标题： 反对基于 charm 以下的$e^+e^- \to$含子的 OPE 简单截断的证据 显示英文标题

标题： Evidence against naive truncations of the OPE from $e^+e^- \to$ hadrons below charm

摘要： 算符乘积展开式（OPE）按维度截断后被应用于许多场合。 一个例子是从基于有限能量求和规则的各种分析方法中利用介子 $\tau$ 衰变数据提取强耦合常数 $\alpha_s$。 在这里，我们重新考虑一种长期使用的方法，该方法通过 OPE 参数化 $I=1$ 向量和轴向真空极化的非微扰贡献，并将几个高维系数设为零以实际实施该方法。 假设这样做对 $\alpha_s$ 的值影响可以忽略不计，等价于假设 OPE 的低维部分随着接近 $\tau$ 质量时维度的增加迅速收敛。 如果这一假设成立，则它在高于 $\tau$ 质量的能量上也必须成立。 由此可知，该方法可以通过从$e^+e^-\to\mbox{hadrons}$获取的数据进行测试，因为它们不受$\tau$衰变的运动学约束限制。 我们利用最近发布的关于$R$比值的高精度综合数据进行了此类研究，认为这可以提供对该策略有效性的一些见解，即使它探测的是一个不同的但相关的通道。 我们发现，基于$e^+e^-$的测试对低维度部分在$\tau$质量附近的 OPE 快速收敛的隐含假设提出了质疑，并强调了将有限能量求和规则分析限制在仅接收 OPE 中较低阶项贡献的可观测量上的必要性。 显示英文摘要

摘要： The operator product expansion (OPE), truncated in dimension, is employed in many contexts. An example is the extraction of the strong coupling, $\alpha_s$, from hadronic $\tau$-decay data, using a variety of analysis methods based on finite-energy sum rules. Here, we reconsider a long-used method, which parametrizes non-perturbative contributions to the $I=1$ vector and axial vacuum polarizations with the OPE, setting several higher-dimension coefficients to zero in order to implement the method in practice. The assumption that doing this has a negligible effect on the value of $\alpha_s$ is tantamount to the assumption that the low-dimension part of the OPE converges rapidly with increasing dimension near the $\tau$ mass. Were this assumption valid, it would certainly have to be valid at energies above the $\tau$ mass as well. It follows that the method can be tested using data obtained from $e^+e^-\to\mbox{hadrons}$, as they are not limited by the kinematic constraints of $\tau$ decays. We carry out such an investigation using a recent high-precision compilation for the $R$-ratio, arguing that it provides insights into the validity of the strategy, even if it probes a different, though related channel. We find that $e^+e^-$-based tests call into question the implied assumption of rapid convergence of the low-dimension part of the OPE around the $\tau$ mass, and thus underscore the need to restrict finite-energy sum-rule analyses to observables which receive only contributions from lower-order terms in the OPE.