[Paper Review] A posteriori inclusion of PDFs in NLO QCD final-state calculations

T. Carli, Gavin P. Salam+1 more|ArXiv.org|Oct 25, 2005

Particle physics theoretical and experimental studies3 citations

TL;DR

This paper presents a method for a posteriori inclusion of parton distribution functions (PDFs) in next-to-leading-order (NLO) QCD calculations by storing perturbative cross-section weights on a 2D grid in $x$ and $Q^2$, using high-order interpolation to enable accurate, flexible PDF reweighting after Monte Carlo integration. The technique achieves 0.01% precision for single-inclusive jet cross-sections at LHC energies, enabling consistent global PDF fits using final-state data.

ABSTRACT

Any NLO calculation of a QCD final-state observable involves Monte Carlo integration over a large number of events. For DIS and hadron colliders this must usually be repeated for each new PDF set, making it impractical to consider many `error' PDF sets, or carry out PDF fits. Here we discuss ``a posteriori'' inclusion of PDFs, whereby the Monte Carlo run calculates a grid (in x and Q) of cross section weights that can subsequently be combined with an arbitrary PDF. The procedure is numerically equivalent to using an interpolated form of the PDF. The main novelty relative to prior work is the use of higher-order interpolation, which substantially improves the tradeoff between accuracy and memory use. An accuracy of about 0.01% has been reached for the single inclusive cross-section in the central rapidity region |y|<0.5 for jet transverse momenta from 100 to 5000 GeV. This method should facilitate the consistent inclusion of final-state data from HERA, Tevatron and LHC in PDF fits, thus helping to increase the sensitivity of LHC to deviations from standard Model predictions.

Motivation & Objective

• To enable consistent inclusion of final-state LHC data in global QCD PDF analyses despite high computational cost of NLO calculations.
• To reduce the need for repeated NLO Monte Carlo runs when testing new PDF sets by decoupling PDF dependence from the core calculation.
• To improve memory efficiency and accuracy in PDF reweighting by using higher-order interpolation on a $x$-$Q^2$ grid.
• To facilitate global PDF fits using data from HERA, Tevatron, and LHC by enabling a posteriori PDF combination with precomputed NLO weights.
• To achieve high-precision NLO predictions (0.01%) for high-$P_T$ jet cross-sections with minimal memory usage.

Proposed method

• Represent PDFs on a 2D grid in transformed variables $y = \ln(1/x)$ and $\tau = \ln\ln(Q^2/\Lambda^2)$ to ensure uniform coverage across $x$ and $Q^2$.
• Store NLO cross-section weights from Monte Carlo integration as a function of $P_T$, $y$, and $Q^2$ on this grid for later PDF reweighting.
• Use $n^{\text{th}}$-order polynomial interpolation (e.g., $n=3$ or $n=5$) on the grid to reconstruct the cross-section for any PDF set.
• Apply a variable transformation to concentrate grid points in regions of high physical relevance (e.g., large $x$) via $y(x) = \ln(1/x) + a(1-x)$.
• Ensure grid boundaries and interpolation support only valid, stored grid points to maintain numerical stability.
• Use the same grid and interpolation scheme to reweight the cross-section for any PDF set without re-running the full NLO simulation.

Experimental results

Research questions

• RQ1 Can NLO QCD cross-sections be computed once and reused for multiple PDF sets with high precision?
• RQ2 How can memory usage be minimized while maintaining sub-0.01% accuracy in PDF reweighting for high-$P_T$ jet cross-sections?
• RQ3 Can higher-order interpolation reduce the required grid size without sacrificing accuracy?
• RQ4 To what extent can a posteriori PDF inclusion improve the sensitivity of LHC data to new physics beyond the Standard Model?
• RQ5 Is it feasible to use a single NLO Monte Carlo run to support global PDF fits across HERA, Tevatron, and LHC datasets?

Key findings

• The method achieves a precision of 0.01% for single-inclusive jet cross-sections in the central rapidity region $|y| < 0.5$ over $100$ transverse momentum bins from 100 to 5000 GeV.
• Using a 30×30 grid in $y$ and $\tau$ with third-order interpolation yields a typical accuracy of 0.01% and a maximum bias of 0.04% at low and high $P_T$.
• Increasing the interpolation order to $n=5$ eliminates the low- and high-$P_T$ bias observed with $n=3$, achieving 0.01% precision without increasing grid size.
• A smaller grid of $10^2 \times 10 \times 100$ with $n=5$ interpolation achieves 0.5% precision using only 10 MB of memory.
• A finer $x$-grid ($50^2$) improves accuracy to within 0.005%, but finer $Q^2$ sampling does not improve precision beyond a certain point.
• The technique enables consistent inclusion of final-state data from HERA, Tevatron, and LHC in global PDF analyses, enhancing sensitivity to new physics.