Three-quark static potential in perturbation theory

We study the three-quark static potential in perturbation theory in QCD. A complete next-to-leading order calculation is performed in the singlet, octets, and decuplet channels and the potential exponentiation is demonstrated. The mixing of the octet representations is calculated. At next-to-next-to-leading order, the subset of diagrams producing three-body forces is identified in Coulomb gauge and its contribution to the potential calculated. Combining it with the contribution of the two-body forces, which may be extracted from the quark-antiquark static potential, we obtain the complete next-to-next-to-leading order three-quark static potential in the color-singlet channel.

Figure 1

Static Wilson loop with edges $x_1=({\mathit{x}}_1,T_W/2)$, $x_2=({\mathit{x}}_2,T_W/2)$, $x_3=({\mathit{x}}_3,T_W/2)$, $y_1=({\mathit{x}}_1,-T_W/2)$, $y_2=({\mathit{x}}_2,-T_W/2)$, $y_3=({\mathit{x}}_3,-T_W/2)$ and insertions of the tensors ${\mathcal{C}}_{ijk}^u$ and ${\mathcal{C}}_{lmn}^{v*}$ in $X=(\mathit{R},T_W/2)$ and $Y=(\mathit{R},-T_W/2)$, respectively.

Figure 2

The three terms contributing to the static potential at order $g^2$. Dashed lines are longitudinal gluons.

Figure 3

A diagram contributing to the mixing of the two octets at LO. The three quark lines represent from above to below the quarks in ${\mathbf{x}}_1$, ${\mathbf{x}}_2$, and ${\mathbf{x}}_3$, respectively.

Figure 4

Diagrams appearing at order $g^4$ in the three-quark potential.

Figure 5

Three-body diagrams at NNLO that contribute, in Coulomb gauge, to the exponentiation of the LO and NLO potential.

Figure 6

Abelian three-body diagrams that have zero amplitude in Coulomb gauge.

Figure 7

Non-Abelian three-body diagrams that have zero amplitude in Coulomb gauge.

Figure 8

The only three-body diagrams that are not exponentiations and that have a nonvanishing amplitude in Coulomb gauge. Dashed lines are longitudinal gluons, and curly lines are transverse ones.

Figure 9

(a) We plot the coefficient $c_{\mathcal{H}}(\theta )$ as defined in Eq. (57) and obtained from the numerical integration of Eq. (b3). (b) We plot $g(\theta )c_{\mathcal{H}}(\theta )$.

Figure 10

The normalized three-body potential, $V_{\mathcal{H}\mathcal{C}}^{\mathrm{tot}}(L,\dots )/(-f_{\mathcal{H}}(\mathcal{C}){\alpha }_{\mathrm{s}}^3)$, plotted as a function of $L$ for the geometry described in (b1).

Figure 11

The normalized three-body potential, $V_{\mathcal{H}\mathcal{C}}^{\mathrm{tot}}(L,\dots )/(-f_{\mathcal{H}}(\mathcal{C}){\alpha }_{\mathrm{s}}^3)$, plotted as a function of $L$ for the geometry described in (b2).