⚛️ phenomenology

Simultaneous production of a $W$ boson and a charmed hadron at the LHC in general-mass variable-flavour-number scheme

This paper investigates the simultaneous production of $W$ bosons and charmed hadrons at the LHC using next-to-leading order QCD calculations in a general-mass variable-flavor-number scheme, finding that while ratio observables effectively reduce theoretical uncertainties and reveal tension between ATLAS data and PDF sets allowing non-zero strangeness asymmetry, the processes show limited sensitivity to intrinsic charm and nuclear PDFs.

Original authors: Ville Alanko, Ilkka Helenius, Hannu Paukkunen

Published 2026-08-17

📖 4 min read🧠 Deep dive

CC BY 4.0

Original authors: Ville Alanko, Ilkka Helenius, Hannu Paukkunen

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine the universe is a giant, bustling kitchen where the most fundamental ingredients are constantly being mixed, smashed, and transformed. In this kitchen, protons—the tiny, positively charged particles that make up the core of every atom in your body—are not solid, indivisible balls. Instead, they are more like chaotic, swirling clouds of even smaller particles called "partons." Inside these clouds, you have quarks (the building blocks of matter) and gluons (the sticky glue holding them together). But here's the tricky part: we can't see these partons directly. We can only guess how many of each are floating around inside a proton by watching what happens when we smash two protons together at nearly the speed of light.

To make sense of these collisions, physicists use a special recipe book called "Parton Distribution Functions" (PDFs). Think of a PDF as a map that tells you the odds of finding a specific type of parton (like a strange quark) at a specific speed inside a proton. For a long time, scientists have been pretty good at mapping the common ingredients, like up and down quarks. But the "strange" quark is a bit of a mystery. It's elusive, and its map is fuzzy. One of the biggest questions is whether there's a perfect balance between strange quarks and their antimatter twins, anti-strange quarks, or if there's a hidden imbalance lurking inside the proton. If there is an imbalance, it could rewrite our understanding of how matter is built.

This is where the story of the paper comes in. The authors, Ville Alanko, Ilkka Helenius, and Hannu Paukkunen, decided to play detective using the world's biggest particle collider, the Large Hadron Collider (LHC). They focused on a very specific, rare event: when a proton collision creates a "W boson" (a heavy particle that carries the weak nuclear force) at the same time as a "charmed hadron" (a particle containing a heavy charm quark). It's like looking for a specific pair of shoes and a hat that appear together in a pile of laundry. By studying how often these pairs appear and comparing the "left-handed" versions to the "right-handed" versions, the team hoped to sharpen the blurry map of the strange quark.

The team performed complex computer simulations to predict what should happen if different theories about the strange quark were true. They compared their predictions against real data collected by the ATLAS experiment at the LHC, which had already spotted these rare W-plus-charm events. The results were telling. When they looked at the data, the predictions from one major theory (called CT18A), which assumes the strange quark and anti-strange quark are perfectly balanced, matched the real-world observations the best. In contrast, two other popular theories (MSHT20 and NNPDF4.0), which allow for an imbalance between strange and anti-strange quarks, seemed to struggle to match the data. The authors suggest that the real universe might be more balanced in its strange quarks than some of the more complex theories allow.

They also checked if the proton might contain "intrinsic charm"—a scenario where charm quarks are a permanent, built-in part of the proton rather than just appearing briefly during a collision. Their calculations showed that this idea doesn't seem to play a significant role in these specific events; the data doesn't need this extra ingredient to make sense. Finally, the team looked ahead to future experiments where protons would collide with heavy lead nuclei. They estimated that with enough time and data, these collisions could be seen, but the statistical noise might be too loud to learn much new about the strange quarks inside the lead nucleus just yet.

In short, by carefully measuring the ratio of these rare particle pairs, the authors found that the simplest picture of the strange quark—where it and its anti-twin are equal in number—fits the current evidence best. While they didn't prove this beyond any doubt, their work suggests that if there is a hidden imbalance in the strange quark world, it's much smaller than some scientists had hoped or feared. It's a small but important step in cleaning up the recipe book of the universe, showing us that sometimes, the most complex mysteries are solved by looking for the simplest balance.

Technical Summary: Simultaneous Production of a W Boson and a Charmed Hadron at the LHC

Problem and Motivation
The determination of the strange quark parton distribution function (PDF) remains a significant challenge in global QCD analyses, typically carrying larger uncertainties than other quark flavors. A specific open question concerns the size of the strangeness asymmetry ( $s - \bar{s}$ ), which is constrained to be zero in some PDF sets (e.g., CT18) but allowed to vary in others (e.g., MSHT20, NNPDF4.0). Additionally, the existence of intrinsic charm ( $c \neq \bar{c}$ ) remains an open inquiry. The simultaneous production of a $W^\pm$ boson and a charmed hadron ( $D^{(*)\pm}$ ) in proton-proton ($pp$) collisions offers a complementary probe to constrain these distributions. While previous studies have utilized $W$ +charm jet data, this work focuses on the specific process where the charm quark is reconstructed as a specific charmed meson, matching recent ATLAS measurements.

Methodology
The authors perform a next-to-leading order (NLO) perturbative QCD calculation within the general-mass variable-flavour-number scheme (GM-VFNS). This approach allows for the inclusion of the full charm-quark mass dependence while resumming large logarithmic terms ( $\sim \log(p_{T,c}/m_c)$ ) arising from collinear gluon radiation into scale-dependent fragmentation functions (FFs).

Theoretical Framework: The calculation utilizes the MCFM code for the partonic cross section ( $pp \to Wc + X$ ) and MadGraph5_aMC@NLO for charm-quark initial states. The partonic results are convoluted with scale-dependent FFs ( $D_{c \to D^{(*)}}$ ) to obtain the hadronic cross section. The authors employ the KKKS08 FFs fitted to OPAL data as the default, with variations using global fits and the SMSKA19 fit.
Subtraction Scheme: To avoid double-counting logarithmic contributions already resummed in the FFs, a subtraction term is applied, effectively removing the leading logarithmic behavior from the fixed-order partonic cross section.
Kinematics and Observables: The study focuses on the OS-SS (opposite-sign minus same-sign) subtraction method, which suppresses background processes and initial-state channels without strange quarks. The primary observables are the integrated and differential cross sections for $W^\mp D^{(*)\pm}$ , and crucially, the charge-conjugate production ratios $R_c^\pm$ .
Reweighting: To assess the impact of the data on PDFs, the authors employ PDF reweighting techniques (Hessian for CT18A/MSHT20 and Bayesian for NNPDF4.0) using the measured ratios as constraints.

Key Results

Cross Section Discrepancy: The NLO GM-VFNS predictions for absolute integrated and differential cross sections are approximately 20% lower than the recent ATLAS data at $\sqrt{s} = 13$ TeV. The authors note that this tension is consistent with zero-mass NNLO calculations, which show a $\sim 15\%$ upward correction, suggesting that residual charm-mass effects are not the primary source of the discrepancy at the studied kinematics ( $p_{T,D} > 8$ GeV).
PDF Sensitivity via Ratios: The production ratios $R_c^\pm$ $R_{c}^{\pm}$ (e.g., $\sigma(W^+ D^-)/\sigma(W^- D^+)$ $σ (W^{+} D^{-}) / σ (W^{-} D^{+})$ ) prove to be powerful discriminators. In these ratios, uncertainties from fragmentation functions, scale variations, and intrinsic charm largely cancel out.
- CT18A: Assumes zero strangeness asymmetry ( $s = \bar{s}$ ) and shows the best agreement with ATLAS data, with the smallest PDF uncertainties.
- MSHT20 and NNPDF4.0: Both allow for non-zero strangeness asymmetry. These sets exhibit greater tension with the data, with NNPDF4.0 showing the largest deviation (high $\chi^2$ ).
Strangeness Asymmetry: The discrepancy in the ratios is traced to the behavior of the $s - \bar{s}$ asymmetry at larger momentum fractions $x$ . PDF reweighting indicates that achieving better agreement with the ATLAS data would require a significant reduction in the strangeness asymmetry ( $s_-$ ) for MSHT20 and NNPDF4.0, or an increase in the total strange density ( $s_+$ ).
Intrinsic Charm: The study finds no significant sensitivity to intrinsic charm content ( $c \neq \bar{c}$ ) in the OS-SS subtracted observables, as the contribution from diagrams with initial-state charm cancels out unless a significant asymmetry exists, which is found to be negligible in the tested scenarios.
Fragmentation Functions: The ratios are found to be insensitive to the choice of FFs (at the permille level), validating their use as robust PDF constraints. However, the absolute cross-section ratios between $D^\pm$ and $D^{*\pm}$ differ from experimental values, suggesting potential needs for modifications in FF fits or an understanding of production fraction differences between $pp$ and $e^+e^-$ environments.

Proton-Lead (pPb) Predictions
The authors provide theoretical predictions for $W^\mp D^{(*)\pm}$ production in pPb collisions at $\sqrt{s} = 8.5$ TeV. Assuming an integrated luminosity of $1.2 \text{ pb}^{-1}$ and realistic detector efficiencies, approximately 100 events are expected. While the statistical uncertainty for the absolute cross section is comparable to current nuclear PDF (nPDF) uncertainties, the statistical uncertainty for the production ratios is significantly larger than the PDF uncertainties. Consequently, the constraining power for nuclear PDFs, particularly regarding strangeness asymmetry in nuclei, is currently limited.

Significance and Claims
The paper claims that the simultaneous measurement of $W$ and charmed hadrons, specifically through charge-conjugate ratios, provides a clean and powerful probe for the strange quark PDF and its asymmetry. The authors demonstrate that current data favors a scenario with zero or very small strangeness asymmetry (as in CT18A) over models with significant asymmetry (MSHT20, NNPDF4.0). They conclude that while the absolute cross sections require higher-order corrections (NNLO) for full agreement, the ratios effectively isolate PDF uncertainties. The study also highlights the potential of these processes to refine D-meson fragmentation functions and suggests that future measurements with lower $p_T$ cuts could further elucidate the differences in charm hadronization between $pp$ and $e^+e^-$ collisions. Finally, the work establishes the feasibility of measuring these processes in pPb collisions at the High-Luminosity LHC, though with limited immediate impact on nPDF constraints due to statistical limitations.

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