In this paper, we show that the Cutkosky cutting rules are still valid term by term in the expansion in powers of 𝜅 of the 𝜅-deformed 1-loop correction to the propagator of the 𝜅-deformed complex scalar field. We first present a general argument which relates each term in the expansion to a nondeformed amplitude containing additional propagators with mass 𝑀>𝜅. We then show the same thing more pragmatically, by reducing the singularity structure of the coefficients in the expansion of the 𝜅-deformed amplitude, to the singularity structure of nondeformed loop amplitudes, by using algebraic and analytic identities. We will explicitly show this up to second order in 1/𝜅, but the technique can be generalized to higher orders in 1/𝜅. Both the abstract and the more direct approach easily generalize to different deformed theories. We will then compute the full imaginary part of the 𝜅-deformed 1-loop correction to the propagator in a specific model, up to second order in the expansion in 1/𝜅, highlighting the usefulness of the approach for the phenomenology of deformed models. This explicitly confirms previous qualitative arguments concerning the behavior of the decay width of unstable particles in the considered model.

中文翻译：

---

Cutkosky 规则和 1-loopκ 变形振幅


在本文中，我们表明，在 κ 变形的 1 环校正的 κ 幂扩展到 κ 变形复标量场的传播器时，Cutkosky 切割规则仍然有效。我们首先提出了一个一般论点，该论点将展开中的每个项与包含质量为 M>κ 的额外传播器的未变形振幅相关联。然后，我们通过使用代数和解析恒等式，将 κ 变形振幅扩展中系数的奇点结构简化为未变形环振幅的奇点结构，从而更实际地展示同样的事情。我们将在 1/κ 中明确显示到二阶，但该技术可以推广到 1/κ 中的更高阶。抽象和更直接的方法都很容易推广到不同的变形理论。然后，我们将计算特定模型中传播子的 κ 变形 1 环校正的完整虚部，直到 1/κ 中的展开二阶，突出该方法对变形模型现象学的有用性。这明确证实了之前关于所研究模型中不稳定粒子衰变宽度行为的定性论点。