S. Munier

Geometric scaling as traveling waves.

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.

Universal behavior of QCD amplitudes at high energy from general tools of statistical physics

Abstract We show that high energy scattering is a statistical process essentially similar to reaction–diffusion in a system made of a finite number of particles. The Balitsky-JIMWLK equations correspond to the time evolution law for the particle density. The squared strong coupling constant plays the role of the minimum particle density. Discreteness is related to the finite number of partons one may observe in a given event and has a sizeable effect on physical observables. Using general tools developed recently in statistical physics, we derive the universal terms in the rapidity dependence of the saturation scale and the scaling form of the amplitude, which come as the leading terms in a large rapidity and small coupling expansion.

Universality and tree structure of high-energy QCD

Using nontrivial mathematical properties of a class of nonlinear evolution equations, we obtain the universal terms in the asymptotic expansion in rapidity of the saturation scale and of the unintegrated gluon density from the Balitskii-Kovchegov equation. These terms are independent of the initial conditions and of the details of the equation. The last subasymptotic terms are new results and complete the list of all possible universal contributions. Universality is interpreted in a general qualitative picture of high-energy scattering, in which a scattering process corresponds to a tree structure probed by a given source.

On the small-x evolution of the color quadrupole and the Weizsäcker–Williams gluon distribution

Abstract Color quadrupoles have been found to be important in the proper description of observables sensitive to the small- x regime in nuclei as well as in the operator definition of the Weizsacker–Williams gluon distribution. In this Letter, we derive the small- x evolution equation of the quadrupole and the Weizsacker–Williams gluon distribution without taking the large N c limit and study the properties of the equation in both dilute and saturation regime. We find that the quadrupole evolution follows the BFKL evolution in the dilute regime and then saturates in the dense region due to nonlinear terms. This leads us to conclude that the Weizsacker–Williams gluon distribution should obey the same geometrical behavior as the dipole gluon distribution as found in the inclusive DIS measurement.

The High energy asymptotics of scattering processes in QCD

High energy scattering in the QCD parton model was recently shown to be a reaction-diffusion process, and thus to lie in the universality class of the stochastic Fisher-Kolmogorov-Petrovsky-Piscounov equation. We recall that the latter appears naturally in the context of the parton model. We provide a thorough numerical analysis of the mean field approximation, given in QCD by the Balitsky-Kovchegov equation. In the framework of a simple stochastic toy model that captures the relevant features of QCD, we discuss and illustrate the universal properties of such stochastic models. We investigate in particular the validity of the mean field approximation and how it is broken by fluctuations. We find that the mean field approximation is a good approximation in the initial stages of the evolution in rapidity.

Diffractive photon dissociation in the saturation regime from the Good-Walker picture

Combining the QCD dipole model with the Good and Walker picture, we formulate diffractive dissociation of a photon of virtuality Q^2 off a hadronic target, in the kinematical regime in which Q is close to the saturation scale and much smaller than the invariant mass of the diffracted system. We show how the obtained formula compares to the HERA data and discuss what can be learnt from such a phenomenology. In particular, we argue that diffractive observables in these kinematics provide useful pieces of information on the saturation regime of QCD.

Geometric scaling in a symmetric saturation model

We illustrate geometric scaling for the photon-proton cross section with a very simple saturation model. We describe the proton structure function F2 at small x in a wide kinematical range with an elementary functional form and a small number of free parameters. We speculate that the symmetry between low and high Q2 recently discovered in the data could be related to a well-known symmetry of the two-gluon- exchange dipole-dipole cross section.

High-energy factorization predictions for the charm structure function F2c at HERA

Abstract High-energy factorization predictions for F 2 c are derived using BFKL descriptions of the proton structure function F 2 at HERA. The model parameters are fixed by a fit of F 2 at small x . Two different approaches of the non-perturbative proton input are shown to correspond to the factorization at the gluon or quark level, respectively. The predictions for F 2 c are in agreement with the data within the present error bars. However, the photon wave-function formulation (factorization at quark level) predicts significantly higher F 2 c than both gluon factorization and a next-leading order DGLAP model.

On parton number fluctuations at various stages of the rapidity evolution

Abstract Starting with the interpretation of parton evolution with rapidity as a branching–diffusion process, we describe the different kinds of fluctuations of the density of partons which affect the properties of QCD scattering amplitudes at moderately high energies. We then derive some of these properties as direct consequences of the stochastic picture. We get new results on the expression of the saturation scale of a large nucleus, and a modified geometric scaling valid at intermediate rapidities for dipole–dipole scattering.

Dense-dilute factorization for a class of stochastic processes and for high energy QCD

Stochastic processes described by evolution equations in the universality class of the Fisher-Kolmogorov-Petrovsky-Piscounov equation may be approximately factorized into a linear stochastic part and a nonlinear deterministic part. We prove this factorization on a model with no spatial dimensions and we illustrate it numerically on a one-dimensional toy model that possesses some of the main features of high energy QCD evolution. We explain how this procedure may be applied to QCD amplitudes, by combining Salams Monte Carlo implementation of the dipole model and a numerical solution of the Balitsky-Kovchegov equation.