I summarise a method to quantitatively assess the consistency of power-counting proposals in Effective Field Theories which are non-perturbative at leading order. It uses the fact that the Renormalisation Group evolution of an observable predicts the functional form of its residual cutoff dependence on the breakdown scale of an Effective Field Theory (EFT), on the low-momentum scales, and on the order of the calculation. Passing this test is a necessary but not sufficient consistency criterion for a suggested power counting whose exact nature is disputed. For example, in Chiral Effective Field Theory (\(\chi \hbox {EFT}\)) with more than one nucleon, a lack of universally accepted analytic solutions obfuscates the relation between convergence pattern and numerical results, and led to proposals which predict different numbers of Low Energy Coefficients (LECs) at the same chiral order, and at times even predicts a different ordering long-range contributions. The method may provide an independent check whether an observable is properly renormalised at a given order, and allows one to estimate both the breakdown scale and the momentum-dependent order-by-order convergence pattern of an EFT. Conversely, it may help identify those LECs (and long-range pieces) which produce renormalised observables at a given order. I also discuss its underlying assumptions and relation to the Wilsonian Renormalisation Group Equation; useful choices for observables and cutoffs; the momentum window in which the test likely provides best signals; its dependence on the values and forms of cutoffs as well as on the EFT parameters; the impact of fitting LECs to data in different or the same channel; and caveats as well as limitations. Since the test is designed to minimise the use of data, it allows one to quantitatively falsify if the EFT has been renormalised consistently. This complements other tests which quantify how an EFT compares to experiment. Its application in particular to the \({}^{3}\mathrm {P}_{0}\) and \({}^{3}\mathrm {P}_{2}\)–\({}^{3}\mathrm {F}_{2}\) partial waves of \({\mathrm {NN}}\) scattering in \(\chi \hbox {EFT}\) may elucidate persistent power-counting issues.

中文翻译：

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根据剩余截止阈值对EFT功率计数进行一致性测试

我总结了一种方法，该方法可以定量地评估有效领域理论中权力计算提案的一致性，而该领域在领先方面是无微不足道的。它利用了一个事实，即可观测对象的重整化组演化预测了其剩余截止的函数形式，该函数形式取决于有效场论（EFT）的分解尺度，低动量尺度和计算顺序。通过此测试是建议的功率计数（其确切性质有争议）的必要但不充分的一致性标准。例如，在手性有效场论（\（\ chi \ hbox {EFT} \））具有一个以上的核子，缺乏普遍接受的解析解决方案混淆了收敛模式与数值结果之间的关系，并导致提出了以相同手性顺序预测不同数量的低能系数（LEC）的建议，有时甚至可以预测订购不同的远程产品。该方法可以提供独立的检查，以观察物是否以给定的顺序正确地重新归一化，并允许人们估计击穿尺度和EFT的动量依赖的逐阶收敛模式。相反，它可以帮助识别那些以给定顺序生成经过重新规范化的可观察物的LEC（和远距离碎片）。我还将讨论其基本假设以及与威尔逊重整化群方程的关系。观测值和临界值的有用选择；测试可能在其中提供最佳信号的动量窗口；它对临界值和形式的依赖以及EFT参数的依赖；将LEC拟合到不同或相同通道中的数据的影响；和注意事项以及限制。由于该测试旨在最大程度地减少数据的使用，因此可以使人们定量地伪造EFT是否已被一致地重新归一化。这是对其他测试的补充，这些测试量化了EFT与实验的比较方式。它的应用尤其适用于 它可以定量地伪造EFT是否始终如一地重新归一化。这是对其他测试的补充，这些测试量化了EFT与实验的比较方式。它的应用尤其适用于 它可以定量地伪造EFT是否始终如一地重新归一化。这是对其他测试的补充，这些测试量化了EFT与实验的比较方式。它的应用尤其适用于\（{} ^ {3} \ mathrm {P} _ {0} \）和\（{} ^ {3} \ mathrm {P} _ {2} \） – \（{} ^ {3} \ mathrm在（\ chi \ hbox {EFT} \）中散射的{{} \ mathrm {NN}} \）{F} _ {2} \）分波可以阐明持续的功率计数问题。