Persistence of tensor superconvergence beyond twist two

Determine whether the tensor superconvergence relation involving the moment of $b_2-3b_4$ persists order-by-order in the twist expansion beyond the target-mass-complete twist-two sector.

Background

The paper derives an all-$Q^2$ tensor superconvergence relation, $\int_0^1 dx\,[b_2(x,Q^2)-3b_4(x,Q^2)]=0$, from an exact Siegert relation combined with an unsubtracted dispersion relation and a high-energy convergence assumption. The authors then compare this relation with Detmold’s target-mass-complete twist-two operator-product-expansion results for the tensor structure functions.

At moment index $n=1$, the target-mass-corrected twist-two relation implies $$\int_0^1 dx\,b_4^{\rm tmc}(x,Q^2)=\frac13\int_0^1 dx\,b_2^{\rm tmc}(x,Q^2)$$, thereby reproducing the tensor superconvergence relation within the target-mass-complete twist-two approximation. The unresolved issue is whether this agreement extends systematically to every order in the full twist expansion, including dynamical higher-twist contributions.