Low x Physics

The following sections are included:

• ACKNOWLEDGEMENTS

• Madrid Workshop on low x Physics

• Madrid low x meeting

• CONTENTS

Four studies performed within the H1 and ZEUS collaborations are described: the gluon determination from a QCD fit to the F₂ data, the neutral current cross-section at very high transfered momentum, the charm structure function and an extraction of the longitudinal structure function F_L.

We discuss the partonic description of the HERA data for F₂, paying particular attention to a recent K_T factorization approach which unifies DGLAP and BFKL effects.

We discuss the status of twist-four corrections to the deep inelastic structure fuction F₂ at small x and small Q². First numerical results are presented which are based upon the double logarithmic approximation of the coupled twist-four evolution equations of those gluonic twist-four operators that dominate at small x.

The impact of the recently evaluated ‘irreducible’ contributions to the resummed next-to-leading logarithmic small-x anomalous dimension γ_gg is evaluated for the unpolarized parton densities and structure functions of the nucleon. These new terms diminish the gluon distribution and are found to overcompensate the contributions caused by the resummed leading logarithmic small-x anomalous dimension and the quarkonic anomalous dimensions beyond next-to-leading order.

We compute the onium-onium scattering amplitude at fixed impact parameter in the framework of the perturbative QCD dipole model. Relying on the conformal properties of the dipole cascade and of the elementary dipole-dipole scattering amplitude, we obtain an exact result for this onium-onium scattering amplitude, which is proved to be identical to the BFKL result, and which is frame invariant. The asymptotic expression for this amplitude at fixed impact parameter agree with previous numerical simulations.

Using the dipole picture describing the 1/N_C limit of QCD at small x and the conformal invariance properties of the BFKL kernel in transverse coordinate space, we show that the 1→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.

On the basis of the available results on the real and virtual corrections to the BFKL equation the “irreducible” part of the next-to-leading BFKL kernel is computed. This corrections are consistent with the two-loop renormalization group equations and yield a negative and sizable contribution to the anomalous dimensions and to the Pomeron intercept. The effect of these corrections is to smooth out the small-x rise of the structure functions at low values of Q², in qualitative agreement with the data. A quantitative analysis will be possible when some additional terms, related to the energy-scale, will be computed.

We show how the sum of a hard singularity, and a soft Pomeron Const, for the singlet piece of the structure function F_2s = F_2H + F_2P for a few GeV², plus a saturating expression for the strong coupling, give an excellent description of experiment i) For small Q², 0≲Q² ≤ 8.5 GeV² and ii) For large Q², 10 ≲ Q² ≲ Q²≤1500GeV² if evolved with QCD. The x range is 6 × 10⁻⁶ ≲ x ≲.04. The description for low Q² implies self-consistent values for the parameters in the exponents of x both for singlet and nonsinglet. One has to have αρ(0)= 0.48 and λ= 0.470 [α_P(0) = 1.470], in uncanny agreement with other determinations of these parameters, and in particular the results of the large Q² fits. We also look for a “triple Pomeron” vertex, for which some evidence is found.

We present very simple non-integral relations between deep inelastic structure functions F_L, F₂ and the gluon distribution at small x based on perturbative QCD which are useful for the phenomenological analysis of data at low x. As an application we extract the deep inelastic scattering cross-sections ratio R = σ_L/σ_T in the range 10⁻⁴ ≤ x ≤ 10⁻² from F₂ HERA data.

ZEUS and H1 collaborations have measured integrated and differential cross sections for charm production in DIS, at the HERA ep collider, in the 1994 and 1995 data via the tagging of D* and D⁰ hadrons. was extracted and compared with theoretical predictions. Both the integrated cross sections and were found to be in agreement with the theoretical predictions. Finally it was measured that the dominating charm production mechanism at small scale Q² is described by boson-gluon fusion.

We show some of the recent advances in calculating objects to three or more loops in perturbative QCD.

I report on the observation of very high Q² events in e⁺p collisions by the two HERA collaborations, H1 and ZEUS.

Measurements of the hadronic final state at HERA are reviewed, which aim at the investigation of the parton dynamics of the proton at small Bjorken x.

The Linked Dipole Chain model interpolates smoothly between the DGLAP and BFKL regimes. It is a reformulation and generalization of results by Ciafaloni, Catani, Fiorani and Marchesini. Some results are presented for E_⊥-flow and multiple hard subcollisions in low-x DIS and photoproduction.

Bose-Einstein correlations are studied with respect to two-particle correlation functions at small distances and possible differences between diffractive and non-diffractive events. The second part deals with the dependence of charged particle fragmentation functions on the four-momentum transfer squared. The observed scaling violations are compared with predictions of pQCD. In the last section the energy dependence of the mean values of event shape variables is investigated. Applying power corrections to account for hadronisation effects, fits to second order pQCD calculations are carried out.

This talks examines the effect of angular ordering on the small-x evolution of the unintegrated gluon distribution, and discusses the characteristic function for the CCFM equation.

The concept of resolved virtual photons in addition to direct deep inelastic ep scattering is used to simulate the 2 + 1 jet - rate and the forward jet cross section, which cannot be described by direct LO/NLO processes. With standard DGLAP evolution of the parton densities of the virtual photon and the proton the HERA data can be described, provided a sufficiently large scale for the hard scattering matrix element is used.

We outline the existing descriptions of the charm component of the deep inelastic proton structure function F₂. We discuss recent approaches to include charm mass effects in the parton evolution equations and the coefficient functions.

We briefly review final state processes in small Bjorken x deep inelastic scattering which may be used to identify ln 1/x dynamics.

We analyse recent data on the production of forward neutrons in deep inelastic scattering at HERA in the framework of a perturbative QCD description for semi inclusive processes, which includes fracture functions.

Studies on the structure of the photon are presented by means of the extraction of a leading order effective parton distribution in the photon and measurements of inclusive jet differential cross sections in photoproduction. Measurements of the internal structure of jets have been performed and are also presented as a function of the transverse energy and pseudorapidity of the jets.

The inclusive measurements of the Spin Muon Collaboration (SMC) with a special focus on the preliminary '96 results are presented. The SMC performed the polarized DIS experiment with the highest beam energy and is therefore the best suited one to study the polarized structure functions at low x, the kinematic region under investigation in this workshop. For Q² > 1 GeV² the x-region down to x = 0.003 can be explored. By dropping the low Q² cut the x-region can be extended to x = 0.0008. This data play an essential role in estimating the high energy contribution to the DHG integral, which turns out not to be better constraint than to 20 - 40 % of the total prediction. At low x, the requirement of an observed hadron in the final state is an efficient tool to select DIS events. Preliminary results for the '96 data with a significant reduction of the statistical error (up to 40 %) using hadron tagged events are presented.

We present a combined next-to-leading order QCD analysis to data on both inclusive and semi-inclusive polarised deep inelastic scattering asymmetries.

Gribov theory is applied to investigate the shadowing effects in the structure functions of nuclei. In this approach a reasonable description of the x, Q² and A-dependence of nuclear shadowing is achieved by using the CKMT model for the diffractive dissociation of a virtual photon.

We discuss some topics on the physics case for using polarised protons and electrons at the ep collider HERA. We show that if a polarisation of 70% can be achieved for the protons, important and partly unique measurements can be made at HERA. These include measurements of the polarised structure function g₁(x) at low x, the polarised gluon distribution in the range 0.001 < x < 0.1, weak asymmetries from charged current events and possible access to polarised parton distributions in the photon. Thus HERA can make a significant contribution to the spin physics program.

We present a brief account of some physics topics which could be addressed at a collider consisting of the HERA proton ring and the future e⁺e⁻ Linear Collider. A few experimental aspects are also discussed.

Recent results on the structure function from H1 and ZEUS are reviewed. A QCD analysis of the H1 measurement is discussed, and the extracted α_P values from H1 and ZEUS are compared.

Using their forward neutron calorimeters and leading proton spectrometers, the ZEUS and H1 collaborations at HERA have explored leading baryon production in deep inelastic scattering to test predictions from pomeron and pion exchange models and of the soft color interaction model.

In the present talk diffraction in deep inelastic scattering is discussed in the framework of the semiclassical approach. The main emphasis is on the possibility of a consistent semiclassical description of both hard and soft colour singlet exchange processes. This approach allows the comparison of hard and soft colour neutralization in diffractive electroproduction of high-p_⊥ jets or heavy quarks.

The proton structure function F₂, the gluon density F_G, and the longitudinal structure function F_L are derived in the QCD dipole picture of BFKL dynamics. We use a four parameter fit to describe the 1994 H1 proton structure function F₂ data in the low x, moderate Q₂ range. Without any additional parameter, the gluon density and the longitudinal structure functions are predicted. The diffractive dissociation processes are also discussed within the same framework, and a new prediction for the proton diffractive structure function is obtained.

Measurements are presented of the system X of the hadronic final state produced in deep-inelastic scattering positron-proton diffractive scattering (e⁺p → eXY) using data collected by the H1 and ZEUS experiments at HERA. Shown are the hadronic energy flow and charged particle spectra in the rest frame of the system X. Also presented is the average value of thrust 〈T〉 as a function of 1/M_X, where M_X is the invariant mass of the system X and the thrust p_T. Also shown is the average sphericity 〈S〉 as a function of M_X and the average transverse momentum squared with respect to the sphericity axis , as a function of M_X. The inclusive open charm cross section is also presented. The data are shown to be consistent with a picture in which the partonic structure of the diffractive exchange (ℙ) is dominated by gluons, and cannot be described if the ℙ is composed primarily of quarks.

Diffractive open charm and high p_⊥-jet production in deep inelastic scattering are discussed within the context of the semiclassical approach. It is expected that the mass and p_⊥-spectra of these final states will reveal the underlying mechanism of colour neutralization. Several phenomenological predictions are presented.

The amplitude for 4 interacting reggeized gluons is studied in the high-colour limit. The leading order amplitude is explicitly shown to reduce to a pair of reggeons, i.e to a single BFKL pomeron. The next-to-leading order diffractive amplitude is found to split into a double pomeron exchange and triple pomeron contribution. The obtained three-pomeron vertex is different from the originally proposed in².

The odderon singularity is studied in perturbative QCD in the framework of the Bartels-Kwieciński-Praszalowicz (BKP) equation. Arguments for the odderon intercept being exactly equal to unity are given. Besides, a variational method based on a complete system of one-gluon functions is presented. For the odderon, the highest intercept calculated by this method is 1 − (N_cα_s/π)0.45. Comparison to other calculations is shown.

Measurements are presented of two-jet cross sections in diffractive photoproduction and diffractive deep inelastic scattering at HERA. A combined QCD analysis of the two-jet cross sections in diffractive photoproduction and diffractive structure function results leads to a picture in which the pomeron has a substantial gluon component with a hard momentum spectrum.

The distribution of an azimuthal angle in the final state of deep inelastic diffraction contains information on the polarisation of the exchanged photon. I discuss the potential of the bounds on the longitudinal part of the diffractive structure function one may obtain from its measurement.

We consider the pomeron-pomeron fusion mechanisms for the double diffractive production of a Higgs boson or large E_T dijet. We determine the suppression of the ‘rapidity gap’ pomeron-pomeron fusion events due to QCD radiative effects. In particular we use leading log techniques to estimate the cross sections for both exclusive and inclusive double-diffractive Higgs production at LHC energies. The same approach is applied to the double-diffractive central production of large E_T dijets.

We present a minimal model for two and three jet events with a large rapidity gap at HERA. Basically the model has only two parameters. In the end we compare the model with some of the HERA data.

The “triple-Regge” analysis of subleading Reggeon contributions to the diffractive structure function in DIS at HERA is presented. The recently published data allow to determine the only free parameter of the analysis related to the ratio of the RRℙ and ℙℙℙ couplings. The large value of this ratio is prefered which is in agreement with analyses of soft hadronic interactions. The role of the subleading Reggeons as well as pions for the fast forward neutron production in diffractive processes is estimated. The forward πN state production is also discussed.

Recent results on vector meson production at HERA are presented. The elastic (ep → eVp) and the proton dissociative (ep → eVN) reactions are discussed for both the photoproduction regime and finite photon virtualities.

This talk summarized the proof of hard-scattering factorization for exclusive deep-inelastic processes, such as diffractive meson production.

Aspects of diffraction studies by the HERA experiments are addressed from an experimental point of view, in the light of discussions during the Workshop.

The following sections are included:

• List of Participants