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

We make a detailed study on the $J\mathrm{/}\psi$ meson longitudinal leading-twist distribution amplitude ${\varphi }_{2;J\mathrm{/}\psi }^{\mathrm{\parallel }}$ by using the QCD sum rules within the background field theory. By keeping all the non-perturbative condensates up to dimension-six, we obtain accurate QCD sum rules for the moments $⟨{\xi }_{n;J\mathrm{/}\psi }^{\mathrm{\parallel }}⟩$. The first three ones are $⟨{\xi }_{2;J\mathrm{/}\psi }^{\mathrm{\parallel }}⟩=0.083(12)$, $⟨{\xi }_{4;J\mathrm{/}\psi }^{\mathrm{\parallel }}⟩=0.015(5)$ and $⟨{\xi }_{6;J\mathrm{/}\psi }^{\mathrm{\parallel }}⟩=0.003(2)$, leading to a single peaked behavior for ${\varphi }_{2;J\mathrm{/}\psi }^{\mathrm{\parallel }}$ which is sharper than the previous ones around the region of $x\sim 0.5$. As an application, we adopt the QCD light-cone sum rules to calculate the $B_c$ meson semileptonic decay $B_c^+ \to J/\psi \ell^+ \nu_\ell$\mathrm{.}Weobtain$\Gamma(B_c^+ \to J/\psi \ell^+ \nu_\ell) = (89.67^{+24.76}{-19.06}) \times 10^{-15}~{\rm GeV}$and$\Re(J/\psi \ell^+ \nu\ell) = 0.217^{+0.069}_{-0.057}$, which agree with the next-to-leading order pQCD prediction and the new CDF measurement within errors.