Error Scaling of Sea Quark Isospin-Breaking Effects

TL;DR

This study investigates the error scaling of sea-quark isospin-breaking effects in lattice QCD, showing that errors do not diverge as expected in the continuum limit within the examined parameter range.

Contribution

The paper provides the first detailed analysis of error scaling for sea-quark isospin-breaking effects using $N_f=3$ Wilson fermions, challenging previous assumptions about divergence.

Findings

Errors do not show the expected $1/a$ or $1/a^2$ divergence.

Errors are consistent with the predicted $ ext{sqrt}(V)$ divergence.

Gauge errors reach the dominant contribution level.

Abstract

Sea-quark isospin-breaking effects (IBE) are difficult to compute since they require the evaluation of all-to-all propagators. However, the quest for high-precision calculations motivates a detailed study of these contributions. There are strong arguments that the stochastic error associated with these quantities should diverge in the continuum and infinite-volume limit, resulting in a possible bottleneck for the method. In this work, we present the study of the error scaling for these quantities using $N_f=3$ $O(a)$-improved Wilson fermions QCD with C-periodic boundary conditions in space, a pion mass $M_{\pi}=400$ MeV, a range of lattice spacings $a=0.05, 0.075, 0.1$ fm, and volumes $L=1.6, 2.4, 3.2$ fm. The analysis of the error as a function of the number of stochastic sources shows that we reach the gauge error for the dominant contributions. The errors do not show the leading order divergence $1/a$ for strong-IBE and $1/a^2$ for electromagnetic IBE, in the considered range of lattice spacings. On the other hand, our data are consistent with the predicted leading divergence $\sqrt{V}$.