Quarkonium Parton Shower in Herwig 7

M.R. Masouminia; P. Richardson

REVIEW 4 major objections 6 minor 1 cited by

Quarkonium Parton Shower in Herwig 7

T0 review · 4 major / 6 minor · reviewed 2026-08-05 · deepseek-v4-flash

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Pith's one-line read This paper claims that a fully automated quarkonium parton shower in Herwig 7, built on NRQCD factorisation with spin-colour projections, reproduces LHC charmonium and bottomonium data better than the default shower and will be released in

desk verdict Useful tool paper: the new thing is the Herwig 7 implementation, not the physics; the splitting functions check out, but the headline 'improved agreement' is partly tuning to the same data, and the octet kernel is an effective constant rather than a real fragmentation function. read the letter →

arxiv 2508.06307 v1 pith:AL5VOTMM submitted 2025-08-08 hep-ph

M.R. Masouminia , P. Richardson This is my paper

classification hep-ph

keywords quarkoniumproductionpartonshowerNRQCDfactorisationcolour-octetfragmentationcharmoniumbottomoniumsplittingfunctionsHerwig7

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

Quarkonium states—bound heavy-quark pairs such as J/ψ, χc, ψ(2S), and Υ—have been hard to simulate inside general-purpose event generators because their production mixes perturbative QCD with non-perturbative binding. This paper claims to close that gap in Herwig 7 by implementing a process-independent parton shower built on non-relativistic QCD (NRQCD) factorisation: perturbative short-distance coefficients for S-, P-, and D-wave splittings are combined with non-perturbative matrix elements, and the resulting splitting functions are inserted into the angular-ordered shower. The implementation covers colour-singlet quark and gluon fragmentation, colour-octet gluon fragmentation, and diquark channels, and it preserves spin correlations and polarisation while including feed-down from excited states. If the claim holds, Herwig 7 can produce fully differential, hadron-level predictions for prompt charmonium and bottomonium at the LHC, and the tuned shower is reported to agree better with measured transverse-momentum spectra than the default model. The code is scheduled to become public with Herwig 7.4.0.

What carries the argument

The key machinery is the set of NRQCD splitting functions embedded in Herwig's existing angular-ordered final-state shower. Each branching $a\to b\,O_c(nL_J)$ is written in quasi-collinear Sudakov kinematics and assigned a spin- and colour-averaged probability obtained from the projected short-distance amplitude; non-perturbative binding enters through the radial wavefunction $R(0)$ (S-wave), $R'(0)$ (P-wave), or $R''(0)$ (D-wave), or through the octet matrix elements $\langle O_8(nL_J)\rangle$ in the fixed-probability $g\to O_8$ channel. The distinctive device is the Landau-Yang surrogate for $g\to g\,O_1({}^3S_1)$, whose matrix element is fixed to unity because the true amplitude vanishes

What would settle it

With the tuned octet matrix elements held fixed, compute the prompt $J/\psi$ polarisation coefficients as functions of $p_T$ at 13 TeV and compare with CMS or ATLAS data; because polarisation is sensitive to the spin structure of the octet channel, a persistent mismatch would falsify the fixed-branching colour-octet model rather than just its normalisation.

Watch

Extended reading notes

Core claim

The central claim is that quarkonium production can be handled as a set of quasi-collinear branchings inside an angular-ordered parton shower, with each allowed $^{2S+1}L_J$ state of the heavy $Q\bar{Q}$ pair assigned a splitting function derived from NRQCD. The authors derive and implement the splitting kernels for colour-singlet $q\to q' O_1$ and $g\to g O_1$ channels for S, P, and D waves, for colour-octet $g\to O_8$ transitions, and for heavy diquarks, using spin-colour projections built from Bethe-Salpeter wavefunctions and the radial wavefunction at the origin (or its derivatives). The $g\to g\,O_1({}^3S_1)$ channel, which vanishes by the Landau-Yang theorem, is replaced by a surrogate

Load-bearing premise

The load-bearing premise is that all colour-octet quarkonium production can be represented by one fixed branching probability per unit shower time, with S- and P-wave structure absorbed into long-distance matrix elements that are later tuned partly to the same LHC data used for validation; if that effective probability misrepresents real fragmentation, the high-transverse-momentum yields would change.

Editorial extensions

If this is right

Fully differential, hadron-level predictions for prompt quarkonium become available in a public general-purpose generator, so LHC measurements of $p_T$ spectra, rapidity, and feed-down can be compared directly with NRQCD-based models.
Spin correlations and polarisation are propagated through splittings and decays, so polarisation observables can be used as a discriminating test of the colour-singlet versus colour-octet decomposition.
The same shower machinery extends to doubly heavy baryons through diquark splittings, giving predictions for $\Xi_{cc}$, $\Xi_{bc}$, and $\Xi_{bb}$ production.
The 30 newly implemented splitting classes cover 103 registered splittings and are runtime-configurable, so users can switch channels on or off and vary mixing angles for $P$- and $D$-wave states without recompilation.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

Because the octet matrix elements are tuned using the same LHC data sets used for validation, the quoted agreement should be read as a test of the shower's kinematic structure rather than as an independent determination of the octet LDMEs. A cleaner check would use polarisation and 13 TeV measurements that played a smaller role in the fit.
The Landau-Yang surrogate is an explicit approximation. If the true $g\to gg\,O({}^3S_1)$ fragmentation is important at high $p_T$, then disabling the surrogate and comparing yields would quantify how much of the quoted agreement comes from this effective channel.
The same splitting-function machinery could be adapted to test the singlet-triplet mixing angles beyond the fixed defaults ($25^\circ$ for $P$, $34.4^\circ$ for $D$), or to study excited $B_c$ states, since the interface exposes those parameters.
Replacing the fixed-probability colour-octet treatment with explicit $g\to O_8$ splitting functions would provide a sharper test of whether the high-$p_T$ octet-dominated regime is captured dynamically rather than absorbed into tuning.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Short version: this is a tool paper, and a decent one. The genuinely new content is the automated Herwig 7 implementation of quarkonium splittings with spin-colour projections, singlet-triplet and D-wave mixing, diquark splitting classes, and a tuned parameter set. The analytic splitting functions are re-derivations that the authors check against Refs [59-63,68]; I did not find obvious discrepancies in the quoted forms. The section 3.6 comparisons, which are explicitly untuned, are useful evidence that the shower produces reasonable shapes even before fitting. Credit where due: this is real work and the community will use it.

The soft spots are real but not fatal. The colour-octet g->O8 splitting is a constant probability per unit evolution time (Eq. 3.50), independent of z and q^2. That is not the differential fragmentation function D_{g->O8}(z) of NRQCD factorisation; it is an effective vertex with the LDME absorbing the dynamics. Since those LDMEs are then tuned to the same ATLAS data used for the 'improved agreement' plots in Section 5, the high-pT validation is partly circular and tells you as much about tuning freedom as about short-distance physics. The Landau-Yang-forbidden g->g O1(3S1) channel is replaced by a unit-ME surrogate; the authors disclose it and state it is numerically negligible, which is honest, but a cross-check would be better. There are also no MC uncertainty bands anywhere, which weakens the agreement claims.

I would not call this a paradigm shift. It is an engineering-plus-phenomenology paper with a clear scope: build the shower, document the splitting functions, report tunings, and make first comparisons. In that frame the simplified octet treatment is acceptable if labelled as an effective model.

Who is this for? Anyone simulating quarkonium production with Herwig at the LHC, and people hunting for new physics with heavy-flavour final states. It deserves a serious referee. The referee should ask for: (1) a clean separation of tuning from prediction, e.g. fit to one subset of data and predict the rest, or a scan showing the sensitivity to the octet parameters; (2) uncertainty bands on the shower predictions; (3) a discussion of the z-dependence of the constant octet kernel versus the known fragmentation function, even if approximate. With those addressed, this becomes a citable reference rather than a promising preprint. I would bring it to the reading group and would cite it once the code ships.

Referee Report

4 major / 6 minor

Summary. The paper presents a new quarkonium parton shower implemented in Herwig 7, based on NRQCD factorisation. It derives and implements splitting functions for colour-singlet q -> q' O1 and g -> g O1 channels for S, P, and D waves, as well as colour-octet g -> O8 transitions and diquark splittings. The shower preserves spin correlations, includes feed-down, and is tuned to LHC data. The authors claim improved agreement with existing LHC data compared to a generic Herwig shower and to a Pythia8 quarkonium shower, and announce public release in Herwig 7.4.0.

Significance. If the central claim is robust, this would be a valuable public tool for heavy-quarkonium production in a general-purpose Monte Carlo generator. The perturbative singlet splitting functions for S, P, and D waves are a substantial technical contribution and appear to follow the standard NRQCD factorisation approach, with many analytic results stated to match earlier literature. The implementation covers a wide range of states and processes. However, the headline phenomenological claim ('improved agreement with existing LHC data') rests on a colour-octet model that is an ad hoc fixed-probability ansatz, with its LDMEs tuned to the very ATLAS data used for validation. This limits the current evidence for the predictive power of the NRQCD implementation, even though the tool itself may still be useful.

major comments (4)

[§3.4, Eq. (3.50)] The g -> O8 colour-octet transition is implemented as a fixed probability per unit shower time, P = π α_S(4m^2)/(24m^3) <O8>, with no dependence on the energy fraction z or the virtuality. NRQCD fragmentation into O8(3S1) has a non-trivial D_{g->O8}(z) (e.g. Braaten & Yuan, Phys. Rev. Lett. 71 (1993) 1673). A constant kernel samples z from a flat distribution, so the quarkonium momentum in the octet channel is not generated according to the short-distance coefficient. Since Section 3.4 states that colour-octet fragmentation dominates at high pT, and since the octet LDMEs in Table 2 are tuned to ATLAS data with pT > 10 GeV (Section 4), the high-pT agreement in Figures 7-12 may be a consequence of the tuning freedom rather than a test of NRQCD. The authors should either implement a z-dependent kernel or quantify the error introduced by the constant ansatz, e.g. by comparing with a z-depend
[§3.3.2] The g -> g O1(3S1) amplitude vanishes by the Landau-Yang theorem, and the paper replaces it with a 'technically convenient surrogate' whose matrix element is fixed to unity. This is an ad hoc element not derived from NRQCD. The text claims the channel is 'numerically negligible in all results presented here', but no evidence is shown: there is no plot isolating the surrogate contribution or a statement of its numerical size. If it is negligible, it should be removed or its contribution demonstrated; if it is not negligible, the unit-normalised surrogate introduces an uncontrolled systematic error into the g -> g O1 channel that is separate from the octet tuning.
[§4 and §5] The colour-octet LDMEs in Table 2 are adjusted by comparing to LHC data (ATLAS at 5.02 and 7 TeV, Refs. [84,85], listed in Tables A1/A2), and the same ATLAS data are then used to validate the 'tuned' shower in Figures 7-12. This is circular for the validation of the central claim: the improved agreement is partly built into the fit. The paper should validate on a subset of data not used in the tuning (e.g. a different experiment, energy, or rapidity range), or at minimum present the number of free octet parameters versus the number of data points and quantify the goodness of fit. Without this, the claim 'improved agreement with the existing LHC data' is not strong evidence for the physical content of the shower.
[Figures 4-12] None of the MC predictions are shown with uncertainty bands, either statistical or systematic (scale, PDF, shower parameter). The MC/Data ratio panels show only the data errors. Since the paper's main quantitative claim is that the tuned shower 'noticeably improves' agreement, the absence of any uncertainty estimate on the predictions makes it impossible to judge whether the residual discrepancies in the low-pT bins or the apparent agreement at high pT are significant. The authors should add at least MC statistical uncertainties and, ideally, a simple scale-choice or LDME variation band.

minor comments (6)

[Eq. (2.6b)] The prefactor 'i√6(m1+m2)' appears to be dimensionally inconsistent with the rank-3 polarisation tensor; likely a missing power of (m1+m2) in the denominator. Please check.
[Eq. (3.32) and related] The denominator is written 'π3M^5' in Eq. (3.32) while other D-wave formulas use π M^5; please verify whether the cube is intentional or a typo.
[Fig. 6, right panel] The y-axis label reads 'd²σ/dy', but the caption describes dσ/dy. Please correct the label.
[Abstract and Appendix B] The abstract says the shower is 'fully automated', but Appendix B states that no quarkonium splittings are enabled by default and the user must read OniumShower.in and register splitting objects. Please clarify the intended workflow so the claim matches the user experience.
[Section 3.6 vs Section 5] Section 3.6 states the figures there are 'untuned', while Section 5 shows tuned results. This is clear in the text but could confuse readers; consider labelling the earlier figures as 'pre-tuning' in captions.
[Section 4, Table 1] The text says 'values shown in bold font are directly extracted using our tuning framework', but in the table as presented no bold formatting is visible. Please ensure the font distinction appears in the published version or use a different marker.

Circularity Check

1 steps flagged · score 6.0 of 10

Improved agreement is a fit-quality statement: octet LDMEs are tuned to the same ATLAS data later shown as validation, and the g->O8 kernel is a fitted constant surrogate.

fitted input called prediction [Section 3.4 (Eq. 3.50), Section 4, Section 5.1/5.2, Tables A1/A2]
"Rather than computing the full production amplitudes and subsequently the splitting functions for octet production, a simplified approach is adopted ... Pg->O8(nLJ) = pi alpha_S(4m^2)/(24m^3) <0|O8(nLJ)|0> ... After the wavefunction parameters were tuned using decay widths, the colour-octet MEs were adjusted by comparing theoretical predictions of shower prompt production cross-sections to LHC data ... These data sets are used in tuning the non-perturbative parts of the octet states MEs in this study."

The g->O8 kernel is a constant with no z or q^2 dependence; all state dependence enters through <O8>. Table 2 sets one <O8> per final state (J/psi, psi(2S), chi_cJ, Upsilon(nS), chi_bJ) by fitting to LHC data listed in Tables A1/A2. Section 5.1/5.2 plots the same ATLAS 5.02/7 TeV data against the resulting 'Tuned Quarkonium Shower'. Consequently the normalisation of every octet-dominated channel is matched to the validation data by construction; the abstract's 'improved agreement with existing LHC data' is a fit-quality statement. The pT shapes are not fully fixed by the constant LDME, so the circularity is partial, but the headline phenomenological result does not independently test the NRQCD octet fragmentation physics.

full rationale

The central splitting-function derivation is largely self-contained: Eqs. (3.14)-(3.49) are computed from NRQCD spin-colour projections and cross-checked against independent published calculations (Refs. [59-63,68]), so those parts are not circular. No load-bearing self-citation chain or imported uniqueness theorem appears; self-citations to earlier Herwig showers are infrastructural. The circularity is confined to the validation of the phenomenological claim. Section 4 tunes colour-octet LDMEs to LHC data enumerated in Tables A1/A2, and Section 5 compares the resulting tuned shower to the same ATLAS datasets, so the claimed 'improved agreement' is an in-sample fit result rather than a prediction. The octet g->O8 implementation is explicitly a fixed-probability surrogate (Eq. 3.50) with all S/P-wave structure absorbed into the fitted MEs, further reducing the high-pT octet tail to a normalisation matched to the data. This is partially mitigated by the untuned comparisons in Section 3.6, which use generic parameters and illustrate the shower's structural impact, and by the fact that each LDME is only an overall normalisation, so the differential shapes are not literally forced. The g->gO1(3S1) Landau-Yang surrogate with ME=1 is an acknowledged ad hoc choice but is stated to be numerically negligible, so it is a limitation rather than a circular step. Overall score 6: one or more phenomenological predictions reduce to fitting the data used for validation.

Assumptions & free parameters 5 free parameters · 5 assumptions · 2 invented entities

The central claim rests on NRQCD factorization and the smoothness of the Bethe-Salpeter wavefunction, plus the kinematic assumptions of the Herwig angular-ordered shower. The listed parameters are non-perturbative inputs fitted to data. The invented entities are effective vertices introduced to avoid computing full amplitudes (surrogate g->g 3S1 and fixed-probability g->O8).

free parameters (5)

Color-singlet radial wavefunctions |R(0)|^2, |R'(0)|^2, |R''(0)|^2 = various GeV^3, e.g., 1.0285 for c cbar 1S (Table 1)
Non-perturbative inputs for splitting functions; some taken from Eichten-Quigg updates, some fitted to decay widths (bold entries in Table 1).
Colour-octet LDMEs <O8(nLJ)> = e.g., 1.09e-4 GeV^3 for g->J/psi (Table 2)
Tuned to LHC production data for J/psi, psi(2S), chi_cJ, Upsilon(nS), chi_bJ (Section 4).
Mixing angles theta_P1, theta_D2 = 25.0 degrees, 34.4 degrees
Chosen default values for 1P1-3P1 and 1D2-3D2 mixing (Sections 3.2.4, 3.2.6, Appendix B).
pT cutoff for tuning = 10 GeV
pT bins below 10 GeV neglected in tuning due to non-perturbative effects (Section 4).
Runtime knobs (EnhancementFactor, PDFmax, Cutoff) = configurable defaults
User-facing parameters in the Herwig interface (Appendix B) that affect splitting probabilities.

assumptions (5)

domain assumption NRQCD factorization: production/decay amplitudes factorize into short-distance coefficients and long-distance matrix elements with power counting in v
Used throughout Section 2 to derive splitting functions; relies on v<<1 for heavy quarkonia.
domain assumption Non-relativistic Bethe-Salpeter wavefunction with smooth k dependence; radial wavefunctions at origin encode bound state
Eqs (2.1), (2.7) and Section 2.
domain assumption Quasi-collinear kinematics and angular ordering of the Herwig shower can be applied to quarkonium splittings
Section 3.1, Eqs (3.1)-(3.6).
standard math Previous fragmentation function calculations cited (Braaten-Cheung-Yuan, Cheung-Yuan, Braaten-Yuan) are correct
Sections 3.2-3.3 state agreement with Refs [59-63,68].
standard math Standard QCD Feynman rules and SU(3) colour algebra
Used in amplitudes, e.g., Eq (3.8).

invented entities (2)

Surrogate g->g O1(3S1) vertex with matrix element fixed to unity
purpose: Stands in for the forbidden g->g O1(3S1) transition (Landau-Yang), representing unresolved extra gluon radiation
Section 3.3.2: the physical amplitude vanishes for on-shell kinematics; authors set the ME to unity as a 'technically convenient surrogate', claiming it is numerically negligible. No external validation of this approximation.
Effective g->O8 branching with fixed probability per unit evolution time
purpose: Model colour-octet quarkonium production without computing full amplitudes
Section 3.4, Eq (3.50): P = pi alpha_S(4m^2)/(24 m^3) <O8>. All structural differences absorbed into fitted LDMEs; no perturbative derivation provided.

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Cite this review

Pith. "Pith review of Quarkonium Parton Shower in Herwig 7." pith.science (2026). https://pith.science/paper/AL5VOTMM

@misc{pith2026250806307,
  author       = {Pith},
  title        = {Pith review of: Quarkonium Parton Shower in Herwig 7},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/AL5VOTMM}},
  note         = {Machine review of arXiv:2508.06307}
}

abstract

We present the implementation of a fully automated quarkonium parton shower in Herwig 7, based on non-relativistic QCD (NRQCD) factorisation with spin-colour projections. The framework systematically incorporates colour-singlet and colour-octet production mechanisms, gluon fragmentation, and diquark production processes. Perturbative short-distance coefficients are combined with non-perturbative NRQCD matrix elements to simulate heavy-quark bound state formation. Splitting functions for $S$-, $P$- and $D$-wave states are explicitly derived and integrated into the angular-ordered shower evolution. The implementation preserves spin correlations and polarisation effects while accurately accounting for feed-down contributions. Results demonstrate improved agreement with the existing LHC data. This quarkonium parton shower will become publicly available with the release of Herwig-7.4.0.

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Forward citations

Cited by 1 Pith paper

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