  We present a lattice QCD study of the neutrinoless double beta decay
involving light sterile neutrinos. The calculation is performed at physical
pion mass using five gauge ensembles generated with the $2+1$-flavor domain
wall fermions. We obtain the low-energy constants
$g_{\text{LR}}^{\pi\pi}(m_\nu)$ with the neutrino mass $m_\nu$ from $0$ GeV to
$3$ GeV. The lattice results are reasonably consistent with the previous
interpolation method with a $\sim20\%$ deviation at small $m_\nu$. We provide
an explanation on the discrepancy at vanishing neutrino mass. At large $m_\nu$,
a good consistence between our results and the previous lattice determination
of $g_{4}^{\pi\pi}(\mu)$ is found at $\mu=m_\nu=3$ GeV.
