%% slacpub7099: page file slacpub7099006u1.tcx.
%% section* Appendix: Tensor Gluon Model [slacpub7099006u1 in slacpub7099006u1: slacpub7099006u2]
%%%% latex2techexplorer block:
%% latex2techexplorer page setup:
\iftechexplorer
\setcounter{section}{5}
\fi
\iftechexplorer
\setcounter{secnumdepth}{2}
\setcounter{tocdepth}{2}
\def\thepart#1{}%
\def\thechapter#1{}%
\newcommand{\partLink}[3]{\docLink{#1.tcx}[part::#2]{#3}\\}
\newcommand{\chapterLink}[3]{\docLink{#1.tcx}[chapter::#2]{#3}\\}
\newcommand{\sectionLink}[3]{\docLink{#1.tcx}[section::#2]{#3}\\}
\newcommand{\subsectionLink}[3]{\docLink{#1.tcx}[subsection::#2]{#3}\\}
\newcommand{\subsubsectionLink}[3]{\docLink{#1.tcx}[subsubsection::#2]{#3}\\}
\newcommand{\paragraphLink}[3]{\docLink{#1.tcx}[paragraph::#2]{#3}\\}
\newcommand{\subparagraphLink}[3]{\docLink{#1.tcx}[subparagraph::#2]{#3}\\}
\newcommand{\partInput}{\partLink}
\newcommand{\chapterInput}{\chapterLink}
\newcommand{\sectionInput}{\sectionLink}
\else
\newcommand{\partInput}[3]{\input{#2.tcx}}
\newcommand{\chapterInput}[3]{\input{#2.tcx}}
\newcommand{\sectionInput}[3]{\input{#2.tcx}}
\fi
\newcommand{\subsectionInput}[3]{\input{#2.tcx}}
\newcommand{\subsubsectionInput}[3]{\input{#2.tcx}}
\newcommand{\paragraphInput}[3]{\input{#2.tcx}}
\newcommand{\subparagraphInput}[3]{\input{#2.tcx}}
\aboveTopic{slacpub7099.tcx}%
\previousTopic{slacpub7099006.tcx}%
\nextTopic{slacpub7099006u2.tcx}%
\bibfile{slacpub7099006u2.tcx}%
\newmenu{slacpub7099::context::slacpub7099006u1}{
\docLink{slacpub7099.tcx}[::Top]{Top}%
\sectionLink{slacpub7099006}{slacpub7099006}{Previous: 6. Acknowledgements}%
\sectionLink{slacpub7099006u2}{slacpub7099006u2}{Next: Bibliography}%
}
%%%% end of latex2techexplorer block.
%%%% code added by add_nav perl script
\docLink{slacpub7099.tcx}[::Top]{Top of Paper}%

\docLink{pseudo:previousTopic}{Previous Section}%
\bigskip%
%%%% end of code added by add_nav
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%% author definitions added by nc_fix
\renewcommand{\baselinestretch}{1.1}
\renewcommand{\baselinestretch}{1.1}
\renewcommand{\thefootnote}{\fnsymbol{footnote}}
\renewcommand{\thefootnote}{\fnsymbol{footnote}}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%% end of definitions added by nc_fix
\section*{\usemenu{slacpub7099::context::slacpub7099006u1}{Appendix: Tensor Gluon Model}}\label{section::slacpub7099006u1}
Since the tensor gluon toy model is new, whereas the
vector and scalar cases have been studied in detail in the
literature, we discuss briefly how Eq.~6 was obtained.
The only wellknown theory involving the exchange of massless, spin2
gauge fields is the quantized version of General
Relativity, which is both highly nonlinear and nonrenormalizable.
To obtain a simple parallel model for tensor gluons,
which couple only to color nonsinglet sources, we
begin by linearizing the theory of quantum gravity based on General
Relativity by keeping only the
lowest order terms in the coupling and by ignoring the tensor field
selfinteractions \cite{29}. Although now linear, the theory remains
nonrenormalizable, as will be the tensor gluon model,
which should be viewed only as a
toy model against which to test the predictions of QCD.
If tensor gluons behaved in the same way as
gravitons one could write down the complete gaugeinvariant
amplitude for the treelevel process
\z0 \ra \qqg. The various contributions arise from a set
of four Feynman diagrams: the usual two
which involve gluon bremsstrahlung from the q or $\overline{\rm q}$
in the final
state, the bremsstrahlung of a tensor gluon from the \z0 in the initial
state, producing an offshell \z0 which
`decays' to q$\overline{\rm q}$,
and finally a new $Z^0$\qqg contact interaction.
We need to remove or modify the $Z^0 Z^0$g piece of the amplitude as
the \z0 is known phenomenologically not to carry a color charge.
We consider two possible approaches to this problem. In the first
instance we surrender the possibility of a gauge symmetry for the
tensor gluon theory and omit the diagram involving the $Z^0 Z^0$g
vertex. (We note that the scalar gluon model is also not
a gauge theory.) In this case, using the Feynman gauge for the tensor
gluon, we arrive at the distribution given
in Eq. 6. A second possibility is to mimic the quantum gravity theory as
far as possible and include the $Z^0 Z^0$g diagram
in a modified form. To do this we extend the particle spectrum of
the Standard Model by introducing a coloroctet partner to the
$Z^0$, $Z^0_8$, which is degenerate with the $Z^0$ and couples to quarks
in exactly the same way as does the $Z^0$,
except for the presence of color generators. The problematic
$Z^0 Z^0$g vertex is now replaced by the $Z^0 Z^0_8\,$g coupling.
In this case we arrive at a form for the tensor distribution
given by \cite{30}:
$$
{1\over\sigma}{d^2\sigma^T\over d x_1d x_2} \quad \propto \quad
{{(x_1+x_21)(x_1^2 + x_2^2)}
\over{(2 x_1 x_2)^2}}\quad +
\quad\quad\quad
\quad\quad\quad
\quad\quad\quad
\quad\quad\quad
$$
\begin{eqnarray}
{{(1x_2)(x_1^2 + (2x_1x_2)^2)}
\over{x_2^2}} \quad + \quad
{{(1x_1)(x_2^2 + (2x_1x_2)^2)}
\over{x_1^2}},
\end{eqnarray}
which, apart from the overall normalisation, is the same as that for
graviton radiation in \z0 decays.
Although algebraically different, this form yields
numerically similar results to Eq.~6 (Fig.~17).
In the analysis presented in the text the comparison of the tensor
model with the data is based on Eq.~6. It is clear from
Fig.~17, however, that our conclusions would not differ
if Eq.~18 had been chosen instead.
%
% References
%
%================
% Bibliography
%================
\vfill
\eject
%%%% code added by add_nav perl script
\bigskip%
\docLink{pseudo:nextTopic}{Next Section}%

\docLink{slacpub7099006u2.tcx}[::Bottom]{Bottom of Paper}%
%%%% end of code added by add_nav