Landau Equations and Physical Singularities

The Landau equations encode the algebraic conditions for a Feynman integration contour to be pinched as masses or external invariants vary. Within the standard real, physical-region hypotheses stated below, they characterize such pinches; outside that theorem class they identify candidate singular loci. In neither case do they by themselves establish that a singularity lies on the physical sheet, survives numerator and diagram cancellations, or has a nonzero discontinuity.

Required background. Anatomy of a Loop Integral supplies propagator denominators and contour pinches, while Analyticity and Crossing for Amplitudes supplies sheets, branch points, and continuation in external invariants.

Helpful background. Feynman and Schwinger Parameters supplies the parameter-domain form in which the stationary conditions are especially transparent.

Pinch conditions for a graph

Let

$$\begin{aligned} D_i(\ell,p)&=q_i^2-m_i^2+i0,\\ q_i&=\sum_r c_{ir}\ell_r+P_i(p). \end{aligned}$$

Introduce one homogeneous parameter $\alpha_i$ for each line; their common scale is irrelevant. They can be chosen real and nonnegative on the ordinary physical Feynman-parameter domain, while the algebraically continued Landau variety can involve other signs or complex values. A leading or reduced-diagram Landau configuration satisfies

$$\alpha_i(q_i^2-m_i^2)=0 \qquad\text{for every }i,$$

and, for each independent loop $r$,

$$\sum_i\alpha_i c_{ir}q_i^\mu=0.$$

Thus every line either goes on shell or has $\alpha_i=0$ and is contracted in the reduced diagram. The second equation is the stationarity condition that prevents a common deformation away from the on-shell surfaces. In parameter language it follows from requiring the kinematic polynomial to vanish with stationary first derivatives on the relevant projective domain.

Landau’s original derivation and graph interpretation appear in Landau 1959, pp. 181–192. A modern formulation in momentum and parameter variables is given in Weinzierl 2022, §6.6, pp. 209–212.

For a first-sheet physical pinch in the ordinary Feynman-parameter domain one additionally seeks $\alpha_i\ge0$, not all zero, with an energy-flow assignment compatible with the physical channel. Negative or complex parameters can describe other sheets or analytically continued loci.

Bubble thresholds derived

Choose bubble momenta

$$q_1=\ell, \qquad q_2=p-\ell.$$

For a leading solution $\alpha_1\alpha_2\ne0$, both lines are on shell and the loop equation is

$$\alpha_1\ell^\mu-\alpha_2(p-\ell)^\mu=0.$$

Hence

$$\begin{aligned} \ell^\mu&=\frac{\alpha_2}{\alpha_1+\alpha_2}p^\mu,\\ p^\mu-\ell^\mu&=\frac{\alpha_1}{\alpha_1+\alpha_2}p^\mu. \end{aligned}$$

Eliminating the parameter ratio while allowing either relative on-shell energy orientation gives the algebraic candidates

$$p^2=(m_1\pm m_2)^2.$$

When $\alpha_1,\alpha_2>0$ and $p$ is future-directed, both coefficients in the decomposition of $p$ lie between zero and one. Both internal energies are then future-directed and only the plus sign remains: this is the normal two-particle threshold. The minus sign requires the opposite relative energy orientation, or equivalently leaves the positive-parameter physical domain; it is the pseudothreshold and does not begin the standard bubble’s physical-sheet two-particle cut. This example shows why solving the polynomial equations is only the first step.

Normal, anomalous, and reduced-diagram loci

A normal threshold corresponds to a familiar set of intermediate particles that can propagate on shell in a physical channel. More complicated graphs can have anomalous thresholds in invariants not reducible to the lowest sum of masses in that channel. Setting selected $\alpha_i=0$ yields subleading Landau singularities associated with contracted subgraphs.

For physical-region solutions, the Coleman–Norton interpretation assigns a spacetime displacement proportional to $\alpha_iq_i^\mu$ to each on-shell line. The loop equations then say that the displacements close consistently into a classical on-shell process with forward propagation Coleman and Norton 1965, pp. 438–442. This interpretation is a physical-sheet test, not a replacement for evaluating the local singular behavior.

What is necessary, sufficient, and still unproved

The common statement that Landau equations are “only necessary” is too coarse. For finite real physical-region points in a specified class of Feynman integrals, Collins proves that the nonnegative Landau condition is necessary and sufficient for the integration contour to be pinched. The hypotheses include real external and integration variables, real-for-real analytic denominators with a common $+i0$, a nonsingular analytic numerator, denominators at most quadratic in the integration variables, and a local same-sign condition on the nonzero quadratic terms along directions whose first-order shifts vanish. Ordinary quadratic propagators and the linear eikonal denominators treated there satisfy these conditions Collins 2020, Definition 6 and Theorems 1–3, pp. 9–10; Eq. (8.15), pp. 27–28, PDF.

The theorem is about a trapped contour, not automatically about a singular value of the integral or amplitude. Even when the contour is pinched:

A Landau solution outside those hypotheses, or a pinch inside them, can fail to produce a singularity of the amplitude because:

• the algebraic solution does not meet the real-domain, sign, sheet, or theorem hypotheses;
• the numerator vanishes on the candidate locus;
• symmetry or gauge-invariant sums cancel the singular coefficient;
• the contour has zero intersection with the relevant vanishing cycle;
• the candidate lies at infinity and needs a separate UV or IR analysis.

Conversely, endpoint singularities and second-type singularities can require a compactified or extended analysis beyond the simplest finite Landau equations. The equations should therefore be reported as candidate-locus evidence together with parameter signs, energy flow, sheet, and a local scaling or discontinuity check.

Exercises

1. In the bubble, why are two signs obtained after squaring? The on-shell equations forget the relative time orientation; positive parameters and future-directed energy flow select the normal threshold.
2. What does $\alpha_i=0$ mean? That line need not be on shell and is contracted in the reduced Landau diagram.

Where a candidate pinch leads

• Cutkosky Cutting Rules adds the physical channel, positive-energy support, and phase-space normalization needed to compute a discontinuity.
• Differential Equations for Master Integrals uses candidate singular loci to guide boundary points and continuation paths, while retaining possible basis-induced singularities as a separate issue.

References

• Coleman, Sidney, and Richard E. Norton. “Singularities in the Physical Region.” Il Nuovo Cimento 38 (1965): 438–442. doi:10.1007/BF02750472.
• Collins, John C. “Feynman Graphs and Other Integrals.” arXiv:2007.04085 [hep-ph] (2020). Open PDF.
• Landau, Lev D. “On Analytic Properties of Vertex Parts in Quantum Field Theory.” Nuclear Physics 13 (1959): 181–192. doi:10.1016/0029-5582(59)90154-3. Open English translation PDF.
• Weinzierl, Stefan. Feynman Integrals: A Comprehensive Treatment for Students and Researchers. Cham: Springer, 2022. doi:10.1007/978-3-030-99558-4. Open PDF.