should remember that the origin of the nontrivial crossover from the PCP to the DP lies in the finiteness of the auxiliary field density at the PCP critical point, while it vanishes at the PCPD critical point. So, if the PCPD belongs to the DP class, then one should expect a trivial crossover with $\phi=1$. Our finding of the nontrivial crossover in [@PP06] implies that the PCPD class is distinct from the DP class. Hence it is likely that the similarity of two crossover exponents be a mere coincidence.

The distinction of the PCPD from the DP can also be evidenced by the study of the crossover model with the hybrid-type reaction dynamics $2A \rightarrow 0$ and $3A \rightarrow 4A$. Without diffusion, this model belongs to the DP${}_\textrm{IM}$ and its crossover to the DP was studied in Sec. \[Sec:dpima\]. Even if particles are allowed to diffuse, this model (called as [tp12]{}) is numerically known to belong to the DP class [@KC03].

![\[Fig:tp12\] (Color online) Semi-log plot of $\rho(t)
t^{\delta}$ vs $t$ with $\delta = 0.1595$ at $p_c
=0.058~427,~0.058~723,~0.058~966$, and $0.059~031$ for $w=10^{-3},~5\times 10^{-4},~10^{-4}$, and $0$ (from left to right), respectively. There is no diverging time scale as $w$ approaches zero. 