Singularities of Feynman integrals

Abstract

In this paper, we study the singularities of Feynman integrals by compactifying the integration domain as well as the ambient space of these integrals, by embedding them in higher-dimensional space. In this compactified space, the singularities occur due to the meeting of compactified propagators at non-general position. The present analysis, which had been previously used only for the singularities of second type, is used to study other kinds of singularities viz threshold, pseudo-threshold and anomalous threshold singularities. We study various one-loop and two-loop examples and obtain their singularities. We also present observations based on results obtained, that allow us to determine whether the singularities lie on the physical sheet or not for some simple cases. Thus, this work at the frontier of our knowledge of Feynman integral calculus sheds insight into the analytic structure.

Appendix: A toy example

In this appendix, we consider a toy example to demonstrate the method. We consider the following one-dimensional version of the bubble integral

$$\begin{aligned} I_{2} = \int _{{\mathbb {R}}} \frac{dk}{(k^{2}-m_{1}^{2})((k-p)^{2}-m_{2}^{2})}. \end{aligned}$$

(A.1)

Here, the integration cycle is ${\mathbb {R}}$ and the ambient space is ${\mathbb {C}}$, so we use the compactification procedure outlined in Sect. 2.2. Compactifying the propagators, we get the following

$$\begin{aligned} S_1&= \frac{x_{2}(-m_{1}^{2}-1)+ x_{3}(-m_{2}^{2}+1)}{x_{2}+x_3},\nonumber \\ S_2&= \frac{-2 p x_{1} +x_{2}(p^{2}-m_{2}^{2}-1)+x_{3}(p^{2}-m_{1}^{2}+1)}{x_{2}+x_3}, \end{aligned}$$

(A.2)

and the new ambient space W is given by

$$\begin{aligned} W= \{(x_{1},x_{2},x_{3})|x_{1}^{2}+x_{2}^{2}-x_{3}^{2}=0 \}\subset \mathbb{C}\mathbb{P}^2. \end{aligned}$$

Similarly, we have $dk \rightarrow \frac{dx_{1}}{x_{2}+x_{3}}$. This gave rise to an effective denominator

$$\begin{aligned} S_{3}= x_{2}+x_{3}. \end{aligned}$$

(A.3)

The singularities of integral (A.1) correspond to the following two cases

• 1. When $S_{1}$ and $S_{2}$ meet in non-general position in W. This case is similar to the case of the Bubble Integral in Sect. 3. We get the two singularities: $p^{2}= (m_{1}+m_{2})^{2}$ and $p^{2}=(m_{1}-m_{2})^2$.

• 2. When $S_{1},S_{2}$ and $S_{3}$ meet in non-general position in W. This case is similar to the Bubble Integral case and we get $p^{2}=0$.

The situation for the $S_{1}$ and $S_{2}$ meeting in non-general position (for real $x_{1}, x_{2}, x_{3}$) is as shown in Fig. 7. We can also look at the situation in the real $(x_{1},x_{2})$-plane (with $x_{3}=1$). The situation of general and non-general positions is shown in Fig. 8. We note a special feature of the intersection of these surfaces is the ‘vanishing cycle’. In the plots shown in Fig. 8, notice that the green circle is divided into four parts in Fig. 8a and into three parts in Fig. 8b. The region that vanishes due to the meeting of the surfaces at non-general position is called the ‘vanishing cycle’. So whenever the surfaces meet at non-general position it corresponds to the vanishing of a cycle.

Fig. 7

$S_{1}$ and $S_{2}$ (in blue and yellow color, respectively) meeting in non-general position in W(in green)

Fig. 8

a $S_{1}$ and $S_{2}$ meeting in general position in W. b $S_{1}$ and $S_{2}$ meeting in non-general position in W, corresponding to singularity $p^{2}=(m_{1}+m_{2})^{2}$. The plots shows the situation shown in Fig. 7 in $(x_{1},x_{2})$-plane with $x_{3}=1$