Baryonic hybrids: Gluons as beads on strings between quarks

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.Comment: 18 pages, revtex4 plus 8 .eps figures in one .tar.gz fil

Three easy exercises in off-shell string-inspired methods

Off-shell string-inspired methods (OSSIM) calculate off-shell QCD Green's functions using Schwinger-Feynman proper-time techniques, always in the background field method (BFM) Feynman gauge for technical convenience, and so far only at one loop. We already know that these results are gauge-invariant, because this gauge realizes the prescriptions of the Pinch Technique (PT), a Feynman-graph formulation for any gauge, but the idea of the first exercise is to show this directly in OSSIM. In this exercise we extend proper-time OSSIM beyond the BFM Feynman gauge so that one can apply PT algorithms, and show that the intrinsic PT is equivalent to resolving ambiguities in OSSIM in other gauges. In the second exercise we use forty-year-old rules of the author and Tiktopoulos for expressing loop integrals with numerator momenta directly in terms of Feynman parameters after momentum integration (the goal of OSSIM) and show that these rules elegantly and with economy of effort give rise, at least at one loop, to standard OSSIM algorithms. In the third exercise we apply world-line techniques to the problem of the breaking of adjoint strings, requiring a non-perturbative treatment that in the end reduces to a variant of the Schwinger result for production of electron-positron pairs in an electric field. This generalizes OSSIM to non-perturbative processes.Comment: 12 pages, 3 figures, talk given at "From quarks and gluons to hadronic matter: A bridge too far?"[QCD-TNT-III], Trento, Italy, Sept. 2-6, 201

Exploring dynamical gluon mass generation in three dimensions

In the d=3 gluon mass problem in pure-glue non-Abelian $SU(N)$ gauge theory we pay particular attention to the observed (in Landau gauge) violation of positivity for the spectral function of the gluon propagator. This causes a large bulge in the propagator at small momentum. Mass is defined through $m^{-2}=\Delta (p=0)$, where $\Delta(p)$ is the scalar function for the gluon propagator in some chosen gauge, it is not a pole mass and is generally gauge-dependent, except in the gauge-invariant Pinch Technique (PT). We truncate the PT equations with a new method called the vertex paradigm that automatically satisfies the QED-like Ward identity relating the 3-gluon PT vertex function with the PT propagator. The mass is determined by a homogeneous Bethe-Salpeter equation involving this vertex and propagator. This gap equation also encapsulates the Bethe-Salpeter equation for the massless scalar excitations, essentially Nambu-Goldstone fields, that necessarily accompany gauge-invariant gluon mass. The problem is to find a good approximate value for $m$ and at the same time explain the bulge, which by itself leads, in the gap equation for the gluon mass, to excessively large values for the mass. Our point is not to give a high-accuracy determination of $m$ but to clarify the way in which the propagator bulge and a fairly accurate estimate of $m$ can co-exist, and we use various approximations that illustrate the underlying mechanisms. The most critical point is to satisfy the Ward identity. In the PT we estimate a gauge-invariant dynamical gluon mass of $m \approx Ng^2/(2.48 \pi)$. We translate these results to the Landau gauge using a background-quantum identity involving a dynamical quantity $\kappa$ such that $m=\kappa m_L$, where $m_L^{-2}\equiv \Delta_L(p=0)$. Given our estimates for $m,\kappa$ the relation is fortuitously well-satisfied for $SU(2)$ lattice data.Comment: 22 pages, 5 figure