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%% subsection 2.1 Scaled Jet Energy Distributions [slac-pub-7099-0-0-2-1 in slac-pub-7099-0-0-2: ^slac-pub-7099-0-0-2 >slac-pub-7099-0-0-2-2]
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\subsection{\usemenu{slac-pub-7099::context::slac-pub-7099-0-0-2-1}{Scaled Jet Energy Distributions}}\label{subsection::slac-pub-7099-0-0-2-1}
Ordering the three jets in \ep \ra \qqg $\;$ according to their energies,
$E_1>E_2>E_3$, and normalising by the c.m. energy $\sqrt{s}$, we
obtain the scaled jet energies:
\begin{eqnarray}
x_i\quad=\quad {2 E_i\over\sqrt{s}}\quad\quad\quad(i=1,2,3),
\end{eqnarray}
where $ x_1 + x_2 + x_3 = 2$.
Making a Lorentz boost of the event into the rest frame of
jets 2 and 3 the Ellis-Karliner angle $\theta_{EK}$ is defined \cite{9}
to be the angle between jets 1 and 2 in
this frame. For massless partons at tree-level:
\begin{eqnarray}
cos\theta_{EK}={{ x_2- x_3}\over{ x_1}}.
\end{eqnarray}
The inclusive differential cross section can be calculated to
$\Oa$ in perturbative QCD incorporating spin-1
(vector) gluons and assuming massless partons \cite{10}:
\begin{eqnarray}
{1\over\sigma}{d^2\sigma^V\over d x_1d x_2} \propto
{{ x_1^3+ x_2^3+(2- x_1- x_2)^3}\over{(1- x_1)(1- x_2)( x_1+ x_2-1)}}.
\end{eqnarray}
One can also consider alternative `toy' models of
strong interactions. For a model incorporating spin-0
(scalar) gluons one obtains at leading order at the \z0 resonance
\cite{11}:
\begin{eqnarray}
{1\over\sigma}{d^2\sigma^S\over d x_1d x_2} \propto
\biggl[{{ x_1^2(1- x_1)+ x_2^2(1- x_2)+(2- x_1- x_2)^2( x_1+ x_2-1)}
\over {(1- x_1)(1- x_2)( x_1+ x_2-1)}}-R \biggr]
\end{eqnarray}
where
\begin{eqnarray}
{R}={10\;{\Sigma_j a_j^2}\over{\Sigma_j (v_j^2+a_j^2)}}
\end{eqnarray}
and $a_j$ and $v_j$ are the axial and vector couplings, respectively,
of quark flavor $j$ to the \z0.
For a model of strong interactions incorporating spin-2
(tensor) gluons (see Appendix) one obtains at leading order:
\begin{eqnarray}
{1\over\sigma}{d^2\sigma^T\over d x_1d x_2} \propto
{{(x_1+x_2-1)^3 + (1-x_1)^3 + (1-x_2)^3}
\over{(1- x_1)(1- x_2)( x_1+ x_2-1)}}.
\end{eqnarray}
Singly-differential cross sections for $x_1$, $x_2$, $x_3$
or cos$\theta_{EK}$ were obtained by numerical
integrations of Eqs.~3, 4 and 5.
These cross sections are shown in Fig.~1; the shapes are different for
the vector, scalar and tensor gluon cases.
%We note, however, that in all four cases the tensor and vector cases
%are more alike than are the scalar and vector cases, so that
%comparison of QCD with the tensor model represents a more stringent
%test than comparison with the scalar model.
It is well known that vector particles coupling to quarks
in either Abelian or
non-Abelian theories allow consistent and renormalizable
calculations to all orders in perturbation theory. However,
the scalar and tensor gluon models have limited
applicability beyond leading order.
In the scalar model no symmetry, such as gauge invariance, exists to
prevent the gluons from acquiring mass. In the tensor case the
model is non-renormalizable (see Appendix), so
that higher order predictions are not physically meaningful. Given
these difficulties we limit ourselves to the leading-order expressions
for 3-jet event production in these two cases. In the vector case we
do consider the influence of higher-order corrections to the
leading-order predictions. We also assume that the transformation of
the partons in 3-jet events into the observed hadrons is independent
of the gluon spin.