  We are interested in the branching capacity of the range of a random walk in
$\mathbb Z^d$.Schapira [28] has recently obtained precise asymptotics in the
case $d\ge 6$ and has demonstrated a transition at dimension $d=6$. We study
the case $d=5$ and prove that the renormalized branching capacity converges in
law to the Brownian snake capacity of the range of a Brownian motion. The main
step in the proof relies on studying the intersection probability between the
range of a critical Branching random walk and that of a random walk, which is
of independent interest.
