Landau Singularities from Whitney Stratifications

Abstract

We demonstrate that the complete and non-redundant set of Landau singularities of Feynman integrals may be explicitly obtained from the Whitney stratification of an algebraic map. As a proof of concept, we leverage recent theoretical and algorithmic advances in their computation, as well as their software implementation, in order to determine this set for several nontrivial examples of two-loop integrals. Interestingly, different strata of the Whitney stratification describe not only the singularities of a given integral, but also those of integrals obtained from kinematic limits, e.g.~by setting some of its masses or momenta to zero.

Figures (6)

• Figure 1: The Whitney stratification of the Whitney cusp, $X=\mathbf{V}(x^2 + z^3-y^2z^2)\subset \mathbb{R}^3$, is given by $X\supset \mathbf{V}(x,z)\supset \{(0,0,0)\}$.
• Figure 2: Two sequences of points, one on the $y$-axis and one along the top of the surface, both approaching the origin. The limiting secant line is the $z$-axis, while the limiting tangent plane is the $(x,y)$-plane, meaning condition B fails at $(0,0,0)$ for the $y$-axis relative to the rest of the surface. Hence, the origin must be placed in a separate strata from the other points on the $y$-axis.
• Figure 3: Plots of the curve defined by \ref{['eq:planarCubic']} for different parameter values $z$; the topology of the curve changes at $z=0$. While the curve in (\ref{['fig: curve smooth']}) is smooth and has two connected components (of different dimensions) in $\mathbb{R}^2$ it is connected and singular, with singularity at $(0,1)$, in $\mathbb{C}^2$.
• Figure 4: The singularity structure of the one-loop bubble is classical and well-known. We show that the Landau variety not only reproduces these results but that the full Whitney stratification provides the singularities of kinematic limits.
• Figure 5: The slashed box is one of the simplest graphs where different approaches to calculating Landau singularities do not coincide. This is discussed in Section \ref{['sec: slashed box']}.

Theorems & Definitions (2)

• Example 1: Topology of Parameterized Cubic
• Definition 1: Landau variety