AI 中文总结
本文针对量子场论中的三角费曼图，验证了对其费曼参数表示作简单替换$p_i^2\to p_i^2+i0$、$m_i^2\to m_i^2-i0$，即可复现类时区域色散关系的严格结果。
AI 中文摘要
在量子场论中，三角费曼图$F(p_1^2,p_2^2,p_3^2|m_1,m_2,m_3)$是其变量的解析函数，其解析结构完全由圈中粒子传播子的奇点位置决定。当所有变量处于欧氏区域（满足$p_i^2<0$，$i=1,2,3$）时，形状因子$F(p_1^2,p_2^2,p_3^2|m_1,m_2,m_3)$易于计算。获取类时区域形状因子的严格方法是利用单重或双重色散关系从欧氏区域进行解析延拓。另一方面，三角图存在一种简单的表示，即对费曼参数积分。本文的目标是证明，对于$F(p_1^2,p_2^2,p_3^2|m_1,m_2,m_3)$，只需进行替换$p_i^2\to p_i^2+i0$和$m_i^2\to m_i^2-i0$（其中$m_i$为圈中传播粒子的质量），就能复现在部分变量$p_i^2$处于物理闵氏区域时色散关系的所有已知严格结果，这种简单替换可恰当考虑色散关系框架下反常割和阈值所给出的所有微妙贡献。
英文摘要
In Quantum Field Theory, triangle Feynman diagram $F(p_1^2,p_2^2,p_3^2|m_1,m_2,m_3)$ is an analytic function of its variables, whose analytic structure is fully determined by the location of singularities of the propagators of particles in the loop. The form factor $F(p_1^2,p_2^2,p_3^2|m_1,m_2,m_3)$ is easily calculable in the Euclidean region of all variables, $p_i^2<0$, $i=1,2,3$. A rigorous way to obtain the form factor in the timelike region is to perform the analytic continuation from the Euclidean region using single or double dispersion representations. On the other hand, there is a simple representation of the triangle as integral over Feynman parameters. The goal of this paper is to demonstrate that all known rigorous results of dispersion representations in the regions where some of the variables $p_i^2$ are in the physical Minkowski region, are reproduced by the Feynman-parameter representation for $F(p_1^2,p_2^2,p_3^2|m_1,m_2,m_3)$ by a mere replacement $p_i^2\to p_i^2+i0$ and $m_i^2\to m_i^2-i0$, where $m_i$ are masses of particles propagating in the loop. This simple replacement takes properly into account all subtle contributions given in the context of dispersion representations by the anomalous cuts and thresholds.
Comments9 pages
