Error Scaling of Sea Quark Isospin-Breaking Effects

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 $1\mathrm{/}a$ for strong-IBE and $1\mathrm{/}{a}^2$ for electromagnetic IBE, in the considered range of lattice spacings. On the other hand, our data are consistent with the predicted leading divergence $\sqrt{V}$.