ISSN 1063-7788, Physics of Atomic Nuclei, 2008, Vol. 71, No. 6, pp. 1077–1087. c  Pleiades Publishing, Ltd., 2008. ELEMENTARY PARTICLES AND FIELDS Theory The Approximation Method for Calculation of the Exponents of the Gluon Distribution, λ g , and the Structure Function, λ S , at Low x * G. R. Boroun ** and B. Rezaie Physics Department, Razi University, Kermanshah, Iran Received November 29, 2007 Abstract—We present a set of formulas using the solution of the QCD Dokshitzer–Gribov–Lipatov– Altarelli–Parisi (DGLAP) evolution equation to extract of the exponents of the gluon distribution, λ g , and structure function, λ S , from the Regge-like behavior at low x. The exponents are found to be independent of x and to increase linearly with ln Q 2 and are compared with the most data from the H1 Collaboration. We also calculated the structure function F 2 (x, Q 2 ) and the gluon distribution G(x, Q 2 ) at low x assuming the Regge-like behavior of the gluon distribution function at this limit and compared them with an NLO-QCD fit to the H1 data, two-Pomeron fit, multipole Pomeron exchange fit, and MRST (A.D. Martin, R.G. Roberts, W.J. Stirling, and R.S. Thorne), DL (A. Donnachie and P.V. Landshoff), and NLO GRV (M. Gl ¨ uk, E. Reya, and A. Vogt) fit results. PACS numbers: 12.40.Nn, 12.38.-t, 14.70.Dj DOI: 10.1134/S1063778808060100 1. INTRODUCTION The knowledge of the deep-inelastic scattering (DIS) structure functions at small values of the Bjorken scaling variable x is interesting for under- standing 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 quantum chromodynamics (pQCD) or a probe of new effects, and also because it is the basic ingredient in many other calculations of different high-energy hadronic processes. The behavior of the proton structure function F 2 (x, Q 2 ) at small x reflects the behavior of the gluon distribution, since the gluon is by far the dominant parton in this regime. At small x, only the structure function F 2 is measured. On the other hand, the gluon distribution cannot be measured directly from experiments. It is, therefore, important to measure the gluon distribution G(x, Q 2 ) indirectly from the proton structure function F 2 (x, Q 2 ) through the transition g → q q. Here, the representation for the gluon distribution G(x)= xg(x) is used, where g(x) is the gluon density. ∗ The text was submitted by the authors in English. ** E-mail: boroun@razi.ac.ir In pQCD, the high-Q 2 behavior of DIS is given by the Dokshitzer–Gribov–Lipatov–Altarelli–Parisi (DGLAP) evolution equations [1]. In the double- asymptotic limit (large energies, i.e., small-x, and large photon virtualities Q 2 ), the DGLAP evolution equations can be solved [2] and the structure function is expected to rise approximately like a power of x towards small x. This steep rise of F 2 (x, Q 2 ) towards low x observed at HERA [3] also indicates in pQCD a similar rise of the gluon towards low x. This sim- ilar behavior predicts a steep power-law behavior for the gluon distribution. Accordingly, the approximate solutions of DGLAP evolution equations have been reported in recent years [4, 5] with considerable phe- nomenological success. The small-x region of DIS offers a unique pos- sibility to explore the Regge limit of pQCD [6, 7]. This theory is successfully described by the exchange of a particle with appropriate quantum numbers and the exchange particle is called a Regge pole. Phe- nomenologically, the Regge pole approach to DIS im- plies that the structure functions are sums of powers in x, modulus logarithmic terms, each with a Q 2 - dependent residue factor. This model gives the fol- lowing parametrization of the DIS structure function F 2 (x, Q 2 ) at small x: F 2 (x, Q 2 )= ∑ i A i (Q 2 )x -λ i , that the singlet part of the structure function is con- trolled by Pomeron exchange at small x. The rapid 1077