Hybrid static potential flux tubes from SU(2) and SU(3) lattice gauge theory

We compute chromoelectric and chromomagnetic flux densities for hybrid static potentials in SU(2) and SU(3) lattice gauge theory. In addition to the ordinary static potential with quantum numbers ${\mathrm{\Lambda }}_{\eta }^{\epsilon }={\mathrm{\Sigma }}_g^{+}$, we present numerical results for seven hybrid static potentials corresponding to ${\mathrm{\Lambda }}_{\eta }^{(\epsilon )}={\mathrm{\Sigma }}_u^{+},{\mathrm{\Sigma }}_g^{-},{\mathrm{\Sigma }}_u^{-},{\mathrm{\Pi }}_g,{\mathrm{\Pi }}_u,{\mathrm{\Delta }}_g,{\mathrm{\Delta }}_u$, where the flux densities of five of them are studied for the first time in this work. We observe hybrid static potential flux tubes, which are significantly different from that of the ordinary static potential. They are reminiscent of vibrating strings, with localized peaks in the flux densities that can be interpreted as valence gluons.

Figure 1

Graphical illustration of lattice field theory quantities needed to determine $\mathrm{\Delta }{E}_{j,{\mathrm{\Lambda }}_{\eta }^{\epsilon }}^2(r;\mathbf{x})$ and $\mathrm{\Delta }{B}_{j,{\mathrm{\Lambda }}_{\eta }^{\epsilon }}^2(r;\mathbf{x})$. Red spheres, black dots, black solid lines, and black dashed lines represent static (anti)quarks, lattice sites, gauge links, and operators $a_{S;{\mathrm{\Lambda }}_{\eta }^{\epsilon }}$, respectively. (a) $\tilde{W}(r,t_2,t_0)·P_{0x}(\mathbf{x},t_1)$ for $r=3a$, $t_2-t_0=4a$, $t_1=(t_2-t_0)/2$ (gray dashed lines parallel to the $t$ axis and the $z$ axis are drawn to guide the eye). (b) The corresponding Wilson loop $\tilde{W}(r,t_2,t_0)$. (c) The symmetrized plaquette $P_{\mu \nu }$.

Figure 2

$\mathrm{\Delta }{F}_{\mathrm{eff};j,{\mathrm{\Lambda }}_{\eta }^{\epsilon }}^2(r,t_2,t_0;\mathbf{x}=\overrightarrow{0},t_1)$ as a function of $t_2-t_0$ for gauge group SU(2); ${\mathrm{\Lambda }}_{\eta }^{\epsilon }={\mathrm{\Sigma }}_g^{+},{\mathrm{\Sigma }}_u^{+},{\mathrm{\Sigma }}_g^{-},{\mathrm{\Sigma }}_u^{-},{\mathrm{\Pi }}_g,{\mathrm{\Pi }}_u,{\mathrm{\Delta }}_g,{\mathrm{\Delta }}_u$; and $Q\overline{Q}$ separation $r=10a$. Plateau fits and fitting ranges $[t_{\min },t_{\max }]$ are indicated by horizontal straight lines. (Top) Temporal links in $\tilde{W}$ are unsmeared. (Bottom) Temporal links in $\tilde{W}$ are HYP2 smeared.

Figure 3

Flux densities $\mathrm{\Delta }{F}_{j,{\mathrm{\Lambda }}_{\eta }^{\epsilon }}^2(r;\mathbf{x}=(0,0,z))$ as a function of $z$ for gauge group SU(3), ${\mathrm{\Lambda }}_{\eta }^{\epsilon }={\mathrm{\Sigma }}_g^{+},{\mathrm{\Sigma }}_u^{-}$ and $Q\overline{Q}$ separation $r=10a$.

Figure 4

Flux densities $\mathrm{\Delta }{F}_{j,{\mathrm{\Lambda }}_{\eta }^{\epsilon }}^2(r;\mathbf{x}=\overrightarrow{0})$ as a function of the $Q\overline{Q}$ separation $r$ for gauge group SU(3) and ${\mathrm{\Lambda }}_{\eta }^{\epsilon }={\mathrm{\Sigma }}_g^{+},{\mathrm{\Sigma }}_u^{-}$.

Figure 5

Flux densities $\mathrm{\Delta }{F}_{j,{\mathrm{\Pi }}_u^{+}}^2(r;\mathbf{x})$, $\mathrm{\Delta }{F}_{j,{\mathrm{\Pi }}_u^{-}}^2(r;\mathbf{x})$, and $\mathrm{\Delta }{F}_{j,{\mathrm{\Pi }}_u}^2(r;\mathbf{x})$ in the mediator plane ($z=0$) for gauge group SU(2) and $Q\overline{Q}$ separation $r=10a$.

Figure 6

Flux densities $\mathrm{\Delta }{F}_{j,{\mathrm{\Delta }}_g^{+}}^2(r;\mathbf{x})$, $\mathrm{\Delta }{F}_{j,{\mathrm{\Delta }}_g^{-}}^2(r;\mathbf{x})$, and $\mathrm{\Delta }{F}_{j,{\mathrm{\Delta }}_g}^2(r;\mathbf{x})$ in the mediator plane ($z=0$) for gauge group SU(2) and $Q\overline{Q}$ separation $r=10a$.

Figure 7

Flux densities $\mathrm{\Delta }{F}_{j,{\mathrm{\Lambda }}_{\eta }^{(\epsilon )}}^2(r;\mathbf{x}=(x,y,0))$, $j=x,y,z$ in the mediator plane for gauge group SU(2), all investigated sectors ${\mathrm{\Lambda }}_{\eta }^{(\epsilon )}={\mathrm{\Sigma }}_g^{+},{\mathrm{\Sigma }}_u^{+},{\mathrm{\Sigma }}_g^{-},{\mathrm{\Sigma }}_u^{-},{\mathrm{\Pi }}_g,{\mathrm{\Pi }}_u,{\mathrm{\Delta }}_g,{\mathrm{\Delta }}_u$ and $Q\overline{Q}$ separation $r=10a$.

Figure 8

Flux densities on the $x$ axis for gauge group SU(2), all investigated sectors ${\mathrm{\Lambda }}_{\eta }^{(\epsilon )}={\mathrm{\Sigma }}_g^{+},{\mathrm{\Sigma }}_u^{+},{\mathrm{\Sigma }}_g^{-},{\mathrm{\Sigma }}_u^{-},{\mathrm{\Pi }}_g,{\mathrm{\Pi }}_u,{\mathrm{\Delta }}_g,{\mathrm{\Delta }}_u$ and $Q\overline{Q}$ separation $r=10a$. (Top) $\mathrm{\Delta }{F}_{j,{\mathrm{\Lambda }}_{\eta }^{(\epsilon )}}^2(r;\mathbf{x}=(x,0,0))$, $j=\perp ,z$. (Bottom) $\mathrm{\Delta }{F}_{j,{\mathrm{\Lambda }}_{\eta }^{(\epsilon )}}^2(r;\mathbf{x}=(x,0,0))-\mathrm{\Delta }{F}_{j,{\mathrm{\Sigma }}_g^{+}}^2(r;\mathbf{x}=(x,0,0))$, $j=\perp ,z$.

Figure 9

Flux densities $\mathrm{\Delta }{F}_{j,{\mathrm{\Lambda }}_{\eta }^{\epsilon }}^2(r;\mathbf{x}=(x,0,z))$, $j=x,y,z$ in the separation plane for gauge group SU(2) and sectors ${\mathrm{\Lambda }}_{\eta }^{\epsilon }={\mathrm{\Sigma }}_g^{+},{\mathrm{\Sigma }}_u^{+},{\mathrm{\Sigma }}_g^{-},{\mathrm{\Sigma }}_u^{-}$. (Left) $Q\overline{Q}$ separation $r=6a$. (Right) $Q\overline{Q}$ separation $r=10a$.

Figure 10

Flux densities $\mathrm{\Delta }{F}_{j,{\mathrm{\Lambda }}_{\eta }}^2(r;\mathbf{x}=(x,0,z))$, $j=x,y,z$ in the separation plane for gauge group SU(2) and sectors ${\mathrm{\Lambda }}_{\eta }={\mathrm{\Pi }}_g,{\mathrm{\Pi }}_u,{\mathrm{\Delta }}_g,{\mathrm{\Delta }}_u$. (Left) $Q\overline{Q}$ separation $r=6a$. (Right) $Q\overline{Q}$ separation $r=10a$.

Figure 11

Flux densities $\mathrm{\Delta }{F}_{j,{\mathrm{\Lambda }}_{\eta }^{(\epsilon )}}^2(r;\mathbf{x}=(x,y,0))$, $j=x,y,z$ in the mediator plane for gauge group SU(3); all investigated sectors ${\mathrm{\Lambda }}_{\eta }^{(\epsilon )}={\mathrm{\Sigma }}_g^{+},{\mathrm{\Sigma }}_u^{+},{\mathrm{\Sigma }}_g^{-},{\mathrm{\Sigma }}_u^{-},{\mathrm{\Pi }}_g,{\mathrm{\Pi }}_u,{\mathrm{\Delta }}_g,{\mathrm{\Delta }}_u$; and $Q\overline{Q}$ separation $r=10a$.

Figure 12

Flux densities on the $x$ axis for gauge group SU(3); all investigated sectors ${\mathrm{\Lambda }}_{\eta }^{(\epsilon )}={\mathrm{\Sigma }}_g^{+},{\mathrm{\Sigma }}_u^{+},{\mathrm{\Sigma }}_g^{-},{\mathrm{\Sigma }}_u^{-},{\mathrm{\Pi }}_g,{\mathrm{\Pi }}_u,{\mathrm{\Delta }}_g,{\mathrm{\Delta }}_u$; and $Q\overline{Q}$ separation $r=10a$. (Top) $\mathrm{\Delta }{F}_{j,{\mathrm{\Lambda }}_{\eta }^{(\epsilon )}}^2(r;\mathbf{x}=(x,0,0))$, $j=\perp ,z$. (Bottom) $\mathrm{\Delta }{F}_{j,{\mathrm{\Lambda }}_{\eta }^{(\epsilon )}}^2(r;\mathbf{x}=(x,0,0))-\mathrm{\Delta }{F}_{j,{\mathrm{\Sigma }}_g^{+}}^2(r;\mathbf{x}=(x,0,0))$, $j=\perp ,z$.

Figure 13

Flux densities $\mathrm{\Delta }{F}_{j,{\mathrm{\Lambda }}_{\eta }^{\epsilon }}^2(r;\mathbf{x}=(x,0,z))$, $j=x,y,z$ in the separation plane for gauge group SU(3) and sectors ${\mathrm{\Lambda }}_{\eta }^{\epsilon }={\mathrm{\Sigma }}_g^{+},{\mathrm{\Sigma }}_u^{+},{\mathrm{\Sigma }}_g^{-},{\mathrm{\Sigma }}_u^{-}$. (Left) $Q\overline{Q}$ separation $r=6a$. (Right) $Q\overline{Q}$ separation $r=10a$.

Figure 14

Flux densities $\mathrm{\Delta }{F}_{j,{\mathrm{\Lambda }}_{\eta }}^2(r;\mathbf{x}=(x,0,z))$, $j=x,y,z$ in the separation plane for gauge group SU(3) and sectors ${\mathrm{\Lambda }}_{\eta }={\mathrm{\Pi }}_g,{\mathrm{\Pi }}_u,{\mathrm{\Delta }}_g,{\mathrm{\Delta }}_u$. (Left) $Q\overline{Q}$ separation $r=6a$. (Right) $Q\overline{Q}$ separation $r=10a$.