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I was reading this Quantum Field Theory book written by Claude Itzykson and Jean-Bernard Zuber. In section 6-3, page 304, the authors introduced analytic properties of Feynman integrals.

Consider the following integral in momentum space

$$I(P)=\underset{\epsilon\rightarrow 0}{\lim}\int\prod_{l=1}^{L}\frac{d^{4}Q_{l}}{(2\pi)^{4}}\prod_{i=1}^{I}\frac{i}{(K_{i})^{2}-(m_{i})^{2}+i\epsilon}$$

where each internal momentum $K_{i}$ can be expressed in terms of loop momenta $Q_{l}$ and external momentum $P$.

It is said that the necessary condition of $I(P)$ being singular is given by Landau:

$$\left\{ \begin{align} &\lambda_{i}\left[(K_{i})^{2}-(m_{i})^{2}\right]=0 \quad \quad i=1,\cdots,I\\ \\ &\underset{i}{\sum}\lambda_{i}K_{i}\frac{\partial K_{i}}{\partial Q_{l}}=0\quad\quad\quad\quad l=1,\cdots,L \end{align} \right. \tag{1} $$

More generally, consider an integral along an $h$-dimensional hypersurface in the complex plane $\mathbb{C}^{h+1}$

$$F(z)=\underset{H\subset\mathbb{C}^{h+1}}{\int}dx^{1}\wedge\cdots\wedge dx^{h}f(x^{1},\cdots,x^{h},z).$$

Here, the boundaries of the integration domain, i.e. the hypercontour $H$, is specified by a set of equations

$$B_{r}(\vec{x},z)=0,\quad r=1,\cdots,R.$$

The possible singularities of the integrand $f(\vec{x},z)$ take place on a submanifold specified by a set of equations

$$A_{s}(\vec{x},z)=0,\quad s=1,\cdots,S.$$

Singularities occur when the hypercontour $H$ is pinched between two or more surfaces of singularities $A_{s}$ or when a surface of singularity meets a boundary $B_{r}$. surface.

Then, the authors claim that the necessary condition of $F(z)$ being singular (also known as Landau variety) is that there exists a set of complex parameters $\lambda_{s}$ and $\mu_{r}$ such that at some point $(\vec{x}_{0},z_{0})\in\mathbb{C}^{h+1}$:

\begin{align} &\lambda_{s}A_{s}(\vec{x}_{0},z_{0})=0 \quad s=1,\cdots,S \\ &\mu_{r}B_{r}(\vec{x}_{0},z_{0})=0 \quad r=1,\cdots,R \\ \tag{2} &\frac{\partial}{\partial x^{i}}\left[\underset{r}{\sum}\mu_{r}B_{r}(\vec{x},z)+\underset{s}{\sum}\lambda_{s}A_{s}(\vec{x},z)\right]\Bigg|_{(\vec{x}_{0},z_{0})}=0 \quad i=1,\cdots,h. \end{align}

Can anybody enlighten me how to derive the Landau conditions (1) and (2)?

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    $\begingroup$ I recommend reading the "The Analytic S matrix" by Eden, Landshoff, Olive and Polkinghorne. They explain in detail the origin of the endpoint and pinch singularities that give rise to the branchpoint singularities. $\endgroup$
    – mike stone
    Commented Jan 22, 2023 at 14:12
  • $\begingroup$ @mikestone Thanks for your comment. I downloaded that book. It's so obscure and I had to give it up. Hopefully, someone can explain it like I am a five year old. $\endgroup$
    – Valac
    Commented Jan 22, 2023 at 15:02
  • $\begingroup$ I suggest that you only look up the bit about Landau-Cutkosky and the pinch and endpoint stuff about how contour intgrals can become singular. The rest is unneccessary. $\endgroup$
    – mike stone
    Commented Jan 22, 2023 at 15:43
  • $\begingroup$ Your link is dead. $\endgroup$ Commented Jan 22, 2023 at 17:58

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