Speaker
Description
The $\rho$ resonance appears as a pole in the two-pion scattering amplitude with quantum numbers $J^{PC}=1^{--}$. Our aim is to determine the properties of the $\rho$ resonance from first principles of quantum chromodynamics using the lattice QCD methodology. On four lattices with lattice spacings ranging from $a=0.114$ fm to $a=0.068$ fm and pion masses ranging from $m_\pi=317$ MeV to $m_\pi=175$ MeV, we simulated gauge fields and quark propagation. From the quark propagators, we constructed two-point correlation functions describing two-pion scattering on the lattice. We projected these onto the irreducible representations of the cubic lattice symmetry group, $\Lambda$, and used them to determine the energy spectrum for all four lattice ensembles. We then fitted the scattering amplitude to this energy spectrum using the quantization condition $ c_{\Lambda,lm} \left[ T_{lm,lm}^{-1}(E) + F_{lm,lm}(E) \right] c^\dagger_{lm,\Lambda}=0.$ Here, we project the scattering amplitude $T$ from the space of spherical harmonics $l,m$ onto the space of irreducible representations $\Lambda$ of the cubic lattice symmetry group.
In fitting the scattering amplitude, we parametrized the dependence of the energy spectra on the lattice spacing, $E(a)$, as well as the dependence of the scattering amplitude on the quark masses, $T(m_\pi,m_{\eta_s})$, where the dependence on the light-quark masses $u,d,s$ is incorporated indirectly through the masses of the $\pi$ and $\eta_s$ mesons. This allows us to extrapolate $T$ to the continuum limit, $a=0$, and to the physical quark masses. From the extrapolated scattering amplitude, we calculated the phase shift $\delta(E)$ for two-pion scattering and determined the pole in the amplitude corresponding to the $\rho$ resonance.