Small-x behavior of the structure function F_2 and its slope partial ln(F_2)/partial ln(1/x) for"frozen"and analytic strong-coupling constants

Using the leading-twist approximation of the Wilson operator product expansion with"frozen"and analytic versions of the strong-coupling constant, we show that the Bessel-inspired behavior of the structure function F_2 and its slope\break partial ln(F_2)/partial ln(1/x) at small values of x, obtained for a flat initial condition in the DGLAP evolution equations, leads to good agreement with experimental data of deep-inelastic scattering at DESY HERA.


Introduction
The experimental data from DESY HERA on the structure function F 2 of deep-inelastic scattering (DIS) [1,2,3,4,5,6,7,8,9,10,11,12,13,14] and its derivatives ∂F 2 /∂ ln Q 2 [4,6,15] and ∂ ln F 2 /∂ ln(1/x) [15,16,17,18] bring us into a very interesting kinematic range for testing theoretical ideas on the behavior of quarks and gluons carrying a very small fraction of the proton's momentum, the so-called small-x region. In this limit, one expects that the conventional treatment based on the Dokshitzer-Gribov-Lipatov-Altarelli-Parisi (DGLAP) equations [19,20,21,22] does not account for contributions to the cross section which are leading in α s ln(1/x); moreover, the parton density functions (PDFs), in particular the one of the gluon, become large, and the need arises to apply a high-density formulation of QCD.
However, reasonable agreement between HERA data and the next-to-leading-order (NLO) approximation of perturbative QCD has been observed for Q 2 ∼ > 2 GeV 2 (see reviews in Ref. [23,24] and references cited therein) indicating that perturbative QCD can describe the evolution of F 2 and its derivatives down to very small Q 2 values, traditionally characterized by soft processes.
The standard program to study the x dependence of quark and gluon PDFs is to compare the numerical solutions of the DGLAP equations with the data and so to fit the parameters of the x profiles of the PDFs at some initial factorization scale Q 2 0 and the asymptotic scale parameter Λ. However, for analyzing exclusively the small-x region, there is the alternative of doing a simpler analysis by using some of the existing analytical solutions of the DGLAP equations in the small-x limit [25,26,27,28]. This was done in Ref. [25], where it was pointed out that the small-x data from HERA can be interpreted in terms of the so-called double-asymptotic-scaling (DAS) phenomenon related to the asymptotic behavior of the DGLAP evolution discovered in Ref. [29] many years ago.
The study of Ref. [25] was extended in Refs. [26,27,28] to include the subasymptotic part of the Q 2 evolution. This led to predictions [27,28] of the small-x asymptotic PDF forms in the framework of DGLAP dynamics starting at some initial value Q 2 0 with flat x distributions: where f a (x, Q 2 ) are the PDFs and A a are unknown constants to be determined from the data. We refer to the approach of Refs. [26,27,28] as generalized DAS approximation. In this approach, the flat initial conditions in Eq. (1) play the basic role of the singular parts of the anomalous dimensions by determining the small-x asymptotics, as in the standard DAS case, while the contributions from the finite parts of the anomalous dimensions and from the Wilson coefficients can be considered as subasymptotic corrections, which are, however, important for better agreement with the experimental data. In the present paper, similarly to Refs. [25,26,27,28], we neglect the contribution from the non-singlet quark component.
The rise of λ(Q 2 ) linearly with ln Q 2 can be traced to strong nonperturbative physics (see Ref. [34] and references cited therein), i.e. λ(Q 2 ) ∼ 1/α s (Q 2 ). However, the analysis of Ref. [35] demonstrated that this rise can be explained naturally in the framework of perturbative QCD (see also Section 3).
The H1 and ZEUS Collaborations [15,17,18] also presented new data for λ(Q 2 ) at quite small values of Q 2 . As may be seen from Fig. 8 of Ref. [15], the ZEUS value for λ(Q 2 ) is consistent with a constant of about 0.1 at Q 2 ∼ < 0.6 GeV 2 , as is expected under the assumption of single-soft-Pomeron exchange within the framework of Regge phenomenology.
It is interesting to extend the analysis of Ref. [35] to the small-Q 2 range with the help of the well-known infrared modifications of the strong-coupling constant. We shall adopt the "frozen" [36] and analytic [37] versions.
This paper is organized as follows. Section 2 contains basic formulae for the structure function F 2 and its slope ∂ ln F 2 /∂ ln(1/x) in the generalized DAS approximation [27,28,35], which are needed for the present study. In Section 3, we compare our results on F 2 and ∂ ln F 2 /∂ ln(1/x) with experimental data. Our conclusions may be found in Section 4.

Generalized DAS approach
The flat initial conditions in Eq. (1) correspond to the case when the PDFs tend to constants as x → 0 at some initial value Q 2 0 . The main ingredients of the results at the leading order (LO) [27,28] include the following. 1 Both, the gluon and quark-singlet PDFs are presented in terms of two components ("+" and "−"), which are obtained from the analytic Q 2 -dependent expressions of the corresponding ("+" and "−") PDF moments. Here, e = ( f i=1 e 2 i )/f is the average charge square and f is the number of active quark flavors. The small-x asymptotic results for the PDFs f ± a are where d + (1) = 1 + 20f /(27β 0 ) and d − (1) = 16f /(27β 0 ) are the regular parts of the anomalous dimensions d + (n) and d − (n), respectively, in the limit n → 1. 2 Here, n is the variable in Mellin space. The functionsĨ ν (ν = 0, 1) are related to the modified Bessel function I ν and the Bessel function J ν bỹ The variables s, σ, and ρ are given by is the strong-coupling constant in the LO approximation, and β 0 is the first term of its β function.
Contrary to the approach of Refs. [25,26,27,28], various groups were able to fit the available data using a hard input at small values of x, of the form x −λ , with different values λ > 0 at small and large values of Q 2 [33,38,39,40,41,42,43,44,45,46,47,48]. At small Q 2 values, there are well-known such results [33]. At large Q 2 values, this is not very surprising for the modern HERA data because they cannot distinguish between the behavior based on a steep PDF input at quite large Q 2 values and the steep form acquired after the dynamical evolution from a flat initial condition at quite small Q 2 values.
As has been shown in Refs. [27,28], the x dependencies of F 2 and the PDFs given by the Bessel-like forms in the generalized DAS approach can mimic power-law shapes over a limited region of x and Q 2 values: In the twist-two LO approximation, the effective slopes have the following forms: The corresponding NLO expressions and the higher-twist terms may be found in Refs. [27,28].
The effective slopes λ eff F 2 and λ eff a depend on the magnitudes A a of the initial PDFs and also on the chosen input values of Q 2 0 and Λ. To compare with the experimental data, it is necessary to use the exact expressions from Eq. (8), but for a qualitative analysis one can use some appropriate approximations. At large values of Q 2 , the "−" components of the PDFs are negligible, and the dependencies of the slopes on the PDFs disappear. In this case, the asymptotic behaviors of the slopes are given by the following expressions: 3 where the symbol ≈ marks the approximation obtained from the expansion of the usual and modified Bessel functions in Eq. (5). One can see from Eq. (9) that the gluon effective slope λ eff,as g is larger than the quark one λ eff,as q , which is in excellent agreement with global analyses [23,24].

Comparison with experimental data
Using the results of the previous sections, we analyze HERA data for the structure function F 2 and its slope ∂ ln F 2 /∂ ln(1/x) at small x values from the H1 and ZEUS Collaborations [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18] The experimental results for the x dependence of F 2 in bins of Q 2 are shown in Fig. 1, while the Q 2 dependence of λ eff  [14]. We fit the combined H1 and ZEUS data on F 2 [1,2,3,4,5,6,7,8,9,10,11,12,13,14] at LO and NLO imposing two different cuts on Q 2 , namely Q 2 > 1.5 GeV 2 and Q 2 > 0.5 GeV 2 . The resulting values for A g , A q , and Q 2 0 are collected in Table 1 together with the values of χ 2 per data point (χ 2 /n.d.f.) achieved. In Fig. 1, the H1 and ZEUS data on F 2 , which come as x distributions in bins of Q 2 , are compared with the NLO result obtained with the cut Q 2 > 0.5 GeV 2 . Furthermore, the Q 2 dependence of λ eff F 2 (x, Q 2 ) as determined by H1 and ZEUS at an average small-x value of 10 −3 is confronted with the result of the NLO fit for Q 2 > 0.5 GeV 2 in Fig. 2. 3 The asymptotic formulae given in Eq.  Results of the LO and NLO fits to H1 and ZEUS data [1,2,3,4,5,6,7,8,9,10,11,12,13,14] for different small-Q 2 cuts. Because the twist-two approximation is only reasonable at Q 2 ∼ > 2.5 GeV 2 [28], as may be seen from Fig. 1, some theoretical improvements are necessary for smaller Q 2 values. In Ref. [28], the higher-twist corrections through twist six were added to find good agreement for Q 2 ∼ > 0.5 GeV 2 . However, the twist-four and twist-six terms increase the number of parameters, which become strongly correlated.
Here, we investigate an alternative possibility, namely to modify the strong-coupling constant in the infrared region. Specifically, we consider two modifications, which effectively increase the argument of the strong-coupling constant at small Q 2 values, in accordance with Refs. [49,50,51,52,53,54]. In the first case, which is more phenomenological, we introduce a freezing of the strong-coupling constant by changing its argument as Q 2 → Q 2 +M 2 ρ , where M ρ is the rho-meson mass [36]. Thus, in the formulae of Section 2 and their NLO generalizations [27,28], we introduce the following replacement where α LO s (Q 2 ) and α MS s (Q 2 ) have the canonical forms dictated by the renormalization group.
The second possibility is based on the idea by Shirkov and Solovtsov [37,55] (see also the recent reviews in Refs. [56,57,58] and the references cited therein) regarding the analyticity of the strong-coupling constant that leads to an additional power dependence. In this case, the one-loop and two-loop coupling constants α LO s (Q 2 ) and α MS s (Q 2 ) appearing in the formulae of the previous sections and their NLO generalizations are to be replaced as where the ellipsis stands for cut terms which give negligible contributions.
We thus repeat the LO and NLO fits discussed above using in turn the frozen and analytic versions of the strong-coupling constant according to the replacements of Eqs. (10) and (11), respectively. The results for F 2 are included in Table 1 and Fig. 1 and those for λ eff F 2 (x, Q 2 ) in Fig. 2. Figure 1, as also Fig. 5 in Ref. [28], only covers the small-Q 2 region, Q 2 < 9.22 GeV 2 , because the three NLO predictions are hardly distinguishable at larger values of Q 2 .
From Fig. 1, we observe that the fits based on the frozen and analytic strong-coupling constants are very similar and describe the data in the small-Q 2 range significantly better than the canonical fit. This is also reflected in the values of χ 2 /n.d.f. listed in Table 1.
The improvement is especially striking at NLO if data with very small Q 2 values, with Q 2 ≥ 0.5 GeV 2 , are included in the fits. Then χ 2 /n.d.f. is almost reduced by a factor of two to assume values close to unity when the canonical version of the strong-coupling constant is replaced by the frozen or analytic ones. The situation is very similar to the case when the higher-twist corrections according to the renormalon model are included [28]. In order to illustrate this, we display in Fig. 1 also the results obtained at NLO in the renormalon model of higher-twist terms, which are taken from Fig. 5 in Ref. [28]. We see that the latter describe the experimental data slighly better for 0.65 GeV 2 ∼ < Q 2 ∼ < 2.0 GeV 2 than the results obtained here, which is also reflected in the values of χ 2 /n.d.f. achieved, namely χ 2 /n.d.f. = 565/658 = 0.86 for renormalon improvement versus χ 2 /n.d.f. = 633/662 = 0.96 and 670/662 = 1.01 for analytic and frozen strong-coupling constants, respectively. However, one should bear in mind that this improvement happens at the expense of introducing four additional adjustable parameters. Figure 2 nicely demonstrates that the theoretical description of the small-Q 2 ZEUS data on λ eff F 2 (x, Q 2 ) by NLO QCD is significantly improved by implementing the frozen and analytic strong-coupling constants. Again, these two alternatives lead to very similar results. For comparison, the linear rise of λ eff F 2 (x, Q 2 ) with ln Q 2 as described by Eq. (2) is also indicated in Fig. 2. For comparison, we display in Fig. 2 also the results obtained by Kaidalov et al. [47] and by Donnachie and Landshoff [48] adopting phenomenological models based on Regge theory. While they yield an improved description of the experimental data for Q 2 ∼ < 0.4 GeV 2 , the agreement generally worsens in the range 2 GeV 2 ∼ < Q 2 ∼ < 8 GeV 2 .
As may be seen from Table 1, the three NLO fits for F 2 (x, Q 2 ) yield Q 2 0 ≈ 0.5-0.7 GeV 2 (see also Ref. [28]). Figure 2 shows that the conventional NLO fit yields λ eff F 2 (x, Q 2 0 ) = 0 as suggested by Eq. (1). The replacements of Eqs. (10) and (11) raise the value of λ eff F 2 (x, Q 2 0 ). In fact, the results for λ eff F 2 (x, Q 2 ) obtained with the frozen and analytic versions of the strong-coupling constant agree much better with the ZEUS data at Q 2 ∼ > 0.5 GeV 2 . Nevertheless, for Q 2 < 0.5 GeV 2 , there is still some disagreement with the data, which needs additional investigation.
In Fig. 2, the NLO results for λ eff F 2 (x, Q 2 ) are evaluated at x = 10 −3 . In Fig. 3, we study the variation with x in the range 10 −5 < x < 10 −2 . For simplicity, we only do this for the case of the frozen strong-coupling constant; the result for the analytic one would be very similar. We observe good agreement between the experimental data and the generalized DAS approach for a broad range of small-x values. At small Q 2 values, λ eff F 2 (x, Q 2 ) is practically independent of x, which is because the variable ρ defined in Eq. (6) takes rather small values there. At large Q 2 values, the x dependence of λ eff F 2 (x, Q 2 ) is rather strong. However, it is well known that the boundaries and mean values of the experimental x ranges [16] increase proportionally with Q 2 , which is related to the kinematical restrictions in the HERA experiments, namely x ∼ 10 −4 × Q 2 (see Refs. [4,6,13,14] and, for example, Fig. 1 of Ref. [15]). From Fig. 3, we see that the HERA data are close to λ eff F 2 (x, Q 2 ) at x ∼ 10 −4 -10 −5 for Q 2 = 4 GeV 2 and at x ∼ 10 −2 for Q 2 = 100 GeV 2 . Indeed, the correlations between x and Q 2 of the form x eff = a × 10 −4 × Q 2 with a = 0.1 and 1 lead to a modification of the Q 2 evolution which starts to resemble ln Q 2 , rather than ln ln Q 2 as is standard [35].