Three-Point Amplitudes

Massless three-point amplitudes are the local seeds of four-dimensional on-shell construction. Momentum conservation makes real Lorentzian three-particle kinematics degenerate, but its complexification has two nontrivial branches. On either branch, one homogeneous little-group equation per leg fixes the spinor dependence up to a coupling; dimensional analysis and the interaction content then decide which seed is present.

Required background. Spinor-Helicity Variables supplies the bispinor factorization, bracket conventions, and little-group weights used throughout.

Why complex three-point kinematics is nontrivial

Take three all-outgoing null momenta,

$$p_1+p_2+p_3=0, \qquad p_i^2=0.$$

Squaring $p_i+p_j=-p_k$ gives

$$2p_i\!\cdot p_j =\langle ij\rangle[ji]=0$$

for every pair. For generic real Lorentzian momenta, angle and square brackets are conjugate, so both vanish and all three momenta are collinear. Complex kinematics removes that conjugation condition. Momentum conservation then has two generic branches:

$$\begin{array}{lll} \text{holomorphic branch:}&[12]=[23]=[31]=0,& \langle12\rangle,\langle23\rangle,\langle31\rangle\ne0,\\[2mm] \text{antiholomorphic branch:}& \langle12\rangle=\langle23\rangle=\langle31\rangle=0,& [12],[23],[31]\ne0. \end{array}$$

The amplitude is defined as an analytic function on one of these branches and later appears as a residue of a higher-point physical amplitude. Complex momenta are therefore part of the analytic continuation, not additional observable states. The two branches and their role in determining three-point seeds are developed in Elvang and Huang 2014, § 2.6, pp. 28–31, PDF.

Solving the little-group equations

On the holomorphic branch, write

$$\mathcal A_3 =g\,\langle12\rangle^{a} \langle23\rangle^{b} \langle31\rangle^{c}.$$

Requiring weight $t_i^{-2h_i}$ on each leg gives

$$a+c=-2h_1, \qquad a+b=-2h_2, \qquad b+c=-2h_3.$$

Hence

$$\boxed{ \mathcal A_3^{\langle\ \rangle} =g\, \langle12\rangle^{h_3-h_1-h_2} \langle23\rangle^{h_1-h_2-h_3} \langle31\rangle^{h_2-h_3-h_1} }.$$

On the antiholomorphic branch,

$$\boxed{ \mathcal A_3^{[\ ]} =\widetilde g\, [12]^{-h_3+h_1+h_2} [23]^{-h_1+h_2+h_3} [31]^{-h_2+h_3+h_1} }.$$

The two coefficients are related by parity in a parity-invariant theory, after phases and the transformation of the particle species are fixed. They need not be equal in a chiral theory.

In four dimensions a stripped three-point amplitude has mass dimension one. The bracket monomial in the angle solution has dimension $-(h_1+h_2+h_3)$, while the square solution has dimension $h_1+h_2+h_3$. Therefore

$$[g]=1+h_1+h_2+h_3, \qquad [\widetilde g]=1-h_1-h_2-h_3.$$

This is the point at which kinematics meets dynamics: the available coupling dimensions and particle content determine which formally covariant structures actually occur.

Representative local seeds

Scalar cubic interaction. For $h_1=h_2=h_3=0$,

$$\mathcal A_3(1^0,2^0,3^0)=g_{\phi^3}, \qquad [g_{\phi^3}]=1.$$

No momentum dependence survives on shell for three massless scalars. Derivative operators proportional to $p_i\!\cdot p_j$ also vanish on three-point kinematics, although field content and masses can change this conclusion.

Yukawa seed. Two equal-helicity Weyl fermions and a scalar give

$$\mathcal A_3(1_a^{-1/2},2_b^{-1/2},3^0) =y_{ab}\langle12\rangle, \qquad \mathcal A_3(1_a^{+1/2},2_b^{+1/2},3^0) =\widetilde y_{ab}[12].$$

Both couplings are dimensionless. For identical Weyl species the antisymmetry of the bracket combines with fermion exchange statistics, so the allowed flavor tensor has the corresponding symmetric component. Hermiticity may relate $y$ and $\widetilde y$, but only after the field and external-state phase conventions are specified.

Yang–Mills seed. For helicities $(--+)$,

$$A_3(1^-,2^-,3^+) =g_{\mathrm{YM}} \frac{\langle12\rangle^3} {\langle23\rangle\langle31\rangle}.$$

The conjugate $(++-)$ seed uses square brackets. The bracket ratio has dimension one, so $g_{\mathrm{YM}}$ is dimensionless. The all-minus and all-plus structures instead carry three powers of momentum,

$$A_3(1^-,2^-,3^-)=c_{F^3}\langle12\rangle\langle23\rangle\langle31\rangle,$$

and its square-bracket conjugate, with $[c_{F^3}]=-2$. They signal a higher-derivative interaction rather than ordinary two-derivative Yang–Mills theory.

Graviton seed. For $(--+)$,

$$\mathcal M_3(1^{-2},2^{-2},3^{+2}) =\kappa\, \frac{\langle12\rangle^6} {\langle23\rangle^2\langle31\rangle^2},$$

whose bracket part has dimension two and whose gravitational coupling has dimension $-1$. The squared exponents reflect helicity two, but this observation alone is not a derivation of a double-copy representation.

Locality, poles, and coupling data

Negative bracket exponents in a three-point helicity expression do not by themselves indicate a propagating pole. Every Mandelstam invariant vanishes on three-point kinematics, and polarization wavefunctions already contain reference-spinor denominators. The relevant locality questions are whether the seed can arise from a local interaction with the declared particle content and coupling dimension, and whether its use in a higher-point amplitude produces only allowed factorization poles.

Conversely, little-group covariance does not fix the numerical coupling, gauge-group tensor, flavor tensor, or whether a candidate vertex is forbidden by permutation statistics. For identical bosons the full seed must have the required exchange symmetry after the kinematic and internal tensors are combined. For identical fermions it must have the corresponding antisymmetry.

Supersymmetric applications can package several seeds into a superamplitude, but the representation, Grassmann variables, and Ward identities must be imported from Supersymmetry and Duality. The construction and factorization of the resulting amplitudes remain on-shell dynamics handled here.

Checks before using a seed

Apply all five checks:

1. Branch: the formula uses only brackets that can be nonzero on the selected complex branch.
2. Weight: each leg has homogeneous weight $-2h_i$.
3. Dimension: the bracket degree plus the coupling dimension equals one.
4. Statistics and internal tensors: the full expression has the required permutation behavior.
5. Factorization compatibility: when glued into a four-point amplitude, its residues occur only in allowed channels and sum over the correct internal states.

The last check distinguishes a merely covariant monomial from consistent interaction data.

Common pitfalls

Concluding that every massless three-point amplitude vanishes. That is true for nonsingular functions on real Lorentzian three-point kinematics. On-shell recursion uses the nontrivial complex branches, whose values determine physical higher-point residues.

Mixing the two branches in one generic solution. On a three-point support, either all square brackets or all angle brackets vanish. Choose the branch before simplifying.

Inferring an interaction from helicity weights alone. Coupling dimension, internal symmetry, statistics, and higher-point consistency still have to be checked.

Exercises

Insert $(h_1,h_2,h_3)=(-1,-1,+1)$ into the general angle solution and determine the coupling dimension.

Solution

The exponents are $3,-1,-1$, so

$$\mathcal A_3 =g\frac{\langle12\rangle^3} {\langle23\rangle\langle31\rangle}.$$

The bracket ratio has dimension one, hence $[g]=0$. The square-branch expression describes the parity-conjugate $(++-)$ configuration, not a second value on the same holomorphic branch.

Where to continue

• Complex Momenta and Factorization shows how these seeds appear in residues of higher-point amplitudes.
• BCFW Recursion assembles the residues when the boundary contribution is controlled.
• Soft Limits as On-Shell Constraints studies another universal factorization regime.

References

• Elvang, Henriette, and Yu-tin Huang. Scattering Amplitudes in Gauge Theory and Gravity. Cambridge: Cambridge University Press, 2015. Open prepublication version. Open PDF.