Establish the all-orders absence of contact anomalies for nonequilibrium operators

Establish an all-orders theorem for the relation between the current-like and local-transport correction-to-scaling exponents, $y_J-y_{\\Theta_1}=-\\eta$, by controlling the composite-operator renormalizations of both $\\mathcal O_\\Theta$ and $\\mathcal O_J$ beyond the present functional truncation and beyond two-loop order, respectively.

Background

The paper derives the local-transport exponent within an LPA′'+A functional truncation and verifies the absence of an additional current contact anomaly through two loops. These results imply the conditional relation yJ−yΘ1=−ηy_J-y_{\Theta_1}=-\eta.

However, the authors explicitly distinguish these finite-order or truncation-based results from an all-orders proof. Establishing the relation generally requires demonstrating that neither the local-transport operator OΘ=ϕ(∂iϕ^)2\mathcal O_\Theta=\phi(\partial_i\hat\phi)^2 nor the current operator OJ=(∂iϕ^)(∂iϕ)∂iϕ\mathcal O_J=(\partial_i\hat\phi)(\partial_i\phi)\partial_i\phi acquires an additional composite-operator contact anomaly.

References

We therefore do not claim an all-orders theorem for $y_J-y_{\Theta_1}=-\eta$ until the composite-operator renormalizations of both $\mathcal O_\Theta$ and $\mathcal O_J$ are controlled, respectively beyond the present truncation and beyond two loops.

— Nonequilibrium corrections to conserved Ising criticality in scalar active matter: Ward identities, spectrum, and long crossovers  (2609.02351 - Zdybel, 2 Sep 2026) in Section 6, “Discussion and conclusions”