Geometric scaling as traveling waves

Abstract

We show the relevance of the nonlinear Fisher and Kolmogorov-Petrovsky- Piscounov (KPP) equation to the problem of high energy evolution of the QCD amplitudes. We explain how the traveling wave solutions of this equation are related to geometric scaling, a phenomenon observed in deep-inelastic scattering experiments. Geometric scaling is for the first time shown to result from an exact solution of nonlinear QCD evolution equations. Using general results on the KPP equation, we compute the velocity of the wave front, which gives the full high energy dependence of the saturation scale.

4 pages, 1 figure. v2: references added

Cited by in corpus (239)

• The Color Glass Condensate
• Exclusive diffractive processes at HERA within the dipole picture
• Saturation Physics and Deuteron--Gold Collisions at RHIC
• Evolution at small x_bj: The Color Glass Condensate
• Prompt neutrino fluxes from atmospheric charm
• Color Glass Condensate at the LHC: hadron multiplicities in $pp$, $pA$ and $AA$ collisions
• Gluon saturation and initial conditions for relativistic heavy ion collisions
• QCD thermalization: Ab initio approaches and interdisciplinary connections
• Traveling wave fronts and the transition to saturation
• Relating high-energy lepton-hadron, proton-nucleus and nucleus-nucleus collisions through geometric scaling
• Solving The High Energy Evolution Equation Including Running Coupling Corrections
• Impact-parameter dependent Color Glass Condensate dipole model and new combined HERA data
• Single Inclusive Hadron Production at RHIC and the LHC from the Color Glass Condensate
• Collinearly-improved BK evolution meets the HERA data
• Universal behavior of QCD amplitudes at high energy from general tools of statistical physics
• On the linearly polarized gluon distributions in the color dipole model
• Small-x physics beyond the Kovchegov equation
• Effect of selection on ancestry: an exactly soluble case and its phenomenological generalization
• Odderon in the Color Glass Condensate
• A phenomenological theory giving the full statistics of the position of fluctuating pulled fronts
• A Langevin equation for high energy evolution with pomeron loops
• Color Glass Condensate and Glasma
• Numerical analysis of the Balitsky-Kovchegov equation with running coupling: dependence of the saturation scale on nuclear size and rapidity
• Saturation QCD predictions with heavy quarks at HERA
• Universality and tree structure of high energy QCD
• Study of the Fundamental Structure of Matter with an Electron-Ion Collider
• Extension of the JIMWLK Equation in the Low Gluon Density Region
• Quark pair production in high energy pA collisions: General features
• Deep Inelastic Scattering in Conformal QCD
• Non-linear evolution in QCD at high-energy beyond leading order
• Deep inelastic scattering at strong coupling from gauge/string duality : the saturation line
• Unintegrated gluon distribution from modified BK equation
• Gluon saturation and inclusive hadron production at LHC
• Transverse-momentum-dependent gluon distributions from JIMWLK evolution
• CGC factorization for forward particle production in proton-nucleus collisions at next-to-leading order
• Back-to-Back Correlations of Di-hadrons in dAu Collisions at RHIC
• Saturation in Deep Inelastic Scattering from AdS/CFT
• Diffusive scaling and the high-energy limit of deep inelastic scattering in QCD at large N_c
• Small x Phenomenology - summary of the 3rd Lund Small x Workshop in 2004
• Exclusive vector meson production at HERA from QCD with saturation
• Balitsky's hierarchy from Mueller's dipole model and more about target correlations
• On the small-x evolution of the color quadrupole and the Weizsäcker-Williams gluon distribution
• High Gluon Densities in Heavy Ions Collisions
• Cronin effect and high-p_T suppression in the nuclear gluon distribution at small x
• Gluon saturation and energy dependence of hadron multiplicity in pp and AA collisions at the LHC
• A unified description of diffractive deep inelastic scattering with saturation
• Geometric scaling in diffractive deep inelastic scattering
• Semi-inclusive Deep Inelastic Scattering at small x
• Saturation and shadowing in high-energy proton-nucleus dilepton production
• Summing Pomeron loops in the dipole approach
• Higher-point correlations from the JIMWLK evolution
• Baryon Stopping as a new Probe of Geometric Scaling
• The non-linear evolution of jet quenching
• The high energy asymptotics of scattering processes in QCD
• Systematics of geometric scaling
• JIMWLK evolution in the Gaussian approximation
• Intrinsic Fluctuations of the Proton Saturation Momentum Scale in High Multiplicity p+p Collisions
• Treading on the cut: Semi inclusive observables at high energy
• Deep inelastic structure functions from supergravity at small x
• A QCD dipole formalism for forward-gluon production
• The survival probability of a branching random walk in presence of an absorbing wall
• Classical and quantum entropy of parton distributions
• Higher Order Effects in Non Linear Evolution from a Veto in Rapidities
• Solution for the BFKL Pomeron Calculus in zero transverse dimensions
• On possible implications of gluon number fluctuations in DIS data
• Gluon splitting in a shockwave
• Dynamical entropy of dense QCD states
• Color Glass Condensate and Glasma
• Fluctuations effects in high-energy evolution of QCD
• Production of Forward Rapidity Photons in High Energy Heavy Ion Collisions
• Next-to-leading and resummed BFKL evolution with saturation boundary
• Pomeron loop and running coupling effects in high energy QCD evolution
• Pomeron loops in zero transverse dimensions
• TMD gluon distributions at small x in the CGC theory
• Baryon stopping and saturation physics in relativistic collisions
• A novel formulation of the unintegrated gluon distribution for DIS
• On the probability distribution of the stochastic saturation scale in QCD
• Exclusive Higgs boson production at the LHC: hard rescattering corrections
• Geometric scaling in high-energy QCD at nonzero momentum transfer
• Hadrons and direct photon in pp and pA collisions at LHC and saturation effects
• The Balitsky-Kovchegov equation in full momentum space
• Geometry of Rank Reduction
• Drell-Yan Lepton-Pair-Jet Correlation in pA collisions
• Traveling waves and geometric scaling at non-zero momentum transfer
• On the Probabilistic Interpretation of the Evolution Equations with Pomeron Loops in QCD
• Quantum chromodynamics at high energy and statistical physics
• High energy amplitude in the dipole approach with Pomeron loops: asymptotic solution
• Scaling Phenomena in Gravity from QCD
• Forward gluon production in hadron-hadron scattering with Pomeron loops
• A zero-dimensional model for high-energy scattering in QCD
• Negative velocity fluctuations of pulled reaction fronts
• Quasi-stationary regime of a branching random walk in presence of an absorbing wall
• Correlation of small-x gluons in impact parameter space
• Geometrical Scaling and the Dependence of the Average Transverse Momentum on the Multiplicity and Energy for the ALICE Experiment
• Geometrical scaling for identified particles
• The Asymptotic Form of Non-Global Logarithms, Black Disc Saturation, and Gluonic Deserts
• Algebraic models for the hierarchy structure of evolution equations at small x
• Solving effective field theory of interacting QCD pomerons in the semi-classical approximation
• Direct photons at forward rapidities in high-energy pp collisions
• Has saturation physics been observed in deuteron-gold collisions at RHIC?
• One-dimensional model for QCD at high energy
• A modified Balitsky-Kovchegov equation
• An Analytical Solution of the Balitsky-Kovchegov Equation with the Homogeneous Balance Method
• Consequences of strong fluctuations on high-energy QCD evolution
• Geometrical Scaling in Inelastic Inclusive Particle Production at the LHC
• ${\gamma }^{\ast }p$ cross section from the dipole model in momentum space
• Investigating the exclusive protoproduction of dileptons at high energies
• Theoretical overview: towards understanding the quark-gluon plasma
• Systematic Analysis of Scaling Properties in Deep Inelastic Scattering
• Probing proton fluctuations with asymmetric rapidity correlations
• CCFM Evolution with Unitarity Corrections
• Reaction-diffusion processes in zero transverse dimensions as toy models for high-energy QCD
• On parton number fluctuations at various stages of the rapidity evolution
• CGC/saturation approach: a new impact-parameter dependent model in the next-to-leading order of perturbative QCD
• Non-linear QCD dynamics and exclusive production in $ep$ collisions
• Analytical asymptotics for hard diffraction
• QCD traveling waves at non-asymptotic energies
• Testing the Gaussian Approximation to the JIMWLK Equation
• Large-displacement statistics of the rightmost particle of the one-dimensional branching Brownian motion
• High-energy dipole scattering amplitude from evolution of low-energy proton light-cone wave functions
• Non-linear equation: energy conservation and impact parameter dependence
• Parton saturation scaling function for exclusive production of vector mesons and Deeply Virtual Compton Scattering
• O(α_s^2)-corrections to JIMWLK evolution from the classical equations of motion
• Geometric Scaling and QCD Dynamics in DIS
• Emergence of fluctuating traveling front solutions in macroscopic theory of noisy invasion fronts
• Nuclear DVCS at small-x using the color dipole phenomenology
• Quantitative Study of Different Forms of Geometrical Scaling in Deep Inelastic Scattering at HERA
• QCD saturation in the dipole sector with correlations
• Correlations in impact-parameter space in a hierarchical saturation model for QCD at high energy
• On the Wess-Zumino term in high energy QCD
• The Color Glass Condensate: an overview
• Page Entropy of Proton System in Deep-Inelastic-Scattering at Small-$x$
• Diffractive electron-nucleus scattering and ancestry in branching random walks
• Rapidity gap distribution in diffractive deep-inelastic scattering and parton genealogy
• Testing Non-linear Evolution with Running Coupling Corrections in $ep$ and $pp$ collisions
• On the longitudinal structure function in the dipole model
• On correlations and discreteness in non-linear QCD evolution
• Lecture notes on "Quantum chromodynamics and statistical physics"
• QCD Saturation Equations including Dipole-Dipole Correlation
• Liouville field theory for gluon saturation in QCD at high energy
• Geometric scaling in ultrahigh energy neutrinos and nonlinear perturbative QCD
• BFKL Pomeron: modeling confinement
• Validity of the Brunet-Derrida formula for the speed of pulled fronts with a cutoff
• Total gluon shadowing due to fluctuation effects
• CGC/saturation approach for high energy soft interactions: ${\mathbf{v}}_2$ in proton-proton collisions
• Incoherent diffractive dijet production in electron DIS off nuclei at high energy
• Fisher Waves: an individual based stochastic model
• Travelling waves and impact-parameter correlations
• Random Energy Model with complex replica number, complex temperatures and classification of the string's phases
• Non linear evolution: revisiting the solution in the saturation region
• Prompt photon production in high-energy $pA$ collisions at forward rapidity
• The suppressions of dijet azimuthal correlations in the future EIC
• High energy scattering in QCD as a statistical process
• Parametric form of QCD travelling waves
• A simple model for the nuclear unintegrated gluon distribution
• Solution to the Balitsky-Kovchegov equation in the saturation domain
• BFKL and CCFM evolutions with saturation boundary
• Color Glass Condensate and its relation to HERA physics
• Geometric Scaling in Mueller-Navelet Jets
• Solution of the Fan Diagram Equation in 2+1 Dimensional QCD
• QCD traveling waves beyond leading logarithms
• Saturation momentum scale extracted from semi-inclusive transverse spectra in high-energy pp collisions
• Fisher Equation for a Decaying Brane
• Novel Cross Section Ratios as Possible Signals of Saturation in UPCs
• Inclusive hadron and photon production at LHC in dipole momentum space
• Violation of Geometrical Scaling in pp Collisions at NA61/SHINE
• Compatibility of phenomenological dipole cross sections with the Balitsky-Kovchegov equation
• Geometric scaling behavior of the scattering amplitude for DIS with nuclei
• QCD at small x and nucleus-nucleus collisions
• Dense-dilute factorization for a class of stochastic processes and for high energy QCD
• Scaling properties of direct photon yields in heavy ion collisions
• Universality of QCD traveling-waves with running coupling
• Semiclassical solution to the BFKL equation with massive gluons
• Universality of traveling waves with QCD running coupling
• Exclusive processes in $ep$ collisions at the EIC and LHeC: A closer look on the predictions of saturation models
• Asymptotics of QCD traveling waves with fluctuations and running coupling effects
• Violation of the geometric scaling behaviour of the amplitude for running QCD coupling in the saturation region
• Analytic solution of Balitsky-Kovchegov equation with running coupling constant using homogeneous balance method
• QCD traveling waves phenomenology revisited
• DIS and the effects of fluctuations: a momentum space analysis
• Exclusive vector meson productions with the analytical solution of Balitsky-Kovchegov Equation
• Saturation and linear transport equation
• Extended Geometric Scaling from Generalized Traveling Waves
• Probing gluon number fluctuation effects in future electron-hadron colliders
• Nuclear scattering configurations of onia in different frames
• An analytical solution of Balitsky-Kovchegov equation using homotopy perturbation method
• The Status of Parton Saturation and the CGC
• On the anomalous dimensions of the multiple pomeron exchanges
• Scattering from an external field in quantum chromodynamics at high energies: from foundations to interdisciplinary connections
• Traveling wave solution of the Reggeon Field Theory
• Classicalization and unitarization of wee partons in QCD and Gravity: The CGC-Black Hole correspondence
• Exclusive vector meson production with leading neutrons in a saturation model for the dipole amplitude in mixed space
• BFKL ansatz for BK equation in conformal basis
• High density QCD and nucleus-nucleus scattering deeply in the saturation region
• Infrared instability from nonlinear QCD evolution
• Strong interactions in air showers
• Traveling waves in discretized Balitsky-Kovchegov evolution
• High energy electroweak DVCS processes in neutrino-nucleon scattering
• Soft processes at high energy without soft Pomeron: a QCD motivated model
• Reaction-diffusion equation for quark-hadron transition in heavy-ion collisions
• Effects of gluon number fluctuations on photon - photon collisions at high energies
• Running coupling and pomeron loop effects on inclusive and diffractive DIS cross sections
• Fragility of reaction-diffusion models to competing advective processes
• Azimuthal angle correlations in hadron-nucleus scattering: enhanced diagrams
• Search for exotic baryons in double radiative capture on pionic hydrogen
• Probabilistic picture for particle number densities in stretched tips of the branching Brownian motion
• Low-$x$ Physics in $pA$ Collisions and at the EIC
• Momentum space saturation model for deep inelastic scattering and single inclusive hadron production
• Evolution equation for elastic scattering of hadrons
• Resummation of projectile-target multiple scatterings and parton saturation
• Twist decomposition of non-linear effects in Balitsky-Kovchegov evolution of proton structure functions
• On the maximal noise for stochastic and QCD traveling waves
• Non-linear evolution of unintegrated gluon density at large values of coupling constant
• Setting Initial Conditions for Inflation with Reaction-Diffusion Equation
• Saturation momentum at fixed and running QCD coupling
• Recent results on saturation and CGC
• Homotopy approach for scattering amplitude for running QCD coupling
• Investigation of proton structure function $F_2^p$ at HERA in light of an analytical solution to Balitsky-Kovchegov equation
• Spin-glass model of QCD near saturation
• From Classical to Quantum Saturation in the Nuclear Gluon Distribution
• Traveling waves and the renormalization group improved Balitsky-Kovchegov equation
• FKPP dynamics mediated by a parent field with a delay
• First correction to JIMWLK evolution from the classical equations of motion
• Monte Carlo study of the tip region of branching random walks evolved to large times
• Possibility of DCC formation in pp collisions at LHC energy via reaction-diffusion equation
• Statistical properties of partonic configurations and diffractive dissociation in high-energy electron-nucleus scattering
• Saturation and geometrical scaling
• Anomalous diffusion in QCD matter
• The dipole picture in DIS: saturation and heavy quarks
• Universality aspects of quantum corrections to transverse momentum broadening in QCD media
• Geometrical Scaling of Direct Photons in Relativistic Heavy Ion and d+Au Collisions
• Forward and Mueller-Navelet Jets
• A geometrical scaling solution to the BFKL Pomeron Calculus in the saturation domain
• Anomalous dimension of transverse momentum broadening in planar $\mathcal{N}=4$ SYM
• Extraction of the color dipole amplitude with physics-informed neural networks
• Non-linear evolution equation for gluon density at large values of coupling constant
• Multiplicity distributions in DIS for heavy nucleus
• High Energy Collisions of Dense Hadrons in Quantum Chromodynamics : LHC Phenomenology and Universality of Parton Distributions
• Beyond BFKL