which we call an RG interface. The idea to associate such interfaces with RG flows was put forward in [@BR] (see also [@FQ]). A concrete proposal for such an interface for $\psi_{1,3}$-flows between neighbouring minimal models was put forward in [@Gaiotto] and pairing (\[conf\_pairing\]) associated with it was considered[^2].

Note that we do not need to have fundamental fields and functional integral representation to define the pairing in terms of the RG interface. To elucidate renormalisation and to relate the above functional integral picture to conformal perturbation and TCSA we first recast (\[perturbed\_psi\]) in the language of conformal perturbation. Wave functionals correspond to a particular basis in state space in which the field operator $\varphi$ is diagonal. We can choose instead the conformal basis of scaling states $|i\rangle$ in ${\rm CFT}_{\rm UV}$. Consider the following amplitude on a cylinder of length $L$ \[Z0i\] Z\_[0,i]{}(L) = 0| | i illustrated on the picture below.

(-2,0) – (6,0) arc ( 270:90: 0.6 and 1.1)– (-2,2.2) arc (90:270: 0.6 and 1.1)–cycle; (-2.25,0)–(6.25,0); (-2.25,2.2)–(6.25,2.2); (-2.25,2.2) arc (90: 270: 0.6 and 1.1); (-2.3,0) arc (-90:90:0.6 and 1.1 ); (-2,2.2) arc (90: 270: 0.6 and 1.1); (-2,0) arc (-90:90:0.6 and 1.1 ); (6,0) arc (270:90:0.6 and 