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Manuscript Title: SHARE: Statistical Hadronization with Resonances
Authors: G. Torrieri, S. Steinke, W. Broniowski, W. Florkowskib, J. Letessier, J. Rafelski
Program title: SHARE, October 2004, version 1.2
Catalogue identifier: ADVD_v1_0
Distribution format: tar.gz
Journal reference: Comput. Phys. Commun. 167(2005)229
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.
Computer: Any computer with an f77 compiler.
Operating system: Linux: RedHat 6.1, 7.2, FEDORA etc. not system dependent.
Keywords: relativistic heavy-ion collisions, particle production, statistical models, decays of resonances.
PACS: 25.75.-q, 25.75.Dw.
Classification: 17.7.

External routines: KERNLIB, MATHLIB and PACKLIB from the CERN Program Library ( see http://cernlib.web.cern.ch for download and installation instructions).

Nature of 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.

Solution method:
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),Κ*, 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 Χ2 function is defined in the standard way. For an investigated quantity f and experimental error Δf,
Χ2 = (fexperiment - ftheory)2 / (Δfstatistical + Δfsystematic)2

NDoF = Ndata points - Nfree parameters

(note that systematic and statistical errors are independent, since the systematic error is not a random variable).
Aside of Χ2, 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 Χ2 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 (NDoF), the statistical significance function depends only on Χ2/NDoF, with 90% statistical significance at Χ2/NDoF ∼ 1, and falling steeply at Χ2/NDoF > 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 Χ2 and NDoF separately. In particular, if NDoF < 20, often for a fit to have an acceptable statistical significance, a Χ2/NDoF significantly less than 1 is required.
The fit routine does not always find the true lowest Χ2 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 Χ2 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 Sect. 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 the creation of 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 the comparison of various similar models and programs which are currently being used.

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.