[Paper Review] Double, Triple and Hidden Charm Production in the Statistical Coalescence Model

A. Kostyuk|ArXiv.org|Feb 1, 2005

Neural Networks and Applications3 citations

TL;DR

This paper develops a statistical coalescence model for double, triple, and hidden charm hadron production in heavy-ion collisions, treating charm quark-antiquark pairs as created early and distributed among final-state hadrons via statistical mechanics. The key contribution is deriving formulas for charm hadroproduction using Poisson-distributed charm pair fluctuations—valid when average charm pair production is small—showing significant differences from canonical approaches, especially for hidden charm states.

ABSTRACT

The production of particles with double, triple and hidden charm in heavy ion collisions is studied in the framework of the statistical coalescence model. According to the postulates of the model, the charm quark-antiquark pairs are created at the initial stage of a heavy ion reaction in hard parton collisions. The amount of charm is assumed to be unchanged at later stages. The charm (anti)quarks are distributed among different hadron species at hadronization according to the laws of statistical physics. Several approaches to the statistical treatment of charm hadronization are considered. The grand canonical approach is appropriate for systems containing large number of charm (anti)quarks. The exact charm conservation and Poissonian fluctuations of the number of charm quark-antiquark pairs should be taken into account, if the average number of these pairs is of oder of unity or smaller. The charm hadronization in a subsystem of a larger system is discussed. It is explained why the canonical approach is not appropriate for the description of charm hadronization. The obtained formulas can be used to calculate the production of charm in heavy ion collisions in a wide energy range.

Motivation & Objective

• To extend the statistical coalescence model (SCM) to describe double, triple, and hidden charm hadron production in heavy-ion collisions.
• To address the challenge of charm hadroization in systems with small numbers of charm quark-antiquark pairs, where equilibrium assumptions break down.
• To clarify the validity of grand canonical vs. canonical vs. Poissonian statistical treatments for charm hadroization, particularly in the context of low average charm pair production.
• To provide a framework for calculating charm hadron yields in subsystems, such as those probed in experimental detectors, without requiring full system thermodynamics.
• To enable quantitative predictions for charm hadron production across a wide energy range, from SPS to LHC, including rare multi-charm states.

Proposed method

• Applies the statistical coalescence model (SCM) postulating that charm quarks are created in initial hard parton collisions and do not recombine later.
• Uses the grand canonical ensemble for systems with large numbers of charm pairs ($\langle N_{c\bar{c}} \rangle \gg 1$), deriving analytical expressions for hadron yields.
• Introduces a Poissonian fluctuation model for the number of $c\bar{c}$ pairs when $\langle N_{c\bar{c}} \rangle \lesssim 1$, based on independent nucleon-nucleon collisions.
• Derives formulas for the yields of hidden charm, double charm, and triple charm hadrons under Poissonian statistics, using the partition function and statistical weights.
• Considers charm hadroization in a subsystem of a larger system, showing that only local thermal parameters and the number of $c\bar{c}$ pairs in the subsystem are needed.
• Demonstrates that the canonical approach is invalid for charm hadroization due to its assumption of thermal equilibration of $c\bar{c}$ pairs at freeze-out, contradicting the SCM's initial production postulate.

Experimental results

Research questions

• RQ1 How can the statistical coalescence model be extended to describe double and triple charm hadron production in heavy-ion collisions?
• RQ2 What statistical ensemble—grand canonical, canonical, or Poissonian—is most appropriate for charm hadroization when the average number of $c\bar{c}$ pairs is small?
• RQ3 How do the predictions for hidden charm production differ between the Poissonian and canonical approaches, especially in the intermediate regime $\langle N_{c\bar{c}} \rangle \sim 1$?
• RQ4 Can the statistical coalescence model be applied to charm hadroization in a subsystem of a larger system without full knowledge of the entire system’s thermodynamics?
• RQ5 What are the quantitative predictions for multi-charm hadron yields at LHC energies, and how do they depend on the choice of statistical treatment?

Key findings

• The grand canonical approach is valid for systems with $\langle N_{c\bar{c}} \rangle \gg 1$, allowing analytical solutions for hadron yields when the number of multi-charm hadrons is small.
• For $\langle N_{c\bar{c}} \rangle \lesssim 1$, the canonical approach is invalid because it assumes thermal equilibration of $c\bar{c}$ pairs at freeze-out, contradicting the SCM’s initial production postulate.
• The Poissonian fluctuation model correctly describes the number of $c\bar{c}$ pairs in small systems, as they arise from independent nucleon-nucleon collisions.
• The Poissonian and grand canonical approaches yield identical results for double and triple charm hadron yields, but differ by up to 10% for hidden charm production in the intermediate regime $\langle N_{c\bar{c}} \rangle \sim 1$.
• In a subsystem, the number of double and triple charm hadrons depends only on the number of $c\bar{c}$ pairs in that subsystem and its local thermal parameters.
• For hidden charm, the total number of $c\bar{c}$ pairs in the entire system is required to compute the yield in a subsystem, even if only local thermal parameters are known.