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Reference-spinor changes are pure gauge

Statement

Let $p \neq 0$ be a complex null momentum and choose its spinor factorization $κ (p)_{a \dot{a}} = λ_{a} {\tilde{λ}}_{\dot{a}}$ , which exists by the null-factorization result on this page. Raise spinor indices and define brackets as in the spinor-helicity definition. For an undotted reference spinor $r$ with $⟨ λ r ⟩ \neq 0$ and a dotted reference spinor $\tilde{r}$ with $[\tilde{λ} \tilde{r}] \neq 0$ , define polarization tensors

$E_{+}^{a \dot{a}} (p; r) = \sqrt{2} \frac{r^{a} {\tilde{λ}}^{\dot{a}}}{⟨ λ r ⟩}, E_{-}^{a \dot{a}} (p; \tilde{r}) = \sqrt{2} \frac{λ^{a} {\tilde{r}}^{\dot{a}}}{[\tilde{λ} \tilde{r}]} .$

The reference spinors have the same inverse-square-root length units as $λ$ and $\tilde{λ}$ , so these polarization tensors are dimensionless.

Associate a vector $ε_{\pm}^{μ}$ to each tensor by the inverse of the raised-index Pauli map $p \mapsto ϵ^{a b} ϵ^{\dot{a} \dot{b}} κ (p)_{b \dot{b}}$ . A polarization change is called pure gauge here when its vector difference is a scalar multiple of $p^{μ}$ . For any two admissible references of the corresponding chirality, the polarization-vector difference is pure gauge. Therefore, for every complex current $J \in C^{4}$ satisfying the explicitly defined conservation condition $h (p, J) = 0$ , the contraction $h (ε_{\pm} (p; r), J)$ is independent of the reference spinor. Current conservation is a separate hypothesis; no general QFT Ward identity is asserted.

Statement references

Complex null momenta and helicity weights

Null momenta admit unique spinor-helicity classes

Elvang and Huang, Scattering Amplitudes, §§2.3–2.4

Britto, Cachazo, Feng, and Witten, Direct Proof of Tree-Level Recursion Relation in Yang-Mills Theory, §3.1

Facts & Assumptions

Given: $p \neq 0$ is complex null, $κ (p) = λ \tilde{λ}$ , all reference denominators used below are nonzero, and $h (p, J) = 0$ .

[F1]

Complex null momenta and helicity weights: a spinor factorization has entries $κ (p)_{a \dot{a}} = λ_{a} {\tilde{λ}}_{\dot{a}}$ .

[F2]

Complex null momenta and helicity weights: $h (p, q) = p^{0} q^{0} - p^{1} q^{1} - p^{2} q^{2} - p^{3} q^{3}$ .

[F3]

Complex null momenta and helicity weights: indices are raised by the alternating tensors, and $⟨ λ r ⟩ = ϵ^{a b} λ_{a} r_{b}$ , $[\tilde{λ} \tilde{r}] = ϵ^{\dot{a} \dot{b}} {\tilde{λ}}_{\dot{a}} {\tilde{r}}_{\dot{b}}$ .

[F4]

Null momenta admit unique spinor-helicity classes: every nonzero null $p$ has a nonzero spinor factorization.

[F5]

Complex null momenta and helicity weights: the explicit inverse formula for $κ$ makes it a linear isomorphism.

[A1]

The two undotted references $r, s$ and the two dotted references $\tilde{r}, \tilde{s}$ have nonzero denominators with the fixed momentum spinors.

[A2]

The current obeys the exact complex-bilinear conservation condition $h (p, J) = 0$ .

Proof

1.1givenF1F3F4A1algebra

The factorization exists by [F4]. For undotted references, put $w^{a} = r^{a} / ⟨ λ r ⟩ - s^{a} / ⟨ λ s ⟩$ . Then $λ_{a} w^{a} = 1 - 1 = 0$ by [F3]. The nonzero covector $λ_{a}$ has a one-dimensional kernel. Since the alternating tensor is invertible, $λ^{a}$ is nonzero and lies in that kernel; hence $w^{a} = c λ^{a}$ for some $c \in C$ . Substitution in the displayed polarization formula gives $E_{+} (p; r) - E_{+} (p; s) = \sqrt{2} c λ^{a} {\tilde{λ}}^{\dot{a}}$ , which is a scalar multiple of the momentum tensor.

1.2givenF1F3F4A1algebra

For dotted references, put ${\tilde{w}}^{\dot{a}} = {\tilde{r}}^{\dot{a}} / [\tilde{λ} \tilde{r}] - {\tilde{s}}^{\dot{a}} / [\tilde{λ} \tilde{s}]$ . Then ${\tilde{λ}}_{\dot{a}} {\tilde{w}}^{\dot{a}} = 1 - 1 = 0$ by [F3]. The kernel is spanned by ${\tilde{λ}}^{\dot{a}}$ because the dotted alternating tensor is invertible, so ${\tilde{w}}^{\dot{a}} = d {\tilde{λ}}^{\dot{a}}$ and $E_{-} (p; \tilde{r}) - E_{-} (p; \tilde{s}) = \sqrt{2} d λ^{a} {\tilde{λ}}^{\dot{a}}$ , again proportional to $p$ .

2.1F2F5A2step 1.1step 1.2algebra∎

The explicit inverse formula in [F5] makes $κ$ a linear isomorphism, and each alternating tensor has determinant one, so the raised-index Pauli map is also an isomorphism. Each tensor identity therefore gives the same proportionality for the associated four-vectors. Pairing either difference with $J$ gives a scalar multiple of $h (p, J) = 0$ under [F2, A2]. Thus both polarization contractions are reference independent.

Remarks

This proposition proves reference independence only after the current-conservation condition is supplied. It does not prove that every model-specific tree current satisfies that condition.

Reference-spinor changes are pure gauge

Statement

Statement references

Facts & Assumptions

Proof

Remarks

Depends on

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