Theorem: Complete classification in the declared class

Every $S$ in Definition def:stressclass has a unique form

Its trace is

Its translation Ward action on the scalar is $\lambda$ times the original translation action. Requiring the original translation normalization and zero trace forces

Proof. Dimension four and the available constant tensor contractions exhaust the monomials as follows.

A degree-two term has two derivatives and is a linear combination of

The last is zero on shell. A degree-four term has no derivatives and is a multiple of ${\eta }_{\mu \nu }:{\varphi }^4:$. A degree-one term would need three derivatives and an invariant constant tensor of odd total rank; none is built from $\eta$. A degree-three term has one derivative and the same odd-rank obstruction. A degree-zero term of dimension four would require a forbidden dimensionful constant; derivatives of the identity vanish. No higher field degree is compatible with dimension four.

Thus, on shell,

Conservation gives

The vacuum-to-four-particle matrix of the last term is nonzero, whereas every quadratic term has zero such matrix. Hence $e=0$. The vacuum-to-two-particle matrix of ${\partial }_{\nu }K$ is nonzero for noncollinear positive-energy momenta, so

Since

the unique parameters are

Uniqueness also follows from the independent quadratic tensor polynomials on noncollinear two-particle momenta.

The trace is $-\lambda K+3c\square Q=(6c-\lambda )K$. For a translation, $I_{\mu \nu }{a}^{\nu }$ is a divergence superpotential. Its compact-slab commutator vanishes under the collar contract. The canonical tensor therefore contributes exactly $\lambda$ times the original translation action. The last assertion follows.