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

Title: 在SU(3)格点规范理论中，从$(1 /m_Q)^1$阶的混合自旋依赖和混合夸克胶子势

Abstract: 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. Show Chinese abstract

Abstract: 我们首次给出了在杂化介子的Born-Oppenheimer有效场理论中，出现于$(1 /m_Q)^1$阶的混合自旋相关和混合夸克onium混合势的结果，该理论涉及胶子自旋$\kappa^{PC} = 1^{+-}$。具体来说，我们计算了四个未知势$V^{sa}_{11}(r)$、$V^{sb}_{10}(r)$，这些势对于重杂化介子谱中的超精细分裂是相关的，以及$V^\text{mix}_{\Sigma_u^-}(r)$和$V^\text{mix}_{\Pi_u}(r)$，这些势描述了重杂化介子与普通夸克onium的混合。我们将这些势与矩阵元相关联，这些矩阵元通过在时间线上插入色磁场所得的广义威尔逊环以及用合适的混合产生算符代替空间线来提取。我们使用梯度流，这有助于矩阵元的重整化，并且额外的好处是显著减少了统计噪声。我们给出了规范群SU(3)和格点间距$a=0.060$飞米的结果。我们的结果表明，在我们的设置中，未来可以进行连续体和零流时间外推的结合，这将是有必要通过耦合通道薛定谔方程可靠地预测重杂化介子的超精细分裂及其与普通夸克onium的混合。