Lattice artifacts of local fermion bilinears up to \(\text {O}(\text {a}^2)\)

Recently the asymptotic lattice spacing dependence of spectral quantities in lattice QCD has been computed to \(\textrm{O}(a^2)\) using Symanzik Effective theory (Husung et al. in Phys Lett B 829:137069, 2022; Husung in Eur Phys J C 83:142, 2023). Here, we extend these results to matrix elements and correlators of local fermion bilinears, namely the scalar, pseudo-scalar, vector, axial-vector, and tensor. This resembles the typical current insertions for the effective Hamiltonian of electro-weak or BSM contributions, but is only a small fraction of the local fields typically considered. We again restrict considerations to lattice QCD actions with Wilson or Ginsparg–Wilson quarks and thus lattice formulations of QCD without flavour-changing interactions realising at least \(\textrm{SU}(N_{\textrm{f}})_\textrm{V}\times \textrm{SU}(N_{\textrm{b}}|N_{\textrm{b}})_\textrm{V}\) flavour symmetries for \(N_{\textrm{f}}\) sea-quarks and \(N_{\textrm{b}}\) quenched valence-quarks respectively in the massless limit. Overall we find only few cases \({\hat{\Gamma }}\), which worsen the asymptotic lattice spacing dependence \(a^n[2b_0{\bar{g}}^2(1/a)]^{{\hat{\Gamma }}}\) compared to the classically expected \(a^n\)-scaling. Other than for trivial flavour quantum numbers, only the axial-vector and much milder the tensor may cause some problems at \(\textrm{O}(a)\), strongly suggesting to use at least tree-level Symanzik improvement of those local fields.