We establish formulae for the asymptotic growth (with respect to the scaling dimension) of the number of operators in effective field theory, or equivalently the number of $S$-matrix elements, in arbitrary spacetime dimensions and with generic field content. This we achieve by generalising a theorem due to Meinardus and applying it to Hilbert series -- partition functions for the degeneracy of (subsets of) operators. Although our formulae are asymptotic, numerical experiments reveal remarkable agreement with exact results at very low orders in the EFT expansion, including for complicated phenomenological theories such as the standard model EFT. Our methods also reveal phase transition-like behaviour in Hilbert series. We discuss prospects for tightening the bounds and providing rigorous errors to the growth of operator degeneracy, and of extending the analytic study and utility of Hilbert series to EFT.

中文翻译：

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电子转帐渐近：操作员简并性的增长

我们建立了有效场论中算子数量（或等价的$ S $-矩阵元素）的渐近增长（相对于缩放维数）的公式，表示任意时空维数并具有通用字段内容。我们通过归纳归因于Meinardus的一个定理并将其应用于希尔伯特级数（Hilbert series）-分区函数来简化算子（子集）来实现。尽管我们的公式是渐近的，但数值实验表明，在极低阶的EFT展开中，包括标准模型EFT等复杂的现象学理论，它们与精确结果具有显着的一致性。我们的方法还揭示了希尔伯特级数中类似相变的行为。我们讨论了收紧界限并为操作人员退化的增长提供严格错误的前景，