BFKL predictions at small x from kT and collinear factorization viewpoints

Hard scattering processes involving hadrons at small x are described by a kT -factorization formula driven by a BFKL gluon. We explore the equivalence of this description to a collinear-factorization approach in which the anomalous dimensions γgg and γqg/αS are expressed as power series in αS log(1/x), or to be precise αS/ω where ω is the moment index. In particular we confront the collinear-factorization expansion with that extracted from the BFKL approach with running coupling included. On leave from the Henryk Niewodniczański Institute of Nuclear Physics, 31-342 Kraków, Poland. Recently there have been several studies [1-5] of the validity and possible modification of the conventional Altarelli-Parisi (or GLAP) description of deep inelastic scattering in the small x region that has become accessible at HERA, x ∼ 10. The relevant modifications are the inclusion of contributions which are enhanced by powers of log(1/x), but which lie outside the leading (and next-to-leading) Altarelli-Parisi perturbative expansion. Formally they correspond to the expansion of the anomalous dimensions γgg and γqg/αS as power series in αS/ω where ω is the moment index. An alternative approach which automatically resums all these leading log(1/x) contributions to γgg and γqg/αS is provided by the BFKL equation coupled with the kT -factorization formula for calculating observable quantities [6, 7]. The main aim of this paper is to explore the connection between these two approaches. To be specific we study the relation between the collinear-factorization formula with log(1/x) terms included and the kT factorization formula based on the solution of the BFKL equation [8] with running coupling αS. We show that both approaches generate the same first few terms in the perturbative expansion of γgg and, more important, of γqg, which are presumably the most relevant contributions for the description of deep inelastic scattering in the HERA range. They differ substantially, however, in the asymptotically small x regime. Deep inelastic unpolarised electron-proton scattering may be described in terms of two structure functions, F2(x,Q ) and FL(x,Q ). As usual, the kinematic variables are defined to be Q = −q and x = Q/2p.q, where p and q are the four-momenta of the incoming proton and virtual photon probe respectively. At small values of x, x < ∼ 10 , these observables reflect the distribution of gluons in the proton, which are by far the dominant partons in this kinematic region. The precise connection between the small x structure functions and the gluon distribution is given by the kT -factorization formula [6, 7], Fi(x,Q ) = ∫ dk T k T ∫ 1 x dx x F γg i ( x x , k T , Q 2 ) f(x′, k T ) (1) with i = 2, L, which is displayed pictorially in Fig. 1. The gluon distribution f(x, k T ), unintegrated over k T , is a solution of the BFKL equation, while F γg i are the off-shell gluon structure functions which at lowest-order are determined by the quark box (and crossed-box) contributions to photon-gluon fusion, see Fig. 1. For sufficiently large values of Q the leading-twist contribution is dominant, and it is most transparent to discuss the Q evolution of Fi(x,Q ) in terms of moments. Then the x convolution of (1) factorizes to give F i(ω,Q ) = ∫ dk T k T F γg i (ω, k 2 T , Q )f(ω, k T ) (2) where the moment function f(ω, k T ) ≡ ∫ 1 0 dx x xf(x, k T ), (3) with similar relations for F i and F γg i . 1 Fixed αS : kT -factorization to collinear-factorization It is illuminating to first consider the case of fixed coupling αS [2]. Then the photon-gluon moments, F γg i are simply functions of τ ≡ Q /k T (and ω), for massless quarks. Hence (2) becomes a convolution in k T which, in analogy with the x ′ convolution, may be factorized by taking moments a second time. In this way we obtain representations for the F i with factorizable integrands F i(ω,Q ) = 1 2πi ∫ c+i∞ c−i∞ dγ F̃ γg i (ω, γ) f̃(ω, γ)(Q ) (4) with c = 1 2 . The (double) moments F̃ γg i and f̃ of the gluon structure functions and the gluon distribution are respectively defined by F̃ γg i (ω, γ) = ∫ dτ τ F γg i (ω, τ) (5) f̃(ω, γ) = ∫ dk T (k 2 T ) −γ−1 f(ω, k T ) (6) where the F̃ γg i are dimensionless, but f̃ carries the dimension (k 2 0) . Representation (4) enables the leading-twist contribution to be identified from a knowledge of the analytic properties of f̃ and F̃ γg i in the complex γ plane. The gluon distribution f(x, k T ) satisfies the BFKL equation, which in moment space has the form f(ω, k T ) = f (ω, k T ) + αS ω ∫ dk T k T K(k T , k ′2 T )f(ω, k ′2 T ) (7) where αS ≡ 3αS/π andK is the usual BFKL kernel. The double-moment function f̃ is therefore given by f̃(ω, γ) = f̃ (ω, γ) 1− (αS/ω)K̃(γ) (8) where K̃(γ) is the eigenvalue of the BFKL kernel corresponding to the eigenfunction proportional to (k T ) . It can be shown that K̃(γ) = 2Ψ(1)−Ψ(1− γ)−Ψ(γ) = 1 γ [