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\subsection{\usemenu{slac-pub-7099::context::slac-pub-7099-0-0-4-1}{Scaled Jet Energy Distributions}}\label{subsection::slac-pub-7099-0-0-4-1}
The measured distributions of the three scaled jet energies $x_1$,
$x_2$, $x_3$, and the Ellis-Karliner angle $\theta_{EK}$, are
shown in Fig. 3. Also shown in Fig.~3 are
the predictions of the
HERWIG 5.7 \cite{22} Monte Carlo program for the simulation of
hadronic decays of $Z^0$ bosons, combined with a simulation of the SLD
and the same selection and analysis cuts as applied to the real data.
The simulation describes the data well.
For each observable $X$,
the experimental distribution $D^{data}_{SLD}(X)$ was then
corrected for the effects of selection cuts,
detector acceptance, efficiency,
resolution, particle decays and interactions
within the detector, and for initial state photon radiation, using
bin-by-bin correction factors $C_D(X)$:
\begin{eqnarray}
C_D(X)_m = \frac{D^{MC}_{hadron}(X)_m}{D^{MC}_{SLD}(X)_m},
\label{cd}
\end{eqnarray}
where: $m$ is the bin index;
$D^{MC}_{SLD}(X)_m$ is the content of bin $m$ of the distribution
obtained from reconstructed clusters in Monte Carlo events
after simulation of the detector; and
$D^{MC}_{hadron}(X)_i$ is that from all generated particles with
lifetimes greater than $3 \times 10^{-10}$~s in Monte Carlo events
with no SLD simulation and no initial state radiation.
The bin widths were chosen from the estimated experimental resolution
so as to minimize bin-to-bin migration effects.
The $C_D(X)$ were calculated from events generated with HERWIG 5.7
using default parameter values \cite{22}.
The {\it hadron level} distributions are then given by
\begin{eqnarray}
D^{data}_{hadron}(X)_m = C_D(X)_m \cdot D^{data}_{SLD}(X)_m.
\end{eqnarray}
Experimental systematic errors arising from uncertainties in
modelling the detector were estimated by varying the
event selection criteria over wide ranges, and by varying the cluster
energy response corrections in the detector simulation \cite{8}.
In each case the correction factors $C_D(X)$, and hence the corrected
data distributions $D^{data}_{hadron}(X)$, were rederived.
The correction
factors $C_D(X)$ are shown in Figs. 4(b)--7(b);
the errors comprise the sum in quadrature of the
statistical component from the finite size of the Monte Carlo event
sample, and the systematic uncertainty. It can be seen that the
$C_D(X)$ are close to unity and slowly-varying, except near the
boundaries of phase-space. The hadron level
data are listed in Tables I--IV, together with statistical
and systematic errors; the central values represent the data corrected
by the central values of the correction factors.
Before they can be compared with parton-level predictions the data must
be corrected for the effects of hadronization.
In the absence of a complete theoretical calculation,
the phenomenological models implemented in JETSET 7.4 \cite{23}
and HERWIG 5.7 represent our best description of the
hadronization process, and are not based upon a particular choice of
the gluon spin. These models have been compared
extensively with, and tuned to, \ep \ra hadrons data at the $Z^0$
resonance \cite{24}, as well as data at $W$ $\sim$ 35 GeV from
the PETRA and PEP storage rings \cite{25}. We find that
they provide a
good description of our data in terms of the observables presented
here (Fig.~3) and other hadronic event shape observables
\cite{26}, and hence employ them to calculate hadronization
correction factors. The HERWIG parameters were left at their default
values. Several of the JETSET parameters were set to values
determined from our own optimisation to hadronic $Z^0$ data; these are
given in Table V.
The hadronization
correction procedure is similar to that described above for
the detector effects. Bin-by-bin correction factors
\begin{eqnarray}
C_H(X)_m = \frac{D^{MC}_{parton}(X)_m}{D^{MC}_{hadron}(X)_m},
\end{eqnarray}
where $D^{MC}_{parton}(X)_m$ is the
content of bin $m$ of the distribution obtained from Monte Carlo events
generated at the parton level, were calculated and applied to
the hadron level data distributions $D^{data}_{hadron}(X)_m$ to
obtain the {\it parton level} corrected data:
\begin{eqnarray}
D^{data}_{parton}(X)_m = C_H(X)_m \cdot D^{data}_{hadron}(X)_m.
\end{eqnarray}
For each bin
the average of the JETSET-- and HERWIG--derived values was used as the
central value of the correction factor, and the difference between this
value and the extrema was assigned as a symmetric hadronization
uncertainty. The correction
factors $C_H(X)$ are shown in Figs. 4(c)--7(c);
the errors comprise the sum in quadrature of the
statistical component from the finite size of the Monte Carlo event
sample, and the systematic uncertainty.
It can be seen that the $C_H(X)$ are within 10\% of unity and are
slowly-varying, except near the boundaries of phase space.
The fully-corrected data are shown in Figs.~4(a)--7(a);
the data points
correspond to the central values of the correction factors,
and the errors shown comprise the
statistical and total systematic components added in quadrature.
These results are in agreement with an analysis of our 1992 data
sample using charged tracks for jet reconstruction \cite{27}.
We first compare the data with QCD predictions from
$\Oa$ and $\Oaa$ perturbation theory, and from parton shower (PS)
models. For this purpose we used the JETSET 7.4
$\Oa$ matrix element, $\Oaa$ matrix element, and PS
options, and the HERWIG 5.7 PS,
and generated events at the parton level.
In each case all parameters were left at their
default values \cite{22,23}, with the exception of the
JETSET parton shower parameters listed in Table V. The QCD scale
parameter values used were $\Lambda$ = 1.0 GeV ($\Oa$), 0.25 GeV
(\oalpsq), 0.26 GeV (JETSET PS) and 0.18 GeV (HERWIG PS).
The shapes of the $x_1$, $x_2$, $x_3$ and cos$\theta_{EK}$
distributions do not depend on $\Lambda$ at $\Oa$, and only weakly so
at higher order. The resulting
predictions for $x_1$, $x_2$, $x_3$ and cos$\theta_{EK}$ are
shown in Figs. 4(a) -- 7(a). These results represent Monte Carlo
integrations of the respective QCD formulae and are hence
equivalent to analytic or numerical QCD results based on the same
formulae; in the $\Oa$ case we have checked explicitly that JETSET
reproduces the numerical results of the analytic calculation
described in Section 2.
The $\Oa$ calculation describes the data reasonably well, although
small discrepancies in the details of the shapes of the distributions
are apparent and the $\chi^2$ for the comparison between data and
MC is poor (Table VI).
The $\Oaa$ calculation describes the $x_1$, $x_2$ and $x_3$ data
distributions better, but the description of the cos$\theta_{EK}$
distribution is slightly worse; this is difficult to
see directly in Figs.~4(a)--7(a), but is evident from the $\chi^2$
values for the data--MC comparisons (Table VI).
Both parton shower calculations describe the
data better than either the $\Oa$ or $\Oaa$ calculations
and yield relatively good $\chi^2$ values (Table VI).
This improvement in the quality of description of the data
between the $\Oa$ and parton shower calculations
can be interpreted as an indication of the
contribution of multiple soft gluon emission to the fine details of the
shapes of the distributions.
In fact for all calculations the largest discrepancies,
at the level of at most 10\%, arise in the regions $x_1$ $>$ 0.98,
$x_2$ $>$ 0.93, $x_3$ $<$ 0.09 and cos$\theta_{EK}$ $>$
0.9, near the boundaries of phase space where soft and collinear
divergences are expected to be large and to require resummation in
QCD perturbation theory \cite{28}; such resummation has not been
performed for the observables considered here.
For each observable we chose a range such that the
detector and hadronization correction factors are close to unity,
$0.80.09$ and
cos$\theta_{EK}<0.9$; they exclude the phase-space boundary regions.
Within these ranges the comparison between data and calculations
yields significantly improved $\chi^2$ values
(values in parentheses in Table VI); the $\Oaa$ calculation has
acceptable $\chi^2$ values and those for both parton shower models are
typically slightly better. These results
support the notion that QCD, incorporating vector gluons, is the
correct theory of strong interactions.
We now consider alternative models of strong interactions,
incorporating scalar and tensor gluons, discussed in Section 2.
Since these model calculations are at leading order in perturbation
theory we also consider
first the vector gluon (QCD) case at the same order.
The data within the selected ranges are shown in Fig.~8;
from comparison with the raw data (Fig.~3) it is apparent
that the shapes of the distributions are barely affected by the
detector and hadronization corrections. The leading-order scalar, vector
and tensor gluon predictions, normalised to the
data within the same ranges, are also shown in Fig.~8.
The vector calculation clearly
provides the best description of the data; neither the scalar nor
tensor cases predicts the correct shape for any of the observables.
The $\chi^2$ values for the comparisons are given in Table VII.
This represents the first comparison
of a tensor gluon calculation with experimental data.
It is interesting to consider whether the data allow an admixture of
contributions from the different gluon spin hypotheses.
For this purpose
we performed simultaneous fits to a linear combination of the
vector (V) + scalar (S) + tensor (T) predictions,
allowing the relative normalisations to vary according to:
\begin{eqnarray}
(1-a-b)\, V \quad + \quad a\, S\quad + \quad b \, T
\end{eqnarray}
where $a$ and $b$ are free parameters determined from the fit.
For the vector contribution we used in turn the \oalp, \oalpsq, JETSET
PS and HERWIG PS calculations. In all cases the fit to the
distribution of each observable yielded a slightly lower $\chi^2$
value than the vector-only fit.
We found that the allowed contributions of scalar and tensor
gluons depend upon the order of the vector calculation used, as well
as on the observable.
The largest allowed scalar contribution was $a$ = 0.11 from the
fit to cos$\theta_{EK}$ using the \oalpsq calculation.
The largest allowed tensor contribution was $b$ = 0.31 from the
fit to $x_1$ using the $\Oa$ calculation.
The smallest allowed contributions were
$a$ and $b$ $<$ 0.001 from the fit to $x_1$ using the HERWIG PS.
Any pair of the observables $x_1$, $x_2$, $x_3$ and cos$\theta_{EK}$
may be taken to be independent variables, subject to the overall
constraint $x_1+x_2+x_3=2$. Therefore, in order to utilise more
information, we also performed fits of Eq.~17 simultaneously to the
$x_2$ and $x_3$ distributions. We found
the relative S, V and T contributions and the
$\chi^2/d.o.f.$ values to be comparable with those from
the fits to $x_2$ alone.
%We first used the leading-order vector calculation; the relative
%contributions of V, S, and T are shown in the second rows of Tables
%VIII, IX, X, and XI for fits to $x_1$, $x_2$, $x_3$, and
%cos$\theta_{EK}$
%respectively. The resulting scalar contribution is below 0.1\%,
%except for the cos$\theta_{EK}$ distribution, where a value of 4.4\%
%is allowed. Tensor contributions of between 1.7\% (cos$\theta_{EK}$)
%and 30.5\% ($x_1$) are allowed. The $\chi^2$/d.o.f. values for these
%fits are 2.5 ($x_1$), 3.1 ($x_2$), 3.1 ($x_3$), and 1.0
%(cos$\theta_{EK}$).
%This exercise was then repeated using in turn
%for the vector case the JETSET $\Oaa$, JETSET PS, and HERWIG PS
%calculations; the results are shown in the third, fourth, and fifth
%rows, respectively, of Tables VIII--XI. The allowed scalar and
%tensor contributions can be seen to vary considerably depending on
%which vector calculation is used and which observable is fitted.
%The largest scalar contribution
%(10.8\%) occurs for the $\Oaa$ vector fit to cos$\theta_{EK}$, and the
%largest tensor contribution (15.7\%) occurs for the JETSET PS vector
%fit to $x_2$. For all four observables the best fits (lowest $\chi^2$)
%were obtained when either of the
%vector parton shower calculations was used; the $\chi^2$ values for
%these fits are slightly better than the $\chi^2$ values for the
%corresponding comparisons with vector parton showers only (Table VI).