Two-body 4He\(\left( \alpha \right)\)-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 \(NJ/\psi\) and \(N\eta _{c}\) interactions. The \(N\text {-}c\bar{c}\) potentials are observed to exhibit an attractive nature across all distances, accompanied by a characteristic long-range tail. It is found that the \(\alpha \text {-}J/\psi\) 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 \(\alpha \text {-}\eta _{c}\), no bound or resonance state (with respect to the \(\alpha \text {-} c\bar{c}\) threshold) was found. The \(\alpha \text {-}c\bar{c}\) correlation function in high-energy collisions is examined to explore the \(N\text {-}c\bar{c}\) interaction. The analysis revealed that variations in spin-dependent \(\alpha \text {-}c\bar{c}\) interactions– spin-3/2 \(\alpha \text {-}J/\psi\), spin-1/2 \(\alpha \text {-}J/\psi\), spin-1/2 \(\alpha \text {-}\eta _c\), and the spin-averaged \(\alpha \text {-}J/\psi\)–produce noticeable differences in the \(\alpha \text {-}c\bar{c}\) 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 \(\alpha \text {-}c\bar{c}\) system. Furthermore, a comparison has been conducted between two density functions of 4He–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 \(\alpha \text {-}J/\psi\) binding energies for the two models are nearly indistinguishable, their corresponding correlation functions demonstrate markedly different behaviors. This divergence could yield valuable insights into the nuclear matter distribution function of the alpha particle.
