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Quantitative predictions of alpha-charmonium correlation functions in high-energy collisions

Faisal Etminan

TL;DR

The paper addresses how to extract information on N–c cbar interactions from correlations between alpha particles and charmonium produced in high-energy collisions. It builds an effective alpha–c cbar potential by single-folding HAL QCD N–c cbar inputs with different 4He density models, solves the Schrödinger equation to obtain binding and scattering parameters, and computes alpha–c cbar momentum correlation functions using the Koonin-Pratt formalism and, for large sources, the Lednicky-Lyuboshits approximation. The results indicate a loosely bound alpha–J/ψ state (central binding ~0.1–0.6 MeV) and no bound alpha–ηc state; the correlation functions show spin- and density-model-dependent signatures, with sensitivity to source size around R ≈ 3 fm and to the alpha density profile. These findings suggest that alpha–c cbar femtoscopy can constrain the short- and long-range components of the N–c cbar interaction and the structure of the alpha cluster, with potential experimental tests at FAIR, NICA, and J-PARC HI in low-energy collisions where alpha and charmonium production coexist.

Abstract

Two-body $ ^{4}\textrm{He}\left(α\right)$-charmonium $ \left(c\bar{c}\right) $ potentials in the single-folding potential (SFP) approach are built by using a first principles HAL QCD low-energy $ NJ/ψ$ and $ Nη_{c} $ interactions. The $N\textrm{-}c\bar{c}$ potentials are observed to exhibit an attractive nature across all distances, accompanied by a characteristic long-range tail. It is found that the $ α\textrm{-}J/ψ$ system appears to be loosely bound with the central binding energy in the range of 0.1-0.6 MeV, while for spin-$ 1/2 $ $α\textrm{-}η_{c}$, no bound or resonance state (with respect to the $ α\textrm{-} c\bar{c} $ threshold) was found. The $ α\textrm{-}c\bar{c} $ correlation function in high-energy collisions is examined to explore the $ N\textrm{-}c\bar{c} $ interaction. The analysis revealed that variations in spin-dependent $α\textrm{-}c\bar{c}$ interactions-spin-$3/2$ $α\textrm{-}J/ψ$, spin-$1/2$ $α\textrm{-}J/ψ$, spin-$1/2$ $α\textrm{-}η_c$, and the spin-averaged $α\textrm{-}J/ψ$-produce noticeable differences in the $α\textrm{-}c\bar{c}$ correlation function, especially when the source size is around $ 3 $ fm. It is found that different results are produced by the Lednicky-Lyuboshits formula at small source sizes. This indicates that a relatively long-range interaction exists for the $ α\textrm{-}c\bar{c} $ system. Furthermore, a comparison has been conducted between two density functions of $ ^{4}\textrm{He}$ the central depression (CD) and the simple single Gaussian (SG) density-both of which share an identical rms radius of 1.56 fm. Although the $α\textrm{-}J/ψ$ binding energies for the two models are nearly indistinguishable, their corresponding correlation functions demonstrate markedly different behaviors.

Quantitative predictions of alpha-charmonium correlation functions in high-energy collisions

TL;DR

The paper addresses how to extract information on N–c cbar interactions from correlations between alpha particles and charmonium produced in high-energy collisions. It builds an effective alpha–c cbar potential by single-folding HAL QCD N–c cbar inputs with different 4He density models, solves the Schrödinger equation to obtain binding and scattering parameters, and computes alpha–c cbar momentum correlation functions using the Koonin-Pratt formalism and, for large sources, the Lednicky-Lyuboshits approximation. The results indicate a loosely bound alpha–J/ψ state (central binding ~0.1–0.6 MeV) and no bound alpha–ηc state; the correlation functions show spin- and density-model-dependent signatures, with sensitivity to source size around R ≈ 3 fm and to the alpha density profile. These findings suggest that alpha–c cbar femtoscopy can constrain the short- and long-range components of the N–c cbar interaction and the structure of the alpha cluster, with potential experimental tests at FAIR, NICA, and J-PARC HI in low-energy collisions where alpha and charmonium production coexist.

Abstract

Two-body -charmonium potentials in the single-folding potential (SFP) approach are built by using a first principles HAL QCD low-energy and interactions. The potentials are observed to exhibit an attractive nature across all distances, accompanied by a characteristic long-range tail. It is found that the system appears to be loosely bound with the central binding energy in the range of 0.1-0.6 MeV, while for spin- , no bound or resonance state (with respect to the threshold) was found. The correlation function in high-energy collisions is examined to explore the interaction. The analysis revealed that variations in spin-dependent interactions-spin- , spin- , spin- , and the spin-averaged -produce noticeable differences in the correlation function, especially when the source size is around fm. It is found that different results are produced by the Lednicky-Lyuboshits formula at small source sizes. This indicates that a relatively long-range interaction exists for the system. Furthermore, a comparison has been conducted between two density functions of the central depression (CD) and the simple single Gaussian (SG) density-both of which share an identical rms radius of 1.56 fm. Although the binding energies for the two models are nearly indistinguishable, their corresponding correlation functions demonstrate markedly different behaviors.

Paper Structure

This paper contains 10 sections, 12 equations, 11 figures, 2 tables.

Table of Contents

  1. Introduction
  2. $N\textrm{-}c\bar{c}$ interactions and $\alpha\textrm{-}c\bar{c}$ SFP
  3. State-of-the-art QCD nucleon-charmonium interactions
  4. $\alpha\textrm{-}c\bar{c}$ SFP
  5. Approximation for when charmonium is outside of alpha
  6. Central depression density distribution
  7. Simple single Gaussian density distribution
  8. Two-particle correlation function formalism
  9. Numerical results and discussion
  10. Summary and conclusions

Figures (11)

  • Figure 1: The $S$-wave $N\textrm{-}c\bar{c}$ potentials (in Eq. \ref{['eq:NCharm_pot']}) as functions of the distance between $N$ and $c\bar{c}$ are shown at the imaginary-time distances $t/a=14$ by parametrization from Ref. LyuPLB2025 as given in Table \ref{['tab:Charm-N-para']}. The spin-$3/2$$NJ/\psi$ is depicted by dashed red line, spin-$1/2$$NJ/\psi$ by dotted blue line, spin-$1/2$$N\eta_{c}$ by solid green line and the spin-averaged $NJ/\psi$ (Eq. \ref{['eq:NJpsi-spin-ave']}) by dash-dotted purple line.
  • Figure 2: The data points show the obtained $\alpha\textrm{-}c\bar{c}$ potentials through solving integral equation and the solid line display the corresponding WS fits for (a) spin-$3/2$$NJ/\psi$, (b) spin-$1/2$$NJ/\psi$, (c) spin-$1/2$$N\eta_{c}$ and (d) the spin-averaged $NJ/\psi$. The different symbols correspond to the different calculation method and rms radii. The data points (filled black circles) show the $U_{\alpha\textrm{-}c\bar{c}}\left(r\right)$ that calculated by Eq. \ref{['eq:V_alfaOmega']} using density function which is given by Eq. \ref{['eq:nucleon-density']}. The corresponding errors for data points are statistical. The obtained potential using Eq. eq:anal_VccbarAlpha and central depression density distribution in Eq. \ref{['eq:ctr_depRHO']} which gives rms matter radius of $^4$He, 1.56 fm is shown by the filled purple triangle. The analytical obtained potential via Eq. eq:anal_VccbarAlpha using simple single Gaussian density distribution (Eq. \ref{['eq:gauss-dist']}) for the values of rms radius: 1.56, 1.70, and 1.84 fm, are indicated by the hollow red circle, green triangle and blue square, respectively. The lines show the fitting of $\alpha\textrm{-}c\bar{c}$. The results of the fit are presented in Table \ref{['tab:Charm-alpha-para']}.
  • Figure 3: The phase shifts $\delta_{0} / \pi$ for the $\alpha\textrm{-}c\bar{c}$ system are depicted as functions of the relative momentum $q = \sqrt{2\mu E}$, based on the obtained potential $U_{\alpha\textrm{-}c\bar{c}}(r)$. Here, $\mu$ represents the reduced mass of the $\alpha\textrm{-}c\bar{c}$ system.
  • Figure 4: The spin-$3/2$$\alpha\textrm{-}J/\psi$ correlation functions for three different source sizes: (a) $R=1$ fm, (b) $R=3$ fm and (c) $R=5$ fm, with models of central depression (Cent. Dep.) density distribution as given by Eq. \ref{['eq:ctr_depRHO']} which gives rms matter radius of $^4$He, 1.56 fm (dash-dotted magenta line) and with the model of simple single Gaussian density distribution as given by Eq. \ref{['eq:gauss-dist']} for the three values of the rms radius: $1.84$ fm (long dashed red line), $1.70$ fm (solid green line), and $1.56$ fm (dotted blue magenta line).
  • Figure 5: The spin-$1/2$$\alpha\textrm{-}J/\psi$ correlation functions for three different source sizes: (a) $R=1$ fm, (b) $R=3$ fm and (c) $R=5$ fm. Symbols have the same description as in Fig. \ref{['fig:cq-kp-JPsi4S32Alpha']}.
  • ...and 6 more figures