Gluon dynamics from an ordinary differential equation

Abstract

We present a novel method for computing the nonperturbative kinetic term of the gluon propagator from an ordinary differential equation, whose derivation hinges on the central hypothesis that the regular part of the three-gluon vertex and the aforementioned kinetic term are related by a partial Slavnov–Taylor identity. The main ingredients entering in the solution are projection of the three-gluon vertex and a particular derivative of the ghost-gluon kernel, whose approximate form is derived from a Schwinger–Dyson equation. Crucially, the requirement of a pole-free answer determines the initial condition, whose value is calculated from an integral containing the same ingredients as the solution itself. This feature fixes uniquely, at least in principle, the form of the kinetic term, once the ingredients have been accurately evaluated. In practice, however, due to substantial uncertainties in the computation of the necessary inputs, certain crucial components need be adjusted by hand, in order to obtain self-consistent results. Furthermore, if the gluon propagator has been independently accessed from the lattice, the solution for the kinetic term facilitates the extraction of the momentum-dependent effective gluon mass. The practical implementation of this method is carried out in detail, and the required approximations and theoretical assumptions are duly highlighted.

Exploring smoking-gun signals of the Schwinger mechanism in QCD

“…6 of Ref. [108], here we only collect the main results. Specifically, we obtain the general expression…”

“…The above considerations become particularly relevant in the case of the three-gluon vertex, because the form factor of its pole-free part has been evaluated rather accurately in recent lattice simulations [104][105][106]. As a result, the displacement originating from the onset of the Schwinger mechanism, to be denoted by C(r 2 ), may be calculated by appropriately combining this form factor with all other constituents that enter into the WI of the three-gluon vertex; all of them are available from lattice simulations, with the exception of a particular partial derivative, denoted by W(r 2 ), related to the ghost-gluon kernel that appears in the STI [107,108].…”

“…Let us finally mention that the principal uncertainty associated with the WI determination originates from the computation of the function W(r 2 ), which is not available from lattice simulations, and has been approximated by a truncated version of the SDE of the ghost-gluon kernel. As was explained in [108], the simulation of this function on the lattice is theoretically conceivable, but practically rather cumbersome.…”

Ghost dynamics from Schwinger-Dyson equations

“…In the present work, we solve the coupled system of SDEs governing the momentum evolution of the ghost propagator, D(p 2 ), and the form factor, denoted by B 1 (r, p, q), of the classical (tree level) tensor structure of the ghost-gluon vertex. For the gluon propagator which appears as ingredient, we capitalize on lattice results [47][48][49], carefully extrapolated to the continuum and to infinite volume [48,49] and displaying the now firmly established IR saturation [40,47,[50][51][52][53][54][55], associated with the dynamical generation of a gluon mass gap [6,7,41,56,57]. In this way, we are left with the three-gluon vertex, Γ αµν (q, r, p), as the most uncertain ingredient, whose nonperturbative structure has only recently begun to be unraveled [18, 20, 31, 32, 34, 35, 37-42, 45, 46].…”

Ghost dynamics in the soft gluon limit

“…In particular, the nonperturbative masslessness of the ghost is responsible for the vanishing of the gluon spectral density at the origin [66][67][68][69], and for the infrared suppression of the three-gluon vertex [52,53,[61][62][63]. In that sense, the ghost dynamics leave their imprint on a variety of fundamental phenomena, such as chiral symmetry breaking and the generation of quark constituent masses [1,[70][71][72][73][74], the emergence of a mass gap in the gauge sector of the theory [9,65,75,76], and the dynamical formation of hadronic bound states [3,20,[77][78][79] and glueballs [80][81][82][83].…”