Lattice calculation of the $\eta_c\eta_c$ and $J/\psi J/\psi$ s-wave scattering length

We calculate the s-wave scattering length in the $0^+$ sector of $\eta_c\eta_c$ and the $2^+$ sector of $J/\psi J/\psi$ using three $N_f=2$ twisted mass gauge ensembles with the lattice spacing $a=0.0667,0.085,0.098$ fm, respectively. The scattering lengths are extracted using the conventional L{\"u}scher finite size method. We observe sizable discretization effects and the results after a continuum extrapolation are $a^{0^+}_{\eta_c\eta_c}=-0.104(09)$ fm and $a^{2^+}_{J/\psi J/\psi}=-0.165(16)$ fm. The slight difference of light quark mass is ignored in this study. Our results indicate that the interaction between the two respective charmonia are repulsive in nature in both cases.