Landau Analysis in Momentum Space with Massless Particles: an Amuse Bouche

We illustrate how methods from Landau analysis that have been developed for studying the properties of massive Feynman integrals in momentum space can be generalized to massless integrals. We consider integrals with both massive and massless propagators in arbitrary dimensions, paying attention to square root branch points. By focusing on a number of well-chosen examples, we show how resolution of singularities (via blow-ups or complex structure deformation) can be used to predict how the behavior of these integrals is modified as different numbers of propagators are chosen to be massless.

💡 Research Summary

The paper extends the well‑established Landau analysis, traditionally applied to Feynman integrals with massive propagators, to the case where some propagators are massless. The authors identify two complementary strategies for handling the new singularities that arise: (i) blow‑up procedures that resolve the conical singularities of light‑cones in loop‑momentum space, and (ii) complex‑structure deformations that either give the massless propagators a small regulator mass or expand around the distance to the Landau locus.

The work begins by reviewing the standard Landau equations, emphasizing that for massive propagators the singular surfaces are smooth and the Hessian matrix at a pinch point is positive‑definite. When a propagator becomes massless, its on‑shell surface turns into a light‑cone whose tip (zero momentum) is a singular point. This prevents a straightforward complex deformation of the integration contour and leads to permanent pinches that generate infrared (IR) divergences.

To cure the singular tip, the authors perform a blow‑up: they re‑parameterize a massless momentum (q) as (q = \rho (1,\hat y)), where (\rho\in\mathbb R) and (\hat y) lies on a unit sphere (S^{D-2}). The light‑cone is thereby replaced by a cylinder (\mathbb R\times S^{D-2}); the singular tip is replaced by the sphere at (\rho=0). In this new coordinate system the on‑shell conditions become non‑singular, and the Landau equations acquire three distinct regimes—soft, collinear, and soft‑collinear—depending on how the Feynman parameters (\alpha_i) scale with (\rho).

The authors apply this machinery to a series of representative integrals:

• Massless bubble (one massive, one massless propagator). After blow‑up the integral separates into a factor ((p^2-m^2)^{D-3}) times a regular integral over (\rho) and (\hat y). The analysis reproduces known asymptotics and clarifies the interplay between the IR divergence at (\rho\to0) and the Landau singularity at (p^2=m^2).

• Sunrise with one massless line. Solving the Landau equations yields a non‑unique pinch where all loop momenta collapse to the cone tip. The blow‑up again introduces (\rho,\hat y) variables, and the resulting differential form lives in a relative cohomology group (H^D(X\setminus S_1\cup S_2, S_0)). The authors discuss how the dimension of the cohomology changes with spacetime dimension (D).

• Massless box and triangle. Multiple blow‑ups are required to resolve intersecting light‑cones. Each blow‑up introduces a new divisor (S_i); the final integration form is a relative cohomology class on the complement of all divisors.

• Monodromy. The paper examines how square‑root branch points survive the blow‑up, leading to a modified monodromy matrix that still reflects the underlying two‑sheeted Riemann surface but with branch points displaced along the (\rho) direction.

• Second‑type singularities. By applying an inversion map that sends the point at infinity to the origin, the authors reduce second‑type Landau singularities (which appear in dimensional regularisation) to ordinary first‑type singularities at the origin. This provides a unified treatment of all singularities within the blow‑up framework.

The second major tool is complex‑structure deformation. Two variants are explored: (a) giving each massless propagator a small mass (\epsilon) and studying the (\epsilon\to0) limit, and (b) expanding in a small parameter (\lambda) measuring the distance to the Landau locus. When the Hessian matrix is singular, the determinant scales as (\det H\sim \lambda^{k}), and the authors show how this scaling reproduces the Pham–Steinmann relations and satisfies the hierarchical principle that constrains the order of discontinuities.

In an extensive appendix the authors present a geometric solution of the Landau equations based on volumes of simplices with fixed edge lengths. By interpreting each propagator mass as a side length, the Landau conditions become statements about the existence of Euclidean, Lorentzian, or mixed‑signature simplices. This perspective, inspired by earlier two‑dimensional work, offers an alternative to the algebraic elimination of (\alpha) parameters.

Overall, the paper delivers a systematic framework for analyzing Landau singularities in the presence of massless particles. By combining blow‑up resolution with controlled complex deformations, it disentangles infrared divergences from genuine Landau pinch singularities, validates known analytic constraints (Pham–Steinmann, Steinmann‑Pham compatibility, hierarchical principle), and paves the way for a Landau bootstrap that includes massless external states. Future directions suggested include extending the method to multi‑loop, multi‑scale integrals and to elliptic or Calabi–Yau type Feynman integrals where the underlying geometry is richer.