Traveling waves in high energy QCD

Saturation is expected to occur when a high density of partons (mainly gluons)- or equivalently strong fields in Quantum Chromodynamics (QCD) - is realized in the weak coupling regime. A way to reach saturation is through the high-energy evolution of an extended target probed at a fixed hard scale. In this case, the transition to saturation is expected to occur from nonlinear perturbative QCD dynamics. We discuss this approach to saturation, which is mathematically characterized by the appearance of traveling wave patterns in a suitable kinematical representation. A short review on traveling waves in high energy QCD and a first evidence of this phenomenon in deep-inelastic proton scattering are presented.Comment: 10 pages, 5 figures, talk given at the XXXVth International Symposium on Multiparticle Dynamics (ISMD05), Kromeriz, Czech Republic, August 9-15 200

Dynamical entropy of dense QCD states

We discuss dense states of QCD matter formed in high-energy hadronic and heavy-ion collisions from the point of view of statistical physics of non-equilibrium processes. For this sake, we first propose a formulation of the dynamical entropy of dense QCD states in the "saturation regime" leading to a color glass condensate (CGC). The statistical physics description amounts to describe the modification of the color correlation length with energy as a compression process for which non equilibrium thermodynamic properties are applicable. We derive an expression of the dynamical entropy in terms of the rapidity evolution of the unintegrated gluon distributions in the colliding nuclei, verifying suitable positivity and irreversibility properties. We extend this approach to the initial pre-equilibrium (glasma) state of an heavy-ion collision. It allows for a definition of the initial entropy before the evolution towards the hydrodynamic regime as a function of the glasma correlation length and an overlap parameter characterizing the low-momentum spectrum of the glasma state. This initial entropy, by extension to the N=4 SYM theory, is then matched as the key input parameter to the strong coupling evaluation of thermalization towards the hydrodynamic regime based on the AdS/CFT correspondence. It thus allows to cast a bridge between the weak and strong coupling phases of an heavy-ion reaction.Comment: Version mildly updated to match publication. Adding a discussion of the k_T factorization formula. Results and conclusions unchange

Gauge/Cosmology Brane-to-Brane Duality

We introduce a duality relation between two distinct branes, a cosmological brane with macroscopic matter and a holographic brane with microscopic gauge fields. Using brane-world cosmology with a single brane in a 5-dimensional AdS5 background, we find an explicit time-dependent holographic correspondence between the bulk metric surrounding the cosmological brane and the N=4 gauge field theory living on the boundary of the Z2-symmetric mirror bulk, identified with the holographic brane. We then relate the cosmic acceleration on the cosmological brane to the conformal anomaly of the gauge theory on the holographic brane. This leads to a dual microscopic interpretation of the number of e-foldings of the cosmological eras on the cosmological brane.Comment: 21 pages, 1 figur

On the positivity of Fourier transforms

Characterizing in a constructive way the set of real functions whose Fourier transforms are positive appears to be yet an open problem. Some sufficient conditions are known but they are far from being exhaustive. We propose two constructive sets of necessary conditions for positivity of the Fourier transforms and test their ability of constraining the positivity domain. One uses analytic continuation and Jensen inequalities and the other deals with Toeplitz determinants and the Bochner theorem. Applications are discussed, including the extension to the two-dimensional Fourier-Bessel transform and the problem of positive reciprocity, i.e. positive functions with positive transforms.Comment: 12 pages, 9 figures (in 4 groups

From "Dirac combs" to Fourier-positivity

Motivated by various problems in physics and applied mathematics, we look for constraints and properties of real Fourier-positive functions, i.e. with positive Fourier transforms. Properties of the "Dirac comb" distribution and of its tensor products in higher dimensions lead to Poisson resummation, allowing for a useful approximation formula of a Fourier transform in terms of a limited number of terms. A connection with the Bochner theorem on positive definiteness of Fourier-positive functions is discussed. As a practical application, we find simple and rapid analytic algorithms for checking Fourier-positivity in 1- and (radial) 2-dimensions among a large variety of real positive functions. This may provide a step towards a classification of positive positive-definite functions.Comment: 17 pages, 14 eps figures (overall 8 figures in the text