Critical Behavior in a Massless Scalar Field Collapse with Self-interaction Potential

We examine a one-parameter family of analytical solutions representing spherically symmetric collapse of a nonlinear massless scalar field with self-interaction in an asymptotically flat spacetime. The time evolution exhibits a type of critical behavior. Depending on the scalar charge parameter $q$ as compared to a critical value $q^{\ast }$, the incoming scalar wave collapses either to a globally naked central singularity if $q<q^{\ast }$ (weak field) or to a scalar-hairy black hole if $q>q^{\ast }$ (strong field), both having finite asymptotic masses. Near the critical evolution, the black hole mass follows a product-logarithmic scaling law: $-M^2\ln M\sim q-q^{\ast }$ with $0<M\ll 1$ and $q>q^{\ast }$. The solution admits no self-similarity and satisfies the null and the strong energy conditions.