Tree Amplitudes and Gauge Consistency

Tree amplitudes make locality, particle exchange, and gauge consistency visible in explicit calculations. This chapter moves from connected amputated graphs to scalar, fermion, and vector examples, using residues and channel relations as independent checks.

Enter this chapter

Begin with Connected Tree Diagrams and Amputated Amplitudes, then work through scalar contact and exchange graphs before introducing external spin.

Topics

1. Connected Tree Diagrams and Amputated Amplitudes — Separates graph connectivity from external-pole removal.
2. Scalar Contact and Exchange Amplitudes — Distinguishes local polynomial interactions from exchanged-particle poles.
3. Fermion and Yukawa Tree Amplitudes — Tracks spinor wave functions, ordering, and spin sums.
4. Vector External States and Ward Checks — Tests complete vector amplitudes for unphysical polarization dependence.
5. Mandelstam Channels and Tree-Level Crossing — Relates channel kinematics through the correct crossed external states.
6. Physical Poles and Tree-Level Factorization — Checks residues against products of lower-point on-shell amplitudes.

How the ideas fit together

An exchange pole appears when an internal particle goes on shell. Its residue factorizes into lower-point amplitudes summed over the appropriate intermediate states, whereas a local contact term has no such pole. Pole checks can therefore verify propagation and couplings but leave polynomial contact information undetermined. See Weinberg 1995, §§ 6.1–6.3.

For gauge particles, cancellation of unphysical polarization dependence generally uses the full set of diagrams. A missing contact graph may spoil that cancellation while every retained exchange graph has a plausible denominator. Crossing also requires the correct state, charge, and spin interpretation; changing a momentum sign alone does not finish the calculation.

Review the chapter

Can two tree amplitudes share all exchange-pole residues but differ physically?

Solution

Yes. Their difference can be a local polynomial contact amplitude allowed by the symmetries. Factorization fixes residues but not such terms. Additional information from the action, power counting, symmetry, or high-energy behavior is required.

Where to go next

Continue to On-Shell Construction to reconstruct amplitudes from these checks, or S-Matrix Unitarity, Analyticity, and Resonances for the constraints beyond tree level.

References

• Steven Weinberg, The Quantum Theory of Fields, Volume I: Foundations, Cambridge University Press, 1995. DOI.