QCD at lowQ 2—A correspondence relation for moments of structure functions

The precocious validity of QCD predictions in deep inelastic lepton nucleon scattering ande ⁺ e ^? annihilation is interpreted as a signal for an underlying “correspondence principle” relating perturbative and nonperturbative physics on theQ ² average. Correspondence relations for nonsinglet moments of deep inelastic structure functions are formulated, discussed and successfully tested against experiment. The relations provide an independent determination of the QCD ?-parameter from lowQ ² data in perfect agreement with results from largeQ ² analyses.     

The applicability of perturbative QCD to the pion form factor and the pionic wavefunction

We reanalyse the pionic form factor by using perturbative QCD theory and contributions from endpoint regions. We find that the perturbative QCD can be applied to the pionic form factor asQ ²>4 GeV² and they become unreliable asQ ²≦4 GeV². Therefore the applicability of perturbative QCD to the form factor is questionable only asQ ²≦4 GeV².     

QCD analysis of the Bjorken sum rule revisited

We present extended analysis of the polarized Bjorken sum rule using the four-loop expression for the coefficient function C _Bj(α_s) available now and the recent low Q ²-data from the Jefferson Lab and COMPASS experiments. We demonstrate that the perturbative series for the function C _Bj gives a hint to its asymptotic nature manifesting itself in the region Q ² ? 1 GeV². It is confirmed by the considered integral model for the perturbative QCD correction. We analyze values of higher-twist terms extracted from the mentioned data and discuss the interplay between higher orders perturbative and higher-twist contributions. We extend our consideration to the Gross-Llewellyn Smith sum rule and investigate the relation between higher twist coefficients in these two sum rules.     

Analysis of the electroproduction structure functions in the lowQ 2 region,combining the vector meson dominance and the parton model with possible scaling violation

The nucleon structure functionF ₂ is constructed and analysed for low values ofQ ² using the generalised vector meson dominance representation with the largeQ ² spectral function calculated from the analytic continuation of the parton model structure function. Various parametrisations of the parton distributions are considered. Possible effects of the largeQ ² scaling violation on the lowQ ² part of the structure function are investigated. The magnitude of the total contribution given by this asymptotic part can be as high as 50% of the vector meson contribution in the low-Q ², low-χ region. The contribution of the valence quarks alone to the structure function at lowQ ² turns out to be at least as important as the corresponding non-Pomeron Regge-pole-like terms coming from the vector meson part. Increase of the structure function with ν coming from the increase of the quark sea in the limit of small χ implied by QCD turns out to be relatively weak at lowQ ². Predictions of the model are compared with available experimental data. The photoproduction cross section and the nuclear effects in the structure function are also briefly discussed.     

The parton distribution functions of the photon and the structure functionF 2 γ (x,Q 2)

We present new parametrisations of the parton distribution functions of the photon including the first parametrisation in next-to-leading order QCD. We take into account some recent theoretical considerations pertaining to the gluon content of the photon,g ^γ. We argue that if an evolution is started at very lowQ ² and a fit to allF ₂ ^γ data performed with no constraints on the gluon distribution, then physically unreasonable gluon distributions may result. Our results support recent indications thatQ ₀ ² ≤1 GeV² is too low a value from which to start a perturbative evolution. Starting our evolution atQ ₀ ² =5.3 GeV², we evolve up inQ ² using a modified version of Rossi's Ansatz. The limited lever arm inQ ² leads to limited sensitivity to the QCD scale parameter Λ, though there is a preference for low values in the 0.1–0.2 GeV range. We also present new parametrisations of the singular asymptotic quark and gluon distribution functions of the photon which we believe are more accurate than those in current use.     

Hadroproduction of heavy quark flavors in QCD

Within the framework of perturbative QCD, we investigate the hadroproduction of charmed particles via two-gluon as well as quark-pair annihilation. The total production cross section is found to be sensitive to the choice of the scale parameter Q² and in general not free from non-perturbative effects. Charmed-particle production in the high-p_T region is also studied.     

DSE Perspective on QCD Modeling,Distribution Amplitudes,and Form Factors

We describe results for the pion distribution amplitude (PDA) at the non-perturbative scale μ = 2 GeV by projecting the Poincaré-covariant Bethe–Salpeter wave-function onto the light-front and use it to investigate the ultraviolet behavior of the electromagnetic form factor, F _π (Q ²), on the entire domain of spacelike Q ². The significant dilation of this PDA compared to the known asymptotic PDA is a signature of dynamical chiral symmetry breaking on the light front. We investigate the transition region of Q ² where non-perturbative behavior of constituent-like quarks gives way to the partonic-like behavior of quantum chromodynamics (QCD). The non-perturbative approach is based on the Dyson–Schwinger equation (DSE) framework for continuum investigations in QCD. The leading-order, leading-twist perturbative QCD result for Q ² F _π (Q ²) underestimates the new DSE computation by just 15 % on \({Q^2\gtrsim 8\,}\) GeV², in stark contrast with the result obtained using the asymptotic PDA.     

Infrared renormalons and the relations between the Gross-Llewellyn Smith and the Bjorken polarized and unpolarized sum rules

It is demonstrated that the infrared renormalon calculus indicates that the QCD theoretical expressions for the Gross-Llewellyn Smith sum rule and for the Bjorken polarized and unpolarized ones contain an identical negative twist-4 1/Q² correction. This observation is supported by the consideration of the results of calculations of the corresponding twist-4 matrix elements. Together with the indication of the similarity of the perturbative QCD contributions to these three sum rules, this observation leads to simple new theoretical relations between the Gross-Llewellyn Smith and Bjorken polarized and unpolarized sum rules in the energy region Q² ≥ 1 GeV². The validity of this relation is checked using concrete experimental data for the Gross-Llewellyn Smith and Bjorken polarized sum rules.     

The curvature of F<Subscript>2</Subscript><Superscript>p</Superscript>(x,Q<Superscript>2</Superscript>) as a probe of the range of validity of perturbative QCD evolutions in the small-x region

Perturbative NLO and NNLO QCD evolutions of parton distributions are studied, in particular in the (very) small-x region, where they are in very good agreement with all recent precision measurements of F₂ ^p(x,Q²). These predictions turn out to be also rather insensitive to the specific choice of the factorization scheme (MS̄ or DIS). A characteristic feature of perturbative QCD evolutions is a positive curvature of F₂ ^p which increases as x decreases. This perturbatively stable prediction provides a sensitive test of the range of validity of perturbative QCD.     

A model of deep inelastic polarized electroproduction

A realistic phenomenological model combining parton/QCD ideas with lower energy SU(6) constraints is proposed for the shape and evolution of the leading spin-dependent structure function G^ep(x, Q²) in polarized electroproduction. Close's broken-SU(6) ansatz is used to relate appropriately defined polarized quark-parton distribution densities to unpolarized ones at the matching momentum scale Q² = Q₀². The differences between spin and helicity distribution densities as well as the complications due to perturbative QCD and parton k_T (with related target-mass) effects are taken into account. Evolution to higher (>10 GeV²) values of Q² (where target-mass effects can be neglected) yields experimentally testable numerical predictions that are presented through various plots. The value of Q₀ is self-consistently determined to be about 0.5 GeV.     

The role of the pion cloud in the interpretation of the valence light-cone wavefunction of the nucleon

The pion cloud renormalises the light-cone wavefunction of the nucleon which is measured in hard, exclusive photon-nucleon reactions. We discuss the leading twist contributions to high-energy exclusive reactions taking into account both the pion cloud and perturbative QCD physics. The nucleon’s electromagnetic form-factor at high Q² is proportional to the bare nucleon probability Z and the cross-sections for hard (real at large angle or deeply virtual) Compton scattering are proportional to Z². Our present knowledge of the pion-nucleon system is consistent with Z = 0.7 ± 0.2. If we apply just perturbative QCD to extract a light-cone wavefunction directly from these hard exclusive cross-sections, then the light-cone wavefunction that we extract measures the three valence quarks partially screened by the pion cloud of the nucleon. We discuss how this pion cloud renormalisation effect might be understood at the quark level in terms of the (in-)stability of the perturbative Dirac vacuum in low energy QCD.     

Measurement of the structure functionsF 2 andxF 3 and comparison with QCD predictions including kinematical and dynamical higher twist effects

The isoscalar nucleon structure functionsF ₂(x, Q ²) andxF ₃(x, Q ²) are measured in the range 0<Q ²<64 GeV², 1.7<W ²<250 GeV²,x<0.7 using ν and \(\bar v\) interactions on neon in BEBC. The data are used to evaluate possible higher twist contributions and to determine their impact on the evaluation of the QCD parameter Λ. In contrast to previous analyses reaching to such lowW ² values, it is found that a low \(\Lambda _{\overline {MS} } \) value in the neighbourhood of 100 MeV describes the data adequately and that the contribution of dynamical higher twist effects is small and negative.     

“Massive” perturbative QCD, Regular in the IR limit

The goal of the research is to devise a modification of the perturbative QCD (pQCD) that should be regular in the low-energy region and could serve as a practical means for the analysis of data below 1 GeV down to the IR limit. Recent observation of the four-loop pQCD series “blow-up” in the region below 1 GeV for the Bjorken Sum Rule gave an impetus to this attempt. The proposed “massive” analytic pQCD has two sources. It can be treated as the common logarithmic pQCD with only one parameter added, the effective “glueball mass” m _gl ? 1 GeV, serving as an IR regulator. At the same time, it looks like a modification of Analytic Perturbation Theory (APT) comprising nonpower perturbative expansion that makes it compatible with linear integral transformations. Figuratively (with minor reservations), the proposed MAPT differs from the minimal APT by simple ansatz Q ² → Q ² + m _g1 ² .     

Proton structure functions in the quasielastic limit

The evolution equation for the nonsinglet distribution of partons is solved in the leading order of perturbative QCD. It is shown that an exact analytic solution to the evolution equations can be found in the quasielastic limit. The Q ² evolution of the structure function for x → 1 is in good agreement with experimental data.     

Nonperturbative corrections in resummed cross sections

We show that the resummation of large perturbative corrections in QCD leads to ambiguities in high energy cross sections that are suppressed by powers of large momentum scales. These ambiguities are caused by infrared renormalons, which are a general feature of resummed hardscattering functions in perturbative QCD, even though these functions are infrared safe order-by-order in perturbation theory. As in the case of the operator product expansion, the contributions of infrared renormalons to coefficient functions may be absorbed into the definition of higher-dimensional operators, which induce nonperturbative corrections that are power-suppressed at high energies. The strength of the suppression is determined by the location of the dominant infrared renormalon, which may be identified explicitly in the resummed series. In contrast to the operator product expansion, however, the relevant operators in factorized hadron-hadron scattering and jet cross sections are generally nonlocal in QCD, although they may be expressed as local operators in an effective theory for eikonalized quarks. In this context, we verify and interpret the presence of 1 /Q corrections to the inclusive Drell-Yan cross section with Q the pair mass. In a similar manner, we find exp (-b² In Q) corrections in the impact parameter space of the transverse momentum distributions of the Drell-Yan process and e⁺6^- annihilation. We also show that the dominant nonperturbative corrections to cone-based jet cross sections behave as 1 /(Qδ), with δ the opening angle of the jet and Q the center of mass energy.     

The effect of quark masses on the running of the QCD coupling constant

The presence of thresholds for finite quark masses alters the behavior of the QCD coupling constant. While the alterations are small they can have significant influence on the determination of the scale parameter Λ. Part of the discrepancy between recent highQ ² determinations of Λ and older lowQ ² determinations can be attributed to charm and bottom thresholds. We present simple, analytic expressions for the mass dependence of α(Q ²) and exploit these to discuss Λ determinations, scale breaking and the phenomenon of temporary freedom when more than 17 quark flavors exist.     

Probing the perturbative NLO parton evolution in the small-<Emphasis Type="Italic">x</Emphasis> region

A dedicated test of the perturbative QCD NLO parton evolution in the very small-x region is performed. We find a good agreement with recent precision HERA data for F ₂ ^p(x,Q ²), as well as with the present determination of the curvature of F ₂ ^p. Characteristically, perturbative QCD evolutions result in a positive curvature which increases as x decreases. Future precision measurements in the very small x-region, x < 10^-4, could provide a sensitive test of the range of validity of perturbative QCD.Received: 6 December 2004, Revised: 1 February 2005, Published online: 9 March 2005