We discuss the latest update of the MRST parton distributions in response
to the most recent data. We discuss the areas where there are hints
of difficulties in the global fit, and compare to some other updated sets of
parton distributions, particularly CTEQ6. We briefly discuss the issue of
uncertainties associated with partons

Martin, AD

Roberts, RG

Stirling, W James

Thorne, Robert S

English

arXiv

We discuss the latest update of the MRST parton distributions in response to the most recent data. We discuss the areas where there are hints of difficulties in the global fit, and compare to some other updated sets of parton distributions, particularly CTEQ6. We briefly discuss the issue of uncertainties associated with partons.We discuss the latest update of the MRST parton distributions in response to the most recent data. We discuss the areas where there are hints of difficulties in the global fit, and compare to some other updated sets of parton distributions, particularly CTEQ6. We briefly discuss the issue of uncertainties associated with partons

Thorne, R.S.

Martin, A.D.

Stirling, W.J.

Roberts, R.G.

CERN Document Server

Update of MRST Parton Distributions

Apollo (Cambridge)

arXiv:hep-ph/0207067v1  4 Jul 2002 Update of MRST Parton Distributions.R.S. ThorneCavendish Laboratory, University of Cambridge†, Madingley Road,Cambridge, CB3 0HE, U.K.A.D. Martin and W.J. StirlingDepartment of Physics, University of Durham, Durham, DH1 3LE, U.K.R.G. RobertsTheory Division, CERN, 1211 Geneva 23, SwitzerlandWe discuss the latest update of the MRST parton distributions in re-sponse to the most recent data. We discuss the areas where there are hintsof difficulties in the global fit, and compare to some other updated sets ofparton distributions, particularly CTEQ6. We briefly discuss the issue ofuncertainties associated with partons.Over the past couple of years there has been a large amount of updateddata both from HERA [1, 2], on small x structure functions, and from theTevatron, on high-ET jet production [3, 4], which has been more accuratethan previous data, and expanded the phase space significantly. This hadled to a number of updated sets of parton distributions [5]-[9]. In this talkwe discuss the most recent updates to the MRST set of parton distributions,highlighting the successes and failings, and also compare to other new setsof distributions.The updated MRST partons were released in 2001 [5]. Compared to pre-vious sets the main improvement was in the accuracy of the determinationof the gluon distribution, which was constrained far more strongly at highx due to the new Tevatron jet data. αS(M2Z) was left as a free parameter inthe fit, and found to be 0.119±0.002(exp)±0.003(theory), where the exper-imental error was determined by letting the χ2 for the global fit increase by20 from the minimum (see [10] for a discussion of the suitable increment in† Royal Society University Research Fellow(1)2χ2 to determine the error in a global fit). The fit was of good quality overall,but struggled a little in some regions. It was hard to provide enough highx gluon to fit the jet data very well, and also to have sufficient moderatex gluon to obtain a large enough value of dF2(x,Q2)/d ln Q2 for x ∼ 0.01.Conversely, the data required the very small x gluon to be small (whichalso helps the previous shortcomings due to the constraint on the totalgluon from the momentum sum rule), and at our input scale Q20= 1GeV2 itwas found to be necessary to expand our parameterization to allow the verysmall x gluon to become negative. This latter point led to a dangerouslysmall prediction for FL(x,Q2) at small x and Q2.MRST 2001 and D0 jet data, αS(MZ)=0.119 , χ2= 106/82 pts00.510 50 100 150 200 250 300 350 400 4500.0 < | η  | < 0.500.510 50 100 150 200 250 300 350 400 4500.5 < | η | < 1.000.510 50 100 150 200 250 300 350 400 4501.0 < | η | < 1.5(Data - Theory) / Theory-0.500.510 50 100 150 200 250 300 350 400 4501.5 < | η | < 2.0ET (GeV)Fig. 1. Comparison of MRST fit to D0 jet data to CTEQ6 fit.Soon afterwards the CTEQ6 set of partons was published [6]. In mostways these are very similar to the MRST01 partons, and produce similarresults. However, there are a number of significant differences, particularlyconcerning the gluon. CTEQ have developed a different type of parameter-ization for the partons, which allows for a different shape at very high x.Whereas MRST were only able to get a completely satisfactory fit to theTevatron jet data if the input gluon is allowed to have a definite kink atx ∼ 0.5 (and with αS(M2Z) = 0.121), CTEQ obtain a very good fit with nosuch modifications.However, this problem of obtaining a very good fit to the jet data de-pends on many issues. CTEQ do indeed obtain a much better fit using3this new parameterization for the gluon (with same NLO prescription theχ2 quality is about 50 better) as seen for D0 data in Fig. 1. However,there are many differences in their approach compared to MRST other thanthe parameterization: CTEQ cut data above Q2 = 4GeV2, compared toQ2 = 2GeV2; they do not use some data sets used in [5], i.e. SLAC andone H1 high-Q2 set; they use (10%) systematic errors (in quadrature) forDrell-Yan data whereas in [5] only statistical errors are used. Addition-ally CTEQ have a positive-definite small x gluon at their starting scaleof Q20= 1.69GeV2, they use a massless charm prescription and there arevarious other minor differences.In order to investigate which initial choices are most important for thequality of the fit to the jet data, or equivalently, which affect the extractedform of the high-x gluon, we performed various fits changing these choices.We found that we can improve the fits to jets within the global fit by variousmodifications. Unexpectedly, allowing one of the parameters controllingthe negative contribution to our gluon at very small x to vary away froma previously fixed value resulted in ∆χ2J∼ −5. The fit to the Drell-Yandata actually competes with that to the jets, and using only statistical errors(the systematic errors being defined a little vaguely) presumably overweightsthese. Adding 5% systematic errors in quadrature to the statistical errors(which is probably the best approach [11]) leads to ∆χ2J∼ −10. Boththese modifications should be performed, and will be implemented in futureMRST fits. The resulting partons are currently denoted MRST⋆. The onlyreal change compared to MRST01 is for the high x gluon.We also discovered that further changes could improve the quality ofthe jet fit. Changing the Q2-cut on the data from the MRST value ofQ2 = 2GeV2 to the CTEQ value of Q2 = 4GeV2 leads to ∆χ2J∼ −10.Fitting to the same data as CTEQ, i.e. omitting the SLAC data and oneH1 high-Q2 data set and increasing the Drell-Yan systematic errors to 10%leads to ∆χ2J∼ −15. The cumulative effect of all these above steps in asingle fit is ∆χ2J∼ −40, which is obtained with a smooth high-x gluon. Wedenote the resulting partons by MRSTCTQ. We conclude that the remain-ing improvement of ∆χ2J∼ −10 seen by CTEQ is due mainly to their newparameterization, but that this is only a relatively minor effect. Indeed, wecompare the gluons from CTEQ6, MRST01, MRST⋆ and MRSTCTQ inFig. 2. Clearly MRSTCTQ has a very similar high-x gluon to CTEQ6, andeven MRST⋆ has gone much of the way in the same direction. However,all the MRST gluons are different from the CTEQ6 gluon at smaller x dueto their freedom to have a negative input distribution. We also note thatalthough we feel the steps producing the MRST⋆ partons should be madein future, the further ones leading to MRSTCTQ are a different matter.Although they improve the quality of the jet fit they are not the best fit40.80.911.11.21.310-410-310-210-1xRatio (CTEQ6,MRSTCTQ,MRST*)/MRST2001for gluon at Q2 = 10 GeV2CTEQ6MRSTCTQMRST*0.80.911.11.21.310-410-310-210-1xRatio (CTEQ6,MRSTCTQ,MRST*)/MRST2001for gluon at Q2 = 104 GeV2Fig. 2. Comparison of MRST2001 gluon distribution to the other distributionsdescribed in the text.when including the data sets omitted by CTEQ and the fit is not good at allfor data with Q2 < 4GeV2. It is certainly true that we should question thenature of our cut on Q2 (as well as on W 2 and x), but this is a complicatedquestion which will be addressed elsewhere [12].The comparison to the other sets of parton distributions obtained by fitsto mainly structure function data, e.g. [7]-[9], are qualitatively the same asthey have been for some time. Since only MRST and CTEQ fit the jet data,it is only these partons which have a direct constraint on the high-x gluon.All other fits always obtain, to varying degrees, a smaller high-x gluon whichconsequently allows both a larger moderate-x gluon to fit the HERA dataand a usually a slightly smaller value of αS(M2Z). Hence, the omission ofthe jet data tends to mask slightly the possible problems encountered intrying to fit the HERA data very well.Recently, many groups have not only obtained partons from a best fitbut, using various methods, have also examined the uncertainty on thesepartons due to experimental errors. MRST have concentrated on the La-grange Multiplier technique [13] in order to obtain uncertainties on physicalquantities and the corresponding extreme sets of partons. Such uncertain-5ties, and partons, are available for W and Higgs production at the Tevatronand LHC, and for charged current cross-sections at x = 0.5 for HERA [14],both for fixed and varying αS . However, we have always believed that the-ory is one of the dominant sources of error. Hence, as well as attemptingto determine the areas where the current theory may require correctionsby investigating the cuts on data [12], we have also produced approximateNNLO parton distributions and predictions [15] (based on the approximatesplitting functions [16] obtained from the known NNLO moments [17]). In-deed, we find, for example, that the NNLO W cross-section at the Tevatronis 4% higher than at NLO, and believe this result is reliable. This change isat least as large as the uncertainty due to experimental errors, and W pro-duction is likely to be subject to smaller theoretical uncertainty than manyother quantities - particularly those directly related to the gluon. Hence, anunderstanding of theoretical uncertainties seems to be a priority at present.REFERENCES[1] H1 Collaboration: C. Adloff et al., Eur. Phys. J. C13 (2000) 609;H1 Collaboration: C. Adloff et al., Eur. Phys. J. C19 (2001) 269;H1 Collaboration: C. Adloff et al., Eur. Phys. J. C21 (2001) 33.[2] ZEUS Collaboration: S. Chekanov et al., Eur. Phys. J. C21 (2001) 443.[3] D0 Collaboration: B. Abbott et al., Phys. Rev. Lett. 86 (2001) 1707.[4] CDF Collaboration: T. Affolder et al., Phys. Rev. D64 (2001) 032001.[5] A.D. Martin, R.G. Roberts, W.J. Stirling and R.S. Thorne, Eur. Phys. J. C23(2002) 73.[6] CTEQ Collaboration: J. Pumplin et al., hep-ph/0201195.[7] H1 Collab: C. Adloff et al., Eur. Phys. J. C21 (2001) 33.[8] S. I. Alekhin, Phys. Rev. D63 (2001) 094022.[9] ZEUS Collab: S. Chekanov et al., in preparation.[10] R.S. Thorne, et al., to be published in “Conference on Advanced StatisticalTechniques in Particle Physics”, March 2002, Durham, hep-ph/0205233.[11] E605 Collaboration: G. Moreno et al., Phys. Rev. D43 (1991) 2815.[12] A.D. Martin, R.G. Roberts, W.J. Stirling and R.S. Thorne, in preparation.[13] D. Stump et al., Phys. Rev. D65 (2002) 014012.[14] A.D. Martin, R.G. Roberts, W.J. Stirling and R.S. Thorne, in preparation.[15] A.D. Martin, R.G. Roberts, W.J. Stirling and R.S. Thorne, Phys. Lett. B531(2002) 216.[16] W.L. van Neerven and A. Vogt, Phys. Lett. B490 (2000) 111.[17] S.A. Larin, P. Nogueira, T. van Ritbergen and J.A.M. Vermaseren, Nucl. Phys.B492 (1997) 338;A. Rétey and J.A.M. Vermaseren, Nucl. Phys. B604 (2001) 281.

609; H1 Collaboration: C. Adloﬀ et al.,

Chekanov et al., in preparation.

to be published

Update of MRST parton distributions.

I present a brief summary of the most important updates in the MRST determinations
of the NLO and NNLO parton distributions. I concentrate on uncertainties
due to input assumptions and possible theoretical corrections, particularly isospin
violation, QED corrections, NNLO and corrections at low Q2 and small x

arXiv:hep-ph/0407311v1  27 Jul 2004Proceedings of the DIS’2004, Štrbské Pleso, SlovakiaUPDATE OF MRST PARTON DISTRIBUTIONSR.S. THORNECavendish Laboratory, University of Cambridge∗, Madingley Road,Cambridge, CB3 0HE, U.K.A.D. MARTIN AND W.J. STIRLINGDepartment of Physics, University of Durham, Durham, DH1 3LE, U.K.R.G. ROBERTSRutherford Appleton Laboratory, Chilton, Didcot, Oxon, OX11 0QX, U.K.I present a brief summary of the most important updates in the MRST determina-tions of the NLO and NNLO parton distributions. I concentrate on uncertaintiesdue to input assumptions and possible theoretical corrections, particularly isospinviolation, QED corrections, NNLO and corrections at low Q2 and small x.1 IntroductionA knowledge of the partonic structure of the proton is an essential ingredient in theanalysis of hard scattering data from pp or p̄p or ep high energy collisions. Much at-tention has recently been devoted to obtaining reliable uncertainties on the partondistributions obtained from global or semi-global fits. One obvious uncertainty isdue to the systematic and statistical errors of the data, i.e the experimental errors.They have been estimated by several groups [1]–[10], working within a NLO (andsometimes NNLO) framework using a variety of different procedures. The generalconclusion is that in all approaches the uncertainties are of the order of 2 − 5%,except for certain partons and regions of phase space, where the uncertainty canbe bigger, particularly the high-x gluons and very high-x down quarks. However,when the partons and predictions for physical quantities using different parton setsare compared, the deviations can be significantly greater than the quoted errors.A recent summary can be found in [11]. This unfortunate discrepancy is likely tobe due to different assumptions made by different groups when performing theirfitting procedures, such as different data sets used, parameterizations for input sets,choices of heavy flavour prescriptions, ordering of perturbative series, etc. However,the different results for partons obtained when making different assumptions alsosuggest that the standard fixed order in the perturbative expansion is not com-pletely adequate to describe the data, with different individual data sets makingdifferent and conflicting demands on the partons and on αS . Indeed, MRST do seecertain difficulties in obtaining the best possible fit to the data with the currenttheoretical treatment – the HERA and NMC data in the range x ∼ 0.01 − 0.001are too steep in Q2; the gluon is negative (or at best valencelike) at small x and∗Royal Society University Research Fellowc© Institute of Experimental Physics SAS, Košice, Slovakia 12 R.S. Thorne et al.Q2; and the Tevatron jet data prefer a different shape gluon from the DIS data.Hence, in order to understand the uncertainties in parton distributions, it is vital toconsider theoretical corrections. These include the possibility of isospin violation;s(x) 6= s̄(x); higher orders (NNLO); QED (comparable to NNLO ? i.e. α3s ∼ α);large x (αns ln2n−1(1 − x)); low Q2 (higher twist); small x (αns lnn−1(1/x)). Theuncertainties associated with many of these possible sources have recently beenconsidered [12]. Here I will highlight some of the main conclusions and some addi-tional results which postdate this analysis.One of the issues we have examined is the possibility of isospin violation. Thisis particularly relevant because of the “NuTeV anomaly” [13]. NuTeV measure,with a 3 − σ discrepancy,R− =12− sin2 θW + (1 −73sin2 θW )[δUv] − [δDv]2[V −].where [δUv] = [Upv] − [Dnv] and [δDv] = [Dpv] − [Unv], [Upv] =∫10xupv(x) dx etc.. Weparameterize the isospin violation byupv(x) = dnv(x) + κf(x), dpv(x) = unv(x) − κf(x),where f(x) is a number conserving function with valence like behaviour at large andsmall x [12]. The variation of the χ2 for the global fit is shown in Fig. 1, and thebest fit leads to κ = −0.2 (which agrees very well with a prediction in [14]), and to areduction of the NuTeV anomaly to ∼ 1.5−σ, i.e. ∆sin2θW ∼ −0.002. However, awide variation in κ is possible, and the anomaly may be reduced further or enlarged.Other issues to which the anomaly is sensitive are the strangeness asymmetry in theproton recently investigated in [15] and electroweak corrections [16]. In each casethe uncertainties may be large as the anomaly, though there is some disagreementregarding the strangeness asymmetry.050100-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1∆χ2κValence quarksFigure 1. Variation of global χ2 with the value of κWe have also recently examined the effects due to QED. The corrections toDIS include those in Fig. 2 and lead to mass singularities when γ ‖ q. The QEDcollinear singularities are universal and can be absorbed into pdfs, exactly as forQCD collinear singularities, leaving finite (as mq → 0) O(α) QED corrections inUpdate of MRST Parton Distributions 3coefficient functions, and QED–improved DGLAP equations. The effect on thegluon distribution is negligible as is that on the quark distributions at small xwhere the gluon contribution dominates DGLAP evolution. Only at large x is theeffect noticeable (O(1%)) as first noted in [17], and more recently verified in [18],and it is equivalent to a slight shift in αS : ∆αS(M2Z) ≃ +0.0003. This is muchsmaller than many sources of uncertainty.Figure 2. Diagrams for QED corrections to DISHowever, these QED effects, although small, lead to isospin violation. upV (x)quarks radiate more photons than dnV (x) quarks. To a rough approximation, thephoton distribution should beγ(x, Q2) =∑je2jα2πln(Q2/m2q)∫1xdyyPγq(y) qj(xy, Q2).So there is more photon momentum in the proton than in the neutron due to high-xup quarks radiating more than high-x down quarks. Momentum conservation leadsto upV (x) < dnV (x) at high x. Hence, [δUv] < 0 as required by the NuTeV anomalyand as found by our best fit above. Our estimates imply that the QED effects leadto ∼ 3/4 of the isospin violation observed by our best fit. Details of this and partondistributions involving QED corrections will soon appear [19].Much more generally important are NNLO corrections. The coefficient func-tions for F2(x, Q2) are known at NNLO [20]. Singular limits x → 0 and large NFlimits are known for the NNLO splitting functions [21]–[23] as well as limited mo-ments [24], and approximate NNLO splitting functions were devised [25] and usedby MRST [26] (and Alekhin [3]) in NNLO fits. Compared to NLO they improvedthe quality of the fit very slightly, and reduced αS to ∼ 0.1155. The predictionslead to reasonable stability order by order for (quark-dominated) W and Z cross-sections. However, this fairly good convergence is largely guaranteed because thequarks are fit directly to data, and it is much worse for gluon-dominated quanti-ties e.g. FL(x, Q2) or Higgs production. Very recently, the full calculation of theNNLO splitting functions was completed [27]. We have confirmed that these alterthe parton distributions obtained from the previous best estimates by only a verysmall amount. Full NNLO partons will appear soon, but also require the completeincorporation of NNLO coefficient functions for as many processes as possible, anda rigorous treatment of heavy flavour thresholds at NNLO [28]. For predictions it4 R.S. Thorne et al.will be useful to have the exact NNLO (O(α3S)) longitudinal coefficient functions,and these, along with the O(α3S) (NNNLO) coefficient functions for F2(x, Q2) willsoon be complete. We have estimated the most likely value, along with uncertain-ties, from the limited information, as had previously been done for the splittingfunctions. The central value and the two “extremes” for the dominant contribu-tion, i.e. the part of C3Lg which is ∝ Nf are respectively (the common factor ofNf (αS/(4π))3 is taken out and L = ln(1/x), L1 = ln(1 − x))410Lx−1212x− 1685 − 54161x− 47131L1 − 11496L21− 867L31,410Lx−0x+ 20298− 106974x− 67527L1 − 16336L21− 1241L31− 14116L,410Lx−2057x+ 30895x3 + 6220L1 − 102L21− 66.6L31+ 1378L2.The failure of the NNLO corrections to solve completely the problems with thequality of the fit leads us to try an alternative approach. In order to investigatethe real quality of the fit and the regions with problems we vary kinematic cuts ondata, i.e. change W 2cut, Q2cut and xcut, re-fit and see if the quality of the fit to theremaining data improves and/or the input parameters change dramatically. Thisis continued until the quality of the fit and the partons stabilize. This is discussedin detail in [12]. Briefly, our W 2 cut of 12.5GeV2 seems reasonable, whereas the fitimproves gradually until Q2cut ∼ 10GeV2 and xcut ∼ 0.005.a This is true at bothNLO and NNLO, but the modification of the partons with changing cuts is muchgreater at NLO. For both NLO and NNLO we have defined “conservative” partons[12] which are obtained using only the data within the conservative cuts.In order to investigate the possible theoretical improvements which could modifythe partons at small x and low Q2 (particularly at high x) we have tried includingknown ln(1−x) resummations, saturation corrections, and purely phenomenologicalhigher twist and ln(1/x) corrections. At high x, higher orders in the coefficientfunctions and higher twist contributions can improve the fit, but are difficult todistinguish. At low x, there is no evidence for phenomenological higher twist.Saturation corrections [30] do not seem to help the NLOb or NNLO fits (theywould help only at LO). There is a distinct improvement in the fit quality when(momentum conserving) ln(1/x) corrections are added to the splitting functions,increasing the small-x evolution. Hence, there is some reason for uncertainty aboutthe correct approach, and hence the correct predictions at very small x. Indeed, acomparison of the LO, NLO, approximate NNLO and a resummed prediction [32]for FL(x, Q2) is shown in Fig. 3, exhibiting the level of uncertainty. Also shownare potential data which could be obtained by low energy running at HERA. It isclear that such data would play a huge role in enabling us to understand and beconfident of predictions involving small x partons.In summary, as well as the uncertainties on partons due to experimental errors,there are uncertainties on partons, and related physical quantities, due to assump-tions in the fit and having to work within an incomplete theoretical framework.aWe note that CTEQ do not seem to reproduce these conclusions [29].bA recent joint analysis with diffractive data suggests some slight improvement [31].Update of MRST Parton Distributions 5Some of the uncertainties are small, but in some cases, e.g. QED corrections ors(x) 6= s̄(x), they can be important for specific quantities. 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