Rho resonance, timelike pion form factor, and implications for lattice studies of the hadronic vacuum polarization

We study isospin-1 $P$-wave $\pi \pi$ scattering in lattice QCD with two flavors of $\mathrm{O}(a)$ improved Wilson fermions. For pion masses ranging from $m_{\pi }=265\mathrm{MeV}$ to $m_{\pi }=437\mathrm{MeV}$, we determine the energy spectrum in the center-of-mass frame and in three moving frames. We obtain the scattering phase shifts using Lüscher’s finite-volume quantization condition. Fitting the dependence of the phase shifts on the scattering momentum to a Breit-Wigner form allows us to determine the corresponding $\rho$ mass $m_{\rho }$ and $g_{\rho \pi \pi }$ coupling. By combining the scattering phase shifts with the decay matrix element of the vector current, we calculate the timelike pion form factor, $F_{\pi }$, and compare the results to the Gounaris-Sakurai representation of the form factor in terms of the resonance parameters. In addition, we fit our data for the form factor to the functional form suggested by the Omnès representation, which allows for the extraction of the charge radius of the pion. As a further application, we discuss the long-distance behavior of the vector correlator, which is dominated by the two-pion channel. We reconstruct the long-distance part in two ways: one based on the finite-volume energies and matrix elements, and the other based on $F_{\pi }$. It is shown that this part can be accurately constrained using the reconstructions, which has important consequences for lattice calculations of the hadronic vacuum polarization contribution to the muon anomalous magnetic moment.

Figure 1

Spectra from the GEVP on the F6 lattice, using the window method, for two irreps: ${\mathit{d}}^2=1,E$ on the left (a typical example for the energy levels we extract) and ${\mathit{d}}^2=2,A_1$ on the right (an example where the plateau does not look as good, particularly the intermediate level). The different levels in the respective irrep are plotted using different colors and the accordingly colored bands are the fit results of the corresponding eigenvalues to fit function allowing for the ground state and an excited state. The width of those bands indicates the statistical error of the fit, and the length shows the chosen fit range. The horizontal lines are the free two-pion levels in the respective moving frame.

Figure 2

Phase shifts on all three ensembles using the window method. The horizontal axis shows the CMF energy of each level and data points of the same color and symbol belong to the same frame and irrep. Error bars follow the curves allowed by the Lüscher zeta functions. The red vertical line indicates the $4m_{\pi }$ threshold in each system; data points above are excluded from the fit and thus shown in grey. The black line is the result of the Breit-Wigner fit to our data by minimizing the ${\chi }^2$ functional defined in (3.5). The ${\chi }^2/\mathrm{d}.\mathrm{o}.\mathrm{f}.$ for each fit is shown in the plots.

Figure 3

Overview of lattice results for the coupling $g_{\rho \pi \pi }$ as a function of the pion mass in the calculation. The upper pane shows the results from simulations with dynamical light and strange quarks [6, 7, 11, 12, 14, 15, 16, 18, 19, 22], while the lower pane shows results with dynamical light quarks only [8, 9, 10, 13, 17]. Where available, the scale-setting uncertainty provided by the authors has been added in quadrature to obtain the errors on the horizontal axis. The value extracted from the physical $\rho$-meson width is indicated by the magenta star and the black dashed line. The results from this work are the red open squares in the lower pane. With the exception of [22] which quotes result from a modified Gounaris-Sakurai parametrization, the plotted couplings refer to the Breit-Wigner parametrization.

Figure 4

The timelike pion form factor on the E5, F6, F7 ensembles (top to bottom). Data points with the same symbol and color belong to the same frame and irreps. The error bars associated with each data point come from a jackknife estimate. The grey curve is the GS representation of $F_{\pi }$, which only takes the fit parameters of the phase-shift fit $m_{\rho },g_{\rho \pi \pi }$ into account—it is not a fit to the data pictured in these plots. The vertical red bars indicate the $4m_{\pi }$ threshold for each lattice.

Figure 5

Left panel: The timelike pion form factor on the E5, F6, F7 lattice (top to bottom), window method. Data points with the same symbol and color belong to the same frame and irreps. The orange curve is the fit to $F_{\pi }$, parametrized via the three-subtracted version of Eq. (4.6). The vertical red bars indicate the $4m_{\pi }$ threshold in each lattice and data points above this threshold have not been included in the fit and are shown in grey for this reason. Right panel: The data which we are actually fitting to. The vertical axis shows $F_{\pi }$ divided by the Omnès integral, i.e., the analytically calculable part of Eq. (4.6), and the fit function is $f(s)=\exp (Ps)$, where $P$ is a first-order polynomial. The vertical axis is displayed on a log scale and the orange curve is the fit function with the jackknife error. Also shown are the ${\chi }^2/\mathrm{d}.\mathrm{o}.\mathrm{f}.$ values of the respective fits, which are quite high.

Figure 6

Left panel: Pion mass dependence of the square radius $⟨r_{\pi }^2⟩$. Right panel: The same for the curvature $c_v$. Our lattice results are compared to the determinations in Refs. [67] and [68], respectively. The inner error bar on the lattice data denotes the statistical uncertainty, while the outer error bar includes the scale setting uncertainty from the conversion to physical units.

Figure 7

The light quark contribution to the integrand of Eq. (5.1) for ensembles E5, F6, F7 (top to bottom). The data points computed in [58] are plotted as black filled circles up to $x_0^{\mathrm{cut}}$ and as open circles above the cut. The bands represent the continuation of the correlator above $x_0^{\mathrm{cut}}$ as discussed in Ref. [58]. Colored symbols denote the data from this work using the reconstructed light-quark correlator $G_{n_{\max }}^{ud}$ from Eq. (5.5) for different values of $n_{\max }$. The vertical lines indicate the value of $x_0^{\mathrm{cut}}$.

Figure 8

The integrand of Eq. (5.1) for ensembles E5, F6, F7 (top to bottom). The meaning of the black filled and open circles is the same as in Fig. 7. Blue triangles represent the integrand reconstructed from the isovector correlator $G_{n_{\max }}^{ud}$ of Eq. (5.5) for the corresponding value of $n_{\max }$. Data corresponding to the isovector correlator constructed from the GS parametrization and the three-subtracted Omnès representation of $F_{\pi }$ are shown as red and blue bands, respectively. The different types of extending the correlator above $x_0^{\mathrm{cut}}$ are used to compute the results for $a_{\mu }^{\mathrm{hvp}}$ presented in Table 9. The difference between the two- and three-subtracted versions of the Omnès representation integral is too small to be seen on this plot.

Figure 9

Pion mass dependence of $a_{\mu }^{\mathrm{hvp}}$ at $\beta =5.3$. Red triangles correspond to the data computed in Ref. [58] with the chiral extrapolation at nonzero lattice spacing represented by the band. The blue circles denote the data points determined from the large-$x_0$ tail of our most precise reconstruction of the correlator, which is the $G_{n_{\max }}^{ud}$ correlator using the matrix elements $|A|$ as an input. Points are slightly shifted for clarity. The leftmost red triangle corresponds to ensemble G8 ($m_{\pi }=185\mathrm{MeV}$), which was not considered in this work. Our determination of the tail of the isovector correlator allows for a significantly more precise determination of $a_{\mu }^{\mathrm{hvp}}$ compared to Ref. [58]. Furthermore, we find that the result for F7 moves closer to the curve when our reconstruction of the large-distance tail is used.