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% My definitions ...
%%%%%%%%%%%%%   my stuff
 
\renewcommand{\thefootnote}{\fnsymbol{footnote}}  % symbols for footnotes
 
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\renewcommand{\Pqt}{\ensuremath{\mathrm{t}}}%        t replace q_t in pennames
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\def\simgr{^>\hskip -2.5mm_\sim}               %  greater than or approx equal
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%%%%%%%%%%%%%  end my stuff
 
 
\begin{document}
\begin{titlepage}
%
\title{Measurement     of Bose-Einstein Correlations \\
       and Reconstruction of the Source Function     \\
       in Hadronic Z-boson Decays} %                 \\
%      using the L3 Detector}
 
\author{ The L3 Collaboration
\\[4mm] prepared by\\[2mm]
\renewcommand{\thefootnote}{\fnsymbol{footnote}}  % symbols for footnotes
  T. Nov\'ak,\footnotemark[1]
  T. Cs\"org\H{o},\footnotemark[3]
  W. Kittel ,
  W.J. Metzger
\\[3mm]
\renewcommand{\thefootnote}{\fnsymbol{footnote}}  % symbols for footnotes
\enote{Add to L3 author list: R. Hakobyan, T. Cs\"org\H{o}\/ \footnotemark[3]}
}
%
% The abstract
 
\begin{abstract}
\vspace{-5mm}
Bose-Einstein correlations of pairs of identical charged pions
produced in hadronic Z decays are analyzed in terms of various parametrizations.
A good description is achieved using a L\'evy stable distribution in conjunction with a
hadronization model where a particle's momentum is highly correlated with its space-time point of
production, the \taumodel.
Using these results, the time evolution of particle emission in two-jet events is reconstructed.
\end{abstract}
 
%\submitted                   % if version to be submitted
 
\footnotetext[1]{Present address: Dept.\ of Business Mathematics and Informatics, K\'aroly R\'obert College, H-3200 Gy\"ongy\"os, Hungary}
\footnotetext[3]{Visitor from MTA KFKI RMKI, H-1525 Budapest 114, Hungary,
sponsored by the Scientific Exchange between Hungary (OTKA) and The Netherlands (NWO),
%project B64-27/N25186.}
project B64-27/N25186;  also supported by Hungarian OTKA grants T49466 and NK73143 and a HAESF Senior Leaders and Scholars Fellowship.}
\end{titlepage}
 
 
\normalsize
\pagenumbering{roman}  \setcounter{page}{2}
%\tableofcontents
%\clearpage
%\listoftables
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%\listoffigures
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\pagenumbering{arabic}
 
\setcounter{page}{1}
 
 
\section{Introduction}\label{sect:intr}
\enote{All figures are working versions and need to be made better for the publication.}
 
In particle and nuclear physics, intensity interferometry provides a direct
experimental method for the determination of sizes, shapes and lifetimes
of particle-emitting sources
(for reviews see ~\cite{Gyulassy:1979,Boal:1990,Baym:1998,Wolfram:Zako2001,Tamas:HIP2002}).
In particular, boson interferometry provides a powerful tool for the
investigation of the space-time structure of particle production processes,
since Bose-Einstein correlations (BEC) of two identical bosons reflect both
geometrical and dynamical properties of the particle radiating source.
 
In \Pep\Pem\ annihilation BEC have been observed\cite{TASSO:1986}
to be maximal when the invariant momentum difference of the bosons,
$Q=\sqrt{-(p_1-p_2)^2}$, is small,
even when one of the relative momentum components is large.
This is not the case in hadron-hadron interactions \cite{NA22q}
or in heavy-ion interactions \cite{Solz:thesis,Solz:paper},
where BEC are found not to depend simply on $Q$, but to
decrease when any of the relative momentum components is large,
a behavior that can be described by hydrodynamical models of the source
\cite{Tamas;Lorstad:1996,Tamas:HIP2002}.
 
The size (radius) of the source in heavy-ion collisions
has been found to decrease with increasing transverse momentum, \pt,
or transverse mass, $\mt=\sqrt{m^2+\pt^2}$, % =\sqrt{E^2-p_z^2}$,
of the bosons.  This effect can also be explained by hydrodynamical models\cite{Wiedemann:96}.
A similar effect has been seen in pp collisions\cite{STAR:QM05}, as well as
in \Pep\Pem\ annihilation~\cite{Smirnova:Nijm96,Dalen:Maha98,OPAL:2007}.
 
 
 
A model for BEC in \Pep\Pem\ annihilation, known as the \taumodel,
which results in BEC depending simply on $Q$, rather than on its components separately,
has been proposed\cite{Tamas;Zimanji:1990}.
In this model the simple $Q$ dependence is a consequence of a strong correlation
between a particle's four-momentum and its space-time point of production.
This strong correlation also leads to a decrease of the observed source size
with increasing \mt\cite{ourTauModel}.
 
On the other hand, at  LEP studies of BEC in \Pep\Pem\ annihilation have found that the source
volume is not spherical, as indicated by a dependence only on $Q$, but rather is elongated along the event
axis~\cite{L3_3D:1999,OPAL3D:2000,DELPHI2D:2000,ALEPH:2004,OPAL:2007}.  At HERA a similar elongation is
observed in neutral current deep inelastic ep scattering~\cite{ZEUS2D:2004}.
 
It is the purpose of this paper to study to what extent BEC in \Pep\Pem\ annihilation depend only on $Q$
and to test the \taumodel.
%
Here we study BEC in hadronic \PZ\ decay.
We investigate various static parametrizations in terms of $Q$
%the four-momentum difference, $Q=\sqrt{-(p_1-p_2)^2}$
and find that none give an adequate description of the Bose-Einstein correlation function.
%However, within the framework of models assuming
%strongly correlated coordinate and momentum space a good description is achieved
Next we investigate the \taumodel\ and find that it provides a good description of BEC,
both for two-jet and three-jet events.
The description becomes even better when elongation of the source region is added \textit{ad hoc} to the
model.
Further, the results for two-jet events,
from fits in terms of $Q$ and the transverse masses of the two pions with respect to the event axis,
are used to reconstruct the complete
space-time picture of the particle emitting source of two-jet events in hadronic Z decay.
 
 
 
 
 
 
\section{Analysis}                     \label{sect:anal}
\subsection{Bose-Einstein Correlation Function}
%\section{Bose-Einstein Correlation Function}
The two-particle correlation function of two particles with
four-momenta $p_{1}$ and $p_{2}$ is given by the ratio of the two-particle number density,
$\rho_2(p_{1},p_{2})$,
to the product of the two single-particle number densities, $\rho_1 (p_{1})\rho_1 (p_{2})$.
Since we are here interested only in the correlation $R_2$ due to Bose-Einstein
interference, the product of single-particle densities is replaced by
$\rho_0(p_1,p_2)$,
the two-particle density that would occur in the absence of Bose-Einstein correlations:
 
\begin{equation} \label{eq:R2def}
  R_2(p_1,p_2)=\frac{\rho_2(p_1,p_2)}{\rho_0(p_1,p_2)} \;.
\end{equation}
This $\rho_2$ is corrected for detector acceptance and efficiency using Monte Carlo events
generated by the \JETSET\  Monte Carlo generator~\cite{JETSET74}
with the so-called \BEz\ simulation of BEC~\cite{BE0}
to which a full detector simulation has been applied \cite{detsim}, by multiplying $\rho_2$, on a bin-by-bin
basis, by the ratio of $\rho_2$ of the generated events to
$\rho_2$ of the generated events after detector simulation.
For the detector-simulated events, as for the data, all charged tracks are used,
whereas for the generator-level events only charged pions are used.
 
An event mixing technique is used to construct $\rho_0$, whereby all tracks of each data
event are replaced by  tracks from  different events having similar multiplicity to the original event.
$\rho_0$ is corrected for detector acceptance and efficiency in the same way as $\rho_2$.
The mixing technique removes all correlations, \eg, resonances and energy-momentum conservation,
not just Bose-Einstein correlations.
Hence, $\rho_0$  is also corrected for this  % ~\cite{tamas:thesis,L3_3D:1999}
by a multiplicative factor which is the ratio of the densities of events to mixed events found using
events generated by \JETSET\ without BEC simulation.
Thus $R_2$ is measured by
\begin{equation} \label{eq:R2cor}
  R_2 = \left(R_\textrm{2\;data} R_\textrm{2\;gen}\right) / \left(R_\textrm{2\;det} R_\textrm{2\;gen-noBE}\right) \;,
\end{equation}
where data, gen, det, gen-noBE refer, respectively, to the data sample, a generator-level Monte Carlo sample,
the same Monte Carlo sample passed through detector simulation and subjected to the same selection procedure as the data,
and a generator-level sample of a Monte Carlo without BEC simulation.
 
%\section{Data} \label{sect:data}
\subsection{Data} \label{sect:data}
%\subsection{Event and Charged-Particle Selection}
The data used in the analysis were collected by the
\Lthree\ detector~\cite{l3:construction,L3:Ecal_calib,L3:Hcal,L3:TEC,l3:SMD}
at an \Pep\Pem\    center-of-mass energy of $\sqrt{s}\simeq 91.2$ \GeV.
Approximately 36 million like-sign pairs of well-measured charged tracks of about 0.8 million
hadronic Z decays are used \cite{tamas:thesis}.
 
%We perform analyses on the complete sample as well as  on two- and three-jet
%We perform analyses on both two- and three-jet samples, which are found using calorimeter clusters with the
Events are classified as two- or three-jet events on the basis of the
Durham jet algorithm~\cite{durham,durham2,durham3}    % ~\cite{durham,durham1,durham2,durham3}
with a jet resolution parameter $\ycut=0.006$, yielding about 0.5 million 2-jet and 0.3 million 3-jet events.
Since there are few events with more than three jets, these events are included in the three-jet sample.
To determine the thrust axis of the event we also use calorimeter clusters.
 
 
\section{Parametrizations of BEC}              \label{sect:param}
 
%Since the mass of the two identical particles of the pair is fixed to the pion mass,
Since the four-momenta of the two identical particles is on the mass-shell,
the correlation function is defined in six-dimensional momentum space.
Since Bose-Einstein correlations can be large only at small four-momentum difference
$Q$,
% $Q=\sqrt{-(p_1-p_2)^2}$,
they are often parametrized in this one-dimensional distance measure.
With a few assumptions~\cite{GGLP:1960,Boal:1990,Tamas:HIP2002},
the two-particle correlation function, \Eq{eq:R2def},
is related to the Fourier transformed source distribution:
 
\begin{equation}  \label{eq:R2fourier}
%     R_2(p_1,p_2) = \gamma \left[ 1 + \lambda |\tilde{f} (Q)|^2 \right]
     R_2(Q)       = \gamma \left[ 1 + \lambda \abs{\tilde{f} (Q)}^2 \right]
                    \left(1 + \epsilon Q \right) \;,
\end{equation}
where $f(x)$ is the (space-time) density distribution of the source,
% $Q$ is the invariant four-momentum difference, $Q=\sqrt{-(p_1-p_2)^2}$
 and
 $\tilde{f}(Q)$ is the Fourier transform of $f(x)$.
The parameter $\lambda$ is introduced as a measure of the strength of the correlation.
It may be less than unity for a variety of reasons such as inclusion of misidentified non-identical particles
% the possible lack of complete incoherence of particle production
or the presence of long-lived resonance decays if the particle emission
consists of a small, resolvable core and a halo with experimentally unresolvable large length scales
\cite{Bolz:1992hc,Csorgo:1994in}.
The parameter $\gamma$ and the $(1 + \epsilon Q)$ term parametrize possible long-range correlations not
adequately accounted for in the reference sample.  While there is no guarantee that $(1 + \epsilon Q)$
is the correct form, we will see that it does provide a good description of $R_2$ in the region $Q>1.5\GeV$.
 
 
There is no reason, however,
to expect the hadron source to be spherically symmetric in jet fragmentation.
Recent investigations have, in fact, found an elongation of the source along the
jet axis~\cite{L3_3D:1999,OPAL3D:2000,DELPHI2D:2000,ALEPH:2004,OPAL:2007}.
While this effect is      well established, the elongation is actually only about 20\%,
which suggests that a parametrization in terms of the single variable $Q$,
may be a good approximation.
 
This is not the case in heavy-ion and hadron-hadron interactions, where BEC are found not to depend simply
on $Q$, but on components of the four-momentum difference separately
\cite{Tamas:HIP2002, Tamas;Lorstad:1996, NA22q,Solz:thesis,Solz:paper,STAR:QM05}.
However, in \Pep\Pem\ annihilation at lower energy~\cite{TASSO:1986} it has been observed that $Q$ is the
appropriate variable.
We checked this and confirm \cite{tamas:thesis} that this is indeed the case, both for all and for two-jet events:
We observe that $R_2$ does not decrease when both $q^2=(\vec{p}_1-\vec{p}_2)^2$ and
$q_0^2=(E_1-E_2)^2$ are large while $Q^2=q^2-q_0^2$ is small, but is maximal for
$Q^2=q^2-q_0^2=0$,
independent of the individual values of $q$ and $q_0$.
Furthermore, two-dimensional fits with $\abs{\tilde{f} (Q)}^2$ parametrized as $\exp \left((rq)^2-(r_0q_0)^2\right)$
find $r$ and $r_0$ to be equal.
The same is observed in a different decomposition: $Q^2=Q_\mathrm{t}^2 + Q_{L,B}^2$, where
$Q_\mathrm{t}^2=(\vec{p}_\mathrm{t1}-\vec{p}_\mathrm{t2})^2$ is the component transverse to the thrust axis
and
$Q_{L,B}^2=(p_\mathrm{l1}-p_\mathrm{l2})^2-(E_1-E_2)^2$ combines the longitudinal momentum and energy differences.
Again, $R_2$ is maximal along the line $Q=0$, as is shown in \Fig{fig:qinv}, and fits find equal radii.
 
We conclude that a parametrization in terms of  $Q$
can be considered a good approximation for the purposes of this \Letter.
We shall return to the question of elongation at the end of this \Letter.
 
 
\begin{figure}
  \centering
%  \includegraphics[width=\figwidth]{\olddirfig q2vsq02_2j.eps}
%  \includegraphics[width=.75\figwidth]{\mydirfig ql_vs_qt_2j_rot.eps}
  \includegraphics[width=.33\figwidth]{\mydirfig ql_vs_qt_2j_bowler.eps}
  \includegraphics[width=.33\figwidth]{\mydirfig ql_vs_qt_2j_bowler_box.eps}
  \includegraphics[width=.33\figwidth]{\mydirfig diag_bowler_2j.eps}
   \caption{(a,b) $R_2$ for two-jet events as function of the squares of the transverse momentum difference,
              $Q_\mathrm{t}^2=(\vec{p}_\mathrm{t1}-\vec{p}_\mathrm{t2})^2$,
                and the combination of longitudinal momentum difference and energy difference,
\enote{Choose between (a) and (b).}
              $Q_\mathrm{L,B}^2=(p_\mathrm{l1}-p_\mathrm{l2})^2-(E_1-E_2)^2$.
            (c)  $R_2$ \vs\ $Q_\mathrm{t}^2$ when $Q_\mathrm{t}^2\approx -Q_\mathrm{L,B}^2$.
%  \caption{For two-jet events:
%           (a) $R_2$ as function of the squares of the 3-momentum difference, $q$,
%               and the energy difference, $q_0$;
%           (b) $R_2$ \vs\ $q^2$ when $q^2\approx q^2_0$;
%      and  (c) $R_2$ as function of the squares of the transverse momentum difference
%               and the combination of longitudinal momentum difference and energy difference.
           \label{fig:qinv}
%\enote{Perhaps a scatter plot would look better.  It is being prepared.}
\enote{Probably (c) is not needed.}
           }
%%%%%%%%    }           ADDING THIS LINE PREVENTS segmentation fault in latexdiff
\end{figure}
 
 
 
 
 
%\begin{equation}
%  \tilde{f} (Q) = \int \mathrm{d} x \exp (iQx) f(x).
%\end{equation}
 
\subsection{Symmetric parametrizations} \label{sect:sym_param}
The simplest assumption is that the source, in the rest frame of a pion pair, has a symmetric Gaussian
distribution.  In this case
$\tilde{f} (Q)=\exp \left(i\mu Q -\frac{(RQ)^2}{2}\right)$ and
 
\begin{equation} \label{eq:gaussR2}
  R_2(Q) = \gamma \left[1 + \lambda \exp \left(-(R Q)^2 \right) \right]
           \left(1 + \epsilon Q \right) \;.
\end{equation}
 
Fits of \Eq{eq:gaussR2} to the data (not shown \enote{That is the intention---for discussion the fits are included: \Fig{fig:gauss}})
result in unacceptably low confidence levels: $10^{-15}$ ($\chisq=246$ for 96 degrees of freedom) for 2-jet and much worse ($\chisq=456$) for 3-jet events.
The fit is particularly bad at low values of $Q$,
where both for two- and three-jet events $R_2$ is much steeper than a Gaussian.
%as is shown in \Fig{fig:gauss_2jet}a for two-jet
%events and in \Fig{fig:gauss_3jet}a for three-jet events,
%from which we conclude that the shape of the source deviates from a Gaussian.
 
\begin{sidewaysfigure}
  \centering
  \includegraphics[width=.45\textwidth]{\mydirfig run_fitQ1G4.ps}
  \includegraphics[width=.45\textwidth]{\mydirfig run_fitQ1G4_3j.ps}
  \caption{The Bose-Einstein correlation function $R_2$ for (a) two- and (b) three-jet events with the
            results of fits of
            the Gaussian parametrization, \Eq{eq:gaussR2}.
           The dashed line represents the long-range part of the fit, \ie, $\gamma(1+\epsilon Q)$.
\enote{This figure is included for discussion --- not to be in paper.}
           \label{fig:gauss}
           }
\end{sidewaysfigure}
 
 
\subsubsection{Edgeworth parametrization} \label{sect:edgeworth_param}
A model-independent way to study deviations from the Gaussian parametrization is to
use~\cite{Tamas:Moriond28,Tamas:Cracow1994,Tamas:HIP2002}
the Edgeworth expansion~\cite{Edgeworth} about a Gaussian. %, for which a Gaussian is the lowest-order term.
Keeping only the first non-Gaussian term, we have
 
\begin{equation} \label{eq:edgeworthR2}
  R_2(Q) = \gamma \left(1 + \lambda \exp \left(-(R Q)^2 \right)
                  \left[1+\frac{\kappa}{3!}H_{3}(RQ)\right]
                  \right)
           \left(1 + \epsilon Q \right) \;,
\end{equation}
where
$\kappa$ is the third-order cumulant moment and
$H_{3}(RQ)\equiv (\sqrt{2}RQ)^{3}-3\sqrt{2}RQ$
is the third-order Hermite polynomial.
Note that the second-order cumulant corresponds to the radius~$R$.
\Eq{eq:edgeworthR2} reduces to \Eq{eq:gaussR2} for $\kappa=0$.
 
A fit of \Eq{eq:edgeworthR2} to the two-jet data, shown in \Fig{fig:gauss_2jet}a,
finds $\kappa=0.74\pm0.07$, about 10 standard deviations from the Gaussian value. %  of zero.
The confidence level is indeed
much better than that of the purely Gaussian fit, but is still poor, approximately $10^{-5}$.
Close inspection of the figure shows that the fit curve is systematically above the data in the region
0.6--1.2\,\GeV\ and that the data for $Q\ge1.5$\,\GeV\ appear flatter than the curve, as is also the case
for the purely Gaussian fit.
Similar behavior is observed for three-jet events
(\Fig{fig:gauss_3jet}a).
%and for all events.
%(\Fig{fig:gauss_all}b).
The inadequacy of this parametrization had gone unnoticed in our previous analysis~\cite{L3_3pi:2002}
where the fit was confined to $Q<1.4\GeV$.
 
 
\begin{sidewaysfigure}
  \centering
%  \includegraphics[width=.75\figwidth]{\olddirfig q_inv_gauss100_linear.eps}
%  \includegraphics[width=.75\figwidth]{\olddirfig q_inv_edge100_linear.eps}
  \includegraphics[width=.45\textwidth]{\mydirfig run_fitQ1E4.ps}
  \includegraphics[width=.45\textwidth]{\mydirfig run_fitQ1SL4.ps}
  \caption{The Bose-Einstein correlation function $R_2$ for two-jet events with the
results of fits of
%           (a) the Gaussian and (b) the Edgeworth parametrizations,
%           \Eqs{eq:gaussR2} and (\ref{eq:edgeworthR2}), respectively.
            (a)                        the Edgeworth, \Eq{eq:edgeworthR2}, and
            (b)                        the symmetric L\'evy, \Eq{eq:symlevR2}, parametrizations.
           The dashed line represents the long-range part of the fit, \ie, $\gamma(1+\epsilon Q)$.
\enote{$\Delta =$ data -- fit; $\sigma=\Delta/$ uncertainty.}
\enote{Perhaps we do not want to show $\Delta$ and/or $\sigma$ --- also in other figures of fits.}
\enote{Add dotted line representing fit with $\lambda=0$ to $Q>1.5$.}
           \label{fig:gauss_2jet}
           }
\end{sidewaysfigure}
 
 
\subsubsection{L\'evy distributed source}         \label{sect:Levy}
%Adopting Nolan's $S(\alpha, \beta=0, \gamma, \delta; 1)$ convention~\cite{Nolan} for the symmetric
%L\'evy stable distribution with rescaling of the scale parameter $\gamma$ to $R$ and
%the location parameter $\delta$ to $x_0$, the Fourier transform (characteristic function)
Another way to depart from the assumption of a Gaussian source is to replace the Gaussian by a
symmetric L\'evy stable distribution, which
is characterized by three parameters: $x_0$, $R$, and $\alpha$.
Its Fourier transform,
$\tilde{f}(Q)$, has the following form:
 
\begin{equation}
% \tilde{f}(Q) = \exp (iQ x_0 - |R Q|^\alpha) \;.
  \tilde{f}(Q) = \exp \left(iQ x_0 - \frac{|R Q|^\alpha}{2} \right) \;,
\end{equation}
where the index of stability, $\alpha$, satisfies the inequality $0<\alpha \leq 2$.
The case $\alpha=2$ corresponds to a Gaussian source distribution with mean $x_0$ and standard deviation $R$.
For more details, see, \eg, \cite{Nolan}.
 
Then $R_2$ has the following, relatively simple, form~\cite{Tamas:Levy2004}:
 
\begin{equation}\label{eq:symlevR2}
    R_2(Q) = \gamma \left[ 1+ \lambda \exp \left(-(RQ)^\alpha \right) \right]
             (1+ \epsilon Q) \;.
\end{equation}
The fit of \Eq{eq:symlevR2} to the two-jet data, shown in \Fig{fig:gauss_2jet}b, finds $\alpha = 1.44\pm0.06$,
far from the Gaussian value of 2.
The confidence level, although
improved compared to the fit of \Eq{eq:gaussR2} % and somewhat better than the fit of \Eq{eq:edgeworthR2},
is still unacceptably low, approximately $10^{-8}$.
Similar behavior is seen for three-jet events (\Fig{fig:gauss_3jet}b).
%and for all events.
%(\Fig{fig:sym_levy_all}).
%\mnote{?}\enote{I don't think we need these values in the actual paper.}
 
 
\begin{sidewaysfigure}
%\begin{figure}
  \centering
%  \includegraphics[width=.75\figwidth]{\olddirfig q_inv_gauss100_linear_3jet.eps}
%  \includegraphics[width=.75\figwidth]{\olddirfig q_inv_edge100_linear_3j.eps}
  \includegraphics[width=.45\textwidth]{\mydirfig run_fitQ1E4_3j.ps}
  \includegraphics[width=.45\textwidth]{\mydirfig run_fitQ1SL4_3j.ps}
  \caption{The Bose-Einstein correlation function $R_2$ for three-jet events with the
           results of fits of
            (a)                        the Edgeworth, \Eq{eq:edgeworthR2}, and
            (b)                        the symmetric L\'evy, \Eq{eq:symlevR2}, parametrizations.
           The dashed line represents the long-range part of the fit, \ie, $\gamma(1+\epsilon Q)$.
           \label{fig:gauss_3jet}
           }
\end{sidewaysfigure}
%\end{figure}
 
 
%\begin{figure}
%  \centering
%  \includegraphics[width=.75\figwidth]{\olddirfig q_inv_levy100_linear.eps}
%  \caption{The Bose-Einstein correlation function $R_2$ for two-jet events. The curve
%           corresponds to the fit of the symmetric L\'evy parametrization, \Eq{eq:symlevR2}.
%           The dashed line represents the long-range part of the fit, \ie, $\gamma(1+\epsilon Q)$.
%           The dot-dashed line represents a linear fit in the region $Q>1.5$\,\GeV.
%           \label{fig:sym_levy_2jet}
%           }
%\end{figure}
%
%\begin{figure}
%  \centering
%  \includegraphics[width=.75\figwidth]{\olddirfig q_inv_levy100_linear_3jet.eps}
%  \caption{The Bose-Einstein correlation function $R_2$ for three-jet events. The curve
%           corresponds to the fit of the symmetric L\'evy parametrization, \Eq{eq:symlevR2}.
%           The dashed line represents the long-range part of the fit, \ie, $\gamma(1+\epsilon Q)$.
%           The dot-dashed line represents a linear fit in the region $Q>1.5$\,\GeV.
%           \label{fig:sym_levy_3jet}
%           }
%\end{figure}
 
Both the symmetric L\'evy parametrization and the Edgeworth parametrization  do a fair job of describing the
region $Q<0.6\,\GeV$, but fail at higher $Q$.  $R_2$ in the region $Q\ge1.5\GeV$ is nearly constant
($\approx1$). However, in the region 0.6--1.5\,\GeV\ $R_2$ has a smaller value,
dipping below unity~\footnote{More correctly, dipping below the value of $\gamma(1+\epsilon Q)$.},
which is indicative of an anti-correlation.
This is clearly seen
%in \Figs{fig:sym_levy_2jet} and~\ref{fig:sym_levy_3jet}
in \Figs{fig:gauss_2jet} and~\ref{fig:gauss_3jet}
by comparing the data in this region  to an extrapolation of a linear fit, \Eq{eq:symlevR2} with
$\lambda=0$, in the region $Q\ge1.5\,\GeV$.
%which is indicated by the dashed line
The inability to describe this dip in $R_2$ is the primary reason for the failure of both the
Edgeworth and symmetric L\'evy parametrizations.
 
 
\subsection{Time dependence of the source}
The parametrizations discussed so far, which have proved insufficient to describe the BEC, all assume a
static source.  The parameter $R$, representing the size of the source as seen in the rest frame of the
pion pair, is a constant.  It has, however, been observed that $R$ depends on the transverse mass,
$\mt=\sqrt{m^2+\pt^2}=\sqrt{E^2-p_z^2}$, of the pions~\cite{Smirnova:Nijm96,Dalen:Maha98,OPAL:2007}.
It has been shown~\cite{Bialas:1999,Bialas:2000} that this dependence can be understood if the produced
pions satisfy, approximately, the (generalized) Bjorken-Gottfried
condition~\cite{Gottfried:1972,Bjorken:SLAC73,Bjorken:1973,Gottfried:1974,Low:1978,Bjorken:ISMD94}, whereby
the four-momentum of a produced particle and the space-time position at which it is produced are linearly
related.
%related:
%
%\begin{equation} \label{eq:GB-corr}
%   x^\mu = d p^\mu \;.
%\end{equation}
Such a correlation between space-time and momentum-energy is  also a feature of the Lund string model,
which, incorporated in \JETSET,
is very successful in describing detailed features of the hadronic final states of \Pep\Pem\ annihilation.
 
 
In the previous section we have seen that BEC depend, at least approximately, only on $Q$ and not on
its components separately.
This is a non-trivial result.
For a hydrodynamical type of source,
on the contrary, BEC decrease when any of the relative momentum components is large~\cite{Tamas:HIP2002, Tamas;Lorstad:1996}.
%
Further, we have seen that $R_2$ in the region 0.6--1.5\,\GeV\ dips below its values at higher $Q$.
 
A model which predicts such $Q$-dependence while incorporating the  Bjorken-Gottfried condition
is the so-called \taumodel, described below.
 
\subsubsection{The \boldmath{$\tau$}-model}  \label{sect:taumodel}
A model of strongly correlated phase-space, known as the \taumodel~\cite{Tamas;Zimanji:1990},
predicts that $R_2$ should depend on $Q$, not on its components separately.
%explains the experimentally found invariant relative momentum
%dependence of Bose-Einstein correlations in $\mathrm{e}^+\mathrm{e}^-$ reactions.
This model also predicts a specific transverse mass dependence of $R_2$,
which we subject to an experimental test here.
 
In this model, it is assumed that the average production point in the overall center-of-mass system,
$\overline{x}=(\overline{t},\overline{r}_x,\overline{r}_y,\overline{r}_z)$, of particles with a given
four-momentum $p$ is given by
 
\begin{equation} \label{eq:tau-corr}
   \overline{x}^\mu (p^\mu)  = a\tau p^\mu \;.
\end{equation}
In the case of two-jet events,
%$\tau=\sqrt{\overline{t}^2-\overline{r}_z^2}$ is the longitudinal proper time and
  $a=1/\mt$
%\begin{equation} \label{eq:d}
%   d=\tau/\mt \;,  %   d = \frac{\tau}{\mt} \;,
%\end{equation}
where
% $\mt=\sqrt{m^2+\pt^2}=\sqrt{E^2-p_z^2}$ is the transverse mass
\mt\ is the transverse mass
and
$\tau = \sqrt{\overline{t}^2 - \overline{r}_z^2}$ is the longitudinal proper time~\footnote{The
terminology `longitudinal' proper time and `transverse' mass seems customary in the literature
even though their definitions are analogous $\tau = \sqrt{\overline{t}^2 - \overline{r}_z^2}$
and                                         $ \mt = \sqrt{E^2            - p_z^2}$.}.
For isotropically distributed particle production, the transverse mass is replaced by the
mass in the definition of $a$ and $\tau$ is the proper time.  %in \Eq{eq:d}, while
In the case of three-jet events the relation is more complicated.
%\mnote{!}\enote{Perhaps phrase this differently.}
 
The second assumption is that the distribution of $x^\mu (p^\mu)$ about its average,
$\delta_\Delta ( x^\mu(p^\mu) - \overline{x}^\mu (p^\mu) )$, is narrower than the
proper-time distribution, $H(\tau)$.
%Then the emission function of the \taumodel\ is
%\begin{equation} \label{eq:source}
%%  S(x,k) = \int_0^{\infty} \mathrm{d}\tau H(\tau)\delta_{\Delta}(x-dk) \rho_1(k) \;,
%  S(x,k) = \int_0^{\infty} \mathrm{d}\tau H(\tau)\delta_{\Delta}(x-a\tau k) \rho_1(k) \;,
%\end{equation}
%where $H(\tau)$ is the (longitudinal) proper-time distribution, the factor
%%$\delta_{\Delta}(x-dk)$ describes the strength of the correlations between
%$\delta_{\Delta}(x-a\tau k)$ describes the strength of the correlations between
%coordinate space and momentum space variables and $\rho_1(k)$ is the experimentally
%measurable single-particle spectrum.
%
%The two-pion distribution, $\rho_2(k_1,k_2)$, is related to $S(x,k)$, in the plane-wave approximation,
%by the Yano-Koonin formula~\cite{Yano}:
%%In the plane-wave approximation, the Yano-Koonin formula~\cite{Yano} gives the following two-pion distribution:
%\begin{equation}  \label{eq:yano}
%   \rho_2(k_1,k_2) = \int \mathrm{d}^4 x_1 \mathrm{d}^4 x_2 S(x_1,k_1) S(x_2,k_2) %\nonumber \\
%                     \left(\strut 1 + \cos\left(\strut\left[k_1-k_2\right]\left[x_1-x_2\right]\right)\strut\right)\;. \end{equation}
%Approximating the function $\delta_\Delta$ by a Dirac delta function,
%\Eq{eq:source} becomes
%\begin{equation} \label{eq:S}
%%  S(x,k) = \int_0^{\infty} \mathrm{d}\tau H(\tau)\delta_{\Delta}(x-dk) \rho_1(k) \;,
%  S(x) = \int_0^{\infty} \mathrm{d}\tau H(\tau)  \rho_1\left(\frac{x}{a\tau}\right) \;,
%\end{equation}
%and the argument of the cosine in \Eq{eq:yano} becomes
%\begin{equation} \label{eq:cos}
%   (k_1 - k_2)(\bar{x}_1 - \bar{x}_2) = - 0.5 (a_1\tau_1 + a_2\tau_2) Q^2 \;.
%\end{equation}
%%where the subscripts 1 and 2 refer to the two pions of the pair.
%Substituting \Eqs{eq:S} and (\ref{eq:cos}) in \Eq{eq:yano} leads to the following approximation of
%the two-particle Bose-Einstein correlation function:
Then   the two-particle Bose-Einstein correlation function is indeed found to depend on the invariant
relative mometum $Q$, rather than on its separate components, as well as on the values of $a$ of the two particles\cite{ourTauModel}:
 
\begin{equation} \label{eq:levyR2}
%   R_2(k_1,k_2) = 1 + \lambda \mathrm{Re} \widetilde{H}\left(\frac{a_1 Q^2}{2}\right)
   R_2(p_1,p_2) = 1 +         \mathrm{Re} \widetilde{H}\left(\frac{a_1 Q^2}{2}\right)
                                          \widetilde{H}\left(\frac{a_2 Q^2}{2}\right) \;,
\end{equation}
where $\widetilde{H}(\omega) = \int \mathrm{d} \tau H(\tau) \exp(i \omega \tau)$
is the Fourier transform (characteristic function) of $H(\tau)$.
Note that $H(\tau)$ is normalized to unity.
%This formula simplifies further if $R_2$ is measured with the restriction
%\begin{equation} \label{eq:a1a2equal}
%     a_1\approx a_2\approx \bar{a} \;.
%\end{equation}
%In that case,
%\begin{equation}     \label{eq:levyR2a}
%   R_2(k_1,k_2) = 1 + \lambda \mathrm{Re} \widetilde{H}^2
%                  \left(  \frac{\bar{a}Q^2}{2} \right) \;,
%\end{equation}
%which for two-jet events becomes
%\begin{equation}     \label{eq:levy2jetR2a}
%   R_2(k_1,k_2) = 1 + \lambda \mathrm{Re} \widetilde{H}^2
%                  \left(  \frac{Q^2}{2 \overline{m}_{\mathrm{t}}} \right) \;,
%\end{equation}
%%Thus for a given average of $a$ of the two particles, $R_2$  is found to
%%depend only on the invariant relative momentum $Q$.
%%Further, the model predicts a specific dependence on $\bar{a}$, which for two-jet events is
%%a specific dependence on  $\overline{m}_\mathrm{t}$.\footnote{In the initial formulation of the \taumodel\
%%this dependence was averaged over \cite{Tamas;Zimanji:1990} due to the lack of \mt\ dependent investigations
%%at that time.}
 
 
Since there is no particle production before the onset of the collision,
$H(\tau)$ should be a  one-sided distribution.
Choosing \cite{ourTauModel} a one-sided L\'evy distribution, which has the characteristic function~\cite{Tamas:Levy2004}
(for $\alpha\ne1$)
 
\begin{equation} \label{eq:levy1sidecharf}
   \widetilde{H}(\omega) = \exp\left[ -\frac{1}{2}\left(\Delta\tau|\omega|\strut\right)^\alpha
          \left( 1 -  i\, \mathrm{sign}(\omega) \tan\left(\frac{\alpha\pi}{2}\right) \strut \right)
       + i\,\omega\tau_0\right]
%                                  \;,\quad \alpha\ne1   \;,
 \; .
\end{equation}
where the parameter $\tau_0$ is the proper time of the onset of particle production
and $\Delta \tau$ is a measure of the width of the proper-time distribution.
For the special case $\alpha=1$, see, \eg, \cite{Nolan}.
Using this characteristic function in \Eq{eq:levyR2} yields
 
\begin{eqnarray} \label{eq:levyR2a}
                                                 \label{eq:levyR2mt}  % also this label if we don't include it as a separate equation
   R_2(Q,a_1,a_2) &=& 1 +
      \cos\left[\frac{\tau_0 Q^2 (a_1+a_2)}{2} +
\tan\left(\frac{\alpha\pi}{2}\right)\left(\frac{\Delta\tau {Q^2}}{2}\right)^{\!\alpha}\frac{a_1^\alpha+a_2^\alpha}{2} \right]  \nonumber \\
    & & \cdot           \exp \left[-\left(\frac{\Delta\tau {Q^2}}{2}\right)^{\!\alpha}\frac{a_1^\alpha+a_2^\alpha}{2} \right]
 \; .
\end{eqnarray}
%Using this characteristic function in \Eq{eq:levy2jetR2a} yields
%\begin{eqnarray} \label{eq:levy2jetR2}
%   R_2(Q,\mtbar) &=&
%       \gamma
%       \left[ 1+\lambda \cos \left( \frac{\tau_0 Q^2}{\mtbar}
%                                   + \tan \left( \frac{\alpha \pi}{2} \right)
%                                     \left( \frac{\Delta\tau Q^2}{2\mtbar}\right)^{\!\alpha} \right)
%   \right. \nonumber \\
%   & & \cdot\left.      \exp \left( -\left( \frac{\Delta\tau Q^2}{2\mtbar} \right)^{\!\alpha} \right)
%        \right] (1+\epsilon Q)   \;.
%\end{eqnarray}
 
 
%\subsubsection{The \boldmath{$\tau$} model for average \boldmath{$\mt$}}  \label{sect:tausimpl}
\subsubsection{The \boldmath{$\tau$}-model for average \boldmath{$a$}}    \label{sect:tausimpl}
%Before proceeding to fits of \Eq{eq:levy2jetR2},
Before proceeding to fits of \Eq{eq:levyR2a},
we first consider a simplification of the equation
obtained by assuming (a) that particle production starts immediately, \ie, $\tau_0=0$,
%and (b) an average $a$-dependence, which is implemented in an approximate way \cite{ourTauModel}
and (b) an average $a$-dependence, which is implemented  % in an approximate way \cite{ourTauModel}
%by defining an effective radius, $R=\sqrt{\Delta\tau/(2\mtbar)}$.
by introducing an effective radius defined by
 
\begin{equation}\label{eq:effR}
    R^{2\alpha} = \left(\frac{\Delta\tau}{2}\right)^{\!\alpha} \frac{a_1^\alpha+a_2^\alpha}{2} \;.
\end{equation}
This results, including the factor $\lambda$ and the long-range parametrization, in
 
\begin{equation}\label{eq:asymlevR2}
    R_2(Q) = \gamma \left[ 1+ \lambda \cos \left(\left(R_\mathrm{a}Q\right)^{2\alpha} \right)
             \exp \left(-\left(RQ\right)^{2\alpha} \right) \right] (1+ \epsilon Q) \;,
\end{equation}
where $R_\mathrm{a}$ is  related to $R$ by
 
\begin{equation}\label{eq:asymlevRaR}
    R_\mathrm{a}^{2\alpha} = \tan\left(\frac{\alpha\pi}{2}\right) R^{2\alpha} \;.
\end{equation}
Fits of
\Eq{eq:asymlevR2} are first performed with $R_\mathrm{a}$ as a free parameter.
The fit results obtained for two- and three-jet events are listed in \Tab{tab:a_levy}
and shown  in \Fig{fig:a_levy_2jet} for two-jet events and in \Fig{fig:a_levy_3jet} for three-jet events.
%, \ref{fig:a_levy_3jet}, and \ref{fig:a_levy_all}, respectively.
They have acceptable confidence levels, describing well the dip below unity in the 0.6--1.5\,\GeV\ region,
as well as the low-$Q$ peak.
 
 
%\begin{figure}
%  \centering
%  \includegraphics[width=.75\figwidth]{\olddirfig alevy_2j.eps}
%  \caption{The Bose-Einstein correlation function $R_2$ for two-jet events. The curve
%           corresponds to the fit of the one-sided L\'evy parametrization, \Eq{eq:asymlevR2}.
%           The dashed line represents the long-range part of the fit, \ie, $\gamma(1+\epsilon Q)$.
%           \label{fig:a_levy_2jet}
%           }
%\end{figure}
%
%\begin{figure}
%  \centering
%  \includegraphics[width=.75\figwidth]{\olddirfig alevy_3j.eps}
%  \caption{The Bose-Einstein correlation function $R_2$ for three-jet events. The curve
%           corresponds to the fit of the one-sided L\'evy parametrization, \Eq{eq:asymlevR2}.
%           \label{fig:a_levy_3jet}
%           The dashed line represents the long-range part of the fit, \ie, $\gamma(1+\epsilon Q)$.
%           }
%\end{figure}
\begin{sidewaysfigure}
  \centering
  \includegraphics[width=.45\textwidth]{\mydirfig run_fitQRR4.eps}
  \includegraphics[width=.45\textwidth]{\mydirfig run_fitQR4.eps}
  \caption{The Bose-Einstein correlation function $R_2$ for two-jet events. The curve
           corresponds to the fit of the one-sided L\'evy parametrization, \Eq{eq:asymlevR2},
           with the parameter $\Ra$ (a) free and (b) constrained by \Eq{eq:asymlevRaR}.
           The dashed line represents the long-range part of the fit, \ie, $\gamma(1+\epsilon Q)$.
           \label{fig:a_levy_2jet}
           }
\end{sidewaysfigure}
 
\begin{sidewaysfigure}
  \centering
  \includegraphics[width=.45\textwidth]{\mydirfig run_fitQRR4_3j.eps}
  \includegraphics[width=.45\textwidth]{\mydirfig run_fitQR4_3j.eps}
  \caption{The Bose-Einstein correlation function $R_2$ for three-jet events. The curve
           corresponds to the fit of the one-sided L\'evy parametrization, \Eq{eq:asymlevR2},
           with the parameter $\Ra$ (a) free and (b) constrained by \Eq{eq:asymlevRaR}.
           The dashed line represents the long-range part of the fit, \ie, $\gamma(1+\epsilon Q)$.
           \label{fig:a_levy_3jet}
           }
\end{sidewaysfigure}
 
 
 
Some fit parameters are rather highly correlated. For two-jet events the estimated correlaton coefficients
from the fit for the important parameters, $\alpha$, $R$ and $\Ra$, are
$\rho(\alpha,R)=-0.62$, $\rho(\alpha,\Ra)=-0.92$, and $\rho(R,\Ra)=0.38$,
while for three-jet events they are
$\rho(\alpha,R)=-0.77$, $\rho(\alpha,\Ra)=-0.95$, and $\rho(R,R_a)=0.61$.
Taking these correlations into account, the fit parameters for the two-jet events satisfy \Eq{eq:asymlevRaR},
the difference between the left- and right-hand sides of the equation being less than a standard deviation.
For three-jet events the difference amounts to about 1.5 standard deviations.
 
 
 
Fit results imposing \Eq{eq:asymlevRaR} are given in \Tab{tab:a_levy_c}.
%and shown in \Figs{fig:a_levy_2jet_c}, \ref{fig:a_levy_3jet_c}, and \ref{fig:a_levy_all_c}.
For two-jet events, the values of the parameters are comparable to those with $\Ra$ free.
For three-jet and all events, the imposition of \Eq{eq:asymlevRaR} results in values of $\alpha$ and $R$
closer to those for two-jet events, but the $\chi^2$ is noticeably worse, though acceptable, than with $\Ra$ free.
 
 
 
 
\begin{table}%[htbp]
\caption{Results of fits of \Eq{eq:asymlevR2}
           for two-jet and three-jet events.
%           for two-jet, three-jet, and all events.
         The first uncertainty is statistical, the second systematic.
\label{tab:a_levy}
         }
%\centering
\begin{center}
\begin{tabular}{ l r@{$\;\pm\;$}l
                   r@{$\;\pm\;$}l
               }
\hline
    parameter               & \multicolumn{2}{c }{2-jet}        & \multicolumn{2}{c }{3-jet}         \\
\hline
 \rule{0pt}{11pt}$\alpha$   & 0.41  & $0.02^{ +0.04}_{ -0.06}$  &  0.35  & 0.01$^{+0.03}_{-0.04}$    \\
 $\lambda$                  & 0.63  & 0.03$^{ +0.08}_{ -0.35}$  &  0.92  & 0.05$^{+0.06}_{-0.48}$    \\
 $R$            (fm)        & 0.79  & $0.04^{+0.09}_{-0.19}$    &  1.06  & 0.05$^{+0.59}_{-0.31}$    \\
 $R_\mathrm{a}$ (fm)        & 0.69  & 0.04$^{+0.21}_{-0.09}$    &  0.85  & 0.04$^{+0.15}_{-0.05}$    \\
 $\epsilon$ (\invGeV)         & 0.001 & 0.002$^{+0.005}_{-0.008}$ &  0.000 & 0.002$^{+0.001}_{-0.007}$ \\
    $\gamma$                & 0.988 & 0.005$^{+0.026}_{-0.012}$ &  0.997 & 0.005$^{+0.019}_{-0.002}$  \\
\hline
  \chisq/DoF                & \multicolumn{2}{c }{91/94} & \multicolumn{2}{c }{84/94} \\
  confidence level          & \multicolumn{2}{c }{57\%}  & \multicolumn{2}{c }{76\%}  \\
\hline
\end{tabular}
\end{center}
\end{table}
 
\begin{table}%[htbp]
\caption{Results of fits of \Eq{eq:asymlevR2} imposing \Eq{eq:asymlevRaR}
           for two-jet and three-jet events.
%           for two-jet, three-jet, and all events.
         The first uncertainty is statistical, the second systematic.
\label{tab:a_levy_c}
         }
 
%\centering
\begin{center}
\begin{tabular}{ l r@{$\;\pm\;$}l
                   r@{$\;\pm\;$}l
               }
\hline
    parameter              & \multicolumn{2}{c }{2-jet}        & \multicolumn{2}{c }{3-jet} \\
\hline
 \rule{0pt}{11pt}$\alpha$  & 0.44  & $0.01^{+0.05}_{-0.02}$    &  0.42  & $0.01^{+0.02}_{-0.04}$   \\
 $\lambda$                 & 0.61  & $0.03^{+0.08}_{-0.26}$    &  0.84  & $0.04^{+0.04}_{-0.37}$    \\
 $R$            (fm)       & 0.78  & $0.04^{+0.09}_{-0.16}$    &  0.98  & $0.04^{+0.55}_{-0.14}$   \\
 $\epsilon$ (\invGeV)        & 0.005 & $0.001\pm0.003$           &  0.008 & $0.001\pm0.005$          \\
    $\gamma$               & 0.979 & $0.002^{+0.009}_{-0.003}$ &  0.977 & $0.001^{+0.013}_{-0.008}$ \\
\hline
  \chisq/DoF               & \multicolumn{2}{c }{95/95} & \multicolumn{2}{c }{113/95} \\
  confidence level         & \multicolumn{2}{c }{49\%}  & \multicolumn{2}{c }{10\%}   \\
\hline
\end{tabular}
\end{center}
\end{table}
 
%\enote{Need systematics.}\mnote{syst}
 
 
 
\subsubsection{The \boldmath{$\tau$}-model with \boldmath{$\mt$} dependence}  \label{sect:taufits}
 
For two-jet events, $a=1/\mt$, while for three-jet events the situation is more complicated.
We therefore limit fits of  \Eq{eq:levyR2a} to the two-jet data.
For each bin in $Q$ the average values of $m_{\mathrm{t}1}$ and $m_{\mathrm{t}2}$ are calculated,
where $m_{\mathrm{t}1}$ and $m_{\mathrm{t}2}$ are the transverse masses of the
two particles making up a pair, requiring  $m_{\mathrm{t}1} > m_{\mathrm{t}2}$.
Using these averages,
and including the factor $\lambda$ and the long-range parametrization,
 \Eq{eq:levyR2a} is fit to $R_2(Q)$.
% \Eq{eq:levyR2a} becomes
%
%\begin{eqnarray} \label{eq:levyR2mt}
%   R_2(Q,\overline{m}_{\mathrm{t}1},\overline{m}_{\mathrm{t}2}) &=& \gamma\left[ 1 + \lambda
%      \cos\left[\frac{\tau_0 Q^2 (\overline{m}_{\mathrm{t}1}+\overline{m}_{\mathrm{t}2})}{2 \overline{m}_{\mathrm{t}1} \overline{m}_{\mathrm{t}2}} +
%\tan\left(\frac{\alpha\pi}{2}\right)\left(\frac{\Delta\tau {Q^2}}{2}\right)^{\!\!\alpha}\,\frac{\overline{m}_{\mathrm{t}1}^\alpha + \overline{m}_{\mathrm{t}2}^\alpha}{2 \overline{m}_{\mathrm{t}1}^\alpha \overline{m}_{\mathrm{t}2}^\alpha} \right]
%\right.
% \nonumber \\
%    & &
%\left.
%     \cdot           \exp \left[-\left(\frac{\Delta\tau {Q^2}}{2}\right)^{\!\!\alpha}\,\frac{\overline{m}_{\mathrm{t}1}^\alpha + \overline{m}_{\mathrm{t}2}^\alpha}{2 \overline{m}_{\mathrm{t}1}^\alpha \overline{m}_{\mathrm{t}2}^\alpha} \right]
%                     (1+\epsilon Q)
%\right]
% \; .
%\end{eqnarray}
The fit results in $\tau_0=0.00\pm0.02$, and we have re-fit with $\tau_0$ fixed to zero.
The results are shown in \Fig{fig:2jetR2} and \Tab{tab:2jetR2}.
The parameters $\alpha$, $\Delta\tau$ and $\lambda$ are highly correlated with estimated correlaton coefficients
$\rho(\alpha,\Delta\tau)=-0.96$, $\rho(\alpha,\lambda)=-0.92$, and $\rho(\Delta\tau,\lambda)=0.95$.
%\Eq{eq:levyR2mt} can be simplified by requiring $m_{\mathrm{t}1}\approx m_{\mathrm{t}2}$.
\Eq{eq:levyR2a} for two-jet events can be simplified by requiring $m_{\mathrm{t}1}\approx m_{\mathrm{t}2}$.
This leads to fit results\cite{tamas:thesis} consistent with those without this simplification.
 
 
 
\begin{figure}
  \centering
  \includegraphics[width=.94\textwidth]{\mydirfig run_fitQ1mtmt.ps}
  \caption{The Bose-Einstein correlation function $R_2$ for two-jet events. The curve
           corresponds to the fit of \Eq{eq:levyR2mt}.
           The dashed line represents the long-range part of the fit, \ie, $\gamma(1+\epsilon Q)$.
           \label{fig:2jetR2}
           }
\end{figure}
 
 
\begin{table}%[htbp]
\caption{Results of the fit of \Eq{eq:levyR2mt} for two-jet events.
         The parameter $\tau_0$ is fixed to zero.
         The first uncertainty is statistical, the second systematic.
\label{tab:2jetR2}
         }
%\centering
\begin{center}
\begin{tabular}{ l r@{$\;\pm\;$}l
               }
\hline
    parameter               & \multicolumn{2}{c }{\ }           \\
\hline
  \rule{0pt}{11pt}$\alpha$  & 0.47  & $0.01^{+0.04}_{-0.02}$    \\
  $\Delta\tau$   (fm)       & 1.56  & $0.12^{+0.32}_{-0.45}$    \\
  $\lambda$                 & 0.58  & $0.03^{+0.08}_{-0.24}$    \\
  $\epsilon$ (\invGeV)        & 0.001 & $0.001\pm0.003$           \\
  $\gamma$                  & 0.988 & $0.002^{+0.006}_{-0.002}$ \\
\hline
  \chisq/DoF                & \multicolumn{2}{c }{90/95}        \\
  confidence level          & \multicolumn{2}{c }{62\%}         \\
\hline
\end{tabular}
\end{center}
\end{table}
 
 
 
 
Since the \taumodel\ describes the \mt\ dependence of $R_2$, an important test of the model
is that its parameters, $\alpha$,  $\Delta\tau$, and $\tau_0$, not depend on \mt.
Therefore we have repeated the fit for various regions of the $m_{\mathrm{t}1}$-$m_{\mathrm{t}2}$ plane.
The values of $\alpha$ and  $\Delta\tau$ found in such regions are shown in \Fig{fig:aR4b}.
At first glance it would appear that the parameters are not independent of \mt.
However, note that the variation in the values roughly follows the major axis of the covariance ellipse.
Further, both of these parameters are highly correlated with $\lambda$.
Hence, this plot does not in itself provide evidence of dependence on \mt.
To adequately test the hypothesis of independence of \mt\ we repeat the fits with  $\alpha$, $\Delta\tau$, or both fixed
to the values found in the fit to the entire \mt\ plane (\Tab{tab:2jetR2}).  The confidence levels of the resulting
fits are compared to those of the fits with these parameters free in \Tab{tab:2jetmtCL}.
While those of the free parameter fits are generally higher, those of the fixed $\alpha$ and  $\Delta\tau$ fits are
acceptable.  The present data thus do not reject the hypothesis of \mt-independence.
 
\begin{figure}
  \centering
  \includegraphics*[width=.80\figwidth,angle=270]{\mydirfig aR_fitQ1mtmt_binsb4.eps}
  \caption{The results of fits of \Eq{eq:levyR2mt} to two-jet data for various  regions of the $m_{\mathrm{t}1}$-$m_{\mathrm{t}2}$ plane.
     The solid point represents the result of the fit to the entire $m_{\mathrm{t}1}$-$m_{\mathrm{t}2}$ plane.
  \label{fig:aR4b}
  }
\end{figure}
 
\begin{sidewaystable}%[htbp]
\caption{Confidence levels of fits of \Eq{eq:levyR2mt} for two-jet events
         in various  regions of the $m_{\mathrm{t}1}$-$m_{\mathrm{t}2}$ plane
         with $\alpha$ and  $\Delta\tau$ free parameters and with them fixed to the result of the fit to the entire plane.
\enote{This table probably contains more than we will actually want in the paper---perhaps keep only the columns left of the double vertical line.}
\label{tab:2jetmtCL}
         }
%\centering
\begin{center}
\begin{tabular}{cc|cc|c|ccc||c|ccc}
\hline
    \multicolumn{2}{c}{\mt\ region (GeV)}   & \multicolumn{2}{c}{average} & \multicolumn{4}{c}{confidence level (\%)} & \multicolumn{4}{c}{$\lambda$}  \\
    $m_{\mathrm{t}1}$  &  $m_{\mathrm{t}2}$ &\multicolumn{2}{c}{\mt\ (GeV)}&  free  & \multicolumn{3}{c}{fixed}&     free      & \multicolumn{3}{c}{fixed} \\
                       &                    &$Q<0.4$&all&        &$\alpha$&$\Delta\tau$&$\alpha$, $\Delta\tau$ &       &$\alpha$&$\Delta\tau$&$\alpha$, $\Delta\tau$ \\
\hline
    0.14 -- 0.26       &   0.14 -- 0.22     & 0.19 & 0.19 &  21    &  22   &  23        & 10    & $0.39\pm0.05$          &$0.36\pm0.02$ &$0.41\pm0.02$ &$0.39\pm0.02$ \\ % 11
    0.14 -- 0.34       &   0.22 -- 0.30     & 0.27 & 0.27 &  55    &  57   &  56        & 48    & $0.80\pm0.08$          &$0.80\pm0.03$ &$0.75\pm0.03$ &$0.76\pm0.03$ \\ % 12
    0.14 -- 0.46       &   0.30 -- 0.42     & 0.37 & 0.37 &  80    &  82   &  80        & 74    & $0.89\pm0.09$          &$0.87\pm0.04$ &$0.82\pm0.03$ &$0.83\pm0.03$ \\ % 13
    0.14 -- 0.66       &   0.42 -- 4.14     & 0.52 & 0.52 &  11    &  13   &  13        & 13    & $0.98\pm0.12$          &$0.99\pm0.05$ &$0.96\pm0.04$ &$0.97\pm0.04$ \\ % 14
    0.26 -- 0.42       &   0.14 -- 0.22     & 0.25 & 0.26 &  23    &  20   &  21        & 22    & $0.71^{+0.19}_{-0.13}$ &$0.53\pm0.03$ &$0.53\pm0.02$ &$0.53\pm0.02$ \\ % 21
    0.34 -- 0.46       &   0.22 -- 0.30     & 0.32 & 0.33 &  49    &  41   &  46        & 33    & $0.67\pm0.08$          &$0.84\pm0.04$ &$0.79\pm0.03$ &$0.80\pm0.03$ \\ % 22
    0.46 -- 0.58       &   0.30 -- 0.42     & 0.43 & 0.44 &  35    &  34   &  36        & 34    & $0.82\pm0.10$          &$0.94\pm0.05$ &$0.91\pm0.04$ &$0.91\pm0.04$ \\ % 23
    0.66 -- 0.86       &   0.42 -- 4.14     & 0.65 & 0.65 &  70    &  64   &  64        & 66    & $0.82\pm0.09$          &$1.00\pm0.06$ &$1.01\pm0.05$ &$1.01\pm0.05$ \\ % 24
    0.42 -- 0.62       &   0.14 -- 0.22     & 0.34 & 0.34 &  53    &  17   &  15        & 17    & $0.16\pm0.03$          &$0.39\pm0.04$ &$0.41\pm0.03$ &$0.41\pm0.03$ \\ % 31
    0.46 -- 0.70       &   0.22 -- 0.30     & 0.41 & 0.41 &  75    &  70   &  74        & 55    & $0.53\pm0.09$          &$0.72\pm0.05$ &$0.65\pm0.03$ &$0.64\pm0.03$ \\ % 32
    0.58 -- 0.82       &   0.30 -- 0.42     & 0.52 & 0.52 &  63    &  56   &  56        & 59    & $0.53\pm0.08$          &$0.70\pm0.05$ &$0.70\pm0.04$ &$0.70\pm0.04$ \\ % 33
    0.86 -- 1.22       &   0.42 -- 4.14     & 0.80 & 0.81 &  65    &  68   &  64        & 24    & $0.84^{+0.20}_{-0.15}$ &$0.86\pm0.08$ &$0.69\pm0.05$ &$0.66\pm0.05$ \\ % 34
%   0.62 -- 4.14       &   0.14 -- 0.22     & 0.48 & 0.57 &  \pho6 & \pho7 & \pho6      & \pho6 & $3.00\pm0.06$          &$0.26\pm0.09$ &$0.21\pm0.05$ &$0.20\pm0.03$ \\ % 41
    0.70 -- 4.14       &   0.22 -- 0.30     & 0.59 & 0.65 &  18    & \pho8 &  11        & \pho4 & $0.19\pm0.06$          &$0.50\pm0.08$ &$0.42\pm0.05$ &$0.37\pm0.04$ \\ % 42
    0.82 -- 4.14       &   0.30 -- 0.42     & 0.71 & 0.76 &  14    &  12   &  13        & 11    & $0.39^{+0.13}_{-0.09}$ &$0.62\pm0.07$ &$0.57\pm0.05$ &$0.56\pm0.05$ \\ % 43
%   1.22 -- 4.14       &   0.42 -- 4.14     & 1.01 & 1.07 &  37    &  36   &  31        & 30    & $1.48\pm1.70$          &$0.51^{+0.21}_{-0.17}$&$0.29\pm0.08$ &$0.27\pm0.07$ \\ % 44
\hline
\end{tabular}
\end{center}
\end{sidewaystable}
 
\subsection{Systematic Uncertainties} \label{sect:syst}
The following sources of systematic uncertainty are investigated:
\begin{description}
  \item[Event and track selection:] The cuts used in selecting hadronic events and well-measured tracks are
varied within reasonable limits.
 
 
  \item[Fit range:] The standard fits were performed in the range $0<Q<4$\,\GeV.
  The fits were repeated in the ranges $0.04<Q<4.00$\,\GeV, $0<Q<3$\,\GeV\ and $0<Q<5$\,\GeV.
 
 
  \item[Mixing:] The event mixing method to construct $\rho_0$ uses tracks from events having similar
multiplicity.  The definition of ``similar'' was varied.
 
 
  \item[Monte Carlo:] Events generated by \JETSET\ with no BEC simulation or by \HERWIG\ \cite{HERWIG59}
  are used
instead of \JETSET\ with BEC for the correction of $\rho_2$ for detector acceptance and efficiency
and \PYTHIA\ \cite{PYTHIAsix} and \HERWIG\ \cite{HERWIGsix} are used\footnote{\HERWIG\ was modified to
use the decay and BEC routines of \PYTHIA. Fragmentation parameters of both \HERWIG\ and \PYTHIA\ were
tuned using \Lthree\ data with BEC simulated by the \BEtt\ Gaussian algorithm \cite{PYBOEI} and the events
generated without BEC simulation.}
instead of \JETSET\ without BEC to correct $\rho_0$.
The root mean square of the changes in parameter values found using these alternative Monte Carlo generators
are taken as the systematic uncertainties, positive and negative deviations being treated separately.
 
\end{description}
For each source the root mean square of the deviations from the standard value is taken
as the systematic uncertainty, positive and negative deviations being treated separately.
The systematic uncertainties from the four sources are added in quadrature to obtain the total systematic uncertainty.
In most cases the largest contributions come from Monte Carlo and/or  mixing.
 
 
 
 
 
 
 
\section{The emission function of two-jet events}  \label{sect:emission2jet}
Within the framework of the \taumodel, we now
reconstruct the space-time picture of the emitting process for two-jet events.
 
The emission function in space-time, $S(x)$,
%is the proper time derivative of the integral over $k$ of $S(x,k)$, which in the
%\taumodel\ is given by \Eq{eq:source}.
%Approximating $\delta_\Delta$ by a Dirac delta function, we find
is, in the \taumodel, given by  \cite{ourTauModel}:
 
\begin{equation}   \label{eq:Sspace}
   S(x) = \frac{\mathrm{d}^4 n}{\mathrm{d}\tau\mathrm{d}^3 x}
        = \left(\frac{\mt}{\tau}\right)^3 H(\tau) N_1\left( p=\frac{\mt x}{\tau} \right) \;,
\end{equation}
where $N_1(p)$ is the single-particle invariant momentum distribution,
 
\begin{equation}   \label{eq:rho1}
   N_1(p) = E \frac{\mathrm{d}^3 n}{\mathrm{d}^3 p}
             = \frac{\mathrm{d}^3 n}{\dd{p_\mathrm{x}}\dd{p_\mathrm{y}}\dd{y}}   \;,
\end{equation}
where $y$ is the rapidity;
$N_1$ is normalized to the mean multiplicity, $\bar{n}$.
Given the symmetry of two-jet events, $S$ does not depend on the azimuthal angle,
and we can write it in cylindrical coordinates as $S(r,z,t)=P(r,\eta)H(\tau)$,
where $\eta$ is the space-time rapidity.
With the strongly correlated phase-space of the \taumodel, $\eta=y$ and $r=\pt\tau/\mt$.
Consequently,
 
\begin{equation}  \label{eq:Preta}
     P(r,\eta) = \left(\frac{\mt}{\tau}\right)^{\!3} N_\mathrm{\pt,y}(r\mt/\tau, \eta) \;,
\end{equation}
where $N_\mathrm{\pt,y}$ is the joint single-particle distribution of \pt\ and $y$.
 
To simplify the reconstruction of $S(x)$ we assume that
$N_\mathrm{\pt,y}$
can be factorized in the product of the single-particle \pt\ and rapidity distributions, \ie,
$     N_\mathrm{\pt,y} = N_\mathrm{\pt}(\pt) N_\mathrm{y}(y)$.
This is found to be a reasonable approximation except at very  high values of $\abs{y}$ or \pt\ \cite{tamas:thesis}.
Then \Eq{eq:Preta} becomes
 
\begin{equation}   \label{eq:Pretafact}
     P(r,\eta) = \left(\frac{\mt}{\tau}\right)^{\!3} N_\mathrm{\pt}(r\mt/\tau) N_\mathrm{y}(\eta) \;.
\end{equation}
So, we can write \Eq{eq:Sspace} as
 
\begin{equation}   \label{eq:Sspacefact}
     S(r,z,t) = H(\tau)  P(r,\eta) =
        H(\tau)  \left(\frac{\mt}{\tau}\right)^{\!3} N_\mathrm{\pt}(r\mt/\tau) N_\mathrm{y}(\eta) \;.
\end{equation}
 
 
%With these assumptions and using
Using
$H(\tau)$ as obtained from the fit of \Eq{eq:levyR2mt} (\Tab{tab:2jetR2}),
which is shown in \Fig{fig:Htau},
together with the inclusive rapidity and \pt\ distributions \cite{tamas:thesis},
the full emission function is reconstructed.
Its integral over the transverse distribution is plotted in \Fig{fig:longemis}.
It exhibits a ``boomerang'' shape with a maximum at low $t$ and $z$
but with tails reaching out to very large values of $t$ and $z$,
a feature also observed in hadron-hadron~\cite{NA22emiss} and heavy ion collisions~\cite{HIemiss:Ster}.
 
\begin{figure}%[thbp]
  \centering
   \includegraphics*[width=.85\figwidth,viewport=0 260 550 512]{\mydirfig htau_v2.eps}
%   \includegraphics*[width=.55\figwidth]{\olddirfig htau_4.eps}
  \caption{The proper-time distribution, $H(\tau)$,
           for $\alpha=0.47$, $\tau_0=0$ and $\Delta\tau=1.56$\,fm.
%\enote{Should be for $\alpha=0.474$, $\tau_0=0$ and $\Delta\tau=1.56$\,fm.}
  \label{fig:Htau}
  }
\end{figure}
 
 
%The transverse profile, which follows from \Eq{eq:Sspace}, has the following form:
%\begin{equation}   \label{eq:transmovie}
%   \frac{\mathrm{d}^2 n}{\mathrm{d}\tau\mathrm{d}r}
%           = \frac{\mt^3}{\tau^3} H(\tau) N_1\left( k=\frac{\mt r}{\tau} \right) \;.
%\end{equation}
The transverse part of the emission function is obtained by integrating over $z$ and azimuthal angle.
%$N_1$ is determined from the inclusive \pt\ distribution.
%The particle production probability is proportional to the proper-time distribution $H(\tau)$.
\Fig{fig:xymovie} shows the transverse part of the emission function for
various proper times. Particle production starts immediately, increases rapidly and decreases slowly.
A ring-like structure, similar to the expanding, ring-like wave created by a pebble in a pond, is observed.
%These pictures together form a movie that ends in about 3.5\,fm, \enote{(New value needed.)}
%making it the shortest movie ever made of a process in nature.
These pictures together form a movie,
perhaps the shortest movie ever made of a process in nature.\footnote{An animated gif file covering the first  0.3\,fm
%(10$^{-24}$\,sec)
\enote{(New value needed.)}
%= 0--10$^{-24}$\,sec
is available~\cite{tamas_movie}.}
In the movie two smoke-ring like structures separate along the thrust axis, their radii increasing.
They thus span the surface of two back-to-back cones whose tips meet at the origin.  The rings are
very faint at the beginning, then quickly brighten as the particle production probability increases to a
sharp peak at $\tau\approx1$\,fm. These separating and expanding smoke-rings the start to fade gradually
with a characteristic duration of $\Delta\tau\approx1.56$\,fm following a power-law tail.
 
\begin{figure}
  \centering
  \includegraphics*[width=.45\figwidth]{\mydirfig l3-zt-a_v3.eps} % \\ % \hfil
\hfil
  \includegraphics*[width=.45\figwidth]{\olddirfig l3-zt-a.eps} % \\ % \hfil
%  \includegraphics*[width=.75\figwidth]{\olddirfig l3-zt-a.eps} % \\ % \hfil
  \caption{The temporal-longitudinal part of the source function normalized to the
           average number of pions per event.
\enote{Which view do we prefer? If the left, labelling needs to be improved like in the right figure.}
\label{fig:longemis}
          }
\end{figure}
 
 
\begin{figure}
  \centering
  \includegraphics[width=.3\figwidth,bb=97 40 362 225,clip]{\mydirfig trans_tau001.eps}
  \includegraphics[width=.3\figwidth,bb=97 40 362 225,clip]{\mydirfig trans_tau002.eps}
  \includegraphics[width=.3\figwidth,bb=97 40 362 225,clip]{\mydirfig trans_tau003.eps}
  \includegraphics[width=.3\figwidth,bb=97 40 362 225,clip]{\mydirfig trans_tau004.eps}
  \includegraphics[width=.3\figwidth,bb=97 40 362 225,clip]{\mydirfig trans_tau005.eps}
  \includegraphics[width=.3\figwidth,bb=97 40 362 225,clip]{\mydirfig trans_tau006.eps}
  \includegraphics[width=.3\figwidth,bb=97 40 362 225,clip]{\mydirfig trans_tau007.eps}
  \includegraphics[width=.3\figwidth,bb=97 40 362 225,clip]{\mydirfig trans_tau008.eps}
  \includegraphics[width=.3\figwidth,bb=97 40 362 225,clip]{\mydirfig trans_tau009.eps}
  \includegraphics[width=.3\figwidth,bb=97 40 362 225,clip]{\mydirfig trans_tau010.eps}
  \includegraphics[width=.3\figwidth,bb=97 40 362 225,clip]{\mydirfig trans_tau011.eps}
  \includegraphics[width=.3\figwidth,bb=97 40 362 225,clip]{\mydirfig trans_tau012.eps}
  \includegraphics[width=.3\figwidth,bb=97 40 362 225,clip]{\mydirfig trans_tau013.eps}
  \includegraphics[width=.3\figwidth,bb=97 40 362 225,clip]{\mydirfig trans_tau014.eps}
  \includegraphics[width=.3\figwidth,bb=97 40 362 225,clip]{\mydirfig trans_tau015.eps}
%  \includegraphics[width=.3\figwidth]{\mydirfig trans_tau001_v3.eps}
%  \includegraphics[width=.3\figwidth]{\mydirfig trans_tau002_v3.eps}
%  \includegraphics[width=.3\figwidth]{\mydirfig trans_tau003_v3.eps}
%  \includegraphics[width=.3\figwidth]{\mydirfig trans_tau004_v3.eps}
%  \includegraphics[width=.3\figwidth]{\mydirfig trans_tau005_v3.eps}
%  \includegraphics[width=.3\figwidth]{\mydirfig trans_tau006_v3.eps}
%  \includegraphics[width=.3\figwidth]{\mydirfig trans_tau007_v3.eps}
%  \includegraphics[width=.3\figwidth]{\mydirfig trans_tau008_v3.eps}
%  \includegraphics[width=.3\figwidth]{\mydirfig trans_tau009_v3.eps}
%  \includegraphics[width=.3\figwidth]{\mydirfig trans_tau010_v3.eps}
%  \includegraphics[width=.3\figwidth]{\mydirfig trans_tau011_v3.eps}
%  \includegraphics[width=.3\figwidth]{\mydirfig trans_tau012_v3.eps}
%  \includegraphics[width=.3\figwidth]{\mydirfig trans_tau013_v3.eps}
%  \includegraphics[width=.3\figwidth]{\mydirfig trans_tau014_v3.eps}
%  \includegraphics[width=.3\figwidth]{\mydirfig trans_tau015_v3.eps}
  \caption{The transverse source function normalized to the average number of pions per event for
           various proper times.
%\enote{Should be for $\alpha=0.474$, $\tau_0=0$ and $\Delta\tau=1.56$\,fm.}
\label{fig:xymovie}
}
\end{figure}
 
%\enote{Add longitudinal, \ie, $z$ \vs\ $r$, ``movie''}
 
 
 
 
\section{Elongation?}                              \label{sect:elongation}
In this section we return to the question of a possible elongation of the source.
The success of the \taumodel\ would suggest a spherical shape, whereas previous
studies \cite{L3_3D:1999,OPAL3D:2000,DELPHI2D:2000,ALEPH:2004,OPAL:2007} have found
a shape elongated along the event (thrust) axis.  In these studies $Q^2$ is split into three
components,
\begin{eqnarray*}   \label{eq:QLCMS}
  \Qsquare &=& \Qslong + \Qsside + \Qsout - (\Delta E)^2   \\
           &=& \Qslong + \Qsside + \Qsout \left(1 - \left(\frac{p_{1\mathrm{out}}+p_{2\mathrm{out}}}{E_1+E_2}\right)^2\right)
\end{eqnarray*}
in the Longitudinal Center of Mass System (LCMS) of the pair.
The LCMS frame is defined as the frame, obtained by a Lorentz boost along the event axis,
where the sum of the three-momenta of the two pions ($\vec{p}_1+\vec{p}_2$)
is perpendicular to the event axis.
In this frame the event axis is referred to as the longitudinal direction;
the direction of $\vec{p}_1+\vec{p}_2$ as the out direction; and
the direction perpendicular to these as the side direction.
\Qlong, \Qout, and \Qside\ are the components of $\abs{\vec{p}_1-\vec{p}_2}$ along these directions.
$\Rsquare\Qsquare$ is then replaced in the Gaussian or Edgeworth parametrizations by
 
\begin{equation}   \label{eq:RQLCMS}
    \Rsquare\Qsquare  = \Rslong\Qslong + \Rsside\Qsside + \Rsout\Qsout \;.
\end{equation}
\Rlong\ and \Rside\ are measures of the longitudinal and transverse size of the source, respectively,
whereas the value of \Rout\ reflects both the transverse the temporal sizes.
 
In our previous analysis~\cite{L3_3D:1999} we found, using all events,  $\Rside/\Rlong=0.80\pm0.02^{+0.03}_{-0.18}$
and $\Rside/\Rlong=0.81\pm0.02^{+0.03}_{-0.19}$ for the Gaussian and Edgeworth parametrizations, respectively.
Since the present analysis uses a somewhat different event sample and mixing algorithm, we have repeated these fits
using the same binning of $0.08\GeV$ for $0.00<Q_i<1.04\GeV$ ($i$=L, s,~o).
We find consistent values~\cite{tamas:thesis} of $\Rside/\Rlong$: 0.76 for both parametrizations.
The elongation is found to be greater for two-jet events than for three-jet events, as has previously been
observed by \OPAL\cite{OPAL3D:2000}.
Using the Edgeworth parametrization we find
(statistical uncertainties only)
$\Rside/\Rlong = 0.65\pm0.02$ for two-jet events,
                $0.76\pm0.02$ for all events,
and
                $0.84\pm0.02$ for three-jet events.
 
 
 
%\enote{How much detail do we want to give?}
 
 
We next investigate whether the \taumodel\ parametrizations would prefer an elongation.
This is done by incorporating \Eq{eq:RQLCMS}
in \Eq{eq:asymlevR2}.  If \Eq{eq:asymlevRaR} is imposed, this results in
 
\begin{equation}\label{eq:asymlevR2BP}
    R_2(Q) = \gamma \left[ 1+ \lambda
                       \cos\left(\!\tan\left(\frac{\alpha\pi}{2}\right) A^{2\alpha} \vphantom{\frac{\Gamma}{2}}
                          \right)
             \exp \left(-A^{2\alpha} \right) \right]
             \left(1+ \epsillong\Qlong + \epsilside\Qside + \epsilout\Qout\right) \;,
\end{equation}
where
 
\begin{equation}\label{eq:asymlevR2BPA}
        A^2 = \Rslong\Qslong + \Rsside\Qsside + \Rsout\Rsout  \;,
\end{equation}
and where we use the same long-range parametrization as in the above Edgeworth and Gaussian parametrizations.
We have attempted the long-range parametrization $(1+\epsilon Q)$, but that results in unacceptable \chisq.
To extend the range of $Q$ included in the fit, the bin size is increased to $0.16\GeV$ for $Q_i>0.88\,\GeV$.
Results for two-jet events of fits for $Q<4\,\GeV$ of \Eq{eq:asymlevR2BP} and of the analogous equation not imposing
\Eq{eq:asymlevRaR} are shown in \Tab{tab:BPlcms}.  A clear preference for elongation ($\Rside/\Rlong<1$) is seen.
 
\begin{table}%[htbp]
\caption{Results
         for two-jet events
         of fits
         of \Eq{eq:asymlevR2BP} and of the analogous equation not imposing \Eq{eq:asymlevRaR}.
%         of \Eq{eq:asymlevR2} with $(RQ)^{2\alpha}=(\Rslong\Qslong + \Rsside\Qsside + \rsout\qsout)^\alpha$
%         and imposing the condition \Eq{eq:asymlevRaR}.
%         with $\epsillong$, $\epsilside$ and $\epsilout$  free parameters.
%         and with them fixed to the result of a fit for $1.5<Q<4.0\GeV$ with $\lambda=0$.
         Uncertainties are statistical only.
\enote{This table may contain more than we will actually want in the paper---perhaps we don't need the fit with $R_\mathrm{a}$.}
\label{tab:BPlcms}
         }
%\centering
\begin{center}
$
\begin{array}{ccc}
\hline
    \alpha                         &   0.38 \pm 0.03  &    0.46 \pm 0.01  \\
    \lambda                        &   0.56 \pm 0.04  &    0.49 \pm 0.02  \\
    \Rlong\text{ (fm)}             &   1.07 \pm 0.11  &     .85 \pm 0.04  \\
    \Rside/\Rlong                  &   0.27 \pm 0.07  &     .61 \pm 0.02  \\
    \Rout/\Rlong                   &   0.60 \pm 0.05  &     .66 \pm 0.02  \\
    \Ralong\text{ (fm)}            &   0.81 \pm 0.07  &                   \\
    \Raside/\Ralong                &   0.72 \pm 0.03  &                   \\
    \Raout/\Ralong                 &   0.65 \pm 0.03  &                   \\
    \epsillong\text{ (\invGeV)}    &  -0.001\pm 0.002 &    0.001\pm 0.001 \\
    \epsilside\text{ (\invGeV)}    &  -0.065\pm 0.004 &   -0.076\pm 0.003 \\
    \epsilout\text{ (\invGeV)}     &  -0.030\pm 0.002 &   -0.029\pm 0.002 \\
    \gamma                         &   1.015\pm 0.005 &    1.011\pm 0.002 \\
\hline
    \chisq/\text{DoF}              &   14815/14918    &   14847/14921     \\
    \text{CL}                      &    \text{72\%}   &   \text{66\%}    \\
\hline
\end{array}%{tabular}
$
\end{center}
\end{table}
 
 
To further test the hypothesis of no elongation in the \taumodel, we use
%However rather than \Eq{eq:RQLCMS}, we use
 
\begin{equation}   \label{eq:RQLCMSm}
    \Rsquare\Qsquare  = \Rslong\Qslong + \Rsside\Qsside + \rsout\qsout \;,
\end{equation}
where  $\qsout=\Qsout - (\Delta E)^2$
rather than \Eq{eq:RQLCMS}.
This has the advantage that if $R_2$ depends on $Q$ only rather than on components of $Q$,
$\Rlong=\Rside=\rout$. Thus a fit of $R_2(\Qlong,\Qside,\qout)$ imposing
$\Rlong=\Rside=\rout$ serves to test the hypothesis of no elongation.
%Hence, $(RQ)^{2\alpha}$ in \Eq{eq:asymlevR2} is replaced by $(\Rslong\Qslong + \Rsside\Qsside +
%\Rsout\qsout)^\alpha$ and $(R_\mathrm{a}Q)^{2\alpha}$ is treated analogously.
%Also, the long-range parameterization is modified to depend on the components of $Q$.
%Thus,
Modifying \Eq{eq:asymlevR2} in this way, imposing \Eq{eq:asymlevRaR}, results in
 
\begin{equation}\label{eq:asymlevR2BPm}
    R_2(Q) = \gamma \left[ 1+ \lambda
                       \cos\left(\!\tan\left(\frac{\alpha\pi}{2}\right) a^{2\alpha} \vphantom{\frac{\Gamma}{2}}
                          \right)
             \exp \left(-a^{2\alpha} \right) \right]
             \left(1+ \epsillong\Qlong + \epsilside\Qside + \epsilout\qout\right) \;,
\end{equation}
where
 
\begin{equation}\label{eq:asymlevR2BPa}
        a^2 = \Rslong\Qslong + \Rsside\Qsside + \rsout\qsout  \;.
\end{equation}
 
%To extend the range of $Q$ included in the fit, the bin size is increased to $0.16\GeV$ for $Q_i>0.88\,\GeV$.
The results of fits of \Eq{eq:asymlevR2BPm}  for various ranges in $Q$ are shown in \Tab{tab:2jetBPRf}
and for $Q<4\,\GeV$ in \Fig{fig:2jetBPRfr4}.
While some parameters show considerable dependence on the $Q$-range of the fit, $\Rside/\Rlong$ and $\rout/\Rlong$ are seen to be
quite stable. The variation is thought to arise from correlations among the parameter values and the inability to determine well the
values of the long-range parameters $\epsillong$,  $\epsilside$ and $\epsilout$ when $Q$ is confined to small values.
The value of the elongation, $\Rside/\Rlong$, is far from unity and agrees well with the value in the fit of \Eq{eq:asymlevR2BP}
imposing \Eq{eq:asymlevRaR} (\Tab{tab:BPlcms}) and with that found using the Edgeworth parametrization.
The value of $\rout/\Rlong$ is also far from unity.
 
 
To test the no elongation hypothesis directly, we repeat the fits with the constraints
$\Rlong=\Rside=\rout$. The results are shown in \Tab{tab:2jetBPRfNOel}   and for $Q<4\,\GeV$ in \Fig{fig:2jetBPRfr4NOel}.
Here too, the values of the parameters are quite stable.
The confidence levels, however, are unacceptably low, whereas they were acceptable for the fits allowing elongation.
As is apparent in \Figs{fig:2jetBPRfr4} and~\ref{fig:2jetBPRfr4NOel}, the shape of the dependence of $R_2$
is different for the different components of $Q$ at small values of these components and can not be described by equal values of
\Rlong, \Rside\ and \rout.
 
\begin{table}%[htbp]
\caption{Results
         for two-jet events
         of fits
         of \Eq{eq:asymlevR2BPm}
%         of \Eq{eq:asymlevR2} with $(RQ)^{2\alpha}=(\Rslong\Qslong + \Rsside\Qsside + \rsout\qsout)^\alpha$
%         and imposing the condition \Eq{eq:asymlevRaR}.
         Uncertainties are statistical only.
\enote{This table probably contains more than we will actually want in the paper---perhaps keep only the $Q<4$ column.}
\label{tab:2jetBPRf}
         }
%\centering
\begin{center}
$
\begin{array}{cccccc}
\hline
    {\ }                           &   Q<1.04\GeV     &   Q<2.00\GeV      &   Q<3.04\GeV     &   Q<4.00\GeV      \\
\hline
    \alpha                         &   0.39 \pm 0.02  &    0.34 \pm 0.02  &   0.40 \pm 0.01  &   0.41 \pm 0.01  \\
    \lambda                        &   0.77 \pm 0.07  &    0.93 \pm 0.12  &   0.69 \pm 0.04  &   0.65 \pm 0.03  \\
    \Rlong\text{ (fm)}             &   0.98 \pm 0.07  &    1.26 \pm 0.16  &   1.00 \pm 0.06  &   0.96 \pm 0.05  \\
    \Rside/\Rlong                  &   0.61 \pm 0.02  &    0.63 \pm 0.02  &   0.62 \pm 0.02  &   0.62 \pm 0.02  \\
    \rout/\Rlong                   &   1.24 \pm 0.04  &    1.28 \pm 0.04  &   1.24 \pm 0.03  &   1.23 \pm 0.03  \\
    \epsillong\text{ (\invGeV)}    &   0.051\pm 0.014 &    0.032\pm 0.008 &   0.008\pm 0.001 &   0.004\pm 0.001 \\
    \epsilside\text{ (\invGeV)}    &  -0.058\pm 0.007 &   -0.059\pm 0.004 &  -0.063\pm 0.003 &  -0.067\pm 0.003 \\
    \epsilout\text{ (\invGeV)}     &   0.014\pm 0.016 &    0.001\pm 0.009 &  -0.020\pm 0.003 &  -0.022\pm 0.003 \\
    \gamma                         &   0.942\pm 0.016 &    0.958\pm 0.013 &   0.993\pm 0.003 &   1.000\pm 0.002 \\
\hline
    \chisq/\text{DoF}              &    1127/1037     &    4727/4617      &      8065/7843   &    10966/10647   \\
    \text{CL}                      &    \text{3\%}    &    \text{13\%}    &      \text{4\%}  &    \text{2\%}    \\
\hline
\end{array}%{tabular}
$
\end{center}
\end{table}
 
 
\begin{table}%[htbp]
\caption{Results
         for two-jet events
         of fits
         of \Eq{eq:asymlevR2BPm}
%         of \Eq{eq:asymlevR2} with $(RQ)^{2\alpha}=(\Rslong\Qslong + \Rsside\Qsside + \rsout\qsout)^\alpha$
%         and imposing the condition \Eq{eq:asymlevRaR}.
         with the constraints $\Rlong=\Rside=\rout$.
         Uncertainties are statistical only.
\enote{This table probably contains more than we will actually want in the paper---perhaps keep only the $Q<4$ column.}
\label{tab:2jetBPRfNOel}
         }
%\centering
\begin{center}
$
\begin{array}{cccccc}
\hline
    {\ }                            &   Q<1.04\GeV     &   Q<2.00\GeV      &   Q<3.04\GeV     &   Q<4.00\GeV      \\
\hline
    \alpha                          &   0.39 \pm 0.02  &    0.35 \pm 0.03  &   0.42 \pm 0.01  &   0.44 \pm 0.01  \\
    \lambda                         &   0.69 \pm 0.06  &    0.81 \pm 0.12  &   0.60 \pm 0.04  &   0.57 \pm 0.03  \\
    \Rlong=\Rlong=\rout\text{ (fm)} &   0.88 \pm 0.07  &    1.10 \pm 0.15  &   0.85 \pm 0.05  &   0.82 \pm 0.04  \\
    \epsillong\text{ (\invGeV)}     &   0.038\pm 0.013 &    0.028\pm 0.009 &   0.006\pm 0.001 &   0.003\pm 0.001 \\
    \epsilside\text{ (\invGeV)}     &  -0.010\pm 0.011 &   -0.038\pm 0.007 &  -0.053\pm 0.003 &  -0.059\pm 0.003 \\
    \epsilout\text{ (\invGeV)}      &  -0.046\pm 0.010 &   -0.022\pm 0.007 &  -0.029\pm 0.003 &  -0.029\pm 0.002 \\
    \gamma                          &   0.951\pm 0.015 &    0.964\pm 0.015 &   0.996\pm 0.003 &   1.003\pm 0.002 \\
\hline
    \chisq/\text{DoF}               &    1512/1039     &    5191/4619      &      8523/7845   &    11430/10649   \\
    \text{CL}                       &    10^{-20}      &    10^{-8}        &      10^{-7}     &    10^{-7}       \\
\hline
\end{array}%{tabular}
$
\end{center}
\end{table}
 
 
\begin{sidewaysfigure}
  \centering
  \includegraphics[width=.30\textwidth]{\mydirfig run_fitBPo3sR4r_1.eps}
  \includegraphics[width=.30\textwidth]{\mydirfig run_fitBPo3sR4r_2.eps}
  \includegraphics[width=.30\textwidth]{\mydirfig run_fitBPo3sR4r_3.eps}
  \caption{The Bose-Einstein correlation function $R_2(\Qlong,\Qside,\qout)$ for two-jet events for various cuts on
           \Qlong, \Qside\ and \qout. The curves
           correspond to the fit of \Eq{eq:asymlevR2BPm} for $Q<4\GeV$.
           \label{fig:2jetBPRfr4}
           }
\end{sidewaysfigure}
 
 
\begin{sidewaysfigure}
  \centering
  \includegraphics[width=.30\textwidth]{\mydirfig run_fitBPo3sR4rNOel_1.eps}
  \includegraphics[width=.30\textwidth]{\mydirfig run_fitBPo3sR4rNOel_2.eps}
  \includegraphics[width=.30\textwidth]{\mydirfig run_fitBPo3sR4rNOel_3.eps}
  \caption{The Bose-Einstein correlation function $R_2(\Qlong,\Qside,\qout)$ for two-jet events for various cuts on
           \Qlong, \Qside\ and \qout. The curves
           correspond to the fit of \Eq{eq:asymlevR2BPm} for $Q<4\GeV$
         with the constraints $\Rlong=\Rside=\rout$.
           \label{fig:2jetBPRfr4NOel}
           }
\end{sidewaysfigure}
 
 
 
%
%Accordingly, we determine these values from a fit to only the long-range parametrization in the region $1.5<Q<4\GeV$,
%and perform the fits again, fixing $\epsillong$,  $\epsilside$ and $\epsilout$ to the values so obtained (see \Tab{tab:2jetBPlr}).
%The results are shown in \Tab{tab:2jetBPRflr} and for $Q<4\GeV$ in \Fig{fig:2jetBPRflr4}.
%All of the parameters are now quite stable,  and the elongation has not changed
%significantly from the values with $\epsillong$,  $\epsilside$ and $\epsilout$ free.  The elongation is far from unity.
%well with that found
%
%To test the no elongation hypothesis directly, we repeat the fits with the constraints
%$\Rlong=\Rside=\rout$. The results are shown in \Tab{tab:2jetBPRflrNOel}   and for $Q<4\,\GeV$ in \Fig{fig:2jetBPRflrNOel4}.
%Here too, the values of the parameters are quite stable.
%The confidence levels, however, are unacceptably low, whereas they were acceptable for the fits allowing elongation.
%
%
%
%\enote{How much detail do we want to give?}
%
%
%
%
%\begin{table}%[htbp]
%\caption{Results
%         for two-jet events
%         of a fit of the long-range parametrization, \ie, $R_2=\gamma (1+ \epsillong\Qlong + \epsilside\Qside + \epsilout\qout)$.
%%         in the region $1.5<Q<4.0\GeV$.
%         Uncertainties are statistical only.
%\label{tab:2jetBPlr}
%         }
%%\centering
%\begin{center}
%$
%\begin{array}{ccc}
%\hline
%                                    &  1.5<Q<4.0\;\GeV & 2.0<Q<4.0\;\GeV \\
%\hline
%    \epsillong\text{ (\invGeV)}     &   0.005\pm 0.002 &  0.011\pm 0.002 \\
%    \epsilside\text{ (\invGeV)}     &  -0.072\pm 0.004 & -0.067\pm 0.005 \\
%    \epsilout\text{ (\invGeV)}      &  -0.027\pm 0.003 & -0.031\pm 0.003 \\
%    \gamma                          &   1.007\pm 0.005 &  0.988\pm 0.008 \\
%\hline
%    \chisq/\text{DoF}               &     9974/10118   &    7205/7400    \\
%    \text{CL}                       &    \text{84\%}   &   \text{95\%}   \\
%\hline
%\end{array}%{tabular}
%$
%\end{center}
%\end{table}
%
%
%\begin{table}%[htbp]
%\caption{Results
%         for two-jet events
%         of fits
%         of \Eq{eq:asymlevR2BPm}
%%         of \Eq{eq:asymlevR2} with $(RQ)^{2\alpha}=(\Rslong\Qslong + \Rsside\Qsside + \rsout\qsout)^\alpha$
%%         and imposing the condition \Eq{eq:asymlevRaR}.
%%         with $\epsillong$, $\epsilside$ and $\epsilout$  free parameters.
%         with $\epsillong$, $\epsilside$ and $\epsilout$
%         fixed to the result of a fit for $1.5<Q<4.0\GeV$ with $\lambda=0$ (\Tab{tab:2jetBPlr}).
%         Uncertainties are statistical only.
%\enote{This table probably contains more than we will actually want in the paper---perhaps keep only the $Q<4$ column.}
%\label{tab:2jetBPRflr}
%         }
%%\centering
%\begin{center}
%$
%\begin{array}{cccccc}
%\hline
%    {\ }                           &   Q<1.04\GeV   &   Q<2.00\GeV    &   Q<3.04\GeV   &   Q<4.00\GeV      \\
%\hline
%    \alpha                         &   0.40 \pm 0.01  &    0.41 \pm 0.01  &   0.41 \pm 0.01  &   0.41 \pm 0.01  \\
%    \lambda                        &   0.69 \pm 0.04  &    0.65 \pm 0.03  &   0.65 \pm 0.03  &   0.67 \pm 0.03  \\
%    \Rlong\text{ (fm)}             &   1.01 \pm 0.07  &    0.96 \pm 0.05  &   0.97 \pm 0.05  &   0.99 \pm 0.06  \\
%    \Rside/\Rlong                  &   0.62 \pm 0.02  &    0.61 \pm 0.02  &   0.61 \pm 0.02  &   0.61 \pm 0.02  \\
%    \rout/\Rlong                   &   1.26 \pm 0.04  &    1.23 \pm 0.03  &   1.23 \pm 0.03  &   1.23 \pm 0.03  \\
%    \epsillong\text{ (\invGeV)}    &   \multicolumn{4}{c}{0.005420\text{ fixed}} \\
%    \epsilside\text{ (\invGeV)}    &  \multicolumn{4}{c}{-0.071554\text{ fixed}} \\
%    \epsilout\text{ (\invGeV)}     &  \multicolumn{4}{c}{-0.027472\text{ fixed}} \\
%    \gamma                         &   0.986\pm 0.002 &    1.000\pm 0.001 &   1.001\pm 0.001 &   1.005\pm 0.002 \\
%\hline
%    \chisq/\text{DoF}              &    1295/1160     &    6611/6264      &     10896/10655  &    13445/13379   \\
%    \text{CL}                      &    \text{0.3\%}  &    \text{0.1\%}   &      \text{5\%}  &    \text{34\%}   \\
%\hline
%\end{array}%{tabular}
%$
%\end{center}
%\end{table}
%
%\begin{sidewaysfigure}
%  \centering
%  \includegraphics[width=.30\textwidth]{\mydirfig run_fitBPo3sR4rBKfix_1.eps}
%  \includegraphics[width=.30\textwidth]{\mydirfig run_fitBPo3sR4rBKfix_2.eps}
%  \includegraphics[width=.30\textwidth]{\mydirfig run_fitBPo3sR4rBKfix_3.eps}
%  \caption{The Bose-Einstein correlation function $R_2(\Qlong,\Qside,\qout)$ for two-jet events for various cuts on
%           \Qlong, \Qside\ and \qout. The curves
%           correspond to the fit of \Eq{eq:asymlevR2BPm} for $Q<4\GeV$.
%         with $\epsillong$, $\epsilside$ and $\epsilout$
%         fixed to the result of a fit for $1.5<Q<4.0\GeV$ with $\lambda=0$ (\Tab{tab:2jetBPlr}).
%           \label{fig:2jetBPRflr4}
%           }
%\end{sidewaysfigure}
%
%\begin{table}%[htbp]
%\caption{Results
%         for two-jet events
%         of fits
%         of \Eq{eq:asymlevR2BPm}
%%         of \Eq{eq:asymlevR2} with $(RQ)^{2\alpha}=(\Rslong\Qslong + \Rsside\Qsside + \rsout\qsout)^\alpha$
%%         and imposing the condition \Eq{eq:asymlevRaR}.
%%         with $\epsillong$, $\epsilside$ and $\epsilout$  free parameters.
%         with $\epsillong$, $\epsilside$ and $\epsilout$
%         fixed to the result of a fit for $1.5<Q<4.0\GeV$ with $\lambda=0$ (\Tab{tab:2jetBPlr})
%         and with the constraints $\Rlong=\Rside=\rout$.
%         Uncertainties are statistical only.
%\enote{This table probably contains more than we will actually want in the paper---perhaps keep only the $Q<4$ column.}
%\label{tab:2jetBPRflrNOel}
%         }
%%\centering
%\begin{center}
%$
%\begin{array}{cccccc}
%\hline
%    {\ }                           &   Q<1.04\GeV   &   Q<2.00\GeV    &   Q<3.04\GeV   &   Q<4.00\GeV      \\
%\hline
%    \alpha                         &   0.42 \pm 0.02  &    0.43 \pm 0.01  &   0.43 \pm 0.01  &   0.42 \pm 0.01  \\
%    \lambda                        &   0.59 \pm 0.04  &    0.58 \pm 0.03  &   0.59 \pm 0.03  &   0.60 \pm 0.03  \\
%    \Rlong=\Rside=\rout            &   0.86 \pm 0.06  &    0.84 \pm 0.04  &   0.85 \pm 0.05  &   0.87 \pm 0.05  \\
%    \epsillong\text{ (\invGeV)}    &   \multicolumn{4}{c}{0.005420\text{ fixed}} \\
%    \epsilside\text{ (\invGeV)}    &  \multicolumn{4}{c}{-0.071554\text{ fixed}} \\
%    \epsilout\text{ (\invGeV)}     &  \multicolumn{4}{c}{-0.027472\text{ fixed}} \\
%    \gamma                         &   0.989\pm 0.002 &    1.002\pm 0.001 &   1.002\pm 0.001 &   1.002\pm 0.001 \\
%\hline
%    \chisq/\text{DoF}              &    1776/1162     &    7092/6266      &     11379/10660  &    13933/13381   \\
%    \text{CL}                      &    10^{-28}      &    10^{-12}       &      10^{-6}     &     10^{-4}      \\
%\hline
%\end{array}%{tabular}
%$
%\end{center}
%\end{table}
%
%\begin{sidewaysfigure}
%%  \centering
%\begin{center}
%  \includegraphics[width=.30\textwidth]{\mydirfig run_fitBPo3sR4rBKfixNOel_1.eps}
%  \includegraphics[width=.30\textwidth]{\mydirfig run_fitBPo3sR4rBKfixNOel_2.eps}
%  \includegraphics[width=.30\textwidth]{\mydirfig run_fitBPo3sR4rBKfixNOel_3.eps}
%  \caption{The Bose-Einstein correlation function $R_2(\Qlong,\Qside,\qout)$ for two-jet events for various cuts on
%           \Qlong, \Qside\ and \qout. The curves
%           correspond to the fit of \Eq{eq:asymlevR2BPm} for $Q<4\GeV$.
%         with $\epsillong$, $\epsilside$ and $\epsilout$
%         fixed to the result of a fit for $1.5<Q<4.0\GeV$ with $\lambda=0$ (\Tab{tab:2jetBPlr})
%         and with the constraints $\Rlong=\Rside=\rout$.
%           \label{fig:2jetBPRflrNOel4}
%           }
%\end{center}
%\end{sidewaysfigure}
 
 
We also investigate elongation in the parametrization of \Eq{eq:levyR2a} by replacing $\Delta\tau Q^2$ by
%In a similar fashion $\Delta\tau Q^2$ in \Eq{eq:levyR2a} is replaced by
$\Delta\tau_\text{long}\Qslong+\Delta\tau_\text{side}\Qsside+\Delta\tau_\text{out}\qsout$.
Fits of the resulting parametrization are performed.  In these fits only one \mt\ bin is used, $0<\mt<4\GeV$.
The results are shown in \Tabs{tab:2jetBPmtf} and~\ref{tab:2jetBPmtfNOel}. % and for $Q<4\GeV$ in \Fig{fig:2jetBPRflr4}.
The conclusions are the same as for the fits of \Eq{eq:asymlevR2BPm}.
Elongation is preferred: The value of $\Deltatauside/\Deltataulong$ agrees with that of $\Rside/\Rlong$.
Constraining the fits by $\Deltataulong=\Deltatauside=\Deltatauout$ results in unacceptable \chisq.
 
\enote{We don't really need all of these tables, but they are there for your information. We need to discuss which tables/figures to keep.}
 
 
 
\begin{table}%[htbp]
\caption{Results
         for two-jet events
         of fits
         of \Eq{eq:levyR2a} with $\Delta\tau Q^2$ replaced by
         $\Delta\tau_\text{long}\Qslong+\Delta\tau_\text{side}\Qsside+\Delta\tau_\text{out}\qsout$
         and long-range parametrization     $1+ \epsillong\Qlong + \epsilside\Qside + \epsilout\qout)$,
         where $\epsillong$, $\epsilside$ and $\epsilout$  are free parameters.
%         and with them fixed to the result of a fit for $1.5<Q<4.0\,\GeV$ with $\lambda=0$.
         Uncertainties are statistical only.
\enote{This table probably contains more than we will actually want in the paper---perhaps keep only the $Q<4$ column.}
\label{tab:2jetBPmtf}
         }
%\centering
\begin{center}
$
\begin{array}{cccccc}
\hline
    {\ }                           &   Q<1.04\GeV   &   Q<2.00\GeV    &   Q<3.04\GeV   &   Q<4.00\GeV      \\
\hline
    \alpha                         &   0.39 \pm 0.02  &   0.45  \pm 0.02  &   0.47 \pm 0.01  &   0.47 \pm 0.01  \\
    \lambda                        &   0.82 \pm 0.08  &   0.61  \pm 0.04  &   0.58 \pm 0.03  &   0.57 \pm 0.03  \\
    \Deltataulong\text{ (fm)}      &   2.50 \pm 0.40  &   1.85  \pm 0.22  &   1.72 \pm 0.16  &   1.74 \pm 0.15  \\
    \Deltatauside/\Deltataulong    &   0.49 \pm 0.04  &   0.64  \pm 0.04  &   0.62 \pm 0.03  &   0.61 \pm 0.03  \\
    \Deltatauout/\Deltataulong     &   1.48 \pm 0.10  &   1.58  \pm 0.09  &   1.50 \pm 0.08  &   1.48 \pm 0.08  \\
    \epsillong\text{ (\invGeV)}    &   0.077\pm 0.019 &   0.006 \pm 0.004 &   0.002\pm 0.001 &   0.000\pm 0.001 \\
    \epsilside\text{ (\invGeV)}    &  -0.086\pm 0.008 &  -0.061 \pm 0.003 &  -0.060\pm 0.003 &  -0.064\pm 0.003 \\
    \epsilout\text{ (\invGeV)}     &   0.026\pm 0.015 &  -0.018 \pm 0.005 &  -0.020\pm 0.002 &  -0.021\pm 0.003 \\
    \gamma                         &   0.927\pm 0.019 &   0.999 \pm 0.005 &   1.004\pm 0.002 &   1.008\pm 0.002 \\
\hline
    \chisq/\text{DoF}              &    1153/1037     &  4803/4617        &      8093/7843   &    10977/10647   \\
    \text{CL}                      &    \text{0.7\%}  &   \text{3\%}      &      \text{3\%}  &    \text{1\%}    \\
\hline
\end{array}%{tabular}
$
\end{center}
\end{table}
 
 
\begin{table}%[htbp]
\caption{Results
         for two-jet events
         of fits
         of \Eq{eq:levyR2a} with $\Delta\tau Q^2$ replaced by
         $\Delta\tau_\text{long}\Qslong+\Delta\tau_\text{side}\Qsside+\Delta\tau_\text{out}\qsout$
         and long-range parametrization     $1+ \epsillong\Qlong + \epsilside\Qside + \epsilout\qout)$,
         where the constraints $\Deltataulong=\Deltatauside=\Deltatauout$ are imposed.
         Uncertainties are statistical only.
\enote{This table probably contains more than we will actually want in the paper---perhaps keep only the $Q<4$ column.}
\label{tab:2jetBPmtfNOel}
         }
%\centering
\begin{center}
$
\begin{array}{cccccc}
\hline
    {\ }                           &   Q<1.04\GeV     &   Q<2.00\GeV      &   Q<3.04\GeV   &   Q<4.00\GeV      \\
\hline
    \alpha                         &   0.46 \pm 0.02  &   0.50  \pm 0.01  &   0.50 \pm 0.01  &   0.50 \pm 0.01  \\
    \lambda                        &   0.59 \pm 0.04  &   0.49  \pm 0.03  &   0.49 \pm 0.02  &   0.49 \pm 0.02  \\
    \Deltataulong=\Deltatauside=\Deltatauout\text{ (fm)}
                                   &   1.53 \pm 0.16  &   1.33  \pm 0.12  &   1.32 \pm 0.10  &   1.32 \pm 0.10  \\
    \epsillong\text{ (\invGeV)}    &   0.042\pm 0.014 &  -0.001 \pm 0.003 &  -0.000\pm 0.001 &  -0.001\pm 0.001 \\
    \epsilside\text{ (\invGeV)}    &  -0.031\pm 0.007 &  -0.049 \pm 0.003 &  -0.051\pm 0.003 &  -0.056\pm 0.002 \\
    \epsilout\text{ (\invGeV)}     &  -0.043\pm 0.009 &  -0.030 \pm 0.004 &  -0.025\pm 0.003 &  -0.025\pm 0.002 \\
    \gamma                         &   0.964\pm 0.013 &   1.006 \pm 0.004 &   1.006\pm 0.002 &   1.010\pm 0.002 \\
\hline
    \chisq/\text{DoF}              &    1394/1039     &  5061/4619        &      8348/7845   &    11239/10649   \\
    \text{CL}                      &    10^{-12}      &   10^{-6}         &      10^{-4}     &    10^{-5}       \\
\hline
\end{array}%{tabular}
$
\end{center}
\end{table}
%
%\begin{table}%[htbp]
%\caption{Results
%         for two-jet events
%         of fits
%         of \Eq{eq:levyR2a} with $\Delta\tau Q^2$ replaced by
%         $\Delta\tau_\text{long}\Qslong+\Delta\tau_\text{side}\Qsside+\Delta\tau_\text{out}\qsout$
%         and long-range parametrization     $1+ \epsillong\Qlong + \epsilside\Qside + \epsilout\qout)$,
%         where $\epsillong$, $\epsilside$ and $\epsilout$  are
%         fixed to the result of a fit for $1.5<Q<4.0\GeV$ with $\lambda=0$ (\Tab{tab:2jetBPlr}).
%         Uncertainties are statistical only.
%\enote{This table probably contains more than we will actually want in the paper---perhaps keep only the $Q<4$ column.}
%\label{tab:2jetBPmtflr}
%         }
%%\centering
%\begin{center}
%$
%\begin{array}{cccccc}
%\hline
%    {\ }                           &   Q<1.04\GeV   &   Q<2.00\GeV    &   Q<3.04\GeV   &   Q<4.00\GeV      \\
%\hline
%    \alpha                         &   0.45 \pm 0.01  &    0.45 \pm 0.01  &   0.44 \pm 0.01  &   0.43 \pm 0.01  \\
%    \lambda                        &   0.61 \pm 0.04  &    0.61 \pm 0.03  &   0.62 \pm 0.03  &   0.64 \pm 0.03  \\
%    \Deltataulong\text{ (fm)}      &   1.94 \pm 0.22  &    2.04 \pm 0.20  &   2.15 \pm 0.21  &   2.29 \pm 0.23  \\
%    \Deltatauside/\Deltataulong    &   0.61 \pm 0.04  &    0.59 \pm 0.03  &   0.57 \pm 0.03  &   0.56 \pm 0.03  \\
%    \Deltatauout/\Deltataulong     &   1.61 \pm 0.09  &    1.51 \pm 0.08  &   1.46 \pm 0.08  &   1.46 \pm 0.08  \\
%    \epsillong\text{ (\invGeV)}    &   \multicolumn{4}{c}{0.005420\text{ fixed}} \\
%    \epsilside\text{ (\invGeV)}    &  \multicolumn{4}{c}{-0.071554\text{ fixed}} \\
%    \epsilout\text{ (\invGeV)}     &  \multicolumn{4}{c}{-0.027472\text{ fixed}} \\
%    \gamma                         &   0.993\pm 0.002 &    1.005\pm 0.001 &   1.005\pm 0.001 &   1.004\pm 0.001 \\
%\hline
%    \chisq/\text{DoF}              &    1316/1160     &   6647/6264       &     10937/10658  &    13503/13379   \\
%    \text{CL}                      &   \text{0.09\%}  & \text{0.04\%}     &      \text{3\%}  &    \text{22\%}   \\
%\hline
%\end{array}%{tabular}
%$
%\end{center}
%\end{table}
%
%
%\begin{table}[htbp]
%\caption{Results
%         for two-jet events
%         of fits
%         of \Eq{eq:levyR2a} with $\Delta\tau Q^2$ replaced by
%         $\Delta\tau_\text{long}\Qslong+\Delta\tau_\text{side}\Qsside+\Delta\tau_\text{out}\qsout$
%         and long-range parametrization     $1+ \epsillong\Qlong + \epsilside\Qside + \epsilout\qout)$,
%         where $\epsillong$, $\epsilside$ and $\epsilout$  are
%         fixed to the result of a fit for $1.5<Q<4.0\GeV$ with $\lambda=0$ (\Tab{tab:2jetBPlr})
%         and the constraints $\Deltataulong=\Deltatauside=\Deltatauout$ are imposed.
%         Uncertainties are statistical only.
%\enote{This table probably contains more than we will actually want in the paper---perhaps keep only the $Q<4$ column.}
%\label{tab:2jetBPmtflrNOel}
%         }
%%\centering
%\begin{center}
%$
%\begin{array}{cccccc}
%\hline
%    {\ }                           &   Q<1.04\GeV   &   Q<2.00\GeV    &   Q<3.04\GeV   &   Q<4.00\GeV      \\
%\hline
%    \alpha                         &   0.49 \pm 0.01  &    0.48 \pm 0.01  &   0.47 \pm 0.01  &   0.46 \pm 0.01  \\
%    \lambda                        &   0.50 \pm 0.03  &    0.52 \pm 0.02  &   0.54 \pm 0.03  &   0.55 \pm 0.03  \\
%    \Deltataulong=\Deltatauside=\Deltatauout\text{ (fm)}
%                                   &   1.42 \pm 0.13  &    1.53 \pm 0.13  &   1.62 \pm 0.14  &   1.71 \pm 0.15  \\
%    \epsillong\text{ (\invGeV)}    &   \multicolumn{4}{c}{0.005420\text{ fixed}} \\
%    \epsilside\text{ (\invGeV)}    &  \multicolumn{4}{c}{-0.071554\text{ fixed}} \\
%    \epsilout\text{ (\invGeV)}     &  \multicolumn{4}{c}{-0.027472\text{ fixed}} \\
%    \gamma                         &   0.997\pm 0.002 &    1.007\pm 0.001 &   1.006\pm 0.001 &   1.005\pm 0.001 \\
%\hline
%    \chisq/\text{DoF}              &    1611/1162     &   6935/6266       &     11225/10660  &    13797/13381   \\
%    \text{CL}                      &    10^{-17}      &   10^{-9}         &      10^{-4}     &   \text{0.6\%}   \\
%\hline
%\end{array}%{tabular}
%$
%\end{center}
%\end{table}
%
 
 
 
 
 
 
\section{Discussion}  \label{sect:discussion}
\enote{The \taumodel\ does not describe the elongation. I think we have to admit this. Can we conclude anything from it?}
\enote{For example, what if the space-momentum correlation is stronger longitudinally than transversely?}
 
When BEC is parametrized in terms of $Q$ the \taumodel\
provides a good description,
much better than the Gaussian and Edgeworth parametrizations,
both in its effective-$R$ formulation, \Eq{eq:asymlevR2}, and in its full formulation for two-jet events, \Eq{eq:levyR2a}.
This suggests that the \taumodel\ formulation is at least                           semi-quantitatively  correct.
%This suggests that the \taumodel\ formulation is at least qualitatively, and likely semi-quantitatively, correct.
 
 
 
 
 
 
%BEC of two- and three-jet events, when parametrized in terms of $Q$,
%are observed to
%be well-described by a L\'evy parametrization
%incorporating strong correlations between configuration- and momentum-space.
% whereas Gaussian and Edgeworth parametrizations fail.
%%\mnote{refs}\enote{Need references.}
The \taumodel, as we have used it, incorporates a L\'evy distribution.
A L\'evy distribution arises naturally from a fractal,
or from a random walk or anomalous diffusion~\cite{metzler},
and the parton shower of the leading log
approximation of QCD is a fractal \cite{Dahlqvist:1989yc,Gustafson:1990qi,Gustafson:1991ru}.
In this case,
the L\'evy index of stability is related to the strong coupling
constant, \alphas, by\cite{alphasLevy,alphasLevy:arx}
 
\begin{equation}
    \alphas = \frac{2\pi}{3} \alpha^2 \;.
\end{equation}
Assuming (generalized) local parton hadron duality \cite{Azimov:lphd,Azimov:lphd1,genlphd},
one can expect that the distribution of hadrons retains the features of the gluon distribution.
For the value of $\alpha$ found in the fit of \Eq{eq:levyR2mt} for 2-jet events (\Tab{tab:2jetR2}) we find
$\alphas=0.46\pm0.01^{+0.04}_{-0.02}$.
This is a reasonable value for a scale of the order of 1\,\GeV,  % 1--2\,\GeV,
which is the scale at which the production of hadrons is thought to take place.
For comparison, from $\tau$ decay, $\alphas(m_\tau\approx1.8\GeV)=0.34\pm0.03$ \cite{PDG:2008},
and from the average value of $\alphas(M_\mathrm{Z})$,  $\alphas(1\GeV)=0.50^{+0.06}_{-0.05}$ \cite{runningalpha}.
That the value of \alphas\ obtained from $\alpha$ is so reasonable provides further support for the physical picture of the
\taumodel.
 
%\enote{Or should we compare to a value from the running of \alphas\ using the current value of $\alphas(M_\mathrm{Z})$?}
%\mnote{!!!!}\enote{This is a huge value if we argue that it should be at the scale of $\sqrt{s}$.
%                   On the other hand, it would seem reasonable for a scale of 1--2\,\GeV.
%                   But at this low scale, does the fractal picture still hold?
%                  }
%At first sight, these values of \alphas\ seem too small.
%However, we note that they are similar to the value,   $ 0.0805 \pm 0.0010$,
%found~\cite{L3:QCDphysrep}
%from a fit of the leading log formula to the \rs\ dependence of the mean multiplicity in
%$\Pep\Pem\rightarrow\mathrm{hadrons}$, while a fit using the next-to-next-to-next-to-leading order
%resulted in a value of $0.129\pm0.003$ consistent with more conventional determinations of $\alphas(M_\PZ)$.
 
 
It is of particular interest to point out the \mt\ dependence of the ``width'' of the source.
In \Eq{eq:levyR2mt} the parameter associated with the width is $\Delta \tau$.
Note that it enters \Eq{eq:levyR2mt} as $\Delta\tau Q^2/\overline{m}_\mathrm{t}$.
In a Gaussian parametrization the radius $R$ enters the parametrization as $R^2Q^2$.
Our observance that $\Delta\tau$ is independent of $\overline{m}_\mathrm{t}$ thus corresponds to
$R\propto1/\sqrt{\overline{m}_\mathrm{t}}$
and can be interpreted as confirmation of the observance~\cite{Smirnova:Nijm96,Dalen:Maha98,OPAL:2007}
of such a dependence of the Gaussian radii in 2- and 3-dimensional analyses of Z decays.
The lack of  dependence of all the parameters of \Eq{eq:levyR2mt} on the transverse mass is in
accordance with the \taumodel.
 
 
 
 
 
 
Given these successes,
the BEC fit results of the \taumodel\ have been used to reconstruct the emission function of two-jet events.
Particle production begins immediately after collision, increases rapidly and then decreases slowly,
occuring predominantly close to the light cone.
In the transverse plane a ring-like structure expands outwards,
which is similar to the picture in hadron-hadron interactions but unlike that of heavy ion collisions.
 
 
However,
the \taumodel\ breaks down when applied in the LCMS frame.
%As seen in the previous section,
The significant elongation previously observed is not accommodated in the \taumodel.
{\it Ad hoc} incorporation of elongation into the \taumodel\ parametrizations does, however,
result in acceptable fits with values of
the elongation consistent with those previously found using Gaussian and Edgeworth parametrizations.
Further theoretical work is thus needed to adapt the \taumodel.
We note that elongation is expected~\cite{AR98a} in the Lund model,
where the transverse and longitudinal components of a particle's momentum arise from different generation
mechanisms.
Therefore a possible improvement of the \taumodel\ could be the incorporation of a different degree of
correlation between a particle's momentum and its space-time production point in \Eq{eq:tau-corr}.
 
 
 
 
% Acknowledgements
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
%\section*{Acknowledgments}
%We thank T. Cs\"org\H{o} for many enlightening and useful discussions.
%We wish to express our gratitude to the CERN accelerator divisions for
%the excellent performance of the LEP machine.
%We acknowledge the contributions of the engineers
%and technicians who have participated in the construction
%and maintenance of this experiment.
 
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% The author list
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
%\newpage
%\section*{Author List}
%\input /l3/paper/namelist213.tex
\newpage
 
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%
%\begin{figure}
%  \centering
%  \includegraphics[width=.75\figwidth]{\olddirfig gauss_all.eps}
%  \includegraphics[width=.75\figwidth]{\olddirfig q_inv_edge100_linear_all.eps}
%  \caption{The Bose-Einstein correlation function $R_2$ for all events with the result of a fit of
%            (a) the Gaussian and (b) the Edgeworth parametrizations,
%%          the Gaussian parametrization,
%            \Eqs{eq:gaussR2} and (\ref{eq:edgeworthR2}), respectively.
%%          \Eq{eq:gaussR2}.
%           The dashed line represents the long-range part of the fit, \ie, $\gamma(1+\epsilon Q)$.
%           \label{fig:gauss_all}
%           }
%\end{figure}
 
 
\end{document}
