AI 中文总结
本文推导了有限温度 PTRG 的闭式热阈值函数，扩展了零温度反常维度构造到有限温度，还首次将改进截断的不动点作为正则化子的连续函数追踪，结果平滑有界无病理点。
AI 中文摘要
我们推导了有限温度固有时间重整化群（PTRG）的热阈值函数的闭式表达式，此前这些函数只能通过逐 Matsubara 模式数值计算得到。对于标准的单参数正则化子族，对 Matsubara 和式进行泊松重求和可得到由修正贝塞尔函数构成的快速收敛的绕数级数，且单个代数恒等式可将所有高阶阈值函数在偏移后的核参数处简化为同一闭式形式。尖锐固有时间正则化子作为该族在 m→∞ 时的精确端点（以可控 O(1/m) 方式趋近），可分解为场依赖项和纯热项。基于此，O(N) 对称理论的局域势近似（LPA）及其改进形式（含跑动反常维度的 LPA'）可简化为已有的结果，所有结果均通过优化正则化子的精确 Wetterich 方程进行了交叉验证。有两项发现超出了简化范畴：其一，本文将已知的零温度反常维度构造扩展到了有限温度；其二，由于该闭式形式适用于任意实正则化子参数，首次可将改进截断的不动点作为正则化子的连续函数进行追踪，而非仅在少数孤立点上计算，结果显示其平滑变化且保持有界，在整个区间内无特殊点或病理点。
英文摘要
We derive closed-form expressions for the thermal threshold functions of the finite-temperature proper-time renormalisation group (PTRG): until now these have been evaluated numerically, Matsubara mode by Matsubara mode. For the standard one-parameter regulator family, Poisson resummation of the Matsubara sum yields a rapidly convergent winding-number series of modified Bessel functions, and a single algebraic identity reduces every higher threshold function to this same closed form at a shifted kernel parameter. The sharp proper-time regulator, recovered as the exact $m\to\infty$ endpoint of the family with a controlled $O(1/m)$ approach, factorises into a field-dependent and a purely thermal piece. Built on them, the local potential approximation (LPA) and its refinement to a running anomalous dimension (LPA$'$) for the $O(N)$-symmetric theory reduce to established results, all cross-checked against the exact Wetterich equation with the optimised regulator. Two findings go beyond reduction. First, the known zero-temperature anomalous-dimension construction is extended here to finite temperature. Second, because the closed form holds for any real regulator parameter, the refined truncation's fixed point can for the first time be tracked as a continuous function of the regulator, rather than at a handful of isolated points, and is found to vary smoothly and remain bounded, with no special or pathological point anywhere on the line.
Comments22 pages, 4 figures
