  Very recently, the Belle and BESIII experiments observed a new
charmonium-like state $X(3823)$, which is a good candidate for the $D$-wave
charmonium $\psi(1^3D_2)$. Because the $X(3823)$ is just near the $D\bar{D}^*$
threshold, the decay $X(3823)\to J/\psi\pi^+\pi^-$ can be a golden channel to
test the significance of coupled-channel effects. In this work, this decay is
considered including both the hidden-charm dipion and the usual quantum
chromodynamics multipole expansion (QCDME) contributions. The partial decay
width, the dipion invariant mass spectrum distribution
$\mathrm{d}\Gamma[X(3823)\to J/\psi\pi^+\pi^-]/\mathrm{d}m_{\pi^+\pi^-}$, and
the corresponding $\mathrm{d}\Gamma[X(3823)\to
J/\psi\pi^+\pi^-]/\mathrm{d}\cos\theta$ distribution are computed. Many
parameters are determined from existing experimental data, so the results
depend mainly only on one unknown phase between the QCDME and hidden-charm
dipion amplitudes.
