Reduced effect of impurities on the universal pairing scale in the cuprates

We consider the effect of non-magnetic impurities on the onset temperature $T^{\ast }$ for the $d-$wave pairing in spin-fluctuation scenario for the cuprates. We analyze intermediate coupling regime when the magnetic correlation length $\xi \mathrm{/}a>1$ and the dimensionless coupling $u$ is O(1). In the clean limit, $T^* \approx 0.02 v_f/a$inthisparameterrange,andweaklydependson$\xi$and$u$. We found numerically that this universal pairing scale is also quite robust with respect to impurities: the scattering rate ${\mathrm{\Gamma }}_{cr}$ needed to bring $T^{\ast }$ down to zero is about 4 times larger than in weak coupling, in good quantitative agreement with experiments. We provide analytical reasoning for this result.