  In this paper, we are concerned with optimal decay rate for the 2-D
generalized Oldroyd-B model with only stress tensor diffusion
$(-\Delta)^{\beta}\tau$. In the case $\beta=1$, we first establish optimal
decay rate in $H^1$ framework and remove the smallness assumption of low
frequencies by virtue of the Fourier splitting method and the Littlewood-Paley
decomposition theory. Furthermore, we prove optimal decay rate for the highest
derivative of the solution by a different method combining time frequency
decomposition and the time weighted energy estimate. In the case $\frac 1 2\leq
\beta <1$, we study optimal decay rate for the highest derivative of the
solution by the improved Fourier splitting method.
