\begin{slide}{}

{\Large VI. \underline{Data Analysis}}\\
\begin{center}
\underline{Charged-Particle Overview}
\end{center}
\begin{enumerate}
\item Inclusive Charged-Particle Spectra
\begin{itemize}
\item Preliminary Analyses Only
\item Reasonable Shapes of
\begin{displaymath}
{{dN}\over{d\eta}} \; \; \&  \; \;{{dN}\over{d\phi}}
\end{displaymath} 
\item Normalization Consistent with\\
Extrapolations from UA5 and CDF
\item Overall Normalization Needs More Work
\end{itemize}
\item \underline{Bottom Line}:
\begin{itemize}
\item No Surprises, So Far,\\
at this Level of the Analysis
\item Many Corrections to Apply
\end{itemize}
\end{enumerate} 
\vfill\eject

\begin{center}
{\Large \underline{DCC Search: Analysis Challenges}}
\end{center}
\begin{itemize}
\item Small Acceptance:\\
Improbable that both $\gamma$'s from a $\pi^{0}$ Seen 
\item Conversion Efficiency per $\gamma \approx 50 \%$
\item Not All $\gamma$'s come from $\pi^{0}$'s 
\item Not All Observed Charged Tracks\\
Come from  $\pi^{\pm}$
\item Detection Efficiencies for Charged Tracks and $\gamma$'s:\\
Momentum Dependent and Unequal 
\item Detection Efficiency Correlations\\
$\sim$ Observed Multiplicity,\\
Background Density, etc.
\end{itemize}

\vfill\eject 

\begin{center}
{\Large \underline{Input to Efficiency Determination}}
\end{center}
\begin{itemize}
\item Varying Thickness of Pb Converter\\
($0, 0.5 X_{0},1.0 X_{0},2.0 X_{0}$)
\item Varying Composition of Pb Converter\\
($1.0 X_{0}, 1.0 Fe$)
\item Varying Window Thickness
\item Different Chamber Configurations:\\
(Original vs. Closely Packed)
\item Widely Varying Running Conditions
\item PYTHIA/GEANT Simulations
\end{itemize}

\vfill\eject 


\begin{center}
\underline{DCC Search Strategy-1}
\end{center}
\begin{itemize}
\item Generating Functions
\item Factorial and Cumulant Moments
\end{itemize}
\begin{itemize}
\item $P_{N} \equiv$ Probability of N $\pi$'s Being Produced
\item Binomial Distribution\\
of Charged \& Neutral Pions;\\
 with Neutral Fraction f:
\begin{displaymath}
P(N_{ch}, N_{0})= f^{N_{0}}(1-f)^{N_{ch}}{{N!} \over {N_{0}!N_{ch}!}}P_{N}
\end{displaymath}
\begin{displaymath}
N = N_{ch} + N_{0}
\end{displaymath}
\end{itemize}  
\vfill\eject


\begin{center}
\underline{DCC Search Strategy-2}
\end{center}
Assume the Parent Pion Distribution\\
$\sim$ Superposition of Poisson Distributions:
\begin{displaymath}
P_{N} = \int_{0}^{\infty}d\mu \, \rho(\mu){{\mu^{N}e^{-\mu}}\over {N!}}
\end{displaymath}
Construct Generating Function:
\begin{eqnarray*}
G(z_{ch}, z_{0}) & = & \sum\limits_{N_{ch},N_{0}=0}^{\infty}z_{ch}^{N_{ch}}z_{0}^{N_{0}}P(N_{ch},N_{0}) \\
                 & = & \int_{0}^{\infty}d\mu \, \rho(\mu)\int_{0}^{1}df \, p(f)e^{\mu(z-1)} 
\end{eqnarray*}
where 
\begin{displaymath}
z = fz_{0} + (1 - f)z_{ch}
\end{displaymath}
\begin{itemize}
\item Assume For Generic Pion Production:
\begin{displaymath}
       p(f) = \delta(f - {{1}\over{3}})
\end{displaymath}
\item Assume For DCC Production:
\begin{displaymath}
      p(f) = {{1}\over{2\sqrt{f}}}
\end{displaymath}
\end{itemize}
\vfill\eject

\begin{center}
{\Large \underline{DCC Search Strategy-3}}
\end{center}
Note:
\begin{itemize}
\item For Generic Production, $G(z_{ch}, z_{0})$ is a function of only one variable 
\item For DCC it is \underline{NOT}
\end{itemize}  
\vfill\eject


\begin{center}
{\Large \underline{DCC Search Strategy-4}}
\end{center}
Define
\begin{itemize}
\item $\epsilon_{ch}=$ Probability of Observing Charged-Pion Track
\item $\epsilon_{m}=$ Probability of Seeing m $\gamma$'s from the decaying $\pi^{0}$
\end{itemize}
\begin{displaymath}
\epsilon_{0}+ \epsilon_{1}+ \epsilon_{2} =1
\end{displaymath}
Assume $\epsilon_{m}$'s are Uncorrelated with Other Parameters\\
Then Generating Function for\\
\underline{Observed} Distribution $G(z_{ch},z_{\gamma})$\\
Obtained by Substitution:
\begin{eqnarray*}
      z_{0} & \rightarrow & \epsilon_{0}+ \epsilon_{1}z_{\gamma}+ \epsilon_{2}z_{\gamma}^{2}\\
     z_{ch} & \rightarrow & (1-\epsilon_{ch})+ \epsilon_{ch}z_{ch}
        \end{eqnarray*}

\vfill\eject


\begin{center}
{\Large \underline{DCC Search Strategy-5}}
\end{center}
Note:
\begin{itemize}
\item For Generic Production ($f = 1/3$) the Phenomenological G is still a Function of One Variable
\item DCC Correlations can Survive into Correlations Between Measured Quantities
\item Useful Quantitites for Phenomenology are \underline{Factorial Moments}:
\begin{displaymath}
      f_{mn} = (m, n)^{th} \; Derivative \, of \, G(z_{ch},z_{\gamma})
\end{displaymath}
at $z_{ch}=z_{\gamma}=1$
\vfill\eject

\begin{center}
{\Large \underline{DCC Search Strategy-6}}
\end{center}
\item \underline{Examples}:
\begin{eqnarray*}
      f_{10} & = & <N_{ch}>\\ 
			   f_{01} & = & <N_{\gamma}>\\
      f_{11} & = & <N_{ch}N_{\gamma}>\\
      f_{20} & = & <N_{ch}(N_{ch}-1)>  \\
      f_{30} & = & <N_{ch}(N_{ch}-1)(N_{ch}-2)>,\\
      f_{21} & = & <N_{ch}(N_{ch}-1)N_{\gamma}>
\end{eqnarray*}
\item These Moments are Easily Extracted from Data
\item Sensitivity to Statistics \& \\
Tail of Distributions
Can be Tested Straightforwardly 
        \end{itemize} 
\vfill\eject

\begin{center}
{\Large \underline{DCC Search Strategy-7}}
\end{center}
\begin{itemize}
\item Within our Assumptions:\\
Ratios of (Normalized)\\
Factorial Moments are:
\begin{itemize}
\item Independent of Efficiency Corrections
\item Unity if No DCC is Present
\item Very Different for Pure DCC
\end{itemize}
\vfill\eject

\begin{center}
{\Large \underline{DCC Search Strategy-8}}
\end{center} 
\item Examples: 
\begin{eqnarray*}
         r_{2} & = & {{f_{11}f_{10}}\over{f_{20}f_{01}}}=  1(Generic)\\ 
               & \,&\; \; \; \; \; \; \; \; \; \;\; \; \;     ={{1}\over{2}}(DCC)\\
         r_{3} & = & {{f_{21}f_{10}}\over{f_{30}f_{01}}} =  1(Generic)\\ 
               & \, & \; \; \; \; \; \; \; \; \; \;\; \; \;   ={{1}\over{3}}(DCC)
\end{eqnarray*}
\item Generalization:\\
If $\epsilon_{2} =0$ ($2\gamma$ Efficiency Vanishes),\\
No Correlations,\\
\& No DCC, then
\begin{displaymath}
	       r_{jk}= {{f_{jk}}\over{f_{(j+k)0}}}({{f_{10}}\over{f_{01}}})^{k}=1
\end{displaymath}
\end{itemize} 
\vfill\eject

 
\begin{center}
{\Large \underline{Very Preliminary Results-1}}
\end{center}
\begin{itemize}
\item \underline{Results for $r_{2}\equiv r_{11}$}\\
 (Based on 1995 Runs ($10^{6}$ Events) 
\begin{eqnarray*}
    r_{2} & = & 0.91 \pm .01 (DATA)\\
    r_{2} & = & 0.96 \pm .05 (Generic)\\
    r_{2} & = & 0.56 \pm .02 (Pure DCC + GEANT)
\end{eqnarray*}
Note: Generic $\sim$ PYTHIA/GEANT\\
Also Checked:
\begin{itemize}
\item No Significant Run-to-Run Dependence
\item No Significant Dependence on Converter\\
Thickness or Composition 
\end{itemize}
\vfill\eject



\begin{center}
{\Large \underline{Very Preliminary Results-2}}
\end{center}
\item \underline{Results for $r_{2} ($1996 Data)} \begin{displaymath}
r_{2} = 0.98 \pm .01
\end{displaymath}
(Normal Trigger;$\sim 10^{6}$ Events)
\begin{displaymath}
r_{2} = 1.08 \pm .06
\end{displaymath}
($x_{F}=0.9$, Upstream Tag; 10\,k Events) \begin{itemize}
\item Some Changes in Tracking Algorithms \item Chamber Configuration Changed;
 Lower Backgrounds \end{itemize}
\vfill\eject

\begin{center}
\underline{Very Preliminary Results-3}
\end{center}
\item \underline{Very Preliminary Results for $r_{ij}$ (1996 Data)}
 \begin{eqnarray*}
r_{11} & = & 0.98 \pm .01 \; \; r_{02}=1.50 \pm .01 \\
 r_{21} & = & 0.99 \pm .01 \; \; r_{12}=1.41 \pm .03 \\
 r_{31} &= & 1.02 \pm .04 \; \; r_{22}=1.40 \pm .07 \\
 r_{03} & = & 2.97 \pm .10 \; \; r_{13}=2.64 \pm .18 \end{eqnarray*}
\vfill\eject




\begin{center}
{\Large \underline{Results So Far: Summary}}
\end{center}
\begin{itemize}
\item Data is Consistent with\\
Only Generic Production Mechanisms 
\item Dependence of $r_{ij}$ on Systematic Effects $\leq$ 5\% 
\item Sensitivity to DCC Admixture is at $10-20\%$ Level\\
(Depends on Modeling, etc.)
\end{itemize}

\end{itemize}

\begin{center}
{\Large Many More Studies Need to Be Done}
\end{center}

\end{slide}{}