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Unbiased determination of the proton structure function F_2^p with faithful uncertainty estimation

The NNPDF Collaboration, Luigi Del Debbio, Stefano Forte, Jose I. Latorre, Andrea Piccione, Joan Rojo

TL;DR

This paper delivers an unbiased, faithful uncertainty determination of the proton structure function $F_2^p(x,Q^2)$ by constructing a Monte Carlo ensemble of neural networks trained on replicas of all available DIS data, including HERA. It introduces a three-phase training pipeline (backpropagation on central values and diag errors, followed by a full covariant GA optimization) to handle large correlated systematics and derive a probability measure over structure functions. The resulting 1000-network ensemble yields a chi^2 per data point around 1.18 and provides one-sigma uncertainty bands that are smaller than many data uncertainties in the measured region, with a public FORTRAN routine to compute $F_2^p$ and its uncertainties. The methodology establishes a foundation for faithful uncertainty estimation in the subsequent determination of parton distributions.

Abstract

We construct a parametrization of the deep-inelastic structure function of the proton F_2 based on all available experimental information from charged lepton deep-inelastic scattering experiments. The parametrization effectively provides a bias-free determination of the probability measure in the space of structure functions, which retains information on experimental errors and correlations. The result is obtained in the form of a Monte Carlo sample of neural networks trained on an ensemble of replicas of the experimental data. We discuss in detail the techniques required for the construction of bias-free parameterizations of large amounts of structure function data, in view of future applications to the determination of parton distributions based on the same method.

Unbiased determination of the proton structure function F_2^p with faithful uncertainty estimation

TL;DR

This paper delivers an unbiased, faithful uncertainty determination of the proton structure function by constructing a Monte Carlo ensemble of neural networks trained on replicas of all available DIS data, including HERA. It introduces a three-phase training pipeline (backpropagation on central values and diag errors, followed by a full covariant GA optimization) to handle large correlated systematics and derive a probability measure over structure functions. The resulting 1000-network ensemble yields a chi^2 per data point around 1.18 and provides one-sigma uncertainty bands that are smaller than many data uncertainties in the measured region, with a public FORTRAN routine to compute and its uncertainties. The methodology establishes a foundation for faithful uncertainty estimation in the subsequent determination of parton distributions.

Abstract

We construct a parametrization of the deep-inelastic structure function of the proton F_2 based on all available experimental information from charged lepton deep-inelastic scattering experiments. The parametrization effectively provides a bias-free determination of the probability measure in the space of structure functions, which retains information on experimental errors and correlations. The result is obtained in the form of a Monte Carlo sample of neural networks trained on an ensemble of replicas of the experimental data. We discuss in detail the techniques required for the construction of bias-free parameterizations of large amounts of structure function data, in view of future applications to the determination of parton distributions based on the same method.

Paper Structure

This paper contains 7 sections, 19 equations, 8 figures, 6 tables.

Table of Contents

  1. Introduction
  2. Experimental data
  3. Fitting technique
  4. Training to the $F_2^p$ data
  5. Results
  6. Conclusion
  7. Statistical estimators

Figures (8)

  • Figure 1: Kinematic range of the experimental data
  • Figure 2: Scatter plot of experimental vs. Monte Carlo central values and errors.
  • Figure 3: Dependence of the $\chi^2$ eq. (\ref{['chi2']}) on the length of training: (left) total training (right) detail of the GA training.
  • Figure 4: Dependence of $\left\langle\sigma^{(\mathrm{net})}\right\rangle_{\mathrm{dat}}$ on the length of training for the BCDMS experiment.
  • Figure 5: Final results for $F_2(x,Q^2)$ compared to data. For the neural net result, the one-$\sigma$ error band is shown.
  • ...and 3 more figures