QCD Effective Charge Research Articles

Experimental Determination of the QCD Effective Charge αg1(Q)

The QCD effective charge αg1(Q) is an observable that characterizes the magnitude of the strong interaction. At high momentum Q, it coincides with the QCD running coupling αs(Q). At low Q, it offers a nonperturbative definition of the running coupling. We have extracted αg1(Q) from measurements carried out at Jefferson Lab that span the very low to moderately high Q domain, 0.14≤Q≤2.18 GeV. The precision of the new results is much improved over the previous extractions and the reach in Q at the lower end is significantly expanded. The data show that αg1(Q) becomes Q-independent at very low Q. They compare well with two recent predictions of the QCD effective charge based on Dyson–Schwinger equations and on the AdS/CFT duality.

QCD effective charges and the structure function F2 at small-x: Higher twist effects

We consider the effect of higher twist operators of the Wilson operator product expansion in the structure function F2(x,Q2) at small-x, taking into account QCD effective charges whose infrared behavior is constrained by a dynamical mass scale. The higher twist corrections are obtained from the renormalon formalism. Our analysis is performed within the conventional framework of next-to-leading order, with the factorization and renormalization scales chosen to be Q2. The infrared properties of QCD are treated in the context of the generalized double-asymptotic-scaling approximation. We show that the corrections to F2 associated with twist-four and twist-six are both necessary and sufficient for a good description of the deep infrared experimental data.

Proton-proton forward scattering at the LHC

Recently the TOTEM experiment at the LHC has released measurements at s=13 TeV of the proton-proton total cross section, σtot, and the ratio of the real to imaginary parts of the forward elastic amplitude, ρ. Since then an intense debate on the C-parity asymptotic nature of the scattering amplitude was initiated. We examine the proton-proton and the antiproton-proton forward data above 10 GeV in the context of an eikonal QCD-based model, where nonperturbative effects are readily included via a QCD effective charge. We show that, despite an overall satisfactory description of the forward data is obtained by a model in which the scattering amplitude is dominated by only crossing-even elastic terms, there is evidence that the introduction of a crossing-odd term may improve the agreement with the measurements of ρ at s=13 TeV. In the Regge language the dominant even(odd)-under-crossing object is the so called Pomeron (Odderon).

Process-Independent Effective Coupling: From QCD Green’s Functions to Phenomenology

This article reports on a very recent proposal for a new type of process-independent QCD effective charge [Phys.Rev.D96(2017)054026] defined, as an anologue of the Gell-Mann-Low effective charge in QCD, on the ground of nothing but the knowledge of the gauge-field two-point Green's function, albeit modified within a particular computational framework; namely, the combination of pinch technique and background field method which makes possible a systematic rearranging of classes of diagrams in order to redefine the Green's function and have them obey linear QED-like Slavnov-Taylor identities. We have here calculated that effective charge, shown how strikingly well it compares to a process-dependent effective charge based on the Bjorken sum rule; and, finally, employed it in an exploratory calculation of the proton electromagnetic form factor in the hard scattering regime.

Cross sections, multiplicity and moment distributions at the LHC

The unitarity of the $S$-matrix requires that the absorptive part of the elastic scattering amplitude receives contributions from both the inelastic and the elastic channels. We explore this unitarity condition in order to describe, in a connected way, hadron-hadron observables like the total and elastic differential cross sections, the ratio of the real to imaginary part of the forward scattering amplitude and the inclusive multiplicity distributions in full phase space, over a large range of energies. We introduce non-perturbative QCD effects in the forward scattering amplitude by using the infrared QCD effective charge dependent on the dynamical gluon mass. In our analysis we pay special attention to the theoretical uncertainties in the predictions due to this mass scale variation. We also present quantitative predictions for the $H_{q}$ moments at high energies. Our results reproduce the moment oscillations observed in experimental data, and are consistent with the behavior predicted by QCD.

QCD effective charge from the three-gluon vertex of the background-field method

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.

QCD effective charge and the structure function [formula omitted] at small-x

We study infrared contributions to the QCD description of the HERA data on the structure function F2 within the generalized DAS approximation. We argue that this approximation is a natural one and consistent with the phenomenon of dynamical mass generation in QCD. The investigation is performed at next-to-leading order by using the leading-twist expansion of F2(x,Q2) and by adopting an effective charge whose finite infrared behavior is constrained by a dynamical gluon mass. We propose one ansatz for the behavior of this effective coupling beyond leading order. The dependence of the experimental data on the infrared value of the effective charge is used in order to study the asymptotic behavior of the running gluon mass. The deep inelastic structure function F2 obtained in this approach shows very good agreement with the experimental data.

QCD effective charges from lattice data

We use recent lattice data on the gluon and ghost propagators, as well as the Kugo-Ojima function, in order to extract the non-perturbative behavior of two particular definitions of the QCD effective charge, one based on the pinch technique construction, and one obtained from the standard ghost-gluon vertex. The construction relies crucially on the definition of two dimensionful quantities, which are invariant under the renormalization group, and are built out of very particular combinations of the aforementioned Green's functions. The main non-perturbative feature of both effective charges, encoded in the infrared finiteness of the gluon propagator and ghost dressing function used in their definition, is the freezing at a common finite (non-vanishing) value, in agreement with a plethora of theoretical and phenomenological expectations. We discuss the sizable discrepancy between the freezing values obtained from the present lattice analysis and the corresponding estimates derived from several phenomenological studies, and attribute its origin to the difference in the gauges employed. A particular toy calculation suggests that the modifications induced to the non-perturbative gluon propagator by the gauge choice may indeed account for the observed deviation of the freezing values.

QCD running coupling in low-energy region

We estimate the QCD effective charge $\alpha_s$ in the low-energy region by exploiting the conventional meson spectrum within a relativistic quantum-field model based on analytic confinement. The ladder Bethe-Salpeter equation is solved for the masses of two-quark bound states. We found a new, independent and specific infrared-finite behavior of QCD coupling below energy scale 1 GeV. Particularly, an infrared-fixed point is extracted at $\alpha_s(0)\simeq 0.757$ for confinement scale $\Lambda=345$ MeV. As an application, we estimate masses of some intermediate and heavy mesons and obtain results in reasonable agreement with recent experimental data.

Non-perturbative QCD effective charges

Using gluon and ghost propagators obtained from Schwinger-Dyson equations (SDEs), we construct the non-perturbative effective charge of QCD. We employ two different definitions, which, despite their distinct field-theoretic origin, give rise to qualitative comparable results, by virtue of a crucial non-perturbative identity. Most importantly, the QCD charge obtained with either definition freezes in the deep infrared, in agreement with theoretical and phenomenological expectations. The various theoretical ingredients necessary for this construction are reviewed in detail, and some conceptual subtleties are briefly discussed.

Nonperturbative comparison of QCD effective charges

We study the nonperturbative behavior of two versions of the QCD effective charge, one obtained from the pinch technique gluon self-energy, and one from the ghost-gluon vertex. Despite their distinct theoretical origin, due to a fundamental identity relating various ingredients appearing in their respective definitions, the two effective charges are almost identical in the entire range of physical momenta, and coincide exactly in the deep infrared, where they freeze at a common finite value. Specifically, the dressing function of the ghost propagator is related to the two form factors in the Lorentz decomposition of a certain Green's function, appearing in a variety of field-theoretic contexts. The central identity, which is valid only in the Landau gauge, is derived from the Schwinger-Dyson equations governing the dynamics of the aforementioned quantities. The renormalization procedure that preserves the validity of the identity is carried out, and various relevant kinematic limits and physically motivated approximations are studied in detail. A crucial ingredient in this analysis is the infrared finiteness of the gluon propagator, which is inextricably connected with the aforementioned freezing of the effective charges. Some important issues related to the consistent definition of the effective charge in the presence of such a gluon propagator are resolved. We finally present a detailed numerical study of a special set of Schwinger-Dyson equations, whose solutions determine the nonperturbative dynamics of the quantities composing the two effective charges.

Convergence and gauge dependence properties of the resummed one-loop quark-quark scattering amplitude in perturbative QCD

The one-loop QCD effective charge α eff for quark-quark scattering is derived by diagrammatic resummation of the one-loop amplitude using an arbitrary covariant gauge. Except for the particular choice of gauge parameter ξ = −3, α eff is found to increase with increasing physical scale, Q, as lnQ or ln2 Q. For ξ = −3, α eff decreases with increasing Q and satisfies a renormalization group equation. Also, except for the case ξ = 19/9, convergence radii of geometric series are found to impose upper limits on Q.

The QCD analytic running coupling and chiral symmetry breaking

We study the dependence on the pion mass of the QCD effective charge by employing the dispersion relations for the Adler D function. This new massive analytic running coupling is compared to the effective coupling saturated by the dynamically generated gluon mass. A qualitative picture of the possible impact of the former coupling on the chiral symmetry breaking is presented.

The QCD effective charge to all orders

Using the pinch technique, we show how to construct the QCD effective charge to all orders.

Renormalization-scale-invariant PQCD predictions forRe+e−and the Bjorken sum rule at next-to-leading order

We discuss the application of the physical QCD effective charge ${\ensuremath{\alpha}}_{V},$ defined via the heavy-quark potential, in perturbative calculations at next-to-leading order. When coupled with the Brodsky-Lepage-Mackenzie prescription for fixing the renormalization scales, the resulting series are automatically and naturally scale and scheme independent, and represent unambiguous predictions of perturbative QCD. We consider in detail such commensurate scale relations for the ${e}^{+}{e}^{\ensuremath{-}}$ annihilation ratio ${R}_{{e}^{+}{e}^{\ensuremath{-}}}$ and the Bjorken sum rule. In both cases the improved predictions are in excellent agreement with experiment.

Can the QCD effective charge be symmetrical in the Euclidean and Minkowskian regions?

We study a possible symmetrical behavior of the effective charges defined in the Euclidean and Minkowskian regions and prove that such symmetry is inconsistent with the causality principle.

The gauge-independent QCD effective charge

It is shown how the QED concept of a gauge-, scale- and scheme-independent one-loop effective charge can be extended directly at the diagrammatic level to QCD, thus justifying explicitly the “naive non-abelianization” prescription used in renormalon calculus. It is first argued that, for onshell external fields and at the strictly one-loop level, the required gluon self-energy-like function is precisely that obtained from S-matrix elements via the pinch technique. The generalization of the pinch technique to explicitly off-shell processes is then introduced. It is shown how, as a result of a fundamental cancellation among conventional perturbation theory diagrams, encoded in the QCD Ward identities, the pinch technique one-loop gluon self-energy i \\ ̂ gP μv ab(q) remains gauge-independent and universal regardless of the fact that the “external” fields in the given process are off-shell. This demonstration involves a simple technique enabling the isolation, in an arbitrary gauge, of i \\ ̂ gP μv ab(q) from subclasses of up to several hundred diagrams at once. Furthermore, it is shown how this one-loop cancellation mechanism iterates for the subclasses of n-loop diagrams containing implicitly the Dyson chains of n one-loop self-energies i \\ ̂ gP μv ab(q) . The gauge cancellation required for the Dyson summation of i \\ ̂ gP μv ab(q) is thus demonstrated explicitly in a general class of ghost-free gauges for all orders n.