Towards a New Global QCD Analysis: Solution to the Non-Linear Equation at Arbitrary Impact Parameter

A numerical solution is presented for the non-linear evolution equation that governs the dynamics of high parton density QCD. It is shown that thesolution falls off as $e^{-b/R}$ at large values of the impact parameter $b$. The power-like tail of the amplitude appears in impact parameter distributions only after the inclusion of dipoles of size larger than the target, a configuration for which the non-linear equation is not valid. The value, energy and impact parameterof the saturation scale $Q_s(y=\ln(1/x),b)$) are calculated both for fixed and running QCD coupling cases. It is shown that the solution exhibits geometrical scaling behaviour. The radius of interaction increases as the rapidity in accordance with the Froissart theorem. The solution we obtain differs from previous attempts, where an anzatz for $b$ behaviour was made. The solutions for running and fixed \as differ. For running \as we obtain a larger radius of interaction (approximately twice as large), a steeper rapidity dependence, and a larger value of the saturation scale.Comment: 26 pages with 19 figures in eps.file

The Iancu-Mueller factorization and high energy asymptotic behaviour

We show that the Iancu-Mueller factorization has a simple interpretation in the Reggeon - like technique based on the BFKL Pomeron. The formula for calculating the high energy asymptotic behaviour for the colour dipole-dipole amplitude is proposed which suggests a procedure to calculate this amplitude through the solution to the Balitsky-Kovchegov non-linear equation. We confirm the Iancu - Mueller result that a specific set of enhanced diagrams is responsible for the high energy behaviour for fixed QCD coupling. However, it is argued that in the case of running QCD coupling, this asymptotic behaviour originates from the Balitsky-Kovchegov non-linear equation. A new solution to the non-linear equation is found which leads to a different asymptotic behaviour of the scattering amplitude even for fixed QCD coupling..Comment: 27 pages, 9 figuure

Examining a reduced jet-medium coupling in Pb+Pb collisions at the Large Hadron Collider

Recent data on the nuclear modification factor $R_{AA}$ of jet fragments in 2.76 ATeV Pb+Pb collisions at the Large Hadron Collider (LHC) indicate that the jet-medium coupling in a Quark-Gluon Plasma (QGP) is reduced at LHC energies and not compatible with the coupling deduced from data at the Relativistic Hadron Collider (RHIC). We estimate the reduction factor from a combined fit to the available data on $R_{AA}(\sqrt{s},p_T,b)$ and the elliptic flow $v_2(\sqrt{s},p_T,b)$ at $\sqrt{s}=0.2,2.76$ ATeV over a transverse momentum range $p_T$ 10-100 GeV and a broad impact parameter, b, range. We use a simple analytic "polytrope" model ($dE/dx=- \kappa E^{a} x^z T^{c}$) to investigate the dynamical jet-energy loss model dependence. Varying a=0-1 interpolates between weakly-coupled and strongly-coupled models of jet-energy dependence while z=0-2 covers a wide range of possible jet-path dependencies from elastic and radiative to holographic string mechanisms. Our fit to LHC data indicates an approximate 40% reduction of the coupling $\kappa$ from RHIC to LHC and excludes energy-loss models characterized by a jet-energy exponent with a>1/3. In particular, the rapid rise of $R_{AA}$ with $p_T$>10 GeV combined with the slow variation of the asymptotic $v_2(p_T)$ at the LHC rules out popular exponential geometric optics models (a=1). The LHC data are compatible with $0\leq a\leq 1/3$ pQCD-like energy-loss models where the jet-medium coupling is reduced by approximately 10% between RHIC and LHC.Comment: 10 pages, 7 figures, version published in Physical Review

High-enegy effective action from scattering of QCD shock waves

At high energies, the relevant degrees of freedom are Wilson lines - infinite gauge links ordered along straight lines collinear to the velocities of colliding particles. The effective action for these Wilson lines is determined by the scattering of QCD shock waves. I develop the symmetric expansion of the effective action in powers of strength of one of the shock waves and calculate the leading term of the series. The corresponding first-order effective action, symmetric with respect to projectile and target, includes both up and down fan diagrams and pomeron loops.Comment: 15 pages, 10 eps figure

From RHIC to EIC: Nuclear Structure Functions

We study the nuclear structure function $F_2^A$ and its logarithmic derivative in the high energy limit (small $x$ region) using the Color Glass Condensate formalism. In this limit the structure function $F_2$ depends on the quark anti-quark dipole-target scattering cross section $N_F (x_{bj}, r_t, b_t)$. The same dipole cross section appears in single hadron and hadron-photon production cross sections in the forward rapidity region in deuteron (proton)-nucleus collisions at high energy, i.e. at RHIC and LHC. We use a parameterization of the dipole cross section, which has successfully been used to describe the deuteron-gold data at RHIC, to compute the nuclear structure function $F_2^A$ and its log $Q^2$ derivative (which is related to gluon distribution function in the double log limit). We provide a quantitative estimate of the nuclear shadowing of $F_2^A$ and the gluon distribution function in the kinematic region relevant to a future Electron-Ion Collider.Comment: 13 pages, 6 figure

More Remarks on High Energy Evolution

We discuss several issues related to recent work on high energy evolution. In particular we show that the Hilbert space for action of the operator of the evolution can be conveniently defined by the space of weight functional introduced recently in \cite{kl4}. These weight functionals automatically implement the correct properties of the charge density correlators, thus eliminating the need in explicit introduction of the Wess Zumino term. We also discuss various aspects of Dense Dilute Duality in the toy dipole model.Comment: 20 pages, 1 figur

The Yin and Yang of High Energy Chromodynamics: Scattering in Black and White

We further discuss the QCD Reggeon field theory (RFT) as it emerges from the JIMWLK/KLWMIJ evolution equation and beyond. We give an explicit expression for the calculation of scattering amplitude in terms of the eigenstates of the RFT Hamiltonian. We point out that the spectrum of RFT is doubly degenerate, the degeneracy being related to the spontaneous breaking of the Dense-Dilute Duality symmetry of RFT. The degeneracy is between the "almost white" states (the Yang sector) which contain a small number of gluons, and "almost black" states (the Yin sector). The excitations above the Yang vacuum have natural interpretation in terms of gluons. Analogously the excitations above the Yin vacuum have natural interpretation as "holes" in the black disk - points at which an incoming gluon does not scatter with unit probability. We discuss in detail the spectrum of the "parton model approximation" to the KLWMIJ evolution introduced in our previous paper, and prove that it is explicitly selfdual. This allows us to find explicitly the counterpart hole states in this approximation. We also present an argument to the effect that the end point of the evolution for any physical state cannot be a "grey disk" but must necessarily be the "black disk" Yin vacuum state. Finally, we suggest an approximation scheme for including the Pomeron loop contribution to the evolution which requires only the solution of the JIMWLK/KLWMIJ Hamiltonian.Comment: 25 pages, 1 figur

Low x saturation at HERA ?

We compare the predictions of two distinct dipole models for inclusive and exclusive diffractive processes. While only one of these dipole models contains perturbative saturation dynamics, we show that the predictions of both models are fully consistent with the available HERA data, indicating no compelling evidence for saturation at present HERA energies.Comment: 8 pages, 4 figures, presented at the 26th Montreal-Rochester-Syracuse-Toronto (MRST) conference held at Concordia University, Montreal, Canada, 12th-14th May 2004. To appear in the proceeding

Non-linear evolution in CCFM: The interplay between coherence and saturation

We solve the CCFM equation numerically in the presence of a boundary condition which effectively incorporates the non-linear dynamics. We retain the full dependence of the unintegrated gluon distribution on the coherence scale, and extract the saturation momentum. The resulting saturation scale is a function of both rapidity and the coherence momentum. In Deep Inelastic Scattering this will lead to a dependence of the saturation scale on the photon virtuality in addition to the usual x-Bjorken dependence. At asymptotic energies the interplay between the perturbative non-linear physics, and that of the QCD coherence, leads to an interesting and novel dynamics where the saturation momentum itself eventually saturates. We also investigate various implementations of the "non-Sudakov" form factor. It is shown that the non-linear dynamics leads to almost identical results for different form factors. Finally, different choices of the scale of the running coupling are analyzed and implications for the phenomenology are discussed.Comment: 37 pages, 21 figure

The MMT API: A Generic MKM System

The MMT language has been developed as a scalable representation and interchange language for formal mathematical knowledge. It permits natural representations of the syntax and semantics of virtually all declarative languages while making MMT-based MKM services easy to implement. It is foundationally unconstrained and can be instantiated with specific formal languages. The MMT API implements the MMT language along with multiple backends for persistent storage and frontends for machine and user access. Moreover, it implements a wide variety of MMT-based knowledge management services. The API and all services are generic and can be applied to any language represented in MMT. A plugin interface permits injecting syntactic and semantic idiosyncrasies of individual formal languages.Comment: Conferences on Intelligent Computer Mathematics (CICM) 2013 The final publication is available at http://link.springer.com