Determine the Symanzik scaling exponent for leading discretization errors in the 4stout lattice setup

Determine the correct value of the exponent n in the α_s^n a^2 scaling of the leading discretization errors for the 4stout lattice action (tree-level Symanzik gauge action with one-link staggered fermions and stout smearing) and for the hadronic vacuum polarization window observables analyzed via the time–momentum representation; this exponent arises from anomalous dimensions in the Symanzik effective theory and directly impacts continuum extrapolations.

Background

In the analysis of window observables for the leading-order hadronic vacuum polarization contribution to a_μ, the authors model lattice-spacing dependence using polynomials either in a^2 or in a taste-violation measure Δ_KS. The theoretical expectation is that leading discretization errors scale as α_s^n a^2 due to anomalous dimensions in the Symanzik effective theory, but the exponent n is not fixed for the specific lattice action and observables considered.

Because n is not established, the authors hedge by performing fits with two different continuum parametrizations (using a^2 and Δ_KS), and they report that taste-symmetry violation decreases approximately as α_s^n a^2 with n≈3 toward finer lattices. Firmly determining n would reduce systematic uncertainty in continuum extrapolations and improve precision.