Tracing Vacuum Hadronization with Conserved Currents

Abstract: We show that color triality constrains the nonperturbative states that screen a Wilson-line endpoint, allowing the joint flows of net electric charge, baryon number, and strangeness to probe QCD vacuum hadronization. Whether evaluated from resolved hadrons in a jet initiated by a quark of flavor $f$ or from the corresponding charge correlators, these flows satisfy the Gell-Mann--Nishijima relation $$\langle Q_{\rm flow}\rangle_f \simeq I_{3,f} +\frac{1}{2}\left( \langle S_{\rm flow}\rangle_f +\langle B_{\rm flow}\rangle_f \right)$$, where $I_{3,f}$ is the third component of the initiating-quark isospin. The net jet baryon number, $$\langle B_{\rm flow}\rangle_f\simeq \frac{r_{qq}}{1+r_{qq}}$$, directly probes the relative probability of a diquark--antidiquark vacuum excitation, with $r_{qq}$ the diquark-to-quark production ratio. Thus, for $r_{qq}\ll1$, the baryon number carried by the jet is substantially suppressed relative to the initiating-quark value $B_f=1/3$. Likewise, the strange-to-light pair-production ratio, defined by $u\bar u:d\bar d:s\bar s=1:1:r_s$, is encoded in the measured jet strangeness $$\langle S_{\rm flow}\rangle_f \simeq S_f+\frac{3r_s}{2+r_s} \left(\frac{1}{3}- \langle B_{\rm flow}\rangle_f\right)$$, predicting a nonzero mean net strangeness even in $u$- and $d$-initiated jets. The conserved-current moments appearing in these relations are independent of the renormalization scale. Their simultaneous measurement therefore provides a direct, flavor-resolved probe of vacuum pair production and quantum-number transport during hadronization.

• The paper develops a factorization framework in which shifts in electric charge, baryon number, and strangeness flow reveal nonperturbative hadronization at a Wilson-line defect, with leading-power scale independence.
• The paper derives relations connecting joint Q, B, and S flows to effective strange-pair and diquark production probabilities, predicting suppressed net baryon number and induced strangeness in light-quark jets.
• The paper validates these predictions with Pythia 8.312 and Herwig 7.3, recovering consistent strange-pair parameters across multiple observables and finding baryon-flow changes from 0.0139 to 0.3298 in a diquark scan.

The paper develops a factorization-based framework in which the net electric charge, baryon number, and strangeness flowing into a collinear sector—such as a jet or a hemisphere—act as direct probes of nonperturbative QCD vacuum hadronization (2608.19708). The central observation is that a gauge-invariant collinear sector must be completed by a Wilson line extending to infinity, and that the endpoint of this Wilson line carries nonzero color triality that must be screened by dynamical matter. The quantum numbers of this screening state are invisible to a detector measuring resolved hadrons, so the shift of the measured flow away from the initiating-parton value isolates the "defect" charge localized at the Wilson-line puncture. Because these shifts are governed by conserved currents, they are renormalization-scale independent at leading power, and their joint flavor structure yields invertible relations to the underlying pair-production probabilities of the vacuum.

Charge flow and the defect sector

For any conserved quantum number $\mathcal{Q} = Q, B, S$, the authors define the flow into a region $\mathcal{R}$ as $$\mathcal{Q}_{\mathcal{R}} = \sum_{h \in \mathcal{R}} \mathcal{Q}_h$$. Unlike momentum-weighted jet charges, the first moment $\langle \mathcal{Q}_{\mathcal{R}} \rangle$ is infrared safe, so once calculable perturbative contributions are separated, any deviation from the initiating-parton charge signals nonperturbative charge transfer. In the $n$-collinear theory, the charge-flow operator is defined as an integral over the celestial sphere with a small disk $D_\epsilon(\bar{n})$ around the antipodal direction removed; this excluded puncture is precisely the endpoint of the collinear Wilson line. The total charge of the sector is conserved, so $$\langle Q_{\rm flow}\rangle_f = Q_f - \langle Q_D\rangle_f$$, where $\langle Q_D\rangle_f$ is the mean charge carried by degrees of freedom bound to the Wilson-line endpoint—the defect sector.

Perturbatively, the puncture carries no charge: charge-weighted radiation projects onto the flavor-nonsinglet channel, whose Regge behavior lacks a nonintegrable singularity at $\bar{n}\cdot q = 0$, so $\langle Q_D\rangle_f^{\rm pert} = 0$ and $\mathcal{R}$0. Nonperturbatively, however, hadronization can correlate the soft quark distribution along the fragmentation chain with the initiating flavor, breaking the $\mathcal{R}$1 cancellation within a fixed quark sector while preserving it between conjugate sectors ($\mathcal{R}$2). A key structural result follows from current conservation: since perturbative ultraviolet dynamics gives the flow moment the vanishing anomalous dimension of the conserved total charge, both $\mathcal{R}$3 and $\mathcal{R}$4 are renormalization-scale independent at leading power. This scale invariance makes the observables clean targets for precision comparison between theory and experiment.

Color triality and screening clouds

Confinement imposes a selection rule on the puncture sector: the anti-fundamental color triality of the Wilson-line endpoint must be screened by matter of conjugate triality. If the screening cloud contains $\mathcal{R}$5 endpoint quarks and $\mathcal{R}$6 endpoint antiquarks, triality requires $\mathcal{R}$7, so the defect baryon number satisfies $\mathcal{R}$8 for integer $\mathcal{R}$9. Since each additional quark-antiquark excitation costs energy, the authors conjecture—this is an assumption, not a derivation—that the puncture is dominated by the minimal one-quark cloud, with the anti-diquark $$\mathcal{Q}_{\mathcal{R}} = \sum_{h \in \mathcal{R}} \mathcal{Q}_h$$0 cloud as the leading correction. Under this two-channel truncation, the measured jet baryon number becomes

$$\mathcal{Q}_{\mathcal{R}} = \sum_{h \in \mathcal{R}} \mathcal{Q}_h$$1

where $$\mathcal{Q}_{\mathcal{R}} = \sum_{h \in \mathcal{R}} \mathcal{Q}_h$$2 is the diquark-to-quark vacuum production ratio. A strong prediction follows immediately: for $$\mathcal{Q}_{\mathcal{R}} = \sum_{h \in \mathcal{R}} \mathcal{Q}_h$$3, the net jet baryon number is substantially suppressed relative to the naive expectation $$\mathcal{Q}_{\mathcal{R}} = \sum_{h \in \mathcal{R}} \mathcal{Q}_h$$4 of the initiating quark. This is a bold claim—it states that a quark-initiated jet carries almost no net baryon number among its resolved hadrons, with the missing $$\mathcal{Q}_{\mathcal{R}} = \sum_{h \in \mathcal{R}} \mathcal{Q}_h$$5 residing invisibly on the Wilson-line endpoint.

Assuming the screening flavors lose memory of the initiating flavor and are drawn independently from a common light-flavor distribution $$\mathcal{Q}_{\mathcal{R}} = \sum_{h \in \mathcal{R}} \mathcal{Q}_h$$6, the charge and strangeness flows obey

$$\mathcal{Q}_{\mathcal{R}} = \sum_{h \in \mathcal{R}} \mathcal{Q}_h$$7

$$\mathcal{Q}_{\mathcal{R}} = \sum_{h \in \mathcal{R}} \mathcal{Q}_h$$8

The latter predicts a nonzero mean net strangeness even in $$\mathcal{Q}_{\mathcal{R}} = \sum_{h \in \mathcal{R}} \mathcal{Q}_h$$9- and $\langle \mathcal{Q}_{\mathcal{R}} \rangle$0-initiated jets, which carry $\langle \mathcal{Q}_{\mathcal{R}} \rangle$1. Combining these relations eliminates $\langle \mathcal{Q}_{\mathcal{R}} \rangle$2 and yields the jet-level Gell-Mann–Nishijima relation,

$\langle \mathcal{Q}_{\mathcal{R}} \rangle$3

with $\langle \mathcal{Q}_{\mathcal{R}} \rangle$4 the initiating-quark isospin. The paper is careful to distinguish this from the exact particle-level identity obtained by summing Gell-Mann–Nishijima over the measured region; the nontrivial content is the isospin-retention relation $\langle \mathcal{Q}_{\mathcal{R}} \rangle$5, which follows from isospin symmetry of the screening cloud alone and requires fewer assumptions than the probability extractions. It thus serves as a model-insensitive closure test.

Monte Carlo validation

The framework is validated with Pythia 8.312 (Lund string fragmentation) and Herwig 7.3 (cluster hadronization) in $\langle \mathcal{Q}_{\mathcal{R}} \rangle$6 annihilation with a thrust cut $\langle \mathcal{Q}_{\mathcal{R}} \rangle$7 selecting the dijet limit. Three results stand out. First, hadronization induces clear, energy-independent shifts of $\langle \mathcal{Q}_{\mathcal{R}} \rangle$8 away from $\langle \mathcal{Q}_{\mathcal{R}} \rangle$9 across $n$0–$n$1 GeV, while the difference $n$2 remains consistent with $n$3 over the full range, as predicted by flavor-symmetry arguments. Second, both generators reproduce the jet Gell-Mann–Nishijima relation at parton and hadron level across all energies. Third, scanning the Pythia parameters probStoUD (strange-pair rate) and probQQtoQ (diquark rate) produces linear families in the $n$4 flow planes whose slopes yield mutually consistent extractions of $n$5: for example, at the default probStoUD $n$6, the three projections give $n$7, $n$8, and $n$9, in good agreement with the input parameter. Herwig exhibits the same behavior under its own parameterization, indicating that the relations are not artifacts of a specific fragmentation algorithm.

The baryon channel behaves as predicted: $D_\epsilon(\bar{n})$0 is nearly independent of the strange-pair parameters, and varying probQQtoQ from 0.010 to 0.800 moves $D_\epsilon(\bar{n})$1 from 0.0139 to 0.3298. Notably, the extracted $D_\epsilon(\bar{n})$2 values do not equal probQQtoQ itself, because that parameter weights diquark production only at eligible steps of the state-dependent fragmentation history—a caveat the authors state explicitly rather than glossing over. An event-level cross-check using the topology estimator $D_\epsilon(\bar{n})$3 agrees with the moment-based extraction within roughly 10% over the full scan, supporting the two-channel minimal-screening approximation while quantifying its residual correction.

Extensions to correlators and heavy flavor

The appendices demonstrate that the formalism extends beyond inclusive hemisphere moments. For the one-point charge correlator (OPCC), the parton-level flavor ratios reduce to ratios of initiating-quark charges, and the hadron-level ratios—for instance $D_\epsilon(\bar{n})$4 versus the naive partonic value $D_\epsilon(\bar{n})$5—agree between the OPCC measurement at 5–10 TeV and the inclusive hemisphere result at 1 TeV, confirming observable independence. A bin-by-bin differential version of the Gell-Mann–Nishijima relation also holds, though its normalization currently borrows the one-point energy correlator at leading-log accuracy, leaving a precise normalization calculation open. The framework further predicts that heavy-flavor jets retain the charm and bottomness of the initiating quark—since nonperturbative heavy-pair excitation is strongly suppressed—while their light-flavor flows mirror those of $D_\epsilon(\bar{n})$6- and $D_\epsilon(\bar{n})$7-jets respectively, subject to generalized Gell-Mann–Nishijima relations including heavy quantum numbers.

Limitations and open questions

Several assumptions constrain the quantitative reach of the framework. The extraction of $D_\epsilon(\bar{n})$8 rests on the conjecture that higher screening clouds (beyond one quark and one anti-diquark) are negligible; the ~10% discrepancy between the moment-based and topology-based estimators measures the residual correction but does not establish the size of genuinely higher-triality configurations. The identification of the conditional endpoint ratio $D_\epsilon(\bar{n})$9 with the homogeneous vacuum ratio $$\langle Q_{\rm flow}\rangle_f = Q_f - \langle Q_D\rangle_f$$0 assumes screening probabilities are uniform over the celestial sphere, and the flavor-independence of screening assumes no correlation with the initiating flavor beyond what isospin symmetry enforces. The extracted $$\langle Q_{\rm flow}\rangle_f = Q_f - \langle Q_D\rangle_f$$1 and $$\langle Q_{\rm flow}\rangle_f = Q_f - \langle Q_D\rangle_f$$2 are effective quantities tied to each generator's internal parameterization—they agree with Pythia's probStoUD but not with probQQtoQ—so their interpretation as fundamental vacuum probabilities requires experimental validation. Finally, the extension to hadronic and nuclear collisions, where color reconnection, medium interactions, recombination, and baryon-junction transport may modify the effective screening weights, remains qualitative; whether the defect-flow redistribution approaches a thermodynamic limit with increasing system size is posed as an open question rather than established.

Conclusion

This work establishes conserved-current flows in jets as renormalization-scale-independent, flavor-resolved probes of vacuum hadronization. By combining the geometric picture of the Wilson-line puncture with the color-triality selection rule, the authors derive invertible relations linking the joint $$\langle Q_{\rm flow}\rangle_f = Q_f - \langle Q_D\rangle_f$$3, $$\langle Q_{\rm flow}\rangle_f = Q_f - \langle Q_D\rangle_f$$4, $$\langle Q_{\rm flow}\rangle_f = Q_f - \langle Q_D\rangle_f$$5 shifts to the strange-to-light and diquark-to-quark vacuum excitation ratios, together with the assumption-light jet Gell-Mann–Nishijima closure relation. Monte Carlo studies with two independent hadronization models reproduce the predicted linear correlations and recover effective screening parameters consistently across three pairwise projections. The measurements required—net quantum-number flows in flavor-tagged jets—are within reach of existing $$\langle Q_{\rm flow}\rangle_f = Q_f - \langle Q_D\rangle_f$$6 and collider capabilities, offering a route to constraining hadronization microphysics complementary to inclusive hadron-yield fits.