Black holes with regular scalar hair in Brans-Dicke gravity via the Herglotz variational principle

Brans-Dicke theory is reformulated within the Herglotz variational principle (HVP), and an exact black hole solution with scalar hair is obtained for ${\omega }_0=0$ and vanishing potential $V(\phi )=0$. The scalar profile is strictly positive and the resulting stealth Schwarzschild solution arises without fixing the otherwise arbitrary Herglotz function $\eta (r)$. Motivated by the weak-field limit, the explicit choice $\eta (r)={\eta }_0(r-2M{)}^k\mathrm{/}{r}^{k+2}$, with ${\eta }_0$ a constant of dimension length, produces a scalar field configuration remaining regular at the black hole horizon. Consequently, the HVP provides a new mechanism for evading standard no-hair theorems in scalar-tensor theories.