$\rho$ meson decay in $2\mathbf{+}1$ flavor lattice QCD

We perform a lattice QCD study of the $\rho$ meson decay from the $N_f=2+1$ full QCD configurations generated with a renormalization group improved gauge action and a nonperturbatively $O(a)$-improved Wilson fermion action. The resonance parameters, the effective $\rho \to \pi \pi$ coupling constant and the resonance mass, are estimated from the $P$-wave scattering phase shift for the isospin $I=1$ two-pion system. The finite size formulas are employed to calculate the phase shift from the energy on the lattice. Our calculations are carried out at two quark masses, $m_{\pi }=410\mathrm{MeV}$ ($m_{\pi }/m_{\rho }=0.46$) and $m_{\pi }=300\mathrm{MeV}$ ($m_{\pi }/m_{\rho }=0.35$), on a ${32}^3\times 64$ ($La=2.9\mathrm{fm}$) lattice at the lattice spacing $a=0.091\mathrm{fm}$. We compare our results at these two quark masses with those given in the previous works using $N_f=2$ full QCD configurations and the experiment.

Figure 1

$\sqrt{s}/m_{\rho }$ for states tabulated in Table 1 on our gauge configurations at $m_{\pi }=410\mathrm{MeV}$ (upper panel) and $m_{\pi }=300\mathrm{MeV}$ (lower panel).

Figure 2

Quark contractions of $\pi \pi \to \pi \pi$, $\pi \pi \to \rho$ and $\rho \to \pi \pi$ components of the matrix of the time correlation function $G(t)$. The time runs upward in the diagrams. The vertices refer to the pion or the $\rho$ meson operator with the momentum at the time slice specified in the diagrams. The momentum $\mathbf{p}$ takes $\mathbf{p}=(2\pi /L)(0,0,1)$ for the ${\mathbf{A}}_2^{-}$ and $\mathbf{p}=(2\pi /L)(1,1,0)$ for the ${\mathbf{B}}_1^{-}$ representation.

Figure 3

Four components of the matrix of the time correlation function $G(t)$ at $m_{\pi }=410\mathrm{MeV}$ for the ${\mathbf{A}}_2^{-}$ (upper panel) and for the ${\mathbf{B}}_1^{-}$ representation (lower panel). Same symbols for the components are used in both panels. The source operators $\overline{(\pi \pi )}(t_s)$ and $\overline{\rho }(t_s)$ are located at $t_s=12$. The pion with the zero momentum in the sink operator $(\pi \pi )(t)$ in (10) is set at $t_1=42$.

Figure 4

Effective masses of the ground state for the ${\mathbf{T}}_1^{-}$ and the ${\mathbf{E}}^{-}$ representation, and the ground and first excited states for the ${\mathbf{A}}_2^{-}$ and ${\mathbf{B}}_1^{-}$ representation at $m_{\pi }=410\mathrm{MeV}$. The source operators $\overline{(\pi \pi )}(t_s)$ and $\overline{\rho }(t_s)$ are located at $t_s=12$. For the ${\mathbf{A}}_2^{-}$ and ${\mathbf{B}}_1^{-}$ representation, we set the pion with the zero momentum in the sink operator $(\pi \pi )(t)$ in (10) at $t_1=42$ and the reference time of the variational method at $t_R=23$. The results of the fitting with 1 standard deviation error band are expressed by solid lines. The dotted lines for the the ${\mathbf{A}}_2^{-}$ and ${\mathbf{B}}_1^{-}$ representation indicate the energies of the two free pions.

Figure 5

Same as Fig. 3 for $m_{\pi }=300\mathrm{MeV}$.

Figure 6

Same as Fig. 4 for $m_{\pi }=300\mathrm{MeV}$.

Figure 7

$k^3/(\sqrt{s}·\tan \delta (k))$ as a function of square of the invariant mass $s$ at $m_{\pi }=410\mathrm{MeV}$ (upper panel) and $m_{\pi }=300\mathrm{MeV}$ (lower panel). Same symbols for four representations are used in both panels. Dotted lines are the finite size formulas for the ${\mathbf{A}}_2^{-}$ and the ${\mathbf{B}}_1^{-}$ representation. A solid line for each quark mass is a fitting line by (30).

Figure 8

Comparison of our results (PACS-CS) obtained in $2+1$ flavor QCD with those by ETMC and Lang et al. in 2 flavor QCD. Upper panel shows the effective coupling constant $g_{\rho \pi \pi }$ and lower is the resonance mass $m_{\rho }$. The systematic uncertainty for the determination of the lattice spacing is added to the statistical error in quadrature.

Figure 9

Comparison of our results (PACS-CS) with those by ETMC and Lang et al. for dimensionless value $r_0{m}_{\rho }$ as a function of $(r_0{m}_{\pi }{)}^2$ with the Sommer scale $r_0$. The error of $r_0$ is added to the statistical error in quadrature.