Origin of the Q 2 -dependence of the DIS structure functions
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rigin of the Q2Dependence of the DIS Structure Functions1 B. I. Ermolaeva, M. Grecob, and S. I. Troyanc a
Ioffe PhysicoTechnical Institute, 194021 St. Petersburg, Russia Department of Physics and INFN, University Rome III, Rome, Italy cSt. Petersburg Institute of Nuclear Physics, 188300 Gatchina, Russia b
Abstract—We consider in detail Q2dependence of the DIS structure functions. Very often this dependence is claimed to be originated by the Q2dependence of the QCD coupling. This leads to the smallx asymptotics of the structure functions with Q2dependent intercepts. We demonstrate that the DGLAP parametrization αs = αs(Q2) is an approximation valid in the region of large x (where 2pq can be approximated by Q2) only, providing the factorization scale is also large. Outside this region, the DGLAP parametrization fails, so αs should be replaced by an effective coupling which is independent of Q2 at small x. As a consequence, inter cepts of the structure functions are independent of Q2. Nevertheless, the smallx asymptotics of the structure functions explicitly depend on Q2, even when the coupling does not depend on it. We also consider the struc ture functions at small Q2 and give a comment on powerQ2 corrections to the structure functions at large and small Q2. DOI: 10.1134/S1063779613020111 1
1. INTRODUCTION
Each of the DIS structure functions has two inde pendent arguments: the invariant energy w = 2pq (with p and q being the hadron target and virtual photon momenta respectively) and virtuality of the incoming photon Q2 = –q2 > 0. Early experiments of 60s at SLAC proved that the structure functions depended on only one argument x = Q2/w. This phenomenon was called scaling. Later, new experiments proved that the scaling was violated i.e. in addition to xdepen dence the DIS structure functions also depended on Q2 straightforwardly. Since that it become conven tional to regard variables x and Q2 as independent arguments of the structure functions. However, one has to realize that those arguments (in contrast to the set of really independent arguments w, Q2) cannot be regarded as independent variables at all times. Indeed, they are independent when Q2 is fixed while w is scanned but they are not independent at fixed w and scanned Q2. The Q2dependence of the structure func tions is the objective of the DGLAP evolution equa tions [1] implementing the total resummation of lnk Q2. In DGLAP and beyond this approach, the Q2dependence of the structure functions is often thought to be induced by the Q2dependence of as: 2
α s = α s ( Q ).
(1)
As this parametrization takes place in the DGLAP equations [1], throughout the present paper we will address it as the DGLAPparametrization. Parametri zation (1), originally deduced for large x, where 2pq 1 The article is published in the original.
can be approximated by Q2, is often exploited to describe DIS in the smallx region and even in the smallx asymptotic of the structure functions, i.e. at x 0 (see e.g. [2–6]). As a result, the smallx asymptotics look l
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