Boundary conformal field theory at the extraordinary transition: The layer susceptibility to $O(\varepsilon)$

TL;DR

This paper analytically calculates the layer susceptibility at the extraordinary transition in boundary conformal field theory, providing explicit formulas up to one-loop order and discussing implications for higher-order calculations and correlation functions.

Contribution

It introduces an analytic method for calculating layer susceptibilities at the extraordinary transition using boundary CFT techniques, valid up to order $O( ext{ extmu})$ in the $ ext{ extmu}$ expansion.

Findings

Explicit expression for layer susceptibility at one-loop order.

Results are valid for arbitrary layer width and position.

Comparison with known results for the ordinary transition.

Abstract

We present an analytic calculation of the layer (parallel) susceptibility at the extraordinary transition in a semi-infinite system with a flat boundary. Using the method of integral transforms put forward by McAvity and Osborn [Nucl. Phys. B 455 (1995) 522] in the boundary CFT we derive the coordinate-space representation of the mean-field propagator at the transition point. The simple algebraic structure of this function provides a practical possibility of higher-order calculations. Thus we calculate the explicit expression for the layer susceptibility at the extraordinary transition in the one-loop approximation. Our result is correct up to order $O(\varepsilon)$ of the $\varepsilon=4-d$ expansion and holds for arbitrary width of the layer and its position in the half-space. We discuss the general structure of our result and consider the limiting cases related to the boundary operator expansion and (bulk) operator product expansion. We compare our findings with previously known results and less complicated formulas in the case of the ordinary transition. We believe that analytic results for layer susceptibilities could be a good starting point for efficient calculations of two-point correlation functions. This possibility would be of great importance given the recent breakthrough in bulk and boundary conformal field theories in general dimensions.