  We analyze the ground state of the heavy-quark hybrid system composed of
three quarks and a gluon. The known string tension K and approximately-known
gluon mass M lead to a precise specification of the long-range non-relativistic
part of the potential binding the gluon to the quarks with no undetermined
phenomenological parameters, in the limit of large interquark separation R. Our
major tool (also used earlier by Simonov) is the use of proper-time methods to
describe gluon propagation within the quark system, which reveals the gluon
Wilson line as a composite of co-located quark and antiquark lines. We show
that (aside from color-Coulomb and similar terms) the gluon potential energy in
the presence of quarks is accurately described via attaching these three
strings to the gluon, which in equilibrium sits at the middle of the Y-shaped
string network joining the three quarks. The gluon undergoes small harmonic
fluctuations that slightly stretch these strings and quasi-confine the gluon to
the neighborhood of the middle. In the non-relativistic limit (large R) we use
the Schrodinger equation, ignoring mixing with l=2 states. Relativistic
corrections (smaller R) are applied with a variational principle for the
relativistic harmonic oscillator. We also consider the role of color-Coulomb
contributions. We find leading non-relativistic large-R terms in the gluon
string energy which behave like the square root of K/(MR). The relativistic
energy goes like the cube root of K/R. We get an acceptable fit to lattice data
with M = 500 MeV. We show that in the quark-antiquark hybrid the gluon is a
bead that can slide without friction on a string joining the quark and
anti-quark. We comment briefly on the significance of our findings to
fluctuations of the minimal surface.
