Progress on neural parton distributions
NNPDF Collaboration, J. Rojo, R. D. Ball, L. Del Debbio, S. Forte, A. Guffanti, J. I. Latorre, A. Piccione, M. Ubiali
aa r X i v : . [ h e p - ph ] J un Progress on Neural Parton Distributions
J. Rojo , R. D. Ball , L. Del Debbio , S. Forte , A. Guffanti ,J. I. Latorre , A. Piccione and M. Ubiali (The NNPDF Collaboration)1.- LPTHE, CNRS UMR 7589, Universit´es Paris VI-Paris VII, F-75252, Paris Cedex 05, France2.- School of Physics, University of Edinburgh, Edinburgh EH9 3JZ, Scotland3.- Dipartimento di Fisica, Universit`a di Milano,and INFN, Sezione di Milano, Via Celoria 16, I-20133 Milano, Italy4.- Departament d’Estructura i Constituents de la Mat`eria,Universitat de Barcelona, Diagonal 647, E-08028 Barcelona, Spain5.- INFN, Sezione di Genova, via Dodecaneso 33, I-16146 Genova, ItalyWe give a status report on the determination of a set of parton distributions based onneural networks. In particular, we summarize the determination of the nonsinglet quarkdistribution up to NNLO, we compare it with results obtained using other approaches,and we discuss its use for a determination of α s . The LHC will require an approach to the search for new physics based on the precisiontechniques which are customary at lepton machines [1, 2]. This has recently led to significantprogress in the determination of parton distribution functions (PDFs) of the nucleon. Themain recent development has been the availability of sets of PDFs with an estimate of theassociated uncertainty [3, 4, 5]. However, the standard approach to the determination ofthe uncertainty on parton distributions has several weaknesses, such as the lack of controlon the bias due choice of a parametrization and, more in general, the difficulty in giving aconsistent statistical interpretation to the quoted uncertainties.These problems have stimulated various new approaches to the determination of PDFs [7],in particular the neural network approach, first proposed in Ref. [6]. The basic idea is tocombine a Monte Carlo sampling of the probability measure on the space of functions thatone is trying to determine [7] with the use of neural networks as universal unbiased interpo-lating functions. In Refs. [6, 8] this strategy was successfully applied to a somewhat simplerproblem, namely, the construction of a parametrization of existing data on the DIS struc-ture function F ( x, Q ) of the proton and neutron. The method was proven to be fast androbust, to be amenable to detailed statistical studies, and to be in many respects superiorto conventional parametrizations of structure functions based on a fixed functional form.The determination of a parton set involves the significant complication of having to gofrom one or more physical observables to a set of parton distributions. Recently [9] most ofthe technical complications required for the construction of a neural parton set have beentackled and solved in the process of constructing a determination of the quark isotripletparton distribution. This work will be reviewed here. Also, based on this work, we willpresent preliminary results on the determination of α s and a determination of the variationin χ which corresponds to a one-sigma variation of the underlying parton distributions.Work to apply the techniques of [9] to the singlet sector is at an advanced stage [10]. DIS 2007
Determination of the nonsinglet quark distribution
The first application of the neural network approach to parton distributions, a determinationof the NS parton distribution q NS ( x, Q ) = ( u + ¯ u − d + ¯ d )( x, Q ) from the DIS structurefunction data of the NMC and BCDMS collaborations, was presented in Ref. [9]. Resultsfor this PDF were obtained at LO, NLO, NNLO for different values of α s ( M Z ).In Ref. [9] we have implemented a new fast and efficient method for solving the evolutionequations up to NNLO. This method combines the advantages of x − space and N − spaceevolution codes: an x dependent Green function (evolution factor) is determined by inverseMellin transformation of the exact N -space expression and stored. Evolution of PDFs isthen performed by convoluting this Green function with any given boundary condition. Theaccuracy of this method has been benchmarked up to NNLO with the help of the tables ofRefs. [1, 2].Also, we have implemented a criterion to determine the convergence of the fitting proce-dure in a way which is free of bias related to the choice of parametrization. To this purpose,the dataset is randomly divided into two sets, of which only one is used in the fit. Con-vergence is achieved when the quality of the fit to data which are not used for minimiztionstops improving.An important feature of our approach is that it is possible to check quantitatively thestatistical features of results using suitable estimators. For example, one can check that theresults do not depend on choices made during the fitting procedure, such as the choice ofarchitecture of neural networks, which is analogous to the choice of parton parametrizationin conventional fits. Namely, we repeat the fit with a different choice, and compute thedistance d [ q ] = vuuut* (cid:16) q (1) i − q (2) i (cid:17) ( σ (1) i ) + ( σ (2) i ) + dat , (1)where q (1) i , q (2) i are the predictions for the i -th data point in the two fits, and σ (1) i , σ (2) i the predictions for the corresponding statistical uncertainties, and the average is performedover all data. The results of the first and second fit are the same if d [ q ] = 1 on average.This also checks that the statistical uncertainties are correctly estimated. One can similarlycheck stability of the uncertainty estimate. In Ref. [9] this comparison has been performedsuccesfully.In Fig. 1 we compare our results for the NS structure function F NS2 to other publisheddeterminations. These results are available through the webpage of the NNPDF Collabo-ration: http://sophia.ecm.ub.es/nnpdf . The large uncertainty that we find is a genuinefeature of the determination of the nonsinglet quark distribution from the data includedin our fit, and, especially at small x , it appears to reflect the current knowledge of thenonsinglet quark distribution. Indeed, for x ≤ .
05 the only data which constrain the q NS combination in global fits are the data used in the determination of Ref. [9]. Hence, ourresults suggest that standard fits might be underestimating PDF uncertainties.In recent work on PDF uncertainties [4, 5] it has been suggested that, mostly because ofinconsistencies between data, the variation of the total χ which corresponds to a one–sigmavariation of the underlying PDFs is of order of ∆ χ ∼
50 for the global fits presented inthose references instead of ∆ χ = 1 of a statistically consistent fit [3]. In our approach, thisquantity can be computed. We get ∆ χ ≈ . DIS 2007 -2 -1
10 1 ) ( x , Q N S F -0.0100.010.020.030.040.050.060.07 < 17 GeV < Q
13 GeVNNPDFCTEQMRSTAlekhinData
Figure 1:The nonsinglet structure function F NS2 as determined by the NNPDF collaboration [9]from 229 NMC and 254 BCDMS data points, compared to data and various otherdeterminations.and BCDMS data are mostly consistent, though some inconsistent data are present [6, 8].An extensive discussion of the way the published [9] and forthcoming [10] fits based on theneural network approach can be used for the determination of physical parameters (such as α s ) and statistical properties of the data (such as ∆ χ ) will be presented in a forthcomingpublication.In [9] the strong coupling α s ( M Z ) was fixed, but we could also extract it from the fit. Theresults of a preliminary analysis, shown in Fig. 2, suggest that nonsinglet data determine α s ( M Z ) with an uncertainty which is rather larger than that (∆ α s ( M Z ) ∼ . The extension of the results described in Ref. [9] to a full global PDF fit has benefited fromthe increased manpower of the NNPDF Collaboration, and is at a rather advanced stage [10].In particular, the evolution formalism of Ref. [9] has been extended to the computation ofa full set of neutral-current and charged-current structure functions and fully benchmarked.A first full neural parton fit is in is in preparation. It will at first be based on DIS data only,including all available F p and F d fixed target data and the full NC and CC HERA reducedcross sections. DIS 2007 (M s a c PRELIMINARY RESULTS
Figure 2:The χ profile for a preliminary NNLO determination of α s ( M Z ) from NS data. Thenumber of data points included in the fit is N dat = 483. References [1] W. Giele et al. The QCD/SM working group: Summary report. hep-ph/0204316. [2] M. Dittmar et al. Parton distributions: Summary report. hep-ph/0511119. [3] Sergey Alekhin. Parton distributions from deep-inelastic scattering data.
Phys. Rev. , D68:014002, 2003.[4] J. Pumplin et al. New generation of parton distributions with uncertainties from global QCD analysis.
JHEP , 07:012, 2002.[5] A. D. Martin, R. G. Roberts, W. J. Stirling, and R. S. Thorne. Uncertainties of predictions from partondistributions. I: Experimental errors.
Eur. Phys. J. , C28:455–473, 2003.[6] Stefano Forte, Lluis Garrido, Jose I. Latorre, and Andrea Piccione. Neural network parametrization ofdeep-inelastic structure functions.
JHEP , 05:062, 2002.[7] W. T. Giele, S. A. Keller and D. A. Kosower. Parton distribution function uncertainties. hep-ph/0104052. [8] Luigi Del Debbio, Stefano Forte, Jose I. Latorre, Andrea Piccione, and Joan Rojo. Unbiased deter-mination of the proton structure function F p with faithful uncertainty estimation. JHEP , 03:080,2005.[9] Luigi Del Debbio, Stefano Forte, Jose I. Latorre, Andrea Piccione, and Juan Rojo. Neural networkdetermination of parton distributions: The nonsinglet case.
JHEP , 03:039, 2007.[10] Richard D. Ball, Luigi Del Debbio, Stefano Forte, Alberto Guffanti, Jose I. Latorre, Andrea Piccione,Juan Rojo, and Maria Ubiali. Neural network determination of parton distributions: The singlet case.
In preparation , 2007.[11] J. Blumlein, H. Bottcher and A. Guffanti. Non-singlet QCD analysis of deep inelastic world data atO(alpha(s)**3). Nucl. Phys. B (2007) 182[12] Stefano Forte, Jose I. Latorre, Lorenzo Magnea, and Andrea Piccione. Determination of α s from scalingviolations of truncated moments of structure functions. Nucl. Phys. , B643:477–500, 2002., B643:477–500, 2002.