Hybrid spin-dependent and hybrid-quarkonium mixing potentials at order $(1 /m_Q)^1$ from SU(3) lattice gauge theory

We present the first lattice gauge theory results for hybrid spin-dependent and hybrid-quarkonium mixing potentials appearing at order $(1 /m_Q)^1$ in the Born-Oppenheimer Effective Field theory for hybrid mesons with gluon spin $\kappa^{PC} = 1^{+-}$. Specifically, we compute the four unknown potentials $V^{sa}_{11}(r)$, $V^{sb}_{10}(r)$, which are relevant for the hyperfine splitting in heavy hybrid meson spectra, as well as $V^\text{mix}_{\Sigma_u^-}(r)$ and $V^\text{mix}_{\Pi_u}(r)$, which describe the mixing of heavy hybrid mesons with ordinary quarkonium. We relate these potentials to matrix elements, which we extract from generalized Wilson loops with a chromomagnetic field insertion along one of the temporal lines and suitable hybrid creation operators replacing the spatial lines. We use gradient flow, which facilitates the renormalization of the matrix elements and has the additional benefit of significantly reducing statistical noise. We present results for gauge group SU(3) and lattice spacing $a=0.060$ fm. Our results demonstrate that a future combined continuum and zero-flow time extrapolation is possible within our setup, which will be necessary to reliably predict the hyperfine splitting of heavy hybrid mesons as well as their mixing with ordinary quarkonium through coupled channel Schr\"odinger equations.