It is shown that flavor mixing of the strange and light quarks allows for existence of a much larger baryonic chemical potential window for the formation of a stable dual chiral-wave state as ...compared to the well-known two-flavor case. In addition, strangeness catalyzes the occurrence of a new branch of nonhomogeneous solutions at moderate densities. This case study is addressed at zero temperature within the SU(3) flavor Nambu-Jona-Lasinio model with the't Hooft determinanta! flavor mixing interaction. The modulation of the chiral condensates in the light quark sector is taken to be one dimensional, while strangeness is embedded as a homogeneous condensate in the spontaneously broken phase of chiral symmetry. A finite current quark mass for the strange quark is incorporated, while the up and down current masses are set to zero. In that case the modulation considered provides an exact analytic solution for the system. Despite the simplicity of the ansatz, the emerging phase diagram displays a very rich structure.
SHARE is a collection of programs designed for the statistical analysis of particle production in relativistic heavy-ion collisions. With the physical input of intensive statistical parameters, it ...generates the ratios of particle abundances. The program includes cascade decays of all confirmed resonances from the Particle Data Tables. The complete treatment of these resonances has been known to be a crucial factor behind the success of the statistical approach. An optional feature implemented is the Breit–Wigner distribution for strong resonances. An interface for fitting the parameters of the model to the experimental data is provided.
Title of the program:
SHARE, October 2004, version 1.2
Catalogue identifier: ADVD
Program summary URL:
http://cpc.cs.qub.ac.uk/summaries/ADVD
Program obtainable from: CPC Program Library, Queen's University of Belfast, N. Ireland
Computer: PC, Pentium III, 512 MB RAM (not hardware dependent)
Operating system: Linux: RedHat 6.1, 7.2, FEDORA, etc. (not system dependent)
Programming language:
FORTRAN77:
g77,
f77 as well as
Mathematica, ver. 4 or 5, for the case of full chemical equilibrium and particle widths set to zero
Size of the package: 645 KB directory including example programs (87 KB compressed distribution archive)
External routines: KERNLIB, MATHLIB and PACKLIB from the CERN Program Library (see
http://cernlib.web.cern.ch for download and installation instructions)
Distribution format: tar.gz
Number of lines in distributed program, including test data, etc.: 15 277
Number of bytes in distributed program, including test data, etc.: 88 522
Computer: Any computer with an f77 compiler
Nature of the physical problem: Statistical analysis of particle production in relativistic heavy-ion collisions involves the formation and the subsequent decays of a large number of resonances. With the physical input of thermal parameters, such as the temperature and fugacities, and considering cascading decays, along with weak interaction feed-down corrections, the observed hadron abundances are obtained. SHARE incorporates diverse physical approaches, with a flexibility of choice of the details of the statistical hadronization model, including the selection of a chemical (non-)equilibrium condition. SHARE also offers evaluation of the extensive properties of the source of particles, such as energy, entropy, baryon number, strangeness, as well as the determination of the best intensive input parameters fitting a set of experimental yields. This allows exploration of a proposed physical hypothesis about hadron production mechanisms and the determination of the properties of their source.
Method of solving the problem: Distributions at freeze-out of both the stable particles and the hadronic resonances are set according to a statistical prescription, technically calculated via a series of Bessel functions, using CERN library programs. We also have the option of including finite particle widths of the resonances. While this is computationally expensive, it is necessary to fully implement the essence of the strong interaction dynamics within the statistical hadronization picture. In fact, including finite width has a considerable effect when modeling directly detectable short-lived resonances (
Λ
(
1520
)
,
K
∗
, etc.), and is noticeable in fits to experimentally measured yields of stable particles. After production, all hadronic resonances decay. Resonance decays are accomplished by addition of the parent abundances to the daughter, normalized by the branching ratio. Weak interaction decays receive a special treatment, where we introduce daughter particle acceptance factors for both strongly interacting decay products. An interface for fitting to experimental particle ratios of the statistical model parameters with the help of
MINUIT
1 is provided. The
χ
2
function is defined in the standard way. For an investigated quantity
f and experimental error Δ
f,
(1)
χ
2
=
(
f
experiment
−
f
theory
)
2
(
Δ
f
statistical
+
Δ
f
systematic
)
2
,
(2)
N
DoF
=
N
data points
−
N
free parameters
.
(note that systematic and statistical errors are independent, since the systematic error is not a random variable). Aside of
χ
2
, the program also calculates the statistical significance
2, defined as the probability that, given a “true” theory and a statistical (Gaussian) experimental error, the fitted
χ
2
assumes the values at or above the considered value. In the case that the best fit has statistical significance significantly below unity, the model under consideration is very likely inappropriate. In the limit of many degrees of freedom (
N
DoF
), the statistical significance function depends only on
χ
2
/
N
DoF
, with 90% statistical significance at
χ
2
/
N
DoF
∼
1
, and falling steeply at
χ
2
/
N
DoF
>
1
. However, the degrees of freedom in fits involving ratios are generally not sufficient to reach the asymptotic limit. Hence, statistical significance depends strongly on
χ
2
and
N
DoF
separately. In particular, if
N
DoF
<
20
, often for a fit to have an acceptable statistical significance, a
χ
2
/
N
DoF
significantly less than 1 is required. The fit routine does not always find the true lowest
χ
2
minimum. Specifically, multi-parameter fits with too few degrees of freedom generally exhibit a non-trivial structure in parameter space, with several secondary minima, saddle points, valleys, etc. To help the user perform the minimization effectively, we have added tools to compute the
χ
2
contours and profiles. In addition, our program's flexibility allows for many strategies in performing the fit. It is therefore possible, by following the techniques described in Section 3.7, to scan the parameter space and ensure that the minimum found is the true one. Further systematic deviations between the model and experiment can be recognized via the program's output, which includes a particle-by-particle comparison between experiment and theory.
Additional comments: In consideration of the wide stream of new data coming out from RHIC, there is an on-going activity, with several groups performing analysis of particle yields. It is our hope that SHARE will allow to create an analysis standard within the community. It can be useful in analyzing the experimental data, verifying simple physical assumptions, evaluating expected yields, as well as allowing to compare various similar models and programs which are currently being used.
Typical running time: For the Fortran code, the computation time with the provided default input files is about 10 minutes on 1 GHz processor. The time may rise significantly (by a factor of 300) if the full-fledged optimization and finite widths are included. In Mathematica, the typical running times are of the order of minutes.
Accessibility: The program is available from:
•
The CPC program library,
•
The following websites:
http://www.ifj.edu.pl/Dept4/share.html or
http://www.physics.arizona.edu/~torrieri/SHARE/share.html,
•
From the authors upon request.
The behavior of the non-perturbative parts of the isovector-vector and isovector and isosinglet axial-vector correlators at Euclidean momenta is studied in the framework of a covariant chiral quark ...model with non-local quark-quark interactions. The gauge covariance is ensured with the help of the P-exponents, with the corresponding modification of the quark-current interaction vertices taken into account. The low- and high-momentum behavior of the correlators is compared with the chiral perturbation theory and with the QCD operator product expansion, respectively. The V-A combination of the correlators obtained in the model reproduces quantitatively the ALEPH and OPAL data on hadronic \(\tau \) decays, transformed into the Euclidean domain via dispersion relations. The predictions for the electromagnetic \( {\rm\pi}^{\pm}- {\rm\pi}^{0}\) mass difference and for the pion electric polarizability are also in agreement with the experimental values. The topological susceptibility of the vacuum is evaluated as a function of the momentum, and its first moment is predicted to be \( {\rm\chi}^{\prime}(0)\approx (50 \mathrm{MeV})^{2}\). In addition, the fulfillment of the Crewther theorem is demonstrated.
We show how ultra-relativistic collisions of light nuclei with heavy targets may be used to record snap-shots of the ground-state configurations and reveal information on cluster correlations. The ...development of collective flow in the formed fireball, which reflects the geometric correlations in the initial state, is essential for the method. As an illustration we analyze the 12C-208Pb collisions.
A hydrodynamic model coupled to the statistical hadronization code Therminator is used to study a set of observables in the soft sector at RHIC. A satisfactory description of the
p
⊥
-spectra and ...elliptic flow is obtained, similarly to other hydrodynamic models. With the Gaussian initial conditions the transverse femtoscopic radii are also reproduced, providing a possible solution of the RHIC HBT puzzle.
THERMINATOR is a Monte Carlo event generator designed for studying of particle production in relativistic heavy-ion collisions performed at such experimental facilities as the SPS, RHIC, or LHC. The ...program implements
thermal models of particle production with
single freeze-out. It performs the following tasks: (1) generation of stable particles and unstable resonances at the chosen freeze-out hypersurface with the local phase-space density of particles given by the statistical distribution factors, (2) subsequent space–time evolution and decays of hadronic resonances in cascades, (3) calculation of the transverse-momentum spectra and numerous other observables related to the space–time evolution. The geometry of the freeze-out hypersurface and the collective velocity of expansion may be chosen from two successful models, the Cracow single-freeze-out model and the Blast-Wave model. All particles from the Particle Data Tables are used. The code is written in the object-oriented
c++ language and complies to the standards of the
ROOT environment.
Program title:
THERMINATOR
Catalogue identifier:ADXL_v1_0
Program summary URL:
http://cpc.cs.qub.ac.uk/summaries/ADXL_v1_0
Program obtainable from: CPC Program Library, Queen's University of Belfast, N. Ireland
RAM required to execute with typical data:50 Mbytes
Number of processors used:1
Computer(s) for which the program has been designed: PC, Pentium III, IV, or Athlon, 512 MB RAM not hardware dependent (any computer with the
c++ compiler and the
ROOT environment R. Brun, F. Rademakers, Nucl. Instrum. Methods A 389 (1997) 81,
http://root.cern.ch
Operating system(s) for which the program has been designed:
Linux: Mandrake 9.0, Debian 3.0, SuSE 9.0, Red Hat FEDORA 3, etc.,
Windows XP with Cygwin ver. 1.5.13-1 and
gcc ver. 3.3.3 (cygwin special)—not system dependent
External routines/libraries used: ROOT ver. 4.02.00
Programming language:
c++
Size of the package: (324 KB directory 40 KB compressed distribution archive), without the
ROOT libraries (see
http://root.cern.ch for details on the ROOT R. Brun, F. Rademakers, Nucl. Instrum. Methods A 389 (1997) 81,
http://root.cern.ch requirements). The output files created by the code need 1.1 GB for each 500 events.
Distribution format: tar gzip file
Number of lines in distributed program, including test data, etc.: 6534
Number of bytes in ditribution program, including test data, etc.:41 828
Nature of the physical problem: Statistical models have proved to be very useful in the description of soft physics in relativistic heavy-ion collisions P. Braun-Munzinger, K. Redlich, J. Stachel, 2003,
nucl-th/0304013.
2. In particular, with a few physical input parameters, such as the temperature, chemical potentials, and velocity of the collective flow, the models reproduce the observed particle abundances P. Koch, J. Rafelski, South Afr. J. Phys. 9 (1986) 8; J. Cleymans, H. Satz, Z. Phys. C 57 (1993) 135,
hep-ph/9207204; J. Sollfrank et al., Z. Phys. C 61 (1994) 659; P. Braun-Munzinger et al., Phys. Lett. B 344 (1995) 43,
nucl-th/9410026; P. Braun-Munzinger et al., Phys. Lett. B 365 (1996) 1,
nucl-th/9508020; J. Cleymans et al., Z. Phys. C 74 (1997) 319,
nucl-th/9603004; F. Becattini, J. Phys. G 23 (1997) 1933,
hep-ph/9708248; G.D. Yen, M.I. Gorenstein, Phys. Rev. C 59 (1999) 2788,
nucl-th/9808012; P. Braun-Munzinger, I. Heppe, J. Stachel, Phys. Lett. B 465 (1999) 15,
nucl-th/9903010; J. Cleymans, K. Redlich, Phys. Rev. C 60 (1999) 054908,
nucl-th/9903063; F. Becattini et al., Phys. Rev. C 64 (2001) 024901,
hep-ph/0002267; P. Braun-Munzinger et al., Phys. Lett. B 518 (2001) 41,
hep-ph/0105229; W. Florkowski, W. Broniowski, M. Michalec, Acta Phys. Polon. B 33 (2002) 761,
nucl-th/0106009, the transverse-momentum spectra W. Broniowski, W. Florkowski, Phys. Rev. Lett. 87 (2001) 272302,
nucl-th/0106050, balance functions W. Florkowski, W. Broniowski, P. Bozek, J. Phys. G 30 (2004) S1321,
nucl-th/0403038.
17; P. Bozek, W. Broniowski, W. Florkowski, Acta Phys. Hung. A 22 (2005) 149,
nucl-th/0310062.
18, or the elliptic flow W. Broniowski, A. Baran, W. Florkowski, AIP Conf. Proc. 660 (2003) 185,
nucl-th/0212053.
19; W. Florkowski, W. Broniowski, A. Baran, 2004,
nucl-th/0412077.
20 in both non-strange and strange sectors. The key element of the approach is the inclusion of the complete list of hadronic resonances, which at the rather high temperature at freeze-out, ∼165 MeV, contribute very significantly to the observed quantities. Their two- and three-body decays, taken from the tables, proceed in cascades, ultimately producing the stable particles observed in detectors. At the moment there exist several codes to compute the abundances of particles (the publicly available programs for this purpose are
SHARE G. Torrieri et al., 2004,
nucl-th/0404083 and
THERMUS S. Wheaton, J. Cleymans, 2004,
hep-ph/0407174), which is a rather simple task, since the abundances are insensitive to the geometry of the fireball and its expansion. On the other hand, the calculation of the transverse-momentum spectra of particles is much more complicated due to the sensitivity to these phenomena.
THERMINATOR deals with this problem, offering the
full information on the space–time positions and momenta of the produced particles. As a result, the program allows to compute very efficiently the transverse-momentum spectra of identified particles and examine implications of the assumed expansion model.
THERMINATOR allows easily for the departure from symmetries typically assumed in other approaches. This opens the possibility to study the dependence of physical quantities on rapidity and the azimuthal angle. The contribution of the resonances to various observables may be traced conveniently, and their role in the statistical approach may be verified. As a Monte Carlo event generator written in the object-oriented
c++ language in the
ROOT R. Brun, F. Rademakers, Nucl. Instrum. Methods A 389 (1997) 81,
http://root.cern.ch environment,
THERMINATOR can be straightforwardly interfaced to the standard software routinely used in the data analysis for relativistic heavy-ion colliders, such as SPS, RHIC, and, in the future, LHC. In this way the inclusion of experimental acceptance, kinematic cuts, or interfacing with other programs poses no difficulty.
Method of solving the problem:
THERMINATOR uses the particle data tables Particle Data Group, K. Hagiwara et al., Phys. Rev. D 66 (2002) 010001 in the universal input form used by the
SHARE G. Torrieri et al., 2004,
nucl-th/0404083 package. The user decides for the thermal parameters and the preferred expansion model. The optimum thermal parameters may be taken, e.g., as those obtained with the help of
SHARE G. Torrieri et al., 2004,
nucl-th/0404083 or
THERMUS S. Wheaton, J. Cleymans, 2004,
hep-ph/0407174. At the moment there are two different expansion models implemented in the code: the model of Ref. W. Broniowski, W. Florkowski, Phys. Rev. Lett. 87 (2001) 272302,
nucl-th/0106050, based on the so-called Buda–Lund T. Csorgo, B. Lorstad, Phys. Rev. C 54 (1996) 1390,
hep-ph/9509213 parameterization, and the Blast-Wave model E. Schnedermann, J. Sollfrank, U.W. Heinz, Phys. Rev. C 48 (1993) 2462,
nucl-th/9307020; F. Retiere, M.A. Lisa, Phys. Rev. C 70 (2004) 044907,
nucl-th/0312024. The positions and velocities of the particles are randomly generated on the hypersurface according to the statistical (Bose–Einstein of Fermi–Dirac) distribution factors. All particles, stable and unstable, are included. The particles move along classical trajectories from their initial positions, with velocities composed of the thermal motion and the collective expansion of the system. Stable particles just stream freely, while the resonances decay after some (randomly generated) time, which is controlled by the particle's lifetime. The decays are two-body or three-body, and their implementation involves simple kinematic formulas. The decays can proceed in cascades, down to the stage where only stable particles are present. All particles have tags indicating their parent. The secondary rescatterings are not considered in this approach. Full history of the event is stored in an output file, allowing for a detailed examination of the space–time evolutions and the calculation of the transverse-momentum spectra.
Additional comment: The ongoing analyses of the SPS and the RHIC data as well as the future heavy-ion program at LHC will certainly benefit from
THERMINATOR as a tool for generating events in a simple statistical model. The Monte Carlo code written in
c++ and using the standard
ROOT R. Brun, F. Rademakers, Nucl. Instrum. Methods A 389 (1997) 81,
http://root.cern.ch environment can be easily adapted to purposes directly linked to experimental data analyses. The space–time tracking capability will allow, in the framework of the statistical approach, to better understand the physics of relativistic heavy-ion collisions.
THERMINATOR calculates the particle spectra and other observables related to the space–time evolution of the system. It provides a
c++ framework which may be easily developed for detailed analyses of more involved observables such as, e.g., correlation functions or HBT radii.
Typical running time: The generation of 500 events from scratch takes about 1 hour 15 minutes on a PC with Athlon-Barthon 2.5 GHz under Red Hat Fedora 3. Each subsequent 500 events take about 1 hour. To store 500 events about 1.1 GB disk storage is needed, depending on the kinematic range. After converting the output to the ROOT TTree format, 900 MB may be freed.
The assumption of simultaneous chemical and thermal freeze-outs of the hadron gas leads to a surprisingly accurate, albeit entirely conventional, explanation of the recently measured RHIC ...p(perpendicular) spectra. The original thermal spectra are supplied with secondaries from cascade decays of all resonances, and subsequently folded with a suitably parametrized expansion involving longitudinal and transverse flow. The predictions of this thermal approach, with various parametrizations for the expansion, are in a striking quantitative agreement with the data in the whole available range of 0 < or = p(perpendicular) < or = 3.5 GeV.
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