On the rise of the proton structure function F 2 towards low x

A measurement of the derivative (∂ ln F 2 /∂ ln x) Q 2 ≡ − λ(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 (cid:1) x (cid:1) 0 . 01 and Q 2 (cid:2) 1 . 5GeV 2 , λ(x,Q 2 ) is found to be independent of x and to increase linearly with ln Q 2 .

p Institute of Experimental Physics, Slovak Academy of Sciences, Košice,Slovak Republic 12,13 Abstract A measurement of the derivative (∂ ln F 2 /∂ ln x) Q 2 ≡ −λ(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 , λ(x, Q 2 ) is found to be independent of x and to increase linearly with ln Q 2 .  2001 Elsevier Science B.V.
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 20 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 Fig. 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 Fig. 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 valencequark region causes a strong dependence of λ on x as indicated by the QCD curves in Fig. 1. Fig. 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 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 20 Note that derivatives at adjacent x values are thus anticorrelated. The data points at Q 2 = 150 GeV 2 are obtained from the H1 measurement [8].
Open access under CC BY license. Table 1 Measurement of the derivative λ = −(∂ ln F 2 /∂ ln x) Q 2 at fixed Q 2 . For the systematic uncertainties the correlations between adjacent x values are taken into account. The total error is the squared sum of the statistical and systematic uncertainties, given as absolute values  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 . Fig. 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 . Table 2 The coefficients c and exponents λ from fits of the form [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. Note that part of the systematic uncertainty introduces correlations between the Q 2 bins and shown in Fig. 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 Fig. 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 F 2 (x, Q 2 ) ∝ x −λ where λ = α P (0) − 1 0.08 is given by the Pomeron intercept independently of x 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. Note that part of the systematic uncertainty is correlated from bin to bin. 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 . 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.