We study the scaling of kaon decay amplitudes with the number of colours, \(N_c\), in a theory with four degenerate flavours, \(N_f=4\). In this scenario, two current-current operators, \(Q^\pm \), mediate \(\Delta S=1\) transitions, such as the two isospin amplitudes of non-leptonic kaon decays for \(K\rightarrow (\pi \pi )_{I=0,2}\), \(A_0\) and \(A_2\). In particular, we concentrate on the simpler \(K\rightarrow \pi \) amplitudes, \(A^\pm \), mediated by these two operators. A diagrammatic analysis of the large-\(N_c\) scaling of these observables is presented, which demonstrates the anticorrelation of the leading \({{\mathcal {O}}}(1/N_c)\) and \({{\mathcal {O}}}(N_f/N_c^2)\) corrections in both amplitudes. Using our new \(N_f=4\) and previous quenched data, we confirm this expectation and show that these corrections are naturally large and may be at the origin of the \(\Delta I=1/2\) rule. The evidence for the latter is indirect, based on the matching of the amplitudes to their prediction in Chiral Perturbation Theory, from which the LO low-energy couplings of the chiral weak Hamiltonian, \(g^\pm \), can be determined. A NLO estimate of the \(K \rightarrow (\pi \pi )_{I=0,2}\) isospin amplitudes can then be derived, which is in good agreement with the experimental value.

我们在具有四种简并味\\（N_f = 4 \）的理论中研究了随着颜色数\（N_c \）缩放的kaon衰减幅度。在这种情况下，两个电流-电流算子\（Q ^ \ pm \）介导\（\ Delta S = 1 \）跃迁，例如\（K \ rightarrow（ \ pi \ pi）_ {I = 0,2} \），\（A_0 \）和\（A_2 \）。特别地，我们专注于由这两个算子介导的更简单的\（K \ rightarrow \ pi \）振幅\（A ^ \ pm \）。大（\（N_c \））的图解分析提出了这些可观测对象的缩放比例，它证明了前导\（{{\ mathcal {O}}}（1 / N_c）\）和\（{{\ mathcal {O}}}}（N_f / N_c ^ 2 ）\）校正两个振幅。使用我们的新\（N_f = 4 \）和先前的淬火数据，我们证实了这一预期，并表明这些校正量自然很大，并且可能是\（\ Delta I = 1/2 \）规则的起源。后者的证据是间接的，基于振幅与手性扰动理论中预测的匹配，由此可以确定手性弱哈密顿量的LO低能耦合\（g ^ \ pm \）。\（K \ rightarrow（\ pi \ pi）_ {I = 0,2} \）的NLO估计 然后可以推导出同位旋振幅，这与实验值非常吻合。