Parton Distributions, Logarithmic Expansions and Kinetic Evolution

Aspects of the QCD parton densities are briefly reviewed, drawing some parallels to the density matrix formulation of quantum mechanics, exemplified by Wigner functions. We elaborate on the solution of their evolution equations using logarithmic expansions and overview their kinetic interpretation. We illustrate how a Fokker-Planck equation can be derived using the master formulation of the same equations and its construction in the case of the transverse spin distributions. A simple connection of the leading order DGLAP equation to fractional diffusion using fractional calculus is also briefly outlined.Comment: Review article to appear in Lecture Notes of SIM (S. Dragomir ed.) 32 pages, 1 figur

Ultra High Energy Cosmic Rays and Air Shower Simulations: a top-bottom view

Stable Superstring Relics (SSR) provide some of the candidates for the possible origin of the Ultra High Energy Cosmic Rays (UHECR). After a brief overview of the motivations for introducing such relics, we address the question whether statistical fluctuations in the formation of the air showers generated by the primary spectrum of protons can be separated from a possible signal of new physics hidden in the first impact with the atmosphere. Our results are generated by using minimal modifications in the cross section of the primaries, and using available simulation codes used by the experimental collaborations. The results indicate that substantial increases in the cross section of the first impact, possibly due to new interactions, are unlikely to be detected in geometrical and/or variations of multiplicities in the cascade.Comment: 6 pages. 2 figures. Standard Latex. Typos corrected. To appear in the proceedings of the XV Incontri sulla Fisica delle Alte Energie (IFAE), Lecce, Italy, 23-26 April 200

Numerical analysis of the one-mode solutions in the Fermi-Pasta-Ulam system

The stability of the one-mode nonlinear solutions of the Fermi-Pasta-Ulam - $\beta$ system is numerically investigated. No external perturbation is considered for the one-mode exact analytical solutions, the only perturbation being that introduced by computational errors in numerical integration of motion equations. The threshold energy for the excitation of the other normal modes and the dynamics of this excitation are studied as a function of the parameter $\mu$ characterizing the nonlinearity, the energy density $\epsilon$ and the number N of particles of the system. The achieved results confirm in part previous results, obtained with a linear analysis of the problem of the stability, and clarify the dynamics by which the one-mode exchanges energy with the other modes with increasing energy density. In a range of energy density near the threshold value and for various values of the number of particles N, the nonlinear one-mode exchanges energy with the other linear modes for a very short time, immediately recovering all its initial energy. This sort of recurrence is very similar to Fermi recurrences, even if in the Fermi recurrences the energy of the initially excited mode changes continuously and only periodically recovers its initial value. A tentative explanation of this intermittent behaviour, in terms of Floquet's theorem, is proposed.Comment: 37 pages, 41 figure

Searching for Extra Dimensions in High Energy Cosmic Rays

We present a study of the decay of an intermediate mini black hole at the first impact of a cosmic ray particle with the atmosphere, in the context of D-brane world scenarios with TeV scale gravity and large extra dimensions. We model the decay of the black hole using the semiclassical approximation and include the corrections coming from energy loss into the bulk. Extensive simulations show that mini black hole events are characterized by essentially different multiplicities and wider lateral distributions of the air showers as a function of the energy of the incoming primary, as compared to standard events. Implications for their detection and some open issues on their possible discovery are also briefly addressed.Comment: 4 pages, 4 figures, Presented by C. Coriano' at the XIII Intl. Symp. on High Energy Cosmic Rays Interactions, Pylos, Greece, 6-12 Sept. 200

Identifying String Relics at AUGER?

The identification of string relics, or of other very massive states, at forthcoming ultra high energy cosmic rays experiments, requires a good reconstruction of the main properties of the extensive air showers produced by the collision of the primary protons with the atmosphere. In particular, the current hadronization models used in the simulations need to incorporate possible new interactions. We briefly discuss these aspects and then proceed by describing some of the observables which characterize the atmospheric shower. The linear growth of the multiplicities - as a function of the energy - for all the main particles in the shower can play an interesting role in an attempt to identify channels of missing energy due to a plausible dark matter component.Comment: 8 pages, 10 figures, presented at String Phenomenology 2003, Durham, U

Precision Studies of the NNLO DGLAP Evolution at the LHC with CANDIA

We summarize the theoretical approach to the solution of the NNLO DGLAP equations using methods based on the logarithmic expansions in x-space and their implementation into the C program Candia 1.0. We present the various options implemented in the program and discuss the different solutions. The user can choose the order of the evolution, the type of the solution, which can be either exact or truncated, and the evolution either with a fixed or a varying flavor number, implemented in the varying-flavour-number scheme (VFNS). The renormalization and factorization scale dependences are treated separately. In the non-singlet sector the program implements an exact NNLO solution.Comment: 31 pages, 1 Fi

NNLO Logarithmic Expansions and Exact Solutions of the DGLAP Equations from x-Space: New Algorithms for Precision Studies at the LHC

A NNLO analysis of certain logarithmic expansions, developed for precision studies of the evolution of the QCD parton distributions (pdf) at the Large Hadron Collider, is presented. We elaborate on their relations to all the solutions of the DGLAP equations that have been hitherto obtained from Mellin space, to which are equivalent. Exact expansions, equivalent to exact solutions of the equations, are constructed in the non-singlet sector. The algorithmic features of our approach are also emphasized, since this method allows to obtain numerical solutions of the evolution equations with the same accuracy of other methods, based on Mellin space, and of brute force methods, which solve the equations by finite differences. The implementation of our analysis allows to compare with existing benchmarks for the evolution of the pdf's, useful for applications at the LHC, and to extend them significantly in a systematic fashion, especially when solutions that retain logarithmic corrections only of a certain accuracy are searched for.Comment: 56 pages, no figure

NNLO Logarithmic Expansions and High Precision Determinations of the QCD background at the LHC: The case of the Z resonance

New methods of solutions of the DGLAP equation and their implementation through NNLO in QCD are briefly reviewed. We organize the perturbative expansion that describes in $x$-space the evolved parton distributions in terms of scale invariant functions, which are determined recursively, and logarithms of the ratio of the running couplings at the initial and final evolution scales. Resummed solutions are constructed within the same approach and involve logarithms of more complex functions, which are given in the non-singlet case. Differences in the evolution schemes are shown to be numerically sizeable and intrinsic to perturbation theory. We illustrate these points in the case of Drell-Yan lepton pair production near the Z resonance, analysis that can be extended to searches of extra $Z^{\prime}$. We show that the reduction of the NNLO cross section compared to the NLO prediction may be attributed to the NNLO evolution.Comment: 5 pages, 2 figures. Talk given at QCD@work 2007, Martina Franca, Italy, 16-20 June 2007. To be published in the American Institute of Physics (AIP) conference proceeding

Direct Solution of Renormalization Group Equations of QCD in x−x-space: NLO Implementations at Leading Twist

We illustrate the implementation of a method based on the use of recursion relations in (Bjorken) $x-$space for the solution of the evolution equations of QCD for all the leading twist distributions. The algorithm has the advantage of being very fast. The implementation that we release is written in C and is performed to next-to-leading order in $\alpha_s$.Comment: 31 pages, 15 figs, replaced with revised final version, to be published by Comp. Phys. Com