{"id":"dbce5aea-6588-4e05-b7ec-2a3ddc335a52","arxiv_id":"2511.08308","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":8,"one_line_summary":"Excited Sigma states of the hydrogen-antihydrogen molecule support near-threshold rovibrational levels and must be included in ground-state H-Hbar collision calculations.","lead":"The paper calculates the energy curves of many excited electronic states of the hydrogen-antihydrogen molecule, including states where the electron and positron form free positronium. It finds many molecular bound states just at the collision threshold energy, so future scattering calculations must include these excited states.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"BO extrapolation below Rc = 0.744 a0 is the load-bearing step; the paper's own four-body check shows the BO approximation is unreliable there, so the near-threshold excited-state spectrum is not yet established.","rationale":"The reader's conditional verdict is well calibrated. I read the paper as a careful BO study whose main new deliverable is the excited Σ curves, including discretized positronium states, with honest convergence tests (Figs. 4, 5, 10, 11) and a benchmark of the ground state against [32]. The independent four-body check of the ground state is a real plus, and the paper explicitly flags its own limitations in Sec. V. My concern coincides with the reader's weakest assumption. What makes it load-bearing is that the near-threshold spectrum of Fig. 9 is not an optional refinement: it is the primary quantitative evidence for the abstract's conclusion. Because the extrapolation region is also where the BO approximation is known to fail—and where the paper's own four-body result demonstrates a qualitative stability error—the conclusion is conditional on the extrapolation being faithful for the excited states. The paper's sensitivity test covers only Q-odd states and only two extrapolation forms; it does not test Q-even states, the ones relevant under CP conservation. A full four-body or complex-scaling resonance calculation would settle the issue. I would keep the verdict CONDITIONAL rather than strengthen to reject: the method is state-of-the-art for BO curves, the tabulated data are valuable, and the qualitative argument is plausible. The missing piece is a quantitative check that the near-threshold level density survives beyond the BO approximation.","tokens_in":23394,"tokens_out":5786,"duration_ms":68024,"concrete_test":"Perform a complex-scaling four-body resonance calculation for H–H̄ (as in Refs. [40,41]) and enumerate all resonances within ~10 mHartree above the ground-state dissociation threshold; compare the resulting resonance density and energies with the BO rovibrational levels of Fig. 9. If the full non-adiabatic spectrum does not show a comparable cluster of near-threshold states, the central claim is not supported.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central conclusion—that excited Σ states must be included in ground-state H–H̄ scattering—rests on the claim (Sec. IV.C, Fig. 9) that these states support a dense set of rovibrational levels just above the −1 a.u. dissociation threshold. That spectrum is obtained in Sec. III.C by (i) extrapolating the leptonic BO curves below R = 0.8 a0 using a simple linear form, and (ii) solving a one-channel hadronic equation for each curve. Both steps are weakest exactly in the region that matters: Rc ≈ 0.744 a0 is below 0.8 a0, reference [33] showed the adiabatic correction diverges there, and the paper's own non-BO four-body check (Sec. IV.C) finds the true ground state above the positronium+protonium threshold while the BO result lies below it. The authors explicitly acknowledge in Sec. V that the calculation is not of high accuracy and that part of the potentials were extrapolated. The sensitivity check in Fig. 9 compares extrapolation forms only for Q-odd states, whereas the states that can couple to the Q-even ground state under CP are Q-even; the Q-even near-threshold density is not tested in the same way. If the true non-adiabatic couplings rearrange or dissolve these levels—e.g. through the avoided crossings with positronium states seen in Fig. 6—the 'plethora of resonances' and the conclusion that single-ground-state-potential treatments are incomplete would not follow from the data shown.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The manuscript computes Born-Oppenheimer potential curves for excited Σ states of the hydrogen-antihydrogen molecule, separately for Q-even and Q-odd symmetries, using a modified version of the explicitly correlated Kolos-Wolniewicz code H2SOLV with a dual basis. The authors identify nearly R-independent states as discretized free positronium states, resolve many avoided crossings, extrapolate the potentials below R=0.8 a0, and solve the one-channel hadronic radial equation on a B-spline grid. They find a dense set of rovibrational levels just above the ground-state dissociation threshold and conclude that excited leptonic states must be included in theoretical treatments of ground-state H–Hbar collisions. The ground-state curve is benchmarked against Strasburger's curve and against a full four-body non-BO calculation, with mixed results.","tokens_in":23796,"tokens_out":3280,"duration_ms":38368,"significance":"If the central claim is correct, it would be an important correction to scattering calculations that use only the ground-state potential curve, and the tabulated excited curves plus the identification of Ps states in a Kolos-Wolniewicz basis would be useful for future nonadiabatic or close-coupling work. The paper has clear strengths: explicit convergence studies in Ω, a benchmark against the best ground-state curve (4.3e-8 relative deviation at R=5.0 a0), a full four-body cross-check, and numerical tables of ten states per symmetry. However, the quantitative claim about a 'plethora of rovibrational states' near threshold is not yet established because it depends on uncontrolled extrapolation below the critical distance and on an inference from single-channel bound-state energies to scattering resonances without calculating couplings.","major_comments":[{"comment":"The near-threshold spectrum is obtained by extrapolating V(R) below R=0.8 a0 with a linear form, while the critical distance is Rc≈0.744 a0 and Ref. [33] showed that the adiabatic correction diverges there. The sensitivity test in Fig. 9, comparing linear and polynomial extrapolations, is shown only for Q-odd states. The states that can couple to the Q-even ground state under conservation of Q/CP are Q-even, so the relevant near-threshold density is not tested. The authors should either provide the same extrapolation sensitivity for Q-even states or give a physical argument why the Q-odd test is representative.","section":"Sec. III.C and Fig. 9"},{"comment":"The four-body check that the authors themselves present shows that the BO treatment is unreliable in the region that matters: the BO ground-state energy (-460.347 a.u.) lies below the positronium+protonium threshold (-459.288 a.u.), whereas the non-BO four-body energy (-459.219 a.u.) lies above it. This is exactly the small-R region used to generate the near-threshold rovibrational levels. The paper acknowledges this in Sec. V, but the central claim that the 'plethora' of levels is real and not an artifact of the BO extrapolation requires a more direct test than a single ground-state energy comparison, since the excited-state curves are less well converged than the ground state.","section":"Sec. IV.C"},{"comment":"The conclusion that excited leptonic states 'need to be considered' in H–Hbar scattering rests on energy degeneracy only. No leptonic coupling matrix elements, nonadiabatic couplings, or widths are computed. The authors note that the coupling decreases with n but give no quantitative estimate, and they invoke finite lifetimes to argue that resonances occur at almost any collision energy. This is an interesting conjecture but not a demonstrated result. To support the abstract's conclusion, the paper should at least estimate the relevant coupling or reframe the conclusion as a motivation for future close-coupling calculations.","section":"Sec. IV.C"}],"minor_comments":[{"comment":"The name 'Ko los-Wolniewicz' appears with a missing space (likely a typographical issue from the source), and 'prositronium' appears in Sec. III.B. Please proofread.","section":"Throughout"},{"comment":"The caption says 'Q-positive states'; this should be 'Q-even states' for consistency with the text.","section":"Appendix B, Fig. 10"},{"comment":"The B-spline calculation uses R0=0 a0 and Rmax=5.0 a0. Please clarify the boundary conditions at both ends, since the potential is extrapolated to finite values near R=0 and the density of states near threshold may depend on the enclosing box.","section":"Sec. III.C"},{"comment":"The inset is hard to read and the energy axis is not labeled with numerical values. Please enlarge and annotate the threshold region.","section":"Fig. 9"},{"comment":"The sentence 'since the Q-odd states should for symmetry reasons not couple to the Q-even ground state, if CP symmetry is conserved' could be made more precise by stating explicitly that Q is a symmetry of the full Hamiltonian, not only of the BO Hamiltonian.","section":"Sec. IV.C"}],"recommendation":"major_revision","confidential_remarks":"The paper presents valuable new data and a plausible physical argument, but the central claim is currently supported by an untested extrapolation for the relevant Q-even sector and by an inference from bound-state energies to scattering resonances without coupling matrix elements. A revision that adds Q-even sensitivity tests and softens or better substantiates the scattering conclusion would make the paper suitable for publication."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The paper does something genuinely new and is mostly honest about where it is approximate. It computes the first Born-Oppenheimer potential curves for excited Sigma states of H–Hbar for both Q symmetries, including discretized positronium states, and finds a dense set of rovibrational levels just above the ground-state dissociation threshold. If that holds, scattering calculations built on a single ground-state curve are incomplete. That is the take-home.\n\nWhat is good: the ground-state curve agrees with Strasburger to 4.3e-8 at R=5.0 a0, the full four-body cross-check gives 2.4e-3 relative agreement, and the convergence tests with increasing Omega are shown. The authors also refute the old claim in [45] that Q-odd states can be ignored, and they tabulate the curves. The dual-base Kolos-Wolniewicz approach that captures free positronium states within a single diagonalization is a real technical step forward.\n\nThe soft spot is the extrapolation below R=0.8 a0. That is exactly where the adiabatic approximation diverges (Rc≈0.744 a0), and the paper's own four-body check shows the BO ground state falls below the positronium+protonium threshold while the true state sits above it. So the near-threshold spectrum is built on shaky ground. That does not kill the qualitative conclusion — many excited curves dive like -1/R, so some near-threshold rovibrational states are expected — but the exact density and positions are not established. The sensitivity check in Fig. 9 only varies the extrapolation for Q-odd states; the states that can couple to the Q-even ground state under CP are Q-even, and that is the spectrum left untested. The authors also admit the excited states are less converged than the ground state, which is fine but worth keeping in mind.\n\nOverall this is a solid computational study with appropriate caveats. The central claim is plausible and not overstated: it is about which states need to be included in future scattering treatments, not about precise resonance energies. I would send it to peer review. The referee should push on the Q-even extrapolation sensitivity and ask for error bars on the near-threshold state density, but the core new results deserve to be in the literature.","headline":"First excited Sigma curves for H–Hbar with honest caveats; the near-threshold density claim is plausible but the extrapolation below Rc is the real soft spot.","tokens_in":24323,"tokens_out":1801,"would_cite":true,"duration_ms":22170,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper shows that excited leptonic Σ states of the hydrogen-antihydrogen molecule support rovibrational levels at the ground-state dissociation threshold, so collision models based on the ground-state potential curve alone are incomplete","keywords":["hydrogen-antihydrogen molecule","Born-Oppenheimer potential curves","excited Sigma states","Q symmetry","positronium discretization","scattering resonances","explicitly correlated basis functions","antihydrogen collisions"],"falsifier":"A full non-relativistic four-body scattering calculation of H-antihydrogen that includes all rearrangement and excited-state channels without the Born-Oppenheimer approximation: if the near-threshold rovibrational levels just above -1.0 Hartree disappear or shift substantially when non-adiabatic couplings are included, the paper's central claim would be refuted. A less direct but observable falsifier would be a low-energy collision experiment showing no resonance structure in the annihilation or inelastic-scattering cross section.","tokens_in":23241,"feed_emoji":"⚛️","tokens_out":5782,"duration_ms":57246,"temperature":0.7,"pith_summary":"The paper calculates Born-Oppenheimer potential curves for a number of excited Σ states of the hydrogen-antihydrogen molecule, for both even and odd Q symmetry, using explicitly correlated Kolos-Wolniewicz-type basis functions. It finds that these excited leptonic states, unlike in H2, plunge to -∞ at small inter-hadronic distances because of the attractive proton-antiproton Coulomb interaction. As a result, they become energetically accessible in zero-energy ground-state H-antihydrogen collisions and support many rovibrational states just above the ground-state dissociation threshold. The authors conclude that any theoretical treatment of ground-state H-antihydrogen scattering that ignores the excited leptonic states is incomplete, since these states can produce scattering resonances at essentially any collision energy. This matters for efforts to cool antihydrogen with ultracold hydrogen and for interpreting future matter-antimatter collision experiments.","feed_headline":"Include excited states in H-antihydrogen collision models","feed_subtitle":"They host rovibrational levels at the ground-state dissociation threshold, so single-curve scattering models miss resonances.","key_machinery":"The central tool is a dual-base set of Kolos-Wolniewicz-type explicitly correlated basis functions, i.e. products of exponentials in prolate-spheroidal coordinates with two different sets of non-integer exponents. This dual-base construction is what reveals (discretized) free positronium states—constant-energy curves that cross the molecular states—alongside the bound molecular curves. The Born-Oppenheimer potential curves are then fed into a radial B-spline solution of the hadronic equation, with the small-R region below 0.8 a0 treated by a linear (or polynomial) extrapolation. The Q-symmetry operation, a composition of mirroring the leptons on a plane bisecting the inter-hadronic axis and","core_discovery":"On the paper's own terms, the central discovery is that the spectrum of excited Σ leptonic states of the H-antihydrogen quasimolecule contains many rovibrational states with energies close to the ground-state dissociation limit of about -1.0 Hartree. Because the proton-antiproton attraction dominates the Born-Oppenheimer potential of every leptonic state at small R, each excited curve falls to -∞ as R→0, so even at zero collision energy a ground-state hydrogen and antihydrogen pair can access these states. The calculated rovibrational spectrum, obtained by extrapolating the potential curves below the critical distance and solving the hadronic Schrödinger equation, shows a dense set of levels","pith_inferences":["If the density of near-threshold resonances is as high as suggested, sympathetic cooling of antihydrogen using ultracold hydrogen may be severely affected by inelastic loss channels, since the atoms would frequently pass through resonant excited states instead of scattering elastically.","A direct test of the paper's extrapolation would be a full non-adiabatic four-body scattering calculation covering the near-threshold region; if the near-threshold level density survives without the Born-Oppenheimer extrapolation, the conclusion is robust, and if not, it pinpoints the limit of the claim.","The dual-base technique for capturing continuum-like positronium states in a bound-state basis could extend to other matter-antimatter systems or to including the protonium-positronium rearrangement channel variationally without separate basis functions.","Because the resonance positions are not computed to spectroscopic accuracy, an experiment looking for enhanced annihilation or inelastic loss at specific collision energies would first need a coupled-channel calculation that includes non-adiabatic couplings to know where to look."],"forward_implications":["Ground-state H-antihydrogen scattering models based on a single Born-Oppenheimer potential curve are incomplete; reliable cross-sections require close-coupling treatments that include excited leptonic states.","A large number of scattering resonances should occur near zero collision energy, and because the rovibrational states have finite lifetimes (annihilation, rearrangement into protonium and positronium, radiative decay), the resonance condition can be met at practically any collision energy.","The Q-odd Σ states cannot be dismissed as monotonically repulsive; they behave similarly to Q-even states at small R and must be included unless symmetry selection rules forbid it.","The demonstration that free positronium states can be captured in the same diagonalization as molecular states paves the way for close-coupling scattering calculations in which rearrangement channels are intrinsically included.","The findings are consistent with earlier full four-body calculations that found resonances near this threshold, reinforcing the need for excited-state treatment."],"fun_headline_variants":["Excited Sigma states host near-threshold resonances","Rovibrational levels near H-Hbar threshold demand new models","Include excited leptonic states in H-Hbar scattering","Excited states reshape H-antihydrogen collision physics"],"cache_read_input_tokens":2304,"weakest_assumption_plain":"The load-bearing premise is that extrapolating the Born-Oppenheimer leptonic potential curves below R=0.8 a0 and then solving the one-channel hadronic equation yields a faithful density of rovibrational states just above the ground-state dissociation threshold, even though the adiabatic approximation is known to break down near the critical distance.","fun_headline_variants_meta":{"raw":{"variants":["Excited Sigma states host near-threshold resonances","Rovibrational levels near H-Hbar threshold demand new models","Include excited leptonic states in H-Hbar scattering","Excited states reshape H-antihydrogen collision physics"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000876,"raw_usage":{"total_tokens":3564,"prompt_tokens":619,"completion_tokens":2945,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":363,"completion_tokens_details":{"reasoning_tokens":2879}},"tokens_in":363,"tokens_out":2945,"duration_ms":20340,"temperature":1.0,"reasoning_tokens":2879,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-03T22:49:48.346579+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"A full non-relativistic four-body scattering calculation of H-antihydrogen that includes all rearrangement and excited-state channels without the Born-Oppenheimer approximation: if the near-threshold rovibrational levels just above -1.0 Hartree disappear or shift substantially when non-adiabatic couplings are included, the paper's central claim would be refuted. A less direct but observable falsifier would be a low-energy collision experiment showing no resonance structure in the annihilation or inelastic-scattering cross section.","supporting_citations":[],"review_version":1}