Ju l 2 01 9 QCD effective charges and the structure function F 2 at small-x : Higher twist effects

We consider the effect of higher twist operators of the Wilson operator product expansion in the structure function F2(x,Q ) at small-x, taking into account QCD effective charges whose infrared behavior is constrained by a dynamical mass scale. The higher twist corrections are obtained from the renormalon formalism. Our analysis is performed within the conventional framework of next-to-leading order, with the factorization and renormalization scales chosen to be Q. The infrared properties of QCD are treated in the context of the generalized double-asymptotic-scaling approximation. We show that the corrections to F2 associated with twist-four and twist-six are both necessary and sufficient for a good description of the deep infrared experimental data.


I. INTRODUCTION
In the description of the structure function F 2 (x, Q 2 ) of the proton, a long-standing question is the extent to which the nonperturbative properties of QCD affect the behavior of F 2 . The necessity of nonperturbative corrections arises as follows: at sufficiently small x the power series in α s ln(1/x) may be resummed via BFKL equation [1]. The result of resumming these leading terms is sensitive to the infrared k T region and, for running α s , it is found that [2] g(x, k 2 where λ ∼ 0.5 andg(x, k 2 T ) is here the unintegrated gluon distribution. The relation betweeng(x, k 2 T ) and g(x, Q 2 ), the standard gluon distribution to be determined by the analysis of the F 2 (x, Q 2 ) data, reads g(x, k 2 T ) = ∂(xg(x, Q 2 )) ∂ ln Q 2 At this point it is clear that nonperturbative contributions are needed. First, the resummation program requires knowledge of the gluon for all k 2 T including the deep infrared region. Unfortunately, in this confinement region the BFKL equation is not expected to be valid. Second, the HERA data on F 2 imply a steep gluon at small-x and, in addition, such a steep behavior can be generated from a flat-x gluon distribution at some initial low Q 2 0 scale, i.e. the data in the small-x region show that F 2 tend to a flat shape with decreasing Q 2 , particularly for low Q 2 . This indicates that the singular behavior x −λ predicted by BFKL must be suppressed by nonperturbative effects. Hence approaching the low Q 2 region from the QCD theory makes evident the problem of how to incorporate in an effective way nonperturbative corrections into the description of the structure function F 2 .
Fortunately, this problem was properly addressed some time ago [3] by considering the possibility that the nonperturbative dynamics of QCD generate a dynamical gluon mass at very slow Q 2 region. This effective mass is intrinsically related to a finite strong coupling constant, and its existence is strongly supported either by QCD lattice simulations [4][5][6] or by phenomenological results [7,8]. In [3] the investigation was focused exclusively on the small-x region since this condition establish the main criteria for the validity of the so called generalized double-asymptotic-scaling (GDAS) approximation [9][10][11], which in turn is particularly relevant to the analysis since it is consistent with the phenomenon of dynamical mass generation in QCD [3,12]. The importance of the GDAS approximation can be understood as follow. By analyzing exclusively the small-x region, some of the simpler existing analytical solutions of the DGLAP evolution equation [13] in the small-x limit can be directly used [14][15][16][17][18]. Within this approach the F 2 data at smallx can be interpreted in terms of the double-asymptoticscaling (DAS) phenomenon [14,15], where small-x nucleon structure functions exhibit scaling in two new variables, provided only that the small-x behavior of the parton distribution functions (PDFs) at some starting point Q 2 0 is sufficiently soft. The resulting analytical solutions can in turn be extended in order to include the subasymptotic part of the DGLAP evolution [9,10,14,19], in what is finally called GDAS approximation, leading to the prediction of flat forms at small-x for parton distributions at some input scale Q 2 0 , namely where A a are unknown parameters to be determined from the data. Note that in the above expression the parton distribution functions are multiplied each by x, that is . In summary, in Ref. [3] the structure function F 2 (x, Q 2 ) was described by means of the GDAS approximation, exploring the close interrelation between the flat behavior of initial parton distributions and the phenomenon of dynamical mass generation in QCD.
In this Letter we extend the previous study in three significant ways. First, we have included deep infrared F 2 data in our global fits. The new dataset includes measurements at Q 2 = 0.2 GeV 2 and Q 2 = 0.25 GeV 2 . The inclusion of deep infrared data is observed to have quite a large impact on the overall result. Second, we have introduced higher twist corrections to F 2 for the case, studied in the previous work, of a flat initial condition for the leading twist QCD evolution. These higher twist corrections are estimated from the infrared renormalon model [20,21]. Third, we have investigated, in addition to the logarithmic and power-law cases, a QCD effective charge built from a particular case of the Curci-Ferrari Lagrangians.
The Letter is organized as follows: in the next section we introduce higher twist contributions to F 2 from the renormalon formalism and present the formalism for analyzing the structure function of the proton in the GDAS approximation. In the Sec. III we discuss the definition of the QCD effective charge and present three versions of holomorphic charges. Our results are presented in the Sec. IV, where we compare our results on the structure function F 2 with experimental data. Our analysis is carried on using the formalism developed in the sections II and III. In Sec. V we draw our conclusions.

II. HIGHER TWIST OPERATORS
Higher twist corrections to deep inelastic scattering (DIS) processes have been studied systematically in the framework of the operator product expansion (OPE) [22]. It has to be admitted at once that the details of this corrections are not yet fully understood, owing to the theoretical difficulties of controlling power corrections in effective theories [23]: the calculation of power corrections requires the evaluation of the matrix elements of higher-twist operators, but in order to cancel renormalon ambiguities it is also necessary to compute the Wilson coefficient functions to sufficiently high orders of the perturbation series. These renormalon ambiguities are of the same order as the power corrections.
In the case of twist-four operators, for example, radiative corrections generate terms of the form of a square of the (dimensionful) ultraviolet cut-off multiplied by the lower-order (dimensionless) matrix element of the twisttwo. These mixing terms make the definition of twist-four contributions ambiguous. However, in the operator product expansion of DIS structure functions, this quadratic ambiguity always cancels against the corresponding ambiguity in the definition of the twist-two contribution. As a result, if the twist-two and twist-four contributions are calculated within the same regularization scheme, the sum of both contributions is unambiguous up to order 1/Q 2 [20,21].
Unfortunately, given that in general only a few terms of the perturbative series are known, and furthermore these series are plagued by similar renormalon ambiguities, it is not clear if the ambiguity of higher twist contributions can also be canceled [23]. However, the subtle relation between the twist-two and the twist-four contributions has inspired the hypothesis that the main contributions to the matrix elements of the twist-four operators are proportional to their (quadratically) divergent parts [20]. This means that in practice we can obtain information about power corrections from the large-order behavior of the corresponding series. This approach is called infrared renormalon model. We shall from now on present the higher-twist contributions to the DIS structure function F 2 (x, Q 2 ) in the framework of the renormalon formalism, as calculated by Illarionov,Kotikov and Parente in [10].
We start by writing down the twist-two term of F 2 (x, Q 2 ) at NLO [9,10,19]: where n f is the effective number of quarks, e = f i e 2 i /n f is the average charge squared, T R = 1/2 is the color factor for g → qq splitting, and with a = q, g. Here the "+" and "−" representation follows from the solution, at twist-two approximation, of the DGLAP equation in the Mellin moment space [9]: where and ρ = d + s +D + p / ln x = σ 2 ln(1/x) ; (13) in the above expressions β 0 (β 1 ) is the first (second) coefficient of the β function of the QCD. The components of the anomalous dimension d − (n) and of the regular (d) and singular (d) parts of d + (n) =d + /(n − 1) +d + (n), in the limit n → 1, are given bŷ and The functionsĨ ν (ν = 0, 1) are functions related to the Bessel function J ν and the modified Bessel function I ν byĨ and the factors A +,− a , as well as the components of the regular and singular parts of the anomalous dimensions D ± are given by and where ζ is the Riemann zeta function, and C A = N and Higher twist estimations (twist-four and twist-six) are known, in the framework of the infrared renormalon formalism, for the nonsiglet case [24] as well to the singlet one [21]. In this approach the higher twist corrections can be expressed in terms of the leading twist expansion of F 2 (x, Q 2 ). As previously indicated, we adopt the formalism and notation put forwarded in [10]. In this case the twist-four (τ 4) correction to F 2 (x, Q 2 ) in the [R]enormalon formalism is given by where f τ 2 a (x, Q 2 ) are the gluon and quark-singlet PDFs in twist-two (τ 2) approximation, whereasμ τ 4 a (x, Q 2 ) are just the functions obtained in [21] by means of the infrared renormalon model. Therefore, once we have found the twist-four correction, we can write the structure function as In Eq. (29) the symbol ⊗ stands for the Mellin convolution where the Mellin transform ofμ τ 4 a (x, Q 2 ) is given by As in the case of the twist-two expansion, F [R]τ 4 2 (x, Q 2 ) can also be split into two parts, namely the "+" and the "-" representations: Thus, applying the technique of transforming the Mellin convolutions (taking the limit n → 1) to simple products at small x [25] (these products replace the convolution of two functions at small x), the "+" and the "-" parts of F [R]τ 4 2 (x, Q 2 ) can be written as [10]: The function Ψ is the derivative of the logarithm of the Γ function. From quark counting rules we have ν q ≈ 3 and ν g ≈ 4, which results in p(ν q ) ≈ 11/6, p(ν g ) ≈ 25/12, p ′ (ν q ) ≈ −49/36, and p ′ (ν g ) ≈ −205/144.
In the framework of the infrared renormalon model the twist-six (τ 6) correction to F 2 (x, Q 2 ) is given by [10]: whereμ τ 6 a (x, Q 2 ) are functions also obtained in [21] by means of the infrared renormalon model. Now, taking into account all higher twist corrections, the structure function F 2 is given by Most importantly, as pointed out in Ref. [10], in the renormalon model the twist-six contribution can be obtained in terms of the twist-four one: Thus, if F [R]hτ 2 (x, Q 2 ) denotes the higher-twist operators (twist-4 and twist-6), we have (39) where the "+" and the "-" representations of F [R]hτ 2 (x, Q 2 ) can each be put into a compact form: with k 4 = 1, k 6 = −8/7, l 4 = 1, and l 6 = 1/2. As applied to the case of QCD, from now on we set N = 3 in order to fix C A (= 3) and C F (= 4/3). With all these definitions we can discuss in the next section the QCD effective charges that we shall use in our calculations.

III. THE QCD EFFECTIVE CHARGE
It is a commonly accepted view nowadays that the nonperturbative dynamics of QCD may generate an effective momentum-dependent mass m(q 2 ) for the gluons while preserving the SU (3) c local invariance [26]. Numerical simulations, in which the space-time continuum is represented as a discrete lattice of points, indicate that such a dynamical mass does arise when the nonperturbative regime of QCD is probed. Specifically, large-volume lattice QCD simulations, both for SU(2) [4] and SU(3) [5], reveal that the gluon propagator is finite in the deep infrared region, both in Landau gauge and away from it [6]. In the continuum, it turns out that the nonperturbative dynamics of the gluon propagator is governed by the corresponding Schwinger-Dyson equations. These equations constitute an infinite set of coupled nonlinear integral equations governing the dynamics of all QCD Green's functions. According to the Schwinger-Dyson equations, a finite gluon propagator corresponds to a massive gluon [27].
Furthermore, we have known for a long time that the vacuum energy in dynamically broken gauge theories is related to a dynamical mass [28]. Recent studies show that the condition for the existence of a global minimum of the vacuum energy for a non-Abelian gauge theory, in the presence of a dynamical mass scale, implies the existence of a fixed point of the β function [26]. This, in turn, indicates that the QCD exhibits nearly conformal behavior at infrared momenta [29].
The QCD effective chargeᾱ(q 2 ) is a nonperturbative generalization of the canonical perturbative running coupling α s (q 2 ) and is intimately related to the phenomenon of dynamical gluon mass generation [30][31][32]. The chargē α(q 2 ) provides the bridge leading from the deep ultraviolet regime to the deep infrared one. It is important to note, however, that the definition ofᾱ(q 2 ) is not unique. The effective charge may be obtained, for example, from the ghost-gluon vertex in the Landau gauge [33], by considering a renormalization-group invariant (independent of the renormalization scale µ) quantityr(q 2 ) defined bŷ where g(µ 2 ) is the gauge coupling, ∆(q 2 ) is the gluon propagator and F 2 (q 2 ) is the ghost dressing function. From this quantity the effective charge may then be defined asᾱ where m(q 2 ) is the gluon dynamical mass. The relation (43) is the definition of theᾱ(q 2 ) which is most commonly used in lattice simulations, since ∆(q 2 ) and F 2 (q 2 ) are quantities measured directly on it. Alternatively, the QCD effective charge can be obtained within the framework of pinch technique [30,34,35]. This process-independent definition, obtained from the gluon self-energy∆(q 2 ) in the background-field method [36], is a direct non-Abelian generalization of the QED effective charge. In this definition the Schwinger-Dyson solutions for∆(q 2 ) are used to form another renormalization-group invariant quantity: The inverse of the above quantity may be written where now 1 Note that here b 0 is precisely the first coefficient of the QCD β function and Λ is the QCD mass scale (of a few hundred MeV). Despite the distinct theoretical origins of α gh (q 2 ) andᾱ pt (q 2 ), they coincide exactly in the deep infrared. The ultimate reason for this is the existence of a non-perturbative identity relating various of the Green functions appearing in their respective definitions [32]. It is worth remarking on the fact that the form of (46) is exactly the same as the form of the leading order (LO) perturbative QCD coupling, namely if q 2 + m 2 (q 2 ) → p 2 in the argument of the logarithm; it is this that will effectively ensure that, in practice, the QCD effective charge can be successfully obtained by saturating the LO perturbative strong coupling α LO s (q 2 ). That is to say, where b 0 = β 0 /4π = (11C A −2n f )/12π. If the Schwinger-Dyson equations preserves the multiplicative renormalizability, a next-to-leading order (NLO) effective charge can be built through the same procedure [3]: We investigate three different types of QCD effective chargeᾱ N LO (q 2 ). They can be constructed from two independent dynamical gluon masses having distinct asymptotic behaviors: the first runs as an inverse power of a logarithm, the second drops as an inverse power of momentum [37]. A logarithmic running of m 2 (q 2 ) had been initially found in studies of linearized Schwinger-Dyson equations, namely m 2 (q 2 ) ∼ ln q 2 −1−γ , with γ > 1 [30,31]. Further studies of a non-linear version of the Schwinger-Dyson equation for the gluon self-energy have shown that m 2 (q 2 ) could be rewritten as [37] where γ 1 = −6(1 + c 2 − c 1 )/5. The parameters c 1 and c 2 are related to the ansatz for the three-gluon vertex used in the numerical analysis of the gluon self-energy. The values of c 1 and c 2 are restricted by a "mass condition" that controls the behavior of the dynamical mass in the ultraviolet region, namely c 1 ∈ [0.15, 0.4] and c 2 ∈ [−1.07, −0.92]. The parameters m g and ρ, which control the behavior of m 2 log (q 2 ) in the infrared region, are also constrained by the mass condition. In particular, they are constrained to lie in the intervals m g ∈ [300, 800] and ρ ∈ [1.0, 8.0] MeV [37]. A power-law running m 2 (q 2 ) exhibit a distinct behavior whereby, in accordance with OPE calculations [38], the most probable asymptotic behavior of the running gluon mass is proportional to 1/q 2 . At the level of an non-linear Schwinger-Dyson equation the asymptotic behavior is written as where γ 2 = (4 + 6c 1 )/5. Here the same mass condition imposes c 1 ∈ [0.7, 1.3] whereas the ρ and m g parameters are constrained to lie in the same interval as before, namely ρ ∈ [1.0, 8.0] and m g ∈ [300, 800] MeV [37]. At this point we may start to define the first two QCD effective charges to be explored in this paper. The first charge is constructed simply by combining the equations (49) and (50), henceforth called "logarithmic charge" and denoted byᾱ log (q 2 ): The second one, hereafter called "power-law charge" and denoted byᾱ pl (q 2 ), is constructed by combining the equations (49) and (51): The third effective charge can be postulated by considering renormalization aspects of Lagrangians containing an explicit mass term for the vector field. As demonstrated by Curci and Ferrari some time ago, the action obtained from these massive Yang-Mills Lagrangians, while also containing a four-vertex for the Faddeev-Popov ghost field, is invariant under a generalized nonlinear gauge transformation [39]. The invariance under this transformation, which is an extended version of the BRST one, guarantees the validity of the Slavnov-Taylor identities. More recently, the two-point correlation functions of gluons and ghosts for the pure Yang-Mills theory in Landau gauge were accurately reproduced for all momenta by using a particular case of the Curci-Ferrari Lagrangians [40,41], i.e. the one-loop calculation successfully reproduces lattice simulations in d = 4. More important, the introduction of a bare gluon mass in the gauge-fixed Lagrangian allows to capture the main effects of Gribov copies in a reliable way [42,43]. Furthermore, two-and three-point correlation functions have been computed within this massive model and also successfully compared with lattice simulations [44]. The introduction of two-loop corrections improve the comparison of ghost and gluon two-point correlation functions with lattice data in the quenched approximation [45]. The Curci-Ferrari Lagrangian is infrared safe, meaning that there exists renormalization schemes without Landau pole. In particular, the one-loop calculation implies that the β function for the coupling constant g in the infrared region behaves like where c 0 = C A /24π. This means that the strong coupling vanishes logarithmically in the infrared, in agreement with some recent lattice results using a renormalization group invariant coupling resulting from a particu-lar combination of the gluon and ghost propagators [46][47][48]. Thus, our Curci-Ferrari effective charge can be constructed as follows: It may be worth emphasizing that the QCD effective chargesᾱ log (q 2 ),ᾱ pl (q 2 ) andᾱ CF (q 2 ) exhibit infrared fixed points as q 2 → 0, i.e. the dynamical gluon mass tames the Landau pole. Moreover, in the limit q 2 ≫ Λ 2 these effective charges match with the canonical perturbative two-loop coupling:ᾱ(q 2 ≫ Λ 2 ) → α s (q 2 ). The analyticity ofᾱ log (q 2 ),ᾱ pl (q 2 ) andᾱ CF (q 2 ) is automatically preserved if the gluon mass scale is set larger than half of the QCD scale parameter, namely m g /Λ > 1/2 [8]. This ratio may be phenomenologically determined [3,8,12] and typically lies in the interval m g /Λ ∈ [1.1, 2]. The canonical coupling α s (q 2 ), on the other hand, has Landau singularities on the spacelike semiaxis 0 ≤ q 2 ≤ Λ 2 . That is, α s has a nonholomorphic behavior at low q 2 [49]. This problem has been worked out by using analytic versions of QCD whose coupling α s (q 2 ) is holomorphic in the entire complex plane except the timelike axis (q 2 < 0) [50], with many applications in hadronic physics [51,52]. In a mathematical sense, the QCD effective charges belong to the same class of holomorphic couplings. Moreover, as pointed out by Cvetič [51], evaluation of renormalization scale-invariant quantities at low Q 2 , in terms of infrared finite couplings, can be effectively done as a series in derivatives of the coupling with respect to the logarithm of Q 2 . This truncated series exhibit significantly better convergence properties.

IV. RESULTS
The nucleon structure function F 2 (x, Q 2 ) has been measured in deep inelastic scattering (DIS) of leptons off nucleons at the HERA collider. In this work we carry out global fits to small-x F 2 (x, Q 2 ) data at low and moderate Q 2 values [53], where x is the Bjorken variable and Q 2 is the virtuality of the photon. We use HERA data from the ZEUS and H1 Collaborations, with the statistic and systematic errors added in quadrature. Specifically, we fit to the structure function at Q 2 = 0. 2,0.25,0.3,0.5,0.65,0.85,1.2,1.3,1.5,1.9,2.0,2.5,3.5,5.0,6.5 and 10 GeV 2 . The global fits were performed using a χ 2 fitting procedure, where the value of χ 2 min is distributed as a χ 2 distribution with ν degrees of freedom. We have adopted an interval χ 2 − χ 2 min corresponding to the projection of the χ 2 hypersurface enclosing 90% of probability. As test of goodness-of-fit we adopt the chi-square per degree of freedom, namelyχ ≡ χ 2 /ν.
A suitable combination of the asymptotic freedom and the factorization properties of QCD results in a systematic expansion of the DIS cross sections in terms of α s (Q 2 ), evaluated at the virtuality scale. Thus the virtuality Q is the natural scale in our calculations. This means that all the couplings α s (Q 2 ) appearing in the Section II are replaced by QCD effective charges evaluated at the virtuality scale: α s (Q 2 ) →ᾱ log (Q 2 ),ᾱ pl (Q 2 ) or α CF (Q 2 ) (or, equivalently, that q 2 is replaced by Q 2 in Eqs. (50)-(55)). In all the fits we fix n f = 4 and Λ = 284 MeV since these choices are not only consistent to NLO procedures, but are also the same ones adopted in Refs. [3] and [11]. Concerning the dynamical masses (50) and (51), in both cases we set ρ = 4; it is the optimal value originally obtained by Cornwall in order to reproduce numerical results of a Schwinger-Dyson equation for the gluon propagator [30]. Moreover, this ρ value is consistent with the expected threshold Q 2 = 4m 2 g for gluons to pop up from the vacuum [54]. As in the previous study [3], we set γ 1 = 0.084 and γ 2 = 2.36 since again the best determinations of F 2 come from the analyses using these values. We have observed this general pattern in all analysis performed in this work, meaning that the magnitudes of γ 1 and γ 2 are not sensitive to twist corrections and changes in the form of the QCD effective charges.
We start our investigation by considering the twist-two (leading-twist) expansion of F 2 (x, Q 2 ). We determine the values of the parameters m g , Q 2 0 , A g and A q using the effective chargesᾱ log (Q 2 ),ᾱ pl (Q 2 ) andᾱ CF (Q 2 ). The values of the fitted parameters are given in Tables I, II and  for the x dependence of F 2 (x, Q 2 ) for specific Q 2 bins, especially in the deep infrared region (data in the region 0.2 GeV 2 Q 2 0.5 GeV 2 ). Some improvement in theχ values is achieved by incorporating the twist-four (τ 4) correction to F 2 . In this case theχ values obtained using logarithmic, power-law and Curci-Ferrari charges are 2.03, 1.51 and 2.01, respectively. In the combination of the leading and the twistfour correction we have 6 free parameters: m g , Q 2 0 , A g , A q , A τ 4 g and A τ 4 q . Their values are given in Tables I, II and III, in the columns headed "τ 2 + τ 4". However, as shown by the dashed curves in Figures 1, 2 and 3, the fits are not so good, especially in the deep infrared region. Thus the improvement in the description of F 2 is not enough, mostly because the flat trend of the infrared data.
Clearly, we can improve on the description of the structure function in the deep infrared region by adding the twist-six (τ 6) correction to the previous case. Now, theχ values obtained using logarithmic, power-law and Curci-Ferrari charges are 1.22, 1.19 and 1.21, respectively. We have in this analysis 8 fitting parameters: m g , Q 2 0 , A g , A q , A τ 4 g , A τ 4 q , A τ 6 g and A τ 6 q . Their values are given in Tables I, II and III, in the columns headed "τ 2 + τ 4 + τ 6". The low values obtained forχ indicate good agreement between the theory and the experimental data, as indeed shown by the solid curves in Figures 1, 2 and 3. In particular, for the mass scales obtained by usingᾱ log (Q 2 ),ᾱ pl (Q 2 ) andᾱ CF (Q 2 ) effective charges, we have m g = 308 ± 43 MeV, m g = 413 ± 68 MeV and m g = 299 ± 42 MeV, respectively. These mass values preserve the analyticity constraint on the strong running coupling, since for the three values we have m g /Λ > 1/2.
In Figure 4 we compare the higher-twist results in the deep infrared region.

V. CONCLUSIONS
In this letter we have investigated the structure function F 2 (x, Q 2 ) of the proton by means of the generalized DAS approximation. We have studied the effect of higher twist operators of the Wilson operator product expansion in F 2 (x, Q 2 ) at small-x. The higher twist corrections have been obtained from the infrared renormalon model in which the higher twist operators can be expressed in terms of the leading-twist F 2 expansion.
In order to describe the infrared F 2 data we have introduced three different QCD effective charges at NLO. The charges have been constructed by using a phenomenological procedure that mimics the process-independent method of obtaining effective charges within the framework of pinch technique. One of the QCD charges, denotedᾱ CF (Q 2 ), is postulated by considering renormalization aspects of massive Yang-Mills Lagrangians and it goes to zero in the limit Q 2 → 0. The other two, namelyᾱ log (q 2 ) andᾱ pl (q 2 ), have distinct infrared behaviors but similar freezing processes when the Q 2 scale goes to 0. The three charges exhibit infrared fixed points as Q 2 → 0, i.e. the dynamical gluon mass m(Q 2 ) tames the Landau pole. Moreover, these charges are examples of holomorphic running couplings, since they preserve the analyticity constraint in any arbitrary scale Q.
We have performed global fits to F 2 data using either its leading twist expansion or its higher-twist version. We have observed that the best fits are obtained when we introduce the higher-twist corrections to F 2 . In fact there is a large improvement in the description of the structure function in the deep infrared region only when the τ 4 and τ 6 corrections are introduced at the same time. In other words, the fitting parameters are very strongly constrained by the deep infrared data sets. As indicated in the previous section, when all higher-twist corrections are taken into account, the gluon masses are found to be equal to m g = 308 MeV, m g = 413 MeV and m g = 299 MeV for the case of logarithmic, power-law and Curci-Ferrari effective charges, respectively. It is noteworthy that these m g values are of the same order of magnitude as the gluon masses obtained in other calculations of strongly interacting processes [3,7,8]. We argue that these results corroborate theoretical analyzes considering the nonperturbative phenomenon of dynamical mass generation in QCD.
FIG. 1. Fits of the x dependence of F2(x, Q 2 ) for specific Q 2 with the logarithmic effective charge. The dotted curves correspond to the twist-two (leading) approximation. The dashed (solid) curves were obtained considering the twist-four (twist-six) correction to F2(x, Q 2 ).