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A few-level Landau-Zener model fitted to the static charmonium spectrum predicts that transient magnetic-field sweeps strongly redistribute quarkonium occupation probabilities through nonadiabatic transitions and Stückelberg interference.

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2026-08-03 13:28 UTC pith:XOTQR6WC

load-bearing objection A new application of LZSM machinery to charmonium spectra, internally careful, but the few-channel mapping is never benchmarked against the full quark-model dynamics, leaving the quantitative claims conditional. the 3 major comments →

arxiv 2512.24072 v2 pith:XOTQR6WC submitted 2025-12-30 hep-ph nucl-th

Landau-Zener-St\"uckelberg-Majorana dynamics of magnetized quarkonia

Ahmad Jafar Arifi , Kei Suzuki This is my paper
classification hep-ph nucl-th
keywords charmoniumLandau-Zener transitionStückelberg interferencetime-dependent magnetic fieldquarkoniumnonadiabatic dynamicsoccupation probabilityavoided level crossing
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 tries to establish that the real-time response of charmonia to a time-dependent magnetic field is dominated by Landau-Zener physics: as the field sweeps through avoided level crossings, probability is transferred between states at a rate set by the sweep speed and the crossing gap. To make this quantitative, the authors fit a two- to five-channel Landau-Zener Hamiltonian, with diagonal energies linear in eB and constant nearest-neighbor couplings, to a static quark-model charmonium spectrum, then solve the time-dependent Schrödinger equation for linear-ramp, Gaussian-decay, and Gaussian-pulse field profiles. They find that fast sweeps leave the system nearly diabatic, slow sweeps adiabatic, and that a symmetric pulse produces Stückelberg interference whose final outcome is highly sensitive to pulse width. A sympathetic reader would care because it says that the static level structure alone, once mapped to LZ parameters, can predict which charmonium state survives a transient field, and it offers concrete benchmark predictions for future real-time lattice simulations.

Core claim

The central claim is that nonadiabatic dynamics, not just static level mixing, controls the occupation probabilities of charmonia in time-dependent magnetic fields. Using a Landau-Zener Hamiltonian whose parameters are extracted from the static quark-model spectrum, the paper shows that a single passage through the J/ψ–η'c avoided crossing transfers population according to the exponential Landau-Zener factor, while a double passage (Gaussian pulse) generates Stückelberg oscillations: the final probability after the second crossing depends sharply on the accumulated dynamical phase, varying from roughly 0.4 to 1.0 when the pulse width changes by less than one fm/c. Initial conditions also mat

What carries the argument

The multi-channel Landau-Zener Hamiltonian H(t) with diagonal elements H_ii = α_i eB(t)+δ_i and constant nearest-neighbor couplings Δ_{i,i+1}. The slopes α_i and couplings are fitted to the static charmonium spectrum, so the magnetic-field dependence enters only through the diagonal; the sweep rate ν_LZ=|α_1−α_2|ν_B then sets the adiabaticity. Its work: it reduces a hadronic avoided-crossing problem to the textbook two-parameter LZ language, with analytic benchmarks P_LZ=exp(−2πλ) for a single passage and the LZS phase-integral formula for double passages, which the paper verifies against numerical solutions of the time-dependent Schrödinger equation.

Load-bearing premise

The calculation rests on assuming that a handful of charmonium states whose energies move linearly with the field and whose mutual couplings stay constant faithfully reproduces the real time-dependent dynamics; if a transient field excites channels outside the retained few, or if the couplings themselves change with B, every quoted occupation probability changes.

What would settle it

Solve the full quark-model time-dependent Schrödinger equation with the same linear-ramp, Gaussian-decay, and Gaussian-pulse profiles and compare the final occupation probabilities (especially the sharp pulse-width dependence of the double-passage result) against the paper's few-channel LZ predictions; disagreement in the interference pattern would falsify the truncation.

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

If this is right

  • Sweep rate becomes a control parameter: ramps faster than about 1 GeV²/fm keep charmonia mostly in their initial diabatic state, while sweeps lasting hundreds of fm/c produce near-complete adiabatic transfer across the avoided crossing.
  • For a Gaussian pulse, the final J/ψ occupation oscillates with pulse width because of Stückelberg interference; small variations (γ_P from 6.5 to 7.2 fm/c) can swing the final probability from about 0.4 to 1.0, so the pulse shape effectively gates the transition.
  • The static spectrum alone suffices to predict real-time dynamics: LZ parameters extracted from the spectrum replace a full time-dependent solution of the underlying quark model, and the same fitting procedure could be applied to spectra from other models or lattice calculations.
  • Multi-channel effects are not small: including ηc and ψ' adds transition pathways, and in the slow-sweep regime the nearby ηc channel causes large-amplitude coherent oscillations in diabatic occupation probabilities that a two-channel model misses.
  • The predicted double-passage interference provides a benchmark observable for future sign-problem-free real-time simulations of gauge theories in time-dependent magnetic fields.

Where Pith is reading between the lines

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

  • If the LZ mapping survives a direct comparison with the full time-dependent Schrödinger evolution of the parent quark model, the paper's interference curves imply that the final charmonium state in a transient field is a sensitive probe of the field's temporal profile — a way to 'reconstruct' the pulse shape from the dilepton or photon yield of J/ψ versus ηc, which the paper does not compute.
  • The strong sensitivity to pulse width at intermediate sweep rates means that in realistic heavy-ion environments, where the magnetic field fluctuates event by event, the charmonium final state may vary strongly from event to event; this is an inference from the paper's LZS formula, not a claim the paper makes.
  • A testable extension: prepare the same few-channel LZ Hamiltonian with parameters fitted to lattice or other model spectra and compare the predicted Stückelberg oscillation period with the quark-model prediction, which would reveal how model-dependent the interference pattern is.
  • The paper's neglect of decay suggests its occupation probabilities describe coherent redistribution within a few bound states; coupling to continuum channels could damp the oscillations, so the LZS pattern is a clean prediction only on time scales shorter than charmonium lifetimes.

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

3 major / 6 minor

Summary. The paper proposes to describe the time evolution of charmonium states in time-dependent magnetic fields by a multi-channel Landau-Zener Hamiltonian. The diagonal energies are assumed linear in eB(t), the off-diagonal couplings are taken constant, and the parameters are fitted to static charmonium spectra obtained from a nonrelativistic quark model. The authors then solve the time-dependent Schrödinger equation for linear ramps, Gaussian decays, and Gaussian pulses, computing occupation probabilities in diabatic and adiabatic bases for two- to five-channel models. They find sweep-rate-dependent Landau-Zener transitions and Stückelberg interference, and conclude that nonadiabatic dynamics strongly influences charmonium occupation probabilities, with implications for future lattice simulations.

Significance. If the LZ mapping were validated against the full quark-model dynamics, the paper would provide a simple and transparent framework for estimating nonadiabatic transition probabilities of magnetized quarkonia. The numerical solver is cross-checked against analytic parabolic-cylinder solutions and the LZS formula in Appendix A, and the static-spectrum fitting is systematic. However, because the mapping is unbenchmarked and the off-diagonal coupling in the parent model is B-dependent, the central physical claim is conditional. The paper is a useful model study but does not yet establish the stated conclusion for actual charmonia.

major comments (3)
  1. [Sec. II.B, Eq. (6) and Eq. (4)] The central mapping from the parent quark model to the multi-channel LZ Hamiltonian is not benchmarked. In the microscopic Hamiltonian, the spin-mixing matrix element is explicitly linear in B and vanishes at B=0 (Eq. (4)), whereas Eq. (6) takes off-diagonal couplings Δ to be independent of B. The fits in Table I are made to static eigenvalues only, and for the two- and three-channel models the fit excludes the gray regions in Fig. 1, so the LZ Hamiltonian is not constrained to reproduce the quark-model dynamics at all B values reached in the time evolution. Since the final probabilities are controlled by λ=Δ²/|ν_eff| in Eq. (A6), they are direct consequences of the assumed Δ, α_i, and δ_i. The paper calls the construction approximate (Sec. II.B) but does not quantify the error; without a comparison to the full TDSE of the parent Hamiltonian (e.g., the works cited as Refs. [57,59,61,62])
  2. [Sec. II.C, Table I, Figs. 3–10] The initial states used in the time evolution are not, at eB=0, the physical charmonium states when the fit parameters are inserted into Eq. (6). For the two-channel model, H(0) has eigenvalues (δ1+δ2)/2 ± sqrt((δ1−δ2)^2/4+Δ^2) ≈ 2.974 and 3.322 GeV, whereas the physical J/ψ and η'_c masses are ≈3.097 and ≈3.637 GeV. The three-channel H(0) similarly does not reproduce the η_c and J/ψ masses because of the nonzero Δ1,2. Consequently, a diabatic initial vector labeled ψ=(1,0,...,0)^T is not a pure physical J/ψ at t=0, and the probabilities plotted as P_dia[J/ψ], P_dia[η_c], etc., cannot be directly interpreted as occupation probabilities of those mesons unless the unitary relation between the LZ diabatic basis and the physical spin basis is specified. This is especially important for the Gaussian-decay and pulse profiles, where the field returns to zero and the final observables are define
  3. [Table I, 4- and 5-channel fits] The multi-channel parameter extraction is not robust: Δ3,4 changes from −0.012 GeV in the four-channel model to +0.206 GeV in the five-channel model, and the corresponding α3, δ3, δ4 also shift substantially. Since the dynamics in Figs. 4–7 depend on these couplings, the lack of stability with respect to adding one channel calls into question the quantitative reliability of the multi-channel results. The authors should discuss this sensitivity and, ideally, provide an estimate of the parameter uncertainties from the fitting procedure, or restrict the conclusions to the two- and three-channel models.
minor comments (6)
  1. [Fig. 1 caption] The last line 'c 1st, J/ 2nd, ′ c 3rd, ′ 4th' is garbled; the level labels should be written out as η_c, J/ψ, η'_c, ψ'.
  2. [Table I] The numerical values run together (e.g., '0.562−0.049' for α1 α2); use separate aligned columns with explicit signs so that negative entries are unambiguous.
  3. [Figs. 3–10] Axis labels such as 'B = 1.00 [GeV2/fm]' should use ν_B for the linear-ramp slope to match Eq. (9) and avoid confusion with the magnetic field itself.
  4. [Sec. III.B.2] 'no adibatic transitions' should be 'nonadiabatic transitions'; the sentence 'no adibatic transitions involving the ψ′ state emerges predominantly at large magnetic fields' is ungrammatical and should be rewritten.
  5. [Sec. II.B] The sentence 'Although one could in principle construct H(t) directly from the full quark-model Hamiltonian, including the explicit time dependence induced by the magnetic field as practiced in the previous study [62]' is grammatically unclear; suggest rewriting.
  6. [Appendix A, Eq. (A10)] It would help to state explicitly that P_LZS is the transition probability between the two diabatic states and to specify the initial state used for the formula, since the text refers to 'the transition probability' without saying transition from which state to which.

Circularity Check

0 steps flagged

No significant circularity: the occupation probabilities are genuine solutions of a fitted effective Hamiltonian, and the self-citations supply input spectra, not the conclusions.

full rationale

The paper's chain is: (1) take an externally computed static charmonium spectrum [28,29]; (2) fit the parameters of the multi-channel Landau-Zener Hamiltonian in Eq. (6) to that spectrum (Table I); (3) solve the time-dependent Schrödinger equation, Eq. (5), with that Hamiltonian for various B(t) profiles; (4) report the resulting occupation probabilities. The dynamical observables -- P_dia and P_adi in Eqs. (10)–(11) -- are not quantities used in the fit; they are solutions of the time-evolution problem and depend additionally on the pulse/ramp profile and initial condition. Thus the final probabilities are not equal by construction to the fitted spectrum. The use of the standard LZ formula Eq. (A6) and LZS formula Eq. (A10) in Appendix A is an internal consistency check of the numerical integrator against known analytic solutions of the same two-level Hamiltonian, not a claim that the charmonium mapping is independently validated. The self-citations [28,29] are load-bearing only as sources of the input static spectrum and quark-model parameters, not as derivations of the dynamical results; no uniqueness theorem or methodological ansatz is imported from those works. The paper explicitly acknowledges its mapping is approximate ('Although approximate, this construction is particularly well suited...') and that decay effects are neglected, which are limitations/correctness risks rather than circular steps. Because no prediction reduces to a fitted parameter by the paper's own equations, and no load-bearing argument is justified solely by a self-citation, the circularity score is 0.

Axiom & Free-Parameter Ledger

4 free parameters · 5 axioms · 0 invented entities

Everything the central claim rests on: up to 15 fitted LZ parameters (5-channel model), the adopted quark-model potential parameters, the untested constant-coupling and linear-slope ansatz of Eq. (6), the nearest-neighbor truncation, and the parent static-spectrum computation from the authors' earlier work. No new physical entities are postulated.

free parameters (4)
  • LZ diagonal slopes α_i (per channel model) = α1=0.562, α2=−0.049 GeV⁻¹ (2ch); 5ch: −0.177, 0.553, −0.065, 0.0017, 0.626
    Fitted to the static quark-model spectrum; they set the LZ sweep rate and hence all transition probabilities via Eq. (A6).
  • LZ level offsets δ_i = δ1=2.999, δ2=3.645 GeV (2ch); 5ch: 2.991, 3.011, 3.798, 3.905, 3.677
    Fitted intercepts that set the crossing positions in eB; the 5-channel offsets are not mass-ordered, suggesting the fit is partly degenerate.
  • LZ nearest-neighbor couplings Δ_i,i+1 = Δ1,2=0.028 GeV (2ch); 5ch: 0.053, 0.041, 0.206, 0.006; 4ch gives Δ3,4=−0.012
    Constant off-diagonal mixings fitted to avoided-crossing gaps; Δ² enters the LZ exponent directly. The 4ch vs 5ch discrepancy for Δ3,4 shows fit instability.
  • Parent quark-model potential parameters = m_c=1.784 GeV, A=0.713 GeV, √σ=0.402 GeV, β=0.4778 GeV, Λ=1.020 GeV², C=−0.5693 GeV
    Adopted (not re-fit) from Refs. [28,29]; they determine the static spectrum that is the sole input to the LZ fit.
axioms (5)
  • domain assumption Nonrelativistic constituent-quark Hamiltonian (Eqs. 1–3) with the Eichten–Barnes–Godfrey–Swanson potential (Eq. 2) and parameters from Refs. [28,29] is the correct parent theory for charmonia in magnetic fields.
    The static spectrum to which the LZ models are fitted comes from this model only; no lattice or experimental cross-check is provided.
  • domain assumption The cylindrical Gaussian expansion (CGEM) spectra of Refs. [28,29] are numerically accurate.
    Sec. II.A adopts these as exact inputs; the present paper does not reproduce them.
  • ad hoc to paper Diabatic energies are linear in eB(t) and off-diagonal couplings are field-independent (Eqs. 6–7).
    The central ansatz of Sec. II.B, labeled 'approximate' by the authors and never checked against the full TDSE; Fig. 1 shows visibly curved levels, so linearity holds only on the fitted windows.
  • domain assumption Truncation to 2–5 nearest-neighbor-coupled channels captures the dynamics; decays and continuum channels can be neglected.
    Stated in Sec. IV ('we neglected realistic decay effects') and Sec. II.B; the error incurred by truncation is never quantified.
  • standard math Standard Landau-Zener analytic solution (parabolic cylinder functions, Eqs. A3–A6) and LZS interference formula (Eq. A10).
    Textbook results used as benchmarks in Appendix A and for the double-passage analysis.

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read the original abstract

The mass spectrum of hadrons in magnetic fields features avoided level-crossing structures arising from the mixing of spin eigenstates. In this work, we investigate the impact of level-crossing dynamics of charmonia subjected to time-dependent magnetic fields, where we particularly focus on the occupation probabilities of two or more states as they undergo transitions at avoided crossings. Using a static spectrum of charmonia in magnetic fields, we construct a multi-channel Landau-Zener Hamiltonian. Within this framework, we analyze the time evolution under several representative magnetic-field profiles, including linear ramps and Gaussian decays corresponding to single-passage dynamics, as well as Gaussian pulses realizing double-passage dynamics, and compute the occupation probabilities over a wide range of sweep rates and initial conditions. Our results show that nonadiabatic dynamics, including Landau-Zener transitions and St\"uckelberg interference, strongly influences the occupation probabilities of charmonia. These findings provide new insights into the real-time dynamics of magnetized hadrons and offer useful guidance for future lattice simulation studies.

Figures

Figures reproduced from arXiv: 2512.24072 by Ahmad Jafar Arifi, Kei Suzuki.

Figure 1
Figure 1. Figure 1: FIG. 1. Comparison between the eigenvalues of the multi-channel LZ Hamiltonians and the quark-model spectrum for the [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. Schematic illustration of the avoided crossings in the two-channel model and the driving magnetic-field profiles [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. Occupation probabilities in the two-channel model for a single passage at three sweep rates with a [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. Occupation probabilities in the three-channel model for a fast sweep, shown for three different initial diabatic states: [PITH_FULL_IMAGE:figures/full_fig_p007_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. Same as Fig [PITH_FULL_IMAGE:figures/full_fig_p008_5.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7. Occupation probabilities in the five-channel model [PITH_FULL_IMAGE:figures/full_fig_p009_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: FIG. 8. Occupation probabilities in the two-channel model [PITH_FULL_IMAGE:figures/full_fig_p010_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: FIG. 9. Diabatic occupation probabilities [PITH_FULL_IMAGE:figures/full_fig_p011_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: FIG. 10. Occupation probabilities in the three-channel model for three different initial states in a Gaussian pulse, showing fast [PITH_FULL_IMAGE:figures/full_fig_p012_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: FIG. 11. Comparison between the analytic Landau-Zener prediction and the numerical solution of the time-dependent [PITH_FULL_IMAGE:figures/full_fig_p013_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: FIG. 12. Double-passage Landau-Zener-St¨uckelberg (LZS) interference for a triangular magnetic-field pulse. (Left panel) [PITH_FULL_IMAGE:figures/full_fig_p014_12.png] view at source ↗

discussion (0)

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Reference graph

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    Figure 3 shows the time evolution for three representative sweep rates, starting from aψ= (1,0) T initial state

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    Multi-channel system In the case of three or more channels, the dynamics become richer due to the presence of multiple avoided 9 0 25 50 75 100 125 150 175 200 Time t [fm/c] 0.0 0.2 0.4 0.6 0.8 1.0Probability B = 0.01 [GeV2/fm] adi = (1, 0)T 0 25 50 75 100 125 150 175 200 Time t [fm/c] 0.0 0.2 0.4 0.6 0.8 1.0Probability adi = (0, 1)T FIG. 6. Occupation pr...

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