Hillmodel: HillModel.tex

File HillModel.tex, 26.4 KB (added by claudeduhr, 7 years ago)

TeX file containing the model and the Feynman rules

Line 
1%
2%
3% This TeX-file has been automatcally generated by FeynRules.
4%
5% C. Duhr, 2008
6%
7%
8
9\documentclass[11pt]{article}
10
11\usepackage{amsfonts}
12\usepackage{amsmath}
13
14\newenvironment{respr}[0]{\sloppy\begin{flushleft}\hspace*{0.75cm}\(}{\)\end{flushleft}\fussy}
15
16\setlength{\topmargin}{-.2 cm}
17\setlength{\evensidemargin}{.0 cm}
18\setlength{\oddsidemargin}{.0 cm}
19\setlength{\textheight}{8.5 in}
20\setlength{\textwidth}{6.4 in}
21
22
23\begin{document}
24
25
26\section{Model description}
27This file contains the Feynman rules for the model \verb+SMHill+.
28The Feynman rules have been generated automatically by FeynRules0.4.
29
30\subsection{Model information}
31
32Author(s) of the model file: \\
33\indent P. Aquino\\
34\indent C. Duhr\\
35Institution(s):\\
36\indent Universite catholique de Louvain (CP3)\\
37Emails:\\
38\indent priscila@fma.if.usp.br\\
39\indent claude.duhr@uclouvain.be\\
40Date: {08. 03. 2008}\\
41References used to build the model file:\\
42\indent "The minimal non-minimal Standard Model", J.J. van der Bij, Phys.Lett.B636:56-59,2006, hep-ph/0603082\\
43
44\subsection{Index description}
45
46\begin{center}\begin{tabular}{|c|c|c|}
47\hline
48Index & Index range & Symbol\\ 
49\hline
50Generation & 1 \ldots 3 & $ f $\\ 
51\hline
52Colour & 1 \ldots 3 & $ i $\\ 
53\hline
54Gluon & 1 \ldots 8 & $ a $\\ 
55\hline
56SU2W & 1 \ldots 3 & N/A
57\\ \hline
58\end{tabular}\end{center}
59\subsection{Particle content of the model}
60
61\begin{enumerate}
62\item 
63\begin{tabular}{ll}
64Class: F(1) = $ \text{vl} $, &  Fieldtype: Dirac Field.\\ 
65\multicolumn{2}{l}{Indices: Spin, Generation.}\\ 
66\multicolumn{2}{l}{Class Members: \text{ve}, vm, vt.}
67\end{tabular}
68\item 
69\begin{tabular}{ll}
70Class: F(2) = $ l $, &  Fieldtype: Dirac Field.\\ 
71\multicolumn{2}{l}{Indices: Spin, Generation.}\\ 
72\multicolumn{2}{l}{Class Members: e, m, tt.}
73\end{tabular}
74\item 
75\begin{tabular}{ll}
76Class: F(3) = $ \text{uq} $, &  Fieldtype: Dirac Field.\\ 
77\multicolumn{2}{l}{Indices: Spin, Generation, Colour.}\\ 
78\multicolumn{2}{l}{Class Members: u, c, t.}
79\end{tabular}
80\item 
81\begin{tabular}{ll}
82Class: F(4) = $ \text{dq} $, &  Fieldtype: Dirac Field.\\ 
83\multicolumn{2}{l}{Indices: Spin, Generation, Colour.}\\ 
84\multicolumn{2}{l}{Class Members: d, s, b.}
85\end{tabular}
86\item 
87\begin{tabular}{ll}
88Class: V(1) = $ A $, &  Fieldtype: Real Vectorfield.\\ 
89\multicolumn{2}{l}{Indices: Lorentz.}\\ 
90\end{tabular}
91\item 
92\begin{tabular}{ll}
93Class: V(2) = $ Z $, &  Fieldtype: Real Vectorfield.\\ 
94\multicolumn{2}{l}{Indices: Lorentz.}\\ 
95\end{tabular}
96\item 
97\begin{tabular}{ll}
98Class: V(3) = $ W $, &  Fieldtype: Complex Vectorfield.\\ 
99\multicolumn{2}{l}{Indices: Lorentz.}\\ 
100\end{tabular}
101\item 
102\begin{tabular}{ll}
103Class: V(4) = $ G $, &  Fieldtype: Real Vectorfield.\\ 
104\multicolumn{2}{l}{Indices: Lorentz, Gluon.}\\ 
105\end{tabular}
106\item 
107\begin{tabular}{ll}
108Class: V(5) = $ \text{Wi} $, &  Fieldtype: Real Vectorfield (Unphysical).\\ 
109\multicolumn{2}{l}{Indices: Lorentz, SU2W.}\\ 
110\end{tabular}
111\item 
112\begin{tabular}{ll}
113Class: V(6) = $ B $, &  Fieldtype: Real Vectorfield (Unphysical).\\ 
114\multicolumn{2}{l}{Indices: Lorentz.}\\ 
115\end{tabular}
116\item 
117\begin{tabular}{ll}
118Class: S(1) = $ \text{h1} $, &  Fieldtype: Real Scalar Field.\\ 
119\multicolumn{2}{l}{Indices: N/A.}\\ 
120\end{tabular}
121\item 
122\begin{tabular}{ll}
123Class: S(2) = $ \text{h2} $, &  Fieldtype: Real Scalar Field.\\ 
124\multicolumn{2}{l}{Indices: N/A.}\\ 
125\end{tabular}
126\item 
127\begin{tabular}{ll}
128Class: S(3) = $ H $, &  Fieldtype: Real Scalar Field (Unphysical).\\ 
129\multicolumn{2}{l}{Indices: N/A.}\\ 
130\end{tabular}
131\item 
132\begin{tabular}{ll}
133Class: S(4) = $ h $, &  Fieldtype: Real Scalar Field (Unphysical).\\ 
134\multicolumn{2}{l}{Indices: N/A.}\\ 
135\end{tabular}
136\end{enumerate}
137
138
139%%
140%% The Lagrangian
141%%
142
143\section{The lagrangian}
144
145
146%
147% HillModel
148%
149
150The lagrangian corresponding to \verb+HillModel+.
151
152\begin{respr}
153-\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}
154
155%%
156%% The Vertices
157%%
158\section{Vertices}
159
160\subsection{ 3-point vertices}
161
162\begin{itemize}
163\item
164Vertex $\{\text{h1},1\} $, $\{\text{h1},2\} $, $\{\text{h2},3\} $
165\begin{respr}
166\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}
167\item
168Vertex $\{\text{h1},1\} $, $\{\text{h1},2\} $, $\{\text{h1},3\} $
169\begin{respr}
170-\frac{3}{4} i c_a^2 \big(-2 f_1 s_a+c_a v\big) \lambda _0\end{respr}
171\item
172Vertex $\{\text{h1},1\} $, $\{\text{h2},2\} $, $\{\text{h2},3\} $
173\begin{respr}
174-\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}
175\item
176Vertex $\{\text{h2},1\} $, $\{\text{h2},2\} $, $\{\text{h2},3\} $
177\begin{respr}
178\frac{3}{4} i s_a^2 \big(2 c_a f_1+s_a v\big) \lambda _0\end{respr}
179\item
180Vertex $\{G,1\} $, $\{G,2\} $, $\{G,3\} $
181\begin{respr}
182g_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}
183\item
184Vertex $\{G,1\} $, $\big\{b^{\dagger },2\big\} $, $\{b,3\} $
185\begin{respr}
186i g_s \gamma ^{\mu _1}{}_{s_2,s_3} T^{a_1}{}_{i_2,i_3}\end{respr}
187\item
188Vertex $\{G,1\} $, $\big\{d^{\dagger },2\big\} $, $\{d,3\} $
189\begin{respr}
190i g_s \gamma ^{\mu _1}{}_{s_2,s_3} T^{a_1}{}_{i_2,i_3}\end{respr}
191\item
192Vertex $\{G,1\} $, $\big\{s^{\dagger },2\big\} $, $\{s,3\} $
193\begin{respr}
194i g_s \gamma ^{\mu _1}{}_{s_2,s_3} T^{a_1}{}_{i_2,i_3}\end{respr}
195\item
196Vertex $\{G,1\} $, $\big\{c^{\dagger },2\big\} $, $\{c,3\} $
197\begin{respr}
198i g_s \gamma ^{\mu _1}{}_{s_2,s_3} T^{a_1}{}_{i_2,i_3}\end{respr}
199\item
200Vertex $\{G,1\} $, $\big\{t^{\dagger },2\big\} $, $\{t,3\} $
201\begin{respr}
202i g_s \gamma ^{\mu _1}{}_{s_2,s_3} T^{a_1}{}_{i_2,i_3}\end{respr}
203\item
204Vertex $\{G,1\} $, $\big\{u^{\dagger },2\big\} $, $\{u,3\} $
205\begin{respr}
206i g_s \gamma ^{\mu _1}{}_{s_2,s_3} T^{a_1}{}_{i_2,i_3}\end{respr}
207\item
208Vertex $\{A,1\} $, $\big\{W^{\dagger },2\big\} $, $\{W,3\} $
209\begin{respr}
210i 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}
211\item
212Vertex $\{\text{h1},1\} $, $\{W,2\} $, $\big\{W^{\dagger },3\big\} $
213\begin{respr}
214\frac{i c_a e^2 v \eta _{\mu _2,\mu _3}}{2 s_w^2}\end{respr}
215\item
216Vertex $\{\text{h2},1\} $, $\{W,2\} $, $\big\{W^{\dagger },3\big\} $
217\begin{respr}
218-\frac{i e^2 s_a v \eta _{\mu _2,\mu _3}}{2 s_w^2}\end{respr}
219\item
220Vertex $\{Z,1\} $, $\{W,2\} $, $\big\{W^{\dagger },3\big\} $
221\begin{respr}
222-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}
223\item
224Vertex $\{\text{h1},1\} $, $\big\{b^{\dagger },2\big\} $, $\{b,3\} $
225\begin{respr}
226-\frac{i c_a \delta _{i_2,i_3} \delta _{s_2,s_3} \text{yd}_3}{\sqrt{2}}\end{respr}
227\item
228Vertex $\{\text{h2},1\} $, $\big\{b^{\dagger },2\big\} $, $\{b,3\} $
229\begin{respr}
230\frac{i s_a \delta _{i_2,i_3} \delta _{s_2,s_3} \text{yd}_3}{\sqrt{2}}\end{respr}
231\item
232Vertex $\{\text{h1},1\} $, $\big\{\text{tt}^{\dagger },2\big\} $, $\{\text{tt},3\} $
233\begin{respr}
234-\frac{i c_a \delta _{s_2,s_3} \text{yl}_3}{\sqrt{2}}\end{respr}
235\item
236Vertex $\{\text{h2},1\} $, $\big\{\text{tt}^{\dagger },2\big\} $, $\{\text{tt},3\} $
237\begin{respr}
238\frac{i s_a \delta _{s_2,s_3} \text{yl}_3}{\sqrt{2}}\end{respr}
239\item
240Vertex $\{\text{h1},1\} $, $\big\{c^{\dagger },2\big\} $, $\{c,3\} $
241\begin{respr}
242-\frac{i c_a \delta _{i_2,i_3} \delta _{s_2,s_3} \text{yu}_2}{\sqrt{2}}\end{respr}
243\item
244Vertex $\{\text{h2},1\} $, $\big\{c^{\dagger },2\big\} $, $\{c,3\} $
245\begin{respr}
246\frac{i s_a \delta _{i_2,i_3} \delta _{s_2,s_3} \text{yu}_2}{\sqrt{2}}\end{respr}
247\item
248Vertex $\{\text{h1},1\} $, $\big\{t^{\dagger },2\big\} $, $\{t,3\} $
249\begin{respr}
250-\frac{i c_a \delta _{i_2,i_3} \delta _{s_2,s_3} \text{yu}_3}{\sqrt{2}}\end{respr}
251\item
252Vertex $\{\text{h2},1\} $, $\big\{t^{\dagger },2\big\} $, $\{t,3\} $
253\begin{respr}
254\frac{i s_a \delta _{i_2,i_3} \delta _{s_2,s_3} \text{yu}_3}{\sqrt{2}}\end{respr}
255\item
256Vertex $\{\text{h1},1\} $, $\{Z,2\} $, $\{Z,3\} $
257\begin{respr}
258\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}
259\item
260Vertex $\{\text{h2},1\} $, $\{Z,2\} $, $\{Z,3\} $
261\begin{respr}
262-\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}
263\item
264Vertex $\{A,1\} $, $\big\{b^{\dagger },2\big\} $, $\{b,3\} $
265\begin{respr}
266-\frac{1}{3} i e \gamma ^{\mu _1}{}_{s_2,s_3} \delta _{i_2,i_3}\end{respr}
267\item
268Vertex $\{A,1\} $, $\big\{d^{\dagger },2\big\} $, $\{d,3\} $
269\begin{respr}
270-\frac{1}{3} i e \gamma ^{\mu _1}{}_{s_2,s_3} \delta _{i_2,i_3}\end{respr}
271\item
272Vertex $\{A,1\} $, $\big\{s^{\dagger },2\big\} $, $\{s,3\} $
273\begin{respr}
274-\frac{1}{3} i e \gamma ^{\mu _1}{}_{s_2,s_3} \delta _{i_2,i_3}\end{respr}
275\item
276Vertex $\{A,1\} $, $\big\{e^{\dagger },2\big\} $, $\{e,3\} $
277\begin{respr}
278-i e \gamma ^{\mu _1}{}_{s_2,s_3}\end{respr}
279\item
280Vertex $\{A,1\} $, $\big\{m^{\dagger },2\big\} $, $\{m,3\} $
281\begin{respr}
282-i e \gamma ^{\mu _1}{}_{s_2,s_3}\end{respr}
283\item
284Vertex $\{A,1\} $, $\big\{\text{tt}^{\dagger },2\big\} $, $\{\text{tt},3\} $
285\begin{respr}
286-i e \gamma ^{\mu _1}{}_{s_2,s_3}\end{respr}
287\item
288Vertex $\{A,1\} $, $\big\{c^{\dagger },2\big\} $, $\{c,3\} $
289\begin{respr}
290\frac{2}{3} i e \gamma ^{\mu _1}{}_{s_2,s_3} \delta _{i_2,i_3}\end{respr}
291\item
292Vertex $\{A,1\} $, $\big\{t^{\dagger },2\big\} $, $\{t,3\} $
293\begin{respr}
294\frac{2}{3} i e \gamma ^{\mu _1}{}_{s_2,s_3} \delta _{i_2,i_3}\end{respr}
295\item
296Vertex $\{A,1\} $, $\big\{u^{\dagger },2\big\} $, $\{u,3\} $
297\begin{respr}
298\frac{2}{3} i e \gamma ^{\mu _1}{}_{s_2,s_3} \delta _{i_2,i_3}\end{respr}
299\item
300Vertex $\{W,1\} $, $\big\{\text{ve}^{\dagger },2\big\} $, $\{e,3\} $
301\begin{respr}
302\frac{i e \big(\gamma ^{\mu _1}.P_-\big){}_{s_2,s_3}}{\sqrt{2} s_w}\end{respr}
303\item
304Vertex $\{W,1\} $, $\big\{\text{vm}^{\dagger },2\big\} $, $\{m,3\} $
305\begin{respr}
306\frac{i e \big(\gamma ^{\mu _1}.P_-\big){}_{s_2,s_3}}{\sqrt{2} s_w}\end{respr}
307\item
308Vertex $\{W,1\} $, $\big\{\text{vt}^{\dagger },2\big\} $, $\{\text{tt},3\} $
309\begin{respr}
310\frac{i e \big(\gamma ^{\mu _1}.P_-\big){}_{s_2,s_3}}{\sqrt{2} s_w}\end{respr}
311\item
312Vertex $\{W,1\} $, $\big\{c^{\dagger },2\big\} $, $\{d,3\} $
313\begin{respr}
314\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}
315\item
316Vertex $\{W,1\} $, $\big\{c^{\dagger },2\big\} $, $\{s,3\} $
317\begin{respr}
318\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}
319\item
320Vertex $\{W,1\} $, $\big\{t^{\dagger },2\big\} $, $\{b,3\} $
321\begin{respr}
322\frac{i e \delta _{i_2,i_3} \big(\gamma ^{\mu _1}.P_-\big){}_{s_2,s_3}}{\sqrt{2} s_w}\end{respr}
323\item
324Vertex $\{W,1\} $, $\big\{u^{\dagger },2\big\} $, $\{d,3\} $
325\begin{respr}
326\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}
327\item
328Vertex $\{W,1\} $, $\big\{u^{\dagger },2\big\} $, $\{s,3\} $
329\begin{respr}
330\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}
331\item
332Vertex $\big\{W^{\dagger },1\big\} $, $\big\{e^{\dagger },2\big\} $, $\{\text{ve},3\} $
333\begin{respr}
334\frac{i e \big(\gamma ^{\mu _1}.P_-\big){}_{s_2,s_3}}{\sqrt{2} s_w}\end{respr}
335\item
336Vertex $\big\{W^{\dagger },1\big\} $, $\big\{m^{\dagger },2\big\} $, $\{\text{vm},3\} $
337\begin{respr}
338\frac{i e \big(\gamma ^{\mu _1}.P_-\big){}_{s_2,s_3}}{\sqrt{2} s_w}\end{respr}
339\item
340Vertex $\big\{W^{\dagger },1\big\} $, $\big\{\text{tt}^{\dagger },2\big\} $, $\{\text{vt},3\} $
341\begin{respr}
342\frac{i e \big(\gamma ^{\mu _1}.P_-\big){}_{s_2,s_3}}{\sqrt{2} s_w}\end{respr}
343\item
344Vertex $\big\{W^{\dagger },1\big\} $, $\big\{b^{\dagger },2\big\} $, $\{t,3\} $
345\begin{respr}
346\frac{i e \delta _{i_2,i_3} \big(\gamma ^{\mu _1}.P_-\big){}_{s_2,s_3}}{\sqrt{2} s_w}\end{respr}
347\item
348Vertex $\big\{W^{\dagger },1\big\} $, $\big\{d^{\dagger },2\big\} $, $\{c,3\} $
349\begin{respr}
350\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}
351\item
352Vertex $\big\{W^{\dagger },1\big\} $, $\big\{d^{\dagger },2\big\} $, $\{u,3\} $
353\begin{respr}
354\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}
355\item
356Vertex $\big\{W^{\dagger },1\big\} $, $\big\{s^{\dagger },2\big\} $, $\{c,3\} $
357\begin{respr}
358\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}
359\item
360Vertex $\big\{W^{\dagger },1\big\} $, $\big\{s^{\dagger },2\big\} $, $\{u,3\} $
361\begin{respr}
362\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}
363\item
364Vertex $\{Z,1\} $, $\big\{b^{\dagger },2\big\} $, $\{b,3\} $
365\begin{respr}
366-\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}
367\item
368Vertex $\{Z,1\} $, $\big\{d^{\dagger },2\big\} $, $\{d,3\} $
369\begin{respr}
370-\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}
371\item
372Vertex $\{Z,1\} $, $\big\{s^{\dagger },2\big\} $, $\{s,3\} $
373\begin{respr}
374-\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}
375\item
376Vertex $\{Z,1\} $, $\big\{e^{\dagger },2\big\} $, $\{e,3\} $
377\begin{respr}
378-\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}
379\item
380Vertex $\{Z,1\} $, $\big\{m^{\dagger },2\big\} $, $\{m,3\} $
381\begin{respr}
382-\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}
383\item
384Vertex $\{Z,1\} $, $\big\{\text{tt}^{\dagger },2\big\} $, $\{\text{tt},3\} $
385\begin{respr}
386-\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}
387\item
388Vertex $\{Z,1\} $, $\big\{c^{\dagger },2\big\} $, $\{c,3\} $
389\begin{respr}
390\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}
391\item
392Vertex $\{Z,1\} $, $\big\{t^{\dagger },2\big\} $, $\{t,3\} $
393\begin{respr}
394\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}
395\item
396Vertex $\{Z,1\} $, $\big\{u^{\dagger },2\big\} $, $\{u,3\} $
397\begin{respr}
398\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}
399\item
400Vertex $\{Z,1\} $, $\big\{\text{ve}^{\dagger },2\big\} $, $\{\text{ve},3\} $
401\begin{respr}
402\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}
403\item
404Vertex $\{Z,1\} $, $\big\{\text{vm}^{\dagger },2\big\} $, $\{\text{vm},3\} $
405\begin{respr}
406\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}
407\item
408Vertex $\{Z,1\} $, $\big\{\text{vt}^{\dagger },2\big\} $, $\{\text{vt},3\} $
409\begin{respr}
410\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}
411\end{itemize}
412
413\subsection{ 4-point vertices}
414
415\begin{itemize}
416\item
417Vertex $\{\text{h1},1\} $, $\{\text{h1},2\} $, $\{\text{h1},3\} $, $\{\text{h1},4\} $
418\begin{respr}
419-\frac{3}{4} i c_a^4 \lambda _0\end{respr}
420\item
421Vertex $\{\text{h1},1\} $, $\{\text{h1},2\} $, $\{\text{h1},3\} $, $\{\text{h2},4\} $
422\begin{respr}
423\frac{3}{4} i c_a^3 s_a \lambda _0\end{respr}
424\item
425Vertex $\{\text{h1},1\} $, $\{\text{h1},2\} $, $\{\text{h2},3\} $, $\{\text{h2},4\} $
426\begin{respr}
427-\frac{3}{4} i c_a^2 s_a^2 \lambda _0\end{respr}
428\item
429Vertex $\{\text{h1},1\} $, $\{\text{h2},2\} $, $\{\text{h2},3\} $, $\{\text{h2},4\} $
430\begin{respr}
431\frac{3}{4} i c_a s_a^3 \lambda _0\end{respr}
432\item
433Vertex $\{\text{h2},1\} $, $\{\text{h2},2\} $, $\{\text{h2},3\} $, $\{\text{h2},4\} $
434\begin{respr}
435-\frac{3}{4} i s_a^4 \lambda _0\end{respr}
436\item
437Vertex $\{G,1\} $, $\{G,2\} $, $\{G,3\} $, $\{G,4\} $
438\begin{respr}
439i 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}
440\item
441Vertex $\{\text{h1},1\} $, $\{\text{h1},2\} $, $\{W,3\} $, $\big\{W^{\dagger },4\big\} $
442\begin{respr}
443\frac{i c_a^2 e^2 \eta _{\mu _3,\mu _4}}{2 s_w^2}\end{respr}
444\item
445Vertex $\{\text{h1},1\} $, $\{\text{h2},2\} $, $\{W,3\} $, $\big\{W^{\dagger },4\big\} $
446\begin{respr}
447-\frac{i c_a e^2 s_a \eta _{\mu _3,\mu _4}}{2 s_w^2}\end{respr}
448\item
449Vertex $\{\text{h2},1\} $, $\{\text{h2},2\} $, $\{W,3\} $, $\big\{W^{\dagger },4\big\} $
450\begin{respr}
451\frac{i e^2 s_a^2 \eta _{\mu _3,\mu _4}}{2 s_w^2}\end{respr}
452\item
453Vertex $\{A,1\} $, $\{A,2\} $, $\{W,3\} $, $\big\{W^{\dagger },4\big\} $
454\begin{respr}
455i 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}
456\item
457Vertex $\{W,1\} $, $\{W,2\} $, $\big\{W^{\dagger },3\big\} $, $\big\{W^{\dagger },4\big\} $
458\begin{respr}
459-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}
460\item
461Vertex $\{A,1\} $, $\{W,2\} $, $\big\{W^{\dagger },3\big\} $, $\{Z,4\} $
462\begin{respr}
463-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}
464\item
465Vertex $\{\text{h1},1\} $, $\{\text{h1},2\} $, $\{Z,3\} $, $\{Z,4\} $
466\begin{respr}
467\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}
468\item
469Vertex $\{\text{h1},1\} $, $\{\text{h2},2\} $, $\{Z,3\} $, $\{Z,4\} $
470\begin{respr}
471-\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}
472\item
473Vertex $\{\text{h2},1\} $, $\{\text{h2},2\} $, $\{Z,3\} $, $\{Z,4\} $
474\begin{respr}
475\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}
476\item
477Vertex $\{W,1\} $, $\big\{W^{\dagger },2\big\} $, $\{Z,3\} $, $\{Z,4\} $
478\begin{respr}
479i 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}
480\end{itemize}
481
482
483\end{document}