10^{14}&0.0259887&{\rm Yes}\\\ \hline\cr
2&F&1&1&0&1&S&3&3&\frac{2}{3}&3.63426\times 10^{14}&0.0272945&{\rm Yes}\\\
\hline\cr 2&F&3&1&\frac{2}{3}&1&S&6&3&1&1.2987\times 10^{16}&0.0690751&{\rm
No}\\\ \hline\cr 2&F&3&2&\frac{1}{2}&1&S&8&1&0&8.1903\times
10^{15}&0.0390761&{\rm Yes}\\\ \hline\cr
2&F&3&2&\frac{1}{2}&1&S&8&1&\frac{1}{3}&3.12815\times 10^{15}&0.0388947&{\rm
Yes}\\\ \hline\cr 2&F&1&2&\frac{1}{6}&2&S&6&2&\frac{1}{2}&1.29764\times
10^{17}&0.034587&{\rm Yes}\\\ \hline\cr
2&F&3&2&\frac{1}{6}&2&S&6&2&\frac{5}{6}&6.99517\times 10^{17}&0.0713552&{\rm
Yes}\\\ \hline\cr\end{array}}$ (344)

##  Appendix G Two loop Beta Function

For Standard Model, in Yukawa sector the beta function are Jones:1981we ;
Arason:1991ic ; Luo:2002ey

$\frac{dY_{u,d,e}}{dt}=Y_{u,d,e}\frac{1}{16\pi^{2}}{\beta}^{(1)}_{u,d,e}+\frac{1}{(16\pi^{2})^{2}}{\beta}^{(2)}_{u,d,e}$
(345)

where one loop contribution are given as

$\displaystyle\beta_{u}^{(1)}$ $\displaystyle=$
$\displaystyle\frac{3}{2}(Y_{u}^{{\dagger}}Y_{u}-Y_{d}^{{\dagger}}Y_{d})+Y_{2}(S)-\Big{(}\frac{17}{20}g_{1}^{2}+\frac{9}{4}g_{2}^{2}+8g_{3}^{2}\Big{)}$
(346) $\displaystyle\beta_{d}^{(1)}$ $\displaystyle=$
$\displaystyle\frac{3}{2}(Y_{d}^{{\dagger}}Y_{d}-Y_{u}^{{\dagger}}Y_{u})+Y_{2}(S)-\Big{(}\frac{1}{4}g_{1}^{2}+\frac{9}{4}g_{2}^{2}+8g_{3}^{2}\Big{)}$
(347) $\displaystyle\beta_{e}^{(1)}$ $\displaystyle=$
$\displaystyle\frac{3}{2}Y_{e}^{{\dagger}}Y_{e}+Y_{2}(S)-\frac{9}{4}(g_{1}^{2}+g_{2}^{2})$
(348)

with

$Y_{2}(S)=Tr(3Y_{u}^{{\dagger}}Y_{u}+3Y_{d}^{{\dagger}}Y_{d}+Y_{e}^{{\dagger}}Y_{e})$
(349)

the two-loop contribution are given as

$\displaystyle\beta_{u}^{(2)}$ $\displaystyle=$
$\displaystyle\frac{3}{2}(Y_{u}^{{\dagger}}Y_{u})^{2}-Y_{u}^{{\dagger}}Y_{u}Y_{d}^{{\dagger}}Y_{d}-\frac{1}{4}Y_{d}^{{\dagger}}Y_{d}Y_{u}^{{\dagger}}Y_{u}+\frac{11}{4}(Y_{d}^{{\dagger}}Y_{d})^{2}+Y_{2}(S)\Big{(}\frac{5}{4}Y_{d}^{{\dagger}}Y_{d}-\frac{9}{4}Y_{u}^{{\dagger}}Y_{u}\Big{)}$
(350) $\displaystyle-$
$\displaystyle\chi_{4}(S)+\frac{3}{2}\lambda^{2}-6\lambda
Y_{u}^{{\dagger}}Y_{u}+\Big{(}\frac{223}{80}g_{1}^{2}+\frac{135}{16}g_{2}^{2}+16g_{3}^{2}\Big{)}Y_{u}^{{\dagger}}Y_{u}$
$\displaystyle-$
$\displaystyle\Big{(}\frac{43}{80}g_{1}^{2}-\frac{9}{16}g_{2}^{2}+16g_{3}^{2}\Big{)}Y_{d}^{{\dagger}}Y_{d}+\frac{5}{2}Y_{4}(S)+\left(\frac{9}{200}+\frac{29}{45}n_{g}\right)g_{1}^{4}$
$\displaystyle-$
$\displaystyle\frac{9}{20}g_{1}^{2}g_{2}^{2}+\frac{19}{15}g_{1}^{2}g_{3}^{2}-\left(\frac{35}{4}-n_{g}\right)g_{2}^{4}+9g_{2}^{2}g_{3}^{2}-\left(\frac{404}{3}-\frac{80}{9}n_{g}\right)g_{3}^{4}$
$\displaystyle\beta_{d}^{(2)}$ $\displaystyle=$
$\displaystyle\frac{3}{2}(Y_{d}^{{\dagger}}Y_{d})^{2}-Y_{d}^{{\dagger}}Y_{d}Y_{u}^{{\dagger}}Y_{u}-\frac{1}{4}Y_{u}^{{\dagger}}Y_{u}Y_{d}^{{\dagger}}Y_{d}+\frac{11}{4}(Y_{u}^{{\dagger}}Y_{u})^{2}+Y_{2}(S)\Big{(}\frac{5}{4}Y_{u}^{{\dagger}}Y_{u}-\frac{9}{4}Y_{d}^{{\dagger}}Y_{d}\Big{)}$
(351) $\displaystyle-$
$\displaystyle\chi_{4}(S)+\frac{3}{2}\lambda^{2}-2\lambda
3Y_{d}^{{\dagger}}Y_{d}+\Big{(}\frac{187}{80}g_{1}^{2}+\frac{135}{16}g_{2}^{2}+16g_{3}^{2}\Big{)}Y_{d}^{{\dagger}}Y_{d}$
$\displaystyle-$
$\displaystyle\Big{(}\frac{79}{80}g_{1}^{2}-\frac{9}{16}g_{2}^{2}+16g_{3}^{2}\Big{)}Y_{u}^{{\dagger}}Y_{u}+\frac{5}{2}Y_{4}(S)-\left(\frac{29}{200}+\frac{1}{45}n_{g}\right)g_{1}^{4}$
$\displaystyle-$
$\displaystyle\frac{27}{20}g_{1}^{2}g_{2}^{2}+\frac{31}{15}g_{1}^{2}g_{3}^{2}-\left(\frac{35}{4}-n_{g}\right)g_{2}^{4}9g_{2}^{2}g_{3}^{2}-\left(\frac{404}{3}-\frac{80}{9}n_{g}\right)g_{3}^{4}$
$\displaystyle\beta_{e}^{(2)}$ $\displaystyle=$
$\displaystyle\frac{3}{2}(Y_{e}^{{\dagger}}Y_{e})^{2}-\frac{9}{4}Y_{2}(S)Y_{e}^{{\dagger}}Y_{e}-\chi_{4}(S)+\frac{3}{2}\lambda^{2}-6\lambda
Y_{e}^{{\dagger}}Y_{e}+\Big{(}\frac{387}{80}g_{1}^{2}+\frac{135}{15}g_{2}^{2}\Big{)}Y_{e}^{{\dagger}}Y_{e}$
(352) $\displaystyle+$
$\displaystyle\frac{5}{2}Y_{4}(S)+\left(\frac{51}{200}+\frac{11}{5}n_{g}\right)g_{1}^{4}+\frac{27}{20}g_{1}^{2}g_{2}^{2}-\left(\frac{35}{4}-n_{g}\right)g_{2}^{4})$

with

$Y_{4}(S)=\Big{(}\frac{17}{20}g_{1}^{2}+\frac{9}{4}g_{2}^{2}+8g_{3}^{2}\Big{)}Tr[Y_{u}^{{\dagger}}Y_{u}]+\Big{(}\frac{1}{4}g_{1}^{2}+\frac{9}{4}g_{2}^{2}+8g_{3}^{2}\Big{)}Tr[Y_{d}^{{\dagger}}Y_{d}]+\frac{3}{4}(g_{1}^{2}+g_{2}^{2})Tr[Y_{e}^{{\dagger}}Y_{e}]$
(353)

and

$\chi_{4}(S)=\frac{9}{4}\Big{(}3(Y_{u}^{{\dagger}}Y_{u})^{2}+3(Y_{d}^{{\dagger}}Y_{d})^{2}+(Y_{e}^{{\dagger}}Y_{e})^{2}-\frac{2}{3}Y_{u}^{{\dagger}}Y_{u}Y_{d}^{{\dagger}}Y_{d}\Big{)}$
(354)

In Higgs sector we present $\beta$ functions for the quartic coupling:

$\frac{d\lambda}{dt}=\frac{1}{16\pi^{2}}\beta^{(1)}_{\lambda}+\frac{1}{(16\pi^{2})^{2}}\beta^{(2)}_{\lambda}$
(355)

where the one loop contribution is given as,

$\displaystyle\beta_{\lambda}^{(1)}=12\lambda^{2}-\left(\frac{9}{5}g_{1}^{2}+9g_{2}^{2}\right)\lambda+\frac{9}{4}\left(\frac{3}{5}g_{1}^{4}+\frac{2}{5}g_{2}^{2}g_{2}^{2}+g_{2}^{4}\right)+4Y_{2}(S)\lambda-4H(S),$
(356)

with

$H(S)=Tr(3(Y_{u}^{{\dagger}}Y_{u})^{2}+3(Y_{d}^{{\dagger}}Y_{d})^{2}+(Y_{e}^{{\dagger}}Y_{e})^{2})$
(357)

and the two loop contribution is given as:

$\displaystyle\beta_{\lambda}^{(2)}$ $\displaystyle=$
$\displaystyle-78\lambda^{3}+18\left(\frac{3}{5}g_{1}^{2}+3g_{2}^{2}\right)\lambda^{2}-\left[\left(\frac{313}{8}-10n_{g}\right)g_{2}^{4}-\frac{117}{20}g_{1}^{2}g_{2}^{2}-\left(\frac{687}{200}+2n_{g}\right)g_{1}^{4}\right]\lambda$
(358) $\displaystyle+$
$\displaystyle\left(\frac{497}{8}-8n_{g}\right)g_{2}^{3}-\frac{3}{5}\left(\frac{97}{24}+\frac{8}{3}n_{g}\right)g_{1}^{2}g_{2}^{4}-\frac{9}{25}\left(\frac{239}{24}+\frac{40}{9}n_{g}\right)g_{1}^{4}g_{2}^{2}-\frac{27}{125}\left(\frac{59}{24}+\frac{40}{9}n_{g}\right)g_{1}^{6}$
$\displaystyle-$ $\displaystyle
64g_{3}^{2}Tr((Y_{u}^{{\dagger}}Y_{u})^{2}+(Y_{d}^{{\dagger}}Y_{d})^{2})-\frac{8}{5}g_{1}^{2}Tr(2(Y_{u}^{{\dagger}}Y_{u})^{2}-(Y_{d}^{{\dagger}}Y_{d})^{2}+3(Y_{e}^{{\dagger}}Y_{e})^{2})-\frac{3}{2}g_{2}^{4}Y_{2}(S)$
$\displaystyle+$ $\displaystyle
10\lambda\left[\left(\frac{17}{20}g_{1}^{2}+\frac{9}{4}g_{2}^{2}+8g_{3}^{2}\right)Tr(Y_{u}^{{\dagger}}Y_{u})+\left(\frac{1}{4}g_{1}^{2}+\frac{9}{4}g_{2}^{2}+8g_{3}^{2}\right)Tr(Y_{d}^{{\dagger}}Y_{d})+\frac{3}{4}\left(g_{1}^{2}+g_{2}^{2}\right)Tr(Y_{e}^{{\dagger}}Y_{e})\right]$
$\displaystyle+$
$\displaystyle\frac{3}{5}g_{1}^{2}\left[\left(-\frac{57}{10}g_{1}^{2}+21g_{2}^{2}\right)Tr(Y_{u}^{{\dagger}}Y_{u})+\left(\frac{3}{2}g_{1}^{2}+9g_{2}^{2}\right)Tr(Y_{d}^{{\dagger}}Y_{d})+\left(-\frac{15}{2}g_{1}^{2}+11g_{2}^{2}\right)Tr(Y_{e}^{{\dagger}}Y_{e})\right]$
$\displaystyle-$ $\displaystyle 24\lambda^{2}Y_{2}(S)-\lambda H(S)-42\lambda
Tr(Y_{u}^{{\dagger}}Y_{u}Y_{d}^{{\dagger}}Y_{d})+20Tr(3(Y_{u}^{{\dagger}}Y_{u})^{3}+3(Y_{d}^{{\dagger}}Y_{d})^{3}+(Y_{e}^{{\dagger}}Y_{e})^{3})$
$\displaystyle-$ $\displaystyle
12Tr\\{Y_{u}^{{\dagger}}Y_{u}(Y_{u}^{{\dagger}}Y_{u}+Y_{d}^{{\dagger}}Y_{d})Y_{d}^{{\dagger}}Y_{d}\\}$

where $n_{g}$ is the number of generation of fermions in SM.

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