Singularities, Cuts, and Integrand Reconstruction

Loop amplitudes are multivalued functions, and several distinct on-shell constructions illuminate that structure. Landau equations identify candidate contour pinches. Cutkosky rules compute physical discontinuities across appropriate branch cuts. Generalized unitarity imposes additional on-shell constraints to determine coefficients or an integrand representative. Leading singularities are maximal multidimensional residues. The methods are related, but none of those four statements may be substituted for another without checking its hypotheses.

Enter the singularity analysis

The most reliable route begins with the question being asked:

Question	Page	Output
Where can a Feynman integral become singular?	Landau Equations and Physical Singularities	Candidate pinch loci, reduced diagrams, and physical-sheet tests
What is the discontinuity across a physical channel cut?	Cutkosky Cutting Rules	Positive-energy on-shell delta functions and a phase-space integral
How can products of trees determine loop data?	Generalized Unitarity and Integrand Reduction	Cut constraints on master coefficients or integrand numerators
What does complete localization of a loop form measure?	Leading Singularities and Integrand Geometry	Oriented maximal residues and their limits

The distinctions can be summarized without conflating their outputs:

• Landau solution: a candidate singular locus;
• physical cut: a discontinuity on a specified sheet;
• generalized cut: on-shell algebraic constraints; and
• maximal residue: local integrand data with a declared contour orientation.

None of the reverse implications is automatic. A Landau solution can lie on another sheet or be canceled by a numerator. A generalized cut can use complex solutions with no interpretation as positive-energy intermediate states. Four-dimensional cuts can miss $(-2\epsilon)$-dimensional and rational information. An integrand can change by a surface term while the integrated amplitude remains fixed.

The useful ordering is

$$\text{denominator surfaces} \longrightarrow\text{candidate pinch} \longrightarrow\text{physical discontinuity} \longrightarrow\text{cut constraints on a representation},$$

with a new hypothesis at every arrow. It is not a chain of equivalences: positivity and real energy flow enter the physical cut, while an ansatz and completeness proof enter reconstruction. Maximal residues may constrain the last step even when they have no direct positive-energy interpretation.

One bubble, four viewpoints

For a scalar bubble with masses $m_1,m_2$ and channel invariant $s$:

1. the Landau equations locate $s=(m_1\pm m_2)^2$ as algebraic candidates;
2. positive Feynman parameters and future-directed internal momenta select the normal physical threshold $s=(m_1+m_2)^2$;
3. the Cutkosky rule gives its discontinuity as two-particle phase space;
4. imposing both propagators on shell gives a double cut that constrains a bubble coefficient in an amplitude decomposition.

At one loop in four variables, a box can additionally be localized by four on-shell equations at isolated complex solutions. That maximal cut is powerful coefficient data, but it is not merely a more exclusive physical decay rate.

The original contour conditions were formulated by Landau 1959, pp. 181–192, and the discontinuity rule by Cutkosky 1960, pp. 429–433. Quadruple-cut coefficient extraction in a specific one-loop supersymmetric setting is developed in Britto, Cachazo, and Feng 2005, §§1–2, pp. 275–285.

Boundaries of the chapter

This chapter works perturbatively with Feynman integrals and loop integrands. It does not claim that every singularity of the exact S-matrix appears graph by graph, nor that diagram singularities survive sums required by gauge invariance. It also does not replace a dispersion relation, a full proof of analyticity, or an infrared-safe observable construction.

Cut reconstruction must preserve the dimensional regulator and external-state scheme when those affect the desired answer. Physical discontinuities of renormalized amplitudes can also require counterterm cuts or subtractions beyond the bare graph rule. Thermal cutting rules, unstable external particles, and nonperturbative spectral reconstruction lie outside the present scope.

Review the chapter

Use the questions below to identify topics worth revisiting.

• Why is $\alpha_i(q_i^2-m_i^2)=0$ weaker than putting every internal line on shell?
• Which extra condition turns an on-shell replacement into a physical channel cut?
• Why can an integrand be fully matched on selected cuts yet remain nonunique?
• What fixes the sign of a multidimensional leading singularity?

Concise answers should mention reduced diagrams, positive-energy flow and sheet, surface terms or cut-free terms, and contour orientation.

References

• Britto, Ruth, Freddy Cachazo, and Bo Feng. “Generalized Unitarity and One-Loop Amplitudes in $\mathcal N=4$ Super-Yang–Mills.” Nuclear Physics B 725 (2005): 275–305. doi:10.1016/j.nuclphysb.2005.07.014. Open PDF.
• Cutkosky, Richard E. “Singularities and Discontinuities of Feynman Amplitudes.” Journal of Mathematical Physics 1 (1960): 429–433. doi:10.1063/1.1703676.
• 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.