Two-Flavor Simulations of $\rho (770)$ and the Role of the $K\overline{K}$ Channel

The $\rho (770)$ meson is the most extensively studied resonance in lattice QCD simulations in two ($N_f=2$) and three ($N_f=2+1$) flavor formulations. We analyze $N_f=2$ lattice scattering data using unitarized chiral perturbation theory, allowing not only for the extrapolation in mass but also in flavor, $N_f=2\to N_f=2+1$. The flavor extrapolation requires information from a global fit to $\pi \pi$ and $\pi K$ phase shifts from experiment. While the chiral extrapolation of $N_f=2$ lattice data leads to masses of the $\rho (770)$ meson far below the experimental one, we find that the missing $K\overline{K}$ channel is able to explain this discrepancy.

Figure 1

Insertion of a $K\overline{K}$ intermediate state in $\pi \pi$ scattering, iterated in the present approach to provide coupled-channel unitarity.

Figure 2

Results for the $N_f=2$ lattice simulations (ordered by pion mass) of Bali et al. (RQCD Collaboration) [22], Guo et al. (GWU Collaboration) [24], Göckeler et al. (QCDSF Collaboration) [18], Lang et al. [20], Feng et al. (ETMC Collaboration) [19], and Aoki et al. ($CP$-PACS Collaboration) [17]. For each result, the left picture shows the lattice data and fit, and the right figure shows the $N_f=2$ chiral extrapolation (blue dashed line or light blue area). Without changing this result, the $K\overline{K}$ channel is then included to predict the effect from the missing strange quark (red solid line or light red area). The experimental data (blue circles from Ref. [51], squares from Ref. [43]) are then postdicted. For the inherent model uncertainties, see the text.

Figure 3

Effect of the $K\overline{K}$ channel in the $(m_{\rho },g)$ plane indicated with arrows, after chiral extrapolation to the physical pion mass. See Fig. 2 for the labeling of the extrapolations. Only statistical uncertainties are shown, and only for the case after including $K\overline{K}$. See the text for further explanations and the Supplemental Material [46] for the effect of the chiral extrapolation.