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 ( ε ) of the ε = 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.

中文翻译：

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超常转变的边界共形场理论：层对 O(ε) 的敏感性

我们提出了在具有平坦边界的半无限系统中异常转变时层（平行）磁化率的解析计算。使用 McAvity 和 Osborn 提出的积分变换方法[Nucl. 物理。B 455 (1995) 522] 在边界 CFT 中，我们导出平均场传播器在过渡点的坐标空间表示。该函数的简单代数结构为高阶计算提供了实用的可能性。因此，我们计算了单环近似中异常转变时层磁化率的显式表达式。我们的结果在 ε = 4 - d 扩展的 O ( ε ) 阶上是正确的，并且适用于层的任意宽度及其在半空间中的位置。我们讨论了我们结果的一般结构，并考虑了与边界算子扩展和（批量）算子乘积扩展相关的限制情况。我们将我们的发现与先前已知的结果和普通过渡情况下不太复杂的公式进行比较。我们相信层磁化率的分析结果可能是有效计算两点相关函数的良好起点。考虑到最近在一般维度的体积和边界共形场理论方面的突破，这种可能性将非常重要。我们相信层磁化率的分析结果可能是有效计算两点相关函数的良好起点。考虑到最近在一般维度的体积和边界共形场理论方面的突破，这种可能性将非常重要。我们相信层磁化率的分析结果可能是有效计算两点相关函数的良好起点。考虑到最近在一般维度的体积和边界共形场理论方面的突破，这种可能性将非常重要。