The transverse size of q q-bar fluctuations of the longitudinal photon is reduced relative to the transverse size of q q-bar fluctuations of the transverse photon. This implies R(W2, Q2) = 0.375 or, equivalently, FL/F2 = 0.27 at x<<0.1 and Q2 sufficiently large, while R(W2, Q2) = 0.5, if this effect is not taken into account. Forthcoming experimental data from HERA will allow to test this prediction.
The transverse size of q q-bar fluctuations of the longitudinal photon is reduced relative to the transverse size of q q-bar fluctuations of the transverse photon. This implies R(W^2, Q^2) = 0.375 or, equivalently, F_L(W^2,Q^2)/F_2(W^2,Q^2) = 0.27 for x = Q^2/W^2 << 1 and Q^2 sufficiently large, while R(W^2, Q^2) = 0.5, if this effect is not taken into account. Forthcoming experimental data from HERA will allow to test this prediction.
The prediction of FL(x,Q2)=0.27F2(x,Q2) in the color-dipole picture based on color-transparency and transverse-size reduction, is consistent with the experimental results from HERA.
Using the QCD dipole picture of BFKL dynamics, the quasi elastic component of the diffractive gamma*-dipole cross-section, dominating at small M^2/Q^2, is calculated.
Starting from a genuinely nonperturbative formulation of photon-proton scattering we discuss which approximations and assumptions are required to obtain the dipole picture of high energy scattering.
Forward differential cross-section for diffraction dissociation of virtual photons on nucleon target is calculated in the QCD dipole picture. The numerical estimates are presented and discussed.
We discuss two aspects of the color dipole picture of high energy photon-proton scattering. First we present bounds on various ratios of deep inelastic structure functions resulting from the dipole picture that, together with the measured data, can be used to restrict the kinematical range of its applicability. The second issue that we address is the choice of energy variable in the dipole-proton cross section.
Compact and well defined formulae for the shadow of the virtual photon interacting with a large nucleus at small $x_{Bj}$ are given in the QCD dipole picture. Two classes of contributions are considered: (a) quasi-elastic interaction of the $q\bar{q}$ dipole and (b) multi-pomeron coupling.
We give a brief representation of the theoretical results from the color dipole picture, covering the total photoabsorption cross section, high-energy $J/ψ$ photoproduction with respect to recent experimental data from the LHCb Collaboration at CERN, and ultra-high energy neutrino scattering, relevant for the ICE-CUBE experiment.
The QCD dipole picture of BFKL dynamics provides an attractive theoretical approach to the study of the QCD (resummed) perturbative expansion of small-x physics and more generally to hard high-energy processes. We discuss applications to the phenomenology of proton structure functions in the HERA range and to the longstanding problem of unitarity corrections, and outline some specific predictions of the dipole picture.
The soft gluon part of a proton wave function is investigated and compared with an onium case. It is argued that at every step of the gluon cascade new color structures appear. Dipole equation kernel emerges when a diquark limit is assumed.
We investigate the mass range of the quark-antiquark fluctuations of the photon that are active in producing the total photoabsorption cross section in the color dipole picture, emphasizing the notion of color transparency and saturation. We consider the implications of measurements at future extensions of the available electron-proton-scattering energy.
We discuss the description of the proton structure function within the dipole factorisation framework. We parametrise the forward dipole amplitude to account for saturation as predicted by the small-x QCD evolution equations. Contrarily to previous models, the saturation scale does not decrease when taking heavy quarks into account. We show that the same dipole amplitude also allows to reproduce diffractive data and exclusive vector meson production.
Using the eikonal approximation, we show that inclusive gluon production is related to the scattering of a gluon-gluon dipole while diffractive gluon production in DIS is related to a two-$q\bar q$ dipole scattering amplitude. Hence diffractive photon dissociation cross-sections are observables that provide potential information on dipole correlations, which represent an open issue in high-energy QCD.
We study the dipole picture for the description of the deep inelastic scattering, focusing on the structure functions which are driven directly by the gluon distribution. One performs estimates using the effective dipole cross section given by the Glauber-Mueller approach in QCD, which encodes the corrections due to the unitarity effects associated with the saturation phenomenon. We also address issues about frame invariance of the calculations when analysing the observables.
We calculate, for nonzero momentum transfer, the dipole formula for the high energy behaviour of elastic and quasielastic scattering of a virtual photon. We obtain an expression of the nonforward photon impact factor and of the nonforward photon wave function, and we give a physical interpretation.
We report on the results obtained in the study of the parton saturation effects, taken into account through the multiscattering Glauber-Mueller approach, applied to the Drell-Yan (DY) process described in the color dipole picture. As a main result, one shows that those effects play an important role in the estimates of the DY differential cross section atRHIC energies.
A personal summary of the discussions which took place at the informal meeting in Amirim, Israel from June 1-4 2000, concerning the dipole picture of small-$x$ physics is presented. The broad aim of the meeting was to address the question ``Has HERA reached a new QCD regime (at small $x$) ?''. The new regime in question is the high-density, but weak-coupling, limit of perturbative QCD.
We pose the question: how far can we push the QCD dipole model and the idea of QCD saturation to the kinematical regions traditionally associated with soft nonperturbative physics? The answer is that it works sufficiently well allowing a smooth transition from hard (BFKL) pomeron to the soft one.
The relation between the two approaches is discussed both in the linear and nonlinear regimes. It is connected to the gauge invariance and Moebius symmetry in LL perturbative QCD. First corrections beyond the large N approximation are discussed for the nonlinear case.
Based upon the color-dipole picture, we provide closed analytic expressions for the longitudinal and the transverse photoabsorption cross sections at low values of the Bjorken variable of x<0.1. We compare with the experimental data for the longitudinal-to-transverse ratio of the (virtual) photoabsorption cross section and with our previous fit to the experimental data for the total photoabsorption cross section. Scaling in terms of the low-x scaling variable eta(W^2,Q^2) is analyzed in terms of the reduced cross section of deep inelastic scattering.
We summarize recent work on low-x deep inelastic scattering. The generalized vector dominance/color-dipole picture (GVD/CDP) implies a scaling behavior for $σ_{γ^* p} (W^2, Q^2)\cong σ_{γ^* p}(η)$, with $η= (Q^2 + m^2_0)/Λ^2 (W^2)$ and yields an excellent representation of the experimental results on $σ_{γ^* p}(η)$.
Using the QCD dipole picture of the BFKL pomeron, the cross-section of single diffractive dissociation of virtual photons at high energy and large diffractively excited masses is calculated. The calculation takes into account the full impact-parameter phase-space and thus allows to obtain an exact value of the triple BFKL Pomeron vertex. It appears large enough to compensate the perturbative 6-gluon coupling factor (alpha/pi)^3 thus suggesting a rather appreciable diffractive cross-section.
We perform an adjust to the most recent structure function data, considering the QCD dipole picture applied to ep scattering. The structure function F2 at small x and intermediate Q2 can be described by the model containing an economical number of free-parameters, which encodes the hard Pomeron physics. The longitudinal structure function and the gluon distribution are predicted without further adjustments. The data description is effective, whereas a resummed next-to-leading level analysis is deserved.
It is argued that the QCD dipole picture allows to build an unified theoretical description -based on BFKL dynamics- of the total and diffractive nucleon structure functions. This description is in qualitative agreement with the present collection of data obtained by the H1 collaboration. More precise theoretical estimates, in particular the determination of the normalizations and proton transverse momentum behaviour of the diffractive components, are shown to be required in order to reach definite conclusions.
The $F_2$, $F_G$, $R = F_L/F_T$ proton structure functions are derived in the QCD dipole picture. Assuming $k_T$ and renormalization-group factorization, we relate deep-inelastic proton scattering to deep-inelastic onium scattering. We get a three-parameter fit of the 1994 H1 data in the low-$x,$ moderate $Q^2$ range. The ratios $F_G/F_2$ and $R$ are predicted without further adjustment. The dipole picture of BFKL dynamics is shown to provide a relevant model for quantitatively describing the proton structure functions at HERA. The predictions for $F_2$ and $F_G$ are compatible with the next-t…
In this contribution we present an exploratory QCD analysis of the neutrino structure functions in charged current DIS using the color dipole formalism. The corresponding dipole cross sections are taken from recent phenomenological and theoretical studies in deep inelastic inclusive production, including nuclear shadowing corrections. The theoretical predictions are compared to the available experimental results in the small-x region.
This is the first of two papers in which we study real and virtual photon-proton scattering in a nonperturbative framework. We classify different contributions to this process and identify the leading contributions at high energies. We then study the renormalisation of the photon-quark-antiquark vertex that occurs in the leading contributions. We find something like the dipole picture in one of these contributions but also find two correction terms which can potentially become large at small photon virtualities. In the second paper we will discuss the additional approximations and assumptions …
The AGL equation includes all multiple pomeron exchanges in the double logarithmic approximation (DLA) limit, leading to an unitarized gluon distribution in the small $x$ regime. This equation was originally obtained using the Glauber-Mueller approach. We demonstrate in this contribution that the AGL equation can also be obtained from the dipole picture. Our conclusion is that the AGL equation is a good candidate for an unitarized evolution equation at small $x$ in the DLA limit.
We pursue the intriguing possibility that larger-size instantons build up diffractive scattering, with the marked instanton-size scale $<ρ> \approx 0.5$ fm being reflected in the conspicuous ``geometrization'' of soft QCD. As an explicit illustration, the known instanton contribution to DIS is transformed into the intuitive colour dipole picture. With the help of lattice results, the $q \bar{q}$-dipole size r is carefully increased towards hadronic dimensions. Unlike pQCD, one now observes a competition between two crucial length scales: the dipole size r and the size $ρ$ of the background ins…
Recently Kovner and Lublinsky proposed a set of equations which can be viewed as dual to JIMWLK evolution. We show that these dual equations have a natural dipole-like structure, as conjectured by Kovner and Lublinsky. In the high energy large N_c limit these evolution equations reduce to equations previously derived in the dipole model. We also show that the dual evolution kernel is scheme dependent, although its action on the weight functional describing a high energy state gives a unique result.
The total $γ^*γ^*$ cross-section is derived in the Leading Order QCD dipole picture of BFKL dynamics, and compared with the one from 2-gluon exchange. The Double Leading Logarithm approximation of the DGLAP cross-section is found to be small in the phase space studied. Cross sections are calculated for realistic data samples at the $e^+e^-$ collider LEP and a future high energy linear collider. Next to Leading order corrections to the BFKL evolution have been determined phenomenologically, and are found to give very large corrections to the BFKL cross-section, leading to a reduced sensitivity …
In this contribution we discuss distinct approaches for the high density QCD (hdQCD) and the demonstration of the equivalence between the AGL approach and the K approach is presented. Our conclusion is that the AGL equation is a good candidate for an unitarized evolution equation at high densities in the DLA limit.
The AGL equation includes all multiple pomeron exchanges in the double logarithmic approximation (DLA) limit, leading to an unitarized gluon distribution in the small x regime. This equation was originally obtained using the Glauber-Mueller approach. We demonstrate in this paper that the AGL equation and, consequently, the GLR equation, can also be obtained from the dipole picture in the double logarithmic limit, using an evolution equation, recently proposed, which includes all multiple pomeron exchanges in the leading logarithmic approximation. Our conclusion is that the AGL equation is a go…
This is the second of two papers in which we study real and virtual photon-proton scattering in a nonperturbative framework. In the first paper we have identified the leading contributions to this process at high energies and have derived expressions for them which take into account the renormalisation of the photon-quark-antiquark vertex. In the present paper we investigate the approximations and assumptions that are necessary to obtain the dipole model of high energy scattering from the results derived in the first paper. We discuss the gauge invariance of different contributions to the scat…
The scaling in $σ_{γ^*p}$ cross sections (for $Q^2/W^2 << 1$) in terms of the scaling variable $η= (Q^2 + m^2_0)/Λ^2 (W^2)$ is interpreted in the generalized vector dominance/color-dipole picture (GVD/CDP). The quantity $Λ^2 (W^2)$ is identified as the average gluon transverse momentum absorbed by the $q \bar q$ state, $<\vec l^{~2}> = (1/6) Λ^2 (W^2)$. At any $Q^2$, for $W^2 \to \infty$,the cross sections for virtual and real photons became universal, $σ_{γ^*p}(W^2,Q^2)/σ_{γp}(W^2) \to 1$.
In this presentation, we obtain the corresponding universal function to the diffractive process and show the cross section exhibits the geometrical scaling. It is observed the diffractive theory according to the color dipole approach at small-$x$ is a convenient framework that reveals the color transparency and the saturation phenomena. Also we calculate the contribution of heavy quark productions in the diffractive cross section for high energy that is determined by the small size dipole configuration. The ratio of the diffractive cross section to the total cross section in the electron-proto…
We discuss two different models for the impact parameter dependent dipole cross section: one based on DGLAP evolution and the other inspired by the Balitsky-Kovchegov equation. The parameters are determined from fits to data on the total gamma* p cross section measured at HERA. The impact parameter dependent saturation scale is extracted. Predictions are then confronted with HERA data on exclusive diffractive vector meson production and deeply virtual Compton scattering. Finally, predictions are given for the cross sections of exclusive photoproduced J/psi and Upsilon mesons, and Z^0 bosons, e…
For photon virtualities of $Q^2 \gsim 20 {\rm GeV}^2$, the results of the (hard Pomeron) fit in the tensor-Pomeron model in terms of the variables $Q^2$ and the virtual-photon-proton energy squared, $W^2$, provide empirical evidence for the validity of the Color Dipole Picture (CDP). Consistency of the CDP with the perturbative QCD (pQCD) improved parton model implies the prediction of $C_2 = ε_0 \cong 0.30$ for the exponent $C_2$ of the energy-squared dependence in agreement with the results of the fits. For $Q^2 \lsim 20 GeV^2$, the CDP yields a parameter-free smooth transition from $Q^2 \gs…
We provide predictions for the rapidity distributions of exclusive photoproduced J/psi and Upsilon mesons, and Z^0 bosons, at the Tevatron and LHC. We use the equivalent-photon approximation with the photon-proton cross sections given by the impact parameter dependent dipole saturation model, which has already been shown to give a good description of a wide variety of HERA data. We derive the quark-antiquark light-cone wave functions of timelike neutral electroweak bosons. An essential difference is pointed out between the amplitude for timelike heavy boson photoproduction and the amplitude fo…
The transverse size of q q-bar fluctuations of a longitudinally polarized photon is reduced relative to the transerve size of q q-bar fluctuation of a transversely polarized photon. This implies a model-independent prediction of the ratio R(W2,Q2)=sigma_L/sigma_T=0.375, or, equivalently, F_L/F_2=0.27, for x=Q2/W2<<1 and Q2 sufficiently large, while R(W2,Q2)=0.50, if this effect is ignored. Experimental data from HERA confirm the transverse-size reduction.
We present a simultaneous analysis, within an impact parameter dependent saturated dipole model, of exclusive diffractive vector meson (J/psi, phi and rho) production, deeply virtual Compton scattering and the total gamma* p cross section data measured at HERA. Various cross sections measured as a function of the kinematic variables Q^2, W and t are well described, with little sensitivity to the details of the vector meson wave functions. We determine the properties of the gluon density in the proton in both longitudinal and transverse dimensions, including the impact parameter dependent satur…
In this work we obtain the DLLA limit of BFKL in the dipole picture and compare it with HERA data. We demonstrate that in leading-logarithmic- approximation, where $α_s$ is fixed, a transition between the BFKL dynamics and the DLLA limit can be obtained in the region of $Q^2 \approx 150 GeV^2$. We compare this result with the DLLA predictions obtained with $α_s$ running. In this case a transition is obtained at low $Q^2$ $(\le 5 GeV^2)$. This demonstrates the importance of the next-to-leading order corrections to the BFKL dynamics. Our conclusion is that the $F_2$ structure function is not the…
In this work we consider the QCD parton saturation models to describe soft interactions at the high energy limit. The total and elastic cross sections as well as the elastic slope parameter are obtained for proton-proton and pion-proton collisions and compared to recent experimental results. The analyses are done within the color dipole formalism taking into account saturation models which have been tested against DIS data. The main point is that the match between soft and hard interaction occurs in the saturation region which can be described by high density QCD approaches. Discussion is perf…
Using the QCD dipole picture of BFKL dynamics and the conformal invariance properties of the BFKL kernel in transverse coordinate space, we show that the 1--to--p dipole densities can be expressed in terms of dual Shapiro- Virasoro amplitudes B{2p+2} and their generalization including non-zero conformal spins. We discuss the possibility of an effective closed string theory of interacting QCD dipoles.
We give a detailed account of the recently formulated generalized vector dominance/colour-dipole picture (GVD/CDP) of deep-inelastic scattering at low $x\cong Q^2/W^2$, including photoproduction. The approach, based on $γ^*(q \bar q)$ transitions, $q \bar q$ propagation and diffractive $(q \bar q)p$ scattering via the generic structure of the two-gluon exchange, provides a unique and quantitatively successful theory for the $γ^* p$ total cross section, $σ_{γ^* p} (W^2,Q^2)$, at low $x$. The GVD/CDP is shown to imply the empirical low-$x$ scaling law, $σ_{γ^* p} (W^2,Q^2)=σ_{γ^* p} (η)$ with $η…
The proton structure function in the diffraction region of small Bjorken-$x$ and $10 {\rm GeV}^2 \le Q^2 \le 100 {\rm GeV}^2$ behaves as $F_2 (x, Q^2) = F_2 (W^2) = f_0 \cdot (W^2)^{C_2}$, where $x = Q^2 / W^2$. The exponent $C_2$ of the $γ^* p$ center-of-mass energy squared, $W^2$, is predicted from evolution of the flavor-singlet quark distribution, $C_2 = 0.29$, and the only free parameter, the normalization $f_0 = 0.063$, is fitted. The evolution of the gluon density multiplied by $α_s (Q^2)$ is dentical to the evolution of the flavor-singlet quark density. This simple picture is at varian…
The pion nucleon sigma-term is extracted on the basis of the soliton picture of the nucleon from the mass spectrum of usual and the recently observed exotic baryons, assuming that they have positive parity. The value found is consistent with that inferred by means of conventional methods from pion nucleon scattering data. The study can also be considered as a phenomenological consistency check of the soliton picture of baryons.
We attempt to describe soft hadron interactions in the framework of saturation models, one based upon the Balitsky-Kovchegov non-linear equation and another one due to Golec-Biernat and Wüsthoff. For $pp$, $Kp$, and $πp$ scattering the relevant hadronic wave functions are formulated, and total, elastic cross-sections, and the forward elastic slope are calculated and compared to experimental data. The saturation mechanism leads to reasonable reproduction of the data for the quantities analyzed, except for the forward elastic slope, where the predicted increase with energy is too moderate.
We show that the recently developed Hamiltonian theory for high energy evolution in QCD in the dilute regime and in the presence of Bremsstrahlung is consistent with the color dipole picture in the limit where the number of colors N_c is large. The color dipoles are quark-antiquark pairs which can radiate arbitrarily many soft gluons, and the evolution consists in the splitting of any such a dipole into two. We construct the color glass weight function of an onium as a superposition of color dipoles, each represented by a pair of Wilson lines. We show that the action of the Bremsstrahlung Hami…
Parton showers have become a standard component in the description of high-energy collisions. Nowadays most final-state ones are of the dipole character, wherein a pair of partons branches into three, with energy and momentum preserved inside this subsystem. For initial-state showers a dipole picture is also possible and commonly used, but the older global-recoil strategy remains a valid alternative, wherein larger groups of partons share the energy--momentum preservation task. In this article we introduce and implement a dipole picture also for initial-state radiation in PYTHIA, and compare w…
An extensive phenomenological study of the Drell-Yan (DY) process in $pp$ collisions at various energies is performed in the color dipole framework. Besides previously studied $γ^*$ production we have also included the $Z^0$ contribution relevant at large dilepton invariant masses. We investigate the DY cross section differential in invariant mass, rapidity and transverse momentum of the dilepton pair in $pp$ collisions at RHIC and LHC. We consider three different phenomenological models for the dipole cross section and found a reasonable agreement with the available data. As a further test of…
We derive correlated bounds on ratios of deep inelastic structure functions from the dipole picture of photon-hadron scattering at high energies. In particular we consider ratios of the longitudinal structure function, the total structure function and the charm part of the latter. We also consider ratios of total structure functions taken at the same energy but at three different photon virtualities. It is shown that by confronting these bounds with experimental data we can significantly constrain the range of validity of the dipole picture.
