[Paper Review] The PDF4LHC Working Group Interim Recommendations

M. Botje, J. M. Butterworth+9 more|arXiv (Cornell University)|Jan 3, 2011

Particle physics theoretical and experimental studiesPhysics and Astronomy19 references488 citations

TL;DR

This paper proposes a practical, consensus-based method for estimating PDF and $α_s$ uncertainties in LHC cross-section predictions by combining results from three leading PDF sets—CTEQ6.6, MSTW2008, and NNPDF2.0—at NLO and extrapolating to NNLO. The key innovation is using the envelope of individual PDF uncertainty bands to form a conservative, experimentally relevant uncertainty estimate, validated through Higgs boson production cross-section calculations at 7 and 14 TeV.

ABSTRACT

This note provides an interim summary of the current recommendations of the PDF4LHC working group for the use of parton distribution functions (PDFs) and of PDF uncertainties at the LHC, for cross section and cross section uncertainty calculations. It also contains a succinct user guide to the computation of PDF uncertainties and correlations using available PDF sets. A companion note (the PDF4LHC Working Group Interim Report) summarizes predictions for benchmark cross sections at the LHC (7 TeV) at NLO using modern PDFs currently available from 6 PDF fitting groups.

Motivation & Objective

• To provide a pragmatic, experimentally viable protocol for estimating PDF and strong coupling constant uncertainties in LHC cross-section predictions.
• To reconcile differences among leading PDF sets (CTEQ, MSTW, NNPDF) without favoring any single set, ensuring robustness for LHC analyses.
• To establish a standardized method for combining PDF and $α_s$ uncertainties that reflects both individual PDF uncertainties and inter-set discrepancies.
• To extend NLO uncertainty estimates to NNLO using a rescaling procedure based on NLO envelope width, given the lack of multiple NNLO PDF sets.
• To support LHC experiments and theorists in achieving consistent, reliable uncertainty estimates for benchmark processes like Higgs production via gluon fusion.

Proposed method

• At NLO, compute Higgs cross sections using CTEQ6.6, MSTW2008nlo68cl, and NNPDF2.0 with their respective preferred $α_s(m_Z)$ values (0.118, 0.1207, 0.119).
• For each PDF set, compute the PDF+$\alpha_s$ uncertainty band using the collaboration-specific rules detailed in the PDF4LHC Interim Report.
• Form the envelope of the three individual uncertainty bands to create a conservative, combined uncertainty estimate.
• Define the central value as the midpoint of the envelope and the uncertainty as the distance from the midpoint to either edge.
• At NNLO, use only MSTW2008nnlo68cl due to the absence of multiple NNLO PDF sets, and rescale its uncertainty band using a ratio derived from the NLO envelope width to MSTW2008nlo68cl width.
• Apply the rescaling factor—dependent on Higgs mass, collider energy, and type—to estimate a NNLO envelope-like uncertainty band.

Experimental results

Research questions

• RQ1 How can a consistent, conservative estimate of PDF and $α_s$ uncertainties be derived when multiple PDF sets show discrepancies in cross-section predictions?
• RQ2 What is the most practical way to combine uncertainties from CTEQ, MSTW, and NNPDF sets without overestimating or underestimating the true uncertainty?
• RQ3 How can NNLO uncertainty estimates be reliably extrapolated from NLO results when only one global NNLO PDF set (MSTW2008nnlo) is available?
• RQ4 To what extent do inter-set differences in PDF central values and uncertainties affect the reliability of LHC cross-section predictions for Higgs production?
• RQ5 What is the impact of varying $α_s(m_Z)$ values on the combined uncertainty band, and how should this be consistently handled across PDF sets?

Key findings

• The envelope of the three NLO PDF+$\alpha_s$ uncertainty bands from CTEQ6.6, MSTW2008nlo68cl, and NNPDF2.0 provides a conservative, experimentally robust estimate of the total uncertainty in Higgs production cross sections.
• The rescaling factor for the MSTW2008nnlo68cl uncertainty band at NNLO is found to be approximately 2, but varies non-trivially with Higgs boson mass, collider energy, and type.
• The combined NLO uncertainty band spans a wider range than any individual PDF set, reflecting both intrinsic PDF uncertainties and inter-set discrepancies.
• The NLO and NNLO uncertainty bands from MSTW2008 are found to be very similar in shape and magnitude, validating the rescaling approach for NNLO extrapolation.
• The method ensures that uncertainty estimates remain connected to individual PDF uncertainties while accounting for systematic differences between PDF sets.
• The recommendation is designed to be practical and backward-compatible with existing LHC experimental practices, particularly those relying on CTEQ and MSTW.