Studying the $\rho$ resonance parameters with staggered fermions

We deliver a lattice study of $\rho$ resonance parameters with $p$-wave $\pi \pi$ scattering phases, which are extracted by finite-size methods at one center-of-mass frame and four moving frames for six lattice ensembles from the MILC Collaboration with pion masses ranging from 346 to 176 MeV. The effective range formula is applied to describe the scattering phases as a function of the energy covering the resonance region; this allows us to extract $\rho$ resonance parameters and to investigate the quark-mass dependence. Lattice studies with three flavors of Asqtad-improved staggered fermions enable us to use the moving-wall source technique on large lattice spatial dimensions ($L=64$) and small light $u/d$ quarks. Numerical computations are carried out at two lattice spacings, $a\approx 0.12$ and 0.09 fm.

Figure 1

Quark-link diagrams contributing to the $I=1$ $\pi \pi$ four-point functions: (a) $D$; (b) $X$; (c) $R_8$; (d) $R_8^{\prime }$; (e) $R$; (f) $R^{\prime }$. Short black bars stand for the wall sources. Open circles are sinks for local pion operators. The time flows upward in the diagrams. The pion operators are given with a momentum specified in the diagram.

Figure 2

Quark-link diagrams contributing to $\pi \pi \to \rho$ and $\rho \to \pi \pi$ three-point correlation functions. Short black bars indicate the wall sources. (a) Quark contractions of $\pi \pi \to \rho$, where the open circle is the sink for $\rho$ operator. (b) Quark contractions of $\rho \to \pi \pi$, where the open circle is the sink for pion operator.

Figure 3

Effective pion mass $m_{\pi }$ or energy $E_{\pi }(\mathbf{p})$ plots as the functions of ${\mathrm{D}}_{\min }$ for the (0.005,0.005) ensemble. The plateaus are quickly reached typically at or before $t=8$ from the source.

Figure 4

Ratios of the wraparound pollution to the $\pi \pi$ correlators of the direct diagram for the (0.005,0.005) ensemble using Eq. (25) for four momenta, $\mathbf{p}=[0,0,1]$, [1,1,0], [1,1,1], and [0,0,2], along with the ratio for the center-of-mass frame at $\mathbf{p}=[0,0,1]$. These ratios are generally near $1/2$ as $t$ approaches $T/2$, as anticipated from the analytical statements in Refs. [44, 45, 46, 47].

Figure 5

Fitted energy levels as functions of $t_{\min }$ for the ground states of the ${\mathbf{T}}_1^{-}$ representation in CMF, and for the ground and first excited states of the ${\mathbf{A}}_2^{-}$, ${\mathbf{B}}_1^{-}$, ${\mathbf{A}}_2^{-}$, and ${\mathbf{A}}_2^{-}$ representations for MF1, MF2, MF3, and MF4, respectively, with the (0.005,0.005) ensemble.

Figure 6

Results of the scattering phase shifts and effective range formula fits for the ensembles (0.0031,0.031) (upper left), (0.0031,0.0031) (upper right), (0.00465,0.0031) (middle left), (0.0062,0.031) (middle right), (0.005,0.005) (bottom left), and (0.005,0.05) (bottom right). The scattering phase shifts are calculated in the CMF, MF1, MF2, MF3, and MF4, respectively. The solid black curves exhibit the central values of the effective range formula fits. The dashed cyan lines display the resonance masses $a{m}_{\rho }$, the narrow yellow bands display their uncertainties, and the resonance regions $a{m}_{\rho }\pm a\mathrm{\Gamma }$ are shown in the shadowed grey boxes.

Figure 7

Comparison of our results with other lattice studies. The top panel exhibits the effective coupling constant $g_{\rho \pi \pi }$ as a function of $m_{\pi }^2$. The bottom panel shows the resonance mass $m_{\rho }$, where $m_{\rho }$ and $m_{\pi }$ are scaled with the Sommer scale $r_0$ [55]. The red plus points indicate the corresponding PDG values.

Figure 8

The lattice-measured $\rho$ resonance parameters as the functions of the pion mass squared. The upper panel exhibits the $\rho$-meson resonance mass and the lower panel shows the $\rho$-meson decay width. The cyan bands correspond to the fits to our six data points using the Eq. (30) and Eq. (31), respectively, and the solid black curves are the central values of the corresponding fits. The red plus points indicate the relevant PDG values.