"No-Scalar-Hair" Theorems for Nonminimally Coupled Fields with Quartic Self-Interaction

Self-gravitating scalar fields with nonminimal coupling to gravity and having a quartic self-interaction are considered in the domain of outer communications of a static black hole. It is shown that there is no value of the nonminimal coupling parameter $\zeta$ for which nontrivial static black hole solutions exist. This result establishes the correctness of Bekenstein ``no-scalar-hair'' conjecture for quartic self-interactions.