We present a calculation of the decays ${\eta }^{{(}^{\prime })}\to {\pi }^{+}{\pi }^{-}{\gamma }^{(*)}$ at the one-loop level up to and including next-to-next-to-leading order (NNLO) in large-$N_c$ chiral perturbation theory. The numerical evaluation of the results is performed successively at LO, NLO, and NNLO, fitting the relevant low-energy constants to the available experimental data. We discuss the widths and decay spectra of ${\eta }^{{(}^{\prime })}\to {\pi }^{+}{\pi }^{-}\gamma$ as well as ${\eta }^{{(}^{\prime })}\to {\pi }^{+}{\pi }^{-}{l}^{+}{l}^{-}$, with $l=e$, $\mu$.

中文翻译：

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η(′)→π+π−γ(*)在大Ncchiral微扰理论中

我们提出了衰减的计算 ${\eta }^{{(}^{\prime })}\to {\pi }^{+}{\pi }^{-}{\gamma }^{(*)}$ 在单循环级别，直到并包括大中的 next-to-next-to-lead order (NNLO)$N_C$手性微扰理论。结果的数值评估依次在 LO、NLO 和 NNLO 下进行，将相关的低能常数与可用的实验数据拟合。我们讨论的宽度和衰减谱${\eta }^{{(}^{\prime })}\to {\pi }^{+}{\pi }^{-}\gamma$ 也 ${\eta }^{{(}^{\prime })}\to {\pi }^{+}{\pi }^{-}{升}^{+}{升}^{-}$， 和 $升=\mathrm{电子}$, $\mu$.