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We make a detailed study on the $J/\psi$ meson longitudinal leading-twistdistribution amplitude $\phi_{2;J/\psi}^\|$ by using the QCD sum rules withinthe background field theory. By keeping all the non-perturbative condensates upto dimension-six, we obtain accurate QCD sum rules for the moments$\langle\xi_{n;J/\psi}^\|\rangle$. The first three ones are$\langle\xi_{2;J/\psi}^\|\rangle=0.083(12)$,$\langle\xi_{4;J/\psi}^\|\rangle=0.015(5)$ and$\langle\xi_{6;J/\psi}^\|\rangle=0.003(2)$, leading to a single peaked behaviorfor $\phi_{2;J/\psi}^\|$ which is sharper than the previous ones around theregion of $x\sim0.5$. As an application, we adopt the QCD light-cone sum rulesto calculate the $B_c$ meson semileptonic decay $B_c^+ \to J/\psi \ell^+\nu_\ell$. We obtain $\Gamma(B_c^+ \to J/\psi \ell^+ \nu_\ell) =(89.67^{+24.76}_{-19.06}) \times 10^{-15}~{\rm GeV}$ and $\Re(J/\psi \ell^+\nu_\ell) = 0.217^{+0.069}_{-0.057}$, which agree with the next-to-leadingorder pQCD prediction and the new CDF measurement within errors.

The longitudinal leading-twist distribution amplitude of $J/\psi$ meson within background field theory