Quarkonium at nonzero isospin density

Abstract

We calculate the energies of quarkonium bound states in the presence of a medium of nonzero isospin density using lattice QCD. The medium, created using a canonical (fixed isospin charge) approach, induces a reduction of the quarkonium energies. As the isospin density increases, the energy shifts first increase and then saturate. The saturation occurs at an isospin density close to that where previously a qualitative change in the behaviour of the energy density of the medium has been observed, which was conjectured to correspond to a transition from a pion gas to a Bose-Einstein condensed phase. The reduction of the quarkonium energies becomes more pronounced as the heavy-quark mass is decreased, similar to the behaviour seen in two-colour QCD at non-zero quark chemical potential. In the process of our analysis, the $\eta_b$-$\pi$ and $\Upsilon$-$\pi$ scattering phase shifts are determined at low momentum. An interpolation of the scattering lengths to the physical pion mass gives $a_{\eta_b,\pi} = 0.0025(8)(6)$ fm and $a_{\Upsilon,\pi} = 0.0030(9)(7)$ fm.