\tilde \sigma -\sigma =
w \sigma,\quad \tilde \xi = w \sigma,\\
&\tilde \lambda - \lambda  \simeq -  w (\eta+ 2 \lambda),\quad
\tilde \eta - \eta \simeq  w (\eta + 2 \lambda - 2 \sigma).
\end{aligned}
\label{Eq:crossBAW}$$ If $w$ is sufficiently small, it is always possible to associate the BAW1 with a stochastic process with spontaneous death in an equivalent way with all nonnegative rates. Since the BAW1 in high dimensions is also known to have a nontrivial transition point [@CCD04], the following discussion is valid in any spatial dimension. The transition points for stochastic systems with small $w$ can be always calculated exactly from Eq.  (approximately from Eq. ), if the transition point of the BAW1 is given.

The conclusions from the above analysis are two-fold. First, it is clear from Eq.  that the phase boundary (critical line) should meet the BAW1 transition point [*linearly*]{} with finite slope, as $w$ vanishes. This implies that there is no additional singularity involved near $w=0$, which is fully expected from the crossover between models with the identical universality class. If one defines the crossover exponent $\phi$ from the shape of the critical line near $w=0$ (see Sec. \[Sec:dpima\]), one can say that $\phi=1$. 