Anatomy of the $ρ$ resonance from lattice QCD at the physical point

Abstract

We propose a strategy to access the $q\bar{q}$ component of the $ρ$ resonance in lattice QCD. Through a mixed action formalism (overlap valence on domain wall sea), the energy of the $q\bar{q}$ component is derived at different valence quark masses, and shows a linear dependence on $m_π^2$. The slope is determined to be $c_1=0.505(3)\,{\rm GeV}^{-1}$, from which the valence $πρ$ sigma term is extracted to be $σ_{πρ}^{(\rm val)}=9.82(6)$ MeV using the Feynman-Hellman theorem. At the physical pion mass, the mass of the $q\bar{q}$ component is interpolated to be $m_ρ=775.9\pm 6.0\pm 1.8$ MeV, which is close to the $ρ$ resonance mass. We also obtain the leptonic decay constant of the $q\bar{q}$ component to be $f_{ρ^-}=208.5\pm 5.5\pm 0.9$ MeV, which can be compared with the experimental value $f_ρ^{\rm exp}\approx 221$ MeV through the relation $f_ρ^{\rm exp}=\sqrt{Z_ρ}f_{ρ^\pm} $ with $Z_ρ\approx 1.13$ being the on-shell wavefunction renormalization of $ρ$ owing to the $ρ-π$ interaction. We emphasize that $m_ρ$ and $f_ρ$ of the $q\bar{q}$ component, which are obtained for the first time from QCD, can be taken as the input parameters of $ρ$ in effective field theory studies where $ρ$ acts as a fundamental degree of freedom.