Fermion masses and mixings, dark matter, leptogenesis and $g-2$ muon anomaly in an extended 2HDM with inverse seesaw

Abstract

We propose a predictive $Q_4$ flavored 2HDM model, where the scalar sector is enlarged by the inclusion of several gauge singlet scalars and the fermion sector by the inclusion of right-handed Majorana neutrinos. In our model, the $Q_4$ family symmetry is supplemented by several auxiliary cyclic symmetries, whose spontaneous breaking produces the observed pattern of SM charged fermion masses and quark mixing angles. The light active neutrino masses are generated from an inverse seesaw mechanism at one loop level thanks to a remnant preserved $Z_2$ symmetry. Our model successfully reproduces the measured dark matter relic abundance and is consistent with direct detection constraints for masses of the DM candidate around $\sim$ 6.3 TeV. Furthermore, our model is also consistent with the lepton and baryon asymmetries of the Universe as well as with the muon anomalous magnetic moment.

Appendices

Appendix A: The product rules for $Q_{4}$

The irreducible representations of the $Q_{4}$ group are four singlets, $\mathbf {1}_{+}{+}$, $\mathbf {1}_{+-}$, $\mathbf {1}_{-+}$ and $\mathbf {1}_{--}$, and one doublet $\mathbf {2}$. The tensor products of the $Q_{4}$ irreducible representation are given by [3]:

$$\begin{aligned}&\left( \begin{array}{c} z \\ \bar{z} \end{array} \right) _{\mathbf {2}}\otimes \left( \begin{array}{c} z^{\prime } \\ \bar{z}^{\prime } \end{array} \right) _{\mathbf {2}} \nonumber \\&\quad =\left( z\bar{z}^{\prime }-\bar{z}z^{\prime }\right) _{\mathbf {1}_{++}}\oplus \left( z\bar{z}^{\prime }+\bar{z}z^{\prime }\right) _{\mathbf {1}_{--}} \nonumber \\&\quad \oplus \left( zz^{\prime }-\bar{z}\bar{z}^{\prime }\right) _{\mathbf {1} _{+-}}\oplus \left( zz^{\prime }+\bar{z}\bar{z}^{\prime }\right) _{\mathbf {1} _{-+}}, \end{aligned}$$

(89)

$$\begin{aligned}&\left( w\right) _{\mathbf {1}_{++}}\otimes \left( \begin{array}{c} z \\ \bar{z} \end{array} \right) _{\mathbf {2}}=\left( \begin{array}{c} wz \\ w\bar{z} \end{array} \right) _{\mathbf {2}},\quad \left( w\right) _{\mathbf {1}_{--}}\otimes \left( \begin{array}{c} z \\ \bar{z} \end{array} \right) _{\mathbf {2}}=\left( \begin{array}{c} wz \\ -w\bar{z} \end{array} \right) _{\mathbf {2}}, \nonumber \\&\quad \left( w\right) _{\mathbf {1}_{+-}}\otimes \left( \begin{array}{c} z \\ \bar{z} \end{array} \right) _{\mathbf {2}}=\left( \begin{array}{c} w\bar{z} \\ wz \end{array} \right) _{\mathbf {2}},\quad \left( w\right) _{\mathbf {1}_{-+}}\otimes \left( \begin{array}{c} z \\ \bar{z} \end{array} \right) _{\mathbf {2}}=\left( \begin{array}{c} w\bar{z} \\ -wz \end{array} \right) _{\mathbf {2}}, \end{aligned}$$

(90)

$$\begin{aligned} \mathbf {1}_{s_{1}s_{2}}\otimes \mathbf {1}_{s_{1}^{\prime }s_{2}^{\prime }}= \mathbf {1}_{s_{1}^{\prime \prime }s_{2}^{\prime \prime }}, \end{aligned}$$

where $s_{1}^{\prime \prime }=s_{1}s_{1}^{\prime }$ and $s_{2}^{\prime \prime }=s_{2}s_{2}^{\prime }$.

Appendix B: Scalar potential for two $Q_{4}$ doublets

The scalar potential for two $Q_{4}$ doublets $\xi$ and $\Phi$ (with $\xi$ real and $\Phi$ complex) has the form

$$\begin{aligned} V= & {} \mu _{\xi }^{2}\left( \xi \xi \right) _{\mathbf {1}_{++}}+\mu _{\Phi }^{2}\left( \Phi \Phi ^{\dagger }\right) _{\mathbf {1}_{++}}+\lambda _{1}\left( \xi \xi \right) _{\mathbf {1}_{++}}\left( \xi \xi \right) _{ \mathbf {1}_{++}}+\lambda _{2}\left( \xi \xi \right) _{\mathbf {1}_{+-}}\left( \xi \xi \right) _{\mathbf {1}_{+-}}+\lambda _{3}\left( \xi \xi \right) _{ \mathbf {1}_{-+}}\left( \xi \xi \right) _{\mathbf {1}_{-+}} \nonumber \\&+\lambda _{4}\left( \xi \xi \right) _{\mathbf {1}_{--}}\left( \xi \xi \right) _{\mathbf {1}_{--}}+\lambda _{5}\left( \Phi \Phi ^{\dagger }\right) _{ \mathbf {1}_{++}}\left( \Phi \Phi ^{\dagger }\right) _{\mathbf {1} _{++}}+\lambda _{6}\left( \Phi \Phi ^{\dagger }\right) _{\mathbf {1} _{+-}}\left( \Phi \Phi ^{\dagger }\right) _{\mathbf {1}_{+-}}+\lambda _{7}\left( \Phi \Phi ^{\dagger }\right) _{\mathbf {1}_{-+}}\left( \Phi \Phi ^{\dagger }\right) _{\mathbf {1}_{-+}} \nonumber \\&+\lambda _{8}\left( \Phi \Phi ^{\dagger }\right) _{\mathbf {1}_{--}}\left( \Phi \Phi ^{\dagger }\right) _{\mathbf {1}_{--}}+\lambda _{9}\left( \xi \xi \right) _{\mathbf {1}_{++}}\left( \Phi \Phi ^{\dagger }\right) _{\mathbf {1} _{++}}+\lambda _{10}\left( \xi \xi \right) _{\mathbf {1}_{+-}}\left( \Phi \Phi ^{\dagger }\right) _{\mathbf {1}_{+-}}+\lambda _{11}\left( \xi \xi \right) _{\mathbf {1}_{-+}}\left( \Phi \Phi ^{\dagger }\right) _{\mathbf {1} _{-+}} \nonumber \\&+\lambda _{12}\left( \xi \xi \right) _{\mathbf {1}_{--}}\left( \Phi \Phi ^{\dagger }\right) _{\mathbf {1}_{--}} \end{aligned}$$

(91)

The above given scalar potential can be rewritten as follows:

$$\begin{aligned} V&= \mu _{\Phi }^{2}\left( \Phi _{1}\Phi _{2}^{\dagger }-\Phi _{2}\Phi _{1}^{\dagger }\right) +\lambda _{2}\left( \xi _{1}^{2}-\xi _{2}^{2}\right) ^{2}+\lambda _{3}\left( \xi _{1}^{2}+\xi _{2}^{2}\right) ^{2}+4\lambda _{4}\xi _{1}^{2}\xi _{2}^{2}+\lambda _{5}\left( \Phi _{1}\Phi _{2}^{\dagger }-\Phi _{2}\Phi _{1}^{\dagger }\right) ^{2} \nonumber \\&+\lambda _{6}\left( \Phi _{1}\Phi _{1}^{\dagger }-\Phi _{2}\Phi _{2}^{\dagger }\right) ^{2}+\lambda _{7}\left( \Phi _{1}\Phi _{1}^{\dagger }+\Phi _{2}\Phi _{2}^{\dagger }\right) ^{2}+\lambda _{8}\left( \Phi _{1}\Phi _{2}^{\dagger }+\Phi _{2}\Phi _{1}^{\dagger }\right) ^{2} \nonumber \\&+\lambda _{10}\left( \xi _{1}^{2}-\xi _{2}^{2}\right) \left( \Phi _{1}\Phi _{1}^{\dagger }-\Phi _{2}\Phi _{2}^{\dagger }\right) +\lambda _{11}\left( \xi _{1}^{2}+\xi _{2}^{2}\right) \left( \Phi _{1}\Phi _{1}^{\dagger }+\Phi _{2}\Phi _{2}^{\dagger }\right) +2\lambda _{12}\xi _{1}\xi _{2}\left( \Phi _{1}\Phi _{2}^{\dagger }+\Phi _{2}\Phi _{1}^{\dagger }\right) \end{aligned}$$

(92)

Due to Hermiticity, the parameters are reals and the minimum conditions are the following

$$\begin{aligned} 0&= v_{\xi _{1}}\left[ \lambda _{2}\left( v^{2}_{\xi _{1}}-v^{2}_{\xi _{2}} \right) +\lambda _{3}\left( v^{2}_{\xi _{1}}+v^{2}_{\xi _{2}}\right) +2 \lambda _{4}v^{2}_{\xi _{2}}+v_{\Phi _{1}}v_{\Phi _{2}}\left\{ \lambda _{10}\cos { (\alpha -\theta )}+i\lambda _{11}\sin {(\theta -\alpha )}\right\} +\lambda _{12} \frac{v_{\xi _{2}}}{2v_{\xi _{1}}}\left( v^{2}_{\Phi _{2}}-v^{2}_{\Phi _{1}} \right) \right] , \nonumber \\ 0 &= v_{\xi _{2}}\left[ -\lambda _{2}\left( v^{2}_{\xi _{1}}-v^{2}_{\xi _{2}} \right) +\lambda _{3}\left( v^{2}_{\xi _{1}}+v^{2}_{\xi _{2}}\right) +2 \lambda _{4}v^{2}_{\xi _{1}}+v_{\Phi _{1}}v_{\Phi _{2}}\left\{ -\lambda _{10}\cos { (\alpha -\theta )}+i\lambda _{11}\sin {(\theta -\alpha )}\right\} +\lambda _{12} \frac{v_{\xi _{1}}}{2v_{\xi _{2}}}\left( v^{2}_{\Phi _{2}}-v^{2}_{\Phi _{1}} \right) \right] , \nonumber \\ 0&= v_{\Phi _{1}}\left[ \frac{\mu ^{2}_{\Phi }}{2}+\lambda _{5}\left( v^{2}_{ \Phi _{1}}+v^{2}_{\Phi _{2}}\right) +2v^{2}_{\Phi _{2}}\left\{ \lambda _{6}\cos ^{2} {(\alpha -\theta )}-\lambda _{7}\sin ^{2}{(\theta -\alpha )}\right\} -\lambda _{8} \left( v^{2}_{\Phi _{2}}-v^{2}_{\Phi _{1}}\right) -\lambda _{12}v_{\xi _{1}}v_{ \xi _{2}}\right. \nonumber \\&\left. +\frac{v_{\Phi _{2}}}{2v_{\Phi _{1}}}\left\{ \lambda _{10}\left( v^{2}_{ \xi _{1}}-v^{2}_{\xi _{2}}\right) \cos {(\alpha -\theta )}+i\lambda _{11} \left( v^{2}_{\xi _{1}}+v^{2}_{\xi _{2}}\right) \sin {(\theta -\alpha )}\right\} \right] , \nonumber \\ 0&= v_{\Phi _{2}}\left[ \frac{\mu ^{2}_{\Phi }}{2}+\lambda _{5}\left( v^{2}_{ \Phi _{1}}+v^{2}_{\Phi _{2}}\right) +2v^{2}_{\Phi _{1}}\left\{ \lambda _{6}\cos ^{2} {(\alpha -\theta )}-\lambda _{7}\sin ^{2}{(\theta -\alpha )}\right\} +\lambda _{8} \left( v^{2}_{\Phi _{2}}-v^{2}_{\Phi _{1}}\right) +\lambda _{12}v_{\xi _{1}}v_{ \xi _{2}}\right. \nonumber \\&\left. +\frac{v_{\Phi _{1}}}{2v_{\Phi _{2}}}\left\{ \lambda _{10}\left( v^{2}_{ \xi _{1}}-v^{2}_{\xi _{2}}\right) \cos {(\alpha -\theta )}+i\lambda _{11} \left( v^{2}_{\xi _{1}}+v^{2}_{\xi _{2}}\right) \sin {(\theta -\alpha )}\right\} \right] . \end{aligned}$$

(93)

where we have considered in general

$$\begin{aligned} \left\langle \xi \right\rangle =\left( v_{\xi _{1} }, v_{\xi _{2} }\right) , \left\langle \Phi \right\rangle =\left( v_{\Phi _{1} } e^{i\theta },v_{\Phi _{2} }e^{i\alpha }\right) . \end{aligned}$$

(94)

According to our purpose, we need the alignment $\left\langle \xi \right\rangle =v_{\xi }\left( 1,0\right)$ ($v_{\xi _{1}}\ne 0$ and $v_{\xi _{2}}=0$), then we use the former two expressions in Eq. ((93)) to obtain

$$\begin{aligned} 0&= v_{\xi }^{2}\left( \lambda _{2}+\lambda _{3}\right) +v_{\Phi _{1}}v_{\Phi _{2}}\left\{ \lambda _{10}\cos {(\alpha -\theta )}+i\lambda _{11}\sin {(\theta -\alpha )}\right\} , \nonumber \\ 0&= v_{\Phi _{1}}\left[ \frac{\mu _{\Phi }^{2}}{2}+\lambda _{5}\left( v_{\Phi _{1}}^{2}+v_{\Phi _{2}}^{2}\right) +2v_{\Phi _{2}}^{2}\left\{ \lambda _{6}\cos ^{2}{(\alpha -\theta )}-\lambda _{7}\sin ^{2}{(\theta -\alpha )}\right\} -\lambda _{8}\left( v_{\Phi _{2}}^{2}-v_{\Phi _{1}}^{2}\right) \right. \nonumber \\&\left. +\frac{v_{\Phi _{2}}}{2v_{\Phi _{1}}}\left\{ \lambda _{10}\cos { (\alpha -\theta )}+i\lambda _{11}\sin {(\theta -\alpha )}\right\} v_{\xi }^{2}\right] , \nonumber \\ 0&= v_{\Phi _{2}}\left[ \frac{\mu _{\Phi }^{2}}{2}+\lambda _{5}\left( v_{\Phi _{1}}^{2}+v_{\Phi _{2}}^{2}\right) +2v_{\Phi _{1}}^{2}\left\{ \lambda _{6}\cos ^{2}{(\alpha -\theta )}-\lambda _{7}\sin ^{2}{(\theta -\alpha )}\right\} +\lambda _{8}\left( v_{\Phi _{2}}^{2}-v_{\Phi _{1}}^{2}\right) \right. \nonumber \\&\left. +\frac{v_{\Phi _{1}}}{2v_{\Phi _{2}}}\left\{ \lambda _{10}\cos { (\alpha -\theta )}+i\lambda _{11}\sin {(\theta -\alpha )}\right\} v_{\xi }^{2}\right] \end{aligned}$$

(95)

As one can notice, in the last to expressions in Eq. (95), there is a symmetry of interchange $v_{\Phi _{1}}\leftrightarrow v_{\Phi _{2}}$. Along with this, we demand that $v_{\Phi _{1}}\ne 0\ne v_{\Phi _{2}}$ therefore $v_{\Phi _{1}}=v_{\Phi _{2}}\equiv v_{\Phi }$ from the last two expressions. Finally, we end up having

$$\begin{aligned} 0&= v_{\xi }^{2}\left( \lambda _{2}+\lambda _{3}\right) +v_{\Phi }^{2}\left\{ \lambda _{10}\cos {(\alpha -\theta )}+i\lambda _{11}\sin { (\theta -\alpha )}\right\} , \nonumber \\ 0&= \frac{\mu _{\Phi }^{2}}{2}+2v_{\Phi }^{2}\left\{ \lambda _{5}+\lambda _{6}\cos ^{2}{(\alpha -\theta )}-\lambda _{7}\sin ^{2}{(\theta -\alpha )} \right\} +\frac{v_{\xi }^{2}}{2}\left\{ \lambda _{10}\cos {(\alpha -\theta )} +i\lambda _{11}\sin {(\theta -\alpha )}\right\} . \end{aligned}$$

(96)

This shows that the VEV pattern of the two $Q_{4}$ doublets $\xi$ and $\Phi$ shown in Eq. (1) is consistent with the minimization conditions of the scalar potential.

Appendix C: Stability of the scalar potential for two $Q_{4}$ doublets

With the aim to determine the stability conditions of the scalar potential for the two $Q_{4}$ doublets $\xi$ and $\Phi$, we proceed to analyze its quartic terms because they will dominate the behavior of the scalar potential in the region of very large values of the field components. To this end, we introduce the following Hermitian bilinear combination of the scalar fields:

$$\begin{aligned} a&= \Phi _{1}\Phi _{1}^{\dagger },\quad b=\Phi _{2}\Phi _{2}^{\dagger },\quad c=\Phi _{1}\Phi _{2}^{\dagger }+\Phi _{2}\Phi _{1}^{\dagger },\quad d=i\left( \Phi _{1}\Phi _{2}^{\dagger }-\Phi _{2}\Phi _{1}^{\dagger }\right) , \nonumber \\ e&= \xi _{1}^{2},\quad f=\xi _{2}^{2} \end{aligned}$$

(97)

and rewrite the quartic terms of the scalar potential for the two $Q_{4}$ doublets $\xi$ and $\Phi$:

$$\begin{aligned} V_{4}&= \lambda _{2}\left( \xi _{1}^{2}-\xi _{2}^{2}\right) ^{2}+\lambda _{3}\left( \xi _{1}^{2}+\xi _{2}^{2}\right) ^{2}+4\lambda _{4}\xi _{1}^{2}\xi _{2}^{2}+\lambda _{5}\left( \Phi _{1}\Phi _{2}^{\dagger }-\Phi _{2}\Phi _{1}^{\dagger }\right) ^{2}+\lambda _{6}\left( \Phi _{1}\Phi _{1}^{\dagger }-\Phi _{2}\Phi _{2}^{\dagger }\right) ^{2} \nonumber \\&+\lambda _{7}\left( \Phi _{1}\Phi _{1}^{\dagger }+\Phi _{2}\Phi _{2}^{\dagger }\right) ^{2}+\lambda _{8}\left( \Phi _{1}\Phi _{2}^{\dagger }+\Phi _{2}\Phi _{1}^{\dagger }\right) ^{2}+\lambda _{10}\left( \xi _{1}^{2}-\xi _{2}^{2}\right) \left( \Phi _{1}\Phi _{1}^{\dagger }-\Phi _{2}\Phi _{2}^{\dagger }\right) \nonumber \\&+\lambda _{11}\left( \xi _{1}^{2}+\xi _{2}^{2}\right) \left( \Phi _{1}\Phi _{1}^{\dagger }+\Phi _{2}\Phi _{2}^{\dagger }\right) +2\lambda _{12}\xi _{1}\xi _{2}\left( \Phi _{1}\Phi _{2}^{\dagger }+\Phi _{2}\Phi _{1}^{\dagger }\right) \end{aligned}$$

(98)

in the following form:

$$\begin{aligned} V_{4}&= \left( \lambda _{2}+\lambda _{3}\right) \left( e^{2}+f^{2}\right) +2\left( \lambda _{3}-\lambda _{2}+2\lambda _{4}\right) ef-\lambda _{5}d^{2}+\left( \lambda _{6}+\lambda _{7}\right) \left( a^{2}+b^{2}\right) +2\left( \lambda _{7}-\lambda _{6}\right) ab \nonumber \\&+\lambda _{8}c^{2}+\lambda _{10}\left( e-f\right) \left( a-b\right) +\lambda _{11}\left( e+f\right) \left( a+b\right) +2\lambda _{12}\sqrt{ef}c \end{aligned}$$

(99)

Defining

$$\begin{aligned} \kappa _{1}=\lambda _{2}+\lambda _{3},\quad \kappa _{2}=2\left( \lambda _{3}-\lambda _{2}+2\lambda _{4}\right) ,\quad \kappa _{3}=\lambda _{6}+\lambda _{7},\quad \kappa _{4}=2\left( \lambda _{7}-\lambda _{6}\right) , \end{aligned}$$

(100)

The above given quartic scalar interactions can be rewritten as follows:

$$\begin{aligned} V_{4}&= \kappa _{1}\left( e^{2}+f^{2}\right) +\kappa _{2}ef-\lambda _{5}d^{2}+\kappa _{3}\left( a^{2}+b^{2}\right) +\kappa _{4}ab+\lambda _{8}c^{2} \nonumber \\&+\lambda _{10}\left( e-f\right) \left( a-b\right) +\lambda _{11}\left( e+f\right) \left( a+b\right) +2\lambda _{12}\sqrt{ef}c \nonumber \\& = \frac{\kappa _{1}}{2}\left[ \left( e-f\right) ^{2}+\left( e+f\right) ^{2} \right] +\frac{\kappa _{3}}{2}\left[ \left( a-b\right) ^{2}+\left( a+b\right) ^{2}\right] \nonumber \\&+\kappa _{2}ef-\lambda _{5}d^{2}+\kappa _{4}ab \nonumber \\&+\lambda _{8}c^{2}+\lambda _{10}\left( e-f\right) \left( a-b\right) +\lambda _{11}\left( e+f\right) \left( a+b\right) +2\lambda _{12}\sqrt{ef}c \nonumber \\&= \left[ \sqrt{\frac{\kappa _{1}}{2}}\left( e-f\right) +\sqrt{\frac{\kappa _{3}}{2}}\left( a-b\right) \right] ^{2}+\left[ \sqrt{\frac{\kappa _{1}}{2}} \left( e+f\right) +\sqrt{\frac{\kappa _{3}}{2}}\left( a+b\right) \right] ^{2} \nonumber \\&+\left( \lambda _{10}-\sqrt{\kappa _{1}\kappa _{3}}\right) \left( e-f\right) \left( a-b\right) +\left( \lambda _{11}-\sqrt{\kappa _{1}\kappa _{3}}\right) \left( e+f\right) \left( a+b\right) \nonumber \\&-\lambda _{5}d^{2}+\kappa _{4}ab+\left[ \sqrt{\kappa _{2}}\sqrt{ef}+\sqrt{ \lambda _{8}}c\right] ^{2}+2\left( \lambda _{12}-\sqrt{\kappa _{2}\lambda _{8}}\right) \sqrt{ef}c \end{aligned}$$

(101)

Following the procedure used for analyzing the stability described in Refs. [74, 75], we find that our scalar potential of two $Q_{4}$ doublets will be stable when the following conditions are fulfilled:

$$\begin{aligned} \lambda _{2}+\lambda _{3}\ge & {} 0,\quad \lambda _{6}+\lambda _{7}\ge 0,\quad \lambda _{10}-\sqrt{\left( \lambda _{2}+\lambda _{3}\right) \left( \lambda _{6}+\lambda _{7}\right) }\ge 0,\quad \lambda _{11}-\sqrt{\left( \lambda _{2}+\lambda _{3}\right) \left( \lambda _{6}+\lambda _{7}\right) }\ge 0, \quad \lambda _{5}\le 0, \nonumber \\ \lambda _{7}\ge & {} \lambda _{6},\quad \lambda _{8}\ge 0,\quad \lambda _{3}-\lambda _{2}+2\lambda _{4}\ge 0,\quad \lambda _{12}\ge \sqrt{2\left( \lambda _{3}-\lambda _{2}+2\lambda _{4}\right) \lambda _{8}}. \end{aligned}$$

(102)

Appendix D: Analytical expressions for the entries of the CKM matrix

Explicitly, the CKM entries are given as

$$\begin{aligned} (\mathbf {V}_{CKM})_{ud}= & {} -\frac{\vert m_{u}\vert }{\vert a_{u}\vert }\sqrt{\frac{\vert m_{u}\vert ^{2} \mathcal {N}_{2}\mathcal {N}_{3} \mathcal {M}_{2}}{\mathcal {D}_{1}}}~\cos {\theta _{d}}+ \sqrt{\frac{\mathcal {N}_{1} \mathcal {M}_{1}\mathcal {M}_{3}}{\mathcal {D}_{1}} }~\sin {\theta _{d}}~e^{-i\bar{\eta }_{c}},\nonumber \\ (\mathbf {V}_{CKM})_{us}= & {} -\left[ \frac{\vert m_{u}\vert }{\vert a_{u}\vert }\sqrt{\frac{\vert m_{u}\vert ^{2} \mathcal {N}_{2}\mathcal {N}_{3} \mathcal {M}_{2}}{\mathcal {D}_{1}}}~\sin {\theta _{d}}+\sqrt{\frac{\mathcal {N}_{1} \mathcal {M}_{1}\mathcal {M}_{3}}{\mathcal {D}_{1}} }~\cos {\theta _{d}}~e^{-i\bar{\eta }_{c}}\right] ,\nonumber \\ (\mathbf {V}_{CKM})_{ub}= & {} \frac{1}{\vert a_{u}\vert }\sqrt{\frac{\mathcal {N}_{1} \mathcal {M}_{2} \mathcal {K}}{\mathcal {D}_{1}}}~e^{-i\eta _{t}},\nonumber \\ (\mathbf {V}_{CKM})_{cd}= & {} -\left[ \frac{ \vert m_{c}\vert }{\vert a_{u}\vert }\sqrt{\frac{\vert m_{c}\vert ^{2}\mathcal {N }_{1}\mathcal {N}_{3} \mathcal {M}_{1}}{\mathcal {D}_{2}}}~\cos {\theta _{d}}+\sqrt{\frac{ \mathcal {N}_{2} \mathcal {M}_{2}\mathcal {M}_{3}}{\mathcal {D} _{2}}}~\sin {\theta _{d}}~e^{-i\bar{\eta }_{c}}\right] ,\nonumber \\ (\mathbf {V}_{CKM})_{cs}= & {} \frac{ \vert m_{c}\vert }{\vert a_{u}\vert }\sqrt{\frac{\vert m_{c}\vert ^{2}\mathcal {N }_{1}\mathcal {N}_{3} \mathcal {M}_{1}}{\mathcal {D}_{2}}}~\sin {\theta _{d}}+\sqrt{\frac{ \mathcal {N}_{2} \mathcal {M}_{2}\mathcal {M}_{3}}{\mathcal {D} _{2}}}~\cos {\theta _{d}}~e^{-i\bar{\eta }_{c}},\nonumber \\ (\mathbf {V}_{CKM})_{cb}= & {} -\frac{1}{\vert a_{u}\vert }\sqrt{\frac{\mathcal {N}_{2} \mathcal {M}_{1} \mathcal {K}}{\mathcal {D}_{2}}}~e^{-i\eta _{t}},\nonumber \\ (\mathbf {V}_{CKM})_{td}= & {} \frac{\vert m_{t}\vert }{\vert a_{u}\vert }\sqrt{\frac{\vert m_{t}\vert ^{2}\mathcal {N}_{1} \mathcal {N}_{2} \mathcal {M}_{3}}{\mathcal {D}_{3}}}~\cos {\theta _{d}}-\sqrt{\frac{ \mathcal {N}_{3} \mathcal {M}_{1}\mathcal {M}_{2}}{ \mathcal {D}_{3}}}~\sin {\theta _{d}}~e^{-i\bar{\eta }_{c}},\nonumber \\ (\mathbf {V}_{CKM})_{ts}= & {} \frac{\vert m_{t}\vert }{\vert a_{u}\vert }\sqrt{\frac{\vert m_{t}\vert ^{2}\mathcal {N}_{1} \mathcal {N}_{2} \mathcal {M}_{3}}{\mathcal {D}_{3}}}~\sin {\theta _{d}}-\sqrt{\frac{ \mathcal {N}_{3} \mathcal {M}_{1}\mathcal {M}_{2}}{ \mathcal {D}_{3}}}~\cos {\theta _{d}}~e^{-i\bar{\eta }_{c}},\nonumber \\ (\mathbf {V}_{CKM})_{tb}= & {} \frac{1}{\vert a_{u}\vert }\sqrt{\frac{\mathcal {N}_{2} \mathcal {M}_{3} \mathcal {K}}{\mathcal {D}_{3}}}~e^{-i\eta _{t}}. \end{aligned}$$

(103)