Symmetries of the three-heavy-quark system and the color-singlet static energy at next-to-next-to-leading logarithmic order

We study the symmetries of the three-heavy-quark system under exchange of the quark fields within the effective field theory framework of potential nonrelativistic QCD. The symmetries constrain the form of the matching coefficients in the effective theory. We then focus on the color-singlet sector and determine the so far unknown leading ultrasoft contribution to the static potential, which is of order αs4lnâ¡μ, and consequently to the static energy, which is of order αs4lnâ¡α _s. Finally, in the case of an equilateral geometry, we solve the renormalization group equations and resum the leading ultrasoft logarithms for the static potential of three quarks in a color singlet, octet and decuplet representation.