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REVIEW 2 major objections 5 minor 148 references

MadSONS automates tree-level production of S- and P-wave quarkonia and leptonia in MadGraph, using dual-number projectors and momentum reshuffling so physical bound-state masses enter the phase space.

Reviewed by Pith at T0; open to challenge. T0 means a machine referee read the full paper against a public rubric. the ladder, T0–T4 →

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2026-07-30 22:27 UTC pith:VKCZ2PUK

load-bearing objection Solid LO automation of P-wave NRQCD/NRQED in MadGraph via dual numbers, with honest validation; the mass-reshuffling claim is useful but process-dependent and oversold in the abstract. the 2 major comments →

arxiv 2607.26739 v1 pith:VKCZ2PUK submitted 2026-07-29 hep-ph hep-ex

Automated NRQCD and NRQED simulations of quarkonium and leptonium production with P-wave states and physical-mass effects

Luca Maxia , Hua-Sheng Shao , Lukas Simon This is my paper
classification hep-ph hep-ex
keywords NRQCDNRQEDquarkonium productionleptoniumP-wave statesautomatic differentiationmomentum reshufflingMonte Carlo event generation
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

This paper delivers a MadGraph module that generates arbitrary tree-level events for non-relativistic quarkonium and leptonium states, including P-wave configurations that earlier automation left out. Orbital angular-momentum projections are done with multivariate dual numbers, so the required derivatives of the amplitude are exact and numerically stable. A momentum-reshuffling step lets the phase space use measured bound-state masses while the matrix element still evaluates on-shell constituents. The authors validate amplitudes and cross sections against an independent generator and show that the physical-mass treatment lowers the LO single-parton prediction for J/ψ+ψ(2S) at the LHC, bringing it into better agreement with LHCb while leaving room for double-parton scattering. The result is a practical, shower-compatible tool for collider studies of heavy quarkonia and exotic leptonic atoms.

Core claim

Fully automated LO event generation for arbitrary numbers of S- and P-wave NRQCD/NRQED bound states is now possible inside MadGraph: colour, spin, orbital, and total-angular-momentum projectors are applied automatically, orbital derivatives are evaluated with dual numbers, and a momentum-reshuffling map puts physical meson or leptonium masses into the phase space without breaking the on-shell conditions required by the short-distance amplitude.

What carries the argument

Multivariate dual numbers for the orbital projector (exact automatic differentiation of the amplitude with respect to relative quark or lepton momenta) together with initial- or final-state momentum reshuffling that maps physical-mass phase space onto constituent-mass kinematics for the matrix element.

Load-bearing premise

The claim that keeping the leading-order matrix element at constituent-sum masses while only fixing the phase space to physical masses captures the main physical-mass and relativistic effects, without large compensating corrections in the amplitude itself.

What would settle it

Recompute the same J/ψ+ψ(2S) fiducial distributions with consistent higher-order relativistic corrections in both the amplitude and the phase space (or with full NLO QCD); if the physical-mass improvement relative to equal-mass LO disappears or reverses against LHCb, the reshuffling approximation fails.

Watch this falsifier — get emailed when new claim-graph text bears on it.

If this is right

  • Any tree-level process with an arbitrary number of S- and P-wave quarkonia or leptonia can be generated as unweighted LHE events and showered with standard multipurpose generators.
  • Physical production thresholds and mass splittings (for example J/ψ versus ψ(2S), or colour-octet intermediates) are respected in the phase space instead of forcing a common 2mQ mass.
  • Residual differences between initial- and final-state reshuffling prescriptions become a quantifiable extra theory uncertainty for phenomenology.
  • Colour-octet P-wave and feed-down channels that matter for inclusive yields and global LDME fits become routinely accessible in automated multi-onium final states.
  • Closed-form NRQED checks for P-wave positronium and ditauonium production are reproduced at the permille level, confirming the projectors for pure QED bound states.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The dual-number design is already shaped so that the same differentiation path can later support NLO real-emission subtractions and higher-L (D-wave) or explicit v² matrix-element corrections with little rewrite.
  • Processes near threshold or with large mass splittings (double excited charmonium, Bc with open heavy flavour) will show the largest practical gain from reshuffling and should be prioritised in the next round of data comparisons.
  • Once NLO automation exists, the same mass-treatment uncertainty band can be carried through to precision LDME extractions and DPS-versus-SPS separations at the LHC and future e⁺e⁻ facilities.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

2 major / 5 minor

Summary. The paper presents MadSONS, an extension of MadGraph5_aMC@NLO that automates LO tree-level event generation for arbitrary numbers of S- and P-wave quarkonia (NRQCD) and leptonia (NRQED). Building on a prior S-wave implementation, it adds orbital and total angular-momentum projectors, evaluates the orbital derivatives with multivariate dual numbers, and introduces a momentum-reshuffling procedure that generates phase space with physical bound-state masses while evaluating the matrix element at constituent-sum on-shell masses. Validation against HELAC-Onia reaches double-precision agreement on |M|^{2} at individual phase-space points and MC-consistent integrated cross sections; independent analytic NRQED cross sections for e^{+}e^{-}→L(2P)+γ are reproduced at the permille level or better. Phenomenological illustrations cover single- and double-quarkonium hadroproduction, charmonium production at B-factory energies, and a dedicated J/ψ+ψ(2S) study at the LHC that includes feed-down and is interfaced to Pythia8, yielding a lower SPS cross section and an improved description of LHCb data relative to conventional equal-mass treatments.

Significance. If the implementation is correct, MadSONS fills a genuine gap: public process-independent tools for P-wave quarkonia are limited (MadOnia is unmaintained and single-onium; HELAC-Onia has multiplicity restrictions), and none are embedded in the modern MG5_aMC ecosystem with dual-number differentiation and physical-mass phase space. The dual-number approach is well chosen for future NLO extensions. The HELAC-Onia and analytic NRQED validations are strong, external, and quantitative. The J/ψ+ψ(2S) application is a concrete demonstration that the mass treatment can matter for data comparison. The work is therefore a solid and useful technical contribution to automated NRQFT phenomenology, even though the mass-reshuffling approximation remains uncontrolled.

major comments (2)
  1. [§3.3, eq. (3.20); §5.1; §5.3/Table 10] The central claim of an “improved mass treatment” (title, abstract, §3.3) rests on the approximation A(N,0)(ṙ)≈A(reshuffled momenta) in eq. (3.20). The paper’s own benchmarks show this is not uniformly beneficial: in §5.3/Table 10 several exclusive e^{+}e^{-} channels (J/ψ+χcJ, χc1+hc) give Δσ_resh smaller than the naive ΣΔMQ/MQ scaling, with FS and IS reshuffling moving the cross section in opposite directions; for the CO channel ψ(2S)+g, Δσ_resh≈1% despite ΔM[8]/M[8]∼18%. In §5.1 the χb0+γ discussion notes that reshuffling can increase quark-mass sensitivity. The abstract and conclusions should state clearly that the procedure is an uncontrolled, process-dependent approximation whose domain of validity is not established, and that the residual FS–IS difference (Δσ_algo) must be treated as an extra theory uncertainty rather than a systematic improvement.
  2. [§5.4, Fig. 3, eqs. (5.3)–(5.5)] In the flagship J/ψ+ψ(2S) application (§5.4, Fig. 3, eqs. 5.3–5.5) the LO scale bands are +190%/−90%. The claim of an “improved description of the LHCb data” is therefore driven almost entirely by the downward shift of the central value under reshuffling. The text should quantify how much of that shift comes from direct production versus feed-down, report the no-reshuffling prediction on the same plots, and avoid overstating agreement given that NLO QCD, full relativistic corrections, and DPS are still absent.
minor comments (5)
  1. [Table 1, footnote 4] Table 1 caption and surrounding text: the B_c1(L/H) naming convention and the statement that they are pure 1P[1]1 / 3P[1]1 without mixing should be cross-checked against the PDG and lattice references already cited, and the distinction from physical mass eigenstates made more prominent.
  2. [§3.2] The dual-number array layout (eqs. 3.3–3.16) is correct but dense; a short explicit example for n=1 (single P-wave) and n=2 would help readers who wish to re-implement or extend the module.
  3. [§3.3, Appendix A] Default CO–CS mass shift of +200 MeV is taken from Pythia8 hadronisation; this choice should be stated as a tunable parameter in the main text (not only in the appendix) because it directly feeds the reshuffling thresholds.
  4. [§5, opening paragraphs] In §5 the LDME mass-rescaling (mc/1.55)^n is acknowledged as simplistic; a sentence on how a full refit would change the quoted Δσ_resh values would strengthen the parametric-uncertainty discussion.
  5. [passim] Minor typos and notation: “resctricted” (p. 5), “pri nc ip al” in Listing 1, and inconsistent use of M_Q vs m_Q+m_Q̄ for the same quantity in a few places in §4.2.

Circularity Check

0 steps flagged

No load-bearing circularity: tool paper validated against independent analytics and a distinct prior code path; mild same-author tooling lineage only.

full rationale

MadSONS is a methods/automation paper. The P-wave projectors (dual-number orbital derivatives, Clebsch–Gordan total-J) and the momentum-reshuffling map are constructive algorithms, not predictions derived from the quantities they are later compared to. Squared amplitudes and integrated cross sections are checked against HELAC-Onia (different internal method: dual numbers vs dedicated P-wave currents) at double-precision / MC-consistent level, and NRQED P-wave e+e− cross sections are checked against closed-form analytic formulae derived in the paper (eqs. 6.3–6.9), agreeing at permille level. LDMEs and default masses are taken as external inputs from the literature (Eichten–Quigg wavefunctions; published CO fits) and rescaled by stated mass powers when masses change—they are not fitted to the showcased LHC or B-factory observables and then re-predicted. The J/ψ+ψ(2S) LHCb comparison uses those fixed inputs plus physical-mass phase space; the lower SPS yield is a consequence of the mass treatment, not a fit to the same data. The only mild self-dependence is sequential: the prior S-wave MadSONS paper (arXiv:2510.26773, overlapping authors) and HELAC-Onia (same lead author). Neither is used as a uniqueness theorem or as the sole justification of a central physical claim; both are normal tooling lineage and cross-code checks. The soft point flagged by the skeptic (approx. A≈A_reshuffled, eq. 3.20) is an uncontrolled approximation about relativistic corrections—correctness risk, not circularity. Score 1 for non-load-bearing same-author code lineage; steps empty of genuine circular reductions.

Axiom & Free-Parameter Ledger

5 free parameters · 5 axioms · 1 invented entities

The work sits entirely inside standard NRQCD/NRQED factorisation. Load-bearing external inputs are published LDMEs, PDFs, and the on-shell projector formalism; the new methodological axioms are dual-number differentiation for PL and the ME/PS mass split under reshuffling. No new particles or forces are postulated.

free parameters (5)
  • Colour-octet and colour-singlet LDMEs ⟨O^B_n⟩ = Literature defaults, e.g. ⟨O^{J/ψ}_{3S[1]_1}⟩=1.16 GeV³; CO sets from Han et al. / Butenschoen-style fits
    Taken from external fits/wavefunctions (Tables 13–15); cross sections scale linearly with them. CO values especially drive several channels (e.g. χb0+γ, J/ψ+ηb).
  • Heavy-quark masses m_c, m_b (and lepton masses) = Defaults m_c=1.55 GeV, m_b=4.7 GeV (Table 11)
    Enter both short-distance ME and LDME rescaling (m/m0)^n; alternative setups in §5 vary them to physical half-meson masses.
  • CO–CS mass shift (+200 MeV) = 200 MeV
    Ad hoc match to Pythia8 hadronisation convention for intermediate CO states; affects reshuffling thresholds.
  • mom_resh_type / φ smoothing parameter ñ = Default mom_resh_type=1, ñ=100
    Chooses among initial-state vs two final-state recoil maps; ñ=100 in smooth φ (eq. 3.40). Difference is promoted to theory uncertainty Δσ_algo.
  • Renormalisation/factorisation scale choice μ_R=μ_F=H_T/2 = H_T/2 with factor-2 variations
    Standard but free; 7-point variation dominates LO error bands in the LHCb comparison.
axioms (5)
  • domain assumption NRQCD/NRQED factorisation: σ = Σ_n ŝ(CC'[n]) × ⟨O^B_n⟩ with velocity power counting up to v^7 (Table 1)
    Foundation of entire framework (§2); IR cancellation subtleties for P-waves assumed under control at LO tree level.
  • domain assumption Covariant projectors P[C], P_S, P_L=(ε*·d/dq)^L, P_J=Clebsch–Gordan, evaluated at relative momentum q=0 (eqs. 2.6–2.10)
    Standard Bodwin–Braaten–Lepage / Petrelli et al. projection method used to build A(N,0)(ṙ).
  • standard math Multivariate dual numbers with nilpotent ε_j give exact automatic derivatives for tree-level P_L without symbolic differentiation (§3.2)
    Mathematical tool choice; correctness checked numerically vs HELAC-Onia.
  • ad hoc to paper Matrix element depends only weakly on small external-momentum shifts, so A≈A(reshuffled on-shell constituents) while PS uses physical masses (eq. 3.20)
    Core justification of momentum reshuffling; authors note it is an incomplete stand-in for relativistic corrections and can fail to reduce mass sensitivity.
  • domain assumption LO α_s (and LO α for leptonia) plus parton shower is sufficient for the qualitative LHCb comparison after feed-down and Br(ψ(2S)→non-J/ψ) reweighting
    Explicit in §5.4; large scale bands and missing NLO/DPS are acknowledged.
invented entities (1)
  • MadSONS module (dual_mode, sm_onia boundstate extensions, mom_resh_type) independent evidence
    purpose: Software layer that automates projectors, LDMEs/masses cards, and reshuffling inside MG5_aMC
    Not a physical entity; engineering contribution. Independent evidence is the validation suite and intended public MG5 integration.

pith-pipeline@v1.2.0-daily-grok45 · 57961 in / 4159 out tokens · 82574 ms · 2026-07-30T22:27:56.249981+00:00 · methodology

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read the original abstract

We present the $\texttt{MadSONS}$ module, which extends the $\texttt{MadGraph5_aMC@NLO}$ framework by providing a fully automated solution for the simulation of arbitrary tree-level production processes involving non-relativistic bound states. More specifically, we focus on the production of quarkonia and leptonia in non-relativistic QCD and QED, respectively. This work constitutes the next step in the programme initiated in arXiv:2510.26773, where the colour and spin projectors required for S-wave states were first incorporated into $\texttt{MadGraph5_aMC@NLO}$. In the present development, we implement the remaining projectors associated with orbital and total angular momentum, thereby enabling the event generation of processes involving P-wave states. The derivatives required for the orbital-angular-momentum projection are evaluated using dual numbers. In addition, we implement a momentum-reshuffling procedure that accounts for physical bound-state mass effects in the phase space. The resulting framework applies to a wide range of collider environments while remaining compatible with standard multi-purpose Monte Carlo event generators for parton showering and hadronisation. As a first phenomenological application, we revisit $J/\psi + \psi(2{\rm S})$ production at the LHC and obtain an improved description of the LHCb data.

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