  In this paper, we study the large time behavior of solutions to the
compressible magnetohydrodynamic equations in the $L^p$-type critical Besov
spaces. Precisely, we show that if the initial data in the low frequencies
additionally belong to some Besov space $\dot{B}_{2,\infty}^{-\sigma_1}$ with
$\sigma_1\in (1-N/2, 2N/p-N/2]$, then the $\dot{B}_{p,1}^0$ norm of the
critical global solutions presents the optimal decay
$t^{-\frac{N}{2}(\frac{1}{2}-\frac{1}{p})-\frac{\sigma_1}{2}}$ for
$t\rightarrow+\infty$. The pure energy argument without the spectral analysis
is performed, which allows us to remove the usual smallness assumption of low
frequencies.
