{"id":"bdc8d472-d1ba-4be3-96c5-9c098250d9e4","arxiv_id":"1908.11699","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":8,"one_line_summary":"Nonperturbative spin-dependent hybrid potentials are expressed as gluonic correlators, fitted to charmonium lattice data, and used to predict bottomonium hybrid spectra.","lead":"Some heavy exotic mesons may be quarkonium hybrids: a heavy quark, an antiquark, and an excited gluonic cloud. This paper derives formulas for the spin-dependent energies of these states and uses charmonium lattice data to predict bottomonium hybrid spin splittings.","discovery_kind":"first_principles","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The mv >> Lambda_QCD factorization is not controlled at the charm scale used for the fit; the bottomonium spin-splitting predictions inherit that unquantified error.","rationale":"The reader's weakest assumption identifies the same load-bearing issue: the derivation assumes mv >> Lambda_QCD, and the numerical analysis uses charmonium where this hierarchy is marginal. I agree with the reader's assessment that this makes the numerical comparison conditional rather than fatal. After reading the full manuscript, including the appendices, I found no additional internal error that would overturn the formal matching calculation; the C, P, T decomposition and the tensor reductions in Appendices A and B are extensive and internally plausible, though I did not re-derive every step. The manuscript itself flags the assumption in Sec. II and notes that the spin structure would be the same for mv ~ Lambda_QCD but the potential would be given by generalized Wilson loops. This self-acknowledged limitation means the central claim is precisely about the factorized regime, and the charm fit does not test that regime. The bottomonium predictions are therefore a prediction of an EFT whose main expansion parameter is not controlled at the extraction point. Because this is an addressable condition (direct lattice computation of the correlators, or use of the generalized Wilson-loop form for the fit), the appropriate verdict remains CONDITIONAL rather than ACCEPT or REJECT. I recommend no change to the reader's verdict, and I aggregate with the reader's weakest assumption rather than proposing a different objection.","tokens_in":36629,"tokens_out":5672,"duration_ms":52080,"concrete_test":"Compute the leading gluonic correlator entering Eq. (48) (that is, \\tilde U_B in Vnp(0)_SK = c_F \\tilde U_B / 2) directly on the lattice at the charmonium scale, using the Wilson-line technology of Refs. [34,40-43] adapted to the correlator in Eq. (43). Evaluate it for heavy-quark source mass near m_c = 1.5 GeV and typical r in the range 0.2-0.4 fm, and compare with the value implied by the Table II fits (Vnp(0)_SK/c_F * 2). If the direct correlator value differs from the fitted value by more than the estimated O(Lambda_QCD/m_c) correction (roughly 30%), the factorized form is not quantitatively reliable at the charm scale and the transferred bottomonium predictions should carry a correspondingly larger uncertainty.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim is that the nonperturbative spin-dependent hybrid potentials factorize into perturbative NRQCD coefficients (c_F, c_s) times flavor-independent gluonic correlators, enabling transfer of coefficients fitted to charmonium lattice data to bottomonium. This factorization and the truncation of the multipole expansion at order r^2 require the hierarchy mv >> Lambda_QCD (r << 1/Lambda_QCD), as stated in Sec. II. The numerical extraction in Sec. IV, however, uses charmonium with m_c^RS(1 GeV)=1.477 GeV and Lambda_QCD=0.5 GeV, and the hybrid wave functions from Ref. [15] have typical interquark distances r ~ 0.2-0.4 fm, so r*Lambda_QCD ~ 0.5-1 and 1/r ~ Lambda_QCD. The eight fitted nonperturbative coefficients can therefore absorb corrections that are not encoded in the factorized form, and their values are not controlled by the claimed EFT hierarchy. This is not an internal inconsistency, but it means the charm fit does not validate the factorized form; the bottomonium predictions in Figs. 5 and 6 are conditional on an assumption that is marginal for the very system used to determine the coefficients. The instability of the fitted coefficients between the two lattice data sets (e.g., Vnp(0)_SLa changes sign from +0.81 to -1.32 and Vnp(0)_S12b from +0.69 to -0.39 in Table II) further indicates that the extraction is not uniquely determined by the current data.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper derives the short-distance matching of the spin-dependent hybrid potentials in the Born-Oppenheimer EFT for quarkonium hybrids from weakly-coupled pNRQCD, expressing the nonperturbative parts of the potentials as products of NRQCD matching coefficients and gauge-invariant gluonic correlators. It then fits eight nonperturbative coefficients to charmonium hybrid lattice spectra from two Hadron Spectrum Collaboration data sets and uses the resulting values to predict spin splittings in the bottomonium hybrid spectrum.","tokens_in":37007,"tokens_out":6923,"duration_ms":65845,"significance":"If the factorization result holds, it is a substantial step: it gives an explicit correlator representation of the spin-dependent hybrid potentials, with the heavy-quark flavor dependence cleanly separated into the NRQCD coefficients c_F and c_s, and it identifies objects that are in principle lattice-computable. The appendices supply the discrete-symmetry identities and angular matrix elements needed to make the calculation reproducible. The numerical section, however, is a calibration rather than a test, and its quantitative control is questionable; this limits the significance of the bottomonium predictions as they currently stand.","major_comments":[{"comment":"The derivation relies on the hierarchy m >> mv >> Lambda_QCD >> mv^2, as stated in Sec. II, but the numerical extraction in Sec. IV uses charmonium with m_c^RS(1 GeV) = 1.477 GeV and Lambda_QCD = 0.5 GeV. For the hybrid wave functions of Ref. [15], typical interquark distances are r ~ 0.2-0.4 fm, giving r Lambda_QCD ~ 0.5-1 and 1/r ~ Lambda_QCD. Hence mv is not parametrically larger than Lambda_QCD for the very system used to determine the eight nonperturbative coefficients, so the multipole expansion and the factorized correlator forms in Eqs. (48) and (74)-(81) are not quantitatively controlled for the charm fit. Since the bottomonium predictions in Figs. 5 and 6 inherit these coefficients, those predictions rest on an assumption that is marginal for the fit. The paper should either present an explicit estimate of the missing O((r Lambda_QCD)^2) corrections or clearly restrict the numerical comparison to an exploratory calibration.","section":"Sec. II; Sec. IV"},{"comment":"The fitted coefficients are not stable across the two lattice data sets: Vnp(0)_SLa/Lambda^3 changes from +0.81 to -1.32 and Vnp(0)_S12b/Lambda^3 changes from +0.69 to -0.39, with sign changes, while the text itself states that the mass hierarchies among the spin-triplet states of H2 and H4 are not firmly determined. With eight parameters fitted to roughly ten spin splittings, chi^2/d.o.f. = 0.999 cannot be taken as strong support for the factorized form; the charm comparison is a calibration, not a validation. The paper should state this limitation explicitly and propagate the resulting uncertainty into the bottomonium figures.","section":"Sec. IV, Table II"},{"comment":"The potentials in Eqs. (34)-(41) are explicitly valid only for 1/r >> Lambda_QCD, and for arbitrary r they should be replaced by generalized Wilson loops. Nevertheless, the Schroedinger equation solved in Sec. IV uses these short-distance forms at all r, and the wave functions from Ref. [15] have support at distances where the expansion parameter is not small. The paper does not estimate the error from integrating the r^2-truncated potentials over the large-r region. A quantitative estimate of this truncation error is needed before the fitted coefficient values and the bottomonium spin splittings can be considered reliable.","section":"Sec. III (after Eq. (41)); Sec. IV"}],"minor_comments":[{"comment":"Table II uses the label Vnp(0)_p×S for the coefficient that Eqs. (34)-(41) and the text call Vnp(0)_SLc; please align the notation.","section":"Sec. IV"},{"comment":"The text contains 'simpliest' instead of 'simplest' before Eq. (42); a similar typo ('vaccuum') appears in Appendix A.","section":"Sec. III"},{"comment":"The sentence referring to 'Figs. 6 and 5' should refer to 'Figs. 5 and 6' for consistency with the figure numbering.","section":"Sec. V"}],"recommendation":"major_revision","confidential_remarks":"The analytic matching calculation is the strongest part of the paper and appears internally consistent. The main risk is in the numerical section, where the charm fit is used both to determine and to illustrate the nonperturbative coefficients under a hierarchy that is marginal for charm. I would be comfortable with publication after the authors add explicit caveats and error estimates for this regime."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Colleague,\n\nThe formal matching result is the real contribution here. Eqs. (48) and (74)-(81) give the first explicit expressions for the nonperturbative spin-dependent hybrid potentials as products of NRQCD coefficients and gauge-invariant gluonic correlators. That is new relative to Ref. [19], which had the operator basis but not these correlator expressions. The C, P, T reduction in Appendix A and the detailed matching in Appendix B are substantial and transparent. This is a genuine step: it makes the hybrid spin potentials computable objects on the lattice and explains the flavor transfer from charm to bottom.\n\nThe numerical section is more of a calibration than a precision prediction. Eight parameters are fitted to roughly ten charmonium spin splittings, with chi^2/d.o.f. around 1, and the extracted coefficients shift notably between the two lattice data sets (e.g., Vnp(0)_SLa changes sign in Table II). That is not fatal, but it means the data are not pinning the coefficients down uniquely. The heavier issue is the hierarchy. The paper assumes mv >> Lambda_QCD, but for charm with m_c ~ 1.5 GeV, Lambda_QCD ~ 0.5 GeV, and typical r ~ 0.2-0.4 fm, you have r*Lambda_QCD ~ 0.5-1. So the charm fit sits in a regime where the factorized form is not parametrically controlled. The paper is honest about this in Sec. II and the conclusions, but the quoted uncertainty does not include that systematic, and the bottomonium predictions in Figs. 5 and 6 inherit it. I would treat them as conditional predictions.\n\nThe matching calculation is not circular: the coefficients are expressed in terms of freshly defined gluonic correlators, and the charm fit is explicitly a calibration. There is no serious citation problem that I can see. The appendices are thorough and the paper is readable.\n\nThis paper deserves a serious referee. The formal part is valuable, and the numerical part is a useful first estimate even though the error budget is not closed. I would send it to review, focusing referee attention on the hierarchy question and the stability of the fit.","headline":"The formal matching result is the real contribution here; the bottomonium spin-splitting predictions are conditional on a hierarchy assumption that is marginal for the charm fit.","tokens_in":37596,"tokens_out":2292,"would_cite":true,"duration_ms":20254,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":["14.40.Pq","14.40.Rt","31.30.-i"],"model":"deepseek-v4-flash","headline":"This paper derives a factorized form for the nonperturbative spin-dependent potentials of heavy quarkonium hybrids, in which the heavy-quark dependence sits entirely in known NRQCD coefficients and the rest is flavor-blind gluonic…","keywords":["quarkonium hybrids","Born-Oppenheimer effective field theory","spin-dependent potentials","NRQCD matching coefficients","gluonic correlators","charmonium hybrid spectrum","bottomonium hybrids","XYZ states"],"falsifier":"Compute one of the gluonic correlators in Eq. (43) or Eqs. (49)-(63) directly on the lattice and compare it with the value extracted from the charmonium fit; disagreement would show that the factorized matching or the flavor-independence claim fails. A lattice computation of the bottomonium hybrid spin splittings compared with the predictions in Figs. 5 and 6 would settle the transfer directly.","tokens_in":36401,"feed_emoji":"⚛️","tokens_out":9114,"duration_ms":77426,"temperature":0.7,"pith_summary":"The paper claims that the spin-dependent forces inside heavy quarkonium hybrids—mesons made of a heavy quark, a heavy antiquark, and a gluonic excitation—separate at short distances into a perturbative piece and a nonperturbative piece that is a product of a known NRQCD matching coefficient and a purely gluonic correlator. Because the gluonic correlator is independent of which heavy quark is present, the same nonperturbative constants fitted to the charmonium hybrid spectrum can be carried over to predict the spin splittings of bottomonium hybrids, which lattice QCD has not yet resolved. This gives an explicit factorized form of the hybrid spin potentials for the first time and a QCD-based route to the spin structure of XYZ states interpreted as hybrids.","feed_headline":"Hybrid meson spin splittings trace to flavor-blind gluonic correlators","feed_subtitle":"Extracted from charmonium lattice data, the same constants predict bottomonium hybrid splittings.","key_machinery":"The load-bearing device is the matching of the gauge-invariant two-point Green's function for a hybrid state between weakly-coupled pNRQCD and the Born-Oppenheimer effective field theory for hybrids. In the short-distance regime $r\\ll 1/\\Lambda_{\\rm QCD}$, the heavy-quark pair sees the lowest gluelump with quantum numbers $K^{PC}=1^{+-}$, whose projected states are labeled by $\\lambda=0,\\pm1$ under the cylindrical symmetry group. The nonperturbative matching coefficients emerge as time-ordered integrals of gluonic field insertions along adjoint Wilson lines between gluelump operators; discrete-symmetry identities reduce these correlators to the few independent tensor components that appear in the final potentials.","core_discovery":"The central discovery is that the nonperturbative parts of the eight spin-dependent hybrid potentials $V_{SK}$, $V_{SKb}$, $V_{SLa}$, $V_{SLb}$, $V_{SLc}$, $V_{S2}$, and $V_{S12b}$ can be written as products of the NRQCD matching coefficients $c_F$ and $c_s$ with gauge-invariant gluonic correlators built from gluelump operators, adjoint Wilson lines, and insertions of chromoelectric or chromomagnetic fields. The paper computes the correlator structures explicitly, using charge conjugation, parity, and time reversal to reduce the pNRQCD two-point functions to the operator basis of the hybrid effective field theory. It also shows that the leading spin effect appears already at order $1/m$: a coupling between the angular momentum of the gluonic excitation and the total spin of the heavy-quark pair, which has no counterpart in standard quarkonium and dominates the splittings. Fitting the resulting constants to two lattice determinations of the charmonium hybrid spectrum yields coefficients of natural size, and the flavor-blindness of the correlators then turns those fits into predictions for the bottomonium hybrid spin splittings.","pith_inferences":["Beyond the paper, a direct lattice evaluation of one of the gluonic correlators in Eqs. (43) and (49)-(63) would either confirm or falsify the factorized form with no need for the charmonium fit.","The paper leaves implicit that the same flavor-blind correlators should control spin splittings in $B_c$ hybrids and, through the generalized BOEFT, possibly the fine structure of doubly heavy tetraquarks.","The two fitted lattice data sets differ in light-quark mass, so the drift in the extracted constants gives a rough map of the light-quark-mass dependence of the correlators; a physical-pion-mass spectrum would test whether that dependence is as mild as assumed."],"forward_implications":["The nonperturbative constants extracted from the charmonium hybrid spectrum can be transferred, with only the one-loop mass dependence of $c_F$ and $c_s$, to predict the spin splittings of bottomonium hybrids before a direct lattice calculation exists.","Hybrid spin multiplets split differently from ordinary quarkonia because the leading spin-dependent operator is suppressed by only one power of the heavy-quark mass, not two.","In the fitted spectrum the nonperturbative contributions reverse the perturbative trend in the spin-triplet states, and the ordering of those states in the higher multiplets is not robustly determined; the paper quantifies that uncertainty.","The same operator structure persists at distances $r\\sim 1/\\Lambda_{\\rm QCD}$, where the factorized potentials become generalized Wilson loops, so the present results are the short-distance limit of a more general description."],"supporting_citations":[{"why":"Builds the hybrid BOEFT and provides the coupled Schrödinger equations whose solutions give the zeroth-order hybrid wave functions used here.","marker":"[15]"},{"why":"Derives the spin-dependent operators of the BOEFT up to $1/m^2$, the operator basis that this paper matches.","marker":"[19]"},{"why":"Formulates weakly-coupled pNRQCD and the multipole-expanded singlet-octet and octet-gluon vertices used in the matching.","marker":"[22]"},{"why":"Supplies the NRQCD matching coefficients $c_F$ and $c_s$ whose heavy-quark mass dependence carries the flavor dependence of the potentials.","marker":"[25]"},{"why":"Supply the gluelump masses used to identify the lightest $1^{+-}$ hybrid multiplet and its short-distance degeneracy.","marker":"[30,31]"},{"why":"Provides the lattice hybrid static energies used for the leading-order hybrid potentials plotted and fitted in this work.","marker":"[33]"},{"why":"Provides a recent comprehensive lattice study of hybrid static energies and the technology available for computing gluonic correlators.","marker":"[34]"},{"why":"Lattice charmonium hybrid spectrum at $m_\\pi\\approx 400$ MeV used as one of the two data sets for the fit.","marker":"[36]"},{"why":"Lattice charmonium hybrid spectrum at $m_\\pi\\approx 240$ MeV used as the second, lighter-quark data set for the fit.","marker":"[37]"}],"fun_headline_variants":["Spin-orbit coupling at 1/m dominates heavy hybrid splittings","Flavor-blind gluonic correlators tie charmonium to bottomonium hybrid spins","Heavy hybrid spin splittings predicted from charmonium lattice data","New 1/m spin operator sets hybrid meson splittings apart from quarkonia","Gluonic correlators from lattice charmonia predict bottomonium hybrids"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The derivation requires the heavy-quark relative momentum $mv$ to be much larger than $\\Lambda_{\\rm QCD}$; with the charmonium parameters used here ($m_c=1.477$ GeV, $\\Lambda_{\\rm QCD}=0.5$ GeV) that separation is numerically marginal, so the factorized fit is not quantitatively controlled for charmonium.","fun_headline_variants_meta":{"raw":{"variants":["Spin-orbit coupling at 1/m dominates heavy hybrid splittings","Flavor-blind gluonic correlators tie charmonium to bottomonium hybrid spins","Heavy hybrid spin splittings predicted from charmonium lattice data","New 1/m spin operator sets hybrid meson splittings apart from quarkonia","Gluonic correlators from lattice charmonia predict bottomonium hybrids"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000645,"raw_usage":{"total_tokens":2982,"prompt_tokens":977,"completion_tokens":2005,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":593,"completion_tokens_details":{"reasoning_tokens":1907}},"tokens_in":593,"tokens_out":2005,"duration_ms":12465,"temperature":1.0,"reasoning_tokens":1907,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T10:08:57.464352+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute one of the gluonic correlators in Eq. (43) or Eqs. (49)-(63) directly on the lattice and compare it with the value extracted from the charmonium fit; disagreement would show that the factorized matching or the flavor-independence claim fails. A lattice computation of the bottomonium hybrid spin splittings compared with the predictions in Figs. 5 and 6 would settle the transfer directly.","supporting_citations":[],"review_version":1}