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

TL;DR

This paper extends the analysis of lattice spacing effects in lattice QCD to local fermion bilinear matrix elements and correlators, identifying specific operators that require improved actions to reduce discretization errors.

Contribution

It provides the first detailed $ ext{O}(a^2)$ analysis of lattice artifacts for local fermion bilinears in lattice QCD, highlighting operators needing improvement.

Findings

Few operators worsen lattice spacing dependence at $ ext{O}(a)$

Axial-vector and tensor operators may cause issues at $ ext{O}(a)$

Tree-level Symanzik improvement recommended for certain operators

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.