On-Shell Construction
On-shell construction asks how much of an amplitude is fixed by its external states, singularities, and symmetries. Spinor variables make those constraints compact, while complex deformations turn factorization into recursion only when the contribution at infinity is controlled.
Enter this chapter
Section titled “Enter this chapter”Begin with On-Shell States and Little-Group Scaling, then use spinor-helicity variables and three-point amplitudes as the massless starting point.
Topics
Section titled “Topics”- On-Shell States and Little-Group Scaling — Determines how amplitudes transform when external-state frames change.
- Spinor-Helicity Variables — Factors null momenta into spinors in four dimensions.
- Three-Point Amplitudes — Classifies local seeds in complexified massless kinematics.
- Massive On-Shell Variables and Little-Group Covariance — Retains the massive little-group indices needed for spin covariance.
- Complex Momenta and Factorization — Turns factorization channels into poles of complex momentum deformations.
- BCFW Recursion — Reconstructs trees from residues when the chosen shift permits it.
- Color Decomposition and Partial Amplitudes — Separates group tensors from ordered kinematic functions.
- Soft Limits as On-Shell Constraints — Uses low-energy emission to constrain hard amplitudes.
- Constructibility, Boundary Terms, and Failure Modes — Identifies contact information and boundary terms left unfixed by recursion.
How the ideas fit together
Section titled “How the ideas fit together”A complex deformation preserves on-shell momenta and conservation while making the amplitude a function of a complex parameter. Cauchy’s theorem expresses the undeformed value through finite factorization residues plus a possible contribution at infinity. The residues are lower-point data; the large-parameter behavior decides whether they are sufficient. See Britto, Cachazo, Feng, and Witten 2005, §§ 2–3, Open PDF.
Little-group covariance constrains the allowed functions but does not prove constructibility. A contact interaction can carry correct weights without any finite deformation pole, so different theories and shifts can require different additional input.
Review the chapter
Section titled “Review the chapter”What does a nonzero constant four-scalar contact amplitude become under a two-line momentum shift, and what does recursion miss?
Solution
It remains constant. There are no finite factorization poles, yet the undeformed amplitude is nonzero. The contour contribution at infinity supplies the missing contact term, showing why residue-only recursion is not automatic.
Where to go next
Section titled “Where to go next”Continue to Loop Integrals and Reduction for integration, or Singularities, Cuts, and Integrand Reconstruction for on-shell loop constraints.