Hillmodel: HillModel.tex

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% This TeX-file has been automatcally generated by FeynRules.
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% C. Duhr, 2008
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\begin{document}


\section{Model description}
This file contains the Feynman rules for the model \verb+SMHill+.
The Feynman rules have been generated automatically by FeynRules0.4.

\subsection{Model information}

Author(s) of the model file: \\
\indent P. Aquino\\
\indent C. Duhr\\
Institution(s):\\
\indent Universite catholique de Louvain (CP3)\\
Emails:\\
\indent priscila@fma.if.usp.br\\
\indent claude.duhr@uclouvain.be\\
Date: {08. 03. 2008}\\
References used to build the model file:\\
\indent "The minimal non-minimal Standard Model", J.J. van der Bij, Phys.Lett.B636:56-59,2006, hep-ph/0603082\\

\subsection{Index description}

\begin{center}\begin{tabular}{|c|c|c|}
\hline
Index & Index range & Symbol\\ 
\hline
Generation & 1 \ldots 3 & $ f $\\ 
\hline
Colour & 1 \ldots 3 & $ i $\\ 
\hline
Gluon & 1 \ldots 8 & $ a $\\ 
\hline
SU2W & 1 \ldots 3 & N/A
\\ \hline
\end{tabular}\end{center}
\subsection{Particle content of the model}

\begin{enumerate}
\item 
\begin{tabular}{ll}
Class: F(1) = $ \text{vl} $, &  Fieldtype: Dirac Field.\\ 
\multicolumn{2}{l}{Indices: Spin, Generation.}\\ 
\multicolumn{2}{l}{Class Members: \text{ve}, vm, vt.}
\end{tabular}
\item 
\begin{tabular}{ll}
Class: F(2) = $ l $, &  Fieldtype: Dirac Field.\\ 
\multicolumn{2}{l}{Indices: Spin, Generation.}\\ 
\multicolumn{2}{l}{Class Members: e, m, tt.}
\end{tabular}
\item 
\begin{tabular}{ll}
Class: F(3) = $ \text{uq} $, &  Fieldtype: Dirac Field.\\ 
\multicolumn{2}{l}{Indices: Spin, Generation, Colour.}\\ 
\multicolumn{2}{l}{Class Members: u, c, t.}
\end{tabular}
\item 
\begin{tabular}{ll}
Class: F(4) = $ \text{dq} $, &  Fieldtype: Dirac Field.\\ 
\multicolumn{2}{l}{Indices: Spin, Generation, Colour.}\\ 
\multicolumn{2}{l}{Class Members: d, s, b.}
\end{tabular}
\item 
\begin{tabular}{ll}
Class: V(1) = $ A $, &  Fieldtype: Real Vectorfield.\\ 
\multicolumn{2}{l}{Indices: Lorentz.}\\ 
\end{tabular}
\item 
\begin{tabular}{ll}
Class: V(2) = $ Z $, &  Fieldtype: Real Vectorfield.\\ 
\multicolumn{2}{l}{Indices: Lorentz.}\\ 
\end{tabular}
\item 
\begin{tabular}{ll}
Class: V(3) = $ W $, &  Fieldtype: Complex Vectorfield.\\ 
\multicolumn{2}{l}{Indices: Lorentz.}\\ 
\end{tabular}
\item 
\begin{tabular}{ll}
Class: V(4) = $ G $, &  Fieldtype: Real Vectorfield.\\ 
\multicolumn{2}{l}{Indices: Lorentz, Gluon.}\\ 
\end{tabular}
\item 
\begin{tabular}{ll}
Class: V(5) = $ \text{Wi} $, &  Fieldtype: Real Vectorfield (Unphysical).\\ 
\multicolumn{2}{l}{Indices: Lorentz, SU2W.}\\ 
\end{tabular}
\item 
\begin{tabular}{ll}
Class: V(6) = $ B $, &  Fieldtype: Real Vectorfield (Unphysical).\\ 
\multicolumn{2}{l}{Indices: Lorentz.}\\ 
\end{tabular}
\item 
\begin{tabular}{ll}
Class: S(1) = $ \text{h1} $, &  Fieldtype: Real Scalar Field.\\ 
\multicolumn{2}{l}{Indices: N/A.}\\ 
\end{tabular}
\item 
\begin{tabular}{ll}
Class: S(2) = $ \text{h2} $, &  Fieldtype: Real Scalar Field.\\ 
\multicolumn{2}{l}{Indices: N/A.}\\ 
\end{tabular}
\item 
\begin{tabular}{ll}
Class: S(3) = $ H $, &  Fieldtype: Real Scalar Field (Unphysical).\\ 
\multicolumn{2}{l}{Indices: N/A.}\\ 
\end{tabular}
\item 
\begin{tabular}{ll}
Class: S(4) = $ h $, &  Fieldtype: Real Scalar Field (Unphysical).\\ 
\multicolumn{2}{l}{Indices: N/A.}\\ 
\end{tabular}
\end{enumerate}


%%
%% The Lagrangian
%%

\section{The lagrangian}


%
% HillModel
%

The lagrangian corresponding to \verb+HillModel+.

\begin{respr}
-\frac{1}{2} f_1^2 h^2 \lambda _0-\frac{H^2 \lambda _0}{16}+\frac{1}{4} f_1 h H^2 \lambda _0-\frac{H^4 \lambda _0}{32}-\frac{1}{8} H v \lambda _0+\frac{1}{2} f_1 h H v \lambda _0-\frac{1}{8} H^3 v \lambda _0+\frac{v^2 \lambda _0}{16}+\frac{1}{4} f_1 h v^2 \lambda _0-\frac{3}{16} H^2 v^2 \lambda _0-\frac{1}{8} H v^3 \lambda _0-\frac{v^4 \lambda _0}{32}+\frac{e^2 H^2 B_{\mu }^2}{8 c_w^2}+\frac{e^2 H v B_{\mu }^2}{4 c_w^2}+\frac{e^2 v^2 B_{\mu }^2}{8 c_w^2}+\frac{1}{2} \partial _{\mu }(h){}^2+\frac{1}{2} \partial _{\mu }(H){}^2-\frac{1}{4} \partial _{\nu }\big(B_{\mu }\big){}^2+\frac{1}{2} \partial _{\nu }\big(B_{\mu }\big) \partial _{\mu }\big(B_{\nu }\big)-\frac{1}{4} \partial _{\mu }\big(B_{\nu }\big){}^2-\frac{1}{4} \partial _{\nu }\big(G_{\mu ,\text{a1}}\big){}^2+\frac{1}{2} \partial _{\nu }\big(G_{\mu ,\text{a1}}\big) \partial _{\mu }\big(G_{\nu ,\text{a1}}\big)-\frac{1}{4} \partial _{\mu }\big(G_{\nu ,\text{a1}}\big){}^2-\frac{1}{4} \partial _{\nu }\big(\text{Wi}_{\mu ,\text{i1}}\big){}^2+\frac{1}{2} \partial _{\nu }\big(\text{Wi}_{\mu ,\text{i1}}\big) \partial _{\mu }\big(\text{Wi}_{\nu ,\text{i1}}\big)-\frac{1}{4} \partial _{\mu }\big(\text{Wi}_{\nu ,\text{i1}}\big){}^2+i \text{dq}^{\dagger }.\gamma ^{\mu }.\partial _{\mu }(\text{dq})+i l^{\dagger }.\gamma ^{\mu }.\partial _{\mu }(l)+i \text{uq}^{\dagger }.\gamma ^{\mu }.\partial _{\mu }(\text{uq})+i \text{vl}^{\dagger }.\gamma ^{\mu }.\partial _{\mu }(\text{vl})+\frac{e B_{\mu } \text{dq}^{\dagger }.\gamma ^{\mu }.P_-.\text{dq}}{6 c_w}-\frac{e B_{\mu } \text{dq}^{\dagger }.\gamma ^{\mu }.P_+.\text{dq}}{3 c_w}-\frac{e B_{\mu } l^{\dagger }.\gamma ^{\mu }.P_-.l}{2 c_w}-\frac{e B_{\mu } l^{\dagger }.\gamma ^{\mu }.P_+.l}{c_w}+\frac{e B_{\mu } \text{uq}^{\dagger }.\gamma ^{\mu }.P_-.\text{uq}}{6 c_w}+\frac{2 e B_{\mu } \text{uq}^{\dagger }.\gamma ^{\mu }.P_+.\text{uq}}{3 c_w}-\frac{e B_{\mu } \text{vl}^{\dagger }.\gamma ^{\mu }.P_-.\text{vl}}{2 c_w}+g_s \text{dq}^{\dagger }.\gamma ^{\mu }.T^a.\text{dq} G_{\mu ,a}+g_s \text{uq}^{\dagger }.\gamma ^{\mu }.T^a.\text{uq} G_{\mu ,a}+\frac{1}{4} g_s \partial _{\nu }\big(G_{\mu ,\text{a1}}\big) f_{\text{a1},\text{a2},\text{a3}} G_{\mu ,\text{a2}} G_{\nu ,\text{a3}}-\frac{1}{4} g_s \partial _{\mu }\big(G_{\nu ,\text{a1}}\big) f_{\text{a1},\text{a2},\text{a3}} G_{\mu ,\text{a2}} G_{\nu ,\text{a3}}+\frac{1}{4} g_s \partial _{\nu }\big(G_{\mu ,\text{a1}}\big) f_{\text{a1},\text{a4},\text{a5}} G_{\mu ,\text{a4}} G_{\nu ,\text{a5}}-\frac{1}{4} g_s \partial _{\mu }\big(G_{\nu ,\text{a1}}\big) f_{\text{a1},\text{a4},\text{a5}} G_{\mu ,\text{a4}} G_{\nu ,\text{a5}}-\frac{1}{4} g_s^2 f_{\text{a1},\text{a2},\text{a3}} f_{\text{a1},\text{a4},\text{a5}} G_{\mu ,\text{a2}} G_{\mu ,\text{a4}} G_{\nu ,\text{a3}} G_{\nu ,\text{a5}}+\frac{e \text{vl}^{\dagger }.\gamma ^{\mu }.P_-.l W_{\mu }}{\sqrt{2} s_w}+\frac{e \text{uq}^{\dagger }.\gamma ^{\mu }.P_-.\text{CKM}.\text{dq} W_{\mu }}{\sqrt{2} s_w}+\frac{e l^{\dagger }.\gamma ^{\mu }.P_-.\text{vl} W^{\dagger }{}_{\mu }}{\sqrt{2} s_w}+\frac{e \text{dq}^{\dagger }.\gamma ^{\mu }.P_-.\text{CKM}^{\dagger }.\text{uq} W^{\dagger }{}_{\mu }}{\sqrt{2} s_w}+\frac{e^2 H^2 \text{Wi}_{\mu ,1}^2}{8 s_w^2}+\frac{e^2 H v \text{Wi}_{\mu ,1}^2}{4 s_w^2}+\frac{e^2 v^2 \text{Wi}_{\mu ,1}^2}{8 s_w^2}+\frac{e^2 H^2 \text{Wi}_{\mu ,2}^2}{8 s_w^2}+\frac{e^2 H v \text{Wi}_{\mu ,2}^2}{4 s_w^2}+\frac{e^2 v^2 \text{Wi}_{\mu ,2}^2}{8 s_w^2}-\frac{e^2 H^2 B_{\mu } \text{Wi}_{\mu ,3}}{4 c_w s_w}-\frac{e^2 H v B_{\mu } \text{Wi}_{\mu ,3}}{2 c_w s_w}-\frac{e^2 v^2 B_{\mu } \text{Wi}_{\mu ,3}}{4 c_w s_w}-\frac{e \text{dq}^{\dagger }.\gamma ^{\mu }.P_-.\text{dq} \text{Wi}_{\mu ,3}}{2 s_w}-\frac{e l^{\dagger }.\gamma ^{\mu }.P_-.l \text{Wi}_{\mu ,3}}{2 s_w}+\frac{e \text{uq}^{\dagger }.\gamma ^{\mu }.P_-.\text{uq} \text{Wi}_{\mu ,3}}{2 s_w}+\frac{e \text{vl}^{\dagger }.\gamma ^{\mu }.P_-.\text{vl} \text{Wi}_{\mu ,3}}{2 s_w}+\frac{e^2 H^2 \text{Wi}_{\mu ,3}^2}{8 s_w^2}+\frac{e^2 H v \text{Wi}_{\mu ,3}^2}{4 s_w^2}+\frac{e^2 v^2 \text{Wi}_{\mu ,3}^2}{8 s_w^2}+\frac{1}{4} g_w \partial _{\nu }\big(\text{Wi}_{\mu ,\text{i1}}\big) \text{ep}_{\text{i1},\text{i2},\text{i3}} \text{Wi}_{\mu ,\text{i2}} \text{Wi}_{\nu ,\text{i3}}-\frac{1}{4} g_w \partial _{\mu }\big(\text{Wi}_{\nu ,\text{i1}}\big) \text{ep}_{\text{i1},\text{i2},\text{i3}} \text{Wi}_{\mu ,\text{i2}} \text{Wi}_{\nu ,\text{i3}}+\frac{1}{4} g_w \partial _{\nu }\big(\text{Wi}_{\mu ,\text{i1}}\big) \text{ep}_{\text{i1},\text{i4},\text{i5}} \text{Wi}_{\mu ,\text{i4}} \text{Wi}_{\nu ,\text{i5}}-\frac{1}{4} g_w \partial _{\mu }\big(\text{Wi}_{\nu ,\text{i1}}\big) \text{ep}_{\text{i1},\text{i4},\text{i5}} \text{Wi}_{\mu ,\text{i4}} \text{Wi}_{\nu ,\text{i5}}-\frac{1}{4} g_w^2 \text{ep}_{\text{i1},\text{i2},\text{i3}} \text{ep}_{\text{i1},\text{i4},\text{i5}} \text{Wi}_{\mu ,\text{i2}} \text{Wi}_{\mu ,\text{i4}} \text{Wi}_{\nu ,\text{i3}} \text{Wi}_{\nu ,\text{i5}}-\frac{H \text{dq}^{\dagger }{}_{\text{s$\$$170},\text{n$\$$170},\text{i$\$$170}}.\text{dq}_{\text{r$\$$170},\text{n$\$$170},\text{i$\$$170}} \big(P_+\big)_{\text{s$\$$170},\text{r$\$$170}} \text{yd}_{\text{n$\$$170}}}{\sqrt{2}}-\frac{v \text{dq}^{\dagger }{}_{\text{s$\$$170},\text{n$\$$170},\text{i$\$$170}}.\text{dq}_{\text{r$\$$170},\text{n$\$$170},\text{i$\$$170}} \big(P_+\big)_{\text{s$\$$170},\text{r$\$$170}} \text{yd}_{\text{n$\$$170}}}{\sqrt{2}}-\frac{H \text{dq}^{\dagger }{}_{\text{r$\$$171},\text{n$\$$171},\text{i$\$$171}}.\text{dq}_{\text{s$\$$171},\text{n$\$$171},\text{i$\$$171}} \big(P_-\big)_{\text{r$\$$171},\text{s$\$$171}} \text{yd}_{\text{n$\$$171}}}{\sqrt{2}}-\frac{v \text{dq}^{\dagger }{}_{\text{r$\$$171},\text{n$\$$171},\text{i$\$$171}}.\text{dq}_{\text{s$\$$171},\text{n$\$$171},\text{i$\$$171}} \big(P_-\big)_{\text{r$\$$171},\text{s$\$$171}} \text{yd}_{\text{n$\$$171}}}{\sqrt{2}}-\frac{H l^{\dagger }{}_{\text{s$\$$170},\text{n$\$$170}}.l_{\text{r$\$$170},\text{n$\$$170}} \big(P_+\big)_{\text{s$\$$170},\text{r$\$$170}} \text{yl}_{\text{n$\$$170}}}{\sqrt{2}}-\frac{v l^{\dagger }{}_{\text{s$\$$170},\text{n$\$$170}}.l_{\text{r$\$$170},\text{n$\$$170}} \big(P_+\big)_{\text{s$\$$170},\text{r$\$$170}} \text{yl}_{\text{n$\$$170}}}{\sqrt{2}}-\frac{H l^{\dagger }{}_{\text{r$\$$171},\text{n$\$$171}}.l_{\text{s$\$$171},\text{n$\$$171}} \big(P_-\big)_{\text{r$\$$171},\text{s$\$$171}} \text{yl}_{\text{n$\$$171}}}{\sqrt{2}}-\frac{v l^{\dagger }{}_{\text{r$\$$171},\text{n$\$$171}}.l_{\text{s$\$$171},\text{n$\$$171}} \big(P_-\big)_{\text{r$\$$171},\text{s$\$$171}} \text{yl}_{\text{n$\$$171}}}{\sqrt{2}}-\frac{H \text{uq}^{\dagger }{}_{\text{s$\$$170},\text{n$\$$170},\text{i$\$$170}}.\text{uq}_{\text{r$\$$170},\text{n$\$$170},\text{i$\$$170}} \big(P_+\big)_{\text{s$\$$170},\text{r$\$$170}} \text{yu}_{\text{n$\$$170}}}{\sqrt{2}}-\frac{v \text{uq}^{\dagger }{}_{\text{s$\$$170},\text{n$\$$170},\text{i$\$$170}}.\text{uq}_{\text{r$\$$170},\text{n$\$$170},\text{i$\$$170}} \big(P_+\big)_{\text{s$\$$170},\text{r$\$$170}} \text{yu}_{\text{n$\$$170}}}{\sqrt{2}}-\frac{H \text{uq}^{\dagger }{}_{\text{r$\$$171},\text{n$\$$171},\text{i$\$$171}}.\text{uq}_{\text{s$\$$171},\text{n$\$$171},\text{i$\$$171}} \big(P_-\big)_{\text{r$\$$171},\text{s$\$$171}} \text{yu}_{\text{n$\$$171}}}{\sqrt{2}}-\frac{v \text{uq}^{\dagger }{}_{\text{r$\$$171},\text{n$\$$171},\text{i$\$$171}}.\text{uq}_{\text{s$\$$171},\text{n$\$$171},\text{i$\$$171}} \big(P_-\big)_{\text{r$\$$171},\text{s$\$$171}} \text{yu}_{\text{n$\$$171}}}{\sqrt{2}}\end{respr}

%%
%% The Vertices
%%
\section{Vertices}

\subsection{ 3-point vertices}

\begin{itemize}
\item
Vertex $\{\text{h1},1\} $, $\{\text{h1},2\} $, $\{\text{h2},3\} $
\begin{respr}
\frac{1}{4} i c_a \big(2 c_a^2 f_1-4 f_1 s_a^2+3 c_a s_a v\big) \lambda _0\end{respr}
\item
Vertex $\{\text{h1},1\} $, $\{\text{h1},2\} $, $\{\text{h1},3\} $
\begin{respr}
-\frac{3}{4} i c_a^2 \big(-2 f_1 s_a+c_a v\big) \lambda _0\end{respr}
\item
Vertex $\{\text{h1},1\} $, $\{\text{h2},2\} $, $\{\text{h2},3\} $
\begin{respr}
-\frac{1}{4} i s_a \big(4 c_a^2 f_1-2 f_1 s_a^2+3 c_a s_a v\big) \lambda _0\end{respr}
\item
Vertex $\{\text{h2},1\} $, $\{\text{h2},2\} $, $\{\text{h2},3\} $
\begin{respr}
\frac{3}{4} i s_a^2 \big(2 c_a f_1+s_a v\big) \lambda _0\end{respr}
\item
Vertex $\{G,1\} $, $\{G,2\} $, $\{G,3\} $
\begin{respr}
g_s f_{a_1,a_2,a_3} \big(p_1^{\mu _3} \eta _{\mu _1,\mu _2}-p_2^{\mu _3} \eta _{\mu _1,\mu _2}-p_1^{\mu _2} \eta _{\mu _1,\mu _3}+p_3^{\mu _2} \eta _{\mu _1,\mu _3}+p_2^{\mu _1} \eta _{\mu _2,\mu _3}-p_3^{\mu _1} \eta _{\mu _2,\mu _3}\big)\end{respr}
\item
Vertex $\{G,1\} $, $\big\{b^{\dagger },2\big\} $, $\{b,3\} $
\begin{respr}
i g_s \gamma ^{\mu _1}{}_{s_2,s_3} T^{a_1}{}_{i_2,i_3}\end{respr}
\item
Vertex $\{G,1\} $, $\big\{d^{\dagger },2\big\} $, $\{d,3\} $
\begin{respr}
i g_s \gamma ^{\mu _1}{}_{s_2,s_3} T^{a_1}{}_{i_2,i_3}\end{respr}
\item
Vertex $\{G,1\} $, $\big\{s^{\dagger },2\big\} $, $\{s,3\} $
\begin{respr}
i g_s \gamma ^{\mu _1}{}_{s_2,s_3} T^{a_1}{}_{i_2,i_3}\end{respr}
\item
Vertex $\{G,1\} $, $\big\{c^{\dagger },2\big\} $, $\{c,3\} $
\begin{respr}
i g_s \gamma ^{\mu _1}{}_{s_2,s_3} T^{a_1}{}_{i_2,i_3}\end{respr}
\item
Vertex $\{G,1\} $, $\big\{t^{\dagger },2\big\} $, $\{t,3\} $
\begin{respr}
i g_s \gamma ^{\mu _1}{}_{s_2,s_3} T^{a_1}{}_{i_2,i_3}\end{respr}
\item
Vertex $\{G,1\} $, $\big\{u^{\dagger },2\big\} $, $\{u,3\} $
\begin{respr}
i g_s \gamma ^{\mu _1}{}_{s_2,s_3} T^{a_1}{}_{i_2,i_3}\end{respr}
\item
Vertex $\{A,1\} $, $\big\{W^{\dagger },2\big\} $, $\{W,3\} $
\begin{respr}
i g_w s_w \big(p_1^{\mu _3} \eta _{\mu _1,\mu _2}-p_2^{\mu _3} \eta _{\mu _1,\mu _2}-p_1^{\mu _2} \eta _{\mu _1,\mu _3}+p_3^{\mu _2} \eta _{\mu _1,\mu _3}+p_2^{\mu _1} \eta _{\mu _2,\mu _3}-p_3^{\mu _1} \eta _{\mu _2,\mu _3}\big)\end{respr}
\item
Vertex $\{\text{h1},1\} $, $\{W,2\} $, $\big\{W^{\dagger },3\big\} $
\begin{respr}
\frac{i c_a e^2 v \eta _{\mu _2,\mu _3}}{2 s_w^2}\end{respr}
\item
Vertex $\{\text{h2},1\} $, $\{W,2\} $, $\big\{W^{\dagger },3\big\} $
\begin{respr}
-\frac{i e^2 s_a v \eta _{\mu _2,\mu _3}}{2 s_w^2}\end{respr}
\item
Vertex $\{Z,1\} $, $\{W,2\} $, $\big\{W^{\dagger },3\big\} $
\begin{respr}
-i c_w g_w \big(p_1^{\mu _3} \eta _{\mu _1,\mu _2}-p_2^{\mu _3} \eta _{\mu _1,\mu _2}-p_1^{\mu _2} \eta _{\mu _1,\mu _3}+p_3^{\mu _2} \eta _{\mu _1,\mu _3}+p_2^{\mu _1} \eta _{\mu _2,\mu _3}-p_3^{\mu _1} \eta _{\mu _2,\mu _3}\big)\end{respr}
\item
Vertex $\{\text{h1},1\} $, $\big\{b^{\dagger },2\big\} $, $\{b,3\} $
\begin{respr}
-\frac{i c_a \delta _{i_2,i_3} \delta _{s_2,s_3} \text{yd}_3}{\sqrt{2}}\end{respr}
\item
Vertex $\{\text{h2},1\} $, $\big\{b^{\dagger },2\big\} $, $\{b,3\} $
\begin{respr}
\frac{i s_a \delta _{i_2,i_3} \delta _{s_2,s_3} \text{yd}_3}{\sqrt{2}}\end{respr}
\item
Vertex $\{\text{h1},1\} $, $\big\{\text{tt}^{\dagger },2\big\} $, $\{\text{tt},3\} $
\begin{respr}
-\frac{i c_a \delta _{s_2,s_3} \text{yl}_3}{\sqrt{2}}\end{respr}
\item
Vertex $\{\text{h2},1\} $, $\big\{\text{tt}^{\dagger },2\big\} $, $\{\text{tt},3\} $
\begin{respr}
\frac{i s_a \delta _{s_2,s_3} \text{yl}_3}{\sqrt{2}}\end{respr}
\item
Vertex $\{\text{h1},1\} $, $\big\{c^{\dagger },2\big\} $, $\{c,3\} $
\begin{respr}
-\frac{i c_a \delta _{i_2,i_3} \delta _{s_2,s_3} \text{yu}_2}{\sqrt{2}}\end{respr}
\item
Vertex $\{\text{h2},1\} $, $\big\{c^{\dagger },2\big\} $, $\{c,3\} $
\begin{respr}
\frac{i s_a \delta _{i_2,i_3} \delta _{s_2,s_3} \text{yu}_2}{\sqrt{2}}\end{respr}
\item
Vertex $\{\text{h1},1\} $, $\big\{t^{\dagger },2\big\} $, $\{t,3\} $
\begin{respr}
-\frac{i c_a \delta _{i_2,i_3} \delta _{s_2,s_3} \text{yu}_3}{\sqrt{2}}\end{respr}
\item
Vertex $\{\text{h2},1\} $, $\big\{t^{\dagger },2\big\} $, $\{t,3\} $
\begin{respr}
\frac{i s_a \delta _{i_2,i_3} \delta _{s_2,s_3} \text{yu}_3}{\sqrt{2}}\end{respr}
\item
Vertex $\{\text{h1},1\} $, $\{Z,2\} $, $\{Z,3\} $
\begin{respr}
\frac{i c_a e^2 \big(c_w^2+s_w^2\big){}^2 v \eta _{\mu _2,\mu _3}}{2 c_w^2 s_w^2}\end{respr}
\item
Vertex $\{\text{h2},1\} $, $\{Z,2\} $, $\{Z,3\} $
\begin{respr}
-\frac{i e^2 s_a \big(c_w^2+s_w^2\big){}^2 v \eta _{\mu _2,\mu _3}}{2 c_w^2 s_w^2}\end{respr}
\item
Vertex $\{A,1\} $, $\big\{b^{\dagger },2\big\} $, $\{b,3\} $
\begin{respr}
-\frac{1}{3} i e \gamma ^{\mu _1}{}_{s_2,s_3} \delta _{i_2,i_3}\end{respr}
\item
Vertex $\{A,1\} $, $\big\{d^{\dagger },2\big\} $, $\{d,3\} $
\begin{respr}
-\frac{1}{3} i e \gamma ^{\mu _1}{}_{s_2,s_3} \delta _{i_2,i_3}\end{respr}
\item
Vertex $\{A,1\} $, $\big\{s^{\dagger },2\big\} $, $\{s,3\} $
\begin{respr}
-\frac{1}{3} i e \gamma ^{\mu _1}{}_{s_2,s_3} \delta _{i_2,i_3}\end{respr}
\item
Vertex $\{A,1\} $, $\big\{e^{\dagger },2\big\} $, $\{e,3\} $
\begin{respr}
-i e \gamma ^{\mu _1}{}_{s_2,s_3}\end{respr}
\item
Vertex $\{A,1\} $, $\big\{m^{\dagger },2\big\} $, $\{m,3\} $
\begin{respr}
-i e \gamma ^{\mu _1}{}_{s_2,s_3}\end{respr}
\item
Vertex $\{A,1\} $, $\big\{\text{tt}^{\dagger },2\big\} $, $\{\text{tt},3\} $
\begin{respr}
-i e \gamma ^{\mu _1}{}_{s_2,s_3}\end{respr}
\item
Vertex $\{A,1\} $, $\big\{c^{\dagger },2\big\} $, $\{c,3\} $
\begin{respr}
\frac{2}{3} i e \gamma ^{\mu _1}{}_{s_2,s_3} \delta _{i_2,i_3}\end{respr}
\item
Vertex $\{A,1\} $, $\big\{t^{\dagger },2\big\} $, $\{t,3\} $
\begin{respr}
\frac{2}{3} i e \gamma ^{\mu _1}{}_{s_2,s_3} \delta _{i_2,i_3}\end{respr}
\item
Vertex $\{A,1\} $, $\big\{u^{\dagger },2\big\} $, $\{u,3\} $
\begin{respr}
\frac{2}{3} i e \gamma ^{\mu _1}{}_{s_2,s_3} \delta _{i_2,i_3}\end{respr}
\item
Vertex $\{W,1\} $, $\big\{\text{ve}^{\dagger },2\big\} $, $\{e,3\} $
\begin{respr}
\frac{i e \big(\gamma ^{\mu _1}.P_-\big){}_{s_2,s_3}}{\sqrt{2} s_w}\end{respr}
\item
Vertex $\{W,1\} $, $\big\{\text{vm}^{\dagger },2\big\} $, $\{m,3\} $
\begin{respr}
\frac{i e \big(\gamma ^{\mu _1}.P_-\big){}_{s_2,s_3}}{\sqrt{2} s_w}\end{respr}
\item
Vertex $\{W,1\} $, $\big\{\text{vt}^{\dagger },2\big\} $, $\{\text{tt},3\} $
\begin{respr}
\frac{i e \big(\gamma ^{\mu _1}.P_-\big){}_{s_2,s_3}}{\sqrt{2} s_w}\end{respr}
\item
Vertex $\{W,1\} $, $\big\{c^{\dagger },2\big\} $, $\{d,3\} $
\begin{respr}
\frac{i e \text{CKM}_{2,1} \delta _{i_2,i_3} \big(\gamma ^{\mu _1}.P_-\big){}_{s_2,s_3}}{\sqrt{2} s_w}\end{respr}
\item
Vertex $\{W,1\} $, $\big\{c^{\dagger },2\big\} $, $\{s,3\} $
\begin{respr}
\frac{i e \text{CKM}_{2,2} \delta _{i_2,i_3} \big(\gamma ^{\mu _1}.P_-\big){}_{s_2,s_3}}{\sqrt{2} s_w}\end{respr}
\item
Vertex $\{W,1\} $, $\big\{t^{\dagger },2\big\} $, $\{b,3\} $
\begin{respr}
\frac{i e \delta _{i_2,i_3} \big(\gamma ^{\mu _1}.P_-\big){}_{s_2,s_3}}{\sqrt{2} s_w}\end{respr}
\item
Vertex $\{W,1\} $, $\big\{u^{\dagger },2\big\} $, $\{d,3\} $
\begin{respr}
\frac{i e \text{CKM}_{1,1} \delta _{i_2,i_3} \big(\gamma ^{\mu _1}.P_-\big){}_{s_2,s_3}}{\sqrt{2} s_w}\end{respr}
\item
Vertex $\{W,1\} $, $\big\{u^{\dagger },2\big\} $, $\{s,3\} $
\begin{respr}
\frac{i e \text{CKM}_{1,2} \delta _{i_2,i_3} \big(\gamma ^{\mu _1}.P_-\big){}_{s_2,s_3}}{\sqrt{2} s_w}\end{respr}
\item
Vertex $\big\{W^{\dagger },1\big\} $, $\big\{e^{\dagger },2\big\} $, $\{\text{ve},3\} $
\begin{respr}
\frac{i e \big(\gamma ^{\mu _1}.P_-\big){}_{s_2,s_3}}{\sqrt{2} s_w}\end{respr}
\item
Vertex $\big\{W^{\dagger },1\big\} $, $\big\{m^{\dagger },2\big\} $, $\{\text{vm},3\} $
\begin{respr}
\frac{i e \big(\gamma ^{\mu _1}.P_-\big){}_{s_2,s_3}}{\sqrt{2} s_w}\end{respr}
\item
Vertex $\big\{W^{\dagger },1\big\} $, $\big\{\text{tt}^{\dagger },2\big\} $, $\{\text{vt},3\} $
\begin{respr}
\frac{i e \big(\gamma ^{\mu _1}.P_-\big){}_{s_2,s_3}}{\sqrt{2} s_w}\end{respr}
\item
Vertex $\big\{W^{\dagger },1\big\} $, $\big\{b^{\dagger },2\big\} $, $\{t,3\} $
\begin{respr}
\frac{i e \delta _{i_2,i_3} \big(\gamma ^{\mu _1}.P_-\big){}_{s_2,s_3}}{\sqrt{2} s_w}\end{respr}
\item
Vertex $\big\{W^{\dagger },1\big\} $, $\big\{d^{\dagger },2\big\} $, $\{c,3\} $
\begin{respr}
\frac{i e \text{CKM}_{2,1}{}^* \delta _{i_2,i_3} \big(\gamma ^{\mu _1}.P_-\big){}_{s_2,s_3}}{\sqrt{2} s_w}\end{respr}
\item
Vertex $\big\{W^{\dagger },1\big\} $, $\big\{d^{\dagger },2\big\} $, $\{u,3\} $
\begin{respr}
\frac{i e \text{CKM}_{1,1}{}^* \delta _{i_2,i_3} \big(\gamma ^{\mu _1}.P_-\big){}_{s_2,s_3}}{\sqrt{2} s_w}\end{respr}
\item
Vertex $\big\{W^{\dagger },1\big\} $, $\big\{s^{\dagger },2\big\} $, $\{c,3\} $
\begin{respr}
\frac{i e \text{CKM}_{2,2}{}^* \delta _{i_2,i_3} \big(\gamma ^{\mu _1}.P_-\big){}_{s_2,s_3}}{\sqrt{2} s_w}\end{respr}
\item
Vertex $\big\{W^{\dagger },1\big\} $, $\big\{s^{\dagger },2\big\} $, $\{u,3\} $
\begin{respr}
\frac{i e \text{CKM}_{1,2}{}^* \delta _{i_2,i_3} \big(\gamma ^{\mu _1}.P_-\big){}_{s_2,s_3}}{\sqrt{2} s_w}\end{respr}
\item
Vertex $\{Z,1\} $, $\big\{b^{\dagger },2\big\} $, $\{b,3\} $
\begin{respr}
-\frac{i e \delta _{i_2,i_3} \big(3 c_w^2 \big(\gamma ^{\mu _1}.P_-\big){}_{s_2,s_3}+s_w^2 \big(\gamma ^{\mu _1}.P_-\big){}_{s_2,s_3}-2 s_w^2 \big(\gamma ^{\mu _1}.P_+\big){}_{s_2,s_3}\big)}{6 c_w s_w}\end{respr}
\item
Vertex $\{Z,1\} $, $\big\{d^{\dagger },2\big\} $, $\{d,3\} $
\begin{respr}
-\frac{i e \delta _{i_2,i_3} \big(3 c_w^2 \big(\gamma ^{\mu _1}.P_-\big){}_{s_2,s_3}+s_w^2 \big(\gamma ^{\mu _1}.P_-\big){}_{s_2,s_3}-2 s_w^2 \big(\gamma ^{\mu _1}.P_+\big){}_{s_2,s_3}\big)}{6 c_w s_w}\end{respr}
\item
Vertex $\{Z,1\} $, $\big\{s^{\dagger },2\big\} $, $\{s,3\} $
\begin{respr}
-\frac{i e \delta _{i_2,i_3} \big(3 c_w^2 \big(\gamma ^{\mu _1}.P_-\big){}_{s_2,s_3}+s_w^2 \big(\gamma ^{\mu _1}.P_-\big){}_{s_2,s_3}-2 s_w^2 \big(\gamma ^{\mu _1}.P_+\big){}_{s_2,s_3}\big)}{6 c_w s_w}\end{respr}
\item
Vertex $\{Z,1\} $, $\big\{e^{\dagger },2\big\} $, $\{e,3\} $
\begin{respr}
-\frac{i e \big(c_w^2 \big(\gamma ^{\mu _1}.P_-\big){}_{s_2,s_3}-s_w^2 \big(\gamma ^{\mu _1}.P_-\big){}_{s_2,s_3}-2 s_w^2 \big(\gamma ^{\mu _1}.P_+\big){}_{s_2,s_3}\big)}{2 c_w s_w}\end{respr}
\item
Vertex $\{Z,1\} $, $\big\{m^{\dagger },2\big\} $, $\{m,3\} $
\begin{respr}
-\frac{i e \big(c_w^2 \big(\gamma ^{\mu _1}.P_-\big){}_{s_2,s_3}-s_w^2 \big(\gamma ^{\mu _1}.P_-\big){}_{s_2,s_3}-2 s_w^2 \big(\gamma ^{\mu _1}.P_+\big){}_{s_2,s_3}\big)}{2 c_w s_w}\end{respr}
\item
Vertex $\{Z,1\} $, $\big\{\text{tt}^{\dagger },2\big\} $, $\{\text{tt},3\} $
\begin{respr}
-\frac{i e \big(c_w^2 \big(\gamma ^{\mu _1}.P_-\big){}_{s_2,s_3}-s_w^2 \big(\gamma ^{\mu _1}.P_-\big){}_{s_2,s_3}-2 s_w^2 \big(\gamma ^{\mu _1}.P_+\big){}_{s_2,s_3}\big)}{2 c_w s_w}\end{respr}
\item
Vertex $\{Z,1\} $, $\big\{c^{\dagger },2\big\} $, $\{c,3\} $
\begin{respr}
\frac{i e \delta _{i_2,i_3} \big(3 c_w^2 \big(\gamma ^{\mu _1}.P_-\big){}_{s_2,s_3}-s_w^2 \big(\gamma ^{\mu _1}.P_-\big){}_{s_2,s_3}-4 s_w^2 \big(\gamma ^{\mu _1}.P_+\big){}_{s_2,s_3}\big)}{6 c_w s_w}\end{respr}
\item
Vertex $\{Z,1\} $, $\big\{t^{\dagger },2\big\} $, $\{t,3\} $
\begin{respr}
\frac{i e \delta _{i_2,i_3} \big(3 c_w^2 \big(\gamma ^{\mu _1}.P_-\big){}_{s_2,s_3}-s_w^2 \big(\gamma ^{\mu _1}.P_-\big){}_{s_2,s_3}-4 s_w^2 \big(\gamma ^{\mu _1}.P_+\big){}_{s_2,s_3}\big)}{6 c_w s_w}\end{respr}
\item
Vertex $\{Z,1\} $, $\big\{u^{\dagger },2\big\} $, $\{u,3\} $
\begin{respr}
\frac{i e \delta _{i_2,i_3} \big(3 c_w^2 \big(\gamma ^{\mu _1}.P_-\big){}_{s_2,s_3}-s_w^2 \big(\gamma ^{\mu _1}.P_-\big){}_{s_2,s_3}-4 s_w^2 \big(\gamma ^{\mu _1}.P_+\big){}_{s_2,s_3}\big)}{6 c_w s_w}\end{respr}
\item
Vertex $\{Z,1\} $, $\big\{\text{ve}^{\dagger },2\big\} $, $\{\text{ve},3\} $
\begin{respr}
\frac{i e \big(c_w^2+s_w^2\big) \big(\gamma ^{\mu _1}.P_-\big){}_{s_2,s_3}}{2 c_w s_w}\end{respr}
\item
Vertex $\{Z,1\} $, $\big\{\text{vm}^{\dagger },2\big\} $, $\{\text{vm},3\} $
\begin{respr}
\frac{i e \big(c_w^2+s_w^2\big) \big(\gamma ^{\mu _1}.P_-\big){}_{s_2,s_3}}{2 c_w s_w}\end{respr}
\item
Vertex $\{Z,1\} $, $\big\{\text{vt}^{\dagger },2\big\} $, $\{\text{vt},3\} $
\begin{respr}
\frac{i e \big(c_w^2+s_w^2\big) \big(\gamma ^{\mu _1}.P_-\big){}_{s_2,s_3}}{2 c_w s_w}\end{respr}
\end{itemize}

\subsection{ 4-point vertices}

\begin{itemize}
\item
Vertex $\{\text{h1},1\} $, $\{\text{h1},2\} $, $\{\text{h1},3\} $, $\{\text{h1},4\} $
\begin{respr}
-\frac{3}{4} i c_a^4 \lambda _0\end{respr}
\item
Vertex $\{\text{h1},1\} $, $\{\text{h1},2\} $, $\{\text{h1},3\} $, $\{\text{h2},4\} $
\begin{respr}
\frac{3}{4} i c_a^3 s_a \lambda _0\end{respr}
\item
Vertex $\{\text{h1},1\} $, $\{\text{h1},2\} $, $\{\text{h2},3\} $, $\{\text{h2},4\} $
\begin{respr}
-\frac{3}{4} i c_a^2 s_a^2 \lambda _0\end{respr}
\item
Vertex $\{\text{h1},1\} $, $\{\text{h2},2\} $, $\{\text{h2},3\} $, $\{\text{h2},4\} $
\begin{respr}
\frac{3}{4} i c_a s_a^3 \lambda _0\end{respr}
\item
Vertex $\{\text{h2},1\} $, $\{\text{h2},2\} $, $\{\text{h2},3\} $, $\{\text{h2},4\} $
\begin{respr}
-\frac{3}{4} i s_a^4 \lambda _0\end{respr}
\item
Vertex $\{G,1\} $, $\{G,2\} $, $\{G,3\} $, $\{G,4\} $
\begin{respr}
i g_s^2 \big(f_{a_1,a_3,\text{a1}} f_{a_2,a_4,\text{a1}} \eta _{\mu _1,\mu _4} \eta _{\mu _2,\mu _3}+f_{a_1,a_2,\text{a1}} f_{a_3,a_4,\text{a1}} \eta _{\mu _1,\mu _4} \eta _{\mu _2,\mu _3}+f_{a_1,a_4,\text{a1}} f_{a_2,a_3,\text{a1}} \eta _{\mu _1,\mu _3} \eta _{\mu _2,\mu _4}-f_{a_1,a_2,\text{a1}} f_{a_3,a_4,\text{a1}} \eta _{\mu _1,\mu _3} \eta _{\mu _2,\mu _4}-f_{a_1,a_4,\text{a1}} f_{a_2,a_3,\text{a1}} \eta _{\mu _1,\mu _2} \eta _{\mu _3,\mu _4}-f_{a_1,a_3,\text{a1}} f_{a_2,a_4,\text{a1}} \eta _{\mu _1,\mu _2} \eta _{\mu _3,\mu _4}\big)\end{respr}
\item
Vertex $\{\text{h1},1\} $, $\{\text{h1},2\} $, $\{W,3\} $, $\big\{W^{\dagger },4\big\} $
\begin{respr}
\frac{i c_a^2 e^2 \eta _{\mu _3,\mu _4}}{2 s_w^2}\end{respr}
\item
Vertex $\{\text{h1},1\} $, $\{\text{h2},2\} $, $\{W,3\} $, $\big\{W^{\dagger },4\big\} $
\begin{respr}
-\frac{i c_a e^2 s_a \eta _{\mu _3,\mu _4}}{2 s_w^2}\end{respr}
\item
Vertex $\{\text{h2},1\} $, $\{\text{h2},2\} $, $\{W,3\} $, $\big\{W^{\dagger },4\big\} $
\begin{respr}
\frac{i e^2 s_a^2 \eta _{\mu _3,\mu _4}}{2 s_w^2}\end{respr}
\item
Vertex $\{A,1\} $, $\{A,2\} $, $\{W,3\} $, $\big\{W^{\dagger },4\big\} $
\begin{respr}
i g_w^2 s_w^2 \big(\eta _{\mu _1,\mu _4} \eta _{\mu _2,\mu _3}+\eta _{\mu _1,\mu _3} \eta _{\mu _2,\mu _4}-2 \eta _{\mu _1,\mu _2} \eta _{\mu _3,\mu _4}\big)\end{respr}
\item
Vertex $\{W,1\} $, $\{W,2\} $, $\big\{W^{\dagger },3\big\} $, $\big\{W^{\dagger },4\big\} $
\begin{respr}
-i g_w^2 \big(\eta _{\mu _1,\mu _4} \eta _{\mu _2,\mu _3}+\eta _{\mu _1,\mu _3} \eta _{\mu _2,\mu _4}-2 \eta _{\mu _1,\mu _2} \eta _{\mu _3,\mu _4}\big)\end{respr}
\item
Vertex $\{A,1\} $, $\{W,2\} $, $\big\{W^{\dagger },3\big\} $, $\{Z,4\} $
\begin{respr}
-i c_w g_w^2 s_w \big(2 \eta _{\mu _1,\mu _4} \eta _{\mu _2,\mu _3}-\eta _{\mu _1,\mu _3} \eta _{\mu _2,\mu _4}-\eta _{\mu _1,\mu _2} \eta _{\mu _3,\mu _4}\big)\end{respr}
\item
Vertex $\{\text{h1},1\} $, $\{\text{h1},2\} $, $\{Z,3\} $, $\{Z,4\} $
\begin{respr}
\frac{i c_a^2 e^2 \big(c_w^2+s_w^2\big){}^2 \eta _{\mu _3,\mu _4}}{2 c_w^2 s_w^2}\end{respr}
\item
Vertex $\{\text{h1},1\} $, $\{\text{h2},2\} $, $\{Z,3\} $, $\{Z,4\} $
\begin{respr}
-\frac{i c_a e^2 s_a \big(c_w^2+s_w^2\big){}^2 \eta _{\mu _3,\mu _4}}{2 c_w^2 s_w^2}\end{respr}
\item
Vertex $\{\text{h2},1\} $, $\{\text{h2},2\} $, $\{Z,3\} $, $\{Z,4\} $
\begin{respr}
\frac{i e^2 s_a^2 \big(c_w^2+s_w^2\big){}^2 \eta _{\mu _3,\mu _4}}{2 c_w^2 s_w^2}\end{respr}
\item
Vertex $\{W,1\} $, $\big\{W^{\dagger },2\big\} $, $\{Z,3\} $, $\{Z,4\} $
\begin{respr}
i c_w^2 g_w^2 \big(\eta _{\mu _1,\mu _4} \eta _{\mu _2,\mu _3}+\eta _{\mu _1,\mu _3} \eta _{\mu _2,\mu _4}-2 \eta _{\mu _1,\mu _2} \eta _{\mu _3,\mu _4}\big)\end{respr}
\end{itemize}


\end{document}