This work explores scattering amplitudes that couple two-particle systems via a single external current insertion, $2+\mathcal{J}\to 2$. Such amplitudes can provide structural information about the excited QCD spectrum. We derive an exact analytic representation for these reactions. From these amplitudes, we show how to rigorously define resonance and bound-state form factors. Furthermore, we explore the consequences of the narrow-width limit of the amplitudes as well as the role of the Ward-Takahashi identity for conserved vector currents. These results hold for any number of two-body channels with no intrinsic spin, and a current with arbitrary Lorentz structure and quantum numbers. This work and the existing finite-volume formalism provide a complete framework for determining this class of amplitudes from lattice QCD.

中文翻译：

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二体跃迁幅度的壳上表示：单个外部电流

这项工作探索了通过单个外部电流插入耦合两个粒子系统的散射幅度， $2+\mathcal{J}\to 2$. 这种振幅可以提供有关激发 QCD 谱的结构信息。我们推导出这些反应的精确解析表示。根据这些幅度，我们展示了如何严格定义共振和束缚态形状因子。此外，我们探讨了幅度窄幅限制的后果以及 Ward-Takahashi 恒等式对守恒矢量电流的作用。这些结果适用于任意数量的没有内在自旋的双体通道，以及具有任意洛伦兹结构和量子数的电流。这项工作和现有的有限体积形式为从晶格 QCD 确定此类振幅提供了一个完整的框架。