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The neural network approach to parton distributions

We introduce the neural network approach to global fits of parton distrubution functions. First we review previous work on unbiased parametrizations of deep-inelastic structure functions with faithful estimation of their uncertainties, and then we summarize the current status of neural network parton distribution fits.

We introduce the neural network approach to global fits of parton distribution functions. First we review previous work on unbiased parametrizations of deep-inelastic structure functions with faithful estimation of their uncertainties, and then we summarize the current status of neural network parton distribution fits.
The requirements of precision physics at hadron colliders, as has been emphasized through this workshop, have recently led to a rapid improvement in the techniques for the determination of parton distribution functions (pdfs) of the nucleon. Specifically it is now mandatory to determine accurately the uncertainty on these quantities, and the different collaborations performing global pdf analysis [1] have performed estimations of these uncertainties using a variety of techniques. The main difficulty is that one is trying to determine the uncertainty on a function, that is, a probability measure in a space of functions, and to extract it from a finite set of experimental data, a problem which is mathematically ill-posed. It is also known that the standard approach to global parton fits have several shortcomings: the bias introduced by choosing fixed functional forms to parametrize the parton distributions (also known as model dependence), the problems to assess faithfully the pdf uncertainties, the combination of inconsistent experiments, and the lack of general, process-independent error propagation techniques. Although the problem of quantifying the uncertainties in pdfs has seen a huge progress since its paramount importance was raised some years ago, until now no unambiguous conclusions have been obtained.
In this contribution we present a novel strategy to address the problem of constructing unbiased parametrizations of parton distributions with a faithful estimation of their uncertainties, based on a combination of two techniques: Monte Carlo methods and neural networks. This strategy, introduced in [2,3], has been first implemented to address the marginally simpler problem of parametrizing deep-inelastic structure functions F (x, Q 2 ), which we briefly summarize now. In a first step we construct a Monte Carlo sampling of the experimental data (generating artificial data replicas), and then we train neural networks to each data replica, to construct a probability measure in the space of structure functions P F (x, Q 2 ) . The probability measure constructed in this way contains all information from experimental data, including correlations, with the only assumption of smoothness. Expectation values and moments over this probability measure are then evaluated as averages over the trained network sample, where The first step is the Monte Carlo sampling of experimental data, generating N rep replicas of the original N dat experimental data, where r are gaussian random numbers with the same correlation as the respective uncertainties, and σ stat , σ sys , σ N are the statistical, systematic and normalization errors. The number of replicas N rep has to be large enough so that the replica sample reproduces central values, errors and correlations of the experimental data.
The second step consists on training a neural network 1 on each of the data replicas. Neural networks are specially suitable to parametrize parton distributions since they are unbiased, robust approximants and interpolate between data points with the only assumption of smoothness. The neural network training consist on the minimization for each replica of the χ 2 defined with the inverse of the experimental covariance matrix, Our minimization strategy is based on Genetic Algorithms (introduced in [4]), which are specially suited for finding global minima in highly nonlinear minimization problems.
The set of trained nets, once is validated through suitable statistical estimators, becomes the sought-for probability measure P F (x, Q 2 ) in the space of structure functions. Now observables with errors and correlations can be computed from averages over this probability measure, using eq. (1). For example, the average and error of a structure function F (x, Q 2 ) at arbitrary (x, Q 2 ) can be computed as A more detailed account of the application of the neural network approach to structure functions can be found in [3], which describes the most recent NNPDF parametrization of the proton structure function 2 .
Hence this strategy can be used also to parametrize parton distributions, provided one now takes into account perturbative QCD evolution. Therefore we need to define a suitable evolution formalism. Since complex neural networks are not allowed, we must use the convolution theorem to evolve parton distributions in x−space using the inverse Γ(x) of the Mellin space evolution factor Γ(N ), defined as The only subtlety is that the x-space evolution factor Γ(x) is a distribution, which must therefore be regulated at x = 1, yielding the final evolution equation, where in the above equation q(x, Q 2 0 ) is parametrized using a neural network. At higher orders in perturbation theory coefficient functions C(N ) are introduced through a modified evolution factor,Γ(N ) ≡ Γ(N )C(N ). We have benchmarked our evolution code with the Les Houches benchmark tables [5] at NNLO up to an accuracy of 10 −5 . The evolution factor Γ(x) and its integral are computed and interpolated before the neural network training in order to have a faster fitting procedure.
As a first application of our method, we have extracted the nonsinglet parton distribution q N S (x, as measured by the NMC [6] and BCDMS [7,8] collaborations. The preliminary results of a NLO fit with fully correlated uncertainties [9] can be seen in fig. 1 compared to other pdfs sets. Our preliminary results appear to point in the direction that the uncertainties at small x do not allow, provided the current experimental data, to determine if q N S (x, Q 2 ) grows at small x, as supported by different theoretical arguments as well as by other global parton fits. However, more work is still needed to confirm these results. Only additional nonsinglet structure function data at small x could settle in a definitive way this issue 3 . Summarizing, we have described a general technique to parametrize experimental data in an biasfree way with a faithful estimation of their uncertainties, which has been successfully applied to structure functions and that now is being implemented in the context of parton distribution. The next step will be to construct a full set of parton distributions from all available hard-scattering data using the strategy described in this contribution.