Baryonic hybrids: Gluons as beads on strings between quarks

In this paper we analyze the ground state of the heavy-quark $qqqG$ system using standard principles of quark confinement and massive constituent gluons as established in the center-vortex picture. The known string tension $K_F$ and approximately-known gluon mass $M$ lead to a precise specification of the long-range nonrelativistic 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, along with some elementary group theory describing the gluon Wilson-line as a composite of colocated $q$ and $\overline{q}$ lines. We show that (aside from color-Coulomb and similar terms) the gluon potential energy in the presence of quarks is accurately described (for small gluon fluctuations) via attaching these three strings to the gluon, which in equilibrium sits at the Steiner point of the Y-shaped string network joining the three quarks. The gluon undergoes small harmonic fluctuations that slightly stretch these strings and quasiconfine the gluon to the neighborhood of the Steiner point. To describe nonrelativistic ground-state gluonic fluctuations at large $R$ we use the Schrödinger equation, ignoring mixing with $l=2$ states. Available lattice data and real-world hybrids require consideration of $R$ values small enough for significant relativistic corrections, which we apply using a variational principle for the relativistic harmonic-oscillator. We also consider the role of color-Coulomb contributions. In terms of interquark separations $R$, we find leading nonrelativistic large-$R$ terms in the gluon excitation energy of the form $\epsilon (R)\to M+\xi [K_F/(MR){]}^{1/2}-\zeta {\alpha }_c/R$ where $\xi ,\zeta$ are calculable numerical coefficients and ${\alpha }_c\simeq$ 0.15 is the color-Coulomb $q\overline{q}$ coupling. When the gluon is relativistic, $\epsilon \sim (K_F/R{)}^{1/3}$. We get an acceptable fit to lattice data with $M=500$ MeV. Although we do not consider it in full detail, we show that in the $q\overline{q}G$ hybrid the gluon is a bead that can slide without friction on a string joining the $q$ and $\overline{q}$. We comment briefly on the significance of our findings to fluctuations of the minimal surface, a subject difficult to understand from the point of view of center vortices.