Gluon Polarization from QCD Sum Rules

The gluon polarization ∆G in a nucleon can be defined in a gauge invariant way as the integral over the Ioffe-time distribution of polarized gluons. We argue that for sufficiently regular polarized gluon distributions ∆G is dominated by contributions from small and moderate values of the Ioffe-time z ∼ 10. As a consequence ∆G can be estimated with 20% accuracy from the first two even moments of the polarized gluon distribution, and its behavior at small values of Bjorken x or, equivalently, at large Ioffe-times z. We employ this idea and compute the first two moments of the polarized gluon distribution within the framework of QCD sum rules. Combined with the color coherence hypothesis we obtain an upper limit for ∆G ∼ 2± 0.5 at a typical scale μ ∼ 1 GeV. Work supported in part by BMBF On leave of absence from N. Copernicus Astronomical Center, Polish Academy of Science, ul. Bartycka 18, PL–00-716 Warsaw (Poland) During the last two decades deep-inelastic lepton-nucleon scattering (DIS) has proven to be one of the most valuable sources of information about nucleon structure. According to the Operator-Product-Expansion (OPE), the Q-dependence of the cross-section can be entirely included in perturbatively calculable Wilson coefficients, and thus completely factorized from the effects of long-distance dynamics, described by a set of twist-2 parton distribution functions which are defined at a certain reference scale. The analysis of experiments with unpolarized lepton beams has provided detailed informations on quark momentum distribution functions and has lead to the conclusion that – even at relatively low scales of few GeV – the nucleon momentum is shared almost equally between quarks and gluons. With the advent of high-quality polarized targets and beams it has become possible to study polarized quark distributions [1]. In particular, combining experimental results with SU(3)-flavor symmetry allows to determine the fraction Σ of the nucleon polarization carried by quarks [2]. Since the latter turns out to be much smaller, Σ ≈ 0.3, than expected from simple quark models [3], it is natural to search for other contributions to the nucleon spin. At a given scale μ the nucleon spin sum rule allows to separate the nucleon spin into gauge-invariant quark and gluon contributions [4]: 1 2 = Jq(μ ) + Jg(μ ) , (1) with both parts calculable as matrix elements of local operators sandwiched between polarized nucleon states. The quark contribution can be split further into gauge-invariant polarization and angular momentum parts: Jq(μ ) = 1 2 Σ(μ) + Lq(μ ). (2) Here the quark polarization Σ(μ) is linked to the matrix element of the flavour singlet axial current operator ∫ dx ψ̄γγ5ψ, and the angular momentum Lq(μ ) is connected to the gauge-invariant angular momentum operator − ∫ dx ψ̄(x× iD)ψ, with the covariant derivative D = (D0,D) = ∂ μ + igA. The separation of the gluon contribution Jg into angular momentum and polarization parts, Jg(μ ) = ∆G(μ) + Lg(μ ), (3) is more involved. The reason is that in QCD appropriate relations between (even) moments of the polarized gluon density and matrix elements of twist-2 operators exist only for l ≥ 2: