The proton structure function in the diffraction region of small Bjorken-$x$
and $10 {\rm GeV}^2 \le Q^2 \le 100 {\rm GeV}^2$ behaves as $F_2 (x, Q^2) = F_2
(W^2) = f_0 \cdot (W^2)^{C_2}$, where $x = Q^2 / W^2$. The exponent $C_2$ of
the $\gamma^* p$ center-of-mass energy squared, $W^2$, is predicted from
evolution of the flavor-singlet quark distribution, $C_2 = 0.29$, and the only
free parameter, the normalization $f_0 = 0.063$, is fitted. The evolution of
the gluon density multiplied by $\alpha_s (Q^2)$ is dentical to the evolution
of the flavor-singlet quark density. This simple picture is at variance with
the standard approach to evolution based on the coupled equations of
flavor-singlet and gluon density.Comment: 11 pages, 4 figure

Schildknecht, Dieter

English

arXiv

Schildknecht D. Perturbative QCD evolution and color dipole picture. arxiv:1104.0850. 2011

Publications at Bielefeld University

arXiv:1104.0850v1  [hep-ph]  5 Apr 2011Perturbative QCD Evolution and Color Dipole PictureDieter SchildknechtFakulta¨t fu¨r Physik, Universita¨t BielefeldD-33501 Bielefeld, GermanyandMax-Planck Institute fu¨r Physik (Werner-Heisenberg-Institut),Fo¨hringer Ring 6, D-80805, Mu¨nchen, GermanyAbstractThe proton structure function in the diffraction region of small Bjorken-x and 10GeV2 ≤ Q2 ≤ 100GeV2 behaves as F2(x,Q2) = F2(W2) = f0 ·(W 2)C2 , where x = Q2/W 2. The exponent C2 of the γ∗p center-of-massenergy squared, W 2, is predicted from evolution of the flavor-singlet quarkdistribution, C2 = 0.29, and the only free parameter, the normalization f0 =0.063, is fitted. The evolution of the gluon density multiplied by αs(Q2) isidentical to the evolution of the flavor-singlet quark density. This simplepicture is at variance with the standard approach to evolution based on thecoupled equations of flavor-singlet and gluon density.At sufficiently low values of the Bjorken variable, x ∼= Q2/W 2 ≤ 0.1, thestructure function for deep inelastic electron-proton scattering (DIS) is ingood approximation of perturbative QCD (pQCD) dominated by the gluondensity, or gluon distribution function, of the proton.The longitudinal structure function of the proton in this approximationof low x and reasonably large Q2 is given by [1]FL(x,Q2) =αs(Q2)3πnf∑qQ2q6Ig(x,Q2) (1)withIg(x,Q2) ≡∫1xdyy(xy)2 (1−xy)G(y,Q2), (2)1where G(y,Q2) ≡ yg(y,Q2),and g(y,Q2) stands for the gluon density. Thesum over the (active) quark charges squared is denoted by∑nfq Q2q . Indepen-dently of the specific form of the gluon distribution, for a wide range of suchdistributions, the integration in (2) yields a longitudinal structure functiondirectly proportional to the gluon distribution, but at a rescaled value ofx→ ξLx,FL(ξLx,Q2) =αs(Q2)3πnf∑qQ2qG(x,Q2). (3)The rescaling factor in (3) has the preferred value of ξL ∼= 0.40.[1].The structure function F2(x,Q2) for x ≤ 0.1 in the DIS scheme is propor-tional to the flavor-singlet quark distribution,∑(x,Q2). For nf = 4 flavorsof quarks, we haveF2(x,Q2) =14nf∑qQ2q · x∑(x,Q2) =518x∑(x,Q2). (4)The evolution of the flavor-singlet quark distribution, and accordingly ofF2(x,Q2), with increasing virtuality, Q2, of the photon, γ∗, is in good ap-proximation determined by the gluon distribution according to [2, 3]∂F2(ξ2x,Q2)∂ lnQ2=αs(Q2)3π∑qQ2qG(x,Q2). (5)In this case of F2(x,Q2) in (5), the rescaling factor is given by ξ2 ∼= 0.50.By writingFL(x,Q2) =12ρ+ 1F2(x,Q2), (6)we introduce the ratio of the structure functions F2(x,Q2) and FL(x,Q2).As long as ρ is allowed to vary with the kinematic variables, ρ = ρ(x,Q2),relation (6) amounts to a definition of the ratio of the longitudinal to thetransverse photoabsorption cross section,σγ∗Lp(x,Q2)σγ∗Tp(x,Q2)=12ρ. (7)By replacing the right-hand side in (5) by (3), upon employing the defini-tion (6), we obtain an evolution equation that solely contains the structurefunction F2(x,Q2) or, equivalently, the flavor-singlet distribution (4),(2ρ+ 1)∂∂ lnQ2F2(ξLξ2x,Q2)= F2(x,Q2). (8)2Similarly, by replacing the left-hand side in (6) by the gluon distributionaccording to (3), and inserting the result into (5), we find an equation forthe gluon distribution that reads∂∂ lnQ2(2ρ+ 1)αs(Q2)G(ξ2ξLx,Q2)= αs(Q2)G(x,Q2). (9)Note that, without loss of generality, ρ = ρ(x,Q2) is allowed, both in (8) and(9).In the CDP [4], at sufficiently large Q2, the structure functions becomefunctions of the γ∗p center-of-mass energy, W [5, 6]1 i.e.F2,L(x,Q2) = F2,L(W2 =Q2x). (10)The dependence on the single variable W 2 for a wide range of photon virtu-alities, Q2, is consistent with the experimental data. Compare fig. 1, wherewe show 2 the experimental data [7] for F2(x,Q2) as a function of 1/W 2 for10GeV2 ≤ Q2 ≤ 100GeV2.In terms of theW dependence from (10), the evolution equation (8) reads(2ρW + 1)∂∂ lnW 2F2(ξLξ2W 2)= F2(W2), (11)where the notation ρ = ρW indicates that a potential W dependence of theratio ρ from (6) and (7) on the kinematical variables, now W 2, is allowed in(11).More specifically, we now assume a power law for the W -dependence ofF2(W2) in (10),F2(W2) = f2 ·(W 21GeV2)C2= f2 ·(Q21GeV2)C2x−C2 , (12)where the normalization f2 and the exponent C2 are constants. With (12),the evolution equation (11) yields,(2ρW + 1)C2(ξLξ2)C2= 1. (13)The conclusion (13) from the evolution equation (11) clearly rests on the Wdependence of F2(x,Q2) = F2(W2 = Q2/x) from the CDP combined with the1The W -dependence in (10) is a consequence of the W -dependence of the color-dipole-proton cross section of the CDP [5, 6].2We thank Prabhdeep Kaur for providing the plot of the experimental data.3power-law ansatz (12). According to (13), a constant value of the exponentC2 implies a constant value of ρW = ρ = const from (6) and (7), and viceversa.In the CDP, the parameter ρ is associated with the enhanced transversesize [8, 9] of qq¯ fluctuations originating from transversely polarized photons,γ∗T , relative to the transverse size of qq¯ fluctuations from longitudinally po-larized photons, γ∗L. The known distributions of the quark (antiquark) trans-verse momentum in transversely versus longitudinally polarized qq¯ fluctu-ations, via the uncertainty principle, imply an enhanced transverse size oftransversely polarized qq¯ fluctuations. The size enhancement of definite mag-nitude yields an enhancement of the transverse relative to the longitudinalphotoabsorption cross section in (7) and (6) that is quantitatively fixed by[8, 9]ρ =43. (14)We note in passing that the factor 2 in (7) is due to the fact that the intensityof qq¯ pairs from transversely polarized photons in DIS at large Q2 is twice aslarge as the one from longitudinally polarized photons. This is in distinctionfrom the factor ρ which is a property of the (qq¯)p interaction cross section.The CDP prediction (14), according to (6), impliesFL(x,Q2) = 0.27 · F2(x,Q2). (15)The prediction (15) is in agreement with the HERAmeasurements of FL(x,Q2)for small x and large Q2. Compare refs. [8] and [9].Substituting the empirically verified prediction (14) of ρ = 4/3 into (13)together with the rescaling factors ξL = 0.40 and ξ2 = 0.50 from (3) and (5),we find that C2 is determined to be equal toC2 =12ρ+ 1(ξ2ξL)C2= 0.29. (16)We note that the result of C2 = 0.29 is fairly insensitive against variation ofthe ratio of the rescaling factors ξ2 and ξL. The (ad hoc) variation of this ratioin the interval of 1 ≤ ξ2/ξL ≤ 1.5 around the preferred value of ξ2/ξL = 1.25,according to (16), yields 0.27 ≤ C2 ≤ 0.313 . Higher energies than the onesthat were available at HERA are needed for a precision determination of C2within this interval.3Note that (16) differs from the result in ref.[10] by taking into account the rescalingfactor ξL = 0.4 as well as ρ = 4/3.4Returning to the experimental data for F2 = F2(W2 = Q2/x), in fig. 1,we show the theoretical result from (12) forF2(W2) = f2 · (W 21GeV2)C2 ≡ 0.063(W 21GeV2)0.29, (17)where C2 = 0.29 is the theoretical result from (16), while the normalization,f2 = 0.063, was determined by an “eye-ball” fit to the experimental data infig. 1.With only a single fitted parameter, f2 = 0.063, we obtained a rep-resentation of the experimental data for F2(x,Q2) over a wide range of10GeV2 ≤ Q2 ≤ 100GeV2. A more complete analysis of the experimen-tal data will be presented in a forthcoming paper [9] where, within the CDPby refining the previous analysis [5], the extension of the description of theexperimental data for F2(x,Q2) to Q2 = 0 and Q2 > 100GeV2 is treated indetail.We turn our attention to the evolution of the gluon density in (9). Forρ = const, compare (14), we have from (9)(2ρ+ 1)∂∂ lnQ2αS(Q2)G(ξ2ξLx,Q2)= αS(Q2)G(x,Q2). (18)Comparison of (18) with (8), taking into account (4), reveals that the evolu-tion of αS(Q2)G(x,Q2), i.e. the evolution of the gluon density multiplied byαS(Q2), coincides with the evolution of the flavor-singlet quark density 4.In the language of quark and gluon distributions, the forward-Compton-scattering amplitude of the CDP in fig.2 corresponds to γ∗ gluon→ qq¯ fusionas shown in fig. 3. According to fig. 3, the evolution of the flavor-singletquark distribution induced by the interacting photon, γ∗, of virtuality Q2 di-rectly measures the evolution of the gluon distribution thus suggesting iden-tical evolutions of the singlet quark distribution and the gluon distributionmultiplied by the strong coupling αS(Q2), as obtained in (18).Combining (3) with (6), and taking into account the W dependence from(10), we can directly deduce the gluon distribution from the (fit to the)experimental data in fig. 1,αS(Q2)G(x,Q2) =3π(2ρ+ 1)∑qQ2qF2(1ξLW 2 =Q2ξLx), (19)4This result is at variance with the usual procedure that supplements the evolutionequation for the flavor-singlet quark distribution by a separate equation for the gluondistribution. See discussion below.5where (17) is to be inserted on the right-hand side 5 together with ρ = 4/3from (14) and∑q Q2q = 10/9 for four active flavors.Before confronting the gluon distributions obtained from (19) with theresults available in the literature [12, 13] we briefly summarize the widelyused procedure usually employed when deducing a gluon-distribution func-tion from the experimental data on the structure function F2(x,Q2).The evolution equation for the flavor-singlet quark distribution is supple-mented by an equation for the gluon distribution, obtained [2] by replacingthe quarks in fig. 3 by (electromagnetically neutral) gluons, disregarding thephoton, and replacing the gluon→ quark splitting in fig.3 by gluon→ gluonsplitting. The resulting well-known coupled DGLAP equations [2], in con-nection with the fits [13, 14, 15, 16] to the experimental data on F2(x,Q2),are then solved numerically.The DGLAP gluon-evolution equation has an approximate analytic solu-tion, the well-know double-asymptotic solution (DAS) [17] that is understood[2, 4] as resummation of a gluon ladder subject to certain “ordering assump-tions” on the gluon momenta, compare fig.4. The DAS corresponds [4] tointroducing an x-dependent generalized dipole cross section into the CDP,thus resolving the lower blob in fig.2. The x-dependence of the DAS of theDGLAP approach is at variance with our requirement of a W -dependentcolor-dipole cross section. It is precisely this W dependence that (withinthe CDP) allows [5, 9] for a transition to the region of Q2 → 0, includingphotoproduction by real photons.Based upon an analysis employing the DAS of the DGLAP equations,the exponent, λ, of the Bjorken-x dependence of the gluon distribution,G(x,Q2) ∼ x−λ, was predicted [18] as λ = 0.32± 0.05. This result is consis-tent with our prediction of C2 = 0.29 from (16), (17) and (19).Application of DGLAP evolution to the “hard Pomeron” part of a Reggefit [13] to the experimental data for F2(x,Q2), led to G(x,Q2) ∼ x−ǫ0 , whereǫ0 = 0.427 is the fit parameter characterizing the (necessary) hard Pomeroncontribution to the structure function, F2(x,Q2) ∼ x−ǫ0. While this x de-5Note that from (19) with (17) the gluon distribution function can be extracted for anypair of values of x and Q2 with W 2 = Q2/x for given W 2. Even though the underlyingrelations (3) and (6) do not hold for Q2 ≤ 10GeV2, the gluon distribution from (19)remains sensible. The relation of the structure function to the gluon distribution becomesmodified for Q2 → 0, while the W -dependent gluon distribution remains the one extractedfrom (19), compare ref.[9].We note that our gluon distribution (19) is manifestly positivein distinction to some result in the literature (compare e.g. ref.[11])6pendence is somewhat stronger, the Q2 dependence of the gluon distributionextracted [13] from the Regge fit is somewhat weaker than ours that coincideswith the x dependence and is determined by αS(Q2)G(x,Q2) ∼ (Q2)C2 =(Q2)0.29.The most elaborate and technically demanding numerical extractions ofvalence-quark as well as sea-quark and gluon distributions were carried outby so-called global fits to the experimental data of the structure functionF2(x,Q2) by several collaborations [14, 15, 16]. Comparing the results of thedifferent collaborations collected in the Durham Data Base [12], one findsa significant spread of the values of the extracted gluon distributions as afunction of x as well as Q2.Our results for the gluon distribution from (19) with (17) lie within therange of the distributions from the hard Pomeron [13] and from refs.[14, 15,16] as given by the Durham Data Base [12]. More details will be presentedin [9].In summary:i) By combining pQCD in the approximation (3) and (5) with the W -dependence of F2(x,Q2) = F2(W2 = Q2/x) from the CDP, we findthat the evolution equation for the flavor-singlet quark distributionpredicts the exponent C2 = 0.29, where F2(W2) ∼ (W 2)C2 . Only onefitted parameter, the normalization of F2(W2), is required to representF2(W2) for x < 0.1 in the wide range of 10GeV2 ≤ Q2 ≤ 100GeV2.Our results for F2(x,Q2) = F2(W2) within the CDP of pQCD can besmoothly extended to Q2 → 0.ii) The evolution of the gluon-distribution function multiplied by αS(Q2)is identical to the flavor-singlet evolution. This result is at variancewith the results from the usual extraction of the gluon distributionthat relies on supplementing the DGLAP evolution of the flavor-singletquark distribution by the gluon-evolution equation.iii) Our gluon distribution is directly related to (the fit to) the experi-mental data for F2(W2) by a known proportionality constant. Ourextraction of the (manifestly positive) gluon distribution from the ex-perimental data is transparent, as far as the underlying theoreticalassumptions and the relation to the experimental data are concerned,and it is straight forward and simple.7iv) The results for the gluon distribution from the analysis of the exper-imental data by different collaborations differ significantly from eachother. Within this wide range, our gluon distribution is compatiblewith the published ones, even though both, our underlying assumptionsand our procedure, differ appreciably from the ones in the literature.AcknowledgementI am grateful to Prabhdeep Kaur for providing the plot of the experimentaldata for the structure function.)-2(GeV21/W-610 -510 -410 -310 -210 -1102F0.20.40.60.811.21.41.61.822=80-100 GeV2Q2=50-80 GeV2Q2=30-50 GeV2Q2=10-30 GeV2Qx-410 -310 -210 -1102F0.20.40.60.811.21.41.61.822=80-100 GeV2Q2=50-80 GeV2Q2=30-50 GeV2Q2=10-30 GeV2QFigure 1: The experimental data on the proton structure function F2(x,Q2)as a function of 1/W 2. The theoretical curve is based on (17). For compari-son, we also show F2(x,Q2) as a function of x ∼= Q2/W 2.8+γ∗γ∗γ∗γ∗Figure 2: The forward Compton amplitude of the CDPγ∗Figure 3: Photon-gluon → qq¯ fusion, equivalent to the CDP from fig.1.γ∗ γ∗Figure 4: Higher order corrections to photon-gluon → qq¯ fusion resolvingthe lower blob in fig.2. 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Perturbative QCD Evolution and Color Dipole Picture

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