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Aguilar, A. C., Binosi, D., Ibañez, D., & Papavassiliou, J. (2014). New method for determining the quark-gluon vertex. Phys. Rev. D, 90(6), 065027–26pp.
Abstract: We present a novel nonperturbative approach for calculating the form factors of the quark-gluon vertex in terms of an unknown three-point function, in the Landau gauge. The key ingredient of this method is the exact all-order relation connecting the conventional quark-gluon vertex with the corresponding vertex of the background field method, which is Abelian-like. When this latter relation is combined with the standard gauge technique, supplemented by a crucial set of transverse Ward identities, it allows the approximate determination of the nonperturbative behavior of all 12 form factors comprising the quark-gluon vertex, for arbitrary values of the momenta. The actual implementation of this procedure is carried out in the Landau gauge, in order to make contact with the results of lattice simulations performed in this particular gauge. The most demanding technical aspect involves the approximate calculation of the components of the aforementioned (fully dressed) three-point function, using lattice data as input for the gluon propagators appearing in its diagrammatic expansion. The numerical evaluation of the relevant form factors in three special kinematical configurations (soft-gluon and quark symmetric limit, zero quark momentum) is carried out in detail, finding qualitative agreement with the available lattice data. Most notably, a concrete mechanism is proposed for explaining the puzzling divergence of one of these form factors observed in lattice simulations.
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Aguilar, A. C., Binosi, D., Ibañez, D., & Papavassiliou, J. (2014). Effects of divergent ghost loops on the Green's functions of QCD. Phys. Rev. D, 89(8), 085008–26pp.
Abstract: In the present work, we discuss certain characteristic features encoded in some of the fundamental QCD Green's functions, for which the origin can be traced back to the nonperturbative masslessness of the ghost field, in the Landau gauge. Specifically, the ghost loops that contribute to these Green's functions display infrared divergences, akin to those encountered in the perturbative treatment, in contradistinction to the gluonic loops, for which perturbative divergences are tamed by the dynamical generation of an effective gluon mass. In d = 4, the aforementioned divergences are logarithmic, thus causing a relatively mild impact, whereas in d = 3 they are linear, giving rise to enhanced effects. In the case of the gluon propagator, these effects do not interfere with its finiteness, but make its first derivative diverge at the origin, and introduce a maximum in the region of infrared momenta. The three-gluon vertex is also affected, and the induced divergent behavior is clearly exposed in certain special kinematic configurations, usually considered in lattice simulations; the sign of the corresponding divergence is unambiguously determined. The main underlying concepts are developed in the context of a simple toy model, which demonstrates clearly the interconnected nature of the various effects. The picture that emerges is subsequently corroborated by a detailed nonperturbative analysis, combining lattice results with the dynamical integral equations governing the relevant ingredients, such as the nonperturbative ghost loop and the momentumdependent gluon mass.
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Ibañez, D., & Papavassiliou, J. (2013). Gluon mass generation in the massless bound-state formalism. Phys. Rev. D, 87(3), 034008–25pp.
Abstract: We present a detailed, all-order study of gluon mass generation within the massless bound-state formalism, which constitutes the general framework for the systematic implementation of the Schwinger mechanism in non-Abelian gauge theories. The main ingredient of this formalism is the dynamical formation of bound states with vanishing mass, which give rise to effective vertices containing massless poles; these latter vertices, in turn, trigger the Schwinger mechanism, and allow for the gauge-invariant generation of an effective gluon mass. This particular approach has the conceptual advantage of relating the gluon mass directly to quantities that are intrinsic to the bound-state formation itself, such as the “transition amplitude'' and the corresponding ”bound-state wave function.'' As a result, the dynamical evolution of the gluon mass is largely determined by a Bethe-Salpeter equation that controls the dynamics of the relevant wave function, rather than the Schwinger-Dyson equation of the gluon propagator, as happens in the standard treatment. The precise structure and field-theoretic properties of the transition amplitude are scrutinized in a variety of independent ways. In particular, a parallel study within the linear-covariant (Landau) gauge and the background-field method reveals that a powerful identity, known to be valid at the level of conventional Green's functions, also relates the background and quantum transition amplitudes. Despite the differences in the ingredients and terminology employed, the massless bound-state formalism is absolutely equivalent to the standard approach based on Schwinger-Dyson equations. In fact, a set of powerful relations allows one to demonstrate the exact coincidence of the integral equations governing the momentum evolution of the gluon mass in both frameworks.
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Binosi, D., Ibañez, D., & Papavassiliou, J. (2013). QCD effective charge from the three-gluon vertex of the background-field method. Phys. Rev. D, 87(12), 125026–10pp.
Abstract: In this article we study in detail the prospects of determining the infrared finite QCD effective charge from a special kinematic limit of the vertex function corresponding to three background gluons. This particular Green's function satisfies a QED-like Ward identity, relating it to the gluon propagator, with no reference to the ghost sector. Consequently, its longitudinal form factors may be expressed entirely in terms of the corresponding gluon wave function, whose inverse is proportional to the effective charge. After reviewing certain important theoretical properties, we consider a typical lattice quantity involving this vertex, and derive its exact dependence on the various form factors, for arbitrary momenta. We then focus on the particular momentum configuration that eliminates any dependence on the (unknown) transverse form factors, projecting out only the desired quantity. A preliminary numerical analysis indicates that the effective charge is relatively insensitive to the numerical uncertainties that may afflict future simulations of the aforementioned lattice quantity. The numerical difficulties associated with a parallel determination of the dynamical gluon mass are briefly discussed.
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Aguilar, A. C., Ibañez, D., & Papavassiliou, J. (2013). Ghost propagator and ghost-gluon vertex from Schwinger-Dyson equations. Phys. Rev. D, 87(11), 114020–14pp.
Abstract: We study an approximate version of the Schwinger-Dyson equation that controls the nonperturbative behavior of the ghost-gluon vertex in the Landau gauge. In particular, we focus on the form factor that enters in the dynamical equation for the ghost dressing function, in the same gauge, and derive its integral equation, in the “one-loop dressed” approximation. We consider two special kinematic configurations, which simplify the momentum dependence of the unknown quantity; in particular, we study the soft gluon case and the well-known Taylor limit. When coupled with the Schwinger-Dyson equation of the ghost dressing function, the contribution of this form factor provides considerable support to the relevant integral kernel. As a consequence, the solution of this coupled system of integral equations furnishes a ghost dressing function that reproduces the standard lattice results rather accurately, without the need to artificially increase the value of the gauge coupling.
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