by the DP relaxation time exponent: $\phi = \nu_\|\simeq 1.733$ (or $\mu_\|=1$) which is compatible with the numerical estimation within error. The reason is as follows: Take the model of Eq.  at $q=0$. The critical line should be determined by the competition between the single-particle annihilation process ($A\rightarrow\emptyset$) parameterized by $w$ and the multi-particle (pair) reaction process ($2A\rightarrow\emptyset$ or $3A$) parameterized by $\Delta\sim(p_0-p)$. We expect both events should appear at the same time scale along the critical line to balance off each other. Since the single-particle (auxiliary field) density is finite at the PCP (DP${}_\textrm{IM}$) point, the time scale for $A\rightarrow\emptyset$ should be simply proportional to $w^{-1}$. The time scale for the pair reaction process should be given by the relaxation time scale $\tau\sim \Delta^{-\nu_\|}$. Consequently, the critical line is determined as $\Delta_c\sim(p_0-p_c(w)) \sim w^{1/\nu_\|}$, which yields $\phi=\nu_\|$ and equivalently $\mu_\|=1$.

Considering the crossover between the DP models with a unique absorbing state, the time scale for the process parameterized by $w$ is proportional to $w^{-\nu_\|}$ like the other competing process because the auxiliary field (particle) density is also vanishing critically as $w$ decreases to zero. In this case, we get $\phi=1$ and equivalently $\mu_\|=\nu_\|$, which is consistent with our 