Linear dependence of peak width in $\chi (\text{bq},\omega )$ vs T_c for YBCO superconductors

It is shown that the momentum space width of the peak in the spin susceptibility, Im$\chi (q,\omega )$, is linearly proportional to the superconducting $T_c$: $T_c=\mathrm{\hslash }{v}^{\ast }\mathrm{\Delta }q$ with $\hbar v^* \simeq 35 meV$\AA. This relation is similar to the linear relation between incommensurate peak splitting and $T_c$ in LaSrCuO superconductors, as first proposed by Yamada et al. (Phys. Rev. B 57, 6165, (1998)). The velocity $\mathrm{\hslash }{v}^{\ast }$ is smaller than Fermi velocity or the spin-wave velocity of the parent compound and remains the same for a wide doping range. This result points towards strong similarities in magnetic state of YBCO and LaSrCuO.