Landau Singularities from Whitney Stratifications

Technical Summary: Landau Singularities from Whitney Stratifications

Problem Statement
The analytic structure of Feynman integrals, which are central to cross-section calculations at the Large Hadron Collider and gravitational wave physics, is characterized by their singularities. These singularities, known as Landau singularities, occur at specific values of kinematic variables where the integral becomes non-analytic. While the rigorous definition of the Landau variety was established by Pham as the critical values of a projection map from the integrand's variety to the kinematic space, directly computing this set has proven challenging.

Existing computational methods often rely on the principal $A$-determinant or its refinements (such as the principal Landau determinant). However, these approaches suffer from two main limitations:

1. Spurious Candidates: They may produce factors that vanish identically when specialized to physical Feynman integral coefficients, requiring ad-hoc removal of factors.
2. Incompleteness: They are known to miss certain Landau singularities, particularly in cases involving specific kinematic limits or non-generic configurations. Furthermore, empirical refinements based on sequential limits lack a general proof of correctness.

Consequently, there is no general, rigorous recipe for computing the complete and non-redundant set of Landau singularities for arbitrary integrals.

Methodology
The authors propose a method to compute the Landau variety directly and completely by leveraging recent algorithmic advances in Whitney stratifications.

1. Mathematical Framework:

   • The Feynman integral is represented using the homogenized Lee-Pomeransky polynomial $G_h = U x_0 + F$, where $U$ and $F$ are Symanzik polynomials.
   • The authors define the variety $X = V(x_0 \cdots x_n G_h)$ in the product space of projective integration variables and kinematic parameters ($\mathbb{P}^n \times \mathbb{C}^m$).
   • They consider the projection map $\pi: X \to \mathbb{C}^m$ onto the kinematic space.
   • Following Pham's definition, the Landau variety is identified as the variety $Y_{m-1}$ appearing in the minimal Whitney stratification of the map $\pi$.

2. Whitney Stratification:

   • A Whitney stratification decomposes a potentially singular variety into smooth manifolds (strata) such that the local geometry is consistent within each stratum and satisfies Whitney's condition B (ensuring well-behaved limits of tangent planes and secant lines).
   • For a map, a stratification ensures that the map restricted to any stratum of the domain is a submersion onto a stratum of the codomain.
   • The authors utilize the Thom's First Isotopy Lemma, which guarantees that the topology of the fiber $\pi^{-1}(q)$ remains constant over any stratum $N$ of the base space. Thus, the boundaries between these strata (the lower-dimensional components of the stratification) correspond exactly to the points where the topology changes—i.e., the singularities.

3. Implementation:

   • The computation is performed using the WhitneyStratifications package (version 2.27) for the Macaulay2 software.
   • The algorithm explicitly computes the minimal Whitney stratification of the projection map defined by the Feynman integral's defining equations.

Key Contributions and Results
The paper provides a "proof of concept" demonstrating that this rigorous definition yields the complete set of Landau singularities for non-trivial examples, including cases where other methods fail.

1. One-Loop Bubble:

   • The method successfully recovers the Landau variety for generic masses and momenta.
   • Crucially, it correctly identifies singularities arising from kinematic limits (e.g., setting masses to zero) directly from the lower-dimensional strata of the stratification, without needing to manually substitute limits into the generic result.

2. Two-Loop Slashed Box (Two-Mass Hard):

   • The computed Landau variety agrees with results from the HyperInt package.
   • It successfully identifies a component ($p_4^2 - s - t = 0$) that is missed by the PLD.jl (Principal Landau Determinant) package. The authors verify this component is a true singularity via explicit analytic expressions.

3. Two-Loop Parachute Integral:

   • This case is noted as problematic for both the naive principal $A$-determinant and the principal Landau determinant, which miss a specific singularity derived in prior literature.
   • While the full stratification for the general 7-parameter case was computationally too demanding, the authors validated the method by restricting kinematics to a specific limit. In this limit, the method correctly identified the singularity ($p_3^2 - 1 = 0$) that other approaches missed.

Significance and Claims
The authors claim that their approach offers a rigorous, algorithmic, and complete method for determining Landau singularities.

• Completeness: Unlike previous methods that may provide supersets (with spurious factors) or subsets (missing singularities), the Whitney stratification of the projection map yields exactly the Landau variety and nothing more.
• Unified Treatment of Limits: The stratification naturally encodes the Landau varieties of integrals obtained via kinematic limits (e.g., massless limits) within its lower-dimensional strata. This provides a unified geometric description of the singularities for the original integral and its limits.
• Physical Relevance: The lower-dimensional strata also relate to the Euler characteristic of the parametric variety, which counts the number of master integrals in the integral family, suggesting a deeper connection between the stratification and the basis of the integral family.

Limitations and Outlook
The paper modestly acknowledges that the current computational route via WhitneyStratifications is not yet as efficient as specialized software for finding subsets of singularities. The method is currently limited by computational complexity for high-dimensional parameter spaces (as seen in the full parachute example). Future work is suggested to improve efficiency by exploiting the specific algebraic structures of Feynman integrals and to explore the direct access to singularities in the physical (real) region via real semi-algebraic set stratifications.