  The Cauchy problem of the compressible Oldroyd-B model without damping
mechanism in R^n$ with $n\ge2$ is considered. The lack of dissipation in
density and stress tensor in the model is compensated by exploiting an
intrinsic structure and introducing new quantities between density, velocity
and stress tensor. Therefore, global solutions to the system with small initial
data in critical Besov spaces are obtained. As a byproduct, optimal time decay
rates of the solutions are derived by using an energy estimation argument. The
results remain valid for the compressible viscoelastic system without the
`div-curl structure assumption and thus improve those given by Hu and Wang [ J.
Differential Equations, {\bf 250}, 1200--1231, 2011] and Qian and Zhang [Arch.
Ration. Mech. Anal., {\bf 198}, 835--868, 2010].
