The Longitudinal Structure Function F L 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 longitudinal deep inelastic structure function FL from the transverse structure function F2 and its derivative dF2/dlnQ 2 at small x. Our expressions are valid for any value of δ, being x the behavior of the parton densities at low x. Using F2 HERA data we obtain FL in the range 10 ≤ x ≤ 10 at Q = 20 GeV. Some other applications of the formulae are pointed out. E-mail:[email protected] E-mail:[email protected] For experimental studies of hadron-hadron processes on the new, powerful LHC collider, it is necessary to know in detail the values of the parton (quark and gluon) distributions (PD) of nucleons, especially at small values of x. The basic information on the quark structure of nucleons is extracted from the process of deep inelastic lepton-hadron scattering (DIS). Its differential cross-section has the form: dσ dxdy = 2παem xQ4 [ ( 1− y + y/2 ) F2(x,Q )− ( y/2 ) FL(x,Q )], where F2(x,Q) and FL(x,Q) are the transverse and longitudinal structure functions (SF), respectively. The longitudinal SF FL(x,Q) is a very sensitive QCD characteristic because it is equal to zero in the parton model with spin−1/2 partons (it is very large with spin−0 partons). In addition, at small values of x, FL data are not yet available, as they require a rather cumbersome procedure (see [3], for example). In the present article we study the behaviour of FL(x,Q) at small values of x, using the HERA data [4], [5] and the method [6] of replacement of the Mellin convolution by ordinary products. By analogy with the case of the gluon distribution function (see [7] and its references) it is possible to obtain the relation between FL(x,Q), F2(x,Q) and dF (x,Q)/dlnQ at small x. Thus, the small x behaviour of the SF FL(x,Q) can be extracted directly from the measured values of F2(x,Q) and its derivative without a cumbersome procedure (see [3]). These extracted values of FL may be well considered as new small x “experimental data” of FL. When experimental data for FL at small x become available with a good accuracy, a violation of the relation will be an indication of the importance of other effects as higher twist contribution and/or of non-perturbative QCD dynamics at small x. We follow the notation of our previous work in ref. [7]. The singlet quark s(x,Q0) and gluon g(x,Q0) parton distribution functions (PDF) 4 at some Q0 are parameterized by (see, for example, [8]): p(x,Q0) = Apx (1− x)p(1 + p √ x+ γpx) (p = s, g) (1) The value of δ is a matter of controversy. The “conventional” choice is δ = 0, which leads to a non-singular behaviour of the PD (as for example the D 0 fit in [8]) when x → 0. Another value, δ ∼ 1 2 , was obtained in the studies performed in ref. [9] as the sum of the leading powers of ln(1/x) in all orders of perturbation theory (PT) (D − fit in [8]). Experimentally, recent NMC data [10] favor small values of δ. This result is also in agreement with present data for pp and pp total cross-sections (see [11]) and corresponds to the model of Landshoff and Nachtmann pomeron [12] with the exchange of a pair of non-perturbative gluons, yielding δ = 0.086. However, the new HERA data [4, 5] prefer δ ≥ 0.2. In the time of preparing this article, the H1 collaboration presented [2] the first (preliminary) measurement of FL at small x We use PDF multiplied by x and neglect the nonsinglet quark distribution at small x.