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REVIEW 3 major objections 5 minor 27 references

Formulation of the Deser formula for the shift of energy levels of hadronic atoms in terms of the effective radius of the strong interaction with the Coulomb wave function of hadronic atoms at the origin

T0 review · 3 major / 5 minor · reviewed 2026-08-05 · deepseek-v4-flash

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Pith's one-line read The paper argues that strong interactions shift but do not spoil the Coulomb n^-3 scaling of hadronic-atom formation probabilities, and recasts the Deser level-shift formula in terms of an effective strong-interaction radius and an n-indepe

desk verdict A plausible but incomplete robustness claim for the n^-3 formation ratio; the Deser reformulation is a true but essentially definitional identity. read the letter →

arxiv 2608.03795 v1 pith:FG7QDRXY submitted 2026-08-04 hep-ph

O. Voskresenskaya This is my paper · ORCID
classification hep-ph
keywords hadronicatomspioniumDeserformulascatteringlengthstronginteractionCoulombwavefunctionperturbationtheoryenergy-levelshifts
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

Hadronic atoms are hydrogen-like bound states of two oppositely charged hadrons, and their strong-interaction level shifts are a precision probe of low-energy hadron scattering. This paper tries to establish that the simplest Coulomb predictions for these atoms are far more stable than the wave functions themselves: even though the strong interaction can distort the s-wave function at the origin by order one, the distortion is essentially the same for every principal quantum number n. If that holds, the ratio of formation probabilities in different ns states keeps the pure-Coulomb value (n2/n1)^3 to an accuracy of about 0.1%, so the strong-interaction uncertainties that plague absolute probabilities largely cancel in the ratios used by pionium-lifetime experiments. The paper also rewrites the standard Deser level-shift formula in terms of the mean radius of the strong-interaction region and the n-independent relative wave-function correction, giving a route to measure that radius from the 2s-2p splitting. The argument is carried by a first-order perturbation method that expresses the correction through the unperturbed wave function of the very state being corrected.

What carries the argument

The mechanism is a small-distance identity of the Coulomb wave functions: the logarithmic derivative of the reduced ns wave function at the origin, |dψ_n0/dr|_{r=0}/ψ_n0(0) = -μ α, is the same for all n, and the first two terms of the small-r expansion of n^{3/2}χ_n0(r) are also n-independent (Eqs. 62, 65). These coincidences make the integrals that define the first-order relative correction R_n0(0) differ from state to state only by terms of order r_s/r_B ~ 10^-3. The derivation uses a perturbation-theory variant that expresses the first-order correction to a discrete state in terms of that same state's unperturbed wave function and a second linearly independent solution, so it bypasses the

What would settle it

Evaluate c_n with the full Coulomb continuum wave functions for n=1,2,3 using a Yukawa strong potential of range 1/m_ρ, and check whether c_2-c_1 remains below 10^-3; alternatively, measure the 2s/1s or 3s/1s pionium formation-probability ratio experimentally at sub-percent precision and test (n2/n1)^3. Either would settle the n-independence claim.

Watch

Extended reading notes

Core claim

The central claim, stated the way the paper's author would state it, is that in first-order perturbation theory the relative strong-interaction correction to the ns Coulomb wave function at the origin, c_n = Δψ_ns(0)/ψ_ns^c(0), is independent of n up to terms of order a_s/r_B ~ 10^-3, for any form of the strong potential. From this it follows that the formation-probability ratio w_n1s/w_n2s remains (n2/n1)^3 + O(10^-3) (Eq. 66) even though the absolute wave functions at short distances are strongly modified. The same n-independence lets the Deser formula be reformulated as ΔE_ns = -(2π/μ) <r>_s c_n |ψ_n0^c(0)|^2 (Eq. 75), where <r>_s = a_s/c_n is the mean radius of the strong-interaction reg

Load-bearing premise

The result depends on the omitted (a_s/r_B) ln(r_B/r_s) ~ 10^-2 contributions—dropped by using plane waves for the continuous spectrum—being independent of n; if those terms varied with n at the 10^-2 level, the claimed 10^-3 accuracy of the formation-probability ratio would fail.

Editorial extensions

If this is right

  • The pure-Coulomb ratio w_{n1s}/w_{n2s} = (n2/n1)^3 survives strong interactions to O(10^-3), so pionium formation-ratio inputs to lifetime analyses can be treated as Coulomb-like even if the short-distance wave functions are badly distorted.
  • Measuring the 2s-2p energy splitting together with the ground-state formation probability (or c_1) determines the mean strong-interaction radius <r>_s, complementing the scattering lengths extracted from the lifetime.
  • The Deser formula's factorization into strong and electromagnetic factors persists in the reformulated version, now with the relative wave-function correction c_n and radius <r>_s carrying the strong part.
  • Free-pair to bound-state formation ratios w_ps/w_ns also retain their Coulomb value to about 10^-3 in the same approximations.
  • Absolute formation probabilities remain model-dependent at the tens-of-percent level; only the ratios are protected by the n-independence.

Reading between the lines

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

  • The n-independence rests on a special property of the Coulomb s-state family near the origin; analogous ratios for non-s states or for states where the wave function vanishes at the origin would not be protected, so the result should not be read as a general statement about all formation channels.
  • The same perturbative machinery should apply to other short-range modifications of the Coulomb problem, e.g. finite nuclear size or vacuum-polarization potentials; the prediction is that their leading correction to formation-probability ratios also cancels at the 10^-3 level.
  • If the full Coulomb-continuum calculation the paper flags as needed were carried out and showed 10^-2-level n-dependence, the central accuracy claim would collapse; this makes a numerical evaluation of c_2-c_1 with exact continuum wave functions the natural check.
  • A dedicated pionium experiment comparing 3s/2s or 3s/1s formation rates directly would be a sharper test of the n-independence than the current lifetime measurement, because it isolates the wave-function ratio without relying on annihilation widths.
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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 paper addresses the effect of the strong interaction on the formation probabilities and energy shifts of hadronic (pionium) atoms. It argues that in first-order perturbation theory the strong interaction modifies the ns Coulomb wave function at the origin by a state-independent relative correction c_n, up to terms of order a_s/r_B ~ 10^-3. Consequently the ratio w_{n1s}/w_{n2s} remains (n2/n1)^3 + O(10^-3), Eq. (66), despite strong-interaction-induced distortions that are O(1). The paper also rewrites the Deser level-shift formula in terms of an effective strong-interaction radius <r>_s and c_n, Eqs. (74)-(75), and gives numerical estimates for the 1s and 2s-2p shifts.

Significance. The standard derivation of the Deser formula (68)-(70) is clean, and the numerical estimates in Eqs. (72)-(73) are consistent with the stated scattering lengths. If the n-independence of c_n were rigorously established, Eq. (66) would provide a useful robustness argument for the DIRAC determination of pion scattering lengths. The paper explicitly identifies the limitation of the plane-wave approximation, which is commendable. However, the central O(10^-3) accuracy claim is not yet demonstrated: it rests on integrals (Eqs. (61)-(63)) whose derivation is not shown. The advertised Deser reformulation is an algebraic identity rather than a new physical relation. The paper is promising but requires a major revision to substantiate the headline claim.

major comments (3)
  1. [Section 2.2, Eqs. (61)-(63)] The central accuracy claim, Eq. (66), rests on the assertion that R_n0(0) in Eq. (61) is n-independent up to O(a_s/r_B). This equation is introduced with 'The calculations lead to the following result' and no derivation. The explanation after Eq. (62) is insufficient: Eq. (62) fixes only the linear term of the Coulomb wave function at r=0, while the integrand in Eq. (61) extends over r ~ r_s, where the polynomial P_{2n}(νr) contributes at O(νr) with coefficients that in principle depend on n. A logarithmically enhanced term of order (a_s/r_B) ln(r_B/r_s) ~ 10^-2 is acknowledged in Section 2.1, and the claim that this term is n-independent is not demonstrated. Please provide the derivation of Eqs. (61) and (63) or an explicit n-dependent error bound.
  2. [Section 3, Eqs. (74)-(75)] Equation (75) is an algebraic identity rather than a new relation: substituting <r>_s = a_s/c_n from Eq. (74) returns exactly the standard Deser formula (70). The paper should state explicitly that this is a parameterization, and that the only nontrivial content is the n-independence of c_n. As written, the advertised 'formulation ... in terms of the effective radius' overstates the novelty. This does not invalidate the rest of the paper, but the presentation should be corrected.
  3. [Section 2.2, Eqs. (63)-(64)] The factorized form ψ_n0(r) = R_n(r)ψ_n0^(0)(r) with R_n independent of n is stated for an arbitrary potential U_s. Equation (63) contains an O(r_s/r_B) error term, and for r comparable to r_B the n-dependence of ψ_n0^(0)(r) may enter the factorization. The numerical support cited as [25] is a preprint covering the first four states and does not establish a universal statement for all n. Please state the domain of validity in r and n, or supply a proof that the remainder is n-independent.
minor comments (5)
  1. [Section 1.1, Eq. (12)] The ratio should read (n2/n1)^3, not (n2/n2)^3.
  2. [Section 2.1, Eq. (41)] The sign of c_n is inconsistent with Eq. (37) and with Eq. (74): Eq. (41) should be c_n = -∫ U_s(r) r dr + O(10^-3) if U_s is defined as in Eq. (36).
  3. [Section 2.1, Eq. (42)] The final equality should be '= (n2/n1)^3 + O(10^-3)', not '= O(10^-3)' as written.
  4. [Section 2.2, Eq. (61)] The integrand appears to contain an extra factor r ('r^2 e^{-2νr/n} r U_s(r)'); please check dimensions and the intended measure.
  5. [Section 3, Eqs. (72)-(73)] Use 'eV' rather than 'эВ' and a decimal point in '-0.45 eV' for consistency with the rest of the text.

Circularity Check

1 steps flagged · score 6.0 of 10

The advertised reformulation of the Deser formula is definitionally circular: Eq. (75) is Eq. (70) with a_s replaced by <r>_s c_n, where Eq. (74) defines <r>_s = a_s/c_n. The main n-independence claim is not circular but relies on unshown estimates.

  1. self definitional [Section 3, Eqs. (74)-(75)]
    "the relation a_s/c_n = -∫ U_s(r)r^2 dr/∫ U_s(r)r dr = <r>_s (74) allows reformulating the Deser equation (70) ... as follows ΔE_{n0}^s = -(2π/μ) <r>_s c_n |ψ_{n0}^c(0)|^2. (75)"

    Equation (75) is obtained from the standard Deser formula (70), ΔE_{n0}^s = -(2π/μ) a_s |ψ_{n0}^c(0)|^2, by substituting a_s = <r>_s c_n, which is exactly the definition given in Eq. (74). Thus the 'new formulation' is the input formula rewritten with a renamed combination of parameters; it carries no independent content. The effective radius <r>_s is not measured independently but is defined through a_s/c_n, so the advertised reformulation reduces by construction to the original Deser formula.

full rationale

The paper's substantive physics claim—that the ns-formation probability ratio remains (n2/n1)^3 + O(10^-3) when strong interaction is included—is not itself circular: it is argued from the Zeldovich perturbation expression (61) and Coulomb-function properties (18), (65), with the n-dependent corrections claimed to be O(a_s/r_B). Whether the unshown integrals behind Eq. (61) really justify the O(10^-3) independence is a verification/correctness issue, not a circularity. The one clearly circular element is the Deser 'reformulation' of Section 3: Eq. (74) defines <r>_s = a_s/c_n, and Eq. (75) is then just Eq. (70) with this substitution, so the reformulation is equivalent to its input by construction. Same-group citations ([22], [25]) are used for numerical confirmation and earlier plane-wave estimates, but the main derivation does not reduce to those citations, so they do not raise the score further. Overall: partial circularity in the advertised reformulation, while the central n-independence claim has independent mathematical content, giving score 6.

Assumptions & free parameters 4 free parameters · 7 assumptions · 0 invented entities

No new physical entities are introduced; c_n and <r>_s are defined quantities (Eqs. (41), (74)), not postulated entities. The central derivation claims validity for any short-range potential U_s, so there are no fitted parameters in this paper; the listed inputs are order-of-magnitude scales and literature values entering the numerical estimates, plus the unshown Zeldovich evaluation that carries the accuracy claim.

free parameters (4)
  • r_s (scale of the strong interaction region) = ~ m_rho^-1
    Order-of-magnitude input; controls the O(10^-3) estimates in (66) and the O(1) correction in (39); the paper proposes extracting it from measurements via (77).
  • <r^2>_p (size of the pion-pair production region) = 3 to 10 fm (estimate from [18])
    Enters the O(<r^2>_p/r_B^2) ~ 10^-3 correction in (19) and the uncertainty of absolute probabilities; only known to within a factor of about 3.
  • a_0, a_2 (pi pi s-wave scattering lengths) = 0.22, -0.044 (m_pi = 1)
    Taken from [27]; used only for the numerical estimates (72)-(73), not in the central derivation.
  • U_s(r) (strong interaction potential) = model-dependent (delta-function, virton-quark, Yukawa)
    The n-independence claim is made for arbitrary U_s, but the magnitude of c_n (~0.28 to 0.77) and the estimates depend on its form ([15], [16], [25], [26]).
assumptions (7)
  • domain assumption Nonrelativistic Schrodinger description of hadronic atoms with V = V_c + V_s
    Eqs. (8)-(9), (20)-(21); the whole analysis is nonrelativistic two-body quantum mechanics.
  • domain assumption Slowly varying production amplitude M(p) factorized at p = 0 (point-source approximation)
    Eqs. (10), (15); reduction of formation probability to |psi(0)|^2; source-size effects enter only as O(<r^2>_p/r_B^2).
  • domain assumption Standard scattering-length normalization of the strong potential, integral U_s r^2 dr = -a_s (Eq. (69))
    Bridges the potential and the Deser formula (68)-(70); inherited from [8] and standard low-energy scattering theory.
  • standard math Small-r expansions of reduced Coulomb s-wave functions have n-independent leading terms (Eq. (65))
    Properties of hydrogen-like Coulomb functions, explicitly displayed for n = 1 to 4; powers the O(10^-3) n-independence claim.
  • ad hoc to paper First-order Zeldovich perturbation solution (57)-(60) evaluates to (61)-(63) with n-dependence only at O(r_s/r_B)
    The decisive step is announced as 'The calculations lead to the following result' (Section 2.2); the derivation is not shown in the text.
  • ad hoc to paper Plane-wave approximation for the Coulomb continuum in the Rayleigh-Schrodinger variant (Eqs. (35)-(41))
    The paper itself flags that this omits (a_s/r_B)ln(r_B/r_s) ~ 10^-2 contributions, 'the accuracy of which requires further study' (Section 2.1 after Eq. (45)).
  • domain assumption The 2p-level strong shift is negligible in the 2p-2s splitting
    Eq. (73) sets Delta E_2p-2s = Delta E_20, implicitly dropping the p-wave contribution without comment.

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Cite this review

Pith. "Pith review of Formulation of the Deser formula for the shift of energy levels of hadronic atoms in terms of the effective radius of the strong interaction with the Coulomb wave function of hadronic atoms at the origin." pith.science (2026). https://pith.science/paper/FG7QDRXY

@misc{pith2026260803795,
  author       = {Pith},
  title        = {Pith review of: Formulation of the Deser formula for the shift of energy levels of hadronic atoms in terms of the effective radius of the strong interaction with the Coulomb wave function of hadronic atoms at the origin},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/FG7QDRXY}},
  note         = {Machine review of arXiv:2608.03795}
}
read the original abstract

It is shown that the relations between probabilities of A_2{\pi}-atoms creation in ns-states, derived with neglecting of strong interaction between pions, remain practically unchanged if the strong interaction is taken into account in the first order of perturbation theory. The formulation of Deser equation for the energy shift of energy levels of hadronic atoms is given in terms of the effective range of strong interaction and the relative correction to the Coulomb wave function of hadronic atoms at the origin, caused by the strong interaction.

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