The smeared $R$-ratio in isoQCD from first-principles lattice simulations

Abstract

The $R$-ratio is a phenomenological observable of great relevance, both in itself and in applications such as the dispersive approach to the muon anomalous magnetic moment. It can be investigated from first-principles with controlled statistical and systematic errors in lattice QCD by introducing an arbitrary smearing kernel and employing spectral reconstruction techniques, such as the well-known Hansen-Lupo-Tantalo method. Improving upon a first study published in 2023, we show preliminary results using the correlation functions produced by ETMC in $N_f = 2+1+1$ lattice simulations at four lattice spacings, different volumes and with higher statistics w.r.t. our previous study. The new correlators, thanks to the implementation of the Low Mode Average technique, allow the determination of the $R$-ratio smeared with Gaussian kernels of widths down to $σ\sim 200$ with phenomenologically relevant precision.

Figures (7)

• Figure 1: Comparison of $R_\sigma(E)$ (blue points) and $R^\mathrm{exp}_\sigma(E)$ (red points) as functions of $E$ for $\sigma=0.63$ GeV and $0.44$ GeV. The relative difference $R_\sigma(E)/R^{\mathrm{exp}}_\sigma (E)-1$ is also shown as a function of the energy. For $\sigma \simeq 0.6$ GeV, the total relative uncertainty is at the level of $\sim 1\%$ and a deviation of about $3\sigma$ with respect to the experimental $e^+e^-$ data can be observed. For $\sigma \simeq 0.4$ GeV, the total relative uncertainty increases to $\sim 3.5\%$, preventing us from resolving statistically significant deviations from the experimental determination.
• Figure 2: Stability analysis of $R_\sigma(E)$ for different values of $d(\mathbf{g})$ on the B64 ensemble at energy $E=0.70$ GeV and $\sigma=0.4$ GeV. As $\lambda \to 0$, the reconstructed kernel approaches the target one, at the price of larger statistical uncertainties. Conversely, as $\lambda \to 1$, the statistical precision improves while the reconstructed kernel deviates significantly from the target, reflecting the competition between resolution and statistical uncertainty in Eq. (\ref{['eq:W-functional']}). Our best estimate for $R_\sigma(E)$ is selected from this statistically dominated regime, corresponding to small values of $d(\mathbf{g})$ where stability within the statistical errors is observed.
• Figure 3: Top panel: Stability of $R_\sigma(E)$ as a function of $d(\mathbf{g})$ for different values of $N_{\mathrm{config}} = 100, 200, 400, 800, 1200$, at $E=0.70~\mathrm{GeV}$ and $\sigma=0.4~\mathrm{GeV}$ on the B64 ensemble. Bottom panel: We show $R_\sigma(E, N_{\mathrm{config}})$ as a function of $N_{\mathrm{config}}$ on the left and the relative statistical error on the right which is observed to scale as $1/\sqrt{N_{\mathrm{config}}}$, as expected.
• Figure 4: Comparison of the stability analysis for $R^{\ell \ell}_{\sigma}(E)$ with TM regularization on the D96 ensemble at energy $E=0.70$ GeV and $\sigma=0.4$ GeV, using both the statistics and setup of Ref. ExtendedTwistedMassCollaborationETMC:2022sta (labelled as Old Setup in the figure) and the new setup (LMA) with increased statistics and the use of the error reduction technique.
• Figure 5: Continuum extrapolation of $R^{\ell\ell,C}_\sigma(E)$ at $E=0.50~\mathrm{GeV}$ and $\sigma=0.2~\mathrm{GeV}$. Blue and red points correspond to OS and TM regularizations. A correlated $\chi^2$ fit is performed at fixed $E$ and $\sigma$ for $L \sim 5~\mathrm{fm}$.