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Assessing the Impact of Fitting Methodology at aN$^3$LO with FPPDF: an Open Source Tool for Extracting Parton Distribution Functions in the Hessian Approach

J. M. Cruz-Martinez, T. Giani, L. A. Harland-Lang

TL;DR

PDF determinations at high precision are increasingly sensitive to methodological choices in fitting. This work presents FPPDF, an open-source Hessian-based fitting framework that mirrors MSHT parameterisation while reusing NNPDF data/theory inputs, enabling direct comparisons with NN-based MC fits. By applying FPPDF to both NNLO and $aN^3$LO fits and benchmarking against NNPDF results, the study demonstrates that the relative impact of ascending to $aN^3$LO and including missing higher-order uncertainties is largely independent of whether a Hessian/polynomial or NN/MC fitting approach is used. This disentangling clarifies the origin of differences seen among PDF sets and provides a transparent tool for validating methodological choices. The open-source nature and the ability to perform like-for-like comparisons make FPPDF a valuable resource for future PDF studies and collider phenomenology.

Abstract

We present a new public code, FPPDF, to perform global fits of parton distribution functions (PDFs). The fitting methodology follows that implemented by the MSHT collaboration, namely applying a fixed polynomial parameterisation of the PDFs and Hessian approach to error propagation, while for data and theory settings the libraries used by the NNPDF collaboration are taken. This therefore complements the already publicly available NNPDF fitting code to enable fits with both neural network and fixed polynomial PDF parameterisations to be performed by the community, with otherwise identical theoretical and experimental inputs. As a first application, we use the new code to compare the PDFs found from fits at both NNLO and aN$^3$LO perturbative orders, but applying these two fitting approaches. We assess the impact of the two different methodologies on the PDFs and their uncertainties, providing results that complement previous comparisons between published PDF sets at NNLO and aN$^3$LO. We in particular find that the relative impact of going to the higher perturbative order and/or including missing higher order uncertainties is rather insensitive to which of these PDF parameterisation methodologies are used.

Assessing the Impact of Fitting Methodology at aN$^3$LO with FPPDF: an Open Source Tool for Extracting Parton Distribution Functions in the Hessian Approach

TL;DR

PDF determinations at high precision are increasingly sensitive to methodological choices in fitting. This work presents FPPDF, an open-source Hessian-based fitting framework that mirrors MSHT parameterisation while reusing NNPDF data/theory inputs, enabling direct comparisons with NN-based MC fits. By applying FPPDF to both NNLO and LO fits and benchmarking against NNPDF results, the study demonstrates that the relative impact of ascending to LO and including missing higher-order uncertainties is largely independent of whether a Hessian/polynomial or NN/MC fitting approach is used. This disentangling clarifies the origin of differences seen among PDF sets and provides a transparent tool for validating methodological choices. The open-source nature and the ability to perform like-for-like comparisons make FPPDF a valuable resource for future PDF studies and collider phenomenology.

Abstract

We present a new public code, FPPDF, to perform global fits of parton distribution functions (PDFs). The fitting methodology follows that implemented by the MSHT collaboration, namely applying a fixed polynomial parameterisation of the PDFs and Hessian approach to error propagation, while for data and theory settings the libraries used by the NNPDF collaboration are taken. This therefore complements the already publicly available NNPDF fitting code to enable fits with both neural network and fixed polynomial PDF parameterisations to be performed by the community, with otherwise identical theoretical and experimental inputs. As a first application, we use the new code to compare the PDFs found from fits at both NNLO and aNLO perturbative orders, but applying these two fitting approaches. We assess the impact of the two different methodologies on the PDFs and their uncertainties, providing results that complement previous comparisons between published PDF sets at NNLO and aNLO. We in particular find that the relative impact of going to the higher perturbative order and/or including missing higher order uncertainties is rather insensitive to which of these PDF parameterisation methodologies are used.
Paper Structure (33 sections, 3 equations, 7 figures, 2 tables)

This paper contains 33 sections, 3 equations, 7 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Overview of fitting methodologies
  3. PDF parameterization and fitting basis
  4. $\chi^2$ optimization
  5. Error propagation and tolerance
  6. The impact of fitting methodologies at up to aN$^3$LO
  7. Published aN$^3$LO sets
  8. Results
  9. $\chi^2$ trends
  10. Impact of aN$^3$LO corrections on NNLO PDF sets
  11. Comparison between aN$^3$LO results
  12. Parton luminosities
  13. Gluon PDF
  14. Charm PDF
  15. Valence and down quark PDFs
  16. ...and 18 more sections

Figures (7)

  • Figure 1: Parton luminosities at NNLO, NNLO+MHOU and aN$^3$LO+MHOU for the (left) fixed parameterisation and (right) NNPDF methodologies. Fixed parameterisation uncertainties are calculated using the dynamic tolerance criterion.
  • Figure 2: Gluon and singlet distributions at NNLO, NNLO+MHOU and aN$^3$LO+MHOU for the (left) fixed parameterisation and (right) NNPDF methodologies. Fixed parameterisation uncertainties are calculated using the dynamic tolerance criterion.
  • Figure 3: Comparison between parton luminosities at aN$^3$LO+MHOU for the (left) fixed parameterisation and (right) NNPDF methodologies. Fixed parameterisation uncertainties are calculated using the dynamic tolerance criterion.
  • Figure 4: Left panel: comparison between the gluon PDF obtained using the fixed parameterisation and NNPDF methodologies. Right panel: comparison between the medium and large-$x$ charm at 1.65 GeV obtained using the MSHT and NNPDF methodologies. Fixed parameterisation uncertainties are calculated using the dynamic tolerance criterion.
  • Figure 5: Valence (left) and down quark (right) PDFs found using the fixed parameterisation and NNPDF methodologies. Fixed parameterisation uncertainties are calculated using the dynamic tolerance criterion.
  • ...and 2 more figures