The discontinuity method in a BCFT

We consider a conformal field theory in the presence of a boundary, and explain how two-point correlators of mixed bulk-local operators can be bootstrapped by exploiting the analytical structure of the conformal blocks. This yields the operator product expansion coefficients in either bootstrap channel. We apply this bootstrap technique to an $O(N)$-model in $d=4-\epsilon$ dimensions to find the $\phi -{\phi }^5$ bulk correlator and the corresponding operator product expansion coefficients upto $\mathcal{O}(\epsilon )$. Parts of this paper was first presented in my thesis.