500\ \mathrm{GeV}, \tag*{(138)}
\end{aligned} 2HDM Type - I ( II ) : 400 ( 670 ) GeV ≤ m H < 1000 GeV , m A = m H ± = m H , 0.99 ( 0.995 ) ≤ s β − α < 1 , 1.5 ( 1 ) ≤ tan β < 10 , 0 ≤ M < m H + 500 GeV , (138) C H T M H S ( L S ) : 0.1 G e V ≤ v ˉ Δ < 18 ( 12 ) G e V , 300 ( 400 ) G e V ≤ m L i g h t e s t < 1000 G e V , m H = m A , m H ± 2 − m A 2 = m H ± ± 2 − m H ± 2 , 0 ≤ m H ± − m L i g h t e s t < 150 G e V , 0 ( − 5 ) ≤ λ 4 < 4 π , (139) \begin{aligned}
\mathrm{CHTM\ HS\ (LS):}\quad&0.1\ \mathrm{GeV} \le\bar{v}_{\Delta} < 18\ (12)\ \mathrm{GeV},\\
&300\ (400)\ \mathrm{GeV} \le m_{\mathrm{Lightest}} < 1000\ \mathrm{GeV},\\
&m_H = m_A,\quad m_{H^\pm}^2 - m_A^2 = m_{H^{\pm\pm}}^2 - m_{H^\pm}^2,\\
&0 \le m_{H^\pm} - m_{\mathrm{Lightest}} < 150\ \mathrm{GeV},\quad0\ (-5) \le\lambda_4 < 4\pi, \tag*{(139)}
\end{aligned} CHTM HS ( LS ) : 0.1 GeV ≤ v ˉ Δ < 18 ( 12 ) GeV , 300 ( 400 ) GeV ≤ m Lightest < 1000 GeV , m H = m A , m H ± 2 − m A 2 = m H ±± 2 − m H ± 2 , 0 ≤ m H ± − m Lightest < 150 GeV , 0 ( − 5 ) ≤ λ 4 < 4 π , (139) where m L i g h t e s t = m A m_{\mathrm{Lightest}} = m_A m Lightest = m A m H ± ± m_{H^{\pm\pm}} m H ±± S T ST S T U = 0 U = 0 U = 0 [103 ] . For the CHTM, we impose the EWPO constraints following the discussion in § 4.2 (see also Ref. [89 ] ). The constraints from the Higgs signal measurements are imposed using Δ κ X \Delta\kappa_X Δ κ X κ λ \kappa_\lambda κ λ § 4.2 .
In the upper panels of Figure 4 , we show the correlations of the deviations in the decay rates of the SM-like Higgs boson. The darkred, orange, red, blue, purple, and green points are the predictions in the RHTM, HSM, 2HDM-I, 2HDM-II, IDM, and CHTM, respectively. In the 2HDM, predictions with c β − α < 0 c_{\beta-\alpha} < 0 c β − α < 0 c β − α > 0 c_{\beta-\alpha} > 0 c β − α > 0 Δ R ( h → τ τ ) \Delta R(h \to\tau\tau) Δ R ( h → τ τ ) Δ R ( h → Z Z ∗ ) \Delta R(h \to ZZ^*) Δ R ( h → Z Z ∗ ) δ Z h h \delta Z_{hh} δ Z hh v ˉ Δ \bar{v}_{\Delta} v ˉ Δ
Figure 4. Correlations between the deviations in the decay rates (upper panels) and branching ratios (lower panels) of the SM-like Higgs boson at NLO EW. The left and right panels show the predictions on the h → τ τ h \to\tau\tau h → τ τ h → Z Z ∗ h \to ZZ^{*} h → Z Z ∗ h → γ γ h \to\gamma\gamma h → γ γ h → Z Z ∗ h \to ZZ^{*} h → Z Z ∗ c β − α < 0 c_{\beta-\alpha} < 0 c β − α < 0 c β − α > 0 c_{\beta-\alpha} > 0 c β − α > 0
RHTM. In the 2HDMs, Δ R ( h → τ τ ) \Delta R(h \to\tau\tau) Δ R ( h → τ τ ) Δ R ( h → Z Z ∗ ) \Delta R(h \to ZZ^{*}) Δ R ( h → Z Z ∗ ) Δ R ( h → Z Z ∗ ) \Delta R(h \to ZZ^{*}) Δ R ( h → Z Z ∗ ) v Δ v_{\Delta} v Δ Δ R ( h → γ γ ) \Delta R(h \to\gamma\gamma) Δ R ( h → γ γ ) Δ R ( h → γ γ ) \Delta R(h \to\gamma\gamma) Δ R ( h → γ γ ) Δ R ( h → Z Z ∗ ) \Delta R(h \to ZZ^{*}) Δ R ( h → Z Z ∗ ) Δ R ( h → γ γ ) \Delta R(h \to\gamma\gamma) Δ R ( h → γ γ ) Δ R ( h → Z Z ∗ ) \Delta R(h \to ZZ^{*}) Δ R ( h → Z Z ∗ )
In the lower panels of Figure 4 , we show the correlations of Δ μ ( h → X Y ) \Delta\mu(h \to XY) Δ μ ( h → X Y ) Δ R ( h → X Y ) \Delta R(h \to XY) Δ R ( h → X Y ) Δ μ ( h → X Y ) \Delta\mu(h \to XY) Δ μ ( h → X Y ) h → b b ˉ h \to b\bar{b} h → b b ˉ Δ μ ( h → Z Z ∗ ) \Delta\mu(h \to ZZ^*) Δ μ ( h → Z Z ∗ ) Δ R ( h → Z Z ∗ ) \Delta R(h \to ZZ^*) Δ R ( h → Z Z ∗ ) h → τ τ h \to\tau\tau h → τ τ h → Z Z ∗ h \to ZZ^* h → Z Z ∗ h → γ γ h \to\gamma\gamma h → γ γ h → Z Z ∗ h \to ZZ^* h → Z Z ∗
Conclusions
In this paper, we investigated the electroweak one-loop corrections to the decay rates of the SM-like Higgs boson in the RHTM. The EW gauge and Higgs sectors are renormalized in the on-shell scheme, where the scalar mixing angles are renormalized in a gauge-independent way using the pinch technique. The renormalized form factors of the effective three-point vertices of the SM-like Higgs boson are constructed from the 1PI vertex functions and the corresponding counterterms. The two-body decays (h → f f ˉ h \to f\bar{f} h → f f ˉ h → V V ∗ h \to VV^* h → V V ∗ h → γ γ h \to\gamma\gamma h → γ γ Z γ Z\gamma Z γ g g gg g g [59 , 60 , 61 ] , and the correlations of the deviations in the decay rates and branching ratios of the SM-like Higgs boson are studied in comparison with various extended Higgs models.
The LO deviations in the RHTM are rather small due to the constraint on v Δ v_\Delta v Δ h → τ τ h \to\tau\tau h → τ τ h → Z Z ∗ h \to ZZ^* h → Z Z ∗ h → γ γ h \to\gamma\gamma h → γ γ
Acknowledgments
SK was supported, in part, by Japan Society for Promotion of Science (JSPS) under the Bilateral Program between Japan and China (JPSBП120267416). KS was supported by the JSPS KAKENHI Grant Numbers 25K17379 and Tsuruoka National College of
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Technology Promotion Association. ST was supported by JST SPRING, Grant Number JPMJSP2138.
Feynman rules
The interaction terms among the scalar and the gauge bosons are given by
L ⊃ g ϕ V 1 V 2 g μ ν ϕ V 1 μ V 2 ν + g ϕ 1 ϕ 2 V ( ∂ μ ϕ 1 ϕ 2 − ϕ 1 ∂ μ ϕ 2 ) V μ + g ϕ 1 ϕ 2 V 1 V 2 g μ ν ϕ 1 ϕ 2 V 1 μ V 2 ν . (140) \mathcal{L} \supset g_{\phi V_1 V_2}g_{\mu\nu}\phi V_1^\mu V_2^\nu+g_{\phi_1\phi_2V}(\partial^\mu\phi_1\phi_2-\phi_1\partial^\mu\phi_2)V_\mu+g_{\phi_1\phi_2V_1V_2}g_{\mu\nu}\phi_1\phi_2V_1^\mu V_2^\nu. \tag*{(140)} L ⊃ g ϕ V 1 V 2 g μν ϕ V 1 μ V 2 ν + g ϕ 1 ϕ 2 V ( ∂ μ ϕ 1 ϕ 2 − ϕ 1 ∂ μ ϕ 2 ) V μ + g ϕ 1 ϕ 2 V 1 V 2 g μν ϕ 1 ϕ 2 V 1 μ V 2 ν . (140) We define the charge flow such that particles (anti-particles) are incoming to (outgoing from) the vertex. The g ϕ V 1 V 2 g_{\phi V_1 V_2} g ϕ V 1 V 2 g ϕ 1 ϕ 2 V g_{\phi_1\phi_2V} g ϕ 1 ϕ 2 V g ϕ 1 ϕ 2 V 1 V 2 g_{\phi_1\phi_2V_1V_2} g ϕ 1 ϕ 2 V 1 V 2 Table 2 , and Table 3 , respectively. The interaction terms among the Higgs bosons and the ghost fields are given by
Table 2. Table 2: The scalar-scalar-gauge vertices.
Table 3. The scalar-scalar-gauge-gauge vertices.
L ⊃ g ϕ c ˉ 1 c 2 ϕ c ˉ 1 c 2 . (141) \mathcal{L} \supset g_{\phi\bar{c}_1c_2}\phi\bar{c}_1c_2. \tag*{(141)} L ⊃ g ϕ c ˉ 1 c 2 ϕ c ˉ 1 c 2 . (141) The coefficients of the scalar-ghost-ghost vertex g ϕ c ˉ 1 c 2 g_{\phi\bar{c}_1c_2} g ϕ c ˉ 1 c 2 Table 4 . The scalar trilinear and quartic couplings are given by
Table 4. The scalar-ghost-ghost vertices.
L ⊃ g ϕ i ϕ j ϕ k ϕ i ϕ j ϕ k + g ϕ i ϕ j ϕ k ϕ l ϕ i ϕ j ϕ k ϕ l . (142) \mathcal{L} \supset g_{\phi_i\phi_j\phi_k}\phi_i\phi_j\phi_k+g_{\phi_i\phi_j\phi_k\phi_l}\phi_i\phi_j\phi_k\phi_l. \tag*{(142)} L ⊃ g ϕ i ϕ j ϕ k ϕ i ϕ j ϕ k + g ϕ i ϕ j ϕ k ϕ l ϕ i ϕ j ϕ k ϕ l . (142) The coefficients of the scalar trilinear coupling are given by
g H + H − h = − 1 2 μ s β ( s α s β − 2 c α c β ) − λ 1 c α s β 2 v ϕ − λ 2 s α c β 2 v Δ + λ 3 ( − c α c β 2 v ϕ − s α s β 2 v Δ ) , (143) g_{H^+H^-h}=-\frac{1}{2}\mu s_\beta(s_\alpha s_\beta-2c_\alpha c_\beta)-\lambda_1c_\alpha s_\beta^2v_\phi-\lambda_2s_\alpha c_\beta^2v_\Delta+\lambda_3(-c_\alpha c_\beta^2v_\phi-s_\alpha s_\beta^2v_\Delta), \tag*{(143)} g H + H − h = − 2 1 μ s β ( s α s β − 2 c α c β ) − λ 1 c α s β 2 v ϕ − λ 2 s α c β 2 v Δ + λ 3 ( − c α c β 2 v ϕ − s α s β 2 v Δ ) , (143) g H + H − H = − 1 2 μ s β ( c α s β + 2 s α c β ) + λ 1 s α s β 2 v ϕ − λ 2 c α c β 2 v Δ + λ 3 ( s α c β 2 v ϕ − c α s β 2 v Δ ) , (144) g_{H^+H^-H}=-\frac{1}{2}\mu s_\beta(c_\alpha s_\beta+2s_\alpha c_\beta)+\lambda_1s_\alpha s_\beta^2v_\phi-\lambda_2c_\alpha c_\beta^2v_\Delta+\lambda_3(s_\alpha c_\beta^2v_\phi-c_\alpha s_\beta^2v_\Delta), \tag*{(144)} g H + H − H = − 2 1 μ s β ( c α s β + 2 s α c β ) + λ 1 s α s β 2 v ϕ − λ 2 c α c β 2 v Δ + λ 3 ( s α c β 2 v ϕ − c α s β 2 v Δ ) , (144) g G + G − h = − 1 2 μ c β ( s α c β + 2 c α s β ) − λ 1 c α c β 2 v ϕ − λ 2 s α s β 2 v Δ + λ 3 ( − c α s β 2 v ϕ − s α c β 2 v Δ ) , (145) g_{G^+G^-h}=-\frac{1}{2}\mu c_\beta(s_\alpha c_\beta+2c_\alpha s_\beta)-\lambda_1c_\alpha c_\beta^2v_\phi-\lambda_2s_\alpha s_\beta^2v_\Delta+\lambda_3(-c_\alpha s_\beta^2v_\phi-s_\alpha c_\beta^2v_\Delta), \tag*{(145)} g G + G − h = − 2 1 μ c β ( s α c β + 2 c α s β ) − λ 1 c α c β 2 v ϕ − λ 2 s α s β 2 v Δ + λ 3 ( − c α s β 2 v ϕ − s α c β 2 v Δ ) , (145) g G + G − H = − 1 2 μ c β ( c α c β − 2 s α s β ) + λ 1 s α c β 2 v ϕ − λ 2 c α s β 2 v Δ + λ 3 ( s α s β 2 v ϕ − c α c β 2 v Δ ) (146) g_{G^{+}G^{-}H}=-\frac{1}{2}\mu c_{\beta}\left(c_{\alpha}c_{\beta}-2s_{\alpha}s_{\beta}\right)+\lambda_{1}s_{\alpha}c_{\beta}^{2}v_{\phi}-\lambda_{2}c_{\alpha}s_{\beta}^{2}v_{\Delta}+\lambda_{3}\left(s_{\alpha}s_{\beta}^{2}v_{\phi}-c_{\alpha}c_{\beta}^{2}v_{\Delta}\right) \tag*{(146)} g G + G − H = − 2 1 μ c β ( c α c β − 2 s α s β ) + λ 1 s α c β 2 v ϕ − λ 2 c α s β 2 v Δ + λ 3 ( s α s β 2 v ϕ − c α c β 2 v Δ ) (146) g G ± H ∓ h = 1 2 μ ( s α c β s β − c α ( c β 2 − s β 2 ) ) + λ 1 c α c β s β v ϕ − λ 2 s α c β s β v Δ + λ 3 c β s β ( s α v Δ − c α v ϕ ) (147) g_{G^{\pm}H^{\mp}h}=\frac{1}{2}\mu\left(s_{\alpha}c_{\beta}s_{\beta}-c_{\alpha}\left(c_{\beta}^{2}-s_{\beta}^{2}\right)\right)+\lambda_{1}c_{\alpha}c_{\beta}s_{\beta}v_{\phi}-\lambda_{2}s_{\alpha}c_{\beta}s_{\beta}v_{\Delta}+\lambda_{3}c_{\beta}s_{\beta}\left(s_{\alpha}v_{\Delta}-c_{\alpha}v_{\phi}\right) \tag*{(147)} g G ± H ∓ h = 2 1 μ ( s α c β s β − c α ( c β 2 − s β 2 ) ) + λ 1 c α c β s β v ϕ − λ 2 s α c β s β v Δ + λ 3 c β s β ( s α v Δ − c α v ϕ ) (147) g G ± H ∓ H = 1 2 μ ( c α c β s β + s α ( c β 2 − s β 2 ) ) − λ 1 s α c β s β v ϕ − λ 2 c α c β s β v Δ + λ 3 c β s β ( c α v Δ + s α v ϕ ) (148) g_{G^{\pm}H^{\mp}H}=\frac{1}{2}\mu\left(c_{\alpha}c_{\beta}s_{\beta}+s_{\alpha}\left(c_{\beta}^{2}-s_{\beta}^{2}\right)\right)-\lambda_{1}s_{\alpha}c_{\beta}s_{\beta}v_{\phi}-\lambda_{2}c_{\alpha}c_{\beta}s_{\beta}v_{\Delta}+\lambda_{3}c_{\beta}s_{\beta}\left(c_{\alpha}v_{\Delta}+s_{\alpha}v_{\phi}\right) \tag*{(148)} g G ± H ∓ H = 2 1 μ ( c α c β s β + s α ( c β 2 − s β 2 ) ) − λ 1 s α c β s β v ϕ − λ 2 c α c β s β v Δ + λ 3 c β s β ( c α v Δ + s α v ϕ ) (148) g G ± H ∓ G 0 = ± 1 2 i μ (149) g_{G^{\pm}H^{\mp}G^{0}}=\pm\frac{1}{2}i\mu \tag*{(149)} g G ± H ∓ G 0 = ± 2 1 i μ (149) g G 0 G 0 h = 1 4 μ s α − 1 2 λ 1 c α v ϕ − 1 2 λ 3 s α v Δ (150) g_{G^{0}G^{0}h}=\frac{1}{4}\mu s_{\alpha}-\frac{1}{2}\lambda_{1}c_{\alpha}v_{\phi}-\frac{1}{2}\lambda_{3}s_{\alpha}v_{\Delta} \tag*{(150)} g G 0 G 0 h = 4 1 μ s α − 2 1 λ 1 c α v ϕ − 2 1 λ 3 s α v Δ (150) g G 0 G 0 H = 1 4 μ c α + 1 2 λ 1 s α v ϕ − 1 2 λ 3 c α v Δ (151) g_{G^{0}G^{0}H}=\frac{1}{4}\mu c_{\alpha}+\frac{1}{2}\lambda_{1}s_{\alpha}v_{\phi}-\frac{1}{2}\lambda_{3}c_{\alpha}v_{\Delta} \tag*{(151)} g G 0 G 0 H = 4 1 μ c α + 2 1 λ 1 s α v ϕ − 2 1 λ 3 c α v Δ (151) g h h h = 1 4 μ c α 2 s α − 1 2 λ 1 c α 3 v ϕ − 1 2 λ 2 s α 3 v Δ − 1 2 λ 3 c α s α ( c α v Δ + s α v ϕ ) (152) g_{hhh}=\frac{1}{4}\mu c_{\alpha}^{2}s_{\alpha}-\frac{1}{2}\lambda_{1}c_{\alpha}^{3}v_{\phi}-\frac{1}{2}\lambda_{2}s_{\alpha}^{3}v_{\Delta}-\frac{1}{2}\lambda_{3}c_{\alpha}s_{\alpha}\left(c_{\alpha}v_{\Delta}+s_{\alpha}v_{\phi}\right) \tag*{(152)} g hhh = 4 1 μ c α 2 s α − 2 1 λ 1 c α 3 v ϕ − 2 1 λ 2 s α 3 v Δ − 2 1 λ 3 c α s α ( c α v Δ + s α v ϕ ) (152) g H h h = 1 4 μ c α ( c α 2 − 2 s α 2 ) + 3 2 λ 1 c α 2 s α v ϕ − 3 2 λ 2 c α s α 2 v Δ + 1 2 λ 3 ( c α ( 2 s α 2 − c α 2 ) v Δ + s α ( s α 2 − 2 c α 2 ) v ϕ ) (153) g_{Hhh}=\frac{1}{4}\mu c_{\alpha}\left(c_{\alpha}^{2}-2s_{\alpha}^{2}\right)+\frac{3}{2}\lambda_{1}c_{\alpha}^{2}s_{\alpha}v_{\phi}-\frac{3}{2}\lambda_{2}c_{\alpha}s_{\alpha}^{2}v_{\Delta}+\frac{1}{2}\lambda_{3}\left(c_{\alpha}\left(2s_{\alpha}^{2}-c_{\alpha}^{2}\right)v_{\Delta}+s_{\alpha}\left(s_{\alpha}^{2}-2c_{\alpha}^{2}\right)v_{\phi}\right) \tag*{(153)} g H hh = 4 1 μ c α ( c α 2 − 2 s α 2 ) + 2 3 λ 1 c α 2 s α v ϕ − 2 3 λ 2 c α s α 2 v Δ + 2 1 λ 3 ( c α ( 2 s α 2 − c α 2 ) v Δ + s α ( s α 2 − 2 c α 2 ) v ϕ ) (153) g H H h = 1 4 μ s α ( s α 2 − 2 c α 2 ) − 3 2 λ 1 c α s α 2 v ϕ − 3 2 λ 2 c α 2 s α v Δ − 1 2 λ 3 ( s α ( s α 2 − 2 c α 2 ) v Δ + c α ( c α 2 − 2 s α 2 ) v ϕ ) (154) g_{HHh}=\frac{1}{4}\mu s_{\alpha}\left(s_{\alpha}^{2}-2c_{\alpha}^{2}\right)-\frac{3}{2}\lambda_{1}c_{\alpha}s_{\alpha}^{2}v_{\phi}-\frac{3}{2}\lambda_{2}c_{\alpha}^{2}s_{\alpha}v_{\Delta}-\frac{1}{2}\lambda_{3}\left(s_{\alpha}\left(s_{\alpha}^{2}-2c_{\alpha}^{2}\right)v_{\Delta}+c_{\alpha}\left(c_{\alpha}^{2}-2s_{\alpha}^{2}\right)v_{\phi}\right) \tag*{(154)} g H H h = 4 1 μ s α ( s α 2 − 2 c α 2 ) − 2 3 λ 1 c α s α 2 v ϕ − 2 3 λ 2 c α 2 s α v Δ − 2 1 λ 3 ( s α ( s α 2 − 2 c α 2 ) v Δ + c α ( c α 2 − 2 s α 2 ) v ϕ ) (154) g H H H = 1 4 μ c α s α 2 + 1 2 λ 1 s α 3 v ϕ − 1 2 λ 2 c α 3 v Δ + 1 2 λ 3 c α s α ( c α v ϕ − s α v Δ ) (155) g_{HHH}=\frac{1}{4}\mu c_{\alpha}s_{\alpha}^{2}+\frac{1}{2}\lambda_{1}s_{\alpha}^{3}v_{\phi}-\frac{1}{2}\lambda_{2}c_{\alpha}^{3}v_{\Delta}+\frac{1}{2}\lambda_{3}c_{\alpha}s_{\alpha}\left(c_{\alpha}v_{\phi}-s_{\alpha}v_{\Delta}\right) \tag*{(155)} g H H H = 4 1 μ c α s α 2 + 2 1 λ 1 s α 3 v ϕ − 2 1 λ 2 c α 3 v Δ + 2 1 λ 3 c α s α ( c α v ϕ − s α v Δ ) (155) The coefficients of the scalar quartic coupling are given by
g G + G − G + G − = − 1 2 λ 1 c β 4 − 1 2 λ 2 s β 4 − λ 3 c β 2 s β 2 (156) g_{G^{+}G^{-}G^{+}G^{-}}=-\frac{1}{2}\lambda_{1}c_{\beta}^{4}-\frac{1}{2}\lambda_{2}s_{\beta}^{4}-\lambda_{3}c_{\beta}^{2}s_{\beta}^{2} \tag*{(156)} g G + G − G + G − = − 2 1 λ 1 c β 4 − 2 1 λ 2 s β 4 − λ 3 c β 2 s β 2 (156) g H + H − H + H − = − 1 2 λ 1 s β 4 − 1 2 λ 2 c β 4 − λ 3 c β 2 s β 2 (157) g_{H^{+}H^{-}H^{+}H^{-}}=-\frac{1}{2}\lambda_{1}s_{\beta}^{4}-\frac{1}{2}\lambda_{2}c_{\beta}^{4}-\lambda_{3}c_{\beta}^{2}s_{\beta}^{2} \tag*{(157)} g H + H − H + H − = − 2 1 λ 1 s β 4 − 2 1 λ 2 c β 4 − λ 3 c β 2 s β 2 (157) g H + H − G + G − = − 2 λ 1 c β 2 s β 2 − 2 λ 2 c β 2 s β 2 − λ 3 ( c β 2 − s β 2 ) 2 (158) g_{H^{+}H^{-}G^{+}G^{-}}=-2\lambda_{1}c_{\beta}^{2}s_{\beta}^{2}-2\lambda_{2}c_{\beta}^{2}s_{\beta}^{2}-\lambda_{3}\left(c_{\beta}^{2}-s_{\beta}^{2}\right)^{2} \tag*{(158)} g H + H − G + G − = − 2 λ 1 c β 2 s β 2 − 2 λ 2 c β 2 s β 2 − λ 3 ( c β 2 − s β 2 ) 2 (158) g G + G − G ± H ∓ = λ 1 c β 3 s β − λ 2 c β s β 3 − λ 3 c β s β ( c β 2 − s β 2 ) (159) g_{G^{+}G^{-}G^{\pm}H^{\mp}}=\lambda_{1}c_{\beta}^{3}s_{\beta}-\lambda_{2}c_{\beta}s_{\beta}^{3}-\lambda_{3}c_{\beta}s_{\beta}\left(c_{\beta}^{2}-s_{\beta}^{2}\right) \tag*{(159)} g G + G − G ± H ∓ = λ 1 c β 3 s β − λ 2 c β s β 3 − λ 3 c β s β ( c β 2 − s β 2 ) (159) g H + H − H ± G ∓ = λ 1 c β s β 3 − λ 2 c β 3 s β + λ 3 c β s β ( c β 2 − s β 2 ) (160) g_{H^{+}H^{-}H^{\pm}G^{\mp}}=\lambda_{1}c_{\beta}s_{\beta}^{3}-\lambda_{2}c_{\beta}^{3}s_{\beta}+\lambda_{3}c_{\beta}s_{\beta}\left(c_{\beta}^{2}-s_{\beta}^{2}\right) \tag*{(160)} g H + H − H ± G ∓ = λ 1 c β s β 3 − λ 2 c β 3 s β + λ 3 c β s β ( c β 2 − s β 2 ) (160) g H ± G ∓ H ± G ∓ = − 1 2 λ 1 c β 2 s β 2 − 1 2 λ 2 c β 2 s β 2 + λ 3 c β 2 s β 2 (161) g_{H^{\pm}G^{\mp}H^{\pm}G^{\mp}}=-\frac{1}{2}\lambda_{1}c_{\beta}^{2}s_{\beta}^{2}-\frac{1}{2}\lambda_{2}c_{\beta}^{2}s_{\beta}^{2}+\lambda_{3}c_{\beta}^{2}s_{\beta}^{2} \tag*{(161)} g H ± G ∓ H ± G ∓ = − 2 1 λ 1 c β 2 s β 2 − 2 1 λ 2 c β 2 s β 2 + λ 3 c β 2 s β 2 (161) g H + H − h h = − 1 2 λ 1 c α 2 s β 2 − 1 2 λ 2 s α 2 c β 2 − 1 2 λ 3 ( c α 2 c β 2 + s α 2 s β 2 ) (162) g_{H^{+}H^{-}hh}=-\frac{1}{2}\lambda_{1}c_{\alpha}^{2}s_{\beta}^{2}-\frac{1}{2}\lambda_{2}s_{\alpha}^{2}c_{\beta}^{2}-\frac{1}{2}\lambda_{3}\left(c_{\alpha}^{2}c_{\beta}^{2}+s_{\alpha}^{2}s_{\beta}^{2}\right) \tag*{(162)} g H + H − hh = − 2 1 λ 1 c α 2 s β 2 − 2 1 λ 2 s α 2 c β 2 − 2 1 λ 3 ( c α 2 c β 2 + s α 2 s β 2 ) (162) g H + H − H H = − 1 2 λ 1 s α 2 s β 2 − 1 2 λ 2 c α 2 c β 2 − 1 2 λ 3 ( s α 2 c β 2 + c α 2 s β 2 ) (163) g_{H^{+}H^{-}HH}=-\frac{1}{2}\lambda_{1}s_{\alpha}^{2}s_{\beta}^{2}-\frac{1}{2}\lambda_{2}c_{\alpha}^{2}c_{\beta}^{2}-\frac{1}{2}\lambda_{3}\left(s_{\alpha}^{2}c_{\beta}^{2}+c_{\alpha}^{2}s_{\beta}^{2}\right) \tag*{(163)} g H + H − H H = − 2 1 λ 1 s α 2 s β 2 − 2 1 λ 2 c α 2 c β 2 − 2 1 λ 3 ( s α 2 c β 2 + c α 2 s β 2 ) (163) g H + H − H h = λ 1 c α s α s β 2 − λ 2 c α s α c β 2 + λ 3 c α s α ( c β 2 − s β 2 ) (164) g_{H^{+}H^{-}Hh}=\lambda_{1}c_{\alpha}s_{\alpha}s_{\beta}^{2}-\lambda_{2}c_{\alpha}s_{\alpha}c_{\beta}^{2}+\lambda_{3}c_{\alpha}s_{\alpha}\left(c_{\beta}^{2}-s_{\beta}^{2}\right) \tag*{(164)} g H + H − H h = λ 1 c α s α s β 2 − λ 2 c α s α c β 2 + λ 3 c α s α ( c β 2 − s β 2 ) (164) g H ± H ∓ G 0 G 0 = − 1 2 λ 1 s β 2 − 1 2 λ 3 c β 2 (165) g_{H^{\pm}H^{\mp}G^{0}G^{0}}=-\frac{1}{2}\lambda_{1}s_{\beta}^{2}-\frac{1}{2}\lambda_{3}c_{\beta}^{2} \tag*{(165)} g H ± H ∓ G 0 G 0 = − 2 1 λ 1 s β 2 − 2 1 λ 3 c β 2 (165) g G + G − h h = − 1 2 λ 1 c α 2 c β 2 − 1 2 λ 2 s α 2 s β 2 − 1 2 λ 3 ( s α 2 c β 2 + c α 2 s β 2 ) (166) g_{G^{+}G^{-}hh}=-\frac{1}{2}\lambda_{1}c_{\alpha}^{2}c_{\beta}^{2}-\frac{1}{2}\lambda_{2}s_{\alpha}^{2}s_{\beta}^{2}-\frac{1}{2}\lambda_{3}\left(s_{\alpha}^{2}c_{\beta}^{2}+c_{\alpha}^{2}s_{\beta}^{2}\right) \tag*{(166)} g G + G − hh = − 2 1 λ 1 c α 2 c β 2 − 2 1 λ 2 s α 2 s β 2 − 2 1 λ 3 ( s α 2 c β 2 + c α 2 s β 2 ) (166) g G + G − H h = λ 1 c α s α c β 2 − λ 2 c α s α s β 2 − λ 3 c α s α ( c β 2 − s β 2 ) , (167) g_{G^+G^-Hh}=\lambda_1 c_\alpha s_\alpha c_\beta^2-\lambda_2 c_\alpha s_\alpha s_\beta^2-\lambda_3 c_\alpha s_\alpha\left(c_\beta^2-s_\beta^2\right), \tag*{(167)} g G + G − H h = λ 1 c α s α c β 2 − λ 2 c α s α s β 2 − λ 3 c α s α ( c β 2 − s β 2 ) , (167) g G + G − H H = − 1 2 λ 1 s α 2 c β 2 − 1 2 λ 2 c α 2 s β 2 − 1 2 λ 3 ( c α 2 c β 2 + s α 2 s β 2 ) , (168) g_{G^+G^-HH}=-\frac{1}{2}\lambda_1s_\alpha^2c_\beta^2-\frac{1}{2}\lambda_2c_\alpha^2s_\beta^2-\frac{1}{2}\lambda_3\left(c_\alpha^2c_\beta^2+s_\alpha^2s_\beta^2\right), \tag*{(168)} g G + G − H H = − 2 1 λ 1 s α 2 c β 2 − 2 1 λ 2 c α 2 s β 2 − 2 1 λ 3 ( c α 2 c β 2 + s α 2 s β 2 ) , (168) g G + G − G 0 G 0 = − 1 2 λ 1 c β 2 − 1 2 λ 3 s β 2 , (169) g_{G^+G^-G^0G^0}=-\frac{1}{2}\lambda_1c_\beta^2-\frac{1}{2}\lambda_3s_\beta^2, \tag*{(169)} g G + G − G 0 G 0 = − 2 1 λ 1 c β 2 − 2 1 λ 3 s β 2 , (169) g H ± G ∓ h h = 1 2 λ 1 c α 2 c β s β − 1 2 λ 2 s α 2 c β s β − 1 2 λ 3 c β s β ( c α 2 − s α 2 ) , (170) g_{H^\pm G^\mp hh}=\frac{1}{2}\lambda_1c_\alpha^2c_\beta s_\beta-\frac{1}{2}\lambda_2s_\alpha^2c_\beta s_\beta-\frac{1}{2}\lambda_3c_\beta s_\beta\left(c_\alpha^2-s_\alpha^2\right), \tag*{(170)} g H ± G ∓ hh = 2 1 λ 1 c α 2 c β s β − 2 1 λ 2 s α 2 c β s β − 2 1 λ 3 c β s β ( c α 2 − s α 2 ) , (170) g H ± G ∓ H H = 1 2 λ 1 s α 2 c β s β − 1 2 λ 2 c α 2 c β s β + 1 2 λ 3 c β s β ( c α 2 − s α 2 ) , (171) g_{H^\pm G^\mp HH}=\frac{1}{2}\lambda_1s_\alpha^2c_\beta s_\beta-\frac{1}{2}\lambda_2c_\alpha^2c_\beta s_\beta+\frac{1}{2}\lambda_3c_\beta s_\beta\left(c_\alpha^2-s_\alpha^2\right), \tag*{(171)} g H ± G ∓ H H = 2 1 λ 1 s α 2 c β s β − 2 1 λ 2 c α 2 c β s β + 2 1 λ 3 c β s β ( c α 2 − s α 2 ) , (171) g H ± G ∓ H h = − λ 1 c α s α c β s β − λ 2 c α s α c β s β + 2 λ 3 c α s α c β s β , (172) g_{H^\pm G^\mp Hh}=-\lambda_1c_\alpha s_\alpha c_\beta s_\beta-\lambda_2c_\alpha s_\alpha c_\beta s_\beta+2\lambda_3c_\alpha s_\alpha c_\beta s_\beta, \tag*{(172)} g H ± G ∓ H h = − λ 1 c α s α c β s β − λ 2 c α s α c β s β + 2 λ 3 c α s α c β s β , (172) g H ± G ∓ G 0 G 0 = 1 2 λ 1 c β s β − 1 2 λ 3 c β s β , (173) g_{H^\pm G^\mp G^0G^0}=\frac{1}{2}\lambda_1c_\beta s_\beta-\frac{1}{2}\lambda_3c_\beta s_\beta, \tag*{(173)} g H ± G ∓ G 0 G 0 = 2 1 λ 1 c β s β − 2 1 λ 3 c β s β , (173) g G 0 G 0 h h = − 1 4 λ 1 c α 2 − 1 4 λ 3 s α 2 , (174) g_{G^0G^0hh}=-\frac{1}{4}\lambda_1c_\alpha^2-\frac{1}{4}\lambda_3s_\alpha^2, \tag*{(174)} g G 0 G 0 hh = − 4 1 λ 1 c α 2 − 4 1 λ 3 s α 2 , (174) g G 0 G 0 H h = 1 2 λ 1 c α s α − 1 2 λ 3 c α s α , (175) g_{G^0G^0Hh}=\frac{1}{2}\lambda_1c_\alpha s_\alpha-\frac{1}{2}\lambda_3c_\alpha s_\alpha, \tag*{(175)} g G 0 G 0 H h = 2 1 λ 1 c α s α − 2 1 λ 3 c α s α , (175) g G 0 G 0 H H = − 1 4 λ 1 s α 2 − 1 4 λ 3 c α 2 , (176) g_{G^0G^0HH}=-\frac{1}{4}\lambda_1s_\alpha^2-\frac{1}{4}\lambda_3c_\alpha^2, \tag*{(176)} g G 0 G 0 H H = − 4 1 λ 1 s α 2 − 4 1 λ 3 c α 2 , (176) g G 0 G 0 G 0 G 0 = − 1 8 λ 1 , (177) g_{G^0G^0G^0G^0}=-\frac{1}{8}\lambda_1, \tag*{(177)} g G 0 G 0 G 0 G 0 = − 8 1 λ 1 , (177) g h h h h = − 1 8 λ 1 c α 4 − 1 8 λ 2 s α 4 − 1 4 λ 3 c α 2 s α 2 , (178) g_{hhhh}=-\frac{1}{8}\lambda_1c_\alpha^4-\frac{1}{8}\lambda_2s_\alpha^4-\frac{1}{4}\lambda_3c_\alpha^2s_\alpha^2, \tag*{(178)} g hhhh = − 8 1 λ 1 c α 4 − 8 1 λ 2 s α 4 − 4 1 λ 3 c α 2 s α 2 , (178) g H h h h = 1 2 λ 1 c α 3 s α − 1 2 λ 2 c α s α 3 − 1 2 λ 3 c α s α ( c α 2 − s α 2 ) , (179) g_{Hhhh}=\frac{1}{2}\lambda_1c_\alpha^3s_\alpha-\frac{1}{2}\lambda_2c_\alpha s_\alpha^3-\frac{1}{2}\lambda_3c_\alpha s_\alpha\left(c_\alpha^2-s_\alpha^2\right), \tag*{(179)} g H hhh = 2 1 λ 1 c α 3 s α − 2 1 λ 2 c α s α 3 − 2 1 λ 3 c α s α ( c α 2 − s α 2 ) , (179) g H H h h = − 3 4 λ 1 c α 2 s α 2 − 3 4 λ 2 c α 2 s α 2 − 1 4 λ 3 ( c α 4 − 4 c α 2 s α 2 + s α 4 ) , (180) g_{HHhh}=-\frac{3}{4}\lambda_1c_\alpha^2s_\alpha^2-\frac{3}{4}\lambda_2c_\alpha^2s_\alpha^2-\frac{1}{4}\lambda_3\left(c_\alpha^4-4c_\alpha^2s_\alpha^2+s_\alpha^4\right), \tag*{(180)} g H H hh = − 4 3 λ 1 c α 2 s α 2 − 4 3 λ 2 c α 2 s α 2 − 4 1 λ 3 ( c α 4 − 4 c α 2 s α 2 + s α 4 ) , (180) g H H H h = 1 2 λ 1 c α s α 3 − 1 2 λ 2 c α 3 s α + 1 2 λ 3 c α s α ( c α 2 − s α 2 ) , (181) g_{HHHh}=\frac{1}{2}\lambda_1c_\alpha s_\alpha^3-\frac{1}{2}\lambda_2c_\alpha^3s_\alpha+\frac{1}{2}\lambda_3c_\alpha s_\alpha\left(c_\alpha^2-s_\alpha^2\right), \tag*{(181)} g H H H h = 2 1 λ 1 c α s α 3 − 2 1 λ 2 c α 3 s α + 2 1 λ 3 c α s α ( c α 2 − s α 2 ) , (181) g H H H H = − 1 8 λ 1 s α 4 − 1 8 λ 2 c α 4 − 1 4 λ 3 c α 2 s α 2 . (182) g_{HHHH}=-\frac{1}{8}\lambda_1s_\alpha^4-\frac{1}{8}\lambda_2c_\alpha^4-\frac{1}{4}\lambda_3c_\alpha^2s_\alpha^2. \tag*{(182)} g H H H H = − 8 1 λ 1 s α 4 − 8 1 λ 2 c α 4 − 4 1 λ 3 c α 2 s α 2 . (182) Renormalized vertex
In this section, we give the analytic expressions for the renormalized three-point vertices of the SM-like Higgs boson.
vertex
The renormalized h f f ˉ hf\bar{f} h f f ˉ [33 ]
Γ ^ h f f ˉ ( p 1 , p 2 , q ) = Γ ^ h f f ˉ S + γ 5 Γ ^ h f f ˉ P + ̸ p 1 Γ ^ h f f ˉ V 1 + ̸ p 2 Γ ^ h f f ˉ V 2 + ̸ p 1 γ 5 Γ ^ h f f ˉ A 1 + ̸ p 2 γ 5 Γ ^ h f f ˉ A 2 + ̸ p 1 ̸ p 2 Γ ^ h f f ˉ T + ̸ p 1 ̸ p 2 γ 5 Γ ^ h f f ˉ P T , (183) \widehat{\Gamma}_{hf\bar{f}}(p_1,p_2,q)=\widehat{\Gamma}^{S}_{hf\bar{f}}+\gamma_5\widehat{\Gamma}^{P}_{hf\bar{f}}+\not{p}_1\widehat{\Gamma}^{V_1}_{hf\bar{f}}+\not{p}_2\widehat{\Gamma}^{V_2}_{hf\bar{f}}+\not{p}_1\gamma_5\widehat{\Gamma}^{A_1}_{hf\bar{f}}+\not{p}_2\gamma_5\widehat{\Gamma}^{A_2}_{hf\bar{f}}+\not{p}_1\not{p}_2\widehat{\Gamma}^{T}_{hf\bar{f}}+\not{p}_1\not{p}_2\gamma_5\widehat{\Gamma}^{PT}_{hf\bar{f}}, \tag*{(183)} Γ h f f ˉ ( p 1 , p 2 , q ) = Γ h f f ˉ S + γ 5 Γ h f f ˉ P + p 1 Γ h f f ˉ V 1 + p 2 Γ h f f ˉ V 2 + p 1 γ 5 Γ h f f ˉ A 1 + p 2 γ 5 Γ h f f ˉ A 2 + p 1 p 2 Γ h f f ˉ T + p 1 p 2 γ 5 Γ h f f ˉ P T , (183) where p 1 μ ( p 2 μ ) p_1^\mu(p_2^\mu) p 1 μ ( p 2 μ ) q μ q^\mu q μ
Γ ^ h f f ˉ X = Γ h f f ˉ X , t r e e + Γ h f f ˉ X , l o o p , ( X = S , P , V 1 , V 2 , A 1 , A 2 , T , P T ) . (184) \widehat{\Gamma}^{X}_{hf\bar{f}}=\Gamma^{X,\mathrm{tree}}_{hf\bar{f}}+\Gamma^{X,\mathrm{loop}}_{hf\bar{f}},\qquad(X=S,P,V_1,V_2,A_1,A_2,T,PT). \tag*{(184)} Γ h f f ˉ X = Γ h f f ˉ X , tree + Γ h f f ˉ X , loop , ( X = S , P , V 1 , V 2 , A 1 , A 2 , T , P T ) . (184) The tree-level form factors are given by
Γ h f f ˉ S , t r e e = − m f v ζ h , Γ h f f ˉ X , t r e e = 0 ( X ≠ S ) . (185) \Gamma^{S,\mathrm{tree}}_{h f \bar f}=-\frac{m_f}{v}\zeta_h,\qquad\Gamma^{X,\mathrm{tree}}_{h f \bar f}=0\quad(X\ne S). \tag*{(185)} Γ h f f ˉ S , tree = − v m f ζ h , Γ h f f ˉ X , tree = 0 ( X = S ) . (185) The one-loop-level parts are decomposed into the contributions from 1PI diagrams and the counterterms,
Γ h f f ˉ X , l o o p = Γ h f f ˉ X , 1 P I + δ Γ h f f ˉ X . (186) \Gamma^{X,\mathrm{loop}}_{h f \bar f}=\Gamma^{X,\mathrm{1PI}}_{h f \bar f}+\delta\Gamma^{X}_{h f \bar f}. \tag*{(186)} Γ h f f ˉ X , loop = Γ h f f ˉ X , 1PI + δ Γ h f f ˉ X . (186) The analytic expressions for the 1PI contributions are given in Appendix C.3.3. The counterterms are given by
δ Γ h f f ˉ S = Γ h f f ˉ S , t r e e ( δ m f m f − δ v v + δ ζ h ζ h + δ Z V f + δ Z h h 2 ) + Γ H f f ˉ S , t r e e δ Z H h 2 = Γ h f f ˉ S , t r e e ( δ m f m f − δ v v + tan β δ β + δ Z V f + δ Z h h 2 ) + Γ H f f ˉ S , t r e e ( δ α + δ Z H h 2 ) , (187) \begin{aligned}
\delta\Gamma^{S}_{h f \bar f}
&=\Gamma^{S,\mathrm{tree}}_{h f \bar f}\left(\frac{\delta m_f}{m_f}-\frac{\delta v}{v}+\frac{\delta\zeta_h}{\zeta_h}+\delta Z^f_V+\frac{\delta Z_{hh}}{2}\right)+\Gamma^{S,\mathrm{tree}}_{H f \bar f}\frac{\delta Z_{Hh}}{2}\\
&=\Gamma^{S,\mathrm{tree}}_{h f \bar f}\left(\frac{\delta m_f}{m_f}-\frac{\delta v}{v}+\tan\beta\delta\beta+\delta Z^f_V+\frac{\delta Z_{hh}}{2}\right)+\Gamma^{S,\mathrm{tree}}_{H f \bar f}\left(\delta\alpha+\frac{\delta Z_{Hh}}{2}\right), \tag*{(187)}
\end{aligned} δ Γ h f f ˉ S = Γ h f f ˉ S , tree ( m f δ m f − v δ v + ζ h δ ζ h + δ Z V f + 2 δ Z hh ) + Γ H f f ˉ S , tree 2 δ Z H h = Γ h f f ˉ S , tree ( m f δ m f − v δ v + tan β δ β + δ Z V f + 2 δ Z hh ) + Γ H f f ˉ S , tree ( δ α + 2 δ Z H h ) , (187) δ Γ h f f ˉ X = 0 ( X ≠ S ) , (188) \delta\Gamma^{X}_{h f \bar f}=0\qquad(X\ne S), \tag*{(188)} δ Γ h f f ˉ X = 0 ( X = S ) , (188) where Γ H f f ˉ S , t r e e = − m f ζ H / v \Gamma^{S,\mathrm{tree}}_{H f \bar f}=-m_f\zeta_H/v Γ H f f ˉ S , tree = − m f ζ H / v
δ ζ h ζ h = ζ H ζ h δ α + tan β δ β . (189) \frac{\delta\zeta_h}{\zeta_h}=\frac{\zeta_H}{\zeta_h}\delta\alpha+\tan\beta\delta\beta. \tag*{(189)} ζ h δ ζ h = ζ h ζ H δ α + tan β δ β . (189) vertex
The renormalized h V V hVV hV V V = Z , W ± V=Z,W^\pm V = Z , W ±
Γ ^ h V V μ ν ( p 1 , p 2 , q ) = [ g μ ν Γ ^ h V V 1 + p 1 ν p 2 μ m V 2 Γ ^ h V V 2 + i ϵ μ ν ρ σ p 1 ρ p 2 σ m V 2 Γ ^ h V V 3 ] ( p 1 2 , p 2 2 , q 2 ) , (190) \widehat{\Gamma}^{\mu\nu}_{hVV}(p_1,p_2,q)=\left[g^{\mu\nu}\widehat{\Gamma}^{1}_{hVV}+\frac{p_1^\nu p_2^\mu}{m_V^2}\widehat{\Gamma}^{2}_{hVV}+i\epsilon^{\mu\nu\rho\sigma}\frac{p_{1\rho}p_{2\sigma}}{m_V^2}\widehat{\Gamma}^{3}_{hVV}\right](p_1^2,p_2^2,q^2), \tag*{(190)} Γ hV V μν ( p 1 , p 2 , q ) = [ g μν Γ hV V 1 + m V 2 p 1 ν p 2 μ Γ hV V 2 + i ϵ μν ρ σ m V 2 p 1 ρ p 2 σ Γ hV V 3 ] ( p 1 2 , p 2 2 , q 2 ) , (190) where p 1 μ p_1^\mu p 1 μ p 2 μ p_2^\mu p 2 μ q μ q^\mu q μ
Γ ^ h V V i = Γ h V V i , t r e e + Γ h V V i , l o o p , ( i = 1 , 2 , 3 ) . (191) \widehat{\Gamma}^{i}_{hVV}=\Gamma^{i,\mathrm{tree}}_{hVV}+\Gamma^{i,\mathrm{loop}}_{hVV},\qquad(i=1,2,3). \tag*{(191)} Γ hV V i = Γ hV V i , tree + Γ hV V i , loop , ( i = 1 , 2 , 3 ) . (191) The tree-level form factors are given by
Γ h V V 1 , t r e e = c V g h V V , Γ h V V 2 , t r e e = Γ h V V 3 , t r e e = 0 , (192) \Gamma^{1,\mathrm{tree}}_{hVV}=c_Vg_{hVV},\qquad\Gamma^{2,\mathrm{tree}}_{hVV}=\Gamma^{3,\mathrm{tree}}_{hVV}=0, \tag*{(192)} Γ hV V 1 , tree = c V g hV V , Γ hV V 2 , tree = Γ hV V 3 , tree = 0 , (192) where c Z = 2 c_Z=2 c Z = 2 c W = 1 c_W=1 c W = 1
Γ h V V i , l o o p = Γ h V V i , 1 P I + Γ h V V i , T a d + δ Γ h V V i . (193) \Gamma^{i,\mathrm{loop}}_{hVV}=\Gamma^{i,\mathrm{1PI}}_{hVV}+\Gamma^{i,\mathrm{Tad}}_{hVV}+\delta\Gamma^{i}_{hVV}. \tag*{(193)} Γ hV V i , loop = Γ hV V i , 1PI + Γ hV V i , Tad + δ Γ hV V i . (193) The analytic expressions for the 1PI diagrams are given in Appendix C.3.2. The tadpole contributions are given by,
Γ h V V 1 , T a d ∣ S T = 0 , Γ h V V 1 , T a d ∣ F J = c V ( 2 g h h V V Δ v h + g h H V V Δ v H ) . (194) \left.\Gamma^{1,\mathrm{Tad}}_{hVV}\right|_{\mathrm{ST}}=0,\qquad\left.\Gamma^{1,\mathrm{Tad}}_{hVV}\right|_{\mathrm{FJ}}=c_V\left(2g_{hhVV}\Delta v_h+g_{hHVV}\Delta v_H\right). \tag*{(194)} Γ hV V 1 , Tad ST = 0 , Γ hV V 1 , Tad FJ = c V ( 2 g hhV V Δ v h + g h H V V Δ v H ) . (194) and Γ h V V 2 , T a d = Γ h V V 3 , T a d = 0 \Gamma^{2,\mathrm{Tad}}_{hVV}=\Gamma^{3,\mathrm{Tad}}_{hVV}=0 Γ hV V 2 , Tad = Γ hV V 3 , Tad = 0
δ Γ h Z Z 1 = Γ h Z Z 1 , t r e e ( δ m Z 2 m Z 2 − δ v v + tan β δ β + δ Z h h 2 + δ Z Z Z ) + Γ H Z Z 1 , t r e e ( δ α + δ Z H h 2 ) , (195) \delta\Gamma^{1}_{hZZ}=\Gamma^{1,\mathrm{tree}}_{hZZ}\left(\frac{\delta m_Z^2}{m_Z^2}-\frac{\delta v}{v}+\tan\beta\delta\beta+\frac{\delta Z_{hh}}{2}+\delta Z_{ZZ}\right)+\Gamma^{1,\mathrm{tree}}_{HZZ}\left(\delta\alpha+\frac{\delta Z_{Hh}}{2}\right), \tag*{(195)} δ Γ h Z Z 1 = Γ h Z Z 1 , tree ( m Z 2 δ m Z 2 − v δ v + tan β δ β + 2 δ Z hh + δ Z Z Z ) + Γ H Z Z 1 , tree ( δ α + 2 δ Z H h ) , (195) δ Γ h W W 1 = Γ h W W 1 , t r e e ( δ m W 2 m W 2 − δ v v + δ Z h h 2 + δ Z W W ) + Γ H W W 1 , t r e e ( δ α + δ Z H h 2 ) − g m W ( s β c α + 2 c β s α ) δ β , (196) \begin{aligned}
\delta\Gamma^{1}_{hWW}&=\Gamma^{1,\mathrm{tree}}_{hWW}\left(\frac{\delta m_W^2}{m_W^2}-\frac{\delta v}{v}+\frac{\delta Z_{hh}}{2}+\delta Z_{WW}\right)+\Gamma^{1,\mathrm{tree}}_{HWW}\left(\delta\alpha+\frac{\delta Z_{Hh}}{2}\right)\\
&\quad-gm_W(s_\beta c_\alpha+2c_\beta s_\alpha)\delta\beta, \tag*{(196)}
\end{aligned} δ Γ hW W 1 = Γ hW W 1 , tree ( m W 2 δ m W 2 − v δ v + 2 δ Z hh + δ Z W W ) + Γ H W W 1 , tree ( δ α + 2 δ Z H h ) − g m W ( s β c α + 2 c β s α ) δ β , (196) and δ Γ h V V 2 = δ Γ h V V 3 = 0 \delta\Gamma^{2}_{hVV}=\delta\Gamma^{3}_{hVV}=0 δ Γ hV V 2 = δ Γ hV V 3 = 0
vertex
The renormalized h h h hhh hhh
Γ ^ h h h ( p 1 2 , p 2 2 , q 2 ) = Γ h h h t r e e + Γ h h h l o o p ( p 1 2 , p 2 2 , q 2 ) , (197) \widehat{\Gamma}_{hhh}(p_1^2,p_2^2,q^2)=\Gamma^{\mathrm{tree}}_{hhh}+\Gamma^{\mathrm{loop}}_{hhh}(p_1^2,p_2^2,q^2), \tag*{(197)} Γ hhh ( p 1 2 , p 2 2 , q 2 ) = Γ hhh tree + Γ hhh loop ( p 1 2 , p 2 2 , q 2 ) , (197) where p 1 μ p_1^\mu p 1 μ p 2 μ p_2^\mu p 2 μ q μ q^\mu q μ
Γ h h h t r e e = 6 g h h h . (198) \Gamma^{\mathrm{tree}}_{hhh}=6g_{hhh}. \tag*{(198)} Γ hhh tree = 6 g hhh . (198) The one-loop parts are decomposed into the contributions from 1PI diagrams, tadpole diagrams, and the counterterms,
Γ h h h l o o p = Γ h h h 1 P I + Γ h h h T a d + δ Γ h h h . (199) \Gamma^{\mathrm{loop}}_{hhh}=\Gamma^{\mathrm{1PI}}_{hhh}+\Gamma^{\mathrm{Tad}}_{hhh}+\delta\Gamma_{hhh}. \tag*{(199)} Γ hhh loop = Γ hhh 1PI + Γ hhh Tad + δ Γ hhh . (199) The analytic expression for the 1PI diagram is given in Appendix § 9 .§ 3.1 . The tadpole contribution is given by,
Γ h h h T a d ∣ S T = 0 , Γ h h h T a d ∣ F J = 24 g h h h h Δ v h + 6 g H h h h Δ v H . (200) \left.\Gamma^{\mathrm{Tad}}_{hhh}\right|_{\mathrm{ST}}=0,\qquad\left.\Gamma^{\mathrm{Tad}}_{hhh}\right|_{\mathrm{FJ}}=24g_{hhhh}\Delta v_h+6g_{Hhhh}\Delta v_H. \tag*{(200)} Γ hhh Tad ST = 0 , Γ hhh Tad FJ = 24 g hhhh Δ v h + 6 g H hhh Δ v H . (200) The counterterm is given by
δ Γ h h h = 6 ( δ g h h h + 3 2 g h h h δ Z h h + 1 2 g H h h δ Z H h ) , (201) \delta\Gamma_{hhh}=6\left(\delta g_{hhh}+\frac{3}{2}g_{hhh}\delta Z_{hh}+\frac{1}{2}g_{Hhh}\delta Z_{Hh}\right), \tag*{(201)} δ Γ hhh = 6 ( δ g hhh + 2 3 g hhh δ Z hh + 2 1 g H hh δ Z H h ) , (201) where
δ g h h h = − ( c α 3 c β − 2 s α 3 s β ) δ m h 2 2 v − c β s α 2 s β ( c β s α + s β c α ) δ m H ± 2 v − g h h h δ v v + { − ( c α 3 s β c β 2 + 2 s α 3 c β s β 2 ) m h 2 2 v + s α 2 [ s β c α + ( 2 + c β 2 s β 2 ) c β s α ] m H ± 2 v } δ β + [ g H h h + ( c α 2 s α c β + 2 s α 2 c α s β ) m h 2 − m H 2 2 v ] δ α . (202) \begin{aligned}
\delta g_{hhh}={}&-\left(\frac{c_\alpha^3}{c_\beta}-\frac{2s_\alpha^3}{s_\beta}\right)\frac{\delta m_h^2}{2v}-\frac{c_\beta s_\alpha^2}{s_\beta}(c_\beta s_\alpha+s_\beta c_\alpha)\frac{\delta m_{H^\pm}^2}{v}-g_{hhh}\frac{\delta v}{v}\\
&+\left\{-\left(\frac{c_\alpha^3s_\beta}{c_\beta^2}+\frac{2s_\alpha^3c_\beta}{s_\beta^2}\right)\frac{m_h^2}{2v}+s_\alpha^2\left[s_\beta c_\alpha+\left(2+\frac{c_\beta^2}{s_\beta^2}\right)c_\beta s_\alpha\right]\frac{m_{H^\pm}^2}{v}\right\}\delta\beta\\
&+\left[g_{Hhh}+\left(\frac{c_\alpha^2s_\alpha}{c_\beta}+\frac{2s_\alpha^2c_\alpha}{s_\beta}\right)\frac{m_h^2-m_H^2}{2v}\right]\delta\alpha. \tag*{(202)}
\end{aligned} δ g hhh = − ( c β c α 3 − s β 2 s α 3 ) 2 v δ m h 2 − s β c β s α 2 ( c β s α + s β c α ) v δ m H ± 2 − g hhh v δ v + { − ( c β 2 c α 3 s β + s β 2 2 s α 3 c β ) 2 v m h 2 + s α 2 [ s β c α + ( 2 + s β 2 c β 2 ) c β s α ] v m H ± 2 } δ β + [ g H hh + ( c β c α 2 s α + s β 2 s α 2 c α ) 2 v m h 2 − m H 2 ] δ α . (202) Loop-induced vertex
The loop-induced h V V ′ hVV^\prime hV V ′ ( ( V , V ′ ) = ( g , g ) , ( γ , γ ) , ( Z , γ ) ) ((V,V^\prime)=(g,g),(\gamma,\gamma),(Z,\gamma)) (( V , V ′ ) = ( g , g ) , ( γ , γ ) , ( Z , γ ))
Γ ^ h V V ′ μ ν ( p 1 , p 2 , q ) = [ g μ ν Γ ^ h V V ′ 1 + p 1 ν p 2 μ Γ ^ h V V ′ 2 + i ϵ μ ν ρ σ p 1 ρ p 2 σ Γ ^ h V V ′ 3 ] ( p 1 2 , p 2 2 , q 2 ) . (203) \widehat{\Gamma}^{\mu\nu}_{hVV^\prime}(p_1,p_2,q)=\left[g^{\mu\nu}\widehat{\Gamma}^{1}_{hVV^\prime}+p_1^\nu p_2^\mu\widehat{\Gamma}^{2}_{hVV^\prime}+i\epsilon^{\mu\nu\rho\sigma}p_{1\rho}p_{2\sigma}\widehat{\Gamma}^{3}_{hVV^\prime}\right](p_1^2,p_2^2,q^2). \tag*{(203)} Γ hV V ′ μν ( p 1 , p 2 , q ) = [ g μν Γ hV V ′ 1 + p 1 ν p 2 μ Γ hV V ′ 2 + i ϵ μν ρ σ p 1 ρ p 2 σ Γ hV V ′ 3 ] ( p 1 2 , p 2 2 , q 2 ) . (203) where p 1 μ p_1^\mu p 1 μ p 2 μ p_2^\mu p 2 μ q μ q^\mu q μ p 2 μ p_2^\mu p 2 μ
Γ ^ h V V ′ 1 ( m V 2 , 0 , m h 2 ) = − p 1 ⋅ p 2 Γ ^ h V V ′ 2 ( m V 2 , 0 , m h 2 ) . (204) \widehat{\Gamma}^{1}_{hVV^\prime}(m_V^2,0,m_h^2)=-p_1\cdot p_2\widehat{\Gamma}^{2}_{hVV^\prime}(m_V^2,0,m_h^2). \tag*{(204)} Γ hV V ′ 1 ( m V 2 , 0 , m h 2 ) = − p 1 ⋅ p 2 Γ hV V ′ 2 ( m V 2 , 0 , m h 2 ) . (204) In the absence of CP violation, we have Γ ^ h V V ′ 3 ( m V 2 , 0 , m h 2 ) = 0 \widehat{\Gamma}^{3}_{hVV^\prime}(m_V^2,0,m_h^2)=0 Γ hV V ′ 3 ( m V 2 , 0 , m h 2 ) = 0
Γ ^ h V V ′ i = Γ h V V ′ i , 1 P I + δ Γ h V V ′ i ( i = 1 , 2 , 3 ) . (205) \widehat{\Gamma}^{i}_{hVV^\prime}=\Gamma^{i,\mathrm{1PI}}_{hVV^\prime}+\delta\Gamma^{i}_{hVV^\prime}\qquad(i=1,2,3). \tag*{(205)} Γ hV V ′ i = Γ hV V ′ i , 1PI + δ Γ hV V ′ i ( i = 1 , 2 , 3 ) . (205) The analytic expressions for the 1PI diagrams are listed in Appendix C.3.4. For ( V , V ′ ) = ( g , g ) , ( γ , γ ) (V,V^\prime)=(g,g),(\gamma,\gamma) ( V , V ′ ) = ( g , g ) , ( γ , γ ) δ Γ h g g i = δ Γ h γ γ i = 0. O n t h e o t h e r h a n d , f o r \delta\Gamma^{i}_{hgg}=\delta\Gamma^{i}_{h\gamma\gamma}=0. On the other hand, for δ Γ h g g i = δ Γ hγ γ i = 0. O n t h eo t h er han d , f or
δ Γ h Z γ 1 = g h Z Z ( δ C Z γ − δ s W 2 c W s W ) = 2 g h Z Z m Z 2 Re Π Z γ 1 P I ( 0 ) . (206) \delta\Gamma^{1}_{hZ\gamma}=g_{hZZ}\left(\delta C_{Z\gamma}-\frac{\delta s_W^2}{c_Ws_W}\right)=\frac{2g_{hZZ}}{m_Z^2}\operatorname{Re}\Pi^{\mathrm{1PI}}_{Z\gamma}(0). \tag*{(206)} δ Γ h Z γ 1 = g h Z Z ( δ C Z γ − c W s W δ s W 2 ) = m Z 2 2 g h Z Z Re Π Z γ 1PI ( 0 ) . (206) Loop functions
In this section, we give the analytic expressions for the 1PI diagram contributions to one-, two-, three- and four-point functions in terms of the Passarino–Veltman functions [117 ] . The 1PI diagrams are calculated in the ’t Hooft–Feynman gauge, i.e., ξ W = ξ Z = 1 \xi_W=\xi_Z=1 ξ W = ξ Z = 1 F F F B B B
One-point functions
The fermionic contributions to the 1PI tadpole diagrams for h h h H H H
Γ h , F 1 P I = ζ h Γ h , F , S M 1 P I , Γ H , F 1 P I = ζ H Γ h , F , S M 1 P I , (207) \Gamma^{\mathrm{1PI}}_{h,F}=\zeta_h\Gamma^{\mathrm{1PI}}_{h,F,\mathrm{SM}},\qquad\Gamma^{\mathrm{1PI}}_{H,F}=\zeta_H\Gamma^{\mathrm{1PI}}_{h,F,\mathrm{SM}}, \tag*{(207)} Γ h , F 1PI = ζ h Γ h , F , SM 1PI , Γ H , F 1PI = ζ H Γ h , F , SM 1PI , (207) where
16 π 2 Γ h , F , S M 1 P I = − ∑ f N c f 4 m f 2 v A ( f ) . (208) 16\pi^2\Gamma^{\mathrm{1PI}}_{h,F,\mathrm{SM}}=-\sum_f N_c^f\frac{4m_f^2}{v}A(f). \tag*{(208)} 16 π 2 Γ h , F , SM 1PI = − f ∑ N c f v 4 m f 2 A ( f ) . (208) The bosonic contributions are given by
16 π 2 Γ h , B 1 P I = g m W ( c β c α − 2 s β s α ) [ 4 A ( W ) − 2 m W 2 ] + g Z m Z 2 c α [ 4 A ( Z ) − 2 m Z 2 ] − g m W ( c β c α − 2 s β s α ) A ( c ± ) − g Z m Z 2 c α A ( c Z ) − g G + G − h A ( G ± ) − g G 0 G 0 h A ( G 0 ) − 3 g h h h A ( h ) − g H + H − h A ( H ± ) − g H H h A ( H ) , (209) \begin{aligned}
16\pi^2\Gamma_{h,B}^{1\mathrm{PI}}={}&gm_W(c_\beta c_\alpha-2s_\beta s_\alpha)\left[4A(W)-2m_W^2\right]+\frac{g_Zm_Z}{2}c_\alpha\left[4A(Z)-2m_Z^2\right]\\
&-gm_W(c_\beta c_\alpha-2s_\beta s_\alpha)A(c^\pm)-\frac{g_Zm_Z}{2}c_\alpha A(c_Z)\\
&-g_{G^+G^-h}A(G^\pm)-g_{G^0G^0h}A(G^0)-3g_{hhh}A(h)\\
&-g_{H^+H^-h}A(H^\pm)-g_{HHh}A(H), \tag*{(209)}
\end{aligned} 16 π 2 Γ h , B 1 PI = g m W ( c β c α − 2 s β s α ) [ 4 A ( W ) − 2 m W 2 ] + 2 g Z m Z c α [ 4 A ( Z ) − 2 m Z 2 ] − g m W ( c β c α − 2 s β s α ) A ( c ± ) − 2 g Z m Z c α A ( c Z ) − g G + G − h A ( G ± ) − g G 0 G 0 h A ( G 0 ) − 3 g hhh A ( h ) − g H + H − h A ( H ± ) − g H H h A ( H ) , (209) 16 π 2 Γ H , B 1 P I = − g m W ( c β s α + 2 s β c α ) [ 4 A ( W ) − 2 m W 2 ] − g Z m Z 2 s α [ 4 A ( Z ) − 2 m Z 2 ] + g m W ( c β s α + 2 s β c α ) A ( c ± ) + g Z m Z 2 s α A ( c Z ) − g G + G − H A ( G ± ) − g G 0 G 0 H A ( G 0 ) − g h h H A ( h ) − g H + H − H A ( H ± ) − 3 g H H H A ( H ) . (210) \begin{aligned}
16\pi^2\Gamma_{H,B}^{1\mathrm{PI}}={}&-gm_W(c_\beta s_\alpha+2s_\beta c_\alpha)\left[4A(W)-2m_W^2\right]-\frac{g_Zm_Z}{2}s_\alpha\left[4A(Z)-2m_Z^2\right]\\
&+gm_W(c_\beta s_\alpha+2s_\beta c_\alpha)A(c^\pm)+\frac{g_Zm_Z}{2}s_\alpha A(c_Z)\\
&-g_{G^+G^-H}A(G^\pm)-g_{G^0G^0H}A(G^0)-g_{hhH}A(h)\\
&-g_{H^+H^-H}A(H^\pm)-3g_{HHH}A(H). \tag*{(210)}
\end{aligned} 16 π 2 Γ H , B 1 PI = − g m W ( c β s α + 2 s β c α ) [ 4 A ( W ) − 2 m W 2 ] − 2 g Z m Z s α [ 4 A ( Z ) − 2 m Z 2 ] + g m W ( c β s α + 2 s β c α ) A ( c ± ) + 2 g Z m Z s α A ( c Z ) − g G + G − H A ( G ± ) − g G 0 G 0 H A ( G 0 ) − g hh H A ( h ) − g H + H − H A ( H ± ) − 3 g H H H A ( H ) . (210) Two-point functions
Scalar boson
The fermionic contributions to the scalar boson two-point functions are given by
Π h h 1 P I ( p 2 ) F = ζ h 2 Π h h 1 P I ( p 2 ) F , S M , Π H H 1 P I ( p 2 ) F = ζ H 2 Π h h 1 P I ( p 2 ) F , S M , (211) \Pi_{hh}^{1\mathrm{PI}}(p^2)_F=\zeta_h^2\Pi_{hh}^{1\mathrm{PI}}(p^2)_{F,\mathrm{SM}},\qquad
\Pi_{HH}^{1\mathrm{PI}}(p^2)_F=\zeta_H^2\Pi_{hh}^{1\mathrm{PI}}(p^2)_{F,\mathrm{SM}}, \tag*{(211)} Π hh 1 PI ( p 2 ) F = ζ h 2 Π hh 1 PI ( p 2 ) F , SM , Π H H 1 PI ( p 2 ) F = ζ H 2 Π hh 1 PI ( p 2 ) F , SM , (211) Π H h 1 P I ( p 2 ) F = ζ h ζ H Π h h 1 P I ( p 2 ) F , S M , (212) \Pi_{Hh}^{1\mathrm{PI}}(p^2)_F=\zeta_h\zeta_H\Pi_{hh}^{1\mathrm{PI}}(p^2)_{F,\mathrm{SM}}, \tag*{(212)} Π H h 1 PI ( p 2 ) F = ζ h ζ H Π hh 1 PI ( p 2 ) F , SM , (212) Π H + H − 1 P I ( p 2 ) F = ζ H ± 2 Π G + G − 1 P I ( p 2 ) F , S M , Π H + G − 1 P I ( p 2 ) F = ζ H ± Π G + G − 1 P I ( p 2 ) F , S M , (213) \Pi_{H^+H^-}^{1\mathrm{PI}}(p^2)_F=\zeta_{H^\pm}^2\Pi_{G^+G^-}^{1\mathrm{PI}}(p^2)_{F,\mathrm{SM}},\qquad
\Pi_{H^+G^-}^{1\mathrm{PI}}(p^2)_F=\zeta_{H^\pm}\Pi_{G^+G^-}^{1\mathrm{PI}}(p^2)_{F,\mathrm{SM}}, \tag*{(213)} Π H + H − 1 PI ( p 2 ) F = ζ H ± 2 Π G + G − 1 PI ( p 2 ) F , SM , Π H + G − 1 PI ( p 2 ) F = ζ H ± Π G + G − 1 PI ( p 2 ) F , SM , (213) where
16 π 2 Π h h 1 P I ( p 2 ) F , S M = − ∑ f N c f 4 m f 2 v 2 [ A ( f ) + ( 2 m f 2 − p 2 2 ) B 0 ( p 2 ; f , f ) ] , (214) 16\pi^2\Pi_{hh}^{1\mathrm{PI}}(p^2)_{F,\mathrm{SM}}=-\sum_fN_c^f\frac{4m_f^2}{v^2}\left[A(f)+\left(2m_f^2-\frac{p^2}{2}\right)B_0(p^2;f,f)\right], \tag*{(214)} 16 π 2 Π hh 1 PI ( p 2 ) F , SM = − f ∑ N c f v 2 4 m f 2 [ A ( f ) + ( 2 m f 2 − 2 p 2 ) B 0 ( p 2 ; f , f ) ] , (214) 16 π 2 Π G + G − 1 P I ( p 2 ) F , S M = − 2 N c { m u 2 + m d 2 v 2 [ A ( u ) + A ( d ) + ( m u 2 + m d 2 − p 2 ) B 0 ( p 2 ; u , d ) ] − 4 m u 2 m d 2 v 2 B 0 ( p 2 ; u , d ) } − 2 m ℓ 2 v 2 [ A ( ℓ ) + ( m ℓ 2 − p 2 ) B 0 ( p 2 ; 0 , ℓ ) ] . (215) \begin{aligned}
16\pi^2\Pi_{G^+G^-}^{1\mathrm{PI}}(p^2)_{F,\mathrm{SM}}
={}&-2N_c\left\{\frac{m_u^2+m_d^2}{v^2}\left[A(u)+A(d)+(m_u^2+m_d^2-p^2)B_0(p^2;u,d)\right]\right.\\
&\left.-\frac{4m_u^2m_d^2}{v^2}B_0(p^2;u,d)\right\}\\
&-2\frac{m_\ell^2}{v^2}\left[A(\ell)+(m_\ell^2-p^2)B_0(p^2;0,\ell)\right]. \tag*{(215)}
\end{aligned} 16 π 2 Π G + G − 1 PI ( p 2 ) F , SM = − 2 N c { v 2 m u 2 + m d 2 [ A ( u ) + A ( d ) + ( m u 2 + m d 2 − p 2 ) B 0 ( p 2 ; u , d ) ] − v 2 4 m u 2 m d 2 B 0 ( p 2 ; u , d ) } − 2 v 2 m ℓ 2 [ A ( ℓ ) + ( m ℓ 2 − p 2 ) B 0 ( p 2 ; 0 , ℓ ) ] . (215) The bosonic contributions to the scalar boson two-point functions are given by
16 π 2 Π h h 1 P I ( p 2 ) B = g 2 m W 2 ( c β c α − 2 s β s α ) 2 [ 4 B 0 ( p 2 ; W , W ) − 2 ] + g 2 2 ( 1 + 3 s α 2 ) [ 4 A ( W ) − 2 m W 2 ] − g 2 2 ( c β c α − 2 s β s α ) 2 [ 2 A ( W ) − A ( G ± ) + ( 2 m G ± 2 − m W 2 + 2 p 2 ) B 0 ( p 2 ; W , G ± ) ] − g 2 2 ( s β c α + 2 c β s α ) 2 [ 2 A ( W ) − A ( H ± ) + ( 2 m H ± 2 − m W 2 + 2 p 2 ) B 0 ( p 2 ; W , H ± ) ] − g 2 m W 2 2 ( c β c α − 2 s β s α ) 2 B 0 ( p 2 ; c ± , c ± ) + g Z 2 m Z 2 2 c α 2 [ 4 B 0 ( p 2 ; Z , Z ) − 2 ] + g Z 2 4 c α 2 [ 4 A ( Z ) − 2 m Z 2 ] \begin{aligned}
16\pi^2\Pi_{hh}^{1\mathrm{PI}}(p^2)_B={}&g^2m_W^2(c_\beta c_\alpha-2s_\beta s_\alpha)^2[4B_0(p^2;W,W)-2]\\
&+\frac{g^2}{2}(1+3s_\alpha^2)[4A(W)-2m_W^2]\\
&-\frac{g^2}{2}(c_\beta c_\alpha-2s_\beta s_\alpha)^2[2A(W)-A(G^\pm)+(2m_{G^\pm}^2-m_W^2+2p^2)B_0(p^2;W,G^\pm)]\\
&-\frac{g^2}{2}(s_\beta c_\alpha+2c_\beta s_\alpha)^2[2A(W)-A(H^\pm)+(2m_{H^\pm}^2-m_W^2+2p^2)B_0(p^2;W,H^\pm)]\\
&-\frac{g^2m_W^2}{2}(c_\beta c_\alpha-2s_\beta s_\alpha)^2B_0(p^2;c^\pm,c^\pm)\\
&+\frac{g_Z^2m_Z^2}{2}c_\alpha^2[4B_0(p^2;Z,Z)-2]+\frac{g_Z^2}{4}c_\alpha^2[4A(Z)-2m_Z^2]
\end{aligned} 16 π 2 Π hh 1 PI ( p 2 ) B = g 2 m W 2 ( c β c α − 2 s β s α ) 2 [ 4 B 0 ( p 2 ; W , W ) − 2 ] + 2 g 2 ( 1 + 3 s α 2 ) [ 4 A ( W ) − 2 m W 2 ] − 2 g 2 ( c β c α − 2 s β s α ) 2 [ 2 A ( W ) − A ( G ± ) + ( 2 m G ± 2 − m W 2 + 2 p 2 ) B 0 ( p 2 ; W , G ± )] − 2 g 2 ( s β c α + 2 c β s α ) 2 [ 2 A ( W ) − A ( H ± ) + ( 2 m H ± 2 − m W 2 + 2 p 2 ) B 0 ( p 2 ; W , H ± )] − 2 g 2 m W 2 ( c β c α − 2 s β s α ) 2 B 0 ( p 2 ; c ± , c ± ) + 2 g Z 2 m Z 2 c α 2 [ 4 B 0 ( p 2 ; Z , Z ) − 2 ] + 4 g Z 2 c α 2 [ 4 A ( Z ) − 2 m Z 2 ] − g Z 2 4 c α 2 [ 2 A ( Z ) − A ( G 0 ) + ( 2 m G 0 2 − m Z 2 + 2 p 2 ) B 0 ( p 2 ; Z , G 0 ) ] − g Z 2 m Z 2 4 c α 2 B 0 ( p 2 ; c Z , c Z ) − 2 g H + H − h A ( H ± ) − 2 g G + G − h A ( G ± ) − 2 g G 0 G 0 h A ( G 0 ) − 2 g H H h A ( H ) − 12 g h h h A ( h ) + g H + H − h 2 B 0 ( p 2 ; H ± , H ± ) + g G + G − h 2 B 0 ( p 2 ; G ± , G ± ) + 2 g H + G − h 2 B 0 ( p 2 ; H ± , G ± ) + 2 g G 0 G 0 h 2 B 0 ( p 2 ; G 0 , G 0 ) + 2 g H H h 2 B 0 ( p 2 ; H , H ) + 18 g h h h 2 B 0 ( p 2 ; h , h ) + 4 g H h h 2 B 0 ( p 2 ; H , h ) , (216) \begin{aligned}
&-\frac{g_Z^2}{4}c_\alpha^2\left[2A(Z)-A(G^0)+\left(2m_{G^0}^2-m_Z^2+2p^2\right)B_0(p^2;Z,G^0)\right]\\
&-\frac{g_Z^2m_Z^2}{4}c_\alpha^2B_0(p^2;c_Z,c_Z)\\
&-2g_{H^+H^-h}A(H^\pm)-2g_{G^+G^-h}A(G^\pm)\\
&-2g_{G^0G^0h}A(G^0)-2g_{HHh}A(H)-12g_{hhh}A(h)\\
&+g_{H^+H^-h}^2B_0(p^2;H^\pm,H^\pm)+g_{G^+G^-h}^2B_0(p^2;G^\pm,G^\pm)+2g_{H^+G^-h}^2B_0(p^2;H^\pm,G^\pm)\\
&+2g_{G^0G^0h}^2B_0(p^2;G^0,G^0)\\
&+2g_{HHh}^2B_0(p^2;H,H)+18g_{hhh}^2B_0(p^2;h,h)+4g_{Hhh}^2B_0(p^2;H,h), \tag*{(216)}
\end{aligned} − 4 g Z 2 c α 2 [ 2 A ( Z ) − A ( G 0 ) + ( 2 m G 0 2 − m Z 2 + 2 p 2 ) B 0 ( p 2 ; Z , G 0 ) ] − 4 g Z 2 m Z 2 c α 2 B 0 ( p 2 ; c Z , c Z ) − 2 g H + H − h A ( H ± ) − 2 g G + G − h A ( G ± ) − 2 g G 0 G 0 h A ( G 0 ) − 2 g H H h A ( H ) − 12 g hhh A ( h ) + g H + H − h 2 B 0 ( p 2 ; H ± , H ± ) + g G + G − h 2 B 0 ( p 2 ; G ± , G ± ) + 2 g H + G − h 2 B 0 ( p 2 ; H ± , G ± ) + 2 g G 0 G 0 h 2 B 0 ( p 2 ; G 0 , G 0 ) + 2 g H H h 2 B 0 ( p 2 ; H , H ) + 18 g hhh 2 B 0 ( p 2 ; h , h ) + 4 g H hh 2 B 0 ( p 2 ; H , h ) , (216) 16 π 2 Π H h 1 P I ( p 2 ) B = − g 2 m W 2 ( c β c α − 2 s β s α ) ( c β s α + 2 s β c α ) [ 4 B 0 ( p 2 ; W , W ) − 2 ] + 3 g 2 2 s α c α [ 4 A ( W ) − 2 m W 2 ] + g 2 2 ( c β c α − 2 s β s α ) ( c β s α + 2 s β c α ) [ 2 A ( W ) − A ( G ± ) + ( 2 m G ± 2 − m W 2 + 2 p 2 ) B 0 ( p 2 ; W , G ± ) ] + g 2 2 ( s β c α + 2 c β s α ) ( s β s α − 2 c β c α ) [ 2 A ( W ) − A ( H ± ) + ( 2 m H ± 2 − m W 2 + 2 p 2 ) B 0 ( p 2 ; W , H ± ) ] + g 2 m W 2 2 ( c β c α − 2 s β s α ) ( c β s α + 2 s β c α ) B 0 ( p 2 ; c ± , c ± ) − g Z 2 m Z 2 2 c α s α [ 4 B 0 ( p 2 ; Z , Z ) − 2 ] − g Z 2 4 s α c α [ 4 A ( Z ) − 2 m Z 2 ] + g Z 2 4 c α s α [ 2 A ( Z ) − A ( G 0 ) + ( 2 m G 0 2 − m Z 2 + 2 p 2 ) B 0 ( p 2 ; Z , G 0 ) ] + g Z 2 m Z 2 4 c α s α B 0 ( p 2 ; c Z , c Z ) − g H + H − H h A ( H ± ) − g G + G − H h A ( G ± ) − g G 0 G 0 H h A ( G 0 ) − 3 g H H H h A ( H ) − 3 g H h h h A ( h ) + g H + H − h g H + H − H B 0 ( p 2 ; H ± , H ± ) + g G + G − h g G + G − H B 0 ( p 2 ; G ± , G ± ) + 2 g H + G − h g H + G − H B 0 ( p 2 ; H ± , G ± ) + 2 g G 0 G 0 h g G 0 G 0 H B 0 ( p 2 ; G 0 , G 0 ) + 6 g H H h g H H H B 0 ( p 2 ; H , H ) + 6 g h h h g H H h B 0 ( p 2 ; h , h ) + 4 g H h h g H H h B 0 ( p 2 ; H , h ) , (217) \begin{aligned}
16\pi^2\Pi_{Hh}^{\mathrm{1PI}}(p^2)_B
={}&-g^2m_W^2(c_\beta c_\alpha-2s_\beta s_\alpha)(c_\beta s_\alpha+2s_\beta c_\alpha)\left[4B_0(p^2;W,W)-2\right]\\
&+\frac{3g^2}{2}s_\alpha c_\alpha\left[4A(W)-2m_W^2\right]\\
&+\frac{g^2}{2}(c_\beta c_\alpha-2s_\beta s_\alpha)(c_\beta s_\alpha+2s_\beta c_\alpha)\left[2A(W)-A(G^\pm)+\left(2m_{G^\pm}^2-m_W^2+2p^2\right)B_0(p^2;W,G^\pm)\right]\\
&+\frac{g^2}{2}(s_\beta c_\alpha+2c_\beta s_\alpha)(s_\beta s_\alpha-2c_\beta c_\alpha)\left[2A(W)-A(H^\pm)+\left(2m_{H^\pm}^2-m_W^2+2p^2\right)B_0(p^2;W,H^\pm)\right]\\
&+\frac{g^2m_W^2}{2}(c_\beta c_\alpha-2s_\beta s_\alpha)(c_\beta s_\alpha+2s_\beta c_\alpha)B_0(p^2;c^\pm,c^\pm)\\
&-\frac{g_Z^2m_Z^2}{2}c_\alpha s_\alpha\left[4B_0(p^2;Z,Z)-2\right]-\frac{g_Z^2}{4}s_\alpha c_\alpha\left[4A(Z)-2m_Z^2\right]\\
&+\frac{g_Z^2}{4}c_\alpha s_\alpha\left[2A(Z)-A(G^0)+\left(2m_{G^0}^2-m_Z^2+2p^2\right)B_0(p^2;Z,G^0)\right]\\
&+\frac{g_Z^2m_Z^2}{4}c_\alpha s_\alpha B_0(p^2;c_Z,c_Z)\\
&-g_{H^+H^-Hh}A(H^\pm)-g_{G^+G^-Hh}A(G^\pm)\\
&-g_{G^0G^0Hh}A(G^0)-3g_{HHHh}A(H)-3g_{Hhhh}A(h)\\
&+g_{H^+H^-h}g_{H^+H^-H}B_0(p^2;H^\pm,H^\pm)+g_{G^+G^-h}g_{G^+G^-H}B_0(p^2;G^\pm,G^\pm)\\
&+2g_{H^+G^-h}g_{H^+G^-H}B_0(p^2;H^\pm,G^\pm)\\
&+2g_{G^0G^0h}g_{G^0G^0H}B_0(p^2;G^0,G^0)\\
&+6g_{HHh}g_{HHH}B_0(p^2;H,H)+6g_{hhh}g_{HHh}B_0(p^2;h,h)+4g_{Hhh}g_{HHh}B_0(p^2;H,h), \tag*{(217)}
\end{aligned} 16 π 2 Π H h 1PI ( p 2 ) B = − g 2 m W 2 ( c β c α − 2 s β s α ) ( c β s α + 2 s β c α ) [ 4 B 0 ( p 2 ; W , W ) − 2 ] + 2 3 g 2 s α c α [ 4 A ( W ) − 2 m W 2 ] + 2 g 2 ( c β c α − 2 s β s α ) ( c β s α + 2 s β c α ) [ 2 A ( W ) − A ( G ± ) + ( 2 m G ± 2 − m W 2 + 2 p 2 ) B 0 ( p 2 ; W , G ± ) ] + 2 g 2 ( s β c α + 2 c β s α ) ( s β s α − 2 c β c α ) [ 2 A ( W ) − A ( H ± ) + ( 2 m H ± 2 − m W 2 + 2 p 2 ) B 0 ( p 2 ; W , H ± ) ] + 2 g 2 m W 2 ( c β c α − 2 s β s α ) ( c β s α + 2 s β c α ) B 0 ( p 2 ; c ± , c ± ) − 2 g Z 2 m Z 2 c α s α [ 4 B 0 ( p 2 ; Z , Z ) − 2 ] − 4 g Z 2 s α c α [ 4 A ( Z ) − 2 m Z 2 ] + 4 g Z 2 c α s α [ 2 A ( Z ) − A ( G 0 ) + ( 2 m G 0 2 − m Z 2 + 2 p 2 ) B 0 ( p 2 ; Z , G 0 ) ] + 4 g Z 2 m Z 2 c α s α B 0 ( p 2 ; c Z , c Z ) − g H + H − H h A ( H ± ) − g G + G − H h A ( G ± ) − g G 0 G 0 H h A ( G 0 ) − 3 g H H H h A ( H ) − 3 g H hhh A ( h ) + g H + H − h g H + H − H B 0 ( p 2 ; H ± , H ± ) + g G + G − h g G + G − H B 0 ( p 2 ; G ± , G ± ) + 2 g H + G − h g H + G − H B 0 ( p 2 ; H ± , G ± ) + 2 g G 0 G 0 h g G 0 G 0 H B 0 ( p 2 ; G 0 , G 0 ) + 6 g H H h g H H H B 0 ( p 2 ; H , H ) + 6 g hhh g H H h B 0 ( p 2 ; h , h ) + 4 g H hh g H H h B 0 ( p 2 ; H , h ) , (217) 16 π 2 Π H + H − 1 P I ( p 2 ) B = − g 2 4 { ( c α s β + 2 s α c β ) 2 [ 2 A ( W ) − A ( h ) + ( 2 p 2 + 2 m h 2 − m W 2 ) B 0 ( p 2 ; W , h ) ] + ( s α s β − 2 c α c β ) 2 [ 2 A ( W ) − A ( H ) + ( 2 p 2 + 2 m H 2 − m W 2 ) B 0 ( p 2 ; W , H ) ] + s β 2 [ 2 A ( W ) − A ( G 0 ) + ( 2 p 2 + 2 m G 0 2 − m W 2 ) B 0 ( p 2 ; W , G 0 ) ] } − g Z 2 4 { c β 2 s β 2 [ 2 A ( Z ) − A ( G ± ) + ( 2 p 2 + 2 m G ± 2 − m Z 2 ) B 0 ( p 2 ; Z , G ± ) ] + ( c 2 W + c β 2 ) 2 [ 2 A ( Z ) − A ( H ± ) + ( 2 p 2 + 2 m H ± 2 − m Z 2 ) B 0 ( p 2 ; Z , H ± ) ] } \begin{aligned}
16\pi^2\Pi_{H^+H^-}^{\mathrm{1PI}}(p^2)_B
={}&-\frac{g^2}{4}\Bigl\{(c_\alpha s_\beta+2s_\alpha c_\beta)^2\left[2A(W)-A(h)+\left(2p^2+2m_h^2-m_W^2\right)B_0(p^2;W,h)\right]\\
&+(s_\alpha s_\beta-2c_\alpha c_\beta)^2\left[2A(W)-A(H)+\left(2p^2+2m_H^2-m_W^2\right)B_0(p^2;W,H)\right]\\
&+s_\beta^2\left[2A(W)-A(G^0)+\left(2p^2+2m_{G^0}^2-m_W^2\right)B_0(p^2;W,G^0)\right]\Bigr\}\\
&-\frac{g_Z^2}{4}\Bigl\{c_\beta^2s_\beta^2\left[2A(Z)-A(G^\pm)+\left(2p^2+2m_{G^\pm}^2-m_Z^2\right)B_0(p^2;Z,G^\pm)\right]\\
&+(c_{2W}+c_\beta^2)^2\left[2A(Z)-A(H^\pm)+\left(2p^2+2m_{H^\pm}^2-m_Z^2\right)B_0(p^2;Z,H^\pm)\right]\Bigr\}
\end{aligned} 16 π 2 Π H + H − 1PI ( p 2 ) B = − 4 g 2 { ( c α s β + 2 s α c β ) 2 [ 2 A ( W ) − A ( h ) + ( 2 p 2 + 2 m h 2 − m W 2 ) B 0 ( p 2 ; W , h ) ] + ( s α s β − 2 c α c β ) 2 [ 2 A ( W ) − A ( H ) + ( 2 p 2 + 2 m H 2 − m W 2 ) B 0 ( p 2 ; W , H ) ] + s β 2 [ 2 A ( W ) − A ( G 0 ) + ( 2 p 2 + 2 m G 0 2 − m W 2 ) B 0 ( p 2 ; W , G 0 ) ] } − 4 g Z 2 { c β 2 s β 2 [ 2 A ( Z ) − A ( G ± ) + ( 2 p 2 + 2 m G ± 2 − m Z 2 ) B 0 ( p 2 ; Z , G ± ) ] + ( c 2 W + c β 2 ) 2 [ 2 A ( Z ) − A ( H ± ) + ( 2 p 2 + 2 m H ± 2 − m Z 2 ) B 0 ( p 2 ; Z , H ± ) ] } − e 2 [ 2 A ( γ ) − A ( H ± ) + ( 2 p 2 + 2 m H ± 2 − m γ 2 ) B 0 ( p 2 ; γ , H ± ) ] + g 2 2 ( 1 + c β 2 ) [ 4 A ( W ) − 2 m W 2 ] + g Z 2 4 ( 4 c W 4 c β 2 + c 2 W 2 s β 2 ) [ 4 A ( Z ) − 2 m Z 2 ] + e 2 [ 4 A ( γ ) − 2 m γ 2 ] + g Z 2 m W 2 s β 2 c β 2 [ 4 B 0 ( p 2 ; W , Z ) − 2 − 1 2 B 0 ( p 2 ; c ± , c Z ) ] + g H + G − h 2 B 0 ( p 2 ; G ± , h ) + g H + H − h 2 B 0 ( p 2 ; H ± , h ) + g H + G − H 2 B 0 ( p 2 ; G ± , H ) + g H + H − H 2 B 0 ( p 2 ; H ± , H ) + ∣ g G + H − G 0 ∣ 2 B 0 ( p 2 ; G ± , G 0 ) − g H + H − h h A ( h ) − g H + H − G 0 G 0 A ( G 0 ) − g H + H − G + G − A ( G ± ) − g H + H − H H A ( H ) − 4 g H + H − H + H − A ( H ± ) , (218) \begin{aligned}
&-e^2\left[2A(\gamma)-A(H^\pm)+\left(2p^2+2m_{H^\pm}^2-m_\gamma^2\right)B_0(p^2;\gamma,H^\pm)\right]\\
&+\frac{g^2}{2}(1+c_\beta^2)\left[4A(W)-2m_W^2\right]+\frac{g_Z^2}{4}\left(4c_W^4c_\beta^2+c_{2W}^2s_\beta^2\right)\left[4A(Z)-2m_Z^2\right]\\
&+e^2\left[4A(\gamma)-2m_\gamma^2\right]\\
&+g_Z^2m_W^2s_\beta^2c_\beta^2\left[4B_0(p^2;W,Z)-2-\frac{1}{2}B_0(p^2;c^\pm,c_Z)\right]\\
&+g_{H^+G^-h}^2B_0(p^2;G^\pm,h)+g_{H^+H^-h}^2B_0(p^2;H^\pm,h)\\
&+g_{H^+G^-H}^2B_0(p^2;G^\pm,H)+g_{H^+H^-H}^2B_0(p^2;H^\pm,H)\\
&+\left|g_{G^+H^-G^0}\right|^2B_0(p^2;G^\pm,G^0)\\
&-g_{H^+H^-hh}A(h)-g_{H^+H^-G^0G^0}A(G^0)-g_{H^+H^-G^+G^-}A(G^\pm)\\
&-g_{H^+H^-HH}A(H)-4g_{H^+H^-H^+H^-}A(H^\pm), \tag*{(218)}
\end{aligned} − e 2 [ 2 A ( γ ) − A ( H ± ) + ( 2 p 2 + 2 m H ± 2 − m γ 2 ) B 0 ( p 2 ; γ , H ± ) ] + 2 g 2 ( 1 + c β 2 ) [ 4 A ( W ) − 2 m W 2 ] + 4 g Z 2 ( 4 c W 4 c β 2 + c 2 W 2 s β 2 ) [ 4 A ( Z ) − 2 m Z 2 ] + e 2 [ 4 A ( γ ) − 2 m γ 2 ] + g Z 2 m W 2 s β 2 c β 2 [ 4 B 0 ( p 2 ; W , Z ) − 2 − 2 1 B 0 ( p 2 ; c ± , c Z ) ] + g H + G − h 2 B 0 ( p 2 ; G ± , h ) + g H + H − h 2 B 0 ( p 2 ; H ± , h ) + g H + G − H 2 B 0 ( p 2 ; G ± , H ) + g H + H − H 2 B 0 ( p 2 ; H ± , H ) + ∣ g G + H − G 0 ∣ 2 B 0 ( p 2 ; G ± , G 0 ) − g H + H − hh A ( h ) − g H + H − G 0 G 0 A ( G 0 ) − g H + H − G + G − A ( G ± ) − g H + H − H H A ( H ) − 4 g H + H − H + H − A ( H ± ) , (218) 16 π 2 Π H + G − 1 P I ( p 2 ) B = g 2 4 { ( c α c β − 2 s α s β ) ( c α s β + 2 s α c β ) [ 2 A ( W ) − A ( h ) + ( 2 p 2 + 2 m h 2 − m W 2 ) B 0 ] ( p 2 ; W , h ) + ( s α c β + 2 c α s β ) ( s α s β − 2 c α c β ) [ 2 A ( W ) − A ( H ) + ( 2 p 2 + 2 m H 2 − m W 2 ) B 0 ] ( p 2 ; W , H ) + c β s β [ 2 A ( W ) − A ( G 0 ) + ( 2 p 2 + 2 m G 0 2 − m W 2 ) B 0 ] ( p 2 ; W , G 0 ) } − g Z 2 4 c β s β { ( c 2 W + s β 2 ) [ 2 A ( Z ) − A ( G ± ) + ( 2 p 2 + 2 m G ± 2 − m Z 2 ) B 0 ] ( p 2 ; Z , G ± ) + ( c 2 W + c β 2 ) [ 2 A ( Z ) − A ( H ± ) + ( 2 p 2 + 2 m H ± 2 − m Z 2 ) B 0 ] ( p 2 ; Z , H ± ) } + g 2 2 s β c β [ 4 A ( W ) − 2 m W 2 ] + g Z 2 4 ( 4 c W 4 − c 2 W 2 ) s β c β [ 4 A ( Z ) − 2 m Z 2 ] − g Z 2 m W 2 s β c β ( s W 2 − s β 2 ) [ 4 B 0 ( p 2 ; W , Z ) − 2 − 1 2 B 0 ( p 2 ; c ± , c Z ) ] + g G + G − h g G + H − h B 0 ( p 2 ; G ± , h ) + g G + H − h g H + H − h B 0 ( p 2 ; H ± , h ) + g G + G − H g G + H − H B 0 ( p 2 ; G ± , H ) + g G + H − H g H + H − H B 0 ( p 2 ; H ± , H ) − g H − G + h h A ( h ) − g H − G + G 0 G 0 A ( G 0 ) − 2 g G + G − G + H − A ( G ± ) − g H − G + H H A ( H ) − 2 g H + H − H − G + A ( H ± ) . (219) \begin{aligned}
16\pi^2\Pi^{1\mathrm{PI}}_{H^+G^-}(p^2)_B
={}&\frac{g^2}{4}\Bigl\{(c_\alpha c_\beta-2s_\alpha s_\beta)(c_\alpha s_\beta+2s_\alpha c_\beta)\left[2A(W)-A(h)+(2p^2+2m_h^2-m_W^2)B_0\right](p^2;W,h)\\
&+(s_\alpha c_\beta+2c_\alpha s_\beta)(s_\alpha s_\beta-2c_\alpha c_\beta)\left[2A(W)-A(H)+(2p^2+2m_H^2-m_W^2)B_0\right](p^2;W,H)\\
&+c_\beta s_\beta\left[2A(W)-A(G^0)+(2p^2+2m_{G^0}^2-m_W^2)B_0\right](p^2;W,G^0)\Bigr\}\\
&-\frac{g_Z^2}{4}c_\beta s_\beta\Bigl\{(c_{2W}+s_\beta^2)\left[2A(Z)-A(G^\pm)+(2p^2+2m_{G^\pm}^2-m_Z^2)B_0\right](p^2;Z,G^\pm)\\
&+(c_{2W}+c_\beta^2)\left[2A(Z)-A(H^\pm)+(2p^2+2m_{H^\pm}^2-m_Z^2)B_0\right](p^2;Z,H^\pm)\Bigr\}\\
&+\frac{g^2}{2}s_\beta c_\beta\left[4A(W)-2m_W^2\right]+\frac{g_Z^2}{4}(4c_W^4-c_{2W}^2)s_\beta c_\beta\left[4A(Z)-2m_Z^2\right]\\
&-g_Z^2m_W^2s_\beta c_\beta(s_W^2-s_\beta^2)\left[4B_0(p^2;W,Z)-2-\frac{1}{2}B_0(p^2;c^\pm,c_Z)\right]\\
&+g_{G^+G^-h}g_{G^+H^-h}B_0(p^2;G^\pm,h)+g_{G^+H^-h}g_{H^+H^-h}B_0(p^2;H^\pm,h)\\
&+g_{G^+G^-H}g_{G^+H^-H}B_0(p^2;G^\pm,H)+g_{G^+H^-H}g_{H^+H^-H}B_0(p^2;H^\pm,H)\\
&-g_{H^-G^+hh}A(h)-g_{H^-G^+G^0G^0}A(G^0)-2g_{G^+G^-G^+H^-}A(G^\pm)\\
&-g_{H^-G^+HH}A(H)-2g_{H^+H^-H^-G^+}A(H^\pm). \tag*{(219)}
\end{aligned} 16 π 2 Π H + G − 1 PI ( p 2 ) B = 4 g 2 { ( c α c β − 2 s α s β ) ( c α s β + 2 s α c β ) [ 2 A ( W ) − A ( h ) + ( 2 p 2 + 2 m h 2 − m W 2 ) B 0 ] ( p 2 ; W , h ) + ( s α c β + 2 c α s β ) ( s α s β − 2 c α c β ) [ 2 A ( W ) − A ( H ) + ( 2 p 2 + 2 m H 2 − m W 2 ) B 0 ] ( p 2 ; W , H ) + c β s β [ 2 A ( W ) − A ( G 0 ) + ( 2 p 2 + 2 m G 0 2 − m W 2 ) B 0 ] ( p 2 ; W , G 0 ) } − 4 g Z 2 c β s β { ( c 2 W + s β 2 ) [ 2 A ( Z ) − A ( G ± ) + ( 2 p 2 + 2 m G ± 2 − m Z 2 ) B 0 ] ( p 2 ; Z , G ± ) + ( c 2 W + c β 2 ) [ 2 A ( Z ) − A ( H ± ) + ( 2 p 2 + 2 m H ± 2 − m Z 2 ) B 0 ] ( p 2 ; Z , H ± ) } + 2 g 2 s β c β [ 4 A ( W ) − 2 m W 2 ] + 4 g Z 2 ( 4 c W 4 − c 2 W 2 ) s β c β [ 4 A ( Z ) − 2 m Z 2 ] − g Z 2 m W 2 s β c β ( s W 2 − s β 2 ) [ 4 B 0 ( p 2 ; W , Z ) − 2 − 2 1 B 0 ( p 2 ; c ± , c Z ) ] + g G + G − h g G + H − h B 0 ( p 2 ; G ± , h ) + g G + H − h g H + H − h B 0 ( p 2 ; H ± , h ) + g G + G − H g G + H − H B 0 ( p 2 ; G ± , H ) + g G + H − H g H + H − H B 0 ( p 2 ; H ± , H ) − g H − G + hh A ( h ) − g H − G + G 0 G 0 A ( G 0 ) − 2 g G + G − G + H − A ( G ± ) − g H − G + H H A ( H ) − 2 g H + H − H − G + A ( H ± ) . (219) Gauge boson
The fermionic contributions to the gauge boson two-point functions are the same as those in the SM. These are given by
16 π 2 Π γ γ 1 P I ( p 2 ) F = e 2 ∑ f 8 N c f Q f 2 B 3 ( p 2 ; f , f ) , (220) 16\pi^2\Pi^{1\mathrm{PI}}_{\gamma\gamma}(p^2)_F=e^2\sum_f 8N_c^fQ_f^2B_3(p^2;f,f), \tag*{(220)} 16 π 2 Π γ γ 1 PI ( p 2 ) F = e 2 f ∑ 8 N c f Q f 2 B 3 ( p 2 ; f , f ) , (220) 16 π 2 Π Z γ 1 P I ( p 2 ) F = e g Z ∑ f N c f [ 2 p 2 ( 2 I f Q f − 4 s W 2 Q f 2 ) B 3 ] ( p 2 ; f , f ) , (221) 16\pi^2\Pi^{1\mathrm{PI}}_{Z\gamma}(p^2)_F=eg_Z\sum_f N_c^f\left[2p^2(2I_fQ_f-4s_W^2Q_f^2)B_3\right](p^2;f,f), \tag*{(221)} 16 π 2 Π Z γ 1 PI ( p 2 ) F = e g Z f ∑ N c f [ 2 p 2 ( 2 I f Q f − 4 s W 2 Q f 2 ) B 3 ] ( p 2 ; f , f ) , (221) 16 π 2 Π Z Z 1 P I ( p 2 ) F = g Z 2 ∑ f N c f [ 2 p 2 ( 4 s W 2 Q f 2 − 4 s W 2 Q f I f + 2 I f 2 ) B 3 − 2 I f 2 m f 2 B 0 ] ( p 2 ; f , f ) , (222) 16\pi^2\Pi^{1\mathrm{PI}}_{ZZ}(p^2)_F=g_Z^2\sum_f N_c^f\left[2p^2(4s_W^2Q_f^2-4s_W^2Q_fI_f+2I_f^2)B_3-2I_f^2m_f^2B_0\right](p^2;f,f), \tag*{(222)} 16 π 2 Π Z Z 1 PI ( p 2 ) F = g Z 2 f ∑ N c f [ 2 p 2 ( 4 s W 2 Q f 2 − 4 s W 2 Q f I f + 2 I f 2 ) B 3 − 2 I f 2 m f 2 B 0 ] ( p 2 ; f , f ) , (222) 16 π 2 Π W W 1 P I ( p 2 ) F = g 2 ∑ f , f ′ N c f [ 2 p 2 B 3 − B 4 ] ( p 2 ; f , f ′ ) (223) 16\pi^2\Pi^{\mathrm{1PI}}_{WW}(p^2)_{F}=g^2\sum_{f,f'}N_c^{f}\left[2p^2B_3-B_4\right](p^2;f,f') \tag*{(223)} 16 π 2 Π W W 1PI ( p 2 ) F = g 2 f , f ′ ∑ N c f [ 2 p 2 B 3 − B 4 ] ( p 2 ; f , f ′ ) (223) The bosonic contributions to the gauge boson two-point functions are given by
16 π 2 Π γ γ 1 P I ( p 2 ) B = e 2 [ B 5 ( p 2 ; H ± , H ± ) + B 5 ( p 2 ; G ± , G ± ) + 2 m W 2 B 0 ( p 2 ; W , G ± ) ] − e 2 [ [ 16 B 22 + p 2 ( 2 B 21 + 2 B 1 + 5 B 0 ) ] ( p 2 ; W , W ) − 6 A ( W ) − ( 2 m W 2 − p 2 ) ] , (224) \begin{aligned}
16\pi^2\Pi^{\mathrm{1PI}}_{\gamma\gamma}(p^2)_{B}
={}&e^2\left[B_5(p^2;H^\pm,H^\pm)+B_5(p^2;G^\pm,G^\pm)+2m_W^2B_0(p^2;W,G^\pm)\right]\\
&-e^2\left[\left[16B_{22}+p^2(2B_{21}+2B_1+5B_0)\right](p^2;W,W)-6A(W)-(2m_W^2-p^2)\right], \tag*{(224)}
\end{aligned} 16 π 2 Π γ γ 1PI ( p 2 ) B = e 2 [ B 5 ( p 2 ; H ± , H ± ) + B 5 ( p 2 ; G ± , G ± ) + 2 m W 2 B 0 ( p 2 ; W , G ± ) ] − e 2 [ [ 16 B 22 + p 2 ( 2 B 21 + 2 B 1 + 5 B 0 ) ] ( p 2 ; W , W ) − 6 A ( W ) − ( 2 m W 2 − p 2 ) ] , (224) 16 π 2 Π Z γ 1 P I ( p 2 ) B = e g Z [ 1 2 ( c 2 W + c β 2 ) B 5 ( p 2 ; H ± , H ± ) + 1 2 ( c 2 W + s β 2 ) B 5 ( p 2 ; G ± , G ± ) − 2 m W 2 ( s W 2 − s β 2 ) B 0 ( p 2 ; G ± , W ) ] − e g Z c 2 W 2 [ [ 16 B 22 + p 2 ( 2 B 21 + 2 B 1 + 5 B 0 ) ] ( p 2 ; W , W ) − 6 A ( W ) − ( 2 m W 2 − p 2 ) ] , (225) \begin{aligned}
16\pi^2\Pi^{\mathrm{1PI}}_{Z\gamma}(p^2)_{B}
={}&eg_Z\left[\frac{1}{2}(c_{2W}+c_\beta^2)B_5(p^2;H^\pm,H^\pm)\right.\\
&\left.+\frac{1}{2}(c_{2W}+s_\beta^2)B_5(p^2;G^\pm,G^\pm)-2m_W^2(s_W^2-s_\beta^2)B_0(p^2;G^\pm,W)\right]\\
&-eg_Zc_{2W}^2\left[\left[16B_{22}+p^2(2B_{21}+2B_1+5B_0)\right](p^2;W,W)-6A(W)-(2m_W^2-p^2)\right], \tag*{(225)}
\end{aligned} 16 π 2 Π Z γ 1PI ( p 2 ) B = e g Z [ 2 1 ( c 2 W + c β 2 ) B 5 ( p 2 ; H ± , H ± ) + 2 1 ( c 2 W + s β 2 ) B 5 ( p 2 ; G ± , G ± ) − 2 m W 2 ( s W 2 − s β 2 ) B 0 ( p 2 ; G ± , W ) ] − e g Z c 2 W 2 [ [ 16 B 22 + p 2 ( 2 B 21 + 2 B 1 + 5 B 0 ) ] ( p 2 ; W , W ) − 6 A ( W ) − ( 2 m W 2 − p 2 ) ] , (225) 16 π 2 Π Z Z 1 P I ( p 2 ) B = g Z 2 4 [ ( c 2 W + c β 2 ) 2 B 5 ( p 2 ; H ± , H ± ) + ( c 2 W + s β 2 ) 2 B 5 ( p 2 ; G ± , G ± ) + 2 s β 2 c β 2 B 5 ( p 2 ; H ± , G ± ) + c α 2 B 5 ( p 2 ; h , G 0 ) + s α 2 B 5 ( p 2 ; H , G 0 ) ] + g Z 2 [ m Z 2 [ c α 2 B 0 ( p 2 ; h , Z ) + s α 2 B 0 ( p 2 ; H , Z ) ] + m W 2 [ 2 c β 2 s β 2 B 0 ( p 2 ; H ± , W ) + 2 ( s W 2 − s β 2 ) 2 B 0 ( p 2 ; G ± , W ) ] ] − g Z 2 c W 4 [ [ 16 B 22 + p 2 ( 2 B 21 + 2 B 1 + 5 B 0 ) ] ( p 2 ; W , W ) − 6 A ( W ) − ( 2 m W 2 − p 2 ) ] , (226) \begin{aligned}
16\pi^2\Pi^{\mathrm{1PI}}_{ZZ}(p^2)_{B}
={}&\frac{g_Z^2}{4}\left[(c_{2W}+c_\beta^2)^2B_5(p^2;H^\pm,H^\pm)\right.\\
&+(c_{2W}+s_\beta^2)^2B_5(p^2;G^\pm,G^\pm)+2s_\beta^2c_\beta^2B_5(p^2;H^\pm,G^\pm)\\
&\left.+c_\alpha^2B_5(p^2;h,G^0)+s_\alpha^2B_5(p^2;H,G^0)\right]\\
&+g_Z^2\left[m_Z^2\left[c_\alpha^2B_0(p^2;h,Z)+s_\alpha^2B_0(p^2;H,Z)\right]\right.\\
&\left.+m_W^2\left[2c_\beta^2s_\beta^2B_0(p^2;H^\pm,W)+2(s_W^2-s_\beta^2)^2B_0(p^2;G^\pm,W)\right]\right]\\
&-g_Z^2c_W^4\left[\left[16B_{22}+p^2(2B_{21}+2B_1+5B_0)\right](p^2;W,W)-6A(W)-(2m_W^2-p^2)\right], \tag*{(226)}
\end{aligned} 16 π 2 Π Z Z 1PI ( p 2 ) B = 4 g Z 2 [ ( c 2 W + c β 2 ) 2 B 5 ( p 2 ; H ± , H ± ) + ( c 2 W + s β 2 ) 2 B 5 ( p 2 ; G ± , G ± ) + 2 s β 2 c β 2 B 5 ( p 2 ; H ± , G ± ) + c α 2 B 5 ( p 2 ; h , G 0 ) + s α 2 B 5 ( p 2 ; H , G 0 ) ] + g Z 2 [ m Z 2 [ c α 2 B 0 ( p 2 ; h , Z ) + s α 2 B 0 ( p 2 ; H , Z ) ] + m W 2 [ 2 c β 2 s β 2 B 0 ( p 2 ; H ± , W ) + 2 ( s W 2 − s β 2 ) 2 B 0 ( p 2 ; G ± , W ) ] ] − g Z 2 c W 4 [ [ 16 B 22 + p 2 ( 2 B 21 + 2 B 1 + 5 B 0 ) ] ( p 2 ; W , W ) − 6 A ( W ) − ( 2 m W 2 − p 2 ) ] , (226) 16 π 2 Π W W 1 P I ( p 2 ) B = g 2 4 [ ( c α c β − 2 s α s β ) 2 B 5 ( p 2 ; h , G ± ) + ( c α s β + 2 s α c β ) 2 B 5 ( p 2 ; h , H ± ) + ( s α c β + 2 c α s β ) 2 B 5 ( p 2 ; H , G ± ) + ( s α s β − 2 c α c β ) 2 B 5 ( p 2 ; H , H ± ) + c β 2 B 5 ( p 2 ; G ± , G 0 ) + s β 2 B 5 ( p 2 ; H ± , G 0 ) ] + g 2 m W 2 [ ( c β c α − 2 s β s α ) 2 B 0 ( p 2 ; h , W ) + ( c β s α + 2 s β c α ) 2 B 0 ( p 2 ; H , W ) ] + g Z 2 m W 2 [ ( s W 2 − s β 2 ) 2 B 0 ( p 2 ; G ± , Z ) + c β 2 s β 2 B 0 ( p 2 ; H ± , Z ) ] + e 2 m W 2 B 0 ( p 2 ; G ± , γ ) + g 2 [ − s W 2 [ 16 B 22 + p 2 ( 2 B 21 + 2 B 1 + 5 B 0 ) ] ( p 2 ; W , γ ) − c W 2 [ 16 B 22 + p 2 ( 2 B 21 + 2 B 1 + 5 B 0 ) ] ( p 2 ; W , Z ) + 3 [ c W 2 A ( Z ) + A ( W ) ] + m W 2 + c W 2 m Z 2 − p 2 ] . (227) \begin{aligned}
16\pi^2\Pi^{\mathrm{1PI}}_{WW}(p^2)_{B}
={}&\frac{g^2}{4}\left[(c_\alpha c_\beta-2s_\alpha s_\beta)^2B_5(p^2;h,G^\pm)+(c_\alpha s_\beta+2s_\alpha c_\beta)^2B_5(p^2;h,H^\pm)\right.\\
&+(s_\alpha c_\beta+2c_\alpha s_\beta)^2B_5(p^2;H,G^\pm)+(s_\alpha s_\beta-2c_\alpha c_\beta)^2B_5(p^2;H,H^\pm)\\
&\left.+c_\beta^2B_5(p^2;G^\pm,G^0)+s_\beta^2B_5(p^2;H^\pm,G^0)\right]\\
&+g^2m_W^2\left[(c_\beta c_\alpha-2s_\beta s_\alpha)^2B_0(p^2;h,W)+(c_\beta s_\alpha+2s_\beta c_\alpha)^2B_0(p^2;H,W)\right]\\
&+g_Z^2m_W^2\left[(s_W^2-s_\beta^2)^2B_0(p^2;G^\pm,Z)+c_\beta^2s_\beta^2B_0(p^2;H^\pm,Z)\right]\\
&+e^2m_W^2B_0(p^2;G^\pm,\gamma) \\
&+ g^{2}\left[-s_{W}^{2}\left[16B_{22}+p^{2}(2B_{21}+2B_{1}+5B_{0})\right](p^{2};W,\gamma)
-c_{W}^{2}\left[16B_{22}+p^{2}(2B_{21}+2B_{1}+5B_{0})\right](p^{2};W,Z)+3\left[c_{W}^{2}A(Z)+A(W)\right]
+m_{W}^{2}+c_{W}^{2}m_{Z}^{2}-p^{2}\right]. \tag*{(227)}
\end{aligned} 16 π 2 Π W W 1PI ( p 2 ) B = 4 g 2 [ ( c α c β − 2 s α s β ) 2 B 5 ( p 2 ; h , G ± ) + ( c α s β + 2 s α c β ) 2 B 5 ( p 2 ; h , H ± ) + ( s α c β + 2 c α s β ) 2 B 5 ( p 2 ; H , G ± ) + ( s α s β − 2 c α c β ) 2 B 5 ( p 2 ; H , H ± ) + c β 2 B 5 ( p 2 ; G ± , G 0 ) + s β 2 B 5 ( p 2 ; H ± , G 0 ) ] + g 2 m W 2 [ ( c β c α − 2 s β s α ) 2 B 0 ( p 2 ; h , W ) + ( c β s α + 2 s β c α ) 2 B 0 ( p 2 ; H , W ) ] + g Z 2 m W 2 [ ( s W 2 − s β 2 ) 2 B 0 ( p 2 ; G ± , Z ) + c β 2 s β 2 B 0 ( p 2 ; H ± , Z ) ] + e 2 m W 2 B 0 ( p 2 ; G ± , γ ) + g 2 [ − s W 2 [ 16 B 22 + p 2 ( 2 B 21 + 2 B 1 + 5 B 0 ) ] ( p 2 ; W , γ ) − c W 2 [ 16 B 22 + p 2 ( 2 B 21 + 2 B 1 + 5 B 0 ) ] ( p 2 ; W , Z ) + 3 [ c W 2 A ( Z ) + A ( W ) ] + m W 2 + c W 2 m Z 2 − p 2 ] . (227) Fermion
The fermion two-point functions are given by
16 π 2 Π f f , S 1 P I ( p 2 ) = − 2 e 2 Q f 2 ( 2 B 0 − 1 ) ( p 2 ; f , γ ) − 2 g Z 2 ( v f 2 − a f 2 ) ( 2 B 0 − 1 ) ( p 2 ; f , Z ) + m f 2 v 2 [ ζ h 2 B 0 ( p 2 ; f , h ) + ζ H 2 B 0 ( p 2 ; f , H ) − ζ G 0 2 B 0 ( p 2 ; f , G 0 ) ] − 2 m f ′ 2 v 2 [ ζ G ± 2 B 0 ( p 2 ; f ′ , G ± ) + ζ H ± 2 B 0 ( p 2 ; f ′ , H ± ) ] , (228) 16\pi^{2}\Pi_{ff,S}^{\mathrm{1PI}}(p^{2})=-2e^{2}Q_{f}^{2}(2B_{0}-1)(p^{2};f,\gamma)-2g_{Z}^{2}(v_{f}^{2}-a_{f}^{2})(2B_{0}-1)(p^{2};f,Z)
+\frac{m_{f}^{2}}{v^{2}}\left[\zeta_{h}^{2}B_{0}(p^{2};f,h)+\zeta_{H}^{2}B_{0}(p^{2};f,H)-\zeta_{G^{0}}^{2}B_{0}(p^{2};f,G^{0})\right]
-2\frac{m_{f'}^{2}}{v^{2}}\left[\zeta_{G^{\pm}}^{2}B_{0}(p^{2};f',G^{\pm})+\zeta_{H^{\pm}}^{2}B_{0}(p^{2};f',H^{\pm})\right], \tag*{(228)} 16 π 2 Π f f , S 1PI ( p 2 ) = − 2 e 2 Q f 2 ( 2 B 0 − 1 ) ( p 2 ; f , γ ) − 2 g Z 2 ( v f 2 − a f 2 ) ( 2 B 0 − 1 ) ( p 2 ; f , Z ) + v 2 m f 2 [ ζ h 2 B 0 ( p 2 ; f , h ) + ζ H 2 B 0 ( p 2 ; f , H ) − ζ G 0 2 B 0 ( p 2 ; f , G 0 ) ] − 2 v 2 m f ′ 2 [ ζ G ± 2 B 0 ( p 2 ; f ′ , G ± ) + ζ H ± 2 B 0 ( p 2 ; f ′ , H ± ) ] , (228) 16 π 2 Π f f , V 1 P I ( p 2 ) = − e 2 Q f 2 ( 2 B 1 + 1 ) ( p 2 ; f , γ ) − g Z 2 ( v f 2 + a f 2 ) ( 2 B 1 + 1 ) ( p 2 ; f , Z ) − g 2 4 ( 2 B 1 + 1 ) ( p 2 ; f ′ , W ) − m f 2 + m f ′ 2 v 2 [ ζ G ± 2 B 1 ( p 2 ; f ′ , G ± ) + ζ H ± 2 B 1 ( p 2 ; f ′ , H ± ) ] , (229) 16\pi^{2}\Pi_{ff,V}^{\mathrm{1PI}}(p^{2})=-e^{2}Q_{f}^{2}(2B_{1}+1)(p^{2};f,\gamma)-g_{Z}^{2}(v_{f}^{2}+a_{f}^{2})(2B_{1}+1)(p^{2};f,Z)-\frac{g^{2}}{4}(2B_{1}+1)(p^{2};f',W)
-\frac{m_{f}^{2}+m_{f'}^{2}}{v^{2}}\left[\zeta_{G^{\pm}}^{2}B_{1}(p^{2};f',G^{\pm})+\zeta_{H^{\pm}}^{2}B_{1}(p^{2};f',H^{\pm})\right], \tag*{(229)} 16 π 2 Π f f , V 1PI ( p 2 ) = − e 2 Q f 2 ( 2 B 1 + 1 ) ( p 2 ; f , γ ) − g Z 2 ( v f 2 + a f 2 ) ( 2 B 1 + 1 ) ( p 2 ; f , Z ) − 4 g 2 ( 2 B 1 + 1 ) ( p 2 ; f ′ , W ) − v 2 m f 2 + m f ′ 2 [ ζ G ± 2 B 1 ( p 2 ; f ′ , G ± ) + ζ H ± 2 B 1 ( p 2 ; f ′ , H ± ) ] , (229) 16 π 2 Π f f , A 1 P I ( p 2 ) = − 2 g Z 2 v f a f ( 2 B 1 + 1 ) ( p 2 ; f , Z ) − g 2 4 ( 2 B 1 + 1 ) ( p 2 ; f ′ , W ) + m f 2 − m f ′ 2 v 2 [ ζ G ± 2 B 1 ( p 2 ; f ′ , G ± ) + ζ H ± 2 B 1 ( p 2 ; f ′ , H ± ) ] . (230) 16\pi^{2}\Pi_{ff,A}^{\mathrm{1PI}}(p^{2})=-2g_{Z}^{2}v_{f}a_{f}(2B_{1}+1)(p^{2};f,Z)-\frac{g^{2}}{4}(2B_{1}+1)(p^{2};f',W)
+\frac{m_{f}^{2}-m_{f'}^{2}}{v^{2}}\left[\zeta_{G^{\pm}}^{2}B_{1}(p^{2};f',G^{\pm})+\zeta_{H^{\pm}}^{2}B_{1}(p^{2};f',H^{\pm})\right]. \tag*{(230)} 16 π 2 Π f f , A 1PI ( p 2 ) = − 2 g Z 2 v f a f ( 2 B 1 + 1 ) ( p 2 ; f , Z ) − 4 g 2 ( 2 B 1 + 1 ) ( p 2 ; f ′ , W ) + v 2 m f 2 − m f ′ 2 [ ζ G ± 2 B 1 ( p 2 ; f ′ , G ± ) + ζ H ± 2 B 1 ( p 2 ; f ′ , H ± ) ] . (230) Three-point functions
vertex
The fermionic contributions to the 1PI h h h hhh hhh
Γ h h h 1 P I ( p 1 2 , p 2 2 , q 2 ) F = ζ h 3 Γ h h h 1 P I ( p 1 2 , p 2 2 , q 2 ) F , S M , (231) \Gamma_{hhh}^{\mathrm{1PI}}(p_{1}^{2},p_{2}^{2},q^{2})_{F}=\zeta_{h}^{3}\Gamma_{hhh}^{\mathrm{1PI}}(p_{1}^{2},p_{2}^{2},q^{2})_{F,\mathrm{SM}}, \tag*{(231)} Γ hhh 1PI ( p 1 2 , p 2 2 , q 2 ) F = ζ h 3 Γ hhh 1PI ( p 1 2 , p 2 2 , q 2 ) F , SM , (231) where
16 π 2 Γ h h h 1 P I ( p 1 2 , p 2 2 , q 2 ) F , S M = − ∑ f N c f 8 m f 4 v 3 { B 0 ( p 1 2 , f , f ) + B 0 ( p 2 2 , f , f ) + B 0 ( q 2 , f , f ) + ( 4 m f 2 − q 2 + p 1 2 + p 2 2 2 ) C 0 ( f , f , f ) } . (232) 16\pi^{2}\Gamma_{hhh}^{\mathrm{1PI}}(p_{1}^{2},p_{2}^{2},q^{2})_{F,\mathrm{SM}}
=-\sum_{f}N_{c}^{f}\frac{8m_{f}^{4}}{v^{3}}\left\{B_{0}(p_{1}^{2},f,f)+B_{0}(p_{2}^{2},f,f)+B_{0}(q^{2},f,f)
+\left(4m_{f}^{2}-\frac{q^{2}+p_{1}^{2}+p_{2}^{2}}{2}\right)C_{0}(f,f,f)\right\}. \tag*{(232)} 16 π 2 Γ hhh 1PI ( p 1 2 , p 2 2 , q 2 ) F , SM = − f ∑ N c f v 3 8 m f 4 { B 0 ( p 1 2 , f , f ) + B 0 ( p 2 2 , f , f ) + B 0 ( q 2 , f , f ) + ( 4 m f 2 − 2 q 2 + p 1 2 + p 2 2 ) C 0 ( f , f , f ) } . (232) The bosonic contributions obtained as
16 π 2 Γ h h h 1 P I ( p 1 2 , p 2 2 , q 2 ) B = 2 g 3 m W ( c β c α − 2 s β s α ) ( 1 + 3 s α 2 ) [ B 0 ( p 1 2 , W , W ) + B 0 ( p 2 2 , W , W ) + B 0 ( q 2 , W , W ) − 3 2 ] + 8 g 3 m W 3 ( c β c α − 2 s β s α ) 3 C 0 ( W , W , W ) − g 3 2 m W ( c β c α − 2 s β s α ) 3 [ C S S S S V V ( G ± , W , W ) + C S S S V S V ( W , G ± , W ) + C S S S V V S ( W , W , G ± ) ] − g 3 2 m W ( c β c α − 2 s β s α ) ( s β c α + 2 c β s α ) 2 × [ C S S S S V V ( H ± , W , W ) + C S S S V S V ( W , H ± , W ) + C S S S V V S ( W , W , H ± ) ] + g 2 2 g G + G − h ( c β c α − 2 s β s α ) 2 [ C S S S V S S ( W , G ± , G ± ) + C S S S S V S ( G ± , W , G ± ) + C S S S S S V ( G ± , G ± , W ) ] + g 2 2 g H + H − h ( s β c α + 2 c β s α ) 2 [ C S S S V S S ( W , H ± , H ± ) + C S S S S V S ( H ± , W , H ± ) + C S S S S S V ( H ± , H ± , W ) ] − g 2 2 g H + G − h ( c β c α − 2 s β s α ) ( s β c α + 2 c β s α ) × [ C S S S V S S ( W , G ± , H ± ) + C S S S S V S ( G ± , W , H ± ) + C S S S S S V ( G ± , H ± , W ) + C S S S V S S ( W , H ± , G ± ) + C S S S S V S ( H ± , W , G ± ) + C S S S S S V ( H ± , G ± , W ) ] − g 3 2 m W 3 ( c β c α − 2 s β s α ) 3 C 0 ( c ± , c ± , c ± ) + g Z 3 m Z c α 3 [ B 0 ( p 1 2 , Z , Z ) + B 0 ( p 2 2 , Z , Z ) + B 0 ( q 2 , Z , Z ) − 3 2 ] + 4 g Z 3 m Z 3 c α 3 C 0 ( Z , Z , Z ) − g Z 3 4 m Z c α 3 [ C S S S S V V ( G 0 , Z , Z ) + C S S S V S V ( Z , G 0 , Z ) + C S S S V V S ( Z , Z , G 0 ) ] + g Z 2 2 g G 0 G 0 h g Z 2 c α 2 [ C S S S V S S ( Z , G 0 , G 0 ) + C S S S S V S ( G 0 , Z , G 0 ) + C S S S S S V ( G 0 , G 0 , Z ) ] − g Z 3 4 m Z 3 c α 3 C 0 ( c Z , c Z , c Z ) + 2 g H + H − h g H + H − h h [ B 0 ( p 1 2 , H ± , H ± ) + B 0 ( p 2 2 , H ± , H ± ) + B 0 ( q 2 , H ± , H ± ) ] + 2 g G + G − h g G + G − h h [ B 0 ( p 1 2 , G ± , G ± ) + B 0 ( p 2 2 , G ± , G ± ) + B 0 ( q 2 , G ± , G ± ) ] + 4 g H + G − h g H + G − h h [ B 0 ( p 1 2 , H ± , G ± ) + B 0 ( p 2 2 , H ± , G ± ) + B 0 ( q 2 , H ± , G ± ) ] + 4 g G 0 G 0 h g G 0 G 0 h h [ B 0 ( p 1 2 , G 0 , G 0 ) + B 0 ( p 2 2 , G 0 , G 0 ) + B 0 ( q 2 , G 0 , G 0 ) ] + 4 g H H h g H H h h [ B 0 ( p 1 2 , H , H ) + B 0 ( p 2 2 , H , H ) + B 0 ( q 2 , H , H ) ] + 12 g H h h g H h h h [ B 0 ( p 1 2 , h , H ) + B 0 ( p 2 2 , h , H ) + B 0 ( q 2 , h , H ) ] + 72 g h h h g h h h h [ B 0 ( p 1 2 , h , h ) + B 0 ( p 2 2 , h , h ) + B 0 ( q 2 , h , h ) ] − 2 g H + H − h 3 C 0 ( H ± , H ± , H ± ) − 2 g G + G − h 3 C 0 ( G ± , G ± , G ± ) − 8 g G 0 G 0 h 3 C 0 ( G 0 , G 0 , G 0 ) − 8 g H H h 3 C 0 ( H , H , H ) − 216 g h h h 3 C 0 ( h , h , h ) − 2 g H + H − h g H + G − h 2 [ C 0 ( G ± , H ± , H ± ) + C 0 ( H ± , G ± , H ± ) + C 0 ( H ± , H ± , G ± ) ] − 2 g G + G − h g H + G − h 2 [ C 0 ( H ± , G ± , G ± ) + C 0 ( G ± , H ± , G ± ) + C 0 ( G ± , G ± , H ± ) ] \begin{aligned}
16\pi^{2}\Gamma_{hhh}^{\mathrm{1PI}}(p_{1}^{2},p_{2}^{2},q^{2})_{B}
={}&2g^{3}m_{W}(c_{\beta}c_{\alpha}-2s_{\beta}s_{\alpha})(1+3s_{\alpha}^{2})\left[B_{0}(p_{1}^{2},W,W)+B_{0}(p_{2}^{2},W,W)+B_{0}(q^{2},W,W)-\frac{3}{2}\right]\\
&+8g^{3}m_{W}^{3}(c_{\beta}c_{\alpha}-2s_{\beta}s_{\alpha})^{3}C_{0}(W,W,W)\\
&-\frac{g^{3}}{2}m_{W}(c_{\beta}c_{\alpha}-2s_{\beta}s_{\alpha})^{3}\left[C_{SSS}^{SVV}(G^{\pm},W,W)+C_{SSS}^{VSV}(W,G^{\pm},W)+C_{SSS}^{VVS}(W,W,G^{\pm})\right]\\
&-\frac{g^{3}}{2}m_{W}(c_{\beta}c_{\alpha}-2s_{\beta}s_{\alpha})(s_{\beta}c_{\alpha}+2c_{\beta}s_{\alpha})^{2}\\
&\qquad\times\left[C_{SSS}^{SVV}(H^{\pm},W,W)+C_{SSS}^{VSV}(W,H^{\pm},W)+C_{SSS}^{VVS}(W,W,H^{\pm})\right]\\
&+\frac{g^{2}}{2}g_{G^{+}G^{-}h}(c_{\beta}c_{\alpha}-2s_{\beta}s_{\alpha})^{2}\left[C_{SSS}^{VSS}(W,G^{\pm},G^{\pm})+C_{SSS}^{SVS}(G^{\pm},W,G^{\pm})+C_{SSS}^{SSV}(G^{\pm},G^{\pm},W)\right]\\
&+\frac{g^{2}}{2}g_{H^{+}H^{-}h}(s_{\beta}c_{\alpha}+2c_{\beta}s_{\alpha})^{2}\left[C_{SSS}^{VSS}(W,H^{\pm},H^{\pm})+C_{SSS}^{SVS}(H^{\pm},W,H^{\pm})+C_{SSS}^{SSV}(H^{\pm},H^{\pm},W)\right]\\
&-\frac{g^{2}}{2}g_{H^{+}G^{-}h}(c_{\beta}c_{\alpha}-2s_{\beta}s_{\alpha})(s_{\beta}c_{\alpha}+2c_{\beta}s_{\alpha})\\
&\qquad\times\left[C_{SSS}^{VSS}(W,G^{\pm},H^{\pm})+C_{SSS}^{SVS}(G^{\pm},W,H^{\pm})+C_{SSS}^{SSV}(G^{\pm},H^{\pm},W)\right.\\
&\qquad\qquad\left.+C_{SSS}^{VSS}(W,H^{\pm},G^{\pm})+C_{SSS}^{SVS}(H^{\pm},W,G^{\pm})+C_{SSS}^{SSV}(H^{\pm},G^{\pm},W)\right]\\
&-\frac{g^{3}}{2}m_{W}^{3}(c_{\beta}c_{\alpha}-2s_{\beta}s_{\alpha})^{3}C_{0}(c^{\pm},c^{\pm},c^{\pm})\\
&+g_{Z}^{3}m_{Z}c_{\alpha}^{3}\left[B_{0}(p_{1}^{2},Z,Z)+B_{0}(p_{2}^{2},Z,Z)+B_{0}(q^{2},Z,Z)-\frac{3}{2}\right]\\
&+4g_{Z}^{3}m_{Z}^{3}c_{\alpha}^{3}C_{0}(Z,Z,Z)\\
&-\frac{g_{Z}^{3}}{4}m_{Z}c_{\alpha}^{3}\left[C_{SSS}^{SVV}(G^{0},Z,Z)+C_{SSS}^{VSV}(Z,G^{0},Z)+C_{SSS}^{VVS}(Z,Z,G^{0})\right]\\
&+\frac{g_{Z}^{2}}{2}g_{G^{0}G^{0}h}g_{Z}^{2}c_{\alpha}^{2}\left[C_{SSS}^{VSS}(Z,G^{0},G^{0})+C_{SSS}^{SVS}(G^{0},Z,G^{0})+C_{SSS}^{SSV}(G^{0},G^{0},Z)\right]\\
&-\frac{g_{Z}^{3}}{4}m_{Z}^{3}c_{\alpha}^{3}C_{0}(c_{Z},c_{Z},c_{Z})\\
&+2g_{H^{+}H^{-}h}g_{H^{+}H^{-}hh}\left[B_{0}(p_{1}^{2},H^{\pm},H^{\pm})+B_{0}(p_{2}^{2},H^{\pm},H^{\pm})+B_{0}(q^{2},H^{\pm},H^{\pm})\right]\\
&+2g_{G^{+}G^{-}h}g_{G^{+}G^{-}hh}\left[B_{0}(p_{1}^{2},G^{\pm},G^{\pm})+B_{0}(p_{2}^{2},G^{\pm},G^{\pm})+B_{0}(q^{2},G^{\pm},G^{\pm})\right]\\
&+4g_{H^{+}G^{-}h}g_{H^{+}G^{-}hh}\left[B_{0}(p_{1}^{2},H^{\pm},G^{\pm})+B_{0}(p_{2}^{2},H^{\pm},G^{\pm})+B_{0}(q^{2},H^{\pm},G^{\pm})\right]\\
&+4g_{G^{0}G^{0}h}g_{G^{0}G^{0}hh}\left[B_{0}(p_{1}^{2},G^{0},G^{0})+B_{0}(p_{2}^{2},G^{0},G^{0})+B_{0}(q^{2},G^{0},G^{0})\right]\\
&+4g_{HHh}g_{HHhh}\left[B_{0}(p_{1}^{2},H,H)+B_{0}(p_{2}^{2},H,H)+B_{0}(q^{2},H,H)\right]\\
&+12g_{Hhh}g_{Hhhh}\left[B_{0}(p_{1}^{2},h,H)+B_{0}(p_{2}^{2},h,H)+B_{0}(q^{2},h,H)\right]\\
&+72g_{hhh}g_{hhhh}\left[B_{0}(p_{1}^{2},h,h)+B_{0}(p_{2}^{2},h,h)+B_{0}(q^{2},h,h)\right]\\
&-2g_{H^{+}H^{-}h}^{3}C_{0}(H^{\pm},H^{\pm},H^{\pm})-2g_{G^{+}G^{-}h}^{3}C_{0}(G^{\pm},G^{\pm},G^{\pm})\\
&-8g_{G^{0}G^{0}h}^{3}C_{0}(G^{0},G^{0},G^{0})-8g_{HHh}^{3}C_{0}(H,H,H)-216g_{hhh}^{3}C_{0}(h,h,h)\\
&-2g_{H^{+}H^{-}h}g_{H^{+}G^{-}h}^{2}\left[C_{0}(G^{\pm},H^{\pm},H^{\pm})+C_{0}(H^{\pm},G^{\pm},H^{\pm})+C_{0}(H^{\pm},H^{\pm},G^{\pm})\right]\\
&-2g_{G^{+}G^{-}h}g_{H^{+}G^{-}h}^{2}\left[C_{0}(H^{\pm},G^{\pm},G^{\pm})+C_{0}(G^{\pm},H^{\pm},G^{\pm})+C_{0}(G^{\pm},G^{\pm},H^{\pm})\right]
\end{aligned} 16 π 2 Γ hhh 1PI ( p 1 2 , p 2 2 , q 2 ) B = 2 g 3 m W ( c β c α − 2 s β s α ) ( 1 + 3 s α 2 ) [ B 0 ( p 1 2 , W , W ) + B 0 ( p 2 2 , W , W ) + B 0 ( q 2 , W , W ) − 2 3 ] + 8 g 3 m W 3 ( c β c α − 2 s β s α ) 3 C 0 ( W , W , W ) − 2 g 3 m W ( c β c α − 2 s β s α ) 3 [ C S S S S V V ( G ± , W , W ) + C S S S V S V ( W , G ± , W ) + C S S S V V S ( W , W , G ± ) ] − 2 g 3 m W ( c β c α − 2 s β s α ) ( s β c α + 2 c β s α ) 2 × [ C S S S S V V ( H ± , W , W ) + C S S S V S V ( W , H ± , W ) + C S S S V V S ( W , W , H ± ) ] + 2 g 2 g G + G − h ( c β c α − 2 s β s α ) 2 [ C S S S V S S ( W , G ± , G ± ) + C S S S S V S ( G ± , W , G ± ) + C S S S S S V ( G ± , G ± , W ) ] + 2 g 2 g H + H − h ( s β c α + 2 c β s α ) 2 [ C S S S V S S ( W , H ± , H ± ) + C S S S S V S ( H ± , W , H ± ) + C S S S S S V ( H ± , H ± , W ) ] − 2 g 2 g H + G − h ( c β c α − 2 s β s α ) ( s β c α + 2 c β s α ) × [ C S S S V S S ( W , G ± , H ± ) + C S S S S V S ( G ± , W , H ± ) + C S S S S S V ( G ± , H ± , W ) + C S S S V S S ( W , H ± , G ± ) + C S S S S V S ( H ± , W , G ± ) + C S S S S S V ( H ± , G ± , W ) ] − 2 g 3 m W 3 ( c β c α − 2 s β s α ) 3 C 0 ( c ± , c ± , c ± ) + g Z 3 m Z c α 3 [ B 0 ( p 1 2 , Z , Z ) + B 0 ( p 2 2 , Z , Z ) + B 0 ( q 2 , Z , Z ) − 2 3 ] + 4 g Z 3 m Z 3 c α 3 C 0 ( Z , Z , Z ) − 4 g Z 3 m Z c α 3 [ C S S S S V V ( G 0 , Z , Z ) + C S S S V S V ( Z , G 0 , Z ) + C S S S V V S ( Z , Z , G 0 ) ] + 2 g Z 2 g G 0 G 0 h g Z 2 c α 2 [ C S S S V S S ( Z , G 0 , G 0 ) + C S S S S V S ( G 0 , Z , G 0 ) + C S S S S S V ( G 0 , G 0 , Z ) ] − 4 g Z 3 m Z 3 c α 3 C 0 ( c Z , c Z , c Z ) + 2 g H + H − h g H + H − hh [ B 0 ( p 1 2 , H ± , H ± ) + B 0 ( p 2 2 , H ± , H ± ) + B 0 ( q 2 , H ± , H ± ) ] + 2 g G + G − h g G + G − hh [ B 0 ( p 1 2 , G ± , G ± ) + B 0 ( p 2 2 , G ± , G ± ) + B 0 ( q 2 , G ± , G ± ) ] + 4 g H + G − h g H + G − hh [ B 0 ( p 1 2 , H ± , G ± ) + B 0 ( p 2 2 , H ± , G ± ) + B 0 ( q 2 , H ± , G ± ) ] + 4 g G 0 G 0 h g G 0 G 0 hh [ B 0 ( p 1 2 , G 0 , G 0 ) + B 0 ( p 2 2 , G 0 , G 0 ) + B 0 ( q 2 , G 0 , G 0 ) ] + 4 g H H h g H H hh [ B 0 ( p 1 2 , H , H ) + B 0 ( p 2 2 , H , H ) + B 0 ( q 2 , H , H ) ] + 12 g H hh g H hhh [ B 0 ( p 1 2 , h , H ) + B 0 ( p 2 2 , h , H ) + B 0 ( q 2 , h , H ) ] + 72 g hhh g hhhh [ B 0 ( p 1 2 , h , h ) + B 0 ( p 2 2 , h , h ) + B 0 ( q 2 , h , h ) ] − 2 g H + H − h 3 C 0 ( H ± , H ± , H ± ) − 2 g G + G − h 3 C 0 ( G ± , G ± , G ± ) − 8 g G 0 G 0 h 3 C 0 ( G 0 , G 0 , G 0 ) − 8 g H H h 3 C 0 ( H , H , H ) − 216 g hhh 3 C 0 ( h , h , h ) − 2 g H + H − h g H + G − h 2 [ C 0 ( G ± , H ± , H ± ) + C 0 ( H ± , G ± , H ± ) + C 0 ( H ± , H ± , G ± ) ] − 2 g G + G − h g H + G − h 2 [ C 0 ( H ± , G ± , G ± ) + C 0 ( G ± , H ± , G ± ) + C 0 ( G ± , G ± , H ± ) ] − 8 g H H h g h h h 2 [ C 0 ( h , H , H ) + C 0 ( H , H , h ) + C 0 ( H , h , H ) ] − 24 g h h h g h h H 2 [ C 0 ( h , h , H ) + C 0 ( H , h , h ) + C 0 ( h , H , h ) ] , (233) -8g_{HHh}g_{hhh}^{2}\left[C_{0}(h,H,H)+C_{0}(H,H,h)+C_{0}(H,h,H)\right]-24g_{hhh}g_{hhH}^{2}\left[C_{0}(h,h,H)+C_{0}(H,h,h)+C_{0}(h,H,h)\right], \tag*{(233)} − 8 g H H h g hhh 2 [ C 0 ( h , H , H ) + C 0 ( H , H , h ) + C 0 ( H , h , H ) ] − 24 g hhh g hh H 2 [ C 0 ( h , h , H ) + C 0 ( H , h , h ) + C 0 ( h , H , h ) ] , (233) where
C S S S S V V ( X , Y , Z ) = − 1 2 B 0 ( p 1 2 ; X , Y ) + 2 B 0 ( p 2 2 ; Y , Z ) − 1 2 B 0 ( q 2 ; X , Z ) + ( p 1 2 − p 2 2 2 + q 2 + 2 m X 2 − m Y 2 2 − m Z 2 2 ) C 0 ( X , Y , Z ) , (234) C_{SSS}^{SVV}(X,Y,Z)=-\frac{1}{2}B_{0}(p_{1}^{2};X,Y)+2B_{0}(p_{2}^{2};Y,Z)-\frac{1}{2}B_{0}(q^{2};X,Z)+\left(p_{1}^{2}-\frac{p_{2}^{2}}{2}+q^{2}+2m_{X}^{2}-\frac{m_{Y}^{2}}{2}-\frac{m_{Z}^{2}}{2}\right)C_{0}(X,Y,Z), \tag*{(234)} C S S S S V V ( X , Y , Z ) = − 2 1 B 0 ( p 1 2 ; X , Y ) + 2 B 0 ( p 2 2 ; Y , Z ) − 2 1 B 0 ( q 2 ; X , Z ) + ( p 1 2 − 2 p 2 2 + q 2 + 2 m X 2 − 2 m Y 2 − 2 m Z 2 ) C 0 ( X , Y , Z ) , (234) C S S S V S V ( X , Y , Z ) = − 1 2 B 0 ( p 1 2 ; X , Y ) − 1 2 B 0 ( p 2 2 ; Y , Z ) + 2 B 0 ( q 2 ; X , Z ) + ( p 1 2 + p 2 2 − q 2 2 − m X 2 2 + 2 m Y 2 − m Z 2 2 ) C 0 ( X , Y , Z ) , (235) C_{SSS}^{VSV}(X,Y,Z)=-\frac{1}{2}B_{0}(p_{1}^{2};X,Y)-\frac{1}{2}B_{0}(p_{2}^{2};Y,Z)+2B_{0}(q^{2};X,Z)+\left(p_{1}^{2}+p_{2}^{2}-\frac{q^{2}}{2}-\frac{m_{X}^{2}}{2}+2m_{Y}^{2}-\frac{m_{Z}^{2}}{2}\right)C_{0}(X,Y,Z), \tag*{(235)} C S S S V S V ( X , Y , Z ) = − 2 1 B 0 ( p 1 2 ; X , Y ) − 2 1 B 0 ( p 2 2 ; Y , Z ) + 2 B 0 ( q 2 ; X , Z ) + ( p 1 2 + p 2 2 − 2 q 2 − 2 m X 2 + 2 m Y 2 − 2 m Z 2 ) C 0 ( X , Y , Z ) , (235) C S S S V V S ( X , Y , Z ) = 2 B 0 ( p 1 2 ; X , Y ) − 1 2 B 0 ( p 2 2 ; Y , Z ) − 1 2 B 0 ( q 2 ; X , Z ) + ( − p 1 2 2 + p 2 2 + q 2 − m X 2 2 − m Y 2 2 + 2 m Z 2 ) C 0 ( X , Y , Z ) , (236) C_{SSS}^{VVS}(X,Y,Z)=2B_{0}(p_{1}^{2};X,Y)-\frac{1}{2}B_{0}(p_{2}^{2};Y,Z)-\frac{1}{2}B_{0}(q^{2};X,Z)+\left(-\frac{p_{1}^{2}}{2}+p_{2}^{2}+q^{2}-\frac{m_{X}^{2}}{2}-\frac{m_{Y}^{2}}{2}+2m_{Z}^{2}\right)C_{0}(X,Y,Z), \tag*{(236)} C S S S V V S ( X , Y , Z ) = 2 B 0 ( p 1 2 ; X , Y ) − 2 1 B 0 ( p 2 2 ; Y , Z ) − 2 1 B 0 ( q 2 ; X , Z ) + ( − 2 p 1 2 + p 2 2 + q 2 − 2 m X 2 − 2 m Y 2 + 2 m Z 2 ) C 0 ( X , Y , Z ) , (236) C S S S V S S ( X , Y , Z ) = B 0 ( p 1 2 ; X , Y ) − B 0 ( p 2 2 ; Y , Z ) + B 0 ( q 2 ; X , Z ) + ( p 1 2 − 2 p 2 2 + q 2 − m X 2 + m Y 2 + m Z 2 ) C 0 ( X , Y , Z ) , (237) C_{SSS}^{VSS}(X,Y,Z)=B_{0}(p_{1}^{2};X,Y)-B_{0}(p_{2}^{2};Y,Z)+B_{0}(q^{2};X,Z)+\left(p_{1}^{2}-2p_{2}^{2}+q^{2}-m_{X}^{2}+m_{Y}^{2}+m_{Z}^{2}\right)C_{0}(X,Y,Z), \tag*{(237)} C S S S V S S ( X , Y , Z ) = B 0 ( p 1 2 ; X , Y ) − B 0 ( p 2 2 ; Y , Z ) + B 0 ( q 2 ; X , Z ) + ( p 1 2 − 2 p 2 2 + q 2 − m X 2 + m Y 2 + m Z 2 ) C 0 ( X , Y , Z ) , (237) C S S S S V S ( X , Y , Z ) = B 0 ( p 1 2 ; X , Y ) + B 0 ( p 2 2 ; Y , Z ) − B 0 ( q 2 ; X , Z ) + ( p 1 2 + p 2 2 − 2 q 2 + m X 2 − m Y 2 + m Z 2 ) C 0 ( X , Y , Z ) , (238) C_{SSS}^{SVS}(X,Y,Z)=B_{0}(p_{1}^{2};X,Y)+B_{0}(p_{2}^{2};Y,Z)-B_{0}(q^{2};X,Z)+\left(p_{1}^{2}+p_{2}^{2}-2q^{2}+m_{X}^{2}-m_{Y}^{2}+m_{Z}^{2}\right)C_{0}(X,Y,Z), \tag*{(238)} C S S S S V S ( X , Y , Z ) = B 0 ( p 1 2 ; X , Y ) + B 0 ( p 2 2 ; Y , Z ) − B 0 ( q 2 ; X , Z ) + ( p 1 2 + p 2 2 − 2 q 2 + m X 2 − m Y 2 + m Z 2 ) C 0 ( X , Y , Z ) , (238) C S S S S S V ( X , Y , Z ) = − B 0 ( p 1 2 ; X , Y ) + B 0 ( p 2 2 ; Y , Z ) + B 0 ( q 2 ; X , Z ) + ( − 2 p 1 2 + p 2 2 + q 2 + m X 2 + m Y 2 − m Z 2 ) C 0 ( X , Y , Z ) . (239) C_{SSS}^{SSV}(X,Y,Z)=-B_{0}(p_{1}^{2};X,Y)+B_{0}(p_{2}^{2};Y,Z)+B_{0}(q^{2};X,Z)+\left(-2p_{1}^{2}+p_{2}^{2}+q^{2}+m_{X}^{2}+m_{Y}^{2}-m_{Z}^{2}\right)C_{0}(X,Y,Z). \tag*{(239)} C S S S S S V ( X , Y , Z ) = − B 0 ( p 1 2 ; X , Y ) + B 0 ( p 2 2 ; Y , Z ) + B 0 ( q 2 ; X , Z ) + ( − 2 p 1 2 + p 2 2 + q 2 + m X 2 + m Y 2 − m Z 2 ) C 0 ( X , Y , Z ) . (239) hVV vertex
The fermionic contributions to the 1PI h Z Z hZZ h Z Z
Γ h Z Z i , 1 P I ( p 1 2 , p 2 2 , q 2 ) F = ζ h Γ h Z Z i , 1 P I ( p 1 2 , p 2 2 , q 2 ) F , S M ( i = 1 , 2 , 3 ) , (240) \Gamma_{hZZ}^{i,\mathrm{1PI}}(p_{1}^{2},p_{2}^{2},q^{2})_{F}=\zeta_{h}\Gamma_{hZZ}^{i,\mathrm{1PI}}(p_{1}^{2},p_{2}^{2},q^{2})_{F,\mathrm{SM}}\qquad(i=1,2,3), \tag*{(240)} Γ h Z Z i , 1PI ( p 1 2 , p 2 2 , q 2 ) F = ζ h Γ h Z Z i , 1PI ( p 1 2 , p 2 2 , q 2 ) F , SM ( i = 1 , 2 , 3 ) , (240) where
16 π 2 Γ h Z Z 1 , 1 P I ( p 1 2 , p 2 2 , q 2 ) F , S M = − ∑ f N c f 4 g Z 2 m f 2 v { ( v f 2 + a f 2 ) [ B 0 ( p 1 2 , f , f ) + B 0 ( p 2 2 , f , f ) + 2 B 0 ( q 2 , f , f ) + ( 4 m f 2 − p 1 2 − p 2 2 ) C 0 ( f , f , f ) − 8 C 24 ( f , f , f ) ] − ( v f 2 − a f 2 ) [ B 0 ( p 1 2 , f , f ) + B 0 ( p 2 2 , f , f ) + ( 4 m f 2 − q 2 ) C 0 ( f , f , f ) ] } , (241) \begin{aligned}
16\pi^{2}\Gamma_{hZZ}^{1,\mathrm{1PI}}(p_{1}^{2},p_{2}^{2},q^{2})_{F,\mathrm{SM}} \\
&=-\sum_{f}N_{c}^{f}\frac{4g_{Z}^{2}m_{f}^{2}}{v}\left\{(v_{f}^{2}+a_{f}^{2})\left[B_{0}(p_{1}^{2},f,f)+B_{0}(p_{2}^{2},f,f)+2B_{0}(q^{2},f,f)+(4m_{f}^{2}-p_{1}^{2}-p_{2}^{2})C_{0}(f,f,f)-8C_{24}(f,f,f)\right]-(v_{f}^{2}-a_{f}^{2})\left[B_{0}(p_{1}^{2},f,f)+B_{0}(p_{2}^{2},f,f)+(4m_{f}^{2}-q^{2})C_{0}(f,f,f)\right]\right\}, \tag*{(241)}
\end{aligned} 16 π 2 Γ h Z Z 1 , 1PI ( p 1 2 , p 2 2 , q 2 ) F , SM = − f ∑ N c f v 4 g Z 2 m f 2 { ( v f 2 + a f 2 ) [ B 0 ( p 1 2 , f , f ) + B 0 ( p 2 2 , f , f ) + 2 B 0 ( q 2 , f , f ) + ( 4 m f 2 − p 1 2 − p 2 2 ) C 0 ( f , f , f ) − 8 C 24 ( f , f , f ) ] − ( v f 2 − a f 2 ) [ B 0 ( p 1 2 , f , f ) + B 0 ( p 2 2 , f , f ) + ( 4 m f 2 − q 2 ) C 0 ( f , f , f ) ] } , (241) ( m Z 2 ) − 1 16 π 2 Γ h Z Z 2 , 1 P I ( p 1 2 , p 2 2 , q 2 ) F , S M = − ∑ f N c f 8 g Z 2 m f 2 v [ ( v f 2 + a f 2 ) ( 4 C 23 + 3 C 12 + C 11 + C 0 ) + ( v f 2 − a f 2 ) ( C 12 − C 11 ) ] ( f , f , f ) , (242) \begin{aligned}
(m_{Z}^{2})^{-1}16\pi^{2}\Gamma_{hZZ}^{2,\mathrm{1PI}}(p_{1}^{2},p_{2}^{2},q^{2})_{F,\mathrm{SM}} \\
&=-\sum_{f}N_{c}^{f}\frac{8g_{Z}^{2}m_{f}^{2}}{v}\left[(v_{f}^{2}+a_{f}^{2})(4C_{23}+3C_{12}+C_{11}+C_{0})+(v_{f}^{2}-a_{f}^{2})(C_{12}-C_{11})\right](f,f,f), \tag*{(242)}
\end{aligned} ( m Z 2 ) − 1 16 π 2 Γ h Z Z 2 , 1PI ( p 1 2 , p 2 2 , q 2 ) F , SM = − f ∑ N c f v 8 g Z 2 m f 2 [ ( v f 2 + a f 2 ) ( 4 C 23 + 3 C 12 + C 11 + C 0 ) + ( v f 2 − a f 2 ) ( C 12 − C 11 ) ] ( f , f , f ) , (242) ( m Z 2 ) − 1 16 π 2 Γ h Z Z 3 , 1 P I ( p 1 2 , p 2 2 , q 2 ) F , S M = ∑ f N c f 16 g Z 2 m f 2 v v f a f ( C 11 + C 12 + C 0 ) ( f , f , f ) . (243) (m_Z^2)^{-1}16\pi^2\Gamma_{hZZ}^{3,1\mathrm{PI}}(p_1^2,p_2^2,q^2)_{F,\mathrm{SM}}
= \sum_f N_c^f\frac{16g_Z^2m_f^2}{v}v_{faf}(C_{11}+C_{12}+C_0)(f,f,f). \tag*{(243)} ( m Z 2 ) − 1 16 π 2 Γ h Z Z 3 , 1 PI ( p 1 2 , p 2 2 , q 2 ) F , SM = f ∑ N c f v 16 g Z 2 m f 2 v f a f ( C 11 + C 12 + C 0 ) ( f , f , f ) . (243) The bosonic contributions are obtained as
16 π 2 Γ h Z Z 1 , 1 P I ( p 1 2 , p 2 2 , q 2 ) B = 2 g 3 m W c W 2 ( c α c β − 2 s α s β ) C h V V 1 V V V ( W , W , W ) + g 3 m W ( c α c β − 2 s α s β ) ( s W 2 − s β 2 ) [ C h V V 1 S V V ( G ± , W , W ) + C h V V 1 V V S ( W , W , G ± ) ] + g 3 m W ( c α s β + 2 s α c β ) s β c β [ C h V V 1 S V V ( H ± , W , W ) + C h V V 1 V V S ( W , W , H ± ) ] − g g Z 2 m W ( c α c β − 2 s α s β ) ( c 2 W + s β 2 ) ( s W 2 − s β 2 ) [ C 24 ( W , G ± , G ± ) + C 24 ( G ± , G ± , W ) ] − g g Z 2 m W ( c α s β + 2 s α c β ) ( c 2 W + c β 2 ) s β c β [ C 24 ( W , H ± , H ± ) + C 24 ( H ± , H ± , W ) ] + g g Z 2 m W ( c α c β − 2 s α s β ) s β 2 c β 2 [ C 24 ( W , H ± , G ± ) + C 24 ( G ± , H ± , W ) ] + g g Z 2 m W ( c α s β + 2 s α c β ) s β c β ( s W 2 − s β 2 ) [ C 24 ( W , G ± , H ± ) + C 24 ( H ± , G ± , W ) ] − 2 g g Z 2 m W ( c α c β − 2 s α s β ) ( s W 2 − s β 2 ) 2 C 0 ( W , G ± , W ) − 2 g g Z 2 m W 3 ( c α c β − 2 s α s β ) s β 2 c β 2 C 0 ( W , H ± , W ) + 2 g G + G − h g Z 2 m W 2 ( s W 2 − s β 2 ) 2 C 0 ( G ± , W , G ± ) + 2 g H + H − h g Z 2 m W 2 s β 2 c β 2 C 0 ( H ± , W , H ± ) − 2 g H + G − h g Z 2 m W 2 s β c β ( s W 2 − s β 2 ) [ C 0 ( H ± , W , G ± ) + C 0 ( G ± , W , H ± ) ] − 2 g 3 m W c W 2 ( c α c β − 2 s α s β ) C 24 ( c ± , c ± , c ± ) − g 3 c W 2 m W ( c α c β − 2 s α s β ) [ 6 B 0 ( q 2 ; W , W ) − 4 ] − g g Z 2 m W ( c α c β s W 2 + 2 c W 2 s α s β ) ( s W 2 − s β 2 ) [ B 0 ( p 1 2 ; G ± , W ) + B 0 ( p 2 2 ; W , G ± ) ] − g g Z 2 m W ( c α s β s W 2 − 2 c W 2 s α c β ) s β c β [ B 0 ( p 1 2 ; H ± , W ) + B 0 ( p 2 2 ; W , H ± ) ] + g Z 3 2 m Z c α 3 [ C 24 ( Z , h , G 0 ) + C 24 ( G 0 , h , Z ) ] + g Z 3 2 m Z c α s α 2 [ C 24 ( Z , H , G 0 ) + C 24 ( G 0 , H , Z ) ] − g Z 3 m Z 3 c α 3 C 0 ( Z , h , Z ) − g Z 3 m Z 3 c α s α 2 C 0 ( Z , H , Z ) + 6 g h h h g Z 2 m Z 2 c α 2 C 0 ( h , Z , h ) + 2 g H H H g Z 2 m Z 2 s α 2 C 0 ( H , Z , H ) − 2 g H H h g Z 2 m Z 2 c α s α [ C 0 ( h , Z , H ) + C 0 ( H , Z , h ) ] − g Z 3 2 m Z c α 3 [ B 0 ( p 1 2 ; h , Z ) + B 0 ( p 2 2 ; h , Z ) ] − g Z 3 2 m Z c α s α 2 [ B 0 ( p 1 2 ; H , Z ) + B 0 ( p 2 2 ; H , Z ) ] − 4 g Z 2 [ 1 2 g H + H − h ( c 2 W + c β 2 ) 2 C 24 ( H ± , H ± , H ± ) + 1 2 g H + H − h c β 2 s β 2 C 24 ( H ± , G ± , H ± ) + 1 2 g G + G − h c β 2 s β 2 C 24 ( G ± , H ± , G ± ) + 1 2 g H + G − h c β s β ( c 2 W + c β 2 ) [ C 24 ( H ± , H ± , G ± ) + C 24 ( G ± , H ± , H ± ) ] + 1 2 g H + G − h c β s β ( c 2 W + s β 2 ) [ C 24 ( H ± , G ± , G ± ) + C 24 ( G ± , G ± , H ± ) ] + 1 2 g G + G − h ( c 2 W + s β 2 ) C 24 ( G ± , G ± , G ± ) + 1 2 g G 0 G 0 h s α 2 C 24 ( G 0 , H , G 0 ) + 1 2 g G 0 G 0 h c α 2 C 24 ( G 0 , h , G 0 ) \begin{aligned}
16\pi^2\Gamma_{hZZ}^{1,1\mathrm{PI}}(p_1^2,p_2^2,q^2)_B
={}&2g^3m_Wc_W^2(c_\alpha c_\beta-2s_\alpha s_\beta)C_{hVV1}^{VVV}(W,W,W)\\
&+g^3m_W(c_\alpha c_\beta-2s_\alpha s_\beta)(s_W^2-s_\beta^2)
[C_{hVV1}^{SVV}(G^\pm,W,W)+C_{hVV1}^{VVS}(W,W,G^\pm)]\\
&+g^3m_W(c_\alpha s_\beta+2s_\alpha c_\beta)s_\beta c_\beta
[C_{hVV1}^{SVV}(H^\pm,W,W)+C_{hVV1}^{VVS}(W,W,H^\pm)]\\
&-gg_Z^2m_W(c_\alpha c_\beta-2s_\alpha s_\beta)(c_{2W}+s_\beta^2)(s_W^2-s_\beta^2)
[C_{24}(W,G^\pm,G^\pm)+C_{24}(G^\pm,G^\pm,W)]\\
&-gg_Z^2m_W(c_\alpha s_\beta+2s_\alpha c_\beta)(c_{2W}+c_\beta^2)s_\beta c_\beta
[C_{24}(W,H^\pm,H^\pm)+C_{24}(H^\pm,H^\pm,W)]\\
&+gg_Z^2m_W(c_\alpha c_\beta-2s_\alpha s_\beta)s_\beta^2c_\beta^2
[C_{24}(W,H^\pm,G^\pm)+C_{24}(G^\pm,H^\pm,W)]\\
&+gg_Z^2m_W(c_\alpha s_\beta+2s_\alpha c_\beta)s_\beta c_\beta(s_W^2-s_\beta^2)
[C_{24}(W,G^\pm,H^\pm)+C_{24}(H^\pm,G^\pm,W)]\\
&-2gg_Z^2m_W(c_\alpha c_\beta-2s_\alpha s_\beta)(s_W^2-s_\beta^2)^2C_0(W,G^\pm,W)\\
&-2gg_Z^2m_W^3(c_\alpha c_\beta-2s_\alpha s_\beta)s_\beta^2c_\beta^2C_0(W,H^\pm,W)\\
&+2g_{G^+G^-h}g_Z^2m_W^2(s_W^2-s_\beta^2)^2C_0(G^\pm,W,G^\pm)
+2g_{H^+H^-h}g_Z^2m_W^2s_\beta^2c_\beta^2C_0(H^\pm,W,H^\pm)\\
&-2g_{H^+G^-h}g_Z^2m_W^2s_\beta c_\beta(s_W^2-s_\beta^2)
[C_0(H^\pm,W,G^\pm)+C_0(G^\pm,W,H^\pm)]\\
&-2g^3m_Wc_W^2(c_\alpha c_\beta-2s_\alpha s_\beta)C_{24}(c^\pm,c^\pm,c^\pm)\\
&-g^3c_W^2m_W(c_\alpha c_\beta-2s_\alpha s_\beta)[6B_0(q^2;W,W)-4]\\
&-gg_Z^2m_W(c_\alpha c_\beta s_W^2+2c_W^2s_\alpha s_\beta)(s_W^2-s_\beta^2)
[B_0(p_1^2;G^\pm,W)+B_0(p_2^2;W,G^\pm)]\\
&-gg_Z^2m_W(c_\alpha s_\beta s_W^2-2c_W^2s_\alpha c_\beta)s_\beta c_\beta
[B_0(p_1^2;H^\pm,W)+B_0(p_2^2;W,H^\pm)]\\
&+\frac{g_Z^3}{2}m_Zc_\alpha^3[C_{24}(Z,h,G^0)+C_{24}(G^0,h,Z)]
+\frac{g_Z^3}{2}m_Zc_\alpha s_\alpha^2[C_{24}(Z,H,G^0)+C_{24}(G^0,H,Z)]\\
&-g_Z^3m_Z^3c_\alpha^3C_0(Z,h,Z)-g_Z^3m_Z^3c_\alpha s_\alpha^2C_0(Z,H,Z)\\
&+6g_{hhh}g_Z^2m_Z^2c_\alpha^2C_0(h,Z,h)+2g_{HHH}g_Z^2m_Z^2s_\alpha^2C_0(H,Z,H)\\
&-2g_{HHh}g_Z^2m_Z^2c_\alpha s_\alpha[C_0(h,Z,H)+C_0(H,Z,h)]\\
&-\frac{g_Z^3}{2}m_Zc_\alpha^3[B_0(p_1^2;h,Z)+B_0(p_2^2;h,Z)]
-\frac{g_Z^3}{2}m_Zc_\alpha s_\alpha^2[B_0(p_1^2;H,Z)+B_0(p_2^2;H,Z)]\\
&-4g_Z^2[\frac{1}{2}g_{H^+H^-h}(c_{2W}+c_\beta^2)^2C_{24}(H^\pm,H^\pm,H^\pm)\\
&\qquad+\frac{1}{2}g_{H^+H^-h}c_\beta^2s_\beta^2C_{24}(H^\pm,G^\pm,H^\pm)
+\frac{1}{2}g_{G^+G^-h}c_\beta^2s_\beta^2C_{24}(G^\pm,H^\pm,G^\pm)\\
&\qquad+\frac{1}{2}g_{H^+G^-h}c_\beta s_\beta(c_{2W}+c_\beta^2)
[C_{24}(H^\pm,H^\pm,G^\pm)+C_{24}(G^\pm,H^\pm,H^\pm)]\\
&\qquad+\frac{1}{2}g_{H^+G^-h}c_\beta s_\beta(c_{2W}+s_\beta^2)
[C_{24}(H^\pm,G^\pm,G^\pm)+C_{24}(G^\pm,G^\pm,H^\pm)]\\
&\qquad+\frac{1}{2}g_{G^+G^-h}(c_{2W}+s_\beta^2)C_{24}(G^\pm,G^\pm,G^\pm)\\
&\qquad+\frac{1}{2}g_{G^0G^0h}s_\alpha^2C_{24}(G^0,H,G^0)
+\frac{1}{2}g_{G^0G^0h}c_\alpha^2C_{24}(G^0,h,G^0)
\end{aligned} 16 π 2 Γ h Z Z 1 , 1 PI ( p 1 2 , p 2 2 , q 2 ) B = 2 g 3 m W c W 2 ( c α c β − 2 s α s β ) C hV V 1 V V V ( W , W , W ) + g 3 m W ( c α c β − 2 s α s β ) ( s W 2 − s β 2 ) [ C hV V 1 S V V ( G ± , W , W ) + C hV V 1 V V S ( W , W , G ± )] + g 3 m W ( c α s β + 2 s α c β ) s β c β [ C hV V 1 S V V ( H ± , W , W ) + C hV V 1 V V S ( W , W , H ± )] − g g Z 2 m W ( c α c β − 2 s α s β ) ( c 2 W + s β 2 ) ( s W 2 − s β 2 ) [ C 24 ( W , G ± , G ± ) + C 24 ( G ± , G ± , W )] − g g Z 2 m W ( c α s β + 2 s α c β ) ( c 2 W + c β 2 ) s β c β [ C 24 ( W , H ± , H ± ) + C 24 ( H ± , H ± , W )] + g g Z 2 m W ( c α c β − 2 s α s β ) s β 2 c β 2 [ C 24 ( W , H ± , G ± ) + C 24 ( G ± , H ± , W )] + g g Z 2 m W ( c α s β + 2 s α c β ) s β c β ( s W 2 − s β 2 ) [ C 24 ( W , G ± , H ± ) + C 24 ( H ± , G ± , W )] − 2 g g Z 2 m W ( c α c β − 2 s α s β ) ( s W 2 − s β 2 ) 2 C 0 ( W , G ± , W ) − 2 g g Z 2 m W 3 ( c α c β − 2 s α s β ) s β 2 c β 2 C 0 ( W , H ± , W ) + 2 g G + G − h g Z 2 m W 2 ( s W 2 − s β 2 ) 2 C 0 ( G ± , W , G ± ) + 2 g H + H − h g Z 2 m W 2 s β 2 c β 2 C 0 ( H ± , W , H ± ) − 2 g H + G − h g Z 2 m W 2 s β c β ( s W 2 − s β 2 ) [ C 0 ( H ± , W , G ± ) + C 0 ( G ± , W , H ± )] − 2 g 3 m W c W 2 ( c α c β − 2 s α s β ) C 24 ( c ± , c ± , c ± ) − g 3 c W 2 m W ( c α c β − 2 s α s β ) [ 6 B 0 ( q 2 ; W , W ) − 4 ] − g g Z 2 m W ( c α c β s W 2 + 2 c W 2 s α s β ) ( s W 2 − s β 2 ) [ B 0 ( p 1 2 ; G ± , W ) + B 0 ( p 2 2 ; W , G ± )] − g g Z 2 m W ( c α s β s W 2 − 2 c W 2 s α c β ) s β c β [ B 0 ( p 1 2 ; H ± , W ) + B 0 ( p 2 2 ; W , H ± )] + 2 g Z 3 m Z c α 3 [ C 24 ( Z , h , G 0 ) + C 24 ( G 0 , h , Z )] + 2 g Z 3 m Z c α s α 2 [ C 24 ( Z , H , G 0 ) + C 24 ( G 0 , H , Z )] − g Z 3 m Z 3 c α 3 C 0 ( Z , h , Z ) − g Z 3 m Z 3 c α s α 2 C 0 ( Z , H , Z ) + 6 g hhh g Z 2 m Z 2 c α 2 C 0 ( h , Z , h ) + 2 g H H H g Z 2 m Z 2 s α 2 C 0 ( H , Z , H ) − 2 g H H h g Z 2 m Z 2 c α s α [ C 0 ( h , Z , H ) + C 0 ( H , Z , h )] − 2 g Z 3 m Z c α 3 [ B 0 ( p 1 2 ; h , Z ) + B 0 ( p 2 2 ; h , Z )] − 2 g Z 3 m Z c α s α 2 [ B 0 ( p 1 2 ; H , Z ) + B 0 ( p 2 2 ; H , Z )] − 4 g Z 2 [ 2 1 g H + H − h ( c 2 W + c β 2 ) 2 C 24 ( H ± , H ± , H ± ) + 2 1 g H + H − h c β 2 s β 2 C 24 ( H ± , G ± , H ± ) + 2 1 g G + G − h c β 2 s β 2 C 24 ( G ± , H ± , G ± ) + 2 1 g H + G − h c β s β ( c 2 W + c β 2 ) [ C 24 ( H ± , H ± , G ± ) + C 24 ( G ± , H ± , H ± )] + 2 1 g H + G − h c β s β ( c 2 W + s β 2 ) [ C 24 ( H ± , G ± , G ± ) + C 24 ( G ± , G ± , H ± )] + 2 1 g G + G − h ( c 2 W + s β 2 ) C 24 ( G ± , G ± , G ± ) + 2 1 g G 0 G 0 h s α 2 C 24 ( G 0 , H , G 0 ) + 2 1 g G 0 G 0 h c α 2 C 24 ( G 0 , h , G 0 ) + 1 2 g H H h s α 2 C 24 ( H , G 0 , H ) + 3 2 g h h h c α 2 C 24 ( h , G 0 , h ) − 1 2 g H H h s α c α [ C 24 ( H , G 0 , h ) + C 24 ( h , G 0 , H ) ] + g Z 2 [ 1 2 g G + G − h ( 4 c W 4 s β 2 + c 2 W 2 c β 2 ) B 0 ( q 2 , G ± , G ± ) + 1 2 g H + H − h ( 4 c W 4 c β 2 + c 2 W 2 s β 2 ) B 0 ( q 2 , H ± , H ± ) + g H + G − h ( 4 c W 4 − c 2 W 2 ) s β c β B 0 ( q 2 , H ± , G ± ) + 1 2 g G 0 G 0 h B 0 ( q 2 , G 0 , G 0 ) + 3 2 g h h h c α 2 B 0 ( q 2 , h , h ) + 1 2 g H H h s α 2 B 0 ( q 2 , H , H ) − g H h h s α c α B 0 ( q 2 , h , H ) ] , (244) \begin{aligned}
&+\frac{1}{2}g_{HHh}s_{\alpha}^{2}C_{24}(H,G^{0},H)+\frac{3}{2}g_{hhh}c_{\alpha}^{2}C_{24}(h,G^{0},h)\\
&-\frac{1}{2}g_{HHh}s_{\alpha}c_{\alpha}\left[C_{24}(H,G^{0},h)+C_{24}(h,G^{0},H)\right]\\
&+g_{Z}^{2}\left[\frac{1}{2}g_{G^{+}G^{-}h}(4c_{W}^{4}s_{\beta}^{2}+c_{2W}^{2}c_{\beta}^{2})B_{0}(q^{2},G^{\pm},G^{\pm})\right.\\
&\quad+\frac{1}{2}g_{H^{+}H^{-}h}(4c_{W}^{4}c_{\beta}^{2}+c_{2W}^{2}s_{\beta}^{2})B_{0}(q^{2},H^{\pm},H^{\pm})\\
&\quad+g_{H^{+}G^{-}h}(4c_{W}^{4}-c_{2W}^{2})s_{\beta}c_{\beta}B_{0}(q^{2},H^{\pm},G^{\pm})+\frac{1}{2}g_{G^{0}G^{0}h}B_{0}(q^{2},G^{0},G^{0})\\
&\quad\left.+\frac{3}{2}g_{hhh}c_{\alpha}^{2}B_{0}(q^{2},h,h)+\frac{1}{2}g_{HHh}s_{\alpha}^{2}B_{0}(q^{2},H,H)-g_{Hhh}s_{\alpha}c_{\alpha}B_{0}(q^{2},h,H)\right], \tag*{(244)}
\end{aligned} + 2 1 g H H h s α 2 C 24 ( H , G 0 , H ) + 2 3 g hhh c α 2 C 24 ( h , G 0 , h ) − 2 1 g H H h s α c α [ C 24 ( H , G 0 , h ) + C 24 ( h , G 0 , H ) ] + g Z 2 [ 2 1 g G + G − h ( 4 c W 4 s β 2 + c 2 W 2 c β 2 ) B 0 ( q 2 , G ± , G ± ) + 2 1 g H + H − h ( 4 c W 4 c β 2 + c 2 W 2 s β 2 ) B 0 ( q 2 , H ± , H ± ) + g H + G − h ( 4 c W 4 − c 2 W 2 ) s β c β B 0 ( q 2 , H ± , G ± ) + 2 1 g G 0 G 0 h B 0 ( q 2 , G 0 , G 0 ) + 2 3 g hhh c α 2 B 0 ( q 2 , h , h ) + 2 1 g H H h s α 2 B 0 ( q 2 , H , H ) − g H hh s α c α B 0 ( q 2 , h , H ) ] , (244) ( m Z 2 ) − 1 16 π 2 Γ h Z Z 2 , 1 P I ( p 1 2 , p 2 2 , q 2 ) B = 2 g 3 m W c W 2 ( c α c β − 2 s α s β ) C h V V 2 V V V ( W , W , W ) + g 3 m W ( c α c β − 2 s α s β ) ( s W 2 − s β 2 ) [ C h V V 2 S V V ( G ± , W , W ) + C h V V 2 V V S ( W , W , G ± ) ] + g 3 m W ( c α s β + 2 s α c β ) s β c β [ C h V V 2 S V V ( H ± , W , W ) + C h V V 2 V V S ( W , W , H ± ) ] − g g Z 2 m W ( c α c β − 2 s α s β ) ( c 2 W + s β 2 ) ( s W 2 − s β 2 ) [ C h V V 2 V S S ( W , G ± , G ± ) + C h V V 2 S S V ( G ± , G ± , W ) ] − g g Z 2 m W ( c α s β + 2 s α c β ) ( c 2 W + c β 2 ) s β c β [ C h V V 2 V S S ( W , H ± , H ± ) + C h V V 2 S S V ( H ± , H ± , W ) ] + g g Z 2 m W ( c α c β − 2 s α s β ) s β 2 c β 2 [ C h V V 2 V S S ( W , H ± , G ± ) + C h V V 2 S S V ( G ± , H ± , W ) ] + g g Z 2 m W ( c α s β + 2 s α c β ) s β c β ( s W 2 − s β 2 ) [ C h V V 2 V S S ( W , G ± , H ± ) + C h V V 2 S S V ( H ± , G ± , W ) ] − 2 g 3 m W c W 2 ( c α c β − 2 s α s β ) C 1223 ( c ± , c ± , c ± ) + g Z 3 2 m Z c α s α [ C h V V 2 V S S ( Z , h , G 0 ) + C h V V 2 S S V ( G 0 , h , Z ) ] + g Z 3 2 m Z c α s α 2 [ C h V V 2 V S S ( Z , H , G 0 ) + C h V V 2 S S V ( G 0 , H , Z ) ] − 4 g Z 2 [ 1 2 g H + H − h ( c 2 W + c β 2 ) 2 C 1223 ( H ± , H ± , H ± ) + 1 2 g H + H − h c β 2 s β 2 C 1223 ( H ± , G ± , H ± ) + 1 2 g G + G − h c β 2 s β 2 C 1223 ( G ± , H ± , G ± ) + 1 2 g H + G − h c β s β ( c 2 W + c β 2 ) [ C 1223 ( H ± , H ± , G ± ) + C 1223 ( G ± , H ± , H ± ) ] + 1 2 g H + G − h c β s β ( c 2 W + s β 2 ) [ C 1223 ( H ± , G ± , G ± ) + C 1223 ( G ± , G ± , H ± ) ] + 1 2 g G + G − h ( c 2 W + s β 2 ) 2 C 1223 ( G ± , G ± , G ± ) + 1 2 g G 0 G 0 h s α 2 C 1223 ( G 0 , H , G 0 ) + 1 2 g G 0 G 0 h c α 2 C 1223 ( G 0 , h , G 0 ) + 1 2 g H H h s α 2 C 1223 ( H , G 0 , H ) + 3 2 g h h h c α 2 C 1223 ( h , G 0 , h ) − 1 2 g H h h s α c α [ C 1223 ( H , G 0 , h ) + C 1223 ( h , G 0 , H ) ] ] , (245) \begin{aligned}
(m_{Z}^{2})^{-1}16\pi^{2}\Gamma_{hZZ}^{2,1\mathrm{PI}}(p_{1}^{2},p_{2}^{2},q^{2})_{B}
={}&2g^{3}m_{W}c_{W}^{2}(c_{\alpha}c_{\beta}-2s_{\alpha}s_{\beta})C_{hVV2}^{VVV}(W,W,W)\\
&+g^{3}m_{W}(c_{\alpha}c_{\beta}-2s_{\alpha}s_{\beta})(s_{W}^{2}-s_{\beta}^{2})\left[C_{hVV2}^{SVV}(G^{\pm},W,W)+C_{hVV2}^{VVS}(W,W,G^{\pm})\right]\\
&+g^{3}m_{W}(c_{\alpha}s_{\beta}+2s_{\alpha}c_{\beta})s_{\beta}c_{\beta}\left[C_{hVV2}^{SVV}(H^{\pm},W,W)+C_{hVV2}^{VVS}(W,W,H^{\pm})\right]\\
&-gg_{Z}^{2}m_{W}(c_{\alpha}c_{\beta}-2s_{\alpha}s_{\beta})(c_{2W}+s_{\beta}^{2})(s_{W}^{2}-s_{\beta}^{2})\left[C_{hVV2}^{VSS}(W,G^{\pm},G^{\pm})+C_{hVV2}^{SSV}(G^{\pm},G^{\pm},W)\right]\\
&-gg_{Z}^{2}m_{W}(c_{\alpha}s_{\beta}+2s_{\alpha}c_{\beta})(c_{2W}+c_{\beta}^{2})s_{\beta}c_{\beta}\left[C_{hVV2}^{VSS}(W,H^{\pm},H^{\pm})+C_{hVV2}^{SSV}(H^{\pm},H^{\pm},W)\right]\\
&+gg_{Z}^{2}m_{W}(c_{\alpha}c_{\beta}-2s_{\alpha}s_{\beta})s_{\beta}^{2}c_{\beta}^{2}\left[C_{hVV2}^{VSS}(W,H^{\pm},G^{\pm})+C_{hVV2}^{SSV}(G^{\pm},H^{\pm},W)\right]\\
&+gg_{Z}^{2}m_{W}(c_{\alpha}s_{\beta}+2s_{\alpha}c_{\beta})s_{\beta}c_{\beta}(s_{W}^{2}-s_{\beta}^{2})\left[C_{hVV2}^{VSS}(W,G^{\pm},H^{\pm})+C_{hVV2}^{SSV}(H^{\pm},G^{\pm},W)\right]\\
&-2g^{3}m_{W}c_{W}^{2}(c_{\alpha}c_{\beta}-2s_{\alpha}s_{\beta})C_{1223}(c^{\pm},c^{\pm},c^{\pm})\\
&+\frac{g_{Z}^{3}}{2}m_{Z}c_{\alpha}s_{\alpha}\left[C_{hVV2}^{VSS}(Z,h,G^{0})+C_{hVV2}^{SSV}(G^{0},h,Z)\right]\\
&+\frac{g_{Z}^{3}}{2}m_{Z}c_{\alpha}s_{\alpha}^{2}\left[C_{hVV2}^{VSS}(Z,H,G^{0})+C_{hVV2}^{SSV}(G^{0},H,Z)\right]\\
&-4g_{Z}^{2}\left[\frac{1}{2}g_{H^{+}H^{-}h}(c_{2W}+c_{\beta}^{2})^{2}C_{1223}(H^{\pm},H^{\pm},H^{\pm})\right.\\
&\quad+\frac{1}{2}g_{H^{+}H^{-}h}c_{\beta}^{2}s_{\beta}^{2}C_{1223}(H^{\pm},G^{\pm},H^{\pm})+\frac{1}{2}g_{G^{+}G^{-}h}c_{\beta}^{2}s_{\beta}^{2}C_{1223}(G^{\pm},H^{\pm},G^{\pm})\\
&\quad+\frac{1}{2}g_{H^{+}G^{-}h}c_{\beta}s_{\beta}(c_{2W}+c_{\beta}^{2})\left[C_{1223}(H^{\pm},H^{\pm},G^{\pm})+C_{1223}(G^{\pm},H^{\pm},H^{\pm})\right]\\
&\quad+\frac{1}{2}g_{H^{+}G^{-}h}c_{\beta}s_{\beta}(c_{2W}+s_{\beta}^{2})\left[C_{1223}(H^{\pm},G^{\pm},G^{\pm})+C_{1223}(G^{\pm},G^{\pm},H^{\pm})\right]\\
&\quad+\frac{1}{2}g_{G^{+}G^{-}h}(c_{2W}+s_{\beta}^{2})^{2}C_{1223}(G^{\pm},G^{\pm},G^{\pm})\\
&\quad+\frac{1}{2}g_{G^{0}G^{0}h}s_{\alpha}^{2}C_{1223}(G^{0},H,G^{0})+\frac{1}{2}g_{G^{0}G^{0}h}c_{\alpha}^{2}C_{1223}(G^{0},h,G^{0})\\
&\quad+\frac{1}{2}g_{HHh}s_{\alpha}^{2}C_{1223}(H,G^{0},H)+\frac{3}{2}g_{hhh}c_{\alpha}^{2}C_{1223}(h,G^{0},h)\\
&\quad\left.-\frac{1}{2}g_{Hhh}s_{\alpha}c_{\alpha}\left[C_{1223}(H,G^{0},h)+C_{1223}(h,G^{0},H)\right]\right], \tag*{(245)}
\end{aligned} ( m Z 2 ) − 1 16 π 2 Γ h Z Z 2 , 1 PI ( p 1 2 , p 2 2 , q 2 ) B = 2 g 3 m W c W 2 ( c α c β − 2 s α s β ) C hV V 2 V V V ( W , W , W ) + g 3 m W ( c α c β − 2 s α s β ) ( s W 2 − s β 2 ) [ C hV V 2 S V V ( G ± , W , W ) + C hV V 2 V V S ( W , W , G ± ) ] + g 3 m W ( c α s β + 2 s α c β ) s β c β [ C hV V 2 S V V ( H ± , W , W ) + C hV V 2 V V S ( W , W , H ± ) ] − g g Z 2 m W ( c α c β − 2 s α s β ) ( c 2 W + s β 2 ) ( s W 2 − s β 2 ) [ C hV V 2 V S S ( W , G ± , G ± ) + C hV V 2 S S V ( G ± , G ± , W ) ] − g g Z 2 m W ( c α s β + 2 s α c β ) ( c 2 W + c β 2 ) s β c β [ C hV V 2 V S S ( W , H ± , H ± ) + C hV V 2 S S V ( H ± , H ± , W ) ] + g g Z 2 m W ( c α c β − 2 s α s β ) s β 2 c β 2 [ C hV V 2 V S S ( W , H ± , G ± ) + C hV V 2 S S V ( G ± , H ± , W ) ] + g g Z 2 m W ( c α s β + 2 s α c β ) s β c β ( s W 2 − s β 2 ) [ C hV V 2 V S S ( W , G ± , H ± ) + C hV V 2 S S V ( H ± , G ± , W ) ] − 2 g 3 m W c W 2 ( c α c β − 2 s α s β ) C 1223 ( c ± , c ± , c ± ) + 2 g Z 3 m Z c α s α [ C hV V 2 V S S ( Z , h , G 0 ) + C hV V 2 S S V ( G 0 , h , Z ) ] + 2 g Z 3 m Z c α s α 2 [ C hV V 2 V S S ( Z , H , G 0 ) + C hV V 2 S S V ( G 0 , H , Z ) ] − 4 g Z 2 [ 2 1 g H + H − h ( c 2 W + c β 2 ) 2 C 1223 ( H ± , H ± , H ± ) + 2 1 g H + H − h c β 2 s β 2 C 1223 ( H ± , G ± , H ± ) + 2 1 g G + G − h c β 2 s β 2 C 1223 ( G ± , H ± , G ± ) + 2 1 g H + G − h c β s β ( c 2 W + c β 2 ) [ C 1223 ( H ± , H ± , G ± ) + C 1223 ( G ± , H ± , H ± ) ] + 2 1 g H + G − h c β s β ( c 2 W + s β 2 ) [ C 1223 ( H ± , G ± , G ± ) + C 1223 ( G ± , G ± , H ± ) ] + 2 1 g G + G − h ( c 2 W + s β 2 ) 2 C 1223 ( G ± , G ± , G ± ) + 2 1 g G 0 G 0 h s α 2 C 1223 ( G 0 , H , G 0 ) + 2 1 g G 0 G 0 h c α 2 C 1223 ( G 0 , h , G 0 ) + 2 1 g H H h s α 2 C 1223 ( H , G 0 , H ) + 2 3 g hhh c α 2 C 1223 ( h , G 0 , h ) − 2 1 g H hh s α c α [ C 1223 ( H , G 0 , h ) + C 1223 ( h , G 0 , H ) ] ] , (245) 16 π 2 Γ h Z Z 3 , 1 P I ( p 1 2 , p 2 2 , q 2 ) B = 0 , (246) 16\pi^{2}\Gamma_{hZZ}^{3,1\mathrm{PI}}(p_{1}^{2},p_{2}^{2},q^{2})_{B}=0, \tag*{(246)} 16 π 2 Γ h Z Z 3 , 1 PI ( p 1 2 , p 2 2 , q 2 ) B = 0 , (246) where the loop functions are defined as [33 ] .
C h V V 1 V V V ( X , Y , Z ) = [ 18 C 24 + p 1 2 ( 2 C 21 + 3 C 11 + C 0 ) + p 2 2 ( 2 C 22 + C 12 ) + p 1 ⋅ p 2 ( 4 C 23 + 3 C 12 + C 11 − 4 C 0 ) ] ( X , Y , Z ) − 3 , (247) C_{hVV1}^{VVV}(X,Y,Z)=\left[18C_{24}+p_1^2(2C_{21}+3C_{11}+C_0)+p_2^2(2C_{22}+C_{12})+p_1\cdot p_2(4C_{23}+3C_{12}+C_{11}-4C_0)\right](X,Y,Z)-3, \tag*{(247)} C hV V 1 V V V ( X , Y , Z ) = [ 18 C 24 + p 1 2 ( 2 C 21 + 3 C 11 + C 0 ) + p 2 2 ( 2 C 22 + C 12 ) + p 1 ⋅ p 2 ( 4 C 23 + 3 C 12 + C 11 − 4 C 0 ) ] ( X , Y , Z ) − 3 , (247) C h V V 2 V V V ( X , Y , Z ) = ( 10 C 23 + 9 C 12 + C 11 + 5 C 0 ) ( X , Y , Z ) , (248) C_{hVV2}^{VVV}(X,Y,Z)=(10C_{23}+9C_{12}+C_{11}+5C_0)(X,Y,Z), \tag*{(248)} C hV V 2 V V V ( X , Y , Z ) = ( 10 C 23 + 9 C 12 + C 11 + 5 C 0 ) ( X , Y , Z ) , (248) C h V V 1 S V V ( X , Y , Z ) = [ 3 C 24 + p 1 2 ( C 21 − C 0 ) + p 2 2 ( C 22 − 2 C 12 + C 0 ) + 2 p 1 ⋅ p 2 ( C 23 − C 11 ) ] ( X , Y , Z ) − 1 2 , (249) C_{hVV1}^{SVV}(X,Y,Z)=\left[3C_{24}+p_1^2(C_{21}-C_0)+p_2^2(C_{22}-2C_{12}+C_0)+2p_1\cdot p_2(C_{23}-C_{11})\right](X,Y,Z)-\frac{1}{2}, \tag*{(249)} C hV V 1 S V V ( X , Y , Z ) = [ 3 C 24 + p 1 2 ( C 21 − C 0 ) + p 2 2 ( C 22 − 2 C 12 + C 0 ) + 2 p 1 ⋅ p 2 ( C 23 − C 11 ) ] ( X , Y , Z ) − 2 1 , (249) C h V V 2 S V V ( X , Y , Z ) = ( 4 C 11 − 3 C 12 − C 23 ) ( X , Y , Z ) , (250) C_{hVV2}^{SVV}(X,Y,Z)=(4C_{11}-3C_{12}-C_{23})(X,Y,Z), \tag*{(250)} C hV V 2 S V V ( X , Y , Z ) = ( 4 C 11 − 3 C 12 − C 23 ) ( X , Y , Z ) , (250) C h V V 1 V V S ( X , Y , Z ) = [ 3 C 24 + p 1 2 ( C 21 + 4 C 11 + 4 C 0 ) + p 2 2 ( C 22 + 2 C 12 ) + 2 p 1 ⋅ p 2 ( C 23 + 2 C 12 + C 11 + 2 C 0 ) ] ( X , Y , Z ) − 1 2 , (251) C_{hVV1}^{VVS}(X,Y,Z)=\left[3C_{24}+p_1^2(C_{21}+4C_{11}+4C_0)+p_2^2(C_{22}+2C_{12})+2p_1\cdot p_2(C_{23}+2C_{12}+C_{11}+2C_0)\right](X,Y,Z)-\frac{1}{2}, \tag*{(251)} C hV V 1 V V S ( X , Y , Z ) = [ 3 C 24 + p 1 2 ( C 21 + 4 C 11 + 4 C 0 ) + p 2 2 ( C 22 + 2 C 12 ) + 2 p 1 ⋅ p 2 ( C 23 + 2 C 12 + C 11 + 2 C 0 ) ] ( X , Y , Z ) − 2 1 , (251) C h V V 2 V V S ( X , Y , Z ) = ( 2 C 11 − 5 C 12 − 2 C 0 − C 23 ) ( X , Y , Z ) , (252) C_{hVV2}^{VVS}(X,Y,Z)=(2C_{11}-5C_{12}-2C_0-C_{23})(X,Y,Z), \tag*{(252)} C hV V 2 V V S ( X , Y , Z ) = ( 2 C 11 − 5 C 12 − 2 C 0 − C 23 ) ( X , Y , Z ) , (252) C h V V 2 V S S ( X , Y , Z ) = ( C 23 + C 12 + 2 C 11 + 2 C 0 ) ( X , Y , Z ) , (253) C_{hVV2}^{VSS}(X,Y,Z)=(C_{23}+C_{12}+2C_{11}+2C_0)(X,Y,Z), \tag*{(253)} C hV V 2 V S S ( X , Y , Z ) = ( C 23 + C 12 + 2 C 11 + 2 C 0 ) ( X , Y , Z ) , (253) C h V V 2 C S S ( X , Y , Z ) = ( C 23 − C 12 ) ( X , Y , Z ) , (254) C_{hVV2}^{CSS}(X,Y,Z)=(C_{23}-C_{12})(X,Y,Z), \tag*{(254)} C hV V 2 C S S ( X , Y , Z ) = ( C 23 − C 12 ) ( X , Y , Z ) , (254) C 1223 ( X , Y , Z ) = ( C 12 + C 23 ) ( X , Y , Z ) . (255) C_{1223}(X,Y,Z)=(C_{12}+C_{23})(X,Y,Z). \tag*{(255)} C 1223 ( X , Y , Z ) = ( C 12 + C 23 ) ( X , Y , Z ) . (255) The fermionic contributions to the 1PI h W W hWW hW W
Γ h W W i , 1 P I ( p 1 2 , p 2 2 , q 2 ) F = ζ h Γ h W W i , 1 P I ( p 1 2 , p 2 2 , q 2 ) F , S M ( i = 1 , 2 , 3 ) , (256) \Gamma_{hWW}^{i,\mathrm{1PI}}(p_1^2,p_2^2,q^2)_F=\zeta_h\Gamma_{hWW}^{i,\mathrm{1PI}}(p_1^2,p_2^2,q^2)_{F,\mathrm{SM}}\quad(i=1,2,3), \tag*{(256)} Γ hW W i , 1PI ( p 1 2 , p 2 2 , q 2 ) F = ζ h Γ hW W i , 1PI ( p 1 2 , p 2 2 , q 2 ) F , SM ( i = 1 , 2 , 3 ) , (256) where
16 π 2 Γ h W W 1 , 1 P I ( p 1 2 , p 2 2 , q 2 ) F , S M = ∑ f , f ′ N c f g 2 m f 2 v [ 1 2 B 0 ( p 1 2 , f , f ′ ) + 1 2 B 0 ( p 2 2 , f , f ′ ) + B 0 ( q 2 , f , f ) − 4 C 24 ( p 1 2 , p 2 2 , q 2 , f , f ′ , f ) + ( m f 2 + m f ′ 2 − p 1 2 2 − p 2 2 2 ) C 0 ( f , f ′ , f ) ] + ( m f ↔ m f ′ ) , (257) 16\pi^2\Gamma_{hWW}^{1,\mathrm{1PI}}(p_1^2,p_2^2,q^2)_{F,\mathrm{SM}}=\sum_{f,f'}N_c^f\frac{g^2m_f^2}{v}\left[\frac{1}{2}B_0(p_1^2,f,f')+\frac{1}{2}B_0(p_2^2,f,f')+B_0(q^2,f,f)-4C_{24}(p_1^2,p_2^2,q^2,f,f',f)+\left(m_f^2+m_{f'}^2-\frac{p_1^2}{2}-\frac{p_2^2}{2}\right)C_0(f,f',f)\right]+(m_f\leftrightarrow m_{f'}), \tag*{(257)} 16 π 2 Γ hW W 1 , 1PI ( p 1 2 , p 2 2 , q 2 ) F , SM = f , f ′ ∑ N c f v g 2 m f 2 [ 2 1 B 0 ( p 1 2 , f , f ′ ) + 2 1 B 0 ( p 2 2 , f , f ′ ) + B 0 ( q 2 , f , f ) − 4 C 24 ( p 1 2 , p 2 2 , q 2 , f , f ′ , f ) + ( m f 2 + m f ′ 2 − 2 p 1 2 − 2 p 2 2 ) C 0 ( f , f ′ , f ) ] + ( m f ↔ m f ′ ) , (257) ( m W 2 ) − 1 16 π 2 Γ h W W 2 , 1 P I ( p 1 2 , p 2 2 , q 2 ) F , S M = − ∑ f , f ′ N c f g 2 m f 2 v ( 4 C 23 + 3 C 12 + C 11 + C 0 ) ( f , f ′ , f ) + ( m f ↔ m f ′ ) , (258) (m_W^2)^{-1}16\pi^2\Gamma_{hWW}^{2,\mathrm{1PI}}(p_1^2,p_2^2,q^2)_{F,\mathrm{SM}}=-\sum_{f,f'}N_c^f\frac{g^2m_f^2}{v}(4C_{23}+3C_{12}+C_{11}+C_0)(f,f',f)+(m_f\leftrightarrow m_{f'}), \tag*{(258)} ( m W 2 ) − 1 16 π 2 Γ hW W 2 , 1PI ( p 1 2 , p 2 2 , q 2 ) F , SM = − f , f ′ ∑ N c f v g 2 m f 2 ( 4 C 23 + 3 C 12 + C 11 + C 0 ) ( f , f ′ , f ) + ( m f ↔ m f ′ ) , (258) ( m W 2 ) − 1 16 π 2 Γ h W W 3 , 1 P I ( p 1 2 , p 2 2 , q 2 ) F , S M = − ∑ f , f ′ N c f g 2 m f 2 v ( C 11 + C 12 + C 0 ) ( f , f ′ , f ) + ( m f ↔ m f ′ ) , (259) (m_W^2)^{-1}16\pi^2\Gamma_{hWW}^{3,\mathrm{1PI}}(p_1^2,p_2^2,q^2)_{F,\mathrm{SM}}=-\sum_{f,f'}N_c^f\frac{g^2m_f^2}{v}(C_{11}+C_{12}+C_0)(f,f',f)+(m_f\leftrightarrow m_{f'}), \tag*{(259)} ( m W 2 ) − 1 16 π 2 Γ hW W 3 , 1PI ( p 1 2 , p 2 2 , q 2 ) F , SM = − f , f ′ ∑ N c f v g 2 m f 2 ( C 11 + C 12 + C 0 ) ( f , f ′ , f ) + ( m f ↔ m f ′ ) , (259) 16 π 2 Γ h W W 1 , 1 P I ( p 1 2 , p 2 2 , q 2 ) B = g 3 c W m Z c α C h V V 1 V V V ( Z , W , Z ) + g 3 c W m W ( c α c β − 2 s α s β ) C h V V 1 V V V ( W , Z , W ) + g 3 s W 2 m W ( c α c β − 2 s α s β ) C h V V 1 V V V ( W , γ , W ) − 1 2 g 3 m W ( c α c β − 2 s α s β ) ( s W 2 − s β 2 ) [ C h V V 1 S V V ( G ± , Z , W ) + C h V V 1 V V S ( W , Z , G ± ) ] 16\pi^2\Gamma_{hWW}^{1,\mathrm{1PI}}(p_1^2,p_2^2,q^2)_B=g^3c_Wm_Zc_\alpha C_{hVV1}^{VVV}(Z,W,Z)+g^3c_Wm_W(c_\alpha c_\beta-2s_\alpha s_\beta)C_{hVV1}^{VVV}(W,Z,W)
+g^3s_W^2m_W(c_\alpha c_\beta-2s_\alpha s_\beta)C_{hVV1}^{VVV}(W,\gamma,W)
-\frac{1}{2}g^3m_W(c_\alpha c_\beta-2s_\alpha s_\beta)(s_W^2-s_\beta^2)\left[C_{hVV1}^{SVV}(G^\pm,Z,W)+C_{hVV1}^{VVS}(W,Z,G^\pm)\right] 16 π 2 Γ hW W 1 , 1PI ( p 1 2 , p 2 2 , q 2 ) B = g 3 c W m Z c α C hV V 1 V V V ( Z , W , Z ) + g 3 c W m W ( c α c β − 2 s α s β ) C hV V 1 V V V ( W , Z , W ) + g 3 s W 2 m W ( c α c β − 2 s α s β ) C hV V 1 V V V ( W , γ , W ) − 2 1 g 3 m W ( c α c β − 2 s α s β ) ( s W 2 − s β 2 ) [ C hV V 1 S V V ( G ± , Z , W ) + C hV V 1 V V S ( W , Z , G ± ) ] + 1 2 g 3 s W 2 m W ( c α c β − 2 s α s β ) [ C h V V 1 S V V ( G ± , γ , W ) + C h V V 1 V V S ( W , γ , G ± ) ] − 1 2 g 3 m W ( c α s β + 2 s α c β ) s β c β [ C h V V 1 S V V ( H ± , Z , W ) + C h V V 1 V V S ( W , Z , H ± ) ] − g Z 3 m Z m W 2 ( s W 2 − s β 2 ) 2 c β c α C 0 ( Z , G ± , Z ) − g Z 3 m Z m W 2 s β 2 c β 2 c α C 0 ( Z , H ± , Z ) − g 3 m W 3 ( c α c β − 2 s α s β ) 3 C 0 ( W , h , W ) − g 3 m W 3 ( c α c β − 2 s α s β ) ( s α c β + 2 c α s β ) 2 C 0 ( W , H , W ) + 6 g h h h g 2 m W 2 ( c α c β − 2 s α s β ) 2 C 0 ( h , W , h ) + 2 g H H h g 2 m W 2 ( s α c β + 2 c α s β ) 2 C 0 ( H , W , H ) − 2 g H h h g 2 m W 2 ( c α c β − 2 s α s β ) ( s α c β + 2 c α s β ) [ C 0 ( h , W , H ) + C 0 ( H , W , h ) ] + g G + G − h g Z 2 m W 2 ( s W 2 − s β 2 ) 2 C 0 ( G ± , Z , G ± ) + g H + H − h g Z 2 m W 2 s β 2 c β 2 C 0 ( H ± , Z , H ± ) + g G + G − h g 2 s W m W 2 C 0 ( G ± , γ , G ± ) − g H + G − h g Z 2 m W 2 s β c β ( s W 2 − s β 2 ) [ C 0 ( H ± , Z , G ± ) + C 0 ( G ± , Z , H ± ) ] + 1 2 g 3 m W ( c α c β − 2 s α s β ) 3 [ C 24 ( W , h , G ± ) + C 24 ( G ± , h , W ) ] + 1 2 g 3 m W ( c α c β − 2 s α s β ) ( s α c β + 2 c α s β ) 2 [ C 24 ( W , H , G ± ) + C 24 ( G ± , H , W ) ] + 1 2 g 3 m W ( c α c β − 2 s α s β ) ( c α s β + 2 s α c β ) 2 [ C 24 ( W , h , H ± ) + C 24 ( H ± , h , W ) ] + 1 2 g 3 m W ( c α s β + 2 s α c β ) ( s α c β + 2 c α s β ) ( s α s β − 2 c α c β ) [ C 24 ( W , H , H ± ) + C 24 ( H ± , H , W ) ] + 1 2 g g Z 2 m W ( s W 2 − s β 2 ) c β c α [ C 24 ( G 0 , G ± , Z ) + C 24 ( Z , G ± , G 0 ) ] + 1 2 g g Z 2 m W s β 2 c β c α [ C 24 ( G 0 , H ± , Z ) + C 24 ( Z , H ± , G 0 ) ] − g 3 c W m Z c α C 24 ( c Z , c ± , c Z ) − g 3 c W 3 m W ( c α c β − 2 s α s β ) C 24 ( c ± , c Z , c ± ) − g 3 s W 2 m W ( c α c β − 2 s α s β ) C 24 ( c ± , c γ , c ± ) − g 3 m W ( c α c β − 2 s α s β ) [ 3 B 0 ( q 2 , W , W ) − 2 ] − g 3 c W m Z c α [ 3 B 0 ( q 2 , Z , Z ) − 2 ] − 1 2 g 3 ( 1 + 3 s α 2 ) m W ( c α c β − 2 s α s β ) [ B 0 ( p 1 2 ; W , h ) + B 0 ( p 2 2 , W , h ) ] + 3 2 g 3 s α c α m W ( s α c β + 2 c α s β ) [ B 0 ( p 1 2 ; W , H ) + B 0 ( p 2 2 , W , H ) ] − 1 2 g g Z 2 m W [ c α c β s W 2 + 2 c W 2 s α s β ] ( s W 2 − s β 2 ) [ B 0 ( p 1 2 ; Z , G ± ) + B 0 ( p 2 2 ; Z , G ± ) ] − 1 2 g g Z 2 m W [ c α s β s W 2 − 2 c W 2 s α c β ] s β c β [ B 0 ( p 1 2 ; Z , H ± ) + B 0 ( p 2 2 ; Z , H ± ) ] − 1 2 g 3 s W 2 m W ( c α c β − 2 s α s β ) [ B 0 ( p 1 2 ; γ , G ± ) + B 0 ( p 2 2 ; γ , G ± ) ] − 4 g 2 { 1 2 g G 0 G 0 h [ c β 2 C 24 ( G 0 , G ± , G 0 ) + s β 2 C 24 ( G 0 , H ± , G 0 ) ] } + 3 2 g h h h [ ( c β c α − 2 s β s α ) 2 C 24 ( h , G ± , h ) + ( s β c α + 2 c β s α ) 2 C 24 ( h , H ± , h ) ] + 1 2 g H H h [ ( c β s α + 2 s β c α ) 2 C 24 ( H , G ± , H ) + ( s β s α − 2 c β c α ) 2 C 24 ( H , H ± , H ) ] \begin{aligned}
&+\frac{1}{2}g^{3}s_{W}^{2}m_{W}(c_{\alpha}c_{\beta}-2s_{\alpha}s_{\beta})\left[C_{hVV1}^{SVV}(G^{\pm},\gamma,W)+C_{hVV1}^{VVS}(W,\gamma,G^{\pm})\right]\\
&-\frac{1}{2}g^{3}m_{W}(c_{\alpha}s_{\beta}+2s_{\alpha}c_{\beta})s_{\beta}c_{\beta}\left[C_{hVV1}^{SVV}(H^{\pm},Z,W)+C_{hVV1}^{VVS}(W,Z,H^{\pm})\right]\\
&-g_{Z}^{3}m_{Z}m_{W}^{2}(s_{W}^{2}-s_{\beta}^{2})^{2}c_{\beta}c_{\alpha}C_{0}(Z,G^{\pm},Z)-g_{Z}^{3}m_{Z}m_{W}^{2}s_{\beta}^{2}c_{\beta}^{2}c_{\alpha}C_{0}(Z,H^{\pm},Z)\\
&-g^{3}m_{W}^{3}(c_{\alpha}c_{\beta}-2s_{\alpha}s_{\beta})^{3}C_{0}(W,h,W)\\
&-g^{3}m_{W}^{3}(c_{\alpha}c_{\beta}-2s_{\alpha}s_{\beta})(s_{\alpha}c_{\beta}+2c_{\alpha}s_{\beta})^{2}C_{0}(W,H,W)\\
&+6g_{hhh}g^{2}m_{W}^{2}(c_{\alpha}c_{\beta}-2s_{\alpha}s_{\beta})^{2}C_{0}(h,W,h)\\
&+2g_{HHh}g^{2}m_{W}^{2}(s_{\alpha}c_{\beta}+2c_{\alpha}s_{\beta})^{2}C_{0}(H,W,H)\\
&-2g_{Hhh}g^{2}m_{W}^{2}(c_{\alpha}c_{\beta}-2s_{\alpha}s_{\beta})(s_{\alpha}c_{\beta}+2c_{\alpha}s_{\beta})\left[C_{0}(h,W,H)+C_{0}(H,W,h)\right]\\
&+g_{G^{+}G^{-}h}g_{Z}^{2}m_{W}^{2}(s_{W}^{2}-s_{\beta}^{2})^{2}C_{0}(G^{\pm},Z,G^{\pm})+g_{H^{+}H^{-}h}g_{Z}^{2}m_{W}^{2}s_{\beta}^{2}c_{\beta}^{2}C_{0}(H^{\pm},Z,H^{\pm})\\
&+g_{G^{+}G^{-}h}g^{2}s_{W}m_{W}^{2}C_{0}(G^{\pm},\gamma,G^{\pm})\\
&-g_{H^{+}G^{-}h}g_{Z}^{2}m_{W}^{2}s_{\beta}c_{\beta}(s_{W}^{2}-s_{\beta}^{2})\left[C_{0}(H^{\pm},Z,G^{\pm})+C_{0}(G^{\pm},Z,H^{\pm})\right]\\
&+\frac{1}{2}g^{3}m_{W}(c_{\alpha}c_{\beta}-2s_{\alpha}s_{\beta})^{3}\left[C_{24}(W,h,G^{\pm})+C_{24}(G^{\pm},h,W)\right]\\
&+\frac{1}{2}g^{3}m_{W}(c_{\alpha}c_{\beta}-2s_{\alpha}s_{\beta})(s_{\alpha}c_{\beta}+2c_{\alpha}s_{\beta})^{2}\left[C_{24}(W,H,G^{\pm})+C_{24}(G^{\pm},H,W)\right]\\
&+\frac{1}{2}g^{3}m_{W}(c_{\alpha}c_{\beta}-2s_{\alpha}s_{\beta})(c_{\alpha}s_{\beta}+2s_{\alpha}c_{\beta})^{2}\left[C_{24}(W,h,H^{\pm})+C_{24}(H^{\pm},h,W)\right]\\
&+\frac{1}{2}g^{3}m_{W}(c_{\alpha}s_{\beta}+2s_{\alpha}c_{\beta})(s_{\alpha}c_{\beta}+2c_{\alpha}s_{\beta})(s_{\alpha}s_{\beta}-2c_{\alpha}c_{\beta})\left[C_{24}(W,H,H^{\pm})+C_{24}(H^{\pm},H,W)\right]\\
&+\frac{1}{2}gg_{Z}^{2}m_{W}(s_{W}^{2}-s_{\beta}^{2})c_{\beta}c_{\alpha}\left[C_{24}(G^{0},G^{\pm},Z)+C_{24}(Z,G^{\pm},G^{0})\right]\\
&+\frac{1}{2}gg_{Z}^{2}m_{W}s_{\beta}^{2}c_{\beta}c_{\alpha}\left[C_{24}(G^{0},H^{\pm},Z)+C_{24}(Z,H^{\pm},G^{0})\right]\\
&-g^{3}c_{W}m_{Z}c_{\alpha}C_{24}(c_{Z},c^{\pm},c_{Z})-g^{3}c_{W}^{3}m_{W}(c_{\alpha}c_{\beta}-2s_{\alpha}s_{\beta})C_{24}(c^{\pm},c_{Z},c^{\pm})\\
&-g^{3}s_{W}^{2}m_{W}(c_{\alpha}c_{\beta}-2s_{\alpha}s_{\beta})C_{24}(c^{\pm},c_{\gamma},c^{\pm})\\
&-g^{3}m_{W}(c_{\alpha}c_{\beta}-2s_{\alpha}s_{\beta})\left[3B_{0}(q^{2},W,W)-2\right]-g^{3}c_{W}m_{Z}c_{\alpha}\left[3B_{0}(q^{2},Z,Z)-2\right]\\
&-\frac{1}{2}g^{3}(1+3s_{\alpha}^{2})m_{W}(c_{\alpha}c_{\beta}-2s_{\alpha}s_{\beta})\left[B_{0}(p_{1}^{2};W,h)+B_{0}(p_{2}^{2},W,h)\right]\\
&+\frac{3}{2}g^{3}s_{\alpha}c_{\alpha}m_{W}(s_{\alpha}c_{\beta}+2c_{\alpha}s_{\beta})\left[B_{0}(p_{1}^{2};W,H)+B_{0}(p_{2}^{2},W,H)\right]\\
&-\frac{1}{2}gg_{Z}^{2}m_{W}\left[c_{\alpha}c_{\beta}s_{W}^{2}+2c_{W}^{2}s_{\alpha}s_{\beta}\right](s_{W}^{2}-s_{\beta}^{2})\left[B_{0}(p_{1}^{2};Z,G^{\pm})+B_{0}(p_{2}^{2};Z,G^{\pm})\right]\\
&-\frac{1}{2}gg_{Z}^{2}m_{W}\left[c_{\alpha}s_{\beta}s_{W}^{2}-2c_{W}^{2}s_{\alpha}c_{\beta}\right]s_{\beta}c_{\beta}\left[B_{0}(p_{1}^{2};Z,H^{\pm})+B_{0}(p_{2}^{2};Z,H^{\pm})\right]\\
&-\frac{1}{2}g^{3}s_{W}^{2}m_{W}(c_{\alpha}c_{\beta}-2s_{\alpha}s_{\beta})\left[B_{0}(p_{1}^{2};\gamma,G^{\pm})+B_{0}(p_{2}^{2};\gamma,G^{\pm})\right]\\
&-4g^{2}\left\{\frac{1}{2}g_{G^{0}G^{0}h}\left[c_{\beta}^{2}C_{24}(G^{0},G^{\pm},G^{0})+s_{\beta}^{2}C_{24}(G^{0},H^{\pm},G^{0})\right]\right\}\\
&+\frac{3}{2}g_{hhh}\left[(c_{\beta}c_{\alpha}-2s_{\beta}s_{\alpha})^{2}C_{24}(h,G^{\pm},h)+(s_{\beta}c_{\alpha}+2c_{\beta}s_{\alpha})^{2}C_{24}(h,H^{\pm},h)\right]\\
&+\frac{1}{2}g_{HHh}\left[(c_{\beta}s_{\alpha}+2s_{\beta}c_{\alpha})^{2}C_{24}(H,G^{\pm},H)+(s_{\beta}s_{\alpha}-2c_{\beta}c_{\alpha})^{2}C_{24}(H,H^{\pm},H)\right]
\end{aligned} + 2 1 g 3 s W 2 m W ( c α c β − 2 s α s β ) [ C hV V 1 S V V ( G ± , γ , W ) + C hV V 1 V V S ( W , γ , G ± ) ] − 2 1 g 3 m W ( c α s β + 2 s α c β ) s β c β [ C hV V 1 S V V ( H ± , Z , W ) + C hV V 1 V V S ( W , Z , H ± ) ] − g Z 3 m Z m W 2 ( s W 2 − s β 2 ) 2 c β c α C 0 ( Z , G ± , Z ) − g Z 3 m Z m W 2 s β 2 c β 2 c α C 0 ( Z , H ± , Z ) − g 3 m W 3 ( c α c β − 2 s α s β ) 3 C 0 ( W , h , W ) − g 3 m W 3 ( c α c β − 2 s α s β ) ( s α c β + 2 c α s β ) 2 C 0 ( W , H , W ) + 6 g hhh g 2 m W 2 ( c α c β − 2 s α s β ) 2 C 0 ( h , W , h ) + 2 g H H h g 2 m W 2 ( s α c β + 2 c α s β ) 2 C 0 ( H , W , H ) − 2 g H hh g 2 m W 2 ( c α c β − 2 s α s β ) ( s α c β + 2 c α s β ) [ C 0 ( h , W , H ) + C 0 ( H , W , h ) ] + g G + G − h g Z 2 m W 2 ( s W 2 − s β 2 ) 2 C 0 ( G ± , Z , G ± ) + g H + H − h g Z 2 m W 2 s β 2 c β 2 C 0 ( H ± , Z , H ± ) + g G + G − h g 2 s W m W 2 C 0 ( G ± , γ , G ± ) − g H + G − h g Z 2 m W 2 s β c β ( s W 2 − s β 2 ) [ C 0 ( H ± , Z , G ± ) + C 0 ( G ± , Z , H ± ) ] + 2 1 g 3 m W ( c α c β − 2 s α s β ) 3 [ C 24 ( W , h , G ± ) + C 24 ( G ± , h , W ) ] + 2 1 g 3 m W ( c α c β − 2 s α s β ) ( s α c β + 2 c α s β ) 2 [ C 24 ( W , H , G ± ) + C 24 ( G ± , H , W ) ] + 2 1 g 3 m W ( c α c β − 2 s α s β ) ( c α s β + 2 s α c β ) 2 [ C 24 ( W , h , H ± ) + C 24 ( H ± , h , W ) ] + 2 1 g 3 m W ( c α s β + 2 s α c β ) ( s α c β + 2 c α s β ) ( s α s β − 2 c α c β ) [ C 24 ( W , H , H ± ) + C 24 ( H ± , H , W ) ] + 2 1 g g Z 2 m W ( s W 2 − s β 2 ) c β c α [ C 24 ( G 0 , G ± , Z ) + C 24 ( Z , G ± , G 0 ) ] + 2 1 g g Z 2 m W s β 2 c β c α [ C 24 ( G 0 , H ± , Z ) + C 24 ( Z , H ± , G 0 ) ] − g 3 c W m Z c α C 24 ( c Z , c ± , c Z ) − g 3 c W 3 m W ( c α c β − 2 s α s β ) C 24 ( c ± , c Z , c ± ) − g 3 s W 2 m W ( c α c β − 2 s α s β ) C 24 ( c ± , c γ , c ± ) − g 3 m W ( c α c β − 2 s α s β ) [ 3 B 0 ( q 2 , W , W ) − 2 ] − g 3 c W m Z c α [ 3 B 0 ( q 2 , Z , Z ) − 2 ] − 2 1 g 3 ( 1 + 3 s α 2 ) m W ( c α c β − 2 s α s β ) [ B 0 ( p 1 2 ; W , h ) + B 0 ( p 2 2 , W , h ) ] + 2 3 g 3 s α c α m W ( s α c β + 2 c α s β ) [ B 0 ( p 1 2 ; W , H ) + B 0 ( p 2 2 , W , H ) ] − 2 1 g g Z 2 m W [ c α c β s W 2 + 2 c W 2 s α s β ] ( s W 2 − s β 2 ) [ B 0 ( p 1 2 ; Z , G ± ) + B 0 ( p 2 2 ; Z , G ± ) ] − 2 1 g g Z 2 m W [ c α s β s W 2 − 2 c W 2 s α c β ] s β c β [ B 0 ( p 1 2 ; Z , H ± ) + B 0 ( p 2 2 ; Z , H ± ) ] − 2 1 g 3 s W 2 m W ( c α c β − 2 s α s β ) [ B 0 ( p 1 2 ; γ , G ± ) + B 0 ( p 2 2 ; γ , G ± ) ] − 4 g 2 { 2 1 g G 0 G 0 h [ c β 2 C 24 ( G 0 , G ± , G 0 ) + s β 2 C 24 ( G 0 , H ± , G 0 ) ] } + 2 3 g hhh [ ( c β c α − 2 s β s α ) 2 C 24 ( h , G ± , h ) + ( s β c α + 2 c β s α ) 2 C 24 ( h , H ± , h ) ] + 2 1 g H H h [ ( c β s α + 2 s β c α ) 2 C 24 ( H , G ± , H ) + ( s β s α − 2 c β c α ) 2 C 24 ( H , H ± , H ) ] − 1 2 g H h h ( c β c α − 2 s β s α ) ( c β s α + 2 s β c α ) [ C 24 ( h , G ± , H ) + C 24 ( H , G ± , h ) ] − 1 2 g H h h ( s β c α + 2 c β s α ) ( s β s α − 2 c β c α ) [ C 24 ( h , H ± , H ) + C 24 ( H , H ± , h ) ] + 1 4 g H + H − h s β 2 C 24 ( H ± , G 0 , H ± ) + 1 4 g H + H − h [ ( s β c α + 2 c β s α ) 2 C 24 ( H ± , h , H ± ) + ( s β s α − 2 c β c α ) 2 C 24 ( H ± , H , H ± ) ] + 1 4 g G + G − h c β 2 C 24 ( G ± , G 0 , G ± ) + 1 4 g G + G − h [ ( c β c α − 2 s β s α ) 2 C 24 ( G ± , h , G ± ) + ( c β s α + 2 s β c α ) 2 C 24 ( G ± , H , G ± ) ] − 1 4 g H + G − h s β c β [ C 24 ( G ± , G 0 , H ± ) + C 24 ( H ± , G 0 , G ± ) ] − 1 4 g H + G − h ( s β c α + 2 c β s α ) ( c β c α − 2 s β s α ) [ C 24 ( G ± , h , H ± ) + C 24 ( H ± , h , G ± ) ] − 1 4 g H + G − h ( s β s α − 2 c β c α ) ( c β s α + 2 s β c α ) [ C 24 ( G ± , H , H ± ) + C 24 ( H ± , H , G ± ) ] + g 2 [ 1 2 g G + G − h ( 1 + s β 2 ) B 0 ( q 2 ; G ± , G ± ) + 1 2 g H + H − h ( 1 + c β 2 ) B 0 ( q 2 ; H ± , H ± ) + g H + G − h s β c β B 0 ( q 2 ; H ± , G ± ) + 1 2 g G 0 G 0 h B 0 ( q 2 ; G 0 , G 0 ) + 3 2 g h h h ( 1 + 3 s α 2 ) B 0 ( q 2 ; h , h ) + 1 2 g H H h ( 1 + 3 c α 2 ) B 0 ( q 2 ; H , H ) + 3 g H h h s α c α B 0 ( q 2 ; H , h ) ] , (260) -\frac{1}{2}g_{Hhh}(c_{\beta}c_{\alpha}-2s_{\beta}s_{\alpha})(c_{\beta}s_{\alpha}+2s_{\beta}c_{\alpha})[C_{24}(h,G^{\pm},H)+C_{24}(H,G^{\pm},h)]\\
-\frac{1}{2}g_{Hhh}(s_{\beta}c_{\alpha}+2c_{\beta}s_{\alpha})(s_{\beta}s_{\alpha}-2c_{\beta}c_{\alpha})[C_{24}(h,H^{\pm},H)+C_{24}(H,H^{\pm},h)]\\
+\frac{1}{4}g_{H^{+}H^{-}h}s_{\beta}^{2}C_{24}(H^{\pm},G^{0},H^{\pm})\\
+\frac{1}{4}g_{H^{+}H^{-}h}[(s_{\beta}c_{\alpha}+2c_{\beta}s_{\alpha})^{2}C_{24}(H^{\pm},h,H^{\pm})+(s_{\beta}s_{\alpha}-2c_{\beta}c_{\alpha})^{2}C_{24}(H^{\pm},H,H^{\pm})]\\
+\frac{1}{4}g_{G^{+}G^{-}h}c_{\beta}^{2}C_{24}(G^{\pm},G^{0},G^{\pm})\\
+\frac{1}{4}g_{G^{+}G^{-}h}[(c_{\beta}c_{\alpha}-2s_{\beta}s_{\alpha})^{2}C_{24}(G^{\pm},h,G^{\pm})+(c_{\beta}s_{\alpha}+2s_{\beta}c_{\alpha})^{2}C_{24}(G^{\pm},H,G^{\pm})]\\
-\frac{1}{4}g_{H^{+}G^{-}h}s_{\beta}c_{\beta}[C_{24}(G^{\pm},G^{0},H^{\pm})+C_{24}(H^{\pm},G^{0},G^{\pm})]\\
-\frac{1}{4}g_{H^{+}G^{-}h}(s_{\beta}c_{\alpha}+2c_{\beta}s_{\alpha})(c_{\beta}c_{\alpha}-2s_{\beta}s_{\alpha})[C_{24}(G^{\pm},h,H^{\pm})+C_{24}(H^{\pm},h,G^{\pm})]\\
-\frac{1}{4}g_{H^{+}G^{-}h}(s_{\beta}s_{\alpha}-2c_{\beta}c_{\alpha})(c_{\beta}s_{\alpha}+2s_{\beta}c_{\alpha})[C_{24}(G^{\pm},H,H^{\pm})+C_{24}(H^{\pm},H,G^{\pm})]\\
+g^{2}\left[\frac{1}{2}g_{G^{+}G^{-}h}(1+s_{\beta}^{2})B_{0}(q^{2};G^{\pm},G^{\pm})+\frac{1}{2}g_{H^{+}H^{-}h}(1+c_{\beta}^{2})B_{0}(q^{2};H^{\pm},H^{\pm})\right.\\
\left.+g_{H^{+}G^{-}h}s_{\beta}c_{\beta}B_{0}(q^{2};H^{\pm},G^{\pm})+\frac{1}{2}g_{G^{0}G^{0}h}B_{0}(q^{2};G^{0},G^{0})\right.\\
\left.+\frac{3}{2}g_{hhh}(1+3s_{\alpha}^{2})B_{0}(q^{2};h,h)+\frac{1}{2}g_{HHh}(1+3c_{\alpha}^{2})B_{0}(q^{2};H,H)\right.\\
\left.+3g_{Hhh}s_{\alpha}c_{\alpha}B_{0}(q^{2};H,h)\right], \tag*{(260)} − 2 1 g H hh ( c β c α − 2 s β s α ) ( c β s α + 2 s β c α ) [ C 24 ( h , G ± , H ) + C 24 ( H , G ± , h )] − 2 1 g H hh ( s β c α + 2 c β s α ) ( s β s α − 2 c β c α ) [ C 24 ( h , H ± , H ) + C 24 ( H , H ± , h )] + 4 1 g H + H − h s β 2 C 24 ( H ± , G 0 , H ± ) + 4 1 g H + H − h [( s β c α + 2 c β s α ) 2 C 24 ( H ± , h , H ± ) + ( s β s α − 2 c β c α ) 2 C 24 ( H ± , H , H ± )] + 4 1 g G + G − h c β 2 C 24 ( G ± , G 0 , G ± ) + 4 1 g G + G − h [( c β c α − 2 s β s α ) 2 C 24 ( G ± , h , G ± ) + ( c β s α + 2 s β c α ) 2 C 24 ( G ± , H , G ± )] − 4 1 g H + G − h s β c β [ C 24 ( G ± , G 0 , H ± ) + C 24 ( H ± , G 0 , G ± )] − 4 1 g H + G − h ( s β c α + 2 c β s α ) ( c β c α − 2 s β s α ) [ C 24 ( G ± , h , H ± ) + C 24 ( H ± , h , G ± )] − 4 1 g H + G − h ( s β s α − 2 c β c α ) ( c β s α + 2 s β c α ) [ C 24 ( G ± , H , H ± ) + C 24 ( H ± , H , G ± )] + g 2 [ 2 1 g G + G − h ( 1 + s β 2 ) B 0 ( q 2 ; G ± , G ± ) + 2 1 g H + H − h ( 1 + c β 2 ) B 0 ( q 2 ; H ± , H ± ) + g H + G − h s β c β B 0 ( q 2 ; H ± , G ± ) + 2 1 g G 0 G 0 h B 0 ( q 2 ; G 0 , G 0 ) + 2 3 g hhh ( 1 + 3 s α 2 ) B 0 ( q 2 ; h , h ) + 2 1 g H H h ( 1 + 3 c α 2 ) B 0 ( q 2 ; H , H ) + 3 g H hh s α c α B 0 ( q 2 ; H , h ) ] , (260) ( m W 2 ) − 1 16 π 2 Γ h W W 2 , 1 P I ( p 1 2 , p 2 2 , q 2 ) B = g 3 c W m Z c α C h V V 2 V V V ( Z , W , Z ) + g 3 c W 2 m W ( c α c β − 2 s α s β ) C h V V 2 V V V ( W , Z , W ) + g 3 s W 2 m W ( c α c β − 2 s α s β ) C h V V 2 V V V ( W , γ , W ) − 1 2 g 3 m W ( c α c β − 2 s α s β ) ( s W 2 − s β 2 ) [ C h V V 2 S V V ( G ± , Z , W ) + C h V V 2 V V S ( W , Z , G ± ) ] + 1 2 g 3 s W 2 m W ( c α c β − 2 s α s β ) [ C h V V 2 S V V ( G ± , γ , W ) + C h V V 2 V V S ( W , γ , G ± ) ] − 1 2 g 3 m W ( c α s β + 2 s α c β ) s β c β [ C h V V 2 S V V ( H ± , Z , W ) + C h V V 2 V V S ( W , Z , H ± ) ] + 1 2 g 3 m W ( c α c β − 2 s α s β ) 3 [ C h V V 2 C V S S ( W , h , G ± ) + C h V V 2 S S V ( G ± , h , W ) ] + 1 2 g 3 m W ( c α c β − 2 s α s β ) ( s α c β + 2 c α s β ) 2 [ C h V V 2 C V S S ( W , H , G ± ) + C h V V 2 S S V ( G ± , H , W ) ] + 1 2 g 3 m W ( c α c β − 2 s α s β ) ( c α s β + 2 s α c β ) 2 [ C h V V 2 C V S S ( W , h , H ± ) + C h V V 2 S S V ( H ± , h , W ) ] + 1 2 g 3 m W ( c α s β + 2 s α c β ) ( s α c β + 2 c α s β ) ( s α s β − 2 c α c β ) [ C h V V 2 C V S S ( W , H , H ± ) + C h V V 2 S S V ( H ± , H , W ) ] + 1 2 g g Z 2 m W ( s W 2 − s β 2 ) c β c α [ C h V V 2 C V S S ( Z , G ± , G 0 ) + C h V V 2 S S V ( G 0 , G ± , Z ) ] + 1 2 g g Z 2 m W s β 2 c β c α [ C h V V 2 C V S S ( Z , H ± , G 0 ) + C h V V 2 S S V ( G 0 , H ± , Z ) ] (m_{W}^{2})^{-1}16\pi^{2}\Gamma_{hWW}^{2,1\mathrm{PI}}(p_{1}^{2},p_{2}^{2},q^{2})_{B}\\
=g^{3}c_{W}m_{Z}c_{\alpha}C_{hVV_{2}}^{VVV}(Z,W,Z)+g^{3}c_{W}^{2}m_{W}(c_{\alpha}c_{\beta}-2s_{\alpha}s_{\beta})C_{hVV_{2}}^{VVV}(W,Z,W)\\
+g^{3}s_{W}^{2}m_{W}(c_{\alpha}c_{\beta}-2s_{\alpha}s_{\beta})C_{hVV_{2}}^{VVV}(W,\gamma,W)\\
-\frac{1}{2}g^{3}m_{W}(c_{\alpha}c_{\beta}-2s_{\alpha}s_{\beta})(s_{W}^{2}-s_{\beta}^{2})[C_{hVV_{2}}^{SVV}(G^{\pm},Z,W)+C_{hVV_{2}}^{VVS}(W,Z,G^{\pm})]\\
+\frac{1}{2}g^{3}s_{W}^{2}m_{W}(c_{\alpha}c_{\beta}-2s_{\alpha}s_{\beta})[C_{hVV_{2}}^{SVV}(G^{\pm},\gamma,W)+C_{hVV_{2}}^{VVS}(W,\gamma,G^{\pm})]\\
-\frac{1}{2}g^{3}m_{W}(c_{\alpha}s_{\beta}+2s_{\alpha}c_{\beta})s_{\beta}c_{\beta}[C_{hVV_{2}}^{SVV}(H^{\pm},Z,W)+C_{hVV_{2}}^{VVS}(W,Z,H^{\pm})]\\
+\frac{1}{2}g^{3}m_{W}(c_{\alpha}c_{\beta}-2s_{\alpha}s_{\beta})^{3}[C_{hVV_{2}}^{CVSS}(W,h,G^{\pm})+C_{hVV_{2}}^{SSV}(G^{\pm},h,W)]\\
+\frac{1}{2}g^{3}m_{W}(c_{\alpha}c_{\beta}-2s_{\alpha}s_{\beta})(s_{\alpha}c_{\beta}+2c_{\alpha}s_{\beta})^{2}[C_{hVV_{2}}^{CVSS}(W,H,G^{\pm})+C_{hVV_{2}}^{SSV}(G^{\pm},H,W)]\\
+\frac{1}{2}g^{3}m_{W}(c_{\alpha}c_{\beta}-2s_{\alpha}s_{\beta})(c_{\alpha}s_{\beta}+2s_{\alpha}c_{\beta})^{2}[C_{hVV_{2}}^{CVSS}(W,h,H^{\pm})+C_{hVV_{2}}^{SSV}(H^{\pm},h,W)]\\
+\frac{1}{2}g^{3}m_{W}(c_{\alpha}s_{\beta}+2s_{\alpha}c_{\beta})(s_{\alpha}c_{\beta}+2c_{\alpha}s_{\beta})(s_{\alpha}s_{\beta}-2c_{\alpha}c_{\beta})[C_{hVV_{2}}^{CVSS}(W,H,H^{\pm})+C_{hVV_{2}}^{SSV}(H^{\pm},H,W)]\\
+\frac{1}{2}gg_{Z}^{2}m_{W}(s_{W}^{2}-s_{\beta}^{2})c_{\beta}c_{\alpha}[C_{hVV_{2}}^{CVSS}(Z,G^{\pm},G^{0})+C_{hVV_{2}}^{SSV}(G^{0},G^{\pm},Z)]\\
+\frac{1}{2}gg_{Z}^{2}m_{W}s_{\beta}^{2}c_{\beta}c_{\alpha}[C_{hVV_{2}}^{CVSS}(Z,H^{\pm},G^{0})+C_{hVV_{2}}^{SSV}(G^{0},H^{\pm},Z)] ( m W 2 ) − 1 16 π 2 Γ hW W 2 , 1 PI ( p 1 2 , p 2 2 , q 2 ) B = g 3 c W m Z c α C hV V 2 V V V ( Z , W , Z ) + g 3 c W 2 m W ( c α c β − 2 s α s β ) C hV V 2 V V V ( W , Z , W ) + g 3 s W 2 m W ( c α c β − 2 s α s β ) C hV V 2 V V V ( W , γ , W ) − 2 1 g 3 m W ( c α c β − 2 s α s β ) ( s W 2 − s β 2 ) [ C hV V 2 S V V ( G ± , Z , W ) + C hV V 2 V V S ( W , Z , G ± )] + 2 1 g 3 s W 2 m W ( c α c β − 2 s α s β ) [ C hV V 2 S V V ( G ± , γ , W ) + C hV V 2 V V S ( W , γ , G ± )] − 2 1 g 3 m W ( c α s β + 2 s α c β ) s β c β [ C hV V 2 S V V ( H ± , Z , W ) + C hV V 2 V V S ( W , Z , H ± )] + 2 1 g 3 m W ( c α c β − 2 s α s β ) 3 [ C hV V 2 C V S S ( W , h , G ± ) + C hV V 2 S S V ( G ± , h , W )] + 2 1 g 3 m W ( c α c β − 2 s α s β ) ( s α c β + 2 c α s β ) 2 [ C hV V 2 C V S S ( W , H , G ± ) + C hV V 2 S S V ( G ± , H , W )] + 2 1 g 3 m W ( c α c β − 2 s α s β ) ( c α s β + 2 s α c β ) 2 [ C hV V 2 C V S S ( W , h , H ± ) + C hV V 2 S S V ( H ± , h , W )] + 2 1 g 3 m W ( c α s β + 2 s α c β ) ( s α c β + 2 c α s β ) ( s α s β − 2 c α c β ) [ C hV V 2 C V S S ( W , H , H ± ) + C hV V 2 S S V ( H ± , H , W )] + 2 1 g g Z 2 m W ( s W 2 − s β 2 ) c β c α [ C hV V 2 C V S S ( Z , G ± , G 0 ) + C hV V 2 S S V ( G 0 , G ± , Z )] + 2 1 g g Z 2 m W s β 2 c β c α [ C hV V 2 C V S S ( Z , H ± , G 0 ) + C hV V 2 S S V ( G 0 , H ± , Z )] − g 3 c W m Z c α C 1223 ( c Z , c ± , c Z ) − g 3 c W m W ( c α c β − 2 s α s β ) C 1223 ( c ± , c Z , c ± ) − g 3 s W 2 m W ( c α c β − 2 s α s β ) C 1223 ( c ± , c γ , c ± ) − 4 g 2 { 1 2 g G 0 G 0 h [ c β 2 C 1223 ( G 0 , G ± , G 0 ) + s β 2 C 1223 ( G 0 , H ± , G 0 ) ] + 3 2 g h h h [ ( c β c α − 2 s β s α ) 2 C 1223 ( h , G ± , h ) + ( s β c α + 2 c β s α ) 2 C 1223 ( h , H ± , h ) ] + 1 2 g H H h [ ( c β s α + 2 s β c α ) 2 C 1223 ( H , G ± , H ) + ( s β s α − 2 c β c α ) 2 C 1223 ( H , H ± , H ) ] − 1 2 g H h h ( c β c α − 2 s β s α ) ( c β s α + 2 s β c α ) [ C 1223 ( h , G ± , H ) + C 1223 ( H , G ± , h ) ] − 1 2 g H h h ( s β c α + 2 c β s α ) ( s β s α − 2 c β c α ) [ C 1223 ( h , H ± , H ) + C 1223 ( H , H ± , h ) ] + 1 4 g H + H − h s β 2 C 1223 ( H ± , G 0 , H ± ) + 1 4 g H + H − h [ ( s β c α + 2 c β s α ) 2 C 1223 ( H ± , h , H ± ) + ( s β s α − 2 c β c α ) 2 C 1223 ( H ± , H , H ± ) ] + 1 4 g G + G − h c β 2 C 1223 ( G ± , G 0 , G ± ) + 1 4 g G + G − h [ ( c β c α − 2 s β s α ) 2 C 1223 ( G ± , h , G ± ) + ( c β s α + 2 s β c α ) 2 C 1223 ( G ± , H , G ± ) ] − 1 4 g H + G − h s β c β [ C 1223 ( G ± , G 0 , H ± ) + C 1223 ( H ± , G 0 , G ± ) ] − 1 4 g H + G − h ( s β c α + 2 c β s α ) ( c β c α − 2 s β s α ) [ C 1223 ( G ± , h , H ± ) + C 1223 ( H ± , h , G ± ) ] − 1 4 g H + G − h ( s β s α − 2 c β c α ) ( c β s α + 2 s β c α ) [ C 1223 ( G ± , H , H ± ) + C 1223 ( H ± , H , G ± ) ] } , (261) \begin{aligned}
&-g^{3}c_{\mathrm W}m_Zc_\alpha C_{1223}(c_Z,c^\pm,c_Z)-g^{3}c_{\mathrm W}m_W(c_\alpha c_\beta-2s_\alpha s_\beta)C_{1223}(c^\pm,c_Z,c^\pm)\\
&-g^{3}s_{\mathrm W}^{2}m_W(c_\alpha c_\beta-2s_\alpha s_\beta)C_{1223}(c^\pm,c_\gamma,c^\pm)\\
&-4g^{2}\left\{\frac{1}{2}g_{G^{0}G^{0}h}\left[c_\beta^{2}C_{1223}(G^{0},G^\pm,G^{0})+s_\beta^{2}C_{1223}(G^{0},H^\pm,G^{0})\right]\right.\\
&\quad+\frac{3}{2}g_{hhh}\left[(c_\beta c_\alpha-2s_\beta s_\alpha)^{2}C_{1223}(h,G^\pm,h)+(s_\beta c_\alpha+2c_\beta s_\alpha)^{2}C_{1223}(h,H^\pm,h)\right]\\
&\quad+\frac{1}{2}g_{HHh}\left[(c_\beta s_\alpha+2s_\beta c_\alpha)^{2}C_{1223}(H,G^\pm,H)+(s_\beta s_\alpha-2c_\beta c_\alpha)^{2}C_{1223}(H,H^\pm,H)\right]\\
&\quad-\frac{1}{2}g_{Hhh}(c_\beta c_\alpha-2s_\beta s_\alpha)(c_\beta s_\alpha+2s_\beta c_\alpha)\left[C_{1223}(h,G^\pm,H)+C_{1223}(H,G^\pm,h)\right]\\
&\quad-\frac{1}{2}g_{Hhh}(s_\beta c_\alpha+2c_\beta s_\alpha)(s_\beta s_\alpha-2c_\beta c_\alpha)\left[C_{1223}(h,H^\pm,H)+C_{1223}(H,H^\pm,h)\right]\\
&\quad+\frac{1}{4}g_{H^{+}H^{-}h}s_\beta^{2}C_{1223}(H^\pm,G^{0},H^\pm)\\
&\quad+\frac{1}{4}g_{H^{+}H^{-}h}\left[(s_\beta c_\alpha+2c_\beta s_\alpha)^{2}C_{1223}(H^\pm,h,H^\pm)+(s_\beta s_\alpha-2c_\beta c_\alpha)^{2}C_{1223}(H^\pm,H,H^\pm)\right]\\
&\quad+\frac{1}{4}g_{G^{+}G^{-}h}c_\beta^{2}C_{1223}(G^\pm,G^{0},G^\pm)\\
&\quad+\frac{1}{4}g_{G^{+}G^{-}h}\left[(c_\beta c_\alpha-2s_\beta s_\alpha)^{2}C_{1223}(G^\pm,h,G^\pm)+(c_\beta s_\alpha+2s_\beta c_\alpha)^{2}C_{1223}(G^\pm,H,G^\pm)\right]\\
&\quad-\frac{1}{4}g_{H^{+}G^{-}h}s_\beta c_\beta\left[C_{1223}(G^\pm,G^{0},H^\pm)+C_{1223}(H^\pm,G^{0},G^\pm)\right]\\
&\quad-\frac{1}{4}g_{H^{+}G^{-}h}(s_\beta c_\alpha+2c_\beta s_\alpha)(c_\beta c_\alpha-2s_\beta s_\alpha)\left[C_{1223}(G^\pm,h,H^\pm)+C_{1223}(H^\pm,h,G^\pm)\right]\\
&\quad\left.-\frac{1}{4}g_{H^{+}G^{-}h}(s_\beta s_\alpha-2c_\beta c_\alpha)(c_\beta s_\alpha+2s_\beta c_\alpha)\left[C_{1223}(G^\pm,H,H^\pm)+C_{1223}(H^\pm,H,G^\pm)\right]\right\}, \tag*{(261)}
\end{aligned} − g 3 c W m Z c α C 1223 ( c Z , c ± , c Z ) − g 3 c W m W ( c α c β − 2 s α s β ) C 1223 ( c ± , c Z , c ± ) − g 3 s W 2 m W ( c α c β − 2 s α s β ) C 1223 ( c ± , c γ , c ± ) − 4 g 2 { 2 1 g G 0 G 0 h [ c β 2 C 1223 ( G 0 , G ± , G 0 ) + s β 2 C 1223 ( G 0 , H ± , G 0 ) ] + 2 3 g hhh [ ( c β c α − 2 s β s α ) 2 C 1223 ( h , G ± , h ) + ( s β c α + 2 c β s α ) 2 C 1223 ( h , H ± , h ) ] + 2 1 g H H h [ ( c β s α + 2 s β c α ) 2 C 1223 ( H , G ± , H ) + ( s β s α − 2 c β c α ) 2 C 1223 ( H , H ± , H ) ] − 2 1 g H hh ( c β c α − 2 s β s α ) ( c β s α + 2 s β c α ) [ C 1223 ( h , G ± , H ) + C 1223 ( H , G ± , h ) ] − 2 1 g H hh ( s β c α + 2 c β s α ) ( s β s α − 2 c β c α ) [ C 1223 ( h , H ± , H ) + C 1223 ( H , H ± , h ) ] + 4 1 g H + H − h s β 2 C 1223 ( H ± , G 0 , H ± ) + 4 1 g H + H − h [ ( s β c α + 2 c β s α ) 2 C 1223 ( H ± , h , H ± ) + ( s β s α − 2 c β c α ) 2 C 1223 ( H ± , H , H ± ) ] + 4 1 g G + G − h c β 2 C 1223 ( G ± , G 0 , G ± ) + 4 1 g G + G − h [ ( c β c α − 2 s β s α ) 2 C 1223 ( G ± , h , G ± ) + ( c β s α + 2 s β c α ) 2 C 1223 ( G ± , H , G ± ) ] − 4 1 g H + G − h s β c β [ C 1223 ( G ± , G 0 , H ± ) + C 1223 ( H ± , G 0 , G ± ) ] − 4 1 g H + G − h ( s β c α + 2 c β s α ) ( c β c α − 2 s β s α ) [ C 1223 ( G ± , h , H ± ) + C 1223 ( H ± , h , G ± ) ] − 4 1 g H + G − h ( s β s α − 2 c β c α ) ( c β s α + 2 s β c α ) [ C 1223 ( G ± , H , H ± ) + C 1223 ( H ± , H , G ± ) ] } , (261) 16 π 2 Γ h W W 3 , 1 P I ( p 1 2 , p 2 2 , q 2 ) B = 0. (262) 16\pi^{2}\Gamma_{hWW}^{3,1\mathrm{PI}}(p_{1}^{2},p_{2}^{2},q^{2})_{B}=0. \tag*{(262)} 16 π 2 Γ hW W 3 , 1 PI ( p 1 2 , p 2 2 , q 2 ) B = 0. (262) h f f ˉ hf\bar{f} h f f ˉ The h f f ˉ hf\bar{f} h f f ˉ
( m f v ) − 1 16 π 2 Γ h f f ˉ S , 1 P I \left(\frac{m_f}{v}\right)^{-1}16\pi^{2}\Gamma_{hf\bar{f}}^{S,\mathrm{1PI}} ( v m f ) − 1 16 π 2 Γ h f f ˉ S , 1PI = − 4 ζ h [ e 2 Q f 2 C h f f ˉ F V F ( f , γ , f ) + g Z 2 ( v f 2 − a f 2 ) C h f f ˉ F V F ( f , Z , f ) ] + m f 2 v 2 [ ζ h 2 C h f f ˉ F S F ( f , h , f ) + ζ H 2 C h f f ˉ F S F ( f , H , f ) − ζ G 0 2 C h f f ˉ F S F ( f , G 0 , f ) ] − 2 m f ′ 2 v 2 [ ζ G ± 2 C h f f ˉ F S F ( f ′ , G ± , f ′ ) + ζ H ± 2 C h f f ˉ F S F ( f ′ , H ± , f ′ ) ] − 4 g Z 3 m Z v ( v f 2 − a f 2 ) c α C 0 ( Z , f , Z ) − m f 2 v { 6 g h h h ζ h 2 C 0 ( h , f , h ) + 2 g H H h ζ H 2 C 0 ( H , f , H ) + 2 g H h h ζ h ζ H [ C 0 ( h , f , H ) + C 0 ( H , f , h ) ] − 2 g G 0 G 0 h ζ G 0 2 C 0 ( G 0 , f , G 0 ) } (263) \begin{aligned}
={}&-4\zeta_h\left[e^{2}Q_f^{2}C_{hf\bar{f}}^{FVF}(f,\gamma,f)+g_Z^{2}(v_f^{2}-a_f^{2})C_{hf\bar{f}}^{FVF}(f,Z,f)\right]\\
&+\frac{m_f^{2}}{v^{2}}\left[\zeta_h^{2}C_{hf\bar{f}}^{FSF}(f,h,f)+\zeta_H^{2}C_{hf\bar{f}}^{FSF}(f,H,f)-\zeta_{G^{0}}^{2}C_{hf\bar{f}}^{FSF}(f,G^{0},f)\right]\\
&-\frac{2m_{f'}^{2}}{v^{2}}\left[\zeta_{G^\pm}^{2}C_{hf\bar{f}}^{FSF}(f',G^\pm,f')+\zeta_{H^\pm}^{2}C_{hf\bar{f}}^{FSF}(f',H^\pm,f')\right]\\
&-4g_Z^{3}m_Zv(v_f^{2}-a_f^{2})c_\alpha C_0(Z,f,Z)\\
&-\frac{m_f^{2}}{v}\left\{6g_{hhh}\zeta_h^{2}C_0(h,f,h)+2g_{HHh}\zeta_H^{2}C_0(H,f,H)+2g_{Hhh}\zeta_h\zeta_H\left[C_0(h,f,H)+C_0(H,f,h)\right]\right.\\
&\qquad\left.-2g_{G^{0}G^{0}h}\zeta_{G^{0}}^{2}C_0(G^{0},f,G^{0})\right\} \tag*{(263)}
\end{aligned} = − 4 ζ h [ e 2 Q f 2 C h f f ˉ F V F ( f , γ , f ) + g Z 2 ( v f 2 − a f 2 ) C h f f ˉ F V F ( f , Z , f ) ] + v 2 m f 2 [ ζ h 2 C h f f ˉ F S F ( f , h , f ) + ζ H 2 C h f f ˉ F S F ( f , H , f ) − ζ G 0 2 C h f f ˉ F S F ( f , G 0 , f ) ] − v 2 2 m f ′ 2 [ ζ G ± 2 C h f f ˉ F S F ( f ′ , G ± , f ′ ) + ζ H ± 2 C h f f ˉ F S F ( f ′ , H ± , f ′ ) ] − 4 g Z 3 m Z v ( v f 2 − a f 2 ) c α C 0 ( Z , f , Z ) − v m f 2 { 6 g hhh ζ h 2 C 0 ( h , f , h ) + 2 g H H h ζ H 2 C 0 ( H , f , H ) + 2 g H hh ζ h ζ H [ C 0 ( h , f , H ) + C 0 ( H , f , h ) ] − 2 g G 0 G 0 h ζ G 0 2 C 0 ( G 0 , f , G 0 ) } (263) + 2 m f 2 v { g G + G − h ζ G ± 2 C 0 ( G ± , f ′ , G ± ) + g H + H − h ζ H ± 2 C 0 ( H ± , f ′ , H ± ) + g G − H + h ζ G ± ζ H ± [ C 0 ( G ± , f ′ , H ± ) + C 0 ( H ± , f ′ , G ± ) ] } − g 2 4 ζ G ± ( c β c α − 2 s β s α ) [ C h f f V F S ( W , f ′ , G ± ) + C h f f S F V ( G ± , f ′ , W ) ] + g 2 4 ζ H ± ( s β c α + 2 c β s α ) [ C h f f V F S ( W , f ′ , H ± ) + C h f f S F V ( H ± , f ′ , W ) ] − g Z 2 8 ζ G 0 c α [ C h f f V F S ( Z , f , G 0 ) + C h f f S F V ( G 0 , f , Z ) ] , (264) \begin{aligned}
&+\frac{2m_f^2}{v}\left\{g_{G^+G^-h}\zeta_{G^\pm}^2 C_0(G^\pm,f',G^\pm)+g_{H^+H^-h}\zeta_{H^\pm}^2 C_0(H^\pm,f',H^\pm)\right.\\
&\left.\quad+g_{G^-H^+h}\zeta_{G^\pm}\zeta_{H^\pm}\left[C_0(G^\pm,f',H^\pm)+C_0(H^\pm,f',G^\pm)\right]\right\}\\
&-\frac{g^2}{4}\zeta_{G^\pm}(c_\beta c_\alpha-2s_\beta s_\alpha)\left[C_{hff}^{VFS}(W,f',G^\pm)+C_{hff}^{SFV}(G^\pm,f',W)\right]\\
&+\frac{g^2}{4}\zeta_{H^\pm}(s_\beta c_\alpha+2c_\beta s_\alpha)\left[C_{hff}^{VFS}(W,f',H^\pm)+C_{hff}^{SFV}(H^\pm,f',W)\right]\\
&-\frac{g_Z^2}{8}\zeta_{G^0}c_\alpha\left[C_{hff}^{VFS}(Z,f,G^0)+C_{hff}^{SFV}(G^0,f,Z)\right], \tag*{(264)}
\end{aligned} + v 2 m f 2 { g G + G − h ζ G ± 2 C 0 ( G ± , f ′ , G ± ) + g H + H − h ζ H ± 2 C 0 ( H ± , f ′ , H ± ) + g G − H + h ζ G ± ζ H ± [ C 0 ( G ± , f ′ , H ± ) + C 0 ( H ± , f ′ , G ± ) ] } − 4 g 2 ζ G ± ( c β c α − 2 s β s α ) [ C h f f V F S ( W , f ′ , G ± ) + C h f f S F V ( G ± , f ′ , W ) ] + 4 g 2 ζ H ± ( s β c α + 2 c β s α ) [ C h f f V F S ( W , f ′ , H ± ) + C h f f S F V ( H ± , f ′ , W ) ] − 8 g Z 2 ζ G 0 c α [ C h f f V F S ( Z , f , G 0 ) + C h f f S F V ( G 0 , f , Z ) ] , (264) ( m f v ) − 1 16 π 2 Γ h f f P , 1 P I = g 2 4 ζ G ± ( c β c α − 2 s β s α ) [ C h f f V F S ( W , f ′ , G ± ) − C h f f S F V ( G ± , f ′ , W ) ] − g 2 4 ζ H ± ( s β c α + 2 c β s α ) [ C h f f V F S ( W , f ′ , H ± ) − C h f f S F V ( H ± , f ′ , W ) ] + 2 g Z v f a f ζ G 0 c α [ C h f f V F S ( Z , f , G 0 ) − C h f f S F V ( G 0 , f , Z ) ] , (265) \begin{aligned}
\left(\frac{m_f}{v}\right)^{-1}16\pi^2\Gamma_{hff}^{P,1PI}
={}&\frac{g^2}{4}\zeta_{G^\pm}(c_\beta c_\alpha-2s_\beta s_\alpha)\left[C_{hff}^{VFS}(W,f',G^\pm)-C_{hff}^{SFV}(G^\pm,f',W)\right]\\
&-\frac{g^2}{4}\zeta_{H^\pm}(s_\beta c_\alpha+2c_\beta s_\alpha)\left[C_{hff}^{VFS}(W,f',H^\pm)-C_{hff}^{SFV}(H^\pm,f',W)\right]\\
&+2g_Zv_fa_f\zeta_{G^0}c_\alpha\left[C_{hff}^{VFS}(Z,f,G^0)-C_{hff}^{SFV}(G^0,f,Z)\right], \tag*{(265)}
\end{aligned} ( v m f ) − 1 16 π 2 Γ h f f P , 1 P I = 4 g 2 ζ G ± ( c β c α − 2 s β s α ) [ C h f f V F S ( W , f ′ , G ± ) − C h f f S F V ( G ± , f ′ , W ) ] − 4 g 2 ζ H ± ( s β c α + 2 c β s α ) [ C h f f V F S ( W , f ′ , H ± ) − C h f f S F V ( H ± , f ′ , W ) ] + 2 g Z v f a f ζ G 0 c α [ C h f f V F S ( Z , f , G 0 ) − C h f f S F V ( G 0 , f , Z ) ] , (265) ( 16 π 2 ) Γ h f f V , 1 P I = − 2 m f 2 v ζ h [ e 2 Q f 2 ( C 0 + 2 C 11 ) ( f , γ , f ) + g Z 2 ( v f 2 + a f 2 ) ( C 0 + 2 C 11 ) ( f , Z , f ) ] − g 2 2 m f 2 v ζ h ( C 0 + 2 C 11 ) ( f ′ , W , f ′ ) − m f 4 v 3 { ζ h 2 ( C 0 + 2 C 11 ) ( f , h , f ) + ζ H 2 ( C 0 + 2 C 11 ) ( f , H , f ) + ζ G 0 2 ( C 0 + 2 C 11 ) ( f , G 0 , f ) } − m f 2 v 3 ( m f 2 + m f ′ 2 ) ζ h [ ζ G ± 2 ( C 0 + 2 C 11 ) ( f ′ , G ± , f ′ ) + ζ H ± 2 ( C 0 + 2 C 11 ) ( f ′ , H ± , f ′ ) ] + 2 g Z 3 m Z ( v f 2 + a f 2 ) c α ( C 0 + C 11 ) ( Z , f , Z ) + g 3 m W 2 ( c β c α − 2 s β s α ) ( C 0 + C 11 ) ( W , f ′ , W ) − m f 2 v 2 { 6 g h h h ζ h 2 ( C 0 + C 11 ) ( h , f , h ) + 2 g H H h ζ H 2 ( C 0 + C 11 ) ( H , f , H ) + 2 g H h h ζ h ζ H [ ( C 0 + C 11 ) ( H , f , h ) + ( C 0 + C 11 ) ( h , f , H ) ] + 2 g G 0 G 0 h ζ G 0 2 ( C 0 + C 11 ) ( G 0 , f , G 0 ) } − m f 2 + m f ′ 2 v 2 { g G + G − h ζ G ± 2 ( C 0 + C 11 ) ( G ± , f ′ , G ± ) + g H + H − h ζ H ± 2 ( C 0 + C 11 ) ( H ± , f ′ , H ± ) + g H + G − h ζ G ± ζ H ± [ ( C 0 + C 11 ) ( G ± , f ′ , H ± ) + ( C 0 + C 11 ) ( H ± , f ′ , G ± ) ] } + g 2 4 m f 2 v { ζ G ± ( c β c α − 2 s β s α ) [ ( 2 C 0 + C 11 ) ( W , f ′ , G ± ) + ( − C 0 + C 11 ) ( G ± , f ′ , W ) ] − ζ H ± ( s β c α + 2 c β s α ) [ ( 2 C 0 + C 11 ) ( W , f ′ , H ± ) + ( − C 0 + C 11 ) ( H ± , f ′ , W ) ] } + g Z 2 8 m f 2 v ζ G 0 c α [ ( 2 C 0 + C 11 ) ( Z , f , G 0 ) + ( − C 0 + C 11 ) ( G 0 , f , Z ) ] , (266) \begin{aligned}
(16\pi^2)\Gamma_{hff}^{V,1PI}
={}&-\frac{2m_f^2}{v}\zeta_h\left[e^2Q_f^2(C_0+2C_{11})(f,\gamma,f)+g_Z^2(v_f^2+a_f^2)(C_0+2C_{11})(f,Z,f)\right]\\
&-\frac{g^2}{2}\frac{m_f^2}{v}\zeta_h(C_0+2C_{11})(f',W,f')\\
&-\frac{m_f^4}{v^3}\left\{\zeta_h^2(C_0+2C_{11})(f,h,f)+\zeta_H^2(C_0+2C_{11})(f,H,f)\right.\\
&\left.\quad+\zeta_{G^0}^2(C_0+2C_{11})(f,G^0,f)\right\}\\
&-\frac{m_f^2}{v^3}(m_f^2+m_{f'}^2)\zeta_h\left[\zeta_{G^\pm}^2(C_0+2C_{11})(f',G^\pm,f')+\zeta_{H^\pm}^2(C_0+2C_{11})(f',H^\pm,f')\right]\\
&+2g_Z^3m_Z(v_f^2+a_f^2)c_\alpha(C_0+C_{11})(Z,f,Z)+\frac{g^3m_W}{2}(c_\beta c_\alpha-2s_\beta s_\alpha)(C_0+C_{11})(W,f',W)\\
&-\frac{m_f^2}{v^2}\left\{6g_{hhh}\zeta_h^2(C_0+C_{11})(h,f,h)+2g_{HHh}\zeta_H^2(C_0+C_{11})(H,f,H)\right.\\
&\left.\quad+2g_{Hhh}\zeta_h\zeta_H\left[(C_0+C_{11})(H,f,h)+(C_0+C_{11})(h,f,H)\right]\right.\\
&\left.\quad+2g_{G^0G^0h}\zeta_{G^0}^2(C_0+C_{11})(G^0,f,G^0)\right\}\\
&-\frac{m_f^2+m_{f'}^2}{v^2}\left\{g_{G^+G^-h}\zeta_{G^\pm}^2(C_0+C_{11})(G^\pm,f',G^\pm)+g_{H^+H^-h}\zeta_{H^\pm}^2(C_0+C_{11})(H^\pm,f',H^\pm)\right.\\
&\left.\quad+g_{H^+G^-h}\zeta_{G^\pm}\zeta_{H^\pm}\left[(C_0+C_{11})(G^\pm,f',H^\pm)+(C_0+C_{11})(H^\pm,f',G^\pm)\right]\right\}\\
&+\frac{g^2}{4}\frac{m_f^2}{v}\left\{\zeta_{G^\pm}(c_\beta c_\alpha-2s_\beta s_\alpha)\left[(2C_0+C_{11})(W,f',G^\pm)+(-C_0+C_{11})(G^\pm,f',W)\right]\right.\\
&\left.\quad-\zeta_{H^\pm}(s_\beta c_\alpha+2c_\beta s_\alpha)\left[(2C_0+C_{11})(W,f',H^\pm)+(-C_0+C_{11})(H^\pm,f',W)\right]\right\}\\
&+\frac{g_Z^2}{8}\frac{m_f^2}{v}\zeta_{G^0}c_\alpha\left[(2C_0+C_{11})(Z,f,G^0)+(-C_0+C_{11})(G^0,f,Z)\right], \tag*{(266)}
\end{aligned} ( 16 π 2 ) Γ h f f V , 1 P I = − v 2 m f 2 ζ h [ e 2 Q f 2 ( C 0 + 2 C 11 ) ( f , γ , f ) + g Z 2 ( v f 2 + a f 2 ) ( C 0 + 2 C 11 ) ( f , Z , f ) ] − 2 g 2 v m f 2 ζ h ( C 0 + 2 C 11 ) ( f ′ , W , f ′ ) − v 3 m f 4 { ζ h 2 ( C 0 + 2 C 11 ) ( f , h , f ) + ζ H 2 ( C 0 + 2 C 11 ) ( f , H , f ) + ζ G 0 2 ( C 0 + 2 C 11 ) ( f , G 0 , f ) } − v 3 m f 2 ( m f 2 + m f ′ 2 ) ζ h [ ζ G ± 2 ( C 0 + 2 C 11 ) ( f ′ , G ± , f ′ ) + ζ H ± 2 ( C 0 + 2 C 11 ) ( f ′ , H ± , f ′ ) ] + 2 g Z 3 m Z ( v f 2 + a f 2 ) c α ( C 0 + C 11 ) ( Z , f , Z ) + 2 g 3 m W ( c β c α − 2 s β s α ) ( C 0 + C 11 ) ( W , f ′ , W ) − v 2 m f 2 { 6 g hhh ζ h 2 ( C 0 + C 11 ) ( h , f , h ) + 2 g H H h ζ H 2 ( C 0 + C 11 ) ( H , f , H ) + 2 g H hh ζ h ζ H [ ( C 0 + C 11 ) ( H , f , h ) + ( C 0 + C 11 ) ( h , f , H ) ] + 2 g G 0 G 0 h ζ G 0 2 ( C 0 + C 11 ) ( G 0 , f , G 0 ) } − v 2 m f 2 + m f ′ 2 { g G + G − h ζ G ± 2 ( C 0 + C 11 ) ( G ± , f ′ , G ± ) + g H + H − h ζ H ± 2 ( C 0 + C 11 ) ( H ± , f ′ , H ± ) + g H + G − h ζ G ± ζ H ± [ ( C 0 + C 11 ) ( G ± , f ′ , H ± ) + ( C 0 + C 11 ) ( H ± , f ′ , G ± ) ] } + 4 g 2 v m f 2 { ζ G ± ( c β c α − 2 s β s α ) [ ( 2 C 0 + C 11 ) ( W , f ′ , G ± ) + ( − C 0 + C 11 ) ( G ± , f ′ , W ) ] − ζ H ± ( s β c α + 2 c β s α ) [ ( 2 C 0 + C 11 ) ( W , f ′ , H ± ) + ( − C 0 + C 11 ) ( H ± , f ′ , W ) ] } + 8 g Z 2 v m f 2 ζ G 0 c α [ ( 2 C 0 + C 11 ) ( Z , f , G 0 ) + ( − C 0 + C 11 ) ( G 0 , f , Z ) ] , (266) ( 16 π 2 ) Γ h f f ˉ V 2 , 1 P I = − 2 m f 2 v ζ h [ q f 2 Q f 2 ( C 0 + 2 C 12 ) ( f , γ , f ) + g Z 2 ( v f 2 + a f 2 ) ( C 0 + 2 C 12 ) ( f , Z , f ) ] − g 2 2 m f ′ 2 v ζ h ( C 0 + 2 C 12 ) ( f ′ , W , f ′ ) − m f 2 v 3 ζ h [ ζ h 2 ( C 0 + 2 C 12 ) ( f , h , f ) + ζ H 2 ( C 0 + 2 C 12 ) ( f , H , f ) + ζ G 0 2 ( C 0 + 2 C 12 ) ( f , G 0 , f ) ] − m f 2 v 3 ( m f 2 + m f ′ 2 ) ζ h [ ζ G ± 2 ( C 0 + 2 C 12 ) ( f ′ , G ± , f ′ ) + ζ H ± 2 ( C 0 + 2 C 12 ) ( f ′ , H ± , f ′ ) ] + 2 g Z 3 m Z ( v f 2 + a f 2 ) c α C 12 ( Z , f , Z ) + g 3 m W 2 ( c β c α − 2 s β s α ) C 12 ( W , f ′ , W ) − m f 2 v 2 { 6 g h h h ζ h 2 C 12 ( h , f , h ) + 2 g H H H ζ H 2 C 12 ( H , f , H ) + 2 g H H h ζ h ζ H [ C 12 ( H , f , h ) + C 12 ( h , f , H ) ] + 2 g G 0 G 0 h ζ G 0 2 C 12 ( G 0 , f , G 0 ) } − m f 2 + m f ′ 2 v 2 { g G + G − h ζ G ± 2 C 12 ( G ± , f ′ , G ± ) + g H + H − h ζ H ± 2 C 12 ( H ± , f ′ , H ± ) + g H + G − h ζ G ± ζ H ± [ C 12 ( G ± , f ′ , H ± ) + C 12 ( H ± , f ′ , G ± ) ] } + g 2 4 m f ′ 2 v { ζ G ± ( c β c α − 2 s β s α ) [ ( 2 C 0 + C 12 ) ( W , f ′ , G ± ) + ( − C 0 + C 12 ) ( G ± , f ′ , W ) ] − ζ H ± ( s β c α + 2 c β s α ) [ ( 2 C 0 + C 12 ) ( W , f ′ , H ± ) + ( − C 0 + C 12 ) ( H ± , f ′ , W ) ] } + g Z 2 8 m f 2 v ζ G 0 c α [ ( 2 C 0 + C 12 ) ( Z , f , G 0 ) + ( − C 0 + C 12 ) ( G 0 , f , Z ) ] , (267) \begin{aligned}
(16\pi^2)\Gamma_{h f\bar f}^{V_2,\mathrm{1PI}}={}&-\frac{2m_f^2}{v}\zeta_h\left[q_f^2Q_f^2(C_0+2C_{12})(f,\gamma,f)+g_Z^2(v_f^2+a_f^2)(C_0+2C_{12})(f,Z,f)\right]\\
&-\frac{g^2}{2}\frac{m_{f'}^2}{v}\zeta_h(C_0+2C_{12})(f',W,f')\\
&-\frac{m_f^2}{v^3}\zeta_h\left[\zeta_h^2(C_0+2C_{12})(f,h,f)+\zeta_H^2(C_0+2C_{12})(f,H,f)+\zeta_{G^0}^2(C_0+2C_{12})(f,G^0,f)\right]\\
&-\frac{m_f^2}{v^3}(m_f^2+m_{f'}^2)\zeta_h\left[\zeta_{G^\pm}^2(C_0+2C_{12})(f',G^\pm,f')+\zeta_{H^\pm}^2(C_0+2C_{12})(f',H^\pm,f')\right]\\
&+2g_Z^3m_Z(v_f^2+a_f^2)c_\alpha C_{12}(Z,f,Z)+\frac{g^3m_W}{2}(c_\beta c_\alpha-2s_\beta s_\alpha)C_{12}(W,f',W)\\
&-\frac{m_f^2}{v^2}\left\{6g_{hhh}\zeta_h^2C_{12}(h,f,h)+2g_{HHH}\zeta_H^2C_{12}(H,f,H)+2g_{HHh}\zeta_h\zeta_H[C_{12}(H,f,h)+C_{12}(h,f,H)]\right.\\
&\left.\qquad+2g_{G^0G^0h}\zeta_{G^0}^2C_{12}(G^0,f,G^0)\right\}\\
&-\frac{m_f^2+m_{f'}^2}{v^2}\left\{g_{G^+G^-h}\zeta_{G^\pm}^2C_{12}(G^\pm,f',G^\pm)+g_{H^+H^-h}\zeta_{H^\pm}^2C_{12}(H^\pm,f',H^\pm)\right.\\
&\left.\qquad+g_{H^+G^-h}\zeta_{G^\pm}\zeta_{H^\pm}[C_{12}(G^\pm,f',H^\pm)+C_{12}(H^\pm,f',G^\pm)]\right\}\\
&+\frac{g^2}{4}\frac{m_{f'}^2}{v}\left\{\zeta_{G^\pm}(c_\beta c_\alpha-2s_\beta s_\alpha)[(2C_0+C_{12})(W,f',G^\pm)+(-C_0+C_{12})(G^\pm,f',W)]\right.\\
&\left.\qquad-\zeta_{H^\pm}(s_\beta c_\alpha+2c_\beta s_\alpha)[(2C_0+C_{12})(W,f',H^\pm)+(-C_0+C_{12})(H^\pm,f',W)]\right\}\\
&+\frac{g_Z^2}{8}\frac{m_f^2}{v}\zeta_{G^0}c_\alpha[(2C_0+C_{12})(Z,f,G^0)+(-C_0+C_{12})(G^0,f,Z)], \tag*{(267)}
\end{aligned} ( 16 π 2 ) Γ h f f ˉ V 2 , 1PI = − v 2 m f 2 ζ h [ q f 2 Q f 2 ( C 0 + 2 C 12 ) ( f , γ , f ) + g Z 2 ( v f 2 + a f 2 ) ( C 0 + 2 C 12 ) ( f , Z , f ) ] − 2 g 2 v m f ′ 2 ζ h ( C 0 + 2 C 12 ) ( f ′ , W , f ′ ) − v 3 m f 2 ζ h [ ζ h 2 ( C 0 + 2 C 12 ) ( f , h , f ) + ζ H 2 ( C 0 + 2 C 12 ) ( f , H , f ) + ζ G 0 2 ( C 0 + 2 C 12 ) ( f , G 0 , f ) ] − v 3 m f 2 ( m f 2 + m f ′ 2 ) ζ h [ ζ G ± 2 ( C 0 + 2 C 12 ) ( f ′ , G ± , f ′ ) + ζ H ± 2 ( C 0 + 2 C 12 ) ( f ′ , H ± , f ′ ) ] + 2 g Z 3 m Z ( v f 2 + a f 2 ) c α C 12 ( Z , f , Z ) + 2 g 3 m W ( c β c α − 2 s β s α ) C 12 ( W , f ′ , W ) − v 2 m f 2 { 6 g hhh ζ h 2 C 12 ( h , f , h ) + 2 g H H H ζ H 2 C 12 ( H , f , H ) + 2 g H H h ζ h ζ H [ C 12 ( H , f , h ) + C 12 ( h , f , H )] + 2 g G 0 G 0 h ζ G 0 2 C 12 ( G 0 , f , G 0 ) } − v 2 m f 2 + m f ′ 2 { g G + G − h ζ G ± 2 C 12 ( G ± , f ′ , G ± ) + g H + H − h ζ H ± 2 C 12 ( H ± , f ′ , H ± ) + g H + G − h ζ G ± ζ H ± [ C 12 ( G ± , f ′ , H ± ) + C 12 ( H ± , f ′ , G ± )] } + 4 g 2 v m f ′ 2 { ζ G ± ( c β c α − 2 s β s α ) [( 2 C 0 + C 12 ) ( W , f ′ , G ± ) + ( − C 0 + C 12 ) ( G ± , f ′ , W )] − ζ H ± ( s β c α + 2 c β s α ) [( 2 C 0 + C 12 ) ( W , f ′ , H ± ) + ( − C 0 + C 12 ) ( H ± , f ′ , W )] } + 8 g Z 2 v m f 2 ζ G 0 c α [( 2 C 0 + C 12 ) ( Z , f , G 0 ) + ( − C 0 + C 12 ) ( G 0 , f , Z )] , (267) ( 16 π 2 ) Γ h f f ˉ A 1 , 1 P I = 4 g Z 2 v f a f m f 2 v ζ h ( C 0 + 2 C 11 ) ( f , Z , f ) + g 2 2 m f ′ 2 v ζ h ( C 0 + 2 C 11 ) ( f ′ , W , f ′ ) − m f ′ 2 v 3 ( m f 2 − m f ′ 2 ) ζ h [ ζ G ± 2 ( C 0 + 2 C 11 ) ( f ′ , G ± , f ′ ) + ζ H ± 2 ( C 0 + 2 C 11 ) ( f ′ , H ± , f ′ ) ] − 4 g Z 3 m Z v f a f c α ( C 0 + C 11 ) ( Z , f , Z ) − g 3 m W 2 ( c β c α − 2 s β s α ) ( C 0 + C 11 ) ( W , f ′ , W ) − m f 2 − m f ′ 2 v 2 { ζ G ± 2 g G + G − h ( C 0 + C 11 ) ( G ± , f ′ , G ± ) + ζ H ± 2 g H + H − h ( C 0 + C 11 ) ( H ± , f ′ , H ± ) + ζ G ± ζ H ± g H + G − h [ ( C 0 + C 11 ) ( G ± , f ′ , H ± ) + ( C 0 + C 11 ) ( H ± , f ′ , G ± ) ] } − g 2 4 m f ′ 2 v { ζ G ± ( c β c α − 2 s β s α ) [ ( 2 C 0 + C 11 ) ( W , f ′ , G ± ) + ( − C 0 + C 11 ) ( G ± , f ′ , W ) ] − ζ H ± ( s β c α + 2 c β s α ) [ ( 2 C 0 + C 11 ) ( W , f ′ , H ± ) + ( − C 0 + C 11 ) ( H ± , f ′ , W ) ] } − 2 g Z 2 v f a f m f 2 v ζ G 0 c α [ ( 2 C 0 + C 11 ) ( Z , f , G 0 ) + ( − C 0 + C 11 ) ( G 0 , f , Z ) ] , (268) \begin{aligned}
(16\pi^2)\Gamma_{h f\bar f}^{A_1,\mathrm{1PI}}={}&4g_Z^2v_fa_f\frac{m_f^2}{v}\zeta_h(C_0+2C_{11})(f,Z,f)+\frac{g^2}{2}\frac{m_{f'}^2}{v}\zeta_h(C_0+2C_{11})(f',W,f')\\
&-\frac{m_{f'}^2}{v^3}(m_f^2-m_{f'}^2)\zeta_h\left[\zeta_{G^\pm}^2(C_0+2C_{11})(f',G^\pm,f')+\zeta_{H^\pm}^2(C_0+2C_{11})(f',H^\pm,f')\right]\\
&-4g_Z^3m_Zv_fa_fc_\alpha(C_0+C_{11})(Z,f,Z)-\frac{g^3m_W}{2}(c_\beta c_\alpha-2s_\beta s_\alpha)(C_0+C_{11})(W,f',W)\\
&-\frac{m_f^2-m_{f'}^2}{v^2}\left\{\zeta_{G^\pm}^2g_{G^+G^-h}(C_0+C_{11})(G^\pm,f',G^\pm)+\zeta_{H^\pm}^2g_{H^+H^-h}(C_0+C_{11})(H^\pm,f',H^\pm)\right.\\
&\left.\qquad+\zeta_{G^\pm}\zeta_{H^\pm}g_{H^+G^-h}[(C_0+C_{11})(G^\pm,f',H^\pm)+(C_0+C_{11})(H^\pm,f',G^\pm)]\right\}\\
&-\frac{g^2}{4}\frac{m_{f'}^2}{v}\left\{\zeta_{G^\pm}(c_\beta c_\alpha-2s_\beta s_\alpha)[(2C_0+C_{11})(W,f',G^\pm)+(-C_0+C_{11})(G^\pm,f',W)]\right.\\
&\left.\qquad-\zeta_{H^\pm}(s_\beta c_\alpha+2c_\beta s_\alpha)[(2C_0+C_{11})(W,f',H^\pm)+(-C_0+C_{11})(H^\pm,f',W)]\right\}\\
&-2g_Z^2v_fa_f\frac{m_f^2}{v}\zeta_{G^0}c_\alpha[(2C_0+C_{11})(Z,f,G^0)+(-C_0+C_{11})(G^0,f,Z)], \tag*{(268)}
\end{aligned} ( 16 π 2 ) Γ h f f ˉ A 1 , 1PI = 4 g Z 2 v f a f v m f 2 ζ h ( C 0 + 2 C 11 ) ( f , Z , f ) + 2 g 2 v m f ′ 2 ζ h ( C 0 + 2 C 11 ) ( f ′ , W , f ′ ) − v 3 m f ′ 2 ( m f 2 − m f ′ 2 ) ζ h [ ζ G ± 2 ( C 0 + 2 C 11 ) ( f ′ , G ± , f ′ ) + ζ H ± 2 ( C 0 + 2 C 11 ) ( f ′ , H ± , f ′ ) ] − 4 g Z 3 m Z v f a f c α ( C 0 + C 11 ) ( Z , f , Z ) − 2 g 3 m W ( c β c α − 2 s β s α ) ( C 0 + C 11 ) ( W , f ′ , W ) − v 2 m f 2 − m f ′ 2 { ζ G ± 2 g G + G − h ( C 0 + C 11 ) ( G ± , f ′ , G ± ) + ζ H ± 2 g H + H − h ( C 0 + C 11 ) ( H ± , f ′ , H ± ) + ζ G ± ζ H ± g H + G − h [( C 0 + C 11 ) ( G ± , f ′ , H ± ) + ( C 0 + C 11 ) ( H ± , f ′ , G ± )] } − 4 g 2 v m f ′ 2 { ζ G ± ( c β c α − 2 s β s α ) [( 2 C 0 + C 11 ) ( W , f ′ , G ± ) + ( − C 0 + C 11 ) ( G ± , f ′ , W )] − ζ H ± ( s β c α + 2 c β s α ) [( 2 C 0 + C 11 ) ( W , f ′ , H ± ) + ( − C 0 + C 11 ) ( H ± , f ′ , W )] } − 2 g Z 2 v f a f v m f 2 ζ G 0 c α [( 2 C 0 + C 11 ) ( Z , f , G 0 ) + ( − C 0 + C 11 ) ( G 0 , f , Z )] , (268) ( 16 π 2 ) Γ h f f ˉ A 2 , 1 P I (16\pi^2)\Gamma_{h f\bar f}^{A_2,\mathrm{1PI}} ( 16 π 2 ) Γ h f f ˉ A 2 , 1PI = 4 g Z 2 v f a f m f 2 v ζ h ( C 0 + 2 C 12 ) ( f , Z , f ) + g 2 2 m f 2 v ζ h ( C 0 + 2 C 12 ) ( f ′ , W , f ′ ) − m f 2 v 3 ( m f 2 − m f ′ 2 ) ζ h [ ζ G ± 2 ( C 0 + 2 C 12 ) ( f ′ , G ± , f ′ ) + ζ H ± 2 ( C 0 + 2 C 12 ) ( f ′ , H ± , f ′ ) ] − 4 g Z 2 m Z v f a f c α C 12 ( Z , f , Z ) − g 3 m W 2 ( c β c α − 2 s β s α ) C 12 ( W , f ′ , W ) − m f 2 − m f ′ 2 v 2 { ζ G ± 2 g G + G − h C 12 ( G ± , f ′ , G ± ) + ζ H ± 2 g H + H − h C 12 ( H ± , f ′ , H ± ) + ζ G ± ζ H ± g H + G − h [ C 12 ( G ± , f ′ , H ± ) + C 12 ( H ± , f ′ , G ± ) ] } − g 2 4 m f 2 v { ζ G ± ( c β c α − 2 s β s α ) [ ( 2 C 0 + C 12 ) ( W , f ′ , G ± ) + ( − C 0 + C 12 ) ( G ± , f ′ , W ) ] − ζ H ± ( s β c α + 2 c β s α ) [ ( 2 C 0 + C 12 ) ( W , f ′ , H ± ) + ( − C 0 + C 12 ) ( H ± , f ′ , W ) ] } − 2 g Z 2 v f a f m f 2 v ζ G 0 c α [ ( 2 C 0 + C 12 ) ( Z , f , G 0 ) + ( − C 0 + C 12 ) ( G 0 , f , Z ) ] , (269) \begin{aligned}
={}&4g_Z^2v_fa_f\frac{m_f^2}{v}\zeta_h(C_0+2C_{12})(f,Z,f)+\frac{g^2}{2}\frac{m_f^2}{v}\zeta_h(C_0+2C_{12})(f',W,f')\\
&-\frac{m_f^2}{v^3}(m_f^2-m_{f'}^2)\zeta_h\left[\zeta_{G^\pm}^2(C_0+2C_{12})(f',G^\pm,f')+\zeta_{H^\pm}^2(C_0+2C_{12})(f',H^\pm,f')\right]\\
&-4g_Z^2m_Zv_fa_fc_\alpha C_{12}(Z,f,Z)-\frac{g^3m_W}{2}(c_\beta c_\alpha-2s_\beta s_\alpha)C_{12}(W,f',W)\\
&-\frac{m_f^2-m_{f'}^2}{v^2}\left\{\zeta_{G^\pm}^2g_{G^+G^-h}C_{12}(G^\pm,f',G^\pm)+\zeta_{H^\pm}^2g_{H^+H^-h}C_{12}(H^\pm,f',H^\pm)\right.\\
&\qquad\left.+\zeta_{G^\pm}\zeta_{H^\pm}g_{H^+G^-h}\left[C_{12}(G^\pm,f',H^\pm)+C_{12}(H^\pm,f',G^\pm)\right]\right\}\\
&-\frac{g^2}{4}\frac{m_f^2}{v}\left\{\zeta_{G^\pm}(c_\beta c_\alpha-2s_\beta s_\alpha)\left[(2C_0+C_{12})(W,f',G^\pm)+(-C_0+C_{12})(G^\pm,f',W)\right]\right.\\
&\qquad\left.-\zeta_{H^\pm}(s_\beta c_\alpha+2c_\beta s_\alpha)\left[(2C_0+C_{12})(W,f',H^\pm)+(-C_0+C_{12})(H^\pm,f',W)\right]\right\}\\
&-2g_Z^2v_fa_f\frac{m_f^2}{v}\zeta_{G^0}c_\alpha\left[(2C_0+C_{12})(Z,f,G^0)+(-C_0+C_{12})(G^0,f,Z)\right], \tag*{(269)}
\end{aligned} = 4 g Z 2 v f a f v m f 2 ζ h ( C 0 + 2 C 12 ) ( f , Z , f ) + 2 g 2 v m f 2 ζ h ( C 0 + 2 C 12 ) ( f ′ , W , f ′ ) − v 3 m f 2 ( m f 2 − m f ′ 2 ) ζ h [ ζ G ± 2 ( C 0 + 2 C 12 ) ( f ′ , G ± , f ′ ) + ζ H ± 2 ( C 0 + 2 C 12 ) ( f ′ , H ± , f ′ ) ] − 4 g Z 2 m Z v f a f c α C 12 ( Z , f , Z ) − 2 g 3 m W ( c β c α − 2 s β s α ) C 12 ( W , f ′ , W ) − v 2 m f 2 − m f ′ 2 { ζ G ± 2 g G + G − h C 12 ( G ± , f ′ , G ± ) + ζ H ± 2 g H + H − h C 12 ( H ± , f ′ , H ± ) + ζ G ± ζ H ± g H + G − h [ C 12 ( G ± , f ′ , H ± ) + C 12 ( H ± , f ′ , G ± ) ] } − 4 g 2 v m f 2 { ζ G ± ( c β c α − 2 s β s α ) [ ( 2 C 0 + C 12 ) ( W , f ′ , G ± ) + ( − C 0 + C 12 ) ( G ± , f ′ , W ) ] − ζ H ± ( s β c α + 2 c β s α ) [ ( 2 C 0 + C 12 ) ( W , f ′ , H ± ) + ( − C 0 + C 12 ) ( H ± , f ′ , W ) ] } − 2 g Z 2 v f a f v m f 2 ζ G 0 c α [ ( 2 C 0 + C 12 ) ( Z , f , G 0 ) + ( − C 0 + C 12 ) ( G 0 , f , Z ) ] , (269) ( m f v ) − 1 16 π 2 Γ h f f ˉ T , 1 P I \left(\frac{m_f}{v}\right)^{-1}16\pi^2\Gamma_{h f\bar f}^{T,1\mathrm{PI}} ( v m f ) − 1 16 π 2 Γ h f f ˉ T , 1 PI = − m f 2 v 2 [ ζ h 2 ( C 11 − C 12 ) ( f , h , f ) + ζ H 2 ( C 11 − C 12 ) ( f , H , f ) − ζ G 0 2 ( C 11 − C 12 ) ( f , G 0 , f ) ] + 2 m f ′ 2 v 2 ζ h [ ζ G ± 2 ( C 11 − C 12 ) ( f ′ , G ± , f ′ ) + ζ H ± 2 ( C 11 − C 12 ) ( f ′ , H ± , f ′ ) ] − g 2 4 { ζ G ± ( c β c α − 2 s β s α ) [ ( 2 C 0 + 2 C 11 − C 12 ) ( W , f ′ , G ± ) + ( C 0 + C 11 − 2 C 12 ) ( G ± , f ′ , W ) ] − ζ H ± ( s β c α + 2 c β s α ) [ ( 2 C 0 + 2 C 11 − C 12 ) ( W , f ′ , H ± ) + ( C 0 + C 11 − 2 C 12 ) ( H ± , f ′ , W ) ] } − g Z 2 8 ζ G 0 c α [ ( 2 C 0 + 2 C 11 − C 12 ) ( Z , f , G 0 ) + ( C 0 + C 11 − 2 C 12 ) ( G 0 , f , Z ) ] , (270) \begin{aligned}
={}&-\frac{m_f^2}{v^2}\left[\zeta_h^2(C_{11}-C_{12})(f,h,f)+\zeta_H^2(C_{11}-C_{12})(f,H,f)-\zeta_{G^0}^2(C_{11}-C_{12})(f,G^0,f)\right]\\
&+\frac{2m_{f'}^2}{v^2}\zeta_h\left[\zeta_{G^\pm}^2(C_{11}-C_{12})(f',G^\pm,f')+\zeta_{H^\pm}^2(C_{11}-C_{12})(f',H^\pm,f')\right]\\
&-\frac{g^2}{4}\left\{\zeta_{G^\pm}(c_\beta c_\alpha-2s_\beta s_\alpha)\left[(2C_0+2C_{11}-C_{12})(W,f',G^\pm)+(C_0+C_{11}-2C_{12})(G^\pm,f',W)\right]\right.\\
&\qquad\left.-\zeta_{H^\pm}(s_\beta c_\alpha+2c_\beta s_\alpha)\left[(2C_0+2C_{11}-C_{12})(W,f',H^\pm)+(C_0+C_{11}-2C_{12})(H^\pm,f',W)\right]\right\}\\
&-\frac{g_Z^2}{8}\zeta_{G^0}c_\alpha\left[(2C_0+2C_{11}-C_{12})(Z,f,G^0)+(C_0+C_{11}-2C_{12})(G^0,f,Z)\right], \tag*{(270)}
\end{aligned} = − v 2 m f 2 [ ζ h 2 ( C 11 − C 12 ) ( f , h , f ) + ζ H 2 ( C 11 − C 12 ) ( f , H , f ) − ζ G 0 2 ( C 11 − C 12 ) ( f , G 0 , f ) ] + v 2 2 m f ′ 2 ζ h [ ζ G ± 2 ( C 11 − C 12 ) ( f ′ , G ± , f ′ ) + ζ H ± 2 ( C 11 − C 12 ) ( f ′ , H ± , f ′ ) ] − 4 g 2 { ζ G ± ( c β c α − 2 s β s α ) [ ( 2 C 0 + 2 C 11 − C 12 ) ( W , f ′ , G ± ) + ( C 0 + C 11 − 2 C 12 ) ( G ± , f ′ , W ) ] − ζ H ± ( s β c α + 2 c β s α ) [ ( 2 C 0 + 2 C 11 − C 12 ) ( W , f ′ , H ± ) + ( C 0 + C 11 − 2 C 12 ) ( H ± , f ′ , W ) ] } − 8 g Z 2 ζ G 0 c α [ ( 2 C 0 + 2 C 11 − C 12 ) ( Z , f , G 0 ) + ( C 0 + C 11 − 2 C 12 ) ( G 0 , f , Z ) ] , (270) ( m f v ) − 1 16 π 2 Γ h f f ˉ P T , 1 P I \left(\frac{m_f}{v}\right)^{-1}16\pi^2\Gamma_{h f\bar f}^{P T,1\mathrm{PI}} ( v m f ) − 1 16 π 2 Γ h f f ˉ P T , 1 PI = g 2 4 { ζ G ± ( c β c α − 2 s β s α ) [ ( 2 C 0 + 2 C 11 − C 12 ) ( W , f ′ , G ± ) − ( C 0 + C 11 − 2 C 12 ) ( G ± , f ′ , W ) ] − ζ H ± ( s β c α + 2 c β s α ) [ ( 2 C 0 + 2 C 11 − C 12 ) ( W , f ′ , H ± ) − ( C 0 + C 11 − 2 C 12 ) ( H ± , f ′ , W ) ] } + 2 g Z 2 v f a f ζ G 0 c α [ ( 2 C 0 + 2 C 11 − C 12 ) ( Z , f , G 0 ) − ( C 0 + C 11 − 2 C 12 ) ( G 0 , f , Z ) ] , (271) \begin{aligned}
={}&\frac{g^2}{4}\left\{\zeta_{G^\pm}(c_\beta c_\alpha-2s_\beta s_\alpha)\left[(2C_0+2C_{11}-C_{12})(W,f',G^\pm)-(C_0+C_{11}-2C_{12})(G^\pm,f',W)\right]\right.\\
&\qquad\left.-\zeta_{H^\pm}(s_\beta c_\alpha+2c_\beta s_\alpha)\left[(2C_0+2C_{11}-C_{12})(W,f',H^\pm)-(C_0+C_{11}-2C_{12})(H^\pm,f',W)\right]\right\}\\
&+2g_Z^2v_fa_f\zeta_{G^0}c_\alpha\left[(2C_0+2C_{11}-C_{12})(Z,f,G^0)-(C_0+C_{11}-2C_{12})(G^0,f,Z)\right], \tag*{(271)}
\end{aligned} = 4 g 2 { ζ G ± ( c β c α − 2 s β s α ) [ ( 2 C 0 + 2 C 11 − C 12 ) ( W , f ′ , G ± ) − ( C 0 + C 11 − 2 C 12 ) ( G ± , f ′ , W ) ] − ζ H ± ( s β c α + 2 c β s α ) [ ( 2 C 0 + 2 C 11 − C 12 ) ( W , f ′ , H ± ) − ( C 0 + C 11 − 2 C 12 ) ( H ± , f ′ , W ) ] } + 2 g Z 2 v f a f ζ G 0 c α [ ( 2 C 0 + 2 C 11 − C 12 ) ( Z , f , G 0 ) − ( C 0 + C 11 − 2 C 12 ) ( G 0 , f , Z ) ] , (271) where the loop functions are defined as [33 , 89 ]
C h f f ˉ F V F ( X , Y , Z ) = [ m f 2 C 0 + p 1 2 ( C 11 + C 21 ) + p 2 2 ( C 12 + C 22 ) + p 1 ⋅ p 2 ( 2 C 23 − C 0 ) + 4 C 24 ] ( X , Y , Z ) − 1 , (272) \begin{aligned}
C_{hf\bar f}^{FVF}(X,Y,Z) \\
&=\left[m_f^2C_0+p_1^2(C_{11}+C_{21})+p_2^2(C_{12}+C_{22})+p_1\cdot p_2(2C_{23}-C_0)+4C_{24}\right](X,Y,Z)-1, \tag*{(272)}
\end{aligned} C h f f ˉ F V F ( X , Y , Z ) = [ m f 2 C 0 + p 1 2 ( C 11 + C 21 ) + p 2 2 ( C 12 + C 22 ) + p 1 ⋅ p 2 ( 2 C 23 − C 0 ) + 4 C 24 ] ( X , Y , Z ) − 1 , (272) C h f f ˉ F S F ( X , Y , Z ) = [ m f 2 C 0 + p 1 2 ( C 11 + C 21 ) + p 2 2 ( C 12 + C 22 ) + 2 p 1 ⋅ p 2 ( C 11 + C 23 ) + 4 C 24 ] ( X , Y , Z ) − 1 2 , (273) \begin{aligned}
C_{hf\bar f}^{FSF}(X,Y,Z) \\
&=\left[m_f^2C_0+p_1^2(C_{11}+C_{21})+p_2^2(C_{12}+C_{22})+2p_1\cdot p_2(C_{11}+C_{23})+4C_{24}\right](X,Y,Z)-\frac{1}{2}, \tag*{(273)}
\end{aligned} C h f f ˉ F S F ( X , Y , Z ) = [ m f 2 C 0 + p 1 2 ( C 11 + C 21 ) + p 2 2 ( C 12 + C 22 ) + 2 p 1 ⋅ p 2 ( C 11 + C 23 ) + 4 C 24 ] ( X , Y , Z ) − 2 1 , (273) C h f f V F S ( X , Y , Z ) = [ p 1 2 ( 2 C 0 + 3 C 11 + C 21 ) + p 2 2 ( 2 C 12 + C 22 ) + 2 p 1 ⋅ p 2 ( C 12 + C 23 ) + 4 C 24 ] ( X , Y , Z ) − 1 2 (274) C_{hff}^{VFS}(X,Y,Z) = \left[p_1^2(2C_0+3C_{11}+C_{21})+p_2^2(2C_{12}+C_{22})+2p_1\cdot p_2(C_{12}+C_{23})+4C_{24}\right](X,Y,Z)-\frac{1}{2} \tag*{(274)} C h f f V F S ( X , Y , Z ) = [ p 1 2 ( 2 C 0 + 3 C 11 + C 21 ) + p 2 2 ( 2 C 12 + C 22 ) + 2 p 1 ⋅ p 2 ( C 12 + C 23 ) + 4 C 24 ] ( X , Y , Z ) − 2 1 (274) C h f f S F V ( X , Y , Z ) = [ p 1 2 ( − C 0 + C 21 ) + p 2 2 ( − C 12 + C 22 ) + 2 p 1 ⋅ p 2 ( C 23 + C 12 − C 0 − C 11 ) + 4 C 24 ] ( X , Y , Z ) − 1 2 (275) C_{hff}^{SFV}(X,Y,Z) = \left[p_1^2(-C_0+C_{21})+p_2^2(-C_{12}+C_{22})+2p_1\cdot p_2(C_{23}+C_{12}-C_0-C_{11})+4C_{24}\right](X,Y,Z)-\frac{1}{2} \tag*{(275)} C h f f S F V ( X , Y , Z ) = [ p 1 2 ( − C 0 + C 21 ) + p 2 2 ( − C 12 + C 22 ) + 2 p 1 ⋅ p 2 ( C 23 + C 12 − C 0 − C 11 ) + 4 C 24 ] ( X , Y , Z ) − 2 1 (275) vertex
The 1PI h g g hgg h g g
Γ h g a g b i ( p 1 2 , p 2 2 , q 2 ) = ζ h Γ h g a g b i ( p 1 2 , p 2 2 , q 2 ) S M ( i = 1 , 2 ) , (276) \Gamma_{h g^{a}g^{b}}^{i}(p_1^2,p_2^2,q^2)=\zeta_h\Gamma_{h g^{a}g^{b}}^{i}(p_1^2,p_2^2,q^2)_{\mathrm{SM}}\qquad(i=1,2), \tag*{(276)} Γ h g a g b i ( p 1 2 , p 2 2 , q 2 ) = ζ h Γ h g a g b i ( p 1 2 , p 2 2 , q 2 ) SM ( i = 1 , 2 ) , (276) where
16 π 2 Γ h g a g b 1 ( p 1 2 , p 2 2 , q 2 ) S M = − ∑ q 4 g s 2 m q 2 v [ 8 C 24 ( q , q , q ) − 2 B 0 ( q 2 ; q , q ) + ( p 1 2 + p 2 2 − q 2 ) C 0 ( q , q , q ) ] δ a b , (277) 16\pi^2\Gamma_{h g^{a}g^{b}}^{1}(p_1^2,p_2^2,q^2)_{\mathrm{SM}}
=-\sum_q\frac{4g_s^2m_q^2}{v}\left[8C_{24}(q,q,q)-2B_0(q^2;q,q)+(p_1^2+p_2^2-q^2)C_0(q,q,q)\right]\delta^{ab}, \tag*{(277)} 16 π 2 Γ h g a g b 1 ( p 1 2 , p 2 2 , q 2 ) SM = − q ∑ v 4 g s 2 m q 2 [ 8 C 24 ( q , q , q ) − 2 B 0 ( q 2 ; q , q ) + ( p 1 2 + p 2 2 − q 2 ) C 0 ( q , q , q ) ] δ ab , (277) 16 π 2 Γ h g a g b 2 ( p 1 2 , p 2 2 , q 2 ) S M = − ∑ q 8 g s 2 m q 2 v q 2 [ C 0 + 4 C 1223 ] ( q , q , q ) δ a b . (278) 16\pi^2\Gamma_{h g^{a}g^{b}}^{2}(p_1^2,p_2^2,q^2)_{\mathrm{SM}}
=-\sum_q\frac{8g_s^2m_q^2}{v}q^2[C_0+4C_{1223}](q,q,q)\delta^{ab}. \tag*{(278)} 16 π 2 Γ h g a g b 2 ( p 1 2 , p 2 2 , q 2 ) SM = − q ∑ v 8 g s 2 m q 2 q 2 [ C 0 + 4 C 1223 ] ( q , q , q ) δ ab . (278) The 1PI h γ γ h\gamma\gamma hγ γ
Γ h γ γ i ( p 1 2 , p 2 2 , q 2 ) F = ζ h Γ h γ γ i ( p 1 2 , p 2 2 , q 2 ) F , S M ( i = 1 , 2 ) , (279) \Gamma_{h\gamma\gamma}^{i}(p_1^2,p_2^2,q^2)_{F}=\zeta_h\Gamma_{h\gamma\gamma}^{i}(p_1^2,p_2^2,q^2)_{F,\mathrm{SM}}\qquad(i=1,2), \tag*{(279)} Γ hγ γ i ( p 1 2 , p 2 2 , q 2 ) F = ζ h Γ hγ γ i ( p 1 2 , p 2 2 , q 2 ) F , SM ( i = 1 , 2 ) , (279) where
16 π 2 Γ h γ γ 1 ( p 1 2 , p 2 2 , q 2 ) F , S M = − ∑ f N c f 4 e 2 Q f 2 m f 2 v [ 8 C 24 ( f , f , f ) − 2 B 0 ( q 2 ; f , f ) + ( p 1 2 + p 2 2 − q 2 ) C 0 ( f , f , f ) ] , (280) 16\pi^2\Gamma_{h\gamma\gamma}^{1}(p_1^2,p_2^2,q^2)_{F,\mathrm{SM}}
=-\sum_fN_c^f\frac{4e^2Q_f^2m_f^2}{v}\left[8C_{24}(f,f,f)-2B_0(q^2;f,f)+(p_1^2+p_2^2-q^2)C_0(f,f,f)\right], \tag*{(280)} 16 π 2 Γ hγ γ 1 ( p 1 2 , p 2 2 , q 2 ) F , SM = − f ∑ N c f v 4 e 2 Q f 2 m f 2 [ 8 C 24 ( f , f , f ) − 2 B 0 ( q 2 ; f , f ) + ( p 1 2 + p 2 2 − q 2 ) C 0 ( f , f , f ) ] , (280) 16 π 2 Γ h γ γ 2 ( p 1 2 , p 2 2 , q 2 ) F , S M = − ∑ f N c f 8 e 2 Q f 2 m f 2 v [ C 0 + 4 C 1223 ] ( f , f , f ) . (281) 16\pi^2\Gamma_{h\gamma\gamma}^{2}(p_1^2,p_2^2,q^2)_{F,\mathrm{SM}}
=-\sum_fN_c^f\frac{8e^2Q_f^2m_f^2}{v}[C_0+4C_{1223}](f,f,f). \tag*{(281)} 16 π 2 Γ hγ γ 2 ( p 1 2 , p 2 2 , q 2 ) F , SM = − f ∑ N c f v 8 e 2 Q f 2 m f 2 [ C 0 + 4 C 1223 ] ( f , f , f ) . (281) The SM boson and charged scalar contributions are obtained as
16 π 2 Γ h γ γ 1 ( p 1 2 , p 2 2 , q 2 ) V = 2 e 2 m W 2 v ( c β c α − 2 s β s α ) × { ( 5 p 1 2 + 5 p 2 2 − 7 q 2 ) C 0 ( W , W , W ) − 6 [ B 0 ( q 2 ; W , W ) − 4 C 24 ( W , W , W ) ] − m h 2 m W 2 [ B 0 ( q 2 ; W , W ) − 4 C 24 ( W , W , W ) ] − m h 2 C 0 ( W , W , W ) } , (282) \begin{aligned}
16\pi^2\Gamma_{h\gamma\gamma}^{1}(p_1^2,p_2^2,q^2)_{V}
={}&\frac{2e^2m_W^2}{v}(c_\beta c_\alpha-2s_\beta s_\alpha)\\
&\times\left\{(5p_1^2+5p_2^2-7q^2)C_0(W,W,W)-6[B_0(q^2;W,W)-4C_{24}(W,W,W)]\right.\\
&\left.\quad-\frac{m_h^2}{m_W^2}[B_0(q^2;W,W)-4C_{24}(W,W,W)]-m_h^2C_0(W,W,W)\right\}, \tag*{(282)}
\end{aligned} 16 π 2 Γ hγ γ 1 ( p 1 2 , p 2 2 , q 2 ) V = v 2 e 2 m W 2 ( c β c α − 2 s β s α ) × { ( 5 p 1 2 + 5 p 2 2 − 7 q 2 ) C 0 ( W , W , W ) − 6 [ B 0 ( q 2 ; W , W ) − 4 C 24 ( W , W , W )] − m W 2 m h 2 [ B 0 ( q 2 ; W , W ) − 4 C 24 ( W , W , W )] − m h 2 C 0 ( W , W , W ) } , (282) 16 π 2 Γ h γ γ 1 ( p 1 2 , p 2 2 , q 2 ) V = 8 e 2 m W 2 v ( c β c α − 2 s β s α ) [ 4 C 0 + 6 C 1223 + m h 2 m W 2 C 1223 ] ( W , W , W ) , (283) 16\pi^{2}\Gamma^{1}_{h\gamma\gamma}(p_{1}^{2},p_{2}^{2},q^{2})_{V}
= \frac{8e^{2}m_{W}^{2}}{v}(c_{\beta}c_{\alpha}-2s_{\beta}s_{\alpha})\left[4C_{0}+6C_{1223}+\frac{m_{h}^{2}}{m_{W}^{2}}C_{1223}\right](W,W,W), \tag*{(283)} 16 π 2 Γ hγ γ 1 ( p 1 2 , p 2 2 , q 2 ) V = v 8 e 2 m W 2 ( c β c α − 2 s β s α ) [ 4 C 0 + 6 C 1223 + m W 2 m h 2 C 1223 ] ( W , W , W ) , (283) 16 π 2 Γ h γ γ 1 ( p 1 2 , p 2 2 , q 2 ) S = − 2 e 2 g H + H − h [ 4 C 24 ( H ± , H ± , H ± ) − B 0 ( q 2 ; H ± , H ± ) ] , (284) 16\pi^{2}\Gamma^{1}_{h\gamma\gamma}(p_{1}^{2},p_{2}^{2},q^{2})_{S}=-2e^{2}g_{H^{+}H^{-}h}\left[4C_{24}(H^{\pm},H^{\pm},H^{\pm})-B_{0}(q^{2};H^{\pm},H^{\pm})\right], \tag*{(284)} 16 π 2 Γ hγ γ 1 ( p 1 2 , p 2 2 , q 2 ) S = − 2 e 2 g H + H − h [ 4 C 24 ( H ± , H ± , H ± ) − B 0 ( q 2 ; H ± , H ± ) ] , (284) 16 π 2 Γ h γ γ 2 ( p 1 2 , p 2 2 , q 2 ) S = − 8 e 2 g H + H − h C 1223 ( H ± , H ± , H ± ) . (285) 16\pi^{2}\Gamma^{2}_{h\gamma\gamma}(p_{1}^{2},p_{2}^{2},q^{2})_{S}=-8e^{2}g_{H^{+}H^{-}h}C_{1223}(H^{\pm},H^{\pm},H^{\pm}). \tag*{(285)} 16 π 2 Γ hγ γ 2 ( p 1 2 , p 2 2 , q 2 ) S = − 8 e 2 g H + H − h C 1223 ( H ± , H ± , H ± ) . (285) The fermion contributions to the 1PI h Z γ hZ\gamma h Z γ
Γ h Z γ i ( p 1 2 , p 2 2 , q 2 ) F = ζ h Γ h Z γ i ( p 1 2 , p 2 2 , q 2 ) F , S M ( i = 1 , 2 ) , (286) \Gamma^{i}_{hZ\gamma}(p_{1}^{2},p_{2}^{2},q^{2})_{F}=\zeta_{h}\Gamma^{i}_{hZ\gamma}(p_{1}^{2},p_{2}^{2},q^{2})_{F,\mathrm{SM}}\quad(i=1,2), \tag*{(286)} Γ h Z γ i ( p 1 2 , p 2 2 , q 2 ) F = ζ h Γ h Z γ i ( p 1 2 , p 2 2 , q 2 ) F , SM ( i = 1 , 2 ) , (286) where
16 π 2 Γ h Z γ 1 ( p 1 2 , p 2 2 , q 2 ) F , S M = − ∑ f N c f v f Q f 4 e g Z m f 2 v [ 8 C 24 ( f , f , f ) − 2 B 0 ( q 2 ; f , f ) + ( p 1 2 + p 2 2 − q 2 ) C 0 ( f , f , f ) ] , (287) 16\pi^{2}\Gamma^{1}_{hZ\gamma}(p_{1}^{2},p_{2}^{2},q^{2})_{F,\mathrm{SM}}
=-\sum_{f}N_{c}^{f}v_{f}Q_{f}\frac{4egZm_{f}^{2}}{v}\left[8C_{24}(f,f,f)-2B_{0}(q^{2};f,f)+(p_{1}^{2}+p_{2}^{2}-q^{2})C_{0}(f,f,f)\right], \tag*{(287)} 16 π 2 Γ h Z γ 1 ( p 1 2 , p 2 2 , q 2 ) F , SM = − f ∑ N c f v f Q f v 4 e g Z m f 2 [ 8 C 24 ( f , f , f ) − 2 B 0 ( q 2 ; f , f ) + ( p 1 2 + p 2 2 − q 2 ) C 0 ( f , f , f ) ] , (287) 16 π 2 Γ h Z γ 2 ( p 1 2 , p 2 2 , q 2 ) F , S M = − ∑ f N c f v f Q f 8 e g Z m f 2 v [ C 0 + 4 C 1223 ] ( f , f , f ) . (288) 16\pi^{2}\Gamma^{2}_{hZ\gamma}(p_{1}^{2},p_{2}^{2},q^{2})_{F,\mathrm{SM}}
=-\sum_{f}N_{c}^{f}v_{f}Q_{f}\frac{8egZm_{f}^{2}}{v}\left[C_{0}+4C_{1223}\right](f,f,f). \tag*{(288)} 16 π 2 Γ h Z γ 2 ( p 1 2 , p 2 2 , q 2 ) F , SM = − f ∑ N c f v f Q f v 8 e g Z m f 2 [ C 0 + 4 C 1223 ] ( f , f , f ) . (288) The bosonic contributions are obtained as
16 π 2 Γ h Z γ 1 ( p 1 2 , p 2 2 , q 2 ) B = e g 2 m W c W ( c α c β − 2 s α s β ) [ 2 C h V V 1 V V V ( W , W , W ) − 6 B 0 ( q 2 ; W , W ) + 4 − 2 C 24 ( c ± , c ± , c ± ) ] + e g g Z m W ( s W 2 − s β 2 ) ( c α c β − 2 s α s β ) × [ C h V V 1 S V V ( G ± , W , W ) + 2 m W 2 C 0 ( W , G ± , W ) − 2 C 24 ( W , G ± , G ± ) + B 0 ( p 1 2 ; G ± , W ) ] − e g 2 m W c W ( c α c β − 2 s α s β ) C h V V 1 V V S ( W , W , G ± ) − 2 e g Z m W 2 ( s W 2 − s β 2 ) g G + G − h C 0 ( G ± , W , G ± ) + e g g Z m W ( c 2 W + s β 2 ) ( c α c β − 2 s α s β ) C 24 ( G ± , G ± , W ) + e g g Z m W ( c α c β s W 2 + 2 s α s β c W 2 ) B 0 ( p 2 2 ; G ± , W ) − 4 e g Z ( c 2 W + s β 2 ) g G + G − h [ C 24 ( G ± , G ± , G ± ) − 1 4 B 0 ( q 2 ; G ± , G ± ) ] + e g g Z m W s β c β ( c α s β + 2 s α c β ) [ C h V V 1 S V V ( H ± , W , W ) − 2 C 24 ( W , H ± , H ± ) ] + 2 e g Z m W 2 s β c β g H + G − h C 0 ( H ± , W , G ± ) − e g g Z m W s β c β ( c α s β + 2 s α c β ) [ C 24 ( H ± , G ± , W ) − B 0 ( p 1 2 ; H ± , W ) ] − 4 e g Z ( c 2 W + c β 2 ) g H + H − h [ C 24 ( H ± , H ± , H ± ) − 1 4 B 0 ( q 2 ; H ± , H ± ) ] − 4 e g Z s β c β g H + G − h [ C 24 ( H ± , G ± , G ± ) + C 24 ( G ± , H ± , H ± ) − 1 2 B 0 ( q 2 ; H ± , G ± ) ] , (289) \begin{aligned}
16\pi^{2}\Gamma^{1}_{hZ\gamma}(p_{1}^{2},p_{2}^{2},q^{2})_{B}
={}&eg^{2}m_{W}c_{W}(c_{\alpha}c_{\beta}-2s_{\alpha}s_{\beta})\left[2C^{VVV}_{hVV1}(W,W,W)-6B_{0}(q^{2};W,W)+4-2C_{24}(c^{\pm},c^{\pm},c^{\pm})\right]\\
&+egg_{Z}m_{W}(s_{W}^{2}-s_{\beta}^{2})(c_{\alpha}c_{\beta}-2s_{\alpha}s_{\beta})\\
&\quad\times\left[C^{SVV}_{hVV1}(G^{\pm},W,W)+2m_{W}^{2}C_{0}(W,G^{\pm},W)-2C_{24}(W,G^{\pm},G^{\pm})+B_{0}(p_{1}^{2};G^{\pm},W)\right]\\
&-eg^{2}m_{W}c_{W}(c_{\alpha}c_{\beta}-2s_{\alpha}s_{\beta})C^{VV S}_{hVV1}(W,W,G^{\pm})\\
&-2eg_{Z}m_{W}^{2}(s_{W}^{2}-s_{\beta}^{2})g_{G^{+}G^{-}h}C_{0}(G^{\pm},W,G^{\pm})\\
&+egg_{Z}m_{W}(c_{2W}+s_{\beta}^{2})(c_{\alpha}c_{\beta}-2s_{\alpha}s_{\beta})C_{24}(G^{\pm},G^{\pm},W)\\
&+egg_{Z}m_{W}(c_{\alpha}c_{\beta}s_{W}^{2}+2s_{\alpha}s_{\beta}c_{W}^{2})B_{0}(p_{2}^{2};G^{\pm},W)\\
&-4eg_{Z}(c_{2W}+s_{\beta}^{2})g_{G^{+}G^{-}h}\left[C_{24}(G^{\pm},G^{\pm},G^{\pm})-\frac{1}{4}B_{0}(q^{2};G^{\pm},G^{\pm})\right]\\
&+egg_{Z}m_{W}s_{\beta}c_{\beta}(c_{\alpha}s_{\beta}+2s_{\alpha}c_{\beta})\left[C^{SVV}_{hVV1}(H^{\pm},W,W)-2C_{24}(W,H^{\pm},H^{\pm})\right]\\
&+2eg_{Z}m_{W}^{2}s_{\beta}c_{\beta}g_{H^{+}G^{-}h}C_{0}(H^{\pm},W,G^{\pm})\\
&-egg_{Z}m_{W}s_{\beta}c_{\beta}(c_{\alpha}s_{\beta}+2s_{\alpha}c_{\beta})\left[C_{24}(H^{\pm},G^{\pm},W)-B_{0}(p_{1}^{2};H^{\pm},W)\right]\\
&-4eg_{Z}(c_{2W}+c_{\beta}^{2})g_{H^{+}H^{-}h}\left[C_{24}(H^{\pm},H^{\pm},H^{\pm})-\frac{1}{4}B_{0}(q^{2};H^{\pm},H^{\pm})\right]\\
&-4eg_{Z}s_{\beta}c_{\beta}g_{H^{+}G^{-}h}\left[C_{24}(H^{\pm},G^{\pm},G^{\pm})+C_{24}(G^{\pm},H^{\pm},H^{\pm})-\frac{1}{2}B_{0}(q^{2};H^{\pm},G^{\pm})\right], \tag*{(289)}
\end{aligned} 16 π 2 Γ h Z γ 1 ( p 1 2 , p 2 2 , q 2 ) B = e g 2 m W c W ( c α c β − 2 s α s β ) [ 2 C hV V 1 V V V ( W , W , W ) − 6 B 0 ( q 2 ; W , W ) + 4 − 2 C 24 ( c ± , c ± , c ± ) ] + e g g Z m W ( s W 2 − s β 2 ) ( c α c β − 2 s α s β ) × [ C hV V 1 S V V ( G ± , W , W ) + 2 m W 2 C 0 ( W , G ± , W ) − 2 C 24 ( W , G ± , G ± ) + B 0 ( p 1 2 ; G ± , W ) ] − e g 2 m W c W ( c α c β − 2 s α s β ) C hV V 1 V V S ( W , W , G ± ) − 2 e g Z m W 2 ( s W 2 − s β 2 ) g G + G − h C 0 ( G ± , W , G ± ) + e g g Z m W ( c 2 W + s β 2 ) ( c α c β − 2 s α s β ) C 24 ( G ± , G ± , W ) + e g g Z m W ( c α c β s W 2 + 2 s α s β c W 2 ) B 0 ( p 2 2 ; G ± , W ) − 4 e g Z ( c 2 W + s β 2 ) g G + G − h [ C 24 ( G ± , G ± , G ± ) − 4 1 B 0 ( q 2 ; G ± , G ± ) ] + e g g Z m W s β c β ( c α s β + 2 s α c β ) [ C hV V 1 S V V ( H ± , W , W ) − 2 C 24 ( W , H ± , H ± ) ] + 2 e g Z m W 2 s β c β g H + G − h C 0 ( H ± , W , G ± ) − e g g Z m W s β c β ( c α s β + 2 s α c β ) [ C 24 ( H ± , G ± , W ) − B 0 ( p 1 2 ; H ± , W ) ] − 4 e g Z ( c 2 W + c β 2 ) g H + H − h [ C 24 ( H ± , H ± , H ± ) − 4 1 B 0 ( q 2 ; H ± , H ± ) ] − 4 e g Z s β c β g H + G − h [ C 24 ( H ± , G ± , G ± ) + C 24 ( G ± , H ± , H ± ) − 2 1 B 0 ( q 2 ; H ± , G ± ) ] , (289) 16 π 2 Γ h Z γ 2 ( p 1 2 , p 2 2 , q 2 ) B (290) 16\pi^{2}\Gamma^{2}_{hZ\gamma}(p_{1}^{2},p_{2}^{2},q^{2})_{B} \tag*{(290)} 16 π 2 Γ h Z γ 2 ( p 1 2 , p 2 2 , q 2 ) B (290) = e g 2 m W c W ( c α c β − 2 s α s β ) [ 2 C h V V 2 V V V ( W , W , W ) − 2 C 1223 ( c ± , c ± , c ± ) − C h V V 2 V V S ( W , W , G ± ) ] + e g g Z m W ( s W 2 − s β 2 ) ( c α c β − 2 s α s β ) [ C h V V 2 S V V ( G ± , W , W ) − 2 C h V V 2 V S S ( W , G ± , G ± ) ] + e g g Z m W ( c 2 W + s β 2 ) ( c α c β − 2 s α s β ) C h V V 2 S S V ( G ± , G ± , W ) − 4 e g Z ( c 2 W + s β 2 ) g G + G − h C 1223 ( G ± , G ± , G ± ) + e g g Z m W s β c β ( c α s β + 2 s α c β ) × [ C h V V 2 S V V ( H ± , W , W ) − C h V V 2 S S V ( H ± , G ± , W ) − 2 C h V V 2 V S S ( W , H ± , H ± ) ] − 4 e g Z ( c 2 W + c β 2 ) g H + H − h C 1223 ( H ± , H ± , H ± ) − 4 e g Z s β c β g H + G − h [ C 1223 ( H ± , G ± , G ± ) + C 1223 ( G ± , H ± , H ± ) ] . (291) \begin{aligned}
={}&eg^{2}m_{W}c_{W}(c_{\alpha}c_{\beta}-2s_{\alpha}s_{\beta})\left[2C_{hVV2}^{VVV}(W,W,W)-2C_{1223}(c^{\pm},c^{\pm},c^{\pm})-C_{hVV2}^{VVS}(W,W,G^{\pm})\right]\\
&+egg_{Z}m_{W}(s_{W}^{2}-s_{\beta}^{2})(c_{\alpha}c_{\beta}-2s_{\alpha}s_{\beta})\left[C_{hVV2}^{SVV}(G^{\pm},W,W)-2C_{hVV2}^{VSS}(W,G^{\pm},G^{\pm})\right]\\
&+egg_{Z}m_{W}(c_{2W}+s_{\beta}^{2})(c_{\alpha}c_{\beta}-2s_{\alpha}s_{\beta})C_{hVV2}^{SSV}(G^{\pm},G^{\pm},W)\\
&-4eg_{Z}(c_{2W}+s_{\beta}^{2})g_{G^{+}G^{-}h}C_{1223}(G^{\pm},G^{\pm},G^{\pm})\\
&+egg_{Z}m_{W}s_{\beta}c_{\beta}(c_{\alpha}s_{\beta}+2s_{\alpha}c_{\beta})\\
&\quad\times\left[C_{hVV2}^{SVV}(H^{\pm},W,W)-C_{hVV2}^{SSV}(H^{\pm},G^{\pm},W)-2C_{hVV2}^{VSS}(W,H^{\pm},H^{\pm})\right]\\
&-4eg_{Z}(c_{2W}+c_{\beta}^{2})g_{H^{+}H^{-}h}C_{1223}(H^{\pm},H^{\pm},H^{\pm})\\
&-4eg_{Z}s_{\beta}c_{\beta}g_{H^{+}G^{-}h}\left[C_{1223}(H^{\pm},G^{\pm},G^{\pm})+C_{1223}(G^{\pm},H^{\pm},H^{\pm})\right]. \tag*{(291)}
\end{aligned} = e g 2 m W c W ( c α c β − 2 s α s β ) [ 2 C hV V 2 V V V ( W , W , W ) − 2 C 1223 ( c ± , c ± , c ± ) − C hV V 2 V V S ( W , W , G ± ) ] + e g g Z m W ( s W 2 − s β 2 ) ( c α c β − 2 s α s β ) [ C hV V 2 S V V ( G ± , W , W ) − 2 C hV V 2 V S S ( W , G ± , G ± ) ] + e g g Z m W ( c 2 W + s β 2 ) ( c α c β − 2 s α s β ) C hV V 2 S S V ( G ± , G ± , W ) − 4 e g Z ( c 2 W + s β 2 ) g G + G − h C 1223 ( G ± , G ± , G ± ) + e g g Z m W s β c β ( c α s β + 2 s α c β ) × [ C hV V 2 S V V ( H ± , W , W ) − C hV V 2 S S V ( H ± , G ± , W ) − 2 C hV V 2 V S S ( W , H ± , H ± ) ] − 4 e g Z ( c 2 W + c β 2 ) g H + H − h C 1223 ( H ± , H ± , H ± ) − 4 e g Z s β c β g H + G − h [ C 1223 ( H ± , G ± , G ± ) + C 1223 ( G ± , H ± , H ± ) ] . (291) Four-point functions
Box diagrams for h → Z f f ˉ h \to Zf\bar{f} h → Z f f ˉ
There are six SM-like diagrams and two diagrams with the charged Higgs boson. The Feynman diagrams are given in [89 ] , and we use the same diagram numbering as in that reference. For the diagrams without a Z Z Z
( 16 π 2 ) 2 Re [ ∑ M t r e e ∗ M B Z i ] = 4 ( v f + a f ) C t r e e B Z i C B Z i B B Z i . (292) (16\pi^{2})2\operatorname{Re}\left[\sum\mathcal{M}_{\mathrm{tree}}^{*}\mathcal{M}_{BZ i}\right]=4(v_{f}+a_{f})C_{\mathrm{tree}}^{BZ i}C^{BZ i}B^{BZ i}. \tag*{(292)} ( 16 π 2 ) 2 Re [ ∑ M tree ∗ M B Z i ] = 4 ( v f + a f ) C tree B Z i C B Z i B B Z i . (292) On the other hand, for the Z Z Z
( 16 π 2 ) 2 Re [ ∑ M t r e e ∗ M B Z 6 ] = 8 ( v f 4 + 6 v f 2 a f 2 + a f 4 ) C t r e e B Z 6 C B Z 6 B B Z 6 . (293) (16\pi^{2})2\operatorname{Re}\left[\sum\mathcal{M}_{\mathrm{tree}}^{*}\mathcal{M}_{BZ6}\right]=8(v_{f}^{4}+6v_{f}^{2}a_{f}^{2}+a_{f}^{4})C_{\mathrm{tree}}^{BZ6}C^{BZ6}B^{BZ6}. \tag*{(293)} ( 16 π 2 ) 2 Re [ ∑ M tree ∗ M B Z 6 ] = 8 ( v f 4 + 6 v f 2 a f 2 + a f 4 ) C tree B Z 6 C B Z 6 B B Z 6 . (293) The tree-level factor is obtained as C t r e e Z = g Z Γ h Z Z 1 , t r e e / ( s − m Z 2 ) C_{\mathrm{tree}}^{Z}=g_{Z}\Gamma_{hZZ}^{1,\mathrm{tree}}/(s-m_{Z}^{2}) C tree Z = g Z Γ h Z Z 1 , tree / ( s − m Z 2 ) B B Z i B^{BZ i} B B Z i [89 ] . The one-loop factor C B Z i C^{BZ i} C B Z i
C B Z 1 = C B Z 2 = − 2 a f g 4 m W c W ( c β c α − 2 s β s α ) , (294) C^{BZ1}=C^{BZ2}=-2a_{f}g^{4}m_{W}c_{W}(c_{\beta}c_{\alpha}-2s_{\beta}s_{\alpha}), \tag*{(294)} C B Z 1 = C B Z 2 = − 2 a f g 4 m W c W ( c β c α − 2 s β s α ) , (294) C B Z 3 = C B Z 4 = a f g 3 g Z m W ( s W 2 − s β 2 ) ( c β c α − 2 s β s α ) , (295) C^{BZ3}=C^{BZ4}=a_{f}g^{3}g_{Z}m_{W}(s_{W}^{2}-s_{\beta}^{2})(c_{\beta}c_{\alpha}-2s_{\beta}s_{\alpha}), \tag*{(295)} C B Z 3 = C B Z 4 = a f g 3 g Z m W ( s W 2 − s β 2 ) ( c β c α − 2 s β s α ) , (295) C B Z 5 = − ( v f ′ + a f ′ ) g 3 g Z m W ( c β c α − 2 s β s α ) , (296) C^{BZ5}=-(v_{f'}+a_{f'})g^{3}g_{Z}m_{W}(c_{\beta}c_{\alpha}-2s_{\beta}s_{\alpha}), \tag*{(296)} C B Z 5 = − ( v f ′ + a f ′ ) g 3 g Z m W ( c β c α − 2 s β s α ) , (296) C B Z 6 = − 2 g Z 4 m Z c α , (297) C^{BZ6}=-2g_{Z}^{4}m_{Z}c_{\alpha}, \tag*{(297)} C B Z 6 = − 2 g Z 4 m Z c α , (297) C B Z 7 = C B Z 8 = a f g 3 g Z m W s β c β ( s β c α + 2 c β s α ) . (298) C^{BZ7}=C^{BZ8}=a_{f}g^{3}g_{Z}m_{W}s_{\beta}c_{\beta}(s_{\beta}c_{\alpha}+2c_{\beta}s_{\alpha}). \tag*{(298)} C B Z 7 = C B Z 8 = a f g 3 g Z m W s β c β ( s β c α + 2 c β s α ) . (298) Box diagrams for h → W f f ˉ ′ h \to Wf\bar{f}' h → W f f ˉ ′
There are eleven SM-like diagrams and two diagrams with the charged Higgs boson. The Feynman diagrams are given in [89 ] , and we use the same diagram numbering as in that reference. The NLO contributions are obtained as
( 16 π 2 ) 2 Re [ ∑ M t r e e ∗ M B W i ] = 4 C t r e e W C B W i Re [ B B W i ] . (299) (16\pi^{2})2\operatorname{Re}\left[\sum\mathcal{M}_{\mathrm{tree}}^{*}\mathcal{M}_{BW i}\right]=4C_{\mathrm{tree}}^{W}C^{BW i}\operatorname{Re}\left[B^{BW i}\right]. \tag*{(299)} ( 16 π 2 ) 2 Re [ ∑ M tree ∗ M B W i ] = 4 C tree W C B W i Re [ B B W i ] . (299) The tree-level factor is given by C t r e e W = g W Γ h W W 1 , t r e e / ( s − m W 2 ) C_{\mathrm{tree}}^{W}=g_{W}\Gamma_{hWW}^{1,\mathrm{tree}}/(s-m_{W}^{2}) C tree W = g W Γ hW W 1 , tree / ( s − m W 2 ) B B W i B^{BW i} B B W i [89 ] . The one-loop factors C B W i C^{BW i} C B W i
C B W 1 = − 4 a f ′ ( v f ′ + a f ′ ) g 2 2 g Z 2 2 c W m Z c α (300) C^{BW1}=-4a_{f'}(v_{f'}+a_{f'})\frac{g_2^2g_Z^2}{\sqrt{2}}c_Wm_Zc_\alpha \tag*{(300)} C B W 1 = − 4 a f ′ ( v f ′ + a f ′ ) 2 g 2 2 g Z 2 c W m Z c α (300) C B W 3 = − 4 a f ( v f + a f ) g 3 g Z 2 c W m W ( c β c α − 2 s β s α ) (301) C^{BW3}=-4a_f(v_f+a_f)\frac{g^3g_Z}{\sqrt{2}}c_Wm_W(c_\beta c_\alpha-2s_\beta s_\alpha) \tag*{(301)} C B W 3 = − 4 a f ( v f + a f ) 2 g 3 g Z c W m W ( c β c α − 2 s β s α ) (301) C B W 5 = − 4 a f e Q f g 3 2 s W m W ( c β c α − 2 s β s α ) (302) C^{BW5}=-4a_feQ_f\frac{g^3}{\sqrt{2}}s_Wm_W(c_\beta c_\alpha-2s_\beta s_\alpha) \tag*{(302)} C B W 5 = − 4 a f e Q f 2 g 3 s W m W ( c β c α − 2 s β s α ) (302) C B W 7 = − 2 a f ( v f + a f ) g 2 g Z 2 2 ( s W 2 − s β 2 ) m W ( c β c α − 2 s β s α ) (303) C^{BW7}=-2a_f(v_f+a_f)\frac{g^2g_Z^2}{\sqrt{2}}(s_W^2-s_\beta^2)m_W(c_\beta c_\alpha-2s_\beta s_\alpha) \tag*{(303)} C B W 7 = − 2 a f ( v f + a f ) 2 g 2 g Z 2 ( s W 2 − s β 2 ) m W ( c β c α − 2 s β s α ) (303) C B W 9 = 2 a f e 2 Q f g 2 2 m W ( c β c α − 2 s β s α ) (304) C^{BW9}=2a_fe^2Q_f\frac{g^2}{\sqrt{2}}m_W(c_\beta c_\alpha-2s_\beta s_\alpha) \tag*{(304)} C B W 9 = 2 a f e 2 Q f 2 g 2 m W ( c β c α − 2 s β s α ) (304) C B W 10 = 2 a f ′ e 2 Q f ′ g 2 2 m W ( c β c α − 2 s β s α ) (305) C^{BW10}=2a_{f'}e^2Q_{f'}\frac{g^2}{\sqrt{2}}m_W(c_\beta c_\alpha-2s_\beta s_\alpha) \tag*{(305)} C B W 10 = 2 a f ′ e 2 Q f ′ 2 g 2 m W ( c β c α − 2 s β s α ) (305) C B W 11 = − 2 ( v f + a f ) ( v f ′ + a f ′ ) g Z 3 g 2 m Z c α (306) C^{BW11}=-2(v_f+a_f)(v_{f'}+a_{f'})\frac{g_Z^3g}{\sqrt{2}}m_Zc_\alpha \tag*{(306)} C B W 11 = − 2 ( v f + a f ) ( v f ′ + a f ′ ) 2 g Z 3 g m Z c α (306) C B W 12 = − 2 a f ( v f + a f ) g 2 g Z 2 2 s β c β m W ( c α s β + 2 s α c β ) (307) C^{BW12}=-2a_f(v_f+a_f)\frac{g^2g_Z^2}{\sqrt{2}}s_\beta c_\beta m_W(c_\alpha s_\beta+2s_\alpha c_\beta) \tag*{(307)} C B W 12 = − 2 a f ( v f + a f ) 2 g 2 g Z 2 s β c β m W ( c α s β + 2 s α c β ) (307) The one-loop factors C B W 2 C^{BW2} C B W 2 C B W 4 C^{BW4} C B W 4 C B W 6 C^{BW6} C B W 6 C B W 8 C^{BW8} C B W 8 C B W 13 C^{BW13} C B W 13 C B W 1 C^{BW1} C B W 1 C B W 3 C^{BW3} C B W 3 C B W 5 C^{BW5} C B W 5 C B W 7 C^{BW7} C B W 7 C B W 12 C^{BW12} C B W 12 f f f f ′ f' f ′
Gauge dependence of scalar self-energies
In this appendix, we briefly review the derivation of the pinch terms for the scalar self-energies, which are given in Eqs. (74) and (75) . The pinch terms for the self-energies of the CP-even scalar bosons are calculated in the same way as in the CHTM [89 ] , except that contributions from loops involving the CP-odd scalar A A A
Cancellation of the gauge dependence
In this subsection, we describe the cancellation of the gauge dependence associated with the gauge-fixing parameters ξ W \xi_W ξ W ξ Z \xi_Z ξ Z ξ γ \xi_\gamma ξ γ [66 ] , we omit its detailed discussion. In order to analytically confirm that the gauge dependence of the H + H − H^+H^- H + H − H + G − H^+G^- H + G − O \mathcal{O} O
O ∣ G . D . ≡ O ∣ R ξ − O ∣ ξ V = 1 , (308) \left.\mathcal{O}\right|_{\mathrm{G.D.}}\equiv\left.\mathcal{O}\right|_{R_\xi}-\left.\mathcal{O}\right|_{\xi_V=1}, \tag*{(308)} O ∣ G.D. ≡ O ∣ R ξ − O ∣ ξ V = 1 , (308) where the first and second terms on the right-hand side are calculated in the R ξ R_\xi R ξ Figure 5 . The gauge-dependent parts of the H + H − H^{+}H^{-} H + H − H + G − H^{+}G^{-} H + G −
Figure 5. Feynman diagrams contributing to the gauge-dependent parts of the charged scalar self-energies.
( ∑ i = 1 11 Π H + H − W − i + ∑ i = 1 8 Π H + H − Z − i ) ∣ G . D . = p 2 − m H ± 2 64 π 2 [ − 2 g 2 ( 1 − ξ W ) ( 1 + c β 2 ) B 0 ( 0 ; W , G + ) − g Z 2 ( 1 − ξ Z ) ( c H + H − Z 2 + s β 2 c β 2 ) B 0 ( 0 ; Z , G 0 ) + g 2 ( 1 − ξ W ) ∑ S = h , H , G 0 c H ± S W ∓ 2 { p 2 + m H ± 2 − 2 δ S m S 2 } C 0 ( p 2 ; W , G + , S ) + g Z 2 ( 1 − ξ Z ) ∑ ( ϕ , X ) = ( H + , H + ) , ( G + , W ) c H ± ϕ − Z 2 { p 2 + m H ± 2 − 2 δ ϕ m H ± 2 } C 0 ( p 2 ; Z , G 0 , X ) ] , (309) \left(\sum_{i=1}^{11}\Pi_{H^{+}H^{-}}^{W-i}+\sum_{i=1}^{8}\Pi_{H^{+}H^{-}}^{Z-i}\right)\bigg|_{\mathrm{G.D.}}
= \frac{p^{2}-m_{H^{\pm}}^{2}}{64\pi^{2}}\left[
-2g^{2}(1-\xi_{W})(1+c_{\beta}^{2})B_{0}(0;W,G^{+})
-g_{Z}^{2}(1-\xi_{Z})(c_{H^{+}H^{-}Z}^{2}+s_{\beta}^{2}c_{\beta}^{2})B_{0}(0;Z,G^{0})
+g^{2}(1-\xi_{W})\sum_{S=h,H,G^{0}}c_{H^{\pm}SW^{\mp}}^{2}\left\{p^{2}+m_{H^{\pm}}^{2}-2\delta_{S}m_{S}^{2}\right\}C_{0}(p^{2};W,G^{+},S)
+g_{Z}^{2}(1-\xi_{Z})\sum_{(\phi,X)=(H^{+},H^{+}),(G^{+},W)}c_{H^{\pm}\phi-Z}^{2}\left\{p^{2}+m_{H^{\pm}}^{2}-2\delta_{\phi}m_{H^{\pm}}^{2}\right\}C_{0}(p^{2};Z,G^{0},X)
\right], \tag*{(309)} ( i = 1 ∑ 11 Π H + H − W − i + i = 1 ∑ 8 Π H + H − Z − i ) G.D. = 64 π 2 p 2 − m H ± 2 − 2 g 2 ( 1 − ξ W ) ( 1 + c β 2 ) B 0 ( 0 ; W , G + ) − g Z 2 ( 1 − ξ Z ) ( c H + H − Z 2 + s β 2 c β 2 ) B 0 ( 0 ; Z , G 0 ) + g 2 ( 1 − ξ W ) S = h , H , G 0 ∑ c H ± S W ∓ 2 { p 2 + m H ± 2 − 2 δ S m S 2 } C 0 ( p 2 ; W , G + , S ) + g Z 2 ( 1 − ξ Z ) ( ϕ , X ) = ( H + , H + ) , ( G + , W ) ∑ c H ± ϕ − Z 2 { p 2 + m H ± 2 − 2 δ ϕ m H ± 2 } C 0 ( p 2 ; Z , G 0 , X ) , (309) ( ∑ i = 1 11 Π H + G − W − i + ∑ i = 1 8 Π H + G − Z − i ) ∣ G . D . = 1 64 π 2 [ − g 2 s β c β ( 1 − ξ W ) { 2 p 2 − m H ± 2 + m W 2 ( 1 + ξ Z ) } B 0 ( 0 ; W , G + ) − g Z 2 s β c β ( 1 − ξ Z ) { ( c 2 W + 1 2 ) ( 2 p 2 − m H ± 2 ) − c W 2 m W 2 ( 1 + ξ W ) } B 0 ( 0 ; Z , G 0 ) + 2 g 2 s β c β m W 2 { ( 1 − ξ W ) − ( 1 − ξ Z ) } B 0 ( p 2 ; W , Z ) + g 2 ( 1 − ξ W ) ∑ S = h , H , G 0 c H ± S W ∓ c G ± S W ∓ { p 4 − ( 2 p 2 − m H ± 2 ) m S 2 δ S } C 0 ( p 2 ; W , G + , S ) + g Z 2 ( 1 − ξ Z ) ∑ ( ϕ , X ) = ( H + , H + ) , ( G + , W ) c H + ϕ − Z c G + ϕ − Z { p 4 − δ ϕ m H ± 2 ( 2 p 2 − m H ± 2 ) } C 0 ( p 2 ; Z , G 0 , X ) − g 2 s β c β m W 2 m Z 2 ( 1 − ξ W ) { ( p 2 − m Z 2 − m G + 2 ) 2 − 4 m Z 2 m G + 2 } C 0 ( p 2 ; W , G + , Z ) + g 2 s β c β m W 2 m Z 2 ( 1 − ξ W ) { ( p 2 − m G + 2 ) 2 − m G 0 4 + 2 p 2 m G 0 2 } C 0 ( p 2 ; W , G + , G 0 ) + g 2 s β c β ( 1 − ξ Z ) { m W 4 + m G + 4 − 2 p 2 ( m W 2 + m G + 2 ) − 2 m W 2 m G 0 2 } C 0 ( p 2 ; Z , G 0 , W ) ] , (310) \left(\sum_{i=1}^{11}\Pi^{W-i}_{H^{+}G^{-}}+\sum_{i=1}^{8}\Pi^{Z-i}_{H^{+}G^{-}}\right)\bigg|_{\mathrm{G.D.}}
\begin{aligned}
={}&\frac{1}{64\pi^{2}}\Bigl[-g^{2}s_{\beta}c_{\beta}(1-\xi_{W})\left\{2p^{2}-m_{H^{\pm}}^{2}+m_{W}^{2}(1+\xi_{Z})\right\}B_{0}(0;W,G^{+})\\
&-g_{Z}^{2}s_{\beta}c_{\beta}(1-\xi_{Z})\left\{\left(c_{2W}+\frac{1}{2}\right)(2p^{2}-m_{H^{\pm}}^{2})-c_{W}^{2}m_{W}^{2}(1+\xi_{W})\right\}B_{0}(0;Z,G^{0})\\
&+2g^{2}s_{\beta}c_{\beta}m_{W}^{2}\left\{(1-\xi_{W})-(1-\xi_{Z})\right\}B_{0}(p^{2};W,Z)\\
&+g^{2}(1-\xi_{W})\sum_{S=h,H,G^{0}}c_{H^{\pm}SW^{\mp}}c_{G^{\pm}SW^{\mp}}
\left\{p^{4}-(2p^{2}-m_{H^{\pm}}^{2})m_{S}^{2}\delta_{S}\right\}C_{0}(p^{2};W,G^{+},S)\\
&+g_{Z}^{2}(1-\xi_{Z})\sum_{(\phi,X)=(H^{+},H^{+}),(G^{+},W)}
c_{H^{+}\phi-Z}c_{G^{+}\phi-Z}\left\{p^{4}-\delta_{\phi}m_{H^{\pm}}^{2}(2p^{2}-m_{H^{\pm}}^{2})\right\}C_{0}(p^{2};Z,G^{0},X)\\
&-g^{2}s_{\beta}c_{\beta}\frac{m_{W}^{2}}{m_{Z}^{2}}(1-\xi_{W})\left\{(p^{2}-m_{Z}^{2}-m_{G^{+}}^{2})^{2}-4m_{Z}^{2}m_{G^{+}}^{2}\right\}C_{0}(p^{2};W,G^{+},Z)\\
&+g^{2}s_{\beta}c_{\beta}\frac{m_{W}^{2}}{m_{Z}^{2}}(1-\xi_{W})\left\{(p^{2}-m_{G^{+}}^{2})^{2}-m_{G^{0}}^{4}+2p^{2}m_{G^{0}}^{2}\right\}C_{0}(p^{2};W,G^{+},G^{0})\\
&+g^{2}s_{\beta}c_{\beta}(1-\xi_{Z})\left\{m_{W}^{4}+m_{G^{+}}^{4}-2p^{2}(m_{W}^{2}+m_{G^{+}}^{2})-2m_{W}^{2}m_{G^{0}}^{2}\right\}C_{0}(p^{2};Z,G^{0},W)\Bigr],
\end{aligned} \tag*{(310)} ( i = 1 ∑ 11 Π H + G − W − i + i = 1 ∑ 8 Π H + G − Z − i ) G.D. = 64 π 2 1 [ − g 2 s β c β ( 1 − ξ W ) { 2 p 2 − m H ± 2 + m W 2 ( 1 + ξ Z ) } B 0 ( 0 ; W , G + ) − g Z 2 s β c β ( 1 − ξ Z ) { ( c 2 W + 2 1 ) ( 2 p 2 − m H ± 2 ) − c W 2 m W 2 ( 1 + ξ W ) } B 0 ( 0 ; Z , G 0 ) + 2 g 2 s β c β m W 2 { ( 1 − ξ W ) − ( 1 − ξ Z ) } B 0 ( p 2 ; W , Z ) + g 2 ( 1 − ξ W ) S = h , H , G 0 ∑ c H ± S W ∓ c G ± S W ∓ { p 4 − ( 2 p 2 − m H ± 2 ) m S 2 δ S } C 0 ( p 2 ; W , G + , S ) + g Z 2 ( 1 − ξ Z ) ( ϕ , X ) = ( H + , H + ) , ( G + , W ) ∑ c H + ϕ − Z c G + ϕ − Z { p 4 − δ ϕ m H ± 2 ( 2 p 2 − m H ± 2 ) } C 0 ( p 2 ; Z , G 0 , X ) − g 2 s β c β m Z 2 m W 2 ( 1 − ξ W ) { ( p 2 − m Z 2 − m G + 2 ) 2 − 4 m Z 2 m G + 2 } C 0 ( p 2 ; W , G + , Z ) + g 2 s β c β m Z 2 m W 2 ( 1 − ξ W ) { ( p 2 − m G + 2 ) 2 − m G 0 4 + 2 p 2 m G 0 2 } C 0 ( p 2 ; W , G + , G 0 ) + g 2 s β c β ( 1 − ξ Z ) { m W 4 + m G + 4 − 2 p 2 ( m W 2 + m G + 2 ) − 2 m W 2 m G 0 2 } C 0 ( p 2 ; Z , G 0 , W ) ] , (310) with δ S = 1 ( 0 ) \delta_{S}=1\ (0) δ S = 1 ( 0 ) S = h , H ( S = G 0 ) S=h,H\ (S=G^{0}) S = h , H ( S = G 0 ) δ ϕ = 1 ( 0 ) \delta_{\phi}=1\ (0) δ ϕ = 1 ( 0 ) ϕ = H + ( ϕ = G + ) \phi=H^{+}\ (\phi=G^{+}) ϕ = H + ( ϕ = G + ) Π G + H − V − i = Π H + G − V − i \Pi^{V-i}_{G^{+}H^{-}}=\Pi^{V-i}_{H^{+}G^{-}} Π G + H − V − i = Π H + G − V − i B 0 ( 0 ; X , Y ) = ( A ( X ) − A ( Y ) ) / ( m X 2 − m Y 2 ) B_{0}(0;X,Y)=(A(X)-A(Y))/(m_{X}^{2}-m_{Y}^{2}) B 0 ( 0 ; X , Y ) = ( A ( X ) − A ( Y )) / ( m X 2 − m Y 2 ) C 0 ( p 2 , X , Y , Z ) ≡ ( B 0 ( p 2 ; X , Z ) − B 0 ( p 2 ; Y , Z ) ) / ( m X 2 − m Y 2 ) C_{0}(p^{2},X,Y,Z)\equiv(B_{0}(p^{2};X,Z)-B_{0}(p^{2};Y,Z))/(m_{X}^{2}-m_{Y}^{2}) C 0 ( p 2 , X , Y , Z ) ≡ ( B 0 ( p 2 ; X , Z ) − B 0 ( p 2 ; Y , Z )) / ( m X 2 − m Y 2 )
To identify the contributions that cancel the gauge dependence of these self-energies, we consider the scattering process u d ˉ → u d ˉ u\bar{d}\to u\bar{d} u d ˉ → u d ˉ M ‾ \overline{\mathcal{M}} M
M = M ‾ ( 2 m u 2 v 2 ) ( d ˉ P R u ) × ( u ˉ P L d ) . (311) \mathcal{M}=\overline{\mathcal{M}}\left(\frac{2m_{u}^{2}}{v^{2}}\right)(\bar{d}P_{R}u)\times(\bar{u}P_{L}d). \tag*{(311)} M = M ( v 2 2 m u 2 ) ( d ˉ P R u ) × ( u ˉ P L d ) . (311) For the self-energies of the charged scalars, the pinch contributions can be extracted from the reduced amplitude which arises from diagrams shown in Figure 6 .
Figure 6. Feynman diagrams contributing to the pinch terms extracted from the u d ˉ → u d ˉ u\bar{d} \to u\bar{d} u d ˉ → u d ˉ
In the processes in which one or more vector bosons are attached to a fermion line, the pinch terms are calculated by canceling the internal fermion propagators. The resulting contributions are referred to as pinched amplitudes. In particular, diagrams containing two vector bosons (W W W Z Z Z
d ˉ [ ( v u − a u ) m u P R + ( v u + a u ) P R k̸ ] u , (312) \bar{d}\left[(v_{u}-a_{u})m_{u}P_{R}+(v_{u}+a_{u})P_{R}\not{k}\right]u, \tag*{(312)} d ˉ [ ( v u − a u ) m u P R + ( v u + a u ) P R k ] u , (312) where k k k u u u v u v_{u} v u a u a_{u} a u
are also contributions from diagrams in which the order of the Z Z Z W W W H ± W ∓ Z H^{\pm}W^{\mp}Z H ± W ∓ Z W Z WZ W Z Figure 6 , such as ( V W Z - 1 ) (VWZ\text{-}1) ( V W Z - 1 ) ( V W Z - 4 ) (VWZ\text{-}4) ( V W Z - 4 ) H + G − H^{+}G^{-} H + G − W Z WZ W Z H + H − H^{+}H^{-} H + H − H + G − H^{+}G^{-} H + G − 8 H ± W ∓ Z H^{\pm}W^{\mp}Z H ± W ∓ Z W W W Z Z Z γ \gamma γ Figure 6 , generate only scalar-like pinch contributions.
We here give explicit expressions for gauge-dependent parts of each vertex, box, and fermion wave-function contributions shown in Figure 6 . The gauge-dependent parts of the scalar-like pinch contributions are given by
M ‾ v e r [ g V , V , S , ϕ ] ∣ G . D . = g V 2 32 π 2 1 − ξ V p 2 − m ϕ 2 [ B 0 ( 0 ; V , G V ) − ( p 2 − m S 2 ) C 0 ( p 2 ; V , G V , S ) ] , (313) \left.\overline{\mathcal{M}}_{\mathrm{ver}}[g_V,V,S,\phi]\right|_{\mathrm{G.D.}}=\frac{g_V^2}{32\pi^2}\frac{1-\xi_V}{p^2-m_\phi^2}\left[B_0(0;V,G_V)-(p^2-m_S^2)C_0(p^2;V,G_V,S)\right], \tag*{(313)} M ver [ g V , V , S , ϕ ] G.D. = 32 π 2 g V 2 p 2 − m ϕ 2 1 − ξ V [ B 0 ( 0 ; V , G V ) − ( p 2 − m S 2 ) C 0 ( p 2 ; V , G V , S ) ] , (313) M ‾ b o x [ g V , V , S ] ∣ G . D . = g V 2 64 π 2 ( 1 − ξ V ) C 0 ( p 2 ; V , G V , S ) , (314) \left.\overline{\mathcal{M}}_{\mathrm{box}}[g_V,V,S]\right|_{\mathrm{G.D.}}=\frac{g_V^2}{64\pi^2}(1-\xi_V)C_0(p^2;V,G_V,S), \tag*{(314)} M box [ g V , V , S ] G.D. = 64 π 2 g V 2 ( 1 − ξ V ) C 0 ( p 2 ; V , G V , S ) , (314) M ‾ v e r + δ [ g V , V , ϕ ] ∣ G . D . = − g V 2 64 π 2 1 − ξ V p 2 − m ϕ 2 B 0 ( 0 ; V , G V ) . (315) \left.\overline{\mathcal{M}}_{\mathrm{ver}+\delta}[g_V,V,\phi]\right|_{\mathrm{G.D.}}=-\frac{g_V^2}{64\pi^2}\frac{1-\xi_V}{p^2-m_\phi^2}B_0(0;V,G_V). \tag*{(315)} M ver + δ [ g V , V , ϕ ] G.D. = − 64 π 2 g V 2 p 2 − m ϕ 2 1 − ξ V B 0 ( 0 ; V , G V ) . (315) For W W W Z Z Z Figure 6 , we obtain
∑ i = 1 4 M ‾ V W − i s c ∣ G . D . = ∑ ϕ + , S ζ ϕ + ζ S c ϕ + S W + M ‾ v e r [ g , W , S , ϕ + ] ∣ G . D . + g 2 32 π 2 ∑ ϕ + ζ ϕ + ζ G 0 c ϕ ± G 0 W ∓ p 2 − m ϕ + 2 [ B 0 ( p 2 ; W , Z ) − B 0 ( p 2 ; W , G 0 ) ] , (316) \begin{aligned}
\left.\sum_{i=1}^{4}\overline{\mathcal{M}}_{VW-i}^{\mathrm{sc}}\right|_{\mathrm{G.D.}}={}&\sum_{\phi^+,S}\zeta_{\phi^+}\zeta_S c_{\phi^+SW^+}\left.\overline{\mathcal{M}}_{\mathrm{ver}}[g,W,S,\phi^+]\right|_{\mathrm{G.D.}}\\
&+\frac{g^2}{32\pi^2}\sum_{\phi^+}\frac{\zeta_{\phi^+}\zeta_{G^0}c_{\phi^\pm G^0W^\mp}}{p^2-m_{\phi^+}^2}\left[B_0(p^2;W,Z)-B_0(p^2;W,G^0)\right], \tag*{(316)}
\end{aligned} i = 1 ∑ 4 M V W − i sc G.D. = ϕ + , S ∑ ζ ϕ + ζ S c ϕ + S W + M ver [ g , W , S , ϕ + ] G.D. + 32 π 2 g 2 ϕ + ∑ p 2 − m ϕ + 2 ζ ϕ + ζ G 0 c ϕ ± G 0 W ∓ [ B 0 ( p 2 ; W , Z ) − B 0 ( p 2 ; W , G 0 ) ] , (316) ∑ i = 1 2 M ‾ B W − i s c ∣ G . D . = ∑ S ζ S 2 M ‾ b o x [ g , W , S ] ∣ G . D . , (317) \left.\sum_{i=1}^{2}\overline{\mathcal{M}}_{BW-i}^{\mathrm{sc}}\right|_{\mathrm{G.D.}}=\sum_S\zeta_S^2\left.\overline{\mathcal{M}}_{\mathrm{box}}[g,W,S]\right|_{\mathrm{G.D.}}, \tag*{(317)} i = 1 ∑ 2 M B W − i sc G.D. = S ∑ ζ S 2 M box [ g , W , S ] G.D. , (317) M ‾ C W − 1 s c ∣ G . D . = 2 ∑ ϕ + ζ ϕ + 2 M ‾ v e r + δ [ g , W , ϕ + ] ∣ G . D . , (318) \left.\overline{\mathcal{M}}_{CW-1}^{\mathrm{sc}}\right|_{\mathrm{G.D.}}=2\sum_{\phi^+}\zeta_{\phi^+}^2\left.\overline{\mathcal{M}}_{\mathrm{ver}+\delta}[g,W,\phi^+]\right|_{\mathrm{G.D.}}, \tag*{(318)} M C W − 1 sc G.D. = 2 ϕ + ∑ ζ ϕ + 2 M ver + δ [ g , W , ϕ + ] G.D. , (318) ∑ i = 1 8 M ‾ V Z − i s c ∣ G . D . = c 2 W ∑ ϕ + , χ + ζ ϕ + ζ χ + c ϕ + χ − Z M ‾ v e r [ g Z , Z , χ + , ϕ + ] ∣ G . D . + g Z 2 c 2 W 32 π 2 ∑ ϕ + ζ ϕ + ζ G + c ϕ ± G ∓ Z p 2 − m ϕ + 2 [ B 0 ( p 2 ; W , Z ) − B 0 ( p 2 ; G + , Z ) ] . (319) \begin{aligned}
\left.\sum_{i=1}^{8}\overline{\mathcal{M}}_{VZ-i}^{\mathrm{sc}}\right|_{\mathrm{G.D.}}={}&c_{2W}\sum_{\phi^+,\chi^+}\zeta_{\phi^+}\zeta_{\chi^+}c_{\phi^+\chi^-Z}\left.\overline{\mathcal{M}}_{\mathrm{ver}}[g_Z,Z,\chi^+,\phi^+]\right|_{\mathrm{G.D.}}\\
&+\frac{g_Z^2c_{2W}}{32\pi^2}\sum_{\phi^+}\frac{\zeta_{\phi^+}\zeta_{G^+}c_{\phi^\pm G^\mp Z}}{p^2-m_{\phi^+}^2}\left[B_0(p^2;W,Z)-B_0(p^2;G^+,Z)\right]. \tag*{(319)}
\end{aligned} i = 1 ∑ 8 M V Z − i sc G.D. = c 2 W ϕ + , χ + ∑ ζ ϕ + ζ χ + c ϕ + χ − Z M ver [ g Z , Z , χ + , ϕ + ] G.D. + 32 π 2 g Z 2 c 2 W ϕ + ∑ p 2 − m ϕ + 2 ζ ϕ + ζ G + c ϕ ± G ∓ Z [ B 0 ( p 2 ; W , Z ) − B 0 ( p 2 ; G + , Z ) ] . (319) ∑ i = 1 8 M ‾ B Z − i s c ∣ G . D . = c 2 W 2 ∑ ϕ ζ ϕ + 2 M ‾ b o x [ g Z , Z , ϕ + ] ∣ G . D . , (320) \left.\sum_{i=1}^{8}\overline{\mathcal{M}}^{\mathrm{sc}}_{BZ-i}\right|_{\mathrm{G.D.}}
=c_{2W}^{2}\sum_{\phi}\zeta_{\phi^{+}}^{2}\left.\overline{\mathcal{M}}_{\mathrm{box}}[gZ,Z,\phi^{+}]\right|_{\mathrm{G.D.}}, \tag*{(320)} i = 1 ∑ 8 M B Z − i sc G.D. = c 2 W 2 ϕ ∑ ζ ϕ + 2 M box [ g Z , Z , ϕ + ] G.D. , (320) ∑ i = 1 3 M ‾ C Z − i s c ∣ G . D . = c 2 W 2 ∑ ϕ + ζ ϕ + 2 M ‾ v e r + δ [ g Z , Z , ϕ + ] ∣ G . D . . (321) \left.\sum_{i=1}^{3}\overline{\mathcal{M}}^{\mathrm{sc}}_{CZ-i}\right|_{\mathrm{G.D.}}
=c_{2W}^{2}\sum_{\phi^{+}}\zeta_{\phi^{+}}^{2}\left.\overline{\mathcal{M}}_{\mathrm{ver}+\delta}[gZ,Z,\phi^{+}]\right|_{\mathrm{G.D.}}. \tag*{(321)} i = 1 ∑ 3 M C Z − i sc G.D. = c 2 W 2 ϕ + ∑ ζ ϕ + 2 M ver + δ [ g Z , Z , ϕ + ] G.D. . (321) As mentioned above, ( V W Z - 1 ) (V_{WZ\text{-}1}) ( V W Z - 1 ) ( V W Z - 4 ) (V_{WZ\text{-}4}) ( V W Z - 4 ) ( B W Z - 1 ) (B_{WZ\text{-}1}) ( B W Z - 1 ) ( B W Z - 4 ) (B_{WZ\text{-}4}) ( B W Z - 4 )
∑ i = 1 4 M ‾ V W Z − i s c ∣ G . D . = − g Z 2 c 2 W 32 π 2 ∑ ϕ = H + , G + ζ ϕ c ϕ ± W ∓ Z p 2 − m ϕ 2 [ m W 2 m Z 2 ( 1 − ξ W ) { ( p 2 + m Z 2 − m G + 2 ) C 0 ( p 2 ; W , G + , Z ) + ( m G + 2 + m G 0 2 − p 2 ) C 0 ( p 2 ; W , G + , G 0 ) } + ( 1 − ξ Z ) ( m W 2 + m G + 2 ) C 0 ( p 2 ; Z , G 0 , W ) ] , (322) \begin{aligned}
\left.\sum_{i=1}^{4}\overline{\mathcal{M}}^{\mathrm{sc}}_{VWZ-i}\right|_{\mathrm{G.D.}}
={}&-\frac{g_Z^{2}c_{2W}}{32\pi^{2}}\sum_{\phi=H^{+},G^{+}}
\frac{\zeta_{\phi}c_{\phi^{\pm}W^{\mp}Z}}{p^{2}-m_{\phi}^{2}}
\Bigg[
\frac{m_{W}^{2}}{m_{Z}^{2}}(1-\xi_{W})
\left\{(p^{2}+m_{Z}^{2}-m_{G^{+}}^{2})C_{0}(p^{2};W,G^{+},Z)\right.\\
&\left.+(m_{G^{+}}^{2}+m_{G^{0}}^{2}-p^{2})C_{0}(p^{2};W,G^{+},G^{0})\right\}\\
&\left.+(1-\xi_{Z})(m_{W}^{2}+m_{G^{+}}^{2})C_{0}(p^{2};Z,G^{0},W)
\right], \tag*{(322)}
\end{aligned} i = 1 ∑ 4 M V W Z − i sc G.D. = − 32 π 2 g Z 2 c 2 W ϕ = H + , G + ∑ p 2 − m ϕ 2 ζ ϕ c ϕ ± W ∓ Z [ m Z 2 m W 2 ( 1 − ξ W ) { ( p 2 + m Z 2 − m G + 2 ) C 0 ( p 2 ; W , G + , Z ) + ( m G + 2 + m G 0 2 − p 2 ) C 0 ( p 2 ; W , G + , G 0 ) } + ( 1 − ξ Z ) ( m W 2 + m G + 2 ) C 0 ( p 2 ; Z , G 0 , W ) ] , (322) ∑ i = 1 4 M ‾ B W Z − i s c ∣ G . D . = g Z 2 c 2 W 2 64 π 2 m W 2 m Z 2 ( 1 − ξ W ) [ C 0 ( p 2 ; W , G + , Z ) − C 0 ( p 2 ; W , G + , G 0 ) ] , (323) \left.\sum_{i=1}^{4}\overline{\mathcal{M}}^{\mathrm{sc}}_{BWZ-i}\right|_{\mathrm{G.D.}}
=\frac{g_{Z}^{2}c_{2W}^{2}}{64\pi^{2}}\frac{m_{W}^{2}}{m_{Z}^{2}}(1-\xi_{W})
\left[C_{0}(p^{2};W,G^{+},Z)-C_{0}(p^{2};W,G^{+},G^{0})\right], \tag*{(323)} i = 1 ∑ 4 M B W Z − i sc G.D. = 64 π 2 g Z 2 c 2 W 2 m Z 2 m W 2 ( 1 − ξ W ) [ C 0 ( p 2 ; W , G + , Z ) − C 0 ( p 2 ; W , G + , G 0 ) ] , (323) where c ϕ ± W ∓ Z c_{\phi^{\pm}W^{\mp}Z} c ϕ ± W ∓ Z c H ± W ∓ Z = s β c β c_{H^{\pm}W^{\mp}Z}=s_{\beta}c_{\beta} c H ± W ∓ Z = s β c β c G ± W ∓ Z = s β 2 − s W 2 c_{G^{\pm}W^{\mp}Z}=s_{\beta}^{2}-s_{W}^{2} c G ± W ∓ Z = s β 2 − s W 2
Using the coupling sum rules in Eqs. (D.28)–(D.31), we organize the pinch contributions into terms proportional to ζ H + 2 \zeta_{H^{+}}^{2} ζ H + 2 ζ H + ζ G + \zeta_{H^{+}}\zeta_{G^{+}} ζ H + ζ G + ζ G + 2 \zeta_{G^{+}}^{2} ζ G + 2 H + H − H^{+}H^{-} H + H − H ± G ∓ H^{\pm}G^{\mp} H ± G ∓ G + G − G^{+}G^{-} G + G − [89 ] . For some scalar-like contributions and the vector-like contributions, we further separate the H + G − H^{+}G^{-} H + G − H + W − H^{+}W^{-} H + W −
Λ G + = i 2 m u v u P L d , Λ W μ = i g 2 u γ μ P L d , (324) \Lambda_{G^{+}}=i\frac{\sqrt{2}m_{u}}{v}uP_{L}d,\qquad
\Lambda_{W}^{\mu}=i\frac{g}{\sqrt{2}}u\gamma^{\mu}P_{L}d, \tag*{(324)} Λ G + = i v 2 m u u P L d , Λ W μ = i 2 g u γ μ P L d , (324) and use the identity
Λ G + p 2 = Λ G + p 2 − m G + 2 − i m W p 2 Λ W μ ( Δ W ) μ ν p ν , (325) \frac{\Lambda_{G^{+}}}{p^{2}}
=\frac{\Lambda_{G^{+}}}{p^{2}-m_{G^{+}}^{2}}
-\frac{i m_{W}}{p^{2}}\Lambda_{W}^{\mu}(\Delta_{W})_{\mu\nu}p^{\nu}, \tag*{(325)} p 2 Λ G + = p 2 − m G + 2 Λ G + − p 2 i m W Λ W μ ( Δ W ) μν p ν , (325) where Δ W \Delta_{W} Δ W W W W H + G − H^{+}G^{-} H + G − H + W − H^{+}W^{-} H + W − M ‾ X Y P T \overline{\mathcal{M}}^{\mathrm{PT}}_{XY} M X Y PT X Y XY X Y
The H + G − H^{+}G^{-} H + G − ( V W Z (V_{WZ} ( V W Z ( V W Z (V_{WZ} ( V W Z
∑ i = 1 4 M ‾ H + G − , V W Z − i P T , v e c ∣ G . D . = g 2 ζ H + ζ G + s β c β 64 π 2 ( p 2 − m H ± 2 ) ( p 2 − m G + 2 ) [ m W 2 ( 1 − ξ W ) ( 1 + ξ Z ) B 0 ( 0 ; W , G + ) − m W 2 ( 1 − ξ Z ) ( 1 + ξ W ) B 0 ( 0 ; Z , G 0 ) − 2 m W 2 { ( 1 − ξ W ) − ( 1 − ξ Z ) } B 0 ( p 2 ; W , Z ) + m W 2 m Z 2 ( 1 − ξ W ) { [ ( p 2 − m Z 2 − m G + 2 ) 2 − 4 m Z 2 m G + 2 ] C 0 ( p 2 ; W , G + , Z ) − [ ( p 2 − m G + 2 ) 2 − m G 0 4 ] C 0 ( p 2 ; W , G + , G 0 ) } − ( 1 − ξ Z ) { m W 4 + m G + 4 − 2 p 2 ( 2 m W 2 + m G + 2 ) − 2 m W 2 m G 0 2 } C 0 ( p 2 ; Z , G 0 , W ) ] . (326) \left.\sum_{i=1}^{4}\overline{\mathcal{M}}^{\mathrm{PT,vec}}_{H^{+}G^{-},VWZ-i}\right|_{\mathrm{G.D.}} = \frac{g^{2}\zeta_{H^{+}}\zeta_{G^{+}}s_{\beta}c_{\beta}}{64\pi^{2}(p^{2}-m_{H^{\pm}}^{2})(p^{2}-m_{G^{+}}^{2})}\left[
m_{W}^{2}(1-\xi_{W})(1+\xi_{Z})B_{0}(0;W,G^{+})
- m_{W}^{2}(1-\xi_{Z})(1+\xi_{W})B_{0}(0;Z,G^{0})
-2m_{W}^{2}\{(1-\xi_{W})-(1-\xi_{Z})\}B_{0}(p^{2};W,Z)
+\frac{m_{W}^{2}}{m_{Z}^{2}}(1-\xi_{W})\left\{\left[(p^{2}-m_{Z}^{2}-m_{G^{+}}^{2})^{2}-4m_{Z}^{2}m_{G^{+}}^{2}\right]C_{0}(p^{2};W,G^{+},Z)-\left[(p^{2}-m_{G^{+}}^{2})^{2}-m_{G^{0}}^{4}\right]C_{0}(p^{2};W,G^{+},G^{0})\right\}
-(1-\xi_{Z})\left\{m_{W}^{4}+m_{G^{+}}^{4}-2p^{2}(2m_{W}^{2}+m_{G^{+}}^{2})-2m_{W}^{2}m_{G^{0}}^{2}\right\}C_{0}(p^{2};Z,G^{0},W)
\right]. \tag*{(326)} i = 1 ∑ 4 M H + G − , V W Z − i PT , vec G.D. = 64 π 2 ( p 2 − m H ± 2 ) ( p 2 − m G + 2 ) g 2 ζ H + ζ G + s β c β [ m W 2 ( 1 − ξ W ) ( 1 + ξ Z ) B 0 ( 0 ; W , G + ) − m W 2 ( 1 − ξ Z ) ( 1 + ξ W ) B 0 ( 0 ; Z , G 0 ) − 2 m W 2 {( 1 − ξ W ) − ( 1 − ξ Z )} B 0 ( p 2 ; W , Z ) + m Z 2 m W 2 ( 1 − ξ W ) { [ ( p 2 − m Z 2 − m G + 2 ) 2 − 4 m Z 2 m G + 2 ] C 0 ( p 2 ; W , G + , Z ) − [ ( p 2 − m G + 2 ) 2 − m G 0 4 ] C 0 ( p 2 ; W , G + , G 0 ) } − ( 1 − ξ Z ) { m W 4 + m G + 4 − 2 p 2 ( 2 m W 2 + m G + 2 ) − 2 m W 2 m G 0 2 } C 0 ( p 2 ; Z , G 0 , W ) ] . (326) Similarly, the vector-like parts of diagrams ( B W Z (B_{WZ} ( B W Z ( B W Z (B_{WZ} ( B W Z
∑ i = 1 4 M ‾ H + G − , B W Z − i P T , v e c ∣ G . D . = g 2 ζ H + ζ G + s β c β 64 π 2 ( p 2 − m G + 2 ) [ m W 2 m Z 2 ( 1 − ξ W ) { − ( p 2 + m Z 2 − m G + 2 ) C 0 ( p 2 ; W , G + , Z ) + ( p 2 + m G 0 2 − m G + 2 ) C 0 ( p 2 ; W , G + , G 0 ) } + ( 1 − ξ Z ) ( m W 2 − m G + 2 ) C 0 ( p 2 ; Z , G 0 , W ) ] . (327) \left.\sum_{i=1}^{4}\overline{\mathcal{M}}^{\mathrm{PT,vec}}_{H^{+}G^{-},BWZ-i}\right|_{\mathrm{G.D.}} = \frac{g^{2}\zeta_{H^{+}}\zeta_{G^{+}}s_{\beta}c_{\beta}}{64\pi^{2}(p^{2}-m_{G^{+}}^{2})}\left[
\frac{m_{W}^{2}}{m_{Z}^{2}}(1-\xi_{W})\left\{-(p^{2}+m_{Z}^{2}-m_{G^{+}}^{2})C_{0}(p^{2};W,G^{+},Z)+(p^{2}+m_{G^{0}}^{2}-m_{G^{+}}^{2})C_{0}(p^{2};W,G^{+},G^{0})\right\}
+(1-\xi_{Z})(m_{W}^{2}-m_{G^{+}}^{2})C_{0}(p^{2};Z,G^{0},W)
\right]. \tag*{(327)} i = 1 ∑ 4 M H + G − , B W Z − i PT , vec G.D. = 64 π 2 ( p 2 − m G + 2 ) g 2 ζ H + ζ G + s β c β [ m Z 2 m W 2 ( 1 − ξ W ) { − ( p 2 + m Z 2 − m G + 2 ) C 0 ( p 2 ; W , G + , Z ) + ( p 2 + m G 0 2 − m G + 2 ) C 0 ( p 2 ; W , G + , G 0 ) } + ( 1 − ξ Z ) ( m W 2 − m G + 2 ) C 0 ( p 2 ; Z , G 0 , W ) ] . (327) We have analytically confirmed that the gauge-dependent contributions cancel separately in the H + H − H^{+}H^{-} H + H − H + G − H^{+}G^{-} H + G −
[ ζ H + 2 ( p 2 − m H ± 2 ) 2 ( ∑ i = 1 11 Π H + H − W − i + ∑ i = 1 8 Π H + H − Z − i ) + M ‾ H + H − , a l l P T , s c ] G . D . = 0 , (328) \left[\frac{\zeta_{H^{+}}^{2}}{(p^{2}-m_{H^{\pm}}^{2})^{2}}\left(\sum_{i=1}^{11}\Pi^{W-i}_{H^{+}H^{-}}+\sum_{i=1}^{8}\Pi^{Z-i}_{H^{+}H^{-}}\right)+\overline{\mathcal{M}}^{\mathrm{PT,sc}}_{H^{+}H^{-},\mathrm{all}}\right]_{\mathrm{G.D.}}=0, \tag*{(328)} [ ( p 2 − m H ± 2 ) 2 ζ H + 2 ( i = 1 ∑ 11 Π H + H − W − i + i = 1 ∑ 8 Π H + H − Z − i ) + M H + H − , all PT , sc ] G.D. = 0 , (328) [ ζ H + ζ G + ( p 2 − m H ± 2 ) ( p 2 − m G ± 2 ) ( ∑ i = 1 11 Π H + G − W − i + ∑ i = 1 8 Π H + G − Z − i ) + M ‾ H + G − , a l l P T , s c + M ‾ H + G − , a l l P T , v e c ] G . D . = 0. (329) \left[\frac{\zeta_{H^{+}}\zeta_{G^{+}}}{(p^{2}-m_{H^{\pm}}^{2})(p^{2}-m_{G^{\pm}}^{2})}\left(\sum_{i=1}^{11}\Pi^{W-i}_{H^{+}G^{-}}+\sum_{i=1}^{8}\Pi^{Z-i}_{H^{+}G^{-}}\right)+\overline{\mathcal{M}}^{\mathrm{PT,sc}}_{H^{+}G^{-},\mathrm{all}}+\overline{\mathcal{M}}^{\mathrm{PT,vec}}_{H^{+}G^{-},\mathrm{all}}\right]_{\mathrm{G.D.}}=0. \tag*{(329)} [ ( p 2 − m H ± 2 ) ( p 2 − m G ± 2 ) ζ H + ζ G + ( i = 1 ∑ 11 Π H + G − W − i + i = 1 ∑ 8 Π H + G − Z − i ) + M H + G − , all PT , sc + M H + G − , all PT , vec ] G.D. = 0. (329) The gauge dependence in the G + H − G^{+}H^{-} G + H −
Pinch terms of the self-energies of charged scalars
In the ’t Hooft–Feynman gauge, the only nonvanishing pinch contributions arise from the vertex correction diagrams ( V W (V_{W} ( V W ( V W (V_{W} ( V W ( V Z (V_{Z} ( V Z ( V Z (V_{Z} ( V Z ( V γ (V_{\gamma} ( V γ ( V γ (V_{\gamma} ( V γ
∑ i = 1 − 4 M ‾ V W − i = − g 2 32 π 2 ∑ s = h , H [ ( ζ H + ζ s c H ± s W ∓ p 2 − m H ± 2 + ζ G + ζ s c G ± s W ∓ p 2 − m G ± 2 ) B 0 ( p 2 ; W , s ) \sum_{i=1-4}\overline{\mathcal{M}}_{VW-i}
=
-\frac{g^{2}}{32\pi^{2}}\sum_{s=h,H}
\left[
\left(
\frac{\zeta_{H^{+}}\zeta_{s}c_{H^{\pm}sW^{\mp}}}{p^{2}-m_{H^{\pm}}^{2}}
+
\frac{\zeta_{G^{+}}\zeta_{s}c_{G^{\pm}sW^{\mp}}}{p^{2}-m_{G^{\pm}}^{2}}
\right)B_{0}(p^{2};W,s)
\right. i = 1 − 4 ∑ M V W − i = − 32 π 2 g 2 s = h , H ∑ [ ( p 2 − m H ± 2 ζ H + ζ s c H ± s W ∓ + p 2 − m G ± 2 ζ G + ζ s c G ± s W ∓ ) B 0 ( p 2 ; W , s ) − g 2 32 π 2 ( ζ H + ζ G 0 c H ± G 0 W ∓ p 2 − m H ± 2 + ζ G + ζ G 0 c G ± G 0 W ∓ p 2 − m G ± 2 ) B 0 ( p 2 ; W , Z ) ] (330) \left.
-\frac{g^{2}}{32\pi^{2}}
\left(
\frac{\zeta_{H^{+}}\zeta_{G^{0}}c_{H^{\pm}G^{0}W^{\mp}}}{p^{2}-m_{H^{\pm}}^{2}}
+
\frac{\zeta_{G^{+}}\zeta_{G^{0}}c_{G^{\pm}G^{0}W^{\mp}}}{p^{2}-m_{G^{\pm}}^{2}}
\right)B_{0}(p^{2};W,Z)
\right] \tag*{(330)} − 32 π 2 g 2 ( p 2 − m H ± 2 ζ H + ζ G 0 c H ± G 0 W ∓ + p 2 − m G ± 2 ζ G + ζ G 0 c G ± G 0 W ∓ ) B 0 ( p 2 ; W , Z ) ] (330) ∑ i = 1 − 8 M ‾ V Z − i = − g Z 2 c 2 W 32 π 2 ( ζ H + c H + H − Z p 2 − m H ± 2 + ζ H + ζ G + c H ± G ∓ Z p 2 − m G ± 2 ) B 0 ( p 2 ; Z , H ± ) \sum_{i=1-8}\overline{\mathcal{M}}_{VZ-i}
=
-\frac{g_{Z}^{2}c_{2W}}{32\pi^{2}}
\left(
\frac{\zeta_{H^{+}}c_{H^{+}H^{-}Z}}{p^{2}-m_{H^{\pm}}^{2}}
+
\frac{\zeta_{H^{+}}\zeta_{G^{+}}c_{H^{\pm}G^{\mp}Z}}{p^{2}-m_{G^{\pm}}^{2}}
\right)B_{0}(p^{2};Z,H^{\pm}) i = 1 − 8 ∑ M V Z − i = − 32 π 2 g Z 2 c 2 W ( p 2 − m H ± 2 ζ H + c H + H − Z + p 2 − m G ± 2 ζ H + ζ G + c H ± G ∓ Z ) B 0 ( p 2 ; Z , H ± ) − g Z 2 c 2 W 32 π 2 ( ζ H + ζ G + c H ± G ∓ Z p 2 − m H ± 2 + ζ G + 2 c G + G − Z p 2 − m G ± 2 ) B 0 ( p 2 ; Z , W ) (331) -\frac{g_{Z}^{2}c_{2W}}{32\pi^{2}}
\left(
\frac{\zeta_{H^{+}}\zeta_{G^{+}}c_{H^{\pm}G^{\mp}Z}}{p^{2}-m_{H^{\pm}}^{2}}
+
\frac{\zeta_{G^{+}}^{2}c_{G^{+}G^{-}Z}}{p^{2}-m_{G^{\pm}}^{2}}
\right)B_{0}(p^{2};Z,W) \tag*{(331)} − 32 π 2 g Z 2 c 2 W ( p 2 − m H ± 2 ζ H + ζ G + c H ± G ∓ Z + p 2 − m G ± 2 ζ G + 2 c G + G − Z ) B 0 ( p 2 ; Z , W ) (331) ∑ i = 1 − 8 M ‾ V γ − i = − e 2 8 π 2 ( ζ H + 2 p 2 − m H ± 2 B 0 ( p 2 ; γ , H ± ) + ζ G + 2 p 2 − m G ± 2 B 0 ( p 2 ; γ , G ± ) ) . (332) \sum_{i=1-8}\overline{\mathcal{M}}_{V\gamma-i}
=
-\frac{e^{2}}{8\pi^{2}}
\left(
\frac{\zeta_{H^{+}}^{2}}{p^{2}-m_{H^{\pm}}^{2}}B_{0}(p^{2};\gamma,H^{\pm})
+
\frac{\zeta_{G^{+}}^{2}}{p^{2}-m_{G^{\pm}}^{2}}B_{0}(p^{2};\gamma,G^{\pm})
\right). \tag*{(332)} i = 1 − 8 ∑ M V γ − i = − 8 π 2 e 2 ( p 2 − m H ± 2 ζ H + 2 B 0 ( p 2 ; γ , H ± ) + p 2 − m G ± 2 ζ G + 2 B 0 ( p 2 ; γ , G ± ) ) . (332) The pinch terms of these self-energies are related to these pinched amplitudes as
∑ i = 1 − 4 M ‾ V W − i + ∑ i = 1 − 8 M ‾ V Z − i + ∑ i = 1 − 8 M ‾ V γ − i → ζ H + 2 ( p 2 − m H ± 2 ) 2 Π H + H − P T ( p 2 ) + ζ G + 2 ( p 2 − m G ± 2 ) 2 Π G + G − P T ( p 2 ) + ζ H + ζ G + ( p 2 − m H ± 2 ) ( p 2 − m G ± 2 ) ( Π H + G − P T ( p 2 ) + Π G + H − P T ( p 2 ) ) (333) \begin{aligned}
\sum_{i=1-4}\overline{\mathcal{M}}_{VW-i}
+
\sum_{i=1-8}\overline{\mathcal{M}}_{VZ-i}
+
\sum_{i=1-8}\overline{\mathcal{M}}_{V\gamma-i}
\to
\frac{\zeta_{H^{+}}^{2}}{(p^{2}-m_{H^{\pm}}^{2})^{2}}\Pi_{H^{+}H^{-}}^{\mathrm{PT}}(p^{2})
+
\frac{\zeta_{G^{+}}^{2}}{(p^{2}-m_{G^{\pm}}^{2})^{2}}\Pi_{G^{+}G^{-}}^{\mathrm{PT}}(p^{2}) \\
&+
\frac{\zeta_{H^{+}}\zeta_{G^{+}}}{(p^{2}-m_{H^{\pm}}^{2})(p^{2}-m_{G^{\pm}}^{2})}
\left(\Pi_{H^{+}G^{-}}^{\mathrm{PT}}(p^{2})+\Pi_{G^{+}H^{-}}^{\mathrm{PT}}(p^{2})\right) \tag*{(333)}
\end{aligned} i = 1 − 4 ∑ M V W − i + i = 1 − 8 ∑ M V Z − i + i = 1 − 8 ∑ M V γ − i → ( p 2 − m H ± 2 ) 2 ζ H + 2 Π H + H − PT ( p 2 ) + ( p 2 − m G ± 2 ) 2 ζ G + 2 Π G + G − PT ( p 2 ) + ( p 2 − m H ± 2 ) ( p 2 − m G ± 2 ) ζ H + ζ G + ( Π H + G − PT ( p 2 ) + Π G + H − PT ( p 2 ) ) (333) where Π H + G − P T ( p 2 ) = Π G + H − P T ( p 2 ) \Pi_{H^{+}G^{-}}^{\mathrm{PT}}(p^{2})=\Pi_{G^{+}H^{-}}^{\mathrm{PT}}(p^{2}) Π H + G − PT ( p 2 ) = Π G + H − PT ( p 2 ) H + W − H^{+}W^{-} H + W −
Π H + H − P T ( p 2 ) = − g 2 32 π 2 ( p 2 − m H ± 2 ) [ ∑ s = h , H c H ± s W ∓ 2 B 0 ( p 2 ; W , s ) + c H + G 0 W − 2 B 0 ( p 2 ; W , Z ) ] \Pi_{H^{+}H^{-}}^{\mathrm{PT}}(p^{2})
=
-\frac{g^{2}}{32\pi^{2}}(p^{2}-m_{H^{\pm}}^{2})
\left[
\sum_{s=h,H}c_{H^{\pm}sW^{\mp}}^{2}B_{0}(p^{2};W,s)
+c_{H^{+}G^{0}W^{-}}^{2}B_{0}(p^{2};W,Z)
\right] Π H + H − PT ( p 2 ) = − 32 π 2 g 2 ( p 2 − m H ± 2 ) s = h , H ∑ c H ± s W ∓ 2 B 0 ( p 2 ; W , s ) + c H + G 0 W − 2 B 0 ( p 2 ; W , Z ) − g Z 2 32 π 2 ( p 2 − m H ± 2 ) [ c H + H − Z 2 B 0 ( p 2 ; Z , H ± ) + c H ± G ∓ Z 2 B 0 ( p 2 ; Z , W ) ] -\frac{g_{Z}^{2}}{32\pi^{2}}(p^{2}-m_{H^{\pm}}^{2})
\left[
c_{H^{+}H^{-}Z}^{2}B_{0}(p^{2};Z,H^{\pm})
+c_{H^{\pm}G^{\mp}Z}^{2}B_{0}(p^{2};Z,W)
\right] − 32 π 2 g Z 2 ( p 2 − m H ± 2 ) [ c H + H − Z 2 B 0 ( p 2 ; Z , H ± ) + c H ± G ∓ Z 2 B 0 ( p 2 ; Z , W ) ] − e 2 8 π 2 ( p 2 − m H ± 2 ) B 0 ( p 2 ; γ , H ± ) . (334) -\frac{e^{2}}{8\pi^{2}}(p^{2}-m_{H^{\pm}}^{2})B_{0}(p^{2};\gamma,H^{\pm}). \tag*{(334)} − 8 π 2 e 2 ( p 2 − m H ± 2 ) B 0 ( p 2 ; γ , H ± ) . (334) The pinch term of the H + G − H^{+}G^{-} H + G − (75) . In the decomposition, we use the following sum rules among the coupling constants;
ζ s = ζ H + c H ± s W ∓ + ζ G + c G ± s W ∓ , (335) \zeta_{s}=\zeta_{H^{+}}c_{H^{\pm}sW^{\mp}}+\zeta_{G^{+}}c_{G^{\pm}sW^{\mp}}, \tag*{(335)} ζ s = ζ H + c H ± s W ∓ + ζ G + c G ± s W ∓ , (335) ζ G 0 = ζ H + c H ± G 0 W ∓ + ζ G + c G ± G 0 W ∓ , (336) \zeta_{G^{0}}=\zeta_{H^{+}}c_{H^{\pm}G^{0}W^{\mp}}+\zeta_{G^{+}}c_{G^{\pm}G^{0}W^{\mp}}, \tag*{(336)} ζ G 0 = ζ H + c H ± G 0 W ∓ + ζ G + c G ± G 0 W ∓ , (336) c 2 W ζ H + = ζ H + c H + H − Z + ζ G + c H ± G ∓ Z , (337) c_{2W}\zeta_{H^{+}}=\zeta_{H^{+}}c_{H^{+}H^{-}Z}+\zeta_{G^{+}}c_{H^{\pm}G^{\mp}Z}, \tag*{(337)} c 2 W ζ H + = ζ H + c H + H − Z + ζ G + c H ± G ∓ Z , (337) c 2 W ζ G + = ζ H + c H ± G ∓ Z + ζ G + c G + G − Z . (338) c_{2W}\zeta_{G^{+}}=\zeta_{H^{+}}c_{H^{\pm}G^{\mp}Z}+\zeta_{G^{+}}c_{G^{+}G^{-}Z}. \tag*{(338)} c 2 W ζ G + = ζ H + c H ± G ∓ Z + ζ G + c G + G − Z . (338) 
