Abstract
We compute the ${\mathrm{\ensuremath{\Pi}}}_{u}$ and ${\mathrm{\ensuremath{\Sigma}}}_{u}^{\ensuremath{-}}$ hybrid static potentials in SU(3) lattice gauge theory using four different lattice spacings ranging from $a=0.040\text{ }\text{ }\mathrm{fm}$ to $a=0.093\text{ }\text{ }\mathrm{fm}$. We provide lattice data points for quark-antiquark separations as small as 0.08 fm, where the $a$-dependent self-energy as well as lattice discretization errors at tree level of perturbation theory and at leading order in ${a}^{2}$ have been removed. We also investigate and exclude possibly present systematic errors from topological freezing, due to the finite spatial lattice volume and from glueball decays. Moreover, we provide corresponding parametrizations of the potentials, which can e.g. be used for Born-Oppenheimer predictions of heavy hybrid mesons.
Highlights
The constituent quark model is quite successful in explaining the properties of a variety of nonexotic hadrons, quark-antiquark pairs or triplets of quarks or antiquarks without gluonic excitations
The gluonic excitation contributes to the quantum numbers of the hybrid meson such that exotic combinations of JPC are allowed, which do not exist in the constituent quark model
In recent years a lot of effort was invested to refine the second step of the Born-Oppenheimer approximation, e.g. by including the mixing of different sectors via coupled channel equations [11–13] and by taking heavy quark spin effects into account [15,16]. These approaches require precise lattice results for hybrid static potentials, in particular at small quark-antiquark separations r to combine them with perturbative predictions valid only at small r or to fix matching coefficients in potential nonrelativistic QCD [12,15,16,22]
Summary
The constituent quark model is quite successful in explaining the properties of a variety of nonexotic hadrons, quark-antiquark pairs or triplets of quarks or antiquarks without gluonic excitations. In recent years a lot of effort was invested to refine the second step of the Born-Oppenheimer approximation, e.g. by including the mixing of different sectors via coupled channel equations [11–13] and by taking heavy quark spin effects into account [15,16] These approaches require precise lattice results for hybrid static potentials, in particular at small quark-antiquark separations r to combine them with perturbative predictions valid only at small r or to fix matching coefficients in potential nonrelativistic QCD (pNRQCD) [12,15,16,22]. To compute the ordinary static potential and the Πu and Σ−u hybrid static potentials we employ optimized operators from our previous work [14] as well as a multilevel algorithm [47] In this way we obtain precise lattice results for these potentials on four ensembles for quark-antiquark separations as small as 0.08 fm (see Sec. II–IV). We provide similar results for gauge group SU(2), which were obtained at an early stage of this work
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