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\title{Particles Not In the Fundamental Representation} % Declares the document's title.
\author{N. Setzer} % Declares the author's name.
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\maketitle
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\section{Particles in the Adjoint Representation}
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\label{Section:Adjoint}
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From the studies in group theory, it is known that
%
\Style{TrParen=p}
$$
f^{ABC} = \frac{-i}{S_0} \Tr{\commutate{\genFun^A}{\genFun^B}\genFun^C}
$$
$$
f^{ABC} = \frac{-i}{S_0} \operatorname{Tr}\inp{\commutate{\genFun^A}{\genFun^B}\genFun^C}
$$
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$$
\KroneckerDelta{A,B} = \frac{1}{S_0} \Tr{\genFun^A \genFun^B}
$$
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$$
\inp{\genAdj }^A_{BC} = i f^{ABC} = \frac{1}{S_0} \Tr{\commutate{\genFun^A}{\genFun^B}\genFun^C}
$$
%
where a subscript zero indicates the fundamental representation and a subscript $\mathcal{A}$ indicates the adjoint representation. Also, $T^A \in G$, $G$ is the set of matricies that obey the group properties, and \Style{DaggerParen=p} $\Dagger{T^A} = T^A$ \Style{DaggerParen=inv}.
\noindent These properties are convenient facts that are now exploited. To begin, make the following definition
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\newcommand{\DeltaVect}{%
\raisebox{-2pt}{%
$\; {%
\raisebox{2pt}{\mbox{\tiny $\mathcal{A}$}} \mkern-14mu \mbox{\Large $\Delta$}%
}%
\!$%
}%
}
\newcommand{\DeltaAdj}[1]{\DeltaVect^{\inp{#1}}}
$$
\DeltaMatrix_{\alpha \beta} \equiv 2 N \DeltaAdj{A} \inp{\genFun}^A_{\alpha \beta}
$$
%
with $\DeltaMatrix$ a matrix ``in the fundamental representation" (that is, it can couple to particles in this represenation) and
$\DeltaVect$ a particle ``in the adjoint representation" (that is, it is a tuplet with the number of particles in this tuplet equal to the dimensionality of the fundamental representation). The $2$ appears by convention.
\noindent The KE in $\mathcal{A}$ is
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\begin{der}
E_K & = \Dagger{\DeltaAdj{m}} \gamma_0
\inp{ \KroneckerDelta{m,n} \delslash - \I g \inp{\genAdj}^A_{m n} \Slash{B}^A}
\DeltaAdj{n}
\\
%====
%====
& = \Dagger{\DeltaAdj{m}} \gamma_0
\inp{ \frac{1}{S_0} \Tr[G]{\genFun^m \genFun^n } \delslash
- \I g \frac{1}{S_0} \Tr{\commutate{\genFun^A}{\genFun^m}\genFun^n}
}
\DeltaAdj{n}
\\
\Style{DaggerParen=p}
%====
%====
& = \frac{1}{S_0} \Tr[G]{\Dagger{\DeltaAdj{m} \genFun^m} \gamma_0 \delslash \genFun^n \DeltaAdj{n}}
\\
& \phantom{=} \hspace{1cm} {}
- \I g \frac{1}{S_0} \Tr[G]{
\commutate{\genFun^A}{\Dagger{\DeltaAdj{m} \genAdj{m}}} \gamma_0 \Slash{B}^A \genFun^n \DeltaAdj{n}
}
\\
%====
%====
\Style{DaggerParen=none}
& = \frac{1}{4 \Conj{N} N S_0}
\inb{
\Tr[G]{\Dagger{\DeltaMatrix\!} \gamma_0 \delslash \DeltaMatrix }
- \I g \Tr[G]{ \commutate{\genFun^A}{\Dagger{\DeltaMatrix\!}} \gamma_0 \Slash{B}^A \DeltaMatrix }
}
\end{der}
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The only thing left is to choose the normalization, $N$ and the obvious choice is
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$$
\Abs{N}^2 = \frac{1}{4 S_0}
$$
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Thus
$$
\DeltaMatrix \equiv \frac{1}{\sqrt{S_0}} \DeltaAdj{A} \inp{\genFun}^A
$$
%
$$
E_K = \Tr[G]{ \parbox[h][1cm]{0pt}{}
\Dagger{\DeltaMatrix\!} \gamma_0 \delslash \DeltaMatrix
- \I g \commutate{\genFun^A}{\Dagger{\DeltaMatrix\!}} \gamma_0 \Slash{B}^A \DeltaMatrix
}
$$
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\subsection{Supersymmetric $D$ Terms}
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$$
\begin{aligned}
D^A & = - g \Conj{\DeltaAdj{m}} \inp{\genAdj}^A_{mn} \DeltaAdj{n} \\
& = - g \Conj{\DeltaAdj{m}}
\frac{1}{S_0} \Tr{\commutate{\genFun^A}{\genFun^m}\genFun^n}
\DeltaAdj{n} \\
& = - g \frac{1}{S_0} \Tr{\commutate{\genFun^A}{\genFun^m \Conj{\DeltaAdj{m}} }\genFun^n \DeltaAdj{n}}
\\
\Style{DaggerParen=inv}
& = - g \frac{1}{S_0}
\Tr{\commutate{\genFun^A}{\Dagger{\genFun^m} \Dagger{\DeltaAdj{m}} }\genFun^n \DeltaAdj{n}}
\\
\Style{DaggerParen=none}
& = - g \Tr{\commutate{\genFun^A}{\Dagger{\DeltaMatrix}} \DeltaMatrix}
\\
& = - g \Tr{\genFun^A \Dagger{\DeltaMatrix} \DeltaMatrix - \Dagger{\DeltaMatrix} \genFun^A \DeltaMatrix}
\\
\end{aligned}
$$
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