On the Rise of the Proton Structure Function F_2 Towards Low x

A measurement of the derivative (d ln F_2 / d lnx)_(Q^2)= -lambda(x,Q^2) of the proton structure function F_2 is presented in the low x domain of deeply inelastic positron-proton scattering. For 5*10^(-5)<=x<=0.01 and Q^2>=1.5 GeV^2, lambda(x,Q^2) is found to be independent of x and to increase linearly with ln(Q^2).

The inclusive cross section for deeply inelastic lepton-proton scattering is governed by the proton structure function F 2 (x, Q 2 ) . Because of the large centre-of-mass energy squared, s ≃ 10 5 GeV 2 , the ep collider HERA has accessed the region of low Bjorken x, x > Q 2 /s > 10 −5 , for four-momentum transfers squared Q 2 > 1 GeV 2 . One of the first observations at HERA was of a substantial rise of F 2 with decreasing x [1]. However, this rise may be limited at very low x by unitarity constraints.
Perturbative Quantum Chromodynamics (QCD) provides a rigorous and successful theoretical description of the Q 2 dependence of F 2 (x, Q 2 ) in deeply inelastic scattering. In the double asymptotic limit, the DGLAP evolution equations [2] can be solved [3] and F 2 is expected to rise approximately as a power of x towards low x. A power behaviour is also predicted in BFKL theory [4]. The rise is expected eventually to be limited by gluon self interactions in the nucleon [5].
Recently the H1 Collaboration has presented [6] a new measurement of F 2 (x, Q 2 ) in the kinematic range 3 · 10 −5 ≤ x ≤ 0.2 and 1.5 ≤ Q 2 ≤ 150 GeV 2 based on data taken in the years 1996/97 with a positron beam energy E e = 27.6 GeV and a proton beam energy E p = 820 GeV. The high accuracy of these data allows the derivative to be measured as a function both of Q 2 and of x for the first time. Use of this quantity for investigating the behaviour of F 2 at low x was suggested in [7].
Here results are presented of a measurement of this derivative in the full kinematic range available. Data points at adjacent values of x and at fixed Q 2 are used [6] taking account of the full error correlations and the spacing between the x values. The results 1 obtained are presented in Table 1. The sensitivity of the derivative to the uncertainty of the structure function F L [6] throughout the measured kinematic range is estimated to be much smaller than the total systematic error at the lowest values of x and is negligible elsewhere.
As can be seen in Figure 1, the derivative λ(x, Q 2 ) is independent of x for x 0.01 to within the experimental accuracy. This implies that the x dependence of F 2 at low x is consistent with a power law, F 2 ∝ x −λ , for fixed Q 2 , and that the rise of F 2 , i.e. (∂F 2 /∂x) Q 2 , is proportional to F 2 /x. There is no experimental evidence that this behaviour changes in the measured kinematic range.
The derivative is well described by the NLO QCD fit to the H1 cross-section data [6], see Figure 1. In DGLAP QCD, for Q 2 > 3 GeV 2 , the low x behaviour is driven solely by the gluon field, since quark contributions to the scaling violations of F 2 are negligible. At larger x the transition to the valence-quark region causes a strong dependence of λ on x as indicated by the QCD curves in Figure 1. Figure 2 shows the measured derivative as a function of Q 2 for different x values. The derivative is observed to rise approximately logarithmically with Q 2 . It can be represented by a function λ(Q 2 ) which is independent of x within the experimental accuracy.
The function λ(Q 2 ) is determined from fits of the form F 2 (x, Q 2 ) = c(Q 2 )x −λ(Q 2 ) to the H1 structure function data, restricted to the region x ≤ 0.01. The results for c(Q 2 ) and λ(Q 2 ) are presented in Table 2 and shown in Figure 3. The coefficients c(Q 2 ) are approximately independent of Q 2 with a mean value of 0.18. As can be seen, λ(Q 2 ) rises approximately linearly with ln Q 2 . This dependence can be represented as λ(Q 2 ) = a · ln[Q 2 /Λ 2 ], see Figure 3. The coefficients are a = 0.0481 ± 0.0013(stat) ± 0.0037(syst) and Λ = 292 ± 20(stat) ± 51(syst) MeV, obtained for Q 2 ≥ 3.5 GeV 2 . The values of λ(Q 2 ) are more accurate than data hitherto published by the H1 [9] and ZEUS [10] Collaborations.
Below the deeply inelastic region, for fixed Q 2 < 1 GeV 2 , the simplest Regge phenomenology predicts that 08 is given by the Pomeron intercept independently of x and Q 2 [11]. When extrapolating the function λ(Q 2 ) into the lower Q 2 region it has the value of 0.08 at Q 2 = 0.45 GeV 2 , see also [10].
To summarise, the derivative (∂ ln F 2 /∂ ln x) Q 2 is measured as a function both of x and of Q 2 and is observed to be independent of Bjorken x for x 0.01 and Q 2 between 1.5 and 150 GeV 2 . Thus the behaviour of F 2 at low x is consistent with a dependence F 2 (x, Q 2 ) = c(Q 2 ) x −λ(Q 2 ) throughout that region. At low x, the exponent λ is observed to rise linearly with ln Q 2 and the coefficient c is independent of Q 2 to within the experimental accuracy. There is no sign that this behaviour changes within the kinematic range of deeply inelastic scattering explored. ∂ ∂ H1 Collaboration Figure 1: Measurement of the function λ(x, Q 2 ): the inner error bars represent the statistical uncertainty; the full error bars include the systematic uncertainty added in quadrature; the solid curves represent the NLO QCD fit to the H1 cross section data described in [6]; the dashed curves represent the extrapolation of the QCD fit below Q 2 = 3.5 GeV 2 . ∂ ∂ H1 Collaboration Figure 2: Measurement of the function λ(x, Q 2 ): the inner error bars represent the statistical uncertainty; the full error bars include the systematic uncertainty added in quadrature; the solid curves represent the NLO QCD fit to the H1 cross section data described in [6]; the minimum Q 2 value of the data included in this fit is Q 2 = 3.5 GeV 2 . Figure 3: Determination of the coefficients c(Q 2 ) (upper plot) and of the exponents λ(Q 2 ) (lower plot) from fits of the form F 2 (x, Q 2 ) = c(Q 2 )x −λ(Q 2 ) to the H1 structure function data [6] for x ≤ 0.01; the inner error bars illustrate the statistical uncertainties, the full error bars represent the statistical and systematic uncertainties added in quadrature. The straight lines represent the mean coefficient c (upper plot) and a fit of the form a ln[Q 2 /Λ 2 ] (lower plot), respectively, using data for Q 2 ≥ 3.5 GeV 2 .    Table 2: The coefficients c and exponents λ from fits of the form F 2 (x, Q 2 ) = c(Q 2 )x −λ(Q 2 ) using H1 F 2 data [6], for x ≤ 0.01, taking into account the systematic error correlations.
Here δ sta denotes the statistical uncertainty and δ tot comprises all uncertainties added in quadrature. The uncertainties are given as absolute values. They are symmetric to very good approximation, apart from the uncertainties of the coefficient c(Q 2 ) at the edges of the Q 2 region.