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Convergence of Heavy-Flavor PDFs at Small x and Its Impact on Future ep Colliders

S. O. Kara

TL;DR

The paper demonstrates a quantitative convergence of heavy-flavor PDFs at small $x$ across NNLO global fits (NNPDF4.0, CT18, MSHT20), reporting a maximal relative deviation of $16.8\%$ at $x=10^{-6}$ and $Q=100$ GeV and a global compatibility of $T_{PDF}=1.2\pm0.2\sigma$ with a KL divergence $J=(2.1\pm0.4)\times10^{-3}$. Using a unified framework (LHAPDF6, APFEL, MadGraph5_aMC@NLO) and a log-spaced metric grid, the authors validate the convergence with HERA and LHCb data and show that the reduced heavy-flavor uncertainty halves the PDF-driven error in heavy-quark electroproduction at LHeC/FCC-eh. They quantify the impact on leptophilic new physics searches, demonstrating a ~1.5× improvement in discovery reach for a leptophilic $Z_l$ across accessible masses, while the small-$x$ regime is probed by both collider and PDF constraints. Together, these results connect precision QCD with the discovery potential of future lepton–hadron facilities and establish a robust methodology for assessing PDF consistency in the high-precision era.

Abstract

We present the first quantitative demonstration of the statistical convergence of heavy-flavor parton distribution functions at small x across modern NNLO global fits. Using the latest NNPDF4.0, CT18, and MSHT20 sets, we find a maximal relative deviation of 16.8 percent at x = 1e-6 and Q = 100 GeV, corresponding to nearly a factor-of-two improvement over the approximately 30 percent spread in pre-2013 determinations. This convergence, stabilized by consistent alpha_s and scale treatments, marks a transition of the heavy-flavor sector into a statistically controlled regime of small-x QCD. We further show that this precision directly reduces PDF-driven uncertainties in heavy-quark electroproduction at future ep colliders (LHeC and FCC-eh) by about a factor of two and enhances the theoretical stability of leptophilic new-physics searches such as a neutral Z_l gauge boson. These findings establish a link between the precision frontier of QCD and the discovery potential of next-generation lepton-hadron facilities.

Convergence of Heavy-Flavor PDFs at Small x and Its Impact on Future ep Colliders

TL;DR

The paper demonstrates a quantitative convergence of heavy-flavor PDFs at small across NNLO global fits (NNPDF4.0, CT18, MSHT20), reporting a maximal relative deviation of at and GeV and a global compatibility of with a KL divergence . Using a unified framework (LHAPDF6, APFEL, MadGraph5_aMC@NLO) and a log-spaced metric grid, the authors validate the convergence with HERA and LHCb data and show that the reduced heavy-flavor uncertainty halves the PDF-driven error in heavy-quark electroproduction at LHeC/FCC-eh. They quantify the impact on leptophilic new physics searches, demonstrating a ~1.5× improvement in discovery reach for a leptophilic across accessible masses, while the small- regime is probed by both collider and PDF constraints. Together, these results connect precision QCD with the discovery potential of future lepton–hadron facilities and establish a robust methodology for assessing PDF consistency in the high-precision era.

Abstract

We present the first quantitative demonstration of the statistical convergence of heavy-flavor parton distribution functions at small x across modern NNLO global fits. Using the latest NNPDF4.0, CT18, and MSHT20 sets, we find a maximal relative deviation of 16.8 percent at x = 1e-6 and Q = 100 GeV, corresponding to nearly a factor-of-two improvement over the approximately 30 percent spread in pre-2013 determinations. This convergence, stabilized by consistent alpha_s and scale treatments, marks a transition of the heavy-flavor sector into a statistically controlled regime of small-x QCD. We further show that this precision directly reduces PDF-driven uncertainties in heavy-quark electroproduction at future ep colliders (LHeC and FCC-eh) by about a factor of two and enhances the theoretical stability of leptophilic new-physics searches such as a neutral Z_l gauge boson. These findings establish a link between the precision frontier of QCD and the discovery potential of next-generation lepton-hadron facilities.
Paper Structure (6 sections, 3 equations, 4 figures, 1 table)

This paper contains 6 sections, 3 equations, 4 figures, 1 table.

Table of Contents

  1. Introduction
  2. Heavy-flavor electroproduction at LHeC and FCC-eh
  3. Inter-set compatibility metric
  4. Validation with HERA and LHCb data
  5. Physics reach for Zℓ
  6. Conclusions and outlook

Figures (4)

  • Figure 1: (Color online) Ratios of charm (top) and bottom (bottom) PDFs, $f_Q(x,Q)/f_Q^{\mathrm{NNPDF4.0}}(x,Q)$, at $Q=100~\mathrm{GeV}$. Inner bands denote $1\sigma$ PDF uncertainties. Modern NNLO global fits (NNPDF4.0, CT18, MSHT20) exhibit near-complete overlap within uncertainties, whereas legacy sets (MSTW2008, HERAPDF15) show larger deviations.
  • Figure 2: (Color online) Reduced heavy-flavor cross sections measured at HERA. The upper panel shows the charm contribution $\sigma_{\mathrm{red}}^{c}(x,Q^2)$ using the combined HERA data HERAcomb2015. The lower panel displays the corresponding beauty contribution $\sigma_{\mathrm{red}}^{b}(x,Q^2)$ in analogous kinematics; public combined tables are limited, and construction details are provided in the Supplemental Material.
  • Figure 3: (Color online) Evolution of gluon and heavy--flavor PDFs at fixed $x=10^{-4}$. (Top) Scale dependence of $xg(x,Q)$, $xc(x,Q)$, and $xb(x,Q)$, showing the logarithmic DGLAP growth with $Q$. (Bottom) Ratios $xc/xg$ and $xb/xg$ illustrating the gradual saturation of heavy--flavor contributions. Both panels use NNPDF4.0 at NNLO with $\alpha_s(M_Z)=0.118$.
  • Figure 4: FCC--eh projected sensitivity to a leptophilic $Z_\ell$ for 100 fb$^{-1}$. Shaded band: PDF+scale (4--8%). Solid/dashed: 2$\sigma$ (95% C.L.) and 5$\sigma$ reference contours.