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

Abstract

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.

Comments: 27 pages + 14 pages appendix and references, 2 figures, 3 tables; supplemental material contains Mathematica package including a notebook with two examples