On the rise of the proton structure function F2 towards low x

A measurement of the derivative (∂ lnF₂/∂ lnx)_Q²≡−λ(x,Q²) of the proton structure function F₂ is presented in the low x domain of deeply inelastic positron–proton scattering. For 5×10⁻⁵x0.01 and Q²1.5 GeV², λ(x,Q²) is found to be independent of x and to increase linearly with lnQ².     

Q 2 dependence of the measured asymmetry A 1: A test of the Bjorken sum rule

We analyze the proton and deutron data on the spin-dependent asymmetry A ₁(x, Q ²), supposing that the DIS structure functions g ₁(x, Q ²) and F ₃(x, Q ²) have a similar Q ² dependence. As a result, we have found that Λ ₁ ^p −Λ ₁ ⁿ =0.190±0.038 at Q ²=10 GeV² and Λ ₁ ^p −Λ ₁ ⁿ =0.165±0.026 at Q ²=3 GeV²; these values are in the best agreement with the Bjorken sum rule predictions. Pis’ma Zh. éksp. Teor. Fiz. 65, No. 1, 9–14 (10 January 1997) Published in English in the original Russian journal. Edited by Steve Torstveit.     

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.     

Moments of polarized parton distribution functions

We present novel results for the first moment of the spin-dependent structure function g ₁(x,Q ²) of the nucleon at small (Q ² < 0.3 GeV²) photon virtuality in the framework of a relativistic formulation of baryon chiral perturbation theory. We perform a next-to-leading order calculation and obtain significant differences to previously found results based on the heavy-baryon approach for the proton and neutron.Received: 30 September 2002, Published online: 22 October 2003PACS: 12.39.Fe Chiral Lagrangians - 14.20.Dh Protons and neutrons     

Measurement and QCD analysis of the diffractive deep-inelastic scattering cross section at HERA

A detailed analysis is presented of the diffractive deep-inelastic scattering process ep→eXY, where Y is a proton or a low mass proton excitation carrying a fraction 1-x_IP>0.95 of the incident proton longitudinal momentum and the squared four-momentum transfer at the proton vertex satisfies |t|<1 GeV². Using data taken by the H1 experiment, the cross section is measured for photon virtualities in the range 3.5≤Q²≤1600 GeV², triple differentially in x_IP, Q² and β=x/x_IP, where x is the Bjorken scaling variable. At low x_IP, the data are consistent with a factorisable x_IP dependence, which can be described by the exchange of an effective pomeron trajectory with intercept α_IP(0)=1.118±0.008(exp.)^+0.029 _-0.010(model). Diffractive parton distribution functions and their uncertainties are determined from a next-to-leading order DGLAP QCD analysis of the Q² and β dependences of the cross section. The resulting gluon distribution carries an integrated fraction of around 70% of the exchanged momentum in the Q² range studied. Total and differential cross sections are also measured for the diffractive charged current process e⁺p→ν̄_eXY and are found to be well described by predictions based on the diffractive parton distributions. The ratio of the diffractive to the inclusive neutral current ep cross sections is studied. Over most of the kinematic range, this ratio shows no significant dependence on Q² at fixed x_IP and x or on x at fixed Q² and β.     

Study of nucleon spin-dependent properties in the resonance region

In this paper, the spin-dependent structure functions of nucleon g ₁, and photoabsorption cross sections σ_1/2, σ_3/2 and σ_T in the resonance region are estimated based on the constituent quark model and the properties of the five phenomenological Breit-Wigner resonances P ₃₃(1232), S ₁₁(1535), D ₁₃(1520), P ₁₁(1440), and F ₁₅(1680). Our results are compared to the recent E143 data of the polarized structure functions g ₁(W ², Q ²) at points Q ²=0.5 GeV² and Q ²=1.2 GeV² and the data of the total inclusive photoabsorption cross sections. Received: 7 October 1997     

On the perturbative stability of the QCD predictions for the ratio R=FL/FT in heavy-quark leptoproduction

We analyze the perturbative and parametric stability of the QCD predictions for the Callan–Gross ratio, R(x,Q ²)=F _L /F _T , in heavy-quark leptoproduction. We consider the radiative corrections to the dominant photon–gluon fusion mechanism. In various kinematic regions, the following contributions are investigated: exact NLO results at low and moderate Q ²≲m ², asymptotic NLO predictions at high Q ²≫m ², and both NLO and NNLO soft-gluon (or threshold) corrections at large Bjorken variable x. Our analysis shows that large radiative corrections to the structure functions F _T (x,Q ²) and F _L (x,Q ²) cancel each other in their ratio R(x,Q ²) with good accuracy. As a result, the NLO contributions to the Callan–Gross ratio are less than 10% in a wide region of the variables x and Q ². We provide compact LO predictions for R(x,Q ²) in the case of low x ≪1. A simple formula connecting the high-energy behavior of the Callan–Gross ratio and low-x asymptotics of the gluon density is derived. It is shown that the obtained hadron-level predictions for R(x→0,Q ²) are stable under the DGLAP evolution of the gluon distribution function. Our analytic results simplify the extraction of the structure functions F ₂ ^c (x,Q ²) and F ₂ ^b (x,Q ²) from measurements of the corresponding reduced cross sections, in particular at DESY HERA.     

Gluon distributions in real and virtual photon and the photon structure function

Gluon distributions in real and virtual photons are calculated using evolution equations in the NLO approximation. The quark distributions in the photon determined on the basis of the QCD sum rule approach in [1] are taken as an input. It is shown that gluon distribution in the photon can be reliably determined up tox=0.03÷0.05, much lower than the corresponding values in the case of quark distributions. Two variants of the calculations are considered: (1) it is assumed that there are no intrinsic gluons in the photon at some low normalization pointQ ²=Q ₀ ² ∼1GeV²; (2) it is assumed that gluonic content of the photon at lowQ ₀ ² is described by gluonic content of vector mesonsρ, ω, ϕ. The gluon distributions in these two variants appear to be different. This fact permits one to clarify the origin of nonperturbative gluonic content of the photon by comparing the results with experiment. Structure functionsF ₂(x) for real and virtual photon are calculated and it is shown that in the regionx≥0.2 where QCD approach is valid, there is a good agreement with experiment.     

Electroproduction of π+ n, π- p andK + Λ,K + 2211; 0 final states above the resonance region

The reactionsΣ _v p→π⁺ n,K ⁺ Λ,K ⁺ ∑ ⁰ andΣ _v n→π⁺ n were studied at invariant hadronic masses around 2.2. GeV forQ ²=0.06, 0.28, 0.70, and 1.35 GeV². The main results are: At small |t| the π⁺ production is dominated by longitudinally polarized photons and can be described by one pion exchange. At low |t| the transverse (π⁺ n) cross section drops steeply withQ ², but remains roughly constant forQ ²≧0.5 GeV². For |t?≧0.8 GeV², (π⁺ n/dt) is almost independent ofQ ². The integrated cross section (π⁺ n) shows a similarQ ²-dependence asσ _tot (γ _v p) forQ ²≧0.28 GeV². The ratioσ(π^- p)/σ(π⁺ n) atQ ²=0.70 and 1.35 GeV² for |t|≧0.6 GeV² is smaller than in photoproduction and close to 1/4. The ratioσ(K ⁺ ∑ ⁰ decreases steeply withQ ² following roughly the predictions of the quark-parton model.     

The longitudinal structure function F L(x, Q 2) and the gluon distribution G(x, Q 2) at low x

We obtain a relation between the longitudinal structure function F _L(x, Q ²), F ₂(x, Q ²) and G(x, Q ²) at small x, using the formalism recently reported by one of the authors [2]. We also obtain a relation between F _L(x, Q ²), F ₂(x, Q ²) and its slope (dF ₂(x, Q ²))/(dlnQ ²). This provides us with the determination of the longitudinal structure function F _L(x, Q ²) from F ₂(x, Q ²) data and hence extract the gluon distribution G(x, Q ²).     

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     

Triplet superfluidity in neutron matter with Skyrme forces at subnuclear and supranuclear densities in a strong magnetic field

Within a generalized non-relativistic Fermi-liquid approach we have found general analytical formulae for phase-transition temperatures T _c,1(n, H) and T _c,2(n, H) (which are nonlinear functions of density, n, and linear of magnetic field, H) for phase transitions in spatially uniform, dense, pure neutron matter from normal to superfluid states with spin-triplet p-wave pairing (similar to anisotropic superfluid phases ³He - A₁ and ³He - A₂) in steady and homogeneous sufficiently strong magnetic field (but |μ _n |H ≪ E _c < ɛ _F (n), where μ _n is the magnetic dipole moment of a neutron, E _c is the cutoff energy and ɛ _F (n)is the Fermi energy in neutron matter). General formulae for T _c,1,2(n,H) are valid for arbitrary parameterization of the effective Skyrme forces in neutron matter. We have used for definiteness the so-called SLy2, Gs and RATP parameterizations of the Skyrme forces with different exponents in their power dependence on density n (at sub- and supranuclear densities) from the interval 0.7 n ₀ ≲ n < n _c (Skyrme)< 2 n ₀, where n ₀ =0.17 fm⁻³ is the nuclear density and n _c (Skyrme)is the the critical density of the ferromagnetic instability in superfluid neutron matter. These phase transitions might exist in the liquid outer core of magnetized neutron stars.     

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.     

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     

A microscopic calculation of dynamic critical exponents for displacive phase transitions

For anO(n)-isotropic lattice dynamicalQ ⁴-model describing displacive phase transitions ind dimensions, we employ a microscopic 1/n-expansion in order to show that over-damped soft-phonon behavior emerges for frequencies smaller than those of the characteristic orderv _c =O(n ^–x ). This is concluded from the fact that the displacement propagatorD(q, v) assumes the time-dependent Ginzburg-Landau (TDGL) form with a damping coefficient=O(n ^–x ), whenv becomes smaller thanv _c . The exponentx is found to bex=4–d for 2<d<3,x=(d–1)/2 for 3<d<5, andx=2 ford>5. The dynamic critical exponents forv _c (q) and forD(0,v) are derived atT=T _c 0 and toO(1/n). Their values are nontrivial for 2<d<4 and, within the TDGL-region, agree with the those appearing already for frequencies ofO(n ⁰) in TDGL-models with nonconserved order parameter andO(n ⁰)-damping coefficient. The latter case was studied by Halperin, Hohenberg, and Ma in 1972. Even in the TDGL-region, the energy conservation does not affect the dynamic exponents for largen(>2, since the specific heat is finite), but an energy diffusion singularity appears in theQ ²-response function which is related to the basic quantity of the 1/n-method, the effective interactionU _eff. By an estimate of order we find that the damping coefficients resulting from the coupling between the relaxation modes contained inU _eff and the critical modes inD are of ordern ^–w withw>x, such that the coupling between weakly damped critical modes is responsible for the crossover to the TDGL-behavior for largen. The exponentz=d/2, known to be generated by the coupling between order parameter and conservedO(n)-densities in TDGL-models, cannot be seen up to the order calculated. We also point out problems of a microscopic-expansion and comment upon differences between microscopic treatments for displacive transitions and those for the Bose condensation.     

The Q-dependence of the generalised Gerasimov-Drell-Hearn integral for the deuteron,proton and neutron

The Gerasimov-Drell-Hearn (GDH) sum rule connects the anomalous contribution to the magnetic moment of the target nucleus with an energy-weighted integral of the difference of the helicity-dependent photoabsorption cross sections. Originally conceived for real photons, the GDH integral can be generalised to the case of photons with virtuality Q². For spin-1/2 targets such as the nucleon, it then represents the non-perturbative limit of the first moment of the spin structure function g ₁ (x,Q ² ) in deep inelastic scattering (DIS). The data collected by HERMES with a deuterium target are presented together with a re-analysis of previous measurements on the proton. This provides an unprecedented and complete measurement of the generalised GDH integral for photon-virtuality ranging over 1.2&lt;Q ² &lt;12.0 GeV² and for photon-nucleon invariant mass squared W² ranging over 1&lt;W ² &lt;45 GeV², thus covering simultaneously the nucleon-resonance and the deep inelastic scattering regions. These data allow the study of the Q²-dependence of the full GDH integral, which is sensitive to both the Q²-evolution of the resonance form factors and contributions of higher twist. The contribution of the nucleon-resonance region is seen to decrease rapidly with increasing Q². The DIS contribution is sizeable over the full measured range, even down to the lowest measured Q². As expected, at higher Q² the data are found to be in agreement with previous measurements of the first moment of g₁. From data on the deuteron and proton, the GDH integral for the neutron has been derived and the proton-neutron difference evaluated. This difference is found to satisfy the fundamental Bjorken sum rule at Q ² = 5 GeV². Received: 20 October 2002 / Published online: 15 January 2003     

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.     

On the possibility of measuringF L (x,Q 2) at HERA using radiative events

It is shown that a significant measurement of the longitudinal structure functionF _L (x, Q ²) can be performed at HERA, forQ ²=2 GeV² andQ ²=5 GeV² and forx around 10^?4, using radiative events with hard photon emission collinear to the incident lepton beam, under the present running conditions and with an integrated luminosity of 10 pb^?1. The influence of experimental conditions is discussed.