\documentclass[epj,referee]{svjour}
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\begin{document}
\hugehead
\title{Measurement of charged pions in $^{12}$C + $^{12}$C collisions at 1 and 2A~GeV with HADES}
\author{
G.~Agakishiev\inst{8},
C.~Agodi\inst{1},
A.~Balanda\inst{3,V},
G.~Bellia\inst{1,I},
D.~Belver\inst{15},
A.~Belyaev\inst{6},
J.~Bielcik\inst{4},
A.~Blanco\inst{2},
A.~Bortolotti\inst{9};
J.~L.~Boyard\inst{13},
P.~Braun-Munzinger\inst{4},
P.~Cabanelas\inst{15},
S.~Chernenko\inst{6},
T.~Christ\inst{11},
R.~Coniglione\inst{1},
M.~Destefanis\inst{8},
J.~D\'{\i}az\inst{16},
F.~Dohrmann\inst{5},
I.~Dur\'{a}n\inst{15},
A.~Dybczak\inst{3},
T.~Eberl\inst{11},
L.~Fabbietti\inst{11},
O.~Fateev\inst{6},
R.~Ferreira-Marques\inst{2,III},
P.~Finocchiaro\inst{1},
P.~Fonte\inst{2,II},
J.~Friese\inst{11},
I.~Fr\"{o}hlich\inst{7},
T.~Galatyuk\inst{4},
J.~A.~Garz\'{o}n\inst{15},
R.~Gernh\"{a}user\inst{11},
A.~Gil\inst{16},
C.~Gilardi\inst{8},
M.~Golubeva\inst{10},
D.~Gonz\'{a}lez-D\'{\i}az\inst{4},
E.~Grosse\inst{5},
F.~Guber\inst{10},
M.~Heilmann\inst{7},
T.~Heinz\inst{4},
T.~Hennino\inst{13},
R.~Holzmann\inst{4},
A.~Ierusalimov\inst{6},
I.~Iori\inst{9,IV},
A.~Ivashkin\inst{10},
M.~Jurkovic\inst{11},
B.~K\"{a}mpfer\inst{5},
K.~Kanaki\inst{5},
T.~Karavicheva\inst{10},
D.~Kirschner\inst{8},
I.~Koenig\inst{4},
W.~Koenig\inst{4},
B.~W.~Kolb\inst{4},
R.~Kotte\inst{5},
A.~Kozuch\inst{3,V},
A.~Kr\'{a}sa\inst{14},
F.~K\v{r}\'{\i}\v{z}ek\inst{14},
R.~Kr\"{u}cken\inst{11},
W.~K\"{u}hn\inst{8},
A.~Kugler\inst{14},
A.~Kurepin\inst{10},
J.~Lamas-Valverde\inst{15},
S.~Lang\inst{4},
J.~S.~Lange\inst{8},
K.~Lapidus\inst{10},
L.~Lopes\inst{2},
M.~Lorenz\inst{7},
L.~Maier\inst{11},
C.~Maiolino\inst{1},
A.~Mangiarotti\inst{2},
J.~Mar\'{\i}n\inst{15},
J.~Markert\inst{7},
V.~Metag\inst{8},
B.~Michalska\inst{9},
J.~Michel\inst{7},
E.~Morini\`{e}re\inst{13},
J.~Mousa\inst{12} \thanks{e-mail: {mousa@ucy.ac.cy}},
M.~M\"{u}nch\inst{4},
C.~M\"{u}ntz\inst{7},
L.~Naumann\inst{5},
R.~Novotny\inst{8},
J.~Otwinowski\inst{3},
Y.~C.~Pachmayer\inst{7},
M.~Palka\inst{4},
Y.~Parpottas\inst{12},
V.~Pechenov\inst{8},
O.~Pechenova\inst{8},
T.~P\'{e}rez~Cavalcanti\inst{8},
P.~Piattelli\inst{1},
J.~Pietraszko\inst{4},
V.~Posp\'{\i}\v{s}il\inst{14},
W.~Przygoda\inst{3,e},
B.~Ramstein\inst{13},
A.~Reshetin\inst{10},
M.~Roy-Stephan\inst{13},
A.~Rustamov\inst{4},
A.~Sadovsky\inst{10},
B.~Sailer\inst{11},
P.~Salabura\inst{3},
P.~Sapienza\inst{1},
A.~Schmah\inst{11},
C.~Schroeder\inst{4},
E.~Schwab\inst{4},
R.~Simon\inst{4},
Yu.G.~Sobolev\inst{14},
S.~Spataro\inst{8},
B.~Spruck\inst{8},
H.~Str\"{o}bele\inst{7},
J.~Stroth\inst{7,4},
C.~Sturm\inst{7},
M.~Sudol\inst{13},
A.~Tarantola\inst{7},
K.~Teilab\inst{7},
P.~Tlust\'{y}\inst{14} \thanks{e-mail: {tlusty@ujf.cas.cz}},
M.~Traxler\inst{4},
R.~Trebacz\inst{3},
H.~Tsertos\inst{12},
V.~Wagner\inst{14},
M.~Weber\inst{11},
M.~Wisniowski\inst{3},
T.~Wojcik\inst{3},
J.~W\"{u}stenfeld\inst{5},
S.~Yurevich\inst{4},
Y.~Zanevsky\inst{6},
P.~Zhou\inst{5},
P.~Zumbruch\inst{4}\\
}
%
%
%\offprints{J.Mousa, P.Tlusty} % Insert a name or remove this line
%\mail{mousa@ucy.ac.cy \\\and tlusty@ujf.cas.cz}
\institute{
\inst{1} Istituto Nazionale di Fisica Nucleare - Laboratori Nazionali del Sud, 95125~Catania, Italy \\
\inst{2} LIP-Laborat\'{o}rio de Instrumenta\c{c}\~{a}o e F\'{\i}sica Experimental de Part\'{\i}culas , 3004-516~Coimbra, Portugal\\
\inst{3} Smoluchowski Institute of Physics, Jagiellonian University of Cracow, 30-059~Krak\'{o}w, Poland\\
\inst{4} Gesellschaft f\"{u}r Schwerionenforschung mbH, 64291~Darmstadt, Germany\\
\inst{5} Institut f\"{u}r Strahlenphysik, Forschungszentrum Dresden-Rossendorf, 01314~Dresden, Germany\\
\inst{6} Joint Institute of Nuclear Research, 141980~Dubna, Russia\\
\inst{7} Institut f\"{u}r Kernphysik, Johann Wolfgang Goethe-Universit\"{a}t, 60438 ~Frankfurt, Germany\\
\inst{8} II. Physikalisches Institut, Justus Liebig Universit\"{a}t Giessen, 35392~Giessen, Germany\\
\inst{9} Istituto Nazionale di Fisica Nucleare, Sezione di Milano, 20133~Milano, Italy\\
\inst{10} Institute for Nuclear Research, Russian Academy of Science, 117312~Moscow, Russia\\
\inst{11} Physik Department E12, Technische Universit\"{a}t M\"{u}nchen, 85748~M\"{u}nchen, Germany\\
\inst{12} Department of Physics, University of Cyprus, 1678~Nicosia, Cyprus\\
\inst{13} Institut de Physique Nucl\'{e}aire (UMR 8608), CNRS/IN2P3 - Universit\'{e} Paris Sud, F-91406~Orsay Cedex, France\\
\inst{14} Nuclear Physics Institute, Academy of Sciences of Czech Republic, 25068~Rez, Czech Republic\\
\inst{15} Departamento de F\'{\i}sica de Part\'{\i}culas, University of Santiago de Compostela, 15782~Santiago de Compostela, Spain\\
\inst{16} Instituto de F\'{\i}sica Corpuscular, Universidad de Valencia-CSIC, 46971~Valencia, Spain\\
\inst{I} Also at Dipartimento di Fisica e Astronomia, Universit\`{a} di Catania, 95125~Catania, Italy\\
\inst{II} Also at ISEC Coimbra, ~Coimbra, Portugal\\
\inst{III} Also at Universidade de Coimbra, ~Coimbra, Portugal\\
\inst{IV} Also at Dipartimento di Fisica, Universit\`{a} di Milano, 20133~Milano, Italy\\
\inst{V} Also at Panstwowa Wyzsza Szkola Zawodowa , 33-300~Nowy Sacz, Poland\\
}
\date{Received: \today / Revised version: \today}
\abstract{
We present the results of a study of charged pion production in $^{12}$C + $^{12}$C
collisions at incident beam energies of 1 $\,$
and 2A~GeV using the HADES spectrometer at GSI. The main emphasis of
of the HADES program of measurements is on the dielectron signal from
the early
phase of the collision. Here we discuss the data with respect to the
emission of charged hadrons, specifically the production of
$\pi^\pm$ mesons, which are related to neutral pions
representing a dominant contribution to the dielectron yield.
We have performed the first large-angular range
measurement of the distribution of $\pi^\pm$ mesons for the C+C collision system
covering a fairly large rapidity interval.
The yields, transverse mass and
angular distributions are compared with calculations with
a transport model as well as with existing data from
other experiments. The anisotropy is systematically analyzed.
\PACS{
{25.75.-q}{ heavy-ion collisions - }
{25.75.Dw}{ charged pion spectra}
} % end of PACS codes
}% end of abstract
\maketitle
%\titlerunning{Measurement of charged pions in $^{12}$C + $^{12}$C collisions at 1 and 2A~GeV with HADES}
%
\section{Introduction}
The investigation of nuclear matter at
high temperature and high density is one of the major research
topics in modern nuclear physics. Nucleus-nucleus
collisions at relativistic energies offer the unique possibility to
create such highly excited nuclear matter in the laboratory
\cite{ref1,ref1a,ref1b}. The
study of particle production as function of beam energy, system size
and the centrality of the collisions has been instrumental in the past
for understanding the approach towards equilibrium and flow phenomena,
as well as for gaining information about the equation of state.
Collisions of the light $^{12}$C+$^{12}$C system represent a link
between the elementary proton-proton reaction and the heavy-ion collisions
of large nuclei. Important physics issues in this context are the
degree of thermalization achieved, the role of the mean field
and collective motion.
In the few-GeV energy range pions are
the only abundantly produced mesons. In heavy-ion collisions their spectra
and yields are affected by collective effects like thermalization,
directed and elliptic flow, as well as by possible modifications of the
properties of the baryon resonances they decay from, in particular
the $\Delta$ \cite{Mosel1,Mosel2}.
The subtle interplay of the phenomena which change the
characteristics of pion production with respect to nucleon-nucleon (N-N)
interactions is indeed a challenge to theoretical interpretations.
Best suited for description of all phases of the complex heavy-ion reaction
are transport models, based on microscopic transport theory.
The reaction is simulated as a set of multiple elementary collisions, with
elementary cross sections and momentum dependent potentials as
input parameters.
These assumptions are then tested by comparing the experimental observables
with the model predictions, and allow to get understanding of the reaction
dynamics.
Transport models achieved remarkable success in description of bulk properties
of the interactions over a large energy and system size scale.
At the same time, they have difficulties in reproducing
the experimental data precisely. For a recent comprehensive discussion
of various
differential pion observables and their comparison with model calculations
in the region of 1A~GeV see \cite{reisdorf}.
The High Acceptance DiElectron Spectrometer (HADES) \cite{ref2}, in
operation at the heavy-ion synchrotron SIS18 at GSI, Darmstadt, is
designed for high-resolution and high-acceptance dielectron
spectroscopy in hadron-hadron, ha\-dron-nucleus, and nucleus-nucleus
reactions at beam energies in the range from 1 to 2A~GeV. Being a
charged particle detector, it is of course also an efficient device for
hadron detection. First results from HADES on dielectron production in
$^{12}$C~+~$^{12}$C have been presented in \cite{ref3,ref3a}.
These rely on a precise knowledge of the differential yields of neutral
pions, which are the source of the bulk of the detected dielectron pairs,
namely the $\pi^{0}$ Dalitz and photon decays. In these analyses
the $\pi^{0}$ yields were inferred from the charged pion yields measured by
HADES in the same $^{12}$C~+~$^{12}$C data samples.
In the present paper we present detailed data on charged pions
obtained from $^{12}$C + $^{12}$C collisions at 1 and 2A~GeV. For the
first time large intervals of rapidity ($\approx \pm0.8$ in $y/y_{beam}$
for 2A~GeV) and of
centre-of-mass angle ($-0.7 < \cos(\theta) < 0.7$) are covered.
Our results are compared to the UrQMD transport-model predictions and
experimental data from other experiments.
\section{Experiment}
HADES \cite{ref2} is a magnetic spectrometer designed as
second-generation device for measurements of $e^+e^-$ pairs. The
spectrometer, schematically depicted in Fig.~\ref{had1}, is
segmented into six identical sectors that cover polar angles between
18 and 85 degrees. Its large (nearly 2$\pi$) azimuthal acceptance
covers between 65\% and 90\% of 2$\pi$ at small and large polar
angles, respectively. The analysis of charged pions presented here
is based on the same detectors as used in \cite{ref3,ref3a}. A fast
hadron-blind Ring Imaging CHerenkov counter (RICH) is used for electron and
positron identification. Four planes of Multi-wire Drift Chambers
(MDC1 - MDC4), together with a superconducting magnet, form the
magnetic spectrometer for track reconstruction and momentum
determination. In the region behind the magnetic field, a set of
electromagnetic PreShower detectors (at polar angles $18^\circ - 45^\circ$)
\cite{presh} and a time-of-flight wall \cite{tof} are installed
which form the META (Multiplicity and Electron Trigger Array). The
time-of-flight detector wall is subdivided into 2 regions: TOF (at
polar angles $45^\circ - 85^\circ$) consisting of 384 scintillator slabs of
varying length, which are read out at both ends with a
time-of-flight resolution of $\sigma$ = 150 ps, and TOFINO (at polar
angles $18^\circ - 45^\circ$) consisting of 24 scintillator plates
readout on one end with a time-of-flight resolution of $\sigma$ =
450 ps.
The TOFINO is placed directly in front of the Pre-Shower detector, which
provides precise position measurement.
The TOF/TOFINO detectors are also used for fast charged particle
multiplicity measurements. Together with the PreShower detectors
they provide additional lepton/hadron discrimination power and track
coordinate measurements with a spatial resolution in the range from
14~to~25~mm.
\begin{figure}[htb]
\vspace*{+.5cm}
% \center \includegraphics*[width=100mm]{figs/hades_cros.eps}
\includegraphics[width=1.\columnwidth]{figs/hades_cros.eps}
\vspace*{-0.1cm}
\caption[]{Cut through two sectors of the HADES spectrometer, except for the magnet
coils which are projected onto the cut plane to visualize the magnetic field.
The average distance
between the target and the outermost detectors is about 210 cm.}
\label{had1}
\end{figure}
A fast data acquisition system is used together with a two-level trigger
scheme \cite{ref9a,ref9b}: (a) LVL1 is based on a fast determination of the
charged particle multiplicity ($M_{ch}$) in the TOF detectors.
(b) LVL2 is based on a real time identification of electron and
positron candidates. All LVL2 trigger accepted events were written to tape,
as well as a part of LVL1 (regardless of LVL2 decision) events
(typically 10\%) for normalization
purposes, hadron analysis and check of the trigger performance.
For the analysis presented here, only LVL1 trigger events
were processed.
In the very first HADES physics run, the detector was operated using only
the following sub-systems: the RICH, the two inner MDC planes, and the META,
i.e. the two outer MDC planes were not operational and the coordinate
measurement of the META were used for tracking.
In this mode the collision system $^{12}$C~+~$^{12}$C at 2A~GeV was
studied with a beam intensity of $I_{beam} = 10^6$ particles/sec
impinging on a segmented carbon target with thickness $2 \cdot 2.5\%$
interaction length.
%In total
%$6.5 \cdot 10^8$ LVL1 triggered events with $M_{ch} \ge 4$ were examined,
%$2.2 \cdot 10^8$ were written to tape, and
$1.67 \cdot 10^7$ LVL1 triggered events with $M_{ch} \ge 4$ were analyzed in this
study.
In the event reconstruction, the track segments measured in the two inner MDC
planes were correlated with hits in the META.
In the second data taking period the $^{12}$C~+~$^{12}$C system was studied at 1A~GeV.
Then, for the first time, a high-resolution tracking mode exploiting
also the outer MDC planes was available. In this measurement, a carbon beam of
$10^6$ particles/sec was focused onto a carbon foil of 3.8\% interaction length.
%In total $1.1 \cdot 10^9$ LVL1 triggered events with $M_{ch} \ge 4$ were examined,
%$2.7 \cdot 10^8$ were written to tape, and
$1.62 \cdot 10^7$ LVL1 triggered events with $M_{ch} \ge 4$ were used in this
analysis.
\section{Data Analysis}
\subsection{Simulation}
Artificial $^{12}$C + $^{12}$C events were generated with the UrQMD (v1.3b)
transport code \cite{ref10,ref11}. The detector response was simulated with
the help of a Geant 3.21 based package \cite{ref12} including the geometry and
characteristics of all HADES detectors. The same LVL1 trigger condition
($M_{ch} \ge 4$) has been applied. The resulting raw data were processed
in exactly the same way as the real data and used for efficiency corrections
as well as to estimate systematic errors. Details on the different
procedures are given in the corresponding subsections.
For 1 GeV we have analyzed $2.14 \cdot 10^7$ (LVL1) UrQMD events, for 2 GeV
$2.07 \cdot 10^7$ events, so comparable to the amount of analyzed real data.
For this sample, the statistical errors of the $\pi$ yield in the region of
interest are negligible.
Additionally, we simulated the $\pi$ meson production using a simple
Monte-Carlo event generator PLUTO \cite{pluto}, which
assumes a thermal source modified by a polar angular distribution.
The used simulation parameters - inverse slopes and
anisotropies - were derived from our measured data, The generated
rapidity distribution has been used to extrapolate the $\pi$ yields
outside our acceptance. Varying the input parameters of the generator
within their errors serves for estimate of
systematic errors of the extrapolation.
\subsection{Momentum reconstruction}
When traversing the spectrometer charged particles are deflected in
the magnetic field, and at the same time they leave "hits" in the
MDCs and META detectors. From this information together with the
known magnetic field their trajectories are constructed and their
momenta are deduced.
Two different tracking methods have been developed and were both
used in the present analysis (see \cite{ref2} for details). The
first one is the ``kickplane'' algorithm which uses the position
information delivered by the inner MDC chambers and the META system.
In this case the momentum resolution $\sigma_{p}/p$, dominated by
the limited position resolution of META, has been determined in
simulations to be $\simeq 2\%$ at a momentum of 150 MeV/c, with a
linear increase up to $22\%$ at 1400 MeV/c. The second method is a
Runge-Kutta based trajectory integration routine \cite{RK} which uses
the information from all four MDC planes (with resolution
$\sigma_{p}/p \simeq$ 3\%). For the results presented in this paper,
the kickplane method has been applied to the 2A~GeV data, and both
methods were used and compared for the 1A~GeV data.
\subsection{Particle identification}
Particle identification in the HADES data analysis (for details see
\cite{ref2}) is based on Bayesian statistics \cite{Bayes,barlow}. The method
allows to evaluate the probability that the reconstructed track can
be related to a certain particle species (e.g. proton, kaon, $\pi$
meson, electron, etc.). It combines several observables from various
sub-detectors (e.g. time-of-flight, energy loss) via probability
density functions (PDF) determined for each observable and for all
possible particle species. The probabilities for different mass
assignments of any given track are calculated from the assumed
abundances of the individual particle species and from the PDFs of
all measured variables. The latter ones are obtained from
simulations. If the assumed abundances differ significantly from the
final results the procedure is repeated with updated input
distributions. It converges typically after one or two iterations. The
performance of the method in terms of efficiency and purity is
evaluated in detailed simulations and simultaneous comparisons with
the real data.
In our case, hadron identification has been performed
using measured momenta and corresponding velocities computed by means of the
time-of-flight.
For more sophisticated analyses, like electron or rare hadron identification,
data from the RICH and PreShower detectors as well as the energy
loss in META and MDCs can be used in addition.
%description of the method
\begin{figure*}[htb]
\vspace*{+.5cm}
%\center \includegraphics*[width=140mm]{figs/momVsBeta.eps}
\includegraphics[width=1.8\columnwidth]{figs/momVsBeta.eps}
\vspace*{-0.1cm}
\caption[]{Velocity vs.\ charge-times-momentum of charged particles as seen by the
HADES detector from $^{12}$C + $^{12}$C collisions at 2A~GeV (left). Projection onto the
velocity axis of positively charged particles with momenta 350$\pm$5~MeV/c and
$\theta$~=~60$^\circ\pm$5$^\circ$ (right). Fitted signal and background distributions
are shown as lines.}
\label{pid1}
\end{figure*}
The method used for Particle IDentification (PID) is illustrated in
Fig.~\ref{pid1} for the case of particle velocity (right) deduced from the
measured time-of-flight and track length (``velocity-vs-momentum''
algorithm). Particles with different mass occupy different regions
in the velocity-vs-momentum distribution (left side); the pronounced
ridges correspond to positive and negative pions, protons and
deuterons. The Bayesian PID method requires the determination of the
probability density functions for each particle species. In the
case of the velocity-vs-momentum algorithm used here, the PDF is the
probability distribution of velocity. For each type of particle it
has been determined in bins of momentum and polar angle. In those
velocity distributions gaussian fits were used to obtain the signal
(i.e. particle yields) and a 2$^{nd}$-order polynomial fit to obtain
the background (i.e. fake tracks). The fitted distributions were
normalized to unity. Fig.~\ref{pid1} (right side) shows as an
example of such fit for the momentum bin 350$\pm$5~MeV/c in the polar angle range
$\theta$ = 60$^\circ\pm$5$^\circ$.
Two quality parameters are used to characterize the performance of
the method \cite{hommez}: the PID efficiency and the PID purity. The
PID efficiency $\varepsilon_{t}(p,\theta)$ is the probability that a
particle with the true type $t$ is identified as type $t$. The PID
purity $\pi_{t}(p,\theta)$ is the probability that a particle that
is identified as type $t$ is truly of type $t$. The PID efficiency
and purity have been studied in detailed simulations with events
generated with the UrQMD model. The critical parameter here is the
time resolution, which is well known. This limits the
region in which we can use the method for $\pi^{+}$ and p
identification to momenta $<1000$ MeV/c because of the moderate time
resolution of the presently installed TOFINO detectors. We have also
checked that varying
particle abundances even by a factor of 2 does not change the results
significantly in the region of interest.
\subsection{Total correction}
\begin{figure*}[htb]
\vspace*{+.5cm}
%\center \includegraphics*[width=140mm]{figs/ToFEffVsPChargeSim.eps}
\includegraphics[width=1.8\columnwidth]{figs/ToFEffVsPChargeSim.eps}
\vspace*{-0.1cm}
\caption[]{Efficiency (top) and purity (bottom) of the PID method
versus momentum for $\pi^\pm$ and protons in two sub-systems of the
HADES detector: TOF (left) and TOFINO+PreShower (right) by using the kickplane
reconstruction algorithm in $^{12}$C~+~$^{12}$C collisions at 2A~GeV.}
\label{pid2}
\end{figure*}
Fig.~\ref{pid2} shows the dependences of the PID efficiency and
purity on momentum for $\pi^+$, $\pi^-$ and protons for the 2A~GeV
data in the TOFINO (right) and TOF (left) regimes. The efficiency of
pion and proton identification is larger than 95\% for all momenta
in the TOF region. In the TOFINO region with its reduced time
resolution the efficiency to identify positively charged pions drops
steeply above 1000 MeV/c due to the ambiguity with the protons. The
purity of pions (lower plots) does not reach unity because about
$10\%$ of tracks identified as pions are muons from
in-flight pion decays. A strong contamination of the positively
charged pions with protons for momenta above 1000 MeV/c are again
due to the low time resolution of TOFINO.
After the particle identification is done for all tracks, the
resulting yields are corrected for efficiency and purity of the PID
method, as well as for the detector and tracking efficiencies. The
detection/tracking efficiency has also been obtained from Monte
Carlo simulated and reconstructed UrQMD events. The total correction
applied to the reconstructed particle yields reads
\begin{equation}
w_t(p,\theta) = \frac{\pi_{t}(p,\theta)}{\varepsilon_{t}(p,\theta)
\times \varepsilon_{t}^{det}(p,\theta)} \label{eq:effcor},
\end{equation}
where $\varepsilon_{t}^{det}(p,\theta)$, the detection efficiency,
subsumes detector, track reconstruction and acceptance losses. It
should be noted that we specify $\varepsilon_{t}^{det}(p,\theta)$ as
function of $\theta$ and $p$, while averaging over the azimuthal
angle. In this way, corrections for missing geometrical acceptance
at some azimuthal angles (namely the spaces occupied by the six
magnet coils) are accounted for as well as the losses due to pion
in-flight decays.
\begin{figure}[htb]
\vspace*{+.5cm}
%\center \includegraphics*[width=100mm]{figs/AccVsMomSim.eps}
\includegraphics[width=1.\columnwidth]{figs/AccVsMomSim.eps}
\vspace*{-0.1cm}
\caption[]{The detection efficiency $\varepsilon_{t}^{det}(p,\theta)$ vs.\ momentum
for $\pi^\pm$ and
protons using the kickplane reconstruction in $^{12}$C+$^{12}$C collisions at 2A~GeV.
}
\label{accVsMom}
\end{figure}
Fig.~\ref{accVsMom} shows the dependence of the detection efficiency
for $\pi^\pm$ and p as a function of momentum. The difference
between proton and $\pi$ efficiencies is again caused by the
$\pi^\pm$ in-flight decay.
The total correction is applied to the data for each momentum and
polar angle bin, and for each individual particle species. This is
done only for bins with sufficiently high efficiency
$\varepsilon_{t}^{det}(p,\theta)>0.35$ in order to avoid large
corrections at the sector boundaries. Data outside of this fiducial
volume were excluded from further analysis.
\begin{figure}[ht]
% \vspace*{+.5cm}
% \center \includegraphics*[width=100mm]{figs/corrTheta_CM_id8.eps}
\includegraphics[width=1.\columnwidth]{figs/corrTheta_CM_id8.eps}
\vspace*{-0.1cm}
\caption[]{Center-of-mass polar angle distributions of $\pi^+$.
The squares represent the pions as generated by UrQMD. Only $\pi$
mesons with $p_{cms}>200{\rm MeV/c}$ have been selected. The open circles
show those generated pions which are detected and identified in the
HADES acceptance. The full circles depict the result of the efficiency
and purity correction to the accepted pions. }
\label{pid3cm}
\end{figure}
Fig.~\ref{pid3cm} presents simulated polar distributions of pions
in the center-of-mass system (cms). It shows the identified
$\pi^+$ before and after applying the total correction, together with
the primordial distribution delivered by the UrQMD model for 2A~GeV
$^{12}$C~+~$^{12}$C collisions.
This self-consistency check quantifies $w_t(p,\theta)$ as a function of
$\cos \theta_{cms}$
and demonstrates the wide coverage of our spectrometer. In Fig.~\ref{pid3cm}
the angular anisotropy of pion emission in the UrQMD
generator is clearly visible.
\subsection{Event selection}
For the present analysis, we used the HADES LVL1 trigger events,
which are characterized by a hit multiplicity M$_{ch} \ge 4$ in the
time-of-flight detectors. The correlation between the LVL1 trigger
condition and the centrality of the reaction has been studied in
Monte-Carlo simulation using the UrQMD and GEANT codes.
Fig.~\ref{imp} shows the simulated impact-parameter distributions. As for
the ``minimum bias'' events corresponding to the total reaction
cross section, we require in UrQMD at least one nuclear interaction
(distribution marked by circles in Fig.~\ref{imp}). Then we pass
these events through our analysis code and require that they fulfill
the LVL1 condition (triangles in Fig.~\ref{imp}).
%Spectator nucleons are excluded from particle propagation
%if the number of participant nucleons in a reaction is below x
%to account for nuclear binding of collision fragments.
We found that the
LVL1-triggered events correspond to 52\% and 60\% of the total
reaction cross section in $^{12}$C + $^{12}$C collisions at 1 and
2A~GeV, respectively. As we did not find a straightforward way how
to extract the average number of participants for our trigger biased
events, we proceeded in the following way. For a minimum bias events
the average number of participating nucleons was estimated from the
geometrical model \cite {gosset}. In our case of symmetric
collisions systems the average number of participants is
$\langle $A$_{part} \rangle $= $A/2$=6.
We deduce the mean $\langle $A$_{part} \rangle$ for reactions accepted by
LVL1 by comparing the pion multiplicity of UrQMD for LVL1 accepted events
to minimum bias events and using $\langle $A$_{part} \rangle$ scaling of
pion production, i.e.
$6 \cdot \langle M_{\pi}^{LVL1} \rangle / \langle M_{\pi}^{min.b.} \rangle$.
The LVL1 trigger effect is significant, and the number of participants
increases by $\approx 40\%$. The average impact parameters, the
average pion multiplicities and average number of participants from
UrQMD are listed in Table.~\ref{tab4}, for the true minimum-bias
events and after applying LVL1 trigger at both 1~and~2A~GeV.
% \begin{figure*}[htb]
\begin{figure*}[!htb]
\vspace*{+.5cm}
% \center \includegraphics*[width=140mm]{figs/impact_parameter_c2c.eps}
\includegraphics[width=1.8\columnwidth]{figs/impact_parameter_c2c.eps}
\vspace*{-0.1cm}
\caption[]{The impact parameter distribution obtained from the UrQMD model
for $^{12}$C~+~$^{12}$C collisions at 1~(left) and 2A~GeV (right).
}
\label{imp}
\end{figure*}
\begin{table}[htb]
\begin{center}
\caption{Average impact parameters, pion multiplicities,
and average number of participating nucleons
from UrQMD calculations for $^{12}$C~+~$^{12}$C at 1~and~2A~GeV
before and after applying the LVL1
trigger condition. \label{tab4}}
%\begin{ruledtabular}
%\setlength{\extrarowheight}{0.1cm}
\begin{tabular}{l c c c c c}
\\
\hline
\multicolumn{5}{c}{Beam energy = 1A~GeV} \\
\hline
& $\langle b \rangle$ & $\langle M_{\pi^{+}} \rangle$ & $\langle M_{\pi^{-}}\rangle$ & $\langle $A$_{part}\rangle$ \\
minimum-bias events & 3.95 fm& 0.36 & 0.36 & 6 \\
LVL1 triggered & 3.01 fm& 0.51 & 0.52 & 8.61 \\
\hline
\multicolumn{5}{c}{Beam energy = 2A~GeV} \\
\hline
& $\langle b \rangle $ & $\langle M_{\pi^{+}} \rangle $ & $\langle M_{\pi^{-}}\rangle $ & $\langle $A$_{part}\rangle$ \\
minimum-bias events & 3.95 fm & 0.83 & 0.83 & 6 \\
LVL1 triggered & 3.18 fm & 1.15 & 1.17 & 8.38 \\
\hline
\end{tabular}\\
%\end{ruledtabular}
\end{center}
\end{table}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
The distributions of the number of reconstructed tracks per LVL1 event of
data and UrQMD simulations are in a reasonable agreement, as shown in
Fig.~\ref{mult}.
This is confirmation that the modelling of the detector and tracking
efficiency as well as of the LVL1 event selection in our simulation is
realistic. From the differences of the measured and simulated distributions
of the number of charged hits in META and number of reconstructed tracks
we estimate the systematic error of the mean number of participants
determination as 7\%.
\begin{figure*}[htb]
\vspace*{+.2cm}
% \center \includegraphics[width=120mm]{figs/track_multiplicity_nov02_aug04.eps}
\includegraphics[width=1.8\columnwidth]{figs/track_multiplicity_nov02_aug04.eps}
\vspace*{-0.1cm}
\caption[]{Distribution of the number of reconstructed tracks
in the data and in the simulation for 1~(left) and 2A~GeV (right) $^{12}$C~+~$^{12}$C collisions.}
\label{mult}
\end{figure*}
\section{Results}
\subsection{Transverse-mass distributions}
\begin{figure*}[htb]
\vspace*{+.01cm}
% \center \includegraphics*[width=140mm]{figs/dNdMpions_c1c.eps}
\includegraphics[width=1.8\columnwidth]{figs/dNdMpions_c1c.eps}
\vspace*{-0.1cm}
\caption[]{Transverse-mass distributions for positive (left) and negative
(right) pions in different slices of rapidity derived from the data
in $^{12}$C~+~$^{12}$C collisions at 1A~GeV (LVL1 ``semicentral'' events).
Full lines show the results of fits of the data using one exponential
function, while dashed lines show fits of the UrQMD distributions using
the same fit function. Error bars (systematic and statistical ones) are not
visible at this scale.}
\label{dNdMpion_aug04rk}
\end{figure*}
\begin{figure*}[htb]
\vspace*{+.01cm}
%\center \includegraphics*[width=140mm]{figs/dNdMpions_c2c.eps}
\includegraphics[width=1.8\columnwidth]{figs/dNdMpions_c2c.eps}
\vspace*{-0.1cm}
\caption[]{Transverse-mass distributions for positive (left) and negative
pions (right) in different slices of rapidity derived from the data
in $^{12}$C~+~$^{12}$C collisions at 2A~GeV (LVL1 ``semicentral'' events).
Full lines show the results of fits of the data using two exponential
functions, while dashed lines show fits of the UrQMD distributions using
one exponential function. Error bars (systematic and statistical ones) are not
visible at this scale.}
\label{dNdMpion}
\end{figure*}
\begin{table*}[htb]
\caption{Inverse slope parameters for $\pi^\pm$ measured at mid-rapidity
derived from the
data (using one and two exponential functions) and UrQMD (using one exponential function) in
$^{12}$C~+~$^{12}$C collisions at 1~and~2A~GeV, in units of MeV.}
\label{tab3}
\center~
%\begin{ruledtabular}
%\setlength{\extrarowheight}{0.1cm}
\begin{tabular}{l c c c c c c}
\\
\hline
\multicolumn{7}{c}{Beam energy = 1A~GeV} \\
\hline
Particle & \multicolumn{2}{c}{Data} & \multicolumn{2}{c}{UrQMD} \\
& T (1 slope)& $\chi^{2}/ndf$& T (1 slope)& $\chi^{2}/ndf$ \\
\hline
$\pi^+$ &57.8 $\pm$ 0.3 & 1.7 & 55.4 $\pm$ 0.3 & 2.2\\
$\pi^-$ & 57.9 $\pm$ 0.3 & 1.4 & 55.4 $\pm$ 0.3 & 2.0\\
\hline
\multicolumn{7}{c}{Beam energy = 2A~GeV} \\
\hline
Particle & \multicolumn{4}{c}{Data} & \multicolumn{2}{c}{UrQMD} \\
& T (2 slopes)& $\chi^{2}/ndf$ & T (1 slope)& $\chi^{2}/ndf$& T (1 slope)& $\chi^{2}/ndf$ \\
\hline
$\pi^+$ & 47.7 $\pm$ 6.2; & 0.9 & 80.9 $\pm$ 0.5 & 4.7 & 86.5 $\pm$ 0.6 & 1.5\\
& 90.6 $\pm$ 3.3 \\
$\pi^-$ & 46.4 $\pm$ 5.2; & 1.2 & 76.7 $\pm$ 0.5 & 4.9 & 86.7 $\pm$ 0.6 & 1.4\\
& 84.4 $\pm$ 2.1 \\
%pi+ range 40-550: pi+: 47.7+-6.2, 90.6+- 3.3 chi2/ndf 14.39/16
\hline
\end{tabular}\\[3pt]
%\end{ruledtabular}
\end{table*}
Figures~\ref{dNdMpion_aug04rk} and \ref{dNdMpion} exhibit the measured and simulated
transverse mass distributions of $\pi^{+}$ and $\pi^{-}$ in different intervals of normalized rapidity $y_0 = (y_{lab}-y_{cms})/y_{cms}$
for $^{12}$C+$^{12}$C at 1A~GeV and 2A~GeV, respectively.
The systematic errors of the data are estimated from the differences
between distributions from the 6 independent HADES sectors as 5\%.
The transverse-mass ($m_{\perp}$) distributions have been fitted for each
rapidity bin using one
or two exponential functions.
The fit with two slopes employs
\begin{equation}
\frac{1}{m_\perp^2} \frac{dN(y)}{dm_\perp} =
C_1(y) \, \exp\left(- \frac{m_\perp}{T_1(y)}\right) +
C_2(y) \, \exp\left(- \frac{m_\perp}{T_2(y)}\right)
\label{eq_mt}
\end{equation}
with $m_{\perp}=(p_{\perp}^2 + m^2)^{1/2}$,
and $p_\perp$ as transverse momentum;
$C_{1,2}$ are normalizations and $T_{1,2}$
the inverse slope parameters.
It describes the experimental data better than a fit with one slope
(i.e., $C_2 \equiv 0$)
for the 2A~GeV data sample ($\chi^2/ndf$ around 1.0 vs.\ 4.8 for the
fit with one exponential).
Fig.~\ref{dNdMpion_aug04rk} clearly
demonstrates that for the lower bombarding energy of 1A~GeV, a fit
with one slope is sufficient for the description of the spectral
shape. The inverse-slope parameters for $\pi$ mesons at
mid-rapidity for 1A~GeV (-0.15 $\le y_{0} \le$ 0.15) and 2A~GeV
(-0.1 $\le y_{0} \le$ 0.1) are summarized in Table~\ref{tab3} using
one or two exponential functions. The slopes of $\pi^+$ and
$\pi^-$ agree within error bars for the single exponential fit.
At 2A~GeV, UrQMD predicts
different spectral shapes (flatter spectra), while at 1A~GeV
agreement of UrQMD with data is better.
\begin{center}
\begin{figure*}
\includegraphics[width=1.\columnwidth]{figs/c1c_pt_hades_taps_liny.eps}
\includegraphics[width=1.\columnwidth]{figs/c2c_pt_hades_taps_liny.eps}
\includegraphics[width=1.\columnwidth]{figs/mt_pi_c1c_hades_taps_inset.eps}
\includegraphics[width=1.\columnwidth]{figs/mt_pi_c2c_hades_taps_inset.eps}
\includegraphics[width=1.\columnwidth]{figs/c1c_pip_hades_kaos_liny.eps}
\hspace*{+.35cm}
\includegraphics[width=1.\columnwidth]{figs/c2c_pip_hades_kaos_liny.eps}
\caption[]{Comparison of transverse-mass and momenta distributions
of $\pi$ mesons measured by a present experiment and the TAPS and KaoS
experiments. Top: The transverse-momenta ($p_{\perp}$)
distributions of $\pi^{+,-,0}$ mesons for 1A GeV (left) and 2A GeV (right).
Center: The transverse-mass ($m_{\perp}$) distributions of
$\pi^{+,-,0}$ mesons for 1A GeV (left) and 2A GeV (right). Full lines
show the results of fits of our data using two exponential
functions, while dashed lines show fits of the $\pi^{0}$ distribution using
one exponential function.
Bottom: The momenta ($p_{lab}$)
distributions of $\pi^{+}$ mesons for 1A GeV (left) and 2A GeV (right).
The HADES data were re-scaled to min. bias yields, see text.
Insets in all plots show the ratio of $\pi$ yields.
}
\label{figtapshades}
\end{figure*}
\end{center}
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\end{document}