Introduction
The scalar sector of the Standard Model (SM) is responsible for electroweak symmetry breaking and the generation of particle masses. Despite its remarkable phenomenological success, the SM leaves several open questions, including the origin of neutrino masses and the stability of the electroweak scale. Extensions of the scalar sector provide a well-motivated framework for exploring physics beyond the SM. Among the simplest and most extensively studied possibilities are Multi-Higgs Doublet Models (NHDMs), which enlarge the scalar content of the theory by having N N N S U ( 2 ) L \mathrm{SU}(2)_{L} SU ( 2 ) L
The addition of scalar doublets drastically enlarges the parameter space of the theory: the two-Higgs Doublet Model (2HDM) [1 , 2 ] involves 14 independent parameters in its most general form, while the three-Higgs Doublet Model (3HDM) already contains 54.1 Z 2 \mathbb{Z}_2 Z 2 [3 , 4 ] . The resulting reduction of free parameters is therefore a consequence of symmetry, not its fundamental purpose. For many years, the global symmetries of the SU ( 2 ) × U ( 1 ) \operatorname{SU}(2) \times\operatorname{U}(1) SU ( 2 ) × U ( 1 ) Higgs-family symmetries and generalised CP symmetries . The former correspond to unitary transformations in the scalar field space,
Φ i ′ = U i j Φ j , i , j = 1 , 2 , U ∈ U ( 2 ) , (1) \Phi'_i = U_{ij}\Phi_j,\qquad i,j=1,2,\qquad U\in\operatorname{U}(2), \tag*{(1)} Φ i ′ = U ij Φ j , i , j = 1 , 2 , U ∈ U ( 2 ) , (1) while the latter involves generalised CP transformations of the form
Φ i ′ = X i j Φ j ∗ , i , j = 1 , 2 , X ∈ U ( 2 ) . (2) \Phi'_i = X_{ij}\Phi_j^*,\qquad i,j=1,2,\qquad X\in\operatorname{U}(2). \tag*{(2)} Φ i ′ = X ij Φ j ∗ , i , j = 1 , 2 , X ∈ U ( 2 ) . (2) It was shown [5 ] that there were only six such possible symmetries – three of the first kind and three of the second. Recently, however, Ferreira, Grzadkowski, Ogreid, and Osland [6 ] identified a previously unnoticed class of symmetries — now referred to as GOOFy symmetries — through an analysis of the renormalization group (RG) evolution of the scalar parameters. They observed that certain relations among the parameters of the scalar potential remain RG invariant to all orders in perturbation theory when both scalar and gauge interactions are taken into account. The origin of these relations was traced to an imaginary rescaling of the Higgs doublet fields. While such a transformation by itself would spoil the invariance of the kinetic terms, this can be compensated by a corresponding imaginary rescaling of the spacetime coordinates, together with a consistent rescaling of the gauge fields, thereby restoring invariance of the full scalar and gauge Lagrangian. An extension of this construction to the Yukawa sector has also been achieved [7 ] . There has been extensive work on this subject, with different interpretations of the cause of these RG invariant relations and their extension to several versions of the 2HDM and even 3HDM [8 , 9 , 10 , 11 , 12 , 13 , 14 , 15 , 16 , 17 , 18 ] .
In this work, we extend the study of GOOFy symmetries to the three-Higgs Doublet Model, with the intention of obtaining a viable Yukawa sector. In Section 2 we review GOOFy symmetries and discuss their implementation within a 3HDM, including the fermionic sector. We also analyse electroweak symmetry breaking in the model and present its renormalization group equations, demonstrating that the GOOFy parameter relations found are preserved under renormalization. Section 3 discusses the softly broken model we must introduce to have a realistic fermion sector. We further discuss the several theoretical and experimental constraints we impose on the model and present results from a numerical scan of the model, including predictions for Higgs phenomenology which may be measurable at the LHC. We conclude in Section 4.
The model
The three-Higgs doublet model (3HDM) is an extension of the Standard Model in which two additional SU ( 2 ) L \operatorname{SU}(2)_L SU ( 2 ) L + 1 +1 + 1 ρ \rho ρ Z 2 × Z 2 \mathbb{Z}_2 \times\mathbb{Z}_2 Z 2 × Z 2 [19 ] , wherein CP violation occurs in the interactions of the charged scalars. Coexisting charge and neutral breaking minima are possible in the 3HDM [20 ] but not in the 2HDM [5 , 21 , 22 ] , so the vacuum structure of the model acquires novel properties. A non-exhaustive list of papers that deal with the 3HDM, including an exploration of its geometric structure involving the bilinear formalism; possible Abelian symmetries; boundedness from below conditions; exploration of the phenomenology of A 4 A_4 A 4 S 4 S_4 S 4 Δ ( 27 ) \Delta(27) Δ ( 27 ) [23 , 24 , 25 , 26 , 27 , 28 , 29 , 30 , 31 , 32 , 33 , 34 , 35 , 36 , 37 , 38 , 39 , 40 , 41 , 42 , 43 , 44 , 45 , 46 , 47 , 48 ] . These works, and many others, considered usual symmetry transformations as the source of symmetries with the 3HDM. Recently, however, a new type of “GOOFy” symmetries were identified in the 2HDM. Let us first review those new symmetries in order to apply them to the 3HDM.
GOOFy symmetries
It is well known that parameter relations resulting from a given symmetry are invariant under renormalization to all orders. However, this statement is not an equivalence, and no demonstration exists that the opposite is also true – that renormalization-group (RG) invariant relations between parameters in a given theory necessarily imply the existence of a symmetry. Such were the findings of Ref. [6 ] where certain relations for scalar potential parameters were found to be RG invariant to all orders of perturbation theory but those relations cannot be reproduced by any field transformations such as (1) or (2). This raised the question of possible new symmetries having been found, corresponding to different field transformations. The work of [6 ] dealt with the 2HDM potential, which in its most generic SU ( 2 ) L × U ( 1 ) Y \operatorname{SU}(2)_{L} \times\operatorname{U}(1)_{Y} SU ( 2 ) L × U ( 1 ) Y
V = m 11 2 Φ 1 † Φ 1 + m 22 2 Φ 2 † Φ 2 − [ m 12 2 Φ 1 † Φ 2 + h . c . ] + 1 2 λ 1 ( Φ 1 † Φ 1 ) 2 + 1 2 λ 2 ( Φ 2 † Φ 2 ) 2 + λ 3 ( Φ 1 † Φ 1 ) ( Φ 2 † Φ 2 ) + λ 4 ( Φ 1 † Φ 2 ) ( Φ 2 † Φ 1 ) + { 1 2 λ 5 ( Φ 1 † Φ 2 ) 2 + [ λ 6 ( Φ 1 † Φ 1 ) + λ 7 ( Φ 2 † Φ 2 ) ] Φ 1 † Φ 2 + h . c . } , (3) \begin{aligned}
V ={}& m_{11}^{2}\Phi_{1}^{\dagger}\Phi_{1} + m_{22}^{2}\Phi_{2}^{\dagger}\Phi_{2} - \left[m_{12}^{2}\Phi_{1}^{\dagger}\Phi_{2} + \mathrm{h.c.}\right] + \frac{1}{2}\lambda_{1}\left(\Phi_{1}^{\dagger}\Phi_{1}\right)^{2} + \frac{1}{2}\lambda_{2}\left(\Phi_{2}^{\dagger}\Phi_{2}\right)^{2} \\
&+ \lambda_{3}\left(\Phi_{1}^{\dagger}\Phi_{1}\right)\left(\Phi_{2}^{\dagger}\Phi_{2}\right) + \lambda_{4}\left(\Phi_{1}^{\dagger}\Phi_{2}\right)\left(\Phi_{2}^{\dagger}\Phi_{1}\right) \\
&+ \left\{\frac{1}{2}\lambda_{5}\left(\Phi_{1}^{\dagger}\Phi_{2}\right)^{2} + \left[\lambda_{6}\left(\Phi_{1}^{\dagger}\Phi_{1}\right) + \lambda_{7}\left(\Phi_{2}^{\dagger}\Phi_{2}\right)\right]\Phi_{1}^{\dagger}\Phi_{2} + \mathrm{h.c.}\right\}, \tag*{(3)}
\end{aligned} V = m 11 2 Φ 1 † Φ 1 + m 22 2 Φ 2 † Φ 2 − [ m 12 2 Φ 1 † Φ 2 + h.c. ] + 2 1 λ 1 ( Φ 1 † Φ 1 ) 2 + 2 1 λ 2 ( Φ 2 † Φ 2 ) 2 + λ 3 ( Φ 1 † Φ 1 ) ( Φ 2 † Φ 2 ) + λ 4 ( Φ 1 † Φ 2 ) ( Φ 2 † Φ 1 ) + { 2 1 λ 5 ( Φ 1 † Φ 2 ) 2 + [ λ 6 ( Φ 1 † Φ 1 ) + λ 7 ( Φ 2 † Φ 2 ) ] Φ 1 † Φ 2 + h.c. } , (3) with all parameters real with the exception of m 12 2 m_{12}^{2} m 12 2 λ 5 , 6 , 7 \lambda_{5,6,7} λ 5 , 6 , 7
m 11 2 + m 22 2 = 0 , λ 1 = λ 2 , λ 6 = − λ 7 (4) m_{11}^{2} + m_{22}^{2} = 0,\qquad\lambda_{1} = \lambda_{2},\qquad\lambda_{6} = -\lambda_{7} \tag*{(4)} m 11 2 + m 22 2 = 0 , λ 1 = λ 2 , λ 6 = − λ 7 (4) remain invariant under RG evolution to all orders of perturbation theory when considering only the scalar and gauge sectors. A general proof of RG invariance considering the Yukawa sector as well was not found in [6 ] , although two (symmetry-imposed) Yukawa matrix textures were found, by direct calculation of the model’s β \beta β
The relations among quartic couplings in (4) can be obtained by a “standard” 2HDM symmetry, namely the CP2 symmetry wherein invariance of the theory under the field transformations Φ 1 → Φ 2 ∗ \Phi_{1} \to\Phi_{2}^{*} Φ 1 → Φ 2 ∗ Φ 2 → − Φ 1 ∗ \Phi_{2} \to-\Phi_{1}^{*} Φ 2 → − Φ 1 ∗ m 11 2 = m 22 2 m_{11}^{2} = m_{22}^{2} m 11 2 = m 22 2 m 12 2 = 0 m_{12}^{2} = 0 m 12 2 = 0 [6 ] that the relations (4) could be obtained under the unorthodox field transformation
Φ 1 → Φ 2 ∗ , Φ 1 † → − Φ 2 T , Φ 2 → − Φ 1 ∗ , Φ 2 † → Φ 1 T . (5) \begin{aligned}
\Phi_{1} &\to\Phi_{2}^{*}, & \Phi_{1}^{\dagger} &\to-\Phi_{2}^{T}, \\
\Phi_{2} &\to-\Phi_{1}^{*}, & \Phi_{2}^{\dagger} &\to\Phi_{1}^{T}. \tag*{(5)}
\end{aligned} Φ 1 Φ 2 → Φ 2 ∗ , → − Φ 1 ∗ , Φ 1 † Φ 2 † → − Φ 2 T , → Φ 1 T . (5) The transformations on the first column are identical to the CP2 case, but notice that the transformation of Φ 1 † \Phi_{1}^{\dagger} Φ 1 † Φ 2 † \Phi_{2}^{\dagger} Φ 2 † minus the hermitian conjugate of the transformation of Φ 1 \Phi_{1} Φ 1 Φ 2 \Phi_{2} Φ 2 ∣ Φ 1 ∣ 2 + ∣ Φ 2 ∣ 2 → − ( ∣ Φ 1 ∣ 2 + ∣ Φ 2 ∣ 2 ) \lvert\Phi_{1}\rvert^{2} + \lvert\Phi_{2}\rvert^{2} \to-(\lvert\Phi_{1}\rvert^{2} + \lvert\Phi_{2}\rvert^{2}) ∣ Φ 1 ∣ 2 + ∣ Φ 2 ∣ 2 → − (∣ Φ 1 ∣ 2 + ∣ Φ 2 ∣ 2 ) (4) . It is also impossible to obtain such a transformation with “normal” HF or GCP field transformations.
To explain the origin of such transformations, Ref. [6 ] proposed that they arise from an imaginary scaling of the real components of the scalar doublets. Writing the doublets in terms of real scalar fields ϕ i \phi_i ϕ i
Φ 1 = 1 2 ( ϕ 1 + i ϕ 2 ϕ 3 + i ϕ 4 ) , Φ 2 = 1 2 ( ϕ 5 + i ϕ 6 ϕ 7 + i ϕ 8 ) , (6) \Phi_1 = \frac{1}{\sqrt{2}}\begin{pmatrix}\phi_1 + i\phi_2 \\ \phi_3 + i\phi_4\end{pmatrix}, \qquad\Phi_2 = \frac{1}{\sqrt{2}}\begin{pmatrix}\phi_5 + i\phi_6 \\ \phi_7 + i\phi_8\end{pmatrix}, \tag*{(6)} Φ 1 = 2 1 ( ϕ 1 + i ϕ 2 ϕ 3 + i ϕ 4 ) , Φ 2 = 2 1 ( ϕ 5 + i ϕ 6 ϕ 7 + i ϕ 8 ) , (6) one may obtain the transformations of eq. (5) by imposing
ϕ 1 → − i ϕ 6 , ϕ 2 → − i ϕ 5 , ϕ 3 → − i ϕ 8 , ϕ 4 → − i ϕ 7 , ϕ 5 → i ϕ 2 , ϕ 6 → i ϕ 1 , ϕ 7 → i ϕ 4 , ϕ 8 → i ϕ 3 . (7) \begin{aligned}
\phi_1 &\to-i\phi_6, \quad\phi_2 \to-i\phi_5, \quad\phi_3 \to-i\phi_8, \quad\phi_4 \to-i\phi_7,\\
\phi_5 &\to i\phi_2, \quad\phi_6 \to i\phi_1, \quad\phi_7 \to i\phi_4, \quad\phi_8 \to i\phi_3. \tag*{(7)}
\end{aligned} ϕ 1 ϕ 5 → − i ϕ 6 , ϕ 2 → − i ϕ 5 , ϕ 3 → − i ϕ 8 , ϕ 4 → − i ϕ 7 , → i ϕ 2 , ϕ 6 → i ϕ 1 , ϕ 7 → i ϕ 4 , ϕ 8 → i ϕ 3 . (7) In order to preserve the invariance of the kinetic Lagrangian under this transformation, a corresponding imaginary scaling of the spacetime coordinates is introduced, x μ → i x μ x_\mu\to ix_\mu x μ → i x μ
B μ → i B μ , W μ 1 → i W μ 1 , W μ 2 → − i W μ 2 , W μ 3 → i W μ 3 , (8) B_\mu\to iB_\mu, \qquad W_\mu^1 \to iW_\mu^1, \qquad W_\mu^2 \to-iW_\mu^2, \qquad W_\mu^3 \to iW_\mu^3, \tag*{(8)} B μ → i B μ , W μ 1 → i W μ 1 , W μ 2 → − i W μ 2 , W μ 3 → i W μ 3 , (8) Analogous imaginary scalings are introduced for the fermionic fields to ensure invariance of the fermion’s kinetic fields, their gauge interactions and the Yukawa terms [7 ] . Considering only quarks, the 2HDM Yukawa lagrangian is given by
− L Y = Q ‾ L Γ 1 Φ 1 n R + Q ‾ L Γ 2 Φ 2 n R + Q ‾ L Δ 1 Φ ~ 1 p R + Q ‾ L Δ 2 Φ ~ 2 p R + h . c . , (9) -\mathcal{L}_Y = \overline{Q}_L\Gamma_1\Phi_1 n_R + \overline{Q}_L\Gamma_2\Phi_2 n_R + \overline{Q}_L\Delta_1\widetilde{\Phi}_1 p_R + \overline{Q}_L\Delta_2\widetilde{\Phi}_2 p_R + \mathrm{h.c.}, \tag*{(9)} − L Y = Q L Γ 1 Φ 1 n R + Q L Γ 2 Φ 2 n R + Q L Δ 1 Φ 1 p R + Q L Δ 2 Φ 2 p R + h.c. , (9) where the n R n_R n R p R p_R p R Q L Q_L Q L 3 × 3 3 \times3 3 × 3 Γ i \Gamma_i Γ i Δ i \Delta_i Δ i 3 × 3 3 \times3 3 × 3 Φ ~ i = i σ 2 Φ i ∗ \widetilde{\Phi}_i = i\sigma_2\Phi_i^* Φ i = i σ 2 Φ i ∗ (7) for the scalars and (8) for the gauge fields are given by
Q L → X α γ 0 C Q L ∗ , Q ‾ L → − i Q L T C X α † , n R → X β γ 0 C n R ∗ , n ‾ R → − i n R T C X β † , p R → X γ γ 0 C p R ∗ , p ‾ R → − i p R T C X γ † , (10) \begin{aligned}
Q_L &\to X_\alpha\gamma^0 C Q_L^*, \qquad\overline{Q}_L \to-iQ_L^T C X_\alpha^\dagger,\\
n_R &\to X_\beta\gamma^0 C n_R^*, \qquad\overline{n}_R \to-in_R^T C X_\beta^\dagger,\\
p_R &\to X_\gamma\gamma^0 C p_R^*, \qquad\overline{p}_R \to-ip_R^T C X_\gamma^\dagger, \tag*{(10)}
\end{aligned} Q L n R p R → X α γ 0 C Q L ∗ , Q L → − i Q L T C X α † , → X β γ 0 C n R ∗ , n R → − i n R T C X β † , → X γ γ 0 C p R ∗ , p R → − i p R T C X γ † , (10) where, as in [49 ] , the three unitary 3 × 3 3 \times3 3 × 3 X θ X_\theta X θ C C C C 2 = − 1 C^2=-1 C 2 = − 1 C − 1 = C † = − C C^{-1}=C^\dagger=-C C − 1 = C † = − C C γ μ C − 1 = − γ μ T C\gamma^\mu C^{-1}=-\gamma^{\mu T} C γ μ C − 1 = − γ μ T (5) , each field and its hermitian conjugate transform independently (in the former case a minus sign distinguishes those transformations, in the latter a factor of “i”). These scalings imply constraints on the Yukawa matrices, which end up being analogous to those obtained with other “regular” symmetries. As shown in [7 ] , with the scalar transformations of eq. (5) it is possible to choose the matrices X θ X_\theta X θ [49 ] .
The initials of the authors of [6 ] , plus the bizarre nature of these field transformations, justified their being dubbed “GOOFy symmetries”. Whether this GOOFy procedure is indeed being caused by a genuine symmetry or by something else (see [8 , 9 , 10 , 11 , 12 , 13 , 14 , 15 , 16 , 17 , 18 ] for alternative explanations), the fact remains that it is an effective procedure to find hitherto unknown RG-invariant regions of parameter space within specific models, with interesting phenomenology. We will now apply these techniques to the case of the 3HDM.
Imposing a GOOFy symmetry in the Three Higgs Doublet Model
The work presented here extends the ideas of the previous section to the three-Higgs Doublet Model. Let us recall that the most general S U ( 2 ) L × U ( 1 ) Y \mathrm{SU}(2)_{L} \times\mathrm{U}(1)_{Y} SU ( 2 ) L × U ( 1 ) Y
V = − m 11 2 ( Φ 1 † Φ 1 ) − m 22 2 ( Φ 2 † Φ 2 ) − m 33 2 ( Φ 3 † Φ 3 ) − [ m 12 2 ( Φ 1 † Φ 2 ) + m 13 2 ( Φ 1 † Φ 3 ) + m 23 2 ( Φ 2 † Φ 3 ) + h . c . ] + λ 1 ( Φ 1 † Φ 1 ) 2 + λ 2 ( Φ 2 † Φ 2 ) 2 + λ 3 ( Φ 3 † Φ 3 ) 2 + λ 4 ( Φ 1 † Φ 1 ) ( Φ 2 † Φ 2 ) + λ 5 ( Φ 1 † Φ 1 ) ( Φ 3 † Φ 3 ) + λ 6 ( Φ 2 † Φ 2 ) ( Φ 3 † Φ 3 ) + λ 7 ( Φ 1 † Φ 2 ) ( Φ 2 † Φ 1 ) + λ 8 ( Φ 1 † Φ 3 ) ( Φ 3 † Φ 1 ) + λ 9 ( Φ 2 † Φ 3 ) ( Φ 3 † Φ 2 ) + { λ 10 ( Φ 1 † Φ 2 ) 2 + λ 11 ( Φ 1 † Φ 3 ) 2 + λ 12 ( Φ 2 † Φ 3 ) 2 + [ λ 13 ( Φ 1 † Φ 1 ) + λ 14 ( Φ 2 † Φ 2 ) + λ 15 ( Φ 3 † Φ 3 ) ] ( Φ 1 † Φ 2 ) + [ λ 16 ( Φ 1 † Φ 1 ) + λ 17 ( Φ 2 † Φ 2 ) + λ 18 ( Φ 3 † Φ 3 ) ] ( Φ 1 † Φ 3 ) + [ λ 19 ( Φ 1 † Φ 1 ) + λ 20 ( Φ 2 † Φ 2 ) + λ 21 ( Φ 3 † Φ 3 ) ] ( Φ 2 † Φ 3 ) + λ 22 ( Φ 1 † Φ 2 ) ( Φ 1 † Φ 3 ) + λ 23 ( Φ 2 † Φ 1 ) ( Φ 1 † Φ 3 ) + λ 24 ( Φ 1 † Φ 3 ) ( Φ 2 † Φ 3 ) + λ 25 ( Φ 1 † Φ 3 ) ( Φ 3 † Φ 2 ) + λ 26 ( Φ 2 † Φ 1 ) ( Φ 2 † Φ 3 ) + λ 27 ( Φ 1 † Φ 2 ) ( Φ 2 † Φ 3 ) + h . c . } (11) \begin{aligned}
V ={}& -m_{11}^{2}(\Phi_{1}^{\dagger}\Phi_{1}) -m_{22}^{2}(\Phi_{2}^{\dagger}\Phi_{2}) -m_{33}^{2}(\Phi_{3}^{\dagger}\Phi_{3}) \\
&-\left[m_{12}^{2}(\Phi_{1}^{\dagger}\Phi_{2}) + m_{13}^{2}(\Phi_{1}^{\dagger}\Phi_{3}) + m_{23}^{2}(\Phi_{2}^{\dagger}\Phi_{3}) + \mathrm{h.c.}\right] \\
&+\lambda_{1}(\Phi_{1}^{\dagger}\Phi_{1})^{2}+\lambda_{2}(\Phi_{2}^{\dagger}\Phi_{2})^{2}+\lambda_{3}(\Phi_{3}^{\dagger}\Phi_{3})^{2}+\lambda_{4}(\Phi_{1}^{\dagger}\Phi_{1})(\Phi_{2}^{\dagger}\Phi_{2})+\lambda_{5}(\Phi_{1}^{\dagger}\Phi_{1})(\Phi_{3}^{\dagger}\Phi_{3}) \\
&+\lambda_{6}(\Phi_{2}^{\dagger}\Phi_{2})(\Phi_{3}^{\dagger}\Phi_{3})+\lambda_{7}(\Phi_{1}^{\dagger}\Phi_{2})(\Phi_{2}^{\dagger}\Phi_{1})+\lambda_{8}(\Phi_{1}^{\dagger}\Phi_{3})(\Phi_{3}^{\dagger}\Phi_{1})+\lambda_{9}(\Phi_{2}^{\dagger}\Phi_{3})(\Phi_{3}^{\dagger}\Phi_{2}) \\
&+\left\{\lambda_{10}(\Phi_{1}^{\dagger}\Phi_{2})^{2}+\lambda_{11}(\Phi_{1}^{\dagger}\Phi_{3})^{2}+\lambda_{12}(\Phi_{2}^{\dagger}\Phi_{3})^{2}\right. \\
&\quad+\left[\lambda_{13}(\Phi_{1}^{\dagger}\Phi_{1})+\lambda_{14}(\Phi_{2}^{\dagger}\Phi_{2})+\lambda_{15}(\Phi_{3}^{\dagger}\Phi_{3})\right](\Phi_{1}^{\dagger}\Phi_{2}) \\
&\quad+\left[\lambda_{16}(\Phi_{1}^{\dagger}\Phi_{1})+\lambda_{17}(\Phi_{2}^{\dagger}\Phi_{2})+\lambda_{18}(\Phi_{3}^{\dagger}\Phi_{3})\right](\Phi_{1}^{\dagger}\Phi_{3}) \\
&\quad+\left[\lambda_{19}(\Phi_{1}^{\dagger}\Phi_{1})+\lambda_{20}(\Phi_{2}^{\dagger}\Phi_{2})+\lambda_{21}(\Phi_{3}^{\dagger}\Phi_{3})\right](\Phi_{2}^{\dagger}\Phi_{3}) \\
&\quad+\lambda_{22}(\Phi_{1}^{\dagger}\Phi_{2})(\Phi_{1}^{\dagger}\Phi_{3})+\lambda_{23}(\Phi_{2}^{\dagger}\Phi_{1})(\Phi_{1}^{\dagger}\Phi_{3})+\lambda_{24}(\Phi_{1}^{\dagger}\Phi_{3})(\Phi_{2}^{\dagger}\Phi_{3}) \\
&\quad+\lambda_{25}(\Phi_{1}^{\dagger}\Phi_{3})(\Phi_{3}^{\dagger}\Phi_{2})+\lambda_{26}(\Phi_{2}^{\dagger}\Phi_{1})(\Phi_{2}^{\dagger}\Phi_{3})+\lambda_{27}(\Phi_{1}^{\dagger}\Phi_{2})(\Phi_{2}^{\dagger}\Phi_{3})+\mathrm{h.c.}\} \tag*{(11)}
\end{aligned} V = − m 11 2 ( Φ 1 † Φ 1 ) − m 22 2 ( Φ 2 † Φ 2 ) − m 33 2 ( Φ 3 † Φ 3 ) − [ m 12 2 ( Φ 1 † Φ 2 ) + m 13 2 ( Φ 1 † Φ 3 ) + m 23 2 ( Φ 2 † Φ 3 ) + h.c. ] + λ 1 ( Φ 1 † Φ 1 ) 2 + λ 2 ( Φ 2 † Φ 2 ) 2 + λ 3 ( Φ 3 † Φ 3 ) 2 + λ 4 ( Φ 1 † Φ 1 ) ( Φ 2 † Φ 2 ) + λ 5 ( Φ 1 † Φ 1 ) ( Φ 3 † Φ 3 ) + λ 6 ( Φ 2 † Φ 2 ) ( Φ 3 † Φ 3 ) + λ 7 ( Φ 1 † Φ 2 ) ( Φ 2 † Φ 1 ) + λ 8 ( Φ 1 † Φ 3 ) ( Φ 3 † Φ 1 ) + λ 9 ( Φ 2 † Φ 3 ) ( Φ 3 † Φ 2 ) + { λ 10 ( Φ 1 † Φ 2 ) 2 + λ 11 ( Φ 1 † Φ 3 ) 2 + λ 12 ( Φ 2 † Φ 3 ) 2 + [ λ 13 ( Φ 1 † Φ 1 ) + λ 14 ( Φ 2 † Φ 2 ) + λ 15 ( Φ 3 † Φ 3 ) ] ( Φ 1 † Φ 2 ) + [ λ 16 ( Φ 1 † Φ 1 ) + λ 17 ( Φ 2 † Φ 2 ) + λ 18 ( Φ 3 † Φ 3 ) ] ( Φ 1 † Φ 3 ) + [ λ 19 ( Φ 1 † Φ 1 ) + λ 20 ( Φ 2 † Φ 2 ) + λ 21 ( Φ 3 † Φ 3 ) ] ( Φ 2 † Φ 3 ) + λ 22 ( Φ 1 † Φ 2 ) ( Φ 1 † Φ 3 ) + λ 23 ( Φ 2 † Φ 1 ) ( Φ 1 † Φ 3 ) + λ 24 ( Φ 1 † Φ 3 ) ( Φ 2 † Φ 3 ) + λ 25 ( Φ 1 † Φ 3 ) ( Φ 3 † Φ 2 ) + λ 26 ( Φ 2 † Φ 1 ) ( Φ 2 † Φ 3 ) + λ 27 ( Φ 1 † Φ 2 ) ( Φ 2 † Φ 3 ) + h.c. } (11) where the parameters m 11 2 m_{11}^{2} m 11 2 m 22 2 m_{22}^{2} m 22 2 m 33 2 m_{33}^{2} m 33 2 λ 1 \lambda_{1} λ 1 λ 2 \lambda_{2} λ 2 λ 3 \lambda_{3} λ 3 λ 4 \lambda_{4} λ 4 λ 5 \lambda_{5} λ 5 λ 6 \lambda_{6} λ 6 λ 7 \lambda_{7} λ 7 λ 8 \lambda_{8} λ 8 λ 9 \lambda_{9} λ 9 Φ i \Phi_i Φ i Z 2 × Z 2 \mathbb{Z}_{2}\times\mathbb{Z}_{2} Z 2 × Z 2 [19 ] , the A 4 A_{4} A 4 [32 , 44 ] or a model [39 , 43 ] with a symmetry inspired by the 2HDM BGL model [50 ] . In all these cases, the global symmetries considered correspond to field transformations analogous to those of eqs. (1) or (2) , which by construction preserve the theory’s kinetic terms and gauge interactions.
Motivated by the structure of the transformations identified in the 2HDM case, we consider the following set of transformations among the scalar doublets:
Φ 1 → Φ 2 ∗ , Φ 1 † → − Φ 2 T , Φ 2 → Φ 3 ∗ , Φ 2 † → − Φ 3 T , Φ 3 → Φ 1 ∗ , Φ 3 † → − Φ 1 T . (12) \begin{aligned}
\Phi_{1} &\to\Phi_{2}^{*}, & \Phi_{1}^{\dagger} &\to-\Phi_{2}^{\mathrm{T}}, \\
\Phi_{2} &\to\Phi_{3}^{*}, & \Phi_{2}^{\dagger} &\to-\Phi_{3}^{\mathrm{T}}, \\
\Phi_{3} &\to\Phi_{1}^{*}, & \Phi_{3}^{\dagger} &\to-\Phi_{1}^{\mathrm{T}}. \tag*{(12)}
\end{aligned} Φ 1 Φ 2 Φ 3 → Φ 2 ∗ , → Φ 3 ∗ , → Φ 1 ∗ , Φ 1 † Φ 2 † Φ 3 † → − Φ 2 T , → − Φ 3 T , → − Φ 1 T . (12) If one were to consider only the transformations of the first column in the equations above, they would describe a 3HDM GCP symmetry, wherein the standard CP transformation (complex conjugation of a field) is taken in tandem with a cyclic permutation between the two fields, Φ 1 → Φ 2 → Φ 3 → Φ 1 \Phi_{1} \to\Phi_{2} \to\Phi_{3} \to\Phi_{1} Φ 1 → Φ 2 → Φ 3 → Φ 1 Φ i † Φ j Φ k † Φ l \Phi_{i}^{\dagger}\Phi_{j}\Phi_{k}^{\dagger}\Phi_{l} Φ i † Φ j Φ k † Φ l (12) will cancel out. Indeed, we see that the impact of the GOOFy symmetry on the quartic couplings will be identical to that of the GCP + cyclical permutation symmetry mentioned above – the differences arise in the quadratic sector. This has in fact been observed in all GOOFy models proposed thus far.
The field transformations of (12) can also be understood as resulting from imaginary scalings of the doublets’ real components (as in the 2HDM case, see eq. (7) ). Requiring that the scalar potential is invariant under these transformations leads to a set of parameter relations that, as we will see further ahead, are stable under RG evolution. The resulting scalar potential is given by
V G O O F y = − M [ Im ( ϕ 1 † ϕ 2 ) + Im ( ϕ 2 † ϕ 3 ) + Im ( ϕ 3 † ϕ 1 ) ] + Λ 0 [ ( ϕ 1 † ϕ 1 ) + ( ϕ 2 † ϕ 2 ) + ( ϕ 3 † ϕ 3 ) ] 2 + Λ 1 [ ( ϕ 1 † ϕ 1 ) 2 + ( ϕ 2 † ϕ 2 ) 2 + ( ϕ 3 † ϕ 3 ) 2 − ( ϕ 1 † ϕ 1 ) ( ϕ 2 † ϕ 2 ) − ( ϕ 1 † ϕ 1 ) ( ϕ 3 † ϕ 3 ) − ( ϕ 2 † ϕ 2 ) ( ϕ 3 † ϕ 3 ) ] + Λ 2 [ Re ( ϕ 1 † ϕ 2 ) 2 + Re ( ϕ 2 † ϕ 3 ) 2 + Re ( ϕ 3 † ϕ 1 ) 2 ] + Λ 3 [ Im ( ϕ 1 † ϕ 2 ) 2 + Im ( ϕ 2 † ϕ 3 ) 2 + Im ( ϕ 3 † ϕ 1 ) 2 ] + Λ 4 [ Re ( ϕ 1 † ϕ 2 ) Re ( ϕ 2 † ϕ 3 ) + Re ( ϕ 1 † ϕ 2 ) Re ( ϕ 3 † ϕ 1 ) + Re ( ϕ 2 † ϕ 3 ) Re ( ϕ 3 † ϕ 1 ) ] + Λ 5 [ Im ( ϕ 1 † ϕ 2 ) Im ( ϕ 2 † ϕ 3 ) + Im ( ϕ 1 † ϕ 2 ) Im ( ϕ 3 † ϕ 1 ) + Im ( ϕ 2 † ϕ 3 ) Im ( ϕ 3 † ϕ 1 ) ] + Λ 6 ( ϕ 1 † ϕ 1 + ϕ 2 † ϕ 2 + ϕ 3 † ϕ 3 ) [ Re ( ϕ 1 † ϕ 2 ) + Re ( ϕ 2 † ϕ 3 ) + Re ( ϕ 3 † ϕ 1 ) ] + Λ 7 [ Re ( ϕ 1 † ϕ 2 ) ( ϕ 1 † ϕ 1 + ϕ 2 † ϕ 2 − 2 ϕ 3 † ϕ 3 ) + Re ( ϕ 2 † ϕ 3 ) ( − 2 ϕ 1 † ϕ 1 + ϕ 2 † ϕ 2 + ϕ 3 † ϕ 3 ) + Re ( ϕ 3 † ϕ 1 ) ( ϕ 1 † ϕ 1 − 2 ϕ 2 † ϕ 2 + ϕ 3 † ϕ 3 ) ] + Λ 8 [ Re ( ϕ 1 † ϕ 2 ) ( ϕ 1 † ϕ 1 − ϕ 2 † ϕ 2 ) + Re ( ϕ 3 † ϕ 1 ) ( ϕ 3 † ϕ 3 − ϕ 1 † ϕ 1 ) + Re ( ϕ 2 † ϕ 3 ) ( ϕ 2 † ϕ 2 − ϕ 3 † ϕ 3 ) ] . (13) \begin{aligned}
V_{\mathrm{GOOFy}}={}&-M\left[\operatorname{Im}(\phi_1^\dagger\phi_2)+\operatorname{Im}(\phi_2^\dagger\phi_3)+\operatorname{Im}(\phi_3^\dagger\phi_1)\right]+\Lambda_0\left[(\phi_1^\dagger\phi_1)+(\phi_2^\dagger\phi_2)+(\phi_3^\dagger\phi_3)\right]^2\\
&+\Lambda_1\left[(\phi_1^\dagger\phi_1)^2+(\phi_2^\dagger\phi_2)^2+(\phi_3^\dagger\phi_3)^2-(\phi_1^\dagger\phi_1)(\phi_2^\dagger\phi_2)-(\phi_1^\dagger\phi_1)(\phi_3^\dagger\phi_3)-(\phi_2^\dagger\phi_2)(\phi_3^\dagger\phi_3)\right]\\
&+\Lambda_2\left[\operatorname{Re}(\phi_1^\dagger\phi_2)^2+\operatorname{Re}(\phi_2^\dagger\phi_3)^2+\operatorname{Re}(\phi_3^\dagger\phi_1)^2\right]+\Lambda_3\left[\operatorname{Im}(\phi_1^\dagger\phi_2)^2+\operatorname{Im}(\phi_2^\dagger\phi_3)^2+\operatorname{Im}(\phi_3^\dagger\phi_1)^2\right]\\
&+\Lambda_4\left[\operatorname{Re}(\phi_1^\dagger\phi_2)\operatorname{Re}(\phi_2^\dagger\phi_3)+\operatorname{Re}(\phi_1^\dagger\phi_2)\operatorname{Re}(\phi_3^\dagger\phi_1)+\operatorname{Re}(\phi_2^\dagger\phi_3)\operatorname{Re}(\phi_3^\dagger\phi_1)\right]\\
&+\Lambda_5\left[\operatorname{Im}(\phi_1^\dagger\phi_2)\operatorname{Im}(\phi_2^\dagger\phi_3)+\operatorname{Im}(\phi_1^\dagger\phi_2)\operatorname{Im}(\phi_3^\dagger\phi_1)+\operatorname{Im}(\phi_2^\dagger\phi_3)\operatorname{Im}(\phi_3^\dagger\phi_1)\right]\\
&+\Lambda_6\left(\phi_1^\dagger\phi_1+\phi_2^\dagger\phi_2+\phi_3^\dagger\phi_3\right)\left[\operatorname{Re}(\phi_1^\dagger\phi_2)+\operatorname{Re}(\phi_2^\dagger\phi_3)+\operatorname{Re}(\phi_3^\dagger\phi_1)\right]\\
&+\Lambda_7\left[\operatorname{Re}(\phi_1^\dagger\phi_2)(\phi_1^\dagger\phi_1+\phi_2^\dagger\phi_2-2\phi_3^\dagger\phi_3)+\operatorname{Re}(\phi_2^\dagger\phi_3)(-2\phi_1^\dagger\phi_1+\phi_2^\dagger\phi_2+\phi_3^\dagger\phi_3)\right.\\
&\qquad\left.+\operatorname{Re}(\phi_3^\dagger\phi_1)(\phi_1^\dagger\phi_1-2\phi_2^\dagger\phi_2+\phi_3^\dagger\phi_3)\right]\\
&+\Lambda_8\left[\operatorname{Re}(\phi_1^\dagger\phi_2)(\phi_1^\dagger\phi_1-\phi_2^\dagger\phi_2)+\operatorname{Re}(\phi_3^\dagger\phi_1)(\phi_3^\dagger\phi_3-\phi_1^\dagger\phi_1)+\operatorname{Re}(\phi_2^\dagger\phi_3)(\phi_2^\dagger\phi_2-\phi_3^\dagger\phi_3)\right]. \tag*{(13)}
\end{aligned} V GOOFy = − M [ Im ( ϕ 1 † ϕ 2 ) + Im ( ϕ 2 † ϕ 3 ) + Im ( ϕ 3 † ϕ 1 ) ] + Λ 0 [ ( ϕ 1 † ϕ 1 ) + ( ϕ 2 † ϕ 2 ) + ( ϕ 3 † ϕ 3 ) ] 2 + Λ 1 [ ( ϕ 1 † ϕ 1 ) 2 + ( ϕ 2 † ϕ 2 ) 2 + ( ϕ 3 † ϕ 3 ) 2 − ( ϕ 1 † ϕ 1 ) ( ϕ 2 † ϕ 2 ) − ( ϕ 1 † ϕ 1 ) ( ϕ 3 † ϕ 3 ) − ( ϕ 2 † ϕ 2 ) ( ϕ 3 † ϕ 3 ) ] + Λ 2 [ Re ( ϕ 1 † ϕ 2 ) 2 + Re ( ϕ 2 † ϕ 3 ) 2 + Re ( ϕ 3 † ϕ 1 ) 2 ] + Λ 3 [ Im ( ϕ 1 † ϕ 2 ) 2 + Im ( ϕ 2 † ϕ 3 ) 2 + Im ( ϕ 3 † ϕ 1 ) 2 ] + Λ 4 [ Re ( ϕ 1 † ϕ 2 ) Re ( ϕ 2 † ϕ 3 ) + Re ( ϕ 1 † ϕ 2 ) Re ( ϕ 3 † ϕ 1 ) + Re ( ϕ 2 † ϕ 3 ) Re ( ϕ 3 † ϕ 1 ) ] + Λ 5 [ Im ( ϕ 1 † ϕ 2 ) Im ( ϕ 2 † ϕ 3 ) + Im ( ϕ 1 † ϕ 2 ) Im ( ϕ 3 † ϕ 1 ) + Im ( ϕ 2 † ϕ 3 ) Im ( ϕ 3 † ϕ 1 ) ] + Λ 6 ( ϕ 1 † ϕ 1 + ϕ 2 † ϕ 2 + ϕ 3 † ϕ 3 ) [ Re ( ϕ 1 † ϕ 2 ) + Re ( ϕ 2 † ϕ 3 ) + Re ( ϕ 3 † ϕ 1 ) ] + Λ 7 [ Re ( ϕ 1 † ϕ 2 ) ( ϕ 1 † ϕ 1 + ϕ 2 † ϕ 2 − 2 ϕ 3 † ϕ 3 ) + Re ( ϕ 2 † ϕ 3 ) ( − 2 ϕ 1 † ϕ 1 + ϕ 2 † ϕ 2 + ϕ 3 † ϕ 3 ) + Re ( ϕ 3 † ϕ 1 ) ( ϕ 1 † ϕ 1 − 2 ϕ 2 † ϕ 2 + ϕ 3 † ϕ 3 ) ] + Λ 8 [ Re ( ϕ 1 † ϕ 2 ) ( ϕ 1 † ϕ 1 − ϕ 2 † ϕ 2 ) + Re ( ϕ 3 † ϕ 1 ) ( ϕ 3 † ϕ 3 − ϕ 1 † ϕ 1 ) + Re ( ϕ 2 † ϕ 3 ) ( ϕ 2 † ϕ 2 − ϕ 3 † ϕ 3 ) ] . (13) where all the parameters are real. These can be expressed in terms of those of the general 3HDM potential in Eq. (11) through the relations
m 12 2 = m 23 2 = m 13 2 = − i M 2 , λ 1 = 2 Λ 0 , λ 4 = 2 Λ 1 , λ 7 = Λ 2 + Λ 3 , λ 10 = Λ 2 − Λ 3 2 , m_{12}^2=m_{23}^2=m_{13}^2=-\frac{iM}{2},\qquad\lambda_1=2\Lambda_0,\qquad\lambda_4=2\Lambda_1,\qquad\lambda_7=\Lambda_2+\Lambda_3,\qquad\lambda_{10}=\frac{\Lambda_2-\Lambda_3}{2}, m 12 2 = m 23 2 = m 13 2 = − 2 i M , λ 1 = 2 Λ 0 , λ 4 = 2 Λ 1 , λ 7 = Λ 2 + Λ 3 , λ 10 = 2 Λ 2 − Λ 3 , λ 13 = − Λ 6 + Λ 7 + Λ 8 , λ 14 = − Λ 6 + Λ 7 − Λ 8 , λ 15 = − Λ 6 − 2 Λ 7 , \lambda_{13}=-\Lambda_6+\Lambda_7+\Lambda_8,\qquad\lambda_{14}=-\Lambda_6+\Lambda_7-\Lambda_8,\qquad\lambda_{15}=-\Lambda_6-2\Lambda_7, λ 13 = − Λ 6 + Λ 7 + Λ 8 , λ 14 = − Λ 6 + Λ 7 − Λ 8 , λ 15 = − Λ 6 − 2 Λ 7 , λ 22 = Λ 4 + Λ 5 2 , λ 23 = Λ 4 − Λ 5 2 (14) \lambda_{22}=\frac{\Lambda_4+\Lambda_5}{2},\qquad\lambda_{23}=\frac{\Lambda_4-\Lambda_5}{2} \tag*{(14)} λ 22 = 2 Λ 4 + Λ 5 , λ 23 = 2 Λ 4 − Λ 5 (14) with all other terms not explicitly written above equal to zero. This potential is in a different basis but corresponds to the GOOFy S 3 S_3 S 3 [16 ] , wherein 3HDM scalar potential symmetries were exhaustively classified, and where general necessary conditions for non-vanishing bi-linear terms were presented.
Let us now consider the invariance of the kinetic terms. Following the procedure outlined in [6 , 7 ] consider the kinetic terms of the doublets, L Φ K \mathcal{L}_{\Phi K} L Φ K
L Φ K = ( ∂ μ Φ 1 ) † ( ∂ μ Φ 1 ) + ( ∂ μ Φ 2 ) † ( ∂ μ Φ 2 ) + ( ∂ μ Φ 3 ) † ( ∂ μ Φ 3 ) (15) \mathcal{L}_{\Phi K}=(\partial_\mu\Phi_1)^\dagger(\partial^\mu\Phi_1)+(\partial_\mu\Phi_2)^\dagger(\partial^\mu\Phi_2)+(\partial_\mu\Phi_3)^\dagger(\partial^\mu\Phi_3) \tag*{(15)} L Φ K = ( ∂ μ Φ 1 ) † ( ∂ μ Φ 1 ) + ( ∂ μ Φ 2 ) † ( ∂ μ Φ 2 ) + ( ∂ μ Φ 3 ) † ( ∂ μ Φ 3 ) (15) Applying the transformations of Eq. (12) , L Φ K \mathcal{L}_{\Phi K} L Φ K L Φ K → − L Φ K \mathcal{L}_{\Phi K}\to-\mathcal{L}_{\Phi K} L Φ K → − L Φ K
x μ → i x μ (16) x_\mu\to ix_\mu \tag*{(16)} x μ → i x μ (16) To account for the gauge boson interactions, the partial derivative ∂ μ \partial_\mu ∂ μ D μ D_\mu D μ
D μ = ∂ μ + i 2 g σ a W μ a − i 2 g ′ Y B μ (17) D_\mu= \partial_\mu+ \frac{i}{2}g\sigma^a W_\mu^a - \frac{i}{2}g'Y B_\mu \tag*{(17)} D μ = ∂ μ + 2 i g σ a W μ a − 2 i g ′ Y B μ (17) where σ a \sigma_a σ a Y Y Y (12) into account, invariance of the full gauge-kinetic lagrangian is achieved if once again the gauge fields are scaled by imaginary factors exactly as in (8) . Indeed, it is found that each covariant derivative transforms as
D μ Φ 1 → i ( D μ Φ 2 ) ∗ , ( D μ Φ 1 ) † → − i ( D μ Φ 2 ) ⊤ D μ Φ 2 → i ( D μ Φ 3 ) ∗ , ( D μ Φ 2 ) † → − i ( D μ Φ 3 ) ⊤ D μ Φ 3 → i ( D μ Φ 1 ) ∗ , ( D μ Φ 3 ) † → − i ( D μ Φ 1 ) ⊤ (18) \begin{aligned}
D_\mu\Phi_1 &\to i(D_\mu\Phi_2)^*, & (D_\mu\Phi_1)^\dagger&\to-i(D_\mu\Phi_2)^\top\\
D_\mu\Phi_2 &\to i(D_\mu\Phi_3)^*, & (D_\mu\Phi_2)^\dagger&\to-i(D_\mu\Phi_3)^\top\\
D_\mu\Phi_3 &\to i(D_\mu\Phi_1)^*, & (D_\mu\Phi_3)^\dagger&\to-i(D_\mu\Phi_1)^\top \tag*{(18)}
\end{aligned} D μ Φ 1 D μ Φ 2 D μ Φ 3 → i ( D μ Φ 2 ) ∗ , → i ( D μ Φ 3 ) ∗ , → i ( D μ Φ 1 ) ∗ , ( D μ Φ 1 ) † ( D μ Φ 2 ) † ( D μ Φ 3 ) † → − i ( D μ Φ 2 ) ⊤ → − i ( D μ Φ 3 ) ⊤ → − i ( D μ Φ 1 ) ⊤ (18) which leaves the full gauge-kinetic terms invariant. And as in the 2HDM case, the imaginary scaling of gauge fields in (8) leaves invariant the purely gauge part of the lagrangian,
L G K = − 1 4 B μ ν B μ ν − 1 4 W μ ν a W a μ ν . (19) \mathcal{L}_{GK} = -\frac{1}{4}B_{\mu\nu}B^{\mu\nu} - \frac{1}{4}W_{\mu\nu}^a W^{a\mu\nu}. \tag*{(19)} L G K = − 4 1 B μν B μν − 4 1 W μν a W a μν . (19) Electroweak symmetry breaking
Before addressing the inclusion of fermions in the theory let us look at how electroweak symmetry occurs in this model. The 3HDM can have minima of different nature, in which vacuum expectation values (vevs) for the scalar fields break the S U ( 2 ) L × U ( 1 ) Y SU(2)_L \times U(1)_Y S U ( 2 ) L × U ( 1 ) Y U ( 1 ) e m U(1)_{em} U ( 1 ) e m
⟨ Φ a ⟩ = 1 2 ( 0 v a e i θ a ) (20) \langle\Phi_a\rangle= \frac{1}{\sqrt{2}}\begin{pmatrix} 0 \\ v_a e^{i\theta_a} \end{pmatrix} \tag*{(20)} ⟨ Φ a ⟩ = 2 1 ( 0 v a e i θ a ) (20) with
v 2 = v 1 2 + v 2 2 + v 3 2 = ( 246 G e V ) 2 (21) v^2 = v_1^2 + v_2^2 + v_3^2 = (246\ \mathrm{GeV})^2 \tag*{(21)} v 2 = v 1 2 + v 2 2 + v 3 2 = ( 246 GeV ) 2 (21) and arbitrary real phases θ a \theta_a θ a a = 1 , 2 , 3 a = 1, 2, 3 a = 1 , 2 , 3 U ( 1 ) Y U(1)_Y U ( 1 ) Y v 1 v_1 v 1
In the case of the GOOFy potential of eq. (13) , the non-existence of the quadratic coefficients m 11 2 m_{11}^2 m 11 2 m 22 2 m_{22}^2 m 22 2 m 33 2 m_{33}^2 m 33 2 v i = 0 v_i = 0 v i = 0
( ⟨ Φ 1 ⟩ , ⟨ Φ 2 ⟩ , ⟨ Φ 3 ⟩ ) = v 6 ( 1 , ω , ω 2 ) , ω = e i 2 π 3 . (22) (\langle\Phi_1\rangle,\langle\Phi_2\rangle,\langle\Phi_3\rangle) = \frac{v}{\sqrt{6}}(1,\omega,\omega^2), \qquad\omega= e^{i\frac{2\pi}{3}}. \tag*{(22)} (⟨ Φ 1 ⟩ , ⟨ Φ 2 ⟩ , ⟨ Φ 3 ⟩) = 6 v ( 1 , ω , ω 2 ) , ω = e i 3 2 π . (22) This vacuum clearly preserves the S 3 S_3 S 3 Φ 1 → Φ 2 → Φ 3 → Φ 1 \Phi_1 \to\Phi_2 \to\Phi_3 \to\Phi_1 Φ 1 → Φ 2 → Φ 3 → Φ 1 M M M
M = v 2 6 3 ( 12 Λ 0 + Λ 2 + 3 Λ 3 + Λ 4 + 3 Λ 5 − 6 Λ 6 ) . (23) M = \frac{v^2}{6\sqrt{3}}(12\Lambda_0+\Lambda_2+3\Lambda_3+\Lambda_4+3\Lambda_5-6\Lambda_6). \tag*{(23)} M = 6 3 v 2 ( 12 Λ 0 + Λ 2 + 3 Λ 3 + Λ 4 + 3 Λ 5 − 6 Λ 6 ) . (23) This yields the value of the potential at the minimum,
V = − 3 8 M v 2 . (24) V=-\frac{\sqrt{3}}{8}Mv^2. \tag*{(24)} V = − 8 3 M v 2 . (24) We can now compute the second derivatives of the potential, evaluate them at the solution of the minimization conditions (22) and compute the squared scalar masses of the model. The results reflect the restrictive symmetry both of the potential and of the vevs: we find, as expected, five different neutral scalars – one which we call h S M h_{SM} h S M h 1 h_1 h 1 h 2 h_2 h 2 H 1 ± H_1^\pm H 1 ± H 2 ± H_2^\pm H 2 ±
m h S M 2 = v 2 6 ( 12 Λ 0 + Λ 2 + 3 Λ 3 + Λ 4 + 3 Λ 5 − 6 Λ 6 ) m h 1 , 2 2 = v 2 48 ( C ± A 2 + 16 ( B 2 + 27 Λ 8 2 ) ) (twice degenerate) m H 1 ± 2 = v 2 6 ( 12 Λ 0 + Λ 2 + Λ 4 − 6 Λ 6 ) m H 2 ± 2 = v 2 12 ( 12 Λ 0 − 2 Λ 2 − 2 Λ 4 + 3 Λ 6 ) (25) \begin{aligned}
m_{h_{SM}}^2 &= \frac{v^2}{6}(12\Lambda_0+\Lambda_2+3\Lambda_3+\Lambda_4+3\Lambda_5-6\Lambda_6) \\
m_{h_{1,2}}^2 &= \frac{v^2}{48}\left(C\pm\sqrt{A^2+16(B^2+27\Lambda_8^2)}\right) \qquad\text{(twice degenerate)} \\
m_{H_1^\pm}^2 &= \frac{v^2}{6}(12\Lambda_0+\Lambda_2+\Lambda_4-6\Lambda_6) \\
m_{H_2^\pm}^2 &= \frac{v^2}{12}(12\Lambda_0-2\Lambda_2-2\Lambda_4+3\Lambda_6) \tag*{(25)}
\end{aligned} m h S M 2 m h 1 , 2 2 m H 1 ± 2 m H 2 ± 2 = 6 v 2 ( 12 Λ 0 + Λ 2 + 3 Λ 3 + Λ 4 + 3 Λ 5 − 6 Λ 6 ) = 48 v 2 ( C ± A 2 + 16 ( B 2 + 27 Λ 8 2 ) ) (twice degenerate) = 6 v 2 ( 12 Λ 0 + Λ 2 + Λ 4 − 6 Λ 6 ) = 12 v 2 ( 12 Λ 0 − 2 Λ 2 − 2 Λ 4 + 3 Λ 6 ) (25) where
A = 24 Λ 0 + 14 Λ 2 − 6 Λ 3 + 5 Λ 4 + 3 Λ 5 − 30 Λ 6 − 36 Λ 7 B = 6 Λ 1 − 2 Λ 2 + Λ 4 − 3 Λ 7 C = 72 Λ 0 + 24 Λ 1 + 10 Λ 2 + 6 Λ 3 − 5 Λ 4 − 3 Λ 5 − 18 Λ 6 − 12 Λ 7 . (26) \begin{aligned}
A &= 24\Lambda_0+14\Lambda_2-6\Lambda_3+5\Lambda_4+3\Lambda_5-30\Lambda_6-36\Lambda_7 \\
B &= 6\Lambda_1-2\Lambda_2+\Lambda_4-3\Lambda_7 \\
C &= 72\Lambda_0+24\Lambda_1+10\Lambda_2+6\Lambda_3-5\Lambda_4-3\Lambda_5-18\Lambda_6-12\Lambda_7. \tag*{(26)}
\end{aligned} A B C = 24 Λ 0 + 14 Λ 2 − 6 Λ 3 + 5 Λ 4 + 3 Λ 5 − 30 Λ 6 − 36 Λ 7 = 6 Λ 1 − 2 Λ 2 + Λ 4 − 3 Λ 7 = 72 Λ 0 + 24 Λ 1 + 10 Λ 2 + 6 Λ 3 − 5 Λ 4 − 3 Λ 5 − 18 Λ 6 − 12 Λ 7 . (26) We find the rotation matrix R R R § 7 . With this we can determine how each of the neutral states couples to the gauge bosons in the usual manner: starting with the scalars’ kinetic terms written in terms of covariant derivatives defined in (17) ,
L Φ K = ∑ a = 1 3 ( D μ Φ a ) † ( D μ Φ a ) , (27) \mathcal{L}_{\Phi K}=\sum_{a=1}^{3}(D_\mu\Phi_a)^\dagger(D^\mu\Phi_a), \tag*{(27)} L Φ K = a = 1 ∑ 3 ( D μ Φ a ) † ( D μ Φ a ) , (27) and rewriting each doublet as
Φ a = 1 2 ( v a e i θ a + ρ a + i η a ) . (28) \Phi_a=\frac{1}{\sqrt{2}}\left(v_a e^{i\theta_a}+\rho_a+i\eta_a\right). \tag*{(28)} Φ a = 2 1 ( v a e i θ a + ρ a + i η a ) . (28) then, for each neutral scalar mass eigenstate S S S S S S Z Z Z L Φ K \mathcal{L}_{\Phi K} L Φ K S Z Z SZZ S Z Z (118) and the vevs (22) , we find
L S Z Z = g 2 + g ′ 2 8 ( η 2 − η 3 + 2 ρ 1 3 − ρ 2 3 − ρ 3 3 ) v Z μ Z μ = g 2 + g ′ 2 4 v h S M Z μ Z μ . (29) \mathcal{L}_{SZZ}=\frac{g^2+g'^2}{8}\left(\eta_2-\eta_3+\frac{2\rho_1}{\sqrt{3}}-\frac{\rho_2}{\sqrt{3}}-\frac{\rho_3}{\sqrt{3}}\right)vZ_\mu Z^\mu=\frac{g^2+g'^2}{4}vh_{SM}Z_\mu Z^\mu. \tag*{(29)} L S Z Z = 8 g 2 + g ′2 ( η 2 − η 3 + 3 2 ρ 1 − 3 ρ 2 − 3 ρ 3 ) v Z μ Z μ = 4 g 2 + g ′2 v h S M Z μ Z μ . (29) This means that the only neutral scalar which has trilinear vertices with Z Z Z h S M h_{SM} h S M W W W h 1 h_1 h 1 h 2 h_2 h 2 a priori might be good dark matter candidates. This is clearly a consequence of the unbroken S 3 S_3 S 3
Including fermions
The GOOFy symmetry now needs to be extended to the fermion sector. As in the original 2HDM GOOFy model, the 3HDM field transformations of eq. (12) are akin to a GCP symmetry, so GCP-like transformations for the fermions will also be considered. In fact, they are identical in form to the ones defined for the 2HDM in eq. (10) for the quark fields (we do not treat leptons explicitly in what follows, but the reasoning is analogous).
We proceed as done in [7 ] to find the conditions that the GOOFy transformation imposes on the Yukawa matrices. We consider only the case of the down quarks, the remaining fermions would be dealt with in the same manner. With the GOOFy transformations on the scalars, eq. (12) , and fermions, eq. (10) , the Yukawa lagrangian transforms as
− L Y , d = Q ˉ L ( Γ 1 Φ 1 + Γ 2 Φ 2 + Γ 3 Φ 3 ) n R + n ˉ R ( Γ 1 † Φ 1 † + Γ 2 † Φ 2 † + Γ 3 † Φ 3 † ) Q L → (30) -\mathcal{L}_{Y,d}=\bar{Q}_{L}\left(\Gamma_{1}\Phi_{1}+\Gamma_{2}\Phi_{2}+\Gamma_{3}\Phi_{3}\right)n_{R}+\bar{n}_{R}\left(\Gamma_{1}^{\dagger}\Phi_{1}^{\dagger}+\Gamma_{2}^{\dagger}\Phi_{2}^{\dagger}+\Gamma_{3}^{\dagger}\Phi_{3}^{\dagger}\right)Q_{L}\to \tag*{(30)} − L Y , d = Q ˉ L ( Γ 1 Φ 1 + Γ 2 Φ 2 + Γ 3 Φ 3 ) n R + n ˉ R ( Γ 1 † Φ 1 † + Γ 2 † Φ 2 † + Γ 3 † Φ 3 † ) Q L → (30) → i Q ˉ L X α T ( Γ 1 ∗ Φ 2 + Γ 2 ∗ Φ 3 + Γ 3 ∗ Φ 1 ) X β ∗ n R + i n ˉ R X β T ( Γ 1 T Φ 2 † + Γ 2 T Φ 3 † + Γ 3 T Φ 1 † ) X α ∗ Q L . (31) \to i\bar{Q}_{L}X_{\alpha}^{\mathsf{T}}\left(\Gamma_{1}^{*}\Phi_{2}+\Gamma_{2}^{*}\Phi_{3}+\Gamma_{3}^{*}\Phi_{1}\right)X_{\beta}^{*}n_{R}+i\bar{n}_{R}X_{\beta}^{\mathsf{T}}\left(\Gamma_{1}^{\mathsf{T}}\Phi_{2}^{\dagger}+\Gamma_{2}^{\mathsf{T}}\Phi_{3}^{\dagger}+\Gamma_{3}^{\mathsf{T}}\Phi_{1}^{\dagger}\right)X_{\alpha}^{*}Q_{L}. \tag*{(31)} → i Q ˉ L X α T ( Γ 1 ∗ Φ 2 + Γ 2 ∗ Φ 3 + Γ 3 ∗ Φ 1 ) X β ∗ n R + i n ˉ R X β T ( Γ 1 T Φ 2 † + Γ 2 T Φ 3 † + Γ 3 T Φ 1 † ) X α ∗ Q L . (31) From this we obtain the following conditions to be satisfied by the 3 × 3 3\times3 3 × 3 Γ \Gamma Γ
Γ 1 ∗ = − i X α † Γ 3 X β \Gamma_{1}^{*}=-iX_{\alpha}^{\dagger}\Gamma_{3}X_{\beta} Γ 1 ∗ = − i X α † Γ 3 X β Γ 2 ∗ = − i X α † Γ 1 X β \Gamma_{2}^{*}=-iX_{\alpha}^{\dagger}\Gamma_{1}X_{\beta} Γ 2 ∗ = − i X α † Γ 1 X β Γ 3 ∗ = − i X α † Γ 2 X β . (32) \Gamma_{3}^{*}=-iX_{\alpha}^{\dagger}\Gamma_{2}X_{\beta}. \tag*{(32)} Γ 3 ∗ = − i X α † Γ 2 X β . (32) Recall that the X θ X_{\theta} X θ 3 × 3 3\times3 3 × 3 [49 ] to determine which choices of X θ X_{\theta} X θ
The Trivial Case: X α = X β = 1 X_{\alpha}=X_{\beta}=1 X α = X β = 1
The simplest possibility is to take all X θ X_{\theta} X θ (32) yield
Γ = Γ 1 = Γ 2 = Γ 3 = e i π / 4 ( a 11 a 12 a 13 a 21 a 22 a 23 a 31 a 32 a 33 ) , a i j ∈ R , j = 1 , 2 , 3. (33) \Gamma=\Gamma_{1}=\Gamma_{2}=\Gamma_{3}=e^{i\pi/4}
\begin{pmatrix}
a_{11} & a_{12} & a_{13}\\
a_{21} & a_{22} & a_{23}\\
a_{31} & a_{32} & a_{33}
\end{pmatrix},
\qquad a_{ij}\in\mathbb{R},\ j=1,2,3. \tag*{(33)} Γ = Γ 1 = Γ 2 = Γ 3 = e iπ /4 a 11 a 21 a 31 a 12 a 22 a 32 a 13 a 23 a 33 , a ij ∈ R , j = 1 , 2 , 3. (33) The e i π / 4 e^{i\pi/4} e iπ /4 Γ \Gamma Γ 3 × 3 3\times3 3 × 3
M n = 1 2 ( Γ 1 v 1 + Γ 2 v 2 e i θ 2 + Γ 3 v 3 e i θ 3 ) (34) M_n=\frac{1}{\sqrt{2}}\left(\Gamma_{1}v_{1}+\Gamma_{2}v_{2}e^{i\theta_{2}}+\Gamma_{3}v_{3}e^{i\theta_{3}}\right) \tag*{(34)} M n = 2 1 ( Γ 1 v 1 + Γ 2 v 2 e i θ 2 + Γ 3 v 3 e i θ 3 ) (34) = v Γ ( 1 + ω + ω 2 ) 6 (35) = \frac{v\Gamma(1+\omega+\omega^2)}{\sqrt{6}} \tag*{(35)} = 6 v Γ ( 1 + ω + ω 2 ) (35) and since ω = e 2 i π / 3 \omega=e^{2i\pi/3} ω = e 2 iπ /3 1 + ω + ω 2 = 0 1+\omega+\omega^2=0 1 + ω + ω 2 = 0 [51 ] that exact Yukawa alignment leads to at least two massless fermions for the most general set of vevs. Even though the mechanisms that lead to mass cancellation are different in both cases, they present the same feature of vanishing masses.
Generic X α X_\alpha X α X β X_\beta X β
Following the methods of [49 ] we use the work of [52 ] where it was shown that in GCP fermionic transformations it is always possible to choose a fermion basis such that, without loss of generality,
X θ = ( cos θ sin θ 0 − sin θ cos θ 0 0 0 1 ) , θ ∈ [ 0 , π / 2 ] . (36) X_\theta=
\begin{pmatrix}
\cos\theta& \sin\theta& 0 \\
-\sin\theta& \cos\theta& 0 \\
0 & 0 & 1
\end{pmatrix},
\qquad\theta\in[0,\pi/2]. \tag*{(36)} X θ = cos θ − sin θ 0 sin θ cos θ 0 0 0 1 , θ ∈ [ 0 , π /2 ] . (36) This simplifies considerably the treatment of eqs. (32) 3
X α Γ 1 ∗ + i Γ 3 X β = 0 X α Γ 2 ∗ + i Γ 1 X β = 0 X α Γ 3 ∗ + i Γ 2 X β = 0. (37) \begin{aligned}
X_\alpha\Gamma_1^*+i\Gamma_3X_\beta&=0\\
X_\alpha\Gamma_2^*+i\Gamma_1X_\beta&=0\\
X_\alpha\Gamma_3^*+i\Gamma_2X_\beta&=0. \tag*{(37)}
\end{aligned} X α Γ 1 ∗ + i Γ 3 X β X α Γ 2 ∗ + i Γ 1 X β X α Γ 3 ∗ + i Γ 2 X β = 0 = 0 = 0. (37) Following [7 ] , we redefine the Yukawa matrices such that Γ a → e − i π / 4 Γ a \Gamma_a\to e^{-i\pi/4}\Gamma_a Γ a → e − iπ /4 Γ a a = 1 , 2 , 3 a=1,2,3 a = 1 , 2 , 3 i i i 4
X α Γ 1 ∗ + Γ 3 X β = 0 X α Γ 2 ∗ + Γ 1 X β = 0 X α Γ 3 ∗ + Γ 2 X β = 0. (38) \begin{aligned}
X_\alpha\Gamma_1^*+\Gamma_3X_\beta&=0\\
X_\alpha\Gamma_2^*+\Gamma_1X_\beta&=0\\
X_\alpha\Gamma_3^*+\Gamma_2X_\beta&=0. \tag*{(38)}
\end{aligned} X α Γ 1 ∗ + Γ 3 X β X α Γ 2 ∗ + Γ 1 X β X α Γ 3 ∗ + Γ 2 X β = 0 = 0 = 0. (38) This represents a set of 18 linear equations (since each matrix has 9 entries, which have real and imaginary parts). The specific form of the X X X (36) helps to simplify the problem: we start by considering the blocks m n mn mn 3 m 3m 3 m n 3 n3 n 3 33 33 33 m , n = 1 , 2 m,n=1,2 m , n = 1 , 2 R e Re R e I m Im I m Γ 1 \Gamma_1 Γ 1 Γ 2 \Gamma_2 Γ 2 Γ 3 \Gamma_3 Γ 3 α \alpha α γ \gamma γ 3 × 3 3\times3 3 × 3 R e ( Γ 1 ) 33 Re(\Gamma_1)_{33} R e ( Γ 1 ) 33 R e ( Γ 3 ) 33 Re(\Gamma_3)_{33} R e ( Γ 3 ) 33 R e ( Γ 3 ) 33 Re(\Gamma_3)_{33} R e ( Γ 3 ) 33 α \alpha α γ \gamma γ (38) are satisfied.3
Table 1. Summary of the conditions for vanishing parameters for each block of the system of equations (38) .
The same does not happen for the imaginary parts of those matrix entries, though, and one finds Im ( Γ 1 ) 33 = Im ( Γ 3 ) 33 = Im ( Γ 3 ) 33 \operatorname{Im}(\Gamma_1)_{33} = \operatorname{Im}(\Gamma_3)_{33} = \operatorname{Im}(\Gamma_3)_{33} Im ( Γ 1 ) 33 = Im ( Γ 3 ) 33 = Im ( Γ 3 ) 33 (38) .
There would appear to be infinite possibilities for the Yukawa textures. However, the above conclusions for the (3, 3) entries of the Γ \Gamma Γ
( M n ) 33 = 1 12 [ Im ( Γ 1 ) 33 + ω Im ( Γ 2 ) 33 + ω 2 Im ( Γ 3 ) 33 ] = Im ( Γ 1 ) 33 12 [ 1 + ω + ω 2 ] = 0. (39) (M_n)_{33} = \frac{1}{\sqrt{12}}\left[\operatorname{Im}(\Gamma_1)_{33} + \omega\operatorname{Im}(\Gamma_2)_{33} + \omega^2\operatorname{Im}(\Gamma_3)_{33}\right] = \frac{\operatorname{Im}(\Gamma_1)_{33}}{\sqrt{12}}\left[1 + \omega+ \omega^2\right] = 0. \tag*{(39)} ( M n ) 33 = 12 1 [ Im ( Γ 1 ) 33 + ω Im ( Γ 2 ) 33 + ω 2 Im ( Γ 3 ) 33 ] = 12 Im ( Γ 1 ) 33 [ 1 + ω + ω 2 ] = 0. (39) Therefore, if either the third column or row of the down quark mass matrix M n M_n M n α ≠ k π / 3 \alpha\ne k\pi/3 α = k π /3 γ ≠ k π / 3 \gamma\ne k\pi/3 γ = k π /3 k ∈ Z k \in\mathbb{Z} k ∈ Z α \alpha α γ \gamma γ α , γ = k π / 3 \alpha,\gamma= k\pi/3 α , γ = k π /3 α = π 3 ± γ + 2 π k 3 \alpha= \frac{\pi}{3} \pm\gamma+ \frac{2\pi k}{3} α = 3 π ± γ + 3 2 π k α = ± γ + 2 π k 3 \alpha= \pm\gamma+ \frac{2\pi k}{3} α = ± γ + 3 2 π k k ∈ Z k \in\mathbb{Z} k ∈ Z (38) that can be grouped into three classes as shown in Table 2 . The third class is the only one that agrees with experimental evidence of three massive quarks, so all
Table 2. Number of massless fermion generations for each pair of angles ( α , γ ) = ( i π / 3 , j π / 3 ) (\alpha,\gamma) = (i\pi/3,j\pi/3) ( α , γ ) = ( iπ /3 , j π /3 )
discussion from this point deals with those specific solutions. The corresponding Yukawa matrices are presented in Appendix A. Here we will only present the one that we use in the analysis of the next chapter, which is the (1,1) case above so that, for α = γ = π 3 \alpha= \gamma= \frac{\pi}{3} α = γ = 3 π Γ 1 \Gamma_1 Γ 1
Γ 1 = ( i a 11 i a 12 a 13 i a 21 i a 22 a 23 a 31 a 32 i a 33 ) . (40) \Gamma_1 =
\begin{pmatrix}
ia_{11} & ia_{12} & a_{13} \\
ia_{21} & ia_{22} & a_{23} \\
a_{31} & a_{32} & ia_{33}
\end{pmatrix}. \tag*{(40)} Γ 1 = i a 11 i a 21 a 31 i a 12 i a 22 a 32 a 13 a 23 i a 33 . (40) We also obtain expressions for Γ 2 \Gamma_2 Γ 2 Γ 3 \Gamma_3 Γ 3 i i i Γ 1 \Gamma_1 Γ 1 { 11 , 12 , 21 , 22 } \{11,12,21,22\} { 11 , 12 , 21 , 22 } ( 3 , 3 ) (3,3) ( 3 , 3 ) 5
Γ 1 = ( a 11 a 12 a 13 a 21 a 22 a 23 a 31 a 32 a 33 ) , Γ 2 = 1 2 ( 1 2 ( a 11 − 3 a 12 − 3 a 21 + 3 a 22 ) 1 2 ( 3 a 11 + a 12 − 3 a 21 − 3 a 22 ) − a 13 + 3 a 23 1 2 ( 3 a 11 − 3 a 12 + a 21 − 3 a 22 ) 1 2 ( 3 a 11 + 3 a 12 + 3 a 21 + a 22 ) − 3 a 13 − a 23 − a 31 + 3 a 32 − 3 a 31 − a 32 2 a 33 ) , Γ 3 = 1 2 ( 1 2 ( a 11 + 3 a 12 + 3 a 21 + 3 a 22 ) 1 2 ( − 3 a 11 + a 12 − 3 a 21 + 3 a 22 ) − a 13 − 3 a 23 1 2 ( − 3 a 11 − 3 a 12 + a 21 + 3 a 22 ) 1 2 ( 3 a 11 − 3 a 12 − 3 a 21 + a 22 ) 3 a 13 − a 23 − a 31 − 3 a 32 3 a 31 − a 32 2 a 33 ) . (41) \Gamma_1 =
\begin{pmatrix}
a_{11} & a_{12} & a_{13} \\
a_{21} & a_{22} & a_{23} \\
a_{31} & a_{32} & a_{33}
\end{pmatrix},
\qquad
\Gamma_2 =
\frac{1}{2}
\begin{pmatrix}
\frac{1}{2}(a_{11}-\sqrt{3}a_{12}-\sqrt{3}a_{21}+3a_{22}) &
\frac{1}{2}(\sqrt{3}a_{11}+a_{12}-3a_{21}-\sqrt{3}a_{22}) &
-a_{13}+\sqrt{3}a_{23} \\
\frac{1}{2}(\sqrt{3}a_{11}-3a_{12}+a_{21}-\sqrt{3}a_{22}) &
\frac{1}{2}(3a_{11}+\sqrt{3}a_{12}+\sqrt{3}a_{21}+a_{22}) &
-\sqrt{3}a_{13}-a_{23} \\
-a_{31}+\sqrt{3}a_{32} &
-\sqrt{3}a_{31}-a_{32} &
2a_{33}
\end{pmatrix},
\qquad
\Gamma_3 =
\frac{1}{2}
\begin{pmatrix}
\frac{1}{2}(a_{11}+\sqrt{3}a_{12}+\sqrt{3}a_{21}+3a_{22}) &
\frac{1}{2}(-\sqrt{3}a_{11}+a_{12}-3a_{21}+\sqrt{3}a_{22}) &
-a_{13}-\sqrt{3}a_{23} \\
\frac{1}{2}(-\sqrt{3}a_{11}-3a_{12}+a_{21}+\sqrt{3}a_{22}) &
\frac{1}{2}(3a_{11}-\sqrt{3}a_{12}-\sqrt{3}a_{21}+a_{22}) &
\sqrt{3}a_{13}-a_{23} \\
-a_{31}-\sqrt{3}a_{32} &
\sqrt{3}a_{31}-a_{32} &
2a_{33}
\end{pmatrix}. \tag*{(41)} Γ 1 = a 11 a 21 a 31 a 12 a 22 a 32 a 13 a 23 a 33 , Γ 2 = 2 1 2 1 ( a 11 − 3 a 12 − 3 a 21 + 3 a 22 ) 2 1 ( 3 a 11 − 3 a 12 + a 21 − 3 a 22 ) − a 31 + 3 a 32 2 1 ( 3 a 11 + a 12 − 3 a 21 − 3 a 22 ) 2 1 ( 3 a 11 + 3 a 12 + 3 a 21 + a 22 ) − 3 a 31 − a 32 − a 13 + 3 a 23 − 3 a 13 − a 23 2 a 33 , Γ 3 = 2 1 2 1 ( a 11 + 3 a 12 + 3 a 21 + 3 a 22 ) 2 1 ( − 3 a 11 − 3 a 12 + a 21 + 3 a 22 ) − a 31 − 3 a 32 2 1 ( − 3 a 11 + a 12 − 3 a 21 + 3 a 22 ) 2 1 ( 3 a 11 − 3 a 12 − 3 a 21 + a 22 ) 3 a 31 − a 32 − a 13 − 3 a 23 3 a 13 − a 23 2 a 33 . (41) where all nine coefficients a i j a_{ij} a ij Table 2 .
If one now uses the vevs from eq. (22) to compute the down-quark mass matrix, eq. (34) , we obtain three massive quarks, and have enough freedom in the choice of parameters to correctly reproduce the experimental values for those masses. The same is true for the up-quark mass matrix M p = ∑ a = 1 3 Δ a v a e − i θ a / 2 M_p = \sum_{a=1}^{3} \Delta_a v_a e^{-i\theta_a}/\sqrt{2} M p = ∑ a = 1 3 Δ a v a e − i θ a / 2 Δ \Delta Δ (41) and now depends on 9 real b i j b_{ij} b ij (22) yield a mass matrix whose eigenvalues can easily be fitted to the know values of the up, charm and top masses.
One may now move on to the calculation of the CKM matrix: as usual, the quark mass matrices must be rotated to the mass basis via unitary transformations on the left and right quark fields,
M d = diag ( m d , m s , m b ) = U L n † M n U R n , M_d = \operatorname{diag}(m_d,m_s,m_b) = U_L^{n\dagger} M_n U_R^n, M d = diag ( m d , m s , m b ) = U L n † M n U R n , M u = diag ( m u , m c , m t ) = U L p † M p U R p , (42) M_u = \operatorname{diag}(m_u,m_c,m_t) = U_L^{p\dagger} M_p U_R^p, \tag*{(42)} M u = diag ( m u , m c , m t ) = U L p † M p U R p , (42) with the CKM matrix depending only on the left rotation matrices. The high level of symmetry both on the Yukawa matrices and the vev leads to a remarkable, if frustrating, result: the left rotation matrices for the up and dwon sector U L U_L U L constant . Indeed, a direct calculation shows that
U L n = ( − i 2 i 2 0 1 2 1 2 0 0 0 1 ) U L p = ( i 2 − i 2 0 1 2 1 2 0 0 0 1 ) . (43) U_L^n =
\begin{pmatrix}
-\frac{i}{\sqrt{2}} & \frac{i}{\sqrt{2}} & 0 \\
\frac{1}{\sqrt{2}} & \frac{1}{\sqrt{2}} & 0 \\
0 & 0 & 1
\end{pmatrix}
\qquad
U_L^p =
\begin{pmatrix}
\frac{i}{\sqrt{2}} & -\frac{i}{\sqrt{2}} & 0 \\
\frac{1}{\sqrt{2}} & \frac{1}{\sqrt{2}} & 0 \\
0 & 0 & 1
\end{pmatrix}. \tag*{(43)} U L n = − 2 i 2 1 0 2 i 2 1 0 0 0 1 U L p = 2 i 2 1 0 − 2 i 2 1 0 0 0 1 . (43) so that the CKM matric V is found to be
V = ( U L p † ) U L n = ( 0 1 0 1 0 0 0 0 1 ) . (44) V = (U_L^{p\dagger})U_L^n =
\begin{pmatrix}
0 & 1 & 0 \\
1 & 0 & 0 \\
0 & 0 & 1
\end{pmatrix}. \tag*{(44)} V = ( U L p † ) U L n = 0 1 0 1 0 0 0 0 1 . (44) and with a trivial basis rotation on the fields, this means that V V V W W W 6 Table 2 , which bring nothing new to this discussion (indeed, it is well possible that many of thoose textures are related via basis changes).
Since the CKM matrix has off-diagonal elements with magnitudes much smaller than its diagonal ones (by themselves with magnitudes close to 1), the simplest way to generalize the model is to softly break the GOOFy symmetry via extra quadratic terms in the potential (13) . This will allow us to obtain a more generic form for the vevs, which we study in § 3 . Before, however, let us address the RG invariance of this GOOFy model.
Renormalization group invariance of GOOFy parameter relations
In the original GOOFy paper [6 ] it was possible to show, using arguments of dimensional analysis and basis invariance, that the relations between parameters of eq. (4) were renormalization group (RG) invariant to all orders of perturbation theory when only scalar and gauge couplings were included. RG invariance up to two loops was proven for specific Yukawa textures as well (those corresponding to a CP2 and CP3 symmetry) by direct computation of the model’s β \beta β [10 , 13 ] , where GOOFy symmetry relations emerge from independent transformations on fields and their hermitian conjugates, and it is argued that gauge kinetic terms which break those transformations are tantamount to soft breakings and therefore do not change the renormalizability of the model. We now wish to analyse whether the GOOFy parameter relations of the 3HDM we built – to wit, the scalar sector relations of eqs. (13) and (14) , and the Yukawa textures of eqs. (33) and (41) – are preserved under renormalization.
To begin with we need not worry about the RG invariance of the dimensionless couplings. Indeed, if we think about the “S 3 S_3 S 3 Φ 1 → Φ 2 ∗ \Phi_1 \to\Phi_2^* Φ 1 → Φ 2 ∗ Φ 2 → Φ 3 ∗ \Phi_2 \to\Phi_3^* Φ 2 → Φ 3 ∗ Φ 3 → Φ 1 ∗ \Phi_3 \to\Phi_1^* Φ 3 → Φ 1 ∗ (14) . Likewise, this symmetry requires that Yukawa matrices are like those of (33) 7 (41) . Thus, since the GOOFy relations on dimensionless couplings are identical to those obtained by a regular symmetry, their RG invariance is automatically guaranteed. This had already been anticipated and explained in the discussion following eq. (12) . Our GOOFy symmetry differs from the regular S 3 S_3 S 3
We computed the one-loop 3HDM β \beta β [53 , 54 , 55 , 56 , 57 ] which give, for the m i j 2 m_{ij}^2 m ij 2
β m 11 2 = 12 λ 1 m 11 2 + 2 ( 2 λ 4 + λ 7 ) m 22 2 + 2 ( 2 λ 5 + λ 8 ) m 33 2 + [ 6 λ 13 m 12 2 ∗ + 6 λ 16 m 13 2 ∗ + 2 ( 2 λ 19 + λ 23 ) m 23 2 ∗ + h . c . ] − 3 2 ( g 1 2 + 3 g 2 2 ) m 11 2 \begin{aligned}
\beta_{m_{11}^2} ={}& 12\lambda_1 m_{11}^2 + 2(2\lambda_4+\lambda_7)m_{22}^2 + 2(2\lambda_5+\lambda_8)m_{33}^2 \\
&+ \left[6\lambda_{13}m_{12}^{2*} + 6\lambda_{16}m_{13}^{2*} + 2(2\lambda_{19}+\lambda_{23})m_{23}^{2*} + \mathrm{h.c.}\right] - \frac{3}{2}(g_1^2+3g_2^2)m_{11}^2
\end{aligned} β m 11 2 = 12 λ 1 m 11 2 + 2 ( 2 λ 4 + λ 7 ) m 22 2 + 2 ( 2 λ 5 + λ 8 ) m 33 2 + [ 6 λ 13 m 12 2 ∗ + 6 λ 16 m 13 2 ∗ + 2 ( 2 λ 19 + λ 23 ) m 23 2 ∗ + h.c. ] − 2 3 ( g 1 2 + 3 g 2 2 ) m 11 2 + 2 [ 3 Tr ( Δ 1 † Δ 1 ) + 3 Tr ( Γ 1 † Γ 1 ) + Tr ( Π 1 † Π 1 ) ] m 11 2 + { [ 3 Tr ( Δ 1 † Δ 2 ) + 3 Tr ( Γ 2 † Γ 1 ) + Tr ( Π 2 † Π 1 ) ] m 12 2 + [ 3 Tr ( Δ 1 † Δ 3 ) + 3 Tr ( Γ 3 † Γ 1 ) + Tr ( Π 3 † Π 1 ) ] m 13 2 + h . c . } (45) \begin{aligned}
&+2\left[3\operatorname{Tr}\left(\Delta_1^\dagger\Delta_1\right)+3\operatorname{Tr}\left(\Gamma_1^\dagger\Gamma_1\right)+\operatorname{Tr}\left(\Pi_1^\dagger\Pi_1\right)\right]m_{11}^2\\
&+\left\{\left[3\operatorname{Tr}\left(\Delta_1^\dagger\Delta_2\right)+3\operatorname{Tr}\left(\Gamma_2^\dagger\Gamma_1\right)+\operatorname{Tr}\left(\Pi_2^\dagger\Pi_1\right)\right]m_{12}^2\right.\\
&\left.+\left[3\operatorname{Tr}\left(\Delta_1^\dagger\Delta_3\right)+3\operatorname{Tr}\left(\Gamma_3^\dagger\Gamma_1\right)+\operatorname{Tr}\left(\Pi_3^\dagger\Pi_1\right)\right]m_{13}^2+\mathrm{h.c.}\right\} \tag*{(45)}
\end{aligned} + 2 [ 3 Tr ( Δ 1 † Δ 1 ) + 3 Tr ( Γ 1 † Γ 1 ) + Tr ( Π 1 † Π 1 ) ] m 11 2 + { [ 3 Tr ( Δ 1 † Δ 2 ) + 3 Tr ( Γ 2 † Γ 1 ) + Tr ( Π 2 † Π 1 ) ] m 12 2 + [ 3 Tr ( Δ 1 † Δ 3 ) + 3 Tr ( Γ 3 † Γ 1 ) + Tr ( Π 3 † Π 1 ) ] m 13 2 + h.c. } (45) β m 22 2 = 2 ( 2 λ 4 + λ 7 ) m 11 2 + 12 λ 2 m 22 2 + 2 ( 2 λ 6 + λ 9 ) m 33 2 + [ 6 λ 14 ∗ m 12 2 + ( 4 λ 17 ∗ + 2 λ 27 ∗ ) m 13 2 + 6 λ 20 m 23 2 + h . c . ] − 3 2 ( g 1 2 + 3 g 2 2 ) m 22 2 + 2 [ 3 Tr ( Γ 2 † Γ 2 ) + 3 Tr ( Δ 2 † Δ 2 ) + Tr ( Π 2 † Π 2 ) ] m 22 2 + { [ 3 Tr ( Γ 2 † Γ 1 ) + 3 Tr ( Δ 1 † Δ 2 ) + Tr ( Π 2 † Π 1 ) ] m 12 2 + [ 3 Tr ( Γ 3 † Γ 2 ) + 3 Tr ( Δ 2 † Δ 3 ) + Tr ( Π 3 † Π 2 ) ] m 23 2 + h . c . } (46) \begin{aligned}
\beta_{m_{22}^2}={}&2(2\lambda_4+\lambda_7)m_{11}^2+12\lambda_2m_{22}^2+2(2\lambda_6+\lambda_9)m_{33}^2\\
&+\left[6\lambda_{14}^*m_{12}^2+(4\lambda_{17}^*+2\lambda_{27}^*)m_{13}^2+6\lambda_{20}m_{23}^2+\mathrm{h.c.}\right]-\frac{3}{2}(g_1^2+3g_2^2)m_{22}^2\\
&+2\left[3\operatorname{Tr}\left(\Gamma_2^\dagger\Gamma_2\right)+3\operatorname{Tr}\left(\Delta_2^\dagger\Delta_2\right)+\operatorname{Tr}\left(\Pi_2^\dagger\Pi_2\right)\right]m_{22}^2\\
&+\left\{\left[3\operatorname{Tr}\left(\Gamma_2^\dagger\Gamma_1\right)+3\operatorname{Tr}\left(\Delta_1^\dagger\Delta_2\right)+\operatorname{Tr}\left(\Pi_2^\dagger\Pi_1\right)\right]m_{12}^2\right.\\
&\left.+\left[3\operatorname{Tr}\left(\Gamma_3^\dagger\Gamma_2\right)+3\operatorname{Tr}\left(\Delta_2^\dagger\Delta_3\right)+\operatorname{Tr}\left(\Pi_3^\dagger\Pi_2\right)\right]m_{23}^2+\mathrm{h.c.}\right\} \tag*{(46)}
\end{aligned} β m 22 2 = 2 ( 2 λ 4 + λ 7 ) m 11 2 + 12 λ 2 m 22 2 + 2 ( 2 λ 6 + λ 9 ) m 33 2 + [ 6 λ 14 ∗ m 12 2 + ( 4 λ 17 ∗ + 2 λ 27 ∗ ) m 13 2 + 6 λ 20 m 23 2 + h.c. ] − 2 3 ( g 1 2 + 3 g 2 2 ) m 22 2 + 2 [ 3 Tr ( Γ 2 † Γ 2 ) + 3 Tr ( Δ 2 † Δ 2 ) + Tr ( Π 2 † Π 2 ) ] m 22 2 + { [ 3 Tr ( Γ 2 † Γ 1 ) + 3 Tr ( Δ 1 † Δ 2 ) + Tr ( Π 2 † Π 1 ) ] m 12 2 + [ 3 Tr ( Γ 3 † Γ 2 ) + 3 Tr ( Δ 2 † Δ 3 ) + Tr ( Π 3 † Π 2 ) ] m 23 2 + h.c. } (46) β m 33 2 = 2 ( 2 λ 5 + λ 8 ) m 11 2 + 2 ( 2 λ 6 + λ 9 ) m 22 2 + 12 λ 3 m 33 2 + [ ( 4 λ 15 ∗ + 2 λ 25 ∗ ) m 12 2 + 6 m 13 2 λ 18 ∗ + 6 λ 21 m 23 2 + h . c . ] − 3 2 ( g 1 2 + 3 g 2 2 ) m 33 2 + 2 [ 3 Tr ( Γ 3 † Γ 3 ) + 3 Tr ( Δ 3 † Δ 3 ) + Tr ( Π 3 † Π 3 ) ] m 33 2 + { [ 3 Tr ( Γ 3 † Γ 1 ) + 3 Tr ( Δ 1 † Δ 3 ) + Tr ( Π 3 † Π 1 ) ] m 13 2 + [ 3 Tr ( Γ 3 † Γ 2 ) + 3 Tr ( Δ 2 † Δ 3 ) + Tr ( Π 3 † Π 2 ) ] m 23 2 + h . c . } (47) \begin{aligned}
\beta_{m_{33}^2}={}&2(2\lambda_5+\lambda_8)m_{11}^2+2(2\lambda_6+\lambda_9)m_{22}^2+12\lambda_3m_{33}^2\\
&+\left[(4\lambda_{15}^*+2\lambda_{25}^*)m_{12}^2+6m_{13}^2\lambda_{18}^*+6\lambda_{21}m_{23}^2+\mathrm{h.c.}\right]-\frac{3}{2}(g_1^2+3g_2^2)m_{33}^2\\
&+2\left[3\operatorname{Tr}\left(\Gamma_3^\dagger\Gamma_3\right)+3\operatorname{Tr}\left(\Delta_3^\dagger\Delta_3\right)+\operatorname{Tr}\left(\Pi_3^\dagger\Pi_3\right)\right]m_{33}^2\\
&+\left\{\left[3\operatorname{Tr}\left(\Gamma_3^\dagger\Gamma_1\right)+3\operatorname{Tr}\left(\Delta_1^\dagger\Delta_3\right)+\operatorname{Tr}\left(\Pi_3^\dagger\Pi_1\right)\right]m_{13}^2\right.\\
&\left.+\left[3\operatorname{Tr}\left(\Gamma_3^\dagger\Gamma_2\right)+3\operatorname{Tr}\left(\Delta_2^\dagger\Delta_3\right)+\operatorname{Tr}\left(\Pi_3^\dagger\Pi_2\right)\right]m_{23}^2+\mathrm{h.c.}\right\} \tag*{(47)}
\end{aligned} β m 33 2 = 2 ( 2 λ 5 + λ 8 ) m 11 2 + 2 ( 2 λ 6 + λ 9 ) m 22 2 + 12 λ 3 m 33 2 + [ ( 4 λ 15 ∗ + 2 λ 25 ∗ ) m 12 2 + 6 m 13 2 λ 18 ∗ + 6 λ 21 m 23 2 + h.c. ] − 2 3 ( g 1 2 + 3 g 2 2 ) m 33 2 + 2 [ 3 Tr ( Γ 3 † Γ 3 ) + 3 Tr ( Δ 3 † Δ 3 ) + Tr ( Π 3 † Π 3 ) ] m 33 2 + { [ 3 Tr ( Γ 3 † Γ 1 ) + 3 Tr ( Δ 1 † Δ 3 ) + Tr ( Π 3 † Π 1 ) ] m 13 2 + [ 3 Tr ( Γ 3 † Γ 2 ) + 3 Tr ( Δ 2 † Δ 3 ) + Tr ( Π 3 † Π 2 ) ] m 23 2 + h.c. } (47) β m 12 2 = 6 λ 13 m 11 2 + 6 λ 14 m 22 2 + 2 ( 2 λ 15 + λ 25 ) m 33 2 + 2 ( λ 4 + λ 7 ) m 12 2 + 12 λ 10 m 12 2 ∗ + 2 λ 19 ∗ m 13 2 + 6 λ 22 m 13 2 ∗ + 4 λ 23 ∗ m 13 2 + 2 λ 17 m 23 2 ∗ + 6 λ 26 ∗ m 23 2 + 4 λ 27 m 23 2 ∗ − 3 2 ( g 1 2 + 3 g 2 2 ) m 12 2 + [ 3 Tr ( Γ 1 † Γ 2 ) + 3 Tr ( Δ 2 † Δ 1 ) + Tr ( Π 1 † Π 2 ) ] ( m 11 2 + m 22 2 ) + [ 3 Tr ( Γ 1 † Γ 1 ) + 3 Tr ( Δ 1 † Δ 1 ) + Tr ( Π 1 † Π 1 ) + 3 Tr ( Γ 2 † Γ 2 ) + 3 Tr ( Δ 2 † Δ 2 ) + Tr ( Π 2 † Π 2 ) ] m 12 2 + [ 3 Tr ( Γ 3 † Γ 2 ) + 3 Tr ( Δ 3 † Δ 3 ) + Tr ( Π 3 † Π 2 ) ] m 13 2 + [ 3 Tr ( Γ 1 † Γ 3 ) + 3 Tr ( Δ 3 † Δ 1 ) + Tr ( Π 1 † Π 3 ) ] m 23 2 ∗ (48) \begin{aligned}
\beta_{m_{12}^2}={}&6\lambda_{13}m_{11}^2+6\lambda_{14}m_{22}^2+2(2\lambda_{15}+\lambda_{25})m_{33}^2+2(\lambda_4+\lambda_7)m_{12}^2+12\lambda_{10}m_{12}^{2*}+2\lambda_{19}^*m_{13}^2\\
&+6\lambda_{22}m_{13}^{2*}+4\lambda_{23}^*m_{13}^2+2\lambda_{17}m_{23}^{2*}+6\lambda_{26}^*m_{23}^2+4\lambda_{27}m_{23}^{2*}-\frac{3}{2}(g_1^2+3g_2^2)m_{12}^2\\
&+\left[3\operatorname{Tr}\left(\Gamma_1^\dagger\Gamma_2\right)+3\operatorname{Tr}\left(\Delta_2^\dagger\Delta_1\right)+\operatorname{Tr}\left(\Pi_1^\dagger\Pi_2\right)\right](m_{11}^2+m_{22}^2)\\
&+\left[3\operatorname{Tr}\left(\Gamma_1^\dagger\Gamma_1\right)+3\operatorname{Tr}\left(\Delta_1^\dagger\Delta_1\right)+\operatorname{Tr}\left(\Pi_1^\dagger\Pi_1\right)\right.\\
&\left.+3\operatorname{Tr}\left(\Gamma_2^\dagger\Gamma_2\right)+3\operatorname{Tr}\left(\Delta_2^\dagger\Delta_2\right)+\operatorname{Tr}\left(\Pi_2^\dagger\Pi_2\right)\right]m_{12}^2\\
&+\left[3\operatorname{Tr}\left(\Gamma_3^\dagger\Gamma_2\right)+3\operatorname{Tr}\left(\Delta_3^\dagger\Delta_3\right)+\operatorname{Tr}\left(\Pi_3^\dagger\Pi_2\right)\right]m_{13}^2\\
&+\left[3\operatorname{Tr}\left(\Gamma_1^\dagger\Gamma_3\right)+3\operatorname{Tr}\left(\Delta_3^\dagger\Delta_1\right)+\operatorname{Tr}\left(\Pi_1^\dagger\Pi_3\right)\right]m_{23}^{2*} \tag*{(48)}
\end{aligned} β m 12 2 = 6 λ 13 m 11 2 + 6 λ 14 m 22 2 + 2 ( 2 λ 15 + λ 25 ) m 33 2 + 2 ( λ 4 + λ 7 ) m 12 2 + 12 λ 10 m 12 2 ∗ + 2 λ 19 ∗ m 13 2 + 6 λ 22 m 13 2 ∗ + 4 λ 23 ∗ m 13 2 + 2 λ 17 m 23 2 ∗ + 6 λ 26 ∗ m 23 2 + 4 λ 27 m 23 2 ∗ − 2 3 ( g 1 2 + 3 g 2 2 ) m 12 2 + [ 3 Tr ( Γ 1 † Γ 2 ) + 3 Tr ( Δ 2 † Δ 1 ) + Tr ( Π 1 † Π 2 ) ] ( m 11 2 + m 22 2 ) + [ 3 Tr ( Γ 1 † Γ 1 ) + 3 Tr ( Δ 1 † Δ 1 ) + Tr ( Π 1 † Π 1 ) + 3 Tr ( Γ 2 † Γ 2 ) + 3 Tr ( Δ 2 † Δ 2 ) + Tr ( Π 2 † Π 2 ) ] m 12 2 + [ 3 Tr ( Γ 3 † Γ 2 ) + 3 Tr ( Δ 3 † Δ 3 ) + Tr ( Π 3 † Π 2 ) ] m 13 2 + [ 3 Tr ( Γ 1 † Γ 3 ) + 3 Tr ( Δ 3 † Δ 1 ) + Tr ( Π 1 † Π 3 ) ] m 23 2 ∗ (48) β m 13 2 = 6 λ 16 m 11 2 + 2 ( 2 λ 17 + λ 27 ) m 22 2 + 6 λ 18 m 33 2 + 2 λ 19 m 12 2 + 6 λ 22 m 12 2 ∗ + 4 λ 23 m 12 2 + 2 λ 5 m 13 2 + 4 λ 8 m 13 2 + 12 λ 11 m 13 2 ∗ + 2 λ 15 m 23 2 + 6 λ 24 m 23 2 ∗ + 4 λ 25 m 23 2 − 3 2 ( g 1 2 + 3 g 2 2 ) m 13 2 + [ 3 Tr ( Γ 1 † Γ 3 ) + 3 Tr ( Δ 3 † Δ 1 ) + Tr ( Π 1 † Π 3 ) ] ( m 11 2 + m 33 2 ) + [ 3 Tr ( Γ 2 † Γ 3 ) + 3 Tr ( Δ 3 † Δ 2 ) + Tr ( Π 2 † Π 3 ) ] m 12 2 + [ 3 Tr ( Γ 1 † Γ 1 ) + 3 Tr ( Δ 1 † Δ 1 ) + Tr ( Π 1 † Π 1 ) ] + 3 Tr ( Γ 3 † Γ 3 ) + 3 Tr ( Δ 3 † Δ 3 ) + Tr ( Π 3 † Π 3 ) m 13 2 + [ 3 Tr ( Γ 1 † Γ 2 ) + 3 Tr ( Δ 2 † Δ 1 ) + Tr ( Π 1 † Π 2 ) ] m 23 2 (49) \beta_{m_{13}^{2}} = 6\lambda_{16}m_{11}^{2} + 2(2\lambda_{17}+\lambda_{27})m_{22}^{2} + 6\lambda_{18}m_{33}^{2} + 2\lambda_{19}m_{12}^{2} + 6\lambda_{22}m_{12}^{2*} + 4\lambda_{23}m_{12}^{2} + 2\lambda_{5}m_{13}^{2} \\
+4\lambda_{8}m_{13}^{2} + 12\lambda_{11}m_{13}^{2*} + 2\lambda_{15}m_{23}^{2} + 6\lambda_{24}m_{23}^{2*} + 4\lambda_{25}m_{23}^{2} - \frac{3}{2}(g_{1}^{2}+3g_{2}^{2})m_{13}^{2} \\
+\left[3\operatorname{Tr}\left(\Gamma_{1}^{\dagger}\Gamma_{3}\right)+3\operatorname{Tr}\left(\Delta_{3}^{\dagger}\Delta_{1}\right)+\operatorname{Tr}\left(\Pi_{1}^{\dagger}\Pi_{3}\right)\right](m_{11}^{2}+m_{33}^{2}) \\
+\left[3\operatorname{Tr}\left(\Gamma_{2}^{\dagger}\Gamma_{3}\right)+3\operatorname{Tr}\left(\Delta_{3}^{\dagger}\Delta_{2}\right)+\operatorname{Tr}\left(\Pi_{2}^{\dagger}\Pi_{3}\right)\right]m_{12}^{2} \\
+\left[3\operatorname{Tr}\left(\Gamma_{1}^{\dagger}\Gamma_{1}\right)+3\operatorname{Tr}\left(\Delta_{1}^{\dagger}\Delta_{1}\right)+\operatorname{Tr}\left(\Pi_{1}^{\dagger}\Pi_{1}\right)\right] \\
+3\operatorname{Tr}\left(\Gamma_{3}^{\dagger}\Gamma_{3}\right)+3\operatorname{Tr}\left(\Delta_{3}^{\dagger}\Delta_{3}\right)+\operatorname{Tr}\left(\Pi_{3}^{\dagger}\Pi_{3}\right)m_{13}^{2} \\
+\left[3\operatorname{Tr}\left(\Gamma_{1}^{\dagger}\Gamma_{2}\right)+3\operatorname{Tr}\left(\Delta_{2}^{\dagger}\Delta_{1}\right)+\operatorname{Tr}\left(\Pi_{1}^{\dagger}\Pi_{2}\right)\right]m_{23}^{2} \tag*{(49)} β m 13 2 = 6 λ 16 m 11 2 + 2 ( 2 λ 17 + λ 27 ) m 22 2 + 6 λ 18 m 33 2 + 2 λ 19 m 12 2 + 6 λ 22 m 12 2 ∗ + 4 λ 23 m 12 2 + 2 λ 5 m 13 2 + 4 λ 8 m 13 2 + 12 λ 11 m 13 2 ∗ + 2 λ 15 m 23 2 + 6 λ 24 m 23 2 ∗ + 4 λ 25 m 23 2 − 2 3 ( g 1 2 + 3 g 2 2 ) m 13 2 + [ 3 Tr ( Γ 1 † Γ 3 ) + 3 Tr ( Δ 3 † Δ 1 ) + Tr ( Π 1 † Π 3 ) ] ( m 11 2 + m 33 2 ) + [ 3 Tr ( Γ 2 † Γ 3 ) + 3 Tr ( Δ 3 † Δ 2 ) + Tr ( Π 2 † Π 3 ) ] m 12 2 + [ 3 Tr ( Γ 1 † Γ 1 ) + 3 Tr ( Δ 1 † Δ 1 ) + Tr ( Π 1 † Π 1 ) ] + 3 Tr ( Γ 3 † Γ 3 ) + 3 Tr ( Δ 3 † Δ 3 ) + Tr ( Π 3 † Π 3 ) m 13 2 + [ 3 Tr ( Γ 1 † Γ 2 ) + 3 Tr ( Δ 2 † Δ 1 ) + Tr ( Π 1 † Π 2 ) ] m 23 2 (49) β m 23 2 = 2 ( 2 λ 19 + λ 23 ) m 11 2 + 6 λ 20 m 22 2 + 6 λ 21 m 33 2 + 2 λ 17 m 12 2 ∗ + 6 λ 26 m 12 2 + 4 λ 27 m 12 2 ∗ + 2 λ 15 ∗ m 13 2 + 6 λ 24 m 13 2 ∗ + 4 λ 25 ∗ m 13 2 + 2 λ 6 m 23 2 + 4 λ 9 m 23 2 + 12 λ 12 m 23 2 ∗ − 3 2 ( g 1 2 + 3 g 2 2 ) m 23 2 + [ 3 Tr ( Γ 1 † Γ 3 ) + 3 Tr ( Δ 3 † Δ 2 ) + Tr ( Π 2 † Π 3 ) ] ( m 22 2 + m 33 2 ) + [ 3 Tr ( Γ 1 † Γ 3 ) + 3 Tr ( Δ 3 † Δ 1 ) + Tr ( Π 1 † Π 3 ) ] m 12 2 ∗ + [ 3 Tr ( Γ 2 † Γ 1 ) + 3 Tr ( Δ 1 † Δ 2 ) + Tr ( Π 2 † Π 1 ) ] m 13 2 + [ 3 Tr ( Γ 2 † Γ 2 ) + 3 Tr ( Δ 2 † Δ 2 ) + Tr ( Π 2 † Π 2 ) ] + 3 Tr ( Γ 3 † Γ 3 ) + 3 Tr ( Δ 3 † Δ 3 ) + Tr ( Π 3 † Π 3 ) m 23 2 . (50) \beta_{m_{23}^{2}} = 2(2\lambda_{19}+\lambda_{23})m_{11}^{2} + 6\lambda_{20}m_{22}^{2} + 6\lambda_{21}m_{33}^{2} + 2\lambda_{17}m_{12}^{2*} + 6\lambda_{26}m_{12}^{2} + 4\lambda_{27}m_{12}^{2*} + 2\lambda_{15}^{*}m_{13}^{2} \\
+6\lambda_{24}m_{13}^{2*} + 4\lambda_{25}^{*}m_{13}^{2} + 2\lambda_{6}m_{23}^{2} + 4\lambda_{9}m_{23}^{2} + 12\lambda_{12}m_{23}^{2*} - \frac{3}{2}(g_{1}^{2}+3g_{2}^{2})m_{23}^{2} \\
+\left[3\operatorname{Tr}\left(\Gamma_{1}^{\dagger}\Gamma_{3}\right)+3\operatorname{Tr}\left(\Delta_{3}^{\dagger}\Delta_{2}\right)+\operatorname{Tr}\left(\Pi_{2}^{\dagger}\Pi_{3}\right)\right](m_{22}^{2}+m_{33}^{2}) \\
+\left[3\operatorname{Tr}\left(\Gamma_{1}^{\dagger}\Gamma_{3}\right)+3\operatorname{Tr}\left(\Delta_{3}^{\dagger}\Delta_{1}\right)+\operatorname{Tr}\left(\Pi_{1}^{\dagger}\Pi_{3}\right)\right]m_{12}^{2*} \\
+\left[3\operatorname{Tr}\left(\Gamma_{2}^{\dagger}\Gamma_{1}\right)+3\operatorname{Tr}\left(\Delta_{1}^{\dagger}\Delta_{2}\right)+\operatorname{Tr}\left(\Pi_{2}^{\dagger}\Pi_{1}\right)\right]m_{13}^{2} \\
+\left[3\operatorname{Tr}\left(\Gamma_{2}^{\dagger}\Gamma_{2}\right)+3\operatorname{Tr}\left(\Delta_{2}^{\dagger}\Delta_{2}\right)+\operatorname{Tr}\left(\Pi_{2}^{\dagger}\Pi_{2}\right)\right] \\
+3\operatorname{Tr}\left(\Gamma_{3}^{\dagger}\Gamma_{3}\right)+3\operatorname{Tr}\left(\Delta_{3}^{\dagger}\Delta_{3}\right)+\operatorname{Tr}\left(\Pi_{3}^{\dagger}\Pi_{3}\right)m_{23}^{2}. \tag*{(50)} β m 23 2 = 2 ( 2 λ 19 + λ 23 ) m 11 2 + 6 λ 20 m 22 2 + 6 λ 21 m 33 2 + 2 λ 17 m 12 2 ∗ + 6 λ 26 m 12 2 + 4 λ 27 m 12 2 ∗ + 2 λ 15 ∗ m 13 2 + 6 λ 24 m 13 2 ∗ + 4 λ 25 ∗ m 13 2 + 2 λ 6 m 23 2 + 4 λ 9 m 23 2 + 12 λ 12 m 23 2 ∗ − 2 3 ( g 1 2 + 3 g 2 2 ) m 23 2 + [ 3 Tr ( Γ 1 † Γ 3 ) + 3 Tr ( Δ 3 † Δ 2 ) + Tr ( Π 2 † Π 3 ) ] ( m 22 2 + m 33 2 ) + [ 3 Tr ( Γ 1 † Γ 3 ) + 3 Tr ( Δ 3 † Δ 1 ) + Tr ( Π 1 † Π 3 ) ] m 12 2 ∗ + [ 3 Tr ( Γ 2 † Γ 1 ) + 3 Tr ( Δ 1 † Δ 2 ) + Tr ( Π 2 † Π 1 ) ] m 13 2 + [ 3 Tr ( Γ 2 † Γ 2 ) + 3 Tr ( Δ 2 † Δ 2 ) + Tr ( Π 2 † Π 2 ) ] + 3 Tr ( Γ 3 † Γ 3 ) + 3 Tr ( Δ 3 † Δ 3 ) + Tr ( Π 3 † Π 3 ) m 23 2 . (50) Let us now write these expressions with the scalar quartic couplings of the GOOFy model, eq. (14) , but keeping generic quadratic parameters and (for the moment) the general form for the Yukawa contributions. This gives us
β m 11 2 = 24 ( Λ 0 − Λ 1 ) m 11 2 + 2 ( 8 Λ 0 + 4 Λ 1 + Λ 2 + Λ 3 ) ( m 22 2 + m 33 2 ) + [ − 6 ( Λ 6 − Λ 7 − Λ 8 ) m 12 2 − 6 ( Λ 6 − Λ 7 + Λ 8 ) m 13 2 + ( Λ 4 − Λ 5 − 4 Λ 6 − 8 Λ 7 ) m 23 2 + h . c . ] − 3 2 ( g 1 2 + 3 g 2 2 ) m 11 2 + 2 [ 3 Tr ( Γ 1 † Γ 1 ) + 3 Tr ( Δ 1 † Δ 1 ) + Tr ( Π 1 † Π 1 ) ] m 11 2 + { [ 3 Tr ( Γ 2 † Γ 1 ) + 3 Tr ( Δ 1 † Δ 2 ) + Tr ( Π 2 † Π 1 ) ] m 12 2 + [ 3 Tr ( Γ 3 † Γ 1 ) + 3 Tr ( Δ 1 † Δ 3 ) + Tr ( Π 3 † Π 1 ) ] m 13 2 + h . c . } (51) \beta_{m_{11}^{2}} = 24(\Lambda_{0}-\Lambda_{1})m_{11}^{2} + 2(8\Lambda_{0}+4\Lambda_{1}+\Lambda_{2}+\Lambda_{3})(m_{22}^{2}+m_{33}^{2}) \\
+\left[-6(\Lambda_{6}-\Lambda_{7}-\Lambda_{8})m_{12}^{2}-6(\Lambda_{6}-\Lambda_{7}+\Lambda_{8})m_{13}^{2} \\
+(\Lambda_{4}-\Lambda_{5}-4\Lambda_{6}-8\Lambda_{7})m_{23}^{2}+\mathrm{h.c.}\right]-\frac{3}{2}(g_{1}^{2}+3g_{2}^{2})m_{11}^{2} \\
+2\left[3\operatorname{Tr}\left(\Gamma_{1}^{\dagger}\Gamma_{1}\right)+3\operatorname{Tr}\left(\Delta_{1}^{\dagger}\Delta_{1}\right)+\operatorname{Tr}\left(\Pi_{1}^{\dagger}\Pi_{1}\right)\right]m_{11}^{2} \\
+\left\{\left[3\operatorname{Tr}\left(\Gamma_{2}^{\dagger}\Gamma_{1}\right)+3\operatorname{Tr}\left(\Delta_{1}^{\dagger}\Delta_{2}\right)+\operatorname{Tr}\left(\Pi_{2}^{\dagger}\Pi_{1}\right)\right]m_{12}^{2} \\
+\left[3\operatorname{Tr}\left(\Gamma_{3}^{\dagger}\Gamma_{1}\right)+3\operatorname{Tr}\left(\Delta_{1}^{\dagger}\Delta_{3}\right)+\operatorname{Tr}\left(\Pi_{3}^{\dagger}\Pi_{1}\right)\right]m_{13}^{2}+\mathrm{h.c.}\right\} \tag*{(51)} β m 11 2 = 24 ( Λ 0 − Λ 1 ) m 11 2 + 2 ( 8 Λ 0 + 4 Λ 1 + Λ 2 + Λ 3 ) ( m 22 2 + m 33 2 ) + [ − 6 ( Λ 6 − Λ 7 − Λ 8 ) m 12 2 − 6 ( Λ 6 − Λ 7 + Λ 8 ) m 13 2 + ( Λ 4 − Λ 5 − 4 Λ 6 − 8 Λ 7 ) m 23 2 + h.c. ] − 2 3 ( g 1 2 + 3 g 2 2 ) m 11 2 + 2 [ 3 Tr ( Γ 1 † Γ 1 ) + 3 Tr ( Δ 1 † Δ 1 ) + Tr ( Π 1 † Π 1 ) ] m 11 2 + { [ 3 Tr ( Γ 2 † Γ 1 ) + 3 Tr ( Δ 1 † Δ 2 ) + Tr ( Π 2 † Π 1 ) ] m 12 2 + [ 3 Tr ( Γ 3 † Γ 1 ) + 3 Tr ( Δ 1 † Δ 3 ) + Tr ( Π 3 † Π 1 ) ] m 13 2 + h.c. } (51) β m 22 2 = 24 ( Λ 0 − Λ 1 ) m 22 2 + 2 ( 8 Λ 0 + 4 Λ 1 + Λ 2 + Λ 3 ) ( m 11 2 + m 33 2 ) + [ − 6 ( Λ 6 − Λ 7 + Λ 8 ) m 12 2 + ( Λ 4 − Λ 5 − 4 Λ 6 − 8 Λ 7 ) m 13 2 − 6 ( Λ 6 − 7 Λ 7 − Λ 8 ) m 23 2 + h . c . ] − 3 2 ( g 1 2 + 3 g 2 2 ) m 22 2 + 2 [ 3 Tr ( Γ 2 † Γ 2 ) + 3 Tr ( Δ 2 † Δ 2 ) + Tr ( Π 2 † Π 2 ) ] m 22 2 + { [ 3 Tr ( Γ 2 † Γ 1 ) + 3 Tr ( Δ 1 † Δ 2 ) + Tr ( Π 2 † Π 1 ) ] m 12 2 + [ 3 Tr ( Γ 2 † Γ 2 ) + 3 Tr ( Δ 2 † Δ 3 ) + Tr ( Π 2 † Π 2 ) ] m 23 2 + h . c . } (52) \begin{aligned}
\beta_{m_{22}^{2}}={}&24\left(\Lambda_0-\Lambda_1\right)m_{22}^{2}+2\left(8\Lambda_0+4\Lambda_1+\Lambda_2+\Lambda_3\right)\left(m_{11}^{2}+m_{33}^{2}\right)\\
&+\left[-6\left(\Lambda_6-\Lambda_7+\Lambda_8\right)m_{12}^{2}+\left(\Lambda_4-\Lambda_5-4\Lambda_6-8\Lambda_7\right)m_{13}^{2}\right.\\
&\left.\qquad-6\left(\Lambda_6-7\Lambda_7-\Lambda_8\right)m_{23}^{2}+\mathrm{h.c.}\right]-\frac{3}{2}\left(g_1^{2}+3g_2^{2}\right)m_{22}^{2}\\
&+2\left[3\operatorname{Tr}\left(\Gamma_2^\dagger\Gamma_2\right)+3\operatorname{Tr}\left(\Delta_2^\dagger\Delta_2\right)+\operatorname{Tr}\left(\Pi_2^\dagger\Pi_2\right)\right]m_{22}^{2}\\
&+\left\{\left[3\operatorname{Tr}\left(\Gamma_2^\dagger\Gamma_1\right)+3\operatorname{Tr}\left(\Delta_1^\dagger\Delta_2\right)+\operatorname{Tr}\left(\Pi_2^\dagger\Pi_1\right)\right]m_{12}^{2}\right.\\
&\left.\qquad+\left[3\operatorname{Tr}\left(\Gamma_2^\dagger\Gamma_2\right)+3\operatorname{Tr}\left(\Delta_2^\dagger\Delta_3\right)+\operatorname{Tr}\left(\Pi_2^\dagger\Pi_2\right)\right]m_{23}^{2}+\mathrm{h.c.}\right\} \tag*{(52)}
\end{aligned} β m 22 2 = 24 ( Λ 0 − Λ 1 ) m 22 2 + 2 ( 8 Λ 0 + 4 Λ 1 + Λ 2 + Λ 3 ) ( m 11 2 + m 33 2 ) + [ − 6 ( Λ 6 − Λ 7 + Λ 8 ) m 12 2 + ( Λ 4 − Λ 5 − 4 Λ 6 − 8 Λ 7 ) m 13 2 − 6 ( Λ 6 − 7 Λ 7 − Λ 8 ) m 23 2 + h.c. ] − 2 3 ( g 1 2 + 3 g 2 2 ) m 22 2 + 2 [ 3 Tr ( Γ 2 † Γ 2 ) + 3 Tr ( Δ 2 † Δ 2 ) + Tr ( Π 2 † Π 2 ) ] m 22 2 + { [ 3 Tr ( Γ 2 † Γ 1 ) + 3 Tr ( Δ 1 † Δ 2 ) + Tr ( Π 2 † Π 1 ) ] m 12 2 + [ 3 Tr ( Γ 2 † Γ 2 ) + 3 Tr ( Δ 2 † Δ 3 ) + Tr ( Π 2 † Π 2 ) ] m 23 2 + h.c. } (52) β m 33 2 = 24 ( Λ 0 − Λ 1 ) m 33 2 + 2 ( 8 Λ 0 + 4 Λ 1 + Λ 2 + Λ 3 ) ( m 11 2 + m 22 2 ) + [ ( Λ 4 − Λ 5 − 4 Λ 6 − 8 Λ 7 ) m 12 2 − 6 ( Λ 6 − Λ 7 − Λ 8 ) m 13 2 − 6 ( Λ 6 − 7 Λ 7 + Λ 8 ) m 23 2 + h . c . ] − 3 2 ( g 1 2 + 3 g 2 2 ) m 33 2 + 2 [ 3 Tr ( Γ 3 † Γ 3 ) + 3 Tr ( Δ 3 † Δ 3 ) + Tr ( Π 3 † Π 3 ) ] m 33 2 + { [ 3 Tr ( Γ 3 † Γ 1 ) + 3 Tr ( Δ 1 † Δ 3 ) + Tr ( Π 3 † Π 1 ) ] m 13 2 + [ 3 Tr ( Γ 3 † Γ 2 ) + 3 Tr ( Δ 2 † Δ 3 ) + Tr ( Π 3 † Π 2 ) ] m 23 2 + h . c . } (53) \begin{aligned}
\beta_{m_{33}^{2}}={}&24\left(\Lambda_0-\Lambda_1\right)m_{33}^{2}+2\left(8\Lambda_0+4\Lambda_1+\Lambda_2+\Lambda_3\right)\left(m_{11}^{2}+m_{22}^{2}\right)\\
&+\left[\left(\Lambda_4-\Lambda_5-4\Lambda_6-8\Lambda_7\right)m_{12}^{2}-6\left(\Lambda_6-\Lambda_7-\Lambda_8\right)m_{13}^{2}\right.\\
&\left.\qquad-6\left(\Lambda_6-7\Lambda_7+\Lambda_8\right)m_{23}^{2}+\mathrm{h.c.}\right]-\frac{3}{2}\left(g_1^{2}+3g_2^{2}\right)m_{33}^{2}\\
&+2\left[3\operatorname{Tr}\left(\Gamma_3^\dagger\Gamma_3\right)+3\operatorname{Tr}\left(\Delta_3^\dagger\Delta_3\right)+\operatorname{Tr}\left(\Pi_3^\dagger\Pi_3\right)\right]m_{33}^{2}\\
&+\left\{\left[3\operatorname{Tr}\left(\Gamma_3^\dagger\Gamma_1\right)+3\operatorname{Tr}\left(\Delta_1^\dagger\Delta_3\right)+\operatorname{Tr}\left(\Pi_3^\dagger\Pi_1\right)\right]m_{13}^{2}\right.\\
&\left.\qquad+\left[3\operatorname{Tr}\left(\Gamma_3^\dagger\Gamma_2\right)+3\operatorname{Tr}\left(\Delta_2^\dagger\Delta_3\right)+\operatorname{Tr}\left(\Pi_3^\dagger\Pi_2\right)\right]m_{23}^{2}+\mathrm{h.c.}\right\} \tag*{(53)}
\end{aligned} β m 33 2 = 24 ( Λ 0 − Λ 1 ) m 33 2 + 2 ( 8 Λ 0 + 4 Λ 1 + Λ 2 + Λ 3 ) ( m 11 2 + m 22 2 ) + [ ( Λ 4 − Λ 5 − 4 Λ 6 − 8 Λ 7 ) m 12 2 − 6 ( Λ 6 − Λ 7 − Λ 8 ) m 13 2 − 6 ( Λ 6 − 7 Λ 7 + Λ 8 ) m 23 2 + h.c. ] − 2 3 ( g 1 2 + 3 g 2 2 ) m 33 2 + 2 [ 3 Tr ( Γ 3 † Γ 3 ) + 3 Tr ( Δ 3 † Δ 3 ) + Tr ( Π 3 † Π 3 ) ] m 33 2 + { [ 3 Tr ( Γ 3 † Γ 1 ) + 3 Tr ( Δ 1 † Δ 3 ) + Tr ( Π 3 † Π 1 ) ] m 13 2 + [ 3 Tr ( Γ 3 † Γ 2 ) + 3 Tr ( Δ 2 † Δ 3 ) + Tr ( Π 3 † Π 2 ) ] m 23 2 + h.c. } (53) β m 12 2 = − 6 ( Λ 6 − Λ 7 ) ( m 11 2 + m 22 2 ) + 6 Λ 8 ( m 11 2 − m 22 2 ) + ( Λ 4 − Λ 5 − 4 Λ 6 − 8 Λ 7 ) m 33 2 + 4 ( 2 Λ 0 + Λ 1 + Λ 2 + Λ 3 ) m 12 2 + 6 ( Λ 2 − Λ 3 ) m 12 2 ∗ + 2 ( Λ 4 − Λ 5 − Λ 6 − 2 Λ 7 ) m 13 2 + 3 ( Λ 4 + Λ 5 ) m 13 2 ∗ + 3 ( Λ 4 + Λ 5 ) m 23 2 + 2 ( Λ 4 − Λ 5 − Λ 6 − 2 Λ 7 ) m 23 2 ∗ − 3 2 ( g 1 2 + 3 g 2 2 ) m 12 2 + [ 3 Tr ( Γ 1 † Γ 2 ) + 3 Tr ( Δ 2 † Δ 1 ) + Tr ( Π 1 † Π 2 ) ] ( m 11 2 + m 22 2 ) + [ 3 Tr ( Γ 1 † Γ 1 ) + 3 Tr ( Δ 1 † Δ 1 ) + Tr ( Π 1 † Π 1 ) + 3 Tr ( Γ 2 † Γ 2 ) + 3 Tr ( Δ 2 † Δ 2 ) + Tr ( Π 2 † Π 2 ) ] m 12 2 + [ 3 Tr ( Γ 3 † Γ 2 ) + 3 Tr ( Δ 2 † Δ 3 ) + Tr ( Π 3 † Π 2 ) ] m 13 2 + [ 3 Tr ( Γ 1 † Γ 3 ) + 3 Tr ( Δ 3 † Δ 1 ) + Tr ( Π 1 † Π 3 ) ] m 23 2 ∗ (54) \begin{aligned}
\beta_{m_{12}^{2}}={}&-6\left(\Lambda_6-\Lambda_7\right)\left(m_{11}^{2}+m_{22}^{2}\right)+6\Lambda_8\left(m_{11}^{2}-m_{22}^{2}\right)+\left(\Lambda_4-\Lambda_5-4\Lambda_6-8\Lambda_7\right)m_{33}^{2}\\
&+4\left(2\Lambda_0+\Lambda_1+\Lambda_2+\Lambda_3\right)m_{12}^{2}+6\left(\Lambda_2-\Lambda_3\right)m_{12}^{2*}\\
&+2\left(\Lambda_4-\Lambda_5-\Lambda_6-2\Lambda_7\right)m_{13}^{2}+3\left(\Lambda_4+\Lambda_5\right)m_{13}^{2*}\\
&+3\left(\Lambda_4+\Lambda_5\right)m_{23}^{2}+2\left(\Lambda_4-\Lambda_5-\Lambda_6-2\Lambda_7\right)m_{23}^{2*}-\frac{3}{2}\left(g_1^{2}+3g_2^{2}\right)m_{12}^{2}\\
&+\left[3\operatorname{Tr}\left(\Gamma_1^\dagger\Gamma_2\right)+3\operatorname{Tr}\left(\Delta_2^\dagger\Delta_1\right)+\operatorname{Tr}\left(\Pi_1^\dagger\Pi_2\right)\right]\left(m_{11}^{2}+m_{22}^{2}\right)\\
&+\left[3\operatorname{Tr}\left(\Gamma_1^\dagger\Gamma_1\right)+3\operatorname{Tr}\left(\Delta_1^\dagger\Delta_1\right)+\operatorname{Tr}\left(\Pi_1^\dagger\Pi_1\right)\right.\\
&\left.\qquad+3\operatorname{Tr}\left(\Gamma_2^\dagger\Gamma_2\right)+3\operatorname{Tr}\left(\Delta_2^\dagger\Delta_2\right)+\operatorname{Tr}\left(\Pi_2^\dagger\Pi_2\right)\right]m_{12}^{2}\\
&+\left[3\operatorname{Tr}\left(\Gamma_3^\dagger\Gamma_2\right)+3\operatorname{Tr}\left(\Delta_2^\dagger\Delta_3\right)+\operatorname{Tr}\left(\Pi_3^\dagger\Pi_2\right)\right]m_{13}^{2}\\
&+\left[3\operatorname{Tr}\left(\Gamma_1^\dagger\Gamma_3\right)+3\operatorname{Tr}\left(\Delta_3^\dagger\Delta_1\right)+\operatorname{Tr}\left(\Pi_1^\dagger\Pi_3\right)\right]m_{23}^{2*} \tag*{(54)}
\end{aligned} β m 12 2 = − 6 ( Λ 6 − Λ 7 ) ( m 11 2 + m 22 2 ) + 6 Λ 8 ( m 11 2 − m 22 2 ) + ( Λ 4 − Λ 5 − 4 Λ 6 − 8 Λ 7 ) m 33 2 + 4 ( 2 Λ 0 + Λ 1 + Λ 2 + Λ 3 ) m 12 2 + 6 ( Λ 2 − Λ 3 ) m 12 2 ∗ + 2 ( Λ 4 − Λ 5 − Λ 6 − 2 Λ 7 ) m 13 2 + 3 ( Λ 4 + Λ 5 ) m 13 2 ∗ + 3 ( Λ 4 + Λ 5 ) m 23 2 + 2 ( Λ 4 − Λ 5 − Λ 6 − 2 Λ 7 ) m 23 2 ∗ − 2 3 ( g 1 2 + 3 g 2 2 ) m 12 2 + [ 3 Tr ( Γ 1 † Γ 2 ) + 3 Tr ( Δ 2 † Δ 1 ) + Tr ( Π 1 † Π 2 ) ] ( m 11 2 + m 22 2 ) + [ 3 Tr ( Γ 1 † Γ 1 ) + 3 Tr ( Δ 1 † Δ 1 ) + Tr ( Π 1 † Π 1 ) + 3 Tr ( Γ 2 † Γ 2 ) + 3 Tr ( Δ 2 † Δ 2 ) + Tr ( Π 2 † Π 2 ) ] m 12 2 + [ 3 Tr ( Γ 3 † Γ 2 ) + 3 Tr ( Δ 2 † Δ 3 ) + Tr ( Π 3 † Π 2 ) ] m 13 2 + [ 3 Tr ( Γ 1 † Γ 3 ) + 3 Tr ( Δ 3 † Δ 1 ) + Tr ( Π 1 † Π 3 ) ] m 23 2 ∗ (54) β m 13 2 = − 6 ( Λ 6 − Λ 7 ) ( m 33 2 + m 11 2 ) + 6 Λ 8 ( m 33 2 − m 11 2 ) + ( Λ 4 − Λ 5 − 4 Λ 6 − 8 Λ 7 ) m 22 2 + 2 ( Λ 4 − Λ 5 − Λ 6 − 2 Λ 7 ) m 12 2 + 3 ( Λ 4 + Λ 5 ) m 12 2 ∗ + 4 ( 2 Λ 0 + Λ 1 + Λ 2 + Λ 3 ) m 13 2 + 6 ( Λ 2 − Λ 3 ) m 13 2 ∗ + 2 ( Λ 4 − Λ 5 − Λ 6 − 2 Λ 7 ) m 23 2 + 3 ( Λ 4 + Λ 5 ) m 23 2 ∗ − 3 2 ( g 1 2 + 3 g 2 2 ) m 13 2 + [ 3 Tr ( Γ 1 † Γ 3 ) + 3 Tr ( Δ 3 † Δ 1 ) + Tr ( Π 1 † Π 3 ) ] ( m 11 2 + m 33 2 ) + [ 3 Tr ( Γ 2 † Γ 3 ) + 3 Tr ( Δ 3 † Δ 2 ) + Tr ( Π 2 † Π 3 ) ] m 12 2 + [ 3 Tr ( Γ 1 † Γ 1 ) + 3 Tr ( Δ 1 † Δ 1 ) + Tr ( Π 1 † Π 1 ) ] + 3 Tr ( Γ 3 † Γ 3 ) + 3 Tr ( Δ 3 † Δ 3 ) + Tr ( Π 3 † Π 3 ) m 13 2 + [ 3 Tr ( Γ 1 † Γ 2 ) + 3 Tr ( Δ 2 † Δ 1 ) + Tr ( Π 1 † Π 2 ) ] m 23 2 (55) \begin{aligned}
\beta_{m_{13}^{2}}={}&-6\left(\Lambda_6-\Lambda_7\right)\left(m_{33}^{2}+m_{11}^{2}\right)+6\Lambda_8\left(m_{33}^{2}-m_{11}^{2}\right)+\left(\Lambda_4-\Lambda_5-4\Lambda_6-8\Lambda_7\right)m_{22}^{2}\\
&+2\left(\Lambda_4-\Lambda_5-\Lambda_6-2\Lambda_7\right)m_{12}^{2}+3\left(\Lambda_4+\Lambda_5\right)m_{12}^{2*}\\
&+4\left(2\Lambda_0+\Lambda_1+\Lambda_2+\Lambda_3\right)m_{13}^{2}+6\left(\Lambda_2-\Lambda_3\right)m_{13}^{2*}\\
&+2\left(\Lambda_4-\Lambda_5-\Lambda_6-2\Lambda_7\right)m_{23}^{2}+3\left(\Lambda_4+\Lambda_5\right)m_{23}^{2*}-\frac{3}{2}\left(g_1^2+3g_2^2\right)m_{13}^{2}\\
&+\left[3\operatorname{Tr}\left(\Gamma_1^\dagger\Gamma_3\right)+3\operatorname{Tr}\left(\Delta_3^\dagger\Delta_1\right)+\operatorname{Tr}\left(\Pi_1^\dagger\Pi_3\right)\right]\left(m_{11}^{2}+m_{33}^{2}\right)\\
&+\left[3\operatorname{Tr}\left(\Gamma_2^\dagger\Gamma_3\right)+3\operatorname{Tr}\left(\Delta_3^\dagger\Delta_2\right)+\operatorname{Tr}\left(\Pi_2^\dagger\Pi_3\right)\right]m_{12}^{2}\\
&+\left[3\operatorname{Tr}\left(\Gamma_1^\dagger\Gamma_1\right)+3\operatorname{Tr}\left(\Delta_1^\dagger\Delta_1\right)+\operatorname{Tr}\left(\Pi_1^\dagger\Pi_1\right)\right]\\
&+3\operatorname{Tr}\left(\Gamma_3^\dagger\Gamma_3\right)+3\operatorname{Tr}\left(\Delta_3^\dagger\Delta_3\right)+\operatorname{Tr}\left(\Pi_3^\dagger\Pi_3\right)m_{13}^{2}\\
&+\left[3\operatorname{Tr}\left(\Gamma_1^\dagger\Gamma_2\right)+3\operatorname{Tr}\left(\Delta_2^\dagger\Delta_1\right)+\operatorname{Tr}\left(\Pi_1^\dagger\Pi_2\right)\right]m_{23}^{2} \tag*{(55)}
\end{aligned} β m 13 2 = − 6 ( Λ 6 − Λ 7 ) ( m 33 2 + m 11 2 ) + 6 Λ 8 ( m 33 2 − m 11 2 ) + ( Λ 4 − Λ 5 − 4 Λ 6 − 8 Λ 7 ) m 22 2 + 2 ( Λ 4 − Λ 5 − Λ 6 − 2 Λ 7 ) m 12 2 + 3 ( Λ 4 + Λ 5 ) m 12 2 ∗ + 4 ( 2 Λ 0 + Λ 1 + Λ 2 + Λ 3 ) m 13 2 + 6 ( Λ 2 − Λ 3 ) m 13 2 ∗ + 2 ( Λ 4 − Λ 5 − Λ 6 − 2 Λ 7 ) m 23 2 + 3 ( Λ 4 + Λ 5 ) m 23 2 ∗ − 2 3 ( g 1 2 + 3 g 2 2 ) m 13 2 + [ 3 Tr ( Γ 1 † Γ 3 ) + 3 Tr ( Δ 3 † Δ 1 ) + Tr ( Π 1 † Π 3 ) ] ( m 11 2 + m 33 2 ) + [ 3 Tr ( Γ 2 † Γ 3 ) + 3 Tr ( Δ 3 † Δ 2 ) + Tr ( Π 2 † Π 3 ) ] m 12 2 + [ 3 Tr ( Γ 1 † Γ 1 ) + 3 Tr ( Δ 1 † Δ 1 ) + Tr ( Π 1 † Π 1 ) ] + 3 Tr ( Γ 3 † Γ 3 ) + 3 Tr ( Δ 3 † Δ 3 ) + Tr ( Π 3 † Π 3 ) m 13 2 + [ 3 Tr ( Γ 1 † Γ 2 ) + 3 Tr ( Δ 2 † Δ 1 ) + Tr ( Π 1 † Π 2 ) ] m 23 2 (55) β m 23 2 = − 6 ( Λ 6 − Λ 7 ) ( m 22 2 + m 33 2 ) + 6 Λ 8 ( m 22 2 − m 33 2 ) + ( Λ 4 − Λ 5 − 4 Λ 6 − 8 Λ 7 ) m 11 2 + 3 ( Λ 4 + Λ 5 ) m 12 2 + 2 ( Λ 4 − Λ 5 − Λ 6 − 2 Λ 7 ) m 12 2 ∗ + 2 ( Λ 4 − Λ 5 − Λ 6 − 2 Λ 7 ) m 13 2 + 3 ( Λ 4 + Λ 5 ) m 13 2 ∗ + 4 ( 2 Λ 0 + Λ 1 + Λ 2 + Λ 3 ) m 23 2 + 6 ( Λ 2 − Λ 3 ) m 23 2 ∗ − 3 2 ( g 1 2 + 3 g 2 2 ) m 23 2 + [ 3 Tr ( Γ 2 † Γ 3 ) + 3 Tr ( Δ 3 † Δ 2 ) + Tr ( Π 2 † Π 3 ) ] ( m 22 2 + m 33 2 ) + [ 3 Tr ( Γ 1 † Γ 3 ) + 3 Tr ( Δ 3 † Δ 1 ) + Tr ( Π 1 † Π 3 ) ] m 12 2 ∗ + [ 3 Tr ( Γ 1 † Γ 1 ) + 3 Tr ( Δ 1 † Δ 2 ) + Tr ( Π 1 † Π 2 ) ] m 13 2 + [ 3 Tr ( Γ 2 † Γ 2 ) + 3 Tr ( Δ 2 † Δ 2 ) + Tr ( Π 2 † Π 2 ) ] + 3 Tr ( Γ 3 † Γ 3 ) + 3 Tr ( Δ 3 † Δ 3 ) + Tr ( Π 3 † Π 3 ) m 23 2 . (56) \begin{aligned}
\beta_{m_{23}^{2}}={}&-6\left(\Lambda_6-\Lambda_7\right)\left(m_{22}^{2}+m_{33}^{2}\right)+6\Lambda_8\left(m_{22}^{2}-m_{33}^{2}\right)+\left(\Lambda_4-\Lambda_5-4\Lambda_6-8\Lambda_7\right)m_{11}^{2}\\
&+3\left(\Lambda_4+\Lambda_5\right)m_{12}^{2}+2\left(\Lambda_4-\Lambda_5-\Lambda_6-2\Lambda_7\right)m_{12}^{2*}\\
&+2\left(\Lambda_4-\Lambda_5-\Lambda_6-2\Lambda_7\right)m_{13}^{2}+3\left(\Lambda_4+\Lambda_5\right)m_{13}^{2*}\\
&+4\left(2\Lambda_0+\Lambda_1+\Lambda_2+\Lambda_3\right)m_{23}^{2}+6\left(\Lambda_2-\Lambda_3\right)m_{23}^{2*}-\frac{3}{2}\left(g_1^2+3g_2^2\right)m_{23}^{2}\\
&+\left[3\operatorname{Tr}\left(\Gamma_2^\dagger\Gamma_3\right)+3\operatorname{Tr}\left(\Delta_3^\dagger\Delta_2\right)+\operatorname{Tr}\left(\Pi_2^\dagger\Pi_3\right)\right]\left(m_{22}^{2}+m_{33}^{2}\right)\\
&+\left[3\operatorname{Tr}\left(\Gamma_1^\dagger\Gamma_3\right)+3\operatorname{Tr}\left(\Delta_3^\dagger\Delta_1\right)+\operatorname{Tr}\left(\Pi_1^\dagger\Pi_3\right)\right]m_{12}^{2*}\\
&+\left[3\operatorname{Tr}\left(\Gamma_1^\dagger\Gamma_1\right)+3\operatorname{Tr}\left(\Delta_1^\dagger\Delta_2\right)+\operatorname{Tr}\left(\Pi_1^\dagger\Pi_2\right)\right]m_{13}^{2}\\
&+\left[3\operatorname{Tr}\left(\Gamma_2^\dagger\Gamma_2\right)+3\operatorname{Tr}\left(\Delta_2^\dagger\Delta_2\right)+\operatorname{Tr}\left(\Pi_2^\dagger\Pi_2\right)\right]\\
&+3\operatorname{Tr}\left(\Gamma_3^\dagger\Gamma_3\right)+3\operatorname{Tr}\left(\Delta_3^\dagger\Delta_3\right)+\operatorname{Tr}\left(\Pi_3^\dagger\Pi_3\right)m_{23}^{2}. \tag*{(56)}
\end{aligned} β m 23 2 = − 6 ( Λ 6 − Λ 7 ) ( m 22 2 + m 33 2 ) + 6 Λ 8 ( m 22 2 − m 33 2 ) + ( Λ 4 − Λ 5 − 4 Λ 6 − 8 Λ 7 ) m 11 2 + 3 ( Λ 4 + Λ 5 ) m 12 2 + 2 ( Λ 4 − Λ 5 − Λ 6 − 2 Λ 7 ) m 12 2 ∗ + 2 ( Λ 4 − Λ 5 − Λ 6 − 2 Λ 7 ) m 13 2 + 3 ( Λ 4 + Λ 5 ) m 13 2 ∗ + 4 ( 2 Λ 0 + Λ 1 + Λ 2 + Λ 3 ) m 23 2 + 6 ( Λ 2 − Λ 3 ) m 23 2 ∗ − 2 3 ( g 1 2 + 3 g 2 2 ) m 23 2 + [ 3 Tr ( Γ 2 † Γ 3 ) + 3 Tr ( Δ 3 † Δ 2 ) + Tr ( Π 2 † Π 3 ) ] ( m 22 2 + m 33 2 ) + [ 3 Tr ( Γ 1 † Γ 3 ) + 3 Tr ( Δ 3 † Δ 1 ) + Tr ( Π 1 † Π 3 ) ] m 12 2 ∗ + [ 3 Tr ( Γ 1 † Γ 1 ) + 3 Tr ( Δ 1 † Δ 2 ) + Tr ( Π 1 † Π 2 ) ] m 13 2 + [ 3 Tr ( Γ 2 † Γ 2 ) + 3 Tr ( Δ 2 † Δ 2 ) + Tr ( Π 2 † Π 2 ) ] + 3 Tr ( Γ 3 † Γ 3 ) + 3 Tr ( Δ 3 † Δ 3 ) + Tr ( Π 3 † Π 3 ) m 23 2 . (56) Let us look at β m 21 2 \beta_{m_{21}^{2}} β m 21 2 m 11 2 = m 22 2 = m 33 2 = 0 m_{11}^{2}=m_{22}^{2}=m_{33}^{2}=0 m 11 2 = m 22 2 = m 33 2 = 0 m 12 2 m_{12}^{2} m 12 2 m 13 2 m_{13}^{2} m 13 2 m 23 2 m_{23}^{2} m 23 2 m 11 2 m_{11}^{2} m 11 2 m 12 2 m_{12}^{2} m 12 2
3 [ Tr ( Γ 2 † Γ 1 ) m 12 2 + Tr ( Γ 1 † Γ 2 ) m 12 2 ∗ ] . (57) 3\left[\operatorname{Tr}\left(\Gamma_2^\dagger\Gamma_1\right)m_{12}^{2}+\operatorname{Tr}\left(\Gamma_1^\dagger\Gamma_2\right)m_{12}^{2*}\right]. \tag*{(57)} 3 [ Tr ( Γ 2 † Γ 1 ) m 12 2 + Tr ( Γ 1 † Γ 2 ) m 12 2 ∗ ] . (57) We now observe that for both Yukawa textures considered – those of eq. (33) , which yield massless quarks, and eq. (41), which yield massive quarks but a CKM matrix equal to the identity – the Γ \Gamma Γ
Tr ( Γ 1 † Γ 1 ) = Tr ( Γ 1 † Γ 2 ) = Tr ( Γ 1 † Γ 3 ) (58) \operatorname{Tr}\left(\Gamma_1^\dagger\Gamma_1\right)=\operatorname{Tr}\left(\Gamma_1^\dagger\Gamma_2\right)=\operatorname{Tr}\left(\Gamma_1^\dagger\Gamma_3\right) \tag*{(58)} Tr ( Γ 1 † Γ 1 ) = Tr ( Γ 1 † Γ 2 ) = Tr ( Γ 1 † Γ 3 ) (58) Tr ( Γ 1 † Γ 2 ) = Tr ( Γ 1 † Γ 3 ) = Tr ( Γ 1 † Γ 3 ) (59) \operatorname{Tr}\left(\Gamma_1^\dagger\Gamma_2\right)=\operatorname{Tr}\left(\Gamma_1^\dagger\Gamma_3\right)=\operatorname{Tr}\left(\Gamma_1^\dagger\Gamma_3\right) \tag*{(59)} Tr ( Γ 1 † Γ 2 ) = Tr ( Γ 1 † Γ 3 ) = Tr ( Γ 1 † Γ 3 ) (59) Tr ( Γ i † Γ j ) = Tr ( Γ j † Γ i ) , i , j = 1 , 2 , 3. (60) \operatorname{Tr}\left(\Gamma_i^\dagger\Gamma_j\right)=\operatorname{Tr}\left(\Gamma_j^\dagger\Gamma_i\right),\quad i,j=1,2,3. \tag*{(60)} Tr ( Γ i † Γ j ) = Tr ( Γ j † Γ i ) , i , j = 1 , 2 , 3. (60) The last of these equations is a trivial consequence of the fact that, as explained earlier, all GOOFy Yukawa matrices are real. Therefore, for the GOOFy-symmetric Yukawa sector of our 3HDM we may rewrite (57) as
3 Tr ( Γ 1 † Γ 1 ) ( m 12 2 + m 12 2 ∗ ) = 0 , (61) 3\operatorname{Tr}\left(\Gamma_1^\dagger\Gamma_1\right)\left(m_{12}^2+m_{12}^{2*}\right)=0, \tag*{(61)} 3 Tr ( Γ 1 † Γ 1 ) ( m 12 2 + m 12 2 ∗ ) = 0 , (61) where the final equality is due to the fact that the GOOFy symmetry implies Re ( m 12 2 ) = 0 \operatorname{Re}(m_{12}^2)=0 Re ( m 12 2 ) = 0 m 13 2 m_{13}^2 m 13 2 β m 11 2 \beta_{m_{11}^2} β m 11 2 β m 22 2 \beta_{m_{22}^2} β m 22 2 β m 33 2 \beta_{m_{33}^2} β m 33 2
m 11 2 = m 22 2 = m 33 2 = Re ( m 12 2 ) = Re ( m 13 2 ) = Re ( m 23 2 ) = 0 ⟹ β m 11 2 = β m 22 2 = β m 33 2 = 0. (62) m_{11}^2=m_{22}^2=m_{33}^2=\operatorname{Re}(m_{12}^2)=\operatorname{Re}(m_{13}^2)=\operatorname{Re}(m_{23}^2)=0\Longrightarrow\beta_{m_{11}^2}=\beta_{m_{22}^2}=\beta_{m_{33}^2}=0. \tag*{(62)} m 11 2 = m 22 2 = m 33 2 = Re ( m 12 2 ) = Re ( m 13 2 ) = Re ( m 23 2 ) = 0 ⟹ β m 11 2 = β m 22 2 = β m 33 2 = 0. (62) Notice that this conclusion holds even without using the first of the conditions of (14), Im ( m 12 2 ) = Im ( m 23 2 ) = − Im ( m 13 2 ) = − M / 2 \operatorname{Im}(m_{12}^2)=\operatorname{Im}(m_{23}^2)=-\operatorname{Im}(m_{13}^2)=-M/2 Im ( m 12 2 ) = Im ( m 23 2 ) = − Im ( m 13 2 ) = − M /2
Let us now consider β m 12 2 \beta_{m_{12}^2} β m 12 2 8 Γ \Gamma Γ
β Im ( m 12 2 ) = 2 ( 4 Λ 0 + 2 Λ 1 − Λ 2 + 5 Λ 3 ) Im ( m 12 2 ) + ( Λ 4 + 5 Λ 5 + 2 Λ 6 + 4 Λ 7 ) [ Im ( m 23 2 ) − Im ( m 13 2 ) ] + [ 6 Tr ( Γ 1 † Γ 1 ) − 3 2 ( g 1 2 + 3 g 2 2 ) ] Im ( m 12 2 ) + 3 Tr ( Γ 1 † Γ 2 ) [ Im ( m 13 2 ) − Im ( m 23 2 ) ] , (63) \begin{aligned}
\beta_{\operatorname{Im}(m_{12}^2)}={}&2(4\Lambda_0+2\Lambda_1-\Lambda_2+5\Lambda_3)\operatorname{Im}(m_{12}^2)\\
&+(\Lambda_4+5\Lambda_5+2\Lambda_6+4\Lambda_7)\left[\operatorname{Im}(m_{23}^2)-\operatorname{Im}(m_{13}^2)\right]\\
&+\left[6\operatorname{Tr}\left(\Gamma_1^\dagger\Gamma_1\right)-\frac{3}{2}(g_1^2+3g_2^2)\right]\operatorname{Im}(m_{12}^2)+3\operatorname{Tr}\left(\Gamma_1^\dagger\Gamma_2\right)\left[\operatorname{Im}(m_{13}^2)-\operatorname{Im}(m_{23}^2)\right], \tag*{(63)}
\end{aligned} β Im ( m 12 2 ) = 2 ( 4 Λ 0 + 2 Λ 1 − Λ 2 + 5 Λ 3 ) Im ( m 12 2 ) + ( Λ 4 + 5 Λ 5 + 2 Λ 6 + 4 Λ 7 ) [ Im ( m 23 2 ) − Im ( m 13 2 ) ] + [ 6 Tr ( Γ 1 † Γ 1 ) − 2 3 ( g 1 2 + 3 g 2 2 ) ] Im ( m 12 2 ) + 3 Tr ( Γ 1 † Γ 2 ) [ Im ( m 13 2 ) − Im ( m 23 2 ) ] , (63) where we have already used eqs. (58)–(60) to simplify the result. Likewise, we find
β Im ( m 13 2 ) = 2 ( 4 Λ 0 + 2 Λ 1 − Λ 2 + 5 Λ 3 ) Im ( m 13 2 ) − ( Λ 4 + 5 Λ 5 + 2 Λ 6 + 4 Λ 7 ) [ Im ( m 12 2 ) + Im ( m 23 2 ) ] − [ 6 Tr ( Γ 1 † Γ 1 ) − 3 2 ( g 1 2 + 3 g 2 2 ) ] Im ( m 13 2 ) + 3 Tr ( Γ 1 † Γ 2 ) [ Im ( m 12 2 ) + Im ( m 23 2 ) ] , \begin{aligned}
\beta_{\operatorname{Im}(m_{13}^2)}={}&2(4\Lambda_0+2\Lambda_1-\Lambda_2+5\Lambda_3)\operatorname{Im}(m_{13}^2)\\
&-(\Lambda_4+5\Lambda_5+2\Lambda_6+4\Lambda_7)\left[\operatorname{Im}(m_{12}^2)+\operatorname{Im}(m_{23}^2)\right]\\
&-\left[6\operatorname{Tr}\left(\Gamma_1^\dagger\Gamma_1\right)-\frac{3}{2}(g_1^2+3g_2^2)\right]\operatorname{Im}(m_{13}^2)+3\operatorname{Tr}\left(\Gamma_1^\dagger\Gamma_2\right)\left[\operatorname{Im}(m_{12}^2)+\operatorname{Im}(m_{23}^2)\right],
\end{aligned} β Im ( m 13 2 ) = 2 ( 4 Λ 0 + 2 Λ 1 − Λ 2 + 5 Λ 3 ) Im ( m 13 2 ) − ( Λ 4 + 5 Λ 5 + 2 Λ 6 + 4 Λ 7 ) [ Im ( m 12 2 ) + Im ( m 23 2 ) ] − [ 6 Tr ( Γ 1 † Γ 1 ) − 2 3 ( g 1 2 + 3 g 2 2 ) ] Im ( m 13 2 ) + 3 Tr ( Γ 1 † Γ 2 ) [ Im ( m 12 2 ) + Im ( m 23 2 ) ] , β Im ( m 23 2 ) = 2 ( 4 Λ 0 + 2 Λ 1 − Λ 2 + 5 Λ 3 ) Im ( m 23 2 ) + ( Λ 4 + 5 Λ 5 + 2 Λ 6 + 4 Λ 7 ) [ Im ( m 12 2 ) − Im ( m 13 2 ) ] + [ 6 Tr ( Γ 1 † Γ 1 ) − 3 2 ( g 1 2 + 3 g 2 2 ) ] Im ( m 23 2 ) + 3 Tr ( Γ 1 † Γ 2 ) [ Im ( m 13 2 ) − Im ( m 12 2 ) ] , (64) \begin{aligned}
\beta_{\operatorname{Im}(m_{23}^2)}={}&2(4\Lambda_0+2\Lambda_1-\Lambda_2+5\Lambda_3)\operatorname{Im}(m_{23}^2)\\
&+(\Lambda_4+5\Lambda_5+2\Lambda_6+4\Lambda_7)\left[\operatorname{Im}(m_{12}^2)-\operatorname{Im}(m_{13}^2)\right]\\
&+\left[6\operatorname{Tr}\left(\Gamma_1^\dagger\Gamma_1\right)-\frac{3}{2}(g_1^2+3g_2^2)\right]\operatorname{Im}(m_{23}^2)+3\operatorname{Tr}\left(\Gamma_1^\dagger\Gamma_2\right)\left[\operatorname{Im}(m_{13}^2)-\operatorname{Im}(m_{12}^2)\right], \tag*{(64)}
\end{aligned} β Im ( m 23 2 ) = 2 ( 4 Λ 0 + 2 Λ 1 − Λ 2 + 5 Λ 3 ) Im ( m 23 2 ) + ( Λ 4 + 5 Λ 5 + 2 Λ 6 + 4 Λ 7 ) [ Im ( m 12 2 ) − Im ( m 13 2 ) ] + [ 6 Tr ( Γ 1 † Γ 1 ) − 2 3 ( g 1 2 + 3 g 2 2 ) ] Im ( m 23 2 ) + 3 Tr ( Γ 1 † Γ 2 ) [ Im ( m 13 2 ) − Im ( m 12 2 ) ] , (64) It then becomes simple to see that if Im ( m 12 2 ) = − Im ( m 13 2 ) = Im ( m 23 2 ) \operatorname{Im}(m_{12}^{2})=-\operatorname{Im}(m_{13}^{2})=\operatorname{Im}(m_{23}^{2}) Im ( m 12 2 ) = − Im ( m 13 2 ) = Im ( m 23 2 ) β Im ( m 12 2 ) = − β Im ( m 13 2 ) = β Im ( m 23 2 ) \beta_{\operatorname{Im}(m_{12}^{2})}=-\beta_{\operatorname{Im}(m_{13}^{2})}=\beta_{\operatorname{Im}(m_{23}^{2})} β Im ( m 12 2 ) = − β Im ( m 13 2 ) = β Im ( m 23 2 ) (14) . It is equally simple to prove that if Im ( m 12 2 ) = − Im ( m 13 2 ) = M 1 \operatorname{Im}(m_{12}^{2})=-\operatorname{Im}(m_{13}^{2})=M_{1} Im ( m 12 2 ) = − Im ( m 13 2 ) = M 1 Im ( m 23 2 ) = M 2 \operatorname{Im}(m_{23}^{2})=M_{2} Im ( m 23 2 ) = M 2 β Im ( m 12 2 ) = − β Im ( m 13 2 ) \beta_{\operatorname{Im}(m_{12}^{2})}=-\beta_{\operatorname{Im}(m_{13}^{2})} β Im ( m 12 2 ) = − β Im ( m 13 2 ) 9
We further point out that all of the above results were were confirmed at two-loops using the symbolic calculation package PyR@TE [58 ] . We do not show the explicit expressions for the β \beta β (14) is easily checked with that code. Likewise, the soft symmetry breaking conditions discussed in the previous paragraph were also found to be RG-invariant up to at least two loops.
The softly broken model
As we saw in the previous section, the 3HDM with the GOOFy symmetry of eq. (12) yields a model where spontaneous electroweak symmetry breaking can be achieved with a vacuum with vevs (22) , which preserve a residual S 3 S_{3} S 3 (32) , and we found several textures satisfying these conditions: (A) a set of three identical real matrices for each fermion of the same charge, eq. (33) ; and (B) a set of solutions (the third class of textures in Table 2 ), an example of which is shown in eq. (41) . However, all such textures clash with experimental data: the class (A) produces massless fermions, and all textures in class (B), though producing three generations of massive fermions, yield a CKM matrix equal to the identity. The only way to improve this situation is to move away from the vacuum of eq. (22) , which implies adding soft breaking terms to the GOOFy scalar potential.
Soft breaking terms and vacuum
We consider the minimal modification of the potential that is needed to produce a valid fermion sector and maintaining a GOOFy-like structure. To wit, the quadratic terms in the scalar potential, V 2 V_{2} V 2 (13) , now become
V 2 → − M 1 [ Im ( ϕ 1 † ϕ 2 ) + Im ( ϕ 3 † ϕ 1 ) ] − M 2 Im ( ϕ 2 † ϕ 3 ) . (65) V_{2}\to-M_{1}\left[\operatorname{Im}(\phi_{1}^{\dagger}\phi_{2})+\operatorname{Im}(\phi_{3}^{\dagger}\phi_{1})\right]-M_{2}\operatorname{Im}(\phi_{2}^{\dagger}\phi_{3}) . \tag*{(65)} V 2 → − M 1 [ Im ( ϕ 1 † ϕ 2 ) + Im ( ϕ 3 † ϕ 1 ) ] − M 2 Im ( ϕ 2 † ϕ 3 ) . (65) We have verified that this choice of soft breaking terms is RG-invariant at least up to two loops in perturbation theory. It is easy to see, for instance, that to all orders of perturbation theory one will certainly have m 11 2 = m 22 2 = m 33 2 = 0 m_{11}^{2}=m_{22}^{2}=m_{33}^{2}=0 m 11 2 = m 22 2 = m 33 2 = 0 m 12 2 m_{12}^{2} m 12 2 m 13 2 m_{13}^{2} m 13 2 m 23 2 m_{23}^{2} m 23 2 β m 11 2 \beta_{m_{11}^{2}} β m 11 2 β m 22 2 \beta_{m_{22}^{2}} β m 22 2 β m 33 2 \beta_{m_{33}^{2}} β m 33 2 (22) , we will now have
( ⟨ Φ 1 ⟩ , ⟨ Φ 2 ⟩ , ⟨ Φ 3 ⟩ ) = 1 2 ( v 1 , v 2 e i α 1 , v 3 e i α 2 ) = v 2 ( c β 2 s β 1 , s β 2 s β 1 e α 1 , c β 1 e α 2 ) , (66) (\langle\Phi_{1}\rangle,\langle\Phi_{2}\rangle,\langle\Phi_{3}\rangle)=\frac{1}{\sqrt{2}}(v_{1},v_{2}e^{i\alpha_{1}},v_{3}e^{i\alpha_{2}})=\frac{v}{\sqrt{2}}(c_{\beta_{2}}s_{\beta_{1}},s_{\beta_{2}}s_{\beta_{1}}e^{\alpha_{1}},c_{\beta_{1}}e^{\alpha_{2}}), \tag*{(66)} (⟨ Φ 1 ⟩ , ⟨ Φ 2 ⟩ , ⟨ Φ 3 ⟩) = 2 1 ( v 1 , v 2 e i α 1 , v 3 e i α 2 ) = 2 v ( c β 2 s β 1 , s β 2 s β 1 e α 1 , c β 1 e α 2 ) , (66) where we defined tan β 1 = v 1 2 + v 2 2 / v 3 \tan\beta_1 = \sqrt{v_1^2+v_2^2}/v_3 tan β 1 = v 1 2 + v 2 2 / v 3 tan β 2 = v 2 / v 1 \tan\beta_2 = v_2/v_1 tan β 2 = v 2 / v 1 s x ≡ sin x s_x \equiv\sin x s x ≡ sin x c x ≡ cos x c_x \equiv\cos x c x ≡ cos x S 3 S_3 S 3 α 1 \alpha_1 α 1 α 2 \alpha_2 α 2
These more generic vevs allow us to obtain a realistic fermionic sector, to wit with Yukawa textures 10 ^{10} 10 Γ 1 = Γ 2 = Γ 3 \Gamma_1=\Gamma_2=\Gamma_3 Γ 1 = Γ 2 = Γ 3
M n = 1 2 ( Γ 1 v 1 + Γ 2 v 2 e i α 1 + Γ 3 v 3 e i α 2 ) = v 2 Γ ˉ e i θ d , (67) M_n = \frac{1}{\sqrt{2}}\left(\Gamma_1v_1+\Gamma_2v_2e^{i\alpha_1}+\Gamma_3v_3e^{i\alpha_2}\right)=\frac{v}{\sqrt{2}}\bar{\Gamma}e^{i\theta_d}, \tag*{(67)} M n = 2 1 ( Γ 1 v 1 + Γ 2 v 2 e i α 1 + Γ 3 v 3 e i α 2 ) = 2 v Γ ˉ e i θ d , (67) where Γ ˉ \bar{\Gamma} Γ ˉ 3 × 3 3\times3 3 × 3 θ d \theta_d θ d M n M_n M n M p M_p M p θ u \theta_u θ u
Scalar spectrum and interactions
With the soft breaking terms considered, for which the m i j 2 m_{ij}^2 m ij 2 i ≠ j i\ne j i = j M 1 M_1 M 1 M 2 M_2 M 2 Λ 4 \Lambda_4 Λ 4 Λ 5 \Lambda_5 Λ 5 Λ 6 \Lambda_6 Λ 6 Λ \Lambda Λ
Tr ( M n e u t r a l 2 ) = ( 8 Λ 0 + 2 Λ 1 + Λ 2 + Λ 3 ) v 2 + 1 2 ( Λ 4 − Λ 5 + 10 Λ 6 + 4 Λ 7 ) [ c α 1 − α 2 v 2 v 3 + c α 1 v 1 v 2 + c α 2 v 1 v 3 ] , \operatorname{Tr}(M_{\mathrm{neutral}}^2) = (8\Lambda_0+2\Lambda_1+\Lambda_2+\Lambda_3)v^2
+\frac{1}{2}(\Lambda_4-\Lambda_5+10\Lambda_6+4\Lambda_7)
\left[c_{\alpha_1-\alpha_2}v_2v_3+c_{\alpha_1}v_1v_2+c_{\alpha_2}v_1v_3\right], Tr ( M neutral 2 ) = ( 8 Λ 0 + 2 Λ 1 + Λ 2 + Λ 3 ) v 2 + 2 1 ( Λ 4 − Λ 5 + 10 Λ 6 + 4 Λ 7 ) [ c α 1 − α 2 v 2 v 3 + c α 1 v 1 v 2 + c α 2 v 1 v 3 ] , Tr ( M c h a r g e d 2 ) = 6 Λ 0 v 2 + 3 Λ 6 [ c α 1 − α 2 v 2 v 3 + c α 1 v 1 v 2 + c α 2 v 1 v 3 ] (68) \operatorname{Tr}(M_{\mathrm{charged}}^2) = 6\Lambda_0v^2+3\Lambda_6\left[c_{\alpha_1-\alpha_2}v_2v_3+c_{\alpha_1}v_1v_2+c_{\alpha_2}v_1v_3\right] \tag*{(68)} Tr ( M charged 2 ) = 6 Λ 0 v 2 + 3 Λ 6 [ c α 1 − α 2 v 2 v 3 + c α 1 v 1 v 2 + c α 2 v 1 v 3 ] (68) where we observe that all the squared scalar masses of the model are therefore of the order ∼ Λ v 2 \sim\Lambda v^2 ∼ Λ v 2 Λ \Lambda Λ M 1 M_1 M 1 M 2 M_2 M 2 ∼ Λ v 2 \sim\Lambda v^2 ∼ Λ v 2 § 3.2 ), the quartic couplings of the model are constrained to be at most of order 10, which gives a rough upper bound of 800 GeV on the masses of the scalars. As we will see in our numerical analysis, they end up being even lighter than that. We thus conclude that this (softly broken) GOOFy 3HDM has no decoupling limit – it is not possible, within the parameter space of the model to make the masses of the scalars arbitrarily large.
As we have already stated, the model is explicitly CP breaking – one way of seeing this is realizing that the complex number i i i 6 × 6 6 \times6 6 × 6 6 × 6 6 \times6 6 × 6 O O O h S M h_{SM} h S M h 1 , 2 , 3 , 4 h_{1,2,3,4} h 1 , 2 , 3 , 4 G G G (28) by the rotation
\begin{pmatrix}
\rho_1\\
\rho_2\\
\rho_3\\
\eta_1\\
\eta_2\\
\eta_3
\end{pmatrix}
=
O
\begin{pmatrix}
h_{SM}\\
h_1\\
h_2\\
h_3\\
h_4\\
G
\end{pmatrix}. \tag*{(69)}\end{pmatrix}
Thus, analysing the triple vertices between a neutral scalar and a pair of Z Z Z (27) , we find that
g h S M Z Z = [ c β 2 s β 1 O 1 , 1 + s β 2 s β 1 ( c α 2 O 2 , 1 + s α 2 O 5 , 1 ) + c β 1 ( c α 3 O 3 , 1 + s α 3 O 6 , 1 ) ] g h Z Z S M g_{h_{SM}ZZ} = \left[c_{\beta_2}s_{\beta_1}O_{1,1} + s_{\beta_2}s_{\beta_1}(c_{\alpha_2}O_{2,1} + s_{\alpha_2}O_{5,1}) + c_{\beta_1}(c_{\alpha_3}O_{3,1} + s_{\alpha_3}O_{6,1})\right]g_{hZZ}^{SM} g h S M Z Z = [ c β 2 s β 1 O 1 , 1 + s β 2 s β 1 ( c α 2 O 2 , 1 + s α 2 O 5 , 1 ) + c β 1 ( c α 3 O 3 , 1 + s α 3 O 6 , 1 ) ] g h Z Z S M g h i Z Z = [ c β 2 s β 1 O i + 1 , 1 + s β 2 s β 1 ( c α 2 O 2 , i + 1 + s α 2 O 5 , i + 1 ) + c β 1 ( c α 3 O 3 , i + 1 + s α 3 O 6 , i + 1 ) ] g h Z Z S M (70) g_{h_iZZ} = \left[c_{\beta_2}s_{\beta_1}O_{i+1,1} + s_{\beta_2}s_{\beta_1}(c_{\alpha_2}O_{2,i+1} + s_{\alpha_2}O_{5,i+1}) + c_{\beta_1}(c_{\alpha_3}O_{3,i+1} + s_{\alpha_3}O_{6,i+1})\right]g_{hZZ}^{SM} \tag*{(70)} g h i Z Z = [ c β 2 s β 1 O i + 1 , 1 + s β 2 s β 1 ( c α 2 O 2 , i + 1 + s α 2 O 5 , i + 1 ) + c β 1 ( c α 3 O 3 , i + 1 + s α 3 O 6 , i + 1 ) ] g h Z Z S M (70) where g h Z Z S M g_{hZZ}^{SM} g h Z Z S M Z Z Z 11 coupling modifiers of the vertices neutral scalar-Z-Z in this model. For the SM-like state, we define κ V = g h S M Z Z / g h Z Z S M \kappa_V = g_{h_{SM}ZZ}/g_{hZZ}^{SM} κ V = g h S M Z Z / g h Z Z S M κ V = g h S M W W / g h W W S M \kappa_V = g_{h_{SM}WW}/g_{hWW}^{SM} κ V = g h S M W W / g hW W S M κ V ≃ 1 \kappa_V \simeq1 κ V ≃ 1
CP violation also affects the Yukawa interactions of the neutral scalars to fermions. Using the down quark sector as an example, the interactions between neutral scalars and down quarks may be obtained from the respective Yukawa lagrangian, eq. (30) . Expressing the doublets as in eq. (28) and extracting the neutral components we obtain
− L Y , d = Q ‾ L Γ a Φ a n R + n ‾ R Γ a † Φ a † Q L ⟶ 1 2 [ ( n ‾ L Γ a n R + n ‾ R Γ a † n L ) ρ a + i ( n ‾ L Γ a n R − n ‾ R Γ a † n L ) η a ] , (71) -\mathcal{L}_{Y,d} = \overline{Q}_L\Gamma_a\Phi_a n_R + \overline{n}_R\Gamma_a^\dagger\Phi_a^\dagger Q_L \longrightarrow\frac{1}{\sqrt{2}}\left[(\overline{n}_L\Gamma_a n_R + \overline{n}_R\Gamma_a^\dagger n_L)\rho_a + i(\overline{n}_L\Gamma_a n_R - \overline{n}_R\Gamma_a^\dagger n_L)\eta_a\right], \tag*{(71)} − L Y , d = Q L Γ a Φ a n R + n R Γ a † Φ a † Q L ⟶ 2 1 [ ( n L Γ a n R + n R Γ a † n L ) ρ a + i ( n L Γ a n R − n R Γ a † n L ) η a ] , (71) where a sum in the a a a n L , R n_{L,R} n L , R
n L = U L d d L , n R = U R d d R , (72) n_L = U_L^d d_L,\qquad n_R = U_R^d d_R, \tag*{(72)} n L = U L d d L , n R = U R d d R , (72) where d L , R ≡ ( d , s , b ) L , R d_{L,R} \equiv(d,s,b)_{L,R} d L , R ≡ ( d , s , b ) L , R Γ a ′ = U L d † Γ a U R d \Gamma'_a = U_L^{d\dagger}\Gamma_a U_R^d Γ a ′ = U L d † Γ a U R d ρ a \rho_a ρ a η a \eta_a η a
− L h 0 d ˉ d = 1 2 d ‾ { 1 2 [ ( Γ a ′ + Γ a ′ † ) O a , 1 + i ( Γ a ′ − Γ a ′ † ) O a + 3 , 1 -\mathcal{L}_{h_0\bar d d} = \frac{1}{\sqrt{2}}\overline{d}\{\frac{1}{2}[(\Gamma'_a+\Gamma_a^{\prime\dagger})O_{a,1}+i(\Gamma'_a-\Gamma_a^{\prime\dagger})O_{a+3,1} − L h 0 d ˉ d = 2 1 d { 2 1 [( Γ a ′ + Γ a ′ † ) O a , 1 + i ( Γ a ′ − Γ a ′ † ) O a + 3 , 1 + 1 2 [ ( Γ a ′ − Γ a ′ † ) O a , 1 + i ( Γ a ′ + Γ a ′ † ) O a + 3 , 1 ] γ 5 } d h S M + 1 2 d ˉ { 1 2 [ ( Γ a ′ + Γ a ′ † ) O a , j + 1 + i ( Γ a ′ − Γ a ′ † ) O a + 3 , j + 1 ] + 1 2 [ ( Γ a ′ − Γ a ′ † ) O a , j + 1 + i ( Γ a ′ + Γ a ′ † ) O a + 3 , j + 1 ] γ 5 } d h j . (73) + \frac{1}{2}[(\Gamma'_a-\Gamma_a^{\prime\dagger})O_{a,1}+i(\Gamma'_a+\Gamma_a^{\prime\dagger})O_{a+3,1}]\gamma_5\}d h_{SM}
+ \frac{1}{\sqrt{2}}\bar d\{\frac{1}{2}[(\Gamma'_a+\Gamma_a^{\prime\dagger})O_{a,j+1}+i(\Gamma'_a-\Gamma_a^{\prime\dagger})O_{a+3,j+1}]
+ \frac{1}{2}[(\Gamma'_a-\Gamma_a^{\prime\dagger})O_{a,j+1}+i(\Gamma'_a+\Gamma_a^{\prime\dagger})O_{a+3,j+1}]\gamma_5\}d h_j . \tag*{(73)} + 2 1 [( Γ a ′ − Γ a ′ † ) O a , 1 + i ( Γ a ′ + Γ a ′ † ) O a + 3 , 1 ] γ 5 } d h S M + 2 1 d ˉ { 2 1 [( Γ a ′ + Γ a ′ † ) O a , j + 1 + i ( Γ a ′ − Γ a ′ † ) O a + 3 , j + 1 ] + 2 1 [( Γ a ′ − Γ a ′ † ) O a , j + 1 + i ( Γ a ′ + Γ a ′ † ) O a + 3 , j + 1 ] γ 5 } d h j . (73) An analogous expression is found for the h 0 u ˉ u h^0\bar u u h 0 u ˉ u h 0 f ˉ ( a + i b γ 5 ) f h^0\bar f(a+i b\gamma_5)f h 0 f ˉ ( a + ib γ 5 ) f a a a b b b h 0 h^0 h 0 b = 0 b=0 b = 0 a = 0 a=0 a = 0 f f f m f m_f m f
− L h S M f ˉ f = m f v f ˉ ( κ f e + i κ f o γ 5 ) f h S M , (74) -\mathcal{L}_{h_{SM}\bar f f}=\frac{m_f}{v}\bar f\left(\kappa_f^e+i\kappa_f^o\gamma_5\right)f h_{SM}, \tag*{(74)} − L h S M f ˉ f = v m f f ˉ ( κ f e + i κ f o γ 5 ) f h S M , (74) with real CP-even couplings κ e \kappa^e κ e κ o \kappa^o κ o
We also observe that the matrices Γ ′ \Gamma' Γ ′ h S M → b ˉ s h_{SM}\to\bar b s h S M → b ˉ s B B B K K K Δ M B d \Delta M_{B_d} Δ M B d Δ M B s \Delta M_{B_s} Δ M B s Δ M K \Delta M_K Δ M K ϵ K \epsilon_K ϵ K
Numerical analysis and phenomenology
Our purpose with this chapter is to investigate whether this model is phenomenologically viable – given that we imposed a very restrictive GOOFy symmetry, it is not guaranteed that an acceptable scalar spectrum, with a Higgs-like particle, is possible to obtain. We also wish to verify whether the model makes any distinctive predictions that might be tested experimentally. We have an immediate problem, in that the complexity of the softly broken model means that analytical results are no longer possible and we must resort to numerical calculations to probe the phenomenology of the sector. In fact we will use a semi-analytical approach: we obtain analytical expressions for the minimization conditions and the scalar mass matrices, in terms of the model’s parameters and the vevs of eq. (66) , but in order to obtain the scalar mass eigenstates we must diagonalize those matrices numerically. Likewise, though we possess analytical expressions for the GOOFy-symmetric Yukawa matrices, their diagonalization is made numerically.
Scanning procedure
The total number of parameters of the model is sufficiently large (despite the very constraining symmetries imposed) to in principle be able to successfully fit all current experimental constraints – indeed, we have a total of 10 parameters in the scalar potential (down from the 46 of the most general 3HDM); 9 parameters for the down quark Yukawa matrices; another 9 for the up quark Yukawa matrices; and another 9 for the charged lepton sector. Our scan of this parameter space was carried out as follows: starting with the minimisation equations, it was possible to obtain conditions that express M 1 M_1 M 1 M 2 M_2 M 2 Λ 4 \Lambda_4 Λ 4 Λ 5 \Lambda_5 Λ 5 Λ 6 \Lambda_6 Λ 6 Λ 0 \Lambda_0 Λ 0 Λ 1 \Lambda_1 Λ 1 Λ 2 \Lambda_2 Λ 2 Λ 3 \Lambda_3 Λ 3
We demand that h S M h_{\rm SM} h SM 0.9 ≤ ∣ κ V ∣ ≤ 1 0.9 \leq|\kappa_V| \leq1 0.9 ≤ ∣ κ V ∣ ≤ 1 κ V \kappa_V κ V 0.8 ≤ ∣ κ f ∣ ≤ 1.2 0.8 \leq|\kappa_f| \leq1.2 0.8 ≤ ∣ κ f ∣ ≤ 1.2 f = b , t f=b,t f = b , t
κ f = ( κ f e ) 2 + ( κ f o ) 2 (75) \kappa_f = \sqrt{(\kappa_f^{e})^2 + (\kappa_f^{o})^2} \tag*{(75)} κ f = ( κ f e ) 2 + ( κ f o ) 2 (75) with the CP-even and odd coupling modifiers defined in eq. (74).
Since the Higgs has been observed at the LHC as behaving very much like a CP-even particle, we must ensure that the CP-odd components of the Yukawa interactions are not too big. Thus we also required, for each fermion, ∣ κ f o / κ f e ∣ ∈ [ 0 , 0.35 ] |\kappa_f^{o}/\kappa_f^{e}| \in[0,0.35] ∣ κ f o / κ f e ∣ ∈ [ 0 , 0.35 ] [59 , 60 , 61 ] .
In parallel, our numerical fit also ensures that the six quarks have their know values and that quark mixing is also in agreement with experimental results – we compared the magnitudes of all elements of the CKM matrix with their measured values and ensured that the resulting value for the Jarlskog invariant J C K M J_{\rm CKM} J CKM J C K M J_{\rm CKM} J CKM [62 ] .
In table 3 we show the ranges considered for the input parameters, which have been optimised for greater computational speed. Indeed, the numerical fit is quite involved, as we are attempting to reproduce simultaneously quantities from the scalar and Yukawa sector. A three-stage minimisation was required, with different cost functions at each stage. In the first stage, the scalar sector is fitted using a L L L h S M h_{\rm SM} h SM t r f trf t r f t r f trf t r f
Note that Table 3 does not comprise values for the Yukawa matrices of the leptonic sector. The scalar-lepton Yukawa lagrangian would be analogous to the down quark one, eq. (30) , with matrices Π a \Pi_a Π a a = 1 , 2 , 3 a = 1, 2, 3 a = 1 , 2 , 3 Γ a \Gamma_a Γ a τ \tau τ Π a \Pi_a Π a Γ \Gamma Γ (41) – having only a non-zero entry, in the ( 3 , 3 ) (3, 3) ( 3 , 3 )
Table 3. Scan ranges of the input parameters.
M ℓ = Π 1 v 1 + Π 2 e i α 1 v 2 + Π 3 e i α 2 v 3 2 ⇒ m τ = π 33 2 ( v 1 + e i α 1 v 2 + e i α 2 v 3 ) . (76) M_\ell= \frac{\Pi_1 v_1 + \Pi_2 e^{i\alpha_1}v_2 + \Pi_3 e^{i\alpha_2}v_3}{\sqrt{2}} \quad\Rightarrow\quad m_\tau= \frac{\pi_{33}}{\sqrt{2}}\left(v_1 + e^{i\alpha_1}v_2 + e^{i\alpha_2}v_3\right). \tag*{(76)} M ℓ = 2 Π 1 v 1 + Π 2 e i α 1 v 2 + Π 3 e i α 2 v 3 ⇒ m τ = 2 π 33 ( v 1 + e i α 1 v 2 + e i α 2 v 3 ) . (76) The Yukawa couplings for scalars and the τ \tau τ π 33 \pi_{33} π 33 h S M h_{SM} h S M κ τ r \kappa_\tau^r κ τ r κ τ i \kappa_\tau^i κ τ i
Experimental and theoretical constraints
Even though the minimisation process already included some experimental constraints, valid parameter points must also satisfy the bounds imposed by many more LHC measurements, as well as guarantee that the theoretical constraints are met.
The quartic couplings of the scalar potential cannot be too large, otherwise unitarity in scalar-scalar scattering processes would be violated. This is a restriction already applied to the SM, and was used, prior to the Higgs’ discovery at the LHC, to establish upper bounds on its mass. For multiscalar models, the procedure to obtain the perturbative unitarity conditions is known and summarised in [63 , 64 ] . Following the procedure outlined in those works one may find the scattering matrices both for the most general 3HDM as well as the ones for the present model. The unitarity conditions are then obtained by computing the eigenvalues, A i A_i A i
∣ A i ∣ < 16 π . (77) \lvert A_i \rvert< 16\pi. \tag*{(77)} ∣ A i ∣ < 16 π . (77) These conditions were implemented numerically as the calculations of the eigenvalues of such large matrices are far too complicated. In practice, these conditions limit the magnitude of the Λ \Lambda Λ (13) and motivate the ranges for these quantities considered in Table 3 .
Boundedness from Below
Since the scalar potential must have a minimum somewhere, it must be bounded from below (BFB), i.e. it cannot tend to − ∞ -\infty − ∞ λ \lambda λ − ∞ -\infty − ∞ [5 ] (see [65 ] for a simple implementation of these conditions). These conditions, which require obtaining the eigenvalues of 4 × 4 4 \times4 4 × 4
However, in the 3HDM the situation is different. To this day, there is no set of necessary and sufficient BFB conditions for the most general 3HDM scalar potential. Specific models with simplifying symmetries may have such BFB conditions – for instance, Ivanov and Faro developed a technique that guarantees BFB for the U ( 1 ) × U ( 1 ) U(1) \times U(1) U ( 1 ) × U ( 1 ) [38 ] . The method consists on finding the copositivity conditions [66 ] for matrices built from the model parameters. For a class of more complicated models, Boto et al built on Faro and Ivanov’s method, in which BFB is guaranteed by introducing a potential that is a lower bound on the one under study, for which the copositivity conditions can be extended [67 ] .
For the GOOFy model under study here, however, these methods become too cumbersome. We can obtain of course some simple necessary conditions for some of the quartic couplings of eq. (13) . Following the same procedure used in the 2HDM to find necessary conditions for the couplings λ 1 \lambda_1 λ 1 λ 2 \lambda_2 λ 2 λ 3 \lambda_3 λ 3 Λ 0 \Lambda_0 Λ 0 Λ 1 \Lambda_1 Λ 1
Λ 0 + Λ 1 ≥ 0 , 4 Λ 0 + Λ 1 ≥ 0. (78) \Lambda_0 + \Lambda_1 \ge0 \qquad,\qquad4\Lambda_0 + \Lambda_1 \ge0 . \tag*{(78)} Λ 0 + Λ 1 ≥ 0 , 4 Λ 0 + Λ 1 ≥ 0. (78) This however leaves out all the remaning Λ \Lambda Λ
Oblique parameters
In any SM extension with extra doublets there may be substantial contributions to W W W Z Z Z W W W Z Z Z S S S T T T U U U [68 , 69 ] , also called oblique parameters . To compute these quantities we followed [70 ] wherein expressions for S S S T T T U U U
With these basic constraints on the scalar sector, the masses we obtain are show in Figure 1 for an extensive scalar of the parameter space which obeyed all constraints listed thus far. One may easily verify that the masses of the different particles are not entirely independent. In fact, the masses of the states h 2 h_2 h 2 h 3 h_3 h 3 h 4 h_4 h 4 h 1 h_1 h 1 h 1 h_1 h 1 h S M h_{SM} h S M ∼ 500 \sim500 ∼ 500 decoupling limit in this model. Indeed, there are only two massive parameters, M 1 M_1 M 1 M 2 M_2 M 2 (66) , implies that one must solve five minimization equations, those parameters are limited in size to be of the order ∼ Λ v 2 \sim\Lambda v^2 ∼ Λ v 2 Λ \Lambda Λ h 1 h_1 h 1 ∼ 280 \sim280 ∼ 280
Figure 1. Mass distributions of the generated parameter points.
Direct searches at LHC
The LHC collaborations have searched for extra scalars beyond the Higgs boson. With comparably low masses for the extra scalar particles in our model, it is necessary to verify that their existence has not been ruled out by current experimental bounds. To do so we looked at h 1 h_1 h 1 h 1 h_1 h 1 [71 ] . Cross section numbers were obtained via SusHi [72 , 73 ] . QCD corrections to scalar decays to quarks were implemented as in [74 , 75 ] . In Figure 2 we show the result of the calculations with the parameters of our scan which survived all the constraints outlined before (both for scalars and fermions) for the quantity
Figure 2. Gluon-fusion production cross-section times branching ratio of the lightest non-SM-like neutral scalar into W W WW W W Z Z ZZ Z Z τ τ \tau\tau τ τ h S M h S M h_{\mathrm{SM}}h_{\mathrm{SM}} h SM h SM [76 , 77 , 78 , 79 ] from the four channels, while the grey points represent the complete generated sample.
σ ( g g → h 1 ) B R ( h 1 → X X ) (79) \sigma(gg \to h_1)BR(h_1 \to XX) \tag*{(79)} σ ( g g → h 1 ) B R ( h 1 → X X ) (79) for the final states W W WW W W Z Z ZZ Z Z τ + τ − \tau^+\tau^- τ + τ − h S M h S M h_{\mathrm{SM}}h_{\mathrm{SM}} h SM h SM m h 1 > 2 m h S M m_{h_1} > 2m_{h_{\mathrm{SM}}} m h 1 > 2 m h SM m h 1 m_{h_1} m h 1 h 1 h_1 h 1 h S M h_{\mathrm{SM}} h SM h 1 h_1 h 1
bs constraints
In theories with extra charged scalars there may be substantial extra contributions to the b → s γ b \to s\gamma b → s γ W W W m H ± − tan β m_{H^\pm} - \tan\beta m H ± − tan β b → s γ b \to s\gamma b → s γ B → X s γ B \to X_s\gamma B → X s γ [80 ] , and we follow the procedure outlined in [81 ] to incorporate QCD corrections and evaluate the corresponding operators at the appropriate scales. We chose a bound of at most 5 × 10 − 5 5 \times10^{-5} 5 × 1 0 − 5 b → s γ b \to s\gamma b → s γ ∣ δ B R ( b → s γ ) ∣ < 5 × 10 − 5 |\delta BR(b \to s\gamma)| < 5 \times10^{-5} ∣ δ B R ( b → s γ ) ∣ < 5 × 1 0 − 5 [82 ] .13 Figure 3 , where the non-SM contributions to the branching ratio of the process b → s γ b \to s\gamma b → s γ b → s γ b \to s\gamma b → s γ b → s γ b \to s\gamma b → s γ H 1 ± H_1^\pm H 1 ± H 2 ± H_2^\pm H 2 ± b → s γ b \to s\gamma b → s γ Figure 4 . As is usually the case in these models, the charged contributions are substantially larger than the neutral ones. This also shows that in this model the FCNC interactions should be under control, though of course their contribution to several sensitive observables (such as B B B
Figure 3. Parameter points that are accepted and rejected by the bound on the new physics contribution to the branching ratio of b → s γ b \to s\gamma b → s γ ∣ δ B R ( b → s γ ) ∣ < 5 × 10 − 5 |\delta BR(b \to s\gamma)| < 5 \times10^{-5} ∣ δ B R ( b → s γ ) ∣ < 5 × 1 0 − 5
Figure 4. Neutral and charged contribution to the branching ratio of b → s γ b \to s\gamma b → s γ
Precision Higgs data
Finally, we must analyse what the model implies for the properties of h S M h_{SM} h S M h S M h_{SM} h S M W W W Z Z Z h S M h_{SM} h S M
R X = σ ( g g → h S M ) BR ( h S M → X X ) [ σ ( g g → h ) BR ( h → X X ) ] S M , (80) R_X = \frac{\sigma(gg \to h_{SM})\operatorname{BR}(h_{SM} \to XX)}{[\sigma(gg \to h)\operatorname{BR}(h \to XX)]_{SM}}, \tag*{(80)} R X = [ σ ( g g → h ) BR ( h → X X ) ] S M σ ( g g → h S M ) BR ( h S M → X X ) , (80) for four different decay channels: h S M → γ γ h_{SM} \to\gamma\gamma h S M → γ γ h S M → Z Z h_{SM} \to ZZ h S M → Z Z h S M → W W h_{SM} \to WW h S M → W W h S M → τ + τ − h_{SM} \to\tau^{+}\tau^{-} h S M → τ + τ − Figure 5 , where we plot the values for the four R X R_X R X σ \sigma σ R X = 1 R_X = 1 R X = 1 h 1 h_1 h 1 b → s γ b \to s\gamma b → s γ
Figure 5. Higgs precision results for the γ γ \gamma\gamma γ γ Z Z ZZ Z Z W W WW W W τ − τ + \tau^{-}\tau^{+} τ − τ + b b ˉ b\bar{b} b b ˉ § 3.3.2 to 3.3.4, while the coloured points lie within the 2 σ 2\sigma 2 σ Z Z ZZ Z Z W W WW W W τ − τ + \tau^{-}\tau^{+} τ − τ + Z Z ZZ Z Z W W WW W W τ − τ + \tau^{-}\tau^{+} τ − τ + b b ˉ b\bar{b} b b ˉ [83 , 84 ] .
In Figure 5 we observe that the GOOFy 3HDM we are studying has a preference for lower values of R γ γ R_{\gamma\gamma} R γ γ R b b R_{bb} R bb R γ γ R_{\gamma\gamma} R γ γ W W W κ V \kappa_V κ V (70) ) and the CP-even top coupling modifier κ t e \kappa_t^e κ t e (74) ) always have the same relative signal than the SM case; and the relative sign of the couplings of h S M h_{SM} h S M H 1 ± H_1^{\pm} H 1 ± H 2 ± H_2^{\pm} H 2 ± h S M → γ γ h_{SM} \to\gamma\gamma h S M → γ γ h S M W W h_{SM}WW h S M W W h S M H i + H i − h_{SM}H_i^{+}H_i^{-} h S M H i + H i − a priori by the model: these vertices can have any signs before theoretical bounds are imposed, but their relative sign is fixed by a combination of the unitarity and BFB bounds. An even more thorough scan of the parameter space would be necessary to confirm that this is indeed a feature of the model.
The green points in Figure 5 are those that comply with three of the signal rates in the ATLAS case, and four in the CMS one. The model has a clash with the γ γ \gamma\gamma γ γ b b bb bb γ γ \gamma\gamma γ γ 3 3 3 σ \sigma σ γ γ \gamma\gamma γ γ Figure 6 .
Figure 6. Higgs precision results for the γ γ \gamma\gamma γ γ Z Z ZZ Z Z W W WW W W τ − τ + \tau^{-}\tau^{+} τ − τ + b b ˉ b\bar{b} b b ˉ § 3.3.2 to 3.3.4, while the coloured points lie within the 3 σ 3\sigma 3 σ Z Z ZZ Z Z W W WW W W τ − τ + \tau^{-}\tau^{+} τ − τ + b b ˉ b\bar{b} b b ˉ [83 , 84 ] .
Conclusions
We constructed a 3HDM with a so-called GOOFy symmetry, which transforms the doublets and respective conjugates differently, with a permutation of the three up to negative signs and conjugations. The respective scalar potential is stable under RG evolution and features a pair of mass degenerate neutral scalars and a very symmetrical VEV, with equal magnitude for the 3 doublets (in the symmetry basis). The Yukawa sector is not realistic, which motivates the soft-breaking of the symmetry. The soft-breaking lifts the degeneracy of the two physical neutral scalars, and allows for a realistic Yukawa sector including a physical CP phase for the quark sector. We studied the RG stability of this complete model, and verified that the GOOFy parameter relations held to at least two loops via a direct inspection of the model’s β \beta β b → s γ b \to s\gamma b → s γ b b bb bb
Further features of the model include soft explicit CP breaking due to the imaginary soft breaking parameters introduced, yielding scalars which are CP admixtures, neither scalar nor pseudoscalar; and FCNC interactions mediated by the neutral scalars. These two properties will have contributions to observables such as the electric dipole moment of the electron and neutron; and to B B B K K K h 2 , 3 , 4 h_{2,3,4} h 2 , 3 , 4
Acknowledgments
IdMV thanks the University of Basel for hospitality. IdMV acknowledges funding from Fundação para a Ciência e a Tecnologia (FCT) through the FCT Mobility program, and through the project CFTP-FCT Unit UID/00777/2025 (https://doi.org/10.54499/UID/00777/2025). PMF is supported by Fundação para a Ciência e a Tecnologia under contract UIDP/00618/2025, UIDB/00618/2025 and through the PRR (Recovery and Resilience Plan), within the scope of the investment “RE-C06-i06 - Science Plus Capacity Building”, measure “RE-C06-i06.m02 - Reinforcement of financing for International Partnerships in Science, Technology and Innovation of the PRR”, under the project with the reference 2024.03328.CERN.
S3-GOOFy conserving Yukawa matrices
In Section 2.4.2 eleven Yukawa matrices capable of generating three generations of massive fermions are mentioned. Here their explicit form is provided. As mentioned before, the fact that these are able to reproduce the same CKM matrix possibly signals the existence of basis changes that might relate some, if not all, of these structures. However, that is left for future work.
The Yukawa matrices are presented in a basis in which all their entries are real, obtained through the rephasing discussed in Section 2.4.2.
Γ 1 = ( a 11 a 12 a 13 a 21 a 22 a 23 a 31 a 32 a 33 ) , (81) \Gamma_{1} =
\begin{pmatrix}
a_{11} & a_{12} & a_{13} \\
a_{21} & a_{22} & a_{23} \\
a_{31} & a_{32} & a_{33}
\end{pmatrix}, \tag*{(81)} Γ 1 = a 11 a 21 a 31 a 12 a 22 a 32 a 13 a 23 a 33 , (81) Γ 2 = 1 4 ( ( a 11 − 3 a 12 − 3 a 21 + 3 a 22 ) ( 3 a 11 + a 12 − 3 a 21 − 3 a 22 ) 2 ( − a 13 + 3 a 23 ) ( 3 a 11 − 3 a 12 + a 21 − 3 a 22 ) ( 3 a 11 + 3 a 12 + 3 a 21 + a 22 ) 2 ( − 3 a 13 − a 23 ) 2 ( − a 31 + 3 a 32 ) 2 ( − 3 a 31 − a 32 ) 4 a 33 ) , (82) \Gamma_{2} = \frac{1}{4}
\begin{pmatrix}
(a_{11} - \sqrt{3}a_{12} - \sqrt{3}a_{21} + 3a_{22}) &
(\sqrt{3}a_{11} + a_{12} - 3a_{21} - \sqrt{3}a_{22}) &
2(-a_{13} + \sqrt{3}a_{23}) \\
(\sqrt{3}a_{11} - 3a_{12} + a_{21} - \sqrt{3}a_{22}) &
(3a_{11} + \sqrt{3}a_{12} + \sqrt{3}a_{21} + a_{22}) &
2(-\sqrt{3}a_{13} - a_{23}) \\
2(-a_{31} + \sqrt{3}a_{32}) &
2(-\sqrt{3}a_{31} - a_{32}) &
4a_{33}
\end{pmatrix}, \tag*{(82)} Γ 2 = 4 1 ( a 11 − 3 a 12 − 3 a 21 + 3 a 22 ) ( 3 a 11 − 3 a 12 + a 21 − 3 a 22 ) 2 ( − a 31 + 3 a 32 ) ( 3 a 11 + a 12 − 3 a 21 − 3 a 22 ) ( 3 a 11 + 3 a 12 + 3 a 21 + a 22 ) 2 ( − 3 a 31 − a 32 ) 2 ( − a 13 + 3 a 23 ) 2 ( − 3 a 13 − a 23 ) 4 a 33 , (82) Γ 3 = 1 4 ( ( a 11 + 3 a 12 + 3 a 21 + 3 a 22 ) ( − 3 a 11 + a 12 − 3 a 21 + 3 a 22 ) 2 ( − a 13 − 3 a 23 ) ( − 3 a 11 − 3 a 12 + a 21 + 3 a 22 ) ( 3 a 11 − 3 a 12 − 3 a 21 + a 22 ) 2 ( 3 a 13 − a 23 ) 2 ( − a 31 − 3 a 32 ) 2 ( 3 a 31 − a 32 ) 4 a 33 ) . (83) \Gamma_{3}=\frac{1}{4}\begin{pmatrix}
(a_{11}+\sqrt{3}a_{12}+\sqrt{3}a_{21}+3a_{22}) & (-\sqrt{3}a_{11}+a_{12}-3a_{21}+\sqrt{3}a_{22}) & 2(-a_{13}-\sqrt{3}a_{23})\\
(-\sqrt{3}a_{11}-3a_{12}+a_{21}+\sqrt{3}a_{22}) & (3a_{11}-\sqrt{3}a_{12}-\sqrt{3}a_{21}+a_{22}) & 2(\sqrt{3}a_{13}-a_{23})\\
2(-a_{31}-\sqrt{3}a_{32}) & 2(\sqrt{3}a_{31}-a_{32}) & 4a_{33}
\end{pmatrix}. \tag*{(83)} Γ 3 = 4 1 ( a 11 + 3 a 12 + 3 a 21 + 3 a 22 ) ( − 3 a 11 − 3 a 12 + a 21 + 3 a 22 ) 2 ( − a 31 − 3 a 32 ) ( − 3 a 11 + a 12 − 3 a 21 + 3 a 22 ) ( 3 a 11 − 3 a 12 − 3 a 21 + a 22 ) 2 ( 3 a 31 − a 32 ) 2 ( − a 13 − 3 a 23 ) 2 ( 3 a 13 − a 23 ) 4 a 33 . (83) Γ 1 = ( a 11 a 12 a 13 a 21 a 22 a 23 a 31 a 32 a 33 ) , (84) \Gamma_{1}=\begin{pmatrix}
a_{11} & a_{12} & a_{13}\\
a_{21} & a_{22} & a_{23}\\
a_{31} & a_{32} & a_{33}
\end{pmatrix}, \tag*{(84)} Γ 1 = a 11 a 21 a 31 a 12 a 22 a 32 a 13 a 23 a 33 , (84) Γ 2 = 1 4 ( ( a 11 + 3 a 12 − 3 a 21 − 3 a 22 ) ( − 3 a 11 + a 12 + 3 a 21 − 3 a 22 ) 2 ( − a 13 + 3 a 23 ) ( 3 a 11 + 3 a 12 + a 21 + 3 a 22 ) ( − 3 a 11 + 3 a 12 − 3 a 21 + a 22 ) 2 ( − 3 a 13 − a 23 ) 2 ( − a 31 − 3 a 32 ) 2 ( 3 a 31 − a 32 ) 4 a 33 ) , (85) \Gamma_{2}=\frac{1}{4}\begin{pmatrix}
(a_{11}+\sqrt{3}a_{12}-\sqrt{3}a_{21}-3a_{22}) & (-\sqrt{3}a_{11}+a_{12}+3a_{21}-\sqrt{3}a_{22}) & 2(-a_{13}+\sqrt{3}a_{23})\\
(\sqrt{3}a_{11}+3a_{12}+a_{21}+\sqrt{3}a_{22}) & (-3a_{11}+\sqrt{3}a_{12}-\sqrt{3}a_{21}+a_{22}) & 2(-\sqrt{3}a_{13}-a_{23})\\
2(-a_{31}-\sqrt{3}a_{32}) & 2(\sqrt{3}a_{31}-a_{32}) & 4a_{33}
\end{pmatrix}, \tag*{(85)} Γ 2 = 4 1 ( a 11 + 3 a 12 − 3 a 21 − 3 a 22 ) ( 3 a 11 + 3 a 12 + a 21 + 3 a 22 ) 2 ( − a 31 − 3 a 32 ) ( − 3 a 11 + a 12 + 3 a 21 − 3 a 22 ) ( − 3 a 11 + 3 a 12 − 3 a 21 + a 22 ) 2 ( 3 a 31 − a 32 ) 2 ( − a 13 + 3 a 23 ) 2 ( − 3 a 13 − a 23 ) 4 a 33 , (85) Γ 3 = 1 4 ( ( a 11 − 3 a 12 + 3 a 21 − 3 a 22 ) ( 3 a 11 + a 12 + 3 a 21 + 3 a 22 ) 2 ( − a 13 − 3 a 23 ) ( − 3 a 11 + 3 a 12 + a 21 − 3 a 22 ) ( − 3 a 11 − 3 a 12 + 3 a 21 + a 22 ) 2 ( 3 a 13 − a 23 ) 2 ( − a 31 + 3 a 32 ) 2 ( − 3 a 31 − a 32 ) 4 a 33 ) . (86) \Gamma_{3}=\frac{1}{4}\begin{pmatrix}
(a_{11}-\sqrt{3}a_{12}+\sqrt{3}a_{21}-3a_{22}) & (\sqrt{3}a_{11}+a_{12}+3a_{21}+\sqrt{3}a_{22}) & 2(-a_{13}-\sqrt{3}a_{23})\\
(-\sqrt{3}a_{11}+3a_{12}+a_{21}-\sqrt{3}a_{22}) & (-3a_{11}-\sqrt{3}a_{12}+\sqrt{3}a_{21}+a_{22}) & 2(\sqrt{3}a_{13}-a_{23})\\
2(-a_{31}+\sqrt{3}a_{32}) & 2(-\sqrt{3}a_{31}-a_{32}) & 4a_{33}
\end{pmatrix}. \tag*{(86)} Γ 3 = 4 1 ( a 11 − 3 a 12 + 3 a 21 − 3 a 22 ) ( − 3 a 11 + 3 a 12 + a 21 − 3 a 22 ) 2 ( − a 31 + 3 a 32 ) ( 3 a 11 + a 12 + 3 a 21 + 3 a 22 ) ( − 3 a 11 − 3 a 12 + 3 a 21 + a 22 ) 2 ( − 3 a 31 − a 32 ) 2 ( − a 13 − 3 a 23 ) 2 ( 3 a 13 − a 23 ) 4 a 33 . (86) Γ 1 = ( a 11 a 12 a 13 a 21 a 22 a 23 a 31 a 32 a 33 ) , (87) \Gamma_{1}=\begin{pmatrix}
a_{11} & a_{12} & a_{13}\\
a_{21} & a_{22} & a_{23}\\
a_{31} & a_{32} & a_{33}
\end{pmatrix}, \tag*{(87)} Γ 1 = a 11 a 21 a 31 a 12 a 22 a 32 a 13 a 23 a 33 , (87) Γ 2 = 1 4 ( ( a 11 − 3 a 12 + 3 a 21 − 3 a 22 ) ( 3 a 11 + a 12 + 3 a 21 + 3 a 22 ) 2 ( − a 13 − 3 a 23 ) ( − 3 a 11 + 3 a 12 + a 21 − 3 a 22 ) ( − 3 a 11 − 3 a 12 + 3 a 21 + a 22 ) 2 ( 3 a 13 − a 23 ) 2 ( − a 31 + 3 a 32 ) 2 ( − 3 a 31 − a 32 ) 4 a 33 ) , (88) \Gamma_{2}=\frac{1}{4}\begin{pmatrix}
(a_{11}-\sqrt{3}a_{12}+\sqrt{3}a_{21}-3a_{22}) & (\sqrt{3}a_{11}+a_{12}+3a_{21}+\sqrt{3}a_{22}) & 2(-a_{13}-\sqrt{3}a_{23})\\
(-\sqrt{3}a_{11}+3a_{12}+a_{21}-\sqrt{3}a_{22}) & (-3a_{11}-\sqrt{3}a_{12}+\sqrt{3}a_{21}+a_{22}) & 2(\sqrt{3}a_{13}-a_{23})\\
2(-a_{31}+\sqrt{3}a_{32}) & 2(-\sqrt{3}a_{31}-a_{32}) & 4a_{33}
\end{pmatrix}, \tag*{(88)} Γ 2 = 4 1 ( a 11 − 3 a 12 + 3 a 21 − 3 a 22 ) ( − 3 a 11 + 3 a 12 + a 21 − 3 a 22 ) 2 ( − a 31 + 3 a 32 ) ( 3 a 11 + a 12 + 3 a 21 + 3 a 22 ) ( − 3 a 11 − 3 a 12 + 3 a 21 + a 22 ) 2 ( − 3 a 31 − a 32 ) 2 ( − a 13 − 3 a 23 ) 2 ( 3 a 13 − a 23 ) 4 a 33 , (88) Γ 3 = 1 4 ( ( a 11 + 3 a 12 − 3 a 21 − 3 a 22 ) ( − 3 a 11 + a 12 + 3 a 21 − 3 a 22 ) 2 ( − a 13 + 3 a 23 ) ( 3 a 11 + 3 a 12 + a 21 + 3 a 22 ) ( − 3 a 11 + 3 a 12 − 3 a 21 + a 22 ) 2 ( − 3 a 13 − a 23 ) 2 ( − a 31 − 3 a 32 ) 2 ( 3 a 31 − a 32 ) 4 a 33 ) . (89) \Gamma_{3}=\frac{1}{4}\begin{pmatrix}
(a_{11}+\sqrt{3}a_{12}-\sqrt{3}a_{21}-3a_{22}) & (-\sqrt{3}a_{11}+a_{12}+3a_{21}-\sqrt{3}a_{22}) & 2(-a_{13}+\sqrt{3}a_{23})\\
(\sqrt{3}a_{11}+3a_{12}+a_{21}+\sqrt{3}a_{22}) & (-3a_{11}+\sqrt{3}a_{12}-\sqrt{3}a_{21}+a_{22}) & 2(-\sqrt{3}a_{13}-a_{23})\\
2(-a_{31}-\sqrt{3}a_{32}) & 2(\sqrt{3}a_{31}-a_{32}) & 4a_{33}
\end{pmatrix}. \tag*{(89)} Γ 3 = 4 1 ( a 11 + 3 a 12 − 3 a 21 − 3 a 22 ) ( 3 a 11 + 3 a 12 + a 21 + 3 a 22 ) 2 ( − a 31 − 3 a 32 ) ( − 3 a 11 + a 12 + 3 a 21 − 3 a 22 ) ( − 3 a 11 + 3 a 12 − 3 a 21 + a 22 ) 2 ( 3 a 31 − a 32 ) 2 ( − a 13 + 3 a 23 ) 2 ( − 3 a 13 − a 23 ) 4 a 33 . (89) Γ 1 = ( a 11 a 12 a 13 a 21 a 22 a 23 a 31 a 32 a 33 ) , (90) \Gamma_{1}=\begin{pmatrix}
a_{11} & a_{12} & a_{13}\\
a_{21} & a_{22} & a_{23}\\
a_{31} & a_{32} & a_{33}
\end{pmatrix}, \tag*{(90)} Γ 1 = a 11 a 21 a 31 a 12 a 22 a 32 a 13 a 23 a 33 , (90) Γ 2 = 1 4 ( a 11 + 3 a 12 + 3 a 21 + 3 a 22 − 3 a 11 + a 12 − 3 a 21 + 3 a 22 2 ( − a 13 − 3 a 23 ) − 3 a 11 − 3 a 12 + a 21 + 3 a 22 3 a 11 − 3 a 12 − 3 a 21 + a 22 2 ( 3 a 13 − a 23 ) 2 ( − a 31 − 3 a 32 ) 2 ( 3 a 31 − a 32 ) 4 a 33 ) , (91) \Gamma_2=\frac{1}{4}\begin{pmatrix}
a_{11}+\sqrt{3}a_{12}+\sqrt{3}a_{21}+3a_{22} & -\sqrt{3}a_{11}+a_{12}-3a_{21}+\sqrt{3}a_{22} & 2(-a_{13}-\sqrt{3}a_{23}) \\
-\sqrt{3}a_{11}-3a_{12}+a_{21}+\sqrt{3}a_{22} & 3a_{11}-\sqrt{3}a_{12}-\sqrt{3}a_{21}+a_{22} & 2(\sqrt{3}a_{13}-a_{23}) \\
2(-a_{31}-\sqrt{3}a_{32}) & 2(\sqrt{3}a_{31}-a_{32}) & 4a_{33}
\end{pmatrix}, \tag*{(91)} Γ 2 = 4 1 a 11 + 3 a 12 + 3 a 21 + 3 a 22 − 3 a 11 − 3 a 12 + a 21 + 3 a 22 2 ( − a 31 − 3 a 32 ) − 3 a 11 + a 12 − 3 a 21 + 3 a 22 3 a 11 − 3 a 12 − 3 a 21 + a 22 2 ( 3 a 31 − a 32 ) 2 ( − a 13 − 3 a 23 ) 2 ( 3 a 13 − a 23 ) 4 a 33 , (91) Γ 3 = 1 4 ( a 11 − 3 a 12 − 3 a 21 + 3 a 22 3 a 11 + a 12 − 3 a 21 − 3 a 22 2 ( − a 13 + 3 a 23 ) 3 a 11 − 3 a 12 + a 21 − 3 a 22 3 a 11 + 3 a 12 + 3 a 21 + a 22 2 ( − 3 a 13 − a 23 ) 2 ( − a 31 + 3 a 32 ) 2 ( − 3 a 31 − a 32 ) 4 a 33 ) . (92) \Gamma_3=\frac{1}{4}\begin{pmatrix}
a_{11}-\sqrt{3}a_{12}-\sqrt{3}a_{21}+3a_{22} & \sqrt{3}a_{11}+a_{12}-3a_{21}-\sqrt{3}a_{22} & 2(-a_{13}+\sqrt{3}a_{23}) \\
\sqrt{3}a_{11}-3a_{12}+a_{21}-\sqrt{3}a_{22} & 3a_{11}+\sqrt{3}a_{12}+\sqrt{3}a_{21}+a_{22} & 2(-\sqrt{3}a_{13}-a_{23}) \\
2(-a_{31}+\sqrt{3}a_{32}) & 2(-\sqrt{3}a_{31}-a_{32}) & 4a_{33}
\end{pmatrix}. \tag*{(92)} Γ 3 = 4 1 a 11 − 3 a 12 − 3 a 21 + 3 a 22 3 a 11 − 3 a 12 + a 21 − 3 a 22 2 ( − a 31 + 3 a 32 ) 3 a 11 + a 12 − 3 a 21 − 3 a 22 3 a 11 + 3 a 12 + 3 a 21 + a 22 2 ( − 3 a 31 − a 32 ) 2 ( − a 13 + 3 a 23 ) 2 ( − 3 a 13 − a 23 ) 4 a 33 . (92) Γ 1 = ( a 11 a 12 a 13 a 21 a 22 a 23 a 31 a 32 a 33 ) , (93) \Gamma_1=\begin{pmatrix}
a_{11} & a_{12} & a_{13} \\
a_{21} & a_{22} & a_{23} \\
a_{31} & a_{32} & a_{33}
\end{pmatrix}, \tag*{(93)} Γ 1 = a 11 a 21 a 31 a 12 a 22 a 32 a 13 a 23 a 33 , (93) Γ 2 = 1 4 ( a 11 − 3 a 12 − 3 a 21 + 3 a 22 3 a 11 + a 12 − 3 a 21 − 3 a 22 2 ( − a 13 + 3 a 23 ) 3 a 11 − 3 a 12 + a 21 − 3 a 22 3 a 11 + 3 a 12 + 3 a 21 + a 22 2 ( − 3 a 13 − a 23 ) 2 ( − a 31 + 3 a 32 ) 2 ( − 3 a 31 − a 32 ) 4 a 33 ) , (94) \Gamma_2=\frac{1}{4}\begin{pmatrix}
a_{11}-\sqrt{3}a_{12}-\sqrt{3}a_{21}+3a_{22} & \sqrt{3}a_{11}+a_{12}-3a_{21}-\sqrt{3}a_{22} & 2(-a_{13}+\sqrt{3}a_{23}) \\
\sqrt{3}a_{11}-3a_{12}+a_{21}-\sqrt{3}a_{22} & 3a_{11}+\sqrt{3}a_{12}+\sqrt{3}a_{21}+a_{22} & 2(-\sqrt{3}a_{13}-a_{23}) \\
2(-a_{31}+\sqrt{3}a_{32}) & 2(-\sqrt{3}a_{31}-a_{32}) & 4a_{33}
\end{pmatrix}, \tag*{(94)} Γ 2 = 4 1 a 11 − 3 a 12 − 3 a 21 + 3 a 22 3 a 11 − 3 a 12 + a 21 − 3 a 22 2 ( − a 31 + 3 a 32 ) 3 a 11 + a 12 − 3 a 21 − 3 a 22 3 a 11 + 3 a 12 + 3 a 21 + a 22 2 ( − 3 a 31 − a 32 ) 2 ( − a 13 + 3 a 23 ) 2 ( − 3 a 13 − a 23 ) 4 a 33 , (94) Γ 3 = 1 4 ( a 11 + 3 a 12 + 3 a 21 + 3 a 22 − 3 a 11 + a 12 − 3 a 21 + 3 a 22 2 ( − a 13 − 3 a 23 ) − 3 a 11 − 3 a 12 + a 21 + 3 a 22 3 a 11 − 3 a 12 − 3 a 21 + a 22 2 ( 3 a 13 − a 23 ) 2 ( − a 31 − 3 a 32 ) 2 ( 3 a 31 − a 32 ) 4 a 33 ) . (95) \Gamma_3=\frac{1}{4}\begin{pmatrix}
a_{11}+\sqrt{3}a_{12}+\sqrt{3}a_{21}+3a_{22} & -\sqrt{3}a_{11}+a_{12}-3a_{21}+\sqrt{3}a_{22} & 2(-a_{13}-\sqrt{3}a_{23}) \\
-\sqrt{3}a_{11}-3a_{12}+a_{21}+\sqrt{3}a_{22} & 3a_{11}-\sqrt{3}a_{12}-\sqrt{3}a_{21}+a_{22} & 2(\sqrt{3}a_{13}-a_{23}) \\
2(-a_{31}-\sqrt{3}a_{32}) & 2(\sqrt{3}a_{31}-a_{32}) & 4a_{33}
\end{pmatrix}. \tag*{(95)} Γ 3 = 4 1 a 11 + 3 a 12 + 3 a 21 + 3 a 22 − 3 a 11 − 3 a 12 + a 21 + 3 a 22 2 ( − a 31 − 3 a 32 ) − 3 a 11 + a 12 − 3 a 21 + 3 a 22 3 a 11 − 3 a 12 − 3 a 21 + a 22 2 ( 3 a 31 − a 32 ) 2 ( − a 13 − 3 a 23 ) 2 ( 3 a 13 − a 23 ) 4 a 33 . (95) Γ 1 = ( a 11 a 12 a 13 a 21 a 22 a 23 a 31 a 32 a 33 ) , (96) \Gamma_1=\begin{pmatrix}
a_{11} & a_{12} & a_{13} \\
a_{21} & a_{22} & a_{23} \\
a_{31} & a_{32} & a_{33}
\end{pmatrix}, \tag*{(96)} Γ 1 = a 11 a 21 a 31 a 12 a 22 a 32 a 13 a 23 a 33 , (96) Γ 2 = 1 4 ( a 11 + 3 a 12 − 3 a 21 − 3 a 22 − 3 a 11 + a 12 + 3 a 21 − 3 a 22 2 ( − a 13 + 3 a 23 ) 3 a 11 + 3 a 12 + a 21 + 3 a 22 − 3 a 11 + 3 a 12 − 3 a 21 + a 22 2 ( − 3 a 13 − a 23 ) 2 ( − a 31 − 3 a 32 ) 2 ( 3 a 31 − a 32 ) 4 a 33 ) , (97) \Gamma_2=\frac{1}{4}\begin{pmatrix}
a_{11}+\sqrt{3}a_{12}-\sqrt{3}a_{21}-3a_{22} & -\sqrt{3}a_{11}+a_{12}+3a_{21}-\sqrt{3}a_{22} & 2(-a_{13}+\sqrt{3}a_{23}) \\
\sqrt{3}a_{11}+3a_{12}+a_{21}+\sqrt{3}a_{22} & -3a_{11}+\sqrt{3}a_{12}-\sqrt{3}a_{21}+a_{22} & 2(-\sqrt{3}a_{13}-a_{23}) \\
2(-a_{31}-\sqrt{3}a_{32}) & 2(\sqrt{3}a_{31}-a_{32}) & 4a_{33}
\end{pmatrix}, \tag*{(97)} Γ 2 = 4 1 a 11 + 3 a 12 − 3 a 21 − 3 a 22 3 a 11 + 3 a 12 + a 21 + 3 a 22 2 ( − a 31 − 3 a 32 ) − 3 a 11 + a 12 + 3 a 21 − 3 a 22 − 3 a 11 + 3 a 12 − 3 a 21 + a 22 2 ( 3 a 31 − a 32 ) 2 ( − a 13 + 3 a 23 ) 2 ( − 3 a 13 − a 23 ) 4 a 33 , (97) Γ 3 = 1 4 ( a 11 − 3 a 12 + 3 a 21 − 3 a 22 3 a 11 + a 12 + 3 a 21 + 3 a 22 2 ( − a 13 − 3 a 23 ) − 3 a 11 + 3 a 12 + a 21 − 3 a 22 − 3 a 11 − 3 a 12 + 3 a 21 + a 22 2 ( 3 a 13 − a 23 ) 2 ( − a 31 + 3 a 32 ) 2 ( − 3 a 31 − a 32 ) 4 a 33 ) . (98) \Gamma_3=\frac{1}{4}\begin{pmatrix}
a_{11}-\sqrt{3}a_{12}+\sqrt{3}a_{21}-3a_{22} & \sqrt{3}a_{11}+a_{12}+3a_{21}+\sqrt{3}a_{22} & 2(-a_{13}-\sqrt{3}a_{23}) \\
-\sqrt{3}a_{11}+3a_{12}+a_{21}-\sqrt{3}a_{22} & -3a_{11}-\sqrt{3}a_{12}+\sqrt{3}a_{21}+a_{22} & 2(\sqrt{3}a_{13}-a_{23}) \\
2(-a_{31}+\sqrt{3}a_{32}) & 2(-\sqrt{3}a_{31}-a_{32}) & 4a_{33}
\end{pmatrix}. \tag*{(98)} Γ 3 = 4 1 a 11 − 3 a 12 + 3 a 21 − 3 a 22 − 3 a 11 + 3 a 12 + a 21 − 3 a 22 2 ( − a 31 + 3 a 32 ) 3 a 11 + a 12 + 3 a 21 + 3 a 22 − 3 a 11 − 3 a 12 + 3 a 21 + a 22 2 ( − 3 a 31 − a 32 ) 2 ( − a 13 − 3 a 23 ) 2 ( 3 a 13 − a 23 ) 4 a 33 . (98) Γ 1 = ( a 11 a 12 a 13 a 21 a 22 a 23 a 31 a 32 a 33 ) , (99) \Gamma_1 =
\begin{pmatrix}
a_{11} & a_{12} & a_{13} \\
a_{21} & a_{22} & a_{23} \\
a_{31} & a_{32} & a_{33}
\end{pmatrix}, \tag*{(99)} Γ 1 = a 11 a 21 a 31 a 12 a 22 a 32 a 13 a 23 a 33 , (99) Γ 2 = 1 4 ( ( a 11 − 3 a 12 + 3 a 21 − 3 a 22 ) ( 3 a 11 + a 12 + 3 a 21 + 3 a 22 ) 2 ( − a 13 − 3 a 23 ) ( − 3 a 11 + 3 a 12 + a 21 − 3 a 22 ) ( − 3 a 11 − 3 a 12 + 3 a 21 + a 22 ) 2 ( 3 a 13 − a 23 ) 2 ( − a 31 + 3 a 32 ) 2 ( − 3 a 31 − a 32 ) 4 a 33 ) , (100) \Gamma_2 = \frac{1}{4}
\begin{pmatrix}
(a_{11}-\sqrt{3}a_{12}+\sqrt{3}a_{21}-3a_{22}) & (\sqrt{3}a_{11}+a_{12}+3a_{21}+\sqrt{3}a_{22}) & 2(-a_{13}-\sqrt{3}a_{23}) \\
(-\sqrt{3}a_{11}+3a_{12}+a_{21}-\sqrt{3}a_{22}) & (-3a_{11}-\sqrt{3}a_{12}+\sqrt{3}a_{21}+a_{22}) & 2(\sqrt{3}a_{13}-a_{23}) \\
2(-a_{31}+\sqrt{3}a_{32}) & 2(-\sqrt{3}a_{31}-a_{32}) & 4a_{33}
\end{pmatrix}, \tag*{(100)} Γ 2 = 4 1 ( a 11 − 3 a 12 + 3 a 21 − 3 a 22 ) ( − 3 a 11 + 3 a 12 + a 21 − 3 a 22 ) 2 ( − a 31 + 3 a 32 ) ( 3 a 11 + a 12 + 3 a 21 + 3 a 22 ) ( − 3 a 11 − 3 a 12 + 3 a 21 + a 22 ) 2 ( − 3 a 31 − a 32 ) 2 ( − a 13 − 3 a 23 ) 2 ( 3 a 13 − a 23 ) 4 a 33 , (100) Γ 3 = 1 4 ( ( a 11 + 3 a 12 − 3 a 21 − 3 a 22 ) ( − 3 a 11 + a 12 + 3 a 21 − 3 a 22 ) 2 ( − a 13 + 3 a 23 ) ( 3 a 11 + 3 a 12 + a 21 + 3 a 22 ) ( − 3 a 11 + 3 a 12 − 3 a 21 + a 22 ) 2 ( − 3 a 13 − a 23 ) 2 ( − a 31 − 3 a 32 ) 2 ( 3 a 31 − a 32 ) 4 a 33 ) . (101) \Gamma_3 = \frac{1}{4}
\begin{pmatrix}
(a_{11}+\sqrt{3}a_{12}-\sqrt{3}a_{21}-3a_{22}) & (-\sqrt{3}a_{11}+a_{12}+3a_{21}-\sqrt{3}a_{22}) & 2(-a_{13}+\sqrt{3}a_{23}) \\
(\sqrt{3}a_{11}+3a_{12}+a_{21}+\sqrt{3}a_{22}) & (-3a_{11}+\sqrt{3}a_{12}-\sqrt{3}a_{21}+a_{22}) & 2(-\sqrt{3}a_{13}-a_{23}) \\
2(-a_{31}-\sqrt{3}a_{32}) & 2(\sqrt{3}a_{31}-a_{32}) & 4a_{33}
\end{pmatrix}. \tag*{(101)} Γ 3 = 4 1 ( a 11 + 3 a 12 − 3 a 21 − 3 a 22 ) ( 3 a 11 + 3 a 12 + a 21 + 3 a 22 ) 2 ( − a 31 − 3 a 32 ) ( − 3 a 11 + a 12 + 3 a 21 − 3 a 22 ) ( − 3 a 11 + 3 a 12 − 3 a 21 + a 22 ) 2 ( 3 a 31 − a 32 ) 2 ( − a 13 + 3 a 23 ) 2 ( − 3 a 13 − a 23 ) 4 a 33 . (101) Γ 1 = ( a 11 a 12 a 13 a 21 a 22 a 23 a 31 a 32 a 33 ) , (102) \Gamma_1 =
\begin{pmatrix}
a_{11} & a_{12} & a_{13} \\
a_{21} & a_{22} & a_{23} \\
a_{31} & a_{32} & a_{33}
\end{pmatrix}, \tag*{(102)} Γ 1 = a 11 a 21 a 31 a 12 a 22 a 32 a 13 a 23 a 33 , (102) Γ 2 = 1 4 ( ( a 11 + 3 a 12 + 3 a 21 + 3 a 22 ) ( − 3 a 11 + a 12 − 3 a 21 + 3 a 22 ) 2 ( − a 13 − 3 a 23 ) ( − 3 a 11 − 3 a 12 + a 21 + 3 a 22 ) ( 3 a 11 − 3 a 12 − 3 a 21 + a 22 ) 2 ( 3 a 13 − a 23 ) 2 ( − a 31 − 3 a 32 ) 2 ( 3 a 31 − a 32 ) 4 a 33 ) , (103) \Gamma_2 = \frac{1}{4}
\begin{pmatrix}
(a_{11}+\sqrt{3}a_{12}+\sqrt{3}a_{21}+3a_{22}) & (-\sqrt{3}a_{11}+a_{12}-3a_{21}+\sqrt{3}a_{22}) & 2(-a_{13}-\sqrt{3}a_{23}) \\
(-\sqrt{3}a_{11}-3a_{12}+a_{21}+\sqrt{3}a_{22}) & (3a_{11}-\sqrt{3}a_{12}-\sqrt{3}a_{21}+a_{22}) & 2(\sqrt{3}a_{13}-a_{23}) \\
2(-a_{31}-\sqrt{3}a_{32}) & 2(\sqrt{3}a_{31}-a_{32}) & 4a_{33}
\end{pmatrix}, \tag*{(103)} Γ 2 = 4 1 ( a 11 + 3 a 12 + 3 a 21 + 3 a 22 ) ( − 3 a 11 − 3 a 12 + a 21 + 3 a 22 ) 2 ( − a 31 − 3 a 32 ) ( − 3 a 11 + a 12 − 3 a 21 + 3 a 22 ) ( 3 a 11 − 3 a 12 − 3 a 21 + a 22 ) 2 ( 3 a 31 − a 32 ) 2 ( − a 13 − 3 a 23 ) 2 ( 3 a 13 − a 23 ) 4 a 33 , (103) Γ 3 = 1 4 ( ( a 11 − 3 a 12 − 3 a 21 + 3 a 22 ) ( 3 a 11 + a 12 − 3 a 21 − 3 a 22 ) 2 ( − a 13 + 3 a 23 ) ( 3 a 11 − 3 a 12 + a 21 − 3 a 22 ) ( 3 a 11 + 3 a 12 + 3 a 21 + a 22 ) 2 ( − 3 a 13 − a 23 ) 2 ( − a 31 + 3 a 32 ) 2 ( − 3 a 31 − a 32 ) 4 a 33 ) . (104) \Gamma_3 = \frac{1}{4}
\begin{pmatrix}
(a_{11}-\sqrt{3}a_{12}-\sqrt{3}a_{21}+3a_{22}) & (\sqrt{3}a_{11}+a_{12}-3a_{21}-\sqrt{3}a_{22}) & 2(-a_{13}+\sqrt{3}a_{23}) \\
(\sqrt{3}a_{11}-3a_{12}+a_{21}-\sqrt{3}a_{22}) & (3a_{11}+\sqrt{3}a_{12}+\sqrt{3}a_{21}+a_{22}) & 2(-\sqrt{3}a_{13}-a_{23}) \\
2(-a_{31}+\sqrt{3}a_{32}) & 2(-\sqrt{3}a_{31}-a_{32}) & 4a_{33}
\end{pmatrix}. \tag*{(104)} Γ 3 = 4 1 ( a 11 − 3 a 12 − 3 a 21 + 3 a 22 ) ( 3 a 11 − 3 a 12 + a 21 − 3 a 22 ) 2 ( − a 31 + 3 a 32 ) ( 3 a 11 + a 12 − 3 a 21 − 3 a 22 ) ( 3 a 11 + 3 a 12 + 3 a 21 + a 22 ) 2 ( − 3 a 31 − a 32 ) 2 ( − a 13 + 3 a 23 ) 2 ( − 3 a 13 − a 23 ) 4 a 33 . (104) Γ 1 = ( a 11 a 12 a 13 a 21 a 22 a 23 a 31 a 32 a 33 ) , (105) \Gamma_1 =
\begin{pmatrix}
a_{11} & a_{12} & a_{13} \\
a_{21} & a_{22} & a_{23} \\
a_{31} & a_{32} & a_{33}
\end{pmatrix}, \tag*{(105)} Γ 1 = a 11 a 21 a 31 a 12 a 22 a 32 a 13 a 23 a 33 , (105) Γ 2 = 1 4 ( a 11 + 3 a 12 + 3 a 21 + 3 a 22 − 3 a 11 + a 12 − 3 a 21 + 3 a 22 2 ( − a 13 − 3 a 23 ) − 3 a 11 − 3 a 12 + a 21 + 3 a 22 3 a 11 − 3 a 12 − 3 a 21 + a 22 2 ( 3 a 13 − a 23 ) 2 ( − a 31 − 3 a 32 ) 2 ( 3 a 31 − a 32 ) 4 a 33 ) , (106) \Gamma_2 = \frac{1}{4}
\begin{pmatrix}
a_{11}+\sqrt{3}a_{12}+\sqrt{3}a_{21}+3a_{22} & -\sqrt{3}a_{11}+a_{12}-3a_{21}+\sqrt{3}a_{22} & 2(-a_{13}-\sqrt{3}a_{23}) \\
-\sqrt{3}a_{11}-3a_{12}+a_{21}+\sqrt{3}a_{22} & 3a_{11}-\sqrt{3}a_{12}-\sqrt{3}a_{21}+a_{22} & 2(\sqrt{3}a_{13}-a_{23}) \\
2(-a_{31}-\sqrt{3}a_{32}) & 2(\sqrt{3}a_{31}-a_{32}) & 4a_{33}
\end{pmatrix}, \tag*{(106)} Γ 2 = 4 1 a 11 + 3 a 12 + 3 a 21 + 3 a 22 − 3 a 11 − 3 a 12 + a 21 + 3 a 22 2 ( − a 31 − 3 a 32 ) − 3 a 11 + a 12 − 3 a 21 + 3 a 22 3 a 11 − 3 a 12 − 3 a 21 + a 22 2 ( 3 a 31 − a 32 ) 2 ( − a 13 − 3 a 23 ) 2 ( 3 a 13 − a 23 ) 4 a 33 , (106) Γ 3 = 1 4 ( a 11 − 3 a 12 − 3 a 21 + 3 a 22 3 a 11 + a 12 − 3 a 21 − 3 a 22 2 ( − a 13 + 3 a 23 ) 3 a 11 − 3 a 12 + a 21 − 3 a 22 3 a 11 + 3 a 12 + 3 a 21 + a 22 2 ( − 3 a 13 − a 23 ) 2 ( − a 31 + 3 a 32 ) 2 ( − 3 a 31 − a 32 ) 4 a 33 ) . (107) \Gamma_3 = \frac{1}{4}\begin{pmatrix}
a_{11}-\sqrt{3}a_{12}-\sqrt{3}a_{21}+3a_{22} & \sqrt{3}a_{11}+a_{12}-3a_{21}-\sqrt{3}a_{22} & 2(-a_{13}+\sqrt{3}a_{23}) \\
\sqrt{3}a_{11}-3a_{12}+a_{21}-\sqrt{3}a_{22} & 3a_{11}+\sqrt{3}a_{12}+\sqrt{3}a_{21}+a_{22} & 2(-\sqrt{3}a_{13}-a_{23}) \\
2(-a_{31}+\sqrt{3}a_{32}) & 2(-\sqrt{3}a_{31}-a_{32}) & 4a_{33}
\end{pmatrix}. \tag*{(107)} Γ 3 = 4 1 a 11 − 3 a 12 − 3 a 21 + 3 a 22 3 a 11 − 3 a 12 + a 21 − 3 a 22 2 ( − a 31 + 3 a 32 ) 3 a 11 + a 12 − 3 a 21 − 3 a 22 3 a 11 + 3 a 12 + 3 a 21 + a 22 2 ( − 3 a 31 − a 32 ) 2 ( − a 13 + 3 a 23 ) 2 ( − 3 a 13 − a 23 ) 4 a 33 . (107) Γ 1 = ( a 11 a 12 a 13 a 21 a 22 a 23 a 31 a 32 a 33 ) , (108) \Gamma_1 = \begin{pmatrix}
a_{11} & a_{12} & a_{13} \\
a_{21} & a_{22} & a_{23} \\
a_{31} & a_{32} & a_{33}
\end{pmatrix}, \tag*{(108)} Γ 1 = a 11 a 21 a 31 a 12 a 22 a 32 a 13 a 23 a 33 , (108) Γ 2 = 1 4 ( a 11 + 3 a 12 − 3 a 21 − 3 a 22 − 3 a 11 + a 12 + 3 a 21 − 3 a 22 2 ( − a 13 + 3 a 23 ) 3 a 11 + 3 a 12 + a 21 + 3 a 22 − 3 a 11 + 3 a 12 − 3 a 21 + a 22 2 ( − 3 a 13 − a 23 ) 2 ( − a 31 − 3 a 32 ) 2 ( 3 a 31 − a 32 ) 4 a 33 ) , (109) \Gamma_2 = \frac{1}{4}\begin{pmatrix}
a_{11}+\sqrt{3}a_{12}-\sqrt{3}a_{21}-3a_{22} & -\sqrt{3}a_{11}+a_{12}+3a_{21}-\sqrt{3}a_{22} & 2(-a_{13}+\sqrt{3}a_{23}) \\
\sqrt{3}a_{11}+3a_{12}+a_{21}+\sqrt{3}a_{22} & -3a_{11}+\sqrt{3}a_{12}-\sqrt{3}a_{21}+a_{22} & 2(-\sqrt{3}a_{13}-a_{23}) \\
2(-a_{31}-\sqrt{3}a_{32}) & 2(\sqrt{3}a_{31}-a_{32}) & 4a_{33}
\end{pmatrix}, \tag*{(109)} Γ 2 = 4 1 a 11 + 3 a 12 − 3 a 21 − 3 a 22 3 a 11 + 3 a 12 + a 21 + 3 a 22 2 ( − a 31 − 3 a 32 ) − 3 a 11 + a 12 + 3 a 21 − 3 a 22 − 3 a 11 + 3 a 12 − 3 a 21 + a 22 2 ( 3 a 31 − a 32 ) 2 ( − a 13 + 3 a 23 ) 2 ( − 3 a 13 − a 23 ) 4 a 33 , (109) Γ 3 = 1 4 ( a 11 − 3 a 12 + 3 a 21 − 3 a 22 3 a 11 + a 12 + 3 a 21 + 3 a 22 2 ( − a 13 − 3 a 23 ) − 3 a 11 + 3 a 12 + a 21 − 3 a 22 − 3 a 11 − 3 a 12 + 3 a 21 + a 22 2 ( 3 a 13 − a 23 ) 2 ( − a 31 + 3 a 32 ) 2 ( − 3 a 31 − a 32 ) 4 a 33 ) . (110) \Gamma_3 = \frac{1}{4}\begin{pmatrix}
a_{11}-\sqrt{3}a_{12}+\sqrt{3}a_{21}-3a_{22} & \sqrt{3}a_{11}+a_{12}+3a_{21}+\sqrt{3}a_{22} & 2(-a_{13}-\sqrt{3}a_{23}) \\
-\sqrt{3}a_{11}+3a_{12}+a_{21}-\sqrt{3}a_{22} & -3a_{11}-\sqrt{3}a_{12}+\sqrt{3}a_{21}+a_{22} & 2(\sqrt{3}a_{13}-a_{23}) \\
2(-a_{31}+\sqrt{3}a_{32}) & 2(-\sqrt{3}a_{31}-a_{32}) & 4a_{33}
\end{pmatrix}. \tag*{(110)} Γ 3 = 4 1 a 11 − 3 a 12 + 3 a 21 − 3 a 22 − 3 a 11 + 3 a 12 + a 21 − 3 a 22 2 ( − a 31 + 3 a 32 ) 3 a 11 + a 12 + 3 a 21 + 3 a 22 − 3 a 11 − 3 a 12 + 3 a 21 + a 22 2 ( − 3 a 31 − a 32 ) 2 ( − a 13 − 3 a 23 ) 2 ( 3 a 13 − a 23 ) 4 a 33 . (110) Γ 1 = ( a 11 a 12 a 13 a 21 a 22 a 23 a 31 a 32 a 33 ) , (111) \Gamma_1 = \begin{pmatrix}
a_{11} & a_{12} & a_{13} \\
a_{21} & a_{22} & a_{23} \\
a_{31} & a_{32} & a_{33}
\end{pmatrix}, \tag*{(111)} Γ 1 = a 11 a 21 a 31 a 12 a 22 a 32 a 13 a 23 a 33 , (111) Γ 2 = 1 4 ( a 11 − 3 a 12 + 3 a 21 − 3 a 22 3 a 11 + a 12 + 3 a 21 + 3 a 22 2 ( − a 13 − 3 a 23 ) − 3 a 11 + 3 a 12 + a 21 − 3 a 22 − 3 a 11 − 3 a 12 + 3 a 21 + a 22 2 ( 3 a 13 − a 23 ) 2 ( − a 31 + 3 a 32 ) 2 ( − 3 a 31 − a 32 ) 4 a 33 ) , (112) \Gamma_2 = \frac{1}{4}\begin{pmatrix}
a_{11}-\sqrt{3}a_{12}+\sqrt{3}a_{21}-3a_{22} & \sqrt{3}a_{11}+a_{12}+3a_{21}+\sqrt{3}a_{22} & 2(-a_{13}-\sqrt{3}a_{23}) \\
-\sqrt{3}a_{11}+3a_{12}+a_{21}-\sqrt{3}a_{22} & -3a_{11}-\sqrt{3}a_{12}+\sqrt{3}a_{21}+a_{22} & 2(\sqrt{3}a_{13}-a_{23}) \\
2(-a_{31}+\sqrt{3}a_{32}) & 2(-\sqrt{3}a_{31}-a_{32}) & 4a_{33}
\end{pmatrix}, \tag*{(112)} Γ 2 = 4 1 a 11 − 3 a 12 + 3 a 21 − 3 a 22 − 3 a 11 + 3 a 12 + a 21 − 3 a 22 2 ( − a 31 + 3 a 32 ) 3 a 11 + a 12 + 3 a 21 + 3 a 22 − 3 a 11 − 3 a 12 + 3 a 21 + a 22 2 ( − 3 a 31 − a 32 ) 2 ( − a 13 − 3 a 23 ) 2 ( 3 a 13 − a 23 ) 4 a 33 , (112) Γ 3 = 1 4 ( a 11 + 3 a 12 − 3 a 21 − 3 a 22 − 3 a 11 + a 12 + 3 a 21 − 3 a 22 2 ( − a 13 + 3 a 23 ) 3 a 11 + 3 a 12 + a 21 + 3 a 22 − 3 a 11 + 3 a 12 − 3 a 21 + a 22 2 ( − 3 a 13 − a 23 ) 2 ( − a 31 − 3 a 32 ) 2 ( 3 a 31 − a 32 ) 4 a 33 ) . (113) \Gamma_3 = \frac{1}{4}\begin{pmatrix}
a_{11}+\sqrt{3}a_{12}-\sqrt{3}a_{21}-3a_{22} & -\sqrt{3}a_{11}+a_{12}+3a_{21}-\sqrt{3}a_{22} & 2(-a_{13}+\sqrt{3}a_{23}) \\
\sqrt{3}a_{11}+3a_{12}+a_{21}+\sqrt{3}a_{22} & -3a_{11}+\sqrt{3}a_{12}-\sqrt{3}a_{21}+a_{22} & 2(-\sqrt{3}a_{13}-a_{23}) \\
2(-a_{31}-\sqrt{3}a_{32}) & 2(\sqrt{3}a_{31}-a_{32}) & 4a_{33}
\end{pmatrix}. \tag*{(113)} Γ 3 = 4 1 a 11 + 3 a 12 − 3 a 21 − 3 a 22 3 a 11 + 3 a 12 + a 21 + 3 a 22 2 ( − a 31 − 3 a 32 ) − 3 a 11 + a 12 + 3 a 21 − 3 a 22 − 3 a 11 + 3 a 12 − 3 a 21 + a 22 2 ( 3 a 31 − a 32 ) 2 ( − a 13 + 3 a 23 ) 2 ( − 3 a 13 − a 23 ) 4 a 33 . (113) Rotation to the Physical Basis of the Scalar Sector
For the potential with the exact GOOFy symmetry, the rotation to the physical basis of the scalar sector is accomplished in two phases: first we move to the Higgs basis, and then we diagonalise the remaining matrix. The full rotation matrix is then given by the following matrix product,
R = U ⋅ U H i g g s (114) R=U\cdot U_{Higgs} \tag*{(114)} R = U ⋅ U H i g g s (114) U H i g g s = ( 1 3 − 1 2 3 − 1 2 3 0 1 2 − 1 2 0 − 1 2 2 1 2 2 0 3 2 2 3 2 2 − 2 3 − 1 2 6 − 1 2 6 0 1 2 2 − 1 2 2 0 − 1 2 2 1 2 2 1 3 − 1 2 3 − 1 2 3 0 − 3 2 2 − 3 2 2 0 − 1 2 2 1 2 2 0 − 1 2 2 1 2 2 − 2 3 − 1 2 6 − 1 2 6 ) (115) U_{Higgs}=
\begin{pmatrix}
\frac{1}{\sqrt{3}} & -\frac{1}{2\sqrt{3}} & -\frac{1}{2\sqrt{3}} & 0 & \frac{1}{2} & -\frac{1}{2}\\
0 & -\frac{1}{2\sqrt{2}} & \frac{1}{2\sqrt{2}} & 0 & \frac{\sqrt{\frac{3}{2}}}{2} & \frac{\sqrt{\frac{3}{2}}}{2}\\
-\sqrt{\frac{2}{3}} & -\frac{1}{2\sqrt{6}} & -\frac{1}{2\sqrt{6}} & 0 & \frac{1}{2\sqrt{2}} & -\frac{1}{2\sqrt{2}}\\
0 & -\frac{1}{2\sqrt{2}} & \frac{1}{2\sqrt{2}} & \frac{1}{\sqrt{3}} & -\frac{1}{2\sqrt{3}} & -\frac{1}{2\sqrt{3}}\\
0 & -\frac{\sqrt{\frac{3}{2}}}{2} & -\frac{\sqrt{\frac{3}{2}}}{2} & 0 & -\frac{1}{2\sqrt{2}} & \frac{1}{2\sqrt{2}}\\
0 & -\frac{1}{2\sqrt{2}} & \frac{1}{2\sqrt{2}} & -\sqrt{\frac{2}{3}} & -\frac{1}{2\sqrt{6}} & -\frac{1}{2\sqrt{6}}
\end{pmatrix} \tag*{(115)} U H i g g s = 3 1 0 − 3 2 0 0 0 − 2 3 1 − 2 2 1 − 2 6 1 − 2 2 1 − 2 2 3 − 2 2 1 − 2 3 1 2 2 1 − 2 6 1 2 2 1 − 2 2 3 2 2 1 0 0 0 3 1 0 − 3 2 2 1 2 2 3 2 2 1 − 2 3 1 − 2 2 1 − 2 6 1 − 2 1 2 2 3 − 2 2 1 − 2 3 1 2 2 1 − 2 6 1 (115) U=
\begin{pmatrix}
1 & 0 & 0 & 0 & 0 & 0\\
0 & -\frac{A\left(\sqrt{A^2+16B^2+432\Lambda_8^2-4B\right)}{A^2+432\Lambda_8^2} & -\frac{12\sqrt{3}\Lambda_8\left(\sqrt{A^2+16B^2+432\Lambda_8^2-4B\right)}{A^2+432\Lambda_8^2} & 0 & 0 & 1\\
0 & -\frac{12\sqrt{3}\Lambda_8\left(\sqrt{A^2+16B^2+432\Lambda_8^2-4B\right)}{A^2+432\Lambda_8^2} & \frac{A\left(\sqrt{A^2+16B^2+432\Lambda_8^2-4B\right)}{A^2+432\Lambda_8^2} & 0 & 1 & 0\\
0 & 0 & 0 & 1 & 0 & 0\\
0 & \frac{A\left(\sqrt{A^2+16B^2+432\Lambda_8^2+4B\right)}{A^2+432\Lambda_8^2} & \frac{12\sqrt{3}\Lambda_8\left(\sqrt{A^2+16B^2+432\Lambda_8^2+4B\right)}{A^2+432\Lambda_8^2} & 0 & 0 & 1\\
0 & \frac{12\sqrt{3}\Lambda_8\left(\sqrt{A^2+16B^2+432\Lambda_8^2+4B\right)}{A^2+432\Lambda_8^2} & -\frac{A\left(\sqrt{A^2+16B^2+432\Lambda_8^2+4B\right)}{A^2+432\Lambda_8^2} & 0 & 1 & 0
\end{pmatrix}}}}}}}}} \tag*{(116)}
where the quantities A A A B B B C C C (26) .
Writing each doublet as
Φ a = ( ϕ a + 1 2 ( v a + ρ a + i η a ) ) (117) \Phi_a=
\begin{pmatrix}
\phi_a^+\\
\frac{1}{\sqrt{2}}(v_a+\rho_a+i\eta_a)
\end{pmatrix} \tag*{(117)} Φ a = ( ϕ a + 2 1 ( v a + ρ a + i η a ) ) (117) the rotation between neutral real and imaginary doublet components and the mass eigenstates is therefore,
\begin{pmatrix}
\rho_1\\
\rho_2\\
\rho_3\\
\eta_1\\
\eta_2\\
\eta_3
\end{pmatrix}
=R
\begin{pmatrix}
h_{SM}\\
H_1\\
H_2\\
G\\
H_3\\
H_4
\end{pmatrix}. \tag*{(118)}\end{pmatrix}
Notice that in this equation H 1 , 2 H_{1,2} H 1 , 2 h 1 h_1 h 1 H 3 , 4 H_{3,4} H 3 , 4 h 2 h_2 h 2
h S M = 1 2 ( η 2 − η 3 + 2 ρ 1 3 − ρ 2 3 − ρ 3 3 ) . (119) h_{SM}=\frac{1}{2}\left(\eta_2-\eta_3+\frac{2\rho_1}{\sqrt{3}}-\frac{\rho_2}{\sqrt{3}}-\frac{\rho_3}{\sqrt{3}}\right). \tag*{(119)} h S M = 2 1 ( η 2 − η 3 + 3 2 ρ 1 − 3 ρ 2 − 3 ρ 3 ) . (119) 
