  MnBi$_2$Te$_4$, the first confirmed intrinsic antiferromagnetic topological
insulator, has garnered increasing attention in recent years. Here we
investigate the energy correction and lifetime of magnons in MnBi$_2$Te$_4$
caused by magnon-magnon interaction. First, a calculation based on the density
functional theory was performed to get the parameters of the magnetic
Hamiltonian of MnBi$_2$Te$_4$. Subsequently, the perturbation method of
many-body Green's function was employed and the first-order self-energy
[$\Sigma^{(1)}(\bm k)$] and second-order self-energy [$\Sigma^{(2)}(\bm
k,\varepsilon_{\bm k})$] of magnon were obtained. Numerical computations reveal
that the corrections from both $\Sigma^{(1)}(\bm k)$ and $\Sigma^{(2)}(\bm
k,\varepsilon_{\bm k})$ strongly rely on momentum and temperature, with the
energy renormalization near the Brillouin zone (BZ) boundary being
significantly more pronounced than that near the BZ center. Furthermore, our
findings indicate the occurrence of dip structures in the renormalized magnon
spectrum near the $\rm K$ and $\rm M$ points. These dip structures are
determined to be attributed to the influence of $\Sigma^{(2)}(\bm
k,\varepsilon_{\bm k})$.
