New attractor mechanism for spherically symmetric extremal black holes

We introduce a new attractor mechanism to find the entropy for spherically symmetric extremal black holes. The key ingredient is to find a two-dimensional (2D) dilaton gravity with the dilaton potential $V(\varphi )$. The condition of an attractor is given by ${\mathrm{\nabla }}^2\varphi =V({\varphi }_0)$ and ${\bar{R}}_2=-V^{\mathrm{\prime }}({\varphi }_0)$ and for a constant dilaton $\varphi ={\varphi }_0$, these are also used to find the location of the degenerate horizon $r=r_e$ of an extremal black hole. As a nontrivial example, we consider an extremal regular black hole obtained from the coupled system of Einstein gravity and nonlinear electrodynamics. The desired Bekenstein-Hawking entropy is successfully recovered from the generalized entropy formula combined with the 2D dilaton gravity, while the entropy function approach does not work for obtaining this entropy.