  We study the $d$-dimensional ($d\geq2$) incompressible Oldroyd-B model with
only stress tensor diffusion and without velocity dissipation as well as the
damping mechanism on the stress tensor. Firstly, based upon some new
observations on the model, we develope the pure energy argument (independent of
spectral analysis) in general $L^p$ framework, and present a small initial data
global existence and uniqueness of solutions to the model. Our results yield
that the coupling and interaction of the velocity and the non-Newtonian stress
actually enhances the regularity of the system. Later, by adding some
additional $L^2$ type conditions on the low frequencies of the initial data
$(u_0,\tau_0)$, %but without any more smallness restrictions, we obtain the
optimal time-decay rates of the global solution $(u,\tau)$. Our result solves
the problem proposed in Wang, Wu, Xu and Zhong \cite{Wang-Wu-Xu-Zhong} ({\it J.
Funct. Anal.}, 282 (2022), 109332.).
