Lemma: Conserved stress class

Every conserved member of ${\mathcal{W}}_4$ has the form

Proof. Dimension and tensor rank restrict a general member to the span of

Indeed the bilinear terms have two derivatives; $:\varphi {\partial }_{\mu }{\partial }_{\nu }\varphi :$ is $\frac{1}{2}{\partial }_{\mu }{\partial }_{\nu }Q-A_{\mu \nu }$. The scalar $:\varphi \square \varphi :$ vanishes on the free equation, and $\square Q=2K_0$. Odd numbers of remaining tensor indices cannot be contracted to a symmetric rank-two tensor using invariant tensors. Any possible antisymmetric epsilon contraction at this derivative order vanishes after symmetrization.

Write a general member as

Its divergence is

Different Wick degrees are independent. Equivalently, coherent displacements scaled by a real parameter distinguish the quadratic and quartic powers. Since ${\partial }_{\nu }:{\varphi }^4:$ is not identically zero, conservation forces $\zeta =0$. Also ${\partial }_{\nu }{K}_0$ is not identically zero, as may be checked on the classical free solution $\varphi =(x^0{)}^2+(x^1{)}^2$. Therefore $\beta =-\alpha /2$, proving (eq:stressclass).