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

Abstract: The simultaneous production of a $W<sup>\pm$ boson and a charmed hadron in proton-proton collisions offers a potential probe for constraining the strange quark parton distribution functions (PDFs). We study these processes at next-to-leading order in perturbative Quantum Chromodynamics within the general-mass variable-flavor-number scheme. By considering ratios of cross-section between oppositely charged mesons, uncertainties associated with unphysical scale choices and fragmentation functions are shown to effectively cancel out, leaving the uncertainty originating from the PDFs as the dominant one. By comparing our calculations with CT18A, MSHT20 and NNPDF4.0 PDFs with the recent ATLAS measurement at $\sqrt{s} = 13\,{\rm TeV}$ we find that CT18A, which imposes zero strangeness asymmetry, agrees best with the data, while MSHT20 and NNPDF4.0, both of which allow for a non-zero strangeness asymmetry, exhibit greater tension with the ATLAS data. The sensitivity to the strangeness asymmetry is further confirmed by the PDF reweighting methods. We also study the impact of possible intrinsic charm content of the proton finding no significant sensitivity. Finally, we explore the possibility of measuring these processes in proton-lead collisions. With the estimated detector efficiencies and projected luminositites at the high-luminosity LHC, these processes should be visible, yet with a rather limited constraining power for nuclear PDFs.

• The paper develops an NLO general-mass variable-flavour-number framework that retains charm-mass effects, resums collinear logarithms through fragmentation functions, and avoids double counting.
• The predictions undershoot ATLAS absolute cross sections by about 20%, while the predicted D±/D*± ratio of roughly 0.9 is below the measured 0.97–0.98, indicating possible fragmentation-function limitations.
• The charge ratio R_c± largely cancels scale, fragmentation, and luminosity uncertainties, strongly favoring CT18A over MSHT20 and NNPDF4.0 and revealing sensitivity to large-x strangeness asymmetry but negligible intrinsic-charm effects.

Motivation and theoretical setting

Associated production of a $W^\pm$ boson and a charmed hadron in proton-proton collisions is, at leading order, dominated by the $sg$ partonic channel, which makes it a direct probe of the strange-quark PDFs of the proton. The paper by Alanko, Helenius and Paukkunen (2608.14268) computes $pp \to W^\mp(\to \ell^\pm\nu) + D^{(*)\pm} + X$ at next-to-leading order in perturbative QCD within the general-mass variable-flavour-number scheme (GM-VFNS), retaining the full charm-quark mass dependence while resumming the collinear logarithms $\log(p_{T,c}/m_c)$ into scale-dependent charm-to-hadron fragmentation functions (FFs). This complements existing zero-mass NNLO calculations [Caletti et al.] and jet-based treatments, avoiding the non-perturbative modeling of multi-parton interactions required for charm-jet observables.

The observable matched to experiment is the ATLAS "OS–SS" combination,

$$\sigma_{\text{OS-SS}}(W^\mp D^{(*)\pm}) = \sigma(W^\mp D^{(*)\pm}) - \sigma(W^\mp D^{(*)\mp}),$$

which cancels backgrounds and all partonic channels without an initial-state strange quark. Under the assumption $c = \bar{c}$ in the PDFs, diagrams with initial-state charm also cancel exactly, so intrinsic charm can enter only through a possible $c - \bar{c}$ asymmetry. Double-parton scattering contributions cancel as well.

The computational framework convolutes MCFM 10.3 partonic cross sections for $Wc$ production with scale-dependent FFs (KKKS08 fitted to OPAL data as default), including the subtraction term that removes the double-counted first-order logarithm. All three scales are set to $\mu = M_W$ by default. The authors verify that the choice of fragmentation variable ($z_- = p_{T,D}/p_{T,c}$ versus $sg$0) changes cross sections by at most $sg$1 at the lowest considered $sg$2 bin, and that imposing the FF fit boundary $sg$3 discards only about 1% of the integrated cross section.

Absolute cross sections and tension with ATLAS data

Compared with the most precise LHC measurement — the ATLAS $sg$4 data with cuts $sg$5, $sg$6, $sg$7, $sg$8 — the NLO predictions fall below the measured absolute cross sections by roughly 20%, even accounting for scale and PDF uncertainties. This deficit is consistent with zero-mass NNLO results showing a nearly constant $sg$9 upward correction, indicating that residual charm-mass effects are weak at these transverse momenta. A practical implication follows: zero-mass NNLO calculations should be reliable for PDF fits at NNLO accuracy in this kinematic regime.

A notable discrepancy concerns the meson composition: the predicted ratio $pp \to W^\mp(\to \ell^\pm\nu) + D^{(*)\pm} + X$0 at NLO is below the experimental value of $pp \to W^\mp(\to \ell^\pm\nu) + D^{(*)\pm} + X$1–$pp \to W^\mp(\to \ell^\pm\nu) + D^{(*)\pm} + X$2, which itself is consistent with LEP charm-hadron production fractions. Since the KKKS08 FFs are anchored to OPAL data, this suggests modifications to the charm-hadron FFs would be needed; conversely, the $pp \to W^\mp(\to \ell^\pm\nu) + D^{(*)\pm} + X$3 spectra probe the large-$pp \to W^\mp(\to \ell^\pm\nu) + D^{(*)\pm} + X$4 region of the FFs and could usefully enter future global FF analyses.

Charge ratios isolate the strangeness asymmetry

The central result of the paper is the charge-conjugate ratio

$pp \to W^\mp(\to \ell^\pm\nu) + D^{(*)\pm} + X$5

in which scale uncertainties, FF uncertainties (at the permille level), fragmentation-variable ambiguities, and luminosity systematics cancel almost completely, leaving PDFs as the dominant uncertainty. To leading order in the CKM suppression factor $pp \to W^\mp(\to \ell^\pm\nu) + D^{(*)\pm} + X$6, the ratio tracks the PDF combination

$pp \to W^\mp(\to \ell^\pm\nu) + D^{(*)\pm} + X$7

so sensitivity is concentrated on the strangeness asymmetry $pp \to W^\mp(\to \ell^\pm\nu) + D^{(*)\pm} + X$8 and total strangeness $pp \to W^\mp(\to \ell^\pm\nu) + D^{(*)\pm} + X$9 at large $\log(p_{T,c}/m_c)$0.

Comparing CT18ANLO (which imposes $\log(p_{T,c}/m_c)$1), MSHT20NLO, and NNPDF4.0NLO against the differential $\log(p_{T,c}/m_c)$2 data constructed from the ATLAS measurement yields a clear hierarchy:

PDF set	$\log(p_{T,c}/m_c)$3, $\log(p_{T,c}/m_c)$4	$\log(p_{T,c}/m_c)$5-value [%]
CT18ANLO	0.7	75.5
MSHT20nlo_as118	1.3	21.7
NNPDF40_nlo_pch_as_01180	4.8	$\log(p_{T,c}/m_c)$6

CT18A agrees best and carries by far the smallest PDF uncertainty — a direct consequence of its imposed $\log(p_{T,c}/m_c)$7 constraint — while NNPDF4.0, which allows a free strangeness asymmetry, is strongly disfavored. Hessian reweighting (with tolerance parameters $\log(p_{T,c}/m_c)$8 and $\log(p_{T,c}/m_c)$9) and Bayesian reweighting confirm the direction of the pull: for CT18A the data favor an increased $$\sigma_{\text{OS-SS}}(W^\mp D^{(*)\pm}) = \sigma(W^\mp D^{(*)\pm}) - \sigma(W^\mp D^{(*)\mp}),$$0, whereas for MSHT20 and NNPDF4.0 the dominant pull reduces $$\sigma_{\text{OS-SS}}(W^\mp D^{(*)\pm}) = \sigma(W^\mp D^{(*)\pm}) - \sigma(W^\mp D^{(*)\mp}),$$1 toward large $$\sigma_{\text{OS-SS}}(W^\mp D^{(*)\pm}) = \sigma(W^\mp D^{(*)\pm}) - \sigma(W^\mp D^{(*)\mp}),$$2. The reweighting conclusions are robust against the choice of FF set. The authors caution that since the data points initially lie outside the PDF error bands, the reweighting becomes partly extrapolative and a full refit would be needed for a conclusive statement; they also note that the required changes in NNPDF4.0 are significant enough that conflicts with other datasets in the global fit cannot be excluded.

Two further negative results deserve emphasis. First, no meaningful sensitivity to intrinsic charm is found: comparing NNPDF4.0 sets with and without a $$\sigma_{\text{OS-SS}}(W^\mp D^{(*)\pm}) = \sigma(W^\mp D^{(*)\pm}) - \sigma(W^\mp D^{(*)\mp}),$$3 asymmetry changes $$\sigma_{\text{OS-SS}}(W^\mp D^{(*)\pm}) = \sigma(W^\mp D^{(*)\pm}) - \sigma(W^\mp D^{(*)\mp}),$$4 only at the level of $$\sigma_{\text{OS-SS}}(W^\mp D^{(*)\pm}) = \sigma(W^\mp D^{(*)\pm}) - \sigma(W^\mp D^{(*)\mp}),$$5. Second, the GM-VFNS resummation is quantitatively essential — omitting it (effectively a fixed-flavour-number calculation) or double-counting the subtraction term alters the $$\sigma_{\text{OS-SS}}(W^\mp D^{(*)\pm}) = \sigma(W^\mp D^{(*)\pm}) - \sigma(W^\mp D^{(*)\mp}),$$6 spectrum by up to nearly a factor of two at large $$\sigma_{\text{OS-SS}}(W^\mp D^{(*)\pm}) = \sigma(W^\mp D^{(*)\pm}) - \sigma(W^\mp D^{(*)\mp}),$$7 and distorts its shape.

Prospects in proton-lead collisions

For $$\sigma_{\text{OS-SS}}(W^\mp D^{(*)\pm}) = \sigma(W^\mp D^{(*)\pm}) - \sigma(W^\mp D^{(*)\mp}),$$8Pb collisions at $$\sigma_{\text{OS-SS}}(W^\mp D^{(*)\pm}) = \sigma(W^\mp D^{(*)\pm}) - \sigma(W^\mp D^{(*)\mp}),$$9 with a projected integrated luminosity of $c = \bar{c}$0, using detector efficiencies inferred from the ATLAS $c = \bar{c}$1 measurement ($c = \bar{c}$2 for $c = \bar{c}$3, $c = \bar{c}$4 for $c = \bar{c}$5), the authors estimate roughly 100 reconstructed events per charge combination. The resulting statistical uncertainty on absolute cross sections is comparable to the 68% C.L. nuclear-PDF uncertainty from EPPS21 and nNNPDF3.0, so the process should be visible but offers limited constraining power. For the ratio $c = \bar{c}$6 the situation is worse: PDF uncertainties largely cancel between numerator and denominator while statistical errors do not, leaving the projected measurement far from competitive. The smallness of the current nuclear-PDF uncertainty here is itself partly an artifact of the $c = \bar{c}$7 assumption made in both nuclear fits, so a dedicated study of strangeness asymmetry effects in nuclear PDFs remains open.

Limitations and open questions

Several caveats qualify the results. The NLO predictions systematically undershoot the ATLAS absolute cross sections by about 20%, and although NNLO corrections are expected to close much of the gap, the paper relies on external zero-mass NNLO estimates rather than a mass-retaining NNLO computation. The mismatch in the $c = \bar{c}$8 production fractions points to deficiencies in the charm-hadron FF treatment, and the explanation offered — modified fragmentation in the denser $c = \bar{c}$9 environment — remains a hypothesis rather than an established mechanism. The reweighting analysis uses ten statistically overlapping data points with an artificially reduced tolerance parameter, and the authors themselves state that a refit would be required for definitive conclusions. Finally, whether a consistent compromise exists between the strangeness-asymmetry reduction favored by these data and the constraints from dimuon neutrino DIS and other datasets in global fits is left unresolved.

Conclusion

This work provides a complete NLO GM-VFNS framework for associated $c - \bar{c}$0 production with full charm-mass retention and consistent logarithmic subtraction, validated against the ATLAS $c - \bar{c}$1 measurement. Its principal finding is that the charge-conjugate ratios $c - \bar{c}$2 cancel essentially all theory and many experimental uncertainties, exposing a clean sensitivity to the strangeness asymmetry: CT18A's zero-asymmetry assumption agrees best with the data, while the freely fitted $c - \bar{c}$3 of NNPDF4.0 creates significant tension that reweighting traces to an overestimated $c - \bar{c}$4 at large $c - \bar{c}$5. The process shows negligible sensitivity to intrinsic charm, and projected $c - \bar{c}$6Pb measurements would be visible but only weakly constraining. The open questions the paper leaves are concrete: a full PDF refit incorporating these observables, improved charm-hadron FFs constrained by the $c - \bar{c}$7 spectra, and the impact of strangeness asymmetry in nuclear PDF fits.