Singularities, Cuts, and Integrand Reconstruction

Loop singularities, physical cuts, generalized cuts, and maximal residues answer related but different questions. This chapter explains what each construction determines and what information remains before a complete amplitude has been reconstructed.

Enter this chapter

Begin with Landau Equations and Physical Singularities to identify candidate pinches. Then compare physical discontinuities with generalized on-shell constraints.

Topics

1. Landau Equations and Physical Singularities — Finds candidate contour pinches and tests their physical-sheet interpretation.
2. Cutkosky Cutting Rules — Relates appropriate physical discontinuities to on-shell intermediate states.
3. Generalized Unitarity and Integrand Reduction — Matches multiple cuts to determine integral coefficients or integrand data.
4. Leading Singularities and Integrand Geometry — Uses maximal multidimensional residues in specified loop-integrand settings.

How the ideas fit together

Landau equations provide candidate singular loci, but a physical singularity also requires a relevant contour pinch, sheet, and nonvanishing contribution. Cutkosky rules then compute appropriate discontinuities using on-shell replacements. A generalized cut may impose a different contour and need not be a physical positive-energy state sum. See Zwicky 2016, §§ 4.1–4.3.

Reconstruction also depends on the chosen function or integral space. Four-dimensional cuts can leave rational contributions or dimension-dependent information undetected. Two integrands can integrate to the same result while differing pointwise, so an integrand identity must specify the allowed equivalences.

Review the chapter

Why does a contribution with no ordinary four-dimensional branch discontinuity evade reconstruction from those discontinuities alone?

Solution

The cut data vanish for that contribution, so adding it does not change any of those constraints. Rational terms provide an important example. Extra information, such as dimensionally regulated cuts, factorization, or boundary conditions, is needed to remove the ambiguity.

Where to go next

Continue to Dispersion, Positivity, and UV Constraints to combine discontinuities with growth and subtraction assumptions, or return to Loop Integrals and Reduction for evaluation.

References

• Roman Zwicky, “A Brief Introduction to Dispersion Relations and Analyticity,” lectures at the Dubna School “Strong Fields and Heavy Quarks” (2016), arXiv:1610.06090. Stable preprint.