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\begin{document} 
\title{
%\begin{flushright}
%\vspace*{-1cm}{\rm CDF/PUB/MIN\_BIAS/PUBLIC/9719}
%\end{flushright}
Minimum Bias Studies at CDF and Comparison with MonteCarlo
} 
\author{Niccol\`o Moggi$^1$ (for the CDF Collaboration)}
\institute{$^1$Istituto Nazionale Fisica Nucelare, Bologna}
\maketitle
\begin{abstract}
Measurements of particle production and inclusive differential cross sections in inelastic 
$p\bar{p}$ collisions are reported. The data were collected with a minimum-bias trigger at 
the Tevatron Collider with the CDF~II experiment. Previous measurements are widely extended 
in range and precision.
A comparison with a {\sc pythia} prediction at the hadron level is performed.
Inclusive particle production is fairly well reproduced only in the low transverse 
momentum range. Final state  correlation measurements are poorly reproduced, but 
favor models with multiple parton interactions.
\end{abstract}

\section{Introduction}
In hadron collisions, hard interactions are theoretically well described as collisions 
of two incoming partons along with softer interactions from the remaining partons. The
so-called “minimum-bias” (MB) interactions, on the contrary, can only be defined through
a description of the experimental apparatus that triggers the collection of the data. Such a
trigger is meant to collect events from all possible inelastic interactions proportionally
to their natural production rate. 
MB physics offers a unique ground for studying both the theoretically poorly understood softer 
phenomena and the interplay between the soft and the hard perturbative interactions. 

The understanding of the softer components of MB is interesting not only in its own right, 
but is also important for precision measurements of hard interactions in which soft effects
need to be subtracted (see, e.g. \cite{skands}). 
The observables that are experimentally accessible in the MB final state represent a 
complicated mixture of different physics effects such that most models could readily be tuned 
to give an acceptable description of each single observable, but not to describe simultaneously 
the entire set. 
Effects due to multiple parton parton interactions (MPI) are essential for an exhaustive
description of inelastic non-diffractive hadron interactions.

\section{The CDF Detector and Data Samples}
\subsection{The Data Collection and Event Selection}
\label{sec:data}
This analysis is based on an integrated luminosity of 506 pb$^{−1}$ collected with the CDF II
detector at $\sqrt{s}=1.96$~TeV during the first Tevatron stores in Run II. 
CDF II is a general purpose detector that combines precision charged particle tracking
with projective geometry calorimeter towers. A detailed description of the detector,
with detailed information about the transverse momentum ($p_T$) and transverse energy ($E_T$) 
resolutions, can be found elsewhere \cite{CDF1,*CDF2,*CDF3}. 

Two systems of gas Cherenkov counters (CLC) \cite{clc}, covering the pseudorapidity
forward regions $3.7 < |\eta| < 4.7$, are used to determine the luminosity. 
The MB trigger is implemented by requiring a coincidence in time of signals in both forward
and backward CLC modules.

%For the transverse energy measurements, only runs with lower initial instantaneous luminosity
%have been used in order to reduce the effects of event pile-up in the calorimeters. 
%The average instantaneous luminosities of the two samples are roughly 
%$17 \times 10^{30}$~cm$^{-2}$s$^{-1}$ for the energy subsample and $20 \times 10^{30}$~cm$^{-2}$s$^{-1}$ 
%for the full sample.
Only runs with lower initial instantaneous luminosity have been used in order to reduce the effects 
of event pile-up. The average instantaneous luminosity of the full MB sample is roughly 
$20 \times 10^{30}$~cm$^{-2}$s$^{-1}$. For measurements where the calorimeter is involved, only a
subsample of average luminosity $17 \times 10^{30}$~cm$^{-2}$s$^{-1}$ was used.

Primary vertices are identified by the convergence of reconstructed tracks along the $z-$axis. 
For vertices reconsructed from less than ten tracks a requirement that they be symmetric is added:
the quantity $|(N^{+} − N^{−})/(N^{+} + N^{−})|$, where $N^{\pm}$ is the number
of tracks in the positive or negative $\eta$ hemisphere, cannot equal one.
Only events that contain one, and only one, primary vertex in the fiducial region
$|Z_{vtx}| \leq 40$~cm centered around the nominal CDF $z = 0$ position are accepted. 
This fiducial interval is further restricted to $|Z_{vtx}| \leq 20$~cm when measurements 
with the calorimeter are involved.


\subsection{The MonteCarlo Sample}
A sample of simulated Monte Carlo (MC) events about twice the size of the data was generated 
with {\sc pythia} version 6.2 \cite{pythia}, with parameters optimized for the best reproduction
of minimum-bias interactions. 
%{\sc pythia} Tune A [2] describes the MB interactions starting
%from a leading order QCD $2 \rightarrow 2$ matrix element augmented by initial- and final-state
%showers and multiple parton interactions [22], folded in with CTEQ5L parton distribution
%functions [23] and the Lund string fragmentation model [24]. 
To model the mixture of hard and soft interactions, {\sc pythia} Tune A \cite{pythia_new}
\cite{tuneA,*tuneA_2} introduces a $p_{T}^{0}$ cut off parameter that regulates the
divergence of the 2-to-2 parton-parton perturbative cross section at low momenta. This
parameter is used also to regulate the additional parton-parton scatterings that may occur
in the same collision \cite{mpi}. Thus, fixing the amount of multiple-parton interactions (i.e., setting
the $p_{T}$ cut-off) allows the hard 2-to-2 parton-parton scattering to be extended all the way
down to $p_{T}(hard) = 0$, without hitting a divergence. The amount of hard scattering in
simulated MB events is, therefore, related to the activity of the so-called underlying event in
the hard scattering processes. The final state, likewise, is subject to several effects such as
the treatments of the beam remnants and color (re)connection effects. 
The pythia Tune A results presented here are the predictions, not fits.

A run-dependent simulation with a realistic distribution of multiple interactions was
employed to compute corrections and acceptance. Events were fully simulated through the 
detector and successively reconstructed with the standard CDF reconstruction chain.
All data is corrected to hadron level.
The definition of primary particles was to consider all particles with mean lifetime 
$\tau > 0.3\times 10^{-10}$~s produced promptly in the $p\bar{p}$ interaction, 
and the decay products of those with shorter mean lifetimes. With this definition strange hadrons 
are included among the primary particles, and those that are not reconstructed are corrected for. 
On the other hand, their decay products (mainly $\pi^{\pm}$ from $K^{0}_{S}$ decays) are excluded, 
while those from heavier flavor hadrons are included.


\section{Results}
\label{sec:result}

\subsection{Efficiency and Acceptance Corrections}
Reconstructed tracks are accepted if they comply with a minimal set of quality selections.
Primary charged particles are selected by requiring that they originate in a fiducial region 
around the $p\bar{p}$ vertex.
%in the plane $(d_{0};\Delta z)$, where $d_{0}$ is the track impact parameter with respect to the
%beam axis and $\Delta z$ is the track mimimum distance to the $z$-coordinate of the 
%$p\bar{p}$ vertex. These selection criteria were determined from MC simulation as the ones 
%that maximize the ratio of primary to secondary particles.
In order to optimize the efficiency and acceptance conditions particles 
are required to have a transverse momentum greater than 0.4 GeV/$c$ and pseudorapidity 
$|\eta| \leq 1$. 

The transverse energy sum ($\sum E_{T}$) is computed in the limited region $|\eta| \leq 1$ 
as the scalar sum over the calorimeter towers of the transverse energies in the electromagnetic 
and hadronic compartments. The calorimeter response has been evaluated with MC. 
The region below about 5 GeV is the most critical. 
The reliability of MC in evaluating the calorimeter response was checked 
%– for charged particles – 
against the single particle response measured from data.
%down to a track $p_{T}$ of about 400 MeV/$c$. 
The simulation of the energy deposition of neutral particles was assumed to be correct.
%A systematic uncertainty as high as 15\% is attributed to the $\sum E_{T}$ measurement in this region.

In the end, all data presented is corrected for the trigger and vertex efficiency, undetected
pile-up, diffractive background and event selection acceptance.
Charged particle measurements are corrected also for the tracking efficiency, contamination 
of secondary particles (particle interaction, pair creation), particle decays and mis-identified tracks.
These quantties are evaluated as a function of $p_{T}$, in different ranges of track multiplicity.
The total correction includes also the smearing correction for very high $p_{T}$ tracks, 
where the small curvature may cause a significant dispersion in the measure of the momentum.
$\sum E_{T}$ measurements are corrected for the calorimeter response and acceptance, and
are unfolded to correct the dispersion due to the finite calorimeter resolution.

\subsection{The charged particle $p_{T}$ spectrum}
\label{sec:pt}
We may write the single-particle invariant $p_T$ differential cross section as:
\begin{equation}
E\frac{d^3\sigma}{dp^3} = \frac{d^{3}\sigma}{p_{T} \Delta\phi \Delta y dp_{T}} =
\frac{N_{pcles}/ (\varepsilon \times A)}{\mathcal{L} p_{T} \Delta\phi \Delta y dp_{T}} \; ,
\end{equation}
where $E$, $p$, $\phi$, and $y$ are the particle energy, momentum, azimuthal angle and rapidity, 
respectively; $N_{pcles}$ is the raw number of charged particles that is to be corrected for all 
efficiencies ($\varepsilon$) and acceptance ($A$). 
$\mathcal{L}$ is the effective time-integrated luminosity of the sample.
The accepted region in $\Delta y$ is calculated from the $\eta$ for each charged track, always
assuming the charged pion mass.
The differential cross section is shown in Fig.~\ref{fig:dsdpt}.

\begin{figure}[t!]
\vspace{-1ex}
  \begin{center}
    \includegraphics[width=8cm,clip]{dsdpt_proceedings.eps}\includegraphics[width=7cm,clip]{dsdpt_vs_pythia_proceedings.eps}
    \caption{{\it Left:} the track $p_T$ differential cross section with statistical uncertainty
    is shown.
    A fit to the functional form used in the 1988 analysys in the region of $0.4<p_T<10$~GeV/$c$ 
    is also shown (dashed line). The fit with the more complex function (Eq.\ref{eq:fit2}) is shown 
    as a continuous line.
    In the plot at the bottom, the systematic and the total uncertainties are shown.
    {\it Right:} comparison with {\sc pythia} Tune A simulation at hadron level. 
    The ratio of data over prediction is shown in the lower plot. Note that these distributions 
    are cut off at 50~GeV/$c$ since {\sc pythia} does not produce particles at all beyond that value.}
    \label{fig:dsdpt}
  \end{center}
\end{figure}

This measure was discussed in~\cite{studimisti,*studimisti2} and last published by the CDF 
collaboration in 1988~\cite{ptpaper}. There is a scale factor of 2 between the 1988 and the 
new measurement, due to different normalization.
Besides this, the new measurement is about 4\% higher than the previous one. 
At least part of this difference may be explained by the increased center-of-mass 
energy of the collisions from 1800 to 1960~GeV.
The new measurement extends the momentum spectrum from 10 to over 100 GeV/$c$, and enables 
verification of the empirical modeling of minimum-bias production up to the high $p_{T}$ 
production region.

We observe that modeling the spectrum with the power-law form used in 1988 to fit the 
distribution, does not account for the high $p_T$ tail observed in this measurement 
(Fig.~\ref{fig:dsdpt}, left).
%The form in Eq.~\ref{eq:fit} is merely empirical, and the $\chi^{2}$s of the 1988 data fits
%were already quite poor. 
Nevertheless, in the limited region up to $p_{T}=10$~GeV/$c$, we obtain,
for the present data, a set of fit parameters compatible with those published in 1988.
%\begin{equation}
%f = A \left(\frac{p_{0}}{p_{T}+p_{0}}\right)^{n}  \; .
%\label{eq:fit}
%\end{equation}
In our measurement, the tail of the distribution is at least three orders of magnitude higher 
than what could be expected by extrapolating to high $p_T$ the function that fits the 
low $p_T$ region. In order to fit the whole spectrum, we introduced a more complex parametrization 
by adding a second term to the function used in \cite{ptpaper} (Eq.\ref{eq:fit2}):
\begin{equation}
f = A \left(\frac{p_{0}}{p_{T}+p_{0}}\right)^{n} + B\left(\frac{1}{p_T}\right)^{s} \; .
\label{eq:fit2}
\end{equation}
\noindent With this empirical function, we obtain a good  $\chi^{2}$ but the 
data are still not well reproduced above about 100~Gev/$c$.

Figure~\ref{fig:dsdpt} (right) shows a comparison with {\sc pythia} simulation at hadron level.
Also in this case, the data show a larger cross section at high $p_T$ starting from about 
20~GeV/$c$. The MC generator does not produce any particles at all beyond 50~GeV/$c$.

\subsection{The dependence of $\langle p_{T} \rangle $ on the particle multiplicity}
\label{sec:ptm}
The dependence of the particle transverse momentum on multiplicity ($\langle p_{T}\rangle (N_{ch})$) 
is computed as the average $p_T$ of all charged primary particles
in events with the same charged multiplicity ($N_{ch}$), as a function of $N_{ch}$.
A study of $\langle p_{T}\rangle (N_{ch})$ was already performed by CDF in Run~I and published 
in~\cite{run1paper}.
This new measurement benefits from the larger statistics obtained with a dedicated 
"high multiplicity" trigger. Data from this trigger are included by merging them into the MB sample. 

This is one of the variables most sensitive to the combination of the various physical effects 
present in MB collisions, and is also the variable most poorly reproduced by the available MC 
generators. 
The rate of change of $\langle p_T\rangle$ versus $N_{ch}$ is a measure of the amount of hard 
versus soft processes contributing to minimum-bias collisions; in simulation the rate is 
sensitive to the modeling of the multiple-parton interactions (MPI)~\cite{skands}.

%The model that currently best reproduces the dependence, {\sc pythia} Tune~A, was tuned to fit
%the activity in the so-called underlying event in high transverse momentum jet production~\cite{Field2}.
%However, it uses the same cut-off parameter $\hat{p}_{T^{0}}$ to regulate the divergence of the primary 
%2-to-2 parton-parton scattering and the number of additional parton-parton interactions in the same 
%collision. In addition, in {\sc pythia} the final state is subject to color (re)connection effects 
%between different parton interactions of the same collision.

\begin{figure}[htb]
  \vspace{-1ex}
  \begin{center}
    \includegraphics[width=9cm,clip]{ptm_pythia_proceedings.eps}
    \caption{The dependence of the average charged particle $p_T$ on the event multiplicity is shown.
    A comparison with various {\sc pythia} tunes at hadron level is shown. Tune~A with
    $\hat{p}_{T^{0}}=1.5$~GeV/$c$ was used to compute the MC corrections in this analysis
    (the statistical uncertainty is shown only for the highest multiplicities where it is significant).
    Tune~A with $\hat{p}_{T^{0}}=0$~GeV/$c$ is very similar to $\hat{p}_{T^{0}}=1.5$~GeV/$c$. 
    The same tuning with no multiple parton interactions allowed (``no MPI'') yields an average 
    $p_T$ much higher than data for multiplicities greater than about 5. 
    The ATLAS tune yields too low an average $p_T$ over the whole multiplicity range.
    The uncertainties shown are only statistical.}
    \label{fig:ptm}
  \end{center}
\end{figure}

If only two processes contribute to the MB final state, one soft, and one hard 
(the hard 2-to-2 parton-parton scattering), then demanding large $N_{ch}$ would preferentially 
select the hard process and lead to a high $\langle p_T\rangle$.  
However, we see from Fig.~\ref{fig:ptm} (Tune~A, no MPI) that with these two processes alone, 
the average $p_T$ increases much too rapidly. MPI provide another mechanism for producing large 
multiplicities that are harder than the beam-beam remnants, but not as hard as the primary 
2-to-2 hard scattering.
By introducing this mechanism, {\sc pythia} in the Tune~A configuration gives a fairly good 
description of $\langle p_{T}\rangle (N_{ch})$ and, although the data are quantitatively not 
exactly reproduced, there is great progress over fits to Run~I data \cite{run1paper}.
The systematic uncertainty is always within 2\%, a value significantly smaller
than the discrepancy with data.

\subsection{The $\sum E_{T}$  spectrum}
\label{sec:et}
%The $\sum E_T$ differential cross section is given by $d^3\sigma / (\Delta\phi \Delta\eta dE_{T})$
%\begin{equation}
%\frac{d^3\sigma}{\Delta\phi \Delta\eta dE_{T}} =
%\frac{N_{ev} / (\varepsilon \cdot A)}{\mathcal{L}\Delta\phi \Delta\eta dE_{T}} \; ,
%\end{equation}
The differential cross section $d^3\sigma / (\Delta\phi \Delta\eta dE_{T})$ 
for $|\eta|<1$ is shown in Fig.~\ref{fig:dsdet}.
The raw and corrected event average transverse energies are $E_{T}=7.350 \pm 0.001 (\mathrm{stat.})$ 
and $E_{T}=10.4 \pm 0.2 (\mathrm{stat.}) \pm 0.7 (\mathrm{syst.})$~GeV, respectively.

The measurement of the event transverse energy sum is new to the field, and
represents a first attempt at describing the full final state including neutral particles. In this
regard, it is complementary to the charged particle measurement in describing the global
features of the inelastic $p\bar{p}$ cross section.

The {\sc pythia} simulation does not closely reproduce the data over the whole $\sum E_{T}$
spectrum. In particular the peak of the MC distribution is slightly shifted to higher energies 
with respect to the data.

\begin{figure}[h]
  \begin{center}
    \includegraphics[width=8cm,clip]{dsdet_pythia_proceedings.eps}
    \caption{The differential $\sum E_T$ cross section in $|\eta|<1$ compared to a {\sc pythia}
      prediction at hadron level. The ratio of data to  {\sc pythia} Tune~A is shown in the lower plot.}
    \label{fig:dsdet}
  \end{center}
\end{figure}

\subsection{Systematic Uncertainties}
\label{sec:sys}
We have detected several possible sources of systematic uncertainties.
The largest ones are the uncertainties on the calorimeter response (up to 15\% at lower $\sum E_{T}$), 
on the pile-up correction,
on the diffractive background, and the uncertainty related to the MC generator used to compute
the various corrections. These uncertainties are, in general, larger in the $\sum E_{T}$ measurement
than in charged particle measurements.

There is an overall global 6\% systematic uncertainty on the effective time-integrated luminosity 
measurement \cite{Xs} that is to be added to all the cross section measurements.

\section{Experimentl Hot Topics}

\subsection{The MB trigger}
\label{sec:trigger}
The acceptance of the MB trigger has been measured by comparing it to a sample of zero-bias events 
collected during the same period. The zero-bias data set is collected without any trigger requirements, 
simply by starting the data acquisition at the Tevatron radio-frequency signal.
The results indicate that the acceptance depends on a number of variables, most of which are, in some way,
related to the number of tracks present in the detector: number of interactions, number of tracks, 
instantaneous luminosity and the CLC calibration.
We parametrized the dependence on these variables so that a correction could be applied
on an event-by-event basis.
The total trigger acceptance therefore increases linearly with the instantaneous luminosity.
As a function of the reconstructed number of tracks, the acceptance is well represented by a typical
turn-on curve starting at about 20\% (few tracks) and reaching its plateau with a value between 97 
and 99\% for about 15 tracks.

\subsection{Pile Up}
\label{sec:pu}
In spite of the low instantaneous luminosity, the selected data sample contains a contamination 
of pile-up events. This is due to multiple interactions when the separation between 
$p\bar{p}$ collisions is less than the vertex resolution in the $z-$coordinate (about 3 cm).

\begin{figure}[h]
  \begin{center}
    \includegraphics[width=8cm,clip]{multlum_proceedings.eps}
    \caption{The raw event average charged particle multiplicity as a function of the instantaneous luminosity.
    The line represents a linear fit (with slope equal to 0.0022$\pm$0.0003). The uncertainty is 
    statistical only.}
    \label{fig:lumsys}
  \end{center}
\end{figure}

The number of undetected events was estimated indirectly  by plotting the average $N_{ch}$ 
as a function of the instantaneous luminosity (Fig.~\ref{fig:lumsys}).
In this plot, the increase in $\langle N_{ch}\rangle$ is due to the increase in number of pile-up events.
We assume that virtually no pile-up is present at a luminosity of 
$\mathcal{L}=1\times 10^{30}$~cm$^{-2}$s$^{-1}$. The difference with respect to the 
$\langle N_{ch}\rangle$ at the average luminosity of the sample yields the estimated number of events 
that went unobserved.
However, although the pile-up probability in the low luminosity sample is small ($<1\%$), it is not 
negligible.  By assuming conservatively an uncertainty on the MB inelastic-non-diffractive cross section 
used by the MC generator of 6~mb, we calculate that this is equivalent to a variation in the sample 
average luminosity of $2.5\times 10^{30}$~cm$^{-2}s^{-1}$. This, in turn, 
corresponds to an uncertainty $<3$\% on the $\sum E_{T}$ distribution and negligible on the charged
particle distributions. 

\section{Conclusions}

%Minimum-bias collisions are a mixture of hard processes (perturbative QCD) and soft processes 
%(non-perturbative QCD) and, therefore, are very difficult to simulate. 
%They contain soft 
%beam-beam remnants, hard QCD 2-to-2 parton-parton scattering, and multiple parton interactions 
%(soft and hard). To simulate such collisions correctly, the correct combination of all the 
%processes involved must be known. 
%The absence of threshold effects in all the distributions studied, and on such a large scale
%as reached by the present measurements, indicate that there is no clear separation of hard and 
%soft processes.
A set of high precision measurements of the final state in minimum-bias interactions is provided
and compared to the best available MC model.

The former power-law modeling of the single particle $p_T$ spectrum is not compatible with
the high momentum tail ($p_{T}\geq 10$~GeV/$c$) observed in data. 
The more recent tunings of the {\sc pythia} MC generator (Tune~A) reproduce the inclusive  
charged particle $p_T$ distribution in data within 10\% up to $p_T\simeq20$~GeV/$c$ but the 
prediction lies below the data at high $p_T$. 

The $\sum E_T$ cross section represents the first attempt to measure the neutral
particle activity in MB. {\sc pythia} Tune A does not closely reproduce the shape of 
the distribution.

%The dependence of the charged particle momentum
%on the event multiplicity seems to be one of the most sensitive variables to the relative
%contributions by several components of MB interactions. 
The mechanism of multiple parton interactions (with strong final-state correlations among them) has 
been shown to be very useful in order to reproduce high multiplicity final states with the correct 
particle transverse momenta. In fact, the data very much disfavor models without MPI, and put strong 
constraints on  multiple-parton interaction models.

The data presented here can be used to improve QCD Monte Carlo models and further our understanding 
of multiple parton interactions.
A detailed understanding of MB interactions is especially important in very high luminosity
environments (such as at the LHC) where a large number of such interactions is expected in the same 
bunch crossing.

\begin{footnotesize}
\bibliographystyle{mpi08} 
{\raggedright
%\bibliography{proceedings}
\bibliography{mpi08}
}
\end{footnotesize}
\end{document}
