Balitsky-Kovchegov Equation Research Articles

D0 meson production in pp collisions at large Qs2

The impact of the nonlinear effects in the QCD on the observables is directly related to the magnitude of the saturation scale Qs, which is predicted to increase with the energy, rapidity, and multiplicity. In this paper, we investigate the D0 meson production in pp collisions at forward rapidities and/or high multiplicities considering the color glass condensate (CGC) formalism and the solutions of the running coupling Balitsky-Kovchegov equation. The contributions of gluon- and charm-initiated processes are taken into account, and a comparison with the current LHCb data is performed. The impact of an intrinsic charm component in the proton’s wave function is also estimated. Predictions for the self-normalized yields of D0 mesons as a function of the multiplicity of coproduced charged hadrons are presented, considering pp collisions at s=13 TeV and different values of the meson rapidity. A comparison with the predictions for the kaon and isolated photon production is performed. Our results indicate that a future experimental analysis of the D0 meson production at forward rapidities and high multiplicities can be useful to probe the CGC formalism and to disentangle the contribution of initial—and final—state effects. Published by the American Physical Society 2024

Inferring the initial condition for the Balitsky-Kovchegov equation

We apply Bayesian inference to determine the posterior likelihood distribution for the parameters describing the initial condition of the small-x Balitsky-Kovchegov evolution equation at leading logarithmic accuracy. The HERA structure function data is found to constrain most of the model parameters well. In particular, we find that the HERA data prefers an anomalous dimension γ≈1 unlike in previous fits where γ>1 which led to, e.g., the unintegrated gluon distribution and the quark-target cross sections not being positive definite. The determined posterior distribution can be used to propagate the uncertainties in the nonperturbative initial condition when calculating any other observable in the color glass condensate framework. We demonstrate this explicitly for the inclusive quark production cross section in proton-proton collisions and by calculating predictions for the nuclear modification factor for the F2 structure function in the EIC and LHeC/FCC-he kinematics. Published by the American Physical Society 2024

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.

Complete next-to-leading order calculation of single inclusive π0 production in forward proton-nucleus collisions

We present the first fully consistent calculation of inclusive π0 production at forward rapidities at next-to-leading order (NLO) accuracy in proton-lead collisions at s=8.16 TeV within the color glass condensate approach. The center-of-mass energy dependence is determined by the Balitsky-Kovchegov equation with the initial condition constrained at NLO accuracy by the proton structure function measurements. We find that the LHCb data further constrain this nonperturbative input, and that both the NLO corrections to the evolution equation and to the impact factor have significant effects on the nuclear modification factor RpPb. The complete NLO calculation is shown to be in qualitative agreement with the LHCb data, and the predictions are found to be sensitive to the details of the applied deep inelastic scattering (DIS) fit. These results demonstrate the need to perform global analyses of DIS and LHC data in order to probe gluon saturation phenomena at precision level at very small momentum fraction x. Published by the American Physical Society 2024

Analytic solution of the Balitsky-Kovchegov equation with a running coupling constant using the homogeneous balance method

Analytic solution of the Balitsky-Kovchegov equation with a running coupling constant using the homogeneous balance method

Solutions to the Balitsky-Kovchegov equation including the dipole orientation

Solutions of the target-rapidity Balitsky-Kovchegov (BK) equation are studied considering, for the first time, the complete impact-parameter dependence, including the orientation of the dipole with respect to the impact-parameter vector. In our previous work [1] it has been demonstrated that the spurious Coulomb tails could be tamed using the collinearly-improved kernel and an appropriate initial condition in the projectile-rapidity BK equation. Introducing a different interpretation of the evolution variable, the target-rapidity formulation of the BK equation brings non-locality in rapidity and a kernel modification, removing the term that previously helped to suppress the Coulomb tails. To address this newly emerged non-locality, three different prescriptions are explored here to take into account the rapidities preceding the initial condition. Two of these approaches induce mild Coulomb tails, while the other is free from this effect within the studied rapidity range. The range is chosen to correspond to that of interest for existing and future experiments. To demonstrate that this set up can be used for phenomenological studies, the obtained solutions are used to compute the F2 structure function of the proton and the diffractive photo- and electro-production of J/ψ off protons. The predictions agree well with HERA data, confirming that the target-rapidity Balitsky-Kovchegov equation with the full impact-parameter dependence is a viable tool to study the small Bjorken-x limit of perturbative QCD at current facilities like RHIC and LHC as well as in future colliders like the EIC.

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.

Kaon production in high multiplicity events at the Large Hadron Collider

The production of the ${K}_{S}^{0}$ meson in high multiplicity $pp$ collisions at $\sqrt{s}=$ 13 TeV is investigated considering the hybrid formalism and the solution of the running coupling Balitsky-Kovchegov (BK) equation. The associated cross section is estimated and compared with the experimental data for the transverse momentum spectrum. Moreover, we analyze the self-normalized yields of ${K}_{S}^{0}$ mesons as a function of the multiplicity of coproduced charged hadrons and demonstrate that a steep increasing is theoretically predicted. A comparison with the ALICE data is presented considering two distinct solutions of the BK equation.

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.

Exclusive vector meson production with the analytical solution of Balitsky-Kovchegov equation * *Supported by the Strategic Priority Research Program of Chinese Academy of Sciences (XDB34030301)

Exclusive vector meson production is an excellent probe for describing the structure of protons. In this study, based on the dipole model, the differential cross sections, total cross sections, and ratios of the longitudinal to transverse cross section of the and productions are calculated with the analytical solution of the Balitsky-Kovchegov (BK) equation. In addition, we consider the influences of two meson wave function models on the results. Our predictions, which are slightly sensitive to meson wave functions, agree with the experimental data. The analytical solution of the BK equation is reliable for description of exclusive vector meson productions in a certain range of .

Hadron production with collinearly-improved unintegrated gluon distributions in high energy proton-proton collisions * *Supported by the National Natural Science Foundation of China (12165004, 12061141008, 11975005), Guizhou Provincial Science and Technology Projects ([2019]5103, [2019]5653), the Education Department of Guizhou Province (KY[2021]131), the National Key Research and Development Program of China (2018YFE0104700, 2018YFE01014800)

The collinearly-improved Balitsky-Kovchegov (ciBK) equation evolved unintegrated gluon distribution (UGD) is used for the first time to study hadron production in high energy proton-proton collisions in order to improve the predictive power of the Color Glass Condensate effective theory. We show that the ciBK equation evolved UGD provides a relatively better description of LHC data on the transverse momentum and integrated multiplicity distributions of charged hadron and neutral pion production for several collision energies compared with the running coupling Balitsky-Kovchegov (rcBK) equation evolved UGD. This is because the ciBK evolved UGD has a sharper transverse momentum distribution than the rcBK UGD. The impact of running coupling prescriptions on hadron production is studied, and it is found that the parent dipole and smallest dipole running coupling prescriptions provide similar depictions of the data. Moreover, the scale dependence of the fragmentation function is investigated by taking three typical values of scale. We find that the differences resulting from the scale dependence of the fragmentation function can be fully absorbed into the normalization factor, which lumps higher order corrections.

Signatures of gluon saturation from structure-function measurements

We study experimentally observable signals for nonlinear QCD dynamics in deep inelastic scattering (DIS) at small Bjorken variable $x$ and moderate virtuality $Q^2$, by quantifying differences between the linear Dokshitzer-Gribov-Lipatov-Altarelli-Parisi (DGLAP) evolution and nonlinear evolution with the Balitsky-Kovchegov (BK) equation. To remove the effect of the parametrization freedom in the initial conditions of both equations, we first match the predictions for the DIS structure functions $F_2$ and $F_{\rm L}$ from both frameworks in a region in $x,Q^2$ where both frameworks should provide an accurate description of the relevant physics. The differences in the dynamics are then quantified by the deviations when one moves away from this matching region. For free protons we find that the differences in $F_2$ remain at a few-percent level, while in $F_{\rm L}$ the deviations are larger, up to $10\,\%$ at the EIC and $40\,\%$ at the LHeC kinematics. With a heavy nucleus the differences are up to $10\,\%$ in $F_2$, and can reach $20\,\%$ and $60\,\%$ in $F_{\rm L}$ for the EIC and the LHeC, respectively.

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.

Running coupling effect in next-to-leading order Balitsky-Kovchegov evolution equations * *Supported by the National Natural Science Foundation of China (12165004, 12061141008, 11975005); the Fund of Science and Technology Department of Guizhou Province ([2019]5653); the Education Department of Guizhou Province (KY[2021]131); the National Key Research and Development Program of China (2018YFE0104700, 2018YFE0104800)

Balitsky-Kovchegov equations in projectile and target rapidity representations are analytically solved for fixed and running coupling cases in the saturation domain. Interestingly, we find that the respective analytic S-matrices in the two rapidity representations have almost the same rapidity dependence in the exponent in the running coupling case, which provides a method to explain why the equally good fits to HERA data were obtained when using three different Balitsky-Kovchegov equations formulated in the two representations. To test the analytic outcomes, we solve the Balitsky-Kovchegov equations and numerically compute the ratios between these dipole amplitudes in the saturation region. The ratios are close to one, which confirms the analytic results. Moreover, the running coupling, collinearly-improved, and extended full collinearly-improved Balitsky-Kovchegov equations are used to fit the HERA data. We find that all of them provide high quality descriptions of the data, and the obtained from the fits are similar. Both the analytic and numerical calculations imply that the Balitsky-Kovchegov equation at the running coupling level is robust and has a sufficiently strong predictive power at HERA energies; however, higher order corrections could be significant for future experiments, such as those at the EIC or LHeC.

Suppressions of dijet azimuthal correlations in the future EIC

Quark-antiquark pair (or dijet) production at the electron-ion collider (EIC) has been argued to be one of most important processes that allowing to access the Weizs\"acker-Williams (WW) gluon distributions at small $x$ limit. Within the framework of Color Glass Condensate (CGC) effective field theory (EFT), we calculated the dijet cross sections and the azimuthal correlations by including the Sudakov resummations, numerical results shown that the back-to-back correlations are significantly suppressed when the Sudakov resummations are taken into account. In addition, by using the solutions of running-coupling Balitsky-Kovchegov (rcBK) equation, the unpolarized and linearly polarized WW gluon distributions both in coordinate and momentum space are presented.

Exclusive and dissociative J/ψ production with collinear-improved Balitsky-Kovchegov equation * *Supported by the National Natural Science Foundation of China (1765005, 12042511), the Fund of Science and Technology Department of Guizhou Province ([2019]5653), Fund of Education Department of Guizhou Province (KY[2021]131), the National Key Research and Development Program of China (2018YFE0104700, CCNU18ZDPY04), the fund of China Postdoctoral Science Foundation (2018M633414)

We extend the hotspot model to include the virtuality dependence and use it to study the exclusive and dissociative production combined with the dipole amplitude in the target rapidity representation. We determined that virtuality takes effect on a number of hotspots, thus providing a better description of the production data at HERA. The collinear improved Balitsky-Kovchegove equation in the target rapidity representation is numerically solved and used to fit the experimental data with a series of hotspot sizes. We infer that virtuality significantly influences the number and size of hotspots. The expression resulting from the fit with the collinear improved dipole amplitude in the target rapidity representation is more reasonable than the corresponding originating from the leading order fit, which indicates that the collinear improved evolution equation in the target rapidity representation can provide a relatively good depiction of the exclusive and dissociative HERA data.

Photonuclear [formula omitted] production at the LHC: Proton-based versus nuclear dipole scattering amplitudes

The coherent photonuclear production of a J/ψ vector meson at the LHC has been computed using two different sets of solutions of the impact-parameter dependent Balitsky-Kovchegov equation. The nuclear dipole scattering amplitudes are obtained either from (i) solutions for this process off proton targets coupled with a Glauber-Gribov prescription, or (ii) from solutions obtained with an initial condition representing the nucleus. These approaches predict different cross sections, which are compared with existing data from ultra-peripheral collisions at the LHC. The latter approach seems to better describe current measurements. Future LHC data should be precise enough to select one of the two approaches as the correct one.

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.

Nuclear scattering configurations of onia in different frames

In the scattering of a small onium off a large nucleus at high center-of-mass energies, when the parameters are set in such a way that the cross section at fixed impact parameter is small, events are triggered by rare partonic fluctuations of the onium, which are very deformed with respect to typical configurations. Using the color dipole picture of high-energy interactions in quantum chromodynamics, in which the quantum states of the onium are represented by sets of dipoles generated by a branching process, we describe the typical scattering configurations as seen from different reference frames, from the restframe of the nucleus to frames in which the rapidity is shared between the projectile onium and the nucleus. We show that taking advantage of the freedom to select a frame in the latter class makes possible to derive complete asymptotic expressions for some boost-invariant quantities, beyond the total cross section, from a procedure which leverages the limited available knowledge on the properties of the solutions to the Balitsky-Kovchegov equation that governs the rapidity-dependence of total cross sections. We obtain in this way an analytic expression for the rapidity-distribution of the first branching of the slowest parent dipole of the set of those which scatter. This distribution provides an estimator of the correlations of the interacting dipoles, and is also known to be related to the rapidity-gap distribution in diffractive dissociation, an observable measurable at a future electron-ion collider. Furthermore, our result may be formulated as a more general conjecture, that we expect to hold true for any one-dimensional branching random walk model, on the branching time of the most recent common ancestor of all the particles that end up to the right of a given position.

Analytical solution of Balitsky-Kovchegov equation with homogeneous balance method

Nonlinear QCD evolution equations are essential tools in understanding the saturation of partons at small Bjorken $x_{\rm B}$, as they are supposed to restore an upper bound of unitarity for the cross section of high energy scattering. In this paper, we present an analytical solution of Balitsky-Kovchegov (BK) equation using the homogeneous balance method. The obtained analytical solution is similar to the solution of a traveling wave. By matching the gluon distribution in the dilute region which is determined from the global analysis of experimental data (CT14 analysis), we get a definitive solution of the dipole-proton forward scattering amplitude in the momentum space. Based on the acquired scattering amplitude and the behavior of geometric scaling, we present also a new estimated saturation scale $Q_s^2(x)$.