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

Recently the asymptotic lattice spacing dependence of spectral quantities in lattice QCD has been computed to $\mathrm{O}(a^2)$ 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 $\mathrm{SU}(N_\mathrm{f})_\mathrm{V}\times\mathrm{SU}(N_\mathrm{b}|N_\mathrm{b})_\mathrm{V}$ flavour symmetries for $N_\mathrm{f}$ sea-quarks and $N_\mathrm{b}$ 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 $\mathrm{O}(a)$, strongly suggesting to use at least tree-level Symanzik improvement of those local fields.