$\displaystyle=-32\pi^{2}\lambda_{\phi}+6\beta_{\lambda_{\phi}}+(i\pi-1)\Big{[}\kappa_{2}^{2}+4\lambda_{\phi}^{2}\Big{]}.$
(D.6)

###  $\bullet$ $Q=1,Y=1/2$ :

$\displaystyle\phi^{0}\xi^{+}\to\phi^{0}\xi^{+}$
$\displaystyle=-32\pi^{2}\lambda_{\phi\xi}+6\beta_{\lambda_{\phi\xi}}+(i\pi-1)\Big{[}3\kappa_{3}^{2}+4\lambda_{\phi\xi}^{2}\Big{]},$
$\displaystyle\phi^{0}\xi^{+}\to\phi^{+}\xi^{0}$
$\displaystyle=-\sqrt{2}(i\pi-1)\kappa_{3}^{2},$
$\displaystyle\phi^{0}\xi^{+}\to\chi^{+}\phi^{0*}$
$\displaystyle=-16\sqrt{2}\pi^{2}\kappa_{3}+3\sqrt{2}\beta_{\kappa_{3}}+\frac{(i\pi-1)}{\sqrt{2}}\Big{[}\kappa_{3}\left(\kappa_{2}+4\lambda_{\phi\xi}+2\lambda_{\phi\chi}\right),$
$\displaystyle\phi^{0}\xi^{+}\to\phi^{-}\chi^{++}$
$\displaystyle=-16\pi^{2}\kappa_{3}+3\beta_{\kappa_{3}}+(i\pi-1)\Big{[}\frac{1}{2}\kappa_{3}\left(3\kappa_{2}+4\lambda_{\phi\xi}+2\lambda_{\phi\chi}\right)\Big{]},$
$\displaystyle\phi^{+}\xi^{0}\to\phi^{0}\xi^{+}$
$\displaystyle=\phi^{0}\xi^{+}\to\phi^{+}\xi^{0},$
$\displaystyle\chi^{+}\phi^{0*}\to\phi^{0}\xi^{+}$
$\displaystyle=\phi^{0}\xi^{+}\to\chi^{+}\phi^{0*},$
$\displaystyle\phi^{-}\chi^{++}\to\phi^{0}\xi^{+}$
$\displaystyle=\phi^{0}\xi^{+}\to\phi^{-}\chi^{++},$
$\displaystyle\phi^{+}\xi^{0}\to\phi^{+}\xi^{0}$
$\displaystyle=-32\pi^{2}\lambda_{\phi\xi}+6\beta_{\lambda_{\phi\xi}}+2(i\pi-1)\Big{[}\kappa_{3}^{2}+2\lambda_{\phi\xi}^{2}\Big{]},$
$\displaystyle\phi^{+}\xi^{0}\to\chi^{+}\phi^{0*}$
$\displaystyle=-(i\pi-1)\kappa_{2}\kappa_{3},$
$\displaystyle\phi^{+}\xi^{0}\to\phi^{-}\chi^{++}$
$\displaystyle=16\sqrt{2}\pi^{2}\kappa_{3}-3\sqrt{2}\beta_{\kappa_{3}}-\frac{1}{\sqrt{2}}(i\pi-1)\Big{[}\kappa_{3}\left(\kappa_{2}+4\lambda_{\phi\xi}+2\lambda_{\phi\chi}\right)\Big{]},$
$\displaystyle\chi^{+}\phi^{0*}\to\phi^{+}\xi^{0}$
$\displaystyle=\phi^{+}\xi^{0}\to\chi^{+}\phi^{0*},$
$\displaystyle\phi^{-}\chi^{++}\to\phi^{+}\xi^{0}$
$\displaystyle=\phi^{+}\xi^{0}\to\phi^{-}\chi^{++},$
$\displaystyle\phi^{-}\chi^{++}\to\phi^{-}\chi^{++}=$
$\displaystyle-16\pi^{2}\left(\lambda_{\phi\chi}+\frac{\kappa_{2}}{2}\right)+3\left(\beta_{\lambda_{\phi\chi}}+\frac{1}{2}\beta_{\kappa_{2}}\right)$
$\displaystyle+(i\pi-1)\Big{[}\frac{3}{4}\kappa_{2}^{2}+3\kappa_{3}^{2}+\kappa_{2}\lambda_{\phi\chi}+\lambda_{\phi\chi}^{2}\Big{]},$
$\displaystyle\phi^{-}\chi^{++}\to\chi^{+}\phi^{0*}=$
$\displaystyle-8\sqrt{2}\kappa_{2}+\frac{3}{\sqrt{2}}\beta_{\kappa_{2}}+\frac{1}{2\sqrt{2}}(i\pi-1)\Big{[}\kappa_{2}^{2}+4\kappa_{3}^{2}+4\kappa_{2}\lambda_{\phi\chi}\Big{]},$
$\displaystyle\chi^{+}\phi^{0*}\to\phi^{-}\chi^{++}$
$\displaystyle=\phi^{-}\chi^{++}\to\chi^{+}\phi^{0*},$
$\displaystyle\chi^{+}\phi^{0*}\to\chi^{+}\phi^{0*}$
$\displaystyle=-16\pi^{2}\lambda_{\phi\chi}+3\beta_{\lambda_{\phi\chi}}+(i\pi-1)\Big{[}\frac{\kappa_{2}^{2}}{2}+2\kappa_{3}^{2}+\lambda_{\phi\chi}^{2}\Big{]}.$
(D.7)

###  $\bullet$ $Q=1,Y=1$ :

$\displaystyle\phi^{+}\phi^{0}\to\phi^{+}\phi^{0}$
$\displaystyle=-32\pi^{2}\lambda_{\phi}+6\beta_{\lambda_{\phi}}+2(i\pi-1)\Big{[}\kappa_{3}^{2}+2\lambda_{\phi}^{2}\Big{]},$
$\displaystyle\phi^{+}\phi^{0}\to\chi^{+}\xi^{0}$
$\displaystyle=0=\chi^{+}\xi^{0}\to\phi^{+}\phi^{0},$
$\displaystyle\phi^{+}\phi^{0}\to\chi^{0}\xi^{+}$
$\displaystyle=-16\pi^{2}\kappa_{3}+3\beta_{\kappa_{3}}+(i\pi-1)\Big{[}\kappa_{3}\left(-\kappa_{1}+2(\lambda_{\phi}+\lambda_{\chi\xi})\right)\Big{]},$
$\displaystyle\phi^{+}\phi^{0}\to\chi^{++}\xi^{-}$
$\displaystyle=-16\pi^{2}\kappa_{3}+3\beta_{\kappa_{3}}+(i\pi-1)\Big{[}\kappa_{3}\left(-\kappa_{1}+2(\lambda_{\phi}+\lambda_{\chi\xi})\right)\Big{]},$
$\displaystyle\chi^{0}\xi^{+}\to\phi^{+}\phi^{0}$
$\displaystyle=\phi^{+}\phi^{0}\to\chi^{0}\xi^{+},$
$\displaystyle\chi^{++}\xi^{-}\to\phi^{+}\phi^{0}$
$\displaystyle=\phi^{+}\phi^{0}\to\chi^{++}\xi^{-},$
$\displaystyle\chi^{+}\xi^{0}\to\chi^{+}\xi^{0}$
$\displaystyle=-32\pi^{2}\left(\lambda_{\chi\xi}+\kappa_{1}\right)+6(\beta_{\lambda_{\chi\xi}}+\beta_{\kappa_{1}})+(i\pi-1)\Big{[}2\kappa_{1}^{2}+4(\kappa_{1}+\lambda_{\chi\xi})^{2}\Big{]},$
$\displaystyle\chi^{+}\xi^{0}\to\chi^{++}\xi^{-}$
$\displaystyle=-16\pi^{2}\kappa_{1}+3\beta_{\kappa_{1}}+(i\pi-1)\Big{[}\kappa_{1}\left(5\kappa_{1}+4\lambda_{\chi\xi}\right)\Big{]},$
$\displaystyle\chi^{+}\xi^{0}\to\chi^{0}\xi^{+}$
$\displaystyle=16\pi^{2}\kappa_{1}-3\beta_{\kappa_{1}}-(i\pi-1)\Big{[}\kappa_{1}\left(5\kappa_{1}+4\lambda_{\chi\xi}\right)\Big{]},$
$\displaystyle\chi^{++}\xi^{-}\to\chi^{+}\xi^{0}$
$\displaystyle=\chi^{+}\xi^{0}\to\chi^{++}\xi^{-},$
$\displaystyle\chi^{0}\xi^{+}\to\chi^{+}\xi^{0}$
$\displaystyle=\chi^{+}\xi^{0}\to\chi^{0}\xi^{+},$
$\displaystyle\chi^{++}\xi^{-}\to\chi^{++}\xi^{-}=$
$\displaystyle-16\pi^{2}(\kappa_{1}+2\lambda_{\chi\xi})+3\left(\beta_{\kappa_{1}}+2\beta_{\lambda_{\chi\xi}}\right)$
$\displaystyle+(i\pi-1)\Big{[}5\kappa_{1}^{2}+\kappa_{3}^{2}+(\kappa_{1}+2\lambda_{\chi\xi})^{2}\Big{]},$
$\displaystyle\chi^{++}\xi^{-}\to\chi^{0}\xi^{+}$
$\displaystyle=32\pi^{2}\kappa_{1}-6\beta_{\kappa_{1}}+(i\pi-1)\Big{[}-5\kappa_{1}^{2}+\kappa_{3}^{2}-8\kappa_{1}\lambda_{\chi\xi}\Big{]},$
$\displaystyle\chi^{0}\xi^{+}\to\chi^{++}\xi^{-}$
$\displaystyle=\chi^{++}\xi^{-}\to\chi^{0}\xi^{+},$
$\displaystyle\chi^{0}\xi^{+}\to\chi^{0}\xi^{+}=$
$\displaystyle-16\pi^{2}\left(2\lambda_{\chi\xi}+\kappa_{1}\right)+3(2\beta_{\chi\xi}+\beta_{\kappa_{1}})$
$\displaystyle+(i\pi-1)\Big{[}5\kappa_{1}^{2}+\kappa_{3}^{2}+(\kappa_{1}+2\lambda_{\chi\xi})^{2}\Big{]}.$
(D.8)

###  $\bullet$ $Q=1,Y=\>3/2$ :

$\displaystyle\phi^{+}\chi^{0}\to\phi^{0}\chi^{+}=$
$\displaystyle-8\sqrt{2}\pi^{2}\kappa_{2}+\frac{3}{\sqrt{2}}\beta_{\kappa_{2}}-\frac{1}{2\sqrt{2}}(i\pi-1)\Big{[}\kappa_{2}\left(\kappa_{2}-4\lambda_{\phi\chi}\right)\Big{]},$
$\displaystyle\phi^{0}\chi^{+}\to\phi^{0}\chi^{+}=$
$\displaystyle-16\pi^{2}\lambda_{\phi\chi}+3\beta_{\lambda_{\phi\chi}}+(i\pi-1)\Big{[}\frac{\kappa_{2}^{2}}{2}+\lambda_{\phi\chi}^{2}\Big{]},$
$\displaystyle\phi^{0}\chi^{+}\to\phi^{+}\chi^{0}$
$\displaystyle=\phi^{+}\chi^{0}\to\phi^{0}\chi^{+},$
$\displaystyle\phi^{0}\chi^{+}\to\phi^{0}\chi^{+}$
$\displaystyle=\phi^{0}\chi^{+}\to\phi^{0}\chi^{+},$
$\displaystyle\phi^{+}\chi^{0}\to\phi^{+}\chi^{0}=$
$\displaystyle-16\pi^{2}\left(\lambda_{\phi\chi}-\frac{1}{2}\kappa_{2}\right)+3\left(\beta_{\lambda_{\phi\chi}}-\frac{1}{2}\beta_{\kappa_{2}}\right)$
$\displaystyle+(i\pi-1)\Big{[}-\kappa_{2}\lambda_{\phi\chi}+\frac{3}{4}\kappa_{2}^{2}+\lambda_{\phi\chi}^{2}\Big{]}.$
(D.9)

###  $\bullet$ $Q=1,Y=2$ :

$\displaystyle\chi^{+}\chi^{0}\to\chi^{+}\chi^{0}$
$\displaystyle=-32\pi^{2}\lambda_{\chi}+6\beta_{\lambda_{\chi}}+4(i\pi-1)\lambda_{\chi}^{2}.$
(D.10)

###  $\bullet$ $Q=2,Y=0$ :

$\displaystyle\chi^{++}\chi^{0*}\to\chi^{++}\chi^{0*}=$
$\displaystyle-32\pi^{2}(\lambda_{\chi}+2\tilde{\lambda}_{\chi})+6\left(\beta_{\lambda_{\chi}}+2\beta_{\tilde{\lambda}_{\chi}}\right)$
$\displaystyle+(i\pi-1)\Big{[}2\kappa_{1}^{2}+4\left(\lambda_{\chi}+2\tilde{\lambda}_{\chi}\right)^{2}\Big{]},$
$\displaystyle\chi^{++}\chi^{0*}\to\frac{1}{\sqrt{2}}(\xi^{+}\xi^{+})=$
$\displaystyle 16\sqrt{2}\pi^{2}\kappa_{1}-3\sqrt{2}\beta_{\kappa_{1}}$
$\displaystyle+\frac{1}{\sqrt{2}}(i\pi-1)\Big{[}-4\kappa_{1}\left(4\lambda_{\xi}+\lambda_{\chi}+2\tilde{\lambda}_{\chi}\right)\Big{]},$
$\displaystyle\frac{1}{\sqrt{2}}(\xi^{+}\xi^{+})\to\chi^{++}\chi^{0*}$
$\displaystyle=\chi^{++}\chi^{0*}\to\frac{1}{\sqrt{2}}(\xi^{+}\xi^{+}),$
$\displaystyle\frac{1}{\sqrt{2}}(\xi^{+}\xi^{+})\to\frac{1}{\sqrt{2}}(\xi^{+}\xi^{+})$
$\displaystyle=-128\pi^{2}\lambda_{\xi}+24\beta_{\lambda_{\xi}}+2(i\pi-1)\Big{[}\left(\kappa_{1}^{2}+32{\lambda_{\xi}}^{2}\right)\Big{]}.$
(D.11)

###  $\bullet$ $Q=2,Y=1/2$ :

$\displaystyle\phi^{+}\xi^{+}\to\phi^{+}\xi^{+}$
$\displaystyle=-32\pi^{2}\lambda_{\phi\xi}+6\beta_{\lambda_{\phi\xi}}+(i\pi-1)\Big{[}\left(\kappa_{3}^{2}+4\lambda_{\phi\xi}^{2}\right)\Big{]},$
$\displaystyle\phi^{+}\xi^{+}\to\chi^{++}\phi^{0*}$
$\displaystyle=-16\pi^{2}\kappa_{3}+3\beta_{\kappa_{3}}+(i\pi-1)\Big{[}\kappa_{3}\left(-\frac{\kappa_{2}}{2}+2\lambda_{\phi\xi}+\lambda_{\phi\chi}\right)\Big{]},$
$\displaystyle\chi^{++}\phi^{0*}\to\phi^{+}\xi^{+}$
$\displaystyle=\phi^{+}\xi^{+}\to\chi^{++}\phi^{0*},$
$\displaystyle\chi^{++}\phi^{0*}\to\chi^{++}\phi^{0*}=$
$\displaystyle-16\pi^{2}\left(\lambda_{\phi\chi}-\frac{1}{2}\kappa_{2}\right)+3\left(\beta_{\lambda_{\phi\chi}}-\frac{1}{2}\beta_{\kappa_{2}}\right)$
$\displaystyle+(i\pi-1)\Big{[}\kappa_{3}^{2}+\left(-\frac{\kappa_{2}}{2}+\lambda_{\phi\chi}\right)^{2}\Big{]}.$
(D.12)

###  $\bullet$ $Q=2,Y=3/2$ :

$\displaystyle\phi^{+}\chi^{+}\to\phi^{+}\chi^{+}$
$\displaystyle=-16\pi^{2}\lambda_{\phi\chi}+3\beta_{\lambda_{\phi\chi}}+(i\pi-1)\Big{[}\frac{1}{2}\kappa_{2}^{2}+\lambda_{\phi\chi}^{2}\Big{]},$
$\displaystyle\phi^{+}\chi^{+}\to\chi^{++}\phi^{0}$
$\displaystyle=-8\sqrt{2}\pi^{2}\kappa_{2}+\frac{3}{\sqrt{2}}\beta_{\kappa_{2}}-\frac{1}{2\sqrt{2}}(i\pi-1)\Big{[}\kappa_{2}\left(\kappa_{2}-4\lambda_{\phi\chi}\right)\Big{]},$
$\displaystyle\chi^{++}\phi^{0}\to\phi^{+}\chi^{+}$
$\displaystyle=\phi^{+}\chi^{+}\to\chi^{++}\phi^{0},$
$\displaystyle\chi^{++}\phi^{0}\to\chi^{++}\phi^{0}=$
$\displaystyle-16\pi^{2}\left(\lambda_{\phi\chi}-\frac{\kappa_{2}}{2}\right)+3\left(\beta_{\lambda_{\phi\chi}}-\frac{1}{2}\beta_{\kappa_{2}}\right)$
$\displaystyle+(i\pi-1)\Big{[}\frac{3\kappa_{2}^{2}}{4}-\kappa_{2}\lambda_{\phi\chi}+\lambda_{\phi\chi}^{2}\Big{]}.$
(D.13)

###  $\bullet$ $Q=2,Y=2$ :

$\displaystyle\chi^{++}\chi^{0}\to\chi^{++}\chi^{0}=$
$\displaystyle-32\pi^{2}\left(\lambda_{\chi}+2\tilde{\lambda}_{\chi}\right)+6\left(\beta_{\lambda_{\chi}}+2\beta_{\tilde{\lambda}_{\chi}}\right)$
$\displaystyle+(i\pi-1)\Big{[}8{\tilde{\lambda}_{\chi}}^{2}+4\left(\lambda_{\chi}+2\tilde{\lambda}_{\chi}\right)^{2}\Big{]},$
$\displaystyle\chi^{++}\chi^{0}\to\frac{1}{\sqrt{2}}(\chi^{+}\chi^{+})=$
$\displaystyle
32\sqrt{2}\pi^{2}\tilde{\lambda}_{\chi}-6\sqrt{2}\beta_{\tilde{\lambda}_{\chi}}+(i\pi-1)\Big{[}-4\sqrt{2}\tilde{\lambda}_{\chi}\left(2\lambda_{\chi}+3\tilde{\lambda}_{\chi}\right)\Big{]},$
$\displaystyle\frac{1}{\sqrt{2}}(\chi^{+}\chi^{+})\to\chi^{++}\chi^{0}$
$\displaystyle=\chi^{++}\chi^{0}\to\frac{1}{\sqrt{2}}(\chi^{+}\chi^{+}),$
$\displaystyle\frac{1}{\sqrt{2}}(\chi^{+}\chi^{+})\to\frac{1}{\sqrt{2}}(\chi^{+}\chi^{+})=$
$\displaystyle-32\pi^{2}\left(\lambda_{\chi}+\tilde{\lambda}_{\chi}\right)+6\left(\beta_{\lambda_{\chi}}+\beta_{\tilde{\lambda}_{\chi}}\right)$
$\displaystyle+4(i\pi-1)\Big{[}2{\tilde{\lambda}_{\chi}}^{2}+\left(\tilde{\lambda}_{\chi}+\lambda_{\chi}\right)^{2}\Big{]}.$
(D.14)

###  $\bullet$ $Q=2,Y=1$ :

$\displaystyle\frac{1}{\sqrt{2}}\left(\phi^{+}\phi^{+}\right)\to\chi^{++}\xi^{0}=$
$\displaystyle
16\pi^{2}\kappa_{3}-3\beta_{\kappa_{3}}+(i\pi-1)\Big{[}\kappa_{3}\left(\kappa_{1}-2\left(\lambda_{\phi}+\lambda_{\chi\xi}\right)\right)\Big{]},$
$\displaystyle\frac{1}{\sqrt{2}}\left(\phi^{+}\phi^{+}\right)\to\chi^{+}\xi^{+}=$
$\displaystyle-16\pi^{2}\kappa_{3}+3\beta_{\kappa_{3}}+(i\pi-1)\Big{[}\kappa_{3}\left(-\kappa_{1}+2\left(\lambda_{\phi}+\lambda_{\chi\xi}\right)\right)\Big{]},$
$\displaystyle\frac{1}{\sqrt{2}}\left(\phi^{+}\phi^{+}\right)\to\frac{1}{\sqrt{2}}\left(\phi^{+}\phi^{+}\right)=$
$\displaystyle-32\pi^{2}\lambda_{\phi}+6\beta_{\lambda_{\phi}}+2(i\pi-1)\Big{[}\kappa_{3}^{2}+2\lambda_{\phi}^{2}\Big{]},$
$\displaystyle\chi^{++}\xi^{0}\to\frac{1}{\sqrt{2}}\left(\phi^{+}\phi^{+}\right)$
$\displaystyle=\frac{1}{\sqrt{2}}\left(\phi^{+}\phi^{+}\right)\to\chi^{++}\xi^{0},$
$\displaystyle\chi^{+}\xi^{+}\to\frac{1}{\sqrt{2}}\left(\phi^{+}\phi^{+}\right)$
$\displaystyle=\frac{1}{\sqrt{2}}\left(\phi^{+}\phi^{+}\right)\to\chi^{+}\xi^{+},$
$\displaystyle\chi^{++}\xi^{0}\to\chi^{++}\xi^{0}$
$\displaystyle=-32\pi^{2}\lambda_{\chi\xi}+6\beta_{\lambda_{\chi\xi}}+(i\pi-1)\Big{[}\kappa_{1}^{2}+\kappa_{3}^{2}+4\lambda_{\chi\xi}^{2}\Big{]},$
$\displaystyle\chi^{++}\xi^{0}\to\chi^{+}\xi^{+}$
$\displaystyle=-16\pi^{2}\kappa_{1}+3\beta_{\kappa_{1}}+(i\pi-1)\Big{[}-\kappa_{3}^{2}+4\kappa_{1}\lambda_{\chi\xi}\Big{]},$
$\displaystyle\chi^{+}\xi^{+}\to\chi^{++}\xi^{0}$
$\displaystyle=\chi^{++}\xi^{0}\to\chi^{+}\xi^{+},$
$\displaystyle\chi^{+}\xi^{+}\to\chi^{+}\xi^{+}$
$\displaystyle=-32\pi^{2}\lambda_{\chi\xi}+6\beta_{\lambda_{\chi\xi}}+(i\pi-1)\Big{[}\Big{.}\begin{aligned}
\kappa_{1}^{2}+\kappa_{3}^{2}+4\lambda_{\chi\xi}^{2}\Big{]}.\end{aligned}$
(D.15)

###  $\bullet$ $Q=3,Y=1$ :

$\displaystyle\chi^{++}\xi^{+}\to\chi^{++}\xi^{+}=$
$\displaystyle-16\pi^{2}\left(2\lambda_{\chi\xi}+\kappa_{1}\right)+3\left(2\beta_{\lambda_{\chi\xi}}+\beta_{\kappa_{1}}\right)$
$\displaystyle+(i\pi-1)\Big{[}4\kappa_{1}\lambda_{\chi\xi}+\kappa_{1}^{2}+4\lambda_{\chi\xi}^{2}\Big{]}.$
(D.16)

###  $\bullet$ $Q=3,Y=3/2$ :

$\displaystyle\chi^{++}\phi^{+}\to\chi^{++}\phi^{+}=$
$\displaystyle-16\pi^{2}\left(\lambda_{\phi\chi}+\frac{\kappa_{2}}{2}\right)+3\left(\beta_{\lambda_{\phi\chi}}+\frac{1}{2}\beta_{\kappa_{2}}\right)$
$\displaystyle+(i\pi-1)\Big{[}\kappa_{2}\lambda_{\phi\chi}+\frac{\kappa_{2}^{2}}{4}+\lambda_{\phi\chi}^{2}\Big{]}.$
(D.17)

###  $\bullet$ $Q=3,Y=2$ :

$\displaystyle\chi^{++}\chi^{+}\to\chi^{++}\chi^{+}$
$\displaystyle=-32\pi^{2}\lambda_{\chi}+6\beta_{\lambda_{\chi}}+4(i\pi-1)\lambda_{\chi}^{2}.$
(D.18)

###  $\bullet$ $Q=4,Y=2$ :

$\displaystyle\frac{1}{\sqrt{2}}\left(\chi^{++}\chi^{++}\right)\to\frac{1}{\sqrt{2}}\left(\chi^{++}\chi^{++}\right)$
$\displaystyle=-32\pi^{2}\lambda_{\chi}+6\beta_{\lambda_{\chi}}+4(i\pi-1)\lambda_{\chi}^{2}\,.$
(D.19)

##  Appendix E Two-loop renormalization group equations

In this appendix, we list the renormalization group equations (RGEs) for the
quartic couplings of the most general GM model, obtained using the public code
PyR@TE Sartore:2020gou . For any coupling $A$, the complete $\beta$ function
up to two-loop order can be sub-divided into leading and next-to-leading order
contributions separately as follows:

$\beta_{A}\equiv\frac{dA}{d\ln\mu}=\frac{1}{16\pi^{2}}\beta_{A}^{(0)}+\frac{1}{(16\pi^{2})^{2}}\beta_{A}^{(1)}\,.$
(E.1)

The two-loop RGEs for the Yukawa couplings are the same as in the SM
Luo:2002ey  and type-I 2HDM Chowdhury:2015yja , because the contributions of
Higgs triplets to the Yukawa Lagrangian are not considered in this analysis
Chiang:2012cn . The RGEs of the gauge couplings up to two-loop order read as,

$\displaystyle\beta_{g_{1}}^{(0)}$ $\displaystyle=\frac{47}{6}g_{1}^{3},\quad$
$\displaystyle\beta_{g_{1}}^{(1)}$
$\displaystyle=\frac{1}{18}g_{1}^{3}\Big{(}415g_{1}^{2}+513g_{2}^{2}+264g_{3}^{2}-15y_{b}^{2}-51y_{t}^{2}-45y_{\tau}^{2}\Big{)},$
$\displaystyle\beta_{g_{2}}^{(0)}$
$\displaystyle=-\frac{13}{6}g_{2}^{3},\quad$
$\displaystyle\beta_{g_{2}}^{(1)}$
$\displaystyle=\frac{1}{6}g_{2}^{3}\Big{(}57g_{1}^{2}+203g_{2}^{2}+72g_{3}^{2}-9y_{b}^{2}-9y_{t}^{2}-3y_{\tau}^{2}\Big{)},$
$\displaystyle\beta_{g_{3}}^{(0)}$ $\displaystyle=-7g_{3}^{3},\quad$
$\displaystyle\beta_{g_{2}}^{(1)}$
$\displaystyle=\frac{1}{6}g_{3}^{3}\Big{(}11g_{1}^{2}+27g_{2}^{2}-156g_{3}^{2}-12y_{b}^{2}-12y_{t}^{2}\Big{)},$

The running of the quartic couplings are given by the following expressions:

$\displaystyle\beta_{\lambda_{\phi}}^{(0)}$
$\displaystyle=+24\lambda_{\phi}^{2}+\frac{3}{8}\Big{(}g_{1}^{4}+2g_{1}^{2}g_{2}^{2}+3g_{2}^{4}\Big{)}-6y_{b}^{4}-6y_{t}^{4}-2y_{\tau}^{4}$
$\displaystyle\quad+\lambda_{\phi}\Big{(}12y_{b}^{2}+12y_{t}^{2}+4y_{\tau}^{2}-3g_{1}^{2}-9g_{2}^{2}\Big{)}+6\lambda_{\phi\xi}^{2}+3\lambda_{\phi\chi}^{2}+\frac{1}{2}\kappa_{2}^{2}+2\kappa_{3}^{2}\,,$
$\displaystyle\beta_{\lambda_{\phi}}^{(1)}$
$\displaystyle=-312\lambda_{\phi}^{3}+\frac{1}{48}g_{1}^{4}\Big{(}60y_{b}^{2}-643g_{2}^{2}-228y_{t}^{2}-300y_{\tau}^{2}\Big{)}+\frac{1}{48}g_{1}^{2}\Big{(}24g_{2}^{2}\Big{(}9y_{b}^{2}+21y_{t}^{2}+11y_{\tau}^{2}\Big{)}$
$\displaystyle\quad+64\Big{(}y_{b}^{4}-2y_{t}^{4}-3y_{\tau}^{4}\Big{)}-373g_{2}^{4}\Big{)}-\frac{3}{4}g_{2}^{4}\Big{(}3y_{b}^{2}+3y_{t}^{2}+y_{\tau}^{2}\Big{)}+\frac{25}{6}g_{1}^{2}y_{b}^{2}\lambda_{\phi}+\frac{45}{2}g_{2}^{2}y_{b}^{2}\lambda_{\phi}$
$\displaystyle\quad+80g_{3}^{2}y_{b}^{2}\lambda_{\phi}-32g_{3}^{2}y_{b}^{4}-42y_{b}^{2}\lambda_{\phi}y_{t}^{2}-6y_{b}^{2}y_{t}^{4}-6y_{b}^{4}y_{t}^{2}-144y_{b}^{2}\lambda_{\phi}^{2}-3y_{b}^{4}\lambda_{\phi}+30y_{b}^{6}$
$\displaystyle\quad+2\kappa_{3}^{2}\Big{(}g_{1}^{2}+10g_{2}^{2}-14\lambda_{\phi}-20\lambda_{\phi\xi}-10\lambda_{\phi\chi}\Big{)}+\kappa_{2}^{2}\Big{(}4g_{1}^{2}+5g_{2}^{2}-7\lambda_{\phi}-10\lambda_{\phi\chi}\Big{)}$
$\displaystyle\quad+2\kappa_{2}\Big{(}5g_{1}^{2}g_{2}^{2}-3\kappa_{3}^{2}\Big{)}+6\lambda_{\phi\chi}^{2}\Big{(}4g_{1}^{2}+8g_{2}^{2}-5\lambda_{\phi}\Big{)}+\frac{761}{24}g_{1}^{4}\lambda_{\phi}+36g_{1}^{2}\lambda_{\phi}^{2}+\frac{39}{4}g_{2}^{2}g_{1}^{2}\lambda_{\phi}$
$\displaystyle\quad+108g_{2}^{2}\lambda_{\phi}^{2}+\frac{59}{8}g_{2}^{4}\lambda_{\phi}+96g_{2}^{2}\lambda_{\phi\xi}^{2}+30g_{2}^{4}\lambda_{\phi\xi}+15\Big{(}g_{1}^{4}+2g_{2}^{4}\Big{)}\lambda_{\phi\chi}+\frac{85}{6}g_{1}^{2}\lambda_{\phi}y_{t}^{2}$
$\displaystyle\quad+\frac{45}{2}g_{2}^{2}\lambda_{\phi}y_{t}^{2}+\frac{25}{2}g_{1}^{2}\lambda_{\phi}y_{\tau}^{2}+\frac{15}{2}g_{2}^{2}\lambda_{\phi}y_{\tau}^{2}-\frac{463g_{1}^{6}}{48}+\frac{221g_{2}^{6}}{16}+80g_{3}^{2}\lambda_{\phi}y_{t}^{2}-32g_{3}^{2}y_{t}^{4}$
$\displaystyle\quad-60\lambda_{\phi}\lambda_{\phi\xi}^{2}-48\lambda_{\phi\xi}^{3}-12\lambda_{\phi\chi}^{3}-144\lambda_{\phi}^{2}y_{t}^{2}-3\lambda_{\phi}y_{t}^{4}+30y_{t}^{6}-48\lambda_{\phi}^{2}y_{\tau}^{2}-\lambda_{\phi}y_{\tau}^{4}+10y_{\tau}^{6},$
$\displaystyle\beta_{\lambda_{\xi}}^{(0)}$
$\displaystyle=+88\lambda_{\xi}^{2}+3g_{2}^{4}-24g_{2}^{2}\lambda_{\xi}+2\kappa_{1}\lambda_{\chi\xi}+\kappa_{1}^{2}+3\lambda_{\chi\xi}^{2}+2\lambda_{\phi\xi}^{2},$
$\displaystyle\beta_{\lambda_{\xi}}^{(1)}$
$\displaystyle=-4416\lambda_{\xi}^{3}+2\lambda_{\phi\xi}\Big{(}2\lambda_{\phi\xi}\Big{(}-3y_{b}^{2}-3y_{t}^{2}+g_{1}^{2}+3g_{2}^{2}-4\lambda_{\phi\xi}-y_{\tau}^{2}\Big{)}+5g_{2}^{4}-4\kappa_{3}^{2}\Big{)}$
$\displaystyle\quad+4\kappa_{1}^{2}\Big{(}2g_{1}^{2}+g_{2}^{2}-12\lambda_{\xi}-8\lambda_{\chi\xi}\Big{)}+4\kappa_{1}\Big{(}4\lambda_{\chi\xi}\Big{(}g_{1}^{2}+2g_{2}^{2}-5\lambda_{\xi}\Big{)}+5g_{2}^{4}-6\lambda_{\chi\xi}^{2}\Big{)}$
$\displaystyle\quad+\lambda_{\xi}\Big{(}\frac{470}{3}g_{2}^{4}+8\kappa_{3}^{2}-80\lambda_{\phi\xi}^{2}\Big{)}+4\Big{(}10g_{2}^{4}-\kappa_{3}^{2}\Big{)}\lambda_{\chi\xi}+24\lambda_{\chi\xi}^{2}\Big{(}g_{1}^{2}+2g_{2}^{2}-5\lambda_{\xi}\Big{)}$
$\displaystyle\quad+1024g_{2}^{2}\lambda_{\xi}^{2}-\frac{83}{3}g_{2}^{6}-12\kappa_{1}^{3}-24\lambda_{\chi\xi}^{3},$
$\displaystyle\beta_{\lambda_{\chi}}^{(0)}$
$\displaystyle=+28\lambda_{\chi}^{2}+12g_{1}^{2}\Big{(}g_{2}^{2}-\lambda_{\chi}\Big{)}-24g_{2}^{2}\lambda_{\chi}+6g_{1}^{4}+9g_{2}^{4}+4\kappa_{1}\lambda_{\chi\xi}+\kappa_{1}^{2}+\frac{1}{2}\kappa_{2}^{2}$
$\displaystyle\quad+6\lambda_{\chi\xi}^{2}+16\tilde{\lambda}_{\chi}^{2}+16\lambda_{\chi}\tilde{\lambda}_{\chi}+2\lambda_{\phi\chi}^{2},$
$\displaystyle\beta_{\lambda_{\chi}}^{(1)}$
$\displaystyle=-384\lambda_{\chi}^{3}+2g_{2}^{2}\Big{(}16\Big{(}8\lambda_{\chi}\tilde{\lambda}_{\chi}+5\tilde{\lambda}_{\chi}^{2}+11\lambda_{\chi}^{2}+3\lambda_{\chi\xi}^{2}\Big{)}+32\kappa_{1}\lambda_{\chi\xi}+5\kappa_{1}^{2}+6\lambda_{\phi\chi}^{2}\Big{)}$
$\displaystyle\quad+g_{2}^{4}\Big{(}128\tilde{\lambda}_{\chi}+30\kappa_{1}+\frac{638}{3}\lambda_{\chi}+80\lambda_{\chi\xi}+20\lambda_{\phi\chi}\Big{)}+\frac{1}{3}g_{1}^{2}\Big{(}6g_{2}^{2}\Big{(}-80\tilde{\lambda}_{\chi}+5\kappa_{2}+68\lambda_{\chi}\Big{)}$
$\displaystyle\quad+3\Big{(}4\Big{(}32\tilde{\lambda}_{\chi}^{2}+32\lambda_{\chi}\tilde{\lambda}_{\chi}+44\lambda_{\chi}^{2}+\lambda_{\phi\chi}^{2}\Big{)}+\kappa_{2}^{2}\Big{)}-958g_{2}^{4}\Big{)}+g_{1}^{4}\Big{(}80\tilde{\lambda}_{\chi}-\frac{1138}{3}g_{2}^{2}$
$\displaystyle\quad+\frac{811}{3}\lambda_{\chi}+10\lambda_{\phi\chi}\Big{)}-\kappa_{2}^{2}\Big{(}3y_{b}^{2}+3y_{t}^{2}+y_{\tau}^{2}\Big{)}-\frac{598}{3}g_{1}^{6}-29g_{2}^{6}-2\Big{(}2\Big{(}80\tilde{\lambda}_{\chi}^{3}+156\lambda_{\chi}\tilde{\lambda}_{\chi}^{2}$
$\displaystyle\quad+88\lambda_{\chi}^{2}\tilde{\lambda}_{\chi}+3y_{b}^{2}\lambda_{\phi\chi}^{2}+\kappa_{3}^{2}\Big{(}-\lambda_{\chi}+2\lambda_{\chi\xi}+2\lambda_{\phi\chi}\Big{)}+15\lambda_{\chi}\lambda_{\chi\xi}^{2}+12\lambda_{\chi\xi}^{3}+5\lambda_{\chi}\lambda_{\phi\chi}^{2}$
$\displaystyle\quad+2\lambda_{\phi\chi}^{3}+3\lambda_{\phi\chi}^{2}y_{t}^{2}+\lambda_{\phi\chi}^{2}y_{\tau}^{2}\Big{)}+8\kappa_{1}^{2}\Big{(}\tilde{\lambda}_{\chi}+\lambda_{\chi}+3\lambda_{\chi\xi}\Big{)}+\kappa_{1}\Big{(}\kappa_{3}^{2}+4\lambda_{\chi\xi}(5\lambda_{\chi}+6\lambda_{\chi\xi}\Big{)}\Big{)}$
$\displaystyle\quad+\kappa_{2}^{2}\Big{(}3\lambda_{\chi}+4\lambda_{\phi\chi}\Big{)}+7\kappa_{1}^{3}+\kappa_{2}\kappa_{3}^{2}\Big{)},$
$\displaystyle\beta_{\tilde{\lambda}_{\chi}}^{(0)}$
$\displaystyle=+12\tilde{\lambda}_{\chi}^{2}+24\tilde{\lambda}_{\chi}\Big{(}\lambda_{\chi}-g_{2}^{2}\Big{)}-12g_{1}^{2}\Big{(}\tilde{\lambda}_{\chi}+g_{2}^{2}\Big{)}+3g_{2}^{4}+\kappa_{1}^{2}-\frac{1}{2}\kappa_{2}^{2},$
$\displaystyle\beta_{\tilde{\lambda}_{\chi}}^{(1)}$
$\displaystyle=-16\tilde{\lambda}_{\chi}^{3}+\frac{1}{3}\Big{(}g_{2}^{4}\Big{(}86\tilde{\lambda}_{\chi}+30\kappa_{1}+72\lambda_{\chi}\Big{)}-6g_{2}^{2}\Big{(}\kappa_{1}^{2}-48\tilde{\lambda}_{\chi}\Big{(}\tilde{\lambda}_{\chi}+2\lambda_{\chi}\Big{)}\Big{)}$
$\displaystyle\quad+g_{1}^{2}\Big{(}-6g_{2}^{2}\Big{(}-84\tilde{\lambda}_{\chi}+5\kappa_{2}+48\lambda_{\chi}\Big{)}-3\Big{(}48\tilde{\lambda}_{\chi}\Big{(}\tilde{\lambda}_{\chi}-2\lambda_{\chi}\Big{)}+\kappa_{2}^{2}\Big{)}+598g_{2}^{4}\Big{)}$
$\displaystyle\quad+g_{1}^{4}\Big{(}331\tilde{\lambda}_{\chi}+778g_{2}^{2}\Big{)}+6\Big{(}-4\kappa_{1}^{2}\Big{(}\tilde{\lambda}_{\chi}+\lambda_{\chi}+2\lambda_{\chi\xi}\Big{)}+\kappa_{1}\Big{(}\kappa_{3}^{2}-20\lambda_{\chi\xi}\tilde{\lambda}_{\chi}\Big{)}$
$\displaystyle\quad-2\tilde{\lambda}_{\chi}\Big{(}4\Big{(}28\lambda_{\chi}\tilde{\lambda}_{\chi}+28\lambda_{\chi}^{2}\Big{)}-\kappa_{3}^{2}+15\lambda_{\chi\xi}^{2}+5\lambda_{\phi\chi}^{2}\Big{)}+\kappa_{2}^{2}\Big{(}3\tilde{\lambda}_{\chi}+2\Big{(}\lambda_{\chi}+\lambda_{\phi\chi}\Big{)}\Big{)}$
$\displaystyle\quad-5\kappa_{1}^{3}+\kappa_{2}\kappa_{3}^{2}\Big{)}+3\kappa_{2}^{2}\Big{(}3y_{b}^{2}+3y_{t}^{2}+y_{\tau}^{2}\Big{)}-245g_{2}^{6}\Big{)},$
$\displaystyle\beta_{\kappa_{1}}^{(0)}$
$\displaystyle=+10\kappa_{1}^{2}-24g_{2}^{2}\kappa_{1}+6g_{2}^{4}+2\kappa_{1}\Big{(}4\tilde{\lambda}_{\chi}-3g_{1}^{2}+8\lambda_{\xi}+2\lambda_{\chi}+8\lambda_{\chi\xi}\Big{)}-2\kappa_{3}^{2},$
$\displaystyle\beta_{\kappa_{1}}^{(1)}$
$\displaystyle=-22\kappa_{1}^{3}+2\kappa_{1}^{2}\Big{(}-60\tilde{\lambda}_{\chi}+5g_{1}^{2}+80g_{2}^{2}-152\lambda_{\xi}-46\lambda_{\chi}-66\lambda_{\chi\xi}\Big{)}$
$\displaystyle\quad+2\kappa_{3}^{2}\Big{(}4\tilde{\lambda}_{\chi}+6y_{b}^{2}+6y_{t}^{2}+g_{1}^{2}+2\kappa_{2}+8\lambda_{\xi}+2\lambda_{\chi}+4\lambda_{\chi\xi}+8\lambda_{\phi\xi}+4\lambda_{\phi\chi}+2y_{\tau}^{2}\Big{)}$
$\displaystyle\quad+\frac{1}{6}\kappa_{1}\Big{(}12\Big{(}-8\tilde{\lambda}_{\chi}\Big{(}g_{2}^{2}+5\lambda_{\chi}+12\lambda_{\chi\xi}\Big{)}-12\tilde{\lambda}_{\chi}^{2}+8\lambda_{\chi\xi}\Big{(}13g_{2}^{2}-36\lambda_{\xi}-12\lambda_{\chi}\Big{)}$
$\displaystyle\quad-4\Big{(}4\lambda_{\xi}\Big{(}g_{2}^{2}+13\lambda_{\xi}\Big{)}+g_{2}^{2}\lambda_{\chi}+4\lambda_{\chi}^{2}\Big{)}-57\lambda_{\chi\xi}^{2}\Big{)}+96g_{1}^{2}\Big{(}4\tilde{\lambda}_{\chi}+2\lambda_{\chi}+\lambda_{\chi\xi}\Big{)}$
$\displaystyle\quad+307g_{1}^{4}+48g_{2}^{2}g_{1}^{2}+136g_{2}^{4}+12\kappa_{2}^{2}+84\kappa_{3}^{2}-12\Big{(}16\lambda_{\phi\xi}\lambda_{\phi\chi}+4\lambda_{\phi\xi}^{2}+\lambda_{\phi\chi}^{2}\Big{)}\Big{)}$
$\displaystyle\quad+\frac{2}{3}g_{2}^{4}\Big{(}60\tilde{\lambda}_{\chi}-45g_{1}^{2}-245g_{2}^{2}+120\lambda_{\xi}+30\lambda_{\chi}+12\lambda_{\chi\xi}\Big{)},$
$\displaystyle\beta_{\kappa_{2}}^{(0)}$
$\displaystyle=+\frac{3}{2}g_{1}^{2}\Big{(}8g_{2}^{2}-5\kappa_{2}\Big{)}-\frac{33}{2}g_{2}^{2}\kappa_{2}+4\kappa_{3}^{2}$
$\displaystyle\quad+2\kappa_{2}\Big{(}-4\tilde{\lambda}_{\chi}+2\lambda_{\chi}+2\lambda_{\phi}+4\lambda_{\phi\chi}+3y_{b}^{2}+3y_{t}^{2}+y_{\tau}^{2}\Big{)},$
$\displaystyle\beta_{\kappa_{2}}^{(1)}$
$\displaystyle=+\frac{1}{2}\kappa_{2}^{3}+6g_{1}^{2}\kappa_{2}^{2}+\frac{1}{3}g_{1}^{2}g_{2}^{2}\Big{(}120\lambda_{\chi}+120\lambda_{\phi}-240\tilde{\lambda}_{\chi}+24\lambda_{\phi\chi}+108y_{b}^{2}+252y_{t}^{2}$
$\displaystyle\quad+132y_{\tau}^{2}-643g_{1}^{2}-463g_{2}^{2}\Big{)}+\kappa_{3}^{2}\Big{(}16\tilde{\lambda}_{\chi}-12y_{b}^{2}+8\kappa_{1}-8\lambda_{\chi}-16\lambda_{\chi\xi}-24\lambda_{\phi}$
$\displaystyle\quad-16\lambda_{\phi\xi}-16\lambda_{\phi\chi}-12y_{t}^{2}-4y_{\tau}^{2}+g_{1}^{2}+47g_{2}^{2}\Big{)}+\frac{1}{48}\kappa_{2}\Big{(}48\Big{(}2\lambda_{\phi\chi}\Big{(}16\tilde{\lambda}_{\chi}-48\lambda_{\chi}$
$\displaystyle\quad-12y_{b}^{2}-12y_{t}^{2}-4y_{\tau}^{2}+5g_{1}^{2}+23g_{2}^{2}\Big{)}+8\Big{(}4g_{1}^{2}\Big{(}\lambda_{\chi}-2\tilde{\lambda}_{\chi}\Big{)}+5g_{2}^{2}\Big{(}\lambda_{\chi}-2\tilde{\lambda}_{\chi}\Big{)}$
$\displaystyle\quad+9\tilde{\lambda}_{\chi}^{2}+6\lambda_{\chi}\tilde{\lambda}_{\chi}-4\lambda_{\chi}^{2}\Big{)}+8\lambda_{\phi}\Big{(}-3y_{b}^{2}-3y_{t}^{2}+g_{1}^{2}-10\lambda_{\phi\chi}-y_{\tau}^{2}\Big{)}-28\lambda_{\phi}^{2}$
$\displaystyle\quad-29\lambda_{\phi\chi}^{2}\Big{)}+10g_{1}^{2}\Big{(}10y_{b}^{2}+489g_{2}^{2}+34y_{t}^{2}+30y_{\tau}^{2}\Big{)}+180g_{2}^{2}\Big{(}3y_{b}^{2}+3y_{t}^{2}+y_{\tau}^{2}\Big{)}$
$\displaystyle\quad+24\Big{(}80g_{3}^{2}\Big{(}y_{b}^{2}+y_{t}^{2}\Big{)}-27\Big{(}y_{b}^{2}-y_{t}^{2}\Big{)}^{2}-9y_{\tau}^{4}\Big{)}+96\Big{(}-2\kappa_{1}\Big{(}\lambda_{\chi\xi}+4\lambda_{\phi\xi}\Big{)}$
$\displaystyle\quad+3\kappa_{1}^{2}-3\Big{(}\lambda_{\chi\xi}^{2}+8\lambda_{\chi\xi}\lambda_{\phi\xi}+\lambda_{\phi\xi}^{2}\Big{)}\Big{)}+3121g_{1}^{4}-911g_{2}^{4}+192\kappa_{3}^{2}\Big{)},$
$\displaystyle\beta_{\kappa_{3}}^{(0)}$
$\displaystyle=-\frac{1}{2}\kappa_{3}\Big{(}9g_{1}^{2}+33g_{2}^{2}\Big{)}+2\kappa_{3}\Big{(}3y_{b}^{2}-\kappa_{1}+\kappa_{2}+2\lambda_{\chi\xi}+2\lambda_{\phi}+4\lambda_{\phi\xi}+2\lambda_{\phi\chi}+3y_{t}^{2}+y_{\tau}^{2}\Big{)},$
$\displaystyle\beta_{\kappa_{3}}^{(1)}$
$\displaystyle=+3\kappa_{3}^{2}+\frac{1}{48}\kappa_{3}\Big{(}1797g_{1}^{4}-911g_{2}^{4}-24\Big{(}\kappa_{2}\Big{(}-16\tilde{\lambda}_{\chi}+4\Big{(}3y_{b}^{2}+3y_{t}^{2}+y_{\tau}^{2}\Big{)}-17g_{1}^{2}-47g_{2}^{2}$
$\displaystyle\quad+8\lambda_{\chi}+16\lambda_{\chi\xi}+24\lambda_{\phi}+16\lambda_{\phi\xi}+16\lambda_{\phi\chi}\Big{)}+2\Big{(}-12\tilde{\lambda}_{\chi}^{2}+8\tilde{\lambda}_{\chi}\Big{(}-\lambda_{\chi}+2\lambda_{\chi\xi}+2\lambda_{\phi\chi}\Big{)}$
$\displaystyle\quad+\lambda_{\phi\xi}\Big{(}8\Big{(}3y_{b}^{2}+3y_{t}^{2}+y_{\tau}^{2}\Big{)}-2g_{1}^{2}-46g_{2}^{2}+160\lambda_{\xi}+56\lambda_{\chi\xi}+24\lambda_{\phi\chi}\Big{)}+4\lambda_{\phi}\Big{(}6y_{b}^{2}+6y_{t}^{2}$
$\displaystyle\quad+g_{1}^{2}+20\lambda_{\phi\xi}+10\lambda_{\phi\chi}+2y_{\tau}^{2})+12y_{b}^{2}\lambda_{\phi\chi}-4g_{1}^{2}\lambda_{\chi\xi}-40g_{2}^{2}\lambda_{\chi\xi}-17g_{1}^{2}\lambda_{\phi\chi}-23g_{2}^{2}\lambda_{\phi\chi}$
$\displaystyle\quad+80\lambda_{\xi}\lambda_{\chi\xi}+32\lambda_{\chi}\lambda_{\chi\xi}-8\lambda_{\chi}^{2}-\lambda_{\chi\xi}^{2}+28\lambda_{\phi}^{2}+34\lambda_{\phi\xi}^{2}+32\lambda_{\chi}\lambda_{\phi\chi}+40\lambda_{\chi\xi}\lambda_{\phi\chi}+10\lambda_{\phi\chi}^{2}$
$\displaystyle\quad+12\lambda_{\phi\chi}y_{t}^{2}+4\lambda_{\phi\chi}y_{\tau}^{2}\Big{)}+4\kappa_{1}\Big{(}-4\tilde{\lambda}_{\chi}+g_{1}^{2}+10g_{2}^{2}-2\kappa_{2}+2\lambda_{\chi}-7\lambda_{\chi\xi}-4\lambda_{\phi\xi}\Big{)}-28\kappa_{1}^{2}$
$\displaystyle\quad-2\kappa_{2}^{2}\Big{)}+10g_{1}^{2}\Big{(}10y_{b}^{2}+45g_{2}^{2}+34y_{t}^{2}+30y_{\tau}^{2}\Big{)}+180g_{2}^{2}\Big{(}3y_{b}^{2}+3y_{t}^{2}+y_{\tau}^{2}\Big{)}$
$\displaystyle\quad+24\Big{(}80g_{3}^{2}\Big{(}y_{b}^{2}+y_{t}^{2}\Big{)}-3\Big{(}y_{b}^{2}-y_{t}^{2}\Big{)}^{2}-y_{\tau}^{4}\Big{)}+3840\lambda_{\xi}^{2}\Big{)},$
$\displaystyle\beta_{\lambda_{\phi\xi}}^{(0)}$
$\displaystyle=+8\lambda_{\phi\xi}^{2}-\frac{33}{2}g_{2}^{2}\lambda_{\phi\xi}+\frac{1}{2}\lambda_{\phi\xi}\Big{(}12y_{b}^{2}+80\lambda_{\xi}+24\lambda_{\phi}+12y_{t}^{2}+4y_{\tau}^{2}-3g_{1}^{2}\Big{)}$
$\displaystyle\quad+3g_{2}^{4}+2\lambda_{\phi\chi}\Big{(}\kappa_{1}+3\lambda_{\chi\xi}\Big{)}+4\kappa_{3}^{2},$
$\displaystyle\beta_{\lambda_{\phi\xi}}^{(1)}$
$\displaystyle=-46\lambda_{\phi\xi}^{3}+\frac{1609}{48}g_{2}^{4}\lambda_{\phi\xi}+640g_{2}^{2}\lambda_{\xi}\lambda_{\phi\xi}+100g_{2}^{4}\lambda_{\xi}+72g_{2}^{2}\lambda_{\phi}\lambda_{\phi\xi}+24g_{1}^{2}\lambda_{\phi}\lambda_{\phi\xi}+30g_{2}^{4}\lambda_{\phi}$
$\displaystyle\quad+22g_{2}^{2}\lambda_{\phi\xi}^{2}+\frac{15}{8}g_{1}^{2}g_{2}^{2}\lambda_{\phi\xi}+2g_{1}^{2}\lambda_{\phi\xi}^{2}+40g_{2}^{4}\lambda_{\phi\chi}+\frac{45}{4}g_{2}^{2}\lambda_{\phi\xi}y_{t}^{2}+\frac{85}{12}g_{1}^{2}\lambda_{\phi\xi}y_{t}^{2}+\frac{15}{4}g_{2}^{2}\lambda_{\phi\xi}y_{\tau}^{2}$
$\displaystyle\quad+\frac{25}{4}g_{1}^{2}\lambda_{\phi\xi}y_{\tau}^{2}+40g_{3}^{2}\lambda_{\phi\xi}y_{t}^{2}-8\kappa_{1}^{2}(\lambda_{\phi\xi}+2\lambda_{\phi\chi})-\frac{3}{2}\kappa_{2}^{2}\lambda_{\phi\xi}-480\lambda_{\xi}\lambda_{\phi\xi}^{2}-800\lambda_{\xi}^{2}\lambda_{\phi\xi}$
$\displaystyle\quad-144\lambda_{\phi}\lambda_{\phi\xi}^{2}-60\lambda_{\phi}^{2}\lambda_{\phi\xi}-12\lambda_{\chi\xi}^{2}(\lambda_{\phi\xi}+2\lambda_{\phi\chi})-3\lambda_{\phi\xi}\lambda_{\phi\chi}^{2}-72\lambda_{\phi}\lambda_{\phi\xi}y_{t}^{2}-24\lambda_{\phi\xi}^{2}y_{t}^{2}$
$\displaystyle\quad-\frac{27}{2}\lambda_{\phi\xi}y_{t}^{4}-24\lambda_{\phi}\lambda_{\phi\xi}y_{\tau}^{2}-8\lambda_{\phi\xi}^{2}y_{\tau}^{2}-\frac{9}{2}\lambda_{\phi\xi}y_{\tau}^{4}+\kappa_{3}^{2}\Big{(}-4\Big{(}3y_{b}^{2}+3y_{t}^{2}+y_{\tau}^{2}\Big{)}+17g_{1}^{2}$
$\displaystyle\quad+23g_{2}^{2}-4\kappa_{2}-80\lambda_{\xi}-28\lambda_{\chi\xi}-40\lambda_{\phi}-34\lambda_{\phi\xi}-12\lambda_{\phi\chi}\Big{)}+\frac{1}{48}\Big{(}689g_{1}^{4}\lambda_{\phi\xi}$
$\displaystyle\quad-4g_{2}^{4}\Big{(}12\Big{(}3y_{b}^{2}+3y_{t}^{2}+y_{\tau}^{2}\Big{)}+45g_{1}^{2}-217g_{2}^{2}\Big{)}\Big{)}+\frac{45}{4}g_{2}^{2}y_{b}^{2}\lambda_{\phi\xi}+\frac{25}{12}g_{1}^{2}y_{b}^{2}\lambda_{\phi\xi}-\frac{27}{2}y_{b}^{4}\lambda_{\phi\xi}$
$\displaystyle\quad+40g_{3}^{2}y_{b}^{2}\lambda_{\phi\xi}-21y_{b}^{2}\lambda_{\phi\xi}y_{t}^{2}-72y_{b}^{2}\lambda_{\phi}\lambda_{\phi\xi}-24y_{b}^{2}\lambda_{\phi\xi}^{2}+3\lambda_{\chi\xi}\Big{(}4\lambda_{\phi\chi}\Big{(}4g_{1}^{2}+8g_{2}^{2}-4\lambda_{\phi\xi}$
$\displaystyle\quad-\lambda_{\phi\chi})+5\Big{(}g_{1}^{4}+2g_{2}^{4}\Big{)}-2\kappa_{2}^{2}\Big{)}+\kappa_{1}\Big{(}4\lambda_{\phi\chi}\Big{(}4g_{1}^{2}+8g_{2}^{2}-4\lambda_{\phi\xi}-\lambda_{\phi\chi}\Big{)}+5\Big{(}g_{1}^{4}+2g_{2}^{4}\Big{)}$
$\displaystyle\quad-2\kappa_{2}^{2}+4\kappa_{3}^{2}-8\lambda_{\chi\xi}\Big{(}\lambda_{\phi\xi}+2\lambda_{\phi\chi}\Big{)}\Big{)},$
$\displaystyle\beta_{\lambda_{\phi\chi}}^{(0)}$
$\displaystyle=+4\lambda_{\phi\chi}^{2}+2\lambda_{\phi\chi}\Big{(}4\tilde{\lambda}_{\chi}+3y_{b}^{2}+8\lambda_{\chi}+6\lambda_{\phi}+3y_{t}^{2}+y_{\tau}^{2}\Big{)}+4\lambda_{\phi\xi}\Big{(}\kappa_{1}+3\lambda_{\chi\xi}\Big{)}$
$\displaystyle\quad+2\kappa_{2}^{2}+4\kappa_{3}^{2}-\frac{15}{2}g_{1}^{2}\lambda_{\phi\chi}-\frac{33}{2}g_{2}^{2}\lambda_{\phi\chi}+3g_{1}^{4}+6g_{2}^{4},$
$\displaystyle\beta_{\lambda_{\phi\chi}}^{(1)}$
$\displaystyle=-13\lambda_{\phi\chi}^{3}-\frac{1061}{12}g_{1}^{6}+\frac{217}{6}g_{2}^{6}+g_{1}^{4}\Big{(}-\frac{165}{4}g_{2}^{2}+5y_{b}^{2}-19y_{t}^{2}-25y_{\tau}^{2}+30\lambda_{\phi}+40\lambda_{\chi}$
$\displaystyle\quad+20\tilde{\lambda}_{\chi}+\frac{6121}{48}\lambda_{\phi\chi}\Big{)}+g_{2}^{4}\Big{(}-6y_{b}^{2}-6y_{t}^{2}-2y_{\tau}^{2}+10\kappa_{1}+60\lambda_{\phi}+80\lambda_{\phi\xi}+\frac{3529}{48}\lambda_{\phi\chi}$
$\displaystyle\quad+80\lambda_{\chi}+30\lambda_{\chi\xi}+40\tilde{\lambda}_{\chi}\Big{)}+\frac{1}{24}g_{1}^{2}\Big{(}-900g_{2}^{4}+\Big{(}96\kappa_{2}+525\lambda_{\phi\chi}\Big{)}g_{2}^{2}+60\kappa_{2}^{2}+24\kappa_{3}^{2}$
$\displaystyle\quad+2\lambda_{\phi\chi}\Big{(}25y_{b}^{2}+85y_{t}^{2}+75y_{\tau}^{2}+288\lambda_{\phi}+60\lambda_{\phi\chi}+768\Big{(}2\lambda_{\chi}+\tilde{\lambda}_{\chi}\Big{)}\Big{)}\Big{)}+\frac{1}{4}g_{2}^{2}\Big{(}46\kappa_{2}^{2}$
$\displaystyle\quad+92\kappa_{3}^{2}+256\lambda_{\phi\xi}\Big{(}\kappa_{1}+3\lambda_{\chi\xi}\Big{)}+\lambda_{\phi\chi}\Big{(}45y_{b}^{2}+45y_{t}^{2}+15y_{\tau}^{2}+288\lambda_{\phi}+44\lambda_{\phi\chi}$
$\displaystyle\quad+512\Big{(}2\lambda_{\chi}+\tilde{\lambda}_{\chi}\Big{)}\Big{)}\Big{)}-6y_{b}^{2}\kappa_{2}^{2}-6y_{t}^{2}\kappa_{2}^{2}-2y_{\tau}^{2}\kappa_{2}^{2}-12y_{b}^{2}\kappa_{3}^{2}-12y_{t}^{2}\kappa_{3}^{2}-4y_{\tau}^{2}\kappa_{3}^{2}$
$\displaystyle\quad-8\kappa_{2}\kappa_{3}^{2}-16\kappa_{1}\lambda_{\phi\xi}^{2}-12y_{b}^{2}\lambda_{\phi\chi}^{2}-12y_{t}^{2}\lambda_{\phi\chi}^{2}-4y_{\tau}^{2}\lambda_{\phi\chi}^{2}-72\lambda_{\phi}\lambda_{\phi\chi}^{2}-80\lambda_{\phi\chi}\lambda_{\chi}^{2}$
$\displaystyle\quad-48\lambda_{\phi\xi}\lambda_{\chi\xi}^{2}-6\lambda_{\phi\chi}\lambda_{\chi\xi}^{2}-120\lambda_{\phi\chi}\tilde{\lambda}_{\chi}^{2}-20\kappa_{2}^{2}\lambda_{\phi}-40\kappa_{3}^{2}\lambda_{\phi}-32\kappa_{1}^{2}\lambda_{\phi\xi}-24\kappa_{3}^{2}\lambda_{\phi\xi}$
$\displaystyle\quad-\frac{27}{2}y_{b}^{4}\lambda_{\phi\chi}-\frac{27}{2}y_{t}^{4}\lambda_{\phi\chi}-\frac{9}{2}y_{\tau}^{4}\lambda_{\phi\chi}+40g_{3}^{2}y_{b}^{2}\lambda_{\phi\chi}+40g_{3}^{2}y_{t}^{2}\lambda_{\phi\chi}-21y_{b}^{2}y_{t}^{2}\lambda_{\phi\chi}-4\kappa_{1}^{2}\lambda_{\phi\chi}$
$\displaystyle\quad-\frac{29}{2}\kappa_{2}^{2}\lambda_{\phi\chi}-20\kappa_{3}^{2}\lambda_{\phi\chi}-60\lambda_{\phi}^{2}\lambda_{\phi\chi}-6\lambda_{\phi\xi}^{2}\lambda_{\phi\chi}-96\lambda_{\phi\chi}^{2}\lambda_{\chi}-72y_{b}^{2}\lambda_{\phi}\lambda_{\phi\chi}-72y_{t}^{2}\lambda_{\phi}\lambda_{\phi\chi}$
$\displaystyle\quad-24y_{\tau}^{2}\lambda_{\phi}\lambda_{\phi\chi}-16\kappa_{1}\lambda_{\phi\xi}\lambda_{\phi\chi}-24\kappa_{2}^{2}\lambda_{\chi}-32\kappa_{3}^{2}\lambda_{\chi}-40\kappa_{3}^{2}\lambda_{\chi\xi}-48\lambda_{\phi\xi}^{2}\lambda_{\chi\xi}-4\kappa_{1}\lambda_{\phi\chi}\lambda_{\chi\xi}$
$\displaystyle\quad-32\kappa_{1}\lambda_{\phi\xi}\lambda_{\chi\xi}-48\lambda_{\phi\xi}\lambda_{\phi\chi}\lambda_{\chi\xi}+8\tilde{\lambda}_{\chi}\Big{(}\kappa_{2}^{2}-2\Big{(}\kappa_{3}^{2}+3\lambda_{\phi\chi}^{2}+5\lambda_{\phi\chi}\lambda_{\chi}\Big{)}\Big{)},$
$\displaystyle\beta_{\lambda_{\chi\xi}}^{(0)}$
$\displaystyle=+8\lambda_{\chi\xi}^{2}+2\lambda_{\chi\xi}\Big{(}4\tilde{\lambda}_{\chi}+20\lambda_{\xi}+8\lambda_{\chi}-3g_{1}^{2}\Big{)}-24g_{2}^{2}\lambda_{\chi\xi}+6g_{2}^{4}+4\kappa_{1}\Big{(}2\lambda_{\xi}+\lambda_{\chi}\Big{)}$
$\displaystyle\quad+2\kappa_{1}^{2}+2\kappa_{3}^{2}+4\lambda_{\phi\xi}\lambda_{\phi\chi},$
$\displaystyle\beta_{\lambda_{\chi\xi}}^{(1)}$
$\displaystyle=-50\lambda_{\chi\xi}^{3}-120\lambda_{\chi\xi}\tilde{\lambda}_{\chi}^{2}+\frac{667}{6}g_{1}^{4}\lambda_{\chi\xi}+10g_{1}^{4}\lambda_{\phi\xi}+20g_{1}^{2}g_{2}^{2}\lambda_{\chi\xi}+100g_{2}^{4}\lambda_{\chi}+20g_{2}^{4}\lambda_{\phi\xi}$
$\displaystyle\quad+\frac{2}{3}g_{2}^{4}\Big{(}-45g_{1}^{2}+79g_{2}^{2}+360\lambda_{\xi}\Big{)}+128g_{1}^{2}\lambda_{\chi}\lambda_{\chi\xi}+8g_{1}^{2}\lambda_{\chi\xi}^{2}+8g_{1}^{2}\lambda_{\phi\xi}\lambda_{\phi\chi}+10g_{2}^{4}\lambda_{\phi\chi}$
$\displaystyle\quad+640g_{2}^{2}\lambda_{\xi}\lambda_{\chi\xi}+\frac{326}{3}g_{2}^{4}\lambda_{\chi\xi}+256g_{2}^{2}\lambda_{\chi}\lambda_{\chi\xi}+32g_{2}^{2}\lambda_{\chi\xi}^{2}+24g_{2}^{2}\lambda_{\phi\xi}\lambda_{\phi\chi}-\kappa_{2}^{2}\lambda_{\chi\xi}-26\kappa_{1}^{3}$
$\displaystyle\quad-4\kappa_{2}^{2}\lambda_{\phi\xi}-480\lambda_{\xi}\lambda_{\chi\xi}^{2}-800\lambda_{\xi}^{2}\lambda_{\chi\xi}-192\lambda_{\chi}\lambda_{\chi\xi}^{2}-80\lambda_{\chi}^{2}\lambda_{\chi\xi}-8\lambda_{\chi\xi}\lambda_{\phi\xi}^{2}-32\lambda_{\chi\xi}\lambda_{\phi\xi}\lambda_{\phi\chi}$
$\displaystyle\quad-8\lambda_{\phi\xi}\lambda_{\phi\chi}^{2}-16\lambda_{\phi\xi}^{2}\lambda_{\phi\chi}-2\lambda_{\chi\xi}\lambda_{\phi\chi}^{2}-24y_{b}^{2}\lambda_{\phi\xi}\lambda_{\phi\chi}-24y_{t}^{2}\lambda_{\phi\xi}\lambda_{\phi\chi}-8y_{\tau}^{2}\lambda_{\phi\xi}\lambda_{\phi\chi}$
$\displaystyle\quad+2\kappa_{1}^{2}\Big{(}-12\tilde{\lambda}_{\chi}+g_{1}^{2}+4g_{2}^{2}-56\lambda_{\xi}-22\lambda_{\chi}-18\lambda_{\chi\xi}\Big{)}+8\tilde{\lambda}_{\chi}\Big{(}2\lambda_{\chi\xi}\Big{(}4g_{1}^{2}+8g_{2}^{2}-5\lambda_{\chi}$
$\displaystyle\quad-6\lambda_{\chi\xi}\Big{)}+5g_{2}^{4}\Big{)}-2\kappa_{3}^{2}\Big{(}4\tilde{\lambda}_{\chi}+g_{1}^{2}+2\kappa_{2}+16\lambda_{\xi}+6\lambda_{\chi}+\lambda_{\chi\xi}+12\lambda_{\phi\xi}+8\lambda_{\phi\chi}+2y_{\tau}^{2}$
$\displaystyle\quad+6y_{b}^{2}+6y_{t}^{2}\Big{)}+\kappa_{1}\Big{(}-4\Big{(}-12g_{2}^{2}\tilde{\lambda}_{\chi}+8\tilde{\lambda}_{\chi}^{2}+8\lambda_{\xi}\Big{(}4\lambda_{\chi\xi}-7g_{2}^{2}\Big{)}+\lambda_{\chi}\Big{(}16\lambda_{\chi\xi}-22g_{2}^{2}\Big{)}$
$\displaystyle\quad+3\lambda_{\chi\xi}\Big{(}4g_{2}^{2}+\lambda_{\chi\xi}\Big{)}+32\lambda_{\xi}^{2}+4\lambda_{\chi}^{2}\Big{)}+20g_{1}^{4}+4g_{1}^{2}g_{2}^{2}+32g_{1}^{2}\lambda_{\chi}+34g_{2}^{4}-\kappa_{2}^{2}\Big{)}.$

For the sake of completeness, the beta functions of the dimensionful
parameters at one-loop are given in Ref. Blasi:2017xmc .

##  Appendix F Planes of quartic couplings and heavy Higgs masses

In the following, we present supplementary figures of the GM (eGM) model
parameter space: the results of the fits to three different BFB conditions in
the quartic coupling planes and the various unitarity conditions in the
$\lambda_{\phi\chi}$ vs. $\kappa_{2}$ (or, $\lambda_{\chi\xi}$ vs.
$\kappa_{1}$) plane in Figures 10 and 11, respectively. At a 99.7% CL, we
observed that the regions, where $\lambda_{\xi}$ is small and
$\lambda_{\phi\xi}\,(\lambda_{\phi\chi})$ has large negative value, are ruled
out by the BFB conditions with all possible 2-field and 3-field directions.
This regions can be viable if we use the BFB conditions considering all
13-field directions at once. The colors in Figure 10 have the same meaning as
in Figure 1.

From Eq. (4.6), the tree-level amplitudes vanish along the lines,
$\kappa_{2}=\lambda_{\phi\chi},\;\kappa_{2}=-2\lambda_{\phi\chi},\;\kappa_{1}=-2\lambda_{\chi\xi}$,
and $\kappa_{1}=-\lambda_{\chi\xi}/2$, which are denoted by cyan dotted lines
in Figure 11. The perturbativity test ($R_{1}^{\prime}<1$ or $R_{1}<1$) fails
for coupling values lying in the neighbourhood region of these lines. In
Figure 11, the effect of such cancellations are clearly visible in the NLO
unitarity with $R_{1}^{\prime}$ contour. The colors in Figure 11 have the same
meaning as in Figure 2.

The masses of the heavy Higgs bosons and their mass differences are shown in
Figures 12 and 13, for the GM and eGM model, respectively. The colors in
Figures 12 and 13 have the same meaning as in Figure 6. In case of the GM
model, we observe that the mass differences of the heavy Higgs bosons are
around $400$ GeV ($250$ GeV) when all the heavy Higgs boson masses are above
$700$ GeV ($1000$ GeV). In contrast, the maximal mass difference is of
$\mathcal{O}(100)$ GeV when all the heavy Higgs boson masses are above $700$
GeV in the eGM model.

Figure 10: All possible quartic coupling planes in the eGM model with various
BFB conditions. The grey (yellow) region shows the stability (BFB) of the
potential in the 2-field (3-field) directions, while the green areas are
allowed if we use BFB conditions on the full potential. Figure 11: 99.7% CL
allowed regions in the $\lambda_{i}$ vs. $\kappa_{j}$ planes. The green, red,
and blue regions have the same meaning as in Figure 2. The brown areas show
the parameter space without violating the perturbativity conditions
($R_{1}^{\prime}<1$) at one-loop level. The cyan dashed lines represent
directions along which the tree-level amplitudes vanish. Figure 12: Allowed
regions in the $m_{i}$ vs. $m_{j}$ ($m_{j}-m_{k}$) planes in the GM model. The
color shaded regions have the same meaning as in Figure 6.  Figure 13: Allowed
regions in the $m_{i}$ vs. $m_{j}$ ($m_{j}-m_{k}$) planes in the eGM model.
The color shaded regions have the same meaning as in Figure 6.

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