{"id":"b908fda0-8af6-4394-acce-c63fb7d52107","arxiv_id":"2606.22231","paper_version":1,"verdict":"UNVERDICTED","confidence":"LOW","novelty_score":4.0,"correctness_risk":"unknown","formal_verification":"none","parameter_count":0,"one_line_summary":"Computes branching ratios for Z to double S-wave quarkonia (VV, VP) and radiative Z to X(QQbar) gamma using Bethe-Salpeter in heavy quark limit, reporting larger rates than NRQCD for charmonium and smaller for bottomonium.","lead":"This paper calculates rare Z boson decays into pairs of S-wave quarkonia and to single quarkonium plus photon using the Bethe-Salpeter formalism in the heavy quark limit, comparing results to NRQCD. A smart generalist might read it to understand how relativistic effects in charmonium versus bottomonium affect predictions for rare collider processes.","discovery_kind":"new_application","skeptic_critique":{"model":"grok-4.3","headline":"Heavy quark limit adopted to simplify calc, yet used to argue charmonium is relativistic (deviates from NRQCD) while bottomonium is not","rationale":"Reader correctly flagged the heavy-quark limit as the weakest assumption. The tension between adopting that limit and then interpreting deviations as proof of relativistic character for charm is the single most load-bearing point for the headline claim; full text would need to show how the limit is implemented without erasing the relativistic content that distinguishes BS from NRQCD.","tokens_in":1783,"tokens_out":360,"duration_ms":9655,"concrete_test":"Extract the typical quark velocity v^2 from the BS wave-function normalization or binding energy for both c and b systems; if v^2(c) ≳ 0.3 while v^2(b) ≪ 0.1, the heavy-quark limit used for charmonium channels is inconsistent with the relativistic interpretation drawn from the NRQCD comparison.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"The central interpretive claim rests on comparing BS results (under heavy-quark limit) to NRQCD: larger for charmonium, smaller for bottomonium, taken as evidence that charm is relativistic and bottom non-relativistic. But the heavy-quark limit is explicitly invoked to simplify the BS amplitudes for both systems (abstract and §2). This creates an internal tension: if the limit is valid enough to trust the numerics, the deviation for charm should be small; if the deviation is large, the limit's applicability to charm is questionable and the comparison cannot cleanly diagnose relativistic effects. No independent check (e.g., extracted v^2 or size of 1/m corrections) is referenced in the abstract to resolve this.","agreement_with_reader":"agree"},"referee_report":{"model":"grok-4.3","summary":"The manuscript investigates rare exclusive decays of the Z-boson into S-wave quarkonia using the Bethe-Salpeter formalism. It analyzes double S-wave quarkonia production (Z→VV, Z→VP) including QCD and QED contributions, introduces a new QED channel for mixed bottomonium-charmonium states, and studies radiative decays Z→X(QQ̄)γ. The heavy quark limit is used to simplify calculations. Results for charmonium processes are larger than NRQCD, while for bottomonium they are smaller, interpreted as indicating that charmonium is relativistic and bottomonium is non-relativistic.","tokens_in":1931,"tokens_out":479,"duration_ms":23783,"significance":"If the results hold, the work offers a relativistic treatment of these rare decays via the Bethe-Salpeter approach, providing a comparison to NRQCD that could highlight the importance of relativistic effects in lighter quarkonia. The introduction of the mixed flavor production channel is a novel contribution. Credit is given for considering both QCD and QED amplitudes and for exploring the implications for quarkonium dynamics.","major_comments":[{"comment":"Abstract and §2: The heavy quark limit is adopted to simplify the Bethe-Salpeter amplitudes for both charmonium and bottomonium. Yet the central interpretive claim uses larger BS results (vs NRQCD) for charmonium processes to conclude that charmonium is relativistic while bottomonium is non-relativistic. This is load-bearing for the conclusion but internally inconsistent: if the limit suffices to trust the numerics for both systems, the deviation for charm should be small; the observed deviation instead questions the limit's applicability to charm without an independent check (e.g., extracted v² or explicit 1/m correction size) referenced in the text.","section":"Abstract and §2"}],"minor_comments":[{"comment":"The abstract would benefit from quoting at least one or two explicit branching-ratio values (or ratios to NRQCD) to make the size of the reported deviations concrete.","section":null},{"comment":"Ensure all NRQCD comparison references are explicitly cited with paper numbers in the text and reference list.","section":null}],"recommendation":"major_revision","confidential_remarks":null},"author_rebuttal":{"model":"grok-4.3","summary":"We thank the referee for the careful reading of the manuscript and the constructive comment. We address the major point below.","responses":[{"response":"The heavy quark limit is applied uniformly to simplify the propagators and reduce the BS amplitudes to their leading covariant structures for both systems. Within this framework the BS equation remains relativistic, so the resulting amplitudes differ from the NRQCD expansion even after the limit is taken. The larger rates obtained for charmonium therefore indicate that relativistic kinematics and binding effects retained by the BS approach are numerically important for charm, while the smaller rates for bottomonium show closer agreement with the non-relativistic expectation. The direct numerical comparison to NRQCD thus functions as the consistency check on the applicability of the limit. We will add a clarifying paragraph in §2 and a corresponding sentence in the abstract stating that the observed deviations themselves test the validity of the heavy-quark approximation for each system.","revision_made":"partial","referee_comment":"[Abstract and §2] Abstract and §2: The heavy quark limit is adopted to simplify the Bethe-Salpeter amplitudes for both charmonium and bottomonium. Yet the central interpretive claim uses larger BS results (vs NRQCD) for charmonium processes to conclude that charmonium is relativistic while bottomonium is non-relativistic. This is load-bearing for the conclusion but internally inconsistent: if the limit suffices to trust the numerics for both systems, the deviation for charm should be small; the observed deviation instead questions the limit's applicability to charm without an independent check (e.g., extracted v² or explicit 1/m correction size) referenced in the text."}],"tokens_in":1489,"tokens_out":355,"duration_ms":16549,"standing_objections":[]},"desk_editor":{"model":"grok-4.3","letter":"The new element is the inclusion of Z decays into one bottomonium and one charmonium state through the QED amplitude, such as Z to J/psi + Upsilon(1S). That channel is not standard in prior NRQCD work on these processes, so the paper supplies first estimates for it along with the usual double-charmonium and radiative modes.\n\nThey carry out the calculation in the Bethe-Salpeter framework under the heavy-quark limit and report that their widths exceed NRQCD results for the charmonium channels while falling below for bottomonium. The numerical comparisons are the main output.\n\nThe interpretive step is the weak point. The abstract states that the heavy-quark limit is adopted to simplify the amplitudes for both systems, yet the larger deviation for charm is taken as evidence that charm is relativistic. If the limit is reliable enough to trust the numerics, the difference should be modest; if the difference is large, the limit's validity for charm is in question and the comparison does not cleanly isolate relativistic effects. No auxiliary check on v^2 or the size of 1/m corrections appears in the available text to resolve this.\n\nThe paper is aimed at collider phenomenologists who need order-of-magnitude rates for these rare Z decays. The mixed-channel numbers could be cited if the full calculation holds up under scrutiny. The central claim about relativistic versus non-relativistic behavior would need tightening before it influences broader discussion.\n\nI would send it to referees. The concrete predictions and the new channel give it enough substance to merit review, even though the discussion of the results requires work.","headline":"They add the mixed bottom-charmonium channel and give concrete rates, but the claim that charmonium is relativistic while bottomonium is not sits in tension with the heavy-quark limit they apply to both.","tokens_in":2420,"tokens_out":412,"would_cite":false,"duration_ms":12905,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"grok-4.3","headline":"Z-boson decays to charmonium pairs and radiative charmonium yield larger branching fractions in the Bethe-Salpeter formalism than in NRQCD, while bottomonium fractions are smaller.","keywords":["Z boson decays","quarkonia","Bethe-Salpeter formalism","charmonium","bottomonium","NRQCD","radiative decays","double production"],"falsifier":"A measured branching ratio for Z to J/psi J/psi or Z to J/psi gamma that matches the NRQCD number within experimental precision while lying outside the Bethe-Salpeter band would falsify the central claim.","tokens_in":2683,"feed_emoji":"","tokens_out":694,"duration_ms":14771,"temperature":0.7,"pith_summary":"The paper computes rare Z-boson decays into pairs of S-wave quarkonia (Z to VV and Z to VP) and into a single S-wave quarkonium plus a photon using the Bethe-Salpeter formalism for the bound states. Leading-order QCD and QED amplitudes are included, along with a new mixed bottomonium-charmonium channel via virtual photon. Results exceed NRQCD predictions for all charmonium modes but fall below them for bottomonium modes. The authors conclude that charmonium must be treated as a relativistic system while bottomonium behaves as a non-relativistic one. The heavy-quark limit is used throughout to reduce the algebra.","feed_headline":"Bethe-Salpeter Z decays to charmonium exceed NRQCD rates","feed_subtitle":"Calculations indicate charmonium is relativistic while bottomonium is not, with new mixed bottomonium-charmonium channels included.","key_machinery":"Bethe-Salpeter wave functions for S-wave quarkonia in the heavy-quark limit, which supply the relativistic bound-state amplitudes for both QCD and QED transition diagrams.","core_discovery":"Within the Bethe-Salpeter formalism and the heavy-quark limit, the calculated branching fractions for double-charmonium and radiative-charmonium Z decays exceed the corresponding NRQCD values, whereas the bottomonium branching fractions are smaller than NRQCD results. This pattern demonstrates that charmonium is a relativistic particle while bottomonium is non-relativistic.","pith_inferences":["If the pattern holds, Bethe-Salpeter calculations would be preferred for any process involving the lighter charmonium system.","Precision measurements at a future Z factory could directly discriminate between the two formalisms for these rare decays.","The same Bethe-Salpeter amplitudes could be reused for related rare decays such as Z to quarkonium plus light meson."],"forward_implications":["The mixed QED channel Z to J/psi plus Upsilon(1S) and its charge-conjugate processes receive explicit leading-order predictions.","Radiative modes Z to eta_c gamma, Z to J/psi gamma, Z to eta_b gamma and Z to Upsilon gamma obtain new numerical estimates that differ from NRQCD.","Charmonium production rates require relativistic corrections beyond the NRQCD approximation.","Bottomonium rates remain consistent with a non-relativistic treatment."],"fun_headline_variants":["Bethe-Salpeter Z decays to charmonium exceed NRQCD","Charmonium Z decays larger under Bethe-Salpeter than NRQCD","Bethe-Salpeter yields lower bottomonium Z rates than NRQCD","New Bethe-Salpeter channels for Z to mixed quarkonia"],"cache_read_input_tokens":2112,"weakest_assumption_plain":"The heavy quark limit is adopted to simplify calculations.","fun_headline_variants_meta":{"raw":{"variants":["Bethe-Salpeter Z decays to charmonium exceed NRQCD","Charmonium Z decays larger under Bethe-Salpeter than NRQCD","Bethe-Salpeter yields lower bottomonium Z rates than NRQCD","New Bethe-Salpeter channels for Z to mixed quarkonia"]},"model":"grok-4.3","cost_usd":0.00857,"raw_usage":{"total_tokens":3905,"prompt_tokens":739,"num_sources_used":0,"completion_tokens":78,"cost_in_usd_ticks":85699500,"prompt_tokens_details":{"text_tokens":739,"audio_tokens":0,"image_tokens":0,"cached_tokens":256},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":3088,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":739,"tokens_out":78,"duration_ms":22790,"temperature":1.0,"reasoning_tokens":3088,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-06-26T11:33:46.728640+00:00","model_set":{"reader":"grok-4.3"},"falsifier":"A measured branching ratio for Z to J/psi J/psi or Z to J/psi gamma that matches the NRQCD number within experimental precision while lying outside the Bethe-Salpeter band would falsify the central claim.","supporting_citations":[],"review_version":1}