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Excited $\Sigma$ states of the hydrogen-antihydrogen molecule

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Pith's one-line read 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

desk verdict 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. read the letter →

arxiv 2511.08308 v1 pith:ZUIFCHCZ submitted 2025-11-11 physics.atom-ph quant-ph

L. Brumm , J. Schürmann , A. Saenz This is my paper · ORCID
classification physics.atom-phquant-ph
keywords hydrogen-antihydrogenmoleculeBorn-OppenheimerpotentialcurvesexcitedSigmastatesQsymmetrypositroniumdiscretizationscatteringresonancesexplicitlycorrelatedbasisfunctionsantihydrogencollisions
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

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.

What carries the argument

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

What would settle it

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.

Watch

Extended reading notes

Core claim

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

Load-bearing premise

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.

Editorial extensions

If this is right

  • 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.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • 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.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 5 minor

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.

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 (3)
  1. [Sec. III.C and Fig. 9] 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.
  2. [Sec. IV.C] 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.
  3. [Sec. IV.C] 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.
minor comments (5)
  1. [Throughout] 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.
  2. [Appendix B, Fig. 10] The caption says 'Q-positive states'; this should be 'Q-even states' for consistency with the text.
  3. [Sec. III.C] 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.
  4. [Fig. 9] The inset is hard to read and the energy axis is not labeled with numerical values. Please enlarge and annotate the threshold region.
  5. [Sec. IV.C] 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.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the near-threshold rovibrational spectrum is obtained by solving the hadronic Schrödinger equation with independently computed Born-Oppenheimer curves, not by fitting the claimed result.

full rationale

The paper's central claim — that excited leptonic Σ states support rovibrational states near the ground-state dissociation threshold — follows from a direct computation: leptonic Born-Oppenheimer curves are obtained by diagonalizing the explicitly correlated Hamiltonian (Eqs. 5–8), and the hadronic rovibrational spectrum is obtained by solving the radial Schrödinger equation with those curves (Sec. III.C). No target quantity (e.g., the resonance density or near-threshold level positions) is used as an input to the calculation. The basis parameters in Table I are variational parameters optimized by energy minimization (Sec. III.B), not by fitting to the resonance spectrum or to the threshold density. The extrapolation of the potential curves below R = 0.8 a0 is a stated approximation, and the authors explicitly test its sensitivity by comparing linear and 6th-order polynomial extrapolations (Fig. 9), reporting only small changes; this is a robustness check, not a circular fit. The four-body check in Sec. IV.C uses an independent non-Born-Oppenheimer code (ATOM-MOL-nonBO, Ref. [54]) and is used only as a cross-check of the ground state, not to impose the excited-state spectrum. Relevant self-citations (e.g., refs. [14], [19], [21], [24]) provide context and earlier method development, but the central derivation does not reduce to those citations; the existence of the near-threshold states is supported by the paper's own diagonalized spectra and by the independent four-body resonance findings of Refs. [40,41]. The paper candidly states limitations ('This calculation is not meant to be of high accuracy, since part of the Born-Oppenheimer potentials were extrapolated and the effects of the strong-force interaction completely ignored'), but these limitations affect accuracy and physical completeness, not circularity. I find no step in which a prediction is identical by construction to an input, no fitted parameter renamed as a prediction, and no load-bearing self-citation chain. Score 0.

Assumptions & free parameters 8 free parameters · 6 assumptions · 0 invented entities

The calculation is a variational basis-set computation; no physical constants are fitted. It depends on several numerical choices: optimized basis exponents, basis truncation, extrapolation functions below Rc, and B-spline parameters. The key physical assumption is that Born-Oppenheimer curves extrapolated below Rc give a reliable enough near-threshold level density. No new particles or forces are postulated.

free parameters (8)
  • Q-even base1 exponents (y,x,u,w) = 0.3, -0.5, 0.5, 0.5
    Optimized by gradient descent minimizing the ground-state energy at R=5.0 a0 (Table I).
  • Q-even base2 exponents (y,x,u,w) = -0.625, -0.65, 0.121875, 0.14375
    Optimized for the 2nd excited state, starting from R=5.0 and iterated to R=0.8 a0 (Table I).
  • Q-odd base1 exponents (y,x,u,w) = -0.1, -0.5, 0.3, 0.5
    Optimized for the 1st excited state at R=5.0 a0 (Table I).
  • Q-odd base2 exponents (y,x,u,w) = -3.86875, -3.2825, 0.14375, 0.1375
    Optimized for the 1st excited state at R=0.8 a0 (Table I).
  • Basis truncation parameter Omega = 10
    Chosen as a compromise between accuracy and resources; convergence checked by increasing Omega.
  • Vext coefficients A, B, C near R=0 = 4.545473e4, 1.294251e5, 3.044522e6
    Tuned so that Vext approximates -1/R in the small-R interval; identical for all states.
  • Small-R extrapolation form below R=0.8 a0 = linear (or 6th-order polynomial)
    Chosen because the basis deviates below 0.8 and the adiabatic approximation fails; sensitivity was checked.
  • B-spline grid parameters = 2048 splines, order 15, Rmax=5.0 a0
    Chosen to converge the ro-vibrational energies.
assumptions (6)
  • domain assumption Born-Oppenheimer factorization of the wavefunction, Eq. (4)
    Used for all leptonic states; known to break down near Rc according to [33], acknowledged in Sec. III.C.
  • domain assumption Q-symmetry separates the Hilbert space into even and odd sectors
    Adopted from [10]; used to reduce basis size and to explain why Q-odd states should not couple to the Q-even ground state.
  • standard math Variational principle applies to excited states obtained by diagonalization
    Strict if all lower states are exact; approximately valid for the lower excited states as discussed in Sec. III.B.
  • standard math Kolos-Wolniewicz basis completeness in the limit Omega -> infinity
    Assumed as in [47]; practical convergence is tested by increasing Omega.
  • standard math Scaling u,w by rho emulates box discretization of continuum states, Eq. (10)
    Stabilization method used to identify free positronium states; validated by the observed quadratic energy scaling.
  • domain assumption Non-relativistic Hamiltonian and neglect of strong interaction and annihilation
    Stated in Sec. I and Sec. V; affects absolute energies and lifetimes, not the qualitative level-density conclusion.

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Pith. "Pith review of Excited $\Sigma$ states of the hydrogen-antihydrogen molecule." pith.science (2026). https://pith.science/paper/ZUIFCHCZ

@misc{pith2026251108308,
  author       = {Pith},
  title        = {Pith review of: Excited $\Sigma$ states of the hydrogen-antihydrogen molecule},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/ZUIFCHCZ}},
  note         = {Machine review of arXiv:2511.08308}
}
abstract

Adopting explicitly correlated Kolos-Wolniewicz-type basis functions, the Born-Oppenheimer potential curves of a number of excited $\Sigma$ states of the hydrogen-antihydrogen system ($\bar{\rm H}$) were calculated for both, even and odd, Q symmetries, including also free positronium states. It is demonstrated that the excited leptonic states support ro-vibrational states with energies close to the ground-state dissociation threshold. As a consequence, the excited leptonic states need to be considered in theoretical treatments of ground-state H-$\bar{\mathrm{H}}$ collisions.

Figures

Figures reproduced from arXiv: 2511.08308 by the authors.

Figure 1
Figure 1. Structure of the Hamiltonian matrix for a [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. Lowest-lying Born-Oppenheimer potential curves of H [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗
Figure 3
Figure 3. As Fig. 2, but showing the leptonic energies [PITH_FULL_IMAGE:figures/full_fig_p008_3.png] view at source ↗
Figures from the paper (6 more)
Figure 5
Figure 5. Figure 5: As Fig. 4, but for [PITH_FULL_IMAGE:figures/full_fig_p008_5.png]
Figure 6
Figure 6. Figure 6: Energy of Q-even states at R = 0.8 as a function of a real scaling parameter ρ. emulated by a simultaneous scaling of all exponents of the basis functions, i. e. of the base. More accurately, it is the radial coordinate that should be scaled, and thus within the here a…
Figure 8
Figure 8. Figure 8: Comparison of selected Born-Oppenheimer potential curves (Σ symmetry) for H2 (literature data) and HH (red curves, this work). For H ¯ 2 a selection of lowest lying X 1Σ + g (black solid lines, [51]) states as well as the b 3Σ + u (black dashed line, [52]) and the B 1Σ…
Figure 9
Figure 9. Figure 9: Spectrum of Q-even (red) and Q-odd (blue) [PITH_FULL_IMAGE:figures/full_fig_p010_9.png]
Figure 10
Figure 10. Figure 10: Relative differences of the Born-Oppenheimer potential curves for Q-positive states at different [PITH_FULL_IMAGE:figures/full_fig_p015_10.png]
Figure 11
Figure 11. Figure 11: as Fig. 10, but for the ten lowest lying Q-odd instead of Q-even states. [PITH_FULL_IMAGE:figures/full_fig_p016_11.png]

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