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Imprint of $α$-Clustering on Ab Initio Correlations in Relativistic Light Ion Collisions

Hadi Mehrabpour

Abstract

This study investigates the influence of $α$-cluster structures in relativistic light nuclear collisions. Using a cluster framework, I extract the characteristics of the nucleonic configurations of $^{16}$O and $^{20}$Ne as predicted by various \textit{ab-initio} models, including Nuclear Lattice Effective Field Theory (NLEFT), Variational Monte Carlo (VMC), and the Projected Generator Coordinate Method (PGCM). Additionally, I analyze configurations derived from a three-parameter Fermi (3pF) density function. The investigation focuses on the effects of cluster parameters on two-point correlators using a rotor model for symmetric collisions ($^{16}$O+$^{16}$O and $^{20}$Ne+$^{20}$Ne) and asymmetric collisions ($^{208}$Pb+$^{16}$O and $^{208}$Pb+$^{20}$Ne). The cluster parameters are determined by minimizing the \textit{chi-square} statistic to align the nucleon distributions with those predicted by the aforementioned theories. The results reveal that perturbative calculations effectively capture the structural features of these nuclei, while comparisons with Monte Carlo simulations validate these findings. Furthermore, the analysis reveals distinct cluster geometries: VMC suggests tetrahedral shapes, while NLEFT, PGCM, and 3pF indicate irregular triangular pyramids. Notably, NLEFT shows a bowling pin-like $α$ cluster structure for $^{20}$Ne. The study also identifies constraints on cluster parameters in the different oxygen structures, with a gradual increase in $\varepsilon_2\{2\}$ for the states of $α$+$^{12}$C. Accurate modeling of asymmetric collisions necessitates a range of nucleons from heavy spherical nuclei, leading to weighted correlators in perturbative calculations. I demonstrate consistency between perturbative calculations and Monte Carlo models, with analytical calculations providing more insights into asymmetric than symmetric collisions.

Imprint of $α$-Clustering on Ab Initio Correlations in Relativistic Light Ion Collisions

Abstract

This study investigates the influence of -cluster structures in relativistic light nuclear collisions. Using a cluster framework, I extract the characteristics of the nucleonic configurations of O and Ne as predicted by various \textit{ab-initio} models, including Nuclear Lattice Effective Field Theory (NLEFT), Variational Monte Carlo (VMC), and the Projected Generator Coordinate Method (PGCM). Additionally, I analyze configurations derived from a three-parameter Fermi (3pF) density function. The investigation focuses on the effects of cluster parameters on two-point correlators using a rotor model for symmetric collisions (O+O and Ne+Ne) and asymmetric collisions (Pb+O and Pb+Ne). The cluster parameters are determined by minimizing the \textit{chi-square} statistic to align the nucleon distributions with those predicted by the aforementioned theories. The results reveal that perturbative calculations effectively capture the structural features of these nuclei, while comparisons with Monte Carlo simulations validate these findings. Furthermore, the analysis reveals distinct cluster geometries: VMC suggests tetrahedral shapes, while NLEFT, PGCM, and 3pF indicate irregular triangular pyramids. Notably, NLEFT shows a bowling pin-like cluster structure for Ne. The study also identifies constraints on cluster parameters in the different oxygen structures, with a gradual increase in for the states of +C. Accurate modeling of asymmetric collisions necessitates a range of nucleons from heavy spherical nuclei, leading to weighted correlators in perturbative calculations. I demonstrate consistency between perturbative calculations and Monte Carlo models, with analytical calculations providing more insights into asymmetric than symmetric collisions.

Paper Structure

This paper contains 11 sections, 71 equations, 8 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Methodology
  3. Charge density
  4. Correlators
  5. Results and discussion
  6. Symmetric Collisions
  7. Asymmetric Collisions
  8. Conclusion
  9. Two-body density distribution
  10. Two-point correlations
  11. Cumulant formulation of fireball energy-density fluctuations

Figures (8)

  • Figure 1: a) Schematic representations of the $\alpha$-cluster structures of $^{16}$O (tetrahedron) and $^{20}$Ne (bowling pin) are shown. Here, $\ell_c$ denotes the side length of the regular triangular pyramid in both structures, while $\ell_h$ indicates the distance from the center of the cluster at the top to the other three clusters in the middle of the bowling pin structure. Panel (b) illustrates two types of nucleon-nucleon correlations within a cluster model, intra-$\alpha$-cluster and inter-$\alpha$-cluster. The colors blue and red represent protons and neutrons, respectively. The top panel depicts the correlations between nucleons within the same clusters, whereas the bottom panel shows the correlations between nucleons from different clusters. These two effects contribute simultaneously to the results; however, for clarity, I distinguish them in this figure.
  • Figure 2: The one-nucleon density distributions are illustrated for $^{16}O$ (using PGCM in panel (a), VMC (b), NLEFT (c), 3pF (d)) and $^{20}Ne$ (using NLEFT (e), 3pF (f)). The red points represent the nuclear radial densities derived from the cluster configurations (labeled by $\alpha$ cl.). The curves for 3pF and PGCM are obtained from sampled configurations that have been re-centered to ensure an apples-to-apples comparison with the NLEFT and VMC results.
  • Figure 3: The values of $\chi^2$/NDF for the best estimates of the cluster parameters for VMC, NLEFT, PGCM, and 3pF are presented. The parameters can be found in the first and third rows of Table \ref{['tab1']}.. The $\chi^2$/NDF values serve as a criterion for selecting the sampled nucleons. The values corresponding to the nucleon configurations for neon are indicated within the shaded area.
  • Figure 4: The results of initial correlators, var($E/\langle E\rangle$) and $\varepsilon_2\{2\}$, for collisions involving $^{16}$O and $^{20}$Ne are presented in panel (a) and (b). The results of Ne+Ne collisions are highlighted by the shaded area in panels (a) through (d). Additionally, the ratios of $\mathcal{O}_{\text{configs}}/\mathcal{O}_{\text{3pF}}$ for different models of the oxygen and neon structures are shown in panels (c) and (d). I identify three types of results: perturbative calculations (represented by red and pink horizontal lines), Monte Carlo results (depicted by green and brown circles), and the results from TRENTo simulations at $\sqrt{s_{NN}}=200$ GeV (illustrated by gray rectangles) for O+O and Ne+Ne collisions. Note that TRENTo model results in panel (a) have been multiplied by 0.32 for better comparison. The results from Eqs.\ref{['varS']} and \ref{['eps2S']} for spherical shapes of oxygen and neon are indicated by cyan starts. The relative variation of $\varepsilon_2\{2\}$ between O+O and Ne+Ne collisions is presented in panel (e). The difference of regular (Fig.\ref{['Fige5']}(a)) and irregular (Fig.\ref{['Fige5']}(b)) triangular pyramid shapes for oxygen in $\varepsilon_2\{2\}$ are illustrated in panel (f). The values of cluster parameters can be found in Table \ref{['tab1']}. Note that statistical error bars are smaller than the symbols used in the figures.
  • Figure 5: The schematics of regular (a) and irregular (b) triangular pyramids for oxygen are presented. The values of cluster parameters for different configurations are shown in the second and third rows of Table \ref{['tab1']}.
  • ...and 3 more figures