Lattice artifacts of local fermion bilinears up to O(a²)

Recently the asymptotic lattice spacing dependence of spectral quantities in lattice QCD has been computed to O(a²) using Symanzik Effective theory [1,2]. 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 SU(Nf)V(Nb|Nb)V flavour symmetries for Nf sea-quarks and Nb quenched valence-quarks respectively in the massless limit. Overall we find only few cases Γ̂, which worsen the asymptotic lattice spacing dependence aⁿ[2b₀ḡ²(1/a)]Γ̂ compared to the classically expected aⁿ-scaling. Other than for trivial flavour quantum numbers, only the axial-vector and much milder the tensor may cause some problems at O(a), strongly suggesting to use at least tree-level Symanzik improvement of those local fields.