HERA data and DGLAP evolution: theory and phenomenology
aa r X i v : . [ h e p - ph ] O c t IFUM-962-FT
HERA data and DGLAP evolution:theory and phenomenology
Fabrizio Caola, Stefano Forte and Juan Rojo,
Dipartimento di Fisica, Universit`a di Milano and INFN, Sezione di Milano,Via Celoria 16, I-20133 Milano, Italy
Abstract:
We examine critically the evidence for deviations from next-to-leading order perturbativeDGLAP evolution in HERA data. We briefly review the status of perturbative small- x resummation and of global determinations of parton distributions. We then show that thegeometric scaling properties of HERA data are consistent with DGLAP evolution, whichis also strongly supported by the double asymptotic scaling properties of the data. Wefinally show that backwards evolution of parton distributions into the low x , low Q regionshows evidence of deviations between the observed behaviour and the next-to-leading orderpredictions. These deviations cannot be explained by missing next-to-next-to-leading orderperturbative terms, but are consistent with perturbative small- x resummation. Perturbative QCD is a quantitatively tested theory which describes in a very accurate waya vast body of data, and it is at the basis of physics at colliders such as the LHC [1]. TheDGLAP equations, i.e. the renormalization-group equations which govern the scale de-pendence of parton distributions are, together with asymptotic freedom and factorization,a cornerstone of the theory, both in terms of phenomenological success and theoreticalfoundation. Specifically, they are the tool which allows us to combine information onnucleon structure from a variety of data, and use it for predictions at the LHC.The current frontier in perturbative QCD is systematically going from the secondto the third perturbative order, namely from next-to-leading (NLO) to next-to-next-to-leading order (NNLO). In view of this, it is of utmost importance for both theory andphenomenology to understand whether in any given kinematic region an order-by-orderperturbative approach is sufficient. There is now considerable theoretical evidence thatat sufficiently high center-of-mass energies this approach may break down, and thus asthis region is approached one should resum to all orders perturbative corrections whichare logarithmically enhanced in the ratio x of the hard scale to center-of-mass energy —the so–called small- x resummation. However, conclusive experimental evidence for suchresummation effects in the data is lacking, partly due to the fact that the relevant effectsare difficult to disentangle from model and theoretical assumptions.It is the purpose of this contribution to review, update and put in context a recentattempt to provide some such evidence. The outline of this contribution is as follows.1n Sect. 2 we briefly review the status of linear small- x resummation. Then in Sect. 3we present the state of the art of fixed-order DGLAP and global PDF analysis and itsimplications for the LHC. Finally in Sect. 4 we present a technique designed to identifydeviations from NLO DGLAP evolution in the data, and apply it to recent global partonfit. We will specifically see that previous evidence of deviations from NLO DGLAP isconsiderably strengthened by the recent precise combined HERA-I determination of deep-inelastic structure functions. x resummation As well known, deep–inelastic partonic cross sections and parton splitting functions receivelarge corrections in the small- x limit due to the presence of powers of α s log x to all ordersin the perturbative expansion [2, 3]. This suggests dramatic effects from yet higher orders,so the success of NLO perturbation theory at HERA has been for a long time very hardto explain. In the last several years this situation has been clarified [4–9], showing that,once the full resummation procedure accounts for running coupling effects, gluon exchangesymmetry and other physical constraints, the effect of the resummation of terms which areenhanced at small- x is perceptible but moderate — comparable in size to typical NNLOfixed order corrections in the HERA region.Figure 1: Ratio of the resummed and NNLO prediction to the NLO fixed order for the singlet F deep–inelastic structure function. The curves are: fixed order perturbation theory NNLO (green,dashed); resummed NLO in Q MS scheme (red, solid), resummed NLO in the MS scheme (blue,dot-dashed). In each case, the three curves shown correspond to fixed x = 10 − , x = 10 − , x = 10 − , with the smallest x value showing the largest deviations. For phenomenology, it is necessary to resum not only evolution equations, but also hardpartonic coefficient functions. The relevant all–order coefficients have been computed forDIS [2,3], and more recently for several LHC processes such as heavy quark production [10],Higgs production [11,12], Drell-Yan [13,14] and prompt photon production [15,16]. Thesecoefficients can be used for a full resummation of physical observables, by suitably combin-ing them with resummed DGLAP evolution and accounting for running–coupling effects,so as to maintain perturbative independence of physical observables of the choice of fac-torization scheme [17]. 2his program has been carried out for the first time for deep–inelastic scattering inRef. [6], and more recently for prompt photon production [18], which makes resummedphenomenology for these processes possible. In particular in Ref. [6] results have been pre-sented for the ratio of resummed to unresummed NLO deep–inelastic structure functions.These ratios are shown in Fig. 1 for two choices of the resummed factorization schemediscussed in Ref. [6], and compared to analogous ratio of the fixed order NNLO to NLO.They were determined under the hypothesis that the structure functions F and F L arekept fixed for all x at Q = 2 GeV: this models the situation in which parton distributionsare determined at the scale Q , and one then sees the change in prediction when goingfrom NLO to NNLO, or from NLO to unresummed. In Sect. 4 we will present evidencefor departures from the NLO which appear to be consistent with this figure. Fixed–order DGLAP evolution is an integral ingredient of any PDF determination. Cur-rently, the most comprehensive PDF sets are obtained from a global analysis of hard-scattering data from a variety of processes like deep–inelastic scattering, Drell-Yan andweak vector boson production and collider jet production. In such global analysis, QCDfactorization and DGLAP evolution are used to relate experimental data to a commonset of PDFs. Three groups produce such global analysis and provide regular updates ofthese: NNPDF [19], CTEQ [20, 21] and MSTW [22]. The typical dataset included in oneof such global analysis is shown in Fig. 2. We also show in Fig. 2 the kinematic rangewhich is available at the LHC as compared to that covered by present experimental data:extrapolation to larger Q from the current data region is possible thanks to DGLAPevolution. x -5 -4 -3 -2 -1
10 1 ] [ G e V T / p / M Q NMC-pdNMCSLACBCDMSHERAI-AVCHORUSFLH108NTVDMNZEUS-H2DYE886CDFWASYCDFZRAPD0ZRAPCDFR2KT = 0.5 cut
A = 1.0 cut
A = 1.5 cut
A = 3.0 cut
A = 6.0 cut A NNPDF2.0 dataset
Figure 2:
Left: Experimental data used in the NNPDF2.0 global analysis; the series of A cut kinematic cuts is discussed in Sect. 4. Right: LHC kinematical region. A significant advance in global PDF analysis in the recent years has been the develop-3 it 2.0 DIS 2.0 DIS+JET NNPDF2.0 χ Table 1:
The χ for individual experiments included in NNPDF2.0 fits with DIS data only, DISand jet data only, and the full DIS, jet and Drell-Yan data set. For each fit, values of the χ fordata not included in the fit are shown in italic. The value of χ in the first line does not includethese data. ment of the NNPDF methodology [19, 23–29]. NNPDF provides a determination of PDFsand their uncertainty which is independent of the choice of data set, and which has beenshown in benchmark studies [30] to behave in a statistically consistent way when data areadded or removed to the fit. Also, because of the use of a Monte Carlo approach, theNNPDF methodology is easily amenable to the use of standard statistical analysis tools.The most updated NNPDF analysis is NNPDF2.0 [19], a global fit to all relevant DIS andhadronic hard scattering data.Comparing the effect of individual datasets on a global fit such as NNPDF2.0 allowsdetailed studies of QCD factorization, DGLAP evolution, and the compatibility betweenDIS and hadronic data. A very stringent test is obtained by comparing the results ofa fit to DIS data only to that of DIS+jet data (Table 1). Indeed, it turns out thatjet data, which are at much higher scale, are well predicted by PDFs determined fromlower scale DIS data. Furthermore, the gluon extracted from the DIS–only fit, which isessentially determined from DGLAP scaling violations, turns out to agree very well withthat determined when jet data are also included (see Fig. 3): upon inclusion of the jetdata, the uncertainty decreases without a significant change in central value.Further consistency checks are obtained by comparing the effect of the inclusion of aspecific dataset (such as Drell-Yan) to different datasets (such as DIS, or DIS+jets). Ifthere was any inconsistency between different sets, the impact of the new data would bedifferent according to whether they are added to data they are or are not consistent with.No such differences are observed (see Fig. 4). Because the various sets are at different scalesand related through DGLAP evolution this also provides a strong check of its accuracy. We have seen that NLO DGLAP is extremely successful in describing in a consistentway all relevant hard scattering data. On the other hand, there are several theoretical4 ) x g ( x , Q -0.100.10.20.30.40.50.60.7 NNPDF2.0 - DISNNPDF1.2 - DIS + JET x -3 -2 -1 ) x g ( x , Q -0.500.511.522.533.5 NNPDF2.0 - DISNNPDF1.2 - DIS + JET
Figure 3:
Impact on the gluon distribution of the inclusion of jet data in a fit with DIS data. indications that at small x and/or at small Q the NLO DGLAP might undergo sizablecorrections due to leading–twist small- x perturbative resummation, or non linear evolution,parton saturation and other higher twist effects. Even if it is unclear in which kinematicalregime these effects should become relevant, it is likely that eventually they should becomerelevant, specifically in order to prevent violations of unitarity.When trying to trace these effects, one should beware of the possibility that putativesignals of deviation might in fact be explained using standard NLO theory. An exampleof this situation is the so–called geometric scaling [31] , which is often thought to provideunequivocal evidence for saturation. This is the prediction, common to many saturationmodels, that DIS cross sections, at small- x depend only on the single variable τ ( x, Q ) = (cid:0) Q /Q (cid:1) · ( x/x ) λ , (1)rather than on x and Q separately. However, it turns out that geometric scaling Eq. (1) isalso generated by linear DGLAP evolution [32]: fixed order DGLAP evolution evolves any(reasonable) boundary condition into a geometric scaling form. Furthermore, the scalingexponent λ Eq. (1) obtained in such way agrees very well with the experimental value.Following Ref. [32], in Fig. 5 we compare the scaling behaviour of the HERA data andthe LO small- x DGLAP evolution of a flat boundary condition: the DGLAP predictionscales even better than data. Note that the accurate combined HERA-I dataset [33] areused here, and that the scaling behaviour persists also at larger x ∼ < .
1, where it is unlikelythat it is related to saturation. Clearly, this shows that geometric scaling is not sufficientto conclude that fixed–order DGLAP fails. However, one may wonder whether small- x LO DGLAP is phenomenologically relevant.To answer this, in Fig. 5 we also show a comparison of the data to the so–calleddouble asymptotic scaling (DAS) [34] form, obtained from the small- x limit of the LODGLAP solution. The agreement between data and theory is so good that one can see thechange in DAS slope when the number of active flavours goes from n f = 4 to n f = 5: so,while on the one hand geometric scaling cannot discriminate between pure DGLAP andsaturation, double asymptotic scaling provide evidence that the data follow the predictedDGLAP behavior in most of the HERA region.However, deviations from DGLAP evolution can be investigated exploiting the morediscriminating and sensitive framework of global PDF fits. The key idea in this kind of5 ) ( x , Q x T ) ( x , Q x T NNPDF2.0NNPDF2.0 DIS+JET x0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 ) x V ( x , Q ) x V ( x , Q NNPDF2.0NNPDF2.0 DIS+JET x0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 ) ( x , Q S D x -0.0200.020.040.060.08 NNPDF2.0 DISNNPDF2.0 DIS+DY x0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 ) ( x , Q S D x NNPDF2.0 DIS+JETNNPDF2.0 x0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 ) ( x , Q - xs -0.02-0.0100.010.020.030.04 NNPDF2.0 DISNNPDF2.0 DIS+DY x0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 ) ( x , Q - xs -0.02-0.0100.010.020.030.04 NNPDF2.0 DIS+JETNNPDF2.0
Figure 4:
Impact of the inclusion of Drell–Yan data in a fit with DIS data only (left), and in afit with DIS and jet data (right). From top to bottom, isotriplet, total valence, ¯ d − ¯ u and s − ¯ s PDFs are shown in each case. -4 -3 -2 -1
10 1 10 p * g t o t s -4 -3 -2 -1 · DAS prediction ( l ) x / (1 GeV = Q t = 0.33 l s R F =4 f nQCD prediction 2) · =5 ( f n 2) · QCD prediction (
Figure 5:
Left: geometric scaling of the HERA-I data and of the (Double Asymptotic Scaling)small– x solution of the LO DGLAP evolution equation for a a flat boundary condition at Q =1 GeV . Only points with Q >
10 GeV are included in the DAS curve, which is offset forclarity. Right: the rescaled proton F structure function plotted as a function of the DAS variables σ ≡ p ln 1 /x ln (ln t/t ), with t = ln Q / Λ QCD and Q = 1 GeV. The points correspond to thesame data and the curves corresponds to the same prediction as in the right plot. The two curvesinclude data with 10 GeV ≥ Q ≥
25 GeV and x < .
01 ( n f = 4) or Q ≥
25 GeV and x < . n f = 5). analysis is to perform global fits only in the large- x , large- Q region, where NLO DGLAPis certainly reliable. This way one can determine “safe” parton distributions which arenot contaminated by possible non-DGLAP effects. These “safe” PDFs are then evolvedbackwards into the potentially “unsafe” low- x and low- Q kinematic region, and used tocompute physical observables, which are compared with data. A deviation between thepredicted and observed behaviour in this region can then provide a signal for effects beyondNLO DGLAP. Since possible deviations are small, this kind of studies is meaningful only onstatistical grounds, hence a reliable estimate of PDFs uncertainties and theoretical biasesis mandatory. The NNPDF framework provides useful tools for this kind of investigations.In [35] such an analysis was performed using the NNPDF1.2 PDF set [28], and itdid provide some evidence for deviations from NLO DGLAP. Here, we update this anal-ysis using the NNPDF2.0 PDF set discussed in Sect. 3. In comparison to NNPDF1.2,NNPDF2.0 also includes hadronic data (fixed target Drell-Yan production, collider weakboson production and collider inclusive jets), and the combined HERA-I dataset [33] re-places previously less accurate data from ZEUS and H1. The inclusion of the very accuratecombined HERA-I dataset has the potential to increase the significance of the observeddeviations from DGLAP, while the presence of hadronic data allows stringent tests of theglobal compatibility of the NLO DGLAP framework, as discussed in Sect. 3. A furtherdifference between NNPDF1.2 and NNPDF2.0 is an improved treatment of normaliza-tion uncertainties based on the so–called t method [29], which avoids the biases of othercommonly used methods to deal with normalization uncertainties.The “safe” region, where non–DGLAP effects are likely to be negligible, is defined as Q ≥ A cut · x λ , (2)with λ = 0 .
3. This definition has the feature of only considering unsafe small- x data iftheir scale is low enough, with the relevant scale raised as x is lowered; its detailed shape7 -4 -3 -2 ) = . G e V ( x , Q F Fit without cuts = 0.5 cut
Fit with A = 1.5 cut
Fit with AData x -4 -3 -2 ) = . G e V ( x , Q F Fit without cuts = 0.5 cut
Fit with A = 1.5 cut
Fit with AData
Figure 6:
The proton structure function F ( x, Q ) computed using NNPDF1.2 (left) or NNPDF2.0(right) PDFs obtained from fits with different values of A cut . is inspired by saturation and resummation studies. We have performed fits with only datawhich pass the cut Eq. (2) included, with a variety of choices for A cut , shown in Fig. 2.Results depend smoothly on A cut .As a first test, we have computed the proton structure function F and compared itwith data (see Fig. 6) at Q = 3 . , where a significant x range falls below the cut(compare with Fig. 2). Clearly, the prediction obtained from backward evolution of thedata above the cut exhibits a systematic downward trend. This deviation, which becomesmore and more apparent as A cut is raised, is visible but marginal when the NNPDF1.2set based on old HERA data is used, but it becomes rather more significant when usingNNPDF2.0 and new HERA data. Interestingly, with old HERA data the uncut fit agreeswell with the data, showing that whatever the possible deviation between data and theory,it is absorbed by the PDFs. This is no longer possible when the more precise combinedHERA data are used: in such case, even when no cut is applied, the theory cannotreproduce the data fully. This suggests that low- x and Q NLO DGLAP evolution isstronger than the scale dependence seen in the data.In order to quantify this observed deviation from NLO DGLAP, we introduce thestatistical distance d stat ( x, Q ) ≡ F data − F fit q σ + σ , (3)i.e. the difference between the observable F data and the NLO DGLAP prediction F fit inunit of their combined uncertainties σ fit , σ data . Note that d stat ∼ x HERA data andthe NNPDF1.2 and NNPDF2.0 fits without cuts (i.e. with A cut = 0) and with the cut A cut = 1 .
5. Again, we see that with NNPDF1.2 if all data are included the fit managesto compensate for the deviation by readjusting the PDFs: the fit lies both above andbelow the data and the mean distances is compatible with zero: from the points plottedin Fig. 7 we obtain h d stat i = 0 . ± .
56. On the other hand using NNPDF2.0, based onthe combined HERA-I data, we get h d stat i = 1 . ± .
7, which shows a systematic tension8 -3 -2 Q D i s t a n ce data > F NLO F data < F NLO F x -3 -2 Q D i s t a n ce data > F NLO F data < F NLO F x -3 -2 Q D i s t a n ce data > F NLO F data < F NLO F x -3 -2 Q D i s t a n ce data > F NLO F data < F NLO F Figure 7:
Statistical distance, Eq. (3), between small- x HERA data and NLO DGLAP predictionfor fits without kinematical cuts (top row) and fits with the cut at A cut = 1 . at the one- σ level between data and theory. When the cut is applied, the discrepancy isapparent: using NNPDF1.2 h d stat i = 0 . ± .
45, while with NNPDF2.0 h d stat i = 2 . ± . σ level.It is interesting to note that the significance of the effect is considerably weakened ifone instead of performing the cut Eq. (2) were to simply cut out the small- x region atall Q . For example, if we consider the region x ≤ .
01 we obtain (from HERA-I dataonly) h d stat i = 1 . ± . A cut = 0) and h d stat i = 1 . ± . A cut = 1 . Q thatdeviations appear, as one would expect of an effect driven by perturbative evolution. Arecent study [36] did find that in the low- x region the distance fluctuations are larger thanexpected, consistent with our conclusions, but no significant deviation of h d i from zerowas found in this less sensitive x ≤ .
01 region for an uncut fit, also consistent with ourconclusion.Evidence for a systematic deviation between data and theory is also provided by study-ing the behaviour of the (cid:10) d stat (cid:11) in different kinematic slices, both without cuts and with A cut = 1 .
5. The results, displayed in Fig. 8, show that data and theory increasingly deviateas one moves towards the small- x , small- Q region. This deviation is already present whenall data are fitted, but it becomes significantly stronger when the cut is applied. However,the discrepancy is concentrated in the region which is affected by the cuts. Indeed, inTab. 2 we compare the χ of various datasets for the cut and uncut fits: the quality of thefit to high–scale hadronic data, unaffected by the cuts, is the same in the two fits.Having strengthened our previous [35] conclusion that there is evidence for deviations9 < d = 1.5 cut Fit with AFit without cuts < 1.0 cut
A < 1.5 cut cut cut cut A Figure 8: (cid:10) d stat (cid:11) computed in the different kinematic slices of Fig. 2: from the NNPDF2.0 fitwithout kinematic cuts (yellow, lower curve) and with the A cut = 1 . Fit All dataset Only fitted points χ The χ of the individual experiments included in NNPDF2.0 for the A cut = 1 . χ computed on all NNPDF2.0 dataset, while Col. 3 the χ computed only on data whichpass the cut. x and Q HERA data one may ask what are possibletheoretical explanations for the observed effect. Because NLO DGLAP overestimates theamount of evolution required to reproduce experimental data, NNLO corrections as apossible explanation are ruled out, as they would lead to yet stronger evolution in thisregion thus making the discrepancy larger. This conclusion was recently confirmed bythe HERAPDF group, which finds that the description of small- x and Q HERA-I dataworsens when NNLO corrections are included [37]. Charm mass effects, not included inNNPDF2.0, could be partly responsible, but they seem [35] too small to account for thedata. This conclusion is borne out by preliminary studies based on the NNPDF2.1 set [18]which does include charm mass effects using the FONLL [38, 39] framework, and whichconfirm the conclusions of the present study.Interestingly, the small- x resummation corrections shown in Fig. 1 and discussed inSect. 2 go in the right direction, and appear to be roughly of the size which is needed toexplain the data. A quantitative confirmation that this is actually the case could comefrom a fully resummed PDF fit, which is doable using current knowledge and it wouldonly require implementation of the resummation in a PDF fitting code. An alternativeinteresting possibility is that the observed slow-down of perturbative evolution may bedue to saturation effects related to parton recombination. However, it is more difficult tosingle out a clear signature for these effects, given that saturation models usually yieldpredictions for the x dependence of structure functions, rather than their scale dependencewhich is relevant in this context.Finally, one may ask whether these deviations, if real, might bias LHC phenomenology.A first observation is that these deviations might explain the well–known fact [1] that the α s value obtained from deep–inelastic scattering tends to be lower that the global average:if the observed evolution is weaker than the predicted one, the value of the coupling isbiased downwards: the value of α s from a fully resummed fit would be higher. A moredirect impact on HERA phenomenology can be assessed by comparing predictions for LHCstandard candles obtained from cut and uncut fits: their difference provides a conservativeupper bound for the phenomenological impact of these deviations. In Table 3 we showresults for W , Z , Higgs and t ¯ t inclusive production at the LHC at 7 TeV center of massenergies, computed with the MCFM code [40]. Even with the largest kinematical cuts, A cut = 1 .
5, the corrections are moderate, below the 1–sigma level (except for t ¯ t ), ofsimilar size of other comparable effects at the precision level, such as α s uncertainties [41]or variations of the charm mass.The impact of the effect we discovered is moderate at present but it might becomesignificant as the accuracy of PDF determination improves. It will be interesting to seewhether further confirmation of the effect comes from other groups. Its full understandingmight lead to a deeper grasp of perturbative QCD.The NNPDF2.0 PDFs (sets of N rep = 100 and 1000 replicas) and the PDF sets based onNNPDF2.0 with various A cut kinematical cuts are available at the NNPDF web site,11bservable NNPDF2.0 without cuts NNPDF2.0 with A cut = 1 . σ ( W + )B l + ν [nb] 5.80 ± ± σ ( W − )B l − ¯ ν [nb] 3.97 ± ± σ ( Z )B l + l − [nb] 2.97 ± ± σ ( t ¯ t ) [pb] 169 ± ± σ ( H, m H = 120 GeV) [pb] 11.60 ± ± LHC observables at 7 TeV computed from the default NNPDF2.0 set and with the fitwith kinematical cut A cut = 1 . http://sophia.ecm.ub.es/nnpdf . NNPDF2.0 is also available through the LHAPDF interface [42].