  Consider a long-range percolation model on $\mathbb{Z}^d$ where the
probability that an edge $\{x,y\} \in \mathbb{Z}^d \times \mathbb{Z}^d$ is open
is proportional to $\|x-y\|_2^{-d-\alpha}$ for some $\alpha >0$ and where $d >
3 \min\{2,\alpha\}$. We prove that in this case the one-arm exponent equals $
\min\{4,\alpha\}/2$. We also prove that the maximal displacement for critical
branching random walk scales with the same exponent. This establishes that both
models undergo a phase transition in the parameter $\alpha$ when $\alpha =4$.
