On the Resummed Gluon Anomalous Dimension and Structure Functions at Small x

The impact of the recently evaluated ‘irreducible’ contributions to the resummed next-to-leading logarithmic small-x anomalous dimension γgg 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 enhancement caused by the resummed leading logarithmic small-x anomalous dimension and the quarkonic contributions beyond next-to-leading order. PACS: 12.38.Cy, 13.60.Hb Phys. Rev. D57 (1998) 1 The structure functions of the proton in unpolarized deep–inelastic scattering (DIS) show a steep rise towards very small Bjorken-x values, which becomes stronger with increasing resolution Q. This behavior is a consequence both of the shape of the non-perturbative quark singlet and gluon initial distributions – xΣ and xg rise roughly like x for a starting scale Q0 ≃ 4GeV 2 – and of the form of the evolution kernels governing the renormalization group equations of the mass singularities. The anomalous dimensions γij for the evolution of the parton densities, as well as the Wilson coefficients Cn of the structure functions, contain large logarithmic small-x contributions. In order to arrive at a reliable theoretical framework at very small x, the resummation of these terms may be necessary to all orders in the strong coupling αs(Q ). For the unpolarized singlet case considered here the dominant contributions take, in Mellin-N space, the form α s /(N − 1) , l ≥ 0. The resummation of γgg and γgq in the leading small-x approximation (Lx), l = 0, was performed long ago [1]. γ gg (N) is obtained as the solution of 1 = αs N − 1 χ0(γ (0) gg ) (1) with αs = CAαs(Q )/π, CA = Nc = 3, CF = 4/3 and χ0(γ) = 2ψ(1)− ψ(γ)− ψ(1− γ) , (2) furthermore γ gq = (CF/CA) γ (0) gg . The quark anomalous dimensions γqq and γqg, on the other hand, receive contributions for l ≥ 1 only. The l = 1 terms were derived in ref. [2], together with the corresponding resummations for the coefficient functions C2 and CL. The large effects of these quantities on the small-x behavior of the DIS structure functions were subsequently studied in detail [3–7]. In those investigations the small-x resummation of the gluon anomalous dimension γgg to next-to-leading order small-x (NLx) accuracy could not be taken into account. This resummation has now been performed for the quarkonic contributions [8–11] proportional to the number of quark flavors Nf . Recently also the ‘irreducible’ gluonic terms ∝ CA have been derived [12], i.e., those contributions which are energy–scale independent in the framework of ref. [10] underlying that calculation. The corresponding terms of γgq, however, still remain to be determined. In this note we investigate the impact of these new resummed contributions to γgg on the evolution of the parton densities and the proton structure functions F2(x,Q ) and FL(x,Q ), for the first time including calculated subleading terms into the renormalization group analysis. Hence the comparison of the results to the findings of previous studies [3–7] should allow for improved estimates of the convergence of the small-x resummation approximation, despite a fully quantitative, scheme–independent NLx analysis not being possible at present. As will be demonstrated below, the effect of the new contributions to the resummed anomalous dimension γgg is very large and opposite to that of the previously known resummed terms. This implies, already at the present stage, considerable changes particularly for gluon–dominated quantities, which partly modify conclusions obtained in previous numerical investigations [3–6]. A detailed account of the solution of the evolution equations in the presence of all-order anomalous dimensions and coefficient functions will be given in a forthcoming publication [13]. As shown in ref. [9] the larger eigenvalue of the singlet anomalous dimension matrix, γ+(N), may be obtained in the Q0 scheme [14] as the solution of 1 = αs N − 1 [χ0(γ+) + αsχ1(γ+)] , (3)