Dipole model analysis of high precision HERA data
aa r X i v : . [ h e p - ph ] D ec Dipole model analysis of high precision HERA data
A. Luszczak , H. Kowalski T. Ko´sciuszko Cracow University of Technology Deutsches Elektronen-Synchrotron DESY, D-22607 Hamburg, Germany
Abstract
We analyse, within a dipole model, the inclusive DIS cross section data, obtainedfrom the combination of the H1 and ZEUS HERA measurements. We show that thesehigh precision data are very well described within the dipole model framework, whichis complemented with a valence quark structure functions. We discuss the propertiesof the gluon density obtained in this way.
Many investigations have shown that HERA inclusive and diffractive DIS cross sections arevery well described by the dipole models [1–3]. Interest in the dipole description emergefrom the fact that dipole picture provides a natural description of QCD reaction in the low- x region. Due to the optical theorem, dipole models allow a simultaneous description ofmany different physics reactions, like inclusive DIS processes, inclusive diffractive processes,exclusive J/ψ , ρ, φ production, diffractive jet production, or diffractive and non-diffractivecharm production. In the dipole picture, all these processes are determined by the same,universal, gluon density [4–6].In the era of the LHC, the precise knowledge of gluon density is very important becausethe QCD-evolved gluon density determines the cross sections of most relevant physics pro-cesses, e.g. Higgs production. Any significant deviation of the predicted cross section fromtheir Standard Model value could be a sign of new physics.The validity of the dipole approach was experimentally established, a decade ago, bya comparison of the dipole predictions with HERA F and diffractive data in the low x region [1], [2]. In the meantime, the precision of data obtained from HERA experimentsincreased substantially. The H1 and ZEUS experiments have combined their inclusive DIScross sections which, due to a substantial reduction of systematic measurements errors, led toan increase of precision by about a factor two [7]. In the same way the quality of the inclusive1harm data was substantially improved [8]. Finally, recently, the exclusive J/ψ productionwas much more precisely measured [9] . All these reaction were used in the past to establishthe dipole approach. It is therefore interesting to re-evaluate these reactions because thedipole picture provides a somewhat different approach to the gluon density than the usualpdf approach. In the usual pdf approach the gluon density contributes to F mainly throughthe evolution of the see quarks, the direct gluon contribution is only of the order of a fewpercent. On the other hand, in the dipole models the gluon density is directly connected tothe see quarks. In the pdf scheme the evolution is evaluated in the collinear approximationwhereas the dipole approach uses the k T factorization.The direct connection between the dipole production and gluon density is particularlyclearly seen in the exclusive J/ψ production, which was therefore proposed as a testingground of the properties of the gluon density [10]. Presently, the exclusive
J/ψ productionis precisely measured in heavy ion collisions at RHIC and LHC. These measurements com-bined with their dipole analysis can become a new source of information about the gluonicstructures of nuclei [12, 13]Another important application of the dipole description is the investigation of the gluonichigh density states. These can be characterized by the degree by which a dipole is absorbedor multiply scattered in such states. The states with the highest gluon densities are producedtoday in the high energy heavy ion scattering at RHIC and LHC. This is now a very livelyfield of saturation investigation [14, 15].The aim of this paper is to investigate the additional information which is containedin the new, combined HERA data. The most precise data where obtained in the regionof higher Q ’s (Q from 3.5 to O(10000) GeV ), where the DGLAP evolution is known todescribe data very well. Therefore, as discussed below, in this investigation we use the socalled BGK dipole model, because it uses the DGLAP evolution scheme.This paper concentrates first on the inclusive DIS measurements in the low x region.Here, the contribution of the valence quarks is small, below 7%, and has therefore beenneglected until now. However, the combined H1 and ZEUS HERA data achieve however aprecision of about 2%, so the contribution of the valence quarks can no longer be neglected.The present paper addresses the question to what extent the contribution of the valencequark and the dipoles are compatible with each others. To do so we use the HARAfitterframework [16] which allows to treat consistently QCD evolution together with the valencequark and dipoles contributions.The paper is organized as follows: in Section 2 we recall the main properties of the dipoleapproach and review various models in order to motivate our choice. In Section 3 we discussthe results of fits and in Section 4 we compare the fits with data. Section 5 contains thesummary. 2 Dipole models
In the dipole picture the deep inelastic scattering is viewed as a two stage process; first thevirtual photon fluctuates into a dipole, which consists of a quark-antiquark pair (or a q ¯ qg or q ¯ qgg ... system) and in the second stage the dipole interacts with the proton [17] , [18–24].Dipole denotes a quasi-stable quantum mechanical state, which has a very long life time( ≈ /m p x ) and a size r , which remains unchanged during scattering. The wave functionΨ determines the probability to find a dipole of size r within a photon. This probabilitydepends on the value of external Q and the fraction of the photon momentum carried by thequarks forming the dipole, z . Neglecting the z dependence, in a very rough approximathion, Q ∼ /r .The scattering amplitude is a product of the virtual photon wave function, Ψ, with thedipole cross section, σ dip , which determines a probability of the dipole-proton scattering.Thus, within the dipole formulation of the γ ∗ p scattering σ γ ∗ pT,L ( x, Q ) = Z dr Z dz Ψ ∗ T,L ( Q, r, z ) σ dip ( x, r )Ψ T,L ( Q, r, z ) , (2.1)where T, L denotes the virtual photon polarization and σ γ ∗ pT,L the total inclusive DIS crosssection.Several dipole models have been developed to test various aspects of the data. They varydue to different assumption made about the physical behavior of dipole cross sections. Inthe following we will shortly review them and motivate our the choice of the model used forthe present investigation. The dipole model became an important tool in investigations of deep-inelastic scatteringdue to the initial observation of Golec-Biernat and W¨uesthoff (GBW) [1], that a simpleansatz for the dipole cross section was able to describe simultaneously the total inclusiveand diffractive cross sections.In the GBW model the dipole-proton cross section σ dip is given by σ dip ( x, r ) = σ (cid:18) − exp (cid:20) − r R ( x ) (cid:21)(cid:19) , (2.2)where r corresponds to the transverse separation between the quark and the antiquark, and R is an x dependent scale parameter which has a meaning of saturation radius, R ( x ) =( x/x ) λ GBW . The free fitted parameters are: the cross-section normalisation, σ , as well as x and λ GBW . In this model saturation is taken into account in the eikonal approximation andthe saturation radius is intimately related to the gluon density in the transverse plane, seebelow. The exponent λ GBW determines the growth of the total and diffractive cross sectionwith decreasing x . For dipole sizes which are large in comparison to the saturation radius,3 , the dipole cross section saturates by approaching a constant value σ , i.e. saturationdamps the growth of the gluon density at low x .The GBW model provided a good description of data from medium Q values ( ≈ ) down to low Q ( ≈ .
1) GeV ). Despite its success and its appealing simplicity themodel has some shortcomings; in particular it describes the QCD evolution by a simple x dependence, ∼ (1 /x ) λBGW , i.e the Q dependence of the cross section evolution is solelyinduced by the saturation effects. Therefore, it does not match with DGLAP QCD evolution,which is known to describe data very well from Q ≈ to very large Q ≈ . The evolution ansatz of the GBW model was improved in the model proposed by Bartels,Golec-Biernat and Kowalski, (BGK) [2], by taking into account the DGLAP evolution of thegluon density in an explicit way. The model preserves the GBW eikonal approximation tosaturation and thus the dipole cross section is given by σ dip ( x, r ) = σ (cid:18) − exp (cid:20) − π r α s ( µ ) xg ( x, µ )3 σ (cid:21)(cid:19) . (2.3)The evolution scale µ is connected to the size of the dipole by µ = C/r + µ . Thisassumption allows to treat consistently the contributions of large dipoles without makingthe strong coupling constant, α s ( µ ), un-physically large.The gluon density, which is parametrized at the starting scale µ , is evolved to largerscales, µ , using LO or NLO DGLAP evolution. We consider here three forms of the gluondensity: • the soft ansatz, as used in the original BGK model xg ( x, µ ) = A g x − λ g (1 − x ) C g , (2.4) • the soft + hard ansatz xg ( x, µ ) = A g x − λ g (1 − x ) C g (1 + D g x + E g x ) , (2.5) • the soft + negative gluon xg ( x, µ ) = A g x − λ g (1 − x ) C g − A ′ g x − λ ′ g (1 − x ) C ′ g , (2.6)The free parameters for this model are σ , µ and the parameters for gluon A g , λ g , C g oradditionally D g , E g , or A ′ g , λ ′ g , C ′ g . Their values are obtained by a fit to the data. The fitresults were found to be independent on the parameter C , which was therefore fixed as C = 4GeV , in agreement with the original BGK fits.4 .3 IIM model Although we do not use the IMM (Iancu, Itacura and Mounier) model in this paper wemention it here because it may take better into account the saturation effects than it isthe case in the BGK or GBW models. The last models use for satuartion the eikonalapproximation, whereas the IIM model uses a simplified version of the Balitsky-Kovchegovequation [27]. The explicit formula for σ dip can be found in [3]. We do not use this modelbecause we concentrate here on the higher Q data, which precise description requires anequally precise transition to the DGLAP regime. The dipole models are valid in the low- x region where the valence quark contribution issmall. Therefore, this contribution was usually neglected which was justified as long as theexperimental errors were relatively large. Theoretically, it is very difficult to treat valencequarks inside the dipole framework because, until now, the dipole amplitudes are not welldefined in the region of high x . The problem may be solved, in future, by the analyticcontinuation of the dipole (or BFKL) amplitudes from the low x to the high x region [28].However, for the purpose of this paper, we propose to take an heuristic approach and justto add the valence quark contribution from the standard pdf’s fits to the dipole predictions.In this approach the dipole contribution plays a role of the see quarks in the standard pdf’s.This procedure is justified by the fact that the see quark contribution disappears at larger x . The HERAfitter project is well suited for this purpose since the dipole model and thevalence quarks contributions are a part of the same framework. No Q σ A g λ g C g C N p χ χ /N p . . . . . σ r for H1ZEUS-NC-(e+p) and H1ZEUS-NC-(e-p)data in the range Q ≥ . x ≤ .
01. NLO fit. RT HF Scheme.
Soft gluon .In this section we investigate how well the dipole model can describe the new, precise,HERA data which were obtained in the region of Q > . . Since the quality of datain the region of Q < was not improved until now we concentrate here on the higherQ region where the valence quark contribution becomes relevant.5 .1 Dipole fits with valence quarks First, we show that it is possible to combine the dipole and valence quark contributionsand obtain a good fit to the data. For the purpose of this investigation we choose the BGKmodel because it uses the DGLAP evolution. The fits were performed within the HERAfitterframework, i.e.; the QCD evolution is the same as in the standard HERAfitter pdf fits. Theresults of the BGK fit, with valence quarks, are shown in Table 1. The fit is performed inthe low x range, x < .
01, for various µ values. The value of µ plays a role of the startingscale of the QCD evolution which is usually denoted by Q in the pdf fits. N p denotes thenumber of measured values of the reduced cross section, σ r , which were used in the fit. Theparameters σ of the dipole model and the parameters for gluon A g , λ g , C g are obtainedfrom the fit at a given value of Q (in GeV ). The value of the parameter C was fixed, asexplained above.The table shows that the BGK model with valence quarks taken from the usual HER-Afitter pdf fit, is describing the precise HERA data very well for all Q value. The fit qualityimprove slightly with diminishing Q . This could indicate that HERA data in the low rangeof Q ∼ . , retain some sensitivity to the saturation effects. In the BGK model thesaturation effects increases with decreasing Q value.In the Table 2 we show results of the standard HERAPDF fits. They are performed inthe same Q range as the dipole fits but in the full x range. The full x range is necessary tofix the contribution of valence quarks.No Q HF Scheme χ N p χ /N p σ r for H1ZEUS-NC-(e+p), H1ZEUS-NC-(e-p) and H1ZEUS-CC-(e+p), H1ZEUS-CC-(e-p) data in the range Q ≥ . x ≤ . Q HF Scheme χ N p χ /N p σ r for H1ZEUS-NC-(e+p), H1ZEUS-NC-(e-p) and H1ZEUS-CC-(e+p), H1ZEUS-CC-(e-p) data in the range Q ≥ . x ≤ . Q σ A g λ g C g C N p χ χ /N p . . . . . σ r for H1ZEUS-NC-(e+p) andH1ZEUS-NC-(e-p) data in the range Q ≥ . x ≤ .
01. NLO fit. RT HF Scheme.
Softgluon .Table 2 shows a very good agreement with data of the standard pdf fit. The agreementis similar as in the dipole fits, if corrected for the number of points and the number of freeparameter, which is N free = 10 for the HARAPDF fit and N free = 4 in case of the BGK fitwith the soft gluon assumption. In difference to the dipole fits, the quality of the HERAPDFfit is deteriorating with decreasing Q scale.Table 3 and 4 show HERAPDF and BGK dipole fits in the higher Q range, Q > . . We see that the quality of fits clearly improves in the higher Q region. In case ofHARAPDF fit the χ /N p improves from 0.97 to 0.85 and in case of the BGK fit from ∼ . Q .The HERAPDF fits do still show some slight deterioration with decreasing Q but the effectis much smaller than seen in Table 2.In Fig. 1 we show a comparison the gluon density obtained in the fits with valence quarksand compare it to the gluon density obtained in the HERAPDF fit. We see that the twogluon densities, at NLO, differ at smaller scales but then start to approach each other athigher scales. It is interesting to observe that the convergence of the two gluon densities ismuch slower in LO, Fig. 2. In this section we investigate whether the more involved forms of the gluon density, eq.(2.5)and eq.(2.6), can improve the data description. In table 5 and 6 we show the fit results forthe fits with soft + hard gluon of eq.(2.5), in the lower Q > . and higher Q > . regions. We observe that the fit quality improves significantly by adding a ”hard”component, D g x + E g x , to a classic soft gluon of eq.(2.4). The value of χ diminishes byabout ∆ χ ≈
20 for Q > . and by about ∆ χ = 15 for Q > . , which is amuch larger drop than the increase of the parameter number (just by 2).In Table 7 we show the fit results for the fits with the soft + negative gluon of eq.(2.6).The fit in the lower Q range is not significantly improved by the addition of the negativegluon term. In the higher Q range, Q > . , the fit improves somewhat, althoughnot so clearly as in the ”hard” case. 7o Q σ A g λ g C g D g E g χ χ /N p . . . . . σ r for H1ZEUS-NC-(e+p) andH1ZEUS-NC-(e-p) data in the range Q ≥ . x ≤ .
01. NLO fit. RT HF Scheme.
Soft+ hard gluon . N p = 201 and C=4.0 GeV .No Q σ A g λ g C g D g E g χ χ /N p . . . . . σ r for H1ZEUS-NC-(e+p) andH1ZEUS-NC-(e-p) data in the range Q ≥ . x ≤ .
01. NLO fit. RT HF Scheme.
Soft+ hard gluon . N p = 162 and C = 4 . . Q Q σ A g λ g C g A ′ g B ′ g C ′ g χ χ /N p . . . . σ r for H1ZEUS-NC-(e+p) andH1ZEUS-NC-(e-p) data. NLO fit. RT HF Scheme. Soft + negative gluon . C = 4 . , N p = 201 for Q > . and N p = 162 for Q > . xg l u o n BGK NLOHERAPDF NLO -3 -2 -1 -3 -2 -1 -3 -2 -1 x Figure 1: Comparison between the dipole (soft) and HERAPDF gluon in NLO.
To better understand the meaning of the fits which are using alternative forms of the gluondensity we performed also fits without valence quarks, and with valence quarks fitted todata. In Table 8 and Table 9 we show fits performed without valence quarks for the soft andsoft+hard forms of the gluon density in the region of Q > . .No Q Q σ A g λ g C g χ χ /N p . . σ r for H1ZEUS-NC-(e+p) andH1ZEUS-NC-(e-p) data in the range Q ≥ . x ≤ .
01. NLO fit. RT HF Scheme.
Softgluon . C = 4 . and N p = 201.The contribution of the valence quarks in the low x region are large enough to be able todetermine them in this region only. In Table 10 we show an example of a fit with parametersof the valence quarks fitted to data together with the parameters of the gluon density. Thefit is performed for Q > . , in the low x range, x < .
01. In Table 11 we give, forcompleteness, the parameters of the valence quarks determined in this way. Note, that thefit with fitted valence quarks is better than the fit with fixed valence quarks of Table 1 and9 xg l u o n BGK LOHERAPDF LO -3 -2 -1 -3 -2 -1 -3 -2 -1 x Figure 2: Comparison between the dipole (soft) and HERAPDF gluon in LO.No Q Q σ A g λ g C g D g E g χ χ /N p . . σ r for H1ZEUS-NC-(e+p) andH1ZEUS-NC-(e-p) data in the range Q ≥ . x ≤ .
01. NLO fit. RT HF Scheme.
Soft+ hard gluon . C = 4 . and N p = 201.it is also better than the fit without valence quarks, Table 8.Fig. 3 shows the comparison between the NLO gluon densities determined with the soft and soft + hard assumptions . The soft gluon density is taken from the fit of Table 1. The soft+hard gluon density shown on the LHS of Fig. 3 is taken from the fit of Table 5 andwas obtained with the fixed valence quark contribution. The RHS of this figure shows the soft+hard gluon density obtained from the fit of Table 10. Here, the contribution of valencequarks is fitted to data together with the gluon density. Both fits, of Table 5 and Table 10,have a very similar quality, the form of gluon densities differs, however, at lower scales inthe high x region; the one with the fixed valence quarks shows a clear bump around x ≈ . soft case. In all fits which weperformed, the bump in the soft+hard gluon density fitted with the fixed valence quarkswas always present. independently of the Q cut or the LO or NLO QCD evolution. Thisbump disappears, however, when the valence quark contribution is fitted. Therefore, we do10ot attribute a physical meaning to this bump, especially that it is in the region which isnot directly tested by data and it contributes only to the low- x region through the QCDevolution. Nevertheless, its existence emphasizes the necessity of a full fit to the data, i.e.of a fit in which the gluon density is fitted together with the valence quarks. xg l u o n soft + hard gluonsoft gluon -3 -2 -1 -3 -2 -1 -3 -2 -1 x xg l u o n soft + hard gluonsoft gluon -3 -2 -1 -3 -2 -1 -3 -2 -1 x Figure 3: Comparison between the NLO gluon densities determined with the soft and soft+ hard assumptions. LHS shows the gluon distribution functions determined with the fixedvalence quark contribution. RHS shows the gluon distribution functions determined withthe contribution of valence quarks fitted to data in the x < .
01 region.No Q Q σ A g λ g C g χ χ /N p . . σ r for H1ZEUS-NC-(e+p)and H1ZEUS-NC-(e-p) data in the range Q ≥ . x ≤ .
01. NLO fit. RT HF Scheme.
Soft gluon . C = 4 . and N p = 201.
In Fig. 4 we show a comparison of the dipole BGK fit with the HERA reduced cross sectiondata. Figure shows an excellent agreement of the fit with data. In Fig. 5 we show acomparison of F l structure function obtained from the dipole BGK fit with HERA data. Inboth figures we use the BGK fit of Table 1, with Q = 1 . .11o Auv Buv Cuv Euv Adv Cdv CU bar ADbar BDbar CDbar σ r for H1ZEUS-NC-(e+p) and H1ZEUS-NC-(e-p) data in the range Q ≥ . x ≤ .
01. NLO fit. RT HF Scheme.
Soft gluon . Parameter cBGK = 4 . N p = 201.
Combined ZEUS and H1 Data
BGK NLO fit s r -4 -2 -4 -2 -4 -2 -4 -2 x Combined ZEUS and H1 Data
BGK NLO fit s r -4 -2 -4 -2 -4 -2 -4 -2 x Figure 4: Comparison of the dipole BGK fit of Table 1 with the reduced cross sections ofHERA data. 12 ombined ZEUS and H1 Data
BGK LO fitBGK NLO fit F L -5 -4 -3 -2 -5 -4 -3 -2 -5 -4 -3 -2 x Combined ZEUS and H1 Data
BGK LO fitBGK NLO fit F L -5 -4 -3 -2 Figure 5: Comparison of F l structure function obtained from the dipole BGK fit of Table 1with HERA data. 13 Summary
We have shown that the k T factorized, DGLAP evolved gluon density, evaluated withinthe BGK model, describe the combined, precise HERA data in the low- x region, very well.The valence quark contribution added to the dipole model improves the fit significantly.Therefore, for precise dipole evaluations the gluon contribution should be complemented byvalence quarks.The resulting gluon density obtained from fits with fitted valence quarks could be usedfor the prediction of LHC cross sections, provided that the dipole amplitude, which is nowonly well defined in the low- x region, can be analytically continued to the high x region [28].As a byproduct of this investigation we observe that the fits of all dipole and pdf typesimprove significantly when the Q cut on data is increased from Q > . Q > . .We have checked this with the dipole model with quarks and without quarks, with variousforms of the gluon density, as well as with the standard HERAPDF1.0 fit. The persistenceof this effect indicate some shortcomings of the theoretical description; it could be due tothe lack of higher order QCD corrections or to saturation effects. We note, that the higherorder corrections diminish logarithmically with increasing Q whereas the saturation effectsdiminish like a power of Q , or faster. In our view, the relatively fast change of χ /N p withthe increased Q cut indicates that the effect is due to saturation, at least to large extent. Inthis way, the increase of precision in HERA data offers a novel testing ground for saturationstudy in the well measured region above Q > . . The study of this type may becomevery interesting when, in the near future, the combined HERA I and HERA II data, withyet further increased precision, is published. We would like to thank P. Belov, A. Glazov, R. Plackacyle and Voica Radescu for introductionto the HERAfitter project and various help with solving problems. We would also like tothank J. Bartels and D. Ross for reading the manuscript and useful comments.
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