Three-Loop Static Potential

We compute the three-loop corrections to the potential of two heavy quarks. In particular, we consider in this Letter the purely gluonic contribution which provides, in combination with our previous fermion corrections, the complete answer at three loops.

Figure 1

Sample diagrams contributing to the static potential at tree-level, one-, two-, and three-loop order. Solid lines and curly lines represent quarks and gluons, respectively. In the case of closed loops the quarks are massless; the external quarks are heavy and treated in the static limit.

Figure 2

One-, two-, and three-loop diagrams. The solid line stands for massless relativistic propagators and the zigzag line represents static propagators.

Figure 3

Three-loop master integrals where the $\mathcal{O}(ϵ)$ part is only known numerically. The label “$-i0$” indicates that instead of the static propagator $1/(p_0+i0)$ there is the propagator $1/(p_0-i0)$.