1.1); (6.25,1.1) circle \[x radius=0.6, y radius = 1.1\]; (6.0,0) arc (-90:90:0.6 and 1.1);

(7.3,1.1) node [$|i \rangle$]{}; (-3.3,1.1) node [$\langle 0|$]{}; (2.3,1.1 ) node [$e^{\lambda^{k}\! \int\! \phi_{k}}$]{}; (2.2, 0.6) node [perturbed theory]{}; (-2,2.25) – (-2, 2.6); (6,2.25)–(6,2.6); (-2, 2.45) – (6,2.45); (2,2.68) node [$L$]{};

We have added little collars (depicted white) of unperturbed theory at both ends where the states $\langle 0|$, $|i\rangle$ are attached. The actual width of the collars is inessential. The perturbative expansion of (\[Z0i\]) is Z\_[0,i]{}(L) = \_[n]{} \_[i\_1,…, i\_[n]{}]{} \_[-Ly\_10]{} d\^2 z\_[1]{} …\_[-Ly\_n0]{} d\^2 z\_[n]{}0| \_[i\_1]{}(z\_1) …\_[i\_n]{}(z\_n) |i The correlators here are correlators in ${\rm CFT}_{\rm UV}$ and can be calculated by mapping the cylinder onto an annulus on the plane and inserting $\phi_{i}$ at infinity. Divergences arise when operators $\phi_{i_{k}}$ collide. Collisions can happen at a point in the bulk of the cylinder (annulus) or at a point on the boundary: at $y=0$ or $y=-L$. The former are handled by the usual renormalisation of perturbed theory while the latter may give rise to additional boundary counter terms of the type discussed in [@Symanzik][^3]. It is clear from the collisions picture that all divergences are local. Assuming they can be renormalised we obtain a 