Diffractive deep-inelastic scattering with a leading proton at HERA

The cross section for the diffractive deep-inelastic scattering process ep→eXp is measured, with the leading final state proton detected in the H1 Forward Proton Spectrometer. The data analysed cover the range x_IP<0.1 in fractional proton longitudinal momentum loss, 0.08<|t|<0.5 GeV^-2 in squared four-momentum transfer at the proton vertex, 2<Q²<50 GeV² in photon virtuality and 0.004<β=x/x_IP<1, where x is the Bjorken scaling variable. For , the differential cross section has a dependence of approximately dσ/dt∝e^6t, independently of x_IP, β and Q² within uncertainties. The cross section is also measured triple differentially in x_IP, β and Q². The x_IP dependence is interpreted in terms of an effective pomeron trajectory with intercept α_IP(0)=1.114±0.018(stat.)±0.012(syst.)^+0.040 _-0.020(model) and a sub-leading exchange. The data are in good agreement with an H1 measurement for which the event selection is based on a large gap in the rapidity distribution of the final state hadrons, after accounting for proton dissociation contributions in the latter. Within uncertainties, the dependence of the cross section on x and Q² can thus be factorised from the dependences on all studied variables which characterise the proton vertex, for both the pomeron and the sub-leading exchange.     

Diffractive photoproduction of D*±(2010) at HERA

Diffractive photoproduction of D^*±(2010) mesons was measured with the ZEUS detector at the ep collider HERA, using an integrated luminosity of 78.6 pb^-1. The D^* mesons were reconstructed in the kinematic range: transverse momentum p_T(D^*) > 1.9 GeV and pseudorapidity |η(D^*)|<1.6, using the decay D^*+→D⁰π⁺ _s followed by D⁰→K^-π⁺(+c.c.). Diffractive events were identified by a large gap in pseudorapidity between the produced hadronic state and the outgoing proton. Cross sections are reported for photon–proton centre-of-mass energies in the range 130 < W < 300 GeV and for photon virtualities Q² < 1 GeV², in two ranges of the Pomeron fractional momentum x_IP<0.035 and x_IP<0.01. The relative contribution of diffractive events to the inclusive D^*±(2010) photoproduction cross section is about 6%. The data are in agreement with perturbative QCD calculations based on various parameterisations of diffractive parton distribution functions. The results are consistent with diffractive QCD factorisation.     

Diffractive photoproduction of dijets in ep collisions at HERA

Diffractive photoproduction of dijets was measured with the ZEUS detector at the ep collider HERA using an integrated luminosity of 77.2 pb^-1. The measurements were made in the kinematic range Q² < 1 GeV², 0.20<y<0.85 and x_IP<0.025, where Q² is the photon virtuality, y is the inelasticity and x_IP is the fraction of the proton momentum taken by the diffractive exchange. The two jets with the highest transverse energy, E_T ^jet, were required to satisfy E_T ^jet>7.5 and 6.5 GeV, respectively, and to lie in the pseudorapidity range -1.5<η^jet<1.5. Differential cross sections were compared to perturbative QCD calculations using available parameterisations of diffractive parton distributions of the proton.     

Elastic J/ψ production at HERA

Cross sections for elastic production of J/ψ mesons in photoproduction and electroproduction are measured in electron proton collisions at HERA using an integrated luminosity of 55 pb^-1. Results are presented for photon virtualities Q² up to 80 GeV². The dependence on the photon-proton centre of mass energy W_γp is analysed in the range 40≤W_γp≤305 GeV in photoproduction and 40≤W_γp≤160 GeV in electroproduction. The W_γp dependences of the cross sections do not change significantly with Q² and can be described by models based on perturbative QCD. Within such models, the data show a high sensitivity to the gluon density of the proton in the domain of low Bjorken x and low Q². Differential cross sections dσ/dt, where t is the squared four-momentum transfer at the proton vertex, are measured in the range |t|<1.2 GeV² as functions of W_γp and Q². Effective Pomeron trajectories are determined for photoproduction and electroproduction. The J/ψ production and decay angular distributions are consistent with s-channel helicity conservation. The ratio of the cross sections for longitudinally and transversely polarised photons is measured as a function of Q² and is found to be described by perturbative QCD based models.     

Tests of QCD factorisation in the diffractive production of dijets in deep-inelastic scattering and photoproduction at HERA

Measurements are presented of differential dijet cross sections in diffractive photoproduction (Q²<0.01 GeV²) and deep-inelastic scattering processes (DIS, 4<Q²<80 GeV²). The event topology is given by ep→eXY, in which the system X, containing at least two jets, is separated from a leading low-mass baryonic system Y by a large rapidity gap. The dijet cross sections are compared with NLO QCD predictions based on diffractive parton densities previously obtained from a QCD analysis of inclusive diffractive DIS cross sections by H1. In DIS, the dijet data are well described, supporting the validity of QCD factorisation. The diffractive DIS dijet data are more sensitive to the diffractive gluon density at high fractional parton momentum than the measurements of inclusive diffractive DIS. In photoproduction, the predicted dijet cross section has to be multiplied by a factor of approximately 0.5 for both direct and resolved photon interactions to describe the measurements. The ratio of measured dijet cross section to NLO prediction in photoproduction is a factor 0.5±0.1 smaller than the same ratio in DIS. This suppression is the first clear observation of QCD hard scattering factorisation breaking at HERA. The measurements are also compared to the two soft colour neutralisation models SCI and GAL. The SCI model describes diffractive dijet production in DIS but not in photoproduction. The GAL model fails in both kinematic regions.     

Measurement of the proton structure function F 2 at low x and low Q 2 at HERA

We report on a measurement of the proton structure functionF ₂ in the range 3.5×10⁻⁵≤x≤4×10⁻³ and 1.5 GeV²≤Q ²≤15GeV² at theep collider HERA operating at a centre-of-mass energy of √s=300GeV. The rise ofF ₂ with decreasingx observed in the previous HERA measurements persists in this lowerx andQ ² range. TheQ ² evolution ofF ₂, even at the lowestQ ² andx measured, is consistent with perturbative QCD. supported by EU HCM contract ERB-CHRX-CT93-0376     

Truncated first moment of the parton distribution – A modified approach

We derive the LO DGLAP evolution equation for the full Mellin moments of the first moment of the nonsinglet parton distribution truncated at x₀. This “moment of moment” approach allows one to determine the small-x₀ behaviour of the truncated first moment. We compare our predictions to results obtained from x-space solutions for parton distributions with use of the Chebyshev-polynomial method and to solutions of the evolution equations for the truncated moments proposed by other authors. The comparison is performed for different input parametrisations for 10^-5≤x₀≤0.1 and 1≤Q²≤100 GeV². We give an example of an application to the determination of the contribution to the Bjorken sum rule. PACS 12.38.Bx; 11.55.Hx     

Saturation in diffractive deep inelastic eA scattering

In this paper, we investigate the saturation physics in diffractive deep inelastic electron-ion scattering. We estimate the energy and nuclear dependence of the ratio σ^diff/σ^tot and predict the x_ℙ and β behavior of the nuclear diffractive structure function F_2,A^D(3)(Q²,β,x_ℙ). Moreover, we analyze the ratio R^diff_A1,A2(Q²,β,x_ℙ)=F_2,A1^D(3)/F_2,A2^D(3), which probes the nuclear dependence of the structure of the pomeron. We show that saturation physics predicts that approximately 37% of the events observed at eRHIC should be diffractive.     

D meson production in deep-inelastic diffractive interactions at HERA

A measurement is presented of the cross section for D^*± meson production in diffractive deep-inelastic scattering for the first time at HERA. The cross section is given for the process ep→eXY, where the system X contains at least one D^*± meson and is separated by a large rapidity gap from a low mass proton remnant system Y. The cross section is presented in the diffractive deep-inelastic region defined by 2<Q²<100 GeV², 0.05<y<0.7, , M_Y<1.6 GeV and |t|<1 GeV². The D^*± mesons are restricted to the range p_T,D^*>2 GeV and |η_D^*|<1.5. The cross section is found to be 246±54±56 pb and forms about 6% of the corresponding inclusive D^*± cross section. The cross section is presented as a function of various kinematic variables, including which is an estimate of the fraction of the momentum of the diffractive exchange carried by the parton entering the hard-subprocess. The data show a large component of the cross section at low where the contribution of the boson–gluon-fusion process is expected to dominate. The data are compared with several QCD-based calculations.     

Renormalization of a Hard-Core Guest Charge Immersed in a Two-Dimensional Electrolyte

This paper is a continuation of a previous one [L. Šamaj, J. Stat. Phys. 120:125 (2005)] dealing with the renormalization of a guest charge immersed in a two-dimensional logarithmic Coulomb gas of pointlike ± unit charges, the latter system being in the stability-against-collapse regime of reduced inverse temperatures 0 ≤ β < 2. In the previous work, using a sine-Gordon representation of the Coulomb gas, an exact renormalized-charge formula was derived for the special case of the pointlike guest charge Q, in its stability regime β |Q| < 2. In the present paper, we extend the renormalized-charge treatment to the guest charge with a hard core of radius σ, which allows us to go beyond the stability border β|Q| = 2. In the limit of the hard-core radius much smaller than the correlation length of the Coulomb-gas species and at a strictly finite temperature, due to the counterion condensation in the extended region β|Q| > 2, the renormalized charge Q _ren turns out to be a periodic function of the bare charge Q with period 1. The renormalized charge therefore does not saturate at a specific finite value as |Q| →∞, but oscillates between two extreme values. In the high-temperature Poisson-Boltzmann scaling regime of limits β→ 0 and Q→∞ with the product β Q being finite, one reproduces the Manning-Oosawa type of counterion condensation with the uniform saturation of β Q _ren at the value 4/π in the region β|Q| ≥ 2. The obtained results disprove the “regularization hypothesis” of the previous work about the possibility of an analytic continuation of the formula for Q _ren from the stability region β |Q| < 2 to β |Q| ≥ 2.     

Dijet production in diffractive deep inelastic scattering at HERA

The production of dijets in diffractive deep inelastic scattering has been measured with the ZEUS detector at HERA using an integrated luminosity of 61 pb^-1. The dijet cross section has been measured for virtualities of the exchanged virtual photon, 5 < Q² < 100 GeV², and γ^*p centre-of-mass energies, 100 < W < 250 GeV. The jets, identified using the inclusive k_T algorithm in the γ^*p frame, were required to have a transverse energy E^* _T,jet > 4 GeV and the jet with the highest transverse energy was required to have E^* _T,jet > 5 GeV. All jets were required to be in the pseudorapidity range -3.5<η^* _jet<0. The differential cross sections are compared to leading-order predictions and next-to-leading-order QCD calculations based on recent diffractive parton densities extracted from inclusive diffractive deep inelastic scattering data.     

Proton dissociative ρ and elastic φ electroproduction at HERA

The electroproduction of ρ mesons with proton diffractive dissociation for Q ² > 7 GeV² and the elastic electroproduction of φ mesons for Q ² > 6 GeV² are studied in e ⁺ p collisions at HERA with the H1 detector, for an integrated luminosity of 2.8 pb^-1. The dependence of the cross sections on P _t ² and Q ² is measured, and the vector meson polarisation obtained. The cross section ratio between proton dissociative and elastic production of ρ mesons is measured and discussed in the framework of the factorisation hypothesis of diffractive vertices. The ratio of the elastic cross section for φ and ρ meson production is investigated as a function of Q ².     

A new numerical method for inverse Laplace transforms used to obtain gluon distributions from the proton structure function

We recently derived a very accurate and fast new algorithm for numerically inverting the Laplace transforms needed to obtain gluon distributions from the proton structure function F₂^gp(x,Q²)F_{2}^{\gamma p}(x,Q^{2}). We numerically inverted the function g(s), s being the variable in Laplace space, to G(v), where v is the variable in ordinary space. We have since discovered that the algorithm does not work if g(s)→0 less rapidly than 1/s as s→∞, e.g., as 1/s ^β for 0<β<1. In this note, we derive a new numerical algorithm for such cases, which holds for all positive and non-integer negative values of β. The new algorithm is exact if the original function G(v) is given by the product of a power v ^β−1 and a polynomial in v. We test the algorithm numerically for very small positive β, β=10⁻⁶ obtaining numerical results that imitate the Dirac delta function δ(v). We also devolve the published MSTW2008LO gluon distribution at virtuality Q ²=5 GeV² down to the lower virtuality Q ²=1.69 GeV². For devolution, β is negative, giving rise to inverse Laplace transforms that are distributions and not proper functions. This requires us to introduce the concept of Hadamard Finite Part integrals, which we discuss in detail.     

Measurement of the Q and energy dependence of diffractive interactions at HERA

Diffractive dissociation of virtual photons, , has been studied in ep interactions with the ZEUS detector at HERA. The data cover photon virtualities 0.17 < Q ² < 0.70 GeV² and 3 < Q ² < 80 GeV² with 3 < M_X < 38 GeV, where M_X is the mass of the hadronic final state. Diffractive events were selected by two methods: the first required the detection of the scattered proton in the ZEUS leading proton spectrometer (LPS); the second was based on the distribution of M_X. The integrated luminosities of the low- and high-Q² samples used in the LPS-based analysis are 0.9 pb^-1 and 3.3 pb^-1, respectively. The sample used for the M_X-based analysis corresponds to an integrated luminosity of 6.2 pb^-1. The dependence of the diffractive cross section on W, the virtual photon-proton centre-of-mass energy, and on Q² is studied. In the low-Q² range, the energy dependence is compatible with Regge theory and is used to determine the intercept of the Pomeron trajectory. The W dependence of the diffractive cross section exhibits no significant change from the low-Q² to the high-Q² region. In the low-Q² range, little Q² dependence is found, a significantly different behaviour from the rapidly falling cross section measured for Q ² > 3 GeV². The ratio of the diffractive to the virtual photon-proton total cross section is studied as a function of W and Q². Comparisons are made with a model based on perturbative QCD. Received: 27 March 2002 / Published online: 9 August 2002     

Diffraction at HERA, Color Opacity and Nuclear Shadowing in DIS off nuclei

The QCD factorization theorem for diffractive processes in DIS is used to derive formulae for the leading twist contribution to the nuclear shadowing of parton distributions in the low thickness limit (due to the coherent projectile (photon) interactions with two nucleons). Based on the current analyzes of diffraction at HERA we find that the average strength of the interactions which govern diffraction in the gluon sector at x≤ 10⁻³, Q ₀= 2 GeV is ∼50mb. This is three times larger than in the quark sector and suggests that applicability of DGLAP approximation requires significantly larger Q ₀ in the gluon sector. We use this information on diffraction to estimate the higher order shadowing terms due to the photon interactions with N≥ 3 nucleons which are important for the scattering of heavy nuclei and to calculate nuclear shadowing and Q ² dependence of gluon densities. For the heavy nuclei the amount of the gluon shadowing: G _A(x,Q ₀ ²) /AG _N(x,Q ₀ ²)_{|x ≤ 10−3}∼ 0.25–0.4 is sensitive to the probability of the small size configurations within wave function of the gluon “partonometer” at the Q ₀ scale. At this scale for A∼ 200 the nonperturbative contribution to the gluon density is reduced by a factor of 4–5 at x≤ 10⁻³ unmasking PQCD physics in the gluon distribution of heavy nuclei. We point out that the shadowing of this magnitude would strongly modify the first stage of the heavy ion collisions at the LHC energies, and also would lead to large color opacity effects in eA collisions at HERA energies. In particular, the leading twist contribution to the cross section of the coherent J/ψ production off A≥ 12 nuclei at s ⁻²≥ 70 GeV is strongly reduced as compared to the naive color transparency expectations. The Gribov black body limit for F _2A(x,Q ²) is extended to the case of the gluon distributions in nuclei and shown to be relevant for the HERA kinematics of eA collisions. Properties of the final states are also briefly discussed. Received: 12 March 1999     

The Asymptotic Limits of Zero Modes of Massless Dirac Operators

Asymptotic behaviors of zero modes of the massless Dirac operator H = α · D + Q(x) are discussed, where α = (α₁, α₂, α₃) is the triple of 4 × 4 Dirac matrices, , and Q(x) = (q _jk (x)) is a 4 × 4 Hermitian matrix-valued function with | q _jk (x) | ≤ C 〈x〉^−ρ, ρ > 1. We shall show that for every zero mode f, the asymptotic limit of |x|² f (x) as |x| → + ∞ exists. The limit is expressed in terms of the Dirac matrices and an integral of Q(x) f (x).      

A measurement of the proton structure function F2(x,Q2)

A measurement of the proton structure function F₂(x, Q²) is reported for momentum transfers squared Q² between 4.5 GeV² and 1600 GeV² and for Bjorken x between 1.8 × 10⁻¹⁴ and 0.13 using data collected by the HERA experiment H1 in 1993. It is observed that F₂ increases significantly with decreasing x, confirming our previous measurement made with one tenth of the data available in this analysis. The Q² dependence is approximately logarithmic over the full kinematic range covered. The subsample of deep inelastic events with a large pseudo-rapidity gap in the hadronic energy flow close to the proton remnant is used to measure the “diffractive” contribution to F₂.     

Dissociation of virtual photons in events with a leading proton at HERA

The ZEUS detector has been used to study dissociation of virtual photons in events with a leading proton, , in e ⁺ p collisions at HERA. The data cover photon virtualities in two ranges, 0.03 < Q ² < 0.60 GeV² and 2 < Q ² < 100 GeV², with M _X > 1.5 GeV, where M _X is the mass of the hadronic final state, X. Events were required to have a leading proton, detected in the ZEUS leading proton spectrometer, carrying at least 90% of the incoming proton energy. The cross section is presented as a function of t, the squared four-momentum transfer at the proton vertex, , the azimuthal angle between the positron scattering plane and the proton scattering plane, and Q ². The data are presented in terms of the diffractive structure function, . A next-to-leading-order QCD fit to the higher-Q ² data set and to previously published diffractive charm production data is presented.Received: 10 August 2004, Revised: 4 October 2004, Published online: 9 November 2004     

pp interactions at 303 GeVc: Analysis of diffraction excitation in the reaction pp → pX

The properties of the diffractively excited system observed at low missing mass in the inclusive pp → pX reaction at $\text{303}\text{GeV}\text{c}$ are presented. For diffraction excitation with |x| > 0.9, the cross section, σ_D=(5.6±0.8) mb, the mean charged multiplicity depends on the mass of the diffractive system approximately at 〈n_c〉 ≈ 0.6M_x + 2, and the slope of the momentum transfer distribution with |t| < 0.5 GeV² is 7.5±0.8 GeV^?2.