Kovchegov Equation Research Articles

Investigation of proton structure function F2p at HERA in light of an analytical solution to the Balitsky–Kovchegov equation

In this paper, the proton structure function F2p(x,Q2) at small-x is investigated using an analytical solution to the Balitsky–Kovchegov (BK) equation. In the context of the color dipole description of deep inelastic scattering (DIS), the structure function F2p(x,Q2) is computed by applying the analytical expression for the scattering amplitude N(k, Y) derived from the BK solution. At transverse momentum k and total rapidity Y, the scattering amplitude N(k, Y) represents the propagation of the quark-antiquark dipole in the color dipole description of DIS. Using the BK solution we extracted the integrated gluon density xg(x, Q 2) and then compared our theoretical estimation with the LHAPDF global data fits, NNPDF3.1sx and CT18. Finally, we have investigated the behavior of F2p(x,Q2) in the kinematic region of 10−5 ≤ x ≤ 10−2 and 2.5 GeV2 ≤ Q 2 ≤ 60 GeV2. Our predicted results for F2p(x,Q2) within the specified kinematic region are in good agreement with the recent high-precision data for F2p(x,Q2) from HERA (H1 Collaboration) and the LHAPDF global parametrization group NNPDF3.1sx.

Twist decomposition of non-linear effects in Balitsky–Kovchegov evolution of proton structure functions

Effects of non-linear small-x evolution of the gluon distribution given by the Balitsky–Kovchegov equation are analyzed within the collinear approximation framework. We perform a twist decomposition of the proton structure functions F2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$F_2$$\\end{document} and FL\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$F_{\ extrm{L}}$$\\end{document} obtained from the Balitsky–Kovchegov equation using the Mellin representation of the scattering cross-sections at high energies. In both the structure functions we find strong corrections coming from the non-linear effects in the gluon evolution at twist 2, and strongly suppressed higher twist effects. This implies that unitarization effects of high energy scattering amplitudes are mostly the leading twist effect. Furthermore we consider the double logarithmic limit of the Balitsky–Kovchegov equation for the collinear gluon distribution, and compare the result to the Gribov–Levin–Ryskin equation. We find that these two equations differ by two powers of the hard scale logarithm for the large scales.

An analytical solution of Balitsky–Kovchegov equation using homotopy perturbation method

An approximate analytical solution of the Balitsky–Kovchegov (BK) equation using the homotopy perturbation method (HPM) is suggested in this work. We carried out our work in perturbative Quantum Chromodynamics (QCD) (pQCD) dipole picture of deep inelastic scattering (DIS). The BK equation in momentum space with some change of variables and truncation of the Balitsky–Fadin–Kuraev–Lipatov (BFKL) kernel can be reduced to Fisher–Kolmogorov–Petrovsky–Piscounov (FKPP) equation. The observed geometric scaling phenomena are similar to the travelling wave solution of the FKPP equation. We solved the BK equation using the HPM. The obtained solution in this work also suggests the travelling wave nature of the measured scattering amplitude [Formula: see text] plotted at various rapidities. We also extracted the saturation momentum, [Formula: see text], from the obtained solution and plotted it against different rapidities. The result obtained in this work can be helpful for various phenomenological studies in high-density QCD.

Computational modeling of the Balitsky–Kovchegov equation and its numerical solution using hybrid B-spline collocation technique

The dynamics of high-energy scattering of hadrons are governed by Quantum Chromodynamics (QCD). In the parton model of QCD, scattering can be represented as a kind of reaction–diffusion process and consequently can be included in the category of the Fisher–Kolmogorov–Petrovsky–Piscounov (FKPP) equation. We perform a numerical study in the mean-field approximation of QCD evolution given by the Balitsky–Kovchegov (BK) equation, using the hybrid B-spline (HB-spline) collocation technique. The Rubin–Graves type linearization process is used to linearize the nonlinear terms. The accuracy and efficiency of the method are checked by taking numerical examples. It is noticed that the present method gives more accurate results than existing methods. The von Neumann stability analysis is also discussed for the discretized system of the BK equation.

Momentum transfer squared dependence of exclusive quarkonia photoproduction in ultraperipheral collisions

In this paper, we study fully differential quarkonia photoproduction observables in ultraperipheral collisions (UPCs) as functions of momentum transfer squared. We employ the dipole picture of the QCD part of the scattering with proton and nucleus targets, with the projectile being a quasi-real photon flux emitted by an incoming hadron. We analyse such observables for ground $J/\psi$, $\Upsilon(1S)$ and excited $\psi'$, $\Upsilon(2S)$ states whose light-front wave functions are obtained in the framework of interquark potential model incorporating the Melosh spin transformation. Two different low-$x$ saturation models, one obtained by solving the Balitsky--Kovchegov equation with the collinearly improved kernel and the other with a Gaussian impact-parameter dependent profile, are used to estimate the underlined theoretical uncertainties of our calculations. The results for the proton target and with charmonium in the final state are in agreement with the available HERA data, while in the case of nucleus target we make predictions for $\gamma A$ and $AA$ differential cross sections at different $W$ and at $\sqrt{s}=5.02$ TeV, respectively.

First measurement of the |t|-dependence of coherent J/ψ photonuclear production

The first measurement of the cross section for coherent J/ψ photoproduction as a function of |t|, the square of the momentum transferred between the incoming and outgoing target nucleus, is presented. The data were measured with the ALICE detector in ultra-peripheral Pb–Pb collisions at a centre-of-mass energy per nucleon pair sNN=5.02TeV with the J/ψ produced in the central rapidity region |y|<0.8, which corresponds to the small Bjorken-x range (0.3−1.4)×10−3.The measured |t|-dependence is not described by computations based only on the Pb nuclear form factor, while the photonuclear cross section is better reproduced by models including shadowing according to the leading-twist approximation, or gluon-saturation effects from the impact-parameter dependent Balitsky–Kovchegov equation. These new results are therefore a valid tool to constrain the relevant model parameters and to investigate the transverse gluonic structure at very low Bjorken-x.

Thermal radiation and inclusive production in the running coupling $$k_T$$-factorization approach

The characteristics of the thermal radiation are investigated using a two - component model, with the hard component being described by the Color Glass Condensate formalism. The inclusive transverse momentum spectra of charged hadrons produced in proton - proton and proton - nucleus collisions at LHC energies and large - $p_T$ are estimated using the running coupling $k_T$ - factorization formula and the solution of the Balitsky - Kovchegov equation. Our results indicate that the thermal term is necessary to describe the experimental data and that the effective thermal temperature has an energy dependence similar to the saturation scale. We demonstrate that the enhancement of the thermal temperature in $pPb$ collisions is consistent with that predicted by the saturation scale.

Contribution of the non-linear term in the Balitsky–Kovchegov equation to the nuclear structure functions

In this paper, we present nuclear structure functions calculated from the impact-parameter dependent solution of the Balitsky-Kovchegov equation with our recently proposed set of nuclear initial conditions. We calculate the results with and without the non-linear term in the BK equation in order to study the impact of saturation effects on the measurable structure functions and nuclear modification factor. The difference of these results rises with decreasing Bjorken $x$ and increasing scale. These predictions are of interest to the physics program at the future ep and eA colliders.

Pomeron Fans for Finite Nuclei

In perturbative QCD, we study a set of pomeron fan diagrams propagating from a projectile to A nucleons in the nucleus target for finite A and investigate its behavior at large A and convergence as A→∞. We find convergence for only very small values of the coupling constant αs, much smaller than the commonly assumed values. We compare the results with the solution of the Balitski–Kovchegov equation.

Transverse momentum dependent parton distributions at small-x

We study the transverse momentum dependent (TMD) parton distributions at small-x in a consistent framework that takes into account the TMD evolution and small-x evolution simultaneously. The small-x evolution effects are included by computing the TMDs at appropriate scales in terms of the dipole scattering amplitudes, which obey the relevant Balitsky–Kovchegov equation. Meanwhile, the TMD evolution is obtained by resumming the Collins–Soper type large logarithms emerged from the calculations in small-x formalism into Sudakov factors.

The collinearly-improved Balitsky–Kovchegov equation

The high-energy evolution in perturbative QCD suffers from a severe lack-of-convergence problem, due to higher order corrections enhanced by double and single transverse logarithms. We resum double logarithms to all orders within the non-linear Balitsky-Kovchegov equation, by taking into account successive soft gluon emissions strongly ordered in lifetime. We further resum single logarithms generated by the first non-singular part of the splitting functions and by the one-loop running of the coupling. The resummed BK equation admits stable solutions, which are used to successfully fit the HERA data at small x for physically acceptable initial conditions and reasonable values of the fit parameters.

Leptons from heavy-quark semileptonic decay in pA collisions within the CGC framework

We study single lepton production from semileptonic decays of heavy flavor hadrons (D,B→l) in pp and pA collisions at RHIC and the LHC within the saturation/Color-Glass-Condensate (CGC) framework. Using the gluon distribution function obtained with the dipole amplitude, whose energy dependence is described by the Balitsky–Kovchegov equation with running coupling effect, we compute the transverse-momentum (p⊥) spectra of the lepton yields at mid and forward rapidities. We find that a large fraction of leptons at low p⊥ stems from the saturation regime of the incoming gluons in the target, especially in pA collisions at the LHC. The resultant p⊥ spectra is slightly harder than the data, but the nuclear modification factor seems consistent with the data within some uncertainty. We also update the nuclear modification factors for J/ψ and D meson at the LHC energy.

Collinearly-improved BK evolution meets the HERA data

In a previous publication, we have established a collinearly-improved version of the Balitsky–Kovchegov (BK) equation, which resums to all orders the radiative corrections enhanced by large double transverse logarithms. Here, we study the relevance of this equation as a tool for phenomenology, by confronting it to the HERA data. To that aim, we first improve the perturbative accuracy of our resummation, by including two classes of single-logarithmic corrections: those generated by the first non-singular terms in the DGLAP splitting functions and those expressing the one-loop running of the QCD coupling. The equation thus obtained includes all the next-to-leading order corrections to the BK equation which are enhanced by (single or double) collinear logarithms. We then use numerical solutions to this equation to fit the HERA data for the electron–proton reduced cross-section at small Bjorken x. We obtain good quality fits for physically acceptable initial conditions. Our best fit, which shows a good stability up to virtualities as large as Q2=400 GeV2 for the exchanged photon, uses as an initial condition the running-coupling version of the McLerran–Venugopalan model, with the QCD coupling running according to the smallest dipole prescription.

Gluon saturation beyond (naive) leading logs

Abstract An improved version of the Balitsky–Kovchegov equation is presented, with a consistent treatment of kinematics. That improvement allows to resum the most severe of the large higher order corrections which plague the conventional versions of high-energy evolution equations, with approximate kinematics. This result represents a further step towards having high-energy QCD scattering processes under control beyond strict Leading Logarithmic accuracy and with gluon saturation effects.

Remarks on diffractive dissociation within JIMWLK evolution at NLO

We discuss the high energy diffractive dissociation in DIS at the Next to Leading Order. In the large Nc dipole limit we derive the NLO version of the Kovchegov–Levin equation. We argue that the original structure of the equation is preserved, that is it coincides with the Balitsky–Kovchegov equation at NLO.

Dipole-dipole scattering in CGC/saturation approach at high energy: summing Pomeron loops

In this paper we demonstrate that the dense system of partons (gluons) can be produced in dilute-dilute system scattering, using the example of dipole-dipole collisions. This increase in density stems from the intensive gluon cascades that can be described by the enhanced BFKL Pomeron diagrams (Pomeron loops). For the first time we found the analytical solution to the equation for diffraction production in the dipole-dense parton system scattering, using the simplified BFKL kernel. Having this solution as well as the solution to Balitsky- Kovchegov equation we developed technique that allowed us to calculate the total cross section, cross sections for single and double diffractions in the MPSI approximation. Calculating inclusive production and two gluon correlations we see that the dense and strongly correlated system of gluons can be produced at high energy in the dipole-dipole scattering.

The NLO inclusive forward hadron production in pA collisions

Abstract Recently, the one-loop factorization for inclusive hadron productions in pA collisions in the saturation formalism has been established via the complete next-to-leading order calculation. The differential cross section is written into a factorization form in the coordinate space at the next-to-leading order, while the naive form of the convolution in the transverse momentum space does not hold. The rapidity divergence with small-x dipole gluon distribution of the nucleus is factorized into the energy evolution of the dipole gluon distribution function, which is known as the Balitsky–Kovchegov equation. Furthermore, the collinear divergences associated with the incoming parton distribution of the nucleon and the outgoing fragmentation function of the final state hadron are factorized into the splittings of the associated parton distribution and fragmentation functions, which allows us to reproduce the well-known DGLAP equation. The hard coefficient function, which is finite and free of divergence of any kind, is evaluated at one-loop order. This result is important, not only for the phenomenological applications to the inclusive hadron production in p-A collisions at RHIC and future LHC experiment, but also for theoretically promoting the rigorous developments towards factorizations in small-x physics.

Heavy quark pair production in high-energy pA collisions: Quarkonium

Quarkonium production in high-energy proton (deuteron)–nucleus collisions is investigated in the color glass condensate framework. We employ the color evaporation model assuming that the quark pair produced from dense small-x gluons in the nuclear target bounds into a quarkonium outside the target. The unintegrated gluon distribution at small Bjorken x in the nuclear target is treated with the Balitsky–Kovchegov equation with running coupling corrections. For the gluons in the proton, we examine two possible descriptions, unintegrated gluon distribution and ordinary collinear gluon distribution. We present the transverse momentum spectrum and nuclear modification factor for J/ψ production at RHIC and LHC energies, and those for ϒ(1S) at LHC energy, and discuss the nuclear modification factor and the momentum broadening by changing the rapidity and the initial saturation scale.

Running coupling and pomeron loop effects on inclusive and diffractive DIS cross sections

Within the framework of a (1+1)-dimensional model which mimics high-energy QCD, we study the behavior of the cross sections for inclusive and diffractive deep inelastic γ ∗ h scattering cross sections. We analyze the cases of both fixed and running coupling within the mean-field approximation, in which the evolution of the scattering amplitude is described by the Balitsky–Kovchegov equation, and also through the pomeron loop equations, which include in the evolution the gluon number fluctuations. In the diffractive case, similarly to the inclusive one, suppression of the diffusive scaling, as a consequence of the inclusion of the running of the coupling, is observed.

Geometric scaling behavior of the scattering amplitude for DIS with nuclei

The main question, that we answer in this paper, is whether the initial condition can influence on the geometric scaling behavior of the amplitude for DIS at high energy. We re-write the non-linear Balitsky–Kovchegov equation in the form which is useful for treating the interaction with nuclei. Using the simplified BFKL kernel, we find the analytical solution to this equation with the initial condition given by the McLerran–Venugopalan formula. This solution does not show the geometric scaling behavior of the amplitude deeply in the saturation region. On the other hand, the BFKL Pomeron calculus with the initial condition at x A = 1 / m R A given by the solution to Balitsky–Kovchegov equation, leads to the geometric scaling behavior. The McLerran–Venugopalan formula is the natural initial condition for the Color Glass Condensate (CGC) approach. Therefore, our result gives a possibility to check experimentally which approach: CGC or BFKL Pomeron calculus, is more satisfactory.