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This paper derives model-independent formulas that correct the smoothness and on-shell approximations in femtoscopy and coalescence, showing that for modern pp and PbPb collision sources the corrections stay at or below one percent.

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2026-08-03 05:13 UTC pith:LCNL3GLJ

load-bearing objection Solid, honest theory paper: general model-independent corrections to smoothness and on-shell approximations in femtoscopy and coalescence; the unverified equal-time approximation is a real caveat but is openly flagged and does not sink the central derivation. the 3 major comments →

arxiv 2602.02810 v2 pith:LCNL3GLJ submitted 2026-02-02 nucl-th hep-exnucl-ex

Corrections to the Smoothness and On-Shell Approximations in Femtoscopy and Coalescence

Isaac G. Smith , Kfir Blum This is my paper
classification nucl-th hep-exnucl-ex PACS 25.75.Gz
keywords femtoscopycoalescencesmoothness approximationon-shell approximationKoonin-Pratt formulaequal-time approximationblast-wave modelfinal-state interactions
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

Two-particle femtoscopic correlations and deuteron coalescence are usually extracted from the standard Koonin-Pratt formula, which assumes the particle emission source barely changes with the pair's relative momentum (the smoothness and on-shell approximations). This paper derives model-independent expansions — Eqs. (26), (43)–(45), and (69) — that give the leading, second-order corrections to those approximations for arbitrary sources and final-state interactions, at essentially the same numerical cost as the leading-order calculation. For angle-averaged correlations the first-order terms vanish by symmetry, so the corrections enter at second order. Tested in a blast-wave source model with realistic proton–proton interactions, the corrections are at or below about one percent for pp correlations and deuteron coalescence — comparable to current experimental uncertainties. The paper also constructs an observable, the difference between the angle-dependent and angle-averaged correlation, that isolates a subset of smoothness corrections and can be measured directly.

Core claim

The central discovery is that the smoothness and on-shell approximations are not ad hoc but are the zeroth-order terms of a controlled expansion in the relative momentum q times a characteristic source length scale ε. Expanding the source function S(r,q) in q and the Wigner-density kernels in partial waves yields explicit first- and second-order correction terms: the smoothness expansion (26), its angle-averaged form (43), and the analogous on-shell expansion (69) obtained by tracking how the true pair rest frame differs from the on-shell frame at order q². A key byproduct is the anisotropy observable (45), which vanishes under the smoothness approximation and therefore provides a model-inde

What carries the argument

The machinery is the Wigner-density expansion. The paper expands the relative-momentum-dependent source as S(r,q) = S(r) + q^i S_i(r) + q^i q^j S_ij(r) + O(q³ε³) and derives the kernels K(r,q) = |φ_q(r)|², K_i, and K_ij from the Schrödinger Wigner density of the scattering or bound wavefunction. Inserting these into the Koonin-Pratt correlation formula yields the smoothness expansion (26); a separate kinematic expansion of the boost between the true pair rest frame and the on-shell frame yields the on-shell expansion (69). The angle-averaged versions (43)–(45) isolate the second-order corrections and define the anisotropy observable that vanishes in the smoothness limit.

Load-bearing premise

The equal-time approximation — that the Bethe-Salpeter amplitude is time-independent and reduces to the Schrödinger wavefunction — must hold for the Koonin-Pratt formula to be the right starting point; the authors state it must be verified for a given model or interaction before their corrections can be trusted.

What would settle it

Compute the full relativistic two-particle correlation without the equal-time approximation for a specified source and final-state interaction and compare it with the Koonin-Pratt expression plus the corrections (26)/(43)–(45)/(69); if the difference exceeds the predicted percent-level corrections for modern-collider-size sources, the equal-time approximation — not smoothness or on-shell — is the dominant error. A cheaper, purely experimental falsifier: measure the anisotropy observable (45) in high-statistics pp collisions and check that it is as small as predicted; a much larger value would

Watch this falsifier — get emailed when new claim-graph text bears on it.

If this is right

  • Femtoscopic analyses can now quote a quantitative bound on smoothness and on-shell systematic errors at the same numerical cost as the leading-order calculation, for any source model and final-state interaction.
  • For blast-wave sources fitted to modern pp and PbPb data, the corrections are at or below one percent, so existing femtoscopy results using these approximations are not invalidated by them.
  • The anisotropy observable (45) gives an experimental handle: measuring C(q) and its angle average separately can reveal a subset of smoothness corrections without assuming a source model.
  • Deuteron coalescence factors inherit the same percent-level corrections, so the coalescence–correlation relation remains viable at this precision.
  • For distinguishable particles such as proton–lambda pairs, non-vanishing first-order anisotropic terms appear, making those systems the most likely place for smoothness corrections to matter.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same expansion could be applied to construct observables that isolate on-shell corrections specifically, since the on-shell correction is isotropic in q and currently only enters the angle-averaged correlation.
  • One could test the convergence of the qε expansion by computing the O(q³) terms for small sources (near R0 = 1 fm), where the paper's plots show corrections growing; without that, the percent-level claim is only validated for the blast-wave parameter sets considered.
  • High-statistics measurements of the anisotropy observable (45) in pp collisions would either confirm the sub-percent smoothness corrections or reveal that the blast-wave model underestimates them, providing a direct falsification path.
  • If the equal-time approximation is violated in a given interaction, these corrections correct the wrong zeroth order; the formulas would then need to be embedded in a relativistic Bethe-Salpeter treatment rather than the Schrödinger-based Koonin-Pratt framework.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 5 minor

Summary. The paper derives model-independent first- and second-order corrections to the smoothness and on-shell approximations in femtoscopy and coalescence, starting from the Koonin-Pratt formalism within the equal-time approximation. It presents explicit expansions for angle-dependent and angle-averaged correlations, an anisotropic observable combination, and numerical illustrations in a blast-wave model with Argonne v18 final-state interactions, finding corrections at or below the percent level for representative LHC parameters.

Significance. If correct, the framework provides a practical, low-cost way to bound two common systematic errors for arbitrary sources and FSI, and the observable in Eq. (45) is a useful empirical diagnostic. The free-particle and Gaussian-coalescence limits provide nontrivial consistency checks, and the paper is clearly written. However, one of the central angle-averaged formulas appears to omit a term of the same q^2 order, and the equal-time approximation explicitly limits the practical reach of the numerical conclusions.

major comments (3)
  1. [Sec. IIIB, Eq. (43)] For the angle-averaged correlation defined in Eq. (29), the denominator expansion must act on the full numerator at a given q^2 order. Writing N = N0 + N2 with N0 = ∫ K̄0,0 S̄0,0 and N2 = ∫ (K̄2,0 S̄2,0 + K̄2,2 S̄2,2), and D = 1 + (q^2/3)tr ε, the O(q^2) expansion is N/D = N0 + N2 − (q^2/3)tr ε (N0 + N2) + O(q^4). Eq. (43) omits the −(q^2/3)tr ε N2 term. This is not a higher-order-in-q term: for interacting wavefunctions Kij in Eq. (24) has a finite q→0 limit, so N2 is O(1) in q. The free-particle check in Sec. IIIC does not expose the omission because there N2 ∝ q^2. Please verify and correct, or state explicitly that N2 is truncated beyond leading order; if the latter, the expansion is not a complete second-order-in-q formula.
  2. [Sec. VI, equal-time approximation] The paper's central quantitative claim—percent-level corrections for blast-wave parameters—is conditional on the equal-time approximation in Eq. (6). The authors correctly state this in Sec. VI, but the abstract's 'corrections at or below the percent level' can easily be read as a bound on the total Koonin-Pratt systematic error. Since the ETA is an input to every formula in the paper, please make the conditionality explicit in the abstract and conclusions, and, if feasible, provide at least an order-of-magnitude test of the ETA for the blast-wave model (e.g., by comparing the time-integrated source with the full relativistic source for the same parameters).
  3. [Sec. V, Figs. 1–3; Eq. (18)] The source expansion (18) is truncated at O(q^3 ε^3) without a quantitative remainder estimate. For the smallest source (R0 = τ0 = 1 fm) and the upper q range shown, qε is not very small, so the size of omitted terms should be assessed. If the 'C(q)' curves in Fig. 1 are exact evaluations of Eq. (7) with the full q-dependent source, please state this explicitly; if they are the truncated second-order expansion, the percent-level claim needs a convergence check against the exact expression.
minor comments (5)
  1. [Sec. IIIB, Eqs. (43)–(45)] The symbol C(q) is used both for the angle-averaged correlation of Eq. (29) and for the angle-dependent correlation in Eq. (45). This makes Eq. (45) appear to be identically zero if taken literally. Please introduce distinct notation, e.g., C_avg(q) and C(q), for the two objects.
  2. [Sec. IIID, Eq. (59)] The symbol O(q^3 ε^3) is confusing in the coalescence context, where there is no external q. The expansion is in the relative-momentum integration variable of the bound-state Wigner density; please write O(k^3 ε^3) or define the third-moment term explicitly.
  3. [Sec. IIIC, Eq. (57)] The free-identical-particle result (57) keeps only the δij part of the angular integral (55). This is correct only for a spherically symmetric source S(r); otherwise the Qij term contracted with εij contributes. Please state this restriction in the text.
  4. [Abstract and Sec. V] The abstract says the corrections are 'at or below the percent level for pp correlations and deuteron coalescence', but the numerical demonstration is in the blast-wave model. Please add 'in the blast-wave model' to the abstract to avoid overgeneralization.
  5. [Sec. VB] The statement that the data in Fig. 1 of Ref. [29] were 'fit to the blast wave model' with the listed parameters gives no fit quality. Please report a χ² or show the fit curve, or say the parameters are representative rather than fitted.

Circularity Check

0 steps flagged

No significant circularity: the correction formulas are derived from the defining correlation/coalescence expressions and a Taylor expansion of the source, with no fitted quantity later renamed as a prediction.

full rationale

The derivation chain is self-contained. The smoothness expansion follows by inserting the Taylor expansion S(r,q)=S(r)+q_i S_i(r)+q_i q_j S_ij(r)+O(q^3 eps^3), Eq. (18), into the ETA correlation function, Eq. (7), and expanding the denominator to second order; the result, Eq. (26), is a direct algebraic consequence of the definitions, not of the target conclusion. Similarly, the angle-averaged results (43)-(45) and the coalescence result (62) follow from the same expansion plus angular integrals. The on-shell expansion, Eq. (69), follows from the kinematic relation P^0 = p^0 + kappa q^2/(2p^0)+..., Eq. (63), and a Taylor expansion of S in the frame boost; no fitted parameter is used as input to derive these formulas. The blast-wave parameters are fitted to ALICE correlation data, but the correction magnitudes are then evaluated at those parameters, not fitted or predicted from the corrections themselves. The self-citations [8,18] are used as background references for the standard Koonin-Pratt and coalescence formalism, while the paper re-derives the relevant expressions from the definitions; there is no load-bearing self-citation chain. The abstract and Sec. VI explicitly acknowledge that the equal-time approximation, Eq. (6), must also be verified: 'we have used the equal-time approximation, which must also be verified in a given model/interaction in order to safely use the Koonin-Pratt formalism.' That is an honest limitation about an input assumption, not circularity, because the corrections are corrections to the same leading-order expression built on that assumption. No step reduces, by construction, to its own input.

Axiom & Free-Parameter Ledger

3 free parameters · 7 axioms · 0 invented entities

The central derivation relies on a Taylor-series ansatz for the source and the standard Koonin-Pratt framework (ETA, sudden approximation, factorization). No new physical entities are introduced; the only free parameters enter through the illustrative blast-wave and Gaussian-deuteron models, and the fitted values are not part of the general derivation.

free parameters (3)
  • Blast-wave parameters fitted to ALICE pp data = R0=τ0=2.0 fm, Δτ=1.5 fm, βS=0.5, n=2, T=150 MeV
    Fitted in this paper to Fig. 1 of ALICE [29]; used for the percent-level correction estimate; no uncertainties reported.
  • Source-size scenarios (R0=τ0=1.0, 2.0, 7.0 fm) = 1.0, 2.0, 7.0 fm
    Chosen to illustrate weak/strong correction regimes; not data-constrained.
  • Deuteron Gaussian wavefunction width b = b≈3.5 fm
    Taken from literature [8] for deuteron; also scanned to show b-dependence.
axioms (7)
  • domain assumption Source function S(r,q) is analytic in q and the series S(r,q)=S(r)+q^i S_i(r)+q^i q^j S_ij(r)+... converges for the q values probed (Eq. 18).
    Underlies the entire expansion; controls the size of the neglected O(q³ε³) terms; not tested for small sources where qε may approach 1.
  • domain assumption Equal-time approximation: the Bethe-Salpeter amplitude is approximated by the nonrelativistic Schrödinger wavefunction (Eqs. 6–8).
    Acknowledged as requiring verification; the paper's corrections are computed inside ETA, so an ETA breakdown would dominate over smoothness/on-shell corrections.
  • domain assumption Sudden freeze-out: pair dynamics after freeze-out described by two-particle interaction alone.
    Standard Koonin-Pratt premise, stated in Sec. II.
  • domain assumption Pair source factorizes as product of single-particle sources in the blast-wave model (Eq. 75).
    Assumes no two-particle correlations in emission; used for the numerical estimates.
  • domain assumption For identical particles, the source is symmetric under (r,q)→(−r,−q) and thus S_i(r) is odd in r with ∫d³r S_i=0.
    Used to eliminate first-order corrections; stated in Sec. IIIA.
  • domain assumption Real spherically-symmetric FSI potential for the kernel identity ∫dΩ_q K_i=0 (Eq. 35).
    Used for angle-averaged correlations; reasonable for central potentials but excludes spin-orbit/tensor forces.
  • standard math Argonne v18 potential provides the pp FSI wavefunctions in CATS.
    External physics input from [32]; standard for pp femtoscopy.

pith-pipeline@v1.3.0-alltime-deepseek · 15689 in / 14636 out tokens · 141088 ms · 2026-08-03T05:13:46.752457+00:00 · methodology

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read the original abstract

Relativistic heavy-ion collisions produce femtometer-scale sources whose space-time structure can be constrained using two-particle femtoscopic correlations. Standard implementations rely on the smoothness and on-shell approximations, which effectively remove the relative momentum dependence of the particle emission function. We explore the validity of these approximations by deriving model-independent expansions that quantify the leading corrections for femtoscopy and coalescence with arbitrary sources and final-state interactions. The resulting first- and second-order correction terms can be evaluated with essentially the same numerical complexity as the usual Koonin-Pratt expressions; for angle-averaged correlations the first-order contributions vanish by symmetry. We illustrate the framework with explicit calculations in a blast-wave source model; for blast-wave parameter sets representative of pp and PbPb fits at LHC energies, the corrections are at or below the percent level for pp correlations and deuteron coalescence. These corrections are potentially subdominant compared to other effects, for example, corrections to the equal time approximation.

Figures

Figures reproduced from arXiv: 2602.02810 by Isaac G. Smith, Kfir Blum.

Figure 1
Figure 1. Figure 1: FIG. 1. Left [PITH_FULL_IMAGE:figures/full_fig_p019_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2 [PITH_FULL_IMAGE:figures/full_fig_p020_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3 [PITH_FULL_IMAGE:figures/full_fig_p021_3.png] view at source ↗

discussion (0)

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Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

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