{"id":"e8099a74-2d18-46a8-97ce-dd8800423ec1","arxiv_id":"2605.28756","paper_version":1,"verdict":"UNVERDICTED","confidence":"LOW","novelty_score":5.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Square well analysis shows Breit-Wigner poles are not eigenvalues and lead to growing spatial waves; PT symmetry gives conjugate pairs E∓=E2∓iΓ2 with stable amplitudes and one physical resonance.","lead":"The paper solves the square well scattering problem to show problems with the standard Breit-Wigner resonance formula, including non-eigenvalue poles and unphysical growing wave functions for decaying states. It resolves these via PT symmetry producing complex conjugate energy pairs that yield time-independent probability amplitudes and only one observable resonance.","discovery_kind":"extension","skeptic_critique":{"model":"grok-4.3","headline":"PT-symmetric resolution claims time-independent probability amplitude for the physical resonance, incompatible with finite-lifetime decay","rationale":"The reader's weakest assumption correctly flags limited representativeness of the square well, but the more load-bearing issue is internal to the strongest claim: the explicit assertion of time-independent amplitude directly contradicts the exponential decay that defines a resonance. This is independent of model details and can be settled by the time-evolution check above. The abstract-only review is noted, but the quoted phrasing is unambiguous.","tokens_in":1792,"tokens_out":349,"duration_ms":35522,"concrete_test":"From the square-well solution section, extract the explicit time-dependent wave function for the retained PT-symmetric state (the one identified as the physical resonance) and compute |ψ(x,t)|² as a function of t; if it remains constant for all t rather than decaying as exp(−Γt), the claimed resolution does not describe resonances.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"The central claim asserts that PT symmetry yields conjugate energy pairs E∓=E2∓iΓ2 (E−≠EBW) whose solutions produce 'a time independent probability amplitude that neither grows nor decays in time or space' and 'leads to just one now observable physical resonance'. A resonance is defined by nonzero width Γ and the associated exponential decay of |ψ(t)|² ∼ e^{-Γt}. Time-independent |ψ(x,t)|² corresponds to a stationary eigenstate with real energy and infinite lifetime. This internal mismatch means the proposed alternative does not reproduce the decaying states that Breit-Wigner is meant to describe, rendering the critique of the standard approach inapplicable to resonance scattering.","agreement_with_reader":"disagree"},"referee_report":{"model":"grok-4.3","summary":"The paper critiques the Breit-Wigner parametrization of resonance scattering by exactly solving the square-well Schrödinger equation. It reports that the form tan δ_BW = Γ1/(E1 - E) does not always match the real-energy scattering amplitude, that Γ1 can be negative, that the complex pole E_BW is not an energy eigenvalue, and that energy-decaying states produce spatially exponentially growing wave functions. These issues are claimed to be resolved by PT symmetry, which produces complex-conjugate eigenvalue pairs E∓ = E2 ∓ i Γ2 (with E- ≠ E_BW) whose associated probability density is time-independent and yields only a single observable physical resonance.","tokens_in":1943,"tokens_out":507,"duration_ms":33554,"significance":"If the central claims were shown to hold beyond the square-well model, the work would challenge the standard identification of Breit-Wigner poles with physical resonances in particle phenomenology. The explicit square-well solutions constitute a reproducible calculation, but the manuscript does not demonstrate that the PT-symmetric construction reproduces the finite-lifetime exponential decay required of resonances.","major_comments":[{"comment":"Abstract (final paragraph): the assertion that PT-symmetric solutions produce 'a time independent probability amplitude that neither grows nor decays in time or space' is incompatible with the defining property of a resonance, namely |ψ(t)|² ∼ e^{-Γ t} with Γ > 0. This internal mismatch means the proposed alternative does not describe the decaying states that Breit-Wigner is intended to model.","section":"Abstract"},{"comment":"Abstract (paragraph on square-well solution): the leap from the square-well results to the general claim that 'E_BW is not in fact an energy eigenvalue (and thus not a physical particle)' and that Breit-Wigner is therefore invalidated rests on a single exactly solvable model without demonstrated robustness under changes to the potential or comparison to standard resonance calculations in the literature.","section":"Abstract"}],"minor_comments":[{"comment":"Notation: the subscripts on E∓, E2, Γ2 are introduced without an explicit definition of the indexing convention used for the PT pair.","section":"Abstract"}],"recommendation":"reject","confidential_remarks":"The manuscript is framed as a hep-ph contribution yet relies exclusively on non-relativistic quantum mechanics with a square-well potential; this raises a scope question for the target journal."},"author_rebuttal":{"model":"grok-4.3","summary":"We thank the referee for their detailed review. We respond to each major comment below and note revisions where appropriate.","responses":[{"response":"The referee correctly notes that our PT-symmetric construction yields time-independent probability densities, which does not reproduce the exponential decay |ψ(t)|² ∼ e^{-Γ t}. The manuscript's focus is on resolving inconsistencies in the Breit-Wigner approach, such as possible negative widths, non-eigenvalue poles, and spatially growing wave functions for decaying states. The PT pairing provides conjugate eigenvalues with conserved probability, avoiding those problems. We agree this creates a distinction from standard resonance modeling and will revise the abstract to clarify that the approach prioritizes eigenvalue structure and probability conservation over temporal decay.","revision_made":"yes","referee_comment":"[Abstract] Abstract (final paragraph): the assertion that PT-symmetric solutions produce 'a time independent probability amplitude that neither grows nor decays in time or space' is incompatible with the defining property of a resonance, namely |ψ(t)|² ∼ e^{-Γ t} with Γ > 0. This internal mismatch means the proposed alternative does not describe the decaying states that Breit-Wigner is intended to model."},{"response":"The square-well potential is selected specifically because it allows exact analytic solutions, making it possible to compute the phase shift explicitly and demonstrate mismatches with the Breit-Wigner form as well as the non-eigenvalue character of E_BW. The underlying issues arise from the analytic continuation of the scattering amplitude and the antilinear PT symmetry of the real-potential Schrödinger equation, features that are not unique to this model. We will add discussion of the generality of these PT-symmetric conjugate pairs and include comparisons to known resonance results in the literature for other potentials.","revision_made":"partial","referee_comment":"[Abstract] Abstract (paragraph on square-well solution): the leap from the square-well results to the general claim that 'E_BW is not in fact an energy eigenvalue (and thus not a physical particle)' and that Breit-Wigner is therefore invalidated rests on a single exactly solvable model without demonstrated robustness under changes to the potential or comparison to standard resonance calculations in the literature."}],"tokens_in":1444,"tokens_out":512,"duration_ms":42329,"standing_objections":["The PT-symmetric construction does not reproduce the finite-lifetime exponential decay |ψ(t)|² ∼ e^{-Γ t} required of resonances, as the probability amplitude is time-independent by construction."]},"desk_editor":{"model":"grok-4.3","letter":"The core takeaway is that this work solves the square well scattering problem and finds that the usual tan delta = Gamma1/(E1-E) form does not always match the exact phase shift, that Gamma1 can turn negative, and that the Breit-Wigner pole is not an eigenvalue of the Hamiltonian. Those points are new in this specific setting and rest on direct solution of the Schrödinger equation rather than fitting.\n\nThe calculation itself is standard and reproducible for the square well. It is useful to see an explicit case where the Breit-Wigner parametrization deviates and where the spatial wave function for the decaying solution grows exponentially outside the well. That flags a real limitation of the model.\n\nThe proposed resolution is weaker. The paper states that PT symmetry supplies conjugate pairs E∓ = E2 ∓ i Gamma2 with E- ≠ EBW and that these yield a time-independent probability amplitude that neither grows nor decays. Resonances are defined by finite lifetime and exponential decay of |psi(t)|^2. A strictly time-independent amplitude corresponds to a stationary state with real energy, not to the unstable states Breit-Wigner is meant to capture. This internal mismatch means the alternative does not reproduce the physics the critique targets.\n\nThe square well is a toy model; nothing in the abstract shows that the same PT structure or the same mismatch appears in realistic potentials used in particle physics. The claim that Breit-Wigner poles are therefore not physical particles in general therefore rests on an untested extrapolation.\n\nThe paper is mainly of interest to people working on PT-symmetric quantum mechanics or on foundational questions about resonance poles. A data analyst fitting Breit-Wigner forms to scattering data will not find immediate practical guidance. It is coherent enough on its own terms to warrant referee time, mainly to check the derivations and to test whether the time-independent claim survives a closer look at the time-dependent wave functions.","headline":"The paper shows concrete mismatches between Breit-Wigner and exact square-well scattering but its PT-symmetric fix produces time-independent amplitudes that do not describe decaying resonances.","tokens_in":2404,"tokens_out":453,"would_cite":false,"duration_ms":23579,"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":"The Breit-Wigner complex pole is not an energy eigenvalue and produces exponentially growing wave functions, while PT symmetry yields conjugate energy pairs with one observable resonance.","keywords":["Breit-Wigner resonance","PT symmetry","square well scattering","complex energy poles","resonance scattering","unstable particles","phase shift"],"falsifier":"Measurement of an exponentially growing spatial wave function for a resonance state, or detection of two distinct observable resonances corresponding to a conjugate energy pair, would test the claims.","tokens_in":2685,"feed_emoji":"","tokens_out":607,"duration_ms":17968,"temperature":0.7,"pith_summary":"The paper solves the square well scattering problem to test the standard Breit-Wigner phase shift form tan delta equals Gamma1 over E1 minus E. This form fails to always match the real-energy amplitude, allows negative Gamma1, places the pole E_BW at a non-eigenvalue, and assigns decaying-energy states spatial wave functions that grow exponentially. PT symmetry of the potential instead requires solutions in conjugate pairs E∓ equals E2 ∓ i Gamma2, with the lower member unequal to E_BW, producing a time-independent probability amplitude that neither grows nor decays and only one observable physical resonance.","feed_headline":"Square well shows Breit-Wigner poles are not physical particles","feed_subtitle":"PT symmetry produces conjugate energy pairs yielding one stable resonance with time-independent amplitude.","key_machinery":"Antilinear PT symmetry of the square well potential, which forces scattering solutions to appear in conjugate energy pairs rather than isolated complex poles.","core_discovery":"Because of its antilinear PT symmetry, solutions to the square well Schrödinger equation appear in complex conjugate energy pairs E∓ = E2 ∓ i Γ2 with E− ≠ E_BW, giving a time-independent probability amplitude that neither grows nor decays in time or space and leading to just one now observable physical resonance.","pith_inferences":["Resonance analyses in other short-range potentials may need to replace isolated complex poles with PT-symmetric conjugate pairs.","Time-independent probability amplitudes allow resonances to be treated as stationary states for probability calculations.","Reinterpretation of existing particle data could reduce the number of reported resonances by half in some channels."],"forward_implications":["The standard identification of the Breit-Wigner pole E_BW with an unstable physical particle is invalid.","Energy-decaying states must be rejected if their spatial wave functions grow exponentially.","Scattering data yield only one observable resonance rather than two from conjugate pairs.","The phase-shift parameter Gamma1 can take negative values without violating unitarity."],"fun_headline_variants":["Square well PT symmetry shows Breit-Wigner poles lack physical status","PT symmetry pairs energies to yield one square well resonance","Breit-Wigner scattering description fails in square well analysis","Complex conjugate energies from PT symmetry fix Breit-Wigner flaws","Square well scattering produces time-independent resonance amplitude"],"cache_read_input_tokens":2112,"weakest_assumption_plain":"The square well potential and its PT symmetry properties represent the essential physics in realistic resonance scattering problems where the Breit-Wigner approach is applied.","fun_headline_variants_meta":{"raw":{"variants":["Square well PT symmetry shows Breit-Wigner poles lack physical status","PT symmetry pairs energies to yield one square well resonance","Breit-Wigner scattering description fails in square well analysis","Complex conjugate energies from PT symmetry fix Breit-Wigner flaws","Square well scattering produces time-independent resonance amplitude"]},"model":"grok-4.3","cost_usd":0.003099,"raw_usage":{"total_tokens":1682,"prompt_tokens":670,"num_sources_used":0,"completion_tokens":76,"cost_in_usd_ticks":30987000,"prompt_tokens_details":{"text_tokens":670,"audio_tokens":0,"image_tokens":0,"cached_tokens":256},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":936,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":670,"tokens_out":76,"duration_ms":7282,"temperature":1.0,"reasoning_tokens":936,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-06-29T11:19:54.118747+00:00","model_set":{"reader":"grok-4.3"},"falsifier":"Measurement of an exponentially growing spatial wave function for a resonance state, or detection of two distinct observable resonances corresponding to a conjugate energy pair, would test the claims.","supporting_citations":[],"review_version":1}