On-Shell Construction

On-shell construction replaces much of the gauge- and field-redefinition-dependent intermediate algebra of Feynman graphs by physical state data, locality, and factorization. It is most effective when the external particles, spacetime dimension, masses, helicities, color basis, and large-complex-momentum behavior are all declared. Those qualifications matter: residues can reconstruct an amplitude only up to information with no factorization pole, such as a local contact interaction or a boundary contribution at infinity.

This chapter develops the four-dimensional massless language first, then shows how massive little-group covariance changes it Arkani-Hamed, Huang, and Huang 2021, §§ 2.1–2.2, pp. 6–10, PDF. Three-point amplitudes provide the local seeds; complex factorization and BCFW recursion assemble higher-point trees when the chosen deformation has no uncontrolled boundary term. Color ordering and soft limits add powerful checks without becoming universal assumptions.

Enter this chapter

Choose the route that matches the missing ingredient in your calculation.

Question	Start here	What a satisfactory result contains
How must an amplitude transform for specified helicities?	On-Shell States and Little-Group Scaling	One homogeneous weight per external leg, correct mass dimension, and no dependence on a polarization reference
How are null momenta represented compactly in four dimensions?	Spinor-Helicity Variables	A convention-fixed bispinor factorization, bracket identities, reality conditions, and invariant checks
Which local three-particle interactions are compatible with helicity?	Three-Point Amplitudes	The correct holomorphic or antiholomorphic branch, little-group exponents, and coupling dimension
What changes for massive external states?	Massive On-Shell Variables and Little-Group Covariance	Explicit SU(2) indices, spin-state counting, normalization, and a controlled high-energy limit
How do physical poles appear after a complex deformation?	Complex Momenta and Factorization	Pole locations, on-shell internal momentum, residue, and the undeformed physical limit
Can residues determine the whole tree amplitude?	BCFW Recursion	A valid shift, all factorization channels, the internal-state sum, and a checked residue at infinity
How can gauge-group algebra be separated from kinematics?	Color Decomposition and Partial Amplitudes	A named generator normalization and basis, ordering relations, and reconstruction of the color-dressed amplitude
What information is forced by a soft external particle?	Soft Limits as On-Shell Constraints	The leading soft factor, charge or momentum-conservation check, and a clear tree/loop scope
Why did a recursion miss a contact term?	Constructibility, Boundary Terms, and Failure Modes	Large-$z$ scaling, the boundary residue, shift dependence, and the local data not fixed by poles

The construction chain

For a massless external leg $i$, little-group covariance requires

$$\mathcal A_n(\ldots,t_i\lambda_i,t_i^{-1}\widetilde\lambda_i,h_i,\ldots) =t_i^{-2h_i}\mathcal A_n(\ldots,\lambda_i,\widetilde\lambda_i,h_i,\ldots).$$

Together with dimensional analysis, this almost fixes three-point amplitudes. Locality and tree-level unitarity then say that a higher-point amplitude may have poles only when an allowed internal momentum $P_I$ becomes on shell. To separate that invariant statement from an overall Feynman-rule phase, write the singular part as

$$\mathcal A_n =\frac{R_I}{P_I^2}+O(1), \qquad R_I\propto \sum_h \mathcal A_L(\ldots,P_I^h)\, \mathcal A_R(-P_I^{-h},\ldots).$$

The proportionality becomes an equality once the amplitude phase, internal-state crossing, and propagator conventions are fixed. This chapter uses the standard analytic-amplitude convention in which those factors are absorbed into the subamplitudes; when translating back to the volume’s $i\mathcal M$ convention, the sign and phase must be restored explicitly.

A complex shift turns these factorization limits into isolated poles of a meromorphic function $\mathcal A_n(z)$. Cauchy’s theorem reconstructs $\mathcal A_n(0)$ from their residues only after the contour at infinity has been evaluated. The exact closing criterion is $B_\infty=-\operatorname{Res}_{z=\infty}\mathcal A_n(z)/z=0$; decay of $\mathcal A_n(z)$ is the standard sufficient condition, not the definition. The original Yang–Mills recursion proof makes this contour argument explicit Britto, Cachazo, Feng, and Witten 2005, § 2, eqs. (2.3)–(2.7), pp. 3–5, PDF. The broader formalism and its scope are developed in Elvang and Huang 2014, §§ 2.2, 2.6, and 3.1–3.3, pp. 9–11, 27–30, and 34–46, PDF.

This chain has two independent filters:

• Kinematic covariance fixes how a candidate expression transforms and which invariants it may use.
• Dynamical input fixes couplings, particle content, color tensors, contact terms, and large-shift behavior.

Passing the first filter never proves the second. An expression can have perfect little-group weights and still have the wrong poles, residues, dimension, permutation symmetry, or boundary term.

Boundaries of the method

The chapter focuses on amplitude construction and its diagnostics, including applications that import supersymmetric state constraints. Supersymmetry and Duality develops on-shell supermultiplet representations, Grassmann packaging, supersymmetric Ward identities, and protection statements. Renormalization and Effective Field Theory supplies the interpretation and matching of independent local contact operators. Loop recursion, rational terms, and generalized cuts belong to the later loop and cut chapters.

The massless spinor-helicity formulas below are four-dimensional. Higher-dimensional massless little groups and arbitrary-spin massive classifications require additional representation data. Even in four dimensions, real Lorentzian three-point kinematics is degenerate; complex momenta are part of the analytic construction, not extra physical external states.

Review the chapter

Use the questions below to identify topics worth revisiting.

1. Take a four-point candidate amplitude and verify, in this order: one little-group weight per leg; total mass dimension; all allowed physical-channel residues; absence of spurious reference-spinor dependence; and its behavior under a useful BCFW shift. A satisfactory diagnosis distinguishes a wrong residue or spurious pole from a polynomial ambiguity that factorization cannot see. Repair the former with Complex Momenta and Factorization and the latter with Constructibility, Boundary Terms, and Failure Modes.
2. Starting from a color-ordered four-gluon partial amplitude, list the poles allowed by that cyclic ordering before restoring color. A satisfactory answer does not add every channel of the color-dressed amplitude to one partial amplitude; use Color Decomposition and Partial Amplitudes to repair the basis or ordering step.
3. Compare a leading photon soft factor with the cancellation of an infrared divergence in a measured rate. A satisfactory answer identifies the first as an amplitude-level factorization statement and the second as a real-virtual, measurement-dependent statement; continue to Soft Limits as On-Shell Constraints and then the infrared chapter if those claims were conflated.

Where to continue

• For a first derivation, begin with little-group scaling and the spinor-helicity dictionary.
• To build a tree recursively, study three-point seeds, complex factorization, and BCFW recursion in order.
• To test loop information using trees, continue to Singularities, Cuts, and Integrand Reconstruction.
• To interpret local terms as EFT interactions, continue to Renormalization and Effective Field Theory.

References

• Arkani-Hamed, Nima, Tzu-Chen Huang, and Yu-tin Huang. “Scattering Amplitudes for All Masses and Spins.” Journal of High Energy Physics 11 (2021): 070. DOI. Open PDF.
• Britto, Ruth, Freddy Cachazo, Bo Feng, and Edward Witten. “Direct Proof of Tree-Level Recursion Relation in Yang–Mills Theory.” Physical Review Letters 94 (2005): 181602. DOI. Open PDF.
• Elvang, Henriette, and Yu-tin Huang. Scattering Amplitudes in Gauge Theory and Gravity. Cambridge: Cambridge University Press, 2015. Open prepublication version. Open PDF.