Precision computation of hybrid static potentials in SU(3) lattice gauge theory

We perform a precision computation of hybrid static potentials with quantum numbers ${\mathrm{\Lambda }}_{\eta }^{\epsilon }={\mathrm{\Sigma }}_g^{-},{\mathrm{\Sigma }}_u^{+},{\mathrm{\Sigma }}_u^{-},{\mathrm{\Pi }}_g,{\mathrm{\Pi }}_u,{\mathrm{\Delta }}_g,{\mathrm{\Delta }}_u$ using SU(3) lattice gauge theory. The resulting potentials are used to estimate masses of heavy $\overline{c}c$ and $\overline{b}b$ hybrid mesons in the Born-Oppenheimer approximation. Part of the lattice gauge theory computation, which we discuss in detail, is an extensive optimization of hybrid static potential creation operators. The resulting optimized operators are expected to be essential for future projects concerning the computation of three-point functions as e.g., needed to study spin corrections, decays or the gluon distribution of heavy hybrid mesons.

Figure 1

Terms appearing in the construction of the trial state via Eq. (7) for an exemplary operator $S(-r/2,+r/2)$ (top left). The columns correspond to rotations of the operator around the separation axis, while the rows correspond to applications of $\mathcal{P}\circ \mathcal{C}$ and ${\mathcal{P}}_x$. Red lines represent gauge link variables, red spheres the quark and the antiquark and black dots lattice sites.

Figure 2

Operators $S$ used to generate trial states $|{\mathrm{\Psi }}_{\text{hybrid}}{⟩}_{S;{\mathrm{\Lambda }}_{\eta }^{\epsilon }}$ according to Eqs. (7)–(9). Each arrow represents a straight path of gauge links. Arrows with the same color (red, green or blue) have the same length, i.e., represent the same number of gauge links. Dotted arrows can have length zero, while solid arrows represent at least one gauge link. If the starting point and the end point of $S$ are marked by black dots, $U(-r/2,r_1)$ and $U(r_2,+r/2)$ in Eq. (2) have the same length, i.e., $r_1-(-r/2)=+r/2-r_2$, else their length can be different. $n\in \{1,2,4\}$ is the number of differently oriented operators (i.e., operators, which cannot be transformed into each other by rotations around the $z$ axis) obtained by applying $\mathcal{P}\circ \mathcal{C}$, ${\mathcal{P}}_x$ and $(\mathcal{P}\circ \mathcal{C}){\mathcal{P}}_x$.

Figure 3

Investigation of the dependence of $V_{\mathrm{eff};S_{\mathrm{I},1}^{E_x,E_z=r/a};{\mathrm{\Pi }}_u}(r,t=a)$ on $E_x$. Red spheres, red arrows, and black dots represent quarks, gauge links and lattice sites, respectively.

Figure 4

Investigation of the dependence of $V_{\mathrm{eff};S_{\mathrm{I},1}^{E_x,E_z};{\mathrm{\Pi }}_u}(r,t=a)$ on $E_z$ ($E_x\in \{2,3\}$, the optimum according to Fig. 3). Red spheres, red arrows, and black dots represent quarks, single gauge links and lattice sites, respectively.

Figure 5

Investigation of the dependence of $V_{\mathrm{eff};S_{\mathrm{I},1}^{1,r/a};{\mathrm{\Pi }}_u}(r,t=a)$ on the number of APE smearing steps $N_{\mathrm{APE}}$.

Figure 6

Effective potentials $V_{\mathrm{eff};{\mathrm{\Lambda }}_{\eta }^{\epsilon }}^{(0)}(r,t,t_0=a)a$, ${\mathrm{\Lambda }}_{\eta }^{\epsilon }={\mathrm{\Sigma }}_g^{+},{\mathrm{\Sigma }}_g^{-},{\mathrm{\Sigma }}_u^{+},{\mathrm{\Sigma }}_u^{-},{\mathrm{\Pi }}_g,{\mathrm{\Pi }}_u,{\mathrm{\Delta }}_g,{\mathrm{\Delta }}_u$ and $V_{\mathrm{eff};{\mathrm{\Sigma }}_g^{+}}^{(1)}(r,t,t_0=a)a$ as functions of $t/a$ together with the corresponding plateau fits for separations $r/a=2$, 5, 8. To allow a straightforward comparison of different sectors, the vertical scale is the same for all nine plots.

Figure 7

The ordinary static potential $V_{{\mathrm{\Sigma }}_g^{+}}(r)r_0$ and the corresponding first excitation $V_{{\mathrm{\Sigma }}_g^{+}}^{\prime }(r)r_0$ as well as the hybrid static potentials $V_{{\mathrm{\Lambda }}_{\eta }^{\epsilon }}(r)r_0$, ${\mathrm{\Lambda }}_{\eta }^{\epsilon }={\mathrm{\Sigma }}_g^{-},{\mathrm{\Sigma }}_u^{+},{\mathrm{\Sigma }}_u^{-},{\mathrm{\Pi }}_g,{\mathrm{\Pi }}_u,{\mathrm{\Delta }}_g,{\mathrm{\Delta }}_u$ as functions of the separation $r/r_0$, where $r_0=0.5\mathrm{fm}$. To allow a straightforward comparison with results from the literature, e.g., with [13, 23], the vertical scale has been shifted by an additive constant such that $V_{{\mathrm{\Sigma }}_g^{+}}(2r_0)=0$.

Figure 8

Parametrizations of lattice field theory results. (left) ${\mathrm{\Sigma }}_g^{+}$ static potential [Eq. (16) with parameters (23)]. (right) ${\mathrm{\Pi }}_u$ and ${\mathrm{\Sigma }}_u^{-}$ hybrid static potentials [Eq. (21) and (22) with parameters (26)]. Vertical dotted lines indicate the data points, which have been considered in the ${\chi }^2$ minimizing fits.

Figure 9

Predictions for heavy hybrid meson masses. (left) ${\mathrm{\Pi }}_u$ hybrid static potential. (right) ${\mathrm{\Sigma }}_u^{-}$ hybrid static potential.

Figure 10

Probability density for the separation $r$. (left) ${\mathrm{\Pi }}_u$ hybrid static potential. (right) ${\mathrm{\Sigma }}_u^{-}$ hybrid static potential.