The Gluon Distribution as a Function of F 2 and dF 2 / dlnQ 2 at small x . The Next - to - Leading Analysis

We present a set of formulae to extract the gluon distribution function from the deep inelastic structure function F2 and its derivative dF2/dlnQ 2 at small x in the leading and next-to-leading order of perturbation theory. The detailed analysis is given for new HERA data. The values of the gluon distribution are found in the range 10 ≤ x ≤ 10 at Q = 20 GeV E-mail:[email protected] E-mail:[email protected] The knowledge of the DIS structure functions at small values of the Bjorken scaling variable x is interesting for understanding the inner structure of hadrons. Of great relevance is the determination of the gluon density at low x, where gluons are expected to be dominant, because it could be a test of perturbative QCD or a probe of new effects, and also because it is the basic ingredient in many other calculations of different high energy hadronic processes. Recently two experiments working in the electron-proton collider HERA at DESY (H1 and ZEUS) have published new data on the structure function F2 [1], [2]. Up to now all the analysis performed of these data [3] [5] found that the gluon distribution rises steeply towards low x (in the moderate Q range of the measurements). This behaviour has been recently connected within the DGLAP evolution equations [6] with the less singular are found at lower values of Q by NMC and E665 experiments. We introduce the standard parameterizations of singlet quark s(x,Q) and gluon g(x,Q) parton distribution functions (PDF) (see, for example, [5]) s(x,Q) = Asx (1− x)s(1 + ǫs √ x+ γsx) ≡ xs̃(x,Q) g(x,Q) = Agx (1− x)g(1 + ǫg √ x+ γgx) ≡ xg̃(x,Q), (1) with Q dependent parameters in the r.h.s.. Note that the behaviour of Eq. (1) with a Q-independent value for δ (δq = δg) obeys the DGLAP equation when x ≫ 1 (see, for example, [8] [10]). If δ(Q0) = 0 in some point Q0 ∼ 1GeV 2 (see [11], [12], [6]) , then the behaviour p(x,Q) ∼ Const (p = (s, g)) is not compatible with DGLAP equation and a more singular behaviour is generated. If we restrict the analysis to a Regge-like form of structure functions, one obtains (see [6]) p(x,Q) ∼ xp2 with next-to-leading order (NLO) δq(Q ) 6= δg(Q) intercept trajectories. Without any restriction the double-logarithmical behaviour, i.e. p(x,Q) ∼ exp ( 2 √