Green's function for cut points of chordal SLE attached with boundary arcs

Let $\kappa \in (4,8)$. Let $\gamma$ be an SLE${}_{\kappa }$ curve in a Jordan domain $D$ connecting $a_1\mathrm{\ne }{a}_2\in \mathrm{\partial }D$. We attach $\gamma$ with two open boundary arcs $A_1,A_2$ of $D$, which share end points $b_1\ne b_2\in\partial D\setminus{a_1,a_2}$,andconsiderforeach$z_0\in D$ the limit $$\underset{r\downarrow 0}{\lim }{r}^{1-\frac{3}{8}\kappa }\mathbb{P}[\gamma \cup A_1\cup A_{2}hasacutpointin\{\mathrm{\mid }z-z_0\mathrm{\mid }<r\}]\mathrm{.}$$ We prove that the limit converges, derive a rate of convergence, and obtain the exact formula of the limit up to a multiplicative constant depending only on $\kappa$.