suggesting that a larger enhancement of the rate constant will be reached with a longer τ c . Eq. 2 provides an excellent agreement with HEOM under this lossy regime, as is verified in the previous work . When τ c further increases, Eq. 2 needs to include phonon broadening effect to avoid divergence when τ c → ∞, resulting in ) which is a convolution between Eq. 2 and the broadening function A ν (ω -ω 0 ) (see Eq. 14), and the fundamental scaling suggested in Eq. 9 is preserved.