322 research outputs found
Non-linear QCD evolution with improved triple-pomeron vertices
In a previous publication, we have constructed a set of non-linear evolution
equations for dipole scattering amplitudes in QCD at high energy, which extends
the Balitsky-JIMWLK hierarchy by including the effects of fluctuations in the
gluon number in the target wavefunction. In doing so, we have relied on the
color dipole picture, valid in the limit where the number of colors is large,
and we have made some further approximations on the relation between scattering
amplitudes and dipole densities, which amount to neglecting the non-locality of
the two-gluon exchanges. In this Letter, we relax the latter approximations,
and thus restore the correct structure of the `triple-pomeron vertex' which
describes the splitting of one BFKL pomeron into two within the terms
responsible for fluctuations. The ensuing triple-pomeron vertex coincides with
the one previously derived by Braun and Vacca within perturbative QCD. The
evolution equations can be recast in a Langevin form, but with a multivariable
noise term with off-diagonal correlations. Our equations are shown to be
equivalent with the modified version of the JIMWLK equation recently proposed
by Mueller, Shoshi, and Wong.Comment: 15 page
Duality and Pomeron effective theory for QCD at high energy and large N_c
We propose an effective theory which governs Pomeron dynamics in QCD at high
energy, in the leading logarithmic approximation, and in the limit where N_c,
the number of colors, is large. In spite of its remarkably simple structure,
this effective theory generates precisely the evolution equations for
scattering amplitudes that have been recently deduced from a more complete
microscopic analysis. It accounts for the BFKL evolution of the Pomerons
together with their interactions: dissociation (one Pomeron splitting into two)
and recombination (two Pomerons merging into one). It is constructed by
exploiting a duality principle relating the evolutions in the target and the
projectile, more precisely, splitting and merging processes, or fluctuations in
the dilute regime and saturation effects in the dense regime. The simplest
Pomeron loop calculated with the effective theory is free of both ultraviolet
or infrared singularities.Comment: 13 pages, 1 figur
On the Relationship between Large Order Graphs and Instantons for the Double Well Oscillator
The double well oscillator is used as a QCD-like model for studying the
relationship between large order graphs and the instanton-antiinstanton
solution. We derive an equation for the perturbative coefficients of the ground
state energy when the number of 3 and/or 4-vertices is fixed and large. These
coefficients are determined in terms of an exact``bounce'' solution. When the
number of 4-vertices is analytically continued to be near the negative of half
the number of 3-vertices the bounce solution approaches the
instanton-antiinstanton solution and detremines leading Borel singularity.Comment: 26 pages, Latex, 6 figures, 1 tabl
On possible implications of gluon number fluctuations in DIS data
We study the effect of gluon number fluctuations (Pomeron loops) on deep
inelastic scattering (DIS) in the fixed coupling case. We find that the
description of the DIS data is improved once gluon number fluctuations are
included. Also the values of the parameters, like the saturation exponent and
the diffussion coefficient, turn out reasonable and agree with values obtained
from numerical simulations of toy models which take into account fluctuations.
This outcome seems to indicate the evidence of geometric scaling violations,
and a possible implication of gluon number fluctuations, in the DIS data.
However, we cannot exclude the possibility that the scaling violations may also
come from the diffusion part of the solution to the BK-equation, instead of
gluon number fluctuations.Comment: 9 pages, 2 figures; references added, minor changes, matches
published versio
A zero-dimensional model for high-energy scattering in QCD
We investigate a zero-dimensional toy model originally introduced by Mueller
and Salam which mimics high-energy scattering in QCD in the presence of both
gluon saturation and gluon number fluctuations, and hence of Pomeron loops.
Unlike other toy models of the reaction-diffusion type, the model studied in
this paper is consistent with boost invariance and, related to that, it
exhibits a mechanism for particle saturation close to that of the JIMWLK
equation in QCD, namely the saturation of the emission rate due to high-density
effects. Within this model, we establish the dominant high-energy behaviour of
the S-matrix element for the scattering between a target obtained by
evolving one particle and a projectile made with exactly n particles.
Remarkably, we find that all such matrix elements approach the black disk limit
S=0 at high rapidity Y, with the same exponential law: ~ exp(-Y) for all
values of n. This is so because the S-matrix is dominated by rare target
configurations which involve only few particles. We also find that the bulk
distribution for a saturated system is of the Poisson type.Comment: 34 pages, 9 figures. Some explanations added on the frame-dependence
of the relevant configurations (new section 3.3
One-dimensional model for QCD at high energy
We propose a stochastic particle model in (1+1)-dimensions, with one
dimension corresponding to rapidity and the other one to the transverse size of
a dipole in QCD, which mimics high-energy evolution and scattering in QCD in
the presence of both saturation and particle-number fluctuations, and hence of
Pomeron loops. The model evolves via non-linear particle splitting, with a
non-local splitting rate which is constrained by boost-invariance and multiple
scattering. The splitting rate saturates at high density, so like the gluon
emission rate in the JIMWLK evolution. In the mean field approximation obtained
by ignoring fluctuations, the model exhibits the hallmarks of the BK equation,
namely a BFKL-like evolution at low density, the formation of a traveling wave,
and geometric scaling. In the full evolution including fluctuations, the
geometric scaling is washed out at high energy and replaced by diffusive
scaling. It is likely that the model belongs to the universality class of the
reaction-diffusion process. The analysis of the model sheds new light on the
Pomeron loops equations in QCD and their possible improvements.Comment: 35 pages, 4 figures, one appendi
On the Probabilistic Interpretation of the Evolution Equations with Pomeron Loops in QCD
We study some structural aspects of the evolution equations with Pomeron
loops recently derived in QCD at high energy and for a large number of colors,
with the purpose of clarifying their probabilistic interpretation. We show
that, in spite of their appealing dipolar structure and of the self-duality of
the underlying Hamiltonian, these equations cannot be given a meaningful
interpretation in terms of a system of dipoles which evolves through
dissociation (one dipole splitting into two) and recombination (two dipoles
merging into one). The problem comes from the saturation effects, which cannot
be described as dipole recombination, not even effectively. We establish this
by showing that a (probabilistically meaningful) dipolar evolution in either
the target or the projectile wavefunction cannot reproduce the actual evolution
equations in QCD.Comment: 31 pages, 2 figure
The Energy Dependence of the Saturation Momentum from RG Improved BFKL Evolution
We study the energy dependence of the saturation momentum in the context of
the collinearly improved Leading and Next to Leading BFKL evolution, and in the
presence of saturation boundaries. We find that the logarithmic derivative of
the saturation momentum is varying very slowly with Bjorken-x, and its value is
in agreement with the Golec-Biernat and Wusthoff model in the relevant x
region. The scaling form of the amplitude for dipole-dipole or dipole-hadron
scattering in the perturbative side of the boundary is given.Comment: 32 page
A Langevin equation for high energy evolution with pomeron loops
We show that the Balitsky-JIMWLK equations proposed to describe non-linear
evolution in QCD at high energy fail to include the effects of fluctuations in
the gluon number, and thus to correctly describe both the low density regime
and the approach towards saturation. On the other hand, these fluctuations are
correctly encoded (in the limit where the number of colors is large) in
Mueller's color dipole picture, which however neglects saturation. By combining
the dipole picture at low density with the JIMWLK evolution at high density, we
construct a generalization of the Balitsky hierarchy which includes the
particle number fluctuations, and thus the pomeron loops. After an additional
coarse-graining in impact parameter space, this hierarchy is shown to reduce to
a Langevin equation in the universality class of the stochastic
Fisher-Kolmogorov-Petrovsky-Piscounov (sFKPP) equation. This equation implies
that the non-linear effects in the evolution become important already in the
high momentum regime where the average density is small, which signals the
breakdown of the BFKL approximation.Comment: 56 pages, 10 figure
Radiation by a heavy quark in N=4 SYM at strong coupling
Using the AdS/CFT correspondence in the supergravity approximation, we
compute the energy density radiated by a heavy quark undergoing some arbitrary
motion in the vacuum of the strongly coupled N=4 supersymmetric Yang-Mills
theory. We find that this energy is fully generated via backreaction from the
near-boundary endpoint of the dual string attached to the heavy quark. Because
of that, the energy distribution shows the same space-time localization as the
classical radiation that would be produced by the heavy quark at weak coupling.
We believe that this and some other unnatural features of our result (like its
anisotropy and the presence of regions with negative energy density) are
artifacts of the supergravity approximation, which will be corrected after
including string fluctuations. For the case where the quark trajectory is
bounded, we also compute the radiated power, by integrating the energy density
over the surface of a sphere at infinity. For sufficiently large times, we find
agreement with a previous calculation by Mikhailov [hep-th/0305196].Comment: 22 page
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