发表机构
Universidad de la República; Ca’ Foscari University of Venice（共和国大学; 威尼斯卡福斯卡里大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文提出函数维度正则化（FDR），一种结合维度正则化与函数RG的新框架，在局域势和导数展开下恢复ε展开并精确计算Ising临界指数，兼具技术简洁与通用性。
AI 中文摘要
我们引入并发展了函数维度正则化（FDR），这是一种新颖的函数重整化群（RG）框架，它将维度正则化的原理扩展到微扰ε展开之外。关键见解在于，将FDR β函数（在任意连续维度d中有效，并以阈值函数为特征）定义为DR中计算出的相应方案无关β函数在所有临界维度上的总和。我们证明了该形式体系具备完整函数RG的所有特征。在局域势近似内，我们进行了普遍的定点分析，恢复了所有多临界模型的ε展开，并建立了与函数标度解、尖峰图和本征微扰谱的直接联系。将该框架扩展到导数展开的二阶，我们计算了d=3中Ising普适类的临界指数，提供了反常维度η和RG谱的估计值，这些估计值快速收敛，并与最先进的结果表现出良好的一致性。我们与非微扰和固有时RG方法进行了详细比较，突出了结构相似性和关键差异。特别是，我们引入了一种“蒸馏”程序，允许从任何其他单环精确RG方案中系统地推导出FDR流。我们的结果确立了FDR作为一个自洽且具有竞争力的RG框架，它结合了维度正则化的技术简洁性与函数方法的通用性。
英文摘要
We introduce and develop Functional Dimensional Regularization (FDR), a novel functional renormalization group (RG) framework that extends the principles of dimensional regularization beyond the perturbative $\varepsilon$-expansion. The key insight is to define the FDR beta functions, valid in any continuous dimension $d$ and characterized by threshold functions, as a sum over all critical dimensions of the corresponding scheme-independent beta functions computed in DR. We demonstrate that this formalism exhibits all the hallmarks of a fully-fledged functional RG. Within the local potential approximation, we perform a general fixed point analysis, recovering the $\varepsilon$-expansion for all multi-critical models and establishing a direct connection to functional scaling solutions, spike plots, and eigen-perturbation spectra. Extending the framework to the second order of the derivative expansion, we compute the critical exponents for the Ising universality class in $d=3$, providing estimates for the anomalous dimension $η$ and RG spectrum that converge rapidly and show favorable agreement with state-of-the-art results. We present a detailed comparison with non-perturbative and proper-time RG approaches, highlighting structural similarities and key differences. In particular, we introduce a ``distillation'' procedure that allows one to systematically derive the FDR flow from any other one-loop-exact RG scheme. Our results establish FDR as a self-contained and competitive RG framework that combines the technical simplicity of dimensional regularization with the versatility of the functional approach.
Comments18 pages, 7 figures
