Gluon Green Function Research Articles

Instanton dominance over αs at low momenta from lattice QCD simulations at Nf = 0, Nf = 2 + 1 and Nf = 2 + 1 + 1

We report on an instanton-based analysis of the gluon Green functions in the Landau gauge for low momenta; in particular we use lattice results for αs in the symmetric momentum subtraction scheme (MOM) for large-volume lattice simulations. We have exploited quenched gauge field configurations, Nf = 0, with both Wilson and tree-level Symanzik improved actions, and unquenched ones with Nf = 2 + 1 and Nf = 2 + 1 + 1 dynamical flavors (domain wall and twisted-mass fermions, respectively). We show that the dominance of instanton correlations on the low-momenta gluon Green functions can be applied to the determination of phenomenological parameters of the instanton liquid and, eventually, to a determination of the lattice spacing. We furthermore apply the Gradient Flow to remove short-distance fluctuations. The Gradient Flow gets rid of the QCD scale, ΛQCD, and reveals that the instanton prediction extents to large momenta. For those gauge field configurations free of quantum fluctuations, the direct study of topological charge density shows the appearance of large-scale lumps that can be identified as instantons, giving access to a direct study of the instanton density and size distribution that is compatible with those extracted from the analysis of the Green functions.

Stability of azimuthal-angle observables under higher order corrections in inclusive three-jet production

Recently, a new family of observables consisting of azimuthal-angle generalised ratios was proposed in a kinematical setup that resembles the usual Mueller-Navelet jets but with an additional tagged jet in the central region of rapidity. Non-tagged minijet activity between the three jets can affect significantly the azimuthal angle orientation of the jets and is accounted for by the introduction of two BFKL gluon Green functions. Here, we calculate the, presumably, most relevant higher order corrections to the observables by now convoluting the three leading-order jet vertices with two gluon Green functions at next-to-leading logarithmic approximation. The corrections appear to be mostly moderate giving us confidence that the recently proposed observables are actually an excellent way to probe the BFKL dynamics at the LHC. Furthermore, we allow for the jets to take values in different rapidity bins in various configurations such that a comparison between our predictions and the experimental data is a straightforward task.

Three-gluon Green functions: low-momentum instanton dominance and zero-crossing

We will report on a some efforts recently made in order to gain a better understanding of some IR properties of the 3-point gluon Green function by following both lattice and continuum QCD approaches.

Inclusive four-jet production at 7 and 13 TeV: Azimuthal profile in multi-Regge kinematics

Recently, new observables in LHC inclusive events with three tagged jets were proposed. Here, we extend that proposal to events with four tagged jets. The events are characterized by one jet in the forward direction, one in the backward direction with a large rapidity distance Y from the first one and two more jets tagged in more central regions of the detector. In our setup, non-tagged associated mini-jet multiplicity is present and needs to be accounted for by the inclusion of BFKL gluon Green functions. The projection of the cross section on azimuthal-angle components opens up the opportunity for defining new ratios of correlation functions of the azimuthal-angle differences among the tagged jets that can be used as probes of the BFKL dynamics.

Solution of the Bartels-Kwiecinski-Praszalowicz equation via Monte Carlo integration

We present a method of solution of the Bartels-Kwiecinski-Praszalowicz (BKP) equation based on the numerical integration of iterated integrals in transverse momentum and rapidity space. As an application, our procedure, which makes use of Monte Carlo integration techniques, is applied to obtain the gluon Green function in the odderon case at leading order. The same approach can be used for more complicated scenarios.

The effect of the infrared phase of the discrete BFKL pomeron on transverse momentum diffusion

Imposing infrared boundary conditions on the BFKL equation with running coupling transforms the complex momentum w-plane cut present in the gluon Green function into an infinite series of positive Regge poles. In addition, a cut on the negative w line remains. We consider a Hermitian kernel at leading order with running coupling and construct the gluon Green function performing the w integration away from the real axis. We find a strong dependence of the asymptotic intercepts and collinear behaviour on the non-perturbative choice of the boundary conditions, in the form of an infrared phase. This is particularly manifest in the asymmetric infrared/ultraviolet structure of the associated diffusion in transverse momentum. We find that random walks into the infrared region are largely reduced in this approach.

The effect of a rapidity gap veto on the discrete BFKL pomeron

We investigate the sensitivity of the discrete BFKL spectrum, which appears in the gluon Green function when the running coupling is considered, to a lower cut-off in the relative rapidities of the emitted particles. We find that the eigenvalues associated to each of the discrete eigenfunctions decrease with the size of the rapidity veto. The effect is stronger on the lowest eigenfunctions. The net result is a reduction of the growth with energy for the Green function together with a suppression in the regions with small transverse momentum.

Monte Carlo study of double logarithms in the smallxregion

We investigate the effect of the resummation of collinear double logarithms in the Balitsky-Fadin-Kuraev-Lipatov (BFKL) gluon Green function using the Monte Carlo event generator BFKLex. The resummed collinear terms in transverse momentum space were calculated by Sabio Vera [Nucl. Phys. B722, 65 (2005)] and correspond to the addition to the next-to-leading order BFKL kernel of a Bessel function of the first kind whose argument contains the strong coupling and a double logarithm of the ratio of the squared transverse momenta of the Reggeized gluons. We discuss how these additional terms improve the collinear convergence of the whole approach and reduce the asymptotic growth with energy of the cross sections. Taking advantage of the Monte Carlo implementation, we show how the new results reduce the diffusion of the gluon ladder into infrared and ultraviolet transverse momentum scales, while strongly affecting final state configurations by reducing the minijet multiplicity.

The next-to-leading order vertex for a forward jet plus a rapidity gap at high energies

We present the results for the calculation of the forward jet vertex associated to a rapidity gap (coupling of a hard pomeron to the jet) in the Balitsky–Fadin–Kuraev–Lipatov (BFKL) formalism at next-to-leading order (NLO). We handle the real emission contributions making use of the high energy effective action proposed by Lipatov, valid for multi-Regge and quasi-multi-Regge kinematics. This result is important since it allows, together with the NLO non-forward gluon Green function, to perform NLO studies of jet production in diffractive events (Mueller–Tang dijets, as a well-known example).

Estimation of saturation and coherence effects in the KGBJS equation — a non-linear CCFM equation

We solve the modified non-linear extension of the CCFM equation - KGBJS equation - numerically for certain initial conditions and compare the resulting gluon Green functions with those obtained from solving the original CCFM equation and the BFKL and BK equations for the same initial conditions. We improve the low transversal momentum behaviour of the KGBJS equation by a small modification.

A study of the diffusion pattern in [formula omitted] SYM at high energies

In the context of evolution equations and scattering amplitudes in the high energy limit of the N=4 super Yang–Mills theory we investigate in some detail the BFKL gluon Green function at next-to-leading order. In particular, we study its collinear behavior in terms of an expansion in different angular components. We also perform a Monte Carlo simulation of the different final states contributing to such a Green function and construct the diffusion pattern into infrared and ultraviolet modes and multiplicity distributions, making emphasis in separating the gluon contributions from those of scalars and gluinos. We find that the combined role of the non-gluonic degrees of freedom is to improve the collinear behavior and reduce the diffusion into ultraviolet regions while not having any effect on the average multiplicities or diffusion into the infrared. In terms of growth with energy, the non-zero conformal spin components are mainly driven by the gluon terms in the BFKL kernel. For zero conformal spin (Pomeron) the effect of the scalar and gluino sectors is to dramatically push the Green function towards higher values.

A comparative study of small x Monte Carlos with and without QCD coherence effects

We compare two Monte Carlo implementations of resummation schemes for the description of parton evolution at small values of Bjorken x. One of them is based on the Balitsky–Fadin–Kuraev–Lipatov (BFKL) evolution equation and generates fully differential parton distributions in momentum space making use of reggeized gluons. The other one is based on the Catani–Ciafaloni–Fiorani–Marchesini (CCFM) partonic kernel where QCD coherence effects are introduced. It has been argued that both approaches agree with each other in the x → 0 limit. We show that this is not the case for azimuthal angle dependent quantities since at high energies the BFKL approach is dominated by its zero conformal spin component while the CCFM gluon Green function receives contributions from all conformal spins even at very small x.

A solvable model for small-xphysics inD> 4 dimensions

I present a simplified model for the gluon Green's function governing high-energy QCD dynamics, in arbitrary space-time dimensions. The BFKL integral equation (either with or without running coupling) reduces to a second order differential equation that can be solved in terms of Bessel and hypergeometric functions. Explicit expressions for the gluon density and its anomalous dimension are derived in MS and Q_0 factorization schemes. This analysis illustrates the qualitative features of the QCD gluon density in both factorization schemes. In addition, it clarifies the mathematical properties and validates the results of the ``gamma-representation'' method proposed by M.Ciafaloni and myself for extracting resummed next-to-leading-log x anomalous dimensions of phenomenological relevance in the two schemes.

NLO inclusive jet production inkT-factorization

The inclusive production of jets in the central region of rapidity is studied in $k_T$-factorization at next-to-leading order (NLO) in QCD perturbation theory. Calculations are performed in the Regge limit making use of the NLO BFKL results. A jet cone definition is introduced and a proper phase--space separation into multi-Regge and quasi-multi-Regge kinematic regions is carried out. Two situations are discussed: scattering of highly virtual photons, which requires a symmetric energy scale to separate the impact factors from the gluon Green's function, and hadron-hadron collisions, where a non--symmetric scale choice is needed.

Small- x: QCD: the (improved) perturbative picture

After a brief introduction to low- x QCD and to resummation approaches, I illustrate the predictions for the gluon Green function and splitting function of a recent renormalization-group improved small- x resummation scheme. The perturbative high-energy exponents are predicted in a phenomenologically interesting range, and significant preasymptotic effects are found. In particular, the gluon splitting function shows a shallow dip in the moderate- x region, followed by the expected small- x power increase.

A Study of the Gluon Ladder in Diffractive Processes

The solution to the non-forward BFKL equation in the Leading Logarithmic approximation is expressed in terms of a sum of iterations of its kernel directly in transverse momentum and rapidity space. Several studies of the non-forward solution are performed both at the level of the gluon Green's function and for a toy cross-section, including an analysis of the diffusion properties as found in this approach. The method developed in this paper allows for a direct inspection of the momenta in the BFKL ladder, and can be applied to solving the non-forward BFKL equation to next-to-leading logarithmic accuracy, when the corresponding kernel is available.

Modified instanton profile effects from lattice Green functions

We trace here instantons through the analysis of pure Yang-Mills gluon Green functions in the Landau gauge for a window of IR momenta (0.4 GeV $< k < 0.9$ GeV). We present lattice results that can be fitted only after substituting the BPST profile in the Instanton liquid model (ILM) by one based on the Diakonov and Petrov variational methods. This also leads us to gain information on the parameters of ILM.

The gluon Green's function in [formula omitted] supersymmetric Yang–Mills theory

The high energy limit of scattering amplitudes in the N = 4 supersymmetric Yang–Mills theory is studied by solving the corresponding BFKL equation in the next-to-leading approximation. The gluon Green's function is analysed using a newly proposed method suitable for investigating the contribution from higher conformal spins. From this new approach complete agreement is obtained with the results of Kotikov and Lipatov on conformal spins and angular dependence.

The gluon Green's function in the BFKL approach at next-to-leading logarithmic accuracy

We investigate the gluon Green's function in the high energy limit of QCD using a recently proposed iterative solution of the Balitsky–Fadin–Kuraev–Lipatov (BFKL) equation at next-to-leading logarithmic (NLL) accuracy. To establish the applicability of this method in the NLL approximation we solve the BFKL equation as originally written by Fadin and Lipatov, and compare the results with previous studies in the leading logarithmic (LL) approximation.

Extending QCD perturbation theory to higher energies

On the basis of the results of a new renormalisation group improved small-x resummation scheme, we argue that the range of validity of perturbative calculations is considerably extended in rapidity with respect to leading log expectations. We thus provide predictions for the energy dependence of the gluon Green function in its perturbative domain and for the resummed splitting function. As in previous analyses, high-energy exponents are reduced to phenomenologically acceptable values. Additionally, interesting preasymptotic effects are observed. In particular, the splitting function shows a shallow dip in the moderate small-x region, followed by the expected power increase.