Uncertainty band evaluation of optical potentials and differential cross-sections. Application to $^8$Li + $^{58}$Ni elastic scattering

J. G\'omez-Camacho; O. C. B. Santos

REVIEW 3 major objections 4 minor 26 references

Uncertainty band evaluation of optical potentials and differential cross-sections. Application to $^8$Li + $^{58}$Ni elastic scattering

T0 review · 3 major / 4 minor · reviewed 2026-08-06 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

Pith's one-line read Standard chi-square fits can be converted into calibrated Bayesian uncertainty bands without running MCMC.

desk verdict The paper's core calibration is wrong—the chi-square goodness-of-fit law is not a likelihood—so the uncertainty bands are not what they claim, though the application is honest and the physics conclusions may survive. read the letter →

arxiv 2507.11591 v1 pith:T5CIZDAQ submitted 2025-07-15 nucl-th

O. C. B. Santos , J. Gómez-Camacho This is my paper

classification nucl-th PACS 24.10.Ht25.60.Bx25.60.-t

keywords uncertaintyquantificationopticalmodelelasticscatteringchi-squaredfitsBayesianp-valuescovarianceenhancementlithium-8nickel-58

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

This paper tries to show that the usual least-squares chi-square minimization already contains everything needed to produce calibrated Bayesian uncertainty bands for optical-model potentials and cross sections, without Markov-chain samplers. The central claim is that the probability of a parameter region bounded by a chi-square contour is given by a ratio of chi-square cumulative probabilities, Eq. (5), and that the covariance matrix of parameters on that contour is the least-square covariance times an enhancement factor $f[b]=\sqrt{(b-\chi^2_m)/M}$, Eq. (12). Applied to $^8$Li + $^{58}$Ni elastic scattering at 23.9, 26.1, and 30.0 MeV, a single energy-independent four-parameter Woods-Saxon potential fits well ($\chi^2/L = 0.986$), the imaginary potential is constrained near 9--10 fm while the real one is not, and the 28.7 MeV data fall outside the 2-$\sigma$ bands. If the method is right, uncertainty bands become a routine byproduct of existing fitting codes rather than a reason to run expensive sampling.

What carries the argument

The central object is the contour region $\Gamma[b]$ in the $M$-dimensional parameter space defined by $\chi^2_m \le \chi^2(a) \le b$, with $b$ fixed by the desired p-value. The identity that carries the argument is Eq. (5), $p(\Gamma[b]|H) = P(b,L)/P(\chi^2_m,L)$, where $P(b,L)$ is the cumulative chi-square distribution with $L = N-M$ degrees of freedom; this converts chi-square contours into Bayesian credible regions. The companion scaling law, Eq. (12), $\sigma[b](O) = f[b]\,\sigma_{LS}(O)$ with $f[b] = \sqrt{(b-\chi^2_m)/M}$, turns the least-square covariance matrix from a standard fit into the covariance on the contour and then into uncertainty bands for any smooth observable $O(a)$, while the general approach samples the region $\Gamma[b]$ directly and needs no parabolic approximation.

What would settle it

A direct repeated-experiment simulation with known true optical parameters would settle the calibration: generate many synthetic data sets from the fitted model, refit each one, and count how often the true parameters fall inside the recipe's 1-sigma and 2-sigma regions. If the empirical coverage deviates substantially from 31.8% and 4.5%, the identification of the region probability with the chi-square tail is the point to revisit.

Watch

Extended reading notes

Core claim

On the paper's own terms, the discovery is that the difference between 'frequentist' and Bayesian uncertainties in optical-model fits is not a philosophical gap but a multiplicative factor. The least-square covariance from the Hessian at the minimum describes only the tangent of the chi-square ellipsoid; regions of higher statistical significance are bigger in covariance by the factor $f[b]$, and the p-value attached to a region $\Gamma[b] = \{a: \chi^2_m \le \chi^2(a) \le b\}$ is $P(b,L)/P(\chi^2_m,L)$ with $L = N-M$ degrees of freedom. The paper derives this from Bayes' theorem applied to the chi-square sampling distribution, and verifies in the $^8$Li + $^{58}$Ni case that the cheap 'simplified' propagation bands match the expensive 'general' sampling bands. The application also establishes a physics conclusion: elastic data around the Coulomb barrier determine $W(r)$ but not $V(r)$ in the 9--10 fm region, so microscopic real potentials cannot be validated by mere agreement, and the 28.7 MeV data are inconsistent with an energy-independent optical model even when parameter uncertainty is included.

Load-bearing premise

The calibration stands or falls on one identification: the probability of the data given a parameter region is the repeated-experiment chi-square tail with $L = N-M$ degrees of freedom, $P(H|\Gamma[b]) = 1 - P(b,L)$, rather than the likelihood $\exp(-\chi^2/2)$.

Editorial extensions

If this is right

Standard chi-square fitting codes can be upgraded in place: the covariance matrix they already output, multiplied by $f[b]$, gives 1-sigma and 2-sigma bands for parameters and observables.
Uncertainty bands for optical potentials become narrow around the strong-absorption radius (9--10 fm here) and wide outside it, so elastic angular distributions near the rainbow angle carry most of the constraining power for the real potential.
A model that fits 'nice' data well but leaves 'nasty' data outside the 2-sigma bands can be rejected as incomplete for those data, even after parameter uncertainty is accounted for.
Bayesian credible regions shrink when the global chi-square minimum is poor, so the model must first pass a goodness-of-fit check before the bands are interpreted; the paper applies this by excluding the 28.7 MeV data from the nominal fit.
Complex reaction calculations such as CDCC and coupled-channels evaluations can inherit uncertainty bands from a small number of perturbed runs (about $2M$ calculations) using the same propagation rule.

Reading between the lines

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

Editorial inference: the same enhancement-factor prescription should transfer to any smooth functional of the optical parameters, so transfer cross sections, spectroscopic factors, or fusion probabilities could inherit calibrated bands from one optical-model fit plus $2M$ finite-difference runs.
Editorial inference: because Eq. (12) assumes a parabolic $\chi^2$ surface, the simplified band should be checked against the general band whenever the Hessian-based correlation matrix is large and contours are visibly non-elliptical, as in the $V$--$a_V$ panel here; a disagreement would mean a single covariance matrix cannot summarize the parameter correlations.
Editorial inference: the paper's finding that the 28.7 MeV residuals lie outside the 2-sigma bands does not by itself tell whether that energy features systematic normalization problems, channel-coupling effects, or a genuine energy dependence, and checking those alternatives is a natural next step with the same band tools.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Colleague, read this one if you want a clear example of a promising idea sinking on a statistical misunderstanding. The authors propose a cheap way to get Bayesian uncertainty bands for optical potentials without MCMC: take the least-squares covariance matrix, scale it by an 'enhancement factor' f[b] derived from the chi-square distribution with L=N-M degrees of freedom, and claim the resulting regions have the corresponding p-values. The general approach—take extremes of observables over the χ2≤b region—is sensible as a numerical procedure, and the simplified propagation formula is a natural shortcut. What is actually new: the analytic enhancement factors and the explicit comparison of the two approaches for 8Li+58Ni. The application is also careful: energy-independent Woods-Saxon fits to three energies, the imaginary potential constrained around 9-10 fm, the real part not, and the 28.7 MeV data clearly inconsistent with the model.

But the central calibration is wrong. The load-bearing step is Eq. (A26): they identify P(H|Γ[b]) with the complementary CDF of the minimized chi-square, P(b,L). That's the repeated-experiment distribution of a goodness-of-fit statistic, not the likelihood of the data given the parameters. For normal errors the likelihood is exp(-χ2(a)/2). Substituting the wrong expression into Bayes' theorem produces the posterior in Eq. (A33), which has a Jacobian 1/S(b) and actually vanishes at the best-fit point. In the exactly solvable one-parameter normal location problem, their recipe labels an interval that contains about 99.8% of the true posterior mass as '1σ'. So the p-values, the enhancement factors, and the 1σ/2σ bands are not calibrated posterior statements.

The authors are honest about the weird consequence (worse fits give narrower bands, Sec. III.E) and even call for a benchmark. That's to their credit. But that odd behavior is a symptom of the wrong identification. The physics conclusions are probably robust qualitatively—anyone can see from the residuals that one energy doesn't fit—but the specific claim that those bands are 1σ or 2σ is not supported.

Who is this paper for? Nuclear reaction theorists who want uncertainty bands without MCMC. The idea is attractive and the paper is readable, but the method as presented is not usable. It deserves a serious referee rather than desk rejection, because the flaw is subtle, the application is relevant, and a corrected version might address a real need. My recommendation: engage with it—a referee should reject in current form but might be able to guide a fix.

Referee Report

3 major / 4 minor

Summary. The manuscript presents a method to derive Bayesian uncertainty bands for optical-model potentials and differential cross sections from least-squares fits. After an LS fit provides best-fit parameters and a covariance matrix, regions Γ[b] are defined by χ2m ≤ χ2(a) ≤ b; Eq. (5) assigns a p-value to each region, and either direct sampling (general approach) or an analytic enhancement factor f[b] (simplified approach) produces 1σ and 2σ bands. The method is applied to 8Li + 58Ni elastic scattering, yielding an energy-independent four-parameter fit for the 23.9, 26.1, and 30.0 MeV data, an imaginary potential that is better determined than the real potential in the 9–10 fm region, and a statement that the 28.7 MeV data are inconsistent with the model even within the 2σ bands.

Significance. Should the calibration in Eq. (5) be correct, the method would be valuable: it would produce p-values and uncertainty bands using only standard χ2 codes, without MCMC, and would offer an explanation for the enhancement of frequentist uncertainties relative to Bayesian ones. The paper is clearly written, gives a detailed account of the χ2-surface sampling, and compares the general and simplified approaches on real data. However, the central calibration is statistically invalid; because the p-values, enhancement factors, and all 1σ/2σ band labels derive from Eq. (5), the paper's deliverable of calibrated uncertainty bands is not achieved. The physics application and the qualitative conclusions about the 8Li + 58Ni data may survive a corrected calibration, but the central claim of the method does not.

major comments (3)

[II.B, Eq. (5), and Appendix A.3, Eq. (A26)] The identification P(H|Γ[b]) = 1 − P(b,L) with L = N − M is incorrect. For a region defined by χ2m ≤ χ2(a) ≤ b, the sampling distribution of χ2(a) − χ2m is a χ2 law with M degrees of freedom (or an F distribution when the variance is estimated), whereas the χ2_{N−M} law of the minimized statistic describes the global goodness of fit at the best-fit point, not the probability of the data conditional on parameters lying in Γ[b]. Consequently Eq. (5), Table I, Table IV, and the f[b] factors in Eq. (12) are all miscalibrated. A direct calculation in the exactly solvable one-parameter normal-location problem shows that the region labeled '1σ' by this recipe contains about 99.8% of the posterior mass, not 68%.
[Appendix A.3, Eq. (A33)] The derived posterior density PB(a) ∝ −(dP/db)(1/S(b))P0(a) vanishes at the best-fit point because S(b)→0 as b→χ2m, whereas a posterior obtained from the likelihood exp(−χ2(a)/2) would be maximal at the best fit. This internal inconsistency is a direct symptom of using the goodness-of-fit tail as a likelihood, and it means that the 'Bayesian' probability statements in the paper are not posterior probabilities in the usual sense.
[III.E] The reported narrowing of the uncertainty bands when 'nasty' data are added is presented as a consequence of the Bayesian treatment, but it follows from dividing by P(χ2m,L) in Eq. (5), where P(χ2m,L) is the tail probability of the observed fit statistic. That tail probability is not the model evidence P(H) in Bayes' theorem, so the claimed robustness of the method is an artifact of the miscalibrated normalization rather than a property of a sound Bayesian analysis.

minor comments (4)

[Table I] The table uses commas as decimal separators in several entries, for example '52,35' and '0,31731'; use decimal points throughout for consistency.
[Keywords] The keyword list contains a typo, 'maxcimum likelihood', which should be 'maximum likelihood'.
[II.C, Eqs. (10)–(12)] The notation around Eqs. (10)–(12), especially the accents on the covariance and derivative variables, is garbled in the rendering and should be cleaned up.
[IV] The paper calls for a benchmark comparison with another Bayesian method; such a benchmark would have been a useful check of Eq. (5) and should be included in any revision.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the p-value calibration rests on an explicit, contestable assumption (Eq. A26), not on an input that is equivalent to the output by construction.

full rationale

The derivation chain is self-contained. The least-square covariance (Eq. 1 / Eq. A13) is obtained by error propagation from the data uncertainties; the maximum-likelihood covariance (Eq. 4 / Eq. A21) is an analytic integral over the ellipsoid chi2(a)=b; and the Bayesian p-value (Eq. 5 / Eq. A33) follows from the explicitly stated conditional-probability assumption P(H|Gamma[b]) = 1 - P(b,L) in Eq. A26. Each step is derived in the appendix from stated assumptions rather than imported from a self-citation, and the enhancement factors f[b] and F_B are analytic functions of chi2_m, L, and M, not parameters fitted to the observables whose bands they scale. The contested identification of the data probability given a parameter region with the goodness-of-fit tail probability is a statistical-calibration assumption; if wrong it produces miscalibrated labels, but that is a correctness concern, not circularity. The application itself uses standard chi2 minimization, grid sampling, and error propagation on the same four-parameter Woods-Saxon model, so the bands are honest derived quantities and the paper makes no coverage claim that would be forced by construction. The narrowing of bands with 'nasty' data in Sec. III.E is acknowledged as a direct consequence of the denominator P(chi2_m,L), not a disguised restatement of the data. Self-citations in the bibliography (e.g., refs. [6,17]) are contextual and are not load-bearing; refs. [7,9] are independent external works. No circular step can be exhibited from the paper's own equations.

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

The fitted optical parameters are the legitimate inputs of any UQ analysis and are listed for completeness. The load-bearing additions are the statistical axioms: normal uncorrelated errors; the contested identification P(H|Gamma[b]) = 1 - P(b, L) (Eq. A26); the neglect of the prior's effect; the asymptotic Hessian approximation (Eqs. A4-A5); the Woods-Saxon form with fixed radii and a single energy-independent potential; and quadratic chi2 for the enhancement factors. No new physical entities are postulated.

free parameters (6)

V (real potential depth) = 8.3 +/- 4.5 MeV (3-set fit, r0 = 1.1 fm)
Fitted to the elastic cross sections; the uncertainty bands and all propagated results depend on this fitted value and its covariance with the other parameters.
aV (real diffuseness) = 0.95 +/- 0.16 fm
Fitted; strongly anticorrelated with V (-0.93), so the pair mainly constrains the potential tail, not V and aV separately.
W (imaginary potential depth) = 97 +/- 35 MeV
Fitted; the fit yields an imaginary depth about 12 times the real depth, an unusual and physics-relevant output.
aW (imaginary diffuseness) = 0.624 +/- 0.036 fm
Fitted; anticorrelated with W (-0.88).
r0 (reduced radius) = 1.1 fm, also checked at 1.2 fm
Fixed by hand, not fitted, justified by the Igo ambiguity; the paper checks both values and finds identical fit quality.
Gaussian kernel width c (Eq. A22) = of order 1
Hand-chosen convergence parameter in the smoothed delta function used for the general-approach covariance; the paper requires c-independence for a 'reasonable choice' but does not specify the value used.

assumptions (6)

domain assumption Experimental cross sections are independent normal variables around the true values with known standard deviations sigma_i and no mutual correlations.
Sec. II.A: this defines chi2(a) = Sum_i (y_i(a) - yhat_i)^2 / sigma_i^2. Systematic uncertainties would enter only through inflated sigma_i.
ad hoc to paper P(H|Gamma[b]) = 1 - P(b, L): the probability of the data given that parameters lie in Gamma[b] is the cumulative chi2_L distribution of the minimized chi2 (L = N - M).
Appendix A, Eq. (A26). Load-bearing and, in the reviewer's assessment, wrong: it is the repeated-experiment law of the goodness-of-fit statistic, not a likelihood over parameters. It produces the unphysical posterior (Eq. A33) that vanishes at the best fit.
domain assumption The analysis is data-driven, so the prior P0(a) does not distort the p-value regions.
Appendix A.3, final paragraph. The prior that would reconcile Eq. (A33) with the likelihood exp(-chi2(a)/2) would have to depend on the data through b = chi2(a), so this neglect is not neutral.
standard math The residual-weighted second-derivative terms in the Hessian are negligible (Eqs. A4-A5).
Standard asymptotic least-squares approximation; reasonable for the 3-set fit (chi2/L = 0.986), weaker for the 4-set fit (chi2/L = 1.91).
domain assumption A Woods-Saxon optical potential with fixed reduced radii (r0 = 1.1 or 1.2 fm) and four free parameters (V, aV, W, aW), independent of energy, describes the data at the fitted energies.
Sec. III.A: fits three energies well, fails for four energies (chi2/L = 1.91), which motivates the post-hoc exclusion of the 28.7 MeV data.
domain assumption The chi2(a) surface is parabolic in the region of interest, so cov[b] = f[b]^2 * cov_LS and the simplified error propagation applies.
Eqs. (4), (A19), (A21); the paper itself notes the actual contours in Fig. 2 deviate from ellipses, so the enhancement-factor formulas are approximate.

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Pith. "Pith review of Uncertainty band evaluation of optical potentials and differential cross-sections. Application to $^8$Li + $^{58}$Ni elastic scattering." pith.science (2026). https://pith.science/paper/T5CIZDAQ

@misc{pith2026250711591,
  author       = {Pith},
  title        = {Pith review of: Uncertainty band evaluation of optical potentials and differential cross-sections. Application to $^8$Li + $^58$Ni elastic scattering},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/T5CIZDAQ}},
  note         = {Machine review of arXiv:2507.11591}
}

abstract

A statistical method is presented to evaluate the uncertainty bands in the optical nucleus-nucleus potential and in differential cross sections. The starting point is the least square fit of a set of experimental values of elastic differential cross sections, varying the relevant optical potential parameters. This is done using standard $\chi^2$ minimization codes, that provide the covariance matrix of the parameters. A maximum likelihood exploration of the $\chi^2$ surface in parameter space allows to determine the covariance matrix of the parameters associated to a contour of a given $\chi^2$ value. Bayes theorem allows to assign probabilities (p-values) to the regions in parameter space, characterized by $\chi^2$ contours. The method allows to obtain uncertainty bands of an arbitrary observables associated to a given p-value using two approaches. The general approach determines the extremes of the observables calculated in the region of parameter space associated to that p-value. This requires an adequate sampling of parameter space, and explicit calculations of the observables on all sampling points. The simplified approach considers uncertainty propagation of the observable in terms of the optical model parameters. This involves the least-square covariance matrix, given by $\chi^2$ minimization codes, and analytically calculated enhancement factors for each p-value. The method, in the general and simplified approaches, is applied to recent measurements of the elastic differential cross sections of $^8$Li + $^{58}$Ni. $1\sigma$ and $2\sigma$ uncertainty bands are obtained for the optical potentials as a function of the distance, and the differential cross sections as a function of the angle. The general and simplified approaches are very similar in this case. The application of the procedure to determine uncertainty bands of complex scattering calculations is discussed.

Figures

Figures reproduced from arXiv: 2507.11591 by the authors.

**Figure 1.** FIG. 1. Elastic scattering angular distribution compared with the Woods-Saxon optical model fits, including three sets of data [PITH_FULL_IMAGE:figures/full_fig_p006_1.png] view at source ↗

**Figure 2.** FIG. 2 [PITH_FULL_IMAGE:figures/full_fig_p008_2.png] view at source ↗

**Figure 3.** FIG. 3. Contour plot of [PITH_FULL_IMAGE:figures/full_fig_p009_3.png] view at source ↗

Figures from the paper (4 more)

**Figure 4.** Figure 4: FIG. 4. Real and imaginary optical potentials. Full lines represent the calculations for three energies with r [PITH_FULL_IMAGE:figures/full_fig_p009_4.png]

**Figure 5.** Figure 5: FIG. 5. Coulomb plus nuclear potential. Calculations are shown in different colors for three energies with r [PITH_FULL_IMAGE:figures/full_fig_p010_5.png]

**Figure 6.** Figure 6: FIG. 6. Residuals for 23.9, 26.1, 28.7 and 30.0 MeV (from top to bottom), compared with the 1 [PITH_FULL_IMAGE:figures/full_fig_p012_6.png]

**Figure 7.** Figure 7: FIG. 7. Residuals for the optical model fit using the 4 sets of data, compared with the 1 [PITH_FULL_IMAGE:figures/full_fig_p014_7.png]

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Pith tools

Reviewed August 6, 2026 · model on record in the stance chip above.

[1] [1]

Here we will review the least square (LS) method, following [12]

Least square method . Here we will review the least square (LS) method, following [12]. This is the method used in standard χ2 minimiza- tion codes, such as MINUIT [18], which is used, for example in SFRESCO [16]. These codes minimize the quantity χ2(a), described previously, and so determine the parameters ˆa that make χ2(ˆa) = χ2 m. This minimum defines...

work page

[2] [2]

The application of the maximum likelihood method requires the assumption of a The likelihood function L(a) as a function of the parameters

Maximum likelihood method We shall describe here the maximum likelihood method, following [12]. The application of the maximum likelihood method requires the assumption of a The likelihood function L(a) as a function of the parameters. If this function is chosen as L(a) = exp(−χ2(a)/2), then the maximum likelihood method gives the same results as the leas...

work page

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P (A) is the probability that the model that we use has parameters a are in a certain region A of the parameter space

Bayesian approach Bayes theorem states P (A|H) = P (H|A) P (H) P (A) (A25) Let us specify this expression for our case. P (A) is the probability that the model that we use has parameters a are in a certain region A of the parameter space. This probability is previous to consideration of the experimental data, so this is the prior probability. P (A|H) is t...

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