BK Equation Research Articles

Conformal BK equation at QCD Wilson-Fisher point

High-energy scattering in pQCD in the Regge limit is described by the evolution of Wilson lines governed by the BK equation [1, 2]. In the leading order, the BK equation is conformally invariant and the eigenfunctions of the linearized BFKL equation are powers. It is a common belief that at d ≠ 4 the BFKL equation is useless since unlike d = 4 case it cannot be solved by usual methods. However, we demonstrate that at critical Wilson-Fisher point of QCD the relevant part of NLO BK restores the conformal invariance so the solutions are again powers. As a check of our approach to high-energy amplitudes at the Wilson-Fisher point, we calculate the anomalous dimensions of twist-2 light-ray operators in the Regge limit j → 1.

Bifurcation Theory, Lie Group-Invariant Solutions of Subalgebras and Conservation Laws of a Generalized (2+1)-Dimensional BK Equation Type II in Plasma Physics and Fluid Mechanics

The nonlinear phenomena in numbers are modelled in a wide range of fields such as chemical physics, ocean physics, optical fibres, plasma physics, fluid dynamics, solid-state physics, biological physics and marine engineering. This research article systematically investigates a (2+1)-dimensional generalized Bogoyavlensky–Konopelchenko equation. We achieve a five-dimensional Lie algebra of the equation through Lie group analysis. This, in turn, affords us the opportunity to compute an optimal system of fourteen-dimensional Lie subalgebras related to the underlying equation. As a consequence, the various subalgebras are engaged in performing symmetry reductions of the equation leading to many solvable nonlinear ordinary differential equations. Thus, we secure different types of solitary wave solutions including periodic (Weierstrass and elliptic integral), topological kink and anti-kink, complex, trigonometry and hyperbolic functions. Moreover, we utilize the bifurcation theory of dynamical systems to obtain diverse nontrivial travelling wave solutions consisting of both bounded as well as unbounded solution-types to the equation under consideration. Consequently, we generate solutions that are algebraic, periodic, constant and trigonometric in nature. The various results gained in the study are further analyzed through numerical simulation. Finally, we achieve conservation laws of the equation under study by engaging the standard multiplier method with the inclusion of the homotopy integral formula related to the obtained multipliers. In addition, more conserved currents of the equation are secured through Noether’s theorem.

Variety interaction solutions comprising lump solitons for a generalized BK equation by trilinear analysis

Variety interaction solutions comprising lump solitons for a generalized BK equation by trilinear analysis

Modulational instability and multiple rogue wave solutions for the generalized CBS–BK equation

An integrable of the generalized Calogero–Bogoyavlenskii–Schiff–Bogoyavlensky–Konopelchenko (CBS–BK) equation is studied, by employing Hirota’s bilinear method the bilinear form is obtained, and the multiple-soliton solutions are constructed. The modified of improved bilinear method has been utilized to investigate multiple solutions. In addition, some graphs including 3D, contour, density, and [Formula: see text]-curves plots of the addressed equation with specific coefficients are shown. Finally, under certain conditions, the asymptotic behavior of the linearization solution is analyzed to prove that the modulation instability is stable for some points.

Sudakov effects in central-forward dijet production in high energy factorization

We discuss central-forward dijet production at LHC energies within the framework of high energy factorization. In our study, we profit from the recent progress on consistent merging of Sudakov resummation with small-x effects, which allows us to compute two different gluon distributions which depend on longitudinal momentum, transverse momentum and the hard scale of the process: one for the quark channel and one for the gluon channel. The small-x resummation is included by means of the BK equation supplemented with a kinematic constraint and subleading corrections. We test the new gluon distributions against existing CMS data for transverse momentum spectra in forward-central dijet production. We obtain results which are largely consistent with our earlier predictions based on model implementation of Sudakov form factors. In addition, we study dijet azimuthal decorrelations for the forward-central jets, which are known to be sensitive to the modeling of soft radiation.

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 fan diagrams in perturbative QCD

Within QCD reggeon field theory we study the formation of two subsequent triple pomeron vertices in the process P→PP→PPP. We make use of an earlier investigation [1] of the six-reggeon amplitude in deep inelastic scattering and show that in the large-Nc limit pomeron fan diagrams emerge with the same triple pomeron vertex in all places. We thus confirm the BK-equation, but we also find additional terms related to the reggeization of the gluon, and we discuss their potential significance. Our analysis also includes the general pomeron → two odderon vertex: a particular version of this vertex has been included into earlier generalizations the BK equation.

Deep inelastic scattering in the dipole picture at next-to-leading order

We study quantitatively the importance of the recently derived NLO corrections to the DIS structure functions at small x in the dipole formalism. We show that these corrections can be significant and depend on the factorization scheme used to resum large logarithms of energy into renormalization group evolution with the BK equation. This feature is similar to what has recently been observed for single inclusive forward hadron production. Using a factorization scheme consistent with the one recently proposed for the single inclusive cross section, we show that it is possible to obtain meaningful results for the DIS cross sections.

Including resummation in the NLO BK equation

We include a resummation of large transverse momentum logarithms in the next-to-leading order (NLO) Balitsky-Kovchegov equation. The resummed evolution equation is shown to be stable, the evolution speed being significantly reduced by NLO corrections. The contributions from NLO terms that are not enhanced by large logarithms are found to be numerically important close to phenomenologically relevant initial conditions. We numerically determine the value for the constant in the resummed logarithm that includes a maximal part of the full NLO terms in the resummation.

Approximate solution to the NLL BK equation in the saturation region

We analytically solve the full next-to-leading logarithmic Balitsky-Kovchegov equation in the saturation regime, which includes corrections from quark and gluon loops, and large double transverse logarithms. The analytic result for the $S$-matrix in the saturation regime shows that the linear rapidity decrease with rapidity of the exponent in running coupling case is replaced by rapidity raised to power of 3/2 decreasing. The collinearly-improved Balitsky-Kovchegov equation are also analytically solved in the saturation regime. It shows that the double collinear logarithms do not contribute to the $S$-matrix and the solution is the same as the one obtained from the leading order Balitsky-Kovchegov equation. The numerical solutions to the leading order and full next-to-leading logarithmic Balitsky-Kovchegov equations are performed in order to test the analytic results derived in the saturation regime.

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.

Resumming large higher-order corrections in non-linear QCD evolution

Linear and non-linear QCD evolutions at high energy suffer from severe issues related to convergence, due to higher order corrections enhanced by large double and single transverse logarithms. We resum double logarithms to all orders 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 resulting collinearly improved 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.

Neutrino-nucleon cross section at ultrahigh energy and its astrophysical implications

We present a quantitative study of the $\nu N$ cross section in the neutrino energy range $10^4<E_{\nu}<10^{14}$ GeV within two transversal QCD approaches: NLO DGLAP evolution using different sets of PDFs and BK small-$x$ evolution with running coupling and kinematical corrections. We show that the non-linear effects embodied in the BK equation yield a slower raise in the cross section for $E_{\nu}\gtrsim 10^{8}$ GeV than the usual DGLAP based calculation. Finally, we translate this theoretical uncertainty into upper bounds for the ultra-high-energy neutrino flux for different experiments.

Resumming double logarithms in the QCD evolution of color dipoles

The higher-order perturbative corrections, beyond leading logarithmic accuracy, to the BFKL evolution in QCD at high energy are well known to suffer from a severe lack-of-convergence problem, due to radiative corrections enhanced by double collinear logarithms. Via an explicit calculation of Feynman graphs in light cone (time-ordered) perturbation theory, we show that the corrections enhanced by double logarithms (either energy-collinear, or double collinear) are associated with soft gluon emissions which are strictly ordered in lifetime. These corrections can be resummed to all orders by solving an evolution equation which is non-local in rapidity. This equation can be equivalently rewritten in local form, but with modified kernel and initial conditions, which resum double collinear logs to all orders. We extend this resummation to the next-to-leading order BFKL and BK equations. The first numerical studies of the collinearly-improved BK equation demonstrate the essential role of the resummation in both stabilizing and slowing down the evolution.

The non-linear evolution of jet quenching

We construct a generalization of the JIMWLK Hamiltonian, going beyond the eikonal approximation, which governs the high-energy evolution of the scattering between a dilute projectile and a dense target with an arbitrary longitudinal extent (a nucleus, or a slice of quark-gluon plasma). Different physical regimes refer to the ratio L/τ between the longitudinal size L of the target and the lifetime τ of the gluon fluctuations. When L/τ ≪1, meaning that the target can be effectively treated as a shockwave, we recover the JIMWLK Hamiltonian, as expected. When L/τ ≫1, meaning that the fluctuations live inside the target, the new Hamiltonian governs phenomena like transverse momentum broadening and radiative energy loss, which accompany the propagation of an energetic parton through a dense QCD medium. Using this Hamiltonian, we derive a non-linear equation for the dipole amplitude (a generalization of the BK equation), which describes the high-energy evolution of jet quenching. As compared to the original BK-JIMWLK evolution, the new evolution is remarkably different: the plasma saturation momentum evolves much faster with increasing energy (or decreasing Bjorken’s x) than the corresponding scale for a shockwave. This widely opens the transverse phase-space for the evolution in the vicinity of the saturation line and implies the existence of large radiative corrections, enhanced by the double logarithm ln2(LT ), with T the temperature of the medium. This confirms from a wider perspective a recent result by Liou, Mueller, and Wu (arXiv:1304.7677). The dominant, double-logarithmic, corrections to the dipole amplitude are smooth enough to be absorbed into a renormalization of the jet quenching parameter $$ \widehat{q} $$ . This renormalization is universal: it applies to all the phenomena, like the transverse momentum broadening or the radiative energy loss, which can be computed from the dipole amplitude.

Improving the kinematics for low-xQCD evolution equations in coordinate space

High-energy evolution equations, such as the BFKL, BK or JIMWLK equations, aim at resumming the high-energy (next-to-)leading logarithms appearing in QCD perturbative series. However, the standard derivations of those equations are performed in a strict high-energy limit, whereas such equations are then applied to scattering processes at large but finite energies. For that reason, there is typically a slight mismatch between the leading logs resummed by those evolution equations without finite-energy corrections and the leading logs actually present in the perturbative expansion of any observable. That mismatch is one of the sources of large corrections at NLO and NLL accuracy. In the case of the BFKL equation in momentum space, that problem is solved by including a kinematical constraint in the kernel, which is the most important finite-energy correction. In this paper, such an improvement of kinematics is performed in mixed-space (transverse positions and $k^+$) and with a factorization scheme in the light-cone momentum $k^+$ (in a frame in which the projectile is right-moving and the target left-moving). This is the usual choice of variables and factorization scheme for the the BK equation. A kinematically improved version of the BK equation is provided, consistent at finite energies. The results presented here are also a necessary step towards having the high energy limit of QCD (including gluon saturation) quantitatively under control beyond strict leading logarithmic accuracy.

First passage of stochastically dynamical system with fractional derivative and power-form restoring force under Gaussian excitation

First passage problem in a dynamical system with power-form restoring force and fractional derivative is studied in this paper. At first, the original system is transformed into an diffusion differential equation by way of generalized Van der Pol transformation and stochastic averaging method. After that , according to the definition of first passage, BK equation and GP equation are constructed and solved. The numerical results tell us that the reliability probabilities are decreased monotonously with respect to time. Higher order of fractional derivative can lead to higher reliability of the system. Boundary value of safe domain can affect the reliability probability greatly. The larger it is, the higher probability value is.

Single inclusive particle production at high energy from HERA data to proton-nucleus collisions

We study single inclusive hadron production in proton-proton and proton-nucleus collisions in the CGC framework. The parameters in the calculation are determined solely by standard nuclear geometry and by electron-proton deep inelastic scattering data, which is fit using the running coupling BK equation. We show that it is possible to obtain a good fit of the HERA inclusive cross section also without an anomalous dimension in the initial condition. We argue that one must consistently use the proton transverse area as measured by a high virtuality probe in DIS also for the single inclusive cross section in proton-proton and proton-nucleus collisions. We show that this leads to a midrapidity nuclear modification ratio R_pA that approaches unity at high transverse momentum independently of sqrt(s), in contrast to most CGC calculations in the literature. We also present predictions for future forward R_pA measurements at the LHC.

Gluon saturation at higher orders and improvement of kinematics

Standard perturbative calculations lead to pathologically large NLO corrections to low-xBj evolution equations like BFKL and BK. Using a more refined treatment of kinematics in mixed-space, relevant when gluon saturation sets on, one obtains an improved version of the BK equation, resumming to all orders the most severe of those large higher order corrections.

Forward dihadron correlations in deuteron–gold collisions with a Gaussian approximation of JIMWLK

We compute dihadron correlations in forward deuteron–gold or proton–gold collisions. The running coupling BK equation is used to calculate the energy dependence of the dipole cross sections and extended to higher-point Wilson line correlators using a factorized Gaussian approximation. Unlike some earlier works we include both the “inelastic” and “elastic” contributions to the dihadron cross section. We show that the double parton scattering contribution is included in our calculation and obtain both an away side peak that roughly agrees with experimental observations and an estimate for the azimuthal angle-independent pedestal. We find that nonlinear effects for momenta close to the saturation scale are clearly visible in the away side peak structure.